Student Name _____________________________________ Student ID______________________ 

______________ Degree in ___________________________________________________________

Thesis Title:

Have you performed research involving human subjects which requires approval from the 

Institutional Review Board (IRB)? ___ Yes ___ No 

IRB Protocol Number ________________

Have you used live animals, animal tissue, or observational animal work which requires approval 

from the Institutional Animal Care and Use Committee (IACUC)? ___ Yes ___ No

IACUC Protocol Number _______________

Attach the final copy of thesis/dissertation for committee review. While formatting changes may 

be requested by the Graduate School, the content of the attached document should be final.

Approvals:

Committee Chair Name ______________________________________________________

Signature _____________________________________ Date Signed _________________

Committee Member Name ______________________________________________________

Signature _____________________________________ Date Signed _________________

The final copy of this thesis has been examined by the signatories, and we find that both the 

content and the form meet acceptable presentation standards of scholarly work in the abovementioned discipline.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69

Doctoral

X

Squeezing and amplification of trapped-ion motion

104352524

Physics

David Wineland

Shaun Burd

X

Dietrich Leibfried

7/27/2020

7/27/2020
Squeezing and amplification of trapped-ion motion

by

Shaun C. Burd

B.S., University of the Witwatersrand, 2010

M.S., University of Natal, 2014

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2020

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
This thesis entitled:

Squeezing and amplification of trapped-ion motion

written by Shaun C. Burd

has been approved for the Department of Physics

Dr. Dietrich Leibfried

Prof. David J. Wineland

Date

The final copy of this thesis has been examined by the signatories, and we find that both the

content and the form meet acceptable presentation standards of scholarly work in the above

mentioned discipline.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
iii

Burd, Shaun C. (Ph.D., Physics)

Squeezing and amplification of trapped-ion motion

Thesis directed by Dr. Dietrich Leibfried and Prof. David J. Wineland

In my thesis work I have explored a variety of ways for using parametric modulation to control the dynamics of trapped atomic ions. Parametric modulation allows the wavefunction of a

quantum oscillator to be reversibly distorted or “squeezed” by redistributing quantum fluctuations

between conjugate observables. This enables tailoring of an oscillator’s wavefunction to enhance the

sensitivity of measurements of observables of interest, and permits amplification of the oscillator’s

motion. In our experiment, parametric modulation is implemented by periodic distortion of the

confining potential of the ion trap. With this tool, we can prepare oscillator states with up to 20

decibels of squeezing in 5 μs, a duration short compared to the oscillator coherence time. Furthermore, parametric modulation implements a unitary squeezing operator, enabling nearly noise-free

amplification of harmonic oscillator displacements. Using this amplification method, we have increased the sensitivity for measurements of weak forces by up to 17 decibels. Unitary squeezing also

enables amplification of phonon-mediated interactions between trapped-ion qubits — allowing us

to accelerate the generation of entanglement between qubits. Additionally, I describe experiments

demonstrating “Hamiltonian amplification”, a general method for using parametric modulation to

accelerate dynamics of quantum systems governed by a wide variety of Hamiltonians. The methods demonstrated here can potentially extend the capabilities of many physical platforms used for

precision measurements, quantum information science, and explorations of fundamental physics, in

which harmonic oscillators are essential components.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Dedication

To my wife Caroline, my son James, and my parents.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
v

Acknowledgements

Firstly, I thank my supervisors Didi Leibfried and Dave Wineland for giving me the opportunity to work in the NIST Ion Storage Group. It has been a great privilege to work with some of

the brightest people in experimental physics. I am very grateful that I was allowed and encouraged

to pursue independent research directions. I feel that this allowed me to grow as a physicist. I am

also indebted to John Bollinger for his continued support and advice, and to Andrew Wilson for

helping to keep everything working.

Special thanks go to the other members of the “μCrew”. To my fellow grad student and best

man Raghu Srinivas, we had many thought-provoking discussions about physics, and we also had

fun outside of lab watching movies and playing board games. Thanks also to David Allcock and

Daniel Slichter, my postdocs during most of my time at NIST, two of the smartest people I have

ever met. Although I couldn’t understand what he was saying at first, David eventually taught me

the art of planning and engineering an experiment, and other important things an experimentalist

should know like how (and how not) to solder, how to connect the inputs and outputs of microwave

amplifiers, and how to find tiny screws dropped into the depths of over-engineered equipment.

Daniel showed me the importance of paying attention to detail and how to write the best possible

paper. To Hannah, and Laurent, thanks for reading my thesis, maybe we will finally trap calcium.

In addition to those mentioned above, I had the opportunity to work with many great physicists in the ion-storage group: David Hume, Chin Wen Chou, Ethan Clements, Steven Erickson, Yong Wan, Yiheng Lin, David Leibrandt, Ryan Bowler, Ting Rei Tan, John Gaebler, Kevin

Gilmore, Elena Jordan, Alejandra Collopy, May Kim, Panyu Hou, Jenny Wu, Dan Cole, Sam

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
vi

Brewer, Matt Affolter, Dalton Chaffee, Kaifeng Cui, Jonas Keller, Robert J ̈ordens, Katie McCormick, Susanna Todaro, Justin Niedermeyer, Justin Bohnet, Brian Sawyer, and Joe Britton. I

would also like to thank several of my mentors from my time at the CSIR in South Africa, Christoph

Bollig, who introduced me to real lasers, Musa “Yoda” Mhlanga my supervisor during my foray

into molecular biology, and Hermann Uys, who introduced me to atomic physics and guided me to

NIST. Finally, I would like to thank my family. Thank you to my beautiful wife Caroline, who has

always stood by me; my mother Annette, who raised me and tolerated my many science experiments; and my father Michael for always supporting me every time I decided to change my career

path.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
vii

Contents

Chapter

1 Introduction 1

1.1 Squeezing and metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Squeezed states in trapped ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Beyond metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 The harmonic oscillator 6

2.1 The quantum harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Wigner function representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 The driven harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Parametric modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Displaced squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.8 Thermal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Spins and Springs 25

3.1 Harmonically confined spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Spin-motion coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Spin-dependent forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
viii

3.4 Geometric phase gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 The Mølmer-Sørensen gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.2 ˆσzσˆz gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Motional decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5.1 Motional dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.2 Motional heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Experimental setup 51

4.1 Surface-electrode ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Vacuum chamber and cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 25Mg+ ion qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 State preparation and detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.1 Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.2 Repumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.3 Resolved sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.4 Imaging and detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.5 Micromotion compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5.1 Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5.2 Doppler cooling, detection, and repumping . . . . . . . . . . . . . . . . . . . 66

4.5.3 Raman lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Electronic drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6.1 Microwave drive system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6.2 Resonant motional drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6.3 Parametric drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Laser-free control of the trapped-ion quantum harmonic oscillator 78

5.1 Motional state analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
ix

5.2 Thermal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Number states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.6 Squeezed cat states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Quantum amplification of trapped-ion motion 89

6.1 Noiseless amplification of displacements . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Experimental implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 Quantum-enhanced displacement measurements . . . . . . . . . . . . . . . . . . . . . 96

7 Hamiltonian amplification 103

7.1 Amplification of displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2 Amplification of the Jaynes-Cummings interaction . . . . . . . . . . . . . . . . . . . 108

8 Parametric amplification of boson-mediated interactions 113

8.1 Boson-mediated interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.2 Parametric amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.3 Experimental implementation with trapped-ions . . . . . . . . . . . . . . . . . . . . 119

8.3.1 Calculation of the gate time and detuning for parametrically amplified gates

without decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.3.2 Calibrating sideband Rabi rates . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.3.3 Mølmer-Sørensen (MS) gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.3.4 Calibrating the parametric drive . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.4 Parametrically amplified entanglement generation . . . . . . . . . . . . . . . . . . . . 126

8.5 Speed up and enhanced fidelity in the presence of qubit decoherence . . . . . . . . . 131

8.6 Phase dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
x

9 Conclusion and outlook 136

9.1 Two-mode operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.1.1 Beam splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.1.2 Two-mode squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.2 Continuous-variable quantum information processing . . . . . . . . . . . . . . . . . . 139

9.3 Improving coherence in trapped-ion systems limited by qubit decoherence . . . . . . 140

Bibliography 142

Appendix

A Magnus Expansion 150

B Thermal state density matrix 152

C Matrix elements of the displacement operator 154

D Vertical external-cavity surface emitting lasers 156

D.1 Single-frequency VECSELs for AMO physics . . . . . . . . . . . . . . . . . . . . . . 156

D.2 Infrared VECSEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

D.2.1 Gain-chip cooling block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

D.2.2 Output coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

D.2.3 Birefringent filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

D.2.4 Etalon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

D.2.5 Pump diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

D.3 Infrared VECSEL alignment procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 165

D.3.1 Alignment of the pump diode onto the gain chip: . . . . . . . . . . . . . . . . 165

D.3.2 VECSEL alignment: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

D.3.3 Tuning the laser to the desired wavelength: . . . . . . . . . . . . . . . . . . . 167

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
xi

D.4 Output power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
xii

Tables

Table

8.1 Two-ion sideband Rabi flopping fitting parameters . . . . . . . . . . . . . . . . . . . 122

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
xiii

Figures

Figure

1.1 Phase-space illustration of a squeezing-enhanced position measurement . . . . . . . . 2

2.1 Wavefunctions of the quantum harmonic oscillator . . . . . . . . . . . . . . . . . . . 9

2.2 Wigner functions of the first few harmonic oscillator number states . . . . . . . . . . 11

2.3 Number state probabilities and Wigner function for a coherent state. . . . . . . . . . 12

2.4 Coherent state in the lab frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Driven oscillator without rotating wave approximation. . . . . . . . . . . . . . . . . . 16

2.6 Detuned oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Number state probabilities and Wigner function for a squeezed state. . . . . . . . . . 18

2.8 Number state probabilities and Wigner function for a displaced squeezed state. . . . 22

2.9 Number state probabilities and Wigner function for a thermal distribution . . . . . . 23

3.1 The spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Harmonically confined spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Energy levels and important transitions of a harmonically confined qubit. . . . . . . 32

3.4 Number state probabilities and Wigner function for a cat state. . . . . . . . . . . . . 36

3.5 Spin-dependent force applied to number states and thermal distributions. . . . . . . 36

3.6 Number state probabilities and Wigner function for a squeezed cat state. . . . . . . 38

3.7 Spin-dependent force applied to squeezed thermal states. . . . . . . . . . . . . . . . . 38

3.8 Geometric phase gate: phase space illustration. . . . . . . . . . . . . . . . . . . . . . 42

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
xiv

3.9 Populations and Bell state-fidelity as a function of the gate detuning for the MølmerSørensen entangling gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Bell-state fidelity as a function of the Mølmer-Sørensen gate detuning for various

values of the initial thermal occupation. . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.11 Simulated fidelity as a function of the Mølmer-Sørensen gate duration in the presence

of motional dephasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.12 Simulated Mølmer-Sørensen gate duration and detuning for maximized fidelity in

the presence of motional dephasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.13 Simulated Bell-state fidelity as a function of Mølmer-Sørensen gate detuning in the

presence of motional heating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.14 Simulated Mølmer-Sørensen gate duration and detuning for maximized fidelity in

the presence of motional dephasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 RF Paul traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Surface-electrode ion trap for implementing near-field magnetic field gradient entangling gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Pillbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Partial CAD rendering of the vacuum chamber and cryostat . . . . . . . . . . . . . . 56

4.5 Annotated photograph of the vacuum chamber, cryostat, and imaging system . . . . 56

4.6 25Mg+ energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.7 Illustration of resolved sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.8 Red and blue sidebands after ground state cooling . . . . . . . . . . . . . . . . . . . 62

4.9 One-ion detection histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.10 Schematic diagram indicating laser access to the vacuum chamber . . . . . . . . . . 65

4.11 Frequency stabilized 559 nm system . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.12 Doppler-free iodine spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.13 AOM switch yard for controlling Doppler cooling, detection, and repump light . . . 68

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
xv

4.14 Raman laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.15 Electronic drive configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.16 Magnetic field near a current carrying wire of infinite length . . . . . . . . . . . . . 72

4.17 Microwave drive system for generating carrier and sideband signals . . . . . . . . . 73

4.18 Parametric drive electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1 Red sideband analysis of a thermal state . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Number states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Sideband Rabi frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Blue sideband analysis of a coherent state . . . . . . . . . . . . . . . . . . . . . . . . 83

5.5 Blue sideband analysis of a squeezed state analysis . . . . . . . . . . . . . . . . . . . 85

5.6 Squeezed cat states in phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.7 Squeezed cat state experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1 Ideally noiseless parametric amplification illustration . . . . . . . . . . . . . . . . . . 91

6.2 Unitary squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.3 Fock state population analysis at each stage of the amplification protocol . . . . . . 94

6.4 Amplification gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5 Bloch sphere illustration of the phase-sensitive red sideband readout scheme . . . . . 97

6.6 Measurement sensitivity enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.1 Hamiltonian amplification pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 Hamiltonian amplification of a coherent displacement . . . . . . . . . . . . . . . . . . 108

7.3 Hamiltonian amplification of the Jaynes-Cummings interaction: Numerical simulations110

7.4 Hamiltonian amplification of the Jaynes-Cummings interaction: Amplification gain . 111

7.5 Hamiltonian amplification of the Jaynes-Cummings interaction: Experimental data . 112

8.1 Phase space illustration of boson-mediated interactions . . . . . . . . . . . . . . . . . 114

8.2 Red sideband flopping on the ground-state cooled rocking mode . . . . . . . . . . . . 121

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
xvi

8.3 Mølmer-Sørensen interaction entangling gate two-ion populations and parity oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.4 Two ion detection histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.5 Red sideband flopping on the squeezed rocking mode . . . . . . . . . . . . . . . . . . 125

8.6 Parametric drive strength g as a function of the rocking mode frequency . . . . . . . 126

8.7 Parametrically amplified Mølmer-Sørensen interaction . . . . . . . . . . . . . . . . . 127

8.8 Bell-state fidelity of an accelerated gate as a function of the interaction time tI and

detuing δ for g = 50(1) kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.9 Bell-state fidelity as a function of t0/tI , the ratio of the gate time without parametric

modulation to the interaction time at which the fidelity is measured . . . . . . . . . 129

8.10 Speedup as a function of the parametric drive amplitude g . . . . . . . . . . . . . . . 130

8.11 Two-ion Ramsey experiment in the presence of induced qubit decoherence: Two-ion

populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.12 Two-ion Ramsey experiment in the presence of induced qubit decoherence: Bright

fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.13 Bell-state fidelity as a function of the parametric coupling strength g in the presence

of qubit dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.14 Interaction duration for maximum fidelity tImax as a function of the parametric drive

phase θ for g/2π = 12.75(4) kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.15 Fidelity as as function of the parametric drive phase θ for fixed parametric coupling

strength g, interaction time tI and detuning δ . . . . . . . . . . . . . . . . . . . . . . 135

D.1 Schematic diagram of a vertical external-cavity surface-emitting laser . . . . . . . . . 157

D.2 Semiconductor gain chip diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

D.3 IRVECSELphoto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

D.4 Cooling block photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

D.5 Birefringent filter photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
xvii

D.6 Etalon oven photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

D.7 Photograph of a 1120 nm gain chip under illumination with light from an 809 nm

pump diode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

D.8 Photograph of an 1120 nm gain chip showing the pump spot of the 809 nm pump

diode and a defect visible as a white spot. . . . . . . . . . . . . . . . . . . . . . . . . 166

D.9 VECSEL output power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 1

Introduction

Control of systems at the quantum level has led to advances such the observation of exotic

states of matter [1], precision measurements of unprecedented accuracy [2], and the development

of quantum computers and simulators [3, 4, 5]. In many physical platforms for quantum science,

harmonic oscillators are essential components and are used directly as sensors or to mediate interactions between other quantum objects. Expanding the toolbox of methods for controlling a quantum

oscillator will likely lead to an improved understanding of quantum physics and to novel applications. In this thesis, I present new methods for controlling quantum states of trapped-ion harmonic

oscillators. While laser-based methods have traditionally been used for high-fidelity manipulation

of both the internal and oscillator degrees of freedom of trapped ions, here I describe methods that

use magnetic and electric fields at much lower frequencies for controlling trapped ion dynamics.

In particular, modulation of electric field gradients enables the generation of squeezed harmonic

oscillator states, which can be used as a resource for quantum-enhanced metrology. This parametric

modulation produces a unitary squeezing operation that has been challenging to implement using

laser methods. Beyond generating metrologically useful states, unitary squeezing enables amplification of interactions between harmonic oscillators and external fields and also enables amplification

of interactions between quantum objects that are mediated by harmonic oscillators, which are essential for quantum-information processing in a variety of physical platforms. While the experiments

described in this thesis were implemented using trapped ions, these techniques should be applicable

to any system where the required interactions can be generated.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
2

a) Coherent state b) Squeezed state

p

x

p

x

∆x ∆xs

x x

Figure 1.1: Phase-space illustration (not to scale) of a squeezing enhanced position measurement.

a) Coherent state with position uncertainty ∆x displaced by x. b) Squeezed state where the

uncertainty ∆xs for measuring the system position has been reduced compared to the coherent

state.

1.1 Squeezing and metrology

Quantum-enhanced metrology aims to use intrinsically quantum effects such as entanglement

and superposition for increasing the sensitivity of measurements beyond what could be achieved

using classical methods. One method for obtaining quantum-enhanced measurement sensitivity is

the method of squeezing. The key concept is illustrated in Figure 1.1. Consider measurements of

two non-commuting observables such as the position x and momentum p of a mechanical oscillator.

The uncertainties for measuring these quantities ∆x and ∆p are constrained by the Heisenberg

uncertainty principle ∆x∆p ≥ ~/2. By squeezing the oscillator, the uncertainties for measuring

these observables can be redistributed, allowing greater sensitivity for measuring one observable at

the expense of increased fluctuations of the other.

The seminal work of Caves [6] was the first detailed proposal describing how squeezed light

can improve the sensitivity of gravitational wave detectors. Work by Wineland et al. [7, 8], and

Kitagawa and Ueda [9] laid out the theory of spin squeezing, paving the way towards quantumenhanced atomic clocks and atom interferometers. Squeezed states are now routinely created in a

variety of physical systems including electromagnetic fields [10], spin systems [11, 12, 13, 14, 15],

micro-mechanical oscillators [16, 17, 18], and the motional modes of single trapped ions [19, 20].

However, exploiting squeezing for enhanced metrology has proven to be challenging. One reason is

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
3

that large squeezed states can be very sensitive to perturbations from the environment that result

in their decoherence. Another more subtle reason is that a detection method with noise below

the squeezed noise is typically required to take advantage of squeezing. These challenges are not

fundamental and are being overcome by technological improvements. Almost forty years after the

original proposal [6], squeezed light was used in LIGO to give a 2.2 dB sensitivity enhancement

[21]. Furthermore spin-squeezed atom interferometers [22] and optical atomic clocks [23] are on the

horizon.

1.2 Squeezed states in trapped ions

Trapped atomic ions form one of the most pristine platforms for experimental quantum

physics. The ions’ motion can be described as an almost ideal quantum harmonic oscillator and

specially chosen pairs of internal electronic states can serve as qubits for quantum computation or

simulation applications. Both squeezed states of the harmonic motion of ion crystals, and spinsqueezed states of hundreds of ions [24] have been realized experimentally. Squeezing of the motional

state has been accomplished using several methods. Historically, the proposal of Heinzen and

Wineland [25] was the first to describe using parametric modulation for squeezing and amplification

of the motion of trapped charged particles. Resonant parametric modulation involves applying an

oscillating potential at twice the motional frequency to squeeze the motional state. While electronic

(laser-free) parametric modulation has been used to squeeze the thermal motion of of single ions [26]

and for phase-sensitive parametric amplification of highly displaced thermal states [27], it has not

previously been implemented with ions cooled close to the motional ground state. Optical forces

can also be used for parametric modulation [19, 28], but decoherence due to photon scattering

and higher order nonlinearities in the optical field have limited the achievable squeezing [19]. An

alternative method known as reservoir engineering [16, 20] relies on dissipative processes to prepare

an oscillator in a squeezed state.

A key enabling technology for the experiments described in this thesis is the implementation

of direct electronic parametric modulation in a surface-electrode ion trap, with ions initially cooled

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
4

close to the motional ground state. With this method we can rapidly prepare and manipulate large

squeezed states. In contrast to methods for preparing squeezed states that rely on dissipation,

parametric modulation is unitary, enabling reversal of the squeezing and ideally noiseless phasesensitive amplification of ion motion. For sensing applications, noiseless amplification can reduce the

requirements of low noise detection. In a force sensing protocol we use squeezing-based amplification

to improve the measurement sensitivity by a factor of 17(3) dB.

1.3 Beyond metrology

The applications of parametric modulation in quantum systems extends far beyond the generation of squeezed states. In trapped-ion and other experimental platforms for quantum information

processing and simulation, interactions between qubits are often mediated by bosons - excitations

of a shared harmonic oscillator coupled to the qubits. These interactions can be amplified by parametrically modulating the shared harmonic oscillator. We demonstrate this by using parametric

driving to speed up the generation of a maximally entangled Bell state between two trapped-ion

qubits. More generally, parametric modulation can be used to increase coupling strengths in a

large class of Hamiltonians involving harmonic oscillator degrees of freedoms, and should enable

exploration of new parameter regimes for fundamental studies of quantum mechanics, and improved

quantum information processors and simulators.

1.4 Thesis structure

Chapter 2 reviews the theory of quantum harmonic oscillators with a focus on parametric

modulation and squeezing in trapped ion systems.

Chapter 3 gives an overview of spin-motion coupling using magnetic field gradients including

the theory of spin-dependent forces and trapped-ion entangling gates.

Chapter 4 presents the experimental system. This includes descriptions of the 25Mg+ ion

internal structure, the surface-electrode ion trap, vacuum system, laser systems, microwave systems,

and parametric drive.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
5

Chapter 5 discusses the preparation and analysis of various harmonic oscillator states including number states, coherent states, and squeezed states, using laser-free methods.

Chapter 6 describes an experimental implementation of phase-sensitive amplification of coherent displacements of a single trapped ion.

Chapter 7 extends the technique discussed in Chapter 6 to allow phase-insenstive amplification and describes an experimental demonstration of “Hamiltonian amplification” a method for

accelerating dynamics for a large class of Hamiltonians.

Chapter 8 describes parametric amplification of phonon-mediated interactions between two

trapped-ion qubits leading to a reduction in the time required to implement an entangling gate.

Chapter 9 summarizes key results of the thesis and discusses prospective outlooks.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 2

The harmonic oscillator

Harmonic oscillators are ubiquitous in nature. Systems as seemingly varied as a mass on a

spring, an electromagnetic field in a cavity, and the vibrations of atoms in molecules can all be

described as harmonic oscillators. The harmonic oscillator is defined as a system that is subject

to a restoring force proportional to the displacement of the oscillator from its equilibrium position.

For a one-dimensional oscillator where the displacement from equilibrium is denoted by x, the

restoring force is given by Hooke’s law: F = −kx, where k is the spring constant. This corresponds

to the conservative potential V (x) = 1

2

kx2. Summing the kinetic and potential energies gives the

Hamiltonian for a classical harmonic oscillator:

H =

1

2m

p

2 +

1

2

mω2x

2

, (2.1)

where m is the mass of the oscillator, ω =

q

k

m

is the oscillator frequency, and p is the momentum of

the oscillator. The total energy of the oscillator E and the position and momentum are constrained

to fall on ellipses in phase space given by

E =

1

2m

p

2 +

1

2

mω2x

2

. (2.2)

The position and momentum oscillate sinusoidally in time according to

x(t) = A sin(ωt + φ),

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
7

and,

p(t) = Amω cos(ωt + φ), (2.3)

where the amplitude A and phase φ depend on the initial conditions. The motion can be described

in terms of the dimensionless variables

x ̃ =

1

2

r

2mω

~

x, (2.4)

and

p ̃ =

1

2

r

2

m~ω

p. (2.5)

In this picture, the oscillator follows circular trajectories in phase space with amplitude α =

p

mw/2~A. Planck’s constant ~ has been used in the definitions of  ̃x and  ̃p in anticipation of

the phase-space description of the quantum harmonic oscillator. However, ~ can be replaced by

any constant with units of angular momentum to give a dimensionless transformation.

In the next sections we move to the quantum mechanical description of the harmonic oscillator. After reviewing the theory of the quantum oscillator, I will describe several important

quantum states of the harmonic oscillator, their properties, and how they can be generated.

2.1 The quantum harmonic oscillator

The Hamiltonian for the quantum harmonic oscillator is obtained from the classical one by

replacing the canonical variables x and p with operators ˆx and ˆp, giving

Hˆ =

1

2m

pˆ

2 +

1

2

mω2xˆ

2

. (2.6)

The canonical operators satisfy the commutation relation [ˆx, pˆ] = i~. The harmonic oscillator

Hamiltonian can be diagonalized by expressing ˆx and ˆp in terms of the raising and lowering operators

aˆ and ˆa

† as ˆx = x0(ˆa + ˆa

†

), where x0 =

q

~

2mω

, and ˆp = ip0(ˆa

† − aˆ), where p0 =

q

~mω

2

. The

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
8

creation and annihilation operators satisfy the commutation relations [ˆa, aˆ

†

] = 1. The Hamiltonian

can then be written as

Hˆ = ~ω



nˆ +

1

2



, (2.7)

where ˆn = ˆa

†aˆ. The energy eigenvalues En = ~ω(n +

1

2

) are evenly spaced, with integer n ≥ 0.

The eigenstates of the harmonic oscillator (also known as number states) are denoted by |ni. One

of the striking features of the quantum oscillator is that even in its lowest possible energy state

|n = 0i, there are fluctuations of the measured position and momentum. To see this consider the

uncertainty in observing ˆx: σ(ˆx) = q

hxˆ

2i − hxˆi

2

. In the motional ground state, h0|xˆ|0i = 0 and

h0|xˆ

2

|0i = x

2

0

giving σ(ˆx) = x0. Similarly, the uncertainty in observing ˆp for an oscillator prepared

in the motional ground state is σ(ˆp) = p0. The uncertainties of any non-commuting observables are

fundamentally linked by the Heisenberg uncertainty principle. The Heisenberg uncertainty relation

for an oscillator in the grounds state is given by

σ(ˆx)σ(ˆp) ≥

1

2

| h[ˆx, pˆ]i | =

~

2

(2.8)

Since σ(ˆx)σ(ˆp) = ~/2, the uncertainty principle is saturated and the ground state is said to be a

minimum uncertainty state. It is also useful to know the underlying probability distribution for

position measurements. The probability of finding the oscillator at position x given that it was

initially in a number state n is given by Px = | hx|ni |2

. The projections of the eigenstates onto

eigenfunctions |xi of the position operator hx|ni are known as wavefunctions. The wavefunctions

of the harmonic oscillator are given by [29]

hx|ni =

1

√

2

nn!



1

2πx2

0

1/4

e

−



x

2x0

2

Hn



x

√

2x0



(2.9)

where Hn are the Hermite polynomials. Several of these wavefunctions are shown in Fig. 2.1. For

the special case of the motional ground state, the wave function is a Gaussian given by

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
9

6 4 2 0 2 4 6

x/x0

0

2

4

6

8

10

E

n

erg

y

/( 2

)

n = 0

n = 1

n = 2

n = 3

n = 4

Figure 2.1: Wavefunctions of the quantum harmonic oscillator.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
10

hx|0i =



1

2πx2

0

1/4

e

−



x

2x0

2

. (2.10)

The probability for measuring the position x when the oscillator was prepared in the ground state

is

P0(x) = | hx|0i |2 =



1

2πx2

0

1/2

e

− 1

2



x

x0

2

, (2.11)

which is a Gaussian of standard deviation x0 as expected.

2.2 Wigner function representation

The states of a quantum harmonic oscillator can also be visualized in phase space using a

quasi-probability distribution function [30]. Although there are several quasi-probability distribution functions, the Wigner distribution [31] is the most widely used and is most suitable for the

quantum states described in this thesis. For an arbitrary density operator ˆρ, the Wigner function

is defined as

W(x, p) ≡

1

2π~

Z ∞

−∞

hx +

1

2

q|ρˆ|x −

1

2

qi e

ipq/~

dq, (2.12)

where |x ±

1

2

qi are position eigenstates. The Wigner function for the n

th number state is given by

[32]

W(n, α) = 2

π

(−1)nLn(4|α|

2

)e

−2|α|

2

, (2.13)

where Ln is a Laguerre polynomial of degree n, and α = x/2x0 +ip/2p0 is a complex dimensionless

phase-space coordinate. Example Wigner functions of pure number states are plotted in Fig. 2.2.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
11

2.5 0.0 2.5

Re( )

4

2

0

2

4

Im( )

n = 0

2.5 0.0 2.5

Re( )

n = 1

2.5 0.0 2.5

Re( )

n = 2

2.5 0.0 2.5

Re( )

n = 3

-0.6

-0.3

0

0.3

0.6

Figure 2.2: Wigner functions of the first few harmonic oscillator number states.

2.3 Coherent states

Coherent states have been described as the most classical-like states of the harmonic oscillator [33]. A coherent state |αi is defined as the eigenstate of the annihilation operator ˆa with

eigenvalue α, mathematically ˆa |αi = α |αi. The coherent state can be expressed in the number

state basis as

|αi = e

−|α|

2

2

X∞

n=0

α

n

√

n!

|ni. (2.14)

The probability of finding the oscillator in state |ni after it was prepared in |αi is given by the

Poisson distribution

Pn = | hn|αi |2 = e

−|α|

2 |α|

2n

n!

, (2.15)

which has a mean value of hnˆi =

P

n nPn = |α|

2

equal to its variance. The probabilities Pn

for an example coherent state are shown in Fig. 2.3 a. The expectation values of the position

and momentum of the coherent state are hα|xˆ|αi = 2x0Re{α} and hα|pˆ|αi = 2p0Im{α}. The

uncertainties for measurements of the position are σ(ˆx) = x0 and σ(ˆp) = p0. The coherent state

therefore has the same phase-space extent as the ground state, but the center of the wavepacket

is shifted from the origin by α. This is clear from the coherent state Wigner function (Fig. 2.3 b)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
12

0 10 20

n

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

P

n

0 1 2 3

Re( )

0

1

2

3

Im( )

| |

0.0

0.2

0.4

0.6

a) b)

Figure 2.3: a) Number state probabilities and b) Wigner function for a coherent state with α = 2+2i

given by [32]

W(α, β) = 2

π

e

−2|α−β|

2

. (2.16)

Coherent states can be generated from the motional ground state by the action of the displacement operator |αi = Dˆ(α)|0i where

Dˆ(α) = e

αaˆ

† − α∗

aˆ

. (2.17)

The displacement operator has some very interesting (and useful!) properties. A displacement

followed by a displacement in the opposition direction in phase space brings the system back to

where it started, i.e. Dˆ†(α)Dˆ(α) = Dˆ(−α)Dˆ(α) = ˆI. The displacement operator is therefore a

unitary operator. The following identities are especially useful:

Dˆ†(α)ˆaDˆ(α) = ˆa + α

Dˆ(α)ˆaDˆ†(α) = ˆa − α, (2.18)

If a displacement by α is followed by a displacement by β,

Dˆ(β)Dˆ(α) = e

1

2

(βα∗−β

∗α)Dˆ(α + β) = e

iIm[βα∗]Dˆ(α + β) (2.19)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
13

This can be proved by using the Baker-Campbell-Hausdorff formula [29]. The phase factor is

a geometric phase that depends on the area circumscribed by the path in phase space. If the

displacements follow a rectangular closed path

Dˆ(−β)Dˆ(−α)Dˆ(β)Dˆ(α) = ˆIe

1

2

(βα∗−β

∗α)

e

1

2

(βα∗−β

∗α) = ˆIe2iIm[βα∗]

, (2.20)

where ˆI is the identity operator, which shows that the geometric phase is equal to the area of

the enclosed shape. The presence of this geometric phase is essential for generating entanglement

between trapped-ion qubits as described in Chapter 3.

It is useful to consider the displacement operator from a slightly different perspective. From

introductory quantum mechanics, the generator of translations in position space is momentum

and conversely, the generator of translations in momentum space is the position. For a position

translation by an amount xd, the translation operator is

Tˆ(xd) = exp 

−ixdpˆ

~



. (2.21)

For the harmonic oscillator, this becomes

Tˆ(xd) = exp xdp0

~

(ˆa

† − aˆ)



= exp 

xd

2x0

(ˆa

† − aˆ)



= Dˆ



xd

2x0



. (2.22)

Which shows that a displacement by α along the real axis in phase space corresponds to a position

translation of xd = 2x0α. A similar line of reasoning applies to the momentum and further motivates

why Dˆ(α) is called a displacement operator.

2.4 The driven harmonic oscillator

We have seen that a coherent state can be generated by the action of the displacement operator on the ground state. I will now describe how the displacement operator can be implemented

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
14

using an oscillating classical force with frequency ωd [33]. If the force can be derived from the

potential Vˆ (t) = −Fxˆ sin(ωdt + φ), the Hamiltonian describing the dynamics is given by

Hˆ = ~ω



aˆ

†

aˆ +

1

2



− Fxˆ sin(ωdt + φ)

= ~ω



aˆ

†

aˆ +

1

2



− F x0(ˆa + ˆa

†

) sin(ωdt + φ). (2.23)

Transforming into the interaction picture [29] with respect to Hˆ

0 = ~ωaˆ

†aˆ gives

Hˆ

I = e

iHˆ0t/~Vˆ (t)e−iHˆ0t/~

= −F x0(ˆae−iωt + ˆa

†

e

iωt) sin(ωdt + φ).

= −

F x0

2i



aˆ(e

i(ωd−ω)t+iφ − e−i(ω+ωd)t−iφ) − aˆ†

(e

i(ω−ωd)t−iφ − ei(ω+ωd)t−iφ)



. (2.24)

where I have used

e

i

~ Hˆ0t

aeˆ

− i

~ Hˆ0t = ˆae−iωt

e

i

~ Hˆ0t

aˆ

†

e

− i

~ Hˆ0t = ˆa

†

e

iωt

. (2.25)

Provided Ω  ω, where Ω = F x0/~, the counter-rotating terms oscillating at ω+ωd can be dropped

(rotating-wave approximation) and the interaction Hamiltonian reduces to

Hˆ

I = −

~Ω

2i

(ˆae−iδt+iφ − aˆ

†

e

iδt−iφ), (2.26)

where the detuning is defined as δ ≡ ωd−ω. It is worth considering the case where the classical drive

is tuned onto resonance with the harmonic oscillator (ω = ωd). With this condition, the interaction

Hamiltonian (Eq. 2.26) commutes with itself at all times and the time evolution operator is given

by

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
15

3 2 1 0 1 2 3

Re( )

3

2

1

0

1

2

3

Im( )

| |

= t

Figure 2.4: Coherent state in the lab frame after applying a resonant drive for a duration td =

2|α|

Ω

to the ground state. The wavepacket follows circular trajectories with angular frequency ω.

Uˆ

I = exp 

−

i

~

Z t

0

Hˆ

I (t

0

)dt0



= exp 

Ωt

2

(ˆaeiφ − aˆ

†

e

−iφ)



= Dˆ



Ωt

2

e

−iφ

, (2.27)

where the displacement amplitude α(t) = Ωt

2

e

−iφ depends linearly on the duration td the drive is

applied. Although the the coherent state has a fixed phase in the interaction picture, in the lab

frame Uˆ(td) = e

i

~ Hˆ0tdUˆ

Ie

− i

~ Hˆ0td = Dˆ



Ωtd

2

e

iωtd−iφ

the wavepacket will rotate about the origin in

phase space at the frequency ω (Fig 2.4).

What about the validity of the rotating wave approximation? It is straightforward to show

that without dropping any fast rotating terms the time evolution operator is (up to an overall phase

factor)

Uˆ

I = Dˆ



Ωt

2

e

−iφ −

Ω

2ω

sin(ωt)e

i(ωt+φ)



, (2.28)

which shows that the effect of the counter-rotating terms is to superimpose a circular motion on the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
16

0.00 0.25 0.50 0.75 1.00

Re( )

0.0

0.2

0.4

0.6

0.8

1.0

I

m

(

)

/ = 0.01

/ = 0.05

/ = 0.1

Figure 2.5: Phase space trajectories of α(t), in the interaction frame rotating at ω, for a resonantly

driven oscillator without invoking the rotating wave approximation for Ωt ∈ [0, 0.5], for various

values of Ω/ω.

linearly increasing amplitude. The amplitude of the fast circular motion is suppressed by a factor

of Ω/ω (see Fig. 2.5). For the experiments described in this thesis, we typically have Ω/ω ∼ 10−3,

allowing us to safely make the rotating wave approximation for all subsequent analysis.

Next I consider the more general case where the driving force is detuned from the oscillator

frequency. As Eq. 2.26 does not commute with itself at all times for arbitrary δ, the time evolution

operator cannot be derived by direct integration of Hˆ

I . As described in Appendix A, the Magnus

expansion method can be used to evaluate the time evolution operator. Since

[Hˆ

I (t), HˆI (t

0

)] = ~

2Ω2

(sin(δ(t − t

0

))) (2.29)

and

[[Hˆ

I (t), HˆI (t

0

)], Hˆ

I (t

00)] = 0, (2.30)

the Magnus expansion terminates at the first order. The time evolution operator is therefore

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
17

0.5 0.0 0.5 1.0

Re( )

0.0

0.5

1.0

I

m

(

)

/ = 0.1

/ = 0.5

/ = 1.0

/ = 1.5

/ = 2.0

Figure 2.6: Phase space trajectories of α(t), in the interaction frame rotating at ω, for the detuned

driven oscillator for Ωt/2π ∈ [0, 0.49], for various values of δ

Ω

for α(0) = 0.

Uˆ

I (t) = exp 

−

Ω

δ

sin 

δt

2

 h

aeˆ

−iδt/2

e

iφ + ˆa†

e

iδt/2

e

−iφi

× exp −i

Ω

2

δ

2

[δt − sin(δt)]

= Dˆ(α(t))e

iΦ(t)

(2.31)

where the time dependent displacement amplitude is given by

α(t) = −

Ω

δ

sin 

δt

2



e

iδt/2

e

−iφ

, (2.32)

and the geometric phase by

Φ(t) = Ω

2

δ

2

[δt − sin(δt)]. (2.33)

From a phase-space perspective, α(t) follows a path along the arc of a circle of diameter Ω

δ

(Fig. 2.6).

The motional state returns to where it started at t = 0 every tn =

2nπ

δ

where n is an integer and

n > 0. The geometric phase Φ(t) is equal to the area circumscribed by the trajectory and a straight

line from the point α(t) back to α(0) = 0.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
18

0 5 10 15 20

n

0.0

0.1

0.2

0.3

0.4

0.5

0.6

P

n

3 2 1 0 1 2 3

Re( )

3

2

1

0

1

2

3

Im( )

0.0

0.2

0.4

a) b)

Figure 2.7: a) Number state probabilities and b) Wigner function for a squeezed state with r = 1,

θ = 0.

2.5 Squeezed states

The previous section described the generation of coherent states of motion by displacing the

ground state of an oscillator with a near-resonant force. In the rotating frame at the harmonic oscillator frequency, resonant driving serves to translate the oscillator’s wavepacket in phase space, but

leaves its shape unchanged. Controllable distortion of the oscillator’s wavepacket can be generated

using the single-mode squeezing operator defined as

Sˆ(ξ) = e

1

2

(ξ

∗aˆ2−ξaˆ†2

)

, (2.34)

where ξ = reiθ is the squeezing parameter. If the squeezing operator is applied to the harmonic

oscillator ground state, the resulting state |ξi = Sˆ |0i, is known as a squeezed ground state or

squeezed vacuum state. For a squeezed state characterized by the complex squeezing parameter

ξ = reiθ, the number state decomposition is given by

|ξi =

1

√

cosh r

X∞

n=0

e

inθ(− tanh r)np

(2n)!

2

nn!

|2ni, (2.35)

showing that the odd Fock state amplitudes are zero (see Fig. 2.7 a). The even number state

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
19

populations are given by

P2n =

(tanh r)

2n

cosh r

(2n)!

(2nn!)2

. (2.36)

The Wigner function for a squeezed ground state is given by [30]

W(r, θ, α) = 2

π

exp −2e

2r Re[e−iθ/2α]2 − 2e−2r

Im[e

−iθ/2α]2



, (2.37)

which is a two-dimensional Gaussian function (see Fig. 2.7 b).

The action of the squeezing operator on the annihilation and creation operators yields the

Boguliubov transformations [34]:

Sˆ†(ξ)ˆaSˆ(ξ) = ˆa cosh r − aˆ

†

e

iθ sinh r (2.38)

Sˆ†(ξ)ˆa

†Sˆ(ξ) = ˆa

†

cosh r − aeˆ

−iθ sinh r (2.39)

Using these expressions, it is easy to show that the expectation value of the number operator

hni = sinh2r, which shows that the expectation value of the energy grows exponentially with r.

Like the ground state, the squeezed ground state is centered on the origin in phase space and

the expectation values of the position and momentum operators are zero i.e. hξ|xˆ|ξi = 0, and

hξ|pˆ|ξi = 0. Perhaps the most striking feature of the squeezed state is that the uncertainties

along orthogonal phase space quadratures can be very different. The uncertainty in measuring the

position is

σ(ˆx) = x0

r

e

2r(1 − cos θ)

2

+

e−2r(1 + cos θ)

2

. (2.40)

When θ = 0, σ(ˆx) = e

−rx0, which is smaller than the width of the ground state probability

distribution. For θ = π, σ(ˆx) = e

rx0, which is larger than the ground state width. Furthermore

σ(ˆp) = p0

r

e

2r(1 + cos θ)

2

+

e−2r(1 − cos θ)

2

, (2.41)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
20

and the uncertainty product is given by

σ(ˆx)σ(ˆp) = ~

2

r

1 +

1

2

(cosh 4r − 1) sin2θ (2.42)

which is equal to ~/2 when θ = nπ, for n = 0, 1, 2.... As the squeezed state can saturate the

Heisenberg uncertainty relation, it is also a minimum uncertainty state.

2.6 Parametric modulation

The squeezing operator described in the previous section can be implemented by modulating

the curvature of the potential of the quantum oscillator at twice its natural frequency. To see how

this arises, consider the Hamiltonian given by

Hˆ =

pˆ

2

2m

+

1

2

mω2xˆ

2 − ~g sin(ωpt − θ)



xˆ

x0

2

,

= ~ω



aˆ

†

aˆ +

1

2



− ~g sin(ωpt − θ)(ˆa

†2 + ˆa2 + 2ˆa†

aˆ + 1), (2.43)

where ωp is the modulation frequency, g is the parametric modulation strength, θ is the phase of

the modulation drive. Transforming into the interaction picture with respect to the bare (g = 0)

harmonic oscillator Hamiltonian gives the interaction Hamiltonian

Hˆ

I = −~g sin(ωpt − θ)(ˆa

†2

e

2iωrt + ˆa2

e

−2iωrt + 2ˆa†

aˆ + 1). (2.44)

If we make the rotating wave approximation by neglecting terms oscillating at ±(2ωr +ωp) and ωp,

Hˆ

I reduces to

Hˆ

I = −~

g

2i



aˆ

2

e

−2iωrt+iωpt−iθ − aˆ†2

e

2iωrt−iωpt+iθ

. (2.45)

If ωp = 2ωr,

Hˆ

I = i~

g

2

(ˆa

2

e

−iθ − aˆ†2

e

iθ), (2.46)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
21

which is the Hamiltonian describing degenerate parametric amplification of a single-mode boson

field [35]. If this Hamiltonian is applied for a duration t, the time evolution operator Uˆ(t) in the

frame rotating at ω is

Uˆ(t) = e

−iHˆ

I

t

~ = e

gt

2

(ˆa

2e−iθ−aˆ†2eiθ)

. (2.47)

Paramteric modulation therefore implements a squeezing operator Sˆ(ξ) with squeezing parameter

ξ = gteiθ.

2.7 Displaced squeezed states

Displacement and squeezing operations can combined to make more complicated states. For

a state that is prepared by first squeezing the ground state with squeezing parameter ξ and then

displaced by α, the state vector is given by

|ξ, αi = Dˆ(α)Sˆ(ξ)|0i, (2.48)

and the probabilities of finding the oscillator in a given number state are [30]

Pn = | hn|α, ξi |2 =

(

1

2

tanh r)

n

n! cosh r











Hn

 

α cosh r + α

∗

e

iθ sinh r

p

e

iθ sinh(2r)

!









2

× exp −|α|

2 −

1

2

(α

∗2

e

iθ + α2

e

−iθ) tanh r



, (2.49)

where Hn are Hermite polynomials. The number state probability distribution and Wigner function

for a particular displaced squeezed state are shown in Fig. 2.8.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
22

0 5 10 15

n

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

P

n

3 2 1 0 1 2 3

Re( )

3

2

1

0

1

2

3

Im( )

0.0

0.2

0.4

0.6

a) b) Figure 2.8: a) Number state probabilities and b) Wigner function for a state that is first squeezed

with r = 1.0, and the displaced with |α| = 1.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
23

0 5 10

n

0.0

0.1

0.2

0.3

0.4

0.5

P

n

3 2 1 0 1 2 3

Re( )

3

2

1

0

1

2

3

Im( )

0.0

0.1

0.2

a) b)

Figure 2.9: a) Number state probabilities and b) Wigner function for a thermal distribution with

n ̄ = 1.0.

2.8 Thermal distribution

The states described in the previous sections are pure quantum states. In experimentally

relevant situations, the oscillator will have some degree of coupling with its environment, and after

an extended period of time will tend to reach thermal equilibrium with its surroundings. The

oscillator will then be in a mixed state with a density matrix given by [29]

ρˆth =

e

−βHˆ

Z

(2.50)

where Z is the partition function and β = 1/kBT. As described in Appendix B, for a one dimensional

harmonic oscillator

ρˆth = (1 − e

−~ωzβ

)

X∞

n=0

e

−β~ωzn

|ni hn| . (2.51)

In typical ion trapping experiments, the quantity that can be measured is  ̄n = Tr[ˆρthnˆ], which is

related to the temperature T by

T =

~ωz

kB

1

ln[ n ̄

1+ ̄n

]

. (2.52)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
24

Substituting into 2.51 gives

ρˆth =

1

n ̄ + 1

X∞

n=0



n ̄

1 +  ̄n

n

|ni hn| (2.53)

and thus the probabilities for finding the oscillator in a given number state (Fig. 2.9 a) are given

by

Pn =

1

n ̄ + 1 

n ̄

1 +  ̄n

n

. (2.54)

The Wigner function of the thermal state is given by[32]

W( ̄n, α) = 2

π

1

1 + 2 ̄n

e

−2|α|

2/(1+2 ̄n)

, (2.55)

which describes a circular Gaussian in phase space, but with a larger width than that of a

coherent state (Fig. 2.9 b).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 3

Spins and Springs

Trapped ions have been used to construct the world’s most accurate atomic clock [36] and for

pioneering experiments in quantum information processing [4], and quantum simulation [24, 37, 38].

For these and other experiments, controllable couplings between the ions’ motion and internal

electronic states are essential.

In this chapter, I describe the theory of coupling two-level systems or “spins” and harmonic

oscillators. In addition to providing a powerful method for harmonic oscillator state analysis, spinmotion coupling is also required for resolved-sideband cooling, and for implementing spin-dependent

forces that can be used to generate interesting non-classical states such as Schr ̈odinger-cat states

and squeezed cat states. Moreover, spin-dependent forces are essential for generating high-fidelity

entanglement between trapped-ion qubits.

3.1 Harmonically confined spins

Consider a two-level system (TLS) with an energy splitting of ~ω0 (see Fig. 3.1) and energy

eigenstates denoted by |↑i and |↓i. It is convenient to represent the TLS as a spin 1/2 system.

If a static magnetic field Bs~ez applied to define the quantization axis, the Hamiltonian for such a

system, up to an additive constant is given by

Hˆ =

~ω0

2

σˆz, (3.1)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
26

ћω

a) b)

z

x

θ

φ y

Figure 3.1: The spin. a) Two levels with energy separation ~ω with energy eigenstates |↓i and

|↑i. b) Bloch sphere representation of the quantum state of a two-level system. A pure state is

represented by a point (red dot) on the surface of the sphere

where ˆσz = |↑i h↑| − |↓i h↓| is a Pauli operator. A pure state of the TLS can be represented as a

point on the Bloch sphere (Fig. 3.1 b) given by

|Ψi = cos 

θ

2



|↓i + e

iφ sin 

θ

2



|↑i. (3.2)

If the TLS is harmonically confined in one spatial dimension, the Hamiltonian, up to an additive

constant, becomes

Hˆ

0 =

~ω0

2

σˆz + ~ωaˆ

†

a. ˆ (3.3)

This can be visualized as each spin state being associated with a set of harmonic oscillator energy

levels (Fig. 3.2). In a trapped-ion system there is typically no significant intrinsic coupling between

the internal states of the ion and its harmonic oscillator states. However, they can be coupled

together by applying external electric or magnetic field gradients. Here I describe the essential

physics of the latter and leave details of the experimental implementation for Chapter 4

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
27

↑

↑

RSB Carrier BSB

Figure 3.2: Energy levels and transitions of a harmonically confined two-level system. The red

sideband (RSB) induces transitions from |↓i |ni ↔ |↑i |n − 1i. The blue sideband (BSB) induces

transitions from |↓i |ni ↔ |↑i |n + 1i. Carrier transitions change the spin state but leave the

motional state unchanged.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
28

3.2 Spin-motion coupling

We now subject the harmonically confined ion to a sinusoidally oscillating magnetic field

B~ = (Bx~ex + By~ey + Bz~ez) cos(ωdt + φ), (3.4)

where ωd and φ are the drive frequency and phase respectively. As described below, the spin can

be coupled to the oscillator provided the components of B~ vary appropriately in space 1. The

oscillating field results in the Hamiltonian

Vˆ (t) = −ˆ~μ · B~ (t). (3.5)

where ~μ is the dipole moment operator of the TLS. Two relevant situations arise depending on the

difference ∆m of the projection of the total angular momentum of the ion on the quantization axis

between the |↓i and |↑i states. For ∆m = 0,

Vˆ (t) = −(μ||σˆx + μzσˆz + μI

ˆI)Bz cos(ωdt + φ). (3.6)

The first term describes transitions between |↓i and |↑i when a π polarized field (where the direction

of oscillation of the magnetic field is parallel to the quantization axis) is applied to the ion. The

second term is a time varying frequency shift of the TLS, where the transition energy is periodically

modulated due to the difference in magnetic dipole moments (μz) of the two states. The last term

(where ˆI is the 2×2 identity operator) represents the common shift of the levels and does not affect

the dynamics. This term will be neglected in later discussions. For the case where ∆m = ±1

V (t) = −(μ⊥(ˆσxBx + ˆσyBy) + μzσˆzBz + μI

ˆI) cos(ωdt + φ), (3.7)

where now the transverse components of the magnetic field drive transitions between |↓i and |↑i.

The remaining terms result in a time varying frequency shift as in the ∆m = 0 case. We will now

1 Spin-motion coupling can also be achieved using static magnetic field gradients [39]

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
29

focus on the ∆m = 0 case, and return to the ∆m = ±1 case later in the chapter. The z component

of the magnetic field can be Taylor expanded along the direction of the unit vector ~q that is parallel

to the direction of oscillation of a normal mode of motion. The displacement from equilibrium is

described by the operator ˆq = q0(ˆa + ˆa

†

), where q0 =

q

~

2mω

, to give Bz = (Bz0 + (∇B.~q)ˆq + ...).

Higher-order contributions to the Taylor expansion are assumed to be negligible. Furthermore we

will assume that the differential magnetic moment μz = 0 (these assumptions will be justified in

Chapter 4). The Hamiltonian describing the oscillating field can then be written as

Vˆ (t) = −μ|| h

Bz0 + (∇B.~q)q0(ˆa + ˆa

†

)

i

σˆx cos(ωdt + φ) (3.8)

and the combined Hamiltonian for the harmonically confined spin in the oscillating magnetic field

is then

Hˆ = Hˆ

0 + Vˆ (t)

'

~ω0

2

σˆz + ~ωaˆ

†

aˆ − μk

h

Bz0 + (∇B.~q)q0(ˆa + ˆa

†

)

i

σˆx cos(ωdt + φ). (3.9)

Insight into the dynamics resulting from Eq. 3.9 can be gained by transforming into the interaction

picture where

Hˆ

I = e

i

~ Hˆ0tVˆ (t)e

− i

~ Hˆ0t

. (3.10)

Using the Baker-Campbell-Hausdorff formula [29] we obtain

e

i

~ Hˆ0tσˆ+e

− i

~ Hˆ0t = ˆσ+e

iω0t

(3.11)

e

i

~ Hˆ0tσˆ−e

− i

~ Hˆ0t = ˆσ−e

−iω0t

. (3.12)

and we can write the interaction Hamiltonian as

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
30

Hˆ

I = −~Ω0(ˆσ+e

iω0t + ˆσ−e−iω0t

) cos(ωdt + φ)

−~Ω(ˆσ+e

iω0t + ˆσ−e−iω0t

)(ˆae−iωt + ˆa

†

e

+iωt) cos(ωdt + φ) (3.13)

where

Ω0 ≡

μ||Bz0

~

(3.14)

and,

Ω ≡

μ||q0B0

~

, (3.15)

where B0 ≡ ∇B.~q . In the experiments described in this thesis, we typically operate in a regime

where ω0, ωd  ω, and |ω0 + ωd|  |ω − ωd|. Terms oscillating at ±(ω0 + ωd) in Equation 3.13 can

safely be dropped (rotating-wave approximation) and the interaction Hamiltonian becomes

Hˆ

I = −

~

2

(ˆσ+e

i(ω0−ωd)t−iφ + ˆσ−e−i(ω0−ω)t+iφ)

×

h

Ω0 + Ω(ˆae−iωt + ˆa

†

e

+iωt)

i

. (3.16)

There are three particular choices of ωd that are worth describing in detail.

i) Carrier interaction

If the drive is tuned close to the qubit frequency so that |ω0 − ωd|  ω, and Ω  ω, the terms

depending on the motion will be off-resonant and can be dropped, and the interaction Hamiltonian

can be written as

Hˆ

I → HˆC = −

~Ω0

2

(ˆσ+e

iδt−iφ + ˆσ−e−iδt+iφ), (3.17)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
31

where the detuning δ = ω0 − ωd. On resonance (δ = 0), HˆC = −

~Ω0

2

(cos(φ)ˆσx + isin(φ)ˆσy). This

Hamiltonian will generate rotations about an axis in the equatorial plane of the Bloch sphere (Fig.

3.1 b). HˆC therefore induces carrier transitions |↓i |ni ↔ |↑i |ni, which change the spin state, but

leave the oscillator state unchanged. This is illustrated in Figs. 3.2 and 3.3.

ii) Red sideband interaction

If the homogeneous part of the magnetic field is negligible (Bz0 = 0), and Ω  ω, tuning ωd

near ω0 − ω gives

Hˆ

I → HˆRSB =

~Ω

2

(ˆσ+aeˆ

iδt−iφ + ˆσ−aˆ†

e

−iδt+iφ), (3.18)

where δ = ω0−ωd−ω. HˆRSB is known as the red-sideband interaction. It is also formally equivalent

to the Jaynes-Cummings Hamiltonian for an atom coupled to a single electromagnetic field mode in

a cavity [30]. To get a feel for this Hamiltonian consider the case where δ = 0. The time evolution

of the state |ni |↑i generated by HˆRSB is given by

|ψ(t)i = e

− i

~ HˆRSBt

|ni |↑i

=

X∞

n=0 "

(−1)n

(2n)! √

n + 1Ωt

2

2n

|ni |↑i − ie−iφ (−1)n

(2n + 1)! √

n + 1Ωt

2

2n+1

|n + 1i |↓i#

= cos √

n + 1Ωt

2



|ni |↑i − ie−iφ sin √

n + 1Ωt

2



|n + 1i |↓i, (3.19)

which describes Rabi oscillations between states |ni |↑i and |n + 1i |↓i with Rabi frequency √n + 1Ω

(Fig. 3.3). Note that the |0i |↓i state is an eigenstate of HˆRSB with eigenvalue 0, therefore

e

− i

~ HˆRSBt

|0i |↓i = |0i |↓i for all t . This property is exploited in resolved sideband cooling (see

Chapter 4), which enables cooling of an oscillator close to its motional ground state [40, 41].

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
32

Ω

Ω

Ω

n-1

n

n+1

↓

↑ Ω(n+1)1/2

Ωn1/2

Ωn1/2

Ω(n+1)1/2

Figure 3.3: Energy levels of a harmonically confined qubit (not to scale). Carrier transitions

indicated by black arrows change the qubit state, but leave the oscillator state unchanged. Blue

and red sideband transitions, depicted as appropriately colored arrows, change both the qubit and

motional state. Rabi frequencies for sideband transitions are indicated on the transition arrows.

iii) Blue sideband interaction

If the drive frequency is tuned close to the sum of the qubit and motional frequencies ωd =

ω0 + ω, with the restriction Ω  ω,

Hˆ

I → HˆBSB =

~Ω

2

(ˆσ+aˆ

†

e

iδt−iφ + ˆσ−aeˆ−iδt+iφ), (3.20)

where δ = ω0 − ωd + ω. HˆBSB is the blue-sideband or the anti-Jaynes Cummings interaction. For

the case where δ = 0, the state |ni |↓i evolves into

|ψ(t)i = e

− i

~ HˆBSBt

|ni |↓i

=

X∞

n=0 "

(−1)n

(2n)! √

n + 1Ωt

2

2n

|ni |↓i − ie−iφ (−1)n

(2n + 1)! √

n + 1Ωt

2

2n+1

|n + 1i |↑i#

= cos √

n + 1Ωt

2



|ni |↓i − ie−iφ sin √

n + 1Ωt

2



|n + 1i |↑i. (3.21)

So the blue sideband induces transitions between the |ni |↓i and |n + 1i |↑i states. The probability

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
33

of measuring the spin state |↓i after applying HˆBSB for a duration t is given by

P↓(t) = | h↓ |ψ(t)i |2 = cos2

√

n + 1Ωt

2



, (3.22)

which describes Rabi oscillations with a frequency √n + 1Ω. If the oscillator is prepared in a

superposition of number states before the blue sideband is applied

|ψ(t)i = e

− i

~ HˆBSBtX∞

n=0

cn |ni |↓i

=

X∞

n=0

cn cos √

n + 1Ωt

2



|ni |↓i − ie−iφX∞

n=0

cn+1 sin √

n + 1Ωt

2



|n + 1i |↑i, (3.23)

and the probability of measuring |↓i is given by

P↓(t) = X∞

n=0

Pn cos2

√

n + 1Ωt

2



, (3.24)

where the Pn = |cn|

2 are the probabilities of finding the harmonic oscillator in state |ni before

applying the BSB drive. It is straightforward to show that Eq. 3.24 also holds if the oscillator

state is initially in a mixed state described by the density matrix ˆρ =

P∞

n=0,m=0 ρnm |ni hm|. P↓(t)

contains different frequency components weighted according to Pn. Therefore Pn for an unknown

oscillator state can be determined by Fourier transforming P↓(t) [19].

Sequences of sideband and carrier interactions can also be used to prepare number states of

the harmonic oscillator [19, 42]. For example, starting in the state |↓i |0i and applying HˆBSB for

time t = π/Ω generates |↑i |n = 1i. A carrier π pulse then gives |↓i |1i. This sequence of BSB

π pulses (with appropriately chose interaction durations) and carrier π pulses can be repeated to

generate |↓i |ni.

3.3 Spin-dependent forces

The next family of spin-motion coupling interactions that we consider are spin-dependent

forces. These interactions are used to generate non-classical harmonic oscillator states and to

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
34

implement geometric-phase gates. A spin-dependent force that acts in a spin basis on the equator

of the Bloch sphere can be implemented by simultaneously applying a blue and a red sideband

[43]. The frequencies of the sideband drives are set to ωd = ω0 + ω + δ and ωd = ω0 − ω − δ for

the blue and red sideband respectively. If the sidebands have equal Rabi frequencies, the resulting

Hamiltonian after making the RWA, is given by

Hˆ

SDF =

~Ω

2



σˆ+aeˆ

iδt+iφ + ˆσ−aˆ†

e

−iδt−iφ

+

~Ω

2



σˆ−aeˆ

iδt+iφ + ˆσ+aˆ†

e

−iδt−iφ

=

~Ω

2

σˆφ(ˆae+iδt + ˆa

†

e

−iδt), (3.25)

where ˆσφ = cos φσˆx − isin φσˆy. For the remainder of this chapter, the phase φ plays no role, so I

will set it to 0. If this Hamiltonian is applied on resonance (δ = 0), the time evolution operator is

equivalent to a spin-dependent displacement operator

Uˆ(t) = e

−i

Ωt

2

σˆx(ˆa + ˆa

†

)

= Dˆ (α(t)ˆσx), (3.26)

where α(t) = −i

Ωt

2

is the time-dependent displacement amplitude. If the system is initially prepared in |↓i |ni =

1

√

2

(|+i − |−i)|ni, where |±i are the eigenstates of ˆσx, the resulting state is

|ψ(t)i =

1

√

2

h

Dˆ(α(t))|ni |+i − Dˆ(−α(t))|ni |−ii, (3.27)

which shows that the |+i and |−i spin states are displaced in opposite directions in phase space.

If the oscillator is initially prepared in |0i, this coherent superposition of separated oscillator

wavepackets is sometimes called a Schr ̈odinger-cat state [32, 44]. The Wigner function (Fig.3.4)

shows interference fringes with regions of negative quasi probability—a signature of the nonclassical nature of the cat state. These features have been termed sub-Planck structure [45] and can

be much narrower than the ground state wave function extent allowing cat states and variations of

cat states to be used for quantum-enhanced metrology [46, 47]. If a measurement is made on the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
35

spin state, the probability of finding the system in the |↓i state is given by

P↓ =

1

2

h

1 + hn|Dˆ(2α(t))|ni

i

. (3.28)

From Appendix C, the matrix element hn|Dˆ(α)|ni = e

−|α|

2/2Ln(|α|

2

). Substituting into 3.32 gives

P↓ =

1

2

h

1 + e

−2|α(t)|

2

Ln(|2α(t)|

2

)

i

. (3.29)

Using the generating function for the Laguerre polynomials (see Appendix C), P↓(t) for a system

initially prepared in a thermal distribution with motional occupation  ̄n is given by

P↓,th =

1

2

h

1 + e

−4( ̄n+1/2)|α(t)|

2

i

. (3.30)

Figure 3.5 shows P↓ for various values of n and  ̄n for the initial oscillator state. For pure number

states, P↓(t) drops faster than for the ground state and there are clear oscillations, resulting from

the phase space structure of Fock states (see Fig.2.2), before P↓ settles at 0.5. For thermal distributions, the oscillations are damped out from the contributions of multiple Lageurre polynomials

of different orders to P↓. Note that the thermal distribution approaches P↓ = 1/2 faster than the

ground state despite the fact that the thermal state has a larger phase space width than the ground

state. This is because P↓ for a thermal distribution is the average of P↓ taken over a distribution

of number states; and the higher number states have a faster initial decay.

Next we consider the case where the oscillator is prepared in a squeezed number state. After

applying the SDF, the state is given by

|ψ(t)i =

1

√

2

h

Dˆ(α(t))Sˆ(ξ)|ni |+i − Dˆ(−α(t))Sˆ(ξ)|ni |−ii, (3.31)

which is known as a squeezed cat state. An example Wigner function, where the initial squeezing

is in the same direction as the displacement, is shown in Fig. 3.6. The sub-Planck structure is

stretched out along the anti-squeezed axis. This highlights another way of thinking about squeezing:

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
36

0 5 10 15

n

0.0

0.1

0.2

0.3

0.4

P

n

3 2 1 0 1 2 3

Re( )

3

2

1

0

1

2

3

Im( )

0.0

0.5

a) b)

Figure 3.4: a) Number state probabilities and b) Wigner function for the cat state 1

√

2

(|αi + |−αi)

with α = 2i.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.25

0.50

0.75

1.00

P

Number states n = 0

n = 1

n = 3

0.0 0.5 1.0 1.5 2.0 2.5 3.0

| |

0.00

0.25

0.50

0.75

1.00

P

Thermal states

n = 0

n = 1

n = 3

Figure 3.5: P↓ as a function of the spin dependent force displacement amplitude |α| for systems

prepared in pure number states (Eq. 3.29) and thermal distributions (Eq. 3.30)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
37

instead of visualizing squeezing as a distortion of the wavepacket, one can instead think of all

of phase space being compressed in one phase-space quadrature and stretched in the orthogonal

quadrature.

As squeezing redistributes quantum uncertainty between orthogonal quadratures, we expect

P↓ given by

P↓ =

1

2

h

1 + hn|Sˆ†(ξ)Dˆ(2α(t))Sˆ(ξ)|ni

i

, (3.32)

to depend both on the squeezing angle and on the direction of the SDF. Applying Eq. 2.39,

Sˆ†(ξ)Dˆ(β)Sˆ(ξ) = Sˆ†(ξ)e

βaˆ

† − β∗

aˆSˆ(ξ)

= Dˆ(β cosh r + β

∗

e

iθ sinh r). (3.33)

The matrix element can now be evaluated using the results from Appendix C,

hn| Dˆ(β cosh r + β

∗

e

iθ sinh r)|ni = Ln(|β cosh r + β∗

e

iθ sinh r|2

)e

− 1

2

|β cosh r+β

∗eiθ sinh r)|2

= Ln(|β|

2

(e

2r

cos2(φ − θ/2) + e

−2r

sin2(φ − θ/2)))

×e

−|β|

2

(e

2r cos2

(φ−θ/2)+e−2rsin2(φ−θ/2))/2

, (3.34)

where Ln(x) is a Laguerre polynomial and β ≡ |β|e

iφ. Replacing β with 2α gives

P↓ =

1

2

(1 + Ln(|2α|

2

(e

2r

cos2(φ − θ/2) + e

−2r

sin2(φ − θ/2)))

×e

−2|α|

2

[e

2r cos2

(φ−θ/2)+e−2rsin2(φ−θ/2)]) (3.35)

For the special case where the ion is initially in the ground state,

P↓,n=0 =

1

2



1 + e

−2|α|

2

[e

2r cos2

(φ−θ/2)+e−2rsin2

(φ−θ/2)]

(3.36)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
38

0 5 10 15

n

0.0

0.1

0.2

0.3

0.4

P

n

3 2 1 0 1 2 3

Re( )

3

2

1

0

1

2

3

Im( )

0.0

0.5

a) b)

Figure 3.6: a) Number state probabilities and b) Wigner function for a squeezed cat state with

ξ = −1.0 and α = 2i

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.6

0.8

1.0

P

Squeezed state r = 1, = 0

r = 1, = /2

r = 1, =

r = 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

| |

0.6

0.8

1.0

P

Squeezed thermal state n = 2.0

r = 1, = 0

r = 1, = /2

r = 1, =

r = 0

a)

b)

Figure 3.7: P↓ as a function of |α| for a squeezed states with r = 1.0, and various values of the

squeezing phase θ, and for different initial thermal occupations ( ̄n = 0 upper panel,  ̄n = 0 lower

panel). The relative angle of the SDF φ = 0. For comparison, the unsqueezed state (r = 0) is also

shown.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
39

P↓ depends on both the angle of the spin-dependent force and on the direction of squeezing (see

Fig. 3.7). The rate at which P↓ tends to 1/2 is maximized when the SDF is applied along the

squeezed axis (θ = 0, φ = 0) and minimized if the SDF is applied along the anti squeezed axis

(θ = π, φ = 0). If the ion is initially in a thermal distribution with hni =  ̄n,

P↓,th =

1

2



1 + e

−|α|

2

(2 ̄n+1)[e

2r cos2

(φ−θ/2)+e−2rsin2(φ−θ/2)]

. (3.37)

The initial thermal occupation results in faster decay of P↓,th to 1/2 than for a system prepared in

the motional ground state (see Fig. 3.7).

3.4 Geometric phase gates

This section describes the implementation of geometric phase gates using spin-dependent

forces. There are two classes of geometric phase gates that operate on different spin bases: the

Mølmer-Sørensen, or ˆσφσˆφ, gate [48, 49, 50] and the ˆσzσˆz gate [51, 52].

3.4.1 The Mølmer-Sørensen gate

The Mølmer-Sørensen gate [48] is implemented by applying a spin dependent force to two

(or more) trapped ions with a shared mode of motion. Without any spin-motion coupling, the

Hamiltonian describing the harmonically confined qubits is

Hˆ

0 =

~ω0

2

σˆz1 +

~ω0

2

σˆz2 + ~ωaˆ

†

a, ˆ (3.38)

where ˆσz1 and ˆσz2 are Pauli operators describing the qubit state of each ion and ˆa(ˆa

†

) is now

the annihilation (creation) operator of the shared motional mode. Following the methods of the

previous section, the interaction picture Hamiltonian for the spin-dependent force is given by

HˆMS =

~Ω

2

Sˆ

x(ˆaeiδt + ˆa

†

e

−iδt) (3.39)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
40

where ˆa is the harmonic oscillator annihilation operator for the shared motional mode. The coupling

strength Ω is assumed to be the same for both ions, Sˆ

x =

P2

i=1 βiσˆxi, where βi

is proportional to the

i

th component of the mode eigenvector, and ˆσxi is the Pauli operator for the ith ion. For a centerof-mass (COM) mode, the mode eigenvector is √

1

2

(1, 1). For convenience, we set Sˆ

x = ˆσx1 + ˆσx2

and let the factor of √

1

2

be absorbed into Ω. For an out-of-phase motional mode the eigenvector is

√

1

2

(1, −1) and Sˆ

x = ˆσx1 − σˆx2. The time evolution operator for Eq. 3.39 can be calculated exactly

using the Magnus expansion method (see Appendix A). The relevant commutators are

[Hˆ (t), Hˆ (t

0

)] = 

~Ω

2

2

Sˆ2

x

[ˆaeiδt + ˆa

†

e

−iδt

, aeˆ

iδt0

+ ˆa

†

e

−iδt0

]

= 2i



~Ω

2

2

Sˆ2

x

sin(δ(t − t

0

)) (3.40)

and

[[Hˆ (t), Hˆ (t

0

)], Hˆ (t

00)] = 0. (3.41)

The time evolution operator is then

Uˆ(t, t0) = e

−C1(t,t0)

e

−C2(t,t0)

(3.42)

where

C1(t, t0) = 1

i~

Z t

t0

Hˆ (t

0

)dt0

=

1

i~

~Ω

2

Sˆ

x

Z t

t0

(ˆaeiδt0+ ˆa

†

e

−iδt0

)dt0

= i

Ω

δ

Sˆ

x sin 

δ(t − t0)

2



(ˆaeiδ(t+t0)/2 − aˆ

†

e

−iδ(t+t0)/2

), (3.43)

and

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
41

C2(t, t0) = 1

2



1

i~

2 Z t

t0

Z t

0

t0

[Hˆ (t

0

), Hˆ (t

00)]dt0

= −i

Ω

2

4

Sˆ2

x

Z t

t0

Z t

0

t0

sin(δ(t

0 − t00))dt00dt0

= i

Ω

2

4δ

2

Sˆ2

x

[δ(t − t0) − sin(δ(t − t0))], (3.44)

giving

Uˆ(t, t0) = Dˆ(β(t, t0)Sˆ

x)e

−iΦ(t,t0)Sˆ2

x (3.45)

where the spin-dependent displacement amplitude is

β(t, t0)Sˆ

x = −i

Ω

δ

sin 

δ(t − t0)

2



e

−iδ(t+t0)/2Sˆ

x, (3.46)

and the spin-dependent geometric phase is given by

Φ(t, t0)Sˆ2

x =

Ω

2

4δ

2

[δ(t − t0) − sin(δ(t − t0))]Sˆ2

x

. (3.47)

If Sˆ

x = ˆσx1 + ˆσx2, the matrix representations of Sˆx and Sˆ2

x

in terms of the two-ion spin states |++i,

|+−i, |−+i and |−−i (where |±i = √

1

2

(|↑i ± |↓i) are the eigenstates of ˆσx) are given by

Sˆ

x =





















2 0 0 0

0 0 0 0

0 0 0 0

0 0 0 −2





















(3.48)

and

Sˆ2

x =





















4 0 0 0

0 0 0 0

0 0 0 0

0 0 0 4





















. (3.49)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
42

+ - Im(α)

Re(α)

Geometric 

phase Φ

- +

Figure 3.8: Phase-space illustration of the oscillator wavepacket trajectories during a geometric

phase gate.

If Sˆ

x = ˆσx1 − σˆx2, the matrix representations are

Sˆ

x =





















0 0 0 0

0 2 0 0

0 0 −2 0

0 0 0 0





















(3.50)

and

Sˆ2

x =





















0 0 0 0

0 4 0 0

0 0 4 0

0 0 0 0





















. (3.51)

The detuned spin-dependent force results in spin-dependent displacements with circular trajectories in phase space. For the out-of-phase mode, the |++i and |−−i states are not displaced.

The |+−i and |−+i states follow circular trajectories and return to the origin when tK = t0+K

2π

δ

,

where K = 1, 2, 3... . The area enclosed by each phase space loop is equal to the geometric phase

accrued at tn, 4Φ(tn) = 2πΩ

2

δ

2

.

To implement an entangling gate with one phase space loop (K = 1), the |+−i and |−+i

states must acquire a phase of π/2 relative to the |++i, and |−−i states. The truth table for the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
43

gate is given by

|++i → |++i

|+−i → e

iπ/2

|+−i

|−+i → e

iπ/2

|−+i

|−−i → |−−i. (3.52)

If the initial state is |Ψi

0 = |↓↓i |0i, applying the Mølmer-Sørensen time evolution operator (Eq.

3.45) to |Ψi

0

, with t0 = 0 and tn = tg =

π

Ω

gives

|Ψ(tg)i = Uˆ(tg)|Ψi

0

=

1

2

exp −i

πSˆ2

x

8

!

(|++i − |+−i − |−+i + |−−i)

= −

e

−iπ/4

√

2

(|↓↓i + i|↑↑i), (3.53)

a maximally entangled Bell state. It is also useful to have an analytical expression for the state

and populations at any time given an arbitrary initial Fock state |ni and spin state |↓↓i.

|Ψn(t)i = Uˆ(t, t0 = 0)|↓↓i |ni

=

1

2

h

Dˆ(2β(t))e

−4iΦ(t)

|++i + Dˆ(−2β(t))e

−4iΦ(t)

|−−i

− |+−i − |−+i

i

|ni (3.54)

The projection onto the |↓↓i state is then

h↓↓ |Ψn(t)i =

1

4

h

Dˆ(2β(t))e

−4iΦ(t) + Dˆ(−2β(t))e−4iΦ(t) + 2i

|ni, (3.55)

and probability of measuring |↓↓i is

P↓↓(t) = 1

8



3 + 4 cos(4Φ(t))Ln(|2β(t)|

2

)e

−

|2β(t)|

2

2 + Ln(|4β(t)|

2

)e

−

|4β(t)|

2

2



. (3.56)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
44

The projection onto the |↓↑i state is

h↓↑ |Ψn(t)i =

1

4

h

Dˆ(2β(t))e

−4iΦ(t) − Dˆ(−2β(t))e−4iΦ(t)

i

|ni, (3.57)

and the probability of measuring |↑↓i is

P↑↓(t) = 1

8



1 − Ln(|4β(t)|

2

)e

−

|4β(t)|

2

2



. (3.58)

The remaining populations can be calculated from P↓↓(t) and P↑↓(t). The probability of measuring

|↓↑i, P↓↑(t) = P↑↓(t), and P↑↑(t) = 1 − P↓↓(t) − P↑↓(t) − P↓↑(t). If the initial state is in a thermal

distribution

P↓↓(t) = 1

8



3 + 4 cos(4Φ(t))e

−( ̄n+1/2)|2β(t)|

2

+ e

−( ̄n+1/2)|4β(t)|

2



(3.59)

and

P↑↓(t) = 1

8



1 − e

−( ̄n+1/2)|4β(t)|

2



. (3.60)

Another quantity of interest is the Bell-state fidelity defined as

F ≡ hΨBell| ρˆsys |ΨBelli, (3.61)

where |ΨBelli = √

1

2

(↑↑ +i|↓↓i), and ˆρsys is the system density matrix. If the system is in a pure

state |Ψn(t)i, the density matrix is ˆρsys = |Ψn(t)i hΨn(t)| and the fidelity is given by

F =

1

8

h

3 + e

−|4β(t)|

2/2Ln(|4β(t)|

2

) + 4 sin(4Φ(t))e

−|2β(t)|

2/2Ln(|2β(t)|

2

)

i

. (3.62)

Figure 3.9 shows the populations and fidelity as a function of detuning for the optimal singleloop gate time. For an initial thermal distribution with mean motional occupation  ̄n, the fidelity

becomes

F =

1

8

h

3 + e

−( ̄n+1/2)|4β(t)|

2

+ 4 sin(4Φ(t))e

−( ̄n+1/2)|2β(t)|

2

i

. (3.63)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
45

4 3 2 1 0 1 2 3 4

Detuning / 0

0.0

0.2

0.4

0.6

0.8

1.0 P

P + P

P

Fidelity

Figure 3.9: Populations and fidelity for generating the Bell-state √

1

2

(|↓↓i + i|↑↑i) as a function of

the gate detuning for the Mølmer-Sørensen entangling gate, where the gate mode was initialized

in the motional ground state. The gate time is tg = π/Ω. Note that the fidelity reaches 1 at a

detuning of δ/Ω = 2. At δ/Ω = −2, the orthogonal bell-state √

1

2

(|↓↓i − i|↑↑i) is generated giving

a fidelity of 0.

1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20

Detuning /

0.5

0.6

0.7

0.8

0.9

1.0

Fidelity

n = 0.0

n = 0.5

n = 3.0

n = 10.0

Figure 3.10: Bell-state fidelity as a function of the gate detuning for various values of the initial

thermal occupation  ̄n. The gate time is tg = 2π/Ω0. Increasing  ̄n increases the sensitivity of the

fidelity to detuning errors.

The initial thermal distribution does not prevent the generation of maximally entangled Bell-states,

however the gate becomes more sensitive to miscalibrations of parameters such as the gate time

and detuning (see Fig.3.10). There are many other mechanisms that can reduce the fidelity of

geometric phase gates. These will be discussed later in this chapter.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
46

3.4.2 σˆzσˆz gate

Geometric phase gates can also be implemented using spin-dependent forces that act in the

z basis. The so called ˆσzσˆz gate was originally proposed by Milburn et al. [51] and was first

demonstrated using lasers at NIST by Leibfried et al. [52]. The interaction picture Hamiltonian

for the ˆσzσˆz interaction is given by

Hˆ

ZZ =

~Ω

2

Sˆ

z(ˆaeiδt + ˆa

†

e

−iδt), (3.64)

where Sˆ

z =

P2

i=1 βiσˆzi. For a two-ion radial center-of-mass mode (COM), Sˆ

z = ˆσz1 + ˆσz2 and for

an out-of-phase mode Sˆ

z = √

1

2

(ˆσz1 − σˆz2). Because the ˆσzσˆz gate acts on the eigensates of Sˆ

z,

generation of entangled states is usually requires Hˆ

ZZ to be applied within a Ramsey sequence.

The derivation of the time evolution operator is very similar to the derivation for the MS gate

described above.

3.5 Motional decoherence

In this section I discuss the dominant noise processes affecting our experiments, motional heating and motional dephasing. Experimentally both dephasing and heating can arise from anomalous

electric field noise on ion trap surfaces [41, 53, 54] and can also result from technical noise. We

account for motional decoherence processes by numerically integrating the master equation[32]

∂ρˆ

∂t = −

i

~

[H, ˆ ρˆ] + 1

2

X

n=0

h

2Cˆ

nρCˆ

†

n − {Cˆ†nCn, ρˆ}

i

, (3.65)

where ˆρ is the system density matrix, Hˆ is the Hamiltonian in the absence of noise, and the Cˆ

n

are Lindblad operators that describe the decoherence processes (the sum runs over all processes

that are considered). This approach requires that the noise can be modeled as a stochastic Markov

process with no memory [55]. A more comprehensive analysis of coherent and incoherent error

sources for MS gates can be found in [56]. For all numerical simulations in the thesis, I used the

QuTip [57] python package.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
47

0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575

Gate duration t/2

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Fidelity

/ = 0.0

/ = 0.02

/ = 0.04

/ = 0.06

Figure 3.11: Simulated fidelity of a Bell state produced by the Mølmer-Sørensen gate as a function

of gate duration Ωt/2π for various values of the motional dephasing rate γ, with all other gate

parameters held at their ideal values. Here t is the gate duration and Ω is the Rabi frequency

(Eq. 3.39). It is assumed that no other sources of decoherence are present. For all simulations, the

gate detuning is set to δ = 2Ω. Circles indicate the duration at which maximum fidelity occurs for

each value of γ. Increasing γ reduces the gate fidelity and also the gate duration at which maximum

fidelity occurs.

3.5.1 Motional dephasing

The first process that we consider results from motional mode frequency fluctuations that are

fast compared to the duration of a coherent operation such as a geometric phase gate. During a gate

these fluctuations dephase the motional state while it is entangled with the qubit states resulting

in reduced fidelity. We model motional dephasing using the Lindblad operator Cˆ

0 =

√γaˆ

†aˆ, where

γ is the motional dephasing rate. For a purely motional density matrix expressed in the number

state basis as ˆρ =

P

n,m ρnm |mi hn|, dephasing results in the decay of the off diagonal elements

(coherences) given by [58]

ρnm(t) = e

−

γ

2

(n−m)

2

t

ρ(0)nm, (3.66)

showing that the off-diagonal elements decay exponentially at a rate proportional to the square of

the number state difference n − m. After integrating Eq. 3.65, we plot the fidelity of a single-loop

two-qubit MS gate as a function of the gate time for various values of γ (Fig. 3.11), with the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
48

0.00 0.02 0.04 0.06

/

0.44

0.46

0.48

0.50

tm

a

x

/

2

0.00 0.02 0.04 0.06

/

0.970

0.975

0.980

0.985

0.990

0.995

1.000

| m

a

x

tm

a

x

/

2 |

0.00 0.02 0.04 0.06

/

0.75

0.80

0.85

0.90

0.95

1.00

Fidelity

a) b) c)

Figure 3.12: Simulated gate duration tmax (a) and product tmaxδmax (b) required for maximum

fidelity of a Bell state produced by the Mølmer-Sørensen gate (c) as a function of the motional dephasing rate γ, with no additional sources of decoherence. Ω is the gate Rabi frequency. Increasing

γ reduces both the fidelity and the gate duration that gives maximum fidelity.

gate detuning δ fixed at −2Ω. As expected, increasing γ reduces the gate fidelity. It is also clear

that shorter gate durations maximize the fidelity as the dephasing rate increases. To investigate

this effect further, we calculate the gate duration tmax and detuning δmax that maximize the

Bell-state fidelity for various values of γ (Fig.3.12). The optimized gate duration falls off linearly

with γ, whereas the fidelity decrease is less than linear. Physically the tradeoff between fidelity

reduction from motional decoherence (which penalizes longer gate durations) and from incorrect

geometric phase acquisition or failure to close the phase space loop (which penalize either shorter

or longer durations than the theoretically optimal duration) results in a shorter gate duration

giving the highest fidelity. This effect also occurs with motional heating, which I discuss next

and is an important factor in our investigation of parametrically accelerated geometric phase gates

(Chapter 8).

3.5.2 Motional heating

Another type of motional decoherence results from uncontrolled heating of the ion crystal.

We model stochastic motional heating using the Lindblad operators Cˆ

1 =

√

n ̇ aˆ and Cˆ

2 =

√

n ̇ aˆ

†

,

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
49

where  ̇n is the heating rate. Theoretically, this kind of decoherence arises when an oscillator is

coupled via its position to a bath consisting of an infinite bath of oscillators at thermal equilibrium

[58]. Results from numerical integration of Eq. 3.65 including only Cˆ

1 and Cˆ2 Lindblad operators

are shown in Fig. 3.13. Increasing the heating rate  ̇n reduces the fidelity as expected. As for

motional dephasing, heating shifts the gate time and detuning required to maximize the fidelity

(Fig. 3.14).

1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20

detuning /

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Fidelity

n/ = 0.0

n/ = 0.1

n/ = 0.2

n/ = 0.3

Figure 3.13: Simulated fidelity of a Bell state produced by the Mølmer-Sørensen gate as a function

of gate detuning δ/Ω, where δ is the detuning and Ω is the Rabi frequency (Eq. 3.39), for various

values of the heating rate  ̇n, assuming no other sources of error. For all simulations, the gate

duration is set to t = π/Ω. Circles indicate the duration at which maximum fidelity occurs for each

value of  ̇n. Increasing  ̇n reduces the gate fidelity and lowers the detuning needed to maximize the

fidelity for a given value of  ̇n.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
50

0.0 0.1 0.2 0.3 0.4

n/

0.40

0.42

0.44

0.46

0.48

0.50

tm

a

x

/

2

0.0 0.1 0.2 0.3 0.4

n/

2.0

2.1

2.2

2.3

2.4

2.5

m

a

x

/

2

0.0 0.1 0.2 0.3 0.4

n/

0.75

0.80

0.85

0.90

0.95

1.00

Fidelity

a) b) c)

Figure 3.14: Simulated Mølmer-Sørensen gate duration tmax (a) and detuning δmax (b) for maximized Bell-state fidelity (c) as a function of the motional dephasing rate  ̇n, assuming no other error

sources are present. Ω is the gate Rabi frequency. Increasing  ̇n reduces both the fidelity and the

gate duration that gives maximum fidelity.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 4

Experimental setup

Our experimental apparatus was designed for trapping ions in close proximity to current

carrying electrodes for implementing high-fidelity “laser-free” entangling gates using oscillating

near-field magnetic field gradients. Furthermore, the geometry of our trap enables relatively strong

oscillating electric-field gradients to be generated at the ions’ position allowing parametric modulation and squeezing of the ions’ motion. This chapter gives an overview of the experimental

hardware we use for trapping ions and for controlling them using these gradients.

4.1 Surface-electrode ion trap

Charged particles cannot be confined in three dimensions using only static electric fields.

However, stable confinement can be achieved using a combination of static and time varying fields.

Figure 4.1 a shows a diagram of a typical 3D linear radio frequency (RF) or Paul trap [59]. Confinement along the trap axis (dotted line) is provided by static potentials applied to the outer DC

electrodes, whereas confinement in the plane normal to the axis (the radial plane) is achieved by

applying an oscillating potential VRF cos(ωRF t) + Vb to two diagonally opposite electrodes. To

leading order, the potential near the trap axis in the radial plane is a quadrupole (Fig. 4.1 a). Provided the amplitude VRF and RF frequency ωRF are chosen correctly [41, 60], a charged particle

will feel a ponderomotive restoring force and a corresponding effective harmonic potential, which

enables stable confinement in the radial plane. Away from locations with zero RF field, motion in

the effective potential (known as “secular motion”) is superimposed with small-amplitude motion

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
52

d

RF

DC

RF

DC

DC

DC

DC

DC

DC RF DC RF DC

RF

DC

DC

RF

d

RF

RF

DC

DC

DC

DC

DC

DC

DC

a) b)

Figure 4.1: Linear radio-frequency (RF) Paul traps. The upper figures show schematic diagrams of

a conventional linear RF Paul trap (a) and a surface-electrode Paul trap (b) with two trapped ions

indicated by green spheres. Dashed lines indicate the trap axis. Lower panels show radial cross

sections perpendicular to the trap axis. Dashed arrows illustrate electric field lines and green disks

the radial equilibrium position of the ions.

at ωRF which is known as micromotion.

One can also imagine unfolding and flattening the 3D Paul trap structure onto a plane

(Fig. 4.1 B). In this case a quadrupole forms above the trap surface and the device is known

as a surface-electrode ion trap [61, 62]. Our current trap is a surface-electrode trap that was

designed and fabricated by Daniel Slichter and David Allcock building on previous NIST designs

[63]. A photograph and electron microscope image of the trap are shown in Figure 4.2. Ions

are trapped at a height of d ' 30 μm above the trap surface. RF potentials with a frequency of

ωRF /2π = 68.608 MHz and amplitude of VRF ' 30 V are applied to the RF electrodes to provide

confinement in the radial (y −z) plane. The amplitude of the RF drive is stabilized using a limiting

amplifier circuit based on a design in [64]. A half-wave coaxial resonator serves to step up the RF

amplitude to the level required at the trap electrodes. Eighteen DC electrodes are used to apply

potentials for axial confinement, for micromotion compensation, for setting the frequencies of the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
53

two secular modes of motion in the y-z plane, and for implementing resonant displacements. The

potentials on these electrodes are controlled using a multi-channel digital-to-analog converter [65].

The axial secular frequency is ωx/2π ' 3 MHz. Typical secular frequencies in the radial plane are

(ωr1, ωr2)/2π ' (6, 7.5) MHz, where r1 and r2 denote orthogonal unit vectors in the y − z plane.

The orientations of the radial modes depend on the values of the DC potentials. The trap also has

three “microwave” electrodes for carrying the oscillating currents required to generate near-field

magnetic field gradients. These electrodes are grounded at one end of the trap to give a current

anti-node near the position of the ions. These microwave electrodes are discussed in more detail in

section 4.6.1

All trap electrodes are 8 μm-thick gold electroplated onto a sapphire substrate. The trap

electrodes are ribbon-bonded onto screen-printed gold electrodes on an alumina circuit board,

which also hosts RC low-pass filters (100 kHz cut-off frequency) to reduce high-frequency voltage

noise on the dc electrodes.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
54

a) Scale: 2 mm

b)

Gold electrodes

Sapphire substrate

DC electrodes

Microwave 

electrodes

RF electrodes

Ribbon bonds

Alumina lter 

board

z

x

y

Figure 4.2: Surface-electrode ion trap designed for implementing near-field magnetic field gradient

entangling gates. a) Photograph of the trap chip. b) Electron microscope image of the trapping

zone with false colors indicating the DC, RF, and microwave electrodes.

4.2 Vacuum chamber and cryostat

The filter board and ion trap are contained in a gold-coated oxygen-free high thermal conductivity (OFHC) copper pillbox (Fig. 4.3). The relatively large thermal diffusivity of the pillbox

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
55

Pillbox

Filter board

Microwave 

SMA cables

Magnesium 

oven

Wires to DC 

electrodes 

Figure 4.3: Annotated photograph of the pillbox.

helps to rapidly transfer heat generated by microwave and RF currents away from the trap. The

pillbox is cryogenically cooled below 20 K through a copper braid connected to a liquid helium flow

cryostat. There are several advantages of operating at low temperatures. First, cryogenic operation

reduces background gas pressure, which improves ion lifetime in the trap. Second, heating due to

anomalous electric field noise is typically suppressed at cryogenic temperatures [66, 67, 68]. Third,

the thermal and electrical conductivity of the gold trap electrodes are larger at cryogenic temperatures than at room temperature enabling larger current densities in the microwave electrodes

than would be possible at room temperature. Larger currents in these electrodes permit stronger

spin-motion coupling (see section 4.6.1).

The pillbox is housed in a 6” octagonal vacuum chamber1see Figs. 4.4 and 4.5). The chamber

has angled viewports which serve to keep view-port reflections of laser beams away from the trap.

Ultra-high vacuum pressure is maintained using a non-evaporable getter pump2.

1 Kimball Physics custom part

2 Agilent Vaclon Plus 20 Star Cell

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
56

Cryostat

Pillbox

Octagon

Cold nger

Trap

Thermal link

Figure 4.4: Partial CAD rendering of the vacuum chamber and cryostat

Flow 

cryostat

He recovery 

line

Camera Imaging 

system

Vacuum 

chamber

PMT

B-eld coil

B-eld coil

Figure 4.5: Annotated photograph of the vacuum chamber, cryostat, and imaging system. PMT:

Photo-multiplier tube

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
57

4.3 25Mg+ ion qubit

In our system we use 25Mg+ ions. With a single valence electron, 25Mg+ has an electronic

structure similar to that of neutral alkali atoms. A diagram with the relevant energy levels is shown

in Figure 4.6. The electronic ground state configuration is 1s

22s22p63s2 2SJ=1/2

, where J = L + S

is the sum of the orbital angular momentum (L = 0) and the electron spin angular momentum

(S = 1/2). The interaction between the nuclear spin (I = 5/2) and the outer electron spin give rise

to hyperfine splitting of the atomic energy levels. In the ground state, there are two hyperfine levels

that are split by ∼ 1.7 GHz and are labeled F = 2 and F = 3, where F = I + J is the total angular

momentum quantum number. The hyperfine levels are further split into Zeeman sublevels labeled

by the magnetic quantum number mF , which is the projection of the total angular momentum

along the direction of an external magnetic field of B ' 212 G. At this field the frequency of the

|F = 3, mF = 1i↔ |F = 2, mF = 1i “clock” transition (ω0/2π = 1.686 GHz) is first-order insensitive

to magnetic field fluctuations [69, 70]. These levels, labeled as |↓i ≡ |3, 1i, and |↑i ≡ |2, 1i, comprise

the qubits for most of the experiments in this thesis. The coherence time of these qubits (measured

using a Ramsey experiment) is longer than 200 ms. For several applications it is convenient to use

a two-level system formed by the |↓si ≡ 2S1/2|3, 3i and |↑si ≡ 2S1/2|2, 2i levels, which have a

frequency splitting of ωs/2π = 1.326 GHz.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
58

2

S1/2

2

P3/2

2

P3/2

B field ~ 212 G

F = 3, mF = 3

2, 1 2, 2

2, 0 2, -1

Detection and Doppler 

cooling (BD, BDD) 

279.635 nm

3,3

4,4

Repumping

(RD)

Raman detuning

~ 50 GHz

ω0/2π =

1.686 GHz

Raman

σ+

σ+

3, 2 3, 1 3, 0

2, -2

3, -1 3, -2 3, -3

ωs/2π =

1.326 GHz

s

s

=

=

=

=

Figure 4.6: Relevant energy level structure (not to scale) of 25Mg+ in a magnetic field of ' 212 G.

Arrows indicate lasers or microwaves used to drive the associated transitions. Qubits are encoded

in |↓i ≡ 2S1/2|3, 1i and |↑i ≡ 2S1/2|2, 1i. The qubit frequency is first-order insensitive to magnetic

field fluctuations.

4.4 State preparation and detection

Any experimental run starts with loading ions into the trap using laser-based photoionization

and Doppler cooling. After loading, we perform experiments to minimize micromotion for optimal

cooling and detection. This is achieved by systematically moving the ion(s) towards the rf null

with DC potentials, based on measurements of the transverse and out-of plane micromotion. After

compensating the micromotion, we perform several procedures to optimize the position of laser

beams on the ion(s) and to measure the frequencies of important microwave transitions in the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
59

2S1/2 ground state manifold. Once the micromotion compensation and calibration experiments are

complete (some iteration may be required), we can proceed to physics experiments.

We typically initialize the ion(s) in the |↓i qubit state and close to the |n = 0i motional state

at the start of each shot of a physics experiment. State preparation begins with a stage of Doppler

cooling to reduce the average motional occupation to  ̄n ∼ 2. This is followed by a sequence of

coherent microwave pulses and dissipative optical repumping to prepare the ion’s internal state in

|3, 3i. Resolved sideband cooling then serves to cool the motional mode(s) used in the experiment

closer to the ground state. State preparation is completed after the internal state is transferred

from |3, 3i to |↓i using coherent microwave pulses. At the end of an experiment we need to measure

our qubits. We apply microwave pulses to transfer or “shelve” the populations of the |↓i and |↑i

states in the |3, 3i and |2, −1i states respectively. Detection of fluorescence from the 2S1/2|3, 3i↔

2P3/2

|4, 4i cycling transition indicates the ion was projected into the |↓i qubit state.

Each of the stages of state preparation, detection, and micromotion compensation are discussed in more detail in the subsections below. The laser systems used to implement these operations are described in section 4.5. The microwave drive chain is discussed in section 4.6.

4.4.1 Doppler cooling

We use two laser beams tuned below resonance with the dipole allowed 2S1/2|3, 3i↔ 2P3/2|4, 4i

cycling transition for Doppler cooling 25Mg+ ions. The center frequency and natural linewidth of

the transition are 1072.084547(5) THz (at 0 B field) and Γ = 2π × 41.3(3) MHz respectively [71].

Both beams are σ

+ polarized and have a 1/e2 beam intensity radius of ∼ 14 μm at the location

of the ions. The wavevectors of both beams have components along every normal mode direction,

allowing Doppler cooling in three dimensions.

The “blue Doppler” (BD) laser is red detuned by ∆ ' Γ/2 from the line center. In the weak

binding regime, where Γ/2  ωm, with ωm the frequency of a particular motional mode, the steady

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
60

state temperature that is achieved by Doppler cooling is given by [60, 72]

TD =

Γ

8kB

(1 + ξ)



(1 + s)

Γ

2∆ +

2∆

Γ



, (4.1)

where the saturation parameter is given by s = 2|Ω|

2/Γ2

, with Ω the resonant Rabi frequency, and

ξ is a geometric factor related to the average component of photon emission recoils along the axis

of oscillation with a value of ξ = 2/5 for dipole radiation [60]. TD is minimized for ∆ = Γ√

s + 1/2.

For a single trapped 25Mg+ ion with a radial trapping frequency of ωm = ∼ 2π × 6 MHz, and

saturation parameter s ' 0.5 we measure an average thermal occupation of  ̄n ' 2.0 after Doppler

cooling and blue sideband analysis (chapter 3).

The second laser, labeled “blue Doppler detuned” (BDD), is shifted by 522 MHz below the

BD frequency. For this beam we use a saturation parameter of s ∼ 10, so that an ion Doppler

shifted into resonance will see a power broadened line. This beam provides cooling for ions that are

excited to high velocities by background gas collisions, where cooling with the BD is not efficient.

Another important function of the BDD is to perform optical pumping, within the F = 2 manifold

(and to some extend the F = 3 manifold as well), towards the towards the |F = 3, mF = 3i state.

4.4.2 Repumping

Doppler cooling on the 2S1/2|3, 3i↔ 2P3/2|4, 4i transition ideally optically pumps the ion

into the 2S1/2|3, 3i state. However, polarization impurities in the BD beam result in population

being transferred to other states within the 2S1/2 manifold, predominantly to the |3, 2i and |2, 2i

states. To transfer this population back to |3, 3i, we use coherent microwave pulses and a repump

(RD) laser resonant with the 2S1/2|2, 2i↔ 2P3/2|3, 3i transition. The RD is σ

+ polarized and is

derived from the same laser source as the BD and BDD beams. The repumping sequence starts with

microwave carrier π pulses on the |3, 3i ↔ |2, 2i and |3, 2i ↔ |2, 2i transitions. This is followed

by an RD pulse, which effectively transfers population from |2, 2i to both |3, 2i and |3, 3i, and a

microwave π pulse on the |3, 3i ↔ |2, 2i transition. This dissipation sequence is repeated several

times to transfer population from the |3, 3i and |2, 2i states to the |3, 2i state. Finally, coherent

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
61

n = 0

RSB

RSB

RSB

Repumping

Repumping

Repumping

Repumping

n = 1

n = 2

n = 3

↑

↑

s

s

Figure 4.7: Ground state cooling. Red sideband (RSB) pulses induce transitions from |↑si |ni

to |↓si |n − 1i. Dissipative repumping returns the internal state to |↓si. Population in |n = 0i is

trapped because a RSB transition from this state is not possible.

microwave pulses transfer the population to |3, 3i.

Dissipative repumping from the |2, 2i to the |3, 3i state is also used for resolved sideband

cooling which is described in the next subsection.

4.4.3 Resolved sideband cooling

After preparing the ion(s) in the 2S1/2|3, 3i state, we use pulsed resolved sideband cooling

[40, 73] to cool the ion(s) close to the motional ground state. The process is illustrated in Fig. 4.7.

A pair of laser beams with a frequency difference of ωs − ωm − ∆ACS, where ωm is the frequency

of the mode being cooled and ∆ACS/2π ∼ 400 kHz is the AC Stark shift from the laser fields,

implement a red sideband (RSB) interaction (see Chapter 3) that induces transitions from |↓si |ni

to |↑si |n − 1i. The Raman beams are derived from the same laser source and both are detuned from

the 2P3/2state by ∼ 50 GHz (Fig. 4.6). This detuning is chosen such that the rate of off-resonant

scattering from the 2P3/2 manifold is much less than the sideband Rabi rate. Interleaving RSB

pulses with dissipative repumping results in a net flow of population towards the |n = 0i motional

state, which is a dark state (eigenstate with eigenvalue 0) of the RSB Hamiltonian.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
62

100 75 50 25 0 25 50 75 100

Detuning (kHz)

0.2

0.4

0.6

0.8

1.0

P

Blue sideband

Red sideband

Figure 4.8: Fluorescence counts as a function of the Raman red and blue sideband detuning after

ground state cooling. From the data we extract  ̄n = 0.03(2).

We measure the motional occupation after sideband cooling by comparing the probabilities of

measuring |↓si, P

RSB

↓s

and P

BSB

↓s

, after excitation with either the red or blue sideband respectively

[40]. To eliminate systematic effects, an effective blue sideband (BSB) is implemented by first

applying a microwave π pulse to flip the internal state from |↓si to |↑si and then applying an RSB

pulse. A final carrier π pulse is applied before detection. As described in [60], the average motional

occupation, assuming that the final state has a thermal distribution, is given by

n ̄ =

R

1 − R

, (4.2)

where R =

P

RSB

↓s

P

BSB

↓s

. From the example data shown in Fig. 4.8, we calculate  ̄n = 0.03(2). Alternatively, we can measure  ̄n by using blue sideband analysis (see Chapter 3).

4.4.4 Imaging and detection

After ground state cooling, the ion is transferred to the |↓i qubit state with resonant microwave π pulses, completing initialization for an experiment. To read out the qubit state after

the experiment, we first apply microwave pulses to “shelve” the populations in |↓i and |↑i to |3, 3i

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
63

and |2, −1i respectively. We then apply a BD pulse resonant with the 2S1/2|3, 3i↔ 2P3/2|4, 4i

cycling transition. Detection of scattered 280 nm light corresponds to projecting the qubit into the

|↓i state, whereas a lack of scattering corresponds to the |↑i qubit state. Scattered photons are

collected using an imaging system with a primary objective with a numerical aperture (NA) of 0.4

and magnification of ∼ 4, followed by a secondary objective with magnification of ∼ 13 and NA of

∼ 0.13. Slits in the primary image plane system allow us to block stray light. We can choose to

direct scattered photons to either an electron-multiplying charged-coupled device (EMCCD) camera3 or to a photomultiplier tube (PMT)4

for counting. In a detection duration of 200 μs with a

saturation parameter of s ∼ 0.5, we record on average ' 50 counts for a single 25Mg+ in the bright

state and '5 counts in the dark state (fig. 4.9) using the PMT. The dark counts result primarily

from detection light scattering off the trap surface.

0 10 20 30 40 50 60 70

Photon counts

0

1000

2000

3000

4000

5000

Number of events

Figure 4.9: Typical detection histogram for a single 25Mg+ ion for a detection time of 200 μs and

a saturation parameter of s ∼ 0.5. Bars correspond to population in |↑i = 2S1/2|2, 1i) (blue) or

|↓i = 2S1/2|3, 1i (orange).

3 Andor DV887ECS-UVB

4 Hammamatsu R7600U-203

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
64

4.4.5 Micromotion compensation

We detect micromotion in the y direction by monitoring the photon scattering rate as a

function of the ion position along the y axis. The ion’s position is systematically adjusting by

changing potentials on the DC electrodes of the trap. As the ion moves away from the trap axis,

the micromotion amplitude increases and the ion will start to spend a significant amount of time

during a detection period Doppler shifted out of resonance with the detection beam, thus reducing

the fluorescence count rate. The in-plane micro-motion is compensated by moving the ion to the

position that maximizes the count rate. This method only allow us to know that we have nulled

the micromotion parallel to the Doppler beam. Because we have some residual axial micromotion,

this compensation technique will end up putting you in a place with some axial and some radial

micromotion, such that the total is normal to the Doppler beam. This will be small, but non-zero.

Out-of-plane micromotion (in the z direction) is detected by applying a microwave tone

at a frequency of ω0 + ωRF [74]. This technique requires an oscillating magnetic field gradient.

The microwave-gradient field will be resonant with the first micromotion sideband, and induce

sideband transitions between the qubit states. By adjusting potentials on DC electrodes, we can

change the ion’s position along z to minimize the transition probability and hence the out-of-plane

micromotion.

4.5 Laser systems

This section describes the laser systems used in the experiment. The orientation of the laser

beams relative to the magnetic field direction and the trap axis are shown schematically in Fig.

4.10. The amplitude and frequency of the laser beams used for manipulation of trapped ions are

controlled using acousto-optic modulators (AOMs). All RF signals used to drive the AOMs are

derived from signals generated using direct digital synthesizers (DDSs)5that are controlled using

a field-programmable gate array (FPGA) board6 programmed using ARTIQ experimental control

5 Analog device AD9914 clocked at 2.4 GHz

6 Xilinx KC705

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
65

software [75].

B

Doppler cooling, 

repumping, σ+, 280 nm Photoionization, 

285 nm

Raman π,

280 nm

Raman σ+,

280 nm

B eld coil

Ions

B eld coil

22.5o

trap axis

Figure 4.10: Schematic diagram indicating laser access to the vacuum chamber.

4.5.1 Photoionization

Our experiment uses photoionization of neutral 25Mg to load ions into the trap. We generate

a thermal beam of neutral magnesium using a resistively heated oven (see Fig. 4.3) containing a

sample that is enriched with 25Mg. Laser beams directed through the trapping region can then

ionize the magnesium atoms via a two-stage process. In the first step, a 285 nm photon excites the

1S0 ↔ 1P1 electronic transition in 25Mg. Subsequently, another photon at 285 nm or one at 280 nm

(from the Doppler cooling laser) can promote the electron to the continuum.

For the experiments described in this thesis, a Raman-fiber-amplifier-based system was used

to generate 285 nm light for photoionization. The fiber amplifier is seeded by an 1140 nm diode laser

and outputs ∼ 2 W which is frequency-doubled using a waveguide frequency doubler to produce

∼ 280 mW at 570 nm. The visible light is converted to 285 nm using a build-up cavity containing

a beta barium borate (BBO) crystal. Although this system is easily able to generate several

hundred mW in the visible, the low efficiency of the subsequent resonant frequency doubling stage

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
66

required for generating 285 nm light typically limits the UV power to ∼ 1 mW. Luckily, this is plenty

for our purposes. The Raman fiber laser system is sufficiently stable that there is no need for active

control of the laser frequency. We typically leave it free running and monitor the frequency on a

wavemeter7, making manual adjustments when necessary (every ∼ 5 hours).

4.5.2 Doppler cooling, detection, and repumping

The source of resonant light is a home-built vertical external-cavity surface-emitting laser

(VECSEL) system which is described in detail in Appendix D. As shown in figure 4.11, the infrared

light from the VECSEL at 1118 nm is converted to 559 nm using a commercial waveguide doubler

(NTT-0559-000-A-B-C). With 700 mW from the VECSEL we typically generate 260 mW at 559 nm.

The 559 nm beam is split into a main beam (∼190 mW), which is directed to a resonant frequency

doubler, and a weaker beam for analysis that is frequency shifted using an AOM in a double pass

configuration and directed to a Doppler-free molecular iodine spectrometer (see Fig. 4.12). The

Doppler-free absorption features of I2 provide an absolute frequency reference, enabling us to lock

the laser frequency to within 2 MHz. The I2 spectrometer is based on the setup described in [76]

and the error signal is generated using the modulation transfer spectroscopy technique [77]. In

contrast to the traditional I2 lock setup used in the Ion Storage group, which uses an acousto-optic

modulator (AOM) for frequency dithering [78], our setup uses an electro-optic modulator (EOM)

to phase modulate the pump beam after it is frequency shifted by an AOM. This allows both a

higher modulation frequency and a larger moduation depth leading to higher signal-to-noise ratio

in the error signal. We typically require ∼ 5 mW of 559 nm to enter the spectrometer for a good

lock. The error signal from the I2 spectrometer provides the input to a proportional-integral (PI)

controller which adjusts the voltage across the VECSEL piezo transducer (PZT) to stabilize the

VECSEL cavity length and thus the laser frequency.

The main 559 nm beam is frequency doubled to 280 nm using an enhancement cavity containing a BBO crystal [79]. As shown in Figure 4.13 the UV light is split and frequency shifted

7 HighFinesse GmbH WS6-200

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
67

330 MHz 1118 nm 

VECSEL

AOM

λ/2

λ/4

PM ber

To BD UV

doubler

Waveguide

SHG

I2 spectrometer

PM ber

Isolator

PM ber

PZT 

input

HV amplier PI controller

Error 

signal

559 nm

Figure 4.11: Frequency stabilized 559 nm system. Light at 1118 nm from a VECSEL laser is

frequency doubled to 559 nm using a waveguide second harmonic generation (SHG) module. A

fraction of the 559 nm light is frequency shifted using a double pass AOM for analysis using a

Doppler-free iodine spectrometer. A proportional-intergal (PI) controller adjusts the VECSEL

piezo (PZT) based on the error signal from the spectrometer. Dotted lines indicate electronic

signals.

AOM

EOM

I2 cell

80 MHz

1 MHz

PBS PBS

λ/2 λ/2

APD

PM ber

φ

Error 

signal

LPF

Mixer

Amplier

PS

Figure 4.12: Doppler-free iodine spectrometer. Light at 559 nm is delivered via a polarization

maintaining (PM) optical fiber. The light is split by polarizing beam splitter (PBS) into probe

and pump beams that overlap and counterpropagate through the iodine cell. The pump beam

is first shifted by 80 MHz using an AOM. The diffracted beam is phase modulated at 1 MHz

using an electro-optical modulator (EOM). Absorption of the probe is detected on an avalanche

photodetector (APD). The photodiode signal is amplified, demodulated and filtered to generate an

error signal. LPF: Low pass filter, PS: phase shifter. Dotted lines indicate electronic signals.

using AOMs to generate the BD, BDD, and RD beams. Roughly 2% of the UV light is picked off

and frequency shifted by 2 × 205 MHz to generate the RD light. The remaining UV light enters

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
68

UV doubler

330 MHz

232 MHz

205 MHz 

AOM

AOM

AOM

AOM

UV ber

- 2 X 205 MHz 

+2 X 330 MHz

559 nm 280 nm

261 MHz

RD

BD+BBD+RD

BDD

glass 

plate

+2 X 261 MHz

Figure 4.13: AOM switch yard for controlling Doppler cooling and detection (BD and BDD), and

repump (RD) light. Dashed lines indicate rf signals for controlling AOMs. The RD, BD, and BDD

beams are coupled into the same solarization resistant UV fiber and directed to the ion trap.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
69

a double-pass AOM setup and is frequency shifted by 2 × 330 MHz. The diffracted light is further frequency shifted by 232 MHz to generate BD light. The residual undifracted light from the

330 MHz double pass AOM is shifted by 2 × 261 MHz to produce BDD light. The BD, RD and

BDD are coupled into the same solarization resistant UV fiber [80] and routed to the ion trap (see

Fig. 4.10).

4.5.3 Raman lasers

As for the resonant laser source, the Raman laser system (see Fig. 4.14) is based a VECSEL

emitting at 1118 nm. Infrared light from the VECSEL is frequency doubled twice to generate

280 nm light. The first stage uses a waveguide doubler to generate 559 nm light and the second

stage uses a BBO enhancement cavity to produce light at 280 nm. From ∼ 290 mW of 559 nm light,

we typically generate ∼ 10 mW at 280 nm. The 280 nm light is split into two beams, each of which

is frequency shifted using an AOM in double-pass configuration to reach the the required frequency

difference. Each Raman beam is coupled into a separate UV fiber and routed to the ion trap (see

Fig. 4.10). Polarization optics near the vacuum chamber set the polarization of the lower-frequency

Raman beam to σ

+ and that of higher-frequency Raman beam to π.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
70

UV doubler

λ/2

341.5 MHz 

321.5 MHz 

AOM

AOM

+2x 321.5 MHz

- 2x 341.5 MHz

To ion trap

To ion trap

559 nm 280 nm

UV ber

UV ber

GLP

1118 nm 

VECSEL

Isolator

Waveguide

SHG

PM ber

Figure 4.14: Raman laser system. Single-frequency 1118 nm light, generated using a VECSEL

laser, is converted to 559 nm using a waveguide SHG module. The 559 nm light is frequency

doubled to 280 nm using an enhancement cavity containing a BBO crystal. The UV light is split

into two beams using a half-wave plate (λ/2) and a polarizing beam splitter (PBS) cube. Each

beam is frequency shifted using acousto-optical modulators (AOMs) in a double pass configuration

and coupled into UV fibers.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
71

4.6 Electronic drives

In our system, we use use magnetic fields for coherent qubit manipulations and magnetic

field gradients for spin-motion coupling. We also employ electric fields and electric field gradients

to manipulate the motional states of trapped ions. All of these fields are generated by direct

digital synthesis. Figure 4.15 shows the relevent portion of the surface trap with connections for

the electronic drives. All drive signals are generated using DDSs8 which are controlled using the

same FPGA board and ARTIQ software as the laser AOMs. This enables precision control of the

amplitude, frequency, and phase of each each signal.

4.6.1 Microwave drive system

Unlike most ion traps, our trap has integrated current-carrying structures for generating

oscillating magnetic fields and near-field gradients[63, 81, 82]. The essential physics can be understood by considering the magnetic field generated by an infinitely long current carrying wire along

the x direction (Fig. 4.16). Applying Ampere’s law, the magnetic field in the y − z plane is given

by

B~ =

μ0I

2π



y~ez − z~ey

y

2 + z2



, (4.3)

where μ0 is the permeability of free space, and I is the current. At (y, z) = (0, d), the components

of the field are By = −

μ0I

2πd and Bz = 0, and the magnetic field gradients are

∂By

∂z =

∂Bz

∂y =

μ0I

2πd2

, and ∂By

∂y =

∂Bz

∂z = 0 (4.4)

Coherent operations for state preparation, detection, and qubit manipulation require tones at

multiple frequencies ranging from 1.326 to ∼ 2.0 GHz. Depending on ∆mF between the levels of

the particular transition, we require magnetic filed components either parallel to (∆mF = 0) or

transverse to (∆mF = ±1) to the static quantization magnetic field. The |↓i ↔ |↑i transition has

8 Analog device AD9914 clocked at 2.4 GHz

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
72

Parametric

drive

~12 MHz

z

x

y d

25Mg+ ions

~6 MHz

Resonant motional

drive

Trap RF

68.608 MHz

Figure 4.15: Schematic diagram of the relevant portion of the surface-electrode ion trap. Microwave

currents (inidicated by blue arrows) in electrodes MW1, MW2 and, MW3 generate magnetic field

gradients for spin-motion coupling. The resonant electric field drive signal is applied to DC electrode

C4. The parametric drive (PD) is applied directly to the RF electrodes of the trap.

|B| ∝ I

d |∇B| ∝ I

d2

d

I x

y

z

B

Figure 4.16: A current I~ex flowing along a wire of negligible transverse extent generates a magnetic

field that curls around the wire. The magnitude of the field falls off like 1/d, and the magnetic field

gradients ∂Bz/∂y and ∂By/∂z fall off like 1/d2.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
73

DAC

Carrier drive

To trap 

electrode

RSB drive

DDS (Analog devices AD9914)

I

Q

IQ Mixer (Analog devices ADL5375)

Power amplifier: 50dB gain 

(Minicircuits ZHL-30W-252-S+)

Bandpass filter: 1.574 -1.785 GHz 

(K&L filters 3C421686/U30-O/O) 

Hybrid coupler (Microlab/FXR CA-J06)

Isolator: 20 dB isolation (Nova 0165IAS) 

Directional coupler (Commscope 

C-20-CPUSE-N-Ai6)

BSB drive

DAC

To interlock

DAC

I

Q

DAC

Low-pass filter 

RF switch (Minicircuits 

ZASW-2-50DR+)

To monitor

Band-pass filter 

(Lorch 8IZ4-1737/A850-S)

Circulator: UTE CT-2006-OT

a b

a

b

Directional coupler (Commscope 

C-10-CPUSE-N-Ai6)

Figure 4.17: Microwaved drive system for generating carrier, red sideband, and blue sideband

signals. This entire system is duplicated three times for each of the trap electrodes. Low pass

filters are homebuilt Gaussian filters with a measured 3 dB cutoff frequency of ∼ 30 MHz.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
74

∆mF = 0 and hence requires π polarized microwaves. As the currents in the trap are at a 67.5

◦

angle to the quantization B field, the Rabi rate (Chapter 3 Eq. 3.14) for carrier transition on the

clock qubit is roughly

Ω0 = cos(67.5

◦

)

μ||By

~

= cos(67.5

◦

)

μ||μ0I

2π~d

, (4.5)

with μ|| ≈ 1μB [83] and μB is the Bohr magneton. Plugging in the numbers shows that carrier

Rabi frequncies of Ω0/2π ∼ 1 MHz can be achieved with ∼ 10 mA current amplitude. For spinmotion coupling, we need much higher currents to generate sufficiently strong near-field oscillating

magnetic field gradients. Oscillating currents at ωRSB = ω0 − ω and ωBSB = ω0 + ω give sideband

interactions (see Chapter 3) 9. Assuming the motional-mode direction ~q is parallel to the y axis

(Fig. 4.15), an order of magnitude estimate of the sideband interaction strength can be obtained

from Eq.3.15

Ω = cos(67.5

◦

)

μk(∇B.~q)q0

~

= cos(67.5

◦

)

μkμ0q0I

2π~d

2

(4.6)

which shows that we would need a factor of d/x0 ≈ 6 × 103 more microwave current to achieve a

Rabi frequency comparable to Ω0. In experiments, we typically measure sideband Rabi rates of

∼ 2 kHz corresponding to current amplitudes of ∼ 0.1 A (from Eq. 4.6 ). Each sideband field also

causes an AC Zeeman shift ∆ACZ ∝ −Ω

2/(ωSB −ω0), where ωSB is the sideband angular frequency

[69]. The AC Zeeman shift can be measured using a Ramsey experiment, where the sideband fields

are applied during the time between the Ramsey carrier π/2 pulses. For this measurement it is

important to detune (by several Ω) the magnetic field gradients away from the sideband resonances

(ω0 ± ω) so as not to excite the ion’s motion. For typical settings we measure |∆ACZ/2π| ∼ 15

kHz for the blue sideband and |∆ACZ/2π| ∼ 5 kHz for the red sideband. Unless otherwise stated,

in subsequent discussions, I will always include the AC Zeeman shift in the definition of the qubit

frequency i.e. ω0 → ω0 + ∆ACZ.

9 Note the difference between the coordinate system used in this chapter and in chapter 3

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
75

A schematic of the microwave chain used to drive currents in one microwave electrode of the

trap is sketched in Figure 4.17. Each chain has components for generating carrier, red sideband,

and blue sideband signals. There is a separate chain for each of the three microwave electrodes.

Each signal tone is generated by a DDS clocked at 2.4 GHz followed by a bandpass filter that

selects the first Nyquist image. The temporal envelope of the sideband signals is modulated using

an IQ mixer that is controlled by a 100 MSPS, arbitrary wave form generator [65]. The signal is

then amplified by ∼47 dB and filtered using a narrow-band cavity filter. The two sideband signals

are then combined with a hybrid coupler. The signal leaving one output port is attenuated and

sent to an interlock circuit that shuts off the input signals to the power amplifiers if the reflected

power from the trap exceeds a preselected value. The signal leaving the other port of the hybrid

is combined with the carrier signal using a directional coupler and capacitively coupled to the trap

through a 50 Ohm characteristic impedance SMA cable. On the trap chip, the microwave signals

are coupled into 50 Ohm strip lines that taper down in the trap region (Fig. 4.2). Immediately

after the trap, the electrodes are grounded to enforce a current anti-node boundary condition near

the trapping region. In this configuration, almost all of the microwave power delivered to the trap

is reflected.

4.6.2 Resonant motional drive

As described in Chapter 2, applying an oscillating force at the resonance frequency of a

harmonic oscillator displaces the oscillator in a frame rotating at ω. We implement an oscillating

force on a trapped ion by applying an oscillating potential (also known as a “tickle”) to one of

the DC electrodes of the surface trap (Fig. 4.15). The potential is generated using a switched,

amplified DDS. The oscillating potential results in an oscillating electric field at the position of the

ion which gives a force F(t) = qE0 sin(ωt+φ), where E0 is the component of the electric field along

the direction of oscillation of the mode of interest. In the frame rotating at ω (after making the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
76

parametric drive

12.6 MHz

rf trap drive

68.6 MHz

λ/2 @ 68.6 MHz

6.5 μH

500 nH

1.5 Ω

330 pF

33 pF

2.3 pF

BPF

BPF

to trap rf 

electrodes

Figure 4.18: Circuit for combining the parametric drive with the trap rf drive. BPF: Band-pass

filter

rotating wave approximation), the Hamiltonian is given by

Hˆ = −

~Ω

2i

(ˆae−iφ + ˆa

†

e

iφ), (4.7)

where Ω = qE0x0

~

. The time evolution operator Uˆ(t) = e

Ωt

2

(ˆae−iφ+ˆa

†eiφ) = Dˆ(α(t)), is a displacement operator with displacement amplitude α(t) = Ωt

2

e

−iφ

.

4.6.3 Parametric drive

Parametric modulation is implemented by applying an oscillating potential near 2ω to the rf

electrodes of the surface trap (Fig. 4.15). This modulates the trapping potential in the radial plane.

Applying the parametric drive to the rf electrodes ensures that the electric field curvature at the

position of the ion can be strongly modulated with minimal residual electric field oscillating at the

parametric drive frequency which might cause off-resonant excitation of the ion secular motion. The

drive signal is filtered using a band-pass filter with a bandwidth of ∼ 2 MHz to prevent unwanted

noise, particularly near ω, reaching the trap. We use a resonant tank circuit with series inductance

(Fig. 4.18) to combine the parametric drive with the trap rf drive before the coaxial resonator.

As described in Chapter 2, modulation of the confining potential of a harmonic oscillator at

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
77

2ω realizes in the interaction Hamiltonian for degenerate parametric modulation

Hˆ =

~g

2

(ˆa

2 + ˆa†2

), (4.8)

where g is the parametric coupling strength. Applying the Hamiltonian for a duration t implements

the squeezing operator Uˆ(t) = e

−igt

2

(ˆa

2+ˆa2†

) = Sˆ(ξ), with the squeezing parameter ξ = igt. From

an electric field simulation of the trap, we estimate that a parametric drive amplitude of ∼ 20 mV

results in g ∼ 50 kHz.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 5

Laser-free control of the trapped-ion quantum harmonic oscillator

The previous chapters described the theoretical and experimental tools required to generate and control non-classical states of trapped-ion motion using laser-free methods. This chapter

presents the experimental implementation of these methods for the generation and analysis of

thermal mixed states, number states, coherent states, squeezed states, and squeezed cat states.

Furthermore, the calibration and analysis procedures described in this chapter are essential for the

experiments presented in later chapters.

5.1 Motional state analysis

Since we cannot measure the motional state of our trapped ions directly, we need to map the

motional state onto the ion’s qubits states. As described in chapter 3 we can accomplish this using

either sideband interactions or spin-dependent forces. Chapter 3 describes the theory of sideband

analysis for the case where there is no decoherence of either the qubit or oscillator state. As the

coherence time of our “clock” qubits is several orders of magnitude larger than the duration of our

experiments (typically a few ms), qubit decoherence is usually negligible. The effects of motional

decoherence are taken into account using either full numerical simulations from a master equation

or by incorporating phenomenological decay factors following [19]. For the latter, Eq 3.24 which

gives the probability of of finding the ion in the |↓i state after applying a blue sideband for duration

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
79

t becomes

P↓(t) = 1

2

 

1 +X∞

n=0

Pne

−γnt

cos √

n + 1Ωt



!

, (5.1)

where we assume the phenomenological decay constants depends on n according to γn = γ0

√

n + 1

[20]. To avoid unnecessary exercise of the fingers of the reader, Pn are the probabilities of measuring

the oscillator in Fock state n (before application of the sideband), and Ω is the Rabi rate. Similarly

for red sideband analysis

P↓(t) = 1

2

 

1 +X∞

n=0

Pne

−γnt

cos √

nΩt



!

. (5.2)

5.2 Thermal distribution

After Doppler cooling, an ion’s motion is (theoretically) thermally distributed. Indeed, redsideband analysis after Doppler cooling a 25Mg+ ion (5.1) shows that the measured population

distribution is in good agreement with a thermal mixed state. Sideband-analysis using magnetic

field gradients also allows us to extract the mean thermal occupation  ̄n. For the data shown in Fig.

5.1,  ̄n = 2.89(3).

Doppler cooling is typically followed by resolved-sideband cooling to bring the motional occupation closer to zero. As  ̄n gets smaller, P↓(t) measured using RSB analysis tends to 1 (Fig.

5.1 a). Furthermore, the high frequency oscillations present in the Doppler-cooled data are largely

absent after ground state cooling, indicating that population in higher-order motional states has

been transferred to the ground state. Numerical simulations suggest that the slow decay of P↓(t)

for the ground-state-cooled data results from motional decoherence during the analysis pulse.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
80

0 2 4 6 8 10

Time (ms)

0.25

0.50

0.75

1.00

P

0 5 10 15

n

0.0

0.1

0.2

0.3

P

n

a) b)

Figure 5.1: Red sideband analysis of a thermal state. a) P↓ as a function of the red sideband

pulse duration t, after Doppler cooling (red points) and following ground sate cooling (blue points).

The solid black curve is a theoretical fit to the data assuming a thermal distribution. The mean

thermal occupation from the fit is  ̄n = 2.89(3). b) Number state populations obtained by fitting

an unconstrained number state distribution to the data in a). The black line indicates Pn for a

thermal distribution with  ̄n = 2.89. Error bars indicate one standard deviation of the mean.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
81

0.0 2.5 5.0

t(ms)

0.00

0.25

0.50

0.75

1.00

P

a) n = 0 n = 1

0.0 0.5 1.0

t(ms)

0.00

0.25

0.50

0.75

1.00 b) n = 9 n = 10

0.0 0.5 1.0

t(ms)

0.00

0.25

0.50

0.75

1.00 c) n = 29 n = 30

Figure 5.2: P↓ as a function of the blue sideband duration t, after preparing the system in Fock

states with n = 0 (a), n = 9 (b), n = 29 (c). The solid red curves are fits of decaying sinusoids (Eq.

5.3) to the data. The fitted Rabi frequencies are Ω0/2π = 1.1439(8) kHz, Ω9/2π = 3.60(1) kHz,

and Ω29/2π = 6.20(2) kHz. Error bars indicate one standard deviation of the mean.

5.3 Number states

Once the ion has been prepared in the motional ground state, we can use sequences of

sideband and carrier pulses to prepare higher-order number states. Starting in the state |↓i |n = 0i,

a sequence k blue sideband π pulses each followed by a carrier π pulse will prepare the state |↓i |ki.

Figure 5.2 shows BSB flopping data after preparing the number states |0i, |9i, and |29i. The Rabi

frequency Ω for each is obtained by fitting to the data an exponentially decaying sinusoid of the

form

P↓(t) = 1

2



1 + ae−γt cos(Ωt + φ)



, (5.3)

where, a, γ, φ, and Ω are fitting parameters. For a sideband interaction generated using magnetic

field gradients, we expect the Rabi frequency for flopping between |ni |↓i and |n + 1i |↑i to depend

on n according to Ωn = Ω0

√

n + 1 (this assumes the magnitude of the magnetic field varies linearly

over the spatial extent of the ion). The experimentally measured Rabi frequencies closely follow

this relationship up to at least n = 50 (Fig. 5.3). The loss of contrast of the Rabi oscillations

as n is increased (Fig. 5.2), results mainly from motional heating and dephasing during the time

required to prepare the number state.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
82

0 10 20 30 40 50

Number state n

0

1

2

3

4

5

6

7

n

+ 1 /

0

n + 1

Data

Figure 5.3: Sideband Rabi frequencies as a function of n, corresponding to the initially prepared

number state |ni. Points indicate experimentally measured values. Error bars are smaller than the

data points. The solid curve plots √n + 1.

5.4 Coherent states

In section 2.4 I described how coherent states of motion can be generated from the ground

state by applying an oscillating electric field at ω for a duration t. Example data for sideband

analysis of a coherent state is shown in Fig. 5.4. P↓(t) clearly shows a revival similar to the

“Jaynes Cummings revival” [30] at ∼ 2.4 ms BSB pulse duration. This revival is a signature of a

two-level system (TLS) coupled to an oscillator which is initially prepared in a coherent state. The

Jaynes Cummings interaction is usually discussed in context of cavity quantum electrodynamics,

where the coupling between an electric field mode and the TLS is mediated by the cavity. Here the

coupling between the trapped-ion mechanical oscillator and its internal states is achieve using the

sideband interaction (strictly speaking the Jaynes Cummings interaction is equivalent to the red

sideband interaction). The coherent state amplitude α is extracted by fitting Eq. 5.1 to the data,

assuming a coherent state distribution for Pn (Eq. 2.15). The Fock state populations extracted

from the fit (Fig. 5.4 b) show good agreement with the theoretically expected Poisson distribution

with amplitude |α| = Ωrt/2.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
83

0.0 2.5 5.0

BSB duration (ms)

0.0

0.25

0.50

0.75

1.00

P

a)

0 5 10

n

0.0

0.1

0.2

0.3

P

n

b)

0 10 20 30 40

Resonant drive duration ( s)

0.0

0.5

1.0

1.5 c)

Figure 5.4: Blue-sideband analysis of a coherent state. a) P↓ as a function of the blue sideband

pulse duration. The solid black curve is a theoretical fit to the data assuming a coherent state

distribution giving α = 1.50(2). b) Number state populations obtained by fitting an unconstrained

number state distribution to the data in a). The black line indicates Pn for a coherent state with

α = 1.50. Error bars indicate one standard deviation of the mean. c) Displacement amplitude |α| as

a function of the resonant drive duration. Points indicate values of α obtained using BSB analysis as

in a). A straight line fit to the data gives the resonant drive coupling strength Ωr/2π = 6.71(7) kHz.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
84

5.5 Squeezed states

Squeezed states are prepared by applying the electronic parametric drive at 2ω for a fixed

duration duration ts. If the ion is initially prepared in the motional ground state, the resulting

state is know as a squeezed ground or vacuum state. A signature of a squeezed ground state is

that it should contain only even Fock state populations (see chapter 2). We see this clearly in

the Fock state populations determined using BSB analysis (Fig5.5 b). BSB flopping data (Fig5.5

a) can be used to extract the magnitude of the squeezing parameter r and also the parametric

coupling strength g. The parametric coupling strength g, determines how fast a squeezed state can

be prepared. Since r = gts, a plot of r vs ts should give a straight line with a slope given by g. An

example calibration plot is shown in Fig 5.5 c.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
85

0.0 2.5 5.0

BSB duration (ms)

0.0

0.25

0.50

0.75

1.00

P

a)

0 10 20

n

0.0

0.1

0.2

0.3

P

n

b)

0.0 2.5 5.0 7.5 10

Resonant drive duration ( s)

0.0

0.5

1.0

1.5

2.0

2.5

r

c)

Figure 5.5: Blue-sideband analysis of a squeezed state. a) P↓ as a function of the blue sideband

pulse duration. The solid black curve is a theoretical fit to the data assuming a squeezed state

distribution giving r = 1.89(2). b) Number state populations obtained by fitting an unconstrained

number state distribution to the data in a). The black line indicates Pn for a squeezed ground state

with r = 1.89. Error bars indicate one standard deviation of the mean. c) Squeezing parameter r as

a function of the resonant drive duration. Points indicate values of r obtained using BSB analysis as

in a). A straight line fit to the data gives the parametric drive coupling strength g/2π = 50(2) kHz.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
86

5.6 Squeezed cat states

While sideband analysis of squeezed states allows us to extract the magnitude of the squeezing

parameter, it does not provide information about the coherence or orientation of squeezed states

in phase space. The phase-sensitive nature of squeezing can be clearly revealed by using spindependent forces (SDFs) to generate superpositions of oscillator wavepackets. Depending of the

orientation of the initial squeezed state and the direction of the SDF, measurements of the qubit

state can have very different outcomes (Chapter 3). For the situation shown in Fig 5.6, a SDF

that acts in the ˆσx basis is applied to an ion initially prepared in |↓i |ξ = reiθi. The final overlap

of the wavepackets associated with the |+i and |−i spin states is larger if the squeezing phase is

θ = π (SDF along the antisqueezed direction) than if θ = 0 (SDF along the squeezed direction).

The probability of finding the ion in |↓i is clearly larger if θ = π.

The generation of squeezed Schr ̈odinger cat states and squeezed-state analysis using SDFs

has been demonstrated by the ETH group [84]. In their experiment, the initial squeezed state was

prepared by a dissipative method [20] and the SDF was generated using lasers. Motivated by these

results, we decided to generate squeezed cat states in our system using laser-free methods. We

prepare an ion in the |↓i |ξ = reiθi, where the squeezing is performed using parametric modulation

of the trapping potential. We then apply the SDF, which is generated using magnetic field gradients,

for a duration t, and finally we detect. As expected (see Chapter 3), the probability of finding the

ion in |↓i depends on the phase of the parametric drive relative to the spin-dependent force (Fig.

5.7). For comparison we also performed the experiment for an initial ground state (r = 0). While

the data for the squeezed and the ground state cases gives good agreement with analytical theory,

the antisqueezed case does not. Numerical simulations suggest that motional decoherence from

fast dephasing and heating (see section 3.5), and shot-to-shot fluctuations of the motional mode

frequency can account for some of the discrepancy. The physical picture is that the wavpacket

overlap is more sensitive to small phase-space rotations if the squeezed wavepackets are nominally

aligned along their major axes. I suspect that the remaining discrepancy is present because the spinDocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
87

- +

Im(α)

Re(α) +

Im(α)

Re(α)

a) c)

+

Im(α)

Re(α)

b)

-

-

Figure 5.6: Squeezed cat states in phase spaces. In each panel an identical spin-dependent displacement (indicated by black arrows) is applied along the real axis to an ion initially prepared in the

|↓i qubit state and in a motional state with a) squeezing along Re(α) axis (θ = 0) , b) the ground

state (r = 0), and c) squeezing along the Im(α) axis (θ = π).

dependent force was not configured correctly when these data were taken. For the data shown here,

the SDF force was generated using the Bermudez scheme [85] where a single sideband is applied

together with a strong carrier field. For reasons we don’t fully understand, we found it difficult to

reliably set the carrier amplitude to give a reproducible spin-dependent force (Numerical analysis

by Harty et al. [86], shows that the fidelity of two-qubit gates implemented using the Bermudez

scheme can be very sensitive to the carrier to sideband Rabi frequency ratio).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
88

b)

a)

Figure 5.7: Squeezed cat state experimental data. a) P↓ as a function of the spin-dependent

force duration tSDF for a Shr ̈odinger cat state with and without squeezing. The solid curves are

analytical theory (Eq. 3.37 ). The value of the squeezing parameter (r = 1.28(2)) was obtained

from the fit of the analytical theory to the red data points (with θ = 0). The dashed curve is from

a numerical model incorporating known motional decoherence mechanisms. b) P↓ as a function of

the parametric drive phase θ at tSDF = 60 μs (dashed black line in a).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 6

Quantum amplification of trapped-ion motion

In this chapter and in [87], I describe a technique for amplifying coherent displacements of a

mechanical oscillator with initial magnitudes well below the zero-point fluctuations. By applying

two orthogonal squeezing interactions, before and after a small displacement, the displacement can

be amplified, ideally with no added quantum noise. We have implemented this protocol with a

trapped-ion oscillator and measured an increase by a factor of up to 7.3(3) in sensitivity to small

displacements. This technique could aid the detection of weak forces and the search for new physics,

in particular a dark matter axion-like field [88, 89].

Mechanical oscillators are essential components in an increasing variety of precision sensing

applications including gravitational wave detection [90], atomic force microscopy [91], cavity optomechanics [92], and measurement of weak electric fields [93]. However, the attainable knowledge

of an oscillator’s motional state is limited by unavoidable quantum fluctuations that are present

even if the oscillator is in its ground state. By squeezing the oscillator’s motional state improved

measurement precision for one observable can be achieved at the expense of increased fluctuations

of the conjugate observable [6]. Although squeezed states have been created in a variety of physical

systems (See Chapter 1), exploiting squeezing for enhanced metrology has been challenging. In

particular, noise added during the detection process limits the metrological enhancement unless it

is smaller than the squeezed noise. The requirement of low-noise detection can be overcome by

increasing the magnitude of the signal to be measured, provided the noise does not significantly

increase. In optical interferometry [94] and in spin systems [95], it has been shown that reversal of

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
90

squeezing interactions can magnify small phase shifts, relaxing detection requirements [96]. Photon field displacements in microwave cavities have also been amplified using similar phase-sensitive

amplification schemes [97, 98].

6.1 Noiseless amplification of displacements

Here I describe a protocol based on reversible squeezing for ideally noiseless phase-sensitive

amplification [6] of mechanical oscillator displacements. This amplification method (Fig. 6.1) is

applicable to any harmonic oscillator where reversible squeezing can be implemented faster than

system decoherence. By first squeezing the motional ground state, quantum fluctuations along a

particular phase space quadrature are suppressed. A small test displacement Dˆ(αi) (to be amplified) is then applied along the squeezed axis. At this stage, although the signal-to-noise ratio for

measuring αi has been improved by squeezing, resolution below the zero-point fluctuations would

require a detection method with yet lower noise. Finally, by reversing the squeezing interaction,

the oscillator returns to a minimum-uncertainty coherent state with a larger amplitude αf = Gαi,

where G is the gain. Ideally, this process adds no noise in either quadrature. For an oscillator

described using creation and annihilation operators ˆa

† and ˆa, the amplification is given by the

expression [99]

Dˆ(αf ) = Sˆ†(ξ)Dˆ(αi)Sˆ(ξ), (6.1)

where Dˆ(α) = exp(αaˆ

† − α∗aˆ) is the displacement operator, and Sˆ(ξ) = exp( 1

2

(ξ

∗aˆ2 − ξaˆ†2

))

is the squeezing operator with the complex squeezing parameter ξ(r, θ) = reiθ (see chapter 2).

For arbitrary orientations of the displacement αi with respect to the initial squeezing axis, αf =

αi cosh(r) + α

∗

i

e

iθ sinh(r). Maximum amplification is achieved if the displacement is along the

squeezed axis (where arg(αi) = θ/2), giving G = e

r

.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
91

squeeze antisqueezedisplace

Im(Į)

Re(Į)

Įi

*Įi

S D( S† Įi)

Figure 6.1: Conceptual illustration of the amplification protocol. Each panel shows a cartoon

phase space quasiprobability distribution in a frame rotating at the oscillator frequency. A displacement αi of an initially squeezed ground state is amplified by subsequent reversed squeezing

(“anti-squeezing”), resulting in a final coherent state with amplitude Gαi with no added noise.

Dashed red circles indicate the characteristic extent of the initial ground state fluctuations.

6.2 Experimental implementation

This amplification technique was demonstrated using a single trapped 25Mg+ ion as the

mechanical oscillator (details of the experimental apparatus are given in chapter 4). Experiments

were performed on a radial motional mode of the ion with frequency ωr ' 2π × 6.3 MHz and

zero-point wavefunction extent of ' 5.7 nm. To analyze the motional state, we use the “clock”

qubit states |↓i ≡ |F = 3, mF = 1i and |↑i ≡ |F = 2, mF = 1i within the 2S1/2 manifold. In each

experiment, the ion was initialized in the electronic ground state |↓i and close to the motional

ground state |n = 0i with optical pumping, resolved-sideband laser cooling, and microwave pulses

as described in chapter 4. The motional state of the ion was analysed using blue sideband analysis

(see chapters 3 and 5). The sideband interactions were generated using oscillating magnetic field

gradients (see chapters 3 and 4). Applying a sideband interaction for various durations results in

qubit Rabi oscillations with multiple frequency components whose amplitudes depend on the Fock

state populations. Squeezing and displacement operations are implemented using electronic drives

as described in chapters 2-5.

To characterize the unitarity of the squeezing operations, we measured the ground state

population using a red sideband π pulse after squeezing and anti-squeezing, which would ideally

yield perfect overlap h0| Sˆ†Sˆ |0i = 1 (Fig. 6.2). Imperfections of the operation lead to a decrease in

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
92

squeeze

S

antisqueeze

S†

Re(α)

Im(α) a)

b)

c)

Prepare

|0|↓ Sˆ(ξ) Sˆ†

(ξ) RSB

π Detect

Figure 6.2: Unitary squeezing. a) Illustration of the experiment. The ion is initially prepared in

the motional ground state. A squeezed state is then generated by parametric modulation. Finally

the inverse squeezing interaction is applied to bring the oscillator back to the ground state. b)

Pulse sequence. After squeezing and anti-squeezing a red sideband (RSB) analysis π pulse can flip

the qubit state from |↓i to |↑i if oscillator is not returned to the ground state. c) Probability of

measuring |↓i after the RSB pulse. Even without squeezing, there is ∼2 % residual population in

|↑i after the RSB pulse due to imperfect state preparation and motional heating during the analysis

pulse. This background is indicated by the red line.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
93

the observed overlap as r is increased. For r < 2, P↓ ≈ 0.98, which is consistent with the measured

value without squeezing and anti-squeezing (r = 0). The population in n = 0 remains above 0.93

for r < 2.37(3), or 20.6(3) dB of squeezing1.

After each operation of the amplification process, the Fock state composition of the ion’s

motional state was analyzed using sideband analysis (Fig. 6.3). The ion is first prepared in the

motional and electronic ground state |0i |↓i. Interactions (Sˆ(ξ), Dˆ(αi), and Sˆ†(ξ)) that transform

the motional state are then applied in sequence. A blue sideband pulse is then applied for a

variable duration and the internal state is detected. Fitting Eq. 5.1 to the data as a function of the

interaction duration, and assuming an underlying distribution for Pn from Eq. 2.15, 2.36, or 2.49

as appropriate, allows us to extract best fit values for parameters characterizing the assumed form

of the motional state. We also separately fit Eq. 5.1 to the data assuming no particular underlying

distribution for Pn, giving the Fock state populations of the motional state being analyzed. Error

bars for fitting parameters are calculated as the square root of the corresponding diagonal elements

of the least-squares covariance matrix. We repeated this process for several different values of

the squeezing strength and calculate the measured amplification gains as the ratio αf /αi. The

measured gain for various values of the squeezing parameter r = gt closely follows the theoretically

expected exponential growth of the coherent state amplitude (Fig. 6.4).

1 The conversion from r to decibels of squeezing is calculated according to dBsqueezing ≡ 20log10(er

) [100].

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
94

Im(α)

Re(α)

Re(α)

Im(α)

Im(α)

Re(α)

αi

Gαi

Figure 6.3: Fock state population analysis. a), b), and c) show BSB Rabi oscillations (left column)

and corresponding fitted Fock state population histograms (right column). Solid red lines are fits

assuming parameterized functional forms of the ideal Fock state populations, yielding values of r, αi,

and as described in the main text. Dashed red lines are parameterized functional forms that do not

include decoherence. Histograms are derived by fitting to an unconstrained population distribution.

Insets show Wigner function illustrations of the corresponding motional states. a) Initial squeezed

motional ground state with r = 2.26(2). b) After displacing this state by αi = 0.200(2). c) Final

coherent state with amplitude αf = 1.83(1), following the reversed squeezing operation. The initial

displacement is amplified by G = αf /αi = 9.17(9). Error bars in the first column indicate one

standard deviation of the mean, and in the second column 68% confidence intervals.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
95

0.0 0.5 1.0 1.5 2.0 2.5

r

0

2

4

6

8

10

G

= f/ i

e

r

Data

Figure 6.4: Measured gain as a function of the squeezing parameter r. The solid line is the

theoretical gain G = e

r

. Error bars in all figures indicate 68% confidence intervals.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
96

6.3 Quantum-enhanced displacement measurements

Using this amplification technique, we achieve increased sensitivity when measuring displacements much smaller than the zero-point fluctuations. To map the final displacements αf onto the

qubit states, we use a phase-sensitive red sideband (PSRSB) method [47] (Fig. 6.6a). For displacements with |α|  1, this method gives measurement sensitivities proportional to |α|, which is the

standard quantum limit for a displacement measurement. The linear scaling enables straightforward measurement of the amplification gain and is desirable for measuring small displacements. I

first give a intuitive description of the PSRSB method that is valid for small displacements, and

then give a derivation that is valid for all orders of |α|. The ion is first prepared in the motional

ground state |ψi = |↓i |0i. A small displacement of the motional ground state (|α|  1) results in

a small population transfer out of the ground state, predominantly to the n = 1 Fock state. In the

interaction picture with respect to the qubit and the harmonic oscillator [41] :

|ψi → Dˆ(α)|0i |↓i (6.2)

= (ˆI + αaˆ

† − α∗

aˆ + ...)|0i |↓i (6.3)

≈ (|0i + α |1i)|↓i. (6.4)

The subsequent red sideband π pulse maps the the population in |1i |↓i onto |0i |↑i, but leaves the

population in |↓i |0i unchanged. The state becomes

|ψi ≈ (|↓i − |α| |↑i)|0i (6.5)

for the case where the phase of the beatnote between the RSB pulse and the displacement has been

set to give a qubit rotation about the φ = 0 axis on the Bloch sphere as shown in Fig 6.5. A carrier

π/2 rotation about the φ axis then gives

|ψi ≈ 1

√

2

h

(1 − |α|e

+iφ)|↓i − (|α| + e−iφ)|↑ii

|0i, (6.6)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
97

Gαi

αi

Carrier RSB 

π φ

φ=0 π

2 Detect

Figure 6.5: Bloch sphere illustration of the phase-sensitive red sideband readout scheme.

showing that the state vector has a phase-sensitive projection onto the polar axis of the Bloch

sphere. The probability of finding the system in |↓i is

P↓ ≈

1

2



1 − |α|e

iφ



2

=

1 + |α|

2

2

− |α| cos(φ)

≈

1

2

− |α| cos(φ). (6.7)

which shows that P↓ sinusoidally oscillates as a function of φ with contrast 2|α|. If the displacement

is amplified, then the contrast becomes 2|Gα| with the requirement that |Gα|  1. Therefore G can

be extracted by measuring the slope of the contrast as a function of |α| provided |Gα| is sufficiently

small.

Next, I derive the state |ψi of the ion at the end of the PSRSB sequence for all orders of |α|.

The state of the ion after the displacement, sideband interaction, and carrier pulse is given by

|ψi = Rˆ(π/2, φ)UˆRSBDˆ(α)|↓i |0i, (6.8)

where Rˆ(π/2, φ) is a spin rotation matrix describing the π/2 carrier pulse and UˆRSB is the resonant

red sideband time evolution operator

UˆRSB = e

− iΩt

2

(ˆσ+aˆ+ˆσ−aˆ

†

)

, (6.9)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
98

where Ω is the sideband Rabi frequency. The duration t of the RSB interaction is set to give

Ωt/2 = π/2. The component in |↓i is

h↓ |ψi = h↓| Rˆ(π/2, φ)UˆRSBDˆ(α)|↓i |0i

=

1

√

2



h↓| + h↑| e

iφ

UˆRSB |αi |↓i. (6.10)

The probability of measuring |↓i is therefore

P↓ =

1

2

h

1 + e

iφ h↓| hα|Uˆ†

RSB |↓i h↑|UˆRSB |αi |↓i

+e

−iφ h↓| hα|Uˆ†

RSB |↑i h↓|UˆRSB |αi |↓ii

. (6.11)

Evaluating the matrix elements gives:

h↑|UˆRSB |αi |↓i =

X∞

l=0

(−i

π

2

)

2l+1(1 + ˆn)l

(2l + 1)! α |αi, (6.12)

and

h↓|UˆRSB |αi |↓i =

X∞

l=0

(i

π

2

)

2l

(ˆn)

l

(2l)! |αi, (6.13)

where ˆn = ˆa

†aˆ is the harmonic oscillator number operator. Setting the phase of the beatnote

between the RSB pulse and the displacement so that the initial rotation is about the φ = 0 axis as

before, and inserting equations 6.12 and 6.13 into 6.11 gives

P↓ =

1

2

(1 − |α|f(|α|)e

iφ − |α|f(|α|)e−iφ)

=

1

2

− |α|f(|α|) cos(φ), (6.14)

where

f(|α|) = X∞

n=0

e

−|α|

2

|α|

2n

n!

 

cos(π

2

√

n) sin( π

2

√

n + 1)

√

n + 1 !

. (6.15)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
99

Prepare

|0| ↓ Sˆ(ξ) Dˆ(αi) Sˆ†(ξ) RSB

π

Carrier

π/2|φ Detect

Aa) Dˆ(Gαi)

b)

c)

d)

e)

Figure 6.6: Measurement sensitivity enhancement for oscillator displacements. a) Pulse sequence

for displacement sensing using phase-sensitive red-sideband (PSRSB) detection. b) Population in

|↓i as a function of the carrier π/2 pulse phase. Blue diamonds are data with no squeezing and red

circles are data with amplification. Solid lines show sinusoidal fits to the data. c) Contrast of the

carrier phase scan, as shown in b), as a function of the squeezing phase θ for a fixed displacement.

d) Contrast as a function of the displacement amplitude |αi| for different initial squeezing pulse

durations. Each data point is calculated from ≈ 104experiments. The data shown in b) and c)

have initial |αi| = 0.0578(6) and a squeezing duration of t = 8 μs (nominally r = 2.54(3)). The solid

black line is the maximum theoretical contrast without squeezing. Dashed lines in c) and d) are

derived from a numerical model that includes motional decoherence. e) Measurement sensitivity

enhancement in the linear small-displacement regime as a function of the ideal gain G = e

gt. For

each squeezing duration, the enhancement is determined by dividing the slope of the contrast for

C ≤ 0.25 (obtained by fitting a straight line to data points in d) with C . 0.25) by the slope of

the 0 dB black line, which represents the standard quantum limit for a displacement measurement.

All error bars represent one standard deviation of the mean.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
100

Equation 6.14 shows that P↓ oscillates sinusoidally with contrast C(|α|) = 2|α|f(|α|) as a

function of φ. The contrast of the oscillation reaches a maximum value of 0.84 for |α|=0.69. If

squeezing is used to amplify the displacement, |αi| → G|αi| = |αf |. For |αf |  1, C(αf ) ≈ 2|αf |.

In comparison, simply measuring the qubit directly after the RSB π-pulse gives a signal P↓ ∝ |αf |

2

.

Without amplification, αf = αi and the PSRSB contrast is C(|αi|). With amplification, the initial

displacement amplitude αiis ideally increased by a factor of G and the PSRSB contrast becomes

C(|Gαi|). This increase in contrast is shown in Fig. 6.6b, where the presence of oscillations for

the state after amplification indicates that it has a well-defined motional phase. The theory curves

in Fig. 6.6b are functions of the form b + a cos(φ), with b and a as the fitting parameters. With

8 μs parametric drive duration for each squeezing pulse and |αi| = 0.055(2), the fitting parameters

are b = 0.503(3) and a = 0.263(4). For the scan with no amplification, the fitting parameters

are b = 0.500(2) and a = 0.055(3). The carrier phase dependence in Fig. 6.6b is a feature of the

PSRSB method, not of the amplification protocol. Fig. 6.6c highlights the phase-sensitive nature

of the amplification protocol by plotting the contrast C of the PSRSB fringe against the squeezing

phase θ for a fixed displacement and with φ = 0. Maximum amplification is achieved when the

displacement is oriented along the squeezed axis of the initial squeezed state in motional phase

space (see Fig. 6.1).

For various parametric drive durations, we measure the contrast of the carrier phase scan

signal (Fig. 6.6b), as a function of the displacement amplitude, with the squeezing phase set to

maximize the contrast (θ = 0). Experimental data and theory curves are shown in Fig. 6.6d. The

displacement amplitude is varied by changing the amplitude of the digital synthesizer that generates

the resonant drive. The resonant drive pulse duration is fixed at 5 μs. For each displacement, the

contrast is defined as C ≡ P↓,max−P↓,min, where P↓,max and P↓,min are the maximum and minimum

values of P↓ of the fringes shown in Fig. 6.6b. The projection noise [101] associated with measuring

the contrast is σ(C) = p

σ(P↓,max)

2 + σ(P↓,min)2, where σ(P↓,max) = p

P↓,max(1 − P↓,max)/Nmax

and σ(P↓,min) = pP↓,min(1 − P↓,min)/Nmin, where Nmax(min)is the number of measurements taken

at the fringe maximum (minimum). The signal-to-noise ratio (SNR) for a displacement measureDocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
101

ment is then s(G) = C(αf )/σ(C(αf )). For the case where |αf |  1, the contrast is C = 2|αf |,

P↓,min = P↓,max ≈ 0.5, and Nmax = Nmin = N/2. The total projection noise is σ(C) = 1/

√

N,

and hence, the signal-to-noise ratio is s(G) = 2|αf |

√

N. This is the SNR for a displacement

measurement at the standard quantum limit [102].

For a given number of experiments, amplification allows the SNR for a displacement measurement to be improved in comparison to the ideal PSRSB measurement with no squeezing (black

solid line in Fig. 6.6d), giving a measurement sensitivity enhancement of s(G)/s(G = 1). For

measurements where C . 0.25, the contrast varies linearly with |αi|, and the gain in contrast

C(Gαi)/C(αi) sets a lower bound (which becomes exact as |αi| → 0) on the measurement sensitivity enhancement, because the projection noise decreases monotonically with increasing contrast.

Increasing the squeezing results in increased contrast for |αf |  1, up to a squeezing time of approximately 8 μs (corresponding to r = 2.54(3), and ideally 22.0(3) dB of squeezing). Here we achieve

a contrast gain of 7.3(3), corresponding to a factor of 53(4) reduction in the number of measurements required to achieve a given SNR. The reduction in contrast for larger squeezing durations is

attributed mainly to motional heating and motional dephasing. The theory curves (dashed lines)

shown in the Figs. 6.6c and d are from numerical simulations incorporating these effects (see section 3.5). The motional heating rate was independently measured to be  ̇n = 20(3) quanta/s−1

and the motional dephasing rate Γ = 18(6) s−1 was obtained from a least-squares fit to the 8 μs

squeezing duration curve shown in Fig. 6.6d. This motional decoherence is not a limitation of the

amplification process or our squeezing method.

The amplification technique described in this chapter can enhance measurement sensitivity

in protocols that use phase-stable displacements, such as photon-recoil spectroscopy [47, 103],

where the phase of momentum kicks from photon absorption can be controlled by modulating

the photon source. Moreover, our method can be extended to amplify displacements of unknown

frequency or phase and for amplifying spin-dependent forces to enhance phonon-mediated spinspin interactions [104, 105], which are used to generate entanglement in quantum simulation and

quantum information processing experiments. These advanced parametric amplification protocols

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
102

are the subject of the next two chapters.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 7

Hamiltonian amplification

The Hamiltonian is an operator of fundamental importance in quantum physics. According

to Schr ̈odinger’s equation,

i~

∂

∂t |ψi = Hˆ |ψi, (7.1)

the Hamiltonian (Hˆ ) determines how a quantum state |ψi will evolve in time. In experimental

quantum physics, it is often desirable to control the rate at which a quantum system evolves from

an initial state to a target state. This is often achieved by directly adjusting the strength of Hˆ .

For example, if we wish to generate a Bell state in a specific duration t starting from the initial

state |↓↓i using the geometric phase gate Hamiltonian (see Chapter 3)

Hˆ =

~Ω

2

(ˆaeiδt + ˆa

†

e

−iδt)(ˆσx1 − σˆx2), (7.2)

we require the coupling strength Ω = 2π

t

. The coupling strength must be set experimentally by

adjusting the intensity of the lasers or the magnitude of the magnetic field gradients that are

used to implement the gate interaction. However, technical limits prevent arbitrary increases in Ω.

Furthermore, the increase in Ω may be accompanied by increased noise that can lead to decoherence

or reduced fidelity of the target state.

The recent proposal of Arenz et al.[105] describes an alternative approach for increasing

coupling strengths in a broad class of Hamiltonians involving harmonic oscillator interactions.

The method uses a sequence of rapid squeezing pulses along alternating quadratures to effectively

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
104

Repeat n times

Uˆ

 t

2n



Uˆ

 t

2n



Original dynamics:

Hamiltonian amplification:

Uˆ(t) = e−i Ht ˆ

h ̄

UˆHA(t) =  e−i cosh(kr)Ht ˆ

h ̄

Hˆ → cosh(kr)Hˆ

Figure 7.1: Hamiltonian amplification pulse sequence. The Hamiltonian to be amplified (Hˆ of the

form given in Eq. 7.3) is applied within a sequence of squeezing interactions. The overall sequence

is repeated n times. If n is sufficiently large, the effective Hamiltonian will converge to cosh(kr)Hˆ .

amplify the Hamiltonian of interest. The operator sequence required to implement the discrete

version of Hamiltonian amplification (HA) is shown in Figure 7.1. Here I will consider Hamiltonians

Hˆ of the form

Hˆ = β1xˆ

k + β2pˆk

, (7.3)

where β1,2 are constants or operators that do not depend on ˆa or ˆa

†

, and k is an integer. For more

complicated Hamiltonians refer to [105]. We will assume that the squeezing pulses are configured

to give a squeezing parameter of ξ = r so that

Sˆ†(r)ˆxSˆ(r) = ˆxer

Sˆ†(r)ˆpSˆ(r) = ˆpe−r. (7.4)

The Hamiltonian amplification process consists of n repetitions of the sequence

Sˆ(ξ)Uˆ



t

2n



Sˆ†(ξ)Sˆ†(ξ)Uˆ



t

2n



Sˆ(ξ), (7.5)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
105

where t is the total duration for which Hˆ is applied and Uˆ(

t

2n

) = e

−iHt/ ˆ 2n~

. The overall process

can be described by the unitary operator

UˆHA =



Sˆ(ξ)Uˆ



t

2n



Sˆ†(ξ)Sˆ†(ξ)Uˆ



t

2n



Sˆ(ξ)

n

=

h

e

− it

2~n

(β1xˆ

ke−r+β2pˆker

)

e

− it

2~n

(β1xˆ

ker+β2pˆke−r

)

in

=



1 −

it

~n

cosh kr β1xˆ

k + β2pˆk



+ ...n

' e

i

~

cosh(kr)Hˆ

, (7.6)

for sufficiently large n, becoming exact as n → ∞ by the Trotter theorem [106]. This shows

that the Hamiltonian (Eq. 7.3) is effectively amplified by a factor of cosh(kr). Moreover, the

amplification factor does not depend on the phase relationship between the squeezing interactions

and the Hamiltonian of interest.

7.1 Amplification of displacements

As described in Chapter 2, displacements of a trapped ion’s motional state can be implemented using an oscillating electric field with a frequency ω equal to the frequency of one of the

ion’s normal modes. After making the rotating wave approximation and transforming into the

interaction picture with respect to Hˆ

0 =

~ω0

2

σˆz +

~ω

2

aˆ

†aˆ, the Hamiltonian for the driven oscillator

is given by

HˆD =

~Ω

2

(ˆae−iθ + ˆa

†

e

iθ), (7.7)

where Ω = qEz0/~ is the coupling strength and θ is the relative phase between the co-rotating part

of the electric field amplitude and the real axis in the interaction picture. If we choose θ = π/2 and

the Hamiltonian is applied for a time ∆t = t/2n, the time evolution operator is given by

Uˆ(t) = e

−i

Hˆ ∆t

~ = e

−i

xpˆ

2 = Dˆ

x

2



, (7.8)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
106

which is a displacement operator giving a translation of x

2 =

Ω∆t

4

along the real axis in phase space.

Applying the squeezing sequence given in Eq. 7.4

Sˆ†(r)e

−i

xpˆ

2 Sˆ(r) = e

−i

xerpˆ

2 , (7.9)

and

Sˆ†(−r)e

−i

xpˆ

2 Sˆ(−r) = e

−i

xe−rpˆ

2 . (7.10)

If we combine these sequences, we get

Sˆ(r)Uˆ(t)Sˆ(−2r)Uˆ(t)Sˆ(r) = Sˆ(r)

†

e

−i

xpˆ

2 Sˆ(r)Sˆ(−r)

†

e

−i

xpˆ

2 Sˆ(−r)

= Sˆ†(r)e

−i

xpˆ

2 Sˆ(r)Sˆ(r)e

−i

xpˆ

2 Sˆ†

(r)

= e

−ix

(e

r + e−r

)

2

pˆ

= e

−ix cosh(r)ˆp

, (7.11)

which is equivalent to a displacement operator Dˆ(cosh(r)x). Note that the amplification is formally

exact for n = 1 and also for any larger values of n. For an arbitrary displacement Dˆ(α) and a

general squeezing interaction Sˆ(ξ = reiφ), we have

Sˆ†(ξ)Dˆ(α)Sˆ(ξ) = Dˆ(γ1), (7.12)

where γ1 = α cosh r + α

∗

e

iφ sinh r. Similarly,

Sˆ†(−ξ)Dˆ(α)Sˆ(−ξ) = Dˆ(γ2), (7.13)

with γ2 = α cosh r − α

∗

e

iφ sinh r. The combined sequence is given by

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
107

Sˆ†(ξ)Dˆ(α)Sˆ(ξ)Sˆ†(−ξ)Dˆ(α)Sˆ(−ξ)

= Dˆ(γ1)Dˆ(γ2)

= Dˆ(γ1 + γ2)e

(γ1γ

∗

2 −γ2γ

∗

1

)/2

. (7.14)

which is an overall displacement of γ1 +γ2 = 2α cosh r, independent of the angle of the squeezing

interaction with respect to the displacement direction. The overall geometric phase factor would

not affect the outcome of measurements of the spin (the only observable that we measure directly

with ions) unless the displacement is spin dependent.

We have performed preliminary experiments with a single trapped ion that demonstrate

Hamiltonian amplification of displacements. The ion is initially prepared close to the ground state

|n = 0i of a radial motional mode with frequency ω/2π × 6.8 MHz and in the |↓i clock qubit state

(details about the experimental setup are given in chapter 4). Displacements are implemented by

applying an oscillating electric field at ω. Squeezing pulses are implemented by modulating the

trapping potential at 2ω, using the electronic parametric drive. The magnitude of the squeezing

parameter is set by varying the squeezing pulse time ts (r = gts) or by changing the amplitude

of the parametric drive. A sequence of squeezing and displacement pulses are applied to give a

Hamiltonian amplification sequence with n = 3 (see Fig. 7.1). At the end of the amplification

sequence, we apply a blue sideband analysis pulse for varying durations to map the resulting

motional state onto the ion’s qubit states (see chapter 3) which we can measure using fluorescence

detection. The sideband is implemented using oscillating magnetic field gradients. We first perform

a control experiment where we apply a HA sequence with n = 3, but with r = 0. For the control,

we measure a coherent state amplitude of α = 0.50(2) (Fig. 7.2 a). When the squeezing is turned

on, with r ∼ 1.9, we measure a displacement amplitude of α = 1.68(3) (Fig. 7.2 b). Note that

the amplitude and total duration of the resonant drive pulses are the same for both experiments

(20 μs). The amplification gain is 3.4(2) in good agreement with the theoretically expected gain of

cosh(1.9) = 3.4.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
108

0 1 2 3 4

0.00

0.25

0.50

0.75

1.00

P

a)

0 2 4 6 8

0.00

0.25

0.50

0.75

Population

0 1 2 3 4

Time (ms)

0.00

0.25

0.50

0.75

1.00

P

b)

0 2 4 6 8

Fock state

0.00

0.25

0.50

0.75

Population

Figure 7.2: Amplification of a coherent displacement. BSB analysis after Hamiltonian amplification

with n = 3 (see Fig. 7.1), with squeezing parameters of r = 0 in a) and r = 1.9 in b). In the left

column, populations in |↓i are plotted as function of the BSB analysis pulse duration. Blue circles

are experimental data. Error bars indicate the standard error of the mean. The solid red curves

are fits of Eq. 5.1 to the data assuming a coherent state Fock state distribution (Eq. 2.15) with the

displacement amplitude α and the phenomenological decoherence rate γ0 as the fitting parameters.

The dashed red lines are plots of Eq. 5.1 using the fitted value of α, but with the decoherence rate

γ0 set to zero. The panels on the right indicate the Fock state populations obtained from fitting

unconstrained Fock state distribution to the data. The solid blue lines indicate the populations of

coherent states expected from the fitted values of α.

7.2 Amplification of the Jaynes-Cummings interaction

The Jaynes-Cummings (JC) model gives a fully quantum-mechanical description of the interaction of a two-level system and a single-mode bosonic field [32]. It is of great theoretical and

experimental interest in quantum optics, atomic physics, and solid-state physics. An important extension is the Tavis-Cummings model [107] which describes the interaction of a multiple two-level

systems with a single bosonic mode. This model describes the generation of entanglement between

the two-level systems mediated by the boson field. As discussed in Chapter 2, JC-type interactions

are used extensively in trapped-ion experiments. Applications include resolved-sideband cooling,

qubit entangling gates, and motional state analysis.

In trapped-ion systems, Jaynes-Cummings dynamics can be implemented by applying the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
109

red sideband interaction Hamiltonian (see chapter 3)

HˆRSB =

~Ω

2

(ˆa

†σˆ− + ˆaσˆ+). (7.15)

The time evolution operator for this Hamiltonian is given by

UˆRSB(∆t) = exp −i

Ω∆t

2

(ˆaσˆ+ + ˆa

†σˆ−)



. (7.16)

Applying squeezing pulses for the Hamiltonian amplification protocol gives

Sˆ†(ξ)UˆRSB(∆t)Sˆ(ξ)

= exp −i

Ω∆t

2



(ˆa cosh r − aˆ

†

e

iφ sinh r)ˆσ+ + (ˆa†

cosh r − aeˆ

−iφ sinh r)ˆσ−





(7.17)

and

Sˆ†(−ξ)UˆRSB(∆t)Sˆ(−ξ)

= exp −i

Ω∆t

2



(ˆa cosh r + ˆa

†

e

iφ sinh r)ˆσ+ + (ˆa†

cosh r + ˆae−iφ sinh r)ˆσ−





. (7.18)

Combined, the overall sequence is

Sˆ†

(ξ)UˆBSB(∆t)Sˆ(ξ)Sˆ†(−ξ)UˆRSB(∆t)Sˆ(−ξ)

= Sˆ†(ξ)UˆRSB(∆t)Sˆ(2ξ)UˆRSB(∆t)Sˆ†(ξ)

= ˆI +



Ω∆t

2

2 cosh r(ˆaσˆ+ + ˆa

†σˆ−)



+ ... (7.19)

With ∆t =

t

2n

, in the limit that n → ∞ becomes

UˆHA(t) = limn→∞ 

ˆI +



Ωt

2n

cosh r(ˆaσˆ− + ˆa

†σˆ+)



+ ...n

= exp −i

cosh rHˆRSBt

~

!

(7.20)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
110

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

t/4

0.0

0.2

0.4

0.6

0.8

1.0

P

r=0

r=0.50

r=1.00

Figure 7.3: Numerical simulations of Hamiltonian amplification of the Jaynes-Cummings interaction

(assuming a √n + 1 scaling of the sideband Rabi frequency) for various values of the squeezing

parameter r. Solid curves are numerical simulations of the population in |↓si after initially preparing

the system in |↑si |0i . In all cases, n = 6.

The Jaynes-Cummings interaction results in Rabi flopping between states |↑i |0i and |↓i |1i

with Rabi frequency Ω. Figure 7.3 shows numerical simulations 1for amplification with n = 6.

The reduction of contrast of the Rabi oscillations for increasing r is a result of the finite value of n.

With Hamiltonian amplification and n sufficiently large, the Rabi frequency increases to cosh(r)Ω

(Fig. 7.4).

We implement the Jaynes-Cummings interaction using stimulated Raman transitions (see

chapter 4). The difference frequency between the Raman lasers is set to ωs − ω − ∆ACS, where

ωs/2π = 1.326 GHz is the frequency of the |↓si ≡ |3, 3i ↔ |↑si ≡ |2, 2i “stretch” transition and

∆ACS/2π ∼ 400 kHz is the AC Stark shift from the laser fields. Hamiltonian amplification is

implemented by interleaving the Raman pulses with squeezing pulses (each with 6 μs duration) as

described previously. With the squeezing parameter r ' 1, we observe an increase in the Rabi

frequency (not including the time required for squeezing) for flopping on the |↑si |0i ↔ |↓si |1i

transition (Fig. 7.5). The increase in Rabi rate by a factor of 1.5 is consistent with the calibrated

value of the squeezing parameter r ' 1.0. With faster squeezing, the increased sideband Rabi

1 Here I am assuming a √

n + 1 scaling of the sideband Rabi frequency which is only valid in the Lamb-Dicke limit

for laser-based sidebands [41].

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
111

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Amplfication gain

simulation

cosh(r)

Figure 7.4: Amplification gain for Hamiltonian amplification of the Jaynes-Cummings interaction

(assuming a √

n + 1 scaling of the sideband Rabi frequency) as a function of the squeezing parameter

r and n = 6 sequences. Blue circles give the ratio of the numerically calculated Rabi rate for a

given value of r to the Rabi rate with r = 0. The solid line is the theoretical gain cosh(r) in the

limit n → ∞.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
112

0 5 10 15 20 25 30

Time (us)

0.0

0.2

0.4

0.6

0.8

1.0

P

r=0

r=1.1

Figure 7.5: Population in |↑si as a function of the total RSB pulse duration (not including the

duration of the squeezing pulses) in a HA sequence with n = 6. The x axis indicates the total

duration of the Raman sideband pulses in the amplification sequence.

frequency could enable faster ground-state cooling, and faster generation of entanglement between

trapped-ion qubits.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 8

Parametric amplification of boson-mediated interactions

Strong and precisely controlled interactions between quantum objects are essential for quantum information processing [49, 108], simulation [24], and sensing [12, 13]. However, in many

experimental platforms for quantum physics, direct interactions between quantum objects can be

weak or difficult to control. Furthermore, increasing the direct coupling of some interactions simultaneously increases coupling to the environment. In many systems strong, controllable interactions

are achieved by coupling the individual quantum objects through a shared harmonic oscillator

mode. The interactions can then be modeled as being mediated by bosons — excitations of the

shared harmonic oscillator. Important physical implementations include photon-mediated interactions between atoms [30, 32], superconducting qubits [109], and color centers in diamond [110], and

phonon-mediated interactions between trapped ions [41, 48, 111] and optical and microwave photons [112]. However, the fidelity of these interactions is typically constrained by both the effective

interaction strength and the ratio of the interaction strength to characteristic rates of decoherence

in the system. Moreover, quantum objects can suffer additional decoherence when coupled to the

bosonic degree of freedom (see section 3.5). Recently, numerous theoretical proposals have described how boson-mediated interactions can be amplified by parametric modulation of the shared

harmonic interaction in systems such as trapped ions [104], cavity optomechanics [113], superconducting circuit QED systems [109], and atom-cavity systems [114, 115]. Crucially, the parametric

drive need not couple directly to the effectively-coupled quantum objects. Parametric amplification

can allow for higher fidelities if the dominant source of decoherence affects the quantum objects

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
114

and not the boson channel.

In this chapter I describe the theory (based on [104]) and experimental demonstration of

parametric amplification of phonon-mediated interactions between trapped ion qubits. The amplified interaction enables a 3.2-fold increase of the effective interaction strength as determined

by measuring the speedup of two-qubit entangling gates. Furthermore, I describe experiments

demonstrating that parametric modulation can improve the gate fidelity if random qubit frequency

fluctuations are the dominant source of decoherence of the system.

8.1 Boson-mediated interactions

The excitations of practically used harmonic oscillators obey the commutation relations

and statistics of bosons (typically photons or phonons). Bosonic interactions have been used to

demonstrate high-fidelity quantum logic gates [116, 117, 118], spin-squeezed states of atoms and

ions [11, 12, 13, 14], and the formation of novel phases of matter [119, 120]. Often, boson-mediated

+ -

Im(α) Parametric 

modulation

a) b)

Im(α)

Re(α)

Geometric 

phase Φ

- +

- +

+ -

Re(α)

2ω+ 2ẟ

Figure 8.1: Phase space illustration of boson-mediated interactions. I show the case of two qubits

(Sˆ = ˆσ

1

x − σˆ

2

x

) coupled by a harmonic oscillator mode with frequency ω. a) Without parametric

modulation, the harmonic oscillator quasi-probability distributions (blue fuzzy disks) associated

with the |+−i and |−+i spin states follow circular trajectories in a frame rotating at ω + δ. The

enclosed area (shown in light blue) is the geometric phase Φ. b) When the harmonic oscillator is

parametrically modulated at a frequency of 2ω + 2δ, in a frame rotating at ω + δ the trajectories

become elliptical and the motional wave packets (represented by grey and black ellipses) are alternately squeezed and anti-squeezed. More geometric phase is acquired per unit time, increasing the

effective interaction strength.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
115

interactions offer longer-range coupling than direct interactions and can be described by a generic

Hamiltonian given in the interaction picture rotating at ω + δ by

HˆM =

~Ω

2

(Sˆ†aˆ + Sˆaˆ

†

) − ~δaˆ

†

a. ˆ (8.1)

Here Ω is the coupling strength between the bosonic degree of freedom, with annihilation and

creation operators ˆa and ˆa

† and frequency ω, and the quantum objects to be coupled described

by Sˆ ≡

P

i=1 βisˆi

, where the ˆsi can be bosonic or fermionic operators on the individual quantum

objects coupled by the interaction. The physical significance of δ, a frequency detuning between the

harmonic oscillator and the coupling described by the first term in Eq. 8.1, depends on the details

of the system. For the case of atoms coupled to a single electromagnetic field mode in a cavity,

sˆi = ˆσ

i

+, where ˆσ

i

+ is the atomic raising operator for the i

th atom and δ is the atom-cavity detuning

[107]. HˆM is then the Tavis-Cummings Hamiltonian [107]. In an optomechanical system, the ˆsi

are bosonic creation or annihilation operators for photon or phonon fields, depending on the exact

implementation. For interactions between trapped-ion qubits, typically ˆsi = ˆσ

i

j—effective fermionic

spin 1/2 Pauli operators—and δ is the detuning of an external drive field from the phonon mode

used to implement the interaction. The βj depend on the eigenvector of the phonon mode and

j = x, y [48, 49] or j = z [51, 60] depending on the specific implementation. To understand how

Eq. 8.1 leads to coupling between the ˆsi, consider two trapped-ion qubits coupled through a shared

mode of motion such that Sˆ = ˆσ

1

x−σˆ

2

x

. HˆM is then the Mølmer-Sørensen (MS) interaction described

in detail in Chapter 3. Applying HˆM will result in spin-dependent displacements in phase space

of the normal mode that is near resonantly driven by lasers or oscillating magnetic fields. In this

case, the two-qubit spin states |++i and |−−i, where |±i ≡ √

1

2

(|↑i ± |↓i), are not displaced and

will remain at the origin, whereas |+−i and |−+i traverse circular trajectories in phase space (Fig.

8.1 a). The area enclosed by these trajectories gives rise to a state-dependent geometric phase

Φ. Applying HˆM for a duration τ = 2π/δ returns the harmonic oscillator to its initial state and

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
116

disentangles it from Sˆ. This results in the unitary transformation[48]

Uˆ(τ ) = exp i

Φ

4

Sˆ2



, (8.2)

which describes an effective spin-spin interaction. When δ = Ω and τ = π/Ω, Φ is π/2, and this

interaction allows the generation of maximally entangled states: Uˆ(τ )|↓↓i → √

1

2

(|↓↓i + i|↑↑i).

8.2 Parametric amplification

Boson mediated interactions can be amplified by parametrically modulating the boson mode

at a frequency 2ω + 2δ simultaneously with HˆM. In a frame rotating at ω + δ, the interaction

Hamiltonian becomes

HˆM →

~Ω

2

(Sˆ†aˆ + Sˆaˆ

†

) − ~δaˆ

†

aˆ

+

~g

2

(ˆa

2

e

−iθ + ˆa†2

e

iθ). (8.3)

The third term in Eq. 8.3 describes a parametric modulation of the boson channel with parametric

coupling strength g and phase θ between the parametric drive and the coupling interaction in

the first term. Insight into the dynamics resulting from Eq. 8.3 can be gained by making the

Bogoliubov transformation

ˆb ≡ aˆ cosh r − aˆ

†

e

iθ sinh r, (8.4)

ˆb

† ≡ aˆ†

cosh r − aeˆ

−iθ sinh r. (8.5)

where r is to be determined. The inverse transformations are

aˆ = ˆb cosh r + ˆb

†

e

iθ sinh r, (8.6)

aˆ

† = ˆb†

cosh r + ˆbe−iθ sinh r. (8.7)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
117

Applying these transformations yields

aˆ

†

aˆ = cosh(2r)

ˆb

†ˆb +

1

2

sinh(2r)[ˆb

†2

e

iθ + ˆb2

e

−iθ] − sinh2

(2r), (8.8)

e

−iθaˆ2 = cosh2

(r)e

−iθˆb2 + sinh2

(r)e

iθˆb†2 +

1

2

sinh(2r)[2ˆb

†ˆb + 1], (8.9)

and

e

iθaˆ†2 = cosh2

(r)e

iθˆb†2 + sinh2

(r)e

−iθˆb2 +

1

2

sinh(2r)[2ˆb

†ˆb + 1]. (8.10)

After substituting Eq. 8.8 to 8.10 into Eq. 8.3, to eliminate terms proportional to ˆb

2 and ˆb†2

, we

require

g

2

cosh(2r) = −

δ

2

sinh(2r). (8.11)

or

tanh(2r) = −

g

δ

. (8.12)

Collecting terms proportional to ˆb

†ˆb gives

δ

0 ≡ δ cosh(2r) + g sinh(2r) = p

δ

2 − g2 (8.13)

Finally, the interaction Hamiltonian becomes

HˆM = ~

Ω

2



Sˆ†[cosh r

ˆb + e

iθ sinh r

ˆb

†

] + Sˆ[cosh r

ˆb

† + e−iθ sinh r

ˆb]



− ~δ

0ˆb†ˆb, (8.14)

where constant terms have been dropped. For cases where Sˆ = Sˆ†, we have

HˆM = ~

Ω

2

Sˆ



ˆbf(r, θ) + ˆb

†

f

∗

(r, θ)



− ~δ

0ˆb†ˆb, (8.15)

where,

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
118

f(r, θ) = cosh r + e

iθ sinh r

=

√

cosh 2r + cos θ sinh 2r eiΦ

=

 

δ − g cos(θ)

p

δ

2 − g2

!1/2

e

iΦ (8.16)

with

Φ = tan−1



sin θ sinh r

cosh r + cos θ sinh r



. (8.17)

Maximum amplification occurs when θ = 0 and the transformed Hamiltonian becomes

HˆM = ~

Ωe

r

2



Sˆ†ˆb + Sˆˆb

†



− ~δ

0ˆb†ˆb, (8.18)

where e

r =



δ − g

δ + g

1/4

. The transformed Hamiltonian has exactly the same form as Eq. 8.1, but

the interaction strength is multiplied by a factor of e

r and δ is replaced by δ0

. We can derive the time

evolution operator Eq. 8.2 in the same manner as before, but the time to acquire a given geometric

phase Φ is multiplied by a factor of e

−r and the detuning required δ0

is e

r

larger. Parametric

modulation causes the |+−i and |−+i states to traverse elliptical, rather than circular, trajectories

in the phase space of ˆa and ˆa

†

rotating at ω + δ (Fig. 8.1 b). For e

r > 1, the area enclosed

by the trajectories is larger for a given duration of HˆM, due to the increased coupling strength.

Physically, the parametric modulation alternately squeezes and anti-squeezes the oscillator quasiprobability distributions as they follow their elliptical trajectories, resulting in amplification of the

spin-dependent displacements [87].

In other physical systems, where Sˆ 6= Sˆ†, the dynamics resulting from parametric modulation

depend on the details of the specific implementation. For the important case of atoms coupled to

a cavity, the terms proportional to sinh(r) in Eq. 8.15 describe excitation of the atom that is

accompanied by the addition of a photon to the cavity field. These processes do not conserve

energy and will typically be greatly suppressed. After dropping off-resonant terms, the interaction

strength is given by Ω → Ω cosh(r), and the amplification does not depend on θ [115].

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
119

8.3 Experimental implementation with trapped-ions

In our experiment, we amplify boson-mediated interactions between two trapped 25Mg+ hyperfine

ion qubits. Details of the experimental apparatus are given in chapter 4. The harmonic oscillator

mode is an out-of-phase radial motional mode (“rocking mode”) with frequency ω ' 2π ×5.9 MHz.

We use qubit states |↓i ≡ |F = 3, mF = 1i and |↑i ≡ |F = 2, mF = 1i (with qubit frequency ω0 '

2π×1.686 GHz). The MS interaction is implemented by applying oscillating magnetic field gradients

at frequencies ω0±(ω+δ), generated by currents in the microwave electrodes of the trap. Generating

the MS interaction in this way allows straightforward phase synchronization with the parametric

modulation at 2ω + 2δ

1

, which is implemented by applying an oscillating potential directly to the

rf trapping electrodes.

I will now show explicitly that the combined MS and parametric interactions can be described

by Eq. 8.3. As described in chapter 3, after transforming into an interaction picture with respect

to

Hˆ

0 = ~ωaˆ

†

aˆ + ~

ω0

2

(ˆσ

1

z + ˆσ

2

z

), (8.19)

the MS Hamiltonian is given by

HˆMS = ~

Ω

2

(ˆaeiδt + ˆa

†

e

−iδt)(ˆσ1

x − σˆ

2

x

), (8.20)

and the parametric drive Hamiltonian by

Hˆ

p =

~g

2

(ˆa

2

e

2iδt−iθ + ˆa†2

e

−2iδt+iθ). (8.21)

Applying the MS fields and parametric drive simultaneously yields:

1 A nifty trick for checking that the phase between the MS fields and the parametric drive is stable is to compare

the parametric drive voltage signal with a signal containing the difference frequency of the MS fields, which is also

at 2ω + 2δ, on an oscilloscope. In our experiment, the difference frequency signal is obtained by from a diode power

detector (which acts as a multiplier) connected to the monitor port shown in Fig. 4.17.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
120

Hˆ

I = HˆMS + HˆP

= ~

Ω

2

(ˆaeiδt + ˆa

†

e

−iδt)(ˆσ1

x − σˆ

2

x

) + ~g

2

(ˆa

2

e

2iδt−iθ + ˆa†2

e

−2iδt+iθ). (8.22)

If we transform into another interaction picture with respect to Hˆ

1 = ~δaˆ

†aˆ, the time dependence

can be eliminated

e

−i

Hˆ

1t

~ Hˆ

Ie

i

Hˆ

1I

t

~ = ~

Ω

2

(ˆa + ˆa

†

)(ˆσ

1

x − σˆ

2

x

) + ~g

2

(ˆa

2

e

−iθ + ˆa†2

e

iθ) − ~δaˆ†

a, ˆ (8.23)

which is equivalent to Eq. 8.3 with Sˆ = ˆσ

1

x − σˆ

2

x

.

8.3.1 Calculation of the gate time and detuning for parametrically amplified gates

without decoherence

The preparation of the Bell state |ψBi = √

1

2

(|↓↓i + i|↑↑i) from the initial state |↓↓i requires

the closure of an integer number of phase space loops and the accumulation of a geometric phase

of Φ = π/2. The closure of a single loop occurs when τ is given by:

τ =

2π

δ

0

=

2π

p

δ

2 − g2

(8.24)

The geometric phase acquired in that loop, Φ, must be equal to π/2:

Φ = 2π



Ω|f(g, δ, θ)|

δ

0

2

= 2π

 

Ω|f(g, δ, θ)|

p

δ

2 − g2

!2

=

π

2

(8.25)

Given g, θ, and Ω0, we can determine the correct values of τ and δ by numerically solving Eqs.

8.24 and 8.25 simultaneously. All values of τ , δ, and δ

0

calculated from analytical theory that are

presented below are calculated in this manner.

8.3.2 Calibrating sideband Rabi rates

To quantify the increased interaction strength resulting from parametric modulation, it is

necessary to characterize the bare MS interaction strength Ω. Preliminary estimates of the MS

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
121

interaction strength are obtained from sideband Rabi flopping with a single ion prepared close

to the motional ground state. These measurements also allow us to adjust the microwave power

delivered to the trap to balance the Rabi rate of each sideband to within 5%. More accurate

calibration of the sideband Rabi rate is performed by sideband analysis with two ions using the

gate mode. The ions are initially prepared close to the |0i motional state using resolved sideband

cooling. A series of microwave pulses is then used to prepare the internal state of the ions in |↑i |↑i.

A red sideband analysis pulse is then applied at ω0 + ∆ACZ + ω, where ∆ACZ is the AC Zeeman

shift from the sideband pulse, for duration tRSB, followed by detection. The process is repeated

several hundred times for each tRSB to gather statistics. By fitting a numerical model to the data,

we can extract the Rabi frequency, the differential AC Zeeman shift of each ion, and the residual

initial thermal occupation  ̄n. Example data are shown in Fig. 8.2. The fitting parameters are

listed in table 8.1.

0.0 0.5 1.0 1.5 2.0 2.5

tRSB (ms)

0.0

0.2

0.4

0.6

0.8

1.0

Population

+

Figure 8.2: Two-ion populations as a function of the red sideband analysis pulse duration. The

ions are initially prepared in |↑i |↑i spin state and the radial rocking mode is cooled close to the

motional ground state using resolved sideband cooling. The solid curves are a least squares fit of a

numerical model to the data.

More accurate measurements of the AC Zeeman shift ∆ACZ that arises when both sidebands

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
122

Table 8.1: Fit parameters for sideband Rabi flopping shown in Fig. 8.2

Parameter Symbol Value

Sideband Rabi rate for ion 1 Ω1/2π (kHz) 1.987(4)

Sideband Rabi rate for ion 2 Ω2/2π (kHz) 2.107(4)

AC Zeeman shift for ion 1 ∆AC1/2π (kHz) 0.15(4)

AC Zeeman shift for ion 2 ∆AC2/2π (kHz) 0.15(4)

Initial average thermal occupation n ̄ 0.057(8)

are applied simultaneously are obtained using a Ramsey experiment (on the clock qubit) with the

MS fields appropriately detuned from the motional resonance.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
123

6 4 2 0 2 4 6

Detuning /2 (kHz)

0.00

0.25

0.50

0.75

1.00

Populations

a)

0.0 0.2 0.4 0.6 0.8 1.0

Parity phase /2

0.00

0.25

0.50

0.75

b)1.00

Figure 8.3: Mølmer-Sørensen interaction. a) Two-ion qubit populations of |↓↓i (blue diamonds),

|↑↑i (magenta squares), and anti-aligned spin states |↓↑i and |↑↓i combined (green circles) as a

function of the gate detuning δ. The gate time is tg = 350 μs. Solid curves are from an analytical

model (Eqs. 3.59 and 3.60) fitted to the data. The fitting parameters are the gate coupling strength

Ω/2π = 2.07(3) kHz and the residual thermal occupation  ̄n = 0.046(3). b) Two-qubit populations

as a function of the parity phase φ. For this data, δ/2π = −2.9 kHz. Solid curves are sinusoidal

fits to the data. Error bars indicate the standard error of the mean.

8.3.3 Mølmer-Sørensen (MS) gate

For gate experiments, the ions are initialized near the motional and electronic ground state

|↓↓i |0i with optical pumping, resolved-sideband laser cooling [121], and microwave pulses. We

then apply the gate pulses for a duration tg and detuning δ. The qubits are read out by detecting

state-dependent ion fluorescence (see section 4.4). A detuning scan is shown in Fig. 8.3. The

data show good agreement with the theoretically expected behavior (Eqs. 3.59, 3.60), with the

coupling strength Ω and the residual thermal occupation as free parameters. The gate ideally

creates the Bell state |ΦiBell = √

1

2

(|↓↓i − i|↑↑i), and the fidelity of the state that is generated

experimentally is characterized by measuring the state fidelity hΦBell| ρˆexp |ΦBelli, where ˆρexp is the

experimentally generated density matrix. The diagonal elements of the density matrix P↓↓ and P↑↑

are determined from independent measurements of the populations at a given interaction time and

detuning. The off-diagonal elements of ˆρexp, can be determined by scanning the phase of an analysis

carrier π/2 pulse before detection (Fig 8.3 b)). The parity amplitude A is determined by fitting

the function A sin(2φ + φ0) to the parity P↓↓ + P↑↑ − P↑↓ − P↓↑. The Bell-state fidelity is then given

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
124

by F = (P↓↓ +P↑↑ +A)/2 [49]. Using this method, the Bell-state fidelity for the data shown in 8.3)

is 0.97(1). While this method is easy to implement, and serves to illustrate the basic concept, it

does not correct for state preparation and measurement (SPAM) errors. A more rigorous method

(which we call the “Keith” method) that uses maximum likelihood to calculate the state fidelity in

the presence of SPAM errors, is described in [122]. Besides parity and population data, the method

requires fluorescence count histograms measured after preparing the ions in different initial spin

states (see Fig.8.4) to evaluate the SPAM error. For the gate data shown in Fig. 8.3, the Keith

method gives a mean fidelity of 0.974 with a 95% confidence interval between 0.968 and 0.976 . All

fidelity measurements in the remainder of this chapter were obtained using the Keith method.

0 20 40 60 80 100 120

Photon counts

0

200

400

600

800

1000

1200

1400

Number of events

|

|

1

2

(| + | + | + | )

Figure 8.4: Histograms used to evaluate state preparation and measurement error contributions to

the state fidelity. Bars indicate fluoresence counts measured after preparing the the two-ion qubit

state as |↑↑i (blue), |↓↓i (orange), and after applying a global π/2 pulse on the |↓↓i state (green).

8.3.4 Calibrating the parametric drive

To investigate the relationship between the increased interaction strength and the parametric

coupling strength g, we need to be able to accurately measure g. The procedure is the same as

described above for measuring the sideband Rabi rates, but in this case, the motional state is

prepared as a squeezed state by applying the parametric drive on resonance (at 2ω) for a duration

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
125

0.0 0.5 1.0 1.5 2.0 2.5

tRSB (ms)

0.0

0.2

0.4

0.6

0.8

1.0

Population

+

[!h]

Figure 8.5: Two-ion populations as a function of the red sideband analysis pulse duration. The

ions are initially prepared in |↑i |↑i spin state and the radial rocking mode is prepared in a squeezed

state using parametric modulation. A least squares fit of a numerical model to the data gives a

squeezing parameter of r = 1.19(2) .

ts before the analysis pulse. The squeezing parameter r can then be extracted by fitting a numerical

model with only r as a free parameter, as all other relevant parameters of the model can be measured

by performing the same experiment without squeezing as described in section 8.3.2. The parametric

coupling strength is then r = gts.

The value of g depends on the frequency of the parametric drive ωP . This is expected, as a

resonant circuit is used to couple the parametric drive to the trap RF source circuit (see Fig. 4.18).

To account for the frequency dependence, we measured g at various values of the rocking mode

frequency ω. Since ωP = 2ω for the calibration experiment, we find that g depends on the mode

frequency (Fig. 8.6). The mode frequency was changed by adjusting the DC shim potentials of

the surface trap. At each frequency, we perform sideband flopping analysis to measure Rabi rates,

AC Zeeman shifts,  ̄n, and g as described above. The frequency dependence of g is well-described

by a quadratic polynomial. The polynomial coefficients are obtained using a least-squares fit. This

calibration function is used in experiments described below to assign g based on measurements of

the rocking mode frequency. This is particularly useful for measuring changes in g resulting from

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
126

5.70 5.75 5.80 5.85 5.90 5.95

Rocking mode frequency (MHz)

40

45

50

55

g

/

2 (k

H

z)

Fit

Data

Figure 8.6: Parametric drive strength g as a function of the rocking mode frequency. The solid line

is a quadratic fit to the data giving ∆ ̄ AC/2π = 4.588(7) and and σAC/2π=0.322(5) kHz. Error bars

indicate 68% confidence intervals. Dashed lines indicates 68% functional prediction bands [123].

drifts of ω, since we measure ω before every gate experiment.

8.4 Parametrically amplified entanglement generation

To determine the enhancement in the interaction strength due to parametric modulation, we

find the maximum Bell state fidelity as a function of the interaction time tI and the detuning δ,

for various values of g. This is ideally equivalent to fixing the area enclosed by the phase-space

trajectories of the oscillator wavepackets (Fig. 8.1) such that a maximally-entangled Bell state is

generated at the end of the interaction. Fig. 8.7 shows example data of the two-qubit populations

and parity analysis for tI = 90μs, and g/2π = 50(1) kHz. The fidelity is measured over a grid of

typically more than 25 tI and δ pairs (Fig. 8.8). The effective detuning δ

0 =

p

δ

2 − g2 is calculated

from δ and the calibrated value of g for each experiment. The time and effective detuning at which

the maximum fidelity occurs is then determined by fitting a low-order two-dimensional polynomial

to the grid of (tI , δ

0

) pairs2(Fig. 8.8), to find the duration tmax at which maximum fidelity occurs.

The maximum measured fidelity, as a function of δ for each tI , for various values of g is

2 We find that using δ0

instead of δ gives better conditioned fits. This is because the transformation δ

0 =

p

δ

2 − g2

suppresses common shifts in g and δ resulting from drifts in the rocking mode frequency

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
127

53 52 51 50

Detuning /2 (kHz)

0.00

0.25

0.50

0.75

1.00

Populations

a)

0.0 0.2 0.4 0.6 0.8 1.0

Parity phase /2

0.00

0.25

0.50

0.75

b) 1.00

Figure 8.7: Parametrically amplified Mølmer-Sørensen interaction. a) Two-ion qubit populations

of |↓↓i (blue diamonds), |↑↑i (magenta squares), and anti-aligned spin states (green circles) as a

function of the gate detuning δ for g/2π = 50(1) kHz. The interaction time is t = 90 μs. b) Twoqubit populations as a function of the parity phase φ. The detuning is δ/2π = −50.3 kHz and the

interaction time is tI = 90 μs. Solid curves are sinusoidal fits to the data. Data points represent

the mean of 200 experimental trials. Error bars indicate the standard deviation of the mean.

shown in Fig. 8.9. As g is increased, the interaction time at which the maximum fidelity occurs

(tImax) is reduced (Fig. 8.10). However, the measured maximum fidelities tend to occur at shorter

durations than would be theoretically predicted in the absence of imperfections. The reduction

results from amplification of motional decoherence by the parametric modulation and we find good

agreement with numerical simulations incorporating motional dephasing3(see section 3.5). The

amplified motional decoherence also causes the maximum fidelity to diminish as g is increased. For

a parametric coupling strength of g/2π = 50(1) kHz, we measure a maximum fidelity of 0.860(8)%

(Fig. 8.7 b) at an interaction time of 90 μs a factor of 3.8(2) lower than the un-amplified gate

duration t0 = 342(3)μs. From the numerical simulation incorporating motional dephasing, this

would correspond to an increase in the interaction strength by a factor of e

r = 3.2.

3 Motional heating does not play a significant role due to the low heating rate of the rocking mode, which was

measured to be ∼ 1 q/s

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
128

9 11 13

0.5

0.6

0.7

0.8

0.9

Bell-state fidelity

tI = 75 s

6 8 10 12

tI = 80 s

7 9 11

0

/2 = 2 g

2

/2 (kHz)

tI = 90 s

6 8 10

tI = 100 s

3 5 7 9

tI = 110 s

Figure 8.8: Fidelity as a function of δ

0/2π for various values of tI for a calibrated value of the

parametric coupling strength of g/2π = 49.7 kHz. Data points are fidelities obtained using the Keith

code 122. Vertical error bars indicate 68 % confidence intervals for the fidelity. Horizontal error

bars indicate 68 % confidence intervals for δ

0/2π calculated from error propagation of the measured

uncertainties in δ and g within a given experiment. Red curves are slices of the 2D polynomial

fitting function at the corresponding interaction times.The maximum of the fitted quadratic occurs

at a duration of 88(1) μs .

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
129











Figure 8.9: Bell state fidelity as a function of t0/tI , the ratio of the gate time without parametric

modulation t0 to the interaction time tI at which the fidelity is measured, for various values of the

parametric coupling strength g. Data points indicate the maximum measured fidelity for that tI .

Error bars indicate 95 % confidence intervals. The dashed black curve is a numerical simulation of

the maximum achievable fidelity for a given tI without parametric modulation and without decoherence. The light blue and navy dotted curves are simulations, including motional decoherence, of

the maximum achievable fidelity at a given tI with and without parametric modulation respectively.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
130

0 10 20 30 40 50

g/2 (kHz)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Measured speedup 

Theory (no decoherence)

Numerical simulation

Data

Figure 8.10: Speedup as a function of the parametric drive amplitude g. The speedup is the ratio of

the initial gate duration t0 to tImax the interaction duration at which we find the maximum fidelity.

Error bars correspond to 95 % confidence intervals. The solid line is the theoretically predicted

speedup in the absence of decoherence. The dashed line is a numerical simulation including motional

decoherence.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
131

8.5 Speed up and enhanced fidelity in the presence of qubit decoherence

As parametric driving only couples to the boson mode and does not directly affect the qubits,

effective spin-spin interactions can be enhanced without increasing qubit decoherence. Furthermore,

in the presence of noise that gives rise to qubit decoherence, parametric amplification can enable

both a reduction in the interaction time and an improvement in the fidelity. To demonstrate this

effect, we deliberately introduce fluctuating qubit frequency shifts to decohere our qubits. Since

the |F = 3, mF = 1i ↔ |F = 2, mF = 1i “clock” transition of 25Mg+ is first-order magnetic field

insensitive at our operating magnetic field, ambient magnetic field fluctuations do little to decohere

our qubits. In the absence of applied noise, we measure no significant decay of contrast in a singleion qubit Ramsey experiment up to a 200 ms Ramsey delay time. To introduce qubit decoherence

we apply a current to one of the trap electrodes that oscillates near the quibt frequency ω0. The

current is sufficiently detuned that the probability of a qubit transition is negligible. However,

there is an AC Zeeman shift ∼

Ω

2

n

δn

(ˆσ

1

z + ˆσ

2

z

), where Ωn is the Rabi rate that would be measured

if the frequency of the current were tuned onto resonance and δn is the detuning. We change

the amplitude of the current every ms so as to produce a Gaussian distribution of AC zeeman

shifts. This is on the order of the duration of one of our gate experiments including cooling, state

preparation and detection. To calibrate the noise we first apply the oscillating current with an

applied amplitude (In) to produce a static AC Zeeman shift which can be measured using a singleion Ramsey experiment. By repeating the experiment for several values of the current amplitude,

we measure the AC zeeman shift as a function of the applied amplitude. Since Ωn ∝ In, to get

a Gaussian distribution of AC Zeeman shifts we set the current In ∝

√

∆ACZ, where ∆ACZ is

randomly chosen from a Gaussian distribution with mean ∆ ̄ AC and standard deviation σAC. We

also perform two-ion Ramsey experiments with the noise on for different settings of σAC (example

data for one setting of σAC is shown in Fig. 8.11 ). The decay in contrast is well described by a

e

− t

2

2τ decay, where τ is the decay time (Fig. 8.5). For the gate experiments described below, the

noise settings were ∆ ̄ AC = 4.59(8) kHz, and σAC = 0.47(2) kHz.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
132

0.0 0.2 0.4 0.6 0.8 1.0

tRamsey (ms)

0.0

0.2

0.4

0.6

0.8

1.0

Population

+

Figure 8.11: Ramsey experiment in the presence of induced qubit decoherence. Two-ion populations are plotted as a function of the Ramsey delay time tRamsey. Solid curves are from a numerical

fit to the data giving ∆ ̄ AC/2π = 4.59(8) and and σAC/2π=0.421(7) kHz. Error bars indicate the

standard deviation of the mean.

In the presence of this decoherence, the maximum fidelity we measure without parametric

modulation is 78.7(8)% at an interaction time of 310 μs, whereas with parametric modulation we

measure a fidelity of 91.2(5)% at an interaction time of 160 μs with a parametric coupling strength

of g/2π = 12.1(3) kHz (Fig. 8.5). The fidelity decreases with further increases in g as parametrically

amplified motional dephasing becomes the dominant error mechanism.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
133

0.0 0.2 0.4 0.6 0.8 1.0

tRamsey (ms)

0.0

0.2

0.4

0.6

0.8

1.0

Bright fraction

e

(t/2 )

2

cos( ACt)

Data

Figure 8.12: Ramsey experiment in the presence of induced qubit frequency decoherence. The

bright fraction is plotted as a function of the Ramsey delay time. The solid red curve is a fit to

the function e

− t

2

2τ cos(∆ ̄ ACt)giving ∆ ̄ AC/2π = 4.61(1) kHz and τ = 0.379(8) ms. The dashed line

indicates the e

− t

2

2τ decay envelope.

0 10 20 30 40 50

Parametric coupling strength g (kHz)

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Bell-state fidelity

Motional dephasing

Data

Figure 8.13: Bell-state fidelity as a function of the parametric coupling strength g in the presence

of qubit dephasing with σAC = 0.47(2) kHz. The light-blue dotted line is a numerical simulation

(including motional decoherence, but not qubit decoherence mechanisms) of the maximum possible

single-loop fidelity (for any time or detuning) as a function of g. Error bars indicate 95% confidence

intervals.

8.6 Phase dependence

For the previous results, the phase of the parametric drive relative to the MS phase was

nominally set to give maximum amplification (θ = 0). To further verify the physics of the parametric

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
134

amplification process, we vary the phase of the parametric drive θ relative to the phase of the MS

fields for a fixed parametric coupling strength of g/2π = 12.3(3) kHz (Fig. 8.14). At each phase

setting we locate the maximum Bell-state fidelity (as before) for each interaction duration tI as a

function of the detuning δ. The parametric amplification process is first-order insensitive to changes

in the parametric drive phase [104] and we measure no significant reduction in the interaction time

at which maximum fidelity occurs or the maximum fidelity over the range θ/2π = 0 to 0.14.

Changing θ further from its optimum value results in an increase in tImax that is well described by

theory. The required increased interaction time is accompanied by a reduction in Fmax from 0.95

to 0.76 as θ/2π changes from 0 to 0.39 as parametric amplification of motional decoherence (which

is independent of θ) affects the process more strongly over a longer duration. Measurements of the

fidelity as a function of θ for fixed g, tI , and δ also show the first-order insensitivity to θ (Fig. 8.15).

However, numerical simulations show that the presence of motional decoherence tends to reduce

the phase insensitivity to some extent.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
135

0.0 0.1 0.2 0.3 0.4 0.5

/2

0.0

0.1

0.2

0.3

0.4

0.5

Interaction time (ms)

Theory, no decoherence

Numerical simulation

Data

Figure 8.14: Interaction duration of maximum fidelity tImax as a function of the parametric

drive phase θ for g/2π = 12.75(4) kHz. The dotted red line indicates the gate duration without

parametric modulation (g = 0). The solid blue line is analytical theory with no decoherence. The

dashed blue line is a numerical simulation including motional dephasing. Error bars indicate 68%

confidence intervals

-0.5 -0.375 -0.25 -0.125 0.0 0.125 0.25 0.375 0.5

/2

0.5

0.6

0.7

0.8

0.9

1.0

Fidelity

Figure 8.15: Fidelity as as function of the parametric drive phase θ for fixed parametric coupling

strength g, interaction time tI and detuning δ. The solid black line is a numerical simulation with

no decoherence and with tI and δ set to give maximum fidelity at θ = 0 for the calibrated value of

g. The dashed blue line is a numerical simulation incorporating motional decoherence. Error bars

indicate 68% confidence intervals.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Chapter 9

Conclusion and outlook

Perhaps the most ubiquitous and elementary quantum-mechanical models are the two-level

system (also known as a spin 1/2 or a qubit) and the harmonic oscillator. Improved control of over

real systems that are adequately described by these models has facilitated advances in quantum

information processing and metrology and led to deeper understanding of quantum mechanics.

Trapped atomic ions, where select pairs of internal electronic states comprise qubits, and the motion of the ions constitutes a harmonic oscillator, form an almost ideal platform for studying these

models and the physics emerging if several such elementary systems are coupled together. In this

thesis I have explored new applications of parametric driving for controlling trapped-ion oscillators

and subsequently altering the coupling between qubit degrees of freedom mediated by these oscillators. Experimentally, parametric modulation was implemented by periodic distortion of the ion

trap potential at MHz frequencies. This laser-free method enables the reversible generation of large

squeezed states of trapped-ion motion which can be useful for quantum-enhanced metrology. Furthermore, I have presented experiments demonstrating that reversible squeezing also permits nearly

noise-free, phase-sensitive amplification of harmonic oscillator displacements, which has allowed us

to resolve displacements well below the zero-point fluctuations of the oscillator. Extensions of this

protocol can be used for amplification of a broad class of interactions involving both harmonic oscillators and spins. In particular, I have presented experiments describing parametric amplification

of phonon-mediated interactions between trapped-ion qubits, as demonstrated by a reduction in

the time required to implement two-ion entangling gates. These new tools could benefit quantum

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
137

metrology, quantum simulation, and quantum information processing in systems where harmonic

oscillators are essential components. There are many possible directions in which this work can

be extended in the future. Below I give a perspective of some interesting possibilities: Two-mode

operations, continuous variable quantum information processing, and improving the performance

of trapped-ion systems that are limited by spontaneous emission errors.

9.1 Two-mode operations

For the squeezing and amplification experiments described in this thesis, the parametric

modulation frequency was set to (or relatively close to) twice the frequency of a normal mode

of motion of a trapped ion crystal (see section 2.6). This is known as degenerate parametric

modulation and implements single-mode squeezing. Coupling of the parametric drive to other

normal modes was neglected due to the large detuning (∼ 5 MHz) of the parametric drive from

the degenerate parametric resonance of the nearest spectator modes. With a parametric coupling

strength of g/2π ∼ 50 kHz, the effect of the parametric drive on the nearest mode is suppressed by

a factor ∼ 10−2. However, parametric driving can also be used to couple different modes together.

If we considered coupling two modes described by annihilation operators ˆa and ˆb with frequencies

ωa and ωb respectively, by appropriately setting the parametric drive frequency, two important

interactions can be implemented: two mode squeezing and beam splitting.

9.1.1 Beam splitting

If we set the parametric drive frequency to ωa − ωb, the resulting interaction in a frame

rotating at ωa − ωb, after making a rotating wave approximation,1can be written as [35]

Hˆ =

~g

2

(ˆaˆb

†

e

iθ − aˆ†ˆbe−iθ), (9.1)

1 For both the beamsplitter and two-mode squeezing interactions, I assume that there is a component of the

potential curvature tensor that allows coupling of the two modes. I also assume that the mode frequencies ωa and

ωb, are chosen so that the rotating wave approximation is valid

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
138

where θ is the relative phase of the parametric drive. Eq. 9.1 is formally equivalent to the Hamiltonian for an optical beam splitter [30]. This interaction enables exchange of quanta between the

modes. If mode a is prepared in the ground state and mode b is prepared in a number state with n

quanta, so that the initial state is |0ia|ni

b

, after an interaction duration t = 2π/g, the final state

will be |nia|0i

b

. The beamsplitter interaction is required for most continuous-variable quantum

information processing protocols as discussed in the next section and has been implemented using

laser-free parametric modulation in various trapped-ion systems [124, 125, 126]

9.1.2 Two-mode squeezing

If we set the parametric drive frequency to ωa + ωb (non-degenerate parametric modulation),

after making a rotating wave approximation, in a rotating frame at ωa + ωb we can realize an

interaction given by

Hˆ =

~g

2

(ˆaˆbeiθ + ˆa

†ˆb†

e

−iθ), (9.2)

If this Hamiltonian is applied for a duration t to an oscillator prepared in the ground state of both

modes, |0ia|0i

b

the resulting state is know as a two-mode squeezed vacuum (TMSV) state:

|χi ≡ Sˆ

2(χ)|0ia

|0i

b = exp(χˆbaˆ − χ

∗

aˆ

†ˆb†

)|0ia|0i

b

, (9.3)

where Sˆ

2(χ) is the two-mode squeezing operator and χ = reiθ, with r = gt, is the two-mode

squeezing parameter. In the number state basis, the TMSV state can be written as [35]

|χi =

1

cosh(r)

X∞

n=0

(−e

iθ tanh r)n

|nia|ni

b

(9.4)

which shows that the two modes are highly correlated (if one mode were measured to have n quanta,

the other mode would also be found to contain n quanta with certainty). Because of this, twomode squeezed states are important for continuous-variable entanglement and for investigations

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
139

the Einstein-Rosen-Podolsky paradox [35, 127]. If only one of the modes is considered (i.e. if we

trace over the other mode), the density matrix for the system is given by

ρˆa =

X∞

l=0

hl|

b

[|χi hχ|] |li

b

=

1

cosh2(r)

X∞

n=0

tanh2n(r)|ni hn| , (9.5)

which describes a thermal distribution. Two-mode squeezed states also appear in the theory of the

Unruh effect [128] where an observer accelerating uniformly through empty space (the vacuum state

of an arbitrary electric field mode according to non-accelerating observers), will instead observe a

thermal distribution of photons. This is because the accelerating observer can only receive signals

from a restricted portion of space-time. The Unruh effect is closely related to Hawking radiation

from black holes [129], which is one of the few connections between quantum field theory and gravity.

While two-mode squeezing has been implemented with optical systems[130] and in superconduting

systems [131], to the best of my knowledge, it has not been demonstrated with trapped ions. I am

confident that laser-free parametric modulation will enable investigations of two-mode squeezing

in trapped ions in the near future.

9.2 Continuous-variable quantum information processing

There are several distinct paradigms for quantum information processing. The main approach

is known as discrete-variable QIP, where the observables of the quantum processor have a discrete

spectrum. Such systems are usually composed of qubits (two level systems), and unitary operations

are used to perform operations on the qubits. Alternatively, the continuous degrees of freedom of

quantum systems can be used for encoding and processing information. This approach is known as

continuous variable (CV) QIP. A requirement for a universal CV quantum processor is a universal

set of continuous quantum operations. For a particular set of transformations, a set of continuous

operations is universal if any transformation in the set can be approached arbitrarily closely by

a finite number of applications of the operations [132]. For a CV quantum computer, where the

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
140

quadrature operators of an oscillator are the observables, a universal set consists of linear (or

Gaussian) operations: displacement, squeezing, beam splitting, and phase-shifting and a single

nonlinear (non-Gaussian) operation [132, 133, 134]. In the context of trapped ions, all of the linear

operations can be implemented using laser-free modulation schemes. The nonlinear operation could

be realized by using higher-order parametric modulation, which would require reasonably strong

nonlinearities in the trapping potential. For example, if the trap potential has an appropriate

cubic component, a Hamiltonian of the form Hˆ ∝ aˆ

3 + a†3

can be implemented by modulating the

potential at 3 times the mode frequency. Alternatively, a nonlinear operation can be realized by

coupling the ion motion to its qubit state, to take advantage of the intrinsic nonlinearity of the two

level system [135].

Another approach for using continuous degrees of freedom for quantum information processors

is to encode a qubit in an higher-dimensional quantum system such as a harmonic oscillator.

This approach has advantages for quantum error correction. One proposed oscillator code due

to Gottesman, Kitaev and Preskill [136] (GKP), encodes qubits on grid states (multi-component

superpositions of displaced squeezed states). This encoding was demonstrated on an approximation

to ideal grid state by the ETH group [137], where preparation of squeezed states was achieved using

laser-based reservoir engineering. The amount of squeezing was found to be a limiting factor for this

experiment. Parametric modulation could enable stronger squeezing, permitting the construction

of more ideal grid states and further investigations of the GKP code for error-corrected QIP[137].

9.3 Improving coherence in trapped-ion systems limited by qubit decoherence

In trapped-ion systems where effective spin-spin interactions are generated using laser beams,

off-resonant light scattering can be the largest source of decoherence [4, 24, 117, 118]. While it is

possible to suppress photon scattering by using more laser power and increasing the detuning of

the laser from atomic resonances, other errors can come into play [138]. In chapter 8, we presented

experiments where parametric modulation was used to improve the fidelity of a two-qubit gate in

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
141

the presence of qubit decoherence [104]. However, we also determined that parametric amplification

can significantly amplify motional decoherence, and that there is an optimal parametric coupling

strength for achieving maximum fidelity. This result motivates further investigation of parametric

amplification in to systems limited by photon scattering. We are planning to implement a parametric drive in the NIST Penning trap, where increased coherence enabled by parametric modulation

could allow improved spin squeezing [14] and sensing [139] with hundreds of ions.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Bibliography

[1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

[2] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Rev. Mod. Phys. 87, 637

(2015).

[3] S. Slussarenko and G. J. Pryde, Appl. Phys. Rev. 6, 041303 (2019).

[4] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Appl. Phys. Rev. 6, 021314

(2019).

[5] A. Blais, S. M. Girvin, and W. D. Oliver, Nat. Phys. , 1 (2020).

[6] C. M. Caves, Phys. Rev. D 23, 1693 (1981).

[7] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Phys. Rev. A

46, R6797 (1992).

[8] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A 50, 67 (1994).

[9] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).

[10] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55,

2409 (1985).

[11] V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J.

Wineland, Phys. Rev. Lett. 86, 5870 (2001).

[12] O. Hosten, N. J. Engelsen, R. Krishnakumar, and M. A. Kasevich, Nature 529, 505 (2016).

[13] K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, Phys. Rev. Lett. 116, 093602

(2016).

[14] J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J.

Bollinger, Science 352, 1297 (2016).

[15] L. Pezz`e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys. 90,

035005 (2018).

[16] E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk,

and K. C. Schwab, Science 349, 952 (2015).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
143

[17] F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Phys. Rev. X 5,

041037 (2015).

[18] J.-M. Pirkkalainen, E. Damsk ̈agg, M. Brandt, F. Massel, and M. A. Sillanp ̈a ̈a, Phys. Rev.

Lett. 115, 243601 (2015).

[19] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett.

76, 1796 (1996).

[20] D. Kienzler, H.-Y. Lo, B. Keitch, L. de Clercq, F. Leupold, F. Lindenfelser, M. Marinelli,

V. Negnevitsky, and J. P. Home, Science 347, 53 (2015).

[21] J. Aasi et al., Nat. Photon. 7, 613 (2013).

[22] Y. Wu, R. Krishnakumar, J. Mart ́ınez-Rinc ́on, B. K. Malia, O. Hosten, and M. A. Kasevich, “Retrieval of cavity-generated atomic spin-squeezing after free-space release,” (2019),

arXiv:1912.08334 [quant-ph] .

[23] B. Braverman, A. Kawasaki, E. Pedrozo-Pe ̃nafiel, S. Colombo, C. Shu, Z. Li, E. Mendez,

M. Yamoah, L. Salvi, D. Akamatsu, Y. Xiao, and V. Vuleti ́c, Phys. Rev. Lett. 122, 223203

(2019).

[24] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J.

Biercuk, and J. J. Bollinger, Nature 484, 489 (2012).

[25] D. J. Heinzen and D. J. Wineland, Phys. Rev. A 42, 2977 (1990).

[26] V. Natarajan, F. DiFilippo, and D. E. Pritchard, Phys. Rev. Lett. 74, 2855 (1995).

[27] N. Yu, H. Dehmelt, and W. Nagourney, J. Appl. Phys 73, 8650 (1993).

[28] G. Maslennikov, S. Ding, R. Habl ̈utzel, J. Gan, A. Roulet, S. Nimmrichter, J. Dai, V. Scarani,

and D. Matsukevich, Nat. Commun. 10, 1 (2019).

[29] J. J. Sakurai and J. Napolitano, Modern quantum mechanics (Pearson Harlow, 2014).

[30] C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press,

2005).

[31] E. Wigner, Phys. Rev. 40, 749 (1932).

[32] S. Haroche and J.-M. Raimond, Exploring the quantum: atoms, cavities, and photons (Oxford university press, 2006).

[33] P. Carruthers and M. Nieto, Am. J. Phys 33, 537 (1965).

[34] N. N. Bogoljubov, Il Nuovo Cimento 7, 794 (1958).

[35] D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

[36] S. Brewer, J.-S. Chen, A. Hankin, E. Clements, C.-W. Chou, D. Wineland, D. Hume, and

D. Leibrandt, Phys. Rev. Lett. 123, 033201 (2019).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
144

[37] K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin,

L.-M. Duan, and C. Monroe, Nature 465, 590 (2010).

[38] J. T. Barreiro, M. M ̈uller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F.

Roos, P. Zoller, and R. Blatt, Nature 470, 486 (2011).

[39] A. Khromova, C. Piltz, B. Scharfenberger, T. F. Gloger, M. Johanning, A. F. Var ́on, and

C. Wunderlich, Phys. Rev. Lett. 108, 220502 (2012).

[40] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403

(1989).

[41] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J.

Res. Natl. Inst. Stand. Technol. 103, 259 (1998).

[42] K. C. McCormick, J. Keller, S. C. Burd, D. J. Wineland, A. C. Wilson, and D. Leibfried,

Nature 572, 86 (2019).

[43] A. Sørensen and K. Mølmer, Phys. Rev. Lett. 82, 1971 (1999).

[44] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, Science 272, 1131 (1996).

[45] W. H. Zurek, Nature 412, 712 (2001).

[46] F. Toscano, D. A. Dalvit, L. Davidovich, and W. H. Zurek, Phys. Rev. A 73, 023803 (2006).

[47] C. Hempel, B. P. Lanyon, P. Jurcevic, R. Gerritsma, R. Blatt, and C. F. Roos, Nat. Photonics

7, 630 (2013).

[48] K. Mølmer and A. Sørensen, Phys. Rev. Lett. 82, 1835 (1999).

[49] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe,

Q. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, Nature 404, 256 (2000).

[50] P. J. Lee, K.-A. Brickman, L. Deslauriers, P. C. Haljan, L.-M. Duan, and C. Monroe, J. Opt.

B 7, S371 (2005).

[51] G. Milburn, S. Schneider, and D. James, Fortschritte der Phys. 48, 801 (2000).

[52] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovi ́c, C. Langer, T. Rosenband, et al., Nature 422, 412 (2003).

[53] M. Brownnutt, M. Kumph, P. Rabl, and R. Blatt, Rev. Mod. Phys. 87, 1419 (2015).

[54] I. Talukdar, D. Gorman, N. Daniilidis, P. Schindler, S. Ebadi, H. Kaufmann, T. Zhang, and

H. H ̈affner, Phys. Rev. A 93, 043415 (2016).

[55] C. Gardiner, P. Zoller, and P. Zoller, Quantum noise (Springer Science & Business Media,

2004).

[56] M. Sepiol, A high-fidelity microwave driven two-qubit quantum logic gate in 43Ca+, Ph.D.

thesis, University of Oxford (2016).

[57] J. Johansson, P. Nation, and F. Nori, Comput. Phys. Commun. 184, 1234 (2013).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
145

[58] Q. A. Turchette, C. J. Myatt, B. E. King, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. A 62, 053807 (2000).

[59] W. Paul, Rev. Mod. Phys. 62, 531 (1990).

[60] D. Leibfried, R. Blatt, C. Monroe, and D. J. Wineland, Rev. Mod. Phys. 75, 281 (2003).

[61] J. Chiaverini, R. B. Blakestad, J. Britton, J. D. Jost, C. Langer, D. Leibfried, R. Ozeri, and

D. J. Wineland, Quant. Inf. Comp. 5, 419 (2005).

[62] S. Seidelin, J. Chiaverini, R. Reichle, J. J. Bollinger, D. Leibfried, J. Britton, J. H. Wesenberg,

R. B. Blakestad, R. J. Epstein, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri,

N. Shiga, and D. J. Wineland, Phys. Rev. Lett. 96, 253003 (2006).

[63] C. Ospelkaus, U. Warring, Y. Colombe, K. R. Brown, J. M. Amini, D. Leibfried, and D. J.

Wineland, Nature 476, 181 (2011).

[64] T. P. Harty, High-fidelity microwave-driven quantum logic in intermediate-field 43Ca+,

Ph.D. thesis, University of Oxford (2013).

[65] R. Bowler, U. Warring, J. W. Britton, B. Sawyer, and J. Amini, Rev. Sci. Instrum. 84,

033108 (2013).

[66] J. Labaziewicz, Y. Ge, D. R. Leibrandt, S. X. Wang, R. Shewmon, and I. L. Chuang, Phys.

Rev. Lett. 101, 180602 (2008).

[67] J. A. Sedlacek, A. Greene, J. Stuart, R. McConnell, C. D. Bruzewicz, J. M. Sage, and

J. Chiaverini, Phys. Rev. A 97, 020302 (2018).

[68] D. An, C. Matthiesen, E. Urban, and H. H ̈affner, Phys. Rev. A 100, 063405 (2019).

[69] C. Ospelkaus, C. E. Langer, J. M. Amini, K. R. Brown, D. Leibfried, and D. J. Wineland,

Phys. Rev. Lett. 101, 090502 (2008).

[70] C. Langer, R. Ozeri, J. D. Jost, J. Chiaverini, B. DeMarco, A. Ben-Kish, R. Blakestad,

J. Britton, D. Hume, W. M. Itano, et al., Phys. Rev. Lett. 95, 060502 (2005).

[71] G. Clos, M. Enderlein, U. Warring, T. Schaetz, and D. Leibfried, Phys. Rev. Lett. 112,

113003 (2014).

[72] D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979).

[73] D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

[74] U. Warring, C. Ospelkaus, Y. Colombe, R. J ̈ordens, D. Leibfried, and D. J. Wineland, Phys.

Rev. Lett. 110, 173002 (2013).

[75] S. Bourdeauducq et al., Zenodo. 10.5281/zenodo.51303 (2016).

[76] M. L. Eickhoff and J. L. Hall, IEEE Trans. Instrum. Meas. 44, 155 (1995).

[77] J. H. Shirley, Opt. Lett. 7, 537 (1982).

[78] J. J. Snyder, R. K. Raj, D. Bloch, and M. Ducloy, Opt. Lett. 5, 163 (1980).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
146

[79] A. C. Wilson, C. Ospelkaus, A. VanDevender, J. A. Mlynek, K. R. Brown, D. Leibfried, and

D. J. Wineland, Appl. Phys. B 105, 741 (2011).

[80] Y. Colombe, D. H. Slichter, A. C. Wilson, D. Leibfried, and D. J. Wineland, Opt. Express

22, 19783 (2014).

[81] D. T. C. Allcock, Surface-electrode ion traps for scalable quantum computing, Ph.D. thesis,

Oxford University, UK (2011).

[82] H. Hahn, G. Zarantonello, M. Schulte, A. Bautista-Salvador, K. Hammerer, and C. Ospelkaus, Npj Quantum Inf. 5, 1 (2019).

[83] U. Warring, C. Ospelkaus, Y. Colombe, K. R. Brown, J. M. Amini, M. Carsjens, D. Leibfried,

and D. J. Wineland, Phys. Rev. A 87, 013437 (2013).

[84] H.-Y. Lo, D. Kienzler, L. de Clercq, M. Marinelli, V. Negnevitsky, B. C. Keitch, and J. P.

Home, Nature 521, 336 (2015).

[85] A. Bermudez, P. O. Schmidt, M. B. Plenio, and A. Retzker, Phys. Rev. A 85, 040302 (2012).

[86] T. P. Harty, M. A. Sepiol, D. T. Allcock, C. J. Ballance, J. E. Tarlton, and D. M. Lucas,

Phys. Rev. Lett. 117, 140501 (2016).

[87] S. Burd, R. Srinivas, J. Bollinger, A. Wilson, D. Wineland, D. Leibfried, D. Slichter, and

D. Allcock, Science 364, 1163 (2019).

[88] C. M. Caves, “Reframing SU(1,1) interferometry,” (2019), arXiv:1912.12530 [quant-ph] .

[89] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark,

Rev. Mod. Phys. 90, 025008 (2018).

[90] B. P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016).

[91] H.-J. Butt, B. Cappella, and M. Kappl, Surf. Sci. Rep. 59, 1 (2005).

[92] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).

[93] M. J. Biercuk, H. Uys, J. W. Britton, A. P. VanDevender, and J. J. Bollinger, Nat. Nanotechnol. 5, 646 (2010).

[94] B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986).

[95] E. Davis, G. Bentsen, and M. Schleier-Smith, Phys. Rev. Lett. 116, 053601 (2016).

[96] D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan,

and M. K. Oberthaler, Phys. Rev. Lett. 117, 013001 (2016).

[97] M. Malnou, D. A. Palken, B. M. Brubaker, L. R. Vale, G. C. Hilton, and K. W. Lehnert,

Phys. Rev. X 9, 021023 (2019).

[98] A. Eddins, S. Schreppler, D. M. Toyli, L. S. Martin, S. Hacohen-Gourgy, L. C. G. Govia,

H. Ribeiro, A. A. Clerk, and I. Siddiqi, Phys. Rev. Lett. 120, 040505 (2018).

[99] M. M. Nieto and D. R. Truax, Fortschr. Phys. 45, 145 (1997).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
147

[100] A. I. Lvovsky, “Squeezed light,” (2014), arXiv:1401.4118 [quant-ph] .

[101] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. Gilligan, D. J. Heinzen, F. Moore, M. Raizen,

and D. J. Wineland, Phys. Rev. A 47, 3554 (1993).

[102] W. J. Munro, K. Nemoto, G. J. Milburn, and S. L. Braunstein, Phys. Rev. A 66, 023819

(2002).

[103] Y. Wan, F. Gebert, J. B. W ̈ubbena, N. Scharnhorst, S. Amairi, I. D. Leroux, B. Hemmerling,

N. L ̈orch, K. Hammerer, and P. O. Schmidt, Nat. Commun. 5, 3096 (2014).

[104] W. Ge, B. C. Sawyer, J. W. Britton, K. Jacobs, J. J. Bollinger, and M. Foss-Feig, Phys.

Rev. Lett. 122, 030501 (2019).

[105] C. Arenz, D. I. Bondar, D. Burgarth, C. Cormick, and H. Rabitz, Quantum 4, 271 (2020).

[106] M. A. Nielsen and I. Chuang, Cambridge University Press (Pearson Harlow, 2000).

[107] M. Tavis and F. W. Cummings, Phys. Rev. 170, 379 (1968).

[108] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio,

D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J.

Schoelkopf, Nature 449, 443 (2007).

[109] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69,

062320 (2004).

[110] R. E. Evans, M. K. Bhaskar, D. D. Sukachev, C. T. Nguyen, A. Sipahigil, M. J. Burek,

B. Machielse, G. H. Zhang, A. S. Zibrov, E. Bielejec, H. Park, M. Lonˇcar, and M. D. Lukin,

Science 362, 662 (2018).

[111] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

[112] A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M.

Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, Nat. Phys. 14, 1038 (2018).

[113] M.-A. Lemonde, N. Didier, and A. A. Clerk, Nat. Commun. 7, 11338 (2016).

[114] Y.-H. Chen, W. Qin, and F. Nori, Phys. Rev. A 100, 012339 (2019).

[115] W. Qin, A. Miranowicz, P.-B. Li, X.-Y. L ̈u, J. Q. You, and F. Nori, Phys. Rev. Lett. 120,

093601 (2018).

[116] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus,

A. G. Fowler, B. Campbell, et al., Nature 508, 500 (2014).

[117] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley,

E. Knill, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 117, 060505 (2016).

[118] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, Phys. Rev. Lett.

117, 060504 (2016).

[119] R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, Science 336,

1570 (2012).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
148

[120] J. L ́eonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, Nature 543, 87 (2017).

[121] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and

P. Gould, Phys. Rev. Lett. 75, 4011 (1995).

[122] A. C. Keith, C. H. Baldwin, S. Glancy, and E. Knill, Phys. Rev. A 98, 042318 (2018).

[123] J. Tellinghuisen, The Journal of Physical Chemistry A 105, 3917 (2001).

[124] K. R. Brown, C. Ospelkaus, Y. Colombe, A. C. Wilson, D. Leibfried, and D. J. Wineland,

Nature 471, 196 (2011).

[125] D. J. Gorman, P. Schindler, S. Selvarajan, N. Daniilidis, and H. H ̈affner, Phys. Rev. A 89,

062332 (2014).

[126] K. Toyoda, R. Hiji, A. Noguchi, and S. Urabe, Nature 527, 74 (2015).

[127] Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).

[128] W. G. Unruh, Phys. Rev. D 14, 870 (1976).

[129] S. W. Hawking, Nature 248, 30 (1974).

[130] A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys.

Rev. Lett. 59, 2555 (1987).

[131] C. Eichler, D. Bozyigit, C. Lang, M. Baur, L. Steffen, J. M. Fink, S. Filipp, and A. Wallraff,

Phys. Rev. Lett. 107, 113601 (2011).

[132] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005).

[133] D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, J. Britton, W. M. Itano, B. Jelenkovi ́c, C. Langer, T. Rosenband, and D. J. Wineland, Phys. Rev. Lett. 89, 247901 (2002).

[134] U. L. Andersen, G. Leuchs, and C. Silberhorn, Laser Photonics Rev. 4, 337 (2010).

[135] L. Ortiz-Guti ́errez, B. Gabrielly, L. F. Mu ̃noz, K. T. Pereira, J. G. Filgueiras, and A. S.

Villar, Opt. Commun. 397, 166 (2017).

[136] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001).

[137] C. Fl ̈uhmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home,

Nature 566, 513 (2019).

[138] H. Uys, M. J. Biercuk, A. P. VanDevender, C. Ospelkaus, D. Meiser, R. Ozeri, and J. J.

Bollinger, Phys. Rev. Lett. 105, 200401 (2010).

[139] K. A. Gilmore, J. G. Bohnet, B. C. Sawyer, J. W. Britton, and J. J. Bollinger, Phys. Rev.

Lett. 118, 263602 (2017).

[140] W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954).

[141] S. Burd, D. Leibfried, A. C. Wilson, and D. Wineland, in

Vertical External Cavity Surface Emitting Lasers (VECSELs) V, Vol. 9349 (International

Society for Optics and Photonics, 2015) p. 93490P.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
149

[142] M. Guina, A. Rantam ̈aki, and A. H ̈ark ̈onen, J. Phys. D 50, 383001 (2017).

[143] S. C. Burd, D. T. C. Allcock, T. Leinonen, J. P. Penttinen, D. H. Slichter, R. Srinivas, A. C.

Wilson, R. J ̈ordens, M. Guina, D. Leibfried, and D. J. Wineland, Optica 3, 1294 (2016).

[144] S. Burd, J.-P. Penttinen, P.-Y. Hou, H. Knaack, S. Ranta, M. M ̈aki, E. Kantola, M. Guina,

D. Slichter, D. Leibfried, et al., arXiv preprint arXiv:2003.09060 (2020).

[145] M. Kuznetsov, F. Hakimi, R. Sprague, and A. Mooradian, IEEE Photon. Technol. Lett. 9,

1063 (1997).

[146] A. Tropper and S. Hoogland, Prog. Quantum. Electron. 30, 1 (2006).

[147] A. Garnache, A. Ouvrard, and D. Romanini, Opt. Express 15, 9403 (2007).

[148] A. Laurain, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, Opt. Lett. 39, 1573

(2014).

[149] A. Laurain, J. Hader, and J. V. Moloney, J. Opt. Soc. Am. B 36, 847 (2019)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Appendix A

Magnus Expansion

The Magnus expansion [140] gives a series solution to differential equations of the form

dXˆ(t, t0)

dt = Aˆ(t)Xˆ(t, t0) (A.1)

If the operator Aˆ(t) does not commute with itself at different times, i.e. if [Aˆ(t), Aˆ(t

0

)] 6= 0, then

the solution can be expressed as

Xˆ(t, t0) = exp X∞

n=1

Cn(t)

!

Xˆ(t0, t0) (A.2)

where

C1(t) = Z t

t0

A(t

0

)dt0

C2(t) = 1

2

Z t

t0

Z t

0

t0

[A(t

0

), A(t

00)]dt0

dt00 (A.3)

C3(t) = 1

6

Z t

t0

Z t

0

t0

Z t

00

t0

([A(t

0

), [A(t

00), A(t000)]] + [A(t000), [A(t00), A(t0

)]])dt0dt00dt000 (A.4)

.

.

.

The Magnus expansion has the useful property that if Ci(t) = 0, all high order contributions

vanish and the series truncates. In quantum mechanics, the time evolution operator is given by the

Schrodinger equation

i~

dUˆ(t, t0)

dt = HˆUˆ(t, t0), (A.5)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
151

which can be solved using the Magnus expansion method with Aˆ → −i

Hˆ

~

.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Appendix B

Thermal state density matrix

The density matrix for a system in thermal equilibrium is given by [29]

ρˆth =

e

−βHˆ

Z

(B.1)

where Z is the partition function and β = 1/kBT. For a one dimensional harmonic oscillator

Hˆ = ~ωz(ˆn + 1/2) and

e

−βHˆ

= e

−

~ωz

2

X∞

n=0

e

−β~ωzn

|ni hn| . (B.2)

With

Z = e

−

~ωz

2

X∞

n=0

e

−β~ωzn = e−

~ωz

2

1

1 − e−~ωzβ

, (B.3)

the thermal density matrix can be written as

ρˆth = (1 − e

−~ωzβ

)

X∞

n=0

e

−β~ωzn

|ni hn| . (B.4)

In typical ion trapping experiments, the quantity that can be measured easily is

n ̄ = T r[ˆρthnˆ]

= (1 − e

−~ωzβ

)

X∞

n=0

ne−β~ωzn

=

1

e

β~ωz − 1

(B.5)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
153

Rearanging gives

e

β~ωz =

1 +  ̄n

n ̄

(B.6)

or

T =

~ωz

kB

1

ln[ n ̄

1+ ̄n

]

(B.7)

Substituting into B.4 gives

ρˆth =

1

n ̄ + 1

X∞

n=0



n ̄

1 +  ̄n

n

|ni hn| (B.8)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Appendix C

Matrix elements of the displacement operator

The diagonal matrix elements for the displacement operator in the number state basis can

be expanded as

hn| Dˆ(α)|ni = e

−|α|

2/2

hn| e

αaˆ

†

e

−α

∗aˆ

|ni

=

X∞

l=0

X∞

k=0

hn|

(αaˆ

†

)

l

l!

(−α

∗aˆ)k

k!

|ni.

Notice that the only terms of the double sum that do not vanish must have l = k. Also the series

will truncate after l = n since terms with n < l will involve annihilation of the oscillator ground

state.

hn| Dˆ(α)|ni = e

−|α|

2/2Xn

l=0

(−1)l(|α|

2

)

l × n × (n − 1) × (n − 2). . . × (n − l))

l!l!

= e

−|α|

2/2Xn

l=0

(−1)l n!

l!

2(n − l)!(|α|

2

)

l

= e

−|α|

2/2Ln(|α|

2

) (C.1)

where Ln(|α|

2

) is the nth Laguerre polynomial. Let us now calculate the expectation value of the

displacement operator when the the initial state is a thermal field.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
155

Tr[ˆρthDˆ(α)] = X∞

l=0

hl| ρˆthDˆ(α)|li

=

X∞

l=0

hl|

1

n ̄ + 1

X∞

n=0



n ̄

1 +  ̄n

n

Dˆ(α)|ni hn|li

=

X∞

n=0

1

n ̄ + 1 

n ̄

1 +  ̄n

n

hn| Dˆ(α)|ni

=

X∞

n=0

1

n ̄ + 1 

n ̄

1 +  ̄n

n

e

−|α|

2/2Ln(|α|

2

) (C.2)

Finally, this expression can be simplified by using the generating function for the Laguerre polynomials,

X∞

n=0

y

nLn(x) = 1

1 − y

e

−

yx

1−y , (C.3)

to give

Tr[ˆρthDˆ(α)] = e

−( ̄n+1/2)|α|

2

(C.4)

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
Appendix D

Vertical external-cavity surface emitting lasers

Vertical external-cavity surface-emitting lasers (VECSELs), also known as optically-pumped

semiconductor lasers (OPSLs), or semiconductor disk lasers (SDLs), offer several advantageous

features for AMO physics applications [141, 142]. In this appendix I describe the infrared (IR)

VECSEL design that I have developed and that we have used extensively for magnesium ion

experiments [143]. The same design was also used to generate all the wavelengths typically required

for beryllium ion experiments [144]. This appendix focuses on construction details and VECSEL

operation. Additional details about VECSEL output power, noise characteristics, and performance

in experiments can be found in the articles [143, 144].

D.1 Single-frequency VECSELs for AMO physics

A schematic diagram of the single-frequency VECSELs we use is shown in Figure D.1. The

linear standing-wave laser cavity is formed by an output coupler (OC) and a semiconductor gain

mirror [145]. The gain mirror (Fig. D.2) consists of multiple stacked quantum wells and a Bragg

reflector that functions as one of the laser cavity mirrors [142, 146]. By positioning the quantum

wells at the anti-nodes of the standing wave of the laser mode, spatial hole burning can be eliminated. The VECSEL emission wavelength is determined by the choice of semiconductor materials

[142] and the geometry of the gain chip structure, and can be tailored to target relevant atomic

or molecular transitions. Although the VECSEL gain material can be designed for emission in

the visible or even ultraviotlet [146], in many cases it is necessary or more convenient to generate

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
157

VECSEL

Heat sink Gain 

mirror

Etalon BRF OC

PZT Pump

Water

cooler IR 

output

Figure D.1: Schematic diagram of a vertical external-cavity surface-emitting laser (VECSEL).

BRF: birefringent filter, OC: Output coupler, PZT: piezoelectric transducer.

Subst

rate o

r

heat sp

reader

Distributed Bragg reflector

Conduction band

Valence band

Quantum well Band gap

Electric field intensity

Active region

Strain compensantion layer

Barrier layers

Laser emission

Window layer

Pump absorption

Figure D.2: Diagram of the gain chip band structure (courtesy of the Optoelectronics Research

Center, Tampere University of technology, Tampere, Finland). The energies of the conduction and

valence band are indicated by black rectangular features. The energy is modulated by suitable

doping of the material. Inversion is created by electrons that are excited from pump absorption

and then decay to the bottom of the wells due to non-radiative energy loss. The layer structure of

the mirror and the external cavity favors stimulated emission from the minima of the conduction

band to the maxima of the valence band (realized in the ”wells”), which defines the lasing modes.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
158

shorter wavelengths via nonlinear frequency conversion of infrared VECSELs, often because the

requirement of a pump with a shorter wavelength than the VESCEL emission wavelength limits

the choice of strong and convenient pump lasers and favors IR VECSELs. The relatively large

gain bandwidth typically enables wavelength tuning over several 10 s of nanometers. This opens

the possibility of using a single laser to target multiple fine structure transitions in a given atomic

species.

The gain mirror is typically pumped with another laser; for VECSELs emitting in the infrared,

this is usually a relatively cheap multimode diode laser. Optical pumping together with the external

cavity enable high output power with almost diffraction-limited beam quality, which is desirable

if nonlinear frequency conversion stages or fiber coupling are required. Moreover, the opticallypumped disk laser geometry is intrinsically amenable to power scaling; the active gain volume can

be increased by increasing the laser mode and pump spot areas on the gain medium [146].

To enforce single-longitudinal mode operation and enable wavelength tuning, a birefringent

filter (BRF) and an etalon, are placed inside the laser cavity. The short upper-state lifetime

(∼1 ns) in the gain medium and the relatively high finesse external cavity (giving intracavity photon

lifetimes ∼50 ns), allow relaxation-oscillation-free operation [147] and free-running linewidths of

a few 100 kHz, limited by technical noise. In contrast, rare-earth-atom doped solid-state lasers

typically have long upper state (∼ 1 μs) lifetimes and realtively low finesse laser cavities giving rise

to relaxation oscillations. With active feedback from a reference cavity, single-frequency VECSELs

have been demonstrated with linewidths of 4 kHz and 15W output power [148].

D.2 Infrared VECSEL

At NIST, we have developed a custom single-frequency VECSEL that has replaced fiber and

dye laser based systems in several experiments that use magnesium ions. An annotated photograph

of one of our lasers is shown in Fig. D.3. The laser cavity is formed by the gain mirror 1 and

1 The gain mirrors are specially designed and fabricated by our collaborators at Tampere University of Technology

(TUT) in Finland.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
159

O-ring seal

Cooling

 tubes

Gain mirror 

inside

Etalon

oven

BRF

OC

PZT

Pump ber Invar 

baseplate

TEC cooling

 block

Electrical 

connections

25 mm

Figure D.3: Photograph the infrared (IR) VECSEL. BRF: birefringent filter, OC: Output coupler,

PZT: piezoelectric transducer, TEC: thermoelectric cooler.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
160

the output coupler, separated by ' 170 mm. A BRF and an etalon are inserted inside the laser

cavity to enforce single-frequency operation and enable wavelength tuning. The laser components

are attached on an invar base plate to suppress frequency drifts resulting from thermal fluctuations.

The baseplate is contained within an o-ring sealed enclosure to suppress air pressure fluctuations

and acoustic noise. In the following subsections, I describe several components of the VECSEL

in detail. To facilitate future VECSEL construction, I have tried to list most of the commercial

parts that we typically use. In many cases there are suitable substitute parts available from other

vendors.

D.2.1 Gain-chip cooling block

The thermal management system for the gain chip is non trivial. Under normal operating

conditions, several Watts of absorbed pump power need to be dissipated. Furthermore, for stable operation, the gain chip temperature should be controlled to within ± 0.01 ◦C by means of a

temperature controller2. To efficiently remove heat, the gain chip is mounted on a diamond heat

spreader that is soldered to a copper plate heat spreader. The temperature of the copper plate near

the gain chip is measured with a thermistor3 glued to the copper plate using UV curing epoxy4.

Heat is pumped away from the copper plate by a thermoelectric cooler (TEC)5. The hot side of

the TEC is in contact with a water cooling plate6. We put layers of non-silicone thermal paste7

on both sides of the TEC to ensure efficient heat transfer. The gain chip plate, TEC, and water

cooling plate are clamped to the cooling block. Insulators machined from low-thermal-conductivity

Mykroy glass-mica ceramic8limit heat conduction with the rest of the laser.

2 Arroyo instruments 5305 TECSource

3 TE Technology MP-2444

4 Norland 61 (Thorlabs part number: NOA61)

5 TE Technology HP-199-1.4-0.8

6 Mikros 02749 Cold plate , Ni Plated. At the time of writing this part is not longer in production

7 Silicone tends to migrate onto the gain chip.

8 McMaster-Carr glass mica ceramic: https://www.mcmaster.com/ceramics/easy-to-machine-glass-mica-ceramic/

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
161

TEC

Clamp

Insulators

Copper heat 

spreader

Water cooling 

plate

Gain chip

Diamond

heat spreader

Thermistor

Cooling block

Figure D.4: 3D rendering of the gain chip cooling block. Waste heat from the gain chip is

dispersed by a diamond heat spreader and a copper plate heat spreader. A thermometric cooler

(TEC) pumps heat from the copper plate to the water cooling plate. The temperature of the copper

plate is measured using a thermistor. Mykroy ceramic insulators limit heat transfer between the

copper plate and the clamp and between the water cooling plate and the cooling block. Tubes for

delivering and removing cooling water are not shown.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
162

D.2.2 Output coupler

The output coupler (OC) allows extraction of a portion of the laser light circulating in the

cavity and determines the intracavity mode geometry. We typically choose OCs with a reflectivity

R ' 98% and a 200 mm radius of curvature 9 which gives a laser mode waist (1/e2intensity

radius)on the gain chip of ∼ 160 μm. For fine tuning the laser frequency, the cavity length can be

adjusted using a ring piezo electric transducer10 connected to the OC. The piezo is bonded to a

metal mount using heat curing epoxy11 and the OC is bonded to the piezo with UV-curing epoxy12

. For future applications requiring higher-bandwidth frequency control, a more carefully designed

mount and a low mass OC may be desirable.

D.2.3 Birefringent filter

The BRF enables coarse tuning of the laser frequency (0.1 THz −1 THz range). It is an

uncoated 3 mm thick quartz plate13 inserted at Brewster’s angle. As shown in Figure D.5 the

BRF is glued into a Thorlabs D - mirror mount14 using UV curing epoxy. To enable temperature

measurement and control a heater15 and temperature sensor 16 are glued to the mount using

heat curing epoxy17 . The mirror mount is screwed onto an adpater plate that is mounted on

a goniomoter18 to allow rotation of the BRF. During normal operation, the BRF temperature is

controlled to within ± 0.05 ◦C using a temperature controller19 . The wavelength of the VECSEL

can be coarsely adjusted by rotating the BRF or by changing the BRF set-point temperature

(∼ 40 GHz/K).

9 Cavity-side radius of curvature of 200 mm, diameter 12.7 mm, back side AR coated.

10 Noliac NAC2123, 3.3 μm maximum free stroke at 200 V

11 Thorlabs part number: 353NDPK

12 Norland 61 (Thorlabs part number: NOA61)

13 Newlight Photonics BIR0030

14 Thorlabs KM05DR

15 Induceramic custom part

16 Thorlabs TH100PT

17 Thorlabs part number: 353NDPK

18 Thorlabs GN1

19 Arroyo instruments 5240 TECSource

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
163

PT100

Sensor

D mirror 

mount

Ceramic 

heater

BRF goes 

here

Figure D.5: Photograph of the birefringent filter (BRF) mount. The BRF plate is glued into a

Thorlabs KM05DR D-shaped mirror mount. Heat is supplied by a ceramic heater glued to the side

of the mount. The temperature is measured using a PT100 temperature sensor.

D.2.4 Etalon

The VECSEL contains a 1 mm thick yttrium aluminum garnet (YAG) etalon20 with a freespectral range of ∼ 80 GHz near a wavelength of 1 μm for intermediate range tuning (1 GHz−100 GHz).

The etalon is glued into an aluminum holder (see Fig. D.6) using UV curing epoxy21 . A thermistor 22 and ceramic heater 23 glued to the holder enable measurement and control of the etalon

temperature. The temperature is maintained to within ± 0.05 ◦C a desired set point by a feedback

controller24 . The etalon holder is contained in an oven made from Macor ceramic that is mounted

on a three-axis rotation stage25 for alignment.

20 Light Machinery OP-3167-1000

21 Norland 61 (Thorlabs part number: NOA61)

22 TE Technology MP-2444

23 Induceramic custom part

24 Arroyo instruments 5240 TECSource

25 Opto Sigma KKD-25C

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
164

(a) (b)

Macor 

housing

Aluminum 

plate

Thermistor

Ceramic

heater

High 

temperature 

epoxy

Figure D.6: Photograph of the etalon oven. (a) shows the aluminium plate that holds the the

etalon (on the reverse side) with an NTC thermistor epoxied into a groove on the plate. The

ceramamic heater is then expoxied onto the aluminum plate as shown in (b).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
165

D.2.5 Pump diode

We use a fiber-coupled multi-mode pump diode 26 with a maximum output power of 50W.

The pump fiber 27 is terminated on the VECSEL side with a focussing unit (“fiber focusser”) which

produces a pump spot of ∼ 200 μm radius at on the gain chip. The focussing unit is secured in a

mirror mount 28 so that the pump laser spot can be positioned on the gain chip. The pump spot

is chosen to be slightly smaller than the laser mode (∼ 160 μm 1/e2intensity radius) to suppress

oscillation of transverse modes [149].

D.3 Infrared VECSEL alignment procedure

D.3.1 Alignment of the pump diode onto the gain chip:

Remember to wear laser safety goggles! Use a camera 29 with an adjustable magnification

lens30 to image the gain chip. The pump diode current should be is set below threshold (typically

to ∼6 A), so that there is spontaneous emission (from the pump diode), but not lasing. Using

the camera to observe the pump spot on the gain chip, position the pump spot near the center of

the gain chip (Fig. D.7) and adjust the position of the fiber focusser to minimize the pump spot

size. Neutral density filters or spectral filters 31 should be used to reduce the amount of pump

light reaching the camera. Next, increase the pump diode current to ∼8 A (just above the pump

threshold). If defects are visible on the gain chip surface within the pump spot, move the pump

spot to a region free of defects (Fig. D.8).

D.3.2 VECSEL alignment:

For coarse alignment, use a low power laser 32 as an alignment guide and two mirrors. Ensure

that there is some space to insert a power meter between the VECSEL housing and the alignment

26 Coherent M1F2S22-807.3-40C-SS2.6T3

27 OzOptics LPF-D4-808-200-240-QM-2.2-12-6.2AS,13.9AS-35-5HPL-3A-1.5

28 Thorlabs: K05

29 Thorlabs DCC1545M

30 Thorlabs MVL7000

31 Thorlabs FEL950 or FEL1000

32 Thorlabs CPS532

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
166

Diamond heat 

spreader

Solder

Semiconductor 

gain chip

Pump spot

Gold mount

Figure D.7: Photograph of an 1120 nm gain chip under illumination with light from an 809 nm

pump diode. The pump spot appears larger than it actually is as the reflected pump light is

saturating the camera.

Pump spot

Defect Gain chip

Figure D.8: Photograph of an 1120 nm gain chip showing the pump spot of the 809 nm pump

diode and a defect visible as a white spot.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
167

mirror closest to the VECSEL. Direct the alignment laser beam so that it is reflected from the

center of the gain chip and that the reflected beam overlaps the incident beam. Next, insert the

output coupler. With the alignment laser on, adjust the output coupler alignment so the the beam

reflected from the cavity side of the output coupler returns to the gain chip. To achieve lasing, use a

high-gain, large-area photodetector 33 to detect laser emission. Use long pass filters to block pump

light, and 1/2” lens tube after the filters to block room light. Switch the detector to the maximum

gain setting and monitor the signal on a multimeter or an oscilloscope. Position the detector so

that the beam from the VECSEL would be incident on the detector active area. Set the pump

diode current to about 1−2 A above where the VECSEL threshold is expected to be 34 . Iteratively

adjust the output coupler orientation to maximize the signal on the photodetector. There should

be a clear maximum in the signal when the laser is close to lasing. When lasing occurs there will

be a distinct jump in the photodiode signal. Once this has happened, remove the photodetector

and insert a thermal power meter. Optimize the output coupler orientation for maximum output

power and for the lowest possible lasing threshold. Next insert the BRF into the cavity. The BRF

is at Brewster’s angle and will displace the beam slightly. Use the photodiode and adjust the OC

orientation to achieve lasing again. It may be necessary to increase the pump power to achieve

lasing if the BRF is selecting frequencies far from the center of the gain curve. Rotate the BRF to

find the maximum output power. Adjust the BRF orientation and the OC orientation to

minimize the pump power required to reach threshold. Finally insert the etalon into the cavity.

Adjust the etalon orientation to achieve lasing again and to optimize the output power.

D.3.3 Tuning the laser to the desired wavelength:

Monitor the VECSEL wavelength using a wavemeter. Rotate the BRF and/or adjust its

temperature to coarsely tune the laser. When the laser is as close as possible to the desired

wavelength, adjust the etalon temperature. Increasing the etalon temperature should increase the

33 Thorlabs PDA50B

34 The threshold current depends on many factors including the gain material, the output coupler reflectivity, and

the pump spotsize on the gains chip. If I do not have any idea what the threshold current should be, I would start

at about 15 A

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
168

laser wavelength. There will be mode hops in roughly ∼ 1 GHz steps (corresponding the the FSR

of the laser cavity). Once the laser is within ∼ 1 GHZ of the desired frequency, the PZT can be

used for finer adjustment. Optimize the output power at the desired wavelength by making fine

adjustments to the etalon and BRF temperatures.

D.4 Output power

Figure D.9 presents example measurements of the output power of an 1140 nm VESCEL.

Above threshold, the output power initially increases linearly with the pump power. At sufficiently

high pump power, thermal gradients in the gain chip will reduce the power conversion efficiency. In

general, the efficiency of a laser depends on the single-pass gain and the cavity losses. For singlefrequency lasers, intra-cavity frequency selective elements can contribute a significant loss. For

the data in Fig. D.9, the slope efficiency drops from 27 % in multimode operation (no frequency

selective elements) to 21 % when the etalon and BRF are inserted into the cavity. The output

coupler reflectively can be optimized to maximize the efficiency of the VECSEL for a given pump

power. For each different gain material we typically try OCs with a few different reflectivity values

R ranging from 0.5 to 3.0 %. For the data in D.9, the single-mode slope efficiency drops by more

than a factor of 2 if the OC is changed to one with R = 0.5 %. I have found that it is usually

worthwhile starting with an OC of R = 2.0 %.

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69
169

2 4 6 8 10 12

Pump power (W)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

VECSEL output power (W)

27%

21%

8%

Multi-mode, R = 2.5%

Single mode, R = 2.5%

Single mode, R = 0.5%

Figure D.9: Output power of an 1140 nm VECSEL as a function of pump power for multi-mode

and single-mode operation for different output coupler reflectivities (R). Data points are measured

pump power. Solid lines are linear fits to the data. Slope efficiencies obtained from the fits are

listed. There is ∼ 5% calibration uncertainty for all power measurements (not shown).

DocuSign Envelope ID: 31F15C07-0A07-40F7-B184-FAFF2BDE1D69