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Diss. ETH No. 16731 Correlations and Counting Statistics of an Atom Laser A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Natural Sciences presented by ANTON W. ÖTTL Dipl-Phys, University of Freiburg, Germany born 12.03.1974 citizen of Germany accepted on the recommendation of Prof. Dr. Tilman Esslinger, examiner Prof. Dr. Vahid Sandoghdar, co-examiner 2006 For the loveliest discovery during my PhD LISA Zusammenfassung Die Experimente, die im Rahmen dieser Doktorarbeit entwickelt und durchgeführt wurden, realiseren kontinuierliche Atomlaser von unerreichter Stabilität und messen deren Intensitätsfluktuationen mit Hilfe von zeitaufgelöster Einzelatomdetektion durch quantisierte Lichtfelder in einem optischen Resonator. Unter Verwendung der experimentellen Technik von Teilchenkorrelationen, welche auf Hanbury Brown und Twiss zurückgeht, untersuchen wir die Kohärenzeigenschaften des Atomlasers. In dem wir die Korrelationsfuntkionen höherer Ordnung und sogar die komplette Zählstatistik aufnehmen, zeigen wir die Kohärenz des Atomlasers im Sinne der Definition von Glaubers Quantentheorie der Kohärenz. Wir finden keine überschüssigen Korrelationen in den Korrelationsfunktionen zweiter und dritter Ordnung und bestätigen, dass der Atomlaser durch einen kohärenten Zustand, welcher einer Poisson verteilten Teilchenstatistik gehorcht, beschrieben wird. Die Resultate für den Atomlaser stehen im Kontext zu den Messungen an pseudothermischen Atomstrahlen, welche Anhäufelung zeigen und eine Bose - Einstein Verteilung der Teilchen aufweisen. Ebenfalls beschrieben wird der konzeptionell neuartige Apparat, in dem wir die experimentelle Vereinigung von einem optischen Resonator höchster Güte mit quantenentarteten atomaren Gasen erreichen. Die Verwendung eines in sich geschlossenen, austauschbaren “Wissenschaftsmoduls” gewährt geräumigen Zugang zum Bose - Einstein Kondensate und dem Atomlaser für unterschiedliche Proben und Analysemethoden. Wir produzieren 87 Rb Kondensate von 2 · 106 Atomen und erzeugen extrem stabile Atomlaser mittels Radiofrequenzauskopplung. Der Atomlaser wird in die Mode des optischen Fabry - Pérot Resonators gerichtet, welcher sich 36 mm unterhalb des Kondensates befindet und über ein utrahochvakuumtaugliches Vibrationsisolierungssystem in dem Wissenschaftsmodul integriert ist. Der Resonator mit einer Finesse von 3 · 105 arbeitet im Bereich starker Kopplung zwischen Atom und Lichtfeld und ermöglicht dadurch den Einzelnachweis von Atomen aus einer quantenentarteten Quelle. Die hohe Detektionseffizienz von ca. 25% für diese Atome zeichnet ihn als empfindliche und dadurch minimalinvasive, zeitaufgelöste Messmethode für ultrakalte atomare Gase aus. Das Leistungsvermögung wird beschrieben und charakterisiert durch Einzelatomnachweise für thermische und quantenentartete Atomgase. Dieser experimentelle Aufbau ermöglicht uns kohärente Atomoptik auf Einzelteilchenniveaus zu studieren und dadurch das neuartige Forschungsgebiet der Quantenatomoptik weiterzuentwickeln. i Abstract The experiments developed and performed within the scope of this thesis realize continuous atom lasers of superior stability and measure their intensity fluctuations by time resolved single atom counting using cavity QED methods. By employing particle correlation measurements of the Hanbury Brown - Twiss technique we investigate the coherence properties of the atom laser. We proof the coherence of atom lasers in the sense of Glauber’s definition of the quantum theory of coherence by measuring higher order correlations and even the full counting statistics. We find the absence of any correlations in the second and third order correlation function and confirm that the atom laser represents a coherent state with a Poissonian atom number distribution. These findings are contrasted with measurements on pseudo-thermal atomic beams that exhibit bunching and a Bose - Einstein distribution of particles. The conceptually novel apparatus in which we achieve the experimental integration of an ultrahigh finesse optical cavity with quantum degenerate atomic gases is also presented and characterized. It grants large scale spatial access to the Bose - Einstein condensate and the atom laser for divers samples and probes via a modular and exchangeable “science platform”. We produce 87 Rb condensates of 2 · 106 atoms and generate ultrastable continuous atom lasers by radio frequency output coupling. The atom laser is directed into the mode of an optional Fabry - Pérot cavity which is situated 36 mm below the condensate. It is mounted on the science platform by means of an ultrahigh vacuum compatible vibration isolation system. The cavity of finesse 3 · 105 works in the strong coupling regime of cavity QED and serves as a quantum optical detector for single atoms from the quantum degenerate source. The high detection efficiency (∼ 25%) for quantum degenerate atoms distinguishes the cavity as a sensitive, time resolved and weakly invasive probe for ultracold atomic clouds. The performance of the setup is presented and characterized by single atom detection measurements for thermal and quantum degenerate atomic gases. This system enables us to study coherent atom optics on a single particle level and to further develop the new field of quantum atom optics. iii Contents 1 Introduction 1 2 Basic Theoretical Framework 7 2.1 Bose - Einstein Condensation in Harmonic Traps. . . . . . . . . . . 8 Ideal Bose Gas • Ground State Properties of the weakly Interacting Bose Gas • Magnetic QUIC Trap 2.2 Atom Lasers . . . . . . . . . . . . . . . . . . . . . . . . . 18 Concept • The Output Coupling Process • Beam Propagation • Coherence Properties 2.3 Cavity Quantum Electrodynamics . . . . . . . . . . . . . . . . . 28 Resonator Basics • Atom Field Interaction • Single Atom Detection 3 Experimental Apparatus 3.1 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . 37 39 Main Chamber • MOT Chamber • Installation 3.2 Magnetic Field Configuration . . . . . . . . . . . . . . . . . . 43 Magnetic Transport • QUIC Trap • Magnetic Shielding • Auxiliary Coils 3.3 Science Platform and Cavity Setup . . . . . . . . . . . . . . . . 49 Cavity Design • Vibration Isolation System • Science Platform Layout 4 Characterization of the System 4.1 Experimental Procedure . . . . . . . . . . . . . . . . . . . . 55 56 Bose - Einstein Condensation • Atom Laser Output Coupling • Cavity Lock 4.2 Single Atom Detection Performance. . . . . . . . . . . . . . . . 61 Signal Analysis • Characteristics of Single Atom Events • Detector Qualities • Detection Efficiency • Atom Laser Beam Profile • Guiding the Atom Laser 4.3 Investigation of Ultracold Atomic Gases . . . . . . . . . . . . . . 72 Thermal Clouds • Quantum Degenerate Gases • Phase Transition v CONTENTS 5 Correlations and Counting Statistics of an Atom Laser 5.1 5.2 77 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . 78 79 First Order Coherence • Hanbury Brown - Twiss and the Invention of Bunching • Glauber and the Quantum Theory of Coherence • Counting Statistics 5.3 5.4 Experimental Methods and Techniques . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cavity QED Detection of Interfering Matter Waves 6.1 6.2 6.3 105 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 106 Quantum Mechanical Measurement Process . . . . . . . . . . . . 106 Buildup of Matter Wave Interference . . . . . . . . . . . . . . . 110 7 Conclusion 113 A Appendix 115 A.1 A.2 A.3 A.4 vi 94 96 Breit - Rabi Formula . . . Physical Properties of 87 Rb D2-Line Energy Levels . . Physical Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 116 117 118 Bibliography 119 Credits 133 Publications 135 Curriculum Vitae 137 1 Introduction The research fields of Bose - Einstein condensation (BEC) [1] in dilute atomic gases and cavity quantum electrodynamics (QED) with single atoms [2] both push forward the understanding, engineering, and harnessing of quantum mechanical states. A Bose - Einstein condensate is a collective quantum state of a large atom sample and provides maximum control over external degrees of freedom. Optical cavity QED in the strong coupling regime allows probing and manipulation of single atoms with the quantized electromagnetic field in the cavity mode. A Bose - Einstein condensate is a fascinating demonstration of the quantum character of matter where indistinguishable, weakly interacting particles populate the motional ground state and establish a macroscopic wave function. Its experimental realization [3, 4] in 1995 sparked an ongoing vivid experimental and theoretical research on this novel quantum phase. Initial experiments highlighted its phase coherence [5], superfluidity [6, 7] and demonstrated the production of atom lasers [8, 9, 10, 11]. Current investigations explore inter alia quantum phase transitions [12, 13], tunable atomic interactions [14, 15] and particle correlations [16, 17, 18]. Similarly, the way to cavity QED in the optical domain was paved by first experiments in the 1990s reaching the strong coupling regime and demonstrating vacuum Rabi splitting of the coupled atom cavity system [19]. In the strong coupling regime of cavity QED the atom field interaction dominates over the dissipative losses of the quantum system. This system was used to demonstrate single atom detection in a thermal atomic beam [20]. Recent experimental progress was made in the observation of the motional dynamics [21, 22] as well as the trapping [23, 24] and cooling [25, 26] of single atoms within the cavity mode. This provides an avenue towards implementation of technologies and concepts for quantum information processing, such as nonclassical light sources [27, 28] and quantum state transfer [29]. The experimental combination of quantum degenerate gases with high finesse optical cavities offers fascinating prospects [30, 31, 32] and develops the emerging field of quantum atom optics, where both matter and light fields are quantized. The first experiments detecting single atoms from a coherent matter wave field with an ultrahigh finesse optical cavity have been performed in the scope of this thesis [17, 33]. A different technique with the potential of single atom detection in quantum degenerate samples has been demonstrated 1 1. INTRODUCTION with metastable Helium atoms [34, 18, 35]. However, cavity QED detection of single atoms is potentially nondestructive on the atomic quantum state and could be used to perform atom interferometry with squeezed states and precision measurements at the Heisenberg limit [36]. In addition, the single atom detection method offers an unprecedented sensitive and weakly invasive probe to investigate physical processes in ultracold atomic clouds in situ and time resolved. On the other hand, Bose - Einstein condensates and atom lasers provide dense and coherent atomic sources with precisely controlled external degrees of freedom for exploring and exploiting cavity mediated atom photon interactions. The integration of a high finesse optical cavity in a Bose - Einstein condensation system, despite being a central goal for atom chips [37, 38], has only recently been achieved with the apparatus described here [39]. The experimental difficulties in merging these two experimental research fields arise mainly from adverse vacuum requisites and sophisticated topological requirements on both of these state-of-the art technologies. For example, limited spatial access prevented the inclusion of a high finesse optical cavity in conventional Bose - Einstein condensation setups. Atom lasers represent a prime example of the dualism in the quantum description of nature. The characteristic feature of bosons is their tendency to clump together and multiply occupy the same phase space volume, thereby amplifying a single quantum mechanical mode to a macroscopic and observable level. This phenomenon is based on the stimulated enhancement factor (n + 1) and is the underlying principle of laser operation. The formal analogy between light quanta and massive bosons, evident in the formalism of second quantization, suggests the straight forward extension of Glauber’s quantum theory of coherence [40, 41] from optical to matter wave fields. Atom lasers, as their optical counterparts, feature ultimate directivity and brightness, monochromaticity and phase coherence [42]. However, all these features could in principle be achieved with a thermal beam upon sufficient filtering. To tell wether or not one has a genuine atom laser, requires to measure at least its second order coherence. This was already pointed out by Kleppner after the first experimental realization of an atom laser [43] and performed within the scope of this thesis [17]. For optical lasers the experimental proof of their coherence was demonstrated by Arecchi [44, 45] shortly after the invention of this extraordinary light source [46]. The quantum mechanical state of a laser is fully characterized by the notion of coherent states as introduced by Glauber. The coherent state comes as close as possible to an ideal classical wave, having an uncertainty of both the conjugate variables phase and amplitude at the quantum limit. Glauber’s theory formulates higher order correlation functions that clearly display the characteristic features of quantum radiation in terms of coherence. A high order coherent state is defined by the factorization of its higher order correlation functions and thus by the absence of any Hanbury Brown - Twiss correlations. Equivalently, the particle statistics of a coherent state obey a Poissonian distribution. Therefore the intensity of a laser is utterly stable and only limited by the shot noise contribution, whereas any thermal beam will exhibit excess fluctuations. 2 One aspect of these fluctuations, the comparison of the variance to the mean value, is grasped in the second order correlation function. This quantity was first measured in the innovative experiments by Hanbury Brown and Twiss, where they observed correlations of intensity fluctuations in two coherent beams of light [47, 48], the so-called bunching effect. Although initially measured with analog signals, in principle it represents the joint probability to detect two particles at distinct positions in space or time. The fact that the Hanbury Brown and Twiss method investigates two-particle correlation or interference effects contrasts any previous interferometric techniques. Their revolutionary measuring technique, intended and successfully applied to measure stellar diameters [49], found its way into many diverse scientific disciplines [50, 51], where the correlations between individual particles reveal insight into the quantum state of a system or the particle statistics governing its behavior. The two-particle correlation measurements can be exploited in the temporal or spatial domain to shed light on the bandwidth or the physical size of the source of particle origin. The latter is of interest in astronomy [52] and high energy physics [53]. For modern quantum optics the Hanbury Brown - Twiss method is a cornerstone and key technology for investigating the photon emission process and identifying nonclassical photon states, which form the essence of most quantum information schemes. Correlations between massive particles in atomic physics have first been measured by Yasuda and Shimizu [54]. They observed the analogous Hanbury Brown - Twiss bunching effect for identical bosons in a thermal atomic beam. Using metastable neon atoms and a segmented multi channel plate they detected temporal correlations of two successive particle detection events. Using the Hanbury Brown - Twiss correlation technique the underlying quantum statistics of fermions were investigated in mesoscopic systems [55, 56] and in a beam of free electrons [57]. These experiments demonstrated the existence of anticorrelations or antibunching for fermions as a consequence of Pauli’s exclusion principle. For quantum degenerate systems the decrease of the three-body recombination rate by the factor 3! as compared to a thermal cloud was predicted by Kagan [58] and it has been observed in the group of Cornell and Wieman [59]. Recently, in a similar experiment to the one by Shimizu [54], Hanbury Brown - Twiss correlations of atomic matter waves were measured in all three spatial dimensions by the use of metastable helium atoms and a spatially resolving fast multichannel plate detector [18]. Furthermore, the researchers in the group of Aspect observed the absence of the two-particle correlations for the case of a quantum degenerate source. These studies, carried out independently and published simultaneously are in agreement with our findings [17] and highlight different aspects. Although all the above experiments employ some kind of beam splitter and make use of several detectors to correlate particle detections at different space-time points, neither is a necessity to perform a Hanbury Brown - Twiss type experiment. Having a fast detector with individual particle sensitivity the temporal second order correlation function can be recorded with a single detector. This was already pointed out by Purcell [60] in his paper supporting the initially contended Hanbury Brown - Twiss experiment. In our experiment 3 1. INTRODUCTION we make use of this fact. The dead time of our single atom detector is short compared to the relevant time scales. We are able to record the detection time of each particle explicitly and find the temporal correlations by analyzing all differential times. In fact this represents a major improvement compared to conventional Hanbury Brown - Twiss experiments that rely on start-stop signals due to limitations set by the dead time of the detectors. Start-stop events only correlate exclusive neighboring particles and are therefore just an approximation to the second order correlation function which is a nonexclusive quantity. This poses the constraint on the maximum count rate that the probability of having more than two particles within the relevant time scale is negligible. The consequence are long integration times, that for instance exceeded 70 hours in the experiment by Shimizu [54] Moreover, employing a single atom detector with a high quantum efficiency permits the determination of correlations beyond the second order correlation function. The knowledge of all higher order correlation functions is equivalent to the full counting statistics or particle distributions within a beam. We reported [17] the first experiment that records the full counting statistics of massive particles, which is a significant quantity in mesoscopic physics [61, 62, 63]. There however, no single particle detectors are at the experimenter’s disposal yet, so only some features of the particle distribution can be inferred from the measurement of certain statistical moments [64, 65]. The knowledge of all moments fully characterizes the statistical distribution functions that usually fall into two distinct categories: sub- and super-Poissonian. The prefix refers to smaller and larger variance, in comparison to the delimiting case of a Poisson distribution which marks the total randomness of a process. So both cases provide excess information about the arrival times of particles and are related to the effects of antibunching and bunching in the second order correlation function, respectively. In the present work the first experiments detecting single atoms from a coherent matter wave field with an ultrahigh finesse optical cavity have been performed [17, 33]. The detection process of our single atom detector is based on cavity QED methods and relies on the fully quantized interaction of the electromagnetic field in the cavity with a single twolevel atom as modeled by Jaynes and Cummings [66]. The partially open quantum system of an atom coupling to the cavity mode has inherent dissipation channels through spontaneous scattering and cavity field decay. These sources of decoherence establish a quantum measurement of the presence of a single atom inside the cavity mode with the consequence of the breakdown of probe light transmission through the optical cavity. The implication of the quantum measurement is the collapse of the longitudinally vastly extended atomic matter wave function of an atom from the atom laser. In an experiment in analogy to Young’s double slit we observe the buildup of an interference pattern from single particle detection events. The fact that we reduce the atom flux to such low levels that on average only single atoms are at a time in the “which way” interferometer, represents a prime example of the wave-particle dualism in quantum mechanics and is at the heart of quantum atom optics. 4 The structure of the thesis is as follows: • In Chapter 2 the theoretical background for the three main ingredients of the experiment, Bose - Einstein condensation, atom lasers and cavity QED, is developed. • Chapter 3 explains design considerations and the technical realization of the experimental apparatus, with the main focus on the novel vacuum system, the magnetic transport and the ultrahigh finesse optical cavity design. • Single atom detection measurements for thermal atomic beams and atom lasers are presented in Chapter 4 to characterize and benchmark the performance of our apparatus. • The main part of the thesis is Chapter 5, which gives an account of our measurements of the second and third order correlation functions and counting statistics of an atom laser compared to pseudo-thermal atomic beams, thereby disclosing the true nature of an atom laser as a coherent state. • Chapter 6 deals with the investigation of the quantum measurement process realized with the cavity QED system and the localization of the longitudinally extended matter wave function. 5 2 Basic Theoretical Framework “Felix die Kirsche zu Manni, Manni Banane, ich Birne - Tor.” - Horst Hrubesch Here, the basic theoretical principles relevant for describing and understanding the physical phenomena presented in this thesis are developed. The framework is based on three pillars of modern quantum optics over which we cast a brief overview. First, the general formalism to describe many aspects of Bose - Einstein condensation in dilute atomic gases and the ground state properties of the macroscopic wavefunction are presented. Secondly, we describe the mechanism to create coherent and continuous atom lasers and illustrate some of their characteristic features. Lastly, the ideas of single atom detection in the strong coupling regime of cavity QED along with the full quantum mechanical treatment are mapped out briefly. 7 2. BASIC THEORETICAL FRAMEWORK 2.1. Bose - Einstein Condensation in Harmonic Traps The idea that a macroscopic number of indistinguishable, noninteracting, integer spin particles (bosons) can, and actually will occupy the same quantum state below a certain critical temperature was put forward by S.N. Bose [67] and extended by A. Einstein [68] to massive particles in the 1920’s. Over many years the evolution of the theoretical description of such a Bose - Einstein condensate (BEC) was connected with other macroscopic quantum phenomena like superfluidity of 4 He and superconductivity [69, 70, 71, 72, 73], where however the strong interactions dominate and deplete the condensate fraction. Only a combination of experimental techniques developed in the 1970’s and 1980’s like laser cooling and evaporative cooling applied to dilute alkali atomic vapors succeeded in 1995 to realize a pure BEC of weakly interacting particles [3]. That breakthrough opened the door to investigate this novel quantum state of matter both experimentally and theoretically, which is still an ongoing and active research field in numerous labs worldwide. The theory of Bose - Einstein condensation is discussed comprehensively in current text books [1, 74] and review articles [75, 76, 77]. 2.1.1. Ideal Bose Gas The phenomenon of Bose - Einstein condensation, i.e., the macroscopic population of the ground state by integer spin particles can be derived in terms of quantum statistics [78]. The quantum statistical behavior of indistinguishable integer spin particles by the name of bosons is governed by Bose - Einstein statistics. In the framework of the grand canonical ensemble, where the temperature T and the chemical potential µ are the natural variables, the distribution function can be found. The mean occupation number n̄ of a single-particle state with energy ²i is then given by its degeneracy gi times n̄ (²i ) = 1 e β(²i −µ) − 1 , (2.1) where β = 1/kB T and kB denotes Boltzmann’s constant. The chemical potential µ(N , T ) is set and fixed by the normalization condition for the total number N of massive particles N= X n̄ (²i ) , (2.2) which is in contrast to photons, where the total number is not a constant quantity. From these formulas it is already evident that the chemical potential µ is bounded above by the ground state energy ²0 , since n̄ has to be positive, and that the occupation number of the ground state is special, since it diverges when µ → ²0 . This defines the critical temperature Tc of the phase transition. To correctly account for the total number of particles, the number of particles N0 being in the ground state is treated separately from the number of thermal atoms Nth . For the latter, the sum in equation (2.2) can be replaced by an integral under 8 2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS the condition that kB T À ∆², where ∆² is the spacing of the discrete energy levels. In this semiclassical approximation the discrete spectrum is treated as a continuum, where the density of states ρ (²) contains the details of the level structure Z N = N0 + Nth = N0 + ρ (²) n̄ (²) d ² . (2.3) In order to calculate the transition temperature the topology and the dimensionality of the system have to be taken into account. Generally, the external trapping potential V (r) for ultracold atoms of mass m is well described by a three dimensional harmonic potential ´ 1 ³ V (r) = m ω2x x 2 + ω2y y 2 + ω2z z 2 2 −→ ρ (²) = ²2 2ħ3 ω̄3 (2.4) with eigenfrequencies ωx , ω y , ωz along orthogonal dimensions x, y, z , which yields a density of states being quadratic in energy: ρ (²) ∼ ²2 . Planck’s constant is denoted by ħ = h /2π and we introduced the geometric mean ω̄ = (ωx ω y ωz )1/3 of the oscillator frequencies. In the thermodynamic limit for large atom numbers the zero point energy ²0 = ħ(ωx + ω y + ωz )/2 of the trap can be neglected. For evaluating the integral in equation (2.3) it is most useful to introduce the general Bose function (or polylogarithm) Z g α (z ) = Γ(α) −1 0 ∞ dx ∞ X x α−1 = z k k −α , z −1 e x − 1 k=1 (2.5) with Γ(α) being the Gamma function. The result for the three dimensional harmonic potential is µ ¶ Nth (T, µ) = kB T 3 g 3 (z ) . ħω̄ (2.6) For temperatures above the critical temperature Tc , when Nth = N is valid, this serves as the normalization condition on the chemical potential, which is expressed in terms of the fugacity z = e βµ . The phase transition to a Bose - Einstein condensate occurs when µ → 0, so the critical temperature Tc is obtained by evaluating equation (2.6) for z = 1 and found to be µ ¶ Tc = ħω̄ N 1/3 ≈ 0.94 ħω̄ N 1/3 , k B ζ(3) (2.7) where the Bose integral in equation (2.3) simplifies to the Riemann zeta function ζ(α) = P∞ −α k . Besides the interaction between particles, there are corrections to the critical k=1 temperature due to the finite particle number and the anisotropy of the potential [1]. For temperatures below Tc , when µ = 0, the number of particles in the ground state increases for lower temperatures and can be found from equations (2.6) and (2.7). The condensate fraction is then given by µ ¶3 N0 T = 1− . N Tc (2.8) 9 2. BASIC THEORETICAL FRAMEWORK An important, since experimentally accessible quantity, is the density and the momentum distribution of the ultracold atomic gas. Similarly, they are best treated for the condensed and thermal fraction separately as n (r) = n 0 (r) + n th (r) and n (p) = n 0 (p) + n th (p) , (2.9) respectively. In the absence of interactions the density of the condensate is simply given by the ground state wavefunction φ0 (r) of the anisotropic harmonic trap and the momentum distribution by its Fourier transform φ(p), which is therefore also anisotropic N n 0 (r) = N |φ0 (r)|2 = 2 2 2 2 2 2 e −x /a x e −y /a y e −z /a z π3/2 a x a y a z 2 2 2 2 2 2 N n 0 (p) = N |φ0 (p)|2 = e −p x /c x e −p y /c y e −p z /c z π3/2 c x c y c z p (2.10) (2.11) p with a i = ħ/mωi and c i = ħmωi being the typical length and momentum scales. Therefore the spatial extension of the ground state wavefunction is on the order of the harmonic oscillator lengths a i . The density and momentum distribution of the thermal cloud can be calculated in the local density approximation by considering the particle distribution function in phase space, which is represented by the Wigner function W (r , p) = 1 1 (2πħ)3 e β(²(r,p)−µ) − 1 (2.12) and integrating W (r , p) over momenta and positions, respectively. The semiclassical local density approximation is valid for de Broglie wavelengths small compared to the size of the cloud or equally the variation of the trapping potential V (r), which can therefore be assumed to be locally homogeneous. The local energy is then given by ²(r , p) = p2 /2m +V (r). In the classical limit for T À Tc the effects of quantum statistics, i.e., the “–1” in the denominator of the Planck formula (2.1) can be neglected. The densities may then be calculated using the classical Boltzmann statistics, resulting in n th (r) = n th (p ) = N 2 π3/2 σx σ y σz N (2πmk B 2 2 e −(x /σx ) e −( y /σ y ) e −(z /σz ) T )3/2 2 e −(p /2mkB T ) . (2.13) (2.14) q The radial extension of a cloud at temperature T is given by σi = 2kB T /mω2i . This shows the isotropy in momentum distribution for thermal atoms, only depending on their temperature. However, when T ≈ Tc the shape of the cloud deviates form a Gaussian and is more appropriately described by a Bose function, but remains isotropic in momentum. Evaluating 10 2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS nth(r) nth(p) y py pz z FIGURE 2.1.: Illustration of the Bose functions for a thermal cloud at the phase transition (µ = 0). The anisotropic density (left) and isotropic momentum (right) distributions exhibit both a sharper peak than a Gaussian. The calculation was done with typical parameters of our experiment (ωz = 4 ω y ). The plot range is 150 a z and 50 c z for the density and momentum distribution, respectively. R W (r , p) d 3 p and R W (r , p) d 3 r yields 3 n th (r) = λ− dB g 3/2(z (r)) (2.15) n th (p ) = (λdB m ω̄)−3 g 3/2(z (p )) , (2.16) 2 where z (r) = e β(µ−V (r)) and z (p ) = e β(µ−p /2m ) . The characteristic behavior of the Bose function g 3/2 is illustrated in Figure 2.1. The thermal de Broglie wavelength λdB for an ideal gas of massive particles at thermal equilibrium is defined as s λdB = 2πħ2 mk B T . (2.17) Compared to a distribution of distinguishable particles, the density of a Bose gas is increased by g 3/2(z )/z . From equation (2.15) the critical phase space density D = nλ3dB can be extracted using the peak density for µ → 0 D = n th (0)λ3dB = g 3/2(1) ' 2.6212 , (2.18) which coincides with the criterion of Bose - Einstein condensation in a uniform Bose gas. This criterion has a simple physical interpretation: a macroscopic population of the ground state happens when there is more than one particle per cubic thermal de Broglie wavelength and so the particle waves necessarily overlap. However, while the condensation in the uniform case only takes place in momentum space for particles with p = 0, the condensation in a harmonic trap also occurs in position space as can be seen from equation (2.10). Experimentally, the discrimination of a Bose - Einstein condensate from the thermal gas is usually manifested in their specific peak density and momentum distribution, which is measured in time of flight technique [79, 75, 80, 81] as shown in Figure 2.5. 11 2. BASIC THEORETICAL FRAMEWORK 2.1.2. Ground State Properties of the weakly Interacting Bose Gas In contrast to the thermal cloud, the condensate part with its increased density can only be well described when taking into account interactions between the particles. Because the condensate part occupies the ground state, the contribution of thermal energy becomes negligible compared to the interaction energy. Therefore we have to shift from an ideal gas description towards a real gas, where the particle interactions are governed by the interatomic potential U (r 0 − r). The reduction to binary collisions is usually justified in dilute atomic gases, where the mean particle distance d = n −1/3 is much larger than the range of the interatomic potential r e ¿ d . Furthermore, all collisions in Bose - Einstein condensates, where the temperature is bepr low the critical temperature, are low momentum collisions (p → 0) satisfying ħe ¿ 1. This implies that the scattering amplitude becomes independent of energy and scattering angle known as s-wave collisions. In addition, only elastic scattering processes that preserve the atomic internal state are considered. In the low energy limit the scattering amplitude is given by its asymptotic value, the s-wave scattering length a . In the first order Born approximation the exact form of the interatomic potential is not relevant and it can be apR proximated by an effective potential with U0 = Ueff (r) d 3 r, where U0 = 4πħ2 a m (2.19) represents the interaction energy between two particles. In order for these approximations to be valid, the so-called diluteness condition n|a|3 ¿ 1 (2.20) for the so-called gas parameter needs to be satisfied. It can be violated using a so-called Feshbach resonance, where the scattering length can be tuned to any desired values. To yield downright stable Bose - Einstein condensates it has to be repulsive with a > 0. The elastic s-wave scattering length a for alkali atoms is typically two orders of magnitude larger their physical size given by the Bohr radius (a 0 = 0.5 Å). In a thermal gas, the total cross section σ of two identical bosons is then σ = 8πa 2 , which is a factor two larger than σ for distinguishable particles. In a microscopic theory the quantum mechanical state of the Bose gas is fully described by the many-body Hamiltonian Ĥ = Ĥkin + Ĥpot + Ĥint in terms of the Bose field operators Ψ̂. Its time evolution is then determined by the Heisenberg relation ∂ Ψ̂(r , t ) = [Ψ̂(r , t ), Ĥ ] . (2.21) ∂t P The field operator can be expanded as Ψ̂(r) = i φi âi , where the âi (â i† ) are the annihilation (creation) operators of a particle in state φi , obeying the Bose commutation relations. iħ 12 2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS However, in the Bogoliubov theory, which provides a good description for the macroscopic phenomena associated with Bose - Einstein condensation, the ground state component is separated and treated as a classical field Ψ̂(r) = Ψ0 (r) + δΨ̂(r) , (2.22) thereby neglecting the quantum mechanical commutation relations. The ground state parp ticle creation/annihilation operators can then be replaced by â 0(†) → N0 and the wavefuncp tion of the condensate is expressed as Ψ0 = N0 φ0 . This ansatz is appropriate when N0 À 1 and yields the prominent Gross - Pitaevskii equation in its time dependent form µ 2 2 ¶ ∂ ħ ∇ 2 i ħ Ψ0 (r , t ) = − + V (r , t ) +U0 |Ψ0 (r , t )| Ψ0 (r , t ) ∂t 2m (2.23) when the quantum fluctuation term δΨ̂ and thermal depletion are completely neglected. The condensate wavefunction Ψ0 (r) plays the role of an order parameter and its time evolution can be derived to be Ψ0 (r , t ) = Ψ0 (r)e −i µt /ħ , where the chemical potential µ = ∂E /∂N is the energy per particle. This quantities is now nonzero in a real Bose - Einstein condensate, due to the interaction energy between the particles. The stationary form of the Gross Pitaevskii equation is thus given by µ ¶ ħ2 ∇2 2 + V (r) +U0 |Ψ0 (r)| Ψ0 (r) = µΨ0 (r) , − 2m (2.24) where the external potential V (r) is usually time independent. The order parameter is norR malized to the total number of particles N0 = |Ψ0 (r)|2 d 3 r and gives the density of the gas n 0 (r) = |Ψ0 (r)|2 . The Gross - Pitaevskii equation is a nonlinear Schrödinger equation where the nonlinear term, being proportional to the particle density, describes the meanfield potential produced by the other bosons in the condensate. It is therefore only valid when Ψ0 (r) varies only weakly on a length scale compared to the scattering length a . The eigenvalue of the condensate is given by the chemical potential µ. For a solution Ψ0 of the Gross - Pitaevskii equation at T = 0, the many-body wavefunction, for a system of N bosons in the ground state, in its symmetrized form ignoring particle correlations can be written as Φ0 (r1 , r2 , . . . , rN ) = N Y 1 p Ψ0 (ri ) . i =1 N (2.25) This illustrates the fact that a Bose - Einstein condensate, although consisting of a large number of particles, is essentially a single wave. However, the solutions of the Gross Pitaevskii equation (2.24), due to its nonlinear character, can in general only be obtained by numerical integration. An important exception arises in the Thomas - Fermi approximation, when the kinetic energy term in the Gross - Pitaevskii equation is neglected compared the mean-field term. 13 2. BASIC THEORETICAL FRAMEWORK p This is the case when N0 a /ā À 1, where ā = ħ/m ω̄. Although the fraction a /ā is generally on the order of 10−3 the condition is usually well fulfilled for p typical atom numbers 4 (N0 > 10 ). In this limit an analytical solution is given by ΨTF (r) = n TF (r), where n TF (r) = [µTF − V (r)]/U0 (2.26) for µTF > V (r) and zero elsewhere [82]. Therefore, in the case of a harmonic trapping potential, the density profile has the shape of an inverted parabola as shown in Figure 2.2. The exact ground state wavefunction can only be found by variational methods and differs only at the sharp boundary slightly from the Thomas - Fermi profile. |Ψ0|2/100 E V(r) |ΨTF|2 μ RTF r 0 FIGURE 2.2.: The harmonic trapping potential V (r ) and resulting density distribution |ΨTF (r )|2 of the condensate wavefunction in the Thomas - Fermi limit, in comparison to the ground state wavefunction |Ψ0 (r )|2 in the absence of interactions, scaled down by a factor 100. The peak density is given by n TF (0) = µTF /U0 and the chemical potential follows from the normalization condition (2.2) µ ¶ ħω̄ 15N a 2/5 µTF = . (2.27) 2 ā The physical content of this approximation is that the energy to add a particle at any point in the condensate is the same everywhere. The Thomas - Fermi result for the total energy per particle is 5/7 of µTF . The boundary of the wavefunction when µTF = V (r) is given by the Thomas - Fermi radii s Ri = 2µTF mω2 i 14 µ = ā 15N a ā ¶1/5 ω̄ . ωi (2.28) 2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS The density profile is anisotropic, as is the momentum distribution, which can be found by taking the squared Fourier transform of the Thomas - Fermi wavefunction ΨTF (r) [82] µ 15 J 2 (p̃ ) n TF (p) = N R̄ 3 3 16ħ p̃ 2 ¶2 (2.29) , q where J 2 is the second order Bessel function, p̃ = (p x R x )2 + (p y R y )2 + (p z R z )2 /ħ and R̄ = (R x R y R z )1/3 . The width of the momentum distribution n TF (p i ) scales as ∼ R i−1 and for large samples would approach a δ-function as in the uniform case. n(x) −10 n(v) 0 x [ μm ] 10 −0.5 0 vx [ mm/s ] 0.5 FIGURE 2.3.: Results of a variational method (operator split-step FFT) showing the exact density (left) and momentum (right) distribution of the condensate wavefunction. The numerical simulation was performed for typical parameters of our experiment, having a 87 Rb condensate of 2 · 106 atoms and a trap frequency ω = 2π · 38 Hz. Going to higher order approximations, the quantum fluctuations absent in the mean-field approach can be taken into account. The Bogoliubov transformation [74] allows the diagonalization of the Hamiltonian Ĥ including first order quantum fluctuations (see Eq. 2.22). This microscopic approach yields for instance the excitation spectrum of a homogenous gas p at zero temperature ²(p ) = 4g nm p 2 + p 4 /2m . The dispersion relation is phonon-like for p ¿ mc and particle-like for p À mc , which defines a natural length scale, the healing length 1 ξ= p . (2.30) 8πna It plays the role of a coherence length and sets the minimal length scale over which the condensate wavefunction can be perturbed. Therefore the sharp boundary of the Thomas Fermi profile is smeared out on the size of the healing length ξ. The microscopic Bogoliubovptheory gives corrections to quantities derived by the meanfield theory on the order of na 3 , which is typically a few percent for common condensates. The most interesting consequence is the prediction of the quantum depletion 15 2. BASIC THEORETICAL FRAMEWORK ³ ´ p ≈ 1.5 · na 3 of a condensate originating from the interactions between its constituting particles. It is the reason for the large condensate depletion in strongly interacting quantum fluids like liquid helium. 2.1.3. Magnetic QUIC Trap The method of choice, besides optical dipole traps, to realize a confinement for ultracold neutral atoms are magnetic traps. According to the fundamental Earnshaw theorem no local maximum of the electromagnetic field can exist in free space. Therefore magnetic traps create a local minimum and thus apply exclusively to atoms in “low field seeking” Zeeman states with a positive magnetic moment µm = g F mF µB > 0, being composed of the Landé factor g F and the magnetic spin quantum number mF , times Bohr’s magneton µB . All magnetic traps in cold atom experiments require a finite offset field to avoid Majorana spin flip losses at their center where the magnetic field would be zero [83, 84, 4]. In this region the atomic spins lack a quantization axis and they will undergo spin flip transition into untrapped or antitrapped Zeeman states. Quite generally these types of magnetic traps and in particular our QUIC trap (quadrupole Ioffe configuration) [85] can be described as Ioffe - Pritchard traps [86] exhibiting an axial symmetry with an offset field B 0 along the symmetry axis. An implementation of the QUIC being the simplest and most stable of the Ioffe - Pritchard type traps is shown in Figure 2.4. A small coil (Ioffe coil) is added in series to a quadrupole coil pair to lift the magnetic zero and add a curvature to the resulting magnetic field. The magnetic potential for the atoms with a magnetic moment µm is then well approximated by 00 V (ρ, y ) ' µm (B ρ00 ρ 2 + B y00 y 2 )/2 with B ρ00 B 02 B y = − , B0 2 (2.31) where our symmetry axis (Ioffe axis) is along the y direction. The q trap frequency in a certain µ p axis is then simply related to the field curvature through ω = mm B 00 and scales as ω ∼ I with the electrical current. Usually Bose - Einstein condensates in Ioffe - Pritchard traps have a cigar shaped form with aspect ratios ranging from 2–100. To investigate the ultracold atomic gas, the trapping potential is switched off abruptly and the cloud is allowed to expand in a free time of flight before resonant laser light casts the shadow image on a CCD camera. Absorption imaging is the standard technique to detect the cold atomic cloud and to identify a Bose - Einstein condensate by its high peak density and its anisotropic expansion reminiscent of the trap geometry. A typical image showing the bimodal distribution of a partly condensed cloud with inverted aspect ratio and an isotropic thermal background is depicted in Figure 2.5. 16 2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS z B y FIGURE 2.4.: Schematic sketch of the arrangement of electromagnetic coils forming the magnetic QUIC trap. The small lateral Ioffe coil lifts the magnetic zero of the quadrupole field and adds curvature. The symmetry axis of the BEC is along the y-direction. y -z 100 μm FIGURE 2.5.: Absorption image of a partly condensed cloud after a time of flight of 22 ms. The Bose - Einstein condensate is clearly distinguished by the anisotropic expansion and increased peak density compared to the surrounding thermal cloud. 17 2. BASIC THEORETICAL FRAMEWORK 2.2. Atom Lasers The idea of an atom laser as a coherent matter wave and strategies on how to convert random de Broglie waves into such a coherent state, was proposed by a number of different authors [87, 88, 89, 90, 91, 92] even before the experimental realization of Bose Einstein condensation. But as soon as the creation of single macroscopic wavefunctions provided ultimate control over matter waves in a trapped state it was just a question about finding a suitable extraction method to create an atom laser beam. After the first demonstration of a coherence preserving pulsed output coupler [8], the extraction methods were extended and refined to give mode locked [9], well collimated [10] and continuous [11] atom lasers. Theoretical models describing the characteristics of atom lasers range from rate equations and mean-field approaches to fully quantum field theory usually under the Born - Markov approximation. A review about these methods can be found in reference [93]. Of foremost interest are the coherence properties of atom lasers, where a major contribution was provided with the measurements within this work [17]. The concept of an atom laser is in close analogy to its optical counterpart and they share the same distinct features. An atom laser can quite generally be defined as a device that emits an extremely bright, highly directional and monochromatic beam of atoms rather than photons [94]. Ultimately, however, an atom laser is defined by its high order coherence, which is elaborated more deeply in Chapter 5. The quantum optical description of lasers can be transferred from photons and light fields to incorporate atoms and matter wave fields because of their formal equivalence in quantum field theory. Atom optics is the field which exploits the analogy between Schrödinger waves and electromagnetic waves. The quantum mechanical treatment of the laser is indispensable [40] although, quite ironically, it requires its output beam to be well approximated by a classical wave of fixed intensity and phase. Consequently, the single mode output field must be highly Bose degenerate to limit quantum fluctuations and approach a classical field. The key mechanism to establish a vastly populated quantum mechanical mode relies on the mode selectivity of a cavity and the stimulated bosonic enhancement factor N0 + 1 which amplifies the population of the selected mode. Therefore a fermionic atom laser will be the counterpart of a light sabre and remain science fiction. The ingredients that make up an optical 18 100 μm 2.2.1. Concept 2.2. ATOM LASERS laser can all be recovered as the elements to realize an atom laser as illustrated in Figure 2.6. Although the fundamental principles of matter and light lasers are alike, their technical realizations naturally have to differ. For instance, number conservation allows the atomic system to be prepared in the absolute ground state of the cavity. The natural population inversion of the thermal spectrum can be transformed into a macroscopic ground state population by means of stimulated scattering events during the processes of evaporative cooling towards Bose - Einstein condensation. The trapping potential serving as the cavity for the matter wave has to be rendered partially penetrable to establish an output coupling of the atom laser beam while preserving its coherence properties. gain medium laser beam cavity output coupler pump evaporative cooling trap BEC thermal atoms atom laser output coupler FIGURE 2.6.: Schematic diagram illustrating the key ingredients and similarities of an optical laser and its matter wave counterpart, the atom laser. Of course the analogy between optical and atom lasers is not exhaustive. Atoms are particles of matter rather than gauge bosons such as photons. That means the matter field can not be directly measured but only bilinear combinations of the atom field are observables. The virtue of that is the fact that atoms will not be annihilated by the detection process in contrast to photon detection. However, an atom laser of massive particles will be affected strongly by gravity and can only propagate undisturbed under very good vacuum conditions. But it are these exceptional features that distinguish an atom laser as unique scientific tool for novel applications and research with atom optics. Atom laser experiences the interactions between its constituting atoms. These interactions render an atom laser highly nonlinear and for instance four-wave mixing has been demonstrated in Bose - Einstein con- 19 2. BASIC THEORETICAL FRAMEWORK densates [95]. Also, atomic beams ill not propagate at a fixed speed (of light) and can be focused much more tightly due to their correspondingly shorter de Broglie wavelengths [96]. Therefore much smaller structures can be resolved and created in coherent atom lithography and microscopy. Further prospects of atom lasers include pushing the limits of resolution of the interferometric methods employed for precision sensors and metrology beyond the standard quantum limit towards the Heisenberg limit [36]. 2.2.2. The Output Coupling Process An output coupler for an atom laser is a mechanism that extracts atoms from the confined quantum state to a propagating mode while preserving or even extending (in the temporal domain) its coherence properties. Quite generally, a coherent process couples the trapped spin state of atoms in the Bose - Einstein condensate into an untrapped state that can escape the trap. The principle is shown in Figure 2.7 where the atoms in the condensate state Ψ are degenerate at the chemical potential and the wavefunction of the freed state Φ are the energy eigenfunctions of the linear gravitational potential plus the “hump” from mean-field repulsion of the remaining Bose - Einstein condensate. Different output coupling mechanisms based on coherently induced spin flips [8, 10, 11] and other methods [9, 97] to couple the condensate wavefunction to free states have been demonstrated. In general, the output coupling process can be described quantum mechanically through a set of coupled nonlinear Schrödinger equations in the rotating wave approximation µ 2 2 ¶ £ ∂ ħ ∇ 2 2¤ iħ Ψ = − + V (r) − mg z +U0 |Ψ| + |Φ| Ψ + ħΩe −i ∆t Φ ∂t 2m µ 2 2 ¶ £ ∂ ħ ∇ 2 2¤ iħ Φ = − − mg z +U0 |Ψ| + |Φ| + ħ∆ Φ + ħΩe i ∆t Ψ , ∂t 2m (2.32) (2.33) where the trapped state Ψ is held by the potential V (r) under the influence of gravity (g is the gravitational acceleration) and the mean-field potential exerted by the sum of the densities of both states [98, 99, 100]. The untrapped state Φ only experiences the linear gravitational potential (in −z direction) as well as the combined mean-field term. The nonlinearity constant U0 is well approximated to be the same for both states because the interstate scattering lengths are equal within a few percent [101]. The coupling term between the states is proportional to the Rabi frequency Ω. For radio frequency output coupling the Rabi frequency is given by the magnetic dipole matrix element µ between the states Ψ → Φ and the magnetic field Brf of the radio frequency Ω = µ · Brf . (2.34) The detuning ∆ in eq (2.33) is the energy difference of the radio frequency photon to the potential energy of the trapped versus the untrapped state, which will be taken up by the atoms. 20 2.2. ATOM LASERS E VΨ(z) mF = -1 ħωrf mF = 0 Φ(z) VΦ(z) -z FIGURE 2.7.: Diagram of the coherent atom laser output coupling mechanism. Trapped degenerate atoms (in state Ψ) that experience the overall trapping potential V (r ) plus their mean-field are coupled via rf photons to the untrapped states Φ being the eigenfunctions Ai of the linear gravitational potential. The mean-field of the remaining BEC produces the minor hump. For the case of strong output coupling when the Rabi frequency dominates the intrinsic energy of the system (Ω À µ), the resulting behavior of the coupled equations (2.32) (2.33) exhibits synchronous Rabi cycling between the trapped and the untrapped state. The Rabi oscillation is faster than the “reaction time” of the untrapped atoms and they are cycled back before they can leave the trap. Therefore, in this mode only pulsed operation (π/2−pulse) p is possible as demonstrated in [8, 102]. However, the effective Rabi frequency Ωeff (r) = Ω2 + ∆2 (r) is spatially dependent due to the inhomogeneous detuning across the trap. The case of weak output coupling (Ω ¿ ωz ) is of foremost interest especially for the creation of continuous atom lasers. The valid assumptions of an undepleted trapped state and a very dilute output state allow perturbative solutions which describe the key aspects of the output field behavior The output state eigenfunctions usually form a continuum of states ΦE labeled by their potential energy E to which the output coupling process can couple the trapped state Ψ. However, the output field is dominated by a spectral filter function which depends on the duration of the coupling time τ through Heisenberg’s uncertainty 21 2. BASIC THEORETICAL FRAMEWORK groundstate Ψ0 E ħωrf ΦE ∆E continuum FIGURE 2.8.: The temporary output coupling process of duration τ results in a Fourier limited spectral width of ∆E = 1/τ in the outgoing energy eigenfunctions. relation ∆E = ħ/τ. Initially the weak output coupler produces an output field that is a copy of the original trapped state but for longer times (τ À 1/ω) the spectral filter narrows and approaches a Dirac δ-function. Eventually the output state wavefunction becomes proportional to the energy eigenfunction of the linear gravitational potential. However, despite the spectral narrowing a steady-state laser operation should only be established after an q initial switching effect on a time scale τ = 2R z /g ≈ 2ms, when the first atoms left the condensate and the system can be modeled by a Markov process. An elegant realization of an output coupler is the radio frequency (rf) or microwave (mw) output coupling process which allows the production of continuous atom laser beams. It is the prime choice to coherently extract atoms from a magnetic trap. An alternative method would be the use of a stimulated two-photon Raman transition being off resonant from the excited state. Whereas Raman output coupling can provide a momentum kick of twice the recoil velocity (ħk /m ≈ 6mm/s on the D2-line, see Appendix A), the transferred momentum of the radio frequency photon is negligible due to the low rf energy and the large atomic mass. The freed atoms are solely accelerated downwards under the influence of gravity. A diagram of the level structure of the hyperfine ground states including the magnetic sub-levels of 87 Rb is shown in Figure 2.9. The trappable magnetic states (“low field seekers”) are marked by (N) whereas the magnetic field insensitive (to first order) Zeeman states are labeled by (Ä). The antitrapped high field seeking states are not specially marked. The hyperfine splitting of the ground state in 87 Rb is an extremely well know frequency [103] and given by ∆E hfs = h · 6.834682GHz. The Zeeman splitting of the magnetic sub-levels of both ground states is given by ∆E Z = g F µB ∆mF B where the Landé g-factor for F = 1 and F = 2 is very close to −1/2 and +1/2, respectively. In the linear Zeeman regime, which is valid for the weak magnetic fields on the order of a few Gauss we usually apply, this yields a splitting of ∼ 700kHz/Gauss. The exact energy dependence of the states on the magnetic field can be calculated using the Breit - Rabi formula (see Appendix A). Different output channels respecting the selection rules (∆mF = ±1) for magnetic dipole transitions are possible from one of the stretched states |F = 1, mF = −1〉 and |F = 2, mF = 2〉 where magnetically trappable Bose - Einstein condensates can be produced in. However, rf 22 2.2. ATOM LASERS mw 6.8 GHz F=2 rf F=1 -2 -1 0 +1 +2 ( mF ) FIGURE 2.9.: Zeeman splitting (not to scale) of the two hyperfine ground states of 87 Rb for a weak magnetic field. Trappable low field seeking states are indicated by (N) whereas untrapped states are marked by (Ä). Two possible output coupling channels for an atom laser from a Bose - Einstein condensate in the |F = 1, mF = −1〉 state are drawn by the microwave and radio frequency photon, respectively. output coupling from the |F = 2, mF = 2〉 state involves additional dynamics from the intermediate level |F = 2, mF = 1〉. Therefore we usually prepare our Bose - Einstein condensate in the |F = 1, mF = −1〉 state which allows well defined direct output coupling. For instance, output coupling into the untrapped state |F = 1, mF = 0〉 is possible with a single radio frequency photon ( ∼ 1MHz). On the other hand, this transition also introduces a weak mutual coupling between all states in the F=1 Zeeman manifold due to its symmetric splitting. Therefore we mostly apply microwave output coupling into the |F = 2, mF = 0〉 state which constitutes a true two-level system (see Figure 2.9). The wavefunctions of the outgoing freed state are the energy eigenfunctions of the linear gravitational potential, neglecting the small contribution of the mean-field potential. Assuming a hard wall boundary far away, e.g., the bottom flange of the vacuum apparatus or absorbing boundary conditions, the eigenfunctions of a linear potential are found [98] to be Airy functions Ai(ζ) of the qdimensionless parameter ζ = (z − E /mg )/l which is scaled by the natural length unit l = 3 ħ2 /2m 2 g . The Airy function, being the eigenfunction of the atom laser is shown in Figure 2.10 in comparison to the Thomas - Fermi wavefunction of the trapped Bose - Einstein condensate in a dressed state picture whiteout the energy of the rf photon (see Figure 2.7). The local wavelength of the atom laser constantly decreases corresponding to the acceleration in the gravitational field. Its wavelength after a propagation of 3.6 mm is only 5 nm. 23 2. BASIC THEORETICAL FRAMEWORK |〈Ψ|Φ〉|2 ΨTF(z) 0 0 Φ0(z) -15 -10 -5 0 5 z [ μm ] 10 -15 -10 -5 0 5 ∆z [ μm ] 10 15 15 FIGURE 2.10.: (left) The spatial overlap in a dressed states picture of the trapped condensate wavefunction with the real part of the Airy function for a centrally output coupled wave. (right) Variation of the Franck - Condon factor with the detuning of the output coupling resonance. The transition probability, according to Fermi’s Golden Rule, depends on the overlap between the bound and free states. The total output coupling rate for a Rabi frequency Ω is then Γ ∝ Ω2 |〈Ψ0 (r)|ΦE (r)〉|2 . (2.35) The Franck - Condon factor also determines the spectral width for which output coupling from the Bose - Einstein condensate is possible. The overlap integral is nonzero only for radio frequency detunings ∆ in the interval of |∆| < g p 2mµ , ħωz (2.36) assuming a Thomas - Fermi profile of the trapped groundstate including gravity as shown in Figure 2.10 b. For continuous radio frequency output coupling with a well defined energy E exactly the wavefunction ΦE having its classical turning point at ζ = 0 is selected out of the continuum of states by the resonance condition ħωrf = V−1 (z res ) − V0 (z res ). Quantum mechanically, the overlap of the Airy function ΦE with the groundstate wavefunction Ψ0 and is only significant in a narrow region (≈ l ) around the classical turning point due to its fast oscillatory behavior. Therefore, in a classical picture, an atom laser can be regarded to emanate from the resonant region z res that fulfills energy conservation ħωrf = µm B (z res ) , (2.37) because the different potentials only differ in the contribution from the magnetic trap. Having this picture in mind it is important to consider all energy contributions. In the vertical direction one has to include the effect of gravity which shifts the overall potential 24 2.2. ATOM LASERS for massive particles downwards. For atoms with a magnetic moment µm = g F mF µB the global potential can be expressed as V (ρ 0 , y ) = 1/2µm |B (ρ, y )| − mg z , (2.38) gravity where the symmetry is retained through the coordinate transform z 0 = z − z sag in comparison to equation (2.31). So the minimum of the potential is shifted downwards by z sag = g /ω2z with respect to the center of the magnetic field, where g denotes the gravitational acceleration. This is illustrated to scale in Figure 2.11. In our case for a very relaxed trap the sag is around 200 µm, with the consequence that areas of equal magnetic field resemble two dimensional planes that intersect the condensate at different vertical positions. This fact permits the output coupling from a very well defined position, i.e., horizontal plane within the Bose - Einstein condensate when output coupling weakly and continuously with a single radio frequency. z z 0 zsag x BEC y FIGURE 2.11.: Equilibrium position and size of the Bose - Einstein condensate in the anisotropic magnetic trap including the gravitational sag (to scale). Therefore the regions where the resonance condition for atom laser is fulfilled, which are the areas of equal magnetic field strength, intersect the BEC as almost horizontal planes. 2.2.3. Beam Propagation Along the axial direction the wavefunction of the atom laser is given by an Airy function. Its propagation direction is governed solely by gravity with a slight influence of magnetic field gradients due to the weak magnetic susceptibility of the atoms in second order Zeeman effect (see Appendix A.1). In the radial directions however, the propagation properties of an atom laser in terms of divergence, brightness and beam profile are mainly determined by the influence of the source, i.e., the Bose - Einstein condensate. Transversely, the atom laser wavefunction will be a copy of the initial ground state wavefunction of the BEC. Being coupled to plane waves it will expand quantum mechanically as a free wavepacket governed by the momentum distribution, i.e., the Fourier transform of the trapped wavefunction. That means a small source size corresponds to a faster expansion and larger divergence. 25 n(x) m s 2. BASIC THEORETICAL FRAMEWORK n(y) n0 T = 2. 5 n0 phase 0 x [ μm ] -75 m s n0 0 y [ μm ] n0 75 n(y) T = n(x) 75 86 -75 phase FIGURE 2.12.: Beam profiles from numerically solving the Gross - Pitaevskii equation of the atom laser after different propagation times. The upper series shows the atom laser density (blue) and its phase (green) along the fast (left) and slow (right) axis in comparison to the Thomas - Fermi profile in the trap (gray). The lower series shows the atom laser when it enters the cavity after a propagation time of 86 ms. Its divergence is due to the initial momentum spread (see Figure 2.3) and the mean-field repulsion of the remaining condensate. 26 2.2. ATOM LASERS Apart form this Heisenberg limited expansion the source exerts a repulsive interaction on the freed atoms on their way through the BEC. The mean-field potential on top of the linear gravitational potential acts as a negative lens for the atom laser beam and increases its divergence with increasing curvature and density of the BEC. The radial expansion of the atom laser wavefunction typical for our experiment over 86 ms is shown in Figure 2.12. It demonstrates the increased spreading for smaller source sizes and the appearance of quantum interferences at its wings due to the side lobes of the ground state Fourier transform as shown in Figure 2.3. These higher components in the momentum distribution are present even in the exact solution and due to the non-Gaussian shape of the condensate wavefunction. 2.2.4. Coherence Properties The coherence of an atom laser strongly depends on the coherence quality of the source and the coherence preservation of the output coupling process. Coherence is a somewhat complex notion in physics and will be discussed in more detail in Section 5.2. Basically it characterizes the ability to interfere (first order) and the noise (higher order) of quantum systems. In general, there exists both spatial and temporal coherence. However they are usually treated separately and independently. In many experiments [5, 104] the spatial coherence of a Bose - Einstein condensate has been proven even to higher order [59] and it can therefore truly be regarded as a single wave with a well defined phase. Output coupling continuous atom lasers from different regions within a BEC and demonstrating their ability to interfere [105] also proved the first order spatial coherence of atom lasers as did the observation of quantum interferences in the atom laser beam profile [106]. The temporal evolution of the condensate wavefunction is given by its internal energy −i µt /ħ e with perturbations stemming from quantum phase diffusion. The experimental investigation of temporal coherence is more difficult than its spatial counterpart because it requires the use of a time delay or a reference phase. Only the first order temporal coherence of atom lasers has been determined to be limited by the duration of output coupling [42] up to 1.5 ms. In the scope of the work presented in this thesis it was possible to proof the full temporal coherence of atom lasers up to several seconds by measuring higher order temporal correlations. 27 2. BASIC THEORETICAL FRAMEWORK 2.3. Cavity Quantum Electrodynamics The elementary system in cavity quantum electrodynamics (QED) is a two-level emitter coupled to a single mode of an electromagnetic resonator [107]. It constitutes the most fundamental instance of light-matter interaction and is a topic of active research in divers fields [2]. Quantum mechanical two-level systems employed range from neutral atoms, ions, atomic Rydberg states and molecules to artificial structures like quantum dots and cooper pair boxes. They can be coupled to confined optical and microwave photons in different types of cavities like Fabry - Pérot resonators, nano-fabricated photonic crystal defects, whispering gallery modes and microwave circuit structures. In the early days of cavity QED it was shown that the quantized electromagnetic field and the correspondingly modified density of states drastically influences the radiative decay time of the excited state of a two-level system. The emission can be enhanced by a number (∼ Q · λ3 /V ) known as the Purcell factor. It is proportional to the quality factor Q of a resonator times the ratio of cubed wavelength λ to the mode volume V . However, the ultimate goal of all these attempts is to increase the coherent atom cavity coupling to exceed the dissipation losses of spontaneous emission and cavity damping. In this so called “strong coupling limit” of cavity QED the coherent exchange of energy between the two-level system and the cavity mode is reversible. It provides the unique possibility to deterministically control the quantum evolution of the system. The following two paragraphs establish the discussion of cavity QED in the strong coupling regime with a single (two-level) atom coupled to an ultrahigh Q optical Fabry - Pérot cavity, which is the system experimentally realized in this work. 2.3.1. Resonator Basics A Fabry - Pérot type cavity is realized by placing two highly reflecting mirrors at a distance L . The electromagnetic field along this axis is quantized by the condition nλ/2 = l for the wavelength λ, where n = {1, 2, 3 . . .} labels the number of nodes of the standing wave. For instance, in our cavity we have 456 nodes of the electromagnetic standing wave. By scanning either the length l of the cavity or the frequency ν of the probe laser the resonance condition can be satisfied and the transmission be maximal. The frequency distance of consecutive cavity resonances is given by the free spectral range νFSR , being the inverse of the round trip time νFSR = c /2l . (2.39) The full width at half maximum (FWHM) ∆ν of each resonance depends on νFSR and the finesse F of the cavity through ∆ν = νFSR /F , 28 (2.40) 2.3. CAVITY QUANTUM ELECTRODYNAMICS free spectral range T laser ∆ν detector n · λ/2 (n+1) · λ/2 FIGURE 2.13.: (left) Schematic drawing of the near planar Fabry - Pérot cavity and the electromagnetic standing wave field inside. (right) Spectrum of the resonator exhibiting narrow Lorentzian cavity lines separated by the free spectral range. where the finesse is a measure that is solely determined by the quality of the cavity mirrors. Since we employ symmetric mirrors of ultrahigh reflectivity R with a transmission coefficient T and a loss factor L on the order of a few ppm satisfying 1 = R + T + L , the finesse can be expressed as p F ' π R π ≈ . 1−R T +L (2.41) It will be dominated by the main leakage of photons out of the cavity, either through transmission or loss channels like scattering, absorption and diffraction. We employ a cavity of finesse F ≈ 350.000 being limited by scattering losses of about L ≈ 7 · 10−6 compared to a transmission coefficient of T ≈ 2 · 10−6 . The finesse furthermore determines the number of reflections (F /π) and the 1/e -lifetime of a photon inside the cavity τc = F l · = (2π∆ν)−1 . π c (2.42) Experimentally, measuring the cavity ring down time τc or the cavity linewidth ∆ν are trusted strategies to determine the finesse of a cavity. However, the length of the cavity has to be determined independently, for example by the mode spacing of higher transverse modes or by simultaneously transmitting two different known wavelengths. Classically, the complete power transmission spectrum through the cavity shown in Figure 2.13 is given by frequency dependent Airy transmission function I (ν) = Tmax 1 + (2F /π)2 sin2 (πν/νFSR ) , (2.43) where the maximum normalized power transmission on resonance for lossy cavities can be calculated considering the complex round trip gain [108]. For a high finesse it approximates a Lorentzian line shape on resonance. Mirror losses reduce the maximum on-resonance transmission to T 2 (1 − L ) Tmax = (2.44) [1 − R (1 − L )]2 29 2. BASIC THEORETICAL FRAMEWORK and therefore also increase the minimal reflection on resonance to a finite value. Only matched mirrors satisfying the input impedance condition R in = R out (1 − L )2 would allow perfect coupling of the laser intensity into and through the cavity. Nevertheless, inside the resonator the intensity is magnified by the factor T /[1−R (1−L )]2 . Comparing the circulating energy over the dissipated energy per cycle yields the so-called quality factor Q . As the name already implies the Q quantifies the losses of the system and can more conveniently be expression as the ratio of the resonance frequency ν0 to the linewidth Q = ν0 /∆ν . (2.45) Since we work at optical frequencies the quality factor can be very large. For a cavity linewidth ∆ν ≈ 2 MHz we get to a Q on the order of 108 . Photon confinement of the cavity mode in the radial direction is provided by the focussing ability of curved mirrors surfaces with radii r i resulting in a stable resonator geometry for 0 ≤ (1 − l /r 1 )(1 − l /r 2 ) ≤ 1. We employ a near planar cavity with symmetric mirrors so the cavityp mode is well approximated to be cylindrical with a Gaussian waist given by p 4 w = l λ/π · r /2l . In our case the radius of curvature r ≈ 77.5 mm of the mirrors is much larger than the length of the cavity l ≈ 178 µm and the waist results to w ≈ 25.5 µm. For near planar cavities different transverse modes (TEM) are not degenerate and their spacing is a practical tool to determine the exact length of a cavity [109]. νl ,m,n = (q + 1) c 1 + (m + n + 1) arccos(1 − l /r ) . 2l π (2.46) These modes can be identified by their transverse mode pattern as shown in Figure 2.15 when observing the transmitted light through the cavity with a camera. (0,0) (1,0) (0,1) (2,0) (0,2) (1,2) (3,0) (0,3) FIGURE 2.14.: Series of pictures taken of different transverse modes excited in our cavity. 2.3.2. Atom Field Interaction With ultrahigh finesse cavities it is possible to advance into the strong coupling regime of cavity QED where the interaction between single quanta of both light and matter significantly alter the behavior of the system. We therefore require the full quantum mechanical 30 2.3. CAVITY QUANTUM ELECTRODYNAMICS description of an atom inside the cavity mode while neglecting its motional degrees of freedom. This was developed by Jaynes and Cummings [66] and constitutes an important quantum mechanical system because it has physical significance and is analytically solvable. The Jaynes - Cummings model consists of a single two-level atom coupled to a quantized ΩR |e〉 |1〉 single mode field represented as a harmonic |g〉 |0〉 oscillator. The coupling between atom and field is characterized by a Rabi frequency ΩR . atom field Loss of excitation in the oscillator appears as a gain of excitation in the atom and vice versa. ¯ Considering a general two-level atom having a ground state ¯g 〉 and an excited state |e〉 separated by the energy ħωA = E e −E g . It is convenient to write the Hamiltonian of the bare atom in its symmetric form as 1 (2.47) Ĥ A = ħωA σz . 2 The Pauli spin matrix σz provides a positive one if it operates on an atom in the upper state ¯ |e〉 and a negative one if it operates on an atom in the lower state ¯g 〉. The quantized field of a single mode is traditionally expressed in terms of the photon creation (â † ) and annihilation (â ) operators. Their product gives the number operator and the bare field Hamiltonian can be expressed as ĤF = ħωC (â † â + 1/2) , (2.48) in analogy to a harmonic oscillator with a set of eigenenergies E n = ħωC (n + 1/2) separated by the photon energy ħωC . The operator for the electric field of the standing wave inside the cavity is then found to be Ê (r) = ²F E 0 (â + â † )χ(r) , (2.49) where E 0 is the maximum electric field established by a single photon in the cavity mode, ²F denotes the field polarization and χ(r) represents the normalized spatial mode function χ(r) = cos(kx ) e −( y 2 +z 2 )/w 2 0 , (2.50) obeying the quantization condition that kl /π is an integer. Here, the x -direction is the symmetry axis of the cavity and z denotes the vertical. By comparing the electromagnetic energy density W = 12 (ε0 E 2 + B 2 /µ0 ) = ε0 E 2 , using E = B /c and c = (ε0 µ0 )−1/2 , with the R energy ħωC of a single photon contained to the cavity mode of volume VC = |χ(r)|2 d r = w 02 l π/4, its maximum field can be simplified to s E0 = ħωC . 2ε0VC (2.51) 31 2. BASIC THEORETICAL FRAMEWORK Clearly, the field strength of a single photon depends on its spatial extent. It can be increased by making the mode volume smaller which is not possible in free space but requires an optical cavity. Quite generally the first order atom field interaction Hamiltonian is given by the coupling of the atomic dipole to the electric field of a photon: Ĥ I = −dˆ · Ê . Introducing the dipole moment d = 〈e|dˆ|g 〉 the dipole operator can be written in terms of the raising (σ̂+ ) ¯ ¯ and lowering (σ̂− ) operators, given by |e〉〈g ¯ and ¯g 〉〈e|, respectively and more conveniently expressed through the Pauli spin matrices σ̂± = 12 (σ̂x ± σ̂ y ). Thus the slightly better looking interaction Hamiltonian is Ĥ I = ħg 0 (r)(σ̂+ + σ̂− )(â + â † ) , (2.52) where we have introduced an important quantity playing the role of a Rabi frequency, the atom field coupling strength g 0 which is given by g 0 (r) = ²A ·²F d E0 χ(r) , ħ (2.53) where the ²’s denote the polarizations of the atom and field, respectively and χ(r) characterizes the field distribution of the cavity mode. It is common to drop the polarization and space dependence and label with g 0 the maximum atom field coupling strength for a given mode volume VC . The complete Jaynes - Cummings Hamiltonian is now given by the sum of the bare atomic and field Hamiltonians plus the interaction term. Commonly, the zero point energy is just an additive constant and disregarded. By applying the rotating wave approximation the energy nonconserving terms σ̂+ â † and σ̂− â can be dropped and the Jaynes - Cummings Hamiltonian consists of two cross terms describing absorption and emission of a photon by the two-level system HJC = 1 ħω A σ̂z + ħωC (â † â ) + ħg 0 (r)[σ̂+ â + â † σ̂− ] . 2 (2.54) The Jaynes - Cummings model is well known in quantum mechanics because it is one of the few systems for which analytical solutions can be derived. ¯ A canonical basis for the unperturbed Hamiltonian are the bare states |e, n〉 and ¯g , n + 1〉 labeling the state of the atom and indexing the number of photons in the closed cavity. However, to describe the coupled system a coherent mixture of the bare states |n, ±, r 〉 = ¯ c i |e, n〉 + c j ¯g , n + 1〉 (called dressed states) is better suited. Since the coupling strength g 0 (r) is in general space dependent, the position is included in the state. In this basis the eigenenergies of the Jaynes - Cummings Hamiltonian result to 1 2 E ± (n ) = ħωC (n + 1/2) ± ħ ΩR (n, ∆, r) , 32 (2.55) 2.3. CAVITY QUANTUM ELECTRODYNAMICS + = z |2,+〉 |3〉 |g,3〉, |e,2〉 |2,-〉 |1,+〉 |g,2〉, |e,1〉 |2〉 |1,-〉 |1〉 |e〉 ωA ωc |0〉 |g〉 ħωAσz |0,+〉 |0,-〉 |g,1〉, |e,0〉 + |g,0〉 ħωC(a†a) = bare states dressed FIGURE 2.15.: Coupling a two-state atom with resonant photons contained in a cavity results p in the Jaynes - Cummings ladder of energy doublets split by the frequency 2ħ n + 1g 0 (r), where the coupling strength g 0 is taken to be spatially dependent as χ(r ). where the Vacuum - Rabi frequency Ω2R = Ω2 + ∆2 contains the resonant n-photon interp action frequency Ω(n, r) = 2 n + 1 g 0 (r) and the detuning ∆ = ωC − ω A . The atom photon interaction results in a splitting of the dressed states by E + − E − = ħΩR being dependent on the spatially varying coupling strength and the detuning. For maximum coupling on resop nance the spacings of the Jaynes - Cummings doublets are given by 2 n + 1 g 0 . The choice of the zero level in the dressed states |0, ±〉 results from the historic wish to let an excited atom interact with the vacuum field of an otherwise empty resonator (n = 0), known as Vacuum Rabi oscillations. The coherent exchange of energy between the atom and the cavity mode leads to an oscillatory behavior of finding the atom in the ground or excited state with a frequency ΩR /2. Plotting the eigenenergies versus the detuning ∆ for maximum coupling exhibits the 33 2. BASIC THEORETICAL FRAMEWORK |g,2〉 E |e,1〉 |1,+〉 |1,-〉 |e,1〉 |g,2〉 |g,1〉 |0,+〉 |e,0〉 |e,0〉 |0,-〉 |g,1〉 |g,0〉 |g,0〉 0 Δ FIGURE 2.16.: Avoided level crossing in the Jaynes - Cummings model for different laser detuning. On resonance the splitting is maximal and the eigenstates have rotated from the bare states into the dressed states. avoided crossing type behavior. The eigenstates rotate from the unperturbed bare states to the dressed states and back. The mixing angle is given by tan θn = Ω/(ΩR − ∆). On resonance θn = π/4 and the dressed states result in ¯ 1 |n, ±, r 〉 = p [|e, n〉 ± ¯g , n + 1〉] . 2 (2.56) The individual properties of atom and field are completely amalgamated and define a new system that is commonly know as the dressed atom or Jaynes - Cummings molecule. 2.3.3. Single Atom Detection The Jaynes - Cummings model considered so far is quite academical in a sense that it applies only to a perfectly closed system which could not be observed. However, all real systems are lossy and interact with their environment which leads to decoherence. Furthermore, in order to perform a quantum mechanical measurement we require a coupling to an external system. The loss mechanisms present in the coupled atom cavity system are the field decay rate κ = ∆ν/2 of the cavity field due to nonideal mirrors and the dipole decay rate γ = Γ/2 of the excited atomic state leading to spontaneously scattered photons out of the cavity mode. A coherent pump rate, in form of a laser field supplies the cavity mode with “fresh” photons to compensate the losses. 34 2.3. CAVITY QUANTUM ELECTRODYNAMICS For a complete description of an open quantum system the influence of the environment has to be taken into account in form of a reservoir. If the interaction with the reservoir is weak it can be treated perturbatively, know as the Born approximation. Under the assumption that the excitations and correlations in the reservoir die out quickly, the system is independent of its past evolution and can be regarded as having no memory in the socalled Markov approximation. The temporal evolution of the open quantum system can be determined by the quantum master equation for its density matrix ρ ρ̇ = i i h ρ, Ĥ + L relax , ħ (2.57) where the operator Lrelax is of the Lindblad form and contains all modes of dissipation [110]. Here, Ĥ represents the total Hamiltonian of the system including a pump term that replenishes dissipated photons in the cavity mode Ĥ = ħ∆ A σ̂+ σ̂− + ħ∆C â † â + ²(â + â † ) + ħg 0 (â σ̂+ + â † σ̂− ) . (2.58) The pump term is proportional to the incident field and to the mirror transmittivity with p ² = κ n . Furthermore, the two detunings between the laser and the atomic transition frequency and between the laser and the cavity resonance have been introduced through ∆ A = ωL − ω A and (2.59) ∆C = ωL − ωC . The Lindblad operator Lrelax contains two dissipation channels: an atom spontaneously emits a photon out of the cavity mode, which this is described by the dissipation operator p p γ σ̂− , or a photon is lost through the cavity mirrors, having a dissipation operator κ â . The relaxation operator thus reads L relax = γ(2σ̂− ρ σ̂+ − σ̂+ σ̂− ρ − ρ σ̂+ σ̂− ) + κ(2âρ â † − â † âρ − ρ â † â ) . (2.60) In general, the stationary solutions of the master equation have to be calculated numerically, for instance with quantum Monte - Carlo methods [111]. However, for weak atomic excitation analytical solutions of two interesting observables have been derived [112, 113, 114]. The expectation value of the intracavity photon number and the mean atomic excitation are given by 〈â † â〉 = ²2 ∆2A + γ2 |Λ|2 and 〈σ̂+ σ̂〉 = ²2 g 02 |Λ|2 , (2.61) respectively with Λ = γκ + g 02 − ∆ A ∆C − i (γ∆C + κ∆ A ) . (2.62) The single atom detection principle relies on the significant change of transmissions of the resonant probe laser by the presence of a single atom in the resonator. Therefore, the 35 2. BASIC THEORETICAL FRAMEWORK T g0 η κ z detector y γ x -g0 0 g0 Δ FIGURE 2.17.: (left) An open cavity containing a single atom resulting in a coupling strength g 0 . The dissipation channels are given by the spontaneous atomic scattering rate Γ = 2γ and the loss rate of photons through the cavity mirrors (2κ). (right) Splitting of the transmission p spectrum by 2 N g 0 for a cavity containing one and two atoms, respectively. The significant reduction of on-resonance transmission is shown compared to the empty cavity resonance. normalized cavity transmission is an essential observable and for the resonance condition ∆ A = ∆C it can be written as " 4κ(γ − i ∆) T = (2∆ + i (γ + κ))2 − 4g 02 N + (γ − κ)2 #2 (2.63) , p where N is the number of atoms in the resonator [19]. The spectrum is split by 2 N g 0 and the on resonance transmission is diminished by the factor (1 + g 02 N /κγ)−2 which becomes significant even for single atoms when g 02 > κγ. However, single atom detection could in principle be attained with bad cavities provided the integration time is long enough. Ultimately, the strong coupling in cavity QED is the regime when an atom exchanges a photon coherently many times with the cavity mode before it is lost due to decoherence through one of the dissipation channels. We can characterize the atom cavity system by the dimensionless parameters of the critical atom number N0 and the critical photon number n 0 which are given by N0 = 2γκ g 02 and n0 = γ2 2g 02 , (2.64) respectively. These are the number of quanta required to significantly alter the atom cavity response. In the strong coupling regime g 0 À {γ, κ} the critical numbers are both less than one, which serves as the definition of strong coupling {n 0 , N0 } ¿ 1. 36 3 Experimental Apparatus “Der Worte sind genug gewechselt, Lasst mich auch endlich Taten sehn.” - Faust, Johann Wolfgang von Goethe Our apparatus overcomes the experimental challenges of integrating an ultrahigh finesse optical cavity into a Bose - Einstein condensation machine with a conceptually novel design as illustrated in Figure 3.1. It provides spacious access to the condensate for divers samples and probes which are modularly integrable on our science platform. This is rendered feasible by means of a nested vacuum chamber design, a high vacuum (HV) enclosure inside the ultrahigh vacuum (UHV) main chamber and a short in vacuo magnetic transport. Two distinct pressure regions are required since the two common stages towards Bose - Einstein condensation, a magneto-optical trap (MOT) for laser cooling and trapping a large number of atoms and evaporative cooling have conflicting requirements on their vacuum environment. We utilize a short magnetic transport [115, 116] to convey the cloud of cold 87 Rb atoms from the MOT to the main chamber, where we perform evaporative cooling to quantum degeneracy. From the Bose - Einstein condensate we output couple a continuous atom laser and direct it into the cavity mode. The ultrahigh finesse optical cavity is integrated on the so-called “science platform” and rests on top of an UHV compatible vibration isolation system which is vital for its stable operation. The cavity is located 36 mm below the BEC and enables us to detect single atoms from a quantum degenerate source. In the following, the modular experimental building blocks of our hybrid BEC and cavity QED apparatus are presented in more detail. This chapter has been published at large in [39]: A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Hybrid apparatus for Bose-Einstein condensation and cavity quantum electrodynamics: Single atom detection in quantum degenerate gases, Rev. Sci. Instr. 77 (2006), 063118. 37 3. EXPERIMENTAL APPARATUS BEC production rig magnetic pump & dispensers transport MOT BEC 10-9 mbar transport atom laser magnetic cavity 10-11 mbar to pumps vibration isolation system science platform FIGURE 3.1.: Schematic sketch of the experimental setup illustrating the nested vacuum chambers, the short magnetic transport, and the “science platform” bearing the ultrahigh finesse optical cavity on top of the vibration isolation system. The atomic cloud captured in the magneto-optical trap (MOT) is transferred through a differential pumping tube into the ultrahigh vacuum region and evaporatively cooled towards quantum degeneracy. We output couple a continuous atom laser from the BEC and direct it to the cavity mode where single atoms are detected. 38 3.1. VACUUM SYSTEM 3.1. Vacuum System The vacuum system presented here is out of the ordinary. It consists of two nested steel chambers, where the higher pressure (high vacuum - HV) MOT chamber is situated inside the lower pressure (ultrahigh vacuum - UHV) main tank. The HV region houses the alkali dispenser source. Both vacuum regions are pumped separately and a differential pumping tube maintains a pressure ratio of 102 . The setup grants multiple optical access for laser cooling as well as for observation and manipulation of the resulting Bose - Einstein condensate. 3.1.1. Main Chamber The objective of the vacuum system shown in Figure 3.2 is to attain an UHV environment at 10−11 mbar for efficient evaporative cooling of the laser precooled atomic cloud. Centerpiece of our vacuum system is the custom-welded, cylindrical main tank of nonmagnetic steel (type 1.4436). It has a diameter of 20 cm and features multiple access (see Figure 3.3) in form of optical grade viewports and electrical feedthroughs with standard CF sealing. The viewports are antireflection coated on both sides. Two custom-made CF 200 cluster flanges cap the main chamber from above and below. The top flange (called “BEC production rig”) features optical and electrical access (see Figure 3.3) since most of the electromagnetic coil configuration is mounted on this flange and placed inside the UHV. In addition a liquid nitrogen compatible feedthrough is supplied for cooling the magnet coils and resistive temperature sensors (PT 100) are used to monitor their temperature. The bottom flange (called science platform) serves as an exchangeable mount for the inclusion of samples and probes into our system. Besides viewports and electrical feedthroughs (see Figure 3.3) to connect to electromagnetic coils, PT100 sensors, and the piezo element of the optical cavity design, it includes a cold finger and a 300 l/s nonevaporable getter (NEG) vacuum pump. The core vacuum pumping is performed by a titanium sublimation pump and a 150 l/s ion getter pump. A right angle valve is included in this pumping section for rough pumping the system. The HV part of the system (see Figure 3.2) connects to the MOT chamber which protrudes into the UHV main chamber and serves as a repository for rubidium atoms. It can be shut off with a gate valve between the MOT chamber and the rubidium dispenser source [117, 118]. The HV region is pumped by an ion getter pump (75 l/s) whose pumping speed can be derated by a rotatable disk inside the tube reducing its conductance. This serves to control the rubidium vapor pressure which is monitored with a wide range pressure gauge. Also a right angle valve is included for rough pumping purposes. Our rubidium repository consists of seven alkali metal dispensers fixed in star shape to the tips of an eight pin molybdenum electrical feedthrough where the center pin serves as the common ground. Beforehand, the conductors were bent by 90 ◦ so that the dispensers aim towards the MOT chamber. Dis- 39 3. EXPERIMENTAL APPARATUS HV valve gauge valve r dispense source uc BEC prod tion rig k main tan latform science p ation Ti-sublim valve UHV p ion pum 150 l/s FIGURE 3.2.: Overview of the complete vacuum system showing the pumping sections for the two nested vacuum regions, high vacuum (HV) and ultrahigh vacuum (UHV), respectively. The overall length is close to 2 m. The main tank offers multiple optical and electrical access and is sealed off by two CF 200 cluster flanges called “BEC production rig” and “science platform”. 40 ion pump 75 l/s 3.1. VACUUM SYSTEM penser operation may be viewed through a viewport mounted from above. The dispensers can easily be exchanged without breaking the ultrahigh vacuum in the main chamber by closing the gate valve between the MOT chamber and the dispenser source. 3.1.2. MOT Chamber The MOT chamber as part of the high vacuum region is situated inside the ultrahigh vacuum main tank as shown in Figure 3.3. However the fact that both pressure regions are well in the molecular flow regime allows for relatively simple sealing techniques. The purpose of the inner chamber is to contain a higher vapor pressure of 87 Rb atoms for an efficient loading of the MOT. Our MOT chamber was milled out of a single block of nonmagnetic steel (type 1.4436). This material was chosen to reduce eddy currents produced by fast switching of the magnetic fields. Bores of 35 mm diameter give optical access for the six pairwise counterpropagating laser beams forming the magneto-optical trap. These bores are sealed off by standard optical grade laser windows (BK 7) with double-sided antireflection coating and clamped to the MOT chamber by stainless steel brackets. At the metal glass interface we use thin (0.2 mm) Teflon rings to protect the windows. Additionally we took precautions in the form of ceramic screens to prevent coating of the windows by the titanium sublimation pump. An additional bore provides the connection of the MOT chamber to the HV pumping section and the dispenser source. This connection is sealed against the UHV main tank with a tight fit stainless steel bushing inside the CF 40 socket. The bushing is tightened to the MOT chamber thereby pressing its circular knife edge into a custom-made annealed copper gasket. A screen to prevent a direct line of sight from the hot dispensers to the center of the MOT is included in the laser cut gasket. The MOT chamber is sandwich mounted between the two magnet coil brackets for the magnetic transport (see Section 3.2) and simultaneously functions as a spacer for the magnet coil assembly. The whole structure is suspended from the top flange by four M8 thread bars and represents our BEC production rig. A differential pumping tube interfaces the MOT chamber with the main tank. It serves for conveying the cloud of cold atoms with the magnetic transport from the MOT into the UHV main chamber. The aluminum differential pumping tube is mounted with a press fit in the MOT chamber and can be exchanged. It has an inner diameter of 6 mm over a length of 45 mm and can maintain a differential pressure of 102 -103 depending on the actual pumping speed in the UHV main chamber. Its conductance for rubidium at room temperature is about 0.3 l/s. 3.1.3. Installation All components of the system were electropolished (the custom-welded parts were pickled afore), cleaned, and air baked at 200 °C before assembly [119]. Additionally, all critical in 41 M8 thread bar MOT axis feedthroughs (liquid) (electrical) BEC + Cavity axis 3. EXPERIMENTAL APPARATUS BEC production rig main tank (UHV) differential pumping tube MOT chamber (HV) BEC axis ● ♦ to pump s ser & dispen gasket, screen & bushing NEG pump MOT axis cavity axis windows (teflon-sealed) science platform FIGURE 3.3.: Section through the UHV system illustrating the realization of the nested chambers design and revealing the details and objectives of the divers optical axes. The position of the BEC and cavity are marked by (•) and (¨), respectively. The high vacuum MOT chamber is suspended from the BEC production rig and sealed by a tight fit bushing against the UHV main tank. The science platform provides space for additional components such as the ultrahigh finesse optical cavity. (Note: For clarity in the illustration the magnet coil configuration (see Figure 3.4) and the optical cavity assembly (see Figure 3.8) are omitted in this figure.) 42 3.2. MAGNETIC FIELD CONFIGURATION vacuo materials such as Stycast 2850 FT and Kapton used for the magnet coil brackets (see Section 3.2), Viton A and Wolfmet utilized for the vibration isolation stack (see Section 3.3) as well as plastic [Teflon, Vespel] and ceramic [Macor, Shapal] parts were externally outgassed by vacuum baking them at 200 °C. The bakeout [120] of the fully assembled system was performed at 120 °C which is the maximum temperature rating of the piezotube used in our optical cavity assembly. The ultimate attainable pressure in the UHV system is 3 · 10−11 mbar. It is measured directly inside the main chamber in close proximity of the magnetic trap for Bose - Einstein condensation. In the HV part we maintain a pressure in the range of 10−9 mbar. 3.2. Magnetic Field Configuration A magnetic transport [115, 116] is a reliable and controlled way to transfer the cold atomic cloud from the MOT to a region of considerably lower background pressure for evaporative cooling. Only an in vacuo magnet coil arrangement in conjunction with nested vacuum domains allows for a short transport design and grants spacious access volume inside the main chamber. However, care must be taken to meet the UHV requirements with the materials chosen for the magnet coil structure. Besides spatial and optical accessibilities the requirement on the magnetic trap is mainly magnetic field stability to enable stable atom laser output coupling. Therefore we employ a magnetic trap in the quadrupole Ioffe configuration [85] (QUIC) because its simplicity allows for a compact design and ensures an easy and stable operation at very low power consumption (∼ 2 W). A magnetic shielding enclosure and additional in vacuo coils for manipulating atoms in connection with the cavity round off the magnetic configuration of the system. 3.2.1. Magnetic Transport The magnetic transport design consists of two partially overlapping electromagnetic coil pairs (called “MOT coils” and “transfer coils”) producing quadrupole potentials and the final QUIC trap coils [Figure 3.5(a)]. The overall potential minimum can be moved over a distance of 82 mm so that the cold atoms in a low field seeking Zeeman state are conveyed from the position of the MOT directly into the final magnetic QUIC trap. The transfer coil pair provides sufficient overlap between the two to achieve a smooth transfer of the magnetic potential without significant heating of the cold atomic cloud. The magnet coils were wound from rectangular copper wire (3 × 1 and 1 × 1 mm2 ) for optimal filling fraction. We choose Kapton film isolated wire which is temperature durable and suitable in the ultrahigh vacuum environment. The coils were integrated in two mirrorinverted, custom-made copper brackets and encapsulated with Stycast 2850 FT, a thermally conductive epoxy. The brackets are slotted in order to suppress eddy currents from switch- 43 3. EXPERIMENTAL APPARATUS towards liquid nitrogen feedthrough MOT coil vibration isolation system (VIS) gradient coil cavity coils (5) offset coil extra windings Ioffe coil Ioffe frame sapphire spacers quadrupole coil gradient coil BEC cavity M8 thread bar differential pumping tube transfer coil window (teflon sealed) transport bracket MOT chamber tight fit bushing gasket & screen ceramic spacer cooling circuit FIGURE 3.4.: Section through the complete assembly inside the main vacuum chamber. It illustrates the arrangement of magnet coils, the inner chamber, and the cavity with respect to each other. Functional units of the magnet coil configuration are the two transport brackets that sandwich the inner chamber and the laterally mounted Ioffe frame (elements between dashed lines). These parts, including the top gradient coil, are fixed to each other and mounted from the top flange. The optical cavity on top of the vibration isolation system, the surrounding coils, and the bottom gradient coil are mounted on the science platform. 44 3.2. MAGNETIC FIELD CONFIGURATION ing the magnetic field. The magnet coil assembly was fixed in a sandwich structure around the MOT chamber and suspended from the top flange by M8 thread bars (see Figure 3.4). The complete assembly including the QUIC trap represents the BEC production rig. A cooling system to remove the heat dissipated by the electromagnetic coils is supplied in form of a copper pipe with 4 mm inner diameter. It is soldered in a loop around each coil bracket and connected to the liquid nitrogen feedthrough. A temperature stabilized recirculating chiller permanently pumps pure ethanol cooled to −90°C through the system. Thereby we maintain a maximum operating temperature below 0 °C. This in turn lowers the power consumption. The surface temperature of the coils is monitored with PT 100 sensors and interlocked to the power supplies. The geometry and arrangement of the magnetic transport coils [Figure 3.5(a)] are dominated mainly by constraints set by the size of the MOT chamber, the required length of the differential pumping tube, and the optical access to the MOT, BEC, and cavity. For instance, the square shape of the MOT coils best achieves a large overlap with the transfer coils while granting optical access to the cavity axis. However, the aspect ratio A /R of the coil separation (2 A ) to the coil radius (R ) could be tuned to a balanced tradeoff between a maximally strong ( A /R = 0.5) and a maximally long ( A /R = 0.87) linear gradient region [86]. Anti-Helmholtz configuration is advantageous for tight confinement and deep trap depths whereas long linear gradients yield large handover regions between two coil pairs. Furthermore, the power consumption of a coil pair for a given field gradient can be minimized by choosing a well matched ratio of axial to radial windings. In order to find an optimum current sequence for the magnetic transport [115] we calculate the magnetic field of the coil configuration analytically and discretize it along the transport axis on a 100 µm grid. The currents needed to transfer the magnetic minimum smoothly from the MOT to the QUIC are then computed numerically in accordance with several constraints. Limited by a maximum available electrical current we optimized the magnetic field gradients and trap depths especially during the handover. Furthermore we tried to minimize deformations of the trapping potential. The resulting spatial sequence of currents per coil is converted into a temporal sequence including an acceleration and deceleration phase by taking into account the limited bandwidth of the current control servo [Figure 3.5(b)]. The magnetic transport sequence initiates with a fast (400 µs) ramp to 20 A in the MOT coils after magneto-optical trapping and optical pumping the cold atoms into a lowfield seeking state. The ramp needs to be fast with respect to the expansion of the cloud but adiabatic on the spin degree of freedom. It is followed by a slow (100 ms) compression of the atomic cloud to the maximum field gradients. Increasing the current in the transfer coils pulls the atoms towards their center and by decreasing the MOT coil current the zero of the potential is handed over [Figure 3.5(b)]. The field of the QUIC is aiding at this point to maintain a constant aspect ratio. The magnetic transport finishes by ramping down the current through the transfer coils in favor of the QUIC coils. The atomic cloud is conveyed 45 3. EXPERIMENTAL APPARATUS (a) M OT co ils Transfer coils ls IC coi QU current [ A ] (b) 180 160 140 120 100 80 60 40 20 0 MOT Transfer QUIC 0 0.2 0.4 0.6 time [ s ] 0.8 1 FIGURE 3.5.: (a) Top view of the arrangement of coils for the magnetic transport. The line denotes the trajectory of the atomic cloud from the MOT (filled circle) into the QUIC trap. (b) Temporal sequence of currents through the different coils to realize the compression of the cold atomic cloud (negative times) and the magnetic transport. through the differential pumping tube directly into the magnetic QUIC trap which stays on for the subsequent evaporative cooling stage. The trajectory of the magnetic transport [Figure 3.5(a)] is slightly bent such that atoms in the final magnetic trap position have no direct line of sight into the higher pressure MOT chamber. This suppresses background gas collisions which would shorten the lifetime of the Bose - Einstein condensate. The bend is achieved by laterally offsetting the center of the QUIC trap by 3 mm from the differential pumping tube. The MOT and transfer coils are powered by a general purpose interface bus (GPIB) controllable 5 kW dc power supplies. However, since their internal current control bandwidth is too slow to sample the time-current sequence for the MOT coils we externally feedback control it by a closed-loop servo. It is implemented with a current transducer and a MOSFET bench. The fast initial ramp to 20 A is additionally supported by current from four large 46 3.2. MAGNETIC FIELD CONFIGURATION capacitors (1 mF) charged to 60 V. The electromagnetic properties of the coils with resulting maximum currents and field gradients are listed in Table 3.1. The magnetic transport is performed over a period of 1 s. We maintain a minimum trap depth of ∼ 70 G equivalent to about 2 mK. The total power required is approximately 2 kW which corresponds to an average power consumption of ∼ 34 W at a duty cycle of 1/60. Resistance [ mΩ ] Inductance [ µH ] Maximum current [A] Maximum field gradient [ G/cm ] MOT Transfer QUIC 200 50 300 1000 70 450 115 170 15 310 290 320 TABLE 3.1.: Electromagnetic Properties of the Magnet Coils 3.2.2. QUIC Trap The magnetic QUIC trap consists of three coils connected in series. This is advantageous to diminish relative current fluctuations and therefore magnetic field fluctuations. Two coils (called “quadrupole coils”) produce a quadrupole field and one smaller coil (called “Ioffe coil”), mounted orthogonally between the quadrupole coils lifts the magnetic zero to a finite value and adds a curvature to the resulting potential [85]. Having a nonzero magnetic minimum is crucial when evaporatively cooling atoms towards quantum degeneracy in order to circumvent losses due to Majorana spin flips. The geometry of the QUIC trap potential is approximately cylindrically symmetric with respect to the Ioffe coil axis. Along this direction the curvature and therefore the confinement is weaker than in the radial directions. In our case this results in cigar shaped Bose Einstein condensates with an aspect ratio of 5:1. The exact position and dimension of the Ioffe coil are very critical to yield the desired magnetic bias field B 0 which should be on the order of a few Gauss. A low bias field is preferential because the trap frequencies scale as p B 0 / B 0 , where B 0 is the magnetic field gradient and high trap frequencies permit faster and more efficient evaporative cooling. The construction of the Ioffe coil is done in the same way as for the transport coils. It is integrated in a slotted copper frame and potted with Stycast. The Ioffe frame is mounted laterally between the transport coil brackets which hold the quadrupole coils. Additionally, the Ioffe frame serves as a spacer for the two transport brackets. The mechanical contact is accomplished with sapphire sheets in order to prevent eddy currents by simultaneously maintaining good thermal conductivity (see Figure 3.4). The large mass of the complete magnet coil structure functions as a thermal low pass filter which contributes to the good temperature stability. 47 3. EXPERIMENTAL APPARATUS In the Ioffe frame we have integrated additional coils on the same axis as the Ioffe coil to be able to manipulate the final trap geometry inside the vacuum system after bakeout. Two few-winding coils are employed to fine-tune the value of the magnetic field minimum B 0 . One larger coil (called “offset coil”) a little further away from the trap center can be used to change the aspect ratio of the trap and make it approximately spherical. Furthermore, the Ioffe frame features a conical bore which allows us to image the BEC through the center of the Ioffe coil. The electrical connections of the coils forming the magnetic QUIC trap are realized outside the vacuum. We have included a 1.4 MHz low pass filter in parallel to the Ioffe coil to avoid any radio frequency (rf) pickup because of its low inductance of 4 µH. The QUIC trap is operated with a 150 W power supply specifically tuned to our inductive load. The average power consumption of the magnetic trap is maximally 60 W but can be as low as 2 W when operated at 3 A. 3.2.3. Magnetic Shielding We clad the main vacuum chamber in a mu-metal shielding depicted in Figure 3.6 to minimize the influence of residual external magnetic field fluctuations on the cold atoms. A magnetically quiet environment is essential for stable continuous wave (cw) operation of the atom laser. Mu-metal is a magnetically soft nickel alloy with a very high magnetic permeability µ ∼ 105 which attenuates magnetic fields inside a cohesive enclosure. The screening effect depends very much on the completeness of the mu-metal box. Magnetic field lines penetrate an opening roughly as far as its diameter. Therefore we have attached a stub around the pumping tube of the main vacuum tank to attain a better aspect ratio at the position of the BEC. The design of the mu-metal hull was aided by computer simulations of the electromagnetic field. The mu-metal was machined and cured as recommended by the manufacturer. After demagnetization we have measured a dc magnetic extinction ratio of ∼ 40 in the vertical and ∼ 100 in the horizontal direction at the position of the BEC. 3.2.4. Auxiliary Coils Since the mu-metal shielding prevents any manipulation of the atoms with external magnetic fields, we have arranged supplementary magnet coils inside the mu-metal enclosure. All extra coils were potted with Stycast either in a slotted copper or Shapal frame for good thermal conductivity and mechanical sturdiness. Two large coils (called “gradient coils”) are included in the main vacuum chamber to compensate the gravitational force for the weakest magnetic sublevel (30.5 G/cm) with 22 A. Their total resistance and inductance is about 0.2 Ω and 0.9 mH, respectively. The gradient coils were mounted inside the vacuum chamber on the transport bracket (see Figure 3.4) and on the science platform around the cavity (see Figure 3.8), respectively. With the latter we should be able to reach the widest Feshbach resonance of 87 Rb (∼ 1008 G) [121] at the 48 3.3. SCIENCE PLATFORM AND CAVITY SETUP μ-metal hull FIGURE 3.6.: Photograph of the preassembled mu-metal hull before it is mounted around the main vacuum tank. It consists of seven large and several small individual pieces. position of the cavity. Around the cavity we have placed two pairs of tiny coils (4 Ω, 0.4 mH) along and perpendicular to the cavity axis (see Figure 3.4). They can be used to create magnetic field gradients of about 200 G/cm (with 1 A) for tomography experiments. In combination with a fifth tiny coil (1 Ω, 0.1 mH) mounted above a magnetic trap at the position of the cavity can be formed. These five small coils (called “cavity coils”) were wound from 0.04 mm2 Kapton isolated copper wire on Shapal frames to be penetrable by radio frequency. In addition to the magnet coils inside the vacuum tank we have wound three mutually orthogonal pairs of large extra coils around the main tank. However, they are still within the mu-metal hull and serve to produce homogeneous magnetic fields, e.g., for optical pumping. 3.3. Science Platform and Cavity Setup We have designed this apparatus with attention to versatile access for samples and probes to the BEC. Therefore we have implemented two independent sections of complementary functionality, i.e., the BEC production rig (see Sections 3.1 and 3.2) and the science platform. The latter is a modular, interchangeable flange, which in the current configuration supports our single atom detector in form of the ultrahigh finesse optical cavity. 49 3. EXPERIMENTAL APPARATUS The design of the cavity was guided by the need for stability, compactness, and ultrahigh vacuum compatibility. It rests on top of a passive vibration isolation stack which can be positioned on the science platform. 3.3.1. Cavity Design The Fabry - Pérot optical cavity is formed by two dielectric Bragg mirrors of ultrahigh reflectivity and ultralow scattering losses . The reflection band is 40 nm wide and centered around 780 nm. We have determined an ultimate quality factor Q = 1.6 · 108 after bakeout from the linewidth of the cavity (∆ν = 2.4MHz). The initial Q immediately after cleaning the mirrors was higher by about a factor of 2. The cylindrical mirrors (3 mm diameter, 4 mm length) having a radius of curvature of 77.5 mm are separated by 178 µm which results in a Gaussian mode waist of w0 = 25.5µm. We precisely measured the length of the near planar cavity by simultaneously transmitting two different known wavelengths (see Section 4.1.3) and determined a free spectral range of νFSR = 0.84 THz from which we derive a finesse of F = 3.5 · 105 . Each mirror was bonded with superglue into a specifically fabricated ceramic (Shapal) ring structure. It positions and fixes the mirror inside the piezoceramic tube [113]. A piezo is required to fine tune the length of the cavity (∼ 0.5V/nm) and as the actuator for the cavity lock (see Section 4.1.3). The 7 mm long piezotube has inner and outer diameters of 5.35 mm and 6.35 mm, respectively. It is equipped with nonmagnetic wraparound electrodes (silver) which allows the inner electrode to be contacted from the outside. Additionally, the piezotube features four radial holes of 1 mm diameter for lateral access of atoms and lasers perpendicular to the cavity axis. The cavity assembly is mounted by a specifically designed compact fixture (called “clamp”) making use of mechanical joints [Figure 3.7(a)]. It was manufactured by spark erosion from titanium in order to be nonmagnetic while having good elastic properties. Further design considerations aimed at high mechanical eigenfrequencies to avoid resonances within the bandwidth of the cavity lock (∼ 40kHz), that means a small size and high stiffness are favorable. We estimate the lowest eigenfrequency of our fixture with a simple mechanical fixed-hinged beam model [122] to be ∼ 50 kHz. Our design of the cavity mount consists of the t-shaped clamp and a baseplate with integrated bearings to which the clamp is tightened with a plate nut. It converts the downward force onto the cavity assembly and firmly holds it together. Moreover it provides the piezo with a load. A hole of 1.2 mm diameter in the baseplate and plate nut grants optical access to the cavity from below. This cavity setup is highly modular and easily interchangeable because it freely rests on the vibration isolation stack [Figure 3.7(b)]. 50 viton baseplate -120 -80 -40 0 0 200 400 800 (d) 1 cm frequency [ Hz ] 600 transfer function [ dB ] (b) science (c) keel platform support vibration isolation stack FIGURE 3.7.: Elements of the optical cavity implementation. (a) Plane cut through the assembled cavity design, where the red arrows indicate optical access. (b) Photograph of the cavity setup. The electrical leads for the piezotube are pinched in a slotted Viton piece to efficiently decouple the cavity from the environment. (c) The cavity assembly resting on top of the vibration isolation stack which is positioned on the science platform. (d) Modeled frequency response of our vibration isolation stack. scale 1:1 joint platenut & screw piezo tube clamp holder mirror (a) 3.3. SCIENCE PLATFORM AND CAVITY SETUP 51 3. EXPERIMENTAL APPARATUS 3.3.2. Vibration Isolation System The aforementioned baseplate simultaneously acts as the top mass of our vibration isolation stack [123] which consists of five layers of massive plates (Wolfmet) with rubber dampers (Viton A) in between [Figure 3.7(c)]. Viton has good vibration damping properties and is suitable for an ultrahigh vacuum environment. The 5 mm diameter Viton pieces rest in hexagonal grooves that are radially arranged in 120 ◦ graduations. Consecutive layers are rotated by 60 ◦ to prevent a direct “line of sound”. Hexagonal shaped grooves best avoid squeezing and creeping of the rubber and provide good lateral stability. Position, angle, and tilt reproducibility of this structure are excellent because of the frustum shaped bottom mass with keel. It centers the stack in an inverted, truncated conelike support and assures mechanical stability by lowering the center of mass below the support points. The complete stack has a central 10 mm bore for vertical optical access to the cavity. Its damping properties can be modeled by regarding the structure as a system of coupled masses and springs [124] and calculating its frequency dependent transfer function [Figure 3.7(d)]. For attenuation at low frequencies large masses and small spring constants are favorable [125, 126]. Therefore we have fabricated the plates from a heavy metal alloy (Wolfmet) and employed short (10 mm) Viton pieces. Our vibration isolation stack works well for acoustic frequencies above 200 Hz. Additional precautions to counter low frequency excitations such as building vibrations include setting up the experiment on a damped rigid optical table in a basement laboratory having its own independent foundation and choosing a position with little floor vibration within this laboratory. The quality of the vibration isolation system is such that we could easily operate the cavity in the vicinity of a turbo-molecular pump. Furthermore the vibration isolation stack kept the cavity in place when the whole optical table accidentally dropped by about 2 cm as we tried to tilt it. 3.3.3. Science Platform Layout The self-contained, interchangeable science platform flange was prepared to support and align the complete cavity mount. Its layout provides manual positioning ability of the cavity mount by ± 2 mm along and perpendicular to the cavity axis, respectively. This is rendered feasible by an octagonal support (nonmagnetic steel) of the vibration isolation stack which can be deterministically moved and fixed in a larger octagonal millout on the flange. The second objective of the support is to erect the arrangement of cavity coils with the gradient coil (see Section 3.2.4) to be positioned around the cavity without direct contact. The coil assembly is mounted on two nonmagnetic steel sustainers which are fixed to the vibration isolation support. In order to remove the dissipated heat by the electromagnetic coils, we have connected the copper bracket of the gradient coil to a power feedthrough serving as a heat bridge, i.e., cold finger. Outside the vacuum the 19 mm diameter copper conductor 52 3.3. SCIENCE PLATFORM AND CAVITY SETUP can be connected to the cooling circuit and cooled to −90°C. The copper rod serves as a the main drain for the heat because of the low thermal conductivity of the steel sustainers and support. The mounting of the independent BEC production rig and science platform within the main vacuum chamber has to be noncontact but within a fraction of a millimeter. This results in a final position of the optical ultrahigh finesse cavity being 36.4 mm below the BEC. The orientation of the cavity axis is at 90 ◦ with the symmetry axis of the magnetic trap (Ioffe axis). cavity magnetic coil structure sustainer cold finger NEG pump VIS MOT viewport support electrical feedthrough FIGURE 3.8.: Photograph of the mounted science platform. The support bears the vibration isolation system (VIS) and the magnetic coil structure which surrounds the optical cavity. 53 4 Characterization of the System “Durch diese hohle Gasse muß er kommen.” - Wilhelm Tell, Friedrich Schiller The features of the unique apparatus developed within the scope of this thesis are explored and characterized. For the first time quantum degenerate gases and cavity QED are brought together experimentally. The apparatus proves to be a very robust and reliable BEC machine for delivering high atom numbers 87 Rb Bose - Einstein condensates (∼ 2 · 106 ) with the experimental procedure described below. We can produce extremely stable atom lasers from a very well defined spatial region within the condensate over long timescales (∼ tens of seconds) thanks to the magnetic shielding in combination with superior mechanical and thermal stability. The precisely controlled atom laser output coupling is characterized in this chapter. Furthermore, we present the locking scheme of the ultrahigh finesse optical cavity and describe the best parameters for single atom detection. The cavity functions as a linear single atom detector of atom fluxes spanning three orders of magnitude. We explain how we aim the atom laser into the cavity mode and show measurements of the atom laser beam profile. The threshold behavior of the atom laser is observed and we utilize the atom laser as a bright and reproducible source of cold atoms for investigations of cavity QED effects. The high detection efficiency of degenerate atoms (∼ 25%) enables us to inspect ultracold atomic clouds noninvasively and therefore time resolved. Parts of this chapter have been published in [39]: A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Hybrid apparatus for Bose-Einstein condensation and cavity quantum electrodynamics: Single atom detection in quantum degenerate gases, Rev. Sci. Instr. 77 (2006), 063118. 55 4. CHARACTERIZATION OF THE SYSTEM 4.1. Experimental Procedure We operate the experiment periodically with a cycle time of 60 s. During each cycle we produce a new BEC from which we output couple an atom laser. It is directed to the high finesse optical cavity situated 36.4 mm below the BEC where single atoms are detected. The cavity is probed by a resonant laser and its length is actively stabilized by an off resonant laser with respect to the atomic transition. The experimental sequence is fully computer controlled by a C++ program. Digital and analog channels interface the computer with the elements of the experimental setup. The experiment is distributed on two self contained optical tables, one for the laser system and one for the vacuum apparatus. They are linked by optical fibers. 4.1.1. Bose - Einstein Condensation We form a Bose - Einstein condensate of 87 Rb in dilute atomic vapor from a dispenser loaded magneto-optical trap by means of rf-induced evaporative cooling [79, 116]. During the first 20 s of each cycle we load the magneto-optical trap with atoms from the pulsed alkali dispenser source [118]. The dispensers are operated at ∼ 7A with a temporal offset of −3s to the actual MOT phase. We work on the D2 line of 87 Rb (5 2 S1/2 → 5 2 P3/2 ) at a wavelength ¯ λ = 780nm. For the cooling transition on the hyperfine ground state |F = 2〉 ↔ ¯F 0 = 3〉 a laser power of 17 mW is employed in each of the six 34 mm diameter MOT beams. For optimum collection efficiency we choose a detuning of 3Γ, where Γ = 2 π· 6 MHz is the linewidth of the cooling transition. In order to be frequency tuneable the laser is offset locked [127] ¯ from the |F = 2〉 → ¯F 0 = 2〉 transition by about 250 MHz and subsequently amplified with a tapered amplifier. An additional laser (called “repumper”) to avoid atomic losses into ¯ the |F = 1〉 dark state is directly locked to the |F = 1〉 → ¯F 0 = 2〉 transition and delivers a power of 1 mW in each MOT beam. All our lasers are home-built external cavity diode laser [128] locked by Doppler-free rf-spectroscopy technique [129] to atomic transitions. For the magneto-optical trap we apply a magnetic field gradient of 10 G/cm by applying a current of 3.5 A to the MOT coils. Due to the mu-metal shielding no earth field compensation is required. We collect about 2 · 109 atoms with the magneto-optical trap before we switch off the magnetic field and sub-Doppler cool the atoms in a 10 ms optical molasses phase. Before magnetically transporting the cold atomic cloud we optically pump the atoms into the low field seeking |F = 1, mF = −1〉 hyperfine state. Optical pumping is performed over 2 ms at a homogeneous magnetic field of ∼ 4 G. All light fields are off when the transport sequence starts with adiabatically compressing the cloud [Figure 3.5(b)]. The magnetic transport conveys the atoms within 1 s through the differential pumping tube over a distance of 82 mm directly into the magnetic QUIC trap. We estimate a transport efficiency of > 90% by transferring the atoms back into the MOT and measuring their fluorescence. The losses are mainly due to background collisions and depend on the pressure in the MOT chamber. 56 4.1. EXPERIMENTAL PROCEDURE We operate the magnetic QUIC trap initially with a maximum current of 15 A. This yields the highest trap frequencies of ωx = ωz = 2π· 135 Hz and ω y = 2π· 28 Hz with a bias field B 0 of 4.7 G and a field gradient B 0 of ∼ 300G/cm. Here ω y and ωx denote the trapping frequencies along and perpendicular to the Ioffe axis, respectively and ωz is in the vertical direction. Over a period of 23 s we perform rf-induced evaporative cooling with an exponential frequency ramp and a radio frequency power of 24 dBm. The radio frequency is radiated by a coil which consists of ten turns of Kapton clad copper wire (1 mm2 ) encircling an area of 3 cm2 . It is mounted 2 cm away from the center of the trap and is oriented at 90 ◦ with respect to the Ioffe axis. This results in a B rf of about 30 mG at the position of the cold atoms. (a) (b) (c) 100 μm FIGURE 4.1.: Absorption images of cold atom clouds, (a) thermal cloud at a temperature T above the critical temperature Tc , (b) bimodal distribution for T < Tc , and (c) “pure” Bose Einstein condensate at T ¿ Tc , taken after 30 ms time of flight with a detuning of 2Γ to avoid saturation. Before reaching the critical phase space density for Bose - Einstein condensation we relax the trap to the final parameters of ωx = 2π· 38.5 Hz, ω y = 2π· 7.3 Hz and ωz = 2π· 29.1 Hz with B 0 = 1.2G and B 0 = 60G/cm by powering the QUIC trap with 3 A. The initial trap symmetry is lifted by the large gravitational sag of about 290 µm. It is given by z sag = −g /ω2z , where g is Earth’s gravitational acceleration. Furthermore, the long axis of the BEC is inclined by about 20◦ with respect to the horizontal plane. The opening of the trap is performed adiabatically (ω̇/ω ¿ ω) over a period of 1 s. During this time a rf shield limits the trap depth to prevent heating of the cold atomic cloud. In the weak trap we further cool the atoms evaporatively over 5 s and achieve pure Bose - Einstein condensates of up to 3 · 106 atoms. The density in the weak trap is considerably lower so the losses due to inelastic collisions are reduced. We have measured a 1/e -lifetime for condensates of about 30 s. The typical size of the Bose Einstein condensate is 12 × 15 × 60µm3 (Thomas - Fermi radii) with a chemical potential µ of about 1 kHz. Resonant absorption imaging of the cold atoms after a free expansion time of 30 ms allows us to extract the number of atoms in the cloud and its temperature. We fit the resulting density distribution with the sum of a Gaussian and a Thomas - Fermi profile. The spatial resolution of our imaging system ( f /10) is limited to 9 µm by the diameter of the windows. We employ a charge-coupled device (CCD) camera with an according pixel size. 57 4. CHARACTERIZATION OF THE SYSTEM 4.1.2. Atom Laser Output Coupling An atom laser is a coherent atomic beam extracted from a Bose - Einstein condensate. The trapped condensate, being in a quantum degenerate state, serves as the source for the freely propagating atom laser. A steady-state output coupling process establishes a coupling between the ground state of the trap and the energy eigenfunctions of the linear gravitational potential and produces a continuous wave (cw) atom laser. The resulting cw atom laser [11], in contrast to optical lasers, consists of interacting massive particles propagating downwards in the gravitational field. But alike optical laser it is a matter wave in a coherent state as defined by Glauber in the quantum theory for optical lasers [40] and exhibits higher order coherence [17]. In order to output couple atoms we locally change their internal spin state from microwave antenna the magnetically trapped |F = 1, mF = −1〉 into the untrapped |F = 2, mF = 0〉 hyperfine state. The spin flip is induced by a coherent microwave field at the hyperfine splitting frequency of 87 Rb (∆E hfs /h = 6.8 GHz) [130]. This microwave output coupling scheme is equivalent to a two-level system because of the Zeeman splitting in the hyperfine niveaus (∼ 1MHz). Therefore it is superior to rf output coupling CF 40 which mutually couples all states from a FIGURE 4.2.: Helix antenna built for Zeeman manifold [102]. The microwave sigatom laser output coupling at 6.8 GHz, nal is produced by a global positioning sysmounted on a vacuum feedthrough. tem (GPS) disciplined synthesizer. We use a home-built resonant helix antenna with 14 dB gain (see Figure 4.2) placed inside the ultrahigh vacuum chamber to radiate the microwave field. The antenna is connected and impedance matched to a commercial microwave feedthrough. The energy conservation for the microwave output coupling resonance condition is only satisfied at regions of constant magnetic field where ∆E hfs −g F mF µB B (r) = hνmw . Here νmw is the microwave frequency, B (r) the magnetic field of the trap at position r and µB the Bohr magneton. The hyperfine Landé g factor g F and the magnetic spin quantum number mF apply to the BEC state. The magnetic moment of the output coupled atoms is zero to first order. The resonant regions for output coupling are ellipsoidal shells with the geometry of the magnetic trap, centered at the minimum of the magnetic potential. However, the center of the actual harmonic trapping potential for massive particles is lowered by the gravitational 58 4.1. EXPERIMENTAL PROCEDURE 100 μm gravity atom laser sag with respect to the magnetic field minimum. For our experimental conditions the BEC gravitational sag is ∼ 290 µm and therefore the resonant output coupling shells intersect the Bose - Einstein condensate almost as horizontal planes. The Rabi frequency Ω of the microwave output coupling process is given by µ12 B mw /ħ, where µ12 is the magnetic dipole matrix element between the two coupled states and B mw the magnetic field of the microwave radiation [131]. The magnetic dipole transition has selection rules ∆mF = ±1. In the weak output coupling regime (Ω ¿ ωz ) an atom leaves the condensate much faster than the Rabi frequency and does not undergo Rabi oscillations [132]. The atom laser output coupling rate depends on the number of atoms in the condensate NBEC and the overlap | 〈ΨBEC |ΦE 〉 |2 between the BEC wave function ΨBEC and the energy eigenfunction ΦE of the free atom laser [99, 93]. For given atom number NBEC and microwave frequency the output coupling rate is proportional to Ω2 and therefore to the incident microwave power [98]. Producing a coherent cw atom laser crucially depends on the temporal stability of the resonance condition. We take experimental care to avoid any fluctuations FIGURE 4.3.: Microwave output coupling of a continuous atom laser. Resonant abor drifts of the magnetic resonance posisorption image with (left) and without tion. A temperature controlled cooling cir(right) repumper to image the remaining cuit for the large mass magnet coil structure |F = 1〉 BEC. The atom laser propagation and a GPS locked synthesizer permit excelis ∼ 2 mm. lently reproducible conditions. The magnetic shielding enclosure together with the hermetic steel vacuum chamber eliminate external electromagnetic field fluctuations (see Sections 3.1.1 and 3.2.3). The only detectable noise source is the low noise current supply powering the magnetic QUIC trap. We have measured a magnetic field stability of better 59 4. CHARACTERIZATION OF THE SYSTEM p than 5µG/ Hz (at 3 kHz) or 50 µG overall (bandwidth: 50 kHz), respectively. This enables us to produce second order coherent atom lasers and output couple a cw atom laser over the duration of the BEC lifetime. Due to the extremely low atom fluxes measurable with the cavity detector we do not have to deplete the condensate significantly. The atom laser freely propagates downwards for 86.1 ms before entering the high finesse optical cavity where single atoms are detected. The cavity is placed 36.4 mm below the BEC which results in a velocity of 0.84 m/s for the atoms traversing the cavity mode. This velocity corresponds to a de Broglie wavelength of about 5 nm which could be useful for applications in coherent atom lithography [133] or as an atom laser microscope [134, 135]. 4.1.3. Cavity Lock In order to engage the optical high finesse cavity as a single atom detector we have to stabilize its length to better than 0.5 · λ/F ≈ 1pm with respect to the wavelength of the probe laser. We choose a cavity locking scheme [136] that allows us to independently adjust the frequencies of the cavity resonance (ωc ) and of the probe laser (ωl ). Furthermore it enables us to keep the cavity permanently locked even during atom detection since the action of a single atom transit on the far-detuned stabilization laser is negligible and vice versa. The cavity lock is realized with a far-detuned master laser at 830 nm and a resonant master laser at 780 nm referenced to a 87 Rb line. They are frequency stabilized by means of Pound - Drever - Hall locks [137] to a transfer cavity having a free spectral rage νFSR of 1 GHz. In order to be freely tunable the actual probe and stabilization slave lasers are phase locked [138] with a frequency offset of 0–500 MHz to their respective master lasers. The length of the science cavity is then actively controlled by a Pound - Drever - Hall lock on the stabilization slave laser with a bandwidth of 38 kHz. We create the necessary sidebands for the lock with a home-built electro-optical modulator [139]. It works at 362 MHz to have the sidebands well off resonant with the cavity because its finesse for 830 nm is 3.8·104 and therefore its linewidth 22 MHz. We actively control the incident powers of the stabilization and probe laser on the cavity to about 2 µW and 3 pW, respectively. In order to have a good spatial overlap, the two lasers are guided through the same optical fiber. Their power ratio of 10−6 is realized with an optical color filter. We can couple about 25% of the incident probe laser power into the cavity TEM00 mode being limited by the nonoptimal impedance matching. The atomic resonance (ωa ) we employ for single atom detection is the cycling transi¯ tion |F = 2〉 → ¯F 0 = 3〉 of the D2 line of 87 Rb. It yields a maximum atom field coupling rate g 0 = d iso |Emax | = 2π · 10.4 MHz, were we have assumed an isotropic dipole q matrix element ¡ ¢ [130] diso and a maximum single photon electric field strength |Emax | = 4ħc / ²0 λw20 l according to our mode volume with a beam waist w0 = 25.5 µm and a cavity length l = 178 µm. The atom field coupling rate g 0 is large compared to the dissipation losses being the cavity 60 4.2. SINGLE ATOM DETECTION PERFORMANCE FM PDH transfer cavity 780 nm slave 830 nm master 830 nm slave PDH stabilization probe frequency PDH PLL PLL 780 nm master cavity locking chain Rb science cavity FIGURE 4.4.: Schematic diagram of the cavity locking scheme employing a transfer cavity to bridge the large frequency difference between the resonant probe laser (780 nm) and the far off resonant stabilization laser (830 nm). Phase locks (PLL) to the master lasers provide full and free tuning flexibility of the probe laser. The cavity locks are Pound - Drever - Hall locks (PDH) and the lock to the rubidium reference is realized with a frequency modulation (FM) lock. field decay rate κ = 2π · ∆ν/2 and the dipole decay rate γ = Γ/2 where Γ = 2π · 6.1MHz is the natural linewidth of the excited state. Furthermore the inverse atom transit time τ−1 is orders of magnitude smaller than the coupling rate which means the atom is always in a quasi steady state with the cavity field during the transit. The relevant parameters of our experiment are thus (g 0 , γ, κ, τ−1 ) = 2π· (10.4, 3.0, 1.2, 3·10−3 ) MHz, which brings us into the strong coupling regime of cavity QED defined by g 0 À (γ, κ, τ−1 ). 4.2. Single Atom Detection Performance Single atom detection with an ultrahigh finesse optical cavity [20] can heuristically be viewed as the refractive index of a single atom being sufficient to significantly shift the cavity resonance. Consequently, the transmission of an initially resonant, weak probe laser is measurably reduced. In quantum mechanical terms the coupling of a single atom with the quantized electromagnetic field in the cavity mode dominates the dissipation losses (strong coupling regime) which means the level splitting of the Jaynes - Cummings model [66, 140] 61 4. CHARACTERIZATION OF THE SYSTEM can be resolved. On the other hand, the quantum mechanical detection process on the longitudinally delocalized atom within the atom laser beam projects and localizes them inside the cavity mode [33]. We can efficiently study these cavity QED interactions of single atoms having an atom laser as an unprecedented bright, controllable, reproducible, and well defined atom source. Here we present experimental results that characterize the performance of our combined BEC and ultrahigh finesse optical cavity system. 4.2.1. Signal Analysis In order to identify single atom transits we record the transmission of a resonant weak probe beam through the cavity with a single photon counting module (SPCM). A typical recording showing single atom transits is presented in Figure 4.5. The light coming from the cavity is filtered with a 780 nm bandpass and a 830 nm notch filter to block the stabilization laser. Their combined relative optical density (OD) for 830 nm is 12. The SPCM is located inside a black box and exhibits an overall photon dark count rate of ∼ 100s−1 . cavity transmission [ photons / 20 μs ] 120 100 80 60 40 20 0 1.645 1.650 1.655 time [ s ] 1.660 1.665 FIGURE 4.5.: Cavity detection recording of an atom laser. The atom flux is about four orders of magnitude lower compared to Figure 4.3. Single atom transits are clearly identified by their reduction of the shot noise limited empty cavity transmission. The cumulative detection probability for intracavity probe photons taking into account losses in the optical system and the quantum efficiency of the SPCM is about 7%. It is mainly limited by the fact that we employ symmetric cavity mirrors with equal transmittivity (∼ 2ppm) and by the scattering losses (∼ 7ppm) of the mirrors. In order to achieve a large signal-to-noise ratio for single atom detection we usually work with an average intracavity probe photon number of about 5 [33]. This level corresponds to an intensity of about 40 times the saturation intensity and yields a photon count rate of 2π·∆ν· 5 · 7% ≈ 5 photons/µs. We integrate the signal from the SPCM over 20 µs with a temporal resolution between 1-4 µs 62 4.2. SINGLE ATOM DETECTION PERFORMANCE and set the criterion for single atom detection events to a reduction of more than four times the standard deviation (σ) of the shot noise limited empty cavity transmission. This reduces false atom detection events to less than 0.5s−1 . 4.2.2. Characteristics of Single Atom Events The coupling of a single atom with the cavity mode can be characterized by the magnitude and duration of the resulting transmission dips. A recorded typical single atom transit is shown in Figure 4.6(a). The 4σ threshold here corresponds to about 50% reduction in the probe light transmission of about 70 photons/µs. We analyze detected events and show histograms in Figures 4.6(b) and 4.6(c) for atom laser data taken in 184 iterations of the experiment. The atom flux was set to ∼ 1 · 103 s −1 so the probability [17] for unresolved multiatom events within the dead time of our detector (∼ 70 µs) is less than 0.3%. The dead time of our detector is reflected in the distribution of coupling times, i.e., the full width half maximum (FWHM) of the transmission dips [Figure 4.6(b)]. It is mainly determined by the radial size of the Gaussian cavity mode and the velocity of the atoms during 2 2 their transit. For a radial coupling strength g(r ) = g0 e −r /w0 with w0 = 25.5 µm and an initial velocity of 84.1 cm/s we expect an average coupling time of 45 ± 12 µs [Figure 4.6(b), gray]. Taking the classical free fall velocity is justified since the induced momentum uncertainty by projecting the longitudinally delocalized atom into the cavity mode is on the order of 10µs. In the numerical simulation we take into account photon shot noise and the features of our peak detect routine, namely the 20 µs sliding average. The effect of the dipole potential on the transit time is negligible because the slight gain in velocity (< 2µs) is counteracted by an effectively stronger and therefore longer coupling [Figure 4.6(d)]. The mean of the measured coupling time distribution [Figure 4.6(b), red] is in accordance with the expected value. However, the distribution deviates from the expected shape and exhibits an excess of short and long transit times. We attribute the shorter transits to optical pumping of atoms into the dark state |F = 1〉 because their number is intensity dependent on the probe light. Longer transit times could be explained by diffraction of the atomic beam, scattering of spontaneous photons or cavity cooling effects, if the cavity axis is slightly nonorthogonal with respect to the atom laser (possibly 10−2 rad) and by unresolved multiatom events. The magnitudes of the cavity transmission dips in Figure 4.6(c) reflect the different maximum coupling strengths for single atom transits. Depending on its radial position an atom will experience a varying peak coupling strength according to the Gaussian profile of the cavity mode. In the axial direction however, the light force is strong enough to channel the atoms towards the intensity maxima of the standing wave [141]. Arbitrarily weak coupling transits cannot be resolved due to the shot noise in the empty cavity transmission. We set the single atom detection threshold to 4σ of the original transmission to achieve a large signal-to-noise ratio. 63 4. CHARACTERIZATION OF THE SYSTEM (a) (b) 0 6 20 4 40 60 2 80 0 100 transmission reduction [ % ] frequency [ x 1000 ] 67500 coupling time [ μs ] 20 30 40 50 60 70 80 30 (c) (d) 30 40 40 50 50 60 60 70 70 80 80 90 90 0 2 4 6 8 frequency [ x 1000 ] transmission reduction [ % ] transmission reduction [ % ] 67300 time [ μs ] 67400 20 30 40 50 60 70 80 coupling time [ μs ] FIGURE 4.6.: Characteristics of detected single atom events. (a) The transit of a single atom significantly reduces the probe light transmission through the cavity. We integrate the signal with a 20 µs sliding average and set the detection threshold to 4σ of the photon shot noise. (b) Distribution of measured coupling times (FWHM) (red) compared to the distribution of simulated events (gray). (c) Distribution of measured transmission reduction magnitudes. An evaluation with a 4σ threshold (red) is compared to a 2σ threshold [gray] revealing the discrimination of the events from the photon shot noise. (d) Dependency of the transmission reduction on the coupling time due to the non-Gaussian shape of the dips. 64 4.2. SINGLE ATOM DETECTION PERFORMANCE The resulting histogram of dip depths is displayed in Figure 4.6(c) [red] compared to data for a lower threshold level of 2σ in Figure 4.6(c) [gray] unveiling the photon shot noise. The weakest detectable single atom events correspond to peak atom field coupling strengths of gmin = 2π · 6.5 MHz. The strongest attainable coupling strengths for our cavity are gmax = 0 0 2π · 10.4 MHz, which would be equivalent to a reduction of 80% in the cavity transmission [33]. We do not observe a sharp cutoff in the histogram but rather an equal distribution of transmission reductions from 50-80% with smeared out edges due to the comparatively large photon shot noise at the minimum of the transmission dip. This is consistent with numerical simulations for single atom events. The dependence of remaining probe light transmission through the coupled atom cavity system is nonlinear with the atom field coupling strengths [33]. Therefore the shape of the transmission dips is not Gaussian as the cavity mode and we observe a dependency of the magnitude in transmission reduction on the coupling time and vice versa, as illustrated in Figure 4.6(d). The knowledge about the signatures of single atom events could facilitate the discrimination of “true” single atom events from “false” shot noise events or unresolved multiatom events, but the broad distributions make it difficult to distinguish two weakly coupling atoms from a strongly coupling one. However, the observed characteristics of the detected events are in good agreement with the theoretical predictions for single atom transits. Moreover, these characteristics remain valid even when reducing the atom flux to very few single atom events. 4.2.3. Detector Qualities Having a BEC and an atom laser as the source for atoms that couple with the cavity mode offers several advantages. For instance, it provides well reproducible starting conditions and allows us to precisely control the flux of atoms over a wide range by varying the microwave output coupling power. The attainable atom flux is orders of magnitude larger than in experiments employing a magneto-optical trap as the cold atom source. We have confirmed that our single atom detector functions as a linear detector on the atom flux over three orders of magnitude (see Figure 4.7). The measured atom count rate is proportional to the output coupling microwave power (see Section 4.1.2). Saturation occurs at a flux of about 5 · 103 atoms per second. At higher rates multiatom arrivals within the dead time of our detector become dominant and single atom events cannot be resolved anymore. At a very low atom flux the error bars become increasingly large due to atom shot noise, i.e., the Poissonian distribution in the atom number determination. Additionally, a very weak atom “dark count” rate without intentional output coupling may be present. It is likely due to stray magnetic or optical fields and depends on the size of the Bose - Einstein condensate. However, the dark count rate is still less than 5 atoms per second on average for a BEC with 2 · 106 atoms, for instance. 65 4. CHARACTERIZATION OF THE SYSTEM atom count rate [ s-1 ] 104 103 102 101 -60 -50 -40 -30 radiated microwave power [ dBm ] -20 FIGURE 4.7.: The ultrahigh finesse optical cavity functions as a linear detector on the output coupling rate, i.e., atom flux over three orders of magnitude. Saturation occurs at a count rate of about 5 · 103 atoms per second. 4.2.4. Detection Efficiency The single atom detection efficiency of the ultrahigh finesse optical cavity strongly depends on the frequencies chosen [114] for the probe laser (ωl ) and the cavity resonance (ωc ) with respect to the atomic transition (ωa ). Furthermore the effective coupling strength g0 and therefore the detection probability are determined by the polarization of the probe light with respect to the quantization axis of the atomic spin. In our experimental configuration we have a residual vertical magnetic field at the position of the cavity of about 16 G which represents the quantization axis for the atoms. The field originates from the magnetic QUIC trap which is on during the single atom detection in the atom laser. We set the probe light to horizontal (within 10 ◦ ) polarization which yields a four times higher atom count rate as vertically (within 10 ◦ ) polarized light. Only these two distinct polarization settings are feasible since we experience a birefringence in the cavity resonance of about twice its linewidth. The horizontal polarization of the probe light produces a higher atom field coupling rate because it drives σ+ and σ− transitions compared to the fewer and weaker π transitions for vertically polarized light. The exact atom field interactions are more complex because of the Zeeman splitting and the resulting optical pumping dynamics inside the cavity. However, for red-detuned probe light the atoms entering the cavity in the |F = 2, mF = 0〉 state will predominantly be pumped into the |F = 2, mF = −2〉 stretched state and undergo cycling transitions driven by the σ− polarization component. Therefore this cycling tran- 66 4.2. SINGLE ATOM DETECTION PERFORMANCE 2.4 2.0 2.2 1.8 1.6 1.4 0 1.2 1.0 0.8 -1.0 atom count rate [ ms-1 ] cavity detuning ∆c [ MHz ] 2.0 1.0 0.6 0.4 0.2 -2.0 0 10 20 30 40 50 probe laser detuning ∆l [ MHz ] FIGURE 4.8.: Dependence of the single atom detection efficiency on the probe laser ∆l and cavity ∆c detunings. The vertical dashed line represents the cycling transition which is Zeeman shifted by 22 MHz from the zero field atomic transition. Best single atom detection is performed with a probe laser red-detuned by about 3Γ from the cycling transition and a cavity detuning of about ∆ν/2, corresponding to the maximum dipole potential created by the probe laser. The second local detection maximum corresponds to a blue-detuned probe laser. Therefore the dipole potential is repulsive and the atom count rate reduced. sition will be the main contribution in the single atom detection process. The imbalance is due to a redshift for the σ− component and a blueshift for the σ+ component of ∼ 22MHz in the magnetic field of 16 G at the cavity. The number of detected atoms critically depends on the atom - probe laser detuning ∆l = (ωa − ωl )/2π and probe laser - cavity detuning ∆c = (ωl − ωc )/2π as illustrated in Figure 4.8. Here ωa refers to the bare atomic transition without magnetic field. For most efficient single atom detection we work with an atom - probe laser detuning ∆l ≈ 30-40 MHz and a probe laser - cavity detuning ∆c ≈ 0.5-1 MHz. By taking into account the 22 MHz Zee¯ man shift of the cycling transition |F = 2, mF = −2〉 ↔ ¯F 0 = 3, mF 0 = −3〉 (vertical dashed line in Figure 4.8) the probe laser red-detuning for optimum single atom detection is about 3Γ. This value corresponds to the maximum of the dipole potential created by the probe laser, that means the dipole force channels the atoms in the axial direction towards the antinodes of the standing wave [141] which are simultaneously the areas of the highest atom field coupling strength. In the radial direction the dipole force is too weak to significantly modify the trajectory of the atoms within the cavity mode. Also the dipole potential created by the stabilization laser is weak compared to the one created by the probe laser. 67 4. CHARACTERIZATION OF THE SYSTEM The second set of parameters in Figure 4.8 where single atom transits are detected is around ∆l ≈ 18 MHz and ∆c ≈ −1 MHz. However, the count rate is reduced considerably because the probe light is blue-detuned from the cycling transition and therefore the dipole potential is repulsive. In the other two quadrants spanned by the resonances of the cavity and the cycling transition of Figure 4.8 (dashed lines), atom transits result in increased probe laser transmission versus the empty cavity transmission [24]. We do not use those events for single atom detection because the efficiency is reduced by about a factor of 2 as compared to evaluating dips. Additionally the peaks exhibit a substructure consisting of single photon bursts which makes it more difficult to discriminate single consecutive atom transits. In order to determine the detection efficiency for single atoms from the Bose - Einstein condensate we make use of the linear behavior of the atom flux on the microwave output coupling power (see Figure 4.7). We output couple a significant number of atoms measurable by absorption imaging while still in the weak output coupling regime. This number is compared to the number of atoms detected by the cavity with the corresponding factor of the output coupling powers. We have calibrated the atom number in absorption imaging with the atom number at the critical temperature which is well known for our trap frequencies. For optimum settings of the cavity and laser detunings we are able to detect (24 ± 5)% of the output coupled atoms with the cavity detector. This number is mainly limited by the spatial overlap of the atom laser beam with cavity mode (see Section 4.2.5). 4.2.5. Atom Laser Beam Profile Obviously, in order to see single atoms with the cavity, the atom laser has to propagate through the cavity mode. However, this was not self-evident because the “BEC production rig” and the “science platform” are completely independent entities of the experimental apparatus and the alignment has to be better than a few millirad without knowing the exact position of the cavity mode. Furthermore, the second order Zeeman effect slightly bends the trajectory of the atom laser in the |F = 2, mF = 0〉 state and modifies its final lateral position by hundreds of micrometers. Although we have aligned the cavity with respect to the BEC position as accurately as possible with plummets during the assembly of the apparatus, the atom laser did not innately hit the cavity mode. We correct these deviations by tilting the whole optical table on which the experiment rests employing its height adjustable legs. The tilt is monitored with a dual-axis inclinometer having its axes aligned along and perpendicular to the cavity axis. With this method we aim the atom laser directly into the cavity mode [Figure 4.9(a)] and maximize the atom count rate. Moreover, tilting the experimental setup deterministically enables us to deduce the diameter of the atom laser after a propagation of 36.4 mm. The active area of the cavity mode is approximately 35 × 150 µm2 . The size in the radial direction is determined by the weak- 68 18 atom count rate [ ms-1 ] (b) Ioffe axis [ mrad ] (a) 18 16 14 12 8 10 12 14 cavity axis [ mrad ] 8 10 12 14 cavity axis [ mrad ] 14 (d) (c) 2 1 0 16 15 0μ m atom count rate [ ms-1 ] 6 1.8 1.6 1.4 1.2 1 2 1.0 0 atom count rate [ ms-1 ] 0.8 0.6 0.4 le ofi r 0.2 p er s la m o t a d e n si ty 20 Ioffe axis [ mrad ] 4.2. SINGLE ATOM DETECTION PERFORMANCE 6 35 μ m FIGURE 4.9.: (a) The detected atom count rate for a constant atom flux is shown with respect to the inclination of the optical table along the two axes. The rectangle represents the active area of the cavity mode and the ellipse is the reconstructed size (1/e -diameter) of the atom laser at the position of the cavity. [(b) and (c)] Fit (red) to the measured data (black) by the convolution of the active size of the cavity mode with a Gaussian beam profile along the Ioffe (b) and cavity (c) axes. It is compared to the expected shape from numerical simulations of the Gross - Pitaevskii equation (gray). (d) Visualization of the extracted two-dimensional atom laser beam profile clipped by the active area of the cavity mode. 69 4. CHARACTERIZATION OF THE SYSTEM est detectable atom transits corresponding to gmin 0 = 2π · 6.5 MHz. In the axial direction it is given by the projection of the cavity length clipped by the curved mirrors. A deconvolution of the measured angle dependent count rates with this active area, assuming a Gaussian atom laser beam profile, yields 1/e -diameters of 80 µm and 110 µm along and perpendicular to the Ioffe axis, respectively [Figure 4.9(b) and Figure 4.9(c), red]. The mapped atom laser, being output coupled from the center of a Bose - Einstein condensate with 1 · 106 atoms, is slightly inverted compared to the trap geometry but almost round at the cavity. Here, its divergence along the fast axis, i.e., cavity axis, is about 2 mrad and less than 0.5 mrad along the Ioffe axis, which makes it the best collimated atom laser to date [11, 10, 142]. The repulsive mean-field interaction from the remaining trapped BEC is considerable only along the fast axis where it acts as a defocusing lens for the atom laser beam. This results in an expansion about four times larger than expected from Heisenberg’s uncertainty principle. Along the weakly confining Ioffe axis the lensing effect is negligible and the size of the atom laser is consistent with a free expansion of the initial ground state in the trap. The atom laser size and therefore its divergence, especially along the fast axis, can be further reduced by output coupling below the center plane of the BEC [142] and by smaller condensates (see Section 4.3.2). We compare the measured atom laser profiles along its symmetry axes with numerical simulations of the time evolution using the Gross - Pitaevskii equation. The resulting density distributions of the atom laser deviate slightly from a Gaussian shape [106], but the measured convolutions with the cavity mode agree very well with the simulated curves [Figure 4.9(b) and Figure 4.9(c), gray]. The overestimated width along the Ioffe axis can be explained be the angle of the BEC axis with respect to the horizontal plane, reducing the spatial width of the output coupling region along the Ioffe axis. Along the cavity axis the slight deviation at the edges is probably due to pointing variations, i.e., transverse oscillations of the atom laser beam. Small collective oscillations in the trap are translated into deflections of the atom laser beam over which we integrate with our detection method. The collective oscillations, mainly center of mass dipole oscillations in the trapped Bose - Einstein condensate, can be excited by radio frequency evaporation or incautious relaxation of the magnetic trap. However, we can exploit this effect to precisely determine the frequencies of excited collective oscillations in the trap by analyzing the Fourier spectrum of the atom count rate. Such a spectrum is shown in Figure 4.10 exhibiting harmonics of the trap frequencies (dipole oscillations) and mutual sidebands. The frequencies can be measured in situ with one and the same experimental implementation of a Bose - Einstein condensate to high precision (millihertz), Fourier limited by the duration of the atom laser recording. 70 4.2. SINGLE ATOM DETECTION PERFORMANCE fourier amplitude [ arb. u. ] 5 4 7.2 Hz 3 29.1 Hz 2 2 x 38.5 Hz 1 0 0 10 20 30 40 50 frequency [ Hz ] 60 70 80 FIGURE 4.10.: The Fourier spectrum of the detected atom laser flux exhibiting the trapping frequencies and their harmonics. A fast and precise tool to measure frequencies of collective oscillation in the trap. 4.2.6. Guiding the Atom Laser The reason for the single atom detection efficiency not being unity is mainly the mismatch of the atom laser and cavity mode sizes [Figure 4.9(d)]. Their overlap is only about 50% assuming a box given by the projected length of the cavity mode and a minimum peak atom field coupling strength of gmin = 2π · 6.5 MHz in the radial direction. Calculating the atom 0 trajectories taking into account the channeling effect of dipole potential we find a maximum single atom detection efficiency of 80% and an averaged efficiency of about 50% within this box. This is in good agreement that we detect about one quarter of the released atoms. In order to increase the overlap and therefore the detectable number of atoms it is possible to funnel the atoms with a dipole potential created by a far red-detuned guiding laser (850 nm, 15 mW, beam waists of 30 × 60µm2 ) into the cavity mode. By doing so we are able to improve the single atom detection efficiency by about a factor of 2 to around 50%. This number still differs from a perfect detection efficiency because the dipole potential formed by the probe laser is simply not strong enough to perfectly localize the atoms in the axial direction at the antinodes of the standing wave. Although we are able to increase the single atom detection efficiency, employing the guiding laser involves some disadvantages. The scattering and heating rate in the dipole potential formed by the guiding laser can cause modifications of the atom arrival time statistics which is undesirable for many experiments [17]. Furthermore, the guiding laser acts on both thermal and quantum degenerate atoms and therefore diminishes a characteristic feature of our detector, namely, the very sensitive discrimination of thermal and condensed atom count rates (see Section 4.3). 71 4. CHARACTERIZATION OF THE SYSTEM 4.3. Investigation of Ultracold Atomic Gases The combination of a Bose - Einstein condensate with an ultrahigh finesse optical cavity enables us to detect single atoms from a quantum degenerate gas with very high sensitivity. Therefore we can employ the cavity as a minimally invasive probe for cold atomic clouds. This allows us to perform nondestructive measurements on the ensemble of cold atoms in situ and time resolved. Assuming a constant, weak output coupling power, the atom count rate of the cavity detector depends on the properties of the source via two factors. First of all the number of output coupled atoms is proportional to the number of atoms fulfilling the resonance condition, i.e., the one-dimensional density at the output coupling plane. And secondly the atom count rate depends on the probability for an output coupled atom to hit the cavity mode. Because of its finite active area the cavity functions as a filter in momentum space. 4.3.1. Thermal Clouds For a thermal cloud the density at the output coupling central plane is proportional to Nth T −1/2 and the probability to hit the detector is proportional to 1/T assuming Gaussian density and momentum distributions. Therefore the thermal atom count rate detected with the cavity is proportional to Nth T −3/2 . This dependency is shown in Figure 4.11. At the critical temperature of Tc ≈ 180 nK for 107 atoms only about 0.6% of the output coupled thermal atoms will fly through the cavity mode and can possibly be detected. atom count rate [ ms-1 ] 8 6 atom laser threshold (onset of BEC) 4 2 0 0 0.2 0.4 0.6 0.8 -3/2 -3/2 NthT [ 1 / (NcTc ) ] 1.0 FIGURE 4.11.: Investigation of atom count rates for thermal beams. The atom flux is proportional to Nth T −3/2 for temperatures above the critical temperature and exhibits a threshold behavior when cooling across the phase transition. Just above Tc the density and momentum distributions of the thermal cloud are governed by a Bose distribution and obey a different scaling law than expected for a Gaussian distribution. 72 4.3. INVESTIGATION OF ULTRACOLD ATOMIC GASES The onset of Bose - Einstein condensation can be clearly seen in the sharp increase of detected atoms [143, 35]. Close to the critical temperature, however, the detected atom flux slightly deviates from the expected behavior because the approximated Gaussian distributions for density and momentum are not valid anymore near Tc . The thermal cloud is described by the more peaked Bose distribution which yields an increased atom detection rate of about 30% near the critical temperature of 180 nK compared to the Gaussian distribution. 4.3.2. Quantum Degenerate Gases atom count rate [ s-1 ] For Bose - Einstein condensates the probability for an atom to hit the cavity mode and therefore the atom count rate detected with the cavity is independent of temperature. The number of resonant atoms participating in the output coupling process is proportional to the density of the BEC and the area of the output coupling plane. This means the atom flux is proportional to N04/5 because the Thomas - Fermi radius of a BEC scales as N01/5 . 104 N04/5 103 102 101 100 103 104 105 N0 106 107 FIGURE 4.12.: Investigation of detected atom count rates for pure quantum degenerate samples. The scaling with the atom number in a pure BEC exhibits three different regimes. The expected N04/5 behavior is only valid for intermediate particle numbers. Very small and very large condensates obey different scaling laws due to an increased Heisenberg limited momentum spread and the mean-field repulsion of the remaining condensate, respectively. However, this dependency is only true for Bose - Einstein condensates of intermediate size and deviates for very small and very large condensates as shown in Figure 4.12. Output coupling from small condensates accounts for a faster quantum mechanical expansion of the initial ground state wave function in the atom laser. Therefore the overlap between the transverse atom laser wave function and the cavity mode is reduced. Large condensates on the other hand exhibit increased divergence because of the mean-field repulsion exerted on the atom laser propagating through the BEC. The condensate acts as an imperfect diverging 73 4. CHARACTERIZATION OF THE SYSTEM lens and displaces the maximum density outward [142, 106, 144]. This results in a weaker scaling of the detected atom flux with the number of atoms in the BEC and possibly a decrease when the atom laser profile becomes more “donut-mode-like”. These three regimes are displayed in Figure 4.12 for measured atom count rates versus the number of atoms in the “pure” BEC. The exact position of the crossover between these regimes depends on the active area of the single atom detector. 4.3.3. Phase Transition Our single atom detector in form of the high finesse optical cavity is extremely sensitive and selective to quantum degenerate atoms not only because of the increased density at the output coupling region but also due to the filtering in transverse momentum space. This means we can more accurately observe the onset of Bose - Einstein condensation as compared to absorption imaging techniques. The exact determination of the critical temperature, in combination with the precisely measured trap frequencies (see Figure 4.10), allows one in turn to calibrate the atom number obtained by the absorption images. Furthermore we are able to survey the density distribution in the trap along the vertical direction by scanning the resonant plane for the output coupling process through the trapped cloud of cold atoms as shown in Figure 4.13. For temperatures close to the critical temperature the density distribution of the thermal cloud already deviates from the Gaussian shape and has to be described by the more peaked Bose distribution [Figure 4.13, (134 nK)]. For temperatures slightly below Tc single atom detection with the cavity allows us to observe and map very small condensates that are not visible in absorption images [Figure 4.13, (128 nK) and (123 nK)]. This could be a valuable tool to study the temporal and spatial evolution of the bosonic gas at the phase transition. 74 4.3. INVESTIGATION OF ULTRACOLD ATOMIC GASES 4 atom count rate [ ms-1 ] 123 nK 3 128 nK 2 1 134 nK 0 -40 -20 0 20 output coupling position [ μm ] 40 absorption images FIGURE 4.13.: Analysis of the density distribution of the trapped ultracold atom gas by output coupling at different vertical positions relative to the center of the BEC and measuring the resulting atom count rate with the cavity. The profiles for three different temperatures around Tc are shown in comparison with the absorption images. The high sensitivity of the cavity detector to quantum degenerate atoms allows for precise observation of the onset of Bose - Einstein condensation and the deviations from a Gaussian profile (gray curve). 75 5 Correlations and Counting Statistics of an Atom Laser “. . . a widely believed fact.” - Fry, Futurama We demonstrate time resolved counting of single atoms extracted from a weakly interacting Bose - Einstein condensate of 87 Rb atoms. The atoms are detected with an ultrahigh finesse optical cavity and single atom transits are identified. An atom laser beam is formed by continuously output coupling atoms from the Bose - Einstein condensate. We investigate the full counting statistics of this beam and measure its second order correlation function g (2) (τ) in a Hanbury Brown and Twiss type experiment. For the monoenergetic atom laser we observe a constant correlation function g (2) (τ) = 1.00 ± 0.01 and an atom number distribution obeying Poissonian statistics, whereas a pseudo-thermal atomic beam shows a bunching behavior and a Bose distributed counting statistics. Alternatively, the two-particle correlations are illustrated by evaluating the exclusive probabilities of particle detection both for the conditioned and the unconditioned case. In addition, we show the increased third order correlations (by a factor ∼ 3! = 6) of the pseudothermal compared to the uncorrelated atom laser. The experimental results of this chapter were published in [17]: A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Correlations and Counting Statistics of an Atom Laser, Phys. Rev. Lett. 95 (2005), 090404. 77 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER 5.1. Introduction Correlations between identical particles were first observed by Hanbury Brown and Twiss in light beams [48]. Their idea was that intensity fluctuations and the resulting correlations reveal information about the coherence and the quantum statistics of the probed system. This principle has found applications in many fields of physics [51] such as astronomy [145], high energy physics [53], atomic physics [54] and condensed matter physics [55, 56]. In optics, the reduced intensity fluctuations of a laser have been observed by Arecchi [44] only a few years after its invention, thereby disclosing the extraordinary properties of this light source. With the realization of Bose - Einstein condensation in dilute atomic gases a novel weakly interacting quantum system is available. The interpretation of a Bose - Einstein condensate representing a single, macroscopic wave function has been supported in numerous experiments highlighting its phase coherence [5, 146, 147, 105]. Correspondingly, atom lasers are atomic beams which are coherently extracted from Bose - Einstein condensates [8, 9, 10, 11]. Their first order phase coherence has been observed both in space [5] and time [42]. However, only the second order coherence reveals whether atom lasers exhibit a truly laser-like behavior. Here we present a measurement of the temporal second order correlation function g (2) (τ) of an atom laser in a Hanbury Brown and Twiss type experiment. Measuring all atom arrival times explicitly with a single detector of high quantum efficiency enables us to extract the full counting statistics of atomic beams. The second order correlation function g (2) (τ) represents the conditional likelihood for detecting a particle a time τ later than a previously detected particle and quantifies second order coherence [40]. For a thermal source of bosons g (2) (τ) equals 2 for τ = 0 and decreases to 1 on the time scale of the correlation time which is given by its energy spread. For a coherent source g (2) (τ) = 1 holds for all times and therefore intensity fluctuations are reduced to the shot noise limit. Higher order coherence in quantum degenerate samples was so far only studied in the spatial domain where atom atom interactions reveal the short distance correlations [59]. In an interferometric measurement g (2) (r ) has been determined for elongated, phase fluctuating condensates [148], and recently spatial correlation effects in expanding atom clouds were observed [16]. We demonstrate the detection of single atoms from a weakly interacting quantum gas by employing an ultrahigh finesse optical cavity [20, 21] as illustrated in Figure 5.8. A different technique with the potential of single atom detection in quantum degenerate samples has been demonstrated with metastable Helium atoms [34]. Detecting the arrival times of all atoms at the cavity explicitly gives access to the full counting statistics that reveals the atom number distribution function and its statistical moments [62, 149]. Determining the full counting statistics goes far beyond a measurement of the intensity correlation function only, because it represents the full statistical information about the quantum state. Despite recent 78 5.2. BACKGROUND progress, especially in mesoscopic electronic systems [64], the full counting statistics has not been measured for massive particles before. For neutral atoms this quantity is of special interest, since the strength of the interaction does not overwhelm the quantum statistics as it is often the case for electrons. 5.2. Background Before we present our results, an overview is given about coherence of first and higher order and its relation to correlation functions. These put into context the bunching effect in two-particle correlations of thermal beams, first observed by Hanbury Brown and Twiss and the quantum theory of coherence developed by Glauber. Lastly, the features of statistical distribution functions are described which contain the complete information about a stochastic quantum system in form of the full counting statistics of particle distributions. The textbook by Paul [150] provides a good introduction to photon statistics and coherence. 5.2.1. First Order Coherence Conventional optical interferometers measure the correlation function 〈E ∗ (r1 , t 1 ) E (r2 , t 2 )〉, which compares the complex values of the electromagnetic field at different points in spacetime. This quantity is called first order coherence or phase coherence and describes the ability of fields to interfere and exhibit interferences fringes. In its normalized form 〈E ∗ (r1 , t 1 ) E (r2 , t 2 )〉 g (1) (r1 , t 1 ;r2 , t 2 ) = rD ¯ ¯ E D¯ ¯ E ¯E (r1 , t 1 )¯2 ¯E (r2 , t 2 )¯2 (5.1) the magnitude determines the contrast of the interference fringes in terms of the visibility V as introduced by Rayleigh |g (1) | = V = I max − I min . I max + I min (5.2) The visibility ranges from zero, for incoherent electric fields, to one, for coherent electric fields. Anything in between is described as partially coherent. The above defined field correlation function refers both to spatially and temporally separated points in space-time and is therefore suited to describe all single-particle interference phenomena, like Young’s double slit experiment or optical interferometers such as the Michelson or Mach - Zehnder type. All these geometries have in common that the electric field is split into two components, one part being spatially offset and/or time delayed with respect to the other part, and finally recombined. The intensity of the resulting field is measured as a function of spatial variation or time delay and exhibits interference fringes as long as the two quantum mechanical paths (amplitudes) are indistinguishable. For theses experiments considered so far the famous quote of Dirac applies that “each photon interferes 79 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER only with itself”. However, this can not be generalized to higher order phenomena like the Hanbury Brown - Twiss effect [151], since not photons interfere but quantum mechanical amplitudes ! In practice, most optical interferometers are concerned with measuring the temporal coherence which allows insights to the spectral properties of the light and its emission process. For stationary fields only the time delay τ = t 2 − t 1 enters g (1) and assuming plane waves the spatial retardation can be included through τ = ξ/c , where ξ = x2 − x 1 . So the discussion simplifies to the temporal case g (1) ∗ ® E (t )E (t + τ) (τ) = D¯ . ¯ E ¯E (t )¯2 (5.3) The angle brackets 〈 〉 denote the ensemble or statistical average, which for nonstationary states, such as pulses, is made up of many shots. For stationary states the statistical properties do not change with time, so the ensemble average is equivalent to a time average ∗ 〈E (t ) E (t + τ)〉 = 1 Z T T 0 E ∗ ( t )E ( t + τ )d t . (5.4) This is equivalent to the autocorrelation function of the electric field and visualized in Figure 5.1. Since a convolution is most easily performed in Fourier space, where it reduces to a multiplication it is obvious that the first order correlation function relates to the normalized power spectrum of the light source via an inverse Fourier transform Z ∞ 1 (1) −1 g (τ) = F [P (ν)] = p P (ν)e i 2πντ d ν , (5.5) 2π −∞ which is a form of the Wiener - Khintchine theorem. time t+τ t FIGURE 5.1.: Schematic illustration of the overlap integral in equation (5.4) for two wavetrains incident on a detector. In the marked region it is ambiguous which of the wavepackets is detected. For classical fields, general properties of the correlation function g (1) can be derived [152]. It is symmetric g (1) (τ)∗ = g (1) (−τ) and maximum for no time delay, i.e., g (1) (0) = 1, dropping off to zero over a timescale τc . This correlation time τc is defined as Z τc = ∞ |g (1) (τ)|2 d τ −∞ and is approximately given by the reciprocal of the spectral bandwidth ∆ω. 80 (5.6) 5.2. BACKGROUND The discussion of first order coherence can equally be applied to matter wave fields (see Chapter 6) in the formalism developed by Glauber which is presented in setion 5.2.3. For output coupling two atom lasers at two different planes within the Bose - Einstein condensate it is the phase coherence between the “slits” that determines the ability of these atom lasers to interfere [105]. The g (1) (r1 , r2 ) of atomic clouds is basically the Wigner function in the phase space representation of the many-particle system [80]. 5.2.2. Hanbury Brown - Twiss and the Invention of Bunching In the mid-fifties Robert Hanbury Brown, in collaboration with Richard Twiss who worked out the theoretical details, invented a revolutionary new measuring technique (see Figure 5.2). It is based on measuring correlations between intensities and fluctuations thereof, rather than correlations of the electromagnetic field. The intensity interferometer was particularly developed to measure stellar diameters, where it overcomes major difficulties faced with conventional Michelson interferometers, because the very delicate phase relation of two signals at different space-time points is discarded. However, the Hanbury Brown - Twiss effect caused considerable uproar and disbelief in the physics community: how could photons coming from opposite edges of a star exhibit correlations although being emitted by independent sources ? The mystery was resolved by Purcell in a notable publication to support the Hanbury Brown - Twiss experiment [60], where he gives an intuitive explanation of the origin and calculates the magnitude of the positive cross-correlations observed by Hanbury Brown and Twiss in two coherent beams of thermal light, known nowadays as “bunching”. Purcell praises the contended Hanbury Brown - Twiss method as “an instructive illustration of the elementary principles of quantum mechanics” and in along with his discovery of modified spontaneous emission [153], this can be regarded as the birth of quantum optics. The Hanbury Brown - Twiss technique found its way into many divers fields of physics where correlations between particles reveal insight into fundamental processes [53, 50, 51] and quantum statistics [54, 55, 56, 57, 17, 18]. The measuring principle of the Hanbury Brown - Twiss technique is based on interrogating and correlating the local intensities simultaneously at two different positions within a beam. A coherent beam of thermal light will show intensity correlations in excess to completely random coincidences of incoherent beams. By varying the separation of the detectors, the coherence angle of a light beam could be mapped out and the size of the source inferred. In the original version [47], the Hanbury Brown - Twiss experiment was carried out in the radio frequency band with two antennas as square law detectors of the electromagnetic field from the same stellar object as illustrated in Figure 5.2. They measured increased correlations in the fluctuations of the two distinct detectors of the form 〈∆I 1 ∆I 2 〉 that vanish when the detectors are separated by more than a certain length scale ξc . The AC term ∆I i is given by I i −〈I i 〉, where 〈I i 〉 is the time averaged intensity. Considering classical electromagnetic waves, the correlations in the intensity fluctuations 81 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER star τ filter antennas filters lamp aperture correlator c beam splitter c FIGURE 5.2.: Schematic illustrations of the renowned Hanbury Brown - Twiss experiments. (left) The original version in the radio frequency band, developed and successfully applied to measure stellar diameters. The intensity correlations from two antennas illuminated by the same rf field are recorded. (right) Table top experiment as a proof of principle in the optical domain. Thermal light from a lamp is filtered spatially (aperture) and energetically (color filter). The beam is divided by a half-silvered mirror and cast onto two symmetric photon counters. The read out is done by an electronic coincidence unit. The spatial and temporal second order correlation function can be measured by varying the lateral position of a detector or the time delay. can relatively easy be explained [154, 51]. Assuming two spherical waves of distant origin, the total intensity at the detectors can be calculated. The correlation term 〈I 1 I 2 〉 depends on the distance ξ between the detectors through a compact spatial function f (ξ) that depends on the size of the source 〈I 1 I 2 〉 = 〈I 1 〉 〈I 2 〉 + I 2 f (ξ). Because the average intensities are independent of detector separation 〈I 1 〉 = 〈I 2 〉 = I there exist correlations 〈∆I 1 ∆I 2 〉 = 〈I 1 I 2 〉 − 〈I 1 〉 〈I 2 〉 ∝ I 2 f (ξ) (5.7) for ξ < ξc . For separations ξ > ξc the function f (ξ) tends to zero and consequently 〈∆I 1 ∆I 2 〉 = 0, as expected for completely uncorrelated signals, where 〈I 1 I 2 〉 factorizes. The Hanbury Brown - Twiss effect can equally be observed in the temporal domain by introducing a delay time τ between intensity measurements at the same position. However, since the excess correlations measured by Hanbury Brown - Twiss are not normalized, it became more practical to represent them in the more universal form of the second order 82 5.2. BACKGROUND correlation function g (2) [155] g (2) (τ) = 〈I (t ) I (t + τ)〉 〈I (t )〉 2 (5.8) . A typical behavior of the second order correlation function exhibiting the bunching effect for thermal fields is illustrated in Figure 5.3. The original Hanbury Brown - Twiss correlations are proportional to g (2) − 1. For thermal light, the stochastic behavior of I (t ) can be described through the first order correlation function g (1) 〈I (t ) I (t + τ)〉 = 〈I 〉2 (1 + |g (1) (τ)|2 ) (5.9) and consequently by its spectral properties [60, 152]. This formula shows the relation between the first and second order coherence. Limiting values for g (2) (τ) of thermal, chaotic light follow for instance from the Cauchy - Schwartz inequality 〈I 2 〉 ≥ 〈I 〉2 yielding g (2) (0) ≥ 1 and g (2) (τ À τc ) → 1 . (5.10) For spatially coherent thermal beams g (2) (0) = 2. This follows from eq. (5.9), or the expression 〈I 2 〉 = 2 〈I 〉2 when summing up all complex electric fields of infinitely many random emitters (Wick’s Theorem), which is the case in thermal or chaotic radiation [152]. g(2) 2 1 τc , ξc τ,ξ FIGURE 5.3.: Characteristic second order correlation function g (2) for thermal radiation. The distinct feature of bunching can be observed both in the temporal and spatial domain. The width of the bunching is determined by the inverse bandwidth and the inverse momentum spread, i.e., divergence angle, respectively. These considerations for classical waves notwithstanding, the scepticism centered on the Hanbury Brown - Twiss experiment in the optical regime [48, 156, 145] illustrated in Figure 5.2. It was intended as a proof of principle for their measuring technique and further developed to be applicable in astronomy as demonstrated with measurement on Sirius [49]. 83 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER In the optical however, unlike analog signals from rf antennas which were subsequently multiplied, more or less digital clicks from photon counters were correlated. This caused people to call for a revision of quantum mechanics, because then the controversy about the photoelectric detection process and the quantization of the electromagnetic field was still vivid. However, starting from a gedanken experiment using a single perfect detector, Purcell in his paper supporting paper [60] showed that these correlations are a natural consequence of the Gaussian fluctuations of the electric field. He calculated the excess coincidences over random events to be proportional to the squared count rate and a reduction factor which arises from the finite time resolution T of the photo counters 〈∆n 1 ∆n 2 〉 = 1 〈n〉2 τc /T . 2 (5.11) This illustrates the fact that the Hanbury Brown - Twiss correlations diminish when integrated over long times compared to the coherence time τc . In fact τc /T was on the order of 1/5000 in the original Hanbury Brown and Twiss experiment and only thanks to the high count rate they succeeded in observing the bunching effect, in contrast to their contenders [156]. a source b 1 detector 2 FIGURE 5.4.: A generic Hanbury Brown - Twiss setup. Independent sources emit particles that are detected by independent detectors. However, when the particles are indistinguishable for the detectors, Bose - Einstein statistics comes into effect. This results in a positive interference term, giving a factor two in the joint count rate as opposed to distinguishable classical particles. Although the Hanbury Brown - Twiss experiment can be understood completely in terms of classical electromagnetism it is a fine verification of the laws of quantum mechanics for identical bosons. What the experiment shows is that the bosonic nature of photons is already contained in the superposition principle obeyed by classical electromagnetic fields. To derive a quantum mechanical interpretation of the Hanbury Brown - Twiss effect, let us regard the generalized gedanken experiment [157] illustrated in Figure 5.4. Two bosonic particles a and b , which do not necessarily have to be photons, are emitted by independent sources and are detected by two independent detectors 1 and 2. The quantum mechanical amplitude of detecting particle a in detector i can be written as 〈a|i 〉. In the same way, the amplitudes 〈b|i 〉 for the second particle are defined. For simplicity, we will assume that the amplitudes of detecting one particle in either one of the detectors is equal, i.e., 〈a|1〉 = 〈a|2〉 ≡ a and 〈b|1〉 = 〈b|2〉 ≡ b . Now, we ask for the probability of detecting exactly 84 5.2. BACKGROUND one particle in each detector. This probability depends on whether or not the particles are in principle distinguishable, e.g., due to their energies, momenta, internal degrees of freedom, etc. • For distinguishable particles, we just have the classical sum of probabilities. The total probability is the probability of detecting a in 1 and b in 2 plus the reversed case: P d = | 〈a|1〉 〈b|2〉 |2 + | 〈a|2〉 〈b|1〉 |2 = 2|a|2 |b|2 (5.12) • In the case of indistinguishable bosonic particles we have constructive interference of the two possible paths which means that in this case the probability amplitudes add: P i = | 〈a|1〉 〈b|2〉 + 〈a|2〉 〈b|1〉 |2 = |2ab|2 = 4|a|2 |b|2 = 2P d (5.13) Consequently, for quantum mechanically indistinguishable particles it is twice as probable to detect two of them simultaneously than for classically distinguishable particles. This effect accounts for the increased correlations in chaotic light sources on short time scales τ < τc and is called bunching, or the Hanbury Brown - Twiss effect. Clearly, this phenomenon is not a property of the source(s) since the emission happens to be completely independent. It is solely the indistinguishability of quantum mechanical amplitudes at each detector that renders this effect possible. Pauli’s exclusion principle prevents two identical fermions to be in the same quantum mechanical state or likewise at the same position in space-time. Their two-particle wave function is stated in its antisymmetric form and therefore the quantum mechanical amplitudes don’t sum up but are subtracted, leading to a complete absence of correlations. This behavior, called antibunching has been observed in three remarkable experiments performing the Hanbury Brown - Twiss technique with confined [55, 56] and free [57] electrons. The notion of identicalness can be further discussed in terms of coherence. Two particles a, b arriving at a detector are quantum mechanically indistinguishable when they are detected within their coherence volume. This virtual volume spanned by the overlap of their extending wavefunctions can be quantified by the spatial coherence length in the transverse to the propagation direction and the longitudinal or temporal coherence. The spatial coherence length ξc is given by the inverse spread of the wave vector and is a measure of collimation of the beam, i.e., the planarity of its wavefronts ξc = 2π ∆k = λ , θ (5.14) where λ is the wavelength of the light and θ = s /l the angular size of the source as seen from the detector. The spatial coherence can be determined in the Hanbury Brown - Twiss technique by varying the separation of the detectors (see Figure 5.2), provided the integration time is short compared to the coherence time. It is interesting to note, that the strategy to 85 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER use a beam splitter solely arose form the requirement to measure towards zero separation, i.e., for completely overlapping detector images, which would otherwise not have been possible due to the physical size of the detectors. The use of a half-silvered mirror as a beam splitter, in contrast to a polarizing beam splitter, is essential to ensure the identicalness of photons in each path. The transverse coherence length ξc is also the characteristic size of the speckle pattern [158], which illustrates that the correlations are diminished for detectors having an active diameter d > 2ξc . So, if either the source or the detector are spatially too extended, the particles become in principle distinguishable and the Hanbury Brown - Twiss effect is lost. The criterion to observe bunching is therefore the “single mode” behavior of a beam which means that all particles of a beam are within a single phase space cell. In momentum space this can be achieved when being in the far field limit or being diffraction limited in other words. This was well granted for observation of stars, but poses a limitation on the experiments with cold atoms. There, the correlation length after a time of flight t is given by ξc = ħt , ms (5.15) where s is the size of the source and m the mass of the particles [18]. The bunching is then reduced by ∼ ξc /d in one dimension, where d is the size of the detector. The coherence time, as already stated in equation (5.9), is related to the bandwidth ∆ω of the light or more generally to the energy spread of the particles τc = 2π ∆ω = ħ . ∆E (5.16) The temporal second order correlation function can be recorded when varying the delay time between particle detections at the same position. When associated with the velocity v of massive particles the temporal coherence can be translated into a longitudinal coherence ξc = v τc . For a continuous beam, the correlation time, or equivalently the longitudinal correlation length, depends only on the velocity width of the source and not on the source size. Thus, the longitudinal and transverse directions are qualitatively different. For thermal atomic gases the second order correlation function is given by [54] g (2) (τ) = 1 + p 1 1 + (∆ωτ)2 with ∆ω = mv s2 2ħ , (5.17) where v s represents the velocity width of a thermal source with a Gaussian velocity distribution. Since Hanbury Brown and Twiss were primarily interested in spatial coherence as a tool to determine stellar diameters, the issue of temporal correlations was only later addressed in separate experiments [159, 160]. Although a single detector would be sufficient in the time domain, the double detector geometry proved useful to overcome the dead time of photon counters for time correlated 86 5.2. BACKGROUND measurements. So the time dependence of the intensity correlation function could be determined by coincidence counting techniques down to zero time delay [45]. However, measuring start-stop signals with two consecutive events only gives an approximate second order correlation function. The photon count rate has to be much less than the inverse coherence time, such that having a third event (of unrecorded time delays) within the relevant correlation time is negligible. τi t time FIGURE 5.5.: Extracting g (2) (τ) from all possible detection time differences rather than simple start-stop signals. When recording single events, their temporal second order correlation function can be equally expressed as the conditional likelihood P c (t |t + τ) to detect a particle at time t + τ after detecting an initial one at time t g (2) (τ) = 〈I (t ) I (t + τ)〉 〈I (t )〉 2 = P c (t |t + τ) . (5.18) The second order correlation function in that sense is a nonexclusive property, because it does not matter if any other particles are detected within the time interval τ. The exact g (2) (τ) is recovered by taking all possible time differences of recorded single particle detection events (see Figure 5.5) and plotting them as a histogram. If the overall recording time T is comparable to the relevant time scales, a correction factor T /(T − τ) must be applied to take into account that longer time differences become less likely. A proper normalization (5.10) of the histogram is obtained by considering the total number of pairs, being N (N − 1)/2 for N single particle events, per available time bins. 5.2.3. Glauber and the Quantum Theory of Coherence Glauber’s theory deals with the quantum theory of electromagnetic fields, in particular with the theory of coherence and photon correlations [40]. It provides the required analytical tools, such as coherent states and P-representation, to describe light fields and the photon detection process in a full quantum mechanical treatment and in terms of correlation functions. This became necessary to explain the photon correlation experiments established by Hanbury Brown and Twiss and through the development of quantum light sources such as lasers. Naturally, it is applicable to nonclassical quantum states of light. In the context of atom optics the quantum theory of coherence finds its natural extension to matter wave fields and successfully incorporates all classical (thermal) and quantum (bosonic and fermionic) states of de Broglie waves. 87 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER Starting with the more customary optical theory, the real electric field E = E + +E − is split into two complex conjugate terms, where E + contains only the positive frequencies ∼ e −i ωt and vice versa. The E ± are physically equivalent and define the electric field operators Ê (±) in the Heisenberg picture. For a single mode we therefore have q ³ ´ Ê (χ) = Ê (χ) + Ê (χ) = ħω/2ε0V â e −i χ + â † e i χ , + − (5.19) in analogy to equation (2.49). The phase angle χ = ωt − kz − π/2 contains the explicit time and position dependence. The â and â † represent the annihilation and creation operators of a photon in the particular mode under consideration a † |n〉 = p n + 1 |n + 1〉 and a |n〉 = p n |n − 1〉 . (5.20) The normalized time dependent second order correlation function in its quantized version then reads 〈â † (t 1 )â † (t 2 )â (t 2 )â (t 1 )〉 (2) . (5.21) g (t 1 , t 2 ) = † 〈â (t 1 )â (t 1 )〉〈â † (t 2 )â (t 2 )〉 Here, the angle brackets 〈 〉 represent quantum mechanical expectation values. For a pure ¯ state ¯φ〉 the observable Ô is given by 〈φ|Ô|φ〉. For a system in a mixed state the average P value is given by the statistical mixture i P i 〈φi |Ô|φi 〉 or equivalently by the trace over the density operator ρ̂ 〈Ô〉 = Tr{ρ̂ Ô} . (5.22) In contrast to its classical form (eq. 5.1), the only limiting relation for the second order correlation function in its quantum form is that it is positive 0 ≤ g (2) (t 1 , t 2 ) ≤ ∞ . (5.23) This follows from the fact that the numerator in equation (5.21) is an expectation value of Hermitian operators. The g (2) (t 1 , t 2 ) represents the joint probability to detect two particles at times t 1 and t 2 , respectively. For stationary light beams, the second order correlation function will only depend on the time difference τ = t 2 −t 1 between two particle detection events. It is because the correlation is defined as the average of a normally ordered product of field operators, that it expresses the correlation of pairs of particles. Normally ordered refers to the sorting of all annihilation operators to the right of the creation operators and time sorting from the “outside in”. The density correlation 〈â † (t 1 )â (t 1 )â † (t 2 )â (t 2 )〉 is equivalent to the second order correlation 〈â † (t 1 )â † (t 2 )â (t 2 )â (t 1 )〉 except for the autocorrelation at t 1 = t 2 due to the bosonic commutation relation [â (t 1 ), â † (t 2 )] = δ(t 1 − t 2 ). The quantum treatment establishes the interpretation of g (2) as particle correlations rather than intensity correlations. Therefore it can be equally applied to massive particle by simply replacing the photon creation/annihilation operators â (†) with the atomic field 88 5.2. BACKGROUND operators Ψ̂(†) . Moreover, Glauber’s theory applies for general correlations of order n and he was constructing a state that allows all higher order correlation functions to factorize according to 〈Ψ̂† (t 1 )Ψ̂† (t 2 ) . . . Ψ̂(t 2 )Ψ̂(t 1 )〉 = ψ∗ (t 1 )ψ∗ (t 2 ) . . . ψ(t 2 )ψ(t 1 ) , (5.24) where ψ is a wave function. This is achieved with a coherent state. Full coherence is then defined as g (n ) ≡ 1, which can only be fulfilled up to order O (N ) of the particle number N . Because of the normal ordering N annihilation operators are applied before the N creation operators. This is in stark contrast to a thermal state. With the gedanken experiment illustrated in Figure 5.4 it is straightforward to calculate the joint detection probabilities of multiple bosonic particles on multiple detectors. Summing up all processes, it turns out that the probability of simultaneously detecting n thermal bosons is enhanced by the factor n ! compared to random coincidences. This was confirmed in measurements of the three-body recombination rates of thermal and Bose condensed atomic clouds [58, 59]. The same result is found when calculating the higher order moments for a thermal light field having a Gaussian distribution of the electric field value n 〈I (t )n )〉 = n ! 〈I (t )〉 . (5.25) Coherent states were introduced by Schrödinger in the treatment of the harmonic oscillator in search for solutions to his equation that satisfy the correspondence principle. Glauber coined their use in quantum optics and other bosonic quantum field theories [40, 41]. The coherent state |α〉 is defined to be an eigenstate of the annihilation operator â â|α〉 = α|α〉 , (5.26) where α is just a complex number. This means the coherent state |α〉 remains unchanged when a quantum is removed. Therefore it is clear that it can not contain a definite number of photons. The representation of a coherent state in the basis of the number or Fock states |n〉 reads 1 |α〉 = e − 2 |α| 2 ∞ αn X p |n〉 . n=0 n ! (5.27) When a mode is in a coherent state the probability of finding n quanta is a Poisson distribution P (n ) = | 〈n|α〉 |2 = |α|2n −|α|2 e , n! (5.28) and the mean number of quanta in this distribution is found to be 〈n〉 = |α|2 . Glauber showed that coherent states can be generated by a displacement of the harmonic oscillator ground state |0〉, which is a coherent state itself with α = 0. He defined the displacement operator † ∗ D (α) = e αâ −α â (5.29) 89 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER that achieves just that |α〉 = D (α) |0〉. Since a coherent state is just a displaced version of the the vacuum state |0〉 it has also the same properties in terms of uncertainty. The only contribution to the quantum noise are vacuum fluctuations. The coherent state of a field is the closest a quantum state can come to the ideal classical limit of simultaneously perfectly well defined amplitude and phase. This is true for any other pair of conjugate variables and illustrated in the phase space representation in Figure 5.6. The coherent states reduce the uncertainty product to its minimum value ∆x · ∆p = ħ/2. p Δφ Δp ħ/2 α Δx x FIGURE 5.6.: Phase space representation of a coherent state |α〉. Glauber is also noted for the introduction of the P- and Q-function representation of the density matrix. These functions are quasi probability distributions in phase space, related to the Wigner function. They are not limited to positive values, like classical probability distributions and apparently well suited to solve complex problems in quantum optics. 5.2.4. Counting Statistics The full counting statistics contain the complete information about a mixed quantum state. Therefore the knowledge of the counting statistics is more fundamental than any particular correlation function. In fact it is equivalent to determining all higher order correlation functions [161]. However, it is experimentally much more difficult to record the full counting statistics, than the second order correlation function. The counting statistics measure the probability p (n, T ) to detect n particles within a certain time window of width T . Of course, this quantity is only of significance when the binning time is shorter then the correlation time (T < τc ), which can only be achieved when the integration and dead time of the detector are also much shorter than the correlation time. For T À τc the contribution 90 5.2. BACKGROUND of the correlations is negligibly small and the statistics will be totally random and therefore always approach a Poissonian. Since the inverse bandwidth of conventional thermal radiation is in general much shorter than the resolution of photon counters, the invention of novel light sources [46] and the development of sophisticated experimental techniques [158] were required to record photon counting statistics. Pseudo thermal, chaotic light with arbitrary correlation time can be produced by scattering of laser light on rapidly and randomly moving objects like a ground glass disc. Arecchi [44, 45] used this technique to contrast the photon correlations and statistical properties of such a pseudo-thermal light beam with that of a laser. In a series of experiments he unveiled the extraordinary properties of laser light by measuring its temporal second order correlation function to be constant and the photon counting distribution to be Poissonian. P A classical probability distribution is positive definite and normalized to unity ∞ 0 p (n ) = 1, where p (n ) is taken to be discrete, here. By means of the probability distribution the expectation values and all higher moments µk can be calculated µk = 〈n k 〉 = ∞ X n k p (n ) . (5.30) 0 The moments provide a useful way to characterize a probability distribution function, which can be expanded in an infinite series of all moments. However, besides these moments defined above there exists a multitude of alternative and better adapted parameters to express the features of a probability distribution function, like standardized, central or factorial moments and cumulants. The moments µk and cumulants κk can be derived from the characteristic function P i λn Θ(λ) = 〈e i λn 〉 = ∞ p (n ), which represents the Fourier transform of the probability dis0 e tribution, through µ d µk = −i dλ ¶k µ Θ(λ)|λ=0 and d κk = −i dλ ¶k ln Θ(λ)|λ=0 , (5.31) respectively. Further, the central moments mk are the moments µk of a distribution centered around its average value µ1 . The central moments or the cumulants define the features of a distribution on the basis of its standardized moments, where the first is simply the mean κ1 = µ1 = 〈n〉 and describes the expectation value. The second is the variance κ2 = Var(n ) = 〈(n − µ1 )2 〉 = σ2 and indicates how far from the expectation value the events typically are, where σ is the standard deviation. The third standardized moment, called skewness is a parameter to characterize the asymmetry of a distribution. It is defined as p m 3 /σ3 or alternatively as κ3 / κ2 3 . When the skewness is positive, the distribution has a longer tail to the right, and vice versa. The (excess) kurtosis is the fourth standardized moment κ4 /κ22 = µ4 /σ4 − 3 and expresses the excess peakedness of a function with respect to a 91 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER Gaussian. A distribution with a positive kurtosis has a sharper peak and longer tails, while a negative kurtosis distribution has a more rounded or flat-top peak. As an example, two important probability distributions can particularly easy be grasped by their cumulants κi . The normal distribution (Gaussian) has only two nonvanishing cumulants, the first being the mean and the second being the variance. A Poissonian distribution is distinguished by the equality of all its cumulants κ1 = κ2 = κ3 = . . .. Therefore, in a Poisson distribution, which describes completely random processes, the standard deviation of a p p measurement of N particles is N . The relative error therefore scales as ∝ 1/ N . This is the shot noise inherent in quantized random processes. However, for discrete distributions, especially with respect to particle correlation functions, it is more convenient to introduce the factorial moments ϕk = 〈n (n − 1)(n − 2) . . . (n − k + 1)〉. They are generated by the alternative characteristic function Q (λ) = 〈(1 − λ)n 〉 in the following way µ ¶ d k Q (λ)|λ=0 . ϕk = − dλ (5.32) The factorial cumulants defined in a way analog to equation (5.31) are of less importance. The interest in the factorial moments becomes obvious when considering the normally ordered correlation functions. Re-sorting the operators â (†) to measure particle numbers n = â † â and respecting the bosonic permutation relation [â, â † ] = 1 yields 〈(â † )k â k 〉 = 〈n (n − 1)(n − 2) . . . (n − k + 1)〉 . (5.33) The knowledge of the complete probability distribution function contains more information as the pure correlation functions. Only from all correlation functions could the counting statistics be reconstructed. However, evaluating the counting statistics p (n, T ) for all different time windows T allows for instance the derivation of the second order correlation function from the second factorial cumulants µ g (2) (T ) − 1 = c 2 ¶ ∂ 2 ϕ2 (T ) , ∂T (5.34) where the factor c 2 contains the squared average count rate. It therefore depends only on the second moment (variance) of a distribution, related to the mean for proper normalization. The proportionality is due to the nonideal detection efficiency. A similar relation holds for the other higher order particle correlation function [162, 163], which also depend only on the cumulants of the same order. This can equally be seen by applying the bosonic commu2 tation relation to the normally ordered second order correlation function 〈â † â † â â〉 / 〈â † â〉 , so then g (2) = 1 + 92 σ2 − 〈n〉 〈n〉2 , (5.35) 5.2. BACKGROUND since 〈n 2 〉 − 〈n〉2 = σ2 ≥ 0. It follows that the condition for the single mode second order correlation function g (2) ≥ 1 − 1/ 〈n〉. So it can be nonclassical, i.e., below one for a mode with very few particles. A number state, or a sub-Poissonian distribution will have σ2 < 〈n〉. time 2 0.4 g(2)(τ) 1.5 0.3 1 0.2 0.5 0.1 0 τc τ 0 P(N) 2 4 6 8 N FIGURE 5.7.: Schematic illustration of the relation between the time series of particle detection events, the second order correlation function and the full counting statistics. Three typical cases with equal mean values are shown for Bose - Einstein distributed thermal states (red), a coherent state having a Poissonian distribution (blue) and a sub-Poissonian distribution exhibiting antibunching (green). The effect of a beamsplitter or attenuator in the beam, which is equivalent to a reduced detection efficiency, can be expressed in terms of the Fano factor f = σ2 / 〈n〉. It is one for a coherent state and zero for a number state. A sub-Poissonian state has a Fano factor between zero and one, and the Fano factor of a super-Poissonian or chaotic state is larger than one and unlimited. It can be shown [164], by applying the rules of counting statistics, that the output state has a Fano factor f o corresponding to f o = 1 − ²(1 − f i ) , (5.36) depending on the input Fano factor f i and the loss coefficient 0 ≤ ² ≤ 1. That means the intensity fluctuations of either sub- or super-Poissonian beams will be diminished in heavily attenuated beams (² → 0) and approach a completely random Poissonian state with f = 1. This might be the reason why we don’t observe the second and third order correlation functions to go to two and six respectively, besides the possibly nonlinear mixing process. Two statistical distribution functions of importance for particle counting experiments in quantum optics are generally the Poisson distribution and the Bose - Einstein distribution. Both are discrete probability distributions and their mean value µ = cT here depends on the 93 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER time window T and the average count rate c per time window. The particle count distribution for a coherent state is a Poissonian distribution p (n ) = µn −µ e . n! (5.37) The characteristic function is Q (λ) = e −λµ and all cumulants are given by the same value p n which is the variance. A Poisson distribution describes totally random processes. It’s p characteristic noise is the shot noise (∼ 1/ n ) which is the quantum limit for quantized observables. A narrower distribution can only be achieved by squeezed states with the consequence of increased uncertainty in the conjugate variable. The Poisson distribution is p p positive skew (κ3 = n ) and more peaked than a Gaussian (κ4 = n ). It is the limes of a binomial distribution for very large sample sizes. On the other hand, a chaotic state such as a thermal bosonic beam is represented by a Bose - Einstein or geometric particle distribution p (n ) = µn (1 + µ)n+1 . (5.38) The characteristic function is Q (λ) = (1 + λµ)−1 . One of its distinguishing features is the monotonically decreasing character, being highest for n = 0. Therefore it becomes very broad for large mean values. The Bose - Einstein distribution can be derived from Planck’s law [152] and is in principle a discrete exponential distribution function. This is evident when considering the electric field of a chaotic radiation, which is a Gaussian distribution 2 ∼ e −E around the mean of zero. Therefore chaotic light is also called Gaussian light. Thermal light is a form of chaotic radiation, as is pseudo-thermal light produced by random scattering of a coherent beam [45]. Consequently, the intensity distribution of Gaussian light has an exponential behavior ∼ e −I . 5.3. Experimental Methods and Techniques Our new experimental design combines the techniques for the production of atomic Bose - Einstein condensates with single atom detection by means of an ultrahigh finesse optical cavity. The apparatus consists of an ultra high vacuum (UHV) chamber which incorporates a separated enclosure with a higher background pressure. Here we collect 109 87 Rb atoms in a vapor cell magneto-optical trap which is loaded from a pulsed dispenser source. After polarization gradient cooling and optical pumping into the |F = 1, mF = −1〉 hyperfine ground state we magnetically transfer the atoms over a distance of 8 cm out of the enclosure into a magnetic trap. All coils for the magnetic trapping fields are placed inside the UHV chamber and are cooled to below 0 °C . In the magnetic trap we perform radio frequency induced evaporative cooling of the atomic cloud and obtain almost pure Bose - Einstein condensates with 1.5 · 106 atoms. After evaporation we relax the confinement of the atoms to the final 94 5.3. EXPERIMENTAL METHODS AND TECHNIQUES BEC 85 90 95 100 80 60 40 20 0 90 91 92 80 60 40 20 0 93 cavity transmission [ photons / 20 μs ] 80 atom laser 36 mm 75 time [ ms ] cavity laser photon counter FIGURE 5.8.: Schematic of the experimental setup. A weak continuous atom laser beam is extracted from a Bose - Einstein condensate. After dropping a distance of 36 mm the atoms enter the ultrahigh finesse optical cavity and single atoms in the beam are detected. A typical signal recorded with the single photon counter after time averaging with 20 µs.(right) trapping frequencies ω⊥ = 2π · 29 Hz and ωk = 2π · 7 Hz, perpendicular and parallel to the symmetry axis of the magnetic trap, respectively. For output coupling an atom laser beam we apply a weak continuous microwave field to locally spin flip atoms inside the Bose - Einstein condensate into the |F = 2, mF = 0〉 state. These atoms do not experience the magnetic trapping potential but are released from the trap and form a well collimated beam which propagates downwards due to gravity [11]. The output coupling is performed near the center of the Bose condensate for a duration of 500 ms during which we extract on the order of 3 · 103 atoms. After dropping a distance of 36 mm the atoms enter the high finesse optical cavity as illustrated in Figure 5.8. Fine tuning of the relative position between the atom laser beam and the cavity mode is obtained by tilting the vacuum chamber. We maintain a magnetic field along the trajectory of the atom laser, which at the position of the cavity is oriented vertically and has a strength of approximately 15 G. The cavity consists of two identical mirrors separated by 178 µm. Their radius of curvature is 77.5 mm resulting in a Gaussian TEM00 mode with a waist of w0 = 26 µm. The cou- 95 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER pling strength between a single Rb atom and the cavity field is g 0 = 2π · 10.4 MHz on the F = 2 → F 0 = 3 transition of the D2 -line. The cavity has a finesse of 3 · 105 and the decay rate of the cavity field is κ = 2π · 1.4 MHz. The spontaneous dipole decay rate of the rubidium atom is γ = 2π · 6 MHz. Therefore we operate in the strong coupling regime of cavity QED. The cavity mirrors are mounted inside a piezo tube which enables precise mechanical control over the length of the resonator [21]. Four radial holes in the piezo element allow atoms to enter the cavity volume and also provide optical access perpendicular to the cavity axis. The cavity resides on top of a vibration isolation mount which ensures excellent passive stability. The cavity resonance frequency is stabilized by means of a far detuned laser with a wavelength of 831 nm using a Pound - Drever - Hall locking scheme. The cavity is probed by a weak, near resonant laser beam, whose transmission is monitored by a single photon counting module. The presence of an atom inside the cavity results in a drop of the transmission as shown Figure 5.8. The stabilization light is blocked from the single photon counter by means of optical filters with an extinction of 120 dB. The probe laser and the cavity are red-detuned from the atomic F = 2 → F 0 = 3 transition by 40 MHz and 41 MHz, respectively. The polarization of the laser is aligned horizontally and the average intracavity photon number is 5. These settings are optimized to yield a maximum detection efficiency for the released atoms which is 23%. This number is primarily limited by the size of the atom laser beam which exceeds the cavity mode cross section. The atoms enter the cavity with a velocity of 84 cm/s giving rise to an interaction time with the cavity mode of typically 40 µs, which determines the dead time of our detector. The dead time of our detector is short compared to the time scale of the correlations, which allows us to perform Hanbury Brown and Twiss type measurements with a single detector [165]. 5.4. Results We record the cavity transmission for the period of the atom laser operation and average the photon counting data over 20 µs(see Figure 5.8). Using a peak detection routine we determine the arrival time of an atom in the cavity, requiring that the cavity transmission drops below its background value by at least four times the standard deviation of the photon shot noise. From the arrival times of all atoms we compute the second order correlation function g (2) (τ) by generating a histogram of all time differences within a single trace and normalizing it by the mean atomic flux. Due to the finite duration of the measurement, long time differences between atom pairs are less likely than short intervals and we account for this by an additional normalization. We average these histograms over many repetitions of the experiment to obtain g (2) (τ) with a high signal-to-noise ratio. Figure 5.9 shows the measured second order correlation function of an atom laser beam. The value of the correlation function is g (2) (τ) = 1.00 ± 0.01 which is expected for a coherent source. The second order correlation function being equal to unity reveals the second order 96 5.4. RESULTS coherence of the atom laser beam and is intimately related to the property that it can be described by a single wave function. Residual deviations from unity could arise from technical imperfections. Magnetic field fluctuations either due to current noise in the magnetic trapping coils or due to external fluctuations could imprint small intensity fluctuations onto the atom laser beam. We employ a low noise current source and magnetic shielding to minimize these effects. In addition, we use a highly stable microwave source which is stabilized to a GPS disciplined oscillator. A further contribution to a potential modification of the second order correlation function could be due to the output coupling process itself. The spatial correlation function of atoms output coupled from a weakly interacting condensate has been studied theoretically in a situation neglecting gravity [100]. The modification from a constant unity second order correlation function was on the order of one percent, which is on the same order of magnitude as the uncertainty in our data. Measuring the second order correlation function requires to detect the particles within their coherence time and coherence volume [165]. The uncertainty of the detection time of an atom must be smaller than the correlation time, because otherwise the correlations vanish [54]. We estimate that the acquired time delays resulting from a possibly misaligned detector are shorter than the dead time of our detector. It has been measured that the coherence time of the atom laser is given by the duration of output coupling at least for durations of 1.5 ms [42]. Moreover, for the total flux of 5.2 atoms per ms, the probability of having two or more atoms arriving within the dead time of our detector is 5%. Therefore multiatom arrivals do not significantly influence the data. Trapped Bose - Einstein condensates have been demonstrated to be phase coherent and to have a uniform spatial and temporal phase [146, 10, 105]. The atom laser beam has been theoretically described by a single wave function [100, 166] and its spatial coherence was observed [5]. Moreover, a full contrast interference pattern was observed between two atom laser beams extracted from separate locations inside a condensate [105]. This indicates a high degree of spatial overlap between the two propagating modes and a negligible distortion of the uniform spatial phase due to interactions with the remaining condensate. From this we conclude that the atom laser leaves the condensate region with a well defined spatial wavefront. Many overlapping spatial modes at the detector wash out the correlations. In our experimental geometry this is the case when output coupling from a thermal source, since we can not resolve a single diffraction limited spatial mode. Therefore we do not observe thermal bunching of noncondensed atoms. Determining the arrival times of all detected atoms explicitly allows us to extract the full counting statistics of the atoms. We choose a time bin length of T = 1.5 ms in which we count the number N of detected atoms and plot the probability distribution p (N ) in Figure 5.10. The distribution is close to a Poissonian distribution p (N ) = 〈n〉N e −〈n〉 /N ! (5.39) 97 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER 2.0 g(2)(τ) 1.5 1.0 0.5 0 0 3 6 9 time interval τ [ ms ] 12 15 FIGURE 5.9.: Second order correlation function of an atom laser beam. The data are binned with a time bin size of 50 µs. The average count number is 2 · 105 per bin. We have omitted the first two data points since they are modified by the dead time of our detector. probability p(N) 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 number of events N 9 10 FIGURE 5.10.: Probability distribution p (N ) of the atom number N detected within a time interval of T = 1.5 ms. The (+) symbols show a Poissonian distribution for the same mean value of 〈n〉 = 1.99 as the measured data. The errors indicate statistical errors. 98 5.4. RESULTS with a mean of 〈n〉 = 1.99. For the measured distribution we have calculated the 2nd, 3rd and 4th cumulant to be κ2 = 1.75, κ3 = 1.34 and κ4 = 0.69, respectively. We realize a direct comparison with a pseudo-thermal beam of atoms by output coupling a beam with thermal correlations from a Bose - Einstein condensate. This is in close analogy to changing the coherence properties of a laser beam by means of a rotating ground glass disc [44]. Instead of applying a monochromatic microwave field for output coupling we have used a broadband microwave field with inherent frequency and intensity noise. We have employed a white noise generator in combination with quartz crystal band pass filters which set the bandwidth of the noise. The filters operate at a frequency of a few MHz and the noise signal is subsequently mixed to a fixed frequency signal at 6.8 GHz to match the output coupling frequency. For atomic beams prepared in such a way we observe bunching with a time constant set by the band pass filter as shown in Figure 5.11. To compare our data with the theoretically expected correlation function we have measured the power spectra of the band pass filters and calculated |g (1) (τ)|2 of the rf field before frequency mixing. In Figure 5.11 we plot 1 + β|g (1) (τ)|2 . The normalization factor β = 0.83 accounts for the deviation of the experimental data from g (2) (0) = 2 due to imperfections in the frequency mixing process. For the pseudo-thermal beam we also calculate the counting statistics and find a significantly different behavior than for the atom laser case. For a filter with a spectral width (FWHM) of 90 Hz we have chosen the time bin length of T = 1.5 ms, smaller than the correlation time. The atomic flux with a mean atom number 〈n〉 = 1.99 is equal to the case of the atom laser. We compare the measured probability distribution to a Bose distribution p (N ) = 〈n〉N /(1 + 〈n〉)1+N , (5.40) which is expected for a thermal sample and find good agreement presented in Figure 5.12. From the distribution we have extracted the 2nd, 3rd and 4th cumulant to be κ2 = 4.6, κ3 = 14.5 and κ4 = 50.6, respectively. In comparison to the nonexclusive second order correlation function it is interesting to evaluate the exclusive probability when the next particle is detected [162]. Two cases can be distinguished. First, the probability P (t ) to detect a particle after a random initial time and secondly the conditional probability P c (0|t ) to detect a particle after an initial particle was detected. For an atom laser in a coherent state, both these cases should be totally random and therefore decrease exponentially P (t ) = ce −ct and P (0|t ) = ce −ct . (5.41) When normalized with the count rate c these exponential probabilities are unity for t = 0 as shown in Figure 5.13(a). The lines are fits with only the count rate c as a free parameter, which agrees with the experimental values. 99 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER 2.0 g(2)(τ) 1.5 1.0 0.5 0 0 3 6 9 time interval τ [ ms ] 12 15 FIGURE 5.11.: Second order correlation functions of pseudo-thermal atomic beams. The square symbols correspond to a filter band width (FWHM) of 90 Hz, the triangles to a bandwidth of 410 Hz, and the circles to a bandwidth of 1870 Hz. The data are binned with a time bin size of 50 µs in which the average count number is 8 · 104 . We have omitted the first two data points since they are modified by the dead time of our detector. The lines are the experimentally determined correlation functions of the broadband microwave fields. probability p(N) 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 number of events N 8 9 10 FIGURE 5.12.: Probability distribution p (N ) of the atom number N within a time interval of T = 1.5 ms for the 90 Hz bandwidth data. The (+) symbols indicate a Bose distribution with the same mean value of 〈n〉 = 1.99. The errors indicate the statistical errors. 100 5.4. RESULTS For a chaotic state, on the other hand, these exclusive probabilities decrease geometrically like ∼ t −2 and ∼ t −3 and are given by P (t ) = c and (1 + ct )2 P (0|t ) = 2c (1 + c t )3 (5.42) , (a) 1 P(0|t)/N P(t)/N respectively. The different exponent in the unconditional and conditional probability simply stem from the Bose - Einstein distribution (eq. 5.38) with n = 1 in the first case and n = 2 for the second. The fact that the conditional probability describes a two-particle phenomenon, in contrast to the unconditional one, is also manifested in the limiting value for t = 0. The conditional probability normalized by the count rate c tends to two, as does the second order correlation function, whereas P (t ) just goes to one as shown in Figure 5.38(b). Here, the lines are also fits with only the count rate c as a free parameter, which agrees with the experimental values. 0.1 0.01 0 (b) 1 0.1 0.5 1.0 1.5 2.0 time to first count t [ ms ] 2.5 0.01 0 0.5 1.0 1.5 2.0 time interval t [ ms ] 2.5 FIGURE 5.13.: Exclusive probabilities (normalized) when to detect the next atom. The data for the atom laser is shown in blue, the data for the pseudo-thermal beam with bandwidth 90 Hz is red. (a) Unconditional probability to detect the next atom after t for a random initial time. (b) Conditional probability to detect two atoms, one at 0 and the next one at t . It is 2 for t → 0 due to bunching. Investigating three particle correlations for thermal beams one expects to find increased correlations by a factor 3! = 6. This effect was predicted [58] and observed [59] in the reduced three body recombination for Bose - Einstein condensates compared to a thermal cloud. However, this only measures equal time and equal position correlations. We investigate the temporal third order correlation function g (3) (τ1 , τ2 ) which depends on two time differences τ1 and τ2 for a triple. We plot the symmetric three particle correlation function given by g (3) (τ1 , τ2 ) = 〈Ψ̂† (τ1 )Ψ̂† (0)Ψ̂† (τ2 )Ψ̂(τ2 )Ψ̂(0)Ψ̂(τ1 )〉 〈Ψ̂† Ψ̂〉 3 . (5.43) In Figure 5.14 the three particle bunching peak is shown for an atomic beam with a pseudothermal character, produced with the 90 Hz filter. The result is in good agreement with 101 5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER the expected shape. Clearly, the symmetry and the lower order correlation functions are visible. The peak value is ∼ 5 < 6, which is probably due to the nonlinear microwave mixing process or maybe due to the reduced Fano factor (eq. 5.36) for imperfect detection efficiency. However, since the second order correlation function also does not reach the ideal value of two, this is not so surprising. In contrast, the atom laser g (3) depicted in Figure 5.15, shows the absence of any third order correlations. This is in agreement with a coherent state, where all correlation functions factorize. 102 5.4. RESULTS 6 5 g(3)(τ1,τ2) 4 3 2 1 0 2 4 6 8 10 time interval τ [ m12 s] 1 14 16 18 20 20 4 0 8 ] 16 12 erval τ2 [ ms t in time FIGURE 5.14.: Symmetric third order correlation function g (3) (τ1 , τ2 ) for a pseudo-thermal atomic beam of 90 Hz bandwidth g(3)(τ1,τ2) 2 1 0 2 4 6 8 10 time interval τ [ m12 14 s] 1 16 18 20 20 4 0 8 ] 16 12 erval τ2 [ ms t time in FIGURE 5.15.: Symmetric third order correlation function g (3) (τ1 , τ2 ) for an atom laser beam 103 6 Cavity QED Detection of Interfering Matter Waves “. . . has in it the heart of quantum mechanics. In reality, it contains the only mystery.” - Lectures on Physics, Richard P. Feynman We observe the buildup of a matter wave interference pattern from single atom detection events in a double slit experiment. The interference arises from two overlapping atom laser beams extracted from a Rubidium Bose - Einstein condensate. Our detector is an ultrahigh finesse optical cavity which realizes the quantum measurement of the presence of an atom and thereby projects delocalized atoms into a state with zero or one atom in the resonator. The experiment highlights the granular nature of matter while simultaneously reveals its wave properties, because the atom flux is so low that on average there is only one particle at a time in the interferometer. This chapter has been published as [33]: T. Bourdel, T. Donner, S. Ritter, A. Öttl, M. Köhl, and T. Esslinger, Cavity QED detection of interfering matter waves, Phys. Rev. A. 73 (2006), 043602. 105 6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES 6.1. Introduction The prediction of the duality between particles and waves by de Broglie [167] is a cornerstone of quantum mechanics. Pioneering experiments have observed interferences of massive particles using electrons [168, 169], neutrons [170], atoms [171, 172, 173] and even large molecules [174]. However, the simple picture that matter waves show interferences just like classical waves neglects the granularity of matter. This analogy is valid only if the detector is classical and integrates the signal in such a way that the result is a mean particle flux. With quantum detectors that are sensitive to individual particles the discreteness of matter has to be considered. The probability to detect a particle is proportional to the square amplitude of the wave function and interferences are visible only after the signal is averaged over many particles. In the regime of atom optics, single atom detection has been achieved for example by fluorescence [175], using a micro-channel plate detector for metastable atoms [176], and high finesse optical cavities [20]. In these experiments the size of the de Broglie wave packet of the particles was much smaller than the detector area. Therefore localization effects during the detection have been negligible. With the realization of Bose - Einstein condensation in dilute gases particles with a wave function of macroscopic dimensions have become experimentally accessible. Only very recently the single atom detection capability has been achieved together with quantum degenerate samples reaching the regime of quantum atom optics [17, 18]. The quantum nature of the measurement opens perspectives for atom interferometry at and below the standard limit [36]. 6.2. Quantum Mechanical Measurement Process For atoms with a spatially extended wave function, such as in a Bose - Einstein condensate or in an atom laser beam, a measurement projects the delocalized atom into a state localized at the detector [177]. This quantum measurement requires dissipation in the detection process. For single atom detection we employ an ultrahigh finesse optical cavity in the strong coupling regime of cavity quantum electrodynamics (QED) [178, 21, 179]. We study this open quantum system including the two sources of dissipation, cavity losses and spontaneous emission. In particular, we calculate the time needed for the localization of an atom in the cavity measurement process. We then experimentally investigate atomic interferences on the single atom level using our detector. A schematic of our experimental setup is shown in Figure 6.1. We output couple two weak atom laser beams from a Bose - Einstein condensate and their wave functions overlap and interfere [105]. The flux is adjusted in such a way that there is on average only one atom at a time in the interferometer. Using the ultrahigh finesse cavity we measure the atom flux in the overlapping beams with single atom resolution. We observe the gradual appearance of 106 6.2. QUANTUM MECHANICAL MEASUREMENT PROCESS a matter wave interference pattern as more and more detection events are accumulated. Single atom detection in an optical cavity can be captured in a classical picture: an atom changes the index of refraction in the cavity and thereby brings it out of resonance from the probe laser frequency. In the absence of an atom, the probe beam is at resonance with the cavity and its transmission is maximal. Experimentally we use a probe power corresponding to five photons on average in the cavity. The cavity transmission noise is found to be dominated by photon shot noise and not by the quality of the lock. The presence of an atom results in a drop of the cavity transmission as shown in Figure 6.2. We set the threshold for an atom detection event to a drop in transmission of four times the standard deviation of the photon shot noise in our 20 µsintegration time. Then the overall detection efficiency of atoms extracted from a Bose - Einstein condensate is measured to be 0.23(8). The probability of an artefact detection is lower than 2 · 10−4 in the measurement time of 0.5 s. 36 mm BEC 100 μm laser cavity photon counter FIGURE 6.1.: Schematic of the experimental setup. From two well defined regions in a Bose Einstein condensate (BEC), we output couple atoms to an untrapped state. The real parts of the resulting atom laser wave functions are sketched on the right hand side. The absorption image showing an interference pattern corresponds to a flux ∼ 106 times larger than the one used for the actual single atom interference experiment. Monitoring the transmission through an ultrahigh finesse optical cavity with a photon counter, single atom transits are detected. 107 6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES cavity probe laser (a) single photon counter atom photon counts per 20 μs 100 (b) 80 60 40 20 0 0 0.2 0.6 0.4 time [ ms ] 0.8 FIGURE 6.2.: (a) Cavity single atom detection principle. An atom detunes the ultrahigh finesse cavity from resonance and the cavity transmission consequently drops. (b) Photon flux through the ultrahigh finesse optical cavity when an atom is detected. The photon count rate is averaged over 20 µs. The detection threshold is set to be 4 times the standard deviation of the photon shot noise (dashed line). Our cavity has been described before (see Section 3.3.1) and we only recall here its main figures of merit. Its length is 178 µm, the mode waist radius is w0 = 26 µm, and its finesse is 3 · 105 . The maximum coupling strength between a single 87 Rb atom and the cavity field g = 2π · 10.4 MHz is larger than the cavity field decay rate κ = 2π · 1.4 MHz and the atom spontaneous dipole decay rate γ = 2π·6 MHz. The probe laser and the cavity are red-detuned as compared to the atomic resonance such that a light force pulls the atoms to regions where the coupling is large, therefore enhancing the detection efficiency. To understand the actual detection process we study the dynamics of the atom cavity quantum system taking into account dissipation. We first consider a classical atom entering a simplified square shaped cavity mode so that its coupling to the cavity field increases suddenly to a constant value g . The cavity field is initially coherent with a few photons. We use a two level approximation for the atom description and assume a 30 MHz red-detuning of the probe laser compared to the atomic resonance. Our probe beam is linearly polarized orthogonally to the magnetic field direction. We therefore do not probe on a cycling transition. The 30 MHz detuning is an effective value chosen to match the experimental conditions. In the case of strong coupling the following dynamics occur. On a short time scale given by 1/g , the atom cavity system exhibits coherent oscillations. It progressively reaches an equilibrium state on a time scale given by 1/κ and 1/γ due to cavity loss and atomic spontaneous emission. These are the two sources of dissipation. In the equilibrium state, the mean photon number in the cavity is reduced and the cavity transmission drops. To evaluate this drop quantitatively, we find the steady-state of the master equation for the density matrix numerically [180, 181, 182, 110]. For our parameters, the transmission as a function of the coupling strength g is plotted in Figure 6.3. For a maximally coupled atom 108 6.2. QUANTUM MECHANICAL MEASUREMENT PROCESS g = 2π · 10.4MHz, the average intracavity photon number is found to be reduced from 5 to 0.9, and the number of detected photons is then reduced by the same ratio. Such a reduction corresponds well to the largest observed transmission drops with an example shown in Figure 6.2. The detection threshold corresponds to a coupling of g = 2π · 6.5MHz. Experimentally, unlike in our model, an atom feels a position dependent coupling as it transverses the mode profile. However the atom transit time through the cavity mode (40 µs) is long compared to the cavity relaxation time scales 1/κ and 1/γ and the atom cavity system adiabatically follows a quasi equilibrium state. Therefore the experimental transmission drops can be compared to the calculated ones. 1.0 (a) 0.8 0.6 0.4 (b) 0.8 coherence transmission 1.0 0.6 0.4 0.2 0.2 0 0 0 2 4 6 8 10 12 coupling g0 [ MHz ] 0 0.1 0.2 0.3 time [ μs ] 0.4 0.5 FIGURE 6.3.: (a) Normalized transmission as a function of coupling strength. The solid line corresponds to our probe strength of 5 photons in the cavity in the absence of an atom. The dashed line is the weak probe limit. The dotted line corresponds to 10 photons in the cavity. (b) Coherence between the states with one and no atom as a function of time. The initial coherence is normalized to 1. Solid line: g = 2π· 10MHz. Dashed line: g = 2π· 6.5MHz. Dotted line: g = 2π · 3MHz. Specific to our experiment is that a spatially extended matter wave and not a classical atom enters the cavity. Our system allows us to realize a quantum measurement of the presence of an atom. For our low atom flux, we can neglect the probability of having more than one atom at a time in the cavity. The incoming continuous wave function is thus projected into a state with one or zero atom in the cavity. This projection necessarily involves decoherence that is introduced by spontaneous scattering and cavity photon loss. The origin of the decoherence can be understood as unread measurements in the environment [181, 177]. For example, if a spontaneously emitted photon is detected, there is necessarily an atom in the cavity and the wave function is immediately projected. Similarly, the more different the light field with an atom in the cavity is from the one of an empty cavity, the more different is the scattered radiation out of the cavity, and the projection occurs correspondingly faster. We now quantify the time needed for the projection to occur. For simplicity, rather than a continuous wave function, we consider a coherent mixture of one and zero atom entering 109 6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES a square shaped cavity at a given time. We take the limit when the probability to have one atom is low. The initial cavity field is the one of an empty cavity. Dissipation effects are studied by computing the time evolution of the density matrix [182]. The degree of projection of the initial state can be extracted from the off-diagonal terms between states with one atom and no atom in the density matrix. More precisely, we define the coherence as the square root of the sum of the square modulus of the off-diagonal terms mentioned above. This quantity is maximal for a pure quantum state with equal probability to have an atom or not. The coherence is zero for a statistical mixture. In Figure 6.2, the evolution of the coherence is plotted. As expected, it decays to zero at long times due to dissipation. The decay time increases as the coupling to the cavity is weakened. In the limit where the coupling vanishes, the coherence is preserved. The atomic wave function then evolves as if there was no cavity. For g > 2π · 6.5MHz, the decoherence time is found to be a fraction of a microsecond. This value is much lower than the 40 µstransit time of an atom through the cavity and for all our detected atomic transits, the wave function is thus well projected to a state with one atom, realizing the quantum projection theorem. Our detection scheme realizes a quantum measurement of the presence of an atom in the cavity. However during an atom transit some photons are spontaneously scattered and the velocity of an atom is slightly modified. 6.3. Buildup of Matter Wave Interference Using our cavity detector, we can observe matter wave interferences on the single atom level. The starting point of the experiment is a quasi pure Bose - Einstein condensate with 1.5 · 106 Rubidium atoms in the hyperfine ground state |F = 1, mF = −1〉 [17]. The atoms are magnetically trapped with frequencies ωk = 2π · 7 Hz axially and ω⊥ = 2π · 29 Hz radially. A weak and continuous microwave field locally spin flips atoms from the Bose - Einstein condensate into the untrapped |F = 2, mF = 0〉 state. This process is resonant for a section of the condensate where the magnetic field is constant. Because the magnetic moment of the spin flipped atoms vanishes they fall due to gravity and form a continuous atom laser [11]. When we apply two microwave fields, we are able to output couple atom laser beams from two well defined regions of the condensate [105]. The two distinct atom laser wave functions overlap and interfere. At the entrance of the cavity, the atomic wave function ψ is well described by the sum of two plane waves with the following time dependence ψ(t ) ∝ exp(i ω1 t ) + exp(i ω2 t + φ) (6.1) ∝ cos((ω2 − ω1 )t /2 + φ) , where ħω1 and ħω2 are the energies of the two laser beams and φ is a fixed phase difference. The radial dependence of the wave function is neglected. The probability to detect 110 6.3. BUILDUP OF MATTER WAVE INTERFERENCE an atom is given by the square norm of the wave function which is modulated in time and behaves like a cosine square. The modulation frequency of the interference signal is given by the energy difference between the two atom lasers. Experimentally, it is determined by the frequency difference of the two microwave fields and is chosen to be ∆ f = 10 Hz, which corresponds to a distance of 5 nm between the two output coupling regions. The two microwave fields are generated such that the interference pattern is phase stable from one experimental run to the other. 1 iterations of the experiment 191 time [ ms ] 100 200 300 400 500 atom counts per 5ms time bin FIGURE 6.4.: Histograms of the atom detected in 5 ms intervals. (a) Single experimental run, (b) sum of 4 runs, (c) sum of 16 runs, and (d) sum of 191 runs, where the line is a sinusoidal fit. Please note the different scales. The results of the experiment are presented in Figure 6.4. Each experimental run corresponds to output coupling from a different condensate and on average ∼ 6 atoms are detected in 0.5 s. After the detection of a few atoms, their arrival times appear to be random (Figure 6.4 a). Nevertheless, after adding the results of several runs, an interference pattern progressively appears (Figure 6.4 4b-d). The atom number fluctuation is found to be dominated by the atomic shot noise and the signal to noise ratio of the interference increases as more data are included. A fit to the histogram leads to a contrast of 0.89(5). The slight reduction of contrast can be attributed to an extremely low uncontrolled atomic flux corresponding to one atom every 3 or 4 runs observed in the absence of output coupling. Those 111 6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES atoms could be output coupled either by a stray microwave field or by scattered light from the cavity mirrors. We work with a flux of one detected atom per 83 ms, which is about the time an atom needs to travel from the condensate region to the cavity. We are thus in a regime where the atoms fall one by one in the interferometer. A single atom behaves both like a wave because its time arrival probability shows an interference pattern and like a particle as single atoms are detected. This can be similarly expressed by saying that each individual atom is released from both slits simultaneously. Our experiment is an atomic counterpart of the Young’s double slit experiment with individual photons. In conclusion we detect matter wave interferences with an ultrahigh finesse optical cavity detector which realizes a quantum measurement of the presence of an atom. It is explained how dissipation plays a crucial role in the detection process and for the localization of the atom inside the cavity. Using this detector, we are able to detect a high contrast atom interference pattern at the single atom level. The coupling of a matter wave to a cavity QED system opens the route to the quantum control not only of the internal state of the atoms but also of their positions [183]. Using the presented detection technique we can probe an atomic gas with a good quantum efficiency and introduce only a minimum perturbation through the measurement. This could facilitate nondestructive and time resolved studies of the coherence of a quantum gas, for example during the formation of a Bose - Einstein condensate. Similar interference experiments between two distinct condensates would permit investigations of their relative phase evolution [184, 185] or diffusion [186]. 112 7 Conclusion Within the scope of this thesis an apparatus was designed, constructed and successfully tested that achieves the fusion of quantum degenerate gases with the single atom detection ability of cavity QED in the strong coupling regime. The challenges to experimentally merge these two fields were overcome by forging new paths for the Bose - Einstein condensation setup and the ultrahigh finesse optical cavity design. The novel system distinguishes itself by a very reliable and reproducible operation for the production of large Bose - Einstein condensates and superior stability for the output coupling of continuous atom lasers. It features flexibility for research on and applications of atom lasers through the vast, free and accessible half-space below the BEC. In particular, we have implemented a modular and interchangeable science platform, which, in the current implementation, incorporates the very compact realization of an ultrahigh finesse optical cavity design on top of a proper UHV compatible vibration isolation system. With this experimental setup we are able to detect single atoms from a quantum degenerate source with high efficiency by aiming the atom laser into the cavity mode. We have demonstrated the coherence of atom lasers by measuring a constant second and third order correlation function in contrast to pseudo-thermal atomic beams, which show bunching behavior. Moreover, we have recorded the full counting statistics of atomic beams, confirming a Poissonian distribution for the atom laser and a Bose - Einstein distribution for the chaotic beam. We detect matter wave interferences with an ultrahigh finesse optical cavity detector which realizes a quantum measurement of the presence of an atom. We explain how dissipation plays a crucial role in the detection process and for the localization of the atom inside the cavity. Using this detector, we are able to detect a high contrast atom interference pattern at the single atom level. This opens the potential to probe the coherence of two distinct Bose - Einstein condensates, the evolution [184, 185] and diffusion [186] of their relative phase and possibly the establishment of phase through the detection process [187]. Since the quantum measurement in the cavity projects the coherent beam onto a number state, the atomic state leaving the cavity is highly squeezed. It would be interesting to exploit this prepared number squeezed state by performing a readout measurement with a second cavity, which could be realized in a new implementation of the modular science platform. 113 7. CONCLUSION Our current system facilitates research in the field of cavity QED, because the atom laser is a source of unprecedented brightness and provides very high and controllable reproduction rates of atoms coupling with the cavity mode. On the other hand, the ultrahigh finesse cavity functions as a single atom detector in coherent matter waves which develops the new field of quantum atom optics. It is an extremely sensitive tool to detect atomic beams for high precision interferometry measurements and to investigate particle correlations. Moreover, the capability of detecting single atoms and their coherence is especially useful to probe cold atomic clouds in situ and time resolved in an essentially nondestructive manner. This allows the observation of kinetics and coherence growth in the process of Bose Einstein condensation and the mapping of the correlation length across a phase transition. This behavior is scale invariant and governed by universal critical exponents which could be measured with our setup. Future prospects with the system include single molecule detection and the setup of a heterodyne detection technique [136] for the presence of an atom inside the cavity mode, which can potentially be nondestructive on the atomic quantum state. Furthermore, we intend to transport the Bose - Einstein condensate into the ultrahigh finesse optical cavity by loading it into the dipole potential of a moving optical standing wave and conveying it downwards over the distance of 36 mm into the cavity mode. There, the arrangement of coils around the cavity could be used to apply magnetic field gradients for tomography experiments. The lateral optical access enables us to create a three dimensional optical lattice inside the ultrahigh finesse optical cavity, which will open the route to study strongly correlated systems with single atom resolution. 114 A Appendix A.1. Breit - Rabi Formula The energy shift of the hyperfine states of an atom with a single valence electron, i.e., alkali atoms, in an external magnetic field B is described by the Hamiltonian HB = g e µB S · B + g I µN I · B + 2C S · I , (A.1) where S corresponds to the electron spin and I to the nuclear spin, and g e and g I denote the g -factor of the electron and the nucleus, respectively. µB is the Bohr magneton and µN the nuclear magneton. C = νhfs /(2 I + 1) is related to the frequency splitting νhfs between the hyperfine states at zero magnetic field. Diagonalizing (A.1) yields the Breit - Rabi formula [188] E (B ) = − ± hνhfs 2(2 I + 1) hνhfs 2 + µK g I B m F s 1+ g e µB − g I µN 2 4mF g e µB − g I µN B +( B) , 2I + 1 hνhfs hνhfs (A.2) which can be used to calculate the energy shift of a hyperfine state in the F = I ± 1/2 manifold with the magnetic quantum number mF at arbitrary magnetic fields. For not too high magnetic fields where F = J + I is a good quantum number, the hyperfine Landé g factor is calculated by gF = g J F (F + 1) + J ( J + 1) − I ( I + 1) 2F (F + 1) − gI F (F + 1) + I ( I + 1) − J ( J + 1) 2F (F + 1) . (A.3) 115 APPENDIX A.2. Physical Properties of 87Rb Mass Natural abundance Nuclear spin I Nuclear g I -factor Landé g J -factor for D transitions Hyperfine ground state splitting νhfs Vacuum Wavelength λD 1 (52 S 1/2 → 52 P 1/2 ) Natural linewidth ΓD 1 (FWHM) Vacuum Wavelength λD 2 (52 S 1/2 → 52 P 3/2 ) Natural linewidth ΓD 2 (FWHM) Saturation intensity I sat F = 2 → F 0 = 3 (σ± -polarized light, D 2 -line) Resonant cross section σ0 F = 2 → F 0 = 3 (σ± -polarized light, D 2 -line) Saturation intensity I sat F = 2 → F 0 = 3 (isotropically polarized light, D 2 -line) Resonant cross section σ0 F = 2 → F 0 = 3 (isotropically polarized light, D 2 -line) Scattering length a S (singlet) Scattering length a T (triplet) van der Waals coefficient C 6 86.9091835(27)u 27.83(2)% 3/2 -0.0009951414(10) 2.00233113(20) +6.83468261090429(9)GHz 794.9788509(8)nm 2π · 5.746(8)MHz 780.241209686(13)nm 2π · 6.065(9)MHz 1.669mW/cm2 2.907 · 10−9 cm2 3.576mW/cm2 1.356 · 10−9 cm2 90(1) a0 106(4) a 0 4691(23) a 06 α2 me c 2 TABLE A.1.: Properties of 87 Rb [130]. 116 A.3. D2-Line Energy Levels F=3 266.650(9) MHz 2 3 gF = 2/3 0 -1 (0.93 MHz/G) -2 -3 F=2 2 1 gF = 2/3 0 -1 (0.93 MHz/G) -2 F=1 1 gF = 2/3 0 -1 (0.93 MHz/G) 2 5 P3/2 1 156.947(7) MHz 72.218(4) MHz F=0 0 780.241 209 686(13) nm 384.230 484 468 5(62) THz 1.589 049 439(58) eV 12 816.549 389 93(21) cm-1 5 2S1/2 F=2 12 gF = 1/2 0 -2-1 (0.70 MHz/G) F=1 -1 0 6.834 682 610 904 29(9) GHz gF = -1/2 1 (-0.70 MHz/G) FIGURE A.1.: D2 line of 87 Rb. Energy level structure of the hyperfine and Zeeman sublevels [130]. 117 APPENDIX A.4. 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Mein Dank gilt: • zuvorderst und ganz herzlich Prof. Dr. Tilman Esslinger für die freundliche Aufnahme in seine Forschungsgruppe und die verwantwortungsvolle Aufgabe, ein so spannendes Experiment von Anfang an mitzugestalten. • Dr. Michael Köhl, der jederzeit mit Rat und Tat zur Seite stand und der mit seinem Elan und Wissen die Sache mächtig vorangetrieben hat. • Stephan Ritter, dem Optikmeister, für die hervorragende Zusammenarbeit über die Jahre hinweg und seine unermüdliche Diskussionsbereitschaft, nicht nur im Labor sondern auch auf dem Balkon. • Tobias Donner, Thomas Bourdel und Ferdinand Brennecke, mit denen es mir eine grosse Freude war zu experimentieren und die Physik der kalten Wolke heiß zu diskutieren. • Thilo Stöferle und Henning Moritz, den “alten Hasen” vom Gitter-Experiment, von deren Erfahrungsschatz und Hilfsbereitschaft wir viel profitieren konnten. • Ken Günter, Bruno Zimmermann und Niels Strohmaier, den “jungen Wilden”, für das tolle Klima, das sie mit ihrem Einsatz innerhalb der Gruppe schaffen. • bei unseren Diplomanden Roger Gehr, Gabriel Puebla-Hellmann und Robert Jördens, daß sie ab und zu auch meinen Rat einforderten. Ein besonderer Dank gebührt dem “TEX-bert” für seine Hilfsbereitschaft und typographischen Anregungen. • Alexander Frank für seine vielen tollen elektronischen Gerätschaften, ohne die wir die Geheimnisse des Atomlasers niemals hätten ergründen können, und seine Neugier, neue Ideen gleich in die Tat umzusetzen. • Jean-Pierre Stucki, Paul Herrmann, Hans-Jürg Gübeli und Peter Brühwiler mit seinen Mannen von der Werkstatt für die phantastische “Hardware”, die sie für unser Experiment geschaffen haben. 133 CREDITS • Veronica Bürgisser für ihre fleißige Arbeit im Büro, die uns unglaublich viele Dinge erleichtert hat, und für ihre Nachsicht bei meinen schlampigen Eintragungen. • Christian Schori, Martin Schultze, Sebastian Slama, Tobias Gemperli, Patrick Maletinsky und allen ehemaligen Mitgliedern der Gruppe, mit denen zu arbeiten mir eine grosse Ehre war. • meinen WG-Mitbewohnern und Liselotte für die gute Stimmung im Haus. • meiner Familie für den wunderbaren Platz den man Daheim nennt, und im Andenken an meine kürzlich verstorbene Oma. • und am allermeisten danke ich meiner Lisa ganz lieb für ihre tolle, aufopferungsvolle Unterstützung und ihre Geduld. 134 Publications • A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Correlations and Counting Statistics of an Atom Laser, Phys. Rev. Lett. 95 (2005), 090404 • T. Bourdel, T. Donner, S. Ritter, A. Öttl, M. Köhl, and T. Esslinger, Cavity QED detection of interfering matter waves, Phys. Rev. A. 73 (2006), 043602 • A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Hybrid apparatus for Bose-Einstein condensa- tion and cavity quantum electrodynamics: Single atom detection in quantum degenerate gases, Rev. Sci. Instr. 77 (2006), 063118 • M. Köhl, A. Öttl, S. Ritter, T. Donner, T. Bourdel, and T. Esslinger, Observing the time interval distribution of atoms in atomic beams, accept. for Appl. Phys. B ◦ S. Ritter, A. Öttl, T. Donner, T. Bourdel, M. Köhl, and T. Esslinger, Observing the Formation of Long-Range Order during Bose-Einstein Condensation, subm. to Phys. Rev. Lett. 135 Curriculum Vitae Personal Details ANTON W. ÖTTL Date of birth: 12. March, 1974 Place of birth: Salzburg, Austria Citizenship: German Rötelstrasse 63 8037 Zürich Switzerland phone: +41 44 63 32346 fax: +41 44 63 31254 [email protected] Education and Degrees 03/2002–07/2006 Swiss Federal Institute of Technology (ETH) Zürich, Switzerland PhD Research in Quantum Optics 09/2001–02/2002 Institute National Polytechnique de Grenoble (INPG), France Internship March 2001 Diplom 09/1997–08/1998 University of Toronto, Canada Exchange Program Baden-Württemberg - Ontario October 1996 Vordiplom 10/1994–03/2001 Albert-Ludwigs University in Freiburg, Germany Studies in Physics (major), Mathematics and Chemistry 07/2003–09/2004 Civil Service June 1993 Abitur 09/1984–07/1993 High School (Karlsgymnasium) Bad Reichenhall, Germany Honors and Awards Government of Canada Award 1999 Research Project at the University of Toronto (01–07/2000) US Patent # 6,552,301 Burst Ultrafast Laser Machining Method P. R. Herman, R. S. Marjoribanks, and A. Öttl 137