Light Storage and Retrieval with Nuclear Spin and a Study of Anisotropic Inelastic Collisions

University of Nevada, Reno

Light storage and retrieval with nuclear spin and a study of anisotropic inelastic collisions

A dissertation submitted in partial fulﬁllment of the

requirements for the degree of Doctor of Philosophy in Physics

Mei-Ju Lu

Dr. Jonathan D. Weinstein/Dissertation Advisor

May 2011 c 2011 - Mei-Ju Lu

THE GRADUATE SCHOOL

We recommend that the dissertation prepared under our supervision by

MEI-JU LU

entitled

Light storage and retrieval with nuclear spin and a study of anisotropic inelastic collisions

be accepted in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Jonathan D. Weinstein, Ph.D., Advisor

Ronald A. Phaneuf, Ph.D., Committee Member

Peter Winkler, Ph.D., Committee Member

Robert S. Sheridan, Ph.D., Committee Member

Joseph I. Cline, Ph.D., Graduate School Representative

Marsha H. Read, Ph. D., Associate Dean, Graduate School

May, 2011

i Abstract

Helium buﬀer-gas cooling is combined with laser ablation to produce large numbers of atoms and molecules at a cryogenic temperature. Absorption spectroscopy is employed to observe the target species and optical pumping is used to manipulate them.

In this thesis, the ﬁrst observation of electromagnetically induced transparency

1 (EIT) in a sample of ground-state ( S0) atomic ytterbium at 6 K is reported. The transparency is produced due to coherence between the optical ﬁeld and the nuclear spin state of the 173Yb nucleus. Because the nuclear spin states interact very weakly with their environment they are resistant to decoherence due to inelastic collisions and inhomogeneous ﬁelds. Consequently, atomic ensembles of pure nuclear spin states may be a superior medium for a variety of nonlinear optics and quantum information experiments. For example, the information carried by a light pulse is stored in the

1 173 nuclear spin of ground-state ( S0) Yb by using EIT. Storage times of hundreds of milliseconds are observed at 4 K in our system, which has a competitive performance compared to other systems using electronic spin.

The second topic of this thesis is the cold atom-helium collisions which are not only important for the fundamental studies, but also help to explore the ﬁeld of ultracold atoms. A combined experimental and theoretical study of angular momentum depolarization in cold collisions of 2P -state atoms is presented. It is shown

2 o that collision-induced Zeeman relaxation of ground-state ( P1/2) gallium and indium

4 2 o atoms in cold He gas is dramatically small compared to atoms in P3/2 states. These

2 o results suggest the feasibility of sympathetic cooling and magnetic trapping of P1/2- state atoms. The inelastic collisions between cold titanium atoms and helium gas

2 2 3 cause transitions between the ﬁne-structure levels of the 3d 4s FJ electronic ground ii state of atomic titanium over a temperature range from 5 to 20 K. The Ti-He inelastic collision cross-section is signiﬁcantly smaller than cross-sections measured for collisions of non-transition-metal atoms with noble gas atoms. The theoretical calculations of the inelastic cross-sections reproduce the magnitude and temperature dependence of the measurements, and attribute the suppression of inelastic collisions to titanium’s “submerged” d-shell valence electrons.

3 Finally, large numbers of X ∆1 TiO molecules at a translational temperature of 5 K are generated. Their cold collisions with helium, including their elastic and

inelastic scattering cross-sections, are investigated and measured. As expected for

3∆ molecules, which have large spin-rotation couplings, TiO’s inelastic m-changing collision cross-section is large: on the same order as its momentum transfer cross- section. iii Citations to Previously Published Work

Portions of this thesis have appeared previously in the following papers:

“Fine-structure changing collisions in atomic titanium”, Mei-Ju Lu, Kyle

S. Hardman, Jonathan D. Weinstein, and Bernard Zygelman, Physical

Review A 77, 060701(R) (2008)

“Cold TiO-He collisions”, Mei-Ju Lu and Jonathan D. Weinstein, New

Journal of Physics 11, 055015 (2009)

2 “Suppression of Zeeman relaxation in cold collisions of P1/2 atoms”, T. V. Tscherbul, A. A. Buchachenko, A. Dalgarno, M.-J. Lu, and J. D. We-

instein, Physical Review A 80, 040701(R) (2009)

“Electromagetically induced transparency with nuclear spin”, Mei-Ju Lu

and Jonathan D. Weinstein, Optics Letters 35, 622 (2010)

“Stopped light with a cryogenic ensemble of 173Yb atoms”, Mei-Ju Lu,

Franklin Jose, and Jonathan D. Weinstein, Physical Review A 82, 061802

(2010) iv

Dedicated to my parents, my sister Helen, and my brother Kevin.

獻給我的、父佳母倫親與。青鴻

v Acknowledgements

It is my pleasure to acknowledge people who are very important to my academic studies.

I am deeply grateful to my advisor, Prof. Jonathan Weinstein, for his teaching, patience, guidance, and providing me excellent research opportunities. With his abundant knowledge and thoughtful personality, I enjoyed every second of being his student.

I would like to acknowledge members of Weinstein Lab for their collaboration with me in this thesis. I thank Kyle Hardman for his work on the cryogenic system and diode laser projects; Vijay Singh, for his work on the titanium experiment; Franklin

Jose, for his work on the Yb sputtering cell; Muir Morrison, for his work on the PID circuits; Matt Karam, for his work on the imaging system. I show appreciation to

Tian Li and Ryan Baker for their work on the Yb experiment, and Aja Ellis for her work on the cold molecule experiment. Without the eﬀort and support from these incredible people, this thesis could not be accomplished.

I show my true gratitude to my teachers and people in the society of physics who have taught and helped me. Finally, I would like to thank my family and friends for their love and encouragements. vi

Contents

Abstract...... i Citations to Previously Published Work...... iii Dedication...... iv Acknowledgements...... v Contents...... vi List of Tables...... x List of Figures...... xi List of Acronyms...... xiv

1 Introduction1 1.1 Buﬀer-gas cooling...... 1 1.2 Collisions...... 5 1.2.1 Elastic collisions...... 5 1.2.2 Inelastic collisions...... 6 1.2.3 Reactive collisions...... 6 1.3 Inelastic collisions of atoms...... 7 1.3.1 Spin-exchange...... 8 1.3.2 Dipolar relaxation...... 9 1.4 Inelastic collisions of diatomic molecules...... 12 1.4.1 Cold and ultracold molecules...... 13 1.4.2 Σ-state molecules...... 13 1.4.3 Non-Σ-state molecules...... 15 1.5 Electromagnetically induced transparency...... 17 1.5.1 EIT application...... 19

2 Apparatus 23 2.1 Design of the cryostat...... 23 2.1.1 The assembly of the cryostat...... 25 vii 2.1.2 Targets...... 30 2.1.3 Improvements...... 30 2.2 Measurements of temperature and pressure...... 31 2.2.1 Temperature measurements...... 31 2.2.2 Pressure measurements...... 33 2.3 Lasers...... 36 2.3.1 Ablation laser...... 36 2.3.2 Diode laser...... 38 2.4 Data acquisition system...... 41

3 Ytterbium 44 3.1 Long-lived coherent optically dense media...... 44 3.1.1 A pure nuclear spin system—J = 0 atoms...... 45 3.2 Yb information...... 46 3.3 Yb spectrum...... 47 3.4 Diﬀusion lifetimes and yields...... 51 3.5 Yb collisional properties...... 54 3.5.1 Optical pumping...... 54

3.5.2 The T1 time measurements...... 57

3.5.3 The T2 time measurements...... 60 3.6 EIT with nuclear spin...... 64 3.6.1 Dark state concepts...... 64 3.6.2 173Yb EIT...... 65 3.6.3 EIT with near-collinear beams...... 67 3.6.4 EIT with collinear degenerate beams...... 68

4 Slow and stopped light with nuclear spin 77 4.1 Atom-photon coupling...... 77 4.2 Frequency stabilization...... 79 4.2.1 Doppler-free DAVLL signal...... 80 4.2.2 Frequency lock...... 83 4.3 Slow light...... 85 4.3.1 Experimental details...... 86

4.3.2 Problems at high OD0 ...... 92 4.4 Stopped light...... 94 4.4.1 Methods...... 95 viii 4.4.2 Light-storage lifetimes...... 97 4.4.3 Storage eﬃciency...... 104 4.4.4 Towards quantum memory...... 105

5 Atom-He collisions 107 5.1 Ga-He and In-He...... 108 5.1.1 Spectroscopic structures...... 108 5.1.2 Inelastic collisions...... 111 5.2 Ti-He collisions...... 121 5.2.1 Signiﬁcance to astrophysics...... 122 5.2.2 Titanium spectrum...... 124 5.2.3 Fine-structure-changing collisions...... 125 5.2.4 Experimental and theoretical results...... 130 5.2.5 Other titanium experiments...... 133

6 TiO-He collisions 136 6.1 Basic information of TiO...... 136 6.1.1 Hund’s case (a)...... 137 6.1.2 TiO level structure...... 138 6.1.3 Spectroscopic structures...... 143 6.1.4 Ablation yields...... 144 6.2 Collisional data...... 145 6.2.1 Diﬀusion cross-section and vibrational relaxation...... 145 6.2.2 Rotational-changing collisions...... 148 6.2.3 m-changing collisions...... 149 6.2.4 Discussion...... 152

7 Summary 153

A EIT calculation 155 A.1 Methods...... 155 A.1.1 Interaction picture...... 155 A.1.2 Density matrix approach...... 157 A.2 Example: A three-level EIT system...... 158 A.2.1 The eﬀective Hamiltonian...... 158 A.2.2 Steady-state solutions...... 161 ix A.2.3 Absorption and the index of refraction...... 162 A.2.4 Simulations...... 164 A.3 Leak absorption of N-type EIT system...... 167 A.3.1 Model...... 168 A.3.2 Results...... 170

B Absorption cross-section 173 B.1 Beer-Lambert law...... 173 B.2 Absorption cross-section...... 174

C Diﬀusion models 175

D Rate coeﬃcient 177

E Transition strength 178

F Atomic/Molecular information 181 F.1 Yb...... 181 F.1.1 Transition strengths of Yb lines...... 182 F.2 Ga and In...... 187 F.3 TiO...... 191

G Miscellaneous 192 G.1 AOM driver...... 192 G.2 Helmholtz coils...... 193 G.3 Gaussian pulse...... 194 G.4 Saturation intensity...... 194

Bibliography 196 x

List of Tables

2.1 Ablation target information...... 30 2.2 Laser diodes information...... 39 2.3 Simpliﬁed speciﬁcations of data acquiring devices...... 43

5.1 Ga and In atomic data...... 108 5.2 Experimental and theoretical rate coefficients of 69Ga-4He and 115In-4He119 5.3 Titanium atomic natural abundance and atomic data...... 124 5.4 Ti-He J = 3 → J = 2 inelastic collision rate coefficients at different temperature T ...... 131 5.5 Thermally averaged Ti-He cross-sections at different temperatures.. 133

C.1 A summary of the thermally-averaged cross-section...... 176

F.1 Atomic data of ytterbium...... 181 F.2 Isotope shift of ytterbium...... 182 F.3 The decaying possibilities of the 69Ga excited states...... 187 F.4 The decaying possibilities of the 115In excited states...... 187 F.5 TiO molecular data...... 191

G.1 List of parts for the AOM driver...... 193 G.2 Information for 2008 bias coils...... 193 G.3 Information for 2010 bias coils...... 194 xi

List of Figures

1.1 The two- and three-level atomic systems...... 17 1.2 The responses of the absorption and refractive index with and without the EIT eﬀect...... 18

2.1 Assembly of the cryostat...... 25 2.2 A transparent view of the cryogenic system...... 26 2.3 A cut-away drawing of the cryogenic system...... 27 2.4 A schematic section view of the window assembly...... 28 2.5 Targets and target holders after laser ablation...... 31 2.6 Cooling-down curves of the cryostat...... 32 2.7 Pressure measurements of the cold cell...... 35 2.8 Optics setup for the pulse laser and its power output graph...... 36 2.9 Performance of Nichia NDV4313 laser diode...... 39 2.10 Typical optics setup for the diode laser...... 40

3.1 Yb absorption spectrum and level structure...... 49 3.2 Pressure broadening of Yb spectrum...... 50 3.3 Yb diﬀusion curve...... 52 3.4 Lifetime vs. helium density...... 53 173 3.5 Experimental setup for the Yb T1 time measurements...... 57

3.6 The T1 time measurement...... 59 173 3.7 T2 measurements of Yb nuclear spin...... 61 3.8 173Yb atomic magnetometer measurements...... 63 3.9 173Yb EIT level scheme...... 66 3.10 A schematic EIT setup with near-collinear beams...... 67 3.11 A schematic EIT setup with collinear degenerate beams...... 68 3.12 The ytterbium spectrum under a EIT condition...... 70 3.13 Types of the frequency detuning in the EIT eﬀect...... 71 xii 3.14 Yb transparency window with a two-photon detuning...... 72 3.15 The width and depth of the transparency window vs. control beam power...... 74 3.16 Yb oﬀ-resonance absorption vs. helium density...... 75

4.1 Doppler-free DAVLL optics setup...... 81 4.2 Yb DFSAS DAVLL diagram...... 82 4.3 Yb frequency lock signal...... 84 4.4 Observation of slow light...... 89 4.5 A series of slow light data...... 90 4.6 Time delay versus two-photon detuning...... 92 4.7 The experimental setup of the stopped light experiment...... 95 4.8 Storage of a classical light pulse for 10 ms...... 96 4.9 Efficiency of pulse storage and retrieval vs storage time...... 98 4.10 Lifetimes of the the stopped light experiment vs. helium density... 99 4.11 A test of the σ±-polarization detection...... 101 4.12 The retrieved efficiency vs. the storage time with the presence of a bias field...... 102

4.13 Delay-bandwidth-transmission product vs. OD0 ...... 106

5.1 Absorption spectrum of Ga alongside a 69Ga level structure...... 110 5.2 Absorption spectrum of indium alongside a 115In level structure... 111 5.3 Diffusion lifetimes of gallium and indium atoms vs. helium densities. 112 5.4 Monitoring of the F = 2 and F = 1 population of 69Ga atoms.... 114 5.5 Measured 69Ga F - and J-relaxation rates 1/τ for different 4He densities 115 5.6 Monitoring the Zeeman relaxation of 69Ga after optical pumping... 117 5.7 Zeeman relaxation of the 69Ga-He and 115In-He...... 117 5.8 A preliminary study of the Ga-Ga m-changing collisions...... 120 5.9 Simplified Ti level diagram and an absorption spectrum...... 125 5.10 Absorption spectroscopy of 48Ti atoms returning to equilibrium... 127 5.11 Fine-structure changing rates of 48Ti-4He vs helium density...... 129 5.12 48Ti diffusion curves for the J = 3 and J = 2 atoms...... 130 5.13 Plot of Ti-He experimental data and theoretical curves vs. temperature 132 5.14 Titanium spin decoherence...... 134 5.15 Ti-He pressure broadening...... 135 xiii 6.1 Vector diagram for Hund’s case (a)...... 137 6.2 A schematic level diagram of 48Ti16O molecules...... 139 6.3 Energy separation of TiO rovibrational states...... 140 6.4 Spectrum of TiO molecules...... 144 6.5 OD of different TiO vibrational states...... 146 6.6 The exponential decay lifetime of TiO...... 147 6.7 Rotational relaxation of TiO...... 148 6.8 A schematic Zeeman sublevel diagram of TiO...... 150 6.9 m-changing measurements of TiO molecules...... 151

A.1 Diagram of the three-level EIT system...... 159

A.2 EIT performance vs. coupling strength Ωc ...... 166

A.3 EIT performance vs. decoherence γg ...... 167 A.4 EIT performance vs. detuning...... 168 A.5 N-type EIT systems in multilevel atoms...... 169 A.6 The simulation of the N-type multilevel atoms...... 172

F.1 Simulated Yb spectra...... 184 173 1 1 0 F.2 Transition strength of Yb | S0,F = 5/2i → | P1,F = 5/2i ..... 185 173 1 1 0 F.3 Transition strength of Yb | S0,F = 5/2i → | P1,F = 3/2i ..... 185 173 1 1 0 F.4 Transition strength of Yb | S0,F = 5/2i → | P1,F = 7/2i ..... 185 171 1 1 0 F.5 Transition strength of Yb | S0,F = 1/2i → | P1,F = 3/2i ..... 186 171 1 1 0 F.6 Transition strength of Yb | S0,F = 1/2i → | P1,F = 1/2i ..... 186 1 1 0 F.7 Transition strength of Yb (I = 0) isotopes | S0,F = 0i → | P1,F = 1i 186 F.8 69Ga (I = 3/2) transition hyperﬁne structure...... 188 F.9 115In (I = 9/2) transition hyperﬁne structure...... 189 F.10 Branching ratios of Ga and In...... 190

G.1 AOM driver circuit setup...... 192 xiv List of Acronyms

Acronym Meaning AOM Acousto-optic modulator BEC Bose-Einstein condensation CPT Coherent population trapping DAVLL Dichroic-atomic-vapor laser lock signal DBW Delay-bandwidth product DBWT Delay-bandwidth-transmission product DFSAS Doppler-free saturation absorption spectroscopy DMA Direct memory access ECDL External cavity diode laser EIT Electromagnetically induced transparency EOM Electro-optic modulator FIFO First-in-ﬁrst-out FPI Fabry-Perot interferometer FWHM Full width at half maximum NIST National institute of standards and technology NMR Nuclear magnetic resonance OD Optical density PD Photodetector PID Proportional-integral-derivative QED Quantum electrodynamics RE Rare-earth atoms TTL Transistor-transistor logic signals 1

Chapter 1

Introduction

There are two main topics discussed in this thesis: the nuclear-spin based coherence experiments and the measurements of inelastic collisions. In the ﬁrst topic, we use nuclear spin states to demonstrate the coherent control of a long-lived atomic ensemble by optical ﬁelds, which is suitable for studying quantum information science. In the second topic, we study collisional properties of atoms and molecules in order to understand mechanisms causing decoherence or depolarization. We are especially interested in the inelastic collisions due to the anisotropic interaction potential.

Chapter1 outlines the contents of this thesis, including the experimental motiva- tion and physical concepts. The experimental apparatus and techniques are reviewed in Chapter2. Nuclear-spin experiments are described in Chapter3 and Chapter4, while the inelastic collisions due to the atomic anisotropy are measured in Chapter5.

Finally, a study of the inelastic collisions in molecules is written in Chapter6.

1.1 Buﬀer-gas cooling

In many atomic, molecular and optical experiments, it is important to have the ability to create a cold or ultracold environment to reveal amazing physics. One 2 simple example is laser spectroscopy because being at low temperature gives a better resolution of transitions.

Various cooling methods have been developed to cool species, and different cooling steps are needed depending on the experimental goals. More than one hundred years ago, people started to design different kinds of cryogenic refrigerators to achieve lower temperature. The refrigerator can have a temperature below 1 Kelvin by boiling off liquid helium or a 3He and 4He mixture. Even lower temperatures have been reached, down to millikelvin, through nuclear magnetic refrigeration [1].

Laser cooling is another approach to reach low temperatures. It can directly cool atoms down to microkelvin [2], and has been widely implemented in the past twenty years. Based on cycling photons between closed energy levels, laser cooling uses scattered photons to take energy away from atoms and cool them. If a colder temperature is needed, evaporative cooling of atoms in a magnetic trap can further lower the temperature [3,4]. Simply speaking, evaporative cooling makes atoms become colder by getting rid of the highest energy atoms in a magnetic trap and uses elastic collisions to thermalize them [5]. Finally, ultracold atoms can be generated by laser cooling and quantum degeneracy is created through the evaporative cooling. As a result of this process, the coldest temperature that has ever been achieved is in the atomic Bose-Einstein condensation whose temperature is on the order of nanokelvin

[6,7].

However, it is not easy to apply laser cooling to most of species in the periodic table, because multiple lasers are needed to have a continuous scattering cycle between the ground and excited state. For example, since the ground state of rubidium is split into two energy levels due to hyperﬁne interaction, there are two optical transitions between the ground and excited state. When atoms are scattered to the excited state by the laser, they may decay back to either hyperﬁne levels. If they do not return 3 to the original state, a second laser, the repumping laser, is used to optical pump them back to the original state. In this way, the laser cooling cycling can continue.

As a result, two lasers are required to perform laser cooling to rubidium atoms.

Adding repumping lasers is not practical for most species, because their internal level structures are complicated and they usually need more than two lasers. For example, it may need more than twenty lasers to laser cooling titanium atoms. In addition, powerful lasers with the transition frequencies also need to exist.

Laser cooling is usually good for speciﬁc atomic elements with simple level structures, like alkali atoms. Powerful lasers for exciting the transitions of alkali atoms are usually available. The atomic densities and numbers that can be obtained by laser-cooled related methods have their limitation. Laser-cooled atoms usually have an atomic density of 1010 − 1012 cm−3, and the atomic number is between 108 and

1010 atoms [8,9].

Normally, it is diﬃcult to study collisional behaviors in the gas phase at low temperature because most species freeze. For example, the densities of the gases like

9 −3 H2,F2,O2, and Ne in equilibrium are less than 10 cm at 4 K [10]. Helium gas holds the highest vapor pressure of all species, and has an equilibrium gas density greater than 1017 cm−3 around 1 K. The vapor pressure of 4He gas at 4 K is 426 Torr, while that of 3He gas is 562 Torr [10].

The De Lucia group has measured the pressure broadening of CO molecules at

4 K by thermalizing them through elastic collisions with a helium gas [11]. Combined with cryogenic refrigerators, helium gas can serve as a buffer gas at the cryogenic temperatures. Using a buffer gas in a cryogenic cell has two benefits. First, the buffer gas acts as a thermal link between the target species and the cryostat. Atoms or molecules elastically collide with helium through collisional thermalization. In this way, the translational temperature of the target species can be cooled down to the 4 cryogenic temperature quickly. Secondly, the helium gas increases the lifetimes of the atoms in the cell, because atoms have to diffuse through the helium gas by elastic collisions. Instead of striking to the wall directly, the interesting species can stay in the cell for a long time before they collide with the cell wall and freeze. In this thesis, we use 4He as the buffer gas, and a typical diffusion lifetime is on the order of a second.

Since helium buffer-gas cooling uses elastic collisions to cool the translational temperature of interesting species, there is no concern about the internal energy levels. In fact, helium buffer-gas is very general to atoms and molecules. For example, the Doyle group in Harvard has cooled more than fifty kinds of different atoms or molecules by helium buffer-gas cooling [12]. In the Weinstein lab, we use helium buffer-gas to cool and observe atoms like Ti, In, Ga, Yb and Li, and molecules such as TiO and CaH.

To introduce atoms or molecules into the cryogenic cell, five methods have been employed. Those are capillary filling [11], laser ablation [13], light-induced atom desorption (LIAD) [14], beam injection [15, 16], and discharge etching [12]. Methods like capillary filling, beam injection, and discharge etching require complicated design of cryogenic apparatus and complex procedures to introduce species. The LIAD method only works for alkali atoms so far. In this thesis, laser ablation is used to generate atoms or molecules. We place a stable solid sample inside the cryostat. A high energy laser pulse is focused onto the sample and ablates it, producing atoms or molecules. By using laser ablation and helium buffer-gas cooling, large numbers of atoms and high atomic densities are achieved. In the titanium experiments, we have observed atomic densities on the order of 1012 cm−3, numbers on the order of 1015, and optical densities greater than 300 [17].

The temperatures obtained through buffer-gas cooling are limited by the vapor 5 pressure of helium gas. Because the equilibrium gas density of helium decreases exponentially with the temperature, there will be not enough collisions to achieve thermal equilibrium before the cooling species disappears. As a result, buffer-gas cooling can not cool species to as low temperatures as laser cooling. However, it can be a good pre-cooling step, and cold atoms or molecules can be then loaded into a magnetic trap [18, 19]. A buffer-gas cooled Bose-Einstein condensate has been realized by buffer-gas loading the metastable helium atoms into a magnetic trap and evaporatively cooling to quantum degeneracy [20].

To study quantum sciences and inelastic collisions covered in this thesis, 4He buﬀer-gas cooling is suﬃcient. There are no further magnetic trapping or other cooling methods applied in this thesis.

1.2 Collisions

Collisions are usually classiﬁed into three categories which are elastic, inelastic, and reactive collisions [12].

1.2.1 Elastic collisions

In elastic collisions, the colliding particles remain in the same internal energy states during the collisional events. The collisions alter the movements of particles without inducing transitions between internal levels. Diffusion in a buffer gas is a typical example of elastic collisions. In general, the diffusion or momentum transfer cross- sections1 are on the order of 10−15 − 10−14 cm2 at 300 K [13]. Our measurements of the atom-helium and molecule-helium, thermally-averaged diffusion cross-section around 5 K, are summarized in Table C.1.

1Cross-section is deﬁned in AppendixB and AppendixC. 6

Note that there is a difference between the diffusion cross-section and the total cross-section. The total cross-section integrates the differential cross-section over all scattering angles, while the diffusion cross-section integrates the differential cross- section weighted by the scattering angle [21, 22, 23]. To separate the inelastic collisions from the total cross-section, the total elastic cross-section is used to describe the elastic collisions specifically.

1.2.2 Inelastic collisions

In inelastic collisions, colliding particles change their internal states with their chemical constitutions remaining the same. For example, a coherent superposition state, which is a composition of energy states with deﬁned phases, can be disturbed by inelastic collisions in a way that the population or the relative phase between energy states are depopulated or dephased. Then, the coherent superposition state becomes individual energy states with random phase. The mechanisms causing the inelastic collisions in atoms and molecules will be discussed in Chapter 1.3 and Chapter 1.4.

Measurements of inelastic collisions are one of the main subjects in this thesis.

Collision-induced Zeeman transitions (Zeeman relaxation or m-changing collisions),

ﬁne-structure changing collisions (J-changing collisions) and hyperﬁne-structure changing collisions (F -changing collisions) are the inelastic collisions of atoms reported in

Chapter3, Chapter4 and Chapter5. Inelastic collisions of molecules, including rotation-changing collisions, vibrational relaxation collisions and Zeeman relaxation, are discussed in Chapter6.

1.2.3 Reactive collisions

Reactive collisions are the chemical reactions in which resulting products after collisions are diﬀerent from the colliding particles. Not only the internal levels change, 7 but also their chemical forms. For example, the reactive collisions between atomic titanium and oxygen, Ti(g) + O2(g) → TiO(g) + O(g) + 1.8 eV, is an exothermic chemical reaction [24]. Cold chemical reaction experiments are another interest in the Weinstein lab, but they are not included in this thesis.

1.3 Inelastic collisions of atoms

As mentioned in Chapter 1.1, evaporative cooling in a magnetic trap is a common way to produce ultracold atoms. However, while elastic collisions enable the evaporative cooling, inelastic collisions cause a loss in a magnetic trap. To estimate whether the evaporative cooling in a system can succeed or not, a ratio γ of the elastic cross- section to the inelastic cross-section is used. In order to let evaporative cooling be practical, the ratio must be large enough [3]. In general, it requires γ ≥ 103 for evaporative cooling, while γ ≥ 102 for sympathetic cooling. It is impossible to achieve evaporative cooling if γ < 10. Additionally, for the buﬀer-gas loading into an evaporatively-cooled magnetic trap, γ needs to be at least greater than 104 for eﬃcient cryogenic cooling and trap loading [12]. A reasonable γ is relatively simple to achieve with most S-state atoms, but not with the non-S-state atoms, because the inelastic collision cross-section in the non-S-state atoms is usually large. The reason can be understood from the mechanisms causing inelastic collisions.

There are two important categories of inelastic collisions: spin-exchange collisions and dipolar relaxation collisions [25]. Some people call dipolar relaxation collisions spin relaxation collisions [3]. The diﬀerence between spin-exchange collisions and dipolar relaxation collisions are addressed in Jonathan Weinstein’s PhD thesis [13],

Spin-exchange collisions preserve the total angular momentum projection of the colliding atoms. Dipolar relaxation collisions do not preserve the total internal angular momentum of the colliding particles (but do con- 8

serve total angular momentum, of course, by coupling this internal angular momentum to the orbital motion of the two atoms [3]).

1.3.1 Spin-exchange

Collisional hyperfine relaxation induced by spin-exchange collisions occurs very rapidly between atoms, and typical rate coefficients2 for spin-exchange collisions in ultracold alkali atoms are on the order of 10−12 cm3 s−1 [3]. The fast inelastic collision rate increases the difficulty of preparing the spin states. However, for a different approach like spin-exchange optical pumping, spin-exchange is a powerful way to polarize the nuclear spin of a noble gas via collisions with optically pumped alkali-metal atoms

[26]. As a comparison, spin-exchange cross-sections between alkali-metal atoms and a noble gas with nuclear spin (for example, 3He and 129Xe) at warm temperature are in a range from 10−24 to 10−20 cm2. The corresponding rate coeﬃcients are roughly on the order of 10−19 − 10−15 cm3 s−1 [26].

Spin-exchange collisions can be turned oﬀ by using fully-polarized atoms3 because the spin states of the fully-polarized atoms can not be changed during spin-exchange collisions. In our atom-helium and molecule-helium experiments, spin-exchange is not important because the main colliding partner—the atomic 4He gas—has a ground-

2 1 state conﬁguration of 1s S0 which contains no internal structures, neither electron spin, nuclear spin, or orbital angular momentum (S = I = L = 0).

Spin-exchange collisions between atom-atom or molecule-molecule exist in our measurements, but they are insigniﬁcant compared to other inelastic mechanisms induced by 4He gas because their atomic densities are relatively low compared to that of helium gas.

2Rate coeﬃcient is deﬁned in AppendixD. 3 It is also called a “stretched” state, with all types of spins aligning along a common quantization axis [27]. 9 1.3.2 Dipolar relaxation

Unlike spin-exchange collisions, which can be shut down by using fully-polarized atoms, dipolar relaxation cannot because their total internal angular momentum is not preserved. It is often referred to as a “bad” collision. For example, it limits the eﬃciency of spin-exchange optical pumping and also causes a loss of atoms from a magnetic trap. (Elastic collisions are “good” collisions because they are the fundamental mechanism of buﬀer-gas cooling and evaporative cooling.) Dipolar relaxation collisions may be caused by magnetic dipole-dipole interaction [3, 28] and anisotropic interaction potential [26, 29].

For two colliding particles with spin, a direct magnetic dipole-dipole interaction is induced by the spin-spin dipolar interaction. The magnetic dipole-dipole interaction4 is described by ~µ · ~µ − 3(ˆr · ~µ )(ˆr · ~µ ) H (~r) = 1 2 1 2 (1.1) dipole r3 where µ1 and µ2 are the magnetic dipole moments of two colliding particles, and r is the distance between them [13, 28]. Diﬀerent alignments of the magnetic dipole moments may give rise to an attractive or repulsive force. Rate coeﬃcients from magnetic dipole-dipole collisions are usually less than 10−13 cm3 s−1 for two atoms with one Bohr magneton (µB). However, the magnetic dipole-dipole interaction becomes important for atoms with large magnetic moments because the rate scales as µ4 in atom-atom collisions

[31]. For atomic chromium having 6 µB, the eﬀect seems to be consistent with the magnetic trap loss in the Cr-Cr experiments [13]. Although atoms with large magnetic moments are good for magnetic trapping, they also suﬀer large trap loss from the large dipole-dipole interaction. The magnetic dipolar relaxation is also possible to be induced by the second-order spin-orbit interaction, which usually occurs in heavy

4Reference [30] gives a nice review of the dipole-dipole interaction. 10

atoms [32]. In our case, the main colliding particle—the ground-state 4He atom—

does not have any spin, so that there is no magnetic dipole-dipole interaction of 4He with atoms or molecules.

As for the interaction anisotropy, it may occur due to spin-spin, spin-orbital, spin- rotation, and orbital-orbital coupling [29, 33]. The anisotropic interaction potential is mainly generated from the electronic shell structures of the colliding particles.

The contributions of the interaction anisotropy can be described from the radial and angular parts of the charge distributions [12].

The angular part of charge distribution is described by the orbital angular momentum. For atoms with non-zero orbital angular momentum, the spin states of atoms are usually strongly coupled to the orbital angular momentum through fine-structure interaction and can be easily flipped by spin-orbital coupling in collisional events. A simple description for this effect is that the orbital angular momentum acts as a lever arm and colliding particles with sufficient kinetic energy provide a torque to change the spin states. S-state atoms have zero orbital angular momentum in their ground states, and have spherically symmetric shell structures. As a result, their electric interaction potential is isotropic, and the spin relaxation rate is expected to be very small when compared to the non-S-state atoms. However, it is possible that S-state atoms encounter spin relaxation collisions because their excited states with orbital angular momentum are mixed into the spin states, causing collisional transitions.

(This is the second-order spin-orbital interaction [32].) The typical spin-relaxation cross-section for alkali-metal atoms colliding with noble-gas atoms around 400 K are on the order of 10−25 −10−23 cm2 [34]. (The corresponding rate coeﬃcients are on the

order of 10−20 − 10−18 cm3 s−1.) A theoretical calculation in potassium-3He collisions

shows that the spin relaxation rate coeﬃcient can be reduced to be 6 orders smaller

with a temperature drop from 600 K to 1 K [35]. 11

The ground states of the non-S-state atoms have non-zero orbital angular momentum, and their shell structures are asymmetric resulting in an anisotropic interaction potential. In general, the electronic interaction anisotropy causes large inelastic col-

2 1 3 1 2 lisions [29]. The prior work in collisions of Al[ P1/2]-Ar[ S0], C[ P0]-He[ S0], Si[ P0]-

1 3 2 He[ S0], O[ P2]-H[ S1/2] gives large inelastic collisional rate coeﬃcients which are on the order of 10−12 − 10−10 cm3 s−1 at a temperature range of 15 − 300 K [36, 37, 38].

Such large inelastic rates make it diﬃcult to perform the evaporative cooling of atoms

with orbital angular momentum in a magnetic trap. A previous experiment shows

that the evaporative cooling of metastable calcium atoms in a magnetic trap is ineﬃ-

∗ 3 ∗ 3 −10 3 −1 cient. The trap loss rate from the Ca [ P2]-Ca [ P2] experiment is 3 × 10 cm s . The large rates are due to the inelastic collisions such as the m-changing, J-changing

or quenching5 collisions [39]. Large inelastic collisions also induce a large trap loss in

∗ 3 ∗ 3 an optical trap of the metastable ytterbium. The Yb [ P2]-Yb [ P2] trap loss rate is 1.0(3) × 10−11 cm3 s−1 due to J-changing or quenching collisions [40].

2P -state atoms

Although atoms with non-zero orbital angular momentum usually hold large inelas-

tic collision rates, there are exceptions. Our studies of the ground-state gallium

2 2 2 2 (4s 4p P1/2) and indium atoms (5s 5p P1/2) colliding with helium gas reveal inelastic collisions rate coeﬃcients less than 10−15 cm3 s−1. The experimental measurements

and theoretical results will be described in Chapter 5.1.

The small inelastic collision rate coeﬃcients have been shown to be general to all

2 P1/2 atoms by theoretical calculations [41]. Surprisingly, the slow relaxation rate

2 makes it possible to cool and trap P1/2-state atoms.

5 In general, quenching refers the downward excitation of the atomic transition induced by collisions. Here, we use it to mean that the relaxation involves the changing of the principal quantum number, n-changing collisions. 12

Submerged-shell atoms

Moreover, a collisional shielding from the radial part of the electron wavefunction has been found in the rare-earth and the transition-metal atoms [42, 43]. In these atoms, the unpaired electrons are hidden inside an outer filled s shell, which is called a submerged-shell structure. The collisional behavior of the submerged shell is similar to a S-state atom, resulting in a small inelastic collision rate. In the case of the rare- earth atoms, the unpaired electrons are under two filled s electron shells and have an inelastic cross-section smaller than 10−19 cm2 [42]. As for the transition-metal atoms like titanium, the unpaired electrons are inside a single filled s electron shell. Recent

2 2 3 experiments measuring Ti[3d 4s F2]-He collisions observe a dramatic suppression of Zeeman relaxation collisions with a rate coeﬃcient of (1.1 ± 0.7) × 10−14 cm3 s−1 at 1.8 K and 3.8 T. It is about three orders smaller than that of other non-S-state atoms [43, 44].

In Chapter 5.2, we experimentally demonstrate this suppression in both m- and

J-changing collisions in titanium atoms, and confirm the origin of the suppression effect through theoretical calculations [45, 46]. Additionally, we study titanium’s fine- structure changing collisions which are relevant to a cooling process of star formation in astrophysics [47, 48]. More details are discussed in Chapter 5.2.1.

1.4 Inelastic collisions of diatomic molecules

A similar collisional picture used in atoms can be applied to molecules. Analogous to the non-S-state atoms, non-Σ-state molecules are expected to have a larger inelastic collision rate than the Σ-state molecules due to their anisotropic interaction potential. 13 1.4.1 Cold and ultracold molecules

It has been shown that exciting science can be explored in cold and ultracold molecules.

Techniques to create and prepare them have been developed. Ultracold molecules have been produced from ultracold atoms by photoassociation and through Feshbach resonances [49, 50]. However, these methods are limited to the atomic species that can be laser cooled, such as alkali atoms.

More general methods, like buﬀer-gas cooling and Stark declaration, produce cold molecules and load them into electrostatic and magnetic traps, but the temperatures achieved by these direct cooling methods are not as low as by photoassociation and

Feshbach resonances [51]. One method for obtaining lower temperatures is by sympathetic cooling. For example, to thermalize cold molecules with laser-cooled atoms

[52]. For sympathetic cooling to be eﬃcient, the elastic-to-inelastic collision ratio

γ needs to be large enough (γ > 100) [12]. However, magnetic and electrostatic trapping require that molecules remain in a low-ﬁeld-seeking state, which is not the lowest energy level. Inelastic collisions, like dipole-dipole and anisotropic interaction, can change molecular spin states and cause a trap loss. Consequently, collisional measurements of molecules are important.

1.4.2 Σ-state molecules

Σ-state molecules are simpler to deal with than other kinds of molecules because they do not have electronic orbital angular momenta. Prior works have focused on the inelastic collisions of the 2Σ-state and 3Σ-state molecules.

2Σ state

The first experiment to load cold molecules into a magnetic trap via buffer-gas cooling measured the Zeeman relaxation rate coefficient of CaH[2Σ]-3He to be 10−17 cm3 s−1 at 14 a temperature of 0.4 K and in a magnetic field of 3 T [13, 19]. Another measurement of 2Σ molecules is the collision-induced Zeeman relaxation of CaF[2Σ]-3He, measured to have a rate coefficient of (7.7 + 5.4/ − 2.5) × 10−15 cm3 s−1 at 2 K and 3.44 T, corresponding to a cross-section of (6.3 + 4.4/ − 2.1) × 10−19 cm2 [53]. In the both collisions of CaH-3He and CaF-3He, γ is greater than 104.

The inelastic collision mechanism in 2Σ molecules is mainly due to the spin- rotational interaction, predicted by Krems and Dalgarno [54]. Their theoretical calculation predicts the Zeeman relaxation cross-section for 2Σ molecule-3He to scale

2 4 as γsr/Be , where γsr is the spin-rotation coupling constant and Be is the rotational constant [55]. The Zeeman relaxation of CaF-3He is about two orders larger than that of CaH-3He because the CaF molecule has a smaller rotational constant6 and it has larger anisotropic interaction coupled from the rotationally excited molecular states.

3Σ state

Inelastic collisions in 3Σ molecules are slightly more complicated than in 2Σ molecules because 3Σ molecules have two unpaired electron spins. In addition to the spin- rotational interaction, there may exist the spin-spin interaction in 3Σ molecules from the coupling of the excited rotational states. The collision-induced Zeeman relax-

3 2 2 ation cross-section of Σ molecules has been predicted to scale as λSS/Be , where λSS is the spin-spin interaction constant [57]. The experimental results of NH[3Σ]-4He

3 4 2 2 and ND[ Σ]- He are consistent with the λSS/Be dependence [58]. However, the experimental measurements of NH-3He and ND-3He only partially support the scaling because the inelastic collisions are modiﬁed by a scattering shape resonance which

6 −1 −3 −1 For CaH molecules, Be is 4.3 cm and γsr = 41.5×10 cm ; For CaF molecules, −1 −3 −1 Be is 0.34 cm and γsr = 1.3 × 10 cm [53, 56]. 15

increases the relaxation rate [12, 58].

Reference [58] has measured the Zeeman relaxation rate coeﬃcient of 14NH-3He at 0.6 K to be (4.5 ± 0.3) × 10−15 cm−3 s−1, and that of 14NH-4He at 0.7 K to be

(1.1 ± 0.1) × 10−15 cm−3 s−1. The m-changing collision is about four times larger in

NH-3He than NH-4He collisions, showing the eﬀect of the shape resonance.

The Zeeman relaxation cross-section in NH-3He collisions has been measured to

be (3.8 ± 1.1) × 10−19 cm2 at 710 mK [59], resulting in a ratio of elastic to inelastic

collision rates of 7 × 104. Theoretical calculation shows that it may be possible to

produce ultracold NH molecules by sympathetic cooling with laser-coold Mg atoms

[52].

1.4.3 Non-Σ-state molecules

Analogous to the S-state versus non-S-state atoms, non-Σ-state molecules which are molecules with electronic orbital angular momenta are expected to have fast Zeeman relaxation rates due to the strong coupling between the electronic angular momentum and the molecular axis. Recent studies in Π-state and ∆-state molecules are discussed below.

2Π state

Researches of 2Π-state molecules are important for astrophysics and spectroscopy

[60]. Various theories and experiments have begun in the cold 2Π-state molecules. A theoretical calculation predicts that inelastic collision rates are on the same order of magnitude as elastic collision rate in OH[2Π]-Rb collisions at sub-Kelvin temperatures

[61]. It indicates that sympathetic cooling of OH molecules with cold Rb atoms is

diﬃcult in a magnetic or electrostatic trap.

Experimentally, the Stark-decelerated molecular beam colliding with a Xe atomic 16

2 beam observes the OH[ Π3/2]-Xe collisions as a function of the collision energy in the range of 50 to 400 cm−1 with an energy resolution of 13 cm−1 [62]. Another

experiment uses supersonic beams of He and D2 colliding with magnetically trapped OH in collision energies from 60 cm−1 to 230 cm−1 and from 145 cm−1 to 510 cm−1, respectively. The measurements give the absolute collisional cross-sections of OH-He

−15 −14 2 and OH-D2 to be on the order of 10 −10 cm , corresponding to rate coeﬃcients on the order of 10−11 − 10−10 cm3 s−1 [63]. Because Sawyer et al. measure the collision rates from the trap loss, they can not distinguish inelastic collisions from elastic collisions [63].

3∆ state

Of the non-Σ-molecular states, the collisional properties of the 3∆ state are of particular interest. 3∆-state molecule has been proposed for use in laser-cooling experiments because the rotational ladder can be closed by choosing the appropriate optical transition, and the only repumping is required for the vibrational states [64]. In addition,

3 ∆1 molecule is predicted to be a good candidate for experiments looking for the electron’s electric dipole moment because of their easy polarizability and a small magnetic g factor [65].

In Chapter6, we study the titanium monoxide (TiO) molecule in its 3∆ ground state. Large numbers of buﬀer-gas-cooled TiO molecules at a translational temperature of 5 K are generated. Chapter6 includes the elastic and inelastic collisions of

TiO[3∆]-He, and presents a ﬁrst measurement of the m-changing rate for the non-Σ- state molecules. The importance of 3∆-state molecules and the interest in the TiO molecule will be described as well. 17 1.5 Electromagnetically induced transparency

The previous sections discuss diﬀerent types of collisional mechanisms of atoms and

molecules. In this part, we introduce a powerful technique—electromagnetically in-

duced transparency.

Consider an ideal case that a probe laser passes through a two-level atomic sys-

tem. The probe photons will be absorbed by the atoms if the probe frequency is on

resonance with the atomic transition, as indicated in Figure 1.1(a). The absorption

of the probe beam and its corresponding index of refraction n are simulated as the

red curves in Figure 1.2. A simulation of the absorption and index of refraction are

described in AppendixA. The red curve in Figure 1.2(a) shows that the absorption of

the probe beam has a maximum when the frequency detuning from resonance is zero;

the red curve in Figure 1.2(b) indicates that the derivative of the refractive index

near resonance is negative. (Only the natural line broadening is considered here.)

2 3 γ γ

Probe Probe Control

1 1 2

Figure 1.1: (a) A two-level system is scattered by a probe beam. (b) A three-level system and two laser ﬁelds. The excited state decay rate in both system is γ.

Instead of the two-level atomic system, consider a probe beam and a control beam passing through a three-level atomic system. The system has an excited state |3i and

two lower energy states |1i and |2i. The lifetimes of the lower states are very long,

and the transition between them is optical forbidden. The probe beam is scanned over 18

D 1.0 . D u . 0.4 . u

a 0.8 . @

a 0.2 0.6 @ 2

0.0 0.4 L 1 -0.2 -

0.2 n H -0.4 Absorption 0.0 -2 -1 0 1 2 -2 -1 0 1 2 D Γ D Γ

Figure 1.2: The responses of the absorption and refractive index with and without the EIT eﬀect are shown@ D in blue and red curves, respectively. (a)@ TheD absorption of the probe beam is plotted as an function of the frequency detuning ∆ in units of the excited-state decay rate γ. (b) The curve indicates the response of the refractive index as a function of ∆. the transition |1i → |3i, while the control beam is on resonance with the transition

|2i → |3i, as illustrated in Figure 1.1(b). Instead of a maximum absorption, the probe’s absorption in this case has a minimum when the probe is on resonance with the atomic transition. The atomic system is transparent to the probe beam near the resonance frequency, resulting in a transparency window or a dip, shown as the blue curve in Figure 1.2(a). The index of refraction also has a very different behavior, since the derivative of the blue curve in Figure 1.2(b) becomes positive and large near the resonance frequency. This dramatic feature is first called electromagnetically induced transparency (EIT) by Harris and his coworkers [66]. Other phenomenas such as dark state [67], coherent population trapping (CPT) [67, 68], Autler-Townes splitting [69] and stimulated Raman adiabatic passage (STIRAP) [70] have the same physics as the EIT effect. The EIT effect can be explained with the concept of the dark state described in Chapter 3.6.1. 19 1.5.1 EIT application

Since the absorption of light has dramatically changed under the EIT condition, its

corresponding dispersion is signiﬁcantly diﬀerent according to the Kramers-Kronig

relations [28]. The mathematical treatment can be found in AppendixA, and the

light-matter properties are simulated there as well.

Because the absorption and dispersion become nonlinear7 due to the EIT eﬀect,

many interesting experiments are rapidly developed. Recent reviews of the EIT eﬀect

and its applications can be found in References [67, 70, 72, 73]. Here, we introduce

our interests in the EIT application in the following sections.

Slow and stopped light

With the dramatically changed dispersion and transparency of light under the EIT

condition, a light pulse can experience a very large group index of refraction ng, discussed in Chapter4. The large ng lets the light pulse pass through the EIT medium with a very slow group velocity vg. In this way, it is possible to slow down a light pulse and furthermore to stop the light pulse in an ensemble of atoms. The details of the theory and experiments of the slow and stopped light experiments are discussed in Chapter4.

Quantum memory

The importance of performing the stopped light experiments is not only the demonstration of “trapping” a light pulse, but also progress toward on quantum memory which is important to quantum communication [73].

For the purpose of quantum communication [74], photons are ideal carriers of

7Defined in Reference [71], the optical properties of the medium do not depend on the light field passing through it in a linear optical process. 20 quantum states because they are fast and easy to transport. With high mobility, they are difficult to be localized. On the other hand, atoms can preserve quantum states for a long coherence time [34, 75]. The basic idea is to use photons to deliver quantum states, and store quantum states in atoms. To achieve this, one has to interact photons with atoms, and to transfer quantum states between them.

In general, the probability for a single atom to scatter a single photon is very small. It is necessary to enhance the coupling between the atom and the photon in order to make the interaction becoming practical. The achievement of strong atom- photon coupling can be obtained through a large ensemble of atoms, while the energy dissipation due to absorption can be avoided by using the EIT technique. A detailed discussion of the strong light-matter coupling is presented in Chapter 4.1.

Photons carrying information (quantum states) are usually transmitted in atmo- sphere or in a ﬁber. A direct long-distance distribution is not practical because the information decays exponentially with distance. Unlike classical information, the quantum states can not be duplicated without performing measurements because of the no-cloning theorem [76], and therefore an ampliﬁer for quantum states is impossible. One way to achieve a long-distance quantum communication is to divide the distance into several segments, and connect each segment with a quantum repeater

[74, 77]. The famous DLZC proposal [78] gives a way to transmit quantum states over a long distance with a near-unit ﬁdelity, and it also points out that the quantum memory—the storage of the quantum states—is essential for successfully connecting segments.

Other important quantum information science, such as quantum networks [79] and linear optics quantum computing [80], also need quantum memory because the gate operations are probabilistic and the quantum states need to be stored until the operations succeed. In addition, the realization of a single-photon on demand [81, 82] 21 and the improvement of precision measurements by squeezed states [83, 84] can be realized with quantum memory.

Overall, the EIT technique provides a way to manipulate the quantum states of light and matter, and has been applied to quantum information experiments such as quantum memory [85, 86, 87], quantum repeaters [78] and deterministic single- photon generation [81, 82]. Review papers [88, 89, 90] discuss the application and characteristic parameters of the recent quantum memory experiments.

Many diﬀerent systems (ultracold atoms, crystal, room temperature alkali vapor, etc.) have been studied in order to have optimal performance. We will discuss the EIT experiments by using ytterbium nuclear spin in Chapter3 and show good atom-light coherence. The storage performance of a classical light pulse in our system compared to other groups will be be discussed in Chapter4 as well.

Metrology

An ultranarrow transparency window can be achieved by EIT, and it has the advantages of large dispersion and vanishing absorption which gives a promising application in metrology. The two-photon resonance is sensitive to the relative energy shift of the two lower energy states, which can be used as a high-sensitivity magnetometer

[91, 92]. It has been theoretically predicted that the response of a magnetometer can be enhanced by applying EIT to the laser beam in a pump-probe atomic magnetometer [93].

Nonlinear optics

EIT also has signiﬁcant importance in nonlinear optics such as four-wave mixing

[94, 95] and the Kerr effect (or cross-phase modulation, XPM) [96]. Four-wave mixing is a process whereby two optical fields interact in a nonlinear medium with a third 22 beam, and this interaction generates a fourth beam. It can be used for the generation of a squeezed state and a phase-conjugation wave [67]. A four-wave mixing experiment combined with EIT can have the advantages of reducing the absorption and distortion for the driving and generated fields. The Kerr effect of optical fields is a photon- photon interaction, where the phase of one optical field can be modified by another optical field [67]. It can be applied for the quantum nondemolition measurements

[96].

Four-wave mixing and the Kerr eﬀect are both based on the nonlinear response of the optical ﬁelds interacting with media, and need to have a large third order nonlinear susceptibility. It has been shown that their performances can be improved by EIT to obtain an enlarged third order nonlinear susceptibility with a reduced linear susceptibility and absorption [72, 97]. 23

Chapter 2

Apparatus

To realize helium buﬀer-gas cooling, a well-designed cryogenic system is needed. In this chapter, the design of the cryostat will be described, along with the measurement methods of temperature and pressure. Optics, laser light sources, and data acquisition will be included.

2.1 Design of the cryostat

The sketches of the cryogenic system are displayed in two diﬀerent drawings as shown in Figure 2.2 and Figure 2.3. The description of the cryostat is divided into three main parts based on their thermal temperatures: a room temperature vacuum chamber, an aluminum shielding box and a copper cell.

A commercial pulse tube cooler PT405 [98] sits on a stainless-steel panel sealed with a Viton o-ring. The designs of the Viton o-ring and its o-ring groove are based on Reference [99]. A 12” diameter aluminum cylinder that is vacuum sealed to the bottom of the stainless-steel panel with a Viton o-ring serves as the room temperature chamber. The chamber ensures a vacuum better than 10−5 Torr in order to isolate heat conduction and convection through air. On the top side of the stainless-steel 24

1 plate, there are nine 1 3 ” CF nipples: one is a vacuum pumping line for the chamber and one is a feedthrough for the copper cell; one measures the chamber pressure by a

chamber gauge [100] and one one attaches to a check valve [101] to release gas when

gas pressure is over the check point. Two are electric feedthroughs for temperature

measurements, and the rest are spares.

The wall of the pulse tube cooler is thin and fragile. Its maximum load is less

than 22 lb on the ﬁrst stage and 11 lb on the second stage [98]. In order to attach the

copper cell to the refrigerator without damaging the pulse tube cooler, a structure

which is built by G-10 rods [10] with an aluminum plate and a copper plate is used

as a mechanical support for mounting the aluminum shielding box and the copper

cell, as displayed in Figure 2.1. The aluminum plate is thermally anchored with the

ﬁrst stage of the pulse tube cooler, while the copper plate is thermally attached to

the second stage. A copper cell is mounted to the copper plate by four copper posts.

G-10 rods are good for supporting and thermal isolation, while cooper posts are used

to have good thermal conduction.

The cell is made of oxygen-free copper alloy 101. It is a 4” cube which has ﬁve

2.5” diameter clear holes sealed with 3” diameter windows and one 3” diameter clear

1 hole mounted with a brass cell lid. On the top of the lid, there is a 8 ” diameter stainless-steel tube connected to the room temperature stainless-steel plate as the

helium gas line. The amount of gas inside the cell is controlled and monitored through

this cell gas line. A brass target holder carrying ablation targets is attached to the

bottom of the cell lid. There are diode sensors placed around the cryostat to measure

temperature. A detailed discussion of temperature and pressure measurements is

presented in Chapter 2.2.

To block the blackbody radiation from the room temperature side, a box made

from an aluminum sheet with a thickness of 0.125” is attached onto the aluminum 25

Figure 2.1: Assembly of the cryostat plate covering the copper cell. The aluminum box is also wrapped with three layers of aluminized Mylar sheets (thickness = 0.0005”, coated on one side) [102] in order to have better shielding from blackbody radiation. To have optical access, the aluminum box is also mounted with ﬁve 3” diameter windows so that they can still block radiation.

2.1.1 The assembly of the cryostat

Using Viton o-rings to make the vacuum sealing at cryogenic temperature is imprac- tical because Viton o-rings tend to crack at low temperature. It is possible to seal with an oxygen-free copper o-ring, but it is diﬃcult to make a seal between glass and metal. We use indium wires for vacuum sealing of the cryogenic parts because indium 26

Figure 2.2: A transparent view of the cryogenic system. 27

Figure 2.3: A cut-away drawing of the cryogenic system. 28

is a soft metal and can be used in vacuum and at low temperature. In addition, there

is no need for designing an o-ring groove for the indium wire.

An indium wire with a diameter of 0.035” [103] is used to vacuum seal the windows

and the cell lid onto the cell body. The indium wire is made into a circle to serve as

the vacuum o-ring. When making a circle, both ends of the indium wire are cut to

have a 45 degree facet. We overlap these ends and press them gently to join them.

The circular indium o-rings have two inner diameters: 2.67” for the windows and

2.99” for the cell lid.

Window Kapton Teflon Window Viton Indium holder ring strip Oring Oring Chamber Shielding Cell

(a) (b)

Figure 2.4: A schematic section view of the window assembly. (a) Symbol deﬁnitions. (b) A section view of the room temperature chamber, the aluminum shielding and the copper cell. A Teﬂon strip is used to center the window position.

The methods used for sealing are diﬀerent for the windows and the cell lid. When

sealing the windows, the indium o-ring sits on one side of the cell body and the

window is placed above it. They are centered on the clear holes of the cell body as

shown in Figure 2.1 and Figure 2.4. After the indium o-ring is placed in between the

window surface and one side of the cell body. A window holder with bolts is used

to keep them in position. We press the indium o-ring down by screwing the bolts

in a sequence around the bolt circle until the indium o-ring is pressed ﬂat and looks 29

like a mirror surface. The ﬁnal sealing step is to wait for ten minutes to let indium

“ﬂow” uniformly and then tighten the screws for one more bolt circle. Note that

when assembling windows to the cryostat, it is important to avoid direct contacts

between metal and glass. A Kapton ring with a 0.005” thickness is inserted between metal and glass to prevent the window from cracking. A schematic section view of window assembly is shown in Figure 2.4. For vacuum sealing the cell lid, we put the indium o-ring around the lid as indicated in Figure 2.5 and compress the o-ring by tightening the screws through the lid. In Figure 2.5, the silver indium ring near the edge of the lid is a trace of the indium sealing. Information about the indium sealing and experimental techniques at cryogenic temperature can be found in References

[10, 104, 105].

The cryostat has five optical windows on the chamber, the aluminum box and the copper cell, so that there are 15 windows in total. The windows used in the cryostat are 3” in diameter with a 0.5” thickness. Thick windows are not easy to break through thermal cycling and help to block blackbody radiation. The material of windows used in the cryostat is BK7, which effectively transmits visible light and blocks infrared light [106]. Both sides of the window are coated with an anti-reflection coating (MgF2), which is specified to a reflect less than 1.5% per surface at 400 nm and 532 nm [107]. We have measured a 1.5% reflection per surface at 400 nm.

On the room temperature chamber, windows can be sealed with Viton o-rings. On the 50 K aluminum shielding, there is no need for vacuum sealing so that windows sit on the aluminum box with a separation by Kapton rings. A Teﬂon strip is used to ensure the window is centered in the middle of the window holder. 30 2.1.2 Targets

Ablation targets are clamped down with an aluminum panel onto a brass holder.

The holder is attached to the cell lid in a way that it is slightly above the edge of the

window hole. There is roughly a 2.5 cm gap between the holder and the lid. Various

target samples used in this thesis are listed in Table 2.1. Figure 2.5 shows the bottom

side the cell lid with target samples on the target holder. The corresponding target

species are as the labeled.

Table 2.1: Ablation target information. dia.= diameter; thk = thick. Target Production Purity [%] Dimension Supplier Yb Yb 99.9 disc (1” dia., 0.125” thk) [108] Ti Ti 99.995 disc (1” dia., 0.125” thk) [109]

Ga90Cu10 Ga disc (1” dia., 0.125” thk) [110] In In 99.995 disc (1” dia., 0.125” thk) [111] 1 TiO TiO 99.9 ∼ 3 ” pellets [112] 1 TiO2 TiO 99.9 ∼ 5 ” pieces [113]

Targets listed in Table 2.1 are stable in air, and do not need special care. The

ablation spot lifetime—how many ablation shots a spot can take and still keep nearly

constant production—depends on the species and the ablation energies. Usually,

strong ablation energy leads to large yields but a short spot lifetime. If the spot gets

“tired”, the ablation pulse can be steered to another new spot on the target by a

mirror. Based on the observation, Yb, Ti, Ga90Cu10, In, and TiO usually have good spot lifetimes, but TiO2 does not.

2.1.3 Improvements

For the current setup, the normal operational temperature of the copper cell is 5 K.

Two optical accesses can be blocked for getting a colder temperature. Moreover, an aluminized Mylar shielding box (thickness=0.005”, both sides coated) [102] attached 31

(a) Ti, Yb, Ga, and In targets (b) TiO targets

Figure 2.5: Targets and the brass target holders after laser ablation. An indium ring around the target holder is for vacuum sealing, not an ablation target. The stainless- steel tube that appears behind the target holder is the cell gas line. Traces and spots on the targets and the target holders are the results of laser ablation. onto the second stage is added in order to isolate the copper cell from blackbody radiation. In this way, the cell temperature is cooled down to 4 K.

Since the base temperature of the pulse tube cooler is 2.8 K, a lower temperature of the cell body can be expected by improving the thermal links and the blackbody radiation shielding.

2.2 Measurements of temperature and pressure

2.2.1 Temperature measurements

To monitor the operation of the cryostat, we use silicon diodes as temperature sensors

[114]. The silicon diodes are calibrated relative to a calibrated diode [115], and mounted in various places within the cryostat. There are also a couple of resistors mounted inside the cryostat as heaters. With sensors and resistors, the thermal 32 resistance of the heat links between the refrigerator and the cold cell is measured.

The typical cooing-down curves of the cryogenic system are shown in Figure 2.6. The cell temperature is determined by the calibrated diode [115]; the others are measured by the silicon diodes [114]. It usually takes about one day for the cell body to be cooled from 300 K to 4 K.

100

1st stage shielding 1st stage 2nd plate Temperature [K] Temperature 10 Cell body 2nd stage

12:00 PM 12:00 AM 12:00 PM 3/22/2010 3/23/2010 Time and date

Figure 2.6: Cooling-down curves of the cryostat. The oscillation of curves at cold temperature is due to the operation of the cryostat.

Another independent way to determine the cell temperature is to measure the translational temperature of atoms. The atomic spectroscopy at cryogenic temperature is usually at an intermediate regime between homogeneous and inhomogeneous broadening. In our case, the transition linewidth is dominated by Doppler and pressure broadening. The convolution of these profiles is a Voigt profile [116]. Absorption spectroscopy is used to probe atoms. Once atoms are detected, their translational temperature T can be measured by fitting the spectra to the expected Voigt line 33

1 shape. In this way, a Gaussian half width νD at 1/e maximum is obtained. If only the linear Doppler eﬀect is considered which is a good approximation at a

temperature of a few Kelvin, the translational temperature T is determined by 2 M[g] νD [Hz] T [K] = −1 (2.1) 2kB[erg/K] ν0/c [cm ]

where ν0 is the resonant frequency, M is the mass of the atom (M = atomic weight ×

−24 −16 1.66 × 10 [g]), and kB is the Boltzmann constant (kB = 1.38 × 10 [erg/K]) [117].

FWHM Considering ytterbium atoms in Figure 3.2 as an example, if νD is measured to be 100 MHz (νD = 60 MHz), a temperature 6 K is measured:

173 × 1.66 × 10−24[g] 60 × 106 [Hz] 2 T [K] = = 6[K] (2.2) 2 × 1.38 × 10−16[erg/K] 25068.22 [cm−1] At the same time, the calibrated diode mounted on the cell body reads 5.2 K. These

two independent temperature measurements give a consistent result within their rel-

ative uncertainties.

2.2.2 Pressure measurements

To understand collisional properties of atoms or molecules inside buﬀer gas, it is

important to measure the gas pressure and density inside the cold cell. From the gas

pressure P and cell temperature T , the gas density n can be determined from the

ideal gas law [118]: P n[cm−3] = N/V = kBT P [Torr] · 0.001316[atm/Torr] · 1033[ g ] · 980[ cm ] = atm cm2 s2 (2.3) 1.38 × 10−16[erg/K] · T [K] P [Torr] = 9.654 × 1018 T [K] where N is the gas number and V is the gas volume.

1 FWHM A full width at half maximum√ of a Gaussian proﬁle is denoted as νD , which FWHM is related to νD as νD = 2 ln 2νD. 34

Weber-Schmidt equation

To measure the cell pressure, a Pirani gauge [100] at 300 K is linked to the cold

cell with a stainless-steel tube that has an inner diameter 0.18 cm. The pressure

measurement setup is displayed schematically in Figure 2.7(a). When the mean-free-

path of gas is shorter than tubing wall, gas flow is viscous and the cell pressure Pc is same as the gauge pressure Pw. At low pressure and in a narrow tube, the mean- free-path of gas is larger than tubing diameter, which means that gas particles collide more often with the tubing wall than with each other. In this case, there is a pressure difference between Pc and Pw due to the thermomolecular effect, when there is a temperature gradient between both ends [119]. Fortunately, Pc can be interpolated

2 from Pw with a conversion based on the Weber-Schmidt equation [119, 120, 121]: P 1 T y + 0.1878 log c = log c +0.18131 log c Pw 2 Tw yw + 0.1878 y + 1.8311 y + 4.9930 + 0.41284 log c −0.15823 log c (2.4) yw + 1.8311 yw + 4.9930 1.147 273.15 RPc,w where yc,w = ( ); R [cm]; Tc,w [K]; Pc,w [µm of Hg]. Tc,w 13.42

R is the radius of the tube and Tw (Tc) is the temperature of the gauge (the cold cell).

The cell pressure Pc is determined from the gauge pressure Pw by Equation (2.4). Finally, the gas density in the cold cell can be found from the cell temperature by

inserting numbers into Equation (2.3). An example of pressure conversion can been

seen in Figure 2.7(b) which plots the helium density at 5.2 K versus the gauge pressure

measured at room temperature.

Quantity of gas

Another method of measuring cell pressure is to use a known gas density. In Figure

2.7(a), there is a reservoir as a reference with its pressure P0 and volume V0. The

2Note that 1 µm of Hg = 0.001 Torr. 35

21 10 Reservoir P V 0 0 19 Gauge 10 P T V w w w 17

1 atm & 300K K 5.2 @ ] 10 -3 Chamber R

[cm 15

He 10 n

13 Cold cell 10 P T V c c c -5 -3 -1 1 10 10 10 10

Pw [Torr] @ 300 K

(a) (b)

Figure 2.7: (a) Schematic diagram of pressure measurements. (b) Helium density at cold cell is poltted as a function of Pw. amount of gas in the reservoir can be added to the copper cell and is controlled by a valve. Before the copper cell is cooled down (Tc = 300K), the volume ratio Vc/V0 can

0 0 be determined by comparing Pc and P0. (Use P0 V0 + Pc Vc = P0 V0 + Pc Vc when the reservoir and the cell are at the same temperature. The apostrophe indicates before opening the valve and pressures are measured by gauges.) In the current setup, the

300 K volume ratio Vc/V0 is measured to be 36.96 so that Pc = P0/36.96. Based on the ideal gas law, the gas pressure of the cell at low temperature is determined by

300 K Tc P0 Tc Pc = Pc = . (2.5) Tw 36.96 Tw

In this way, we determine the density of the gas in the cold cell from the quantity diﬀerence in the reservoir and the measured temperatures.

Experimentally, two independent measurements of cell pressure, the Weber-Schmidt equation and measuring the quantity of gas, are consistent within 10% at helium densities ranging from 4 × 1016 cm−3 to 2 × 1017 cm−3. 36 2.3 Lasers

Two main type of optical ﬁelds are employed in these thesis: a pulsed laser and diode lasers. The pulsed laser executes laser ablation, while diode lasers perform absorption spectroscopy and optical pumping.

2.3.1 Ablation laser

A commercial Nd:YAG3 pulsed laser, Surelite I-10, serves as the ablation laser [122].

The laser pulse width is ∼5 ns and the pulse repetition rate is up to 10 Hz. The output wavelengths are 1064 nm and 532 nm after frequency-doubling the 1064 nm photons.

140 Block 140 Glass 120 120 PD Output Photodiode [mV] 100 Diffuser 100 C R + 80 532 nm 80 filter 60 60 40

Mirror [mJ] Pulseenergy 40 Nd:YAG 20 20 HB 0 -50mm +250mm +350mm To cell lens lens lens 200 250 300 350 400 Q-switch delay [µs]

(a) (b)

Figure 2.8: (a) The optics setup for the Nd:YAG pulse laser. See the text for a description of the setup. (b) Pulse energy of the Nd:YAG laser versus Q-switch delay time. On the left axis is the pulse energy of the Nd:YAG laser as a function of the Q-switch delay time, labeled as the red triangles. The corresponding signal monitored by a photodiode corresponds to the right axis (the blue circle).

A schematic drawing of the optics setup is illustrated in Figure 2.8(a). The 532 nm beam is displayed in green, and the 1064 beam in red. A Nd:YAG harmonic

3 Nd:YAG stands for neodymium-doped yttrium aluminum garnet, Nd:Y3Al5O12. 37

beamsplitter (HB) [123] is placed right after the laser output to reﬂect the 1064 nm

photons and transmit the 532 nm photons. Since 532 nm photons are in the visible

regime, the 532 nm beam is used for laser ablation and the 1064 nm infrared light

is blocked for safety reasons. Wearing proper goggles is required when operating the

pulsed laser.

The second surface of the harmonic beamsplitter reﬂects a tiny amount of 532 nm

photons. We use a piece of glass to direct these 532 nm photons into a photodiode to

monitor the laser output power. Before entering the photodiode, the 532 nm beam

passes through a frosted glass and a 532 nm color ﬁlter [124]. The frosted glass is

used as a diﬀuser to reduce the beam power. The photodiode is connected in parallel

with a 30 kΩ resistor and a 1 µF capacitor in order to have a proper time response to the fast light pulse.

The output pulse energy is controlled by varying the Q-switch delay time [122], and its maximum power output is up to 470 mJ at 1064 nm and 240 mJ at 532 nm. The operating Q-switch delay time is usually in the range from 200 µs to 400

µs which gives an output energy from 140 mJ to a few mJ at 532 nm as shown in

Figure 2.8(b). The relation between the photodiode signal and the pulse energy as a function of the Q-switching delay time is plotted in Figure 2.8(b). The output intensity has a shot-to-shot inconsistency ∼10%.

To prevent damaging optics, the transmitted 532 nm laser pulse is expanded to have a 0.8” beam diameter by a beam expander [123]. The laser pulse is then focused by a convex lens with a focal length of 350 mm [123] before it enters into the chamber.

The types of the lenses are chosen to balance the aberration [106]. By changing the position of the 350 mm lens, we can determine the relative distance between the beam focus and the target in order to optimize the ablation yield. A mirror after the 350 mm lens is used to steer the laser pulse to the proper position on the target. Note 38 that the optics setup in Figure 2.8(a) is on two separate optical tables. The mirror before the cell and the 350 mm lens are on the cryogenic table, while the rest of optics is on the main table.

2.3.2 Diode laser

A diode laser is a relatively cheap coherent light source compared to a dye laser or Ti:sapphire4 laser. It is easy to use for probing and manipulating atoms and molecules. Thanks to the recent development of blue light techniques, we are able to have laser diodes with a center wavelength around 400 nm.

We have both commercial [125] and home-built grating-feedback external cavity diode lasers (ECDL) [116, 126]. The detailed design of the home-made ECDL is well- written in Kyle Hardman’s Senior thesis [127]. Depending on the observed atoms and molecules, we use diﬀerent laser diodes and proper optics in order to cover their transition lines. Table 2.2 lists some information about diode lasers used in this project. The range of the lasing wavelength is measured by varying the temperature of the laser diode and the angle of the grating. The maximum output power Pmax is measured after the grating feedback. Pmax is a safe limit to operate the laser without damaging the diode. The details of laser diodes for the Ga, In and TiO experiments can be found in Reference [127]. Note that the laser diode used to probe

TiO molecules can also access O2 transitions. Of all the laser diodes, we show the performance of Yb diode laser in Figure 2.9 as an example of the home-made diode laser. Figure 2.9(a) shows the dependence of the lasing wavelength on the temperature of laser diode. The upper and lower limit of the lasing frequencies are obtained by adjusting the feedback grating angle.

A frequency tuning range of ∼4 nm is obtained by adjusting the grating angle, while

4 Ti:sapphire stands for titanium-sapphire, Ti:Al2O3. 39

Table 2.2: Laser diodes information. Pmax is the maximum output power after the grating feedback. No. refers to EYP-RWE-0790-04000-0750-SOT01-0000. FP: Fabry-Perot type laser diode. AR: laser diode with anti-reﬂection coating.

Laser diodes Range [nm] Type Pmax [mW] Experiment Nichia NDV4313 397.9∼402.8 FP 30 Yb Sanyo DL-5146-152W 403.2∼408.0 FP 13 Ga Sanyo DL-5146-152W 407.6∼411.1 FP 13 In Topica #LD-0405-0030-2 397.9∼400.3 FP 13 Ti Eagleyard No. 750∼790 AR 32 TiO an extra ∼0.6 nm range is added by varying the diode temperature. Choosing proper grating feedback angle and diode temperature, both Ti and Yb transition lines can be covered. Extrapolating data to higher temperature, Ga transition lines can also be included by heating the laser diode. Figure 2.9(b) shows the laser output power as a function of the diode operating current. The output power is measured after the grating feedback.

25200 40

] 25100

-1 30

25000 Upper limit Ti 20 Lower limit Yb Ga 24900 Frequency [cm Frequency 10 Output power [mW] power Output

24800 0 10 12 14 16 18 20 22 0 10 20 30 40 50 60 70 Diode temperature [C] Diode current [mA]

(a) (b)

Figure 2.9: Performance of Nichia NDV4313 laser diode. (a) The lasing wavelength vs. temperature tuning. We specify Ti, Yb and Ga atomic transitions as references. (b) Laser output power as a function of diode operating current at a diode operating temperature of 11 ◦C.

A typical diode laser setup is shown in Figure 2.10. We have a Faraday isolator 40

[128] right after the laser to prevent any reﬂected light going back to the ECDL because the performance of the laser diode is sensitive to the feedback light. After the isolator, the laser beam passes through a half-wave plate [129], and its polarization becomes perpendicular to the optical table.

W Pump Symbol Device ECDL FI W2 AOM beam FI Faraday isolator FPI W2 Half-wave plate WM W Beam splitter DAVLL beam FPI Fabry-Perot interferometer AOM AOM WM Wavelength meter

AOM Acousto-optic modulator Probe beam

Figure 2.10: Typical optics setup for the diode laser. Various mirrors and lenses have been omitted to simplify the drawing.

In order to monitor the lasing modes of diode lasers, we pick up a small portion of the laser light into a home-made Fabry-Perot interferometer (FPI) [130] and a commercial wavelength meter [131]. The type of the FPI is confocal [116]. We prefer to use the confocal FPI because it is easier to align than a plane FPI5. The homemade

FPI has a 50 mm cavity length which gives a free-spectral range of 1.5 GHz, ﬁnesse

150, and a resolution linewidth 10 MHz. By monitoring the signal from the FPI, we ensure that the laser operates in a single-frequency lasing mode. The wavelength meter is used to give the wavelength of the laser. In this way, we can obtain a single- frequency mode laser at the atomic transition frequency by proper controlling laser conditions including the diode operating temperature, the diode operating current, the grating feedback angle, and the current feed-forward [127].

Except for the two beams sent to the FPI and the wavelength meter, the laser

5 The free-spectral range for a confocal FPI is c/4d, while that for a plane FPI is c/2d. c is the speed of light and d is the cavity length. 41 beam after the Faraday isolator and a half-wave plate is divided into three other beams to be the pump, probe, and DAVLL beams as shown in Figure 2.10. The probe beam is weak enough that we use it to probe the atomic population without perturbing it; the pump beam is strong enough that we use it to perturb the atomic population. The details of the probe and pump beams will be described in the following chapters. The

DAVLL beam is used to frequency stabilize the diode laser which will be discussed in Chapter 4.2. The DAVLL beam exists only in the ytterbium experiment, while the probe and pump beams appear generally in all experiments. These three beams are sent into individual acousto-optic modulators6 (AOMs) [132], and their diﬀracted beams are used. In this way, we manipulate the power and frequency shift of the beams. More details of the optics setups will be examined in speciﬁc chapters.

2.4 Data acquisition system

In our experiments, the timing of when events happen needs to be precise. To control the devices, we send two diﬀerent voltage signals: 0 and 5 Volts. (0 refers to a low level; 5 Volts refers to a high level.) They are deﬁned as transistor-transistor logic

(TTL) signals [133]. By changing the voltage level to the device, we determine the action of the device. For example, a high (low) level TTL signal sent to the shutter controller makes the shutter open (close). All events in the experiment are controlled by hardware timing, rather than software-based timing.

A digital I/O system—DIO 64 board [134]—has an internal clock scan rate 20 MHz which is used to control the timing sequences. Because it usually takes two signals from the rising edges of the TTL pulses to alter the status of a device, the minimum time interval determined by the DIO 64 board between each action is 100 ns. This

6See Appendix G.1 for the setup of the AOM drivers. 42

100 ns time interval is suﬃciently short for our application. An example of the timing control can be seen from the Q-switch delay time of the Nd:YAG laser. We use a 10

µs TTL signal to turn on the ﬂashlamp in order to excite the gain medium of the laser. After a user-determined delay time (usually 200 µs to 400 µs), another 10 µs

TTL signal opens the Q-switch device to output the laser pulse. In this way, the pulse energy is determined by the time interval between two TTL signals (Q-switch delay time).

In addition to the DIO 64 board, the data acquisition system also includes an analog input board—NI PCI-6143 [133] and an analog output board—NI PCI-6733

[133]. The NI PCI-6143 board records the voltage signals with time information, such as the signals from the photodetectors; the NI PCI-6733 board can output user- deﬁned waveforms, similar to a voltage function generator. The speciﬁcations of these boards are partly listed in Table 2.3. In our setup, the NI PCI-6733 board operates based on the clock of the DIO 64 board and the NI PCI-6143 board runs on its own clock. The DIO 64 board triggers the NI PCI-6143 board and the NI PCI-6733 board with TTL signals in order to synchronize them. For example, the DIO 64 board triggers the NI PCI-6143 board to read signals 100 ms before the two TTL signals for the Q-switch delay time, and it is programmed to record signals for 2 seconds, so that 100 ms background data and 1900 ms signal data can be obtained as shown in

Figure 3.3.

We program these TTL signals with various time intervals using Labview software

[133]. When executing the Labview program, a script of data encoding is generated with arbitrary time steps based on the user settings. The data stream of the timing script is then sent into the DIO 64 board by direct memory access (DMA), which allows transfer of data between the device and the computer memory without using the computer processor [133]. It also executes in a first-in-first-out (FIFO) mode, 43 which is a data buffering technique allowing the first incoming data to come out first

[133]. Simply speaking, this is one kind of buffered acquisition which transfers data between the onboard FIFO memory of the device and the computer buffer using DMA before it is transferred to the system memory. Finally, the DIO 64 board synchronizes the NI PCI-6143 board to acquire data, and also triggers the NI PCI-6733 board to output voltage waveforms. In this way, all the actions of all the devices can be synchronized, and performed with a specified timing.

In slow and stopped light experiments, another data acquisition system—NI PXIe-

6366 board [133]—is employed. The NI PXIe-6366 board includes the same functions as the first set of data acquisition systems mentioned above. It provides a relative easier way to generate arbitrary waveforms, like a Gaussian pulse, than the first data acquisition system. The simplified specification of the NI PXIe-6366 board is also listed in Table 2.3.

Table 2.3: Simpliﬁed speciﬁcations of data acquiring devices. AI: analog voltage signal input. AO: analog voltage signal output. DO: digital output. Ch. refers to the number of channels that supports the DMA data transfers. Board Function Ch. Supplier DIO-64 DO, 20 MHz (50 ns) 64 [134] NI PCI-6143 AI, 16 bits, 250k Samples/s, ±5V. 8 [133] NI PCI-6733 AO, 16 bits, 1M Samples/s, ±5V 8 [133] NI PXIe-6366 DO, 10 MHz (100 ns) 8 [133] AI, 16 bits; 2M Samples/s, ±10V 8 [133] AO, 16 bits; 3.33M Samples/s, ±10V 2 [133]

In addition to taking data, the written Labview program can also automatically save ﬁles in plain text format, and load them into IGOR Pro software [135]. We write procedures in the IGOR Pro program to load ﬁles automatically, and analyze them. 44

Chapter 3

Ytterbium

In modern physics experiments, a variety of interesting quantum phenomena are

based on atom-atom coherence or atom-light coherence. In this chapter, we ﬁrst

introduce a cryogenically-cooled atomic vapor cell that has a large number of atoms

and a long atomic lifetime. Then, we demonstrate good atomic coherence in the

nuclear spin of the 173Yb isotope. Finally, by using standard electromagnetically- induced-transparency (EIT) techniques with large numbers of the J = 0 ground-state

ytterbium atoms, a strong coherent coupling between photons and nuclear spin states

is obtained as well as a long atom-light coherence time. Later, we extend this atom-

photon coherence of Yb EIT to the slow and stopped light experiments in Chapter4.

3.1 Long-lived coherent optically dense media

Atomic magnetometers [136] and slow/stopped light experiments [137, 138] need large

numbers of atoms or high optical densities1 (OD) to obtain good performance. How-

1We use the atomic physics convention: optical density or optical depth (OD) is related to the transmittance (T ) of the sample, which is T = e−OD. In optics, the OD of neutral density filters is defined as T = 10−OD. 45 ever, it is usually difficult to have a large number of atoms while still keeping good coherence times because inelastic collision rates increase with the density of atoms

[17, 22], and the atomic coherence can be destroyed by inelastic collisions.

Alkali atoms, such as rubidium and cesium, are the most common species for performing atomic coherence experiments because of their collisional properties. They usually have high vapor pressures and laser-accessible line transitions, so that they are easily generated and probed. A room temperature vapor cell or a laser-cooled atomic sample is usually used for these experiments. The room temperature vapor cell can generate large numbers of atoms easily, but suﬀers from inelastic collisions and atomic motion. Inelastic collisions include atom-atom and atom-wall collisions; fast moving atoms lead to large Doppler-broadened transition lines. Laser-cooling methods usually generate cold atomic samples with µK temperature which gives small

Doppler broadening. However, it is usually diﬃcult to operate coherence experiments with OD greater than 10 by laser cooling methods [139]. An alternative way is to use a cryogenically-cooled vapor cell which was ﬁrst examined by Hong et al. [139].

They maintain a buﬀer-gas-cooled rubidium vapor below 7 K with a maximum OD

> 70, showing ground hyperﬁne decoherence rates on the order of 10 kHz.

A cryogenically-cooled system has already shown a great advantage of slow atoms with small Doppler broadening and large atom numbers. We believe that the performance can be further improved by switching from alkali atoms to J = 0 atoms, as discussed below [75, 140].

3.1.1 A pure nuclear spin system—J = 0 atoms

Atomic coherence experiments usually operate in atomic hyperﬁne states F or their sublevels mF . Unfortunately, hyperﬁne states involve electron spin, and electronic angular momentum is highly susceptible to decoherence from inelastic collisions and 46

inhomogeneous magnetic ﬁelds. On the other hand, it has been long known from

nuclear magnetic resonance experiments that long coherence times can be obtained

with pure nuclear spin states [141]. To seek a long-lived coherent medium, instead of

using hyperﬁne states, we choose atoms with a ground state whose total electronic

angular momentum equals zero (J=0) and the nuclear spin is nonzero (I 6= 0).

Because the total electronic angular momentum is zero, there is no coupling between electronic spin and nuclear spin. In this way, pure nuclear spin is protected by a closed electronic shell from its environment and there is almost no eﬀect from inelastic collisions. Furthermore, nuclear spin states are less sensitive to magnetic

ﬁeld gradients than hyperﬁne states because the nuclear magneton is about 3 orders smaller than the electron magneton2.

Among the atoms in the periodic table, naturally occurring ground state J = 0

atoms with non-zero nuclear spin are the noble gases, the alkaline earth metals, the

carbon group, the zinc group, palladium, samarium, ytterbium, and tungsten. We

choose atomic ytterbium for preparing coherent media because of its laser-accessible

wavelength 398.9 nm; atomic ytterbium has naturally occurring isotopes with non-

zero nuclear spin and a suﬃcient natural abundance.

3.2 Yb information

Ytterbium atoms are popular in atomic physics experiments because of their rich iso-

topes and atomic transitions. Among the seven stable isotopes, there are two fermions

(171Yb and 173Yb) and ﬁve bosons (168Yb, 170Yb, 172Yb, 174Yb and 176Yb). The nat-

urally occurring isotopes of ytterbium and their natural abundances are listed in Ta-

ble F.1. The basic atomic data of ytterbium is presented in Appendix F.1 including

2 µN = me = 1 [117]. µe mp 1836 47

its transition strengths. The Bose-Einstein condensates (BEC) of 170Yb, 174Yb, and

176Yb have been realized [142, 143], so as have the quantum degenerate Fermi gases

of 171Yb and 173Yb [144, 145]. Various mixtures of BECs and Fermi gases are possible with ytterbium atoms. The ground-state electronic conﬁguration of ytterbium atoms

2 2 6 2 6 10 2 6 10 2 6 14 2 1 is 1s 2s 2p 3s 3p 3d 4s 4p 4d 5s 5p 4f 6s S0. The ﬁlled electronic shell gives very good collisional properties for atomic ytterbium. The doubly-forbidden3 tran-

2 1 3 sition of ytterbium (6s S0 → 6s6p P0) with a wavelength of 578.4 nm has a very narrow natural linewidth (∼ 10 mHz [146]) which is suited as a clock transition. An

optical lattice-based ytterbium atomic clock gives a promise of a new generation clock

standard with an expected fractional uncertainty better than 10−17 [147, 148]. More-

over, ytterbium holds beneﬁts for measuring atomic parity violation [149] and spin

squeezing because of its level structure and resistance from decoherences [150, 151].

3.3 Yb spectrum

In our lab, there are two ways to generate ytterbium atoms. One is using accelerated

argon ions to sputter a ytterbium metal plate in a room temperature vapor cell [152],

and the other is to laser ablate a ytterbium metal target inside a helium buﬀer gas

in a cryogenically-cooled vapor cell. The sputtered cell is used as an atomic reference

in Chapter 4.2, and the cryogenic cell is used for resolving diﬀerent transition lines of

Yb atoms.

A few-mJ laser pulse (∼5 ns) is focused onto a Yb plate. The high-intensity

pulse is injected into the target and liberates lots of ytterbium atoms. Note that the

ytterbium target has a high purity as listed in Table 2.1, but clusters or a small amount

of impurities may be produced by laser ablation. Although some other species and

3Spin forbidden and parity forbidden. 48 clusters may exist in the cell, there is no evidence showing any effect on the behavior of atomic ytterbium. If there is no buffer gas filling the cell, hot atoms will fly into the cell wall directly and disappear instantly. With the presence of helium buffer gas, hot atoms will elastically collide with helium atoms and transfer momentum to them.

After many thermalization collisions, atoms are cooled down in approximately a half millisecond. Cold atoms continue elastically colliding with helium buﬀer gas. In this way, they diﬀuse around and stay in the cell until they collide with the cell wall and freeze away.

A blue diode laser described in Chapter2 is used to probe the ﬁrst optically-

2 1 1 o allowed transition 6s S0 → 6s6p P1 of ytterbium. This strong optical transition has a fast spontaneous decay rate of 1.92×108 s−1 [153] which is proﬁtable for optical

1 pumping. Although there are metastable states between the ground state S0 and the

1 o P1 excited state [153], the decay probabilities to the metastable states are suﬃciently

2 1 1 o small [154] that the transition 6s S0 → 6s6p P1 can be treated as a nearly-closed “cycling” transition, so within our experiments there is no need for a repumping laser when preparing the initial states.

A probe beam is sent through the atomic cloud for taking an absorption spectroscopy of ytterbium. The probe beam is weak enough (a few microwatts) not to perturb the atoms. A typical spectrum of atomic ytterbium in the cryostat is shown in Figure 3.1 taken at a helium density of 2 × 1016 cm−3, 400 ms after a 1.3 mJ ablation pulse. Seven Yb isotopes are apparent and labeled in the spectrum. Their peak heights are measured to be consistent with their natural abundances. A simulated Yb spectrum with each individual transition line is displayed in Figure F.1. The peaks of 172Yb and 174Yb are distorted in Figure 3.1 owing to saturation4 of our absorption

4OD∼5 corresponds to a transmittance of 0.7%. At this low light level, the electronic noise of the detector, the oﬀ-resonance light from the laser, and room light contribute to the saturation. 49

spectroscopy at OD∼5. Among all ytterbium isotopes, the even isotopes do not have

173 1 o nuclear spin (I = 0). Yb has nuclear spin I = 5/2, and its excited state P1 splits into F 0 = 3/2,F 0 = 5/2, and F 0 = 7/2 due to the hyperﬁne interaction. As

for 171Yb, the excited state divides into F 0 = 1/2 and F 0 = 3/2 because 171Yb has

nuclear spin I = 1/2. The corresponding hyperﬁne transition lines are also labeled in

Figure 3.1. Since the hyperﬁne splitting of the excited state is large enough, the 173Yb

(F = 5/2 → F 0 = 5/2) transition peak is resolved from 173Yb (F = 5/2 → F 0 = 3/2)

and 173Yb (F = 5/2 → F 0 = 7/2) peaks, sitting between isotopes 176Yb and 174Yb.

6 |F’=7/2> 1 7 3 (F'=3/2) 72 MHz 174 172 |F’=3/2> 173 (F'=7/2) o 5 6s6p 1P 1 769 MHz

|F’=5/2> 4 173 y

t (F'=5/2) s i n e

d 176

l 3 171 c a

i (F'=3/2) t

751.53 THz p

O 171 (F'=1/2) (~399 nm) 2 170

168 0 2 1 6s S0 |F=5/2> 0.0 1.0 2.0 Frequency [GHz]

1 1 o Figure 3.1: Yb absorption spectrum taken on the S0 → P1 transition, alongside a simplified 173Yb level diagram [155]. The spectrum’s frequency offset is 25068 cm−1. The peaks are labeled according to isotope and excited-state hyperfine level.

Because the transition line broadening of ytterbium spectrum is dominated by

the Doppler eﬀect and pressure broadening, the Yb spectrum is ﬁtted to a Voigt 50

proﬁle which is a convolution of Lorentzian and Gaussian proﬁles [116, 117]. In

Figure 3.2, a multi-peak Voigt proﬁle5 is applied to ﬁnd Gaussian and Lorentzian

linewidths. Each corresponding Voigt peak can been seen in Figure 3.2, and their

overall sum matches with the Yb spectrum. The Gaussian linewidth is observed to

be independent of the cell pressure over a helium density range from 3 × 1016 to

3 × 1017 cm−3. From the Gaussian width we measure a Doppler temperature of 6

K, which is consistent with the cell body temperature measured from a temperature calibrated diode. Pressure broadening from the helium buﬀer gas is also observed.

The Lorentzian FWHM linewidth increases linearly with helium gas density over the same range, with a pressure broadening coeﬃcient of (1.7 ± 0.2) × 10−19 GHz/cm−3.

The oﬀset of a linear ﬁt to the Lorentzian data is 30.4 MHz which matches with its own natural linewidth (30.6 MHz). We note that the pressure broadening of the Yb spectrum has been previously measured at high temperature [156].

1.2 100 Data 1.0 Voigt Fit Fit Sum 80 0.8 60 0.6 OD

0.4 40

FWHM[MHz] GauFWHM LorFWHM 0.2 20 Fit Fit 0.0 0 17 -0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0x10 -3 Frequency [GHz] He density [cm ]

(a) Voigt ﬁt (b) Fit results

Figure 3.2: (a) Yb absorption spectrum at a helium density of 3 × 1017 cm−3. We fit its profile with the Voigt function. (b) GauFWHM: width from Gaussian part; LorFWHM: width from Lorentzian part. These linewidths measured from the Voigt fit are plotted as a funtion of helium density. Lines on the data are linear fits.

5A built-in function of IGOR Pro software [135]. 51 3.4 Diﬀusion lifetimes and yields

Instead of scanning the laser frequency to measure a spectrum, we ﬁx the laser frequency on the 173Yb (F = 5/2 → F 0 = 5/2) transition to observe the diﬀusion behaviors of ytterbium atoms at the cryogenic temperature. Note that the 173Yb

(F = 5/2 → F 0 = 5/2) transition is the main transition for the Yb experiments in this thesis. A typical atomic diffusion curve is shown in Figure 3.3, plotted in OD as a function of time after laser ablation. We prefer to use optical density rather than transmittance because OD is proportional to the atomic density when the absorption length and cross-section are fixed6. On the bottom axis of Figure 3.3, time is set to zero when the ablation laser fires. At the beginning, the increasing of OD shows that atoms are being loaded into the probe beam. The flat part in Figure 3.3 occurs because the absorption measurements saturate at OD∼5.

Soon after ablation, the diffusion behaviors of atoms are complicated and exist in many higher order diffusion modes. The high order diffusion modes decay quickly with time, so at later time only the lowest diffusion mode7 is left. The atomic density changing with time in the lowest diffusion mode can be described by an exponential decay function [22]:

−t/τD n(t) = npe (3.1) where np is the peak atomic density and τD is a time constant. Here, τD represents the diffusion lifetime. The diffusion lifetimes of atoms are measured by fitting the diffusion curve to Equation (3.1) at later time after ablation where the only diffusion mode is the lowest diffusion mode. In Figure 3.3, the exponential fit looks like a straight line when the ordinate is a logarithmic scale.

6See AppendixB for more details of the absorption cross-section. 7Diﬀusion models are described in AppendixC. 52

1 OD

Data Fit

0.1 0.0 0.5 1.0 1.5 2.0 Time [s]

Figure 3.3: Plot of 173Yb optical density as a function of time in the cryogenic cell, taken at a helium density of 9 × 1016 cm−3. The diﬀusion lifetime is 400 ms obtained from an exponential ﬁt.

In the case that atoms diﬀusing to the cell wall is the dominant atom loss mech-

anism, the decay lifetime is expected to scale linearly with the density of helium, as

indicated in AppendixC. In Figure 3.4, diﬀusion lifetimes with low ablation energies

and at low helium densities are linearly increasing with the helium density. By repeat-

ing the measurements at helium densities from 2 × 1016 cm−3 to 1 × 1017 cm−3 and at

low ablation energies (few mJ), a thermally averaged 173Yb-He diﬀusion cross-section8

of (1.1 ± 0.4) × 10−14 cm2 is extracted.

Unfortunately, at high helium densities, diﬀusion is no longer the only loss mech-

anism, so the lifetime of the ytterbium atoms begins to deviate from the expected

curve. The diﬀusion lifetimes at high helium densities, as shown in Figure 3.4, are

shorter at high ablation energies than at low ablation energies. The additional atom

8The diﬀusion cross-section is described in AppendixC. 53

1.4

1.2 140 Data 120 1.0 Fit Ablation energy [mJ] energy Ablation 100

0.8 80

0.6 60

40 0.4 20 Diffusion lifetime [s] lifetimeDiffusion 0.2

0.0 17 0.0 1.0 2.0 3.0x10 -3 Helium density [cm ]

Figure 3.4: The diﬀusion lifetime of ytterbium atoms, colored with diﬀerent ablation pulse energy, is plotted as a function of helium density at 6 K. loss mechanisms are not clear, but one possible reason may be due to the stirred gas

flow. When a high energy pulse liberates a lot of atoms, a strong atom flux is injected into the helium gas, “dragging” a gas flow and resulting in more atom-wall collisions.

Another explanation is that atoms form clusters or dimers at high helium densities, as either helium atoms provide a third body in collisions or increase atom-atom combining collisions because of “trapping” by helium atoms. The exact mechanism limiting the diﬀusion lifetime is not conclusively understood [14, 157]. The longest diﬀusion lifetime we observed in the Yb experiments is 1.3 s.

Increasing the production of atoms at high helium density and understanding the loss mechanisms are important issues for the technique of laser ablation combining with buﬀer-gas cooing. We have performed a two-ablation-pulse experiment in order to increase the production at high helium density; however, no obvious improvement in the atomic yields resulted [157]. In our experiment, a single laser pulse (∼ 5 ns, 54

∼ 100 mJ) generates N ∼ 3×1013 cold 173Yb ground-state atoms at a peak density of

1 × 1010 cm−3. It leads to an OD∼80 on the 173Yb F = 5/2 → F 0 = 5/2 transition peak.

3.5 Yb collisional properties

Because the level structure of 173Yb atoms is suited for performing experiments like electromagnetically induced transparency, we will focus on this isotope and discuss its inelastic collisions. We use optical pumping and polarization absorption spectroscopy to measure the spin depolarization time (T1) and the spin decoherence time (T2) as an investigation of the decoherence mechanisms in our cryogentically-cooled Yb vapor cell.

3.5.1 Optical pumping

At thermal equilibrium, the ground-state 173Yb atoms are unpolarized, meaning that

9 they have equal populations of the degenerate ground-state sublevels (F = 5/2, mF ) . To prepare the initial state of the 173Yb atoms, we use the optical pumping technique

[158] to perturb the atomic population. The direct optical pumping of nuclear spin states has been previously demonstrated with J = 0 atoms [157, 158].

Circularly-polarized σ+ light10 of high intensity is used to polarize atoms using the

F = 5/2 → F 0 = 5/2 transition. Suppose that one atom is initially in the ground- state Zeeman sublevel mF , and then it is excited into the exited Zeeman sublevel

9The ground state of 173Yb (I = 5/2) can be labeled as F or I and its sublevels are labeled as mF or mI . 10σ±-polarized light denotes that photons carrying ±~ angular momentum can 0 change the Zeeman sublevel quantum number by one (∆mF = mF − mF = ±1). π-polarized light has linear polarization which does not change the quantum number, ∆mF = 0. [4] 55

+ mF 0+1 by the σ optical pumping beam. After an excited-state lifetime, it returns to the ground state through spontaneous emission. Based on the transition selection

rules [4], it either returns to the original state or decays into the mF +1 or mF +2 state, if they exist. In this way, if there are suﬃcient cycles of light scattering, all atoms will be pumped into the ground-state Zeeman mF = +5/2 level. In the case of optical pumping with σ+ light at the F = 5/2 → F 0 = 5/2 transition, the ground-

+ + state Zeeman sublevel mF = +5/2 does not absorb the σ light because the σ light

0 cannot excite atoms in the mF = +5/2 state through the F = 5/2 → F = 5/2

0 transition. The mF = +5/2 state only decays to the ground-state mF = +3/2 or mF = +5/2 levels. Once atoms decay to the mF = +5/2 state, they stay there;

+ once atoms decay to the mF = +3/2 level, they will be excited again by the σ light until they enter the mF = +5/2 state. The mF = +5/2 state is called the dark

+ state, as it does not see the σ light. On the other hand, the mF = +5/2 state is a bright state at the F = 5/2 → F 0 = 7/2 transition, because it can absorb the σ+ light. Fortunately, the hyperﬁne splitting between F 0 = 5/2 and F 0 = 7/2 states is large enough (841 MHz), so there is little oﬀ-resonance scattering of light through the

F 0 = 7/2 state.

After exciting atoms with a σ+-polarized beam, the population of the degenerate ground-state atoms is perturbed to be unequal. In this way, atoms are said to be polarized. The eﬃciency of optical pumping unpolarized atoms to be polarized is related to the optical pumping duration, beam intensity, the transition strengths and depolarization. To transfer atoms from one mF state to the dark state, several scattered photons are usually needed, depending on the excited state decay probabilities and the transition strengths.

Let us ﬁrst consider a case with no external depolarizing mechanism, such as no inelastic collisions transferring atoms from the one mF state to others. With the 56

F = 5/2 → F 0 = 5/2 transition, atoms will be eventually trapped in the dark state

+ + mF = +5/2 by the σ light. In this case, even if the σ light is very weak, all atoms will be eventually pumped into the dark state as long as the optical pumping duration is long enough so that there are suﬃcient photons to scatter atoms. On the other hand, the optical pumping duration can be reduced if the beam intensity is strong enough to provide suﬃcient photons in a short time period. If the combination of beam intensity and pump duration already provides enough photons to pump atoms into the dark state, stronger beam intensity or longer pump duration do not help to pump atoms into the dark state.

On the other hand, if there exist depolarizing mechanisms such as inelastic collisions, they can cause depolarization in atoms. The depolarization is unwanted because it reduces the optical pumping efficiency and destroys the prepared state. In addition to inelastic collisions, atomic diffusion in and out of optical pumping volume is a depolarization mechanism because this polarized atoms may diffuse out and unpolarized atoms diffuse in. To reduce this effect, a large pumping beam size is helpful.

An inhomogeneous magnetic field also cause depolarization because it generates different alignments of the quantization axes distributed in the cell and then the spin alignment of atoms may flip when moving to a different part of the cell [117]. To effectively polarize atoms, the optical pumping effect needs to be stronger than the depolarizing mechanisms.

We use optical pumping techniques to prepare atoms in the condition of interest.

For example, polarized atoms in the depolarization experiment, coherent superposition states in the spin coherence experiments, and the atom-light coherence in the

EIT experiment are all prepared by optical pumping. 57

3.5.2 The T1 time measurements

The T1 time is simply a measurement of the atoms changing their states due to

inelastic collisions. There is no coherence detected in the T1 measurements. The

11 experimental setup of the T1 time measurements is shown in Figure 3.5. Laser beams are from the same diode laser as shown in Figure 2.10. After passing through

its individual AOM, the ﬁrst order diﬀracted beam is picked up to be the probe

(pump) beam. The AOM on the pump beam acts as a fast shutter in order to control

the amount of time perturbing atoms. However, because there is still some scattering

light from the pump AOM when the pump AOM is oﬀ, a mechanical shutter is placed

after the pump AOM to physically block the laser light. The AOM on the probe beam

is used to match the laser frequency shift. In this way, both beams can be adjusted

to have the same frequency shift (110 MHz) induced by the AOMs.

B field

PBS Probe AOM P1 W4 PD2

W4 AOM Pump P

Figure 3.5: A schematic of the experimental setup for the T1 measurements. P1 is a polarizer [159], W4 is a quarter-wave plate [129] and PBS is a polarizer beam splitter [160]. Photodetectors (PD1 and PD2) are with color ﬁlters [161]. Angles between probe and pump beams are for clarifying. The real angle between them are less than 5 degree.

The probe beam has a weak beam power (< 1 µW), and passes through a polarizer becoming a linear-polarized beam which is a combination of 50% σ+-polarized light and 50% σ−-polarized light. The collinear σ+ and σ− lights propagate through the

11In this thesis, 1 refers to the spin relaxation rate or spin depolarization rate. T1 58

cell to probe the atoms. After exiting from the cell, the probe beams are separated

by a polarization detection device which consists of a quarter-wave plate, a polariza-

tion beam splitter, and home-built ampliﬁed photodetectors. The quarter-wave plate

rotates the σ±-polarized light to yield vertical- and horizontal-polarized light, and

the beam splitter directs them to two detectors. Before entering the cell, the strong

(∼2 mW) pump beam with a linear polarization is rotated by a quarter-wave plate

to give a σ+-polarized light. Then, it is expanded to ﬁll up most of the cell volume

with a beam area of ∼25 cm2.

A uniform magnetic ﬁeld (few Gauss) parallel to the probe beams is generated

12 by a pair of Helmholtz coils in order to split the degeneracies of the mF levels and deﬁne a quantization axis. Since the Zeeman splitting is very small compared to the

transition linewidth at 6 K, the atomic samples are essentially unpolarized in thermal

equilibrium.

A typical T1 time measurement is shown in Figure 3.6 where the optical densities detected by the σ±-polarized probe beams are plotted at time after the ablation.

The overall decay shape is similar to the diﬀusion curve shown in Figure 3.3. The optical pump beam is turned on from 1.10 to 1.11 s after the ablation pulse. The noise near the optical pumping time is due to the vibration of the mechanical shutter aﬀecting the diode laser. Before the pump beam is on, atoms are unpolarized so that both probe beams detect the same optical densities. During the optical pumping,

+ the σ -polarized pump beam polarizes atoms and pumps them into the mF = +5/2 dark state. As a result, we see a reduction of the σ+ probe signal. The ratio of the optical densities seen by the σ± lights after the optical pumping is nearly a

constant. The signals on both beams at later time are gradually getting closer due

to the atomic diﬀusion in and out of the optical pumping volume. From ﬁtting the

12The design of the coils are described in AppendixG. 59

+ σ σ− 1 OD

0.1

0.0 1.0 2.0 3.0 Time [s]

17 −3 Figure 3.6: The T1 time measurement, taken at helium density 2 × 10 cm with an 1 mJ ablation pulse.

± difference between the OD’s of the σ light to an exponential day, we deduce a T1 time to be 0.62 s. The measurements are repeated at different helium densities and at different ytterbium densities, respectively. Since the measured data do not have linear dependence with helium or ytterbium densities, the results indicate that the T1 time is not affected by inelastic Yb-He and Yb-Yb collisions. In our case, the time scale are limited by the atomic diffusion and atom loss. Based on this observation, we set an

173 −18 3 −1 upper limit on the Yb-He spin depolarization rate coeﬃcient kT1 < 8×10 cm s

−22 2 and an inelastic collisional cross-section σT1 < 5 × 10 cm . Although this upper limit on Yb-He spin relaxation is similar to measurements of alkali-helium spin relaxation [34, 35], we expect that the actual inelastic collision rate coeﬃcient is many orders of magnitude lower, based on the prior measurements

199 199 1 of Hg atoms. Similar to Yb atoms, Hg also has a ground state S0.A T1 time up to 240 s has been measured in a room temperature 199Hg vapor cell [162]. It is 60 mostly limited by atom-wall collisions and magnetic ﬁeld gradients. The long spin relaxation time makes it practical to search for a permanent electric dipole moment

1 [163]. Since the ground state of Yb atoms is a S0 state, the collisional property of atomic ytterbium is expected to be excellent.

3.5.3 The T2 time measurements

How long a defined phase relation for a coherent superposition state can be maintained is measured by the T2 time. To measure the T2 time, we use a similar setup as shown in Figure 3.5. Instead of using the longitudinal magnetic field, we apply a transverse magnetic field which is perpendicular to the probe and pump beams.

Typical T2 time measurements are shown in Figure 3.7 where the overall atomic loss background is due to atom diﬀusion. An optical pumping pulse with a 1 ms duration is sent at 0.6 s after laser ablation; it polarizes atoms into a superposition state of mF energy eigenstates. Once atoms are pumped into this superposition state, they start to evolve with time and the atomic polarization precesses along the magnetic

field. The precession frequency is proportional to the magnitude of the transverse magnetic field. The relation between them will be soon discussed at the end of this section. Consequently, the absorption of the σ+ and σ− probe beams oscillates with an exponentially decaying diffusion background. We analyze the oscillating signals by subtracting one from the other. In this way, we can take out the diffusion background and extract the oscillating frequency by fitting the signal to

−t/T2 ∆OD = A sin(ωL + φ)e (3.2) where A is the amplitude, φ is the phase, and ωL = 2πνL is the Larmor frequency

[117, 164]. The T2 time in Equation (3.2) accounts for the fact that atoms diﬀuse away. If the rate of the optical pumping is fast compared to the atomic precession frequency, most atoms are pumped into the superposition state, resulting in a large oscillating 61

amplitude of the σ+ and σ− signals. If the optical pumping is slow compared to the

Larmor precession, a long pump duration does not help increase the amplitude of the precession vector.

1 σ+ σ- σ+ σ- 1 OD OD

0.1 0.1

0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Time [s] Time [s]

Figure 3.7: On the left figure, the fast oscillation is measured to have νL = 13.7 Hz, while the slow oscillation on the right figure has νL = 0.9 Hz. Based on Equation (3.4), the experienced magnetic fields are 66 mG and 4 mG, respectively.

The decoherence mechanisms for the atomic precession may be induced by the inelastic collisions (Yb-He and Yb-Yb), diffusion in and out of pumping area and inhomogeneous magnetic fields. Since the T2 time does not decrease with an increase of He density or Yb density, the inelastic collisions do no affect the coherence time.

If the subtraction of σ± signals is normalized by their sum, the fact that the number of atoms goes away due to diﬀusion can be take out. Thus, the T2 time is on the order of seconds which indicates our systematic environment is suﬃciently coherent.

It can be found in Figure 3.7 that the signals keep oscillating until they disappear, showing that inhomogeneous magnetic ﬁeld do not play a role in spin decoherence.

In fact, the atomic coherence is limited by atom loss due to diﬀusion. Since the T2 time does not decrease with a He density ranged from 8×1016 cm−3 to 2×1017 cm−3,

the upper limit on the 173Yb-He collisional decoherence rate coeﬃcient is measured

−17 3 −1 −22 2 to be kT2 < 1×10 cm s with a corresponding cross-section σT2 < 6×10 cm . 62

Although the T2 time in our system is mainly limited by the atom diﬀusion lifetime (∼1 s), it is signiﬁcantly longer than what has been previously measured for optically

trapped 171Yb [165]. However, it is still shorter than what has been achieved with the J = 0 ground state of atomic mercury in a vapor cell [163].

173Yb atomic magnetometer

The T2 time measurement is an interesting experiment since it can be used as an atomic magnetometer to measure magnetic ﬁelds by observing the atomic precession.

We use the value of 173Yb nuclear magneton µ listed on the NIST atomic handbook

[153] and describe its relationship to the gryomagnetic ratio γ as follows:

µ 762[Hz] µ = −0.6776µ = −0.6776 B = −0.6776 N 1836 [G] (3.3) = µN gN I

= γI

where gN is the Land´eg-factor and I is the nuclear spin. From the deﬁnition of the −→ magnetic moment −→µ = γ I , the gryomagnetic ratio γ can be rewritten as

ω 2πν γ = µ g = L = L N N B B µ = −0.6776 N = −0.271µ (3.4) 5/2 N = −206.5[Hz/G]

where B is the magnetic ﬁeld [117].

We measure the Larmor frequencies of the T2 experiments under different transverse magnetic fields, and plot the results in Figure 3.8. The magnitude of the transverse magnetic field abscissa is obtained from the applied current to the coils, while the ordinate is the measured Larmor frequency at the corresponding applied field.

By ﬁtting data in Figure 3.8 to an expected linear dependence, a slope of 210 Hz/G 63

is determined which is about 2% oﬀ the value in Equation (3.4). Based on this dis-

cussion, we measure the magnetic ﬁelds in Figure 3.7 to be 66 mG and 4 mG for the

fast and slow oscillations, respectively.

400

300

200

Data 100 Fit Larmor frequency [Hz] frequency Larmor

0 0.0 0.4 0.8 1.2 B_applied [G]

Figure 3.8: 173Yb atomic magnetometer measurements.

There are two important applications of this atomic magnetometer. One is to

determine how uniform the magnetic field is by measuring the T2 time from the precession oscillation. A uniform magnetic field gives a long T2 time. Note that it is expected to be more sensitive to the field gradients if the magnitude of the magnetic

field is small. The other application is to “zero” out the magnetic field experienced by the atoms. We adjust the bias magnetic fields in three dimensions until the T2 oscillation is as slow as possible while still has a large amplitude. A near zero magnetic

ﬁeld is important for the light storage and retrieval experiments. 64 3.6 EIT with nuclear spin

Chapter 3.5 discusses the collisional behaviors of the 173Yb nuclear spin and shows a good atomic coherence time. Moreover, a long atom-light coherence time is demonstrated in this section. We show that the standard electromagnetically-induced- transparency (EIT) techniques can be employed in atoms with a J = 0 ground state, creating coherence between the optical ﬁelds and pure nuclear spin states.

3.6.1 Dark state concepts

Before presenting the details of the Yb EIT experiments, the transparency of light due to the EIT effect is explained as follows. In the case of a three-level atom and two laser fields shown in Figure 1.1(b), both laser fields are on resonance with the atom. For simplicity, the excited-state decay and the ground-state decoherence are not considered. An effective Hamiltonian of this system, which has been derived in

AppendixA, is presented as Equation (A.22). The normalized eigenstates of this system are written as [72]:

−Ω Ω |a0i = c |1i + p |2i (3.5) p 2 2 p 2 2 Ωc + Ωp Ωc + Ωp 1 −Ω Ω |a+i = √ p |1i − c |2i + |3i (3.6) p 2 2 p 2 2 2 Ωc + Ωp Ωc + Ωp 1 Ω Ω |a−i = √ p |1i + c |2i + |3i (3.7) p 2 2 p 2 2 2 Ωc + Ωp Ωc + Ωp with the corresponding eigenvalues to be

E0 = 0 (3.8) q E+ = ~ Ω2 + Ω2 (3.9) 2 c p − q E− = ~ Ω2 + Ω2 (3.10) 2 c p where Ωc and Ωp are the Rabi frequencies for the control and probe beams. 65

Of all the eigenstates, |a0i does not include the excited state |3i, so that there is no excited state population which leads to no scattering of light. |a0i is called the dark state because it does not absorb light. On the other hand, eigenstates |a±i include the excited state population interacting with light and are called the bright states. (The dark state mentioned in Chapter 3.5.1 can be treated as a special case of the dark state discussed here.)

If initially an atom is in the bright state, it will be excited by the laser light and spontaneous decays back to the ground state. If atom decays to the bright state, it will be excited again. If atom decays to the dark state, it stays in the dark state. In this way, atoms will eventually be optically pumped into the dark state.

The transparency of light in an EIT system is because the atom stays in the dark state, and does not scatter by light. As indicated in Equation (3.5), the dark state is a superposition coherent state of the two lower energy states. The relative population of the lower energy states is determined by the relative intensities of control and probe beams.

3.6.2 173Yb EIT

To examine the EIT eﬀect and to create a medium with a long coherence time, we choose the ground-state nuclear spin of 173Yb atoms to demonstrate the atom-light

2 1 coherence. We use the Zeeman sublevels of the ground state |6s S0 F = 5/2i and the

1 o 0 + − excited state |6s6p P1 F = 5/2i. Two laser ﬁelds with σ - and σ -polarization act as the control and probe beams to form a 173Yb EIT scheme, illustrated in Figure 3.9.

Note that 173Yb, due to its half-integer spin, does not have a true coherent dark state in the scheme shown in Figure 3.9. For example, in the case that the σ+ control beam is stronger than the σ− probe beam, it forms a two-chain EIT with a leak on the

− 0 mF = −3/2 state because the σ probe beam can excite atoms into the mF = −5/2 66 excited state. This leak is a decoherence mechanism because it causes absorption.

The imperfect transparency induced by this kind of “N-type” EIT level structure is theoretically discussed in AppendixA. The leak can be avoided, if the transition

2 1 1 o 0 |6s S0 F = 5/2i → |6s6p P1 F = 3/2i is chosen. However, this transition

172 2 1 is overlapped with the isotope Yb and only 72 MHz away from the |6s S0 F =

1 o 0 5/2i → |6s6p P1 F = 7/2i transition. Fortunately, we expect to achieve a dark state

2 1 1 o 0 with a negligible leak via the |6s S0 F = 5/2i → |6s6p P1 F = 5/2i transition in the limit that the intensity of the control beam is much larger than that of the probe beam. That is theoretically shown in AppendixA. Working with the control beam intensity greater than 10 times the probe beam intensity, we see no evidence of problems due to this leak.

1 m -5/2 -3/2 -1/2 +1/2 +3/2 +5/2 = F' 6s 6 p P1 F ' 5/ 2

σ+ σ−

2 1 = 6s S0 F 5/ 2 m F -5/2 -3/2 -1/2 +1/2 +3/2 +5/2

Figure 3.9: Relevant level structure of 173Yb, with a σ+ control beam and σ− probe beam.

In order to understand the properties of the atom-light coherence, a measurement of the EIT transparency spectrum obtained by frequency detuning is necessary. The frequency detuning can be achieved by either scanning the laser frequency over the atomic transition as described in Chapter 3.6.3 or applying an external magnetic ﬁeld to frequency shift the ground-state Zeeman sublevels as shown in Chapter 3.6.4. 67 3.6.3 EIT with near-collinear beams

We first set up the Yb EIT experiment as shown in Figure 3.10. The control and probe beams are the first-order diffracted beams of their respective AOMs. (Similar to

Figure 2.10.) A polarization beam splitter combines these two beams before the cell, followed by a quarter-wave plate to rotate them to become σ+- and σ−-polarization, respectively. In general, this setup makes it easy to adjust the individual frequency of the control and probe beams. However, atoms may see different frequencies of both beams because the atomic motion in a thermal gas induces a Doppler effect. The overlap of the control and probe beams needs to be good, so that the atom sees both optical fields at the same time. The angle between both beams needs to be small, so that the two-photon Doppler shift is small. The two-photon Doppler shift is about

θ · 100 MHz/rad at 6 K. To eliminate the residual Doppler eﬀect, one either uses a very cold atomic sample to reduce atomic motion or works on the beam geometry.

Co-propagating beams are recommended for a Λ− or N-type EIT scheme, which is the scheme used in this thesis; counter-propagating beams are suggested for a cascade

EIT system [166].

PBS PBS Probe W4 L L W4 PD2

Control 1

Figure 3.10: A schematic EIT setup with near-collinear beams. L is a combination of lenses.

The Yb EIT transparency proﬁle has been observed based on this near-collinear scheme. However, due to the linewidth of the AOM drivers, the narrowest transparency linewidth is limited to 2 kHz which is not good enough to measure the atom- light coherence time. In addition, a small leak of control beam mixes into the probe 68

beam detector because of the imperfect polarization optics, resulting in a frequency

beat noise on the probe beam. The frequency beating complicates the analysis. We

then swap the setup to a collinear beam EIT as described below.

3.6.4 EIT with collinear degenerate beams

The experimental setup is similar to that of Philips et al. [85] and is illustrated in

Figure 3.11. A linear polarization beam passes through an AOM with its ﬁrst diﬀrac-

tion beam directed to an electro-optic modulator (EOM) [167]. The birefringence of

the EOM can be varied by an external electric ﬁeld. In this way, we can rapidly vary

the relative angle between the linear polarization of light and the optical axis of the

quarter-wave plate via the EOM. With the combination of the EOM and the quarter-

wave plate, we turn the linear-polarized beam into a σ+-polarized control beam and

a σ−-polarized probe beam. In addition, their relative intensity is controlled by the

EOM. Since the σ+ and σ− beams are from the same laser, their overall intensity is determined by the AOM. In our experiments, the intensity ratio between the control and probe beams is at least 10:1.

PBS AOM EOM W4 L L W4 PD2

Figure 3.11: A schematic EIT setup with collinear degenerate beams.

Between the EOM and the quarter-wave plate, we insert a 50 µm pin hole (not

shown) in order to spatially ﬁlter the laser beams. The beams are expanded to

fully ﬁll the cell to reduce the decoherence due to diﬀusion. The beam diameter is

approximately 2.5 cm. The polarization detection is the same as described in the 69

T1 time experiments. Two photodetectors monitor the control and probe beams, respectively. Three pairs of Helmholtz coils around the cell are used to zero out the

magnetic ﬁeld. Based on the T2 time measurements, the magnitude of the residual magnetic ﬁeld is less than 4 mG. The value is an upper limit of the measurement.

When needed, an external magnetic ﬁeld is applied along the beam direction in order to frequency shift the ground-state sublevels of atoms. In this way, we obtain the

EIT transparency proﬁle. The setup with collinear degenerate beams is the main experimental scheme in this chapter and Chapter4.

Two-photon resonance

Because the σ+ control beam and σ− probe beam are from the same laser, they have the same frequency. At near-zero magnetic ﬁeld, the ground-state Zeeman sublevels are nearly-degenerate. We ﬁrst scan the laser frequency over the usual transition

2 1 1 6s S0 → 6s6p P1 and obtain a Yb EIT spectrum, as illustrated in Figure 3.12. Comparing to the regular Yb spectrum in Figure 3.1, there is no transition peak of

173Yb F = 5/2 → F 0 = 5/2 appearing in Figure 3.12.

The whole 173Yb F = 5/2 → F 0 = 5/2 transition peak disappears as a result

of the EIT transparency eﬀect. Generally, transparency occurs when the frequency

diﬀerence between the control and probe beams matches that of the two lower energy

states [73]. Because we scan the laser frequencies of the control and probe beams

at the same time, the two-photon resonance condition is automatically satisﬁed over

the transition peak. This kind of detuning away from the atomic resonance is called

single-photon detuning, as shown in Figure 3.13(a). With the experimental setup

discussed above, only the 173Yb F = 5/2 → F 0 = 5/2 transition can satisfy the EIT

condition and its peak is transparent to the light.

The altered absorption heights around the peaks of 171Yb and 173Yb are the result 70

2.5 173(F'=3/2) 174 172 σ+ 2.0 173(F'=7/2) σ−

1.5 OD 171 1.0 176 (F'=3/2) 171 (F'=1/2) 170 0.5

168 Yb 0.0 -0.5 0.0 0.5 1.0 1.5 2.0 [GHz]

Figure 3.12: The ytterbium spectrum under a EIT condition. The 173Yb F = 5/2 → F 0 = 5/2 transition peak satisﬁes the two-photon resonance, so it “disappears.” ∆ is the single-photon detuning. of optical pumping. Note that the σ+ beam is at least 10 times stronger than the

σ− beam. In the Yb EIT spectrum, isotopes with I = 0 have the same optical

density, which is what is expected for the σ+ and σ− beams. As for the transition peak of 171Yb F = 1/2 → F 0 = 1/2, the strong σ+-polarized light pumps atoms

+ into the F = 1/2, mF = +1/2 dark state resulting in a reducing absorption on σ light, but an increasing absorption on the σ− light. The residual OD near the 171Yb

F = 1/2 → F 0 = 1/2 peak on the σ+ signal is due to the overlap with the isotope

170Yb. A similar discussion can be applied to the absorption curves of the 171Yb

F = 1/2 → F 0 = 3/2 peaks where there is a increasing OD on the σ+ signal and a

decreasing OD on the the σ− signal. Note that if there is only optical pumping and

no EIT eﬀect on the 173Yb F = 5/2 → F 0 = 5/2 transition, the signals should also

have a reduced absorption of σ+ light, but an increasing absorption of σ− light. In 71 fact, we see the whole peak disappears, which is an evidence of the EIT eﬀect.

3 3 ∆

Probe Control Probe Control

δ 1 2 1 2

(a) (b)

Figure 3.13: Schematic structures showing (a) the single-photon detuning ∆ and (b) the two-photon detuning δ of the EIT system.

The FWHM and depth of EIT windows

Instead of scanning the laser frequency over the spectrum, the frequency of the EIT beam is tuned to be on resonance with the 173Yb F = 5/2 → F 0 = 5/2 transition. To create the EIT dark resonance, the unpolarized atoms need to be optically-pumped into a coherent superposition state [72, 158]. The details of the optical pumping have been discussed in Chapter 3.5.1. To illustrate the EIT transparency window, we introduce a two-photon detuning δ. The two-photon detuning is induced by applying an external magnetic field to shift the energy of ground-state Zeeman sublevels as shown in Figure 3.13(b). The magnetic filed also shifts the single-photon detuning of the excited state, but by an amount that is small compared with the excited-state linewidth. Usually, the Zeeman shift of the excited states is around a few MHz while the Doppler width is about 100 MHz. With the external magnetic field, the two- photon resonance condition is no longer satisfied and the transparency effect will be reduced when |δ| increases, as shown in Figure 3.14. 72

To avoid confusion, we now define OD0 as the OD in absence of EIT. By fixing the laser frequency on the peak of the 173Yb F = 5/2 → F 0 = 5/2 transition and scanning the external magnetic field, the ratio of OD to OD0 is plotted as a function of the two-photon detuning δ in Figure 3.14. The abscissa is calibrated by the applied current to the longitudinal Helmholtz coils. We determine the two-photon detuning δ by the magnitude of the applied magnetic field B, with a convention that

δ [Hz] = 2 × 206.5[Hz/G]×B[G]. The factor of two accounts for that fact that the two lower levels in the EIT scheme have a quantum number diﬀerence of ∆mF = 2.

1.0

0.8 0 0.6

OD/OD 0.4 4 uW 70 uW 0.2 Fit Fit 0.0 -200 0 200 δ [Hz]

Figure 3.14: The ratio of the OD with EIT to OD0 without EIT is plotted as a function of the two-photon detuning δ.

The transparency profile is sensitive to various parameters, such as the Rabi frequency of the control beam, the optical density of atoms, Doppler broadening, and other mechanisms [168, 169, 170, 171, 172]. Since our Yb EIT is a thermal gas for which the Doppler effect is dominant, the EIT profile is analyzed by the expected functional form for a Doppler-broadened gas. In the limit that the EIT linewidth 73

is much smaller than the Doppler broadening, the EIT dip can be approximated as

a Lorentzian [171]. In Figure 3.14, beam powers of 70 µW and 4 µW pass through

an OD0 = 0.5 cloud, giving EIT FWHM linewidths of FWHM/2π = 200 Hz and FWHM/2π = 10 Hz, respectively. These transparency dips are very narrow and deep

features, appearing on the top of the Doppler-broadened linewidth (100 MHz).

There are only three axis bias coils, and no magnetic shielding is used in Yb

EIT experiments. With a relative simple setup, the Yb EIT dip width is already

narrower than that for a typical alkali EIT system, which usually has a linewidth in

a range from kHz to MHz [139, 170, 173]. Although the state-of-the-art transparency

window width, 2π ×1.3 Hz, is achieved in a rubidium atomic vapor cell, it is placed in carefully-designed four-layer magnetic shielding and surrounded by three pairs of bias coils [91]. The dip width measurements of the Yb EIT can be improved by adding magnetic shielding, improving the stability of the power supply of the coils, etc. The details of the dip width and depth in the Yb EIT will be discussed below.

The width and depth of the transparency window have a strong dependence on the control beam strength. By varying the control beam power, the width and depth of the transparency window were measured and plotted in Figure 3.15. Data is ﬁt to a model based on References [171, 174]:

2 Ωc Dip FWHM ΓEIT = 2γg + (3.11) WD + γ OD 1 Dip depth = (1 − d) · + d (3.12) Ω2 OD0 1 + c 2γg(WD+γ)

where γg is the ground-state decoherence rate, γ is the excited-state decay rate, WD

is the FWHM of the Doppler-broadened line and Ωc is the Rabi frequency of the control beam. The constant d is not included in References [171, 174]. We add it into

Equation (3.12) to account for the oﬀ-resonance absorption from the other ytterbium

isotopes. This modeling assumes that all atoms are already optically pumped into 74

the dark state.

In Figure 3.15(a), the dip width decreases linearly with the control beam power,

and is ﬁnally limited by the decoherence. The experimental results13 suggests a

ground-state decoherence rate smaller than 10 Hz and an atom-light coherence time

better than 100 ms in the Yb pure nuclear spin system. The linewidth measurements

are technically limited by the applied magnetic ﬁeld. With a stable power supply

and laser frequency stabilization, a narrower measurement result is expected. On the

other hand, although a weak control beam leads to a narrow linewidth, it is possible

that there are not suﬃcient photons to scatter atoms into the dark state, resulting in

an imperfect transparency. To obtain a narrow linewidth with a weak control beam,

we optical pump atoms into the dark state with strong control beam power ﬁrst, and

then measure the linewidth along with weak control light.

1.0

3 Data 10 Fit 0.8 Data Fit

0 0.6 [Hz] 2π

/ 2

10 OD/OD 0.4 ΕΙΤ Γ

0.2

1 10 0.0 0.1 1 10 100 1000 0.1 1 10 100 1000 Control beam power [W] Control beam power [µW]

(a) EIT width vs. control power (b) EIT depth vs. control power

Figure 3.15: The width and depth of the transparency window as a function of control beam power.

Although a weak control beam gives a narrow dip width, the depth of the trans-

13Redesigned bias coils were used for the measurements in Figure 3.14, resulting in a smaller decoherence than for those in Figure 3.15. 75

parency window is small at weak control power in Figure 3.15(b) because the ground-

state decoherence term becomes important. On the other hand, a strong control beam

gives a deep transparency depth if the absorption is mainly due to the decoherence.

Our best measurements of the depth still have 20% absorption by using the strong

beam power. The imperfect transparency is partially due to oﬀ-resonance absorption

by other isotopes (176Yb and 174Yb). Based on the simulation in Appendix F.1, there

would be an absorption of 0.03 OD0 from the natural linewidth alone. This is made worse by Doppler broadening eﬀect. If the Yb spectrum is simulated with a Doppler- broadened linewidth of 0.1 GHz and a natural linewidth of 0.03 GHz, there would be an absorption of 0.09 OD0 from the other isotopes. In addition, the oﬀ-resonance from other isotopes increases with the helium density due to pressure broadening. In the limit of high control beam power, a measurement of the dip depth is plotted as a function of helium density in Figure 3.16.

30 [%] 0 20 OD/OD 10 Data Fit

0 17 0 1 2 3 4x10 -3 Helium density [cm ]

Figure 3.16: Measurement of the dip depth versus helium density in the limit of high control beam power. The oﬀset is 9% and the slope is 7% per cm−3.

It may be possible to further suppress oﬀ-resonance absorption by optically pump-

176 174 14 3 0 ing the Yb and Yb isotopes into the metastable 4f 6s6p P0,2 states, which 76 have radiative lifetimes of 140 s and 15 s, respectively [175, 176]. Also, the oﬀ- resonance absorption observed from other isotopes can be greatly reduced by using an isotopically-enriched Yb target.

Yb EIT discussion

Because the nuclear spin is weakly interacting with the environment, it is usually diﬃcult to be manipulated. Through the method of the EIT technique applied to the nuclear spin of Yb atoms, the nuclear spin can be strongly coupled by the optical

ﬁelds. In this way, it seems to be possible to have the ability to control the nuclear spin, while still maintaining its good coherence.

Finally, we have demonstrated the ability to create a cryogenically cooled cell of atomic Yb with high optical density and long atomic coherence times. We have also demonstrated coherent coupling between light and nuclear spin using EIT, showing a long atom-photon coherence time. 77

Chapter 4

Slow and stopped light with nuclear spin

The previous chapter describes the experimental details and results of Yb EIT. In this chapter, we investigate the propagation of a classical light pulse through an EIT medium as the ﬁrst step for the future work of quantum information storage. Light storage and retrieval in pure nuclear spin are demonstrated here, and storage times of hundreds of milliseconds are observed. It shows that the Yb EIT system is promising for realizing quantum memory in the future.

4.1 Atom-photon coupling

There are two main strategies employed to achieve the strong atom-photon coupling.

One method to deterministically control the atom-photon interaction is by cavity quantum electrodynamics (cavity QED) [177]. A built-up atom-cavity resonance greatly enhances the radiative emission. Cavity QED enables reversible radiation to pass through a single atom multiple times, resulting in an enhanced atom-photon interaction. Although cavity QED provides a strong coupling between the atom and 78 light, it still has its technical challenges [73]. The second method uses an atomic ensemble which has large numbers of atoms to interact strongly with photons. For example, photons can be completely absorbed within an optically-thick atomic cloud.

As a result, an optically-thick atomic ensemble can construct the strong light-matter coupling. Even though the absorption of light by atoms is considered a decoherent process, dissipation can be avoided by providing the EIT condition [73]. A large number of atoms gives a signiﬁcant collective interaction such that the coherent coupling between atoms and light scales linearly with the number of atoms [73, 178]. The collective excitation of matter makes it possible to realize an EIT-based quantum memory [78].

An ideal quantum memory for quantum information processing is expected to have properties that quantum states are well-preserved from the environment, and a strong interaction is accessible to operate these quantum states. However, these two expectations are conﬂicting and usually diﬃcult to satisfy at the same time. Nuclear spin has been shown to have a long coherence time due to its very weak interaction with environment. Techniques have been developed to interact with nuclear spin, such as spin-exchanging optical pumping of nuclear spin by polarized alkali atoms

[26]. However, spin-exchanging optical pumping does not provide the coherent manipulation between atoms and light. Other methods of applying nuclear spin for quantum information science have been proposed or developed in 3He gas [179], in a silicon electronic device [180], and with nitrogen-vacancy centers in diamond [181].

Whether or not these systems could have the best performance is not clear.

In the ground state of atomic ytterbium, there is no coupling between the nuclear spin state and the electron angular momentum. It is therefore a good medium for storing quantum information. To prepare the nuclear spin state, we optically pump the atoms through a J = 1 excited state, in which there is hyperﬁne coupling between 79

the electron and the nuclear spin. The hyperﬁne link only turns on during the excited-

state lifetime (∼ 5 ns). Since there is only a short time window to interact with the environment, nuclear spin states can still preserve a good coherence. Once atoms are optically pumped into the coherent dark states through the EIT two-photon resonance, no excited-state population is involved. In this way, optically pumping the nuclear spin with EIT techniques provides the ability to have a strong coupling between atoms and photons while still keeping a long coherence time.

4.2 Frequency stabilization

Before starting the demonstration of slow and stopped light experiments, a technical approach of stabilizing the laser frequency is addressed here. As described in Chap- ter 3.6.4, the EIT transparency proﬁle is sensitive to the laser frequencies and the energy level structures. To obtain the optimal transparency and dispersion of the refractive index for the light pulse, the laser frequency needs to stay at two-photon resonance and single-photon resonance, when the light pulse is passing through the atomic cloud.

The stabilization of the laser frequency is based on the dichroic-atomic-vapor laser lock signal (DAVLL) [182] from a Doppler-free saturation absorption spectroscopy

(DFSAS) [4]. A room temperature sputtered Yb cell, ﬁlled with argon gas, is built as an atomic reference [152]. Argon ions are accelerated by high voltage and strike a Yb metal plate to produce atomic ytterbium. The design of the sputtering cell,

Doppler-free spectroscopy, and pre-DAVLL signals of ytterbium atoms are described in Franklin Jose’s Senior thesis [152]. Here, we examine a Doppler-free DAVLL signal, and use it as a frequency lock error signal. 80 4.2.1 Doppler-free DAVLL signal

The optics setup for the Doppler-free DAVLL experiment is illustrated in Figure 4.1.

A 420 µW pump beam with linear polarization is focused down to a diameter of

∼ 0.5 mm in the center of the sputtered cell in order to saturate the absorption of

atoms. After passing through an absorptive polarizer, the returned beam is a weak

(∼20 µW) linearly-polarized probe beam. The probe beam overlaps with the pump

beam and intersects with it in the center of the cell. The intersection between these

two beams has an angle less than 1◦.

Similar to Chapter3, the weak linear-polarized probe beam consists of a σ+-

polarized beam and a σ−-polarized beam with the same intensities. The Yb saturated

absorption spectrum is probed by these σ+ and σ− beams. They are directed to a

polarization detection device, including a quarter-wave plate and a polarization beam

splitter. In this way, they are directed to two photodetectors. Each photodetector

observes an individual DFSAS signal. By subtracting these two DFSAS signals, a

Doppler-free DAVLL signal is obtained.

DFSAS is based on the principle that the counter-propagating probe and pump

beams interact with a small number of atoms which are near zero velocity, so no

Doppler shift1 is induced due to the atomic motion. The strong pump beam strongly

perturbs the population of ground-state atoms, and causes a reduced absorption of the

1 According to the relativistic Doppler formula [23, 116], s 1 ∓ v/c v 1 v2 ω = ω = ω ∓ ω + ω + ..., 1 ± v/c 0 0 c 0 2 c2 0 the second term on the right hand side is the linear Doppler shift or the ﬁrst-order Doppler shift, which can be eliminated by Doppler-free techniques. However, the third term is independent of the direction of the velocity v and does not depend on the direction, and cannot be eliminated by Doppler-free techniques. It is called the quadratic Doppler eﬀect or the second-order Doppler shift. 81

probe beam. Consequently, small peaks (∼4% increase in transmittance) show up on the background of a room temperature Doppler-broadened absorption signal, shown in Figure 4.2(a) (the blue curve), along with a Yb absorption spectrum taken in a cold cell (the red curve). In Figure 4.2, raw data is collected from the photodetectors as a function of time after laser ablation; the laser frequency is scanned linearly as a function of time. In Figure 4.2, two apparent peaks, 176Yb and 174Yb, are labeled for comparison. There is a relative frequency diﬀerence of 110 MHz between the sputtered and cold atom signals from their respective AOMs.

RF DAVLL L1 AOM Iris Symbol Device beam L1234 50, 30, 40, 75 cm lens C1 Yb sputtering cell B B Permanent magnet P1 L4 C1 L3 L2 P1 Polarizer

W4 Quarter-wave plate DFSAS PD Photodetector DAVLL - PD2 signal PD1 PBS W4 Mirror

Figure 4.1: Doppler-free DAVLL optics setup.

A permanent magnet placed below the cell generates a magnetic ﬁeld (∼0.001 T in

2 7 the cell) and induces a frequency shift of ±2×10 Hz on the mF 0 = ±1 excited energy levels by the Zeeman eﬀect, as indicated in Figure 4.11. Two DFSAS signals detected

by the σ+- and σ−-polarization probe beams are subtracted to be a DAVLL signal

as shown in Figure 4.2(b) (the blue curve) [182]. The Doppler-free DAVLL peaks

are from the diﬀerences between red-shifted and blue-shifted DFSAS peaks. They

should have a frequency separation of 4 × 107 Hz induced by the external magnetic

2 1 The gryomagnetic ratio of the Yb P1 state is 1.4 MHz/G. 82

field. Their background is also a DAVLL signal which is from the difference of the red-shifted and blue-shifted Doppler-broadened absorption profiles.

4 4

176 Yb 176 Yb 174 3 2 Yb 174 Yb

0 174 2 Yb Signal[V] Signal[V]

176 -2 Yb 1

Signal at 4 K -4 Signal at 4 K DFSAS DFSAS DAVLL 0 1.545 1.550 1.555 1.560 1.565 1.570 1.490 1.500 1.510 Time [s] Time [s]

(a) Yb DFSAS (b) Yb DFSAS DAVLL

Figure 4.2: (a) DFSAS signal in a room temperature sputtering cell, obtained from one of the photodetectors in Figure 4.1. It is plotted along with an absorption signal of cold atoms. (b) DFSAS-DAVLL signal in a sputtering cell, obtained from the subtraction between PD1 and PD2 in Figure 4.1. It is shown along with an absorption signal of cold atoms Note that probe frequencies of the red and blue curves have a relative frequency shift of 110 MHz due to the AOMs. The bottom axis records time after laser ablation; laser scanning frequency is linearly proportional to the time.

To have an optimal contrast of the Doppler-free DAVLL signal in the sputtering cell, the following experimental conditions are suggested:

2 • pump beam intensity & the saturation intensity 63 mW/cm • good alignment (as much overlap between pump and probe beams as possible) 83

• the sputtered3 atomic optical density, OD∼1

• The induced Zeeman shift is approximately the natural linewidth

4.2.2 Frequency lock

In the Yb experiments, the Doppler-free DAVLL signal is not only a good atomic reference, but is also used for the frequency stabilization. Because the isotope 174Yb

has a strong contrast signal, its Doppler-free DAVLL signal is chosen as a feedback

reference for the diode laser. Two AOMs are used to frequency shift the relative

frequency between the DAVLL beam at the sputtering cell and the main beam going to

the cold atoms. The −1 order diﬀraction beam from the EIT AOM gives a −110 MHz

frequency shift, while the +1 order diﬀraction beam from the DAVLL AOM has a

frequency shift of +130 MHz. With a relative laser frequency diﬀerence of 240 MHz,

174Yb DAVLL signal is aligned with the 173Yb (F = 5/2 → F 0 = 5/2) cold atom

absorption peak as illustrated in Figure 4.3. The frequency separation between these

two transitions is 253 MHz [155]. This deviation is possible due to the background

oﬀset of the DAVLL signal. The contrast of 174Yb Doppler-free DAVLL gives a slope

of 15 MHz per volt.

The frequency stabilization method is the standard proportional-integral-derivative

(PID) feedback technique4. When slightly adjusting the angle of quarter-wave plate

(see Figure 4.1), the DAVLL signal level will move up or down. In this way, a

frequency lock point is determined by comparing the DAVLL sigal with a zero volt-

age, as displayed in Figure 4.3. The lock point is usually set to be at the middle

of the 174Yb contrast of the DAVLL signal, which is corresponding to the 173Yb

3The applied current for sputtering is 0.38 mA. 4The details of the PID control theory and the home-built electric circuits are not discussed in this thesis. See references [116, 183]. 84

176 4 Yb 174 Yb 176 2 Yb

0 174 Yb PD[V]

-2

Signal at 4K DFSAS DAVLL -4 Marker

1.725 1.730 1.735 1.740 1.745 1.750 1.755 [sec]

Figure 4.3: After shifting the laser frequencies with the AOMs, the 174Yb DFSAS- DAVLL signal on the blue curve is aligned with the 173Yb (F = 5/2 → F 0 = 5/2) transition peak on the red curve. The dashed lines are reference guides. The inversed DAVLL signal comparing to Figure 4.2(b) is due to the inversion of electronics used to stabilize the laser.

(F = 5/2 → F 0 = 5/2) transition peak. To frequency stabilize the diode laser, the laser frequency is fixed at the lock point. When laser frequency drifts from the lock point, a voltage difference (an error signal) between the DAVLL signal and the zero voltage is produced. This voltage difference is delivered to the PID circuit. The circuit outputs a feedback voltage to the laser control box, and corrects the laser frequency until the voltage difference returns to zero, indicating that the laser is back to the target frequency. Based on the PID control theory, the electric circuit is able to have a proper time response to frequency stabilize the diode laser.

During the frequency stabilization, the laser line width is ∼10 MHz and can stay on the transition line as long as the laser mode does not change. When choosing a diﬀerent lock point within a range of 50 MHz away from the center frequency of the 173Yb F = 5/2 → F 0 = 5/2 transition, the performance of EIT and slow light 85

experiments do not seem to be aﬀected.

4.3 Slow light

In the postulate of Einstein’s special relativity, the speed of light in vacuum is a

physical constant5 [28]. It is deﬁned as c = 299792458 m/s, or approximately c =

3 × 1010 cm/s. When light, or an electromagnetic wave, travels through a medium,

its speed will be slower than the speed of light because of the index of refraction n.

As a result, the speed of a monochromatic electromagnetic plane wave traveling in a medium is v = c/n. Common materials, like glass and water, usually have the index of refraction around 1.3-1.6, so that there is no signiﬁcant diﬀerence in the speed of the electromagnetic wave.

When considering a light pulse, which is a superposition of plane waves with different frequencies, one quantity is not enough to describe the relative phases between plane waves and the group behavior of the light pulse. Especially, waves with diﬀerent frequencies have diﬀerent responses to the medium. The motion of a light pulse traveling in a dispersive medium with an index of refraction n(ω) is therefore described by the phase velocity vphase and the group velocity vgroup:

c v = (4.1) phase n(ω) c c vgroup = dn(ω) = (4.2) ng n(ω) + ω dω

where ω = 2πf is the angular frequency and ng is the group index of refraction [28]. In 1992 Harris et al. [185] pointed out that the index of refraction n is unity and

the linear dispersion dn/dω is positive and large in an ideal EIT three-level system

5The international standard meter is deﬁned by the distance light travels in 1/299792458 of a second. The eﬀort to make precision measurements of a second, using devices such as an atomic or optical clock, is extremely important for modern applications and fundamental physics [184]. 86

with single-photon and two-photon resonance (∆ = δ = 0). (The details of the nonlinear response of a EIT medium is described in Chapter 1.5 and AppendixA.)

According to Equations (4.1) and (4.2), the phase velocity then equals the speed of light and the group velocity becomes very small when the light pulse passes through an EIT medium. Moreover, the absorption of light is zero, resulting in no dissipation loss. All these conditions are favorable for realizing slow light with an EIT medium.

The idea of slow light is based on the very large and positive slope of the dispersion

(dn/dω > 0 and large) and no absorption.

Observations of slow light pulses based on this idea have been realized in diﬀerent systems. The ﬁrst slow light experiment was demonstrated in a lead (208Pb) vapor cell

[186], yielding a pulse velocity as slow as c/165 (1818 km/s) with 55% transmission.

A remarkable ultraslow light experiment was realized in Bose-Einstein condensation of sodium atoms, achieving a group velocity of 17 m/s [173]. A similar result was also observed with a group velocity of 8 m/s in a hot rubidium vapor cell [187]. Also, a

45 m/s slow light was realized in a solid of Pr doped Y2SiO5 [188]. Having the ability to slow down a light pulse gives an opportunity to “stop” light.

It is a way to realize quantum memory in that the quantum states carried by the light pulse can be stored by and retrieved from the atoms. Unlike the usual systems, we demonstrate the slow and stopped light experiments in a cryogenic Yb vapor cell and show a competitive performance with other systems. The properties and performances of the Yb system will be discussed below.

4.3.1 Experimental details

When a light pulse travels through a medium with a length of L, it is delayed by a time of

τdelay = L(1/vgroup − 1/c) (4.3) 87

where τdelay is the time delay. Because the cold Yb atoms are a thermal gas, the time delay of the light pulse depends upon the Doppler eﬀect. According to Reference

[174] used in Chapter3, the time delay in a Doppler-broadened medium is described by 2 (WD + γ)Ωc τdelay = OD0 2 2 (4.4) [2γg(WD + γ) + Ωc ] where Equation (4.4) has the same assumption and deﬁnitions as Equation (3.11)

and Equation (3.12). From Equation (4.4), there is a maximum of the time delay of

max 2 τdelay = OD0/8γg when choosing Ωc = 2γg(WD + γ). Under this condition, a medium with a large optical density and a small decoherence rate leads to a long delay time

2 of the light pulse. In our experiments, Ωc is usually much greater than 2γg(WD + γ), which gives good transmittance. So, reducing the control beam intensity also gives a

2 longer time delay until it reaches the limit of Ωc = 2γg(WD + γ). When sending the light pulse through the EIT medium, it is better to make the pulse width (1/τG) small compared to the EIT linewidth (ΓEIT). In this way, the light pulse can ﬁt inside the transparency window in the frequency domain, so that

there is no distortion of the pulse from the absorption by atoms. The whole pulse

experiences a similar value of dn/dω and diﬀerent parts of the pulse obtain the same

delay. To achieve this condition, we either send a pulse with a narrow pulse width

(small 1/τG) or use a wide transparency window (large ΓEIT). A narrow pulse in the frequency domain corresponds to a long pulse width in the time domain. (The

time-frequency conversion based on the Fourier transform is listed in Appendix G.3.)

In fact, a narrow pulse width in frequency domain is not practical to use. However,

a wide transparency window needs a strong control beam power, which shortens the

time delay according to Equation (3.11). 88

Delay-bandwidth product

To satisfy the long time delay and let the pulse fit inside the EIT dip, a relation of 1 · τdelay < OD0 (4.5) τG 1 is obtained from Equations (3.11) and (4.4). We define the quantity of · τdelay to τG be the delay-bandwidth product (DBW) which is the ratio of the pulse’s time delay to its 1/e half width. If DBW is greater than one, it means that the majority of the light pulse fits inside the transparency window; it is a desirable condition for the stopped light experiment. The upper limit of the delay-bandwidth product is the optical density in the absence of the EIT effect. Equation (4.5) suggests that an

1 optically-thick medium is preferred for getting a larger product of · τdelay. τG

Data

The optics setup is similar to Figure 3.11. To observe slow light we use the EOM to modulate the power of the probe beam to create a Gaussian pulse, and send it into the Yb EIT medium as described earlier. An example of slow light can been seen in Figure 4.4. The 1/e half width of the input pulse is 60 µs and the control beam power is 3 mW in a 2 cm diameter beam. A reference pulse is sent through the cell in the absence of Yb atoms; the vertical axis is normalized to the reference pulse intensity and the time axis has an oﬀset to its peak. Another light pulse is sent with the presence of Yb atoms, and has a time delay of 12 µs relative to the reference pulse. Because τdelay is related to the control beam intensity, the time delay of the light pulse can be as long as 1 ms if a small control power is used (∼10 µW).

See Figure 4.6. The 1 ms delay time corresponds to a group velocity of 100 m/s or

3 × 10−7c.

We repeat the slow light experiments by using diﬀerent widths of the light pulse, and analyze them. Figure 4.5 presents a series of the slow light data. Since the 89

1.0

0.8 Without atoms With atoms

0.6

0.4 Intensity [normalized] 0.2

0.0

-200 -100 0 100 200 300 Time [µs]

Figure 4.4: Observation of slow light. The black curve is a reference pulse; the red curve shows a slow light pulse with a transmission of 45% and a 12 µs delay relative to the reference pulse. Data taken at a helium denisty of 1.6 × 1017 cm−3, with OD0 = 1.8.

pulse width is small (20∼80 µs) compared to the diﬀusion lifetime (∼200 ms), the

exponential decay of the atom number is not important here. We ﬁt the light pulse

to a Gaussian proﬁle and compare it with its corresponding reference pulse. In Fig-

ure 4.5, data with diﬀerent colors indicate diﬀerent time length of the pulse width.

Figure 4.5(a) shows the experimental results of time delay as a function of OD0 with diﬀerent pulse widths, while Figure 4.5(b) shows the transmittances of these light pulses. For the same OD0, a long pulse in the time domain gives a longer time delay and a better transmittance because it is closer to the condition of 1/τG < ΓEIT than a short pulse in the time domain. Initially, the time delay linearly increases with the

OD0, which is consistent with the theoretical prediction. However, when OD0 & 4, this is no longer the case. The transmittance also becomes worse with the increasing of OD0, partially due to the oﬀ-resonance scattering of other isotopes. The poor 90

performance at high OD0 will be discussed in Chapter 4.3.2.

40 1.0

0.8 30 s] µ 0.6 20 0.4

Time delay [ Timedelay 10 0.2 Transmission[X100%]

0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 173 173 OD0 of Yb OD0 of Yb

(a) (b)

1.0 0.16 80 0.14 0.8 70 Pulse width [ 0.12 60

0.6 0.10 50 µ

40 s] 0.08 DBW 30 0.4 DBWT 0.06 20

0.2 0.04 0.02 0.0 0.00 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 OD of 173Yb 173 0 OD0 of Yb

Figure 4.5: A series of slow light data. (a) Time delay and (b) transmittance of slow light, along with their (c) DBW and (d) DBWT analyses. Data color scales with the pulse width. Helium density = 3 × 1016 cm−3.

Based on the input pulse width and the measured τdelay, the DBW is plotted versus

OD0 in Figure 4.5(c). A delay-bandwidth product close to 1 is achieved with a 20 µs pulse. For the later discussion of the stopped light experiments, another important quantity is the delay-bandwidth-transmission product (DBWT). It is equal to the delay-bandwidth product of the light pulse multiplied by the transmission of the 91

Gaussian pulse. In the limit that DBW is small, a large DBWT indicates a small loss

of the pulse due to absorption by atoms. In the case of DBWT 1, the whole light

pulse can be stored and retrieved with very little distortion. It is an ideal condition

for a stopped light experiment. The measured DBWT is plotted in Figure 4.5(d),

where the maximum DBWT is 0.16.

For the purpose of slow light, a light pulse with a large width in time tends to

give a good delay time. As for the application of the stopped light experiment, the

transmission and the time delay are both important. In other words, a large DBWT

is favorable. Therefore, a proper pulse width can be chosen for a given control beam

power with the analysis of Figure 4.5. The optimal OD0 can be known as well. In the

case of Figure 4.5, a pulse width of 60 µs at OD0=2 is a good condition for performing the stopped light experiment.

Time delay versus two-photon detuning

The above observation is on the single-photon and two-photon resonance (∆ = δ = 0).

Because slow light is based on the nonlinear properties created by the coherence

between laser ﬁelds and atomic states, it is sensitive to the two-photon detuning of

the EIT medium.

To investigate the slow light eﬀect and also to verify the EIT condition, an external

magnetic ﬁeld is applied to induce a two-photon detuning δ. In Figure 4.6, the measured time delay is plotted as a function of the two-photon detuning. The time delay has its maximum when δ is zero; τdelay becomes small as δ increases. When the two-photon detuning is large enough, no obvious time delay is observed and the transmittance of the pulse is low as expected (not shown in Figure 4.6). Note that the delay time approaches to zero when δ is near 100 Hz which is comparable to the width of the transparency window. 92

1.6 1.4 1.2 1.0 0.8 0.6

Time delay [ms] Timedelay 0.4 0.2 0.0

0 100 200 300 400 500 δ, two-photon detuning [Hz]

Figure 4.6: Time delay versus two-photon detuning. The 1/e pulse width is 6 ms. 16 −3 Data is taken at OD0 = 3, and a helium density 6 × 10 cm .

4.3.2 Problems at high OD0

The good performance of the slow light experiments requires a high atomic optical density. Also, to have strong atom-light couping with the atomic ensemble, an optically-thick medium is needed. However, the Yb experiment does not work properly at high optical density. The possible reasons for the insufficient performance in our system are insufficient optical pumping power, off-resonance scattering from other isotopes, and radiation trapping.

Initially, atoms are thermally-populated in the Zeeman sublevels, and a suﬃcient laser power is needed to optical pump all atoms into the coherent superposition state.

It is possible that there are not enough photons to scatter atoms at high optical density. Varying the power of the control beam does not seem to improve the performance of EIT at high OD0, which reveals that laser power is not the problem. In general, a large beam intensity is preferred for performing slow and stopped light 93 experiments, so that atoms see the ﬁeld more uniformly.

Because ytterbium atoms are produced from a natural-isotopic-abundance target by laser ablation, the generated Yb atoms include all the stable isotopes, which can be confirmed by the Yb spectrum in Figure 3.1. The off-resonance scattering from other isotopes becomes important when the optical density is large and at high helium density, which can be seen in Figure 3.2(a) and Figure 3.16. Fortunately, off-resonance scattering should be largely reduced by switching to an isotopically- enriched ytterbium target.

In an optically-thin6 medium, spontaneously emitted photons are rarely re-absorbed by atoms and leave the atomic ensemble easily. However, in an optically-thick medium, spontaneously emitted photons may be re-absorbed by atoms multiple times before they leave the atomic cloud. This eﬀect is called radiation trapping [158, 189].

Radiation trapping is an issue for experiments working at high optical density, which has been observed before in alkali atoms [189, 190, 191]. The spontaneously emitted photons have random polarization, and will diminish the ground-state coherence.

They also reduce the ability to optically pump atoms into the dark state. The problem of radiation trapping is worse in ytterbium because of the off-resonant scattering by other isotopes which cannot be pumped into a dark state. The tails from other isotopes overlapping with the EIT transition produce many photons with random polarization, destroying the ground-state coherence and degrading the optical pumping efficiency. If an isotopically-enriched target is used, the off-resonant scattering should be greatly reduced. To overcome radiation trapping, techniques has been developed in alkali vapor cells such as elongated cell geometries to let spontaneously emitted photons escape or using quenching gases to provide non-radiative quenching of excited state atoms [191, 192]. In the cryostat, the high OD0 performance should be

6 Usually, OD0 < 1. 94 improved by using an elongated cell geometry. Whether or not a quenching gas can be incorporated into the cryogenic environment remains an open question.

4.4 Stopped light

In Chapter 4.3, a delay of light pulse was described. However, with the static EIT setup, it is not possible to use a very weak control beam with constant power to have the group velocity reduce to zero. On the other hand, a light pulse can completely stop in the EIT medium by coherently ramping the power of the control beam. The process of the pulse propagation in the EIT media can be explained as a dark state polariton [178], which is a superposition state of the photonic and spin states. Review articles like References [72, 73, 193] cover theoretical and experimental details of slow and stopped light. The review letter written by Walsworth et al. [193] gives an elaborate description of the atom-ﬁeld interaction during the pulse propagation, such as the spatial compression of a light pulse, the energy dissipation, and the collective property.

Classical light storage and retrieval has been experimentally realized in ultracold atomic sodium [194]. It also has been observed in a warm rubidium vapor cell [85] and in a solid [188]. It is amazing to see that the dark state polariton based on EIT can be performed under various conditions. So far, different systems hold different prospective benefits. Their comparison and performance will be addressed in Chap- ter 4.4.4. Here, we present a cryogenic pure nuclear spin system, and demonstrate it to be a suitable system for storing quantum states with remarkable storage lifetimes.

To our knowledge, it is the ﬁrst development to perform stopped light involving pure nuclear spin of an atomic ensemble. 95 4.4.1 Methods

The experimental setup of stopped light is shown in Figure 4.7, and the method to stop light is similar to Reference [85]. A half-wave plate before the EOM aligns the polarization of the incoming beam with the axis of the EOM, so that the EOM has the optimal modulation of light polarization. The EOM rotates the polarization of the laser beam to generate a weak Gaussian pulse in the signal polarization. Because the power ratio of the control beam intensity to the peak of the Gaussian pulse is at least ten, the control beam power does not change much when the EOM generates the signal pulse. More polarization optics are used here to correct a birefringence problem in the optics and to have well-deﬁned polarized light. More details on polarization optics and the birefringence problems will be discussed in Chapter 4.4.2. Instead of using a polarization beam splitter in the polarization detection system, a Wollaston polarizer is used for a better extinction ratio7. The spatial mode of the laser beam is made to be a Gaussian beam proﬁle by passing it through a 50 µm pinhole.

RF V e Pump L3 AOM Iris L3 S W2 EOM W2 Symbol Device beam

EOM Electro-optic modulator L5

L5 10 cm lens P W0 W4 E Cell E W4 W4 P 50 µm pin hole L L, E Lens combination PD1 To DAQ W0 Wollaston polarizer PD2

Figure 4.7: Optics and experimental setup for the stopped light experiment.

7Usually, a Wollaston polarizer has an extinction ratio better than 10−5 while a polarizer beam splitter is only better than 10−2. 96

To perform stopped light, the intensities of the control and probe beams are ramped down by the AOM while the signal pulse passes through the atomic cloud.

The AOM driver is controlled by a voltage of an error-function form. In our case, the time to ramp down or turn on the laser beam by the AOM is usually 1 µs. The

information (pulse shape, polarization, quantum states, etc) of the light pulse can

be smoothly transferred into the spin states if the adiabatic following conditions are

satisﬁed [178]. The adiabatic following conditions require that the atomic ensemble

contains the whole pulse (DBW 1) before the control beam is turned oﬀ, and the

1 pulse width ﬁts inside the transparency window in the frequency domain (ΓEIT ) τG

[193]. The rate to ramp the control can be fast if vgroup c [73]. A ramping time a

1 is preferred when γ a τG [194], where γ is the spontaneous decay rate.

0.5

0.4

0.3

0.2 Intensity [normalized] 0.1

0.0 -100 0 100 9900 10000 10100 Time [µs]

Figure 4.8: Storage of a classical light pulse for 10 ms. The density of the helium 17 −3 buﬀer gas is 1.6 × 10 cm ; OD0 = 1.5.

Figure 4.8 shows the signal of the light storage and retrieval. Time zero is set 97 to the peak of a reference pulse which enters the cell without the presence of atoms.

Data is normalized by the height of the reference pulse. The control beam is turned off 20 µs after the peak of a pulse with 60 µs width enters Yb EIT media. This is a similar condition as the slow light experiment in Figure 4.4. Since the DBW is smaller than one, a part of pulse escapes from the atomic cloud before the beam is turned off. However, a portion of the pulse is stored in the atomic ensemble in the form of dark state polariton [178]. After 10 ms, the control beam is turned back on and a retrieved light pulse with an efficiency of 3.6% appears. The efficiency is defined as the energy of the retrieved pulse divided by the energy of the total input pulse, not including optics loss.

Due to imperfections in our polarization optics, a small fraction of the strong beam leaks into the weak beam detector as a background. This background is subtracted in Figure 4.8. Note that the background level is around 5% of the peak height of the Gaussian reference pulse. The leaked control beam may interfere with the probe beam and change the amplitude of the retrieved pulse [195]. By using different input pulse power, the efficiency of the retrieved pulse stays constant. It indicates that interference effects are negligible for the data shown here.

4.4.2 Light-storage lifetimes

Data of central interest in this work is the lifetime of the stored light pulse. The longer information is stored, the greater the chance that it is lost because of various decoherence mechanisms such as inelastic collisions, atomic motion (thermal diffusion) and inhomogeneous magnetic field. Figure 4.9 plots the efficiency of pulse storage and retrieval as a function of storage time. The efficiency of the retrieved pulse decays exponentially in time, as expected from the diffusion of the atomic ytterbium through the helium buffer gas [196]. The data is fit to an exponential function which gives 98

a light-storage lifetime8 of 0.11 s. From the discussion for J = 0 atoms, inelastic

collisions between atoms are not a problem in our system. In addition, there is no

evidence of decoherence due to collisional processes; from the light storage lifetime an

upper limit on the 173Yb-He collisional spin decoherence rate coeﬃcient is determined as 5 × 10−18 cm3 s−1 at 4 K.

0.04

Data Fit 0.03

0.02 Efficiency

0.01

0.00 0.00 0.05 0.10 0.15 0.20 0.25 Storage time [sec]

Figure 4.9: Eﬃciency of pulse storage and retrieval vs. storage time. Data are taken with a helium density of 1.6 × 1017 cm−3 and a near-zero magnetic ﬁeld.

Various systems have been developed to reduce the atomic motion, such as using

Bose-Einstein condensation [197], an optical lattice [87, 198], a Mott insulator [199], and a solid [200]. In the case of buﬀer-gas cooled ytterbium atoms, the atomic motion can be reduced by lowering the cell temperature or increasing the helium density. In Figure 4.10, the storage lifetime is examined as a function helium density.

Triangles are the diﬀusion lifetime of Yb atoms as a function of helium density. The

8The storage time is how long a pulse has been stored; the storage lifetime is a constant to describe how the stored eﬃciency decays with the storage time. 99

0.1 Lifetime Lifetime [s] 0.01

Diffusion Big beam & bias field Big beam & zero field Small beam 0.001

16 17 10 -3 10 Helium density [cm ]

Figure 4.10: Triangles show the diffusion lifetime of Yb atoms in the cryogenic cell. The other symbols show the 1/e light storage lifetime. Lines are linear fits to the data at helium densities < 1017 cm−3. The data points are reproducible to within ±10%. other symbols are the 1/e light storage lifetime. The dependence of the diffusion lifetime on the helium density is similar as what has been discussed in Chapter 3.4.

Unfortunately, the study of the storage lifetimes is complicated by a few technical problems as discussed below.

Birefringence

Birefringence is observed in the cryostat optics, which is attributed to stress-induced birefringence in the cryogenic BK7 windows. The extra retardance in optics rotates the polarization of the incoming control and probe beams, and also causes defective polarization detection. The signal detector may intercept a part of the control beam resulting in an error. A combination of wave plates are used in Figure 4.7 in order to compensate for the retardance. To ensure that the polarization detection system plus the retardance of the window can precisely detect the σ±-polarized light interacting 100 with atoms, a polarization test is performed by applying a strong magnetic ﬁeld

(∼100 G). In the 168Yb transition, the σ+ (σ−) light only excites atoms through

0 mF = 0 → mF = +1 (−1), as shown in Figure 4.11. If the polarization detection does not work correctly, the detector will see a mixed signal of σ+ and σ− light. In this way, the angle between light polarization and wave plates can be precisely adjusted to decompose σ±-polarized light seen by atoms. Consequently, a blue-shifted (red- shifted) spectrum is generated when probing atoms by scanning the laser frequency of the σ+ (σ−) light. Figure 4.11 is an example of a proper σ±-polarized light detection when applying high ﬁeld. Note the strong magnetic ﬁeld is only applied when doing optics alignments.

Since the birefringence of windows is not uniform, a small-diameter beam (a Gaus- sian beam with a 16 mm waist9, apertured to a 15 mm diameter) is used to reduce the beam area on windows. However, this reduced beam diameter results in signiﬁcantly shorter storage lifetimes because atoms diﬀuse in and out of the beam area quickly.

Data taken in this way are shown as the square points in Figure 4.10.

Data is also taken with a large-diameter beam (16 mm waist). With the large beam, we aperture the light with an iris after it passes through the ytterbium cell.

This reduces the problems from non-uniform birefringence while allowing the longer lifetimes associated with a larger beam diameter. This set of data is shown as solid circular points in Figure 4.10. The light storage lifetime at near-zero magnetic ﬁeld increases linearly in the helium density at low densities, as expected from diﬀusion

[196], but deviates at higher densities. We attribute this behavior to an inhomogeneous magnetic ﬁeld because better lifetimes are observed with the improvements discussed below.

2 9 I(r) −2r The waist w of a Gaussian beam is deﬁned as = e w2 form the intensity I0 proﬁle, where r is the distance from the center of the beam (r2 = x2 + y2). 101

2.5 170 σ+ −1 0 +1 Yb σ 171 - Yb (F'=1/2) 2.0

− + 1.5

σ σ OD 168 1.0 Yb

0.5 0 0.0 0.204 0.206 0.208 0.210 0.212 Time [s]

Figure 4.11: A test of the σ±-polarization detection. On the left is a simpliﬁed 168Yb 0 level structure. The sublevels mF = ±1 have been shifted up and down, respectively, due to the applied magnetic ﬁeld. When Yb atoms are probed by the σ±-polarized light, the corresponding signal has its blue (red) frequency shift. The bottom axis is the time after laser ablation; the laser scanning frequency is linear in time.

Magnetic ﬁeld

The atom-light coherence in our system is built on nuclear spin, whose magnetic

momentum is about three orders smaller than that of the electronic spin. For the

same volume of atomic cloud, the nuclear spin system is aﬀected less by the stray

magnetic ﬁeld compared to the electronic spin. Because our current cell size (∼1000

cm3) is larger than other alkali system in general, it is not surprising that light-storage

lifetime may be limited by the inhomogeneous magnetic ﬁeld at high helium density.

There are only three pairs of Helmholtz coils used to zero out the magnetic ﬁeld at

the ﬁrst step of the stopped light experiments.

To obtain a longer storage lifetime, the magnetic ﬁeld gradient is improved in a

couple of ways. First, two layers of magnetic shielding foils [201] are used to cover

the side and the bottom of the cryostat chamber. (There is no room for the shielding

on the top of the chamber.) With the shielding only, the earth’s field is reduced by 102 a factor of 3. The other way is to apply a bias field (∼0.3 G) attempting to make the field more uniform. If the bias field is applied transversely to the field gradient, the ratio of the field gradient to the field magnitude decreases. However, the time evolution of the dark state becomes complicated with the presence of the bias field.

Because the dark-state polariton will precess around the magnetic field, the efficiency of the retrieved pulse will be affected if the precessing polariton does not align with the control beam during the revival. This kind of dark-state polariton collapses and revivals have been previously predicted and observed [202, 203].

0.07

0.06

0.05

0.04

0.03 Efficiency 0.02

0.01

0.00 0.0 0.1 0.2 0.3 0.4 0.5 Storage time [s]

Figure 4.12: A plot of retrieved efficiency versus storage time with a bias field (∼0.3 G) at a helium density of 2.4 × 1017 cm−3. The oscillating efficiency is due to the recurrence of the dark-state polariton.

Figure 4.12 shows the retrieved signal at different storage time when the bias field is perpendicular to the laser beams. The recurrence of the dark-state polariton gives an oscillating efficiency of the retrieved pulse at subsequent revival time. With these approaches, storage lifetimes longer than hundreds of milliseconds are achieved by applying a bias field. In Figure 4.10, data with a large-diameter beam and bias field 103 has significantly longer lifetimes than data with a large-diameter beam and zero field at high helium densities. We expect this same performance could be accomplished in future experiments at near-zero field by incorporating magnetic shielding into the apparatus. Finally, diffusion should become the ultimate limitation of the storage lifetime in our system.

Diﬀusion

Although the storage lifetime is ultimately limited by atomic diﬀusion, a lower cell temperature and a high helium density help to slow down the atom motion and increase the lifetimes. With an improvement of the blackbody shielding and heat exchangers in the current apparatus, a cell temperature of 4 K is achievable. (Note that the cell temperature is 6 K in experiments reported in Chapter 3.3.) A Voigt multi-peak ﬁt to the absorption spectrum is measured to have a Doppler-broadening

FWHM of 8 × 107 Hz which is consistent with the 4 K cell temperature measured by the temperature diode.

As seen in Figure 4.10, at low helium densities both the atom lifetime and light storage lifetime scale linearly with the helium density. Unfortunately, at higher he-

17 −3 lium densities (& 10 cm ), the atom lifetime in the cell is shorter than would be expected from diﬀusion alone and depends on the ablation power used as discussed in Chapter 3.4. We restrict our observations of the light-storage lifetime to helium densities < 3 × 1017 cm−3, as shown in Figure 4.10.

Using large-diameter control and probe beams (irised after the cell) and an applied bias field, an efficiency of 1% at a storage time of 0.4 s is obtained, not including optics and iris losses. While the iris reduces the actual efficiency significantly from this level, and the magnetic field restricts the times at which accurate pulse retrieval can be obtained, we believe these numbers are representative of what could be achieved once 104

the technical problems of nonuniform window birefringence and insuﬃcient magnetic

shielding are remedied.

4.4.3 Storage eﬃciency

From the delay-bandwidth-transmission product measurements in Chapter 4.3.1, the

eﬃciency of storing and retrieving a light pulse can be approximately estimated by √ DBWT/ π if τdelay is small comparing to τG. (It is derived in Appendix G.3.) In our current apparatus we are able to achieve large optical densities OD0 > 50, but the DBWT does not actually increase with OD0. As shown in Figure 4.5(d), the

DBWT increases linearly with OD0 at low OD0, but decreases at high OD0. From √ this analysis, we would expect to achieve an eﬃciency up to 9% (0.16/ π). However,

the measured peak light storage eﬃciency is only 6%. The current limitations of

the storage eﬃciency are believed to be due to radiation trapping and oﬀ-resonance

absorption from other isotopes. The possible improvements and solutions are the

same as for the high OD0 discussion in Chapter 4.3.2. We note that as the helium buﬀer-gas density is increased, the maximum achiev-

able eﬃciency decreases. At a helium density of 3 × 1017 cm−3, the eﬃciency has

decreased to 3%. We attribute this decrease to oﬀ-resonance absorption by the other

Yb isotopes, which increases at high helium density due to pressure broadening [75].

(See Figure 3.2 and Figure 3.16.) At a 3 × 1017 cm−3 helium density, the imperfect

173 transparency has the OD = 0.3 of the OD0 of Yb. (See Figure 3.16.) The stored eﬃciency is reduced in two ways. First, the oﬀ-resonance absorption decreases the

transmittance of the light pulse. Second, the scattered light causes decoherence and

pumps 173Yb atoms out of the dark state. The second eﬀect is exacerbated by ra-

diation trapping. These deleterious eﬀects could be improved dramatically in future

work by employing an isotopically-enriched ytterbium sample. 105

There are already some techniques developed to improve the storage eﬃciency in

the warm alkali vapor cell [190]. For example, the shape of the input signal can be

optimized to a particular form to obtain a maximum eﬃciency [137, 138, 192]. We

expect the same approaches can be applied to the cryogenic ytterbium atomic vapor.

4.4.4 Towards quantum memory

A graph of our data and selected state-of-the-art experimental results are presented

in Figure 4.13, showing the eﬃciency and the storage lifetimes. Although only the

system C [87] represents the quantum light storage (an example of quantum memory)

and the others are classical light storage and retrieval, we treat them as the best

achievements on these diﬀerent approaches. The point B [192] provides a procedure to

improve the storage eﬃciency by optimizing the pulse shape. It achieves an eﬃciency

up to 50% by using a warm rubidium vapor cell, but its storage time is not practical.

(How long a storage time is suﬃcient depends on the application. Take the quantum

communication as an example. Since it takes about 20 ms for a photon traveling

across the North America at the speed of light, the storage time needs to be at

least longer than 100 ms to be useful.) The system A [197] realizes the light storage

in a sodium Bose-Einstein condensate, and shows a signiﬁcant long storage lifetime

up to 1.5 s. However, its eﬃciency (0.5%) is too low to be practical. The method

to obtain long lifetimes is complicated, and achieving a higher eﬃciency would be

diﬃcult. The system E [199] uses a Mott insulator to hold ultracold 87Rb atoms in

a deep three-dimensional optical lattice. In this way, it has a storage time longer

than hundreds of milliseconds, but limited by heating from the laser ﬁelds and the

tunneling between each site. More eﬀorts are needed to improve its eﬃciency. The

system D [198] traps 87Rb atoms in a “magic” optical lattice, using a clock transition which is insensitive to the magnetic field. Compared to the other experiments, the 106 system D shows a relative good performance. There are also other approaches in solids studying light storage and retrieval, such as using atomic frequency comb (AFC) [204] and controlled reversible inhomogeneous broadening (CRIB) [205]. Although it seems that high efficiencies [206, 207] or long storage times [200] can be achieved in solids, it is still difficult for solid systems to have a high efficiency while still maintaining a reasonable storage time. (For example, Reference [206] has an efficiency of 25% with a 800 ns storage time by the atomic frequency comb protocol.)

Our Yb data [140] outperforms the system E and is competitive with the system D.

With a relative simpler scheme than other experiments, a cryogenically-cooled Yb vapor cell is expected to be a good technique for quantum memory.

1 B 17 -3 UNR; nHe ~ 1.5X10 cm bias field

0.1 C

0.01 D A

Efficiency E

E 0.001

0.0001 0.001 0.01 0.1 1 Storage time [s]

Figure 4.13: Points from A to E are the state-of-the-art experiments. References are corresponding to A [197], B [192], C [87], D [198], and E [199]. Colored points are data from this thesis. Line is the ﬁt to to an exponential function. 107

Chapter 5

Atom-He collisions

Another major focus in this thesis is on the measurements of the inelastic collisions between atoms. In Chapter3, we discussed that ytterbium atoms could hold the coherent superposition states for a long time because they have excellent collisional properties; in Chapter4, we have also observed long storage lifetimes in nuclear spin states. The basic concepts of collisions and the common collisional mechanisms have been discussed in Chapter1. In addition to the importance of atom-atom or atom-ﬁeld coherences, understanding the mechanisms of inelastic collisions is not only interesting to fundamental studies of collisional properties, but also helps the development of generating ultracold atoms and molecules.

In Chapter1, we introduce two categories of atoms which have dramatically sup-

2 o pressed inelastic collision rates: the P1/2 atoms and the submerged-shell atoms. The

2 o ﬁrst part of this chapter will examine the observation of inelastic collisions in the P1/2

2 o atoms, gallium and indium. A comparison of P1/2 atoms and the submerged-shell atoms will be addressed. The last part of this chapter reports studies of ﬁne-structure changing collisions which have been predicted as a cooling mechanism in the universe.

Other collisional properties of the submerged-shell atomic titanium are also measured. 108 5.1 Ga-He and In-He

Similar to the Yb experiments, gallium (indium) atoms are produced by laser ablation of Ga-Cu alloy (indium metal) targets, and subsequently cooled to a translational temperature of 5 K with a cryogenic 4He gas. When studying Ga-He collisions, the ablation pulse is directed onto the Ga-Cu target. To measure indium-helium collisions, we steer the ablation laser pulse onto the indium metal with a mirror.

5.1.1 Spectroscopic structures

2 2 o The ground-state electronic conﬁgurations of atomic Ga and In are 4s 4p P1/2 and

2 2 o 69 71 5s 5p P1/2, respectively. Gallium has two stable isotopes: Ga and Ga; indium also has two stable isotopes: 113In and 115In. Their natural abundances are listed in

Table 5.1. We concern ourselves primarily with 69Ga and 115In for they are the most

2 2 o naturally-abundant isotopes. Laser absorption spectroscopy on the ns np P1/2 →

2 2 ns (n+1)s S1/2 transitions at 403 and 410 nm is used to state-selectively monitor the ground ﬁne-structure-state populations of Ga and In, respectively [208, 209]. The low translational temperature allows us to spectrally resolve the isotopes and hyperﬁne states.

Table 5.1: Ga and In atomic data [153]. I is the nuclear spin; µ is the magnetic moment, expressed in units of the nuclear magneton µN (762 Hz/Gauss) [117].

Isotopes Abundance [%] I µ [µN ] 69Ga 60.11 3/2 2.01659 71Ga 39.89 3/2 2.56227 113In 4.3 9/2 5.5229 115In 95.7 9/2 5.534 109

Ga spectrum

A typical spectrum of atomic gallium is shown in Figure 5.1 alongside a simplified level structure. The detailed transition hyperfine structure of gallium atoms can be found in Figure F.8. The ground 4p state of gallium splits into two fine structure levels,

2 o 2 o −1 P1/2 and P3/2, with a separation of 826 cm due to the spin-orbital interaction.

2 o Since both isotopes of gallium have a nuclear spin 3/2, the P1/2 state has hyperﬁne

2 o levels F = 1 and F = 2 while the P3/2 state has four levels from the hyperﬁne interaction [4]. The 4s25s excited state also has two hyperﬁne levels F 0 = 1 and

F 0 = 2. Four optically-allowed transitions are drawn in Figure 5.1(a) obeying the electric dipole transition selection rules. The corresponding transition peaks for each isotope can be seen in Figure 5.1(b).

In spectrum

Atomic indium has a similar electronic shell structure to gallium, so that their atomic properties are alike. Because indium is heavier than gallium, the ground 5p state of indium has a large ﬁne-structure splitting of 2213 cm−1 due to the strong spin-orbital interaction. Two indium isotopes have a large nuclear spin of 9/2. A simpliﬁed level structure of the isotope 115In labeled with optical transitions is displayed in Figure

5.2(a). The detailed transition hyperﬁne structure of indium atoms is illustrated in

Figure F.9.

Unfortunately, the grating-feedback external cavity diode laser used for measurements with indium cannot be hop-free scanned widely enough to cover all indium hyperﬁne peaks, as with the gallium spectrum. Therefore, only part of indium spectrum is shown in Figure 5.2. Being on resonance for the desired optical transitions is not a problem. On the side of the 115In transition peak, a small contribution from the of 113In isotope is observed. The relative height between these two isotopes is 110

0.8 2 -1 F '= 2 S 24789 cm 69Ga 5s 1/ 2 71Ga 2-2' F '= 1 2-2'

0.6

2-1'

1-2'

0.4 OD

F "= 3 2-1' 1-2'

2 o 826 cm -1 F "= 2 P 3/ 2 0.2 1-1' F "= 1 1-1' F "= 0 4 p

0.0 2 o 0 cm -1 F = 2 P 1/ 2 -3 -2 -1 0 1 2 3 F =1 Frequency [GHz]

(a) Ga level structure (b) Ga spectrum

Figure 5.1: Absorption spectrum of Ga alongside a 69Ga level structure. The peaks are labeled according to their isotopes and hyperﬁne transitions. OD is the optical density. The frequency oﬀset is 24789 cm−1 or 403 nm. consistent with the natural abundances of indium.

Diﬀusion of Ga and In atoms

Similar to the description in Chapter 3.4, we measure the diffusion lifetimes of gallium and indium atoms by fitting their diffusion curves to an exponential function in Equation (3.1) when the lowest diffusion mode is dominant. The measured diffusion lifetimes are plotted versus the corresponding helium densities in Figure 5.3.

By measuring the diffusion lifetime as a function of helium density, we determine the thermally averaged diffusion cross-section of gallium to be (7.5 ± 2.0) × 10−15 cm2 and of indium (9.1 ± 2.5) × 10−15 cm2. Note that these diffusion cross-sections are 111

4 2 24373 cm -1 F '= 5 6 S 113In s 1/ 2 ' 4 5-5' F = 115In

4-4' 2 OD

F "= 6

-1 " 5 2 o 2213 cm F = P 3/ 2 F "= 4 1 F "= 3 5 p 5-5' 4-4' 0 0 cm -1 F = 5 2Po -3 -2 -1 0 1 2 3 1/ 2 F = 4 Frequency [GHz]

(a) In level structure (b) In spectrum

Figure 5.2: Absorption spectrum of indium alongside a 115In level structure. The frequency offset is 24373 cm−1 or 410 nm. measured at low helium densities (< 1017 cm−3), where the dominant mechanism of the atom loss is from diffusion. The mechanisms affecting the atomic lifetimes at high helium densities (> 1017 cm−3) were mentioned in Chapter 3.4.

5.1.2 Inelastic collisions

To measure inelastic collisions, the internal-state distribution of the atoms is perturbed by optical pumping as described in Chapter 3.5.1. A strong pump beam with few mW is used to perturb the population of atoms, and a weak probe with few µW is used to probe their population evolutions. Both pump and probe beams are tuned to the same transitions as used for the absorption spectroscopy. 112

2.0

1.5 Gallium data Indium data

1.0

0.5 Diffusion lifetime [s] lifetimeDiffusion

0.0 17 0 1 2 3 4 5x10 -3 Helium density [cm ]

Figure 5.3: The diﬀusion lifetime of gallium and indium atoms are plotted as a function of helium density at 5 K. The ablation energy is less than 10 mJ.

Unlike the ytterbium atom, which is a two-level-like atom, gallium and indium atoms have slightly more complicated level structures. Because gallium and indium atoms have similar level structures, we take gallium atoms as an example of the measurements of inelastic collisions.

Both of the pump and probe beams are on resonance with the optical transition

2 2 o 2 2 0 |4s 4p P1/2,F = 2i → |4s 5s S1/2,F = 1i of gallium. According to the branching ratios summarized in Appendix F.2, the excited F 0 = 1 state has a 67% population

2 o decaying to the ground P3/2 state, a 6% excited state population to the ground

2 o 2 o P1/2 F = 1 state, and a 28% back to the original P1/2 F = 2 state. Although most populations decay to the other J and F states, those atoms eventually return back to thermal equilibrium. Note that at thermal equilibrium, the population of atoms in each energy state is determined by the Boltzmann factor [4],

N2 g2 − ∆E = · e kB T (5.1) N1 g1 113

where g1 and g2 are the degeneracies (the statistical weights of the energy levels),

and ∆E = ~ω is the energy splitting. So that, the population ratios of the gallium hyperﬁne-structure levels and ﬁne-structure levels at 5 K and in thermal equilibrium

can be written as N N F=2 = 1.6 and J=3/2 ≈ 0 (5.2) NF=1 NJ=1/2 By using the optical pumping technique, we perturb the thermal population of gallium

atoms at diﬀerent helium densities and observe the population return to thermal

equilibrium due to the inelastic collisions with helium gas.

The possible relaxation mechanisms due to the inelastic collisions can be through

2 o o ﬁne-structure (or J-changing) collisions from the P3/2 state to the 2P1/2 state, or through hyperﬁne (or F -changing) collisions from the F = 1 state back to the F = 2

state. The ﬁrst task is to distinguish these two collisional mechanisms.

F - and J-changing collisions

2 2 o 2 2 0 In the case of optical pumping from |4s 4p P1/2,F = 2i to |4s 5s S1/2,F = 1i, both populations of the F = 2 and F = 1 states should increase after the optical

pumping if the J-changing relaxation rate is faster than the F -changing rate. On the

other hand, if the J-changing rate is slower than the F -changing rate, we expect that

the population of the F = 2 state would increase while that of the F = 1 state would

decrease.

A typical optical pumping data is shown in Figure 5.4(a) where the population

of the F = 2 state is plotted in optical density versus time. A 2 ms pump pulse is

sent at 1.2 s to pump atoms out of the F = 2 state into the other F or J states

based on the selection rules. After the pump pulse ﬁnishes, we see a return of the

atoms due to collisions. The overall atom loss is due to the fact that atoms diﬀuse

away. At the same time, we also monitor the population of the F = 1 state. On 114

1.0

0.9

0.1 0.8 0.7 Ratio OD 0.6 population of |F=2> 0.5 |F=1> data |F=2> data 0.4

1.20 1.30 1.40 1.50 1.20 1.22 1.24 1.26 1.28 1.30 Time after ablation [s] Time after ablation [s]

(a) (b)

Figure 5.4: Monitoring of the F = 2 and F = 1 populations of 69Ga atoms. (a) The population of the F = 2 state as a function of time. (b) The population ratio as a function of time after optical pumping. The details are discussed in the text. the left axis of Figure 5.4(b), we normalize the population of both the F = 1 and

F = 2 states by their populations at thermal equilibrium. At time after 1.2 s in

Figure 5.4(b), the ratio of the F = 2 state is very small because the population has

been depleted by the optical pumping pulse. As for the ratio of the F = 1 state,

it is greater than 1 as a result of optical pumping. Then, a decrease of the F = 1 data and an increase of the F = 2 data conﬁrm that the return to equilibrium is dominated by the F -changing collisions, rather than the J-changing collisions . In

this way, we distinguish these two processes by monitoring the population of both

hyperﬁne states in the J = 1/2 manifold. By measuring the return of the atomic

population to equilibrium and ﬁtting it to an exponential function, we obtain the

rates for F -changing and J-changing transitions in 69Ga-4He collisions.

To determine the J-changing and F -changing rate coeﬃcients, kJ and kF , we

measure these rates 1/τ over a range of helium densities nHe, as shown in Figure 5.5. With a helium density greater than 1017 cm−3, the relaxation rates are linearly in-

creasing with helium density which indicates that the inelastic collisions are with 115

100

80 ]

-1 60 F-changing rate

[s J-changing rate τ Fit 1/ 40

0 18 0.0 0.2 0.4 0.6 0.8 1.0x10 -3 Helium density [cm ]

Figure 5.5: Measured 69Ga F - and J-relaxation rates 1/τ for diﬀerent 4He densities. Fits are based on Equation (5.3).

helium atoms. At low helium density, the increasing rate with decreasing helium

density occurs because the atoms move in and out of the pump beam volume, as

discussed in Chapter 3.5.1. We model both of these eﬀects by

1 C = k · nHe + (5.3) τ nHe

where the term C/nHe accounts for the diﬀusion of atoms in and out of the pump beam and C is a constant. From the ﬁtting results in Figure 5.5, we measure the

−17 3 −1 J-changing collision rate coeﬃcient to be kJ = (1.0 ± 0.3) × 10 cm s , and the

−17 3 −1 69 F -changing collision rate coeﬃcient to be kF = (5.3 ± 1.3) × 10 cm s for Ga atoms.

For the measurements with 115In atoms, a similar procedure was applied. In

an attempt to improve the performance of the indium laser diode, we removed a

piece of glass in front of the diode in order to avoid feedback of light from the glass.

Unfortunately, the output power of the indium ECDL decreases from 13 mW to a few 116 mW after a few days of operation. The removal of the glass may let the blue laser diode react with air, and it aﬀects the functioning of the diode. Due to the failure of the laser diode, we do not have suﬃcient measurements to determine the values for the In-He collisions. However, we can still set the upper limits of In-He collisions to

−17 3 −1 −17 3 −1 be that kJ < 8 × 10 cm s and kF < (2.3 ± 1.4) × 10 cm s , based on our observations.

Zeeman relaxation collisions

To measure the Zeeman relaxation collisions, we use polarization spectroscopy with optical pumping as discussed in Chapter 3.5.2. We use the same experimental setup as shown in Figure 3.5.

Figure 5.6 shows the time dependence of the optical density for 69Ga in the

2 o P1/2 F = 2 hyperﬁne state. Prior to the pump pulse, there is no measurable polarization of the atoms. After pumping a large polarization is induced, as indicated by the relative absorption of the σ+ and σ− probe beams. This polarization decays over time. During the time 0.6−0.7 s after ablation, the diﬀerence in populations between the data with and without optical pumping is because of the slow J- and F -changing relaxation.

By fitting the difference in the optical densities to the functional form e−t/τ and repeating this measurement over a range of helium densities nHe, we plot the m- changing rates for both 69Ga and 115In atoms in Figure 5.7. At low helium density, the relaxation rates are consistent with the model in Equation (5.3). However, the rates tend to “flatten” at high helium densities, with no linear dependence on helium densities. Unfortunately, it is not clear why the m-changing rate behaves nonlinearly with increasing helium density. Although the non-linear result indicates the presence of additional mechanisms, upper limits on the Zeeman relaxation can be set to 117 OD

σ+ 0.1 σ- σ+ No pump σ- No pump

0.50 0.55 0.60 0.65 0.70 Time after ablation [s]

Figure 5.6: Optical density of 69Ga as observed by the σ± probe beams at a helium density of 6.4 × 1016 cm−3 and a bias ﬁeld of 3 G. A strong σ+ pump beam is turned on for 1 ms at 0.55 s. Also shown is the optical density in the absence of optical pumping; its level is normalized to compensate for shot-to-shot inconsistencies in ablation yield.

600 80 500

400 60 ] ] -1 -1 [s 300 [s τ τ 40 1/ 1/ 200 20 2 4 6 8 10 100 pump duration [ms]

0 0 17 18 0 1 2 3 4 5x10 0.0 0.2 0.4 0.6 0.8 1.0 1.21.4x10 -3 -3 Helium density [cm ] Helium density [cm ]

(a) Ga m-changing rates (b) In m-changing rates

Figure 5.7: Zeeman relaxation of the 69Ga-He and 115In-He 118

−15 3 −1 69 −16 3 −1 115 km < 3 × 10 cm s for Ga and km < 5 × 10 cm s for In, based on the measurements.

Experimental and theoretical results

To interpret the experimental observations, a theoretical calculation was performed by T. V. Tscherbul, A. A. Buchachenko, and A. Dalgarno [41]. They identify three main contributions to the total Zeeman relaxation rate arising from the m-, F - and

J-changing transitions. The calculation indicates that the m-changing rate of atoms

2 o in P3/2 states occurs at a large rate: the rate is comparable to that for elastic

2 o collisions [39, 40], and the ﬁne-structure changing rate of the P3/2 state is slow. In

2 o contrast, Zeeman transitions in collisions of P1/2-state Ga and In atoms are strongly suppressed over a wide range of collisions energies.

The theoretical analysis also shows that Zeeman transitions in collisions of 2P atoms occur due to couplings between diﬀerent Zeeman sublevels induced by the anisotropic part of the interaction potential [41, 210]. The inelastic collisions due to the anisotropic interaction have been discussed in Chapter 1.3. Simply speaking, the spin states can be aﬀected through the coupling to the orbital angular momentum, and the interaction is usually large for atoms with aspherical electric shell structures.

2 o Since the electron-density distribution of atoms in P1/2 electronic states is spherically

2 o symmetric and that of P3/2 atoms is not, the m-changing rate is much faster for

2 o 2 o P3/2-state atoms than P1/2-state atoms.

2 o Zeeman relaxation in collisions of P1/2 atoms can occur indirectly via couplings to

2 o the P3/2-excited state because the interaction with the helium atom couples the two

2 o ﬁne-structure states and distorts the spherical symmetry of the P1/2 state, leading to Zeeman relaxation. This process slows down dramatically with an increase in the

2 o 2 o energy separation (the fine-structure splitting) between the P1/2 and P3/2 terms. For 119 example, indium atoms (with a fine-structure splitting of 2213 cm−1) are predicted to have a slower m-changing rate than gallium atoms (whose fine-structure splitting is 826 cm−1). On the other hand, inelastic collisions also depend on the molecular interaction potential of two colliding parters. For example, the molecular potential of

Ga-Ga collisions may be deeper than Ga-He collisions, expectedly leading to a faster inelastic collision rate in Ga-Ga than Ga-He collisions.

Table 5.2: Experimental and theoretical rate coefficients of 69Ga-4He and 115In-4He at 5 K and 3 Gauss (in units of 10−17 cm3 s−1). γ is the ratio of the cross-sections for diffusion and inelastic relaxation. The values in parentheses are calculated with the interaction anisotropy multiplied by 1.2. Atom 69Ga 115In Rate coefficient Experiment Theory Experiment Theory

km < 300 0.8 (2.3) < 50 3.8

kF 5.3 ± 1.3 2.3 (6.6) < 2.3 ± 1.4 0.1

kJ 1.0 ± 0.3 0.03 (1.3) < 8 0.0004 γ > 4.2 × 104 (2.6 × 106) > 3.0 × 105 6.7 × 106

Table 5.2 summarizes the calculated and measured rate coeﬃcients for Zeeman relaxation of Ga and In atoms in a buﬀer gas of 4He at 5 K temperature and a

3 Gauss ﬁeld. Although the calculated m-changing rates are consistent with the measured upper bounds for both 69Ga-4He and 115In-4He, the theoretical rates for

F -changing and J-changing transitions for 69Ga-4He are too small, which indicates that the ab initio calculations may underestimate the Ga-He interaction anisotropy.

Table 5.2 shows that the increase in the interaction anisotropy leads to quantitative agreement of both F -changing and J-changing rates with experimental measurements.

The calculated elastic-to-inelastic ratios γ are large and consistent with the measured lower bounds. Thus, both experiment and theory suggest that cryogenic cooling and magnetic trapping of Ga and In atoms at low helium buﬀer gas densities would be eﬃcient. 120

Discussions and the Ga-Ga collisions

2 o Among the non-S-state atoms, the P1/2-state atoms show a large suppression of inelastic collisions, comparable to the submerged-shell atoms. Although previous

experiments and theoretical calculation reveal the eﬃcient suppression of inelastic

collisions between the rare-earth atoms (RE) and helium [12, 42], RE-RE collisions

have large inelastic rates. Large spin relaxation rates have been measured in Er-

Er and Tm-Tm to be 3.0 × 10−10 cm3 s−1 and 1.1 × 10−10 cm3 s−1, respectively

12 2 3 [31]. (The ground-state electronic conﬁgurations for Er and Tm are [Xe]4f 6s H6

13 2 2 and [Xe]4f 6s F7/2, respectively.) Evaporative cooling of submerged atoms in a magnetic trap will be ineﬃcient due to inelastic collisions.

160 140 120 100 ] -1

[s 80 τ

1/ 60 40 20 1 2 3 4 5 6 OD of J=1/2 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Estimated OD of J=3/2

Figure 5.8: A preliminary study of the Ga-Ga m-changing collisions.

2 o To investigate whether the P1/2-state atoms still have good collisional properties when colliding with other 2P -state atoms, we perform a preliminary measurement on

Ga-Ga collisions. Similar to the Ga-He experiments, large numbers of Ga atoms are 121 produced by using a 200 mJ ablation pulse at a low helium density (2×1016 cm−3). An

2 2 o 2 2 0 optical pump pulse drives the |4s 4p P1/2,F = 1i → |4s 5s S1/2,F = 1i transition,

2 o and the same for the probe beam. Since most of atoms are stuck in the P3/2 state after the optical pumping, we expect the main colliding particles to be the J = 3/2 and J = 1/2 atoms. Figure 5.8 shows the m-changing rates as a function of the

2 optical density of P3/2 state atoms. Because we do not have a laser diode for the

2 2 o 2 2 4s 4p P3/2 → 4s 5s S1/2 transition, the OD of the J = 3/2 atoms is estimated from the population of J = 1/2 atoms and the branching ratios. Although it is not

−10 3 −1 conclusive, a rate coeﬃcient of km ∼ 10 cm s is preliminarily determined from the slope of the inelastic collision rate to the OD of J = 3/2 atoms.

We believe that the large rate is due to ether spin-exchange collisions of the unpolarized atoms, or the anisotropic J = 3/2 atoms. Because of the ineﬃcient optical pumping of all Ga isotopes, Zeeman changing collisions can occur via spin exchange and depolarize the atomic polarization of J = 1/2 atoms. Due to a large portion of the J = 3/2 atoms after the optical pumping, the strong anisotropic

2 interaction from the P3/2 atoms also contributes significantly to inelastic collisions. For future experiments, a laser with sufficient power is needed to polarize all Ga isotopes in order to turn off the spin-exchange collisionss. In addition, the anisotropic interaction should be reduced by adding an second laser to pump atoms out of the

2 J = 3/2 state. It is also possible to investigate the collisions between P1/2-state atoms with the S-state atoms, such as lithium. Indeed, more eﬀorts are needed to determine rate coeﬃcients for inelastic collisions of the non-S-state atoms.

5.2 Ti-He collisions

In addition to an intrinsic interest in the submerged-shell structure of translation metal atoms, we study titanium atoms to measure the ﬁne-structure changing colli- 122 sions for astrophysical applications.

5.2.1 Signiﬁcance to astrophysics

The collisional excitation of atomic ﬁne structure levels plays an important role in the evolution of our universe. The interstellar medium, which contains ordinary matter, cosmic rays, and magnetic ﬁelds, is the birthplace of solar systems [47]. Only the densest and coldest molecular regions are favorable for star formation. In the process of star forming, large and heavy interstellar clouds start to contract due to the attractive gravitational force. However, many types of mechanisms, such as photoelectric heating, cosmic ray heating, and motional acceleration, heat up the clouds and the star formation terminates [47].

In order to keep contracting the clouds and obtain a sufficient density to form a star, there must exist a cooling mechanism to dissipate the energy. It is believed that the dominant cooling process is collisionally excited spectral line emission [48]. The atomic and molecular gases, which are ready to form stars, usually have a typical temperature of tens of Kelvin [47]. At this low temperature, it is possible to induced a transition via inelastic collisions between fine-structure states. (The energy separation of the fine-structure splitting is usually a few hundred of wavenumbers1.) As a comparison, the hyperfine-structure splitting (few hundreds of MHz to a few GHz, typically) is so small compared to the ∼ 3 K cosmic microwave background that there is little difference in population. (The coldest temperature in space is ∼ 3 K.) Elec- tronic transitions between levels whose energy separation is a few eV are inaccessible at such a low temperature.

The cooling process due to ﬁne-structure changing transitions is believed to be

+ mainly from C -H2 collisions [48]. Carbon ions do not have nuclear spin, and only

1 1 cm−1 = 30 GHz = 1.43 K; 1 eV = 8000 cm−1. 123

2 2 −1 have ﬁne structure due to P1/2 and P3/2 states with a 157 µm (64 cm ) energy separation. (12C has the highest natural abundance, up to 99% [153].) When carbon

2 ions collide with molecular hydrogen, they can be excited from the P1/2 state to the

2 2 2 P3/2 state. The P3/2 state then relaxes to the P1/2 state and emits a photon, as indicated below: + 2 + 2 C [ P1/2] + H2 → C [ P3/2] + H2 (5.4) + 2 + 2 C [ P3/2] → C [ P1/2] + photon

+ Because the photon carries away energy, the interstellar clouds is cooled. The C -H2 ﬁne-structure changing collisions have been predicted to have a rate coeﬃcient of the order of 10−10 cm3 s−1 [48].

Although there are many theoretical and experimental studies of this cooling mechanism, there is little experimental data at low temperature (< 100 K) [36]. For

2 1 3 1 example, Al[ P1/2]-Ar[ S0] collisions were studied at 44 − 137 K; C[ P0]-He[ S0] and

3 1 Si[ P0]-He[ S0] collisions were studied at 15 − 49 K [36, 37]. The lack of experimental data at low temperature makes it diﬃcult to test the theoretical calculations.

With the cryogenic techniques, we could easily generate an environment with a temperature from 4 K to 300 K. Unfortunately, the wavelengths for probing the fine- structure populations of the carbon ion transitions fall in the ultraviolet range, which is difficult experimentally. Instead of studying C+, we measure the fine-structure relaxation of titanium atoms due to collisions with helium. Although titanium is not an astrophysically-important2 atom, we start with titanium as an initial study in order to test the theoretical calculations.

2 According to Reference [47], the composition of interstellar matter: 90.8% by number (70.4% by mass) of hydrogen, 9.1% (28.1%) of helium, and 0.12% (1.5%) of heavier elements. 124 5.2.2 Titanium spectrum

The neutral titanium atom has ﬁve isotopes: 46Ti, 47Ti, 48Ti, 49Ti and 50Ti. Among these isotopes, the odd ones have nuclear spin, but the even ones do not. Their natural abundances and atomic data are listed in Table 5.3. The ground-state orbital

2 2 3 conﬁguration of atomic titanium is 3d 4s a FJ . This state is coupled by the spin- orbital interaction and splits into the ﬁne-structure levels J =2, 3, and 4 at energies

0, 170, and 387 cm−1, respectively [153]. A simpliﬁed Ti level structure is illustrated in Figure 5.9(a).

Table 5.3: Atomic titanium information [153]. I is the nuclear spin; µ is the magnetic moment, expressed in units of the nuclear magneton µN (762 Hz/Gauss) [117].

Isotopes Abundance [%] I µ [µN ] 46Ti 8.0 0 47Ti 7.4 5/2 -0.7885 48Ti 73.8 0 49Ti 5.5 7/2 -1.0417 50Ti 5.4 0

As usual, we create and cool titanium atoms via laser ablation and cryogenic buﬀer-gas cooling. The atoms are observed to translationally thermalize on a time scale less than a half millisecond. A blue diode laser with a center wavelength of

2 2 3 2 3 400 nm is used to probe the strong optical transition 3d 4s a F2 → 3d 4s4p y F2,

2 3 where the 3d 4s4p y F2 excited state has a reported spontaneous lifetime of 23 ns in Reference [153]. By scanning the laser frequency, a titanium absorption spectrum is obtained and shown in Figure 5.9(b). Since 47Ti and 49Ti have nuclear spin, their hyperﬁne structures can be seen in Figure 5.9(b). The absorption spectrum indicates that the titanium target consists of naturally-occurring titanium isotopes. Since 48Ti occupies the most natural abundance, and only has ﬁne-structure levels in its ground state, it is the main colliding isotope used in this section. 125

J = 4 25388.331 cm -1 3 2.5 48 y F J = 3 25227.220 cm -1 Ti J = 2 25107.410 cm -1 2.0

1.5

Optical density Optical 1.0

0.5 46 4 386.875 cm -1 Ti 50 J = 47 Ti Ti 49 3 J = 3 170.134 cm -1 Ti a F 0.0 2 0 cm -1 J = -2 -1 0 1 2 Frequency [GHz]

(a) Ti level structure (b) Ti spectrum

Figure 5.9: Simplified Ti level diagram and an absorption spectrum, taken on 2 2 3 2 3 3d 4s a F2 → 3d 4s4p y F2 transition. The frequency offset in the spectrum is 25107 cm−1; the hyperfine lines are labeled according to isotope.

The translational temperature of the titanium atoms is measured to be 5.2 K by

ﬁtting the measured spectra to the expected Voigt line shape. It is consistent with the cell temperature measured by a calibrated silicon diode.

5.2.3 Fine-structure-changing collisions

The collisional process that we are interested in is the titanium atoms changing their

ﬁne-structure levels due to collisional events with helium atoms:

3 3 Ti[a F3] + He → Ti[a F2] + He (5.5) 126

Optical pumping

The primary method of measuring inelastic collisions is to perturb the titanium atom distribution by optical pumping and observe their return to thermal equilibrium through inelastic collisions. The experimental setup is similar to the previous description in Chapter 3.5.2.

A strong pump beam and a weak probe beam are on resonance with the optical

2 2 3 2 3 transition of 3d 4s a F2 → 3d 4s4p y F2. The strong pump beam excites the

3 3 ground-state a F2 atoms to the y F2 excited state, and then the excited atoms can decay back to either the J = 3 or J = 2 state based on the selection rules. It is also possible that the excited-state atoms decay to other metastable states, since there

2 2 3 are roughly 22 allowed optical transitions between the 3d 4s a F2 state and the

2 3 3d 4s4p y F2 state [211]. Based on Reference [211], we compute the branching ratios

3 3 to be that about 80% excited y F2-state atoms decays to the ground a F2 state, and

3 15% decays to the a F3 state. The remaining 5% decay to the other intermediate metastable states3. If atoms return to the J = 2 state, they will be re-excited by the pump beam until they end up in the J = 3 state. In this way, the population of the

J = 2 state is transferred to the J = 3 state. Although there are a large number of metastable states between the a 3F ground state and y 3F excited state, decay back to the a 3F ground state is the dominant decay path. (There are more than 50 energy

3 3 states lying between the a F2 and y F2 states.) By using a pumping duration which is short compared to the collisional relaxation time, we ensure the majority of atoms

3 3 which are removed from the a F2 are pumped into the a F3.

48 3 Figure 5.10 plots an absorption spectroscopy of Ti atoms in the a F2 states,

3 3 3 In decay from the y F3 state, 12% of the atoms decay to the a F4 state, 76% 3 3 3 to the a F3 state, and 8% to the a F2 state. In decay from the y F4 state, 88% of 3 3 the atoms decay to the a F4 state and 7% to the a F3 state. These branching ratios are estimated from the NIST Atomic Spectra Database [211]. 127

0.5

0.4 Data Fit 0.3

0.2 Optical density Optical

0.1 0.24 0.26 0.28 0.30 0.32 Time after ablation [s]

Figure 5.10: Absorption spectroscopy of 48Ti atoms, showing a return to equilibrium following an optical pumping pulse. showing a return to equilibrium following an optical pumping pulse. Time is shown relative to the ablation pulse. The overall exponential decay in absorption is due to diﬀusion to cell walls, where the titanium atoms adsorb. The sudden decrease in optical density at 0.25 s is due to the optical pumping beam, which was on from

249 ms to 250 ms. The increase in optical density following optical pumping is due to

3 3 Ti-He collisions returning atoms from the a F3 state to the a F2 state. By measuring the return to equilibrium, as shown in Figure 5.10, we determine the J = 3 → J = 2 inelastic collision rate 1/τ.

3 3 We model the return to equilibrium from the a F3 state to the a F2 state by N N N˙ = − 3 − 3 3 τ τ D 3→2 (5.6) ˙ N2 N3 N2 = − + τD τ3→2 where the NJ indicates the population for the corresponding J state and τD is the diﬀusion time constant. 1/τ3→2 is the relaxation rate from the J = 3 to J = 2 state, 128

which is also 1/τ. The J = 4 population is ignored in Equation (5.6) because after

many thermalization times and by the time we perform the optical pumping, it is

negligibly small. At 5 K and thermal equilibrium, the population of the J = 4 state

is very small according to the Boltzmann factor. In addition, nearly-no atoms are

pumped into the J = 4 state by optical pumping.

3 3 Another mechanism for atoms to return from the a F3 state to the a F2 state is through the spontaneous decay. Since the relaxation time scales inversely with the

−1 3 cube of the frequency splitting (170 cm ), the lifetime of the a F2 state is estimated to be longer than 1 minute [117], which indicates that the spontaneous decay from

3 3 the a F3 state can be ignored. It is also possible that the a F2-state atoms can be

3 excited to the a F3 state through colliding with helium atoms. Since the translational

48 temperature of Ti is so low, 1/τ2→3 should be very small compared to 1/τ3→2. At

−21 5 K, 1/τ2→3 is 10 times slower than 1/τ3→2. Once 1/τ3→2 is measured, 1/τ2→3 can be determined form 1/τ3→2 according to the principle of detailed balance [21]. By analyzing the return rate in Figure 5.10 according to Equation (5.6), we measure the inelastic collision rates and repeat the measurements in diﬀerent helium densities, as illustrated in Figure 5.11. As seen in Figure 5.11, the return rate scales linearly with the helium density, so we conclude the return to J = 2 is due to collisions

with helium.

To conﬁrm this measurement, we measured the return to J = 2 under diﬀerent

experimental conditions. 1/τ3→2 is found to be independent of the ablation energy (measured over the range 5 mJ to 50 mJ) and independent of the time delay with respect to the ablation pulse (measured over 0.1 s to 1.5 s), indicating that the collisions are not with substances produced by the ablation pulses. Similarly, 1/τ3→2 is independent of pumping beam direction and diameter, as long as the pumping

beam is suﬃciently large that diﬀusion in and out of the pumping beam volume plays 129

1000

800

] 600 -1 [s τ 1/ 400

200 Initial decay data Optical pumping data 0 0.0 0.5 1.0 1.5 2.0 2.5 17 -3 3.0x10 Helium density [cm ]

Figure 5.11: 1/τ as a function of helium gas density, at a cell temperature of 5 K.

a small role on the time scale of the collisional return to equilibrium. 1/τ3→2 is also independent of the pumping duration, as long as the pumping time is not signiﬁcantly longer than the collisional relaxation time.

Initial decay

As a secondary measurement technique, we directly monitor the J = 3 popula-

2 2 3 2 3 tion with absorption spectroscopy on the 3d 4s a F3 → 3d 4s4p y F3 transition (25057 cm−1 and 399.1 nm). Large numbers of the J = 3 atoms are produced in the initial ablation pulse, but relax to the J = 2 level due to inelastic fine-structure relaxation collisions, as displayed in Figure 5.12. By comparing the diffusion lifetimes of J = 3 atoms and J = 2 atoms, and fitting to the model in Equation (5.6), we can extract values of 1/τ3→2. Data obtained through this “initial decay” method is shown along with optical pumping data in Figure 5.11.

Both optical pumping and the initial decay data are consistent within our exper- 130

J=2 1 J=3 OD

0.1

0.01 0.0 0.1 0.2 0.3 0.4 Time after ablation [sec]

Figure 5.12: 48Ti diffusion curves for the J = 3 and J = 2 atoms. imental error. Error bars in Figure 5.11 are a combination of statistical error and an estimate of systematic error due to diffusion of atoms in and out of the volume of the pumping beam. The line in Figure 5.11 is a least-squares fit to 1/τ = kJ · nHe.(nHe is the helium density.) In this way, we measure the inelastic collision rate coefficient kJ for the fine-structure changing collisions.

5.2.4 Experimental and theoretical results

By applying an electric current through a resistor mounted on the cell body as a heater, we vary the cell temperature. In this way, the Ti-He ﬁne-structure changing collisions are measured repeatedly at diﬀerent temperatures. These experimental results are listed in Table 5.4.

Table 5.4 compares a theoretical prediction to the experimental measurements.

The theoretical calculation performed by Bernard Zygelman [45, 46] uses the ab initio

Ti-He potential curve of Klos et al [212]. The results of this theoretical calculation 131

are also plotted as a blue dashed curve in Figure 5.13.

Table 5.4: Ti-He J = 3 → J = 2 inelastic collision rate coeﬃcients at diﬀerent temperature T [45]. Experimental error is a combination of statistical error and estimates of systematic error, which is primarily due to uncertainty in the helium density. Theory Experiment 3 −1 3 −1 T [K] k3→2 [cm s ] k3→2 [cm s ] 5.2 1.86 × 10−15 (4.4 ± 0.7) × 10−15 9.9 2.74 × 10−15 (5.3 ± 0.8) × 10−15 15.6 4.50 × 10−15 (7.7 ± 1.2) × 10−15 19.9 6.18 × 10−15 (9.8 ± 1.5) × 10−15

The rate coeﬃcients for the Ti-He ﬁne-structure changing collisions are found to

be surprisingly small compared to other atoms with orbital angular momentum, as

described in Chapter 1.3. It is similar to a previous discussion on the Zeeman relax-

ation collisions of the rare-earth atoms [42]. Moreover, a recent experiment measuring

Ti-He collisions observes a dramatic suppression of m-changing collisions due to tita-

nium’s submerged shell structure, because the ground-state Ti atom has two valence

electrons hidden inside a spherical s orbit [43, 44]. The ab initio Ti-He potential

calculations of Klos et al. suggest that titanium may exhibit a similar suppression of

ﬁne-structure-changing collisions [212]. A calculation with a rescaled potential of Ti-

He ﬁne-structure changing collisions based on the results of Reference [43] is plotted as a red dash-dotted line in Figure 5.13.

Although the magnitudes of these predicted curves plotted in Figure 5.13 are outside the experimental uncertainty, the order of magnitude the predicted rate co- eﬃcients and their collision energy dependence are consistent with the experimental data. A better agreement with the experimental data can be found in Reference [46], by introducing a small rescaling of the ab initio potentials. The theoretical results of

Reference [46] are shown as a solid line in Figure 5.13. 132 ] ]

[ ]

Figure 5.13: Ti-He experimental data and theoretical curves are plotted versus temperature. Graph is from Reference [46]. Plotted points are the experimental data. The details are discussed in the text.

Diﬀusion cross-section

From our data, we also determine the diﬀusion cross-section for Ti-He collisions at diﬀerent temperatures, as listed in Table 5.5 alongside theoretically calculated values

[45]. We determine the diffusion cross-sections from the diffusion lifetimes measured at helium densities ranging from 7×1015 cm−3 to 2×1017 cm−3. The first and second columns in Table 5.5 are theoretical estimates for the total elastic and diffusion cross- sections, respectively. The third column lists experimental values for the diffusion cross-section. The dominant source of error is due to approximations made in the modeling of diffusion in the cell geometry [22], as discussed in AppendixC. 133

Table 5.5: Thermally averaged Ti-He cross-sections at diﬀerent temperatures. σe is the total elastic cross-section; σd is the diﬀusion cross-section [45]. Theory Theory Experiment 2 2 2 T [K] σe [cm ] σd [cm ] σd [cm ] 5.2 5.10 × 10−14 1.50 × 10−14 (1.1 ± 0.3) × 10−14 9.9 3.72 × 10−14 1.05 × 10−14 (8.6 ± 2.3) × 10−15 15.6 2.99 × 10−14 8.68 × 10−15 (7.7 ± 2.1) × 10−15 19.9 2.67 × 10−14 7.95 × 10−15 (7.3 ± 2.0) × 10−15

Discussion

Experimentally, we have demonstrated the suppression in the J-changing collisions,

which is conﬁrmed by theoretical calculation.

The experimental technique—a combination of laser ablation, buﬀer-gas cooling,

and optical pumping—should be applicable to measuring ﬁne-structure-changing col-

lisions between atoms and helium for any spectroscopically addressable species. We

have shown its eﬃcacy in an atom without a closed optical transition.

5.2.5 Other titanium experiments

m-changing collisions

The previous measurement on the Ti-He Zeeman relaxation collisions found an m-

−14 3 −1 changing rate coeﬃcient of km = (1.1 ± 0.7) × 10 cm s at 1.8 K and 3.8 Tesla [43].

Similar to the measurement in Chapter 3.5.2, our repetition of Zeeman relaxation

measurements at 5 K and fields of a few Gauss indicates a rate coefficient of km = (1.2 ± 0.6) × 10−13 cm3 s−1 for Ti-He collisions. The difference of the rate coefficients

in both experiments indicates that there may be a magnetic ﬁeld dependence on the

Zeeman changing collisions. Further study is needed to extract the answer. 134

Titanium EIT

The previous discussion only involves the inelastic relaxations by diﬀerent mechanisms

(the T1 time). The spin decoherence of titanium atoms due to inelastic collisions with

helium (the T2 time) is also of interest. However, the Larmor precession of titanium atoms is so fast that it is hard to optically pump titanium atoms into a coherent

superposition state. In order to measure the spin decoherence of titanium atoms due

to inelastic collisions with helium (the T2 time), we set up an EIT experiment which is similar to that reported in Chapter 3.6.3.

0.29 12

0.28 8

OD 6 0.27 4 Data EIT dip width [kHz] widthdip EIT Fit 2 0.26 0 17 -60 -40 -20 0 20 40 60 0.0 0.5 1.0 1.5 2.0x10 -3 Frequency [kHz] He density [cm ]

(a) Ti EIT dip (b) EIT width vs. helium density

Figure 5.14: Ti-He spin decoherence time measured by the EIT method.

A typical EIT dip of titanium atoms is shown in Figure 5.14(a), where the size

of the dip in the optical density is plotted as a function of the two-photon frequency

detuning. Referring to Equation (3.11), we can extract the decoherence by measuring

the width of the transparency window. We plot the measured dip width as a func-

tion of the helium density, and ﬁnd the dip width has a linear dependence on the

helium density. Based on this observation, a linear ﬁt gives a Ti-He decoherence rate

coeﬃcient of 5 × 10−14 cm3 s−1. The oﬀset in Figure 5.14(b) indicates that there

are mechanisms other than Ti-He collisions causing the decoherence. We believe that 135 the decoherence is mainly from the frequency noise of the laser beams induced by the

AOMs. The linewith of the AOM is about few kHz.

Titanium DFSAS

Instead of fitting the spectra with a Voigt profile, we apply Doppler-free saturation absorption spectroscopy of titanium atoms in order to eliminate the Doppler broadening. We take the Doppler-free spectrum at different helium densities, and measure their linewidths. The results are plotted versus helium density in Figure 5.15. Data

fitting to a line gives a pressure broadening coefficient of 2 × 10−18 GHz/cm−3 and an offset of 34 MHz. The offset is larger than the natural linewidth (7 MHz) of Ti atoms because of power broadening.

40 FWHM[MHz] 20 Data Fit

0 16 0.0 1.0 2.0 3.0x10 -3 He density [cm ]

Figure 5.15: The FWHM of transition line as a function of helium density. The line is a linear ﬁt to the data. 136

Chapter 6

TiO-He collisions

In previous chapters, we study collisions of non-S-state atoms to understand their collisional properties, which may be of use in future ultracold atom experiments. As techniques for generating atomic quantum degeneracy have become mature, the developments of cold and ultracold molecules have started to draw much attention. The complex structures of molecules, especially polar diatomic molecules, are promising for quantum information processing [213], the search of the electron electric dipole moment [214], and the dipole-dipole interaction of a degenerate quantum gas [30].

Chemical reactions with energy barriers, whose reaction rates are supposed to exponentially decay with decreasing temperature in classical theory, have been predicted to occur rapidly at low temperature through a quantum tunneling eﬀect [215]. Recent review articles [30, 51, 216] have described the research interests and experimental methods for cold and ultracold molecules.

6.1 Basic information of TiO

In the Weinstein lab, we study titanium monoxide as the first step of the cold molecule experiments. We produce a large amount of cold TiO molecules by a combined 137 technique of laser ablation and buffer gas cooling. Since 48Ti and 16O have no nuclear spin and are the most abundant isotopes [153], 48Ti16O is spectroscopically resolved and used for measuring elastic and inelastic collisions with helium atoms at 5 K in order to avoid complications due to hyperfine structure.

6.1.1 Hund’s case (a)

The ground state of the TiO molecule is X 3∆ [56], and its coupling case is Hund’s case (a), as illustrated in Figure 6.1. The deﬁnition of Hund’s cases in this thesis is based on Reference [60]. The electronic orbital angular momentum L and the spin angular momentum S precess around the internuclear axis, and their projection of the angular momentum on the axis are denoted as Λ and Σ, respectively. The sum of

Λ and Σ is written as Ω, i.e. Ω = Λ+Σ. The vector Ω pointing along the internuclear axis is coupled to the rotational angular momentum of the nuclei R, and form the total angular momentum J which satisﬁes that J ≥ Ω and J = |Ω|, |Ω|+1, |Ω|+2, ··· .

R is perpendicular to the internuclear axis.

R J

L S

Λ Σ

Figure 6.1: Vector diagram for Hund’s case (a) [60].

The individual couplings of L to the internuclear axis and S to the internuclear axis are the strongest. In other words, the precession of L and S about the internuclear 138

axis is much faster than that of J. (For example, the nearest level to X 3∆ is a 1∆

state at ∼3000 cm−1 away; the nearest level to X 3∆ with a diﬀerent value of Λ is

d 1Σ at ∼5000 cm−1 away [56].)

Λ-doubling

Since the precession of L and S about the internuclear axis can be clockwise or

counterclockwise, their projections have magnitudes ±Λ and ±Σ. As a result, a

total projection of ±Ω results in a two-fold degeneracy, called Λ-doubling for a ﬁxed

molecule. When the molecule rotates in space, the degeneracy of the Λ-doubling is

split by a small amount in energy due to the spin-orbital and rotational mixing with

low-lying excited electronic states. Then, the rotational state becomes two states

with opposite parity and a small energy splitting [60]. This feature of the splitting is

called the Λ-doublet, which exists commonly in Π-state and ∆-state molecules.

6.1.2 TiO level structure

The ground X 3∆ state has Σ = 1, Λ = 2, and |Ω| = 1, 2, 3, so that it has three

3 3 3 ﬁne structure levels X ∆1, X ∆2 and X ∆3 due to the spin-orbital interaction, as shown in Figure 6.2. Similarly, the A 3Φ excited state also splits into three states:

3 3 3 A Φ2, A Φ3 and A Φ4. Note that the vibrational energy Tv labeled in Figure 6.2

1 has an energy diﬀerence of 2 ~ω from the bottom of its energy potential well. Taking 3 00 1 the X ∆1, v = 0 state as an example, its corresponding energy is Tv = 0 and

−1 3 00 Te = 504.5 cm ; the energy diﬀerence is from the X ∆1, v = 0 to the bottom of the potential well [217].

Each ﬁne-structure has a pattern of rotational levels, with relative energies deter-

1 In molecules, the double apostrophe, 00, denotes the energy levels of the ground electronic state; the single apostrophe, 0, indicates those of the excited state. 139

3 -1 A Φ4 14361 cm 3 -1 A Φ3 14193 cm v '= 0 3 -1 A Φ2 14020 cm

~768 nm

3 -1 X 3 1198 cm 3 -1 X 2 1097 cm v"= 1 3 1000 cm -1 X 1

3 -1 X 3 198 cm 3 97 cm -1 X 2 v"= 0 3 X 1 0

Figure 6.2: A schematic level diagram of 48Ti16O molecules (not to scale) [218]. The 3 00 red arrow indicates an optical transition from the ground electronic state X ∆1, v = 3 0 1 to the excited state A Φ2, v = 0. Not shown are the rotational states, Λ-doublets, higher-lying vibrational states, and other electronic states. 140

14400 Α3Φ 4 v'=0 3 Α Φ v'=0 14300 3 Α3Φ 2 v'=0 14200

14100

14000

3 X 3 v"=1 3 X 2 v"=1 ] 1200 3 -1 X 1 v"=1 1100

1000 Frequency [cm Frequency 250 3 X 3 v"=0 3 X 2 v"=0 200 3 X 1 v"=0 150 100 50 0 0 1 2 3 4 5 6 7 8 J

Figure 6.3: The energy separation of TiO rovibrational states is plotted as a function of the total angular momentum J. The energy spacing is calculated based on Equation (6.1) and Reference [218]. 141 mined by the corresponding rotational constants. Based the data in Reference [218], the corresponding frequency of each rotational energy level is calculated, and the results are plotted in Figure 6.3 to show the rovibrational levels of the X 3∆ ground states and the A 3Φ excited states to scale. The calculated energy spacing in Fig- ure 6.3 uses the approximated energy expression:

E(J) ≈ Tv + BvJ(J + 1) (6.1)

where Tv and Bv of particular states can be found in Table F.5. The approximation in Equation (6.1) is good when J is small; as J increases, the energy level can be adjusted by adding more correction terms which can be found in Reference [218].

(Our calculated energy level and that of Reference [218] have an agreement better than 0.01 cm−1.)

The Λ-doubling of both X 3∆ and A 3Φ state is negligible, especially for small J, and the splitting is very small compared to the rotational spacing. For example, the

3 nearly degenerate Λ-doublet of the J = 1 rotational level in the X ∆1 state has a splitting smaller than 1 MHz [219].

3 ∆1-state molecules

3 A molecule in the ∆1 state is preferred in experiments searching for the electron’s electric dipole moment [65]. The ∆ symmetry guarantees a small Λ-doubling in the level structure, so it is easy to apply a small electric ﬁeld to mix the doublet with

2 3 states of opposite parity. Also, because the magnetic g factor of ∆1 molecule is very small, it is insensitive to magnetic ﬁeld noise. With a large polarizability and a

3 small magnetic g factor, a ∆1 molecule is sensitive to the electron’s electric dipole moment [65].

2 3 gJ ∼ (gLΛ + gsΣ) ∼ (1 · 2 − 2 · 1) ∼ 0, for ∆1 state with Λ = 2 and Σ = −1. 142

Laser cooling

3 Some ∆1 molecules has been proposed for use in laser-cooling experiments [64]. They are preferred because the rotational ladder can be closed by a proper excitation and

3 the only repumping needed is for the vibrational states. The easy polarizability of ∆1 molecules is also favorable in a electric trap. Especially, 48Ti16O molecule has been theoretically conﬁrmed to satisfy these conditions [64]. Both nuclei of 48Ti and 16O

have zero nuclear spin, so the 48Ti16O molecule is relative simple with no hyperﬁne

structure. In addition, they are the isotopes with the dominant abundances.

Cold chemical reactions

Classically, the chemical reaction rate k of a chemical reaction with an activation

energy barrier Ea can be described by the the Arrhenius equation [220]:

− Ea k = A exp kB T (6.2)

where A is a constant. The reaction rate of a chemical reaction with activation energy

Ea falls exponentially as the reactant temperature T approaches zero. On the other hand, the Wigner threshold law predicts that this kind of reaction can continue at a

ﬁnite nonzero rate at cold or ultracold temperature by quantum-mechanical tunneling

through the energy barrier [215].

Titanium atom is favorable for studying tunneling-driven cold chemical reactions.

Its reaction with oxygen molecules has an activation energy barrier Ea ∼0.5 eV with a reaction rate coeﬃcient of 2 × 10−12 cm3 s−1 at 300 K:

Ti(g) + O2(g) → TiO(g) + O(g) (6.3)

which is exothermic by ∼2 eV (1 eV = 96 kJ/mole) [56] . We can compare the

lifetimes of Ti atoms with and without the presence of O2 in order to observe the 143 cold chemical reaction. Although the chemical reaction can be determined from the measurements of Ti atoms alone, the developed techniques for detecting TiO molecules help to understand the condition. Since large numbers of Ti atoms and

TiO molecules have been spectroscopically detected at ∼5 K, our experimental setup should be suitable for measuring this kind of tunneling-driven gas phase reaction rate coeﬃcient at cryogenic temperatures.

6.1.3 Spectroscopic structures

We detect the ground-electronic state TiO molecules through laser absorption spectroscopy on the X 3∆ → A 3Φ transition. This transition is experimentally convenient because its wavelength is located in the near infrared and optical strengths

3 are strong. The excited lifetimes of A Φ2 state are τv0=0 = (103.3 ± 3.6) ns and

τv0=1 = (112.9 ± 1.8) ns [221]. The probe laser is generated by an AR-coated laser diode stabilized with grating feedback, as mentioned in Chapter2. Unfortunately, because of the available diode wavelengths, this restricts us to probing transitions with v0 = v00 − 1. Consequently, we do not detect the ground vibrational state (v00 = 0). Fortunately, the vibrational relaxation cross-section for TiO-He collisions is suﬃciently small that this restriction does not prevent us from measuring collisions of interest, as discussed in Chapter 6.2.

The rotational transition of TiO molecules can be presented as the P (J 00), Q(J 00) and R(J 00) lines3. For example, a Q(2) line indicates an electronic dipole transition from the ground state J 00 = 2 rotational level to an excited state with J 0 = 2 rotational level. A typical TiO absorption spectrum detected on the transition

3 00 3 0 X ∆1, v = 1 → A Φ2, v = 0 is shown in Figure 6.4. Both Q(2) and Q(3) rotational lines are observed. Calibration of the scan width is obtained from the rotational line

3 P -branch is J 0 = J 00 − 1, Q-branch is J 0 = J 00, and R-branch is J 0 = J 00 + 1 [23]. 144 spacing [218]. When the TiO spectrum is fit to Voigt profiles, the fits indicate that the linewidth is dominated by the Gaussian component. From the Gaussian width, we determine the translational temperature of the TiO to be (6 ± 2) K. The uncertainty in the temperature is primarily due to nonlinearities in the laser scan. This temperature measurement is consistent with the translational temperature of the molecules having reached thermal equilibrium with the cell body, which is measured with a calibrated diode to be 5.2 K.

0.020 Q(2) 0.015 0.015 Q(3) 0.010 0.010

OD 0.005

0.005 0.000 1800 2000 0.000

-2000 -1000 0 1000 2000 Frequency [MHz]

3 00 3 0 Figure 6.4: Spectrum of TiO molecules detected on the X ∆1, v = 1 → A Φ2, v = 0, taken at a helium density of 1 × 1016 cm−3, 30 ms after the ablation pulse. The frequency oﬀset of the scan is 13019.5 cm−1 (768.08 nm). The insert shows a pure Gaussian ﬁt to the Q(2) line.

6.1.4 Ablation yields

TiO molecules are produced by laser ablation of a sintered TiO2 target as well as by ablation of a TiO pellet. (See Table 2.1.) We note that at low helium densities

17 −3 (nHe . 10 cm ), the TiO2 target provides a higher yield of molecules, while at 145

17 −3 high helium densities (nHe & 10 cm ), the TiO target has a superior yield. At a 16 −3 helium density of 1 × 10 cm , a single ablation pulse (∼15 mJ) on the TiO2 target

11 3 00 produces 1×10 TiO molecules in a single rovibrational state of X ∆1, v = 1, with a peak density of 4 × 108 cm−3.

6.2 Collisional data

Once TiO molecules are introduced into the cell, they slowly disappear, as shown in

Figure 6.5. The loss results from a combination of diﬀusion to the cold cell walls, where the molecules adsorb, and inelastic collisions and radiative decay, which both transfer population to lower energy states.

6.2.1 Diﬀusion cross-section and vibrational relaxation

Due to the accessibility of laser diode wavelengths, we measure the diﬀusion lifetime

3 00 3 0 of TiO by laser absorption spectroscopy on the X ∆1, v = 1 → A Φ2, v = 0,R(1) transition. The detection beam is located in the center of the cell, 3 cm away from the ablation target. The early time behavior varies with the ablation conditions. At long times after ablation, the molecular density distribution in the cell is expected to be described by the lowest-order diﬀusion mode. As seen in Figure 6.5, the long- time behavior of the TiO population ﬁts well to the expected exponential form. The measured exponential decay lifetime τ is plotted as a function of helium density in

Figure 6.6.

3 On the other hand, the similar behaviors of atom loss for different X ∆1 vibrational states in Figure 6.5 indicate a small vibrational relaxation rate comparing to the diffusion lifetime. Because of the slow thermalization of the vibrational levels, significant population in the v00=1, 2, and 3 levels is observed. 146

3 3 A Φ2 (v'=0) - X 1 (v"=1) R(1) 3 3 A Φ2 (v'=1) - X 1 (v"=2) R(1) 3 3 A Φ2 (v'=2) - X 1 (v"=3) R(1) Fit

0.01 OD

0.001 0.00 0.04 0.08 0.12 Time [s]

3 Figure 6.5: Optical densities of the X ∆1-state TiO molecules in different vibrational states. The transitions used for probing the different vibrational states are as labeled. At long times after laser ablation, data is fit to a single exponential function.

3 00 In Figure 6.6, the lifetime of the X ∆1, v = 1 state initially increases with the helium density, as is the expectation for diﬀusion to the cell walls. However, at high

helium density this is no longer the case, and the lifetime is shorter than would be

expected from diﬀusion alone. This is because of loss from spontaneous emission and

inelastic collisions [19, 222]. Because the state examined is the lowest Ω and J state

of the v00 = 1 vibrational level of the X state, the only possible decay is vibrational

relaxation to the v00 = 0 state. We model the TiO loss rate as

1 1 = + kv · nHe + A10. (6.4) τ D · nHe

The ﬁrst term on the right-hand side represents diﬀusion, the second term is due to

3 00 inelastic TiO-He collisions, and A10 is the radiative decay rate of the X ∆1, v = 1 level. Data in Figure 6.6 are ﬁt to Equation (6.4), which shows reasonable agreement.

By ﬁtting the lifetime data, the thermally-averaged TiO-He diﬀusion cross-section is 147 determined to be (1.5 ± 0.7) × 10−14 cm2.

0.1

Data Fit Lifetime [s]

0.01

15 16 17 10 10 -3 10 Helium density [cm ]

3 00 Figure 6.6: Exponential decay lifetimes of X ∆1, v = 1 TiO molecules plotted as a function of helium density.

−16 3 −1 The ﬁt also determines an inelastic rate coeﬃcient of kv = (1.9±0.4)×10 cm s and a radiative lifetime of 1/A10 = (53 ± 6) ms. However, this is not a conclusive measurement of these vibrational relaxation processes. Because the observations are

3 00 made at short times after ablation, it is possible that as the X ∆1, v = 1 molecules

3 00 relax through collisions and spontaneous emission, the X ∆1, v = 1 level is simul- taneously being repopulated by decay from an unobserved, higher-lying metastable state of TiO.

The uncertainty arises from a combination of statistical error, approximations made to the cell geometry when modeling diﬀusion, and uncertainty in the helium density. When ablating the sintered TiO2 target, the gas pressure reading of the cell temporarily increases following ablation. We attribute this increase to the liberation of helium absorbed onto the large surface area of the porous TiO2 sample. Once the 148

sample recools, it slowly cryopumps the released helium. Accurate measurements of

these transient pressure changes are limited by the small conductance between the

cryogenic cell and room-temperature pressure gauge.

6.2.2 Rotational-changing collisions

The relative populations of the low-lying X 3∆ rotational J 00 states are measured from

3 00 3 0 00 the optical densities of their corresponding X ∆1, v = 1 → A Φ2, v = 0,R(J ) transitions, as shown in Figure 6.7. By the time we are able to observe molecules in the

cell—typically a few milliseconds after the ablation pulse—the relative populations

of the J 00 levels are not observed to change with time.

3 X ∆1 (v"=1, J"=1) 3 X ∆1 (v"=1, J"=2) 0.1 3 X ∆1 (v"=1, J"=3) 3 X ∆1 (v"=1, J"=4) 0.01 OD OD 0.01

R(J") Fit

0.001 0.001

-3 0 20 40 60 80x10 1 2 3 4 5 6 Time [s] J"

(a) OD vs. time (b) OD vs. J 00

Figure 6.7: Optical densities of diﬀerent TiO rotational levels, showing the TiO rotational temperature.

The rotational temperature of TiO is determined by ﬁtting the rotational state populations to a Boltzmann distribution: 0.53[cm−1] J 00(J 00 + 1) OD = a · (2J 00 + 1) exp − (6.5) kBT where a is a free parameter. The rotational temperature of the TiO is measured to

have a rotational temperature of (6.1 ± 1.3) K, which is in thermal equilibrium with 149 its translational temperature. The optical strength for the individual transition is ignored, resulting in an imperfect ﬁt.

We attempted to measure rotation-changing inelastic collisions by using optical pumping to disturb the rotational populations from thermal equilibrium and monitoring their return to equilibrium through inelastic collisions. Unfortunately, the collision process was too fast to observe, and we are only able to place a lower limit

−11 3 −1 on the rate coeﬃcient: kJ ≥ 1 × 10 cm s . The large rate is not surprising, as rate coeﬃcients for rotational quenching are expected to be on the same order as momentum-transfer collisions [223].

6.2.3 m-changing collisions

For measuring the Zeeman relaxation, the same approach we use with atoms is ap-

3 00 00 plied to TiO molecules. We monitor the polarization of the X ∆1, v = 1,J = 1

+ − 3 00 state by measuring the relative absorption of σ and σ light on the X ∆1, v =

3 0 1 → A Φ2, v = 0,R(1) transition, in the presence of a few-Gauss bias ﬁeld. This is a favorable transition for monitoring polarization, as the mJ00 = ±1 states have a relative absorption cross-section of 6-to-1 for σ± light. See Figure 6.8. After ablation there is no observable TiO polarization. We would not expect an observable polarization, as the splitting between the Zeeman levels is much smaller than kBTtr. Ttr is the translational temperature.

To induce a polarization, we optically pump the Zeeman levels of the TiO molecules using a short pulse of high-intensity σ+ light on the same transition. Note that the pumping transition is not a closed transition. Atoms are not pumped from one mJ00 state into the other, rather the population of one mJ00 level is depleted relative to the other. The Franck-Condon factors for the A − X transition [221] indicate that the majority of the excited molecules will decay into the v00 = 0 state: 72% of the 150

excited molecules are pumped into the v00 = 0 state, while only 23% return to v00 = 1,

as illustrated in Figure 6.8. Once pumped, the v00 = 0 molecules will have negligible

collisional return to the v00 = 1 state, as the vibrational spacing is much larger than

kBTtr. In this way, we expect to introduce polarization in the mJ00 states.

m 2 1 0 1 2 A3 v J ' − − Φ2 , ' = 0

23 % 1 3 6 72 %

X3 v m 1 , " = 1 J '' −1 0 1

X3 v 1 , " = 0

Figure 6.8: A schematic Zeeman sublevel diagram of TiO. Solid arrows indicate the σ+ transitions, labeled with the relative absorption cross-section. The Franck-Condon factors of the 00 → 000 and 00 → 100 transition are 72% and 23%, respectively.

To verify that the pumping and detection optics were working properly, a sample

of atomic rubidium was initially used, and the lasers are tuned to 85Rb’s nearby 780

nm transition. We were able to induce a large polarization in the F = 3 ground state

and observe its decay. Using the same optics with TiO, we were able to pump the

3 00 00 X ∆1, v = 1,J = 1 state of TiO, but no polarization was observed. To within

our signal-to-noise, the populations of the mJ00 = ±1 states were equally depleted, indicating that inelastic m-changing collisions are occurring at a faster rate than the optical pumping.

For example, at a helium density of 2.3 × 1015 cm−3 a 5 µs pulse of pumping light

3 00 00 removed ∼30% of the molecules in the X ∆1, v = 0,J = 1 state. Figure 6.9(a) plots the average signals from two individual photodetectors as shown in Figure 3.5. Each 151 data is an average of ten sets of signals. The dramatic changes in both signals are due to optical pumping. The arrows indicates the removed portion of molecules due to the optical pumping, and the portion is measured 11 µs after the optical pumping due to the response time of the photodetectors. From the Clebsch-Gordan coefficients, we calculate this would produce a population imbalance in the mJ00 = ±1 levels of the remaining molecules of ∼30%. However, the normalized difference between the two average signals shown in Figure 6.9(b) does not indicate any polarization difference.

The noise level of our atomic polarization measurements under these conditions would enable us to detect a population difference as small as 6% (measured from 16 to 28 µs after the pump pulse begin), but no population difference was observed. Figure 6.9(b) shows that a population difference would be created, however, it relaxes to equal population in a rate faster than our detection system can record.

0.010 0.8

0.008 0.6 (siga- sigb)/(siga+ sigb)

0.006 0.4 0.004 Signal [a.u.] Signal Signal [a.u.] Signal No obvious polarization difference avg. of siga 0.2 0.002 avg. of sigb

0.000 0.0 4.98 5.00 5.02 5.04 5.06 5.08 5.10 4.98 5.00 5.02 5.04 5.06 5.08 5.10 Time [ms] Time [ms]

(a) (b)

Figure 6.9: m-changing measurements of TiO molecules. (a) The arrows indicate the change in population due to optical pumping, measured after the response time of the photodetectors. (b) The normalized population diﬀerence. After optical pumping, there is already no polarization diﬀerence.

From this observation, and other measurements taken over a range of pumping times and helium densities, we set a lower limit on the inelastic rate coeﬃcient: 152

−11 3 −1 −15 2 km ≥ 3 × 10 cm s . Expressed as a cross-section, this is σm ≥ 2 × 10 cm . The measurement is limited by the available pumping laser power and the helium density. Although the relaxation rate can be reduced by lowering the helium density, low helium densities unfortunately lead to a poor ablation yield of TiO molecules and consequently poor signal-to-noise on measurements of molecular polarization.

6.2.4 Discussion

We are able to produce large numbers of translationally and rotationally cold TiO molecules by laser ablation and buﬀer-gas cooling, a technique which may be of use for experiments seeking to laser cool TiO [64].

One would naively expect ground-state helium atoms to be the most inert of collisional partners, due to helium’s low polarizability, shallow interaction potential, and lack of internal structure. However, even for this inert partner, the inelastic rate coeﬃcient of TiO is many orders of magnitude larger than what has been measured for Σ-state molecules.

We have placed a lower limit on the rate coeﬃcient for TiO-He m-changing collisions; to our knowledge this is the ﬁrst such measurement for a non-Σ-state molecule.

The inelastic m-changing cross-section is very large: it is measured to be within an order of magnitude of the diﬀusion cross-section, as theoretically predicted for non-

Σ-state molecules [27]. This large rate is within an order of magnitude of estimates of the Langevin cross-section [224]. The Langevin cross-section assumes that the inelastic collision occurs for each scattering event, so it is an estimation of the worst case.

We note that both rotation-changing and m-changing rate coeﬃcients are within one order of magnitude of this estimated “maximum” rate. 153

Chapter 7

Summary

A cryogenic system with optical windows is well-constructed and functionally operating at 4 K. By using laser ablation and buﬀer-gas cooling, a large number of atoms or molecules are produced in the cryogenically-cold vapor cell. We manipulate the atoms or molecules by optical pumping, and observe them with laser spectroscopy.

We have made the first measurements of fine-structure changing collisions in atomic titanium at cold temperature, as well as the first measurement of indium and gallium collisions. In the measurements of inelastic collisions between atoms, the fine-structure changing collisions between titanium atoms and helium gas show a dramatically slow rate, due to titanium’s submerged-shell structure. In addition, surprising slow inelastic collisions are observed in gallium and indium atoms when they separately collide with 4He. These results suggest that it may be possible to cool

2 the P1/2-state atoms by evaporative cooling or sympathetic cooling. We have also made the ﬁrst measurements of the inelastic collisional properties of a 3∆ molecule at cold temperature. The Zeeman relaxation measurement (TiO) is determined to have a similar magnitude as its elastic collisions, as expected for non-Σ state molecules.

We have also perform ensemble-based quantum information experiments in the system. Strong atom-light coupling and long atom-light coherence times are ﬁrst 154

1 demonstrated in the ground-state ( S0) ytterbium atoms by using electromagnetically induced transparency with pure nuclear spin. By performing a stopped light experiment in Yb atoms, storage times of hundreds of milliseconds is achieved. From these observations, atomic ensembles of pure nuclear spin states may be a superior medium for a variety of nonlinear optics and quantum information experiments. 155

Appendix A

EIT calculation

The goal of this chapter is to develop a theoretical calculation of the atom-light interaction. In Chapter A.1, we apply the interaction picture to transform the time base of the system, and use the density matrix to obtain the ensemble average. To clarify the theoretical method, an example of calculating a Λ-type EIT system is presented in Chapter A.2. The application of the theoretical calculation is relevant to the 173Yb EIT scheme which has a “leak” absorption from its N-type level structure. Four diﬀerent types of level structures are used to estimate the transparency of the N-type EIT system, as presented in Chapter A.3.

A.1 Methods

A.1.1 Interaction picture

In the semiclassical theory, the radiation ﬁeld is described by a classical electric wave

E = E0 cos(ωt − kz). According to the dipole approximation, we can assume kz ' 0 as long as the ﬁeld wavelength is much greater than the atom size [67, 116], and the

field can be rewritten as E = E0 cos(ωt). Consider that an atom with an excited state energy ~ω3 is interacted by the field. The total Hamiltonian to describe this atom- 1 field system is written as H(t) = H0 + H1(t) , where H0 is the Hamiltonian operator for a bare atom and H1(t) is the atom-light interaction. H1(t) can be presented as −p · E H (t) = −p · E cos(ωt) = 0 (eiωt + e−iωt) (A.1) 1 0 2

1The operators do not have hats, for example, H ≡ Hb 156 where p = −ex is the dipole operator. If considering the matrix element between the ground state |gi and the excited state |ei, Equation (A.2) can be rewriten as

Ω H (t) = −~ (eiωt + e−iωt) (A.2) 1 2

ehg|x|ei|E0| where Ω = is the Rabi frequency, and µge = ehg|x|ei is the dipole matrix ~ element. In this chapter, the Planck constant ~ has been set to be one. The Schr¨odinger equation [21] of the atom in the presence of the driving ﬁelds can be written as

2 2 2 where time-dependent coeﬃcients satisfy |c1(t)| + |c2(t)| + ··· + |cn(t)| = 1. We introduce a unitary time-evolution operator U(t) = e−iGt, where G is written as   u1 0 0 0    0 u2 0 0  G =   , (A.5)  ...   0 0 0  0 0 0 un and u1, u2, ··· , un are real and time-independent parameters. (The method to determine the parameter ui will soon be discussed.) In addition, U(t) and G are both Hermitian and commute with each other. The matrix form of U(t) is written as   e−itu1 0 0 0

 −itu2   0 e 0 0  U(t) =  .  (A.6)  ..   0 0 0  0 0 0 e−itun

U(t) is a unitary operator satisfying that U −1 = U † or U †U = UU † = 1, and its 157 adjoint matrix is   eitu1 0 0 0  0 eitu2 0 0  †   U (t) =   . (A.7)  ...   0 0 0  0 0 0 eitun The unitary operator U(t) is then used to transform the time base of |ψ(t)i, so that a new eigenstate |ψI (t)i can be presented as

In Equation (A.11), a Schrödinger-like equation with an effective Hamiltonian Heff and an eigenstate |ψI (t)i is derived. With this approach, Equation (A.3) is converted from the Schrödingerpicture into the interaction picture [21, 67]. In this way, once the unitary operator U(t) is properly assigned, time-dependent problems can be simplified as shown in Chapter A.2 and Chapter A.3.

A.1.2 Density matrix approach

Suppose that a measurement is made on a mixed ensemble of an observable Q in a given physical system. The ensemble average of Q, which is the average measured 158

value of Q, is deﬁned by hQiensemble:

hQiensemble = trace(ρQ)[21] (A.12)

where ρ is the density matrix and is written as

ρ(t) = |ψ(t)ihψ(t)| (A.13)

Equation (A.12) is a useful relation for measuring the ensemble average of Q because

hQiensemble is equal to the trace and the trace is independent of the representations

[21, 67]. In this way, hQiensemble can be evaluated by using any convenient basis [21]. To know the time evolution of the system, the ensemble average can be evaluated by the density matrix in a way that the density matrix is solved from its equation of motion which is derived from Equation (A.3) and Equation (A.13): dρ(t) = −i[H(t), ρ(t)] (A.14) dt Equation (A.14) is often called the Liouville or Von Neumann equation of motion [67]. To include the excited state decay due to the spontaneous emission, we rewrite Equation (A.14) as dρ(t) 1 = −i[H(t), ρ(t)] − Γρ(t) + ρ(t)Γ−Λ (A.15) dt 2 where Γ is a relaxation matrix with the excited-state decay rate γ and Λ is the repopulation matrix [67, 71]. With both Γ and Λ, we ensure the population is conserved.

We can apply the interaction picture to Equation (A.15) with ρI (t) = |ψI (t)ihψI (t)|, so it becomes dρ (t) 1 I = −i[H(t), ρ (t)] − Γρ (t) + ρ (t)Γ−Λ (A.16) dt I 2 I I With this approach, it is easy to solve the time-dependent problems in the multilevel system interacting with radiation ﬁelds. An example of the calculation is presented in Chapter A.2.

A.2 Example: A three-level EIT system

A.2.1 The eﬀective Hamiltonian

Consider that there is an atom-light system consisting of a three-level atom and two electric ﬁelds, as shown in Figure A.1. The atom has two lower energy states |1i 159

and |2i with energy eigenvalue ~ω1 and ~ω2, respectively. The lower states have very

long lifetimes. A decoherence term γg describes the decoherence rate of the coherence

between |1i and |2i. The excited state |3i of the atom has a frequency of ω3 and a

spontaneous decay rate γ. A probe ﬁeld with frequency ωp is applied on the transition

|1i → |3i, and a control ﬁeld with frequency ωc excites the transition |2i → |3i. We

deﬁne the single-photon detunings as ∆1 = ω3 − ω1 − ωp and ∆2 = ω3 − ω2 − ωc, and

set ~ = 1 in the following sections. Note that positive ∆1,2 presents red detuning. 3 1 γ 2 ω p ωc

γ 1 g 2

Figure A.1: ωp is the angular frequency of the probe beam and ωc is the angular frequency of the control beam. γ is the excited-state decay rate. γg is the decoherence term between states |1i and |2i. Detunings are ∆1 = ω3−ω1−ωp and ∆2 = ω3−ω2−ωc.

As discussed in Chapter A.1, the total Hamiltonian H(t) is the sum of the bare atom system and the atom-light interaction. Similar to Equation (A.2), the time- dependent Hamiltonian of this system can be written as   ω1 0 −Ωp cos(ωpt)   H(t) =  0 ω2 −Ωc cos(ωct)  (A.17) −Ωp cos(ωpt) −Ωc cos(ωct) ω3 where Ωc,p are the Rabi frequencies of the control and probe beams, respectively. The relative phase between these ﬁelds has set to zero, without loss of generality. Based on the methods shown in Chapter A.1, we use Equations (A.3) and (A.4) to obtain  0    c (t) ω1c1(t) − Ωpc3(t) cos(ωpt) dψ(t) 1 =  c0 (t)  = (−i)  ω c (t) − Ω c (t) cos(ω t)  dt  2   2 2 c 3 c  0 c3(t) ω3c3(t) − Ωcc2(t) cos(ωct) − Ωpc1(t) cos(ωpt) (A.18) 160

Also from Equations (A.7), (A.8), and (A.11), we get

 itu1 itu1 0  ie u1c1(t) + e c1(t) dψI (t) =  ieitu2 u c (t) + eitu2 c0 (t)  (A.19) dt  2 2 2  itu3 itu3 0 ie u3c3(t) + e c3(t)

0 0 0 After plugging Equation (A.18) into Equation (A.19) to replace c1(t), c2(t) and c3(t) in Equation (A.19), it becomes

 iu1t  −e [(u1 − ω1)c1(t) + Ωpc3(t) cos(ωpt)] dψI (t) = (−i)  −eiu2t[(u − ω )c (t) + Ω c (t) cos(ω t)]  dt  2 2 2 c 3 c  iu3t −e [(u3 − ω3)c3(t) + Ωcc2(t) cos(ωct) + Ωpc1(t) cos(ωpt)]

it(u −u ) iu t ω1 − u1 0 −Ωp cos(ωpt)e 1 3 c1(t)e 1 it(u −u ) iu t =(−i) 0 ω2 − u2 −Ωc cos(ωct)e 2 3 c2(t)e 2 −it(u −u ) −it(u −u ) iu t −Ωp cos(ωpt)e 1 3 −Ωc cos(ωct)e 2 3 ω3 − u3 c3(t)e 3

=(−i)Heff|ψI (t)i (A.20) Comparing Equation (A.20) with Equation (A.8), we find the form of the effective

Hamiltonian Heﬀ. Moreover, if we let

u1 = ω1

u2 = ω1 + ωp − ωc

u3 = ω1 + ωp and define the two-photon detuning δ as the difference between the single-photon detunings (δ = ∆1 − ∆2), we can rewrite Heff as

 Ωp −i2ωpt  0 0 − 2 (1 + e ) H =  0 δ − Ωc (1 + e−i2ωct)  (A.21) eﬀ  2  Ωp i2ωpt Ωc i2ωct − 2 (1 + e ) − 2 (1 + e ) ∆1 by using cos(ωt) = (eiωt + e−iωt)/2. Equation (A.21) does not have any analytic solutions. However, as long as the detuning is small and 2ωp,c are very large compared to the detuning, we can use the rotating wave approximation [23] to ignore the high frequency term (e±2ωp,ct → 0). We can do this because on average the fast oscillation makes almost no contribution to the time evolution. Finally, after applying the dipole approximation and the rotating wave approximation within the interaction picture, the eﬀective Hamiltonian in the rotating frame 161

of a three-level atom-ﬁeld system is derived:

 Ωp  0 0 − 2 H =  0 δ − Ωc  (A.22) eﬀ  2  Ωp Ωc − 2 − 2 ∆1 The same approach can be applied to obtain the eﬀective Hamiltonian of the multilevel systems, as demonstrated in Chapter A.3.

A.2.2 Steady-state solutions

In order to understand the dynamics of atom-light mechanisms, such as absorption and the refractive index, we need to solve Equation (A.15) to have steady-state solutions and obtain the ensemble average by using Equation (A.12). This is diﬃcult to achieve directly because both the operator and wave functions contain time variables. Fortunately, it is more convenient to use Equation (A.16) to transform the time base as discussed in Chapter A.1. The corresponding terms in Equation (A.16) will be discussed as follows.

The density matrix ρI (t) = |ψI (t)ihψI (t)|, in the same system as shown in Fig- ure A.1, can be written as   ρ11 ρ12 ρ13   ρI (t) =  ρ21 ρ22 ρ23  (A.23) ρ31 ρ32 ρ33 where ρij is the density matrix elements [67]. Since the system has a spontaneous decay rate γ in the excited state |3i, the relaxation matrix Γ of this system is written as  0 0 0    Γ =  0 0 0  (A.24) 0 0 γ and the repopulation matrix Λ is written as   −γρ33/2 γgρ12 0   Λ =  γgρ21 −γρ33/2 0  (A.25) 0 0 0

1 to ensure that the diagonal terms of the matrix elements of 2 ΓρI (t) + ρI (t)Γ +Λ is summed up to be one. In other words, the population is preserved. 162

By inserting Equations (A.22), (A.23), (A.24), and (A.25) into Equation (A.16), a series of master equations is obtained: γρ iρ Ω iρ Ω ρ˙ = 33 − 13 p + 31 p 11 2 2 2 γ ρ iρ Ω iρ Ω ρ˙ = − g 12 + i∆ ρ − i∆ ρ − 13 c + 32 p 12 2 1 12 2 12 2 2 γρ iρ Ω iρ Ω iρ Ω ρ˙ = − 13 + i∆ ρ − 12 c − 11 p + 33 p 13 2 1 13 2 2 2 γ ρ iρ Ω iρ Ω ρ˙ = − g 21 − i∆ ρ + i∆ ρ + 31 c − 23 p 21 2 1 21 2 21 2 2 γρ iρ Ω iρ Ω ρ˙ = 33 − 23 c + 32 c (A.26) 22 2 2 2 γρ iρ Ω iρ Ω iρ Ω ρ˙ = − 23 + i∆ ρ − 22 c + 33 c − 21 p 23 2 2 23 2 2 2 γρ iρ Ω iρ Ω iρ Ω ρ˙ = − 31 − i∆ ρ + 21 c + 11 p − 33 p 31 2 1 31 2 2 2 γρ iρ Ω iρ Ω iρ Ω ρ˙ = − 32 − i∆ ρ + 22 c − 33 c + 12 p 32 2 2 32 2 2 2 iρ Ω iρ Ω iρ Ω iρ Ω ρ˙ = −γρ + 23 c − 32 c + 13 p − 31 p 33 33 2 2 2 2 Equation (A.26) describes the time behaviors of an ensemble of three-level atoms interacting with two light ﬁelds including the excited-state relaxation and decoherence. Properties like absorption and refractive index are usually characterized when the system reaches equilibrium—being constant with time. In this case, the steady- state solution can be applied to ﬁnd these density matrix elements. To keep the population conserved, the sum of ρ11, ρ22 and ρ33 needs to add up to one. Finally, the density matrix elements are solved by lettingρ ˙ij(t) = 0 and ρ11(t)+ρ22(t)+ρ33(t) = 1.

A.2.3 Absorption and the index of refraction

Before performing a simulation with the solved density matrix elements, let us review the absorption coeﬃcient and the refractive index from References [28, 225]. Considering a plane wave traveling in a medium, its complex form is written as

E˜(z, t) = Ee˜ ikz˜ e−iωt = Ee˜ i(k+iα/2)ze−iωt = Ee˜ ikze−αz/2e−iωt (A.27) where E˜ is the amplitude and z is the distance. k˜ = k+iα/2 is the complex wavenumber, and α is known as the absorption coeﬃcient [28]. The intensity of this electric ﬁeld is written as I = |E˜(z, t)|2 = |E˜|2e−αz = |E˜|2e−OD. (A.28) 163

Equation (A.28) describes the amount of light attenuated when passing through the medium. The amount of attenuation is characterized by either the absorption coefficient α or the optical density OD. From the relation between them, the absorption coefficient is defined as the optical density per unit length: OD α = (A.29) z Since the wave velocity v = ω/k = c/n, the index of refraction n is c n = k (A.30) ω

According to Reference [225], the complex wavenumber k˜ has a relationship with the complex dielectric constant ˜r as ω p ω k˜ = ˜ = p1 +χ ˜ (A.31) c r c e

whereχ ˜e is the complex susceptibility and ˜ = 0˜r = 0(1 +χ ˜e) is the complex

permittivity. 0 is the electric permittivity in vacuum. Sinceχ ˜e is usually small for √ gases, Equation (A.31) can be approximated by 1 +χ ˜e ≈ 1 +χ ˜e/2 and rewritten as

ω 1 ω 1 1 ω k˜ = 1 + χ˜ = 1 + Re[˜χ ] + Im[˜χ ] (A.32) c 2 e c 2 e 2 c e

where Re[˜χe] indicates the real part ofχ ˜e and Im[˜χe] represents the imaginary part ˜ ofχ ˜e. Comparing Equation (A.32) with k = k + iα/2 and Equation (A.30), we have the following results: 1 index of refraction n = 1 + Re[˜χ ] (A.33) 2 e ω absorption coefficient α = Im[˜χ ] (A.34) c e In the case of the probe field in Figure A.1, the atomic polarization P with N atoms in a volume V induced by the optical fields can be written as

N X hµii N P = = trace(ρµ) (A.35) V V i by using the relation in Equation (A.12). With Equation (A.35), the linear suscepti- bilityχ ˜e [67, 72] can be deﬁned in a way that N P = χ˜ E˜ (z, t) = ρ µ (A.36) 0 e p V 31 31 164

Based on the discussion above and considering the probe beam with µ31Ep = ~Ωp, we obtain the following equations: 2 |µ31| N Re[˜χe] = Re[ρ31] (A.37) ~Ωp 0V 2 |µ31| N Im[˜χe] = Im[ρ31]. (A.38) ~Ωp 0V

The permittivity is described by the density matrix element ρ31 and it can be solved from Equation (A.26). After plugging Equations (A.37) and (A.38) into Equa- tions (A.33) and (A.34) respectively, the index of refraction and the absorption coef-

ficient experienced by the probe beam is determined by the density matrix ρ31 and the Rabi frequency of the field with some constants: 1 |µ |2 N Re[ρ ] index of refraction n = 1 + 31 31 (A.39) 2 ~ 0V Ωp ω |µ |2 N Im[ρ ] absorption coefficient α = 31 31 (A.40) c ~ 0V Ωp A.2.4 Simulations

The simulation is based on the same system shown in Figure A.1. From the results in Equation (A.39) and Equation (A.40), the absorption of the probe light is proportional

to Im[ρ31]/Ωp, and the corresponding refractive index is related to Re[ρ31]/Ωp for a

given ensemble with a ﬁxed size. The quantity of the density matrix element ρ31 can be found from the the steady-state solution in Equation (A.26). The steady-state

solution of ρ31 is a function of the laser frequency detuning ∆. We let ∆1 = −∆ in order to deﬁne the sign of detunings: ∆ > 0 is a blue frequency detuning; ∆ < 0 is a red frequency detuning. In the following discussion, the properties of a three-level

system are simulated by using Ωp = 0.01γ. The decoherence, frequency detunings, and the Rabi frequencies are in the units of the excited-state decay rate γ. We analyze the properties of this Λ-type EIT system in three cases. In each case,

the absorption coefficient α versus the frequency detuning ∆ is shown by Im[ρ31]/Ωp 2 2 in the unit of ω |µ31| N since the factor ω |µ31| N is a composite of constants. Com- c ~ 0V c ~ 0V bining Equation (A.29) and Equation (A.40), we write down the optical density in the following forms: ω |µ |2 N Im[ρ ] OD = 31 31 z (A.41) c ~ 0V Ωp and 2 ω |µ31| N OD0 = z (A.42) c ~ 0V 165 where OD0 is the optical density in the absence of the EIT effect. The ratio of OD to OD0 is OD Im[ρ ] = 31 (A.43) OD0 Ωp which indicates that OD equals to OD0 when Im[ρ31]/Ωp = 1. As for the index of refraction n, we plot Re[ρ31]/Ωp as a function of the frequency detuning ∆ to represent 2 (n − 1) in the unit of ω |µ31| N , according to Equation (A.39). c ~ 0V The graphs on the left-hand side of Figure A.2, Figure A.3 and Figure A.4 indicates the absorption coefficient versus probe frequency detuning under different conditions. The graphs on the right-hand side of Figure A.2, Figure A.3 and Figure A.4 shows (n − 1) versus ∆ which is related to the index of refraction.

Case 1

In the case of a three-level system with no ground-state decoherence and the control

field is on resonance with atoms (γg = 0 and ∆2 = 0), the simulation results are plotted in Figure A.2. We notice that a transparency profile appears on the absorption curve. Especially, there is a 100% transparency of the probe light when it is on resonance with atoms, which is different than a regular absorption curve. (See Figure 1.2.) In addition, a significant change on the dispersion curve can be seen. These features are due to the EIT effect. The nonlinear behaviors of the EIT medium depend strongly on the Rabi frequency of the control beam Ωc. In the example of Figure A.2, a strong control beam gives a wide transparency window width but a less steep slope on the dispersion curve near the atomic resonance frequency. The on-resonance results from the calculation match the expectation of a three-level Λ- type EIT system from Chapter 3.6.1. Because the response of the EIT medium can be greatly changed and controlled by the control beam power, it is a powerful way to alter the absorption coefficient and the index of refraction.

Case 2

In the case of a non-zero decoherence rate (γg 6= 0), the transparency window is not perfect anymore and the derivative of Re[ρ31]/Ωp around the atomic resonance becomes smaller, as shown in Figure A.3. In this analysis, the presence of the decoherence term obviously aﬀects the performance of EIT. As a result, a system with long coherence times is preferable when performing the EIT experiment, such as the 166 1.0 0.4 p p 0.8 W W

0.2 D D 0.6 31 31 0.0 Ρ Ρ @ 0.4 @ -0.2 Im 0.2 Re -0.4 0.0 -2 -1 0 1 2 -2 -1 0 1 2 D Γ D Γ

1.0 @ D 0.4

p @ D p 0.8 W W

0.2 D D 0.6 31 31 0.0 Ρ Ρ @ 0.4 @ -0.2 Im 0.2 Re -0.4 0.0 -2 -1 0 1 2 -2 -1 0 1 2 D Γ D Γ

Figure A.2: EIT performance vs. coupling strength Ωc. Common parameters: @ D @ D ∆2 = γg = 0. The upper graphs are with Ωc = γ; the bottom graphs hold Ωc = 0.2γ. ytterbium atoms. For the same amount of decoherence, the proﬁle has a better transparency when the control beam power is stronger. This indicates that even though a decoherence mechanism exists in the system, the transparency can be near perfect as long as the control beam power is very strong compared to the decoherence

(Ωc γg).

Case 3

Instead of having any decoherence, the frequency of the control beam is detuned with

∆2 = 0.4γ. The simulated result is displayed in Figure A.4. The perfect transparency shows up only when it satisﬁes two-photon resonance (∆1 = ∆2 = −∆ = 0.4γ)[73].

Note that the positive ∆1,2 represents the red frequency detuning (∆ < 0). In general, the transparency can exist as long as the frequency difference between two light fields matches the energy difference of those two lower states. In other words, when there is no two-photon detuning (δ = ∆1 − ∆2 = 0). 167 1.0 0.4 p p 0.8 W W

0.2 D D 0.6 31 31 0.0 Ρ Ρ @ 0.4 @ -0.2 Im 0.2 Re -0.4 0.0 -2 -1 0 1 2 -2 -1 0 1 2 D Γ D Γ

1.0 @ D 0.4

p @ D p 0.8 W W

0.2 D D 0.6 31 31 0.0 Ρ Ρ @ 0.4 @ -0.2 Im 0.2 Re -0.4 0.0 -2 -1 0 1 2 -2 -1 0 1 2 D Γ D Γ

Figure A.3: EIT performance vs. decoherence γg. Common parameters: ∆2 = 0 and @ D @ D γg = 0.1γ. The upper graphs are with Ωc = γ; the bottom graphs hold Ωc = 0.2γ.

A.3 Leak absorption of N-type EIT system

A Λ -type EIT system, like Figure A.1, forms a dark state resulting from a complete cancellation of the excitation probabilities due to the destructive quantum interference. A detailed calculation has been developed in Chapters A.1 and A.2. On the other hand, a “N-type” EIT system does not have a true dark state because light fields can still excite atoms into the excited state. For example, consider the four-level system in Figure A.5 (a). The optical field can excite atoms through the transition |2i → |4i. The excitation population results in a “leak” absorption on the probe beam. In addition, the probe field also pump atoms out of the coherent superposition state. This “leak” absorption is a decoherent mechanism which reduces the EIT effect and the transparency window is therefore no longer perfect. Since 173Yb (F = 5/2) has a N-type level scheme, it is important to understand how the leak affects the EIT performance and how to reduce its influence. In this section, we will estimate the leak absorption with four different level structures. The leak absorption is found related to the atomic level schemes and the relative intensity ratio of the fields. We start with a four-level system—the simplest N-type EIT 168 1.0 0.4 p p 0.8 W W

0.2 D D 0.6 31 31 0.0 Ρ Ρ @ 0.4 @ -0.2 Im 0.2 Re -0.4 0.0 -2 -1 0 1 2 -2 -1 0 1 2 D Γ D Γ

1.0 @ D 0.4

p @ D p 0.8 W W

0.2 D D 0.6 31 31 0.0 Ρ Ρ @ 0.4 @ -0.2 Im 0.2 Re -0.4 0.0 -2 -1 0 1 2 -2 -1 0 1 2 D Γ D Γ

Figure A.4: EIT performance vs. detuning. Common parameters: ∆2 = 0.4γ, @ D @ D γg = 0. The upper graphs are with Ωc = γ; the bottom graphs hold Ωc = 0.2γ. system—and use a six-level system as a comparison to show the eﬀect of a multi- chain EIT. The 8-level and 12-level systems show that the atoms are scattered by the σ+-polarized probe beam and the σ−-polarized control beam. We use a 12-level system to demonstrate the case of 173Yb (F = 5/2) and study an 8-level system as a comparison.

A.3.1 Model

Using the same method discussed in Chapters A.1 and A.2, we extend the three-level system to the case of multilevel atoms. The level scheme diagrams of multilevel atoms are shown in Figure A.5. We model all the excited states as having the same energy eigenvalues ~ω0 and excited decay rates γ. All the lower energy states are degenerate and have extremely long lifetimes. Since the developments of the calculation are the same as discussed above, we only list the corresponding eﬀective Hamiltonians in the following equations. Hn with the subscript n = 4, 6, 8, 12 is the eﬀective Hamiltonian for the four, six, eight, and 169

3 4 4 5 6 γ 1 2 1 Ω Ω p c p c p

1 2 2 3 1

(a) 4-level (b) 6-level

5 6 7 8 7 8 9 10 11 12

Ωc Ω p

Ω p Ωc

2 3 4 1 1 2 3 4 5 6

Figure A.5: N-type EIT systems in multilevel atoms. The dash line shows the spontaneous decay. twelve-level system, respectively.

 Ωp  0 0 − 2 0  Ωc Ωp   0 ∆1 − ∆2 − 2 − 2  H4 =   (A.44)  − Ωp − Ωc ∆ 0   2 2 1  Ωp 0 − 2 0 2∆1 − ∆2

 Ωp  0 0 0 − 2 0 0  Ωc Ωp   0 ∆1 − ∆2 0 − − 0   2 2   0 0 2 (∆ − ∆ ) − Ωc − Ωc − Ωp  H =  1 2 2 2 2  6  Ωp Ωc   − − 0 ∆1 0 0   2 2   0 − Ωp − Ωc 0 2 (∆ − ∆ ) 0   2 2 1 2  Ωp 0 0 − 2 0 0 3∆1 − 2∆2 (A.45) 170

 Ωp  0 0 0 0 0 − 2 0 0  Ωc Ωp   0 0 0 0 − 0 − 0   2 2   0 0 ∆ − ∆ 0 0 − Ωc 0 − Ωp   1 2 2 2   Ωc   0 0 0 ∆1 − ∆2 0 0 − 2 0  H8 =    0 − Ωc 0 0 ∆ 0 0 0   2 2   Ωp Ωc   − 0 − 0 0 ∆1 0 0   2 2   0 − Ωp 0 − Ωc 0 0 ∆ 0   2 2 1  Ωp 0 0 − 2 0 0 0 0 2∆1 − ∆2 (A.46)

 Ωp  0 0 0 0 0 0 0 − 2 0 0 0 0  0 0 0 0 0 0 − Ωc 0 − Ωp 0 0 0   2 2   Ωc Ωp   0 0 ∆1 − ∆2 0 0 0 0 − 0 − 0 0   2 2   0 0 0 ∆ − ∆ 0 0 0 0 − Ωc 0 − Ωp 0   1 2 2 2   Ωc Ωp   0 0 0 0 2 (∆1 − ∆2) 0 0 0 0 − 0 −   2 2   0 0 0 0 0 2 (∆ − ∆ ) 0 0 0 0 − Ωc 0   1 2 2  H12 =  Ωc   0 − 0 0 0 0 ∆2 0 0 0 0 0   2   − Ωp 0 − Ωc 0 0 0 0 ∆ 0 0 0 0   2 2 1   Ωp Ωc   0 − 0 − 0 0 0 0 ∆1 0 0 0   2 2   0 0 − Ωp 0 − Ωc 0 0 0 0 2∆ − ∆ 0 0   2 2 1 2   Ωp Ωc   0 0 0 − 2 0 − 2 0 0 0 0 2∆1 − ∆2 0  Ωp 0 0 0 0 − 2 0 0 0 0 0 0 3∆1 − 2∆2 (A.47) By inserting these eﬀective Hamiltonians into Equation (A.16) with their corresponding relaxation and repopulation matrices, the steady-state solutions of their density matrix elements can be found.

A.3.2 Results

The absorption coeﬃcient α of the probe beam in the multilevel N-type EIT system illustrated in Figure A.5 can be presented by the density matrix elements:

N 2 OD0 X α = Im[ρi,i+ N ]/Ωp, for 4- and 6-level (A.48) z 2 i=1 and N 2 −1 OD0 X α = Im[ρi,i+ N +1]/Ωp, for 8- and 12-level (A.49) z 2 i=1

We model the simulation with both ﬁelds on resonance (∆1 = ∆2 = 0) and do not include either the Doppler eﬀect or the collisional broadening. There is also no

decoherence term (γg = 0). 171

The density matrix elements are solved from the steady-state solutions in the different level structures. We sum over the corresponding Im[ρ]/Ωp to show the absorption coefficient α of the probe beam and plot the numerical calculation data of the different level structures on the left axis in Figure A.6. The bottom axis is the relative intensity Ic/Ip of the control beam and the probe beam. For simplicity, the transition strengths are assumed to have the same transition probability in these analyses. This assumption should not affect the overall functional forms. Even if the different transition strengths are included, it only has a slight difference on the value of the absorption coefficient. In the case of a Λ-type EIT as shown in Figure A.1, the leak absorption is zero and independent of Ic/Ip because it has a true dark state. However, for the N-type EIT schemes, the leak absorption is dependent on the intensity ratio.

Ic/Ip < 1

The 4-level and 6-level atoms with bright states have unitary absorption in the limit that Ic/Ip is small. (The unitary absorption indicates that OD equals to OD0.)

The absorption coefficient of 8-level and 12-level systems declines as Ic/Ip becomes small. We fit the curves to a power function to find their functional form. In the small ratio limit (Ic/Ip → 0), the absorption coefficient of the 8-level system is proportional 2 to (Ic/Ip) , while the absorption coefficient of the 12-level system is proportional to 3 (Ic/Ip) .

Ic/Ip > 1

Because the scheme of 4-level and 8-level systems are identical in the limit of the large intensity ratio (Ic/Ip → large), their absorption curves are overlapped. The same reason is applied for the curves of the 6-level and 12-level systems. A ﬁt of the −1 power function gives a slope of the 8-level system to be proportional to (Ic/Ip) and −2 a slope of the 12-level system to be proportional to (Ic/Ip) when Ic/Ip is large.

Discussion

In the 8-level system, it has a combination of a Λ-type EIT scheme and a leak; in the 12-level system, the interference is from a M-type EIT scheme with a leak. From the different curves of the 8-level systems and 12-level system, we find that the absorption coefficient decays faster in the 12-level system than in the 8-level system in both limits 172

of Ic/Ip → large and Ic/Ip → small. It suggests that a multi-chain EIT scheme helps reduce the leak absorption and makes the system closer to a dark state. Although the N-type system has a bright state, the absorption can be reduced by using a large ratio of the control beam to the probe beam. When the control beam intensity is 10 times stronger than the probe beam, it leads to a leak absorption coeﬃcient of 3% (OD = 0.03 OD0) in the 12-level system.

To reduce the leak absorption in the Yb EIT experiment, a large ratio of Ic/Ip is preferred. However, a large intensity ratio may make it diﬃcult in the polarization detection. Based on the signal-to-noise ratio of our system, an intensity ratio between 10 to 100 should give a reasonable “nearly-dark” state.

0 10

-2 10

-4 10

-6 10

-8 10

-10 4-level 10 6-level 8-level

Absorption coefficient [a.u.] coefficient Absorption -12 10 12-level

-14 10

-8 -6 -4 -2 0 2 4 6 8 10 10 10 10 10 10 10 10 10 Ιc / Ιp

Figure A.6: The simulation of the N-type multilevel atoms. Parameters: Ωp = 0.1γ and ∆1 = ∆2 = γg = 0. All axes uses the logarithm scale. (a.u. = arbitrary unit.) 173

Appendix B

Absorption cross-section

B.1 Beer-Lambert law

Consider a flux of N particles passing through a small distance dz in a medium containing n atoms per unit volume. The probability P of the particles colliding with atoms in the distance dz is given by dz P = = nσdz (B.1) l where l is the mean-free-path between particles and atoms; σ is defined as the cross- section [4, 22]. If a collision causes loss, then the fractional loss dN of particles hitting the medium within an infinitesimal distance dz is

dN = −Nnσdz (B.2)

After integrating Equation (B.2), it becomes

−nσz N = N0e (B.3) where N0 is the number of particles before collisions. In the case of a beam of photons passing through an atomic cloud, the beam intensity decreasing with distance due to absorption by atoms is given by

−nσz I = I0e (B.4) where σ is the absorption cross-section and z is the length of the atomic cloud1. The relation between the absorption cross-section, optical density, and the transmittance

1 −1 A quantity of l0 = (nσ) is commonly deﬁned as the absorption length [117]. 174

can be written as I T = = e−nσz = e−OD (B.5) I0 and OD = nσz (B.6)

Note that if the absorption A = 1 − T is small enough (A 1), the OD is approximately proportional to the absorption:

OD = − ln(1 − A) ≈ A (B.7)

B.2 Absorption cross-section

Considering a group of unpolarized atoms with no broadening, the absorption cross- section for a two-level closed transition can be written as

2 λ ge σ0 = (B.8) 2π gg

where ge and gg are the degeneracies of the excited and ground states, respectively [117]. In this case, it only depends on the wavelength λ of the optical ﬁeld and the degeneracy of energy states. For a thermal gas, the transition line broadens because of the Doppler eﬀect. The

Doppler-broadening eﬀect half width νD at 1/e maximum at temperature T can be written as s ν 2 k [erg/K] T [K] ν [Hz] = 0 [cm−1] B (B.9) D c M[g]

where ν0 is the resonance frequency and M is the atomic mass. In the case that νD is much greater than the radiation decay rate A, the absorption cross-section for a Doppler-broadened line of the unpolarized atoms can be written as

√ 2 −1 2 π σ0 [cm ] A [s ] σD [cm ] = (B.10) 2 νD [Hz] 2π For real atoms, the excited state may decay to more than one energy state. When

it decays to diﬀerent states, σD in Equation (B.10) needs to be multiplied by its

branching ratio. The branching ratio is discussed in AppendixE. Once the σD is known, the atomic density can be measured by Equation (B.6). 175

Appendix C

Diﬀusion models

Reference [22] provides detailed models of atomic diffusion as a function of time and space. In order to understand the behaviors of atoms diffusing through helium gas, two suitable atomic density models from Reference [22] are listed based on the geometry of the copper cell which is approximated as a cylinder or a cube. Since only the lowest mode is left at later times of diffusion, the following cases consider only the lowest modes. Note that the discussion in this section is aimed at a mixture of two gases inside the copper cell. In this example, Yb atoms diffuse inside a helium buffer gas where the atomic density of Yb atoms is small compared to the atomic density of helium atoms. The following equations, n(x, y, z, t) and n(r, z, t), describe the density distribution of Yb atoms as a function of time and space in the cube and cylinder model, respectively. The parameter np is a constant which indicates a peak atomic density.

• A cubic cell with a length L πx πy πz n(x, y, z, t) = n cos( ) cos( ) cos( )e−t/τD (C.1) p L L L

the time constant τD is presented by 1 3π2 = D 2 (C.2) τD L • A cylinder cell of radius r and height H πz n(r, z, t) = n cos e−t/τD (C.3) p H with " # 1 2.40482 π 2 = D + (C.4) τD r H 176

The diffusion coefficient D in the helium buffer gas cell can be written as s 3 1 2πk T D = B (C.5) 16 nHeσd µ

1 where µ is the reduced mass of target atoms and helium gas and σd is the thermally- averaged diﬀusion cross-section, also called the thermally-averaged momentum transfer cross-section. From the above discussion, the diﬀusion cross-section is determined in these two cases by using Equation (C.5) together with Equations (C.2) and (C.4):

• σ in the cubic cell d √ 2 s 9π 2π kBT τD σd = 2 (C.6) 16L µ nHe

• σd in the cylinder cell √ s " 2 2# 3 2π kBT 2.4048 π τD σd = + (C.7) 16 µ r H nHe

A summary of the experimental results of σd is listed in Table C.1. They are determined from Equations (C.6) and (C.7) by measuring their temperature T , diﬀusion lifetime τD, and helium density nHe.

Table C.1: A summary of measured thermally-averaged cross-sections. The dominant source of error is due to approximations made in the modeling of diﬀusion in the cell geometry. 2 Mixture Temperature [K] σd [cm ] 173Yb-He 6 [75] (1.1 ± 0.4) × 10−14 69Ga-He 5 [41] (7.5 ± 2.0) × 10−15 115In-He 5 [41] (9.1 ± 2.5) × 10−15 48Ti-He 5.2 [45] (1.1 ± 0.3) × 10−14 TiO-He 5 [224] (1.5 ± 0.7) × 10−14

1µ = matommHe . matom+mHe 177

Appendix D

Rate coeﬃcient

Two quantities, the cross-section σ and the rate coefficient k, are used to describe collisions. In the case of atom-helium collisions, the rate that the atoms leave their energy states due to the inelastic collisions with helium gas can be described as dn 1 = − n (D.1) dt τ where n is the atomic density and 1/τ is the inelastic collision rate. The rate is related to the rate coefficient as 1 = kn (D.2) τ He and the rate coefficient is related to the cross-section σ as

k = σvr (D.3)

when σ is velocity independent. vr is the mean relative velocity, which is given by r 8kBT matom + mHe vr = (D.4) π matommHe

where T is the temperature of the gases, and matom,He are the masses of the atom and helium gas, respectively [22]. In general, for elastic collisions occurring in the limit of T → 0, the cross-section

σ tends to be a constant while the collisional energy (k = σvr → 0) approaches zero

[226, 227]. On the other hand, k is near a constant and σ is proportional to 1/vr in the case of the exothermic inelastic collisions. As a result, the cross-section is used for describing the diﬀusion behaviors and the rate coeﬃcient is used for inelastic collisions. 178

Appendix E

Transition strength

The coupling between the light field and an optical transition is usually described by the Rabi frequency, Ω = µgeE/~, where µge is the dipole matrix element and E is the magnitude of the electric field. For real atoms, different optical transitions have their corresponding µge. The magnitude of µge depends on the the orientation of the atomic dipole moment with respect to the polarization of light [228].

The Wigner-Eckart theorem separates µge into two parts, a reduced matrix element and an angular coefficient. The reduced matrix element is independent of the projection the angular momentum on the quantization axis, while the angular coefficient includes all the various quantum numbers of the coupled states [71, 228]. The angular coefficient is usually described by the Clebsch-Gordan coefficient or a 3j symbol1 depending on the type of coupling [71]. 2 In the case of hyperfine interaction, µge can be presented by two 6j symbols and

1 The 3j symbol is zero unless it satisﬁes that |J1 − J2| ≤ J ≤ J1 + J2 and m1 + m2 + m = 0. Wigner 3j symbol, written as 2 × 3 matrices in parentheses: 1 J1 J2 J J1−J2−m = (−1) √ hJ1m1J2m2|J, −mi m1 m2 m 2J + 1

2 Wigner 6j symbol, written as 2 × 3 matrices in curly brackets:

J1+J2+J3+J J3 JJa (−1) = p hJaJ3J|JbJ1Ji (E.1) J1 J2 Jb (2Ja + 1)(2Jb + 1) 179

one 3j symbol as described in Reference[228]:

1+L0+S+J+J0+I−m0 0 0 µge = e(−1) F hα L ||r||αLi × p(2J + 1)(2J 0 + 1)(2F + 1)(2F 0 + 1) ( )( ) ! (E.2) L0 J 0 S J 0 F 0 I F 1 F 0 × 0 JL 1 FJ 1 mF q −mF here, L is the orbital quantum number, S is the electron spin quantum number, J is the total electronic angular momentum quantum number, I is the nuclear spin

quantum number, F is the total angular momentum quantum number, mF is the projection of the total angular momentum F , e is the electronic charge and hα0L0||r||αLi is a reduced matrix element. The apostrophe symbol, 0, refers to the excited state, and otherwise indicates the ground state. q is related to the polarization of light: q = ±1 for the σ±-polarized light and q = 0 for the π-polarized light. Equation (E.2) 0 is zero if the relation mF + q = mF is not satisﬁed. According to the Fermi’s golden rule [117], the transition rate (the transition 2 probability per unit of time) is proportional to |µge| . For transitions with the same reduced matrix element hα0L0||r||αLi, we can compare their transition rates by their relative transition strengths [71]:

2 0 0 Transition rate ∝ |µge| ∝ (2J + 1)(2J + 1)(2F + 1)(2F + 1) ( )( ) ! 2 L0 J 0 S J 0 F 0 I F 1 F 0 (E.3) × 0 JL 1 FJ 1 mF q −mF The Wolfram Mathematica software [229] is used to compute the transition strengths and find the relative ratios for the specific transitions. The calculation results of different transitions are listed in AppendixF. The transition in Figure F.5 is an example of the calculation result: the numbers labeled in Figure F.5 are the relative + transition strengths. When sending the σ -polarized light into the atoms, the mF =

+1/2 → mF 0 = +3/2 transition has a transition probability three times greater than

the mF = −1/2 → mF 0 = +1/2 transition. The possibility of other transitions is zero. When sending the π-polarized light, the transition probability for both of the

mF = −1/2 → mF 0 = −1/2 transition and the mF = +1/2 → mF 0 = +1/2 transition are equal and for the others is zero.

If summing over all the transition strengths for diﬀerent mF levels, the branching ratio which indicates the decay probability from the excited state to the diﬀerent lower energy states can be found [71]. For example, the numbers labeled in Figure F.10(a) 180

2 0 give the branching ratios: the probability of the | S1/2,F = 1i state decaying to 2 o the | P3/2,F = 0, 1, and 2i states are 2/18, 5/18, and 5/18, respectively, and the 2 0 2 o probability of the | S1/2,F = 1i state decaying to the | P1/2,F = 1 and 2i states are 1/18 and 5/18, respectively. The overall decaying probability sums up to one.

• An example of the Mathematica program code in case of Figure F.5(a) is shown as follows:

In[168]:= Clear II II = 1 2; S = 0; L = 0; J = 0; F = 1 2; @ D L1 = 1; J1 = 1; F1 = 3 2; q = 1;

Table mf1, mf, Abs -1 ^ 1 + L1 + S + J + J1 + II + mf1 * Sqrt 2 * J + 1 * 2 * J1 + 1 * 2 * F + 1 * 2 * F1 + 1 * SixJSymbol L1, J1, S , J, L, 1 * @8SixJSymbol J1, F1, II , F, J, 1 * ThreeJSymbol F, mf , 1, q , F1, -mf1 ^2 , mf1, -3 2, 3 2 , mf, -1 2, 1 2 H @H L H L @H L H L H L H LD @8 < 8

C;1 1 ? 8 < ;3 3 ?G ClebschGordan::phy : ThreeJSymbol , , 1, 1 , , is not physical. 2 2 2 2

C;1 1? 8 < ; 3 1?G ClebschGordan::phy : ThreeJSymbol , - , 1, 1 , , is not physical. 2 2 2 2

General::stop : Further output of ClebschGordanC; ? 8 ::phy< ;will be?Gsuppressed during this calculation. 3 1 3 1 1 1 1 1 1 1 1 1 1 3 1 3 1 1 Out[171]= - , - , 0 , - , , 0 , - , - , 0 , - , , 0 , , - , , , , 0 , , - , 0 , , , 2 2 2 2 2 2 2 2 2 2 9 2 2 2 2 2 2 3

::: > : >> :: > : >> :: > : >> :: > : >>> 181

Appendix F

Atomic/Molecular information

F.1 Yb

The natural abundances of the seven stable atomic ytterbium isotopes are listed in Table F.1. The relative frequency shifts of the Yb absorption lines are listed in Table F.2. The transition strengths of the hyperﬁne transitions of 171Yb and 173Yb can be found in Equation (F.1) and Equation (F.2) which are calculated based on the discussion in AppendixE.

14 2 1 Table F.1: The ground state of Yb is 4f 6s S0. Data is from Reference [153].

Isotopes Abundance [%] I µ [µN ] 168Yb 0.13 0 170Yb 3.05 0 171Yb 14.3 1/2 0.4919 172Yb 21.9 0 173Yb 16.12 5/2 -0.6776 174Yb 31.8 0 176Yb 12.7 0

From the information above, we simulate the line broadening of the ytterbium spectra due to the Doppler effect, the natural line broadening and the pressure broadening. The line broadening induced by the Doppler effect is a Gaussian profile, while the broadening induced by the natural and pressure broadening is described by a Lorentzian function. As a result, we calculate the Yb spectrum by using the Voigt profile which is a convolution of the Gaussian and Lorentzian functions [116]. 182

Table F.2: The frequency oﬀset is zero at the 176Yb isotope. The relative frequency shift is referred by Reference [155]. Isotope Shift [MHz] 176Yb 173Yb (F = 5/2) 256 174Yb 509 173Yb (F = 3/2) 1025 172Yb 1042 173Yb (F = 7/2) 1097 171Yb (F = 3/2) 1341 171Yb (F = 1/2) 1663 170Yb 1701 168Yb 2396

Two examples of the simulated Yb spectra are shown in Figure F.1. First, we simulate a Yb spectrum at 6 K with a Doppler FWHM of 0.1 GHz and a natural line broadening of 0.03 GHz in Figure F.1(a). The individual transition lines and the sum of all these transition peaks can been seen. The overall sum of all peaks is shifted up to distinguish from the individual transitions. Second, the parameters in Figure F.1(b) are calculated for a cell temperature of 300 K which has a Doppler FWHM of 0.7 GHz and a Lorentzian FWHM of 0.03 GHz. Note that because of the multiple Yb isotopes, the whole spectrum at 300 K is not quite a Gaussian proﬁle.

F.1.1 Transition strengths of Yb lines

This part of the appendix lists the transition strengths of the Yb transitions and the calculation is based on the discussion in AppendixE. Each level structure of the Yb isotopes is shown with the diagrams of the σ+ and π transitions. The number labeled on each transition indicates the relative transition strength. The transition strength for σ− polarization is same as the σ+ transition by replacing the Zeeman sublevels with a minus sign. Diagrams of the 173Yb isotope are listed in Figure F.2, Figure F.3, and Figure F.4; diagrams of the 171Yb are displayed in Figure F.5 and Figure F.6. Level structures of other Yb isotopes with no nuclear spin are illustrated in Figure F.7. If summing over all the Zeeman sublevel transition probabilities for σ± and π tran- 183 sitions on diﬀerent hyperﬁne levels, we obtain a relative transition ratio of transitions 1 173 decaying to the | S0,F = 5/2i state of Yb as

|F = 5/2 → F 0 = 5/2i : |F = 5/2 → F 0 = 3/2i : |F = 5/2 → F 0 = 7/2i (F.1) = 3 : 2 : 4 while the similar approach for 171Yb as

|F = 1/2 → F 0 = 3/2i : |F = 1/2 → F 0 = 1/2i = 2 : 1 (F.2) 184

176 100 Yb 173 Yb (F'=5/2) 174 Yb 173 80 Yb (F'=3/2) 172 Yb 173 Yb (F'=7/2) 60 171 Yb (F'=3/2) 171 Yb (F'=1/2) 170 40 Yb

Height [a.u.] Height 168 Yb A sum of all peaks 20

0 -1 0 1 2 3 4 Frequency [GHz]

(a) Gaussian FWHM = 0.1 GHz; Lorentzian FWHM = 0.03 GHz.

176 Yb 100 173 Yb (F'=5/2) 174 Yb 173 80 Yb (F'=3/2) 172 Yb 173 Yb (F'=7/2) 171 60 Yb (F'=3/2) 171 Yb (F'=1/2) 170 Yb 40 168 Height [a.u.] Height Yb A sum of all peaks 20

0 -1 0 1 2 3 4 Frequency [GHz]

(b) Gaussian FWHM = 0.7 GHz; Lorentzian FWHM = 0.03 GHz.

Figure F.1: Simulated Yb spectra. The details are described in the text. 185

− 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2 − 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2

2 16 6 16 2 5 3 1 1 3 5 21 105 35 105 21 21 35 105 105 35 21

− 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2 − 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2

(a) σ-transition (b) π-transition

173 1 1 0 Figure F.2: Yb | S0,F = 5/2i → | P1,F = 5/2i

− 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2

2 2 1 1 4 2 2 4 9 15 15 45 45 15 15 45

− 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2 − 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2

(a) σ-transition (b) π-transition

173 1 1 0 Figure F.3: Yb | S0,F = 5/2i → | P1,F = 3/2i

− 7 / 2 − 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2 + 7 / 2 − 7 / 2 − 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2 + 7 / 2

1 1 2 10 5 1 2 10 4 4 10 2 63 21 21 63 21 3 21 63 21 21 63 21

− 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2 − 5/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 + 5/ 2 (a) σ-transition (b) π-transition

173 1 1 0 Figure F.4: Yb | S0,F = 5/2i → | P1,F = 7/2i 186

− 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2 − 3/ 2 − 1/ 2 + 1/ 2 + 3/ 2

1 3 2 2 9 9 9 9

− 1/ 2 + 1/ 2 − 1/ 2 + 1/ 2

(a) σ-transition (b) π-transition

171 1 1 0 Figure F.5: Yb | S0,F = 1/2i → | P1,F = 3/2i

− 1/ 2 + 1/ 2 − 1/ 2 + 1/ 2

2 1 1 9 9 9

− 1/ 2 + 1/ 2 − 1/ 2 + 1/ 2

(a) σ-transition (b) π-transition

171 1 1 0 Figure F.6: Yb | S0,F = 1/2i → | P1,F = 1/2i

− 1 0 + 1 − 1 0 + 1

1 1 3 3

0 0 (a) σ-transition (b) π-transition

1 1 0 Figure F.7: Yb (I = 0) isotopes | S0,F = 0i → | P1,F = 1i 187 F.2 Ga and In

2 2 Table F.3 summarizes the decaying possibility of the excited states 4s 5s S1/2 for gallium atoms; Table F.4 summarizes the decaying possibility of the excited states 2 2 5s 6s S1/2 for indium atoms.

Table F.3: The decaying possibilities of the 69Ga excited states. Transition wavenumber [cm−1] Same F , J Same J, otherF Other J F = 1 → F 0 = 1 24788.54 5.6 % 27.8 % 66.7 % F = 1 → F 0 = 2 24788.61 16.7 % 16.7 % 66.7 % F = 2 → F 0 = 1 24788.46 27.8 % 5.6 % 66.7 % F = 2 → F 0 = 2 24788.53 16.7 % 16.7 % 66.7 %

Table F.4: The decaying possibilities of the 115In excited states. Transition wavenumber [cm−1] Same F , J Same J, otherF Other J F = 4 → F 0 = 4 24373.01 8.8 % 24.4 % 66.7 % F = 4 → F 0 = 5 24373.29 20 % 13.3 % 66.7 % F = 5 → F 0 = 4 24372.63 24 % 8.8 % 66.7 % F = 5 → F 0 = 5 24372.91 13.3 % 20 % 66.7 %

Figure F.8 and Figure F.9 are the transition hyperﬁne structures for 69Ga and 115In (not to scale). Figure F.10 are the branching ratios of Ga and In. The labeled numbers are computed according to AppendixE. 188

F '= 2 0.80 GHz 4s2 5 s 2 S 1/ 2 2.14 GHz 1.34 GHz F '= 1

23962.34 cm-1 (417 nm)

24788.53 cm-1 (403.41 nm)

F "= 3 634.9 MHz F "= 2 4s2 4 p 2 P o 3/ 2 319.1 MHz F "= 1 128.3 MHz F "= 0

826.19 cm-1 F = 2 0.67 GHz 4s2 4 p 2 P o 1/ 2 2.68 GHz 1.67 GHz F = 1

Figure F.8: 69Ga (I = 3/2) transition hyperﬁne structure [153, 208]. 189

F '= 5 3.80 GHz 2 2 5s 6 s S1/ 2 8.43 GHz 4.63 GHz F '= 4

22160.36 cm-1 (451.26 nm)

24372.96 cm-1 (410.29 nm)

F "= 6 1753 MHz F "= 5 5s2 5 p 2 P o 3/ 2 1117 MHz = 669 MHz F " 4 F "= 3

2212.60 cm-1 F = 5 5.13 GHz 5s2 5 p 2 P o 1/ 2 11.40 GHz 6.27 GHz F = 4

Figure F.9: 115In (I = 9/2) transition hyperﬁne structure [209, 230]. 190

2 F '= 2 S1/ 2 F '= 1 5 3 42 18 90 90

2 5 15 18 18 90 F = 3 2P o F = 2 3/ 2 F =1 F = 0 5 15 18 90 1 15 18 90

2 o P F = 2 1/ 2 F = 1

(a) Gallium branching ratio

2 F '= 5 S1/ 2 F '= 4 22 36 135 495 99 195 35 33 495 495 135 135 F = 6 2P o 3/ 2 F = 5 F = 4 F = 3 33 66 135 495 12 99 135 495

2 o P F = 5 1/ 2 F = 4

(b) Indium branching ratio

Figure F.10: Branching ratios of Ga and In. 191 F.3 TiO

3 Table F.5: Vibrational energy Tv and the rotational constant Bv for the X ∆ and A 3Φ states of TiO, summarized from Reference [218]. State Constant v = 0 v = 1 3 X ∆1 Tv 0 1000.0197

Bv 0.5280 0.5250 3 X ∆2 Tv 96.7496 1096.7955

Bv 0.5341 0.5310 3 X ∆3 Tv 198.2874 1198.3429

Bv 0.5394 0.5363 3 A Φ2 Tv 14019.7356 14879.8417

Bv 0.5029 0.4997 3 A Φ3 Tv 14192.6109 15052.4118

Bv 0.5057 0.5025 3 A Φ4 Tv 14361.3523 15220.8859

Bv 0.5086 0.5054 192

Appendix G

Miscellaneous

G.1 AOM driver

A setup of the AOM driver is shown in Figure G.1. The voltage controlled oscillator (VCO) generates a sine wave into a coupler, and the frequency of the sine wave is tuned by a voltage Vtune. The coupler splits a small amount of signal into a counter

V tune VCO Attenuator

V Mixer V L R Coupler L X R Amplifier

V Counter x

Figure G.1: A schematic graph for the AOM driver setup. to measure the oscillating frequency which can be adjusted by a voltage Vtune, and the main signal goes to the local input of a frequency mixer. An external voltage Vx is mixed with the local input VL = A cos(ωLt). The output of the mixer VR can be described mathematically as follows:

VR = VL ∗ VX (G.1) = A cos(ωLt) ∗ VX 193

1 In this way, the output voltage VR can be controlled by the functional form of VX . The power of the radio wave after the frequency mixer is enhanced by an ampliﬁer. The output power can be varied by proper attenuators before going to the AOM.

Table G.1: List of parts for the AOM driver. Component Part number Manufacture VCO ZX95-200+ Mini-circuits Coupler ZX30-20-4-S+ Mini-circuits Counter Model 103 BK precision Mixer ZP-3LH+ Mini-circuits Ampliﬁer ZHL-3A Mini-circuits

G.2 Helmholtz coils

Around the cold cell, there are usually three pairs of Helmholtz coils which are UD, NS, and EW coils named by their coordinates. Helmholtz coils are deﬁned as a pair of coaxis coils with a separation equal to their radius, so that there will be a uniform magnetic ﬁeld in the center.

Table G.2: The old coils design. G/A = Gauss/Ampere. Coils Radius Turns/coil Cal. Exp. [inch] [G/A] [G/A] UD 7 5 0.25 NS 12 8 0.24 0.24 EW 12 20 0.59

In our experiment, two sets of coils have been used, and they are listed in Table G.2 and Table G.3. Coils are made of single conductor copper wires with American wire gauge 14. The old design serves for Yb, Ti, In, Ga, and TiO experiments. The new design of coils are for Yb experiments. The theoretical calculation is obtained by the BiotSavart software [231], and the experimental values listed in the table are

measured from the T2 time experiments.

1 For example, VX can be a voltage of a step-function form or an error-function form. 194

Table G.3: The new coils design. Resistance is measured with a pair of coils. Coils Radius Turns/coil Cal. Exp. [inch] [G/A] [G/A] UD 6.8 5 0.26 0.28 NS 11.3 10 0.32 0.33 EW 12.3 10 0.29

G.3 Gaussian pulse

2 − x If we deﬁne that a Gaussian proﬁle is y(x) = e ω2 and ω is the half width at 1/e maximum, then the conversions between time and frequency are listed as follows: √ 1. FWHM = 2 ln 2 ω

2. FWHM(2πf)× FWHM(t) = 8 ln 2

3. ω(2πf) × ω(t) = 2

The retrieved eﬃciency of stopped light

The area of a Gaussian pulse can be presented by:

Z ∞ 2 − x √ e ω2 dx = ω · π (G.2) −∞

If the τdelay ω in the stopped light experiments, then τ the fractional delayed area = delay√ (G.3) ω · π The eﬃciency of storing this Gaussian pulse can be written as τ τ max. eﬃciency = transmission · delay√ = 0.56 · transmission · delay (G.4) ω · π ω

With the assumption of τdelay ω, the maximum retrieved eﬃciency can be estimated by 0.56 · DBWT.

G.4 Saturation intensity

Saturation intensity Isat is a quantity to indicate how strong the light interacting with atoms. We consider a two-level atom, which does not have optical pumping. If the 195 intensity of the light field I is near or stronger than Isat, the field can strongly perturb the atomic system. For a two-level system, the population of the ground state and the excited state is near equal when I ∼ Isat. On the other hand, if I Isat, the atomic population is all on the ground state and the light field can be treated not to perturb the atomic population. According to Reference [4], the saturation intensity for a two-level atom has the following relations: π hc I = (G.5) sat 3 λ3τ and I 2Ω2 = 2 (G.6) Isat γ where γ = 1/τ and τ is the lifetime of the excited state. 196

Bibliography

[1] Frank Pobell. Matter and Methods at Low Temperatures. Springer, 2002.

[2] William D. Phillips. Nobel Lecture: Laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70(3), 721–741 Jul (1998).

[3] Wolfgang Ketterle and N.J. Van Druten. Evaporative Cooling of Trapped Atoms. Advances In Atomic, Molecular, and Optical Physics 37, 181–236 (1996).

[4] Christopher J. Foot. Atomic Physics (Oxford Master Series in Atomic, Optical and Laser Physics). Oxford University Press, USA, 2005.

[5] Harald F. Hess, Greg P. Kochanski, John M. Doyle, Naoto Masuhara, Daniel Kleppner, and Thomas J. Greytak. Magnetic trapping of spin-polarized atomic hydrogen. Phys. Rev. Lett. 59(6), 672–675 Aug (1987).

[6] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science 269, 198–201 (1995).

[7] Wolfgang Ketterle. Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser. Rev. Mod. Phys. 74(4), 1131–1151 Nov (2002).

[8] Wolfgang Ketterle, Kendall B. Davis, Michael A. Joﬀe, Alex Martin, and David E. Pritchard. High densities of cold atoms in a dark spontaneous-force optical trap. Phys. Rev. Lett. 70(15), 2253–2256 Apr (1993).

[9] Andrew J. Kerman, Vladan Vuleti´c,Cheng Chin, and Steven Chu. Beyond Optical Molasses: 3D Raman Sideband Cooling of Atomic Cesium to High Phase-Space Density. Phys. Rev. Lett. 84(3), 439–442 Jan (2000).

[10] Jack Ekin. Experimental Techniques: Cryostat Design, Material Properties and Superconductor Critical-Current Testing. Oxford University Press, USA, 2006.

[11] J. K. Messer and Frank C. De Lucia. Measurement of Pressure-Broadening Parameters for the CO-He System at 4 K. Phys. Rev. Lett. 53(27), 2555–2558 Dec (1984). 197

[12] Roman V. Krems, William C. Stwalley, and Bretislav Friedrich. Cold Molecules: Theory, Experiment, Applications. CRC Press, 2009. [13] Jonathan David Weinstein. Magnetic trapping of atomic chromium and molecular calcium monohydride. PhD thesis, Harvard University, 2001. [14] A. Hatakeyama, K. Enomoto, N. Sugimoto, and T. Yabuzaki. Atomic alkali- metal gas cells at liquid-helium temperatures: Loading by light-induced atom desorption. Phys. Rev. A 65(2), 022904 Jan (2002). [15] Dima Egorov, Thierry Lahaye, Wieland Schöllkopf, Bretislav Friedrich, and John M. Doyle. Buffer-gas cooling of atomic and molecular beams. Phys. Rev. A 66(4), 043401 Oct (2002). [16] S. E. Maxwell, N. Brahms, R. deCarvalho, D. R. Glenn, J. S. Helton, S. V. Nguyen, D. Patterson, J. Petricka, D. DeMille, and J. M. Doyle. High-Flux Beam Source for Cold, Slow Atoms or Molecules. Phys. Rev. Lett. 95(17), 173201 Oct (2005). [17] Mei-Ju Lu, Vijay Singh, and Jonathan D. Weinstein. Inelastic titanium- titanium collisions. Phys. Rev. A 79(5), 050702 May (2009). [18] Jinha Kim, Bretislav Friedrich, Daniel P. Katz, David Patterson, Jonathan D. Weinstein, Robert DeCarvalho, and John M. Doyle. Buffer-Gas Loading and Magnetic Trapping of Atomic Europium. Phys. Rev. Lett. 78(19), 3665–3668 May (1997). [19] Jonathan D Weinstein, Robert deCarvalho, Thierry Guillet, Bretislav Friedrich, and John M. Doyle. Magnetic trapping of calcium monohydride molecules at millikelvin temperatures. Nature 395, 148–150 (1998). [20] S. Charles Doret, Colin B. Connolly, Wolfgang Ketterle, and John M. Doyle. Buffer-Gas Cooled Bose-Einstein Condensate. Phys. Rev. Lett. 103(10), 103005 Sep (2009). [21] J. J. Sakurai. Modern Quantum Mechanics (Revised Edition). Addison Wesley, 1993. [22] J. B. Hasted. Physics of Atomic Collisions. Second edition. American Elsevier, 1972. [23] B.H. Bransden and C.J. Joachain. Physics of Atoms and Molecules (2nd Edi- tion). Benjamin Cummings, 2003. [24] Hideo Sakurai and Shigeki Kato. Theoretical Study of the Metal Oxidation Reaction Ti + O2 → TiO + O: Ab Initio Calculation of the Potential Energy Surface and Classical Trajectory Analysis. The Journal of Physical Chemistry A 106, 4350–4357 (2002). 198

[25] H. T. C. Stoof, J. M. V. A. Koelman, and B. J. Verhaar. Spin-exchange and dipole relaxation rates in atomic hydrogen: Rigorous and simpliﬁed calculations. Phys. Rev. B 38(7), 4688–4697 Sep (1988).

[26] Thad G. Walker and William Happer. Spin-exchange optical pumping of noble- gas nuclei. Rev. Mod. Phys. 69(2), 629–642 Apr (1997).

[27] Alexandr V. Avdeenkov and John L. Bohn. Ultracold collisions of oxygen molecules. Phys. Rev. A 64(5), 052703 Oct (2001).

[28] John David Jackson. Classical Electrodynamics Third Edition. Wiley, 1998.

[29] R. V. Krems, G. C. Groenenboom, and A. Dalgarno. Electronic interaction anisotropy between atoms in arbitrary angular momentum states. The Journal of Physical Chemistry A 108(41), 8941–8948 (2004).

[30] T Lahaye, C Menotti, L Santos, M Lewenstein, and T Pfau. The physics of dipolar bosonic quantum gases. Reports on Progress in Physics 72(12), 126401 (2009).

[31] Colin B. Connolly, Yat Shan Au, S. Charles Doret, Wolfgang Ketterle, and John M. Doyle. Large spin relaxation rates in trapped submerged-shell atoms. Phys. Rev. A 81(1), 010702 Jan (2010).

[32] Paul J. Leo, Eite Tiesinga, Paul S. Julienne, D. K. Walter, Steven Kadlecek, and Thad G. Walker. Elastic and Inelastic Collisions of Cold Spin-Polarized 133Cs Atoms. Phys. Rev. Lett. 81(7), 1389–1392 Aug (1998).

[33] Andrei Derevianko, Sergey G. Porsev, Svetlana Kotochigova, Eite Tiesinga, and Paul S. Julienne. Ultracold Collision Properties of Metastable Alkaline-Earth Atoms. Phys. Rev. Lett. 90(6), 063002 Feb (2003).

[34] Thad G. Walker, Joseph H. Thywissen, and William Happer. Spin-rotation interaction of alkali-metal–He-atom pairs. Phys. Rev. A 56(3), 2090–2094 Sep (1997).

[35] T. V. Tscherbul, P. Zhang, H. R. Sadeghpour, A. Dalgarno, N. Brahms, Y. S. Au, and J. M. Doyle. Collision-induced spin depolarization of alkali-metal atoms in cold 3He gas. Phys. Rev. A 78(6), 060703 Dec (2008).

[36] S. D. Le Picard, B. Bussery-Honvault, C. Rebrion-Rowe, P. Honvault, A. Canosa, J. M. Launay, and B. R. Rowe. Fine structure relaxation of aluminum by atomic argon between 30 and 300 K: An experimental and theoretical study. The Journal of Chemical Physics 108, 10319 (1998).

[37] S. D. Le Picard, P. Honvault, B. Bussery-Honvault, A. Canosa, S. Laub´e,J.- M. Launay, Bertrand Rowe, D. Chastaing, and I. R. Sims. Experimental and 199

theoretical study of intramultiplet transitions in collisions of C(3P ) and Si(3P ) with He. The Journal of Chemical Physics 117, 10109 (2002). [38] E. Abrahamsson, R. V. Krems, and A. Dalgarno. Fine-Structure Excitation of O I and C I by Impact with Atomic Hydrogen. The Astrophysical Journal 654, 1171 (2007). [39] Dirk Hansen and Andreas Hemmerich. Observation of Multichannel Collisions of Cold Metastable Calcium Atoms. Phys. Rev. Lett. 96(7), 073003 Feb (2006). [40] A. Yamaguchi, S. Uetake, D. Hashimoto, J. M. Doyle, and Y. Takahashi. In- elastic Collisions in Optically Trapped Ultracold Metastable Ytterbium. Phys. Rev. Lett. 101(23), 233002 Dec (2008). [41] T. V. Tscherbul, A. A. Buchachenko, A. Dalgarno, M.-J. Lu, and J. D. Wein- 2 stein. Suppression of Zeeman relaxation in cold collisions of P1/2 atoms. Phys. Rev. A 80(4), 040701 Oct (2009). [42] Cindy I. Hancox, S. Charles Doret, Matthew T. Hummon, Linjiao Luo, and John M. Doyle. Magnetic trapping of rare-earth atoms at millikelvin temperatures. Nature 431, 281–284 (2004). [43] Cindy I. Hancox, S. Charles Doret, Matthew T. Hummon, Roman V. Krems, and John M. Doyle. Suppression of Angular Momentum Transfer in Cold Colli- sions of Transition Metal Atoms in Ground States with Nonzero Orbital Angular Momentum. Phys. Rev. Lett. 94(1), 013201 Jan (2005). [44] R. V. Krems, J. K los, M. F. Rode, M. M. Szcz¸e´sniak,G. Cha lasiński,and A. Dalgarno. Suppression of Angular Forces in Collisions of Non-S-State Tran- sition Metal Atoms. Phys. Rev. Lett. 94(1), 013202 Jan (2005). [45] Mei-Ju Lu, Kyle S. Hardman, Jonathan D. Weinstein, and Bernard Zygel- man. Fine-structure-changing collisions in atomic titanium. Phys. Rev. A 77(6), 060701 Jun (2008). [46] Bernard Zygelman and Jonathan D. Weinstein. Theoretical and laboratory study of suppression effects in fine-structure-changing collisions of Ti with He. Phys. Rev. A 78(1), 012705 Jul (2008). [47] Katia M. Ferrière.The interstellar environment of our galaxy. Rev. Mod. Phys. 73(4), 1031–1066 Dec (2001).

+ 2 [48] D R Flower. Inelastic collisions between C ( P ) and H2. Journal of Physics B: Atomic, Molecular and Optical Physics 21(15), L451 (1988). [49] T. Mukaiyama, J. R. Abo-Shaeer, K. Xu, J. K. Chin, and W. Ketterle. Dis- sociation and Decay of Ultracold Sodium Molecules. Phys. Rev. Lett. 92(18), 180402 May (2004). 200

[50] Peter Staanum, Stephan D. Kraft, J¨orgLange, Roland Wester, and Matthias Weidem¨uller. Experimental Investigation of Ultracold Atom-Molecule Colli- sions. Phys. Rev. Lett. 96(2), 023201 Jan (2006).

[51] J. Doyle, B. Friedrich, R. V. Krems, and F. Masnou-Seeuws. Editorial: Quo vadis, cold molecules? The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics 31, 149–164 (2004). 10.1140/epjd/e2004-00151-x.

[52] Alisdair O. G. Wallis and Jeremy M. Hutson. Production of Ultracold NH Molecules by Sympathetic Cooling with Mg. Phys. Rev. Lett. 103(18), 183201 Oct (2009).

[53] Kenneth Maussang, Dima Egorov, Joel S. Helton, Scott V. Nguyen, and John M. Doyle. Zeeman Relaxation of CaF in Low-Temperature Collisions with Helium. Phys. Rev. Lett. 94(12), 123002 Mar (2005).

[54] R. V. Krems and A. Dalgarno. Quantum-mechanical theory of atom-molecule and molecular collisions in a magnetic ﬁeld: Spin depolarization. Journal of chemical physics 120, 2296 (2004).

[55] R. V. Krems, A. Dalgarno, N. Balakrishnan, and G. C. Groenenboom. Spin- ﬂipping transitions in 2Σ molecules induced by collisions with structureless atoms. Phys. Rev. A 67(6), 060703 Jun (2003).

[56] P.J. Linstrom and Eds W.G. Mallard. NIST Chemistry WebBook. National Institute of Standards and Technology, Gaithersburg MD, 20899, 2010.

[57] H. Cybulski, R. V. Krems, H. R. Sadeghpour, A. Dalgarno, J. Kos, G. C. Groenenboom, A. van der Avoird, D. Zgid, and G. Chaasi´nski. Interaction of NH with He: Potential energy surface, bound states, and collisional Zeeman relaxation. Journal of chemical physics 122, 094307 (2005).

[58] Wesley C. Campbell, Timur V. Tscherbul, Hsin-I Lu, Edem Tsikata, Roman V. Krems, and John M. Doyle. Mechanism of Collisional Spin Relaxation in 3Σ Molecules. Phys. Rev. Lett. 102(1), 013003 Jan (2009).

[59] Wesley C. Campbell, Edem Tsikata, Hsin-I Lu, Laurens D. van Buuren, and John M. Doyle. Magnetic Trapping and Zeeman Relaxation of NH (X 3Σ−). Phys. Rev. Lett. 98(21), 213001 May (2007).

[60] John M. Brown and Alan Carrington. Rotational Spectroscopy of Diatomic Molecules (Cambridge Molecular Science). Cambridge University Press, 2003.

[61] Manuel Lara, John L. Bohn, Daniel Potter, Pavel Sold´an,and Jeremy M. Hut- son. Ultracold Rb-OH Collisions and Prospects for Sympathetic Cooling. Phys. Rev. Lett. 97(18), 183201 Nov (2006). 201

[62] Joop J. Gilijamse, Steven Hoekstra, Sebastiaan Y. T. van de Meerakker, Ger- rit C. Groenenboom, and Gerard Meijer. Near-Threshold Inelastic Collisions Using Molecular Beams with a Tunable Velocity. Science 313, 1617 (2006).

[63] Brian C. Sawyer, Benjamin K. Stuhl, Dajun Wang, Mark Yeo, and Jun Ye. Molecular Beam Collisions with a Magnetically Trapped Target. Phys. Rev. Lett. 101(20), 203203 Nov (2008).

[64] Benjamin K. Stuhl, Brian C. Sawyer, Dajun Wang, and Jun Ye. Magneto- optical Trap for Polar Molecules. Phys. Rev. Lett. 101(24), 243002 Dec (2008).

[65] Edmund R. Meyer and John L. Bohn. Prospects for an electron electric-dipole moment search in metastable ThO and ThF+. Phys. Rev. A 78(1), 010502 Jul (2008).

[66] Stephen E. Harris. Electromagnetically Induced Transparency. Physics Today 50(7), 36–42 (1997).

[67] Marlan O. Scully and M. Suhail Zubairy. Quantum Optics. Cambridge Univer- sity Press, 1997.

[68] G. Alzetta, A. Gozzini, L. Moi, and G. Orriols. An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour. Il Nuovo Cimento B (1971-1996) 36, 5–20 (1976). 10.1007/BF02749417.

[69] S. H. Autler and C. H. Townes. Stark Eﬀect in Rapidly Varying Fields. Phys. Rev. 100(2), 703–722 Oct (1955).

[70] K. Bergmann, H. Theuer, and B. W. Shore. Coherent population transfer among quantum states of atoms and molecules. Rev. Mod. Phys. 70(3), 1003–1025 Jul (1998).

[71] Marcis Auzinsh, Dmitry Budker, and Simon Rochester. Optically Polarized Atoms: Understanding light-atom interactions. Oxford University Press, USA, 2010.

[72] Michael Fleischhauer, Atac Imamoglu, and Jonathan P. Marangos. Electromag- netically induced transparency: Optics in coherent media. Reviews of Modern Physics 77(2), 633 (2005).

[73] M. D. Lukin. Colloquium: Trapping and manipulating photon states in atomic ensembles. Reviews of Modern Physics 75, 457 (2003).

[74] H.-J. Briegel, W. D¨ur,J. I. Cirac, and P. Zoller. Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication. Phys. Rev. Lett. 81(26), 5932–5935 Dec (1998). 202

[75] Mei-Ju Lu and Jonathan D. Weinstein. Electromagetically induced transparency with nuclear spin. Opt. Lett. 35(5), 622–624 (2010).

[76] William K. Wootters and Wojciech H. Zurek. The no-cloning theorem. Physics Today 62(2), 76–77 (2009).

[77] Nicolas Sangouard, Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin. Quantum repeaters based on atomic ensembles and linear optics. arXiv:0906.2699 .

[78] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413 (2001).

[79] H. J. Kimble. The quantum internet. Nature 453, 1023–1030 (2008).

[80] Pieter Kok, W. J. Munro, Kae Nemoto, T. C. Ralph, Jonathan P. Dowling, and G. J. Milburn. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys. 79(1), 135–174 Jan (2007).

[81] C. W. Chou, S. V. Polyakov, A. Kuzmich, and H. J. Kimble. Single-Photon Generation from Stored Excitation in an Atomic Ensemble. Phys. Rev. Lett. 92, 213601 (2004).

[82] D. N. Matsukevich, T. Chaneliere, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich. Deterministic Single Photons via Conditional Quantum Evo- lution. Physical Review Letters 97(1), 013601 (2006).

[83] T. Fernholz, H. Krauter, K. Jensen, J. F. Sherson, A. S. Sørensen, and E. S. Polzik. Spin Squeezing of Atomic Ensembles via Nuclear-Electronic Spin En- tanglement. Phys. Rev. Lett. 101(7), 073601 Aug (2008).

[84] C. Gross, T. Zibold, E. Nicklas, J. Esteve, and M. K. Oberthaler. Nonlinear atom interferometer surpasses classical precision limit. Nature 464, 1165–1169 (2010).

[85] D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin. Storage of Light in Atomic Vapor. Phys. Rev. Lett. 86(5), 783–786 Jan (2001).

[86] Brian Julsgaard, Jacob Sherson, J. Ignacio Cirac, Jaromir Fiurasek, and Eu- gene S. Polzik. Experimental demonstration of quantum memory for light. Nature 432, 482 (2004).

[87] R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich. Long-lived quantum memory. Nature Physics 5, 100 Dec (2009). 203

[88] Alexander I. Lvovsky, Barry C. Sanders, and Wolfgang Tittel. Optical quantum memory. Nature Photonics 3, 706–714 (2009).

[89] C. Simon, M. Afzelius, J. Appel, A. Boyer de la Giroday, S. J. Dewhurst, N. Gisin, C. Y. Hu, F. Jelezko, S. Kröll, J. H. Müller,J. Nunn, E. S. Polzik, J. G. Rarity, H. De Riedmatten, W. Rosenfeld, A. J. Shields, N. Sköld,R. M. Stevenson, R. Thew, I. A. Walmsley, M. C. Weber, H. Weinfurter, J. Wrachtrup, and R. J. Young. Quantum memories. A Review based on the European Inte- grated Project Qubit Applications (QAP). The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics 58(1), 1–22 May (2010).

[90] W. Tittel, M. Afzelius, T. Chaneli´ere,R.L. Cone, S. Kr¨oll,S.A. Moiseev, and M. Sellars. Photon-echo quantum memory in solid state systems. Laser and Photonics Reviews 4, 244–267 (2010).

[91] Dmitry Budker, Valeriy Yashchuk, and Max Zolotorev. Nonlinear Magneto- optic Eﬀects with Ultranarrow Widths. Phys. Rev. Lett. 81(26), 5788–5791 Dec (1998).

[92] Marlan O. Scully and Michael Fleischhauer. High-sensitivity magnetometer based on index-enhanced media. Phys. Rev. Lett. 69(9), 1360–1363 Aug (1992).

[93] G. Katsoprinakis, D. Petrosyan, and I. K. Kominis. High Frequency Atomic Magnetometer by Use of Electromagnetically Induced Transparency. Phys. Rev. Lett. 97(23), 230801 Dec (2006).

[94] Danielle A. Braje, Vlatko Bali´c,Sunil Goda, G. Y. Yin, and S. E. Harris. Frequency Mixing Using Electromagnetically Induced Transparency in Cold Atoms. Phys. Rev. Lett. 93(18), 183601 Oct (2004).

[95] Hoonsoo Kang, Gessler Hernandez, and Yifu Zhu. Slow-Light Six-Wave Mixing at Low Light Intensities. Phys. Rev. Lett. 93(7), 073601 Aug (2004).

[96] Hoonsoo Kang and Yifu Zhu. Observation of Large Kerr Nonlinearity at Low Light Intensities. Phys. Rev. Lett. 91(9), 093601 Aug (2003).

[97] S. E. Harris, J. E. Field, and A. Imamo˘glu.Nonlinear optical processes using electromagnetically induced transparency. Phys. Rev. Lett. 64(10), 1107–1110 Mar (1990).

[98] PT405, Cryomech Inc., http://www.cryomech.com/.

[99] Parker O-ring ORD 5700, ParkerParker, http://www.parker.com. [100] Cell gauge: MKS 925c-41, chamber gauge: MKS 925, reservior gauge: MKS 910-42, MKS instruments, http://www.mksinst.com/.

[101] ss-4c-VCR-18, Swagelock, http://www.swagelok.com/. 204

[102] Nielsen Enterprises, Hydroponic Gardening/REFLECTIVE FILMS , http://www.mirrorsheeting.com/.

[103] ESPICorp Inc., Lot# Q9279, purity 5N, http://www.acialloys.com/.

[104] Robert C. Richardson and Eric N. Smith. Experimental Techniques In Con- densed Matter Physics At Low Temperatures (Advanced Books Classics). West- view Press, 1998.

[105] Guy K. White and Philip Meeson. Experimental Techniques in Low- Temperature Physics (Monographs on the Physics and Chemistry of Materials, 59). Oxford University Press, USA, 2002.

[106] Refer to the CVI support, http://www.cvimellesgriot.com/.

[107] MaF2 coating, ESCO Corp., http://www.escocorp.com/. [108] ACI Alloys, Inc., Lot# 0615, http://www.acialloys.com/.

[109] Kurt J. Lesker, Part# EJTTIXX451A22, Lot# TI497026IF1/583866, http://www.lesker.com/.

[110] ACI Alloys, Inc., Lot# 0615, http://www.acialloys.com/.

[111] ACI Alloys, Inc., Lot# 2008-03, http://www.acialloys.com/.

[112] Alfa Aesar, Stock# 42940, Lot# D03S039, http://www.alfa.com/.

[113] Alfa Aesar, Stock# 36200, Lot# G18T028, http://www.alfa.com/.

[114] Lake Shore Cryotronics, DT-670.

[115] Lake Shore Cryotronics, DT-670-CU-1.4D.

[116] Wolfgang Demtrder. Laser Spectroscopy. Springer, 2002.

[117] Dmitry Budker, Derek F. Kimball, and David P. DeMille. Atomic Physics: An Exploration through Problems and Solutions. Oxford University Press, USA, 2004.

[118] Richard Wolfson and Jay M. Pasachoﬀ. Physics for Scientists and Engineers (3rd Edition). Addison Wesley, 1999.

[119] S. Weber and G. Schmidt. Leiden. Comm. 246c (1936).

[120] Thomas R. Roberts and Stephen G. Sydoriak. Thermomolecular Pressure Ra- tios for 3He and 4He. Physical Review 102(2), 304 (1956). 205

[121] Reed A. Watkins, William L. Taylor, and Walter J. Haubach. Thermomolecular Pressure Diﬀerence Measurements for Precision Helium-3 and Helium-4 Vapor- Pressure Thermometry. The Journal of Chemical Physics 46(3), 1007–1018 (1967).

[122] Continuum, Surelite I-10, http://www.continuumlasers.com/. [123] Newport Corporation, Nd:YAG harmonic beamsplitter: 10QM20HB.12; plano-concave lens, f = -50 mm: KPC040AR.14; plano-convex lens, f= +250 mm: KPX109AR; plano-convex lens, f=+350 mm: KPX114AR.14,http://www.newport.com/.

[124] Thorlabs, Inc., Bandpass coloer ﬁlter, FL532-10, http://www.thorlabs.com/.

[125] Toptica photonics, DL 100 L, http://www.toptica.com/. [126] Cal Tech Physics Ph 77. Diode Laser Characteristics. Advanced Physics Labo- ratory -Atomic and Optical Physics. CalTech 3 .

[127] Kyle S. Hardman. Laser design for wide scan atom and molecule spectroscopy. Bachelor’s thesis, University of Nevada, Reno, 2008.

[128] Electro-Optical Technology, Inc., PN: 110-10168-0001, SN: 04-405-00005, http://www.eotech.com/.

[129] Casix, PN: WPZ1215-λ/4-400+M, http://www.casix.com/. [130] Mark Fox. Quantum Optics: An Introduction (Oxford Master Series in Physics, 6). Oxford University Press, USA, 2006.

[131] WaveMaster laser wavelength meter, Coherent, http://www.continuumlasers.com/. [132] Isomet Corporation, PN: 1206C-833 and PN: 1206C-833-1, SN: 113573 and PN: 113915, http://www.isomet.com/.

[133] Data aquision cards, National instruments, http://www.ni.com/.

[134] Data aquision cards, Viewpoint systems, http://www.viewpointusa.com/.

[135] IGOR Pro 6. http://www.wavemetrics.com/. [136] D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis. Resonant nonlinear magneto-optical eﬀects in atoms. Rev. Mod. Phys. 74(4), 1153–1201 Nov (2002).

[137] Alexey V. Gorshkov, Axel Andr´e,Michael Fleischhauer, Anders S. Sørensen, and Mikhail D. Lukin. Universal Approach to Optimal Photon Storage in Atomic Media. Phys. Rev. Lett. 98(12), 123601 Mar (2007). 206

[138] Nathaniel B. Phillips, Alexey V. Gorshkov, and Irina Novikova. Optimal light storage in atomic vapor. Phys. Rev. A 78(2), 023801 Aug (2008). [139] Tao Hong, Alexey V. Gorshkov, David Patterson, Alexander S. Zibrov, John M. Doyle, Mikhail D. Lukin, and Mara G. Prentiss. Realization of coherent optically dense media via buﬀer-gas cooling. Phys. Rev. A 79(1), 013806 Jan (2009). [140] Mei-Ju Lu, Franklin Jose, and Jonathan D. Weinstein. Stopped light with a cryogenic ensemble of 173Yb atoms. Phys. Rev. A 82(6), 061802 Dec (2010). [141] L. M. K. Vandersypen and I. L. Chuang. NMR techniques for quantum control and computation. Rev. Mod. Phys. 76(4), 1037–1069 Jan (2005). [142] Yosuke Takasu, Kenichi Maki, Kaduki Komori, Tetsushi Takano, Kazuhito Honda, Mitsutaka Kumakura, Tsutomu Yabuzaki, and Yoshiro Takahashi. Spin-Singlet Bose-Einstein Condensation of Two-Electron Atoms. Phys. Rev. Lett. 91(4), 040404 Jul (2003). [143] Takeshi Fukuhara, Seiji Sugawa, and Yoshiro Takahashi. Bose-Einstein condensation of an ytterbium isotope. Phys. Rev. A 76(5), 051604 Nov (2007). [144] Takeshi Fukuhara, Yosuke Takasu, Mitsutaka Kumakura, and Yoshiro Taka- hashi. Degenerate Fermi Gases of Ytterbium. Phys. Rev. Lett. 98(3), 030401 Jan (2007). [145] Yosuke Takasu and Yoshiro Takahashi. Quantum Degenerate Gases of Ytter- bium Atoms. Journal of the Physical Society of Japan 78(1), 012001 (2009). [146] Sergey G. Porsev, Andrei Derevianko, and E. N. Fortson. Possibility of an 1 3 o 171,173 optical clock using the 6 S0 → 6 P0 transition in Yb atoms held in an optical lattice. Phys. Rev. A 69(2), 021403 Feb (2004). [147] Z. W. Barber, J. E. Stalnaker, N. D. Lemke, N. Poli, C. W. Oates, T. M. Fortier, S. A. Diddams, L. Hollberg, C. W. Hoyt, A. V. Taichenachev, and V. I. Yudin. Optical Lattice Induced Light Shifts in an Yb Atomic Clock. Phys. Rev. Lett. 100(10), 103002 Mar (2008). [148] Masao Takamoto, Feng-Lei Hong, Ryoichi Higashi, and Hidetoshi Katori. An optical lattice clock. Nature 435, 321–324 (2005). [149] K. Tsigutkin, D. Dounas-Frazer, A. Family, J. E. Stalnaker, V. V. Yashchuk, and D. Budker. Observation of a Large Atomic Parity Violation Eﬀect in Ytterbium. Phys. Rev. Lett. 103(7), 071601 Aug (2009). [150] T. Takano, M. Fuyama, R. Namiki, and Y. Takahashi. Spin Squeezing of a Cold Atomic Ensemble with the Nuclear Spin of One-Half. Phys. Rev. Lett. 102(3), 033601 Jan (2009). 207

[151] Tetsushi Takano, Shin-Ichi-Ro Tanaka, Ryo Namiki, and Yoshiro Takahashi. Manipulation of Nonclassical Atomic Spin States. Phys. Rev. Lett. 104(1), 013602 Jan (2010). [152] Franklin Jose. Optical Pumping of Nuclear Spin in a Glow Discharge Sputtering Source. Bachelor’s thesis, University of Nevada, Reno., 2010. [153] J. E. Sansonetti, W. C. Martin, and S. L. Young. Hand- book of Basic Atomic Spectroscopic Data. NIST, 2006. http://physics.nist.gov/PhysRefData/Handbook/. [154] K. Honda, Y. Takahashi, T. Kuwamoto, M. Fujimoto, K. Toyoda, K. Ishikawa, and T. Yabuzaki. Magneto-optical trapping of Yb atoms and a limit on the 1 branching ratio of the P1 state. Phys. Rev. A 59(2), R934–R937 Feb (1999). [155] Dipankar Das, Sachin Barthwal, Ayan Banerjee, and Vasant Natarajan. Abso- lute frequency measurements in Yb with 0.08 ppb uncertainty: Isotope shifts 1 1 and hyperfine structure in the 399 nm S0 → P1 line. Phys. Rev. A 72(3), 032506 Sep (2005). [156] D. F. Kimball, D. Clyde, D. Budker, D. DeMille, S. J. Freedman, S. Rochester, J. E. Stalnaker, and M. Zolotorev. Collisional perturbation of states in atomic ytterbium by helium and neon. Phys. Rev. A 60(2), 1103–1112 Aug (1999). [157] A. O. Sushkov and D. Budker. Production of long-lived atomic vapor inside high-density buffer gas. Phys. Rev. A 77(4), 042707 Apr (2008). [158] WILLIAM HAPPER. Optical Pumping. Rev. Mod. Phys. 44(2), 169–249 Apr (1972). [159] Melles Griot, PN: 03 FPG 019. [160] Melles Griot, PN: 03 PBS 203. [161] Ealing Catalog, Inc., PN: CG 25.4D, BG-3, 2T, http://192.220.33.191/. [162] M. V. Romalis and L. Lin. Surface nuclear spin relaxation of 199Hg. The Journal of Chemical Physics 120, 1511 (2004). [163] W. C. Griffith, M. D. Swallows, T. H. Loftus, M. V. Romalis, B. R. Heckel, and E. N. Fortson. Improved Limit on the Permanent Electric Dipole Moment of 199Hg. Phys. Rev. Lett. 102(10), 101601 Mar (2009). [164] F. Bloch. Nuclear Induction. Phys. Rev. 70(7-8), 460–474 Oct (1946). [165] M. Takeuchi, T. Takano, S. Ichihara, Y. Takasu, M. Kumakura, T. Yabuzaki, and Y. Takahashi. Paramagnetic Faraday rotation with spin-polarized ytterbium atoms. Applied Physics B: Lasers and Optics 83, 107–114 (2006). 10.1007/s00340-006-2136-y. 208

[166] Julio Gea-Banacloche, Yong-qing Li, Shao-zheng Jin, and Min Xiao. Electro- magnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment. Phys. Rev. A 51(1), 576–584 Jan (1995).

[167] Conoptics Inc., Model: M370-LA, SN: 34729, http://www.conoptics.com/.

[168] M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully. Spectroscopy in Dense Coherent Media: Line Narrowing and Interference Eﬀects. Phys. Rev. Lett. 79(16), 2959–2962 Oct (1997).

[169] Ali Javan, Olga Kocharovskaya, Hwang Lee, and Marlan O. Scully. Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium. Phys. Rev. A 66(1), 013805 Jul (2002).

[170] Yanhong Xiao, Irina Novikova, David F. Phillips, and Ronald L. Walsworth. Diﬀusion-Induced Ramsey Narrowing. Phys. Rev. Lett. 96(4), 043601 Feb (2006).

[171] E. Figueroa, F. Vewinger, J. Appel, and A. I. Lvovsky. Decoherence of electromagnetically induced transparency in atomic vapor. Opt. Lett. 31(17), 2625– 2627 (2006).

[172] M. V. Pack, R. M. Camacho, and J. C. Howell. Electromagnetically induced transparency line shapes for large probe ﬁelds and optically thick media. Phys. Rev. A 76(1), 013801 Jul (2007).

[173] Lene Vestergaard, S. E. Harris, Zachary Dutton, and Cyrus H. Behroozi. Light speed reduction to 17 metres per second in an ultracold atomic gas. Nature 397, 594–598 (1999).

[174] F. Goldfarb, J. Ghosh, M. David, J. Ruggiero, T. Chaneliere, J.-L. Le Gouet, H. Gilles, R. Ghosh, and F. Bretenaker. Observation of ultra-narrow electromagnetically induced transparency and slow light using purely electronic spins in a hot atomic vapor. EPL (Europhysics Letters) 82(5), 54002 (6pp) (2008).

[175] Martin M. Boyd, Tanya Zelevinsky, Andrew D. Ludlow, Seth M. Foreman, Sebastian Blatt, Tetsuya Ido, and Jun Ye. Optical Atomic Coherence at the 1-Second Time Scale. Science 314(5804), 1430–1433 (2006).

[176] A. Yamaguchi, S. Uetake, and Y. Takahashi. A diode laser system for spectroscopy of the ultranarrow transition in ytterbium atoms. Applied Physics B: Lasers and Optics 91, 57–60 (2008). 10.1007/s00340-008-2953-2.

[177] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi. Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network. Phys. Rev. Lett. 78(16), 3221–3224 Apr (1997). 209

[178] M. Fleischhauer and M. D. Lukin. Dark-State Polaritons in Electromagnetically Induced Transparency. Phys. Rev. Lett. 84(22), 5094–5097 May (2000).

[179] A. Dantan, G. Reinaudi, A. Sinatra, F. Lalo¨e,E. Giacobino, and M. Pinard. Long-Lived Quantum Memory with Nuclear Atomic Spins. Phys. Rev. Lett. 95(12), 123002 Sep (2005).

[180] B. E. Kane. A silicon-based nuclear spin quantum computer. Nature 393, 133–137 (1998).

[181] Philipp Neumann, Johannes Beck, Matthias Steiner, Florian Rempp, Helmut Fedder, Philip R. Hemmer, Jorg Wrachtrup, and Fedor Jelezko. Single-Shot Readout of a Single Nuclear Spin. Science 329(5991), 542–544 (2010).

[182] Kristan L. Corwin, Zheng-Tian Lu, Carter F. Hand, Ryan J. Epstein, and Carl E. Wieman. Frequency-Stabilized Diode Laser with the Zeeman Shift in an Atomic Vapor. Appl. Opt. 37(15), 3295–3298 (1998).

[183] Kiam Heong Ang, G. Chong, and Yun Li. PID control system analysis, design, and technology. Control Systems Technology, IEEE Transactions on 13(4), 559 – 576 (2005).

[184] R. Holzwarth Th. Udem and T. W. H¨ansch. Optical frequency metrology. Nature 416, 233–237 (2002).

[185] S. E. Harris, J. E. Field, and A. Kasapi. Dispersive properties of electromagnetically induced transparency. Phys. Rev. A 46(1), R29–R32 Jul (1992).

[186] A. Kasapi, Maneesh Jain, G. Y. Yin, and S. E. Harris. Electromagnetically Induced Transparency: Propagation Dynamics. Phys. Rev. Lett. 74(13), 2447– 2450 Mar (1995).

[187] D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk. Nonlinear Magneto-optics and Reduced Group Velocity of Light in Atomic Vapor with Slow Ground State Relaxation. Phys. Rev. Lett. 83(9), 1767–1770 Aug (1999).

[188] A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer. Observation of Ultraslow and Stored Light Pulses in a Solid. Phys. Rev. Lett. 88(2), 023602 Dec (2001).

[189] A. B. Matsko, I. Novikova, M. O. Scully, and G. R. Welch. Radiation Trapping in Coherent Media. Phys. Rev. Lett. 87(13), 133601 Sep (2001).

[190] Mason Klein, Yanhong Xiao, Alexey V. Gorshkov, Michael Hohensee, Cleo D. Leung, Mark R. Browning, David F. Phillips, Irina Novikova, and Ronald L. Walsworth. Optimizing slow and stored light for multidisciplinary applications. Advances in Slow and Fast Light 6904(1), 69040C (2008). 210

[191] M. Klein, Y. Xiao, M. Hohensee, D. F. Phillips, and R. L. Walsworth. Stored light and EIT at high optical depths. arXiv:0812.4939v1 (2008).

[192] Irina Novikova, Alexey V. Gorshkov, David F. Phillips, Anders S. Sørensen, Mikhail D. Lukin, and Ronald L. Walsworth. Optimal Control of Light Pulse Storage and Retrieval. Phys. Rev. Lett. 98(24), 243602 Jun (2007).

[193] Ronald Walsworth, Susanne Yelin, and Mikhail Lukin. The Story Behind ”Stopped Light”. Opt. Photon. News 13(5), 50–54 (2002).

[194] Chien Liu, Zachary Dutton, Cyrus H. Behroozi, and Lene Vestergaard Hau. Observation of coherent optical information storage in an atomic medium using halted light pulses. Nature 409, 490–493 (2001).

[195] A. Mair, J. Hager, D. F. Phillips, R. L. Walsworth, and M. D. Lukin. Phase coherence and control of stored photonic information. Phys. Rev. A 65(3), 031802 Feb (2002).

[196] Claudia Mewes and Michael Fleischhauer. Decoherence in collective quantum memories for photons. Phys. Rev. A 72(2), 022327 Aug (2005).

[197] Rui Zhang, Sean R. Garner, and Lene Vestergaard Hau. Creation of Long- Term Coherent Optical Memory via Controlled Nonlinear Interactions in Bose- Einstein Condensates. Phys. Rev. Lett. 103(23), 233602 Dec (2009).

[198] Y. O. Dudin, R. Zhao, T. A. B. Kennedy, and A. Kuzmich. Light storage in a magnetically dressed optical lattice. Phys. Rev. A 81(4), 041805 Apr (2010).

[199] U. Schnorrberger, J. D. Thompson, S. Trotzky, R. Pugatch, N. Davidson, S. Kuhr, and I. Bloch. Electromagnetically Induced Transparency and Light Storage in an Atomic Mott Insulator. Phys. Rev. Lett. 103(3), 033003 Jul (2009).

[200] J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson. Stopped Light with Storage Times Greater than One Second Using Electromagnetically Induced Transparency in a Solid. Phys. Rev. Lett. 95(6), 063601 Aug (2005).

[201] Magnetic shielding foil, 0.004” thick, The EMF safety superstore, http://www.lessemf.com/.

[202] D. N. Matsukevich, T. Chaneli`ere,S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich. Observation of Dark State Polariton Collapses and Revivals. Phys. Rev. Lett. 96(3), 033601 Jan (2006).

[203] S. D. Jenkins, D. N. Matsukevich, T. Chaneli`ere,A. Kuzmich, and T. A. B. Kennedy. Theory of dark-state polariton collapses and revivals. Phys. Rev. A 73(2), 021803 Feb (2006). 211

[204] Mikael Afzelius, Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin. Multimode quantum memory based on atomic frequency combs. Phys. Rev. A 79(5), 052329 May (2009). [205] B. Kraus, W. Tittel, N. Gisin, M. Nilsson, S. Kröll,and J. I. Cirac. Quantum memory for nonstationary light fields based on controlled reversible inhomogeneous broadening. Phys. Rev. A 73(2), 020302 Feb (2006). [206] M. Sabooni, F. Beaudoin, A. Walther, N. Lin, A. Amari, M. Huang, and S. Kröll. Storage and Recall of Weak Coherent Optical Pulses with an Effi- ciency of 25 %. Phys. Rev. Lett. 105(6), 060501 Aug (2010). [207] Morgan P. Hedges, Jevon J. Longdell, Yongmin Li, and Matthew J. Sellars. Efficient quantum memory for light. Nature 465, 1052–1056 (2010). [208] O.M. Marago, B. Fazio, P.G. Gucciardi, and E. Arimondo. Atomic gallium laser spectroscopy with violet/blue diode lasers. Applied Physics B: Lasers and Optics 77, 809–815 (2003). 10.1007/s00340-003-1332-2. [209] H. Leinen, D. Glässner, H. Metcalf, R. Wynands, D. Haubrich, and D. Meschede. GaN blue diode lasers: a spectroscopists view. Applied Physics B: Lasers and Optics 70, 567–571 (2000). 10.1007/s003400050863. [210] R. V. Krems and A. Dalgarno. Disalignment transitions in cold collisions of 3P atoms with structureless targets in a magnetic field. Phys. Rev. A 68(1), 013406 Jul (2003). [211] Kramida A.E. Reader J. Ralchenko, Yu. and NIST ASD Team (2010). NIST Atomic Spectra Database. National Institute of Standards and Technology, Gaithersburg, MD, 2010. [212] J. Kos, M. F. Rode, J. E. Rode, G. Chalsainski, and M. M. Szczesniak. In- teractions of transition metal atoms with He. The European Physical Jour- nal D - Atomic, Molecular, Optical and Plasma Physics 31, 429–437 (2004). 10.1140/epjd/e2004-00165-4. [213] D. DeMille. Quantum Computation with Trapped Polar Molecules. Phys. Rev. Lett. 88(6), 067901 Jan (2002). [214] M. G. Kozlov and D. DeMille. Enhancement of the Electric Dipole Moment of the Electron in PbO. Phys. Rev. Lett. 89(13), 133001 Sep (2002). [215] N. Balakrishnan and A. Dalgarno. Chemistry at ultracold temperatures. Chem- ical Physics Letters 341(5-6), 652 – 656 (2001). [216] Lincoln D Carr, David DeMille, Roman V Krems, and Jun Ye. Cold and ultracold molecules: science, technology and applications. New Journal of Physics 11(5), 055049 (2009). 212

[217] Jeﬀrey I. Steinfeld. Molecules and Radiation - 2nd Edition: An Introduction to Modern Molecular Spectroscopy. The MIT Press, 1985.

[218] R. S. Ram, P. F. Bernath, M. Dulick, and L. Wallace. The A3Φ- X3∆ System (γ Bands) of TiO: Laboratory and Sunspot Measurements. The Astrophysical Journal Supplement Series 122(1), 331 (1999).

[219] Timothy C. Steimle and Wilton Virgo. The permanent electric dipole moments of the X 3∆; E 3Π; A 3Φ and B 3Π states of titanium monoxide, TiO. Chemical Physics Letters 381, 30–36 (2003).

[220] Linus Pauling. General Chemistry. Dover Publications, 1988.

[221] I.M. Hedgecock, C. Naulin, and M. Costes. Measurements of the radiative lifetimes of TiO (A 3Φ,B 3Π,C 3∆, c 1Φ, f 1∆,E 3Π) states. Astronomy and Astrophysics 304, 667–677 (1995).

[222] Wesley C. Campbell, Gerrit C. Groenenboom, Hsin-I Lu, Edem Tsikata, and John M. Doyle. Time-Domain Measurement of Spontaneous Vibrational Decay of Magnetically Trapped NH. Phys. Rev. Lett. 100(8), 083003 Feb (2008).

[223] Christopher D. Ball and Frank C. De Lucia. Direct observation of Λ-doublet and hyperﬁne branching ratios for rotationally inelastic collisions of NO-He at 4.2 K. Chemical Physics Letters 300(1-2), 227 – 235 (1999).

[224] Mei-Ju Lu and Jonathan D Weinstein. Cold TiO(X3∆) − He collisions. New Journal of physics 11, 055015 (2009).

[225] David J. Griﬃths. Introduction to Electrodynamics (3rd Edition). Benjamin Cummings, 1999.

[226] Eugene P. Wigner. On the Behavior of Cross Sections Near Thresholds. Phys. Rev. 73(9), 1002–1009 May (1948).

[227] John Weiner, Vanderlei S. Bagnato, Sergio Zilio, and Paul S. Julienne. Experi- ments and theory in cold and ultracold collisions. Rev. Mod. Phys. 71(1), 1–85 Jan (1999).

[228] Harold J. Metcalf, Peter van der Straten, and Peter Straten. Laser Cooling and Trapping (Graduate Texts in Contemporary Physics). Springer, 1999.

[229] Wolfram Mathematica 7, 1988-2009. http://www.wolfram.com/.

[230] U. Rasbach, J. Wang, R. dela Torre, V. Leung, B. Kl¨oter, D. Meschede, T. Varzhapetyan, and D. Sarkisyan. One- and two-color laser spectroscopy of indium vapor in an all-sapphire cell. Phys. Rev. A 70(3), 033810 Sep (2004).

[231] BiotSavart. http://www.ripplon.com/BiotSavart/.