Entangled Mechanical Oscillators

by

John D. Jost

B.S., University of Colorado, 2001

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulllment

of the requirements for the degree of

Doctor of Philosophy

College of Arts and Sciences Department of Physics

2010
This thesis entitled:

Entangled Mechanical Oscillators

written by John D. Jost

has been approved for the College of Arts and Sciences Department of Physics

Dave Wineland

Jun Ye

Date

The nal copy of this thesis has been examined by the signatories, and we nd that

both the content and the form meet acceptable presentation standards of scholarly

work in the above mentioned discipline.
iii

Jost, John D. (Ph.D., Physics)

Entangled Mechanical Oscillators

Thesis directed by Dr. Dave Wineland

Quantum entanglement has been the subject of considerable research, in part due

to its non-intuitive nature and ubiquitous presence in quantum information processing.

For this reason, it is of interest to study entanglement in a variety of systems. The work

discussed in this thesis focuses on the demonstration of deterministic entanglement in

a system pervasive in nature: mechanical oscillators. Here, the mechanical oscillators

are composed of the vibrations of two ion pairs in spatially separated locations. The

techniques demonstrated in this experiment are likely to form core components of largescale trapped-ion quantum information processing.
Dedication

To my parents, for always encouraging me no matter what path I chose

and

To my sister, for always putting up with me
Acknowledgements

Many people deserve thanks and credit for the work discussed here and I apologize

in advance for leaving anyone out.

Looking back now it is interesting to see the path that has led me to this point,

and I feel very lucky in the opportunities I have been given. About the time I was

nishing high school, I decided I was interested in physics. It seemed so precise (It

took me awhile to get used the fact that is not always the case, but now I like that)

and explained the world around me. I showed up for orientation at the University of

Colorado at Boulder as an engineering physics major. Fortunately for me, my major was

not very popular and I was the only one there that day with my major. Dana Anderson

was the head of the engineering physics program at the time and gave me a nice personal

laser demonstration (I have always been fascinated by light). I expressed some interest

in this area, and he oered to have one of this grad students (Pete Tucker) show me

around his optics lab. I had never been near a real lab before and was quite impressed.

I started college taking the introductory physics courses. One of the classes required

was an experimental physics class, which I did not like. After a couple of weeks I was

wondering if I should switch majors. One day I was in the gym an ran into Pete again.

He mentioned that Dana was looking for someone to work in his optics lab. I went to

Dana's oce and told him of my interest. He asked me if I had any lab experience. All I

could saw was that I had taken chemistry and physics in high school. Despite this Dana

hired me to work in his lab. This was a turning point for me, and I owe Dana a lot
vi

gratitude for giving me that opportunity. I quickly realized that experimental physics

was not as dull as the entry level physics labs. I really enjoyed that everyday there were

new problems to solve. During the years I worked in Dana's group, I was supervised by

Leslie Czia who was always patient and helpful with all my questions. Also while there I

was able to get help from several kind grad students at the time Edeline Fotheringham,

Dirk Muller,and Peter Schwindt. During this time I also spent many long hours in the

machine shop. I should thank all of the JILA machinists for taking the time to teach

me many things that have proven very valuable. It is a nice feeling to know that you

can take a chunk of metal and make it into almost anything

After several years working for Dana Anderson, I decided I wanted to try something dierent for my undergraduate senior thesis research. I came across Jun Ye's

research with ultra-fast lasers and thought that sounded really cool. He oered to let

me work in his lab. I had no idea how great of an opportunity this would be. Jun's

excitement about my experiment was infectious and motivating. I remember nights

where Jun would be sitting on the oor with me at 3 am turning knobs trying to get the

experiment to work. While working in Jun's Lab I was able to work with many great

people. The guys in the JILA electronics shop taught me a lot about circuits and Mark

Notcutt was always willing to help me with my experiment. I also feel fortunate that

I was able to work with John Hall and was always impressed by the unending ow of

ideas he had.

I stayed on working with Jun after graduating from college to nish the experiment

I had started. It was during this time when I was deciding where to go to grad school

that Jun recommended Dave Wineland's group at NIST. I did not really know anything

about trapping ions or quantum information, but it sounded interesting. Eventually I

settled on staying in Boulder for graduate and joined Dave Wineland's group. I had no

idea at the time what a great research group Dave had. Dave has been a great advisor.

Not only in the physics he knows but in how one goes about doing science. He was
vii

patient enough to let us try and gure things out in the lab on our own, but always had

the answer when we got stuck. I will always look back having enjoyed my time with

Dave and his group.

Another great part of Dave's group is all the intelligent people I have been able

to work with over the years. There are a few people that I own special thanks to.

Jonathan Home joined the group shortly after we started on the work discussed here.

Without his knowledge, drive, and excitement I would still be in the lab trying to

get the experiment to work. I also need to thank Jonathan for his friendship without

which I do not think all the hours we spent side by side in the lab would have gone

so well. I need to thank Chris Langer and Roee Ozeri for taking the time to teach me

almost everything about the experimental apparatusa and for their great contributions

to the experiment. In addition, in another one of those twists of fate, for convincing

me to use the ion trap I built in their experiments. This lead me to be being able to

do or some really interesting physics. Experimental physics has a way of keeping you

humble. I will never forget the ve months Roee and I spent tracking down a problem

that turned out to be a broken piece of equipment. However we learned a lot in the

process. David Hanneke joined our experiment. In a very short amount of time, he

was up to speed and was a great help getting everything accomplished. Also Jason

Amini was an invaluable resource and solved every computer control problem we had

with amazing speed. These are by no means the only people who came through Dave's

group in my time there that contributed in some way or spent time answering my dumb

questions. I have been very fortunate to learn from all these people: Murray Barrett,

Jim Bergquist, Sarah Bickman, Mike Biercuk, Brad Blakestad, John Bollinger, Brana

Jelenkovic, Joe Britton, Kenton Brown, Anders Brusch, John Chiaverini, James Chou,

Yves Colombe, Brian DeMarco, Bob Drullinger, Ryan Epstein, David Hume, Wayne

Itano, Marie Jensen, Manny Knill, Jeroen Koelemeij, Dietrich Leibfried, Rober Maiwald,

Wendell Oskay, Christian Ospelkaus, Rainer Reichle, Till Rosenband, Signe Seidelin,
viii

Tobias Schaetz, Pete Schmidt, Nobuyasu Shiga, Hermann Uys, Aaron VanDevender,

and Janus Wesenberg

Finishing a Ph.D. requires much more than just having great colleagues to work

with. It also requires having people outside of the lab who are willing to be there for

you. Without such a support network, my life would have been a lot less full and I would

not have been able to accomplish this task. I have been very fortunate to have so many

friends over the years not all of which I can mentioned here, but I want thank a few

for tolerating me: Jake Dillow, Elizabeth Chick, Natalie Colatosti, Brian Fetters, Seth

Fletcher, Chris Keller, Quinn Killin, Brian Landon, Michael Pickering, and Juan Pino

This has been a great experience I owe a lot of thanks to everyone who helped me

get here.
ix

Contents

1 Introduction 1

1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Apparatus 8

2.1 Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Ion Trap Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Ion Trap Construction . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Producing the DC and RF Potentials . . . . . . . . . . . . . . . . . . . . 20

2.2.1 DC Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 RF Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Loading Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 24Mg+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Magnetic-Field Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Laser Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8 Controlling the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 43
x

3

9Be+ Qubit 48

3.1 Energy Level Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Hyperne and Zeeman Interactions . . . . . . . . . . . . . . . . . . . . . 53

3.3 State Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Raman Transitions in 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Sympathetic Cooling and 24Mg+ 63

4.1 Sympathetic Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Options for Sympathetic Cooling . . . . . . . . . . . . . . . . . . 64

4.1.2 Issues with Sympathetic Cooling . . . . . . . . . . . . . . . . . . 65

4.2 24Mg+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.1 Energy Level Structure . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Doppler Cooling 73

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Doppler Cooling of Trapped Ions . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Doppler Cooling Setup for 24Mg+. . . . . . . . . . . . . . . . . . . . . . 79

5.4 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4.1 The System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 85

5.4.2 Calculating Populations . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 Beam Orientations and the Doppler Shift . . . . . . . . . . . . . . . . . 95

5.5.1 Co-linear Cooling Beams . . . . . . . . . . . . . . . . . . . . . . . 96

5.5.2 Orthogonal Cooling Beams . . . . . . . . . . . . . . . . . . . . . 97

5.6 Demonstration of Doppler Cooling with 24Mg+. . . . . . . . . . . . . . 101

6 Raman Transitions and Sideband Cooling 109

6.1 Basic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
xi

6.2 Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3 Rabi Rates - Multiple Modes . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 Stimulated Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . 118

6.4.1 Co-propagating Beam Transitions . . . . . . . . . . . . . . . . . . 124

6.5 Motional State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.6 Sideband Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.7 Sideband Cooling Experiments . . . . . . . . . . . . . . . . . . . . . . . 139

7 Phase Gate 154

7.1 Entanglement Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.2 Optical Dipole Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.3 Coherent States and the Forced Harmonic Oscillator . . . . . . . . . . . 163

7.3.1 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.3.2 Forced Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 164

7.4 Geometric Phase and Coherent State Amplitude . . . . . . . . . . . . . 165

7.5 Ideal Gate Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.6 Phase Gate Entanglement Experiment . . . . . . . . . . . . . . . . . . . 173

7.6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.6.2 Determining the Phase - Ion Spacing . . . . . . . . . . . . . . . 175

7.6.3 Determining the Detuning . . . . . . . . . . . . . . . . . . . . . . 176

7.6.4 Entanglement in a Mixed Species Crystal . . . . . . . . . . . . . 177

8 Entangled Mechanical Oscillators 183

8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
xii

Bibliography 194

A Calculating Rabi Frequencies 210

B Determining the Average Motional State 213

C Pulse Sequence for the Entangled Mechanical Oscillators Experiment 215

D Ion Trap Wafer Dimensions 219

E Other Contributions 223
xiii

Tables

2.1 Table of the visible laser systems . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Table of the UV laser systems . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Populations for the three level system. . . . . . . . . . . . . . . . . . . 89
xiv

Figures

2.1 Photo of the experimental apparatus at NIST used for the entangled mechanical oscillators experiment . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Illustration of a linear Radio-Frequency (RF) Paul trap used to trap ions 10

2.3 Multi-zone alumina wafer style linear RF Paul trap . . . . . . . . . . . . 14

2.4 Multi-zone alumina wafer style linear RF Paul trap . . . . . . . . . . . . 15

2.5 Views of the ion trap and electrode labeling . . . . . . . . . . . . . . . . 19

2.6 Side view of the ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Schematic (top view) of one of the trap wafers . . . . . . . . . . . . . . . 21

2.8 Schematic of the electrode structure . . . . . . . . . . . . . . . . . . . . 22

2.9 Schematic of the vacuum system . . . . . . . . . . . . . . . . . . . . . . 26

2.10 Laser beams and optics near the ion trap . . . . . . . . . . . . . . . . . . 29

2.11 Image of 9Be+ − 24Mg+ − 24Mg+ −9 Be+ chain of ions taken with a UV

sensitive charge-coupled device camera . . . . . . . . . . . . . . . . . . . 31

2.12 The Blue Doppler beam lines . . . . . . . . . . . . . . . . . . . . . . . . 36

2.13 The Red Doppler beam lines . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.14 The 9Be+ Raman beam lines . . . . . . . . . . . . . . . . . . . . . . . . 46

2.15 The 24Mg+beam lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Level structure for 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 9Be+ levels at a magnetic eld of 119.4 Gauss with tuning of laser beams

indicated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
xv

3.3 Orientation of the RD, repumper, BD, and BD detuned laser beams with

respect to the trap axis and magnetic eld direction . . . . . . . . . . . 57

4.1 Splittings of the 2P3

2

levels of 24Mg+as a function of the applied magnetic

eld. The g-factor gJ '

4

3

. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Splittings of the 2P1

2

level of 24Mg+as a function of the applied magnetic

eld. The g-factor gJ '

2

3

. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Splittings of the 2S1

2

level of 24Mg+as a function of the applied magnetic

eld. The g-factor gJ '2. . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Atomic level structure for 24Mg+at eld of ∼ 119.65 Gauss . . . . . . . . 72

5.1 This gure shows n ̄INP H for two-level Doppler cooling of the INPH mode

of a 9Be+ − 24Mg+ion pair with 24Mg+on the 2S1

2

to 2P3

2

transition

versus the detuning from resonance . . . . . . . . . . . . . . . . . . . . . 80

5.2 This gure shows n ̄OOP H for two-level Doppler cooling of a OOPH mode

of a 9Be+ − 24Mg+ion pair with 24Mg+on the 2S1

2

to 2P3

2

transition

versus the detuning from resonance for the OOPH mode . . . . . . . . . 81

5.3 The 24Mg+internal levels used for Doppler cooling . . . . . . . . . . . . 82

5.4 Orientation of the 24Mg+laser beams for Doppler cooling with respect to

the trap axis and the magnetic eld . . . . . . . . . . . . . . . . . . . . . 83

5.5 The steady state population of state |ci for the three-level model for

24Mg+ Doppler cooling for various detunings ∆ of laser beam 2 . . . . . 90

5.6 The steady state population of state |bi for the three-level model for

24Mg+ Doppler cooling for various detunings ∆ of laser beam 2 . . . . . 91

5.7 The steady state population of state |ai for the three-level model for

24Mg+ Doppler cooling for various detunings of ∆ of laser beam 2 . . . . 92

5.8 Excited state |ci population versus the Rabi frequency Ω of both laser

beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
xvi

5.9 Populations as function of time for detunings ∆ = δ = 0 . . . . . . . . . 93

5.10 Populations as function of time for detunings ∆ = δ = 0 . . . . . . . . . 94

5.11 Population as function of time for detunings ∆ = 3

1

4 √

2

Ω2 and δ = 0 . . . . 94

5.12 Population as function of time for detunings ∆ = 3

1

4 √

2

Ω2 and δ = 0 . . . . 95

5.13 Excited state population in the co-linear Doppler cooling conguration . 97

5.14 The excited state population versus the velocity along the trap axis for

dierent values of Ω2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.15 Excited state population versus velocity for the case of orthogonal Doppler

cooling beams with dierent values of ∆. . . . . . . . . . . . . . . . . . . 99

5.16 Excited state population versus velocity for the case of orthogonal and

co-linear Doppler cooling beams with dierent values of ∆ . . . . . . . . 100

5.17 Excited state population versus velocity for orthogonal cooling beams

using dierent values of Ω2 . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.18 Sideband opping on the INPH mode on a pair of 9Be+and 24Mg+ions

after 24Mg+ Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.19 Sideband opping on the OOPH mode on a pair of 9Be+and 24Mg+ions

after 24Mg+ Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.20 Flopping on a motion sensitive carrier transition on a pair of 9Be+and

24Mg+

ions after 24Mg+ Doppler cooling . . . . . . . . . . . . . . . . . . 106

5.21 The n ̄INP H of the INPH mode of motion achieved after Doppler cooling

with use of a two-level system in 24Mg+versus the detuning . . . . . . . 107

5.22 The expected n ̄OOP H of the OOPH mode of motion achieved after Doppler

cooling with use of a two-level system in 24Mg+versus the detuning . . 108

6.1 Dierent types of sideband transitions . . . . . . . . . . . . . . . . . . . 115

6.2 Energy level diagram illustrating stimulated Raman sideband transitions 120
xvii

6.3 This gure depicts a few of the dierent types of stimulated Raman transitions that are possible . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4 The two main types of transitions used in Raman sideband cooling . . . 130

6.5 Probability for nding the ion in Fock state |ni, assuming the ion's motional population follows a thermal distribution equation (6.63)with n ̄ = 15133

6.6 The largest motional state |ni that needs to be cooled in order to get X

% of the motional population in the ground state for a range of n ̄ . . . . 134

6.7 This gure shows the rst zeros in the Laguerre polynomials as function

of the Lamb-Dicke parameter η . . . . . . . . . . . . . . . . . . . . . . . 135

6.8 This gure shows the rst zeros in the Laguerre polynomials as function

of the Lamb-Dicke parameter η . . . . . . . . . . . . . . . . . . . . . . . 135

6.9 The value of the Laguerre Polynomials L

|n

0−n|

n



η

2



versus n for rst sideband transitions, where |n

0 − n| = 1 . . . . . . . . . . . . . . . . . . . . 136

6.10 The value of the Laguerre Polynomials L

|n

0−n|

n



η

2



versus n for a sideband

rst stimulated-Raman transition, where |n

0 − n| = 1 . . . . . . . . . . . 137

6.11 The scaled Rabi frequencies Ωn0

,n/Ω from equation 6.23 versus |ni for the

rst and second sidebands where η =0.28 . . . . . . . . . . . . . . . . . 137

6.12 The scaled Rabi frequencies Ωn0

,n/Ω from equation 6.23 versus |ni for the

rst and second sidebands where η =0.08 . . . . . . . . . . . . . . . . . 138

6.13 This gure depicts the relevant energy levels for Raman cooling with 24Mg+140

6.14 Laser Beams used for Raman cooling of 24Mg+ ions in traps A and B . . 141

6.15 Experimental outline for performing and detecting sympathetic cooling

on two pairs of trapped 9Be+ − 24Mg+ ions . . . . . . . . . . . . . . . . 145

6.16 Frequency scan over the resonance of the MA transition on the INPH

mode of the 9Be+ − 24Mg+ pair of ions in trap A . . . . . . . . . . . . . 146

6.17 Frequency scan over the resonance of the MS transition on the INPH

mode of the 9Be+ − 24Mg+ pair of ions in trap A . . . . . . . . . . . . . 147
xviii

6.18 Frequency scan over the resonance of the MA transition on the INPH

mode of the 9Be+ − 24Mg+ pair of ions in trap B . . . . . . . . . . . . . 148

6.19 Frequency scan over the resonance of the MS transition on the INPH

mode of the 9Be+ − 24Mg+ pair of ions in trap B . . . . . . . . . . . . . 149

6.20 Frequency scan over the resonance of the MA transition on the OOPH

mode of the 9Be+ − 24Mg+ pair of ions in trap A . . . . . . . . . . . . . 150

6.21 Frequency scan over the resonance of the MS transition on the OOPH

mode of the 9Be+ − 24Mg+ pair of ions in trap A . . . . . . . . . . . . . 151

6.22 Frequency scan over the resonance of the MA transition on the OOPH

mode of the 9Be+ − 24Mg+ pair of ions in trap B . . . . . . . . . . . . . 152

6.23 Frequency scan over the resonance of the MS transition on the OOPH

mode of the 9Be+ − 24Mg+ pair of ions in trap B . . . . . . . . . . . . . 153

7.1 This shows possible paths in phase space for the states of a product state

when a geometric phase gate has been applied . . . . . . . . . . . . . . . 157

7.2 Phase gate details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3 Phase gate pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.4 Experimental level diagrams for a phase gate on 9Be+ ions . . . . . . . . 172

7.5 Experimental pulse sequence for performing the geometric phase gate on

the four ion mixed species chain 9Be+ − 24Mg+ − 24Mg+ −9 Be+ . . . . 178

7.6 Parity opping on the two 9Be+ ions in the four ion chain 9Be+ −24Mg+ −

24Mg+ −9 Be+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.1 9Be+ − 24Mg+ mechanical oscillators . . . . . . . . . . . . . . . . . . . . 185

8.2 Creation of entangled mechanical oscillators . . . . . . . . . . . . . . . . 185

8.3 Entanglement demonstration through parity oscillation . . . . . . . . . . 192
Chapter 1

Introduction

1.1 Motivations

This dissertation presents two experiments involving entangled mechanical oscillators [Jost 09]. The rst experiment entangles the internal spin of a trapped ion with a

spatially separated mechanical oscillator. The second experiment entangles the motion

of two spatially separated mechanical oscillators. In these experiments, the mechanical

oscillators consist of pairs of trapped ions held in dierent locations. A pair of trapped

ions behaves like a system of masses connected by springs, a canonical example of a

mechanical oscillator.

Creating novel entangled states of mechanical oscillators serves as a test of quantum mechanics in a new regime. The techniques developed to reach this end formed

key components in demonstrating a complete methods set for scalable ion trap quantum information processing (QIP) [Home 09] and realizing the rst multi-qubit quantum

processor [Hanneke 10].

Quantum mechanics is a very successful theory for explaining many aspects of

nature. However, it is perhaps better known for some of the counter intuitive predictions

it makes about the world, such as quantum mechanical superpositions [Schrödinger 35]

and entanglement [Einstein 35, Bell 64]. A superposition state is one where a system

is in two or more possible states at the same time. Entanglement manifests itself as a

correlation between two systems that is stronger than anything described by classical
2

physics. These eects have been observed in many systems, but at size scales or in

systems not commonplace outside of a lab. One of the unresolved questions in physics is

why do we not experience these eects in everyday life, such as cats in a superposition

of being both dead and alive [Schrödinger 35]. Perhaps it might depend on the number

particles that can be put in a superposition state or entangled [Bassi 03, Leggett 02a]

or maybe only certain degrees of freedom can experience these eects.

Many experiments have probed superpositions and entanglement in atomic, photonic, condensed matter systems [Aspelmeyer 08a, Southwell 08]. For the case of ions

a Schrödinger Cat  superposition state [Schrödinger 35] has been created using the

internal spin state of a 9Be+ ion and a coherent state of its motion [Monroe 96]. Several dierent experiments have entangled the internal states of trapped ions [Sackett 00,

Leibfried 04, Leibfried 05, Roos 04, Riebe 04, Häner 05]. Also entanglement has been

demonstrated between a trapped 111Cd+ ion and a photon [Blinov 04], and between

the internal states of two 171Yb+ ions held in separate traps approximately 1m apart

[Moehring 07]. With neutral atoms, the internal spin states of two clouds of Cesium

atoms have been entangled [Julsgaard 01]. Each cloud contained ∼ 1012 atoms. Using

the solid-state system of a semiconductor quantum-dot, two excitons have been entangled [Chen 00]. In addition, entanglement has been demonstrated between the electron

spins of two separate quantum dots [Petta 05]. Another demonstration of entanglement

in a solid-state system involved the currents in two superconducting circuits [Steen 06].

One of the goals of this thesis was to test the limits of quantum mechanics by

probing entanglement in mechanical oscillators. Mechanical oscillators are ubiquitous

in nature. The same equations that govern the motion of a pendulum in a grandfather

clock also dictate the motion of ions in an ion trap. The demonstration of entanglement

of mechanical oscillators on a small size scale formed from a few atoms (bottom up

approach), might imply the possibility of entangling them on a larger scale.

Currently there is a research eort taking a top down approach, focusing on ob-
3

serving quantum mechanical states in the motion of high Q nano- and micro-mechanical

resonators. Only recently has a mode of a nano-mechanical resonator been cooled

to the ground state [O`Connell 10]. There are many other forms of nano- and micro mechanical resonators being studied: harmonically suspended gram sized mirrors

[Corbitt 07], micro-mirrors [Arcizet 06, Gigan 06], SiN3 membranes held in an optical

cavities [Thompson 08], toroidal optical micro cavities [Kippenberg 05], superconducting microwave resonators coupled to a nano-mechanical beam [Regal 08], and dielectric spheres in a optical cavities [Chang 10]. For a review of the eld see references

[Kippenburg 08] and [Aspelmeyer 08b]. These systems will oer a good testing ground

for the eects of quantum mechanics in mesoscopic systems.

The oscillating mechanical systems used in this thesis are composed of two pairs of

9Be+ −24Mg+

ions held in separate locations. The two pairs of ions each vibrate in a way

similar to two balls connected by a spring, and it this motion that is entangled. Since

each oscillator is comprised of only two atoms, this is one of the simpler mechanical

system that could be studied, which makes it an elemental starting point for testing

entanglement of distinct mechanical systems. The systems described above contain

many more atoms. However, the size scales of the systems are comparable. The extent

of one of the 9Be+ − 24Mg+oscillators is ∼ 4 μm, which is comparable to the size scale

of some of the nano-mechanical resonators [Regal 08]. The spacing between the two

oscillators in the experiment was ∼ 240 μm. This may not seem like a large spacing,

but relative to the other distance scales, it is. The spatial extent of the ground state

wavefunction of 9Be+ is ∼ 10 nm [Wineland 98]. If the ground state wavefunction was

scaled up to the size of a soccer ball (diameter = 22 cm), then the two soccer ball sized

9Be+ and 24Mg+

ions that comprise one mechanical oscillator would be ∼ 100 m apart.

The other mechanical oscillator would be over 5 Km away. This work represents the

rst demonstration of entanglement between two such separated mechanical systems.

Another focus of the entangled mechanical oscillators experiment was the devel-
4

opment of the techniques that have important applications for trapped ion based QIP.

The simplest component in classical information theory is the bit, which can be either

0 or 1. In QIP, the analogy to the bit is called the quantum bit or qubit. The qubit is

a two-level quantum system, which has states that can be labeled with 0 or 1 just like

a bit. However, in contrast to a classical bit a qubit can also exist in a superposition

state, where it can be thought of as carrying the information of both 0 and 1 at the

same time, until it is measured [Schrödinger 35, Nielsen 00]. R. Feynman was the rst

to propose that a quantum mechanical systems could be used to perform simulations of

other quantum systems [Feynman 81, Feynman 85]. David Deutsch extended some of

these ideas to universal computations [Deutsch 89]. For a classical computer any computation can be performed using NAND gates [Nielsen 00], which is called a universal

gate. An important realization was that a universal gate set for quantum computation

could be constructed with one and two qubit gates [Barenco 95, Sleator 95]. The eld

gained signicant momentum when P. Shor published his algorithm for factoring numbers with a quantum information processor [Shor 94]. Factoring numbers on a classical

computer with known algorithms scales exponentially with the size of the problem, but

Shor's algorithm scaled polynomially. This received a lot of attention as a result of the

signicant increase in speed it implied, and its potential use for breaking encryption

schemes. Another important algorithm developed by L. Grover involved searching for

an item in an unstructured list [Grover 97].

It was clear early on that it would take a large number of qubits to implement

an algorithm that would outperform a classical computer (for an example see reference [Steane 03a]). Qubit superpositions are sensitive to interactions with their environment. These interactions, which cause superpositions to decohere, degrade or

destroy the information encoded in the qubit. For most systems this would make

it very dicult to perform any signicant computation. However, it was discovered

by P. Shor [Shor 95] and A. Steane [Steane 96] that quantum error correcting codes
5

could be used to correct errors in qubits provided the errors were below a certain level

[Shor 96, Cirac 96b, Zurek 96, Kitaev 97a, Kitaev 97b, Steane 03b, Knill 05].

How one goes about building a useful quantum information processor is still an

open question. One of the early proposals for implementing QIP was for trapped ions

[Cirac 95]. This was followed shortly afterward by the rst demonstration of a quantum

gate with a trapped ion [Monroe 95a]. Since then many systems have been proposed

as a potential qubits [Ladd 10]: infrared photons, trapped ions, trapped neutral atoms,

electron spins in GaAs quantum dots, electron spins bound to 31P:28Si , 29Si nuclear spins

in 28Si, Nitrogen vacancy centers in diamond, and superconducting circuits. Although

it is not precisely determined what conditions qubits must satisfy in order to build a

useful quantum information processor, a reasonable set of criteria was put forth by D.

DiVincenzo [DiVincenzo 01]:

(1) A scalable physical system with well-characterized qubits

(2) The ability to initialize the state of the qubits to a simple ducial state

(3) Long relevant decoherence times, much longer than the gate operation time

(4) A universal set of quantum gates

(5) A qubit-specic measurement capability

To date, trapped ions are one of the most advanced architectures, and the basic DiVincenzo criteria have been demonstrated [Home 09]. It could be argued that it is not

yet a scalable system, since it has not been scaled to more than a few qubits; however,

there appear to be no fundamental limitations to doing this. One of the proposals for

scaling up the trapped ion QIP relies on using a multi-dimensional multi-zone trapping architecture [Kielpinski 02a, Steane 03a, Wineland 98]. An important component

of such architecture would be moving the qubit ions from one location to another. Some

of these processes, if not performed correctly, can excite the ion`s motion [Rowe 02].
6

Electric eld noise can contribute to this motional excitation. Reducing the amount

of motional excitation is important, because multiqubit gates require the motion to be

well-controlled. Gate interactions are mediated by the Coulomb interaction. This interaction has the result that multiple ions trapped together are coupled through their

motion, which acts as a data bus [Cirac 95]. If the ions have a uncontrolled motional

excitation, this can lead to errors. One solution to this problem is to laser cool the

ion`s motion [Wineland 98]. However, this cooling cannot be performed on the qubit

ions. Laser cooling techniques require dissipation, which destroys information encoded

in the internal states. One solution is to simultaneously trap a type of ion dierent from

the qubit ions, called a sympathetic cooling ion, which can be used for laser cooling

[Blinov 02, Rohde 01, Kielpinski 00, Barrett 03].

The work discussed in this thesis and in reference [Jost 09] represents the rst

demonstration of the ability to distribute entangled qubit ions in an ion trap array and

sympathetically recool them. Following the recooling, single qubit gates were performed.

These techniques paved the way for an experimental demonstration of performing multiple two qubit gates with transport and sympathetic cooling [Home 09]. They also

enabled the demonstration of the rst two-qubit quantum processor [Hanneke 10].

1.2 Thesis Organization

This thesis describes the most important components for performing the entangled mechanical oscillators experiments [Jost 09]. The thesis begins with an overview

of the apparatus (chapter 2), where the construction of the ion trap and the vacuum

system are discussed. Other subjects covered in this chapter are magnetic eld coils,

the imaging system, the laser sources, and the experimental control. The next chapter

(chapter 3) discusses the 9Be+ qubit ion used in the experiments. In addition to being a component of the mechanical oscillators experiment, the 24Mg+

ion is also used

as a sympathetic cooling ion, which is discussed in chapter 4. The ability to re-cool
7

the two mechanical oscillators using 24Mg+ions is a crucial part of the experiment.

Chapter 5 discusses Doppler cooling, which is the rst stage of cooling used in the experiments. This chapter also shows the results of using Doppler cooling of a 24Mg+

ion

to sympathetically cool a 9Be+ ion. The entangled mechanical oscillators experiments

required the additional ability of cooling the mechanical oscillator's motion to the ground

state. This is accomplished with Raman cooling, which is described in chapter 6. This

chapter also covers the basics of Raman transitions. It concludes with the experimental demonstration of ground state cooling of the motion of two pairs of trapped ions

held in dierent locations. This served as a tune-up experiment for the work discussed

in the nal chapter. The next chapter (chapter 7) discusses how the internal states

of two 9Be+ ions can be entangled. This entanglement technique is used to generate

the entanglement that will be transferred to the mechanical motion of the ions. This

chapter concludes with the demonstration of entanglement between two 9Be+ ions in a

9Be+ − 24Mg+ − 24Mg+ −9 Be+ mixed species ion chain. The nal chapter (chapter 8)

discusses the entangled mechanical oscillators experiment.
Chapter 2

Apparatus

This chapter describes the apparatus used in the experiments discussed in this

dissertation. A photo of part of the apparatus appears in gure (2.1). In the photo

some of the optics and the ion trap can be seen. This experiment includes ve frequency

doubled dye lasers, hundreds of optical elements, dozens of servo loops, and a multizone ion trap. This chapter attempts to highlight some of the salient components of

the experiment. A simple description of ion trapping and the ion trap used begins the

chapter. This is followed by sections related to the ion trap: a brief discussion of the

sources of the potentials applied to the ion trap, loading ions into the ion trap, and the

ultra high vacuum system containing the ion trap. The next sections describe a few

components external to the ion trap: the magnetic eld coils and imaging system. The

laser sources and dierent laser beams used are then discussed. This chapter ends with

a short discussion of the experimental control used to run the experiments.

2.1 Ion Trap

2.1.1 Ion Trap Theory

One diculty in studying quantum systems comes from their strong coupling

to the environment, which often causes decoherence of the quantum states. Trapped

ions are a good system in which to study quantum mechanical eects [Meekhof 96,

Rowe 01, Barrett 04, Leibfried 05, Jost 09], because they can be reasonably decoupled
9

Ion Trap

Experimental Apparatus for the 

Entangled Mechanical Oscillators Experiment 

Figure 2.1: Photo of the experimental apparatus at NIST used for the entangled mechanical oscillators experiment [Jost 09]. This apparatus has also been used in several quantum information processing experiments [Langer 05, Ozeri 05, Home 09, Hanneke 10].

Part of the ion trap apparatus is visible in the upper right hand corner of the photo.

from the environment [Wineland 98, Langer 05, Bollinger 91]. Long coherence times,

exceeding ten minutes, have been demonstrated for the internal states of trapped ions

[Bollinger 96, Langer 05]. This is not to say that trapped ions are immune to decoherence

(Reference [Wineland 98] discusses many of the sources of decoherence.).

Ions can be well isolated from the environment by conning them in an ion trap

which is held inside a vacuum system. The ions have a charge, which allows them to be

strongly conned using electric elds. However, from Earnshaw's Theorem we know A

charge acted on by electrostatic forces cannot rest in stable equilibrium in an electric

eld [Foot 05]. One way this can be understood is with Gauss's Law, which is ∇·E = 0
10

Linear RF Paul Trap

RF electrode High DC voltage

control electrode

Low DC voltage

control electrode

RF

DC Trap Frequencies

Axial : ~ 5 MHz

Radial: ~ 15 MHz

Radial Freq μ 1/Mass

Positive ions

Radial Directions

Trap Axis

X Y

Z

Uo

Vocos(ΩTt)

Figure 2.2: Illustration of a linear Radio-Frequency (RF) Paul trap used to trap ions.

The gure on the left shows the electrodes used in an ion trap with one trapping zone. In

this style of ion trap the electrodes are made from conducting rods. There are two types

of electrodes RF and static (DC). The RF electrodes have a large amplitude oscillating

electric potential applied to them This provides a quadrupole conning eld resulting

in radial connement for the ions. The radial directions labeled x and y are the two

directions orthogonal to the long axis (trap axis) of the system. The trap axis is labeled

with the coordinate z. A end on view of the electrodes is shown in the right part of

the gure. The quadrupole eld lines are shown in this picture. The DC electrodes are

segmented so that dierent voltages can be applied to create a conning potential along

the trap axis. The connement is made weaker along the trap axis causing the ions to

form a linear chain in this direction.

for a space with no free charge. Therefore, Gauss's law requires that all the electric eld

lines that enter a region of space must leave, which implies that along some directions,

the electric force must be outward. This prevents any stable connement of charged

particles.

There are a couple of solutions to this problem. A combination of static electric

elds and a magnetic eld can be used (Penning Trap). Combining static electric elds

with an oscillating electric eld provides a stable trap (Paul trap). The linear Paul trap

uses oscillating electric elds to conne a charged particle in two dimensions. This can

be visualized using a classical mechanics example. Imagine placing a ball on a saddle

potential. The ball can sit in the middle of the saddle, but it is in an unstable equilibrium

in the XY plane. Rotating the saddle about the vertical axis (z axis, gravity acts in the

z direction in this example) can create a stable equilibrium. If the ball starts to roll

down the saddle it is pushed back up as the saddle, rotates. In the case of a charged
11

particle the z direction is still not conned. However, connement in this dimension can

be provided with static electric elds.

A linear radio frequency (RF) Paul trap is used in experiments discussed in this

dissertation. The design of the trap is similar to the original design by Drees and Paul

[Drees 64], but is closer to the one described by Raizen in reference [Raizen 93]. A

review of ion traps can be found in an article by Paul [Paul 90] and the books by Gosh

[Ghosh 95] and Major et al. [Major 05]. A schematic of a linear RF Paul trap (will be

referred to as an ion trap) is shown in gure (2.2).

The left side of the gure shows all of the relevant electrodes for creating an ion

trap with a single trapping zone. There are two types of electrodes, which are made from

conducting rods: RF electrodes and DC static potential electrodes. The DC electrodes

are coupled with a capacitor to ground, but static DC potentials can be applied to them.

The radial directions are shown in the gure on the right. To provide connement in these dimensions, a potential Vo cos(ΩT t) is applied to the RF electrodes. The

frequency of the oscillating potential is ΩT , and for the trap discussed below is ∼ 150

MHz. The magnitude of the potential is given by Vo and can be a few hundred volts in

typical operating conditions. This gives rise to the potential [Wineland 98]

Φ '



Vo cos (ΩT t)

2

 1 +

x

2 − y2

R2



. (2.1)

The distance from the trap axis to the electrodes is R, and for the trap described

below R ∼ 140 μm. To account for the electrode surfaces not lying at quadrupole

equipotentials, a multiplicative geometric factor of order one can be used. This is the

oscillating saddle potential described above. This does not provide connement along

the trap axis (z direction). This can be accomplish by applying a static voltage Uo to

the outer DC electrodes. It is assumed the central DC electrodes are held at 0 V. This
12

gives rise to the potential [Wineland 98]

Φs = κUo



αz2 −

1

2



βx2 + γy2





(2.2)

=

m

2q

ω

2

z



z

2 −

1

2



β

α

x

2 +

γ

α

y

2

 . (2.3)

The constants α, β, and γ are determined by the electrode geometry. The potential

must satisfy Laplace's equation, which puts the constraint 2α − β − γ = 0 on the

coecients. The axial trap frequency ωz =

q

2καqUo

m

, where κ is a geometric factor, m

is the ion's mass, and q is the charge of the ion. The expression for the static potential

satises Laplace's equation and the constant  will depend on the ion trap geometry.

This assumes that only a single ion is being trapped. Trapping more than one ion does

not change the potential but will change the resonant frequencies of the ions. The total

potential is then

ΦT =



Vo cos (ΩT t)

2

 1 +

x

2 − y2

R2



+

m

2q

ω

2

z



z

2 −

1

2



β

α

x

2 +

γ

α

y

2

 . (2.4)

The total potential in the z direction has only a harmonic term. The other two dimensions do not have such a simple solution for the motion. The ions motion in these

directions is given by the Mathieu equations [Wineland 98]

d

2x

dς2

= [ax + 2qx cos (2ς)] x = 0 (2.5)

d

2y

dς2

= [ay + 2qy cos (2ς)] x = 0. (2.6)

The following substitutions have been made:

ς =

ΩT t

2

, (2.7)

ax =



4q

mΩ2

T



(κβUo), (2.8)

ay =



4q

mΩ2

T



(κγUo) (2.9)

and

qx = −qy =

2qVo

Ω2

T mR2

. (2.10)
13

In the regime where ai < q2

i  1, i ∈ x, y, the Mathieu equations can be solved using

the general Floquet solutions [Wineland 98, Ghosh 95] giving the equations of motion

for the x and y directions.

ui (t) ' Ai



cos (ωit)

h

1 +

qi

2

cos (ΩT t)

i , (2.11)

where ui (t) can either be x (t) or y (t). The constant Ai depends on the initial conditions.

The frequency ωi =

q

ai +

q

2

i

2

, where i ∈ x, y, and this frequency is lower than ΩT . If

the second term oscillating at ΩT is neglected, then in the two radial directions the ion's

motion behaves as if conned in a harmonic potential. This is referred to as the pseudo

potential ΦP

ΦP =

1

2q

mω2

r



x

2 + y2



, (2.12)

where ωr ' √

qVo

2ΩT mR2

. The expression for ωr assumes the eects of the static potential

are negligible. Thus, the ion can be described as experiencing harmonic connement

in all three dimensions. For the traps discussed here, the connement in the radial

directions is stronger than the connement in the axial direction. For a single 9Be+ ion,

typical radial frequencies are ωr = 2π ×(10 − 15) MHz, and typical axial frequencies are

ωz = 2π × (1 − 5) MHz. This imbalance in the conning potential causes the ions to

form a linear chain along the trap axis. One important feature of the pseudo potential is

its mass dependence. The frequency of the pseudo potential is proportional to 1

m

. Thus,

heavier mass ions see a weaker radial potential. This dependence can be exploited to

reorder mixed species ion crystals [Jost 09].

2.1.2 Ion Trap Construction

The construction of a linear RF Paul trap can take many dierent forms. For

quantum information processing it will be important to have many dierent trapping

regions [Kielpinski 02b], and some designs are better suited for this than others. A trap

constructed from conducting rods as depicted in gure (2.2) is not well suited to this
14

1.5 cm

Trap Waffers

Filter Boards

RF Electrode

DC Electrodes

Figure 2.3: Multi-zone alumina wafer style linear RF Paul trap. This ion trap has gold

electrodes deposited on laser machined alumina electrodes. The trap is constructed from

several alumina wafers. Visible on the right side of the gure is the lter board, which

electrically lters the DC voltages applied to the electrodes.

task, because it is typically dicult to scale this down in size. One construction technique that is potentially scalable uses laser machined alumina wafers with patterned

gold electrodes [Rowe 02]. Although, it would be challenging to use this fabrication

technique to create a trap with thousands of trapping zones, no fundamental limitations exist. Currently one of the most advanced ion traps of this style has 18 different trapping zones [Blakestad 09], and has been used to transport an ion around

a corner in a 2D junction. The ion trap used in the work discussed in this dissertation is depicted in gures (2.3) and (2.4), and was built by John Jost and much of

the design was done by Murray Barrett. It is a six-zone linear RF Paul trap. This

particular ion trap has been used in several experiments. It was rst used in a experiment demonstrating eld-independent qubits in 9Be+ [Langer 05]. This was fol-
15

Trap Waffers

DC Electrodes RF Electrodes

Filter Boards

Capacitor

Resistor Figure 2.4: Multi-zone alumina wafer style linear RF Paul trap. This is the same trap

depicted in gure (2.3).

lowed by an experiment that studied hyperne coherence in the presence of spontaneous

scattering in 9Be+ [Ozeri 05]. The next publication using this trap was the demonstration of entangling two mechanical oscillators [Jost 09], which is also the focus of

this dissertation. Two quantum information processing (QIP) protocols have since been

performed in this ion trap. One demonstrated a complete methods state for scalable

QIP [Home 09]. The second QIP experiment was the rst demonstration of a univer-
16

sal two-qubit quantum processor [Hanneke 10]. A nearly identical trap has been used

in several other experiments, see references [Barrett 04, Chiaverini 04, Chiaverini 05,

Chiaverini 06, Leibfried 04, Schaetz 04, Schaetz 05, Leibfried 05, Reichle 06b].

2.1.2.1 Construction

Pictured in gures (2.3) and (2.4) is the ion trap that will be described here. There

are two main parts seen in the gures: the ion trap and the lter boards. The ion trap

consists of two rectangular (1.5 cm x 1.8 cm x 125 μm) alumina wafers stacked together.

Stacked below the ion trap are two additional wafers called the bias board wafer, and

mask wafer (1.5 cm x 1.8 cm x 125 μm). The purpose of these wafers is described below.

The wafers are made from polished alumina from Coors Tek. Additional alumina spacers

set the spacing between the various wafers (see gure (2.6) for a side view). The stack

of wafers is held together with two 18-8 series stainless steel screws, which have been

annealed to reduce their magnetization.

Trap wafers

In gure (2.3) one of the trap wafers is visible. The two types of electrodes can

be seen, which are the RF and DC electrodes. There is another trap wafer 200 μm

below the one seen in the gure with an identical set of electrodes patterned on it. The

electrodes on the two trap wafers are on opposites sides, see gure 2.5. The slits in the

trap wafer, which segmented the DC electrodes, are created with laser machining. A

schematic top view of one of the trap wafers with the dimensions is shown in gure (2.7).

Gold electrodes are made with a ion assisted E-beam deposition, which sputter-coated

gold onto the alumina wafers (There is a titanium adhesion layer between the gold and

the alumina.). The gold layer has a thickness of ∼ 2.5 μm. The sputter-coating allowed

for gold to be deposited in the 20 μm slits between the electrodes. This is important

because it helps reduce the amount of insulating surface near the ions. Insulating surfaces

are problematic, because they can become charged, which leads to uncontrolled electric
17

elds.

Figures (2.7) and (2.5) also show the DC and RF electrodes. The DC electrodes are

segmented, so that potentials can be applied independently to these electrodes. There

are eight electrodes on each of the trap wafers. The RF electrode is not segmented.

One feature visible in the gures is that distance between the DC and RF electrodes is

dierent in two regions. The region where the distance is 400 μm is called the loading

zone, and region where the distance is 200 μm is called the experiment zone. To load

ions into the trap neutral 9Be and 24Mg atoms are emitted form a thermal source (see

section (2.3)). As a result of this process the electrodes can become coated with the

neutral atoms. There is some experimental evidence that this can lead to noisy electric

elds [Turchette 00, Rowe 02]. It is desirable for experiments to have the electric elds

well controlled. Thus, the ion trap has a loading region and an experiment region that

is shielded from the neutral atom ux by the mask board.

In the experiment region (see gures (2.7) and (2.5)) there are dierent sizes of

electrodes. Ideally, all of the electrodes would be as small as possible to provide the most

accurate control over the potential. In addition, small electrodes are useful for separating

ions [Rowe 02]. The separation process works by bringing an electrical potential wedge

up between two ions, which forces them apart. Typical ion spacings are a few microns,

so small electrodes allow for ner control of this separation process. The ion trap has

one electrode that is smaller than the rest, which is used for separation. It is 100 μm

wide, and this size is limited by the current capabilities of laser machining.

For all the experiments discussed in this dissertation, there are three main areas

of the ion trap that are relevant. There are two areas used for trapping during the

experiments, and the electrodes nearest these two regions are labeled with A and B. The

third area is the separation region, which is near the separation electrode (labeled SE).

Bias and Mask Wafers

The two additional wafers depicted in gure (2.6) are the bias wafer and the mask
18

wafer. Both of these wafers are made from polished alumina and are coated with gold on

both sides to create one large electrode. The schematic of the bias wafer can be found

in the appendix (D). It has a rectangular opening (600 μm x 1.3 mm). This slit allows

for laser access to the ions and prevents the metallic neutral atom sources from shorting

out the electrodes. It is called the bias wafer, because a potential bias can be applied

to the electrode on the wafer. The electric eld from the bias wafer is orthogonal to the

trap axis (2.2) and at 45 degrees to the both radial directions. This eld is used to help

null stray electric elds [Meekhof 96].

The mask wafer`s main function is to prevent any ux from the neutral atom

sources depositing on the electrodes in the experiment region. It has two rectangular

openings, which allow the laser beam access to the dierent regions of the trap while at

the same time shielding the experiment region from neutral atom ux. A picture of the

mask wafer appears in appendix (D). This wafer has also been gold plated on both sides

to create one large electrode. This enables an additional bias potential to be applied.

However, it is not typically used for this, and is held at 0V to prevent it from charging

and creating stray electric elds.

Filter boards

One of the two lter boards is visible in gures (2.3) and (2.4), but both can be

seen in gure (2.6). The lter boards are also made of alumina and patterned with gold

traces (Here the traces were silk screened onto the lter boards.). The lter boards are

used to deliver the DC electrical potentials to the trap wafers, and provide a RF ground

for the DC electrodes. Each electrode on the trap wafer has a corresponding trace on

the lter board. Also, each trace on the lter board has a low-pass lter created with a

820 pF capacitor and a 240 Ω resistor, which has a corner frequency of ∼ 800 KHz. This

helps lter high frequency noise from the voltage sources that provide the DC potentials.

This is important because an ion's motion will be excited by electric eld noise at the

resonant frequency of the motion.
19

AB

SE

Loading 

Region

RF Electrode

DC Electrodes

Views of the Ion Trap and Electrode Labeling

Trap Wafer

Trap Wafer

RF 

Electrode

DC 

Electrodes

DC 

Electrodes

(on bottom trap wafer)

Figure 2.5: Views of the ion trap and electrode labeling. The top two pictures show

two dierent perspectives of the ion trap. The bottom picture is a top view of one of

the trap wafers. The main DC electrodes are labeled. Electrodes A and B are the main

trapping regions used in the experiments. The narrow electrode is used for separation

of ion and is labeled SE. The region used for loading ions is labeled the loading region.
20

Mask Wafer

Bias Wafer

Trap Wafer

Trap Wafer

125

125

125

125

200

400

1.5 mm

Spacer

Spacer

Spacer

Filter Board

Filter Board

Neutral Atom 

Source

Side View of the Ion Trap

Figure 2.6: Side view of the ion trap. The gure shows how the two trap wafers, the bias

board wafer, and the mask wafer are stacked together to make the ion trap assembly.

There are additional alumina spacers that set the spacing between the various wafers.

These wafers are also made from polished alumina.

2.2 Producing the DC and RF Potentials

2.2.1 DC Potentials

The segmented DC electrodes provide the connement for the ions along the trap

axis. A trapping area is created by applying the appropriate potentials to the DC
21

1.250 mm

1.010 mm 1.250 mm

1.010 mm

6.000 mm

6.000 mm

7.500 mm

0.400 mm

0.200 mm 3.

0

6

0

m

m 2.

9

4

0

m

m

3.5

0

0

m

m

1.

500

m

m

6.

500

m

m

8.5

0

0

m

m

2.200

m

m

10.0

0

0

m

m

Dimiensions of a Trap Wafer (Top View)

Experiment Region

Loading Region

Screw Hole

Alignment Hole

DC Electrodes

Figure 2.7: Schematic (top view) of one of the trap wafers. The slits in the middle

region, which form the electrodes when gold platted, are made with laser machining.

The other holes were created when the wafers were made by the manufacturer. Two of

the holes are for the 18-8 stainless steel screws that hold the trap assembly together.

The other two holes are used when aligning the wafers.
22

Close up of Electrode Structure (Top View)

1mm

2.000 m

m

0.500 m

m

0.500 m

m

0.360 m

m

0.200 m

m

0.100 m

m

0.200 m

m

2.000 m

m

0.400 mm

0.200 mm

0.020 m

m

Loading 

Region

Experiment 

Region

Figure 2.8: Schematic of the electrode structure. This is a close up view with dimensions

of the trap wafer shown in gure 2.7.

electrodes to create a potential minimum. Typically, the same voltage is applied to the

corresponding electrodes on each trap wafer. The source of the DC potentials is a 16-bit

analog output card from National instrument (NI PCI 6733 with 1MS/s). The applied
23

potential range is from -10 V to 10 V. This is a large enough potential to allow for

trapping frequencies of several MHz along the axial direction for 9Be+ and 24Mg+ions.

The voltages can be changed in a time-dependent fashion to move and separate the ions

[Home 06a, Reichle 06a, Rowe 02, Blakestad 09].

As noted in section (2.1.2.1) it is important to lter the DC potentials applied to

the DC electrodes to prevent excitation of the motion from electrical noise. On the lter

boards inside the vacuum system (section (2.4)) there is a low-pass resistor capacitor

lter for each electrode (see section (2.1.2.1)). External to the vacuum system there

are two in-series low-pass resistor/capacitor lters (R= 820 Ω and C = 1 nF, corner

frequency ∼ 200 KHz).

2.2.2 RF Potentials

The radial connement is provided by an oscillating RF electric eld, which is

applied to the RF electrodes as discussed in section (2.1). The large electric potential

at the trap is generated by a quarter wave coaxial RF resonant transformer [Jeerts 95],

and is constructed out of copper. The resonator has a resonant frequency of ∼ 150 MHz

and Q ∼ 300. An RF signal generator producing 1-2 W of power is inductively coupled

to the resonator. Radial trapping frequencies of ∼ 10 MHz are observed for a single

9Be+ ion when ∼ 1.5 W of RF power is coupled into the resonator.

2.3 Loading Ions

2.3.1 9Be+

The experiments discussed in this thesis involve two types of ions 9Be+ and 24Mg+.

Both types of ions are generated from neutral sources. The 9Be source is created from

50 μm diameter 9Be wire and wrapping it tightly around a 100 μm diameter Tungsten

wire (The 9Be wire comes from the Goodfellow corporation and is 99.7% pure. Note
24

that 9Be is the only stable one of the Beryllium isotopes. The Tungsten wire comes

from Alfa Aesar and is 99.9% pure). This 9Be wrapped tungsten wire is then coiled to

resemble a incandescent light bulb lament, where the coils have a diameter of a few

millimeters and about 6 turns are made. A current of 1.56 A at 0.99 V is sent through

the Tungsten lament, which heats up and evaporates some of the 9Be atoms. With this

current, the central region of the coiled is glow orange in color.

The neutral 9Be atoms are emitted in all directions, and some of the ux of atoms

pass through the loading region of the ion trap. The atoms must be ionized while in

the loading region in order to be trapped. To ionize the neutral 9Be atoms an electrongun is used. The electron gun used here is made from a ∼ 75 μm diameter thoriated

Tungsten wire (This wire comes from H. Cross Co.). A length of this wire ∼ 1.5 cm

long is bent at the middle to create a eld emitter. A current of 1.63 A at 1.10 V is

passed through the wire. The wire is biased to ∼ −110 V with respect to the resonator.

This provides an acceleration voltage for the electrons, which are directed toward the

loading region of the ion trap. These parameters create an emission current of ∼ 30

μA, where the emission current corresponds to the total number of electrons emitted.

The ∼ 120 Gauss magnetic eld used during the experiments is turned o for loading

9Be+ ions. These parameters yield a loading rate of one 9Be+ ion in ∼ 45 s. During

loading we apply a 9Be+ Doppler cooling laser beam in the loading region. This cools

the ion`s motion and causes them to uoresce. This uorescence is imaged onto a UV

sensitive charge-coupled device camera, which allows us to visually determine that an

ion has been loaded into the ion trap.

One major disadvantage of using an electron-gun for loading ions is that it does

not discriminate what atoms or molecules are ionized. Thus, occasionally other types

of ions are loaded into the trap. Typically, when this occurs all the ions are released

from the trap and it is re-loaded. Photo-ionization provides a way to selectively ionize

the 9Be atoms, and it is typically more ecient than ionization with an electron gun in
25

apparatuses similar to the one discussed in this thesis. Photo-ionization of 9Be was used

in reference [Blakestad 09].

2.3.2 24Mg+

To load 24Mg+into the ion trap a similar process to that described for 9Be+ is

used. The 24Mg atoms also come from a thermal source. It is fabricated by taking ∼ 5

mm length 18 Gauge tubing (outer diameter = 1.27 mm, inner diameter = 0.965 mm)

made of 316 stainless steel and cutting a slit perpendicular to the long axis of the tube.

The slit was ∼ 600 μm x ∼ 150 μm and created with a diamond saw. The tube is lled

with isotopically enriched 24Mg from Oak-Ridge National Labs. Isotopically enriched

24Mg is used, because 24Mg's natural abundance is only ∼ 80%. The other isotopes

25Mg and 26Mg would be trapped in our apparatus. However, we only want to work

with one of the isotopes. The stainless steel tube is sealed by mechanically crimping at

both ends, so 24Mg can only be emitted from the slit. A curre nt of 1.01 A at 1.22 V is

passed through the tube. This is sucient to evaporate some of the 24Mg in the tube.

The resulting neutral atom ux is directed toward the loading region of the ion trap.

As for the 9Be atoms, the 24Mg atoms need to be ionized in the trapping region

of the ion trap. This has been accomplished with the electron-gun described above.

However, photo ionization of 24Mg [Epstein 07, Seidelin 06, Madsen 00] was found to

be more ecient. The photo-ionization is a resonant two-photon process using 285 nm

light. The rst step excites the transition 3s

2 1S0 → 3s3p1P1 in 24Mg [Madsen 00]. The

second photon that excites the electron into the continuum can come either from the

285 nm photo-ionization source or from the 280 nm source used for Doppler cooling of

24Mg+

. Did you know Pirates Arrrr Cool! About 1 mW of power at 285 nm is used, and

the beam waist in the loading region of the trap is < 100 μm. With these parameters,

it takes about 45 s to load a 24Mg+ion into the ion trap.
26

Ion Trap Vacuum system

Ion Pump

Ion Gauge

Titanium-Subilmation

Pump

All Metal Valve

Quartz Cell

Trap Support

Electrical Feedthrough, 

25-pin 

Electrical Feedthrough, 7-pin 

RF Resonantor

RF Step Up 

Transformer

Figure 2.9: Schematic of the vacuum system discussed in section (2.4), where the major

components are highlighted. Two solid metal con-at tubes are shown as transparent,

so the titanium sublimation pump can be seen.

2.4 Vacuum System

Keeping the ion trap in an ultra-high-vacuum (UHV) environment (∼ 10−9 Pa)

is critical for the experiments discussed in this dissertation. This is important because

if the trapped ions undergo a single collision with the background gas this is enough to

disrupt an experiment. Typically, the ion is not expelled from the ion trap, but gains
27

enough energy to require being re-cooled before further experiments can be performed.

More importantly, 9Be+ and 24Mg+can undergo chemical reactions with H2 to for form

9BeH+ [King 99, Wineland 98] and 24MgH+ [Bertelsen 04, Mølhave 00]. These ions are

still trapped, but the absorption spectrum changes, and they are no longer useful. In the

apparatus used in the experiments discussed here, lifetimes of several hours for a single

9Be+ ion were typical (The longest lifetime observed was ∼ 42 hours.). A rough estimate

of the collision rate was made by looking at the rate at which a pair of 9Be+ −24Mg+ions

switched their order in the ion trap. It is assumed that a collision event imparts enough

energy to the ion pair that their order can change. The ions were observed to switch

places approximately once every couple of minutes, which gives a reasonable estimate

of the collision rate. An estimate of the collision rate could be made from knowing the

pressure in the vacuum system; however, it was unclear if the pressure gauge used in

the system (ion gauge) gave an accurate reading of the pressure for the typical vacuums

obtained.

The major components of the vacuum system are shown in gure (2.9): an ion

pump, a titanium-sublimation pump, an all metal valve, electric feedthroughs, an ion

gauge, a RF resonator, and a glass envelope. The rest of the vacuum system consists

of UHV grade con-at anges. Standard UHV vacuum techniques are used to assemble

the vacuum system [O'Hanlon 89]. After assembly the vacuum system was evacuated

through the all metal valve using a turbo pump, taking the pressure into the 10−5 Pa

regime. The system then was baked out at a temperature between 200oC and 250oC for

several days, and the gas was pumped out with a large external ion-pump connected to

the system via the all metal valve (Varian, part number: 9515027). The pressure was

in the 10−4 Pa range during the bakeout . An additional ion pump (Varian Plus 20

Starcell, 20L/s), which is permanently part of the system, was also turned on near the

end of the bake out. This ion-pump runs continuously during normal operation. The

pressure measured by the ion-gauge after the bake out was in the mid 10−10 Torr range.A
28

titanium sublimation pump (Varian, part number: 916-0061) is used to evaporate a

titanium getter coating on the inside walls of a section of the vacuum system. This

getter coating also acts as pump by trapping residual gas in the system. The pressure

in the vacuum system was measured with a nude Bayard-Alpert type ionization gauge

(Granville-Phillips). The lowest pressure measured by the ion-gauge was 1.33×10−8 Pa.

Other ion traps used in the past at NIST with similar or worse ion lifetimes measured

pressures < 1 × 10−10 Pa. This observation lead to the conclusion that the ion gauge

used might not be functioning properly, so it was turned o and not used on a regular

basis. Two electrical feedthroughs were used to bring DC currents and potentials into

the vacuum system. One 25 pin feedthrough (MDC, part number 633002) was used to

deliver all of the DC potentials to the segmented DC electrodes of the ion trap. Another

7 pin feedthrough (MDC, part number 647054) was used to supply the large currents

(1-2 A) used for the 9Be+ and 24Mg+sources (see section (2.3)).

The RF resonator used to deliver the RF potentials to the trap was described in

section (2.2.2). This resonator had an additional special feature. It was coated with

a special TiZrV getter coating [Benvenuti 01] (Coating done by the CERN technology

transfer, contact C. Benvenuti). It was dicult to evaluate the performance of the

coating, since it was believed the ion gauge was faulty. However, this vacuum system

has demonstrated comparative ion lifetimes to other traps, so it can be concluded that

it did not make things worse. Evidence that it may have had a positive inuence is that

this vacuum system holds the record for keeping a single 9Be+ ion in the ion trap of

∼ 40 hrs. The ion trap is mounted inside the resonator, which also has holes for optical

access. Surrounding the resonator is a quartz cell that includes UV grade fused silica

windows that provide optical access for the laser beams.
29

RF resonator

quartz envelope

ion

ion trap

magnetic field coils

f/1 imaging lens

ion pump,

titanium sublimation

p ump ,

ion gauge

f = 15 cm

Glan-laser

λ/4

polarizer

f = 15 cm

λ/4

λ/4

λ/2

f = 15 cm

λ/4

trap axis

B

R-CO

Mg-COa

Mg-COb

Beams Polarization#

BD Detuned

RD

Repumper

Mg load

Mg Photo-Ionization

Loading Beams*

* B=0 for loading

 Detection and 

Doppler Cooling Beams

BD

BD Detuned

RD

Repumper

 and Raman 

R-90

Mg-90a

Mg-90b

 and Raman 

# Vertical polarization points normal to the page

Beams and Optics Near The Ion Trap

V

V

V

f = 15 cm

Figure 2.10: Laser beams and optics near the ion trap. The 9Be+ and 24Mg+ Raman

beams are listed in the top portion of the gure. There are two 24Mg+ Raman beams for

each of the experiment regions A (Mg-COa and Mg-90a) and B (Mg-COb and Mg-90b) see

gure (2.5). The 9Be+ Raman beams ( R-CO and R-90) can be directed to either regions

A or B, using acousto-optic modulators (see section (2.7.0.1)). The 9Be+ detection and

Doppler cooling beams are only directed to experiment region A. The loading beams are

only applied to the loading region. This gure is modied from a gure appearing in

[Langer 06].
30

2.5 Magnetic-Field Coils

The ion trap is placed inside a Helmholtz coil. This created an approximately a

uniform magnetic eld oriented at 45 degrees to the trap axis that provides the quantization axis for the ions' internal states. The experiments discussed here used a magnetic

eld of ∼ 120 Gauss. This value of the eld was chosen, since it provides a magnetic

eld-independent transition in 9Be+ [Langer 05] (see chapter (3)). The two coils that

make up the Helmholtz pair are each made from nine turns of 1/8 inch outer diameter

hollow copper tubing. Each coil has a radius of 3.75 inches and they are placed symmetrically around the center of the trap one inch apart see gure (2.10). The copper tubing

is hollow to allow for water-cooling of the coils. Approximately 100 A at 5.5 V applied

to the coils.

2.6 Imaging System

Part of the imaging system is depicted schematically in gure (2.10). The imaging

system collects the ions` uorescence from trapping region A (see gure (2.5)) and can

either image it onto a ultra-violet (UV) sensitive charge coupled device (CCD) camera

or on to a photo multiplier tube (PMT). The CCD camera is used when loading the ions

into the ion trap, and the PMT is used for photon counting. Looking at the number

of photons collected in a 200 μs detection time distinguishes the qubit states from each

other (see chapter (3)). In a 200 μs detection period about 10 photons are collected

from a single 9Be+ ion with a background of ∼ 0.1 photons. The main component of

the imaging system is a ∼ f/1 lens. This lens is placed ∼ 40 mm from the ion trap,

outside of the vacuum system. The lens forms an image with a magnication of x5 at

an image plane. A pinhole with a diameter between 600 μm and 800 μm is placed at

this rst image plane to lter out stray light. A UV microscope objective is used as a

second stage lens to provide an additional magnication of ∼ 25 for a total magnication
31

Figure 2.11: Image of 9Be+ − 24Mg+ − 24Mg+ −9 Be+ chain of ions taken with a UV

sensitive charge-coupled device camera. The two outer ions in the image are the 9Be+

ions, and the imaging system and laser beams are congured to view only the 9Be+ ions.

Two 24Mg+ions are also present but not visible and have been drawn in the gure. The

distance between the 9Be+ ions is ∼ 10 μm.

of ∼ 125. This is enough to distinguish two 9Be+ ions that are about 4 μm apart on

the UV CCD camera. Figure (2.11) shows an image from the UV CCD camera of a

9Be+ − 24Mg+ − 24Mg+ −9 Be+ chain of ions. Only the 9Be+ ions (outer most ions)

uoresce. The two 24Mg+ions have been drawn into the gure. The spacing between

the two 9Be+ ions is ∼ 10 μm. The image can be either directed to the UV CCD camera

or the PMT with the use of a mechanical ipper mirror.

2.7 Laser Sources

The internal electronic states of trapped ions can be manipulated using electricdipole transitions. Typical these transitions are made between the ground and rst

excited electronic states. For the 9Be+ and 24Mg+ions used in the experiments discussed

in this dissertation this corresponded to making a transition between the S and P internal

levels. To excite these transitions requires a photon with and energy of ∼ 4 eV, which
32

Dye-Lasers ∼ Visible Color (nm) ∼ Visible Power (mW)

9Be+ Raman Laser 626.340 770

9Be+ Red Doppler (RD) Laser 626.393 540

9Be+ Blue Doppler (BD) Laser 626.264 605

24Mg+

Laser 560.710 400

24Mg+

Photo-ionization Laser 570.591 200

Table 2.1: Table of the visible laser systems. There are ve frequency doubled dye-laser

systems, four of which are which are used in the QIP experiments. The fth, which is

the 24Mg+photo-ionization laser is only used when loading the ions. This table shows

the visible wavelengths used, and table (2.2) shows the UV wavelengths after doubling

Dye-Lasers ∼ UV color (nm) ∼ UV power (mW)

9Be+ Raman Laser 313.17 100

9Be+ Red Doppler (RD) Laser 313.2 20

9Be+ Blue Doppler (BD) Laser 313.13 30

24Mg+

Laser 280.355 14

24Mg+

Photo-ionization Laser 285.3 4

Table 2.2: Table of the UV laser systems. There are ve frequency doubled dye-laser

systems, four of which are which are used in the QIP experiments. The fth, which is

the 24Mg+photo-ionization laser is only used when loading the ions. This table shows

the UV wavelengths used, and table (2.1) shows the visible wavelengths before doubling.

corresponds to a wavelength in the UV. The transition can be made in 9Be+ with a ∼ 313

nm photon and in 24Mg+with a ∼ 280 nm photon. To drive all the necessary transitions

in 9Be+ requires three dierent wavelengths (see tables (2.1) and (2.2)). One transition

is from 2S1

2

level to the 2P1

2

level and the transition occurs at wavelength of a 313.2

nm (The laser system used for this transition is referred to as Red Doppler (RD). see

chapter (3)). Another laser system is resonant with the 2S1

2

to 2P3

2

transition, and the

transition occurs at a wavelength of 313.13 nm (The laser system used for this transition

is referred to as Blue Doppler (BD). see chapter (3)). A third laser at a wavelength of

313.17 nm is ∼ 80 GHz blue detuned from the 2S1

2

to 2P1

2

transition (This laser system

is referred to as the Raman laser. see chapter (3)). To excite transitions in 24Mg+, one

laser system at a wavelength of 280.355 nm was used. It was ∼ 9.2 GHz red detuned from

the 2S1

2

to 2P1

2

transition (see chapters (4), (5), and (6)). A fth laser system was used
33

in photo-ionizing 24Mg to create 24Mg+[Epstein 07, Seidelin 06, Madsen 00], which uses

a ∼ 285.3 nm photon to excite the transition 3s

2 1S0 → 3s3p1P1 in 24Mg [Madsen 00].

Tables (2.1) and (2.2) shows a list of the dierent lasers used, their wavelength in the

visible and UV and the optical powers in the visible and UV.

To date there are no commercially available laser sources in the UV with sucient

optical powers to meet the experimental requirements. All ve laser systems were custom built. The laser systems used consist of a visible continuous-wave (CW) dye laser

that is frequency doubled [Boyd 68] using a resonant second harmonic generation cavity

[Steinbach 96] to create a UV source.

The CW dye lasers used are a 4-mirror unidirectional ring laser [Hollberg 90],

which were designed by Jim Bergquist. There are four mirrors used to create the ring

cavity (sometimes called a bow-tie cavity). The pump sources for all of the dye laser

were Coherent Verdi lasers, which are doubled Nd:Yag lasers emitting 532 nm light. The

pump powers used ranged from 3W - 6W.

Dierent types of dye were used for the 9Be+ and 24Mg+dye lasers. The 9Be+

lasers used Kiton Red dye made by Exciton. The dye was dissolved into ethylene glycol

at a concentration of ∼ 1g of dye per 1 L of ethylene glycol. The 24Mg+lasers used

200 mg of Rhodamine 575 dye dissolved in 50 mL of Benzyl Alcohol, which is then

mixed with 350 mL of ethylene glycol. For all the laser systems, a thin stream of dye is

created by forcing dye under pressure through a thin nozzle. The dye is then caught and

recirculated. The dye stream is at Brewster's angle with respect to the lasing direction.

Each of the dye lasers has several inter cavity elements. There is an optical diode

to make the laser lase in only one direction. Three frequency selective elements in the

cavity narrow the lasers linewidth: a birefringent lter, a thin etalon, and a thick etalon

(consists of two etalons). The laser frequency was set by aligning the transmission

maxima of all of the lters. This allows for tuning the laser in steps of < 0.005 nm.

The spacing between the two etalons that make up the thick etalon can be varied
34

with a piezo-electric transducer (PZT) to track the center frequency of the laser. This

is accomplished by modulating the PZT. The signal can be demodulated with a lockin-amplier to create a dispersive error signal with a zero crossing that corresponds to

maximum in transmission for that particular etalon mode. This signal can be used in a

servo loop. To stabilize the frequency of the lasers one of the cavity mirrors is mounted

on a PZT. Changing the voltage applied to the PZT shifts the frequency of the laser.

This PZT serves as the transducer in a servo loop. The feedback for this servo loop

is derived from a external Fabry-Perot cavity. For the 9Be+ lasers the cavity is made

from a 10 cm long INVAR support with two mirrors (a 10 cm radius of curvature and

reectivity of 95% at 626 nm) mounted on the ends. The 24Mg+lasers use a similar

external cavity. A Hansch-Couillaud lock [Hansch 80] is used to create the error signal

for the feedback to the PZT.

All of the lasers use an absolute frequency reference except for the 9Be+ Raman

laser. This laser is ∼ 80 GHz detuned from the 2P1

2

transition, so small deviations in

the frequency of the laser have a negligible eect on their function. The other lasers use

Doppler-free saturated absorption spectroscopy of 127I2, which is referred to as an Iodine

lock, to reference the lasers to a transition in 127I2 (The 24Mg+lasers used both 127I2

and 129I2.). The fast frequency noise of the laser is servoed using the Hansch-Couillaud

lock to the external reference cavity, and slow frequency drifts are removed with the

Iodine lock. One of the cavity mirrors in the reference cavity is mounted on a PZT,

which allows the cavity length to be changed. The servo signal from the Iodine lock is

feedback to this PZT. This changes the resonant frequency of external cavity, which in

turn aects the error signal of the Hansch-Couillaud lock. This causes the error signal

sent to the PZT inside the laser cavity to track this change, and thus changes the laser's

frequency. The laser`s frequency can be changed relative to the transitions in iodine,

using AOMs. The laser's linewidth is typically less than ∼ 500 KHz.

To produce the required UV laser light, the visible light is frequency doubled
35

[Boyd 68] inside a second harmonic generation cavity [Steinbach 96]. The cavity is a

4-mirror ring cavity (bow-tie cavity), allowing for the resonant build up of the visible

light. One of the mirrors used is partially transmissive to the visible light, which allows

the visible light to couple into the cavity. The second harmonic generation is done in

a non-linear β-barium borate (BBO) crystal [Boyd 68]. The UV light produced in the

BBO crystal passes out of the cavity through one of the cavity mirrors. This mirror

has a special anti-reection coating that is highly reective in the visible and allows

the UV light to be transmitted. The cavity is stabilized using a Hansch-Couillaud lock

[Hansch 80].

2.7.0.1 Beam lines

Although there are ve laser systems used in the experiment, there are 14 laser

beams generated from theses ve laser systems. Several of these beams were generated

with beam splitters, acousto-optic modulators (AOM) and an electro-optic modulator.

The absolute frequency of the lasers is stabilized to resonances in 127I2 and 129I2; however,

these resonances do not occur at the needed frequencies. Thus, AOMs and EOMs can

be used to shift the laser's frequency by the appropriate amounts. This section briey

highlights the systems used to generate the necessary laser beams.

Blue Doppler

The 9Be+ Blue Doppler (BD) beam line consists of three laser beams used for

Doppler cooling (see chapters and (3) and (5)) and state dependent resonance uorescence detection (see chapter (3)). The BD beam is approximately resonant with

|

2S1

2

, F = 2, mF = 2i → |2P3

2

, mI =

3

2

, mJ =

3

2

i transition in 9Be+ (see chapter (3)). It

is used for state-dependent resonance uorescence when tuned on resonance. It can also

be detuned by −Γ/2 relative to the 9Be+,

2P3

2

excited state of resonance for optimum

two-level Doppler cooling (Γ = 2π × 19.4 MHz) . The BD detuned beam and the BD

load beam are derived from the same beam but have been split by a beam splitter. They
36

vert. pol.

BD Split

+200

BD

Detuned

Sw

+200

BD SW

x2

+200

50/50

RD, Repumper

0

+200

+600

BD

+200 BD detuned

+600 BD

BD doubler

BBO

SW1

SW2 + 200

0

0

- 200

λ/2

PBS

BD Detuned, RD, Repumper

Load beam line

BD, BD Detuned, RD, Repumper

Experiment beam line

Blue Doppler (BD) Beam line: BD and BD Detuned

Figure 2.12: The Blue Doppler (BD) beam lines. This beam line generates the BD and

the BD Detuned beam lines. Also shown is the beam splitter where the beams RD and

repumper are combined with the BD and BD detuned beams. See the text for a detailed

description.

are sent to dierent locations in the ion trap (This is done after the beams propagate

through the AOMs discussed below). These beams are used for far-detuned Doppler

cooling. The BD load beam is used in the loading region of the ion trap to Doppler cool
37

ions as they are loaded. The BD detuned beam is directed into the experiment region

to perform far detuned Doppler cooling on the ions in that region. This far-detuned

cooling is useful for cooling very hot ions, because the motion of the ions can Doppler

shift their transitions out of resonance with the BD beam.

The BD detuned and BD load beams are ∼ 400 MHz lower in frequency than the

(3) BD beam. This frequency dierence is created with AOMs (see gure (2.12) for the

labeling of the AOMs). The BD detuned beam is taken from the 0th order of the BD

split AOM. It is then deected into the +1 order of the 200 MHz BD detuned SW AOM

(SW stands for switch). The BD beam is generated from the +1 diraction order of the

200 MHZ BD split AOM. This beam is then sent through a double pass AOM (BD SW)

that is tunable around a center frequency of 200 MHz. This allows the BD beam to be

tuned from resonance to a detuning of −Γ/2 without shifting the beam position. This

beam is combined with BD detuned beam by overlapping the beam with the +1 order

of the 200 MHz BD detuned SW AOM. The combining technique works because the BD

beam enters the AOM not at the Bragg diraction angle. It is not diracted like the

BD detuned beam. The beams then pass through a 50/50 beam splitter, which allows

them to be overlapped with the two Red Doppler beams discussed below.

These beams can be shut o by turning the power o to the AOMs. However,

there can be some transmitted leakage light, which causes excess scattering of photons

by the 9Be+ ions. To help solve this problem, two additional 200 MHz AOMs were added

in series after the beams have been combined. The rst 200 MHz AOM (SW1) diracts

the light into the +1 order. The second 200 MHz AOM(SW2) diracts the beams into

the -1 order, resulting in no net frequency shift of the beams. The 0th orders of the

beams are blocked. The beams are then sent through a beam splitter, which allows some

of the BD detuned light to be picked o and sent to the loading region of the ion trap.

RD Doppler

The 9Be+ red Doppler (RD) beam line consists of two beams RD and repumper.
38

Both beams are used in state initialization of 9Be+ (see chapter (3)) and for repumping

during Raman-sideband cooling (see chapter (3)). The RD beam is resonant with the

|

2S1

2

, F = 1, mF = 1i → |2P1

2

, F = 2, mF = 2i transition in 9Be+. The repumper beam

is resonant with the |

2S1

2

, F = 2, mF = 1i → |2P1

2

, F = 2, mF = 2i transition. There is

a frequency dierence between these beams of ∼ 1120 MHz, which is generated with

AOMs (see gure (2.13)).

To generate the frequency dierence between the two beams the RD beam is taken

from the 0th order of the 636 MHz RD split1 AOM (see gure (2.13) for AOM labels).

This beam is then diracted into the +1 order of the 636 MHz RD split2 AOM, giving

a frequency shift of +636 MHz from the output of the frequency doubling cavity. The

repumper beam is taken from the -1 order of the RD split1 AOM. It is then directed

through another AOM, which shifts the frequency by +152 MHz. This gives a total

frequency shift from the output of the frequency doubling cavity of -484 MHz. This

beam is recombined with the RD beam by sending it through RD split2 AOM at angle

that does not diract, which allows it to be overlapped with the diracted RD beam.

This generates the required 1120 MHz frequency dierence between the beams.

Both beams are then sent through a 200 MHz double pass AOM (RD SW). This

shifts the overall frequency of both beams by the same amount. The RD SW AOM

switch helps to minimize any leakage light. The beams are then combined with the

blue Doppler beam line (see section (2.7.0.1)) on a 50/50 beam splitter. As described

in the previous section the beams then pass through two more AOMs that provide no

net frequency change, but help reduce any leakage light. The beams are then directed

to the experiment region of the ion trap.

9Be+ Raman Beams

The 9Be+ Raman beam line generates the beams used for two-photon stimulatedRaman co-carrier rotations (see chapter (3)), two-photon stimulated-Raman sideband

transitions (see chapter (3) and (6)), and the geometric phase gate (see chapter (7)).
39

There are two main types of Raman transitions, which are designated co-carrier and

90otransitions. For co-carrier transitions the two beams that make up the stimulated

Raman transition [Wineland 98] are co-linear with k-vectors perpendicular the quantizing magnetic eld axis. These transitions implement motion independent internal state

rotations [Wineland 98]. The experiments discussed in this dissertation used co-carrier

stimulated-Raman transitions that obey the selection rule ∆F = 0 and ∆mF = ±1 for

the 2S1

2

hyperne ground states of 9Be+ (see chapter (3) for a description of the internal

states of 9Be+). The other type of transitions are referred to as 90otransitions, because

the two beams have k-vectors orthogonal to each other. One beam is has a k-vector parallel to the magnetic eld and the other has its k-vector perpendicular to the magnetic

eld. These beams are sensitive to the ions' motion along the trap axis and can be used

to drive stimulated-Raman sideband transitions (see chapter (3) and (6)). They are also

employed to excite a coherent state of the ions motion for use in creating a geometric

phase gate (see chapter (7))

Figure (2.14) is a schematic of the 9Be+ Raman beams lines. After the UV light

leaves the frequency doubler it passes through a spatial lter with a 75 μm pinhole.

This helps lter the spatial mode of the beam and is used as an alignment tool. Next

the beam passes through a double pass AOM labeled ∆F in gure (2.14), which is not

used in the experiments discussed in this dissertation. This is a ∼ 600 MHz AOM that

would allow ∆F = ±1 transitions to be driven in the 2S1

2

hyperne ground states of

9Be+. These transitions are used when working with the magnetic eld-independent

levels of 9Be+ [Langer 05, Langer 06, Home 09, Hanneke 10]. This part of the beam

path is common to all the Raman beams.

First, the co-carrier transitions will be discussed. All of the ∆F = 0 and ∆mF =

±1 transitions in the 2S1

2

hyperne ground states of 9Be+ have frequencies of ∼ 70−100

MHz at a magnetic eld of ∼ 120 Gauss. Thus, the two beams used to make these twophoton stimulated-Raman transitions need to have a frequency dierence equal to these
40

transition frequencies. To generate these frequency dierences the laser beam is sent

through a double pass AOM called the R-CO AOM. This AOM has a bandwidth of

160-240 MHz, and two frequencies are sent into the AOM at one time. This generates

the desired frequency dierence in the beams. For example to create 100 MHz frequency

dierence, a 175 MHz and 225 MHz signal could be sent into the AOM. Assuming the

AOM is aligned so that beams are diracted into the +1 order of the AOM, then there

will be two beams exiting the AOM. One is shifted by +175 MHz and the other by

+225 MHz with respect to the 0th order beam, which is blocked. These beams are retro

reected and diracted a second time and now have the frequency shifts of +350 MHz

and +450 MHz with respect to the 0th order. This gives the desired 100 MHz frequency

dierence. These two co-linear beams are then sent to the ion trap. There is another

AOM used to steer the beams, which is discussed below.

The 90o beams are also generated with AOMs. The dierence frequency required

between the two beams is also ∼ 70−100 MHz, since the same ∆F = 0 and ∆mF = ±1

transitions in the 2S1

2

hyperne ground states of 9Be+ are being driven. To generate the

required frequency dierence and create the two beams necessary, two AOMs are used,

which are labeled R-CO and R-90. The rst beam is generated from the R-CO AOM;

however, only one frequency (+175 MHz) is applied to the AOM. This AOM is used in a

double pass conguration, giving a net frequency shift of +350 MHz to this beam. It is

then sent to the ion trap in a direction perpendicular to the magnetic eld. The second

beam is derived from taking the 0th order of the R-CO AOM and directing it to the

R-90 AOM. This AOM is used in a double pass conguration with a frequency of ∼ 225

MHz. This results in a total frequency shift of the second beam of ∼ 450 MHz, giving a

net frequency dierence between the beams of ∼ 100 MHz. The R-90 AOM also has a

bandwidth of ∼ 160 − 240 MHz, which allows the frequency dierence between the two

beams to be tuned. The second beam is then sent to the ion trap in direction such that

its k-vector is parallel to the quantizing magnetic eld.
41

In order to conserve laser resources, it is advantageous to use the same Raman

laser beams to address ions in dierent locations of the ion trap (The waist of the Raman

beams is ∼ 30μm at the position of the ions, which is smaller than the distance between

the dierent trapping regions). The laser beams can be distributed to the dierent trapping regions using the two AOMs labeled DEFL-CO and DEFL-90 (These AOMs have

a bandwidth of ∼ 160 − 240 MHz. ) in gure (2.14). The AOMs are used in a single

pass conguration, and thus when the frequency of the AOM drive changes the angle of

the diracted beam changes. This allows the beams to be quickly steered to dierent

locations, in a time limited by the time it takes the acoustic wave to travel across the

beam. For a for 5mm diameter beam this takes ∼ 1 μs, which is relatively short compared to the time scale of the experiments (∼ 10 ms) discussed in this work. A problem

can arise when scanning the frequencies of the DEFL-CO and DEFL-90 AOMs. If the

frequencies are not shifted by the same amount to move the beams in the ion trap, then

this changes the dierence frequency between the beams. This can be compensated for

by changing the frequency applied to the R-90 AOM.

24Mg+ Beam line

The 24Mg+beam line has several dierent beams all derived from one source: the

24Mg+

stimulated-Raman beams (see chapter (6)), the 24Mg+ Doppler cooling beams

(see chapter (5) ), and the 24Mg+repump beam (see chapter (6)). The UV output of

the 24Mg+frequency doubler is ∼ −9.2 GHz detuned from the 2S1

2

to 2P1

2

transition in

24Mg+

(see chapter (4)). This is used as the detuning of the 24Mg+stimulated-Raman

beams from resonance. The electro-optic modulator (EOM) before the doubler is used

to shift the frequency of the 24Mg+ Doppler cooling and repump beams and is discussed

below. All of the beams discussed in this section rst pass through a single pass 200 MHz

AOM (The AOM is called Mg SW, and the beam is diracted into the +1 order) at the

output of the frequency doubler. This is used as an optical switch that helps minimize

any leakage light reaching the ion trap when the laser beams are ideally switched o.
42

For the 24Mg+ions, no co-carrier stimulated-Raman transitions are used as in

9Be+. Only 90o

stimulated-Raman transitions are utilized, and these beams are used

for stimulated-Raman sideband cooling (see chapter (6)). The frequency dierence between the two beams needs to be ∼ 334 MHz, because this is the splitting between

the |

2S1

2

, mJ =

−1

2

i and |

2S1

2

, mJ =

1

2

i levels. The two beams used for the transitions

will be referred to as Mg-CO and Mg-90, where at the ion trap Mg-CO has a k-vector

perpendicular to the quantizing magnetic eld and Mg-90 has a k-vector parallel to the

quantizing magnetic eld. The Mg-90 beam is generated from the -1 order of an 80 MHz

AOM labeled Mg Split in the gure (2.15). This beam is then directed toward the ion

trap. The Mg-CO beam is derived from the 0th order of the Mg Split AOM. This beam

is then sent through the -1 order of the double pass Mg DBL AOM, which is driven at a

frequency of ∼ 207 MHz (This AOM has a bandwidth of 160-240 MHz.). This gives the

beam a frequency shift of -414 MHz relative to the frequency of the laser at the input of

the MG Split AOM. This beam is then directed toward the ion trap. The net frequency

dierence between the beams is then the desired 334 MHz.

To implement stimulated-Raman sideband cooling requires the use of the 24Mg+

Raman beams described above is required. However, a 24Mg+repumping beam is also

required (see chapter (6)). The 24Mg+repumping beam used in this experiment is resonant with |

2S1

2

, mJ =

−1

2

i → |2P1

2

, mJ =

1

2

i transition. The 24Mg+laser is nominally

detuned from this transition by ∼ −9.2 GHz. To shift the frequency of the UV output,

a visible 9.2 GHz EOM is placed in front of the frequency doubling cavity. When a 9.2

GHz RF signal is applied to the EOM this creates 9.2 GHz sidebands on the visible

laser light. The frequency doubling cavity is a resonant cavity, and any light not resonant will be reected. The cavity for the 24Mg+

frequency doubling was designed to

have a free-spectral range with a multiple of 9.2 GHz. This allows both the carrier and

the sidebands to be resonant in the cavity. The UV output then consists of the carrier

frequency with 9.2 GHz sidebands. The positive frequency sideband is used to create
43

the light resonant with the |

2S1

2

, mJ =

−1

2

i → |2P1

2

, mJ =

1

2

i transition. The beam used

to do is the same as the Mg-90 described above. The -9.2 GHz o resonant light is

still present but does not scatter a signicant number of photons in the time it takes to

repump the transition (a few μs).

The beams used for 24Mg+ Doppler cooling are derived from the same beam

as those for the 24Mg+stimulated-Raman transitions, but with the EOM turned on.

The frequency of the Mg-Co beam can be changed to suppress any Raman transitions.

The Mg-CO and Mg-90 beams are both split with a beam splitter and sent to the

two dierent trapping locations A and B (see gure (2.5)). These beams are labeled

Mg-COa, Mg-COb, Mg-90a, and Mg-90b. An additional laser beam is derived from the

Mg-90 beam, using a beam splitter. This beam is directed to the loading region and can

be used for Doppler cooling of 24Mg+at zero magnetic eld (The 120 Gauss magnetic

eld is turned o for loading ) see reference [Seidelin 06]. For loading the 24Mg+ions,

the 24Mg+laser could be tuned near resonance with the 2S1

2

to 2P1

2

transition. However,

it was found to be easier to not Doppler cool the 24Mg+ions in the loading region after

being photo-ionized, which meant the magnetic eld could be left on. Because the 9Be+

ions were always loaded rst, the uncooled 24Mg+ions in the loading region could be

transferred to the experiment region and be sympathetically cooled by the 9Be+ions.

2.8 Controlling the Experiment

The computer control used in the experiments described in this dissertation was

discussed in depth in the thesis by Chris Langer [Langer 06]; therefore, only the relevant

features will be highlighted here. All of the experiments described in this thesis have

similar features. A given experiment consists of many laser pulses at dierent frequencies

(sometimes moving the ions in between pulses) followed by the collection of uorescence

from 9Be+ ions. Thus, basic function of the controller is to set the frequencies of the

laser pulses, turn on and o the laser pulses, move the ions, and collect uorescence. For
44

example a single iteration of the entangled mechanical oscillators experiment (discussed

in chapter (8)) lasted ∼ 14 ms and ∼ 600 laser pulses were applied to the ions. This

would be dicult using just a desktop computer.

The experiment is controlled by a eld programmable gate array (FPGA)1. This

is a chip that can be thought of as a recongurable digital logic device. The chip can

be congured to include memory, registers, buer drivers, counters and other functions.

These can be programed using a hardware description language. For the experiments

here the FPGA can be programmed to set the frequency output of direct digital synthesizers (these control the AOMs frequencies), toggle TTL's, and store data in memory.

To run an experiment, the desired pulse sequence is written in a custom text

based language, which can be interpreted by a control computer. The computer then

downloads the pulse sequence to the FPGA. The FPGA runs the pulse sequence and

records the data into its memory. The control computer can then read the data from

the FPGA at a much slower rate than the experiments are run. The control computer

can also perform additional processing of the data that is needed without aecting any

ongoing experiments. The FPGA is also used to control the DC voltages sources, which

enable the ions to be moved during the experiment. A sequence of voltages can be

stored in the on board memory of an analog output card (NI PCI 6733 with 1MS/s).

The FPGA can issue a series of TTL pulses at the appropriate time that tell the analog

output card to step through the voltages stored in its memory.

1 The FPGA card used was the Xlinx Xtreme DSP kit with a Virtex IV chip.
45

RD doubler

BBO

vert. pol. 50/50

BD, BD detune

-636

+636

+152

+636 RD

-4 84 repumper

to trap load

and experiment

zones

RD 

split1

RD 

split2

RD SW

x2

+200

Red Doppler (RD) Beam Lines: RD and Repumper

Figure 2.13: The Red Doppler (RD) beam lines. The AOMs in this beam line generate

the RD and the repumper beam lines. See the text for a detailed description.
46

R-90

x2

+(160 - 240)

+(500 - 740)

λ/4λ/2

Glan-laser

polarizer

λ/4λ/2

Glan-laser

polarizer

∆F DEFL-CO

+(160 - 240)

DEFL-90

+(160 - 240)

trap

axis

B

photo

diode

photo

diode

photo

diode

photo

diode

Telescope

not to scale

Telescope

not to scale

ion positions

in trapping regions

A and B

Raman Beam Lines

Figure 2.14: The 9Be+ Raman beam lines. This beam line generates the R-CO and R-90

beam lines. See the text for a detailed description. This gure is modied from a gure

appearing in [Langer 06].
47

Beam lines: Mg-CO, Mg-90, and Mg Load Mg DBL x2 +200 Mg Split -80 9.2 GHz EOM Mg Doubler BBO Mg SW +200 Mg-COa Mg-COb Mg-90b Mg-90a Mg Load Beam Splitter Beam Splitter Beam Splitter

0 0

+1 0

-1

-1

Figure 2.15: The 24Mg+beam lines. This beam line generates the Mg-CO, Mg-90, and

Mg Load beam lines. The Mg-CO and Mg-90 are split into two beams each and sent

to trapping regions A and B. These beams are labeled Mg-COa, Mg-COb, Mg-90a, and

Mg-90b. See the text for a detailed description.
Chapter 3

9Be+ Qubit

The entangled mechanical oscillators experiment [Jost 09] discussed in chapter (8)

uses two types of ions: a sympathetic cooling ion 24Mg+ (see chapter (4)) and a logical

ion 9Be+ (also referred to as a qubit (two-level quantum system)). The logical ion serves

as an information carrier. In the entangled mechanical oscillator experiment, this enables

the distribution of entanglement (of internal states of the two logical ions) to dierent

parts of an ion trap array. Using sympathetic cooling ions and logical ions, together

for experiments is not unique to the work discussed here, but is considered important

for scaling up quantum information processing (QIP) using trapped ions [Kielpinski 00,

Kielpinski 02b, Blinov 02, Rohde 01, Barrett 03, Häner 08, DiVincenzo 01]

On the surface, the entangled mechanical oscillators experiment might not appear to have much to do with QIP. However, several of the techniques demonstrated

in this experiment are important for QIP, and have enabled several subsequent QIP

experiments [Home 09, Hanneke 10]. Thus, the 9Be+ ion will be discussed in terms of

its advantageous properties for QIP, and these advantages also apply to the entangled

mechanical oscillator experiment.

There are several choices that could serve as logical ions [Ozeri 07], and it is

still unclear what the best ion is for QIP. The 9Be+ ion has several desirable properties

[Ozeri 07] and meets many of the requirements of the DiVincenzo criteria [DiVincenzo 01]

(Although not absolute, these are the widely accepted criteria that must be full lled for
49

scalable QIP). The 9Be+ ion is "1.) A scalable physical system with well characterized

qubits.[DiVincenzo 01]". The internal levels are discussed more in section (3.1). The

internal levels should have "3.) Long relevant decoherence times, much longer than the

gate operation time [DiVincenzo 01].". In 9Be+ the hyperne ground states are used,

which have very long lifetimes [Wineland 98], and internal states can be chosen that

are insensitive to magnetic eld noise [Langer 05, Langer 06] (Magnetic eld noise is a

common source of decoherence in trapped ions). This is discussed briey in section (3.2),

but a detailed description is the subject of the thesis by Langer [Langer 05, Langer 06].

Through the use of optical pumping techniques, which is covered in section (3.3), the

9Be+ ion meets the criteria of "2.) The ability to initialize the state of the qubits to a simple ducial state [DiVincenzo 01]." There exist techniques to accurately determine the

spin state of 9Be+ through state dependent resonance uorescence, which also meets the

DiVincenzo criteria of "5.) A qubit-specic measurement capability [DiVincenzo 01]."

3.1 Energy Level Structure

There are many possible choices of ions to use for QIP. One of the main factors

in choosing a particular atomic species to work with is the simplicity of the internal

level structure. The internal level structure is usually considered simple if the internal

levels have resonances with energies that are compatible with current laser technologies.

Another factor is how hard is it to prepare the states used to form the qubit. If it

takes numerous lasers to optically pump to the desired internal level, then although

technically possible it is not practically feasible for most experiments. There are several

elements scattered throughout the periodic table (see reference [Ozeri 07] for a more

complete list) that make good candidates for QIP. Several of the atomic species used

for ion trap quantum information processing are alkaline-earth elements. These are

common choices since the neutral atoms have two valence electrons. When ionized they

have a similar level structure to the alkali neutral elements, which are commonly used in
50

9

9Be+ Level Diagram

2p 2P1/2

2p 2P3/2

2s 2S1/2

313 nm

F = 2

F = 1

F = 1

F = 2

mF = –2

mF = –1

mF = 0

mF = 1

mF = 2

mF = –2 mF

 = –1 mF

 = 0 mF

 = 1 mF

 = 2

mJ = 3/2

mJ = 1/2

mJ = –1/2

mJ = –3/2

Hyperfine < 1 MHz

197.2 GHz

237 MHz

1.25 GHz

Figure 2.1: Energy level diagram of 9Be+. The fine structure splitting of the excited P

orbital is 197.2 GHz. The hyperfine splitting is 237 MHz for the P1/2 manifold and less

than 1 MHz for the P3/2 manifold. The hyperfine splitting of the ground S1/2states is

1.25 GHz. The S ↔ P transition is electric dipole allowed at 313 nm.

Figure 3.1: Level structure for 9Be+. This gure is reproduced from reference

[Langer 06]. The ground and excited states are coupled via an electric dipole transition with a wavelength of approximately 313 nm. The ground 2S1

2

state has two

hyperne states F = 2 and F = 1 with magnetic sub-levels labeled with mF . Note

that the F labeling of the states is only valid at low magnetic eld. The excited state

2P1

2

also has two hyperne states F = 2 and F = 1 with magnetic sub-levels labeled

with mF . The 2P3

2

excited state also has hyperne levels. but their splitting is < 1MHz

[Poulsen 75]. Since the splitting is small the states are labeled with their mJ quantum

numbers where J = 3/2. For each mJ there will be four mI levels, since I = 3/2

(mI = −I, −I + 1, ..., I − 1, I).
51

neutral atom trapping experiments. In addition, these ions can have ground to excited

state transitions that are excited with photons from the visible spectrum. Although

most of the transition wavelengths are in the UV, they are still accessible with current

laser technology.

Both of the ions species used in the experiments discussed in this dissertation are

alkaline-earth elements. Beryllium is used as a logical ion and Magnesium is used as a

sympathetic cooling ion (see chapter (4) for more details on Magnesium). Beryllium is

the lightest of the alkaline earth elements with a mass of ∼9 atomic mass units. The only

isotope that is stable is 9Be with nuclear spin I=3/2. There is a rare 10Be isotope with

nuclear spin I =0 that has a half-life of approximately 1 million years. The neutral 9Be

is ionized using electron impact ionization to create the ion 9Be+. Figure (3.1) shows the

relevant internal level structure for 9Be+. The electron conguration is 1s

22s1

; therefore

there is one electron in the level labeled by the principle quantum number n = 2. There

are three relevant internal states for the work here; using the Russell-Sanders notation

n

2S+1Lj [Metcalf 99], these levels are the 22S1

2

, 2

2P1

2

, and 2

2P3

2

. The 2 in front of

the states corresponds to the principal quantum number n and will be dropped in the

remaining discussion. The ground state is the 2S1

2

state, and 2P1

2

and 2P3

2

are the rst

excited states. The excited state linewidth is Γ = 2π × 19.4 MHz. The main energy

splitting of the 2S1

2

state from the 2P1

2

and 2P3

2

states is a result of accounting for the

quantum defect in the the energy level calculations [Foot 05]. The 2S1

2

levels are coupled

to the 2P1

2

and 2P3

2

levels via an electric dipole transition of wavelength of ∼ 313 nm.

The electron's orbital angular momentum L couples to its spin, which is known as the

ne structure interaction [Foot 05]. The ne structure shifts the levels 2S1

2

,

2P1

2

, and

2P3

2

, and is the main the source of the splitting of the 2P1

2

, and 2P3

2

levels, which has

been measured to be 197 GHz [Bollinger 85].

One of the reasons for choosing 9Be+ over the isotope 10Be+ is that 9Be+ has a

non-zero nuclear spin of I = 3/2. This has the important implication that 2S1

2

,

2P1

2

,
52

and 2P3

2

levels have hyperne structure (see section (3.2) for more details and reference

[Langer 06] for even more details). Hyperne ground states can make good qubits (see

below). The hyperne interaction comes about from the coupling of the nuclear spin I

to the angular momentum J (J = L + S). The hyperne states are characterized by the

total angular momentum F (F = I + J). At low magnetic elds F and mF are good

quantum numbers and are used to label the states. For convenience these states will

be used to label the energy levels when discussing the internal states at intermediate

magnetic elds, where the only good quantum number is mF . For the 2S1

2

state there

are two hyperne levels F = 2 and F = 1 with a splitting of 1.25 GHz [Wineland 83],

where the F = 2 state is the lower energy state (In Beryllium the g-factor gF has the

opposite sign from that seen in most neutral alkalies, because the g-factor gI is negative.

As a result the F = 2 (gF ' −1

2

) manifold has lower energy than the F = 1 (gF '

1

2

)

manifold.). In the 2P1

2

state there are also two hyperne levels F = 2 and F = 1 with

a splitting of 237 MHz [Bollinger 85]. The 2P3

2

state also has hyperne structure, but

the splitting between the levels is small < 1 MHz [Poulsen 75]. In gure (3.1) the states

in this manifold are labeled with their mJ quantum numbers where J = 3/2. For each

mJ there will be four mI levels since I = 3/2 (mI = −I, −I + 1, ....I − 1, I). Again this

labeling is only correct at low values of the magnetic eld. Figure (3.1) also shows the

magnetic sub-levels for each of the hyperne states. These will be split by the Zeeman

eect, which is discussed in the next section.

It can be seen now how 9Be+ meets the DiVincenzo criteria of "1.) A scalable

physical system with well characterized qubits.[DiVincenzo 01]". Many possible states

can be distinguished and used as a qubit. It will be discussed in the next section how

some of these levels make a better choice than others for forming a qubit.
53

3.2 Hyperne and Zeeman Interactions

As mentioned in the previous section 9Be+ has a nuclear spin of I =

3

2

, and this

couples to the angular momentum J to via the hyperne interaction

H = hA~I · J. ~ (3.1)

The hyperne constant is represented by A for 9Be+ it has a value of ∼ −625 MHz

[Langer 06], h is Planck's constant. At low magnetic elds the good quantum numbers

are the total angular moment F~ = ~I +J~ and its projection mF on the quantization axis.

This is why the states are labeled with |F, mF i. For low magnetic elds the shift in the

energies of the magnetic sub-levels is E = −gF μBmF B [Foot 05]. The Bohr-magneton

is represented by μB, and B represents the value of the magnetic eld. The rst constant

gF is the g-factor. (In Beryllium the g-factor gF has the opposite sign than that seen in

most neutral atom experiments.).

Several experiments [Leibfried 03b, Barrett 04, Chiaverini 04, Chiaverini 05] [Leibfried 04,

Reichle 06a]have been performed using the hyperne ground states of 9Be+ at low magnetic elds (< 20 Gauss). These states have very long lifetimes measured in millions

of years [Wineland 98]. Most level splittings will have a rst order dependence on the

magnetic eld. (An exception is the |

2S1

2

, F = 2, mF = 0i and |

2S1

2

, F = 1, mF = 0i

transitions, which is called a clock transition.) The splitting is rst-order independent

of magnetic eld only at B = 0. A magnetic eld must be introduced to break the

degeneracy of the other levels, which means this level splitting will have some rst order

dependence on the magnetic eld.) This makes a superposition of these levels sensitive

to decoherence from changes in the magnetic eld. Typical decoherence times for these

transitions in the laboratory are . 1ms. This is on the same time scale as the length

of the experiments in references [Leibfried 03b, Barrett 04, Chiaverini 04, Chiaverini 05,

Leibfried 04, Reichle 06b]. Spin echo techniques [Ramsey 63, Vandersypen 04] are used

to extend this time, but they are most likely not sucient to meet the DiVincenzo crite-
54

ria of "3.) Long relevant decoherence times, much longer than the gate operation time

[DiVincenzo 01]."

In the low eld limit where the hyperne interaction is large compared to the Zeeman interaction the splitting of the levels is linear in the magnetic eld except for the

clock transitions. This is also true in the high eld regime where the Zeeman interaction

is much larger than the hyperne interaction. In the intermediate eld regime where

the two interactions are of comparable energies, the splitting between some of the levels

is no longer linear, but has curvature. For particular values of the magnetic eld there

are eld independent transitions [Bollinger 91, Wineland 98, Langer 05, Langer 06],

where the splittings between two levels is no longer depends to rst order on the magnetic eld, but only to second order. The Breit-Rabi formula allows for the calculation

of the eld-independent transitions. References [Langer 05, Langer 06] discuss this in

detail for the system used in this dissertation. There are two eld-independent transitions in 9Be+ that occur near a magnetic eld of ∼ 120 Gauss: |

2S1

2

, F = 2, mF = 0i →

|

2S1

2

, F = 1, mF = 1i and |

2S1

2

, F = 2, mF = 1i → |2S1

2

, F = 1, mF = 0i. In reference

[Langer 05] a coherence time of ∼ 15 seconds was demonstrated on the |

2S1

2

, F = 2, mF = 0i →

|

2S1

2

, F = 1, mF = 1i transition.

The experiments discussed in this dissertation do not make use of the eldindependent transitions. However, the experimental apparatus was set up around with

B = 120 Gauss, so that future experiments [Home 09, Hanneke 10] could use the eldindependent transitions with fewer changes. The two logical levels (qubit levels) in this

dissertation are the |

2S1

2

, F = 2, mF = 2i and the |

2S1

2

, F = 2, mF = 1i. Not using the

eld-independent transitions simplied certain aspects of the experiment.

3.3 State Preparation

As was discussed in the previous section, 9Be+ has several possible states that

could serve as good qubit states. However there needs to be a way to prepare the 9Be+
55

ion in one of these states. This requirement is also one of the DiVincenzo criteria "2.)

The ability to initialize the state of the qubits to a simple ducial state [DiVincenzo 01]."

Through a combination of optical pumping and stimulated Raman transitions (see chapter (4)) it is possible to prepare a 9Be+ ion's internal state in any of the hyperne ground

states. To do this we rst initialize the population to particular level via optical pumping. It can then be transferred to other levels with stimulated Raman transitions.

For the experiments discussed in this dissertation, the states |

2S1

2

, F = 2, mF = 1i

and |

2S1

2

, F = 2, mF = 2i are used as the qubit states. Using optical pumping techniques, it is possible to rst prepare the population in the state |

2S1

2

, F = 2, mF = 2i.

The beams used for optical pumping are shown in gure (3.2). Their orientation with

respect to the trap axis and the magnetic eld are shown in gure (3.3). All the beams

ideally have σ

+ polarization, and this is adjusted with a λ/4 waveplate and magnetic eld

coils that allow for ne tuning of the magnetic eld direction. However, the polarization

is never perfect, which must be taken into account.

After loading an ion in the ion trap, it must be assumed that the internal state

population is distributed over any of the 2S1

2

ground states. It is necessary that the

optical pumping eciently transfer all of the population to the |

2S1

2

, F = 2, mF = 2i

state. A single beam with perfect σ

+ polarization tuned to near resonance with the

2S1

2

→2 P3

2

transition would eventually transfer all the population to this level. However,

the 1.25 GHz splitting of F = 1 and F = 2 ground state manifolds makes it dicult

to accomplish this in a reasonable amount of time (∼ few ms) with one laser beam,

since the excited state linewidth is 19.4 MHz and some of the transitions would be o

resonant. Thus, several beams are used.

Two beams are tuned near resonance with 2P1

2

level. One beam is called the repumper and is tuned to be resonant with the |

2S1

2

, F = 2, mF = 1i → |2P1

2

, F = 2, mF = 2i

transition. These beams eciently pump the population from |

2S1

2

, F = 2, mF = 1i to

|

2S1

2

, F = 2, mF = 2i. Once the population is in the |

2S1

2

, F = 2, mF = 2i state ion
56

F = 2

F = 1

mF = –2

mF = –1

102.8 86.7 76.4 69.1

86.5 76.2

1207.495,843

F = 2

F = 1

mF = –2

mF = –1

39.9 28.5 23.2 20.1

28.2 23.1

261.8

2

P3/2

197 GHz

2

P1/2

313.20 nm

2

S1/2

mJ= 3/2

mJ= 1/2

mJ= –1/2

mJ= –3/2

mF = –3

mF = +3

223.4

BD detuned

BD

RD

repumper

Figure 3.2: 9Be+ levels at a magnetic eld of 119.4 Gauss with tuning of laser beams

indicated. This gure was reproduced from reference [Langer 06]. It depicts the approximate level splittings for the levels used in the experiments discussed in this dissertation.

The exact magnetic eld varied slightly < 1 Gauss from the eld used in this gure.

The level splittings are in MHz. The laser beams used for optical pumping and Doppler

cooling with 9Be+ are shown.
57

B-Field

Trap axis:

RD, repumper

BD, BD detuned

Figure 3.3: Orientation of the RD, repumper, BD, and BD detuned laser beams with

respect to the trap axis and magnetic eld direction. All beams are used for the initial

state preparation of the state |

2S1

2

, F = 2, mF = 2i. The RD and repumper beams are

also used for Raman cooling with 9Be+. The BD detuned beam is used for Doppler

cooling. The BD beam is used for near-resonance Doppler cooling and state dependent

resonance uorescence.

scattering from the RD and repumper beams ceases, because there is no transition coupling the |

2S1

2

, F = 2, mF = 2i ground state to the 2P1

2

state. These beams are also

used as the repumping beams for a Raman cooling cycle on the |

2S1

2

, F = 2, mF = 2i →

|

2S1

2

, F = 2, mF = 1i transition (see chapter (6)). The other beam near resonance with

the 2P1

2

level is called RD (RD stands for Red Doppler, and is somewhat of a historical

name). It is tuned near resonance with the |

2S1

2

, F = 1, mF = 1i → |2P1

2

, F = 2, mF = 2i

transition. Although this beam is farther from resonance with the other levels in the F =

1 ground state manifold, it can help optically pump them to the |

2S1

2

, F = 2, mF = 2i

state. The RD beam is also necessary for Raman cooling. When repumping the

|

2S1

2

, F = 2, mF = 1i state via the |

2P1

2

, F = 2, mF = 2i level the population can spontaneous decay to the |

2S1

2

, F = 1, mF = 1i state, and the RD beam helps pump this

population to |

2S1

2

, F = 2, mF = 2i. The optical power for these beams is ideally kept
58

such that the transitions are not saturated. If the transitions are saturated and the

polarization is not perfect, then this will lead to depumping of the |

2S1

2

, F = 2, mF = 2i

state.

Two other beams aid in state initialization. These are labeled in gure (3.2)

as BD and BD detuned (BD stands for Blue Doppler). Both beams have σ

+ polarization. The BD detuned beam is red detuned from the |

2S1

2

, F = 2, mF = 2i →

|

2P3

2

, mI =

3

2mj =

3

2

i cycling transition [Metcalf 99] by ∼ 400 MHz. The laser intensity

in this beam is several times the saturation intensity, so transitions help in optically

pumping all F = 2 manifold ground states to the state |

2S1

2

, F = 2, mF = 2i. This

beam is also used for Doppler cooling of the 9Be+ ions (see chapter (5) for more on

Doppler cooling). This beam helps pre-cool ions that are very hot. When ions are initially loaded in the trap, they are quite hot, since they are loaded into the trap from a

thermal source of neutral Beryllium atoms. In addition, the ions can undergo collisions

with the background gas, which can impart energy to the ions. The BD beam also

helps with state preparation, but its main purpose is for Doppler cooling and detection

(see section (3.4)). For Doppler cooling it is red detuned from the resonance by ∼

Γ

2

(Γ = 2π ×19.4 MHz) and the intensity is below saturation, which provides the optimum

two-level Doppler cooling (see chapter (5)).

Doppler cooling was discussed along with the state preparation of 9Be+ (see section 3.3), because these to things work together in the experiments. If the internal state

population is not completely optically pumped to the |

2S1

2

, F = 2, mF = 2i state, the cycling transition (|

2S1

2

, F = 2, mF = 2i → |2P3

2

, mI =

3

2

, mJ =

3

2

i) can not be eciently

driven. A typical 9Be+ state preparation/Doppler cooling sequence starts with ∼ 3.5 ms

of RD, repumper, BD and BD detuned. This is followed by ∼ 500 μs of RD, repumper,

and BD. This stage helps achieve the lowest possible Doppler temperature (see chapter

(5)). The last pulse is ∼ 25 μs of RD and repumper.
59

3.4 Detection

The ability to able to distinguish the states of a qubit is important in QIP,

and is one of the DiVincenzo criteria "5.) A qubit-specic measurement capability

[DiVincenzo 01]." For trapped ions, this is typically done by implementing state dependent resonance uorescence. This technique was originally proposed by Dehmelt et al.

[Dehmelt 75] and demonstrated experimentally in references [Wineland 80, Nagourney 86]

[Sauter 86, Bergquist 86]. For QIP applications it is not adequate to simply distinguish the two qubit states. The error rates need to be as low as possible. Fault tolerant thresholds [Knill 05, Steane 03b, Kielpinski 02b] that set minimum tolerable errors

rates for QIP. High detection eciencies have been demonstrated for ions in references

[Hume 07, Myerson 08, Burrell 09].

Resonance uorescence detection of the internal states of 9Be+ used in this dissertation was a major focus of the dissertation by Langer [Langer 06], and only the relevant

portions will be highlighted here. (- A Joke: Why did the cucumber blush? The answer

is somewhere in this dissertation. -) State-dependent resonance uorescence detection

works by having one of the qubit states strongly uoresce (called the bright state, which

is the |

2S1

2

, F = 2, mF = 2i state) and the other uoresce negligibly (called the dark

state, which here is the |

2S1

2

, F = 2, mF = 1i state). The bright state is coupled to the

|

2P3

2

, mI =

3

2mj =

3

2

i state, excited via a closed cycling transition [Metcalf 99] and can

scatter millions of photons per second. This is accomplished with the BD beam shown

in gure (3.2), with the frequency is tuned a few MHz red of resonance with an intensity

below the saturation intensity. The scattering rate is limited by the linewidth of the

excited state, which is Γ = 2π × 19.4 MHz. Assuming the saturation is s = 1/2, the

scattering rate [Metcalf 99] is Rs =

Γ

6

. A 200 μs detection time is used, and in that

time N ∼ 4000 photons are scattered. An F ' 1 imaging system collects some of the

scattered photons and focuses them on to a photo-multiplier tube (The apparatus for
60

collecting the photons is discussed in more detail in chapter (2).). Only a small fraction

of the photons are collected in this time, nd ∼ 10. The overall collection eciency is

then ηd ∼ 2.5 × 10−3. Assuming there is no background and the dark state scatters no

photons, the probability of the ion being in the bright state but not being detected is

[Wineland 98] PN (0) = (1 − ηd)

N ' e

−nd . For nd = 10 then PN (0) ' 5 × 10−5

, which

is a relatively small error.

One source of error in detection is due to the dark state not being completely dark

and scattering photons. The dark state |

2S1

2

, F = 2, mF = 1i is shifted by only ∼ 100

MHz from the |

2S1

2

, F = 2, mF = 2i state. For s = 1/2, approximately 60 photons would

be scattered in the detection time of 200 μs and ∼ 0.2 would be collected by the imaging

system, not an insignicant error. In addition, once the dark state is pumped to the

bright state it will scatter more photons, contributing to a false signal.

This can error can be made smaller by electron shelving [Dehmelt 75]. In our

experiment the dark state |

2S1

2

, F = 2, mF = 1i can be transferred to another auxiliary state |

2S1

2

, F = 2, mF = −2i, which scatters less photons. See reference [Langer 06]

for more details on the error rates. Shelving can be accomplished by using co-carrier

Raman pulses (see section (3.5)) to transfer the population. (Note that a π-pulse denotes complete transfer from one state to another.) This is accomplished by performing a π-pulse on the |

2S1

2

, F = 2, mF = 1i → |2S1

2

, F = 2, mF = 0i transition followed

by two more sequential π-pulses that make the transitions |

2S1

2

, F = 2, mF = 0i →

|

2S1

2

, F = 2, mF = −1i and |

2S1

2

, F = 2, mF = −1i → |2S1

2

, F = 2, mF = −2i.

If a 9Be+ ion is in the bright state, then in a single 200 μs detection period

approximately 10 photons should be detected. Repeating the same experiment many

times gives a histogram of the number of photons collected. The histogram will follow

a Poissonian distribution with a mean of ∼ 10 photons. A single data point in a scan

of some parameter is the average of many single experiments (typically between 100-

500 experiments). When the ion is in the dark state, no photons should be collected;
61

however, a background of ∼ 0.1 counts on the photomultiplier tube is observed. Also

the mean number of photons detected can drift in time. For several of the experiments

discussed (for example see section (6.7)) in this dissertation it sucient to know the

average number of photons received for each data point. For analyzing the data from

the geometric phase gate (7.6) and the entangled mechanical oscillators experiment 8

it is necessary to carefully distinguish the brig detected provides. In the case of the

geometric phase gate (chapter (7)) where two 9Be+ ions are detected, for a given data

point (A data point is the average of many experiments) the state of the qubits might

be in a superposition of having 0, 1, and 2 ions bright, and it is necessary to determine

the relative probabilities. To accomplish this, the data is summed together from a given

experiment creating a histogram of the number of photons collected. This data is then

t to a sum of poissonian distributions, and the resulting means are found. Typically,

the means are approximately 0.1, 10, and 20 photons, which corresponds to detecting 0,

1, or 2 ions in the bright state. For a single data point, the histogram data can be t

to a sum of poissonian distributions with xed means to the nd the relative weights of

each distribution. This gives the relative probabilities of 0, 1, and 2 ions being in the

bright state.

3.5 Raman Transitions in 9Be+

Two photon stimulated Raman transition are discussed in detail in chapter (6).

A simple overview of the levels and the beams used for Raman transitions will be given

in this section. Stimulated Raman transitions are performed with two laser beams

that are blue detuned from the 2S1

2

→2 P1

2

transitions by ∼ 80 GHz. The frequency

dierence between these beams can be tuned from ∼ 0 to ∼ 100 MHz. Stimulated

Raman sideband transitions (see chapter (6)) are driven between |

2S1

2

, F = 2, mF = 2i

and |

2S1

2

, F = 2, mF = 1i states. One beam has π polarization and its k-vector is perpendicular to the magnetic eld. The second beam propagates along the magnetic eld
62

direction with linear polarization. This means both σ

+ and σ− polarization components

are present. Only the σ

+ component contributes to the transition. The σ− component

is o resonance, and is used to balance Stark shifts. The two beams are perpendicular to

each other and have a projection along the trap axis. Thus they can be used for Raman

cooling with 9Be+, for the motion adding, and motion subtracting sidebands, which are

discussed in chapter (8). Although not used on all the transitions, these beams could

drive any ∆F=0 and ∆mF = 1 transitions in the 2S1

2

state.

By making the two Raman beams co-linear, motion insensitive transitions can be

driven between any ∆F = 0 and ∆mF = 1 transitions in the 2S1

2

manifold. These

transitions perform a rotation between two internal states of a 9Be+ ion [Wineland 98].

This is often described as a single qubit gate [Nielsen 00]. The k-vectors of both beams

are oriented perpendicular to the magnetic eld with a polarization such that both beams

have π, σ

+, and σ− polarization components. The frequencies and polarizations of the

two beams are such that ∆F = 0 and ∆mF = 1 transitions in the 2S1

2

state are driven

with both σ

+/π and σ−/π combinations of polarizations.
Chapter 4

Sympathetic Cooling and 24Mg+

This chapter has a discussion of sympathetic cooling and the Magnesium ion. The

rst section is a general discussion of the motivations and issues surrounding sympathetic cooling for quantum information processing. This chapter does not experimental

demonstrations of sympathetic cooling, which is covered elsewhere in this dissertation.

The second section provides an overview of the 24Mg+ ion and its structure, which is

used as the sympathetic cooling ion in the experiments discussed in this dissertation. A

simplied overview is given of how the internal structure comes about.

4.1 Sympathetic Cooling

One of the main goals of quantum information processing is to perform fault

tolerant operations [Nielsen 00] in any system, there will be multiple sources of errors,

and these errors will need to be reduced to below the appropriate fault tolerant threshold.

In ion trap based quantum information processing, one of the fundamental operations

for forming a universal gate set is the two-qubit entangling gate. Most proposals for

implementing such an operation involve using the motional modes of the trapped ions,

where it is assumed that the ions are in the Lamb-Dicke regime [Wineland 98]. The

motion is the natural choice since this provides the dominate coupling between ions

trapped in the same potential well. Most gate implementations involve the axial modes

of motion [Cirac 95, Wineland 98]; however, radial modes can also be used [Zhu 06,
64

Lin 09, Kim 09]. These gates require the ability to excite the motion of the ions in a

state dependent way. Several schemes allow the ions motion to be in a thermal state

[Sørensen 99, Sørensen 00, Kirchmair 09], but all of these schemes require the ions to

be in the Lamb-Dicke regime. Unless the ions can be placed in a completely noise

free environment there will always be some need to recool them. Sympathetic cooling

provides this capability of recooling the ions used for the two qubit gates.

In the context of trapped ions, sympathetic cooling is the process of using one

or more ions to cool other ions. Typically, information is stored in the internal state

of a logic ion and the information needs to be preserved throughout the computation.

Laser cooling techniques applied to the logic ion would destroy this information, since

they rely on manipulating the ions` internal states. When there is more than one ion in

the ion trap, there are multiple normal modes of motion. Since the motion is usually

shared between all the ions it is sucient to only cool one of the ions in the chain.

The primary reason the motional state changes in time in an uncontrolled way

can be attributed to heating of the motional modes from noisy ambient electric elds.

Eorts can be made to minimize the amount of electric eld noise the ions experience.

This can include placing an ion trap in a cryogenic environment [Antohi 09] or ltering

all electrical signals going into the apparatus. Another source of the heating results

from having imperfect waveforms, which are used in transport and separation of the

ions. This can lead to modulations of the trapping potential, which drives the ions'

motion [Blakestad 09].

4.1.1 Options for Sympathetic Cooling

There are many dierent proposals for sympathetic cooling with trapped ions.

Most of the proposals involve having a chain of ions, where some of the ions serve

as logical qubits and others serve as coolant ions. One scheme uses a homogeneous ion

crystal and relies on individual addressing of particular ions to cool the chain [Rohde 01].
65

There are two proposals that involve using a dierent types of ions as the cooling ions.

One makes use of having a dierent isotope of the same type of ion [Blinov 02]. The other

uses a dierent species of ion altogether [Kielpinski 00, Barrett 03]. Another proposal

involves having a long string of ions and cooling some of the ions on the end of the chain

[Herrmann 09].

It maybe possible to use some ions in the chain to remove energy from the other

by separating them o. In the experiment performed in ref [Barrett 04] one ion was

separated o from a chain of three ions leaving the remaining two ions with relatively

low motional excitation. The third ion was presumably hot. A protocol for this might

involve having a reservoir of cold ions where one or a few are merged with the hot chain of

logical ions. The motional energy should thermalize across the normal modes of the new

crystal. Then these ions could be separated o in a way such the ions being separated

from the chain carry away some fraction of the motional energy.

Although schemes exist that allow gates to be preformed at Doppler temperatures.

This alone will not be enough to do away with the need of sympathetic cooling. One

alternative architecture might involve only storing information in the logical qubits for

a time short compared to the heating rate. This would require the ability to bring new

cold ions in at fast rate and transfer the information from the ion that is getting hot to

the one that is cold.

4.1.2 Issues with Sympathetic Cooling

Our experiments involve using a dierent species of ion as the coolant ion. This

scheme has several advantages over others. One of the main ones is negligible spontaneous scattering errors on the logical qubits from the laser light used to cool the coolant

ions. This results from the fact that if the logical ion is a dierent species from the

coolant ion, then its transition frequency will typically be signicantly dierent from

that of the qubit. This will be important, because a large-scale ion based quantum
66

information processor will probably require substantial sympathetic cooling.

The ions experience micromotion from the oscillating radio-frequency (RF) eld

when they are forced o of the RF eld minimum of the trap. This will degrade the

performance of some of the relevant quantum operations [Berkeland 98]. Typically, the

ions are forced o of the RF null by stray electric elds, which can have a number of

sources. If the elds are small and do not change quickly, then the possibility does

exist to compensate for them. However, we required some way to detect the presence

of stray elds. This can be accomplished by detecting depumping1 of the logical ions

[Berkeland 98]. We need observe for this depumping in two non-coplaner x-y directions

to null the micromotion in the X and Y directions (radial directions, which are not

along the trap axis). This can be accomplished with the use of the logical qubits, but

requires two separate laser beams. Often, the detection beam (see chapter (3)) can

be used for one, but another beam is required, which further complicates the setup.

Having a coolant ion with a dierent mass allows for another method of probing the

local electric eld environment. The RF pseudo-potential depends on the mass of the

ion [Wineland 98, Leibfried 03a]. Two ions of dierent mass will be pushed o axis

by dierent amounts from the same electric eld. This has a number of eects. An

ion crystal with more than two ions that was linear will no longer be linear. If the

stimulated Raman beams [Wineland 98] (see chapter 6) were set up to probe along the

axial direction, then it will no longer probe only axial modes. Components from the

radial modes will now also show up when scanning the frequency of a sideband pulse.

This has both advantages and disadvantages. If these modes are not spectrally will

resolved from the axial modes, then the radial modes presence will increase the errors

1 The RF micromotion puts sidebands on a the transitions. A laser beam can be tuned near resonance

with one of these sideband transitions, and when there is micromotion this laser will scatter photons.

If the appropriate transition is chosen, then this scattering can optically pump the ion's state to other

levels. This will cause a decrease in uorescence when using state dependent resonance uorescence. In

addition, if a cycling transition is being used to scatter photons, then the micromotion will broaden this

transition, which leads to a decreased scattering rate, and this can be detected with state-dependent

resonance uorescence.
67

that result from o resonantly driving the radial transitions. However, being sensitive

to these modes can also be viewed as a tool for nulling out the micromotion. The

micromotion can be nulled by scanning a compensating electric elds to minimize the

coupling.

The use of dierent mass ions potentially gives rise to more overlaps of the normal

mode resonant frequencies. Here we assume a typical linear RF Paul trap, where the

radial connement is higher than the axial connement. With a small uniform chain of

a few ions, the frequency of the radial modes do not typically overlap with axial modes.

If the two types of ions have a dierent mass, then depending on the mass dierence,

some of the radial mode can be lower in frequency then some of the axial modes. This

is not a fundamental problem, but does create technical diculties. These diculties

arise from the fact that the closer you have a radial mode near an axial mode in the

frequency domain, the more carefully the stray electric elds need to be nulled. If not

well compensated, then this will lead to increased o resonant coupling when trying to

drive sideband transitions or other operations that involve exciting the motion.

Another consideration for the coolant ion is the simplicity of its internal level

structure. In a large-scale quantum information processing devices cooling will most

likely be needed in several regions. Laser beams will have to be distributed to these

locations. Thus if the coolant ion requires multiple beams to be cooled, then this will

only further complicate the setup.

4.2 24Mg+

4.2.1 Energy Level Structure

The sympathetic cooling ion used in our experiments is the alkali earth element

Magnesium. There are three stable isotopes 26Mg, 25Mg, and 24Mg, which have natural

abundances of 11%, 10%, and 79% respectively [MgI ]. The experiments use 24Mg which
68

use zero nuclear spin, which simplies the internal level structure. We ionize neutral

24Mg to create 24Mg+ . This creates an ion with similar internal properties to alkali

neutral atoms that have one valence electron. In 24Mg+ the valence electron is in the

n = 3 shell. Its ground state is an S orbital labeled 2S1

2

. The rst excited state is a

P orbital. These levels are split by ∼ 280 nm [Herrmann 09]. We excite the electron

through an electric dipole transition using a laser beam with a wavelength approximately

equal to 280 nm.

The excited P orbital has an additional level splitting called ne structure, which

splits the excited state into the two levels 2P1

2

and 2P3

2

. Two eects give rise to the

ne structure for the excited P state: the relativistic shift, and spin orbit coupling

[Brandsden 03, Foot 05]. The ground state will experience an additional shift from the

Darwin term [Brandsden 03]. The relativistic correction comes about by taking into

account the relativistic kinetic energy of the electron. The spin orbit correction arises

from the coupling of the magnetic moment of the electron to the magnetic eld arising

from the electron's motion in the electric eld of the atomic core. Its motion relative to

the nucleus. These eects give rise to a splitting of ∼ 2.75 THz [Yu. 08].

The ground and excited states also have magnetic sub levels, which are degenerate

at zero magnetic eld. Each of the energy levels will have 2J+1 degenerate magnetic sublevels where J is the total angular momentum J = L+S (The orbital angular momentum

is represented by L, and S represents the spin angular momentum.). Breaking the

degeneracy of these levels, is useful for quantum information processing and laser cooling.

Applying a static magnetic eld B~ breaks the degeneracy, which is known as the Zeeman

eect. How the energy shift from the Zeeman eect is calculated depends on the strength

of the interaction compared to the spin-orbit coupling. There are three regimes: the

weak, the intermediate, and the strong. The strength of the external magnetic eld

when compared to the internal magnetic eld determines the regime. In the weak-eld

limit, the Zeeman eect is a perturbation on the spin-orbit correction and treated as a
69

perturbation, using the base kets of spin orbit interaction. For the strong eld limit, the

base kets of the Zeeman Hamiltonian are used and the spin-orbit is the perturbation. In

the intermediate regime both are treated on the same level and the resulting Hamiltonian

has to be diagonalized to nd the energies. The experiments discussed in this dissertation

are performed at a magnetic eld of 120 Gauss, which is in the weak eld regime. The

Hamiltonian that describes this eect is

Hzm = −μbgJ J~ · B. ~ (4.1)

The Bohr magneton is μb. The Landè g-factor gj is given by [Brandsden 03]

gj = 1 +

J(J + 1) + S(S + 1) − L(L + 1)

2J(J + 1) . (4.2)

The gJ factors are: for the 2S1

2

states gJ ' 2, for the 2P1

2

states gJ '

2

3

, and for the 2P3

2

states gJ '

4

3

. The change in energy for a given magnetic sub-level mj is

∆E = −μbgjmjB. (4.3)

As function of the applied magnetic eld the splitting of the energy levels is shown

in gures (4.1) and (4.2), and (4.3). Figure (4.4) shows the internal level structure of

24Mg+ at a magnetic eld of 119.65 Gauss. The splittings of the magnetic sub levels

is determined from calculations. The splitting of the 2S1

2

and 2P3

2

levels is determined

from experimental values. The ne structure splitting of 2.75 THz comes from reference

[Yu. 08].
70

0 50 100 150 200

-200

-100

0

100

200

Magnetic Field HGaussL

Frequency HMHzL

Mg+ Zeeman Effect:

2P32 Levels vs Magnetic Field

ÈmJ=-32\

ÈmJ=-12\

ÈmJ=12\

ÈmJ=32\

Figure 4.1: Splittings of the 2P3

2

levels of 24Mg+as a function of the applied magnetic

eld. The g-factor gJ '

4

3

.

0 50 100 150 200

-50

0

50

Magnetic Field HGaussL

Frequency HMHzL

Mg+ Zeeman Effect:

2P12 Levels vs Magnetic Field

ÈmJ=-12\

ÈmJ=12\

Figure 4.2: Splittings of the 2P1

2

level of 24Mg+as a function of the applied magnetic

eld. The g-factor gJ '

2

3

.
71

0 50 100 150 200

-200

-100

0

100

200

Magnetic Field HGaussL

Frequency HMHzL

Mg+ Zeeman Effect:

2S12 Levels vs Magnetic Field

ÈmJ=-12\

ÈmJ=12\

Figure 4.3: Splittings of the 2S1

2

level of 24Mg+as a function of the applied magnetic

eld. The g-factor gJ '2.
72

334.9 MHz

 111.6 MHz

280.353 nm

B-Field ~ 119.65 Gauss

 223.3 MHz

223.3 MHz

223.3 MHz

2.75 THz* Figure 4.4: Atomic level structure for 24Mg+at eld of ∼ 119.65 Gauss. The splitting

of 280.353 nm of the 2S1

2

and 2P1

2

levels taken from experimentally determined values.

The gJ factors are: for the 2S1

2

states gJ ' 2, for the 2P1

2

states gJ '

2

3

, and for the 2P3

2

states gJ '

4

3

. * This value for the ne structure splitting is taken from the value for

25Mg+ from reference [Yu. 08].
Chapter 5

Doppler Cooling

This chapter provides an overview of Doppler cooling and examines some issues

with its use to cool 24Mg+. The beginning section discusses the basics of Doppler cooling

in a two-level system. However, for our implementation of Doppler cooling with 24Mg+it

is necessary to consider a more complicated three-level system, where dark resonances

can occur. This is examined with a master equation and covered in section (5.4). The

sections following this discuss the issues from having two dierent types of laser beam

orientations. The last section (5.6) discusses an experimental implementation of 24Mg+

Doppler cooling, which is used to sympathetically cool a 9Be+ ion. This technique was

developed to enable the entangled mechanical oscillators experiment [Jost 09].

5.1 Background

Laser cooling has been used for many years to cool the motion of atoms and ions.

Early theoretical proposals for laser cooling were described by Hänsch and Schawlow

[Hänsch 75] as well as Wineland and Dehmelt [Wineland 75] in 1975. The rst experimental demonstration came a few years later by Wineland et. al. [Wineland 78],

concurrently with another set of experiments by Neuhauser et al. [Neuhauser 78]. A

good review of the history of laser cooling can be found in an article by Stenholm

[Stenholm 86].

One of the more common techniques is Doppler cooling. This technique makes
74

use of the Doppler shift of an internal transition in ions (or atoms). If an ion absorbs a

photon it receives a momentum kick ~k from the laser light in the direction the of the

laser propagation. If an atom were stationary then it would start moving in the direction

of the light, which is obviously not an example of cooling. However, if the laser light

is tuned to a lower frequency than the atomic transition, then the stationary ion will

not absorb any light. This assumes the detuning is suciently large. Now, if the atom

is moving toward the laser source, then it will experience the light as Doppler-shifted

to higher frequencies. This brings the light into resonance with the atomic transition,

causing the atom to absorb a photon. In this case, the atom receives a momentum kick

in the direction opposite to its motion, causing it to slow down. In the case where the

atom was moving away from the laser source, it sees the light as being shifted even

further from resonance.

Doppler cooling slows down motion of the ion, which is often referred to as cooling

the ion. However, there is a problem discussing the temperature of an ion [Metcalf 99].

The thermodynamic denition of the temperature of a system requires it to be in thermal

contact and equilibrium with a bath. A trapped ion is not in thermal contact with

any bath. However, it is still convenient to discuss the temperature of a system by

relating its average kinetic energy to a temperature E=

1

2

kBT, where kB is the Boltzmann

constant, and T is the temperature. It should be noted there are innitely many velocity

distributions that can lead to the same average kinetic energy. However, Doppler cooling

leaves the ions' motion in a thermal distribution [Stenholm 86], and the concept of

temperature is valid.

It appears from the above simple description of Doppler cooling that by controlling

the detuning of the laser an ion could be cooled to zero temperature. This would work

by changing the frequency of the detuning, as the ion gets colder until the last recoil

kick removes the nal bit of energy from the motion. At this point, the laser is shut

o. Unfortunately, this is not the case, and there is a limit to how cold the ions become
75

because of recoil upon absorption and re-emission. For a two-level system that has an

excited state line-width of Γ the Doppler cooling limit [Metcalf 99, Wineland 79] is

TD =

~Γ

kB2

. (5.1)

This limit comes from a balance between the cooling and heating.

5.2 Doppler Cooling of Trapped Ions

Most treatments of Doppler cooling treat the particle as a free-particle, which is

not true for a trapped ion. Assuming the 1D case, then the ion sees a potential

V =

1

2

mω2z

2

. (5.2)

The mass of the ion is m, angular frequency is ω, and z is the displacement from

equilibrium. This is a harmonic potential, so the velocity of the ion has the form v (t) =

vo cos [(ωt)]. As long as the trap frequency ω < Γ the system is said to be in the weak

binding limit [Stenholm 86], and the ion can be treated like a free particle. In this case,

the time between absorption and re-emission of a photon is short compared to the period

2π

ω

. This is an oversimplied treatment. It does not take into account micromotion

[Leibfried 03a] in Paul traps, which has a term in the potential that typically oscillates

with a frequency near to or higher than the spontaneous emission rate. Nevertheless the

eects of micromotion are typically small and the simple theory suces.

Following the discussions in references [Leibfried 03a, Wineland 78, Wineland 87,

Neuhauser 78], the cooling rates, heating rates, and nal temperature can be calculated.

The damping force from scattering photons is dependent on the on the velocity of the

ion. The force results from the ion's change in momentum from a scattering event.

The rate of momentum kicks is determined by the linewidth Γ and the excited state

population ρcc (v), giving a force of

Fa = ~kΓρcc (v). (5.3)
76

The magnitude wavevector of the laser is represented by k. The Doppler shift is given

by ~k · ~v (t), and comes into the equations via the excited state population. The Doppler

shift is modeled as changing the eective detuning from resonance, which aects the

excited state population. If the velocities are small, then the force can be linearized by

expanding ρcc (v) in terms of the velocity around v = 0 and keeping terms up to O (v).

This gives a linearized force of

Fa = Fo + βdv, (5.4)

where Fo is a velocity-independent force and βd is the eective damping coecient. It

is worth noting that in this approximation, the damping coecient will not depend on

the velocity and will depend only on the other parameters.

The average cooling rate

·

Ec is given by the force times the velocity averaged over

many cycles of the ions motion.

·

Ec = hFav (t)i = Fo hv (t)i + βd




v

2

(t)



(5.5)

The term proportional to hv (t)i will time average to 0. The quadratic term will

average to 1

2

βdv

2

0

, where vo is the peak velocity. The nal temperature reached will

come from a balance of the cooling with the heating rate. Part of the heating results

from the spontaneous scattering of photons in random directions by the ion. We will

be considering a dipole transition, where the radiation pattern is symmetric resulting

in an average momentum kick hpi = 0 for re-emission. However 
p

2



6= 0 since the

ion is undergoing a random walk in momentum space. Imagine the ion started out at

rest. After undergoing many random momentum kicks, the average velocity would still

be zero, but the velocity would follow some distribution with a nite value of 
v

2



.

Thus, the heating rate from the emission of a photon will be related to the square of the

momentum kick. Another heating mechanism comes from the absorption of photons.

If the ion starts with zero velocity and absorbs photons, then it receives momentum

at random times that the motion. The heating rate is determined by the time rate of
77

change of the kinetic energy, which is proportional to 
p

2



.

·

Eh =

1

2m

d




~p

2



dt =

·

Eabs +

·

Eemis +

·

Eo. (5.6)

The heating rate from absorption

·

Eabs is proportional to the average momentum kick

squared times the rate of absorption at zero velocity, ~~k

2

Γρee (v = 0). The heating

rate from emission

·

Eemis is proportional to this with a constant ξ is proportional to

this. The constant depends on the projection of the dipole radiation pattern along the

axis of motion. It will be seen in the more specic case studied later that

·

Eemis can

be composed of more than one term. This occurs if there are multiple spontaneous

decay paths. Other heating mechanisms are accounted for with the term

·

Eo. One such

example is the heating of the ion's motion from noisy stray electric elds.

·

Eh =

·

Eabs (1 + ξ) +

·

Eo =

1

2m



~~k

2

Γρee (v = 0) (1 + ξ) +

·

Eo. (5.7)

The Doppler cooling temperature limit can be found from setting the sum of (5.5) and

(5.7) to zero. This gives

m




v

2



= kBT = −

(~k)

2

2

Γρee (v = 0)

βd

(1 + ξ) −

m

·

Eo

βd

. (5.8)

Equations (5.5) and (5.8) can be used to optimize Doppler cooling. Before this can be

done, an expression for the excited state population needs to be found, and this will

depend on the system under study.

The typical example used in Doppler cooling is a closed two-level system, which

has been worked out in several papers and books [Metcalf 99, Leibfried 03a]. The result

for the temperature assuming

·

Eo= 0 is given in reference [Leibfried 03a] as

kBT =

~Γ

8

(1 + ξ)



(1 + s)

Γ

2d

+

2d

Γ



. (5.9)

The variable s = 2|Ω|

2

Γ2 and is called the saturation parameter, where Ω is the resonant

Rabi rate. The detuning of the laser from the excited state of the two-level system is
78

given by d. For this case the minimum temperature

Tmin =

~Γ

√

1 + s

4kB

(1 + ξ) (5.10)

is obtained when the detuning d =

Γ

√

1+s

2

. In the limit where ξ = 1 and s = 0, which

corresponds to emission along the ion's axis of motion and low laser beam intensity, this

reproduces the Doppler temperature limit (5.1).

The following sections discuss a more complicated system involving three internal

levels. It will be useful to compare the more complicated setup to two-level Doppler

cooling, using a specic example. The experiments discussed later in section (5.6) involve

a pair of 9Be+ −24Mg+ions conned to a linear RF Paul trap. Along the axis of weakest

connement, there are two normal modes: the in-phase (INPH) and out-of-phase mode

(OOPH) with normal mode frequencies 1.89 MHz and 4.05 MHz respectively (see section

(5.6) for more details).

24Mg+

has a cycling transition on the 2S1

2

to 2P3

2

transitions, which has a linewidth

of ∼ 42 MHz [Herrmann 09]. It should be noted that this transition is not used in any

of the experiments discussed here, but since the linewidth of the 2S1

2

→2 P1

2

transition

(∼ 41 MHz) will be nearly the same, it will be used to compare the experimental results

to the more typical Doppler cooling limits. If Doppler cooling was to be performed on

the cycling transition of 24Mg+, then a laser with σ

+ polarization would be oriented

parallel to the quantizing magnetic eld and at a 45 degree angle to the axis of weakest

connement.

To determine the Doppler cooling temperature for this particular two-level case,

the value of ξ need to be calculated. For the case of spontaneous decay via a ∆m =

−1 transition, which emits σ

− polarized light, the dipole radiation pattern is given in

spherical coordinates by [Itano 82, Jackson 99]

P

σ−

d dΩ = 3

16π



1 + cos2(θ)



dΩ. (5.11)

The probability for emitting a photon in a solid angle dΩ is P

σ−

d

. It is assumed that the
79

dipole axis is aligned with the magnetic eld, which determines the quantization axis.

The probability of the ion receiving a momentum kick along the trap axis will be given

by [Itano 82]



σ−ta =

ˆ

P

σ−

d

ˆk

2

tadΩ. (5.12)

The unit wavevector corresponding to a photon emitted along the trap axis is ˆkta, which

can be written in terms of Cartesian coordinates ˆkta = √

1

2

(ˆx + ˆz). Switching to spherical

coordinates and integrating gives the following result



σ

+ta =

3

16π

1

2

ˆ 2π

0

ˆ π

0



1 + cos2(θ)



sin (θ) (sin (θ) cos (φ) + cos (θ))2dθdφ =

7

20

.

(5.13)

Using equation (5.10) a minimum temperature can be calculated for this case.

Assuming s  0 this gives an average motional occupation of the normal modes of

nINP H = 7 and nOOP H = 3.0. This assumes a thermal distribution of the ion's motion,

see appendix (B). These temperatures are lower than what would be expected for

the typical Doppler cooling limit equation (5.1), which would give nINP H = 10.6 and

nOOP H = 4.7. Figures (5.1) and (5.2) show plots of the average motional state for

both modes versus the detuning of the laser for dierent saturation parameters. The

gures show the quantum number n ̄ dropping below the typical Doppler cooling limit

equation (5.9). This is a result of the parameter ξ < 1. In the rest of this chapter unless

stated otherwise, the Doppler temperature will refer to the two-level Doppler cooling

temperature of equation (5.9).

5.3 Doppler Cooling Setup for 24Mg+

The level structure used in modeling the 24Mg+ Doppler cooling can been seen in

gure (5.3). The states are labeled as follows: |

2S1

2

, mj = −

1

2

i ≡ |ai, |

2S1

2

, mj =

1

2

i ≡

|bi, and |

2P1

2

, mj =

1

2

i ≡ |ci. Each state has a corresponding energy ~ωa, ~ωb, and ~ωc

associated with it. Note that ~ωa could be chosen to be zero but is left as a variable. The
80

-2.0 -1.5 -1.0 -0.5 0.0

0

2

4

6

8

10

12

14

Detuning HGL

-

nINPH

24Mg+

two level Doppler cooling

on the 3S12 -> 3P32 - INPH mode

Doppler

s=0

s=1

Figure 5.1: This gure shows n ̄INP H for two-level Doppler cooling of the INPH mode of

a

9Be+ − 24Mg+

ion pair with 24Mg+on the 2S1

2

to 2P3

2

transition versus the detuning

from resonance. The detuning is in units of the linewidth Γ = 2π × 42 MHz. The nal

temperature is plotted for dierent values of the saturation parameter s. The horizontal

dashed line is the traditional two-level Doppler cooling limit given by equation (5.1).

The vertical line is an asymptote.

fourth level in the system is |

2P1

2

, mj =

−1

2

i. It is shown in the gure but not included in

the model, which simplies the calculations. It should be reasonable to ignore this level,

since there are no near resonant laser beams connecting it to any other levels. In the

experiments there is a σ

− polarized beam connecting |2S1

2

, mj =

1

2

i to |

2P1

2

, mj =

−1

2

i,

but it is o resonance by ∼ 440 MHz, which is why it can be neglected for this simplied

model. Only the two relevant laser beams are included in this simplied model gure

(5.3). The laser connecting states |ai → |ci is σ

+ polarized and has a frequency given

by ω1, which is detuned from the excited state by an amount δ. Laser light with π

polarization and frequency ω2 connects the states |bi → |ci. The light is detuned from

this transition by an amount ∆. Each laser beam has a Rabi frequency given by Ω1 and

Ω2 respectively. Figure (5.3), shows the two possible spontaneous emission decay paths,
81

-2.0 -1.5 -1.0 -0.5 0.0

0

2

4

6

8

10

12

14

Detuning HGL

-

nOOPH

24Mg+

two level Doppler cooling

on the 3S12 -> 3P32 - OOPH mode

Doppler

s=0

s=1

Figure 5.2: This gure shows n ̄OOP H for two-level Doppler cooling of a OOPH mode of a

9Be+−24Mg+

ion pair with 24Mg+on the 2S1

2

to 2P3

2

transition versus the detuning from

resonance for the OOPH mode. The detuning is in units of the linewidth Γ = 2π × 42

MHz. The nal temperature is plotted for dierent values of the saturation parameter

s. The horizontal dashed line is the traditional two-level Doppler cooling limit given by

equation (5.1). The vertical line is an asymptote.

which are labeled Γa for |ci → |ai and Γb for |ci → |bi. Figure (5.4) shows the beam

orientations relative to the magnetic eld and trap axis for the two cases considered

below. The wavevectors for both beams will have a reduced projection along the trap

axis, given by ~k1ta = √

1

2

~k1 and ~k2ta = √

1

2

~k2. Also, there is only a small dierence between

|k1| and |k2| so it will be assumed that |k1| ≈ |k2| ≡ |k| and |k1ta| ≈ |k2ta| ≡ |kta|. The

case where the two beams are orthogonal in gure (5.4) is what used in the experimental

setup. The co-linear case is discussed rst, so it can be compared to the orthogonal case.

5.4 Master Equation

Typically, Doppler cooling is performed on a closed cycling transition, and this

allows the system to be modeled as a two-level system. There are no closed transitions
82

~ 334 MHz

~ 110 MHz

~ 280 nm

B-Field ~ 120 Gauss

Doppler Cooling Beams Spontaneous Emission

Figure 5.3: The 24Mg+internal levels used for Doppler cooling. The three relevant

internal levels are labeled |ai, |bi, and |ci. The gure on the left shows the dierent laser

beams. Two dierent laser beams are used. Beam one is labeled with −→k1 and has σ

+

polarization with a detuning δ from state |ci. Beam two is labeled with

−→k2 and has π

polarization with a detuning ∆ from state |ci. The gure on the right shows the dierent

spontaneous decay paths Γa and Γb from the excited state |ci.

between the 2S1

2

and 2P1

2

states of 24Mg+, when the ion is in a nite magnetic eld,

requiring a more complicated model to be used. A three-level system will be used to

model the lasers used for Doppler cooling in 24Mg+. Two photons will be required

to prevent the population getting trapped and as result, this system exhibits dark

resonances when the condition s  1 is not satised, which can potentially be used to

enhance the cooling [Lindberg 86, Blatt 90]. A master equation approach can be used

to nd the populations and the scattering rates. The disadvantage to using a simple

rate equation approach is that coherent eects such as dark resonances do not show up

in results of the calculations for large values of s.
83

Co-linear Beams Case: Orthogonal Beams Case:

B-Field

B-Field

90 o

45o

Trap axis: Trap axis:

Figure 5.4: Orientation of the 24Mg+laser beams for Doppler cooling with respect to

the trap axis and the magnetic eld. There are two relevant laser beams labeled with

wave vectors

−→k1 and−→k2 with polarizations σ

+ and π. Both laser beams are oriented at

45 degrees with respect to the trap axis. Their projections along the trap axis (ta) are

labeled

−−→k1ta and−−→k2ta respectively. In the co-linear case, their projections along the trap

axis are in the same direction and the beams are perpendicular to the magnetic eld.

In this case there is a σ

− polarization component, but it is o resonance and can be

neglected to a good approximation. In the orthogonal beam case, one beam is parallel

to the magnetic eld and one is orthogonal. The projection of the two wavevectors along

the trap axis are in opposite directions.

The goal of the master equation is to describe the evolution of a system interacting

with the environment [Haroche 06, Cohen-Tannoudji 92, McDonnell 03]. The environment is too complicated to model completely. However, it can often be simplied to a

few relevant interactions. The eect of the environment that will be modeled here is

an atom's interaction with electromagnetic vacuum uctuations. This results in spontaneous emission from the excited state of the atom. It is a assumed the system is

Markovian, meaning the systems future state depends only on its current state.

The Linbald form of the master equation is used [Haroche 06], which describes
84

the time evolution of the density matrix ρ of the system.

∂ρ

∂t = −

ı

~

[H, ρ] +X

i6=0



LiρL†

i −

1

2

L

†

iLiρ −

1

2

ρL†

iLi



(5.14)

The Hamiltonian of the system is given by H, and Li represent the dierent

Linbald operators. This equation results in a set of coupled dierential equations that

need to be solved to nd the time evolution of the density matrix. The calculations are

usually simplied if the resulting set of dierential equations is linear and homogeneous

in time. This is often accomplished by transforming the Hamiltonian to the interaction

picture by going to the appropriate rotating frame. The steady state case where ∂ρ

∂t = 0

is the easier of the cases to be solved. For more complicated dynamics and systems it

can be computationally convenient to convert the density matrix into a column vector

format [McDonnell 03, Tan 99]. For a two-level system this would be

ρ =







ρ11 ρ12

ρ21 ρ22





 =⇒ ρ ̃ =





















ρ11

ρ21

ρ12

ρ22





















, (5.15)

where ∼ over a variable will be used to denote a column vector. In order to convert equation (5.14) into this form, an operator A that acts on ρ needs to be transformed to act on ρ ̃. The transformation will depend on whether or not the operator is

pre or post multiplying the density matrix. For pre multiplication the transformation

[McDonnell 03, Tan 99] is

B = Aρ ⇒ B ̃ = (I ⊗ A)  ̃ρ. (5.16)

The transformation for post multiplication of the operator is

C = ρA ⇒ C ̃ =



A

T ⊗ I



ρ.  ̃ (5.17)

The superscript T represents taking the matrix transpose, and I is the identity matrix
85

with the same dimensions as ρ. In this format equation (5.14) can be written as

∂ρ ̃

∂t = Mρ.  ̃ (5.18)

The solution for this dierential equation is

ρ ̃(t) = e

 ́

Mdtρ ̃(0) = eM tρ ̃(0). (5.19)

The matrix M is time-independent. Routines exist to perform the matrix exponentiation

of M t [Tan 99, Matlab , Mathematica ].

5.4.1 The System Hamiltonian

A three-level system is used to study Doppler cooling of 24Mg+, and the basis

states are |ai, |bi, and |ci (see gure 5.3). the density matrix for the system will take

the form

ρ =















ρaa ρab ρac

ρba ρbb ρbc

ρca ρcb ρcc















. (5.20)

There are two components to the Hamiltonian

H = Ha + Hl. (5.21)

The Hamiltonian Ha represents the internal states of the atom, and Hl represents the

atom-laser interaction. The Hamiltonian for the internal state is given by the energy of

the three possible levels, which is written in terms of the angular frequencies.

Ha = ~















ωa 0 0

0 ωb 0

0 0 ωc















. (5.22)

The interaction of the laser with the ion's is through an electric dipole transition

[Cohen-Tannoudji 92, Wineland 98].

Hl = − ~μe · E~ (t). (5.23)
86

This Hamiltonian describes the coupling of ions electric dipole moment ~μe to the electric

eld E~ (t) of the laser. The dipole moment is written as

~μe = −e hj| ~r |ki. (5.24)

When the atom has a single optically-active electron (as with 24Mg+), the vector ~r is the

electron's position relative to the nucleus and e the charge of the electron. The states

|ji and |ki are the internal states connected by the laser. The electric eld of laser is

E~

i(t) = Eoˆi cos (ωit + φi). (5.25)

The index i sums over the dierent laser beams, which have an angular frequency ωi,

phase φi, and polarization ~i. Substituting equation (5.25) into equation (5.23) gives

Hl = ~Ω1 cos (ω1t + φ1)|ci ha| + ~Ω2 cos (ω2t + φ2)|ci hb| + h.c.. (5.26)

The Rabi frequency Ωi describes the rate at which transitions are made and is given by

Ωi = −

e

~

Eo hj| ~r · ~ |ki. (5.27)

The atom-laser interaction Hamiltonian for the system in gure (5.3) can be written in

matrix form

Hl = ~













0 0 Ω1

2



e

i(ω1t−φ1) + e−i(ω1t−φ1)



0 0 Ω2

2



e

i(ω2t−φ2) + e−i(ω2t−φ2)



Ω1

2



e

i(ω1t−φ1) + e−i(ω1t−φ1)

 Ω2

2



e

i(ω2t−φ2) + e−i(ω2t−φ2)



0













,

(5.28)

where the cosine terms have been written in terms of exponentials. It is convenient to

rewrite the laser frequencies in terms of the atomic energy levels and the detunings:

ω1 = ωc − ωa + δ (5.29)

ω2 = ωc − ωb + ∆. (5.30)

The Hamiltonian is time dependent, which makes the system more dicult to solve. If

the interaction picture is used the time dependence of the equations can be removed.
87

This is accomplished by transforming into a series of rotating frames and dropping

rapidly-oscillating terms. To do this rewrite equation (5.21) as two parts

H = Ho + V, (5.31)

where Ho is often chosen as the unperturbed Hamiltonian of the system. In this case it

is convenient to use a slightly dierent form. It is chosen such that it removes the time

dependence from the interaction Hamiltonian. To accomplish this, Ho is for this system

Ho = ~















ωa − δ 0 0

0 ωb − ∆ 0

0 0 ωc















. (5.32)

This leaves all of the time dependence in V , which is given by.

V = ~













δ 0

Ω1

2



e

i(ω1t−φ1) + e−i(ω1t−φ1)



0 ∆ Ω2

2



e

i(ω2t−φ2) + e−i(ω2t−φ2)



Ω1

2



e

i(ω1t−φ1) + e−i(ω1t−φ1)

 Ω2

2



e

i(ω2t−φ2) + e−i(ω2t−φ2)



0













.

(5.33)

The transformation [Sakurai 94] to the interaction picture is

H0 = e

i

~ Ho V e− i~ Ho

. (5.34)

Substituting equations (5.29) and (5.30) for the laser frequencies, gives

H

0 = ~













δ 0

Ω1

2

e−ıφ1



1 + e−2i(ωc−ωa+δ)t+2iφ1



0 ∆ Ω2

2

e−ıφ2



1 + e−2i(ωc−ωb+∆)t+2iφ2



Ω1

2

e−iφ1



1 + e

2i(ωc−ωa+δ)t+2iφ1

 Ω2

2

e−ıφ2



1 + e

2i(ωc−ωb+∆)t+2iφ2



0













(5.35)

for the interaction picture Hamiltonian. The rotating wave approximation [Cohen-Tannoudji 92]

can be made that allows exponential terms having a frequency near twice the transition

frequencies to be dropped. This is reasonable since these terms rapidly average to zero.

In addition if we chose the time origin appropriately, we can set φ1 = 0 and φ2 = 0,

then the interaction Hamiltonian takes on the much simpler form

H0 = ~















δ 0

Ω1

2

0 ∆ Ω2

2

Ω1

2

Ω2

2

0















. (5.36)
88

For the model of 24Mg+the only Linbald operators taken into account are those for

spontaneous emission, see gure (5.3). They can be written down by inspection. They

couple the excited state to one of the ground states at rates given by the spontaneous

decay rate,

L1 =

p

Γ (1 − βr)|ai hc| (5.37)

L2 =

p

Γβr |bi hc| . (5.38)

The branching ratio for the |ci → |bi decay path is given by βr, and for 24Mg+βr =

1

3

.

This value comes from the Clebsch-Gordon coecients [Griths 95] for the coupling of

the ground and excited states in the 2S1

2

⇒2 P1

2

system.

5.4.2 Calculating Populations

5.4.2.1 Steady State Solutions

The set of dierential equations is found by plugging equations (5.20), (5.36),

(5.37), and (5.38) into equation (5.14). This gives the following set of equations

∂ρaa

∂t = (Γ − βrΓ)ρcc −

1

2

i(ρac − ρca) Ω1 (5.39)

∂ρbb

∂t = βrΓρcc −

1

2

i(ρbc − ρcb) Ω2 (5.40)

∂ρcc

∂t =

1

2

i((ρac − ρca) Ω1 + (ρbc − ρcb) Ω2) − Γρcc (5.41)

∂ρab

∂t =

∂ρ?

ba

∂t =

1

2

i(2(δ − ∆)ρab + ρcbΩ1 − ρacΩ2) (5.42)

∂ρac

∂t =

∂ρ?

ca

∂t = −

1

2

((Γ − 2iδ)ρac + i(ρaaΩ1 − ρccΩ1 + ρabΩ2)) (5.43)

∂ρbc

∂t =

∂ρ?

cb

∂t =

1

2

(−(Γ − 2i∆)ρbc − i(ρbaΩ1 + (ρbb − ρcc) Ω2)). (5.44)

The steady state solution is often the only relevant one, where ∂ρ

∂t = 0. This can be

found by solving the set of linear dierential equations. The constraint the T r[ρ] = 1

is imposed on the system, which guarantees the populations sum to 1. The solutions
89

Table: Populations

1 ρaa =

16δ

4−32∆δ3+4(4∆2−Ω2

)δ

2+3Ω4+4Γ2

(δ−∆)2+4∆2Ω2

2(8δ

4−16∆δ3+2(6∆2+Ω2)δ2−2(4∆3+3Ω2∆)δ+4∆4+3Ω4+3Γ2(δ−∆)2+4∆2Ω2)

2 ρbb =

8∆4−16δ∆3−2Ω2∆2+3Ω4+2Γ2(δ−∆)2+2δ

2

(4∆2+Ω2)

2(8δ

4−16∆δ3+2(6∆2+Ω2)δ2−2(4∆3+3Ω2∆)δ+4∆4+3Ω4+3Γ2(δ−∆)2+4∆2Ω2)

3 ρcc =

3(δ−∆)2Ω2

8δ

4−16∆δ3+2(6∆2+Ω2)δ2−2(4∆3+3Ω2∆)δ+4∆4+3Ω4+3Γ2(δ−∆)2+4∆2Ω2

4 ρab = ρ

?

ba =

Ω2

(4δ

2−6∆δ+2∆2−3Ω2−3iΓ(δ−∆))

2(8δ

4−16∆δ3+2(6∆2+Ω2)δ2−2(4∆3+3Ω2∆)δ+4∆4+3Ω4+3Γ2(δ−∆)2+4∆2Ω2)

5 ρac = ρ

?

bc =

(δ−∆)Ω(8δ

2−8∆δ−3Ω2−4iΓ(δ−∆))

2(8δ

4−16∆δ3+2(6∆2+Ω2)δ2−2(4∆3+3Ω2∆)δ+4∆4+3Ω4+3Γ2(δ−∆)2+4∆2Ω2)

6 ρbc = ρ

?

cb =

(δ−∆)Ω(−4∆2+4δ∆+3Ω2−2iΓ(δ−∆))

2(8δ

4−16∆δ3+2(6∆2+Ω2)δ2−2(4∆3+3Ω2∆)δ+4∆4+3Ω4+3Γ2(δ−∆)2+4∆2Ω2)

Table 5.1: Populations for the three level system.

are quite complicated but can be simplied if a few assumptions are made. Taking

Ω1 = Ω2 = Ω and βr =

1

3

results in solutions shown in table (5.1).

These equations can be used to help nd the optimum parameters for Doppler

cooling. There are a few experimental constraints. Laser beam 1 will ideally be tuned

to resonance on the |ai → |ci transition. This is because this beam is also used for

resonant repumping during Raman cooling. The parameters that can be adjusted are

the detuning and the Rabi frequencies of the two beams.

It will be useful to examine the results of the above equations for some real experimental parameters. The radiative linewidth of the 2S1

2

to 2P1

2

transition [Herrmann 09,

Jost 09] of 24Mg+is 41 MHz, which gives Γ = 2π × 41 MHz. Laser beam 1 couples

|

2S1

2

, mj = −

1

2

i → |2P1

2

, m0

j =

1

2

i with σ

+ polarized light. A typical waist of the beam

is about 15 μm and there is about 0.5 μW of power in the laser beam1. The Rabi

frequency for this transition can be found using equation (A.1) to be Ω = 2π × 18 MHz.

1 Technical aside: Typically we have about ≈ 100μW in this beam but most of it is o resonance

by 9.2 GHz and the polarization is ~ =

1

2





σ+

+ 

σ−



, which gives ≈ 50μW at the right polarization.

An electro-optic modulator is used to shift the beam near resonance. The usual power setting for the

electro-optic modulator puts the sidebands at about 1% of the power of the carrier, or 0.5μW.
90

-100 -50 0 50 100

0.00

0.02

0.04

0.06

0.08

0.10

D HMHzL

Population

Excited state È c \ population vs. detuning D

Figure 5.5: The steady state population of state |ci for the three-level model for 24Mg+

Doppler cooling for various detunings ∆ of laser beam 2. The detuning δ = 0 and

Ω = 2π × 18 MHz.

Laser beam 2 couples |

2S1

2

, mj = +1

2

i → |2P1

2

, m0

j = +1

2

i with π polarized light. This

beam has a similar amount of laser power2 and waist. Figures (5.5), (5.6), and (5.7)

show the excited state population as a function of the detuning ∆ of laser beam 2.

There is a major dierence between the three-level system and a two-level system,

which is exhibited in gure (5.5). The dip in the excited state population at ∆=0 is

the result of a coherent process of the two laser beams interfering, when they have the

same detuning from the excited state. This is know as coherent population trapping

[McDonnell 03, Dalton 82] or a dark resonance [Blatt 90, Lindberg 86]. It leads to a

suppression of the excited state population and thus the scattering rate

dN

dt = Γρcc. (5.45)

This condition needs to be avoided for Doppler cooling. If the scattering rate goes zero

2 This beam typically has 300μW of power and a polarization of ~ =

1

2





σ+

+ 

σ−



+

1

2



π

. As

described in footnote1 the electro-optic modulator puts about 1% of the power in the near resonant

sideband. This gives a power of ≈ 1.5μW in beam 2. For the sake of the model the two laser beam

powers will be considered to be the same
91

-100 -50 0 50 100

0.0

0.2

0.4

0.6

0.8

D HMHzL

Population

State È b \ population vs. detuning D

Figure 5.6: The steady state population of state |bi for the three-level model for 24Mg+

Doppler cooling for various detunings ∆ of laser beam 2. The detuning δ = 0 and

Ω = 2π × 18 MHz.

there will be no Doppler cooling, since the ion will scatter no photons. The width of the

dip in gure (5.5) depends only on the Rabi frequencies and the branching ratio of the

excited state. This can be seen by nding the maximum of the excited state populations,

where dρcc

dt = 0 for arbitrary Ω1, Ω2, and β. It is still assumed that δ= 0. The maximum

of the excited state population occurs at detunings of

∆ = ±

4

q

Ω2

1 + Ω22

2

βrΩ2

1 − (βR − 1)Ω22



2

√4 βr

√

Ω1

. (5.46)

The solution simplies for Ω1 = Ω2 = Ω and βr =

1

3

to

∆ = 3

1

4

√

2

Ω. (5.47)

If this detuning is used, then the excited state population becomes (still assuming δ = 0)

ρcc =

1

2

√

3 − 1



βr +

Γ2

Ω2 + √

2

3

+ 2

. (5.48)

A plot of the excited state population versus the Rabi frequency can be seen in gure

(5.8).
92

-100 -50 0 50 100

0.0

0.1

0.2

0.3

0.4

0.5

D HMHzL

Population

State È a \ population vs. detuning D

Figure 5.7: The steady state population of state |ai for the three-level model for 24Mg+

Doppler cooling for various detunings of ∆ of laser beam 2. The detuning δ = 0 and

Ω = 2π × 18 MHz.

0 50 100 150 200

0.00

0.05

0.10

0.15

0.20

0.25

W HMHzL

Population

Excited state È c \ population vs. Rabi Frequency

Figure 5.8: Excited state |ci population versus the Rabi frequency Ω of both laser beams.

The detunings are δ= 0 and ∆ = 3

1

4 √

2

Ω. The value for βr and Γ is the same as in gure

(5.5)
93

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

Time HΜsL

Population

Populations: W2=3*W1, ∆=D=0

Ρaa

Ρbb

Ρcc

Figure 5.9: Populations as function of time for detunings ∆ = δ = 0. The Rabi frequencies used are Ω1 = Ω = 2π × 18 MHz and Ω2 = 3 × Ω1. The linewidth and branching

ratio are Γ = 2π × 41 MHz and βr = 1/3.

5.4.2.2 Time Dependent Solutions

The time evolution of the populations can be useful to understand the dynamics

at short time scales. Inserting equations (5.36), (5.37), and (5.38) into (5.14) gives the

same set of equations described above. Now if we convert it to a more convenient form

using equations (5.15), (5.16), (5.17), and (5.18). Then the time dependent solution

is given by (5.19). To nd this solution the matrix exponential must be evaluated at

each point in time. This can be calculated using several dierent mathematical software

applications, but in this case Mathematica was used [Mathematica ].

Two cases are shown in gures (5.9) and (5.10), where both lasers are tuned to

resonance and there exists coherent population trapping. Initially the atom is in the state

√

1

2

|ai + √

1

2

|bi. In the steady-state there will be zero population in the excited state

and some other distribution of populations in the ground state |ai and |bi, depending on

the other parameters. The interesting thing to note is it takes a nite amount of time

before this dark state is formed.
94

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.1

0.2

0.3

0.4

0.5

Time HΜsL

Population

Populations: W2=W1, ∆=D=0

Ρaa

Ρbb

Ρcc

Figure 5.10: Populations as function of time for detunings ∆ = δ = 0. The Rabi

frequencies used are Ω1 = Ω2 = 2π × 18 MHz. The linewidth and branching ratio are

Γ = 2π × 41 MHz and βr = 1/3.

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time HΜsL

Population

Populations: W2=3*W1, ∆=0, D=-

3

14

2

W2

Ρaa

Ρbb

Ρcc

Figure 5.11: Population as function of time for detunings ∆ = 3

1

4 √

2

Ω2 and δ = 0. The

Rabi frequencies used are Ω1 = Ω = 2π × 18 MHz and Ω2 = 3 × Ω1. The linewidth and

branching ratio are Γ = 2π × 41 MHz and βr = 1/3.

Other cases are considered is where the detuning is chosen to give the maximum

population in the excited state. These are shown in gures (5.12) and (5.11). The
95

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.1

0.2

0.3

0.4

0.5

Time HΜsL

Population

Populations: W2=W1, ∆=0, D=-

3

14

2

W2

Ρaa

Ρbb

Ρcc

Figure 5.12: Population as function of time for detunings ∆ = 3

1

4 √

2

Ω2 and δ = 0. The

Rabi frequencies used are Ω1 = Ω2 = Ω = 2π × 18 MHz. The linewidth and branching

ratio are Γ = 2π × 41 MHz and βr = 1/3.

populations reach steady state values in ≈ 50 ns. An ion's motional oscillation frequency

of ∼ 3 MHz corresponds to a period of ∼ 300 ns. This is longer than the settling time,

so it should be reasonable to assume the system is close to the steady state. However

for a precise description a more detailed calculation is needed to take this into account.

5.5 Beam Orientations and the Doppler Shift

The above discussion gives the populations for an ion at rest, which is not the

case in the experiments. To try and better understand how the dark resonances where

aect by the ions motion, the Doppler shift can be incorporated into the calculation of

the excited state populations. It is assumed that the ions are in the steady state as

discussed in section (5.4.2.2). It was found that the orientation of the two beams used

in Doppler cooling aect the dark resonance.

Two cases will be analyzed. In the rst case both beams are co-linear and propagating along a direction orthogonal to the quantization axis, and at 45 degrees to the
96

trap axis (see gure (5.4)). The second case is what is used in the actual experiment and

also depicted in gure (5.4). Here, the two beams used are orthogonal to each other. The

one with σ

+ polarization is parallel to the magnetic eld, and the beam that creates the

π polarized light is perpendicular to the magnetic eld. Both beams are at 45 degrees

with respect to the trap axis. The main dierence between the two cases will be how

the Doppler shift from the ions' motion aects the eective detuning of the laser beams.

In the rst case, to incorporate the Doppler shift, the substitutions ∆ → ∆ − ~k · ~v and

δ → δ − ~k · ~v need to be made (Positive velocity is taken to be when the ion is moving

to the left in gure 5.4). Here the Doppler shift acts as a common detuning change for

both the beams. For the second case the substitution ∆ → ∆ − ~k · ~v and δ → δ + ~k · ~v

will be made. In this instance, the Doppler shift acts as a dierential shift between the

two beams.

5.5.1 Co-linear Cooling Beams

When the two beams are co-linear, they must be oriented orthogonal to the quantization axes in order to obtain π- polarization. In this orientation it is not possible to

only have σ

+ polarized light without having a σ− polarization as well. However in this

model the σ

− component will be ignored as explained above. Substituting the Doppler

shift into equation (3) of table (5.1) and considering the 1D case along the trap axis,

the excited state population can be plotted as a function of the velocity as seen in gure

(5.13).

The shift of the maximum population to the negative velocities is a result of the

detuning chosen. As can been seen in gure (5.13) for certain values of the detuning

there are larger values of the excited state population. In the case plotted in gure

(5.13) the Doppler shift is changing the eective detuning, which moves the center of

the peak depending on the values chosen for δ and ∆. The Rabi frequencies also aect

the amplitude of the excited state population. The excited state population for dierent
97

-40 -20 0 20 40

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Velocity HmsL

Ρcc Population

Excited state population for for various values of D

D= -2G

D= -G

D= -.5G

Figure 5.13: Excited state population in the co-linear Doppler cooling conguration.

The population is plotted for dierent values of the detuning ∆. The Rabi frequencies

are chosen to be Ω1 = Ω2 = 0.5 Γ. The detuning δ = 0.

Rabi frequencies Ω2 is plotted as a function of the velocity in gure (5.14). .

5.5.2 Orthogonal Cooling Beams

In this scenario one of the beams is orthogonal to the quantization axis (see

gure (5.4)), which allows for π polarization. The second beam propagates along the

quantization axis and has pure σ

+ polarization. In the experiment, this beam also has

an equal amount of σ

− polarization to balance out stark shifts, but is not considered in

this model. Both beams are oriented at 45 degrees to the trap axis, so the k-vectors from

both beams will have a √

1

2

projection along that axis. The main dierence between this

case and the co-linear one considered early is the Doppler shift has the opposite sign for

the two beams. This makes the Doppler shift appear as a dierential shift between the

beams and not a common one as in the co-linear case.

A interesting feature of this case is that beams are not orthogonal with respect to
98

-40 -20 0 20 40

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Velocity HmsL

Ρcc Population

Excited state population for for various values of W2

W2= 2G

W2= G

W2= .5G

Figure 5.14: The excited state population versus the velocity along the trap axis for

dierent values of Ω2. The detunings are held xed at δ = 0 and ∆ = −0.5 Γ. The Rabi

frequency Ω1 = 0.5 Γ.

each other in the xˆ and yˆ directions of the trap axes (zˆ is trap axis), which are referred

to as the radial axes (This is dierent from the xˆ

0 and yˆ0 directions determined by the

quantization axis depicted in gure (5.4), where zˆ

0

is along the magnetic eld). This

implies that the Doppler cooling will be dierent between the trap axis and the radial

axes.

As above, to incorporate the Doppler shift into the results for the excited state

population (equation 3 table (5.1)) we make the substitutions ∆ → ∆ − ~k · ~v and

δ → δ + ~k · ~v. The excited state population is plotted in gure (5.15) as a function of

velocity along the trap axis for dierent detunings of ∆. The results here are dierent

than that obtained from those in the co-linear case shown in gure (5.13). For the

co-linear case the expression for the excited state population (see equation 3 of table

(5.1)) has the factor (δ − ∆)2in the numerator. Thus when δ = ∆ the excited state

population (see equation 3 of table (5.1)) goes to zero, which is what gives rise to the dark
99

-40 -20 0 20 40

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Velocity HmsL

Ρcc Population

Excited state population for for various values of D

D= -2G

D= -G

D= -.1G

Figure 5.15: Excited state population versus velocity for the case of orthogonal Doppler

cooling beams with dierent values of ∆. The detuning δ is held at zero, and the Rabi

frequency Ω1 = 0.5 Γ and Ω2 = 0.5 Γ.

resonance. The case of orthogonal beams has the factor 2

~k · ~v + δ − ∆

2

. Thus, the

dark resonance now depends on velocity and no longer occurs when the two detunings

are equal. The width of this dark resonance depends on the Rabi frequency. The excited

state population is plotted as a function of velocity for dierent Rabi frequencies Ω2 in

gure (5.17).
100

-40 -20 0 20 40

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Velocity HmsL

Ρcc Population

Excited state population for various values of D:

Co-linear and Orthogonal cases

Co-lin.:D= -2G

Ortho.:D= -2G

Co-lin.:D= -.5G

Orhto.:D= -.5G

Figure 5.16: Excited state population versus velocity for the case of orthogonal and

co-linear Doppler cooling beams with dierent values of ∆. The detuning δ is held at

zero, and the Rabi frequency Ω1 = 0.5 Γ and Ω2 = 0.5 Γ.

-40 -20 0 20 40

0.00

0.05

0.10

0.15

0.20

Velocity HmsL

Ρcc Population

Excited state population for for various values of W2

W2= 2G

W2= G

W2= .5G

Figure 5.17: Excited state population versus velocity for orthogonal cooling beams using

dierent values of Ω2. The detuning δ is held at zero, and ∆ = 0.5 Γ. The Rabi frequency

Ω1 = 0.5 Γ.
101

There is clear dierence between the co-linear and orthogonal beam cases. The

orthogonal beam case exhibits a velocity dependent dark resonance. This resonance

could potentially be exploited to obtain colder temperatures than is possible in the

two-level case. In the simplied two-level model discussed in section (5.2) the damping

coecient is determined by βd =

dρcc(v)

dv |v=0 . Thus the steeper slope suggests that

the damping can be larger. A simple model was initially considered to evaluate the

nal the temperatures that could be reached in this case, and it did appear that lower

temperature could be reached. However, it was determined the model was too simple

because it neglected stimulated-emission and dressed-state eects which are absent in the

simple treatment of Doppler cooling. Therefore, a more complete theoretical treatment

will be needed to describe this case. Even though we did not have an accurate model of

the cooling expected for the three level system, experiments were performed to explore

Doppler cooling in this system. The results of which are discussed in the next section.

5.6 Demonstration of Doppler Cooling with 24Mg+

We performed experiments to observe Doppler cooling of the axial modes of a

9Be+ − 24Mg+

ion pair as discussed in section (5.5.2). With the goal of observing

better cooling than the two-level Doppler limit. Our apparatus is not setup to detect

the motional state of a single 24Mg+ion, and thus 24Mg+is used to sympathetically

cool a 9Be+ion, which can be used to detect the motional state of the two ions. The

experiment is performed by trapping a 9Be+ and a 24Mg+ion together in a linear radio

frequency Paul trap [Jost 09]. There are six possible normal modes of motion; however,

the experiment only probes the two normal modes along the axis of weakest connement.

These are called the in-phase mode (INPH) (finph∼ 1.89 MHz ) and the out-of-phase

mode (OOPH) (fooph∼ 4.05 MHz). At the start of each experiment, the ions are placed

in the same order. This is accomplished by changing the trap potentials in a way

similar to that outlined in reference [Jost 09]. Both 9Be+ and 24Mg+ Doppler cooling
102

are performed during this reordering. This is followed by 24Mg+ Doppler cooling. The

cooling time used is > 1 ms, which is sucient time for the ions to thermalize. The

detunings of the beams were approximately δ/2π = −12 MHz and ∆/2π = −5 MHz.

The absolute electric eld from the lasers at the 24Mg+ion is not known precisely, since

the beam waists were not measured (They should have a waist between 10 μm and

100 μm. There was control of the overall optical power of the beams and the ratio of

the power in the two beams. These powers were varied until an optimum was found.

For reference, the experimental results will be compared to the theoretical outcome of

Doppler cooling on a two-level system.

To measure n ̄, a time scan of a two-photon stimulated Raman sideband transition

is performed on the 9Be+ ion, which is discussed in section (6.5). The sideband transition

is performed on the |F = 2, mF = 1i → |F = 1, mF = 0i transition (This is the labeling

of the states at low magnetic eld. The experiment, like the others discussed in this

thesis, are performed at a eld of ∼ 120 Gauss.). Following cooling those two levels

are then shelved and readout as discussed in reference [Jost 09]. An estimation of the

average motional occupational level n ̄, assuming a thermal distribution, can be obtained

by tting the data to the function that describes the probability Pb of nding the 9Be+

ion in the |

2S1

2

, F = 2, mF = 2i state, which is the bright state for resonance uorescence

detection,

Pb (t) = 1

2

"

1 +X∞

n=0

X∞

k=0

PnPke

−γt× (5.49)

cos (2Ωn+1,n (Ω, n, ηn) Ωk,k (k, ηk)t)] (5.50)

Pn =

n ̄

n

(1 +  ̄n)

n+1 (5.51)

Pk =

 ̄k

k



1 +  ̄k

k+1 (5.52)
103

Ωn0

,n = Ωe

−η

2

2



n<!

n>!

1/2

η

|n

0−n|L

|n

0−n|

n<



η

2



(5.53)

(see section (6.5) for more details). It is assumed in the above model that the energies

of the two motional modes are given by a thermal distribution labeled by Pn and Pk.

The symbols with a horizontal - over them are the average motional state for a mode

of motion, assuming a thermal distribution. The mode on which a transition is being

driven is represented by n and the other mode which is a spectator mode [Wineland 98]

is represented by k. For the spectator mode, the value of Ωk,k (k, ηk) is the Debye-Waller

factor for that mode [Wineland 98, King 98, King 99]. Decay during the sideband transition is introduced into the model with the phenomenological constant γ. The model

is t by oating the parameters n ̄,

 ̄k, Ω, and γ, performing a weighted nonlinear regression t in Matlab [Matlab ]. The tting routine uses a trust-region-reective algorithm

[Matlab ].

The data is obtained by rst taking resonance uorescence histograms [Langer 06]

for the 9Be+ion. For each experiment the histograms are averaged together and t to a

weighted sum of two Poissonian distributions. The two mean values correspond to the

number of scattered photons when the ion is in the bright state and dark state. For each

data point, the weights of each distribution are found by tting, where the means are

xed from the t values obtained from the average of all the data. The weight of the

bright state is then plotted for each data point. The errors on the weights for each data

point are used in the weighted t of the sideband scan, and are one-standard-deviation

error bars on the plotted data.

Figure (5.18) shows a scan of the motion adding sideband transition time on the

INPH mode of motion after Doppler cooling with 24Mg+, where each data point consists

of 600 experiments. By tting the data, the average motional occupation of the modes

was n ̄INP H = 5.39 ± 0.43 and n ̄OOP H = 2.55 ± 0.22. The other parameters found from

the t were Ω/2π = 0.885 ± 0.003 MHz and 1/γ = 163 ± 40 μs. The solid line in the
104

0 20 40 60 80 100 120

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Scan of the duration of the motion adding Raman sideband transition on the INPH mode of 9Be+−

24Mg+

Time (us)

1 ion weight

Data

Fit 

Fit Result:

n ̄INPH = 5.39 ±0.43

n ̄OOPH = 2.55 ±0.22

Figure 5.18: Sideband opping on the INPH mode on a pair of 9Be+and 24Mg+ions

after 24Mg+ Doppler cooling. The solid line is the t to the experimental data. Each

data point consists of 600 experiments.

gure is the expected theoretical curve using the values of n ̄ obtained from the t.

Figure (5.19) depicts the sideband opping on the OOPH mode of motion after

the same cooling described above. The t result are n ̄INP H = 5.57 ± 0.32, n ̄OOP H =

2.29 ± 0.11 and Ω/2π = 0.874 ± 0.003 MHz. In this case the decay parameter was held

xed at 1/γ = 294 μs.

Figure (5.20) plots the opping on the motion sensitive carrier transition. In this

case the two Rabi frequencies in equation (5.50) would be written as Ωn,n and Ωk,k,

since no motion changing transition is made. The t gives the following values of the

parameters n ̄INP H = 4.18 ± 0.34, n ̄OOP H = 3.27 ± 0.14 and Ω/2π = 0.86 ± 0.04 MHz

The decay parameter was xed at a value of γ = 100 μs. The results for the n ̄ for each

mode do not agree well the other scans. This may be due to the carrier transition not
105

0 10 20 30 40 50 60 70

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Scan of the duration of the motion adding Raman sideband transitionon the OOPH Mode of 9Be+−

24Mg+

Time (us)

1 ion weight

Data

Fit Data

Fit Result:

n ̄INPH = 5.57 ±0.32

n ̄OOPH = 2.29 ±0.11

Figure 5.19: Sideband opping on the OOPH mode on a pair of 9Be+and 24Mg+ions

after 24Mg+ Doppler cooling. The solid line is the t to the experimental data. Each

data point consists of 600 experiments.

being as sensitive to the motion.

Two-level Doppler cooling would typically be performed on a closed cycling transition in a two-level system. In this case the nal temperature will be given by equation

(5.1). The value of ξ plays an important role in determining the nal temperature

reached, where ξ is the projection of the scattered photons momentum along the axis

being considered. In our experimental setup a typical Doppler cooling beam would be at

45 degrees to the trap axis and have σ

+- polarization. This is the orientation assumed

for the theoretical calculation. The projection onto the trap-axis of the scattered photons momentum is ξ =

7

20 , which comes from equation (5.13). Figures (5.21) and (5.22)

show the nal temperature that would be reached as a function of the detuning from the

excited 2P3

2

state for 24Mg+two-level Doppler cooling on the INPH and OOPH modes.

The temperature is plotted in terms of the n ̄ reached for each mode, which is given by
106

0 5 10 15 20 25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Scan of the duration of the carrier Raman sideband transition of 9Be+−

24Mg+

Time (us)

1 ion weight

Data

Fit Data

Fit Result:

n ̄INPH = 4.18 ±0.34

n ̄OOPH = 3.27 ±0.14

Figure 5.20: Flopping on a motion sensitive carrier transition on a pair of 9Be+and

24Mg+

ions after 24Mg+ Doppler cooling. The solid line is the t to the experimental

data. Each data point consists of 600 experiments.

equation (B.9). The two-level Doppler temperature (5.1) that assumes ξ = 1, is plotted

as a horizontal line in each gure, which for each mode corresponds to n ̄INP H = 10.6

and n ̄OOP H = 4.7. It can be seen that the two-level Doppler cooling curves drop below

this limit. This is a result of ξ < 1. The t results n ̄INP H = 5.57 and n ̄OOP H = 2.29

from gure (5.18) and (5.19) are plotted as a horizontal lines in gures (5.21) and(5.22)

respectively.

The temperatures reached using the three-level cooling Doppler cooling is approximately a factor of 2 lower than the typical temperature reached with two-level Doppler

cooling. However, when taking the dipole-radiation patterns projection along the trap

axis into account and assuming s = 0 as in gures (5.21) and (5.22) the three-level

Doppler cooling on both modes shows a slightly lower temperature than is possible with
107

-2.0 -1.5 -1.0 -0.5 0.0

0

2

4

6

8

10

12

14

Detuning HGL

-

nINPH

24Mg+

two level Doppler cooling

on the 3S12 -> 3P32 - INPH mode

Exp Data

Doppler

s=0

s=1

Figure 5.21: The n ̄INP H of the INPH mode of motion achieved after Doppler cooling

with use of a two-level system in 24Mg+versus the detuning. The curved lines are for

two dierent values of the saturation parameter and take into account the projection

of the scattered light along the trap axis. The typical 1-D Doppler limit is plotted

as a horizontal line n ̄INP H = 10.6. The result of the t to the experimental data for

n ̄INP H = 5.57 from gure (5.18) is shown.

two-level Doppler cooling, including the projection.
108

-2.0 -1.5 -1.0 -0.5 0.0

0

2

4

6

8

10

12

14

Detuning HGL

-

nOOPH

24Mg+

two level Doppler cooling

on the 3S12 -> 3P32 - OOPH mode

Exp Data

Doppler

s=0

s=1

Figure 5.22: The expected n ̄OOP H of the OOPH mode of motion achieved after Doppler

cooling with use of a two-level system in 24Mg+versus the detuning. The curved lines are

for two dierent values of the saturation parameter and take into account the projection

of the scattered light along the trap axis. The typical 1-D Doppler limit is plotted

as a horizontal line n ̄OOP H = 4.7. The result of the t to the experimental data for

n ̄OOP H = 2.29 from gure (5.19) is shown.
Chapter 6

Raman Transitions and Sideband Cooling

Most physical systems are very dicult to model quantum mechanically. However, trapped ions with one valence electron can be modeled with a few simplications.

The internal states of the ions' (9Be+ and 24Mg+) discussed here can be treated as

two level systems, and the ions' motion in the ion trap is that of a quantum harmonic

oscillator [Wineland 98]. The interaction of a trapped ion with a laser can lead to a coupling between the internal and the motional degrees of freedom, which provides a means

to control the quantum mechanical motion of trapped ions [Wineland 98, Cirac 96a].

This has been exploited to make several non-classical motional states of trapped ions

[Meekhof 96, Leibfried 96, Monroe 96, Wineland 98]. To create these various types of

non-classical states it is typically necessary to cool the motion to near the quantum

mechanical ground state. Through a combination of Doppler and stimulated-Ramantransition cooling this can be accomplished [Monroe 95b, King 98]. Cooling the motion

of a multiple ions of one type to near the ground state using another type of ion (sympathetic cooling) is benecial for QIP with trapped ions, since the shared motion serves

as an information bus [Kielpinski 00, Barrett 03, Blinov 02, Rohde 01]. Minimizing the

amount of energy in the quantized motion can help to reduce errors in some QIP operations. Also, the ability to cool logical ion qubits with sympathetic cooling ions is a necessary component for some ion trap based QIP schemes [Kielpinski 00, Kielpinski 02b].

As another example, in the experiment to entangle two mechanical oscillators discussed
110

in chapter (8), the motion of two pairs of trapped ions must be cooled to near ground

state to before creating the desired entangled motional state.

This chapter describes how the internal and motional states of trapped ions can be

manipulated with laser beams. The rst section discusses a Hamiltonian representation

for the internal state and motion of a trapped ion. Then it is explained how these

two systems can be coupled, using a magnetic eld in the interaction picture. When

working with multiple ions there are many motional states that need to be accounted

for, which is covered in section (6.3). In the experiments discussed in this dissertation,

stimulated-Raman transitions are used to couple the internal states to the motion. In

the following section (6.4) it is shown how this gives rise to the same coupling as the

magnetic eld discussed in section (6.2). This is followed by a brief discussion on how

this interaction can be used for motional state analysis. The details of how stimulatedRaman transitions can be used to the cool the motion of trapped ions is then covered

in section (6.6). Finally a demonstration experiment is discussed in section (6.6). This

experiment involves the sympathetic cooling of two pairs of 9Be+ − 24Mg+ ions in two

dierent locations of a multi-zone ion trap array. These results and the results from

reference [Jost 09] represent the rst demonstration of the ability to sympathetically

recool trapped ions after separating and transporting them in multi zone ion trap array.

6.1 Basic Hamiltonian

There can be many internal states for an ion with a single valence electron. For

the discussion below it is assumed that only electric dipole transitions will be considered.

This allows for the simplication that only two internal levels will be coupled at a time,

assuming the levels are well resolved. The Hamiltonian for the internal state is given by

Ha = ~ω1 |1i h1| + ~ω2 |2i h2| (6.1)
111

The energy for each level is given by ~ω1 and ~ω2. It will simplify the treatment to to

rewrite the Hamiltonian in terms of a spin 1/2 system, where the Pauli spin operators

and the energy dierence (ωo = ω2 −ω1) between the two levels are used [Leibfried 03a].

Ha = ~ωoSz, (6.2)

where a constant term has been omitted. For a single trapped ion there will be three

normal modes of motion. Each mode of motion can be treated independently with its

own Hamiltonian, so only the z direction will be considered here.

Hm = ~ωz(n + 1/2). (6.3)

The mode of motion along this axis has an oscillation frequency ωz. The Hamiltonian

can be rewritten in terms of the raising and lowering operators (a and a

†

)

Hm = ~ωza

†

a. (6.4)

The zero point energy has been left out since it is a constant.

6.2 Interaction Hamiltonian

The interaction that couples the internal and motional states in the experimental

apparatus is a stimulated-Raman transition. This is a coherent two photon process

that makes use of electric dipole transitions. This more specic case will be discussed

in section (6.4). However, it is sucient to consider a single photon electric dipole

transition that has a Hamiltonian of the form Hed = −μe · E(z, t) . This has the same

functional form as a magnetic dipole transition[Wineland 98], which will be used for the

derivation in this section. The Hamiltonian for the interaction of a spin 1/2 magnetic

dipole with a magnetic eld is

Hl = −~μ · (B~

static + B~ (z, t)). (6.5)
112

The magnetic dipole moment is ~μ = γS~ . The gyromagnetic ratio is represented by

γ, and S~ = S~

x + S~y + ~Sz. The static magnetic eld component B~static is gives rise to

the energy splitting in this spin 1/2 analogy represented by Ha in equation (6.2). Since

eventually the light eld of a laser beam will be considered by analogy, an oscillating

magnetic eld propagating in the zˆ direction with a polarization in the xˆ direction and

a magnitude Bo will be used.

B~ (z, t) = B0xˆ cos (k · z − ωt + φ). (6.6)

The wave vector of the radiation is given by k and z is the position operator (about the

ion's equilibrium), which can be written in terms of the raising and lowering operators

as z = zo(a + a

†

). Where zo is the spread in the ground state wave function of the ion

zo =

r

~

2mωz

. (6.7)

The Hamiltonian for the interaction can now be rewritten as

Hl = −

γBo

4

(S

+ + S−)



e

i(k·z−ωt+φ) + e−i(k·z−ωt+φ)



. (6.8)

It useful to use the interaction picture for calculating the time evolution of the system.

The time-independent Hamiltonian is given by Ho = Ha + Hm and the time-dependent

interaction Hamiltonian is VI = Hl. This is transformed into a Hamiltonian in the

interaction picture by the following

HI = e

i

Ho

~

tVIe−i

Ho

~

t

. (6.9)

This can be multiplied out using the Baker-Hausdor theorem. Another method is to

use the Heisenberg equations of motion ( dA

dt =

1

i~

[A, H]) to nd the interaction picture

version of all of the all the operators, which gives

S

+ → S+eiωot

(6.10)

S

− → S−e−iωot

(6.11)

z → zo(ae−iωzt + a

†

e

iωzt

). (6.12)
113

The end result for the interaction a Hamiltonian is

HI = −

γBo

4

h

S



e

i(η(ae−iωzt+a

†eiωzt

)−(ω−ωo)t+φ)

(6.13)

+e

−i(η(ae−iωzt+a

†eiωzt

)−(ω+ωo)t+φ)



(6.14)

+S

−



e

i(η(ae−iωzt+a

†eiωzt

)−(ω+ωo)t+φ)

(6.15)

+e

−i(η(ae−iωzt+a

†eiωzt

)−(ω−ωo)t+φ)

i. (6.16)

The parameter η is the Lamb-Dicke factor, and is given by η ≡ kzo. This is the ratio

of the physical extent of the ion's motional ground state wave packet to the wavelength

of the radiation divided by 2π. The terms in the above equation that contain e

±i(ω+ωo)t

are fast oscillating terms that average to zero. They can be dropped from Hamiltonian,

and making this approximation is called the rotating-wave approximation. The resulting

Hamiltonian is

HI = ~Ω

h

S

+e

i(η(ae−iωzt+a

†eiωzt

)−δt+φ) + S

−e

−i(η(ae−iωzt+a

†eiωzt

)−δt++φ)

i

, (6.17)

where δ ≡ ω − ωo and ~Ω = −

γBo

4

. For the case of an electric dipole transition, the

substitution ~Ω = −

μeE

2 would be made. The electric dipole moment is μe = e h2| r |1i

and E is the electric eld of the radiation. The wave function for the states coupled by

this interaction can be written

Ψ (t) = X

i=1,2

X∞

n=0

Ci,n (t)|ii |ni. (6.18)

The subscripts i and n represent the internal and motional states. The coecients

describing the time evolution of the wave function are given by the Schrodinger equation

i~

∂Ψ

∂t = HIΨ. (6.19)

Only near resonant transitions will be considered, and these are occur when δ ' (n

0 − n) ωz.

For a treatment of o resonant transitions see reference [Wineland 98]. The result is the
114

coupled dierential equations [Wineland 98]

·

C1,n0 = −i

(1+|n

0−n|)

e

iφΩn0

,nC2,n (6.20)

·

C2,n = −i

(1−|n

0−n|)

e

−iφΩn0

,nC1,n0. (6.21)

The term Ωn0

,nis the Rabi frequency and is given by [Wineland 98, Wineland 79, Kahill 69]

Ωn0

,n ≡ Ω





hn

0

| e

iη(a+a

†

)

|ni





 , (6.22)

and this evaluates to

Ωn0

,n = Ωe

−η

2

2



n<!

n>!

1/2

η

|n

0−n|L

|n

0−n|

n<



η

2



. (6.23)

The result does depend on the starting and nal motional state n and n

0

. The values

of n< and n> are the lesser and greater of n and n

0

. The factor L

a

n

is the generalized

Laguerre polynomial [Wineland 98]

L

a

n

(x) = X∞

m=0

(−1)m



n + a

n − m



x

m

m!

. (6.24)

The coupled dierential equations (6.21) can be solved using Laplace transforms to give

the solution for the Schrodinger equation [Wineland 98]

Ψ (t) =







cos Ωn0

,nt



−iei[φ+ π

2

|n

0−n|]

sin Ωn0

,nt



−ie−i[φ+ π

2

|n

0−n|]

sin Ωn0

,nt



cos Ωn0

,nt







 Ψ (0). (6.25)

The two component wave function is Ψ (0) =







C1,n (0)

C2,n0 (0)





.

This is the general result for on resonance transitions. The Hamiltonian in equation (6.17) can be simplied if the amplitude of the motion is small compared to the wavelength of the radiation. The is called the Lamb-Dicke limit and is dened [Wineland 98]

as 

hΨmotion| k

2

z

2

|Ψmotioni

1/2  1. The expectation value of z

2

is taken because the

expectation value of z is zero. This is more restrictive than what is sometimes incorrectly

referred to as the Lamb-Dicke limit where η  1. The Lamb-Dicke parameter maybe
115

Types of Raman Sideband Transitions

Motion Adding (MA) Sideband Motion Subtracting (MS) Sideband

n = 0

n = 1

n = 2

Carrier Transition

n = 0

n = 1

n = 2

n = 0

n = 1

n = 2

Figure 6.1: This gure depicts a few of the dierent types of sideband transitions.

When one quanta of motional energy is added this can be called a motion adding (MA)

transition. A transition where one quanta of energy is removed is called a motion

subtracting (MS) transition. It is also possible to change the internal state without

aecting the motion, which is called a carrier transition.

small; however, if the motional state is large, then the amplitude of the motion can still

be considerably large when compared to the wavelength of radiation. In the Lamb-Dicke

limit, things simplify. The interaction Hamiltonian can be expanded to rst order in the

η to give

HLD

I = ~Ωe

−i(δt−φ)S+

h

1 + iη(ae−iωzt + a

†

e

iωzt

)

i

+ H.c. (6.26)

There are several types of transitions that can be made (see gure (6.1)). A carrier

transition is one where n = n

0 and δ = 0. In this case only the internal state is changed,
116

which leads to a Hamiltonian where

Hc = ~Ωe

−i(φ)S+

h

1 + iη(ae−iωzt + a

†

e

iωzt

)

i

+ H.c.. (6.27)

However, the resonant terms will be dominant, which means any terms oscillating in

time can be dropped to a good approximation if Ω  ωz. This is equivalent to making

another rotating wave approximation, which gives

Hc = ~Ω



S

+e−i(φ) + S−e+i(φ)



. (6.28)

The above Hamiltonian is independent of the motion, which is only strictly true in the

limit that η → 0. In general, for most experimental parameters the motion will play a

role. For example the Rabi frequency equation for the (6.23) n = 0 to n

0 = 0 carrier

transition is Ω0,0 = Ωe

−η

2

2 . This exponential factor is known as the Debye-Waller factor

[Wineland 79]. This will show up again when multiple modes of motion are considered

in section (6.3).

Another resonant transition that is important is the motion subtracting (MS)

transition where δ = −ωz. This is sometimes called a red-sideband in the literature,

because if the initial internal state is the lower energy state, then the radiation that

does the coupling will have a lower frequency than the carrier. However, if the initial

state is the higher energy internal state, then the motion subtracting transition will be

blue detuned with respect to the carrier, that is where δ = ωz. For this reason, sideband

transitions will only be referred to as either motion adding (MA) or motions subtracting

(MS). In the Lamb-Dicke limit, the Rabi frequency for the MS transition simplies to

Ωn−1,n = η

√

nΩ. The interaction Hamiltonian equation (6.26)becomes

HMS = ~Ωe

+iφS+



e

iωzt + iη(a + a†

e

2iωzt

)



(6.29)

+~Ωe

−iφS−



e

−iωzt − iη(ae−2iωzt + a†

)



. (6.30)

Again if we assume that only near-resonant processes are being considered, any fast

oscillating terms average to zero. This is same as making another rotating wave approx-
117

imation. Also the phase φ can be arbitrarily chosen for the rst application of HI . If

the phase is chosen to be φ = −

π

2

, then the Hamiltonian simplies to

HMS = ~ηΩ



S

+a + S−a†



. (6.31)

This is also known as the Jaynes-Cummings Hamiltonian from quantum optics [Jaynes 63].

For the MA transition, the detuning is δ = ωz, and in the Lamb-Dicke limit the Rabi

frequency is Ωn0

,n = η

√

n + 1Ω. The interaction Hamiltonian simplies to

HMS = ~ηΩ



S

+a† + S−a



. (6.32)

It is also possible to drive transitions where more than one quanta of motional energy is

added or subtracted. For some of the experiments discuss later, sideband transitions are

used where two quanta of energy are removed. The Rabi frequency of these transitions

will dier.

Ω1,0 = Ωe

−η

2

2 η (6.33)

Ω2,0 = Ωe

−η

2

2 2

−1/2

η

2

(6.34)

Thus the second sideband Rabi frequency is a factor 2

−1/2η slower than the Rabi frequency for the rst sideband.

6.3 Rabi Rates - Multiple Modes

The above results can be generalized for multiple ions. Here it is assumed that

the laser beam only addresses one of the ions in a chain, and only the axial modes will

be considered. If there are L ions, then there will be L normal modes along the axis. If

all the other normal modes where accounted for, then there would be 3L normal modes,

which is treated in reference [Wineland 98]. The time-independent Hamiltonian for the

the j

th ion (j is the index of the ion be addressed, where j runs from 1 to L) is

Ho,j = ~ωoSz,j +

X

L

k=1

~ωz,kaka

†

k

. (6.35)
118

The index k in the sum represents the dierent normal modes. The Hamiltonian describing the laser interaction (6.8) with the j

th ion becomes

Hl,j = −

γBo

4

(S

+

j + S

−

j

)



e

i(k·zj−ωt+φ) + e−i(k·zj−ωt+φ)



. (6.36)

The zj term can be expanded in terms of the normal modes. The interaction Hamiltonian

now becomes

HI,j = ~ΩS

+

j

e

i

PL

k=1 η

j

k

(ae

−iωz,kt+a

†e

iωz,kt

)−δt+φ



+ H.c.. (6.37)

The index k corresponds to the k

th normal mode along the trap axis. The Lamb-Dicke

parameter η

j

k

correspond to the Lamb-Dicke parameter for the j

th ion on the kth mode.

One consequence of having multiple modes is the Rabi frequency will be aected by the

presence of the other modes. The Rabi frequency becomes [Wineland 98]

Ωn

0

k

,nk ≡ Ω











hnp6=k, n0

k

|

Y

L

l=1

e

iηl(a+a

†

)

|np6=k, nki











. (6.38)

The subscript p represents all the other modes not being addressed by the resonant laser

interaction. For example, the Rabi frequency for a MA transition of the rst mode in a

two ion crystal would be.

Ωn

0

1=1,n1=0,n02=0,n2=0 = Ωe

−η

2

1

2 η1e

−η

2

2

2 . (6.39)

The second exponential factor is the Debye-Waller factor for the second mode, which

eectively reduces the Rabi frequency. For larger ion crystals, each additional mode will

contribute another Debye-Waller factor.

6.4 Stimulated Raman Transitions

In the above, the coupling between the internal states and the ions' motion was

derived assuming an oscillating magnetic eld excites transitions between two levels.

The splitting for most ion qubit systems based on hyperne structure range from a
119

few MHz to a few GHz. Using an oscillating magnetic eld to drive the transitions,

would provide a very small Lamb-Dicke parameter as a result of the long wavelength. In

addition, ion traps are usually built on the centimeter size scale, so it would be dicult

to isolate other ions from being aected by the radiation. One way around this is to have

a magnetic eld gradient, creating ions with dierent resonant frequencies as a function

of their position [Mintert 01, Wineland 98, Wang 09]. It might also be possible to use a

combination of magnetic eld gradients and oscillating magnetic elds to drive MA and

MS transitions [Ospelkaus 08]. Currently the most common technique is to use lasers to

couple the internal states to the motion. This can be done with a single laser beam when

the internal qubit levels are split by an optical frequency, and the linewidth of the excited

state is small compared to ion's motional frequency [J. C. Bergquist 87]. The technique

used in this work is a coherent two-photon process called stimulated-Raman transitions

[Monroe 95b, Heinzen 90]. The two photons have a dierence frequency tuned near the

qubit transition (typically in the MHz to GHz range), and couple the two levels via a

virtual excited state. This technique makes a large eective ~k vector along the trap

axis, which creates a larger Lamb-Dicke parameter than in the case of direct excitation

with radio-frequency magnetic elds. It was discussed earlier that η  1 to be in the

Lamb-Dicke regime. However, there is a trade o. The sideband Rabi frequencies are

proportional to η, so a vanishing η as in the radio-frequency/microwave case leads to a

vanishing Ωn0

,n. Another advantage of using lasers is the ability to individually address

the ions. Typical inter ion spacings are on the order of a few microns. With diraction

limited optics the diameter of the beam at the focus can be of order λ, which at optical

wavelengths is sucient to individually addresses ions in most chains. This has been

demonstrated in 43Ca+ ions where the optical transitions used have a wavelength of

729 nm [Nägerl 99]. The experiments discussed in this work use UV photons, so in

theory a much smaller waist should be obtainable. However, current optics are not

sucient to generate diraction limited spots, but it is possible to individually address
120

n = 0

n = 1

n = 2

Three-Level Stimulated-Raman Transitions

Not to scale

Figure 6.2: Energy level diagram illustrating stimulated Raman sideband transitions.

The transitions are driven with two laser beams, which have frequencies ωL1 and ωL2 .

Both beams are detuned from the excited state |3i by approximately ∆. We consider

two internal ground states labeled |1i and |2i, which have energies ~ω1 and ~ω2. The

frequency dierence between the the two beams is ωL1 − ωL2 + δ. Superimposed on

top of the two internal ground states are ladders of harmonic oscillator states, which

represent the quantized motion. The values of n represents the Fock states. When there

are multiple modes of motion then several sets of states must be superimposed; however,

only one mode will be considered here.

ions spaced a few 10's of microns apart. Lasers oer the additional advantage that they

can be steered with Acoustic Optic Modulators and move-able mirrors, which enables

addressing dierent parts of the ion trap.

This section discusses how the two-photon stimulated-Raman transition gives the

same results for coupling the two internal (qubit) levels to the motions as the pre-
121

vious section. This is also worked out in the thesis by Jonathan Home [Home 06b,

Wineland 04]. Figure (6.2) shows the setup. There are three internal levels labeled

|1i, |2i, and |3i, which have energies ~ω1, ~ω2, and ~ω3 respectively. The splitting between levels |1i and |2i is on the order of 100 MHz or a 1 GHz for the hyperne levels

typically used in 9Be+ at a magnetic eld of ∼ 120 Gauss. The splitting between |1i

and |3i or |2i and |3i is approximately 313 nm. For 24Mg+ the splitting between levels

|1i and |2i is about 300 MHz at a magnetic eld of ∼ 120 Gauss, and around 280 nm

for the dierence between the excited state and ground states. The exact values for the

splitting of the two ground states will depend on the value of the magnetic eld used in

the experiments.

One laser beam with a frequency ωL1 couples the states |1i → |3i with a detuning

∆. The second laser has a frequency ωL2. It couples the states |2i → |3i with a detuning

∆ from state |3i and a detuning δ from state |2i, where δ  ∆  ωL1, ωL2. It should

also be noted that for the case considered here ∆  Γ with Γ being the radiative

line width of the excited state. The electric eld of the lasers is given by E~

i(t) =

ˆiEo,i cos (ki· z − ωL,it + φi). Where the index i ∈ 1, 2 denotes the laser beam. The

polarization is represented by ˆ and the electric eld magnitude by Eo,i. The radiation

couples the ground to the excited states via an electric dipole transition given by

HL = −~μed(E~

1(t) + E~2(t)). (6.40)

Substituting the oscillating electric eld gives,

HL = ~g13 cos (k1 · z − ωL1t + φ1)|1i h3| + ~g23 cos (k2 · z − ωL2t + φ2)|2i h3| + H.c..

(6.41)

The coupling strengths g13 and g23 are

g13 = − h3| ~μed · ˆ1 |1i

Eo,1

~

(6.42)

g23 = − h3| ~μed · ˆ2 |2i

Eo,2

~

. (6.43)
122

The Hamiltonian for the internal state can be written in the form

Ha = ~ω1 |1i h1| + ~ω2 |2i h2| + ~ω3 |3i h3| . (6.44)

As before it is convenient to move to an interaction picture where Ho = Ha and VI = HL.

The laser frequencies can be written in terms of the internal state energies and detunings

as ωL1 = ω3 − ω1 + ∆ and ωL2 = ω3 − ω1 + ∆ − δ. Transforming to the interaction

picture using equation (6.9) gives

HI =

~g13

2

e

i(

~k1·~z+φ1)

e

−i∆t



1 + e

i2ωL1t



|1i h3| (6.45)

+

~g23

2

e

i(

~k2·~z+φ2)

e

−i(∆−δ)t



1 + e

i2ωL2t



|2i h3| + H.c.. (6.46)

Looking at the time dependent terms, there is one that oscillates at ∆ and another

that oscillates at a much higher frequency 2ωL,i. For typical experimental parameters

∆ might range from 10 GHz to 100 GHz where ωl,i is in the PHz regime. The position

operator z has not been transformed to the interaction picture yet but this will be done

later.

The wave function for the internal states can be written as

Ψ (t) = X

3

i=1

Ci (t)|ii. (6.47)

Using this wave function and the interaction Hamiltonian (6.45), the Schrodinger equation (6.19) can be solved to nd the time dependence of the coecients Ci (t). This

gives

·

C1 = −i

g13

2

e

i(

~k1·~z+φ1)

e

−i∆tCc (6.48)

·

C2 = −i

g23

2

e

i(

~k2·~z+φ2)

e

−i(∆−δ)tC3 (6.49)

·

C3 = −i

g

?

13

2

e

−i(

~k1·~z+φ1)

e

+i∆tC1 + −i

g

?

23

2

e

−i(

~k1·~z+φ1)

e

+i(∆−δ)tC2. (6.50)

This is coupled dierential equation for the three level system. The goal is to make

this set of equations look like that for a two level system. To do this it is necessary to
123

eliminate the third level. An approximate solution for

·

C3 can be found by integrating

by parts twice to give

C3 = −i

g

?

13

2

e

−i(

~k1·~z+φ1)





C1

i∆

e

i∆t −

1

i∆





·

C1

i∆

e

i∆t −

ˆ ··

C1

i∆

e

i∆t

dt







 (6.51)

−i

g

?

23

2

e

−i(

~k2·~z+φ2)



C2

i (∆ − δ)

e

i(∆−δ)t



(6.52)

−i

g

?

23

2

e

−i(

~k2·~z+φ2)



−

1

i (∆ − δ)





·

C2

i (∆ − δ)

e

i(∆−δ)t −

ˆ ··

C2

i (∆ − δ)

e

i(∆−δ)t

dt







 (6.53)

An approximation can be made here to simplify things. Terms of order O



g13

∆2



and

O



g123

(∆−δ)

2



can be dropped (note: g13, g23  ∆) as long as C1 and C2 oscillate slowly.

This means dropping the two terms in the square brackets, which gives

C3 = −

g

?

13

2∆e

−i(

~k1·~z+φ1)

e

i∆tC1 −

g

?

23

2 (∆ − δ)

e

−i(

~k2·~z+φ2)

e

i(∆−δ)tC2. (6.54)

This equation can be plugged into the dierential equations (6.50) to get a set of coupled

dierential equations that do not depend on the population in the excited state.

·

C1 = i

|g13|

2

4∆ C1 + i

g

?

23g13

4 (∆ − δ)

e

−iδte

i(∆~k·~z+∆φ)C2 (6.55)

·

C2 = i

|g23|

2

4 (∆ − δ)

C2 + i

g23g

?

13

4∆ e

iδte

−i(∆~k·~z+∆φ)C1. (6.56)

Now the terms in the exponentials depend on the dierence of the wave vectors ∆k =

k1 − k2 and on the dierence of the phases of the two laser beams ∆φ = φ1 − φ2. From

this dierential equation, it can be determined that the Hamiltonian that gives the same

evolution is

HI = −~

|g13|

2

4∆ |1i h1| − ~

|g23|

2

4 (∆ − δ)

|2i h2| (6.57)

−~

g

?

23g13

4 (∆ − δ)

e

−iδte

i(∆~k·~z+∆φ)

|1i h2| (6.58)

−~

g23g

?

13

4∆ e

iδte

−i(∆~k·~z+∆φ)

|2i h1| . (6.59)

The rst two terms in the Hamiltonian are the Stark shifts of the levels as a result of the

light elds. These do not depend on time and can be absorbed into the denition of Ho,

which has the aect of redening the energy of the two levels. Another approximation
124

that can be made is ∆ −δ → ∆, which can be made for δ  ∆. With these assumptions

the interaction Hamiltonian can be written as.

HI = ~Ωe

−iδte

i(∆~k·~z+∆φ)

|1i h2| + ~Ω

?

e

iδte

−i(∆~k·~z+∆φ)

|2i h1| . (6.60)

This has the same form as the Hamiltonian found for the case treated early where an

oscillating magnetic eld was use to couple the two levels. The dierence here is that

Ω = −

g

?

23g13

4(∆) , and the redenition of ~k → ∆~k and φ → ∆φ.

The same types of transitions (MA, MS, and carrier) exist for the three level

stimulated Raman transitions as those for the two level case depicted in gure (6.1).

Figure (6.3) shows several of the possible transitions for the three level system. One

additional transition not depicted in gure (6.1) is the second sideband transition. The

main dierence between this type of transition and the MA and MS transitions previously

discussed is that two quanta of energy are added or removed. This is done by tuning

the dierence frequency to be δ = 2ωz. The Rabi frequency equation (6.23) will also be

dierent for these transitions.

6.4.1 Co-propagating Beam Transitions

In the above discussion of stimulated-Raman transitions, it was assumed that two

beams are orthogonal to each other. The orientation of the two beams has important

implications for whether or not the transitions are sensitive to the ions' motion. The

sensitivity to motion enters through the term ∆~k·~z , where ~k is the eective wavevector

and ~z represents the ions displacement from equilibrium. For the stimulated-Raman

sideband transitions discussed above the two beams used are at UV wavelengths with

wavevectors ~k1 and ~k2. The two beams are orthogonal to each other and at 45 degrees

with respect to the trap axis. Only the transition's sensitivity to motion along the trap

axis is considered. This is valid if ∆~k = √

1

2



~k1 − ~k2



lies along the trap axis. The

other case to be considered is when the two beams are co-propagating and at 45 degrees
125

Types of Stimulated Raman Sideband Transitions

n = 0

n = 1

n = 2

Motion Adding (MA) Sideband Motion Subtracting (MS) Sideband

n = 0

n = 1

n = 2

Carrier Transition Second Sideband Motion Subtracting

n = 0

n = 1

n = 2

n = 0

n = 1

n = 2

Figure 6.3: This gure depicts a few of the dierent types of stimulated Raman transitions that are possible. When one quanta of motional energy is added this called a

motion adding (MA) transition. A transition where one quanta of energy is removed

is called a motion subtracting (MS) transition. It is also possible to change the internal state without aecting the motion, which is called a carrier transition. The second

sideband motion subtracting transition is relevant for the sideband cooling discussed in

section (6.6). This transition removes two quanta of energy from the motion.

with respect to the trap axis. Now the magnitude of ∆~k is the small. For example,

a single trapped 9Be+ ion with a motional frequency of ω = 2π × 3 × 106 Hz, the

Lamb-Dicke parameter η ∼ 0.4 for the orthogonal beam case and η ∼ 3 × 10−8for the
126

co-propagating case (It is assumed the stimulated-Raman transitions were made between

the levels |

2S1

2

, F = 2, mF = 2i → |2S1

2

, F = 2, mF = 1i at a eld of ∼ 120 Gauss giving

rise to a splitting between the levels of ∼ 100 MHz). The small Lamb-Dicke parameter

for the co-propagating beams makes theses transitions insensitive to the motional state

of the trapped ions. Thus, they can be used to perform coherent rotations between the

internal states of the ions. These rotations are termed single qubit gates [Nielsen 00].

6.5 Motional State Analysis

The sensitivity of the Rabi frequency to the motional state and the presence of

other modes provide a tool for determining the distribution motional energy states. Several types of distributions have been experimentally prepared. Reference [Meekhof 96]

gives examples of sideband opping (Sideband opping refers to a scan measuring

the uorescence versus the duration of a MA or MS transition) on a single ion that

has had its motional state prepared in either a thermal distribution, coherent state,

or a squeezed state. Discussed the section on Doppler cooling of this dissertation

is an example where sideband opping is used to estimate the temperature of the

two axial modes of a 9Be+ − 24Mg+ ion pair. Also discussed in this section is another technique for estimating the temperature of a thermal distribution. This technique relies on measuring the relative probability of making a MA and MS transition

[Monroe 95b, Turchette 00, Home 06b, Diedrich 89].

If the duration of the laser interaction is scanned and the internal state populations

are measured, then a Rabi sideband opping curve is obtained. For a single ion initialized

in the state |1, ni, the probability that the ion is found in state |1i is given by

P|1i(t) =



cos 

Ωn0

,nt





2

. (6.61)

This can be re-expressed as P|1i =

1

2



1 + cos 2Ωn0

,nt

. Typically the ion's initial

motional state will follow some distribution, where each motional state in the Fock basis
127

has some probability Pn of being occupied. The ion's motion is harmonic, so for a

particular sideband transition all transitions between all the dierent Fock states will be

driven. For example driving a MA transition causes all the transitions n = 0 → 1, n =

1 → 2 , etc.... The only dierence between all the transitions is that each has a dierent

Rabi rate. Thus, each transition needs to be included in calculating the probability, but

they will be weighted by a distribution.

P|1i =

1

2



1 + X∞

nk=0

Pnkcos 2Ωn

0

k

,nk

t





 . (6.62)

A common distribution that occurs with trapped ions is a thermal distribution. Most of

our ion experiments involve Doppler laser cooling, which to a good approximation leaves

the ions motional state in a thermal distribution.

P

thermal

n =

n ̄

n

(1 +  ̄n)

n+1 . (6.63)

The average motional state is n ̄, and n represents the a particular Fock state. If there are

multiple modes of motion, then the other modes will aect the Rabi rate as a result of

the their Debye-Waller factors. A more general expression for the probability of nding

the ion in state |1i is

P|1i (t) = 1

2



1 +





YL

m=1,m6=k

X∞

nm=0

Pnm





X∞

nk=0

Pnk× (6.64)

cos



2Ω













hnm6=k, n

0

k| e

iηk(a+a

†

)





YL

m=1,m6=k

e

iηm(a+a

†

)



 |mp6=k, nki













t







 (6.65)

The terms in the brackets are the contributions from the other modes, which are represented by the index m, where m 6= k. The rst bracket is the product of the sums over

all the probability distributions. Each distribution is summed over all the Fock states

for that mode. The second bracket, which appears inside the cosine term, is the product

of all the Debye-Waller factors for the other modes.

Equation (6.64) can be used as a model for performing motional state analysis.

This can be accomplished by tuning the laser interaction to the resonance of a particular

mode of motion and scanning the interaction time. The resulting data can be t to the
128

above model. Dierent probability distributions can be tried until an acceptable t for

the experimental data is obtained.

If it is known the ion's motion is in a thermal state, which it is to a good approximation after Doppler cooling, then there is another technique for estimating the

temperature. This method relies on measuring the probability of making a MA and MS

transition with xed interaction time. If a ion is prepared in the |1i state, then the

probability of making either a MS or MA transition to state |2i is given by equation

P

MS

|2i =

X∞

n=1

Pn sin2(Ωn−1,ntd) (6.66)

P

MA

|2i =

X∞

n=0

Pn sin2(Ωn+1,ntd). (6.67)

Note that the sums run over dierent n values. This is because the lowest |ni that a MS

can be made from is the |n = 1i state. Taking the ratio of these two probabilities gives

R =

P∞

n=1 Pn sin2

(Ωn−1,ntd)

P∞

n=0 Pn sin2

(Ωn+1,ntd)

, (6.68)

where R is the ratio of the sideband amplitudes. Using the thermal distribution equation

(6.63)for Pn and making the substitution n → n+1 for the motion subtracting transition,

allows the ratio to be rewritten as

R =

P∞

n=0

n ̄

n+1

(1+ ̄n)

n+2 sin2

(Ωn,n+1td)

P∞

n=0

n ̄n

(1+ ̄n)

n+1 sin2

(Ωn+1,ntd)

. (6.69)

Noting that Ωn,n+1 = Ωn+1,n, the ratio simplies to

R =

n ̄

n ̄ + 1

(6.70)

, which gives the expression for the n ̄

n ̄ =

R

R + 1

. (6.71)

In order to measure the ratio, the signal strengths two sideband are measured by applying

a xed length Raman sideband pulse and scanning the frequency of the pulse δs across
129

the MA and MS resonances. This curve can be t to the expected result for a two level

system given by[Metcalf 99][Home 06b]

P|2,n0i = a

Ω

2

n0,n

Ω2

n0,n + δ

2

s

sin2





q

Ω2

n0,n + δ

2

s

2

t



 + b. (6.72)

After tting experimental data and guring out the amplitude and the baseline for each

transition, the signal strength of the transition can be found. Taking the ratio of the

two signal strengths gives the value of R. If the ion's motion is near the ground state of

motion, then the signal strength of the MS transition will be quite small. In the ideal

case where |n = 0i then no MS transition can be driven. However, a MA transition can

always be driven.

6.6 Sideband Cooling

One of the interesting features of trapped ions is the ability to accurately control

the motional states. Most experiments require the ion to start in the motional ground

state before some special motional state is prepared. Upon initially trapping an ion

there is typically a large energy in the motion. We therefore perform Doppler cooling as

an initial cooling stage, which can often cool the ions down to the level of a few motional

quanta. For example, a single 9Be+ ion with an axial frequency of 3 MHz would be

Doppler cooled to an average motional Fock state n ̄ ∼ 3 with a thermal distribution. It

should be noted that depending on the parameters an average motional state of a few

tens of quanta is possible. Cooling to the Doppler limit, is sucient for some quantum

mechanical gate operations [Kirchmair 09]. However gates many require the motion to

be initialized in the ground state to work [Cirac 95] or to minimize errors in the gate

operations [Leibfried 03b]. In addition, one of the main focuses of this work, which is

the entangled mechanical oscillators experiment discussed in chapter (8), requires the

motion of two spatially separated pairs of 9Be+ −24Mg+ ions to be cooled to the ground

state of motion. Stimulated Raman sideband cooling (Raman cooling) is the technique
130

Types of transitions used for Raman Sideband Cooling

n = 0

n = 1

n = 2

Motion Substracting Sideband Resonant repumping

n = 0

n = 1

n = 2

Figure 6.4: The two main types of transitions used in Raman sideband cooling. On the

left part of the gure, the motion subtracting transition removes one quanta of energy

from the motion. It should be noted that this transition was drawn dierently from the

one depicted in gure (6.3), where the only dierence is the starting state. In this gure

the ion's internal state starts in state |2i, and in gure (6.3) it starts in state |1i. On

the right part of the gure, the resonant repumping transition transfers internal state

population from |1i to |2i by scattering only a few photons, which leads to negligible

heating. State |2i scatters no photons from this laser beam.

used in this work to cool the motion of trapped ions to very near the ground state.

In a simplied picture, the transitions used for Raman cooling are illustrated

in gure (6.4). There are two types of transitions used: the MS sideband transition

discussed in section (6.9) and a resonant electric dipole transition that performs optical

pumping. The procedure begins with the ion being prepared in state |2i by optical

pumping, which will also have some distribution of motional states. A MS transition

is then performed, using a two-photon stimulated Raman transition. The time of the

interaction is chosen such that all the population is transferred from |2i |ni → |1i |n − 1i.,

which is is called a π-pulse (See below for a discussion on choosing which |ni → |n − 1i

transition is driven.). The next step is apply a resonant interaction, which couples

|1i → |3i, referred to as repumping. The excited state will spontaneously emit a photon
131

and decay to either state |1i or |2i. If it decays to state |2i, then optical pumping

shuts o, since it is no longer resonant or it is not possible to drive a transition with the

polarization used. If the decay was back to state |1i, then it will only take a few scattering

events until all the population is in state |2i. The scattering from repumping will cause

the ions' motion to heat up but on average this is less than a motional quanta, because

(~k)

2

2m < ~ωz. Here the momenta of the absorbed/scattered photon are represented by ~k

and ωzis the motional frequency. Another way to say this is if the Lamb-Dicke parameter

is less than one, then there will not be much heating from scattering. This is because

η

2 =

(~k)

2

2m

/~ωz. This process is repeated many times until all the motion is cooled to

the ground state. The sequence of a MS sideband followed by a repumping pulse and

spontaneous emission will be referred to as a single Raman cooling cycle.

Other issues complicate the cooling beyond this simple picture. After Doppler

cooling, the ions' motional state will be in a thermal distribution. This means that

is possible for some of motional population to occupy Fock states with large values

on |ni. From equation (6.23) it can be seen that the Rabi frequency depends on the

particular |ni state. Thus if the time of the interaction is chosen to drive a π-pulse

on the |n = 1i to |n = 0i transition this will not drive a full π-pulse on the |n = 10i

to |n = 9i. A partial solution to this is to estimate the maximum Fock state with

a non negligible population and start with a π- pulse that is optimized to make the

transition |nmaxi → |nmax − 1i. Then for the next cycle, the time is chosen to optimize

the |nmax − 1i → |nmax − 2i transition. During each pulse, partial π-pulses are driven

between the other transitions between motional states. This process can be continued

until reaching the |n = 1i → |n = 0i transition.

Trapped ions are sensitive to electric eld noise, which couples to the ions motion

and causes heating. One common source of noise is Johnson noise from the electrodes.

In addition, sources of electric eld noise are observed that have yet to be fully understood. This noise is commonly referred to as anomalous [Turchette 00, Epstein 07,
132

Labaziewicz 08a, Labaziewicz 08b]. For Raman cooling to be ecient, the heating rate

in quanta per cooling-cycle duration must be much less than one. A typical heating rate

for the ion trap discussed in this dissertation is on the order of 0.25 quanta/ms for a

single 9Be+ ion with a trap frequency of ∼ 3 MHz. A typical Raman cooling cycle in the

experiment requires a few ten's of microseconds. Several of the experiments discussed

in this dissertation involve multiple ions where it is necessary to cool all axial modes. It

would be possible to have separate dedicated laser beams to cool all the dierent modes

in parallel, but this is not an ecient use of laser resources. Typically, a single set of

Raman beams are used, and the frequency is changed to address the dierent modes in a

serial fashion. One way to cool all the modes would be to cool them serially This has the

disadvantage that many Raman cooling cycles are required to reach the ground state.

If one mode is cooled completely before cooling another mode, then the cold mode may

heat up during the time it takes to cool the other modes due to photon recoil. A better

option is to interlace the Raman cooling cycles, applying one Raman cooling cycle to

mode 1 then one Raman cooling cycle to mode 2 and so on. This prevents one mode

from heating while the others are being cooled.

Another potential problem arises from the dependence of the Rabi frequency equation (6.23) on the motional state |ni, which in part comes about through the generalized

Laguerre polynomials L

|n

0−n|

n<



η

2



. For a a xed value of η and |n

0 − n| the Laguerre

polynomials will have a value equal or near to 0 for some value of nL0. This means that

Rabi frequency will go to zero for this particular nL0 state and no transition can be

driven. Thus, any motional population above nL0 will not be cooled to the ground state

but will tend to collect in the |nL0i state. This will not be a problem if nL0 is signi-

cantly larger the n ̄ reached after Doppler cooling. However the motional state follows

a thermal distribution after Doppler cooling and there can be a signicant amount of

motional state population occupying large n states.

Figure (6.5) shows the thermal distribution of motional states equation (6.63) as-
133

0 10 20 30 40 50

0.00

0.02

0.04

0.06

0.08

0.10

Fock state Èn>

Probablity

Motional state probablity versus Èn>: nbar = 15

Figure 6.5: Probability for nding the ion in Fock state |ni, assuming the ion's motional

population follows a thermal distribution equation (6.63)with n ̄ = 15.

suming a n ̄ = 15. Typical Doppler cooling temperatures for 9Be+ can be much lower

than this temperature. However, (in chapter (5) Doppler cooling with 24Mg+ is discussed) it is possible after 24Mg+ Doppler cooling to have an n ̄ ∼ 15. It should be noted

from the gure, that there is a signicant population for n > 15. For the Raman cooling

technique discussed earlier a particular initial |ni state is chosen and cooling proceeds

until all the motional population with n equal to or less than this value is brought to

the |n = 0i state. To capture 100% of motional population in theory it would require

starting at |n = ∞i. Obviously, this is not physically possible. Instead, some threshold

needs to be established regarding what percentage of the motional population is required

to be in the ground state. This will depend on the experiment, and what level of errors

can be tolerated.

Figure (6.6) shows the largest |ni state that would have be cooled to get 95%,

99%, and 99.99% for a given n ̄. This assumes that the highest |ni is the starting state

for Raman cooling and that the cooling continues from there, stepping down from that
134









































o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

0 5 10 15 20

0

50

100

150

200

nbar HThermal DistributionL

Motional state Èn

>

Highest Èn> value needed for Raman Cooling to get

X% of Population in Motional ground state

o 99.99%

 99%

95%

Figure 6.6: The largest motional state |ni that needs to be cooled in order to get X %

of the motional population in the ground state for a range of n ̄. The gure shows the

values for requiring 95%, 99%, and 99.99% of motional population to be cooled.

|ni all the way to the ground state. For example assuming n ̄ = 15, the highest |ni state

that would need to be cooled to capture 95%, 99%,or 99.99% of the motional population

would be |n = 46i, |n = 71i, or |n = 151i respectively.

Now that it is clear that it is necessary to start cooling from reasonably high values

of |ni, let's examine where the zeros in the Laguerre polynomials occur. Figures (6.7) and

(6.8) show the |ni values of the rst zeros in the Laguerre polynomials as function of the

Lamb-Dicke parameter for both rst (|n

0 − n| = 1) and second (|n0 − n| = 2) sidebands.

For small values of η, the value of n for this rst zero can be quite large. However, values

of η > 0.1 are common in the trapped ion experiments discussed here. The zeros start to

approach the regime, where they may cause problems for Raman cooling. For example

the Lamb-Dicke parameters for the two axial modes of a 9Be+ − 24Mg+ pair discussed

in the next section have the values η ∼ 0.28 and η ∼ 0.08 . This would give the zeros

for the rst sidebands at n ∼ 46 and n ∼ 156 respectively and for the second sidebands
135







                

0.0 0.2 0.4 0.6 0.8 1.0

0

500

1000

1500

2000

2500

3000

Lamb-Dicke Η

Motional state Èn

>

First Zero Crossing for Laguerre Polynomial

 2nd Sideband

1st Sideband

Figure 6.7: This gure shows the rst zeros in the Laguerre polynomials as function of

the Lamb-Dicke parameter η . The zeros are shown for both rst (|n

0 − n| = 1) and

second (|n

0 − n| = 2) sidebands. Note that the zeros of the second sidebands always

occur at a larger value of |ni than the rst sidebands.













          

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200

Lamb-Dicke Η

Motional state Èn

>

First Zero Crossing for Laguerre Polynomial

 2nd Sideband

1st Sideband Figure 6.8: This gure is the same as gure (6.7), but with a dierent vertical scale

at n ∼ 83 and n ∼ 1029. (- The answer to the joke: It saw the salad dressing. The rst

part of the joke is somewhere in this thesis -) It was mentioned above that to cool 99% of

the motional population to the ground state for n ̄ = 15 the cooling would have to start
136





























































0 10 20 30 40 50

0

10

20

30

40

50

Fock state Èn>

Ln

1

HΗ

2

L

Laguerre Polynomial versus Èn> for different Η: 1st Sideband

 Η=0.08 OOPH

Η=0.28 INPH

Figure 6.9: The value of the Laguerre Polynomials L

|n

0−n|

n



η

2



versus n for rst sideband

transitions, where |n

0 − n| = 1. The values of η = 0.28 and η = 0.08 are plotted.

with |n = 71i state. For a mode with a Lamb-Dicke parameter of η = 0.28 not all of

the motional population could be cooled using rst sideband transitions alone. For this

case, rst sideband cooling would leave behind ∼ 5% of the motional population. This

problem can be solved if second sideband cooling is used in addition to rst sideband

cooling, since the location of the zeros in the Laguerre polynomials is suciently high.

As an example gures (6.9) and (6.10) show the value of the Laguerre polynomials

for both the rst and second sidebands where ηINP H ∼ 0.28 and ηOOP H ∼ 0.08. Also

gures (6.11) and (6.12) show the scaled Rabi frequency Ωn0

,n/Ω from equation (6.23)

for these two cases. There are regions in these last two plots where the Rabi frequency

will be higher for second sideband transitions than for rst sideband transitions for the

same value of |ni. Thus it may be possible to nd mixture of rst and second sideband

cooling to optimize the amount of time it takes to Raman cool.
137































































0 10 20 30 40 50

0

200

400

600

800

1000

1200

Fock state Èn>

Ln

1

HΗ

2

L

Laguerre Polynomial versus Èn> for different Η: 2nd Sideband

 Η=0.08 OOPH

Η=0.28 INPH

Figure 6.10: The value of the Laguerre Polynomials L

|n

0−n|

n



η

2



versus n for a sideband

rst stimulated-Raman transition, where |n

0 − n| = 1. The values of η = 0.28 and

η = 0.08 are plotted.

0 20 40 60 80 100

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fock state Èn>

Wn',nW

Rabi Frequency for Different Sidebands versus Èn>: Η= 0.28

2

nd Sideband

1

st Sideband

Figure 6.11: The scaled Rabi frequencies Ωn0

,n/Ω from equation 6.23 versus |ni for the

rst and second sidebands where η =0.28
138

0 200 400 600 800 1000

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fock state Èn>

Wn',nW

Rabi Frequency for Different Sidebands versus Èn>: Η= 0.08

2

nd Sideband

1

st Sideband

Figure 6.12: The scaled Rabi frequencies Ωn0

,n/Ω from equation 6.23 versus |ni for the

rst and second sidebands where η =0.08
139

6.7 Sideband Cooling Experiments

One of the approaches in ion trap quantum computing includes having the ability to move ions around an ion trap-array and to sympathetically recool them to the

ground state of motion. This section describes an experiment demonstrating this capability. In addition this experiment served as a calibration experiment for a the entangled

mechanical oscillators experiment[Jost 09] discussed in chapter (8).

The goal in the experiments described here is to take two 9Be+ions that start

in the same trapping region, separate them transfer them to dierent locations in an

ion trap array, and show that their axial modes can be sympathetically recooled to

ground state of motion after the transport and separation (separation tends to heat the

ions). The recooling is performed with another species of ion 24Mg+ that accompany

the two 9Be+ions, which allows any quantum information stored in the 9Be+ions to be

maintained. Only the details relevant to discussing the sideband cooling of the two pairs

of 9Be+ − 24Mg+ ions will be elaborated on here, and the other details are discussed

elsewhere in this dissertation. Figure (6.15) shows the basic outline of the experiments.

Several dierent experiments are required to demonstrate the sympathetic cooling of the

axial modes of the two 9Be+ −24Mg+ pairs. Experiments will have to be done where MA

and MS sideband transitions are performed on the 9Be+ ions in both trapping locations,

which allow the temperature after sympathetic recooling with 24Mg+ to be determined.

Due to technical limitations, the experiments for measuring the temperature in traps A

and B are performed separately.

The experiment involves four trapped ions. These are loaded serially into the trap.

The order of the ions is important, since it determines the normal mode frequencies of

the ions [Jost 09]. Independent of the order they start in the ions are ordered [Jost 09]

into the spatial sequence 9Be+ − 24Mg+ − 24Mg+ −9 Be+ as shown in step 1. This is

accomplished by applying a series of control potentials [Jost 09] combined with Doppler
140

~ 9.2 GHz

~ 334 MHz

~ 110 MHz

~ 280 nm

B-Field ~ 120 Gauss

R2

R1

Repump

Figure 6.13: This gure depicts the relevant energy levels for Raman cooling with 24Mg+.

The two beams used to make a MS sideband transition are labeled R1 and R2. The beam

labeled repump is used for optical pumping, and the wavy line represents spontaneous

emission. Superimposed on to the ground state internal levels |ms =

1

2

i and |m = −

1

2

i

are the harmonic oscillator levels for the INPH and OOPH modes (see text) for a pair

of 9Be+ − 24Mg+ ions.

cooling on both the 9Be+ and 24Mg+ ions. In step 2 the four ions are Doppler cooled

using a combination of both 24Mg+ and 9Be+ Doppler cooling. Combining these two

types of cooling helps ensure all modes are well-cooled. For example, if the 9Be+ ions

have a small motional amplitude on a mode they will not be eciently cooled by 9Be+

Doppler cooling, but the 24Mg+ ions will have large amplitude and eciently cooled.

Next, the four axial modes are Raman cooled to very near the ground state by driving

MS sidebands on the 9Be+ ions. (Subsequent experiments [Home 09, Hanneke 10] have

performed this operation with only 24Mg+ cooling for both the Doppler component and

the for the MS sidebands. )

The ions are then transported from the position near trap A in the gure to

the separation region in front of electrode S.E.. This is accomplished by changing the
141

Trap A S.E. Trap B

B-Field

Beams:

R1 and Repump

Beams:

R1 and Repump

Beam:

R2

Beam:

R2

Figure 6.14: Laser Beams used for Raman cooling of 24Mg+ ions in traps A and B. The

beams R1 and R2 comprise the two beams used for MS sideband transitions on 24Mg+.

The repump beam is used for optical pumping. The two trapping locations A and B are

located ∼ 240 μm apart. The boxes represent the some of the electrodes, where S.E.

stands for separation electrode.

control voltages on the electrodes [Rowe 02, Barrett 04, Reichle 06a]. The four ions can

be separated into two pairs by changing the control voltages on the electrodes such that

a potential wedge is inserted between the four ions to form two separate potential wells.

The ion pairs are then moved to trap A and B, which are separated by about 240 μm.

This is represented in the gure as step 3. The separation process typically heats the

ions [Rowe 02, Barrett 04].

The two separate potential wells were designed to have approximately the same

trap frequency. There are two normal modes along the axial direction: the in-phase

(INPH) and the out-of-phase (OOPH). They have a frequency of ∼ 2.3 MHz [0.37, 0.93]

and ∼ 4.9 MHz [-0.93, 0.37] respectively. The terms in the brackets are the normal

mode amplitudes for the 9Be+ − 24Mg+ions respectively. In the experiment, the two

trapping locations have slightly dierent axial frequencies. The INPH modes dier in

frequency by ∼ 8 KHz, and the OOPH modes dier in frequency by ∼ 30 KHz. The
142

Lamb-Dicke parameters for 24Mg+ for the two modes of motion in each trapping location

are ηINP H = 0.28 and ηOOP H = 0.08.

Sympathetic cooling is performed to recool the ions in both locations, which is

shown in step 5. The ions in both traps are simultaneously Doppler cooled for ∼ 400μs

using the 24Mg+ ions, which is shown in step 4 . The Doppler cooling used is discussed in

chapter (5). The stimulated Raman sideband transitions on 24Mg+ are performed on the

|ms =

1

2

i → |ms = −

1

2

i transition see gure (6.13) of the 2S1

2

ground state. A magnetic

eld of ∼ 120 Gauss is applied to the ions, which causes the two ground states to be

split by ∼ 334 MHz. There is some uncertainty of the nal temperature reached after

Doppler cooling with 24Mg+. It is possible that the temperature will reach a level that

rst sideband cooling is not sucient to cool all the motional population. Thus second

sideband cooling is initially performed. 40 Raman cooling cycles of second sideband

cooling were applied to the 24Mg+ ions in both trapping locations simultaneously. For

the rst sideband cooling, both traps are also simultaneously Raman cooled. However,

the frequency of the MS sideband transitions are chosen to be resonant with the motional

mode frequencies of the dierent traps. This means that four dierent Raman cooling

cycles are applied to all the ions with the parameters optimized for the INPH mode in

trap A, the INPH mode in trap B, the OOPH mode in trap A, and the OOPH mode in

trap B. In total 30 interlaced Raman cooling cycles of rst sideband cooling with 24Mg+

are applied for each of the four modes. Both types of Raman cooling are performed in

step 5.

At this point, both normal modes in both traps have been cooled to near the

ground state. Due to technical limitations closed cycling transition in 24Mg+ between

the 2S1

2

→2 P3

2

levels is not used, the 24Mg+ uorescence cannot be probed directly.

To diagnose the temperature, the ratio of driving MA and MS transitions on the 9Be+

ions in each trapping location is measured, see section (6.5). To measure the temperature in trap A of the INPH and OOPH modes, both a MA and MS Raman side-
143

band transitions are driven (These two dierent probe pulses are performed in separate experiments on each mode) on the |F = 2, mF = 2i → |F = 2, mF = 1i transitions

in 9Be+(probe pulse). Any population in the |F = 2, mF = 1i level is transferred to

the |F = 2, mF = −2i state prior to performing resonance uorescence detection on the

9Be+ion, using co-propagating Raman π-pulses. This is included in step 6a of gure

(6.15) . The uorescence is then plotted versus the frequency of the sideband pulse for

both MA and MS transitions see gures (6.16), (6.17), (6.20), and (6.21), where each

scan is a separate experiment (see the gures for the number of experiments performed).

The ions are recombined in trap A after the detection and the experiments are repeated

(step 7a). From the ratio of the two sidebands, n ̄ for each mode is estimated to be

n ̄INP H = 0.095 ± .022 and n ̄OOP H = 0.063 ± 0.018, where the errors are one standard

deviation.

Measurement of the temperature of the 9Be+ − 24Mg+ ions in trapping region B,

requires a slightly dierent procedure than above. This is due to the technical limitation

of not being able to measure resonance uorescence of the 9Be+ ions in that location.

The probe pulse is performed while the ion pair is still in trap B. To do the measurement,

the ions have to be transferred back to trapping region A. The state of the 9Be+ ion

in trapping region A does not matter. The internal state of the 9Be+ in trap A is

optically pumped back to the bright state |F = 2, mF = 2i. Both of these processes

are represented by step 6b. In step 7b the ions are recombined into trapping region A.

Now resonance uorescence detection is performed on both ions 9Be+. The ion that was

in trapping region A only contributes extra background photons to the data. Figures

(6.18), (6.19), (6.22), and (6.23) show the result of scanning the frequency of the probe

pulse in trapping region B over the MA and MS transition resonance on both modes,

where each data point is a separate experiment repeated many times (see the gures

for the number of experiments performed). By tting the depths of the sidebands, the

temperature is estimated to be n ̄INP H = 0.03 ± 0.049 and n ̄OOP H = 0.016 ± 0.024,
144

where the errors are one standard deviation.

The average n ̄ for all the modes is very near the ground state. The nite temperature could result from a number of imperfections. The beam that performs the

repumping has a σ

− polarization component (not depicted in gure (6.13)) that is o

resonant, but could optically pump the internal state of the 24Mg+ ions. This would

mean excess scattering events could occur that could heat the ions. Also if the optical power of the σ

− repumping beam is above saturation then this will increase the

scattering rate of the o resonant σ

+component. The INPH mode is more sensitive to

heating from ambient electric eld noise than the OOPH [Kielpinski 00]. As a result,

if the Raman cooling process takes longer than the time to gain one motional quantum

from heating for this mode, then it will be dicult to cool to the ground state. An

additional problem can arise from the separation process. During the separation the

trap frequency decreases; this makes the ions more likely to acquire motional energy

from ambient electric eld noise. To separate the ions, a potential bump is brought up

between the ions to create a double well potential. If the location of this potential bump

is not located exactly between the ions, then one or more of the ions can eectively ride

up this potential hill before eventually being dumped into one of the wells. This can

leave the ions with a large amount of motional energy, which can be dicult to remove.
145

Trap A S.E. Trap B

1) Order the ions while Doppler 

cooling with 9Be+ and 24Mg+ 9B

2) Doppler Cool with 9Be+ and 

24Mg+, Raman Cool with 9Be+

3) Move and separate the ions

4) Doppler Cool with 24Mg+

5) Raman Cool with 24Mg+

6a) Probe and detect the 

temperature of 9Be+ in trap A

7a) Recombine the ions and 

repeat steps 1-5, 6a, and 7a

6b) Probe temperature on 9Be+

in trap B. Repump 9Be+ in trap A

7a) Recombine then detect Both 

the 9Be+ ions, repeat 1-5, 6b, 

and 7b

Measuring the temperature in:

Trap A: Steps 1-5, 6a, 7a

Trap B: Steps 1-5, 6b, 7b

Figure 6.15: Experimental outline for performing and detecting sympathetic cooling on

two pairs of trapped 9Be+ − 24Mg+ ions. (More details are provided in the text.) This

gure shows the steps for two dierent experiments. Steps 1-5, 6a, and 7a describes

the procedure for sympathetic cooling of a pair of 9Be+ − 24Mg+ ions in trap A and

measuring the temperature of the two axial modes. Steps 1-5, 6b, and 7b describe the

procedure for doing the same thing but with the pair of ions in trap B. The boxes

represent one bank of electrodes in the ion trap. There are two main trapping locations

separated by ∼ 240 μm, labeled trap A and B. There is a separation electrode (S.E.)

used for separating the four ions into two pairs.
146

225.04 225.05 225.06 225.07 225.08 225.09 225.1 225.11 225.12 225.13

0

2

4

6

8

10

Frequency of scanning AOM (MHZ)

Photon counts

INPH mode of 9Be+ − 

24Mg+

 in trap A: 

Motion Adding transition on 9Be+

Figure 6.16: Frequency scan over the resonance of the MA transition on the INPH mode

of the 9Be+ − 24Mg+ pair of ions in trap A. The horizontal axis is the frequency of an

acoustic-optic modulator that is used to scan the frequency of the laser beam. The real

frequency scale is is twice the value shown on the axis. The vertical axis shows the

number of photons collected, during a single detection period of 200 μs. The line is the

t of a sinc function to the experimental data. Each data point is the average of 70

experiments.
147

227.3 227.31 227.32 227.33 227.34 227.35 227.36 227.37 227.38 227.39

0

2

4

6

8

10

12

Frequency of scanning AOM (MHZ)

Photon counts

INPH mode of 9Be+ − 

24Mg+

 in trap A: 

Motion Subtracting transition on 9Be+

Figure 6.17: Frequency scan over the resonance of the MS transition on the INPH mode

of the 9Be+ − 24Mg+ pair of ions in trap A. The lack of strong resonance implies that

the temperature is near the motional ground state. The horizontal axis is the frequency

of an acoustic-optic modulator that is used to scan the frequency of the laser beam. The

real frequency scale is is twice the value shown on the axis. The vertical axis shows the

number of photons collected, during a single detection period of 200 μs. The line is the

t of a sinc function to the experimental data. Each data point is the average of 70

experiments, and two scans were performed.
148

225.06 225.07 225.08 225.09 225.1 225.11 225.12 225.13

0

2

4

6

8

10

12

14

16

18

20

Frequency of scanning AOM (MHZ)

Photon counts

INPH mode of 9Be+ − 

24Mg+

 in trap B: 

Motion Adding transition on 9Be+

Figure 6.18: Frequency scan over the resonance of the MA transition on the INPH mode

of the 9Be+ − 24Mg+ pair of ions in trap B. The vertical axis shows the number of

photons collected, during a single detection period of 200 μs. There is a ∼ 10 count

background on the signal from the presence of another 9Be+ ion, see the text for more

detail. The horizontal axis is the frequency of an acoustic-optic modulator that is used

to scan the frequency of the laser beam. The real frequency scale is is twice the value

shown on the axis. The line is the t of a sinc function to the experimental data. Each

data point is the average of 210 experiments.
149

227.1 227.15 227.2 227.25 227.3 227.35 227.4 227.45 227.5 227.55

0

2

4

6

8

10

12

14

16

18

20

INPH mode of 9Be+ − 

24Mg+

 in trap B: 

Motion Subtracting transition on 9Be+

Frequency of scanning AOM (MHZ)

Photon counts

Figure 6.19: Frequency scan over the resonance of the MS transition on the INPH mode

of the 9Be+ − 24Mg+ pair of ions in trap B. The lack of an apparent strong resonance

implies that n ̄INP H is near the ground state. The vertical axis shows the number of

photons collected. There is a ∼ 10 count background on the signal from the presence of

another 9Be+ ion, see the text for more detail. The horizontal axis is the frequency of

an acoustic-optic modulator that is used to scan the frequency of the laser beam. The

real frequency scale is is twice the value shown on the axis. The line is the t of a sinc

function to the experimental data. Each data point is the average of 70 experiments,

and two scans were performed.
150

223.72 223.74 223.76 223.78 223.8 223.82 223.84 223.86

0

2

4

6

8

10

Frequency of scanning AOM (MHZ)

Photon counts

OOPH mode of 9Be+ − 

24Mg+

 in trap A: 

Motion Adding transition on 9Be+

Figure 6.20: Frequency scan over the resonance of the MA transition on the OOPH

mode of the 9Be+ −24Mg+ pair of ions in trap A. The horizontal axis is the frequency of

an acoustic optic modulator that is used to scan the frequency of the laser beam. The

real frequency scale is is twice the value shown on the axis. The vertical axis shows the

number of photons collected, during a single detection period of 200 μs. The line is the

t of a sinc function to the experimental data. Each data point is the average of 100

experiments.
151

228.58 228.6 228.62 228.64 228.66 228.68 228.7 228.72

0

2

4

6

8

10

12

Frequency of scanning AOM (MHz)

Photon counts

OOPH mode of 9Be+ − 

24Mg+

 in trap A: 

Motion Subtracting transition on 9Be+

Figure 6.21: Frequency scan over the resonance of the MS transition on the OOPH mode

of the 9Be+ − 24Mg+ pair of ions in trap A. The lack of strong resonance implies that

n ̄OOP H is near the ground state is near the motional ground state. The horizontal axis

is the frequency of an acoustic-optic modulator that is used to scan the frequency of

the laser beam. The real frequency scale is is twice the value shown on the axis. The

vertical axis shows the number of photons collected, during a single detection period of

200 μs. The line is the t of a sinc function to the experimental data. Each data point

is the average of 70 experiments, and three scans were preformed.
152

225.06 225.07 225.08 225.09 225.1 225.11 225.12 225.13

0

2

4

6

8

10

12

14

16

18

20

Frequency of scanning AOM (MHZ)

Photon counts

INPH mode of 9Be+ − 

24Mg+

 in trap B: 

Motion Adding transition on 9Be+

Figure 6.22: Frequency scan over the resonance of the MA transition on the OOPH

mode of the 9Be+ − 24Mg+ pair of ions in trap B. The vertical axis shows the number

of photons collected, during a single detection period of 200 μs. There is a ∼ 10 count

background on the signal from the presence of another 9Be+ ion, see the text for more

detail. The horizontal axis is the frequency of an acoustic-optic modulator that is used

to scan the frequency of the laser beam. The real frequency scale is is twice the value

shown on the axis. The line is the t of a sinc function to the experimental data. Each

data point is the average of 100 experiments.
153

228.55 228.6 228.65 228.7

0

2

4

6

8

10

12

14

16

18

20

Frequency of scanning AOM (MHz)

Photon counts

OOPH mode of 9Be+ − 

24Mg+

 in trap B: 

Motion Subtracting transition on 9Be+

Figure 6.23: Frequency scan over the resonance of the MS transition on the OOPH

mode of the 9Be+ − 24Mg+ pair of ions in trap B. The lack of strong resonance implies

that n ̄OOP H is near the ground state. The vertical axis shows the number of photons

collected, during a single detection period of 200 μs. There is a ∼ 10 count background

on the signal from the presence of another 9Be+ ion, see the text for more detail. The

horizontal axis is the frequency of an acoustic-optic modulator that is used to scan the

frequency of the laser beam. The real frequency scale is twice the value shown on the

axis. The line is the t of a sinc function to the experimental data. Each data point is

the average of 70 experiments, and three scans were performed.
Chapter 7

Phase Gate

Two qubit gates are one of the essential components of QIP, since they are necessary along with single qubit rotations to make a universal gate set [Nielsen 00]. One

of the early proposals for two qubit gates involved ions in a chain coupled via their

Coulomb interaction [Cirac 95]. This proposal was followed quickly by the rst demonstration of a two qubit gate using the internal state of a single ion and its quantized

motion as the two qubits [Monroe 95a]. Although the rst demonstration involved entanglement between the internal and external degrees of freedom, there have been many

proposals to perform gates that entangle the internal states of multiple ions [Mølmer 99,

Wang 01, Solano 99, Milburn 01, Cirac 95]. This entanglement is always mediated by

the Coulomb interaction between the ions, since it is the strongest coupling. Several

experimental demonstrations have also been implemented [Sackett 00, Leibfried 03b,

Leibfried 04, Leibfried 05, Haljan 05, Häner 05, Kirchmair 09, Roos 04]. These experiments used dierent types of ion species, but the demonstrations of entanglement were

always between one type of ion. There also has been an experimental demonstration of

the entanglement of the internals states of two dierent isotopes of Calcium (40Ca+and

43Ca+)[Home ]. One of the goals of ion trap QIP is to incorporate sympathetic cooling

ions with the logical qubits [Kielpinski 00, Barrett 03, Rohde 01, Blinov 02]. For large

scale QIP that incorporates sympathetic cooling ions, it will most likely be necessary to
155

perform entangling operations in the presence of these cooling ions. Prior to reference

[Jost 09], which is the focus of this dissertation; this had not been demonstrated before. There was followed by experiments, which have also performed such entanglement

[Home 09, Hanneke 10]. The experimental implementation, which discussed in section

(7.6), is a tune-up experiment for the experimental work discussed in chapter (8) and

reference [Jost 09].

The rest of this chapter provides a more detailed description of the geometrical

phase gate [Leibfried 03b, Home 06b] used in reference [Jost 09]. The rst section provides a general overview of entanglement and the phase gate. Discussed second is how a

stimulated-Raman transition can be used to create a state dependent optical dipole force.

A particular example is used to show the resulting force on two ions. Next properties

of coherent states and the forced harmonic oscillator will be discussed. This is relevant,

since the optical dipole force when acting on a Fock state behaves like a forced harmonic

oscillator, creating a coherent state. Next, the amplitude of the coherent state and the

phase, arising from the geometrical phase gate, will be derived. This followed by a step

through of the implementation of a ideal geometric phase gate. The resulting quantum

mechanical states at each point, and the analysis of the entanglement are highlighted.

The last section goes through the experimental details for implementing the geometrical

phase gate on two 9Be+ ions in the four-ion chain 9Be+ − 24Mg+ − 24Mg+ −9 Be+.

7.1 Entanglement Overview

An entangled state is a quantum mechanical state of two or more systems that

can not be written as a product state [Nielsen 00]. Let us consider two spin 1

2

systems

labeled A and B, where the possible states of each system are ΨA = |↑Ai , ΨA = |↓Ai,

ΨB = |↑Bi, and ΨB = |↓Bi. It is possible to place each system in a quantum mechanical

superposition Ψ = |↑i+|↓i (The normalization constant is omitted.). The product state
156

of both systems can be written

ΨA,B = ΨA ⊗ ΨB =

1

2



|↑↑iA,B + |↑↓iA,B + |↓↑iA,B + |↓↓iA,B

. (7.1)

With product states the wave function can always be written as the outer product of

the individual wave functions for each system. However, if a phase is introduce between

some of the elements then the overall state is an entangled state, for example

ΨA,B =

1

2



|↑↑iA,B + i|↑↓iA,B + i|↓↑iA,B + |↓↓iA,B. (7.2)

This state cannot be written as the outer product of two individual wave functions. The

systems can longer be thought of as separate and distinct.

Product states of the internal states of trapped ions are signicantly easier to

generate in the lab than entangled states. The product states are generated with an

appropriately tuned co-carrier Raman pulse, see chapter (6). To create an entangled

state of two trapped ions of the form in equation (7.2), a state dependent phase factor

needs to be created. This is done in the experiments discussed in this dissertation, using

a geometric phase gate [Leibfried 03b]. This gate uses a state dependent optical dipole

force to excite a coherent state of motion of the ions in the selected states. The force

is adjusted such that it drives the states around a loop in phase space see gure (7.1).

This gure shows a possible path in phase space for the states of a product state when

a geometric phase gate has been applied. The states where motion has been excited

acquire a geometric phase that is proportional to area enclosed by the loop in phase

space. In this case the states |↑↓iA,B and |↓↑iA,B, acquire a phase e

i

π

2 , where the states

|↑↑iA,B and |↓↓iA,B acquire no phase. This creates the state (7.3).

7.2 Optical Dipole Force

The optical dipole force is a conservative force that is uniform in space. As a result

the potential giving rise to such a force depends on the position, so the Hamiltonian for
157

Figure 7.1: This shows possible paths in phase space for the states of a product state

when a geometric phase gate has been applied. The states where motion has been

excited acquire a geometric phase that is proportional to area enclosed by the loop in

phase space. In this case the states |↑↓iA,B and |↓↑iA,B acquire a phase e

i

π

2 , where the

states |↑↑iA,B and |↓↓iA,B acquire no phase (from reference [Leibfried 03b]).

a dipole like force in the linear approximation, takes the form

Hdp = −zf (t). (7.3)

The position is represented by the coordinate z, since only one dimension will be considered here. The force can have a time dependence or be a constant and is represented

by f (t). The optical dipole force considered here will have a time dependence. This

force arises from the ions interaction with the laser beams used for stimulated Raman

transitions.
158

From section (6.4) equation (6.60) the Hamiltonian for a stimulated Raman transition is

HI = ~Ωe

−iδtei(∆k·z+∆φ)

|1i h2| + H.C.. (7.4)

It is assumed that the system is in the Lamb-Dicke regime. This allows the Hamiltonian

to be simplied to

HI = ~Ωe

−i(ωt−φ)

(1 + iηzˆ)|1i h2| + H.C.. (7.5)

To simplify the notation a few replacements have been made. The dierence phase ∆φ

between the two laser beams has been replaced with φ. The term ∆k·z has been rewritten

as ηzˆ, where η = ∆k zo and zˆ =



a + a

†



. The dierence frequency of the Raman beams

δ in section (6.4) has been rewritten as ω to emphasis that a dierent type of transition

is driven. For the Raman interaction considered in this section ω = ωz + δ, where ωz is

the axial frequency of one of the normal modes and δ is a small frequency detuning. In

section (6.4) on stimulated Raman transitions, the transitions always involved driving a

spin-ip transition. Here, no spin ip transitions are driven. The frequency dierence

between the two laser beams will be close to one of the motional frequencies (see gure

(7.2)). This means the matrix elements |1i h2| in equation (7.5) need to be replaced with

|1i h2| → |1i h1| , |2i h2| . Making these substitutions and rewriting equation (7.5), gives

HI = ~Ωe

−i(ωt−φ)

(1 + iηzˆ) [|1i h1| + |2i h2|] + H.C.. (7.6)

Also to make the notation a little less confusing since multiple ions will be considered, the following replacements will be made |1i → |↓i and |2i → |↑i. Making these

changes and collecting all the terms together, allows equation (7.6) to be rewritten as

Hg = ~Ω cos (ωt − φ) (|↓i h↓| + |↑i h↑|) − ~Ωηzˆsin (ωt − φ) (|↓i h↓| + |↑i h↑|). (7.7)

The rst two terms, which do not depend on the position operator zˆ, behave as a Stark

shift that oscillates in time. The shift can be canceled out by choosing the time the of

interaction appropriately. Another possibility is to embed the gate interaction in the two
159

arms of a spin-echo sequence, which cancels the shift (see section 7.5). It will be ignored

in the rest of the derivation. The last two terms show a position dependent force on

both spin states. In general either the Rabi frequency or the phase will be dierent for

the dierent spin states. The above Hamiltonian is for one ion, considering the general

case for two ions gives

Hg =

X

j=1,2

X

mj=↑,↓

−~Ω

mj ηjzˆj sin 

ωt − φmj



|mj i hmj | . (7.8)

This Hamiltonian can be used to understand the force on multiple ions for dierent

parameters and laser beam setups. The next section will go through a simple example.

A more complicated example will discussed in the experimental section.

7.2.1 Example

One particular scenario is depicted in gure (7.2). This case is also discussed nicely

in the dissertation by Jonathan Home [Home 06b]. In this case the two ground states

are coupled via a two level excited state. The two Raman laser beams have a detuning

∆ from the excited state, where ∆ is much larger than the splittings between the ground

state levels or the excited state levels. The two ground state levels are labeled by |↑i and

|↓i. The |↓i state is coupled to the excited state only with σ

+ polarized light, and the |↑i

state is coupled with σ

− light. It has been assumed the Rabi frequency Ω is the same for

all the dierent transitions and for both ions. The interference between two laser beams

with orthogonal polarizations gives rise to the state dependent force. The interference

creates a walking standing wave of the polarization. This is similar to the lin ⊥ lin

conguration discussed in reference [Metcalf 99]. In this case the resulting electric eld is

E (z, t) = Eo [ˆx cos (ωlt − k · z) + ˆy cos (ωlt + ωt − k · z)]. Where Eo is the electric eld

amplitude of both beams and ωlis the optical frequency of the laser beams. If ω = 0,

then a standing wave is created, where the polarization cycles from linear to circular

to orthogonal linear to orthogonal circular over a distance of λef f /2. The eective
160

{ {

{ A)

B)

C)

Trap axis

Horizontal polarization Vertical polarization

Figure 7.2: A) The level diagram for the phase gate considered in section (7.2.1). It is

a four level system, where the coupling laser beams are detuned by ∆ from the excited

state, which is much larger than any of the other level splittings. Each of the ground

states are coupled to one of the excited states with a stimulated-Raman transition. The

dierence frequency between the two beams that drive each transitions is ωz + δ. The

lower heavier dashed lines represent the one of the motional states of the pair of trapped

ions. In this case it will represent the stretch mode of motion. B) This gure shows the

polarization the laser beams used. C) The resulting polarization interference pattern

from the two lasers beams used for the gate drive. They set up what is called a walking

standing wave of the polarization.

wavelength λef f is related to the actual wavelength of the laser by λef f = √

λ

2

. The √

1

2

factor comes from the projection of the laser's k-vector on to the trap axis, which is

oriented at 45 degrees with respect to the trap axis. When the two lasers have a nite

detuning ω 6= 0 then polarization gradient changes at a rate ωt. This is what gives rise
161

to the walking part of the walking standing wave.

The two ions are coupled via the Coulomb interaction and as such behave as two

masses connected by a spring conned in an external harmonic well. Thus the position

of ions can instead be written in terms of the their normal mode coordinates

z1 = zc + zs (7.9)

z2 = zc − zs. (7.10)

The normal mode coordinates zc and zs represent the center-of-mass and the stretch

mode respectively. Substituting the normal mode coordinates into equation (7.8) and

collecting like terms in the position coordinate gives

Hg = −~Ω[ηczˆc (sin (ωt − φ↑1)|↑1i h↑1| + sin (ωt − φ↓1)|↓1i h↓1|) (7.11)

−ηszˆs (sin (ωt − φ↑1)|↑1i h↑1| + sin (ωt − φ↓1)|↓1i h↓1|) (7.12)

+ηczˆc (sin (ωt − φ↑2)|↑2i h↑2| + sin (ωt − φ↓2)|↓2i h↓2|) (7.13)

+ηszˆs (sin (ωt − φ↑2)|↑2i h↑2| + sin (ωt − φ↓2)|↓2i h↓2|)]. (7.14)

The normal mode coordinates equation can be rewritten as zc → zczˆc and zs → zszˆs.

The terms with theˆrepresent the position operator a + a

†



for that particular mode,

and the other component is the magnitude. The form of the Hamiltonian can be made

more intuitive rewriting it in terms of the basis states for two spin 1

2

systems (two qubits):

|↑↑i, |↓↓i, |↑↓i, and |↓↑i. Note that when two spins are represented in the same ket it is

assumed the rst ion corresponds to the spin state of ion 1 and the second spin is for ion

2. This helps reduce the number of subscripts, but they will be added if it helps with

the clarity of a particular expression. The Hamiltonian in this basis can be determined

from the above Hamiltonian by calculating all the matrix elements such as h↑↑| Hg |↑↑i.
162

The Hamiltonian in this form is

Hg = −2~Ω[ηczˆc sin (ωt)|↑↑i h↑↑| (7.15)

+ηczˆc sin (ωt − ∆φ)|↓↓i h↓↓| (7.16)

+ηczˆc cos 

∆φ

2



sin (ωt − ∆φ) [|↑↓i h↑↓| + |↓↑i h↓↑|] (7.17)

+ηszˆs sin 

∆φ

2



cos (ωt − ∆φ) [|↑↓i h↑↓| − |↓↑i h↓↑|] . (7.18)

The phase dierence between the spin states has been dened as ∆φ≡ φ↑ − φ↓, and it is

adjusted to be the same for both ions. The rst three lines represent a time dependent

optical dipole force on the center-of-mass mode of motion. The last line represents a

time dependent force on the stretch mode of motion. Note the sign dierence between

the matrix elements |↑↓i h↑↓| and − |↓↑i h↓↑|. This implies that the sign of the force

is dierent for the two dierent states. The frequency of the force will be chosen such

that it is near the resonance with the stretch mode. This means all the terms in the

Hamiltonian involving the center-of-mass mode of motion to a good approximation, be

neglected, since they will be o resonance. The Hamiltonian can now be written as

Hg = −2~Ωηszˆs sin 

∆φ

2



cos (ωt − ∆φ) [|↑↓i h↑↓| − |↓↑i h↓↑|] . (7.19)

Transforming the position operator zˆs into the interaction picture gives

zˆs = ase

−iωst + a†

s

e

−iωst

, (7.20)

where as and a

†

s are the annihilation and the creation operators for the stretch mode of

motion. The frequency of the this mode is ωs. Taking equation (7.20) and inserting into

equation (7.19) gives a version of the Hamiltonian that will be convenient for seeing the

connection to coherent states discussed in a later section. A rotating-wave approximation

is made to simplify the Hamiltonian, where terms that oscillate a ω + ωz are dropped.

Hg = −~ηsΩ sin 

∆φ

2



ase

i(ωt−

∆φ

2

)

e

−iωst

[|↑↓i h↑↓| − |↓↑i h↓↑|] + H.c.. (7.21)
163

7.3 Coherent States and the Forced Harmonic Oscillator

In this section some basic properties of coherent states [Howard 87, Walls 94] and

the quantum forced harmonic oscillator [Carruthers 65, Merzbacher 98] will be described.

This is relevant because the oscillations of a pair of trapped ions are that of a harmonic

oscillator. Also applying the interaction given by equation (7.21) to the stretch mode of

a pair of ions, is equivalent to driving a harmonic oscillator with a time dependent force.

In the rst section a few of the relevant properties of coherent states will be mentioned.

Then in the next section, the results of a forced harmonic will be given.

7.3.1 Coherent States

Typically when talking about the motion of trapped ions for quantum information processing, the Fock states (number basis) are used, since they form a complete

orthonormal set of states for the harmonic oscillator. Also some quantum algorithms

[Cirac 95] require the ability to put in a single quanta of energy into one of the motional

modes, changing the Fock state by one, in order to perform a two qubit gate. The

geometric phase gate [Leibfried 03b] can be used on any motional state as long as the

ions are in the Lamb-Dicke regime. In the geometric phase gate experiments discussed

in this dissertation, the ions' motion is cooled to the ground state and then an optical

dipole force is applied. This has the eect of creating a coherent state of motion. A

coherent state |αi can be expanded in the Fock state basis [Walls 94]

|αi = e

|α|

2

2

X∞

n=0

α

n

n!

1

2

|ni. (7.22)

A coherent state can be generated by applying a displacement operator to |n = 0i Fock

state. The form of the displacement operator is [Walls 94]

D (α) = e

αa†−α

?a?

. (7.23)
164

Applying this operator to the ground state, results in the creation of the coherent state

|αi = D (α)|0i. (7.24)

7.3.2 Forced Harmonic Oscillator

As mention earlier the Hamiltonian equation (7.21) derived in section (7.2) is

the same Hamiltonian that gives rise to a forced harmonic oscillator (FHO). This section describes a few of the FHO's relevant properties. The text book by Merzbacher

[Merzbacher 98] has a good discussion of the subject, and this section follows that discussion closely. The Hamiltonian for a FHO is

Hfho =

p

2

2m

+

1

2

mωz2 − zf (t). (7.25)

Using the annihilation and creation operator form of the position and momentum operators and going to an interaction picture where Ho = ~ω(a

†a +

1

2

) the interaction

Hamiltonian is

HI

fho (t) = −f (t) ae−iωt − f

?

(t) a

†

e

iωt

. (7.26)

To understand the time development of this system, the Schrodinger equation for the

time-evolution operator will be used [Merzbacher 98]

i~

∂T(t, t1)

∂t = HI

fho (t) T (t, t1), (7.27)

where the solution for time-evolution operator is

T(t, t1) = −

i

~

ˆ

t

t1

dt0HI

fho (t) T (t, t1). (7.28)

The solution depends on the structure of the Hamiltonian HI

fho (t). It will be seen later

that for the case considered here the Hamiltonian at dierent times does not commute

with itself. This leads to following solution [Merzbacher 98] for T(t, t1).

T (t, t1) = e

h

−i

~

 ́ t

t1

dt0



f(t

0

)ae−iωt0

+f

?(t0

)a

†eiωt0

i

(7.29)

×e

−1

2~2

 ́ t

t1

dt0

 ́ t

t1

dt00[HI

fho(t

0

),HI

fho(t

00)]

. (7.30)
165

The rst term has the form of the displacement operator, which can be seen if the

following substitutions are made

α (t) = −i

~

ˆ t

t1

dt0f



t

0



e

−iωt0

(7.31)

and

α

?

(t) = i

~

ˆ t

t1

dt0f

?



t

0



e

iωt0

. (7.32)

The argument of the second term of equation (7.30) is the phase acquired during the application of the interaction, which will be called Φ(t). The commutator h

HI

fho (t

0

), HI

fho (t

00)

i

can be evaluated using equation (7.26) and the commutation relations for the a and a

†

[Sakurai 94]. Working through the algebra gives for the phase

Φ (t) = −1

2~

2

ˆ t

t1

dt0

ˆ t

t1

dt00 f



t

0



f

?



t

00

e

iω(t

00−t0

) − f?



t

0



f



t

00

e

−iω(t

00−t0

)



. (7.33)

Now the time evolution operator equation (7.30) can be rewritten in the form

T (t, t1) = e

(α(t)a

†−α?(t)a?

)

e

Φ(t)

. (7.34)

From this it can been seen what happens to harmonic oscillator subject to time dependent dipole force. If the system starts in a Fock state, then it will evolve to coherent

state that acquires a phase Φ (t).

7.4 Geometric Phase and Coherent State Amplitude

Now that the general form of the coherent state amplitude and phase acquired

during the phase gate interaction has been derived, the results can applied to the specic

case of the Hamiltonian in equation (7.21). The coherent state amplitude as a function

of time is given by equation (7.31). By inspection of equation (7.21) the time dependent

force can be written as

f(t) = −~ηs sin 

∆φ

2



e

i(ωt−

∆φ

2

)

. (7.35)
166

The frequency ω used in the general expression for α (t) will be replaced by ωz, making

these substitutions leads to

α (t) = iηsΩ

ˆ t

t1

dt0sin 

∆φ

2



e

i(ωt0−

∆φ

2

)

e

−iωst

0

. (7.36)

Integrating this expression and making the denition δ≡ ω − ωs gives the expression

α (t) = 2ηsΩ sin 

∆φ

2



e

i(

δ

2

t−

∆φ

2

)

sin 

δ

2

t



δ

. (7.37)

There are a few interesting things to note about this expression. The coherent state

amplitude oscillates in time, and after a time t =

2nπ

δ

(n = 0, 1, 2...) the coherent state

amplitude returns to zero. This is important, because if some other time is chosen

for the length of the interaction, then the spin states are entangled with the motion.

This is undesirable in some circumstances and should be avoided for the phase gate. It

could lead to excess decoherence as a result of the motion being sensitive to decoherence

from ambient electric eld noise. Also it should be noted the detuning δ inuences the

magnitude of the coherent state obtained. Smaller detunings lead to larger amplitude

coherent states, which follows from the fact that as the detuning decreases the drive

frequency approaches the resonance of the motion.

The next important term is the geometrical phase equation (7.33) acquired during

the gate interaction. To calculate the phase the commutator



HI

fho 

t

0



, HI

fho 

t

00 (7.38)

needs to be calculated using equation (7.21). Using the commutator relations for a and

a

†

[Sakurai 94], gives the result



Hg



t

0



, Hg



t

00 = 2i~2

η

2

sΩ

2

sin2



∆φ

2



sin δ



t

00 − t0

 . (7.39)

The phase as function of the interaction time can now be written as, plugging equation

(7.39) into equation 7.33,

Φ (t) = i

2

η

2

sΩ

2

sin2



∆φ

2

 ˆ t

t1

dt0

ˆ t

t1

dt00 sin δ



t

00 − t0

 . (7.40)
167

Integrating this expression gives

Φ (t) = −η

2

sΩ

2

sin2



∆φ

2



(sin (δt) − δt)

δ

2

. (7.41)

Now if the time is chosen such that the coherent state returns to its starting state at

t =

2nπ

δ

then the phase acquired will be

Φ



t =

2nπ

δ



= η

2

sΩ

2

sin2



∆φ

2



(2nπ)

δ

2

. (7.42)

In order to get Φ = π

2

, the experiment parameters such as the phase ∆φ, the Rabi

frequency Ω, and the detuning δ can be adjusted.

7.5 Ideal Gate Implementation

This section describes how the geometrical phase gate is implemented. The eect

of the phase gate is to imprint a phase on some of the states of a superposition. However

this phase cannot be directly observed in our experiments. With trapped ions, the

measurable quantity is the ion's uorescence. By embedding the phase gate drive into

a Ramsey experiment [Ramsey 63] enables the creation of states with dierent parity,

which can be distinguished via resonance uorescence techniques.

This implementation assumes that two ions are conned in a linear radio-frequency

Paul trap, and the motion along the trap axis has been cooled to the ground state. Figure

(7.3) shows the sequence of pulses used in performing the gate operations. The rst step

is to prepare the ions via optical pumping techniques [Wineland 98] to create the state

Ψ = |↑↑i. (7.43)

The motional state has been left out of the wave function. The next step is to create

a equal superposition of all the states. This is accomplished by applying a co-carrier

rotation to both ions. Applying the rotation R



π

2

, 0



(note σx = iR (π, 0) and σy =

iR π, π

2



), creates the superposition

Ψ = 1

2

(|↑↑i − i|↑↓i − i|↓↑i − |↓↓i). (7.44)
168

State

Prep

Phase Gate Pulse Sequence

G G

Figure 7.3: Phase gate pulse sequence. The state preparation includes everything required to create state (7.43). This includes things such as Doppler cooling, Raman

cooling, etc.. The phase gate operator is represented by G, and the phase acquired during each application is adjusted to be e

i

π

4 . The drive is embedded in a spin echo pulse

sequence composed of the rotations R



π

2

, 0



, R (π, 0), and R



π

2

, 0



. Theses rotations

are co-carrier stimulated Raman transitions applied to both ions. The last pulse is the

analysis pulse for the phase gate given by R



π

2

, θA



.

At this point, the phase gate interaction could be applied to imprint the necessary phase.

However the phase gate drive is embedded in a spin echo sequence [Vandersypen 04].

This has a number of advantages. It makes the gate resistant to uniform magnetic eld

changes that are slow compared to the length of the arms of the spin echo sequence[Leibfried 03b].

By splitting up the phase gate interaction so that half the phase is acquired in the rst

half of the spin-echo sequence and the rest acquired in the second half, residual stark

shifts resulting from the gate interaction are canceled. One possibility is to choose the

detuning such that each half of the gate drive takes you half way around a loop in

phase space. This has the disadvantage that there is additionally time where the spin

and motion are in entangled. If the motional decoherence rate is larger than the spin

decoherence, which is often the case, then the decoherence will be larger. Therefore it is

more desirable for each half of the gate drive to make one complete smaller loop in phase

space. Thus at the end of each drive the spin and motion are unentangled. The gate

is applied for a time that imprints the phase e

i

π

4 . In particular this would be applied

near the resonance of the stretch mode of motion for the two ions. This is equivalent to
169

applying the operation

G

π

4



=





















1 0 0 0

0 e

i

π

4 0 0

0 0 e

i

π

4 0

0 0 0 1





















(7.45)

to the spin states of the two ions. This changes the state to

Ψ = 1

2



|↑↑i − iei

π

4 |↑↓i − iei

π

4 |↓↑i − |↓↓i

. (7.46)

The next pulse is the refocusing pulse. This is implemented by applying the operation

R (π, 0) to both ions, which gives

Ψ = 1

2



i|↑↑i − e

i

π

4 e |↑↓i − e

i

π

4 |↓↑i − i|↓↓i

. (7.47)

In the second half of the spin-echo sequence, the second part of the phase gate interaction

is applied, so the another loop in phase space is driven. This gives the state

Ψ = 1

2

(|↑↑i − |↑↓i − |↓↑i − |↓↓i). (7.48)

Next in the sequence, is the nal pulse in the Ramsey experiment. The rotation applied

is R



π

2

, 0



.

Ψ = 1

√

2

(|↑↑i + i|↓↓i). (7.49)

The state at this point is a more obvious entangled state and happens to be similar

to one of the Bell states[Nielsen 00] (It will be referred to as a Bell state)ξ. However

just measuring the resonant uorescence at this point would not give any information

about the entanglement phase. In order show the entanglement, a nal analysis pulse

R



π

2

, θA



with a variable phase is applied, which results in creating

Ψg =

1

2

e

iθA (cos (θA) − sin (θA))|↑↑i (7.50)

+

1

2

(cos (θA) + sin (θA))|↑↓i (7.51)

+

1

2

(cos (θA) + sin (θA))|↓↑i (7.52)

√

2

4

e

i

5π

4



e

−i

3π

2 + e

i2θA



|↓↓i. (7.53)
170

It may be dicult to see how this state makes it possible to analyze the entanglement.

Let's examine the resulting state for a couple choices of the phase θA.

Ψg



θA =

π

4



=

1

√

2

(|↑↓i + |↓↑i) (7.54)

Ψg



θA =

3π

4



=

1

√

2

(|↑↑i + i|↓↓i) (7.55)

Ψg



θA =

5π

4



=

−1

√

2

(|↑↓i + |↓↑i) (7.56)

Ψg



θA =

7π

4



=

1

√

2

(|↑↑i + i|↓↓i). (7.57)

The result of varying the phase θA between 0 and 2π is that the state oscillates between

the state (7.54) and the state (7.55) and a rate of 2θA. The state in equation (7.54)

is an example of a decoherence-free subspace (DFS) state [Kielpinski 01] (also called a

triplet state). The DFS state is very useful for its resistance to decoherence from uniform

magnetic eld noise, and will be used in the entangled mechanical oscillators experiment

discussed in chapter (8). These two dierent states can be distinguished via resonance

uorescence, since they have a dierent parity. In this context the parity refers to the

number of ions in the |↑i state (strongly uorescing state). The Bell state equation

(7.55) has even parity, since a measurement of this state will always result in measuring

two or zero ions uorescing. The DFS state is an odd parity state since only one ion is

measured in the |↑i. The populations P (for example P|↑↑i = hΨg |↑↑i h↑↑| Ψgi) of the

for the various spin states are

P|↑↑i = P|↓↓i =

1

4

(1 − sin (2θA)) (7.58)

P|↑↓i = P|↓↑i =

1

4

(1 + sin (2θA)). (7.59)

The parity signal Π (θA) is dened as

Π (θA) = P|↑↑i + P|↓↓i −



P|↑↓i + P|↓↑i(7.60)

= − sin (2θA). (7.61)

The parity signal can be used to verify that entanglement is present by looking at the
171

amplitude of the oscillation. The parity can be written as Π (θA) = −C sin (θA), where C

characterizes the contrast of the oscillation. In the ideal case C = 1. Having a contrast

greater than 50% shows that entanglement is present [Sackett 00, Leibfried 03b]. Also

the contrast is a lower bound on the delity F, which is a standard measure of the

amount of entanglement[Nielsen 00].

F = hΨideal| ρ |Ψideali. (7.62)

This can be understood as the overlap of the created state with the ideal state. The

density matrix of the system is represented by ρ. The delity of creating the Bell state

equation (7.55) is

F =

1

2



P|↑↑i + P|↓↓i+ |ρ↑↑,↓↓| . (7.63)

The density matrix element |ρ↑↑,↓↓| = hΨideal |↑↑i h↓↓| Ψideali and also note that ρ↑↑,↓↓ =

ρ

?

↓↓,↑↑. The delity of creating the DFS state (7.54) is

F =

1

2



P|↑↓i + P|↓↑i+ |ρ↑↓,↓↑| . (7.64)

Assuming the delity of creating either state is the same, then equations (7.64) and

(7.63) can be inserted into equation to (7.60) to give

Π (θA)

2

= −

C

2

sin (θA) = |ρ↑↓,↓↑| − |ρ↑↑,↓↓| . (7.65)

Thus when the phase is chosen such that the Bell state is created then contrast can

be related to o diagonal density matrix elements |ρ↑↑,↓↓| =

C

2

. This allows for a more

experimentally convenient denition of the delity

F =

1

2



P|↑↑i + P|↓↓i+

C

2

. (7.66)

The contrast can be experimentally determined from a parity measurement. The populations can be determined from a separate experiment, where the analysis pulse is omitted

and the populations are measured.
172

n = 0

n = 1

n = 2

~100 MHz ~ 313 nm

= 197.2 GHz

Figure 7.4: Experimental level diagrams for a phase gate on 9Be+ ions. This gure

shows the relevant levels. Only two of the S1

2

ground state hyperne levels play a role in

the gate; these are split by ∼ 100 MHz. The ne structure splitting of the excited states

is 197.2 GHz [Langer 06]. Each of the spin states is o resonantly coupled to both of the

excited states with σ

− and σ+ polarized light, except there is no state in the P1

2

level

for |↑i to couple to with σ

+ polarized light. The detuning of the Raman laser beams

from the P1

2

is given by ∆, which is ∼ 80 GHz. Superimposed on top of the spin states

are the harmonic oscillator states for one of the normal modes, which is labeled by ωz.

The detuning of the dierence frequency of the Raman beams from being on resonance

with the motional states is δ.
173

7.6 Phase Gate Entanglement Experiment

7.6.1 Theory

The theory for the experimental implementation of the phase gate discussed in

section (7.6) is more complicated than the example discussed above. In this section some

of the dierences are highlighted. The phase gate will be between two 9Be+ ions, whose

relevant energy levels are shown in gure (7.4). The two ground state levels are coupled

to the excited states with both types of circular polarizations. For each ion the |↑i and

the |↓i states will not have the same Rabi frequency and each ion will see dierent laser

beam phases. The stimulated Raman transition Rabi frequency for each state will be

given by [Wineland 03]

Ω

mi

i =

e

i(φ1−φ2)

4~

2

X

j

hmi|

~d · E~

n |ji hj|

~d · E~

c |mii

∆j

. (7.67)

In this expression i represents the ion number and miis the spin state for ion i. It is often

convenient to use the low magnetic eld notation for the labeling of the spin states. In

general the states at intermediate magnetic elds will be a superposition of these states.

The experiment discussed in the next section uses the states |↓i = |F = 2, mF = 1i and

|↑i = |F = 2, mF = 2i states of 9Be+ at a eld of ~ 120 Gauss. The two levels are split

by ∼ 100 MHz. In the |mI , mj i basis we have |↓i = α |mI =

1

2

, mJ =

1

2

i + β |

3

2

,

−1

2

i and

|↑i = |mI =

3

2

, mJ =

1

2

i. These ground state levels are coupled to the excited states by

the electric elds E~

n = En



ˆσ+ + ˆσ−



and E~

c = Ec



−ˆσ+ + ˆσ−



see part B and C of

gure (7.2). The term in the denominator ∆iis the detuning from the excited states,

and the sum is over all the possible excited states, and in 9Be+these would be the P1

2

and P3

2

states. Both Raman beams can be considered to have the same detuning from

an excited state if the detuning ∆ is much larger than the splitting between the spin

states. In the experiment discussed below the splitting between the excited states is

197.2 GHz [Langer 06] and the two Raman beams are detuned 80 GHz from the P1

2
174

level.

The phase gate Hamiltonian (7.8) can be used to write the Hamiltonian for this

case. Again it will be assumed there are only two ions being used and two axial normal

modes, which can be written in terms of the individual ion positions as equation (7.9).

The Hamiltonian becomes

Hg = −~[ηczˆc



sin (ωt − φ1)



Ω

↑

|↑1i h↑1| + Ω↓|↓1i h↓1|

 (7.68)

ηszˆs



sin (ωt − φ1)



Ω

↑

|↑1i h↑1| + Ω↓|↓1i h↓1|

 (7.69)

+ηczˆc



sin (ωt − φ2)



Ω

↑

|↑2i h↑2| + Ω↓|↓2i h↓2|

 (7.70)

−ηszˆs



sin (ωt − φ2)



Ω

↑

|↑2i h↑2| + Ω↓|↓2i h↓2|

]. (7.71)

This Hamiltonian can be written in terms of the two qubit basis states: |↑↑i, |↓↓i, |↑↓i,

and |↓↑i as

Hg = −~[2ηczˆc sin ωt −

1

2

(φ1 + φ2)



cos 

−∆φ

2



(7.72)

×

h

Ω

↑

|↑↑i h↑↑| + Ω↓|↓↓i h↓↓|i(7.73)

+ηczˆc sin (ωt − φ1)

h

Ω

↓

|↓↑i h↓↑| + Ω↑|↑↓i h↑↓|i(7.74)

+ηczˆc sin (ωt − φ2)

h

Ω

↑

|↓↑i h↓↑| + Ω↓|↑↓i h↑↓|i(7.75)

+2ηszˆs cos ωt −

1

2

(φ1 + φ2)



sin 

−∆φ

2



(7.76)

×

h

Ω

↑

|↑↑i h↑↑| + Ω↓|↓↓i h↓↓|i(7.77)

+ηszˆs sin (ωt − φ1)

h

Ω

↓

|↓↑i h↓↑| + Ω↑|↑↓i h↑↓|i(7.78)

−ηszˆs sin (ωt − φ2)

h

Ω

↑

|↓↑i h↓↑| + Ω↓|↑↓i h↑↓|i. (7.79)

This Hamiltonian simplies when looking at a few special cases. The interaction is

usually tuned to be near resonance with only one of the motional modes. Most commonly

this will be the stretch mode, since it has a reduced sensitivity to heating from stray

electric elds [Kielpinski 00]. Choosing the ion spacing see part C of gure (7.2) such

that the phase dierence between two ions is an integer multiple of 2π and that one ion
175

has a phase of zero, (i.e. φ1 = 0 and φ2 = φ1 + 2π), the Hamiltonian simplies to

Hg = −~ηszˆs sin (ωt)



Ω

↑ − Ω↓



[|↓↑i h↓↑| − |↑↓i h↑↓|] . (7.80)

In this case only the states |↑↓i, and |↓↑i will acquire a motional dependent phase,

which is proportional to the dierence in the Rabi frequency between the spin states.

Another case worth considering is the one where the ions are spaced by λef f /2 in the

interference pattern, which corresponds to having a phase dierence between the ions of

π (i.e. φ1 = 0 and φ2 = φ1 + π)

Hg = −~ηszˆs sin (ωt) [2Ω↑|↑↑i h↑↑| + 2Ω↓|↓↓i h↓↓| (7.81)

+



Ω

↑ + Ω↓



|↑↓i h↑↓| − Ω

↑ + Ω↓



|↓↑i h↓↑| . (7.82)

Here all the states are driven and pick up a phase. If the Rabi frequencies are dierent,

then a dierential phase between the states is produced, which leads to entanglement.

However, if Ω

↑ = Ω↓

then the force on all the states is the same and no entanglement

will occur.

Well if you read this part then you must be really into ion trapping. Send me an

email at john.d.jost@gmail.com and let me know you got this far. (It would be cool to

know if after a few years have past since the writing of this thesis if anyone is actually

still looking at it.)

7.6.2 Determining the Phase - Ion Spacing

Determining the phase of the ions in the interference pattern is crucial to being able

to apply a known state dependent force. A calibration experiment can be performed to

determine the relative phase and adjust it. This calibration experiment will be explained

in the context of dealing with two 9Be+ions, but the same procedure applies to the four

ion mixed species crystal entanglement discussed in section (7.6.4). After cooling the

two ions to the ground state see chapter (3) the ions are prepared in the |↑↑i state using
176

optical pumping. Then the laser beams that give rise to the gate interaction equation

(7.79) are applied for a xed length of time. In the examples discussed above it is

assumed that the dierence frequency of the laser is detuned by δ from the resonance of

the one of the modes. In this experiment the dierence frequency is set to be resonant

with the stretch mode of motion. If the two ions have a spacing such that there is a phase

dierence of 2π, then the ions experience the Hamiltonian given by equation (7.80). In

this case the ions experience no force. However if the phase dierence is π then they

do experience a force given by equation (7.82). This the force is on resonance with the

mode of motion, and the ions' motional state is driven to a coherent state see section

7.3.2. Then a motion subtracting (MS) Raman transition (see section (6)) is performed.

If the ions' motion has been excited, then this transition gets driven, but no transition is

driven if the motional state is still in the ground state. When a transition is driven the

ions internal state is put into a non-uorescing state. Thus using resonance uorescence

detection [Langer 06] it can be determined if the motional state was excited. The ion

spacing can be varied by changing the potentials on the electrodes, which also changes

the mode frequencies. This can be done and the experiment repeated until the correct

spacing is obtained. The goal is to apply the Hamiltonian in equation (7.80), which

occurs when no force is applied to the ions in this calibration experiment.

7.6.3 Determining the Detuning

It is often dicult to know the exact magnitude the of Rabi frequency and other

parameters that will determine the required detuning for the phase gate, see equation

(7.3). This means it is necessary to determine it experimentally, and a separate tuneup experiment accomplishes this. In the gate sequence depicted in gure (7.3) the

gate interaction is chosen such that a total phase of e

i

π

2 is acquired on two of the two

states |↑↓i and |↓↑i. If the gate interaction is chosen to induce a phase factor of e

iπ

,

then the result of the gate sequence is not an entangled state, but the state |↓↓i. The
177

experiment works by applying the gate sequence (7.3) and scanning the detuning δ.

The interaction time is proportional to the inverse of the detuning. For each value of

the detuning an experiment is performed and then the internal state populations are

measured using resonance uorescence [Langer 05]. When the detuning is such that the

both ions internal state populations are measured to be in the state |↓↓i , then the right

detuning has been found.

7.6.4 Entanglement in a Mixed Species Crystal

This section discusses experimentally demonstrating entanglement of the internal

spin states of two ions in a four ion mixed species chain 9Be+ − 24Mg+ − 24Mg+ −9 Be+

[Jost 09]. The ions are conned in one trapping zone of the linear radio frequency Paul

trap as discussed in chapter (2). The ions to be entangled are two 9Be+ ions, which could

serve as logical qubits in QIP applications. The other two ions are 24Mg+ ions, which are

used as sympathetic cooling ions. Although the theory in the previous section was for

two ions, it applies here as well. There are several possible orders of the four ion mixed

species crystal. This experiment uses one particular order: 9Be+−24Mg+−24Mg+−9Be+.

There are four axial modes, which we designate with these names: in-phase mode (INPH)

(frequency ∼ 2.0 MHz, mode vector: [0.32, 0.63, 0.63, 0.32]), the out-of-phase mode

(OOPH) (∼ 4.1 MHz, [-0.47, - 0.53, 0.53, 0.47]), the Egyptian mode (∼ 5.5 MHz, [0.63,

-0.32, -0.32, 0.63]) and the Funk (∼ 5.7 MHz, [0.53, -0.47, 0.47, -0.53]). The amplitudes

given in the mode vectors (written in ion order from left to right) are related to each

ion's root-mean-squared ground state wave function size by multiplying by p

~/2Mω,

with M the mass of the relevant ion and ω the mode frequency in angular units. The

gate drive will use the OOPH mode, which resembles the stretch like motion of two ions,

where the 9Be+ ions move in opposite directions but with equal amplitudes. All other

modes will be o resonance with the gate drive. The gate interaction will also only

address the internal states of the 9Be+ ions and these levels are the same as discussed
178

Experimental Phase Gate Pulse Sequence

Reorder

ions

Doppler

Cooling

and

Doppler

Cooling

Raman

Cooling

State

Prep

G G Det.

Figure 7.5: Experimental pulse sequence for performing the geometric phase gate on the

four ion mixed species chain 9Be+ − 24Mg+ − 24Mg+ −9 Be+. The sequence is explained

in more detail in the text. The rst line is everything involved in the initial state

preparation: Ordering the ions to create 9Be+ − 24Mg+ − 24Mg+ −9 Be+ , a sequence of

Doppler and Raman cooling to initialize the axial motion to near the ground state, and a

state preparation pulse that optically pumps the 9Be+ ions to the |↑↑i state. The second

line contains the phase gate sequence. The phase gate drive is represented by G, and

the phase acquired during each drive is e

i

π

4 . The drive is embedded in a spin echo pulse

sequence composed of the rotations R



π

2

, 0



, R (π, 0), and R



π

2

, 0



. Theses rotation

are co-carrier stimulated Raman transitions applied to both ions. The last pulse is the

analysis pulse for the phase gate given by R



π

2

, θA



. The last pulse is the resonance

uorescence detection.

in section (7.6). This means the 24Mg+ ions internal states will not be aected by the

gate drive, and the Hamiltonian derived in section (7.6)still applies. The 24Mg+ions are

used when initially Doppler cooling (see chapter (2)) the four ion crystal.

The experimental pulse sequence is depicted in gure (7.5). If the ions have a

dierent order from 9Be+ − 24Mg+ − 24Mg+ −9 Be+, then the normal mode frequencies

will change. Thus it is necessary to ensure the ions always start in this particular order.

The ions can be forced into this order using time varying potentials and laser cooling

[Jost 09], which is done before each experiment.

After the ordering, Doppler cooling is performed using a combination of both
179

9Be+ and 24Mg+(see chapter (5)) Doppler cooling. Both types of cooling are used

because the Doppler cooling only eciently cools the modes where that type of ion

has large motional amplitude. The 24Mg+cooling is especially important; because there

are radial modes with frequencies comparable to the axial ones. If these modes have

a large amount of energy, then this can lead to o resonant coupling of the radial

modes to the axial modes of motion if micromotion is not well compensated. The

Doppler cooling used on 9Be+ is a combination of both far o resonance cooling (∼ −400

MHz) and cooling where the frequency is tuned to near the optimum −

Γ

2

point. The

far o resonance cooling is used to cool instances where the ions are highly excited

(from for example a background gas collision) and cool these ions to a regime where

the near resonance cooling is eective. This combined Doppler cooling lasts for 3500

μs. When this experiment was initially performed it was believed the 9Be+ Doppler

cooling would provide the lowest possible temperatures. Also in subsequent experiments

[Home 09, Hanneke 10] the Doppler cooling of the four crystal has been performed with

only 24Mg+ Doppler cooling. The combined Doppler cooling is followed by 500 μs. of

near resonance Doppler cooling with 9Be+. This brings the temperature of the axial

modes to a low enough n to be eciently Raman cooled to the ground state.

Before Raman cooling is performed, the 9Be+ ions are optically pumped to the

|↑↑i state (The same is done with the 24Mg+ ions, but this just part of a protocol

that is performed for subsequent experiments). After Doppler cooling, the state of the

9Be+ions should be well initialized, but the last optical pumping step helps to ensure

this. Raman cooling (6) is performed using the 9Be+ states |↓i = |F = 2, mF = 1i and

|↑i = |F = 2, mF = 2i. Twenty interlaced cooling cycles are performed on each of the

axial modes of motion, which takes 2800 μs in total. After this cooling all the motional

modes have been cooled to n < 0.2[Jost 09].

Now that the 9Be+ ions have been prepared, the gate sequence discussed in section (7.5) can be implemented, see gure (7.5). The co-carrier Raman transitions that
180

perform the single qubit rotations R (θ, φ) are implemented on the same internal transitions as the Raman cooling. The rst step is to implement the rotation R



π

2

, 0



( takes

approximately 1.5 μs), which prepares the two 9Be+ ions in the superposition state given

by equation (7.44). It should be noted that the internal states of 24Mg+ ions are omitted

since they no longer play a role. The next step in the sequence is to apply the rst half

of the gate drive. The detuning of the gate drive is 40 KHz, which makes a one loop

in phase space in a duration of ∼ 25μs. This leaves the system in the state (7.46). The

next step is to apply the refocusing pulse of the spin echo sequence, where a co-carrier

pulse implements the rotation R (π, 0) (takes approximately 3 μs) on both 9Be+ ions,

which produces the state (7.47). Now the second half of the gate drive is applied, which

drives the motion around another loop in phase space to give (7.48). The nal pulse in

the spin echo sequence applies the rotation R



π

2

, 0



and creates the Bell like state (7.49)

. Next the variable phase analysis pulse R



π

2

, θA



applied to both 9Be+ ions, which

creates the phase dependent state (7.52).

After the nal analysis pulse, shelving is performed see chapter (3) on the |↓↓i

state. This transfers the population of both ions in the state |↓i = |F = 2, mF = 1i to

the state |F = 2, mF = −2i, which helps distinguish the uorescing state |↑i from the

other spin state [Langer 06]. Finally a 200 μs pulse of resonant detection light is applied

to the 9Be+ ions and photons are collected and counted. The resulting histograms are

t to a sum of Poissonian distributions to extract the populations for a given data point,

where the data in gure (7.6) is composed of 500 experiments per point. From the

these populations, the parity (7.60) for a given phase can be determined. The scan of

analysis phase θA is shown in gure (7.6). The resulting curve is t using a non-linear

curve tting routine in Matlab to the function C2 cos (2θA + φ2)+C1 cos (θA + φ1)+C0.

From the t, the values for the amplitudes are C2 = 0.84 ± 0.015, C1 = 0.03 ± 0.015,

and C0 = −0.012 ± 0.011 1. The value of C2 is greater than the 0.5 necessary to show

1 Parity data was taken on 10/10/08 at 19:17:28
181

0 1 2 3 4 5 6

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Parity

Phase (rads)

Mixed species phase gate:

Figure 7.6: Parity opping on the two 9Be+ ions in the four ion chain 9Be+ − 24Mg+ −

24Mg+ −9 Be+. A parity of value of 1 corresponds to the state √

1

2

(|↑↑i + i|↓↓i), and a

value of -1 corresponds to creating the state √

1

2

(|↑↓i + |↓↑i). The opping signal data is

t to a C2 cos (2θA + φ2) +C1 cos (θA + φ1) +C0 function and is shown by the solid line.

The amplitude C2 = 0.84 ± 0.015 is the important quantity, since a value > 0.5 shows

there is entanglement. Each data point is composed of 500 experiments. The vertical

error bars represent one standard deviation.

entanglement.

In order to obtain the populations required for calculating the delity a separate

almost identical experiment is performed, where the only dierence is that the analysis

pulse is excluded2. The resulting histograms are and t to a sum of Poissonian distributions to extract the populations for P|↑↑i = 0.46 ± 0.02, P|↓↓i = 0.45 ± 0.02, and

P|↑↓i+|↓↑i = 0.09 ± 0.01. The delity from equation (7.63) is found to be F = 0.876 ±

0.015. There are several possible sources of the errors in the phase gate that prevent the

delity from being higher. Spontaneous emission from the nite detuning of the Raman

beams from the excited state is one such source of error [Leibfried 03b, Ozeri 07, Jost 09].

2 Population data was taken on 10/10/08 at 19:20:03
182

Many laser pulses are required to implement the phase gate and the rotations. These

pulses often have intensity noise leading to a loss in delity [Leibfried 03b, Jost 09].
Chapter 8

Entangled Mechanical Oscillators

This chapter discusses the experiments [Jost 09] performed to entangle two spatially separated mechanical oscillators and to entangle the internal electronic state of an

ion with a spatially separate mechanical oscillator.

8.1 Motivation

Hallmarks of quantum mechanics include superposition and entanglement. In

the context of large complex systems, these features should lead to situations like

Schrödinger's cat [Schrödinger 35], which exists in a superposition of alive and dead

states entangled with a radioactive nucleus. Such situations are not observed in nature.

This may simply be due to our inability to suciently isolate the system of interest from

the surrounding environment [Ball 08, Schlosshauer 08]  a technical limitation. Another possibility is some as-of-yet undiscovered mechanism that prevents the formation

of macroscopic entangled states [Bassi 03]. Such a limitation might depend on the number of elementary constituents in the system [Leggett 02b] or on the types of degrees of

freedom that are entangled. Tests of the latter possibility have been made with photons,

atoms, and condensed matter devices [Southwell 08, Aspelmeyer 08a]. One system ubiquitous to nature where entanglement has not been previously demonstrated is distinct

mechanical oscillators. Here we demonstrate deterministic entanglement of separated

mechanical oscillators, consisting of the vibrational states of two pairs of atomic ions
184

held in dierent locations. We also demonstrate entanglement of the internal states of

an atomic ion with a distant mechanical oscillator. These results show quantum entanglement in a degree of freedom that pervades the classical world. Such experiments

may provide pathways towards generation of entangled states of larger-scale mechanical oscillators [Mancini 02, Schwab 05, Kippenburg 08], and oer possibilities for testing

non-locality with mesoscopic systems [Milman 05]. In addition, the control developed in

these experiments is an important ingredient to scale up quantum information processing

based on trapped atomic ions [Wineland 98, Cirac 00, Kielpinski 02a].

Mechanical oscillators pervade nature; examples include the vibrations of violin

strings, the oscillations of quartz crystals used in clocks, and the vibrations of atoms

in a molecule. Independent of the size of the system, each mode of vibration can be

described by the same equations that describe the oscillations of a mass attached to

a xed object by a spring. For very low energy oscillations, quantum mechanics is

needed for a correct description: the energy is quantized and the motion can be described generally by superpositions of wavefunctions corresponding to each quantum

level. Coherent states behave very much like classical oscillators, while other states have

properties with distinctly non-classical features [Schleich 01]. Quantum mechanics also

permits superposition states of multiple systems called entangled states, where the measured properties of the systems are correlated in ways that defy our every-day experience

[Aspect 02, Southwell 08, Pan 08, Matsukevich 08, Aspelmeyer 08a]. When extended to

macroscopic scales, situations akin to Schrödinger's cat should appear. Our inability to

produce such macroscopic entanglement may be just a question of technical diculty.

However, there might be a more fundamental cause, such as the inability to entangle

certain types of degrees of freedom.
185

Figure 8.1: 9Be+ − 24Mg+ mechanical oscillators. Simplied depiction of the two mechanical oscillators indicating motion in the stretch mode of each 9Be+ − 24Mg+ ion

pair, held in separate locations (not to scale). The pairs  spaced by ' 0.24 mm  each

behave as two masses spaced by ∼ 4 μm, connected by a spring.

Create state

Move and separate ions

into two pairs - state

Sympathetically recool with

24Mg+- state

Create spin - motion

entanglement - state

Create entangled mechanical

oscillators - state

24Mg+ 9

Be+

A SE B

Creation of Entangled Mechanical Oscillators

Figure 8.2: Creation of entangled mechanical oscillators. Schematic showing the ions'

positions with respect to the ion trap electrodes (A, X, and B) and the quantum states

at key points in the experiments (not to scale). After entangling the 9Be+ ions' spins

in a single well, the ions are separated into two pairs by electrode X and distributed

to dierent wells. Laser cooling of 24Mg+ removes motional excitation incurred during

separation. A 9Be+ laser pulse in well A entangles the motion in well A with the 9Be+

spin in well B. A subsequent pulse in well B entangles the two mechanical oscillators.

8.2 Experiment

To explore the latter territory in a new regime, we demonstrate entanglement of

two separated mechanical oscillators. Here each oscillator is comprised of a pair of ions -
186

one 9Be+ and one 24Mg+ - conned in a potential well. In the context of the experiment

described below, each pair behaves like two masses connected by a spring of length ∼ 4

μm, undergoing vibrational motion (gure (8.1)). The two pairs are separated by 0.24

mm such that the coupling between them can be neglected. To create the entangled state

of the oscillators, we start with all four ions in one location and entangle the internal

states of the two 9Be+ ions [Leibfried 03b]. We then separate the four ions into two pairs,

each containing one of the entangled 9Be+ ions. Finally, we transfer the entanglement

from the 9Be+ ions' internal states to the motion of the separated ion pairs, creating the

desired motional entanglement.

Initially, all the ions are held in a single potential well of a multi-zone linear Paul

trap [Rowe 02, Barrett 04]. The potential well is congured to locate the ions along a line

corresponding to the axis of weakest connement, which we call the axial direction. We

will be concerned only with motional modes along this axis. While applying continuous

laser cooling, we initialize the ions in a particular order, 9Be+ −24Mg+ −24Mg+ −9 Be+,

by rst increasing the axial connement until no linear arrangement is stable. The axial

potential is independent of ion mass while the radial potential strength scales inversely

with the mass [Wineland 98], thus there exist axial potentials where the heavier 24Mg+

ions are displaced from the axis and must reside between the 9Be+ ions. We then relax

the axial connement giving the desired order [Rosenband ]. The four ions are located

in one well in the conguration 9Be+ − 24Mg+ − 24Mg+ −9 Be+, which has four axial

modes of motion. In order of ascending frequency, we designate them as the in-phase

mode (frequency ' 2.0 MHz, mode vector: [0.32, 0.63, 0.63, 0.32]), the out-of-phase

mode (4.1 MHz, [-0.47, -0.53, 0.53, 0.47]), a third mode (5.5 MHz, [0.63, -0.32, -0.32,

0.63]) and the Funk mode (5.7 MHz, [0.53, -0.47, 0.47, -0.53]). The amplitudes given

in the mode vectors (written in ion order from left to right) are related to each ion's

root-mean-squared ground state wavefunction size by multiplying by p

~/(2Mω), with

M the mass of the relevant ion and ω the mode frequency in angular units.
187

In addition to comprising part of the mechanical oscillators, the 24Mg+ ions serve

as a tool to provide sympathetic cooling of the 9Be+ ions [Barrett 03]. Doppler cooling

of 24Mg+ is accomplished by driving transitions between the ground 2S1/2states and

excited 2P1/2states, which have a radiative linewidth of 41 MHz [Herrmann 09]. In the

0.012 T applied magnetic eld, the ground Zeeman states |mJ = ±1/2i are split by 334

MHz, hence ecient Doppler cooling requires an additional repump beam to prevent

optical pumping.

Lasers provide control of the ions' motion and internal states through laser cooling

and stimulated-Raman carrier or sideband transitions [King 98, Wineland 98] [Southwell 08].

Doppler cooling is performed on the 9Be+ and 24Mg+ ions. There are radial modes that

have small amplitudes for 9Be+ but large amplitudes for 24Mg+. This means 9Be+ cooling is inecient for these modes, hence we also cool these modes using 24Mg+ Doppler

cooling. This is followed by sideband cooling on the 9Be+ ions, which prepares the motion of each of the four axial normal modes to an average motional occupation of hni ≤

0.17. By applying a magnetic eld of 0.012 T, we spectrally isolate two internal (hyper-

ne) states in each 9Be+ ion, which we call spin" states, and label |↑i ≡ |F = 2, mF = 2i

and |↓i ≡ |F = 2, mF = 1i. These states are split by 102 MHz. Using a spin rotations

and a geometric phase gate [Leibfried 03b] operation, which implements a 9Be+ statedependent motional displacement on the out-of-phase mode, we create the decoherencefree-subspace entangled state

|Ψ1i =

1

√

2

[|↑↓i + |↓↑i] (8.1)

of the two 9Be+ ions. This state is resistant to decoherence from spatially uniform

magnetic eld noise [Kielpinski 01].

Time-varying axial potentials move and separate [Rowe 02, Barrett 04] the four

ions into two 9Be+ − 24Mg+ pairs in dierent wells, which are spaced by ∼ 0.24 mm

(see gure (8.2)). Each pair of ions has two axial normal modes: the stretch" mode
188

(frequency ' 4.9 MHz, mode vector ' [-0.93, 0.37]) in which the two ions oscillate outof-phase and the common" mode (frequency ' 2.3 MHz, mode vector ' [0.37, 0.93]

MHz) where they oscillate in-phase. The experiment involves the ground |n = 0i

j

and

rst excited |n = 1i

j

states of the stretch modes, where j ∈ {A, B} refers to the well.

In general, the separation process excites the motional modes into unknown states. The

wavefunction of the 9Be+ spin states after separation is \k

|Ψ2i =

1

√

2

h

|↑iA|↓iB + e

iξ(t)

|↓iA|↑iB

i

(8.2)

where ξ(t) is a phase that accumulates through the course of the experiment due to a

small dierence in magnetic eld between wells A and B.

To create the motional entangled state we rst prepare the stretch modes close to

|0iA|0iB(see appendix (C) for a detailed pulse sequence). For this, Doppler and sideband laser cooling on the 24Mg+ ions in both wells sympathetically cools [Barrett 03] the

9Be+ ions. One cycle of the pulsed 24Mg+sideband cooling [Barrett 03] uses stimulatedRaman transitions on a motional sideband of the |+1/2i → |−1/2i ground state transition (duration ∼ 5 μs), followed by application of the repumping beam to reprepare

|+1/2i (∼ 2 μs). The 24Mg+ sideband laser cooling uses 40 cooling cycles per mode on

the second motional sideband and then 60 cycles per mode on the rst sideband. This

prepares the stretch modes to mean occupation numbers of hnAi = 0.06(2) and hnBi =

0.02(2) where the uncertainties are standard errors (s.e.m.). We also cool the common

mode in each well to hni ≤ 0.13. The cooling does not aect the spin states of the 9Be+

ions [Barrett 03], thereby approximating the state

|Ψ3i =

1

√

2

h

|↑iA|↓iB + e

iξ(t)

|↓iA|↑iB

i

|0iA|0iB(8.3)

We transfer the entanglement from the spin to the motion with a sequence of laser

pulses on the 9Be+ ions. Carrier transitions (labeled with superscript c, duration ' 4

μs) only aect the spin states, and sideband transitions (superscript m, referred to as
189

spin ↔ motion transfer pulses, duration ' 13 μs) couple the spin and motion. These

can be described as generalized rotations:

R

c,m

j

(θ, φ) =





cos θ

2 −ie

−iφsinθ

2

−ie

iφsinθ

2

cos θ

2



 , (8.4)

where j ∈ {A, B}. Carrier transitions correspond to rotations in the basis





1

0



 = |↑i,





0

1



 = |↓i,

and sideband transitions correspond to rotations in the basis





1

0



 = |↑i |1i,





0

1



 = |↓i |0i.

The rotation angle θ is proportional to the intensity and duration of the pulses, and

the phase φ is determined by the phase dierence between the two optical Raman elds

[Wineland 98, King 98] at the position of the ion. We individually address the 9Be+ ions

in each well using acousto-optic modulators to shift the positions of the laser beams.

Applying Rm

A

(π, 0) to state (8.3) entangles the 9Be+ − 24Mg+ motion in well A

with the 9Be+ spin in well B, creating the state

|Ψ4i = √

1

2

|↑i h|↓i |0iA − ieiξ(t)|↑i |1iA

i

|0iB. (8.5)

After this spin → motion transfer, the spin in well B is sensitive to decoherence from uctuating magnetic elds. To minimize this eect, we apply a spin-echo pulse [Vandersypen 04],

Rc

B

(π, 0), T ' 40 μs after the previous pulse. After a second delay T, we apply a second

spin → motion transfer pulse Rm

B

(π, 0) in well B, producing the state

|Ψ5i = √

1

2

|↑i |↑i h|0iA|0iB − e

iξ(t)

|1iA|1iB

i

. (8.6)

This state is an entangled superposition of both stretch modes in the ground and rst

excited states. The entanglement now resides only in the mechanical oscillator states of

both wells. We leave the system in this state for ∼ 50 μs before beginning our analysis.
190

We are not able to directly measure the entangled motional state. The analysis

proceeds by basically reversing the steps used to create state (8.6) and characterizing

the resulting spin state. We transfer the motional state back into the spins using the

pulse sequence: Rm

B

(π, 0), T, Rc

B

(π, 0), T, Rm

A

(π, φA). We then recombine all the ions

into a single potential well, to ideally reproduce the state |Ψ1i, having chosen φA to

compensate for the phase ξ(t).

Imperfect creation of the state (8.6) could leave entanglement in the spin states,

which could mimic motional entanglement in the analysis. To prevent this, we transfer

residual populations A,B of states |↓iA,B into auxiliary internal (hyperne) states prior

to performing the motion → spin transfers (discussed more below). Moreover, since

all experiments are used in the analysis, this transfer process does not constitute postselection and cannot enhance the deduced entanglement.

Our detection relies on analyzing the state |Ψf i = √

1

2

[|↑↑i+i|↓↓i], which we create

by applying a common rotation Rc(

π

2

, −

3π

4

) to both spins. We verify the entanglement

created in state (8.6) by measuring the o-diagonal element |ρ↓↓,↑↑| = | h↓↓| ρf |↑↑i | of the

density matrix ρf corresponding to our approximation to the state |Ψf i. We determine

|ρ↓↓,↑↑| by applying a nal analysis pulse, Rc(

π

2

, φp), to both 9Be+ ions with a phase φp

and measuring the parity[Sackett 00], P↓↓ + P↑↑ − (P↓↑ + P↑↓), for dierent values of φp,

where P↓↓, P↑↑, P↓↑, and P↑↓ are the populations of the spin states |↓↓i, |↑↑i, |↓↑i, and

|↑↓i. The entanglement is revealed by the component of the parity signal that oscillates

as C2 cos(2φp), where C2 = |ρ↓↓,↑↑|. A value of C2 > 0.5 veries the spin entanglement

of |Ψf i and thus the motional entanglement in state (8.6).

To deduce the spin populations, we use state-dependent resonance uorescence

[Wineland 98, Southwell 08]. The |↑i state strongly uoresces. Prior to nal spin population measurement, the |↓i population is transferred to the dark state |F = 2, mF = −2i

using carrier π pulses R(π, 0) on the sequence of transitions |2, 1i → |2, 0i, |2, 0i →

|2, −1i, |2, −1i → |2, −2i. The number of photons measured per 9Be+ ion if all the pop-
191

ulation were in the |2, −2i dark state during the 200 μs detection period approximates

a Poisson distribution with a mean of 0.2. For the uorescing state |↑i, we observe a

Poisson distribution, with a mean number of photons ' 10 per 9Be+ ion.

As described in the main text, we move populations A,B of the spin states |↓iA,B

to the auxiliary hyperne state |2, 0i, so they do not contribute to entanglement veri-

cation. To ensure that the A,B populations end in dark states for the measurements, we

precede the transfer pulses described in the previous paragraph with transfer of the |2, 0i

populations to |2, −2i using a sequence of carrier π pulses on the |2, 0i → |2, −1i and

|2, −1i → |2, −2i transitions. Since the last pulse of the nal transfer sequence is also

a carrier π pulse on the |2, −2i ↔ |2, −1i transition, this leads to the populations A,B

ending in |2, −1i. If all the population is in this state, it would give a mean uorescence

value per 9Be+ ion of ∼ 1 photon during detection. This uorescence falsely contributes

to P↓↓ but in a way that does not depend on φp, and hence does not contribute to C2.

We t the data in the top plot of gure (8.3) with C2cos(2φp +φ2)+C1cos(φp +φ1)+C0

and extract C2 = 0.57(2). This demonstrates that entanglement was present in the

motion after the steps to create state (8.6).

The intermediate state (8.5) is itself a novel spin  motion" entangled state,

where the spin state of the 9Be+ ion in well B is entangled with the motion of the

stretch mode of the ion pair in well A. We characterize this state in a separate set of

experiments. After creating state (8.5), we allow it to persist for 176 μs. Following

the analysis described above (omitting the spin ↔ motion transfer steps in well B), we

measure the parity (bottom plot in gure (8.3)) and nd C2 = 0.65(2).

To provide partial checks of the spin → motion transfer steps, we perform separate

experiments to determine the spin populations after the transfer. In the rst check experiment, we follow the steps used to create state (8.5) then implement the above hyperne

state transfer sequences (omitting the B population transfer process) and measure the

spin populations. The populations are determined to be P↑↑ = 0.47(1), P↓↓ = 0.04(1),
192

−1

−0.5

0

0.5

1

a

−1

−0.5

0

0.5

1

Á

p

b

0 π/2 π 3π/2 2π

 Parity [ 

P↓↓ + 

P↑↑ − (

P↓↑ + 

P↑↓) ]

Figure 8.3: Entanglement demonstration through parity oscillation. Parity data obtained from a, the entangled mechanical oscillators and b, the spin  motion entanglement experiments. Each point is calculated using the maximum-likelihood method on

the uorescence data from running the experiment 500 times and is plotted with standard error bars (s.e.m.). The solid curve is a t to the data. Two-ion entanglement is

veried by an amplitude greater than 0.5 for the component of the parity signal that

oscillates at twice the analysis pulse phase φp [Sackett 00]. For the data shown, this

amplitude is a, 0.57(2) and b, 0.65(2).

and P↓↑ + P↑↓ = 0.49(2). Ideally we would expect P↑↑ = 1/2, P↓↓ = 0, and P↓↑ + P↑↓ =

1/2. Similarly, after the step used to create state (8.6), and following the transfer procedure, we determine P↑↑ = 0.86(2), P↓↓ = 0.01(1), and P↓↑ + P↑↓ = 0.13(2). Ideally we

should nd P↑↑ = 1.

Signicant sources of indelity are spontaneous photon scattering [Ozeri 07] and

motional decoherence. The delity with which we initially create |Ψ1i is ' 0.88. Motional state superpositions of the stretch mode in each well were independently measured

to have a coherence time of ∼ 800 μs, which is consistent with a model of decoherence due

to couplings to thermally occupied radial modes [Roos 08]. In the entangled mechanical

oscillators experiment, the motional superpositions are occupied for ' 250 μs and ' 50
193

μs in wells A and B respectively; we estimate a decrease in C2 from this source to be ∼

5 %. In the spin  motion entanglement experiment, we estimate a decrease in C2 of ∼

3 % from this source. Non-zero temperature also reduced the delity of motional state

initialization. We estimate that this would reduce the delity for producing the states

(8.6) and (8.5) by 8 % and 6 % respectively. Intensity uctuations at the few percent

level reduce the accuracy of all rotations.

The Coulomb coupling between the ion pairs in wells A and B could lead to an

entangled state of their stretch modes. However, the resonant exchange rate is 5 Hz,

which leads to negligible entanglement for the experimental time scales. Furthermore,

the stretch mode frequencies in wells A and B dier by ∼ 25 kHz, which would yield

negligible entanglement for all time scales.

In summary, we have created two novel entangled states of separated systems

involving mechanical oscillators, extending the regime where entanglement has been observed in nature. Implementing these experiments required deterministic ion ordering

and the ability to separate and recool ions while preserving entanglement and performing subsequent coherent operations. This is the rst demonstration of these techniques

combined. Some of these methods could apply to similar experiments with nano- and

micro-mechanical resonators [Schwab 05, Kippenburg 08, Mancini 02]. The states created could be used to extend tests of nonlocality in ion traps in a manner analogous

to that proposed for the electromagnetic elds of separated cavities [Milman 05]. The

control developed for these experiments also represents an important step towards largescale trapped-ion quantum information processing [Kielpinski 02a, Wineland 98].

Well thanks for actually looking at my thesis. I would be curious to know who

found this work interesting so send me an email if you like at john.d.jost@gmail.com.

Also it would be cool to know if after a few years have past since the writing of this

thesis that there is someone actually still looking at it. Thanks.
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Appendix A

Calculating Rabi Frequencies

The Rabi frequency equation (5.27), which is rewritten here for convenience as

Ω = −

e

~

Eo he| ~r · ~ |gi, (A.1)

is a parameter that commonly shows up in atomic physics calculations. Here the states

|gi and |ei represent the ground and excited states respectively. The Rabi frequency is

usually dened in terms of the electric eld strength of the laser beams and the dipole

matrix operator. However this not all that useful of a quantity for experimentalists. Typically the information an experimentalist has at hand is the laser power, the polarization,

a measurement of the beam waist, the excited state linewidth, and some Clebsch-Gordon

tables. Equation (5.27) can be rewritten such that only that information is needed to

come up with a value for the Rabi frequency to plug into our calculations.

The electric eld for a Gaussian laser beam [Verdeyen 95] is

E~ (x, y, z, t) = Eo~ ωo

ω (z)

e

−



r

ω(z)

2

e



−ikz+i tan

z

z0



e

−i

kr2

2R(z) e

iωt

, (A.2)

where

ω

2

(z) = ω

2

o

 

1 + 

z

zo

2

!

(A.3)

R (z) = z



1 + zo

z

2



(A.4)

zo =

πnωo

λ

. (A.5)
211

This describes the electric eld and the phase in the radial direction of the beam as

it travels in the z direction. The beam waist ωo is dened as the 1

e

point, so this is a

radius. The wavelength and the index of refraction are given by λ and n respectively.

The polarization vector is given by ~. The intensity in the beam can be calculated from

the poynting vector[Hecht 04] S~

I =

D

S~

E

T

= c

2

o







E~ × B~





 , (A.6)

where DS~

E

T

represents the time average. Substituting equation (5.25) and B~ =

E~

c

into

the above equation gives

I =

D

S~

E

T

= coE

2

o

ωo

ω (z)

e

−



2r

ω(z)

2

*

e



−ikz+i tan

z

z0



e

−i

kr2

2R(z) e

iωt2

+

T

. (A.7)

With an optical power-meter the instantaneous power is not measured but rather the

time averaged power. The time average of the oscillating eld[Hecht 04] is 1

2

. This gives

a more convenient expression for the intensity.

I = co

E2

o

2

ωo

ω (z)

e

−



2r

ω(z)

2

. (A.8)

However with the lab tools you are not going to measure the intensity but rather the

total power in the beam and its waist. Assuming the ion is at the beam waist, ω (z) = ωo

the power in the beam can be expressed as

P =

ˆ

IdA = co

E2

o

2

ˆ∞

0

ˆ

2π

0

re

−



2r

ωo

2

drdφ = co

E2

o

2

πω2

o

2

. (A.9)

Now the electric eld can be written in terms of things measurable in the lab

Eo =

s

4P

coπω2

o

. (A.10)

The next thing that needs to be cast in a more useful form is the dipole matrix

element he| ~r · ~ |gi. The Wigner-Eckart theorem can be used to simplify this, using the
212

convention of Edmonds [Edmonds 96]

hγ

0

, j0, m0| T (k, q)|γ, j, mi = (−1)j

0−m0





j

0 k j

m0q m



 (A.11)

×




γ

0

j

0





 T (k, q) kγji (A.12)

= (−1)k−j+j

0 hk, q, j, m| k, q, j0

, m0i

√

2j

0 + 1

× (A.13)

hγ

0

j

0kT(k, q) kγji. (A.14)

The matrix





j

0 k j

m0q m



 is the Wigner 3-j symbol and hk, q, j, m| k, q, j0

, m0i represents

the Clebsch-Gordon coecient. The values can be looked up in tables or calculated using

several common math programs. The term hγ

0

j

0k T (k, q) kγji is the reduced density

matrix element, where T (k, q) is the tensor operator that connects the two states. The

rank of the tensor is given by k, which is 1 for an electric dipole transition. The parameter

q represents the quantum number, and for electric dipole transitions it can take on values

of -1, 0, and 1, which corresponds to light with polarization of σ

−, π, and σ+ polarizations

respectively. The value for the reduced density matrix can only be calculated for a

few special cases. However, it can be related to the spontaneous decay rate, which

is a measurable quantity in most systems. Reference [King 08] has a derivation and

discussion of this. It gives as a result the relation for the reduced density matrix element

to the spontaneous emission rate as




γ

0

j

0





 T (k, q) kγji = (−1)j+j>

p

2j

0 + 1r

3o~λ3Γ

8π

2e2

, (A.15)

where j> is the larger of j and j

0

. The wavelength of the transition is given by

λ, and the spontaneous decay rate is in Γ (units of radians per second). Equations

(A.1),(A.10),(A.11), and(A.15)can be combined to give the following expression for the

Rabi frequency

Ω = s

3λΓP

2~cπ3ω

2

0

(−1)j

0+j+j>−m p

2j

0 + 1





j

0 k j

m0q m



 (A.16)

.
Appendix B

Determining the Average Motional State

It is not strictly correct to refer to the temperature of an ion since it is not in contact with a thermal reservoir but it is still commonplace to do so. If we assume an ion's

motion has a thermal distribution, an expression can be found for the mean occupation

number hni (This is equivalent to n ̄ in the text) of the ion's quantized harmonic motion

for a given temperature. The resulting distribution is a Bose-Einstein distribution. Here

the case of a harmonic oscillator including the zero-point energy is derived.

The energy of a harmonic oscillator Fock state is given by E = ~ω(n + 1/2).

Where ω is the frequency of the harmonic oscillator and n is the quantized motional

state of the oscillator. The average energy for a particular mode is

hEi = ~ω hn + 1/2i. (B.1)

The average for a distribution of states can be calculated by summing over all the possible

n states times the probability pn of being in that state for a given energy, which is given

by

hn + 1/2i =

X∞

n=0

(n + 1/2) pn. (B.2)

The probability for being in a particular state is

pn =

e

−En

kbT

Z

=

e

−~ω(n+1/2)

kbT

Z

. (B.3)
214

The partition function Z is a sum over the energy states, which gives

Z =

X∞

n=0

e

−

En

kbT = e

− ~ω

2kbT X∞

n=0

e

− n~ω

2kbT . (B.4)

This series can be summed using a geometric series to give

Z =

e

− ~ω

2kbT

1 − e

− ~ω

kbT

=

1

2

csch 

~ω

2kbT



. (B.5)

Plugging equations (B.5) and (B.3) into (B.2) gives

hn + 1/2i =

1

1

2

csch 

~ω

2kbT



X∞

n=0

(n + 1) e

−~ω(n+1/2)

kbT . (B.6)

The sum in the above equation can be evaluated using the trick

X

n

ne−nx = −

d

dx

X

n

e

−nx = −

d

dx 

1

1 − e−x



. (B.7)

Making use of this trick and simplifying give the following expression for equation on

the preceding page

hn + 1/2i =

1

2

coth 

~ω

2kbT



. (B.8)

Subtracting 1/2 from each side of the above expression gives

hni =

1

2

coth 

~ω

2kbT



−

1

2

. (B.9)

This is the same expression obtained it is assumed there is no zero point energy to begin

with, which is a slightly simpler calculation.
Appendix C

Pulse Sequence for the Entangled Mechanical Oscillators Experiment

The following table is the pulse sequence for the entangled mechanical oscillators

experiment. This table is reproduced from reference [Jost 09]. As a NIST publication it

is not subject to US Copyright.
216

Supplementary Table 1: Detailed procedure for the entangled mechanical oscillators experiment. Rotations are defined in

the text; those without superscripts are carrier rotations between hyperfine pairs other than {|↑i , |↓i}. These auxiliary states are

indicated with |F, mF i. Times are rounded to the nearest microsecond. Steps without an explicit operation typically involve laser

frequency changes and intensity stabilization. The procedure for the spin–motion entanglement experiment is identical except the

seven steps between the spin-echo pulses in well B are replaced with a single 90 μs delay.

Operation Ideal State after Operation Time (μs) Notes

Order ions – 935 Order the ions to

9Be+ –24Mg+ –24Mg+ –9Be+

Lock the laser

intensity – 380 –

– – 26 –

Doppler cool

(

9Be+ & 24Mg+ )

– 3500 –

Doppler cool

(

9Be+ only) – 500 –

– – 2 –

Repump 24Mg+ – 2 –

Repump 9Be+ |↑↑i 25 –

9Be+ sideband cool |↑↑i |0i 2753

Ground-state cool the four axial modes

(20 cooling pulses per mode). From this

point on, all 9Be+ interactions are coherent.

Prepare |Ψ1i √1

2

h

|↑↓i + |↓↑ii

266 –

Separate √1

2

h

|↑iA |↓iB + e

iφg(t)

|↓iA |↑iB

i

|ΛiA |ΥiB 819 Unknown motional state after separation (state |Ψ2i)

24Mg+ Doppler cool – 400 Cool in both wells simultaneously;

9Be+ coherence undisturbed

24Mg+ 2nd sideband

cool – 539

20 cooling pulses on the second sideband of each of the two axial modes;

cool both wells simultaneously

24Mg+ 2nd sideband

cool – 539 Repeat the previous step.

24Mg+ sideband cool √1

2

h

|↑iA |↓iB + e

iφg(t)

|↓iA |↑iB

i

|0iA |0iB 1277

Account for the slight difference in

mode frequencies between the two

wells by applying 30 cooling pulses resonant with each mode (4 modes total).

Final hni < 0.1 on the stretch modes.

(state |Ψ3i)

Continued on next page
217

Supplementary Table 1: (continued)

Operation Ideal State after Operation Time (μs) Notes

– – 22 –

Rm

A (π, 0) √1

2

|↑iA

h

|↓iB |0iA − ieiφg(t)|↑iB |1iA

i

|0iB 12 Spin→motion transfer pulse in well A

(state |Ψ4i)

– – 14

At this point, we have entangled a mechanical oscillator with the spin of a

separated ion.

RA(π, 0) A |↓iA ⇒ A |2, 0iA 3

Shelve residual amplitude A of |↓iA

(A 6= 0 only if there is an error in the

spin→motion transfer)

– – 22 This step and the previous two constitute T of the first spin-echo sequence.

Rc

B(π, 0) √1

2

|↑iA

h

|↑iB |0iA − ieiφg(t)|↓iB |1iA

i

|0iB 4 First spin-echo pulse in well B

– – 38 Second T delay of the first spin-echo

sequence

Rm

B (π, 0) √1

2

|↑iA |↑iB

h

|0iA |0iB − e

iφg(t)

|1iA |1iB

i

14 Spin→motion transfer pulse in well B

(state |Ψ5i)

– – 24 At this point, we have entangled separated mechanical oscillators.

RB(π, 0) B |↓iB ⇒ B |2, 0iB 4

Shelve residual amplitude B of |↓iB.

As above, this removes any residual

spin entanglement for the remainder of

the experiment.

– – 24 –

Rm

B (π, 0) √1

2

|↑iA

h

|↑iB |0iA + ieiφg(t)|↓iB |1iA

i

|0iB 14 Motion→spin transfer pulse in well B

– – 38 First T delay of the second spin-echo

sequence

Rc

B(π, 0) √1

2

|↑iA

h

|↓iB |0iA + ieiφg(t)|↑iB |1iA

i

|0iB 4 Second spin-echo pulse in well B

– – 39 Second T delay of the second spin-echo

sequence

Rm

A (π, φA) √1

2

h

|↑iA |↓iB + e

i(φg(t)+φA)

|↓iA |↑iB

i

|0iA |0iB 11 Motion → spin transfer pulse in well A

Recombine √1

2

h

|↑↓i + |↓↑ii1219 Recombine all ions to the same well

24Mg+ Doppler cool √1

2

h

|↑↓i + |↓↑ii

400 –

– – 22 –

R(π, 0) AB |2, 0i |2, 0i ⇒ AB |2, −1i |2, −1i 3

Further shelve the residual spin amplitudes

Continued on next page
218

Supplementary Table 1: (continued)

Operation Ideal State after Operation Time (μs) Notes

– – 22 –

R(π, 0) AB |2, −1i |2, −1i ⇒ AB |2, −2i |2, −2i 4

Further shelve the residual spin amplitudes

– – 22 –

Rc(

π

2

, φD) √1

2

h

|↑↑i + i|↓↓ii1

Rotate into the detection basis

(φD ≈ − 3π

4

)

– – 6 –

Rc(

π

2

, φp)

1

2

h

(cos φp − sin φp)



e−iφp |↑↑i − e

iφp |↓↓i

+

(cos φp + sin φp) (|↑↓i + |↓↑i)

i 1 Analysis pulse with variable phase φp

– – 79 –

R(π, 0) |↓i ⇒ |2, 0i 3 Shelve any |↓i (dark) population

– – 22 –

R(π, 0) |2, 0i ⇒ |2, −1i 3 Further shelve the dark population

– – 22 –

R(π, 0) |2, −1i ⇔ |2, −2i 4

Further shelve the dark population (and

make any residual spin population A

and B less dark)

– – 43 –

Detect P↑↑, P↓↓, P↑↓ + P↓↑ 200 Determine the spin populations

Total Time ≈14 ms ≈600 laser pulses
Appendix D

Ion Trap Wafer Dimensions

1.010 mm

1.25 mm

1.010 mm

1.25 mm1.

5

0

m

m

1.50 mm

4.0

0

0

m

m

6.000 mm

1.20 mm

2.40 mm

1.5

0

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n 1.25 mm

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50 m

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Appendix E

Other Contributions

This section contains the other work not highlighted in this disseration where I

was one of the lead co-authors.

• Realization of a programmable two-qubit quantum processor , Hanneke

D., Home J. P., Jost J. D., Amini J. M., Leibfried D. & Wineland D. J. , Nature

Physics, 2010, 1, 13-16

• Complete Methods Set for Scalable Ion Trap Quantum Information

Processing, Home J. P., Hanneke D., Jost J. D., Amini J. M., Leibfried D., &

Wineland D. J., Science, 2009, 308, 997-1000

• Long-lived qubit memory using atomic ion, Langer C., Ozeri R., Jost J.

D., Chiaverini J., DeMarco B., Ben-Kish A., Blakestad R. B., Britton J., Hume

D. B., Itano W. M., Leibfried D., Reichle R., Rosenband T., Schaetz T., Schmidt

P. O., & Wineland D. J., Phys. Rev. Lett., 2005, 95, 060502-1-4

• Hyperne coherence in the presence of spontaneous photon scattering,

Ozeri R., Langer C., Jost J. D., DeMarco B. L., Ben-Kish A., Blakestad B. R.,

Britton J., Chiaverini J., Itano W. M., Hume D., Leibfried D., Rosenband T.,

Schmidt P, & Wineland D. J., Phys. Rev. Lett., 2005, 95, 030403-1-4

This section contains the other work not highlighted in this disseration where I was one

of the co-authors.
224

• Randomized benchmarking of quantum gates, E. Knill, D. Leibfried, R.

Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin,

and D. J. Wineland, Phys. Rev. A 77, 012307-1 - 012307-7 (2008).

• Towards scaling up trapped ion quantum information processing, D.

Leibfried, D. J. Wineland, R. B. Blakestad, J. J. Bollinger, J. Britton, J. Chiaverini, R. J. Epstein, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, R.

Reichle, S. Seidelin, N. Shiga, and J. H. Wesenberg, Hyperne Interactions 174,

1 - 7 (2007). Proc. 4th Int. Conf. Trapped Charged Particles and Fundamental

Physics (TCP 2006), Parksville, Canada 3-8 Sept. 2006.

• Simplied motional heating rate measurements of trapped ions, R. J.

Epstein, S. Seidelin, D. Leibfried, J. H. Wesenberg, J. J. Bollinger, J. M. Amini,

R. B. Blakestad, J. Britton, J. P. Home, W. M. Itano, J. D. Jost, E. Knill, C.

Langer, R. Ozeri, N. Shiga, and D. J. Wineland, Phys. Rev. A 76, 033411-1 -

033411-5 (2007).

• Fluorescence during Doppler cooling of a single trapped atom, J. H.

Wesenberg, R. J. Epstein, D. Leibfried, R. B. Blakestad, J. Britton, J. P. Home,

W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, S. Seidelin, and D. J.

Wineland, Phys. Rev. A 76, 053416-1 - 053416-11 (2007).

• Errors in trapped-ion quantum gates due to spontaneous photon scattering, R. Ozeri, W. M. Itano, R.B. Blakestad, J. Britton, J. Chiaverini, J.D.

Jost, C. Langer, D. Leibfried, R. Reichle, S. Seidelin, J. H. Wesenberg, and D.J.

Wineland, Phys. Rev. A 75, 042329-1 - 042329-14 (2007).

• Trapped atomic ions and quantum information processing, D. J. Wineland,

D. Leibfried, J. C. Bergquist, R. B. Blakestad, J. J. Bollinger, J. Britton, J. Chiaverini, R. J. Epstein, D. B. Hume, W. M. Itano, J. D. Jost, M. Knill, J. C.
225

J. Koelemeij, C. Langer, R. Ozeri, R. Reichle, T. Rosenband, T. Schaetz, P.

O. Schmidt, S. Seidelin, N. Shiga, and J. H. Wesenberg, in Atomic Physics 20,

Proceedings of the International Conference (ICAP 2006), Innsbruck, Austria,

16-21 July 2006, edited by C. Roos, H. Haner, and R. Blatt, AIP COnf. Proc.

869 (American Institute of Physics, 2006) p. 103 - 110.

• Transport dynamics of single ions in segmented microstructured Paul

trap arrays, R. Reichle, D. Leibfried, R.B. Blakestad, J. Britton, J.D. Jost, E.

Knill, C. Langer, R. Ozeri, S. Seidelin, and D.J. Wineland, Fortschr. Phys. 54,

666 - 685 (2006).

• Experimental purication of two-atom entanglement, R. Reichle, D.

Leibfried, E. Knill, J. Britton, R.B. Blakestad, J. Jost, C. Langer, R. Ozeri,

S. Seidelin, and D.J. Wineland, Nature. 443, 838 - 841 (2006).

• A microfabricated surface-electrode ion trap for scalable quantum information processing, S. Seidelin, J. Chiaverini, R. Reichle, J.J. Bollinger, D.

Leibfried, J. Britton, J. Wesenberg, R.B. Blakestad, R. J. Epstein, D. Hume, W.

M. Itano, J. D. Jost, C. Langer, R. Ozeri, N. Shiga, and D.J. Wineland, Phys.

Rev. Lett. 96, 253003-1 - 253003-4 (2006).

• Recent experiments in trapped-ion quantum information processing

at NIST, J. Chiaverini, M. D. Barrett, R. B. Blakestad, J. Britton, W. Itano,

J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, T. Schaetz, and D.J.

Wineland, Proc. SPIE 6256, 625610-1 - 625610-12 (2006).

• Quantum control, quantum information processing, and quantumlimited metrology with trapped ions, D. J. Wineland, D. Leibfried, M.

D. Barrett, A. Ben-Kish, J. C. Bergquist, R. B. Blakestad, J. J. Bollinger, J.

Britton, J. Chiaverini, B. DeMarco, D. Hume, W. M. Itano, M. Jensen, J. D.
226

Jost, E. Knill, J. Koelemeij, C. Langer, W. Oskay, R. Ozeri, R. Reichle, T.

Rosenband, T. Schaetz, P. O. Schmidt, and S. Seidelin, in Laser Spectroscopy,

Proceedings of the 17th International Conference, Aviemore, Scotland, 19-24

June 2005, edited by E. A. Hinds, A. Ferguson, and E. Riis (World Scientic,

Singapore 2005) p. 393 - 402.

• Creation of a six-atom 'Schroedinger cat' state, D. Leibfried, E. Knill, S.

Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano,

J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, Nature. 438,

639-642 (2005).

• Surface-electrode architecture for ion-trap quantum information processing, J. Chiaverini, R. B. Blakestad, J. Britton, J. D. Jost, C. Langer, D.

Leibfried, R. Ozeri, and D.J. Wineland, Quantum Information and Computation

5, 419-439 (2005).

• Implementation of the semiclassical quantum Fourier transform in a

scalable system, J. Chiaverini, J. Britton, D. Leibfried, E. Knill, M. D. Barrett,

R. B. Blakestad, W.M. Itano, J.D. Jost, C. Langer, R. Ozeri, T. Schaetz, and

D.J. Wineland, Science 308, 997-1000 (2005).

• Enhanced quantum state detection eciency through quantum information processing, T. Schaetz, M. D. Barrett, D. Leibfried, J. Britton, J.

Chiaverini, W.M. Itano, J.D. Jost, E. Knill, C. Langer, and D.J. Wineland,

Phys. Rev. Lett. 94, 010501-1 - 010501-4 (2005).

• Quantum information processing with trapped ions, M. D. Barrett, T.

Schaetz, J. Chiaverini, D. Leibfried, J. Britton, W. M. Itano, J. D. Jost, E. Knill,

C. Langer, R. Ozeri, and D.J. Wineland, in Atomic Physics 19, Proceedings

of the International Conference (ICAP 2004), edited by L. G. Marcassa, K.
227

Helmerson, and V. S. Bagnato (American Institute of Physics, 2005) p. 350 -

358.

• Towards a scalable quantum computer/simulator based on trapped

ions, T. Schaetz, D. Leibfried, J. Chiaverini, M. D. Barrett, J. Britton, B.

DeMarco, W. M. Itano, J.D. Jost, C. Langer, and D.J. Wineland, Appl. Phys.

B 79, 979-986 (2004).

• Realization of quantum error correction, J. Chiaverini, D. Leibried, T.

Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W.M. Itano, J.D. Jost, E.

Knill, C. Langer, R. Ozeri, and D.J. Wineland, Nature 432, 602-605 (2004).

• Deterministic quantum teleportation of atomic qubits, M. D. Barrett, J.

Chiaverini, T. Schaetz, J. Britton, W.M. Itano, J.D. Jost, E. Knill, C. Langer,

D. Leibfried, R. Ozeri, and D.J. Wineland, Nature 429, 737-739 (2004).

• Quantum dense coding with atomic qubits, T. Schaetz, M. D. Barrett, D.

Leibfried, J. Chiaverini, J. Britton, W.M. Itano, J.D. Jost, C. Langer, and D.J.

Wineland, Phys. Rev. Lett. 93, 040505-1 - 040505-4 (2004).

• Toward Heisenberg-limited spectroscopy with multiparticle entangled

states, D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W.M.

Itano, J.D. Jost, C. Langer, and D.J. Wineland, Science 304, 1476-1478 (2004).

• Building blocks for a scalable quantum information processor based

on trapped ions, D. Leibfried, M. D. Barrett, A. Ben-Kish, J. Britton, J.

Chiaverini, B. DeMarco, W.M. Itano, B. Jelenkovic, J. D. Jost, C. Langer, D.

Lucas, V. Meyer, T. Rosenband, M. A. Rowe, T. Schaetz, and D.J. Wineland,

in Laser Spectroscopy, Proceedings of the 16th International Conference, Palm

Cove, Australia, 13-18 July, 2003, edited by P. Hannaford, A. Sidorov, H. Bacher,

and K. Baldwin (World Scientic, Singapore, 2004) p. 295 - 303.
228

• Sympathetic cooling of 9Be+ and 24Mg+ for quantum logic, M. Barrett, B.L. DeMarco, T. Schaetz, V. Meyer, D. Leibfried, J. Britton, J. Chiaverini,

W.M. Itano, B.M. Jelenkovic, J.D. Jost, C. Langer, T. Rosenband, and D.J.

Wineland, Phys. Rev. A 68, 042302-1 - 042302-7 (2003).