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Generation of Spin Squeezing in an Ensemble of Cold Rubidium 87 Marco Koschorreck Dissertation at the Institute of Photonic Sciences (ICFO) in the group of Prof. Dr. Morgan W. Mitchell and Univeristat Politècnica de Catalunya Thesis advisor: Prof. Dr. Morgan W. Mitchell co-advisor: Prof. Dr. Jürgen Eschner The work described in this thesis was carried out at ICFO - The Institute of Photonic Sciences Mediterranean Technology Park Av. Canal Olimpic s/n 08860 Castelldefels (Barcelona), Spain c Marco Koschorreck 2010 To my parents... Abstract At the beginning of the 20th century measurement precision was related exclusively to the capabilities of the experimenter and the apparatus. Any imprecision was attributed to imperfect devices for the measurement. Later on, when the quantumness of objects like photons, electrons, or atoms, was recognized, it became clear that the measurement and the object to be measured could no longer be separated. Both follow the laws of quantum physics which impose limits on the accuracy of any measurement. In the last three decades, a lot of research has been dedicated to find strategies to overcome these limitations. This new field of quantum metrology is where this work is placed. This thesis describes the generation of inter-particle entanglement in form of a squeezed spin state in a cloud of cold 87 Rb atoms by performing non-destructive spin measurements. Towards this goal, we have implemented a shot-noise limited polarization probing and detection system, atom counting by means of absorption imaging, spin state preparation via optical pumping and non-destructive spin state detection. The quantum noise limited spin read-out with polarized light is characterized in two complementary ways in terms of sensitivity and all possible quantum and classical noise sources are measured. To address the rich opportunities provided in the multilevel structure of 87 Rb with dispersive probing, we have developed and implemented a frequency offset-lock which allows a large tuning range of several GHz for the dispersive probing. Working with a magnetically sensitive atomic system triggered the development of a magnetic field imaging technique. This applies the elongated atomic sample and internal atomic states as a way of measuring magnetic fields over several√millimeters with micro-meter resolution and sensitivities down to a few tens of pT/ Hz. We developed a new method for tomographic measurements of the atomic density matrix. Based on our ability of shot noise limited detection of polarization rotations we can in principle, characterize the whole atomic spin state. First, measurements are shown and possible improvements are discussed. An in depth characterization of the light-atom interaction between polarized pulses of light and collective atomic states are the basis for measurements at the quantum level. On one side, we apply classical measurements of spin polarized atoms to measure the strength of the light-atom interaction. On the other side, we use quantum measurements of unpolarized atomic states to characterize all noise sources both technical and quantum. One of the major results of this thesis is the realization of quantum non-demolition measurements in a large-spin system. We realized limitations to the performance of the QND measurement implied by higher order light shifts in multilevel atoms, which are often neglected for these kind of systems. We develop a technique which suppresses the detrimental effects of higher order light shifts and recovers an ideal QND measurement. The thesis culminates in the demonstration of spin squeezing in the magnetically sensitive ground state of 87 Rb. We achieve a quantum noise reduction by 3 dB, where metrological sensitivity is improved by 2 dB beyond the standard quantum limit of a coherent spin state with the same spin polarization. We address the question how spatial and temporal inhomogeneities both in light and atoms influence the preparation of non-classical states. An existing covariance matrix approach is extended and we apply it to study effects of detector time resolution, spatial inhomogeneities and atomic motion. Resumen A principios de S.XX la precisión en la medición se asoció únicamente a la capacidad del experimentador y el aparato. Cualquier imprecisión se atribuía a la imperfección de los dispositivos de medición. Más tarde, cuando se se reconocieron las propiedades cuánticas de objetos como los fotones, electrones o átomos, se puso de manifiesto que la medición y el objeto a ser medido ya no podían ser separados. Ambos siguen las leyes de la física cuántica que imponen límites a la exactitud de la medición. En las últimas tres décadas, mucha investigación se ha dedicado a encontrar estrategias para superar estas limitaciones. En este nuevo campo de la metrología cuántica es donde este trabajo se encuentra. Esta tesis describe la generación de entrelazamiento entre las partículas de una nube de átomos fríos de 87 Rb en forma de un estado de espín apretado mediante la realización de medidas de espín no destructivas. Con este objetivo, hemos implementado un sistema de sondeo y de detección de luz polarizado limitado solo por ’shot-noise’, contando átomos por medio de imágenes de absorción, preparación del estado de espín a través de bombeo óptico y de detección no destructiva de estado de espín. La detección con luz polarizada se caracteriza de dos maneras complementarias en términos de sensibilidad y de todos los posibles fuentes de ruido de origen cuántico y clásico. Para hacer frente a las oportunidades que proporciona la estructura de varios niveles en 87 Rb en el sondaje con dispersión, hemos desarrollado e implementado un offset-lock de frecuencia que permite un gran rango de configuración de varios GHz. Trabajar con un sistema atómico magnéticamente sensible provocó el desarrollo de una técnica de resonancia magnética de campo. Esto se aplica a la muestra alargada atómica y a estados internos atómicos como una forma de medir los campos magnéticos sobre varios milímetros √ con una resolución de micro-metros y la sensibilidad a unas pocas décimas de pT/ Hz. Hemos desarrollado un método para la medición tomográfica de la matriz de densidad atómica. Sobre la base de nuestra capacidad de detección de rotaciones de polarización que puede, en principio, caracterizar el estado atómico de espin. En primer lugar, las mediciones se muestran y las posibles mejoras de estas comunidades. Una caracterización en profundidad de la interacción entre los pulsos polarizados de la luz con el estado colectivo de los atómos son la base para las mediciones a nivel cuántico. Por un lado, aplicamos las medidas clásicas de los átomos de spin polarizado para medir la magnitud de la interacción luz-átomo. Por otro, utilizamos mediciones cuánticas de estados atómicos no polarizados para caracterizar todas las fuentes de ruido tanto técnicas como cuánticas. Uno de los principales resultados de esta tesis es la realización de las mediciones cuánticas no destructivas en un sistema de gran espín. Nos dimos cuenta de las limitaciones para la realización la medición QND implicadas por efectos de orden superior en los sistemas con varios niveles, que a menudo son olvidadas para este tipo de sistemas. Desarrollamos una técnica que elimina los efectos perjudiciales de orden superior en al AC Stark shift y se recupera una medida QND ideal. La tesis culmina en la manifestación del spin-squeezing en el estado fundamental de 87 Rb, que es magnéticamente sensible. Logramos una reducción del ruido cuántico en un 3 dB, donde se mejora la sensibilidad metrológica en un 2 dB por encima del límite cuántico estándar de un estado de spin coherente. Abordamos la cuestión de cómo heterogeneidades espaciales y temporales tanto en la luz como en los átomos influyen en la preparación de los estados no-clásicos. Un método de la matriz de covarianza existente se amplió y aplicó al estudio de los efectos de la resolución del detector en tiempo, heterogeneidades espaciales y del movimiento atómico. Contents Abstract i Contents I 1. Introduction 1 2. Light-atom interfaces - background 2.1. Continuous variables for atoms . . . . . . . 2.2. Continuous variables for light . . . . . . . . 2.3. Light-atom interaction . . . . . . . . . . . . 2.4. Spin squeezing . . . . . . . . . . . . . . . . 2.5. Quantum non-demolition measurements . . 2.6. Simulation of collective quantum properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 9 10 16 19 27 3. Experimental apparatus and techniques 3.1. Atomic trap . . . . . . . . . . . . . . . . 3.2. Atom number measurement . . . . . . . 3.3. Dipole trap characterization . . . . . . . 3.4. Spin state preparation . . . . . . . . . . 3.5. Probing atomic spins . . . . . . . . . . . 3.6. Shot-noise-limited polarization detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 31 39 47 58 64 74 . . . . . . . . . . . 4. Dispersive Spin Measurements 87 4.1. Classical characterization of QND measurement . . . . . . . . . . . . 88 4.2. Measuring Zeeman coherences . . . . . . . . . . . . . . . . . . . . . . 94 4.3. Dispersive atom number measurement . . . . . . . . . . . . . . . . . 101 4.4. Relation between scattering cross-section and observed depolarization 102 5. Magnetic Field Measurements 105 5.1. Spatially integrating magnetic field measurements . . . . . . . . . . . 106 5.2. Spatially resolved magnetic field measurements . . . . . . . . . . . . 110 6. Spin State Tomography 121 6.1. Stern-Gerlach in free fall . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.2. Measuring ground-state populations of trapped atomic sample . . . 124 7. Spin Squeezing 7.1. Calibration of spin detector . . . . . . . . . . . . . . . . . . . . . . . 7.2. QND measurement of large-spin ensembles by dynamical decoupling 7.3. Spin squeezing of magnetically sensitive atomic state . . . . . . . . . 133 134 143 152 8. Summary and Outlook 157 8.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 158 A. Appendix I: Gaussian state picture for light and atoms A.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Physical system and mathematical model . . . . . . . . . . . . . . . A.3. Unified description of physical processes . . . . . . . . . . . . . . . . A.4. Instructive examples . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6. Coherent and incoherent transport processes described by N-port unitaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 162 162 165 170 176 177 B. Appendix II: Numeric simulation of multi-level atoms 179 B.1. Scattering cross section for different hyperfine ground states . . . . . 180 B.2. AC Stark shifts and the limit of the rotating wave approximation . . 180 B.3. Numerical simulation techniques . . . . . . . . . . . . . . . . . . . . 183 Abbreviations Bibliography 185 i List of Publications xv Acknowledgements xix 1. Introduction The quest to perform more and more precise measurements has driven humans for centuries and even millennia to improve the present technology towards a better one. At the advent of quantum mechanics it was perceived that there exists, in fact, a limitation implied by nature, in recent times referred to as standard quantum limit [1]. Many decades later, it was realized that quantum mechanics itself gives a solution to this problem. Quantum mechanical superpositions and entanglement are concepts which provide strategies to go beyond classical limits. This gave birth to a new class of ’Q-technologies’ like quantum information processing (QIP), quantum communication (QC), and quantum metrology (QM). Measurements performed with a system of N identically prepared particles, e.g., photons or atoms, will give a distribution of outcomes and show quantum noise, √ −1 which scales as δN ∝ N . These fluctuations are not the fault of an unskilled experimenter but rather a manifestation of the Heisenberg uncertainty principle and will ultimately limit the accuracy in any precision measurement [2]. The standard quantum limit is not fundamental in nature, but rather a result of an non-optimal measurement strategy. Introducing non-classical, i.e., quantum, correlations between the N particles, the measurement sensitivity can be improved towards the fundamental Heisenberg limit scaling as δN ∝ N −1 [3]. This thesis describes the generation of inter-particle entanglement in an ensemble of atoms which results in reduced quantum fluctuations - below the standard quantum limit - in one of its quantum variables. We use a measurement based approach, by applying optical quantum non-demolition (QND) measurements, to reduce the quantum uncertainty and generate inter-particle entanglement. In quantum information processing and precision metrology atomic ensemble systems of meso- or macroscopic number of atoms have proven very useful. Key to their utility, atomic ensembles are simultaneously macroscopic, with many atoms contributing coherently to a given effect, and simple, having only a few relevant quantum degrees of freedom. Seminal experiments with vapor cells demonstrated light-induced entanglement between ensembles [4], quantum memory for light [5], teleportation from light to matter [6], and projection noise limited magnetometry [7]. We are exploring a new generation of a magnetically sensitive quantum light-atom interface - optically trapped cold atomic ensembles - suitable for high precision metrology and quantum information processing. Ensembles of laser cooled and optically trapped 87 Rb atoms provide a very sensitive magnetic field sensor [8]. Generating a non-classical state improves the sensitivity further towards more fundamental limits. Cold atomic ensembles provide a number of advantages over room temperature vapor 2 1. Introduction cells. Firstly, they reach the same interaction strength with considerably fewer particles, i.e., 106 compared to 1012 . This is important in terms of technical noise. This allows us to work directly with rotation signals, without the modulation and demodulation typically used in the large-number vapor cell experiments. Secondly, because our atoms are effectively stationary, we can have much faster interactions, without need for averaging of thermal motion. Thirdly, non-classical states of light, e.g., single photons [9] or Schrödinger cat states [10, 11], or arbitrary two photon states [12] have been generated with bandwidth compatible with microsecond interaction times. Macroscopic ensembles of atoms and light are conveniently described in terms of collective spin operators. In Chapter 2, which collects the important theoretical concepts, we give a summary of their basic properties which establishes the ’language’ for the rest of the work. Further we review the light-atom interaction and give a description in terms of atomic spin and the optical Stokes operator which facilitates the understanding of the process. We also review important calculation techniques, e.g., density matrix and master equation, frequently used throughout the work to understand and predict the behavior of the atoms and the light. The concept of spin squeezing is explained and important criteria reviewed. Quantum non-demolition measurements, as the cornerstone of the state preparation, are described at the end of this chapter and there relation to spin squeezing is explained. Chapter 3 summarizes the experimental apparatus and the essential improvements over the previous setup. The optical trap as central tool to produce well-localized and confined atomic clouds is characterized in terms of trap lifetime, atom temperature and number, and loading characteristics. Absorption imaging as atom number counting technique important in the quantitative characterization of the QND interaction is reviewed and its implementation discussed. The coherent spin state preparation and probing are discussed and characterized. The polarization detection is measured in terms of its noise characteristics. The dispersive spin probing, which provides the quantum non-demolition spin measurement, is investigated in Chapter 4. The light-atom interaction strength is characterized with the help of polarized spin states and absorption imaging. We introduce a dispersive atom number measurement (DANM) which provides a way of non-destructive atom counting necessary for measurements in later chapters. In Chapter 5 we discuss efforts to produce a magnetic field free environment of the atomic ensemble. This is important to achieve a long coherence time for the coherent spin states and crucial for the spin squeezing experiments. Two techniques to measure constant stray magnetic fields are presented and experimental results shown. To measure magnetic field gradients and higher order moment we designed a novel technique which uses the extended atomic sample as a way of measuring magnetic field with spatial resolution. This magnetic field imaging technique is applied to cancel the existing magnetic field gradient in the vicinity of the atomic sample. The magnetic field imaging technique is published in [...]. In Chapter 6 we briefly review the concept of spin state tomography as characterization tool for the atomic state. Important requirements for a complete characterization of the full density matrix of a spin state are discussed. We propose a novel 3 tomographic technique based on state-selective population transfer and sensitive dispersive atom number measurement on a trapped atomic ensemble. We present first results and discuss possible improvements. In Chapter 7 we present measurements results of the preparation of a squeezed spin state through measurement induced entanglement. We start by calibrating the lightatom interaction using a thermal spin state. In this way, a reliable characterization is possible with a test state that shows with known quantum noise but which is robust against technical noise and depolarization noise. The results of this subsection are published in [13]. For the QND measurements of a quantum states in the hyperfine ground state of rubidium 87 we identify an important obstacle which impedes the generation of spin squeezing in such a system. A technique which surpasses this limitations based on the concept of dynamical decoupling is developed and implemented. In this way we are able to recover a pure QND measurement even in a large spin system. These results are published in [14]. Finally, first results on spin squeezing in the F = 1 hyperfine ground state of 87 Rb are shown. The quantum noise of the coherent input state is reduced by 3 dB on an ensemble of 700000 atoms. The metrological improvement compared to the standard quantum limit of the coherent spin state is 2 dB. The atom light interaction between optically trapped atomic ensembles and a Gaussian laser beam is far from being uniform. In Appendix A we study the effects of spatial and temporal inhomogeneities by applying the covariance matrix method. The results of this chapter are published as [15]. 2. Light-atom interfaces - background Contents 2.1. Continuous variables for atoms . . . . . . . . . . . . . . . 2.1.1. Spin states in 87 Rb 6 . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2. Collective atomic spin variables . . . . . . . . . . . . . . . . 8 2.2. Continuous variables for light . . . . . . . . . . . . . . . . 2.2.1. Polarized light . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 2.2.2. Stokes operators . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3. Light-atom interaction . . . . . . . . . . . . . . . . . . . . 10 2.3.1. Dipole interaction . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2. Polarizability Hamiltonian . . . . . . . . . . . . . . . . . . 12 2.3.3. Master equation . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4. Spin squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.2. Spin squeezing criteria . . . . . . . . . . . . . . . . . . . . 18 2.5. Quantum non-demolition measurements . . . . . . . . . 19 2.5.1. Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2. Atomic quantum non-demolition measurements via dispersive probing . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.3. Non-destructive magnetization measurements in 87 Rb . . . 22 2.5.4. Squeezing via quantum non-demolition measurements . . . 23 2.6. Simulation of collective quantum properties . . . . . . . 27 6 2. Light-atom interfaces - background The chapter summarizes the main theoretical framework to describe the interaction of a collection of atoms and polarized light. We focus on the presentation of the main models, ideas, and concepts and give reference to more detailed texts on the subject. We begin by introducing the concept of collective spin variables for an ensemble of atoms and pulses of light. In this, a macroscopic number of particles, i.e., atoms or photons, is described as a single large spin state. This will facilitate the description of the state for atoms and light, which otherwise would be almost intractable. We present the Hamiltonian for the light-atom interaction in the dipole approximation. For the special case of off-resonant light we derive the polarizability Hamiltonian which is the cornerstone in the description of the experiments of probing the quantum noise of the atomic spin state. The master-equation, fundamental for simulations during the course of the thesis is discussed. The concept of spin squeezing is reviewed and the two main relevant spin squeezing criteria are discussed. Quantum non-demolition measurements are introduced and their implementation in the context of an ensemble of rubidium atoms is discussed. In the end we sketch the main idea of the experiment to be implemented and give the spin squeezing criteria in terms of measurable quantities. 2.1. Continuous variables for atoms We review important aspects of the description of collective properties of atomic ensembles. At the beginning we briefly recall important properties of single atom angular momentum operators. Then we mention the hyperfine structure of 87 Rb and the important transitions. At the end we introduce collective spin operators which will be the basis of our description of the atomic state. 2.1.1. Spin states in 87 Rb The D lines of Rubidium are perhaps the best studied transitions in atomic physics. A very complete summary of the physical data are found in references [16, 17, 18, 19]. Here we present the spin and energy structure relevant to this thesis. We are mainly concerned about rubidium 87 atoms (87 Rb) in their hyperfine ground and excited states. The energy-eigenstate of atoms in free space are characterized through a set of quantum numbers, both the ground and excited state of interest have a principal quantum number n = 5 and a nuclear angular momentum quantum number i = 3/2. Its single valence electron has an angular momentum quantum number of l = {0, 1} = {S, P } for the ground and excited state, respectively, and a spin quantum umber of s = 1/2. Through spin orbit coupling the total electronic angular momentum is j = 1/2 for the ground and j = 3/2 for the excited state. In spectroscopic notation we can write the ground and excited on the D2 transition as as 5S1/2 and 5P3/2 . 2.1. Continuous variables for atoms 7 The coupling of nuclear spin, i, and total electronic spin, j, leads to the total angular momentum f = j + i, reflected in the hyperfine splitting of ground and excited states, respectively. Here and throughout, we use bold letters to indicate vector quantities. In the following we will use atomic spin when we refer to the total angular momentum of an atom. The projection of f onto a given direction, conventionally called quantization axis, can take the following values mf = {−f, −f + 1, ..., f }. In the following we are mainly concerned about quantum states and transitions which are unambiguously specified by two numbers f and mf , and we write |f, mf i. The multilevel structure with all hyperfine levels is shown in Fig. 2.1 and an almost complete list of properties for 87 Rb can be found in [19]. Figure 2.1.: Level structure of D2 transition of 87 Rb. Single spin operators The above mentioned physical quantities, for example the total angular momentum f , are represented by operators f̂ which we indicate with a hat. Spin operators have certain properties which are briefly reviewed. More details can be found in the text books by J.J. Sakurai [20] or A. Messiah [21]. The spatial components fˆx , fˆy and fˆz of the total angular momentum f̂ obey the following commutation relations [fˆλ , fˆµ ] = i~λ,µ,ν fˆν (2.1) where λ,µ,ν is the Levi-Civita symbol and the Greek symbols run over {x, y, z}. As a result of the commutation relations Eq. (2.1) we find the following eigenvalue equations for f̂ 2 and fˆz f̂ 2 |f, mf i = f (f + 1)~2 |f, mf i fˆz |f, mf i = mf ~ |f, mf i . (2.2) (2.3) As quantum mechanical quantities, spin operators obey the Heisenberg uncertainty principle expressed in the Robertson-Schrödinger relation iE2 1 Dh var(Â) var(B̂) ≥ Â, B̂ , (2.4) 4 8 2. Light-atom interfaces - background where  and B̂ can be any of the three angular momentum operators along the Cartesian coordinates x, y, and z. The variance of an operator Ô is denoted as var(Ô) and calculated as var(Ô) = h Ô2 i − h Ô i2 , where h.i is the expectation value. More properties can be found in [20] and references therein. 2.1.2. Collective atomic spin variables For an ensemble of atoms we can define collective angular momentum operators and states. If we assume symmetry under particle exchange we can write F̂ ≡ NA X f̂ (i) . (2.5) i=1 Here we sum the individual single atom operators f̂ (i) over the number of atoms NA . The total angular momentum of the collective state in the symmetric space1 will be F = NA f . For the collective operators there exist the same commutation and uncertainty relations as for single atom operators [F̂λ , F̂µ ] =i~λ,µ,ν F̂ν ~2 D E2 var(F̂λ ) var(F̂µ ) ≥ F̂ν , 4 (2.6) (2.7) where λ 6= µ 6= ν. The eigenstates for the collective operators F̂2 and F̂z , along the quantization axis, are called Dicke states and were introduced in the context of super-radiance [27]. Interestingly, the Dicke states have many similarities to Fock states in the optical domain [28]. There exist a constructive definition [21] using the raising and lowering operators defined as F̂+ = F̂x + iF̂y and F̂− = F̂x − iF̂y : 1 |F, MF i = (MF + F )! 2F MF + F − 12 F̂+F +MF |F, −F i , (2.8) which applies the raising operator F̂+ successively to generate a state with eigenvalue MF from the lowest energy (ground) state |F, −F i defined through F̂− |F, −F i = 0 . (2.9) Hence, any state |F, MF i has a well defined projection along z and a completely undefined phase. Their uncertainty orthogonal to z is large for |MF | F . Of the Dicke states, only those with |MF | = F are minimum uncertainty states in the sense that their uncertainty orthogonal to z gives an equality sign in Eq. (2.7). 1 The limitation to the symmetric subspace is sufficient in many case where the individual particle is not detectable. This was used in the past to describe many experiments [4, 22, 23, 24, 25, 26]. The dimension of the symmetric sub-space scales only linearly with the system size, compared to the full Hilbert space, i.e., 2NA f + 1 (2f + 1)NA . 2.2. Continuous variables for light 9 There exist a class of states which is very similar to Glauber’s optical coherent states, called spin coherent states [29] or atomic coherent states [28]. An intuitive definition was given by Arecchi et al. [28] |θ, ϕi = e−iϕF̂z e−iθF̂y |F, −F i . (2.10) The state |F, −F i is rotated about the y- and z-axis. The state |θ, ϕi is completely characterized by the two angles θ and ϕ or equivalently by one complex number ξ = 2θ e−iϕ , and the number of atoms NA . Note that the coherent spin state, as defined above, is always minimum uncertainty for all angles θ and ϕ . 2.2. Continuous variables for light Polarized light plays an important role in the manipulation and detection of atomic spin states. For example, narrow-band on-resonant light is brought into interaction with atoms to prepare certain spin states, or off-resonant light to probe spin states dispersively. It is, therefore, important to review some properties of polarized light. The experiments use light from quiet frequency stabilized lasers, which for our purposes can be described as coherent states. 2.2.1. Polarized light For many practical purposes we are interested in this thesis, it is possible to neglect the spatial properties of the light orthogonal to the propagation direction and describe the field as a plane wave. This one-dimensional description is valid if the beam extension is much larger than the wavelength of the light. A light field traveling along z with a single frequency ω and pure polarization is described in classical terms [30, 31] as E(z, t) = Ec X eq aq eikz−iωt + c.c. (2.11) q=−1,1 where Ec is the classical real electric field amplitude in SI units V/m, aq is a complex dimensionless amplitude of polarization mode q, and eq are unit polarization vectors [32]. The c.c. indicates the complex conjugate of the leading expression. The electric field of a plane wave in second quantization is given by [30] Ê(z, t) = Ê+ (z, t) + Ê− (z, t) , (2.12) where Ê+ (z, t) ≡ Eq X eq âq eikz−iωt (2.13) e∗q â†q e−ikz+iωt . (2.14) q=−1,1 − Ê (z, t) ≡ Eq X q=−1,1 10 2. Light-atom interfaces - background The operator Ê+/− is the positive/negative-frequency part of the field and contains only the annihilation/creation operators. The electric field amplitude is Eq = p ~ω/2ε0 V defined in the quantization volume V , where ε0 is the permittivity of free space. The complex mode amplitude of the classical description is replaced by the annihilation operator âq of the polarization mode q. 2.2.2. Stokes operators For the experiments on dispersive measurements of the atomic spin polarization it is practical to describe the light in terms of its polarization properties through the Stokes operators. For this, it is convenient to use a traveling-wave description discussed in [33], and a wave-packet mode of duration τ of a paraxial beam with area A [34], which defines the quantization volume V ≡ Aτ . We find Ŝx = ~ † â σ x â 2 Ŝy = ~ † â σ y â 2 Ŝz = ~ † â σ z â . 2 (2.15) Here, â ≡ [â+ , â− ]T and â+ , â− are annihilation operators for left and right circular polarization, respectively and σ x , σ y , and σ z are the Pauli matrices. The Stokes operators obey angular momentum commutation relations: [Ŝx , Ŝy ] = i~Ŝz and cyclic permutations. Bold indices are used for the components of the Stokes operators to distinguish them from Cartesian coordinates. The expectation value of the Stokes operators, and therefore polarized light, can be conveniently visualized as a point on a sphere, called Poincarè sphere, which recalls the Bloch sphere for spin 1/2 particles. We will usually work with highly-polarized states, in which one component dominates and for NL 1 we can substitute the operator Ŝx by its expectation value Sx ≡ h Ŝx i. Thereafter, we will denote expectation values of operators as the same symbol without hat. Only the orthogonal components Ŝy and Ŝz contain quantum features and have to be treated as operator quantities. For example, with horizontally polarized laser light: 1 h Ŝx i = ~NL ≡ Sx 2 and h Ŝy i , h Ŝz i = 0 (2.16) where NL is the average number of photons. The variances are 1 var(Ŝx ) = var(Ŝy ) = var(Ŝz ) = ~2 NL . 4 (2.17) The commutation relations are effectively [ Ŝy , Ŝz ] = i~Sx and [ Ŝx , Ŝy/z ] = 0. 2.3. Light-atom interaction We briefly recall the dipole approximation and the resulting Hamiltonian as a starting point of the discussion about light-atom interactions. More detailed treatments can be found in many standard textbooks on quantum optics [30, 35]. Then we 2.3. Light-atom interaction 11 elaborate in more detail on the polarizability Hamiltonian which is at the heart of the light-atom interaction in this thesis. Thereafter, we rewrite the dipole interaction Hamiltonian in a semi-classical way, which serves as the basis for the numerical simulations we carry out along the course of the thesis. The last point summarizes the density matrix and master-equation formalism as working horse for the theoretical understanding in this experimental work. 2.3.1. Dipole interaction The interaction of a single atom with a single electromagnetic field mode can be described in the so called dipole approximation as atoms being point-like emitters compared to the wavelength of the light [36, 30]. We consider a single mode radiation field with fixed polarization which has a wavelength much longer than the extension of the atom. Usually, this is a safe assumption, knowing that atoms have normally a size in the order of a few Ångstroms (10−10 m) and the light a wavelength of a few hundreds of nanometers2 . We follow the approach described in [35, 30, 36]. There, the light is described by the quantum field of Eq. (2.12), and the atoms and light-atom interaction are described by the Hamiltonian Ĥ = ĤA + Ĥint . (2.18) The first term in Eq. (2.18) is the Hamiltonian of the unperturbed atom which can be written [30] as ∞ X ĤA = ~ ωi |Ψi i hΨi | , (2.19) i=1 where ~ωi are energy-eigenvalues and |Ψi i the corresponding energy-eigenstates. Hy 0 perfine ground |gi = |f, mf i and excited state |ei = f , mf 0 on the D2 transition are examples of such energy-eigenstates. The second term in Eq. (2.18) describes the dipole interaction Ĥint = −d̂ · Ê , (2.20) where dipole operator d̂ in the Schrödinger picture is defined as d̂ ≡ q X dij |Ψj i hΨi | . (2.21) i,j=1 The dipole matrix elements dij = e hΨi | r |Ψj i give the amplitude of a transition between |Ψi i and |Ψj i. 2 Maybe things are different if we consider Rydberg atoms and microwave radiation. 12 2. Light-atom interfaces - background 2.3.2. Polarizability Hamiltonian Now we discuss a very important way of representing the coherent part of the lightatom-interaction. The terminology we introduce here will be the basis for many discussion in later chapters in connection with the spin squeezing experiment. The multilevel structure of alkali metal atoms produces a light-atom-interaction with tensorial character. That is, light interacting with an atom in a polarized atomic state can change its polarization state, i.e., the medium is said to be optically active [37]. This was recognized already a long time ago in connection with optical pumping of atomic vapors [38, 34]. As shown in [38, 39], when second-order time-dependent perturbation is applied in the limit of small excitation [39, 34, 40, 41, 42], with ĤA as the unperturbed Hamiltonian and Ĥint as the perturbation, the second-order contribution to the energy is (−) (+) Ĥint,pol = −Ei Ej αij , (2.22) where αij is the atomic dynamic polarizability tensor at the frequency of the laser light ωL : 1 X P̂g d̂P̂e d̂† P̂g α=− , (2.23) ~ g,e ∆eg where P̂g = |f, mf i hf, mf | and P̂e = f 0 , mf 0 f 0 , mf 0 are projection operators P onto P the ground and excited state subspace, respectively. The summation g,e ≡ f,f 0 ,m,m0 goes over all magnetic substates of both ground and excited states. The detuning of the light from the atomic resonance is ∆eg = ωL − ωeg . The Eq. (2.22) has the same form as the expression for the potential energy of a small polarizable particle in an electric field oscillating or static [37]. The above expression (2.22) describes a light mediated connection of two states in the ground state manifold via a state in the excited state. From a physical point of view, the interaction will cause a change of the field through a spin-state dependent refractive index and of the atoms through a polarization dependent light shift. To see this effect more clearly it is practical to decompose the polarizability tensor into its irreducible components α̂ ≡ α̂(0) ⊕ α̂(1) ⊕ α̂(2) , (2.24) where ⊕ is the tensor sum. After some considerable amount of math, we arrive at a Hamiltonian that splits into different parts Ĥint,pol = K̂0 Ŝ0 + K̂1 Ŝx + K̂2 Ŝy + K̂3 Ŝz . (2.25) where the operator K̂k are atomic observables [34]. K̂0 = g 2 (0) α Îf + α(2) 3 NA X i fˆz(i) 1 − f (f + 1)Îf 3 !! , (2.26) 2.3. Light-atom interaction 13 N K̂1 ≡ gα(2) A 1X fˆx(i) fˆx(i) − fˆy(i) fˆy(i) , 2 N K̂2 ≡ gα(2) i A 1X fˆx(i) fˆy(i) + fˆy(i) fˆx(i) , 2 i (2.27) N K̂3 = gα(1) A 1X fˆz(i) , 2 (2.28) i α(k) where are the polarizability components [39, 40], Îf the identity on a space of dimension 2f + 1, and g = ωL /(20 V ) is a constant which contains the interaction volume V . Having casted the interaction into the form of Eq. (2.25) we can appreciate the symmetry between atoms acting on the light and the light acting on the atomic spin. The individual terms of Eq. (2.25) have the following interpretation: The term proportional to S0 is the well known AC Stark shift. It is independent of the internal atomic state and reveals no information about the atomic spin state nor introduces spin rotations. Hence, we can neglect it for further considerations. The other terms, from the perspective of the atoms, represent spin rotations proportional to the polarization components Ŝx , Ŝy , and Ŝz . For the quantum non-demolition measurement P A ˆ(i) fz is very important. It will prothe last term which is proportional to F̂z ≡ N i duce a polarization rotation of the light proportional to the spin polarization along z. Likewise, the terms ∝ Ŝx and ∝ Ŝy can be used to measure the observables P A ˆ(i) ˆ(i) P A ˆ(i) ˆ(i) (i) (i) (i) (i) fx fx − fˆy fˆy and F̂x F̂y + F̂y F̂x ≡ N fx fy + fˆy fˆx . The F̂x2 − F̂y2 ≡ N i i final form of the interaction Hamiltonian is Ĥpol = i ~1 ~1 h 2 G1 Ŝz F̂z + G2 Ŝx F̂x − F̂y2 + Ŝy F̂x F̂y + F̂y F̂x , τ2 τ2 (2.29) where G1 and G2 are light-atom coupling constants proportional to the vector α(1) and tensor α(2) polarizability components, respectively. They can be interpreted as the per-atom rotation of the incoming light polarization. The coupling constants for atoms in F = 1 on the D2 line are 1 Γλ2 4 5 5 G1 (∆, A) ≡ − − + , A 16π δ0 (∆) δ1 (∆) δ2 (∆) 1 Γλ2 4 5 1 G2 (∆, A) ≡ − + , A 16π δ0 (∆) δ1 (∆) δ2 (∆) (2.30) (2.31) where δf 0 = ∆ − ∆0f 0 for each hyperfine excited state f 0 , where ∆0f 0 is the frequency difference between the lowest excited hyperfine level, i.e., f 0 = 0, and the level with f 0 . The detuning from the resonance F = 1 → F 0 = 0 is ∆, where ∆ < 0 is referred to as red detuning. We characterize the interaction of light and atoms by an effective interaction area A. It is the transverse area over which we could spread atoms and light homogenously and achieve the same interaction strength, i.e., Gi , as we actually measure. The excited state linewidth is Γ. 14 2. Light-atom interfaces - background |G1|, |G2| ï6 10 ï4 10 ï8 10 ï6 10 ï1 ï0.5 ï0.05 6 [GHz] ï8 10 ï10 10 ï2 ï1.5 ï1 ï0.5 6 [GHz] 0.5 1 Figure 2.2.: Coupling constants for vector and tensor light shift G1 (solid line) and G2 (dashed line) for the interesting region of detunings ∆ from transition F = 1 → F 0 = 0. The inset shows the characteristic scaling with detuning for G1 ∝ ∆−1 and G2 ∝ ∆−2 . For the interaction area we chose A = 4 × 10−9 m2 , which is a realistic value as we will see further below in Chap. 4. The polarizability components α(1/2) and therefore the coupling constants G1/2 show different scaling with detuning from resonance ∆, as we can see in Fig. 2.2. It is possible to find spectral positions where one or the other vanishes, or where they are equal. This spectral feature can be used to design the Hamiltonian at wish for different applications. In reference [40] we discuss a number of proposals for applications in quantum cloning, quantum memory and atom number measurement. 2.3.3. Master equation Often it is important to model dissipative processes like absorption or spontaneous emission or describe a system which is in a statistical mixture. The most appropriate description is the density operator formalism [30] which can be used to develop a master equation [43]. The density operator is ρ̂ = X ρij |Ψi i hΨj | (2.32) i,j and ρij is the density matrix [20]. The evolution without dissipation under Ĥ is 2.3. Light-atom interaction 15 described by the von Neumann equation i ∂ ρ̂ i h = − Ĥ, ρ̂ ∂t ~ (2.33) which is equivalent to the Schrödinger equation. If we allow for Markovian dissipative processes, as for instance spontaneous emission, the system described by ρ̂ is no longer closed. It is coupled to a reservoir, which, for spontaneous emission, are a number of electromagnetic field modes. The master equation can be used to describe its dynamics i ∂ ρ̂ i h (2.34) = − Ĥ, ρ̂ + Ldiss (ρ̂) , ∂t ~ where Ldiss (ρ̂) is a super-operator acting on ρ̂. Under the imposed Markovian approximation, the relaxation operator Ldiss is local in time, i.e., ∂∂tρ̂ only depends on ρ̂ at the same time. All possible dissipative processes can be included in Ldiss . We can write i 1 Xh † † † Ldiss (ρ̂) = − Ĉm Ĉm ρ̂ + ρ̂Ĉm Ĉm − 2Ĉm ρ̂Ĉm , (2.35) 2 m with Ĉm being the Lindblad operators for different dissipative processes acting on the system. The operator describing, for example, spontaneous emission has the form √ Cse = Γdrel, ge |Ψg i hΨe | (2.36) where Γ is the excited state decay rate and |Ψg i and |Ψe i are hyperfine ground and excited states, respectively, and drel, ge is the branching ratio or relative oscillator strength3 of the hyperfine transition |Ψe i → |Ψg i. We give the expression for drel, ge in Appendix B.3. The master equation (2.34) can be rewritten as a linear differential equation ∂ ρ̂ = Ltot (ρ̂) , ∂t (2.37) where we can define another super-operator, sometimes called Liouvillian or Liouville operator Ltot (ρ̂) = − i 1 Xh i i h † † † Ĥ, ρ̂ − Ĉm Ĉm ρ̂ + ρ̂Ĉm Ĉm − 2Ĉm ρ̂Ĉm . ~ 2 m (2.38) It contains both the coherent and incoherent part of the evolution. Rewriting the density matrix as a vector of density matrix elements, e.g., ρ ~ = (ρ11 , ..., ρqq ) and Ltot as a matrix L, the master equation is ∂~ ρ = L~ ρ, ∂t where L has dimension q 2 × q 2 for a q-dimensional system. 3 In a simple two level atom this would be one. (2.39) 16 2. Light-atom interfaces - background In the case of a time-independent Hamiltonian Ĥ, the solution of Eq. (2.39) is a simple exponential function of L ρ ~(t) = eLt ρ ~(0) . (2.40) Hence, once the matrix L is determined, we can calculate the solution of the master equation at any time from the initial state ρ ~(0). Throughout the thesis we use this method to calculate the time behavior of the atomic state in processes like optical puming atoms into a coherent spin state (cf. Sect. 3.4) or answer the question of the de-polarization in dispersive spin measurements (cf. Sect. 4.4). 2.4. Spin squeezing Spin squeezing is at the heart of this thesis and we give here a short review of important spin squeezing criteria which are used to quantify the amount of noise reduction and the implied multipartite entanglement. 2.4.1. General remarks The concept of squeezing - the redistribution of quantum fluctuations between conjugate observable - was conceived when people started to think about the limitations for an interferometric measurement with light [44, 45, 46] as for example in the detection of gravitational waves. Caves [46] at the beginning of the 1980’s showed that, if the secondary input port of a conventional interferometer, for example a Mach-Zehnder interferometer, is fed with squeezed vacuum state, the random vacuum fluctuations could be suppressed. Short after, it was shown theoretically by Yurke et al. [47] that an interferometer, or better any device having two input and output ports, i.e., a two mode device, can be represented by the SU(2) group. This insight, seems to have inspired to ’think the other way around’ and formulate the concept of spin squeezed states by Kitagawa and Ueda [48] and at the same time by D. Wineland et al. [49]. That is, the two modes in the optical case can be identified as the two ground states of a spin 1/2 half system. We note, a decade later, the group around J.P. Dowling introduced the formal equivalence between a Mach-Zehnder interferometer, the Ramsey spectroscopy, and a generic quantum logic circuit and called it the ’quantum rosetta stone”. They have shown that there exist a close connection between interferometry and quantum computation [50]. Here, we are describing the correspondence between the Mach-Zehnder interferometer and the Ramsey spectroscope containing N spin one-half particles. This will facilitate the understanding of spins squeezing in general and the conditional spin squeezing we are going to generate. In a classical interferometer as shown in Fig. 2.3 a) the input mode is coherently split into the modes u and d by a first beam-splitter. This corresponds to the coherent rotation of the spin state |⇑i ≡ |↑i⊗N by π/2 about an axis lying in the equatorial plane. 2.4. Spin squeezing 17 Figure 2.3.: a) Correspondence between Mach-Zehnder interferometer and a spin onehalf system, here demonstrated in a Ramsey sequence. The single mode input of the interferometer corresponds to a spin in one of the eigenstate, here|↑i. The first beamsplitter prepares a coherent superposition between path u and d, which is equivalent to a coherent rotation of the spin state into a state in the equatorial plane. The lower mode d experiences a phase shift ϕ which is a rotation in the spin picture about the z axis. Finally, the two modes are over lapped on the second beams-splitter. At the two output ports we receive a measurable signal proportional to the introduced phase shift. In the spin picture we measure the population difference between |↑i and |↓i. b) Phase sensitivity can be improved when the two modes u and d are entangled. In this case the spin state is squeezed. If we chose the axis to be along y, the state is rotated into the a coherent superposition 1 (|↑i + |↓i)⊗N . Both modes continue independently and one experiences a phase 2N/2 ⊗N 1 |↑i + eiϕ |↓i . The phase shift ϕ which changes the superposition state into 2N/2 shift in the interferometer is equivalent to a rotation of the coherent spin state about the z axis. The phase difference is read out by overlapping both modes on a second beam splitter which results in a measurable signal m ≡ (Nu − Nd ) /2 = N2 sin ϕ, where Nu and Nd are the population of the modesP u and d, respectively. This has its correspondence in the observable MJ of Jˆz = 21 i (|↑i h↑| − |↓i h↓|) which is the difference in the number of spins pointing up and down. ∆m , The error in phase estimation according to estimation theory [51], ∆ϕ = ∂m/∂ϕ is proportional to the error in population difference ∆m scaled by the slope of the 18 2. Light-atom interfaces - background interferometer signal ∂m/∂ϕ. For N uncorrelated particles the uncertainty in the population difference is ruled by counting statistics for a collection of two state √ systems ∆m = N /2 cos ϕ. The slope of the interferometer signal is ∂m/∂ϕ = N/2 √ cos ϕ. Hence, the phase sensitivity for a classical coherent state state is ∆ϕc = 1/ N . Although it can be explained completely in terms of classical statistics [52], it was given the name standard quantum limit by Braginsky and Vorontsov [1]. Introducing non-classical correlations between the N particles in the interferometer allows to surpass the SQL and achieves a much better sensitivity which scales as ∆ϕ = 1/N . There exist a number of possibilities to reach a scaling which is better than the SQL by designing an appropriate input state to the interferometer or changing the linear beam-splitter into a non-linear one as depicted in Fig. 2.3 b). A comprehensive overview about appropriate input states in different kind of systems can be found in [50]. The ’non-linear beam-splitter approach was proposed already in the very beginning by Kitagawa and Ueda [48]. They showed theoretically that a non-linear interaction of the form Jˆz2 will produce a spin squeezed state, where the system evolves unitarily from a coherent into a squeezed spin state. This interaction Hamiltonian is sometimes called “one-axis” twisting, referring to the transformation of the quantum uncertainty from a circle into an ellipse. A third way of creating the necessary non-classical correlations are measurement based strategies using quantum non-demolition measurements. This technique which is applied in our experiment is explained in more detail in Sect. 2.5. 2.4.2. Spin squeezing criteria The direct measurement of quantum correlations, i.e., on a microscopic level, is difficult. The non-classical correlations between a particle and any other of the N −1 particles in the system is small for collective states with large N . Fortunately, it is sufficient to measure a finite set of macroscopic quantities to determine the degree of squeezing and entanglement. Metrological spin squeezing The most stringent criterion, due to Wineland et al. [49] related the question of squeezing to the improvement in measurement sensitivity. Any N -particle spin state that provides a smaller phase estimation error than a coherent spin state of same particle number is squeezed. The phase estimation error can be expressed in first and second order moments of spin variables as q var(Jˆzs ) ∆ϕs ≡ , h Jˆxs i (2.41) where var(Jˆzs ) is the variance of the squeezed state and h Jˆxs iits expectation value. Compared to the best classical case var(Jˆzc ) = h Jˆxc i /2 of a coherent spin state, the 2.5. Quantum non-demolition measurements 19 phase measurement is then improved by ξm q N var(Jˆzs ) ∆ϕs ≡ = . ∆ϕc h Jˆxs i (2.42) Note, most of the experiments that demonstrated spin squeezing will state the squeez2 which is readily computed from the variances var(Jˆs ). ing in terms of ξm z c ˆ Later in the experimental section, we will measure var(Jz ) of the input state to verify that we have a coherent spin state. This will serve as a reference level for the squeezed state var(Jˆzs ). We will also describe how a QND measurement can reduce var(Jˆz ) from an initial, coherent state value var(Jˆzc ) to a reduced value var(Jˆzs ) < var(Jˆzc ). In practice, the QND measurement will in addition reduce h Jˆx i from the coherent (in) (out) (in) state value h Jˆx i = NA /2 to h Jˆz i = (1 − η) h Jˆx i (in the experiment, η will be related to the fraction of atoms that suffer a spontaneous light-scattering event). The operation results in a squeezed state, i.e ξm < 1 if var(Jˆz )s < (1 − η)2 var(Jˆz )c (2.43) The above criterion implies non-classical correlations which was proven in reference [53]. The value of ξm can be strictly related to the depth of entanglement, i.e., the minimum number of particles forming multi-particle entangled states in the system. For decreasing ξm the number of particles involved in the entanglement is increasing [54]. Kitagawa-Ueda-criterion A weaker criterion than the above-mentioned was introduced by Kitagawa and Ueda [48]. They define spin squeezing as the reduction of spin uncertainty below the coherent state level q var(Jˆzs ) ξe = 2 (2.44) h Jˆx i or more closely related to our experiments var(Jˆzs ) < (1 − η) var(Jˆzc ) . (2.45) Also this criterion implies entanglement. 2.5. Quantum non-demolition measurements This section has the aim to review important aspects of quantum non-demolition (QND) measurements. After a historical excursion we formally describe criteria for a QND measurement. In Sect. 2.5.2 we explain the QND measurement strategy for atomic spins via dispersive probing. The next section applies this idea to the case of 87 Rb. At the end, we give the connection between spin squeezing and QND measurement and define the signal-to-noise ratio as central figure of merit in dispersive QND measurements. 20 2. Light-atom interfaces - background 2.5.1. Concept Interestingly the initial idea for quantum non-demolition (QND) measurements was developed in the context of gravitational wave detection, i.e., the same field which initiated the idea of squeezing, we discussed above. The first introducing the concept was Bragisnky in the mid 1970’s [1]. The main idea is to generate an indirect measurement which gives information about a ’system’ using a ’meter’ without changing the system variable. An ideal QND measurement can be viewed a realization of a von-Neumann measurement. That is, the system is projected via the measurement into one of its eigenstates corresponding to the measured eigenvalue. Any subsequent measurement of the same variable will give the same results reflecting the fact that the system, once projected, remains in the eigenstate [55, 56]. The QND measurement is an indirect way of gathering information about an observable  of the system S coupling it to a meter M. Before interaction, the combined M + S system is in a separable state |ψS i ⊗ |ψM i, where |ψS i and |ψM i are the respective eigenstates of S and M before interaction. The interaction changes the combined quantum state from a separable into an entangled one |ψS i ⊗ |ψM i → |ψSM i (2.46) At this stage, information about S is carried by M and vice versa. At the instant when M is detected information about  is acquired which collapses the wavefunction of S according to the measurement outcome a. There exist a necessary and sufficient condition [55] which ensures the QND character of the measurement the observable  h i Â, Û |ψM i = 0 , (2.47) where Û is the operator of the joint evolution of S and M. This condition is very general and in many situations hard to verify and for many experimental situations it can be replaced by a simpler one. Often the system Hamiltonian of S + M can be written as a sum Ĥ = ĤS + ĤM + Ĥint , (2.48) where HS and HM are the Hamiltonian for the system and the meter and Hint describes the interaction of both. With Eq. (2.48) two conditions about the QND character of the variable  and the interaction Ĥint can be formulated [44]. The observable  is called a QND variable if it is unchanged by the free evolution of the system S. h i ĤS ,  = 0 (2.49) That means,  is a constant of the free evolution of S. This condition ensures that any excess noise which is introduced into the conjugate variable is not fed back into Â. The interaction between M and S is called QND interaction if it does not perturb the measured observable  h i Ĥint ,  = 0 (2.50) We have a QND measurement, if both conditions (2.49) and (2.50) are fulfilled simultaneously [57]. 2.5. Quantum non-demolition measurements 21 2.5.2. Atomic quantum non-demolition measurements via dispersive probing Dispersive atom-light interaction can be used to acquire information about atomic properties in a non-destructive way. The concept relies on off-resonant interactions of light with atomic quantum systems. In this, the population of certain atomic states can be read out relative to each other by comparing the phase shifts acquired through the dispersive interaction. This idea was introduced by Takahashi et al. [58] and Kuzmich et al. [59] and assumes that we can couple two orthogonal electromagnetic field modes u and d to two distinct atomic states |↑i and |↓i and their respective excited states. As in the discussion about spin squeezing, the correspondence between a two-mode oscillator and the elements of the SU(2) group is used here. The properties of the proposal by Kuzmich et al. can be discussed without specifying the details of the actual physical implementation of |↑i and |↓i. In an abstract way, we can associate a pseudo-spin operator to the two spin states, which has the components ĵx = ~ † b σxb 2 ĵy = ~ † b σy b 2 ĵz = ~ † b σz b , 2 (2.51) h iT where b ≡ b̂↑ , b̂↓ are bosonic operators with |↑ / ↓i = b̂†↑/↓ |0i. A collective spin operator for an ensemble of NA atoms is defined as Ĵ = NA X ĵ(k) , k=1 where ĵ(k) is the spin operator for the kth atom. We are interested in the population difference given by Jˆz . For all relevant cases we can assume that the population of the hyperfine ground state is constant when the system evolves freely. Therefore, we can call Jˆz a QND variable. Also, the two electromagnetic field modes, represented by their photon annihilation operators âu and âd , correspond to a spin angular momentum via the Schwinger representation [60] of a two-mode oscillator Ŝx = ~ † a σxa 2 Ŝy = ~ † a σy a 2 Ŝz = ~ † a σz a , 2 (2.52) where a ≡ [âu , âd ]T . Each e.m. field mode âu/d interacts dispersively with one of the spin states |↑ / ↓i which is described by a Hamiltonian of the form Ĥ ∝ n̂u NA X k=1 |↑ik h↑|k − n̂d NA X |↓ik h↓|k = Ŝz Jˆz , (2.53) k=1 where n̂u/d are the number operators for the modes u/d. The second criterion for a QND measurement (2.50) is fulfilled for the observable Jˆz . Thereafter, we will call any Hamiltonian of the form (2.53) a QND Hamiltonian, because it describes the QND interaction we are interested in. 22 2. Light-atom interfaces - background 2.5.3. Non-destructive magnetization measurements in 87 Rb Now, we focus on actual implementation of the Takahashi-Kuzmich proposal for the measurement of the magnetization in the hyperfine ground state for alkali metal atoms. We will restrict the discussion to the relevant case for this thesis which are QND measurements in the F = 1 manifold of the 5S1/2 . The extension to higher F manifolds is theoretically straightforward but might require slightly different experimental realizations than discussed here. To measure the magnetization h F̂z i in the F = 1 manifold, it is sufficient to limit the description to the ground states with mF 6= 0. An effective spin one-half system is defined by |↑ / ↓i ≡ |1, ±1i . This is physically reasonable, since atoms in |1, 0i are not contributing any magnetization along the quantization axis z, i.e., h1, 0| F̂z |1, 0i = 0. The set of pseudo-spin operators for this case is 1 Jˆx = 2 i Jˆy = 2 1 Jˆz = 2 NA X i=1 NA X i=1 NA X [|↓i h↑| + |↑i h↓|]i (2.54) [|↓i h↑| − |↑i h↓|]i (2.55) [|↑i h↑| − |↓i h↓|]i . (2.56) i=1 Linearly polarized off-resonant light propagating along z serves as a meter for the spin magnetization Jˆz = F̂z /2. The left and right circular components of the linear probe light play the role of the two orthogonal electromagnetic field modes we introduced above. By identifying âu = â+ and âd = â− , we recognize that the abstract Schwinger operators (2.52) are in this particular case the Stokes operators (2.15). The off-resonant interaction of polarized light with alkali metal atoms was discussed further above in Sect. 2.3.2. We found a very compact description in terms of irreducible tensor components (2.29) which can be restated here using the pseudo-spin operators (2.54-2.56) ~ ~ Ĥpol = G1 Ŝz Jˆz + G2 Ŝx Jˆx + Ŝy Jˆy . (2.57) τ τ If we neglect the tensor term for a moment and set G2 = 0, the Hamiltonian describes a perfect QND interaction for the measurement of Jˆz . In Chapter 7, we return to the full Hamiltonian and discuss the limitation of the assumption of vanishing tensor polarizability. There, we propose and demonstrate a method to recover the pure QND interaction ∝ Ŝz Jˆz , even in the case of G2 6= 0. In the Heisenberg picture we can study the evolution of the spin operators for light and atoms. For a first qualitative statement it is sufficient to study the evolution 2.5. Quantum non-demolition measurements 23 of the operators to lowest order in Ĥpol . This gives simple input-output relations between operators before (in) and after (out) interaction. A more detailed model is presented in Appendix A, which allows to study the exact evolution. For infinititsimal changes the operator after interaction can be written as τ h (in) i Ô(out) = Ô(in) + Ô , Ĥ . (2.58) i~ After QND interaction Hpol light and atom operators are (in) (out) (in) Ŝx Ŝx 1 −G1 Jˆz 0 (out) (in) (in) Ŝy = G1 Jˆz 1 0 Ŝy (out) (in) 0 0 1 Ŝz Ŝz (2.59) (in) (out) (in) Jˆx Jˆx 1 −G1 Ŝz 0 ˆ(in) ˆ(out) (in) = G1 Ŝz Jy 1 0 Jy (in) (out) ˆ 0 0 1 Jˆz Jz (2.60) In the above representation the symmetry between light and atoms becomes apparent. Both systems contain information about the other part. If the input states are pure, the interaction will entangle atoms and light. As requested, the z components for both systems remain unchanged by the interaction, i.e., they are QND variables. Information about the magnetization is transferred onto linear components of the probe light, i.e, Ŝx and Ŝy . 2.5.4. Squeezing via quantum non-demolition measurements We explain the role of QND measurement for the generation of spin squeezing and introduce the notion of conditional variance following [61, 57]. The experimental proposal of Kuzmich for our case is explained. We relate the theoretical spin squeezing criteria introduced above to experimental parameters we actually measure in the experiment. There exist a close connection between QND measurement and squeezing. An ideal QND measurement, i.e., a projective measurement in the sense of von Neumann 20, will produce a post-measurement state which is known through the recorded measurement outcome M1 on the meter system M. A second measurement M2 on the same quantum system will be perfectly correlated with the first one, M2 = M1 . Also in the realistic case of imperfect QND measurements, i.e., if the state is not projected with certainty into an eigenstate, a second measurement will show some degree of correlation with the first one. More formally, the amount of correlation can be parametrized by a correlation coefficient, which can be measured directly and has the definition [57] κ≡ cov(M1 , M2 ) , var(M ) (2.61) 24 2. Light-atom interfaces - background Figure 2.4.: Metrology scheme for conditional spin squeezed spin states. We show the part of the interferometer after and before the initial and final beam-splitter, respectively. The coherent spin state is measured in a first QND measurements which gives a result M1 . This state is now exposed to a field which introduces a rotation Ry (ϕ). A second QND measurement gives a result M2 . with cov(M1 , M2 ) = h M1 M2 i − h M1 i h M2 i, where M1/2 are the results of a first and second measurement of the meter on the system S [61, 57] and we have assumed that var(M ) ≡ var(M1 ) = var(M2 ). In the case of an ideal QND measurement we have κ = 1 and less for imperfect QND measurements. If the first QND measurement is omitted, i.e., M1 is unknown and no prior information about the system is recorded, we have κ = 0. Using the language of an interferometer as introduced in Sect. 2.4.1 the relation between squeezing and QND measurement becomes apparent. We concentrate on the part of the interferometer where the actual phase measurement takes place (cf. 2.3) which is shown in Fig. 2.4. The coherent input state is probed in a first QND measurement which converts it into a squeezed state with certain offset in Jˆz direction ∝ M1 . The system is now subjectto a field which introduces a phase shift ϕ, which ˆ is in Fig. 2.4 a rotation R̂y = exp −iϕJy about y. The second QND measurement will give a result M2 . Combining the first and second QND measurement outcomes, the phase shift is determined better than the intrinsic quantum uncertainty of a coherent spin state would allow. The amount of metrological enhancement in the sense of Wineland et al. (2.42) can be calculated directly from the statistics of the two measurement outcomes for ϕ ∼ 0 via the conditional variance var(M2 − κM1 ) 2 ξm = var(M2 − κM1 ) = 1 − κ2 . var(M2 ) (2.62) We define a conditionally squeezed state as the one which shows experimentally var(M2 − κM1 ) < var(M ) , (2.63) where we have to assume that the measured variances contain mostly quantum fluctuations4 . 4 It is also important that var(M1/2 ) is governed mostly by quantum noise. Classical correlations 2.5. Quantum non-demolition measurements 25 Experimental proposal For the experiments described later, we use linearly polarized light pulses with average number of photons NL that have the following characteristics before interaction iE N Dh h i N L L Ŝx , Ŝy , Ŝz = , 0, 0 var Ŝx , Ŝy , Ŝz = [1, 1, 1] . (2.64) 2 4 The ensemble of NA atoms will be polarized in a coherent superposition state (|↑i + |↓i)⊗NA which represents a pseudo-spin state polarized along x with i N iE N h Dh A A ˆ ˆ ˆ Jx , Jy , Jz = , 0, 0 var Jˆx , Jˆy , Jˆz = [0, 1, 1] . (2.65) 2 4 Note the difference between in Eq. (2.64) and (2.65) in the fluctuation along the polarized direction for light and atoms. This is because the number of atoms in the sample is a fixed quantity, which in principle could be measured precisely [28, 62, 29]. In experiments, of course, the number of atoms is also fluctuating due to technical noise through the trap loading process. In contrast the photon number is in general governed by quantum fluctuations. Via the interaction, information about Jˆz is transferred to Ŝy , i.e., Ŝy is the meter for Jˆz , which is measured many times on re-prepared atomic states, but otherwise under the same conditions. If we compute the variance of the measurement outcomes, we find (out) (in) (in) var(M1 ) ≡ var(Ŝy ) = var(Ŝy ) + G21 Sx2 var(Jˆz ) , (2.66) where for large photon numbers we can assume Sx2 var(Ŝx ). The above equation can be used to quantify the sensitivity of the QND measurement of Jˆz . The first term is the shot noise of the probe light, which places an ultimate limit on the QND measurement sensitivity. The second term contains the atomic quantum information which is amplified by the product of the coupling constant and the number of photons. The different scaling of the first and second term with photon number allows us to “amplify” the quantum signal to an amount comparable or larger than the limiting (in) light shot noise var(Ŝy ). We define a ratio of atomic and light quantum noise and call it the signal-to-noise ratio of the QND measurement. ς≡ (in) G21 Sx2 var(Jˆz ) (in) var(Ŝy ) = G21 NL NA , 4 (2.67) where the second equality holds if all light and atomic input states are quantum (in) (in) noise limited, i.e., var(Ŝy ) = N4L and var(Jˆz ) = N4A . The state after the first QND measurement, conditioned on the measurement outcome M1 , will show a reduced quantum fluctuations as derived in [63] (in) var(Jˆz ) (out) var(Jˆz |M1 ) = , 1+ς (2.68) can easily lead to a large correlation coefficient and on the same time to large var(M1/2 ). This issue will be discussed in the measurement of atomic quantum noise in Chapter 7. 26 2. Light-atom interfaces - background which can be derived also by classical multivariate Gaussian statistics [64]. For the final judgment on the success of the spin squeezing generation via QND measurements we have to introduce the conditional spin variance (2.71) into one of the squeezing criteria (2.43) or (2.45) and we find spin squeezing in the respective definitions by Wineland et al. or Kitagawa et al. iff (out) (in) |M1 ) < (1 − η)2 var(Jˆz ) (2.69) (in) |M1 ) < (1 − η)var(Jˆz ) . (2.70) var(Jˆz or (out) var(Jˆz Experimentally, we can verify the spin squeezing in the first QND measurement by performing a second QND measurement. This is in complete analogy to the interferometric scheme we introduced above (cf. Fig. 2.4), except that we don’t allow for a rotation of the state, i.e., ϕ = 0. More concretely, we perform a double pulse experiment, i.e., two QND measurements, on a coherent spin state polarized along h Jˆx i = NA /2, where the first and the second probe pulse interact in the same way with the atomic sample. The experiment is repeated many times under the same experimental conditions and we gather a list of measurement values {M1 , M2 }. The conditional variance and the fluctuation of the input state, necessary for the spin squeezing criteria (2.69) and (2.70), is calculated form the set measurement outcomes by (out) var(Jˆz |M1 ) = (a) (a) (e) var M2 − κM1 − var M2 G21 NL and (a) (e) var M1 − var M1 (in) var(Jˆz ) = G21 NL , , (2.71) (2.72) where we indicate the presence of atoms with (a) and an empty trap with (e). The number of photons NL is measured by photo-detection, the number of atoms NA by absorption imaging, the coupling constant G1 by observing average rotation for a known number of atoms in a polarized state, and the depolarization is characterized by measuring the decay of the polarization as a function of probe photons. The polarization of the atomic state after QND measurement is (out) Jx (in) = Jx exp(1 − η) , (2.73) where we can model the depolarization by a single parameter η [41, 63, 65]. It describes the probability that an atom absorbs a photon out of the pulse of NL incoming photons. In the limit of small destruction, we can write η= σ(∆) NL , A (2.74) where σ(∆) is the off-resonant scattering cross section, A the effective interaction area between light and atoms and NL the number of probe photons. The exact 2.6. Simulation of collective quantum properties 27 expression for σ(∆) is given in Appendix B.1. Both, σ(∆) and SNR are proportional to NL ∆−2 . Thus, the ratio between them is constant and there exist no preferred detuning for the probe laser. 2.6. Simulation of collective quantum properties A technique which has proven to be very useful in the simulation of the evolution of the quantum state during QND measurements is the covariance matrix approach [66, 67, 68, 65, 69, 70, 71]. Much of the intuition gained during the work on spin squeezing came through simulation of the system using this technique. Even so this thesis is based on experiments, we were able to contribute some insight on the influence of spatial and temporal inhomogenieties on the degree of spin squeezig. The results are published in: “Unified description of inhomogeneities, dissipation and transport in quantum light-atom interfaces” by M. K. and Morgan W. Mitchell in J. Phys. B 42, 195502 (2009) [15]. In Appendix A we present the covariance matrix technique and discuss the extension to incorporate inhomogenieties in a natural way. We give examples which illuminate the basic questions as for instance the influence of the detector time resolution or thermal motion to the degree of spin squeezing. 3. Experimental apparatus and techniques Contents 3.1. Atomic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1. Trap overview . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.2. Cooling and trapping lasers . . . . . . . . . . . . . . . . . . 33 3.1.3. Preparation of cold atomic ensembles . . . . . . . . . . . . 35 3.2. Atom number measurement . . . . . . . . . . . . . . . . . 39 3.2.1. Fluorescence methods . . . . . . . . . . . . . . . . . . . . . 39 3.2.2. Absorption imaging . . . . . . . . . . . . . . . . . . . . . . 40 3.3. Dipole trap characterization . . . . . . . . . . . . . . . . . 3.3.1. Trap depth and sample shape 47 . . . . . . . . . . . . . . . . 47 3.3.2. Loss through light scattering . . . . . . . . . . . . . . . . . 49 3.3.3. Trap lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.4. Atom temperature . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.5. Axial shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4. Spin state preparation . . . . . . . . . . . . . . . . . . . . 58 3.4.1. Optical pumping into dark states . . . . . . . . . . . . . . 58 3.4.2. Loss of spin polarization . . . . . . . . . . . . . . . . . . . 61 3.5. Probing atomic spins . . . . . . . . . . . . . . . . . . . . . 64 3.5.1. Polarization control and photon-number referencing . . . . 65 3.5.2. Frequency shifting and pulse generation . . . . . . . . . . . 66 3.5.3. Frequency Offset-lock . . . . . . . . . . . . . . . . . . . . . 67 3.5.4. Experimental details on probe lasers . . . . . . . . . . . . . 72 3.6. Shot-noise-limited polarization detection . . . . . . . . . 74 3.6.1. Polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6.2. Linear birefringence and optical losses in detection . . . . . 76 3.6.3. Operation Principle of balanced detector . . . . . . . . . . . 78 3.6.4. Detector Calibration . . . . . . . . . . . . . . . . . . . . . . 79 3.6.5. Noise of the detection system . . . . . . . . . . . . . . . . . 82 30 3. Experimental apparatus and techniques This chapter summarizes the essential parts of the experimental apparatus and explains the important techniques. The cooling and trapping setup was described several times in previous doctoral theses [72, 73, 74]. For this reason, we concentrate on extensions and improvements of the existing setup which are important steps towards a well controlled atom-light interface. A variety of laser sources are the main tool to trap, cool, and manipulate external and internal atomic properties, as for instance temperature and spin polarization. In Fig. 3.1 we give a functional diagram of all involved laser light sources and their interplay. This will facilitate the understanding of particular experimental details in later chapters. Details of particular parts of the laser systems are explained in detail in according chapters. The central part is the atomic trap which we describe in Sect. 3.1. In the functional overview we divided this into its three parts 2D-MOT, 3D-MOT and dipole trap. Single lasers are often subdivided into different tasks which are marked as rounded rectangles close to the laser. Manipulations of laser light in frequency and intensity F=2 laser Yb:YAG laser F=2 depletion cooling seeding referencing imaging F=1 laser repumping opt. pumping transfer referencing tensor probe vector-probe laser h polarized v polarized F=2 slave cooler push beam 2D MOT crossed beam 3D MOT mode matching single beam FORT Figure 3.1.: Functional diagram of laser sources for experiment. Dark red boxes are lasers tuned near the D2 line of 87 Rb and the purple box is a far-detuned laser for optical atom trapping. The lasers are subdivided into functions marked in white rounded rectangles. Free space and fiber transfer of laser light is marked in dotted and solid lines, respectively. For details see text. 3.1. Atomic trap 31 is done by means of acousto-optic modulators (AOM). The F=1 and F=2 lasers are referenced to atomic transitions in 87 Rb. All laser light is transferred to the atom trap apparatus by polarization-maintaining (PM) single mode optical fibers. This ensures a controlled delivery of laser radiation and stable working conditions on a daily basis. In addition it makes the trap itself movable. In the future, experiments with a non-classical light source are planned. For this purpose the trap and the non-classical light source would have to be combined on a separate table. 3.1. Atomic trap In the last 30 years, trapping and cooling of neutral atoms has become a standard technique in atom and molecular physics. The first atom beam slowing experiments were demonstrated in the 1980’s, followed by neutral atom traps in 1985 using an optical molasses [75] and a purely magnetic trap [76]. The first cooling and trapping in a magneto optical trap was reported 1987 AT&T Bell labs [77]. Only ten years later the three pioneering groups of Steven Chu (Stanford), William D. Phillips (NIST, Gaithersburg) and Claude Cohen-Tannoudji (Paris) were awarded with the Nobel Prize in physics. We describe the laser system for the atom trapping apparatus and the preparation of cold atomic ensembles. Full details can be found in references [72, 73, 74]. We assume a familiarity with basic concepts of trapping and cooling. Detailed explanations of cooling and trapping atoms can be found in several textbooks and review publications, for example [78, 17, 79]. 3.1.1. Trap overview We use a double stage magneto-optical trap (MOT) setup depicted in Fig. 3.2. Atoms are accumulated in a two-dimensional MOT which continuously feeds a threedimensional MOT. A single-beam optical dipole trap is overlapped with the 3D MOT. After loading, the gradient fields and trapping beams of the 2D and 3D MOT are turned off, leaving the atoms in an all-optical trap. The two stage atom loading has several advantages. Using two MOT systems, separated by a differential pumping stage allows us to have a high pressure system for fast loading and a low pressure system for long lifetimes. The 2D MOT is used to load atoms from a background pressure of 10−8 mbar into an elongated cloud which is only cooled and trapped in two dimensions allowing atoms to move freely in the third dimension. This elongated cloud of atoms is constantly illuminated by a ’push-beam’ along the untrapped direction from top to bottom in order to assist the transfer of atoms into the 3D MOT. In this way, we are able to transfer ∼ 1 × 107 atoms per second. In the 3D MOT we have very pure vacuum of < 10−11 mbar background pressure. We use a standard three dimensional MOT arrangement, which allows us 32 3. Experimental apparatus and techniques Figure 3.2.: Magneto-optical and all-optical trap scheme: The upper 2D MOT work at a background gas pressure of 10−8 mbar. Cooling and trapping beams (red arrows) collect atoms into an elongated sample which is constantly transferred into the lower chamber. The chambers are separated by a differential pumping stage which allows a much smaller background pressure of only 10−11 mbar for the lower one. There, atoms are collected into a 3D MOT formed by six counter-propagating laser beams. A gently focused laser beam, the far off-resonant trap (FORT), is superimposed with the atomic cloud of the 3D MOT. After extinguishing the quadrupole magnetic field and the cooling lasers, atoms are held in the FORT for further experiments. 3.1. Atomic trap 33 to cool the atoms below the Doppler limit to about 25 µK. A schematic drawing of the combined MOT setup is shown in Fig. 3.2. For further manipulation we extinguish the MOT light beams and the quadrupole magnetic field. Trapping atoms only by means of a focused laser beam offers a way to store them in a state independent way. This is in strong contrast to magnetooptical trapping where atoms change constantly their internal state while absorbing and re-emitting photons. 3.1.2. Cooling and trapping lasers We summarize the most important features and parameters of the laser setup for magneto-optical and optical trapping. MOT lasers To cool and trap alkali metal atoms in a magneto optical trap two laser fields are needed. The first, often called cooler, is tuned slightly below the cycling transition in F = 2 → F 0 = 3. It is responsible for the trapping force. The second, called repumper, transfers atoms once they have escaped from the cycling transition back into F = 2. This laser is tuned close to the resonance F = 1 → F 0 = 2. The cooling light for the 2D and 3D MOT is supplied by two separate laser diodes. This provides us with sufficient optical power for fast loading. The F=2 laser, which feeds the 3D MOT, and all derived beams is schematically shown in 3.3. It is a commercial ECDL1 that is stabilized to an atomic transition in 87 Rb by means of saturated absorption spectroscopy (cf. Fig. 3.3 a)). We blue shift the laser by 60 MHz before it enters into the spectroscopy setup (cf. Fig. 3.3 b)) to have a large range for the cooler detuning of many natural linewidths. In this way we effectively lock the laser closer to F 0 = 2 and not as typically done at the crossover transition between F 0 = 2 and F 0 = 3 2 . The large detuning range is important for subDoppler cooling and increased the number of atoms in the dipole trap by factor of two compared to previous experiments [74]. The 2D MOT is supplied by the F=2 slave laser. This is a Fabry-Perot diode laser which is injection locked to the F=2 laser. For injection we tune the light 10 MHz to the red of the transition F = 2 → F 0 = 3. This light is a small fraction of the light used for spectroscopy of the F=2 laser. It is shifted in a double-pass AOM configuration by 2 × 62 MHz. Reliable injection is achieved at a power of 1.5 mW. The output of the slave is split in two parts. One, the majority of the light, is used 1 2 The used model is DL100 from Toptica AG, Germany At the crossover transition the signal is larger compared to the direct transitions. This is readily explained by the fact that for a cross-over transition not the zero velocity class is selected but two non-zero velocity classes. Atoms which co- or counter-propagate relative to the pump beam with a velocity which bridges half of the hyperfine excited state splitting by means of Doppler shift will absorb a photon on one transition and appear saturated for the other. 34 3. Experimental apparatus and techniques Figure 3.3.: Schematic layout of F=2 laser. a) is the spectroscopy setup to perform saturated absorption spectroscopy. b) to e) are acousto optical modulator setups to shift the laser frequency. Details are given in the text. The output beams are labeled from 1 to 5. 3.1. Atomic trap 35 for the 2D MOT beams. The other part is used for the vertically push-beam (cf. Fig. 3.2) We use a portion of the F=1 laser to provide the repumper light for both magnetooptical traps. The laser is a home built ECDL which is directly locked to the transition from F = 1 to the cross over transition between F 0 = 1 and F 0 = 2 . The light is shifted by a single pass AOM to the resonance F = 1 → F 0 = 2. Dipole trap laser The laser for the far off-resonant optical trap is a diode pumped Yb:YAG thin disk laser3 . It is capable of producing 20 W of continuous wave power in a single longitudinal mode. The center wavelength is 1030 nm at a linewidth of < 5 MHz. Single-mode operation is provided by an intracavity etalon. We monitor the single mode operation constantly using a scanning Fabry-Perot cavity. The wavelength is adjustable via an intracavity Lyot filter in a range between 1000 and 1060 nm. Both the etalon and the Lyot filter are adjusted from time to time to ensure single mode operation at 1030 nm. The laser itself has proven to perform best in terms of single mode behavior and long term power stability when optimized at high powers. Over long measurement times of more than 12 hours the relative change in wavelength is < 3 × 10−4 and in power < 4 × 10−2 . As earlier stated, all light that arrives at the trapping setup is fiber coupled into single mode fibers. Since normal single mode fibers can not handle tens of Watts of optical power easily, we are using a large mode area photonic crystal fiber4 . It has a solid core of 25 µm diameter. The single mode light propagation is enabled through the photonic crystal structure surrounding the solid core in a hexagonal arrangement. The photonic band-gap supports only a single spatial mode over a large wavelength range. The intensity at the surface is reduced because the focus is translated into the fiber-material. Nevertheless, greatest care has to be taken to avoid power damages on the input facet. We included a mode cleaning spatial filter consisting of a pair of lenses and a pinhole of 50 µm diameter in the focus. This spatial filter rejects all but a single TEM00 mode. We achieve coupling efficiencies of over 60 %. At the location of the MOT we have a maximum power of 6.0 W. The laser power can be switched or changed continuously with an AOM. This allows us to switch the trapping light in a computer controlled way. 3.1.3. Preparation of cold atomic ensembles We load the optical trap in a standardized scheme which is the same for all experiments in this work. The procedure can be divided into three main steps, which are illustrated in Fig. 3.4. First we produce cold atoms in a MOT. Second, we cool atoms below the Doppler limit to increase the number of atoms in the trap. Third, 3 The company producing the product was originally called ELS. It has since been purchased by another laser company called SLT. 4 The model is LMA-25 from Crystal Fibre (now part of NKT Photonics). 36 3. Experimental apparatus and techniques Figure 3.4.: Dipole trap loading cycle. We show the three most important parameters for the trap loading. The MOT loading phase can vary between a few hundreds of ms to 4 s. During the thermalization period of 400 ms untrapped atoms will escape from the trapping region and the atoms thermalize in the trap. we allow untrapped atoms to escape from the trapping region by waiting a period of 400 ms before we start to use the laser cooled atoms. The light of the dipole trap is present from the beginning. In our experience this increases the number of atoms in the trap. Atoms which are cold enough already in the MOT loading phase will stay in the dipole trap potential. There, the light scattering from the MOT beams is reduced due to the AC Stark shift induced by the trapping light. In this way we continuously start loading sufficiently cold atoms from the MOT cloud. After an atomic cloud of ∼ 4 × 107 is produced in about 4 s, the sub-Doppler cooling phase starts. To reduce the temperatures below the Doppler cooling temperature (TD = 144 µK for 87 Rb), we slowly ramp down the magnetic field gradient and the start to reduce the light intensity of the cooling and repumper light. At the same time, the detuning of the cooler light is increased. The process which allows sub-Doppler temperatures is called polarization gradient cooling. In brief, the main cooling mechanism is the differential scattering of light from two counter-propagating laser beams. The differential scattering is caused by the imbalance of the ground state population which results from its non-adiabatic following to the local light polarization [80, 78]. The final phase of 400 ms is used to allow untrapped atoms to escape from the trapping region and a thermalization of the trapped atoms along the weakly confined direction with 4 Hz trapping frequency. Several parameters are important in the loading process. First, we have to guarantee enough light power over the whole detuning range of the cooling laser. In Fig. 3.5 we plot the measured total power into the MOT beams as a function of the cooler detuning (set by the RF frequency on the double-pass AOM, cf. Fig. 3.29). By tilting this AOM in the double-pass configuration (cf. Fig. 3.29) we can shift the position of the maximum slightly and equalize the power which is used for cooling. 3.1. Atomic trap 37 Second, the amount of repumper light in the last 112 ms of the sub-Doppler cooling phase has to be adjusted very carefully. Small differences can change the number of atoms in the dipole trap by a factor of two. In this last phase we have ∼ 4.8 mW/m2 injected through all six MOT beams. Third, the position of the atomic cloud from the MOT is fine-adjusted by correcting the compensation coil currents. We see a strong dependence in x and y direction due to the small transverse dimension of the dipole trap. 60 P [mW] 50 40 30 20 Detuning Range 10 0 ï100 ï80 ï60 ï40 6 cooler [MHz] 0 Figure 3.5.: Measured power into the 3D MOT apparatus as a function of detuning from the F = 2 → F 0 = 3 transition. For a trapping-beam waist of ∼ 6 mm and a coupling efficiency of 75 the intensity in the trap center is ∼ 9 W/m2 per 1 mW. We indicate the range over which the laser is detuned during the sub-Doppler phase. The sub-Doppler phase is adjusted for the best dipole trap loading. However, no detailed study was carried out since we are able to load more than > 106 atoms in about 2 s into the trap by adjusting the three major parameters described above. The evolution of the number of atoms in the dipole trap is measured as a function of the MOT loading time. In Fig. 3.6 we number of atoms measured with absorption imaging, which we explain in Sect. 3.2. The data are fitted by the solution of simple rate equation model for the transfer of atoms from the MOT into the dipole trap NA (t) = NA,∞ (1 − exp (−kt)) , (3.1) where NA,∞ is the final number of atoms in the dipole trap and k is the trap loading rate. The gives NA,∞ = 1.21(5) × 106 and k = 0.85(9) s−1 . 38 3. Experimental apparatus and techniques Figure 3.6.: Evolution of atom number NA in dipole trap as a function of MOT loading time. The time constant is k = 0.85 s−1 3.2. Atom number measurement 39 3.2. Atom number measurement For the characterization of light-atom interactions it is important to have, in addition to shot noise limited photon detectors, also a reliable measurement for atom number. There are various ways to measure the number of atoms present in an ensemble. The most primitive, used in atomic vapor cell experiments, is to use ideal gas theory and calculate the number of atoms knowing temperature and pressure. This works well for homogeneous distribution. For atoms in a potential of a focused laser beam, things are a bit more difficult. The solution we prefer is absorption imaging. The necessary considerations and the experimental setup are explained in Sect. 3.2.2. Before, we present in Sect. 3.2.1 a brief summary of fluorescence techniques which were used in the experiment to measure temperature (cf. Sect. 3.3.4) and trap lifetime (cf. Sect. 3.3.3). Figure 3.7.: Different atom number counting strategies. a) Fluorescence detection with calibrated photo-diode or CCD camera. The amount of collected fluorescence photons is determined by the numerical aperture of the imaging lenses L1 and L3, respectively. b) The absorption image is taken with linearly polarized light. The atomic absorption casts a shadow which is detected by the CCD camera. 3.2.1. Fluorescence methods The principle of atom-detection with fluorescence is very simple. It relies on the measurement of spontaneously emitted photons from previously excited atoms. It is possible, at least in principle, to get a one-to-one correspondence between the 40 3. Experimental apparatus and techniques number of detected photons and the number of atoms present. To find the relation between atoms and photon-number, we have to know the following parameters: 1. The absorption cross section, σ(∆), for the atomic transition we want to probe. For this, it is important to chose a transition which is closed. That means, the atom cycles between one ground and one excited state. In many situations this requirement is not fulfilled because of the complex structure in multilevel atoms. 2. The detected solid angle which determines the part of the full solid we have excess to. This can be estimated from the imaging optics used. A slight modification has to be taken into account if the emitted radiation has an anisotropic radiation pattern. This is the case if we would observe, for instance, the emission of π-polarization under a small angle to the quantization axis. 3. The most critical parameter is the light intensity at the position of the atoms. Since the amount of emitted light is directly related to the number of incident photons we have to rely on a calibrated illumination. Light fluctuations would be transferred immediately to the recorded signal. For quantitative atom number measurements, all above parameters have to be characterized with good accuracy. For the first two, this poses no major obstacle. However, the stability in light intensity is the major problem in quantitative atom detection with fluorescence. There are experimental situations were fluorescence imaging is the only and, at the same time, a very good method. The shelving technique [81] in ion detection is a perfect tool which allows a high-fidelity state detection with fluorescence imaging. All measurement of a small number of atoms can be carried out with fluorescence techniques. There, atoms are detected by discrete changes in the detected light. This allows simple calibration and a larger tolerance to intensity fluctuations. [82, 83, 84, 85]. We use two methods of detecting atoms with fluorescence as depicted in Fig. 3.7 a). One uses a calibrated photodiode which records the light collected by a lens close to the atomic cloud. The other images the fluorescence light onto a CCD camera. 3.2.2. Absorption imaging We use absorption imaging (AI) as the primary method of measuring the atom number. The technique is widely used in the field of ultracold atomic physics and there exist several in-depth treatments of the physical implementation and the data analysis [86, 87]. Here we summarize the important concepts. Principle Absorption imaging could be seen as a spatially-resolved transmission measurement of the atomic cloud. Resonant laser light is sent through an atomic cloud. Atoms scatter photons out of the beam so that the transmission is reduced as sketched in 3.2. Atom number measurement 41 Fig. 3.7 b). The relation between incoming and outgoing light in the limit of no saturation is given by Lambert-Beer’s law I (out) (x, z) = I (in) (x, z)e−d0 (x,z) (3.2) The optical density, d0 , is directly related to the absorption cross section introduced in App. B.1 and the atomic density distribution ZL d0 (x, z) = σ(∆) n(x, y, z)dy = σ(∆)nc (x, z) , (3.3) 0 where we simplified the expression by using the column density nc . Measuring the transmission as the ratio between output and input intensity reveals the column density distribution of the atoms. One advantage, compared to fluorescence detection, is that measuring transmission in the low saturation regime is independent of the intensity used. Therefore, slight intensity fluctuations can be neglected. Realization In the ideal situation it seems sufficient to take a single picture and calculate the transmission. For high quality absorption images, however, it is necessary to acquire three pictures. A shadow image Iatoms which contains the atomic cloud, a reference image Ilight without atoms, but under the same conditions as the first, and a dark image Idark with neither imaging beam nor atoms. There are several reasons for that. First, calculating the transmission as a ratio greatly reduces the sensitivity to intensity fluctuations. Second, the imaging light, I (in) (x, y), is not a smooth plane wave or Gaussian distribution but is rather distorted, e.g., by diffraction effects from dust particles etc. (cf. Fig. 3.8 b)). That is, we can not extrapolate the light intensity from the values of I (in) in the vicinity of the atomic cloud. Third, the bias image gives a calibration of the dark currents in the CCD array and accounts for stray light which is not coming from the probe. It is important to take the three pictures as close as possible in time to ensure the same conditions for all of them. In Fig. 3.8 a), b) and c) we show the three images. The transmission and optical depth are than calculated as Tmeas = e−d0,meas = Iatoms − Idark . Ilight − Idark (3.4) The result is shown in the right most image in Fig. 3.8. The are two main systematic errors in absorption imaging. One is the saturation of the atomic absorption by power broadening. However, to avoid this, we use imaging intensities two orders of magnitude below saturation intensity. The second, and more important systematic effect, is a limited observable optical depth. Probe light that does not interact with the atomic cloud, but reaches the 42 3. Experimental apparatus and techniques Figure 3.8.: a)-c) Shadow, Reference and Dark image. The dark image is scaled by a factor of 100. Red color is maximum intensity. d) retrieved transmission image. Black color is small transmission. CCD, will reduce the apparent optical depth. Two reasons can be found. First, light which is off-resonant will not experience any absorption. Sometimes, laser diodes have a broad pedestal which is not part of the main frequency mode. This light is produced by amplified spontaneous emission and observed as an incoherent background. It has a very small power density (power per frequency window) of . 10−4 compared to the main frequency mode, but a large spectrum over the whole gain profile of the laser diode (up to 50 nm). Second, probe light, resonant or not, which is scattered indirectly into the CCD and carries no information about the atomic cloud. We can find a correction for the finite observable optical density d0,max by applying a simple model. The intensity of the light reaching the CCD chip is a sum of light which experienced atomic absorption and a fraction ε which reached the CCD without interaction. Both are proportional to the incident light intensity Ilight Iatoms = (1 − ε)Ilight e−d0,real + Ilight ε . (3.5) We measure a transmission of Tmeas = Iatoms = (1 − ε)e−d0,real + ε Ilight (3.6) In the limit of large real optical density, d0,real → ∞, we would observe a remaining minimum transmission of Tmin = e−d0,max = ε (3.7) The real optical depth can be retrieved from the measured transmission via 1 − e−d0,max . d0,real = ln Tmeas − e−d0,max (3.8) Practically, the maximum observable optical depth is measured by using a dense cloud of atoms with a known density distribution. Implementation In the present experimental apparatus we implemented absorption imaging as shown in Fig. 3.9. Owing to the large on-axis optical depth, we record absorption images 3.2. Atom number measurement 43 Figure 3.9.: a) Schematic of absorption imaging of an elongated atom cloud. b) Resonant light fields for absorption imaging. only in the transverse direction. In this direction the maximum optical depth is approximately one, and straightforward to measure. For imaging we use linearly polarized light, resonant with the cycling transition F = 2 → F 0 = 3. Atoms can be probed many times before they are lost from the F = 2 manifold5 . The choice of linear light is not optimal. The polarization is defined by the optical pumping light, which shares the same optical path. We describe the optical pumping in Sect. 3.4. Compared to circular light, where atoms are driven in a closed transition, we observe a smaller scattering cross section than the maximum λ2 on the closed transition of 3 2π . To determine the exact scattering cross section, we have to use a model which includes optical pumping. It would not be sufficient to calculate the scattering cross section assuming equal population in F = 2. During the time the imaging light is on, atoms will be pumped into a steady state distribution among the hyperfine ground states in F = 2. In Fig. 3.10 we plot the results of a master-equation simulation. The left panel shows the evolution of the population in F = 2. In the right panel we compare the number of scattered photons. The black is the result of the ME and the red is calculated via the scattering cross section for this λ2 ' 1.356×10−13 m2 . The later would transition assuming equal population, σ0 = 75 2π overestimate the number of photons via Eq. (3.3). We find σ0 ' 1.587 × 10−13 m2 (Isat = 30.5 W/m2 ) which is ∼ 17 % larger than the former. 5 The probability to make a transition from F = 2 → F = 1 due to one scattered photon on the imaging transition is < 1.4 × 10−5 . This number can be deduced from the calculations shown in Fig. 3.10 knowing that 30 scattered photons cause 4 × 10−4 atoms to end up in F = 1. 44 3. Experimental apparatus and techniques 0.5 Populations 35 mF=0 Scattered Photons 30 ME calculations Theory weak excitation 0.4 25 0.3 mF=ï1,+1 20 15 0.2 10 0.1 mF=ï2,+2 0 0 25 50 75 t [µs] 5 0 0 25 50 75 t [µs] Figure 3.10.: Left: Population evolution during the imaging pulse. We plotted only the population of the five F = 2 magnetic substates, because the total population in F = 1 is only ∼ 4 × 10−4 . Right: Number of scattered photons as a function of time. The black solid line is the result of the master-equation simulations and the red dashed one the theoretical result for equal distribution of population in F = 2. Geometrically, the imaging beam is a large Gaussian beam of ∼ 5 mm waist. It is expanded in the horizontal direction by a one-axis telescope consisting of two cylindrical lenses to ∼ 8 mm. In this way, the whole sample is covered with light. The intensity drops from the center to the edges from ∼ 0.43 W/m2 to ∼ 0.14 W/m2 . Any power broadening can be neglected since the intensity is only a few percent of the saturation intensity over the whole sample. The timing of the imaging light is as follows. For each picture we shoot a rectangular pulse of constant power and a duration of 300 µs. The separation between the pictures is 80 ms. This time is only limited by the frame rate of the camera. Before the shadow image is taken we start to pump atoms from F = 1 to F = 2 by applying the repumper laser for 100 µs. It is also kept on during the shadow image. Characterization Here we present the two main results for the characterization of the absorption imaging. i.e., the maximum measurable optical depth and linewidth of the imaging laser. Both are important to guarantee the quantitative aspect of the absorption imaging. Maximum OD We produce a dense cloud of atoms by loading the 3D MOT for 5 s. Then we take the absorption images as explained above. The raw data for a single experiment 3.2. Atom number measurement 45 2D Fit Data Residual Figure 3.11.: Left: Transmission image of dense atomic cloud. Center: 2D fit using a rotated Gaussian, where the rotation angle was a free parameter. Right: Residual of the fit. and the deduced transmission image are plotted in Fig. 3.8. This measurement is repeated 20 times to acquire statistics. The results are shown in Fig. 3.11. The atomic cloud has an optical depth which is larger than d0,max and, therefore we see the saturation of the transmission signal. We deduce d0,max by assuming a Gaussian density distribution of the atomic cloud 2 −2B(x−x d0,real (x, y) = d0,real (0)e−A(x−x0 ) 2 0 )(y−y0 )−C(y−y0 ) , (3.9) with A = cos2 θ sin2 θ + 2σx2 2σy2 (3.10) B = sin(2θ) sin(2θ) − 4σy2 4σx2 (3.11) C = sin2 θ cos2 θ + 2σx2 2σy2 (3.12) and express the observed transmission as Tmeas (x, y) = e−d0,real (x,y) 1 − e−d0,max + e−d0,max . (3.13) For the Gaussian density distribution we assume a very general two-dimensional distribution with different width in two orthogonal directions σx and σy which can be rotated by an angle θ. The center plot in Fig. 3.11 shows the result of the twodimensional fitting. We find a maximum optical depth of d0,max = 3.10(4). This value will be used throughout the rest of the thesis to correct absorption images. If the imaging setup is changed, the calibration of d0,max has to be repeated. Linewidth measurement For the second essential characterization, we measure the linewidth and absolute frequency of the imaging laser. The knowledge about the absolute frequency is 46 3. Experimental apparatus and techniques 1.5 Optical Depth 1 0.5 0 −10 −5 0 5 Detuning from F=2 −> F‘=3 [MHz] 10 Figure 3.12.: Linewidth measurement with atomic cloud. The real optical depth d0,real is plotted and we deduce a linewidth of Γmeas = 6.18(85) MHz and a peak position at ∆0 = 0.24(25) MHz. crucial for quantitative atom number measurement. Another important aspect is dispersive effects in the absorption imaging. If the imaging laser is not exactly on resonance, the atomic sample would act as lens6 . This can lead to image distortions, which lead to systematic errors [88]. The linewidth is interesting as a measure on its own. It will give important insight into the quality of the laser lock used. For the experiment we load the 3D MOT as before, but stop after 2s. This ensures a modest optical depth of ∼ 1, well below d0,max . The detuning of the imaging beam is changed between −6 to 6 MHz around the estimated resonance and the d0,real is determined by the three-image procedure as described above. The results are plotted in Fig. 3.12. The linewidth and center position are deduced by fitting a power broadened Lorentzian line to it. The saturation parameter s is deduced from the local intensity which is measured directly with the reference images Ilight . The average saturation parameter over the whole atomic ensemble is I/Isat = 0.014. The measured line width is ∼ 110 kHz larger than the natural line width. We take this amount as the linewidth of the imaging laser. The center-position is shifted 240 kHz to higher frequencies compared to the resonance-position we deduced form absorption spectroscopy. All absorption imaging measurements reported in what follows were made at this adjusted line center. 6 The atomic cloud effectively acts as a gradient-index lens. 3.3. Dipole trap characterization 47 3.3. Dipole trap characterization We use a single strong far off-resonant laser which is gently focused to trap pre-cooled atoms state-independently in a small spacial region. The concept of the AC Stark or light shift can be found in many quantum optics textbooks [30, 36]. Here, we collect a set of parameters which are important for later interpretation and understanding of other experimental results. 3.3.1. Trap depth and sample shape We use the simplest realization for a dipole trap, a gently focused laser beam. In the case that the laser is red detuned and has a Gaussian profile, atoms are attracted to the point of largest intensity. We will only discuss this case here, whereas other configurations of blue detuned light could be imagined using Laguerre-Gaussian beams to trap atoms at an intensity minimum. We furthermore restrict the discussion to linearly polarized light, for which all Zeeman substates stay degenerate. However, later in Sect. 3.4.2, we will discuss the influence of circularly polarized trapping light as a decoherence mechanism for spin polarized atoms. Dipole potential The shape of the dipole potential follows the intensity distribution, I(r), of the focused Gaussian beam 2 exp − w2ρ 2 (z) I(r) = I(ρ, z) = I0 (3.14) 2 . 1 + zzR Where, ρ and z are transverse and longitudinal coordinates which both have their origin in the focal point. The Rayleigh range zR is a measure of the length of the r 2 focus7 . The function w(z) = w0 1 + zzR describes the waist size at position z starting at the minimum waist w0 . The peak intensity for a Gaussian beam with 2P given power, P , is I0 = πw 2 . 0 We write the potential as the product of the light shift at peak intensity and the spatial characteristics of the Gaussian beam 2 exp − w2ρ 2 (z) U (ρ, z) = ς(λ)I(ρ, z) = ς(λ)I0 (3.15) 2 . z 1 + zR In Appendix B.2 we have calculated ς(λ) for a large spectrum of frequencies. Here, we are only interested at the value for λ = 1030 nm, which is ς = −3.4573 × 7 More precisely, it measures the distance at which the intensity on axis drops to one half of the intensity at the focus. 48 3. Experimental apparatus and techniques 10−3 Hzm2 /W. The dipole trap laser has a minimum waist in the focus of w0 = 52 µm and we usually use P = 6 W, which results in a maximum potential depth U0 /h = U (0, 0)/h = −4.9 MHz. Expressed in thermal energy this corresponds to trap depth of U0 /kB = 234 µK. Sample shape and harmonic approximation The shape of the sample, or better, the distribution of atoms in space, can be deduced from the trap potential. Under the assumption of a thermalized atom cloud, we can write the number density using the Boltzmann factor ! Ũ (ρ, z) n(ρ, z) = n0 exp − , (3.16) kB T e (ρ, z) = U (ρ, z) + |U0 | is the shifted trap potential which ensures zero powhere U tential energy in the focus of the dipole beam. Equation (3.16) has two unknown parameters, the peak density, n0 , and the temperature, T , of the atoms. If the atoms have a temperature much smaller than the potential depth, i.e., kB T |U0 |, we can e as a harmonic potential. Usually this is a good approximation for approximate U atomic samples with similar extension in all direction, as for instance in a magnetic trap. However, great care has to be taken when the sample is very elongated and trapped in a gently focused laser beam, as in our case. In axial direction, the shape of the atomic sample is Lorentzian rather than Gaussian. The axial shape of the sample is measured in Sect. 3.3.5. Nevertheless, for the radial direction we can make the approximation of a harmonic potential and find e2D harm. (ρ, z) = U0 U 2ρ2 w(z)2 1+ + z zR z2 2 zR 2 . (3.17) It is interesting to compare the full and the approximated potential and the resulting number density distribution. In Fig. 3.13 we plot the equipotential line with the thermal energy for both models in blue. The resulting density distributions, drawn as black lines, are rather similar. In the experiments of Sect. 3.3.4 we observe a Gaussian density distribution in transverse direction. Atomic Waist It is useful to introduce a parameter ζ which relates the thermal energy of the atoms to the potential depth. In literature this is also referred to as the filling factor . ζ≡ kB T . U0 (3.18) 3.3. Dipole trap characterization 49 1.2 1.2 Non−trapping Region Non−trapping Region 1 Trapping Region ρ [w0] 0.8 0.6 1 0.8 Trapping Region 0.01 0.6 0.01 0.4 0.1 0.2 0.5 1 1.5 2 2.5 0 0.01 0 0 0.2 0. 5 0.01 5 0. 0.2 0.4 0.1 0. 2 0.5 1 1.5 2 0 2.5 z [zR] z [zR] Figure 3.13.: We show in black the normalized atomic density and in blue the equipotential line for the trapping potential when it equals the thermal energy. The left plot is for the full potential from the Gaussian beam (3.15) and the right for the 2D harmonic approximation (3.17). On one hand, we have a comparison of energy scales, which is always handy. On the other hand, we can use ζ to verify the approximation of a harmonic potential. For the verification we introduce ζ into the expression for the number density (3.16), under the harmonic approximation (3.17), and at position z = 0: 1 2ρ2 . (3.19) n(ρ, 0) = n0 exp − ζ w02 For atomic sample we find now a characteristic extension, which is similar to the waist of a Gaussian beam. In the future we will call this the atomic waist 8 given by p wa ≡ ζw0 . (3.20) 3.3.2. Loss through light scattering In order to estimate the influence of photon scattering on the trap lifetime, we calculate the scattering rate and from this the heating rate for our trap parameters. The scattering rate for a two level system is calculated as (see for example [17]) Rsc, 2 level = 1 2 Γ 2Ω , 1 2 2 ∆ + 2 Ω2 + 14 Γ2 (3.21) where Ω is the Rabi frequency, Γ the excited state line width, and ∆ = ωL − ωge the detuning from resonance. The laser frequency is ωL and the resonance of the atom two level atom is ωge . In the limit of large detuning, ∆ Ω, Γ, the denominator can 8 The relation to the full width at half maximum (FWHM) is given by wFWHM = 1.18wa . √ 2 ln 2wa ' 50 3. Experimental apparatus and techniques be approximated as ∆2 . For very large detunings, as we find them in our dipole trap, the RWA is no longer applicable and we have to modify the detuning. Therefore, we use expression (B.8) introduced in Appendix B.2. In addition, a correction for the DC limit, i.e., ωL → 0, has to be introduced, which ensures that the scattering rate approaches zero. Similar to the argument in Appendix B.2, the classical Lorentz oscillator can be applied and gives a prefactor of (ωL /ωge )3 . For the multilevel case we sum over all excited states X ωL 3 Γf Ω2if Rsc = , ˜2 ωif 4∆ f 6=i if ˜ if is defined in Eq. (B.8). where i and f label the initial and final states, and ∆ For the Yb:YAG laser we use in the experiment, the scattering rate as a function of intensity is Rsc = I × 2π × 1.14 × 10−10 s−1 m2 /W, where I is the intensity of the laser. The atomic sample is heated due to absorption and re-emission of photons from the trap laser. Absorption occurs in the propagation direction of the laser and is therefore anisotropic. Emission happens isotropically if we average over all polarizations. If we sum up, the sample is heated in longitudinal direction ~ ωrec + 31 ωrec and in the two transverse directions each with 31 ~ωrec , where ~ωrec = ~2π × 3.6325 kHz is the energy of one recoil, or expressed in temperature units 348.66 nK.. In total, each scattered photon leaves a mean energy of 2~ωrec to the atom. Hence, we can express the heating rate in term of the scattering rate by Rheat = 2~ωrec Rsc . For our trap parameters we get a heating rate of 112 nK/s at the center of the trap. This is negligible compared to the trap depth of 234 µK and implies a loss rate due to photon scattering of 112 nKs−1 /234 µK ' 4.8 × 10−4 s−1 . This number should be the same in the whole trap, since scattering rate and trap depth have the same scaling with intensity. Here we summarize some auxiliary measurements which are standard tools in any atom trap laboratory. They are performed after major changes in the trap setup, as for instance after changing the single mode fiber for the high power trapping laser or optimization of the MOT-loading. 3.3.3. Trap lifetime Measuring the lifetime of optically trapped atoms not only gives an idea how much time the experimenter has to ask his questions, but also reveals import parameters like the background pressure of the vacuum setup. There are different mechanism which limit the lifetime. One can categorize them by their scaling with atom number. Independent of particle number are collisions of trapped atoms with background gas atoms or molecules. Equally, inelastic scattering of the trapping light is an atom- 6 Number of Recaptured Atoms [x10 ] 3.3. Dipole trap characterization 51 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 40 60 Trapping Time [s] 80 100 Figure 3.14.: Recaptured number of atoms as a function of trapping time. We fit the data with Eq. (3.23) plus an offset Noffset . The fit parameters are N0 = 7.6(1) × 105 , α = 4 × 10−6 s−1 (upper bound 3.3 × 10−4 s−1 ), β = 7.9(6) s−1 , and an offset of Noffset = 1.3 × 103 (±1.3 × 104 ). number independent loss mechanism9 . Density dependent loss mechanism are twoand three-body collisions between trapped atoms. In general, we can describe the atom number dynamics with a rate equation with different terms dN = −αN − βN 2 − γ N 3 . dt (3.22) The coefficient α , β and γ describe the back-ground, two-body and three-body losses. To measure the trap loss, we load atoms for about three seconds as described in 3.1.3. After a variable trapping time t we measure the remaining number of atoms in the trap by recapturing them into a MOT. Fluorescence measurements from MOT atoms give a signal proportional to the number of trapped atoms, while fluorescence measurements of absolute atom number are difficult to calibrate, here we are interested in the relative atom number. To gather enough statistics, we repeat the trap loading several dozen times. For every other measurement, we inhibit the MOT loading and and dipole trapping to measure the background fluorescence signal. In Fig. 3.14 we plot the number of recaptured atoms as a function of trapping time. We extract the loss parameters by fitting the solution of Eq. (3.22) to the data points, neglecting three-body collisions, i.e., γ = 0. For our atom densities of n ∼ 1011 cm−3 this is a safe assumption . The solution for arbitrary α and β is β 1 β −1 N (t) = eαt + − . (3.23) α N0 α In the limit of short times or dominant density dependent losses, α → 0, this reduces to N0 N (t) = . (3.24) 1 + N0 βt 9 One can imagine that at very high atomic densities the re-absorption makes light scattering density dependent. However, for the atomic densities and the elongated geometry, we do not expect this to be the case. 52 3. Experimental apparatus and techniques To out surprise, we see negligible linear losses and almost no exponential decay. The loss of atoms from the trap can be described solely with two body collisions. The solid line in Fig. 3.14 uses Eq. (3.23) with the parameters mentioned in the caption. The linear loss coefficient is tiny and it would be possible to fit the data points only with Eq. (3.24). If we compare the loss rate α = 4 × 10−6 s−1 to the estimated loss due to inelastic photon scattering from the dipole laser of 4.8 × 10−4 s−1 , calculated in Sect. 3.3.2, it becomes clear that it is hard to adjoin any physical meaning to the found α. Nevertheless, we can use the upper bound of the error in the fitted value αmax = 3.3 × 10−4 s−1 to estimate an upper limit for the back ground pressure in the 3D MOT. At very low pressure the background gas contains mostly H2 and He. Helium has a lower collision-rate with 87 Rb than hydrogen. Therefore, we take the collisional rates calculated in [89] and estimate an upper bound for the back ground gas pressure. We find that the background pressure is as low as 0.7 × 10−11 mbar. 3.3.4. Atom temperature Another interesting parameter for us is the temperature of the dipole trapped atoms. To measure the temperature we use the time-of-flight (TOF) technique [86]. It relies on the very definition of temperature for a thermalized ensemble of particles, i.e., the kinetic energy. When atoms are released from the trap, i.e., the trapping potential is removed, they will expand ballistically depending on its initial position and momentum at time t = 0. Theoretically, one can show that the characteristic size of the ensemble is described by [86] r 4kB T 2 wa (t) = wa2 (0) + t . (3.25) m This expression contains at most two free parameters: the initial extension of the ensemble wa (0) and its temperature T . We will limit ourselves to the measurement of the transverse extension wa of the atom cloud. In longitudinal direction the measurement is more difficult, since atoms at micro Kelvin temperatures, expand very little compared to the sample length of several millimeters. Strictly speaking, the measured temperature is primarily connected to the kinetic energy in transverse direction. However, by assuming that atoms were loaded from an isotropically cooled MOT cloud and are not further cooled in the dipole trap, both transverse and longitudinal dimensions have the same kinetic energy. To measure the extension directly, we have to image the atom cloud while it is expanding. Usually, the measurement is done by imaging the atoms, either in fluorescence or absorption. Both of these techniques are in general destructive or have a slow frame rate. Therefore, we have to repeat the experiment several times for different expansion times. For the experiment we load atoms into the dipole trap, as described in 3.1.3. We use 6 W of trapping light which produces a maximum trap depth of 234 µK. After some thermalization time of 1.2 s we switch the trapping laser off and take a fluorescence image after time t. For the fluorescence image, the atoms are illuminated by the 3.3. Dipole trap characterization 53 MOT beams for a short period of a few tens of microseconds. The imaging system collects the fluorescence light onto a charge-coupled device (CCD). Figure 3.15.: Fluorescence images of time-of-flight measurement. The atomic cloud is released after 1.2 s of thermalization. Expansion times are indicated in the lower right frame. In Fig. 3.15 we plot the fluorescence images for different expansions times from 0 to 4 ms. Each frame is an average of 10 individual images. The vertical stripes, mainly visible for longer expansion times, are an artifact from the video electronics. For the analysis, we integrate each frame along the longitudinal direction (horizontal) and determine the center and the characteristic extension of the Gaussian profile. In Fig. 3.16 we plot the profiles in a semi-logarithmic plot. The distribution is clearly Gaussian with expanding width and moving center. Free falling cloud The center position of the free falling cloud, can be used to check the imaging magnification by measuring the gravitational acceleration g. In the left of Figure 3.17 we plot the center positions as a function of expansion time. We determine the gravitational acceleration g by fitting s(t) = 12 g (t − t0 )2 + y0 to the data points, where t0 Integrated Fluorescence Signal [arb. units] 54 3. Experimental apparatus and techniques 3 10 2 10 ï500 ï400 ï300 ï200 ï100 0 100 Vertical Position [µm] 200 300 400 Figure 3.16.: Integrated fluorescence profiles. Note the transverse dimension is measured from top to bottom, i.e., increasing time of flight. is the time delay for switching off the dipole trap, and y0 the exact initial position. We find g to be 9.98(+0.39/ − 0.40) ms−2 . This is less than two percent of error to the accepted value of 9.81 ms−2 . For the delay we find 41(+78/ − 78) µs and for the initial position y0 = 0.15(+57/ − 58) µm. The error in g can be mainly attributed to the calibration of the imaging system. Expanding cloud The expanding width of the atomic cloud contains the information about the temperature of the atoms in the trap. The characteristic extension of the atomic cloud wa is plotted in the right of Figure 3.17 as a function of time. The data points are fitted by expression (3.25), where we substituted t by t − t0 , to account for the uncertainty of the trap opening. As initial waist we use the waist measured at times close to zero wa (0) = 16.65(7) µm as a fixed parameter. For the initial delay we find 38(±27) µs, which is very close to the value we deduced from the center-of-mass trajectory. The temperature of the atomic sample is 25.7(6) µK 10 . From the temperature we can calculate the filling ratio of the optical trap, which we defined in Eq. (3.18) comparing the thermal energy kB T and the potential depth U0 , which is calculated from the laser power and the measured waist of the trapping laser w0 . We find that the filling ratio is ζ = 11.0(3) % of the total trap depth. Note, we have repeated the same measurement for a shorter thermalization time of only 0.3 s. The temperature we find is slightly smaller at 24.3(10) µK. Since both values are similar we have not investigated the evolution of the atom temperature as a function of trapping time. For all experiments we will perform on the atoms the trapping times will be between 0.3 and 1.2 seconds. 10 If we use wa (0) as free parameter, the fitting gives very similar results. We find t0 = 35(+34/ − 33) µs, wa (0) = 16.3(+4.4/ − 4.5) µm, and T = 25.6(7) µK. 3.3. Dipole trap characterization 55 450 0 400 −10 350 Atomic Waist wa(t) [µm] Vertical Position [µm] −20 −30 −40 −50 −60 300 250 200 150 −70 100 −80 50 −90 0 1 2 3 Time of Flight [ms] 0 0 4 1 2 3 Time of flight [ms] 4 Figure 3.17.: Left: Center-of-mass position of the expanding cloud. Right: Characteristic width of the expanding cloud. 3.3.5. Axial shape Through the imaging of the atomic cloud, either with fluorescence or absorption imaging, we are able to deduce important parameters, as for example temperature and transverse dimension of the atomic ensemble. The axial shape of the atomic ensemble does not carry as much information as the transverse dimension, but it is still interesting to have a closer look. In Fig. 3.18 we plot the integrated fluorescence signal over the transverse dimension ρ of the sample. The black solid line is a fit with a Lorentzian of the form L(z) = A , 0 2 1 + ( z−z za ) where A is the amplitude of the observed signal, z0 is the position of the peak density and za is the half width at half maximum (HWHM) of the sample. We define the length of the atomic sample √ as La = 2za . We expect a relation between Rayleigh range and La of La = 2 ζzR , similar to the one we found for the beam waist w0 and the atomic waist wa in Eq. (3.20). Alternatively, we use the trapping potential in its 2D harmonic approximation (3.17) to calculate an analytic expression of the signal from the axial sample shape, which is shown as the orange dashed line. the imaging integrates over the two transverse dimensions. The observed signal is proportional to the integral of the atomic density (3.16) over ρ Z ∞ S(z) = 2π ρn(ρ, z)dρ . (3.26) 0 The atomic density n(ρ, z) is calculated from Eq. (3.16) the trapping potential in 56 3. Experimental apparatus and techniques Fluorescence [norm.] 1 0.75 0.5 0.25 0 0 2 4 6 8 z [mm] 10 Figure 3.18.: Axial shape of the trapped atomic ensemble measured with fluorescence imaging. The black solid line is a Lorentzian fit and the dashed line a fit with the analytic solution of the integrated atomic density. the 2D harmonic approximation (3.17) 1 1 z2 n2D (ρ, z) = n0 exp − 2 exp − 2 ζ z + zR ζ 2ρ2 . 2 2 z 2 w0 1 + zR (3.27) This leads to the measurable signal π S(z) = ζ w02 2 1+ z zR 2 !2 1 z2 exp − 2 2 ζ z + zR . (3.28) For the two fits in Figs. 3.18 we get the following result. On one hand, the simple Lorentzian, shown as solid black line in Fig. 3.18, √ gives a sample length (FWHM) of La = 3.0(3) mm, where we would calculate 2 0.11 × 8.2 mm = 5.4 mm from the filling ratio ζ = 0.11 and the Rayleigh range of the trapping beam (zR = 8.2 mm). Going the opposite direction by calculating the trapping beam properties from the 0 = 4.8(1) mm for a measured signal S(z) gives a hypothetical Rayleigh range of zR fixed filling ratio ofζ = 0.11. We measure the longitudinal distribution in the trap. Unlike the transverse dimension, the imaging shows a distribution which is more compact than the predicted Boltzmann distribution. This is plausible, since the MOT which loads the trap is smaller than the Boltzmann distribution and because the trap frequency is low. If we would allow the sample to thermalize longer it would probably lead to a more extended sample, which, at the same time, will have less atoms due to the trap loss. 3.3. Dipole trap characterization 57 We note that the actual longitudinal sample shape has no major consequences for the subsequent experiments. In Chap. 4, for example, we measure an optical depth which is among the largest for these kind of system. However, in the future it might be interesting to explore possibilities of loading even more atoms into the trap. This could be achieved by translating the MOT cloud along the dipole trap potential to load atoms over a larger spatial range. 58 3. Experimental apparatus and techniques 3.4. Spin state preparation We discuss now the possibilities of creating polarized spin states via optical pumping. A ’basis’ set of polarized spin states, with either orientation or alignment, can be created through the proper choice of polarization, frequency and propagation direction of the light field used for optical pumping. In Sect. 3.4.1 the principle of optical pumping is explained and the evolution of the single atom density matrix is simulated for realistic experimental conditions. At the end we discuss polarization errors in the dipole as a possible decoherence mechanism. 3.4.1. Optical pumping into dark states For given F , F 0 , and given polarization, there may be states in the F manifold that are not coupled to any state in the F 0 by the given polarization. These do not scatter light and are called dark states. For example, for F = 1 and F 0 = 1, with σ + polarization, by angular momentum addition rules, the state with mF = 1 would be coupled to the state F 0 = 1, mF 0 = mF + 1 = 2. But there is no such state, so mF = 1 is coupled to no F 0 = 1 state by σ + polarization and is thus a dark state. If there is only one dark state for a given polarization it is possible to produce highly polarized spin states using optical pumping which is an incoherent processes. If we limit ourselves to the polarization basis states of σ + , σ − , and π light, we can give necessary conditions for the existence of a dark state in the ground state |F, mF i of a particular multilevel atom which is coupled to an excited state manifold with F 0 . For circular polarization σ ± , mF is a dark state, if F 0 < 1 ∓ mF . (3.29) In the case of linear polarization along the polarization axis, i.e., π-light , a dark state exists, if mF = 0 and F = F 0 . (3.30) The two states we are interested in are the following ones: First, the z-polarized state: |Φi ≡ |1, ±1i, which has the maximum projection on F̂z . From criterion (3.29) we see that |Φi is only a dark state if the laser is resonant to the transition F = 1 → F 0 = 1. √ Second, the x-polarized state: |Ξi ≡ {|1, −1i − |1, +1i} / 2, which is polarized along the pseudo-spin direction Jˆx . The latter state is an example of a dark state for linear polarization. If we define x as the quantization axis and use π light resonant to F = 1 → F 0 = 1, the dark state is |1, 0ix , where the subscript indicates the choice of quantization axis. Changing into the ’normal’ frame of reference, i.e., with the quantization axis points along z, we find Ry (−π/2) |1, 0ix = |Ξi, where Ry (−π/2) is a rotation of the system about y by an angle of −90◦ . Experimentally, we apply pumping light on the F = 1 → F 0 = 1 transition either along the sample axis in σ polarization or orthogonal to the sample axis in linear polarization. At the same time we apply light resonant to the transition F = 2 → F 0 = 2. This transfers atoms back into F = 1 which were scattered into F = 2. We 3.4. Spin state preparation 59 Figure 3.19.: a) Geometrical configuration for x and z optical pumping. The light for pumping atoms into the state |Φi(see text) is circularly polarized and sent along the longitudinal direction. For the state|Ξi (see text) the light comes from the side orthogonal to the long sample axis. b) The pumping light is resonant to transition F = 1 → F 0 = 1 and the light to deplete F = 2 is resonant to F = 2 → F 0 = 2. call the laser F=2 depletion. It is applied through the same optical path as the beams for the MOT. This ensures two things. First, the illumination from six different directions will create a polarization pattern which has a sub-wavelength structure. This is very important to avoid a pumping into a dark state of F = 2, which exist for the case of linear polarization, as we can see in Eq. (3.30). Thermal motion or oscillations of atoms in the trap will produce an effectively unpolarized light field. Atoms will move randomly through the polarization pattern and experience all kinds of polarizations in a random order. If the motion of atoms is fast compared to the inverse of the scattering rate from the F=2 depletion laser, atoms never enter into a dark state. Even if an atom was pumped into a dark state, it will be repumped an instant later when it has changed place and the local polarization is different. Second, the mechanical action of absorption and re-emission is balanced and atoms are not moved on average. In Fig. 3.19 we show the experimental implementation of the two pumping schemes. The z-polarized state is generated by sending circularly polarized light along the long sample axis. For the coherent superposition state we send linearly polarized light from the side. 60 3. Experimental apparatus and techniques Simulations We perform numerical simulations of the pumping dynamics from a completely mixed state in F = 1 into a dark state without and with the F=2 depletion laser. We are interested in the time-scale of the pumping process for realistic experimental parameters, the number of scattered photons and the maximum fidelity which can be reached. The main tool for the simulations are explained in Appendix B.3. As a figure of merit we use the mixed state fidelity which is introduced in Appendix B.3 in Eq. (B.13). The general dynamics will be similar for the states |Φi and |Ξi and we concentrate on the case of x-polarized state |Ξi. The parameters for the simulation are chosen close to the experimental conditions. The intensities for pumping and F=2 depletion are Ipump = 0.44 W/m2 and Idepl. = 3.6 W/m2 which correspond to the measured values. The unpolarized depletion laser is modeled by changing the polarization randomly between two orthogonal states, which are in our case x and y polarization. The actual choice of the depletion polarization in unimportant for the simulations. A small bias field of Bx = 0.5 G is included as well. Population 0.5 mF=±1 0.4 0.3 0.2 0.5 mF=0 0.1 Re(l) 0 0 Error ï0.5 ï1 0.1 F=1 0+1 ï2 0.01 ï1 F=2 0.001 0 20 40 60 80 Time [µs] 0 +1 +2 ï1 0 +1 +2 ï1 0 F=2 +1 ï2 F=1 Figure 3.20.: Left-Top: Evolution of hyperfine ground state population during optical pumping into the dark state |Ξi starting from a completely mixed state in F = 1. No F = 2 depletion laser is applied. Left-Bottom: Error in state preparation calculated for all eight hyperfine ground states using a mixed state fidelity [90]. Right: Final density matrix (real part) of the hyperfine ground state. The Blue and dark red parts are the blocks for the two hyperfine manifolds F = 1 and F = 2, respectively. In Fig. 3.20 we plot the evolution of the populations of the eight hyperfine ground states, the fidelity, and show the final density matrix in the case of no F = 2 depletion laser. We see that we can reach a maximum fidelity of ∼ 80 % . The missing ∼ 20 % reflect directly the amount of atoms which are pumped into F = 2. If we instead apply light for the depletion of F = 2 at the same time as the optical pumping light, the situation changes considerably. In Fig. 3.21 we can see that the error in the state preparation is reduced to ∼ 10−3 . In the main graph of Fig. 3.22 we plot the number of scattered photons for a single atoms. We can see that an atom reaches 3.4. Spin state preparation 61 the desired state by scattering fewer than two photons on average. That means, the energy deposited in the atomic cloud is rather small and won’t heat atoms too much while pumping them. As we calculated in Sect. 3.3.2 the scattering of a photon will increase the thermal energy of an atom by two recoil energies (Erec /kB = 0.35 µK), which is negligible compared to the trap depth. Population 0.5 mF=±1 0.4 0.3 0.2 0.5 mF=0 0.1 Re(l) 0 0 Error ï0.5 ï1 0.1 F=1 0+1 ï2 0.01 ï1 F=2 0.001 0 20 40 60 80 Time [µs] 0 +1 +2 ï1 0 +1 +2 ï1 0 F=2 +1 ï2 F=1 Figure 3.21.: Left-Top: Evolution of hyperfine ground state population during the optical pumping into the dark state |Ξi starting from a completely mixed state in F = 1 The F = 2-depletion light field is applied simultaneously tot the pumping light F = 1 → F 0 = 1. Left-Bottom: Error in state preparation calculated for all eight hyperfine ground states using a mixed state fidelity [90]. Right: Final density matrix (real part) of the hyperfine ground state. The Blue and dark red parts are the blocks for the two hyperfine manifolds F = 1 and F = 2, respectively. The amount of photon scattering, once the atoms are pumped into the dark state, is reduced by two orders of magnitude as we can see in the inset of 3.22. Off-resonant absorption is responsible for the remaining scattering, which could be reduced if the optical pumping would be performed on the D1 transition of 87 Rb, where the excited state hyperfine splitting is with 814 MHz, much larger than on the presently used D2 transition. 3.4.2. Loss of spin polarization The spin polarization can be reduced due to decoherence and dephasing mechanisms. Here, we discuss some of the main mechanisms for the loss of spin polarization which could be relevant for our system. One important mechanism for the loss of spin polarization are inhomogeneous magnetic fields. This issue will be discussed in great detail in Chap. 5. 62 3. Experimental apparatus and techniques No. scatt. photons 2 0 1.5 20 40 60 80 t [µs] 5 10 R sc [sï1] 1 4 10 0.5 3 10 0 0 20 40 60 80 Time [µs] Figure 3.22.: Average number of photons scattered by an atom in the course of optical pumping as simulated in Fig. 3.21. The inset shows the scattering rates of pumping (dotted black line) and F = 2 depletion laser (dahsed blue line). Number of scattered photons and scattering rate are calculated as described in Appendix B.3. Dephasing due to elliptical trapping light Any elliptically polarized trapping light, which couples differently to individual magnetic substates mF , will break the degeneracy of the hyperfine ground states. The coherent superposition state |Ξi will start to precess about z. The same effect is described by the QND part of the Hamiltonian (2.53), which is ∝ G1 Ŝz Jˆz . The coupling constant is proportional to the vectorial part of the polarizability tensor, which describes the AC Stark effect. We can make an estimate of the effect by calculating the energy shifts of all hyperfine ground states for a completely circularly polarized dipole trap. In Fig. 3.23 we plot the results for the ground state as the difference compared to the light shift from linear trapping light. We can see that the energy difference between |1, −1i and |1, +1i is about 100 kHz in the focus of the dipole trap. The state |Ξi would precess at this frequency about the z axis. We get the same result if we would calculate the coupling constant G1 for from the interaction of the trapping laser on the D1 and D2 lines. We use trapping light which is mostly linear, with < 10−5 of circular contributions. This is estimated from the specifications of the polarization dependent components in the setup and the output of the laser. A maximum precession frequency of 10−5 × 100 kHz = 1 Hz could be introduced by this amount of circular light. We can conclude that the effect of hypothetical circular polarization of the trapping light will not cause a major problem for the polarized spin state. Decoherence via spin-exchange collisions A decoherence mechanism for spin-polarized atoms are spin changing collisions [91]. In a binary collision event atoms with initial magnetic quantum number mF,1 and mF,2 are transferred into a final state characterized by the quantum numbers mF,3 and mF,4 . The total spin projection on the quantization axis is a constant, i.e., 3.4. Spin state preparation 63 6 Em+ [kHz] 100 0 ï100 F=2 100 F=1 0 ï100 ï2 ï1 0 1 2 mF Figure 3.23.: Level shifts of magentic substates in the two hyperfine manifolds at the focus of a completely circular dipole trap. mF,1 + mF,2 = mF,3 + mF,4 . For example we can have |1, 0i + |1, 0i → |1, −1i + |1, +1i . This leads to a decoherence of the initially polarized spin state 11 . From the cross-section of exchange as [94] 87 Rb-87 Rb collision we can calculate the rate of spin - ΓSE = nσSE v̄ , where n ∼ 1011 cm−3 is the atomic density, σSE = 1.9 × 10−14 cm−2 [95], and v̄ is the average relative velocity of atoms. For our temperatures of T = 25 µK, we find v̄ = 2vrms = 16 cm s−1 . The expected spin exchange rate for our systems is therefore, ΓSE = 0.03 s−1 . 11 In contrast, condensed ultracold quantum gases can show coherent collisions both in an extended BEC [92] and in the Mott insulator phase [93]. 64 3. Experimental apparatus and techniques 3.5. Probing atomic spins An essential part of the experiment are dispersive spin measurements. In Chap. 2 we discussed the spectral properties of the vectorial and tensorial polarizabilities, which are the main handles to get information about the atomic spin state. To address them individually we have to have very good control over the polarization properties of the light and the possibility to tune the laser within a large range around the atomic resonances in on the D2 -line. There are two interesting spectral regions for us. In Fig, 3.24 we marked them blue and green. To measure the coherent spin state polarization along the pseudo-spin direction Jˆx (blue region) we use the tensorial part of the effective Hamiltonian (A.17). The tensorial polarizability very quickly decays away from the resonances as ∆−2 . Therefore, we want to tune the laser only a few hundred MHz to the red of the F = 1 → F 0 = 0 transition. Hereafter, we call the laser which probes the Jˆx -component the tensor-probe. The QND measurement of the Jˆz -component relies on the vectorial part of the polarizability tensor. Since, the relative strength of vectorial to tensorial part increases linearly with the detuning, we want to put the probe laser rather far away from resonance at a distance of 0.5 to 2.5 GHz to the red of F = 1 → F 0 = 0 (green region). We call this probe the vector-probe. Figure 3.24.: Spectral probe region for tensor- and vector-probe marked in blue and green related to the level scheme of 85 Rb and 87 Rb. The gray area indicates the spectral region accessible by locking to F = 2 → F 0 transition of 85 Rb. The four black Gaussian peaks show the Doppler-broadened absorption spectrum for 87 Rb and 85 Rb. 3.5. Probing atomic spins 65 3.5.1. Polarization control and photon-number referencing The heart of the spin measurement is a polarization rotation measurement on light. Therefore, it is of utmost importance to have precise control over the polarization state of the probe laser. At the same time, we also need a reliable measurement of the number of photons that reach the atomic sample. The following description applies for both the tensor- and the vector-probe beams. In Fig. 3.25 we show the essential parts of the polarization control and photon-number referencing setup. It is separated into five parts, which are explained now. Figure 3.25.: Polarization cleaning and preparation scheme for vector- and tensorprobe beams. a) Linearly polarized light is coupled through a polarizing beams splitter cube into a polarization-maintaining single mode fiber. b) At the output side the light is projected into a well defined linear polarization by transmitting it through a second polarizing beam-splitter. c) To monitor the photon-number we install a reference detector which is measuring a portion of the probe light. The splitting ratio can be adjusted freely with a half wave-plate and a PBS. d) The polarization is cleaned a last time and a combination of half and quarter waveplate can be used to compensate optical elements which come farther down stream. e) Finally, the probe light is mode matched over two mirrors onto the dipole trap beam and overlapped on a dichroic beam-splitter cube (DCC). In the very beginning of this chapter we mentioned that all light that reaches the atoms is coupled with polarization-maintaining (PM) single mode optical fibers. This is of huge advantage and makes the setup stable and flexible. As we learned over time, it is also important to use the PM fibers in the right way. For all fibers we install a PBS in front of the input coupler and align its axis either orthogonal or parallel to the slow axis of the PM fiber. This is shown in Fig. 3.25a. Usually we measure extinction ratios of more than 30 dB with a fiber coupled polarimeter (Thorlabs PAX5710IR1). Fibers prepared in this way show very little polarization distortion when put under stress by bending. 66 3. Experimental apparatus and techniques When the light leaves the PM fiber we clean it with a second PBS, shown in Fig. 3.25b. This is to project the light into a well defined linear polarization state of high purity. For the photon-number referencing we use a combination of half-wave-plate (HWP) and PBS to split the beam into two modes (cf. Fig. 3.25c). One part propagates towards the atomic cloud and the other serves as a reference. We record it with a fast photo-detector 12 and save the output on a digital storage oscilloscope. The waveplate-PBS combination allows a flexible ratio between reference and probe beam. The calibration is performed on a regular basis by measuring the amount of light close to the position of the atoms and relating it to the voltage reading of the photodetector. We see very little change over the course of many weeks, which we ascribe (from previous experience) to the elaborate polarization control before and after the fiber. Before the light can interact with the atomic ensemble, it is overlapped with the trapping light beam on a dichroic beam-splitter cube shown in Fig. 3.25e. This allows us to use the same optics for the dipole trapping and for the probing. In this way, we achieve a well defined mode overlap of atoms and light. However, the dichroic cube and other components in the beam path, change the polarization state of the probe light. For this reason we have to manipulate the polarization of the probe light beforehand to account for the changes down-stream. This is done with a half- and a quarter wave-plate. In this combination we can produce any polarization state we wish. In Fig. 3.25d we see that an additional PBS is used to project the light once again into a pure linear polarization. For this projection we use a thin-film polarizer cube which has an extinction ratio of 105 : 1. The compensation wave-plates in Fig. 3.25d are adjusted according to the measurement of the polarization state of the light right after it has passed the atomic cloud and left the glass cell again. The polarization state is determined in two ways. First, with a commercial polarimeter (Thorlabs PAX5710IR1), which has an accuracy of 0.2◦ in the Poincaré sphere. Second, with the polarimeter we built to measure the atomic quantum noise which is much more precise. It will be explained in Sect 3.6.1. 3.5.2. Frequency shifting and pulse generation We make broad use of acousto-optic modulators (AOM)13 , both for frequency shifting in the range of 30 to 200 MHz and for pulse generation. Since, we almost have a dozen AOMs in the experimental setup, it is worth spending a few lines to explain the way we are using them. We were able to double the number of trapped atoms in the dipole trap by changing to the configuration described below. From the electronic point of view we use them in the following configuration. A voltage controlled oscillator (VCO) serves as radio frequency (RF) source and is controlled via a fixed DC voltage or a variable voltage coming from one of the data 12 13 We use PDA10A or DET36A from Thorlabs. More correctly these devices are frequency shifters. However, depending on the provider terms are interchanged. 3.5. Probing atomic spins 67 acquisition cards (DAC). The RF signal passes through a voltage controlled attenuator (VCA) which is used to select the amount of RF power that reaches the AOM. After the VCA we send the RF through a switch which generates RF pulses with rising times of a few nanoseconds. It is driven with the digital output of the DAC or the output of a pulse generator. Before, the RF reaches the AOM, it is amplified to power-levels of 1 − 2 W. In this way we can control the amount of light sent to a certain part of the experiment and the can generate pulses down to a few tens of nanoseconds. Double-pass configuration In the case we need tunability, the AOMs are used in a double-pass (cat’s eye) configuration [96]. This ensures the maximum in conversion efficiency and perfect alignment over the whole detuning range. In Fig. 3.29 (further below Sect. 3.5.4) we show the typical configuration we use in many places. As a first step we reduce the size of the laser beam with a telescope to fit the aperture size of the AOM crystal. We use a Galilean telescope to save space. It consists of a convex lens of 100 mm focal length and a concave lens of 50 or 75 mm focal length, depending on the required de-magnification. The laser is then sent through the crystal of the AOM where it is diffracted by the traveling density wave. To optimize the diffraction efficiency we mount the AOM on a kinematic mount to change its angle about a vertical and horizontal axis (both orthogonal to the propagation direction of the laser beam). We always optimize the AOM for the single pass efficiency. The diffracted light is then collected by a focal lens and focused onto a mirror. The mirror and AOM are both placed one focal length away from the lens. This is the optimal configuration to ensure maximal double-pass efficiency and best possible injection efficiency into a single mode fiber. The double-pass efficiency is optimized by adjusting the mirror. We reach efficiencies of above 70 % for the center frequency. The double-pass scheme is also very flexible. If we, for example, decide to change the AOM frequency or the diffraction order we want to use, i.e., from +1 to −1, almost no realignment is necessary. Even if AOMs are fantastic devices which are easy to use and give a reliable frequency tuning, their bandwidth is limited. The rule of thumb is that the bandwidth is at maximum half the center frequency, e.g., for a model with center frequency 80 MHz, we can have at most shifts between 60 and 100 MHz. Moreover, the conversion efficiency is not constant over the band-width, but decreases rather strongly towards the edges. Therefore, we describe the technique we are using to detune the laser over a large range of up to several GHz. 3.5.3. Frequency Offset-lock To overcome the limitations of the AOMs there are different techniques to achieve a large detuning from a given reference, e.g., an atomic resonance. Starting from a laser which is referenced to an atomic resonance, which is called master, there are different ways to frequency lock a second laser, called slave. One is for example, 68 3. Experimental apparatus and techniques to use the discrete frequency spectrum of a Fabry-Perot cavity to separate master and slave by an integer of the free-spectral range of the cavity [97, 98, 99, 100]. The technique we use, which is better suited for for frequency differences between a few hundred MHz and several GHz, is to interfere both lasers on a beam splitter and use the beat-note signal detected by a fast photo-detector. The upper limit for the frequency-difference is set only by the bandwidth of the photo-detector used to measure the beat-note. Analog offset-locks The beat-note signal, which can go up to several GHz, is usually reduced to a smaller frequency in the MHz-range by mixing it with a local oscillator of known frequency. Then there exist several ways to convert this frequency information into a measurable signal, e.g., voltage. One approach proposed by Rutt [101] and realized for external cavity diode lasers by Schünemann et al. [102] uses an electronic delay-line, with delay time τ , to produce a frequency dependent phase-shift Φ = (ωbeat − ωV CO ) τ . This is converted into an amplitude signal by an electronic phase detector. Despite its simplicity it has some serious drawbacks. First, there are multiple locking points separated by the inverse of the delay time τ . Second the frequency precision and the capture range are not independently variable. Another approach by Ritt et al. [103] used a sharp electronic RF high pass filter as frequency-to-amplitude converter. There, the frequency is fixed by locking to a certain attenuation given by the frequency response of the high pass filter. This approach has the advantage to have only one locking point, i.e., capture ranges of several hundreds of MHz are possible. Furthermore, the precision is given only by the steepness of the high pass filter, which can be arbitrarily large. Digital offset-locks A very simple frequency offset-lock using digital components is described by Stace et al. [104]. They use a digital frequency-to-voltage converter (FVC). In their design they have to digitize the beat-note signal using a voltage comparator. Furthermore, it is necessary to divide the resulting TTL signal down to the narrow band-width of the FVC of only 500 kHz. To generate an error-signal with zero crossing, a negative control voltage is added to the output voltage of the FVC. In this way, the locking point can be chosen by selecting an appropriate control voltage. Limitations of this design are the finite gain and drifts in the output voltage of the FVC. Implemented technique We apply an idea by Günter [105] for a digital offset-lock. It is based on a commercial digital phase-locked loop (DPLL) synthesizer, which is in-expensive and straight forward to use. Usually, the DPLL is designed for frequency- and phase-locking of a VCO in GSM devices and works up to several GHz. If the VCO signal is now 3.5. Probing atomic spins 69 substituted for the beat-note of the master and slave laser, we are able to control the frequency- and the phase-difference of both lasers. The whole digital offset lock can be divided into three parts. First, a fast photodetector is needed with a bandwidth larger than the frequency difference we want to establish between slave and master. Second, the above mentioned digital PLL. Third, a servo controller to act on the grating and the current of the slave laser. Figure 3.26.: Functional block diagram of digital offset lock using a digital phase-locked loop. Fast photo-detector As a fast photo-detector we use a fiber coupled photo-receiver which has a bandwidth of 9.5 GHz (model PT10GC from Bookham14 ) usually used in telecommunication applications. It consists of an InGaAs-PIN photo-diode which is pre-amplified on chip and connectorized to a single-mode fiber pigtail. Digital PLL The DPLL contains several functional blocks which we show schematically in Fig. 3.26. The beating signal runs through a pre-scaler which is a fractional N-divider that reduces the beat-note frequency to something close to the reference frequency of the local oscillator. The local oscillator is either provided by a precision crystal oscillator on the used evaluation board or by an external frequency source up to 250 MHz. For the presented experiments, we use the temperature compensated crystal oscillator of 10 MHz from the evaluation board. The reference frequency can be optionally doubled. Both the local oscillator and the pre-scaled beat signal are fed into the “heart” of the DPPL, the phase frequency detector (PFD), which generates a signal proportional to the frequency- and phase-difference. The PFD is an electronic element, which can have various implementations [106, 107]. In our case, this is a circuit with two D-type flip flops U1 and U2, shown in Fig. 3.27. A D-type flip flop is a 1-bit memory. If it is clocked in rising edges, then it changes its output state Q from low to high, if the input IN made a transition from low to 14 www.bookham.com 70 3. Experimental apparatus and techniques high. The output is reset to low if the CLR input receives high. The feedback over the AND gate (U3) brings both outputs Q1 and Q2 back to low if they were both high. To understand the principle better, we plot the output for two case. First, neither frequency nor phase are locked. Second, the frequency of beat signal and reference are the same but the phase is not. The output of the PFD are current pulses with a maximum current of 4.375 mA, where the voltage goes from V − = 0 V to V + = 4.88 V. The maximum repetition rate is limited by the local oscillator frequency, which is the maximum comparison frequency. Figure 3.27.: a) Phase frequency detector as used in the DPLL. U1 and U2 are D-type flip flops, U3 an AND gate. The two outputs Q1 and Q2 control the current on the output OUT. b) Example of different frequencies on the IN- and IN+ which result in a large output signal. c) Example of same frequency but different phase. The output is a periodic signal which eventually drives the two oscillator in phase. All three figures are taken from [106]. To illustrate the principle of the frequency offset lock, we record the integrated error signal and compare it to the simultaneously recorded saturated absorption spectrum with open feedback loop. We lock the reference laser to the cross-over transition F = 1 → 20 |10 (see figure) and set the offset lock frequency to 1 GHz. In Fig. 3.28 we show the error signal after integration (monitor output of PID). The measurement is taken with the laser scanning over the F = 1 → F 0 transitions of 87 Rb. We fit the spectrum including the Doppler-broadened profile and the saturated lines. Closing the feedback loop The frequency of an external cavity diode laser can be controlled in various ways. One is to change the temperature of the laser diode, which will change the cavity length of the diode. It is very slow and usually not used to stabilize a laser to an external reference. The second way is to change the external cavity feedback. If the external cavity is built by a diffraction grating than its angle is changed 3.5. Probing atomic spins 71 Figure 3.28.: Top: Error signal after low pass filtering in the PID controller of the probe laser driver. The white area is the forward scanning direction of the PZT, the gray one the backward direction. Bottom: Transmission signal (blue dots) of the saturated absorption spectroscopy. The solid black line is a fit, see text for details. In the inset we show a close-up of the hyperfine transitions F = 1 and their cross-over transitions. The spectrum is grayed for the backward movement of the PZT. [108]. Alternative designs use frequency selective elements in the external cavity [109, 110]. Having moving mechanical parts limits the bandwidth of this feedback by the resonance frequency of the mechanical components. The typical bandwidth is a few kHz, mostly limited by mechanical resonances of the grating holder. The fastest way of applying feedback is on the current of the laser diode. Here one can achieve bandwidth of several MHz. The output of the PFD is split into two branches. For fast feedback we connect the PFD directly to the FET input of the laser diode. The laser diode has a large bandwidth and allows, in principle, to use the locking system to phase-lock the master and the slave. Unfortunately, we were not yet able to realize this. For slow feedback we send the second branch into the PID controller of the laser system which actuates on the piezo. The piezo feedback is intrinsically slow and limited to the mechanical resonances. The maximum frequency is in the order of few kHz. With this we can correct for acoustic vibrations and slow drifts introduced by temperature. 72 3. Experimental apparatus and techniques 3.5.4. Experimental details on probe lasers Before we continue, we close the section by explaining how the probe lasers are locked in the experiment and give relevant details. Vector probe The probe laser for the QND measurements of Jˆz is an external cavity diode laser of the type DL100 from Toptica. In Fig. 3.29 we can see that it is split into different beam paths. After a Faraday isolator, to prevent optical feedback, we use a wedge and split off two small portions of 3−4 %. One part, which corresponds to ∼ 2.5 mW, goes to a non-polarizing beam splitter cube and is overlapped with light from either the F = 1 or the F = 2 laser. Both beams are then coupled into the same single mode fiber which sends them to the offset lock box. The other portion is attenuated and sent into a saturated absorption setup, which allows to have a control over the position of the laser before it is locked. The continuing portion of light (∼ 56 mW) has a frequency of νm − νbn , where νm is the frequency of the master laser15 and νbn is the beat-note frequency we want to lock to. The light propagates now through a telescope to a double-pass AOM configuration, described above. We use the double-pass configuration also here, even if we never scan the AOM frequency. This allows us to compensate the lower locking frequency of the beat-note lock, which is νbn.min = 300 MHz and achieve a continuous tuning range for the vector probe laser resonance F = 1 → F 0 = 0 down to more than 2.5 GHz red detuning. The final vector probe detuning as a function of the beat-note frequency is ∆vect = νm − νF =1 + 2νAOM − νbn = −(0...2.7) GHz . As we will see further below, for most of the experiments we are working with trains of probe pulses. It is fundamental to change have access to the pulse duration, repetition rate, number of pulses and light power. The later is adjustable with the RF amplitude of the AOM. Where the former are dialed with a function generator which is pre-programmed or alternatively can be remotely programmed to define the pulse width and repetition time. The number of pulses is determined by the duration of the pulse train, which could be one to many pulses. The timing precision of the function generator, which has a resolution of 1 ns and a jitter of < 400 ps is very small. Tensor probe For the measurements of Jˆx we use only a double-pass AOM configuration and no offset-lock, because we probe much closer to resonance (cf. Fig. 3.24). The light 15 The master laser could be F=1 or the F=2 laser. The former is locked to the cross-over transition between F 0 = 1 and F 0 = 2, and has a frequency of νm = νF 0 =0 + 150.7 MHz − νF =1 . The latter is locked 60 MHz below the cross-over between F 0 = 2 and F 0 = 3, and has a frequency νm = νF 0 =2 + 73.3 MHz − νF =2 . 3.5. Probing atomic spins 73 Figure 3.29.: Vector probe setup: a) Spectroscopy setup with rubidium vapor cell b) laser light from the vector probe and from either F = 1 or F = 2 laser are overlapped on a beam splitter and one output port is coupled into a single mode fiber which goes to the frequency off-set lock box. c) the pulse generation is achieved by means of an AOM in double-pass configuration. The output light is coupled into a single mode fiber and sent to the atom trap. from the F=1 laser is shifted by an AOM with a center frequency of 200 MHz and bandwidth ∼ 80 MHz. This gives us a range of detunings from the F = 1 → F 0 = 0 transition of ∆tens = 150.7 MHz − 2νAOM = −(170...330) MHz . The pulses are also generated with a pulse generator in gated trigger mode. 74 3. Experimental apparatus and techniques 3.6. Shot-noise-limited polarization detection We measure atomic spin polarization stroboscopically by sending pulses of polarized light and record their polarization rotation in a polarimeter. For the range of probe light detunings we are working in (c.f. previous section), the pulsed interogation is necessary for technical reasons. On one side, the photon flux we can use to gather information about the atomic polarization is limited through off-resonant light scattering which destroys the atomic state. On the other side, intrinsic noise of the detection electronic and low frequency noise (intensity fluctuations) of the probe laser can be rejected by using pulses instead of a continuous wave. We measure polarization rotations of light with a pulsed polarimeter down to very small angles below 10−4 rad. For this we set up a polarimeter that measures the 45◦ -component of the Stokes operator, i.e., Ŝy . Here, we explain the main optical elements for the polarimeter and discuss the influence of polarization dependent losses in Sect. 3.6.1 and 3.6.2. Further, we describe the operation principle of the balanced detector (3.6.3), discuss the calibration in the time domain (3.6.4) and determine its performance at the quantum level (3.6.5). 3.6.1. Polarimeter We use a simple polarimeter consisting of a polarizing beams-splitter (PBS) (Wollaston or normal cube), a HWP, and a differential photodetector, shown in the right part of Fig. 3.31. In this way we are sensitive to any Stokes component on the equator of the Poincaré sphere, depending on the angle of the HWP. This polarimeter setup is most sensitive when the photo-detector is working in the balanced regime, i.e., when the differential photo-current is zero. Any rotation of the input polarization about the Ŝz axis of the Poincaré sphere will cause an imbalance which is recorded as an electrical signal. The Stokes component we are most interested in is Ŝy . Hence, we balance the two output ports of the PBS by rotating the incoming light into the 45◦ -basis. The balanced regime is not only the most sensitive, but also suppresses unwanted technical noise in the probe light, e.g. from intensity fluctuations. This can be seen if we write down the intensity of the light emerging from the two output ports of Figure 3.30.: Polarization rotation of a) electric field vector in real space, and b) Stokes vector in the Poincaré sphere. The rotation angles have the relation ϕ = 2α. 3.6. Shot-noise-limited polarization detection 75 ◦ the PBS. As shown in Fig. 3.30 a), the in electric field components the +45 and π π ◦ −45 basis are E+45◦ = E0 cos 4 + α and E−45◦ = E0 sin 4 + α , respectively, where E0 is the envelope of the electric field and α the angle between the electric field vector and the +45◦ axis. The intensities at the two output ports are I+45◦ I−45◦ I0 (1 + sin(2α)) 2 I0 = |E−45◦ |2 = (1 − sin(2α)) , 2 = |E+45◦ |2 = (3.31) (3.32) where we write the intensity at the input as I0 = |E0 |2 . We can separate the intensity I0 into a common mean intensity I 0 , and into fluctuating parts δI±45◦ = δIc + δIu,±45◦ which are different for the two output ports. They consist of a correlated δIc and an uncorrelated term δIu,±45◦ . The differential photo-detector will measure a signal which is proportional to the difference of the two intensities (3.31) and (3.32) and we get D∝ 1 δIu,+45◦ − δIu,−45◦ + 2I 0 + 2δIc + δIu,+45◦ + δIu,−45◦ sin(2α) , (3.33) 2 where we assume equal gain of the two photo-diodes. The signal of the polarimeter (3.33) is directly proportional to the rotation α of the electric field vector. For the balanced case, i.e., α = 0, the mean value of the polarimeter output is zero h D i = 0 and the uncertainty is δD = δIu , (3.34) where we assume that the uncorrelated fluctuations are equal in magnitude for both output ports δIu ≡ δIu,+45◦ = δIu,−45◦ . If the uncorrelated fluctuations are completely quantum, the polarimeter is said to be shot noise limited [111]. For an imbalanced polarimeter, i.e., α 6= 0, the uncertainty is increased and correlated noise, e.g., intensity fluctuations, enter into the signal δD = δIu + sin(2α) (δIu + δIc ) , (3.35) and the polarimeter is limited by classical fluctuations. In Sect. 3.6.5 we measure the noise behavior for a large range of photon-number and identify the individual noise contributions by their respective scaling with photonnumber. In the experiment, as shown in Fig. 3.31, we measure the rotation angle of the electric field α, or more precisely, of the Stokes vector, which is ϕ = 2α. If the analyzer waveplate is set to 22.5◦ , we measure Ŝy . For vertically polarized light entering the setup, (in) i.e., h Ŝx i = NL /2, the rotation angle of the Stokes vector can be calculated by ! (meas) h Ŝy i ϕ ≡ arcsin . (3.36) (in) h Ŝx i (meas) where h Ŝy i is the measured mean photon-number imbalance. 76 3. Experimental apparatus and techniques Figure 3.31.: Polarization probing of atomic spin. The dipole trapping and probing beams are overlapped to maximize atom-light interaction. The probe beam has acquired a polarization rotation after passing through the sample. The birefringence of the dichroic mirror (DCM) is corrected by a quarter- and a half-waveplate. The polarization is analyzed in the 45◦ -basis by a differential photo detector. 3.6.2. Linear birefringence and optical losses in detection In our experimental setup probing and trapping light are overlapped/separated on dichroic optical elements. This permits a complete overlap of the atomic cloud and probing light, but at the same time introduces problems of birefringence and polarization dependent losses. Birefringence The dichroic mirror (DCM) which is used to separate the dipole from the probing light (cf. Fig. 3.31) shows (linear) birefringence. Combined with a second DCM of the same kind, which compensates the walk-off introduced by the first, they act similar to a quarter wave-plate16 . To compensate this, we use a combination of quarter- and half-waveplate before the light enters into the polarimeter setup (cf. Fig. 3.31). To make a full polarimetry and to determine the Müller matrix [112] of the optical component between the atomic cloud and the Ŝy -measurement is rather difficult in terms of available space. Instead, we use crossed polarizers which are placed around the DCMs and the compensation wave-plates. The first polarizer in front of the DCM will define a pure linear polarization (v or h). A second crossed polarized after the combination of quarter and half wave-plate serves as an analyzer. Both wave-plates are adjusted until the transmitted amount of light is minimized. In this way, we can guarantee the compensation of the DCMs by the wave-plates. 16 Before building them in the optical properties of the dichroic mirrors (DCM) were measured separately outside. They have shown to behave very similarly to a quarter waveplate. 3.6. Shot-noise-limited polarization detection 77 Polarization dependent losses In the presence of polarization dependent losses (PDL) the relation between observed (meas) photon-number difference h Ŝy i and the rotation angle of the Stokes vector ϕ is slightly more complicated than in Eq. (3.36). We have to take into account that the presence of PDL can change the angle of the electric field vector. To model the influence of PDL, we measure the attenuation of horizontally and vertically polarized light propagating through the DCMs and compensation waveplates. We found transmission of vertical (s-polarized) and horizontal (p-polarized) light of Tv = 74.5(2) % and Th = 86.6(2) %. The large difference can be explained with the usual observation that a dielectric mirror has a better reflection for s than for p polarization. The relation between the actual and the measured polarization rotation is derived in the following way. The incoming light is vertically polarized and receives a rotation through the circular birefringence. Neglecting the time and spatial behavior we can write for the input 1 Ein = √ E0 {e+ − e− } , (3.37) 2 where e± are the spherical basis vectors, defined in the textbook by Edmonds on angular momentum algebra 32. The output reads 1 Eout = √ E0 eiϕ+ e+ − eiϕ− e− , (3.38) 2 where ϕ± are phase shifts on the left- and right-circular light. Transforming from circular into the linear polarization basis (see for instance 32) we find 1 1 Eout = E0 eiϕ− ei∆ϕ + 1 ex + E0 ieiϕ− ei∆ϕ − 1 ey , 2 2 = Av ex + Ah ey (3.39) where ∆ϕ = ϕ+ − ϕ− it the phase difference between the two circular polarization components, which is the measure of interest. Note that this quantity is twice the rotation angle of the electric field. Av and Ah carry the amplitude of the vertical (horizontal) polarization component. The reduction of the vertical (x) and horizontal (y) amplitudes changes the electric field into p p E0out = Tv Av ex + Th Ah ey . (3.40) Since the polarimeter measures the Ŝy component we have to transform the above √ expression into the +45◦ / − 45◦ basis using e±45◦ = (ex ± ey ) / 2 ! ! r r r r Tv Th Tv Th 0 Eout = Av + Ah e+45◦ + Av − Ah e−45◦ . (3.41) 2 2 2 2 We find a similar expression for rotation angle as in the ideal case, which has corrections for the introduced loss ! (meas) Sy . (3.42) ∆ϕ = arcsin (in) √ Sx Tv Th 78 3. Experimental apparatus and techniques 3.6.3. Operation Principle of balanced detector The heart of our detection system is a differential, integrating photo-detector. It is designed to detect pulses between 10 ns and 200 µs with a very low electronic noise. The version of the detector we use was designed and made in the group of Eugene Polzik at the NBI in Copenhagen by Patrick Windpassinger and Jöreg Helge Müller. The design is adapted from [113]. Previous detectors following the same design where developed together in collaboration between ICFO and NBI [74]. Since we are only user of the device, we limit the explanation to the operation principle to get an idea of the main concept and refer the reader to [114, 115]. Nevertheless, to fully understand the functionality of the device and use it appropriately in the experiment, a series of characterizations is necessary. The detector is built to sense the difference photo charge of two photo-diodes (HAMAMATSU S388317 ) which are connected in series. This signal will correspond to the imbalance in photon-number between the two detection modes. We bias the photodiodes in reverse direction to improve their time response by lowering the photo-diode capacitance. The differential signal is AC coupled to the first essential part of the detector, the integrator. It consists of a operational amplifier which has in its feedback branch a capacitor Ci and a (discharge) resistor Ri . Together they determine the time constant τ = Ri Ci of the integrator. An input charge signal which is faster than τ will be converted into a voltage output, where the gain is set by Ci−1 . For this reason these devices are also sometimes called charge sensitive amplifiers. In Fig. 3.32 we see how a train of squared charge pulses are converted into a step-like voltage signal with linear ramps connecting the steps. The integrated signal will decay exponentially with a time constant τ. The integrated voltage signal is fed into the second important part, the pulse shaper. Here, a combination of high- and low-pass filter with identical time-constant τd = Rd Cd converts the step-like signal into √ a pulse of almost Gaussian shape. The output pulse has a duration (FWHM) of 2 2 ln 2 τd . The combination of filters can also be interpreted as an active differentiator followed by a low pass filter. It is sometimes called Gaussian shaping amplifier. One additional refinement is necessary to make the detector work properly. The fact that the output of the integrator is not a pure step function, but decays with τ to zero, makes the pulse shaper output have an undershoot. The reason is that the transfer function of the integrator stage has a pole. A zero in the transfer function of the high-pass filter can compensate this pole. This is achieved by adding a resistor RP/Z in parallel to the capacitor Cd . The value has to be chosen so that RP/Z Cd = τ = Ri Ci . In the present detector the integrator and the pulse shaper are self contained components provided by Cremat, Inc. 18 The integrator CR-110 is a module which contains the input FET and the feedback network. For the shaping amplifier a series of modules with fixed shaping time τd is available. The detector is equipped with module CR-200 with 250 ns shaping time. Only the resistor for the pole-zero-cancellation 17 18 The diodes have a quantum efficiency of 92 % at 780 nm. http://cremat.com 3.6. Shot-noise-limited polarization detection 79 Figure 3.32.: Schematic of detector circuit [114]. From left to right: Photon number τ light = Ri reaching Ci differences between upper or lower photo diode is measured as the differential photo current in between the two diodes. The (differential) photo electron pulses RP/ZFor further processing the steps are converted into are integrated to a step-like signal. a pulsed signal of fixed pulse width and a height proportional to the number of detected photons. has to be added externally. An additional AC coupled buffer amplifier was added between integrator and pulse shaper to add more gain to the detector. 3.6.4. Detector Calibration For the characterization of the detector we limit ourselves to the time domain. We will measure the response for short pulses below 25 ns, compare the output when only one photo-diode is illuminated at a time, and measure the gain, integration window, and shot-noise performance for different pulse lengths between 250 ns and 5 µs. “Delta” pulse response τ = 1.46 As a first test we measure the intrinsic response of the detector. That means we send a pulse which is much shorter than the pulse shaping time of the shaping amplifier, which is in our case is 250 ns. From the specifications we should expect to observe a Gaussian pulse of 590 ns (FWHM) out of the shaping amplifier. Using an AOM we can produce pulses with durations of 15 ns (FWHM). To create an observable signal we illuminate only one of the photo-diodes to create a differential photo current. In Fig. 3.33a) we plot the input to the balanced detector recorded with a 150 MHz band-width amplified silicon reference photo-detector (Thorlabs PDA10A). In the inset we show the zoom around t = 0, which has almost a Gaussian shape with a duration of 15 ns. Compared to the pulse shaping time the input pulse represents a good approximation of a delta input. The response of the balanced photo-detector plotted in Fig. τ3.33b) has a duration of = Ri C Ri of 587.5 ns. This is the intrinsic pulseCshape i i the shaping amplifier and corresponds to the specified 590 ns for this module. We present this measurement here, because it shows the principle of the detector in a nice way. An input pulse is recorded and an output pulse, with different shape, is τ produced. The numberCof i detected charges is, as we will see later, proportional to 80 3. Experimental apparatus and techniques the area of the output pulse. The difference between input and output pulse shape requires that we calibrate the detector for each pulse length we want to use. In the following we will present the results of this calibration for a set of pulse lengths. Figure 3.33.: Short pulse response of balanced detector. a) Input pulse with 25 ns FWHM. b) Response of balanced detector. The gray area is the integration window over which we average the signal. c) The delta pulse response as given in the spec sheet. Integration window Before we calibrate the gain and characterize the shot noise behavior of the detector, we should study the relation between input- and output-pulse length. This is important because the signal we are interested in, is the pulse area, which is proportional to the photon-number difference. Later, we will use the detector always together with a digital storage oscilloscope. On this, we record the pulse shape and integrate it numerically during the data analysis. To ensure a good low noise performance, the integration window has to be chosen depending on the pulse length τpulse used. We chose the integration window to contain the whole pulse shape. It seems to be an obvious choice, which is at the same time rather conservative. In Fig. 3.34 we plot the integration windows for pulse shapes between 250 ns and 5 µs. We find that the integration window follows Tint (τpulse ) = T0 + τpulse , (3.43) 3.6. Shot-noise-limited polarization detection 81 where T0 is the initial integration windows due to the shaping time of the pulse shaping amplifier. For our module with 250 ns shaping time we have T0 = 1.3 µs. If we compare T0 to the pulse shape in Fig. 3.33 for the delta pulse response, we see that the integration window would cover the whole pulse. 7 Integration window [µs] 6 5 4 3 2 1 0 0 1 2 3 4 5 6 Pulse length [µs] Figure 3.34.: Manually chosen integration windows for different pulse lengths. The solid black line is directly proportional to the pulse length and offset by 1.3 µs. Calibration of photon-number gain We define the photon-number gain of the balanced detector as the conversion between observed voltage U (t), averaged over the integration window Tint , Z t0 +Tint 1 Uavg = U (t)dt . (3.44) Tint t0 and the photon-number difference ∆NL , which caused the signal: Gγ−V ≡ ∆NL , Uavg (3.45) where the integration window starts at t0 . In this way, we can relate the observed averaged voltage signal to the photon-number imbalance and, hence, to the observable of Ŝy . 2 h Ŝy i ≡ ∆NL = Gγ−V Uavg (3.46) The gain as a function of pulse length is measured by sending pulses of known photonnumber ∆NL = NL onto one of the photo-diodes and integrating the output signal over the corresponding integration window. In the left panel of Fig. 3.35 we plot the averaged output signal as a function of photon-number difference ∆NL . As expected, we observe a linear relation between photon-number imbalance and output signal. The inverse of the linear slopes, i.e., Gγ−V , are plotted in the right panel of Fig. 3.35 for different pulse lengths. The 82 3. Experimental apparatus and techniques 1.2 1 −1 U avg 0.25 µs 0.50 µs 1 µs 1.50 µs 2 µs 3 µs 4 µs 5 µs −2 10 −3 10 3 10 4 5 10 10 Photon number imbalance ∆NL Gγ−V [Photon/µV] [V] 10 0.8 0.6 0.4 0.2 0 0 2 4 Pulse length [µs] 6 Figure 3.35.: Left: Averaged voltage over the integration window for different input pulse lengths as a function of pulse energy in number of photons NL . Right: The photon-number gain Gγ−V deduced as the slope of measurement shown in the left panel. The blue line is a linear fit with a slope of 0.151(7) photons/(µVµs) and a zero at t0 = 1.5(1) s. This time is very close to T0 . almost linear relation between integration window Tint and Gγ−V , is related to definition (3.45). By fitting a line we get a constant ratio ∆NL photons . = 0.151(7) R t0 +Tint µV µs U (t)dt t0 (3.47) We didn’t expect this fixed ratio between photon-number and integrated voltage, in the beginning, and see that it also changes for pulses shorter than 500 ns. Hence, we decided to keep the definition of Gγ−V as introduced, throughout the remainder of the thesis. In practice, we will always work with a constant pulse length of 1 µs, for reasons which become clear in the next section. 3.6.5. Noise of the detection system Now, we characterize the capability of the detector to measure quantum noise of the light. For the preparation of non-classical spin states via QND measurements it is of utmost importance to have a detection system which is only limited by quantum noise. Therefore, we will measure now the smallest number of photons we have to send in a coherent state, so that their shot noise equals the intrinsic electronic noise of the device [111]. We call this number the noise equivalent photon-number (NEPN) NL,SN . From this we can deduce the equivalent noise charge (ENC) as pthe number of photo-electrons necessary to overcome the electronic noise qSN = ηQ NL,SN . The quantum efficiency of the photo-diodes in the detector is ηQ ∼ 0.92. As a reference we use coherent states of light, which is a natural choice to calibrate a photo-detector because of its well defined quantum properties. Here, we measure the 3.6. Shot-noise-limited polarization detection 83 quantum noise in the Ŝy component, because we use the polarimeter setup in this way for later measurements. In order to suppress any classical intensity fluctuations we chose as input state light which is polarized vertically, i.e.,h Ŝx i = NL /2 and h Ŝy i = 0 and var(Ŝy ) = NL /4. Data analysis The way we analyze the data for this calibration measurement is the same as later for the quantum noise measurements. Therefore, we will explain it in more detail here and refer later to this section. First, we always convert the measured voltage into an photon-number by means of Gγ−V . This photon-number will correspond to the observable for the 45◦ -component of the Stokes operator, i.e., Ŝy . We can write the y Stokes component for an individual pulse Z t0 +Tint Gγ−V 1 sy,i = U (t)dt , (3.48) 2 Tint t0 where t = 0 at the moment when the light pulse is triggered. The shift t0 is a delay to place the integration window right. Delays occur due to the electronics for the RF generation and the finite speed of sound in the AOM crystal. Once, t0 is measured, it will stay the same until something essential in the experiment has changed. Usually, t0 is in the order of a few hundreds of nanoseconds. Second, to suppress low frequency fluctuations, e.g., a drifting baseline due to imperfectly compensated base line pulling, we integrate also about a region before the pulse and subtract this from the signal. This is very similar to the correlated double sampling used in charge coupled devices (CCD) to determine the added charge more precisely. In this way we ensure that a wrong baseline, which is a kind of memory effect from previous pulses, does not lead to classical correlations between adjacent pulses. The expression for the individual measurement outcome in the case of correlated double sampling is scds y,i Gγ−V 1 = 2 Tint Z t0 +Tint Z tb +Tint U (t)dt − t0 U (t)dt , (3.49) tb where tb is the starting point for the background window, which has the same width, Tint , as the actual integration windows for the pulse. From the set of scds y,i we calculate the variance as n n i=1 i=1 1 X cds 1 X cds var(Ŝy ) = sy,i − sy,i n−1 n where n is the number of samples. !2 , (3.50) 84 3. Experimental apparatus and techniques Performance We send a series of probe pulse with pulse lengths between 250 ns and 5 µs and a period of 20 µs into the polarimeter and vary the number of photons per pulse. For each pulse length and photon number, we send a train of 500 pulses and evaluate var(Ŝy ). In Fig. 3.36 we plot the variance as a function of photon-number for a pulse length of 1 µs. We clearly can distinguish two different scalings. The first one for small photonnumber below NL = 104 shows no dependence on photon-number and represents the electronic noise level of the detector. The value we can read off Fig. 3.36 as the noise equivalent photon-number is NL,SN = 3 × 105 . This will be referred to as the photon-number at which the detector starts to be shot-noise-limited. We have to notice that the correlated double sample we perform by subtracting a base line from√ each pulse (3.49) adds one unit of electronic noise. The resulting NEC is qSN = 0.92 0.5 ∗ 3 × 105 ' 360. This coincides with the noise estimation given by the Cremat, Inc. for a similar combination of CSA and pulse shaping amplifier. For the detector it is also important to stay shot-noise-limited for large photonnumbers. We can see in Fig. 3.36 that this is fulfilled for whole measured range which represents the range of interesting photon-numbers for all later experiments. We repeated this calibration also for other pulse lengths and plot the result for the NL,SN in the left panel of Fig. 3.37. We can see that there exist an optimal pulse length in terms of shot-noise-limit. For increasing pulse lengths one could expect a better performance if the noise of the detector would be white noise. An average over a longer time would decrease the electronic noise proportional to the integration time [116]. This behavior is nicely visible in the right panel of Fig. 3.37, where we plot the electronic noise versus pulse length. In the conversion from voltage to photon-number via Gγ−V , the noise in terms of photon-number will increase linearly 9 10 8 4Var(Sy) 10 7 10 6 10 5 10 3 10 4 10 5 10 6 10 Photon number NL 7 10 8 10 9 10 Figure 3.36.: Variance of detector signal as function of photonnumber. The pulse length is 1 µs. 3.6. Shot-noise-limited polarization detection 85 5 5 x 10 4 Electronic noise [x10−6 V2] 3.5 NL,SN [photons] 4 3 2 1 3 2.5 2 1.5 1 0.5 0 0 2 4 Pulse length [µs] 6 0 0 2 4 6 Pulse length [µs] Figure 3.37.: Left: Noise equivalent photon-number as function of input pulse length. The black line is not a fit, but the product of the fit to the electronic noise and the photon-number gain (squared). Right: Electronic noise in integration window as function of pulse length. The solid black line is a fit ∝ (τpulse − τ0 )−1 , where τ0 ∼ 230 ns is a small offset. with pulse length, since Gγ−V is a linear function in time visible in Fig. 3.35. The minimum results from the fact that the electronic noise is proportional to the pulse length (up to a shift of 230 ns) and the photon-number gain to the integration window. If we multiply fit functions for the electronic gain and the square of Gγ−V , we get the black line shown in the left panel of Fig. 3.37. 4. Dispersive Spin Measurements Contents 4.1. Classical characterization of QND measurement . . . . . 88 4.1.1. Interaction strength . . . . . . . . . . . . . . . . . . . . . . 88 4.1.2. Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.1.3. Spin resolution . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2. Measuring Zeeman coherences . . . . . . . . . . . . . . . 94 4.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.2. How to measure Zeeman coherences in the presence of vector light shifts? . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2.3. Calibration of G2 . . . . . . . . . . . . . . . . . . . . . . . . 98 4.2.4. Optical pumping dynamics . . . . . . . . . . . . . . . . . . 99 4.3. Dispersive atom number measurement . . . . . . . . . . 101 4.4. Relation between scattering cross-section and observed depolarization . . . . . . . . . . . . . . . . . . . . . . . . . 102 88 4. Dispersive Spin Measurements In this chapter we characterize the atom-light interaction for the vector- and the tensor-probe, which are used to measure the observables Jˆz and Jˆx , respectively. Both coupling constants G1 and G2 are measured over a range of probe light detunings. For this we prepare atoms in the according spin state, i.e., along Jˆz or Jˆx , via optical pumping as explained in Sect. 3.4. Simultaneous measurement of macroscopic rotations and the number of atoms in the sample allows us to obtain an absolute calibration of G1 and G2 . The effective interaction area for both probe beams and atomic sample is determined using the theoretical expressions for G1 and G1 , as given in Sect. 2.3.2. At the end we explain how we use the dispersive measurements to count the number of atoms in the atomic ensemble. 4.1. Classical characterization of QND measurement We measure the per-atom rotation for the QND interaction ∼ Ŝz Jˆz and the depolarization introduced. Based on this classical measurement we estimate the sensitivity of the quantum non-demolition measurement for Jˆz . 4.1.1. Interaction strength We can measure the atom-light interaction for the QND measurement of Jˆz given by the coupling constant G1 in the effective interaction Hamiltonian (2.22). The coupling constant G1 can also be viewed as a per-atom rotation of the probe light polarization. We use a completely polarized atomic spin state along the propagation direction of the probe light. In terms of atomic operators this corresponds to h Jˆz i = NA 1 2 h F̂z i = 2 . A pulse of off-resonant polarized light will experience a large rotation of its polarization vector, compared to its quantum components. If the rotation angle is small, i.e., φ 1, the input-output relation for the 45◦ -component of the Stokes operator is (out) (in) (in) (in) (4.1) Ŝy = Ŝy + G1 Ŝx Jˆz . The per-atom rotation is calculated from the measured rotation angle and the number of atoms by 2φ G1 ≡ , (4.2) NA (out) where the rotation angle is defined as φ ≡ h Ŝy (in) i / h Ŝx i = G1 N2A , assuming the (in) input light is polarized completely along the vertical direction, i.e., h Ŝx i = NL /2 (in) and h Ŝy i = 0. The number of atoms is determined using absorption imaging as explained in Sect. 3.2. For the experiments, we load atoms into the dipole trap as described in Sect. 3.1.3. The ensemble is polarized by illuminating atoms along the quantization axis with circularly polarized light, which is resonant to the transition F = 1 → F 0 = 1. A small guiding magnetic field along z defines the quantization axis and prevents the depolarization of the state through stray magnetic fields. To avoid the accumulation of atoms in the F = 2 manifold, we apply light resonant to the transition F = 2 → 4.1. Classical characterization of QND measurement 89 ï7 G1×10 2 1 0.5 ï2 ï1 ï0.5 6 [GHz] Figure 4.1.: Per-atom rotation for different detunings calculated from the rotation angle and atom number. The black solid line is a fit using the theoretical expression for G1 (∆, A) and leaving the interaction area as only free parameter. We find A = 3.7(1) × 10−9 m2 , where the given error indicates one standard deviation. F 0 = 2 via the MOT beams. Optical pumping into dark states is avoided through the thermal motion of the atoms in the sub-wavelength polarization pattern produced by the six MOT beams. For the dispersive probing we send pulses of vertically polarized light, i.e., h Ŝx i = NL /2, along the trap axis and record their polarization rotation, determined in Sect. 3.6.1 according to Eq. (3.36). The detuning of the probe laser is changed in various steps to measure the spectral dependence of the atom-light coupling. In Fig. 4.1 we plot the measured coupling constants as a function of detuning ∆ from the transition F = 1 → F 0 = 0. We can see the ∆−1 dependence of the interaction strength, which is typical for the vector polarizability. The solid black line is a fit using the theoretical expression for G1 (∆, A) as defined in Sect. 2.3.2 in Eq. (2.30). The only free parameter is the effective interaction area A between light and atoms. We find a value of 3.7(1) × 10−9 m2 . The concept of the effective interaction area is very useful to simplify the inhomogeneous interaction of the Gaussian density of atoms and photons. Assuming a top p hat distribution of light and atoms we can associate a hypothetical radius of A/π = 34 µm to the measured effective interaction area A. Optical depth There is an alternative way of characterizing the amount of interaction strength [63, 117, 118]. To state the amount of interaction independent on the detuning used, we can define the on-resonant optical depth d0 . In a multilevel atom ’onresonance’ is ill-defined and needs some explanation. In our case, we have three 90 4. Dispersive Spin Measurements relevant excited state hyperfine levels which are important for the interaction, which are F = 1 → F 0 = 0, 1, 2. In the limit of large detuning, the main transition for the dispersive interaction is F = 1 → F 0 = 0. The other two give contributions to the vector polarizability of same magnitude and opposite sign. their sum vanishes for detunings much larger than the excited state hyperfine splitting. The absorptive part of the interaction is described in terms of the scattering cross section. In Appendix B.1 we derive the scattering cross section in terms of atomic quantities as a function of detuning [17]. We find the on-resonance scattering cross-section for atoms in F = 1 2 of σ0 = λπ , where we assume that the excited states are degenerate and interact with linearly polarized light. We define the effective on-resonance optical depth d0 in analogy with the usual definition for a homogeneous medium: d0 = σ0 NA . A (4.3) Note that for our system, which is far from homogeneous, and in addition has three resolved spectral lines, the effective optical depth does not describe the transmission characteristics of the medium. It is an important figure of merit, however, for comparing dispersive effects, and allows direct comparison with other atom geometries. If we use the parameters we deduced from the per-atom rotation measurement above, we can calculate the on-resonance optical depth for our atomic ensemble. We find OD = 57(3) (we used NA = 1.09(3) × 106 , A = 3.7(1) × 10−9 m2 , and σ0 = 1.94 × 10−13 m2 ). This optical depth is among the largest observed for such systems. Other groups working with cold atomic samples have measured OD ∼ 16 [119] or OD ∼ 4 [120]. Higher optical depth of above 1000 have been observed in BEC. 4.1.2. Depolarization Probing a polarized atomic state consecutively with a large number of pulses allows us to monitor the decrease of spin polarization as a function of the number of probe photons. In the simplest case the spin polarization is exponentially decaying as (out) h Jˆz (in) (NL ) i = h Jˆz i e−ηγ NL , (4.4) where ηγ is the depolarization per probe photon. We can define a total damage for a number of photons NL as η ≡ ηγ NL . (4.5) We measure the depolarization for the same range of detunings as in Fig. 4.1. For this, we send a train of 900 pulses, each containing ∼ 2.5 × 106 photons, and record the rotation signal. The left plot of Fig. 4.2 shows the measured ϕ for all detunings. For the simulations, we start, as in the experiment, with a polarized atomic state in F = 1 that is probed by a train of pulses. The shown traces are the rotation signals calculated as the sum of the rotation signal from atoms in F = 1 and F = 2, i.e., the signal we actually measure. The results are plotted in the right of Fig. 4.2 4.1. Classical characterization of QND measurement q [mrad] q [mrad] 100 100 80 80 60 60 40 40 20 20 0 0 91 200 400 600 Number of Pulses 800 0 0 200 400 600 Number of Pulses 800 Figure 4.2.: Angle of polarization rotation as function of probe pulses for different probe detunings between −490 and −1990 MHz (from top to bottom) in steps of 100 MHz. Left: Measured rotation signal. Right: Numerical simulation of rotation signal for NA = 1.1 × 106 , A = 3.7 × 10−9 m2 and NL = 3.1 × 106 per pulse. The number of atoms NA ∼ 1.1 × 106 and the interaction area A = 3.7 × 10−9 m2 are fixed to the experimental values and the photon number was adjusted to achieve comparable decay constants. This fitting finds a photon number larger by ∼ 24 % compared to the photon-number in the experiment, when measured after the vacuum cell. A small systematic error in the measurement of the actual number of photons, interacting with the atomic cloud, is present due to the additional reflection of probe light on the window of the vacuum cell. Although, this effect can only account for ∼ 4% of difference. It is also possible that spontaneously emitted light is reabsorbed in the atomic ensemble, which is often referred to as radiation trapping. The simulations is not taking such effects into account. The measured and the simulated signal both show a behavior which is not a single exponential we have postulated in Eq. (4.4), but has some offset. It becomes apparent in the crossing of the curves at larger times. This can be understood as the transfer of atoms from F = 1 into F = 2 by the probing light. Since, atoms where polarized in F = 1 they remain partly polarized in F = 2. Atoms in F = 2 will not be completely polarized, but a large population imbalance is still present. The red detuned probe beam for the transition F = 1 → F 0 = 0, will be blue detuned for the transition F = 2 → F 0 = 0. For increasing red detuning, we reduce the blue detuning on the other transition. This is the reason why we see the almost flat traces for the larges detuning of ∆ = −1990 MHz. This corresponds to a blue detuning of +4850 MHz for atoms in F = 2. The depolarization can be extracted from the measured decay of the signal by assuming the following model: Atoms contribute some signal from each hyperfine manifold. The polarization in F = 1 will exponentially decay and a polarization in F = 2 will be built up at the same rate. The measurable signal is then described as the sum of 92 4. Dispersive Spin Measurements d [×10ï4] 10 5 2 depolar. in F=1 polar. in F=2 modified decay sgl. exp. decay 1 ï2 ï1 ï0.5 6 [GHz] Figure 4.3.: Results of the simulations of the decay of atomic polarization in F = 1 as a function of the probe detuning. Conditions are the same as for the simulations in the right plot of Fig. 4.2. The black circles show the depolarization of atoms in F = 1. The red squares indicate the growth of polarization in F = 2. The blue crosses are determined from the modified decay of the rotation signal as described in the text. The black dots are the result of fitting a simple single exponential decay for small photon numbers. For more details see text. the decaying and growing signals of different amplitude ϕ(p) = A1 exp (−ηγ NL ) + A2 (1 − exp (−ηγ NL )) = A2 + (A1 − A2 ) exp (−ηγ NL ) . (4.6) To gain some intuition, we apply the above model for the modified decay on the simulated data. There we have access to the depolarization of atoms in F = 1 and the build-up of polarization in F = 2. In Fig. 4.3 we plot the extracted total depolarization η = ηγ NL using the above model and the loss of polarization in F = 1 for the simulated signals. The decay of the polarization in F = 1 is plotted as black circles and the growth of the polarization in F = 2 as red squares. For detunings close to resonance, i.e., small |∆|, we can measure the depolarization directly from the exponentially decaying signal shown as black dots. This starts to fail for larger detunings, where the depolarization in F = 1 is hidden by the growing polarization in F = 2. This model neglects the loss of trap atoms which is negligible at this photon number. For another experiment we measured as rate of ∼ 3 × 10−12 per photon. 4.1. Classical characterization of QND measurement 93 d [×10ï4] 10 5 2 1 expected measured ï2 ï1 ï0.5 6 [GHz] Figure 4.4.: Measured and expected polarization decay of the data shown in the left plot of Fig. 4.2. The depolarization rate from the measurements (cf. left of Fig. 4.2) are shown in Fig. 4.4. We used the single exponential decay of the first part of the signal. This gives results which are close to the depolarization rate calculated with the master equations for the range of detunings we are interested in, i.e. −1.5 < ∆ < −0.5 GHz. Fitting with the modified decay model (4.6) gives results with large error bars for large detunings, where the signal is almost flat. 4.1.3. Spin resolution Once we have characterized the atom-light interaction in terms of the per-atom rotation and the destruction, we can estimate the sensitivity of the QND probing from this measurement of mean values, which we call a classical measurement. For a dispersive spin measurement the sensitivity is in the best case limited by the quantum noise of the probe light. The minimum angular resolution we get from a coherent polarization state, as opposed to a squeezed polarization state, is ∆φmin = √ 1 . NL (4.7) 94 4. Dispersive Spin Measurements Introducing this expression into (4.2), we can find the smallest number of spins that can be distinguished from the light shot noise of the dispersive probing 1 . ∆NA = √ 2 . NL G1 (4.8) A QND measurement is considered sensitive if it allows to resolve the quantum noise of the measured state. In terms of the signal-to-noise ratio, introduced in the theory chapter in Eq. √ (2.67), this corresponds to ζ = 1.√A coherent spin state has a quantum noise of NA . Hence, starting from ∆NA = NA the QND measurement reaches an appreciable sensitivity. However, this definition only makes sense if we keep depolarization effects in mind. Therefore, it is important at which level of destruction η the QND measurement starts to be sensitive. In Fig. 4.5 we plot ∆NA as a function of the depolarization η, changing the number of probe photons. For each point we calculate the ∆NA from the number of photons NL sent via Eq. (4.8). The depolarization is computed from the amount of decay observed in the rotation signal (cf. Fig. 4.2) for this particular number of photons. For example, for the data at ∆ = −490 MHz (black circles) we can have a sensitivity of ∆NA ' 900 atoms at NL ' 1.5 × 108 photons, which cause a damage of 6 %. We can derive a relation between η and ∆NA by introducing Eq. (4.8) into Eq. (4.5) and get 4 η(∆NA ) = ηγ . (4.9) (G1 ∆NA )2 In Fig. 4.5 we show η(∆NA ) for detuning of −490 and −590 MHz. In addition we −2 plot a curve with η ∝ d−1 0 (∆NA ) , where the black line is for d0 = 57, the red and the blue for d0 = 25 and d0 = 100, respectively. For our atomic ensemble of NA = 1 × 106 we indicate the projection noise level with a vertical gray line at ∆NA = 1 × 103 . This level of sensitivity can be achieved with only ∼ 5 % of destruction. As we have seen in Appendix A by simulating the quantum noise behavior for our experimental parameters, a damage of about 20 % is the optimum trade-off between the information gain and state destruction for the best spin squeezing. The measurements here confirm that we have enough dispersive interaction at a reasonable amount of absorptive interaction to perform sensitive QND measurements. Nevertheless, this experiment is not yet proof of having a spin detector which is quantum noise limited and allows us to produce a squeezed spin state and the associated macroscopic entanglement. This characterization is performed in the Sect. 7.1. 4.2. Measuring Zeeman coherences In this section we present measurements of the per-atom rotation caused by the tensorial term of the polarizability tensor G2 . We will use this interaction to measure 1 For a photon detector this would be the noise equivalent photon number to overcome the electronic noise, which is in our case the light shot noise. 4.2. Measuring Zeeman coherences 95 Figure 4.5.: Sensitivity: measured depolarization versus minimum detectable number of spins ∆NA for detunings close to resonance. The circles and the squares show the data for −490 and −590 MHz, respectively. The black line has the expected ∆NA−2 -dependence for the measured optical depth of 57(3). As comparison we plot lines for the case of smaller (25) and larger (100) optical depth as compared to the present situation. the spin polarization of the coherent spin state we are using for the spin squeezing measurements in Chap. 7. 4.2.1. Introduction Using the tensorial term in the dipole interaction Hamiltonian is the same as using the vectorial, with the difference that it allows to measure the coherences between the two hyperfine ground states |1, −1i and |1, +1i. In the case of the Jˆz measurements, the physical effect was circular birefringence or paramagnetic Faraday rotations. Now, we want to use the effect of linear birefringence as in a wave-retarder, for instance. A spin state along Jˆz has a symmetry axis along z. The coherent superposition state, we want to measure now, has a symmetry axis which is orthogonal to the z in a direction which depends on the way we prepared the superposition (cf. Sect. 3.4). Experimentally, we can make very sensitive measurements of Ŝy , so it is convenient (in) (in) to use circular light for the probing, i.e., h Ŝz i = NL /2 and h Ŝy i = 0. This circular light will acquire an Ŝy -component proportional to the Jˆx component of the state. In terms of Stokes vectors, we can write the input-output relation of the light entering and leaving the atomic cloud as (out) Ŝy (in) = Ŝy (in) (in) (in) (in) + G1 Ŝx Jˆz + G2 Ŝz Jˆx , (4.10) (in) where the second term for the case of h Jˆx i = NA /2 and circular polarization is much bigger than the first. Also, for a detuning where G1 = 0 we can neglect the first term. The measured signal is then directly proportional to Jˆx and we can write (out) Ŝy (in) = Ŝy (in) ˆ(in) Jx + G2 Ŝz , (4.11) Analogous to the case of the vector probe we can measure the per-atom rotation or coupling constant G2 by recording both the rotation angle of the light polarization 96 4. Dispersive Spin Measurements 0 s [rad] 3 s [rad] 6 s [mrad] 5 ï0.05 2 4 ï0.1 1 ï0.15 0 3 2 1 ï0.2 ï500 ï400 ï300 ï200 ï100 6 [MHz] ï1 0 100 200 6 [MHz] 300 400 0 432 462 6 [MHz] 492 Figure 4.6.: Spectrum of tensor interaction. The solid black line is calculated through the tensor polarizability α(2) and the blue circles are numerically calculated values for the linear-birefringence of the atomic sample. Both show a reasonably good agreement which ensures that the approximations applied in deriving α(i) are correct. We plot the polarization rotation obtainable with realistic experimental parameters of NA = 1×106 and A = 4 × 10−9 m2 . Left: Region we use for the experiments which has reasonable dispersive interaction. The large green circle at −200 MHz marks the point where we simulated the time behavior in Fig. 4.7 Center: Region between the excited state hyperfine levels. The point, marked with the square, is the position where G1 = 0. There, dispersive effects due to G2 are strong but so is absorption. Right: Region around the point of vanishing G1 (∆ = 461.7 MHz). This would be ideal place to probe Jˆx with a purely tensorial interaction. Unfortunately, the dispersive interaction very small. Note, the difference in the scale for the rotation angles. and the number of atoms in the ensemble. The rotation angle is now defined as the (out) (in) ratio of Ŝy - and Ŝz -components: ψ ≡ h Ŝy i / h Ŝz i = G2 N2A . Ideally, we would like to measure at a spectral position where the vector contribution is vanishing. There exist two points in the spectrum where we have G1 = 0 and G2 6= 0. One of them is located between the resonance F = 1 → F 0 = 0 and F = 1 → F 0 = 1 at ∆ = 35.84 MHz. The other is situated at ∆ = 461.7 MHz. In the two right plots of Fig. 4.6 we show the spectrum of G2 around these two points. For the first one at 35.84 MHz the dispersive interaction is very large. Simultaneously, the vicinity of the resonances implies a large absorptive interaction. Hence, we cannot use this point for a non-destructive measurement of the state. For the point at 461.7 MHz the situation is better in terms of absorption, but also the dispersive interaction is particularly small. The reason is that also G2 has a zero at about 501.8 MHz. Much better is the region below the resonance F = 1 → F 0 = 0. This is shown in the left panel of Fig. 4.6. There, an appreciable dispersive interaction can be obtained at reasonably low photon scattering rates. The price we have to pay is that the vector interaction is not vanishing and we get contributions from the term ∼ Ŝz Jˆz . We explain how we can overcome this problem in the next section. 4.2. Measuring Zeeman coherences 97 4.2.2. How to measure Zeeman coherences in the presence of vector light shifts? A circularly polarized probe pulse would cause a precession of the atomic state about Jˆz which interchanges Jˆx and Jˆy . Physically, we can understand this precession as the result of a different AC Stark shift of the magnetic substates in the ground state from the circular probe light. They lift the degeneracy of the two components of the coherent superposition state, i.e., |1, −1i and |1, +1i, and the state starts to precess at a rate related to this energy splitting. We could assign a fictitious magnetic field to the energy shifts and interpret the interaction as being purely magnetic [121]. One possibility to avoid this problem is to probe the atomic state successively with pulses of left and right circular light. In effect this will “balance” the spin state along Jˆx and no effective precession is taking place. Though very elegant, the solution demands a separate control of an additional light field with exactly orthogonal polarization. Later, when we measure the projection noise of a the coherent spin state in presence of non-vanishing tensor light shifts, we will apply this strategy. Now, since we are interested only in the magnitude of the large spin polarization, it is much simpler to apply a magnetic field along the symmetry axis of the state. A magnetic field which is large enough to dominate the fictitious magnetic field will stabilize the prepared coherent superposition while we measure its magnitude Jˆx . We simulate both situations, i.e., with and without a magnetic field, and compare their results. For the simulations we start with a coherent superposition of |1, −1i and |1, +1i which has a mean value of h Jˆx i = NA /2, where we assume NA = 1 × 106 and an interaction area A = 4 × 10−9 m2 , as in the calculations shown in Fig. 4.6. In the left of Fig. 4.7 we plot the rotation signal as a function of probing time. In the case of no magnetic field, the state precesses due to the vectorial light shifts. With the magnetic field, we suppress the influence of the vector light shifts and can measure the signal over a long time scale. Both situations are equivalent in terms of the information deduced. The decay we observe is the depolarization introduced through the probe light. In the right plot, we show the evolution of the pseudo-spin projections. In the case of no magnetic field we see the expected precession of the state about the z axis. The Jˆz component which is increasing over time is the result of optical pumping during the probing process with circularly polarized light. In the experiment we probe atoms over much shorter timescales to avoid the reduction of the atomic polarization. Here, we show the evolution for longer time to see all the processes. The precession of the state due to the influence of the vectorial polarizability can also be used to produce a state along any direction in the Bloch sphere. For this, one would have to change the illumination, in order to have a constant intensity over the whole sample. For our geometry this might be a difficult undertaking. In smaller ensembles this manipulation is already demonstrated [122, 123]. 98 4. Dispersive Spin Measurements 40 s [mrad] 0.5 Jx [NA] 30 20 10 0 0 ï10 ï20 Jx ï30 ï40 0 Jy Time [ms] 0.25 ï0.5 0 Jz 0.25 Time [ms] Figure 4.7.: Simulation of rotation signal ψ and pseudo-spin projections as a function of time (photon number) with and without guiding magnetic field B = (0, 0.5, 0) mG. The tensor-probe light is tuned ∆ = −200 MHz away form the F = 1 → F 0 = 0 transition. Left: The green solid and the blue dotted line are the respective signals with and without magnetic field along y. Right: Projection along the pseudo-spin directions Jˆx , Jˆy , and Jˆz for the case of no bias field. The blue dotted line is the projection along Jˆx when the magnetic field is applied. In this case the magnitude of Jˆy ∼ Jˆz ∼ 0 and not shown in the plot. 4.2.3. Calibration of G2 Now we measure G2 by polarizing the atomic sample along Jˆx in a guiding magnetic field. The circular probe light detuning is changed over one order of magnitude. The number of atoms is measured via absorption imaging. In this way we can get a precise measure of G2 as the per-atom rotation. The atomic state is prepared as explained in Sect. 3.4 by optical pumping atoms with x-polarized light propagating orthogonal to the sample axis. The light is resonant to F = 1 → F 0 = 1 and transfers atoms optically into the hyperfine ground state |1, 0ix which is a dark state for this polarization and detuning. The index x indicates the quantization axis is along x, parallel to the light polarization. In the usual notation, i.e., with the quantization axis along z, the state is the coherent superposition between |1, +1i and |1, −1i |Ξi = 1 2NA /2 [|1, −1i − |1, +1i]⊗NA (4.12) In terms of pseudo spin operators we find hΞ| Jˆx |Ξi = −NA /2. After loading the atomic ensemble we switch on a small magnetic field of a Bx ∼ 200 mG along x and send linearly polarized light through the imaging optics to the atomic cloud. We choose a pumping time of 300 µs, which is sufficiently long to ensure a high degree of optical pumping. Once the coherent spin state is prepared, 4.2. Measuring Zeeman coherences 99 we send a train of circular probe pulses along the atomic ensemble and record their polarization rotation ψ. The number of atoms is measured by absorption imaging after the probing. With Eq. (4.11) we can calculate the coupling constant G2 , which is plotted in Fig. 4.8 for all measured detunings. G [×107] 2 10 1 0.1 ï500 ï300 ï200 ï100 ï70 6 [MHz] Figure 4.8.: Measured tensorial coupling strenth G2 as a function of probe detuning. The black line is a fit using expression (4.13). The free parameters are interaction area A = 3.0(3) × 10−9 , a frequency off-set due to light shifts ∆0 = 4(6) MHz, and an initial imbalance of the polarimeter G2,offset = 6(2) × 10−9 . Using Eq. (4.11) we can construct the fitting function G2,meas = G2 (∆ − ∆0 , A) + G2,offset , (4.13) where G2 (∆, A) is the theoretical expression of the tensorial interaction strength given in Eq. (2.31), with an effective interaction area A, a frequency offset ∆0 , and (in) an offset G2,offset caused by an initial imbalance of the polarimeter h Ŝy i = 6 0. The frequency offset ∆0 might be caused by the AC Stark shift of the trapping light. The result of the fitting reveals an interaction area A = 3.0(3) × 10−9 m2 which is comparable to the one we measured for the vector probe. The shift in the detuning is ∆0 = 4(6) MHz. This in good agreement with the light shifts, we measured for example in the state tomography in the trap (cf. Chap. 6) or the optical pumping dynamics in the next section. The initial imbalance of the polarimeter corresponds to an interaction strength of G2,offset = 6(2) × 10−9 . The data points follow over the whole range of detunings the typical ∆−2 behavior of the tensorial polarizability. We conclude that the experiment successfully shows a measurement of the pseudo-spin component Jˆx or equivalently the coherences between magnetic substates by using the tensor term in the dipole interaction Hamiltonian. 4.2.4. Optical pumping dynamics We apply the measurement of spin state polarization to monitor the evolution of the optical pumping into the coherent superposition state. For this we vary the 100 4. Dispersive Spin Measurements duration of the optical pumping light and measure the amount of spin polarization. As already mentioned, we are able to record the pumping light with a calibrated CCD camera which gives a measure of the local intensity. The time evolution of the spin polarization together with numerical simulations can be used to estimate the degree of optical pumping. The measurement proceeds in the same way as before, except we stop the optical pumping after a variable time t and measure the signal of the tensor probe. We model the pumping process by a simple three level system where we pump atoms from an initial into a final state by exciting an intermediate level, which decays into both the final and the initial levels. This gives an exponentially saturating growth of spin polarization ψ(t) = ψ∞ (1 − exp (−k(t − t0 ))) , (4.14) where k is the rate of optical pumping, t0 an offset in pumping time, and ψ∞ the final amplitude of the signal. In the left plot of Fig. 4.9 we show the measured spin polarization as a rotation signal of the tensor probe and a fit to the data points using Eq. (4.14). We find the following parameters: k = 4.7(3) × 104 s−1 , ψ∞ = 17.1(4), and t0 = 1.7(5) µs. The offset in pumping time is an experimental artifact of the preparation of the pumping light, i.e., a delay in the switching of the AOM used. After ∼ 100 µs the spin polarization reaches a steady state. 20 s [arb. units] J [norm.] x 1 15 0.8 0.6 10 0.4 6=0MHz 6=ï1MHz 6=ï2MHz 6=ï3MHz 6=ï6MHz norm. data 5 0.2 0 0 50 100 150 t [µs] 0 0 50 100 150 t [µs] Figure 4.9.: Left: Measured rotation signal from tensor probe as a function of optical pumping time t. Right: Master equation simulations of the optical pumping for different detunings of the pumping laser from resonance. We used: Ipump = 0.44 Wm−2 and Ideplete = 3.5 Wm−2 . For a better understanding of the pumping process, we use our knowledge of the local light intensity as an input parameter for numerical master equation calculations. In this way, we can estimate the amount of optical pumping from the shape of the 4.3. Dispersive atom number measurement 101 measured rotation signal. We also have access to the amount of light used for the depletion of F = 2 during the pumping. The only free parameter in the calculation is the detuning of the pumping laser. This is only known within a few MHz since we prepare atoms in the dipole trap, where they experience an AC Stark shift. In the right of Fig. 4.9, we show the results of the simulations for a set of detunings. We find the best overlap of simulation and normalized rotation signal for a detuning of −3 MHz of the pumping light. The good agreement between simulations and measured data is a first indication of a pumping efficiency close to unity for the atoms in F = 1. However, there are other effects which can not be revealed by this measurement. Our main concern is the depletion laser of F = 2 which could pump atoms into a dark state in F = 2. This effect should be small since the depletion light is applied from six different directions and forms a polarization pattern on the sub-wavelength scale. Atoms will move through this pattern due to their thermal energy. If we estimate the rate of light scattering for atoms in F = 2 we get values of a few percent of the natural linewidth Γ = 2π × 6.066 MHz. If this is large compared to the time an atom needs to travel a distance to experience a different local polarization, e.g., ∼ 100 nm, the atom cannot adiabatically follow. For our temperatures, we estimate this time to be around 1 µs. From this we conclude that the thermal motion of atoms is sufficient to largely suppress the pumping into dark states. For an order of magnitude estimate, it is important to know that in the worst case, i.e., a single fixed polarization for the depletion laser, at maximum of 10 % of the atoms are pumped into a dark state in F = 2. In summary, we conclude that the effect of ’trapping’ atoms in a dark state in F = 2 is a small effect of a few percent of the total number of atoms. Hence, the measured degree of optical pumping reflects realistic values close to unity. 4.3. Dispersive atom number measurement Throughout the work we will use paramagnetic Faraday rotation measurements as a fast and nondestructive way of determining the number of atoms. We will call this technique dispersive atom number measurement (DANM). For this we measure the per-atom rotation G1 as described above, through the macroscopic rotation from a spin polarized atomic state and the atom number by absorption imaging. This allows us to have a quick and precise reference of the number of atoms in the atomic ensemble without the destructiveness of absorption imaging. In a single trap loading, we are able to perform many experiments and measure the number of atoms present at each of them. Before or after any large experimental run, we repeat the calibration to account for daily fluctuations of light intensities and alignments. The DANM has the following pulse sequence: the sample is spin-polarized along z by on-axis optical pumping with a 50 µs pulse of circularly-polarized light tuned to the F = 1 → F 0 = 1 transition. At the same time, light resonant to the F = 2 → F 0 = 2 transition (via the MOT beams) prevents accumulation of atoms in F = 2. We define a quantization axis by applying a small bias field of ∼ 100 mG along z. 102 4. Dispersive Spin Measurements Probe pulses with NL = 1 × 105 − 5 × 105 tuned 500 − 1000 MHz to the red of the F = 1 → F 0 = 0 transition are used to measure the rotation angle φ = NA G1 . No appreciable atomic depolarization could be observed experimentally after 20 probe pulses, which is supported by calculations that give a destruction of 0.08 % for this number of pulses. 4.4. Relation between scattering cross-section and observed depolarization Sometimes in the literature [63, 124, 125, 117, 118, 126] the depolarization η, we have measured above in Sect. 4.1.2, is related to the atomic scattering cross-section σ(∆) and an effective interaction area A . The relation used is η= σ(∆) NL , A (4.15) where we can relate the depolarization per photon to the ratio of scattering crosssection and effective interaction area ηγ = σ(∆) . A (4.16) Making this straight connection is not always valid. A scattering event will not inevitably destroy the coherence of the state and reduce the polarization. Using the master equation we simulate polarization loss ηME of atoms in F = 1 from light scattering out of a probe pulse with NL photons at a detuning ∆ and with an effective interaction area A. This we compare to the value η which is calculated from expression (4.15) for the same set of parameters. The ratio between the value (in) (out) for the master equation simulations ηME ≡ ln h Jˆi i − ln h Jˆi i, for small NL and different i = x, z and Eq. (4.15) are shown in Fig. 4.10 for a polarized spin state along Jˆz , i.e., |1, 1i, and along Jˆx , probed with linearly and circularly polarized light, respectively. We see that there is a large difference between the estimated depolarization from the scattering cross-section and the actual value. In the case of the polarized state along z we see that only every third scattering event will lead to a polarization loss of the atomic state. In order to use the scattering cross section to predict the depolarization of an atomic ensemble one has to apply a correction. The correction factor is the ratio κ ≡ ηME /η we plot in Fig. 4.10. The corrected expression for the depolarization is σ(∆) η = κ(∆) NL , A where the correction has to be calculated depending on the spin state. Also the scattering cross-section σ(∆) is different for different states. 4.4. Relation between scattering cross-section and observed depolarization 1 103 g 0.8 0.6 0.4 0.2 0 ï2 ï1.5 ï1 ï0.5 6 [GHz] Figure 4.10.: Ratio of depolarization rates ηME and η for master equation simulations and using the scattering cross section. The upper and the lower curve show the ratio for a spin state along Jˆx and Jˆz , respectively. 5. Magnetic Field Measurements Contents 5.1. Spatially integrating magnetic field measurements . . . 106 5.1.1. Magnetic field measurement via MOT position . . . . . . . 106 5.1.2. Magnetic field measurements via collective Larmor precession107 5.2. Spatially resolved magnetic field measurements . . . . . 110 5.2.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2.2. Spatially-resolved Larmor precession . . . . . . . . . . . . . 111 5.2.3. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2.4. Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 106 5. Magnetic Field Measurements Atoms are highly sensitive to magnetic fields. This can be useful, because an atomic cloud can be used as a highly sensitive magnetometer. It also means that experiments such as spin squeezing require good control over the magnetic field. We use three different methods to determine the magnetic field around the atomic ensemble. All of them use the atomic sample as magnetic field probe, but in different ways. We start with a very simple measurement of the MOT position as a function of applied bias fields in Sect. 5.1.1. This method gives an accuracy of tens of mG. In Sect. 5.1.2 we use Larmor precession, measured via paramagnetic Faraday rotations. This increases the sensitivity dramatically to sub-mG precision. Finally, in Sect. 5.2 we develop a √ method to measure the field spatially resolved with sensitivities down to ∼ 150 nT/ Hz and at a resolution of 50 µm. 5.1. Spatially integrating magnetic field measurements 5.1.1. Magnetic field measurement via MOT position The main idea of this approach is to measure uniform stray magnetic fields from the position of the atomic cloud in the MOT. One example of a stray field is the earth magnetic field, which we have to compensate for. In this, we assume that the stray magnetic field and the compensation field, we apply, are both uniform over the size of the atomic cloud 1 . Furthermore, we assume that the magnetic field, produced by the anti-Helmholtz coils, has a constant gradient, i.e., grows linearly going away from the center in any spatial direction. The light fields of the MOT are suppossed to produce equal radiation pressure in all directions. Thus, no systematic error in the MOT position due to the light fields is present.2 We can write for the field along a single direction ∂Bz Bz (z) = (z − z0 ) + Bz,0 , (5.1) ∂z z where ∂B ∂z is the gradient of Bz along z, Bz,0 a constant offset, and z0 the displacement from the geometrical center of the quadrupole coils. If the light fields of the MOT beams are well balanced, then the position of the atomic cloud in a MOT potential is the place of vanishing magnetic field. We can write zM OT = z0 − Bz,0 ∂Bz ∂z . (5.2) The offset field can be expressed as the sum of a stray-field Bz,s , we want to determine, and a known contribution Bz,c from the compensation coils, Bz,0 = Bz,s +Bz,c . z Hence, measuring the MOT position as a function of the gradient ∂B ∂z and the applied compensation field Bz,c , will reveal the unknown offset field. At Bz,0 = 0 the position of the atomic cloud becomes independent of the applied gradient. 1 We also have to ensure that the center of the quadrupole coils and the dipole trap are close together. If not, we just measure the stray field at the geometric center of the quadrupole coils. 2 This assumption was tested epxerimentally by optimizing the MOT alignment for a stable optical molasse. 5.1. Spatially integrating magnetic field measurements 107 Figure 5.1.: Fluorescence images of atomic cloud position in MOT for different gradients (indicated as current in the MOT coils in the left upper corner). Horizontal and vertical axes show the z and x direction in pixel, where one pixel= 9.4 µm. We plot in Fig. 5.1 fluorescence images taken along the y-direction for different applied gradient fields, to illustrate the method. The atomic cloud position depends on gradient current due to an uncompensated x field. The magnetic field gradient is reduced from left to right by 50 % of its initial value. To study the effect in more detail, we take a series of measurements for different gradients and scan the compensation fields along x and z. For values of the gradient coil current between 3 and 6 A, we plot the results in Fig. 5.2. As expected from Eq. (5.2) the center of the atomic cloud is a function of applied field. The slope is inversely proportional to the applied gradient. We make an estimate on the accuracy of this method from Fig. 5.2. For our setup, the accuracy of this method is limited by the precision of the current in the coils. With uncertainty of about 10 mA for both coils. This converts into a magnetic field sensitivity of several tens of mG. In summary, we see that this method can only give a coarse estimate of the bias field amplitude. 5.1.2. Magnetic field measurements via collective Larmor precession More precision can be gained by using internal atomic states. This, will be the first application, in this thesis, of the atomic ensemble as a sensitive magnetic field probe - a magnetometer. For this purpose we use our ability to polarize the atomic ensemble along the quantization axis z, described in Sect. 3.4, and the non-destructive measurement of spin polarization along z, described in Chap. 4. This gives us a method to determine the fields orthogonal to the quantization axis z with high accuracy. The compensation field Bk,c , necessary to cancel the existing stray field Bk,s for k = {x, y}, can be found by scanning the coil current (compensation field) and measure the Larmor frequency as a function of Ik,c . For this, we polarize the atomic sample along z and send a train of hundreds of far detuned probe pulse through the cloud, as demonstrated in Chap. 4. The rotation signal will be modulated in time due to the Larmor precession of the polarized spin state about the magnetic field axis. 108 5. Magnetic Field Measurements 525 335 330 475 Z MOT Center [pixel] X MOT Center [pixel] 500 450 425 400 320 315 310 375 350 −30 325 0 30 60 X Coil Current [mA] 90 305 −30 0 30 60 Z Coil Current [mA] 90 Figure 5.2.: MOT position as function of compensation coil current for x-compensation coils (left) and z-compensation coils (right). The observation direction is y. A single pixel on the CCD corresponds to 9.4 µm at the position of the atoms. The MOT coil current is changed from 3 A (circle), 4 A (square), 5 A (cross) and 6 A (triangle), which corresponds to a change from 21 G/cm to 42 G/cm. Exemplary, we show the linear fit for the case of 6 A in the black solid line. The light parameters are chosen in a way that we minimize the damage of the atomic spin polarization, but probe rather non-destructively. This allows to observe the spindynamics in real time. Practically, we us NL ∼ 106 at a detuning |∆| > 1 GHz as in Chap. 4. For the optical puming we use a rather short pulse of ∼ 10 µs to have an instantaneous preparation on the time scale of the precession frequency. The amount of optical pumping is of minor importance for this kind of measurement. The expectation value of the spin component along the quantization axis will describe a damped oscillation Bz2 + Bx2 + By2 cos (γB t) e−ηt h F̂z i = NA , (5.3) B2 where Bk are the field amplitudes along the k-direction, γ is the gyromagnetic ratio3 , and η describes the loss of collective spin polarization. The collective spin polarization could be reduced due to inhomogeneous magnetic fields which would dephase different parts of the atomic sample to each other. The observed signal can be written as h Ŝy i = h Ŝy i∞ + h Ŝy iosc cos (γB t) e−ηt , (5.4) where h Ŝy i∞ ∝ Bz2 /B 2 is the rotation signal after the oscillations have ceased and h Ŝy iosc ∝ (Bx2 + By2 )/B 2 is the amplitude of the oscillation at t = 0. 3 For atoms in the F = 1 manifold γ = 2π700 kHz/G. 5.1. Spatially integrating magnetic field measurements 109 Precession Frequency [kHz] 12 10 0 8 6 4 0 2 0 35 37 39 41 Y Coil Current [mA] 43 45 0 0.1 0.2 0.3 0.4 Time [ms] 0.5 Figure 5.3.: Left: Larmor precession frequency measurement for the y compensation coil. We deduce a conversion between current and magnetic field for q the y coil of 2 + B2 3.62(5) G/A. For the stray field we find By,s = 145.6(2) mG and Bx,s z,s = +1.2 1.2 −1.1 mG. Right: Time traces of Larmor precession for two different compensation fields along y. Here, they serve only for illustration and were not taken at the same time as the data for the left figure. Nevertheless, the bias field components By are the same as used for the corresponding points in the left plot, indicated by the colored squares. The deduced z bias field is therefore slightly different than what is shown in the left plot. We find Bz,s = 4.3(4) mG and Bz,s = 3.1(3) mG for the upper and lower plot using Eq. (5.6). At the right side of Fig. 5.3, we plot two examples of the Larmor precession signals when applying a compensation field along y. The fast decay of the oscillation contrast is the result of magnetic field inhomogeneity over the length of the atomic sample. One reason could be the inhomogeneity produced by the compensation coil itself which are far from being in a Helmholtz configuration. Another reason could be the existence of gradients from external sources, which will be discussed in the next section in greater detail. The different phases in the observed signal, i.e., the red data points start from above and the green from below, are due to the different delay time between the preparation and the probing. From the measurement of the Larmor frequency as a function of the compensation fields, as plotted in the left panel of Fig. 5.3, we get several pieces of information. To understand them better, we can write the expression for the frequency as function of field components, determined from (5.3) q γ Bx2 + By2 + Bz2 fL = . (5.5) 2π First, we can deduce the relation between applied current and resulting magnetic field from the linear slope on either arm of the “V”. The slope is linear in a region where 110 5. Magnetic Field Measurements By Bx , Bz , which gives a direct measures of By . Second, the compensation field for the scanned direction, i.e., By = By,c + By,s = 0, can be found from the position of the minimum in fL . Third, we can estimate the uncompensated p field in the plane orthogonal to the scanned direction, i.e., in the present case Bx2 + Bz2 . By fitting the data with the theoretical expression (5.5), we can deduce all three quantities. In Fig. 5.3 we show the resulting fitted function for i) the case of no remaining orthogonal components (Bx2 + Bz2 = 0) and ii) the exact expression, in the solid and the dotted lines, respectively. The precision in the determined compensation field in i) and ii) is 0.1 %, or in absolute terms, 200 µG for the presented data in Fig. 5.3. The orthogonal components are less well resolved. The experimental errorp in measuring very low frequencies is large and prevents a reliable measurement of Bx2 + Bz2 . For this reason, we use this method in an iterative manner, compensating x and y several times. To determine the remaining z component we compute s h Ŝy i∞ Bz = Bx , (5.6) h Ŝy iosc where we assumed Bx2 By2 . The accuracy of this method is also rather poor, compared to the compensation field scan and measurement the fL . The stray fields we found for the example in Fig. 5.3 are given in its caption. In summary, we have seen that Larmor precession measurement gives a precise measure of the magnetic field. We can reach sub-mG sensitivities in the field component which is scanned. Furthermore, we are still limited by magnetic field inhomogeneities which are not easily resolvable with this method. A possible way to measure the inhomogeneity of the field would be a Fourier analysis of the precession signal as proposed in [127]. Instead, we are using another method to measure the magnetic fields with spatial resolution, which will be explained in the next section. 5.2. Spatially resolved magnetic field measurements We have talked about the inhomogeneity of the magnetic field several times in the course of this chapter. On our journey to the realization of spin squeezing (Chap. 7) we had to learn that the magnetic field environment of the trap apparatus is more complicated than expected and contains spatially changing magnetic fields. Our atomic sample has a length of ∆z ∼ 10 mm and dephasing happens at a rate4 of dφ/dt == ∆z2γ∂B/∂z, where γ is the gyromagnetic ratio. Practically, our experiments will take about 100 µs, so that it will be sensitive to gradients of the order of ∂B/∂z = 2π(100 µs 1cm 2 2π700kHz/G)−1 ∼ 7 mG/cm, where we assumed a complete dephasing for a rate of dφ/dt = 2π/100 µs. This demands a well controlled magnetic field environment in terms of spatial field properties. Otherwise, a spin polarized atomic state would dephase very fast and the time for its use, e.g., as a 4 The above expression for the dephasing rate assumes an extended sample spread over a length ∆z which is exposed to a magnetic field gradient of ∂B/∂z. The phase difference between the beginning and the end of the atomic sample after a time t is φ = 2ωL t = 2γ∆z∂B/∂z t, where ωL is the Larmor precession frequency. 5.2. Spatially resolved magnetic field measurements 111 magnetometer, is limited to unnecessarily short periods. At the same time, the extended atomic ensemble is an attractive system for measuring magnetic field with high spatial resolution over a large spatial range. Here, we present a technique which can measure magnetic fields down to µG over several millimeter with micrometer resolution. In this section we present a method which uses the atomic cloud to map the magnetic field along the sample. We further show measurements in which we apply this method to cancel the magnetic field gradient present in the trapping setup. Part of this chapter is published in 128. 5.2.1. Background There exist a number of approaches to measure magnetic fields with high sensitivity and spatial resolution. Scanning Hall probes can reach sub-micron resolution with µT sensitivity [129, 130, 131]. On the other hand, superconducting quantum interference devices (SQUID) reach sensitivities down to 0.1 pT but are limited in spatial resolution to tens of microns [132, 133, 134]. Ultra-cold atomic ensembles seem to bridge the gap by offering high magnetic field sensitivity (10−10 − 10−11 T) together with micron size spatial resolution over length scales of a few hundred microns [135, 136, 137]. The most sensitive magnetometers today use room temperature vapor cells and measure √ Larmor precession of spin polarized atoms. They can reach sensitivities of 0.5 fT/ Hz at a very poor spatial resolution of tens of millimeters. A more complete overview can be found in [138] . 5.2.2. Spatially-resolved Larmor precession Spatially varying magnetic fields are detected measuring Larmor precession by means of sensitive absorption imaging. Similar to room temperature vapor cell magnetometers [139], Larmor precession give a sensitive and direct measure of local magnetic fields. However, micro Kelvin cold atoms held in a dipole trap, compared to hot vapor cells, are much better localized (< 20 µm) and their thermal motion is reduced, which allows for spatial resolution within the atomic ensemble. Locally, the measurement is similar to a Ramsey spectroscopy sequence: The initial state for a single atom ρo ≡ |Ψd i hΨd | is a magnetically sensitive coherent superposition of hyperfine ground states. It is produced via optically pumping atoms into a dark state |Ψd i. The density matrix ρ(t) evolves under Ĥmag = γB · F, where γ is the gyromagnetic factor, and possible decoherence mechanisms, including atomic motion and collisions, for a time t. Atoms which leave the state |Ψd i are detected by state-selective absorption imaging. For an extended atomic sample with the density n(~r) we find the absorption imaging signal to be Z ∞ S(x, z, t) ≡ K n(~r) (1 − Tr [ρ0 ρ(t)]) dy , (5.7) −∞ where y is the imaging direction and K is a constant. For elongated thin atomic samples, as ours, there is little information in x-direction. Essentially, we are left 112 5. Magnetic Field Measurements with a 1+1 dimensional map of the magnetic field Z S(z, t) ≡ S(x, z, t) dx (5.8) along the sample and in time. Here we integrate only over the region of the signal S(x, z, t) which contains atomic information. In the experiment, this will be a few lines of the CCD image. For alkali metal atoms the magnetic Ramsey sequence can be implemented in the following way. In Fig. 5.4 we show the sequence for the special case of 87 Rb and the D2 line. The existence of two hyperfine ground states with total angular momentum F± = I ± 1/2, where I is the nuclear spin quantum number, allows us to split the task of spin state preparation and detection. An initial state is prepared in the F− manifold and evolves for some time t under the influence of Ĥmag . Atoms are detected by state-selectively transfer to F+ from where they can be counted by means of absorption imaging, which is highly sensitive if performed on a cycling transition with F+ → F 0 = F+ + 1. The state preparation on the F− → F 0 = F− transition with π-polarized light produces an initial state by pumping atoms into the dark-state |Ψd i ≡ |F− , 0i. This state has a symmetry axis s which is parallel to the quantization axis, i.e., the electric field direction of the pump light in this case [140]. For the case of F− = 1, and no decoherence, the signal can be directly calculated using 2 Tr[ρ0 ρ(t)] = hψd | exp[−iĤmag t/~] |ψd i (5.9) 2 S(z, t) = 1 − 1 + sin2 θ (cos (ωL t) − 1) , (5.10) to get where cos θ ≡ s · B(z)/ |B(z)| and ωL ≡ γB(z) is the Larmor frequency. Vector Magnetometer For one initial state |Ψd i we acquire full information about the magnitude and some information about the direction through θ of the magnetic field. To identify the magnetic field direction unambiguously, it is sufficient to repeat the magnetic field imaging with another dark state |Ψ0d i along a different symmetry axis s0 ∦ s. The process should be supplemented by applying small bias fields along three orthogonal direction to distinguish between Bk and −Bk , for k = x, y, z. In more detail, we can implement the vector field measurement protocol as illustrated in Fig. 5.5: a) A single magnetic field image (MFI) would reveal the the angle θ up to a phase of π. That means that a single measurement would reduce the possible direction of the magnetic field from 4π to two cones with an opening angle of θ and θ + π. b+c) Applying a magnetic field parallel or anti-parallel to the symmetry axis s, we find out in which hemisphere the magnetic field is pointing. d) A second dark state with symmetry axis s0 is used. Experimentally, it is desirable to use another 5.2. Spatially resolved magnetic field measurements 113 Figure 5.4.: Magnetic Ramsey sequence: a) The atoms are prepared into a dark state for linearly polarized light. resonant laser light empties the F = 2 manifold. b) Local magnetic field will rotate the state. c) The state selective transfer is performed with pumping light. d) Absorption imaging is used to count the number of transferred atoms. linear polarization orthogonal to the first one. This prepares a dark state which has its symmetry axis 90◦ flipped. We are still left with four possibilities for the magnetic field vector. e) As for the first dark state we apply a magnetic field along its symmetry axis s0 g) To distinguish between the last two possibilities we apply a magnetic field along the remaining axis which is orthogonal to s and s0 . Spatial Information The magnetic field imaging technique provides a convenient way to measure gradients and higher derivatives, which often give better signal to noise in the presence of fluctuating backgrounds [141]. If we write B(z) = B0 + B1 z + θ(z) = θ0 + θ1 z + B2 2 z + O(z 3 ) , 2 θ2 2 z + O(z 3 ) , 2 (5.11) (5.12) where Bn = ∂ n B(z0 )/∂z n and θn = ∂ n θ(z0 )/∂z n , we see the gradients cause a spatial modulation of the magnetic field image S(z) at any given time τ . In addition, B2 , the magnetic field curvature will produce a linear chirp of the spatial frequency. 114 5. Magnetic Field Measurements Figure 5.5.: Visualization of vector magnetometer protocol. a) Magnetic imaging with dark state along x. b) Discrimination between θx and θx + π, by application of small field along x. c) Magnetic field is pointing in upper hemisphere. d) Magnetic imaging with dark state along y. e) Discrimination between θy and θy + π, by application of small field along y. e) The magnetic field has positive values for Bx and By . g) Discrimination between positive and negative value for Bz , by application of small field along z. h) We measured the amplitude and the direction of the unknown magnetic field unambiguously. More quantitative information can be deduced from (spatial) Fourier transformation of the measured signals S̃(k) ≡ F (S(z)) at any given time t = τ , where k is the spatial frequency. We also assume the angle of the magentic field is a constant over the length of the sample. shape of the atom distribution along z, i.e., the R The ~ atomic line-density n(z) = n(r) dxdy, will influence the Fourier spectrum of the √ signal. For a Gaussian distribution, n(z) ≡ n0 exp −z 2 /σ 2 /(σ π), with a FWHM√ length 2 ln 2σ. Expanding the trigonometric function cos(ωL t) in Eq. (5.10) into complex exponential function the measurable signal reads 2 X S(z, t) = n(z) Ap exp −itγp(B1 z + B2 z 2 /2) + c.c. , p=0 where A0 = sin2 θ − 43 sin4 θ, A1 = sin4 θ − sin2 θ, and A2 = − 14 sin4 θ. We can identify three parts Sp of the signal, which have the Fourier transformations (k−pB1 τ γ)2 exp − 4/σ 2 +i2pB τ γ 2 S̃p (k) = Ap p , (5.13) 2π + iπB2 pτ γσ 2 where we only write the positive frequency part for k > 0. Later, we compute the power spectrum S̃p S̃p∗ of the Fourier transformation from the measured data. This 5.2. Spatially resolved magnetic field measurements 115 can contain up to three Gaussian peaks at kp = pB1 τ γ (5.14) with a FWHM of r √ 2 B2 ∆kp = 2 ln 2 + 2 σ 2 τ 2 γ 2 p2 , (5.15) 2 σ 2 where we observe only two in the case of θ = π/2 because A0 = 1/4, A1 = 0, and A2 = −1/4. No peaks are observed for the trivial case of θ = 0. 5.2.3. Measurements We present two sets of measurements. The first one shows the main features of the technique by following the spatially modulated signal over time for an inhomogeneous background magnetic field. The second illustrates the use of the method for in-situ monitoring of magnetic field changes. We show how a gradient field is canceled and how the method can serve as a quantitative characterization tool. Figure 5.6.: a) Laser frequency for pump-, transfer-, imaging-, and depletion-lasers. b) The atomic sample is held in a dipole trap and interrogated with linear pump, transfer and imaging light from the side. The depletion laser is applied through the MOT beams. Proof of principle The measurement proceeds as follows: we load the atomic trap for about 4 s and wait 400 ms for untrapped atoms to escape from the trapping region. Then the dark 116 5. Magnetic Field Measurements state is prepared by sending a 10 µs pulse of x-polarized light into the direction of the camera. At the same time, F = 2 is depleted by resonant light applied through the MOT beams. After a variable waiting time ∆t of 50 to 1000 µs we send a second 10 µs pulse to transfer atoms from the bright state to F = 2. The absorption image is taken with 100 µs of light resonant to the F = 2 → F 0 = 3 transition. At each waiting time we acquire 40 images under the same conditions, averaged them and integrate over four horizontal lines of the CCD which contain the atomic absorption signal. Knowing the on-resonant scattering cross-section (cf. Appendix B.1), we convert the signals from relative absorption into atom number. To suppress high frequency noise in the images, we apply a moving average over 7 adjacent pixels. The sensitivity in atom number per pixel is better than hundred atoms, as we can see from Fig. 5.8. A small background, due to atoms that remain in F = 2 during the preparation, is subtracted using a reference image taken immediately after preparation. Atoms that remained F = 2 do not influence the measurement of the magnetic field amplitude. However, it is an indication that the used preparation time of 10 µs or the applied depletion laser are not sufficient for effective state transfer. The measured signals for all delay times are plotted in Fig. 5.7. The model for fitting the data is: A(z, t) = n(z) [p (1 − C(z, t)) + C(z, t)S(z, t)] , (5.16) where 1 − p ≡ Tr [ρtherm ρ0 ] = 1/(2F + 1) is the overlap of an unpolarized (thermal) R ~ spin state and the dark state and n(z) = n(r) dxdy is the atomic line density which has approximately Lorentzian shape as we measured in Sect. 3.3.5. The local coherence is modeled by C(z, t) = exp −ηL2 (z)t (5.17) where L(z) is a Lorentzian function to account for density dependent variations in the coherence times. The entire data set, shown in Fig. 5.7, is fitted simultaneously to find parameter values given in the caption. The origin of the additional decoherence in the trap center was not investigated further. One possible explanation are spin changing collisions which cause a rapid decoherence of the spin state [38, 142]. In Sect. 3.4 we calculated the rate of spinchanging collisions to be ∼ 0.03 s−1 for realistic densities of n = 1011 cm−3 . This is too slow to explain the rapid decoherence within ∼ 900 s−1 . Another possibility are hyperfine changing collisions between the remaining atoms in F = 2. Further experiments are necessary to resolve this issue and are beyond the scope of the presented work. To see temporal Larmor precession, we plot the number of transferred atoms versus time for specific positions along the sample in Fig. 5.8. We chose two points which are located on either side of the trap center. There, we see that the local coherence time of the spin state is many milliseconds. The influence of the angle θ between the magnetic field and symmetry axis of the dark state is visible in the bimodal 5.2. Spatially resolved magnetic field measurements 117 Figure 5.7.: Spatial distribution of transferred population into F = 2. The horizontal axis covers a field of view of ∆z = 13.2 mm. Delay times are indicated for each frame. The solid yellow line is the fitted profile using the expression discussed in the text. The deduced parameters are for z0 = 6.6 mm: B0 = 5.7(2) mG, ∂B/∂z = 19.4(4) mG/cm, ∂ 2 B/∂z 2 = 7.9(8) mG/cm2 , θ = 81(2)◦ . The additional decoherence in the trap center has a FWHM ∼ 1.3(3) mm. structure of the oscillation signal. The observation of the long local coherence time shows that we would be able to produce coherent spin states with ms coherence times, by removing the magnetic field inhomogeneities. In-situ magnetic field gradient compensation The method described above permits us to shape the magnetic field environment. Our main goal is to compensate all magnetic field gradients along the sample axis, ∂By ∂Bz x i.e., ∂B ∂z , ∂z , and ∂z . We have seen that the field is sufficiently inhomogeneous to de-phase the collective polarization of a spin state in less than 50 µs (cf. first frame of Fig. 5.7). In the spin squeezing measurement we conduct in Chap. 7, it will be important to nullify the magnetic field over the whole sample length of more than 10 mm and increase the coherence time to > 100 µs, which is the necessary measurement time. The main sources for the gradient observed above are the permanent magnets built into the two ion-getter pumps of the vacuum system. They are mounted at a distance of 60 − 70 cm to the atomic trap and not magnetically shielded. At the time of the experiments, we were not able to provide a good magnetic shielding. The method for the gradient compensation is simple but laborious. We mount a pair of magnetic field sensors close to the trap apparatus at a distance of about 5 − 8 cm to each side of the trap along the sample axis. In this way, we monitor to first order the gradient ∂By ∂z . To compensate this gradient, we use a permanent magnet which was placed 118 5. Magnetic Field Measurements NA [per pixel] 200 0 NA [per pixel] 200 0 0 0.2 0.4 0.6 Time [ms] 1 Figure 5.8.: Larmor precession at fixed position. The upper trace is taken at z = 0.04 mm and the lower trace at z = 2.89 mm. The black solid line is a least-square fit of a damped oscillation. The fitted angles are θ = 80◦ (20◦ ) and θ = 80◦ (6◦ ), for uper and lower trace, respectively. The exponential decay of the contrast corresponds to a local coherence time ∼ 5 ms. Errorbars indicate statistical fluctuations in the sample of 40 measurements. in a ’strategic’ position5 to counteract the magnetic field gradient produced by the pump magnets. The permanent magnet is mounted on a rotation stage which allows to change the magnetic field at the position of the atoms in controlled way. With ∂B x this method we compensate ∂zy and partly ∂B ∂z by turning the permanent magnet. ∂Bz For the remaining gradient ∂z we use the compensation coils along z. For this, we independently adjust the current through each of the z compensation coils. The resulting magnetic field is the sum of a constant and a gradient field. For the currents in the left and right coil Il and Ir we can write:. Bz = Kc Il + Ir Il − Ir + Kg z, 2 2 (5.18) where Kc (Kg ) is the conversion factor between applied current and constant (gradient) magnetic field, which has units of G/A (G/cm/A). For the in-situ gradient cancellation at the position of the atoms, we fix the delay time to τ = 500 µs and record the spatial profile as an externally-applied field is adjusted stepwise by changing Il and Ir . In Fig. 5.9 a) and b) we plot the oscillating part of the signal, i.e., the Lorentzian atom distribution is subtracted, before and after compensation. In Fig. 5.9 c) we plot the power spectrum of the Fourier transformation of the signals for the individual adjustment steps. The oscillations 5 The atomic sample, the permanent and the pump magnet form a isosceles triangle, where the two magnets are placed symmetrically. 5.2. Spatially resolved magnetic field measurements 119 Figure 5.9.: Fourier analysis of gradient compensation. In a) and b) we plot the differential transmission ∆T ∝ S before and after the compensation, respectively. The constant atom number distribution n(z) was subtracted. In plot c) the FFT signal shows a successive reduction in the gradient from ∼ 20 mG/cm down to < 5 mG/cm. The width of the resonant peaks suggests a gradient resolution of ∼ 2 mG/cm. 120 5. Magnetic Field Measurements 2πk frequency is converted into a field gradient by ∂B ∂z = 2γτ , where k the frequency of the FFT. We start at ∼ 20 mG/cm and reduce the gradient down to < 5 mG/cm. The width of the resonant peaks suggests a gradient resolution of ∼ 2 mG/cm. 5.2.4. Sensitivity Motivated by recent experiments demonstrating projection noise limited atom number counting by absorption imaging [52, 143], we estimate the magnetic field sensitivity in the projection noise limit following Ref. [8]. The sensitivity is δB = −1 1p zres nτ T γ (5.19) , where zres is the spatial resolution, n the line density, T the total measurement time. The coherent evolution time τ is either the duration of the measurement τm , limited by atomic motion, or the spin coherence time τc , depending on the context. In our conditions, the measurement time is limited by atomic motion through τm = zres /v, where v is the mean atomic velocity along z. For √ zres = 50 µm v = 0.08 m/s, and 8 n = 10 atoms/m we find a sensitivity ∼ 15 pT/ Hz. In optical lattices the thermal motion along z is suppressed. This improves the sensitivity since we are limited only by the spin coherence time, which can reach 60 s [144] or about 105 times the τm considered above [145] giving√an improvement by ∼ 300. This would lead to a sensitivity of the order of 50 fT/ Hz. 6. Spin State Tomography Contents 6.1. Stern-Gerlach in free fall . . . . . . . . . . . . . . . . . . . 122 6.1.1. Falling atom clouds . . . . . . . . . . . . . . . . . . . . . . 123 6.1.2. Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2. Measuring ground-state populations of trapped atomic sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2.1. Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2.2. State selective transfer . . . . . . . . . . . . . . . . . . . . . 125 6.2.3. Re-distribution and dispersive probing . . . . . . . . . . . . 127 6.2.4. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 128 122 6. Spin State Tomography There exist an asymmetry in quantum physics. Given a quantum state, we can, at least in principle, easily find the expectation value of a given operator. However, going the other way around to determine the quantum state from a set of measurement outcomes is non-trivial. Wolfgang Pauli has recognized this already in the early years of quantum mechanics [146] and since, we refer to it as the “Pauli problem”. In the advent of the quantum computation and information era the problem has again gained considerable importance. Even so, there exist no general solution to it, many fields in quantum physics have developed their own techniques. The “drosophila” in this respect is probably the homodyne detection of quantum states of continuous light [147]. It allows one to measure the phase space distribution and therefore a complete reconstruction of the state. For other quantum systems, as for example polarization-entangled photon pairs [148], center-of-mass motion in ion traps [149], ions [150], molecular vibrations [151], and atomic beams [152] techniques have been demonstrated. In the field of atomic ensemble experiments there has been a lot of pioneering work in the last decade, especially in the group of P. Jessen with theoretical input from I. Deutsch [34]. Complete characterization of angular momentum states of arbitrary magnitude, ranging from destructive Stern-Gerlach type of measurements [153] to non-destructive dispersive measurements [122, 123]. In this chapter we discuss the possibility to apply a Stern-Gerlach technique, demonstrated by Klose et al. [153], with our setup in Sect. 6.1. In Sect. 6.2, we introduce an alternative approach for spin state tomography which translates spatial SternGerlach into the time domain. We report first results and discuss improvements to elevate the technique onto a practical level. 6.1. Stern-Gerlach in free fall We study the possibility to perform a Stern-Gerlach experiment (SGE) of a free falling atomic cloud in a gradient field. In this, atoms in different magnetic substates are separated during the free-fall by application of a magnetic gradient. We follow a demonstration of this technique for thermal atomic clouds by Klose et al. [153]. They have measured the ground state density matrix for angular momentum F = 4 in a cold (T = 3.5 µK) cloud of Cesium. This technique is rather temping because it gives a direct measurement of the diagonal elements as 2F + 1 atomic clouds, so to say ’ad oculos’ 1 . We concentrate on the question of the SGE alone and make a feasibility calculation for our experimental conditions. The SGE in Ref. [153] is performed in a time-of-flight (TOF) measurement. On their free fall atoms pass through a light-sheet which is placed below the atomic cloud and produce a time dependent absorption or fluorescence signal. The temporal shape will have, in the ideal case, 2F + 1 maxima, which represent the populations of the different diagonal elements of the density matrix. 1 ad oculos = for the eyes; This expression was used by Arnold Sommerfeld in 1922 in the revised version of his classic book “Atomic Structure and Spectral Lines ” in reaction to Stern and Gerlach’s experiment [154]. 6.1. Stern-Gerlach in free fall 123 In previous chapters we characterized all relevant experimental parameters of our system which are necessary for the SGE experiment. 6.1.1. Falling atom clouds Performing a SGE on a falling atomic cloud in a gradient field relies on the different acceleration of atoms with different spin quantum numbers mF due to the magnetic interaction. The force an a free atom in a gradient field is given by F = mg + ∂B ~γmF , ∂x where g is the gravitational acceleration (which we measured in Chap. 3) and m is the mass of an atom. If the gradient is homogeneous over the region of the atoms, we can describe the movement of the whole cloud by their distribution. In Sect. 3.3.4 we measured the radial distribution in TOF and approximated it by a Gaussian of the form (cf. Eq. 3.19) 2x2 n(x) = n0 exp − 2 , (6.1) wa where n0 is the peak density and wa the atomic waist. After releasing the atomic cloud from the trap it will expand. We measured the trap waist as a function of time using (cf. Eq. 3.25) r 4kB T 2 wa (t) = wa2 (0) + t , (6.2) m from which we deduced the temperature T = 25 µK and the initial waist wa (0) = 17 µm. The center of mass of the atomic cloud would describe a parabola with a trajectory of 1 ∂Bx ~γmF x(t) = x0 − g+ t2 (6.3) 2 ∂x m We can write a normalized version of Eq. (6.1) where we substitute x → x(t) and wa → wa (t) 2 (t) exp − 2x w2 (t) ñ(x, t) = p π a . (6.4) 2 wa (t) 6.1.2. Simulations The relevant experimental quantities are the magnetic field gradient we can apply and the distance atoms can fall freely, i.e., the space below the dipole trap. From the measurements in Sect. 5.1, when we characterized the magnetic field environment via the position of the MOT, we deduce a gradient of ∂B ∂x = 4 G/cm for 1 A in the MOT ∂B coils. This results in ∂x max = 24 G/cm if we apply the maximum current of 6 A. For the maximum distance the atomic cloud can fall, we assume ∆xmax = 15 mm. If we use these parameters we can simulate the signal shape, which are shown in Fig. 6.1. We see three distinct peaks which are almost not overlapping. By postprocessing the individual populations could be retrieved with good precision even 124 6. Spin State Tomography in the case of a slight overlap. Hence, we can measure the (2F + 1) populations in a single experimental run. In this way, atom number fluctuations, e.g., from trap loading, will not influence the accuracy of the results. SGE signal [arb. units] mF=ï1 m =0 F m =+1 F 0 0.05 0.1 Time [s] 0.2 Figure 6.1.: Simulations of SternGerlach experiment in gradient of quadrupole field of MOT coils. The used parameters are: ∂B ∂x = 2400 G/m, xmax = 15 mm, T = 25 µK, and wa (0) = 17 µm. Including realistic beams diameters of ∼ 1 mm renders virtually the same result. 6.2. Measuring ground-state populations of trapped atomic sample We have the ability to measure atomic polarization with high precision as we have seen in Chap. 4. This triggered the idea to use dispersive spin measurements also as the detection mechanism in a tomography experiment. The non-destructiveness of the dispersive probing would allow to measure all (2F + 1) populations of the magnetic substate |F, mF i on a single trapped atomic ensemble. We discuss the general idea which is applicable to all alkali-metal atoms. Three possible implementations are briefly presented. One of these is examined in more detail and first measurements are presented. At the end we discuss possible improvements. 6.2.1. Proposal The main idea is to transfer the population of each magnetic substate |F− , mF i individually into the hyperfine manifold F+ = I + 1/2, by applying a homogeneous magnetic field to lift the ground state degeneracy. To this process we will refer in the future as state-selective transfer . Once transferred, the population is then measured non-destructively via paramagnetic Faraday measurements. Measuring in different bases, by manipulating the state with magnetic field prior to the measurement as demonstrated in Ref. [153], would extend this method into a full tomographic technique. The complete experimental sequence, displayed in Fig. 6.2, would consist of 2F + 1 blocks, each having the following three steps: 1. T: State-selective transfer of atoms from |F = 1, mF i into F = 2. 6.2. Measuring ground-state populations of trapped atomic sample 125 2. S: Re-distribution (shuffling) of atoms in F = 2 to guarantee an asymmetric population. 3. P: Probing the transferred population with paramagnetic Faraday rotations from F = 2. Figure 6.2.: General scheme of transfer, shuffle, and probing to perform a Stern-Gerlach-like measurement on alkali metal atoms. Small magnetic fields allows state selectivity. A single block has three steps and is repeated 2F + 1 times. All three steps are discussed in detail in the next three sub-sections. 6.2.2. State selective transfer For the state-selective transfer (SST) we can imagine three different scenarios. Each of them has a different implementation of the transfer mechanism and a different minimal requirements on the magnetic fields applied: • For small magnetic fields starting from a few mG we could use micro-waves (MW) to transfer atoms directly from a specific state F− , mF− into F+ , mF+ . Small Zeeman shifts of a few tens of kHz are sufficient since hyperfine transitions are magnetic dipole transitions with narrow natural linewidth less than kHz. In Fig. 6.3 a) we show the transfer with linearly polarized microwaves from the F = 1 into F = 2 manifold. • For magnetic fields starting from a few hundreds of mG we can use a stimulated Raman processes. For this, two phase-locked light fields separated in frequency by the hyperfine plus the Zeeman splitting of the specific initial state F− , mF− and the specific final state F+ , mF+ state transfer the population, as shown in Fig. 6.3 b). Different Raman schemes exist, which can achieve a complete transfer in the case of stimulated adiabatic Raman passage (STIRAP) [155, 156, 157], or a partial transfer ( ≤ 50 %) in a single-pulse scheme [158, 159]. • For magnetic fields starting from a few tens of Gauss, when the induced Zeeman shift is larger than the excited state linewidth, simple optical pumping can be used to transfer population from a specific |F− , mF i into the F = 2 manifold. This is possible because the Landé factor for the ground state F = 1 and the excited states F 0 have different signs. In this way we can address transitions from |1, mF i → |F 0 , mF 0 i individually. 126 6. Spin State Tomography We pursue the third option of transferring atoms incoherently via optical pumping in fields of a few tens of Gauss. We note, the following derivation can be applied to all other alkali metal atoms. In general, it is favorable to use the D2 transition, because the Zeeman shifts of the excited states are larger as compared to the D1 excited states [19, 160, 161, 162]. We give here numbers for the D2 transition in 87 Rb . The approach takes advantage of the fact that the magnetic moment of ground state hyperfine manifold F− = I − 1/2 has a different sign than the excited state magnetic moments. On the D2 transition in 87 Rb we have gF =1 = − 12 and gF 0 = 23 . This situation is depicted in Fig. 6.4 a). Transitions from a specific ground state |F− , mF i to an excited state |F 0 , mF 0 i become energetically separated. For polarized light in one of the basis states σ ± or π, the transitions we have a spacing of (gF 0 − gF ) µB B = 2π 67 1.4 MHz/G. Already for some tens of Gauss we reach the situation where the energy splitting is larger than the natural line width and off-resonant excitations Rsc (∆) ∝ Γ2 / 4∆2 + Γ2 are suppressed quadratically in the detuning ∆. For the transfer we will use circularly polarized light which ensures that we can transfer atoms out of one |F− , mF i state without interfering with the population in the other ones. In Fig. 6.4 b) we show the decay channels of a number of excited states which get populated during the transfer. There we can see that it is possible to measure the ground state population in each |F− , mF i state when transferring first the state with the largest magnetic quantum number mF in the case of σ + light and vice versa. By calibrating the system with a known reference state, we can account for atoms which are not transferred into F = 2. Figure 6.3.: Possible alternative transfer schemes: a) Transfer of population directly from F = 1 to F = 2 by micro-waves. b) Raman transitions driven by a pump and a Stokes laser field Ωp and Ωs . The detuning for the single and two-photon transition are ∆ and δ, respectively. Other frequency-polarization combinations are also possible. 6.2. Measuring ground-state populations of trapped atomic sample 127 Figure 6.4.: a) Transfer of atoms with σ + light in a magnetic field. The different sign of the magnetic moment for F = 1 manifold is visible as the opposite slope to the rest of the magnetic substates. b) Different decay channel for σ + transfer light. The numbers in the gray boxes indicate the line strengths ×12. 6.2.3. Re-distribution and dispersive probing For the paramagnetic Faraday rotation measurement, we need an asymmetric distribution of atoms among the magnetic substates in F = 2, i.e., h F̂z i = 6 0. For some decay channels, for example from mF 0 = 0, this is not given and we get h F̂z i = 0 (cf. Fig. 6.4 b)). For this reason, we apply another laser field which re-distributes atoms in F = 2 before we measure the rotation signal. Furthermore, the re-distribution equalizes the measurement strength for all mF substates and guarantees a state independent detection. The re-distribution light ought to be tuned to the cycling transition F = 2 → F 0 = 3 to avoid scattering atoms back into F = 1. One possibility is to apply σ ± -light which keeps atoms in a closed transitions and moves them to mF = ±2. This requires sending light along the sample axis, which is the direction of maximum optical depth. To avoid problems of shadowing and light trapping, another possibility is to apply linearly polarized light perpendicular to the sample axis. If we chose the quantization axis along the magnetic field and electric field vector orthogonal to it, the atoms experience a coherent superposition of σ − and σ + light. The transition frequencies for left- and right circular light, which are degenerate for zero magnetic field, will experience different shifts in the presence of a field. In the left graph of Fig. 6.5 we show the shift of the transition frequencies from |2, mF i to their corresponding excited states for σ ± light. Taking advantage of this non-degeneracy, we are able to use linear light to pump atoms into a state where we have h F̂z i = 6 0 for F = 2. The situation is slightly more complicated when the light shift of the dipole trap is taken into account. Ground and excited states experience an additional shift. For reasons we discuss in Appendix B.2 the excited states experience a different light 128 30 6. Spin State Tomography 6i±(mF) [MHz] 40 20 30 10 20 0 10 ï10 0 ï20 mï ï10 6i±(mF) [MHz] mï + m+ m ï30 ï2 ï1 mF +1 +2 ï20 ï2 ï1 mF +1 +2 Figure 6.5.: Effect of Zeeman shifts and AC-Stark shifts on transition frequencies for the transition F = 2 → F 0 = 3. Left: Transition frequency shifts ∆ν for σ ± light as a function of mF in F = 2 for a magnetic field of B = 20 G. Right: Transition frequency shifts including Zeeman and AC-Stark shifts. The light intensity corresponds to the value we find in the focus of the dipole laser at 6 W optical power. shifts depending on mF . In the right of Fig. 6.5 we plot the energy shifts from magnetic field and the light shift due to the trapping laser. For the dispersive paramagnetic Faraday rotation measurement from F = 2 we use light which is blue detuned compared to the transition F = 2 → F 0 = 3 to maximize the vector polarizability2 . It is sent along the sample axis and detected in the same way as in Chap. 4. 6.2.4. Measurements As a first calibration we measure the spectrum of the transfer operation by scanning the transfer laser over the three expected resonances and measure the rotation signal. This is important to locate the exact position of the transfer resonances for the tomography. at the same time it gives us the strength of the magnetic field, which is actually applied, and the amount of light shift the atoms experience in the dipole trap. In the measurement we load atoms in the dipole trap as described in Sect. 3.1.3. Then we ramp up an additional magnetic field of 15 − 20 G along z. This provides the field strength to split the atomic resonances sufficiently to be measured individually. For the transfer, we use light from the F = 1 laser (cf. Chap. 3) by sending it through 2 When probing atoms in F = 2 the spectrum of the vector polarizability is qualitative the same as for F = 1, when interchanging red for blue detuning. That means, we have the same behavior for blue detuned light on F = 2 as for red detuned light on F = 1. See for instance Fig. 2.2. 6.2. Measuring ground-state populations of trapped atomic sample Transfer Signal [norm.] 1000 a 3000 a 1 0.8 0.6 0.4 0.2 0 ï60 ï40 ï20 6i [MHz] 20 40 129 Figure 6.6.: Transfer spectrum: normalized rotation signal vs. transfer-laser detuning. The transfer laser is scanned from blue to red detuning. We change the number of photons in the transfer pulse between 1000 and 3000 and renormalize the rotation signal accordingly. Note, we flipped the sign of the rotation signal. Usually, the signal is negative for the whole spectrum since we observe rotations from as state with mF > 0 and the laser is blue detuned. two AOMs. This gives us the possibility to scan over all expected resonances in a range of −68.5..31.5 MHz around the field-free resonance F = 1 → F 0 = 2. For the off-resonant probing we frequency off-set lock a laser to the F = 2-laser and shift it above the resonance F = 2 → F 0 = 0. The whole spectrum is measured on the same atomic ensemble in a single run. For this we interrogated with blocks of three pulses, which consists of transfer-, probe- and depletion light, at different detunings of the transfer laser. The transfer laser has right circular polarization, i.e., σ − -light, and is scanned in steps of 2 MHz. The magnetic field is pointing along the positive z-axis. The probe laser is locked +570 MHz to the blue of the transition F = 2 → F 0 = 0. We send 130 probe pulses with NL = 6 × 106 per pulse which are 1 µs long probe pulse and have a period of 10 µs. The large number of probe photons and the relatively close detuning make the application of a re-distribution laser unnecessary. The depletion light is resonant from F = 2 → F 0 = 3. A single block of transfer-, probe-, and depletion light will do the following: The transfer laser pumps a fraction of the population from F = 1 into F = 2, depending on its spectral position. The probe will re-distribute the atoms into a steady state distribution among the magnetic substates in F = 2. We take only the last pulses of the train of probe pulses which measure the steady state. After the probing, we send light which heats the atoms in F = 2 out of the dipole trap to reset the rotation to zero. A background trace is taken under the same conditions as for the transfer but without applying transfer light. We recognized that this is necessary since there are atoms transferred from F = 1 due to the probe laser. Even for the large detuning of more than 6 GHz, we get off-resonant scattering and the depletion laser is not removing these atoms completely. In Fig. 6.6 we show the result of the experiment for two different intensities of the transfer laser. For the trace with blue circles we used about 1000 photons per transfer pulse. Increasing the number of photons by a factor of three gives proportionally 130 6. Spin State Tomography more signal as we can see in the green squares, which were rescaled by 1/3. The scanning direction for both traces is from blue to red detuning. We fitted the two curves by a the sum of three Voigt functions. This function is often used in spectroscopy for structures which are convolutions of Gaussian and Lorentzian signals. As analytic expression for the Voigt function we used the PearsonVII function, which approximates the Voigt function well [163]. For comparison, we also show the best fit for a sum of three power broadened Lorentzian as thin solid lines. We interpret the shift of the peak position for higher photon numbers as a depletion effect from mF = −1 and mF = 0. While the transfer laser slowly scans over the resonances too many atoms are transferred into F = 2 , which empties the according magnetic substates. A maximum in transfer is achieved when a substantial depletion has taken place. This is not necessarily the position of the actual transition resonance, as we can best appreciate in the left-most peak in Fig. 6.6. The general broadening of the transitions is most probably related to the distribution of atoms in the dipole trap. Depending on the position, atoms see smaller or larger AC Stark shifts. Quantitatively, we can estimate the effect by computing the atom number distribution as a function of energy. For a 3D harmonic potential, we find nA () = 2 1 , 2~3 ωρ2 ωz exp −µ − 1 kB T where the first term is the density of states for the 3D harmonic potential and the second the Bose distribution [164]. The energy in the harmonic potential is , ωρ and ωz are the radial and axial trap frequency and µ the chemical potential. The temperature is measured in Chap. 3, the trap frequencies can be calculated from the trap parameter and the chemical potential can be found R ∞ by normalizing the density to the total number of atoms in the sample NA = 0 dnA (). In Fig. 6.7 we see that for our atomic sample we have a rather broad distribution of energies in the range of ∼ 8 MHz. This is consistent with the observed linewidth of the transfer spectrum. In summary, we see that it is possible to measure individual ground state populations by incoherent atom transfer via optical pumping. The observed spectra show the expected three peak structure as we would assume from an unpolarized atomic ground state. Nevertheless, the peak position slightly depends on the amount of transfer. On the other hand, if we could use other transfer techniques like microwaves or Raman processes, the position in the dipole trap would not lead to such large broadening of the transition. 6.2. Measuring ground-state populations of trapped atomic sample 131 2.5 n(¡) ___ NA 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 U0 Figure 6.7.: The atom number distribution as a function of energy in the trap is calculated for NA = 1 × 106 , T = 25 µK, and U0 /h = 4.9 MHz. To get the whole frequency brodening one has to multiple the U0 with a factor of three to get the shift of the transition frequency F = 1 → F 0 = 2 7. Spin Squeezing Contents 7.1. Calibration of spin detector . . . . . . . . . . . . . . . . . 134 7.1.1. Learning from photons . . . . . . . . . . . . . . . . . . . . . 134 7.1.2. Noise scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.1.3. Experimental details . . . . . . . . . . . . . . . . . . . . . . 136 7.1.4. Trap loss in thermal state preparation . . . . . . . . . . . . 139 7.1.5. Projection noise limited spin detection . . . . . . . . . . . . 140 7.2. QND measurement of large-spin ensembles by dynamical decoupling . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.1. Convenient fiction . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.2. Dynamical decoupling in QND measurements . . . . . . . . 146 7.2.3. Two-polarization meta pulses . . . . . . . . . . . . . . . . . 148 7.2.4. Experimental verification of dynamical decoupling idea . . . 149 7.3. Spin squeezing of magnetically sensitive atomic state . 152 7.3.1. Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.3.2. Spin squeezing pulse sequence . . . . . . . . . . . . . . . . . 153 134 7. Spin Squeezing In this chapter we present measurements of spin squeezing in a magnetically sensitive atomic system. In the first part we calibrate our spin detection using a thermal spin states. Analogous to the calibration of a photo detector, we use thermal spin states as robust noise references to determine the amount quantum and technical noise sources in the measurement. In the second part we present a novel technique which allows us to perform quantum non-demolition measurements in large-spin atoms. For atoms with spins larger than 1/2 the dipole interaction is more complicated than the simplified paramagnetic Faraday rotation ∝ Ŝz Jˆz and involves coherent Raman processes (also referred to as tensor light shifts, alignment effects [165], and non-linear effects [166]). The desired QND character of the light atom interaction interaction is hindered. We propose a technique to overcome this fundamental limitation and present results verifying our expectation. The latter parts lead directly to the first results on the generation of spin squeezing in a magnetically sensitive state. 7.1. Calibration of spin detector An important step towards creation of measurement induced squeezing is the thorough characterization of the spin detection method. Measuring the sensitivity of the spin detector as a function of the introduced damage, as we did in Chap. 4, is one part. But the ultimate characterization of any detector, i.e., for atoms or photons, is the measurement of all contributing noise sources. In the past, there has been a big controversy about the demonstration of spin squeezing in a system like ours, because a proper calibration of the measurement at the quantum level was not provided and two publications [167, 168] from high profile journals had to be retracted [169]. We think, therefore, that it is of utmost importance to ensure a proper calibration of the spin measurement, before using the spin detector. Here, we demonstrate sub-projection-noise sensitivity of our spin detection system. The calibration procedure employs as known reference state, the maximum-entropy or ‘‘thermal’’ spin state, and quantitative imaging-based atom counting to identify electronic, quantum, and technical noise in both the probe and spin system. The measurement achieves a sensitivity 2.8 dB better than thermal state quantum noise. Most of the content in this section is published as "Sub-projection-noise sensitivity in broadband atomic magnetometry" in Phys. Rev. Lett. 104, 093602 (2010) [13] 7.1.1. Learning from photons In quantum optics, the use of noise reference states for the calibration of the detection system is a well established method [111]. There, thermal, vacuum, or coherent states of light are used as robust references. They can be produced in a reliable and reproducible way and their noise properties are not influenced by the measurement itself. The characteristic scaling of the quantum noise as a function of the input size 7.1. Calibration of spin detector 135 of the state, i.e., the photon number for a coherent or thermal state of light, can be used as benchmarks. The quantum and classical (including technical) noise sources can be distinguished by their respective scaling with the size of the system. In Sect. 3.6 we applied this strategy already for the calibration of the balanced photo-detector used in the polarimeter. There, we could distinguish between the electronic noise and the light shot noise due to their distinct scaling for photon number of the coherent input state. Thermal spin states To characterize our spin detector, we use thermal spin states (TSS). For NA atoms with spin quantum number F , their density matrix is given by A ρTSS = ρ⊗N cms , (7.1) where ρcms = (2F + 1)−1 I2F +1 is the completely mixed state of dimension 2F + 1 and Im the identity of dimension m . In terms of collective (real) spin F̂ the TSS has zero average value, and a noise of var(F̂n ) = V1 NA , (7.2) where F̂n is any spin component and V1 = F (F + 1)/3 is the variance of a single atom. Thermal spin states provide a robust tool to characterize optical QND measurements and have several advantages compared to coherent spin states. First, photon scattering during the dispersive probing leaves their quantum noise properties largely unchanged due to the complete mixedness. This is very important and distinguishes thermal from coherent spin states. The latter are very vulnerable to any photon scattering, as we have seen in Sect. 4.1.2. Second, TSS are less prone to systematic errors, e.g., due to imperfect spin polarization. Third, the TSS are less susceptible to magnetic field perturbations because of their rotational symmetry (cf. Eqs. (7.1) and (7.2)). 7.1.2. Noise scaling The expected noise behavior of the measurement signal can be calculated from the effective interaction Hamiltonian (A.17), written in terms of the collective real spin operator F̂ e1 ~G Ĥ = Ŝz F̂z , (7.3) τ where we neglect the terms proportional to the tensor part of the polarizability tensor. In the case of a completely un-polarized atomic state, their effect is negligible. For e 1 = G1 /2, where G1 is the coupling convenience, we define the coupling constant as G constant defined in Eq. (2.30). 136 7. Spin Squeezing (in) (in) (in) For linearly polarized probe pulses h Ŝx i = NL /2 and h Ŝy i = h Ŝz i = 0 without (in) (in) initial correlation between Ŝy and F̂z , we find the variance of the light signal after interaction (out) var(Ŝy 2 (in) e 1 NL var(F̂z(in) ) . ) = var(Ŝy ) + G 4 (7.4) The first term, the input optical polarization, in general has a variance (in) var(Ŝy ) = NL + αNL2 , 4 (7.5) where the first part is light shot noise and the second technical noise due to variations in the optical state preparation of strength α. Similarly, the second term of (7.4) contributes a variance related to atomic quantum and technical noise, respectively, i.e., var(F̂z(in) ) = NA V1 + βNA2 V1 , where β is the amount of technical noise. Finally, we must add a constant “electronic noise” VE from the detector, and arrive to the measurable signal (meas) var(Ŝy ) = VE + N2 N2 NL + αNL2 + G2 V1 L NA + βG2 V1 L NA2 . 4 4 4 (7.6) Equation (7.6) contains the essential elements of the calibration technique. All terms have distinct scaling with photon and atom number, and can thus be separately iden(meas) tified if var(Ŝy ) is measured as a function of NL and NA . The terms proportional 2 to NL and NL NA correspond to quantum noise of light and atoms, respectively. Together they provide an absolute calibration of the gain of the detection system and e 1 . The remaining terms represent various noise sources. the atom-light coupling G Only if these are simultaneously small relative to the atomic quantum noise, quantum signals will be detectable. The spin detector is said to be “projection-noise limited” if the atomic projection noise is the dominant contribution in the observed signal, i.e., bigger than the sum of the remaining parts. 7.1.3. Experimental details Measuring the quantum noise requires that we prepare and measure the system many times and compute the variance of the measurement results var(Ŝy ). Each spin measurement has to be performed on a freshly prepared atomic state, which has a random F̂z . To observe the scaling with photon and atom number we have to change NL and NA and repeat the spin measurement for each pair (NL , NA ). A single experiment can be divided into three blocks which are illustrated in Fig. 7.1. 1. Preparation of thermal spin state 2. QND probing with NL linearly polarized photons 3. measurement of number of atoms NA in the thermal spin state 7.1. Calibration of spin detector 137 Figure 7.1.: Three major blocks in the spin projection noise measurement pulse sequence. Details are given in the corresponding sections. For the atom number measurement we use the dispersive atom number measurement, explained in Sect. 4.3. In this way, we can take advantage of the long atomic sample lifetime by repeating the experiment on the same trap loading several times. The dispersive atom number measurement is non-destructive and leaves the number of atoms in the trap unchanged. The intrinsic loss rate of the optical trap, measured in Sect. 3.3.3, is too low to sample in a practical way many different atom numbers. Therefore, we remove atoms by heating them out with resonant light or they are lost ’naturally’ during the preparation stage. Here, in the calibration measurement, we have a non-negligible loss rate due to the way we prepare the thermal spin state. Later, in the spin squeezing experiments, we remove atoms by shining resonant light to heat them out. We repeat experiments on the same atomic sample up to 20 times. With a reasonable loss rate of a few tens of percent we sample about an order of magnitude in atom numbers. Meta pulse concept - simultaneous probing of different photon numbers In order to measure the noise behavior as a function of photon number one could change the power or the duration of the probe pulse. A more convenient way is to sum the signal from multiple pulses in a metapulse, containing a larger total number of photons. As we are in the linear regime, a metapulse will have the same information as a single higher-energy pulse. If the outcome of a single probe pulse with nL photons is (out) ŝy,j (in) = ŝy,j + G1 nL (in) F̂ , 2 z (7.7) we can define the result of combining p pulses with NL = pnL photons as M= p X (out) ŝy,j (7.8) j=1 (in) If the individual Stokes components ŝy,j were not correlated before interaction with (in) (in) the atomic system, i.e., cov(ŝy,j , F̂z ) = 0, and there is no correlation between light 138 7. Spin Squeezing (in) (in) pulses, i.e., cov(ŝy,j , ŝy,k ) = 0, the variance of the metapulse is var(M) = p X (in) var(ŝy,j ) + p2 G21 j=1 =p n2L var(F̂z(in) ) . 4 n2 nL + p2 G21 L var(F̂z(in) ) 4 4 (7.9) (7.10) For a single pulse with NL = pnL photons we find (out) var(Ŝy (in) ) = var(Ŝy ) + G21 = NL2 var(F̂z(in) ) 4 NL (pnL )2 + G21 var(F̂z(in) ) 4 4 We see that the information in a single and a metapulse is identical. In the experiment we send a train of 1 µs long pulses with 10 µs period to the atoms. Each pulse contains 25 × 106 photons, vertically polarized and tuned 800 MHz to the red of the F = 1 → F 0 = 0. Forced thermalization It could be expected that atoms which are loaded into the optical trap are unpolarized. To our surprise, this is not the case for the loading cycle we apply (cf. Sect. 3.1.3). A small net polarization was present after loading the trap and a thermalization period of 400 ms. Most probably, this is due to an unbalanced light field in the last sub-Doppler cooling phase. The optical beams for the MOT are all retroreflected in order to recycle the light. That is, the light passes twice through the atomic cloud with opposite circular polarization (but same helicity). The imbalance can be either caused by static losses on optical elements or due to atoms. The latter is occurs because we retro-reflect the MOT beams once thy have passed the atomic cloud. Hence, the second pass will, in general, have less intensity than the first one. We actively produce thermal spin states in F = 1 by repeatedly optically pumping atoms from F = 1 to F = 2 and back, using lasers tuned to the F = 1 → F 0 = 2 and F = 2 → F 0 = 2 transitions, and applied via the MOT beams (cf. Fig. 7.2). Each pumping cycle takes 300 µs. To avoid any residual polarization, we apply bias fields of a few hundreds of mG along z, y, and x during the three back-and-forth cycles. Finally, the F = 2 manifold is further depleted with a 100 µs pulse of resonant light on the F = 2 → F 0 = 2 transition with zero magnetic field. After these steps, no remaining mean polarization along z is observed. This procedure is designed to transfer disorder from the thermalized center-of-mass degrees of freedom to the spin state. Illumination from six directions produces a polarization field with sub-wavelength structure, in which the atoms are randomly distributed. Possible net imbalances in the pump polarizations are scrambled by the application of different bias fields. The success of the thermalization procedure is directly measurable through the noise scaling and will be discussed further below. 7.1. Calibration of spin detector 139 Figure 7.2.: (a) Atomic ensemble with probing, pumping, and imaging light fields. The polarimeter measures in the 45 basis, i.e., the Stokes component Ŝy . (b) Atomic transitions for probing, preparation, and imaging light fields. Full pulse sequence For the calibration measurement we prepare atomic ensembles of 106 atoms as explained in Sect. 3.1.3. The full experimental sequence is illustrated in Fig. 7.3. After loading the trap we perform 20 experiments which we call blocks. In each block we prepare a thermal spin state, perform the QND probing of the spin state, and measure the atom number NA via DANM, as described above and in Fig. 7.3. The atom number from one block to the next is reduced by ∼ 15 %. After removing the atoms from the trap at the end, we measure the bias of the polarimeter by sending a train of probe pulses. We load the dipole trap for more than 500 times to acquire enough statistics. Figure 7.3.: Full pulse sequence for projection noise measurement of thermal spin state. a) Atom trap loading, b) 20 repetition of the spin state measurements, c) atoms are released from dipole trap, and d) bias of polarimeter is measured. This whole cycle is repeated more than 500 times. 7.1.4. Trap loss in thermal state preparation The loss of atoms from the dipole trap during the state preparation can be estimated from the evolution of the atom number as a function of number of blocks. In Fig. 7.4 we plot the measured number of atoms. For this we convert the rotation signal measured in the DANM at the end of each block with the independently measured 140 7. Spin Squeezing N A 6 10 5 10 1 6 11 16 [Block #] Figure 7.4.: Atom number for different blocks measured in the DANM after the spin measurement. The single exponential decay rate is k = 0.169(2)/block. That is, we lose 15.5 % in each state-preparation/probing block. Error bars (within the circles) indicate the 95 % confidence interval of the fitted atom number. e 1,DANM = 6.6(2) × 10−8 . This was determined from measuring coupling constant G the rotation signal of a fully polarized atomic sample and for a known atom number. For more details see Sect. 4.3. The data points follow a single exponential decay NA (p) = NA,0 exp (−k p), where k = 0.169(2) is the decay constant. The loss per block can be calculated from 1 − e−k and is 15.5(2) %. The loss of atoms from one block to the other can be attributed almost completely to the preparation of the thermal state. The other loss mechanism where measured by repeating the cycle without thermalization and found to be negligible. 7.1.5. Projection noise limited spin detection Experimental data for atom numbers between 4×104 and 8×105 and photon numbers up to 109 are shown in Fig. 7.5. The data are fitted with the theoretical expression e 1 = 6.65(3) × (7.6) which is shown as a surface. The deduced coupling constant is G −8 5 10 and the electronic noise level is VE = 4.9×10 . The coefficients for the technical noise are α = 4.3(1)×10−11 and β = 3.1(7)×10−7 . Atomic quantum noise dominates over other technical and quantum noise sources for a large range of NL and NA , as seen in the vertical panels of Fig. 7.5. For the maximum number of photons NL = 1 × 109 , the noise scaling with atom number is highlighted in Fig. 7.6. For the largest atom number measured, i.e., NA = 7.6 × 105 , the light shot noise, atomic technical noise, light technical noise and electronic noise are respectively 3.5, 6.3, 11.2, and 30 dB below the quantum noise level of the thermal reference state. From Fig. 7.6 we can read a spin readout noise which is factor of ∼ 1.9 (2.8 dB) better than the spin noise for the thermal spin state at NA = 7.6 × 105 with spin F = 1. Comparing to the coherent spin state, which defines the projection noise level, we measure a reduction of 1.6 dB. In absolute terms, this corresponds to a sensitivity of δNA = 515 spins. The light technical noise may be due to small imbalance of the polarization analyzer 7.1. Calibration of spin detector 141 Figure 7.5.: Measured variance of Ŝy plotted as black dots and a fit to the data using Eq.(7.6) as colored surface. The distance of the data points from the fitted surface is indicated as vertical gray lines. The left plot shows the atom-number-scaling for NL = 109 . See Fig. 7.6 for more details. The right plot shows the photon-numberscaling for NA = 7.6 × 105 . In the left and right plot curves indicate (from top to bottom): a) total noise, b) quantum spin noise plus light noise, c) light shot and technical noise, and d) light shot noise. 142 7. Spin Squeezing and thermal birefringence produced by the dipole laser. Active stabilization of the balancing could improve and reduce the light technical noise considerably. Atomic technical noise may come from classical fluctuations in the lasers during optical pumping. Extrapolating the technical noise of atoms and light, both remain below their respective quantum noise terms up to NA,qn ≡ β −1 = 3.2 × 106 and NL,qn ≡ (4α)−1 = 5.8 × 109 , respectively. It would thus be possible to increase the number of atoms in the trap while remaining projection-noise limited. Figure 7.6.: Measured variance of Ŝy for NL = 109 as a function of atom-number. The errobars indicate the statistical uncertainty in Ŝy and the atom-number fluctuations. Dashed curve: Fit using expression 7.6. Solid Line: Pure spin quantum noise. Dotted Line: Shot noise and technical light noise, Thin solid line: Light shot noise. The electronic noise is not plotted because it is negligible for this number of photons. 7.2. QND measurement of large-spin ensembles by dynamical decoupling 143 7.2. QND measurement of large-spin ensembles by dynamical decoupling Now we present a theoretical proposal and its experimental verification for a technique which allows QND measurement with polarized light of large-spin systems. In Sect. 4.1 and the previous section, we have calibrated the spin detector with a classical measurement on a polarized spin state, and a quantum measurement on a completely unpolarized state. For both calibration techniques we were able to neglect the influence of the tensorial part of the interaction Hamiltonian ∝ G2 (Ŝx Jˆx + Ŝy Jˆy ). This is different for measurements of the quantum properties of polarized spin states, e.g., h Jˆx i = NA /2 and h Jˆy i = h Jˆz i = 0. First, we discuss the a problem that arises in optical QND measurements on a multilevel system. It is a direct result of the multilevel structure of the atomic system used and caused by an accumulative built up of Raman coherences. We present the problem in detail, propose a solution and demonstrate its viability. This section is partly published as “Quantum Nondemolition measurement of largespin ensembles by dynamical decoupling” in Phys. Rev. Lett. 105, 093602 (2010). [14] 7.2.1. Convenient fiction Spin QND measurements in the realm of the Takahashi-Kuzmich proposal [58, 59] on atomic systems has proven to be a powerful tool in the last decade. In the original proposals by Takahashi and Kuzmich the authors are concerned with real or effective spin 1/2 systems. The influence of the multilevel structure present in, for example, alkali metal atoms is neglected using an argument of large detuning [58, 59]. The argument goes as follows: The different scalings for vector and tensor polarizability, i.e., G1 ∝ ∆−1 and G2 ∝ ∆−2 , seems to suggest that for detunings large compared to the excited hyperfine splitting, the full polarizability Hamiltonian Ĥpol = ~ ~ G1 Ŝz Jˆz + G2 Ŝx Jˆx + Ŝy Jˆy τ τ (7.11) can be reduced to the ideal QND Hamiltonian ĤQND = ~ G1 Ŝz Jˆz . τ (7.12) We argue that in experiments that intrinsically rely on large spin ensembles, i.e., F > 1/2, as the one presented here, or novel proposals for quantum polarization spectroscopy in spinor gases and the detection of exotic quantum phases [125, 170, 171], the large-detuning argument fails. In Fig. 7.7 we illustrate the effect that each of the Hamiltonian terms has on a polarized spin state h Jˆx i = Jx = NA /2 for h Ŝx i = NL /2. 144 7. Spin Squeezing Figure 7.7.: Polarized probe light h Ŝx i = NL /2 interacts with a polarized spin state along Jˆx . Left: The QND interaction introduces noise along Jˆy through the quantum fluctuations in the probe light var(Ŝz ) = NL /4. Center: The tensorial term ∝ Ŝx Jˆx rotates the introduced noise into Jˆz . Right: the tensorial term ∝ Ŝy Jˆy introduces light quantum noise var(Ŝy ) = NL /4 into Jˆz . The last effect is much smaller than the first two, because for our choice of light and atom polarization h Ŝy i = h Jˆy i = 0. It becomes most obvious that the detuning argument breaks down if we study the input-output relations for the QND variable Jˆz and its conjugate Jˆy after interaction. To lowest order in Ĥpol we have Jˆz0 = Jˆz + G2 Ŝx Jˆy − G2 Ŝy Jˆx . Jˆy0 = Jˆy − G1 Ŝz Jˆx − G2 Ŝx Jˆz , (out) Ŝy = (in) Ŝy + (in) G1 Ŝx Jˆz − (in) G2 Ŝz Jˆx (7.13) (7.14) (7.15) and the variance of the output variable is var(Jˆz0 ) = var(Jˆz ) + G22 Sx2 var(Jˆy ) + G22 var(Ŝy )Jx2 , var(Jˆy0 ) = var(Jˆy ) + G21 Jx2 var(Ŝz ) + G22 Sx2 var(Jˆz ) (7.16) (7.17) where we assumed Jy = Jz = 0 and Sy = Sz = 0. The quantum fluctuations of both variables Jˆy and Jˆz are coupled. We get a first order estimation of the tensorial effects if we compare the pure QND measurement dynamics of var(Jˆz0 )QND and the additional noise in Jˆz that is introduced. In the ideal case of G2 = 0 and no decoherence effects the uncertainty in Jˆz is var(Jˆz0 )QND = var(Jˆz ) , 1+ς (7.18) where ς = G21 Sx2 NA /4 is the signal-to-noise ratio of the QND measurement defined in Sect. 2.5.4. The conjugate variable receives the equivalent amount of quantum noise and reads var(Jˆy0 )QND = (1 + ς) var(Jˆy ) . (7.19) 7.2. QND measurement of large-spin ensembles by dynamical decoupling 1.2 1 Standard Quantum Limit var(Jz) 0.8 0.6 0.4 0.2 0 ï2 covariance matrix first order estimate ï1.5 ï1 Detuning 6 [GHz] ï0.5 145 Figure 7.8.: Simulation results for optimal squeezing in the presence of tensorial effects. The solid line is the minimal variance from a full covariance matrix calculation and the dotted the simple first order estimation Eq.˜(7.23). The first order model overestimates the noise reduction, because we introduce NL,opt into var(Jˆz )QND and not in the exact solution. The number of atoms in the simulations is 106 and the interaction area A = 4 × 10−9 m2 We estimate the additional noise in Jˆz due to tensorial effects using Eq. (7.16) and Eq. (7.19) var(Jˆz0 )add ≡ var(Jˆz0 ) − var(Jˆz0 )QND = G2 S 2 (1 + ς) var(Jˆy ) , 2 x (7.20) (7.21) where the much smaller contribution G22 var(Ŝy )Jx2 is neglected. We can estimate the maximum squeezing in the presence of tensorial effects, if we assume there exist an optimal interaction time, or equivalently number of probe photons NL,opt , when var(Jˆz0 )add = var(Jˆz0 )QND . From this we find the optimum number of photons p G22 + 2G21 G2 NA − G2 2 (7.22) NL,opt = G21 G2 NA The minimum uncertainty in Jˆz in this simple first order estimation is var(Jˆz )min = 2var(Jˆz ) p . 1 + 1 + 2G21 NA /G2 (7.23) Now, it becomes evident why the detuning argument is not valid. The factor G21 /G2 in the denominator is in good approximation independent of the detuning, i.e., a constant. To verify this simple model, we use the covariance matrix calculation technique of Appendix A and apply the full polarizability Hamiltonian Ĥpol . The time steps are chosen sufficiently short to simulate the time evolution realistically. In Fig. 7.8 we plot the simple and the covariance calculation for a set of typical system parameters in a range of interesting detunings. 146 7. Spin Squeezing There is another effect which is related to the second tensorial term ∝ Ŝy Jˆy that is responsible for the last term in Eq. (7.16). It introduces quantum noise from Ŝy into Jˆz which is amplified by the coherent spin state polarization Jx . Fortunately, the effect is small compared to the initial variance of the atomic state. For realistic experimental parameters we can compute the additional noise introduced to the atomic state, which is G22 Jx2 var(Ŝy )/var(Jˆz ) ∼ 10−2 . For this term the detuning argument works, and we could suppress its contribution even further by using larger detuning. Another important aspect is technical noise. Imperfectly prepared atomic or light states for instance with Jy 6= 0 or Sy 6= 0, will lead to technical noise in Jˆz . The additional coupling through the tensor terms will cause the technical noise be redistributed among all atomic spin components, e.g., Eq. (7.13). We note that there are experiments in multilevel alkali metal atoms, which are affected little by the tensorial effects. For effective spin 1/2 systems on clock transition the discussed effects are not present [172, 173, 119]. Others experiments circumvent the problem “naturally” in using composite systems of two oppositely polarized ensembles and study joint properties of the system [4]. The tensor terms can also be used for creating non-linear single atom interactions as demonstrated in Ref. [174, 22]. 7.2.2. Dynamical decoupling in QND measurements We approach this problem using the methods of dynamical decoupling [175, 176, 177], which allow us to effectively cancel the non-QND terms in the Hamiltonian while retaining the QND term. To our knowledge, this is the first application of this method to quantum non-demolition measurements. Dynamical decoupling has been extensively applied in magnetic resonances [178, 179], used to suppress collisional decoherence in a thermal vapor [180], to extend coherence times in solids [181], in Rydberg atoms [182], and with photon polarization [183]. The main idea behind dynamical decoupling (DD) is that unwanted properties of open quantum systems (OQS), e.g., decoherence through interaction with a bath, can be controlled with time-varying control fields which act on the system on time scales shorter than the memory time of the environment [175]. The method allows one to engineer many effective OQS evolutions which are, in principle, immune to noise and decoherence. In our case, where we are interested in suppressing certain parts of the light-atom Hamiltonian, this strategy seems very promising. The open quantum system is the system variable Jˆz which is coupled to a degree of freedom, Jˆy , which is neither system nor meter in the QND measurement and can be considered the bath. This coupling introduces noise into the system variable, and decoherence into the state of the ensemble. To remove the decoherence associated with this coupling G2 Ŝx Jˆy , we adopt the strategy of “bang-bang” dynamical decoupling [175, 176, 177]. In this method, a unitary Ûb and its inverse Ûb† are alternately and periodically applied to the system p times during the evolution, so that the total evolution is Ûtot = [Ûb† ÛH (t/2p)Ûb ÛH (t/2p)]p , (7.24) 7.2. QND measurement of large-spin ensembles by dynamical decoupling 147 where ÛH (t) describes unitary evolution under Ĥpol for a time t. With this evolution, those system variables that are unchanged by Ûb continue to evolve under Ĥpol , while others are rapidly switched from one value to another, preventing coherent evolution. For large p, the system evolves under a modified Hamiltonian Ĥ 0 = P̂ Ĥ , (7.25) where P̂ projects onto the commutant of (i.e., the set of operators which commute with) {Ûb , Ûb† } [177]. To eliminate G2 (Ŝx Jˆx +Ŝy Jˆy ), while keeping G1 Ŝz Jˆz we choose a Ûb which commutes with Jˆz , but not with Jˆx or Jˆy , namely a π rotation about Jˆz , Ûb = exp[iπ Jˆz ]. This leaves Jˆz unchanged, but inverts Jˆx and Jˆy . By the symmetry of Ĥpol , this is equivalent to inverting Ŝx and Ŝy , which suggests a practical implementation: probe with pulses of alternating Ŝx , and define a ‘meter’ variable taking into account the inversion of Ŝy . Two-polarization probing (1) We consider sequential interaction of the ensemble with a pair of pulses, with Ŝx = (2) (diff) (1) (2) −Ŝx = NL /4p. We define also the new ‘meter’ variable Sy ≡ Ŝy − Ŝy . We describe the atomic variables before, between, and after the two pulses with superscripts (in), (mid), (out), respectively. We apply Equations (7.13) and (7.14) to find: (mid) (in) (1) (in) Jˆz = Jˆz + G2 Ŝx Jˆy (1) (in) (mid) (in) (1,in) ˆ Jx − G2 Ŝx Jˆz Jˆy = Jˆy − G1 Ŝz (1,out) (1,in) (1) (in) Ŝy = Ŝy + G1 Ŝx Jˆz (7.26) (7.27) (7.28) and (out) (in) Jˆz = Jˆz (diff,out) Ŝy = (diff,in) Ŝy (7.29) + (1) (in) 2G1 Ŝx Jˆz (7.30) plus terms in G1 G2 Ŝx Ŝz Jˆx , G22 Ŝx2 Jˆz and G1 G2 Ŝx2 Jˆy which become negligible in the limit of large p. The ideal QND form is recovered by the dynamical decoupling. We verify the method in a more realistic modeling using the covariance matrix. There also the influence of incoherent effects, i.e., photon scattering are taken into account. A homogeneous atomic ensemble is probed by a single pulse or by a train of p alternately polarized pulses. In Fig. 7.9 we show the results for the covariance matrix simulation (cf. Appendix A) applying the technique to realistic experimental parameters, which are given in the caption. We plot the amount of spin squeezing −ξe2 (cf. Eq. (2.44)) as a function of normalized interaction time t0 , which is proportional to the probe photon number. The black solid line is the simulation with 148 5 7. Spin Squeezing j2 [dB] K 4 3 2 1 0 0 2 4 6 8 Time [t ] 0 Figure 7.9.: Amount of spin squeezing as a function of probing time in units of normalized interaction time t0 (cf. Chap. A). The black line shows the case of G2 = 0, i.e., no tensor effects. The red dotted line and the blue circles show is the case of the full interaction Hamiltonian for a single and alternating polarization, respectively. For the simulations we used ∆ = −600 MHz, G1 = 1.27×−7 , G2 = −6.86 × 10−9 and NA = 1 × 106 . Note, simulations included decoherence effects from photon scattering, which is the reason for the finite squeezing of ∼ 4.5 dB. The oscillatory behavior for G2 6= 0 shows the rotation of the uncertainty ellipse about Jˆx . Hamiltonian (7.11) with G2 = 0. The red dotted line and the blue circles are the results for the full interaction Hamiltonian Ĥpol probed with a single polarization and alternating pulses, respectively. In the simulation the normalized interaction time t0 was subdivided into 200 time steps. In theory, the method can restore the spin squeezing and effectively compensate the effect of the tensorial terms in the interaction Hamiltonian. The oscillations of the curve for the full interaction without DD show nicely the character of the Ŝx Jˆx term in the interaction Hamiltonian, which leads to a precession-like deformation of the uncertainty ellipse about Jˆx . It is important to note that the two polarization technique naturally removes also inhomogeneous effects of the unwanted effects due to the tensorial terms. All effects we are concerned about, are proportional to either the number of photons or atoms in the sample. In a dipole trapped atomic sample interrogated with a Gaussian laser beam the tensorial effects are different over the size of the sample. This would, if not compensated by reversing the dynamics, lead to an dephasing of the spin state. In this aspect, the TPP method is similar to a spin echo technique. 7.2.3. Two-polarization meta pulses We adapt the concept of meta-pulses to the case of alternating probe pulses. This is important to account for experimental limitations. In theory, the coupling constant G1 is the same for vertical and horizontal probe light, if the spatial properties of both probe beams and the coupling to the atoms are identical. Experimentally, there can be slight differences due to alignment issues. Furthermore, probe pulses for vertical and horizontal polarization can contain different photon numbers. All these experimental issues can be taken into account by defining the measurement 7.2. QND measurement of large-spin ensembles by dynamical decoupling outcome as M ≡ (NL /4) −1/2 2p X (out) (−1)j+1 sy,j , 149 (7.31) j (out) where sy,j is the polarimeter output for j vertical (odd j) and horizontal (even j) probe pulses and p is the number of pairs of v − h probe pulses. The total number of photon sent is NL = NL,v + N L,h , where NL,v ≡ pnL,v . The normalization constant in front of the above sum was chosen so that we get var(M) = 1. In the presence of atoms and for the absence of technical noise in light and atoms we expect (cf. (2.66)) the following outcome for the variance in the measurement results var(M) = 1 + G21 NL var(Jˆz ) , (7.32) where the second term is the signal-to-noise ratio ς. In the case of arbitrary number of probe photons per pulse nL,v (nL,h ) and interaction strength G1,v (G1,h ) for vertical (horizontal) polarization, we get var(M) = 1 + 4 (G1,v NL,h + G1,h NL,h )2 var(Jˆz ) . NL (7.33) If we compare Eqs. (7.32) and (7.33) we find the relation between the individual and the effective interaction strength G1 = 2 G1,v NL,v + G1,h NL,h . NL (7.34) In the case of equal photon number for both polarizations we get G1 = G1,v + G1,h , as we should expect. 7.2.4. Experimental verification of dynamical decoupling idea The experimental sequence is shown schematically in Fig. 7.10. In each measurement cycle the atom number NA is first measured by a dispersive atom-number measurement (DANM) as introduced in Chap. 4. A Jˆx -polarized coherent spin state (CSS) is then prepared and probed with pulses of alternating polarization. Immediately after, h Jˆx i is measured to quantify depolarization of the sample and any atoms having made transitions to the F = 2 manifold are removed from the trap, reducing NA for the next cycle and allowing a range of NA to be probed on a single loading. This sequence of state preparation and probing is repeated ten times for each loading of the trap. The trap is loaded 350 times to acquire statistics. In Fig. 7.11 we plot the measured noise versus atom number, which confirms the linear scaling characteristic of the QND measurement. The red squares indicate the variance var(Ŝy ) normalized to the optical polarization noise, i.e., var(M). Independent measurements confirm the polarimetry is shot-noise limited in this regime. The black solid line is the expected projection noise scaling 4var(Ŝy )/NL = 150 7. Spin Squeezing Figure 7.10.: Experimental sequence for projection noise measurement. The CSS is prepared once and its magnitude h Jˆx i is measured. This serves as a measure of the spin polarization prior to the QND probing. We prepare the CSS a second time and assume it has the same spin polarization as in the first preparation. The state is probed with a train of pulses of alternating polarization. The complete train of probe pulses scatters about 16 % of atoms into F = 2, which are removed from the trap with resonant light in order to reduce the number of atoms in the trap. The whole cycle is repeated 10 times during one trap loading. For more details see text. 1 + G21 NL var(Jˆz ), calculated from the independently measured interaction strength G1 = 1.27(5) × 10−7 and number of probe photons NL = 8 × 108 . Also shown are results of covariance matrix calculations simulating the DD technique, including loss and photon scattering. The scenarios considered include the naive QND measurement, i.e., with a single polarization, and the “bang-bang” or twopolarization QND measurement, with p = 1, 2, 5. These show a rapid decrease in the quadratic component with increasing p. This confirms the removal of G2 due to the dynamical decoupling. Also included in these simulations is the term Ŝy Jˆy which introduces noise into Jˆz proportional to G22 var(Ŝy ) h Jˆx i2 . For our experimental parameters this term leads to an increase of var(Jˆz ) of less then 2 % and as noted above could be reduced with increased detuning. A measurement without applying the DD technique is plotted in the blue circles. The data points show a quadratic scaling which exceeds the predicted one from the covariance matrix simulations (solid gray line). Classical noise is responsible for the large amount of additional noise. This successfully confirms the strength of the method to suppress not only the unwanted quantum noise redistribution but also classical noise sources coupled via G2 . At the largest atom number of NA = 8.8 × 105 , the QND measurement achieves projection-noise limited sensitivity, i.e., the measurement noise is 5.7(6) dB below the projection noise. Within the precision of the measurement, no technical noise from atoms or light is observed. 7.2. QND measurement of large-spin ensembles by dynamical decoupling 151 Figure 7.11.: Variance of polarimeter signal as a function of atom number, comparing naive probing, i.e., a single input polarization, to “bang-bang” dynamically-decoupled probing of different orders p. Grey curves indicate simulation results for: naive probing (solid), and decoupled probing with p = 1 (widely dashed), p = 2 (dashed), and p = 5 (dotted). The black solid line shows the expected projection noise for p → ∞, or the ideal QND interaction G2 = 0. All curves are calculated using the independently measured interaction strength G1 = 1.27(5) × 10−7 and have no free parameters. Red squares are measured data using dynamical decoupling with p = 5. Error bars indicate statistical uncertainties: plus/minus one standard deviation of atom number (horizontal) or of var(Ŝy ) (vertical). Blue circles are measured data with naive probing, which show a quadratic scaling (dash-dotted line). Technical noise from laboratory fields dominates the naive probing results, and pushes them above the theoretical curve, while technical noise is suppressed in the two-polarization probing. 152 7. Spin Squeezing 7.3. Spin squeezing of magnetically sensitive atomic state Having devised a technique to perform spin QND measurements on large-spin systems we are prepared for the generation of (conditional) spin squeezing in a magnetically sensitive spin state on the F = 1 hyperfine ground state of 87 Rb. At the beginning we will characterize the de-polarization of the coherent spin state as a function of photon number in the probing pulse. Then the experimental sequence is detailed and finally the experimental results are presented and discussed. 7.3.1. Depolarization To claim spin squeezing it is not only important to measure the reduction of spin noise, but also the influence of the QND measurement on the state, i.e., its polarization Jˆx . The experimental spin squeezing criteria, presented in Chap. 2 (2.69) and (2.70), require that we measure the polarization of the coherent spin state after the QND interaction. Here, we measure the reduction in spin polarization Jx as a function of number of photons NL in the vector probe (cf. Sect. 3.5.4). The experimental sequence, shown in Fig. 7.12, consists of two spin polarization measurements with the tensor probe (cf. Sect. 3.5.4). In the first, we measure the spin polarization Jx (0) immediately after preparation. In the second, Jx (NL ) is measured after a certain number of probe photons NL was sent to the atomic sample. The coherent spin state is prepared as explained in Sect. 3.4. During the whole measurement we apply a small bias field along x of Bx ∼ 100 mG. The guiding field has two purposes. First, it stabilizes the state against un-compensated stray magnetic fields. Second, it prevents precession of the spin state induced by the vector part of the interaction Hamiltonian (cf. Sect 4.2.2). The tensor probe is tuned 190 MHz to the red of F = 1 → F 0 = 0, where the vectorial term has a large coupling strength of G1 ∼ 6 × 10−7 . The vector probe is tuned 800 MHz to the red of the transition F = 1 → F 0 = 0. This detuning is in the commonly used range for our experiments. Later we will tune the laser slightly closer, to 600 MHz. We send a train of pulses which have a duration of 1 µs and a period of 10 µs. Each pulse has a fixed number of photons and we change the total number of photons, NL , by varying the number of probe pulses. Figure 7.12.: Scheme for characterization of depolarization of coherent spin state. "Pump Jx ": atoms are prepared in the coherent spin state via optical pumping, "Probe Jx ": atoms are disperisvely probed with circular light, and the blue pulses indicate the QND probing with NL photons. 7.3. Spin squeezing of magnetically sensitive atomic state 153 In Fig. 7.13 we plot the normalized spin polarization as a function of photons NL . The black line is a single exponential fit Jx (NL ) = Jx (0) exp(−ηγ NL ) , (7.35) where ηγ is the depolarization probability for a single probe photon. We see that the spin polarization is well described by an exponential decay. The depolarization per probe photon is ηγ = 1.69(3) × 10−10 . In later experiments we determine ηγ from a two-point measurement. For this we prepare the atomic ensemble twice and measure its polarization. The first time the atoms are not probed and the second time we send NL0 vector probe photons. From Eq. (7.35) we can compute ηγ by 1 Jx (0) ηγ = 0 ln . (7.36) NL Jx (NL0 ) 7.3.2. Spin squeezing pulse sequence Atoms are loaded into the dipole trap as described in Sect. 3.1.3. We start the experiment with NA ∼ 106 cold atoms in the dipole trap. Before the QND measurement, we measure the photon scattering probability per probe photon ηγ in a two point measurement as explained above. In Fig. 7.14 we show the following step in the experimental sequence. For the QND measurements, we prepare the coherent spin state twice. The first time, we apply a small guiding magnetic field Bx ∼ 100 mG along x and measure the spin polariza(in) tion. This value serves as the initial spin polarization value Jx for this particular measurement. The second time, for the actual QND measurement, the coherent spin state is prepared in magnetic-field-free conditions. Immediately after, we send Jx(NL) [normalized] 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 2 4 6 N [×109] L Figure 7.13.: Decaying spin polarization as a result of photons scattering. The black line is a fit to the data points (blue squares) with Jx (NL ) = Jx (0) exp(−ηγ NL ). We find the decay constant ηγ = 1.69(3) × 10−10 . The error bars indicate one standard deviation. 154 7. Spin Squeezing a train of 10 pairs of x − y-polarized probe pulses. Each of them is 1 µs long and carries nL ∼ 8.84(2) × 107 photons. The probe laser is tuned 600 MHz to the red of the F = 1 → F 0 = 0 resonance. To sample different atom numbers in a single trap loading, we remove atoms from the trap after the QND measurement. This is possible since a non-negligible fraction of atoms is transferred into F = 2 by the probing light. From simulations we estimate that in ∼ 40 % of the scattering events that lead to a depolarization of the CSS the atom is transferred into F = 2. We apply resonant light from F = 2 → F 0 = 3, which should heat out atoms that are F = 2 without scattering them back into F = 1. The light is applied for 5 µs through MOT beams. The preparation, probing and cleaning steps are repeated ten times on the same trap loading. After that, we remove all remaining atoms from the trap by switching the trapping light off for a few tens of milliseconds. A last train of probe pulse is sent to measure the shot noise of the polarimeter in absence of atoms, i.e., var(M (e) ). The whole sequence is repeated ∼ 350 times to acquire statistics. Characterization of depolarization and coupling constants In the calibration measurement of ηγ right after trap loading we measure a depolarization of 41(2) % for a total of NL = 20 × nL = 1.77 × 109 photons. From this, we calculate the depolarization per probe photon ηγ = 3.0(1) × 10−10 . Note, we are working slightly closer to resonance compared to the measurement of the depolarization presented in Sect. 7.3.1. If we calculate the ratio of the scattering cross section Figure 7.14.: Important steps in the experimental spin squeezing sequence. After loading the dipole trap and measurement of the depolarization rate per photon ηγ , which are not shown here, we send the following pulse sequence: “Pump Jx ” and “Probe Jx ” determines the number of spin polarized atoms. The atoms are re-prepared in the second “Pump Jx ” without bias magnetic field and the QND measurements M1/2 are performed. Different atom numbers are sampled by removing atoms which have been scattered into F = 2 with resonant light (“clean F = 2”). The sequence is repeated ten times before the trap is emptied completely to measure the light shot noise. 7.3. Spin squeezing of magnetically sensitive atomic state 155 according to expression (B.2) for −800 MHz and −600 MHz, we get σ (−600 MHz) ' 1.665 , σ (−800 MHz) (7.37) which is close to the ratio between ηγ (−600 MHz) ∼ 1.775 ηγ (−800 MHz) (7.38) we measure. For the calculation of the amount of spin squeezing, we will use ηγ (−600 MHz) and compute the depolarization for meta-pulses of different photon number NL . The heating of atoms out of the trap after the QND measurement removes 12.8(4) % in each cycle. Knowing the depolarization of 41(2) % (measured) and the transfer probability of 40 % (calculated) we would expect ∼ 16 % to be transferred to F = 2. The slight difference may be due to insufficient optical power. However, the reduction by almost 13 % is sufficient for the purpose of reducing the atom number in the trap to scan a large range of NA . The coupling constants for the two probe polarizations are measured independently. For this we polarized the atomic ensemble along z in a small bias field of Bz ∼ 100 mG and sent linearly polarized vector probe light with the same detuning as in the QND measurements. The number of atoms is measured via absorption imaging. We find slightly different results for G1,v = 1.34(5) × 10−7 and G1,h = 1.20(4) × 10−7 . We explain this by a different mode matching for both probe beams. Using Eq. (7.34), we can compute the effective coupling strength G1 = 1.27(5) × 10−7 . Observation of spin squeezing The detected signals of the train of probe pulses are converted into meta-pulses M as explained in Sect. 7.2.3. We combine two pairs of v and h polarized pulses into one meta-pulse for the first and second QND measurement M1 and M2 , respectively. (in) Then, we group results according to measurement of Jx . For each set of atom (in) (out) numbers we compute var(Jˆz ) and var(Jˆz |M1 ) via Eq. (2.72) and (2.71), respectively. In Fig. 7.15 we plot the variance of the initial coherent spin state as blue squares. The solid line is the projection noise level for the coherent spin state, (in) i.e., var(Jˆz ) = NA /4. The linear scaling of var(Jˆz ) with NA along the theoretical prediction, indicates that we are measuring exclusively the projection noise of the coherent spin state. (out) In the red squares we plot the conditional variance var(Jˆz |M1 ), calculated via Eq. (2.71). At an atom number of NA = 6.7 × 105 the spin noise is reduced by 2.9(+1.3 −1.0 ) dB compared to the projection noise level. The amount of metrologically relevant spin squeezing [49] is calculated from the amount of depolarization during the squeezing interaction via (out) 2 ξm var(Jˆz |M1 ) = , (in) var(Jˆz )(1 − η)2 (7.39) 156 7. Spin Squeezing var(J ) [x105] z 2 1.5 1 0.5 0 0 2 4 N [x105] 6 A (in) Figure 7.15.: Measured variance of coherent spin state var(Jˆz ) (blue dots) and conditional variance var(Jˆz |M1 ) (red squares) as a function of atom number. The solid line is the expected spin projection noise calculated from the independently measured coupling constant G1 . The dashed line is the projection noise level of the depolarized (in) spin state calculated as (1 − η)2 var(Jˆz ). In the dotted line we show the conditional variance expected from the measured coupling strength for the ideal situation, i.e., (in) var(Jˆz )/(1 + ζ). The errorbars in horizontal direction indicate one standard deviation of the atom number fluctuations and in the vertical direction the statistical uncertainty of the variance. where η = 1−exp(−ηγ NL ). For the combination of four individual probe pulses (two 2 = −2.0(+1.3 ) dB v and two h) the depolarization is η = 9.9(4) %. This leads to ξm −1.0 of metrologically relevant spin squeezing. As described in Sect. 2.4.2, this implies entanglement between atoms in the ensemble [53, 54]. We conclude, that the dynamical decoupling via two polarization probing is restoring the QND measurement and can be used to produce measurement induced squeezing. 8. Summary and Outlook 8.1. Summary Quantum entanglement is one of the most surprising discoveries of the last 100 years, and physicists are still trying to understand both its meaning and the practical effects it may have. In recent years entanglement has mostly been studied in small numbers of particles, and for purposes of information processing. In contrast, this thesis describes experiments to generate entanglement of millions of atoms, and aims at practical application in precision measurement. The measurement induced generation of entanglement, i.e., spin squeezing, relies on the preparation and detection of quantum states of light and atoms with a minimum of classical noise. Furthermore, it is important to have sufficient interaction to surpass the inevitable light shot noise of the dispersive spin read out. We characterized the light-atom interaction by measuring macroscopic rotations from a polarized atomic ensemble. These gave, in connection with quantitative absorption imaging a “classical” estimate of the QND interaction strength. From this we established a technique which substitutes the quantitative, but destructive absorption imaging, by dispersive measurement atom number measurement (DANM). In this way, we are able to perform many experiments on a single atomic cloud by re-preparation and measurement without the need for reloading the trap every time. Further, we extended this method towards the measurement of Zeeman coherence via tensor light shifts, i.e., pseudo spin components Jˆx or Jˆy by measuring of polarization rotation of circularly polarized light. Working with a magnetically sensitive system demanded the thorough characterization of the magnetic field environment at the position of the atomic cloud. We developed and demonstrated a method which uses the atomic cloud as a sensitive magnetometer with spatial resolution in the order of a few tens of microns. Furthermore, the magnetic field imaging technique potentially provides a full vector-field measurement. We performed in-situ gradient control with 2 mG/cm resolution. Further reduction of atomic motion, √ for instance in an optical lattice, could in principle give sensitivities of ∼ 50 fT/ Hz. It is paramount for the generation of squeezing to have a knowledge of the input state. We proposed a spin tomography technique for alkali metal atoms which is based on dispersive paramagnetic Faraday rotation measurements. First promising results were presented. Combined with better magnetic field control this technique could serve to measure the ground state density matrix. Trying to change the quantum properties of a spin state has the pre-requisite to have knowledge about all of the noise contribution to the measurement signal of the 158 8. Summary and Outlook spin detection method. We use noise scaling and a thermal spin state to obtain an absolute quantification of the measurement noise. The results are confirmed by independent quantification of the QND measurement gain, i.e., the atom-light interaction strength. The method detects different noise sources, i.e., atomic and light quantum and technical noise, and the electronic noise floor, by their respective scaling with atom and photon number. In this measurement we were able to show that the sensitivity of the spin detection is ∼ 500 spins at a moderate destruction of 20 %. Working in the hyperfine ground state of an alkali-metal atoms, like 87 Rb, offers a lot of interesting physics. At the same time, it is quite different from the simple two-level atoms of many quantum optics textbooks. We apply dynamical decoupling to remove some of this complexity and perform optical quantum non-demolition measurement in a large-spin system with F = 1. We first identify an often-overlooked impediment to this goal: the tensorial polarizability causes decoherence of the measured variable and prevents (pure) QND measurement except in the limit of large optical depth. We then design an appropriate two-polarization probing strategy to cancel the tensorial components of the effective Hamiltonian. The dynamically-decoupled QND measurement achieves a sensitivity 5.7(6) dB better than the projection noise level. Finally, all the previous steps were leading us to the successful generation of spin squeezing in an ensemble of 7 × 105 laser cooled and optically trapped 87 Rb atoms. A spin noise reduction of 2.9 dB was observed at a destruction of < 10 %. The noise reduction and the destruction corresponds to an improvement in measurement sensitivity by 2 dB compared to a coherent state of the same spin polarization. 8.2. Conclusions and Outlook The interaction of light and matter is, on one side, an everyday problem which, for instance, allows us to read this text printed on the sheet of paper in our hands in from of us. One the other side, it is one of the most fundamental and intriguing phenomena which still provides interesting insights and large range possibilities for new developments. In this thesis, I presented results on the interplay between light and matter as quantum mechanical systems and the manipulation of the state of an ensemble of atoms through measurements. This research was motivated from two directions. On one end, the manipulation of the collective quantum state of atomic system that might serve as a very sensitive magnetic field sensor can potentially improve measurements which can give new insights in fundamental and applied problems. A magnetometer reaching sensitivities beyond current technologies might open new ways to the direct detection of magnetic fields from the human heart and brain, or might allow tests of the fundamental symmetries of nature, to name only a few. On the other end, the controlled interaction of atoms with light is a fundamental pre-requisite in the world of quantum information processing (QIP) and quantum computation (QC). There, the light as transmitter of information and matter (in a well controlled quantum state) as the stationary memory. 8.2. Conclusions and Outlook 159 Controlling both systems in a way that they behave as quantum objects was one of the main goals for this thesis. Trapping atoms in the focus of a strong laser is one way to approach this goal. It allows to trap atoms in a state-independent way with well protected from the environment. On the other side, we interrogate the atomic spin system in a dispersive measurement which provides enough interaction to observe quantum effects in the atomic system, but at the same time, introduces only little disturbance. Managing the magnetic field environment is another important issue in a system which is intrinsically highly sensitive to magnetic fields. For this purpose, we implemented a technique that makes direct use of the atomic sample to measure the magnetic field at the position of the atoms with high spatial resolution. It allowed us to identify one impediment to produce long-lived atomic spin state due to magnetic field gradients. The technique itself has the potential to be applied also in other contexts, as for example, imaging of biomedical fields. Making predictions about the future is always hard and mostly wrong. But the system has the potential to allow for the investigation of interesting questions in the near future. The spin squeezing could be extended from the present pseudo- to a “real”-spin system for spin-polarized atomic states along F̂x or F̂y which makes it a magnetometer in the common sense [8]. The ability to perform QND measurements on large-spin atoms using dynamical decoupling techniques will enable the experimental realization of novel theoretical proposals for quantum polarization spectroscopy in spinor gases and the detection of exotic quantum phases that intrinsically rely on large-spin systems [125, 170, 171]. Furthermore, the implementation of quantum memory protocols requires strong atom light interaction. The demonstration of projection noise limited spin detection is one step towards this goal. Spin squeezing via QND measurements is also interesting for unpolarized atomic states. In contrast to polarized ensembles there is no significant measurement backaction, which allows simultaneous squeezing of all spin components. The resulting state will approach a macroscopic singlet. Spin squeezing in such systems implies entanglement of a macroscopic number of atoms with arbitrary spin. Macroscopic singlets are interesting in different areas of physics. First, they are interesting for condensed matter physics since they are the solution to fundamental spin models. Second, they play an important role in quantum information processing. Their invariance under rotations makes them interesting to encode quantum information in decoherence free subspaces [184] and for sending information independent of a reference direction [185]. Third, magnetic field gradients or fluctuations can be detected with high spatial resolution [186]. A. Appendix I: Gaussian state picture for light and atoms Contents A.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 162 A.2. Physical system and mathematical model . . . . . . . . . 162 A.2.1. Linearized continuous variables and covariance matrix . . . 163 A.2.2. Spatially and temporally extended systems . . . . . . . . . 164 A.3. Unified description of physical processes . . . . . . . . . 165 A.3.1. Coherent effects . . . . . . . . . . . . . . . . . . . . . . . . 166 A.3.2. Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 A.3.3. Transport processes . . . . . . . . . . . . . . . . . . . . . . 169 A.3.4. Projective Measurement . . . . . . . . . . . . . . . . . . . . 169 A.3.5. Combining effects . . . . . . . . . . . . . . . . . . . . . . . 170 A.4. Instructive examples . . . . . . . . . . . . . . . . . . . . . 170 A.4.1. Detector time-resolution . . . . . . . . . . . . . . . . . . . . 171 A.4.2. Spatial inhomogeneities . . . . . . . . . . . . . . . . . . . . 173 A.4.3. Atomic motion . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 A.6. Coherent and incoherent transport processes described by N-port unitaries . . . . . . . . . . . . . . . . . . . . . . 177 A.6.1. Transport processes . . . . . . . . . . . . . . . . . . . . . . 177 162 A. Appendix I: Gaussian state picture for light and atoms This chapter extends the covariance matrix model for collective light-atom interfaces, developed in [70, 71, 187, 188, 65, 189, 63], to describe the quantum properties of continuous variable states of light and atoms as Gaussian states. The main interest is to model real world effects such as the finite temporal resolution of the detector, spatial inhomogeneities in light-atom interaction and atomic mixing. Furthermore, we introduce the tools to describe the evolution of entanglement in the presence of tensor light shifts present in multilevel atoms. The content of this chapter is published as: “Unified description of inhomogeneities, dissipation and transport in quantum light-atom interfaces” by M. K. and Morgan W. Mitchell in J. Phys. B 42, 195502 (2009) [15] A.1. Introduction A variety of atomic ensemble systems for spin squeezing and continuous variable quantum memories have been proposed or demonstrated, including hot atoms in cells and cold atoms in magneto-optical traps and in optical dipole traps. The conditions vary greatly from system to system, notably in the number of atoms, from ∼ 1012 in cells to ∼ 106 in dipole traps, in temperature, from ∼ 300 K in cells to ∼ 30 µK in atom traps, and in the time-scale of the interactions, from milliseconds in cells to microseconds in cold atoms. At the same time, a variety of other physical effects, such as loss and decoherence of atoms, scattering and diffraction of light, and inhomogeneous coupling of the light and atomic variables are present to varying degrees in these systems. Modeling of quantum interfaces requires addressing this diversity of scales and physical effects without sacrificing the simplicity that motivates the collective variables approach. Fortunately, many important problems involve Gaussian states [190], which allows the quantum variables to be described very simply in terms of covariance matrices. The covariance matrix formalism also naturally handles partitioning of an ensemble into sub-ensembles, a technique pioneered by Madsen et al. [65], which we use to describe both temporal and spatial inhomogeneities. Previous work on inhomogeneity has included mode matching [191] and introduction of weighted variables [192] in order to describe their influence. In some cases decoherence effects due to inhomogeneous coupling have been identified [193]. To extend the model of Madsen et al. [65], we work with physical variables, for example the Stokes (S) or atomic spin (J) operators in their extrinsic (as opposed to canonical) form. This way, we show that transport and dissipation can also be treated via the covariance matrix formalism. A.2. Physical system and mathematical model We describe atomic ensemble systems and continuous-wave light fields in terms of Gaussian continuous-variables [63, 189]. This treatment is a well established method A.2. Physical system and mathematical model 163 and applies to the majority of operations used in experiments, e.g., light atom interaction, homodyne detection etc. [68]. Furthermore, it allows us to describe nonclassical properties, e.g., squeezing, entanglement, etc. [194, 195, 196]. A.2.1. Linearized continuous variables and covariance matrix Mesoscopic ensembles of atoms and photons are described to good approximation by Gaussian states, which are characterized completely by their first and second moments [190]. While first moments describe the expectation values of the system, the second moments contain the quantum properties of the state and classical fluctuations. The theoretical framework of covariance matrices used to describe many experimental situations, e.g., coherent and incoherent light-atom interaction and measurement induced state preparation is well developed [187, 65, 15, and references therein]. However, all of the treatments rely on well polarized systems. In this limit the description can be reduced from a three dimensional phase space, for SU(2), to the harmonic oscillator phase space. Formally, this is referred to as the contraction from SU(2) to the Heisenberg-Weyl group [28]. In this picture the quantum and the classical variables are separated and can be treated independently. For a general description, the restriction to a subspace is not satisfactory. The twodimensional approximation is not capable of treating, for instance, noise in the main polarization or cases where no main polarization exists at all, i.e., for a thermal atomic state. Here, we will extend the covariance matrix formalism to arbitrary spin systems by linearizing their dynamics [111]. We split the angular momentum operator into a mean value and a fluctuating term Jˆi = Ji + K̂i and Ŝj = Sj + T̂j , (A.1) where Ji ≡ h Jˆi i (Sj ≡ h Ŝj i) is a real number representing the mean value, and K̂i (T̂j ) describe both quantum and classical fluctuations of Jˆi (Ŝj ). Further, we combine variables for atoms and light into a single vector v̂ = (Jˆx , Jˆy , Jˆz , Ŝx , Ŝy , Ŝz )T , (A.2) sometimes referred to as displacement vector , which splits up into a vector of mean values and fluctuating variables v̂ ≡ u + ŵ , (A.3) u ≡ h v̂ i = (Jx , Jy , Jz , Sx , Sy , Sz )T , (A.4) where and the fluctuating term is naturally defined as ŵ = v̂ − u = (K̂x , K̂y , K̂z , T̂x , T̂y , T̂z )T The covariance matrix is defined as E 1D γ= ŵ ∧ ŵ + (ŵ ∧ ŵ)T − hŵi ∧ hŵi . 2 (A.5) (A.6) 164 A. Appendix I: Gaussian state picture for light and atoms where ∧ is the outer product between two vectors [71, 68]. For our purpose of examining entanglement and squeezing, only the second moments are of interest. We can write the covariance matrix of the joint atom-light system as A C γ= , (A.7) CT L where C describes correlation between atoms (A) and light (L). A.2.2. Spatially and temporally extended systems A central goal of present chapter is to include spatial inhomogeneities in a description of the light-atom interaction. In cell experiments, the atomic ensemble has a constant number density while a cold trapped sample can be highly inhomogeneous. In almost all experiments, the light distribution is also inhomogeneous, e.g. from a Gaussian beam. The idea is to subdivide the physical system into locally homogeneous parts, which are described by a single set of continuous variables. We partition the inhomogeneous ensembles of atoms and light into several segments. That is, we define angular 0 momentum variables for the atom segments Ĵ(k,l) , and for the light segments Ŝ(k,l ) , with (k,l) (k,l) (m,n) (A.8) [Jˆλ , Jˆµ ] = i~λµν Jˆν δkm δln and (k,l) [Ŝλ (m,n) , Ŝµ (k,l) ] = i~λµν Ŝν δkm δln . (A.9) Segments orthogonal to the direction of light propagation are called transverse segments (first index) and along the direction of light propagation, longitudinal segments (second index) (cf. Fig. A.1). The segmentation of the light in the transverse direction is a major difference to [65] and the pre-requisite to include any inhomogeneity of light and/or atoms which would appear in realistic experiments. Note that in experiments the destruction of the atomic state will always be non-negligible. Hence, any light inhomogeneity will be mapped onto the atoms even if they were evenly distributed at the beginning. This implies that we need to model the spatial dependence of both atoms and light. Here, we describe the segmentation in two dimensions only; the extension to the third is straightforward. The continuous variable description remains valid so long ^ (1;l0 ) S ::: ^ (1;2) S ^ (1;1) S ^ (2;1) S .. . .. . ::: ^ (1;2) : : : J ^ (1;l) ^ (1;1) J J ::: ^ (2;1) J ^ (k;l0 ) S .. . ::: S^µ ::: ::: .. . channel 1 ^ (k;1) S .. . ^ (k;1) J .. . .. . ::: ^ (k;l) J channel k Figure A.1.: Schematic of partitioned atom-light interface. See text for details. A.3. Unified description of physical processes 165 (k,l) (k,l0 ) as the particle number in each segment is itself large, i.e., NA 1 and NL 1. The total angular momenta for atoms and light are X X 0 Ĵ = Ĵ(k,l) and Ŝ = Ŝ(k,l ) . (A.10) k,l0 k,l Further, we define a channel as the set of segments (light and atoms) which have the same transverse index k. We note that by using the physical angular momenta, we are working with extensive quantities, as opposed to the canonical variables, which can be defined for √ the conˆy / ~Jx , P̂ ≡ tracted systems in the limit of large classical components, e.g., X̂ ≡ J √ Jˆz / ~Jx . That is, sums such as those given above are also angular momenta with the expected quantum properties. This greatly facilitates the partitioning, and also makes it possible to study situations in which Jx changes, for example through loss, decoherence, or transport. In a common phase space for atoms and light, we define an overall phase space vector in terms of angular momentum operators as h iT 0 v̂ = Ĵ(1,1) , ..., Ĵ(k,l) , Ŝ(1,1) , ..., Ŝ(k,l ) (A.11) which is readily rewritten as the direct sum of phase space vectors of the sub-systems for atoms (A) and light (L) v̂ = v̂A ⊕ v̂L . = uA ⊕ uL + ŵA ⊕ ŵL (A.12) (A.13) Analogous to the unsegmented case, we can write the displacement vector v̂ as the sum of a vector of mean values u and a vector carrying the quantum and classical fluctuations ŵ, see Eq. (A.5). The covariance matrix γ is completely defined via ŵ as in the unsegmented case (A.6). A.3. Unified description of physical processes We will describe a variety of physical processes by their effect on v̂ and γ in the Heisenberg picture. Some processes, such as light-matter interaction, can be described by Hamiltonian evolution, while others, such as photo-detection, cannot. To handle both cases, we calculate the dynamics by difference equations which describe small but finite changes between time-steps. This is a well established method which is applied, for instance, in a Monte-Carlo wave-function approach by Mølmer et al. [197] or, as here, in the context of covariance matrices described in Ref. [189]. It allows modeling of coherent light-atom interactions, losses and decoherence, measurement and transport processes in a consistent way. The phase space vector and covariance matrix are updated in finite time-steps τ as v̂(t + τ ) = Fτ (v̂(t)) . (A.14) 166 A. Appendix I: Gaussian state picture for light and atoms After we introduce the main coherent and incoherent effects, we will specify limits for the time-step in Sec. A.3.5. The discretization of the continuous light-atom dynamics is one reason for the segmentation of light into longitudinal segments of length L = cτ . We note that in most experiments the atomic sample is much shorter than the coherence time of the light. That means, in each channel only one longitudinal light segment will overlap with the ensemble at most times. In addition, the effect of several longitudinal atomic segments is, from the light’s perspective, sequential: Ŝ(n,1) interacts with Ĵ(n,1) then with Ĵ(n,2) , and so forth. During the time (m − 1) τ = t < mτ , Ŝ(n,m) interacts with all atomic segments Ĵ(n,.) in the n-th channel. A.3.1. Coherent effects For effects described by a Hamiltonian Ĥ which is linear in the elements of the phase-space vector v̂, the phase space-vector evolves (to lowest order in τ ) as, i iτ h v̂(t + τ ) = v̂(t) − v̂, Ĥ ≡ Tτ v̂(t) . (A.15) ~ The last equality, which expresses the change in v̂ in terms of a matrix Tτ , is possible because of the linearity of the Hamiltonian and the linearization of the atomic and light variables (A.1). The covariance matrix evolves as γ(t + τ ) = Tτ γ(t)TTτ . (A.16) Atom-light interaction The most important unitary evolution of the system is due the dipole interaction of light and atoms. This we will use to gather information about the atomic spin state. At the same time the light influences the atomic variables. For a homogeneous system of light and atoms, off-resonant interaction gives rise to an effective Hamiltonian [165, 39]. For the F = 1 pseudo-spin system defined in Chap. 2, it has the form o ~n Ĥeff = G1 Ŝz Jˆz + G2 Ŝx Jˆx + Ŝy Jˆy , (A.17) τ where coupling constants G1 and G2 are proportional to the vector and tensor part of the atomic polarizability. In (A.17) we neglect terms describing state independent AC Stark shifts, which leave both the probe polarization and the spin state unchanged. Using Eq. (A.15), we can calculate the evolution of the displacement vector v under Ĥeff to first order in time, i.e., for τ being shorter than any time scale in the interaction process. We find separate transformation matrices acting on the mean value u and the fluctuating term ŵ v̂(t + τ ) = u(t + τ ) + ŵ(t + τ ) QS O3 QS RJ = u(t) + ŵ(t) , O3 QJ RS QJ (A.18) (A.19) A.3. Unified description of physical processes 167 where O3 is the 3 × 3 zero matrix. The transformation matrices Q and R depend on the mean values u(t). 1 −G1 Sz G2 Sy 1 −G1 Jz G2 Jy 1 −G2 Sx , QJ = G1 Jz 1 −G2 Jx QS = G1 Sz −G2 Sy G2 Sx 1 −G2 Jy G2 Jx 1 (A.20) 0 G2 Sz −G1 Sy 0 G2 Jz −G1 Jy 0 G1 Sx , RJ = G2 Jz 0 G1 Jx (A.21) RS = G2 Sz G2 Sy −G2 Sx 0 G2 Jy −G2 Jx 0 The dynamics of the classical components u are particularly simple and can be calculated explicitly. In QND measurements of atomic spin degrees of freedom, we can, without lack of generality, assume that the mean values for the light, uL , is practically unchanged on its passage through the atomic cloud. We can calculate the dynamics of the classical atomic components uA as uA (t) = exp (QS t) uA (0) , (A.22) where QS = (QS − I) /τ and I the identity matrix. We can substitute the Stokes components Si defined before for a pulse of length τ in QS (cf. Eq. (A.20)) by Si = τ ΦSi , where Φ is the photon flux and Si the normalized Stokes vector component of the probe pulse. The dynamics of the quantum variables follows directly from (A.19) and we get for the update formula of the covariance matrix (A.16) γ(t + τ ) = QS RJ RS QJ γ(t) QS RJ RS QJ T . (A.23) A.3.2. Noise In addition to Hamiltonian evolution, loss, transport, and decoherence of atoms and/or photons can be described. These processes introduce extra noise into the system. A fully general description of a noisy Gaussian process is the Gaussian completely-positive map [198, 71], which acts on the covariance matrix as γ 0 = MγMT + N , (A.24) where the real matrix M transforms the phase space vector and the real symmetric matrix N describes added noise. These must obey [198] N + iΣ0 − iMΣMT ≥ 0 , (A.25) where iΣpq ≡ [vp , vq ] and Σ0 , similarly defined, are commutation matrices before and (k,l) after the transformation and vp is an element of the phase space vector v̂, e.g, Sy . Note that the commutation relations (A.8) and (A.9), which include the “classical” components Jx and Sx can change due to loss and decoherence. For example, loss of 168 A. Appendix I: Gaussian state picture for light and atoms a fraction of the photons implies a corresponding reduction in Sx , and consequently in Σ0 . This places a lower limit on the noise introduced. Specifically, N = |iΣ0 − iMΣMT | , (A.26) where | · | indicates the matrix absolute value, is the minimal symmetric matrix to satisfy (A.25). Interestingly, there is a more physical picture for completely positive maps (A.24). The map can be modeled as a unitary transformation between different modes of the system, e.g., segments or a reservoir. In App. A.6 we derive Eq. (A.24) using n-port unitary transformations and phase averaging over input modes. Loss and decoherence from photon scattering Inevitably, the coherent interaction of equation (A.17) will be accompanied by spontaneous emission of photons, producing also incoherent changes in the atomic state. We use equation (A.24) to calculate the effect of loss and decoherence of atoms and photons. Here, “loss” of atoms refers to the decay of atoms into meta-stable states which do not interact with the light. Decay of atoms into the |↑i- and |↓i-states can cause decoherence of the spin state. While loss is not present in the ideal spin1/2 system proposed by Kuzmich et al. [59], in alkali metal atoms both processes are observed. For light there is no decoherence process since spontaneously emitted photons scatter into all possible spatial modes and are counted as losses. The covariance matrix transforms as γ(t + τ ) = Mτ γ(t)MTτ + Nτ , (A.27) where the decay is described by Mτ = (1 − ητ )I2 ⊕ (1 − ε)I2 . (A.28) Here, ητ is the probability for an atom to scatter a photon and ε the probability for the complementary process. Noise will have the form Nτ = Nτ,loss + Nτ,dec . With Eq. (A.26) we get Nτ,loss = ητ (1 − ητ ) ~2 ~2 NA I2 ⊕ ε(1 − ε) NL I2 . 4 4 (A.29) and Nτ,dec = ρητ ~2 NA I2 ⊕ O2 . 4 (A.30) Here, I2 is the identity matrix in two dimensions, O2 is the zero matrix, and ρ is the fraction of the scattered atoms which return to the system, assumed to be in a fully mixed state. This last assumption is convenient [63] but not required. It introduces the maximum possible amount of noise. For example, scattering in optical pumping does not produce fully mixed states. For all simulations that follow in section A.4, we assume to have exclusively atomic decoherence and no loss, i.e.,ρ = 1. A.3. Unified description of physical processes 169 A.3.3. Transport processes When atoms move from one segment to another, for example by diffusion, they transport with them a fraction of the collective spin. The effect of the transport is described by a linear transformation of the phase space vector v̂(t + τ ) = Dτ v(t). (A.31) The covariance matrix transforms according to equation (A.24) with M = Dτ and N = iΣ0 − iDτ ΣDTτ . (A.32) That is, the movement of atoms does not introduce extra noise beyond what is required to maintain the commutation relations. Other transport processes, as for example collisions, could be treated as well. They can be coherent or incoherent in nature and are relevant for interacting systems like quantum gases of bosons or fermions. A.3.4. Projective Measurement The next class of operations we can apply are measurements of atomic or light variables. While a measurement will collapse the value of an observable in a way that is fundamentally random, the resulting variances change in a way that is completely predictable: the variance of the measured observable becomes zero, the variance of the conjugate observable becomes large or infinite. The variances of other observables may also be reduced if they are correlated with the measured observable. A measurement can be described by a projection matrix P. For example, to measure (n,i) (n,i) (n,i) a polarization component of the (n, i) light segment, Ŝθ ≡ cos θŜy +sin θŜz ≡ (n,i) T (n,i) (n,i) (pθ ) · v, the projector would be the outer product P = pθ ∧ pθ . In practical situations, measurement of a light variable also implies that a light segment has reached a detector and thus is removed from the problem, reducing the dimension of the vector v. Upon measurement, the covariance matrix becomes γ 0 = γ − γ(PγP)− γ T (n,i) . (A.33) where (...)− indicates the Moore-Penrose pseudoinverse and |...|(n,i) removes the column and row corresponding to the measured, and no longer existing, light segment (n, i). Equation (A.33) is well known in mathematical statistics to compute the conditional covariance matrix of multivariate normal distributions [64]. A more detailed introduction of Gaussian operations on Gaussian states can be found in [71, 68]. For the calculations in Sect. A.4, we consider a large-area detector, i.e., one which does not distinguish between different channels (see Fig. A.1). Therefore, we define the measured light variable to be X (k,l) (l) Ŝθ = Ŝθ . (A.34) k 170 A. Appendix I: Gaussian state picture for light and atoms A.3.5. Combining effects Before we explain how to incorporate simultaneous effects, we want to give some boundaries for the time-step τ . We can identify lower and upper limits as follows: The close connection between τ and the length of a light segment leads to a lower bound. A longitudinal light segment should contain a sufficiently large number of photons in order to justify a continuous variable description (see Sect. A.2.2), i.e., (k,l0 ) NL = Φ(k) τ 1 where Φ(k) is the flux in the k-th channel. The limit is then τ min[Φ(k) ]−1 . At the same time, an upper limit comes from the requirement that the difference equations Eq. (A.15) and Eqs. (A.27),(A.28) accurately model what is in fact a continuous evolution. This limit is more difficult to specify a priori, but in general it should be sufficient that the change of v and γ in one time step are small relative to v and γ themselves. For example, to limit the decoherence introduced per time step, we require ητ = NL σ(∆)/A 1, or τ A/(σ(∆)Φ), where A is the interaction area, σ(∆) the off-resonant scattering cross-section and Φ the total photon flux. In practice, an appropriate τ , for a given scenario, can be found by calculating with decreasing values of τ until the results of the simulation converge. In the appendix we compare upper and lower limits for the conditions used in the simulations. After we have introduced several different coherent and incoherent effects in the above sections, a natural question arises: How to incorporate effects which are present simultaneously? For example, coherent Hamiltonian evolution (A.16) and incoherent photon scattering (A.27) could be modeled as γ(t + τ ) = Mτ Tτ γ(t)TTτ Mτ + Nτ (A.35) γ(t + τ ) = Tτ [Mτ γ(t)Mτ + Nτ ] TTτ , (A.36) or depending on which effect is applied first. Naturally, if the effects are simultaneous, any difference between these is not physical. Fortunately, the difference vanishes as τ becomes small and, we conclude that different simultaneous effects can be modeled if the time-step is sufficiently short. A.4. Instructive examples Now, we give three examples how the model can be applied in the context of atomic spin squeezing. Here, spin squeezing is studied because it serves as a benchmark for more sophisticated applications like atom-light entanglement, atomic quantum memory and other quantum information tasks. They could as well be analyzed with the same technique. For all simulations we consider a cold ensemble of rubidium 87 atoms in an optical dipole trap. The set of used parameters can be found in the appendix. It is well known that for large detunings from resonance we can reduce the dipole interaction Hamiltonian (A.17) to A.4. Instructive examples 171 ~g (k,l) Ĥeff ŝ(k,j) , ĵ(k,l) = Ŝz(k,j) Jˆz(k,l) . τ (A.37) The coupling constant g (k,l) is proportional to the vector polarizability α(1) and defined in the appendix. Note, the Hamiltonian does not explicitly depend on τ because the Stokes operators are proportional to the flux of photons multiplied by τ . As initial states we assume a pulse of horizontally polarized light, i.e., Sx = NL ~/2 and a coherent superposition of the Zeeman substates |F = 1, mF = −1i and |F = 1, mF = 1i, i.e., Jx = NA ~/2. The effect of the Hamiltonian (A.37) on the light is a rotation of Ŝx about the z-axis by an amount proportional to Jˆz . At the same time, Jˆz is not altered in this process. A projective measurement of Ŝy provides information about, and thus reduces the uncertainty of, Jˆz . If both input states are minimum uncertainty states, spin squeezing is obtained. To monitor the evolution of this process we evaluate 2Var ( Jˆz ) /Jx which is also known as the spin squeezing parameter [48]. For squeezed states it will become less than unity. The smaller the spin squeezing parameter the higher the degree of spin squeezing. There are other criteria, for instance by Wineland et al. [49] derived in the context of metrology. Regardless which of the definitions is applied, we obtain the same qualitative results. To make the comparison between different experimental situations clearer, we normalize the timescale. We can define a time when the rotation of the light polarization due to the atom-light interaction (A.37) exceeds the shot noise of the photons, i.e., when the signal-to-noise ratio becomes one. We want this time to be characteristic for the system as a whole. Therefore, we define t0 ≡ 4 1 . 4 2 ~ G NA Φ (A.38) We assume an atom-light interface with a number of atoms, NA , homogeneously distributed over an effective interaction area A. Here, G is an effective interaction strength and Φ the photon-flux. Details on the derivation are given in the appendix. A.4.1. Detector time-resolution As a first example, we study the influence of the detector time-resolution on the amount of spin squeezing and show the importance of correct modeling of pulsed experiments even for pulses much shorter than the detector time resolution. We define an ideal detector as one capable of detecting individual light segments. The covariance matrix would be updated in accordance to (A.33) each time a light segment hits the detector. In contrast, we say a detector has no time-resolution if it detects all segments at the same time. Mathematically, the measured variable is the sum of all n light segments n X Ŝy(1) = Ŝy(1,l) , (A.39) l=1 172 A. Appendix I: Gaussian state picture for light and atoms 2var(Jz)/Jx 1 c) 0.8 b) 0.6 a) 0.4 0.2 0 2 4 Figure A.2.: The spin squeezing achieved by using an ideal detector (a) and a detector with no temporal resolution (b) are shown. The parameters are given in appendix A. Two zero-dimensional calculations are given for comparison. In curve (c) the noise due d) to decoherence and loss was 8 6 10 added after and in (d) before Pulse Duration [units of t0] the interaction. and we apply the transformation (A.33) to the whole covariance matrix. (For simplicity we assume only one atomic segment. Nonetheless, we keep the transverse index to avoid confusion.) The projector has the form P = O2 ⊕ 1 Un ⊗ I2 . n (A.40) Where Un is the unit matrix of rank n. In Fig. A.2 we show the results for both having (a) perfect and (b) no temporal resolution. In the case of no temporal resolution, the achievable spin squeezing is reduced at longer timescales. This is understood if we compare the information carried by different longitudinal light segments. Early light segments interact with the initial atomic state and later ones with a noisier version of it. If the detector is lacking temporal resolution all this different information is mixed. Now we compare the results to calculations which neglect all dynamics during the pulse duration, e.g., in [192]. We call this type of model “zero-dimensional” because it treats the light-atom interaction as a point-like event in time. Therefore, we assume that the light pulse is not partitioned into longitudinal segments. Curve (c) and (d) in Fig. A.2 show the results if the noise is added after (cf. Eq. A.35) and before (cf. Eq. A.36) the interaction, respectively. It becomes obvious that even in the case of a pulsed experiment it is important to model light as a stream of sufficiently short segments. The difference for the two extreme cases of adding the noise after (curve c) and before (curve d) interaction and measurement (IM) can be understood in the following way. If the noise is added before interaction and measurement it can actually be detected and squeezed at the same time, which leads to a much larger squeezing. For the opposite case of addition of the noise and the end, the process has no time to “learn” about the presents of the introduced noise and we see less spin squeezing. A.4. Instructive examples 173 A.4.2. Spatial inhomogeneities In many experiments inhomogeneities in light or atomic distributions are present. We give two examples for typical situations that can arise. As the simplest test model we assume an atomic ensemble which consists of two equally sized transverse segments. For all the following calculations we assume an ideal detector. Furthermore, the total number of atoms NA , the photon flux Φ, the total interaction cross section A, and all other parameters are fixed and stated in the appendix. The figure of merit is the variance of Jˆz for the complete atomic ensemble ! X (i) var Jˆz = var Jˆz . (A.41) i To verify the validity of the segmentation model, we consider first a homogeneous atom distribution either as a single or as two transverse segments. In both cases we observe the same results in the presence of atom-light interaction, loss and decoherence and transport processes, independent of the segmentation. They produce the solid line (a) in Fig. A.2. The first example reflects the situation we would find for inhomogeneous light fields interacting with homogeneously distributed atoms. We model this with two channels of equal interaction cross-section A/2. We assume light is only present in one of the channels. The result is plotted as the dotted curve in Fig. A.3. The overall spin squeezing is reduced. To explain this, we can evaluate both channels independently. One channel contains all the photons and the maximal obtainable amount of squeezing will be the same as for the homogeneous distribution (solid line in Fig. A.2). This reflects a very important property in atomic spin squeezing. The achievable amount of squeezing does not depend on the intensity of light (as long as it is not zero) but rather on the optical depth of the atomic ensemble. For the second channel, without light, we expect no change in the atomic state. If we combine these two results, we get exactly the dotted curve shown in Fig. A.3. The second example is the inverse situation. The light beam has a larger cross-section than the atomic ensemble. We model this case by assuming all NA atoms are in one of the channels and light is homogeneously distributed over both. The result, plotted as the solid line in Fig. A.3, seems surprising. We see exactly the same dynamics as for the homogeneous case. In this situation two effects are compensating each other. The optical depth of the atoms is twice as large as in the previous examples and leads to larger spin squeezing. On the other hand, the light which does not interact with the atoms is also detected, and contributes noise but no additional information about the atoms. The two examples give some intuition about the influence of inhomogeneities. It is now straightforward to apply it to more interesting and complicated experimental cases. 174 A. Appendix I: Gaussian state picture for light and atoms 2var(Jz)/Jx 1 0.8 0.6 0.4 0 2 4 8 6 10 Pulse Duration [units of t0] Figure A.3.: Two exemplary cases are compared to the homogeneous situation where light and atoms are evenly distributed over space. For more information see text. A.4.3. Atomic motion As a last application, we ask what happens when atoms can change places and go from one segment to another. This is a relevant question for comparing different experimental situations. Atoms in vapor cells, for instance, have approximately room temperature. This corresponds to a root-mean-square (rms) velocity of hundreds of meters per second as opposed to a few tens of millimeters per second for lasercooled atoms. Typical timescales, t0 , for the light-atom interaction are milliseconds and microseconds, respectively. Hence, atoms in vapor cells have moved around half a meter (effectively) whereas the atoms in the trap move less than one hundred nanometers in a time t0 . This suggest that spatial inhomogeneities can be transferred from the light to cold atoms. To find a more quantitative description, we consider a two-channel atomic system, and introduce a mixing probability mτ per time-step τ . The system is assumed to be in equilibrium, so that an equal fraction of atoms move in each direction. From Sect. A.3.3, the transport matrix is 1 − mτ mτ Dτ = ⊗ I2 , (A.42) mτ 1 − mτ so that ~2 N = mτ (1 − mτ ) NA 4 1 −1 −1 1 ⊗ I2 . (A.43) We note that while transport can increase the variances of individual spin components, the total spin components are unchanged. Interestingly, the same results for N and M can be found by considering a unitary transformation matrix Uij exp[i(φi − φj )] which acts on the bosonic operators aj of different segments. The mixing matrix is then Mij = |Uij |2 and the noise matrix N above follows from averaging over the φi . This suggests that the above model describes incoherent transport, but could possibly be extended to include coherent transport such as diffraction of light or the motion of condensed atoms. A.4. Instructive examples a) 175 b) c) 0.01 0.01 0.1 0.4 0 2 4 0.1 0.4 1 1 6 Pulse Duration [units of t0] 8 10 10 0.01 0.1 0.4 8 10 10 1 8 10 10 Mixing rate r [units of 1/t0] Figure A.4.: Atomic motion is simulated with different mixing rates. a) shows the degree of spin squeezing for the whole ensemble, b) for the illuminated and c) for the un-illuminated segment. The discussion is given in the text. As a concrete example we assume we have an ideal gas of atoms and derive the mixing probability from kinetic gas theory. The number √ of collisions per area and time in an ideal gas is known to be R = N vrms / V 6π . Where, N is the number of atoms in volume V , and their rms velocity is vrms . From this we can calculate the 0 rate at which individual atoms cross a surface of area A0 , r = RA √ /N . Furthermore, 0 we assume that the atoms occupy a box of volume V = A A, where A is the interaction cross section of a segment. The rate can therefore be written as vrms r=√ . 6πA (A.44) In the limit of small τ we can define a mixing probability as mτ = rτ . vrms τ . mτ = √ 6πA (A.45) p Where vrms = 3kB T /m is the rms velocity of atoms at temperature T . This model is not an exact treatment of the different physical situations we find in vapor cells and atomic traps. Nevertheless, it suggests how atomic motion influences the formation of spin squeezing. In Fig. A.4 we plot the squeezing factor for different mixing probabilities. For mτ /τ → 0 we have the same situation as in the dotted curve of Fig. A.3 For increasing mixing probability we see that the squeezing is improved and the full amount (compared to the homogeneous situation, i.e., solid curve of Fig. A.2) is achieved again. In this limit, when the inverse mixing rate becomes the same order of magnitude as t0 , sufficient atomic movement is present that all the atoms get enough interaction to be uniformly squeezed. This suggest that approaches of matched variables, e.g., by Kuzmich et al. [192] are more relevant for cold atoms than for hot vapor cell experiments. If we instead focus our attention only on the segment of atoms which is illuminated we see in plot b) of Fig. A.4 that the spin squeezing for this segment reduces when the mixing probability increases. One can interpret this as a decoherence mechanism for the smaller segment [118]. 176 A. Appendix I: Gaussian state picture for light and atoms A.5. Conclusion We have presented a model to compute the dynamics of interacting light and atomic ensembles in the framework of Gaussian states. The model is based on covariance matrices for the quantum components of collective angular momentum operators and employs segmentation of the light and atom systems. The use of extrinsic variables, as opposed to derived canonical variables [65], greatly facilitates partitioning and inhomogeneity calculations. Furthermore, partitioning of the light beam, which was not considered in previous treatments, allows us to see the effects of inhomogeneity in the optical beam. This is expected to be important in many realistic situations. As a general point, optimal interactions, for example the optimal probe strength for spin squeezing, balance the desired coherent interaction against scattering noise and generally do not happen in the low-scattering regime. Outside of this regime, the interaction is essentially non-linear and power distributions must be considered. When properly considered, an inhomogeneous distribution of light is mapped to the atoms and vice-versa. The partitioning of the light also allows us to model thermal motion and could be applied to multi-spatial-mode entanglement. The difference between loss and decoherence in the atomic states was pointed out. This becomes important when the atomic state is different from the spin 1/2 system. Modeling of atomic motion addressed a fundamental question for existing atom light interfaces in hot vapor cells and for new experiments with dipole trapped samples. Employing this model, we have made the following observations: The dynamics of spin squeezing requires time-dependent modeling, even when the atoms interact with optical pulses which are shorter than the detection system can resolve. At the same time, the detector time resolution has only a minor effect on the degree of spin squeezing under realistic conditions. The effects of spatial inhomogeneities in the light and atoms have non-equivalent effects on spin squeezing: concentration of the light into a sub-region of the atoms produces equal squeezing of the sub-region, but less squeezing of the entire ensemble, while concentration of the atoms into one subregion of the light gives equal squeezing of the spin ensemble for equal segments. Finally, we observe that atomic motion between an illuminated region and a nonilluminated one tends to degrade squeezing of the illuminated region while increasing squeezing of the entire atomic ensemble. This suggests that high-fidelity experiments should use probe pulses which are either much shorter, or much longer, than the time-scale of the atomic motion. The model can straight-forwardly be adapted to more complicated experimental situations, for example a cold thermal cloud in a focused laser beam. Also, application to multi-pass schemes as proposed by Takeuchi et al. [199] or Sherson et al. [200] is possible. A.6. Coherent and incoherent transport processes described by N-port unitaries177 A.6. Coherent and incoherent transport processes described by N-port unitaries Guassian positive maps as used in Appendix A to describe coherent Hamiltonian evolution (cf. Eq. (A.16)) and incoherent evolution due to loss and decoherence (cf. Eq. (A.24)) or atomic motion (cf. Eq. (A.31)) can be derived in a more physical picture using N-port unitary transformations. A.6.1. Transport processes Redistribution of atoms among segments, diffraction of light among channels, and scattering processes which add or remove atoms or photons can all be considered as transport processes, possibly involving additional reservoir modes. Transport processes can be modeled by considering a unitary transformation U which produces N output modes ĉ0 from the N input modes ĉ as ĉ0i = N X Uij ĉj , (A.46) j=1 where we have defined a phase space vector in terms of the bosonic operators for atoms b and light a as iT h 0 0 . (A.47) ĉ = b(1,1) , ..., b(t,l) , a(1,1) , ..., a(t ,l ) We restrict our discussion to unitaries of the form UA ⊕ UL where UA acts on the atomic modes and UL acts on the photonic modes. Note that the transport process in itself does not change the polarization states, although more general transport processes are easily constructed by adding single-species (+/ − or ↑ / ↓) rotations before and/or after the transport. The components v̂i of the phase space vector v̂ transforms as σx σx X ~ ~ † σ y ĉ0i = v̂i0 = ĉ0 Uij∗ Uik (ĉj )† σ y ĉk 2 i 2 j,k σz σz σx σx X ~X ∗ ~ Uij Uij (ĉj )† σ y ĉj + Uij∗ Uik (ĉj )† σ y ĉk = 2 2 j j6=k σz σz X = Mij v̂j + ξˆi (A.48) j σx X where Mij ≡ Uij∗ Uij and ξ̂ i ≡ ~2 Uij∗ Uik (ĉj )† σ y ĉk . The matrix M describes j6=k σz the mixing of observables associated with different segments. In contrast, ξ̂ describes the influence of coherence between different segments. These coherences are 178 A. Appendix I: Gaussian state picture for light and atoms not included in the phase-space description and it can be easily verified that their D E expectation values will vanish, i.e., ξ̂ = 0. However, we will see they contribute noise to the state. f ≡ M⊗I2 In order to compute the covariance matrix from (A.6) we define a matrix M which has the same dimension as γ. I2 is the identity matrix in two dimensions. If we employ the symmetry of M and the fact that the mean values of ŵ and ξ̂ vanish, we find E D E D E D f + ξ̂ ∧ ξ̂ . f M f+M f ŵ ∧ ξ̂ + ξ̂ ∧ ŵ M (A.49) γ 0 = Mγ The above description is appropriate to coherent transport processes such as diffraction of coherent light or movement of Bose-condensed atoms. For incoherent processes such as movement of thermal atoms, we can average over unitaries with different phases, i.e., we take Uijφ = Uij exp(iφj ) and average over {φ}. D E ξ̂ i {φ} = + σx † i(φl −φk ) ∗ σ y âl e =0 Uik Uil (âk ) k6=l σz {φ} * ~ X 2 while ξ̂ ∧ ξ̂ ij {φ} = ~2 4 = ≡ σx ∗ Uil (âk )† σ y âl Uik k6=l σz + σx X † ∗ i(φl −φk +φn −φm ) σ y ân e × Ujm Ujn (âm ) m6=n σz {φ} σx σx ~2 X ∗ Uik Uil (âk )† σ y âl Ujl∗ Ujk (âl )† σ y â (A.50) k 4 k6=l σz σz * X Nij2 As a result for an incoherent transport, the covariance matrix changes as f M f + N2 , γ 0 = Mγ which is equivalent to the Guassian completely positive map [198, 71]. (A.51) B. Appendix II: Numeric simulation of multi-level atoms Contents B.1. Scattering cross section for different hyperfine ground states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 B.2. AC Stark shifts and the limit of the rotating wave approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 180 B.2.1. AC Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . 180 B.3. Numerical simulation techniques . . . . . . . . . . . . . . 183 180 B. Appendix II: Numeric simulation of multi-level atoms This appendix summarizes the important technical details for numerical simulations of the density matrix and the light shifts of the atoms in the dipole trap. B.1. Scattering cross section for different hyperfine ground states The scattering cross section σ connects the microscopic description of the interaction of two quantum particles, e.g., photons and atoms, to macroscopic observable effects as the probability of absorption [17]. The size of the scattering cross-section has little relation to the actual physical size of the atom. For a two level atom interacting with monochromatic light the scattering cross-section is λ2 σ(∆) = 3 2π Γ2 4 Γ2 4 + ∆2 . (B.1) For a multi-level atom as 87 Rb, where we have a number of excited state hyperfine levels the total scattering cross section is the sum over all the possible transitions. For atoms in F = 1 we find λ2 Γ2 1 σ(∆) = 2π 32 " 4 Γ2 4 + (∆ − ∆00 )2 + 5 Γ2 4 + (∆ − ∆10 )2 + 7 Γ2 4 + (∆ − ∆20 )2 # , (B.2) where ∆F 0 = νF 0 − ν00 is the energy splitting between F 0 = 0 and the higher excited hyperfine states. B.2. AC Stark shifts and the limit of the rotating wave approximation B.2.1. AC Stark shift Light shifts play an important role in this work, because we optically trapped atoms to work with. Here, we want to derive some important expressions for the level shift and the scattering rate of atoms in far-detuned laser fields. We don’t want to restrict the treatment to the ground state light-shift but also discuss the shift of the excited hyperfine levels, since this will be important for the experiments in Chapt. 6 and is interesting in itself1 . There are different possibilities to derive the light shift of an atomic transition that is subject to a detuned laser field. One is a completely classical Lorentz oscillator model assuming a dipole and a classical field (for details see [201, 202]). Another applies second order time-independent perturbation theory [203]. 1 Some times it is important to find a wavelength for the trapping laser at which the ground state light shift is the same as the excited state one. These wavelengths are sometimes called magic. B.2. AC Stark shifts and the limit of the rotating wave approximation 181 within RWA Let us assume we have only a two level system and the semi-classical Hamiltonian has the form 1 0 Ω 2 H=~ 1 . (B.3) 2 Ω −∆ The combined atom-light system will have new eigenstates and eigen-energies, the so-called dressed states [121]. By diagonalizing (B.3) we find the new eigen-energies p 1 ω e± = (B.4) ∆ ± ∆2 + Ω2 . 2 In the limit of large detuning and weak excitation, |∆| Ω, the expression (B.4) simplifies to Ω2 Ω2 ω e+ = − ∆ + ,ω e− = . 4∆ 4∆ For all practical purposes, it is possible to achieve |∆| Ω by choosing an intensity such that the Rabi frequency fulfills the assumption. Using the linearity of the light shift, one can scale to the intensity of interest. We find that the initial energy levels, of the bare atomic system, have changed by ∆e ω=± Ω2 . 4∆ (B.5) Note that ∆ is negative for red detunings. Hence, for a red detuned laser the energy of the ground state is lowered in the same way as the excited state energy is raised. In a multilevel atom, like 87 Rb, the level structure is more complicated and involves many ground and excited states. One way to calculate the light shift of any atomic state is to write the atom-light Hamiltonian for the multitude of states and calculate the new eigenvalues. If we consider all characterized [204, 205, 206, 207, 208] transitions from the 5S1/2 ground and from the 5P1/2 and 5P3/2 excited state we get 26 fine structure levels and 84 hyperfine levels. If we are interested in the light shift of individual magnetic substates, the matrix can be very large (384 × 384 entries at maximum) but feasible to diagonalize. Alternatively, the light shift in multi-level atoms can be calculated with Eq. (B.5) for the two level system and summed over all relevant transitions ∆e ωi = X Ω2if f 6=i 4∆if , (B.6) where the detuning is defined as ∆if = ωL − ωif , with i and f labeling initial and final atomic energy levels. beyond RWA For very large detunings, when |∆| ω0 is no longer valid, i.e., the detuning gets comparable to the atomic transition frequency, the RWA stops being applicable. It 182 B. Appendix II: Numeric simulation of multi-level atoms 12 w/o RWA w/ RWA Ac Stark shift [MHz] 10 8 6 4 2 ground state 0 excited state manifold ï2 ï4 ï6 ï1 0 +1 ï2 ï1 0 +1 +2 0 ï1 0 +1 ï2 ï1 0 +1 +2 ï3 ï2 ï1 0 +1 +2 +3 Figure B.1.: AC Stark shift in dipole trap of magnetic substates in the ground and the excited state manifold calculated without and with RWA. The light intensity corresponds to the value we expect in the focus of the dipole trap for 6 W of laser power. is no longer true that the counter rotating terms disappear. Using the Lorentz model for a classical oscillator, one can arrive at expression which has the same form as Eq. (B.6) [202]. ∆e ωi = X Ω2if , ˜ if 4∆ (B.7) f 6=i where ˜ if ≡ − ∆ 1 1 + ωif − ωL ωif + ωL −1 . (B.8) The first term is the detuning term we used in the expression before. The second is the counter-rotating term which is usually neglected when the RWA is applied. The error in the light shift we make in neglecting the counter-rotating term is ∆e ωerr = ωL − ωif ∆e ωnoRWA − ∆e ωRWA − = . ∆e ωnoRWA ωL + ωif (B.9) For red detuning, i.e., ωL < ωif , we under- and for blue detuning we overestimate the light shifts. In practice, for the used dipole laser with λ = 1030 nm and the D1 and D2 transitions at 795 nm and 780 nm, the error is around 12 %. Therefore, we will use Eq. (B.6) and the corrected detuning (B.8) to calculate the light-shift in the dipole trap. In Fig. B.1 we plot the AC Stark shift for the ground and the excited hyperfine ground state manifold without and with applying the RWA. This clacualtion indicates another important issue which is not immidiately obvious. The light shifts for the different magnetic substates in the excited state manifold are not equal as they are in the ground state. The reason is that the states in the excited state are coupled to hyperfine state manifold which have also smaller J quantum numbers. The coupling are therefore not equal to all of them. B.3. Numerical simulation techniques 183 All considered transitions Transition 5S 1 2 5S 1 2 5P 1 2 5P 3 2 5P 1 2 5P 3 2 5P 3 2 5P 1 2 5P 3 2 5P 1 2 5P 3 2 5P 3 2 5P 1 2 5P 3 2 5P 1 2 5P 3 2 5P 3 2 5S 1 2 5S 1 2 5S 1 2 5S 1 2 5S 1 2 5S 1 2 → 5P 1 2 → 5P 3 2 → 6S 1 2 → 6S 1 2 → 4D 3 2 → 4D 3 2 → 4D 5 2 → 7S 1 2 → 7S 1 2 → 5D 3 2 → 5D 3 2 → 5D 5 2 → 8S 1 2 → 8S 1 2 → 6D 3 2 → 6D 3 2 → 6D 5 2 → 6P 1 2 → 6P 3 2 → 7P 1 2 → 7P 3 2 → 8P 1 2 → 8P 3 Wavelength [nm] |hn0 J 0 kDk nJi| [10−29 Cm] fnJ,n0 J 0 AnJ,n0 J 0 [MHz] rnJ,n0 J 0 6 [10 s−1 mW−1/2 ] 794.9749 780.2374 1323.8743 1366.8700 1475.6393 1529.2558 1523.3608 728.1962 741.0173 762.0990 776.1531 775.9745 607.2394 616.1389 620.7985 630.0925 630.0026 421.672 420.298 359.259 358.807 335.178 334.968 3.5787 5.0497 3.4922 5.0980 6.6530 3.0013 9.0159 0.8088 1.1463 1.3701 0.6725 1.9788 0.4271 0.6020 1.0004 0.4731 1.4057 0.2823 0.4587 0.0975 0.1713 0.0500 0.0941 0.3404 0.6905 0.1946 0.2009 0.6338 0.0622 0.5615 0.0190 0.0187 0.0520 0.0061 0.0533 0.0064 0.0062 0.0341 0.0038 0.0331 0.0040 0.0106 0.00056 0.0017 0.00016 0.00056 35.926 37.783 7.408 14.343 9.707 1.775 10.675 2.388 4.551 2.989 0.671 3.937 1.149 2.183 2.948 0.630 3.712 1.498 3.994 0.289 0.895 0.094 0.332 3.8010 5.3633 3.7091 3.8287 7.0662 2.2541 6.7711 0.8591 0.8609 1.4552 0.5011 1.4862 0.4538 0.4521 1.0626 0.3553 1.0557 0.3000 0.4874 0.1036 0.1820 0.0532 0.1000 2 Table B.1.: All 17 transitions used to calculate light shifts of the 5S 21 , 5P 12 , and 5P 23 . The reduced matrix elements |hn0 J 0 kDk nJi| are found in Ref. [206, 205, 204, 208, 207]. The oscillator strength fnJ,n0 J 0 and the spontaneous emission rate AnJ,n0 J 0 are given. The last column shows the Rabi frequency divided by the square root of the intensity. B.3. Numerical simulation techniques Throughout the thesis we have gained a lot of insight into the physical processes by simulating the evolution of the single atoms density matrix. As we explained in Sect. 2.3.3 we use the master equation to evolve the density matrix under the influence of external fields. The system is modeled with all magnetic substates of the two hyperfine ground and the 4 hyperfine excited state. In total we have a density 184 B. Appendix II: Numeric simulation of multi-level atoms matrix which has the dimension 24 × 24. Time-dependent fields are simulated by dividing the evolution into short time steps, in which we assume the Hamiltonian to be constant. The duration of the time step is chosen to much shorter than the fastest timescale involved. √ The jump operators Cse ≡ Γdrel, ge in Eq. (2.35) for the spontaneous emission are calculated from the relative matrix element 0 p F0 1 F J F0 I I+J+1+2F 0 −mF 0 0 0 drel, ge ≡ (−1) [J ][F ][F ] . −mF 0 q mF F J 1 (B.10) Other dissipative effects, e.g., the influence of the laser linewidth are not included, but are in general feasible. Despite the existence of the wonderful “Quantum Optics Toolbox for Matlab” [209] we developed most of the routines ourselves to be able to incorporate special requirements. Nevertheless, we use a set of routines from the QO-toolbox which allow us to perform the pre- and post multiplication of the density matrix by super-operators as required by Eq. (2.35). Scattering rate number of scattered photons From the evolution of the density matrix we can deduce interesting parameters from a physical point of view. For instance, we can calculate the instantaneous scattering rate in the light-atom interaction be evaluating the following expression Rsc (t) = 2πΓ X ρjj (t) , (B.11) j6=g where Γ is the excited state decay rate and ρjj (t) is the population if an excited state at time t and we sum over all excited states. From the scattering rate we can calculate the number of scattered photons for each atom. Z t X nsc = Rsc (t)dt = ∆t Rsc,k , (B.12) 0 k where the second equality is the one we actually use since we evolve the density matrix in discrete time steps ∆t in which we calculate the scattering rate Rsc,k at time t = k∆t. Mixed state fidelity In Chapter 3 when simulating the optical pumping efficiencies, we use the mixedstate fidelity [210, 90]: q √ √ 2 F = Tr ρ2 ρ1 ρ2 , (B.13) where ρ1/2 are density matrices describing arbitrary states. The above is the generalization of the pure state state overlap |hΨ1 | |Ψ2 i|2 probability [21]. Abbreviations AOM c.c. CSS ECDL e.m. h.c. 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Koschorreck, M. Napolitano, B. Dubost, and M.W. Mitchell, Phys. Rev. Lett. 105, 093602 (2010) Chapter 7 Squeezed-light Optical Magnetometry, F. Wolfgramm, A. Cerè, F.A. Beduini, A. Predojević, M. Koschorreck, and M.W. Mitchell, Phys. Rev. Lett. 105, 053601 (2010) content not part of thesis Sub-Projection-Noise Sensitivity in Broadband Atomic Magnetometry, M. Koschorreck, M. Napolitano, B. Dubost, and M. W. Mitchell, Phys. Rev. Lett. 104, 093602 (2010) Chapter 7 Unified description of inhomogeneities, dissipation, and transport in quantum lightatom interfaces, M. Koschorreck and M.W. Mitchell, J. Phys. B: At. Mol. Opt. Phys 42, 195502 (2009) Appendix A Polarization-based light-atom quantum interface with an all-optical trap, M. Kubasik, M. Koschorreck, M. Napolitano, S.R. de Echaniz, H. Crepaz, J. Eschner, E.S. Polzik, M.W. Mitchell, Phys. Rev. A 79, 043815 (2009) Chapter 4 Ultra low-noise differential ac-coupled photodetector for sensitive pulse detection applications, P.J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E.S. Polzik, J.H. Müller, Meas. Sci. Technol. 20, 055301 (2009) Chapter 3 Hamiltonian design in atom-light interactions with rubidium ensembles: A quantuminformation toolbox, S.R. de Echaniz, M. Koschorreck, M. Napolitano, M. Kubasik, M.W. Mitchell, Phys. Rev. A 77, 032316 (2008) Chapter 2 Conditions for spin squeezing in a cold 87 Rb ensemble, S.R. de Echaniz, M.W. Mitchell, M. Kubasik, M. Koschorreck, H. Crepaz, J. Eschner, E.S. Polzik, J. Opt. B 7, S548 (2005) overall relevant for thesis Conference Contributions Oral presentations Spin Squeezing via Quantum Non-demolition Measurements in a Cold 87 Rb Atomic Ensemble M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell CLEO/QELS 2010, San Jose, USA Ultra-sensitive Spin Measurements below the Standard Quantum Limit M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell ICSSUR 2009, Olomouc, Czech Republic Ultra-sensitive Faraday Rotation Measurements from an Atom-Light Quantum Interface M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell CLEO/EQEC 2009 Munich, Germany Ultra-sensitive Faraday Rotation Measurements from an Atom-Light Quantum Interface M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell CLEO/IQEC 2009 Baltimore, USA Quantum-Enhanced Measurements of Atomic Spin A. Predojević, M. Koschorreck, M. Napolitano, F. Wolfgramm, B. Dubost, Y. de Icaza, N. Behbood, A. Cerè, and M. W. Mitchell CLEO/IQEC 2009 Baltimore, USA State Tomography with Ultra-sensitive Faraday Rotation Measurements M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell QOIT meeting 2009 Madrid, Spain Spin squeezing experiments in a cold ensemble of 87 Rb M. Kubasik, M. Koschorreck, H. Crepaz, S.R. de Echaniz, E.S. Polzik, M.W. Mitchell CLEO/IQEC 2007 Munich, Germany Conditions for Spin Squeezing in Laser-Cooled Rubidium S.R. de Echaniz, M.W. Mitchell, M. Kubasik, M. Koschorreck, H. Crepaz, J. Eschner and E.S. Polzik ICSSUR 2005 Besancon, France Posters Better-than-Heisenberg Scaling of Sensitivity with Light and Cold Atomic Ensemble M. Napolitano, N. Behbood, A. Cerè, B. Dubost, M. Koschorreck and M. W. Mitchell CLEO/EQEC 2009 Munich, Germany Towards spin squeezing in a cold Rb ensemble M. Kubasik, M. Koschorreck, M. Napolitano, S.R. de Echaniz and M.W. Mitchell QOIT meeting 2008 Barcelona, Spain Next-generation atomic ensemble for quantum information M.W. Mitchell, A. Cerè, M. Kubasik, M. Koschorreck, M. Napolitano, E.S. Polzik, J. Eschner, A. Predojević and F. Wolfgramm QCMC 2008 Calgary, Canada Inhomogeneities in atom-light interfaces and spin squeezing dynamics M. Koschorreck, M. Kubasik, S.R. de Echaniz and M.W. Mitchell CLEO/IQEC 2007 Munich, Germany Quantum Light Matter Interaction with Cold Atoms M. Kubasik, S.R. de Echaniz, M. Koschorreck, E.S. Polzik and M.W. Mitchell CQO9 2007 Rochester, USA Conditions for Spin Squeezing in Laser-Cooled Rubidium S.R. de Echaniz, M.W. Mitchell, M. Kubasik, M. Koschorreck, H. Crepaz, J. Eschner and E.S. Polzik CLEO/EQEC 2005 Munich, Germany Acknowledgments Here, I wish to express my deep appreciation for all those who have contributed to the success of this work. I thank Morgan W. Mitchell who directed this work in his own special way. I had the pleasure to work with him as a great teacher and a never ending source of insights into physics. It was a great experience to work under his supervision which extended beyond physics and science. I’d like to express special gratitude to the people who shared many hours in the lab with me. First of all, I enjoyed the companionship of Marcin Kubasik on this journey towards the ambitious goal of creating a low noise quantum state in cold atomic ensembles in the heat of the Spanish sun. Mario Napolitano joined the experiment a bit later but contributed a lot to the success of the work. I’m grateful for his friendship and commitment to offer help whenever it is needed. Many thanks to Brice Dubost for his never ending hunger for answers, which triggered many interesting discussions. I’m thankful to Herbert Crepaz and Sebastian de Echaniz who spent many hours in the lab and getting the experiment running in the first place. I owe special thanks to Rob Sewell who joined at the very end. Despite our short overlap, his contribution to this manuscript and the number of papers as a ghostreader, is invaluable. In his hands, together with Naeimeh Behbood, the experiment will produce many interesting results in the future. Many thanks to the ’light-side’ of the quantum-light-atoms group. It was always a pleasure to have you around to share ideas, cakes and cava. Thanks to the people of the Niels Bohr Institute in Copenhagen, I had the pleasure to work with, especially Eugene Polzik, Jörg-Helge Müller, Patrick Windpassinger, Andrew Hilliard, Franziska Kaminsky and Rodolph Le Targat. I also want to thank Jürgen Eschner who always had an open ear for my problems and needs. It was a pleasure to discuss physics and to do things beyond science with him. I thank all the people in ICFO giving technical support, especially Juli Céspedes i Capdevila, Jose-Carlos Cifuentes, Ricardo Saiz, and Xavier Menino. They were always helpful in solving all kinds of technical issues. I would also like to thank all the people which crossed my way in Barcelona and which became close friends. I’m glad to have you met and apologize not to list you here. Fortunately, the list would be too long. Thanks to my family, especially my parents, who supported me and my ideas invariably, and my brother who has been always an inspiration. The biggest gratitude is reserved for my love, Katharina, who convinced me to come to Spain. All this would have never happened without your loving support at all times. You have brought me always back to the ’real’ world and showed me the beauty of discovering new things. Thank you ...