RICE UNIVERSITY

Creating Strontium Rydberg Atoms

by

Xinyue Zhang

A Thesis Submittedin Partial Fulfillment of the

Requirements for the Degree

Master of Science

Approved, Thesis Committee:

F.B. Dunning, AdvisorProfessor of Physics and Astronomy

T.C. Killian, Vice AdvisorProfessor and Chair of Physics andAstronomy

Douglas NatelsonProfessor of Physics and Astronomy andProfessor in Electrical and ComputerEngineering

Houston, Texas

April, 2013

ABSTRACT

Creating Strontium Rydberg Atoms

by

Xinyue Zhang

Dipole-dipole interactions, the strongest, longest-range interactions possible be-

tween two neutral atoms, cannot be better manifested anywhere else than in a Ryd-

berg atomic system. Rydberg atoms, having high principal quantum numbers n 1

and dipole moments that scale as n2, provide a powerful tool to examine dipole-

dipole interactions. Therefore, we have studied the production and production rates

of strontium Rydberg atoms created using two-photon excitation and have explored

their properties in two distinct experiments. In the first experiment, very-high-n

(n ∼ 300) Rydberg atoms are produced in a tightly collimated atomic beam allowing

spectroscopic studies of their energy levels and their Stark effects. Simulations using a

two-active-electron model, developed by our theoretical collaborators, allow detailed

analysis of the results and are in remarkable agreement with the experimental results.

The high density of Rydberg atoms achieved, ∼ 5× 105 cm−3, in this experiment will

allow studies of strongly interacting Rydberg-Rydberg systems. The second exper-

iment, in which a cold strontium Rydberg gas is excited in a magneto-optic trap,

features an imaging technique offering both spatial and temporal resolution. We use

this technique to observe and study the evolution of an ultra-cold strontium Ryd-

berg gas which reveals the importance of Rydberg-Rydberg interactions in the early

stages of this evolution. A strongly interacting Rydberg gas provides an opportunity

iii

to realize a very strongly-correlated ultra-cold plasma.

Contents

Abstract ii

List of Illustrations vii

List of Tables ix

1 Acknowledgment 1

2 Introduction 4

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Manybody Physics . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 Ultracold Neutral Plasma . . . . . . . . . . . . . . . . . . . . 11

2.1.5 Detecting and imaging ultracold Rydberg atoms . . . . . . . . 12

2.2 The Strontium Rydberg System . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Strontium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Theoretical Results and Background 18

3.1 Traditional Treatment of Two-electron System . . . . . . . . . . . . . 18

3.2 Two Electron model . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Rydberg Atoms in a electric field . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Classical Picture . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.2 Sr Stark Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

v

4 Experiment Setups and Techniques 35

4.1 Frequency Locked Diode Laser System . . . . . . . . . . . . . . . . . 35

4.1.1 Frequency Double High Power Diode Laser System[57] . . . . 36

4.1.2 HeNe Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.3 Scanning Fabry-Perot Interferometer . . . . . . . . . . . . . . 39

4.1.4 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . 40

4.1.5 Locking Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.6 Limitations and Alternatives . . . . . . . . . . . . . . . . . . . 41

4.2 Strontium Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Other Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.2 Interaction Region . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.1 Selective Field Ionization . . . . . . . . . . . . . . . . . . . . . 49

4.4.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Sr Rydberg Atoms in a Collimated Atomic Beam 53

5.1 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.1 Even Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1.2 the odd isotope 87Sr . . . . . . . . . . . . . . . . . . . . . . . 58

5.1.3 Stray Fields Impact On Spectra . . . . . . . . . . . . . . . . . 62

6 Ultracold Rydberg Gas Evolution 66

6.1 Experimental Setup Overview . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Principal Processes in Probing Sr Cold Rydberg Gas . . . . . . . . . 69

6.2.1 Autoionization . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2.2 Electron-Rydberg Collisions . . . . . . . . . . . . . . . . . . . 72

6.2.3 Penning Ionization . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2.4 Blackbody Radiation induced Ionization . . . . . . . . . . . . 75

vi

6.3 Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4 Results Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7 Conclusion and Outlook 85

Bibliography 87

Illustrations

2.1 Crossover to Collective Many-body States . . . . . . . . . . . . . . . 8

2.2 Ultracold Neutral Plasma Creation Setup . . . . . . . . . . . . . . . . 12

2.3 Natural Linewidths and Transition Wavelengths of Principle Sr Levels 14

2.4 Sr+ Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Cooling Transitions towards Quantum Degeneracy . . . . . . . . . . . 16

3.1 Measured and calculated quantum defects in the single-electron

excitation of strontium . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Calculated Excitation Spectra . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Hydrogen Rydberg Atoms in a Electric Field . . . . . . . . . . . . . 27

3.4 NonHydrogen Rydberg Atoms in a Electric Field . . . . . . . . . . . 28

3.5 Stark Map of Strontium Rydberg Atoms . . . . . . . . . . . . . . . . 32

3.6 Parabolic States Distribution of Strontium Stark States . . . . . . . . 34

4.1 TOPTICA Diode Laser System Schematics . . . . . . . . . . . . . . . 36

4.2 Vapor Pressure Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Schematic diagram of the oven assembly . . . . . . . . . . . . . . . . 45

4.4 Interaction Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5 Stark Map for Sodium . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Excitation Spectra For Sr Isotopes . . . . . . . . . . . . . . . . . . . 55

5.2 N ∼ 312 Spectra for Overlapping 86Sr and 88Sr . . . . . . . . . . . . 59

viii

5.3 N ∼ 335 spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Anticrossing of 87Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 Stay Field Limited High N Spectra . . . . . . . . . . . . . . . . . . . 65

6.1 Experiment Schematic, Diagram and Timing . . . . . . . . . . . . . . 67

6.2 Sheet Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Autoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.4 l mixing schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.5 Laser-induced fluorescence imaging . . . . . . . . . . . . . . . . . . . 76

6.6 Dependence of the LIF signal on Rydberg excitation time . . . . . . . 77

6.7 LIF images of spontaneous evolution . . . . . . . . . . . . . . . . . . 79

6.8 Visible Ions in Spontaneous Evolution . . . . . . . . . . . . . . . . . . 80

6.9 Collision Time Vs Initial Inter-Rydberg atoms Distance . . . . . . . . 83

Tables

2.1 Scaling Laws for Rydberg Atoms . . . . . . . . . . . . . . . . . . . . 5

2.2 Principal Isotopes of Strontium . . . . . . . . . . . . . . . . . . . . . 14

5.1 Sr Quantum Defects [65] . . . . . . . . . . . . . . . . . . . . . . . . . 58

1

Chapter 1

Acknowledgment

When I asked Dr. Barry Dunning if I had the honor to join his group three years ago,

I thought I had small chance to be admitted. I had clear idea of what I had to offer:

my B.S. was in geophysics, my English was terrible, my experimental experience was

zero and there were very limited number of equipments that I could lift in the lab.

Yet, he decided to give me a chance. Because of my poor background, Barry had to

painstakingly teach me the basic experimental skills and to make me useful in the

lab. Over the past years, there was not even once has he lost his patience or said one

harsh word to me for the various mistakes I made (everyone who has worked with me

should understand how difficult that was since I can be very bullheaded and cheeky

sometimes). On the contrary, he is always encouraging me, helping me to improve

and never giving up on me. I never ever thought it is possible for a person to be

so nice and kind before I knew Barry and I am so grateful everyday for having him

as my advisor. In addition to his guidance on experiments, he has made so much

contribution to the writing of this thesis. He revised it from structure, contents to

grammar and even punctuations for more than ten times and he managed to upgrade

it from a mindless talking to a professional, well-written thesis. For that, I couldn’t

thank him enough!

I feel very lucky to have Dr. Tom Killian as my second advisor who is also a

remarkable teacher and a very kind person. He spent so much time in teaching me

techniques on laser, fiber, optics and electronics from scratch with great patience. At

2

early times, he would generously devote many hours of his day in just helping me

with optical setup and alignments and making sure that I am not doing anything

foolish. Moreover, he is always trying his best to help me understand concepts in

atomic physics even if that means he has to explain one thing for quite a few times in

different perspectives. Just like Barry, he tolerated my ignorance, weak background,

my sometimes very annoying personalities and has been trying to build something

out of me. His efforts greatly helped me make through my research and are very

much appreciated. Both Barry and Tom are such dedicated scientists who have been

great role models for me to look up to. Having the opportunities to work with both

of them makes me smile in my dream.

All the people in Barry’s and Tom’s groups have also played important roles

in helping me complete this work and lightening up my every day at school. My

senior students, Shuzhen Ye and Patrick McQuillen, who suffered the worst of me

and yet they are still my great friends and teachers. Mi Yan, Brian DeSalvo and

Trevor Strickler can always put aside their work and happily discuss all my whimsy

questions. Changhao Wang, Yu Pu, Ying Huang and Micheal Kelley never said no

to help me. Francisco Caremo is always a delight to work with. My buddies here at

Rice including Ernie Yang, Yang-Zhi Zhou, Zhentao Wang, Sidong Lei, Alicia Chang,

Ksenia Bets, Jason Ball and many more offered me their friendships that I have always

cherished. Every each of these people made my life here simply lovely and I’d love to

express my gratitude for all of them.

At last, I would also like to thank Dr. Han Pu, Dr. Stan Dodds, Dr. Huey W.

Huang, Dr. Anthony Chan and Dr. Randy Hulet for their helps, trusts and insights

and I want to thank Dr. Douglas Natelson for keeping setting aside his time to make

my thesis defense possible. I’d like to end this part with my greatest appreciation to

3

my grandmother whom I will always love unconditionally.

4

Chapter 2

Introduction

Atoms in which one electron is excited to a state of large principal quantum number n,

termed Rydberg atoms, have been studied extensively because they possess extreme

physical characteristics unlike those normally associated with atoms in ground or low-

lying excited states. This is illustrated in Table 2.1 which lists a number of atomic

properties, their n dependencies and their values for selected n levels of interest in

this work. Since the classical Bohr radius of an atom scales as n2, Rydberg atoms

are physically very large and many of their properties can be described in terms

of the classical Bohr model of the atom. Their binding energies, which scale as

n−2 are very small. Because of their large size and weak binding, Rydberg atoms

can be strongly perturbed and even ionized by modest external electric fields, the

threshold for ionization scales as n−4. The classical electron orbital period which

increases as n3, is large allowing, for example, application of electric field pulses

whose duration is much smaller than the electron’s orbital period. At high n the

spacing between adjacent levels which decreases as n−3 becomes very small. The

radiative lifetimes of Rydberg atoms are also very large resulting in very narrow

spectral features. Table 2.1 also includes other atomic properties pertinent to the

present study including their polarizability, dipole moment and 〈r−4〉,〈r−6〉 (which

determine the relative contributions to the polarization energy Wpol from dipole and

quadrupole core polarization).

The dipole moments listed in the Table 2.1 are strictly speaking transition dipole

5

Table 2.1 : Scaling Laws for Rydberg Atoms

Property Scaling

(a.u.)

Na(10d) n = 50d n = 312d

orbital radius n2 147a0 0.368µm 14.3µm

orbital period n3 0.15ps 18.8 ps 4.56 ns

binding energy 1/n2 0.14eV 5.6meV 0.14meV

energy spacing 1/n3 0.023 eV 0.18meV 0.75µeV

ionization field 1/n4 30 kV/cm 48V/cm 32 mV/cm

radiative lifetime n3 1µs 125µs 30.4ms

dipole moment〈nd|er|nf〉 n2 143 ea0 3.58×103ea0 139× 103ea0

polarizability MHz cm2/V2 n7 0.21 1.64× 1010 6.04× 1015

〈r−4〉 3n2−l(l+1)2n5(l+3/2)(l+1)(l+1/2)l(l−1/2)

〈r−6〉 35n4−5n2[6l(l+1)−5]+3(l+2)(l+1)l(l−1)8n7(l+5/2)(l+2)(l+3/2)(l+1)(l+1/2)l(l−1/2)(l−1)(l−3/2)

6

moments, not a permanent electric dipole moment. Atoms in zero field don’t have

permanent electric dipole moments! In a classical picture, high lm states have near

circular orbits, and a near zero net dipole moment. For low l states, the highly

elliptical orbits will precess around the nucleus due to core scattering resulting again

a vanishing net permanent dipole moment. However, there are a number of ways

to create Rydberg atoms with a large permanent dipole moment. One novel way,

as suggested in [36] is to create trilobite molecules in a Bose-Einstein Condensate

which represent a class of ultra-long-range homonuclear diatomic Rydberg molecules

that possess a permanent electric dipole moment in the order of kilodebye and some

experimental success in this direction has been achieved [37].

Another approach to create quasi-one-dimensional states is to selectively excite

extreme red-shifted Stark states in the presence of a DC field. The selection rules

only allow creation of low-l Rydberg states through photo-excitation. For the alkali

and alkaline-earth metals, such states are difficult to polarize due to core scattering

and initially only display quadratic Stark effects in a DC field, indicating a small

induced dipole moment. However at higher fields these states can mix with the

extreme strongly-polarized components of the Stark manifold and can themselves

become very polarized.

Upon obtaining a good quality quasi-one-dimensional (quasi-1D) state, it is straight-

forward to convert it to a near-circular, two-dimensional, “Bohr-like” state by appli-

cation of an appropriate electric field pulse perpendicular to the atomic axis.

2.1 Motivation

In the present work we are developing techniques to create strontium Rydberg atoms

as a first step towards new projects involving strongly-coupled manybody systems

7

and the creation of long-lived two-electron excited states. Many of the proposed

experiments will take advantage of dipole blockade.

Rydberg blockade [1] is a manifestation of the strong, long range interaction be-

tween Rydberg atoms. This interaction could be of the Van der Waals type which

varies with the inter-particle distance R as C6/R6, whereas for Rydberg states with

permanent dipole moments the interaction will be of dipole-dipole type with form

C3/R3. Because of these interactions, excitation of one Rydberg atom shifts the en-

ergy levels of neighboring Rydberg atoms and prevents their excitations using the

same narrow linewidth laser. The resulting “dipole blockade” radii can be large

(∼ 5µm at n ∼ 50 and ∼ 100µm at n ∼ 300). Although initially proposed as a

means to create fast quantum gates for neutral atoms [2], Rydberg blockade has been

shown to be an extremely versatile tool with many applications in areas such as con-

densed matter physics, plasma physics, nonlinear optics and quantum information. In

the past decade, exciting progress has been made in each field that could be paradigm

changing in the future.

2.1.1 Manybody Physics

Rydberg atoms, blessed with their large electric dipole moments, interact strongly

permitting the simulation of a wide variety of condensed matter systems. Further-

more, as pointed out in early theoretical work [3], quantum information processing

could be based on the collective states of mesoscopic atomic ensembles due to Rydberg

dipole blockade effects.

Rydberg blockade effects have been the subject of a lot of experimental interest.

One of the first demonstrations [6] involved tuning an np state to the middle of the

adjacent ns and (n+ 1)s states through Stark shifts induced by application of a DC

8

field. A 30% suppression of Rydberg excitation was observed at a Forster resonance.

Direct van der Waals blockade was also observed [7]. In 2009, two independent groups

demonstrated [9, 8] the collective excitation of two blockaded Rydberg atoms [10].

It was shown theoretically that full control of the strength, shape and character of

the interaction potential is possible by weakly dressing Rydberg atoms contained in

a Bose-Einstein Condensate [11]. In addition, by adjusting experimental parameters

like the detuning, it is experimentally feasible to crossover from two body interactions

to many body interactions; see Figure 2.1.

Figure 2.1 : Crossover to Collective Many-body States. (a) ground states |g〉 dressedwith Rydberg states |r〉 which are excited by a two-photon transition via the interme-diate state |p〉. The total detuning for this two-photon transition is ∆, ∆ = ∆p + ∆r.(b) Diagram for the crossover from two-body to many-body interaction, Ω = Ωr +Ωp.Figure adapted from [11].

The quest for exotic quantum phases can also be realized in a blockaded ultracold

Rydberg ensemble. Inspection of the experimental data revealed the existence of a

dimensionless parameter and an algebraic scaling law (characteristics of a second or-

der phase transition) for an ultracold Rydberg gas. In other words, a frozen Rydberg

system can be employed to study phase transitions in a precise, controllable manner.

9

Thus the capabilities that have been developed to coherently engineer the interactions

in a many-body system, and the abilities to address and manipulate these “super-

atoms” individually due to their huge size, demonstrate the potential of Rydberg

systems as quantum simulators [16]. Here “superatom” is a term used to symbol-

ize the large spherical volume formed by a Rydberg atom and all the consequently

blockaded ground state atoms within its dipole blockade radius.

The crystalline phase can be explored with Rydberg atoms. Dipole blockade was

proposed as a means to obtain dynamical crystallization through the use of a chirped

laser pulse [12], the Rydberg excitation number being predicted to display a staircase

structure. The mechanism is easy to understand (assuming the Rydberg interaction

is repulsive, i.e. C6 > 0): with the excitation laser initially red detuned from the

Rydberg atom transition, the collective many-body ground state for the ensemble

will be one in which every atom is in its one-body ground state, as in the system’s

Fock number state |0〉. As the laser chirps towards resonance with the Rydberg level,

a single atom in the ensemble will be excited to a Rydberg state whereupon further

excitation will be prevented by dipole blockade, so the system jumps to number

state |1〉. When the laser is blue detuned enough to compensate the smallest dipole-

dipole energy shift, which is that between the first Rydberg atom and the furthest

ground state atom, that particular ground atom will be excited, the system jumping

to number state |2〉, . . . As chirping continues, ground state atoms, in well-ordered

positions will be excited one by one resulting in a crystalline structure comprising an

array of dipole blockade superatoms.

Creation of a supersolid, a novel phase simultaneously displaying crystal rigidity

and dissipation-less flow, has been an experimental challenge for decades and so far

has not been achieved. This peculiar phase requires a repulsive two-body potential

10

that softens at short distances and a long system lifetime to allow formation and

observation of this phase. Theoretically, both requirements can be met in an ultracold

atomic ensemble in the Rydberg blockade regime [14, 15]. The artificial potential can

be mimicked and controlled by subjecting the Rydberg atoms to a homogeneous

electric field in which Rydberg atoms possess large permanent dipole moments. Use

of an off-resonant two-photon transition to properly “Rydberg dress” the ground state

atoms can significantly reduce the photon scattering, thereby increasing the lifetime

of the system.

2.1.2 Photonics

Photons don’t interact with each other. Thus entanglement between photons does

not come naturally. So far, experimentalists have resorted to spontaneous paramet-

ric down-conversion to make pairs of entangled photons [17]. There has been a lot

experimental and theoretical work suggesting an effective mapping of the strong Ry-

dberg interactions in the collective ultracold ensemble to photons. For instance, the

electromagnetically induced transparency (EIT ) obtained by driving transition to a

Rydberg level [18] is non-linearly influenced by the character of the Rydberg-Rydberg

interactions [19]. Recent work has also shown [20] the potential of Rydberg system

to applications like a fast single photon source, quantum teleportation, and the fast

entangling of spin waves. The photon retrieved from a superatom has instant second

order auto-correlation g(2)(0) as small as 0.075 and a single photon generation effi-

ciency of 10%, not far from that of its well-developed quantum dot counterpart [21].

11

2.1.3 Quantum Gates

One of the very first proposed applications of Rydberg blockade was the implemen-

tation of quantum gates in an ultracold neutral gas [2]. The strong, long-range

interaction between Rydberg atoms can be coherently turned on and off in a short

time which will result in very fast, high fidelity quantum gates. Two advantages

of quantum gates using neutral atoms are their straightforward scalability to create

multi-qubit registers and their weak coupling to external field noises. The controlled-

Z gate and CNOT gate which form a complete set of universal gates for quantum

computing have already been realized in the neutral system [22].

2.1.4 Ultracold Neutral Plasma

Ultracold neutral plasmas in which the Coulomb potential energy of interaction be-

tween its constituents ECoulomb, are greater than their thermal energies EThermal,

represent an exciting new frontier in plasma physics. Such plasmas can be created by

near-threshold photo-ionization of atoms contained in a cold cloud (see Figure 2.2).

The ionized photo-electrons, which have energies ∼ 1K, begin to escape the cloud and

leave their ion cores behind. This process quickly terminates as the cloud builds up

a net positive charge creating a Coulomb potential well from which further electrons

cannot escape. Photoionization, however, produces ions distributed throughout the

cold atom cloud, and the ions are therefore disordered. As the ions relax to a more

ordered state, they are heated on a timescale of 100 ns, resulting in disorder induced

heating (DIH) and ion temperatures of a few Kelvins rather than the mK temper-

atures characteristic of the parent laser-cooled neutral atoms. The plasma coupling

parameter τ = ECoulomb/EKinetic is thus dramatically reduced [34]. This can be mit-

igated by exploiting dipole blockade to create an ordered cloud of Rydberg atoms

12

Figure 2.2 : Ultracold Neutral Plasma Creation Setup Figure adapted from [33].

and then ionizing these through pulsed electric field ionization [32]. This will allow

creation of plasmas with much larger and controllable values of τ and the exploration

of a new plasma physics regime.

2.1.5 Detecting and imaging ultracold Rydberg atoms

Over the past few years, two major techniques have been employed to image ultracold

Rydberg atoms. The first exploits traditional electric field induced ionization. To

obtain an image, a position sensitive multi-channel plate detector (MCP ) is generally

required [23]. Higher resolution can be achieved by using Field Ion Microscopy as

in [24]. The magneto-optical trap (MOT ) is located at the center of a closed cage

made of 10 independent electrodes that are used to minimize stray fields. The imaging

electrode tip (125µm in radius) projects the field ionized Rydberg atoms onto the

MCP detector. By accumulating many images and studying their auto-correlation,

Rydberg blockade can be seen.

A second approach is optical in situ fluorescence imaging of the Rydberg atoms

by stimulated deexcitation and has produced the first observation of the spatially

13

ordered components of the Rydberg-blockade-induced many-body states that formed

inside of a mesoscopic system [25]. The images’ temporal and spatial resolution is

unprecedented.

2.2 The Strontium Rydberg System

Since strontium Rydberg atoms are the focus of the present work, their particular

properties are now discussed.

2.2.1 Strontium

Strontium, an alkaline earth metal with two valence electrons possessing both sin-

glet and triplet levels, has been the subject of numerous studies in the literature.

The singlet 1S0 ground state makes it immune from magnetic Zeeman splitting. In

addition, strontium has a wealth of bosonic (88Sr,86 Sr,84 Sr) and fermionic (87Sr)

isotopes. The relative natural abundances of these isotopes are listed in Table 2.2

together with their nuclear spins (only the 87Sr isotope has a nuclear spin) and the

isotope shifts for the principal 5s21S0 → 5s6p1P1 transition. Transition wavelengths

and natural linewidths for transitions between its lowest lying levels are shown in

Figure 2.3. Excitation to a high-l Rydberg state leaves an optically active core ion

that behaves much as an independent ion. The energy level structure of the core ion is

shown in Figure 2.4. Absorption/Fluorescence on the 2S1/2 → 2P1/2,2P3/2 transition

can then be used to image and manipulate strontium Rydberg atoms.

The properties of strontium highlighted above have enabled a number of interest-

ing experiments that are outlined below.

14

Figure 2.3 : Natural Linewidths and Transition Wavelengths of Principle Sr Levels.Intercombination lines are 1S0 → 3P . Figure adapted from [31].

Table 2.2 : Principal Isotopes of Strontium

Isotope Atomic

Mass

Natural

Abundance(%)

I F 1S0 → 1P1

Shift(MHz)

Scattering

Length(a0)

84Sr 83.913 0.56 0 - -270.8 122.7

86Sr 85.909 9.86 0 - -124.5 823

7/2 -9.7

87Sr 86.908 7.00 9/2 9/2 -68.9 96.2

11/2 -51.9

88Sr 87.905 82.58 0 - 0 -1.4

15

Figure 2.4 : Sr+ Transitions

Frequency Standards Due to hyperfine mixing, the strongly forbidden transition

(∆s 6= 0, ∆j = 0), 5s2 1S0 → 5s5p 3P0 is weakly allowed for 87Sr. The natural

linewidth of this transition is only 1mHz allowing its use as an atom frequency stan-

dard [40]. The atoms must be cooled to µK to reduce the Doppler shifts and to allow

trapping of a large number of atoms in an optical lattice. Trapping atoms in the

antinodes of the lattice results in an ac Stark shift on the clock transition. However,

a magic wavelength exists [39] for cancellation of the upper and lower Stark shifts and

results in a negligible light shift. Currently, the strontium optical lattice clock is the

best optical atomic frequency standard and has been used to measure fundamental

constants.

Quantum Degenerate Gases For the spinless strontium singlet ground state 1S0,

the traditional technique of evaporative cooling in a magnetic trap is no longer appli-

cable. Nonetheless, quantum degeneracy has already been achieved using all the

16

Figure 2.5 : Cooling Transitions towards Quantum Degeneracy. Solid lines aredriven by lasers, dashed lines are the spontaneous decay path. Figure adapted from[30].

principle isotopes of strontium [27, 28, 29]. The procedures employed are simi-

lar and all-optical [27] and can be understood by reference to Figure 2.5. A blue

laser (461nm) red detuned from the 1S0 → 1P1 is used to Zeeman slow and two-

dimensionally collimate the atomic beam. Atoms are then further cooled in a 461nm

MOT. With repeated cycling, some atoms start to accumulate in 3P2 level (through

path(5s5p)1P1 → (5s4d)1D2 → (5s5p)3P2 transitions ). When sufficient atoms have

been trapped, a 3-micron laser pulse is used to transfer the 3P2 atoms back to ground

state via the transitions (5s5p)3P2 → (5s4d)3D2 → (5s5p)3P1 → (5s2)1S0. The

461nm blue MOT is then extinguished, and a red MOT operating on the 1S0 → 3P1

transition is turned on to further cool the atoms prior to loading into an 1.06µm

optical dipole trap (ODT). The atoms are then further cooled by lowering the trap

depth, evaporative cooling resulting in degeneracy.

17

2.3 Thesis Outline

The main part of this thesis will focus on the results of two recent experiments [35, 50]

designed to study the excitation of very-high-n (n ∼ 300) Rydberg states and to

explore the evolution of cold Rydberg gases towards an UNP by imaging the core

ions.

18

Chapter 3

Theoretical Results and Background

3.1 Traditional Treatment of Two-electron System

Compared with the simple hydrogen atom, alkali Rydberg atoms are more compli-

cated in the sense that the closed-shell-core can be penetrated and polarized. The

resulting effects can be well characterized by one l-dependent quantum defect δl.

However, for alkaline-earth elements, things are far more complicated due to the in-

teraction of the two valence electrons. Even though one of the electrons is promoted

to a Rydberg level, the strong short-range scattering with the one-active-electron core

leads to strong configuration mixing. For each term SL, this interaction among con-

figurations can be described in a set of parameters in Multichannel Quantum Defect

Theory (MQDT) [49, 48].

When two electrons are close r12 < r0, they can exchange angular momentum, spin

and energy via their Coulomb interaction 1/r12 without violating their overall con-

servation. In this regime of free-energy-exchange, a proper set of basis wavefunctions

can only be obtained by diagonalizing a scattering matrix S. This yields a set of Φα

“eigenchannels” with eigenvalues µα. These eigenchannels are formed from a mixture

of different configurations. Energy exchange becomes negligible once r12 > r0 due to

the diminishing overlap of the wavefunctions of the two electrons and the falloff of

their interaction 1/r12. The outer electron can then be described as a superposition

of collision channels. Each eigen-collision-channel is then a pure configuration labeled

19

by quantum number νi,

νi =√R/(Ii − E), (3.1)

where Ii is the ionization limit of the ion core in this collision channel, R is the mass

corrected Rydberg constant and E is the energy of the system(νi can be regarded as

the quantum defect for the ith channel).The eigenenergies of the system for any r12 are

found by connecting the two sets of wavefunctions in regimes r12 < r0 and r12 > r0

via a transformation matrix Uiα and applying appropriate boundary conditions at

infinity. The nontrivial solution requires

Det|Uiα sinπ(νi + µα)| = 0. (3.2)

The bound eigenenergies of the system are found by adjusting the µα and Uiα that

simultaneously satisfy Equation 3.1 and Equation 3.2 until agreement with exper-

imental data is reached. This method can also determine the admixtures of the

different configurations. For example, for Sr J=2 bound states, it has been shown

that the most important channels are 5snd1D2, 5snd 3D2, 4dns 1D2, 4dns 3D2, and

5pnp 1D2. For the 5s15d 1D2 state, there is almost a 40% admixture from 5snd 3D2.

The semi-empirical techniques of MQDT have been very successful and very widely

used since they encapsulate the complex spectra, and configuration interactions, into

a number of parameters. They have also motivated the search for ab initio methods

to calculate short range scattering. There has also been great success in combining

the eigenchannel R-matrix method with MQDT. The R-matrix method is a way to

variationally calculate the set of eigenchannels Φα inside of a volume r < r0 in a

given configuration space. Essentially, this ab initio method requires solving the

time-independent Shrodinger equation with trial wavefunctions.

To construct a proper set of trial wavefunctions, the foremost thing is to find the

20

appropriate Hamiltonian

H = −∆21

2− ∆2

2

2+ V (r1) + V (r2) +

1

r12. (3.3)

In the above Hamiltonian, V (r) is not known. Nevertheless it’s not hard to imagine

this potential should be some kind of l-dependent screening potential. Since a lot of

orbitals are extremely sensitive to this potential, there has been a lot of work trying

to optimize this model potential for different alkaline earth elements. It has been

shown that by using the optimized potentials, accurate spectra can be obtained. One

optimized model potential for strontium is the following,

V (r) = −1

r2 + (Z − 2)exp(−αl1r) + αl2rexp(−αl3r),

α1 = 3.551, α2 = 6.037, α3 = 1.439.

In the Section 3.2, another original ab initio method to calculate strontium Ry-

dberg spectra will be presented. It also employs an l-dependent model potential.

However, while it is not an R-matrix method as described above, it does give accu-

rate spectra that match to our experimental results. This method was developed to

help analyze our experimental results by our collaborators in Vienna.

3.2 Two Electron model

To analyze the excitation spectra of strontium, we employ a two-active-electron

model. The Hamiltonian is written as

H =p212

+p222

+ Vl(r1) + Vl(r2) +1

|~r1 − ~r2|(3.4)

As we are mainly interested in single-electron excitation, it is practical to reduce

the number of configurations so that the eigenenergies can be evaluated efficiently

21

by numerically diagonalizing the Hamiltonian. The basis states are constructed from

the excited states of the Sr+ ion

H =p2

2+ Vl(r) (3.5)

Hion |φni,li,mi〉 = Eion

nlm |φni,li,mi〉

The eigenstates 〈φni,li,mi| and the eigenenergies Eni,li,mi

can be obtained numeri-

cally using the generalized pseudo-spectral method. The generalized pesudospectral

method [60] is a numerical procedure for solving equations such as equation 3.5. It

used for optimal grid discretization of the radial coordinates and is especially well

suited for problems involving a Coulomb singularity. It requires a smaller number of

grid points yet provides higher accuracy. It also introduces a split-operator technique

in the energy representation that allows the wavefunctions to propagate efficiently

in time (It has been widely applied in Floquet studies of atomic processes in strong

fields.). The calculated energies agree quite well with those measured for the ion.

The matrix elements of the two-electron Hamiltonian 3.4 are evaluated using the

basis states defined by

|n1l1n2l2;LM〉 =∑

m1+m2=M

[C(l1,m1; l2,m2;L,M) |φn1,l1,m1〉 |φn2,l2,m2〉√

2(1 + δn1,n2δl1,l2δm1,m2)

± C(l2,m2; l1,m1;L,M) |φn2,l2,m2〉 |φn1,l1,m1〉√2(1 + δn1,n2δl1,l2δm1,m2)

] (3.6)

where L is the total angular momentum, M is a projection, and the Clebsch-

Gordan coefficients are given by

22

C(l1,m1; l2,m2;L,M) = (−1)−l1+l2−M√

2L+ 1

l1 l2 L

m1 m2 −M

(3.7)

The basis states symmetric (antisymmetric) with respect to the exchange of two

electrons are used to calculate the eigenenergies in the singlet (triplet) sector. For

singlet excitation spectra the quantum numbers (n1, l1) of the outer electron may

vary over the range of the whole excitation spectrum but those (n2, l2) of the inner

electron can be limited to near the ground state. Using such a truncated basis set, the

eigenvalues of the active-two-electron system can be evaluated. Since the principal

quantum numbers n1, n2 of the basis describe the excited states of Sr+ ion and not

those of neutral strontium, the correct quantum number n of the Rydberg electron has

to be assigned to the calculated eigenstates of the two interacting electron according

to the known excitation series ( including perturber states ) in each L sector, i.e.

|nLM〉 =∑n1,l1

∑n2,l2

cn1,n2,l1,l2 〈n1l1n2l2;LM | (3.8)

This is not straightforward as some states are hard to identify. For example, there

is a 4d5p state in the 1P1 sector, yet, the calculation shows no state with dominant

4d5p character. For strontium only few perturbers affect the Rydberg series for single

electron excitation. Since they have relatively small energy, the highly excited states

are not directly affected. For the singlet sector the quantum defects of singly-excited

low-L states are plotted in Figure 3.1. The calculated results, which includes the

6 configurations (5s, 4d, 5p, 6s, 5d, and 6p ) for the inner electron, are compared

with the previous studies based on MQDT (Quantum defects for L > 3 are negligibly

small ). The calculations agree well with the measured results, which is expected

as the model potential employed is known to yield the correct quantum defect using

23

Figure 3.1 : Measured and calculated quantum defects in the single-electron excitationof strontium (singlet). A two-active electron model is used with 6 configurations ofthe inner electron. Measured results are marked by circles.

R-matrix theory. Only a small disagreement is seen for the P-and D-states where the

calculated values slightly underestimate the measured quantum defects. We also note

that, as seen in Figure 3.1, the quantum defects slowly increase with the principal

quantum number n especially for P- and D-states. The eigenstates for highly excited

states have contributions from the inner electron that are almost exclusively from the

5s state. Even a very small overlap with the other inner electron configurations shifts

the phase of the wave function near the origin greatly affecting the quantum defect.

The numerical method can be tested by comparing the zero-field excitation spec-

24

trum with the measured data (Figure 3.2). The measured spectrum is taken at

n ∼ 280 and the calculations at lower n, n ∼ 50 and n ∼ 30. To compare the

spectra of two different n the frequency axis is scaled so that the energy difference

between two adjacent levels (n and n− 1) becomes invariant for different values of n.

The calculated spectrum is derived from the dipole transition | 〈5s5p| z |5snl〉 |2 and

convoluted with a Gaussian to match the measured linewidth. The positions of the

n 1S0 states relative to the two adjacent degenerate n levels are observed to be invari-

ant as the quantum defects of n 1S0 states are nearly n-independent. On the other

hand, the peak positions of the n 1D2 states vary with n mirroring the n-dependent

quantum defect. For example, the quantum defect is δd ≈ 2.31 for n = 50 and that

extrapolated for the limit of n → ∞ is δd = 2.38. Another interesting observation

is that the relative intensity of the n 1D2 state to the (n + 1) 1S0 state increases

with n. The excitation strength is sensitive to the quantum defect as it phase-shifts

the wave function near the origin and modifies the overlap with the 5s5p state. In

this case, the quantum defect around δd = 2.38 appears to maximize the relative

intensity and be suppressed away from it. This suggests that the relative intensity

can be used to confirm the size of the quantum defect. We note that the underes-

timate of the calculated quantum defect for 32d overemphasizes this effect slightly.

Calculations using a single-active-electron model with a model potential similar to

that described in [61] have also been performed. In this model, a single electron is

moving in a model potential that is numerically fit from known and extrapolated

quantum defects. The eigenenergies as well as quantum defects can be obtained quite

accurately while the oscillator strength fails to reproduce the measured spectra due

to an inaccurate description of 5s5p state.

25

Figure 3.2 : Comparison between measured (a) and calculated (b, c) excitation spectrain zero field. (a) Measured excitation spectrum recorded at n ∼ 283. Results of two-electron calculations at n ∼ 50(b) and n ∼ 30(c) employing six inner electron states(4s, 4d, 5p, 6s, 5d, and 6p). The energy axis is scaled such that E0 = 1 correspondsto the energy difference between neighboring n and n− 1 manifolds

26

3.3 Rydberg Atoms in a electric field

3.3.1 Classical Picture

As n becomes very large, the quantum mechanical behavior of the excited electron in

a Rydberg atom can be described by the classical Bohr theory. In a hydrogen Rydberg

atom, the electron follows an elliptical orbit that is given by r = L2/(1 + ε cos θ) in

polar coordinates, where ε is the eccentricity and ~L = ~r×~p is the angular momentum.

The hydrogen atom is a special case because the energy levels are highly degenerate

in l and m which is a manifestation of the 1/r character of the Coulomb potential.

Correspondingly, in the classical picture, hydrogen Rydberg atoms have one more

physical quantity that is conserved, the Runge-Lenz vector ~A = ~p× ~L− r. In atomic

units, the magnitude of the Runge-Lenz vector is the eccentricity ε. On the other

hand, alkali or alkaline earth atoms, do not have this “accidental degeneracy” due to

core penetration and polarization. Their energies, characterized by E = −1/2(n−δl)2,

can be viewed as perturbed by the core. As a result, their Keplerian elliptical orbits

will precess about the nucleus just as Mercury’s perihelion precesses about the Sun.

For non-penetrating cases, the frequency of this precession is ∼ 5n3lδl.

The differences between hydrogen and non-hydrogenic Rydberg atoms are magni-

fied in a electric field. For hydrogen, quantum mechanically, application of degenerate

time-independent perturbation theory will lead to the linear Stark effect. The Stark

map of hydrogen is simply like a fan; see Figure 3.3. The Shrodinger equation can

be solved analytically in parabolic coordinates for a hydrogen atom in a electric field.

The parabolic eigenstates are labeled by n, n1, n2,m and these quantum numbers are

related by n = n1 + n2 + |m| + 1. As shown in Figures 3.3, the dipole moments

are the largest for the extreme Stark states which have parabolic quantum numbers

27

Figure 3.3 : Hydrogen Rydberg Atoms in a Electric Field Left: the fanlikeStark map of hydrogen showing linear Stark shifts for |m| = 0 states. Every clusterof l states is often called a hydrogenic manifold; adapted from [56]. The classicalionization limit Wc = −2

√E is shown by a heavy curve where the Stark states begin

to be broadened by field ionization. Quasi-discrete states with lifetime τ > 10−6s(solid line), field broadened states 5 × 10−10s < τ < 5 × 10−6s (bold line), and fieldionized states τ < 5 × 10−10s (broken line). Right: The charge density distributionof hydrogen atoms in a electric field |m| = 0. Each figure is a parabolic eigenstatewhich is a superposition of many l states of hydrogen. Moving from the left toright, top to bottom, these figures are designated by the parabolic quantum numbersk = n1 − n2 = 7 to −7 which are the extreme blue components to the extreme redcomponents; figure adapted from [62].

28

Figure 3.4 : NonHydrogen Rydberg Atoms in a Electric Field Left: Pre-cession of a nearly Keplerian elliptical orbit of a Rydberg electron about the coreion in an electric field. The precession is produce by adding an induced dipole term−αd/2r4 to the Coulomb potential which corresponds to the effects of polarizationinduced in the core. The top figure is with a negative α while the bottom one iswith a positive α; figure adapted from [63]. Right: The Stark map for potassium|m| = 0 states, the anti-crossings that appear near level intersections are obvious.Figure adapted from [56].

29

k = n1 − n2 ∼ n. Also because of the charge distribution, it is easier to ionize the

electron in the extreme red state (in the last subfigure).

For non-hydrogenic Rydberg atoms, the l-degeneracy is lifted by the interaction

with the core. For low-l states non-degenerate time-independent perturbation theory

results in a quadratic Stark shift. Though the time-average of the precession dimin-

ishes the existence of a permanent dipole moment in zero field, a small dipole moment

can be induced by, and interact with, the electric field applied. This behavior can

be visualized as a non-uniform precession of the Keplerian orbital; see Figure 3.4.

However, this effect is only apparent for the low-l states since the interaction with the

core falls off quickly with increasing l. The high l states are still essentially degen-

erate, so in the Stark map, display a near linear Stark shift. One subtle difference,

compared with hydrogen Rydberg atoms, is the anti-crossings that appear between

different Stark states as the electric field is increased. In the following subsection, a

calculated description of the behavior of Sr Rydberg atoms in an electric field will be

presented together with the experimental measurements.

3.3.2 Sr Stark Map

Figure 3.5 shows the calculated eigenenergies for singly-excited strontium (n ' 50)

states as a function of the strength, Fdc, of a dc field applied along the z axis. The

high-l states which are nearly degenerate at Fdc = 0 exhibit a linear Stark shift and

Stark states of two adjacent n manifolds first cross at a field strength of

Fcross '1

3n5. (3.9)

As explained previously, for the low angular momentum (1P1,1D2) states only the

quadratic Stark shift can be observed. In Figure 3.5 the measured excitation spectra

30

of strontium around n = 310 are also plotted. In these measurements orthogonal

polarizations of the 461 nm and 413 nm were used to avoid excitation of the 1S0 states

and simplify the excitation spectrum (The dc field is parallel to the polarization of

the 461 nm laser). This setup yields Rydberg states with the total magnetic quantum

number M = ±1. In order to compare the spectra for different values of n, the energy

axis is scaled by En − En−1 ' n3 and the field axis is by Fcross. Using two-photon

excitation, only n1D2 states can be excited at Fdc = 0. With increasing strength

of the dc field, the nD states become coupled with other angular momentum states

and these l-mixed states have smaller oscillator strengths than the unperturbed D-

states. As the state merges with the linear Stark manifold, the l-mixing is so strong

that the effect of the core scattering becomes negligible. Thus the state can become

strongly polarized and almost indistinguishable from the extreme red-shifted strongly

polarized Stark states. The behavior of the “312D” level mirrors that observed in

earlier studies [61] at lower n, n ∼ 80 which data are also included in Figure 3.5.

Slight shifts of the energy levels seen in the calculated 52P and 52D states are due

to the underestimated quantum defects. This evolution of the n1D2 states can be

visualized by plotting the distribution of the parabolic quantum number k

ρ(k) =∑n

|H 〈n, k,m| |nStark〉Sr |2 (3.10)

or, equivalently, the distribution of Az (Az is the z-component of the Runge-Lenz

vector) as k corresponds to the quantized action of −nAz. Here, |n, k,m〉H are the

parabolic states of the hydrogen atom and |nStark〉 is the outer electron state for an

eigenstate of strontium in a dc field (The inner valence electron is almost exclusively

in the 5s state). Figure 3.6 displays the evolution of the k-distributions as a function

of Fdc for the state which is the 52D state (M = 1) at Fdc = 0. For weak fields,

the k-distribution spreads over a wide range between −n and n. This indicates that

31

the state is unpolarized. Since the Runge-Lenz vector indicates the orientation of

the Kepler ellipse in classical dynamics, a wide distribution of Az implies an ensem-

ble of Kepler ellipses (with l ∼ 2) whose orientations are broadly distributed. For

non-hydrogenic atoms, such a distribution is formed by core scattering which changes

the orientation of the ellipse while keeping the eccentricity. A node near k = 0 is

also noticeable in the plot which mirrors a node of the spherical harmonic Y m=1l=1 .

With increasing Fdc the node is shifting towards the negative k side and the biased k-

distribution indicates that state is becoming increasingly polarized. Near the merging

with the neighboring Stark manifold the k-distribution becomes very narrow indicat-

ing a convergence towards a single parabolic state. The 52P state, on the other hand,

does not show any hints of polarization. This is because its dipole-coupled partners,

S- and D-states, are also hard to polarize. The 52D state is dipole coupled to the

50F state which merges with the Stark manifold at relatively weak Fdc and becomes

polarized. The polarization of the 52D state is, therefore, caused by the coupling

with this polarized “50F” state.

The evolution of the calculated dipole moment, 〈z1 + z2〉, of several eigenstates

around the “52D-state” is shown in Figure 3.6. The states nearly degenerate at Fdc

= 0 becomes polarized even for very weak fields and dipole moments are given by

〈(z1 + z2)〉 = (3/2)nk. For the isolated low-L states, the dipole moment grows almost

linearly in Fdc, i.e.

〈z1 + z2〉 = −αFdc (3.11)

when Fdc ' 0. These linear shifts lead to the energy shift ∆E = −(1/2)αF 2dc quadratic

in Fdc. The polarizability α can be approximated using second-order perturbation

32

Figure 3.5 : Stark Map of Strontium Rydberg Atoms Evolution of the ex-citation spectrum with increasing applied dc field in the vicinity of n ∼ 310( thickred line). The thin solid lines indicate the calculated eigenenergies of singly-excitedstrontium (n ' 50) in a dc field while the dashed blue lines denote the correspondingexcitation spectrum. The squares are the results of earlier measurements at lower n,n ∼ 80 from [61]. Fdc is normalized to the crossing field strength Fcross ∼ 1/(3n5)and the energy is normalized by En − En−1 ' 1.

33

theory as

α = 2∑n′

∑L′=L±1

| 〈nLM | (z1 + z2) |n′L′M〉 |2

En′L′M − EnLM. (3.12)

Numerical calculations show that the polarizability α is dominated by a single term

in the summation for the 50F state as the dipole-coupled state (50G) is almost de-

generate in energy due to its almost vanishing quantum defect. The resulting large

polarizability leads to sizable energy shifts. For the 52P state, similarly, the coupling

to the 52D state dominates the summation in Equation 3.12. However, the large

energy difference (see Figure 3.5 ) due to the quantum defect suppresses the values

of α and, therefore, the state is hardly polarized. The 52D state is found between

two dipole coupled states, 52P and 50F, and is slightly closer to the 50F. Therefore,

the coupling with the 50F dominates over that with 52P leading to the polarization

towards the downhill side resulting in a larger polarizability than that for the 52P

state. We note that, judging from the quantum defect, the (n + 2)D state is found

slightly closer to the midpoint between the (n + 2)P state and nF state for higher

values of n. In fact, the polarizability of the D state appears to be smaller for the

312D state as the measured spectrum (Figure 3.5) shows a smaller energy shift than

that of the calculation for n = 50. With increasing field the growth of the dipole mo-

ment becomes non-linear in Fdc, implying the non-negligible role of the higher-order

perturbation terms, i.e., strong mixing with higher L states. Thus the states can be-

come polarized through a superposition with those high-L states and, as seen for 50F

and 52D, their polarizations approach the maximum value of 〈z1 + z2〉 = 1.5n2a.u..

34

Figure 3.6 : Parabolic State Distribution of Strontium Stark States Lefthand panels show the probability distribution of the parabolic quantum number k(=−nAz) as a function of the dc field strength Fdc. The evolution of the states whichare, at Fdc = 0, the 52D state, 52P state and 50F state are plotted. The distributionfor the 52P state is truncated where it merges into a Stark manifold. On the righthand side, the average dipole moment of selected states including 52D, 52P, 50F aswell as the downhill and uphill Stark states are plotted. Fdc is normalized to thecrossing field strength Fcross.

35

Chapter 4

Experiment Setups and Techniques

For very-high-n strontium Rydberg atom creation in a thermal beam, most of the

equipment is the same as that employed in previous, successful Rydberg experiments

on potassium which jump-started this experimental exploration of strontium. This

left us with only two major new construction projects, the laser system and the stron-

tium oven. Therefore this chapter will concentrate on these new pieces of apparatus

and only make short comments on its other components. Finally, the experimental

techniques employed will be summarized.

4.1 Frequency Locked Diode Laser System

We use two Frequency Doubled High Power Diode Laser Systems from TOPTICA

PHOTONICS to drive the two-photon Rydberg atom excitation. In our applications,

both of them are required to be locked on a specific frequency with MHz accuracy

for 8 hours continuously. This is accomplished by locking them with respect to a

commercial frequency stabilized Helium-Neon laser via a scanning Fabry-Perot in-

terferometer. Their absolute wavelengths are determined using a commercial high

resolution wavemeter which produces GHz accuracy. In the following, the major

components of the locking system will be described as an introduction to explain-

ing the locking scheme later. In the last subsection, system limitations and a few

alternative locking schemes that might provide better performance will be discussed.

36

Figure 4.1 : TOPTICA Diode Laser System Schematics adapted from toptica.com

4.1.1 Frequency Double High Power Diode Laser System[57]

This tunable diode laser system is very compact and rugged. Its narrow linewidth

(MHz over millisecond timescales), high and stable output power (100mW after

doubling) and high tunability makes it perfectly suited for our applications. The

whole laser consists of an diode laser system coupled to an electronic control system.

The electronic control system contains plug-in modules for the DC and HV power, the

diode current, temperature control, the crystal temperature control, laser modulation

37

and regulation, and external interfaces in a 19” unit.

As shown in Figure 4.1, the laser source is a grating stabilized external cavity

diode laser system based on the Littrow-Hansch scheme. With a cavity formed by

the front facet of the diode and a holographic optical grating next to the rear facet of

the diode, this scheme offers a much smaller linewidth (1MHz) as compared with a

bare diode(100MHz). Besides the wavelength can be tuned easily over a large range.

Coarse tuning is achieved by adjusting the angle of the grating via a micrometer

screw and fine tuning is obtained by scanning the cavity length via the piezo element

attached to the grating. Mode hop free tuning is achieved by feedforwarding a current

proportional to the scanning voltage to the diode. To maintain single mode operation,

both the temperature and current of the diode head need to be adjusted together with

the grating.

Upon leaving the master laser diode, the collimated infrared laser beam( 40mW

approximately) is focused, mode matched into another diode, the tapered amplifier

(TA) to achieve more power than is possible with a single-mode laser diode. The gain

bandwidth of the tapered amplifier is usually of order of some tens of nanometers.

Due to the diodes’ sensitivity to feedback, high-suppression-ratio optical isolators are

integrated in the optical path to avoid reflection. Finally the output from the TA

(300mW )is mode matched to couple it into the bow-tie-ring resonator for frequency

doubling. Phase matching of the crystal is sensitive to both the alignment and the

temperature. Second harmonic output powers of 100mW are obtained.

The stabilization of the doubling cavity is achieved mainly via two feedback

loops. Their common error signal is generated by applying the Pound-Drever-Hall

scheme [58]. An RF modulation fed into the current of the master laser head pro-

duces two sidebands which, together with the carrier are all sent to the doubling

38

cavity. Their reflection from the cavity is collected by a fast photodiode (shown in

Figure 4.1) whose output is then fed into the PDD 110 module (the Pound-Dever-Hall

Detector). PDD mixes this signal with the same RF local oscillator used before to

extract their DC phase information which is the output error signal. The error signal

is then fed into two loops. The slower feedback loop (a few kHz) is closed by the

PID regulator stabilizing the cavity length with the piezo element attached to one of

the mirrors in the doubling cavity. The fast loop (5MHz) is closed through adjusting

the master diode’s current to lock the laser frequency to the doubling cavity. In the

fast loop, the doubling cavity is the reference cavity. The combination of these two

loops gets rid of thermal and acoustic noise as well as fast frequency jitter and can

maintain a narrow linewidth.

The electronic plug-in modules can also communicate via the backplane of the

rack if the jumpers are set accordingly. The feedforward function, for instance, is

achieved by sending a portion of the scanning voltage to the current control module

of the diode head(DCC). Thus this voltage/current is added to the current set-

value on the DCC and output to the diode head. The external input of the SC 110

module, which controls scanning of the grating of the master laser diode, is connected

to the same line as the BNC input/output of the computer analog interface DCB.

Therefore, the input from the BNC connector to the DCB module will be added to

whatever voltage is set on the SC module and then output to the grating in the master

laser diode. This particular feature will be used in our locking scheme discussed later.

4.1.2 HeNe Laser

Frequency/Intensity stabilized HeNe reference laser at 632.816nm is a commercially

available unit. Frequency stabilization is attained by comparing the intensity balance

39

of two orthogonally polarized longitudinal modes and can offer a ±2MHz frequency

stability on an eight hour timescale. The laser is very sensitive to any retroreflections.

Given the lack of an optical isolator, we use a 20dB neutral density filter to block

reflections from the Fabry-Perot Interferometer (will be discussed later). In addition,

the interferometer must be intentionally misaligned slightly to prevent reflections back

to the HeNe laser.

4.1.3 Scanning Fabry-Perot Interferometer

The finesse of a Fabry-Perot Interferometer, a measure of the interferometer’s ability

to resolve spectral features, is essentially determined by the reflectivity of the two

mirrors on each end of the cavity F = nR/(1−R2). Thorlabs offers scanning confocal

Fabry-Perot Interferometers (etalons) but not with optical coatings suitable for our

needs. In order to achieve a finesse F = 200 for 633nm, 412nm and 461nm laser

light, we ordered an SA200 with customized coatings on both (confocal) mirrors.

The free spectral range(FSR) of this etalon (≈ c/(4L),where L is the mirror spacing)

is 1.5GHz. With better alignment, when the laser beam is on the optical axis, the

free spectral range becomes c/(2L).

This customized SA200 is driven by the matching SA201 Spectrum Analyzer Con-

troller. This controller provides a saw-tooth waveform with adjustable ramp ampli-

tude and rise time and can be externally triggered. In our experiment, we set the

amplitude of the ramp to just cover one free spectral range so that two peaks of

the HeNe laser can be observed on the output spectrum. We scan the etalon at

50Hz which given the FSR of the etalon will yield peaks having a temporal width of

∼ 100µs.

The optical length of the Fabry-Perot Cavity is also sensitive to air pressure,

40

airflow and temperature change. Since we are using this cavity to lock the laser

frequencies, it is important to make it as stable as possible. Therefore, we put the

whole scanning etalon into an aluminum housing specially made for this purpose.

This enclosure is air-sealed. Laser light enters and exists the two AR coated N-BK7

windows sealed by O-rings. A small hole running cables is sealed by vacuum epoxy

glue.

4.1.4 Data Acquisition System

Laser control is accomplished using Labview and an NI PCle-6341 Xseries DAQ board.

This particular DAQ board has eight analog inputs with a total sampling rate of

500kS/s and two analog output channels. There are also plenty of digital lines with

even higher reading and writing rates.

4.1.5 Locking Scheme

The essence of our laser locking is that by continuously scanning the Fabry-Perot

interferometer, the relative frequency difference between the target laser and the

reference HeNe laser can be monitored and can be locked to a fixed value by feedback.

Here is how this scheme implemented in our system: Three laser beams (the HeNe, the

461nm, and 413nm beams) are all sent into the etalon, and the resulting transmission

peaks from each laser are detected by three independent photo-diodes. The signals

from the photo-diodes are fed into three analog input channels of the DAQ board.

One more analog input channel is used to display the ramp signal from the SA201.

The Labview program first locks the Fabry-Perot cavity to the HeNe laser. This is

done by feedback; any change in HeNe peak position (with respect to the ramp) is

compensated by changing the offset of the ramp voltage from the SA201. This step

41

is necessary due to the fact that the Fabry Perot cavity drifts due to the thermal

expansion or electrical drifts even though it is somewhat thermally isolated in the

aluminum enclosure. The next step is to continuously compare the relative position

of the HeNe peak and the 461nm/412nm laser peaks and generate feedback signals to

compensate for any changes to the diode laser systems to make their relative position

stay at the set value. In this step, the feedback voltage is sent to the BNC input

of the DCB module of the corresponding laser system. So, ultimately, the feedback

is to the position of the grating. In addition, both lasers can be locked or scanned

anywhere over the whole ramp which is simply realized by modifying the set value of

their positions relative to the HeNe peak.

Our locking scheme as described is able to lock both diode laser system to a

range of ±2MHz stably over the course of a day. And it effectively compensates the

long-term thermal drift the diode laser system suffers and successfully satisfies our

experimental needs.

4.1.6 Limitations and Alternatives

There are a couple of factors limiting the bandwidth of our stabilization scheme like

the processing rate of the Labview program and the sampling rate of the DAQ board.

But the major limitation is from the necessity of scanning the Fabry-Perot Cavity

over a whole Free Spectral Range which requires 20ms. Faster feedback is possible

by increasing the scanning rate and, accordingly, the sampling rate.

Locking a laser, like the 412nm laser, that is not associated with direct atomic

transitions or very close to any reference laser (up to GHz frequency difference), is dif-

ficult. Otherwise, either saturation absorption could be performed and the linewidth

of the locking could be cut to 100kHz or, two lasers’ beat signal could be utilized

42

to perform a fast heterodyne locking scheme. There is one other approach as de-

scribed in [59]. They first find a cavity length that coincides with the transmission

maximas of the target laser and the reference laser by generating a sideband from

current modulating the reference laser. Then they lock the cavity to the reference

laser and lock the target laser to the cavity by analog circuitry. The scan of the target

laser is obtained by scanning the sideband of the reference laser. This approach has

achieved sub-MHz linewidth. Another approach is an analog of saturation absorp-

tion spectroscopy. Using the EIT signal from the coherent two-photon transition, the

linewidth should also be greatly reduced.

4.2 Strontium Oven

While several strontium atom beam sources have been described, these are typically

designed to optimize beam fluxes rather than to achieve tight beam collimation. Al-

though beam divergences can be reduced by use of multicapillary arrays, these alone

cannot provide the required collimation. Here beam divergence is controlled through

the use of a small oven aperture and tight beam collimation. Because the oscilla-

tor strengths for excitation of high-n Rydberg states are small, sizeable atom beam

densities are required to achieve reasonable Rydberg photoexcitation rates. Given

that the oven aperture is small, this requires the vapor pressure in the oven be high

necessitating operation at temperatures of ∼ 500 − 650C . Such temperatures can

be reached using commercial resistive heating elements provided that thermal losses

are minimized. While the present source design builds on earlier practice, its use

of a commercial heater allows a particularly simple design that is straightforward to

fabricate. The source has proven reliable in operation and, with appropriate collima-

tion, can provide an atomic beam with a divergence of ∼5mrad FWHM and densities

43

Figure 4.2 : common elements vapor pressure chart, downloaded online

44

approaching 109cm−3 sufficient to enable a wide variety of experiments with high-n

strontium Rydberg atoms.

The present atom source is shown in Figure 4.3. Its central component is a

cylindrical stainless steel oven that is loaded with granular strontium metal from the

rear. Atoms emerge from the oven through an orifice in the form of a cylindrical canal

that is ∼ 0.5mm in diameter and ∼ 1.2mm long. The oven fits inside a commercial

spiral wound coil heater.This heater comprises a heating element that is electrically

isolated inside a stainless steel sheath using MgO. The heater includes an unheated

section at one end through which the heater leads enter and exit, and also contains

an internal thermocouple to monitor its internal temperature. In air, the heater is

rated for 350 W and operation at temperatures up to ∼ 820C. The spacing of the

heater coils is adjusted such that they are closer together near the front of the oven

than at its rear. This ensures that the front of the oven is maintained at a higher

temperature than its rear to prevent the exit canal from becoming blocked through

condensation. To further limit such condensation, the exit orifice is offset towards the

side of the oven and the heating coils are extended well forward of the front of the oven

to provide it with a strong radiant heat bath. The heating coil fits inside a polished

cylindrical copper jacket. The low emissivity of copper minimizes the heater power

required to reach a given operating temperature. Although hot copper reacts with

strontium, the jacket is not exposed directly to strontium vapor and no problems with

such reactions have been encountered. The oven and heater are positioned within the

jacket using a single small screw;see Figure 4.3. The temperatures of the front and

back of the oven, and of the jacket, are monitored using thermocouples.

To ensure good thermal isolation, the copper jacket is held in place using a number

of small stainless steel mounting screws that pass through ceramic bushings. Two

45

Figure 4.3 : Schematic diagram of the oven assembly

46

pairs of horizontally-opposed screws position the jacket vertically within a “U”-shaped

aluminum support bracket. Horizontal positioning is achieved with the aid of a single

vertical screw that passes through a second mounting bracket that runs up and over

the front of the jacket. To allow for expansion upon heating a small clearance is

included between the mounting brackets and the heater jacket, and the mounting

screws are free to slide within their bushings. The support bracket is held in place

by an arm that is connected via a bellows to an x-y translation stage located outside

the vacuum region. This stage is used to position the oven orifice during initial

beam alignment. The whole oven assembly is mounted inside a water-cooled copper

enclosure. To limit the temperature rise of the support bracket and of the power

input end of the heater, these are each cooled by connecting them via copper braids

to the enclosure. A 4 mm-diameter aperture located ∼ 4cm from the oven orifice is

used for initial beam collimation. Final collimation is provided by a 0.5 mm-diameter

aperture ∼ 10cm from the oven. No significant changes in beam pointing have been

observed as a result of day-to-day cycling of the oven temperature.

The oven is typically operated at temperatures in the range ∼ 500− 650C. The

power input to the heater required to reach these temperatures is ∼ 25−50 W and the

internal temperature within the heater remains below 800C. While the strontium

atom beam density cannot be simply measured directly, it can be inferred from earlier

measurements (using a hot wire ionizer) of potassium atom beam densities produced

by an oven having a similar orifice and operating at similar metal vapor pressures;

see Figure 4.2. These earlier measurements suggest that at operating temperatures

above ∼ 600C strontium beam densities approaching 109cm−3 should be obtained.

Measurements of the photoexcitation of strontium Rydberg states demonstrate that

sizeable beam densities are produced. While much of the increase in Rydberg produc-

47

tion can be attributed to differences in oscillator strengths, the data do indicate that

sizeable strontium atom beam densities are obtained and that large photoexcitation

rates can be achieved.

The present atom source has proven reliable in operation and even with day-to-day

cycling of the oven temperature no heater failure has occurred. While the lifetime

of an oven charge depends on the required beam density, its capacity has proven

more than sufficient to allow several months of operation before reloading becomes

necessary. However, if required, longer operational periods could be achieved by

simply scaling up the design (a wide range of suitable coil heaters are available).

4.3 Other Experimental Apparatus

4.3.1 Vacuum System

The whole vacuum system consists of two relatively separate sections, the oven sec-

tion where the oven is mounted and the interaction region section, where ground

state strontium atoms are photo-excited to Rydberg levels and subsequently field

ionized and detected. These two sections are connected by a 500µm-diameter aper-

ture through which the atomic beam passes. Therefore, for efficient pumping, the two

sections are differentially pumped using diffusion pumps, a Varian VHS-4 for the oven

section and VHS-6 for the interaction region section. Their pressures are measured

by two Bayard-Alpert ionization gauges and are typically ∼ 10−7 torr.

The laser beams enter the vacuum system through two Ø1” AR coated Thorlabs

N-BK7 broadband precision windows traveling in opposite directions. The windows

are mounted slightly off normal to the beams to avoid retroreflections. The vacuum

seal is accomplished by Parker O-rings. The 412nm laser beam is focused by a Ø1”

48

Figure 4.4 : Interaction Region

f =40cm Thorlabs N-BK7 Plano-Convex Lens to the beam waist, 170µm, in the

center of the interaction region.

4.3.2 Interaction Region

The interaction region is bounded by three pairs of 10cm×10cm electrodes (see fig-

ure 4.4). Except the bottom plate, 5 of them can be biased independently to cancel

the stray electric fields in three orthogonal directions so that local electric field can

be reduced to ≤ 50µV/cm. A circular electrode disk mounted from the top plate is

used for applying fast electric pulses (often called half-cycle pulses) in the z direction

and shares its bias potential with the rest of the plate. Sometimes, half-cycle pulses

(HCP) are also applied to the side-electrode as shown in Figure 4.4. For detection,

the Rydberg atoms are field ionized by a voltage ramp which is applied to the bottom

49

plate. The resulting electrons exit through the 1” inch aperture on the bottom plate

covered with fine copper mesh to be detected by the funnel shaped aperture of a Dr.

Sjuts channel electron multiplier KBL 25RS mounted beneath the mesh.

4.4 Experimental Techniques

As mentioned earlier, we use a ramped electric field to field ionize the Rydberg atoms

and subsequently collect the resulting electrons. Field ionization enables state selec-

tive detection and is widely employed in Rydberg atom experiments.

4.4.1 Selective Field Ionization

Classical ionization occurs when the electron’s potential energy becomes less than its

total energy. For an electron of energy −1/(2n2) subjected to an electric field ~E, the

total potential energy is −1/r + ~E · ~r which scales as −1/n2 + En2. The classical

ionization limit can be heuristically obtained by equating them, −1/n2 + En2 =

−1/(2n2), thus the ionization field scales as 1/n4. Calculation of saddle point in the

potential yields 1/(16n4) as the classical ionization limit which is shown in the Stark

map for sodium in Figure 4.5. Rydberg atoms are indeed ionized around this limit.

However, before being completely ionized, the energy level of the electron is first

Stark shifted or/and Stark mixed according to its l and the rise time of the applied

field, and these effects determine the exact field at ionization. Consider the Stark map

for sodium in Figure 4.5. Whereas the high l manifolds display linear Stark shifts, the

low-l states display quadratic Stark shifts just as was discussed in Chapter 3. As the

field increases, different Stark states experience avoided crossings. If the electric field

strength increases slowly enough, these crossings are traversed adiabatically, the states

following the paths indicated by the solid lines and ionizing at fields ∼ 1/(16n4). On

50

Figure 4.5 : Stark map for Sodium showing evolution of the m = 0 states in anincreasing field. Color red represents d states (δd = 0.01), color green representsp states (δp = 0.86) and color purple represents the s states (δs = 1.35). In thisparticular case, d states are very close to the manifold which are the whole cluster ofhigh l states. In this graph, the manifold states are represented by the 5 states justabove d states. The dotted lines are the diabatic paths for s,p,d states to ionize whilethe adiabatic paths are the solid black serpentine lines for each states. The coloreddots on the ionization curve are the ionization points for the adiabatic passages ofeach state. This graph is adapted from [56].

51

the other hand, a rapid rise in the applied electric field will result in diabatic passage,

states will follow a linear energy shift after mixing with the extreme l states in the

manifold and they will be ionized in a order that is completely different from that

of their zero field energies; see the dotted lines in Figure 4.5. In our experiment,

ionization can be regarded as diabatic because the energy separation at the avoided

crossings are very small for n ∼ 300.

No matter if the ionization is diabatic or adiabatic, the electric field required, to a

good approximation, scales as 1/(n4). If a ramped field is applied, different states will

ionize at different field and thereby different times. The resulting ionization signal

vs. electric field will provide information about the Rydberg states initially present.

With careful analysis through “selective field ionization” (SFI), especially for n≤100

when the available resolution is high, it is possible to obtain considerable information

on the initial n distribution. However, when n ∼ 300, it is only possible to infer the

approximate n from SFI spectrum.

To generate a SFI spectrum, we measure two quantities using techniques adapted

from previous experiments on potassium for which the excitation rate of Rydberg

atoms was low. Following each laser pulse, there is only one or zero atoms excited

to the Rydberg level. So single particle detection can be employed using a channel

electron multiplier. This amplifies the incident electron and produces a large number

of secondary electrons that generate one large signal pulse for detection. Besides this

signal, we also measure the time duration from the start of the ramp to the time

when we get the signal from the channel electron multiplier. Since we know the time

evolution of the ramp, with these two quantities, we can obtain the electric fields

needed to ionize that Rydberg atom.

Although we don’t use the SFI spectrum directly in the beam experiments. It

52

produces a valuable detection scheme. In our strontium experiment, we found a ramp

that rises from 0 to 5 Volts in 5 µs is more than enough to ionize all the Rydberg

atoms. .

4.4.2 Data Acquisition

The very-high-n Rydberg experiment is triggered by a master pulser which imme-

diately triggers a 461nm laser pulse for Rydberg excitation. About 10µs later, the

ramped field is applied to the bottom plate. Right before the ramp, an ORTEC 566

time-to-amplitude converter (TAC) is started by a trigger pulse and it is stopped by a

signal from the channeltron. The output pulse from the channeltron is then converted

into a TTL signal by a charge sensitive amplifier and is fed into a computer. This

computer also controls, scans the 412nm laser frequency, adjusts the bias voltages

on the plates of interaction region and controls the multiple half-cycle-pulses (HCPs)

that are used for engineering quasi-one-dimensional states in some other experiments.

53

Chapter 5

Sr Rydberg Atoms in a Collimated Atomic Beam

Initial experiments focused on the production of Sr Rydberg atoms, their spec-

troscopy, their photo excitation rates, and their excitation in a weak dc field.

Strontium atoms contained in a tightly-collimated beam are excited to the de-

sired high-n (singlet) state using the crossed outputs of two frequency-doubled diode

laser systems. The two-photon excitation scheme employed utilizes the intermediate

5s5p1P1 state and radiation at 461 nm and 413 nm. The laser beams, which typ-

ically have the same linear polarization, cross the atom beam traveling in opposite

directions. Since their wavelengths are comparable, the use of counter-propagating

light beams can largely cancel Doppler effects associated with atom beam divergence

resulting in very narrow effective experimental linewidths. As noted earlier, residual

stray fields in the experimental volume are reduced to ≤ 50µV cm−1 by application

of small offset potentials to the electrodes that surround it. Measurements are con-

ducted in a pulsed mode. The output of the 461 nm laser is chopped into a series

of pulses of ∼200 ns to 1 µs duration and 20 kHz pulse repetition frequency using

an acousto-optic modulator (The 413 nm radiation remains on at all times). Follow-

ing each laser pulse, the probability that a Rydberg atom is created is determined

by state-selective field ionization for which purpose a slowly-rising (risetime ∼1µs)

electric field is generated in the experimental volume by applying a positive voltage

ramp to the lower electrode. Product electrons are accelerated out of the interaction

region and are detected by a particle multiplier. The probability that a Rydberg atom

54

is created during any laser pulse is maintained below 0.4 to limit saturation effects.

This is accomplished by reducing the strontium atom beam density by operating the

oven at a lower temperature, and/or by reducing the laser powers.

5.1 Spectroscopy

Figure 5.1 shows excitation spectra recorded in the vicinity of n = 282 for various

detunings of the blue 461 nm laser selected to optimize the transitions 5s2 1S0 →

5s5p 1P1 in the different strontium isotopes. The relative frequency of these tran-

sitions together with other properties of naturally-occurring strontium are listed in

Table 2.2. The isotope shifts and hyperfine splittings in both Table 2.2 and Fig-

ure 5.1 are quoted relative to the dominant 88Sr isotope. The blue 461 nm laser

beam was unfocused and had a diameter of ∼3 mm. Its intensity, ∼10 mW cm−2,

was selected to limit line shifts and broadening due to effects such as the ac Stark

shift and Autler-Townes splitting. Its pulse width, ∼ 0.5µs, was selected because

for shorter pulse durations the widths of the spectral features become increasingly

transform limited. The “purple” 413 nm laser beam was focused to a spot with a

full width at half maximum (FWHM) diameter of ∼ 170µm, resulting in an intensity

of ∼ 250Wcm−2. The frequencies of both lasers were stabilized and controlled with

the aid of an optical transfer cavity locked to a polarization-stabilized HeNe laser as

described in Section 4.1. This cavity allows uninterrupted tuning of the lasers over

frequency ranges of up to ∼800 MHz. The frequency axis in Figure 5.1 shows the

sum of the blue and purple photon energies. The spectrum obtained with the blue

laser tuned to optically excite the dominant 88Sr isotope displays a series of sharp

peaks associated with the excitation of 1D2 states with the n values indicated. The

spectrum also contains a series of smaller peaks associated with the production of 1S0

55

Figure 5.1 : Excitation spectra recorded in the vicinity of n = 283 for differentdetunings of the 461nm laser. These detunings, specified relative to the 88Sr 5s21S0 →5s5p1P1transition, are indicated in the figures. The frequency axis shows the sum ofthe 461 nm and 413 nm photon energies. The horizontal bars beneath the dataidentify the features associated with excitation of 5snd1D2 Rydberg states in the88Sr, 86Sr and 84Sr isotopes, together with the positions of features associated withexcitation of 87Sr Rydberg states. Two red arrows in the top subplot point to the3D2 state of 88Sr.

56

states. The widths of these features, ∼5 MHz FWHM, is attributed to a combination

of transit time broadening (the transit time of an atom through the purple laser spot

is ∼ 300 ns ) and fluctuations in the laser frequencies during the ∼ 1 s required to

accumulate data at each point in the spectrum. The probability that a Rydberg atom

was created at the peak of the 1D2 features during each laser pulse was limited to

∼ 0.5 by substantially reducing the strontium atom beam density by operating the

oven at ∼ 500C. Tests were undertaken at higher operating temperatures in which

the 413 nm laser beam was attenuated using neutral density filters to limit the ex-

citation rate. These tests showed that, with the oven operating at 630C and using

the full 413 nm laser power (∼ 70 mW), ∼10-15 Rydberg atoms can be produced per

laser pulse in the excitation volume of ∼ 5× 10−5cm3. This corresponds to a typical

inter-Rydberg spacing of ∼ 130µm which is approaching those at which effects due

to Rydberg-Rydberg interactions such as blockade become important.

5.1.1 Even Isotopes

When the blue laser is tuned on resonance with the 1S0 → 1P1 transition for 88Sr,

as shown in the top subplot in Figure 5.1, the two 1D2 of 88Sr peaks dominate the

spectra. If we normalize the frequency separation between singlet 1D2 states with

consecutive n as 1, then the nearest 1S0 state to a 1D2 should be separated by

|(δ1S0mod1)− (δ1D2

mod1)| = |0.269− 0.381| = 0.112,

where δ is the quantum defect of the corresponding states; see Table 5.1. Inspection

of the figure shows that there is an S state lying 0.11 to the right of every 1D2 state

in accordance with the quantum defect reported previously. The same calculation

can also be carried out for the 3D2 state, which shows that its separation from the

57

nearest 1D2 should be

|(δ3D2mod1)− (δ1D2

mod1)| = |0.636− 0.381| = 0.245.

As marked on the top subplot of Figure 5.1 by red arrows, a small signal from 3D2

states can also be discerned at the expected relative positions. The small size is due to

the fact that ∆S 6= 0 transitions are forbidden in the Russell-Saunders LS-coupling

scheme. For heavy atoms like strontium, however, LS-coupling breaks down and

strong spin-orbit interactions mix the wavefunctions from singlet series and triplet

series. So intercombination transitions are weakly allowed for Sr. But the spin-orbit

interaction strength or equivalently, the fine structure (EFS ∼ (n∗)−3) diminishes

quickly as n increases and, this mixing gets substantially weaker for very high-n

Rydberg states. Therefore, the 3D2 states are barely excited.

As the blue laser is red detuned from resonance with the 1S0 → 1P1 transition

in 88Sr, the size of the 88Sr features in the excitation spectra decreases steadily

and new features emerge associated with the excitation of Rydberg states of the

other isotopes. At a detuning of ∼ −122 MHz, which optimizes the 1S0 → 1P1

transition for 86Sr, the excitation spectrum is dominated by the creation of 86Sr 1D2

states, although some residual excitation of 88Sr isotopes remains. The observed

88Sr → 86Sr isotope shift in the series limit, +210 ± 5 MHz, is consistent with that

reported in earlier spectroscopic studies at lower n [45]. At a blue laser detuning of ∼

−273MHz which optimizes the 1S0 → 1P1 transition in the 84Sr isotope, the excitation

of 84Sr1D2 Rydberg states becomes apparent. The observed 88Sr → 84Sr isotope shift

in the series limit, +440± 8MHz, is again consistent with earlier measurements [45].

However, because the fractional abundance of the 86Sr and 84Sr isotopes in the beam

shown in Table 2.2, are much less than for the 88Sr isotope, many fewer Rydberg

atoms are created.

58

Table 5.1 : Sr Quantum Defects [65]

Series δ

5sns1S0 3.269

5snp1P1 2.730

5snd1D2 2.381

5snf 1F3 0.089

5sns3S1 3.371

5snd3D3 2.630

5snd3D2 2.636

5snd3D1 2.658

At blue laser detunings of ∼ −90 MHz, the sizes of the features associated with

the excitation of 88Sr and 86Sr Rydberg states are nearly equal, resulting in the

appearance of two separate interleaved Rydberg series. The 88Sr to 86Sr isotope

shift, ∼ 210MHz, matches the energy spacing between adjacent Rydberg levels at

n ∼ 313, indicating that in the vicinity of n ∼ 313 the two interleaved Rydberg series

should overlap, the 312 1D288Sr Rydberg level, say, overlapping with the 313 1D2

86Sr

Rydberg level. Spectra recorded in the vicinity of n = 312 (see Figure 5.2) under

these conditions do indeed display only a single series of Rydberg features.

5.1.2 the odd isotope 87Sr

Several new features are also observed in the excitation spectra at blue laser detunings

chosen to favor excitation of the 87Sr isotope. These features do not conform to a

59

Figure 5.2 : Spectra recorded with the detunings ∆ shown in the vicinity of N ∼ 312.In (a), 88Sr is dominant. In (b), the detuning ∆ favors the 1S0 → 1P1 transition ofboth 87Sr and 86Sr. The most discernible peaks for 87Sr are highlighted by greenarrows and they will be explained in Subsection 5.1.2. The 1D2 peaks for n86Sr atthe first three ns are distinguishable from those resulting from residual excitation of(n + 1)88Sr. At higher n, they merge resulting in a single sharp spectral feature. In(c), where the detuning mostly favors 86Sr, small peaks from 88Sr can be observedon the left of the first two dominant n86Sr peaks.

60

simple Rydberg series having the expected values of n. Similar behavior has been

observed previously in studies at lower n and assigned to a combination of strong state

mixing, hyperfine-induced singlet-triplet mixing, and interactions between states of

different n [47]. These complexities of 87Sr Rydberg series are mainly caused by its

nuclear spin I = 9/2 which is absent for all the even isotopes. Major consequences of

the nuclear spin are now discussed.

Hamiltonian with Hyperfine Interaction

The total Hamiltonian, that includes the fine and hyperfine interactions, for a Rydberg

atom with two valence electrons n1l1s1, n2l2s2 and a nuclear spin I, can be written as

H = H1 +H2 +e2

r12+ β1 ~s1 ·~l1 + β2 ~s2 ·~l2 + A1

~I ·~j1 + A2~I ·~j2, (5.1)

where Hi is the Coulomb interaction between the valence electron i and the Sr2+ core,

e2/r12 is the Coulomb repulsion between the two valence electrons which leads to the

singlet triplet splitting, βi~s · ~li is the spin-orbit interaction of the ith electron and

Ai~I ·~ji, ~ji = ~si +~li is its hyperfine interaction. For a strontium Rydberg atom of con-

figuration 5snl, we can use the subscript 1(2) to denote the nl(5s) Rydberg (ground)

electron. The spin-orbit interaction and hyperfine interaction for the Rydberg elec-

tron are negligible when n is large since both the spin-orbit constant β and hyperfine

structure constant A scale as n−3. Because the 5s electron has l2 = 0, j2 = s2 = 1/2,

the spin-orbit interaction is also zero and the hyperfine interaction is reduced to

A2~I · ~s2. Now we can rewrite the Hamiltonian as

H = H1 +H2 +e2

r12+ A2

~I · ~s2. (5.2)

Now, the quantum number describing the total angular momentum is F (for even

isotopes, only the first three terms remain and the Hamiltonian is characterized by J),

61

the quantum number ~F being given by ~F = ~I+ ~J . The last term in this Hamiltonian

will be referred as the Fermi contact interaction.

Rydberg Series Limit Shift

For the even isotopes of strontium, different Rydberg series like 3S1,1S0,

3D2,1D2 will

converge to one common limit, the ion’s ground state 5s Sr+ 2S 12. This is described as

Enl = Iion−R/(n−δl)2 where Iion is the ionization potential and δl is the corresponding

quantum defect. Therefore, the shifts between even isotopes are their normal mass

shifts as shown in last section. However, due to the Fermi contact interaction which

is essentially the hyperfine interaction of the ion, the ion state 5s Sr+ 2S 12

splits into

F = 4 and F = 5 hyperfine states and they are about 2 − 3GHz [47] away from the

ion energy without a hyperfine structure i.e., 87Iion − 88Iion ∼ ±2 − 3GHz. These

shifts, much greater than the energy spacing between the n and n + 1 levels when

n ∼ 300, contribute to the movements of 87Sr peaks relative to the peaks of the even

isotopes in the measured spectra.

Hyperfine Induced Mixings

The Fermi contact interaction becomes increasingly important as n gets higher. When

60 < n < 100, this interaction (∼ 2 − 3 GHz) is comparable to the e2/r12 term

in equation 5.2 but smaller than the energy spacing between two consecutive ns,

and can cause mixing between triplet and singlet levels of the same F within the

same n (hyperfine induced singlet-triplet mixing within the same n). Consequently,

their energies are shifted and triplets can be populated as strongly as singlets. As n

continues to increase, the Fermi contact interaction becomes larger than En − En−1

(the energy spacing between two adjacent ns), and states of the same F but different

62

Figure 5.3 : N ∼ 335 spectra The 3D2 states of 88Sr are circled by red ovals inthe top figure. The blue laser is tuned to favor the 1S0 → 1P1 transition for 87Sr inthe second and the third subplots. Based on our experiences, the 87Sr features aremost apparent at a detuning around −69MHz. The last subplot is of a detuningthat favors excitation of 86Sr.

ns can be mixed. The higher the n is, then states with ever increasing differences in

ns can be mixed. At n ∼ 300 , hyperfine induced n-mixing can lead to “anti-crossing”

in the spectra since the energy differences between states of different n can be tuned

semi-continuously by changing n as shown in Figure 5.3 and 5.4.

5.1.3 Stray Fields Impact On Spectra

Figure 5.5 shows Rydberg excitation spectra recorded, with the blue laser tuned

to the 88Sr5s21S0 → 5s5p1P1 transition in the 88Sr isotope, at successively higher

63

Figure 5.4 : Anticrossing of 87Sr This graph is the detailed, break-down plots ofthe third plot of Figure 5.3. Every subplot of this graph starts from a 1D2 peak of88Sr and ends at next 1D2 peak of 88Sr. At the anticrossing point, where the energymismatch is very small, one of the state features disappears due to the cancellationof the oscillator strength. The red line and green line follow the peak positions of twomerging states.

64

values of n. As expected, as n increases the peak number of Rydberg atoms created

decreases dramatically as a result of both the decrease in the oscillator strength and

the increasing width of the spectra features. For values of n ≤ 350, two well-resolved

Rydberg series are seen, corresponding to excitation of 1D2 and 1S0 states. With

further increases in n the spectra features begin to broaden significantly, their widths

having approximately doubled by n ∼ 400. For even larger values of n the background

level begins to increase significantly, but a strong, well-resolved Rydberg series is still

evident for values of n up to ∼ 460. For n > 500, however, it becomes increasingly

difficult to discern any Rydberg series. This degradation in the Rydberg spectrum

with increasing n can be attributed to the presence of stray background fields which

lead to Stark shifts and broadening. These effects become particularly important at

fields approaching those at which states in adjacent Stark manifolds first cross, given

by Fcross ∼ 1/(3n5)a.u., i.e., ∼ 50µV cm−1 at n ∼ 500. This suggests that stray fields

of ∼ 50µV cm−1 remain in the excitation volume which is consistent with earlier

estimates of their size as described in Section 4.3.2.

65

Figure 5.5 : Excitation spectra recorded near the values of n indicated. The 461nmlaser is tuned to the 88Sr5s21S0 → 5s5p1P1 transition. The frequency axis shows therelative frequency of the 413nm laser during each scan

66

Chapter 6

Ultracold Rydberg Gas Evolution

Ultracold Neutral Plasmas (UNPs) and Ultracold Rydberg Gases are inter-related.

The free electrons and ions in a plasma can form Rydberg atoms via three body re-

combination (e−+e−+R+ = R∗∗+e−, “R∗∗” is the abbreviation for a Rydberg atom).

Conversely, an ultracold Rydberg gas can spontaneously evolve (ionize) into plasma.

Both systems have been studied extensively over the last decade. Yet the mechanism

for some processes are still intriguing. Use of traditional selective field ionization or

simple electron detection prohibits direct investigation towards the “pre-ionization”

stage of the evolution of a Rydberg gas into a UNP. The unique imaging capability

offered by strontium’s optically-active core effectively mitigates this problem. This

chapter, based on paper [35], will describe the techniques we employed in probing cold

Sr Rydberg gas dynamics along with a short introduction to the operative processes.

Our results, in contrast to earlier studies, stress the role played by Rydberg-Rydberg

interactions in the initial phases of evolution of a Rydberg gas.

6.1 Experimental Setup Overview

As shown in Figure 6.1 and 2.2, about a billion strontium atoms are captured in a

magneto-optical trap after the Zeeman slower and cooled to ∼7mK. Both the cooling

lasers and the laser driving the 461nm 5s21S0 → 5s5p1P1 transition are derived from a

frequency doubled Ti-Sapphire laser stabilized by saturated absorption spectroscopy.

67

Figure 6.1 : a).Rydberg excitation beams and UNP ionization beams are shown withrespect to the imaging system. The fluorescence imaging beam not shown in thefigure is parallel to the Rydberg excitation beams. b). Pertinent energy levels for thetwo photon transition to Rydberg level and the core transition used for imaging. c).Timing for the experiment. Figure adapted from [35].

68

The atomic density has a Gaussian profile with a radius of ∼1mm. The Rydberg

atoms are excited by a two-photon transition. The 461nm laser is red detuned by

430MHz from the intermediate 5s5p1P1 state and a 413nm laser is used for further

promotion (see detailed description in Chapter 4). The diameters of both the 461nm

and 413nm laser beams used for Rydberg excitation are much larger than the size of

the MOT so that the density of Rydberg atoms should follow the MOT profile. The

413 nm light remains on all the time (see the time sequence in Figure 6.1) while the

461nm laser light is pulsed on during the excitation.

In order to count the number of Rydberg atoms excited, we directly create a

UNP by photoionizing a small fraction of the strontium atoms after the Rydberg

excitation. The resulting electrons rapidly l change and ionize the Rydberg atoms

allowing fluorescence detection of the core ions. The photoionization is produced by

a 10ns dye laser pulse at 412nm. The 355nm light from a Nd:YAG laser pumps the

dye laser which can be tuned to set the velocity (energy) of the ionized electrons. For

our purposes, we set the frequency just above the ionization threshold so that only a

small but sustainable plasma is produced in the MOT.

For imaging the core ions, we utilize a 422nm laser beam to drive the 5s2S1/2 →

5p2P1/2 core transition. This laser is frequency doubled from a 844nm infrared diode

laser. It is locked to a scanning Fabry Perot cavity that is referenced to the 922nm

laser beam that is frequency doubled (461nm) and is stabilized by saturation absorp-

tion spectroscopy as mentioned previously. The conversion from scanning voltage to

the real frequency of the cavity is calibrated by using the 422nm laser for saturated

absorption spectroscopy in a 85Rb cell. This is possible because the strongest peak in

the 85Rb spectra (5s 2S1/2(F = 2) → 6s 2P1/2(F = 3)) is only 440MHz red-detuned

from the 88Sr+ 5s 2S1/2 → 5p 2P1/2 transition. This 440MHz offset is easy to span

69

by an acousto optic modulator. This light is used for imaging the bare ions since it

directly drives an optical transition in every ion (we can saturate this transition ).

We can either image the light absorption or image the laser induced fluorescence from

the atoms. Frequently, interest centers on the local density of the ions. So instead of

imaging a whole cloud, we image a thin sheet of atoms that has well defined geometry;

see Figure 6.2. Of course, its finite size needs to be taken into consideration for the

final ion density calculation. We argue that this technique can also image the high-l

strontium Rydberg atoms because, a Rydberg electron in a high-l orbit has negligible

overlap with the core ion allowing the core ion to behave as an independent particle.

In contrast, as described in the next section, low-l(l < 6) Rydberg atoms do not yield

a fluorescence signal.

6.2 Principal Processes in Probing Sr Cold Rydberg Gas

Among a complex array of processes occurring in our experiment, the following are

the most influential ones despite the ongoing debates as to their relative importance.

6.2.1 Autoionization

Excitation of the |5s〉 core electron in a Rydberg atom can lead to autoionization.

Autoionization is a special demonstration of multichannel coupling in the language of

MQDT. It is an inherent property of a two electron atom resulting from the interaction

between the two valence electrons. The term H12 = 1/r12 in the Hamiltonian not

only mixes the configurations of bound Rydberg states that are nearly degenerate (as

shown in red arrows in Figure 6.3 ) but also couples the doubly excited Rydberg series

|5pnl〉 to the Rydberg continuum |5sεl′〉; see blue arrows in Figure 6.3. The rate of this

particular transition, i.e. autoionization rate, is determined by | 〈5pnl|H12 |5sεl′〉 |2

70

Figure 6.2 : Optical probes for UNP using light resonant with the Sr+ transition.(a) Absorption Imaging: the laser beam is absorbed by the ions in the UNP creatinga shadow that is recorded by a CCD camera. A complete absorption image can beconstructed by a weighted integration of images taken over many different frequenciesacross the resonance to account for ions having different velocities. The ions’ absorp-tion profile is a Voigt distribution. (b) Fluorescence Imaging of a sheet UNP: theimaging laser beam propagates perpendicularly to the the camera and imaging axis.A complete image construction of all the ions is the same as for absorption imaging.Figure adapted from [69].

71

Figure 6.3 : Sr Rydberg atom autoionization The short solid lines are the stron-tium Rydberg levels converging to different series limits. The shaded regions are thecorresponding continua. The red arrows denote the coupling between bound, nearlydegenerate Rydberg levels in different configurations. The blue arrows denote thecoupling to the continuum of a Rydberg series that has lower series limit. FigureAdapted from [56].

72

which obeys the Rydberg scaling law ∼ 1/n3. Its l dependence is complicated and can

be understood in a scattering picture: the outer electron scatters the inner electron

to the ground state while itself gaining energy and being ionized. To facilitate this,

the outer electron must approach the inner electron and be moving rapidly. In low-l

states, the Rydberg electron moves in a elliptical orbital, the lower l is, the closer

the perigee is and the faster the electron moves around the perigee and of course the

stronger the autoionization is. As a result, the oscillator strength generally decreases

very rapidly as l increases. However, for very low l states, the overlapping of the outer

electron with the inner electron is also strongly affected by the coupling between the

core and the outer electron since it governs the phase of the wavefunction near the

core.

In fact, for very low l states, the transition rate is very high (∆t ≈ 0). So according

to the Heisenberg principle, ∆E∆t ∼ ~, the transition linewidth ∆E is very large. For

strontium in particular, the autoionization transition for 5s48s 1S0 and 5s47s 1D2 is as

broad as tens of GHz [35]. Since the imaging laser we are using is of narrow linewidth

∼ 1MHz, the effective autoionization, having a rate about 1MHz/10GHz ∼ 0.01%

of the total autoionization rate, is negligible for low-l states during the 500ns imaging

time. For high-l states, overlap with the core, i.e., scattering, becomes negligible.

6.2.2 Electron-Rydberg Collisions

Electrons with velocities comparable to the Rydberg electron can be produced easily

in many collision processes. Collisions between electrons and Rydberg atoms can lead

to l and m changing and even n changing if the incoming electron is energetic enough.

In comparison, collisions between the ions and Rydberg atoms are unlikely due to

their low velocities. Therefore the most important collision is the l changing collision

73

Figure 6.4 : l-mixing due to Rydberg-electron collision. Step A shows the mixing intothe manifold as the electron approaches. In step B, the population is randomizedamong the manifold. Step C is the ionization from the manifold. Figure adaptedfrom [68].

between the Rydberg atom and the electron. Here is how it works (see figure 6.4): as

the electron approaches, the low-l Rydberg state follows the quadratic Stark shift and

gradually mixes with the high-l manifold. Depending on the magnitude of the electric

field, it can mix strongly into the nearby manifold states. Thus when the electron

has passed by, it has a good chance to stay in one of the manifold states as the field

decreases. Even after collision the atom is subject to a very small electric field which

can lead to a redistribution among neighboring states in the manifold. As a result,

the original low-l state is further l-mixed into l > n/2 states on average. Though

m is a good quantum number for Rydberg atoms in an electric field, every electron

is incident in a random direction, whereupon m has to be referenced to a new axis.

So m mixing is happening all the time. The n-changing collisions are possible if the

74

electron passes close to the Rydberg electron. In the electron’s strong field manifolds

belonging to different n will cross, and the original state could mix into nearby ns.

6.2.3 Penning Ionization

Collisions between two Rydberg atoms can lead to Penning ionization

R∗∗(n) +R∗∗(n)→ R∗∗(n′) +R+ + e−. (6.1)

Penning ionization can be facilitated if the two Rydberg atoms are attracted to each

other by attractive dipole-dipole interactions or attractive Van der Waals interac-

tions. Once the interparticle distance R is small enough that the interaction energy

is comparable to the binding energy of the Rydberg atom, one of the Rydberg elec-

trons will be ionized and the other atom will be deexcited into a more deeply bound

state. Simulations in [70] show that in almost all likelihood, the ionized electron will

be energetic enough to escape on a nanosecond timescale. The binding energy of the

resulting Rydberg atom almost doubles its initial binding energy. The rest energies

will make the ion and Rydberg atom leave each other in high relative velocities but

small center of mass velocity since their total momentum is negligible initially (cold

atoms). There are also cases where, even for attractive interaction, Penning ioniza-

tion doesn’t happen. As the initial attractive interaction draws two Rydberg atoms

together, their dipole moments or equivalently their Runge-Lenz vectors can precess

to directions such that the interaction changes to repulsive. Consequently, they begin

to move apart from each other. Indeed Penning ionization will almost never happen

if the Rydberg-Rydberg interactions are isotropically repulsive.

Low-l Rydberg atoms don’t possess a permanent dipole. Nevertheless, a dipole-

dipole interaction which scales as ±C3/R3, C3 ∝ n4 can be induced if there is a

75

resonance like a Forster Resonance nl + nl ↔ n1l1 + n2l2. In the absence of both

a permanent dipole and a resonance, the interaction between two Rydberg atoms is

of the Van der Waals type and scales as ±C6/R6, C6 ∝ n11 which is the interaction

between instantaneous dipole moments. This force is termed London dispersion or the

instantaneous dipole-induced dipole force. For the initially excited low-l strontium

Rydberg atoms, the interactions are all Van der Waals interactions. The values of

C6s for different n are calculated in [65].

6.2.4 Blackbody Radiation induced Ionization

Rydberg atoms, having small binding energies, are generally sensitive to the 300K

thermal environment since the blackbody radiation (BBR) can not only drive dipole

allowed transitions to nearby states but also photoionize them. Both effects will limit

the lifetime of Rydberg atoms with BBR decay rates that scale as ∼ kT/n2. For

5s48s 1S0 and 5s47s 1D2 states, the BBR induced n,l changing rates are ∼ 1×104s−1

and the BBR induced photoionization rates are ∼ 1 × 102s−1. Note, however, that

it takes many dipole allowed transitions to induce large changes in l and thus this

process may be unimportant in the present experiments.

6.3 Imaging Technique

A direct laser induced fluorescence imaging of all the Rydberg atoms can be made by

exciting a dilute plasma (based on its function, we call it a seed plasma) immediately

after Rydberg excitation as shown in Figure 6.5. The free electrons trapped in the

MOT by the plasma cause repetitive l-mixing collisions and the low-l Rydberg atoms

are rapidly converted into high-l states which are visible to the imaging light. This

process can happen on a nanosecond time scale enabling us to count the Rydberg

76

Figure 6.5 : Laser-induced fluorescence imaging of UNP and Rydberg atom cloudsusing the 5s 2S1/2 → 5p 2P1/2 core-ion transition at 422 nm. (a) Image after Rydbergexcitation to the 5s48s 1S0 state for texc ≈ 3 µs, which yields ∼ 8 × 105 Rydbergatoms. Notice that the fluorescence signal is very small and the scale bar represents1 mm. (b) Image after exciting the same Rydberg population as in panel (a) butwith superposition of a seed UNP containing ∼ 2× 105 ions and electrons. Note theincreased fluorescence from the cigar-shaped region of Rydberg excitation. (c) Imageof a seed UNP identical to that in panel (b), but with no Rydberg excitation. (d)Signal due to Rydberg excitation obtained by subtracting the signal due to ion coresin the seed UNP. Figure adapted from [35].

atoms. If we vary the Rydberg excitation time, we should expect a linear increase

in the Rydberg excitation due to the fixed excitation rate determined by the Rabi

frequency. This behavior is observed in Figure 6.6.

6.4 Results Discussion

As we can see in Figure 6.6, without a seed plasma, there are no visible ion cores

below excitation times texc ∼ 2.5µs implying all the excited Rydberg atoms are still

in their low-l states. If we stop the excitation at the time, and let the Rydberg gas

77

Figure 6.6 : Dependence of the LIF signal from parent 5s48s 1S0 Rydberg atoms onthe excitation time texc. Left: each image is shown as a function of time with (top)and without (bottom) a seed UNP present. (Contributions to the LIF signals from theUNP are subtracted and the scale bar represents 1 mm). Right:Number of visible ioncores versus Rydberg excitation time for parent 5s48s 1S0 states. Data recorded bothwith and without a seed UNP present are included together with results obtainedwhen only the seed UNP was created. Qualitatively similar results are seen for the5s47d 1D2 state. Figure adapted from [35].

78

evolve by itself, what will happen? We examined this using both the 5s47d 1D2 state

and 5s48s 1S0 state with the same initial density and number of Rydberg atoms.

Figure 6.7 shows that in both cases the core ions gradually become more and more

visible. However, though the initial conditions are very similar, these two states

display rather different time evolutions. These differences can be seen in the total

number of visible ion cores as shown in Figure 6.8. These differences can be explained

by noting that both l-mixed Rydberg atoms and true ions are seen by the imaging

beam. Since a very weak seed plasma (see image (c) in Figure 6.5 and notice that it

has almost the same color bar as Figure 6.7) is able to provide a sufficient number

of trapped free electrons to l-mix all the Rydberg atoms and make them visible, the

majority of the visible ions in Figure 6.7 for 4.1µs evolution time cannot be true ions

otherwise the attendant electrons should have l-mixed almost all the Rydberg atoms

in the central region and made them visible in nanoseconds. Therefore, it is clear that

almost all the visible ions we see for 4.1µs evolution time should be l-mixed Rydberg

atoms resulting from collisions with some early electrons.

Now the question becomes where do the initial electrons (which collide with the

Rydberg atoms and further lead to l-mixing) come from? Whatever the mechanism is,

we can see the early electrons are produced first in the places where the initial Rydberg

atom density is highest. This is shown unambiguously in Figure 6.7. For each state,

the visibility is brighter in the higher Rydberg atom density region in each image and

the visibility propagates to the lower density region with increased evolution time.

Both BBR ionization and Penning ionization will lead to such density dependent

ionization. However, the BBR-induced ionization is slow and cannot explain the

difference seen between the behavior of S and D state. Penning ionization, on the

other hand, would.

79

Figure 6.7 : LIF images showing the spontaneous evolution of an ultracold gas of 1S0

(top) and 1D2 (middle) Rydberg atoms. The evolution time is indicated above eachimage in µs and the scale bar represents 1 mm. The initial numbers and densities forboth states are identical, 8 × 105 and 2.2 × 108 cm−3, respectively. Notice how theS-state population evolves more quickly in both space and time. The bottom panelsshow one-dimensional plots of the density integrated along the vertical direction.The spatial development of the S state (red) leads that of the D state (blue). Figureadapted from [35].

80

Figure 6.8 : Evolution of the number of visible core ions for 5s48s 1S0 and 5s47d 1D2

Rydberg atoms. The circles represent number calibrations performed by scanning theimaging laser through resonance for a complete construction of the “ion” numbers.In most of the experiments, the imaging laser is set on resonance and only one imageis recorded. Such single images are converted to the total “ion” number by using aconversion factor since the “ion” number is varying in a rather fixed profile againstthe imaging frequency. As we can see from the calibrated points, this simple trickworks very well. At late times the number of visible ion cores seen, ∼ 8× 105, agreesreasonably well with the number of parent Rydberg atoms initially excited, ∼ 7×105,as determined using a seed UNP. Figure adapted from [35].

81

In reference [65], it is theoretically predicted that S states should have isotropically

attractive Van der Waals interactions while the interactions between D states should

be mostly repulsive and only for some restricted range of orientations will they attract

weakly. So Penning ionization should be slow for D states but should happen a lot

more frequently for S states. The explains our LIF images very well, especially the

image at an evolution time of 4.1µs. For S states, collisions produce electrons, initially

at the center of the cloud where the Penning ionization is the strongest which will

collide and l-mix with Rydberg atoms before they escape from the cloud. However,

free electron production for D states mostly from BBR ionization is much slower so

the number of the early electrons and the l-changing rate, is much less resulting in

reduced visibility for the core ions. As we can see from Figure 6.7 and 6.8, S states

lead to a faster evolution of the overall visibility.

This contrast in visibility would be more easy to see for even shorter evolution

times tevol < 4µs since the D state would be completely dark. However, this is not

the case because it will take a long time for two nearest Rydberg atoms to Penning

ionize in the Van Der Waals interaction regime. Assuming that the potential energy

of a pair Rydberg atoms that experience an attractive force is completely converted

to kinetic energy as they accelerate towards each other, then the collision time can

be approximated using [70] as

T =

∫ R0

Rc

1√(C6

R6 − C6

R60)× 4/M

dR. (6.2)

where R0 is the initial separation between the pair of Rydberg atoms and Rc is the

lower limit of their separation which can be approximated as the diameter of the

Rydberg atom ∼ 482a.u. = 2304a.u.. Since Rc R0, Rc can be treated as zero. For

82

strontium, we have [65]

C6 = 15× n11 = 15× 4811 = 4.675× 1019a.u., (6.3)∫ 1

0

x3√1− x6

dx ≈ 0.43,M = 16.04× 104a.u.. (6.4)

Equation 6.2 can be reduced to

T =R4

0

2×√

16.04× 104

4.675× 1019× 0.43a.u. (6.5)

T ≈ 0.039× (R0 in µm)4µs. (6.6)

Since collision time is very sensitive to the change in the R0 as shown in Figure 6.9,

it is important to correctly estimate R0. If we use the nearest neighbor distribution

for the initial condition in Figure 6.7 as described in reference [35], it is possible

to have hundreds of Rydberg atom pairs separated by R0 < 3.5µm which yields

collision times that allow electron production on micro-second time scales. However,

if Rydberg blockade is important, the initial separation between two Rydberg atoms

can be no less than the blockade radius Rvdw which can be calculated assuming an

overall linewidth of 1MHz

C6

R6vdw

= 1MHz = 2.419× 10−11a.u. (6.7)

Rvdw = 1.116× 105a.u. = 5.91µm. (6.8)

This restriction will yield a minimum collision time of 47µs! Apparently, Rydberg

blockade effects don’t appear to be dominant. There are multiple possible explana-

tions. For instance, equation 6.2 assumes the initial velocities of the Rydberg atoms

is zero. While this assumption holds in the sense that the initial 7mK thermal veloc-

ities have no fixed direction, it is possible, in some cases, to have two Rydberg atoms

83

Figure 6.9 : Collision Time Vs Initial Inter-Rydberg atoms Distance

84

moving towards each other with their thermal velocity ∼ 1m/s. This initial veloc-

ity, about 1µm per microsecond, can greatly reduce the total collision time. On the

other hand, the overall linewidth for the particular two-photon transition employed

can be a lot larger than the pure laser linewidth. Moreover since we don’t know the

exact detuning of the transition, it is possible that the interaction induced energy

shift between two very close Rydberg atoms matches the detuning which facilitate

closer Rydberg pair excitation. In fact, the strong AC Stark shifts due to 5s2 → 5s5p

transition could lead to an anti-blockade effect [71] that makes Rydberg blockade

inefficient. Also, stray fields may further complicate the Rydberg blockade.

After the very first phases of the dynamics, the initial differences in the behavior of

S states and D states due to their different interactions will be smeared out by other

effects. For example, l-mixed Rydberg atoms possess permanent dipole moments

and their Penning ionization rates will be larger than those expected for Penning

ionization due to the Van der Waals interaction. Following formation of the electron

trap, the l-mixed Rydberg atoms will dominate in the cloud and all memory of the

initial state will be lost.

In the previous studies [67], [66] and [68], it has been suggested that almost all early

electrons are generated by the 300K blackbody radiation and the Penning ionization

plays a negligible role in the dynamics until the l-mixed Rydberg atoms dominate

the Rydberg gas. However, according to our study, we believe Penning ionization in

the early stage of the evolution is at least as significant as BBR induced ionization in

producing the early electrons.

85

Chapter 7

Conclusion and Outlook

We have created strontium Rydberg atoms in two environments and demonstrated

several of the key experimental capabilities they offer which will directly enable a

variety of future studies.

We produced very-high-n (n ∼ 300) strontium Rydberg atoms in a collimated

atomic beam using two-photon excitation. Spectroscopically, we observed the normal

mass shifts of the even isotopes 86Sr, 88Sr, 84Sr and the hyperfine induced mixings of

the odd isotope 87Sr. By exciting Rydberg atoms in a DC field, we studied the Stark

shifts of various Rydberg states. A two-active-electron model was used to analyze the

data and provided results in excellent agreement with experiment. The derived “nD”

Stark state possesses a large permanent dipole moment in DC fields approaching

those at which states in neighboring manifolds first cross. This allows production of

a quasi-one-dimensional state and that can be engineered into circular or elliptical

states. In other words, the dipole moment of this Rydberg atom is not only very large

(when it is a quasi-one-dimensional state) but is also tunable (when it is manipulated

into states of varying ellipticity). Also, the ability to near simultaneously produce

many Rydberg atoms opens the opportunity to study Rydberg-Rydberg interactions.

The Van der Waals interaction between Rydberg atoms of such high n has not yet

been investigated either experimentally or theoretically. Additionally, we can create

planetary states by exciting the second valence electron and examine the interactions

between two excited electrons within one Rydberg atom.

86

We also studied ultracold strontium Rydberg gases by imaging light scattered

from the core ions of l-mixed Rydberg atoms, i.e. the laser induced fluorescence.

To induce rapid l-mixing and rapidly image all the Rydberg atoms, a weak ultracold

neutral plasma is utilized to introduce free electrons that collide with Rydberg atoms.

The temporal and spatial resolution of this imaging technique permits the study of

the evolution of an ultracold Rydberg gas. In the early stages of this evolution, Pen-

ning ionization as well as blackbody radiation induce photoionization and introduce

initial electrons that l-mix the low-l Rydberg atoms and make them visible to the

imaging light. We observed faster dynamics for Rydberg states that have attractive

interactions than for states that have mostly repulsive interactions which highlights

the importance of Penning ionization. In the future, we plan to exploit the strong

interactions between cold strontium Rydberg atoms to form a more highly correlated

ultracold neutral plasma.

87

Bibliography

[1] Daniel Comparat and Pierre Pillet, Dipole blockade in a cold Rydberg atomic

sample, JOSA B, 27, 6, 2010

[2] D. Jaksch, J. I. Cirac, P. Zoller, S.L. Rolston,R. Cotand M. D. Lukin, Fast

Quantum Gates for Neutral Atoms, Phys. Rev. Lett., 85, 10 (2000)

[3] M.D. Lukin, M. Fleischhauer, and R. Cote, Dipole Blockade and Quantum Infor-

mation Processing in Mesoscopic Atomic Ensembles, Phys. Rev. Lett. 87, 037901

(2001)

[4] Immanuel Bloch,Jean Dalibard,Wilhelm Zwerger, Many-body physics with ultra-

cold gases, Rev. Mod. Phys. 80, (2008)

[5] R. M. W. van Bijnen, S. Smit, K. A. H. van Leeuwen, E. J. D. Vredenbregt, S.

J. J. M. F. Kokkelmans, Adiabatic Formation of Rydberg Crystals with Chirped

Laser Pulses, J. Phys. B: At. Mol. Opt. Phys. 44 (2011)

[6] Thibault Vogt, Matthieu Viteau, Jianming Zhao, Amodsen Chotia, Daniel Com-

parat, and Pierre Pillet, Dipole Blockade at Forster Resonances in High Resolu-

tion Laser Excitation of Rydberg States of Cesium Atoms, Phys. Rev. Lett. 97.

083003 (2006)

[7] Ulrich Raitzsch, Vera Bendkowsky, Rolf Heidemann, Bjorn Butscher, Robert

Low, and Tilman Pfau, Echo Experiments in a Strongly Interacting Rydberg

88

Gas, Phys. Rev. Lett. 100, 013002 (2008)

[8] Alpha Gaetan, Yevhen Miroshnychenko, Tatjana Wilk, Amodsen Chotia,

Matthieu Viteau, Daniel Comparat, Pierre Pillet, Antoine Browaeys and

Philippe Grangier, Observation of collective excitation of two individual atoms

in the Rydberg blockade regime, Nature Phys. 5, 115118 (2009)

[9] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker

and M. Saffman, Observation of Rydberg blockade between two atoms, Nature

Phys. 5, 111-114 (2009)

[10] T. Wilk, A. Gaetan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier,

and A. Browaeys, Entanglement of Two Individual Neutral Atoms Using Rydberg

Blockade, Phys. Rev. Lett. 104, 010502 (2010)

[11] Jens Honer, Hendrik Weimer, Tilman Pfau, and Hans Peter Buchler, Collective

Many-Body Interaction in Rydberg Dressed Atoms, Phys. Rev. Lett. 105.160404,

(2010)

[12] T. Pohl, E. Demler, and M. D. Lukin, Dynamical Crystallization in the Dipole

Blockade of Ultracold Atoms, Phys. Rev. Lett. 104. 043002, (2010)

[13] Robert Low, Hendrik Weimer, Ulrich Krohn, Rolf Heidemann, Vera Bendkowsky,

Bjorn Butscher, Hans Peter Buchler, and Tilman Pfau, Universal scaling in a

strongly interacting Rydberg gas, Phys. Rev. A 80, 033422 (2009)

[14] F. Cinti, P. Jain, M. Boninsegni, A. Micheli, P. Zoller, and G. Pupillo, Supersolid

Droplet Crystal in a Dipole-Blockaded Gas, Phys. Rev. Lett. 105.135301,(2010)

89

[15] N. Henkel, R. Nath, and T. Pohl, Three-Dimensional Roton Excitations and

Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates, Phys. Rev.

Lett. 104.195302 (2010)

[16] Hendrik Weimer, Markus Muller, Igor Lesanovsky, Peter Zoller and Hans Peter

Buchler, A Rydberg quantum simulator, Nature Phys. 6, 1614 (2010)

[17] Juan Yin, Ji-Gang Ren, He Lu, Yuan Cao, Hai-Lin Yong, Yu-Ping Wu, Chang

Liu, Sheng-Kai Liao, Fei Zhou, Yan Jiang, Xin-Dong Cai, Ping Xu, Ge-Sheng

Pan, Jian-Jun Jia, Yong-Mei Huang, Hao Yin, Jian-Yu Wang, Yu-Ao Chen,

Cheng-Zhi Peng and Jian-Wei Pan, Quantum teleportation and entanglement

distribution over 100-kilometre free-space channels, Nature 488.11332, (2012)

[18] S. Sevinc.Li, N. Henkel, C. Ates, and T. Pohl, Nonlocal Nonlinear Optics in Cold

Rydberg Gases, Phys. Rev. Lett. 107. 153001, (2011)

[19] J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones,

and C. S. Adams, Cooperative Atom-Light Interaction in a Blockaded Rydberg

Ensemble, Phys. Rev. Lett. 105.193603 (2010)

[20] Y. O. Dudin and A. Kuzmich, Strongly Interacting Rydberg Excitations of a Cold

Atomic Gas, Science, 336, (2012)

[21] Matthew Pelton, Charles Santori, Jelena Vuckovic, Bingyang Zhang, Glenn S.

Solomon, Jocelyn Plant, and Yoshihisa Yamamoto, Efficient Source of Single

Photons: A Single Quantum Dot in a Micropost Microcavity, Phys. Rev. Lett.

89.23, (2002)

[22] M. Saffman, T. G. Walker and K. Mølmer, Quantum information with Rydberg

atoms, Rev. Mod. Phys. 82. 2313, (2010)

90

[23] M. Viteau, M. G. Bason, J. Radogostowicz, N. Malossi, D. Ciampini, O. Morsch,

and E. Arimondo, Rydberg Excitations in Bose-Einstein Condensates in Quasi-

One-Dimensional Potentials and Optical Lattices, Phys. Rev. Lett. 107, 060402,

(2011)

[24] A. Schwarzkopf, R. E. Sapiro, and G. Raithel, Imaging Spatial Correlations of

Rydberg Excitations in Cold Atom Clouds, Phys. Rev. Lett. 107.103001, (2011)

[25] Peter Schauβ, Marc Cheneau, Manuel Endres, Takeshi Fukuhara, Sebastian

Hild, Ahmed Omran, Thomas Pohl, Christian Gross, Stefan Kuhr and Immanuel

Bloch, Observation of spatially ordered structures in a two-dimensional Rydberg

gas, Nature, 491, 87 (2012)

[26] Y. O. Dudin, L. Li, F. Bariani and A. Kuzmich, Observation of coherent many-

body Rabi oscillations, Nature Phys. 8, 2413 (2012)

[27] Y. N. Martinez de Escobar, P. G. Mickelson, M. Yan, B. J. DeSalvo, S. B. Nagel,

and T. C. Killian, Bose-Einstein Condensation of 84Sr, Phys. Rev. Lett. 103.

200402, (2009)

[28] . G. Mickelson, Y. N. Martinez de Escobar, M. Yan, B. J. DeSalvo, and T.

C. Killian Bose-Einstein condensation of 88Sr through sympathetic cooling with

87Sr, Phys. Rev. A. 81.051601, (2010)

[29] B. J. DeSalvo, M. Yan, P. G. Mickelson, Y. N. Martinez de Escobar, and T. C.

Killian, Degenerate Fermi Gas of 87Sr Phys. Rev. Lett. 105. 030402, (2010)

[30] P G Mickelson, Y N Martinez de Escobar, P Anzel, B J DeSalvo, S B Nagel, A J

Traverso, M Yan and T C Killian, Repumping and spectroscopy of laser-cooled Sr

91

atoms using the (5s5p)3P2(5s4d)3D2 transition J. Phys. B: At. Mol. Opt. Phys.

42 235001, (2009)

[31] G. Ferrari, P. Cancio, R. Drullinger, G. Giusfredi, N. Poli, M. Prevedelli, C.

Toninelli, and G. M. Tino, Precision Frequency Measurement of Visible Inter-

combination Lines of Strontium Phys. Rev. Lett. 91. 243002 (2003)

[32] G. Bannasch, T.C. Killian,and T. Pohl, Strongly Coupled Plasmas via Rydberg-

Blockade of Cold Atoms, arXiv: 1302.6108v1 (2013)

[33] C. E. Simien, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson,

S. B. Nagel, and T. C. Killian, Using Absorption Imaging to Study Ion Dynamics

in an Ultracold Neutral Plasma Phys. Rev. Lett. 92.143001, (2004)

[34] Thomas C.Killian, Ultracold Neutral Plasmas Science, 316, (2007)

[35] P. McQuillen, X. Zhang, T. Strickler, F. B. Dunning, and T. C. Killian, Imaging

the evolution of an ultracold strontium Rydberg gas Phys. Rev. A. 87.013407,

(2013)

[36] Chris H. Greene, A.S. Dickinson and H. R. Sdeghpour, Creation of Polar and

Nonpolar Ultra-Long-Range Rydberg Molecules, Phys. Rev. Lett. 85, 12, (2000)

[37] Vera Bendkowsky, Bjorn Butscher, Johannes Nipper, James P. Shaffer, Robert

Low and Tilman Pfau, Observation of ultralong-range Rydberg molecules, Nature,

458, 07945, (2009)

[38] G. Wilpers, T. Binnewies, C. Degenhardt, U. Sterr, J. Helmcke, and F. Riehle,

Optical Clock with Ultracold Neutral Atoms, Phys. Rev. Lett. 89. 23, (2002)

92

[39] Hidetoshi Katori, Masao Takamoto,V.G. Palchikov and V. D. Ovsiannikov, Ul-

trastable Optical Clock with Neutral Atoms in an Engineered Light Shift Trap,

Phys. Rev. Lett. 91. 17, (2003)

[40] S. Blatt, A. D. Ludlow, G. K. Campbell, J. W. Thomsen, T. Zelevinsky, M. M.

Boyd, J. Ye, X. Baillard, M. Fouche, R. Le Targat, A. Brusch, P. Lemonde, M.

Takamoto, F. L. Hong, H. Katori and V. V. Flambaum, New Limits on Coupling

of Fundamental Constants to Gravity Using 87Sr Optical Lattice Clocks Phys.

Rev. Lett. 100.140801, (2008)

[41] Robin Santra, Ennio Arimondo, Tetsuya Ido, Chris H. Greene, and Jun Ye,

High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,

Phys. Rev. Lett. 94.173002, (2005)

[42] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schau, C. Gross, L.

Mazza, M. C. Ban, L. Pollet, I. Bloch, and S. Kuhr, Observation of Correlated

Particle-Hole Pairs and String Order in Low-Dimensional Mott Insulators, Sci-

ence 334, 200 (2011)

[43] R. Mukherjee, J. Millen, R. Nath, M. P. A. Jones and T. Pohl, Many-body physics

with alkaline-earth Rydberg lattices, J. Phys. B: At. Mol. Opt. Phys. 44, (2011)

[44] S. E. Anderson, K. C. Younge, and G. Raithel, Trapping Rydberg Atoms in an

Optical Lattice, Phys. Rev. Lett. 107.263001, (2011)

[45] C. J. Lorenzen, K. Niemax and L. R. Pendrill, Isotope shifts of energy levels in

the naturally abundant isotopes of strontium and calcium, Phys. Rev. A 28.4,

(1982)

93

[46] R. Beigang, K. Lucke, A. Timmermann and P.J. West, Determination of absolute

level energies of 5sns 1S0 and 5snd1D2 Rydberg series of Sr, Optics Communi-

cations, 42.1, (1982)

[47] R. Beigang, High-resolution Rydberg-state spectroscopy of stable alkaline-earth

elements, J. Opt. Soc. Am. B, 5.12, (1988)

[48] Mireille Aymar, Chris H. Greene and Eliane Luc-Koenig, Multichannel Rydberg

spectroscopy of complex atoms, Rev. Mod. Phys. 68. 4, (1996)

[49] Peter Esherick, Bound, even-parity J = 0 and J = 2 spectra of Sr, Phys. Rev.

A, 15. 5, (1976)

[50] S. Ye, X. Zhang, T. C. Killian, F. B. Dunning, M. Hiller, S. Yoshida, and J.

Burgdorfer, Production of very-high-n strontium Rydberg atoms, in preparation.

[51] W. E. Cooke, T. F. Gallagher, S. A. Edelstein, and R. M. Hill, Doubly Excited

Autoionizing Rydberg States of Sr, Phys. Rev. Lett. 40. 3, (1977)

[52] W.E.Cooke and T.F.Gallagher, Measurement of 1D2 →1 F3 microwave transi-

tions in strontium Rydberg states using selective resonance ionization, Opt. Lett.

4. 6, (1979)

[53] I. I. Beterov, I. I. Ryabtsev, D. B. Tretyakov, and V. M. Entin, Quasiclassical

calculations of blackbody-radiation-induced depopulation rates and effective life-

times of Rydberg nS, nP, and nD alkali-metal atoms with n680, Phys. Rev. A.

79.052504, (2009)

[54] Emily Y. Xu, Yifu Zhu, Oliver C. Mullins, and T. F. Gallagher, Sr 5p1/2ns1/2

and 5p3/2ns1/2 J=1 autoionizing states, Phys. Rev. A 33. 04, (1985)

94

[55] I.C.Percival, Planetary Atoms, Proc. R. Soc. Lond. A, 353, (1977)

[56] Thomas F. Gallagher, Rydberg Atoms, Cambridge University Press (1994)

[57] All the manuals and specs provided by TOPTICA PHOTONICS

[58] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J.

Munley, and H. Ward, Laser Phase and Frequency Stabilization Using an Optical

Resonator, Appl. Phys. B 31, (1983)

[59] P. Bohlouli-Zanjani, K. Afrousheh, and J. D. Martin, optical transfer cavity stabi-

lization using current-modulated injection-locked diode lasers, Rev. Sci. Instrum.

77. 093105, (2006)

[60] Xiao-Min Tong and Shih-I Chu, Time-dependent density-functional theory for

strong-field multiphoton processes: Application to the study of the role of dy-

namical electron correlation in multiple high-order harmonic generation, Phys.

Rev. A 57. 01, (1998)

[61] J Millen, G Lochead, G R Corbett, R M Potvliege and M P A Jones, Spectroscopy

of a cold strontium Rydberg gas, J. Phys. B: At. Mol. Opt. Phys. 44. 184001,

(2011)

[62] Daniel Kleppner, Michael G. Littman and Myron L. Zimmerman, Highly Excited

Atoms, Sci. Amer.

[63] T. P. Hezel, C. E. Burkhardt, M. Ciocca, L-W. He, and J. J. Leventhal, Clas-

sical view of the properties of Rydberg atoms: Application of the correspondence

principle, Am. J. Phys. 60, 4, (1992)

95

[64] M. T. Frey, X. Ling, B. G. Lindsay, K. A. Smith, and F. B. Dunning, Use of the

Stark effect to minimize residual electric fields in an experimental volume, Rev.

Sci. Inst. 64, 3649, (1993).

[65] C L Vaillant, M P A Jones and R M Potvliege, Long-range Rydberg-Rydberg

interactions in calcium, strontium and ytterbium, J. Phys. B: At. Mol. Opt.

Phys. 45, (2012)

[66] B. Knuffman and G. Raithel, Emission of fast atoms from a cold Rydberg gas,

Phys. Rev. A 73, 020704 (2006)

[67] A. Walz. Flannigan, J. R. Guest, J. H. Choi, and G. Raithel, Cold-Rydberg-gas

dynamics, Phys. Rev. A.69. 063405 (2004)

[68] S. K. Dutta, D. Feldbaum, A. Walz-Flannigan, J. R. Guest, and G. Raithel,

High-Angular-Momentum States in Cold Rydberg Gases, Phys. Rev. Lett. 86,

3993 (2001)

[69] J Castro, H Gao and T C Killian, using sheet fluorescence to probe ion dynamics

in an ultracold neutral plasma, Plasma Phys. Control. Fusion 50, (2008)

[70] F Robicheaux, Ionization due to the interaction between two Rydberg atoms, J.

Phys. B: At. Mol. Opt. Phys. 38 (2005)

[71] C. Ates, T. Pohl, T. Pattard, and J. M. Rost, many-body theory of excitation

dynamics in an ultracold Rydberg gas, Phys. Rev. A. 76.013413, (2007)