Atom Interferometry with Laser-Cooled Lithium by Kayleigh Cassella Doctor of Philosophy in Physics University of California, Berkeley Professor Holger M¨Uller,Chair

Hot Beats and Tune Outs: Atom Interferometry with Laser-cooled Lithium

Kayleigh Cassella

A dissertation submitted in partial satisfaction of the requirements for the degree of

Doctor of Philosophy

Physics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Holger M¨uller,Chair Professor Dan Stamper-Kurn Professor Jeﬀrey Bokor

Spring 2018 Hot Beats and Tune Outs: Atom Interferometry with Laser-cooled Lithium

Abstract

Hot Beats and Tune Outs: Atom Interferometry with Laser-cooled Lithium by Kayleigh Cassella Doctor of Philosophy in Physics University of California, Berkeley Professor Holger M¨uller,Chair

Ushered forth by advances in time and frequency metrology, atom interferometry remains an indispensable measurement tool in atomic physics due to its precision and versatility. A sequence of four π/2 beam splitter pulses can create either an interferometer sensitive to the atom’s recoil frequency when the momentum imparted by the light reverses direction between pulse pairs or, when constructed from pulses without such reversal, sensitive to the perturbing potential from an external optical field. Here, we demonstrate the first atom interferometer with laser-cooled lithium, advantageous for its low mass and simple atomic structure. We study both a recoil-sensitive Ramsey-Bordéinterferometer and interferometry sensitive to the dynamic polarizability of the ground state of lithium. Recoil-sensitive Ramsey-Bordéinterferometry benefits from lithium’s high recoil frequency, a consequence of its low mass. At an interrogation time of 10 ms, a Ramsey-Bordé lithium interferometer could achieve sensitivities comparable to those realized at much longer times with heavier alkali atoms. However, in contrast with other atoms that are used for atom interferometry, lithium’s unresolved excited-state hyperfine structure precludes the the cycling transition necessary for efficient cooling. Without sub-Doppler cooling techniques. As as result, a lithium atomic gas is typically laser cooled to temperatures around 300 µK, above the Doppler limit, and well above the recoil temperature of 6 µK. This higher temperature gas expands rapidly during the operation of an atom interferometer, limiting the experimental interrogation time and preventing spatially resolved detection. In this work, a light-pulse lithium matter-wave interferometer is demonstrated in spite of these limitation. Two-photon Raman interferometer pulses coherently couple the atom’s spin and momentum and are thus able to spectrally resolve the outputs. These fast pulses drive conjugate interferometers simultaneously which beat with a fast frequency component proportional to the atomic recoil frequency and an envelope modulated by the two-photon detuning of the Raman transition. We detect the summed signal at short experimental times, preventing perturbation of the signal from vibration noise. This demonstration of a sub-recoil measurement with a super-recoil sample opens the door to similar scheme with other particles that are difficult to trap and cool well, like electrons. 2

An interferometer instead composed of π/2-pulses with a single direction of momentum transfer, can be sensitive to the dynamic polarizability of the atomic ground state. By scanning the frequency of an external driving field, such a measurement can be used to determine the atom’s tune-out wavelength. This is the wavelength at which the frequency- dependent polarizability vanishes due to compensating ac-Stark shifts from other atomic states. Lithium’s simple atomic structure allows for a precise computation of properties with only ab initio wave functions and spectroscopic data. A direct interferometric measurement of lithium’s red tune-out wavelength at 670.971626(1) nm, is a precise comparison to existing ‘all-order’ atomic theory computations. It also provides another way to experimentally determine the S− to P − transitions matrix elements, for which large correlations and small values complicate computations. Finally, a future measurement of lithium’s ultraviolet tune- out wavelength of at 324.192(2) nm would be sensitive to relativistic approximations in the atomic structure description. Atom interferometry simultaneously verifies existing atomic theory with measurements of atomic properties and searches for exotic physics lurking in plain sight. The techniques developed here broaden the applicability of interferometry and increase measurement sensitivity by simplifying cooling, increasing atom number and reducing the cycle time. Overcom- ing the current experimental limitations on interrogation time would allow for ultra-precise measurements of both the tune-out wavelength and the fine structure constant. i

To my mom and step-dad, who ﬁlled me with enough resolve to do hard things. To my sisters, who grew with me and tethered me to real things. To my husband, who unfolded all the crumpled parts of me, again and again. To my children, my greatest teachers, who sprinkled light in all the dark places.

I dedicate this work to you. ii

Contents

Contents ii

List of Figures vi

List of Tables viii

1 Outward bound 1 1.1 Corpuscular and undulatory ...... 2 1.1.1 Waves of matter ...... 4 1.2 α, the ﬁne structure constant ...... 5 h 1.2.1 M measurement ...... 8 1.3 α, the polarizability ...... 8 1.3.1 Dynamic polarizability ...... 10 1.4 Previous measurements ...... 12 1.4.1 λto measurement ...... 13 1.5 Overview of this thesis ...... 15

2 Atom interferometry 16 2.1 Light oﬀ ...... 19 2.1.1 The free evolution phase ...... 22 2.2 Light on ...... 23 2.2.1 Raman scattering ...... 23 2.2.1.1 Dressed states ...... 28 2.2.2 The interaction phase ...... 30 2.2.3 The separation phase ...... 31 2.3 The total phase ...... 31 π π π π 2.4 Conjugate interferometers with the 2 - 2 - 2 - 2 ...... 33 2.5 The Ramsey-Bord´einterferometer ...... 33 2.5.1 cRBI phase computation ...... 36 2.6 The copropagating interferometer ...... 38 2.6.1 cCPI phase computation ...... 41 iii

3 Lithium, the smallest alkali 42 3.1 Lithium, the lightest alkali ...... 42 3.2 Lithium, the simplest alkali ...... 45 3.2.1 The Hylleraas basis ...... 46 3.3 Dynamic polarizability ...... 48 3.4 Lukewarm Lithium ...... 50 3.4.1 Lithium atom interferometry in the space domain ...... 53 3.5 Advantages of light-pulsed interferometry with lithium ...... 54

4 Experimental Methods 56 4.1 Lithium Spectroscopy ...... 58 4.1.1 Modulation Transfer Spectroscopy ...... 59 4.1.2 The cascade of frequency generation ...... 62 4.1.2.1 Tapered ampliﬁers ...... 63 4.2 Cooling and trapping ...... 64 4.2.1 2D MOT frequency generation ...... 67 4.2.2 3D MOT frequency generation ...... 67 4.2.3 Vacuum system and optics ...... 68 4.2.3.1 2D MOT chamber ...... 70 4.2.3.2 3D MOT chamber ...... 72 4.2.4 Experimental sequence ...... 73 4.3 State preparation ...... 74 4.3.1 Frequency generation for optical pumping light ...... 76 4.3.2 Optical pumping optics ...... 78 4.3.2.1 Quantization axis ...... 78 4.4 Interferometry ...... 78 4.4.1 Frequency generation for Raman beams ...... 78 4.4.2 Raman optics ...... 82 4.5 Detection ...... 83 4.5.1 Absorption imaging ...... 83 4.5.2 Wollaston prism technique ...... 84 4.5.3 Time-of-ﬂight imaging ...... 84

5 Hot Beats 87 5.1 Super-recoil lithium ...... 87 5.1.1 Large bandwidth pulses ...... 88 5.1.2 k-reversal ...... 88 5.2 Simultaneous and conjugate ...... 89 5.3 Overlapped, simultaneous and conjugate ...... 91 5.3.1 Hot beats ...... 92 5.3.2 Time-domain ﬁtting ...... 92 5.3.3 Frequency-domain ﬁtting ...... 94 iv

5.4 Phase noise ...... 95 5.5 Outlook ...... 97 5.5.1 Vibration immunity ...... 98

6 Tune-outs 100 6.1 Previous polarizability measurements ...... 101 6.1.1 The diﬀerential Stark shift ...... 101 6.1.2 Space-domain atom interferometry ...... 102 6.2 Light-pulsed interferometric lithium tune outs ...... 103 6.2.1 φto, the tune-out phase ...... 103 6.2.2 The tune-out beam ...... 106 6.2.3 Experimental Sequence ...... 106 6.2.4 Detection & Analysis ...... 108 6.2.4.1 Principal component analysis ...... 110 6.3 Towards tune-out ...... 111 6.3.1 Precision ...... 111 6.3.1.1 Single-photon scattering ...... 112 6.3.1.2 Beam shaping ...... 113 6.3.2 Accuracy ...... 113 6.4 Hyperﬁne dynamic polarizabilities ...... 113

7 Conclusion 117 7.1 Outlook for recoil-sensitive interferometry with super-recoil samples . . . . . 117 7.1.1 h/me measurement ...... 117 7.2 Outlook for tune-out interferometric measurements in lithium ...... 119 7.2.1 Beyond the red ...... 119 7.2.2 Investigation of nuclear structure between isotopes ...... 121 7.3 Atom interferometry with lukewarm lithium ...... 122 7.3.1 Sisyphus cooling ...... 123 7.3.2 Gray molasses ...... 123 7.4 Onward ...... 123

A Properties of lithium 125 A.1 The level spectrum ...... 129 A.2 Interaction with static fields ...... 129 A.3 Interaction with dynamic fields ...... 131 A.3.1 Reduced Matrix Elements in Atomic Transitions ...... 131 A.4 Clebsch-Gordan coefficients for D–line transitions ...... 132

B Two-Level System 136 B.1 Flip-ﬂop ...... 138 B.1.1 On resonance ...... 138 v

B.1.2 Almost on resonance ...... 139

C Bloch sphere 141 C.1 Simulations of interferometry ...... 144 C.1.1 Mathematica code ...... 144

D Magneto-optical traps 147 D.0.1 Optical molasses ...... 147 D.0.2 Magnetic trapping ...... 148

E α0, the static polarizability 151 E.1 Nonrelativistic α(0) ...... 151

F Hyperpolarizability 154 F.1 Positive and negative frequency components ...... 156

G Matlab simulation of thermal cloud 158 G.0.1 Code ...... 158 G.0.2 Matlab functions ...... 161 G.0.2.1 Preprecompute.m ...... 161 G.0.2.2 simulatef0is1.m ...... 162

Bibliography 166 vi

List of Figures

1.1 Optical Michelson-Morley and Mach-Zehnder configurations ...... 3 7 1.2 Plot of dynamic polarizability for Li’s 2S2 level...... 14 2.1 Recombination of the superposition at the last π/2-pulse results in interference . 18 2.2 Interferometers in the π/2-π/2-π/2-π/2 geometry ...... 19 2.3 Trajectories of atom in configuration space ...... 22 2.4 Effective wave vector and momentum coupling ...... 24 2.5 Three-level system ...... 25 2.6 Conjugate interferometers ...... 33 2.7 Interaction geometry for the lower Ramsey-Bordéinterferometer ...... 34 2.8 Interaction geometry for the upper Ramsey-Bordéinterferometer ...... 36 2.9 Interaction geometry for the lower copropagating interferometer ...... 39 2.10 Interaction geometry for the upper copropagating interferometer ...... 40

3.1 Comparison of RBI for 7Li and 133Cs ...... 43 3.2 The computed dynamic polarizability for the lithium ground state ...... 51 3.3 The Maxwell-Boltzmann distributions for atoms at the recoil temperature (blue) and at 300 µK (red)...... 52

4.1 Experimental sequence and settings ...... 57 4.2 Lithium spectroscopy ...... 61 4.3 Circuit schematic of master ECDL frequency lockbox ...... 62 4.4 Experimental frequencies ...... 65 4.5 Frequency generation for the 2D MOT...... 68 4.6 Frequency generation for the 3D MOT and pusher beam ...... 69 4.7 2D MOT chamber ...... 71 4.8 2D MOT optics ...... 73 4.9 3D MOT chamber ...... 74 4.10 Microwave spectrum of the |F = 2, mF i ground state...... 75 4.11 Magnetic ﬁeld gradient decay...... 76 4.12 Energy level diagram showing the frequencies for optical pumping...... 77 4.13 Schematic of optical pumping frequency oﬀset lock electronics ...... 77 vii

4.14 Optical pumping optics ...... 79 4.15 Two-photon Raman transition level diagram ...... 80 4.16 Optical set-up for generating the Raman frequencies...... 81 4.17 Optics set-up for Raman beams ...... 82 4.18 The beam path for the imaging light as it transverse the vacuum apparatus. . . 85

5.1 Comparison of pulse bandwidth to temperature of atom cloud ...... 89 5.2 Vacuum tube switch ...... 90 5.3 Spatially overlapped interferometer outputs ...... 91 5.4 Data and fits for a range of two-photon detunings δ ...... 93 5.5 Fit data showing both the amplitude modulation as well as the fast frequency component, the recoil frequency ...... 94 5.6 Fast fourier transform of beating interferometers ...... 95 5.7 Fourier transformed data for various δ’s...... 96 5.8 A plot of the standard deviation resulting from fits of the Fourier-transformed data in the time- and frequency-domain at different two-photon detunings. . . . 98

6.1 Scheme for tune-out measurement in thermal atom interferometer ...... 102 6.2 Orientation of the tune-out beam with respect to the atom cloud ...... 104 6.3 ...... 105 6.4 A gaussian beam’s spatial dependence ...... 105 6.5 Tune-out frequency generation and optics ...... 107 6.6 Tune-out measurement sequence and settings ...... 108 6.7 A comparison between the analysis performed without (top) and with (bottom) the tune-out pulse. The lower principal component analysis only contains that dependent upon the extra light, switched on during the T 0 time step in the interferometer...... 109 6.8 Premliminary tune-out plot ...... 112 6.9 Anamorphic prism pair transforms beam shapes between circular and elliptical. 113 6.10 PC breakdown with elliptical beam...... 114 6.11 Scalar and tensor dynamic polarizabilities for hyperﬁne ground state levels in 7Li. 115 6.12 Comparison of dynamic polarizability between lithium’s hyperﬁne ground states 116

7.1 Ramsey-Bord´einterferometer for electrons ...... 118 7.2 Dynamic polarizability of lithium’s ground level ...... 120 7.3 Lithium’s UV tune-out ...... 120 7.4 Sisyphus cooling frequencies ...... 122

A.1 Vapor pressure of lithium ...... 127

C.1 Model of trajectory of state on the Bloch sphere...... 146

D.1 Magneto-optical trapping ...... 149 viii

List of Tables

1.1 Tune-out measurements to-date, method of measurement and reference...... 12

2.1 Trajectories for lower Ramsey-Bord´einterferometer ...... 35 2.2 Trajectories for upper Ramsey-Bord´einterferometer ...... 37 2.3 Trajectories for lower copropagating interferometer ...... 39 2.4 Trajectories for upper copropagating interferometer ...... 40

3.1 Comparison of mass and single photon recoil velocity (frequency) for lithium, rubidium and cesium’s D2-lines ...... 42 3.2 Angular momentum conﬁgurations for the S, P , D states of lithium...... 48 3.3 Scalar polarizabilities and Stark shift values for 7Li ...... 51

4.1 Experimental detunings ...... 70

5.1 Fitting parameters for Fig. 5.5...... 95

A.1 Physical properties of lithium ...... 125 A.2 Physical properties of lithium ...... 127 7 A.3 Li D2 (2S1/2 → 2P3/2) Transition Properties ...... 128 7 A.4 Li D1 (2S1/2 → 2P1/2) Transition Properties ...... 128 A.5 Clebsch-Gordan coefficients for the D2-line transition with σ+-polarized light such 0 that mF = mF + 1...... 133 A.6 Clebsch-Gordan coefficients for the D2-line transition with σ−-polarized light such 0 that mF = mF − 1...... 133 A.7 Clebsch-Gordan coefficients for the D2-line transition with π-polarized light such 0 that mF = mF ...... 134 A.8 Clebsch-Gordan coefficients for the D1-line transition with σ+-polarized light such 0 that mF = mF + 1...... 134 A.9 Clebsch-Gordan coefficients for the D1-line transition with σ−-polarized light such 0 that mF = mF − 1...... 135 A.10 Clebsch-Gordan coefficients for the D1-line transition with π-polarized light such 0 that mF = mF ...... 135 ix

7 E.1 Scalar polarizability diﬀerences α0(nPJ ) − α0(nS) in a.u. for Li...... 151 x

Acknowledgments

The work presented here on atom interferometry with laser-cooled lithium is the culmina- tion of the efforts of many. I want to acknowledge those who fought or continue to fight ‘the red devil’: Eric Copenhaver, Robert Berghaus, Geena Kim, Paul Hamilton, Chen Lai, Pro- fessor Yanying Feng, Quinn Simmons, Simon Budker, Hunter Akins, Biswaroop Mukherjee, Dennis Schlippert, Daniel Tiarks and Trinity Joshi. I am grateful to have had the oppor- tunity to learn from and work alongside Eric Copenhaver most recently, during which the projects presented in Chapters 5 and 6 were born. The constant amongst these generations of the lithium project is my advisor Holger Müller.Holger continues to fearlessly lead all of us through extraordinary part per billion experimental endeavors - whether by aligning the tricky double-pass AOM, using aluminum foil to successfully impedance match everything or helping you navigate the subtleties of noise, sensitivity and precision - Holger has been instrumental on many occasions in helping me find that epsilon. I am deeply grateful to Holger and an OK experimentalist because of his guidance, support and patience. I would have been lost without the group members of the early days including: Justin Brown, Brian Estey, Paul Hamilton, and Geena Kim. I am grateful to have worked with some of best interferometeers around: Phillip Haslinger, Chenghui Yu, Weicheng Zhong, Richard Parker, Xuejian Wu, Osip Schwartz, Jeremy Axelrod, Matt Jaffe, Victoria Xu, and Jordan Dudley. To the next generation going forth in Physics - Zachary Pagel, Joyce Kwan, Robert Berghaus, Randy Putnam, Ryan Bilotta, Dalila Robledo, and Bola Malek - may entropy be on your side. I look forward to learning of all that you discover. I would like to thank my committee members, Jeffrey Bokor and Dan Stamper-Kurn, who patiently nudged me towards scientific maturity but all the while requiring I stand on my own. I am grateful to the administrators in the department who have helped me navigate the fine print, find my lost child on Cal-Day, and advocate for my needs: Ann Takizawa, Claudia Trujillo, Donna Sakima, Joelle Miles, Amanda Dillon, Eleanor Crump, Anthony Vitan, Carlos Bustamante, Amin Jazaeri, and Rachel Winheld. The faculty in the Biology, Chemistry and Physics departments at my undergraduate college, Indiana University South Bend were the first faces of academia I saw many years ago and sent me off to Berkeley with enough momentum to get me through. Thank you to Ann Grens, Monika Lynker, Bill Feighery, Doug McMillen, Matthew Marmorino, Jerry Hin- nefeld, Ilan Levine, and Henry Scott, who taught and inspired me. In building a community around the undergraduates at IUSB, they created a space that protected and cultivated my curiosity. I have only the utmost gratitude for Rolf Schimmrigk, who not only challenged me to understand my own questions, but to reach. He also instilled in me a love for good, consistent notation which I have tried to implement here. My journey to this point has been because of invaluable friends supporting me throughout and telling me exactly what I needed to hear, even though I often could not listen. To my friends and in particular: Austin Hedeman, Hilary Jacks, Trinity Joshi, Kate Kamdin, Ming xi

Yi, and Shun Wu for being the quantization axis when things got muddled and pointing me forward. My deepest gratitude to Emily Grace whose advice and support has helped me (and continues to) navigate the most diﬃcult days. Thanks to our family’s UC Village comrades and close friends for being a constant source of support, friendship, and drinking solidarity. My family’s love and support enabled me to strive towards what I needed; to my parents, Debbie, Don, Kirsten, Bruce and Frank, my sisters, Taryn, Jordan and Isla, and my nephew Lucian, thank you. I am deeply grateful to our co-parents, Jenni and Frank Almeida and Meryl and Rob McCarthy, and to their parents, who are the village for my children. Thank you for your understanding, encouragement, friendship and support. None of this work would have been possible without Arran, who has kept me whole throughout graduate school, and the children I partly call mine, Finn, Emily, Maggie, Alex, to whom I have dedicated this work. Thank you for your patience and love. 1

Chapter 1

Outward bound

Today, Physics is faced with many known unknowns. The Standard Model of particle physics provides a theoretical framework for the electromagnetic, weak and strong forces. It successfully incorporates experimental data for the fundamental particles composing the known matter in our universe [1] and is even able to predict the electron magnetic moment to a part per trillion [2]. However, its shortcomings are glaring. It neglects the major known constituents of the universe - dark matter [3] and dark energy [4, 5] - and fails to provide an explanation for the observed baryon asymmetry [6] or the fine-tuning of θ in quantum chromodynamics [7] and completely omits the last fundamental interaction, gravity. While the Standard Model (SM) is too incomplete to be the final unifying theory of physics, a consensus as to what higher energy theory must underly it has yet to be reached [8– 12]. This realization motivates some to look outward, away from the high energy, subatomic regime of particle physics, and instead to lower energy atomic systems. In these systems, an unprecedented level of experimental control [13–15] and sensitivity [16, 17] has been realized only over the last few decades. Massive particle colliders, such as the Large Hadron Collider (LHC) at CERN, may be able to search for yet undiscovered particles but such particles often are anticipated to produce observable effects in low-energy precision measurements [7]. For example, time- reversal symmetry violation could explain the origin of baryon asymmetry [6] and would manifest as a measured electric dipole moment (EDM). Extensions made to the SM for the purposes of incorporating gravity into its theoretical framework often result in violations to Lorentz symmetry and CPT invariance, both which could be observed in a low-energy system [18–20]. Such effects arising from whatever exotic physics lay lurking beyond the SM needs to be measurable in the atomic systems we are championing, comparable to theoretical calculations such that a ‘sufficiently significant and robust discrepancy’ [7] can be demonstrated. A measurement needs a ruler, a reference with the same dimensions allowing for a comparison between two systems. Fundamental constants become a focus of many because their role is central in physical theories. Physical constants have an intrinsic theory-dependent existence and determine the magnitude of physical processes but simultaneously have values that CHAPTER 1. OUTWARD BOUND 2

cannot be predicted theoretically.

(They are) constants whose value we cannot calculate with precision in terms of more fundamental constants, not just because the calculation is too complicated but because we do not know of anything more fundamental.

Steven Weinberg [21]

Another focus of metrology is the determination of atomic properties. A comparison of experimentally determined atomic properties with those that have been theoretically computed can inform modern computational methods. Demanding atomic structure calculations can be made more tolerable with experimentally measured values, such as energies or transition matrix elements, as input.

1.1 Corpuscular and undulatory

When two propagating waves originating from the same source are incident at the same point in space, the resultant amplitude is the sum of the individual waves’ amplitudes. Each wave, characterized by crests and troughs or portions of positive and negative amplitude, will have a final amplitude ranging in magnitude from the sum to difference of the individual amplitudes. This superposition of waves which can be either constructive (sum) or destructive (difference) is known as the wave-like phenomenon of interference. In the early 19th century, interference fringes of light were demonstrated Thomas Young’s double slit interferometer in which a sunlit small hole illuminated two subsequent small holes. From the spatial separation of the observed fringes, Young was able to estimate the wavelength of different colors in the spectrum as mentioned below in a quotation from the chapter Experiments and Calculations Relative to Physics Optics of Ref. [22].

In making some experiments on the fringes of colors accompanying shadows, I have found so simple and so demonstrative a proof of the general law of the interference of two portions of light, which I have already endeavored to establish, that I think it right to lay before the Royal Society a short statement of the facts which appear to me so decisive. The proposition on which I mean to insist at present is simply this - that fringes of colors are produced by the interference of two portions of light; and I think it will not be denied by the most prejudiced that the assertion is proved by the experiments I am about to relate, which may be repeated with great ease whenever the sun shines, and without any other apparatus than is at hand to everyone.

Thomas Young, 1804 CHAPTER 1. OUTWARD BOUND 3

Michelson-Moreley

mirror

Intensity x

detector

mirror mirror

beam splitter

a beam splitter source d

source beam splitter detector

mirror

Mach-Zehnder Intensity x

Figure 1.1: (Left) An optical interferometer in the Michelson-Morley geometry, in which the distance d transversed by some movable mirror will be detectable as a phase shift in the overall signal detected at the output. (Right) An optical interferometer in the Mach-Zehnder geometry, in which the diﬀerential phase shift between two arms φ is made to vary.

Young’s experiment played a major role in the acceptance for the wave-like nature of light, which at the time was contrary to Newton’s corpuscular theory. Following Young’s double slit interferometer, optical interferometry became and still remains an indispensable tool. Such optical interferometers, like the Michelson and Mach-Zehnder interferometers depicted in Fig. 1.1, utilize a ‘beam splitter’ or half-silvered mirror to split the initial wave into two beams E1,2 = E1,2 cos φ1,2 − ωt . (1.1) These beams propagate along diﬀerent paths and are eventually directed back to one another with mirrors and recombined with the same (Michelson) or additional (Mach-Zehnder) beam splitter. The detected light has an intensity given by

2 2 I ∝ E1 + E2 + 2E1E2 cos(φ2 − φ1). (1.2)

A difference in path length between the wave functions produces a difference in phase in the interference fringes at the output. In the Michelson interferometer depicted in Fig. 1.1, CHAPTER 1. OUTWARD BOUND 4 the change in phase difference is 2π δ(φ − φ ) = 2d, (1.3) 2 1 λ for light of wavelength λ. This relation reduces to an equation for measurement displacement d λ d = N (1.4) 2 where N is the number of interference fringes. Being able to discern minute changes, on the order of a fraction of a wavelength, over the much larger distance of the beams path translates into extraordinary measurement sensitivity and precision. This demonstrates the power of interferometry as a tool for precision measurement; such qualities poise interferometry as an indispensable tool for metrology.

1.1.1 Waves of matter The resolution of the “ dispute over two viewpoints on the nature of light: corpuscular and undulatory” [23] with the development of quantum theory had implications that extended beyond ‘light atoms’ and into matter. In the early 20th century, quantum mechanics cat- alyzed experimental explorations into the wave-like nature of matter and an acceptance for the wave-particle duality. In the non-relativistic limit, the dynamics of matter waves are described by the time-dependent Schr¨odingerequation

2 ∂ψ(r, t) − ~ ∇2 + V (r, t) ψ(r, t) = i (1.5) 2M ~ ∂t or for the simpliﬁed case of a time-independent potential, V (r)

2M ∇2 + (E − V (r)) ψ(r) = 0. (1.6) ~2 Hence, for a particle with mass M and total energy E, its local wavenumber k (the magnitude of its wave vector) in a potential V (r) is 1 k(r) = p2M(E − V (r)). (1.7) ~ Quantum mechanically, such a particle with velocity v, can be characterized by a wave with mean de Broglie wavelength of λdB = 2π/k(r) = 2π~/Mv, where ~ is the reduced Planck constant. Control over the atom’s velocity or momentum translates directly to control of the mean de Broglie wavelength. The coherent atom optics designed to prolong the quantum coherence of atomic beams [24] and isolate the atom source from the environment laid a foundation of techniques instrumental to the realization of interference and eventually interferometers CHAPTER 1. OUTWARD BOUND 5

with atoms. Gratings of light akin to mechanical gratings imparted momentum upon the atomic wave packets [25], putting the atoms into superposition of momentum states. Ex- ploiting light’s complementary role as a refractive, reflective, absorptive structure to matter and the coupling of such interaction to the atom’s internal energy state was crucial to the advancement of atomic physics; from frequency standards [16], quantum information science to atom interferometry [26, 27]. Analogous to optical interferometry, in an atom interferometer atoms are coherently excited into a superposition of quantum states and allowed to propagate along alternative paths in either the space- or time-domain. The interference that results after recombining the wave function reveals a phase shift arising from a difference between paths as experienced by the wave packet of the evolving superposition. Since this relative phase translates into the detection probability for the atom in a particular interferometer output port, the phase difference is evident in a measurement of atom flux at the end of the interferometer. Despite matter’s tendency to interact strongly with other matter and its short coherence length, atom interferometry is a crucial tool in searches for beyond the Standard Model physics due to its versatility and sensitivity. Compared to light, matter is susceptible to a larger range of phenomena, including gravity, and offer advantages stemming from the wide selection of available atomic properties. Applications of atom interferometers include: ac- celerometry [28–31] , gravity gradiometry [32–35], rotation sensing [36, 37], fifth force searches connected to dark energy [38] and dark matter [39], and measurements of fundamental constants [17, 40–42] to measurements of atomic properties, like the static and dynamic polarizability [43, 44]. The continued advances in time and frequency metrology [16] and the consequently extraordinary accuracy with which laser frequencies can be measured means that manipulating atoms with optics built from light offers a route to even higher sensitivity and precision than that obtainable with crystal structures. Here, we demonstrate the first atom interferometer with laser-cooled lithium. Lithium is advantageous to metrology because of both its low mass and low electron number. Our atom interferometer utilizes coherent atom optics that entangle the internal and external atomic states, coupling the atom’s internal energy state to its momentum. We study recoil-sensitive interferometry, relevant in determinations of the fine structure constant with a measurement of h/M, and interferometry sensitive to the dynamic polarizability of lithium’s ground state 2S1/2.

1.2 α, the ﬁne structure constant

In 1916, Sommerfeld introduced the constant α to quantify the relativistic correction, known as ‘ﬁne structure’, to the spectrum of the Bohr atom []. Sommerfeld’s original interpretation 1 of this dimensionless number, approximately equal to 137 , was that it quantiﬁed the ratio of the velocity of the electron in a Bohr atom to the speed of light [45], CHAPTER 1. OUTWARD BOUND 6

1 q2 α = e (1.8) 4π0 ~c where qe is the charge of an electron, 0 is the permittivity of the vacuum, ~ = h/2π is the reduced Planck constant, and c is the speed of light. While the fine structure constant is prevalent throughout many various subfields of physics [46, 47], from the Josephson-junction oscillations in condensed matter [48] to the spectrum of muonium [49] to the Lamb shift in atomic physics [50], it can be fundamentally defined as the coupling constant for the electromagnetic force or the affinity for which charged particles couple to electromagnetic fields at low energies. Additionally, in particle physics the fine structure constant can be related to the magnetic moment of the electron µ = µSˆ. This value has been measured as µ/µB = -1.001 159 652 180 73 (28) [51]. The numerical value of the electron’s magnetic moment can be calculated from the Standard Model. In this calculation, µ/µB, where µB = qe~/2m, acquires contributions from several aspects of the Standard Model, expressed as follows: −µ = 1 + aQED + aQCD + aweak. (1.9) µB

The quantum chromodynamic correction aQCD contributes at two parts per trillion and quantiﬁes the electron’s interaction with hadron-antihadron pairs. The electroweak in- 13 teraction correction aweak is smaller than the measurement precision of 2.8 parts in 10 . Both these contributions are determined from measured values. The largest factor in the above expansion of the the electron’s anomalous magnetic moment is the QED contribution ge−2 aQED ≡ 2 , at 0.1%. This term comes from contributions to the moment arising from loops made of virtual photons and leptons. The QED contribution aQED can be expanded perturbatively in α n X α α α 2 α 3 α 4 α 5 aQED = C = C + C + C + C + C + ... e 2n π 2 π 4 π 6 π 8 π 10 π n (1.10)

The renormalizability of QED ensures that the factors C2n in the above expansion converge to a ﬁnite value. Redeﬁning these factors as follows [2]

2n 2n 2n C2n = A1 + A2 (me/mµ) + A2 (me/mτ ) 2n +A3 (me/mµ, me/mτ ), (1.11) where me/mµ and me/mτ are the electron-muon and electron-tau mass ratios, allows the expansion to be re-expressed as a sum of like-contributions for a given order

QED X kαn X 2n αn X 2n αn a = A + A (m /m 0 ) + A (m /m 0 , m /m 00 ) . e 1 π 2 e ` π 3 e ` e ` π n≥1 n≥2 n≥3 CHAPTER 1. OUTWARD BOUND 7

(1.12) At increasing order, computing these corrections requires increasingly complicated applications of quantum electrodynamics. For example, even at third order in α, O(α3), there are 72 diagrams to consider. Most recently, the tenth-order QED contribution to anomalous moment was determined [2] and with the previously mentioned measurement yielded a value at the 0.25 ppb level for the fine structure constant of α−1 = 137.035999173(35). (1.13) Other determinations of α, particularly those independent from the QED and Standard Model framework, are needed to test its theoretical formalism. Because α is defined in terms of parameters that cannot yet be independently calculated, a determination of the fine structure constant must be pursued through either a direct measurement or determined indirectly through the measurement of other quantities. We can define the fine structure constant not in terms of an expansion in QED but instead relative to other fundamental constants [52] 2hR α2 = ∞ , (1.14) mec

where me is the mass of the electron and R∞ is the Rydberg constant [53, 54]. Historically, the Rydberg constant arises in the context of the Bohr atom and is defined as 4 1 1 1 meqe 1 1 = R∞ 2 − 2 = 2 3 2 − 2 , (1.15) λ n1 n2 8ε0h c n1 n2 and has already been determined spectroscopically to great accuracy, with a relative uncer- −12 tainty of 7 × 10 . Therefore, a determination of α is possible via a measurement of h/me. Recognizing that free electrons are difficult to trap and coherently manipulate, we expand h/me again, redefining α now as 2R h 2R u M h α2 = ∞ = ∞ c me c me u M (1.16)

where u is the atomic mass unit. The relative mass of the electron me/u [55, 56] and the relative mass of an atom M/u [57] can both be determined extraordinarily precisely. Re- ported values are known to better than 2.0 × 10−10. Therefore, for the above deﬁnition of h α, what remains left to be determined is the quantity M . As discussed in the next section, atom interferometry is able to determine this ratio independently of the SM, from a measurement of the atom’s recoil splitting and knowledge of the photon frequency [17, 40, 41, 58, 59]. Comparing the results obtained for α with atom interferometry to that determined through QED computations, in conjunction with the measurement of the electron’s anomalous magnetic moment, is a consistency check of the Standard Model in particle physics but from a measurement made at the resolution of an atom! CHAPTER 1. OUTWARD BOUND 8

h 1.2.1 M measurement h The ratio of Planck’s constant to the mass of an atom, M , is intimately connected to the kinetic energy imparted to an atom by a photon. For an atom with initial ground internal energy state |ai and velocity v0 in the presence of an external resonant ﬁeld with frequency, ωL = ωe − ωa where ωe − ωa is the energy splitting the ground and excited state |ei, the initial total energy and momentum for the system is Mv2 E = 0 + ω + ω 0 2 ~ a ~ L p0 = Mv0 + ~k. (1.17)

Absorbing a photon from the field (λ) consequently imparts momentum pγ = h/λ to the atom, in the direction of the photon, and will recoil with a velocity of ∆v = pγ/M = h/λM = ~k/M. The atomic resonances will also be Doppler-shifted as ∆ω/2π = ∆v/λ = 1/λ2(h/m). The final energy and momentum of this atom-light system are M E = |v + δv|2 + ω f 2 0 ~ e pf = M(v0 + δv). (1.18) The resonant laser frequency for the field is imposed via energy conservation as k2 ω = v · k + ~ + (ω − ω ) (1.19) L 0 2M e a which depends upon the energy level difference between the atomic states, the first-order Doppler shift, v0 ·k, and a similar-in-spirit recoil shift arising from the change in momentum accompanying photon absorption. An accurate knowledge of the laser wavelength in combination with a measurement of the frequency shift leads to a measurement of h/M and therefore a determination of α. As will be discussed in Chapter 2 the sensitivity of an interferometer depends upon several things but a particular interferometer geometry that is sensitive to the recoil frequency of the atoms in the Ramsey-Bordéinterferometer which consists of four beam splitter with the second pair having a reverse effective wave-vector compared to the first two pulses. An atom interferometer becomes sensitive to the recoil frequency of the atom when during interferometry, a difference in energy exists for the coherent superposition resulting from the kinetic energy obtained after absorption (and emission) of a photon. The coherent superposition spends an unequal amount of time recoiling or with a kinetic energy imparted to it by the absorption and emission of a photon. This difference in energy will show up in the quantum phase difference read out at the end of interferometry.

1.3 α, the polarizability

The field of cold-atomic physics exists due to the ability to manipulate atoms (trapping, laser cooling) with electromagnetic fields. The first order response of an atom to an applied CHAPTER 1. OUTWARD BOUND 9 electric field is its polarizability. Classically, when an external electric field is applied to matter, the charged particles in the object are rearranged. The polarizability characterizes the response of the atomic charge cloud to this perturbing field [60]. For a perfectly conducting sphere of radius r0 in a uniform electric field E, the resultant electric field at a position in space given by r > r0 is

3 3 E − ∇(E· rr0/r ). (1.20)

This expression is equivalent to replacing the perfectly conducting sphere by an electric dipole with a dipole moment given by

d = αE (1.21)

3 where α = r0 is the polarizability of the sphere. Quantum mechanically, a system of particles with positions ri and electric charges qi exposed to an applied, uniform electric ﬁeld (E = Eεˆ) can be described by following Hamil- tonian 0 H = H0 + H = H0 − Eεˆ· d (1.22) where the electric dipole d is a sum over the individual particle dipoles X P = qiri. (1.23) i For an atom, this summation is over all charged particles of the atom, including the nucleus. Treating the ﬁeld strength E as a perturbation parameter and expanding the energy and wave function leads to the following:

2 E = E0 + EE1 + E E2 + ... (1.24) 2 |Ψi = |Ψ0i + E|Ψ1i + E |Ψ2i + ... (1.25) such that H0Ψ0 = E0Ψ0. The atomic polarizability α is identiﬁed from the energy-level shifts for the state |Ψi, which up to the second order energy correction via perturbation theory is

ˆ 2 X |hΨ|E· d|Ψki| 1 δE = hΨ|E·ˆ d|Ψi + = − E 2α (1.26) E0 − Ek 2 k where |Ψki label all other atomic states. Assuming |Ψ0i is an eigenfunction of parity, then the ﬁrst-order shift vanishes. CHAPTER 1. OUTWARD BOUND 10

1.3.1 Dynamic polarizability The frequency-dependent or dynamic polarizability quantifies the response of an atom to the presence of an external off-resonant optical field. As derived previously, the atom-field interaction energy or termed ac Stark shift, is given to first order as 1 U = − αhE 2i (1.27) 2 with electric field intensity, |E(ω)|2 and the kets indicate a time average. This energy shift of an atomic energy level can be written explicitly in terms of the electric dipole transition matrix elements with initial state |ii and excited states |ki in the presence of a monochromatic field E as [61]:

2 2 X 2ωik|hi|ˆ· d|ki| |E| δE = − (1.28) i (ω2 − ω2) k ~ ik

where ωik := (Ek − Ei)/~. The dynamic polarizability of the atom is deﬁned as follows

2 X 2ωik|hi|ˆ· d|ki| α(ω) = (1.29) (ω2 − ω2) k ~ ik

and can be broken down into contributions from core electrons, αc, a modiﬁcation resulting from core-valence interactions, αvc, and a contribution from the valence electron, αv

α(ω) = αv + αc + αvc. (1.30) The contribution from the valence electron dominates the above sum, especially in the case presented here: a measurement of the tune-out wavelength between the D1 and D2 lines in lithium. The expression can be reduced into a sum only over the excited electronic states |ki coupled to the initial (ground) state of the atom |ii by the off-resonant external optical field. The reduced dipole matrix elements in the definition of the polarizability can be obtained from oscillator strengths fgk, transition probability coefficients Akg and line strengths Sgk which lead to the following alternative definitions for the polarizability.

2 q X fgk α(ω) = e (1.31) M ω2 − ω2 k6=g gk −2 X Akgωgk g α(ω) = 2π c3 × k (1.32) 0 ω2 − ω2 g k6=g gk g

1 X Sgkωgk α(ω) = (1.33) 3 ω2 − ω2 ~ k6=g gk CHAPTER 1. OUTWARD BOUND 11

Considering an initial state given in terms of the hyperfine-basis, |ii = |nLF mF i, the 0 0 0 sum then is over final states |ki = |nL F mF i and the polarizability is defined as 0 0 0 0 X 2ωF 0F hF mF |dν|F mF ihF mF |dν|F mF i αµν(ω) = 2 2 . (1.34) (ω 0 − ω ) 0 0 ~ F F F mF Decomposing the polarizability into irreducible tensors leads to the following expression for the ac Stark shift 1 (∗ × ) · F δE(F, m ; ω) = − |E|2 α(0) − iα(1) F 4 2F 3(∗ · F)( · F) + ( · F)(∗ · F) − 2F2 +α(2) (1.35) 2F (2F − 1) in terms of the scalar, vector and tensor polarizabilities defined as

0 2 (0) 2 X ωF 0F |hF kdkF i| α (F ; ω) = 2 3 (ωF 0F − ω ) F 0 ~ r 0 2 (1) X F +F 0+1 6F (2F + 1) 1 1 1 ωF 0F |hF kdkF i| α (n, J, F ) = (−1) 0 2 F + 1 FFF (ωF 0F − ω ) F 0 ~ s 0 2 (2) X F +F 0 40F (2F + 1)(2F − 1) 1 1 2 ωF 0F |hF kdkF i| α (n, J, F ) = (−1) 0 FFF 0 2 0 3(F + 1)(2F + 3) ~(ωF F − ω ) . F (1.36) Precise calculations of atomic polarizabilities have implications in many areas of physics, from fundamental searches to quantum information processes [62, 63], and also in optical cooling and trapping [64]. For instance, parity nonconservation experiments (PNC) in heavy atoms search for new physics beyond the electroweak sector of the standard model through the precise evaluation of the weak charge or parity violation in the nucleus with nuclear anapole moment evaluations [65, 66]. These experiments require detailed studies of parity- conserving quantities, like atomic polarizabilities, to accurately determine the uncertainty in the theoretical value [67, 68]. As experimental capabilities continue to grow, the requirements for greater precision and accuracy have necessitated a greater understanding of the corrections for the effects of the electromagnetic fields used to manipulate the atoms. This is evident in the ‘next-generation’ of atomic clocks which have recently renewed an interest in polarizability [69]. These standards are significantly impacted by a displacement of the atom’s energy levels resulting from blackbody radiation shifts (BBR) [70–72]. BBR shifts are the universal ambient thermal fluctuations of the electromagnetic field, given by 2 ∆E = − (απ)3α (0)T 4(1 + η) (1.37) 15 0 CHAPTER 1. OUTWARD BOUND 12

λto [n]m Method Reference K 768.9712(15) AIFM [43] Rb 789.85(1) BEC diﬀraction [76] Rb 790.032388(32) BEC AIFM [77] Rb 790.01858(23) BEC scattering in OL [78] Rb 423.018(7), 421.075(2) BEC diﬀraction [79] He∗ 413.0938(9)(20) Trapping potential [80]

Table 1.1: Tune-out measurements to-date, method of measurement and reference.

where α is the fine structure constant, α0(0) is the static scalar polarizability, T is the temperature, and η is a correction factor containing the frequency dependence of the polarizability [73] 40π2T 2 η ≈ − S(−4) (1.38) 21αd(0) with the following sum rule: (1) X fgk S(−4) = . (1.39) (∆E )4 n gk The differential Stark shift caused by external electromagnetic fields leads to a temperature- dependent shift in the transition frequency of the two states involved in the clock transition.

1.4 Previous measurements

The tune-out wavelengths for several of the alkali-atoms have been measured, see Fig. 1.1, but measurements of polarizabilities are less abundant than theoretical determinations. For lithium, there are currently only indirect Stark shift measurements between the ground and excited states [74, 75] and a static polarizability measurement made with thermal atom interferometry [44], as discussed in Chapter 3. A direct measurement of lithium’s tune- out wavelength between the 2P1/2 and 2P3/2 excited levels with atom interferometry is the pursuit of ongoing work here. The status and project outlook is the focus of Chapter 6. Presently, many of the best estimates of atomic polarizabilities are derived from a composite analysis which blends first principle calculations of atomic properties with experimental measurements. This sum-over states method is widely applied to systems with one or two valence electrons and can be combined with oscillator strengths or matrix elements derived from experimentally measured values [81]. While ideally the total uncertainty of the theoretical value should provide an estimate as to how far away an observed value is from the actual exact result, without knowledge of the exact value the evaluation of the complete theoretical uncertainty is non-trivial. It ultimately requires the knowledge of a quantity that is not known beforehand nor can be determined by the adopted methodology. The most common CHAPTER 1. OUTWARD BOUND 13 numerical uncertainties are associated with the choice of basis sets, configuration space, ra- dial grid or termination of iterative procedures after achieving some convergence tolerance. For example, in a Hylleraas calculation, expectation values are expected to converge as 1/Ωp where Ω = j1 + j2 + j3 + j12 + j13 + j23 (1.40) is the summed polynomial power for the correlated wave function. Varying appropriate parameters and tabulating the results may allow for an estimate on uncertainties in various atomic properties [82, 83] but numerical constraints resulting from measured values can ensure that these intense computations be performed within a reasonable amount of time. A second class of uncertainties are those associated with the particular theoretical computation methodology, such as the uncertainty associated with halting a perturbation theory treatment. Developing hybrid theoretical approaches may be the key towards better computations of atomic structure properties. While directly incorporating the Dirac Hamilto- nian into orbital-based calculations is standard, this is not the norm for calculations with correlated basis sets. Correlated basis set computations are unmatched in their realized accuracies. Comparing such calculations with both relativistic and non-relativistic orbital- based ones could be used to estimate relativistic corrections to the Hylleraas calculations and greatly increase the obtained precision [84, 85]. Lithium’s simple atomic structure allows for a precise computation of properties with only ab initio wave functions, those derived from first principles in quantum mechanics, and spectroscopic data. Lithium’s polarizability could be pivotal in metrology [81]; a measured value would constrain the calculated dynamic polarizabilities and thereby refine the method of computation. Furthermore, a multispecies interferometer with lithium would be capable of normalizing another atom’s polarizability to that of lithium’s [24]. This could lead to a new accuracy benchmark for many elements in conjunction with a definitive calculation of α0. Hylleraas polarizability calculations could serve as standard for coupled-cluster type calculations applied to larger atoms, like cesium.

1.4.1 λto measurement

The tune-out wavelength, λto, is deﬁned as the wavelength at which the dynamic polarizability vanishes, α(ωto) = 0. (1.41) Contrary to conventional spectroscopic methods which measure the energy of a particular atomic electronic states indicated by poles in the frequency response of the atom to the external optical ﬁeld [66], the tune-out is a zero in the atom’s frequency response. The dynamic polarizability is made up largely by contributions from the valence electron. This work aims to measure the red tune-out wavelength in lithium, located between the 2P1/2 and 2P3/2 levels. Between a nearby pair of such dipole allowed transitions, a zero in the dynamic polarizability will occur where the opposite signs of detuning of the light with CHAPTER 1. OUTWARD BOUND 14

Dynamic polarizability of 2S2 state in lithium 2000

1000 5 a.u.] 4

670.9714 670.9718 670.9724 670.9728

-5 polarizability [10 polarizability -1000

-10

-2000 670.955 670.960 670.965 670.970 670.975 670.980 wavelength [nm]

7 Figure 1.2: A plot of the dynamic polarizability α(ω) for Li’s ground state 2S2. respect to the atomic transition will perfectly cancel out. Due to this cancellation, no net energy shift will be experienced by the ground level.

S ω ω α(ω ) = 1 D1 + R D2 + α (ω ) = 0 (1.42) to 3 ω2 − ω2 ω2 − ω2 rem to ~ D1 to D2 to where αrem accounts for remaining contributions to the dynamic polarizability (core, core- valence) and R = S2/S1 is a ratio of the line strengths for the D1- and D2-lines. The location of the zero depends primarily on the ratio of the matrix elements of the two states. An interferometer composed of π/2-pulses, all with a single direction of momentum transfer, is no longer sensitive to the recoil frequency of the atoms. This interferometer geometry can be made sensitive to an external optical potential [86] and thus used instead to determine the atom’s tune-out wavelength. By pulsing on light during the atom’s free evolution, the atom’s response as the frequency is swept over the anticipated tune-out point can be tracked. As the dynamic polarizability goes to zero, the eﬀect on the atoms from the additional pulse of light will diminish as well. As the frequency is moved away from tune-out, the atoms will again be perturbed by the interaction with the light. A direct interferometric measurement of lithium’s red tune-out wavelength at 670.971626(1) nm, is a precise comparison to existing ‘all-order’ atomic theory computations. The location of the zero in the atom’s frequency response depends primarily on the ratio of the matrix elements of the two states, providing a route toward a precise determination of the S− and P − transitions matrix elements, providing independent veriﬁcation of QED predictions for such transition rate ratios [87]. A precise determination of matrix elements is also necessary in experimental endeavors including: measurements of parity violation [88], the characterization of Feshbach resonances [89, 90], and estimation of blackbody shifts [69, 91]. CHAPTER 1. OUTWARD BOUND 15

Perturbations to the core or higher level contributions is the polarizability could be observable at experimental precisions of 0.1 ppb [92]. Furthermore, a future measurement of lithium’s ultraviolet tune-out wavelength, predicted to occur at 324.192(2) nm, would have increased sensitivity to relativistic approximations made in the atomic structure description.

1.5 Overview of this thesis

Here, the versatility and utility of atom interferometry, both in verifying existing atomic theories with measurements of atomic properties, like the tune-out wavelength, and in searches for exotic physics, like with a recoil-sensitive interferometer is demonstrated. Chapter 2 discusses the theory behind the coherent atom-light interactions comprising the beam splitters and mirrors used in interferometry as well as the phase calculation for the recoil-sensitive Ramsey-Bordéand tune-out sensitive schemes. Chapter outlines the advantages of using lithium, the smallest alkali in both mass and electron number, in atom interferometry. It also reviews previous interferometry measurements with lithium relying on thermal atomic beams and diffraction gratings of standing light. A summary of the experimental details required to prepare the atom source for interferometry is included in Chapter 4. Previous work on this experiment, as detailed in Ref. [93] resulted in a new cooling technique for atom interferometry with lithium [94]. However, since joining the Müllergroup in the Fall of 2013, the focus shifted to interferometry and without such additional cooling. I, Professor Yanying Feng and Chen Lai first explored inertially- sensitive interferometry with a Mach-Zehnder interaction geometry. From mid-2015 through 2016, Eric Copenhaver and I have been focused instead on recoil-sensitive interferometry with lithium, discussed in Chapter 5. Following that demonstration and a diversion into modeling the system’s intrinsic and mysterious imperfection, we shifted toward a similar interferometer scheme as detailed in Chapter 6. This second project exploits the simplicity of lithium’s electronic structure, the last nicety of lithium. This work is ongoing presently and Chapter 6 discusses the set-up thus far and also exhibits preliminary data. The outlook and future of atom interferometry with laser-cooled lithium is explored in the final chapter. 16

Chapter 2

Atom interferometry

This chapter describes in detail the theoretical underpinnings behind the manifestation of interference between the components of the atom’s wave function as it transverses an interferometer. Phase is accrued both in the absence and presence of the external light field and if a spatial separation exists between wave packets at recombination. Details of the atom-light interaction for the case of two-photon Raman transitions is outlined here. In the last sections, the total phase difference between upper and lower trajectories for the four π/2-pulse configurations is derived algebraically. A discussion of interferometry begins at the experimental end, with interference. As seen in the optical interferometers discussed in Chapter 1, an atom interferometer proceeds by ‘beam splitting’ an initial matter-wave |Ψi either via slit or grating, optically or mechanically, such that the total wave function is now defined as a linear combination of two or more different states X |Ψi = |ψni. (2.1) n The split matter-wave propagates freely, eventually interacting with a second beam splitter fracturing each of the wave packets of the superposition. For the optical interferometers discussed previously, a second beam splitter recombined the light which had propagated along different paths, yielding two output ports. Detection after recombination projects the final matter-wave probabilistically determined by the phase difference accrued along the paths of propagation. Each component is traveling along a different path in spacetime and has accumulated quantum phase from period of evolution and as with optically interferometry, a difference in the final phases will be heralded by interference at the output. Consider the simplified circumstance that the interaction turned on to create this superposition couples the initial state to only one other state |ψ2i. The atom after a first ‘beam splitter’ is given by 1 i √ |ψ1i + √ |ψ2i. (2.2) 2 2 After the matter-wave has evolved for a time T and each component has accrued a phase φi, a second beam splitter is used to recombine the wave function. At the interferometer’s CHAPTER 2. ATOM INTERFEROMETRY 17 output the total wave function is given by 1 i |ψ i = eiφ1 − eiφ2 |ψ i + eiφ1 + eiφ2 |ψ i. (2.3) out 2 1 2 2

A measurement of the |ψ1i population results in interference as seen in the following detected intensity I 1 φ1 − φ2 I = |hψ |ψ i|2 = |hψ |eiφ1 − eiφ2 |ψ i|2 = sin2 . 1 out 4 1 1 2 (2.4)

The above consideration proceeded assuming perfect splitting, but under a more realistic approximation, the amplitudes of the states after recombination are scaled by c1 and c2 with the interferometer’s contrast C deﬁned as

Imax − Imin 2c1c2 C = = 2 2 . (2.5) Imax + Imin c1 + c2 The phase difference accumulated over the course of the atom interferometer is a sum of contributions arising from a free evolution phase, an atom-light interaction or ‘laser’ phase, and a separation phase that occurs if the wave packets of the superposition are separated spatially at the last pulse: φtot = φfree + φγ + φsep. (2.6) In this chapter, the origin of these phases will be discussed. An understanding of these phases arises out of an understanding of the dynamics of the quantum system, particularly one in which the total Hamiltonian has a contribution from an interaction Hamiltonian which characterizes a perturbation to the atom from an external electromagnetic field. For an atom with internal energy states |ai and |bi such that Ea < Eb, the dynamics are described by the following Hamiltonian ˆp2 Hˆ = Hˆ (0) + Hˆ (1) = + Hˆ − d ·E + Hˆ (1) (2.7) tot 2M int where ˆp is the momentum operator in the center-of-mass frame. The internal degrees of free of the atom are described by the operator Hînt defined explicitly as ˆ ˆ Hint|ai = Ea|ai = ~ωa|ai and Hint|bi = Eb|bi = ~ωb|bi. (2.8) The coupling to the electromagnetic field is given by d·E, the projection of the dipole operator d along the direction of the external electric field. The additional term Hˆ (1) represents any additional perturbative interactions such as those resulting from an external disturbance or potential present during interferometry. The sequence of pulses for a particular interferometer geometry results in a unique total phase difference between wave packets of the superposition. In this chapter, the phase for CHAPTER 2. ATOM INTERFEROMETRY 18

mirror

detector a

source

beam splitter beam splitter

mirror Intensity

Mach-Zehnder atom interferometer

Figure 2.1: The coherently split wave function transverses an upper and lower trajectory simultaneously. Upon recombination at a final pulse τπ/2, with effective wave vector k, interference occurs with a phase given by the phase difference between paths traveled by the superpositon wave packets, φu. CHAPTER 2. ATOM INTERFEROMETRY 19

z z |a > |b > |b > |b > |b > |a >

|a > |a > |a >

|a > |b > |b > |b > |b > |b > |b > |b > τ τ τ τ τ τ τ τ |a π/ 2 π/ 2 π/ 2 π/ 2 π/ 2 π/ 2 π/ 2 π/ 2 >

Figure 2.2: Two interferometer geometries can be realized when utilizing four sequential π/2-pulses. When the effective k wave vector of the light is maintained throughout the pulses the parallelogram geometry is realized (left). or (reversed) as shown here. Reversing the momentum direction creates the trapezoidal geometry (right) and has the added feature of building in a sensitivity to the recoil frequency ωr of the atoms. an interferometer comprised of four beam splitter pulses will be derived, in the cases with and without a k-reversal mid-interferometer. It will be shown that reversing the direction of momentum transfer between the pulses pairs builds in a sensitivity to the recoil frequency, called a Ramsey-Bordéinterferometer. Without such reversal, the phase resulting from the interferometer is independent of the recoil frequency but by turning on an additional external optical field between the pulse pairs, the interferometer’s phase will have a dependence on the dynamic polarizability of the atom.

2.1 Light oﬀ

Quantum mechanics allows for the state of an atom at time t to be described by its state at ˆ an earlier time t0 < t with the evolution operator U(t, t0), ˆ |ψ(t)i = U(t, t0)|ψ(t0)i. (2.9)

Projecting this state vector onto the position basis leads to the following conﬁguration- space representation or wave function of the atom at (t, z(t)) ˆ ψ(z(t), t) = hz|ψ(t)i = hz|U(t, t0)|ψ(t0)i Z ˆ = dz0hz|U(t, t0)|z0ihz0|ψ(t0)i, (2.10)

in terms of the wave function at an earlier time t0, considering all possible starting points. The probability amplitude that the atom, starting at (t0, z0), will be found at z after a time t − t0 is known as the ‘conventional’ quantum propagator K. The quantum propagator is deﬁned as ˆ K(zt, z0t0) ≡ hz|U(t, t0)|z0i. (2.11) CHAPTER 2. ATOM INTERFEROMETRY 20

Therefore, the total amplitude for a particular evolution between two points in space-time can also be calculated by considering the amplitude for a trajectory first to an intermediate position z(t)(t ∈ [t0, tf ]), followed by the evolution from this intermediate time z(t) to the final time tf . This is known as the composition property of the quantum propagator. An evolving atom has infinitely many possible paths between its initial (t0, z0) and a final position (tf , zf ). Feynman defined the quantum propagator as a sum of contributions from all possible paths connecting the initial and final points expressed as

X iSΓ/~ K(zf tf , z0t0) = N e . (2.13) Γ Expressing the quantum propagator as a functional integral over all possible paths Γ,

Z zf iSΓ/~ K(zf tf , z0t0) = Dz(t) e , (2.14) z0

connects the initial (t0, z0) and ﬁnal (tf , zf ) positions of the state. The path integral method links the traditional formulations of quantum mechanics to the more intuitive principles of wave mechanics. The phase factor, here equal to the action SΓ scaled by ~, can be rewritten in terms of the integral of the system’s Lagrangian L(z, z˙). The Lagrangian is a function of position z and velocityz ˙ over a path Γ = z(t) from initial point (z0, t0) to ﬁnal point (zf tf ) given by

Z tf SΓ = dt L[z(t), z˙(t)]. (2.15) t0 In the limit that the phase evolution of the atomic wave function is

SΓ/~ 1, (2.16) as is often the case in atom interferometers, the rapidly oscillating phases of neighboring paths will destructively interfere. This will not occur if the paths are close to the extrema of the action. CHAPTER 2. ATOM INTERFEROMETRY 21

For paths close to which the action is extremal, rather than conspiring to cancel, the slowly varying terms will be the dominant contribution to the integral. In this limit called the ‘classical’ limit, only paths close to the classical one Γcl contribute to the phase. The classical path is the one for which the action is extremal, given by

Z tf ∂L ∂L δS = δz(t) + δz˙(t) = 0 t0 ∂z ∂z˙ tf Z tf ∂L ∂L d ∂L = δz(t) + − δz(t)dt = 0. (2.17) ∂z˙ ∂z dt ∂z˙ t0 t0 Above, the ﬁrst term vanishes due to the boundary conditions

δz(t0) = δz(tf ) = 0. (2.18)

Imposing that the second term must also equal zero as well yields the classical equations of motion, the Euler-Lagrange equations: ∂L d ∂L − = 0. (2.19) ∂z dt ∂z˙ In the absence of an electromagnetic ﬁeld, the Lagrangian for an atom with mass M and internal energy ~ωi in a uniform potential V (z) is a quadratic function of position z and velocityz ˙ Mz˙2 L = − V (z) − ω (2.20) 2M ~ i with the following (classical) solutions

z˙cl(t) = v0 − g(t − t0) (2.21) g z (t) = z + v (t − t ) − (t − t )2. (2.22) cl 0 0 0 2 0 Therefore, in an atom interferometer the above path integral for the accrued quantum phase during free evolution will be dominated by the atom’s classical trajectories. An arbitrary path z(t) can be parameterized in terms of its deviation, ξ(t), from the classical one z(t) = zcl(t) + ξ(t). (2.23) The action is Z tf M ˙2 S[zcl(t) + ξ(t)] = Scl(zf tf , z0t0) + dt ξ . (2.24) t0 2 and the quantum propagator expressed in terms of the deviation ξ, is

Z ξ(tf ) i K(zf tf , z0t0) = Dξ(t) exp S[zcl(t) + ξ(t)] ξ(t0) ~ CHAPTER 2. ATOM INTERFEROMETRY 22

(z f,t f)

(t) )+ξ z cl(t zcl (t)

(z 0,t 0)

Figure 2.3: The quantum propagator is the sum of contributions from all possible paths connecting the initial and ﬁnal points. The atom’s path in conﬁguration-space is dominated by the classical action, shown in red. An arbitrary path, shown in blue, can be parameterized in terms of its deviation ξ(t), from the classical path.

Z 0 i = Dξ(t) exp S[zcl(t) + ξ(t)] 0 ~ i = exp Scl(zf tf , z0t0) ~ Z 0 i Z tf M × Dξ(t) exp dt ξ˙2 . 0 ~ t0 2 (2.25)

2.1.1 The free evolution phase

The functional integral in Eq. 2.25 is independent of both z0 and zf and is denoted in the literature as F (tf , t0) [95]. The propagator simpliﬁes to

i K(zf tf , z0t0) = F (tf , t0) exp Scl(zf tf , z0t0) . (2.26) ~ CHAPTER 2. ATOM INTERFEROMETRY 23

The wave function after some time tf − t0 of free evolution is Z i ψ(tf , zf ) = F (tf , t0) dz0 exp Scl(zf tf , z0t0) ψ(t0, z0) (2.27) ~

where the classical action Scl(zf tf , z0t0) for the Lagrangian in Eq. 2.1 is explicitly determined to be Z tf Mv(t)2 Scl(zf tf , z0t0) = dt − Mgz(t) − ~ωi t0 2 2 M (zf − z0) Mg = − (zf + z0)(tf − t0) 2 tf − t0 2 Mg2 − (t − t )3 − ω (t − t ) (2.28) 24 f 0 ~ i f 0 and reduces to a function only of the path’s endpoints Scl ≡ Scl(zf tf , z0t0). Therefore, a determination of the initial and ﬁnal points for the particular step of free evolution is all that is needed in order to calculate the phase acquired by the atom as it evolves freely!

2.2 Light on

The atom also acquires a phase shift from the laser pulses used to coherently split, manipulate and recombine the matter-waves. Two diﬀraction mechanisms that are commonly used in light-pulse atom interferometry are Bragg and Raman scattering. In Bragg scattering, a pair of counter-propagating laser beams induce a two-photon transition between momentum states, leaving the atom’s internal state unchanged. In Raman scattering, a pair of laser beams interact with the atom but the two-photon transition is accompanied by a transition to another internal state. Here, two-photon Raman transitions are discussed in more detail since the work in this thesis utilizes Raman scattering as the mechanism behind interferometry with laser-cooled lithium.

2.2.1 Raman scattering A stimulated Raman transition couples the atom’s momentum to its internal energy state. It can be formally defined as a two-photon transition from one ground state to another and between different motional states, mediated by an excited state. Appendix B reviews the example of the two-level system. Consider a three-level atomic system with ground states |ai and |bi, separated by an energy difference of ~ωba = ~(ωb − ωa), and excited state |ei, at energy ~ωe. The general state vector of the atom at a time t is

|ψi = ca(t)|ai + cb(t)|bi + ce(t)|ei. (2.29) CHAPTER 2. ATOM INTERFEROMETRY 24

E + p2/2m

nb = -1 nb = 1 p

na = 2 na = -2

na = 0

Figure 2.4: For an atom starting in state |a, p = 0i absorption of ω1 (blue) followed by emission of ω2 (red) will move the atom up or down in momentum given the direction of keﬀ = k1 − k2, the eﬀective wave vector of the light.

The atom is irradiated by two monochromatic ﬁelds given by +i(k1·r−ω1t) ∗ −i(k1·r−ω1t) +i(k2·r−ω2t) ∗ −i(k2·r−ω2t) E(r, t) = ˆ1 E1e + E1 e + ˆ2 E2e + E2 e (2.30)

where E1,2 = ˆ1,2E1,2 with ˆ1,2 representing the unit field polarization vectors and k1,2 and ω1,2 are the wave vector and frequency for the external fields. The dynamics of the system is described by the time-dependent Schrödingerequation, ∂ i |ψ(t)i = Hˆ |ψ(t)i, (2.31) ~∂t

ωe e , p + ħk | 1> ∆ b2

∆ = ∆ a1 = ∆ b2

Ωb1 ∆ a2

Ωb2 Ωa1

ω2 Ωa2

ω1

δ ωb b , p + ħ (k - k ) | 1 1 > ωba

ωa | a , p >

The strength of the coupling of level |ni to the excited level |ei through the ﬁeld Ei is described by the Rabi frequency given by

hn|d ·Ei|ei µˆne ·Ei Ωni := − = − . (2.35) ~ ~

The Hamiltonian in the rotating frame is

ˆ ∗ ∗ H = −~ ∆1|eihe| + δ|bihb| + Ω1|eiha| + Ω2|eihb| + Ω1|aihe| + Ω2|bihe| (2.37) with ∆1,2 equal to the single photon detunings from excited state |ei, the two-photon detuning is given by δ = ∆1 − ∆2 and the energy of |ai and the excited state is set to zero. The two beams are detuned by many line widths from single photon resonance and therefore the detuned resonant excitation to |bi will dominate over incoherent spontaneous emission from the excited state |ei. The Rabi frequency determines the time period at which the atoms will ‘flop’ between the two ground states due to the interaction with the driving field. By the dipole interaction, a laser with frequency ωn will couple a state n (|ni) to the intermediate or excited state (|ei). If the beam is detuned by a largely sufficient amount ∆n Γ from single photon resonance then the occupation of |ei will be negligible. A pair of frequencies with a common detuning ∆1 = ∆2 ≡ ∆ induces a coherent transfer between the ground states |ai and |bi. This type of transition defines a two-photon Raman transition. Because a photon posses momentum dictated by its wave vector, ~k, it is possible to define a momentum basis for this particular atomic transition. CHAPTER 2. ATOM INTERFEROMETRY 27

The rotating frame is deﬁned by the following state vector

|ψi =c ˜a|ai +c ˜b|bi + ce|ei (2.38)

for which the ground states |ni have been eﬀectively boosted by energy ~ωn. The dynamics of the system is now described by the rotating-frame free atomic Hamiltonian,

|ψi = ca|a, pi + cb|b, p + ~(k1 − k2)i + ce|e, p + ~k1i. (2.41) Projecting the state vectors onto the conﬁguration basis and boosting all energies by −~∆ produces the following equations of motion:

2 p Ωa1 Ωb2 i ∂ ψ = ψ + ~ eik1zψ + ~ eik2zψ − ∆ψ (2.42) ~ t e 2m e 2 a 2 b ~ e 2 p Ωa1 i ∂ ψ = ψ + ~ e−ik1zψ + (∆ − ∆)ψ (2.43) ~ t a 2m a 2 e ~ 1 a 2 p Ωb2 i ∂ ψ = ψ + ~ e−ik2zψ + (∆ − ∆)ψ . (2.44) ~ t b 2m b 2 e ~ 2 b This two-photon process can be viewed as an effective one-photon process with effective k-vector given by keff = k1 − k2 = 2k (2.45) and an effective frequency that is twice what was realized with the one photon process. Adiabatically eliminating the excited state gives

Ωa1 Ωb2 ψ = eik1zψ + eik2zψ (2.46) e 2∆ a 2∆ b for the excited state wave function and two coupled equations of motion for the ground states

2 p ΩR i ∂ ψ = ψ + ~ ei(k2−k1)zψ + (∆ + ΩAC )ψ (2.47) ~ t a 2m a 2 b ~ 1 a a CHAPTER 2. ATOM INTERFEROMETRY 28

2 p ΩR i ∂ ψ = ψ + ~ ei(k1−k2)zψ + (∆ + ΩAC )ψ (2.48) ~ t b 2m b 2 a ~ 1 b b where Ω Ω Ω := a1 b2 (2.49) R 2∆ is the two-photon or Raman Rabi frequency and the ac Starks shifts are given by

2 X |Ωni| ΩAC := , n = a, b. (2.50) n 4∆ i=1,2 ni

An atom initially in state |ai will be excited by a rightward traveling photon (~k1) with frequency ω1 to a virtual level, linewidths away from the excited state |ei. The likelihood of spontaneous emission from the excited state is small and the atom is stimulated down to |bi by a leftward traveling photon (~k2) of frequency ω2. The following is the Raman resonance condition, conserving energy and momentum, for such a process

(ω − ω ) − ω = v · (k − k ) ± ~ (k − k )2. (2.51) 1 2 ab 1 2 2m 1 2

2.2.1.1 Dressed states The eﬀective two level system that falls out of the three level system in the ‘adiabatic limit’, for which the fast dynamics of the excited state average to zero, has the following simple Hamiltonian " # ΩAC ΩR ei(δt−φL) Hˆ = 1 2 . (2.52) ΩR∗ −i(δt−φL) AC 2 e Ω2

By transforming the wave function as

0 −i(ΩAC +ΩAC )t/2 |ψ i = e 1 2 I|ψi, (2.53)

AC AC the energies are shifted by −~(Ω1 + Ω2 )/2. Then, rotating the wave function by the detuning from two-photon Raman resonance δ given by

2 2 (p + ~(k1 − k)) (p) δ = + ωb − + ωa − (ω2 − ω1) (2.54) 2m~ 2m~ through the operator Rˆ " # e−iδt/2 0 Rˆ = eiσzδt/2 (2.55) 0 eiδt/2 CHAPTER 2. ATOM INTERFEROMETRY 29 yields the (time-independent eﬀective) Hamiltonian for the system

" AC # (δ −δ) ΩR −iφL ˆ 2 2 e Heﬀ = − AC (2.56) ~ ΩR∗ iφL (δ −δ) 2 e − 2

~ p 2 AC 2 with eigenvalues ± 2 |ΩR| + (δ − δ) and wave function

AC AC AC AC −i(Ω +Ω )t/2 −iδt0/2 −i(Ω +Ω )t/2 iδt0/2 |ψ(t0)iR = ca(t0)e 1 2 e |ai + cb(t0)e 1 2 e |bi. (2.57)

In the presence of the external field for an interaction duration τ = t − t0, the atom will ˆ evolve according to the time-dependent Schrödingerequation but with Hamiltonian Heff and time-evolution operator

−ΩR −(δAC −δ) ΩR tan Θ = cos Θ = ˜ sin Θ = . (2.61) AC ΩR ˜ (δ − δ) ΩR

The probability amplitudes ca(t0 + τ) and cb(t0 + τ) are ˜ i(ΩAC +ΩAC )τ/2 iδτ/2 i(δt −φ ) ΩRτ c (t + τ) = e 1 2 e c (t )e 0 L i sin Θ sin a 0 b 0 2 ! Ω˜ τ Ω˜ τ + c (t ) cos R − i cos Θ sin R (2.62) a 0 2 2

˜ i(ΩAC +ΩAC )τ/2 −iδτ/2 −i(δt −φ ) ΩRτ c (t + τ) = e 1 2 e c (t )e 0 L i sin Θ sin b 0 a 0 2 ! Ω˜ τ Ω˜ τ + c (t ) cos R + i cos Θ sin R . (2.63) b 0 2 2

The squared amplitudes yield the probabilistic populations of atoms in the corresponding ˜ state for a pulse of time τ.A π/2-pulse satisﬁes ΩRτ = π/2 and yields the following amplitudes

iπ(ΩAC +ΩAC )/4Ω˜ iπδ/4Ω˜ e 1 2 R e R i(δt0−φL) ca(t0 +τ) = √ cb(t0)e i sin Θ + ca(t0) 1 − i cos Θ 2 CHAPTER 2. ATOM INTERFEROMETRY 30

iπ(ΩAC +ΩAC )/4Ω˜ −iπδ/4Ω˜ e 1 2 R e R −i(δt0−φL) cb(t0 + τ) = √ ca(t0)e i sin Θ + cb(t0) 1 + i cos Θ 2 ˜ and a ‘mirror’ pulse or π-pulse satisﬁes ΩRτπ/2 = π with the following amplitudes: AC AC iπ(Ω +Ω )/2Ω˜ R iπδ/2Ω˜ R i(δt0−φL) ca(t0 + τ) = e 1 2 e cb(t0)e i sin Θ − ca(t0) cos Θ

AC AC iπ(Ω +Ω )/2Ω˜ R −iπδ/2Ω˜ R −i(δt0−φL) cb(t0 + τ) = e 1 2 e ca(t0)e i sin Θ + ca(t0) i cos Θ .

In the absence of the external optical ﬁeld coupling internal atomic states, the Rabi frequencies which are proportional to the transition dipole matrix elements will vanish Ω1 = Ω2 = ΩR = 0, as will all the terms resulting from the light shift. Therefore, when the atom is in the dark the generalized Rabi frequencies and Θ parameter becomes ˜ ΩR = −δ cos Θ = −1 sin Θ = 0

and the amplitudes corresponding to this free evolution are as follows: iδτ/2 ca(τ + T ) = e ca(τ) cos −δT/2 + i sin −δT/2 = ca(τ) −iδτ/2 cb(τ + T ) = e cb(τ) cos −δT/2 − i sin −δT/2 = cb(τ).

2.2.2 The interaction phase From the equations given in the previous section, for an atom-light interaction of time τ, a phase ±φ is imprinted onto the matter-wave. The probability amplitude is a function of the system’s parameters as well as the initial amplitude prior to turning on the light. The driving ﬁelds for the three-level atomic system are

i(k1·r−ω1t+φ1) i(k2·r−ω2t+φ2) E(r, t) = ˆ1E1e + ˆ2E2e (2.64)

which for the initial condition φ1(t = 0) − φ2(t = 0) = φ0, an eﬀective phase φ is a function of the atom’s position in spacetime given by

φ(r, t) = (k1 − k2) · r − (ω1 − ω2)t + φ0 (2.65)

The phase is evaluated at each position and the corresponding time is referenced within the overall pulse sequence. Setting t = 0 at the beginning of the ﬁrst pulse in the interferometer implies that t corresponds to the absolute time. CHAPTER 2. ATOM INTERFEROMETRY 31

2.2.3 The separation phase The superposition of states at the interferometer’s end may not overlap perfectly in position- or momentum-space at the last π/2–pulse. This discrepancy may arise due to perturbations in either the atom-light interaction or potential during the sequence. This results in an additional phase shift of the ﬁnal wave function given by

∆φsep = p · δr (2.66) where p is the average canonical momentum of the detected atomic wave function and δr is the separation between the two wave packets. From here on out, we neglect this contribution to the phase.

2.3 The total phase

An interferometer’s geometry or sequence of pulses will determine the final phase difference between the atomic superposition. Consider the interferometer shown in Figure 2.7 with 0 periods of evolution, T , T and T . At the last pulse, the amplitude cb of the final state vector at time tf = (tf −τ) + τ is a function of the amplitudes for each initial state with respect to the last pulse τ given by

iφAC τ/2 −iδτ/2 e e (4) −iδ(tf −τ)+φ cb tf = √ ca(tf −τ)e L i sin Θ + cb(tf −τ) 1 + i cos Θ . 2 Consider the probability for an atom to be found in the |bi state after the pulse sequence, in the approximation where the laser is close to resonance, the following approximations can be made δAC − δ 1 → cos Θ ≈ 0 and sin Θ ≈ 1 (2.67) ΩR yielding 1 1 |c (t )|2 = |c |2 + |c |2 + ic c e−iφ − eiφ = |c |2 + |c |2 − 2c c sin φ . b f 2 a b a b 2 a b a b The wave packets interfere and an atom that completes the interferometer will probabilistically be projected into a state that is a function of the phase difference acquired between the interferometer arms 1 P = |eiΦu + eiΦ` |2. (2.68) |bi 4 The total phase difference is computed by considering the total trajectory in spacetime as a piecewise evolution of phase differences. The periods of free evolution and the periods during which the atom-light interaction is turned on are considered separately. CHAPTER 2. ATOM INTERFEROMETRY 32

u,` u,` X 1 Φ = S[zi, pi,Ti] + φγ[zi] (2.69) a,b i ~ a,b where S[zi, pi,Ti] is the action along the classical path beginning at zi with momentum pi for a time of evolution Ti and φγ[zi] is the laser phase imprinted on the atom during the ith pulse. The free evolution phase is determined by evaluating the action for the time of evolution Ti Z Ti S[z(t0), p(t0),Ti] = L[z(t), p(t)]dt (2.70) t0 where z(t) and p(t) are solutions to the Euler-Lagrange equation with initial position z(t0) and momentum p(t0). The Lagrangian for an atom in state ~ωn moving in a gravitational potential is p2 1 L = − Mgy + Mγy2 − ω . (2.71) 2M 2 ~ n 3 Here, g is the gravitational acceleration along the y-axis and γ = 2GMe/Re is the gravity gradient. Going forward the gradient is set to zero and in subsequent chapters gravity will also be neglected due to the orientation of the interferometer. Therefore, the free evolution phase with respect to the motion of the particle along the direction set by the interferometry beam zˆ is

Z Ti 1 X φf z(t), p(t),Ti = L[z(t), p(t)]dt ~ i 0 2 1 X Z Ti M ∂z(t) = − Mg cos θz(t) − ω dt 2 ∂t ~ n ~ i 0 2 2 3 1 X M z(Ti) − z0 Mg cos θ z(Ti) + z0 Ti Mg T = − − i − ω T 2 T 2 24 ~ n i ~ i i (2.72) with cos θ quantifying the projection of gravity along the interferometry axis zˆ. The initial time has been set to zero, t0 = 0. This computation is performed for each segment of the trajectory, using the ﬁnal position and velocity for the previous path as the starting point of the atom’s motion entering the current segment.

The laser phase φ(zi)γ is evaluated similarly, as a piecewise function along the atom’s trajectory given by

φ(zi)γ = (k1 − k2) × z − (ω1 − ω2) × ti + φ0, (2.73) where ti corresponds to the absolute time for the atom at position zi in the interferometer. For the counter-propagating Raman scheme employed here, the wave vector of the external ﬁeld are for our purposes are equal in magnitude and oppositely directed k1 = −k2. CHAPTER 2. ATOM INTERFEROMETRY 33

⟩ eff ħk 2 + |b,|b, p

Figure 2.6: (Left) The conjugate interferometer geometry for the Ramsey-Bord´efour π/2- pulse sequence. (Right) The conjugate interferometer geometry for the copropagating four π/2-pulse sequence.

π π π π 2.4 Conjugate interferometers with the 2 - 2 - 2 - 2 π π π π A consequence of the 2 - 2 - 2 - 2 is that the outputs of the second pulse not contributing to the lower interferometer may, if addressed at the third and forth pulse, close a second interferometer. This second interferometer is referred to as the conjugate in a double scheme and is realized with and without k-reversal. Phase calculations for both the lower and upper interferometers with and without ﬂipped the eﬀective wave vector of the Raman light for the second pulse pair, is detailed in the last sections of this chapter.

2.5 The Ramsey-Bord´einterferometer

An iconic paper by Bord´eexplored an atom interferometer described as an ‘optical Ramsey’ interferometer [96], which consisted of four (optical) beam splitter pulses such that the initial state was split similarly as in a Ramsey interferometer but the momentum was also addressed, put into a superposition given by

|a, pi → sin θ|a, pi + cos θ|b, p + 2~ki where p is the initial momentum and k is the wave vector of the light. In a Ramsey-Bordéinterferometer (RBI), an equal time of evolution denoted by T separates the pulses in each pair and a second time of duration (not necessarily equal to the first) T 0 separates the two pairs of pulses. The effective wave vector which determines the direction of momentum transfer is reversed for the second pulse pair and a final phase difference is 2 produced that is proportional to the recoil frequency of the atom ωr = ~k /(2m). CHAPTER 2. ATOM INTERFEROMETRY 34

|b , p ⟩ τ π 2 ⟩ |a f ef , p 1 ħk - + 1 2 ħk 2 , p e |a ff ⟩ 2 2 1 1 z

|b , p ⟩ t T T’ T

Figure 2.7: A Ramsey-Bordéinterferometer’s interaction geometry is shown here. This interferometer is built from four π/2-pulses (beam splitters) separated by three periods of free evolution: T , T 0, and T , respectively. the effective k-vector of the light is switched between the pulse pairs, reversing the direction of momentum imparted to the atoms and building sensitivity to the kinetic energy of recoiling that is imparted to the atom after absorption and emission in the two-photon transition. Prior to the last recombination pulse in this geometry, the interferometer arms are in states |b, pi and |a, p − ~keffi.

For an interferometer with pulses that induce a transition between internal states of the atom, the lifetimes need to be at least comparable to the transit time of the atom through the interferometer. Otherwise, spontaneous decay will destroy coherence. Such interferometers historically have operated in the space-domain, utilizing single photon transitions in atoms such as magnesium or calcium that have such long-lived metastable states. The spatial resolution of the two paths enjoyed by such schemes allows for sensitivity to field gradients as well as inertial displacements. Here, we use two-photon transitions for the interferometry pulses. Transitioning the atoms between hyperfine ground states, allows us to completely disregard any spontaneous decay. Both states are for our experimental purposes considered to be infinitely long-lived with lifetimes much longer than the interrogation time in the interferometer. The relative phase shift between the two paths is the total difference in phase acquired along each path given by

u ` 1 ∆Φf = Φf − Φf = ∆S12 + ∆S23 + ∆S34 ~

where the time ti is constant between the upper and lower trajectories but the position zi and momentum pi may differ. Here, the difference between actions for the paths at a particular time interval ∆Snm = u ` Snm − Snm and the differences in phase for each segment are computed below. CHAPTER 2. ATOM INTERFEROMETRY 35

Table 2.1: Positions and velocities for trajectories in the lower RBI.

Time segment Lower path, (zi`, vi`) Upper path, (ziu, viu)

g 2 g 2 [t1, t2] = T z0 + v0T − 2 T z0 + (v0 + vr)T − 2 T v0 − gT v0 − gT + vr

0 0 g 0 2 2 0 g 0 2 2 [t2, t3] = T z2` + v0T − 2 (T + T ) − T z2u + v0T − 2 (T + T ) − T 0 0 v0 − g(T + T ) v0 − g(T + T )

g 0 2 0 2 g 0 2 0 2 [t3, t4] = T z3` + v0T − 2 (2T + T ) − (T + T ) z3u + (v0 − vr)T − 2 (2T + T ) − (T + T ) 0 0 v0 − g(2T + T ) v0 − g(2T + T ) − vr

1 u ` S12 − S12 = kT 2v0 + vr − 2gT − ωabT (2.74) ~ 1 u ` 0 S23 − S23 = −2gkT T (2.75) ~ 1 u ` 0 S34 − S34 = kT − 2v0 + vr + 2g(T + T ) − ωabT (2.76) ~ A total phase diﬀerence from the free evolution is given by the following expression:

∆Φf = 2kvrT − 2ωabT. (2.77) The lower arm or path in a normal Ramsey-Bord´einterferometer never changes state nor acquires momentum and thus will receive no phase contribution from the interaction with the laser. The upper arm acquires a laser phase given by

0 0 ∆Φγ = φγ1(t0) − φγ2(t0 + T ) + φγ3(t0 + T + T ) − φγ4(t0 + 2T + T )

= (2kz0 − ω12t0) − (2kz2 − ω12t2) + ((−2k)z3 − ω12t3) − ((−2k)z4 − ω12t4) 2 0 = −4kvrT − kg 2T + 2TT + 2ω12T Therefore, the total phase diﬀerence is 0 Φtot = −2kvrT − 2kg T + T T − 2 ωab − ω12 T. (2.78)

The recoil velocity vr results from the transfer of 2~k of momentum. Inputting the 2 deﬁnition of the recoil frequency into the above expression ωr = ~k /2m leads to 4 k2T Φ = − ~ − 2kgT + T 0T − 2ω − ω T tot m ab 12 0 = −8ωrT − 2kg T + T T − 2δT (2.79) where the deﬁnition for the two-photon detuning is used to rewrite ωab − ω12 = δ. CHAPTER 2. ATOM INTERFEROMETRY 36

⟩ eff ħk + 2 p , |b 2

1 τ π 2 ⟩ f 1 ħk ef + 1 , p ⟩ |a f ħk ef 2 2 + z |a , p

|b , p ⟩ t T T’ T

Figure 2.8: A (conjugate) Ramsey-Bordéinterferometer’s interaction geometry is shown here. This interferometer is built from four π/2-pulses (beam splitters) separated by three periods of free evolution: T , T 0, and T , respectively. the effective k-vector of the light is switched between the pulse pairs, reversing the direction of momentum imparted to the atoms and building sensitivity to the kinetic energy of recoiling that is imparted to the atom after absorption and emission in the two-photon transition. Prior to the last recombination pulse in this geometry, the interferometer arms are in states |b, p + 2~keffi and |a, p + ~keffi.

2.5.1 cRBI phase computation The phase difference between the upper and lower trajectories can be computed by considering again a piecewise evolution for the atom during interferometry. These phase differences are the same as in the normal RBI for the first two periods of free evolution T and T 0 but differ for the final T . In each interferometer, the probability of detection at a output depends CHAPTER 2. ATOM INTERFEROMETRY 37

Table 2.2: Positions and velocities for trajectories in a conjugate RBI.

Time segment Lower path, (zi`, vi`) Upper path, (ziu, viu)

g 2 g 2 [t1, t2] = T z2` = z0 + v0T − 2 T z2u = z0 + (v0 + vr)T − 2 T v0 − gT v0 − gT + vr

0 0 g 0 02 0 g 0 02 [t2, t3] = T z3` = z2` + (v0 + vr)T − 2 2TT + T z3u = z2u + (v0 + vr)T − 2 2TT + T 0 0 v0 − g(T + T ) + vr v0 − g(T + T ) + vr

g 2 0 g 2 0 [t3, t4] = T z4` = z3` + (v0 + 2vr)T − 2 3T + 2TT z4u = z3u + (v0 + vr)T − 2 3T + 2TT 0 0 v0 − g(2T + T ) + 2vr v0 − g(2T + T ) + vr

on the phase difference between the arms of the interferometer. A nicety of the simultaneous conjugate RBI scheme is that accelerations add equally to the phase shifts for the normal and upper interferometer phases. Phase fluctuations of the lasers or vibrations of the system can be canceled by considering both interferometers. Repeating the procedure outlined above but now for the upper conjugate Ramsey-Bordé interferometer yields

1 u ` S12 − S12 = kT 2v0 + vr − 2gT − ωabT (2.80) ~ 1 u ` 0 S23 − S23 = −2gkT T (2.81) ~ 1 u ` 0 S34 − S34 = kT − 2v0 − 3vr + 2g(T + T ) − ωabT (2.82) ~ and the total phase difference from the free evolution of the coherent superposition is given by ∆Φf = −2kvrT − 2ωabT. (2.83) For the upper RBI, assume that during the first pulse, the upper arm has been imprinted with the laser phase during the atom-field interaction but then that all subsequent pulses only affect the lower arm of the upper interferometer:

0 0 ∆Φγ = φγ1(t0) − φγ2(t0 + T ) − φγ3(t0 + T + T ) + φγ4(t0 + 2T + T )

= (2kz0 − ω12t0) − (2kz2` − ω12t2) + (−2kz3` − ω12t3) − (−2kz4` − ω12t4)

= 2k(z4` − z3` − z2`) − ω12(t0 − t2 + t3 − t4) 2 0 0 0 = 2k 2vrT − g(T + TT ) − ω12 − T + T + T − 2T − T 0 = 4kvrT − 2kg(T + T )T + 2ω12T (2.84) CHAPTER 2. ATOM INTERFEROMETRY 38

Therefore, the cRBI has a total phase diﬀerence given by

c 0 0 Φtot = 2kvrT − 2kg(T + T )T − 2δT = +8ωrT − 2kg(T + T )T − 2δT. (2.85)

2.6 The copropagating interferometer

If instead of reversing the effective k-vector between the pulse pairs, the direction is maintained then the copropagating scheme (CPI) depicted in Fig. 2.9 will be realized. The phase shift of this interferometer, contrary to a Ramsey-Bordéinterferometer, will not depend on the frequency of the laser generating the pulses and therefore be less sensitive to frequency fluctuations. This loss of dependence arises because each component of the superposition will spend an equal amount of time, T , recoiling. The copropagating interferometer has the same dependence in sensitivity as a Mach- π π Zehnder configuration characterized by a 2 − π − 2 pulse sequence, as can be seen from the above computation. If an additional potential term is turned on in the total Hamiltonian during the T 0 period of free evolution, a resulting phase difference will be realized. When measuring the tune-out wavelength here, we are unable to spatially resolve the arms of the interferometer as has been done to-date. We focus on the spatial gradient of the interaction instead and the implement detection and analysis are discussed more thoroughly in Chapter 6. The total phase in the presence of a ‘tune-out’ beam switched on during the evolution time T 0 is given by

α(ω)sT 0 ∂I φto = × (2.86) 2ε0c~ ∂z where s = 2vrT is the separation between the arms of the interferometer, which can be defined in terms of the recoil velocity of the atom and the intensity of the beam with waist w is given by −2r2/w2 I(r, ω0) = I0e (2.87) 2 where I0 = 2P/(πw0). The phase difference between the upper and lower trajectories during free evolution is 1 0 0 ∆Φf = kT (2v0 + vr − 2gT ) − ωabT − 2gkT T − kT (2v0 + vr − 2g(T + T )) + ωabT = 0. ~ The phase difference between trajectories resulting from the atom-light interaction is ∆Φγ = +φγ1(t0) − φγ2(t2) − φγ3(t3) − φγ4(t4) = 2k z0 −z2u −z3` +z4` − ω12 t0 −t2 −t3 + t4 = −2kgT + T 0T. (2.88) CHAPTER 2. ATOM INTERFEROMETRY 39

|b , p ⟩ τ π 2 ⟩ f 1 ħk ef 1 + 1 1 |a , p 2 2 2 2 z |b , p ⟩ t T T’ T

Figure 2.9: A (lower) copropagating interferometer’s interaction geometry is shown here. This interferometer is built from four π/2-pulses (beam splitters) separated by three periods of free evolution: T , T 0, and T , respectively. The eﬀective k-vector of the light is maintained between the pulse pairs and the lack of k-reversal is evident in that both the upper and lower arms spend equivalent durations of time in each momentum state (compared to each other). Prior to the last recombination pulse in this geometry, the interferometer arms are in states |b, pi and |a, p + ~keﬀi.

Table 2.3: Positions and velocities for the trajectories in the lower copropagating interferometer scheme.

Time segment Lower path, (zi`, vi`) Upper path, (ziu, viu)

g 2 g 2 [t1, t2] = T z2` = z0 + v0T − 2 T z2u = z0 + (v0 + vr)T − 2 T v0 − gT v0 − gT + vr

0 0 g 0 02 0 g 0 02 [t2, t3] = T z3` = z2` + v0T − 2 2TT + T z3u = z2u + v0T − 2 2TT + T 0 0 v0 − g(T + T ) v0 − g(T + T )

g 2 0 g 2 0 [t3, t4] = T z4` = z3` + (v0 + vr)T − 2 3T + 2TT z4u = z3u + v0T − 2 3T + 2TT 0 0 v0 − g(2T + T ) + vr v0 − g(2T + T ) CHAPTER 2. ATOM INTERFEROMETRY 40

|b , p ⟩

2 τ π 1 2 1 1 1 ⟩ f 2 ħk ef 2 2 2 + , p ⟩ |a f ħk ef + , p z |a |b , p ⟩ t T T’ T

Figure 2.10: A (upper) copropagating interferometer’s interaction geometry is shown here. This interferometer is built from four π/2-pulses (beam splitters) separated by three periods of free evolution: T , T 0, and T , respectively. The eﬀective k-vector of the light is maintained between the pulse pairs and the lack of k-reversal is evident in that both the upper and lower arms spend equivalent durations of time in each momentum state (compared to each other). Similar to the lower interferometer discussed in the previous section, prior to the last recombination pulse in this geometry, the interferometer arms are in states |b, pi and |a, p + ~keﬀi.

Table 2.4: Positions and velocities for the trajectories of the upper copropagating interferometer

Time segment Lower path, (zi`, vi`) Upper path, (ziu, viu)

g 2 g 2 [t1, t2] = T z2` = z0 + v0T − 2 T z2u = z0 + (v0 + vr)T − 2 T v0 − gT v0 − gT + vr

0 0 g 0 02 0 g 0 02 [t2, t3] = T z3` = z2` + (v0 + vr)T − 2 2TT + T z3u = z2u + (v0 + vr)T − 2 2TT + T 0 0 v0 − g(T + T ) + vr v0 − g(T + T ) + vr

g 2 0 g 2 0 [t3, t4] = T z4` = z3` + (v0 + vr)T − 2 3T + 2TT z4u = z3u + v0T − 2 3T + 2TT 0 0 v0 − g(2T + T ) + vr v0 − g(2T + T ) CHAPTER 2. ATOM INTERFEROMETRY 41

2.6.1 cCPI phase computation An additional (conjugate) interferometer is also formed in the co-propagating geometry, simply resulting from the use of π/2-pulses. The lower and upper interferometers interferometers share similarities in the phase accrued during the T evolution time even though the upper interferometer is offset in space to the lower one. Details for the free evolution phase can be found in Table 2.4. These interferometers differ as to the state of the atom during the T 0 evolution period. This becomes important because it is precisely during this time in which the perturbing tune-out light is flashed on the atoms. The hyperfine ground states of lithium have dynamic polarizabilities that differ by almost 1 GHz, a result of differing transition matrix element Clebsh-Gordan coefficients. See Chapter 6 for more details. 42

Chapter 3

Lithium, the smallest alkali

Lithium is the third element (Z = 3) of the periodic table and the lightest of the alkali atoms. Its smallness make it an advantageous choice for atom interferometry, particularly its small mass and low electron number, as will be discussed more explicitly. Refer to Appendix A for the general physical and optical properties of lithium.

3.1 Lithium, the lightest alkali

Table 3.1 shows a comparison of lithium to the heavier alkalis, rubidium and cesium, both used atom interferometry determinations of the ﬁne structure constant α through an h/M measurement [17, 40]. As discussed in Chapter 1, a measurement of h/M in combination with an accurate determination the k-vector or frequency of the light, leads to a non-QED determination of the ﬁne structure constant α (independently of g − 2) by way of the following relation

2R u M h α2 = ∞ . c me u M (3.1)

The quantity h/M, is proportional to the recoil frequency ωr of the atom after interacting

Element Z Mass Recoil velocity Recoil frequency Lithium 3 7.0160 u [57] 8.5682 cm/s 2π × 63.8498 kHz Rubidium 37 86.909 u [97] 5.8845 mm/s 2π × 3.7710 kHz Cesium 55 132.905 u [97] 3.5225 mm/s 2π × 2.0663 kHz

Table 3.1: Comparison of mass and single photon recoil velocity (frequency) for lithium, rubidium and cesium’s D2-lines CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 43

3 6.941 Li Lithium

55 132.9054 Cs Cesium

t Figure 3.1: Lithium’s light mass and consequently higher recoil velocity means that for the same amount of time, a lithium atom will transverse a larger spacetime area during interferometry than will a heavier alkali atom, like cesium, depicted here.

with a photon of wave vector k, k2 ω = ~ . (3.2) r 2m For the two-photon Raman transitions utilized as the beam splitters and mirror in this work, two driving fields are present. The atom undergoes stimulated emission from |ei (following absorption to the excited state via the ‘first’ photon) and transition to another state |bi. This leads to the following equations for the momentum ‘kick’ and resonant photon frequency, given conservation of momentum and energy in the atom-light system for an atom of mass M, with velocity v0 in the presence of external driving fields characterized by frequencies ω1 = c/k1 and ω2 = c/k2:

δv = ~ k = ~ (k − k ) (3.3) M eﬀ M 1 2 ωL = (ωb − ωa) + v0 · k ± ωr. (3.4)

2 ~k Above, a positive recoil shift, ωr = 2M , corresponds to absorption and a negative shift corresponds to emission. A Ramsey-Bord´einterferometer, as discussed in Chapter 2, is sensitive to the kinetic energy obtained by the atom from interaction with a photon. It realizes a phase diﬀerence CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 44

∆φ± given by

± 0 ∆φ = ±8ωrT + 2kazT (T + T ) + 2δT (3.5) where here ± denotes either the upper (normal) or lower (conjugate) interferometer, k is the effective wave vector, az denotes accelerations along the interferometry axis and δ is the two-photon detuning. The atomic recoil frequency given by k2 ω = ~ . (3.6) r M It is no surprise that this imparted energy of the recoiling atom is inversely proportional to its mass. Therefore, lithium’s lightness corresponds to a higher phase accrued than compared to heavier atoms in recoil-sensitive Ramsey-Bordéinterferometers. Assuming a fixed phase uncertainty given by shot-noise 1 δφ± = √ (3.7) N where N is the total number of atoms participating in the interferometer per experimental shot. The overall sensitivity to the recoil frequency δωr is δφ± δωr = × ωr (3.8) 8ωrT Comparing to state-of-the-art h/M measurement, a cesium Ramsey-Bordéinterferometer utilizing Bragg diffraction and Bloch oscillations with a phase difference at its output of

± 0 δφ = ± 8n(n + N)ωrT − nωmT + nkgT (T + T ) (3.9)

where n is the Bragg diﬀraction order, counting the number of 2n~k kicks the atom receives and N is the number of Bloch oscillations. Neglecting large momentum transfer techniques like Bloch oscillations, a cesium interferometer with n = 5, T = 100 ms, and N = 105 due to a reduction from velocity selection, has a project sensitivity to ωr of √ Cs 1/ N −8 δωr = 2 × ωr ≈ 4 × 10 ωr. (3.10) 8n ωrT At an interrogation time of only 10−ms, a Ramsey-Bord´elithium interferometer could achieve comparable sensitivities √ Li 1/ N −8 δωr = × ωr ≈ 6 × 10 ωr, (3.11) 8nωrT where N = 107 since we forgo velocity selection and address the majority of the trapped atoms via fast interferometry pulses. CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 45

3.2 Lithium, the simplest alkali

Moving away from hydrogenic systems, atomic structure computations increase in complexity and decrease in accuracy due to the presence of electron correlations and the difficulty in approximating relativistic corrections. The Dirac equation looses utility after one electron, with Dirac-like methods lacking the computational precision needed to be compared with experimental measurements [98–101]. Computations based on the relativistic coupled-cluster method and relativistic many-body perturbation theory do not include the electron correlations accurately enough to be competitive with methods based on the NRQED approach and explicitly correlated functions [102]. Ultimately, quantum electrodynamics is needed to consistently describe these relativistic, correlated electrons in a many-electron system. Perturbation theory affords a description in an extension of QED through a perturbative expansion in powers of the fine structure constant α referred to as nonrelativistic quantum electrodynamics (NRQED). Based upon the foundational nonrelativistic, Schrödinger picture, NRQED allows for the determination of relativistic and QED corrections perturbatively. For example in NRQED, relativistic and QED corrections to the energy levels are accounted for in the following expansion:

E(α) = Mα2ε(2) + Mα4ε(4) + Mα5ε(5) + Mα6ε(6) + ... (3.12)

where the expansion coeﬃcients ε(i) are expressed in terms of the ﬁrst- and second-order matrix elements of the nonrelativistic wave function Ψ

i 2 X |hΨ|O |φji| ε(i) = hΨ|Oi|Ψi + . (3.13) E − E j Ψ φj The wave function Ψ is determined from the time-independent Schr odinger equation

HΨ = EΨ

where H is the nonrelativistic Hamiltonian (inﬁnite mass system) given by

X p2 X Zqi X qi qj H = i − e + e e (3.14) 2M r r i i i i>j ij

i where ri is the position of electron i with charge qe from the nucleus (Z).The electron-electron replusion is given by the term 1/rij with the inter-electronic separation rij = |ri − rj|. The quality of Ψ dictates the accuracy of the numerical calculations. The presence of the repulsion term 1/rij precludes separability for the Hamiltonian and consequently an exact solution to the Schr odinger equation. However, in 1929 Hylleraas proposed when considering helium, to expand the wave function in terms of an explicitly correlated variational basis set [103], X i j k −αr1−βr2 M Ψ∞(r1, r2) = aijkr1r2r12e Yl1l2L(ˆr1,ˆr2) (3.15) i,j,k CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 46 where YM is a vector-coupled product of spherical harmonics with total angular momentum l1l2L L and magnetic quantum number M. The coefficients aijk are linear variational parameters whereas α and β are nonlinear parameters, imposing the distance scale for the wave function. For up to three-electrons there exists such an explicitly correlated variational basis for which analytical computational methods have been developed [104]. This so-called Hylleraas basis has been fully implemented in the two- and three-electron case [105–107] is precluded for use in systems with more than three electrons due to the difficulties of performing the Hylleraas multi-center integrals. Therefore, lithium’s three electron positions it right at the threshold of accessibility, for which these explicitly correlated variational wavefunctions in Hylleraas coordinates can be applied. Calculations for the lithium atom can be done very accurately within NRQED because numerical solutions of the Schrödingerequation can be obtained accurately for the low-lying states of three-electron atoms.The high accuracy of these calculations can be used in conjunction with the high precision experiments to build unique measurement tools such as the testing of higher-order relativistic and QED corrections to the transition frequencies. These first principle calculations can serve as atomic-based standards for quantities that are not amenable to precision measurement. Measurements of the polarizability ratio between an atom and lithium can be measured to a higher degree of precision than can be done solely with the individual atom. Measurements of the ratio in conjunction with high-precision ab initio calculations could lead to a new level of accuracy in polarizability measurements for atomic species commonly used in cold-atom physics.

3.2.1 The Hylleraas basis The internal dynamics of lithium can be described by (3.14) for three-electrons. Transforming to the center-of-mass frame gives

3 3 3 3 1 X 1 X X qi X qiqj H = − ∇2 − ∇ · ∇ + Z + (3.16) 0 2µ i m i j r r i=1 0 i>j≥1 i=1 i i>j≥1 ij

with reduced atom mass µ = mm0/(m + m0), electron mass m, and nuclear mass m0. The charge of the electron’s is now denoted by q. Variationally solving the energy eigenvalue equation or stationary Schr odinger equation

H0Ψ0(r1, r2, r3) = E0Ψ0(r1, r2, r3) (3.17)

yields a variational wave function as a linear combination of the following terms

Ψ0 = A(φ(r1, r2, r3)) (3.18)

in terms of the explicitly correlated basis functions in Hylleraas coordinates

j1 j2 j3 j12 j23 j31 −αr1−βr2−γr3 LM φ(r1, r2, r3) = r1 r2 r3 r12 r23 r31 e Y(`1`2)`12,`3 (ˆr1,ˆr2,ˆr3)χ1 (3.19) CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 47 and the three-particle anti-symmetrizing operator A,

A = (1) − (12) − (13) − (23) + (123) + (132), (3.20) which ensures that the wave function satisﬁes the Pauli principle. The three-electron-spin wave function is denoted by

χ1 = α(1)β(2)α(3) − β(1)α(2)α(3) (3.21)

1 with spin angular momentum equal to 2 . This can be interpretted as that electrons 1 and 2 first couple to a spin singlet state with S12 = 0 and then this state is coupled to the final 1 electron to produce a final state with total spin of 2 [107]. The product of spherical harmonics for the three-electron case, YLML , is a given by (`1`2)`12,`3 X YLML (ˆr ,ˆr ,ˆr ) = ` m ` m ` m ` m ` m LM (`1`2)`12,`3 1 2 3 1 1 2 2 12 12 12 12 3 3 L mi

×Y`1m1 (ˆr1)Y`2m2 (ˆr2)Y`3m3 (ˆr3). (3.22) The angular momentum coupling schemes for a three-body system

(`1, `2)`12`3; LM , `1(`2, `3)`23; LM , (`1, `3)`13`2; LM (3.23) such as lithium are not unique but should be physically equivalent since unitary transforma- tion exist among them.

`1+`2+`3+L X p 0 `1(`2, `3)`23; LM = (−1) (2`12 + 1)(2`23 + 1) 0 `12 ( 0 ) `1 `2 `12 0 × (`1, `2)`12`3; LM . (3.24) `3 L `23

Only a state that satisﬁes the following conditions

L `1 + `2 + `3 = L for parity (−1) , L+1 `1 + `2 + `3 = L + 1 for parity (−1) , needs to be included as part of the Hylleraas bases. The outline of the procedure for such a computation can be found in Ref. [107–109]. To summarize, in the Hylleraas method a series of calculations are performed of increasing dimension while keeping non-linear parameters (α, β, γ) constant. All terms are included such that they satisfy j1 + j2 + j3 + j12 + j23 + j31 ≤ Ω (3.25) where here Ω is an integer that is progressively increased until the energy eigenvalue converges in value [87]. For a particular state, including additional angular momentum conﬁgurations, which albeit redundant physically, has been shown to increase the rate of convergence[107]. CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 48

Table 3.2: Angular momentum conﬁgurations for the S, P , D states of lithium.

state no. `1 `2 `12 `3 L M S 1 0 0 0 0 0 0 1 0 0 0 1 1 0 P 2 0 1 1 0 1 0 3 1 0 1 0 1 0 1 0 0 0 2 2 0 2 0 1 1 1 2 0 3 0 2 2 0 2 0 D 4 1 0 1 1 2 0 5 2 0 2 0 2 0 6 1 1 2 0 2 0

3.3 Dynamic polarizability

In Chapter 1, the ac-Stark of an initial state |ii is defined as a sum over all other states |ki given by 2 (+) 2 X 2ωki|hi|ˆ· d|ki| |E (r)| ∆E = − 0 (3.26) |ii (ω2 − ω2) k ~ ki with ωki := (Ek − Ei)/~. 0 0 From this expression, for a sum over hyperfine states, |ki = |F mF i, the polarizability can be defined in terms of the following two-tensor

0 0 0 0 X 2ωF 0F hF mF |dν|F mF ihF mF |dν|F mF i αµν(ω) = 2 2 . (3.27) (ω 0 − ω ) 0 0 ~ F F F mF A decomposition of the polarizability leads to an expression for the ac-Stark shift of the energy for hyperﬁne state |F mF i: m ∆E(F, m ; ω) = −α(0)(F ; ω)|E (+)|2 − α(1)(F ; ω)(iE (−) × E (+)) F F 0 0 0 z F 3|E (+)|2 − |E (+)|23m2 − F (F + 1) −α(2)(F ; ω) 0z 0 F (3.28) 2 F (2F − 1)

with the following scalar, vector and tensor polarizabilities:

0 2 (0) X 2ωF 0F |hF kdkF i| α (F ; ω) = 2 2 (3.29) 3 (ω 0 − ω ) F 0 ~ F F CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 49

r 0 2 (1) X F +F 0+1 6F (2F + 1) 1 1 1 ωF 0F |hF kdkF i| α (F ; ω) = (−1) 2 2 (3.30) F + 1 F F F’ (ω 0 − ω ) F 0 ~ F F s 0 2 (2) X F +F 0+1 40F (2F + 1)(2F − 1) 1 1 2 ωF 0F |hF kdkF i| α (F ; ω) = (−1) 2 2 3(F + 1)(2F + 3) F F F’ (ω 0 − ω ) F 0 ~ F F (3.31) Ultimately, when we think about an atom’s polarizability, we think about its interaction with an external field. Such an atom-field interaction corresponds to the interaction Hamiltonian Hint = HStark = −d · E (3.32) with energy eigenvalue arising from the expectation value of the effective Hamiltonian applied to the |F mF i state

∆E|F mF i(ω) = hF mF |HStark|F mF i (3.33)

with HStark given by F H = −α(0)(F ; ω)|E (+)|2 − α(1)(F ; ω)(iE (−) × E (+)) z Stark 0 0 0 z F 3|E (+)|2 − |E (+)|2 3F 2 − F2 −α(2)(F ; ω) 0z 0 z (3.34) 2 F (2F − 1)

with the scalar α(0), vector α(1) and tensor α(2) polarizabilities deﬁned as above. In Ref. [61] a basis-independent description for this shift is given by way of the tensor polarizability operator αµν F 3α(2)(ω) 1 F2δ α (ω) = α(0)(ω)δ + α(1)(ω)i σ + (F F + F F ) − µν µν µν σµν F F (2F − 1) 2 µ ν ν µ 3 (3.35)

with eﬀective Hamiltonian (−) (+) HStark = −αµν(ω)E0µ E0ν . (3.36) The Kramers-Heisenberg formula expresses the polarizability in terms of the dipole matrix elements as

X 2ωi0hg|dµ|iihi|dν|gi α (ω) = (3.37) µν (ω2 − ω2) i ~ i0

which for a spherically symmetric atom, possessing identical dipole components dµ, the scalar polarizability becomes 2 X 2ωi0|hg|dµ|ii| α(ω) = . (3.38) (ω2 − ω2) i ~ i0 CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 50

A variety of deﬁnitions exist for the dynamic polarizability in terms of atomic properties −2 3 X Akiωik gk α(ω) = 2π0c 2 2 (3.39) ω − ω gi k6=i ik 2 2 X |hk|qer|ii| ωik α(ω) = (3.40) 3 ω2 − ω2 ~ k6=i ik

1 X Sikωik α(ω) = . (3.41) 3 ω2 − ω2 ~ k6=i ik (3.42) A number of different computational methods for atomic polarizability exist since most that are used to determine the atomic wave function and energy levels can be adapted to a polarizability computation. An overview but not exhaustive list of such approaches can be found in Ref. [81]; their list emphasizes methods which have achieved the highest accuracy or utility. The relativistic single-double all-order many-body perturbation theory (MBPT- SD) calculation of the dynamic polarizability of lithium [98] is fully relativistic and treats correlations effects to a high level of accuracy. However, it does not achieve the same level of precision and has a less exact treatment compared to the variational calculation [110] nor does the MBPT-SD calculation consider finite mass effects. In this particular methodology, the computational uncertainty related to the convergence of the basis set can be determined. The difference between calculated and experimental binding energies yields an estimate for the size of the relativistic correction to the polarizability. Rewriting Eq. 3.41 in terms of the dipole matrix elements and oscillator strengths fik yields 2 2 X |hk|d|ii| ωik X fik α(ω) = = . (3.43) 3 ω2 − ω2 (ω − ω )2 − ω2 ~ k6=i ik k6=i ~ k i This description is in the spirit of Ref. [92] which derives the polarizabilities for the 2S level of lithium based upon another calculation, both utilizing the variational Hylleraas method. Uncertainty in the ground state polarizability creeps in primarily due to the uncertainty in the h2skdk2pi matrix element. A comparison to this matrix element derived from an analysis of the ro-vibrational spectrum of the Li2 dimer still yields an overestimation according to the authors due to the complexity of the spectrum. The work presented in this thesis focuses on the dynamic polarizability for lithium’s ground state, 2S. Highly accurate results for lithium’s atomic properties can be obtained with only ab initio wavefunctions due to its simple electron structure.

3.4 Lukewarm Lithium

0 0 In lithium, the hyperﬁne splitting between the F = 3 and F = 2 excited states of the 2P3/2 state is only 1.6Γ, where natural linewidth (of the D lines) is Γ = 2π × 5.9 MHz. Despite CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 51

Computed dynamic polarizability for the Li ground state 3000 Hylleraas MBPT-SD 2000 TDGI CI-Hylleraas

1000

0.05 0.10 0.15 ω [a.u.] -1000

dynamic polarizability [a.u.] polarizability dynamic

-2000

-3000

Figure 3.2: A plot of the results from various computational methods for the dynamic polarizability in the Li ground state.

Method Year Value [a.u.] Expt. 164.2(11) Hylleraas [∞Li] 2010 164.112(1) Finite mass [7Li] 2010 164.161(1) Relativistic [7Li] 2010 164.11(3) Expt. (2s − 2p1/2) -37.14(2) Hylleraas [7Li] 2013 -37.14(4) RLCCSDT -37.104

Table 3.3: Computed and measured scalar polarizabilities and Stark shift values for lithium. CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 52

Maxwell-Boltzmann Speed Distribution for Super- and Sub-recoil Atoms

6 μK (T rec ) 300 μK

Probability density Probability

0 0.5 1.0 1.5 2.0 speed [m/s]

Figure 3.3: The Maxwell-Boltzmann distributions for atoms at the recoil temperature (blue) and at 300 µK (red). the inverted structure of the level, this small excited state splitting leads to off-resonant excitations of the F = 2 → F 0 = 2 transition and hence subsequent, frequent decay down into the F = 1 lower energy ground state. The presence of repumping light, light from the F = 1 lower energy ground state to the F 0 = 2 excited state, is therefore crucial to trapping and cooling lithium; atoms must be pumped back into F = 2 the state being addressed by the cooling light. The small separation between hyperfine states of the excited 2P3/2 state in lithium thwarts efficient polarization gradient cooling ultimately resulting in an atom source, prior to interferometry, that is at approximately 300 µK. The inability to cool further given our experimental constraints means that not only are the atoms on average at a higher velocity but the velocity width of the sample is much broader, as can be seen in Figure 3.3. While some experiments employ a velocity selection to reduce this width, doing so would result in a dramatic loss of atoms participating in interferometry. From the Maxwell-Boltzmann probability density distribution for atoms at a temperature T r 3 M 2 2 −Mv /2kB T fv(T ) = 4πv e , (3.44) 2πkBT the probability of finding an atom at a velocity up to the recoil atomic velocity at a temperature of 300 µK can be determined by integrating the density distribution up to this CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 53 bound Z vr fv(300 µK)dv. (3.45) 0 Doing so yields that about 0.1% of atoms have a velocity below or equal to the recoil velocity of lithium. If we were to use this as a cut-off in velocity selection, we would exclude 103 atoms of the 107 atoms initially trapped. This translates into almost a reduction in experimental sensitivity of two orders of magnitude!

3.4.1 Lithium atom interferometry in the space domain Previously, atom interferometry with lithium had been performed with a supersonic beam atomic beam seeded in argon gas (contrary to the light-pulsed atom interferometry discussed in Chapter 2). In this instance, the split atomic wave function is allowed to propagate over the space-domain, contrary to the time-domain utilized in light-pulsed atom interferometry schemes. For an atomic beam with average velocity vˆz, diffraction is generated transverse to the propagation direction of the atomic beam by a periodic diffraction grating, potentially generating many momentum components for the scattered wave. In the interferometry experiments with lithium, elastic Bragg diffraction on standing waves of light (λ = 671 nm) performed the function of a beam splitters and mirror. A matterwave given by ψ(r) = exp[ik · r] (3.46) incident on the light-grating Gj, with diffraction period d and wave vector kj = 2π/d, will produce a diffraction order p given by

ψp(r) = cj(p) exp[ik · r + ipkj · (r − rj)] (3.47) where rj is the position of the grating and cj(p) is the amplitude of order diffracted wave. For the n-th momentum component, the average momentum transferred to the waves is characterized by a diffraction angle in the far field given by

δpn λdB θn ≈ = . (3.48) pbeam d In the far field, the resolution of different diffraction orders becomes contingent on the transverse momentum distribution of the incoming beam being smaller than the transverse momentum imparted to it by the grating, δpn. This is the same as requiring that the transverse coherence length be larger than a few grating periods (accomplished via collimating the beam). A Mach-Zehnder configuration utilized in the thermal atom interferometry experiments with lithium was realized via an output with an upper path created by orders p, −p, and 0 and lower path created by orders 0, p, and −p, as shown in Fig. These trajectories interfere to create a resultant wave function given by

ψu/`(r) = cu/` exp[i(k · r + φu/`)] (3.49) CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 54

where cu/` is the product of the amplitudes at each grating and φu/` is the laser phase. Fringe visibility of V = 84.5 ± 1% was observed with the first diffraction order. The ability to interrogate one arm of the interferometer, which are spatially resolved and in the same internal energy state, is a benefit of performing atom interferometry in this way. However, when a static perturbation is applied to one beam for interferometry performed in the space domain, typically the phase shift resulting will depend upon the atom velocity, v. For an atom entering a region with a nonzero electric field E, the phase shift of the atomic wave is Z 2π0α 2 φ= E (z)dz (3.50) ~v where α is the static polarizability of the atom. A measurement of the static polarizability of lithium, determined to be

α = (24.33 ± 0.16) × 10−30m3 = 164.19 ± 1.08au, (3.51)

performed by applying an electric field to one of the trajectories in via an inserted septum [44], confirms that the velocity is a significant systematic in these experiments. The uncertainty, equal to 0.66%, is dominated by the uncertainty in the most probable velocity. Only a the Aharonov-Casher or He-McKellar-Wilkens phase shift is an exception to this dependency, being independent of the atom velocity in the beam. However, measurements of the AC phase, while confirming the velocity independence, only reached statistical uncertainties of approximately 3% [111]. This is a result of the phase’s dependence on the hyperfine Zeeman sublevel of the atom. The experimental challenges in optically pumping a supersonic atomic beam, particularly in controlling residual magnetic fields, are nontrivial [112].

3.5 Advantages of light-pulsed interferometry with lithium

The beneﬁt aﬀorded with a thermal atom interferometer in the enjoyed spatial separation between the interferometer arms is countered by a reduction in experimental sensitivity that results from a necessary velocity selective process of the atom source. The coherence length of an atomic beam is given by 1 hλdBi hpi `coh = = (3.52) σk 2π σp

where σp is the width of the beam’s momentum distribution. The size of the optical path difference between interferometer arms or realized contrast of the detected flux is dependent upon the longitudinal coherence length of the interferometer. Additionally, a highly collimated atomic beam is necessary with thermal atom interferometry since the atom optics superimpose only momentum states. This velocity selectivity strongly reduces the flux of atoms and the sensitivity of the experiment. CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 55

Light-pulsed atom interferometry offer advantages stemming from the ease at which atoms can be manipulated by light but also the extent at which the interaction of atoms and light are understood. In such interferometry experiments, tagging the momentum states with the internal energy states of the atom is a consequence of two photon Raman transitions but such a coupling then allows for state selective detection of the outputs. For interferometry with a thermal atomic beam, such a coupling can be achieved albeit only with a single-photon process. Atoms like calcium and magnesium have been used in thermal Ramsey-Bordéinterferometry to realize the same configurations as mentioned in Chapter 2. Such experiments are limited by the lifetime of the atom’s metastable state however. In comparison, the hyperfine ground state which lithium is transitioned to (or any alkali for that matter) is considered infinitely long-lived. There is not really a branching for decay back into the other hyperfine ground state and thus interferometry done with these atoms will not be limited by the lifetime of the second state. This coupling or momentum to atomic spin which is discernible with absorption imaging allows us to refrain from the lossy velocity selection required with lithium for atom interferometry in the space domain. Light-pulsed atom interferometry in general allows for greater experimental sensitivity to be achieved in recoil-sensitive interferometry since two-photon transitions are possible. 56

Chapter 4

Experimental Methods

This chapter describes experimental methods used to prepare the atom sample for interferometry as well as the procedures, techniques, and hardware common to both recoil- and polarizability-sensitive measurements detailed in Chapters 5 and 6, respectively. In light- pulse atom interferometry, the sensitivity scales with the pulse separation time during interferometry, the time during which the atom exists in a superposition of states accruing a phase difference projected at detection. Thus, an experiment’s precision is increased as the duration of this time is increase, contrary to space-domain atom interferometers whom require a lengthening in space and consequently of the actual experimental apparatus. Lower temperatures allow these experimental times to be extended by hindering the ballistic expansion of the atoms out of the interferometry beam. However, it is important to avoid additional measurement systematics and maintain a dilute, noninteracting sample. The experimental techniques of laser cooling and trapping are commonplace in modern atomic physics experiments and here, I will only discuss details pertinent to this experiment which implements Doppler cooling with magneto-optical trapping. A more substantial explanation can be found in Appendix D and in many other papers [113–115] and textbooks [116–118]. Prior to performing interferometry with lithium, the atom source must be carefully and precisely created. An experiment cycle consists of the following steps: 1. Atomic beam creation. 2. Two-dimensional cooling and trapping in a magneto-optical trap. 3. Loading and subsequent cooling and trapping in a three-dimensional magneto-optical trap. 4. Further cooling by ramping settings to and holding in a compressed magneto-optical trap. 5. Confinement of atoms in optical molasses while experimental (magnetic) fields stabilize.

6. Preparation of atoms into the magnetically insensitive |F = 2, mF = 0i ground state by optically pumping oﬀ the D1-line. CHAPTER 4. EXPERIMENTAL METHODS 57

Experimental Sequence and Settings

cMOT Raman image image dwell MOT load ramp cMOT hold OM hold pump repump pulses atoms background 10 ms 0.8 s 50 ms 2 ms 20 μs 5 ms 70 μs 160 μs 150 ms 150 ms

BMOT

3D ftrap

3D frepump

analog control analog 3D Ptrap

3D Prepump

Bbias

2D shutter OPEN pusher shutter OPEN 3D trap AOM OPEN 3D repump AOM OPEN 3D TA switch OPEN Raman pulse gen. OPEN

digital control digital τ opt. pumping AOM OP

τ τ imaging AOMs img img

Figure 4.1: The time sequence and analog and digital settings for an experimental run are shown here with respective time durations (not to scale). The majority of experimental time is devoted to obtaining and preparing the atom source.

7. Interferometry with counter-propagating two-photon Raman transitions pulsed on in π π π π a 2 − 2 − 2 − 2 sequence. Momentum reversal between pulse pairs is optional. 8. Absorption imaging of the |F = 1i hyperfine ground state. Lithium’s low vapor pressure at room temperature requires that the atoms be heated in an oven in order to realize an atomic beam. A first stage of cooling in a two-dimensional magneto-optical trap catches atoms from the effusive oven and pushes them down a differential pumping stage into a three-dimensional trap. Once trapped, the atoms are optically pumped into the magnetically insensitive mF = 0 sublevel of the F = 2 ground hyperfine state. Then, interferometry follows, during which we flash on four π/2-pulses, either with or without a reversal of momentum between the two pairs. An image of the atoms is taken of those atoms in the |F = 1i ground state following this sequence. A Wollaston prism placed immediately before our camera allows for two images to be made during a single exposure of CHAPTER 4. EXPERIMENTAL METHODS 58 the camera. From these absorption images, a picture of the cloud is discerned and the normalized atom number, spatial extent, position and density of the cloud can be determined. Interference fringes can be traced out in time by varying a parameter, like the duration of a free evolution time-step or frequency difference of the Raman laser, and plotting the atom population.

4.1 Lithium Spectroscopy

The experiment begins with a homemade vapor cell of lithium. This 60-cm long pipe is capped with two AR-coated windows, wound in heating tape and thermal insulation, and filled with inert argon gas to a pressure of 200 mTorr. Inside, is a chunk of lithium has been placed, which when heated to 350◦ C, performs the crucial function of the frequency reference in the experiment. The frequency of a first ‘master’ external cavity diode laser (ECDL) is referenced to this spectroscopy source. All other lasers in the experiment are referenced to the master ECDL either directly through injection locking or indirectly by injection or frequency-offset locking to a directly referenced laser. The lithium’s low vapor pressure requires atoms to be heated considerably above room temperature, to approximately 350◦C, in order to create an appropriate atom vapor for the spectroscopy reference. At such a high temperature, atom speeds can be as fast as a few kilometers per second, resulting in spectral line broadening of a transition by as much as a few GHz. The comparatively small hyperfine splitting of the ground state at just under 1 GHz( 0.803 GHz) necessitates that Doppler-free spectroscopy be employed. Doppler-free spectroscopic methods interact only with a single velocity class in the sample and hence, are immune to the unavoidable broadening of the hot sample. Many Doppler-free spectroscopic techniques utilize methods based upon the saturation of absorption, a two beam pump-probe scheme. In saturated absorption spectroscopy, the population difference between an upper (N2) and lower (N1) energy level, connected by an optical transition, is modified a strong ‘pump’ laser beam. For beams with frequency ω and counter-propagating through the vapor cell, atoms with velocity v = (ω−ω0)/k that Doppler shifts them into resonance with the pump light will thus be excited to the upper level of the reference transition. This reduces the population difference of the lower energy, ground state effectively ‘burning a hole’ in the level. This ‘hole burning’ or reduction in population, caused by a beam with intensity I, has a line width given by

I 1/2 ∆ωhole = Γ 1 + (4.1) Isat where Γ and Isat are the line width and saturation intensity of the transition. Close to resonance ω ω0 the pump reduces the absorption of a probe beam and thus results in a narrow peak of the transmitted probe intensity. In this experiment, the ‘master’ ECDL if frequency referenced to the cross-over (crossover resonance) between the transitions to the 2P3/2 excited level from the ground hyperﬁne CHAPTER 4. EXPERIMENTAL METHODS 59

states, |2S1/2,F = 1i and |2S1/2,F = 2i. Cross-over resonances occur when the pump beam burns two holes in the velocity distribution, producing two peaks in the spectrum for which the laser frequency corresponds to the two transition frequencies. An additional peak appears if the hole burnt by one transition reduces the absorption for the other transition.

4.1.1 Modulation Transfer Spectroscopy Modulation transfer spectroscopy (MTS) is a Doppler-free spectroscopic method that achieves sub-Doppler resolution, a requirement for laser frequency-locking. Like in standard saturation absorption or hyperfine pumping spectroscopy, MTS employs two counter-propagating pump and probe laser beams, but with powers set to be approximately equal. While MTS is able to obtain high sub-Doppler resolution, it does so at the expense of a limited capture range (< 100 MHz) or the system’s tolerable frequency excursion.Single beam techniques, while hindered by the Doppler-broadening of features, are however able to tolerate a much larger excursion, often 100s of MHz. Two clear advantages do set MTS apart from other frequency locking methods. First, MTS generates dispersive-like line shapes atop a zero background, removing the extra complex and costly demodulation step required in frequency modulation spectroscopy (FM). The background free signal also means that the zero-crossings of the MTS signals are accurately centered on the corresponding atomic transitions and the error signal is immune to locking frequency drifts potentially caused by reference level fluctuations [119]. Secondly, the signals are dominated by contributions from closed atomic transitions, especially beneficial when the atomic spectrum contains several closely spaced transitions. In MTS, an intense monochromatic pump beam with frequency ωc passes through an electric-optical modulator (EOM), driven at a frequency ωm. The transmitted light gets phase-modulated in this process [120],

E = E0 sin(ωct + δωmt) (4.2) and when written more explicitly in terms of the modulation index δ and the nth order Bessel function Jn(δ) is given by ∞ X n+1 E = E0 Jn(δ) sin(ωc + nωm)t + (−1) Jn+1(δ) sin(ωc − (n + 1)ωm)t . (4.3) n=0

The pump beam has a carrier wave (n = 0) at ωc flanked by first-order sidebands (n = ±1) at frequencies ωc ± ωm. The modulation index is typically less than one and thus the energy carried in sidebands at higher order is negligible compared to that of the carrier and first- orders and will not be considered here. The probe beam does not transverse the EOM but passes through the lithium vapor cell directly. The phase-modulated pump beam, after it passes through the EOM as discussed, is aligned co-linearly to the probe beam. If the sub-Doppler resonance condition is satisfied, then sufficiently nonlinear interactions between these two beams result in a four-wave CHAPTER 4. EXPERIMENTAL METHODS 60 mixing, transferring modulation to the probe beam. Four-wave mixing occurs when the non- linearity or third order susceptibility (χ(3)) of the hot lithium gas (the absorber) mediates a combination of the two frequency components of the pump with the counter propagating probe. This mixing generates a fourth wave, a sideband on the probe beam, and occurs for each sideband of the pump beam. Because this transfer of modulation occurs only if the sub-Doppler resonance condition is satisfied, the error signal generated by MTS is stable against fluctuations in polarization, temperature and beam intensity. Additionally, the position of the zero-crossing remains on the center of resonance, unaffected by many systematics that plague other techniques like magnetic field or polarization shifts. A photodetector detects the probe beam after it passes through the vapor cell. The sidebands produced in the atomic vapor beat with the probe carrier frequency and produce alternating signals at the modulation frequency ωm. The beat signal is given by

∞ C X S(ωm) = Jn(δ)Jn−1(δ) L(n+1)/2 + L(n−2)/2 cos(ωmt + φ) (4.4) p 2 2 Γ + ωm n=−∞

×(D(n+1)/2 + D(n−2)/2) sin(ωmt + φ)] (4.5)

with Γ2 Ln = 2 2 (4.6) Γ + (∆ − nωm) and Γ(∆ − nωm) Dn = 2 2 (4.7) Γ + (∆ − nωm) and where Γ is the natural line width, ∆ is the detuning from the line’s center, φ is the detector’s phase with respect to the applied modulation ﬁeld and C is a constant representing all other properties of the atom medium and probe beam. For a strong carrier wave, the above equation simpliﬁes to

C S(ωm) = J0(δ)J1(δ) (L−1 − L−1/2 + L1/2 − L1) cos(ωmt + φ) p 2 2 Γ + ωm ×(D1 − D1/2 − D−1/2 + D−1) sin(ωmt + φ) . (4.8)

As shown in Figure , light out of the ‘master’ ECDL is split with a series of half-wave plates and polarizing beam splitting cubes (PBS). The first PBS reflect a majority of the laser power to a mirror and second PBS which then send the beam through an acoustic- optical modulator. The beam is doubled passed and shifted down in frequency by 400 MHz. The beam passes through a quarter wave plate both exiting the AOM and again after being reflected by the retro-mirror. This results in a λ/2 rotation in total such that when again encountering the PBS at the beginning of the optical path, the beam in now transmitted through. It is then coupled into a fiber and sent to the spectroscopy board such that the CHAPTER 4. EXPERIMENTAL METHODS 61

lithium vapor cell

hwp

PBS

EOM

PBS

hwp

PBS

pinhole

qwp

200 MHz AOM

2D injection lock

pinhole

3D injection lock

hwp hwp

master ECDL

hwp hwp PBS

PBS PBS prism pair

Figure 4.2: Modulation transfer spectroscopy set-up. Light originates from a commercial Toptica external cavity diode laser and is sent through a series of half-wave plates and beam splitters. Light split oﬀ here is used to injection lock the FP diode laser further downstream in the optical path. The majority of the beam is doubled passed through an 200 MHz acoustic optical modulator. This frequency shifted light is ﬁber-coupled and sent to the spectroscopy set-up, as shown. Locking the down-shifted light to the 2S,F = 2 to 2P3/2 state results in locking this master laser to the crossover between the ground states. CHAPTER 4. EXPERIMENTAL METHODS 62

MON

470pF 20k 10k IN 10k

OP27 1M 200k 10k 1k 100 OP27 OP27 OP27 20k

10k

To Current

30k

100k OP27

1k 470pF 10k 1M 20k 1k 100k OP27

10k

OP27 10k 10k To PZT OP27

-12V +12V 10k

OP27

Figure 4.3: Circuit schematic of master ECDL frequency lockbox. resulting pump and probe are approximately equal in power; 0.5 mW is realized in each beam with waists of 1.4 mm. The set-up is similar to that of Ref. [121]. The pump beam is first steered through the electric-optical modulator, modulated at a frequency of 13 MHz, before passing through the lithium vapor cell. After the EOM, the pump beam consists of a carrier wave flanked by first-order sidebands at the modulation frequency. The initially unmodulated probe beam is sent directly into the vapor cell, with the beams aligned so as to overlap centrally in the cell but remain spatially separated at the mirrors immediately after the windows. After the vapor cell, the probe passes through another PBS and is detected via a photodiode as shown. A homemade feedback circuit is connected to the current and piezo of the master ECDL. A Schmitt trigger inside the circuit generates an internal ramp, sweeping the diode current and thereby scanning the laser frequency. There are PI feedback loops for the diode current and piezo which function to fix the laser frequency to the desired locking point. From the error signal generated, we lock the frequency-shifted light to the |F = 2i to D2-line which due to the initial down shifting of its frequency by approximately 400 MHz translates into ultimately locking the master ECDL to the cross-over between |F = 2i and |F = 1i to 2P3/2 state. The four parameters responsible for realizing the correct, mode-hop free wavelength in the master ECDL are: temperature, current, grating angle, and piezoelectric actuator voltage.

4.1.2 The cascade of frequency generation The small proportion of light picked off from the master ECDL injection locks two semi- conductor Fabry-Perot (FP) laser diodes. These lasers essentially are intermediate optical CHAPTER 4. EXPERIMENTAL METHODS 63 amplifiers of the well-defined, but weak, reference light from the initial ECDL. Light produced by these FP lasers is frequency shifted further to generate the desired frequencies for trapping, repumping, pushing and imaging in both the 2D and 3D cooling set-ups. The light from the master ECDL is aligned to the FP laser diodes so as to ‘seed’ or be injected into the diode’s cavity. If the injected photons dominate over the other frequency modes in the resonator, the seed beam will be amplified resulting in the FP laser lases at the same frequency as the reference light. The maximum power of commercially available FP laser diodes varies for different wavelengths and unfortunately, both lithium’s D1- and D2-lines require light for which commercially available diodes only are rated to output maximum powers of 20 mW. Therefore, heated 660 nm FP diodes are used in the experiment. Heating these commercially available diodes to 60-70◦C allows us to achieve the wavelengths (671 nm) required for lithium. In order to realized thermal stability at such high temperatures, two layers of boxes surround the FP laser, both designed to have separable walls in order to facilitate easier access to the diode. The laser diode is mounted on the typical f = 4.5 mm collimation tube and underneath the tube holder a piece of indium foil is placed for thermal conduction and then a thermoelectric cooler (TEC) for heating. Indium foil is used in place of thermal paste which has been observed to evaporate at high temperatures and condense on the view port of the housing. This assembly is enclosed with a black Delrin box with an AR-coated window on the front face for the laser beam output and a laser diode protection circuit on the back wall. This enclosure is then enclosed in an aluminum box, also made of separable faces. As reported in Ref. [93], the double box design has demonstrated greater stability in comparison to a single layer enclosure. A thermistor is used in conjunction with the TEC to inform a commercial temperature controller (Wavelength electronics PTCxK-CH Series) and stabilize the temperature of the diode to the desired value.

4.1.2.1 Tapered amplifiers The FP laser diodes typically output less than 100 mW of power in a single Gaussian spatial mode when heated. After the maze of optics needed to shift the reference frequency for cooling and trapping, we realize only at most 10 mW split between repumping and cooling frequencies. This necessitates the use of a tapered amplifier chip to amplify the reduced laser power, to realize the appropriate intensities for employing optical molasses. Contrary to the FP diodes, a tapered amplifier is a broad area laser diode in front of a straight narrow waveguide acting as a modal filter at the back. The wide output facet of the TA, 100s of µm wide and a few µm thin, means that the beam output from the TA diverges faster in the vertical compared to the horizontal axis. To mitigate this and collimate the vertical axis of the beam, a short focal-length aspheric lens is mounted directly after the TA. A cylindrical lens is then placed further along in the optical path, and outside the TA housing, to collimate the beam horizontally. Unfortunately, the short focal lengths of the collimation lenses used here produce a resulting beam shape that is very sensitive to the placement (position and angle) of the aforementioned lenses. CHAPTER 4. EXPERIMENTAL METHODS 64

injected beam

entrance facet

gain region output facet

Tapered ampliﬁer

This precise lens placement after replacement of a TA chip (necessary after wear, tear and heating to a balmy temperature) requires a 3D-translation stage and constant monitoring of the output power while the glue or epoxy is drying. We have found that when replacing the TA chip, coupling the beam into an optical fiber after all of the collimation optics allows for not only power monitoring during gluing and drying but assesses the stability of the beam shape as well. Expansion or contraction of the epoxy used when gluing these optics should be considered in the replacement process. In our experiment, we seed tapered amplifiers with a range of powers, anywhere between 5 - 20 mW. Depending on the particular TA chip, we measure approximately 200 to 500 mW of amplified output power. When operating a TA at a high current, care is required to prevent the diode chip from becoming unseeded for extended periods of time. Doing so, can result in degradation and damage to the laser. Observed symptoms of degradation include decreased output power, shittier output beam shape, and fast oscillations or repeated spikes and dips in output power at particular current settings. Such fast spikes average to a lower power when observed with a less-responsive commercial power meter compared to a fast photodiode.

4.2 Cooling and trapping

Following the oven, first a two- and then three-dimensional stage of cooling via a magneto- optical trap (MOT) are used to first catch and cool down the hot atoms. A MOT is a hybrid trap combining the position-dependent absorption cross section generated by a spatially- dependent magnetic field B = B(z) =≡ Az with a velocity-dependent radiation scattering force resulting from a red-detuned optical field. Optical molasses is the Doppler-cooling technique characterizing the process implemented via the optical field; orthogonal pairs of counter-propagating laser beams red-detuned from an atomic transition result in a force CHAPTER 4. EXPERIMENTAL METHODS 65

F= 0 F= 1

2 F= 2 18.1 MHz P3/2 F= 3 10.056 GHz 670.961 561 nm

F= 2

2 P1/2 91.8 MHz

F= 1 (a) (b) (c) (d) (e) (f) 670.976 658 nm

F= 2

2 803.5 MHz S1/2

F= 1

Figure 4.4: Experimental frequencies broadly referenced to the fine-structure splitting in lithium for (b) MOT cooling and (a) repumping, (c) optical pumping, (d) imaging, and Raman (e) ω1 and (f) ω2. CHAPTER 4. EXPERIMENTAL METHODS 66 damping the motion of the atoms [113, 122]. Introducing a spatially varying magnetic quadrupole field to the aforementioned optical molasses with a particular orientation of polarization (σ+/σ−), is all that is needed to make a MOT. Trapping happens here because as the atom moves away from the center of the trap and the magnetic field zero, the opposing beam will still be Doppler-shifted but now the energy levels will get Zeeman-shifted. For the simplified case of the two-level transition, such as |F = 0i → |F 0 = 1i shown in 0 Appendix D, as the atom moves rightward away from the trap center, the excited mF = −1 level is Zeeman-shifted into resonance. The optical molasses beam impinging on the atoms from the right is (conveniently!) of σ− polarization and thus the atom is allowed to and will 0 0 absorb a photon from this direction, transitioning to the |F = 1, mF = −1i excited state. It will also get a ‘kick’ of photon momentum, opposing its motion and directed back to the trap center. At low intensity, the total force on the atoms in terms of the ratio of the incident to saturation intensity I/I0, detuning from resonance ∆ and line width Γ becomes kΓ I/I F = ±~ sat (4.9) ± 2 2 1 + I/Isat + 2(∆±)/Γ

where the detuning for each beam, defined in terms of the effective magnetic moment µ0 ≡ (geme − ggmg)µB is given by µ0B ∆± = ∆ ∓ k · v ± . (4.10) ~ Depending upon the branching ratio of the excited state, an atom may absorb a photon but then decay to an energy level other than the ground state of the atomic transition. If this occurs, the atom will no longer be addressed by the optical molasses. The full scattering potential will not be realized by the atom; it will be cooled only to the point at which its leaves the two-level system. This hurdle to cooling can be overcome by incorporating additional frequencies of light to address atoms which have fallen into these other energy levels, not of the cooling transition. This ‘repumping’ light re-excites atoms back to the upper energy level and in doing so, in repumping them, the undesired decay is undone. The atom now has another chance to behave and decay appropriately and thus maintain it participation in cooling. Repumping is crucial to laser-cooling lithium. Unlike its sister alkali’s, rubidium and cesium, which both enjoy a spectrally well-resolved cycling transition on the D2 line, the hyperfine splittings of the 2P3/2 state in lithium are comparable to its natural line width of Γ/2π = 5.8 MHz. Even with an inverted level structure, this small excited state splitting 0 results in off-resonant excitations of the |2S1/2,F = 2i → |2P3/2,F = 2i transition leading to subsequent (and frequent) decay down into the |2S1/2,F = 1i ground state, almost 1 GHz below that addressed by the laser-cooling light. The lack of a closed cooling transition for lithium requires that an appreciable amount of repumping light be included in the MOTs, compensating for the atom leakage to the dark state manifold. CHAPTER 4. EXPERIMENTAL METHODS 67

4.2.1 2D MOT frequency generation Half-wave plates and PBS’s, shown in Fig. 4.2, divide light originating from the master ECDL amongst the spectroscopy set-up and FP laser diodes for eventual 2D and 3D MOT frequency generation. At the second PBS, the reflected beam is fiber coupled after which it is then steered backwards through the rejected port of an optical isolator (after the appropriate polarization alignment) so as to injection lock the FP laser used in the generation of the 2D MOT cooling and repumping frequencies. This optical scheme is shown in Fig. 4.5. Prior to passing through the AOMs, a portion of the laser’s light is picked off for frequency referencing another FP laser diode and second homemade ECDL used in the optical pumping scheme. The remainder of power is split, each portion sent to a 200 MHz acoustic-optical modulator. Light used for cooling (repumping) is frequency-shifted approximately -(+) 400 MHz by double-passing the AOMs in a ‘cat’s eye’ configuration, necessary for robust alignment that remains independent of the beam diffraction angle and actual frequency shift or sweep imposed by the modulator [123]. The detuning used for an experimental sequence are given in Table 4.1. The AOMs are driven by amplified voltage controlled oscillators, controlled with the Cicero and Atticus duo. After double-passing the AOMs, the outgoing light is orthogonally polarized relative to its initial state and is picked out of the incoming beam path at the PBS adjacent to the mirrors steering light through the AOMs. The beams, which are also orthogonally polarized relative to the other, are overlapped at another PBS, sent through a half-wave plate and final PBS such that both frequencies are of the same polarization upon entering the optical fiber seeding the 2D tapered amplifier.

4.2.2 3D MOT frequency generation Returning once more the the master ECDL, the remaining power transmitted through the second PBS is sent in free space and back coupled through the rejected port of another optical isolator (after aligning its polarization appropriately) to injection lock the FP laser used in the frequency generation of the 3D MOT cooling and trapping stage, as well as the pusher beam and the light for absorption imaging. Light from this FP laser passes through an isolator and telescope prior to being split at a PBS with a majority fiber-coupled and sent to the optics used in generating the frequencies for the pusher beam and 3D MOT optical molasses. The set-up is shown in Fig. 4.6. The remaining power is reflected at the PBS and coupled into a fiber sent to the imaging AOMs. The details of the imaging optics will be described in more detail later in the chapter. Fig. 4.6 shows the optical set-up used for generating the cooling and repumping 3D MOT frequencies as well as the pusher and imaging beams. As in the case of the 2D MOT, the total power (now out of a fiber) is split by a series of half-wave plates and PBSs and double-passed in the ‘cat’s eye’ geometry. Contrary to what was discussed in the context of the 2D MOT, this is done through a series of three 200 MHz AOMs, shifting the light either up (repumping) or down (cooling) by approximately 400 MHz (see Table 4.1 for detuning details). Power balances can be tuned by adjusting the half-wave plate prior to the PBSs feeding the AOMs. CHAPTER 4. EXPERIMENTAL METHODS 68

PBS PBS

pinhole qwp 150 mm hwp

200 MHz AOM 150 mm PBS hwp PBS hwp

qwp pinhole hwp to 2D TA

100 mm PBS 200 MHz AOM 100 mm

hwp hwp PBS

hwp from hwp Master ECDL hwp

to Raman to OP ECDL diode laser

Figure 4.5: The set-up for the frequency generation for the 2D MOT optical molasses is shown. The frequencies for cooling and repumping light are generated with two double- passed 200 MHz AOMs, shifting either up or down, following initial optics separating a portion of the light to serve as a frequency reference for both interferometry and optical pumping. The frequency-shifted light exiting the double-pass is orthogonally-polarized with respect to the incoming light and the other beam. The beams are spatially overlapped at a PBS and ﬁber-coupled to a tapered ampliﬁer which further increase the light intensity for cooling and trapping in the 2D MOT.

After transversing these optics, the light from the repumping and cooling paths is combined by overlapping the orthogonally polarized beams at a PBS and fiber-coupled, sent to seed another tapered amplifier. After amplification in the TA, the bichromatic light is again fiber-coupled and sent to a homemade 1:6 fiber-splitter, eventually becoming the six arms of the 3D MOT.

4.2.3 Vacuum system and optics This section describes in detail the experiment’s vacuum system set-up as well as the optics used for the 2D and 3D MOT cooling and trapping stages, including the set-ups for beam intensity amplification. The 2D MOT is based off a previous design [124], also with lithium. This is contrary to other cold atoms experiments with lithium employing instead a Zeeman slower as an intermediate between the hot atom source and the three-dimensional trap [125– 128]. Advantages of a 2D MOT over a Zeeman slower include: compactness, large flux, CHAPTER 4. EXPERIMENTAL METHODS 69

to imaging

hwp

hwp PBS

hwp hwp

from hwp PBS Master ECDL 75 mm -25 mm

to 2D MOT

hwp

qwp pinhole to 3D TA

hwp PBS PBS hwp 100 mm 200 MHz AOM 100 mm

hwp PBS

pinhole qwp 150 mm

200 MHz AOM 150 mm PBS

qwp pinhole

150 mm 200 MHz AOM 150 mm

Figure 4.6: The set-up for the frequency generation for the 3D MOT optical molasses and pusher beam is shown. These frequencies for cooling, repumping and pushing are generated with three double-passed 200 MHz AOMs, shifting either up or down as indicated. The frequency-shifted cooling and repumping light exiting the double-pass is orthogonally- polarized with respect to the incoming light and the other beam. The beams are spatially overlapped at a PBS and fiber-coupled to a tapered amplifier which further increase the light intensity for cooling and trapping in the 3D MOT. The pushing light is transmitted through the last PBS, sent to a mirror and then periscope to be aligned through the 2D MOT chamber, along the symmetry axis of the magnetic field. CHAPTER 4. EXPERIMENTAL METHODS 70

Beam Transition Detuning [Γ] 2D MOT trap F = 2 → F 0 = 3 -1.89 2D MOT repump F = 1 → F 0 = 2 0.52 pusher F = 2 → F 0 = 3 0.16 3D MOT trap F = 2 → F 0 = 3 -3.87 3D MOT repump F = 1 → F 0 = 2 0.92 cMOT trap F = 2 → F 0 = 3 -1.88 cMOT repump F = 1 → F 0 = 2 -0.22 imaging F = 1 → F 0 = 2 -3.55 molasses trap F = 2 → F 0 = 3 -1.88 molasses repump F = 2 → F 0 = 2 -0.22

Table 4.1: Frequencies detunings in units of line width Γ for both the 2D and 3D MOT, compressed MOT, molasses hold and imaging. The hyperﬁne states for the transition |2S1/2i → |2P3/2i are indicated. and the absence of additional residual magnetic ﬁelds present which may perturb the atoms (quantization axis) during state selection or interferometry.

4.2.3.1 2D MOT chamber A cylindrical lithium oven (51 cm3), wrapped in heating tape, is heated to just shy of 400◦C, is the starting point for the high flux, cold atom source. The oven is made primarily from stainless steel (304) except at the flange where a nickel gasket is used in conjunction with a variation of stainless steel more robust to corrosion (316LN), both a consequence of lithium’s reactivity. It 6 cm hangs below the two-dimensional MOT vacuum apparatus, a stainless steel (304) six-way cross. Four ConFlat flanges supporting AR-coated view ports are configured at 45◦ while two more (one with a view port for the pusher beam) are aligned to the symmetry axis of a 2D quadrupole magnetic field. The magnetic field is created by two sets of permanent bar magnets (Nd2Fe14B) with gradient of 50 G/cm. This region of the experimental apparatus is separated from an ultra-high vacuum portion by a differential pumping tube with inner diameter 4.5 mm, emerging off the skewed ‘X’ of 2D MOT-chamber and extending 6 inches towards the larger side of the vacuum chamber, ending at a gate valve just prior to the chamber. A differential pumping stage is required to restrict the conductance of particles from the hot atom-vapor thereby maintaining low pressures in the larger trapping region with pressures observed to typically differ by more than 500:1 [93]. A flange cross also extends outward, perpendicular to the differential pumping stage, and is connected to an ion gauge, metal-valve and turbo pump, allowing for the 2D MOT chamber to be efficiently pumped while baking. Two pairs of retro-reflected laser beams with σ+/σ− polarization and red-detuned to lithium’s 2P3/2 excited state are steered into the chamber as shown in Fig. 4.7. They cross orthogonally above the neck of the oven, capturing atoms in a column along the symmetry CHAPTER 4. EXPERIMENTAL METHODS 71

bar magnet

qwp qwp retro-mirror retro-mirror oven

Figure 4.7: The portion of the vacuum chamber that is the 2D MOT is shown here as well as the path of the beams into the chamber. A cylindrical lithium oven hangs below the six-way cross, which is capped with AR-coated view ports held by 2.75 inch stainless steel Conflat flanges. A 50 G/cm magnetic field gradient is generated with sets of permanent magnets (Nd2Fe14B); quadrupole field lines shown in drawing. The beams are steered to mirrors placed above the upper view ports which direct the light through the chamber, through a quarter-wave place and to a mirror. This mirror then retro-reflects the light, back through the chamber, now of opposite circular polarization given its double pass through the quarter-wave plate. CHAPTER 4. EXPERIMENTAL METHODS 72 axis. The 2D MOT beams originate from a commercial tapered amplifier (TOptica BoosTA), which when seeded with 2.2 mW of cooling and 5.0 mW of repumping light produces approximately 200 mW of total bichromatic laser power as measured prior to the shutter shown in Fig. 4.7. Following the shutter, a telescope expands the beam to a 1/e2 5-mm sized waist. A mirror the directs the beam to a 50/50 beam splitter cube which reflects the light upwards, to a mirror placed above the closest view port, and transmits under the oven and 2D MOT chamber to another mirror which also directs the light to an upper mirror positioned above the second upper view port. The two beams are steered into the chamber and through the lower opposite view port on the other side of the symmetry axis. After exiting the chamber, the light passes through a quarter-wave plate, hits a mirror and is retro-reflected back through the chamber. Optimum trapping defined by the number of atoms captured in the 3D MOT either with real-time fluorescence detection or absorption imaging occurs when the retro-reflected beam is not overlapped with the incoming beam. Atoms are trapped and cooled radially along the symmetry axis of the 2D quadrupole field. A pusher beam with approximately 1 mW of power in a waist of 1.2 mm and red- detuned from the 2P3/2 level kicks atoms down the differential pumping tube into 3D MOT chamber. It is aligned so as to more effectively load atoms but also not disturb the real time trapping in the 3D chamber.

4.2.3.2 3D MOT chamber The three-dimensional magneto-optical trap (3D MOT) is a spherical steel octagon (r = 8 inches), capped by two large (r = 8 inch) view ports with six smaller AR-coated ones for beam access positioned as shown in Fig. 4.9. This part of the vacuum chamber is connected to the differential pumping stage through a gate valve, open during normal operation but helpful in modular baking of the chamber. An ion pump (Varian StarCell; 40 L/s), ion gauge, and titanium sublimation pump (TSP) are connected to the chamber on the side opposite the 2D MOT chamber via flange crosses, as shown. For 30 A of current, hollow rectangular magnetic coils wound 64 times per side in an anti-Helmholtz configuration generate a 20 G/cm magnetic field gradient at the center of the chamber. Approximately 1.2 mW of cooling and 2 mW of repumping light seed a commercial tapered amplifier (TOptical TApro), coupled from the optical scheme shown in Fig. 4.6. The amplified output at approximately 150 mW, passes through an optical isolator, telescope for beam shaping and PBS which sends a small fraction of the light to a cavity for monitoring. The remainder of the beam is coupled into an optical fiber and sent into a homemade fiber splitter. The 1:6 fiber splitter distributes the total power evenly amongst six optical fibers. The light out of these fibers, 10 mW of power in each, is telescoped to achieve a 6.5-mm beam waist for each MOT arm and then directed to the atoms. CHAPTER 4. EXPERIMENTAL METHODS 73

50 mm

-25 mm 2D FP laser hwp 2D MOT S to N arm

optical isolator 2D MOT hwp BS 2D TA N to S arm 100 mm

shutter 500 mm

Figure 4.8: The optical scheme intended to amplify the intensity of the 2D MOT cooling and repumping frequencies. Light fiber-coupled after being frequency shifted in the AOM maze of Fig. 4.5 seeds a tapered amplifier as shown. Its steered through a telescope, optical isolator and shutter used to control 3D MOT loading. The light passes through another telescope that expands it to a 5 mm Gaussian beam waist. It is directed to a 50/50 beam splitter which reflects one portion of the beam upwards and transmits the second portion under the oven to a mirror which then directs it upwards.

4.2.4 Experimental sequence Once atoms are loaded into the 3D MOT, they are further cooled in a compressed magneto- optical trap (cMOT) stage. This additional step is necessary because the experimental settings optimized for loading the maximum number of atoms are not optimal for cooling the collective; both the number density and temperature dependent independently on several parameters such as the magnetic field gradient, light intensity, detunings, and number of atoms [129]. For a single atom, a change in the radiation forces arises from the presence of other atoms which create a background field of scattered photons [130]. Therefore, in order to obtain lower temperatures while capturing as many atoms as possible for interferometry, the 3D MOT cooling and repumping light are ramped closer to resonance with the 2P3/2 state over a 5-ms duration. This is accomplished by changing the frequencies of the AOMs through the control voltages sent to the driving VCOs. The intensity in each frequency is simultaneously reduced by ramping up the attenuation of the VCO output and the magnetic field is lowered. After ramping the frequencies, attenuations and magnetic field, a ‘cMOT hold’ time-step of 500 ms is implemented. Time-of-flight images taken immediately following loading of the 3D MOT indicate that the atoms are at CHAPTER 4. EXPERIMENTAL METHODS 74

TSP

2D MOT ion pump pusher

permanet magnet

oven

3D MOT

Figure 4.9: The 3D MOT chamber as part of the larger vacuum apparatus which consists additionally of a lithium oven, 2D MOT chamber, differential pumping stage, an ion pump and TSP pump. The mirror and 50/50 BS of the 2D MOT optics, the periscope for the pusher beam and the steering mirrors for one of the six 3D MOT arms are shown. temperatures of 1 mK whereas after compression, taking a temperature has shown the atoms to be at approximately 200-300 µK, almost at the Doppler cooling limit. A ‘molasses hold’ time-step follows the compressed MOT, during which the 3D MOT magnetic field is completely shut off while the bias magnetic fields are maintained on. The purpose of this step is to create a delay between the slow decay of the eddy currents induced in the vacuum chamber and the beginning of optical pumping, for which a defined quantization axis needs to be set. An unexpected consequence is a launch of the atoms in the z direction, resulting from both the presence of the laser light and bias magnetic fields. It has yet to be determined whether this due an imbalance of intensity resulting from alignment.

4.3 State preparation

Figure 4.10 shows a microwave spectrum taken of the atoms with and without optical pumping. Following the loading and cooling of atoms in the 3D MOT and the optical molasses hold, atoms can be found distributed among magnetic sublevels in the |F = 2i ground state. Scanning a microwave pulse through the a frequency range centered on the |F = 2, mF = 0i → |F = 1, mF = 0i transition around 803.5 MHz and imaging the F = 1 population allows us to determine the approximate distribution of atoms. It conﬁrms that following the molasses timestep, atoms are distributed among the magnetic sublevels of the CHAPTER 4. EXPERIMENTAL METHODS 75

MW spectrum with and without optical pumping

0.5

with pumping 0.4 without pumping

0.3

0.2

Normalized population Normalized 0.1

800 801 802 803 804 805 806 807 Frequency [MHz]

Figure 4.10: Microwave spectrum of the |F = 2, mF i ground state.

|F = 2i hyperfine ground state. Preparing the atoms into the mF = 0 magnetic sublevel is crucial. Eddy currents in the steel vacuum chamber persist for 10s of milliseconds even after the magnetic field coils have been shut off. If the interferometer phase has a dependence on the difference in internal energies, as ours does when employing the Ramsey-Bordégeometry, then the gradient produced by these Eddy currents will Zeeman shift the magnetic sublevels by varying amount, dephasing our interferometers, and reducing contrast. Preparing the atoms into a particular magnetic sublevel prior to interferometry can mitigate this effect. Particularly, optically pumping the atoms into the mF = 0 magnetic sub-level which is insensitive to the first order Zeeman shift, allows us to overcome the limitations of these persistent Eddy currents. We still observe an effect from the quadratic Zeeman shift. After the optical molasses step lasting on the order of a few -ms, during which the quadrupole decays, the atoms are found distributed among all five magnetic sublevels of the |F = 2i ground-state manifold. After 1.5 ms of optical molasses, when the magnetic field persists from the quadrupole field eddy currents has decayed to below 1 G/cm. CHAPTER 4. EXPERIMENTAL METHODS 76

Magnetic Field Gradient Decay 3.0

2.5

2.0

1.5

1.0 Gradient [G/cm] Gradient

0.5

0 500 1000 1500 2000

Time [µs]

Figure 4.11: The decay of the magnetic ﬁeld gradient over a 2 ms duration shows that the gradient from eddy currents induced by the quadrupole ﬁeld persists well into the experimental stages following the compressed MOT.

We optically pump the atoms with linearly polarized light from the ground state F = 2 to excited 2P1/2 level. We optically pump the atoms on the D1–line because the D2–line in 0 lithium is unresolved and consequently more lossy. Selection rules forbid mF = 0 → mF = 0 0 for a F → F = F transition making the mF = 0 ground sublevel dark to the optical pumping light. 0 ± However, excited mF magnetic sublevels can decay into the mF = 0 ground state by σ transitions. The absence of an excitation path out of the |F = 2, 0i state means that atoms will build up in this magnetically insensitive ground state since atoms that decay into it will be uncoupled from the excited level. In each of the six 3D MOT beams, we use 1.5 mW of 0 D2 MOT repumping light, detuned from |F = 1i → F = 2i of the 2P3/2 state, to recover atoms that decay to F = 1, the lower hyperﬁne ground state of lithium.

4.3.1 Frequency generation for optical pumping light

The frequency of the reference ECDL is resonant with the crossover to the 2P3/2 state in lithium. In order to generate the frequency need for optical pumping on the D1 line, an external cavity diode laser is frequency offset locked to the 2D MOT FP diode laser prior to any frequency shifting for trapping or repumping light, so resonant essentially with the crossover. The offset lock needs to be at approximately 10 GHz, the fine-structure splitting CHAPTER 4. EXPERIMENTAL METHODS 77

2P 3/2 -3 +3

2π × 10 GHz

2P F= 2 1/2 2π × 92 MHz

F= 1

F= 2 2S 1/2 2π ×803.5 MHz

F= 1

-2 -1 0 +1 +2 m F

Figure 4.12: Energy level diagram showing the frequencies for optical pumping.

LO V bias delay line spectrum analyzer

servo Figure 4.13: Light from the lasers’ beat note is incident on a fast photodetector and mixed with a local oscillator at 9894 MHz. The signal is filtered and amplified in various stages, finally power split prior to a frequency mixer. One part of the signal goes directly into the mixer and the other transverses a delay line first, set so as to induce a phase shift and consequently error signal for the servo. CHAPTER 4. EXPERIMENTAL METHODS 78 in lithium. The optical pumping laser is beat against the reference laser and sent to a fast photodetector (GaAs MSM Photodetector G4176) as shown in note is subsequently mixed with a local oscillator (Vaunix LabBrick LMS) at 9894 MHz and 10 dB of power. A schematic of the locking electronics is shown in Fig. 4.13. A 600 MHz delay-line (trombone lock) generates the error signal sent to the locking electronics that subsequently stabilizes the laser current and piezo voltage.

4.3.2 Optical pumping optics This light directly out of the optical pumping ECDL is coupled into a ﬁber and 3 mW of light tuned to within a line width (Γ = 2π × 5.87 MHz) of the |F = 2i to |F 0 = 2i transition on the well-resolved D1 line (2P1/2 state) is sent to the 3D MOT chamber. The light exiting the ﬁber passes through an AOM which acts as a switch, through a telescope expanding the beam to a 3.6 -mm Gaussian waist, periscope, polarizing beam splitter cube and waveplate. The beam is polarized alongz ˆ realize a π-polarization relative to the set quantization axis, required in our pumping scheme. After 50 s of optical pumping, greater than 80% of the atoms occupy the dark state.

4.3.2.1 Quantization axis The quantization axis is defined by three pairs of bias magnetic field coils which frame the anti-Helmholtz coils of our trap, generating a zero field to within 10 mG alongx ˆ andy ˆ and approximately 1 G along thez ˆ. The quadrupole field remains appreciable for milliseconds due to eddy currents in the steel vacuum chamber, despite the relatively quick 250 µs decay of the current in the MOT coils after the current is shut off. To limit the thermal expansion of the atom cloud while the eddy currents decay prior to interferometry, the 3D MOT beams are left on as optical molasses. Unfortunately, the small detuning of these beams from the unresolved excited 2P3/2 state in lithium thwarts polarization gradient cooling during this step.

4.4 Interferometry

Two-photon Raman transitions comprise the atom optics utilized here for interferometry with laser-cooled lithium. Fig. 4.15 depicts the frequencies we are using for interferometry, including the single- and two-photon detunings, ∆ and δ, respectively.

4.4.1 Frequency generation for Raman beams

Light resonant with the crossover to excited 2P3/2 state injection, picked oﬀ from the 2D FP laser diode, is sent to injection lock a third FP laser. This light is single passed through an acoustic-optical modulator which shifts the light down in frequency by an additional 200 CHAPTER 4. EXPERIMENTAL METHODS 79

quantizaton axis

π-polarization

PBS retro-mirror

pinhole 80 MHz AOM hwp

hwp

hwp reference

hwp beat note detector optical isolator BS PBS

hwp

h wp optical isolator

Figure 4.14: The optical pumping beam optics and frequency-offset locking scheme. An external cavity diode laser is beat against reference light, here light resonant with the crossover to which spectroscopy is referenced. The majority of the light is coupled into an optical fiber and sent to a fiber port placed near the 3D MOT portion of the vacuum chamber. The beam passes through an AOM, used to control the switching or duration of the pulse sent to the atoms, telescope, to a periscope and through a PBS which helps ensure that the polarization of the light is along the quantization axis (π), crucial for our optical pumping scheme. CHAPTER 4. EXPERIMENTAL METHODS 80

|e p + ħ k > 1 ∆

ω1 ω2

δ |b, p + ħ (k - k )> 1 2 ωba

|a, p >

Figure 4.15: The two-photon Raman transitions that are the optics for the atoms during interferometry consist of two frequencies ω1 and ω2 red-detuned from the excited state by ∆. The two-photon detuning is δ and quantiﬁes how oﬀ-resonance these two frequencies are from the red-detuned frequency value.

MHz (will become our single-photon detuning). Prior to passing through the AOM, a small fraction of this beam is picked off via a polarizing beam splitter cube and sent to the tune-out ECDL, eventually becoming the reference light for our second frequency offset. After being frequency shifted, this beam is fiber coupled and sent to a homemade tapered amplifier (EYP-TAP-0670-00500) for amplification to approximately 400 mW. We realize between 180 - 200 mW prior to an optical fiber at the end of this optical path, downstream from the TA optics. This light is then sent to the two 400 MHz AOMs depicted in Fig. 4.16. A serial controlled four-channel Direct Digital Synthesizer (DDS) sources the AOMs used to frequency shift the light, 200 MHz red-detuned to the crossover of lithium’s ground state. Output frequencies are set and controlled via Cicero and Atticus, the word generator and server respectively, which are responsible for executing the commands sent to various CHAPTER 4. EXPERIMENTAL METHODS 81

to FP cavity

hwp PBS

PBS hwp hwp PBS EOM to the wall

hwp 400 MHz AOM hwp h hwp 400 MHz AOM wp

Raman beam, k2

Raman beam, k1 Raman TA

Driving Electronics for 400 MHz Raman AOMs

200 MHz 11 dBm

13 dBm 300 MHz 2) f0

250 MHz 200 MHz 600 MHz

7 dBm f1

Figure 4.16: Optical set-up for generating the Raman frequencies. Light red-detuned from the cross-over is amplified in a homemade tapered-amplifier and sent to the above set-up. Two 400 MHz AOMs are utilized to achieve the 800 MHz splitting needed. The zero order exiting the first AOM is picked off and sent to the second as shown. The polarizations are orthogonal at the recombination PBS and through the EOM. The frequencies are split at the PBS following the EOM and sent to two optical fibers which will take the light to opposite sides of the vacuum chamber, creating the counter-propagating Raman beams for interferometry. CHAPTER 4. EXPERIMENTAL METHODS 82

hwp Raman beam, k1

bias coils

viewport 2D MOT atom cloud

MOT coil

PBS h Raman beam, k2 wp Top View

Figure 4.17: The Raman beams horizontally transverse the 3D MOT portion of the vacuum chamber after being expanded with 1:4 telescopes. The beams pass through a PBS and wave plate combination to clean up their polarization so as to be lin⊥lin. hardware (GPIB, digital, analog, RS232) during an experimental run.

4.4.2 Raman optics The optical scheme for the interferometry beams is shown in Fig. 4.17. The single-photon detuning of our Raman pair is ∆ = 2π × 210 MHz red-detuned from the D2 line. As discussed in the prior section, a pair of 400 MHz acoustic-optical modulators tunes the frequency difference of the two interferometry beams to be ωA − ωB + δ. At the 2 mm atom cloud, the pair coincides in beams of 2.1 mm Gaussian waists. A lin⊥lin configuration in implemented such that one beam is polarized alongx ˆ and the second beam polarized along yˆ. Since the two-photon Rabi frequency scales as 1/∆, one of the single-photon detuning, we achieve a high Rabi frequency ΩR ∼ 2π × 1.6 MHz or larger and thus require a short pulse duration in the experiment. For beam powers of approximately 30 mW of ω1 and 15 mW of ω2, we drive a π pulse in 320 ns with about 30% efficiency, addressing a considerable fraction of the atoms whose two-photon resonance conditions are Doppler broadened through the thermal velocity spread. CHAPTER 4. EXPERIMENTAL METHODS 83

The interferometry axis, zˆ, is almost completely perpendicular to the force of gravity, gyˆ. The beams comprising the counter-propagating Raman transitions horizontally transverse the vacuum apparatus as shown in Fig. 4.17. We have measured the eﬀect of gravity on an inertially sensitive Mach-Zehnder interferometer.

4.5 Detection

The atom number, temperature, and cloud density can be determined via absorption imaging the atom cloud at the end of an experimental cycle. During such imaging, the shadow cast by the cloud, generated by sending light resonant to the |F = 1i → 2P3/2 transition, is imaged onto a CCD camera (Pixelﬂy PCO). The number density of the atoms can be measured by the observing the absorption of a resonant probe beam with the input of the average scattering cross-section of photons by the atoms.

4.5.1 Absorption imaging

Consider a laser with intensity I0(x, z) propagating along the y-axis in the 3D MOT chamber such that it is aligned to intersect the atom clouds at the end of the experimental sequence. The intensity at a particular position in space, transmitted through the atom cloud, is given by the following expression −OD(x,z) I(x, z) = I0(x, z)e (4.11) where the optical depth proﬁle of the atomic sample, OD(x,z), is equivalent to the column density of the sample at position (x, z) multiplied by the absorption cross section of the transition σtot σ0 σtot = 2 2 . (4.12) 1 + 4∆ /Γ + I0/Isat Here, ∆ is the detuning from resonance, Γ is the natural line width of the transition, Isat is the saturation intensity, and σ0 is the resonant cross section given by 2 Γ~ω0 3λ σ0 = = (4.13) Isat 2π

where ω0 and λ are the resonant frequency and wavelength, respectively. Comparing the probe intensity both with and without the atoms present allows the transmittance or absorption to be determine and consequently the optical depth or the number density proﬁle can be computed, I (x, z) OD(x, z) = n(x, z) × σ = ln 0 . (4.14) I(x, z) Integrating over the optical depth produces the atom number N ZZ 1 ZZ N = n(x, z)dxdz = OD(x, z)dxdz. (4.15) σ0 CHAPTER 4. EXPERIMENTAL METHODS 84

4.5.2 Wollaston prism technique Normalizing our detection scheme is crucial for eliminating the observed fluctuations of the total atom number per experimental cycle. To do this, a novel technique was developed where we take two absorption images of the atom cloud during a single exposure of the CCD camera (PCO pixelfly). The two images are state selectively detected using two overlapped but orthogonally polarized and independently switched beams with 9-mm waists tuned to the D2 transition for |F = 1i atoms. The first image measures the population in the |F = 1i state and the second measures the total population in both the hyperfine ground states, |F = 1i and |F = 2i states. The imaging scheme is simple: after illuminating the atoms with either of the imaging beams, the light passes through a Wollaston prism positioned before the camera. Light incident on the Wollaston prism will be deflected afterwards based upon the polarization of the beam; orthogonal polarizations will be sent to opposite sides of the CCD array. Therefore, each imaging beam is coupled to a particular image generated, based on the designated polarization. During the first 90 s of the exposure, we illuminate the atoms with one beam and image only the population in the |F = 1i state. After a 10-s delay, we switch on the orthogonally polarized beam of the same frequency for 90 s while simultaneously turning on the 3D MOT cooling light. The cooling light depumps the atoms from |F = 2i to |F = 1i via the D2 line and therefore we detect the sum of their populations or all atoms present regardless of their internal energy state. Due to its deflection at the Wollaston prism, this second image forms on the other side of the CCD. Finally, we allow the atoms to disperse and take a second (background) exposure with the same pulse sequence to generate side-by-side absorption images of the entire cloud and the |F = 1i population. The ratio of the two absorption imaging signals gives PF = 1i, or the normalized population in the |F = 1i state.

4.5.3 Time-of-ﬂight imaging Releasing the atoms from optical molasses allows the cloud to ballistically expand. The atoms ﬂy away from each other since the sample has a (very) non-zero thermal velocity. Assuming the atoms are thermalized during cooling, the velocity of the atoms can be described by a Maxwell-Boltzmann distribution given by r M 2 f(v) = e−Mv /2kB T (4.16) 2πkBT

where kB is the Boltzmann constant and T is the temperature of the atom ensemble. The spatial distribution of atoms at a time after release, referred to as the ‘time of ﬂight’, is determined by considering the overlap of the atom’s initial density proﬁle ρ0(z) with the velocity distribution Z ∞ 0 0 0 f(z, v, t) = f(v)ρ0(z )δ(z − z − vt)dz (4.17) −∞ CHAPTER 4. EXPERIMENTAL METHODS 85

Wollaston prism PCO.pixelfly

PBS hwp

200 MHz AOM hwp

PBS

hwp

200 MHz AOM

qwp qwp from 3D FP diode laser

Figure 4.18: The beam path for the imaging light as it transverse the vacuum apparatus. CHAPTER 4. EXPERIMENTAL METHODS 86 where δ(z) is the Dirac delta function. Assuming an initial Gaussion density given by

2 2 −1/2 z /2σz ρ0 = N (2πσz ) e (4.18) simpliﬁes the integral to the following expression

r 2 2 0 N M −Mv (z − vt) f (z, v, t) = exp − 2 (4.19) 2πσ kBT 2kBT 2σ where N is a normalization factor. The atom number density is found by integrating with respect to velocity

Z ∞ N z2 ρ(z, t) = f 0(z, v, t)dv = exp − (4.20). p 2 2 2 2 −∞ 2π(σ + kBT t /M) 2(σ + kBT t /M) The distribution of the atom cloud’s density in two dimensions is

1 z2 + x2 ρ(x, z, t) = exp − (4.21) 2πσ2(t) 2σ2(t)

with q 2 2 σ(t) = σ0 + (kBT/M)t (4.22) Comparing images taken at variable durations of the time of ﬂight allows for the determination of the temperature of the atoms. Plotting the ﬁnal density with respect to time t, allows for the temperature to be determined

2 2 2 Mv (σf − σ0) T = 2 2 . (4.23) (tf − t0) 87

Chapter 5

Hot Beats

The scaling of the atom’s recoil frequency ωr with the inverse of its mass positions atoms, like lithium, as advantages candidates in recoil-sensitive interferometers, offering an automatic increase in sensitivity and measurement precision for the same interrogation time compared to a heavier atom. In addition to our measurement sensitivity benefiting from lithium’s high recoil frequency of ωr = 2π × 63 kHz, the lack of both additional time-consuming cooling and lossy velocity selection steps [131] boost our measurement sensitivity. However, further cooling beyond the standard Doppler temperature, obtained in a magneto- optical trap, offer advantages in detection at the interferometer outputs. In this chapter, the details of recoil-sensitive interferometry with a super-recoil sample of lithium is discussed including how the lack of spatial resolution in detection can be overcome.

5.1 Super-recoil lithium

While this is the first demonstration of atom interferometry with laser-cooled lithium, or any atom lighter than sodium-23, lithium has been used in supersonic atomic-beam interferometers. Cooling lithium below the recoil temperature Tr, where the average thermal speed equals the recoil velocity, is difficult due to its small excited state 2P3/2 hyperfine splitting, as discussed in Chapter 3. In lithium, the realization of Sisyphus cooling is contingent on that the light is detuned by an amount sufficiently greater than the hyperfine structure. This rather large detuning ensures that the fine structure of the atomic energy levels, rather than the hyperfine structure, dominate the atom’s response to the radiation. Achieving sub-Doppler temperatures directly in standard ‘D2’ optical molasses cooling is impossible. Without employing any additional sub-Doppler cooling techniques [94, 132], the atoms prior to interferometry are limited in temperature to the Doppler temperature of 140 µK set by the cooling transition to the 2P3/2 state. Previously, our experimental set-up did successfully implemented Sisyphus cooling, achieving 1D temperatures of 40 µ K [94]. Unfor- tunately, due to the limited availability of laser power, we cannot simultaneously implement both Sisyphus cooling and interferometry. Prior to interferometry, as measured with time- CHAPTER 5. HOT BEATS 88 of-flight imaging techniques, the atoms are at super-recoil temperatures of approximately 300 µK, fifty times the recoil limit 50Tr. After the 2D and 3D MOT trapping and cooling stages, 10 million lithium atoms are optically pumped into the magnetically insensitive sublevel, mF = 0, of the |F = 2i ground state which is required to mitigate dephasing effects arising from persistent Eddy currents in the vacuum chamber even after the quadrupole coils have long ceased. Interferometry follows; in a Ramsey-Bordéinterferometer, four sequential π/2 pulses split, redirect and π π π π recombine the atom wavefunction. The sequence 2 − 2 − 2 − 2 , consisting of three periods of free evolution (T , T 0 and T respectively), reveals a phase difference Φ+(−) for the upper (lower) conjugate interferometer given by

± 0 Φ = ±8ωrT − 2kazT (T + T ) − 2δT. (5.1)

From the above expression, it is evident that the phase diﬀerence is proportional to the 2 recoil frequency of the atom ωr = ~k /(2m), the amount of kinetic energy the atom gains after recoiling from the emission and absorption of a photon. Here, the second term arises from any accelerations az, like gravity or vibrations, along the axis of the laser beam with k = (k1 + k2)/2 being the average wavenumber of the counter-propagating beams, and the third term from the detuning of the laser beams from two-photon resonance in the absence of ac-Stark shifts, δ = ω1 − ω2 − (ωA − ωB).

5.1.1 Large bandwidth pulses

At the atom cloud, 30 mW of ω1 and 15 mW of ω2 red-detuned by ∆ = 2π×200 MHz intersect in waists of 2.1-mm. The Raman Rabi frequency for the two-photon process is measured to be ΩR ∼ 2π × 1.6 MHz or larger during interferometry, corresponding to π-pulses being driven in 320 ns (with approximately 30% transfer eﬃciency). A plot of the probability of transfer as a function of momentum for these pulse parameters is shown in Fig. 5.1 and compared to the Maxwell-Boltzmann velocity distribution along the interferometry axis. Even while broadly spread in terms of velocity, the majority of the atom cloud is addressed or has an appreciable probability of transfer at the short interferometer pulse durations.

5.1.2 k-reversal As seen in the previous section, time is of the essence. Short pulse durations are necessary in order to address an appreciate fraction of the trapped population and the overall time duration of interferometry is limited by speed at which atoms are exiting the interferometer beams. Since the sensitivity of this particular measurement scales with T , the first and last free evolution times, it is ideal to reduce the length of T 0 as much as possible. A distinguishing factor of the Ramsey-Bordéinterferometer compared to the alternative co-propagating scheme discussed in the next chapter is that a reversal of the effective wave CHAPTER 5. HOT BEATS 89

Maxwell-Boltzmann distribution vs. pulse bandwidth MB

-v +3v transfer probability transfer z z

−2 −1 0 1 2

v z [m/s]

Figure 5.1: A comparison of the bandwidth of the interferometry pulses to the Maxwell- Boltzmann velocity spread along the axis of interferometry. Raman resonances for the lower (red) and upper (blue) conjugate Ramsey-Bord´einterferometers are indicated.

vector direction of the light builds in the sensitivity to the atom’s recoil frequency. This flip must occur between the pulse pairs during the 10 µs free evolution time step, T 0. Experimentally, this is achieved by first orthogonally polarizing and then spatially overlapping the two Raman frequencies ω1 and ω2 and passing them through an electro-optical modulator (EOM). The EOM sits prior to two optical fibers into which the frequencies are coupled, ending on either side of the 3D MOT portion of the vacuum chamber. The EOM acts as a voltage-controlled wave plate which when quickly switched from 0 V to 215 V during T 0 (switch shown in Figure 5.2), the polarizations of the frequencies is rotated by 90◦.A polarizing beam splitter separates the two frequencies according to their polarization before ◦ the fibers and thus switching the polarization by 90 reverses the Raman wave-vectors keff. Therefore, switching the voltage on the EOM allows us to alternate which frequency is sent to a particular fiber, flipping keff and effectively closing the Ramsey-Bordéinterferometer.

5.2 Simultaneous and conjugate

In chapter 2, it was shown that a consequence of the four π/2-pulse interferometer scheme is the creation of a second conjugate interferometer. The normal and conjugate interferometers share the ﬁrst and second beam splitter pulses; the trajectories emerging from these transition comprise both the upper and lower interferometers. However for the third and fourth pulse, the lower interferometer requires a transition coupling |F = 2, p = 0i → |F = 1, p = −2~ki while the upper interferometer requires the coupling |F = 1, p = +2~ki → |F = 2, p = CHAPTER 5. HOT BEATS 90

+214V

18k, 10W

EO (100pF)

+6.3V +6.3V

Figure 5.2: Circuit schematic of the vacuum tube switch needed to quickly switch the electric-optical modulator between pulse pairs in a Ramsey-Bordéinterferometer. A pulse is sent to the gate of a field effect transistor in series with a vacuum tube as shown. Opening or closing the gate results in a voltage drop across the 18k sense resistor connected to the top of the 100 pF electric-optical modulator.

+4~k. In principle, a Doppler shift resulting from the diﬀerence in speed between lower and upper interferometers distinguishes the Raman resonance conditions by 8ωr. Low in bandwidth pulses typical to other experiments resolve this frequency diﬀerence and only excite a single transition. Hence, only one of the interferometers is closed unless an additional frequency component is incorporated into the third and fourth pulses to address the second interferometer. Our high bandwidth pulses at π/2-pulse durations of 160 ns, which are required to address the large velocity width of the thermal cloud, also drive the transitions in both conjugate interferometers. Without incorporating an additional frequency component, both the |F = 2, p = 0i → |F = 1, p = −2~ki and |F = 1, p = +2~ki → |F = 2, p = +4~k transitions are coupled and we unavoidably close both conjugate interferometers simultaneously. CHAPTER 5. HOT BEATS 91

170 μm

T T’ T

Figure 5.3: The output ports of the upper and lower conjugate interferometer are only separated by approximately 170 µm at experimental time scales.

5.3 Overlapped, simultaneous and conjugate

Given the common and diﬀerential phase components between conjugate interferometers, comparison of the two can allow for the phase contribution arising from unwanted accelerations or even gravity to be discerned [133, 134], leaving just to phase resulting from the recoil frequency

± Φ = ±φr − φg − φdetuning. (5.2) Phase extraction methods for direct rejection of such common-mode inertial signals like gravity and vibrations rely on being able to discriminate between the interferometer pair in detection [135]. Here, the thermal cloud of atoms expands ballistically during interferometry. This is responsible for smearing out the positions of the interferometer outputs with one and also between the upper and lower conjugates. Four interferometer outputs are overlapped and spatially unresolved when imaging. The probability for atoms to detected in the imaged ground state, denoted here by |bi, after the interferometer sequence is

− + P|bi = D 1 − C− cos(Φ ) − C+ cos(Φ ) (5.3) CHAPTER 5. HOT BEATS 92

where C± are the fringe contrasts for each interferometer and D is the overall oﬀset in atom population. If the contrasts for each interferometer are approximately equal, C− = C+ ≡ C/2, the signal can be simpliﬁed to the following

D D P = 2 − C cos Φ− + cos Φ+ = 2 − C cos(−φ + φ ) + cos(φ + φ ) |bi 2 2 r c r c D = 2 − C cos(φ ) cos(φ ) − sin(φ ) sin(φ ) + cos(φ ) cos(φ ) + sin(φ ) sin(φ ) 2 r c c c r c c c 0 = D 1 − C cos(8ωrT ) cos(2δT + 2kazT (T + T )) . (5.4)

For near equal contrasts, the phase component φc common to both Ramsey-Bord´einterferometers is separated from the diﬀerential phase component φr, proportional to the recoil frequency. Because we perform interferometry horizontally through our vacuum chamber and are perpendicular to gravity, at short interrogation times the δ-term contribution to the interferometer phase φc dominates over that resulting from gravity.

5.3.1 Hot beats Interferometry pulses, which couple the momentum state to an internal energy state, allows for us to discern the signal of interest even though we create and close both the normal and conjugate interferometers simultaneously and spatially overlapped. By varying the separation time T (and maintaining that T 0 = 10 µs), we trace out the interference fringes. The two-photon detuning δ, a consequence of the Raman interferometer pulses, is kept constant and small compared to the recoil frequency ωr. Therefore, the summed signals in the limit that the contrasts from each is approximately equal, can be described by a fast oscillation frequency at 8ωr within an envelope function oscillating more slowly at a frequency set by the Raman detuning, δ, and accelerations present on the atoms az. Even without the ex- clusion of gravity, the eﬀect of vibrations and a nonzero two-photon detuning will act to modulate the amplitude of the interference fringe signal whereas the recoil frequency of the atoms will act as the fast the frequency component and remain essentially untouched by these perturbations to the signal.

5.3.2 Time-domain ﬁtting As mentioned in chapter 4, the Wollaston prism normalization technique in our detection scheme allows two images to be captured during a single exposure. We detected the summed interference fringes from the simultaneous conjugate interferometers, taken over 100s of µs of T -evolution. The data presented in Figure 5.4 shows the summed interference signal for setting of the two-photon detuning, ﬁt with a least squares functional form given by Fit = D 1−exp(−T/τ) C− cos (−8ωr + 2δ)T + φ− + C+ cos (8ωr + 2δ)T + φ+ (5.5) CHAPTER 5. HOT BEATS 93

ω1-ω2=803.506 MHz

ω1-ω2=803.508 MHz

ω1-ω2=803.510 MHz

ω1-ω2=803.512 MHz

Normalized population in F=1 ground state F=1 ground state in population Normalized ω1-ω2=803.514 MHz

ω1-ω2=803.516 MHz

ω1-ω2=803.518 MHz

50 100 150 200 250 300 T [μs]

Figure 5.4: Data and ﬁts for a range of two-photon detunings δ. CHAPTER 5. HOT BEATS 94

0.105

0.1

F=1 0.095 P

0.09

0.085 50 100 150 200 250 300 T (μs)

0.105

0.1

F=1 0.095 P

0.09

100 105 110 115 120 125 130 T (μs)

Figure 5.5: The probability of detecting atoms in the |F = 1i state oscillates, beating due to a nonzero two-photon detuning, δ = 2π × 4.3 kHz. Each point is the average of ﬁve experimental shots and error bars have been omitted for clarity. Fitting (in green) yields ωr = 2 × 63.165 ± 0.002 kHz. Closer inspection of the long time scans reveals the fast recoil component of the fringes. Table 5.1 shows results of the ﬁt with 1 − σ precision.

Values and 1-σ uncertainties are shown below in Table 5.1, resulting from the fit data in Figure 5.5. This fit yields a confidence interval that constitutes a 32 ppm recoil measurement performed over only 2 hours. Averaging across 10 such data sets, each with a different δ results in an attained precision of 10 ppm. This phase sensitivity of the fit corresponds to 50 times larger than the shot-noise limit.

5.3.3 Frequency-domain fitting In Figure 5.6, the fast Fourier transform (FFT) of the same data fit in Figure 5.5 has been taken. From the FFT of the averaged data, we are able to resolve the two frequency components, ω = 8ωr2δ and ω+ = 8ωr +2δ, that constitute the sum of conjugate interference fringes as considered previously. The two peaks differ by 4δ. CHAPTER 5. HOT BEATS 95

Table 5.1: Fitting parameters for Fig. 5.5. −T/τ Fit = D 1 − e C− cos(−8ωrT + 2δT + φ−) + C+ cos(8ωrT + 2δT + φ+)

D τ C− C+ ωr/2π δ/2π φ− φ+ 0.09595(2) 297(8) µs 0.069(1) 0.067(1) 63.165(2) kHz -4.312(8) kHz -0.72(2) 0.37(2)

10 -8 10

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0 400 450 500 550 600

Frequency [kHz]

Figure 5.6: A fast Fourier transform (blue) of the data from Fig. 5.5. A ﬁt given by the sum of two Lorentzians is shown in pink.

The Fourier transforms were ﬁtted with a sum of two Lorentzians

A+Γ+ A−Γ− Fit = 2 2 + 2 2 (5.6) (ω − ω+) + Γ+ (ω − ω−) + Γ− and such ﬁts for a range of two-photon detunings are shown in Fig.5.7 with the indicated δ’s. As the the two-photon detuning is tuned through zero, the peaks of the FFT converge and then once again separate in frequency space.

5.4 Phase noise

2 In the simplest case the interferometer phase will have statistical variance denoted by σΦ due to shot noise or Poissonian counting statistics in detection 1 σ2 ≡ h(Φ − hΦi2)i = (5.7) Φ C2N CHAPTER 5. HOT BEATS 96

Comparison of FFTs for Various δ

0.9 δ/2π =8.28(3) kHz

0.8 δ/2π =4.35(2) kHz

0.7 δ/2π =12.10(4) kHz

0.6 ] 7

0.5 δ/2π =0(15) kHz Power [×10 0.4

0.3

δ/2π =3.49(3) kHz 0.2

0.1

0 460 480 500 520 540 Frequency [kHz]

Figure 5.7: Plot of the Fourier transformed data for a range of two-photon detuning δ as indicated. The central peak, at which δ ≈ 0 yields a large amount of variation in the fitted parameters. CHAPTER 5. HOT BEATS 97 where N is the total number of counted atoms and C is the interferometer’s contrast given by ∗ ∗ ∗ ∗ Imax − Imin ψaψb + ψb ψa C = = 2 2 (5.8) Imax + Imin |ψa| + |ψb| 2 2 for a mean interferometer detection intensity hIi = |ψa| + |ψb| with superposition wave packets ψa and ψb along trajectories a and b, respectively. Phase noise σΦ in an interferometer can be a result of noise arising from any of the parameters on which the phase depends ∂Φ σ = σ × , Φ P ∂P where P = P (Φ) = I0 1 − V cos(Φ) is the probability amplitude with visibility V and intensity I0 can be determined from the interferometer’s sensitivity function g(t). This function quantities the dependence of the transition probability δP at a time t on the relative phase shift of the laser φ δP (δφ, t) g(t) ≈ lim δφ→0 δφ Rewriting Eq. 5.9 making explicit the dependence upon the single photon detuning ∆ for the Raman transition with Rabi frequency Ω and pulse time τ yields ∂Φ ∂P −τΩ sin(Ωτ/2) cos(Ωτ/2) π d∆ σΦ = σ∆ × × = σ∆ × = × (5.9) ∂P ∂∆ ∆(I0V ) sin Φ V ∆ We suspect that noise observed in the data is mostly a result of laser noise. We have confirmed this via numerical simulation which were adapted from previous studies in the research group of noise in Ramsey-Bordéinterferometers. The line width of the Raman laser is appreciable compared to its single-photon detuning, measured with an optical fiber interferometer to have a FWHM of approximately 2 MHz.

5.5 Outlook

The coherence time of the interferometer is not yet limited by the thermal expansion out of the Raman beam but instead by magnetic dephasing of the mF = 0 atoms due to the quadratic Zeeman shift [136] 2 2 (gJ − gI )µBB δEqzs = , (5.10) 6hAs

where gJ and gI are the electronic and nuclear g-factors respectively, µB is the Bohr magne- ton, B is the amplitude of the magnetic field and As [137] is the hyperfine constant. Even when performing interferometry 2 ms after the compressed, the magnetic field gradient that survives produces inhomogeneous quadratic Zeeman shifts of the atoms in the ‘magnetically- insensitive’ sublevel. Therefore, the interferometer’s phase depends upon the atoms spatial CHAPTER 5. HOT BEATS 98

Uncertainty in ωrec /2π for various detunings δ/2π Uncertainty in ωrec /2π for various detunings δ/2π 63.3 63.2

63.19 63.25

63.18

63.2

/2π /2π

rec 63.17 rec

63.16 63.15

Recoil frequency, ω frequency, Recoil

63.15 ω frequency, Recoil 63.1

63.14

63.05 63.13

63.12 -2 0 2 4 6 8 10 12 14 63 0 2 4 6 8 10 12 14 Two-photon detuning, δ/2π Two-photon detuning, δ/2π

Figure 5.8: (Left) A plot of the standard deviation resulting from fits of the data over the time-domain at different two-photon detunings. (Right) A plot of the standard deviation resulting from fits of the Fourier-transformed data in the frequency-domain at different two- photon detunings. position in the vacuum chamber. Extending the optical molasses hold step to 5 ms reduces this effect; it has been observed that the gradient reduces by 50% compared to the 2 ms hold duration. The interferometer contrast indeed decays at half the rate. Magnetic gradient compensation would lead to longer coherence times and improved sensitivity. At a conservatively projected√T = 1 ms, we estimate the shot-noise-limited sensitivity with 107 atoms to be 100 ppb/ Hz. Sub-Doppler cooling would reduce the temperature of the atoms to approximately 40 µK or 8Tr [94, 132] and would improve the experimental sensitivity by p50/8 3, the techniques presented here and in [138] would still be required.

5.5.1 Vibration immunity Lithiums high recoil frequency allows us to take sensitive data at T < 10 ms, and therefore to make full use of the common-mode rejection of vibration-induced signals. Phase shifts from vibrations cancel when the fringes are summed in our detection scheme since they enter CHAPTER 5. HOT BEATS 99 the conjugate interferometers with opposite sign. The only eﬀect of vibrations is then an amplitude modulation of the fringes. Consider phase equation for a Ramsey-Bord´einterferometer with a stochastic, Gaussian- distributed acceleration along the quantization axis az, with a zero mean and a standard deviation σ. Such vibrations modulate the interference contrast for

φvib = 2kσT (T + T )π, (5.11)

which decreases proportionally to az due to aliasing. Other interferometers in the group operating on a similar optical tables without vibration isolation [38, 139] accrue phase shifts much less than π due to vibrations, even at T = 10 ms. 100

Chapter 6

Tune-outs

The tune-out wavelength (λto) is the wavelength at which an atom’s dynamic polarizability goes vanishes. Alternatively, it can be thought of as the point at which the energy shift resulting from the presence of an external optical ﬁeld is zero. Between atomic resonances, where light is red-detuned from one level and blue from the other, these opposing contributions conspire to produce a root in the energy shift spectrum. This can be seen more explicitly by writing out the polarizability for an alkali as a sum primarily of contributions

Dynamic polarizability of 2S2 state in lithium

2×10 7

1×10 7

D -line 2 D1-line polarizability [AU] polarizability

-1×10 7

-2×10 7 670.955 670.960 670.965 670.970 670.975 670.980 670.985 wavelength [nm] CHAPTER 6. TUNE-OUTS 101

from the D1- and D2-lines 1 ωD1 ωD2 α(ω) = SD1 2 2 + R 2 2 + αrem (6.1) 3~ ωD1 − ω ωD2 − ω

where R is the ratio of the line strengths of the D2- and D1-lines, SD2 and SD1 , given by 2 SD2 hψ2Skdkψ2P3/2 i R = = 2 (6.2) SD1 hψ2Skdkψ2P1/2 i and αrem represents the contributions from core electron excitations, higher energy transitions of the valence electrons, and the coupling of the valence and core electrons.

6.1 Previous polarizability measurements

The increasing precision and accuracy attained and required in experimental atomic physics necessitates a better understanding of the interactions between the atoms and external optical ﬁelds. For lithium, there are currently only indirect Stark shift measurements for the dynamic polarizability between the ground and excited state [74, 75] and a static polarizability determination made with thermal atom interferometry [44, 140], as discussed in Chapter 3. A direct measurement of lithium’s tune-out wavelength between the 2P1/2 and 2P3/2 with atom interferometry is the pursuit of ongoing work here. The status and project outlook is the focus of this Chapter.

6.1.1 The differential Stark shift The Stark shift predates the modern formalism of quantum mechanics [60] but does allow for measurements of polarizability differences. In such a measurement, the shift in frequency of an atomic spectral line is measured as a function of either the static or dynamic electric field strength, effectively measuring the polarizability difference between the the two atomic states involved in the transition. Of the few experimental measurements of the ac-Stark shift at optical frequencies, a recent experiment was actually performed with lithium. Varying laser intensity and tracking the transition frequency dependence allowed a determination of the ac-Stark shift [74, 75]. A difficulty exists in the interpretation of the ac-Stark shift experiments and a lack of precise knowledge about the overlap of the laser beam with atoms in the interaction region. A high- precision measurement of the 2s − 3s transition in lithium has uncertainty resulting from ac-Stark shifts at the frequencies of the pump and probe laser of a two-photon resonance transition between the states. A composite method involving the high-precision measurement of the (2s − 3s) transition in conjunction with a Hylleraas method determination of lithium’s ground state 2S1/2 dynamic polarizability and a configuration interaction plus core polarization CICP method CHAPTER 6. TUNE-OUTS 102

TO laser atom beam

z nanograting

Figure 6.1: In an atom interferometer uses a atomic beam as its source, the wave functions must be split by an appreciable amount in the x-domain to make a measurement. Due to this, it is possible to incorporate a laser experimentally such that it will irradiates only one of the interferometer arms.

calculation of lithium’s excited 3S1/2 state dynamic polarizability was performed to assess the reliability of the experiment’s determination. This method obtained an overall uncertainty of only slightly better than 1% [92].

6.1.2 Space-domain atom interferometry

Lithium’s static polarizability α0 was measured with a thermal Mach-Zehnder atom interferometer [44, 140]. A uniform electric ﬁeld was applied to one of the interferometer’s separated arms, shifting its energy by the Stark potential

α E 2 U = − 0 . (6.3) 2 The phase shift observed in the output depends upon this added potential energy. A determination of the static polarizability was made by studying the dependence of the resulting phase on the voltage V applied to generate the external electric ﬁeld

φ D2 α0 = 2 (2~v) (6.4) V Leff where D is the distance between electrode and septum, v is the mean velocity of the atom beam, and Leff is the effective length of the interaction region. An accurate determination of the polarizability requires precise knowledge of not only the phase shift as a function of the applied voltage V but also the geometry of the interaction region and the velocity of the atomic beam. CHAPTER 6. TUNE-OUTS 103

A nicety of a tune-out interferometry measurement of the dynamic polarizability rather than the static polarizability is that while the phase φ depends on the frequency of the tune-out beam given for a thermal atom interferometer as follows:

α(ω) Z d φ(ω) = s I(ω, z)dz (6.5) 20c~v dx where I(ω, z) is the intensity of the ‘tune-out’ beam, v is the atom’s velocity, and s is the spatial separation of the arms, the frequency at which the polarizability vanishes or φ(ω) = 0 will be independent of both the laser and atomic beam properties [141].

6.2 Light-pulsed interferometric lithium tune outs

To measure lithium’s tune-out wavelength, we employ an interferometer scheme consisting of four beam-splitter (π/2) pulses with three periods of free evolution T , T 0, and T . Contrary to a Ramsey-Bord´escheme, here this interferometer is without the reversal of the momentum transfer for the last two pulses. Hence, the electric-optical modulator implemented to achieve this purpose, has been removed from the Raman board.

6.2.1 φto, the tune-out phase As discussed in Chapter 2, an interferometer composed of four co-propagating beam splitter pulses is no longer sensitive to the recoil frequency of the atoms but can be made sensitive to a phase induced by the interaction of the atom’s wave function with potential of a driving field turned on during the T 0 free evolution time. During this time of free evolution, the wave functions of the coherent superposition are in the same internal energy state, allowing for a determination not of a differential tune-out between hyperfine levels, but instead the tune-out or for a particular state. This interferometer geometry [142] has been utilized previously with an atomic beam of calcium to make a measurement of the static polarizability [143]. It is similar to a Mach- Zehnder configuration except with the π-pulse split into two π/2-pulses as evident by the interferometer’s phase difference. The phase of the aforementioned interferometer in the presence of a ‘tune-out’ beam pulsed on during the free evolution time T 0 is given by

α(ω)sT 0 ∂I φto = × (6.6) 2ε0c~ ∂z

where s = 2vrT is the separation between the arms of the interferometer, which can be deﬁned in terms of the atomic recoil velocity vr and the intensity of the beam with waist w is given by −2r2/w2 I(r, ω0) = I0e (6.7) CHAPTER 6. TUNE-OUTS 104

Figure 6.2: The tune-out beam is aligned centrally to the cloud of atoms. Atoms on either side undergoing interferometry will see an intensity gradient with opposite sign over the arm separation distance of s.

2P where I0 = 2 . πw0 Unlike in thermal atom interferometers, the lack of spatial separation between the interferometer arms precludes the ability to interrogate only a single arm with the tune-out light. However, by positioning the beam such that it is central to the cloud of lithium atoms, atoms on either side will ‘see’ an opposite relative gradient over the superposition’s trajectories during interferometry. This results in atoms that accrue a phase shift equal in magnitude for the same starting positions (aside from position) but opposite in sign. By measuring the phase accrued on either side of the tune-out frequency, the slope through zero is determined and thus the tune-out frequency to within a given precision. The given precision is set by the sensitivity of the experiment, dependent on the experimental parameters of the tune-out beam and of the atom cloud, particularly in the thermal dephasing of the atoms during interferometry. Varying the frequency of the tune-out laser allows us to track the dependence on the CHAPTER 6. TUNE-OUTS 105

⟩ ħk eff + |a , p

|b , p ⟩

⟩ ħk eff + |a , p

|b , p ⟩ T T’ T z

t Figure 6.3

Spatial Dependence of Tune-out Beam Intensity

I(x,z)

] 1.0 0 I [

∂zI(x,z)

0.5

0 -2 -1 0 1 2

0.5 Normalized Amplitude Amplitude Normalized

1.0

z [ w]

Figure 6.4: The spatial dependence of the intensity of a Gaussian beam varies in units of beam waist. For positions on either side of the center, movement in the same direction will be at equal and opposite intensity gradients. CHAPTER 6. TUNE-OUTS 106 phase variation across the cloud as a function of frequency. For a particular beam waist which corresponds to a particular intensity in the chamber as seen by the atoms, we predict a given contrast slope of the tune-out signal. A further complication occurs due to thermal dephasing that occurs resulting from the signiﬁcant spread of the atoms in momentum space. An atomic cloud at a temperature of 300 µK corresponds to a average thermal speed of approximately 0.6 m/s. Therefore, 0 it is possible for an atom to transverse a distance of vth 2T + T during the time of the interferometer.

6.2.2 The tune-out beam The tune-out beam originates from a commercial Topitca ECDL () that is frequency-offset locked from the cross-over frequency between the hyperfine ground states to the 2P3/2 state. Light from the diode laser seeding the tapered-amplifier for interferometry is the reference light in this locking scheme. The local oscillator is a frequency-doubled Agilent function generator which is mixed with the laser beat note and sent to a servo similar to what is utilized in both our spectroscopy and optical pumping schemes. The majority of the light from the tune-out ECDL is sent through an AOM after which the -1 order is coupled into an optical fiber and sent to below the vacuum chamber of the 3D MOT. Here, it is telescoped to achieve a 150 µm waist at the atoms. The waist is measured via single-photon scattering by moving the frequency of the beam closer to resonance with either the D1- or D2-line. The beam passes through a half-wave plate and PBS before it is steered with two mirrors such that it propagates upwards, along the same path as the imaging light. After transversing the chamber, the beam is picked out of the imaging path with a ‘D’-shaped mirror, sent to another mirror mounted above the chamber and retro-reflected. Retro-reflecting increases the intensity at the atoms.

6.2.3 Experimental Sequence During the second free evolution time step T 0, if considering a single interferometer then both components of the superposition are in the same momentum and internal energy state. Because each hyperfine has a different tune-out wavelength resulting from varying transition matrix elements, pulsing on the tune-out beam during this time step is crucial. The tune-out beam is switched on in the middle of this time step as shown in Fig. 6.6. We also turn on imaging light for a pulse duration of 60 µs to decohere the upper interferometer unavoidably generated in the conjugate scheme due to the fast beamsplitter pulses. Rather than scanning the free evolution time between pulses and tracing out interference fringes over an extended duration, for a fixed time of evolution, the frequencies from the DDS function generator driving the Raman AOMs is stepped between the pulse pairs. Ex- perimentally, by switching which output port from the DDS is mixed with a third frequency also originating from the DDS, the result is a modulation inducing an interferometer phase during the experimental cycle. CHAPTER 6. TUNE-OUTS 107

hwp PBS

hwp

reference light

beat note detector pinhole BS

80 MHz hwp AOM

PBS

hwp

Figure 6.5: The optics used to frequency-oﬀset lock and steer the tune-out beam and the path transversed through the vacuum chamber. CHAPTER 6. TUNE-OUTS 108

Experimental Sequence and Settings for Tune-out

cMOT Raman image image dwell MOT load ramp cMOT hold OM hold pump repump pulses atoms background 10 ms 0.8 s 50 ms 2 ms 20 μs 5 ms 70 μs 160 μs 150 ms 150 ms

BMOT

3D ftrap

3D frepump

3D Ptrap

3D Prepump

Bbias

2D shutter OPEN pusher shutter OPEN 3D trap AOM OPEN 3D repump AOM OPEN 3D TA switch OPEN Raman pulse gen. OPEN digital controldigital control analog τ opt. pumping OP

τ τ τ imaging AOMs LKB img img τ tune-out AOM to

Figure 6.6: The experimental sequence and settings utilized for a measurement of 7Li’s tune-out wavelength is shown here. Two additional pulses, that of the tune-out light being switched but also a pulse of imaging light diﬀer from the recoil-sensitive interferometry detailed in the previous chapter.

Data is taken for the tune-out measurement at the point for which the signal slope is the highest and the interferometer is thus most sensitive. For a given frequency of the tune-out beam, an experimental cycle is run with a pulse of tune-out light and without such a pulse. While there are always two runs for every frequency of the beam, data has been taken both at a ﬁxed frequency for a length of time as well as while scanning the frequency many times. Optimization of data taking is still currently being pursued.

6.2.4 Detection & Analysis The atoms are detected with absorption imaging again implementing the Wollaston cube which allows for two images to be taken during one exposure. A series of images at a particular frequency of the tune-out beam are taken, alternating between switching the CHAPTER 6. TUNE-OUTS 109

Figure 6.7: A comparison between the analysis performed without (top) and with (bottom) the tune-out pulse. The lower principal component analysis only contains that dependent upon the extra light, switched on during the T 0 time step in the interferometer. tune-out light either on or off. This allows us to normalize to the observed drift in atom cloud position, seen over the course of the experiment, as well as other experimental defects. Both the pulsed and unpulsed data are analyzed with a method of statistical image analysis called principal component analysis. Doing so allows us to discern only the significant features different between the sets which is being experimentally imposed to be the presence (or lack of) tune-out light. Normalizing the pulsed to the unpulsed in the final pulsed image results in a principal component corresponding to the spatial effect of the tune-out light. A plot of the projection of the pulsed to unpulsed basis determines the ‘magnitude’ or significance of the tune-out signal and for a scan of frequencies is used to determine the experimental precision; we expect at tune-out that no difference we be apparent between bases. CHAPTER 6. TUNE-OUTS 110

6.2.4.1 Principal component analysis

The goal of principal component analysis (PCA) is to represent an image Xi as a linear combination of a set of basis images

Xi = M + Y1iu1 + Y2iu2 + ... + Ypiup (6.8) where M is the mean image for the set given by 1 M = X + X + ... + X . (6.9) N 1 2 N

Each image Xi is a p-dimensional vector, where p is the number of pixels. Given a set of N images, PCA constructs a new matrix of dimension p × N such that each column of this matrix is a vector given by the diﬀerence between a particular image of the N size set and the mean image

ˆ ˆ ˆ ˆ X = X1 − MX2 − M ... XN − M = X1 X2 ... XN . (6.10)

From this matrix in mean-deviation form, a covariance matrix (p × p) is calculated 1 S = Xˆ Xˆ T (6.11) N − 1

where the diagonal element Sjj, for an arbitrary pixel j, is the sample variance

N 1 X S = σ2 = Xˆ 2 (6.12) jj jj N − 1 ji i=1 and the total variance is found by computing the trace of the matrix, tr(S). The oﬀ-diagonal 2 elements are the covariances between pixels. In the instance that σjk = 0, then the pixels are said to be uncorrelated. This covariance matrix is used to determine the new basis for the set of N images

Xˆ = PYˆ (6.13)

where Pˆ is the orthonormal matrix composed of principal components

ˆ P = u1 u2 ... upˆ (6.14)

and the matrix Y is an N ×N matrix of coeﬃcients or weights. The mean-subtracted image can be represented as a linear combination in terms of the new PCA basis as ˆ Xi = Y1iu1 + Y2iu2 + ... + YNiuN . (6.15) CHAPTER 6. TUNE-OUTS 111

PCA calculates the basis images such that they are orthonormal with the goal to find a new better basis for the data using a limited number of principal components. PCA requires that for a set of images, the coefficients for a particular image vector basis j be statistically uncorrelated with the set of coefficients of any other basis image k. This condition is met if and only if the sample covariance matrix of the PC basis D is diagonal 1 D = YYT . (6.16) N − 1 This is the same as ensuring that the PCs are unit eigenvectors of S. Therefore, given this analysis of principal components an image of the set i can be written in the new basis as Xi = M + Y1iu1 + Y2iu2 + ... + YNiuN . (6.17) When performing this analysis on similar images, the most significant features of an image can be reconstructed with a PC basis of reduced dimensionality q < N,

Xi ≈ M + Y1iu1 + Y2iu2 + ... + Yqiuq. (6.18) The dimensionality of the image is reduced by p pixels to q strongest PC weights. This reductions allows speciﬁc experimental parameters to be correlated with the image weights.

6.3 Towards tune-out

The ﬂip side to the nicety of lithium’s simple electron structure allowing for its atomic properties to be computed with out standard quantum mechanical techniques, it that such computations have reached an extraordinary level of precision from the perspective of a precision metrologist. Furthermore, at this resolution of optical frequencies, one must now use a frequency comb as a reference; the intrinsic accuracies of frequencies used experimental for trapping, pumping, and interferometry are known to values at least an order of magnitude above the intended tune-out measurement.

6.3.1 Precision While large-scale calculations done with correlated Hylleraas basis sets can attain a degree of precision not possible for calculations based on orbital basis sets. Such a computation [92], utilizing solutions to the norelativistic Schr¨odingerequation along with experimental energy values computed the tune-out wavelength for lithium’s ground state to be at 670.971 626(1) nm. An inclusion to the dynamic polarizability of an remainder term αrem(ω) estimated to 3 be 2.333 824 a0 or likewise an perturbation to the ratio of reduced dipole matrix elements R deﬁned as

2 |hψ2S(r)kdkψ2P3/2 i| 2 = (2 + R) (6.19) |hψ2S(r)kdkψ2P1/2 i| CHAPTER 6. TUNE-OUTS 112

Magnitude of Tune-out PC vs. Frequency

-3.5

-4.0 ] 15 -4.5 projection [×10 projection

to -5.0 PC

-5.5

3300 3320 3340 3360 3380 3400 3420 3440 3460 3480 3500

Frequency [MHz]

Figure 6.8: A plot of the magnitude of the projection on the TO principal component over a span of frequencies.

results in the above uncertainty to the wavelength and a needed precision of approximately 500 kHz. Currently, we are measuring 20 MHz within the predicted tune-out frequency within a period of hours. A plot of the overlap onto the principal component correlated with the tune-out pulse for various frequencies is shown in Fig. 6.8.

6.3.1.1 Single-photon scattering Increasing the intensity of the tune-out beam is a route towards higher experimental sensitivity however an issue arises due to the single-photon scattering limit in lithium at detunings which correspond to the tune-out wavelength frequency. In the case of π polarization, the scattering rate for the hyperﬁne state F is given by [61] 1 X X 0 p 0 00 R = Ω 0 hF m |F m ; 1 0i Γ 0 00 hF m |F m − q; 1qi sc 4 FF F F F F F F qF 00 F 0 ω 3/2 1 1 2 × − (6.20) ωF 0F ω − ωF 0F ω + ωF 0F CHAPTER 6. TUNE-OUTS 113

Figure 6.9

with a decay-rate for small hyperﬁne splittings given by the following in terms of total decay 0 rate of level J denoted by ΓJ0 , 0 00 p p F 00+J0+1+I p 00 0 J J 1 Γ 0 00 = Γ 0 (−1) (2F + 1)(2J + 1) . (6.21) F F J F 00 F 0 I

6.3.1.2 Beam shaping Altering the profile of the beam will increase or decrease the sensitivity of the measurement by altering the intensity and thereby phase scaling ITT 0 δφ ≈ . (6.22) w0 An anamorphic prism pair inserted after the beam transforms the profile from circular to elliptical as shown in Figure 6.9. Elongating the beam allows us to reduce the area but maintain the same intensity, advantageous to avoid the single-photon scattering limit at the required detunings. A plot of the overlap on the tune-out PC for various Agilent frequencies is shown in Fig. with the fitting parameters resulting from a linear fit applied to the presented data.

6.3.2 Accuracy An optical frequency comb is necessary to determine the wavelength for tune-out with the required accuracy. We plan to use a commercial Menlo Systems (FC8004) optical frequency synthesizer that is based on a femtosecond laser frequency comb, with a specified comb frequency spacing of approximately 200 MHz and accuracy of 10−14. This system consists of a fs-laser, a nonlinear photonic crystal fiber and nonlinear interferometer intended to measure the offset frequency of the comb by interfering the spectral parts around 532 nm and 1064 nm. By beating the tune-out beam against the comb, one can accuracy determine the wavelength of the light.

6.4 Hyperﬁne dynamic polarizabilities

The hyperfine ground states of lithium do have different dynamic polarizabilities resulting from differing values of dipole matrix elements. In order to measure the absolute tune-out CHAPTER 6. TUNE-OUTS 114

PC 1, max=0.158, min=-0.183 PC 2, max=0.146, min=-0.141 PC 3, max=0.214, min=-0.395 250 5

10 200 15

20 150

PC 4, max=0.180, min=-0.329 PC 5, max=0.127, min=-0.191 PC 6, max=0.375, min=-0.234 100 5

10 50 15

5 10 15 20 5 10 15 20 5 10 15 20

Figure 6.10: Image taken with elliptical beam, after broken down into principal components.

Magnitude of Tune-out PC vs. Frequency

m = 324215719246765.4; σm = 5982647790464.469;

6 xint = 2396.0814; σXint = 1.9603;

5 ] 16 4

3 projection [×10 projection to 2 PC

-1 2350 2400 2450 2500 2550 2600

Frequency [MHz] CHAPTER 6. TUNE-OUTS 115

T state F α1 (a.u.) α1 (a.u.) 7 Li 2S1/2 1 161.983 -9.78[-7] 7 Li 2S1/2 2 161.984 5.94[-6]

Figure 6.11: Scalar and tensor dynamic polarizabilities for hyperﬁne ground state levels in 7Li.

wavelength for only the F = 2i ground state, imaging light resonant with the F = 1to2P3/2 is flashed on during the T 0 free evolution step of the interferometer. This light decoheres the conjugate interferometer that is created in this scheme. To compute the scalar the dynamic hyperfine polarizability, first the dipole matrix elements are determined with respect to the different hyperfine levels. Using the Wigner-Eckart theorem, the transition amplitude is rewritten in terms of a reduced matrix element

hF kerkF 0i ≡ hJIF kerkJ 0I0F 0i 0 0 JJ 1 = hJkerkJ 0i(−1)F +J+1+I p(2F 0 + 1)(2F + 1)(2J + 1) F 0 FI (6.23)

depending only upon L, S, and J quantum numbers. k The oscillator strength fgi for a dipole transition (k = 1) is given by

2|hJIF kerkJ 0I0F 0i|2ω f k = gi (6.24) gi 3(2F + 1)

where ~ωgi is the energy required to excite the |gi → |ii transition between states denoted here |gi and |ii. From the oscillator strength, the dynamic polarizability is determined with the oscillator- strength sum rules, (1) X fgi α(ω) = . (6.25) ω2 − ω2 i gi Here, the sum over intermediate states includes all allowed ﬁne-structure and hyperﬁne- structure allowed transitions. In the limit that the frequency goes to zero, the static polarizability should be recovered. The tensor component of the dipole polarizability (for states in which F > 1/2) is given by

1/2 0 (1) 5F (2F − 1)(2F + 1) X 0 F 1 F fgi αT (ω) = 6 × (−1)F +F . 1 6(F + 1)(2F + 3) 1 F 2 ω2 − ω2 i gi (6.26) CHAPTER 6. TUNE-OUTS 116

Figure 6.12: This plot shows how the dynamic polarizability of each hyperﬁne ground state (orange is the F=1 state and green is the F=2 state) and the diﬀerence in α1(ω) (blue) varies with frequency.

Plotting the dynamic polarizabiity, α(ω), for each state and solving for the roots of the expression or when α(ω) = 0, we ﬁnd that the |F = 1i ground state has a tune-out at 670.961 nm (446.8039412 THz) while the |F = 2i ground state has a tune-out at 670.962 nm (446.8031617 THz). These diﬀer from the 2S1/2 tune-out wavelength (670.972 nm) by 2π × 488 MHz and −2π × 292 MHz, respectively. 117

Chapter 7

Conclusion

This chapter explores implications and potential applications of the techniques demonstrated int the preceding two chapters, as well as the outlook for atom interferometry with lukewarm lithium.

7.1 Outlook for recoil-sensitive interferometry with super-recoil samples

With non-zero two-photon detuning for the Ramsey-Bord´einterferometer, the interference fringes allow for the determination of the recoil frequency independent of two-photon detuning and vibrations. Our results relax cooling requirements for recoil interferometry, allowing for increased precision through high experimental repetition rates [144]. This demonstration of interferometry with a sample of atoms at ‘super-recoil’ temperatures opens the door to recoil measurements with other particles that are diﬃcult to cool to subrecoil temperatures, such as electrons.

7.1.1 h/me measurement Electrons have recoil frequencies on the order of GHz. They are susceptible to relativistic effects [145] and consequently a recoil-sensitive measurement can be used to measure Lorentz contraction [146]. A measurement of h/me also allows for a more direct determination of the fine structure constant α; allowing one to neglect measurements of reduced electron and atom masses [2]. While Kapitza-Dirac scattering has been proposed to realize matterwave beam splitters for electrons in a Ramsey-Bordéinterferometer [147], any vibrations or nonzero two-photon detuning will modify the phase ∆φ− for a single Ramsey-Bordé, 4 ∆ϕ = k2 T 0 − 2ak TT 0. (7.1) m~ L L CHAPTER 7. CONCLUSION 118

z -! -! +! +! rec rec rec rec

Δz

T T’ T

Figure 7.1: Ramsey-Bord´einterferometer for electrons. Four bichromatic laser pulses, composed of two counterpropagating waves detuned by multiples of the recoil shift ωrec. These waves are used to split and recombine the electron beam as shown, producing a total of eight partial beams.

As we have shown, the inclusion of the simultaneous conjugate interferometer (∆φ+) recovers the recoil phase independently of a two-photon detuning even when the outputs of conjugate interferometers are spatially unresolved, as would the case for electron plasmas in a Penning-Malmberg trap [148]. The required spectral resolution for detection could be achieved with bichromatic Kapitza- Dirac pulses. Bichromatic pulses with very large intensity have been proposed to impart momentum to an electron while inducing a spin ﬂip [149] and hence couple the electrons external and internal degrees of freedom. With such beam splitters acting on a spin-polarized sample and spin-dependent detection, the techniques we demonstrate in this work pave the way for a recoil-sensitive electron interferometer. CHAPTER 7. CONCLUSION 119

7.2 Outlook for tune-out interferometric measurements in lithium

A direct interferometric measurement of lithium’s red tune-out wavelength at 670.971626(1) nm, is a precision comparison to existing ‘all-order’ atomic theory computations. A determination of lithium’s dynamic polarizability would be pivotal in metrology [81]. In addition to informing existing computational methods, a next generation experiment, in which another atomic species would be included in the same apparatus as lithium, would allow for a direct comparison and normalization, overcoming systematics resulting from different experimental environments. This could lead to a new accuracy benchmark for many elements. Hylleraas polarizability calculations could serve as standard for coupled-cluster type calculations applied to atoms larger than lithium, like cesium. Additionally, it provides another way to determine the S− to P − transitions matrix elements for which large correlations and small values complicate computation. Aside from informing atomic structure calculations, applications of tune-out wavelength measurements include making state-selective and multi-species traps for ultra-cold atomic physics experiments and quantum computing, spin- dependent dispersion compensation for an atom interferometer gyroscope, and can be useful PNC experiments. The lack of spatial resolution during interferometry to measure lithium’s dynamic polarizability is overcome by aligning the tune-out beam to the center of the atom cloud so that for atoms on either side, an equal and opposite ac-Stark shift will be induced. This results in atoms on opposite side of the beam center accruing opposite phase shifts, evident in the normalized atom number (corresponds to probability amplitude for state) during imaging as a corresponding excess or absence of population. An imaging technique called principal component analysis allows for this variation through the tune-out frequency to be observed by splitting up the obtained image into a simplified basis and then allowing images taken with the tune-out light to be projected onto images taken without the tune-out light. This differential imaging scheme, while not simultaneous, does allow for the determination in the midst of varying atom cloud position.

7.2.1 Beyond the red

A second tune-out frequency in lithium occurs right before the 3S → 3P1/2, 3P3/2 excitations, at 324.192(2) nm [98]. The attained uncertainty in its computed value is much larger than the ﬁrst tune-out frequency in the red part of the spectrum, in part due to a greater dependence upon the details of the atomic structure description [92]. Recalling the deﬁnition of the dynamic polarizability in terms of oscillator strengths

X fgk α(ω) = 2 2 (7.2) (Ek − Eg) − ω k CHAPTER 7. CONCLUSION 120

7 Dynamic polarizability of Li atom in its ground state ( 2S 1/2 )

1000

red uv ω0 ω 500 0

ω [au]

α [au] 0.04 0.06 0.08 0.10 0.12

-500

-1000

7 Figure 7.2: Dynamic polarizability of the 2S1/2 level in Li. Tune-out wavelengths have been indicated at values of 0.067 906 526 572 and 0.140 907 329.

7 Dynamic polarizability of Li atom in its ground state ( 2S 1/2 )

ω [au]

α [au] 324.16 324.18 324.20 324.22 324.24

-2

-4

Figure 7.3: Lithium’s second tune-out wavelength at 324.192(2) nm has a larger computation uncertainty due to the greater impact of the atomic structure description on the value. CHAPTER 7. CONCLUSION 121 or more explicitly for the UV tune-out

f2p f2p (2 + R) f3p f3p (2 + R) α(ω) = 1/2 + 1/2 + 1/2 + 1/2 + α (ω) ∆E2 − ω2 ∆E2 − ω2 ∆E2 − ω2 ∆E2 − ω2 rem 2p1/2 2p3/2 3p1/2 3p3/2 (7.3) where R is deﬁned as

2 2 |hψ2s(r)kdkψ2p3/2 (r)i| = |hψ2s(r)kdkψ2p1/2 (r)i| (2 + R). (7.4)

The energies for the excited states needed in the computation can be referenced to experimental data. However, the value of R which was found to be R = 0.00096199, the scaling of the matrix elements, must be determined from a relativistic model potential calculation [92]. The inclusion of R impacted the determined value at the part per million level contrary to the dependence of the red wavelength which has little to dependence on R. However, inclusion of αrem, referred to in the literature as the background polarizability, impacted the value for the UV tune-out wavelength at the forth signiﬁcant digit. A future measurement of lithium’s ultraviolet tune-out wavelength would be more sensitive to relativistic approximations in the atomic structure description.

7.2.2 Investigation of nuclear structure between isotopes Improved computation methods, guided by a comparison of measured and calculated atomic properties can inform theories beyond even that surrounding atom-light interaction. The relativistic and QED corrections have been calculated for the hyperfine splitting of the 2S1/2 ground state of lithium [150], exactly accounting numerically for electronic correlations. This theoretical computation can be compared with experimental values determine atomic nuclear properties such as nuclear charge radius between isotopes. A significant difference in magnetic moment distributions has been computed between lithium’s isotopes. It is claimed this may signal the existence of some unknown spin-dependent short-range force between hadrons and the lepton. In an atom, the magnetic moment of the nucleus interacting with the magnetic moment of the electrons, produces the hyperfine splitting of energy levels. In a many electron atom, theoretical predictions are limited by correlations between the multiple electrons which are difficult to account for using relativistic formalism based on the Dirac Hamiltonian. Explic- itly correlated basis sets are able to accurately account for electron correlations, achieving a few ppm accuracy. A nonrelativistic QED approach perturbatively treats both relativistic and QED effects [150]. All corrections are expanded in powers of the fine structure constant and expressed in terms of the effective Hamiltonian. CHAPTER 7. CONCLUSION 122

7.3 Atom interferometry with lukewarm lithium

The limiting factor in the current experiment is time. The inability to wait for the Eddy currents induced in the vacuum chamber to decay hinders the interferometer’s obtainable contrast. Extending the interferometer’s interrogation time to one at which a competitive h/M measurement would be possible is also not allowed at the current temperature of the atoms. Further cooling, beyond that achievable with only optical molasses, is a route towards both decreased thermal dephasing and resulting systematics and increased experimental sensitivity and hence precision in measurement [151].

2 P3/2 δ1

δ δ 10.056 GHz

δ1 δ2 670.961 561 nm F= 2

δ2 2 P1/2 91.8 MHz

F= 1 670.976 658 nm

F= 2

2 S1/2 803.5 MHz

mJ -3/2 -1/2 +1/2 +3/2

F= 1

Sisyphus cooliing Gray molasses cooling

Figure 7.4: Laser frequencies for the implementation of sub-Doppler (left) Sisyphus cooling and (right) gray molasses cooling in lithium. CHAPTER 7. CONCLUSION 123

Two sub-Doppler cooling techniques that would allow for an order of magnitude reduction in temperature are mentioned in the following sections. To experimentally pursue either, an additional FP diode laser and tapered ampliﬁer would be needed at minimum. Each scheme utilizes light close to lithium’s D1 line, already generated experimentally for optically pumping and tune-out light. The additional lasers would allow for more power around D1 would could then be split among optical pumping, tune-out, and sub-Doppler cooling. Frequency shifting further in each of these set-ups could be achieved with either acoustic- optical modulators and electric-optical modulators.

7.3.1 Sisyphus cooling A simple route to sub-Doppler cooling would be to implement on the experiment the previously demonstrated Sisyphus cooling [94]. Sisyphus cooling achieved temperatures as low as 40 µK in one dimension for up to 45% of the cooled atom fraction. This method uses polarization gradient cooling, but detuned from the 2P3/2 excited state by 19 GHz; such detunings are still produced using in the experiment for both optical pumping and the tune-out. The cooling process operates on a timescale of milliseconds, requiring 180 mW of total power split into three retro-reflected 0.7-mm waist beams, one sent along the axis of the MOT coils and two overlapped with the MOT beams in the plane of the MOT coils. The cooling beams are linearly polarized and reflected through a quarter-wave plate. This cooling scheme is robust against ambient background magnetic fields of up to 100 mG.

7.3.2 Gray molasses

A sub-Doppler cooling technique using three-dimensional bichromatic laser cooling on the D1 line has been implemented in lithium, obtaining temperatures of 60 µK with upwards of 108 trapped atoms after only 1.5 - 2.0 ms of cooling [132]. This method combines a gray molasses cooling scheme on the same transition we use to optically pump, the |F = 2ito|F 0 = 2i transition while simultaneously addressing the |F = 2ito|F 0 = 1i transition, albeit phase- coherently such that a velocity selective coherent population trapping of atoms is realized [152]. The beams have intensities of approximately I ≥ 45Isat, created with a total of approximately 150 mW in 3.4-mm waists of σ+ − σ− counterpropagating pairs.

7.4 Onward

In conclusion, recoil-sensitive Ramsey-Bordéinterferometry and interferometry sensitive to the atomic dynamic polarizability has been demonstrated with laser-cooled lithium-7 at a lukewarm 300 µK (50 Tr). The large Doppler spread of the sample is addressed with fast pulses, driving simultaneous conjugate interferometers in both instances. We suppress first- order magnetic dephasing and extend the coherence time by optically pumping the atoms to the magnetically insensitive |F = 2, mF = 0i state using lithiums well-resolved D1-line. We CHAPTER 7. CONCLUSION 124 overcome the necessity of spatial resolution by instead spectrally resolving the interferometer arms via two-photon Raman transitions. In the case of the tune-out measurement, an analysis of image data such that cloud movement and variations are normalized to can allow for an identification of the phase shift from the tune-out beam. 125

Appendix A

Properties of lithium

Lithium is the lightest alkali atom and the third element (Z = 3) of the periodic table. As a solid, lithium is a soft, shimmery metal. However, moisture and nitrogen in the air quickly corrode lithium to a dull gray and eventually black tarnish. Due to this reactivity, in nature lithium can only be found in compounds and never freely. When in its ground state, lithium’s three electrons are found in the conﬁguration 1s22s1. Two electrons occupy the lowest s-orbital and a lone valence electron sits outside the closed shell, characteristic of the alkali atoms located in the ﬁrst column of the periodic table. The physical properties of bulk lithium are listed in the table below. Lithium’s vapor pressure as a function of temperature is given with Antoine parameters from NIST’s tabulated atomic data 7918.984 log (P ) = 4.98260 − (A.1) 10 (T − 9.52) where the pressure calculated is in the units of bar and here T is in units of K. These

Table A.1: Physical properties of lithium

Property Symbol Value Reference Atomic number Z 3 [153] Nuclear lifetime τn stable [154] Atomic weight Ar,std 6.941 [153] −3 Density (300 K) ρm 0.534g·cm [154] Melting point TM 453.69 K [154] −1 Heat of Fusion QF 2.99 kJ·mol [154] Boiling point TB 1615 K [154] −1 Heat of Vaporization QV 134.7 kJ·mol [154] Vapor pressure (300 K) Pv [155] Ionization limit EI 5.391 7149 5(4) eV [156] APPENDIX A. PROPERTIES OF LITHIUM 126

F= 0 F= 1 11.097 MHz 8.335 MHz F= 2 2 2.516 MHz P3/2 6.955 MHz F= 3

10.056 GHz F=1/2 F= 2 2.875 MHz F=3/2 34.454 MHz 2 1.162 MHz 2 P1/2 P3/2 1.733 MHz 57.424 MHz F=5/2 F= 1

10.050 GHz

F=3/2 8.701 MHz 2 P1/2 17.402 MHz

F=5/2

F= 2 F=3/2 301.314 MHz 76.068 MHz 2 2 S1/2 S1/2 152.137 MHz 502.190 MHz F=1/2

F= 1 6Li 7Li

coeﬃcients were calculated by NIST from the author’s data taken over temperatures from 298.14 to 1599.99 K. Gehm’s data sheet on lithium reports functions of pressure with respect to temperature for both the solid and liquid phase as 6450.944 log P = −54.87864 − − 0.01487480T + 24.82251 log T (solid) 10 V T 10 8345.574 log P = 10.34540 − − 0.00008840T − 0.68106 log T (liquid) 10 V T 10 (A.2) where here contrary to the aforementioned vapor pressure function, pressure in in units of Torr (mmHg) and again temperature is in K. Lithium has two naturally abundant and stable isotopes, 7Li and 6Li with four and three neutrons respectively. General isotopic properties can be found in the table but the experiments presented in this thesis only use the bosonic isotope, lithium-7. APPENDIX A. PROPERTIES OF LITHIUM 127

10 -4

10 -5

10 -6

10 -7

10 -8

Vapor Pressure (Torr) Pressure Vapor 10 -9

10 -10 200 250 300 350 400 Temperature (°C)

Figure A.1: Vapor pressure of lithium.

Table A.2: Physical properties of lithium

Property Symbol 7Li 6Li Reference Atomic number Z 3 3 [153] Total nucleons Z+N 7 6 Natural abundance η 92.5% 7.5% [154] Atomic mass m 7.016003 amu 6.0151214 amu [153] Nuclear spin I 3/2 1 [157] Magnetic moment µ +3.25644 +0.822056 [157] APPENDIX A. PROPERTIES OF LITHIUM 128

7 Table A.3: Li D2 (2S1/2 → 2P3/2) Transition Properties Property Symbol Value Reference Frequency ω0 2π × 446.810183 THz [158] −7 Transition energy ~ω0 2.94128 × 10 eV Wavelength (vacuum) λ 670.961561 nm Wave number k 9.36445 × 104 cm−1 Lifetime τ 27.1 ns [155] Natural line width Γ = 1/τ 2π × 5.87 MHz Oscillator strength f [108] Recoil frequency ωr 2π × 63.312 kHz Recoil energy vr 8.49594 cm/s Recoil temperature Tr 6.07695 µK Doppler temperature TD 142 µK

7 Table A.4: Li D1 (2S1/2 → 2P1/2) Transition Properties Property Symbol Value Reference Frequency ω0 2π × 446.800130 THz [158] −7 Transition energy ~ω0 2.94122 × 10 eV Wavelength (vacuum) λ 670.976658 nm Wave number k 9.36424 × 104 cm−1 Lifetime τ 27.1 ns [155] Natural line width Γ = 1/τ 2π × 5.87 MHz Oscillator strength f [108] Recoil frequency ωr 2π × 63.309 kHz Recoil energy vr 8.49574 cm/s Recoil temperature Tr 6.07668 µK Doppler temperature TD 142 µK

Below are tabulated values for 7Li optical properties for the D-line transitions. The 2 ~k ~k recoil velocity vr = M and frequency ωr = 2M for the transition corresponds to one ~k of photon momentum being imparted to the atom. The recoil temperature is deﬁned as 2 2 ~ k Tr = 2Erec/kB = where kB is the Boltzmann constant. The Doppler temperature is the MkB theoretical minimum temperature for a given transition, determined by the line width of the ~Γ transition: TD = 2 . APPENDIX A. PROPERTIES OF LITHIUM 129

A.1 The level spectrum

In an atom, the effects of relativity and spin on the dynamics of the electron results in fine structure, defined as the coupling between the electron’s orbital angular momentum, L, and its spin angular momentum, S. Fine structure enlarges the Hilbert space of the system,

ε = εorb ⊗ εspin (A.3) and the total electron angular momentum, J, is a sum of its spin and orbital momentum given by J = L + S (A.4) p with |J| = J(J + 1)~, bounded by |L − S| ≤ J ≥ L + S. (A.5) In an alkali atom, the transition in which the valence s-orbital electron is excited to the p-orbital (L = 0 → L = 1), is fractured by fine structure into a doublet of spectral lines, the D1 and D2 lines from 2S1/2 − 2P1/2 and 2S1/2 − 2P3/2, respectively. The nonzero nuclear angular momentum I of lithium (3/2 for lithium-7), couples to the electron’s total angular momentum J and each of these fine structure levels is further split and has additional hyperfine splitting. The total atomic angular momentum F is F = J + I (A.6) and where F can have values |J − I| ≤ F ≥ J + I. (A.7) There exist 2F + 1 magnetic sublevels for each hyperfine level F in the the atom. These sublevels determine the angular distribution of the electron and furthermore, in the presence of an external magnetic field, break the degeneracy of the hyperfine energy level. The states are labeled by the quantum numbers (mF ), associated with the Fz operator.

A.2 Interaction with static ﬁelds

The hyperfine magnetic quantum numbers mF , which satisfy −F ≤ mF ≤ F , are degenerate in the absence of an external field. An externally applied magnetic field couples to the magnetic moments and if the energy shift due to this field is small compared to the fine structure splitting then the interaction Hamiltonian is the fine-structure interaction plus the magnetic-dipole interaction of the nuclear magnetic moment with the magnetic field.

(hfs) (fs) µB HB = HB − µI · B = (gJ Jz + gI Iz)Bz (A.8) ~ and ﬁrst order perturbation theory leads to the following energy shift

(hfs) ∆EB = µBgF mF B (A.9) APPENDIX A. PROPERTIES OF LITHIUM 130

and the Landé gF factor (making the approximation that gI gJ ) is F (F + 1) − I(I + 1) + J(J + 1) g ∼ g (A.10) F J 2F (F + 1) Static electric fields also shift the fine- and hyperfine-structure in an atom, much like how a static magnetic field shifts the energy levels. Consider the following atom-field interaction Hamiltonian

HE = d ·E (A.11) where E is a static electric ﬁeld and d is the atomic dipole operator. The energy level shift for an arbitrary state |αi is

2 X |hα|HE|βji| ∆E = hα|H |αi + (A.12) α E E − E j α βj

with |βji labeling all other atomic states and Eα and Eβj representing the corresponding energy levels. Because the dipole operator can only couple states of opposite parity, the first-order shift to the energy vanishes leaving only the second-order term. The effect is therefore second order in E and is called the quadratic Stark effect. Defining an ‘effective’ Stark interaction Hamiltonian as

X HE|βjihβj|HE X dµ|βjihβj|dν H := = E E (A.13) Stark E − E E − E µ ν j α βj j α βj

makes the energy shift look ‘first-order’ (albeit it is really a second order effect from second order perturbation theory). The Stark Hamiltonian has the form of a rank-2 tensor operator, contracted twice with the electric field vector.

2 X |hJkdkJ 0i|2 α(0)(J) := − (A.17) 3 EJ − EJ0 J0 APPENDIX A. PROPERTIES OF LITHIUM 131 with the energy shift given by 1 ∆E(0) = − α(0)(J)E 2. (A.18) J 2 The tensor polarizability is defined as s 8J(2J − 1) α(2)(J) := −hJkS(2)kJi (A.19) q 3(J + 1)(2J + 2) with the energy shift given by 1 3m2 − J(J + 1) ∆E(2) = − α(2)(J)(3E 2 − E 2) J . (A.20) |JmJ i 4 z J(2J − 1) When the atom has hyperfine structure, the effective hyperfine Stark Hamiltonian is 1 1 3F 2/ 2 − F (F + 1) H (F ) = − α(0)(F )E 2 − (F )(3E 2 − E 2) z ~ (A.21) Stark 2 z 2 z F (2F − 1) and the quadratic Stark shift in energy is 1 1 3m2 − F (F + 1) ∆E = − α(0)(F )E 2 − α(2)(F )(3E 2 − E 2) F . (A.22) |F mF i 2 z 4 z F (2F − 1)

A.3 Interaction with dynamic ﬁelds

A.3.1 Reduced Matrix Elements in Atomic Transitions The electric-dipole transition matrix elements quantify the interaction between internal atomic states and an external optical field that is nearly resonant with an energy splitting of the states. While magnetic and even higher order multipole transitions are possible, these higher-order effects are much less influential and not considered here. The dipole operator, a rank k = 1 tensor, governs atomic electric-dipole transitions.For 0 0 a transition between hyperfine states |F mF i → |F mF i the probability amplitude is related to the Rabi frequency by

(i) 1 0 0 0 ΩF m F 0m = − hα F mF |d · Ei|αF mF i (A.23) F F 0 ~ where Ei is the electric component for a particular driving ﬁeld in a Raman system and d = er is the atom’s dipole moment. Decomposing the ﬁeld and position vectors into irreducible tensor operators yields

X q d · Ei = eEi (−1) rq−q (A.24) q APPENDIX A. PROPERTIES OF LITHIUM 132

with i being the polarization vector for the electric ﬁeld and q ∈ −1, 0, 1. Applying the Wigner-Eckart theorem gives the following result r 0 0 0 F 0−F +m0 −m 2F + 1 0 0 hF m |d |F m i = hF kdkF i(−1) F F hF m |F m ; 1 − qi. F q F 2F 0 + 1 F F (A.25)

From the above expression, it is apparent that when considering the dependence on two magnetic sublevels of a matrix element, the entirety of the angular dependence is given simply by a Clebsch-Gordan coefficient. The reduced hyperfine matrix element can be decomposed in terms of the fine-structure reduced matrix element

hF kdkF 0i = hJIF kdkJ 0I0F 0i 0 J J’ 1 = hJkdkJ 0i(−1)F +J+1+I p(2F 0 + 1)(2J + 1) . (A.26) F’ F I

Furthermore, the ﬁne-structure reduced matrix element can be factored into a reduced matrix element depending only upon the atom’s orbital angular momentum, quantum number L, as

hJkdkJ 0i = hJSJkdkL0S0J 0i 0 L L’ 1 = hLkdkL0i(−1)J +L+1+Sp(2J 0 + 1)(2L + 1) . (A.27) J’ J S

A.4 Clebsch-Gordan coeﬃcients for D–line transitions APPENDIX A. PROPERTIES OF LITHIUM 133

Table A.5: Clebsch-Gordan coeﬃcients for the D2-line transition with σ+-polarized light 0 such that mF = mF + 1.

Table A.6: Clebsch-Gordan coeﬃcients for the D2-line transition with σ−-polarized light 0 such that mF = mF − 1. APPENDIX A. PROPERTIES OF LITHIUM 134

Table A.7: Clebsch-Gordan coeﬃcients for the D2-line transition with π-polarized light such 0 that mF = mF .

|F = 2, mF = 2i p p |F = 1, mF = −1i − 1/12 − 1/12 p p |F = 1, mF = 0i − 1/4 − 1/12 p |F = 1, mF = 1i − 1/2

Table A.8: Clebsch-Gordan coeﬃcients for the D1-line transition with σ+-polarized light 0 such that mF = mF + 1. APPENDIX A. PROPERTIES OF LITHIUM 135

0 0 0 σ+-polarization |F = F + 1i |F = F i |F = F − 1i

Table A.9: Clebsch-Gordan coeﬃcients for the D1-line transition with σ−-polarized light 0 such that mF = mF − 1.

Table A.10: Clebsch-Gordan coeﬃcients for the D1-line transition with π-polarized light such 0 that mF = mF . 136

Appendix B

Two-Level System

When the atom is in the presence of a weak external field, nearly on resonance with a single atomic transition, the physics simplifies to a two-level system. In this simplification, transitions to other states are negligible. Only a ‘ground’ |gi state and ‘excited’ |ei state are connected by absorption and perhaps stimulated emission of the field. The field given by

+i(k·r−ωt) ∗ −i(k·r−ωt) E(r, t) = ˆE0e + ˆE0 e (B.1)

is monochromatic with angular frequency ω with ˆ denoting the unit polarization vector. The lowest order contribution in the multipole expansion of the atom-field interaction, the dipole approximation, assumes that the wavelength of the field is much longer than the spatial extent of the atom. Any variations of the field over the atom system are ignored. This is generally true for cold atomic systems since atomic dimensions are of order an Angstrom and optical transitions are hundreds of -nms or thousands of Angstroms. The total Hamiltonian of this system is the sum of the free atomic Hamiltonian HA

HA = ~ω0|eihe| (B.2) (in which we have set the ground-state energy to zero) and the atom-ﬁeld interaction Hamil- tonian HAF given by HAF = −d ·E. (B.3)

The atomic dipole operator d is deﬁned in terms of the electron position re as

d = −qere (B.4)

The dipole operator can be decomposed with

I = |eihe| + |gihg| (B.5)

and simpliﬁed assuming that hg|d|gi = he|d|ei = 0, (B.6) APPENDIX B. TWO-LEVEL SYSTEM 137

d = hg|d|ei|gihe| + he|d|gi|eihg|

The dipole matrix elements are deﬁned asµ ˆab = ha|d|bi. The total Hamiltonian is given by the following expression

produces a pair of coupled diﬀerential equations for the probability amplitudes cg(t) and ce(t):

−iωt ∗ +iωt i~c˙e(t) = ~ωce(t) − µˆeg E0e + E0e cg(t) (B.9) −iωt ∗ +iωt i~c˙g(t) = −µˆge E0e + E0e ce(t). (B.10) In the rotating frame as deﬁned by

iωt ce(t) =c ˜e(t)e (B.11)

yields the following for these coupled equations

∗ +2iωt i~c˙e(t) = −~∆˜ce(t) − µˆeg E0 + E0e cg(t) (B.12) ∗ −2iωt ∗ i~c˙g(t) = −µˆeg E0e + E0 c˜e(t) (B.13) with detuning given by ∆ = ω − ω0. ±2iωt ±i(ω+ω0)t In the above expression, the terms E0e will oscillate rapidly as e compared to the others oscillating as e±i∆t. The slowly oscillating terms will dominate the dynamics of the system. In the rotating wave approximation, we chose to focus on the slow dynamics or alternatively renormalize to a coarse grained fs-time scale, assuming that |ω −ω0| << ω +ω0. Terms oscillating at optical frequencies are replaced by their zero average value. This is a reasonable approximation to make given the physical response time of modern optical detectors. APPENDIX B. TWO-LEVEL SYSTEM 138

With the rotating wave approximation implemented, the equations for the probability amplitudes become Ω c˜˙ (t) = i∆˜c (t) − i R c (t) (B.14) e e 2 g Ω∗ c˜˙ (t) = −i R c˜ (t) (B.15) g 2 e where the Rabi frequency Ω has been deﬁned as

2E0µ 2E0he|d|gi ΩR := − = − . (B.16) ~ ~

For a Gaussian beam of beam waist w0, the intensity at the beam’s center is given by P I = 2 πw0 where P is the total beam power. Rewriting this in terms of the electric field amplitude E0 gives cE 2 I = 0 0 2 Rewriting the field amplitude in terms of the total power yields the following expression for the Rabi frequency: µ r 2P Ω = ge . (B.17) ~w0 π0c The same above equations are alsp generated by an effective Hamiltonian for the atom in the rotating-frame: ˜ HA = ~∆|eihe|. The effective interaction Hamiltonian is then H = −~ Ω∗ |gihe| + Ω |eihg| (B.18) AF 2 R R

B.1 Flip-ﬂop

B.1.1 On resonance Solving the coupled equations for the amplitudes of the atomic states in the rotating frame yields the driven dynamics of this two-level system. When the light is exactly on resonance with the splitting of the atomic states (∆ = 0), the coupled equations for the probability amplitudes are Ω c˙ = −i c˜ g 2 e APPENDIX B. TWO-LEVEL SYSTEM 139

Ω c˜˙ = −i c . e 2 g (B.19)

Decoupled these reveal the final solution for the amplitudes in terms of the initial conditions for a field on resonance with the energy splitting of the atomic states (∆ = 0), Ωt Ωt c (t) = c (0) cos − ic˜ (0) sin g g 2 e 2 Ωt Ωt c˜ (t) =c ˜ (0) cos( ) − ic (0) sin( ). (B.20) e e 2 g 2 The square of the amplitudes gives the probability of finding the atom in either state as a function of time. If the atom is prepared so as to be in the ground state at t = 0 then we have the following ground- and excited-state populations as function of time Ωt 1 P (t) = |c (t)|2 = cos2 = (1 + cos Ωt) g g 2 2 Ωt 1 P (t) = |c˜ (t)|2 = sin2 = (1 − cos Ωt) e e 2 2 where the probability for either state oscillates as a function of interaction time. For an atom beginning in the ground state, if the field is pulsed on for a duration of t = π/Ω, then the atom will have unit probability to transition to the excited state. This type of pulse is referred to as a π-pulse and acts as the ‘mirror’ in our atom interferometer. Similarly, a beamsplitter pulse or π/2-pulse is realized when the field is pulsed on for a duration (t = π/2Ω) = such that the atom has a 50% probability of being transferred to the excited state.

B.1.2 Almost on resonance For a nonzero detuning, ∆ 6= 0, we begin again with the coupled diﬀerential equations for the probability amplitudes Ω c˙ = −i c˜ g 2 e Ω c˜˙ = i∆˜c − i c . e e 2 g Decoupling, solving and factoring produces

∆ Ω ∆ Ω˜ ∂ − i + i ∂ − i − i c = 0 t 2 2 t 2 2 g ∆ Ω ∆ Ω˜ ∂ − i + i ∂ − i − i c˜ = 0 t 2 2 t 2 2 e APPENDIX B. TWO-LEVEL SYSTEM 140

where now we can deﬁne the generalized Rabi frequency as √ Ω˜ := Ω2 + ∆2. (B.21)

General solutions to the above equations can be found in terms of the initial probability amplitudes at time t = 0 ˜ ˜ i∆t/2 Ωt i Ωt cg(t) = e cg(0) cos − [∆cg(0) + Ωce(0)] sin 2 Ω˜ 2 ˜ ˜ i∆t/2 Ωt i Ωt c˜e(t) = e c˜e(0) cos + [∆˜ce(0) − Ωcg(0)] sin (B.22) 2 Ω˜ 2 Therefore, for an atom initially in the ground state, the probability of excitation to the higher energy state is

2 ˜ 2 Ω 2 Ωt Ω 1 1 Pe(t) = sin = − (B.23) Ω˜ 2 2 Ω˜ 2 2 2 cos Ω˜t At nonzero detuning, we observe both a reduction in amplitude and an increase to the generalized Rabi frequency. 141

Appendix C

Bloch sphere

The Bloch sphere is a geometric interpretation of the two-level atomic system, interacting with an external optical field with Hamiltonian given by a sum of the free-atomic Hamiltonian and the interaction Hamiltonian ˆ ˆ ˆ H = HA + Hint. (C.1) Assuming that the driving field’s frequency is close to the splitting between states, at any instant the wave function for the atom must be a linear combination of the two energy eigenstates |ψi = c1|1i + c2|2i or written in the configuration basis the wave function for the system is the following:

Ψ(r, t) = c1(t)ψ1(r, t) + c2(t)ψ2(r, t). Substitution of the above general wave function into the time-dependent Schr¨odinger equation leads to the following expression for the coeﬃcients c1 and c2, dc i 1 = h1|Hˆ |1ic + exp(−iω t)h1|Hˆ |2ic (C.2) ~ dt int 1 0 int 2 dc i 2 = h2|Hˆ |1ic + exp(iω t)h2|Hˆ |1ic (C.3) ~ dt int 1 0 int 1

This two-dimensional state space warrants a description in terms of the Pauli matrices. Writing down the density matrixρ ˆ which describes the quantum system as a statistical ensemble generically given by X ∗ ρˆ = cncm|nihm| (C.4) n,m

for pure state |n, mi with amplitude cn,m. For the above two-level system, the density matrix is

" 2 ∗ # " # |c1| c1c2 ρ11 ρ12 ρˆ = |ΨihΨ| = ∗ 2 = (C.5) c2c1 |c2| ρ21 ρ22 APPENDIX C. BLOCH SPHERE 142

given in terms of the off-diagonal ‘coherences’, ρ12 and ρ21, which describe the coupling between levels due to the interaction with the field and the diagonal ‘populations’, ρ11 and ρ22, which detail the probabilities for the atom to be found in the associated pure state. The dynamics of the density matrix is described by the (Liouville variant form) time- dependent Schrödingerequation ∂ρˆ i = [H,ˆ ρˆ] (C.6) ~ ∂t with the following two-level Hamiltonian " # AC ΩR i(δt−φγ ) Ω1 2 e Hˆ = − ∗ . (C.7) ~ ΩR −i(δt−φγ ) AC 2 e Ω2 In the rotating frame, defined by

−iδt/2 c˜1 = c1e (C.8) iδt/2 c˜2 = c2e (C.9) the equations of motion for the populations and coherences are as follows ∂ρ˜ Ω Ω∗ i 11 = −~ R e−iφγ ρ˜ + ~ R eiφγ ρ˜ (C.10) ~ ∂t 2 21 2 12 ∂ρ˜ Ω∗ Ω i 22 = −~ R eiφγ ρ˜ + ~ R e−iφγ ρ˜ (C.11) ~ ∂t 2 12 2 21

∂ρ˜ Ω i 12 = (δ − δAC )˜ρ + ~ R e−iφγ (˜ρ − ρ˜ ) (C.12) ~ ∂t ~ 12 2 11 22 ∂ρ˜ Ω∗ i 21 = − (δ − δAC )˜ρ − ~ R eiφγ (˜ρ − ρ˜ ). (C.13) ~ ∂t ~ 21 2 11 22

The density operator can be expanded in the complete basis of Pauli matrices (σx, σy, σz) as 1 ρˆ = 1 + (ρ − ρ )σ + (ρ + ρ )σ + (ρ − ρ )σ . (C.14) 2 22 11 z 21 12 x 21 12 y where σx, σy, and σz are deﬁned as

0 1 0 −i 1 0 σ = σ = σ = (C.15) x 1 0 y i 0 z 0 -1

The expectation value of an arbitrary operator Aˆ can be computed using the trace formula

hAˆi = Tr[ˆρAˆ] = hψ|Aˆ|ψi. APPENDIX C. BLOCH SPHERE 143

The Bloch vector is deﬁned as

R = uxˆ + vyˆ + wzˆ (C.16)

for which the expectation values of the Pauli spin matrices deﬁne the amplitudes along the axes:

u = Tr(σxρ) =ρ ˜12 +ρ ˜21

v = Tr(σyρ) = −i(˜ρ21 − ρ˜12)

w = Tr(σzρ) =ρ ˜11 − ρ˜22. (C.17)

The dynamics of the expectation values are described in the absence of spontaneous emission by the following coupled diﬀerential equations: ∂u i = −(δ − δAC )v − Ω e−iφγ − Ω∗ eiφγ w (C.18) ∂t 2 R R ∂v 1 = (δ − δAC )u − Ω e−iφγ + Ω∗ eiφγ w (C.19) ∂t 2 R R ∂w = −iΩ∗ eiφγ ρ˜ − Ω e−iφγ ρ˜ . (C.20) ∂t R 12 R 21

∗ For a real Rabi frequency, ΩR = ΩR, the equations become ∂u = −(δ − δAC )v − Ω sin φ w (C.21) ∂t R γ ∂v = (δ − δAC )u + Ω cos φ w (C.22) ∂t R γ ∂w = −Ω cos φ v + Ω sin φ u. (C.23) ∂t R γ R γ The time-evolution of R can be represented as torque produced by the field Ω written as dR = R × Ω (C.24) dt with the field vector defined as follows:

AC Ω = Ωr cos φγxˆ + ΩR sin φγyˆ − (δ − δ )zˆ. (C.25)

The rate at which the Bloch vector R rotates around Ω is given by q 2 AC 2 |Ω| = ΩR + (δ − δ ) , (C.26) which is equal to the generalized two-photon Rabi frequency. Incorporating spontaneous emission would add a time-dependence to the magnitude of the vector. APPENDIX C. BLOCH SPHERE 144

C.1 Simulations of interferometry

The Bloch sphere provides a nice visualization in the context of atom interferometry, particularly when considering light-pulses that may or may not induce the expected rotation.

C.1.1 Mathematica code ketΨ[ν , ϕ ]:={Cos[ν/2], Exp[I ∗ ϕ]Sin[ν/2]} Ψ00 = ketΨ[0, 0]; (*Initiallyin(1, 0)state*)

(*timeevolutionoperatorwhereθ = Ωr ∗ t, φisthelaserphaseandαknowsaboutthe2 − photondetuning*) U[θ , φ , α ]:=Cos[θ/2] ∗ ({{1, 0}, {0, 1}})− I ∗ Sin[θ/2] ∗ (PauliMatrix[1] ∗ Cos[φ] ∗ Cos[α] + PauliMatrix[2] ∗ Sin[φ] ∗ Cos[α]+ PauliMatrix[3] ∗ Sin[α])

Utilde[OMr , OM , t , φ ]:= Cos[OMr ∗ t/2] ∗ ({{1, 0}, {0, 1}})− I ∗ Sin[OMr ∗ t/2]∗ (PauliMatrix[1] ∗ Cos[φ] ∗ OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗ OM/OMr+ PauliMatrix[3] ∗ Sqrt[OMr∧2 − OM∧2]/OMr)

UUtilde[θ , φ , δ , OM , OMr ]:= Cos[θ/2] ∗ ({{1, 0}, {0, 1}})− I ∗ Sin[θ/2]∗ (PauliMatrix[1] ∗ Cos[φ] ∗ OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗ OM/OMr− PauliMatrix[3] ∗ δ/OMr) (*Hopfﬁbration; mappingSU(2)toSO(3)*) SPINORmapR3[spinor ]:=Module[{new, x1, x2, x3, x4, z1, z2, z3}, new = {}; x1 = Re[spinor[[1]]]; x2 = Im[spinor[[1]]]; x3 = Re[spinor[[2]]]; x4 = Im[spinor[[2]]]; z1 = 2 ∗ (x1 ∗ x3 + x2 ∗ x4); z2 = 2 ∗ (x2 ∗ x3 − x1 ∗ x4); z3 = (x1)∧2 + (x2)∧2 − (x3)∧2 − (x4)∧2; new = {z1, z2, z3}; out = new ] APPENDIX C. BLOCH SPHERE 145

points1 = Table[SPINORmapR3[Utilde[Pi/2 ∗ OM, Pi/2, 1, 0].Ψ00], {OM, 1, 10, 0.1}]; points1v2 = Table[SPINORmapR3[Utilde[Pi/2 ∗ (1 + ) ∗ OM, Pi/2 ∗ (1 + ), 1, 0].Ψ00], {, 0, 0.25, 0.01}, {OM, 1, 2, 0.1}]; points2 = Table[SPINORmapR3[Utilde[Pi/2 ∗ OM, Pi/2, 1, 0].Ψ00], {OM, 1, 2, 0.05}]; points3 = Table[SPINORmapR3[Utilde[Pi ∗ OM, Pi, 1, 0].Ψ00], {OM, 1, 10, 0.1}]; points4 = Table[SPINORmapR3[Utilde[Pi ∗ OM, Pi, 1, 0].Ψ00], {OM, 1, 2, 0.05}];

Graphics3D[{{Opacity[0.1], Sphere[]}, {Dashed, Line[{{−1.25, 0, 0}, {1.25, 0, 0}}]}, {Dashed, Line[{{0, −1.25, 0}, {0, 1.25, 0}}]}, {Dashed, Line[{{0, 0, −1.25}, {0, 0, 1.25}}]}, {Arrowheads[Medium], Arrow[{{1.25, 0, 0}, {1.35, 0, 0}}]}, {Arrowheads[Medium], Arrow[{{0, 1.25, 0}, {0, 1.35, 0}}]}, {Arrowheads[Medium], Arrow[{{0, 0, 1.23}, {0, 0, 1.35}}]}, Text[Style[x, 14], {1.45, 0, 0}], Text[Style[y, 14], {0, 1.45, 0}], Text[Style[z, 14], {0, 0, 1.45}], {{PointSize[Medium], Pink, Point[points1]}}, {PointSize[Medium], Blue, Point[points2]}}, Boxed → False] Graphics3D[{{Opacity[0.1], Sphere[]}, {Dashed, Line[{{−1.25, 0, 0}, {1.25, 0, 0}}]}, {Dashed, Line[{{0, −1.25, 0}, {0, 1.25, 0}}]}, {Dashed, Line[{{0, 0, −1.25}, {0, 0, 1.25}}]}, {Arrowheads[Medium], Arrow[{{1.25, 0, 0}, {1.35, 0, 0}}]}, {Arrowheads[Medium], Arrow[{{0, 1.25, 0}, {0, 1.35, 0}}]}, {Arrowheads[Medium], Arrow[{{0, 0, 1.23}, {0, 0, 1.35}}]}, Text[Style[x, 14], {1.45, 0, 0}], Text[Style[y, 14], {0, 1.45, 0}], Text[Style[z, 14], {0, 0, 1.45}], {{PointSize[Medium], Black, Point[points3]}, {PointSize[Medium], Red, Point[points4]}}}, Boxed → False] Graphics3D[{{Opacity[0.1], Sphere[]}, {Dashed, Line[{{−1.25, 0, 0}, {1.25, 0, 0}}]}, {Dashed, Line[{{0, −1.25, 0}, {0, 1.25, 0}}]}, {Dashed, Line[{{0, 0, −1.25}, {0, 0, 1.25}}]}, {Arrowheads[Medium], Arrow[{{1.25, 0, 0}, {1.35, 0, 0}}]}, {Arrowheads[Medium], Arrow[{{0, 1.25, 0}, {0, 1.35, 0}}]}, {Arrowheads[Medium], Arrow[{{0, 0, 1.23}, {0, 0, 1.35}}]}, Text[Style[x, 14], {1.45, 0, 0}], Text[Style[y, 14], {0, 1.45, 0}], Text[Style[z, 14], {0, 0, 1.45}], {{PointSize[Small], Red, Point[points1v2[[1, All]]]}, {PointSize[Small], Black, Point[points1v2[[2, All]]]}, {PointSize[Small], Black, Point[points1v2[[4, All]]]}, {PointSize[Small], Black, Point[points1v2[[5, All]]]}, {PointSize[Small], Black, Point[points1v2[[6, All]]]}, {PointSize[Small], Black, Point[points1v2[[7, All]]]}, {PointSize[Small], Black, Point[points1v2[[8, All]]]}, APPENDIX C. BLOCH SPHERE 146

Trajectory of probability amplitude on the Bloch sphere

Figure C.1: Model of trajectory of state on the Bloch sphere.

{PointSize[Small], Black, Point[points1v2[[9, All]]]}, {PointSize[Small], Black, Point[points1v2[[10, All]]]}, {PointSize[Small], Black, Point[points1v2[[3, All]]]}}}, Boxed → False] ListPlot[{0.5 ∗ (1 − points1[[1;;20, 3]]), 0.5 ∗ (1 − points3[[1;;20, 3]])}, PlotRange → Full] Plot[{(Pi/2) ∗ Sqrt[1 + δ∧2], (Pi) ∗ Sqrt[1 + δ∧2]}, {δ, 0, 10}] 147

Appendix D

Magneto-optical traps

When an atom scatters a photon, a deﬁned momentum with a particular direction and magnitude is imparted to the atom as it absorbs the photon. However, if spontaneous emission then follows, repeating this process will ultimately result in a net force or kick of the atom in the direction of the light. Momentum conservation is not the only principle at play in laser cooling atoms but it is the foundation upon which the rest of the subtleties reside. By carefully choosing the attributes of the laser light, optical molasses results so named because of the vicious damping force imposed by it on the atoms.

D.0.1 Optical molasses Consider an atom with nonzero velocity given by v irradiated by a collimated monochromatic laser beam. If the laser is tuned to be near resonant with an atomic transition, the atom will absorb photons from the beam and will scatter light from the with a rate given by 2 Γ Ω /2 Γ I/Isat Rs = 2 2 2 = 2 2 . (D.1) 2 δ + Ω /2 + Γ /4 2 1 + I/Isat + 4δ /Γ

Here, δ = ω − ω0 + k · v is the diﬀerence between the laser frequency ω and Doppler- shifted atomic resonance ω0 + k · v. The isotropicity of emission results in the atom being preferentially ‘kicked’ in the direction of the absorbed photon’s momentum. The magnitude of the resulting force upon the atom is as follows

Γ I/Isat Fs = ~k 2 2 . (D.2) 2 1 + I/Isat + 4δ /Γ

The Rabi frequency Ω and saturation intensity Isat are related by the natural line width of the transition as I 2Ω2 = 2 . (D.3) Isat Γ In the limit of inﬁnite intensity I → ∞, the maximum scattering force is realized as kΓ F = ~ (D.4) max 2 APPENDIX D. MAGNETO-OPTICAL TRAPS 148 and the maximum acceleration of an atom with mass M is kΓ v a = ~ = r (D.5) max 2M 2τ where τ is the lifetime of the excited state and vr = ~k/m is the velocity at which the atom recoils after absorbing and emitting a photon with wavenumber k. Optical molasses is a Doppler-cooling technique which exploits the velocity-dependence of the radiation scattering force on atoms. Optical molasses consists of orthogonal pairs of counter-propagating laser beams with frequency ω, that are red-detuned or tuned below atomic resonance of a cycling transition, ω < ω0. As an atom moves towards a beam, it sees the beam’s frequency Doppler shifted, closer to resonance with the transitions δ± = δ ± kv. The force due to radiation pressure from these counter-propagating beams is given by 3 1 1 I Frad = ~k(Γ/2) 2 2 − 2 2 . (D.6) (δ − kv) + (Γ/2) (δ + kv) + (Γ/2) Isat For small velocities, the force becomes viscous and ‘molasses-like’; it is proportional to v,

~k2γ3 δ I Frad = 2 2 2 v, (D.7) 2 (δ + (γ/2) ) Isat with a velocity capture range of |∆| γ γλ ± = ± . (D.8) k 2k 4π The lower limit to cooling in this picture is referred to as the Doppler temperature,

~Γ TD = (D.9) 2kB and is in principle limited by the line-width of the transition.

D.0.2 Magnetic trapping The velocity-dependent force of optical molasses confines the atoms in momentum space but not position. While the atoms are sludging through the optical molasses, the preferred direction for absorption combined with isotropic emission of momentum will cause the atom to eventually diffuse outward. To remedy this, a position-dependent force is created by coupling a magnetic quadrupole field to the aforementioned optical molasses in a hybrid trap, referred to as a magneto-optical trap (MOT). The quadrupole field imposes a quantization axis and creates spatially varying Zeeman shifts of the atomic energy levels,

∆EF,mF = µBgF mF B (D.10) APPENDIX D. MAGNETO-OPTICAL TRAPS 149

mF -1 +1

0 0 Δ

+1 -1 ω ω 0

σ+ σ-

0 B(z) z

Figure D.1: The combined effect of quadrupole field and optical molasses, here orthogonally polarized, confines the atoms spatially in addition to cooling them to lower temperature. The hybrid trap is called a magneto-optical trap.

where gF is the Landég-factor, mF is the magnetic quantum number, B is the field magnitude and µB is the Bohr magnetron. The energy shift changes sign through the trap center at which |B| = 0. The magneto-optical trap is created by adding counter-propagating laser beams, of opposite circular polarization and red-detuned from resonance to the quadrupole field. At low intensity, the total force on the atoms becomes

~kγ s0 F± = ± 2 . (D.11) 2 1 + s0 + (2δ±/γ)

The detuning δ± now has a contribution from the Zeeman shift given by

0 δ± = δ ∓ k · v ± µ B/~, (D.12)

0 e e g g where µ ≡ (gF mF − gF mF )µB is the eﬀective magnetic moment of the transition. For an atom moving outward from the trap center, the magnetic ﬁeld will tune either ∆mF = 1 or APPENDIX D. MAGNETO-OPTICAL TRAPS 150

g,e ∆mF = −1 closer to resonance, depending upon the sign gF . Fortunately, the transition not tuned closer will shift further from resonance, with the roles being reversed on the other side of the trap’s zero. An appropriate choice of beam polarization, will push the atoms back towards the center of the trap. For example, σ− polarization incident on an atom moving 0 rightward at z > 0, for which the transition ∆mF = −1 is tuned closer, will push the atoms back to the center. The density of trapped atoms is limited by outward radiation pressure or the re-absorption of scattered photons. 151

Appendix E

α0, the static polarizability

Lithium’s atomic properties can be computed with high accuracy, only utilizing ab initio wave functions. This is a result of lithium’s simple electron structure. The Stark shift measurements [74] are currently the most stringent test of the polarizability calculations for lithium. This measurement gave a polarizability diﬀerence of -37.14(2) a.u. for the 2s−2p1/2 transition and is in excellent agreement with the 7Li Hylleraas diﬀerence computed to be -37.14(4) a.u.

E.1 Nonrelativistic α(0)

Beginning with a nonrelativistic Hamiltonian for an inﬁnitely heavy lithium atom in the presence of an external electric ﬁeld E

X pa X Z X 1 X i i H = − + − E ra (E.1) 2 ra rab a a a>b a

Transition Value Method 2s − 2p1/2 Th. Hylleraas 37.14(3) Th. CICP 37.26 Th. RLCCSDT 37.104 Expt. 37.146(17) Expt. 37.11(33) 2s − 2p3/2 Th. RLCCSDT 37.089 MBPT Expt. 37.30(42)

7 Table E.1: Scalar polarizability diﬀerences α0(nPJ ) − α0(nS) in a.u. for Li. APPENDIX E. α0, THE STATIC POLARIZABILITY 152

the nonrelativistic electric dipole polarizability α0 can be evaluated with second-order perturbation theory as 2 X i 1 i α0 = − hφ0|ra rb|φ0i (E.2) 3 E0 − H0 a,b where φ0 and E0 are the ground-state wave function and nonrelativistic energy, respectively. Here, H0 represents the Hamiltonian in the absence of the external ﬁeld. Modifying H0 by δH to include corrections from the ﬁnite mass of the nucleus, relativity and QED given by

2 3 δH = λHMP + α Hrel + α HQED (E.3) where λ = −µ/M is the ratio of the reduced electron mass to the nucleus mass and MP denotes the mass polarization correction. With δH, a perturbative formula for the electric dipole polarizability yields

αE = α0 + δαE (E.4) with 3 X 1 i 1 i − δαE = 2hφ0|δH ra rb|φ0i 2 (E0 − H0) E0 − H0 a,b i 1 1 i +hφ0|ra (δH − hδHi) rb|φ0i . (E.5) E0 − H0 E0 − H0

The components of δH are given by [102]

X i i HMP = − papb, (E.6) a

hP pi (H − E ) ln[2(H − E )] P pi i ln k ≡ a a 0 0 0 0 b b (E.9) 0 P 3 2πZ chδ (rc)i APPENDIX E. α0, THE STATIC POLARIZABILITY 153

3 and P (1/rab) is given by 1 Z 1 hφ|P |ψi = lim d3rφ∗(r) Θ(r − a) r3 a→0 r3 +4πδ3(r)(γ + ln a) ψ(r). (E.10) 154

Appendix F

Hyperpolarizability

The energy shift that results due to the presence of an electric field of strength E can be computed perturbatively ∆E = ∆E2 + ∆E4. (F.1) The first and third order corrections are zero because of parity selection rules. Considering a static field, the second order energy is given by E 2 3M 2 − L(L + 1) ∆E = − [α(0) + α(2) ] (F.2) 2 2 L(2L − 1)

1 (0) (2) for L 6= 0, 2 , where α and α are the scalar and tensor dipole polarizabilities. The fourth-order correction E 4 ∆E = − [γ + γ g (L, M) + γ g (L, M)] (F.3) 4 24 0 2 2 4 4

is given in terms of γ0, the scalar hyperpolarizability, and γ2 and γ4, the tensor hyperpolarizabilities which is deﬁned as [82]

2 2L 128π 1 X γ0 = (−1) √ G0(L, La,Lb,Lc)T (La,Lb,Lc) (F.4) 3 2L + 1 LaLbLc s 128π2 L(2L − 1) X γ = (−1)2L G (L, L ,L ,L )T (L ,L ,L ) (F.5) 2 3 (2L + 3)(L + 1)(2L + 1) 2 a b c a b c LaLbLc s 128π2 L(2L − 1)(L − 1)(2L − 3) X γ = (−1)2L G (L, L ,L ,L )T (L ,L ,L ) 4 3 (2L + 5)(L + 2)(2L + 3)(L + 1)(2L + 1) 4 a b c a b c LaLbLc (F.6)

Here, the functions T (La,Lb,Lc) and GΛ(L, La,Lb,Lc) are given by the following expressions and parameterize the sum over intermediate states and the coupling as deﬁned by Clebsch-Gordarn coeﬃcients, respectively. APPENDIX F. HYPERPOLARIZABILITY 155

X hn0LkT1kmLaihmLakT1knLbihnLbkT1kkLcihkLckT1kn0Li T (La,Lb,Lc) = [Ek(Lc) − E0(L)][Em(La) − E0(L)][En(Lb) − E0(L)] kmn 2 2 2L−La−Lc X |hn0LkT1kmLai| X |hn0LkT1kkLci| −δ(Lb,L)(−1) [Em(La) − E0(L)] [Ek(Lc) − E0(L)] m k (F.7)

X 1 1 K 1 1 K K K Λ G (L, L ,L ,L ) = (Λ,K ,K ) 1 2 1 2 Λ a b c 1 2 0 0 0 0 0 0 0 0 0 K1K2 1 1 K 1 1 K K K Λ × 1 2 2 1 LLb La LLb Lc LLLb (F.8)

In the case that L = 0, one only needs to consider to terms 128π2 1 2 γ = T (1, 0, 1) + T (1, 2, 1) . (F.9) 0 3 9 45 Explicitly these are deﬁned as

X h2SkT1kmP ihmP kT1knSihnSkT1kkP ihkP kT1k2Si T (1, 0, 1) = [Ek − E0][Em − E0][En − E0] kmn 2 2 X |h2SkT1kmP i| X |h2SkT1kkP i| − [Em − E0] [Ek − E0] m k (F.10) and

X h2SkT1kmP ihmP kT1knDihnDkT1kkP ihkP kT1k2Si T (1, 2, 1) = . [Ek − E0][Em − E0][En − E0] kmn (F.11)

However, the previous expressions do not take into account any frequency dependence of the hyperpolarizability. Following the methodology outline in Ref. [159] and Ref. [160] the frequency dependence of the hyperpolarizability can be determined. The polarization response of an atom in an applied oscillating electric ﬁeld E(t) is characterized by the polarizability (linear) and hyperpolarizability (nonlinear) seen in an expansion of the induced dipole moment given by the following

µα(ωσ) = ααβ(−ωσ; ω1)Eβ(ω1) APPENDIX F. HYPERPOLARIZABILITY 156

1 + β (−ω ; ω , ω )E (ω )E (ω ) 2 αβγ σ 1 2 β 1 γ 2 1 + γ (−ω ; ω , ω , ω )E (ω )E (ω )E (ω ) 6 αβγδ σ 1 2 3 β 1 γ 2 δ 3 (F.12) where each term above representative of an induced dipole at a frequency for a particular set of field components X ωσ = ωi. (F.13) i Each combination of polarization (subscripts) and frequencies (arguments) for the applied field components corresponds to a particular nonlinear process. For the instance that the external field is optical, then the second term in the above expansion is zero due to parity selection rules. Therefore, the dynamic (second) hyperpolarizability is given by

γαβγδ(−ωσ; ω1, ω2, ω3) = (F.14)

F.1 Positive and negative frequency components

For an atom in state |gi in the presence of an external optical ﬁeld, the ac-Stark shift is given by 2 2 2 ~|Ω(r)| |hg|εˆ· d|ei| |E0(r)| ∆Eg = = (F.15) 4∆ ~(ω − ω0) for a monochromatic ﬁeld of the form

(+) −iωt E(r) =ε ˆE0 (r)e + c.c. (F.16)

Adding in the energy shift due to the counter-rotating term, undoing the rotation way approximation, leads to the following expression

2 2 2 2 2 2 |hg|εˆ· d|ei| |E0(r)| |hg|εˆ· d|ei| |E0(r)| 2ω0|hg|εˆ· d|ei| |E0(r)| ∆Eg = − = 2 2 , (F.17) ~(ω − ω0) ~(ω + ω0) ~(ω0 − ω ) which produces the scaling with ω0 from previous expressions. APPENDIX F. HYPERPOLARIZABILITY 157

Previous formalism takes this into account explicitly by considering both the total response tensor, a sum of the ‘positive’ (nonsecular) and ‘negative’ (secular) frequency tensors given by Xn = Xn(+) + Xn(−) (F.18) where each is deﬁned as

n(+) −n X X X X = ~ ... h0|zˆ|a1i a1 a2 an X ×ha1|zˆ|a2i...han|zˆ|0i [(ωa1 − ωσ) P −1 ×(ωa2 − ωσ + ω1)...(ωan − ωn)] (F.19) and the other component for the second hyperpolarizability is

3(−) −3 X X 2 2 X = −~ |h0|zˆ|a1i| |h0|zˆ|a3i| a1 a3 X −1 × [(ωa1 − ωσ)(ωa3 − ω2)(ωa3 + ω3)] (F.20) P

The polarizability α(ω) can similarly be expressed in terms of an Ln dependence as

2 8π X |hn0LkT1kmLai| α(L ) = (F.21) a 9(2L + 1) [E (L ) − E (L)] m m a 0 which can be revised to maintain the frequency dependence of α as

2 8π X (Em(La) − E0(L))|hn0LkT1kmLai| α (L , ω) = . (F.22) 1 a 9(2L + 1) [(E (L ) − E (L))2 − ω2] m m a 0

At the tune-out frequency ω = ωto, this term vanishes and reinterpreting the contributions to the hyperpolarizability in this approximation (at the tune-out) allows for a simpli- ﬁcation in that the T (1, 0, 1) term will transform into a sum that looks very much like the T (1, 2, 1) term, except with (n, Lb) = (n, S) instead of (n, D). 158

Appendix G

Matlab simulation of thermal cloud

At a temperature of 300 µK, the velocity width of the cloud prevents a closed functional form for the anticipated contrast of the interference fringes. A Monte Carlo simulation is used to infer the eﬀect of the such a warm cloud.

G.0.1 Code The following script sets the parameters for the ‘precompute’ function given an intensity and pulse time pair.

BiCubicFUNisHAPPENing = 0; %do we interpolate? HOTcloud = 0; %are we using atoms >0 K

if HOTcloud HOTtau = true; else HOTtau = false; end flipFLOP = 1; %if true, determine pi/2 via plot&fit PROject = 0; %true to QPN, false to not STARKshift = 1; %ac Stark shifts SUM = 1; %sum all atoms in state CONJUgate = 0; PROcess = 0; nSims = 100; w0 = 2.1; sigr= 0.5; TEMP = 300; %in K APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 159 sV =@(TEMP) sV(TEMP)/1000; %in mm/us dL = -4.00e-3; %MHz pow1= 0.013; pow2= 0.030; %power in Raman beams Delta0 = -200; %MHz sigD = 1.0; %line width

% Define pulse shape pS = @(int)square(int); % T-Values TValues = 160:0.30:230; if HOTtau tauROOT= findtau(pow1,pow2,sV(300),sigr, w0, Delta0, dL, 100, STARKshift); else tauROOT= findtau(pow1, pow2, 0, 0, w0, Delta0, dL, 100, STARKshift); end

%% determine pi/2 pulse length w/ plotting && fitting %% if flipFLOP TAUs= 0:0.002:0.500; out= zeros(5,length(TAUs)); ii=1; if HOTtau for ii=1:5 out(ii,:)= arrayfun(@(TAU) rabiFLOP(TAU,pow1,pow2,sV(300),... sigr,w0,Delta0,dL,100,STARKshift),TAUs); end else for ii=1:5 out(ii,:)= arrayfun(@(TAU) rabiFLOP(TAU,pow1,pow2,0,0,... w0,Delta0,dL,100,STARKshift),TAUs); end end

tauFIT = fitTAU(out,TAUs,pow1,pow2);

else fprintf(’\n NO FLIP--FLOP, tau is %f\n’,tauROOT); APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 160

end

% Format file name to save data to fileName = formattedFileName(tauROOT,dL,sigD,PROject,HOTcloud,SUM); if BiCubicFUNisHAPPENing

int1= 2*pow1/pi/w0/w0*10^(-6); %J/mm/us int2= 2*pow2/pi/w0/w0*10^(-6); %J/mm/us intR = int2/int1;

diary(’precompute/preprecomputeDiary.txt’); fprintf(’*** prePrecomputes Started %s ***\n’, datestr(now,’yyyy-mm-dd HH:MM:SS.FFF’)); startTime = clock;

preprecompute(int1,intR,Delta0,sigD,dL,tau,fileName) fprintf(’\n*** precompute Complete %s ***\n’, datestr(now,’yyyy-mm-dd HH:MM:SS.FFF’)); fprintf(’Time elapsed is %f minutes\n’,etime(clock,startTime)/60); diary(’off’); end

% Format file name to save data to fileName = formattedFileName(tauFIT,dL,sigD,PROject,HOTcloud,SUM); jj=1; startTime0 = clock; output= zeros(5,length(TValues)); for jj=1:5 diary(’simulationLogFile.txt’); fprintf(’*** Simulations Started %s ***\n’, datestr(now,’yyyy-mm-dd HH:MM:SS.FFF’)); startTime = clock; % Simulate over T values output(jj,:)= arrayfun(@(T) simulatef0is1(fileName,tauFIT,pow1,pow2,T,w0,... sV(300),sigr,Delta0,sigD,dL,... nSims,PROject,HOTcloud,SUM,CONJUgate),TValues); fprintf(’*** Simulations Complete %s ***\n’, APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 161

datestr(now,’yyyy-mm-dd HH:MM:SS.FFF’)); fprintf(’Time elapsed is %f minutes \n’, etime(clock,startTime0)/60); diary(’off’); pause(0.1); end

G.0.2 Matlab functions G.0.2.1 Preprecompute.m This function precomputes the simultaneous conjugate interferometers for a particular choice of parameters. function preprecompute(int0,intR,Delta0,sigD, ... delta,tau,pS,fileName) if (sigD>0) dDelta = (-4*sigD:sigD/10:4*sigD)+Delta0; else dDelta = (-2:0.1:2)+Delta0; end int = (0.00:0.05:2.00)*int0; ndD = numel(dDelta); nInt = numel(int); fprintf(’\nStarted precomputing %s.mat at %s\n’,... fileName,datestr(now,’yyyy-mm-dd HH:MM:SS.FFF’));

%precompute map = zeros(ndD,nInt,10); for i= 1:nInt redmap= zeros(ndD,10); for j= 1:ndD redmap(j,:)= SCIk(int(i),intR,dDelta(j),delta,tau,pS); end map(:,i,:)=redmap; end

%test precompute APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 162 randDelta = normrnd(Delta0,sigD); I0 = int0*exp(-2*4); val = abs(raman(I0,intR,randDelta,delta,tau,0,0,2,2,pS)^2 ... -bicubic(dDelta,int,map(:,:,8),randDelta,I0)^2); if ~exist(’precompute’,’dir’) mkdir(’precompute’); end

% Save data to .mat file save([’precompute/’,fileName],’map’,’dDelta’,’int’,’int0’);

% Calculate precompute time precompTime = toc; endTime = now; endDateFormatted = datestr(endTime,’yyyy-mm-dd’); endTimeFormatted = datestr(endTime,’HH:MM:SS.FFF’);

% Output to command line fprintf(’\nSaved %s.mat with [map]= %sx%sx10.\n’,fileName,num2str(ndD), num2str(nInt));

% Open log file to track and record progress logFileName = sprintf(’precompute/precomputeLogFile%s.csv’,endDateFormatted);

% Create header of log file if new file if ~exist(logFileName) logFile = fopen(logFileName,’w’); fprintf(logFile,’tau,precompTime,endTime,fileName,endTimeFormatted\n’); else logFile = fopen(logFileName,’a’); end

% Write to log file fprintf(logFile,’%d,%d,%f,%f,%f,%f,%s,%s\n’,... tau,precompTime,endTime,fileName,endTimeFormatted); fclose(logFile);

G.0.2.2 simulatef0is1.m function out= APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 163 simulatef0is1(fileName,tau,pow1,pow2,T,waist,sigV,sigr,... Delta0,sigD,dL,nSims,project,HOTcloud,SUM,CONJugate)

omrec = 63.13e-3; %MHz vr= 0.85e-4; %mm/us k = 9364.23; int1= 2*pow1/(pi*waist^2)*10^(-6); %J/mm/us int2= 2*pow2/(pi*waist^2)*10^(-6); %J/mm/us exSUMMed = complex(zeros(1,nSims)); exCONJugate = complex(zeros(1,nSims)); exNORMal = complex(zeros(1,nSims)); exALL = complex(zeros(1,nSims)); exOUT = complex(zeros(1,nSims));

D = Delta0+randn(1,4)*sigD; Tp=10; rMax= 2.0; x0= 0; y0= 0; cup = @(T) exp(1i*commPhi(T,dL) + 1i*diffPhi(T,omrec)); cdwn = @(T) exp(1i*commPhi(T,dL) - 1i*diffPhi(T,omrec)); tic for j=1:nSims detectable= false; while ~detectable if HOTcloud vx= randn()*sigV; vy= randn()*sigV; vz= randn()*sigV/vr; % initial position is offset so that first pulse is centered x0= randn()*sigr; y0= randn()*sigr; z0= randn()*sigr; else vx= 0;vy= 0;vz= 0; %mm/us x0= 0;y0= 0; end

xf(j)= x0+(2*T+Tp)*vx;yf(j)= y0+(2*T+Tp)*vy;

if xf(j)^2+yf(j)^2 < rMax^2 detectable= true; APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 164

end end

% first pulse x1 = x0; y1 = y0; a = exp(-2*(x1^2+y1^2)/waist^2); C1a= DOPPraman(k,vz,a*int1,a*int2,D(1),dL,tau,0,0,1,1); C2a= DOPPraman(k,vz,a*int1,a*int2,D(1),dL,tau,0,2,1,2);

% second pulse x2 = x0+vx*T; y2 = y0+vy*T; a = exp(-2*(x2^2+y2^2)/waist^2); C1b= DOPPraman(k,vz,a*int1,a*int2,D(2),dL,tau,0,0,1,1); C2b= DOPPraman(k,vz,a*int1,a*int2,D(2),dL,tau,0,2,1,2); C3b= DOPPraman(k,vz,a*int1,a*int2,D(2),dL,tau,2,0,2,1); C4b= DOPPraman(k,vz,a*int1,a*int2,D(2),dL,tau,2,2,2,2);

% third pulse x3 = x0+vx*(T+Tp); y3 = y0+vy*(T+Tp); a = exp(-2*(x3^2+y3^2)/waist^2); C1c= DOPPraman(-k,vz,a*int1,a*int2,D(3),dL,tau,0,0,1,1); C4c= DOPPraman(-k,vz,a*int1,a*int2,D(3),dL,tau,2,2,2,2); C5c= DOPPraman(-k,vz,a*int1,a*int2,D(3),dL,tau,0,-2,1,2); C6c= DOPPraman(-k,vz,a*int1,a*int2,D(3),dL,tau,2,4,2,1);

% fourth pulse x4 = x0+vx*(2*T+Tp); y4 = y0+vy*(2*T+Tp); a = exp(-2*(x4^2+y4^2)/waist^2); %C1d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,0,0,1,1); C4d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,2,2,2,2); C5d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,0,-2,1,2); %C6d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,2,4,2,1); C7d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,4,2,1,2); %C8d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,4,4,1,1); C9d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,-2,-2,2,2); %C10d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,-2,0,2,1);

%amplitudes for each path (interfering) exSUMMed(j)= interFEAR(C2a*C4b*C4c*C4d,C1a*C2b*C6c*C7d,cup(T))... +interFEAR(C1a*C1b*C1c*C5d,C2a*C3b*C5c*C9d,cdwn(T));

exCONJugate(j)= interFEAR(C2a*C4b*C4c*C4d,C1a*C2b*C6c*C7d,cup(T))... APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 165

+conj(C1a*C1b*C1c*C5d)*(C1a*C1b*C1c*C5d)... +conj(C2a*C3b*C5c*C9d)*(C2a*C3b*C5c*C9d);

exNORMal(j)= interFEAR(C1a*C1b*C1c*C5d,C2a*C3b*C5c*C9d,cdwn(T))... +conj(C2a*C4b*C4c*C4d)*(C2a*C4b*C4c*C4d)... +conj(C1a*C2b*C6c*C7d)*(C1a*C2b*C6c*C7d); %amplitudes for each path (non-interfering) exBACK(j)= conj(C2a*C3b*C1c*C5d)*(C2a*C3b*C1c*C5d)... +conj(C1a*C1b*C5c*C9d)*(C1a*C1b*C5c*C9d)... +conj(C2a*C4b*C6c*C7d)*(C2a*C4b*C6c*C7d)... +conj(C1a*C2b*C4c*C4d)*(C1a*C2b*C4c*C4d); if SUM exALL(j)= exSUMMed(j)+exBACK(j); elseif (~SUM) && CONJugate exALL(j)= exCONJugate(j)+exBACK(j); elseif (~SUM) && (~CONJugate) exALL(j)= exNORMal(j)+exBACK(j); end

if project pp= rand; if (pp>=0) && (pp=exALL(j)) exOUT(j)=0; end else exOUT(j)= exALL(j); end end OUTput = [T, sum(exOUT)]; out = sum(exOUT); fileName1 = [fileName,’_’,num2str(nSimulations)]; save_data(’results/’,fileName1,OUTput); elapsed= toc; disp([’T = ’,num2str(T),’ in ’,num2str(elapsed),’ seconds’]); end 166

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