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Transcript
RICE UNIVERSITY Creating Strontium Rydberg Atoms by Xinyue Zhang A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree Master of Science Approved, Thesis Committee: F.B. Dunning, Advisor Professor of Physics and Astronomy T.C. Killian, Vice Advisor Professor and Chair of Physics and Astronomy Douglas Natelson Professor of Physics and Astronomy and Professor in Electrical and Computer Engineering Houston, Texas April, 2013 ABSTRACT Creating Strontium Rydberg Atoms by Xinyue Zhang Dipole-dipole interactions, the strongest, longest-range interactions possible between two neutral atoms, cannot be better manifested anywhere else than in a Rydberg atomic system. Rydberg atoms, having high principal quantum numbers n 1 and dipole moments that scale as n2 , provide a powerful tool to examine dipoledipole interactions. Therefore, we have studied the production and production rates of strontium Rydberg atoms created using two-photon excitation and have explored their properties in two distinct experiments. In the first experiment, very-high-n (n ∼ 300) Rydberg atoms are produced in a tightly collimated atomic beam allowing spectroscopic studies of their energy levels and their Stark effects. Simulations using a two-active-electron model, developed by our theoretical collaborators, allow detailed analysis of the results and are in remarkable agreement with the experimental results. The high density of Rydberg atoms achieved, ∼ 5 × 105 cm−3 , in this experiment will allow studies of strongly interacting Rydberg-Rydberg systems. The second experiment, in which a cold strontium Rydberg gas is excited in a magneto-optic trap, features an imaging technique offering both spatial and temporal resolution. We use this technique to observe and study the evolution of an ultra-cold strontium Rydberg gas which reveals the importance of Rydberg-Rydberg interactions in the early stages of this evolution. A strongly interacting Rydberg gas provides an opportunity iii to realize a very strongly-correlated ultra-cold plasma. Contents Abstract ii List of Illustrations vii List of Tables ix 1 Acknowledgment 1 2 Introduction 4 2.1 2.2 2.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Manybody Physics . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.4 Ultracold Neutral Plasma . . . . . . . . . . . . . . . . . . . . 11 2.1.5 Detecting and imaging ultracold Rydberg atoms . . . . . . . . 12 The Strontium Rydberg System . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Strontium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Theoretical Results and Background 18 3.1 Traditional Treatment of Two-electron System . . . . . . . . . . . . . 18 3.2 Two Electron model . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Rydberg Atoms in a electric field . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Classical Picture . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.2 Sr Stark Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 v 4 Experiment Setups and Techniques 4.1 35 Frequency Locked Diode Laser System . . . . . . . . . . . . . . . . . 35 4.1.1 Frequency Double High Power Diode Laser System[57] . . . . 36 4.1.2 HeNe Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1.3 Scanning Fabry-Perot Interferometer . . . . . . . . . . . . . . 39 4.1.4 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . 40 4.1.5 Locking Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1.6 Limitations and Alternatives . . . . . . . . . . . . . . . . . . . 41 4.2 Strontium Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Other Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . 47 4.3.1 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3.2 Interaction Region . . . . . . . . . . . . . . . . . . . . . . . . 48 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.4.1 Selective Field Ionization . . . . . . . . . . . . . . . . . . . . . 49 4.4.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 5 Sr Rydberg Atoms in a Collimated Atomic Beam 5.1 53 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.1.1 Even Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1.2 the odd isotope Sr . . . . . . . . . . . . . . . . . . . . . . . 58 5.1.3 Stray Fields Impact On Spectra . . . . . . . . . . . . . . . . . 62 87 6 Ultracold Rydberg Gas Evolution 66 6.1 Experimental Setup Overview . . . . . . . . . . . . . . . . . . . . . . 66 6.2 Principal Processes in Probing Sr Cold Rydberg Gas . . . . . . . . . 69 6.2.1 Autoionization . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2.2 Electron-Rydberg Collisions . . . . . . . . . . . . . . . . . . . 72 6.2.3 Penning Ionization . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.4 Blackbody Radiation induced Ionization . . . . . . . . . . . . 75 vi 6.3 Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.4 Results Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7 Conclusion and Outlook Bibliography 85 87 Illustrations 2.1 Crossover to Collective Many-body States . . . . . . . . . . . . . . . 8 2.2 Ultracold Neutral Plasma Creation Setup . . . . . . . . . . . . . . . . 12 2.3 Natural Linewidths and Transition Wavelengths of Principle Sr Levels 14 2.4 Sr+ Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Cooling Transitions towards Quantum Degeneracy . . . . . . . . . . . 16 3.1 Measured and calculated quantum defects in the single-electron excitation of strontium . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Calculated Excitation Spectra . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Hydrogen Rydberg Atoms in a Electric Field . . . . . . . . . . . . . 27 3.4 NonHydrogen Rydberg Atoms in a Electric Field . . . . . . . . . . . 28 3.5 Stark Map of Strontium Rydberg Atoms . . . . . . . . . . . . . . . . 32 3.6 Parabolic States Distribution of Strontium Stark States . . . . . . . . 34 4.1 TOPTICA Diode Laser System Schematics . . . . . . . . . . . . . . . 36 4.2 Vapor Pressure Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Schematic diagram of the oven assembly . . . . . . . . . . . . . . . . 45 4.4 Interaction Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Stark Map for Sodium . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1 Excitation Spectra For Sr Isotopes . . . . . . . . . . . . . . . . . . . 55 5.2 N ∼ 312 Spectra for Overlapping 86 59 Sr and 88 Sr . . . . . . . . . . . . viii 5.3 N ∼ 335 spectra 5.4 Anticrossing of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.5 Stay Field Limited High N Spectra . . . . . . . . . . . . . . . . . . . 65 6.1 Experiment Schematic, Diagram and Timing . . . . . . . . . . . . . . 67 6.2 Sheet Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.3 Autoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.4 l mixing schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.5 Laser-induced fluorescence imaging . . . . . . . . . . . . . . . . . . . 76 6.6 Dependence of the LIF signal on Rydberg excitation time . . . . . . . 77 6.7 LIF images of spontaneous evolution . . . . . . . . . . . . . . . . . . 79 6.8 Visible Ions in Spontaneous Evolution . . . . . . . . . . . . . . . . . . 80 6.9 Collision Time Vs Initial Inter-Rydberg atoms Distance . . . . . . . . 83 87 Tables 2.1 Scaling Laws for Rydberg Atoms . . . . . . . . . . . . . . . . . . . . 5 2.2 Principal Isotopes of Strontium . . . . . . . . . . . . . . . . . . . . . 14 5.1 Sr Quantum Defects [65] . . . . . . . . . . . . . . . . . . . . . . . . . 58 1 Chapter 1 Acknowledgment When I asked Dr. Barry Dunning if I had the honor to join his group three years ago, I thought I had small chance to be admitted. I had clear idea of what I had to offer: my B.S. was in geophysics, my English was terrible, my experimental experience was zero and there were very limited number of equipments that I could lift in the lab. Yet, he decided to give me a chance. Because of my poor background, Barry had to painstakingly teach me the basic experimental skills and to make me useful in the lab. Over the past years, there was not even once has he lost his patience or said one harsh word to me for the various mistakes I made (everyone who has worked with me should understand how difficult that was since I can be very bullheaded and cheeky sometimes). On the contrary, he is always encouraging me, helping me to improve and never giving up on me. I never ever thought it is possible for a person to be so nice and kind before I knew Barry and I am so grateful everyday for having him as my advisor. In addition to his guidance on experiments, he has made so much contribution to the writing of this thesis. He revised it from structure, contents to grammar and even punctuations for more than ten times and he managed to upgrade it from a mindless talking to a professional, well-written thesis. For that, I couldn’t thank him enough! I feel very lucky to have Dr. Tom Killian as my second advisor who is also a remarkable teacher and a very kind person. He spent so much time in teaching me techniques on laser, fiber, optics and electronics from scratch with great patience. At 2 early times, he would generously devote many hours of his day in just helping me with optical setup and alignments and making sure that I am not doing anything foolish. Moreover, he is always trying his best to help me understand concepts in atomic physics even if that means he has to explain one thing for quite a few times in different perspectives. Just like Barry, he tolerated my ignorance, weak background, my sometimes very annoying personalities and has been trying to build something out of me. His efforts greatly helped me make through my research and are very much appreciated. Both Barry and Tom are such dedicated scientists who have been great role models for me to look up to. Having the opportunities to work with both of them makes me smile in my dream. All the people in Barry’s and Tom’s groups have also played important roles in helping me complete this work and lightening up my every day at school. My senior students, Shuzhen Ye and Patrick McQuillen, who suffered the worst of me and yet they are still my great friends and teachers. Mi Yan, Brian DeSalvo and Trevor Strickler can always put aside their work and happily discuss all my whimsy questions. Changhao Wang, Yu Pu, Ying Huang and Micheal Kelley never said no to help me. Francisco Caremo is always a delight to work with. My buddies here at Rice including Ernie Yang, Yang-Zhi Zhou, Zhentao Wang, Sidong Lei, Alicia Chang, Ksenia Bets, Jason Ball and many more offered me their friendships that I have always cherished. Every each of these people made my life here simply lovely and I’d love to express my gratitude for all of them. At last, I would also like to thank Dr. Han Pu, Dr. Stan Dodds, Dr. Huey W. Huang, Dr. Anthony Chan and Dr. Randy Hulet for their helps, trusts and insights and I want to thank Dr. Douglas Natelson for keeping setting aside his time to make my thesis defense possible. I’d like to end this part with my greatest appreciation to 3 my grandmother whom I will always love unconditionally. 4 Chapter 2 Introduction Atoms in which one electron is excited to a state of large principal quantum number n, termed Rydberg atoms, have been studied extensively because they possess extreme physical characteristics unlike those normally associated with atoms in ground or lowlying excited states. This is illustrated in Table 2.1 which lists a number of atomic properties, their n dependencies and their values for selected n levels of interest in this work. Since the classical Bohr radius of an atom scales as n2 , Rydberg atoms are physically very large and many of their properties can be described in terms of the classical Bohr model of the atom. Their binding energies, which scale as n−2 are very small. Because of their large size and weak binding, Rydberg atoms can be strongly perturbed and even ionized by modest external electric fields, the threshold for ionization scales as n−4 . The classical electron orbital period which increases as n3 , is large allowing, for example, application of electric field pulses whose duration is much smaller than the electron’s orbital period. At high n the spacing between adjacent levels which decreases as n−3 becomes very small. The radiative lifetimes of Rydberg atoms are also very large resulting in very narrow spectral features. Table 2.1 also includes other atomic properties pertinent to the present study including their polarizability, dipole moment and hr−4 i,hr−6 i (which determine the relative contributions to the polarization energy Wpol from dipole and quadrupole core polarization). The dipole moments listed in the Table 2.1 are strictly speaking transition dipole 5 Table 2.1 : Scaling Laws for Rydberg Atoms Property Scaling Na(10d) n = 50d n = 312d (a.u.) orbital radius n2 147a0 0.368µm 14.3µm orbital period n3 0.15ps 18.8 ps 4.56 ns binding energy 1/n2 0.14eV 5.6meV 0.14meV energy spacing 1/n3 0.023 eV 0.18meV 0.75µeV ionization field 1/n4 30 kV/cm 48V/cm 32 mV/cm radiative lifetime n3 1µs 125µs 30.4ms dipole momenthnd|er|nf i n2 143 ea0 3.58×103 ea0 139 × 103 ea0 polarizability MHz cm2 /V2 n7 0.21 1.64 × 1010 6.04 × 1015 hr−4 i 2n5 (l+3/2)(l+1)(l+1/2)l(l−1/2) hr−6 i 35n4 −5n2 [6l(l+1)−5]+3(l+2)(l+1)l(l−1) 8n7 (l+5/2)(l+2)(l+3/2)(l+1)(l+1/2)l(l−1/2)(l−1)(l−3/2) 3n2 −l(l+1) 6 moments, not a permanent electric dipole moment. Atoms in zero field don’t have permanent electric dipole moments! In a classical picture, high lm states have near circular orbits, and a near zero net dipole moment. For low l states, the highly elliptical orbits will precess around the nucleus due to core scattering resulting again a vanishing net permanent dipole moment. However, there are a number of ways to create Rydberg atoms with a large permanent dipole moment. One novel way, as suggested in [36] is to create trilobite molecules in a Bose-Einstein Condensate which represent a class of ultra-long-range homonuclear diatomic Rydberg molecules that possess a permanent electric dipole moment in the order of kilodebye and some experimental success in this direction has been achieved [37]. Another approach to create quasi-one-dimensional states is to selectively excite extreme red-shifted Stark states in the presence of a DC field. The selection rules only allow creation of low-l Rydberg states through photo-excitation. For the alkali and alkaline-earth metals, such states are difficult to polarize due to core scattering and initially only display quadratic Stark effects in a DC field, indicating a small induced dipole moment. However at higher fields these states can mix with the extreme strongly-polarized components of the Stark manifold and can themselves become very polarized. Upon obtaining a good quality quasi-one-dimensional (quasi-1D) state, it is straightforward to convert it to a near-circular, two-dimensional, “Bohr-like” state by application of an appropriate electric field pulse perpendicular to the atomic axis. 2.1 Motivation In the present work we are developing techniques to create strontium Rydberg atoms as a first step towards new projects involving strongly-coupled manybody systems 7 and the creation of long-lived two-electron excited states. Many of the proposed experiments will take advantage of dipole blockade. Rydberg blockade [1] is a manifestation of the strong, long range interaction between Rydberg atoms. This interaction could be of the Van der Waals type which varies with the inter-particle distance R as C6 /R6 , whereas for Rydberg states with permanent dipole moments the interaction will be of dipole-dipole type with form C3 /R3 . Because of these interactions, excitation of one Rydberg atom shifts the energy levels of neighboring Rydberg atoms and prevents their excitations using the same narrow linewidth laser. The resulting “dipole blockade” radii can be large (∼ 5µm at n ∼ 50 and ∼ 100µm at n ∼ 300). Although initially proposed as a means to create fast quantum gates for neutral atoms [2], Rydberg blockade has been shown to be an extremely versatile tool with many applications in areas such as condensed matter physics, plasma physics, nonlinear optics and quantum information. In the past decade, exciting progress has been made in each field that could be paradigm changing in the future. 2.1.1 Manybody Physics Rydberg atoms, blessed with their large electric dipole moments, interact strongly permitting the simulation of a wide variety of condensed matter systems. Furthermore, as pointed out in early theoretical work [3], quantum information processing could be based on the collective states of mesoscopic atomic ensembles due to Rydberg dipole blockade effects. Rydberg blockade effects have been the subject of a lot of experimental interest. One of the first demonstrations [6] involved tuning an np state to the middle of the adjacent ns and (n + 1)s states through Stark shifts induced by application of a DC 8 field. A 30% suppression of Rydberg excitation was observed at a Förster resonance. Direct van der Waals blockade was also observed [7]. In 2009, two independent groups demonstrated [9, 8] the collective excitation of two blockaded Rydberg atoms [10]. It was shown theoretically that full control of the strength, shape and character of the interaction potential is possible by weakly dressing Rydberg atoms contained in a Bose-Einstein Condensate [11]. In addition, by adjusting experimental parameters like the detuning, it is experimentally feasible to crossover from two body interactions to many body interactions; see Figure 2.1. Figure 2.1 : Crossover to Collective Many-body States. (a) ground states |gi dressed with Rydberg states |ri which are excited by a two-photon transition via the intermediate state |pi. The total detuning for this two-photon transition is ∆, ∆ = ∆p + ∆r . (b) Diagram for the crossover from two-body to many-body interaction, Ω = Ωr + Ωp . Figure adapted from [11]. The quest for exotic quantum phases can also be realized in a blockaded ultracold Rydberg ensemble. Inspection of the experimental data revealed the existence of a dimensionless parameter and an algebraic scaling law (characteristics of a second order phase transition) for an ultracold Rydberg gas. In other words, a frozen Rydberg system can be employed to study phase transitions in a precise, controllable manner. 9 Thus the capabilities that have been developed to coherently engineer the interactions in a many-body system, and the abilities to address and manipulate these “superatoms” individually due to their huge size, demonstrate the potential of Rydberg systems as quantum simulators [16]. Here “superatom” is a term used to symbolize the large spherical volume formed by a Rydberg atom and all the consequently blockaded ground state atoms within its dipole blockade radius. The crystalline phase can be explored with Rydberg atoms. Dipole blockade was proposed as a means to obtain dynamical crystallization through the use of a chirped laser pulse [12], the Rydberg excitation number being predicted to display a staircase structure. The mechanism is easy to understand (assuming the Rydberg interaction is repulsive, i.e. C6 > 0): with the excitation laser initially red detuned from the Rydberg atom transition, the collective many-body ground state for the ensemble will be one in which every atom is in its one-body ground state, as in the system’s Fock number state |0i. As the laser chirps towards resonance with the Rydberg level, a single atom in the ensemble will be excited to a Rydberg state whereupon further excitation will be prevented by dipole blockade, so the system jumps to number state |1i. When the laser is blue detuned enough to compensate the smallest dipoledipole energy shift, which is that between the first Rydberg atom and the furthest ground state atom, that particular ground atom will be excited, the system jumping to number state |2i, . . . As chirping continues, ground state atoms, in well-ordered positions will be excited one by one resulting in a crystalline structure comprising an array of dipole blockade superatoms. Creation of a supersolid, a novel phase simultaneously displaying crystal rigidity and dissipation-less flow, has been an experimental challenge for decades and so far has not been achieved. This peculiar phase requires a repulsive two-body potential 10 that softens at short distances and a long system lifetime to allow formation and observation of this phase. Theoretically, both requirements can be met in an ultracold atomic ensemble in the Rydberg blockade regime [14, 15]. The artificial potential can be mimicked and controlled by subjecting the Rydberg atoms to a homogeneous electric field in which Rydberg atoms possess large permanent dipole moments. Use of an off-resonant two-photon transition to properly “Rydberg dress” the ground state atoms can significantly reduce the photon scattering, thereby increasing the lifetime of the system. 2.1.2 Photonics Photons don’t interact with each other. Thus entanglement between photons does not come naturally. So far, experimentalists have resorted to spontaneous parametric down-conversion to make pairs of entangled photons [17]. There has been a lot experimental and theoretical work suggesting an effective mapping of the strong Rydberg interactions in the collective ultracold ensemble to photons. For instance, the electromagnetically induced transparency (EIT ) obtained by driving transition to a Rydberg level [18] is non-linearly influenced by the character of the Rydberg-Rydberg interactions [19]. Recent work has also shown [20] the potential of Rydberg system to applications like a fast single photon source, quantum teleportation, and the fast entangling of spin waves. The photon retrieved from a superatom has instant second order auto-correlation g (2) (0) as small as 0.075 and a single photon generation efficiency of 10%, not far from that of its well-developed quantum dot counterpart [21]. 11 2.1.3 Quantum Gates One of the very first proposed applications of Rydberg blockade was the implementation of quantum gates in an ultracold neutral gas [2]. The strong, long-range interaction between Rydberg atoms can be coherently turned on and off in a short time which will result in very fast, high fidelity quantum gates. Two advantages of quantum gates using neutral atoms are their straightforward scalability to create multi-qubit registers and their weak coupling to external field noises. The controlledZ gate and CNOT gate which form a complete set of universal gates for quantum computing have already been realized in the neutral system [22]. 2.1.4 Ultracold Neutral Plasma Ultracold neutral plasmas in which the Coulomb potential energy of interaction between its constituents ECoulomb , are greater than their thermal energies ET hermal , represent an exciting new frontier in plasma physics. Such plasmas can be created by near-threshold photo-ionization of atoms contained in a cold cloud (see Figure 2.2). The ionized photo-electrons, which have energies ∼ 1K, begin to escape the cloud and leave their ion cores behind. This process quickly terminates as the cloud builds up a net positive charge creating a Coulomb potential well from which further electrons cannot escape. Photoionization, however, produces ions distributed throughout the cold atom cloud, and the ions are therefore disordered. As the ions relax to a more ordered state, they are heated on a timescale of 100 ns, resulting in disorder induced heating (DIH) and ion temperatures of a few Kelvins rather than the mK temperatures characteristic of the parent laser-cooled neutral atoms. The plasma coupling parameter τ = ECoulomb /EKinetic is thus dramatically reduced [34]. This can be mitigated by exploiting dipole blockade to create an ordered cloud of Rydberg atoms 12 Figure 2.2 : Ultracold Neutral Plasma Creation Setup Figure adapted from [33]. and then ionizing these through pulsed electric field ionization [32]. This will allow creation of plasmas with much larger and controllable values of τ and the exploration of a new plasma physics regime. 2.1.5 Detecting and imaging ultracold Rydberg atoms Over the past few years, two major techniques have been employed to image ultracold Rydberg atoms. The first exploits traditional electric field induced ionization. To obtain an image, a position sensitive multi-channel plate detector (M CP ) is generally required [23]. Higher resolution can be achieved by using Field Ion Microscopy as in [24]. The magneto-optical trap (M OT ) is located at the center of a closed cage made of 10 independent electrodes that are used to minimize stray fields. The imaging electrode tip (125µm in radius) projects the field ionized Rydberg atoms onto the MCP detector. By accumulating many images and studying their auto-correlation, Rydberg blockade can be seen. A second approach is optical in situ fluorescence imaging of the Rydberg atoms by stimulated deexcitation and has produced the first observation of the spatially 13 ordered components of the Rydberg-blockade-induced many-body states that formed inside of a mesoscopic system [25]. The images’ temporal and spatial resolution is unprecedented. 2.2 The Strontium Rydberg System Since strontium Rydberg atoms are the focus of the present work, their particular properties are now discussed. 2.2.1 Strontium Strontium, an alkaline earth metal with two valence electrons possessing both singlet and triplet levels, has been the subject of numerous studies in the literature. The singlet 1 S0 ground state makes it immune from magnetic Zeeman splitting. In addition, strontium has a wealth of bosonic (88 Sr,86 Sr,84 Sr) and fermionic (87 Sr) isotopes. The relative natural abundances of these isotopes are listed in Table 2.2 together with their nuclear spins (only the 87 Sr isotope has a nuclear spin) and the isotope shifts for the principal 5s21 S0 → 5s6p1 P1 transition. Transition wavelengths and natural linewidths for transitions between its lowest lying levels are shown in Figure 2.3. Excitation to a high-l Rydberg state leaves an optically active core ion that behaves much as an independent ion. The energy level structure of the core ion is shown in Figure 2.4. Absorption/Fluorescence on the 2 S1/2 → 2 P1/2 , 2 P3/2 transition can then be used to image and manipulate strontium Rydberg atoms. The properties of strontium highlighted above have enabled a number of interesting experiments that are outlined below. 14 Figure 2.3 : Natural Linewidths and Transition Wavelengths of Principle Sr Levels. Intercombination lines are 1 S0 → 3 P . Figure adapted from [31]. Table 2.2 : Principal Isotopes of Strontium Isotope Atomic Natural I Mass Abundance(%) F 1 S 0 → 1 P1 Scattering Shift(MHz) Length(a0 ) 84 Sr 83.913 0.56 0 - -270.8 122.7 86 Sr 85.909 9.86 0 - -124.5 823 7/2 -9.7 9/2 -68.9 11/2 -51.9 - 0 87 88 Sr Sr 86.908 87.905 7.00 82.58 9/2 0 96.2 -1.4 15 Figure 2.4 : Sr+ Transitions Frequency Standards Due to hyperfine mixing, the strongly forbidden transition (∆s 6= 0, ∆j = 0), 5s2 1 S0 → 5s5p 3 P0 is weakly allowed for 87 Sr. The natural linewidth of this transition is only 1mHz allowing its use as an atom frequency standard [40]. The atoms must be cooled to µK to reduce the Doppler shifts and to allow trapping of a large number of atoms in an optical lattice. Trapping atoms in the antinodes of the lattice results in an ac Stark shift on the clock transition. However, a magic wavelength exists [39] for cancellation of the upper and lower Stark shifts and results in a negligible light shift. Currently, the strontium optical lattice clock is the best optical atomic frequency standard and has been used to measure fundamental constants. Quantum Degenerate Gases For the spinless strontium singlet ground state 1 S0 , the traditional technique of evaporative cooling in a magnetic trap is no longer applicable. Nonetheless, quantum degeneracy has already been achieved using all the 16 Figure 2.5 : Cooling Transitions towards Quantum Degeneracy. Solid lines are driven by lasers, dashed lines are the spontaneous decay path. Figure adapted from [30]. principle isotopes of strontium [27, 28, 29]. The procedures employed are similar and all-optical [27] and can be understood by reference to Figure 2.5. A blue laser (461nm) red detuned from the 1 S0 → 1 P1 is used to Zeeman slow and twodimensionally collimate the atomic beam. Atoms are then further cooled in a 461nm MOT. With repeated cycling, some atoms start to accumulate in 3 P2 level (through path(5s5p)1 P1 → (5s4d)1 D2 → (5s5p)3 P2 transitions ). When sufficient atoms have been trapped, a 3-micron laser pulse is used to transfer the 3 P2 atoms back to ground state via the transitions (5s5p)3 P2 → (5s4d)3 D2 → (5s5p)3 P1 → (5s2 )1 S0 . The 461nm blue MOT is then extinguished, and a red MOT operating on the 1 S0 → 3 P1 transition is turned on to further cool the atoms prior to loading into an 1.06µm optical dipole trap (ODT). The atoms are then further cooled by lowering the trap depth, evaporative cooling resulting in degeneracy. 17 2.3 Thesis Outline The main part of this thesis will focus on the results of two recent experiments [35, 50] designed to study the excitation of very-high-n (n ∼ 300) Rydberg states and to explore the evolution of cold Rydberg gases towards an UNP by imaging the core ions. 18 Chapter 3 Theoretical Results and Background 3.1 Traditional Treatment of Two-electron System Compared with the simple hydrogen atom, alkali Rydberg atoms are more complicated in the sense that the closed-shell-core can be penetrated and polarized. The resulting effects can be well characterized by one l-dependent quantum defect δl . However, for alkaline-earth elements, things are far more complicated due to the interaction of the two valence electrons. Even though one of the electrons is promoted to a Rydberg level, the strong short-range scattering with the one-active-electron core leads to strong configuration mixing. For each term S L, this interaction among configurations can be described in a set of parameters in Multichannel Quantum Defect Theory (MQDT) [49, 48]. When two electrons are close r12 < r0 , they can exchange angular momentum, spin and energy via their Coulomb interaction 1/r12 without violating their overall conservation. In this regime of free-energy-exchange, a proper set of basis wavefunctions can only be obtained by diagonalizing a scattering matrix S. This yields a set of Φα “eigenchannels” with eigenvalues µα . These eigenchannels are formed from a mixture of different configurations. Energy exchange becomes negligible once r12 > r0 due to the diminishing overlap of the wavefunctions of the two electrons and the falloff of their interaction 1/r12 . The outer electron can then be described as a superposition of collision channels. Each eigen-collision-channel is then a pure configuration labeled 19 by quantum number νi , νi = p R/(Ii − E), (3.1) where Ii is the ionization limit of the ion core in this collision channel, R is the mass corrected Rydberg constant and E is the energy of the system(νi can be regarded as the quantum defect for the ith channel).The eigenenergies of the system for any r12 are found by connecting the two sets of wavefunctions in regimes r12 < r0 and r12 > r0 via a transformation matrix Uiα and applying appropriate boundary conditions at infinity. The nontrivial solution requires Det|Uiα sin π(νi + µα )| = 0. (3.2) The bound eigenenergies of the system are found by adjusting the µα and Uiα that simultaneously satisfy Equation 3.1 and Equation 3.2 until agreement with experimental data is reached. This method can also determine the admixtures of the different configurations. For example, for Sr J=2 bound states, it has been shown that the most important channels are 5snd1 D2 , 5snd 3 D2 , 4dns 1 D2 , 4dns 3 D2 , and 5pnp 1 D2 . For the 5s15d 1 D2 state, there is almost a 40% admixture from 5snd 3 D2 . The semi-empirical techniques of MQDT have been very successful and very widely used since they encapsulate the complex spectra, and configuration interactions, into a number of parameters. They have also motivated the search for ab initio methods to calculate short range scattering. There has also been great success in combining the eigenchannel R-matrix method with MQDT. The R-matrix method is a way to variationally calculate the set of eigenchannels Φα inside of a volume r < r0 in a given configuration space. Essentially, this ab initio method requires solving the time-independent Shrödinger equation with trial wavefunctions. To construct a proper set of trial wavefunctions, the foremost thing is to find the 20 appropriate Hamiltonian H=− ∆21 ∆22 1 − + V (r1 ) + V (r2 ) + . 2 2 r12 (3.3) In the above Hamiltonian, V (r) is not known. Nevertheless it’s not hard to imagine this potential should be some kind of l-dependent screening potential. Since a lot of orbitals are extremely sensitive to this potential, there has been a lot of work trying to optimize this model potential for different alkaline earth elements. It has been shown that by using the optimized potentials, accurate spectra can be obtained. One optimized model potential for strontium is the following, 1 V (r) = − {2 + (Z − 2)exp(−α1l r) + α2l rexp(−α3l r)}, r α1 = 3.551, α2 = 6.037, α3 = 1.439. In the Section 3.2, another original ab initio method to calculate strontium Rydberg spectra will be presented. It also employs an l-dependent model potential. However, while it is not an R-matrix method as described above, it does give accurate spectra that match to our experimental results. This method was developed to help analyze our experimental results by our collaborators in Vienna. 3.2 Two Electron model To analyze the excitation spectra of strontium, we employ a two-active-electron model. The Hamiltonian is written as H= p21 p22 1 + + Vl (r1 ) + Vl (r2 ) + 2 2 |~r1 − ~r2 | (3.4) As we are mainly interested in single-electron excitation, it is practical to reduce the number of configurations so that the eigenenergies can be evaluated efficiently 21 by numerically diagonalizing the Hamiltonian. The basis states are constructed from the excited states of the Sr+ ion H= p2 + Vl (r) 2 (3.5) ion |φni ,li ,mi i Hion |φni ,li ,mi i = Enlm The eigenstates hφni ,li ,mi | and the eigenenergies Eni ,li ,mi can be obtained numerically using the generalized pseudo-spectral method. The generalized pesudospectral method [60] is a numerical procedure for solving equations such as equation 3.5. It used for optimal grid discretization of the radial coordinates and is especially well suited for problems involving a Coulomb singularity. It requires a smaller number of grid points yet provides higher accuracy. It also introduces a split-operator technique in the energy representation that allows the wavefunctions to propagate efficiently in time (It has been widely applied in Floquet studies of atomic processes in strong fields.). The calculated energies agree quite well with those measured for the ion. The matrix elements of the two-electron Hamiltonian 3.4 are evaluated using the basis states defined by |n1 l1 n2 l2 ; LM i = X m1 +m2 =M [ C(l1 , m1 ; l2 , m2 ; L, M ) |φn1 ,l1 ,m1 i |φn2 ,l2 ,m2 i p 2(1 + δn1 ,n2 δl1 ,l2 δm1 ,m2 ) ± C(l2 , m2 ; l1 , m1 ; L, M ) |φn2 ,l2 ,m2 i |φn1 ,l1 ,m1 i p ] (3.6) 2(1 + δn1 ,n2 δl1 ,l2 δm1 ,m2 ) where L is the total angular momentum, M is a projection, and the ClebschGordan coefficients are given by 22 L √ l1 l2 C(l1 , m1 ; l2 , m2 ; L, M ) = (−1)−l1 +l2 −M 2L + 1 m1 m2 −M (3.7) The basis states symmetric (antisymmetric) with respect to the exchange of two electrons are used to calculate the eigenenergies in the singlet (triplet) sector. For singlet excitation spectra the quantum numbers (n1 , l1 ) of the outer electron may vary over the range of the whole excitation spectrum but those (n2 , l2 ) of the inner electron can be limited to near the ground state. Using such a truncated basis set, the eigenvalues of the active-two-electron system can be evaluated. Since the principal quantum numbers n1 , n2 of the basis describe the excited states of Sr+ ion and not those of neutral strontium, the correct quantum number n of the Rydberg electron has to be assigned to the calculated eigenstates of the two interacting electron according to the known excitation series ( including perturber states ) in each L sector, i.e. |nLM i = XX cn1 ,n2 ,l1 ,l2 hn1 l1 n2 l2 ; LM | (3.8) n1 ,l1 n2 ,l2 This is not straightforward as some states are hard to identify. For example, there is a 4d5p state in the 1 P1 sector, yet, the calculation shows no state with dominant 4d5p character. For strontium only few perturbers affect the Rydberg series for single electron excitation. Since they have relatively small energy, the highly excited states are not directly affected. For the singlet sector the quantum defects of singly-excited low-L states are plotted in Figure 3.1. The calculated results, which includes the 6 configurations (5s, 4d, 5p, 6s, 5d, and 6p ) for the inner electron, are compared with the previous studies based on MQDT (Quantum defects for L > 3 are negligibly small ). The calculations agree well with the measured results, which is expected as the model potential employed is known to yield the correct quantum defect using 23 Figure 3.1 : Measured and calculated quantum defects in the single-electron excitation of strontium (singlet). A two-active electron model is used with 6 configurations of the inner electron. Measured results are marked by circles. R-matrix theory. Only a small disagreement is seen for the P-and D-states where the calculated values slightly underestimate the measured quantum defects. We also note that, as seen in Figure 3.1, the quantum defects slowly increase with the principal quantum number n especially for P- and D-states. The eigenstates for highly excited states have contributions from the inner electron that are almost exclusively from the 5s state. Even a very small overlap with the other inner electron configurations shifts the phase of the wave function near the origin greatly affecting the quantum defect. The numerical method can be tested by comparing the zero-field excitation spec- 24 trum with the measured data (Figure 3.2). The measured spectrum is taken at n ∼ 280 and the calculations at lower n, n ∼ 50 and n ∼ 30. To compare the spectra of two different n the frequency axis is scaled so that the energy difference between two adjacent levels (n and n − 1) becomes invariant for different values of n. The calculated spectrum is derived from the dipole transition | h5s5p| z |5snli |2 and convoluted with a Gaussian to match the measured linewidth. The positions of the n 1 S0 states relative to the two adjacent degenerate n levels are observed to be invariant as the quantum defects of n 1 S0 states are nearly n-independent. On the other hand, the peak positions of the n 1 D2 states vary with n mirroring the n-dependent quantum defect. For example, the quantum defect is δd ≈ 2.31 for n = 50 and that extrapolated for the limit of n → ∞ is δd = 2.38. Another interesting observation is that the relative intensity of the n 1 D2 state to the (n + 1) 1 S0 state increases with n. The excitation strength is sensitive to the quantum defect as it phase-shifts the wave function near the origin and modifies the overlap with the 5s5p state. In this case, the quantum defect around δd = 2.38 appears to maximize the relative intensity and be suppressed away from it. This suggests that the relative intensity can be used to confirm the size of the quantum defect. We note that the underestimate of the calculated quantum defect for 32d overemphasizes this effect slightly. Calculations using a single-active-electron model with a model potential similar to that described in [61] have also been performed. In this model, a single electron is moving in a model potential that is numerically fit from known and extrapolated quantum defects. The eigenenergies as well as quantum defects can be obtained quite accurately while the oscillator strength fails to reproduce the measured spectra due to an inaccurate description of 5s5p state. 25 Figure 3.2 : Comparison between measured (a) and calculated (b, c) excitation spectra in zero field. (a) Measured excitation spectrum recorded at n ∼ 283. Results of twoelectron calculations at n ∼ 50(b) and n ∼ 30(c) employing six inner electron states (4s, 4d, 5p, 6s, 5d, and 6p). The energy axis is scaled such that E0 = 1 corresponds to the energy difference between neighboring n and n − 1 manifolds 26 3.3 3.3.1 Rydberg Atoms in a electric field Classical Picture As n becomes very large, the quantum mechanical behavior of the excited electron in a Rydberg atom can be described by the classical Bohr theory. In a hydrogen Rydberg atom, the electron follows an elliptical orbit that is given by r = L2 /(1 + ε cos θ) in ~ = ~r ×~p is the angular momentum. polar coordinates, where ε is the eccentricity and L The hydrogen atom is a special case because the energy levels are highly degenerate in l and m which is a manifestation of the 1/r character of the Coulomb potential. Correspondingly, in the classical picture, hydrogen Rydberg atoms have one more ~ = p~ × L ~ − r̂. In atomic physical quantity that is conserved, the Runge-Lenz vector A units, the magnitude of the Runge-Lenz vector is the eccentricity ε. On the other hand, alkali or alkaline earth atoms, do not have this “accidental degeneracy” due to core penetration and polarization. Their energies, characterized by E = −1/2(n−δl )2 , can be viewed as perturbed by the core. As a result, their Keplerian elliptical orbits will precess about the nucleus just as Mercury’s perihelion precesses about the Sun. For non-penetrating cases, the frequency of this precession is ∼ 5 δ. n3 l l The differences between hydrogen and non-hydrogenic Rydberg atoms are magnified in a electric field. For hydrogen, quantum mechanically, application of degenerate time-independent perturbation theory will lead to the linear Stark effect. The Stark map of hydrogen is simply like a fan; see Figure 3.3. The Shrödinger equation can be solved analytically in parabolic coordinates for a hydrogen atom in a electric field. The parabolic eigenstates are labeled by n, n1 , n2 , m and these quantum numbers are related by n = n1 + n2 + |m| + 1. As shown in Figures 3.3, the dipole moments are the largest for the extreme Stark states which have parabolic quantum numbers 27 Figure 3.3 : Hydrogen Rydberg Atoms in a Electric Field Left: the fanlike Stark map of hydrogen showing linear Stark shifts for |m| = 0 states. Every cluster of l states is often called√a hydrogenic manifold; adapted from [56]. The classical ionization limit Wc = −2 E is shown by a heavy curve where the Stark states begin to be broadened by field ionization. Quasi-discrete states with lifetime τ > 10−6 s (solid line), field broadened states 5 × 10−10 s < τ < 5 × 10−6 s (bold line), and field ionized states τ < 5 × 10−10 s (broken line). Right: The charge density distribution of hydrogen atoms in a electric field |m| = 0. Each figure is a parabolic eigenstate which is a superposition of many l states of hydrogen. Moving from the left to right, top to bottom, these figures are designated by the parabolic quantum numbers k = n1 − n2 = 7 to −7 which are the extreme blue components to the extreme red components; figure adapted from [62]. 28 Figure 3.4 : NonHydrogen Rydberg Atoms in a Electric Field Left: Precession of a nearly Keplerian elliptical orbit of a Rydberg electron about the core ion in an electric field. The precession is produce by adding an induced dipole term −αd /2r4 to the Coulomb potential which corresponds to the effects of polarization induced in the core. The top figure is with a negative α while the bottom one is with a positive α; figure adapted from [63]. Right: The Stark map for potassium |m| = 0 states, the anti-crossings that appear near level intersections are obvious. Figure adapted from [56]. 29 k = n1 − n2 ∼ n. Also because of the charge distribution, it is easier to ionize the electron in the extreme red state (in the last subfigure). For non-hydrogenic Rydberg atoms, the l-degeneracy is lifted by the interaction with the core. For low-l states non-degenerate time-independent perturbation theory results in a quadratic Stark shift. Though the time-average of the precession diminishes the existence of a permanent dipole moment in zero field, a small dipole moment can be induced by, and interact with, the electric field applied. This behavior can be visualized as a non-uniform precession of the Keplerian orbital; see Figure 3.4. However, this effect is only apparent for the low-l states since the interaction with the core falls off quickly with increasing l. The high l states are still essentially degenerate, so in the Stark map, display a near linear Stark shift. One subtle difference, compared with hydrogen Rydberg atoms, is the anti-crossings that appear between different Stark states as the electric field is increased. In the following subsection, a calculated description of the behavior of Sr Rydberg atoms in an electric field will be presented together with the experimental measurements. 3.3.2 Sr Stark Map Figure 3.5 shows the calculated eigenenergies for singly-excited strontium (n ' 50) states as a function of the strength, Fdc , of a dc field applied along the z axis. The high-l states which are nearly degenerate at Fdc = 0 exhibit a linear Stark shift and Stark states of two adjacent n manifolds first cross at a field strength of Fcross ' 1 . 3n5 (3.9) As explained previously, for the low angular momentum (1 P1 , 1 D2 ) states only the quadratic Stark shift can be observed. In Figure 3.5 the measured excitation spectra 30 of strontium around n = 310 are also plotted. In these measurements orthogonal polarizations of the 461 nm and 413 nm were used to avoid excitation of the 1 S0 states and simplify the excitation spectrum (The dc field is parallel to the polarization of the 461 nm laser). This setup yields Rydberg states with the total magnetic quantum number M = ±1. In order to compare the spectra for different values of n, the energy axis is scaled by En − En−1 ' n3 and the field axis is by Fcross . Using two-photon excitation, only n1 D2 states can be excited at Fdc = 0. With increasing strength of the dc field, the nD states become coupled with other angular momentum states and these l-mixed states have smaller oscillator strengths than the unperturbed Dstates. As the state merges with the linear Stark manifold, the l-mixing is so strong that the effect of the core scattering becomes negligible. Thus the state can become strongly polarized and almost indistinguishable from the extreme red-shifted strongly polarized Stark states. The behavior of the “312D” level mirrors that observed in earlier studies [61] at lower n, n ∼ 80 which data are also included in Figure 3.5. Slight shifts of the energy levels seen in the calculated 52P and 52D states are due to the underestimated quantum defects. This evolution of the n1 D2 states can be visualized by plotting the distribution of the parabolic quantum number k ρ(k) = X |H hn, k, m| |nStark iSr |2 (3.10) n or, equivalently, the distribution of Az (Az is the z-component of the Runge-Lenz vector) as k corresponds to the quantized action of −nAz . Here, |n, k, miH are the parabolic states of the hydrogen atom and |nStark i is the outer electron state for an eigenstate of strontium in a dc field (The inner valence electron is almost exclusively in the 5s state). Figure 3.6 displays the evolution of the k-distributions as a function of Fdc for the state which is the 52D state (M = 1) at Fdc = 0. For weak fields, the k-distribution spreads over a wide range between −n and n. This indicates that 31 the state is unpolarized. Since the Runge-Lenz vector indicates the orientation of the Kepler ellipse in classical dynamics, a wide distribution of Az implies an ensemble of Kepler ellipses (with l ∼ 2) whose orientations are broadly distributed. For non-hydrogenic atoms, such a distribution is formed by core scattering which changes the orientation of the ellipse while keeping the eccentricity. A node near k = 0 is m=1 also noticeable in the plot which mirrors a node of the spherical harmonic Yl=1 . With increasing Fdc the node is shifting towards the negative k side and the biased kdistribution indicates that state is becoming increasingly polarized. Near the merging with the neighboring Stark manifold the k-distribution becomes very narrow indicating a convergence towards a single parabolic state. The 52P state, on the other hand, does not show any hints of polarization. This is because its dipole-coupled partners, S- and D-states, are also hard to polarize. The 52D state is dipole coupled to the 50F state which merges with the Stark manifold at relatively weak Fdc and becomes polarized. The polarization of the 52D state is, therefore, caused by the coupling with this polarized “50F” state. The evolution of the calculated dipole moment, hz1 + z2 i, of several eigenstates around the “52D-state” is shown in Figure 3.6. The states nearly degenerate at Fdc = 0 becomes polarized even for very weak fields and dipole moments are given by h(z1 + z2 )i = (3/2)nk. For the isolated low-L states, the dipole moment grows almost linearly in Fdc , i.e. hz1 + z2 i = −αFdc (3.11) 2 when Fdc ' 0. These linear shifts lead to the energy shift ∆E = −(1/2)αFdc quadratic in Fdc . The polarizability α can be approximated using second-order perturbation 32 Figure 3.5 : Stark Map of Strontium Rydberg Atoms Evolution of the excitation spectrum with increasing applied dc field in the vicinity of n ∼ 310( thick red line). The thin solid lines indicate the calculated eigenenergies of singly-excited strontium (n ' 50) in a dc field while the dashed blue lines denote the corresponding excitation spectrum. The squares are the results of earlier measurements at lower n, n ∼ 80 from [61]. Fdc is normalized to the crossing field strength Fcross ∼ 1/(3n5 ) and the energy is normalized by En − En−1 ' 1. 33 theory as α=2 X X | hnLM | (z1 + z2 ) |n0 L0 M i |2 . En0 L0 M − EnLM n0 L0 =L±1 (3.12) Numerical calculations show that the polarizability α is dominated by a single term in the summation for the 50F state as the dipole-coupled state (50G) is almost degenerate in energy due to its almost vanishing quantum defect. The resulting large polarizability leads to sizable energy shifts. For the 52P state, similarly, the coupling to the 52D state dominates the summation in Equation 3.12. However, the large energy difference (see Figure 3.5 ) due to the quantum defect suppresses the values of α and, therefore, the state is hardly polarized. The 52D state is found between two dipole coupled states, 52P and 50F, and is slightly closer to the 50F. Therefore, the coupling with the 50F dominates over that with 52P leading to the polarization towards the downhill side resulting in a larger polarizability than that for the 52P state. We note that, judging from the quantum defect, the (n + 2)D state is found slightly closer to the midpoint between the (n + 2)P state and nF state for higher values of n. In fact, the polarizability of the D state appears to be smaller for the 312D state as the measured spectrum (Figure 3.5) shows a smaller energy shift than that of the calculation for n = 50. With increasing field the growth of the dipole moment becomes non-linear in Fdc , implying the non-negligible role of the higher-order perturbation terms, i.e., strong mixing with higher L states. Thus the states can become polarized through a superposition with those high-L states and, as seen for 50F and 52D, their polarizations approach the maximum value of hz1 + z2 i = 1.5n2 a.u.. 34 Figure 3.6 : Parabolic State Distribution of Strontium Stark States Left hand panels show the probability distribution of the parabolic quantum number k(= −nAz ) as a function of the dc field strength Fdc . The evolution of the states which are, at Fdc = 0, the 52D state, 52P state and 50F state are plotted. The distribution for the 52P state is truncated where it merges into a Stark manifold. On the right hand side, the average dipole moment of selected states including 52D, 52P, 50F as well as the downhill and uphill Stark states are plotted. Fdc is normalized to the crossing field strength Fcross . 35 Chapter 4 Experiment Setups and Techniques For very-high-n strontium Rydberg atom creation in a thermal beam, most of the equipment is the same as that employed in previous, successful Rydberg experiments on potassium which jump-started this experimental exploration of strontium. This left us with only two major new construction projects, the laser system and the strontium oven. Therefore this chapter will concentrate on these new pieces of apparatus and only make short comments on its other components. Finally, the experimental techniques employed will be summarized. 4.1 Frequency Locked Diode Laser System We use two Frequency Doubled High Power Diode Laser Systems from TOPTICA PHOTONICS to drive the two-photon Rydberg atom excitation. In our applications, both of them are required to be locked on a specific frequency with MHz accuracy for 8 hours continuously. This is accomplished by locking them with respect to a commercial frequency stabilized Helium-Neon laser via a scanning Fabry-Perot interferometer. Their absolute wavelengths are determined using a commercial high resolution wavemeter which produces GHz accuracy. In the following, the major components of the locking system will be described as an introduction to explaining the locking scheme later. In the last subsection, system limitations and a few alternative locking schemes that might provide better performance will be discussed. 36 Figure 4.1 : TOPTICA Diode Laser System Schematics adapted from toptica.com 4.1.1 Frequency Double High Power Diode Laser System[57] This tunable diode laser system is very compact and rugged. Its narrow linewidth (MHz over millisecond timescales), high and stable output power (100mW after doubling) and high tunability makes it perfectly suited for our applications. The whole laser consists of an diode laser system coupled to an electronic control system. The electronic control system contains plug-in modules for the DC and HV power, the diode current, temperature control, the crystal temperature control, laser modulation 37 and regulation, and external interfaces in a 19” unit. As shown in Figure 4.1, the laser source is a grating stabilized external cavity diode laser system based on the Littrow-Hänsch scheme. With a cavity formed by the front facet of the diode and a holographic optical grating next to the rear facet of the diode, this scheme offers a much smaller linewidth (1M Hz) as compared with a bare diode(100M Hz). Besides the wavelength can be tuned easily over a large range. Coarse tuning is achieved by adjusting the angle of the grating via a micrometer screw and fine tuning is obtained by scanning the cavity length via the piezo element attached to the grating. Mode hop free tuning is achieved by feedforwarding a current proportional to the scanning voltage to the diode. To maintain single mode operation, both the temperature and current of the diode head need to be adjusted together with the grating. Upon leaving the master laser diode, the collimated infrared laser beam( 40mW approximately) is focused, mode matched into another diode, the tapered amplifier (T A) to achieve more power than is possible with a single-mode laser diode. The gain bandwidth of the tapered amplifier is usually of order of some tens of nanometers. Due to the diodes’ sensitivity to feedback, high-suppression-ratio optical isolators are integrated in the optical path to avoid reflection. Finally the output from the TA (300mW )is mode matched to couple it into the bow-tie-ring resonator for frequency doubling. Phase matching of the crystal is sensitive to both the alignment and the temperature. Second harmonic output powers of 100mW are obtained. The stabilization of the doubling cavity is achieved mainly via two feedback loops. Their common error signal is generated by applying the Pound-Drever-Hall scheme [58]. An RF modulation fed into the current of the master laser head produces two sidebands which, together with the carrier are all sent to the doubling 38 cavity. Their reflection from the cavity is collected by a fast photodiode (shown in Figure 4.1) whose output is then fed into the PDD 110 module (the Pound-Dever-Hall Detector). PDD mixes this signal with the same RF local oscillator used before to extract their DC phase information which is the output error signal. The error signal is then fed into two loops. The slower feedback loop (a few kHz) is closed by the PID regulator stabilizing the cavity length with the piezo element attached to one of the mirrors in the doubling cavity. The fast loop (5MHz) is closed through adjusting the master diode’s current to lock the laser frequency to the doubling cavity. In the fast loop, the doubling cavity is the reference cavity. The combination of these two loops gets rid of thermal and acoustic noise as well as fast frequency jitter and can maintain a narrow linewidth. The electronic plug-in modules can also communicate via the backplane of the rack if the jumpers are set accordingly. The feedforward function, for instance, is achieved by sending a portion of the scanning voltage to the current control module of the diode head(DCC). Thus this voltage/current is added to the current setvalue on the DCC and output to the diode head. The external input of the SC 110 module, which controls scanning of the grating of the master laser diode, is connected to the same line as the BNC input/output of the computer analog interface DCB. Therefore, the input from the BNC connector to the DCB module will be added to whatever voltage is set on the SC module and then output to the grating in the master laser diode. This particular feature will be used in our locking scheme discussed later. 4.1.2 HeNe Laser Frequency/Intensity stabilized HeNe reference laser at 632.816nm is a commercially available unit. Frequency stabilization is attained by comparing the intensity balance 39 of two orthogonally polarized longitudinal modes and can offer a ±2MHz frequency stability on an eight hour timescale. The laser is very sensitive to any retroreflections. Given the lack of an optical isolator, we use a 20dB neutral density filter to block reflections from the Fabry-Perot Interferometer (will be discussed later). In addition, the interferometer must be intentionally misaligned slightly to prevent reflections back to the HeNe laser. 4.1.3 Scanning Fabry-Perot Interferometer The finesse of a Fabry-Perot Interferometer, a measure of the interferometer’s ability to resolve spectral features, is essentially determined by the reflectivity of the two mirrors on each end of the cavity F = nR/(1−R2 ). Thorlabs offers scanning confocal Fabry-Perot Interferometers (etalons) but not with optical coatings suitable for our needs. In order to achieve a finesse F = 200 for 633nm, 412nm and 461nm laser light, we ordered an SA200 with customized coatings on both (confocal) mirrors. The free spectral range(FSR) of this etalon (≈ c/(4L),where L is the mirror spacing) is 1.5GHz. With better alignment, when the laser beam is on the optical axis, the free spectral range becomes c/(2L). This customized SA200 is driven by the matching SA201 Spectrum Analyzer Controller. This controller provides a saw-tooth waveform with adjustable ramp amplitude and rise time and can be externally triggered. In our experiment, we set the amplitude of the ramp to just cover one free spectral range so that two peaks of the HeNe laser can be observed on the output spectrum. We scan the etalon at 50Hz which given the FSR of the etalon will yield peaks having a temporal width of ∼ 100µs. The optical length of the Fabry-Perot Cavity is also sensitive to air pressure, 40 airflow and temperature change. Since we are using this cavity to lock the laser frequencies, it is important to make it as stable as possible. Therefore, we put the whole scanning etalon into an aluminum housing specially made for this purpose. This enclosure is air-sealed. Laser light enters and exists the two AR coated N-BK7 windows sealed by O-rings. A small hole running cables is sealed by vacuum epoxy glue. 4.1.4 Data Acquisition System Laser control is accomplished using Labview and an NI PCle-6341 Xseries DAQ board. This particular DAQ board has eight analog inputs with a total sampling rate of 500kS/s and two analog output channels. There are also plenty of digital lines with even higher reading and writing rates. 4.1.5 Locking Scheme The essence of our laser locking is that by continuously scanning the Fabry-Perot interferometer, the relative frequency difference between the target laser and the reference HeNe laser can be monitored and can be locked to a fixed value by feedback. Here is how this scheme implemented in our system: Three laser beams (the HeNe, the 461nm, and 413nm beams) are all sent into the etalon, and the resulting transmission peaks from each laser are detected by three independent photo-diodes. The signals from the photo-diodes are fed into three analog input channels of the DAQ board. One more analog input channel is used to display the ramp signal from the SA201. The Labview program first locks the Fabry-Perot cavity to the HeNe laser. This is done by feedback; any change in HeNe peak position (with respect to the ramp) is compensated by changing the offset of the ramp voltage from the SA201. This step 41 is necessary due to the fact that the Fabry Perot cavity drifts due to the thermal expansion or electrical drifts even though it is somewhat thermally isolated in the aluminum enclosure. The next step is to continuously compare the relative position of the HeNe peak and the 461nm/412nm laser peaks and generate feedback signals to compensate for any changes to the diode laser systems to make their relative position stay at the set value. In this step, the feedback voltage is sent to the BNC input of the DCB module of the corresponding laser system. So, ultimately, the feedback is to the position of the grating. In addition, both lasers can be locked or scanned anywhere over the whole ramp which is simply realized by modifying the set value of their positions relative to the HeNe peak. Our locking scheme as described is able to lock both diode laser system to a range of ±2MHz stably over the course of a day. And it effectively compensates the long-term thermal drift the diode laser system suffers and successfully satisfies our experimental needs. 4.1.6 Limitations and Alternatives There are a couple of factors limiting the bandwidth of our stabilization scheme like the processing rate of the Labview program and the sampling rate of the DAQ board. But the major limitation is from the necessity of scanning the Fabry-Perot Cavity over a whole Free Spectral Range which requires 20ms. Faster feedback is possible by increasing the scanning rate and, accordingly, the sampling rate. Locking a laser, like the 412nm laser, that is not associated with direct atomic transitions or very close to any reference laser (up to GHz frequency difference), is difficult. Otherwise, either saturation absorption could be performed and the linewidth of the locking could be cut to 100kHz or, two lasers’ beat signal could be utilized 42 to perform a fast heterodyne locking scheme. There is one other approach as described in [59]. They first find a cavity length that coincides with the transmission maximas of the target laser and the reference laser by generating a sideband from current modulating the reference laser. Then they lock the cavity to the reference laser and lock the target laser to the cavity by analog circuitry. The scan of the target laser is obtained by scanning the sideband of the reference laser. This approach has achieved sub-MHz linewidth. Another approach is an analog of saturation absorption spectroscopy. Using the EIT signal from the coherent two-photon transition, the linewidth should also be greatly reduced. 4.2 Strontium Oven While several strontium atom beam sources have been described, these are typically designed to optimize beam fluxes rather than to achieve tight beam collimation. Although beam divergences can be reduced by use of multicapillary arrays, these alone cannot provide the required collimation. Here beam divergence is controlled through the use of a small oven aperture and tight beam collimation. Because the oscillator strengths for excitation of high-n Rydberg states are small, sizeable atom beam densities are required to achieve reasonable Rydberg photoexcitation rates. Given that the oven aperture is small, this requires the vapor pressure in the oven be high necessitating operation at temperatures of ∼ 500 − 650◦ C . Such temperatures can be reached using commercial resistive heating elements provided that thermal losses are minimized. While the present source design builds on earlier practice, its use of a commercial heater allows a particularly simple design that is straightforward to fabricate. The source has proven reliable in operation and, with appropriate collimation, can provide an atomic beam with a divergence of ∼5mrad FWHM and densities 43 Figure 4.2 : common elements vapor pressure chart, downloaded online 44 approaching 109 cm−3 sufficient to enable a wide variety of experiments with high-n strontium Rydberg atoms. The present atom source is shown in Figure 4.3. Its central component is a cylindrical stainless steel oven that is loaded with granular strontium metal from the rear. Atoms emerge from the oven through an orifice in the form of a cylindrical canal that is ∼ 0.5mm in diameter and ∼ 1.2mm long. The oven fits inside a commercial spiral wound coil heater.This heater comprises a heating element that is electrically isolated inside a stainless steel sheath using MgO. The heater includes an unheated section at one end through which the heater leads enter and exit, and also contains an internal thermocouple to monitor its internal temperature. In air, the heater is rated for 350 W and operation at temperatures up to ∼ 820◦ C. The spacing of the heater coils is adjusted such that they are closer together near the front of the oven than at its rear. This ensures that the front of the oven is maintained at a higher temperature than its rear to prevent the exit canal from becoming blocked through condensation. To further limit such condensation, the exit orifice is offset towards the side of the oven and the heating coils are extended well forward of the front of the oven to provide it with a strong radiant heat bath. The heating coil fits inside a polished cylindrical copper jacket. The low emissivity of copper minimizes the heater power required to reach a given operating temperature. Although hot copper reacts with strontium, the jacket is not exposed directly to strontium vapor and no problems with such reactions have been encountered. The oven and heater are positioned within the jacket using a single small screw;see Figure 4.3. The temperatures of the front and back of the oven, and of the jacket, are monitored using thermocouples. To ensure good thermal isolation, the copper jacket is held in place using a number of small stainless steel mounting screws that pass through ceramic bushings. Two 45 Figure 4.3 : Schematic diagram of the oven assembly 46 pairs of horizontally-opposed screws position the jacket vertically within a “U”-shaped aluminum support bracket. Horizontal positioning is achieved with the aid of a single vertical screw that passes through a second mounting bracket that runs up and over the front of the jacket. To allow for expansion upon heating a small clearance is included between the mounting brackets and the heater jacket, and the mounting screws are free to slide within their bushings. The support bracket is held in place by an arm that is connected via a bellows to an x-y translation stage located outside the vacuum region. This stage is used to position the oven orifice during initial beam alignment. The whole oven assembly is mounted inside a water-cooled copper enclosure. To limit the temperature rise of the support bracket and of the power input end of the heater, these are each cooled by connecting them via copper braids to the enclosure. A 4 mm-diameter aperture located ∼ 4cm from the oven orifice is used for initial beam collimation. Final collimation is provided by a 0.5 mm-diameter aperture ∼ 10cm from the oven. No significant changes in beam pointing have been observed as a result of day-to-day cycling of the oven temperature. The oven is typically operated at temperatures in the range ∼ 500 − 650◦ C. The power input to the heater required to reach these temperatures is ∼ 25−50 W and the internal temperature within the heater remains below 800◦ C. While the strontium atom beam density cannot be simply measured directly, it can be inferred from earlier measurements (using a hot wire ionizer) of potassium atom beam densities produced by an oven having a similar orifice and operating at similar metal vapor pressures; see Figure 4.2. These earlier measurements suggest that at operating temperatures above ∼ 600◦ C strontium beam densities approaching 109 cm−3 should be obtained. Measurements of the photoexcitation of strontium Rydberg states demonstrate that sizeable beam densities are produced. While much of the increase in Rydberg produc- 47 tion can be attributed to differences in oscillator strengths, the data do indicate that sizeable strontium atom beam densities are obtained and that large photoexcitation rates can be achieved. The present atom source has proven reliable in operation and even with day-to-day cycling of the oven temperature no heater failure has occurred. While the lifetime of an oven charge depends on the required beam density, its capacity has proven more than sufficient to allow several months of operation before reloading becomes necessary. However, if required, longer operational periods could be achieved by simply scaling up the design (a wide range of suitable coil heaters are available). 4.3 4.3.1 Other Experimental Apparatus Vacuum System The whole vacuum system consists of two relatively separate sections, the oven section where the oven is mounted and the interaction region section, where ground state strontium atoms are photo-excited to Rydberg levels and subsequently field ionized and detected. These two sections are connected by a 500µm-diameter aperture through which the atomic beam passes. Therefore, for efficient pumping, the two sections are differentially pumped using diffusion pumps, a Varian VHS-4 for the oven section and VHS-6 for the interaction region section. Their pressures are measured by two Bayard-Alpert ionization gauges and are typically ∼ 10−7 torr. The laser beams enter the vacuum system through two Ø1” AR coated Thorlabs N-BK7 broadband precision windows traveling in opposite directions. The windows are mounted slightly off normal to the beams to avoid retroreflections. The vacuum seal is accomplished by Parker O-rings. The 412nm laser beam is focused by a Ø1” 48 Figure 4.4 : Interaction Region f =40cm Thorlabs N-BK7 Plano-Convex Lens to the beam waist, 170µm, in the center of the interaction region. 4.3.2 Interaction Region The interaction region is bounded by three pairs of 10cm×10cm electrodes (see figure 4.4). Except the bottom plate, 5 of them can be biased independently to cancel the stray electric fields in three orthogonal directions so that local electric field can be reduced to ≤ 50µV /cm. A circular electrode disk mounted from the top plate is used for applying fast electric pulses (often called half-cycle pulses) in the ẑ direction and shares its bias potential with the rest of the plate. Sometimes, half-cycle pulses (HCP) are also applied to the side-electrode as shown in Figure 4.4. For detection, the Rydberg atoms are field ionized by a voltage ramp which is applied to the bottom 49 plate. The resulting electrons exit through the 1” inch aperture on the bottom plate covered with fine copper mesh to be detected by the funnel shaped aperture of a Dr. Sjuts channel electron multiplier KBL 25RS mounted beneath the mesh. 4.4 Experimental Techniques As mentioned earlier, we use a ramped electric field to field ionize the Rydberg atoms and subsequently collect the resulting electrons. Field ionization enables state selective detection and is widely employed in Rydberg atom experiments. 4.4.1 Selective Field Ionization Classical ionization occurs when the electron’s potential energy becomes less than its ~ the total energy. For an electron of energy −1/(2n2 ) subjected to an electric field E, ~ · ~r which scales as −1/n2 + En2 . The classical total potential energy is −1/r + E ionization limit can be heuristically obtained by equating them, −1/n2 + En2 = −1/(2n2 ), thus the ionization field scales as 1/n4 . Calculation of saddle point in the potential yields 1/(16n4 ) as the classical ionization limit which is shown in the Stark map for sodium in Figure 4.5. Rydberg atoms are indeed ionized around this limit. However, before being completely ionized, the energy level of the electron is first Stark shifted or/and Stark mixed according to its l and the rise time of the applied field, and these effects determine the exact field at ionization. Consider the Stark map for sodium in Figure 4.5. Whereas the high l manifolds display linear Stark shifts, the low-l states display quadratic Stark shifts just as was discussed in Chapter 3. As the field increases, different Stark states experience avoided crossings. If the electric field strength increases slowly enough, these crossings are traversed adiabatically, the states following the paths indicated by the solid lines and ionizing at fields ∼ 1/(16n4 ). On 50 Figure 4.5 : Stark map for Sodium showing evolution of the m = 0 states in an increasing field. Color red represents d states (δd = 0.01), color green represents p states (δp = 0.86) and color purple represents the s states (δs = 1.35). In this particular case, d states are very close to the manifold which are the whole cluster of high l states. In this graph, the manifold states are represented by the 5 states just above d states. The dotted lines are the diabatic paths for s,p,d states to ionize while the adiabatic paths are the solid black serpentine lines for each states. The colored dots on the ionization curve are the ionization points for the adiabatic passages of each state. This graph is adapted from [56]. 51 the other hand, a rapid rise in the applied electric field will result in diabatic passage, states will follow a linear energy shift after mixing with the extreme l states in the manifold and they will be ionized in a order that is completely different from that of their zero field energies; see the dotted lines in Figure 4.5. In our experiment, ionization can be regarded as diabatic because the energy separation at the avoided crossings are very small for n ∼ 300. No matter if the ionization is diabatic or adiabatic, the electric field required, to a good approximation, scales as 1/(n4 ). If a ramped field is applied, different states will ionize at different field and thereby different times. The resulting ionization signal vs. electric field will provide information about the Rydberg states initially present. With careful analysis through “selective field ionization” (SFI), especially for n≤100 when the available resolution is high, it is possible to obtain considerable information on the initial n distribution. However, when n ∼ 300, it is only possible to infer the approximate n from SFI spectrum. To generate a SFI spectrum, we measure two quantities using techniques adapted from previous experiments on potassium for which the excitation rate of Rydberg atoms was low. Following each laser pulse, there is only one or zero atoms excited to the Rydberg level. So single particle detection can be employed using a channel electron multiplier. This amplifies the incident electron and produces a large number of secondary electrons that generate one large signal pulse for detection. Besides this signal, we also measure the time duration from the start of the ramp to the time when we get the signal from the channel electron multiplier. Since we know the time evolution of the ramp, with these two quantities, we can obtain the electric fields needed to ionize that Rydberg atom. Although we don’t use the SFI spectrum directly in the beam experiments. It 52 produces a valuable detection scheme. In our strontium experiment, we found a ramp that rises from 0 to 5 Volts in 5 µs is more than enough to ionize all the Rydberg atoms. . 4.4.2 Data Acquisition The very-high-n Rydberg experiment is triggered by a master pulser which immediately triggers a 461nm laser pulse for Rydberg excitation. About 10µs later, the ramped field is applied to the bottom plate. Right before the ramp, an ORTEC 566 time-to-amplitude converter (TAC) is started by a trigger pulse and it is stopped by a signal from the channeltron. The output pulse from the channeltron is then converted into a TTL signal by a charge sensitive amplifier and is fed into a computer. This computer also controls, scans the 412nm laser frequency, adjusts the bias voltages on the plates of interaction region and controls the multiple half-cycle-pulses (HCPs) that are used for engineering quasi-one-dimensional states in some other experiments. 53 Chapter 5 Sr Rydberg Atoms in a Collimated Atomic Beam Initial experiments focused on the production of Sr Rydberg atoms, their spectroscopy, their photo excitation rates, and their excitation in a weak dc field. Strontium atoms contained in a tightly-collimated beam are excited to the desired high-n (singlet) state using the crossed outputs of two frequency-doubled diode laser systems. The two-photon excitation scheme employed utilizes the intermediate 5s5p1 P1 state and radiation at 461 nm and 413 nm. The laser beams, which typically have the same linear polarization, cross the atom beam traveling in opposite directions. Since their wavelengths are comparable, the use of counter-propagating light beams can largely cancel Doppler effects associated with atom beam divergence resulting in very narrow effective experimental linewidths. As noted earlier, residual stray fields in the experimental volume are reduced to ≤ 50µV cm−1 by application of small offset potentials to the electrodes that surround it. Measurements are conducted in a pulsed mode. The output of the 461 nm laser is chopped into a series of pulses of ∼200 ns to 1 µs duration and 20 kHz pulse repetition frequency using an acousto-optic modulator (The 413 nm radiation remains on at all times). Following each laser pulse, the probability that a Rydberg atom is created is determined by state-selective field ionization for which purpose a slowly-rising (risetime ∼1µs) electric field is generated in the experimental volume by applying a positive voltage ramp to the lower electrode. Product electrons are accelerated out of the interaction region and are detected by a particle multiplier. The probability that a Rydberg atom 54 is created during any laser pulse is maintained below 0.4 to limit saturation effects. This is accomplished by reducing the strontium atom beam density by operating the oven at a lower temperature, and/or by reducing the laser powers. 5.1 Spectroscopy Figure 5.1 shows excitation spectra recorded in the vicinity of n = 282 for various detunings of the blue 461 nm laser selected to optimize the transitions 5s2 1 S0 → 5s5p 1 P1 in the different strontium isotopes. The relative frequency of these transitions together with other properties of naturally-occurring strontium are listed in Table 2.2. The isotope shifts and hyperfine splittings in both Table 2.2 and Figure 5.1 are quoted relative to the dominant 88 Sr isotope. The blue 461 nm laser beam was unfocused and had a diameter of ∼3 mm. Its intensity, ∼10 mW cm−2 , was selected to limit line shifts and broadening due to effects such as the ac Stark shift and Autler-Townes splitting. Its pulse width, ∼ 0.5µs, was selected because for shorter pulse durations the widths of the spectral features become increasingly transform limited. The “purple” 413 nm laser beam was focused to a spot with a full width at half maximum (FWHM) diameter of ∼ 170µm, resulting in an intensity of ∼ 250W cm−2 . The frequencies of both lasers were stabilized and controlled with the aid of an optical transfer cavity locked to a polarization-stabilized HeNe laser as described in Section 4.1. This cavity allows uninterrupted tuning of the lasers over frequency ranges of up to ∼800 MHz. The frequency axis in Figure 5.1 shows the sum of the blue and purple photon energies. The spectrum obtained with the blue laser tuned to optically excite the dominant 88 Sr isotope displays a series of sharp peaks associated with the excitation of 1 D2 states with the n values indicated. The spectrum also contains a series of smaller peaks associated with the production of 1 S0 55 Figure 5.1 : Excitation spectra recorded in the vicinity of n = 283 for different detunings of the 461nm laser. These detunings, specified relative to the 88 Sr 5s21 S0 → 5s5p1 P1 transition, are indicated in the figures. The frequency axis shows the sum of the 461 nm and 413 nm photon energies. The horizontal bars beneath the data identify the features associated with excitation of 5snd1 D2 Rydberg states in the 88 Sr, 86 Sr and 84 Sr isotopes, together with the positions of features associated with excitation of 87 Sr Rydberg states. Two red arrows in the top subplot point to the 3 D2 state of 88 Sr. 56 states. The widths of these features, ∼5 MHz FWHM, is attributed to a combination of transit time broadening (the transit time of an atom through the purple laser spot is ∼ 300 ns ) and fluctuations in the laser frequencies during the ∼ 1 s required to accumulate data at each point in the spectrum. The probability that a Rydberg atom was created at the peak of the 1 D2 features during each laser pulse was limited to ∼ 0.5 by substantially reducing the strontium atom beam density by operating the oven at ∼ 500◦ C. Tests were undertaken at higher operating temperatures in which the 413 nm laser beam was attenuated using neutral density filters to limit the excitation rate. These tests showed that, with the oven operating at 630◦ C and using the full 413 nm laser power (∼ 70 mW), ∼10-15 Rydberg atoms can be produced per laser pulse in the excitation volume of ∼ 5 × 10−5 cm3 . This corresponds to a typical inter-Rydberg spacing of ∼ 130µm which is approaching those at which effects due to Rydberg-Rydberg interactions such as blockade become important. 5.1.1 Even Isotopes When the blue laser is tuned on resonance with the 1 S0 → 1 P1 transition for as shown in the top subplot in Figure 5.1, the two 1 D2 of 88 88 Sr, Sr peaks dominate the spectra. If we normalize the frequency separation between singlet 1 D2 states with consecutive n as 1, then the nearest 1 S0 state to a 1 D2 should be separated by |(δ1 S0 mod1) − (δ1 D2 mod1)| = |0.269 − 0.381| = 0.112, where δ is the quantum defect of the corresponding states; see Table 5.1. Inspection of the figure shows that there is an S state lying 0.11 to the right of every 1 D2 state in accordance with the quantum defect reported previously. The same calculation can also be carried out for the 3 D2 state, which shows that its separation from the 57 nearest 1 D2 should be |(δ3 D2 mod1) − (δ1 D2 mod1)| = |0.636 − 0.381| = 0.245. As marked on the top subplot of Figure 5.1 by red arrows, a small signal from 3 D2 states can also be discerned at the expected relative positions. The small size is due to the fact that ∆S 6= 0 transitions are forbidden in the Russell-Saunders LS-coupling scheme. For heavy atoms like strontium, however, LS-coupling breaks down and strong spin-orbit interactions mix the wavefunctions from singlet series and triplet series. So intercombination transitions are weakly allowed for Sr. But the spin-orbit interaction strength or equivalently, the fine structure (EF S ∼ (n∗)−3 ) diminishes quickly as n increases and, this mixing gets substantially weaker for very high-n Rydberg states. Therefore, the 3 D2 states are barely excited. As the blue laser is red detuned from resonance with the 1 S0 → 1 P1 transition in 88 Sr, the size of the 88 Sr features in the excitation spectra decreases steadily and new features emerge associated with the excitation of Rydberg states of the other isotopes. At a detuning of ∼ −122 MHz, which optimizes the 1 S0 → 1 P1 transition for 86 Sr, the excitation spectrum is dominated by the creation of 86 Sr 1 D2 states, although some residual excitation of 88 Sr → 86 88 Sr isotopes remains. The observed Sr isotope shift in the series limit, +210 ± 5 MHz, is consistent with that reported in earlier spectroscopic studies at lower n [45]. At a blue laser detuning of ∼ −273MHz which optimizes the 1 S0 → 1 P1 transition in the 84 Sr isotope, the excitation of 84 Sr1 D2 Rydberg states becomes apparent. The observed 88 Sr → 84 Sr isotope shift in the series limit, +440 ± 8M Hz, is again consistent with earlier measurements [45]. However, because the fractional abundance of the 86 Sr and 84 Sr isotopes in the beam shown in Table 2.2, are much less than for the atoms are created. 88 Sr isotope, many fewer Rydberg 58 Table 5.1 : Sr Quantum Defects [65] Series δ 5sns1 S0 3.269 5snp1 P1 2.730 5snd1 D2 2.381 5snf 1 F3 0.089 5sns3 S1 3.371 5snd3 D3 2.630 5snd3 D2 2.636 5snd3 D1 2.658 At blue laser detunings of ∼ −90 MHz, the sizes of the features associated with the excitation of 88 Sr and 86 Sr Rydberg states are nearly equal, resulting in the appearance of two separate interleaved Rydberg series. The 88 Sr to 86 Sr isotope shift, ∼ 210M Hz, matches the energy spacing between adjacent Rydberg levels at n ∼ 313, indicating that in the vicinity of n ∼ 313 the two interleaved Rydberg series should overlap, the 312 1 D2 88 Sr Rydberg level, say, overlapping with the 313 1 D2 86 Sr Rydberg level. Spectra recorded in the vicinity of n = 312 (see Figure 5.2) under these conditions do indeed display only a single series of Rydberg features. 5.1.2 the odd isotope 87 Sr Several new features are also observed in the excitation spectra at blue laser detunings chosen to favor excitation of the 87 Sr isotope. These features do not conform to a 59 Figure 5.2 : Spectra recorded with the detunings ∆ shown in the vicinity of N ∼ 312. In (a), 88 Sr is dominant. In (b), the detuning ∆ favors the 1 S0 → 1 P1 transition of both 87 Sr and 86 Sr. The most discernible peaks for 87 Sr are highlighted by green arrows and they will be explained in Subsection 5.1.2. The 1 D2 peaks for n86 Sr at the first three ns are distinguishable from those resulting from residual excitation of (n + 1)88 Sr. At higher n, they merge resulting in a single sharp spectral feature. In (c), where the detuning mostly favors 86 Sr, small peaks from 88 Sr can be observed on the left of the first two dominant n86 Sr peaks. 60 simple Rydberg series having the expected values of n. Similar behavior has been observed previously in studies at lower n and assigned to a combination of strong state mixing, hyperfine-induced singlet-triplet mixing, and interactions between states of different n [47]. These complexities of 87 Sr Rydberg series are mainly caused by its nuclear spin I = 9/2 which is absent for all the even isotopes. Major consequences of the nuclear spin are now discussed. Hamiltonian with Hyperfine Interaction The total Hamiltonian, that includes the fine and hyperfine interactions, for a Rydberg atom with two valence electrons n1 l1 s1 , n2 l2 s2 and a nuclear spin I, can be written as H = H1 + H2 + e2 + β1 s~1 · ~l1 + β2 s~2 · ~l2 + A1 I~ · ~j1 + A2 I~ · ~j2 , r12 (5.1) where Hi is the Coulomb interaction between the valence electron i and the Sr2+ core, e2 /r12 is the Coulomb repulsion between the two valence electrons which leads to the singlet triplet splitting, βi~s · ~li is the spin-orbit interaction of the ith electron and Ai I~ · ~ji , j~i = s~i + ~li is its hyperfine interaction. For a strontium Rydberg atom of configuration 5snl, we can use the subscript 1(2) to denote the nl(5s) Rydberg (ground) electron. The spin-orbit interaction and hyperfine interaction for the Rydberg electron are negligible when n is large since both the spin-orbit constant β and hyperfine structure constant A scale as n−3 . Because the 5s electron has l2 = 0, j2 = s2 = 1/2, the spin-orbit interaction is also zero and the hyperfine interaction is reduced to A2 I~ · s~2 . Now we can rewrite the Hamiltonian as H = H1 + H2 + e2 + A2 I~ · s~2 . r12 (5.2) Now, the quantum number describing the total angular momentum is F (for even isotopes, only the first three terms remain and the Hamiltonian is characterized by J), 61 ~ The last term in this Hamiltonian the quantum number F~ being given by F~ = I~ + J. will be referred as the Fermi contact interaction. Rydberg Series Limit Shift For the even isotopes of strontium, different Rydberg series like 3 S1 , 1 S0 , 3 D2 , 1 D2 will converge to one common limit, the ion’s ground state 5s Sr+ 2 S 1 . This is described as 2 2 Enl = Iion −R/(n−δl ) where Iion is the ionization potential and δl is the corresponding quantum defect. Therefore, the shifts between even isotopes are their normal mass shifts as shown in last section. However, due to the Fermi contact interaction which is essentially the hyperfine interaction of the ion, the ion state 5s Sr+ 2 S 1 splits into 2 F = 4 and F = 5 hyperfine states and they are about 2 − 3GHz [47] away from the ion energy without a hyperfine structure i.e., 87 Iion − 88 Iion ∼ ±2 − 3GHz. These shifts, much greater than the energy spacing between the n and n + 1 levels when n ∼ 300, contribute to the movements of 87 Sr peaks relative to the peaks of the even isotopes in the measured spectra. Hyperfine Induced Mixings The Fermi contact interaction becomes increasingly important as n gets higher. When 60 < n < 100, this interaction (∼ 2 − 3 GHz) is comparable to the e2 /r12 term in equation 5.2 but smaller than the energy spacing between two consecutive ns, and can cause mixing between triplet and singlet levels of the same F within the same n (hyperfine induced singlet-triplet mixing within the same n). Consequently, their energies are shifted and triplets can be populated as strongly as singlets. As n continues to increase, the Fermi contact interaction becomes larger than En − En−1 (the energy spacing between two adjacent ns), and states of the same F but different 62 Figure 5.3 : N ∼ 335 spectra The 3 D2 states of 88 Sr are circled by red ovals in the top figure. The blue laser is tuned to favor the 1 S0 → 1 P1 transition for 87 Sr in the second and the third subplots. Based on our experiences, the 87 Sr features are most apparent at a detuning around −69M Hz. The last subplot is of a detuning that favors excitation of 86 Sr. ns can be mixed. The higher the n is, then states with ever increasing differences in ns can be mixed. At n ∼ 300 , hyperfine induced n-mixing can lead to “anti-crossing” in the spectra since the energy differences between states of different n can be tuned semi-continuously by changing n as shown in Figure 5.3 and 5.4. 5.1.3 Stray Fields Impact On Spectra Figure 5.5 shows Rydberg excitation spectra recorded, with the blue laser tuned to the 88 Sr5s21 S0 → 5s5p1 P1 transition in the 88 Sr isotope, at successively higher 63 Figure 5.4 : Anticrossing of 87 Sr This graph is the detailed, break-down plots of the third plot of Figure 5.3. Every subplot of this graph starts from a 1 D2 peak of 88 Sr and ends at next 1 D2 peak of 88 Sr. At the anticrossing point, where the energy mismatch is very small, one of the state features disappears due to the cancellation of the oscillator strength. The red line and green line follow the peak positions of two merging states. 64 values of n. As expected, as n increases the peak number of Rydberg atoms created decreases dramatically as a result of both the decrease in the oscillator strength and the increasing width of the spectra features. For values of n ≤ 350, two well-resolved Rydberg series are seen, corresponding to excitation of 1 D2 and 1 S0 states. With further increases in n the spectra features begin to broaden significantly, their widths having approximately doubled by n ∼ 400. For even larger values of n the background level begins to increase significantly, but a strong, well-resolved Rydberg series is still evident for values of n up to ∼ 460. For n > 500, however, it becomes increasingly difficult to discern any Rydberg series. This degradation in the Rydberg spectrum with increasing n can be attributed to the presence of stray background fields which lead to Stark shifts and broadening. These effects become particularly important at fields approaching those at which states in adjacent Stark manifolds first cross, given by Fcross ∼ 1/(3n5 )a.u., i.e., ∼ 50µV cm−1 at n ∼ 500. This suggests that stray fields of ∼ 50µV cm−1 remain in the excitation volume which is consistent with earlier estimates of their size as described in Section 4.3.2. 65 Figure 5.5 : Excitation spectra recorded near the values of n indicated. The 461nm laser is tuned to the 88 Sr5s21 S0 → 5s5p1 P1 transition. The frequency axis shows the relative frequency of the 413nm laser during each scan 66 Chapter 6 Ultracold Rydberg Gas Evolution Ultracold Neutral Plasmas (UNPs) and Ultracold Rydberg Gases are inter-related. The free electrons and ions in a plasma can form Rydberg atoms via three body recombination (e− +e− +R+ = R∗∗ +e− , “R∗∗ ” is the abbreviation for a Rydberg atom). Conversely, an ultracold Rydberg gas can spontaneously evolve (ionize) into plasma. Both systems have been studied extensively over the last decade. Yet the mechanism for some processes are still intriguing. Use of traditional selective field ionization or simple electron detection prohibits direct investigation towards the “pre-ionization” stage of the evolution of a Rydberg gas into a UNP. The unique imaging capability offered by strontium’s optically-active core effectively mitigates this problem. This chapter, based on paper [35], will describe the techniques we employed in probing cold Sr Rydberg gas dynamics along with a short introduction to the operative processes. Our results, in contrast to earlier studies, stress the role played by Rydberg-Rydberg interactions in the initial phases of evolution of a Rydberg gas. 6.1 Experimental Setup Overview As shown in Figure 6.1 and 2.2, about a billion strontium atoms are captured in a magneto-optical trap after the Zeeman slower and cooled to ∼7mK. Both the cooling lasers and the laser driving the 461nm 5s21 S0 → 5s5p1 P1 transition are derived from a frequency doubled Ti-Sapphire laser stabilized by saturated absorption spectroscopy. 67 Figure 6.1 : a).Rydberg excitation beams and UNP ionization beams are shown with respect to the imaging system. The fluorescence imaging beam not shown in the figure is parallel to the Rydberg excitation beams. b). Pertinent energy levels for the two photon transition to Rydberg level and the core transition used for imaging. c). Timing for the experiment. Figure adapted from [35]. 68 The atomic density has a Gaussian profile with a radius of ∼1mm. The Rydberg atoms are excited by a two-photon transition. The 461nm laser is red detuned by 430MHz from the intermediate 5s5p1 P1 state and a 413nm laser is used for further promotion (see detailed description in Chapter 4). The diameters of both the 461nm and 413nm laser beams used for Rydberg excitation are much larger than the size of the MOT so that the density of Rydberg atoms should follow the MOT profile. The 413 nm light remains on all the time (see the time sequence in Figure 6.1) while the 461nm laser light is pulsed on during the excitation. In order to count the number of Rydberg atoms excited, we directly create a UNP by photoionizing a small fraction of the strontium atoms after the Rydberg excitation. The resulting electrons rapidly l change and ionize the Rydberg atoms allowing fluorescence detection of the core ions. The photoionization is produced by a 10ns dye laser pulse at 412nm. The 355nm light from a Nd:YAG laser pumps the dye laser which can be tuned to set the velocity (energy) of the ionized electrons. For our purposes, we set the frequency just above the ionization threshold so that only a small but sustainable plasma is produced in the MOT. For imaging the core ions, we utilize a 422nm laser beam to drive the 5s2 S1/2 → 5p2 P1/2 core transition. This laser is frequency doubled from a 844nm infrared diode laser. It is locked to a scanning Fabry Perot cavity that is referenced to the 922nm laser beam that is frequency doubled (461nm) and is stabilized by saturation absorption spectroscopy as mentioned previously. The conversion from scanning voltage to the real frequency of the cavity is calibrated by using the 422nm laser for saturated absorption spectroscopy in a 85 Rb cell. This is possible because the strongest peak in the 85 Rb spectra (5s 2 S1/2 (F = 2) → 6s 2 P1/2 (F = 3)) is only 440MHz red-detuned from the 88 Sr+ 5s 2 S1/2 → 5p 2 P1/2 transition. This 440MHz offset is easy to span 69 by an acousto optic modulator. This light is used for imaging the bare ions since it directly drives an optical transition in every ion (we can saturate this transition ). We can either image the light absorption or image the laser induced fluorescence from the atoms. Frequently, interest centers on the local density of the ions. So instead of imaging a whole cloud, we image a thin sheet of atoms that has well defined geometry; see Figure 6.2. Of course, its finite size needs to be taken into consideration for the final ion density calculation. We argue that this technique can also image the high-l strontium Rydberg atoms because, a Rydberg electron in a high-l orbit has negligible overlap with the core ion allowing the core ion to behave as an independent particle. In contrast, as described in the next section, low-l(l < 6) Rydberg atoms do not yield a fluorescence signal. 6.2 Principal Processes in Probing Sr Cold Rydberg Gas Among a complex array of processes occurring in our experiment, the following are the most influential ones despite the ongoing debates as to their relative importance. 6.2.1 Autoionization Excitation of the |5si core electron in a Rydberg atom can lead to autoionization. Autoionization is a special demonstration of multichannel coupling in the language of MQDT. It is an inherent property of a two electron atom resulting from the interaction between the two valence electrons. The term H12 = 1/r12 in the Hamiltonian not only mixes the configurations of bound Rydberg states that are nearly degenerate (as shown in red arrows in Figure 6.3 ) but also couples the doubly excited Rydberg series |5pnli to the Rydberg continuum |5sεl0 i; see blue arrows in Figure 6.3. The rate of this particular transition, i.e. autoionization rate, is determined by | h5pnl| H12 |5sεl0 i |2 70 Figure 6.2 : Optical probes for UNP using light resonant with the Sr+ transition. (a) Absorption Imaging: the laser beam is absorbed by the ions in the UNP creating a shadow that is recorded by a CCD camera. A complete absorption image can be constructed by a weighted integration of images taken over many different frequencies across the resonance to account for ions having different velocities. The ions’ absorption profile is a Voigt distribution. (b) Fluorescence Imaging of a sheet UNP: the imaging laser beam propagates perpendicularly to the the camera and imaging axis. A complete image construction of all the ions is the same as for absorption imaging. Figure adapted from [69]. 71 Figure 6.3 : Sr Rydberg atom autoionization The short solid lines are the strontium Rydberg levels converging to different series limits. The shaded regions are the corresponding continua. The red arrows denote the coupling between bound, nearly degenerate Rydberg levels in different configurations. The blue arrows denote the coupling to the continuum of a Rydberg series that has lower series limit. Figure Adapted from [56]. 72 which obeys the Rydberg scaling law ∼ 1/n3 . Its l dependence is complicated and can be understood in a scattering picture: the outer electron scatters the inner electron to the ground state while itself gaining energy and being ionized. To facilitate this, the outer electron must approach the inner electron and be moving rapidly. In low-l states, the Rydberg electron moves in a elliptical orbital, the lower l is, the closer the perigee is and the faster the electron moves around the perigee and of course the stronger the autoionization is. As a result, the oscillator strength generally decreases very rapidly as l increases. However, for very low l states, the overlapping of the outer electron with the inner electron is also strongly affected by the coupling between the core and the outer electron since it governs the phase of the wavefunction near the core. In fact, for very low l states, the transition rate is very high (∆t ≈ 0). So according to the Heisenberg principle, ∆E∆t ∼ ~, the transition linewidth ∆E is very large. For strontium in particular, the autoionization transition for 5s48s 1 S0 and 5s47s 1 D2 is as broad as tens of GHz [35]. Since the imaging laser we are using is of narrow linewidth ∼ 1M Hz, the effective autoionization, having a rate about 1M Hz/10GHz ∼ 0.01% of the total autoionization rate, is negligible for low-l states during the 500ns imaging time. For high-l states, overlap with the core, i.e., scattering, becomes negligible. 6.2.2 Electron-Rydberg Collisions Electrons with velocities comparable to the Rydberg electron can be produced easily in many collision processes. Collisions between electrons and Rydberg atoms can lead to l and m changing and even n changing if the incoming electron is energetic enough. In comparison, collisions between the ions and Rydberg atoms are unlikely due to their low velocities. Therefore the most important collision is the l changing collision 73 Figure 6.4 : l-mixing due to Rydberg-electron collision. Step A shows the mixing into the manifold as the electron approaches. In step B, the population is randomized among the manifold. Step C is the ionization from the manifold. Figure adapted from [68]. between the Rydberg atom and the electron. Here is how it works (see figure 6.4): as the electron approaches, the low-l Rydberg state follows the quadratic Stark shift and gradually mixes with the high-l manifold. Depending on the magnitude of the electric field, it can mix strongly into the nearby manifold states. Thus when the electron has passed by, it has a good chance to stay in one of the manifold states as the field decreases. Even after collision the atom is subject to a very small electric field which can lead to a redistribution among neighboring states in the manifold. As a result, the original low-l state is further l-mixed into l > n/2 states on average. Though m is a good quantum number for Rydberg atoms in an electric field, every electron is incident in a random direction, whereupon m has to be referenced to a new axis. So m mixing is happening all the time. The n-changing collisions are possible if the 74 electron passes close to the Rydberg electron. In the electron’s strong field manifolds belonging to different n will cross, and the original state could mix into nearby ns. 6.2.3 Penning Ionization Collisions between two Rydberg atoms can lead to Penning ionization R∗∗ (n) + R∗∗ (n) → R∗∗ (n0 ) + R+ + e− . (6.1) Penning ionization can be facilitated if the two Rydberg atoms are attracted to each other by attractive dipole-dipole interactions or attractive Van der Waals interactions. Once the interparticle distance R is small enough that the interaction energy is comparable to the binding energy of the Rydberg atom, one of the Rydberg electrons will be ionized and the other atom will be deexcited into a more deeply bound state. Simulations in [70] show that in almost all likelihood, the ionized electron will be energetic enough to escape on a nanosecond timescale. The binding energy of the resulting Rydberg atom almost doubles its initial binding energy. The rest energies will make the ion and Rydberg atom leave each other in high relative velocities but small center of mass velocity since their total momentum is negligible initially (cold atoms). There are also cases where, even for attractive interaction, Penning ionization doesn’t happen. As the initial attractive interaction draws two Rydberg atoms together, their dipole moments or equivalently their Runge-Lenz vectors can precess to directions such that the interaction changes to repulsive. Consequently, they begin to move apart from each other. Indeed Penning ionization will almost never happen if the Rydberg-Rydberg interactions are isotropically repulsive. Low-l Rydberg atoms don’t possess a permanent dipole. Nevertheless, a dipoledipole interaction which scales as ±C3 /R3 , C3 ∝ n4 can be induced if there is a 75 resonance like a Förster Resonance nl + nl ↔ n1 l1 + n2 l2 . In the absence of both a permanent dipole and a resonance, the interaction between two Rydberg atoms is of the Van der Waals type and scales as ±C6 /R6 , C6 ∝ n11 which is the interaction between instantaneous dipole moments. This force is termed London dispersion or the instantaneous dipole-induced dipole force. For the initially excited low-l strontium Rydberg atoms, the interactions are all Van der Waals interactions. The values of C6 s for different n are calculated in [65]. 6.2.4 Blackbody Radiation induced Ionization Rydberg atoms, having small binding energies, are generally sensitive to the 300K thermal environment since the blackbody radiation (BBR) can not only drive dipole allowed transitions to nearby states but also photoionize them. Both effects will limit the lifetime of Rydberg atoms with BBR decay rates that scale as ∼ kT /n2 . For 5s48s 1 S0 and 5s47s 1 D2 states, the BBR induced n,l changing rates are ∼ 1 × 104 s−1 and the BBR induced photoionization rates are ∼ 1 × 102 s−1 . Note, however, that it takes many dipole allowed transitions to induce large changes in l and thus this process may be unimportant in the present experiments. 6.3 Imaging Technique A direct laser induced fluorescence imaging of all the Rydberg atoms can be made by exciting a dilute plasma (based on its function, we call it a seed plasma) immediately after Rydberg excitation as shown in Figure 6.5. The free electrons trapped in the MOT by the plasma cause repetitive l-mixing collisions and the low-l Rydberg atoms are rapidly converted into high-l states which are visible to the imaging light. This process can happen on a nanosecond time scale enabling us to count the Rydberg 76 Figure 6.5 : Laser-induced fluorescence imaging of UNP and Rydberg atom clouds using the 5s 2 S1/2 → 5p 2 P1/2 core-ion transition at 422 nm. (a) Image after Rydberg excitation to the 5s48s 1 S0 state for texc ≈ 3 µs, which yields ∼ 8 × 105 Rydberg atoms. Notice that the fluorescence signal is very small and the scale bar represents 1 mm. (b) Image after exciting the same Rydberg population as in panel (a) but with superposition of a seed UNP containing ∼ 2 × 105 ions and electrons. Note the increased fluorescence from the cigar-shaped region of Rydberg excitation. (c) Image of a seed UNP identical to that in panel (b), but with no Rydberg excitation. (d) Signal due to Rydberg excitation obtained by subtracting the signal due to ion cores in the seed UNP. Figure adapted from [35]. atoms. If we vary the Rydberg excitation time, we should expect a linear increase in the Rydberg excitation due to the fixed excitation rate determined by the Rabi frequency. This behavior is observed in Figure 6.6. 6.4 Results Discussion As we can see in Figure 6.6, without a seed plasma, there are no visible ion cores below excitation times texc ∼ 2.5µs implying all the excited Rydberg atoms are still in their low-l states. If we stop the excitation at the time, and let the Rydberg gas 77 Figure 6.6 : Dependence of the LIF signal from parent 5s48s 1 S0 Rydberg atoms on the excitation time texc . Left: each image is shown as a function of time with (top) and without (bottom) a seed UNP present. (Contributions to the LIF signals from the UNP are subtracted and the scale bar represents 1 mm). Right:Number of visible ion cores versus Rydberg excitation time for parent 5s48s 1 S0 states. Data recorded both with and without a seed UNP present are included together with results obtained when only the seed UNP was created. Qualitatively similar results are seen for the 5s47d 1 D2 state. Figure adapted from [35]. 78 evolve by itself, what will happen? We examined this using both the 5s47d 1 D2 state and 5s48s 1 S0 state with the same initial density and number of Rydberg atoms. Figure 6.7 shows that in both cases the core ions gradually become more and more visible. However, though the initial conditions are very similar, these two states display rather different time evolutions. These differences can be seen in the total number of visible ion cores as shown in Figure 6.8. These differences can be explained by noting that both l-mixed Rydberg atoms and true ions are seen by the imaging beam. Since a very weak seed plasma (see image (c) in Figure 6.5 and notice that it has almost the same color bar as Figure 6.7) is able to provide a sufficient number of trapped free electrons to l-mix all the Rydberg atoms and make them visible, the majority of the visible ions in Figure 6.7 for 4.1µs evolution time cannot be true ions otherwise the attendant electrons should have l-mixed almost all the Rydberg atoms in the central region and made them visible in nanoseconds. Therefore, it is clear that almost all the visible ions we see for 4.1µs evolution time should be l-mixed Rydberg atoms resulting from collisions with some early electrons. Now the question becomes where do the initial electrons (which collide with the Rydberg atoms and further lead to l-mixing) come from? Whatever the mechanism is, we can see the early electrons are produced first in the places where the initial Rydberg atom density is highest. This is shown unambiguously in Figure 6.7. For each state, the visibility is brighter in the higher Rydberg atom density region in each image and the visibility propagates to the lower density region with increased evolution time. Both BBR ionization and Penning ionization will lead to such density dependent ionization. However, the BBR-induced ionization is slow and cannot explain the difference seen between the behavior of S and D state. Penning ionization, on the other hand, would. 79 Figure 6.7 : LIF images showing the spontaneous evolution of an ultracold gas of 1 S0 (top) and 1 D2 (middle) Rydberg atoms. The evolution time is indicated above each image in µs and the scale bar represents 1 mm. The initial numbers and densities for both states are identical, 8 × 105 and 2.2 × 108 cm−3 , respectively. Notice how the S-state population evolves more quickly in both space and time. The bottom panels show one-dimensional plots of the density integrated along the vertical direction. The spatial development of the S state (red) leads that of the D state (blue). Figure adapted from [35]. 80 Figure 6.8 : Evolution of the number of visible core ions for 5s48s 1 S0 and 5s47d 1 D2 Rydberg atoms. The circles represent number calibrations performed by scanning the imaging laser through resonance for a complete construction of the “ion” numbers. In most of the experiments, the imaging laser is set on resonance and only one image is recorded. Such single images are converted to the total “ion” number by using a conversion factor since the “ion” number is varying in a rather fixed profile against the imaging frequency. As we can see from the calibrated points, this simple trick works very well. At late times the number of visible ion cores seen, ∼ 8 × 105 , agrees reasonably well with the number of parent Rydberg atoms initially excited, ∼ 7×105 , as determined using a seed UNP. Figure adapted from [35]. 81 In reference [65], it is theoretically predicted that S states should have isotropically attractive Van der Waals interactions while the interactions between D states should be mostly repulsive and only for some restricted range of orientations will they attract weakly. So Penning ionization should be slow for D states but should happen a lot more frequently for S states. The explains our LIF images very well, especially the image at an evolution time of 4.1µs. For S states, collisions produce electrons, initially at the center of the cloud where the Penning ionization is the strongest which will collide and l-mix with Rydberg atoms before they escape from the cloud. However, free electron production for D states mostly from BBR ionization is much slower so the number of the early electrons and the l-changing rate, is much less resulting in reduced visibility for the core ions. As we can see from Figure 6.7 and 6.8, S states lead to a faster evolution of the overall visibility. This contrast in visibility would be more easy to see for even shorter evolution times tevol < 4µs since the D state would be completely dark. However, this is not the case because it will take a long time for two nearest Rydberg atoms to Penning ionize in the Van Der Waals interaction regime. Assuming that the potential energy of a pair Rydberg atoms that experience an attractive force is completely converted to kinetic energy as they accelerate towards each other, then the collision time can be approximated using [70] as Z R0 T = Rc 1 q C6 (R 6 − C6 ) R06 dR. (6.2) × 4/M where R0 is the initial separation between the pair of Rydberg atoms and Rc is the lower limit of their separation which can be approximated as the diameter of the Rydberg atom ∼ 482 a.u. = 2304a.u.. Since Rc R0 , Rc can be treated as zero. For 82 strontium, we have [65] C6 = 15 × n11 = 15 × 4811 = 4.675 × 1019 a.u., Z 0 1 √ x3 dx ≈ 0.43, M = 16.04 × 104 a.u.. 1 − x6 (6.3) (6.4) Equation 6.2 can be reduced to R4 T = 0× 2 r 16.04 × 104 × 0.43a.u. 4.675 × 1019 T ≈ 0.039 × (R0 in µm)4 µs. (6.5) (6.6) Since collision time is very sensitive to the change in the R0 as shown in Figure 6.9, it is important to correctly estimate R0 . If we use the nearest neighbor distribution for the initial condition in Figure 6.7 as described in reference [35], it is possible to have hundreds of Rydberg atom pairs separated by R0 < 3.5µm which yields collision times that allow electron production on micro-second time scales. However, if Rydberg blockade is important, the initial separation between two Rydberg atoms can be no less than the blockade radius Rvdw which can be calculated assuming an overall linewidth of 1M Hz C6 = 1M Hz = 2.419 × 10−11 a.u. 6 Rvdw (6.7) Rvdw = 1.116 × 105 a.u. = 5.91µm. (6.8) This restriction will yield a minimum collision time of 47µs! Apparently, Rydberg blockade effects don’t appear to be dominant. There are multiple possible explanations. For instance, equation 6.2 assumes the initial velocities of the Rydberg atoms is zero. While this assumption holds in the sense that the initial 7mK thermal velocities have no fixed direction, it is possible, in some cases, to have two Rydberg atoms 83 Figure 6.9 : Collision Time Vs Initial Inter-Rydberg atoms Distance 84 moving towards each other with their thermal velocity ∼ 1m/s. This initial velocity, about 1µm per microsecond, can greatly reduce the total collision time. On the other hand, the overall linewidth for the particular two-photon transition employed can be a lot larger than the pure laser linewidth. Moreover since we don’t know the exact detuning of the transition, it is possible that the interaction induced energy shift between two very close Rydberg atoms matches the detuning which facilitate closer Rydberg pair excitation. In fact, the strong AC Stark shifts due to 5s2 → 5s5p transition could lead to an anti-blockade effect [71] that makes Rydberg blockade inefficient. Also, stray fields may further complicate the Rydberg blockade. After the very first phases of the dynamics, the initial differences in the behavior of S states and D states due to their different interactions will be smeared out by other effects. For example, l-mixed Rydberg atoms possess permanent dipole moments and their Penning ionization rates will be larger than those expected for Penning ionization due to the Van der Waals interaction. Following formation of the electron trap, the l-mixed Rydberg atoms will dominate in the cloud and all memory of the initial state will be lost. In the previous studies [67], [66] and [68], it has been suggested that almost all early electrons are generated by the 300K blackbody radiation and the Penning ionization plays a negligible role in the dynamics until the l-mixed Rydberg atoms dominate the Rydberg gas. However, according to our study, we believe Penning ionization in the early stage of the evolution is at least as significant as BBR induced ionization in producing the early electrons. 85 Chapter 7 Conclusion and Outlook We have created strontium Rydberg atoms in two environments and demonstrated several of the key experimental capabilities they offer which will directly enable a variety of future studies. We produced very-high-n (n ∼ 300) strontium Rydberg atoms in a collimated atomic beam using two-photon excitation. Spectroscopically, we observed the normal mass shifts of the even isotopes 86 Sr, 88 Sr, 84 Sr and the hyperfine induced mixings of the odd isotope 87 Sr. By exciting Rydberg atoms in a DC field, we studied the Stark shifts of various Rydberg states. A two-active-electron model was used to analyze the data and provided results in excellent agreement with experiment. The derived “nD” Stark state possesses a large permanent dipole moment in DC fields approaching those at which states in neighboring manifolds first cross. This allows production of a quasi-one-dimensional state and that can be engineered into circular or elliptical states. In other words, the dipole moment of this Rydberg atom is not only very large (when it is a quasi-one-dimensional state) but is also tunable (when it is manipulated into states of varying ellipticity). Also, the ability to near simultaneously produce many Rydberg atoms opens the opportunity to study Rydberg-Rydberg interactions. The Van der Waals interaction between Rydberg atoms of such high n has not yet been investigated either experimentally or theoretically. Additionally, we can create planetary states by exciting the second valence electron and examine the interactions between two excited electrons within one Rydberg atom. 86 We also studied ultracold strontium Rydberg gases by imaging light scattered from the core ions of l-mixed Rydberg atoms, i.e. the laser induced fluorescence. To induce rapid l-mixing and rapidly image all the Rydberg atoms, a weak ultracold neutral plasma is utilized to introduce free electrons that collide with Rydberg atoms. The temporal and spatial resolution of this imaging technique permits the study of the evolution of an ultracold Rydberg gas. In the early stages of this evolution, Penning ionization as well as blackbody radiation induce photoionization and introduce initial electrons that l-mix the low-l Rydberg atoms and make them visible to the imaging light. 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