High-Fidelity Entangling Gates with Trapped-Ions

by

Ting Rei Tan

B.S., National University of Singapore, 2009

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

2016
 

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This thesis entitled:

High-Fidelity Entangling Gates with Trapped-Ions

written by Ting Rei Tan

has been approved for the Department of Physics

David J. Wineland

Emanuel Knill

Date

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

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iii

Tan, Ting Rei (Ph.D., Physics)

High-Fidelity Entangling Gates with Trapped-Ions

Thesis directed by Dr. David J. Wineland

Quantum entangling logic gates are key ingredients for the implementation of a quantum

information processing device. In this thesis, we focus on experimental implementations of three

types of entangling geometric-phase gates with trapped ions, which rely on the effective spin-spin

interactions generated with state-dependent forces. First, a mixed-species entangling gate is demonstrated using a 9Be+ and a 25Mg+ ion to create a Bell state with a fidelity of 0.979(1). Combined

with single-qubit gates, we use this mixed-species gate to implement controlled-NOT and SWAP

gates. Second, we implement a high-fidelity universal gate set with 9Be+ ions. Single-qubit gates

with error per gate of 3.8(1) × 10−5 are achieved. By creating a Bell state with a deterministic

two-qubit entangling gate, we deduce a gate error as low as 8(4) × 10−4. Third, a novel two-qubit

entangling gate with dynamical decoupling built-in is demonstrated with a fidelity of 0.974(4).

This gate is robust against qubit dephasing errors and offers simplifications in experimental implementation compared to some other gates with trapped ions. Errors in the above implementations

are evaluated and methods to further reduce imperfections are discussed. In a separate experiment, correlated measurements made on pairs of ions violate a “chained” Bell inequality obeyed

by any local-realistic theory. The lowest chained Bell inequality parameter determined from our

measurements is 0.296(12), this value is significantly lower than 0.586, the minimum value derived

from a perfect Clauser-Horne-Shimony-Horne (CHSH) Bell inequality experiment. Furthermore,

our CHSH Bell inequality results provide a device-independent certification of the deterministically

created Bell states.
Dedication

To the NIST ion storage group, and the trapped-ion community
v

Acknowledgements

Thanks to many people along the way, my Ph.D. journey has been an incredible experience

and probably one of the best that anyone can hope for.

First and foremost, I would like to thank Dr. David J. Wineland for giving me the opportunity

to work in the ion storage group at the National Institute of Standards and Technology (NIST). I

appreciate the responsibility that was given to me to grow myself, and to take charge of the research

and experiments that interest me.

It is my privilege to work in the ion storage group, which is formed by some of the brightest

people I have ever met. I am grateful for being a team member of one of the most prominent, high

quality and efficient experimental atomic physics group in the community, and the chance to work

with some of the towering, established figures in the field.

Special thanks for John Gaebler, the post-doctoral researcher in my experiments during most

of my Ph.D. career, we formed a good team together and I have always enjoyed bouncing ideas off

one another. Team members of the experiments I have closely worked with also includes Yiheng

Lin, and Ryan Bowler. The four of us form the last crews of the “2108” legacy experiment1 , and

the first crews of the “Quantum II” experiment2 .

I am indebted to the previous members of the 2108 experiments, who undertook the enormous

task of building up a very complicated experiment. I have had the opportunities to work with some

of them, especially David Hanneke and Jonathan Home, they have patiently guided me during the

earlier part of my Ph.D. career. The experience and knowledge we gained in the 2108 experiment

1 the room number where the experiment setup was located 2 the 2108 experiment was retired in March of 2013
vi

setup is instrumental in the success of the post-2108, i.e. the “Quantum II” experiment. Quantum

II is later joined by Yong Wan, Stephen Erickson, and Daniel Kienzler. They have contributed

enormously to the continuation of the experiments. I wish them the best of luck and hope that the

laboratory and the experiment setup will serve them well in the future. Furthermore, collaborating

with members of the quantum information theory group at NIST, led by Emanuel Knill, has been

an enlightening experience. Their contributions are crucial for the success of many experiments.

In addition to those mentioned above, I had the great opportunity to work with these amazing physicists: Robert J ̈ordens, Ulrich Warring, Yves Colombe, John Jost, Christian Ospelkaus,

Dietrich Leibfried, Andrew Wilson, Scott Glancy, Adam Keith, Peter Bierhorst, David Allcock, Joe

Britton, Shaun Burd, John Bollinger, Samuel Brewer, Brian Sawyer, Justin Bohnet, Jwo-Sy Chen,

David Leibrandt, Shon Cook, Till Rosenband, James Bergquist, Daniel Slichter, Shlomi Kotler,

James Chou, Katharine McCormick, Susannah Todaro, Kyle McKay, David Hume, Aaron Hankin, Raghavendra Srinivas, and Christoph Kurz. The works presented in this thesis would not be

possible without them, as well as every past and present members of the ion storage group.

I would also like to thank our professional friendly rivals working in the trapped-ion group

at University of Oxford. Discussions with Prof. David Lucas, Chris Ballance, and Tom Harty have

been very helpful in the pursue of high-fidelity two-qubit entangling gate.

My sincere gratitude to Murray Barrett, who ignited my interests in experimental atomic

and laser physics, and guided me in my first two years working in a research laboratory when I was

pursuing my undergraduate degree at the National University of Singapore.

I cannot thank my parents and my family enough, who supported my decision to take up the

challenge for pursuing a Ph.D. career in a foreign country, all by myself. I must also thank all my

good friends I have in my life, both human beings and creatures. A big thanks to D. J. Wineland,

R. Srinivas, S. Burd, D. Kienzler, S. Erickson, E. Jordan, Becky McKay and my sister Rok Ting

“nuoz” Tan, who assisted in reviewing my thesis during preparation.

Furthermore, I am grateful for what Boulder and Colorado offered these past few years: the

flatiron, the fresh air and the bright blue sky, the mountains, the distinctive four seasons weather,
vii

and most importantly, the genuine and friendly people who I have encountered. My seven years in

Boulder, Colorado is definitely one of the highlights in my life. That being said, this journey is not

without bumps on the road. There have been moments when I doubted myself about the decisions

that I have made, when everything feels trapped in the shadow cast by the many problems exist

inside and outside of the laboratory. Running, cycling, and hiking in the beautiful Colorado, as

well as immersing myself in the kitchen trying out and perfecting my Bolognese recipes have helped

me clear those doubts and carried on. I am still amazed by how so many enlightening ideas and

solutions to problems in the laboratory came about when I was outside of the laboratory.

Last but not least, I would like to express my sincerest appreciation to my thesis committee

members: David J. Wineland, Emanuel Knill, Ana Maria Rey, Jun Ye, Eric Cornell, and Kelvin

Wagner. And of course, the readers of this thesis.
Contents

Chapter

1 Introduction 1

1.1 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 9Be+ and 25Mg+ Ion Qubits 14

2.1 9Be+ Ion Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Repumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.3 Stimulated-Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.4 Microwave Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.5 Sideband Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.6 Qubit State Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.7 Qubit State Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 25Mg+ Ion Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 Repumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.3 Coherent Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.4 Sideband Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.5 Qubit State Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.6 Qubit State Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ix

3 Apparatus 39

3.1 Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Magnetic Field Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 9Be+ Ion Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.1 Blue Doppler Laser (2S1/2 ↔ 2P3/2 Transition) . . . . . . . . . . . . . . . . . 49

3.4.2 Red Doppler Laser (2S1/2 ↔ 2P1/2 Transition) . . . . . . . . . . . . . . . . . 51

3.4.3 Qubit Manipulation Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.4 Photo-Ionization Laser and the Loading of 9Be+ Ions . . . . . . . . . . . . . 55

3.5 25Mg+ Ion Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5.1 Resonant Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5.2 Qubit Manipulation Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.3 Photo-Ionization Laser and Loading of 25Mg+ Ions . . . . . . . . . . . . . . . 62

3.6 Experiment Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.7 Arbitrary Waveform Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.8 High-Speed Laser Power Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Two-Qubit Entangling Gates 68

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Cirac-Zoller Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Geometric Phase Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Types of Geometric Phase Gates . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.2 σˆzσˆz Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.3 σˆφσˆφ Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.4 Mølmer-Sørensen Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.5 Bermudez Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Error Sources for Geometric Phase Gates . . . . . . . . . . . . . . . . . . . . . . . . 81
x

4.4.1 Schr ̈odinger and Master Equation Simulations . . . . . . . . . . . . . . . . . . 82

4.4.2 Spontaneous Scattering of Photons . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.3 Motional Frequency Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4.4 Motional Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4.5 Off-Resonance Coupling to Spectator Modes . . . . . . . . . . . . . . . . . . 91

4.4.6 Imperfect Lamb-Dicke Approximation . . . . . . . . . . . . . . . . . . . . . . 94

4.4.7 Rabi Rate Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4.8 Rabi Rate Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.9 Laser Phase Drift and Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.10 Qubit Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Mixed-Species Entangling Gate 98

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.1 Optimizing the Strength of Spin-Dependent Forces . . . . . . . . . . . . . . . 103

5.2.2 Calibration Procedure for the Mixed-Species MS Gate . . . . . . . . . . . . . 104

5.2.3 Laser Beam Phase Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.4 Calibration Procedure for Phase Gate Gb . . . . . . . . . . . . . . . . . . . . . 106

5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3.1 Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.2 Mixed-Species Bell Inequality Test . . . . . . . . . . . . . . . . . . . . . . . . 113

5.4 Mixed-Species Controlled-NOT Gates . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.4.1 Quantum Non-Demolition Measurement . . . . . . . . . . . . . . . . . . . . . 117

5.5 SWAP Gate for Quantum Information Mapping . . . . . . . . . . . . . . . . . . . . . 122

5.6 Mixed-Species Cirac-Zoller Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.7 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
xi

6 9Be+ High Fidelity Universal Gate Set 128

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4 Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.4.1 Spontaneous Emission Induced by Raman Laser Beams . . . . . . . . . . . . 138

6.4.2 Motional Mode Frequency Fluctuations . . . . . . . . . . . . . . . . . . . . . 139

6.4.3 Rabi Rate Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.4.4 Smaller Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.5 Average Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.6 State Detection and Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.6.1 Assumptions with the ML Method . . . . . . . . . . . . . . . . . . . . . . . . 149

6.6.2 Imperfect Optical Pumping and Lower Bound of Bell State Fidelity . . . . . 151

6.7 9Be+ Single-Qubit Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7 Dynamical Decoupling Two-Qubit Geometric Phase Gate 160

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.2 Experiment Setup and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.3 Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8 Chained Bell Inequality Experiment 178

8.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.1.1 “Loopholes” and Previous Experiments . . . . . . . . . . . . . . . . . . . . . 182

8.2 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.3 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.4 Self-Testing of Bell States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
xii

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9 Outlook 197

9.1 Better Trap design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9.1.1 Axial Micromotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9.1.2 Segmented-Stray-Fields Compensating Electrodes . . . . . . . . . . . . . . . . 198

9.2 Cooling of Trap Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.3 Rapid Loading of Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.4 Improved Microwave Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.5 Switchable Noise Filtering for Control Electrodes . . . . . . . . . . . . . . . . . . . . 201

9.6 Electronically-Induced-Transparency Cooling with an Atomic Tripod System . . . . 202

Bibliography 204

Appendix

A Details of 9Be+ 221

B Details of 25Mg+ 225

C Atom-light Interaction Hamiltonian 229

C.1 Two-Level Atoms - Light Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

C.2 Stimulated-Raman Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

C.2.1 Rabi Rate of Stimulated-Raman Transitions . . . . . . . . . . . . . . . . . . . 233

C.3 Spin-Motion Stimulated-Raman Transition . . . . . . . . . . . . . . . . . . . . . . . 233

D Matrix Representations of State Vectors and Qubit Operations 236

D.1 Qubit State Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

D.2 Qubit Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
xiii

D.3 Fock State Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

D.4 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

D.5 Atom and Motion Compound System . . . . . . . . . . . . . . . . . . . . . . . . . . 238

D.6 Summary of Qubit Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

D.7 Three-Level Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

D.8 Qubit Operators in a Three-Level Atom . . . . . . . . . . . . . . . . . . . . . . . . . 240
Tables

Table

3.1 Values of resistors, capacitors, and their corresponding cut-off frequencies (Eq. 3.1)for

the three low-pass filtering stages connected between a multi-channel AWG (Sec. 3.7)

and the control electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Laser beams used for the manipulations of 9Be+ ion and the ionization 9Be atom.

Each beam can be individually switched on/off. Doppler cooling, detection, repumping and qubit manipulations of the 9Be+ ions are detailed in Chap. 2. The laser

beam alignment onto the ions is shown in Fig. 2.3. . . . . . . . . . . . . . . . . . . 52

3.3 Laser beams used for the manipulation and loading of the 25Mg+ ion. Each beam

can be individually switch on/off. Doppler cooling, detection, repumping and qubit

manipulations of the 25Mg+ ions are detailed in Chap. 2. The laser beam alignments

onto the ion are shown in Fig. 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1 Correlation values for the four measurement setting combinations in the CHSH Bell

inequality experiment performed on a mixed-species entangled state. . . . . . . . . 115
xv

6.1 Individually evaluated errors for the entangling gate at a Raman detuning of Δ '

−2π × 900 GHz, and a gate duration of approximately 30 μs. Off-resonant coupling

includes coupling of the qubit states to other hyperfine states and their sidebands,

which can includes motional mode sidebands and micromotion sidebands. The last

(transfer) error reduces the ML-Bell-state fidelity but should minimally affect the

gate fidelity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.1 Individually evaluated errors that contribute to the imperfect creation of the Bell

state |Φ+i state with carrier excitation driven by (i) microwave field and (ii) laser

field. See Sec. 4.4 for detail discussions on error sources in the implementation of

two-qubit gates and their respective evaluation strategies. . . . . . . . . . . . . . . . 174

8.1 Results from CHSH experiments without the fair-sampling assumption. The table

shows each experiment’s measured CHSH parameter BCHSH with one standard deviation uncertainty from the references and self-testing fidelity lower bounds at the

50 % (F50%) and 95 % (F95%) confidence levels determined according to Eq. 8.14. . 196 L L

9.1 Typical durations required for elementary operations using the X-Junction trap. The

actual durations of these operations vary, as they depend on motional mode structure, transition excitation methods, and ion location in the trap array. “Shelving”

refers to the pulses that transfer ions’ states from the computation qubit to the measurement qubit before making fluorescence measurement (see Sec. 2.1.7 and Sec.

2.2.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

A.1 Relevant properties of the 9Be atom and 9Be+ ion which are important for this thesis.

The references from which the properties are extracted are shown in the last column.

See also Ref. [Shiga 11] for the 9Be+ ion’s ground state hyperfine constant that takes

the diamagnetic correction into account. The hyperfine energy level splittings at a

field B = 0 is calculated by solving the Breit-Rabi equation at a vanishing field. . . 222
xvi

A.2 Magnetic-field insensitive transitions and their respective magnetic-field strengths in

the 2S1/2 electronic ground state for the 9Be+ ion in the intermediate field regime.

The transition frequencies and the second-order magnetic-field sensitivities are also

given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

B.1 Relevant properties of the 25Mg+ ion and neutral magnesium. The references from

which the properties are extracted are shown in the last column. See also Ref.

[Drullinger 80] and Ref. [Ansbacher 89] for precision spectroscopy measurements in

Mg+ ions. The hyperfine energy level splittings at a field B = 0 is calculated by

solving the Breit-Rabi equation at a vanishing field. . . . . . . . . . . . . . . . . . . 226

B.2 The magnetic-field insensitive transitions and their respective magnetic-field strengths

in the 2S1/2 electronic ground state for the 25Mg+ ion in the intermediate field regime.

The transition frequencies and the second-order magnetic-field sensitivities are also

given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Figures

Figure

1.1 One proposal to scale up the QIP device with a trapped-ion system: a quantum

charge coupled device (QCCD) architecture [Wineland 98, Kielpinski 02] consists of

multiple, interconnected trapping zones. Each trapping region could be dedicated for

certain operations. This enables parallel processing and handling of a small subset of

qubits at any trapping location. The information transport is achieved by physically

shuttling the ions between different trapping regions by applying well-controlled,

time-varying electric fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Relevant energy level structure (not to scale) of the 9Be+ ion. Transitions to the

electronic excited states are used for Doppler cooling, repumping, and qubit state

measurement as described in the text. The “blue Doppler detuned” (BDD) beam,

which is red shifted ' 400 MHz with respect to the “blue Doppler” (BD) beam (see

Sec. 2.1.1), is not shown in this figure. We label the qubit |1, 1i = |↑i and the

|2, 0i = |↓i states as the “computational qubit manifold”. The “measurement qubit

manifold” consists of the |2, 2i as the “bright” state, and the |1, −1i or |1, 0i state

as the “dark” state (see Sec. 2.1.7). The laser for stimulated-Raman transitions is

red detuned from the 2S ↔ 2P1/2 transition. The decay rate of the 2P3/2 state

is 2π × 19.4(5) MHz [Poul

1/2

sen 75]. See Fig. A.1 for details energy levels diagram in

the 9Be+ electronic ground state. The state labels correspond to the states at low

magnetic field, which evolve to the states indicated at non-zero magnetic field. . . . 16
xviii

2.2 Coherence time of the 9Be+ qubit is investigated with a Ramsey sequence. We

repeat the experiments with and without the presence of the 25Mg+ laser beams,

which have a wavelength of approximately 280 nm. In this experiment, one 9Be+

and one 25Mg+ ion are trapped together in a harmonic well with a separation of

' 5 μm; the laser beams (with beam waists ' 30 μm.) illuminate both ions. We

do not observe a significant difference between these two sets of experiments, which

indicates that the 25Mg+ laser radiation causes negligible additional decoherence. . 17

2.3 Schematic diagram indicating laser beams access into the ion trap apparatus. Four

ports are available, each aligned ∼ 45◦ relative to the trap (z) axis. The external

magnetic field (provided by coils in the Helmholtz-like configuration) is set to be

45◦ with respect to the trap axis. The radial confinement of the ions is in the

plane normal to the z axis. Laser beams for the Doppler cooling, repumping, and

fluorescence detection of the 9Be+ and 25Mg+ ions are coaligned with the magnetic

field direction, and are set to be σ+ polarized. With this configuration, motional

modes in all three directions can be cooled by the Doppler beams. Each species

Raman laser beams are sent into the vacuum chamber via two ports after being

combined with dichroic mirrors. For 9Be+, the laser beam exiting the UV fiber of

path 1 (2) depicted in Fig. 3.8 is sent into the port labeled as “9Be+ 90” (9Be+ Co)

in this figure. For the 25Mg+, the laser beams labeled as path 1 (2) in Fig. 3.10

corresponds to “25Mg+ 90” (“25Mg+ Co”) in this figure. The laser beam entering the

bottom-right port (i.e., the Mg PI beam in this figure) and directed toward zone L is

blocked by an oven shield installed behind the alumina wafers after passing through

the center of the trap (details in Ref. [Blakestad 10]). The boundary of the vacuum

chamber is indicated by blue lines. Details of apparatus are described in Chap. 3. . 19
xix

2.4 The laser beams setup to drive stimulated-Raman transitions. Laser beams are focused onto ions from two paths, which intersect at 90◦ such that the difference in

their k vectors, Δk, is aligned along the axial direction. These two beams corresponds to the Co and 90 beams in Fig. 2.3. This figure is simplified for the relevant

illustrations in this section. Details of Raman beam lines setup are discussed in Sec.

3.4.3, Sec. 3.5.2, and the respective chapters describing implementation of different

experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 A typical single 9Be+ ion detection histogram for the population to be either in the

|2, 2i state (blue) or the |1, −1i state (red). With a detection duration of 330 μs and

a saturation parameter s of ∼ 0.5, we detect on average ' 30 photon counts for the

ion in the |2, 2i state, and ' 2 counts when the ion is in the |1, −1i state. Overlap

of the two distributions is shown in dark color. . . . . . . . . . . . . . . . . . . . . . 29

2.6 Relevant energy level structure (not to scale) for the 25Mg+ ion at an external magnetic field of approximately 119.446 G. The “blue Doppler detuned” (BDD) beam,

which is red shifted ' 500 MHz with respect to the “blue Doppler” (BD) beam (see

Sec. 2.2.1), is not shown in this figure. The 25Mg+ ion qubit is encoded in the

|3, 1i = |↑i state and |2, 0i = |↓i state, they constituent the “computational qubit

manifold”. The qubit transition frequency has a magnetic-field sensitivity of ' 43

kHz/G at the applied magnetic field. The “measurement qubit manifold” consists

of the |3, 3i state as the “bright” state and the |2, −2i state as the “dark state”

(see Sec. 2.2.6). The decay rate of the 2P3/2 state is 2π × 41.3(3) MHz [Clos 14].

See Appendix B.1 for a more detailed energy level diagram in the 25Mg+ electronic

ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7 Coherence time of the 25Mg+ qubit investigated with a Ramsey sequence with and

without the presence of the λ ' 313 nm laser beam used for the manipulations of

the 9Be+ ion. The shorter qubit coherence time compared to the 9Be+ qubit is due

to the non-zero linear component of magnetic field sensitivity. . . . . . . . . . . . . 32
xx

2.8 The ratio of spin-motion transition Rabi rates between different Fock states (Ωn,n0) to

carrier Rabi rate Ω is plotted as a function of the Fock state number n for different

0 values of Lamb-Dicke parameters. The first-order (n = n − 1) and second-order

0 (n = n − 2) sideband transition rates are shown. The Rabi rates of the first-order

sideband transitions vanish at relatively low values of n, which means that spinmotion transitions between these different Fock states cannot be driven. This can

be problematic for the motional cooling to the ground state if only the first-order

sideband transitions are used in the cooling sequence. At the Rabi rate vanishing

points of the first-order sideband transitions, the second-order sideband transition

Rabi rates are non zero. Thus, they can be inserted to improve the cooling. . . . . 35

2.9 A typical detection histogram of a 25Mg+ ion for the population to be either in

the |3, 3i state (blue) or the |2, −2i state (red). The fluorescing “bright” state corresponds to the qubit |↑i while the “dark” state corresponds to the the |↓i state.

Overlap of the two distributions is shown in dark color. . . . . . . . . . . . . . . . . 38

3.1 The schematic of the X-junction trap. The trap is constructed with two gold-coated,

stacked wafers. (b) Top view of the trap showing the load zone L and experiment

zone E. Ions are transported from L to E with time-varying potentials applied to

the segmented control electrodes (colored orange hues). The positions of RF and

control electrodes are exchanged in the lower layer (a). The details of the junction

are illstrated in (c). Coherent manipulations are implemented on ions confined in E.

See Ref. [Blakestad 11] for details of this trap. . . . . . . . . . . . . . . . . . . . . . 40
xxi

3.2 Mock up of the vacuum chamber housing the X-junction trap. A glass envelope

with fused-silica view ports is attached to a stainless-steel vacuum chamber. The

fused-silica view ports are used for the access of laser beams and collection of fluorescence photons emitted by ions. In addition to an ion pump which maintains the

system under ultra-high vacuum condition of ∼ 3×10−11 torr (measured with an ion

gauge), the vacuum system includes a titanium sublimation pump which is turned on

infrequently (approximately once per month). Multiple electrical feedthroughs are

used to connect to RF and control electrodes, and neutral beryllium and magnesium

resistively-heated ovens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Schematic of the magnetic field coils that are used to provide a static magnetic

field of ' 0.0119446 T in a Helmholtz-like configuration. The main coils (orange

color) which have an approximate rectangular dimension of 40 cm × 20 cm, and

are separated by approximately 8 cm, are held in position by custom parts made

of anodized aluminum. The field coils assembly also consists of compensation coils

which allow fine tuning of magnetic field in three dimensions. Wire in compensation

coils are not shown. See also Fig. 3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Schematic of the Schwarzchild objective. The first element is a lenses pair made of

CaF2 and fused silica, it is designed to compensate for the chromatic aberration for

313 nm and 280 nm caused by the vacuum windows. Photons collected are then

bounced of the primary mirror followed by the secondary mirror, which is cemented

onto the lenses pair, before being sent of to the rest of the imaging system and

also providing a common focus for the two wavelengths (see Fig. 3.5). The overall

magnification of this objective lens is ∼ 10×. All components are held together by an

enclosure which is not shown in this figure. The refractions of rays passing through

different materials are not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
xxii

3.5 Schmatic of the imaging system. The light collected from the Schwarzschild objective

is focused onto an intermediate image point which is located at the center of the bore

hole of a mirror. With a relay mirror of a focal length of 50 mm, the intermediate

image is then imaged onto a PMT. A flipper mirror is used to direct the light onto a

EMCCD camera, as needed. The relay optics are setup with a magnification factor of

5×, which results in a magnification of the entire imaging system of 50×. The optics

depicted inside the dashed box are held in place together with custom machined

parts which are attached to a three-dimensional translation stage. . . . . . . . . . . 47

3.6 The picture showing the vacuum system containing the trap, and the apparatus

surrounding it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7 Three UV laser sources are generated with four fiber lasers in the infrared wavelengths by employing sum frequency generation (SFG) and second harmonic generation (SHG) stages. The frequency of the Raman laser can be tuned from ∼ −2π×10

GHz to ∼ −2π × 900 GHz with respect to the 2S1/2 ↔ 2P1/2 electronic transitions. This tuning range is provided by the tuning capabilities of the fiber lasers.

The SFGs are accomplished with magnesium-oxide doped periodically-poled lithiumniobate (MgO:PPLN) crystals while barium borate (BBO) crystals are used for the

SHGs. The frequency of the blue Doppler and red Doppler lasers are stabilized to

iodine transitions with Doppler-free saturated-absorption spectroscopy setups. . . . 50
xxiii

3.8 Schematic of the laser beam line setup used to drive stimulated-Raman transitions in

9Be+ ions. The 313 nm generated with SFG followed by SHG is split into two paths,

sent through AOMs, coupled into fibers, and aligned onto the ions. Path 2 contains a

double-passed 600 MHz AOM that, when switched on, produces an additional beam

shifted by approximately the qubit frequency (∼ 1.2 GHz), that is coaligned with

the unshifted beam for high-fidelity single-qubit gates. Another AOM tunable in the

range of 260 to 360 MHz is used to shift the relative laser frequency in path 2 with

respect to beam 1 for spin-motion stimulated-Raman transitions. For the MølmerSørensen gate, two RF tones with relative frequency difference close to twice the

frequency of the addressed motional mode are injected into the 200 MHz AOM in

path 2. In combination with the beam in path 1, these two beams simultaneously

produce blue and red sideband transitions. A pickoff on the output of each fiber

directs a small fraction of the light onto a photodiode, which is used for active power

stabilization. Each beam is centered on the ions with a motorized mirror mount

before the final lens that focuses the beam on the ions. . . . . . . . . . . . . . . . . 53

3.9 Schematic of the setup to generate laser beams for the optical pumping, Doppler

cooling, and repumping of the 25Mg+ ion. The 280 nm laser beam is generated with

a SHG using a dye laser source at ∼ 560 nm. Three individually controllable UV

beams with different laser frequencies are generated: (i) a blue Doppler (BD) beam

for Doppler cooling and fluorescence detection, (ii) a blue Doppler detuned (BDD)

beam for Doppler cooling of hot ions, and (iii) a repumper beam. All three beams

are overlapped before they are coupled into a UV fiber, which directs the light to

the location of the ions. The power of the BD beam is actively stabilized. . . . . . . 58
xxiv

3.10 Schematic of the laser beams setup for inducing stimulated-Raman transitions in

the 25Mg+ ions. A polarizing beam splitter is used to split the UV output beam

after the SHG stage. The frequency of the laser beam in each path is shifted by two

AOMs before injected into an UV fiber. The relative frequency difference between

these two beams can be tuned with a range of approximately ± 200 MHz. This

enables us to induce spin-motion transitions on (i) the |3, 3i ↔ |2, 2i sidebandcooling transitions and (ii) the |3, 1i ↔ |2, 0i qubit transitions. Two frequency tones

can be injected into the single-passed AOM (with 200 MHz center frequency) in path

1. This configuration is used for the application of the mixed-species entangling gate

as described in Chap. 5. A small fraction of laser power is sampled at each UV

fiber’s output for power stabilization. Each beam is combined with the 9Be+ ion’s

Raman laser beam with a dichroic mirror as depicted in Fig. 2.3. . . . . . . . . . . . 61

3.11 Schematic of the laser power stabilization and shaping of laser pulses employed in

the 9Be+ qubit manipulation laser beam lines. Each laser beam is sampled at the

output of the UV fiber with a photodiode (see Fig. 3.8). A digital servo is then used

to provide the feedback for the power stabilization by adjusting the radio frequency

power level injected into an AOM positioned upstream of the fiber. One advantage of

using the fiber is that laser beam pointing fluctuations occuring before the fiber are

translated into laser power fluctuations at the fiber output, which is then corrected

with the servo system. With the feedback engaged, the time profile of the laser

beam is dynamically modulated by adjusting the servo reference potential using the

high-speed arbitrary waveform generator described in Sec. 3.7. . . . . . . . . . . . . 66

4.1 Quantum circuit using the Cirac-Zoller protocol to produce the controlled-Z gate.

A blue-sideband (BSB) π pulse is applied to the “control” qubit before and after a

red-sideband (RSB) 2π pulse is applied to the “target” qubit. The RSB transition

couples one of the target’s qubit states to an auxiliary (Aux) state. . . . . . . . . . 69
xxv

4.2 Quantum circuit using the Cirac-Zoller protocol to produce the controlled-NOT gate.

The overall pulse sequence implements the unitary operation given in Eq. 4.9. The

notation R(θ, φ) is given in Eq. 2.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 By applying an effective spin-spin interaction, the motional wavepacket is displaced

in phase space dependent on the internal qubit states. If displacements enclose a

loop, the qubit states pick up a geometric phase, Φ, proportional to the area of the

enclosed loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 Orientation of Raman laser beams used to drive two-qubit entangling gates. Beams

1 and 2 are arranged such that their k vectors intersect at 90◦ and their Δk vector

is aligned along the axial direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 This figure shows the different scattering probabilities and errors caused by spontaneous scattering of photons due to Raman laser beams for a two-qubit gate driven

with the Mølmer-Sørensen protocol on 9Be+ ions. Spontaneous scattering of photons can be categorized into Raman and Rayleigh scattering, and they contribute

errors to a two-qubit gate differently. The qubit’s coherence is lost after each Raman

scattering event, while this is not the case for Rayleigh scattering. Both Raman and

Rayleigh process contribute to motional dephasing error through a random trajectory in the motional phase space caused by a uncontrolled momentum kick for each

scattering event. The horizontal axis indicates the detuning of the Raman laser with

respect to the 2S1/2 ↔ 2P1/2 transition of the 9Be+ ion. The second peak at ∼ +200

GHz Raman detuning corresponds to scattering from the 2P3/2 electronic excited

state. The dips between the two fine structure lines are caused by the cancellation

effect as indicated in Eq. 4.29; the Raman laser is blue detuned relative to the 2P1/2

levels while red detuned relative to the 2P3/2 levels. This figure is plotted with a

Lamb-Dicke parameter of η = 0.19, NLoop = 1, and assuming equal laser intensity in

each of the two laser paths as depicted in Fig. 4.4. The polarization of the k1 (k2)

is set to be pure σ+ (π). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
xxvi

4.6 This figure is analogous to Fig. 4.5 but for 25Mg+ ions and with η = 0.165. The

Raman detuning, Δ is plotted with respect to the 2P3/2 excited state, where the

frequency of our 25Mg+ Raman laser is tuned near to. The two peaks correspond

to the fine structure splitting between the 2P1/2 and 2P3/2 electronic excited states,

which is ∼ 2.75 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1 Configuration of laser beams for the mixed-element entangling gate. For the 9Be+

ion, 313 nm laser beams (in red) simultaneously induce near-resonant red and blue

sidebands transitions. Similarly, for 25Mg+, 280 nm beams (in green) induce sideband transitions. When all beams are applied simultaneously, this implements

the Mølmer-Sørensen spin-spin interaction [Sørensen 99, Sørensen 00]. Each set

of qubit addressing laser beams is set up such that the wave vector differences

Δkj,r = kj,90 −kj,Co1 and Δkj,b = kj,90 −kj,Co2 (j = 1, 2) are aligned in the same

direction along the trap axis. With this configuration, only the motional modes along

this axis can be excited. The frequencies of the 9Be+ ion’s Raman beams are set to

drive the spin-motion transition on the second order micromotion sideband, this is

to maximize transition Rabi rates in the presence of axial micromotion (details see

Sec. 2.1.3. Spin-motion transitions in the 25Mg+ ion are driven on the micromotion

carrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2 A Ramsey sequence implemented to remove laser phase sensitivity of the MS interaction. The “90 Carrier” π/2 pulses are implemented by Raman laser beams induced

by kj,Co1 and kj,90 (see Fig. 5.1). The overall sequence produces a phase gate Gb

which is given in Eq. 5.7. The rotation R(θ, φ) is defined in Eq. 2.13. . . . . . . . . 106

5.3 Population evolution as we apply the MS interaction starting from the |↑↑i state.

The Bell state |ΦBelli (Eq. 5.6), which corresponds to (ideally) equal population in

the |↑i |↑i Be Mg and |↓i |↓i Be Mg, is created at tgate ' 33μs. Error bars correspond to

standard error of the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
xxvii

5.4 The pulse sequence using the Gb gate to create the entangled state √

1 (|↑↑i − i|↓↓i) 2

(up to a global phase of eiπ/4) with both ions initialized to their respectively |↑i

state. The Bell state |Φ+i = √

1 (|↑↑i + |↓↓i) (up to a global phase of eiπ/2) can 2

be created by applying a Rz(π/2) rotation to one of the qubits at the end of this

sequence. Likewise, applying a Rz(−π/2) rotation to one of the qubits at the end of

the sequence creates the Bell state |Φ−i = √

1 (|↑↑i − |↓↓i). The definition of Rz 2 is

given in Eq. D.5. The notation “μWv” is used to represents rotations implemented

with a microwave field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5 Parity (P↑↑ + P↓↓ − P↑↓+↓↑) flopping of the Bell state created by the sequence shown

in Fig. 5.4. To demonstrate laser-phase insensitivity of our entangling operation

and the deterministic creation of entanglement, we use microwave fields to induce

the “analysis” pulses with variable phases in this parity oscillation experiment. The

phases of the microwave fields are not synchronized with those of the laser fields. . 111

5.6 Pulse sequence for the CHSH-type Bell inequality experiment on the mixed-species

Bell state. The black dashed box denotes the “black box” which prepares the two

particles of different species. The measurement settings αi, βj are applied (with microwave fields) before making fluorescence measurements. The laser pulse sequence

for the implementation of Gb is shown in Fig. 5.2. The rotations R(θ, φ) and Rz(ξ)

are defined in Eq. 2.13 and Eq. D.5, respectively. Rz(ξ) rotation is accomplished by

adjusting the phase of the DDS which is used to produce the microwave fields. . . . 113

5.7 (a) Pulse sequence for the CNOT gate (Eq. 4.9) with 9Be+ qubit as the “control”

and 25Mg+ qubit as the “target”. (b) Pulse sequence for the CNOTinvert gate (Eq.

5.12), where the two qubits change roles, i.e. the 25Mg+ qubit as the “control” and

the 9Be+ qubit as the “target”. The rotations R(θ, φ) and Rz(ξ) are defined in

Eq. 2.13 and Eq. D.5, respectively. Here, the Rz(ξ) rotations are accomplished by

adjusting the phase of the DDS used to induce microwave transitions (see footnote

2), which can be absorbed into the microwave-induced R(π

2 , −π ) pulses. 2 . . . . . . . 116
xxviii

5.8 Pulse sequences of performing Rabi flopping on the 9Be+ (spectroscopy) ion as detected with the 25Mg+ (logic) ion using (a) conventional quantum logic spectroscopy

[Schmidt 05], and (b) the controlled-NOT gate based procedure. The CNOT gate

implemented here is defined by the matrix in Eq. 4.9, with the pulse sequence of

the Gb gate shown in Fig. 5.2. Ground state cooling which was applied prior to the

qubit state preparations, are not shown in this figure. . . . . . . . . . . . . . . . . . 118

5.9 Rabi flopping of the 9Be+ ion detected on the 25Mg+ ion with the motional modes

cooled to near the ground state ( ̄n ' 0.05). P(|↑iMg) is the probability of finding the

25Mg+ qubit in the |↑i state. In this case, both the QLS and our controlled-NOT

gate mapping procedures performed approximately equally. Error bars correspond

to standard error of the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.10 Rabi flopping of the 9Be+ ion detected on the 25Mg+ ion with the motional modes

cooled to the Doppler temperature of n ̄ ' 4. The controlled-NOT mapping technique, which makes use of the mixed-species gate, performed better than the original

QLS procedure due to the relative low sensitivity to motional excitations [Sørensen 99,

Sørensen 00]. Error bars correspond to standard error of the mean. . . . . . . . . . 121

5.11 The pulse sequence for a Ramsey experiment with a SWAP gate. We used three

controlled-NOT gates to construct a SWAP gate. The first and third controlledNOT gates were implemented with the CNOT gate (Eq. 4.9) with the 25Mg+ ion as

the target, with pulse sequence indicated in Fig. 5.7.(a). The second controlled-NOT

gate was implemented as depicted in Fig. 5.7.(b), with the 9Be+ ion as the target of

the CNOTinvert gate. The overall operation inside the red dashed box implements

the SWAP gate as shown in Eq. 5.16. . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.12 The Ramsey fringes measured on the 25Mg+ ion after applying the SWAP gate as

depicted in Fig. 5.11. This shows that the phase information is preserved between

the two species and transferred from the 9Be+ ion to the 25Mg+ ion. Error bars

correspond to standard error of the mean. . . . . . . . . . . . . . . . . . . . . . . . 125
xxix

1 5.13 Pulse sequence for creating a Bell state √ (|↑↑i − i|↓↓i) using a combination of 2

single-qubit pulses and the Cirac-Zoller gate [Cirac 95]. For high fidelity, this protocol requires the motional mode to be prepared in its ground state. The notations “BSB” and “RSB” denote blue and red-sideband transitions, respectively. The

controlled-phase gate and the controlled-NOT gate are defined in Eq. 4.6 and Eq.

4.9, respectively. The definition of the R(θ, φ) rotation is in Eq. 2.13. Except for

the “RSB to Aux” pulse, all other pulses are driven between the qubit |↑i and |↓i

states. The pulse sequence corresponding to ground state cooling of the motional

modes (applied prior to the operations shown) are not shown in this figure. . . . . . 125

5.14 Parity oscillation of the Bell state created by the Cirac-Zoller gate [Cirac 95]. The

analysis pulses were microwave-induced π/2 pulses with variable phase φ. Error bars

correspond to standard error of the mean. . . . . . . . . . . . . . . . . . . . . . . . 126

6.1 Laser beam geometry for stimulated-Raman transitions. Each of the k2a and k2b

beams when paired with the k1 beam give a wave vector difference Δk aligned along

the axial z direction, and separately excite the blue and red sideband transitions on

the axial stretch mode. All three beams are applied to implement the MølmerSørensen interaction on the second micromotion sideband. Copropagating beams

k2a and k2b are used to implement single-qubit gates described in Sec. 6.7. Details

of laser beam setup is given in Sec. 3.4.3. . . . . . . . . . . . . . . . . . . . . . . . . 131
xxx

6.2 ML-Bell-state error (red dots) as a function of gate duration tgate for a constant

Raman beam detuning Δ ' −2π × 730 GHz. The black line shows the separately

determined error and uncertainty (gray shade) due to the microwave pulses used for

|2, 2i ↔ |↑i state transfer. The three dashed lines show the sum of the expected gate

errors including photon scattering and mode frequency fluctuations (which are slow

compared to gate durations shown) for three different r.m.s. magnitudes of mode

frequency fluctuations. Errors due to these errors are discussed below. The gate

error increases quadratically with increasing tgate due to such frequency fluctuations;

however, for tgate ' 30 μs the error due to frequency fluctuations is approximately

1 × 10−4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3 ML-Bell-state error (red dots), plotted as a function of −2π/Δ for a constant gate

duration of approximately 30 μs, where Δ is detuning of the Raman laser beams

relative to the 2S1/2 ↔ 2P1/2 electronic transitions. The simulated contributions

to the Bell state error from Raman and Rayleigh scattering (for details see Sec.

4.4.2) are shown with the blue and purple dashed lines respectively. For large |Δ|,

the Raman scattering error approaches zero, however, the Rayleigh scattering error

remains approximately constant at 1.7×10−4. The black line is the sum of the Raman

and Rayleigh scattering errors, and the composite microwave pulses used for qubit

state preparation and detection (uncertainty indicated by the gray band). Error

bars for the measured Bell state fidelity are determined from parametric bootstrap

resampling [Efron 93] of the data and represent a 1-σ statistical confidence interval. 136
xxxi

6.4 The photon-counts histogram for two 9Be+ ions prepared into an equal superposition

of |↑i and |↓i states of each ion before making joint fluorescence measurements. (a)

Data is plotted in a linear scale. (b) Same data plotted in a log scale. Count

distributions corresponding to the |↑↓i or the |↓↑i states (on average 30 photon

counts) and the |↑↑i state (on average 60 photon counts) overlap significantly. These

overlapping distributions cause errors in the determination of the quantum state

using a single detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.5 Typical photon-counts histograms for two 9Be+ ions prepared in the (a.) |2, 2i

state and (b.) the “shelved” states of |1, −1i or |1, 0i state. Both distributions

exhibit non-standard photon-count distribution (e.g., the Poissonian distribution).

For (a), this is caused by depumping process due to imperfect detection laser beam

polarization. For (b), finite frequency separation between the atomic states allows

population to leak from the dark state to the bright state (via optical pumping); in

which case the ions start to fluoresce. See Fig. 2.1 for 9Be+ energy level structure;

see also Fig. 6.6 for the detection behavior of each of the Zeeman states in the

9Be+ ion’s electronic ground state. These issues render simple fitting models, e.g.

to Poissonians, incapable of extracting accurate quantum state information. . . . . 147
xxxii

6.6 Simulated detection histograms for a 9Be+ ion initialized to each of the eight Zeeman states in the 2S1/2 electronic ground state of the 9Be+ ion (see Fig. 2.1). Each

histogram is obtained with 1, 000 detection events. Except for the |2, 2i state, all

other figures are shown in log scale. Each state responds differently and uniquely

when the detection beam is applied. Besides the |2, 2i state, all other Zeeman states

are detected as dark to a high degree. Furthermore, the F = 1 manifold states

(|F = 1, mF = 0, ±1i) have smaller fluorescence compared to the F = 2 manifold

states. The bright state of the measurement qubit is the |2, 2i state (which corresponds to the qubit |↑i state), and the dark state is the |1, −1i and the |1, 0i

states (which corresponds to the qubit |↓i state). See Sec. 2.1.7 for details on the

fluorescence detection of the 9Be+ ions. The simulation here uses the saturation

parameter, s = I/Isat = 0.5 and assumes that the detection laser beam is pure σ+

with its frequency tuned on resonance with the 2S1/2|2, 2i ↔ 2P3/2|3, 3i transition.

The detection efficiency (which experimentally relates to the sum of the quantum

efficiency of the PMT and the collection efficiency of the imaging system) is set so

that the |2, 2i state fluorescence counts equal to 30 on average in our simulation. . . 153

6.7 Average fidelity for single-qubit-gate randomized benchmarking sequences, plotted

as a function of sequence length. We determine the average error per computational

gate to be gate = 3.8(1) × 10−5 and state preparation and measurement error to be

SPAM = 2.0(3) × 10−3 for these data sets. Error bars show the standard error of the

mean for each point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
xxxiii

6.8 Two-qubit gate errors (or the errors of the created Bell state) reported in the literature over time. This figure is not comprehensive but shows the best two-qubit

gate results, as well as two other type of two-qubit gate results presented in this

thesis. Gate implementations in three platforms are shown, i.e. (i) laser-driven

gates on ion qubits [Turchette 98, Sackett 00, Rowe 01, Leibfried 03, Benhelm 08,

Tan 13, Tan 15, Ballance 16, Gaebler 16], (ii) microwave-driven gates on ion qubits

[Ospelkaus 11, Weidt 16, Harty 16], and (iii) superconducting qubits [Steffen 06,

DiCarlo 09, Chow 12, Barends 14]. The highest fidelity achieved with ion qubits

using a microwave-based implementation is 0.997(1) [Harty 16] and 0.9944(5) with

superconducting qubits [Barends 14]. Data in this figures are of courtesy from Prof.

David Lucas (University of Oxford) who collected and consolidated the data. . . . . 159

7.1 The electrodes of the ion trap used for the experiment described in this chapter. The

ions are loaded in zone L and transported to zone E for the entangling gate to be

implemented. See Ref. [Jost 10] for details of the design and construction of this

trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.2 Laser beam setup to drive the two-qubit gate with the Bermudez protocol in the case

of a microwave-induced carrier excitation. Two lasers beams labeled as kCo1 and k90

are used to excite the detuned spin-motion sideband transitions. Their wave vector

difference Δk is aligned such that only the axial motional mode can be coupled. . . 164

7.3 The laser beam setup where carrier excitation is induced by stimulated-Raman process. This setup is very similar to that depicted in Fig. 7.2. Here, the microwave

antenna is turned off and an additional beam labeled as kCo2 is adjusted such that

its frequency is shifted ω0 relative to beam kCo1. Together, these two beams excite

carrier transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
xxxiv

7.4 Simulated population evolution of Hamiltonian in Eq. 7.1 with both qubits initialized

to the |↑i state. Although a maximally entangled state can be created at t = tgate,

due to the finite ratio between the carrier transition Rabi rate ΩC and the sideband

transition Rabi rate Ω = Ω1 = Ω2, the populations undergo oscillations at the

carrier Rabi frequency, making the target state highly sensitive to the interaction

time. Here, the simulation parameter is ΩC = 15Ω. . . . . . . . . . . . . . . . . . . 169

7.5 Pulse timing sequence for the microwave-induced-carrier gate. A π rotation with

a π/2 phase with respect to the previous pulse refocuses the fast spin population

oscillations induce by the carrier excitation (see Fig. 7.4). This pulse can suppress

miscalibrations and errors in the gate duration and detuning, δ [Hayes 12]. . . . . . 169

7.6 Pulse timing sequence for the laser-induced-carrier gate. The phase of the carrier

excitation is shifted by π during the second half of the gate. . . . . . . . . . . . . . 170

7.7 Evolution of the populations of |↓↓i (blue points), |↑↑i (red) and anti-aligned spin

states (green) as a function of the duration of simultaneous application of laserinduced carrier and detuned sideband excitations. The phase of the carrier is

shifted by π at half of the interrogation time for each point (see Fig. 7.6). The

gate time for this case is approximately 105 μs, at which point the Bell state

1 |Φ+i = √ (|↓↓i + |↑↑i) (in the ideal case) is created. The solid lines show the 2

results of numerical master equation simulation that include contributions from (i)

spontaneous scattering of photon induced by the Raman laser beams, and (ii) state

preparation and detection error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.8 The coherence of the state produced by the microwave-induced-carrier gate is investigated by applying an analysis π/2-pulse with a variable phase. The contrast of the

parity oscillation is determined by fitting A cos(2φ + φ0) + B to the data points. . . 173
xxxv

7.9 Master equation simulations are used to study the Bell state fidelity as a function of

carrier Rabi rate (ΩC) to sideband Rabi rate (Ω) ratio. The Hamiltonian described

in Eq. 7.1 and the motional dephasing Lindblad term described by Eq. 4.39 are

included. Four different motional dephasing rates, Γ, are considered and color coded

in this figure. Two groups of the gate implementations are shown: (i) solid lines

(marked with the asterisk symbol (∗) in the legend) indicate a spin-echo sequence

using a π-pulse rotation with π/2 phase shift as depicted in Fig. 7.5, and (ii) dashed

lines represent spin-echo sequence implemented by shifting the phase of the carrier

drive at the middle of the gate, as shown in Fig. 7.6. Each simulation begins with

the ions in the |↑↑i state and with the interactions turned on for a gate time of

tgate. The duration of the single-qubit gate for the π-pulse rotation used in spin-echo

sequence (i) is not included in tgate to provide a straightforward comparison. For (i),

δ 1 Ω is set to be and the fidelity of the created Bell state is evaluated at tgate = δ . 2

For (ii), the spin-echo sequence produces a two-loop gate. In this case, Ω is set to

δ 2 be √ and tgate = δ . A total of 11 Fock states are used in the simulation. . . . . . 176 2 2
xxxvi

8.1 (a) Illustration of a Bell inequality experiment. A source emits two systems a and

b, here two 9Be+ ions. After choosing measurement settings ak and bl, the experiment implements Hilbert-space rotation operations (which are controlled with

classical variables) on the ions respectively. Then a standard fluorescence based

measurement in a fixed basis is applied to each ion. This is equivalent to choosing

the measurement basis for the state that is present before the measurement settings

are applied. Each system’s measurement outcome is labeled B for “bright” or D

for “dark”, corresponding to the observation of fluorescence or not. From the joint

measurement we record “c = 1” if the outcomes are the same and “c = 0” if they

are not. (b) “Chaining” of the measurement settings for the Nth CBI experiment.

The measurement settings can be visualized as a chain where akbk and ak+1bk+1

are linked by akbk+1, and the chain is closed by the settings aN b1. The CHSH Bell

inequality experiment corresponds to the special case of N = 2. . . . . . . . . . . . 181

8.2 Layout of the relevant segmented trap electrodes. Each CBI experiment begins

with one ion located in zone E and the other in zone E0

. The blue dots, which

indicate the ions, are overlaid on a photograph showing the trap electrodes (gold).

By transporting the ions in and out of zone S, we individually implement settings

and measure each ion sequentially. The ions are separated by at least ∼ 340 μm

when settings akbl are applied, a distance much larger than the laser beams waist of

∼ 25 μm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
xxxvii

1 8.3 Pulse sequence for generating the entangled state |Φ+i = √ (|↑↑i + |↓↓i). The 2

notation R(θ, φ) represents the rotation with angle θ about an axis in the x-y plane

of the Bloch sphere, and φ is the azimuthal angle of the rotation axis. Operation

Rz(ξ) is the rotation with an angle ξ around the z axis of the Bloch sphere. The

angle θ is adjusted by varying the length of the laser pulse, and φ and ξ are adjusted

with the phases of the RF signal driving the AOMs that controls the laser beams.

Laser pulses in the dashed box all use the same set of laser beams, which makes

the sensitivity of the created state to slow phase drifts between the two Raman

beam paths negligible [Lee 05, Tan 15]. The two laser beams used to drive the

stimulated-Raman transitions for the pulses outside of the dashed box (labeled as

“Co Carrier”) are copropagating (the laser beams labeled as k2a and k2b in Fig. 6.1)

which eliminates phase drifts due to path length differences in the beams [Gaebler 16]

(see also Sec. 6.7). The Rz rotation is implemented by shifting the phase of the

direct digital synthesizer controlling the laser pulses that implement the subsequent

measurement settings bl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.4 Experimental sequence for one trial. The notation ELS refers to transport to place

the ion b in zone EL and ion a in zone S. Similarly for the operation SER. The

entangled state is generated as shown in Fig. 8.3 with the ions located in zone S.

Time-varying potentials are applied to control electrodes for the separation, shuttling

and recombination of the ions [Blakestad 11, Bowler 12]. The total duration of the

entire sequence shown here is approximately 3 ms. The initial optical pumping,

Doppler cooling and ground-state cooling which are applied prior to the Bell state

generation are not shown here. The total duration of one trial is approximately 8 ms. 190
xxxviii

8.5 Typical detection photon histograms that were obtained when we apply the statedependent fluorescence detection on each ion sequentially. During the state measurement of ion a (b) at zone S, ion b (a) is located in zone EL (ER) such that it

does not interact with the detection laser beam. We choose the threshold of 6 for

the differentiation of the “bright” and “dark” measurement outcomes. . . . . . . . . 191

8.6 Experimentally measured values b and I

bA IN N as a function of N. Data represented by

black and blue dots are obtained with two 9Be+ ions, with black (blue) dots corresponding to tests on |Φ+i (|Φ−i). These two data sets were obtained approximately

six months apart. The difference between them and the finer features within each

data set are probably due to miscalibrations and our inability to reproduce exact

experimental conditions. Orange dots are data from test on |Φ+i prepared on a

9Be+-25Mg+ pair. The dashed line indicates the lowest upper bound on the local

content attainable in a perfect CHSH-type experiment. . . . . . . . . . . . . . . . . . 193

9.1 Schematic showing the cross section of a potential linear Paul trap with multiple

shim electrodes. Our X-junction trap (see Sec. 3.1) consists of two pairs of RF

electrodes and two pairs of control electrodes (inside the grey dashed box), as well as

a single bias electrode. The control electrodes are segmented along the z direction.

In the proposed design, in addition to the electrodes inside the dashed box, two

segmented “shim” electrodes that run parallel to the RF and control electrodes

along the z axis will also be included. This design enables excess micromotion at

different locations along the z-axis to be better compensated simultaneously. In

addition to being important for the implementations of motional mode cooling and

fluorescence detection, better compensation of excess micromotion is particularly

crucial for implementing two-qubit entangling gates and shuttling of ions. Shuttling

operations include transporting, separating, and recombining ions. . . . . . . . . . . 199
xxxix

A.1 The electronic ground state of the 9Be+ ion at a magnetic field of 0.0119446 T,

where the transition frequency of the |F = 2, mF = 0i ↔ |1, 1i transition is firstorder insensitive to magnetic-field fluctuations. The frequency differences between

adjacent levels, and the transition frequencies of certain transitions are shown in

bold. The magnetic-field sensitivities of some relevant transitions are also shown

in lighter print. The states are designated by the states they evolve from at low

magnetic field strength where the total angular momentum F and its projection

along the quantization axis mF are both good quantum numbers. . . . . . . . . . . 224

B.1 The electronic ground state of the 25Mg+ ion at the magnetic-field-insensitive point

for the 9Be+ ion |F = 2, mF = 0i ↔ |1, 1i transition. The 25Mg+ |F = 3, mF = 1i ↔

|2, 0i transition is first-order insensitive to magnetic-field fluctuations at B ' 109.464

G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

C.1 A two-level system with a detuned interaction. . . . . . . . . . . . . . . . . . . . . . 229

C.2 A three-level system with two detuned interactions. For simplicity, we assume that

the light field labeled as ωa does not couple the |2i and |3i states, and the light field

labeled as ωb does not couple the |1i and |3i states. . . . . . . . . . . . . . . . . . . . 231
Chapter 1

Introduction

Physical systems that cannot be modeled efficiently with even the most powerful supercomputers exist in many different fields of science (e.g., quantum chemistry, condensed matter physics,

cosmology, quantum field theory, cryptography, statistical mechanics). This is because the number of resources and parameters needed to keep track of the entire system grows exponentially

with the system size. A many-body quantum mechanical system is an example of such a complex

system. Therefore, if we have access to another well-controlled quantum system, we can offset

this exponential explosion and use such a “quantum simulator” to model other complex systems

[Feynman 82].

An important general goal is to realize a universal quantum information processing (QIP)

device — the so-called “quantum computer,” which could be used for algorithms having a quantum

advantage [Shor 97, Grover 97] over processing on conventional bits as well as to simulate other

quantum systems of interest [Feynman 82, Deutsch 85, Lloyd 96]. By using a small set of universal

operations that are well controlled, a quantum computer can simulate other systems and execute

computations by simple programming of such a processor [Deutsch 85].

Many experimental platforms are currently being investigated for building a large-scale

quantum computer [Ladd 10] (e.g., photon systems, nuclear magnetic resonances, neutral atoms,

trapped-ions, nitrogen vacancy (NV) centers, quantum dots, and superconducting circuits). Each

of these physical platforms offers different advantages as well as drawbacks. For example, compared

to other systems, photons are robust against decoherence even over long distances. However, imple-
2

menting interactions between photons for deterministic entangling operations has proven challenging. A trapped-ion system currently offers the highest fidelity of universal elementary operations,

with a number of architecture proposals being pursued to tackle the enormous task of scaling up.

Solid-state based systems, such as quantum dots, NV center, and superconducting circuits, may

eventually offer better prospects to be scaled up. However, the performance of elementary operations demonstrated with these platforms are still arguably insufficient for realizing a practical QIP

device. One dominant issue is the imperfections due to the intrinsic bulk properties of the material

used for the realization of quantum bits (qubit).

In recent years the possibilities of hybrid architectures to be used for QIP and quantum

networks have been discussed in various proposals (see for example Ref. [Wallquist 09] and Ref.

[Kurizki 15]). By interfacing and integrating different platforms, unique advantages of each platform

can be exploited in a single setting and different QIP tasks can be delegated to the best suited

system. Any such hybrid implementation is an ambitious challenge. Efforts are being made to tackle

this challenge [Daniilidis 09, Kotler 16]; nevertheless, quantum information processing is still in its

infancy at its current state. Various challenges remain to be addressed within each platform before

a larger hybrid system and a working QIP device can be realized.

My PhD career has been concentrated on developing and improving the techniques for the

realization of a QIP device with trapped-ions. By applying a combination of static and radiofrequency electric fields to provide a pseudo-harmonic potential in three dimensions [Paul 90,

Wineland 98], charged ions can be trapped and stored over long periods of time. The physics

of ion trapping is well understood, and the motional degree of freedom provided by the harmonic

potential, as well as the internal (atomic) states of the ions, can be controlled with high precision

[Wineland 98].

QIP based on trapped-ion systems started in the mid-90s with the first experimental realization of a quantum logic gate [Monroe 95a] on individual qubits based on the proposal by Cirac and

Zoller [Cirac 95]. Taking advantage of the expertise and techniques developed in atomic and laser

physics, particularly those associated with the developments of atomic clocks [Chou 10, Ludlow 15],
3

QIP based on trapped-ions has since emerged as one of the most promising platforms.

Fundamental building blocks needed to build a large-scale quantum computer have been

demonstrated in trapped-ion systems. The remaining challenges of building a useful, practical

quantum computer are (i) reducing and controlling the errors of the elementary operations and

(ii) combining all elementary operations in a scalable system while maintaining the performance of

each operation. For QIP to be practical, the error per operation has to be below certain threshold

[Preskill 98, Knill 10, Ladd 10], often taken to be 10−4 probability per gate, so that error correction

schemes can be implemented [Nielsen 00]. Among all the elementary operations, the most technically challenging operation is the implementation of a deterministic quantum entangling gate —

which produces, “on demand,” an entangled state of two separate qubits. It requires high precision coherent control of the two degrees of freedom inside a trapped-ion system (i.e., the internal

atomic states degree of freedom and the external motional degree of freedom). Both are subjected

to noise, which includes environmental noise and noise present in the control fields used for the

manipulations of qubits.

A number of trapped-ion architectures for the scaling of operations have been proposed and

pursued [Wineland 98, Kielpinski 02, Monroe 14]. The traps used in this thesis work are designed to

implement the quantum charge coupled device (QCCD) architecture [Wineland 98, Kielpinski 02].

Figure 1.1 provides an illustration of the working principle of this architecture in which a large

number of ions can be trapped in a trap array with multiple trapping zones. Ions can be shuttled

between different zones by applying time-varying control potentials. Certain zones can be delegated

for certain tasks (e.g., state measurement, addressing of single-qubits, and multi-qubit entangling

gates). With this architecture, one only has to deal with a small subset of ions and operations at

a given locations, and parallel processing can be performed in different zones.

Using two different QCCD traps, this thesis focuses on the implementations of two-qubit

entangling gates based on effective spin-spin interactions driven by stimulated-Raman transitions

[Wineland 98]. Three chapters are dedicated to describing three different two-qubit entangling gate

experiments.
4

Figure 1.1: One proposal to scale up the QIP device with a trapped-ion system: a quantum charge

coupled device (QCCD) architecture [Wineland 98, Kielpinski 02] consists of multiple, interconnected trapping zones. Each trapping region could be dedicated for certain operations. This

enables parallel processing and handling of a small subset of qubits at any trapping location. The

information transport is achieved by physically shuttling the ions between different trapping regions

by applying well-controlled, time-varying electric fields.
5

Chapter 5 describes the implementation of a mixed-species entangling gate, demonstrated

with a hybrid system consisting of a 9Be+ and a 25Mg+ ion. In trapped-ion QIP, a hybrid architecture can be implemented by using different species of ions; this enables extra degrees of

freedom to be introduced that can be exploited to expand and refine the control of the system.

Trapped ions of different elements vary in mass, internal atomic structure, and spectral properties,

features that make certain species suited for particular tasks such as storing quantum information, high-fidelity readout, fast logic gates, or interfacing between local processors and photon

interconnects. Ions of different elements have previously been used in QIP experiments for sympathetic cooling [Barrett 03], creation of entanglement through dissipation [Lin 13a], and quantum

non-demolition (QND) measurement of one species by using another [Hume 07]. As a multi-qubit

quantum logic gate is one of the most important elementary operations for QIP, adding a mixedspecies entangling gate between multiple species of ions provides one important step toward a

large-scale trapped-ion QIP device. Furthermore, a mixed-species entangling gate can also potentially serve as an important building block for quantum networking [Moehring 07], precision

spectroscopy [Schmidt 05], metrology [Schulte 16], and quantum simulation. The mixed-species

entangling gate is realized through an effective spin-spin interaction generated by state-dependent

forces [Sørensen 99, Milburn 00, Solano 99, Leibfried 03] using the Mølmer-Sørensen (MS) protocol

[Sørensen 99, Sørensen 00]. The fidelity of creating a mixed-species Bell state with our gate scheme

was evaluated to be 0.979(1) [Tan 15]. Error sources are investigated to study the possibilities of

improving the gate performance.

Operations in the high-fidelity regime are important for the realization of a practical quantum

information processor. Individual quantum gate errors must be reduced below a certain threshold

to achieve fault-tolerance [Preskill 98, Knill 10, Ladd 10] without excessive overhead in the number

of physical qubits. In the trapped-ion QIP community, this error threshold is generally agreed to be

around 10−4. This level has been achieved in some experiments for elementary operations including

single-qubit gates [Brown 11, Harty 14, Ballance 16, Gaebler 16] and state preparation and readout

[Myerson 08], with the exception of two-qubit entangling gates, emphasizing the importance of
6

improving the fidelity of multi-qubit gates.

Chapter 6 describes experiments that push the fidelity of laser-induced gates, especially

a two-qubit entangling gate, toward the fault-tolerant goal. With a “clock” qubit composed of

hyperfine states in a 9Be+ ion, which has a long coherence time (> 1 s) due to its robustness

against fluctuating magnetic fields [Langer 05], a state-dependent force based on the MS protocol

is applied directly to the qubits to implement an entangling geometric phase gate. The error of the

Bell state created by the entangling gate scheme is measured to be 8(4) × 10−4 [Gaebler 16], this

result is currently the highest two-qubit gate fidelity across all platforms. This result is about a

factor of ∼ 50 improvement compared to the previous best result of an entangling gate applied on

a magnetic-field insensitive qubit [Tan 13] (see also Chapter 7). This improvement is partly made

by reducing various noise sources and taking advantage of a number of recent developments in laser

technology [Wilson 11, Colombe 14] and digital electronics [Bowler 13, Leibrandt 15].

Similar two-qubit gate performance has also been demonstrated by the Oxford trapped-ion

group [Ballance 16]. In this case, the gate is applied on a magnetic-field sensitive qubit state using

a different implementation of a geometric phase gate [Leibfried 03]. The Oxford group is working

toward implementing the Mølmer-Sørensen protocol on a clock qubit so that the qubit dephasing

error can be suppressed (private communication). Chapter 4 gives a brief overview of different

two-qubit gate schemes and discusses their respective advantages and disadvantages.

Section 6.7 describes characterization of a single-qubit gate using a randomized benchmarking

technique [Knill 08]. Single-qubit gates are implemented with the same apparatus and errors are

evaluated to be 3.8(1) × 10−5 [Gaebler 16], this error is lower than the 10−4 fault-tolerant threshold. This result appears to be one of the highest single-qubit gate fidelity based on laser-driven

stimulated-Raman transitions [Ballance 16].

One of the most important aspects of implementing high-fidelity gates is the study of error

mechanisms. Here, these errors are investigated with a combination of analytic methods, calibration

measurements, and numerical simulations. This allows for better strategies and novel gate schemes

to be developed that target certain error mechanisms for the further improvement of quantum
7

gates.

As qubit dephasing errors are one of the most important error sources, certain techniques

and gate schemes targeting this error have been proposed in the recent years. One such example

is the technique of dynamical decoupling (DD) [Viola 98]; based on an extension of spin-echos

[Hahn 50]. Dynamical decoupling relies on applying certain operations to the quantum system

of interest that provides protection against environment noise such as magnetic field fluctuations.

Most implementations of DD schemes have been focused on the protection of quantum memories

against external noise [Du 09, de Lange 10, Ryan 10, Szwer 11]. However, as most quantum gates

require the application of external controlled fields, a decoupling scheme that is compatible with

quantum gate is a non-trivial challenge. These DD-compatible quantum gates should be designed to

be immune to fields associated with environmental noise but still able to be implement interactions

induced by the controlled fields required to apply the coherent gate operations.

Chapter 7 describes an experimental demonstration of an entangling gate based on statedependent forces with built-in continuous dynamical decoupling. This implementation is a modified

scheme based on the theoretical proposal presented by Bermudez et al. [Bermudez 12]. By applying a strong carrier excitation that continuously flips the qubit states, the qubit is dynamically

decoupled from certain qubit dephasing mechanisms such as external magnetic field fluctuations,

fluctuating AC Stark shifts, or AC Zeeman shifts. Simultaneously, a spin-motional “sideband” excitation is applied. In the limit that the carrier Rabi rate is significantly greater than the sideband

Rabi rate, the combined interaction is a state-dependent force that can be engineered to implement a geometric phase gate. This protocol shares certain similarities with the Mølmer-Sørensen

protocol in the sense that they can both be directly applied to a clock state qubit. In addition

to the built-in feature of dynamical decoupling, this gate scheme has certain further advantages

compared to the Mølmer-Sørensen protocol, which are discussed in Chapter 4. With this DD gate

scheme, a gate fidelity of 0.974(4) was achieved. Gate errors were dominated by sources associated

with a legacy experimental setup used for this demonstration, which can be suppressed in the futre.

Together with other quantum entanglement generation schemes, it provides an additional tool kit
8

to the study of quantum physics.

The properties of quantum entanglement are at the heart of the promising power of QIP.

The peculiar properties of quantum entanglement’s spukhafte Fernwirkungen, which is translated as

“spooky action at a distance” [Einstein 71], prompted heated philosophical debates and motivated

various interpretations of quantum physics. Spooky action at a distance refers to the counterintuitive implication that quantum entanglement seems to allow faster-than-light communication

between the partners of the entangled state. This is in direct conflict with local-realistic theory, as

well as human beings’ experiences with the surrounding physical world.

John Bell formulated a relatively simple mathematical inequality describing the incompatibility between quantum physics and local realistic theories [Bell 64]. This was followed by proposals

enabling the experimental examination of Bell inequality, notably the one proposed by Clauser,

Horne, Shimony, and Holt (CHSH) [Clauser 69], which is performed by making joint measurements

on a pair of entangled particles to draw inferences about the nonlocal nature of quantum physics.

Since the pioneering work in the 70s performed with photon pairs [Freedman 72] and a number

of experiments using various platforms in the following decades made possible by technological

advancements, the quest for confirming the non-local nature of quantum mechanics reached an

important milestone with three different “loophole-free” CHSH-type Bell inequality experiments

performed in 2015 [Hensen 15, Shalm 15, Giustina 15]. See Ref. [Brunner 14] for a general review

on Bell nonlocality.

However, the quest to fully understand local/non-local properties of quantum physics is far

from finished. Although the loophole-free Bell inequality experiments have rejected with high

confidence theories of local realism, they are limited in the extent to which their data differs from

local realism. In general, a CHSH-type Bell inequality experiment is limited in the extent to which

the experimental data may assert nonlocality exhibited by a quantum state. Therefore, one of

the next quests in the study of the foundation of quantum physics is to experimentally show that

quantum state can exhibit maximum nonlocal properties. One such experiment is the chained

Bell inequality (CBI) [Pearle 70] test, which can be used to further quantify and assert a larger
9

departure of quantum physics from local realism.

Elitzur, Popescu, and Rohrlich [Elitzur 92] described a model of the distribution of outcomes

measured from a quantum state as a mixture of a local-realistic distribution, which obeys Bell’s

inequalities, and another distribution which does not. A perfectly executed Bell inequality experiment of the CHSH type, with two particles and two measurement settings per particle, could

only show with a maximally entangled state exhibits a nonlocal properties with pnonlocal ' 0.414

(maximum nonlocality corresponds to pnonlocal = 1). A CBI experiment can be viewed as a generalized version of the CHSH-type Bell inequality test. There is a hierarchy in which the Nth

CBI experiment involves 2N different combinations of measurement settings. The N = 2 CBI

experiment is equivalent to the CHSH Bell inequality experiment. In the limit of N → ∞ and with

perfect experimental conditions, CBI experiments can be used to show a quantum state’s complete

departure from local realism [Barrett 06, Bierhorst 16].

Similar to a CHSH-type experiment, a CBI experiment may be subject to “loopholes”

[Brunner 14,  ̊A. Larsson 14] that, in principle, allow a local system to show violation of the inequality. Previous CBI experiments [Pomarico 11, Aolita 12, Stuart 12, Christensen 15] employed entangled photon pairs with the highest nonlocal properties shown to be pnonlocal ' 0.874 [Christensen 15].

However, the detectors’ efficiencies in these experiments are low, such that the fair sampling assumption has to be made, thus not closing the so-called detection loophole [Pearle 70, Clauser 74].

To my knowledge, all previous CBI experiments with N ≥ 3 suffer from the locality [Bell 85],

detection, and memory loopholes [Barrett 02].

The detection efficiency of a trapped-ion system can be near 100 %, making it an excellent

platform to perform Bell inequality experiments while closing the detection loophole. The detection

loophole was first closed in a CHSH-type Bell inequality experiment performed with a trapped-ion

system [Rowe 01]. Other CHSH Bell inequality experiments performed with trapped-ion systems

include Ref. [Matsukevich 08, Pironio 10, Ballance 15, Tan 15], with each of them successfully

closing the detection loophole.

Chapter 8 presents detection-loophole-closing CBI experiments that are realized by employ-
10

ing techniques developed for trapped-ion QIP, particularly the high-fidelity deterministic generation of Bell states with an entangling gate and single-qubit gate, robust state preparation and

measurement, long qubit memory storage time, and shuttling and transport of ions for the separate

addressing and measurement of individual ions.

The CBI experiments presented here employ up to 15 different measurement settings per

particle. The highest nonlocal fraction attainable by a quantum distribution is determined to be

pnonlocal ' 0.704(12) for the N = 9 CBI experiment. Although the experiments here close the

detection loophole, the CBI experiments are performed with each ion’s measurement inside the

lightcone of the event where the other ion’s measurement setting choice is made, thus not able to

close the other important loophole, i.e. locality or lightcone loophole.

Historically, a Bell inequality experiment is primarily designed and performed to investigate the incompatibility between quantum physics and local-realistic theory. The possibilities of using a Bell inequality experiment for certain applications had also been discussed. For

example, while the Bell inequality experiments presented here are made possible by the technique developed for QIP with trapped-ions, the experimental protocols can in turn provide a

“black box” certification of the created Bell state [Bardyn 09, Bancal 15, Kaniewski 16] as well

as a device-independent, single-number, conservative benchmark for the QIP device. Such a

characterization with minimal assumptions on the physical system and measurements is formalized by the self-testing framework [Mayers 04, McKague 12], which enables an experimentalist

to quantify the quality of an entangled state given the amount of violation of a Bell inequality

[Bardyn 09, Yang 14, Bancal 15, Kaniewski 16].

Our N = 2 measurements corresponds to a CHSH inequality parameter (sum of correlations)

of BCHSH = 2.80(2). According to Kaniewski’s formulation [Kaniewski 16], the self-tested Bell

state fidelity is inferred to be 0.980, with lower bounds of ∼ 0.958 at the 95% confidence level. This

result appears to be the highest self-tested fidelity on a deterministically created Bell state across

all platforms that are being considered for QIP, demonstrating a trapped-ion system’s capabilities

for the realization of a QIP device.
11

During the course of my PhD, I have worked with two trapped-ion apparatuses in two different

laboratories, chronologically. The first apparatus, which is referred in this thesis as the “legacy

system,” was retired in the spring of 2013. The second system was set up in a laboratory inside

the Katherine Gabbie building at National Institute of Standards and Technology, and became

operational in the fall of 2013. Although the two systems share many similarities, unless specified

otherwise, this thesis refers to the newer system.

The three two-qubit entangling gate experiments described in this thesis resulted in the

following peer-reviewed publications:

(i) “Multi-element logic gates for trapped-ion qubits”

T. R. Tan, J. P. Gaebler, Y. Lin, Y. Wan, R. Bowler, D. Leibfried, and D. J. Wineland,

Nature 528, 380 (2015).

(ii) “High-fidelity universal gate set for 9Be+ ion qubits”

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. Keith, K. Coakley, E. Knill, D.

Leibfried, and D. J. Wineland, Phys. Rev. Lett. 117, 060505 (2016).

(iii) “Demonstration of a dressed-state phase gate for trapped ions”

T. R. Tan, J. P. Gaebler, R. Bowler, Y. Lin, J. D. Jost, D. Leibfried, and D. J. Wineland,

Phys. Rev. Lett. 110, 263002 (2013).

The manuscript for the chained Bell inequality experiment is being prepared at the moment.

(iv) “Chained Bell inequality experiment with high-efficiency measurement”

T. R. Tan, Y. Wan, S. Erickson, P. Bierhorst, D. Kienzler, S. Glancy, E. Knill, D. Leibfried,

and D. J. Wineland, in preparation.

I was primarily in charge of the planning, designing, and assembling the infrastructure of the

new laboratory, as well as heavily involved in a number of other experiments using the apparatus

described in this thesis. These other experiments were led by J. P. Gaebler, R. Bowler, Y. Lin, and

Y. Wan, some of which resulted in the following publications:
12

(i) “Randomized-benchmarking of multi-qubit gates”

J. P. Gaebler, A. M. Meier, T. R. Tan, R. Bowler, Y. Lin, D. Hanneke, J. D. Jost, J. P.

Home, E. Knill, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 108, 260503 (2012).

(ii) “Coherent diabatic ion transport and separation in a multizone trap array”

R. Bowler, J. P. Gaebler, Y. Lin, T. R. Tan, D. Hanneke, J. D. Jost, J. P. Home, D.

Leibfried, and D. J. Wineland, Phys. Rev. Lett. 109, 080502 (2012).

(iii) “Sympathetic electromagnetically-induced-transparency laser cooling of motional modes in

an ion chain”

Y. Lin, J. P. Gaebler, T. R. Tan, R. Bowler, J. D. Jost, D. Leibfried, and D. J. Wineland,

Phys. Rev. Lett. 110, 153002 (2013).

(iv) “Dissipative production of a maximally entangled steady state of two quantum bits”

Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. Sørensen, D. Leibfried, and

D. J. Wineland, Nature 504, 415 (2013).

(v) “Preparation of entangled states by Hilbert space engineering”

Y. Lin, J. P Gaebler, F. Reiter, T. R. Tan, R. Bowler, Y. Wan, A Keith, E. Knill, S.

Glancy, K. Coakley, A. S. Sørensen, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett.

117, 140502 (2016).

1.1 Thesis Organization

 9 + 25 + Chapter 2 describes the use of Be and Mg ions as qubits, including the qubit state

preparation, measurement and coherent manipulations of the qubits. The apparatus relevant to

this thesis is summarized in Chapter 3. This includes the ion trap, magnetic-field coil, laser systems,

imaging system, and certain electronics.

Chapter 4 gives a brief overview of two-qubit entangling gates with trapped-ions. Different gate schemes will be discussed, and the relevant error mechanisms that prevent the perfect

implementations of entangling gates are summarized. The objective of this chapter is to provide
13

background information for the two-qubit entangling gate experiments described in Chapters 5, 6,

and 7.

Chapters 5 and 6 describe experimental implementations of a mixed-species entangling gate

and a universal gate set with 9Be+ ions, respectively. This is followed by Chapter 7, which describes

the demonstration of a geometric phase gate with dynamic decoupling built-in. The chained Bell

inequality experiment is presented in Chapter 8. Chapter 9 discusses prospective outlooks of the

experiments presented in this thesis.

Appendix A and B include the detailed hyperfine energy level diagrams of the 2S1/2 electronic

ground states of the 9Be+ ion and the 25Mg+ ion, respectively. Certain atomic properties and their

respective magnetic-field independent points of these species are also included. Appendix C includes

Hamiltonians describing the atom-light interactions frequently used throughout the thesis. Lastly,

Appendix D gives the matrix representations of state vectors and operators used in the numerical

simulations of quantum systems.
Chapter 2

9Be+ and 25Mg+ Ion Qubits

The 9Be+ ion, and two isotopes of magnesium ions (24Mg+ and 25Mg+) are used in this thesis

work. The 24Mg+ isotope was used in a legacy system to take advantage of its relatively simple

electronic structure. With the absence of nuclear spin, and thus the absence of hyperfine structure,

the laser system required to manipulate the 24Mg+ isotope is relatively simple and straightforward.

The 25Mg+ isotope was then used after the legacy system was retired. With a nuclear spin of

5/2, which gives rise to hyperfine structure, the 25Mg+ isotope is more favorable to be used as a

quantum bit (qubit) in terms of memory storage time, qubit state readout fidelity, and application

of quantum gates. However, the laser system required to fully manipulate the 25Mg+ isotope is

relatively more demanding than for 24Mg+.

The electronic structures of 9Be+ and 25Mg+ are qualitatively very similar. The wavelength

of the lasers to manipulate 9Be+ are all centered around 313 nm, while the laser wavelengths for

the 25Mg+ are centered around 280 nm. This chapter describes the electronic structure of 9Be+

and 25Mg+, the qubit preparation and readout, and the coherent operations for the manipulation

of these ions. Detailed information about the 24Mg+ isotope can be found in Ref. [Jost 10].

2.1 9Be+ Ion Qubit

The only stable isotope of beryllium is 9Be. Being an alkaline earth atom, the neutral Be

atom has two valence electrons. By stripping off one of the outer electrons, the 9Be+ ion’s electronic

structure is very similar to alkali atoms. The electronic ground state is the 1s22s2SJ=1/2 state. The
15

two electronic excited states relevant to this thesis are the 1s22p2P1/2 state and the 1s22p2P3/2 state.

The electronic transitions from the 2S1/2 state to either excited state have wavelengths around 313

nm. The fine structure splitting between the two excited states is approximately 198 GHz.

The nuclear spin, I of the 9Be+ ion is 3/2. The interactions between the nuclear and electronic spin give rise to hyperfine structure. The hyperfine levels in the electronic ground state of

2S1/2 are labeled with the total angular momentum, F. The hyperfine splitting between the F = 1

and F = 2 levels is approximately 1.2 GHz. We apply an external magnetic field of ∼ 119.446

G (0.0119446 T) to further lift the degeneracy of different Zeeman levels. This magnetic field is

chosen such that the |F = 2, mF = 0i ↔ |1, 1i transition is first-order insensitive to magnetic field

fluctuations [Langer 05]. Furthermore, at this “intermediate magnetic field” regime, the combination of hyperfine and Zeeman effects is such that the transition frequencies of all transitions in the

2S1/2 manifold are nondegenerate. Figure 2.1 shows the relevant energy levels of the 9Be+ ion. See

Fig. A.1 for additional details of the 9Be+ electronic ground state.

For the quantum information experiments described in this thesis, the qubit is stored in the

magnetic-field insensitive states labeled as |2, 0i = |↑i and |1, 1i = |↓i. The coherence time of this

qubit is investigated with a Ramsey sequence and the results are shown in Fig. 2.2. Due to the

complicated mechanisms of dephasing sources, which can be correlated, the decay of the Ramsey

contrast does not exhibit simple exponential behavior. Therefore, we choose the delay time in which

the Ramsey contrast equal to 1 − 1/e ' 0.632 as the representative qubit coherence time. With

this convention, the coherence time of the 9Be+ qubit is measured to be ∼ 1.5 s. The second-order

magnetic field sensitivity of the qubit transition is approximately 6.097 kHz/G2.

Relevant atomic properties of the 9Be+ ion are shown in Table A.1. Table A.2 lists the

magnetic-field insensitive transitions in the electronic ground state of the 9Be+ ion.

2.1.1 Doppler Cooling

Doppler cooling of 9Be+ ions is achieved with a σ+ polarized laser beam with frequency red

detuned by approximately Γ/2 from the 2S1/2|2, 2i ↔ 2P3/2|3, 3i transition at ∼ 313.132 nm (see
16

| ↓ ⟩|22, 00⟩

|22, 11⟩ |FF = 22, mm ⟩ FF = 22

Detection and 

Doppler cooling

313.132 nm

|11, −11⟩

|22, −11⟩

|22, −22⟩

State initialization

Shelving

2SS1/2

2PP1/2

2PP3/2

σσ+

σσ+

σσ+

|33, 33 ⟩ Fine structure splitting

≃ 198 GHz

B field ≃ 0.0119446 T

(119.446 G)

|22, 22 ⟩

Qubit frequency

≃ 1207.496 MHz

|11, 00⟩ | ↑ ⟩|11, 11⟩

Raman detuning

≃ 500 GHz Repump

313.196 nm

“RD red”

“RD blue”

“BD”

Figure 2.1: Relevant energy level structure (not to scale) of the 9Be+ ion. Transitions to the

electronic excited states are used for Doppler cooling, repumping, and qubit state measurement

as described in the text. The “blue Doppler detuned” (BDD) beam, which is red shifted ' 400

MHz with respect to the “blue Doppler” (BD) beam (see Sec. 2.1.1), is not shown in this figure.

We label the qubit |1, 1i = |↑i and the |2, 0i = |↓i states as the “computational qubit manifold”.

The “measurement qubit manifold” consists of the |2, 2i as the “bright” state, and the |1, −1i or

|1, 0i state as the “dark” state (see Sec. 2.1.7). The laser for stimulated-Raman transitions is red

detuned from the 2S1/2 ↔ 2P1/2 transition. The decay rate of the 2P3/2 state is 2π × 19.4(5) MHz

[Poulsen 75]. See Fig. A.1 for details energy levels diagram in the 9Be+ electronic ground state.

The state labels correspond to the states at low magnetic field, which evolve to the states indicated

at non-zero magnetic field.
17

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

Ramsey time (s)

Ramsey Contrast

Without 25Mg+ Laser

With 25Mg+ Laser

Figure 2.2: Coherence time of the 9Be+ qubit is investigated with a Ramsey sequence. We repeat

the experiments with and without the presence of the 25Mg+ laser beams, which have a wavelength

of approximately 280 nm. In this experiment, one 9Be+ and one 25Mg+ ion are trapped together in a

harmonic well with a separation of ' 5 μm; the laser beams (with beam waists ' 30 μm.) illuminate

both ions. We do not observe a significant difference between these two sets of experiments, which

indicates that the 25Mg+ laser radiation causes negligible additional decoherence.
18

Fig. 2.1), where Γ ' 2π×19.4(5) MHz is the decay rate of the 2P3/2 state [Poulsen 75]. This beam,

which is labeled as the “blue Doppler” (BD) beam, has a k-vector direction such that it can cool

all modes of the ions’ motion. The BD beam is also used for the fluorescence detection of the 9Be+

ions (see Sec. 2.1.7).

To maximize efficiency during Doppler cooling and detection in the presence of axial micromotion (see Sec. 3.1 and Ref. [Blakestad 10]), we apply a differential voltage of approximately

± 0.15 V to the two control electrodes centered on zone E (Fig. 3.1). This shifts the ions away

from the radial micromotion null point (trap axis) such that the vector sum of the radial and axial

micromotions is perpendicular to the Doppler cooling beam’s wavevector. With this Doppler cooling scheme, the lowest motional temperature is attainable when the heating and cooling process

reach an equilibrium [Itano 82]. Heating mechanisms include recoil momentum kicks during photon

absorption and emission, and background heating [Turchette 00a].

In the weak binding regime, where Γ  ων, with ων as the frequency of the motional mode,

the temperature at the Doppler cooling limit, TD is given by [Wineland 79, Itano 82]

  ~Γ Γ 2δ TD = (1 + χ) (1 + s) + , (2.1) 8kB 2δ Γ

where kB is the Boltzmann constant, s and δ are the saturation parameter and the detuning of

the laser beam associated with the cooling transition, and χ is a geometric factor which is related

to the beam direction with respect to the motional modes’ vibrational directions. In our case, the

polarization of the laser beam is set to be σ+ and with the alignment shown in Fig. 2.3, χ = 0.35

for the axial modes, and ranges from 0.3 to 0.35 for radial modes [Itano 82].

For a single 9Be+ ion with an axial trapping frequency near 2π × 3.6 MHz, we measure

the average axial occupation number after Doppler cooling to be approximately n ̄ ' 4. Our

Raman laser beam geometry does not permit direct diagnostics of the motional modes in the radial

directions (see Sec. 2.1.3 and Sec. 2.1.5). To estimate the Doppler cooling limit for the 9Be+ ion’s

radial motional modes, we first note that the radial trap frequency for one 9Be+ is relatively high,

such that the weak binding condition is not rigorously satisfied. Using Eq. 2.1 in this case could
19

9Be+ Co beam

(π, σ+, σ-)

9Be+ 90 beam

(σ+, σ-)

• 9Be+ Doppler beam (BD)

• 9Be+ far detuned Doppler beam (BDD)

• 9Be+ repump beams

• 25Mg+ Doppler beam (BD)

• 25Mg+ far detuned Doppler beam (BDD)

• 25Mg+ repump beam

• 9Be+ Photo-ionizing laser (9Be+ PI)

*All beams besides PI beams are σ+ polarized

Imaging System

RF 

resonator

B

Axial direction �zz

25Mg+ Photo-Ionizing laser

To PMT 

or camera

25Mg+ 90 beam

(π, σ+, σ-)

Dichroic mirror

25Mg+ Co beam

(σ+, σ-)

Dichroic mirror

Figure 2.3: Schematic diagram indicating laser beams access into the ion trap apparatus. Four

ports are available, each aligned ∼ 45◦ relative to the trap (z) axis. The external magnetic field

(provided by coils in the Helmholtz-like configuration) is set to be 45◦ with respect to the trap

axis. The radial confinement of the ions is in the plane normal to the z axis. Laser beams for the

Doppler cooling, repumping, and fluorescence detection of the 9Be+ and 25Mg+ ions are coaligned

with the magnetic field direction, and are set to be σ+ polarized. With this configuration, motional

modes in all three directions can be cooled by the Doppler beams. Each species Raman laser beams

are sent into the vacuum chamber via two ports after being combined with dichroic mirrors. For

9Be+, the laser beam exiting the UV fiber of path 1 (2) depicted in Fig. 3.8 is sent into the port

labeled as 9Be+ “ 90” (9Be+ Co) in this figure. For the 25Mg+, the laser beams labeled as path 1

(2) in Fig. 3.10 corresponds to “25Mg+ 90” (“25Mg+ Co”) in this figure. The laser beam entering

the bottom-right port (i.e., the Mg PI beam in this figure) and directed toward zone L is blocked

by an oven shield installed behind the alumina wafers after passing through the center of the trap

(details in Ref. [Blakestad 10]). The boundary of the vacuum chamber is indicated by blue lines.

Details of apparatus are described in Chap. 3.
20

overestimate the cooling efficiency and erroneously give a lower motional temperature. Doppler

cooling in the intermediate binding regime can be estimated by [Lindberg 84, Javanainen 84]

~ 1 TInter = ×

4δkB ων

2 (δ2 + γ2 + 6κ2) + 4γ2 (δ2 + γ2 + 2κ2)

h   � 2 � 2 ξ ω2 δ2 + 5γ2 + 4κ2 − ω2 + 4γ2 δ2 + γ2 + 2κ2 − 2ω2 ν ν ν

�  �  +ω2 δ2 + γ2 + 2κ2 δ2 + 5γ2 + 8κ2 + ω2 ν ν

  i � 2 � +4γ2 δ2 + γ2 + 2κ2 + 2κ2 δ2 − 3γ2 + 8κ2 − 3ω2 , (2.2) ν

where γ = Γ/2 and κ is the Rabi rate of the laser-induced electronic cooling transition.

Another laser beam, labeled as “blue Doppler detuned” (BDD), is derived from the same

laser source as the BD beam but with its frequency red shifted by ' 400 MHz from the BD

beam (details in 3.4). The power of the BDD beam is ∼ 1 mW, which power broadens the

transition linewidth when the transition is shifted into resonance. This beam provides strong

cooling for high temperature ions that might not be efficiently cooled by the BD beam. These

occasional high temperature events are likely due to background gas collisions which occur at a

rate of approximately one per minute. It was found that the dark ion (which does not fluoresce

under the illumination of either the BD or the BDD beams) creation rate becomes faster when we

increase the duty cycle of the BDD beam, likely due to BeH+ formation when the 9Be+ ion is in

the electronic excited state [Wineland 98]. The BDD beam also provides optical pumping to assist

initialize the population to the |2, 2i state.

2.1.2 Repumping

The 2S1/2 ↔ 2P3/2 (BD) Doppler cooling and detection laser beam will optically pump the

9Be+ ions to the 2S1/2|2, 2i state as long as the beam has a pure σ+ polarization with respect to the

applied magnetic field. To mitigate the effects of polarization impurity and to speed up the pumping

process, two 2S1/2 ↔ 2P1/2 laser beams are added for the initial optical pumping to the 2S1/2|2, 2i

state. All three beams are first applied, and the final stage of pumping uses only the 2S1/2 ↔ 2P1/2

beams. These beams are formed from one beam that is split into two, which are also σ+ polarized
21

(details in Sec. 3.4). One beam is tuned near the 2S1/2|2, 1i ↔ 2P1/2|2, 2i transition, and the other

is tuned near the 2S1/2|1, 1i ↔ 2P1/2|2, 2i transition (see Fig. 2.1). These two beams are labeled as

the “RD red” beam and “RD blue” beam, respectively. To suppress electromagnetically-induced

transparency effects that could lead to coherent trapping of populations in the 2S1/2|2, 1i and the

2S1/2|1, 1i states when these two beams are applied simultaneously, one of these beams is detuned

from the atomic resonance by approximately −Γ/2.

These beams also serve to repump the population to the 2S1/2|2, 2i state during Raman

sideband cooling [Monroe 95b] (see Sec. 2.1.5). During the repumping process in the sideband

cooling sequence, we do not apply the differential voltage on control electrodes that maximizes the

detection efficiency in the presence of axial micromotion. This is to minimize motional heating that

might occur when transverse micromotion is induced.

2.1.3 Stimulated-Raman Transitions

Coherent manipulation of qubits are accomplished with stimulated-Raman transitions [Wineland 98]

and microwave transitions. Although there are multiple energy levels present in the ions’ atomic

structures (Fig. 2.1 and Fig. 2.6), we can restrict the coherent operations such that, to a high

degree, they interact only with the two energy levels corresponding to the qubit |↑i and |↓i states.

Another degree of freedom which is present in the trapped-ion system is the motion, which for each

normal mode, can be described as a harmonic oscillator. The unperturbed Hamiltonian describing

a single two-level ion in a harmonic well can be written

X 

3

H0 = ~

ω0 (|↑ih↑| − |↓ih↓|) + ~ων,iaˆ†

aˆi, (2.3) i 2

i

where ω0 is the qubit frequency, and the second term represents the three orthogonal motional

modes, each with a harmonic oscillator frequency of ων,i, and aˆi represents the harmonic oscillator

annihilation operator for the ith mode (see also Appendix C).

The laser beams for qubit manipulation of 9Be+ ions with stimulated-Raman transitions

(with wavelength λ ' 313 nm) are red detuned from the 2S1/2 ↔ 2P1/2 electronic transitions. To
22

kk22 kk11

Axial direction ̂zz

ωω2 ωω1

B field ≃ 0.0119 T 

ΔΔkk = kk11-kk22

25Mg 9Be+ +

(π, σ+, σ-) (σ+, σ-

)

Figure 2.4: The laser beams setup to drive stimulated-Raman transitions. Laser beams are focused

onto ions from two paths, which intersect at 90◦ such that the difference in their k vectors, Δk, is

aligned along the axial direction. These two beams corresponds to the Co and 90 beams in Fig. 2.3.

This figure is simplified for the relevant illustrations in this section. Details of Raman beam lines

setup are discussed in Sec. 3.4.3, Sec. 3.5.2, and the respective chapters describing implementation

of different experiments.
23

drive spin-motion transitions with stimulated-Raman excitations, two beams propagate along paths

1 and 2 (see Fig. 2.4) and are focused and intersect at 90◦ such that the difference in their k vectors,

Δk, is aligned along the axial (z) direction, in which case only the axial motion can be coupled

with the Raman laser fields at the position of the ion. If we set the frequency difference of the two

laser fields (Δω = ω1 − ω2) near to the qubit frequency of ω0, we can excite the carrier transitions

|↓, ni ↔ |↑, ni that induce spin-flip oscillations without changing the motional Fock state quantum

number n. Following the derivations in Appendix C, the interaction-frame Hamiltonian describing

a detuned carrier excitation after dropping high-frequency terms has the form of Eq. C.20:

  H i(δt+Δφ) Carrier = ~Ω0 σˆ+e−i(δt+Δφ) + ˆσ−e , (2.4)

σ+

conjugate is σˆ− = |↓ih↑|, δ = ω1−ω2−ω0 and Δφ are the detuning from the transition frequency and

the phase difference between the two Raman laser fields, respectively. Resonant carrier excitations

can be driven by setting δ = 0.

If we tune the relative frequency difference of the two laser fields, Δω, to be the ω0+ωz where

ωz is the frequency of the normal mode along the z direction, we can drive blue-sideband transitions

|↓, ni → |↑, n + 1i that drives |↓i to |↑i while adding a quantum of motion to the motional harmonic

oscillator. After neglecting high-frequency terms, the blue-sideband excitation in the interaction

frame of both the qubit and motion can be described by the Hamiltonian:

where Ω0 is the transition Rabi rate which can be computed with Eq. C.11, ˆ = |↑ih↓| and its

  H † i(δt+Δφ) BSB = ~ΩBSB σˆ+aˆ e−i(δt+Δφ) + ˆσ−aˆe . (2.5)

Likewise, we can also drive a red-sideband transition |↓, ni → |↑, n − 1i, described by the Hamiltonian

  H † i(δt+Δφ) RSB = ~ΩRSB σˆ+aˆe−i(δt+Δφ) + ˆσ−aˆ e . (2.6)

The strength of the spin-motion coupling, and the sideband transitions Rabi rate are parametrized

with the Lamb-Dicke parameter η such that ΩBSB and ΩRSB ' ηΩ0 (see Appendix C). For a single

ion,

η = |Δk|z0, (2.7)
24

p where z0 = ~/(2mωz), is the zero-point amplitude of the ion’s motion, with m as the mass of the

ion. For a single 9Be+ ion, η ' 0.35 for our laser beam setup.

Due to inherent micromotion along the axial direction (see Sec. 3.1 and Ref. [Blakestad 10]),

the actual carrier and spin-motion sideband Rabi rates are reduced for the laser beam geometry

depicted in Fig. 2.4. For our typical applied RF trap potentials, the modulation index due to the

micromotion-induced Doppler shift is approximately 2.9 along the z axis for 9Be+ ion such that

the strongest Raman-laser-induced interaction is provided by the second micromotion sideband.

This is a factor of J2(2.9) ' 0.48 smaller compared to a carrier in the absence of micromotion.

Because of this, interactions induced by this laser beam geometry (Fig. 2.4) are driven on the

second micromotion sidebands. We tune the frequency difference of the Raman beams to be near

ω + 2ωRF or ω − 2ωRF , where ω is the spin-motion transition frequency of interest, and ωRF is the

radio frequency for the trapping of ions. We drive the |2, 2i ↔ |1, 1i (sideband cooling) transitions

on the +2nd-order micromotion sideband, and the spin-motion qubit transition |1, 1i ↔ |2, 0i

transitions on the −2nd-order micromotion sideband.

When the spin-motion transitions are applied to multiple ions at the same time, the motional

mode amplitudes of each ion for a given normal mode have to be taken into account. For example,

for a blue-sideband excitation applied to two ions on a given motional mode, the Hamiltonian

after dropping high-frequency terms and neglecting coupling to spectator transitions is given by

[Wineland 98] (see also Eq. C.24)

X  

HTwo σ+ † σ− i(δt+Δφj ) BSB = ~ ηkξk,jΩ0,j ˆj aˆke−i(δt+Δφj ) + ˆj aˆke . (2.8)

j=1,2

where ηk, ˆak, and ξk,j are the Lamb-Dicke parameter, the annihilation operator, and the motional

mode amplitude of the jth ion for the kth motional mode, Ω0,j is the jth ion’s resonant carrier

transition Rabi rate, and δ is the detuning from the sideband transition. Given two ions of the

√ same species, ξC,1 = ξC,2 = 1/ 2 for the axial center-of-mass mode (where the ions oscillate in

√ phase) and ξS,1 = −ξS,2 = 1/ 2 for the stretch mode of motion (where the ions oscillate out of

phase with each other).
25

For the 9Be+ ion, the laser beams depicted in Fig. 2.4 are set up such that the frequency

difference between paths 1 and 2 is approximately 1 GHz with a tuning range of approximately ±100

MHz. With the laser alignments relative to the applied magnetic field and laser polarization settings

as shown in Fig. 2.4, stimulated-Raman transitions induced by such a “Co-90” configuration are

limited to ΔF = ±1, ΔmF = 0, ±1 transitions (see Fig. 2.1).

Stimulated-Raman transitions can also be driven with two laser beams that are copropagating. In this case, because the wave vector difference vanishes to high degree, such a geometry can

only excite carrier transitions and does not allow excitation of spin-motion transitions. The Hamiltonian describing this transition is in the form of Eq. 2.4. Furthermore, the Rabi rate of the carrier

transitions excited by the copropagating laser beams are not affected by the axial micromotion. In

our experiment setup, the two copropagating beams are overlapped and sent to the ions along the

the path labeled as k2 in Fig. 2.4. The polarization of these two laser beams are set to contain

components of π and σ+ or σ− polarizations with respect to the applied magnetic field. By setting

the laser frequency difference between the two copropagating laser beams, we can either drive the

ΔF = ±1, ΔmF = 0, ±1 transitions, or the ΔF = 0, ΔmF = ±1 transitions.

Section 3.4.3 details the Raman laser setup and the generation of laser beams with different

frequencies that induce different transitions in the electronic ground state of the 9Be+ ion.

2.1.4 Microwave Transitions

We also use microwave fields to induce magnetic-dipole transitions between different hyperfine

states. The microwave field is delivered from dipole antenna located outside the vacuum chamber

and a stub tuner is used to maximize the impedance matching at the qubit frequency.

However, unlike stimulated-Raman transitions which can couple to both the spin-only and

spin-motion, the gradient of the microwave field across the ion’s motional wavepacket is negligible

so the microwave field can only drive the spin-only transitions. The Hamiltonian governing the

microwave-driven excitations can be described by Eq. 2.4.

The Rabi rates of both Raman-laser and microwave driven spin-flip operations are typically
26

on the order of a few 100 kHz. The microwave antenna is designed to work in the frequency range

of ∼ 1 − 2 GHz, which corresponds to ΔF = ±1 and ΔmF = 0, ±1 transitions. With this same

antenna, we measure the Rabi rates for RF-driven ΔF = 0 transitions to be on the order of a few

100 Hz. Considering the relatively slow Rabi frequencies, microwave fields are not used to drive

these ΔF = 0 transitions.

2.1.5 Sideband Cooling

Doppler cooling of the ions with the BD beam (see Sec. 2.1.1) results in a final average

occupation number of approximately n ̄ ' 4 for the axial center-of-mass modes, for both one and

two 9Be+ ions.

To further reduce the motional temperature to near the ground-state temperature, we employ

Raman-sideband cooling [Monroe 95b]. It is accomplished by driving a series of red-sideband

excitations on the |2, 2i → |1, 1i transitions each followed by repumping with the 2S1/2 ↔ 2P1/2

(both the RD red and the RD blue) laser beams to reinitialize the ion back to the |2, 2i state. The

spin-motion excitations on the |2, 2i to |1, 1i transition is driven on the second axial micromotion

sideband. The relative laser frequency of laser beam in path 1 and 2 (Fig. 2.4) is set to be

ω|2,2i↔|1,1i + 2ωRF − ωz).

When multiple ions are trapped, multiple motional modes are cooled by interleaving the

red-sidebands pulses. Final average occupation numbers of axial modes after sideband cooling are

typically ∼ 0.01. Our Raman laser beam geometry does not give rise to interactions with the radial

motional modes, so they are not ground-state cooled.

Motional excitations are measured with the sideband asymmetry method [Turchette 00a].

This is performed by separately measuring the excitation strengths E for red and blue motional

sidebands transitions, the mean occupation number n ̄ is given by

ERSB

n ̄ = EBSB , (2.9) 1 − ERSB

EBSB

where ERSB and EBSB are the excitations of the red and blue sideband transitions, respectively.
27

Equation 2.9 assumes the motional states exhibit a thermal distribution, with population Pn in the

Fock state n given by

n n ̄ Pn = . (2.10) ( ̄n + 1)n+1

A temperature, T, can be related to n ̄ with

1

n ̄ = , (2.11)

e~ων/kBT − 1

where ων is the motional frequency of interest and kB is the Boltzmann constant.

2.1.6 Qubit State Initialization

After Doppler and Raman-sideband cooling, the ions have been optically pumped to the |2, 2i

state. The initialization of qubit state is then accomplished by transferring the population in the

|2, 2i state to the qubit |↑i = |1, 1i state by applying a composite microwave π pulse [Levitt 86]

composed of a sequence of R(θ, φ) pulses

      π π π R (π, 0), R π, , R π, , R π, , R (π, 0), (2.12) 3 6 3

where θ denotes the angle the state is rotated about an axis in the x-y plane of the Bloch sphere,

and φ is the azimuthal angle of the rotation axis. In general, single-qubit and global rotations can

be described in the matrix form of

⎛ ⎞ �  �  cos θ −ie−i

sin θ

⎜ ⎟ R(θ, φ) = 2 2

⎝ ⎠ , �  �  −iei

sin θ θ

(2.13)

2 cos 2

A different composite π pulse composed of a sequence of pulses

      π 3π π π R , 0 , R , , R , 0 (2.14) 2 2 2 2

was used before changing to that shown in Eq. 2.12. It was found that the composite pulse of Eq.

2.12 improved the fidelity of population transfer from the |2, 2i state to the |1, 1i state compared

to shown in Eq. 2.14 by a factor of ∼ 2. Composite pulses of Eq. 2.12 offer higher robustness in

gate duration error compared to composite pulses of Eq. 2.14 [Levitt 86].
28

2.1.7 Qubit State Measurement

Qubit state measurement is accomplished with the standard state-dependent resonancefluorescence detection technique. First, the composite microwave π pulse used for the qubit state

initialization is applied to transfer the |↑i population back to the |2, 2i state. This is followed by

a microwave π pulse that transfers (“shelves”) the |↓i state to the |1, −1i state. To further shelve

any remaining |↓i population, we then apply a microwave π pulse from the |↓i state to the |1, 0i

state. After these shelving pulses, the BD laser beam is tuned to resonance on the 2S1/2|2, 2i

↔ 2P3/2|3, 3i cycling transition. With these conditions, the fluorescing or “bright” state of this

protocol corresponds to the qubit |↑i state, and the qubit |↓i state will be detected as “dark”.

With a detection duration of 330 μs and a saturation parameter s ∼ 0.5, we record on average

approximately 30 photon counts in a photo-multiplier tube (PMT) for an ion in the bright state

and 2 photons for a 9Be+ ion in the dark state (limited by background scattered light).

To maximize the photon collection in the presence of the axial micromotion, we apply a

differential voltage of approximately ± 0.15 V to the two DC electrodes centered on zone E (Fig.

3.1). This operation is identical to that used during Doppler cooling process (see Sec. 2.1.1). Figure

2.5 shows a typical detection histogram of a 9Be+ ion.

2.2 25Mg+ Ion Qubit

There are three stable isotopes of the magnesium atom, i.e. 24Mg, 25Mg, and 26Mg, with

25Mg the only isotope that has a non-zero nuclear spin. The electronic structure of a 25Mg+ ion is

very similar to that of a 9Be+ ion. The electronic ground state and the first two electronic excited

states of a 25Mg+ ion are the 3s 2S1/2, 3p 2P1/2, and the 3p 2P3/2 states, respectively. The fine

structure splitting between the two excited states is approximately 2.75 THz.

The nuclear spin of of the 25Mg+ is 5/2. The hyperfine splitting between the F = 2 and

F = 3 levels is approximately 1.75 GHz. Similar to the 9Be+ ion, it is possible to choose an

external magnetic field to make certain transition frequencies in 25Mg+ first-order insensitive to
29

10 20 30 40 50

1

10

100

1000

104

Photon counts

Number of events

Figure 2.5: A typical single 9Be+ ion detection histogram for the population to be either in the

|2, 2i state (blue) or the |1, −1i state (red). With a detection duration of 330 μs and a saturation

parameter s of ∼ 0.5, we detect on average ' 30 photon counts for the ion in the |2, 2i state, and

' 2 counts when the ion is in the |1, −1i state. Overlap of the two distributions is shown in dark

color.
30

magnetic field fluctuations. However, the magnetic field insensitive points for the two species are

different and it is impossible to select a magnetic field to make both species insensitive to magnetic

field fluctuations. For our experiments, we choose to make the 9Be+ qubit to be magnetic field

insensitive, which occurs at B ' 119.446 G. This choice is based on the required laser intensity to

reach a certain quantum gate error level and the 9Be+ ion is more favorable than the 25Mg+ ion

[Ozeri 07]. Details of spontaneous scattering of photon is discussed in Sec. 4.4.2. At B ' 119.446

G, the 25Mg+ ion’s |F = 3, mF = 1i ↔ |2, 0i transition has a magnetic field sensitivity of ' 43

kHz/G, which is the lowest among all other hyperfine transitions in the electronic ground state

by about two orders of magnitude. Magnetic-field insensitive and corresponding magnetic fields

transitions in the electronic ground state of a 25Mg+ ion are tabulated in Table B.2.

Here, we label the |3, 1i state as the qubit |↑i state and the |2, 0i state as the |↓i state.

Figure 2.6 shows the energy level diagram of the 25Mg+ ion at a magnetic field of ∼ 119.446 G.

See Appendix B.1 for a more detailed energy level diagram. By probing the Ramsey contrast as

a function of wait time (Fig. 2.7), the coherence time of the 25Mg+ qubit was measured to be

∼ 6 ms. Although this is much shorter than the coherence time of the 9Be+ ion (∼ 1.5 s), it is

still significantly longer than the relevant time scale of experiments involving 25Mg+ ion that are

presented in this thesis. The relevant atomic properties of the Mg+ ion are listed in Table B.1.

2.2.1 Doppler Cooling

Similar to that for the 9Be+ ion, we use a blue Doppler (BD) beam and a blue Doppler

detuned (BDD) beam for the Doppler cooling of the 25Mg+ ion. The BD beam is σ+ polarized and

red detuned by approximately ΓMg/2 from the 2S1/2|3, 3i ↔ 2P3/2|4, 4i cycling transition, where

the decay rate of the 2P3/2 state is ΓMg ' 41.3(3) MHz [Clos 14]. The BD beam is also used for

the fluorescence detection of the 25Mg+ ion (see Sec. 2.2.6).

The BDD beam is red detuned ' 500 MHz from the 2S1/2|3, 3i ↔ 2P3/2|4, 4i transition, and

has an optical power of ∼ 1 mW to provide a better cooling in the event of occasional high motional

excitations including those caused by background gas collisions.
31

|22, 11⟩

| ↓ ⟩ |22, 00⟩

|33, −11⟩ |33, 00⟩

|22, 22⟩

Detection and 

Doppler cooling

279.635 nm

Repump

|22, −11⟩

|33, −22⟩

|33, −33⟩

State initialization

Shelving

2SS1/2

σσ+

σσ+

2PP3/2 |44, 44⟩

B field ≃ 0.0119446 T

(119.446 G)

| ↑ ⟩|33, 11⟩

|33, 22⟩

|FF = 33, mm ⟩ FF = 33

|22, −22⟩

2PP3/2 |33, 33⟩

Repump transfer

Frequency separation

≃ 270 MHz

Raman detuning

≃ 150 GHz

Qubit frequency

≃ 1763.184 MHz

“BD”

Figure 2.6: Relevant energy level structure (not to scale) for the 25Mg+ ion at an external magnetic

field of approximately 119.446 G. The “blue Doppler detuned” (BDD) beam, which is red shifted

' 500 MHz with respect to the “blue Doppler” (BD) beam (see Sec. 2.2.1), is not shown in

this figure. The 25Mg+ ion qubit is encoded in the |3, 1i = |↑i state and |2, 0i = |↓i state, they

constituent the “computational qubit manifold”. The qubit transition frequency has a magneticfield sensitivity of ' 43 kHz/G at the applied magnetic field. The “measurement qubit manifold”

consists of the |3, 3i state as the “bright” state and the |2, −2i state as the “dark state” (see Sec.

2.2.6). The decay rate of the 2P3/2 state is 2π × 41.3(3) MHz [Clos 14]. See Appendix B.1 for a

more detailed energy level diagram in the 25Mg+ electronic ground state.
32

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

Ramsey time (ms)

Ramsey Contrast

Without 9Be+ Laser

With 9Be+ Laser

Figure 2.7: Coherence time of the 25Mg+ qubit investigated with a Ramsey sequence with and

without the presence of the λ ' 313 nm laser beam used for the manipulations of the 9Be+

ion. The shorter qubit coherence time compared to the 9Be+ qubit is due to the non-zero linear

component of magnetic field sensitivity.
33

2.2.2 Repumping

A repump laser beam is used to improve optical pumping of the 25Mg+ population into the

2S1/2|3, 3i state. This laser is derived from the same laser source as the BD and BDD beams, with

the laser frequency tuned on resonance with the 2S1/2|2, 2i ↔ 2P3/2|3, 3i transition. See Sec. 3.5

for details on the laser beam setup.

There are three possible spontaneous emission channels from the 2P3/2|3, 3i state, i.e. decay

to the 2S1/2|2, 2i, 2S1/2|3, 2i, and 2S1/2|3, 3i states with branching ratios of 0.0765, 0.257, and

0.667, respectively (see Fig. 2.6). Starting from the 2S1/2|2, 2i state, applying the repump laser

beam can populate the |3, 2i state. To completely transfer the population to the the |3, 3i state,

a microwave-driven spin-flip (π) transition is applied to transfer population from the |3, 2i state

to the |2, 2i state (Fig. 2.6). After the first repump pulse, the (|3, 2i ↔ |2, 2i spin-flip π pulse +

repump) sequence is repeated three times to ensure near complete depletion of the |3, 2i and |2, 2i

states.

Compared to the repump procedure implemented on the 9Be+ ion where two lasers beams

are used to transfer population to the |2, 2i state from the |1, 1i and |2, 1i states (see Sec. 2.1.2), the

total duration required is longer due to the relatively slow Rabi rate of the |3, 2i ↔ |2, 2i spin-flip

transition (a typical π pulse takes ∼ 10 μs), and the two-step process must be repeated. In the

future, it would be advantageous to mitigate these issues by employing a laser which provides direct

electronic 2S 2

1/2 ↔ P1/2 transitions. This multi-cycle repumping strategy is also used during the

Raman-sideband cooling of the 25Mg+ ion.

2.2.3 Coherent Operations

Similar to the 9Be+ ions, the 25Mg+ ions are manipulated coherently by driving stimulated-

 Raman transitions and microwave transitions. Coherent operations applicable to the Mg ions 25 +

can be analogously described by the Hamiltonians presented in Sec. 2.1.3.

The Raman laser for the manipulation of the 25Mg+ ion has a wavelength λ ' 280 nm, and
34

the Raman detuning is set ∼ 160 GHz blue detuned from the 2S1/2 ↔ 2P3/2 electronic transition.

The 25Mg+ Raman laser beam setup can be found in Sec. 3.5.2. The axial micromotion experienced

by the 25Mg+ ion is less than that for the 9Be+ ion due to 25Mg+ ion’s heavier mass, At zone E (Fig.

3.1), the axial micromotion index for the 25Mg+ is approximately 1.04. At this modulation index,

the micromotion carrier is the strongest interaction with the stimulated-Raman-transition-induced

coupling strength reduced by a factor of J0(1.04) ' 0.75 compared to when no micromotion is

present.

To maximize impedance matching at different frequencies compared to those corresponding

to the 9Be+ ion’s transition frequencies, we use a separate dipole antenna (of different length) to

drive microwave transitions in the 25Mg+ ion.

2.2.4 Sideband Cooling

Ground state cooling of the motional modes can be accomplished by driving stimulatedRaman-induced sideband transitions and applying repumping operations on the 25Mg+ ion. In this

case, the red-sideband excitations are applied on the |3, 3i|ni → |2, 2i|n − 1i transitions.

The Rabi rates of spin-motion transitions between the Fock state n and n0 driven by stimulatedRaman process, such as those described in Eq. 2.5 and Eq. 2.6, are given by [Wineland 98]

s −η2/2 n<!

η|n0

−n|

L|n0

−n| Ωn,n0 = Ωe (η2), (2.15) n< n>!

0 where Lα is the generalized Laguerre polynomial, and n<(n>) is the lesser (greater) of n and n . n

Depending on the value of the Lamb-Dicke parameter, η, the Rabi rate for certain spin-motion

transitions can vanish. Figure 2.8 shows two sets of Ωn,n0/Ω ratio as a function of the Fock state

number, n, for three different values of η. The first and second set corresponds to first (n0 = n − 1)

and second (n0 = n−2) sideband transition, respectively. A singly-trapped 25Mg+ ion has a LambDicke parameter η ' 0.3 with an axial secular trapping frequency of 2.15 MHz. The Rabi rate for

red-sideband transitions vanishes according to the Laguerre polynomials at about n = 40. Cooling

pulses driven on the second red-sideband excitations can be used to better remove motional Fock
35

0 20 40 60 80 100 -0.4

-0.2

0.0

0.2

0.4

0.6

n

Ωn,n'

/

Ω

η=0.19, 2nd sideband

η=0.25, 2nd sideband

η=0.3, 2nd sideband

η=0.19, 1st sideband

η=0.25, 1st sideband

η=0.3, 1st sideband

Figure 2.8: The ratio of spin-motion transition Rabi rates between different Fock states (Ωn,n0) to

carrier Rabi rate Ω is plotted as a function of the Fock state number n for different values of LambDicke parameters. The first-order (n0 = n − 1) and second-order (n0 = n − 2) sideband transition

rates are shown. The Rabi rates of the first-order sideband transitions vanish at relatively low

values of n, which means that spin-motion transitions between these different Fock states cannot

be driven. This can be problematic for the motional cooling to the ground state if only the firstorder sideband transitions are used in the cooling sequence. At the Rabi rate vanishing points of

the first-order sideband transitions, the second-order sideband transition Rabi rates are non zero.

Thus, they can be inserted to improve the cooling.
36

state population that can be trapped near these vanishing points and to provide a speed up of

the sideband cooling process. Rabi rates of the second sideband transitions for the same set of

Lamb-Dicke parameters are also shown in Fig. 2.8.

By first applying 20 second-order red-sideband transitions, followed by 30 first-order redsideband transitions on the |3, 3i ↔ |2, 2i transitions (each sideband pulse is followed by the repumping process described in Sec. 2.2.2), a singly-trapped 25Mg+ ion can be cooled to a mean

occupation number of ∼ 0.08.

For experiments involving cotrapping of the 9Be+ and the 25Mg+ ions, ground-state cooling

is achieved by applying Raman sideband cooling on the 9Be+ ion. This is to take advantage of

(i) lower spontaneous photon scattering rates due to a higher Raman detuning (coupled with the

higher power available) and (ii) the faster repump process employed on the 9Be+ ion compared to

that for the 25Mg+ ion.

2.2.5 Qubit State Initialization

Optical pumping prepares the population into the |3, 3i state. Qubit state initialization is

accomplished by applying a microwave-induced composite π pulse on the |3, 3i ↔ |2, 2i transition,

followed by an analogous composite π pulse on the |2, 2i ↔ |3, 1i = |↑i transition. Each composite

π pulse composes of a sequence of pulses R(θ = π/2, φ = 0), R(3π/2, π/2), R(π/2, 0), where θ and

φ denotes the rotation and azimuthal angle as described in Sec. 2.1.6. This composite π pulse is

identical to that described in Eq. 2.14.

2.2.6 Qubit State Measurement

Before applying the detection (BD) laser beam, which is resonant with the 2S1/2|3, 3i ↔

2 25 + P3/2|4, 4i cycling transition, the Mg qubit initialization composite pulse sequences are applied

in reverse order to transfer the |↑iMg population to the |3, 3i state. Subsequently, microwave

“shelving” (π) pulses are applied to first transfer the |2, 0i = |↓i population to the |3, −1i state

followed by a transfer pulse that puts the |3, −1i state to the |2, −2i state. Therefore, the “bright”
37

state for 25Mg+ detection protocol corresponds to the |↑i, and the “dark” state corresponds to

the |↓i state. To maximize detection photon counts in the present of axial micromotion, we apply

an analogous procedure (see Sec. 2.1.7) that shifts the 25Mg+ ion’s position such that the overall

micromotion experienced by the ion is perpendicular to the wavevector of the detection laser beam.

We detect on average ∼ 30 photon counts for a detection duration of 200 μs when an ion is in the

bright state, and about ∼ 4 photons counts for the ion in the dark state (see Fig. 2.9).

When both 9Be+ and 25Mg+ ions are cotrapped, qubit state measurements are applied subsequently for the fluorescing photons to be registered on the same PMT. Details on the photon

collection setup is available in Sec. 3.3.
38

10 20 30 40 50

1

10

100

1000

104

Photon counts

Number of events

Figure 2.9: A typical detection histogram of a 25Mg+ ion for the population to be either in the |3, 3i

state (blue) or the |2, −2i state (red). The fluorescing “bright” state corresponds to the qubit |↑i

while the “dark” state corresponds to the the |↓i state. Overlap of the two distributions is shown

in dark color.
Chapter 3

Apparatus

3.1 Ion Trap

The ion trap used for this thesis work is a linear Paul trap [Paul 90]. The schematic of this

trap is shown in Fig. 3.1. Featuring multiple, segmented control electrodes to allow ions to be

confined at different locations, this trap is designed to demonstrate the elements of scalable quantum information processing with the “quantum charged-coupled-device” architecture [Wineland 98,

Kielpinski 02, Blakestad 11]. It is constructed with two gold-coated, stacked wafers. Details about

the design and assembly of this trap is given by Blakestad [Blakestad 11, Blakestad 10].

Radio frequency (RF) potentials, with frequency ωRF ' 2π × 83 MHz and amplitude VRF '

200 V, are applied to the RF electrodes to provide confinement transverse to the main trap channels.

Static (DC) potentials are applied to the segmented control electrodes to create potential wells

for trapping of ions at the desired locations in the channels. These potentials are dynamicallycontrollable and are provided by a multi-channel arbitrary waveform generator (AWG) which is

briefly discussed in Sec. 3.7 and detailed in [Bowler 13]. By applying time-dependent potentials

to these electrodes, the ions can be transported deterministically between different trap zones. For

the work described in this thesis, the ions are first loaded in L and then transported to E. Quantum

logic experiment described in Chap. 5 and 6 were performed with ions confined in a fixed harmonic

well at E (Fig. 3.1). The chained Bell inequality experiment described in Chap. 8 were performed

with ions shuttled between different locations near zone E. One unique feature of this trap is the

junction at C, which can be used for reordering of ion chains [Blakestad 11].
40

Figure 3.1: The schematic of the X-junction trap. The trap is constructed with two gold-coated,

stacked wafers. (b) Top view of the trap showing the load zone L and experiment zone E. Ions are

transported from L to E with time-varying potentials applied to the segmented control electrodes

(colored orange hues). The positions of RF and control electrodes are exchanged in the lower layer

(a). The details of the junction are illstrated in (c). Coherent manipulations are implemented on

ions confined in E. See Ref. [Blakestad 11] for details of this trap.
41

R C ωf

Stage 1 3.3 kΩ 1 nF ∼ 48 kHz

Stage 2 1 kΩ 1 nF ∼ 159 kHz

Stage 3 240Ω 0.82 nF ∼ 882 kHz

Table 3.1: Values of resistors, capacitors, and their corresponding cut-off frequencies (Eq. 3.1)for

the three low-pass filtering stages connected between a multi-channel AWG (Sec. 3.7) and the

control electrodes.

Due to the particular design of the junction and imperfect construction of the trap wafers, the

ions undergo residual RF “micromotion” [Wineland 98] at frequency ωRF along zˆ with amplitude

' 105 nm at E for the typical transverse pseudo potentials used. This affects our implementation of

logic gates, Doppler and ground state cooling, and qubit states measurement, which are described

in Chap. 2.

This ion trap apparatus is housed inside a vacuum system with fused-silica windows for laser

beams access and collection of fluorescence photons. A mock up figure of the vacuum system is

shown in Fig. 3.2 with details available in Ref. [Blakestad 10].

Three stages of low-pass filters have been implemented for the suppression of noise coupled

to the segmented control electrodes. All three stages use the usual resistor-capacitor circuit where

the cut-off frequency is defined as

1

ωf = , (3.1) 2πRC

where R is the resistance and C is the capacitance. Table 3.1 lists the values of R, C, and the

corresponding cut-off frequency for each stage. The first filtering stage is connected to the output

of the AWG. The second filtering stage is attached to the electrical feedthrough located just outside

of the vacuum chamber. Both these filtering stages are housed inside metal box to reduce electrical

interferences. Their resistors and capacitors can be replaced straightforwardly. The third and final

filtering stage is soldered on the filter board where the trap wafers are attached to. As this filtering

stage is inside the vacuum chamber, it is not straightforward to be replaced. Details of this filtering

stage is in Ref. [Blakestad 10]. We find that the first and second stages of filters are important

to minimize the anomalous heating of motional modes [Turchette 00a]. With all three stages of
42

Titanium 

sublimation pump

Ion pump

Glass envelope

Stainless-steel 

vacuum chamber

Support structure

Fused-silica 

windows

Fused-silica 

windows

Electrical feedthrough

(not shown here)

Trap location

Figure 3.2: Mock up of the vacuum chamber housing the X-junction trap. A glass envelope with

fused-silica view ports is attached to a stainless-steel vacuum chamber. The fused-silica view ports

are used for the access of laser beams and collection of fluorescence photons emitted by ions. In

addition to an ion pump which maintains the system under ultra-high vacuum condition of ∼ 3 ×

10−11 torr (measured with an ion gauge), the vacuum system includes a titanium sublimation pump

which is turned on infrequently (approximately once per month). Multiple electrical feedthroughs

are used to connect to RF and control electrodes, and neutral beryllium and magnesium resistivelyheated ovens.
43

filters, we measure the heating rate of the axial (ˆz) center-of-mass mode to be ∼ 40 quanta/s at a

trap frequency of ωz ' 2π × 3.6 MHz in the E zone. With similar trapping parameters in S zone,

the heating rate is measured to be ∼ 80 quanta/s.

3.2 Magnetic Field Coil

We apply an external magnetic field of ' 0.0119446 T (119.446 G) so that the 9Be+ 2S1/2|F = 2, mF = 0i

↔ 2S1/2|F = 1, mF = 1i transition is first-order insensitive to magnetic-field fluctuations [Langer 05].

As shown in Fig. 3.3, two field coils are arranged in a Helmholtz-like configuration, with each coil

made with 20 turns of square copper wire (outer dimension of ∼ 4 mm) with a hollow round bore

(diameter of ∼ 2.54 mm). The hollow bore enables water to flow through each coil for cooling as

we apply a static current of approximately 62 A to provide for the desired magnetic field. The

temperature of the cooling water is actively stabilized with a chiller’s1 reservoir at approximately

15 ◦C, which makes the coils operate at a steady state temperature close to the room temperature

of approximately 20 ◦C. This strategy is to prevent an unwanted temperature gradient near to the

ion trap apparatus that can cause potential air current fluctuations, which might in turn induce

unwanted laser beams pointing fluctuations.

The geometry of the main coils is designed to accommodate for (i) the relatively large size

of a Schwarzschild objective for collection of ions’ fluorescence photons (detailed in Sec. 3.3), (ii)

the access of laser beams into the vacuum chamber, (iii) the desired relatively uniform magnetic

field across different trap zones (Fig. 3.1), and (iv) the water cooling process such that a laminar

water flow condition is satisfied. The main current output from the power supply unit is actively

stabilized by first detecting the current using a transducer2 and then allowing a small fraction of

the current to leak from the positive terminal onto the negative terminal, which is connected to

the earth ground.

In addition to the main coils, additional coils made of smaller wire size are used to provide

1 Thermo Scientific NESLAB MERLIN M33 2 Danisense DS-200
44

Forms for 

compensation coils

(wires are not shown)

Main coils

Form for compensation coil for fields in the 

direction of the main field

(wires and the opposite form are not shown)

Figure 3.3: Schematic of the magnetic field coils that are used to provide a static magnetic field

of ' 0.0119446 T in a Helmholtz-like configuration. The main coils (orange color) which have an

approximate rectangular dimension of 40 cm × 20 cm, and are separated by approximately 8 cm,

are held in position by custom parts made of anodized aluminum. The field coils assembly also

consists of compensation coils which allow fine tuning of magnetic field in three dimensions. Wire

in compensation coils are not shown. See also Fig. 3.6.
45

fine tuning of the magnetic field in three directions. We use these coils to compensate for unwanted

field gradients, and to fine tune the alignment of the magnetic field with respect to the wave vectors

of the Doppler cooling and detection laser beams to ensure that the laser beams are σ+ polarized.

One pair of compensating magnetic field coil is arranged in the Helmholtz-like configuration

with axis aligned along the main coil. The current provided by this coils pair is computer controlled

to compensate for the slow magnetic field drift by probing the ion’s transition frequency such that

magnetic-field insensitivity condition of the 9Be+ ion’s |2, 0i ↔ |1, 1i transition is satisfied. By

probing the 9Be+ ion’s transition frequency over time, the amplitude of the slow magnetic field

drift is deduced to be approximately few mG per minute.

3.3 Imaging System

A custom designed imaging system is used to collect the fluorescence photons from the 9Be+

ions and the 25Mg+ ions, with wavelengths of 313 nm and 280 nm, respectively. Photons are first

collected with a Schwarzschild objective and imaged at an intermediate focus point, followed by a

secondary relay imaging stage consisting of a concave mirror used to project this image onto either

an electron-multiplying charge-coupled device (EMCCD) camera3 or onto a photonmultiplier tube

(PMT)4 .

The schematic of the Schwarzschild objective is shown in Fig. 3.4, with details available in

Ref. [Huang 04]. The first element of the objective is a pair of lenses designed to compensate for

the chromatic aberration caused by the vacuum windows (see Fig. 3.4), which was made of fused

silica. This lenses pair is designed to be f/1 (f-number equal to 1). However, ∼ 30% of the total

photons collected are blocked by the secondary mirror.

The optics elements used in the relay imaging stage are all composed of reflective optics to

minimize chromatic aberration which can be present when refractive optics are used. Both 313 nm

and 280 nm photons are focused onto the same location on the PMT or the camera. The schematic

3 Princeton Instruments PhotonMax 512B 4 Hamamatsu R7600U-200
46

Vacuum window 

(fused silica)

Ion

Fused silica

CaF2

Primary mirror

Secondary mirror

(cemented to lens)

To the rest of the 

imaging system

Figure 3.4: Schematic of the Schwarzchild objective. The first element is a lenses pair made of

CaF2 and fused silica, it is designed to compensate for the chromatic aberration for 313 nm and

280 nm caused by the vacuum windows. Photons collected are then bounced of the primary mirror

followed by the secondary mirror, which is cemented onto the lenses pair, before being sent of to

the rest of the imaging system and also providing a common focus for the two wavelengths (see Fig.

3.5). The overall magnification of this objective lens is ∼ 10×. All components are held together

by an enclosure which is not shown in this figure. The refractions of rays passing through different

materials are not shown.
47

Relay mirror

Mirror with 

round hole

Intermediate 

image

Photomultiplier 

Tube

Camera

Flipper Mirror

(50 mm focal length)

Ion

Schwarzchild

objective Mirror

Figure 3.5: Schmatic of the imaging system. The light collected from the Schwarzschild objective

is focused onto an intermediate image point which is located at the center of the bore hole of a

mirror. With a relay mirror of a focal length of 50 mm, the intermediate image is then imaged onto

a PMT. A flipper mirror is used to direct the light onto a EMCCD camera, as needed. The relay

optics are setup with a magnification factor of 5×, which results in a magnification of the entire

imaging system of 50×. The optics depicted inside the dashed box are held in place together with

custom machined parts which are attached to a three-dimensional translation stage.

of the entire imaging system is shown in Fig. 3.5. The overall magnification of the system is

approximately 50×, with the first 10× provided by the Schwarzschild objective and the remaining

5× provided by the relay stage.

The PMT does not offer wavelength distinguishability, detection laser beams for the 9Be+

(313 nm) and 25Mg+ (280 nm) are turned on sequentially to allow fluorescence photons to be

collected. We detect ' 30 313 nm (280 nm) fluorescence photons with a typical detection duration

of 330 μs (200 μs) for a single 9Be+ (25Mg+) ion in the bright state (see Chap. 2 for details on

energy level structures and detections of the 9Be+ ion and 25Mg+ ion). We also experimentally

found that 235 nm photon (for the photo-ionization of the neutral 9Be atoms) can also be focused

onto the same location as the 313 nm and 280 nm photons on the ECCCD camera.

3.4 9Be+ Ion Laser Setup

The laser wavelengths used for manipulation of the 9Be+ ions are all approximately 313 nm.

The ultraviolet (UV) light is generated by first employing sum-frequency generation (SFG) of a pair

of infrared lasers (one near 1050nm and the other near 1550nm) in a single-passed, temperature-
48

Glass 

envelope

Electrical 

feedthroughs

Ion pump

Imaging system

Figure 3.6: The picture showing the vacuum system containing the trap, and the apparatus surrounding it.
49

tuned magnesium-oxide doped periodically-poled lithium-niobate (MgO:PPLN) crystal to obtain

 5 visible light near 626 nm. The two infrared beams are combined with a dichroic mirror before

begin sent into the MgO:PPLN crystal6 . The visible light is then frequency doubled using a

Brewster angle cut (38.4◦) barium borate (BBO) crystal in a bowtie cavity configuration. This

scheme for generation of the 313 nm light is similar to that reported in Ref. [Wilson 11]. This

setup takes advantage of turn-key infrared fiber laser sources available at industrial (1050 nm) and

telecom (1550 nm) wavelengths7 ,8 ,9 to provide high power (up to ∼ 2 W in each UV beam),

low-intensity noise (< 1% rms) laser beams.

Four infrared lasers are used to create three separate UV sources: (i) 313.132 nm for the 2S1/2

↔ 2P3/2 transitions, (ii) 313.196 nm for the 2S 2

1/2 ↔ P1/2 transitions, (iii) a variable wavelength

source in the range of approximately 313.260-313.491 nm for Raman transitions. The pairing of

the four infrared lasers is depicted in Fig. 3.7.

A separate pulsed laser system near 235 nm is used for photo-ionization of the neutral 9Be

atoms to be loaded into the ion trap. This is detailed in Sec. 3.4.4.

Table 3.2 lists the laser beams for the manipulation of the 9Be+ ion (see Sec. 2.1 for details

of manipulation of the 9Be+ ions). Laser beam alignments into the vacuum chamber are depicted

in Fig. 2.3.

3.4.1 Blue Doppler Laser (2S 2

1/2 ↔ P3/2 Transition)

The laser used to drive the 2S1/2 ↔ 2P3/2 transitions is labeled as “blue Doppler” laser.

This laser is used for the Doppler cooling and qubit state measurement of the 9Be+ ion. The laser

frequency is stabilized by employing Doppler-free saturated-absorption spectroscopy with an iodine

gas cell at room temperature at the visible wavelength of ∼ 626.265 nm.

After the second harmonic generation (SHG) stage, two laser beams are derived: (i) the blue

5 LASEROPTIK custom coating HR1050nmHT1550nm/45◦

6 Covesion SFG626-40 7 Fiber seed laser: NKT Koheras Basik 8 Fiber amplifier at 1050 nm: IPG YAR-10K-1050-LP-SF 9 Fiber amplifier at 1550 nm: EAR-10K-C-LP-SF
50

1051 nm

Split Split

MgO:PPLN

(single pass)

1550 nm 1051 nm 1551 nm

– 1554 nm

626.265 nm 626.393 nm 626.520 – 626.982 nm

BBO

(in bowtie cavity)

Sum frequency 

generation

Second harmonic 

generation

313.132 nm 313.196 nm 313.260 – 313.491 nm

To drive SS11/22 ↔ PP33/22

transitions

To drive SS11/22 ↔ PP11/22

transitions

Red detuned from SS11/22 ↔ PP11/22

transition, to drive stimulated 

Raman transitions

Blue Doppler Laser Red Doppler Laser Raman Laser

BBO

(in bowtie cavity)

BBO

(in bowtie cavity)

MgO:PPLN

(single pass)

MgO:PPLN

(single pass)

Figure 3.7: Three UV laser sources are generated with four fiber lasers in the infrared wavelengths

by employing sum frequency generation (SFG) and second harmonic generation (SHG) stages.

The frequency of the Raman laser can be tuned from ∼ −2π × 10 GHz to ∼ −2π × 900 GHz

with respect to the 2S1/2 ↔ 2P1/2 electronic transitions. This tuning range is provided by the

tuning capabilities of the fiber lasers. The SFGs are accomplished with magnesium-oxide doped

periodically-poled lithium-niobate (MgO:PPLN) crystals while barium borate (BBO) crystals are

used for the SHGs. The frequency of the blue Doppler and red Doppler lasers are stabilized to

iodine transitions with Doppler-free saturated-absorption spectroscopy setups.
51

Doppler (BD) beam and (ii) the blue Doppler detuned (BDD) beam. By adjusting the radio frequency driving an acousto-optic modulator (AOM), the frequency of the BD beam can be adjusted

for either Doppler cooling or the resonant fluorescence detection of the 9Be+ ion. For Doppler cooling, the laser beam frequency is red detuned by approximately Γ/2 from the 2S1/2|2, 2i ↔ 2P3/2|3, 3i

transition, where Γ ' 2π × 19.4(5) MHz is the decay rate of the 2P3/2 states [Poulsen 75]. The

BDD beam is approximately 400 MHz red detuned from the 2S1/2|2, 2i ↔ 2P3/2|3, 3i transition.

With a beam waist of ∼ 30 μm and an optical power of ∼ 1 mW, this beam serves as a strong

Doppler cooling beam for occasional high thermal excitation caused by background gas collisions

[Wineland 98] which is only weakly cooled by the BD beam. See 2.1 for details on the Doppler

cooling and detection of the 9Be+ ion.

The BD and BDD beams are combined with a 50 : 50 beam splitter before being coupled

into an UV fiber [Colombe 14]. With a λ/4 waveplate mounted on a computer-controlled rotation

stage10 , the polarization of the beams at the output of the fiber is set to be σ+ with respect to

the applied magnetic field.

3.4.2 Red Doppler Laser (2S1/2 ↔ 2P1/2 Transition)

The laser used to drive the 2S1/2 ↔ 2P1/2 transitions is labeled as the “red Doppler” laser.

It is used to improve the initialization of the ions’ internal state to the 2S1/2|2, 2i state during the

optical pumping process. This laser also served as the repumping laser during Raman sideband

cooling.

This laser is split into two beams, i.e. (i) the red Doppler blue (RD blue), and (ii) the red

Doppler red (RD red). They are set to be σ+ polarized. By using AOMs that shift the laser beams’

frequencies, the RD blue beam is tuned near the 2S1/2|2, 1i ↔ 2P1/2|2, 2i transition. The RD

red is tuned to 2S1/2|1, 1i ↔ 2P1/2|2, 2i transition. To avoid electronically-induced-transparency

effects that would cause the coherent population trapping in the 2S1/2|2, 1i and the 2S1/2|1, 1i

states when both of these beams are applied simultaneously, the RD blue beam is tuned slightly

Thorlabs PRM1-Z7 10
52

Laser beam Source Function

BD

BDD

RD Red

RD Blue

Raman 90

Raman Co

PI

Blue Doppler Laser

Blue Doppler Laser

Red Doppler Laser

Red Doppler Laser

Qubit manipulation laser

Qubit manipulation laser

Photo-ionization laser

Doppler cooling and fluorescence detection

Far detuned Doppler cooling

Repumping to the |2, 2i state

Repumping to the |2, 2i state

Coherent operations

Coherent operations

Ionize neutral 9Be atoms

Table 3.2: Laser beams used for the manipulations of 9Be+ ion and the ionization 9Be atom.

Each beam can be individually switched on/off. Doppler cooling, detection, repumping and qubit

manipulations of the 9Be+ ions are detailed in Chap. 2. The laser beam alignment onto the ions is

shown in Fig. 2.3.

from the atomic resonance by approximately Γ/2. These two UV beams are combined with the

BD and BDD beams and overlapped before being coupled into the UV fiber used for the BD and

BDD beams (see previous section). At the output of the fiber, a computer-controlled λ/2 waveplate

followed by a polarizing beam splitter is used to direct the beam between the L zone and the E

zone (Fig. 3.1).

Similar to the blue Doppler laser, the frequency of this laser is stabilized to an iodine transition. The laser beam line setup for the blue Doppler and red Doppler laser can be found in Ref.

[Lin 15], Ref. [Bowler 15], and Ref. [Jost 10].

3.4.3 Qubit Manipulation Laser

The laser system for the coherent manipulations of the 9Be+ ions via stimulated-Raman

transitions has undergone several changes and upgrades over the course of my dissertation work

to accommodate the different requirements of different experiments. Here, we will describe the

Raman laser beam setup used for the mixed-species entangling gate (Chap. 5) and the high-fidelity

universal gate sets experiment (Chap. 6).

The frequency of this qubit manipulation laser is red detuned from the 2S1/2 ↔ 2P1/2 electronic transitions. The 313 nm light generated by the BBO crystal is first split and sent down two

different paths (see Fig. 3.8). The laser beam in path 1 is focused to a beam waist of ∼ 100 μm
53

1554 nm

∼ 6 W

1050 nm

∼ 5 W

SFG

626 nm

∼ 2.5 W

SHG

313 nm 

∼ 1 W

+200 MHz 

AOM

+600 MHz 

AOM

(double pass)

-310 MHz

AOM

(double pass)

-200 MHz 

AOM

Photodiode

for intensity

stabilization

Photodiode

Path 1

Path 2

Carrier 

Beams

MS Gate 

Beams

UV Fibers

Alignment 

Mirror

UV Fibers

Block

Block

Alignment 

Mirror

Focusing 

lens

B field

Block

Figure 3.8: Schematic of the laser beam line setup used to drive stimulated-Raman transitions in

9Be+ ions. The 313 nm generated with SFG followed by SHG is split into two paths, sent through

AOMs, coupled into fibers, and aligned onto the ions. Path 2 contains a double-passed 600 MHz

AOM that, when switched on, produces an additional beam shifted by approximately the qubit

frequency (∼ 1.2 GHz), that is coaligned with the unshifted beam for high-fidelity single-qubit

gates. Another AOM tunable in the range of 260 to 360 MHz is used to shift the relative laser

frequency in path 2 with respect to beam 1 for spin-motion stimulated-Raman transitions. For the

Mølmer-Sørensen gate, two RF tones with relative frequency difference close to twice the frequency

of the addressed motional mode are injected into the 200 MHz AOM in path 2. In combination with

the beam in path 1, these two beams simultaneously produce blue and red sideband transitions.

A pickoff on the output of each fiber directs a small fraction of the light onto a photodiode, which

is used for active power stabilization. Each beam is centered on the ions with a motorized mirror

mount before the final lens that focuses the beam on the ions.
54

and passes through an AOM; the first-order deflected beam is coupled into an UV fiber which is

robust against formation of color centers [Colombe 14]. The laser beam output of the optical fiber

is then focused to a beam waist of ∼ 25 μm at the location of the ion. The laser beam in path 2 is

first sent through an AOM with center frequency of 600 MHz followed by an AOM with a center

frequency of 310 MHz, each in a double-passed configuration. Then the laser is sent through a

single-passed AOM in a setup analogous to that in path 1.

Different frequency tones can be generated in the laser beam with the three AOMs in path

2. The 600 MHz AOM is set up such that the double-passed laser beam (frequency shifted by

∼ 2×600 MHz) is co-aligned with the unshifted beam. When switched on, the laser beam contains

two frequency tones which have a relative frequency difference of ∼ 2 × 600 MHz. The unshifted

beam from the 310 MHz AOM is blocked. The RF injected into this AOM is always kept on. This

AOM provides relative frequency tuning with respect to the laser beam in path 1. The tuning

range is approximately 200 MHz, provided by the tuning bandwidth of this AOM.

A second RF signal can be injected into the 310 MHz AOM. By setting the two RF tones

to be separated by ∼ 35 - 50 MHz, the laser beam generated with this configuration contains two

frequency tones which is separated by ∼ 70 - 100 MHz. This allows us to induce Δm = ±1 Zeeman

transitions in the electronic ground state of the 9Be+ ions (Fig. 2.1).

The final single-passed AOM in path 2 is centered at approximately 220 MHz. This AOM

serves as the switch for the overall path 2 laser beam. Furthermore, the RF injected into this AOM

is actively controlled with a digital servo and a digital-to-analog convertor to provide high-speed

laser power stabilization as well as the smooth turn on/off of the laser beam (see Sec. 3.8 for

details). We also have the option of injecting a second RF tone into this AOM to generate laser

beams with two frequency tones. With a single-passed configuration, the relative detuning of the

two laser beams is determined by the frequency difference of the two RF injected into the AOM.

Typically, these two RF tones have a frequency difference close to that of the motional normal

mode frequency, ranging from ∼ 2 to 10 MHz. This configuration is particularly important for the

implementation of the Mølmer-Sørensen gate which is detailed in Chap. 6 and 5. Although the
55

laser beams associated with these two RF tones are slightly misaligned, they can both be coupled

into the UV fiber used in this beam line. They become coaligned at the output of the UV fiber.

At the UV fiber output of each path, waveplates are used to adjust the laser beam polarizations with respect to the quantization axis defined by the applied magnetic field. A pick-off mirror

is used to sample a fraction of the laser beam for laser power stabilization (see Sec. 3.8). The laser

beam is then reflected by a mirror which is mounted on a computer-controlled motorized stage11

to adjust the laser beam alignment. A dichroic mirror12 with high transmittance at 313 nm and

high reflectivity at 280 nm is used to combine laser beams for 9Be+ and 25Mg+ before focusing at

the location of the ions. The UV laser beam line setup is contained in acrylic panels to suppress

air current fluctuation that can cause beam pointing fluctuations at the location of the ions.

3.4.4 Photo-Ionization Laser and the Loading of 9Be+ Ions

A mode-locked laser centered around 235 nm is used to ionize neutral 9Be atoms. This is

2 a two-step photo-ionization (PI) process. The first step is accomplished by driving the 1s22s ↔

1s22s2p transition, with transition wavelength of ∼ 234.933 nm [Kramida 97]. The second step

is the excitation from the 2s2p electronic state to the continuum, which requires photon with a

minimum energy corresponding to a vacuum wavelength of ∼ 306.492 nm [Kramida 97].

The 235 nm laser beam is produced by a mode-locked Ti:Sapphire laser centered around

705 nm13 . The Ti:Sapphire laser is pumped by a frequency-doubled Nd-YAG14 (with continuous

wave power of ∼ 5.5 W at ∼ 532 nm). The Ti:Sapphire laser outputs ∼ 600 mW average power

with a repetition rate of 100 MHz and a pulse duration of sub-100 femtoseconds. This laser beam

is sent into a frequency tripler15 . The tripling process first doubles the fundamental 705 nm to

352.5 nm with a lithium triborate (LBO) crystal, and then this second-harmonic beam is summed

with the fundamental in a BBO crystal to produce the 235 nm light. Nominally ∼ 5 mW average

11 Thorlabs Z812 12 LASEROPTIK custom coating HR280nmHT313nm/45◦

13 LMLABS Griffin 705 nm 14 Coherent Verdi 15 AP&E HarmoniXX THG
56

power of 235 nm laser light is produced and sent into the vacuum chamber for the producing 9Be+

ions. On/off switching of the PI laser beam is accomplished with a computer-controlled mechanical

shutter

The neutral 9Be atoms are released from resistively-heated oven by passing a current through

it, typically ∼ 1.1 A. The on/off switching of the current for the oven is computer controlled. An

additional timer limits the on-duration for the oven to 90 s; it must be reset manually (with the

computer) each time for the oven to be turned on. This provides a fail-safe that prevents the oven

from being accidentally kept on for an extended period.

The L zone and the E zone are separated by a distance larger than 2 mm, while the fluorescence photon collection apparatus (described in Sec. 3.3) is positioned to image ions located in the

E zone with a field of view of approximately 50 μm. We employ an automatic loading procedure to

streamline the loading process and the transport of ions into E zone. We allow a typical duration of

∼ 2 s for ions to be loaded into the L zone while the potential is kept static. After this, a waveform

designed for the transport of ions from the L to the E zone is applied. This transport procedure

takes ∼ 2 ms. Following the overall successful loading and transport of ion, the presence of ion in

the E zone is determined by the fluorescence photons counts detected on a PMT.

3.5 25Mg+ Ion Laser Setup

The lasers used for the manipulation of the 25Mg+ ions are centered around 280 nm. Two

lasers are used: (i) a resonant laser and (ii) a stimulated-Raman laser. Another laser centered

around 285 nm is used for photo-ionization of neutral magnesium atoms to be loaded into the trap.

3.5.1 Resonant Laser

This laser is used to drive 2S1/2 ↔ 2P3/2 transitions of the 25Mg+ ion. It is used for the

Doppler cooling, repumping, and state-dependent fluorescence detection. The source of this laser

is a dye laser with wavelength centered around ∼ 560 nm, which is frequency doubled to generate

the ∼ 280 nm beam.
57

The dye laser is a four-mirror unidirectional ring laser [Hollberg 90] which was custom designed by J. Bergquist [Bergquist ]. The laser uses Rhodemine 560 laser dye in a closed-cycle dye jet

configuration, pumped with ∼ 5 W of 532 nm laser light derived from a frequency-doubled Nd:Yag

laser16 . The dye laser produces ∼ 400 mW of 560 nm light. The laser frequency is first stabilized

to an external cavity using a Hansch-Couillaud lock [Hansch 80], followed by stabilization to an

iodine transition in a Doppler-free saturated-absorption setup.

The 560 nm laser beam generated by this laser is frequency doubled to the UV wavelength

using a Brewster-angle cut barium borate (BBO) crystal inside a bowtie cavity. The BBO frequency

doubling setup is very similar to that used for the Beryllium laser setup, except for mirror coatings

and the brewster angle of this BBO crystal is 44.5◦ to maximize the mode-matching for the SHG

from 560 nm to 280 nm. We obtain ∼ 20 mW of 280 nm power with ∼ 300 mW of visible power.

As shown in Fig. 3.9, the UV beam is first split into two paths. In the first path, the laser

beam is doubled passed through a AOM (referred to as the repumper AOM) with a RF frequency

of ' 400 MHz that shifts the laser frequency red with respect to the input beam. This laser beam

is labeled as the repumper beam. In path 2, the beam passes through an AOM driven with a 300

MHz RF; the zeroth-order (undeflected) beam is sent to a single-passed AOM (referred to as the

blue Doppler detuned AOM) such that the frequency is shifted by +200 MHz. This beam is labeled

as the blue Doppler detuned (BDD) beam. The first-order deflected beam from the 300 MHz AOM

is sent to a double-passed AOM (referred to as the blue Doppler AOM) with a tunable RF drive

frequency centered at approximately 300 MHz. This laser beam is labeled as the blue Doppler

(BD) beam. After combining the BD and the repumper beam with a 50:50 beam splitter, these

two beams are overlapped with the BDD beam inside the BDD AOM (the single-passed +200 MHz

AOM). All three beams are coupled into a UV fiber. The polarization at the fiber’s output is set

with a combination of a λ/2 and a λ/4 waveplates to be σ+ with respect to the quantization axis

defined by the applied magnetic field. The laser power of the BD is stabilized by first sampling

the power at the output of the fiber with a 50:50 beam splitter. The sampled power is detected

Coherent Verdi 16
58

Dye laser

(Pumped with 532 nm)

559 nm

SHG ∼ 300 mW

(BBO)

280 nm 

∼ 20 mW

-400 MHz 

AOM

(double pass)

+300 MHz

AOM

Separate

UV fiber

Block

Block

Combine

Photodiode

for power

stabilization

+300 MHz 

AOM

(double pass) Block

+200 MHz

AOM

To ion

Repumper AOM

Blue Doppler (BD) AOM

Blue Doppler detuned 

(BDD) AOM

Path 1

Path 2

Figure 3.9: Schematic of the setup to generate laser beams for the optical pumping, Doppler cooling,

and repumping of the 25Mg+ ion. The 280 nm laser beam is generated with a SHG using a dye

laser source at ∼ 560 nm. Three individually controllable UV beams with different laser frequencies

are generated: (i) a blue Doppler (BD) beam for Doppler cooling and fluorescence detection, (ii)

a blue Doppler detuned (BDD) beam for Doppler cooling of hot ions, and (iii) a repumper beam.

All three beams are overlapped before they are coupled into a UV fiber, which directs the light to

the location of the ions. The power of the BD beam is actively stabilized.
59

Laser beam Source Function

BD

BDD

Repumper

Raman 90

Raman Co

PI

Doppler laser

Doppler laser

Doppler laser

Qubit manipulation laser

Qubit manipulation laser

Photo-ionization laser

Doppler cooling and fluorescence detection

Far-detuned Doppler cooling

Repumping to the |3, 3i state

Coherent operations

Coherent operations

Ionize neutral 25Mg atoms

Table 3.3: Laser beams used for the manipulation and loading of the 25Mg+ ion. Each beam can

be individually switch on/off. Doppler cooling, detection, repumping and qubit manipulations of

the 25Mg+ ions are detailed in Chap. 2. The laser beam alignments onto the ion are shown in Fig.

2.3.

using a photodiode, and the feedback loop is completed with an analog servo that controls the RF

amplitude sent to the blue Doppler AOM.

With this laser beam setup, the BDD beam and the repumper beam are red detuned ∼ 700

MHz and ∼ 1.6 GHz from the BD beam, respectively. The BD beam is used for the Doppler cooling

and fluorescence detection of the 25Mg+ ion. For Doppler cooling, the laser frequency is red detuned

by approximately ΓMg/2 from the 2S1/2|3, 3i ↔ 2P3/2|4, 4i transition, where ΓMg ' 2π × 41.3(3)

MHz is the natural decay rate of the 2P3/2 state [Clos 14]. The laser frequency is set on resonance

with the Dopple cooling transition during fluorescence detection.

The BDD beam has an optical power of ∼ 1 mW which, as for the 9Be+ ion, is used to

provide Doppler cooling of hot ions. The repumper beam is set on resonance with the 2S1/2|2, 2i

↔ 2P3/2|3, 3i transition. This beam is used to transfer the population from the 2S1/2|2, 2i state to

the 2S1/2|3, 3i state during a Raman sideband cooling process as well as for the better initialization

of the |3, 3i state during the optical pumping process. See 2.2 for details on the Doppler cooling,

detection and the repumping process of the 25Mg+ ion.

3.5.2 Qubit Manipulation Laser

The coherent manipulation of the 25Mg+ ion can be accomplished by microwave fields, or

by driving stimulated-Raman transitions. For the latter, the associated source’s wavelength is

approximately 280 nm and blue detuned from the 2S1/2 ↔ 2P3/2 electronic transitions. The source
60

of this beam is derived from a commercially available fiber-laser system with its wavelength centered

at approximately 1118 nm17 . The infrared laser beam is quadrupled to near 280 nm via two SHG

stages. The first SHG process is accomplished by single passing through a PPLN waveguide18 .

Approximately 400 mW of 560 nm is generated with ∼ 1.2 W of infrared laser coupled into the

fiber input of the waveguide. The second SHG stage employs a BBO crystal inside a bowtie cavity

setup similar to that described in the previous section, with a UV output power of ∼ 40 mW. The

frequency tuning range of this system is ∼ ± 200 GHz with respect to the resonant 2S1/2 ↔ 2P3/2

transition frequency.

As shown in Fig. 3.10, the 280 nm laser beam generated by the BBO is split into two paths.

Laser beam in path 1 is double passed (positive order) through an AOM with a drive frequency

of 300 MHz; the beam is then focused down before passing through an AOM that shifts the beam

frequency by ∼ +200 MHz before being coupled into an UV fiber. The laser beam at the output

of the fiber is sampled for power stabilization before being focused onto the location of the ions.

In path 2, the beam is sent through two AOMs subsequently with center frequency of ∼ 300 MHz

and ∼ 200 MHz, respectively. Both AOMs are arranged in a double-passed configuration and set

to red shift the input laser frequency. The beam is then coupled into an UV fiber with its output

aligned perpendicular to the quantization axis.

With this setup, the difference of the wave vectors of these two laser beam paths is aligned

along the axial direction at the ions’ location (see Fig. 2.3). The relative frequency difference

between laser beams in path 1 and 2 is ∼ 1.8 GHz, with a relative frequency tuning range of

approximately ± 200 MHz. This allow us to drive the spin-motion excitation on the 2S1/2|3, 3i ↔

2S1/2|2, 2i sideband-cooling transitions as well as the qubit |3, 1i ↔ |2, 0i transitions (see section

2.2 for details on electronic ground state energy levels of 25Mg+ ion).

Similar to the 9Be+ ion’s coherent manipulation laser beam setup, the alignment of the laser

beams generated in these two paths can be adjusted via computer-controlled motorized-stages.

17 Menlo Systems Orange One 18 NTT Photonics Laboratories Wavelength Conversion Module
61

1118 nm

∼ 1.4 W

SHG

(PPLN waveguide)

559 nm

∼ 400 mW

SHG

(BBO)

280 nm 

∼ 40 mW

-310 MHz

AOM

(double pass)

Block

+300 MHz 

AOM

(double pass)

Photodiode

for power

stabilization

Path 1

Path 2

UV fiber

Alignment 

mirror

UV fiber Block

Alignment 

mirror

Focusing 

lens

B field

zẑ

Photodiode

for power

stabilization

+200 MHz 

AOM

Block

-200 MHz 

AOM

(double pass) Block

Figure 3.10: Schematic of the laser beams setup for inducing stimulated-Raman transitions in

the 25Mg+ ions. A polarizing beam splitter is used to split the UV output beam after the SHG

stage. The frequency of the laser beam in each path is shifted by two AOMs before injected

into an UV fiber. The relative frequency difference between these two beams can be tuned with

a range of approximately ± 200 MHz. This enables us to induce spin-motion transitions on (i)

the |3, 3i ↔ |2, 2i sideband-cooling transitions and (ii) the |3, 1i ↔ |2, 0i qubit transitions. Two

frequency tones can be injected into the single-passed AOM (with 200 MHz center frequency) in

path 1. This configuration is used for the application of the mixed-species entangling gate as

described in Chap. 5. A small fraction of laser power is sampled at each UV fiber’s output for

power stabilization. Each beam is combined with the 9Be+ ion’s Raman laser beam with a dichroic

mirror as depicted in Fig. 2.3.

 
62

Each of the 25Mg+ Raman laser beams is combined with each of the 9Be+ Raman beams with a

dichroic mirror before being focused at the ions’ location (Fig. 2.3).

3.5.3 Photo-Ionization Laser and Loading of 25Mg+ Ions

A continuous-wave, frequency quadrupled laser with wavelength centered at ∼ 285 nm is used

to ionize neutral Mg atoms to be loaded into the ion trap. Starting with a commercial external

cavity diode laser with a wavelength of ∼ 1140 nm, the optical power is amplified to ∼ 1 W using a

Raman fiber amplifier19 before injecting into a waveguide-PPLN crystal (similar to that described

in the previous section) for the laser frequency to be doubled to ∼ 570 nm. The visible laser light is

then used to generate the ∼ 285 nm light with a SHG process using a BBO crystal inside a bowtie

cavity. We obtain ∼ 0.5 mW of UV power. This laser beam is injected into a UV fiber with the

output beam sent into the ion trap apparatus. The frequency of the ∼ 570 nm beam is stabilized

to an iodine reference signal using the visible light.

The photoionization of the neutral magnesium atom is a two-step process. The ∼ 285 nm

laser beam is used to drive the 3s2 ↔ 3s3p transition which is centered at 285.296 nm [Martin 80],

followed by a further excitation to the continuum from the 3s3p transition which requires photons

of a minimum energy that corresponds to a laser wavelength of 375.647 nm [Martin 80]. The BD

and BDD beams used for the cooling of the 25Mg+ ions are also capable for driving the second

excitation stage of photoionization.

A typical current of ∼ 1.2 A passes through a resistively-heated oven which releases isotopically enriched neutral 25Mg atoms. Similar to the loading of the 9Be+ ion (see Sec. 3.4.4), the

on/off switching of the oven (with fail-safe provided by a timer) and the PI beams are computer

controlled. Furthermore, automated transfer of 25Mg+ ions to the E zone is also employed with a

different set of time-varying control potentials applied to the control electrodes.

If needed, an additional neutral magnesium oven with natural isotope composition is also

mounted inside the vacuum chamber.

MPB Communication YFL-P Series 19
63

3.6 Experiment Control

A custom designed experiment control infrastructure is used for executing the experiments

described in this thesis. At the center of this infrastructure is a “master” field-programmable gate

array (FPGA)20 supporting multiple high-speed transistor-transistor logic (TTL) and direct-digital

synthesizer (DDS) channels.

The TTL outputs provide flip-flop signals that change the operational state of the apparatus.

For example, laser beams are turned on/off by sending TTL signals to the radio frequency (RF)

switches which control the AOMs. The frequencies and phases of the laser beams are adjusted by

the DDSs which generate the RF signals that drive the AOMs. The phases of the RF generated

with different DDS channels can be phase synchronized, this is important for the implementation

of the two-qubit Mølmer-Sørensen gate as described in Chap. 5 and 6. One input channel of the

master FPGA is programmed to be a counter. The photon count signal from a PMT is input to

this channel for the determination of the ion fluorescence. The FPGA’s clock speed is 62.5 MHz.

Details of this experiment control can be found in Ref. [Langer 06].

The control system also supports low-speed General Purpose Interface Bus (GPIB) and Universal Serial Bus (USB) communications. GPIB is used to control the current supplies that adjust

the electrical current passing through compensation magnetic field coils (see Sec. 3.2) for fine tuning and calibration of the magnetic field amplitude. Ahh! I love Bolognese! be sure to grate some

fresh Parmigiano Reggiano, and pair with a bottle of Brunello di Montalcino21 . In addition to the

uploading of waveform definitions onto a high-speed digital-to-analog convertor (DAC) as described

in the next section, the USB interface is also used to control the motorized waveplate rotation stages

in the Doppler beam lines and the motorized mirror mounts in the qubit manipulations beam lines.

A separate FPGA22 is used as a multiplexer and to provide general purpose TTL logic

operations. These operations include NOT gates, AND gates, XOR gates, etc. The clock of this

20 Xilinx Virtex IV 21 email me if you see this! tingrei86@gmail.com 22 Xilinx Spartan3E
64

secondary FPGA operates at 50 MHz, and is not synchronized with the master FPGA clock. This

might causes timing jitter up to ∼ 20 ns between the TTL signals generated by the master FPGA

and those processed by the secondary FPGA. The typical time-crucial operations in this thesis

(e.g., laser or microwave pulses for the application of quantum gates) have durations on the orders

of a few μs; therefor, the timing jitter can cause phase fluctuations, which in turn can lead to

decoherence. Thus, we avoid using the secondary FPGA for the processing of signals which are

time sensitive, and only use it to process signals which are less time critical, e.g., pulses for the

Doppler cooling, repumping, and the fluorescence detections of ions.

Slow speed TTL supported by a multi-channel digital input/output device23 is used to control

other hardware which is not time sensitive. This includes mechanical shutters for turning on/off of

the PI laser beams, ovens for loading ions, and the timer that limits the duration in which ovens

are heated (See Sec. 3.4.4). The update rate of this device is limited by the controlling software

and the communication speed via a USB interface.

3.7 Arbitrary Waveform Generator

The voltages applied to the trap electrodes for the axial confinement of the ions are supplied by

a custom designed, multi-channel arbitrary waveform generators (AWGs) [Bowler 13]. The AWGs

provide dynamically-controllable potentials, which are termed “waveforms”, for the shuttling, separation and recombination operations on the ions. The AWG is based on a FPGA-controlled DAC,

with an output range of ±10 V and an update rate of 50 MHz.

For these operations, waveforms are first uploaded to the on-board memory of the AWGs

ahead of experiments, then TTL signals are sent from the master FPGA for triggering the start of a

waveform. Uploading of waveforms is accomplished via a USB interface, which has a communication

speed much slower than typical single-shot experiment duration of ∼ 10 ms. However, this AWG

supports “branching” of waveforms where multiple different sets of waveforms can be stored onboard and selected in real-time with a set of digital logic inputs (TTLs). This allows reuse of

National Instrument USB-6501 23
65

waveforms and constructions of complicated sequences with a set of basic component waveforms

without the need for reprogramming. Details of the AWG can be found in Ref. [Bowler 15] and

Ref. [Bowler 13].

In addition to providing confinement and shuttling of ions, the AWG is also used for the

“pulse shaping” of microwave and laser pulses which are applied for the coherent manipulations of

ions. For the pulse shaping of microwave pulses, the amplitude of the microwave field is modulated

by one channel of the AWG’s output with a vector multiplier24 before being amplified. A similar

scheme is used for the pulse shaping of laser pulses, which is described in the next section.

3.8 High-Speed Laser Power Stabilization

A high-speed stabilization scheme is used to reduce noise in the 9Be+ Raman laser beams

shown in Fig. 3.8. As depicted in Fig. 3.11 this is accomplished by a FPGA-based digital servo

[Leibrandt 15] that controls the RF signal amplitude driving the AOM centered at ∼ 200 MHz. The

laser power is adjusted by varying the RF signal injected into the AOM with a vector multiplier25

. The servo loop is completed by sampling a fraction of the laser power with a photodiode whose

photocurrent is sent to the digital servo. The AOM used in this scheme26 is chosen for its fast

rise time of ∼ 20 ns. We estimate the loop bandwidth of this servo to be ∼ 1 MHz, limited by the

digital-signal latency of the servo.

The vector multiplier for controlling the RF signal amplitude applied to the AOM is carefully

chosen. Ideally, the phase of the vector multiplier’s output RF signal should remain constant as

the RF level is modulated for the laser power stabilization. The constancy of the RF phase as a

function of the RF power is important for coherent operations applied to the ion qubits. Varying

RF phases can be viewed as varying rotation angles on a Bloch sphere as the coherent operations

are applied.

Pulse shaping of laser beams is important for the implementation of the high-fidelity two24 Analog device ADL 5390 25 Analog device ADL 5390 26 Brimrose CQM-200-40-313
66

To ion

Pick off mirror

Master 

FPGA

DDS

Multiplier

AOM

200 MHz

(Single pass) 313 nm

Laser 

Switch

UV fiber

RF

RF

AWG

(FPGA based)

D

Digital Servo

(FPGA based)

photodiode

TTL TTL

USB interface

Figure 3.11: Schematic of the laser power stabilization and shaping of laser pulses employed in the

9Be+ qubit manipulation laser beam lines. Each laser beam is sampled at the output of the UV

fiber with a photodiode (see Fig. 3.8). A digital servo is then used to provide the feedback for the

power stabilization by adjusting the radio frequency power level injected into an AOM positioned

upstream of the fiber. One advantage of using the fiber is that laser beam pointing fluctuations

occuring before the fiber are translated into laser power fluctuations at the fiber output, which is

then corrected with the servo system. With the feedback engaged, the time profile of the laser beam

is dynamically modulated by adjusting the servo reference potential using the high-speed arbitrary

waveform generator described in Sec. 3.7.
67

qubit entangling gate described Chap. 6. For the smooth on/off (pulse shaping) of a laser beam,

a precomputed waveform generated from the AWG (see Sec. 3.7) is input into one channel of the

digital servo. This modulates the set point of the loop filter, effectively shaping the time profile of

the laser beam. Typical rise and fall durations of laser beams are ∼ 1 μs.

The power stabilization of the 25Mg+ Raman laser beams is accomplished with an analog

servo27 and the laser beams’ time profiles are rectangular. Pulse shaping was not employed due

to the unavailability of high-speed AOMs at the time when the mixed-species entangling gate

experiment (Chap. 5) was performed. With similar analog servos, the laser powers of the BD

beams for the 9Be+ and 25Mg+ ion are separately stabilized to minimize photon count fluctuations

during fluorescence detections.

New Focus LB1005 27
Chapter 4

Two-Qubit Entangling Gates

4.1 Introduction

In trapped-ion systems, entanglement of the internal states of multiple ions is usually achieved

via an interaction mediated by the Coulomb force between the ions. In this case, a motional degree

of freedom provided by the mutual repulsion and the confinement of the ions serves as an information

bus, allowing the internal states of different ions to interact. Motivated by the first entangling gate

proposal presented by Cirac and Zoller [Cirac 95], now there exists a number of entangling gate

schemes and their experimental implementations.

One promising category of gate schemes uses geometric phase [Sørensen 99, Sørensen 00,

Milburn 00, Solano 99, Leibfried 03, Lee 05], where a state-dependent force is applied to create

an effective spin-spin interaction which can be engineered to create unitary entangling operations.

Compared to the Cirac-Zoller gate, one advantage of the geometric phase gates is that it is more

robust against thermal excitation of the motional degree of freedom, as long as the Lamb-Dicke

criterion is satisfied. Geometric phase gates have been the most successful in terms of the highest

experimentally demonstrated gate fidelity.

Depending on the basis states in which the state-dependent force is applied, as well as other

implementation details, there are a number of variations of the geometric phase gate. Each variation

offers different advantages and disadvantages. For example, geometric phase gates that apply statedependent forces in the σˆφ = cosφσˆx − sinφσˆy basis can be directly implemented with a clock-state

qubit [Langer 05, Lee 05], while geometric phase gates in the σˆz basis rely on the differential AC
69

Controlled-Z Gate

equivalent to

ZZ

control

target

BSB

RR(ππ, 0) Motion |nn = 0⟩

control

target

BSB

RR(ππ, 0)

RSB to Aux

RR(22 , 0)

Figure 4.1: Quantum circuit using the Cirac-Zoller protocol to produce the controlled-Z gate. A

blue-sideband (BSB) π pulse is applied to the “control” qubit before and after a red-sideband

(RSB) 2π pulse is applied to the “target” qubit. The RSB transition couples one of the target’s

qubit states to an auxiliary (Aux) state.

Stark shift of the qubit states and can offer technically simpler laser beam line setups, but cannot

be applied directly on the clock state qubit. In addition, different gate schemes differ in their

sensitivity to error sources.

The objective of this chapter is to provide a background and an overview of the experimental

implementations of two-qubit entangling gates to be presented in Chap. 5, 6, and 7, especially

for readers unfamiliar with geometric phase gates. We first describe the original Cirac-Zoller gate

scheme to illustrate the basic mechanism of a two-qubit gate. This is followed by introducing the

two subcategories of geometric phase gates, i.e. the σˆzσˆz gates and the σˆφσˆφ gates. Then, two

implementations of the σˆφσˆφ gate: (i) Mølmer-Sørensen (MS) protocol, and (ii) Bermudez protocol

will be discussed in Sec. 4.3.4 and 4.3.5. In Sec. 4.4, we discuss error mechanisms that limit the

perfect implementation of these two-qubit gates.

4.2 Cirac-Zoller Gate

The Cirac-Zoller gate [Cirac 95] relies on the blockade of the collective motional harmonic

oscillator’s ground state to provide the quantum logic operation. Without loss of generality, this

protocol requires inducing a blue-sideband (motion-adding) transition between the |↑i and |↓i states

of the control qubit and a red-sideband (motion-subtracting) transition between the target qubit’s

|↑i state and an auxiliary state, |Ai, which has a higher energy than the |↑i state. The pulse

sequence to generate a controlled-Z gate using the Cirac-Zoller gate is shown in Fig. 4.1.
70

To provide a simple picture of the Cirac-Zoller protocol, we write down the action of the

blue-sideband transitions (as described by Eq. 2.5 for a single ion) on control qubit states:

BSB BSB |↓i|n = 0i −→ −i|↑i|1i −→ −|↓i|0i,

BSB |↑i|0i −→ |↑i|0i, (4.1)

BSB where n represents the Fock state quantum number, and each of the symbol −→ indicates a resonant

blue-sideband π pulse. Similarly for the red-sideband transitions applied on the target qubit states:

RSBA RSBA |↑i|1i −→ −i|Ai|0i −→ −|↑i|1i,

RSBA |↑i|0i −→ |↑i|0i,

RSBA |↓i|0i −→ |↓i|0i,

RSBA |↓i|1i −→ |↓i|1i, (4.2)

RSBA where the symbol −→ indicates a resonant red-sideband π pulse applied to the |↑i and the |Ai

states. The Hamiltonian describing this interaction takes the form of Eq. 2.6, and can be written

as

  H † RSBA = ~ΩRSBA |Aih↑|aˆ + |↑ihA|aˆ , (4.3)

where ΩRSBA represents the transition Rabi rate, the transition frequency detuning, δ, is set to

zero for resonant excitation and we have dropped the phase term Δφ in Eq. 2.6 for simplicity.

For convenience, we write |↑icontrol|↑i = |↑↑i and similarly for three other two-qubit target

states. Now we consider the action of the pulse sequence on the four basis states when the motional

mode is prepared in the ground state:

RSB2π BSB A BSB |↑↑i|0i −→ |↑↑i|0i −→ |↑↑i|0i −→ |↑↑i|0i,

RSB2π BSB A BSB |↑↓i|0i −→ |↑↓i|0i −→ |↑↓i|0i −→ |↑↓i|0i,

RSB2π BSB A BSB |↓↑i|0i −→ −i|↑↑i|1i −→ i|↑↑i|1i −→ |↓↑i|0i,

RSB2π BSB A BSB |↓↓i|0i −→ −i|↑↓i|1i −→ −i|↑↓i|1i −→ −|↓↓i|0i, (4.4)
71

RSB2π

where the symbol −→A indicates a red-sideband 2π pulse (with interaction described by Eq. 4.3)

is applied to the target qubit. The net effect of this operation on the spin states is

|↑↑i → |↑↑i

|↑↓i → |↑↓i

|↓↑i → |↓↑i

|↓↓i → −|↓↓i, (4.5)

which is a controlled-Z gate. With the first qubit as the control qubit and the second qubit as the

target qubit. It can be written in the form of unitary matrix:

⎛ ⎞ ⎜

1 0 0 0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0 1 0 0

⎟ Controlled−Z = ⎜ ⎟ ⎜ ⎟ , (4.6)

⎜ 0 ⎟ ⎜

0 0 1

⎟ ⎝ ⎠ 0 0 0 −1

with the basis states as the logical states of the two qubits:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1

⎜ ⎜ ⎟

0

⎟

0

⎜ ⎟ ⎜

0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0

⎟ ⎜

1

⎟ ⎜

0

⎟ ⎜

0

⎟ |↑↑i = ⎜ ⎟ ⎜ ⎜ ⎟ , |↑↓i = ⎟ ⎜ ⎟ , |↓↑i = ⎜ ⎟ |↓↓i ⎜ ⎟ ⎜ ⎟ , = ⎜ ⎟ . (4.7)

⎜⎜

0 ⎟ ⎜ ⎟ ⎜0 ⎟ ⎜ ⎟ ⎜1 ⎟ ⎜ ⎟ ⎟ ⎜

0

⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0 1

By applying two additional single-qubit pulses as shown in Fig. 4.2, the Cirac-Zoller gate

produces the controlled-NOT gate with the truth table of

|↑↑i → |↑↑i

|↑↓i → |↑↓i

|↓↑i → |↓↓i

|↓↓i → |↓↑i, (4.8)
72

equivalent to

control

target

Motion |nn = 0⟩

RR(

ππ

22

,− ππ

22

)

control

target

BSB

RR(ππ, 0)

BSB

RR(ππ, 0)

RSB to Aux

RR(22 , 0)

Controlled-NOT Gate

RR(

ππ

22

, 

ππ

22

)

Controlled-Z Gate

Figure 4.2: Quantum circuit using the Cirac-Zoller protocol to produce the controlled-NOT gate.

The overall pulse sequence implements the unitary operation given in Eq. 4.9. The notation R(θ, φ)

is given in Eq. 2.13.

Or, in the matrix form:

⎛ ⎞ 0

⎜

1 0 0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0 1 0 0

⎟ CNOT = ⎜ ⎟ ⎜ ⎟ . (4.9)

⎜ ⎟ ⎜

0 0 0 1

⎟ ⎝ ⎠ 0 0 1 0

In addition to the requirement for single-qubit addressing, the Cirac-Zoller protocol also

requires preparing the motional state in the ground state, and the performance of the protocol

is directly tied to how well this is accomplished. The Cirac-Zoller protocol was first realized in

Ref. [Monroe 95a] with the control and target qubits encoded in two motional Fock states and two

Zeeman states of a trapped 9Be+ ion, respectively. Two-ion realization of the Cirac-Zoller gate

was first implemented with two 40Ca+ ions [Schmidt-Kaler 03], where single-qubit addressing was

accomplished by tightly focused laser beams. We implemented the Cirac-Zoller controlled-NOT

gate with a 9Be+ and a 25Mg+ ion, which is described in Sec. 5.6.

4.3 Geometric Phase Gates

Unlike the Cirac-Zoller gate where the logic operation is accomplished by the blockade of the

motional ground state, a geometric phase gate creates a forced harmonic oscillator interaction that

drives motional wavefunction around a trajectory in the phase space which is dependent on the

internal qubit states. As shown in Fig. 4.3, if the different displacements enclose a loop, the qubit

states pick up a geometric phase proportional to the area of the enclosed state-dependent loop. An
73

entangling gate can be engineered by choosing an appropriate geometric phase difference between

different qubit states. In this section we decribe some of the basic mechanisms of the gates; more

detailed explanations are given when separate experiments are discussed in Chap. 5, 6, and 7.

Geometric phase gates can be engineered such that the phase-space displacement driven by

the state-dependent force encloses multiple loops in phase space. This strategy has the advantage

of increasing the robustness of the geometric phase gate against timing miscalibration and motional

dephasing [Hayes 12].

4.3.1 Types of Geometric Phase Gates

Depending on the basis states in which the spin-dependent force applies, geometric phase

gate can be separated into two subcategories: (i) the σˆzσˆz gate where the spin-dependent force

acts on the σˆz logical basis, and (ii) the σˆφσˆφ gate where the spin-dependent force acts on the σˆφ

rotated basis.

In this thesis, the laser beam geometry to drive the geometric phase gate is shown in Fig.

4.4. Laser beams 1 and 2, with laser (angular) frequency ω1 and ω2, respectively, are used to drive

stimulated-Raman transitions between hyperfine qubit states. These laser beams intersect at 90◦

such that the difference in their k vectors, Δk, is aligned along the axial direction, in which case

only the axial motional modes will couple to the internal states [Wineland 98].

4.3.2 σˆzσˆz Gate

The σˆzσˆz gate relies on the differential AC Stark shift between the two states of the qubit

to generate a spin-dependent force that applies an effective σˆzσˆz spin-spin interaction. By setting

the frequency difference between the two laser beams in Fig. 4.4 to be ωz + δ, where ωz and δ are

the frequency of a selected motional mode and the detuning of the laser-induced interaction, the

σˆzσˆz-gate Hamiltonian in the interaction frame of qubit and motion after dropping high-frequency
74

p

z

|↑↑ → |↑↑ 

|↓↓ → |↓↓ 

|↑↓ → e 

iΦ|↑↓ 

|↓↑ → e 

iΦ|↓↑ 

Figure 4.3: By applying an effective spin-spin interaction, the motional wavepacket is displaced

in phase space dependent on the internal qubit states. If displacements enclose a loop, the qubit

states pick up a geometric phase, Φ, proportional to the area of the enclosed loop.

k2

k1

Axial direction z 

ω2 ω1

B field ≃ 0.0119 T 

Ions

Figure 4.4: Orientation of Raman laser beams used to drive two-qubit entangling gates. Beams 1

and 2 are arranged such that their k vectors intersect at 90◦ and their Δk vector is aligned along

the axial direction.
75

terms for two qubits in the Lamb-Dicke limit can be written as

    H zz = ~Ω1σˆz,1 aˆe−i(δt−φ1) + aˆ†

ei(δt−φ1) + ~Ω † σˆ aˆe−i(δt−φ2) + aˆ ei(δt−φ2) 2 z,2

X    

= ~ Ω |↑i h↑| − |↓i h↓| aˆe−i(δt−φj ) + aˆ†

ei(δt−φj ) j j j j j , (4.10)

j=1,2

with Ωj = ΩAC,jηξj , where ΩAC,j is the differential AC Stark shift between the two qubit states

induced by the applied laser beams on the jth ion, ξj is the mode amplitude of the jth ion for a

motional mode with a Lamb-Dicke parameter of η, and aˆ is the annihilation operator of the selected

motional mode which the laser-induced interaction couples to. The phase of the laser interaction is

φj = ΔkX0,j + Δφj , where Δk and Δφj are the differences in wave vectors and phases of the laser

fields, and X0,j is the equilibrium position for the jth ion. σˆz is the usual Pauli matrix as shown

in Eq. D.14.

After setting Ω1 = Ω2 = Ω and Δφj = 0 for simplicity, and assuming that the ions’ spatial

separation is set appropriately [Leibfried 03, Lee 05], Eq. 4.10 can be written in a relatively simple

form:

X    

Hzz = ~ − −iδt † iδt Ω |↑i h↑| |↓i h↓| j j j j aeˆ + aˆ e , (4.11)

j=1,2

with Ω as the gate Rabi rate. For t = 2π/δ and Ω = δ/4, the geometric phases acquired by the

qubit states with opposite parity differ by π/2 (Fig. 4.3), and the truth table for the two qubits is

the following:

|↑↑i → |↑↑i

|↑↓i → i|↑↓i

|↓↑i → i|↓↑i

|↓↓i → |↓↓i. (4.12)
76

Or, in matrix form:

⎛ ⎞ ⎜

1 0 0 0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0 i 0 0

⎟ Gˆz = ⎜ ⎟ ⎜ ⎟ . (4.13)

⎜⎜

0 0 i 0 ⎟⎟ ⎝ ⎠ 0 0 0 1

One advantage of the σˆzσˆz gate is that the laser setup required is relatively straightforward. The

relative frequency difference of the laser beams in path 1 and path 2 (in Fig. 4.4) is approximately

the motional frequency, which is usually a few MHz. However, because the σˆzσˆz gate relies on a

differential AC Stark shift on the two logical states of the qubit, the σˆzσˆz gate is not applicable to

a qubit whose transition frequency is first-order independent of magnetic field changes [Lee 05].

The σˆzσˆz gate was first realized in Ref. [Leibfried 03] with two 9Be+ ions, the created Bell

state fidelity was 0.97(2). More recently, a gate fidelity of 0.999(1) was achieved using two 43Ca+

ions [Ballance 16], which is currently one of the best deterministic entangling gates yet reported

[Ballance 16, Gaebler 16].

4.3.3 σˆφσˆφ Gate

The above restriction on qubit transition’s magnetic field sensitivity does not apply for a

state-dependent force acting on the rotated basis, |+i and |−i defined as the eigenstates of σˆφ =

cosφσˆx − sinφσˆy:

  1 |+i = √ |↑i + eiφ|↓i , 2

  1 |−i = √ |↑i − eiφ|↓i , (4.14) 2

The σˆφσˆφ gate is applicable to any qubit state pairs, including the magnetic-field insensitive qubits.

The phase space trajectory driven by the σˆφσˆφ gate is very similar to that of the σˆzσˆz gate (Fig.

4.3), except that |+−i and |−+i states pick up a geometric phase of Φ, while |++i and |−−i states

remain unchanged (assuming the optical phases and ions’ spatial separation are set appropriately).
77

With σˆ+ = |↑ih↓| as shown in Eq. D.14, one simple form of the σˆφσˆφ gate interaction-frame

Hamiltonian in the Lamb-Dicke limit after dropping high-frequency terms can be written as

X  

H = ~Ω σˆ+ − † iδt

φφ j ae iδt ˆ + aˆ e + h.c., (4.15)

j=1,2

where the gate Rabi rate Ω is assumed to be equal on both ions. For simplicity, we have dropped

terms related to the wave vectors and phases of optical fields in Eq. 4.15. The exact form of a

σˆφσˆφ Hamiltonian depends on experimental setup and the influences of these phase terms will be

discussed in detail for different implementations of the σˆφσˆφ gates (next two sections), and in the

respective chapters presenting different experimental realization of the σˆφσˆφ gates.

Using the rotated basis definition given by Eq. 4.14, we can rewrite the Eq. 4.15 to be:

X     ~ H −iδt † φφ = Ω |+i h h−| j +| − |−i j j j aeˆ + aˆ eiδt , (4.16)

j=1,2

this shows the similarity of this expression with Eq. 4.11, illustrating the basis states in which the

spin-dependent forces are applied for the σˆφσˆφ and σˆzσˆz gate. The truth table in the rotated basis

for the σˆφσˆφ gate is

|++i → |++i

|+−i → i|+−i

|−+i → i|−+i

|−−i → |−−i. (4.17)

The unitary operation of the σˆφσˆφ gate written in terms of the basis states of Eq. 4.7 is

⎛ ⎞ ⎜

1 0 0 −i

⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜

0 1 1 0

⎟ Gˆφφ = √ ⎜ ⎟ ⎜ ⎟ . (4.18) 2

⎜ ⎟ ⎜

0 1 1 0

⎟ ⎝ ⎠ i 0 0 1

Compared to the σˆzσˆz gate, the laser setup for σˆφσˆφ gate is relatively more complicated,

primarily because the relative frequency difference of the laser beams in path 1 and path 2 (Fig.
78

4.4) is separated by approximately the qubit frequency, which is usually a few GHz. Depending

on the choice of qubit states and the exact implementation of the σˆφσˆφ gate, careful consideration

is needed for the generation of deterministic entanglement [Lee 05]. In the next two sections, we

introduce two different protocols for the implementation of the σˆφσˆφ gate: the Mølmer-Sørensen

protocol and the Bermudez protocol.

4.3.4 Mølmer-Sørensen Protocol

The Mølmer-Sørensen (MS) protocol [Sørensen 99, Sørensen 00, Milburn 00, Solano 99] requires simultaneous excitation of a blue-sideband transition with a detuning of δ and a red-sideband

transition with a detuning of −δ for a selected motional mode. Following the spin-motion Hamiltonian described in Sec. 2.1.3, the Hamiltonian of the MS protocol in the interaction picture is

[Sørensen 99, Sørensen 00]

X   X  

σ+ −i(δj t−φj,r) σ H + † i(δj t+φj,b) MS = ~ j Ωr,j ˆ ˆae + ~ Ωb,j ˆ aˆ e + h.c., (4.19) j

j=1,2 j=1,2

with Ωr(b),j = Ω0,r(b),jηξj where Ω0,r(b),j is the Rabi rate of the resonant carrier excitation induced

by the laser beams responsible for the red (r) and blue (b) sideband excitations, η is the Lamb-Dicke

parameter of the selected motional mode with ξj as the mode amplitude of the jth ion. If we set

Ω0,r,j = Ω0,b,j = Ω0,j then Eq. 4.19 becomes

X  

H † MS = ~ Ωjσˆj

+ aˆe−i(δj t−φj,r) + ˆa ei(δj t+φj,b) + h.c., (4.20)

j=1,2

with Ωj = Ω0,jηξj . The phases of the red (r) and blue (b) sideband interactions are φj,r(b) =

Δkj,r(b)X0,j + Δφj,r(b) where Δkj,r(b) and Δφj,r(b) are the differences in wave vectors and phases of

the optical fields driving the red and blue sideband transitions respectively, and X0,j is the equilibrium position for the jth ion. By setting t = 2π/δ and Ω1 = Ω2 = Ω = δ/4, the computational
79

truth table for the Mølmer-Sørensen gate is the following:

 P  1 −i( 1

√ + e (φ + |↑↑i → |↑↑i j=1,2 j,b φj,r)+π/2 2 )|↓↓i 2

1 |↑↓i → √ (|↑↓i + |↓↑i) 2

1 |↓↑i → √ (|↓↑i + |↑↓i) 2

 P  1 |↓↓i → √ |↓↓i + ei( 1 (φ +φ )+π/2 j=1,2 j,b j,r 2 )|↑↑i . (4.21) 2

Depending on the laser beams geometry that is used to implement the MS gate, the phase

difference between the two parity components of the produced entangled state can fluctuate, in

turn causing decoherence [Lee 05]. Issues of the MS gates’ laser phase sensitivity will be discussed

in detail for the laser beam geometry used in the mixed-species entangling gate experiment and the

high-fidelity two-qubit gate experiment, described in Chap. 5 and Chap. 6, respectively.

4.3.5 Bermudez Protocol

Another protocol to implement the σˆφσˆφ geometric phase gate is given by Bermudez et

al. [Bermudez 12]. It requires a resonant carrier Rabi oscillation excitation and a single spinmotion sideband spin-flip excitation |↓, ni ↔ |↑, n + 1i or |↑, n − 1i with a detuning of δ on a

selected motional normal mode. This gate scheme features built-in dynamical decoupling [Viola 98,

Lidar 14] that can protect the qubit against certain decoherence errors during the application of

the gate [Bermudez 12].

In the interaction frame for both the spin and motion, and after dropping high-frequency

terms, the Hamiltonian for two ions driven by Bermudez protocol with a blue-sideband excitation

in the Lamb-Dicke limit is

X  

~ + iφ + 0

HB = ΩCσˆj e + iΩjσˆj aˆ†e−iδteiφj + h.c., (4.22)

j=1,2

where ΩC denotes the Rabi rate of the carrier excitation that provides the dynamical decoupling,

and Ωj = Ω0,jηξj , where Ω0,j is the Rabi rate of the resonant laser-driven carrier excitation, η is

the Lamb-Dicke parameter for the selected motional normal mode with ξj as the mode amplitude
 

 

80

of the jth ion, and φ and φ0

j are the respective phases of the carrier and sideband excitation for

the jth ion.

The key concept of this gate scheme is to use a strong carrier excitation to create dressed

states of the qubits, and apply a spin-dependent force in the dressed-state basis using the single

spin-motion sideband excitation. The dressed state created by the strong carrier excitation results

is an effective dynamical decoupling mechanism that reduces the sensitivity of the qubits to certain

dephasing errors. As a simple illustration, consider a two-level system with an unperturbed qubit

frequency of ω0. Under the influence of a certain mechanism that shifts the qubit frequency by ω0

,

the system Hamiltonian can be written as

⎛ ⎜⎝ 

1 0 ⎞ ⎟⎠ +

1

~ω0

⎛ ⎜⎝ 

1 0 ⎞ ⎟⎠ . (4.23) 1 H = ~ω0 2 2 0 −1 0 −1

Now, if we apply a continuous spin-flip excitation to the system with a Rabi rate of ΩC which

satisfies the condition of ω0  ΩC  ω0

, the Hamiltonian in the interaction frame of the qubit

becomes

⎛ ⎜⎝ 

1 0 ⎞ ⎟⎠ + ~ΩC

⎛ ⎜⎝ ⎞ ⎟⎠ . (4.24)

0 1

HI = 1~ω0

2 0 −1 1 0

The eigenvalues, ω0

0 , of this system, which represent the effective shift of the unperturbed system’s

qubit frequency, are obtained by diagonalizing the Hamiltonian:

1

ω0

0 = ±

2

q

ω02 + Ω2

C

s ω02

Ω2

C

1 = ± ΩC 1 + 2

  ! 2

ω02 ω02 1 1

≈ ± ΩC 1 + + + · · · 2 Ω2 2 Ω2

C C

  ! 2 1 ω02 1 ω02

= ± ΩC + + ΩC + · · · . (4.25) 2 ΩC 2 Ω2

C

It becomes apparent that the influence of ω0 becomes a second order effect and decreases as ΩC increases. However, now the overall frequency shift, ω0

0 , depends linearly on ΩC. Therefore, although
81

dynamical decoupling can provide protections against uncontrolled environmental noise (ω0

), it

requires the control field to be constant.

Since the interesting case is when ΩC  |Ωj |, we shift to the interaction frame, where the

states are dressed by the carrier excitation, and consider the effects of the sideband term. For

simplicity, we set φ = φ0

j = 0 since they are not important for the description below. In the |+i, |−i

basis according to Eq. 4.14, the Hamiltonian in Eq. 4.22 becomes [Bermudez 12, Tan 13]

X     Ωj † −iδt − ˆ H iδt

B = i~ |+ij h+|

j − |−ij h−|j aˆ e ae

2

j

    −iδt +i~ + ˆ X Ωj |−ij h+|

je−2iΩC t − |+ij h−|je2iΩC t aˆ†

e aeiδt . (4.26) 2

j

The second term induces off-resonant transitions between the dressed states |+i and |−i and can

be neglected under the assumption that ΩC  |Ωj |. The first term is identical to Eq. 4.16, which

describe a Hamiltonian of a geometric phase gate with the spin-dependent force acting on the

rotated basis. In contrast to the MS protocol, the entangled states produced by Bermudez protocol

are insensitive to optical path length changes of the non-copropagating laser beams (Fig. 4.4)

occurring on a time scale that is long compared to the gate duration. This offers technical simplicity

in terms of implementation. The laser phase insensitivity and experimental implementations of the

Bermudez gate using two 9Be+ ions will be described in Chap. 7. Furthermore, compared to the

MS protocol, error associated with the Rabi rate imbalance of the detuned-blue and detuned-red

sideband is removed with this implementation of the Bermudez protocol.

4.4 Error Sources for Geometric Phase Gates

One of the most important aspects of implementing two-qubit entangling gates is the study

of imperfections and error mechanisms. These studies provide insights into whether these limiting

factors are of a fundamental nature, or technical challenges. In addition, understanding the sources

of imperfection provides motivation to innovate alternative techniques for the further improvement

of gate performance.

Here, we describe error mechanisms that are considered for the experimental implementations
82

of two-qubit gates presented in Chap. 5, 6, and 7. These experiments implement the σˆφσˆφ gates;

therefore the discussions below focuses on error sources associated with them, but is not limited

to the σˆφσˆφ gate. As mentioned before, the objective here is to provide an overview. Further

discussions and evaluations of error sources for different experiments will be presented in their

individual chapter in the remaining of this thesis.

A combination of analytic methods, Monte Carlo simulations, Schr ̈odinger equation simulations, and master equation simulations are used for the study of these error mechanisms, with the

latter used heavily in this thesis.

4.4.1 Schr ̈odinger and Master Equation Simulations

In this dissertation work, we use the Schr ̈odinger equation to study the dynamics of quantum

systems of interest when dissipation is not considered. We also use the Schr ̈odinger equation to

study coherent errors, e.g., errors due to slow fluctuations in laser intensity, timing, frequency

offsets and calibration drifts. The Schr ̈odinger equation is

d|Ψi i~ = H|Ψi, (4.27) dt

where |Ψi is the state vector representing the system, and H is the Hamiltonian describing the

coherent interaction.

The dynamics of an open-quantum system can be simulated with a master equation [Gardiner 00],

which allows multiple dissipative error mechanisms and coherent operations to be combined and

simulated. The master equation used here is in the Lindblad form [Lindblad 76]

X  dρ 1 ˆ ˆ ˆ ˆ † † 1 1 † = [H, ρ] + LjρLˆ − L Ljρ − ρLˆ Lj , (4.28) j j j dt i~ 2 2

j

where ρ is the density matrix of the system, and Lˆj is the Lindblad operator (or quantum jump

operator) describing the jth decoherence/dissipative process. Readers unfamiliar with the master

equation will find Ref. [Dyrting 95, Schneider 98, Puri 01, Harty 13] useful.

Master equation simulations are more computer-resource demanding. For a quantum system

represented by a matrix of dimension M × 1, the density matrix required for the equivalent master
83

equation simulation has a dimension of M ×M. This is precisely the problem faced by conventional

computation; that is, the number of parameters needed to keep track of a quantum system grows

exponentially as the system size increases. For example, in a system of N qubits, M scales as

2N . Nevertheless, master equation simulations are used intensively to study the system dynamics

under the influences of dissipative error sources, such as spontaneous scattering of photons, heating

of motional modes, qubit dephasing, and motional mode dephasing. Furthermore, we have also

combined Schr ̈odinger equations and master equations with Monte Carlo simulations.

The quantum systems included for the simulations of two-qubit gates are the spin of each ion,

and harmonic oscillators to represent the ions’ motion. The harmonic oscillator energy eigenstates

span an infinite dimensional Hilbert space. However, since most of the relevant dymanics occur

close to the ground state, we select a motional system size which includes the ground state up to

the Fock state |Nmaxi that strikes a balance between the required computational resources and

an adequate representation of our system. Depending on which dynamics and properties are of

interest, Nmax ranges from ' 10 to 100.

Due to the internal atomic state structures, our choices of hyperfine states as qubits, and the

fluorescence detection scheme for the 9Be+ and 25Mg+ ions (see Sec. 2.1.7 and Sec. 2.2.6), there

are times where the states outside of the qubit |↑i and |↓i manifold must be included (see Fig. 2.1

and Fig. 2.6 for energy level diagrams for 9Be+ and 25Mg+, respectively). For example, population

can leak from the qubit manifold into other hyperfine states through spontaneous Raman scattering

during the gate. Furthermore, due to the finite fidelity of (i) optical pumping and (ii) population

transfer between computational and measurement qubit manifolds, these operations result in nonzero populations outside the qubit manifold. These populations do not interact with laser-induced

operations on the qubit transition and are predominantly detected as “dark” (see Sec. 2.1.7). To

better capture the dynamics and effects due to these populations, we include a single auxiliary

state (for each ion) in our model for numerical simulation, which consolidates all population in the

non-qubit-manifold. The simplification is made because the population in each of the other Zeeman

states is small (on most scenarios considered). We label this state as |Ai and it is modeled to have
84

the same detection outcome as the |↓i state. In our model, unless stated otherwise, the only spin

states that interact with the applied coherent operations are the qubit |↑i and |↓i states.

Here, most of the numerical simulations on quantum systems are performed using matrix

representations of states and operators (see Appendix D) with the Mathematica software.

4.4.2 Spontaneous Scattering of Photons

One of the dominant decoherence sources when driving stimulated-Raman transitions is the

error caused by spontaneous emission of photons [Wineland 98]. With the initial state of |ii = |↑i

or |↓i state, the spontaneous scattering rate into the final state |fi via the intermediate states |ji

is given by the Kramer-Heisenberg equation [Cline 94]

X E2

m











 XX 

j k

Γj

hf|d · σˆk|jihj|d · σˆk |ii q L,m

Δj











 

2

Γi→f = , (4.29) 4~2

m

where Em is the electric field amplitude of the mth laser beam, d is the electric-dipole operator,

and Δj is the detuning of the Raman laser from each of the i ↔ j transitions. Here, σˆk

L,m represents

the polarization vectors of the mth laser beam and k indexes the different polarization components.

They are expressed in terms of linear (ˆπ), left (ˆσ−) and right (ˆσ+) circular polarizations such that

σˆL,m σ− σ = + − ˆ +0 πˆm ++ ˆ with |− m m m m m m|

2

+

0

m





2

+|+

m|

2 = 1. Similarly, σˆq contains the three possible

polarization-dependent decay channels from the jth excited state to the fth final state. Note that

coupling between different states of the same polarization can interfere with each other, but there

is no interference between coupling of different polarizations. For 9Be+ and 25Mg+ the jth excited

states are the different hyperfine Zeeman states in the P1/2 and the P3/2 electronic excited states,

and we can approximate the natural decay rate of each jth state to be Γj = Γ.

The total probability of spontaneous emission PS.E for a laser-driven spin-flip transition is

X 1 PS.E. = (Γ↑→f + Γ↓→f ) τπ, (4.30) 2

f

where τπ is the duration needed for the π pulse, which is defined by Ωτ = π/2. Now, Eq. 4.30

becomes

P π f (Γ↑→f + Γ↓→f ) PS.E. = . (4.31) 2 2Ω
85

Depending on the final state after each scattering event, spontaneous emission can be separated

into Raman and Rayleigh scattering, and the error mechanisms in which they cause error can be

different [Ozeri 05, Ozeri 07].

4.4.2.1 Raman Scattering

Raman scattering processes are inelastic and project an ion’s internal state to one of the

Zeeman states which is different from the initial state, destroying coherence. Raman scattering

processes can be further separated into two processes: (i) out-of-manifold scattering, and (ii) bitflip scattering. The Lindbald operators describing the Raman spontaneous emission out-of-manifold

scattering are

p Lˆ↑A(Γ↑A) = Γ↑Aσˆ↑A, (4.32)

p Lˆ↓A(Γ↓A) = Γ↓Aσˆ↓A, (4.33)

where σˆ↑A = |Aih↑| and σˆ↓A = |Aih↓|, Γ↑A (Γ↓A) indicates the scattering rate from the |↑i (|↓i)

state to the |Ai state, respectively (see Appendix D for the matrix representations of σˆ↑A and σˆ↓A).

In actuality, there are multiple final states that can be populated by an out-of-manifold scattering

event with different scattering rate (see Fig. 2.1 for energy level diagram of the 9Be+ ion). In our

model, the scattering rates Γ↑A and Γ↓A are the sum of scattering rates from each of the qubit

states to other non-qubit-manifold states. The Lindblad operators for the bit-flip scattering are

[Puri 01]

p ˆ σ− L↑↓(Γ↑↓) = Γ↑↓ ˆ , (4.34)

p Lˆ↓↑(Γ↓↑) = Γ↓↑σˆ+, (4.35)

where Γ↑↓ represents the scattering rate starting from the |↑i state and ending in the |↓i state.

Similarly for Γ↓↑. Both the out-of-manifold and bit-flip scattering events destroy coherence. However, in contrast to the out-of-manifold scattering, the bit-flip scattering event does not result in

qubit manifold population leakage. Raman scattering processes contribute large portions of errors
86

to the entangling gates implemented in Chap. 5, 6, and 7, these scattering errors are accounted for

and are discussed further in their individual chapter.

Another process which is not considered here is the events where population from the nonqubit manifold enters the qubit manifold via Raman scattering. Usually the population in the

non-qubit manifold is small; therefore this process is a second order process that we assume to have

negligible effect on the dynamics of the system under studies.

4.4.2.2 Rayleigh Scattering

Rayleigh scattering processes are elastic, i.e. the scattering event puts the atomic state back

to its original state. Depending on the choice of atomic states for the qubit, Rayleigh scattering

processes do not necessarily cause spin decoherence. In general, different atomic states have unequal

spontaneous scattering rates. Unequal Rayleigh scattering rates for the two qubit states produce an

overall spin-dephasing error [Ozeri 07, Uys 10], which can be described by the Lindblad operator

describing qubit decoherence [Uys 10]:

p LˆR(ΓR) = ΓRσˆz, (4.36)

with the Rayleigh qubit dephasing rate, ΓR given by [Uys 10]

   !

2

X

2

XX h↑|d · σˆk k  E q |jihj|d · σˆ |↑i h↓|d · σˆk

q |jihj|d · σˆk |↓i  m 

L,m L,m Γ  R = . (4.37)



Γ

 ~ j − 2 m  

Δj Δj j k  Rayleigh scattering does not cause a spin-dephasing error if a magnetic-field insensitive states pair

is used as qubit. This is because the Rayleigh scattering amplitude for each qubit state is equal,

thus the two terms in Eq. 4.37 cancel each other, resulting in a vanishing decoherence rate. The

entangling gates presented in this thesis are all implemented on magnetic-field insensitive qubits;

therefore the Rayleigh dephasing error does not apply. Although Rayleigh scattering does not

directly cause spin decoherence to a magnetic-field-insensitive qubit, it can cause errors to a twoqubit gate based on geometric phase via motional decoherence [Ozeri 07].
87

4.4.2.3 Motional Dephasing due to Spontaneous Scattering

The direction of each scattered photon is random, which results in a random recoil momentum

kick to the ion’s motion. This distorts the phase-space trajectory as the state-dependent force is

applied, and causes phase errors in the final state. The worst-case error is given by Eq. 57 in Ref.

[Ozeri 07]:

5 η2

Recoil = PS.E., (4.38) 24 Nloop

where η is the Lamb-Dicke parameter of the motional bus mode, and Nloop is the number of phase

space loops used for the geometric phase gate. The total scattering probability, PS.E. is described

by Eq. 4.30 that sums over all possible final states.

The scattering probabilities of Raman, Rayleigh and their sum for a two-qubit MølmerSørensen gate are shown in Fig. 4.5 and Fig. 4.6 for a 9Be+ and 25Mg+ ion, respectively. The

dephasing errors associated with the momentum kicks caused by spontaneous scattering are also

shown in these two figures.

Motional dephasing due to spontaneous scattering of photons contributes negligible errors

(compared to other dominant errors) to the mixed-species entangling gate (Chap. 5) and the

dynamical decoupling gate (Chap. 7). However, this error is non-negligible in the experiments

presented in Chap. 6, where a Raman detuning as high as Δ ∼ −2π × 900 GHz is used. Further

discussions are in their individual chapters.

Although Raman scattering can be reduced by increasing |Δ|, the Rayleigh scattering error

reaches an asymptotic value as |Δ| is increased. This error is proportional to the Lamb-Dicke

parameter and thus could be reduced by increasing the trap frequency; it can also be reduced by

using multiple loops in phase space [Ozeri 07, Hayes 12] (see Eq. 4.38). However, these methods

reduce the gate Rabi rate and thus increase the Raman scattering error.

Depending on the ion species, Rayleigh scattering can set an absolute lower limit for the

error of a geometric phase gate implemented with stimulated-Raman transitions. This is more

pronounced in the case for ion species which have a low-lying D electronic excited state [Ozeri 07].
88

Figure 4.5: This figure shows the different scattering probabilities and errors caused by spontaneous scattering of photons due to Raman laser beams for a two-qubit gate driven with the

Mølmer-Sørensen protocol on 9Be+ ions. Spontaneous scattering of photons can be categorized

into Raman and Rayleigh scattering, and they contribute errors to a two-qubit gate differently.

The qubit’s coherence is lost after each Raman scattering event, while this is not the case for

Rayleigh scattering. Both Raman and Rayleigh process contribute to motional dephasing error

through a random trajectory in the motional phase space caused by a uncontrolled momentum

kick for each scattering event. The horizontal axis indicates the detuning of the Raman laser with

respect to the 2S1/2 ↔ 2P1/2 transition of the 9Be+ ion. The second peak at ∼ +200 GHz Raman

detuning corresponds to scattering from the 2P3/2 electronic excited state. The dips between the

two fine structure lines are caused by the cancellation effect as indicated in Eq. 4.29; the Raman

laser is blue detuned relative to the 2P1/2 levels while red detuned relative to the 2P3/2 levels. This

figure is plotted with a Lamb-Dicke parameter of η = 0.19, NLoop = 1, and assuming equal laser

intensity in each of the two laser paths as depicted in Fig. 4.4. The polarization of the k1 (k2) is

set to be pure σ+ (π).
89

Figure 4.6: This figure is analogous to Fig. 4.5 but for 25Mg+ ions and with η = 0.165. The Raman

detuning, Δ is plotted with respect to the 2P3/2 excited state, where the frequency of our 25Mg+

Raman laser is tuned near to. The two peaks correspond to the fine structure splitting between

the 2P1/2 and 2P3/2 electronic excited states, which is ∼ 2.75 GHz.
90

4.4.3 Motional Frequency Fluctuations

As the geometric phase gates rely on the coherent evolution of both the spin and motional

degrees of freedom, motional frequency fluctuations affect the gate’s performance in a similar fashion

as qubit dephasing. Motional frequency fluctuation can be described by the Lindblad operator of

[Turchette 00b]

p ˆ a† Lmotion(Γmotion) = Γmotionˆ a, ˆ (4.39)

Physical sources of frequency fluctuation include (i) fluctuations in the control potentials

applied to electrodes for trapping, (ii) fluctuating electric-field gradients from uncontrolled charging

of electrode surfaces [Harlander 10], and (iii) for gates applied on the axial stretch mode, the nonlinear coupling to radial “rocking” modes [Roos 08, Nie 09]. For (iii), to a good approximation,

the shift of the stretch mode frequency, δωS from the thermal excitation of these modes is given by

δωS = χ(nx +ny +1) where χ is the non-linear coupling term, nx and ny are Fock state occupation

numbers of the two transverse (radial) rocking modes. As these fluctuations are dependent on the

occupation numbers of the radial modes, the error can be suppressed by cooling the radial modes

to near the ground state.

Entangling gate errors caused by motional frequency fluctuations are studied closely in the

implementations with two 9Be+ ions (see Chap. 6); motional frequency fluctuations cause negligible

errors to the mixed-species entangling gate and the dynamical decoupling gate presented in Chap.

5 and Chap. 7, respectively.

4.4.4 Motional Heating

Although the geometric phase gate is robust against thermal excitation (the motional ground

state is incompletely populated) [Sørensen 99, Sørensen 00], changes in motion states during the

application of the gate cause decoherence, which leads to errors. This heating can be modeled as

an exchange of energy between the motional harmonic oscillator and a thermal bath with much

higher temperature. Uncontrolled couplings between these two systems consist of a heating and a
91

cooling process, and they can be described by the following Lindblad operators [Sørensen 00]:

ˆ p † Lheat(Γh,c) = Γh,caˆ , (4.40)

p Lˆcool(Γh,c) = Γh,ca, ˆ (4.41)

where Γh,c is the heating rate. One of the dominating heating effects is due to the uncontrolled

electric field noise from the trapping electrodes [Turchette 00a]. The exact mechanism of this

“anomalous heating” is currently not understood, and there is active research to study this problem,

see for examples Ref. [Allcock 11, Hite 13, Daniilidis 11, Brownnutt 15, McConnell 15]. Lindblad

operators given in Eq. 4.40 and Eq. 4.41 are used in master equations (Eq. 4.28) to model the

dynamics of entangling gates presented in Chap. 5, 6, and 7; we find good agreements between

experiment data and our model.

4.4.5 Off-Resonance Coupling to Spectator Modes

The Hamiltonians shown in Eq. 4.11 and Eq. 4.15 describe geometric phase gates with

the spin-spin interaction implemented by coupling to a single mode of motion. As other motional

modes can also be driven off-resonantly by the applied laser fields, this can induce errors due to

the imperfect implementation of the desired Hamiltonians.

We consider the case with two ions driven by the Mølmer-Sørensen interaction using the

laser beams geometry depicted in Fig. 4.4. Following Eq. C.19, the interaction frame Hamiltonian

describing a blue-sideband excitation and a red-sideband excitation before dropping high-frequency

terms is

" # X X   �  H † † iωkt −iΔωBt −iΔωRt

MS = ~ Ω0,jσˆjeiω0t

Exp i ηkξk,j aˆke−iωkt + ˆake e + e

j=1,2 k

+h.c., (4.42)

where k indexes the different motional normal modes with mode frequency ωk that can interact with

laser geometry setup, Ω0,j and ξk,j are the resonant carrier Rabi frequency and the mode amplitude

for the kth normal mode of the jth ion, and ΔωB,R are the relative frequency differences between
92

the Raman laser fields responsible for the detuned blue (B) and the detuned red (R) sideband

excitation, respectively. Assuming that the ion crystal is perfectly aligned with the Δk of the laser

fields such that only normal modes along the axial direction can be driven, and considering the

Lamb-Dicke expansion up to the first order in ηk (Eq. C.17), the Hamiltonian described in Eq.

4.42 becomes

X h  

H † iω0t † iωC t

MS = ~ Ω0,jσˆje 1 + iηCξC,j aˆCe−iωC t + ˆaCe

j=1,2

  i �  † iωSt −iΔωBt −iΔωRt +iηSξS,j aˆSe−iωSt + ˆaSe + . . . e + e + h.c., (4.43)

where ηC,S, ωC,S, and aˆC,S are the Lamb-Dicke parameter, the normal mode frequency, and the

annihilation operator for the center-of-mass (C) mode and stretch (S) mode, respectively.

To generate a spin-dependent force mediated by the center-of-mass mode, we set the relative

frequency differences between the two set of Raman laser beams to be ΔωB = ω0 + ωC − δ and

ΔωR = ω0 − ωC + δ. With this, the Hamiltonian (before dropping high-frequency terms) is

X h 

H † −i(ωC −δ)t i(ωC −δ)t

MS = ~ Ω0,jσˆ e + e j

j=1,2

  † † iδt +iηCξC,j aˆCe−iδt + ˆaCe−i(2ωC −δ)t + ˆaCei(2ωC −δ)t + ˆaCe

 −i(ωS−ωC +δ)t +iηSξS,j aˆSe−i(ωS+ωC −δ)t + ˆaSe

 i † † i(ωS+ωC −δ)t +ˆaSei(ωS−ωC +δ)t + ˆaSe + . . . + h.c.. (4.44)

−iδt † iδt The aˆCe and aˆCe terms (in the second term inside the square bracket) represent the MS

interaction acting on the center-of-mass mode. All other terms represent off-resonant excitations.

For example, the first term in the square bracket represents off-resonant excitation of the car-

−i(ωS−ωC +δ)t † i(ωS−ωC +δ)t rier transition, the aˆSe and aˆSe terms (in the third term inside the square

bracket) represent off-resonant excitation of the stretch mode with a detuning approximately equal

to the mode frequency difference of the center-of-mass mode and the stretch mode. These highfrequency terms are usually dropped (ωC, ωS  δ) to give a reasonable approximation to the gate

[Tan 13, Tan 15]. However, it is important to consider the effects of these off-resonant excitations
93

for the high-fidelity implementation of the MS gate. Usually an off-resonant carrier excitation has

a bigger impact than the off-resonant stretch mode excitation, this is because the strech mode

excitation is scaled by the Lamb-Dicke parameter (ηS) compared to the carrier excitation.

Furthermore, the Raman laser beams could be imperfectly aligned such that the Δk vector

has components in the radial directions. For such cases, Eq. 4.44 can be further generalized to

include the additional motional modes. The coupling strength to motional mode is parametrized by

the Lamb-Dicke parameter which quantifies the overlap of the laser beams’ Δk and the motional

modes. Although the coupling strengths to the radial modes are (usually) relatively small given

reasonable laser beam alignment and the transition frequency detuning is relatively large, certain

higher-order coupling (second or higher-order terms in the Lamb-Dicke expansion, see Eq. C.17)

terms could have a relatively small detuning that induce additional errors.

In addition to coupling to spectator motional modes, other atomic transitions can also be

off-resonantly driven, especially in the presence of micromotion. The two-qubit gates presented in

Chap. 5 and Chap. 6 are implemented inside an ion trap where the excessive axial micromotion

could not be compensated (see Sec. 2.1.3 and Sec. 3.1). The frequency of the trap RF, ωRF , is

comparable to the Zeeman splittings of 9Be+ ion such that spectator atomic transitions could be

coupled on other micromotion sidebands when driven by the laser beam geometry depicted in Fig.

4.4

Off-resonant coupling to spectator transitions can be suppressed by employing laser pulse

shaping (see Sec. 3.8). By smoothing out the rise and fall profile of the laser beam, Fourier

components at the frequencies of spectator transition can be made sufficiently small. With such

dynamic shaping of laser pulses, gate errors contributed by off-resonant coupling are made negligible

for the entangling gates presented in Chap. 6. Laser pulse shaping is not implemented for the gates

presented in Chap. 5 and Chap. 7 due to the unavailability of high-speed apparatus (see Sec. 3.8)

when these two experiments were performed. However, errors caused by off-resonant coupling are

small compared to other dominant errors in these two implementations.
94

4.4.6 Imperfect Lamb-Dicke Approximation

Geometric phase gates are robust against thermal excitation in the Lamb-Dicke limit [Sørensen 99,

Sørensen 00]. However, depending on the ion species and the confinement parameters, this condition might not be rigorously satisfied. Because of this, the realistic MS interaction (used here as

an example) applied to the ions deviates from the desired Hamiltonian of Eq. 4.15.

One such effect is the dependence of sideband transition Rabi rates on the motional Fock

states. In general, the Rabi rate of spin-motional transitions between Fock states labeled by their

respective quantum number n and n0 is given in Eq. 2.15. The non-ideal implementation of the

desired Hamiltonian due to the different Rabi rates between different Fock states has been studied

in Ref. [Sørensen 00] for the MS gate, with error

π2

LD = η4V ar(n), (4.45) 4

π2

= η4n ̄( ̄n + 1), (4.46) 4

where the assumption of a thermal distribution (Eq. 2.10) with mean occupation number of n ̄ for

the motional mode is made in Eq. 4.46. For the parameters used in Chap. 5, 6, and 7, the gate

errors caused by this mechanism are usually small compare to other error sources.

4.4.7 Rabi Rate Fluctuations

Errors can be caused by fluctuations in the Rabi rates of the geometric phase gates, which

cause fluctuations in the state-dependent forces. Physical sources includes (i) fluctuations in the

laser beam intensities at the ion location, (ii) fluctuations of the ion’s micro-motion amplitude, and

(iii) fluctuations in the Debye-Waller factors of spectator motional modes that cause fluctuations

in the Rabi rate of spin-motion transitions.

For most cases, the gate Rabi rate fluctuations occur at a time scale slower than the gate

speed. Thus it can be modeled with trial-to-trial fluctuations. Through numerical simulations by

solving the Schr ̈odinger equation combined with a Monte Carlo simulation, we derived an empirical
95

formula of

 2 δΩ

Ω ' 2.5 × , (4.47) Ω

describing the error of creating a Bell state with a single-loop geometric phase gate due to Rabi

rate fluctuations (this agrees with the expression in Ref. [Benhelm 08]).

Debye-Waller coupling is the effect where the Rabi rate of a particular spin-motion transition

is dependent on the excitations of other motional modes. This is another effect that causes the

implemented Hamiltonian to deviate from the desired Hamiltonian of Eq. 4.15.

Consider two ions interacting with two Raman laser fields arranged in the geometry shown

in Fig. 4.4 where only the two axial motional modes can be driven. The Rabi rate of a spin-motion

transition driven resonantly on the stretch mode between different Fock states is generalized from

Eq. 2.15 and given by [Wineland 98]:

  
  † †

Ω 0  iηS(aˆS+aˆ ) iη (aˆ +aˆ ) S C C C  n ,n0 = Ω0 

nC, nS e e | S nC, nSi , (4.48) S

where Ω0 is the Rabi rate of the resonant carrier transition, nC,S, ηC,S and aˆC,S are the Fock state

quantum number, Lamb-Dicke parameter, and the annihilation operator for the center-of-mass (C)

and stretch (S) mode, respectively. Thus, the Rabi rate deviations for given values n 0 S and nS on

the stretch mode are given by [Wineland 98]

s   2

δΩn ,n0 = Ωn ,n0 − h n 0 i S S Ω , (4.49) ,n S S S S

q

= hΩ2 i − hΩ 0 i2 0 nS,n . (4.50) nS,nS S

Assuming that the center-of-mass mode follows a thermal distribution, with Eq. 4.50, the fractional

Rabi rate fluctuations can be simplified and approximated, given by [Wineland 98]:

δΩ 0 q nS,nS η ≈ 4

Cn ̄C (n ̄C + 1). (4.51) ΩnS,n0

S

Analogously, the Rabi rate of the center-of-mass mode is also dependent on the stretch mode. The

errors contributed to the entangling gate can then be computed with Eq. 4.47.

Equation 4.50 can be easily generalized to include the contributions from radial modes, where

applicable depending on the laser beams geometry. In general, given a thermal distribution for each
96

of the multiple spectator modes, the r.m.s. Rabi rate fluctuation of the mode of interest can be

described by

δΩ sX ≈ η4n ̄k ( ̄nk + 1), (4.52) k Ω

k

where ηk and n ̄k are the Lamb-Dicke parameter and the mean occupation number for the kth

spectator mode [Wineland 98].

Errors caused by Rabi rate fluctuations are more pronounced in the dynamical decoupling

entangling gate presented in Chap. 7 due to a less robust laser beams setup in a legacy system.

With an upgraded experimental setup (see Sec. 3), the mixed-species entangling gate (Chap. 5)

and the entangling gate with two 9Be+ (Chap. 6) suffer less Rabi rate fluctuation errors.

4.4.8 Rabi Rate Imbalance

Two cases of Rabi rate imbalance are considered here. The first is the Rabi rate imbalance

between the two ions, i.e., Ω1 6= Ω2 in Eq. 4.10 or in Eq. 4.20. This can be caused by (i) unequal

illumination of laser beams due to misalignment for two ions of the same species, or (ii) separate

illumination of each ion with individual sets of laser beams (e.g., the mixed-species gate described

in Chap. 5). Second, for the Mølmer-Sørensen protocol, the detuned blue and red sidebands have

different Rabi rates (i.e. Ωr,j 6 Ωb,j = in Eq. 4.19). By using numerical simulation solving the

Schr ̈odinger equation, it was found that the gate error caused by these two sources are usually

small, typically on the order of ∼ 10−5 if the imbalance is on the few percent level. In most of our

experiments, the Rabi rate difference between the blue and red sidenband transitions can usually

be calibrated to be less than one percent.

The σˆzσˆz gate and the σˆφσˆφ gate based on the Bermudez protocol are not affected by coupling

strength imbalance between the blue and red sideband excitations.

4.4.9 Laser Phase Drift and Fluctuations

Taking the example of the Mølmer-Sørensen gate described in Eq. 4.20, as seen in Eq. 4.21,

the entanglement created depends on the phase terms φj,r(b) = Δkj,r(b)X0,j+Δφj,r(b), where Δφj,r(b)
97

is the phase difference between the optical fields labeled as k1 and k2 in Fig. 4.4. Phase fluctuations

due to relative length changes between paths 1 and 2 lead to decoherence. Because this phase is in

the same exponent as the gate detuning δ (see Eq. 4.20), such phase fluctuations can be modeled as

an effective motional mode frequency fluctuation using the Lindblad operators shown in Eq. 4.39.

Compared to other errors, entangling gate errors caused by this mechanism are small with

the gate durations used in the experimental implementations presented in this thesis.

4.4.10 Qubit Decoherence

Qubit decoherence or dephasing can be caused by external magnetic-field fluctuations, or

fluctuating AC Stark shifts induced by the laser beams that drive the coherent operations. Such

effects can be described by random rotations around the z axis of the Bloch sphere, thus can be

modeled as the following Lindblad operator [Uys 10]

p Lˆz(Γz) = Γzσˆz, (4.53)

where Γz represents the qubit dephasing rate. This equation is in the same form as Eq. 4.36.

In our experiments, gate durations are typically ∼ 50 μs whereas qubit coherence times are

∼ 1.5 s for a 9Be+ ion (Fig. 2.2) and ∼ 6 ms for a 25Mg+ ion (Fig. 2.7); they contribute negligible

errors to the entangling gates presented in Chap. 5, 6, and 7. The errors associated with qubit

dephasing caused by fluctuating AC Stark shifts in these implementations are also small compared

to other errors.
Chapter 5

Mixed-Species Entangling Gate

5.1 Introduction

In the field of quantum information processing (QIP) and quantum networking, various

proposals discuss the possibility of hybrid architectures (see for examples, Ref. [Wallquist 09,

Kurizki 15, Kotler 16]), where specific tasks are delegated to the most suitable subsystem. For

example, in quantum networks, it may be advantageous to transfer information from a subsystem

that has long quantum memory storage time to another subsystem that is more efficient at transporting information between nodes in the network. For trapped ions, a hybrid system formed of

different atomic species introduces extra degrees of freedom that can be exploited to expand and

refine the control of the system.

Ions of different elements vary in mass, internal atomic structure and spectral properties,

features that can make certain species better suited for particular tasks such as storing quantum

information, high-fidelity readout, fast logic gates, or interfacing between local processing units and

photon interconnects. One important advantage of a hybrid system incorporating trapped ions of

different species is the ability to manipulate and measure one ion species using laser beams with

negligible effects on the other since the relevant transition wavelengths typically differ substantially.

Using different ion species also provides individual ion addressing without tight focusing of laser

beams nor shuttling [Blakestad 11, Bowler 12]. When scaling trapped-ion systems to greater numbers and density of ions, it will be advantageous to perform fluorescence detection on individual

qubits without inducing decoherence on neighboring qubits due to uncontrolled photon scattering
99

or stray scattered light.

To provide this function in a hybrid system, one can use an entangling gate to transfer

the qubit states to another ion species which is then detected without perturbing the qubits.

This readout protocol could be further generalized to error correction schemes by transferring

the error syndromes to the readout species while the computational qubits remain in the code.

Another application could be in building photon interconnects between trapped-ion devices. In

this application, one species may be better suited for memory while the other is more favorable for

coupling to photons [Monroe 14, Moehring 07].

A mixed-species gate can also improve the readout in quantum logic spectroscopy (QLS)

[Schmidt 05]. In conventional quantum logic readout, the state of the clock or qubit ion is first

transferred to a motional state and in turn transferred to the detection ion, which is then detected

with state-dependent fluorescence. In this case, the transfer fidelity directly depends on the purity

of the motional state. In contrast, transfer utilizing the gate discussed here can be insensitive to the

motion, as long as the ions are in the Lamb-Dicke regime [Wineland 98]. This advantage extends

to entanglement-assisted quantum non-demolition readout of qubit or clock ions, which can lower

the overhead in time and number of readout ions as the number of clock ions increases [Schulte 16].

Ions of different species have previously been used in QIP experiments for sympathetic cooling

[Barrett 03], creation of entanglement through dissipation [Lin 13a], and quantum non-demolition

(QND) measurement of one species with another [Hume 07]. In this chapter, we focus on an

experimental implementation of a mixed-species geometric-phase gate demonstrated using a 9Be+

and a 25Mg+ ion. The geometric phase gate is realized through an effective spin-spin interaction

generated by state-dependent forces [Sørensen 99, Sørensen 00, Milburn 00, Solano 99, Leibfried 03]

induced with laser beams. Combined with single-qubit gates and same-species entangling gates,

this mixed-species entangling gate provides a complete set of gates over such a hybrid system for

universal QIP [Barenco 95, Bremner 02, Zhang 03]. Using a sequence of such gates, we demonstrate

a Controlled-NOT (CNOT) gate and a SWAP gate [Nielsen 00]. In Sec. 5.4, we show the robustness

of these gates against thermal excitation and demonstrate improved detection in quantum logic
100

spectroscopy (QLS) [Schmidt 05]. Furthermore, we also subject the mixed-species entangled state

created by our logic gate to a Bell inequality test [Clauser 69]; the result is reported in Sec. 5.3.2.

Lastly, in Sec. 5.6 we briefly describe an implementation of the original Cirac-Zoller gate [Cirac 95]

using the two species.

5.2 Experiment Setup

In our experiment, we use a beryllium (9Be+) ion and a magnesium (25Mg+) ion separated

by approximately 4 μm along the trap axis at the E zone shown in Fig. 3.1. The Coulomb coupling

between the ions gives rise to two shared motional normal modes along the trap axis: the lower

frequency mode (ωz = 2π × 2.5 MHz1 ), where the ions oscillate in phase, and the higher frequency

mode (2π × 5.4 MHz), where the ions oscillate out of phase.

The addressing laser beams for each ion (λ ' 313 nm for 9Be+ and λ ' 280 nm 25Mg+) illuminate both ions. The qubits are encoded in hyperfine states of the ions. We choose |F = 2, mF = 0i =

|↓iBe and |1, 1i = |↑iBe as the 9Be+ qubit states and |2, 0i = |↓iMg and |3, 1i = |↑iMg for the 25Mg+

qubit at a magnetic field of 119.446 G (see Sec. 2.1 and Sec. 2.2 for details on energy level

structures).

Our entangling gate scheme is implemented through the Mølmer-Sørensen (MS) protocol

[Sørensen 99, Sørensen 00] driven by stimulated-Raman transitions [Wineland 98]. The two-color

laser-beam setup to drive the entangling gate is shown in Fig. 5.1. The laser beam configurations

to induce coherent Raman transitions are analogous for each species; for brevity, we will only

describe the configuration for 9Be+ (red color in Fig. 5.1). Three laser beams, labeled by k1,Co1,

k1,Co2 and k1,90, are derived from a single laser with wavelength λ ' 313 nm. Beams k1,Co1 and

k1,Co2 are copropagating such that their wave vector difference with respect to the k1,90 beam are

aligned along the trap axis. In this configuration, only the axial modes of motion interact with

the laser beams. With this laser beams configuration, the Lamb-Dicke parameter for the 9Be+

(25Mg+) ion is 0.156 (0.265) and 0.269 (0.072) for the axial in-phase mode and the out-of-phase

1 angular frequency
101

mode, respectively. The two copropagating beams induce detuned blue and red sideband Raman

transitions, respectively, when paired with the k1,90 beam to implement the MS interaction. The

radio-frequency (RF) electric fields driving the AOMs to generate the beams kj,Co1 and kj,Co2

(where j = 1, 2) are produced by direct digital synthesizers (DDS) which are controlled by a fieldprogrammable gate array (FPGA). The phases of these four RF fields are synchronized; this is

important for the deterministic generation of entanglement and is discussed in Sec. 5.2.1 and Sec.

5.2.3. Due to the strong modulation of the ions’ motion (micromotion) caused by an axial radio

frequency field (see Sec. 3.1 and Ref. [Blakestad 10]), spin-motion transitions in the 9Be+ ion are

driven on the second micromotion sideband to maximize transition Rabi rates for the given laser

intensity. Spin-motion transitions in the 25Mg+ ion are driven on the micromotion carrier. Details

of Raman laser beam lines setup are described in Sec. 3.4.3 and Sec. 3.5.2.

With this setup, the Mølmer-Sørensen interaction Hamiltonian after transforming into the

respective interaction frames of both qubits as well as that of the shared motional mode of motion

(here, we use the in-phase mode) and dropping high-frequency terms, in the Lamb-Dicke limit is

(Eq. 4.19)

X   X  

H † −i(δj t−φj,r) † † i(δj t+φj,b) = ~ Ωr,jσˆ aˆe + ~ Ωb,jσˆ aˆ e + h.c., (5.1) j j

j=1,2 j=1,2

with Ωr(b),j = Ω0,r(b),jηjξj where Ω0,r(b),j is the Rabi rate of the resonant carrier excitation induced

by the laser beams responsible for the red (r) and blue (b) sideband excitations, ηj is the LambDicke parameter of the in-phase mode with ξj as the mode amplitude of the jth ion. If we set

Ω0,r,j = Ω0,b,j = Ω0,j then Eq. 5.1 becomes

X   † −i(δj t−φj,r) + ˆ H † i(δj t+φj,b) = ~ Ωjσˆ aˆe a e + h.c., j

j=1,2

X  

= † † i(δj t+(Δkj,bX0,j+Δφj,b)) ~ Ωjσˆj aˆe−i(δj t−(Δkj,rX0,j+Δφj,r)) + ˆa e + h.c., (5.2)

j=1,2

with Ωj = Ω0,jηjξj where Ω0,j is the carrier resonant Rabi frequency. The Lamb-Dicke parameter ηj

p is equal to Δkjz0,j , where z0,j = ~/2mjωz, mj is the mass and ωz is the frequency of the in-phase

mode. The phases of the red (r) and blue (b) sideband interactions are φj,r(b) = Δkj,r(b)X0,j +
102

9Be+ 25Mg+

ωωBe − 2ωωRRRR − ωω0,Be + ωωzz + δδ

ωωBe

ωωMg + ωω0,Mg + ωωzz + δδ

ωωMg

ωωBe − 2ωωRRRR − ωω0,Be − ωωzz − δδ

ωωMg + ωω0,Mg − ωωzz − δδ

kk22,9999

kk22,CCCCCC

kk ,CCCC kk22,CCCC

kk ,CCCCCC

kk ,9999

ΔΔkkCC,rr

ΔΔkkCC,bb

ωωMg: Frequency of beam kk22,9999

ωωBe: Frequency of beam kk ,9999

ωω0,Mg: 25Mg+ qubit frequency, 2ππ × 1.763 GHz

ωω0,Be: 9Be+ qubit frequency, 2ππ × 1.207 GHz

ωωzz: Motional mode frequency

ωωRRRR: Radio frequency for trapping of ions

δδ: Detuning from sideband transitions

Carrier transitions driven 

by beams kkjj,CCCCCC and kkjj,9999

Figure 5.1: Configuration of laser beams for the mixed-element entangling gate. For the 9Be+

ion, 313 nm laser beams (in red) simultaneously induce near-resonant red and blue sidebands

transitions. Similarly, for 25Mg+, 280 nm beams (in green) induce sideband transitions. When

all beams are applied simultaneously, this implements the Mølmer-Sørensen spin-spin interaction

[Sørensen 99, Sørensen 00]. Each set of qubit addressing laser beams is set up such that the wave

vector differences Δkj,r = kj,90 − kj,Co1 and Δkj,b = kj,90 − kj,Co2 (j = 1, 2) are aligned in the

same direction along the trap axis. With this configuration, only the motional modes along this axis

can be excited. The frequencies of the 9Be+ ion’s Raman beams are set to drive the spin-motion

transition on the second order micromotion sideband, this is to maximize transition Rabi rates in

the presence of axial micromotion (details see Sec. 2.1.3. Spin-motion transitions in the 25Mg+ ion

are driven on the micromotion carrier.
103

Δφj,r(b) where Δkj,r(b) and Δφj,r(b) are the differences in wave vectors and phases of the optical

fields driving the red and blue sideband transitions respectively, and X0,j is the equilibrium position

for the jth ion. As the Hamiltonian (Eq. 5.2) depends on these wave vectors and optical phases, it

is important to consider their effects for creating deterministic entanglement.

In our implementation, Δkj,r = kj,90 − kj,Co1 and Δkj,b = kj,90 − kj,Co2 for j = 1, 2.

Beams kj,Co1 and kj,Co2 are generated in the same acousto-optic modulator (AOM), one for each

ion species, and travel nearly identical paths. With this setup, the relative optical phase drift

between these two beams is negligible. However, the kj,90 beams take a substantially different

path to reach the ions’ locations (see Sec. 3.4.3 and 3.5.2 for detailed beam line setups). The

resulting slow phase drifts can be taken account of as described below.

5.2.1 Optimizing the Strength of Spin-Dependent Forces

The basis states |+ij and |−ij in which the spin-dependent forces generated by the MS

interaction are applied is defined as the eigenstates of

σˆφ,j = cos((φj,r + φj,b)/2) σˆx,j − sin ((φj,r + φj,b)/2) σˆy,j . (5.3)

With the notation of |+iBe|+iMg = |+, +i, the truth table after applying the MS gate is

|+, +i → eiφ++ |+, +i,

|+, −i → eiφ+− |+, −i,

|−, +i → eiφ−+ |−, +i,

|−, −i → eiφ−− |−, −i. (5.4)

After setting Ω1 = Ω2 = Ω (calibration procedure in the next section) and δ1 = δ2 = δ, and

writing φM,j = (φj,r − φj,b) /2, the geometric phases accumulated after a duration of tgate = 2π/δ

are [Tan 15]

  8πΩ2

2 φM,1 − φM,2

φ++,−− = cos , δ2 2

  8πΩ2 φM,1 − φM,2

φ+−,−+ = sin2 . (5.5) δ2 2
104

From the definitions of the different phase terms, it can be shown that φM,1 − φM,2 depends on the

relative phase differences between the two copropagating beams kj,Co1 and kj,Co2 for each ion. For

each species, the optical phase difference between these two beams is set by the RF injected into

the AOM that generates them. Thus, it is important that the phases of the four RF signals (two

for each species) are synchronized. By adjusting the phases of the RF signals driving the AOMs

that generate these laser beams, the difference in the geometric phases for the different parity qubit

states in Eq. 5.5 can be optimized. In our experiment, this optimization is performed by adjusting

the phase of one of the RF signals while keeping the remaining three fixed when we apply the MS

interaction at the duration of tgate = 2π/δ with δ fixed; we adjust for the phase that equalizes the

populations in the |↑↑i and |↓↓i states with a minimal amount of laser intensity.

5.2.2 Calibration Procedure for the Mixed-Species MS Gate

The MS protocol shown in Eq. 5.1 requires the Rabi rates of the red and blue sideband

transitions of the two species to be equal, i.e. Ωr,1 = Ωb,1 = Ωr,2 = Ωb,2. To accomplish this, we first

minimize the red and blue sideband Rabi rate imbalance for each species by performing sideband

Rabi flopping experiments. We differentially adjust the laser power in beams kj,Co1 and kj,Co2

(Fig. 5.1) while keeping the total laser power in this path fixed. In our procedure, the beam that

is responsible for the red sideband excitation is first detuned from any transition when we calibrate

the Rabi rate of the blue sideband transition. Analogously, we detune the beam responsible for the

blue sideband when the Rabi rate of the red sideband transition is calibrated. This strategy is to

ensure AC Stark shifts change negligibly during calibration. After setting Ωr,j = Ωb,j for each of

the two species, we adjust the laser power in the k1,90 and k2,90 beams so that Ωr(b),1 = Ωr(b),2.

This can change the AC Stark shifts on each ion, which can be compensated by changing the mean

transition frequency of the red and blue sidebands. This process is iterated to finally achieve the

correct tunings and equalize all four sideband Rabi rates.

Starting in the |↑↑i state, a correctly tuned MS gate with Ω = δ/4 ideally produces the
105

entangled state

1  P

( 1 

Φ = √ + e−i (φj,b+φj,r)+π/2) | Belli |↑↑i j=1,2 2 |↓↓i , (5.6) 2

after a duration of tgate = 2π/δ. The complete truth table of the MS gate is given in Eq. 4.21.

5.2.3 Laser Beam Phase Sensitivity

As shown in Eq. 5.3, the basis states in which the spin-dependent force is applied depends

on (φj,r + φj,b), which can be shown to be dependent on the relative phase differences between

the non-copropagating beams kj,90 − kj,Co1 and kj,90 − kj,Co2 (Fig. 5.1). Temperature drift and

acoustic noise can cause changes in the lengths of different beam paths, which can lead to phase

fluctuations in the MS interaction. These fluctuations are slow on the timescale of a single gate but

substantial over the course of many experiment repetitions. To suppress these effects, we embed the

MS interaction in a Ramsey sequence (Fig. 5.2) implemented with two π/2 carrier pulses induced

by kj,Co1 and kj,90 for each qubit [Lee 05, Tan 15]. Without loss of generality, (see next section)

�  we choose the phases of the first π/2 pulses to implement the rotations R θ = π , φ = π . This 2 2

maps the |↑i and |↓i states of each qubit onto the |+ij and |−ij states (defined in Eq. 5.3), whose

�  phases are synchronized with the MS interaction. The final set of R π , −π pulses undoes this 2 2

mapping such that the action of this sequence is independent of the beam path length differences

as long as the differences are constant during the entire sequence. With this Ramsey sequence, the

overall sequence produces a deterministic phase gate Gb that implements (up to a global phase of

eiπ/4) |↑↑i → |↑↑i, |↑↓i → i|↑↓i, |↓↑i → i|↓↑i, and |↓↓i → |↓↓i or, in matrix form:

⎛ ⎞ ⎜

1 0 0 0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0 i 0 0

⎟ Gb = ⎜ ⎟ ⎜ ⎟ , (5.7)

⎜ ⎟ ⎜

0 0 i 0

⎟ ⎝ ⎠ 0 0 0 1
106

GG�

9Be+

25Mg+

MS

Interaction

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 )

Figure 5.2: A Ramsey sequence implemented to remove laser phase sensitivity of the MS interaction.

The “90 Carrier” π/2 pulses are implemented by Raman laser beams induced by kj,Co1 and kj,90

(see Fig. 5.1). The overall sequence produces a phase gate Gb which is given in Eq. 5.7. The

rotation R(θ, φ) is defined in Eq. 2.13.

with basis states of the two qubits given by

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0

⎟ ⎜

0

⎜

0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0

⎟ ⎜

1

⎟ ⎜

0

⎟ ⎜

0

⎟ |↑i |↑i ⎜ ⎟ ⎜ ⎟ , |↑i |↓i Be Mg = = ⎜ ⎟ , |↑i = ⎜ ⎟ ⎜ ⎟ ⎜ , |↓i |↓i = ⎟ ⎜

|↓i

⎟ ⎜ ⎟ . (5.8)

⎜ ⎟

Be Mg

⎟

Be

0 ⎜

Mg

⎟ ⎜

0 ⎟

Be Mg

⎜ ⎜ ⎟ ⎜ ⎟ ⎜

1

⎟ ⎜

0

⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0 1

5.2.4 Calibration Procedure for Phase Gate Gb

To produce the phase gate Gb, the phases of the π/2 pulses in the Ramsey sequence (Fig 5.2)

must be referenced to the basis states of the MS interaction defined by the optical phases. The

phases must also account for the AC Stark shifts induced by the laser beams that are used for the

MS interaction (AC Stark shifts are effective Rz rotations on the qubits).

To calibrate these phases, we first perform the pulse sequence shown in Fig. 5.2 with the MS

interaction pulses detuned far off-resonant from the red and blue sideband transitions such that

they only induce AC Stark shifts on the qubits. Starting with the input state |↑↑i, we set the

phases of the final π/2 laser pulses such that the action of this pulse sequence returns each qubit

to the |↑i state. Then, we perform this sequence with the MS interactions correctly tuned and vary

the phases of the MS interactions. Again, in this case we look for the phase that maps the input
107

state |↑↑i back to itself. We verify the action of this Gb operation by creating a Bell state with the

pulse sequence shown in Fig. 5.4.

5.3 Experimental Results

Before applying the gate, the ions are first Doppler cooled in all three directions. The

axial vibrational modes are further cooled to near the ground state by stimulated-Raman sideband

cooling on the 9Be+ ion [Monroe 95b] (details see Sec. 2.1.5). For state initialization, transfer of

the 9Be+ |2, 2i state to the |1, 1i = |↑iBe state is accomplished by a microwave-driven composite

pulse sequence that is robust against transition detuning errors. The sequence consists of resonant

�  �  �  π 3π π π R 2 , 0 , R 2 , , R 2 , 0 pulses [Levitt 86], where the definition of the R(θ, φ) rotation is given 2

in Eq. 2.13. With analogous sequence, we first transfer the 25Mg+ ion from the |3, 3i state to the

|2, 2i state, and then to the |3, 1i = |↑iMg state.

State-dependent fluorescence detection is accomplished with an achromatic lens system designed for 313 nm and 280 nm (see Sec. 3.3). We sequentially image each ion’s fluorescence onto

a photomultiplier tube. Before applying the detection laser beams, we reverse the initial qubit

state preparation procedures to put the |↑i states back to the |2, 2i and |3, 3i state for 9Be+ and

25Mg+, respectively. The |↓i state of each qubit is transferred to |1, −1i and |2, −2i for the 9Be+

and 25Mg+, respectively, with microwave carrier π pulses. We detect on average approximately 30

photons for each ion when they are in the bright state and approximately 2 photons when they are

in the dark state. The qubit state is determined by choosing a photon count threshold such that

the states are maximally distinguished. Details on state initialization and readout is described in

Chap. 2. After each repetition, we measure one of the possible states: |↑↑i, |↑↓i, |↓↑i, or |↓↓i.

Here, we label the probabilities of measuring |↑↑i and |↓↓i as P↑↑ and P↓↓, respectively. The sum

of the probabilities of measuring P↑↓ and P↓↑ is labeled as P↑↓+↓↑.

In a first experiment, after calibrating the gate as described in Sec. 5.2.2 and Sec. 5.2.1,

we apply the MS interaction described by Eq. 5.2 and record the populations as a function of

interaction time. As shown in Fig. 5.3, at a gate time tgate ' 33 μs, a Bell state |ΦBelli (Eq. 5.6)
108

Figure 5.3: Population evolution as we apply the MS interaction starting from the |↑↑i state. The

Bell state |ΦBelli (Eq. 5.6), which corresponds to (ideally) equal population in the |↑i |↑i Be Mg and

|↓i |↓i Be Mg, is created at tgate ' 33μs. Error bars correspond to standard error of the mean.
109

is created. The performance of the gate is characterized by measuring the Bell state fidelity, which

is given by hΦBell|ρexp|ΦBelli where the density matrix ρexp describes the experimentally produced

state. The ↑↑ and ↓↓ diagonal elements of the density matrix are determined from the sum of P↑↑

and P↓↓, which we measure to be 0.984(1). The coherence of this Bell state is investigated with a

parity (P↑↑ + P↓↓ − P↑↓+↓↑) oscillation experiment by applying “analysis” pulses [Sackett 00]. The

analysis pulses are laser carrier transitions induced by the non-copropagating laser beams kj,Co1

and kj,90 such that the relative phase defining the basis states of MS interaction is stable with

respect to that of the analysis pulses for each experiment repetition. The off-diagonal elements

ρ↓↓,↑↑ are determined from the contrast of the parity oscillation (A) to be 0.974(1). The overall

Bell state fidelity is (P↑↑ + P↓↓ + A)/2, which is deduced to be 0.979(1).

To test the Gb operation and its laser phase insensitivity, we prepare a Bell state by applying

microwave carrier π/2 pulses on each qubit before and after the operation Gb (described in Sec.

5.2.4) as depicted in Fig. 5.4. Using a similar procedure, we determine the total population in the

P↑↑ and P↓↓ states to be 0.976(1), and A = 0.955(1), which corresponds to a Bell state fidelity of

0.964(1). Figure 5.5 shows the parity oscillation of the Bell state created by the Gb operation. The

analysis pulses are driven by microwave fields whose phases are not synchronized with those of the

lasers, demonstrating the laser-phase insensitivity of Gb gate and the deterministic creation of the

mixed-species Bell state.

5.3.1 Imperfections

We investigated the imperfect creation of the entangled states through calibration measurements and numerical simulations.

We estimated the error from imperfect state preparation and measurement (SPAM error) to

be 5×10−3. This is given by the false determinations of the bright and dark states set by the photon

count thresholds (see previous section) chosen by analyzing the histograms obtained after preparing

the ions to their respective bright and dark states in separate experiments. To also account for

the errors caused by qubit state initialization, the bright histogram of each ion is collected as
110

9Be+

25Mg+

|↑⟩

|↑⟩

μWv

RR(

ππ

2 , 0)

μWv

RR(

ππ

2 , 0)

μWv

RR(

ππ

2 , 0)

μWv

RR(

ππ

2 , 0)

GG�

MS

Interaction

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 )

Figure 5.4: The pulse sequence using the Gb gate to create the entangled state √

1 (|↑↑i − i|↓↓i) 2

(up to a global phase of eiπ/4) with both ions initialized to their respectively |↑i state. The Bell

state |Φ+i = √

1 (|↑↑i + |↓↓i) (up to a global phase of eiπ/2) can be created by applying a Rz(π/2) 2

rotation to one of the qubits at the end of this sequence. Likewise, applying a Rz(−π/2) rotation

to one of the qubits at the end of the sequence creates the Bell state |Φ i 1 − = √ (|↑↑i − |↓↓i). 2

The definition of Rz is given in Eq. D.5. The notation “μWv” is used to represents rotations

implemented with a microwave field.
111

Figure 5.5: Parity (P↑↑ + P↓↓ − P↑↓+↓↑) flopping of the Bell state created by the sequence shown in

Fig. 5.4. To demonstrate laser-phase insensitivity of our entangling operation and the deterministic

creation of entanglement, we use microwave fields to induce the “analysis” pulses with variable

phases in this parity oscillation experiment. The phases of the microwave fields are not synchronized

with those of the laser fields.
112

follows: we apply the qubit state initialization procedure to prepare each ion to its computational

|↑i state (see previous section), wait for ∼ 1.5 ms, then reverse the initialization sequence and apply

shelving pulses to transfer the qubit population to the measurement qubit manifold (see Fig. 2.1

and Fig. 2.6 for the energy level diagrams of the 9Be+ ion and 25Mg+ ion, respectively). Finally,

the ions’ states were measured with state-dependent fluorescence detection. The wait time (∼ 1.5

ms) between the transfer from computational to measurement qubit manifolds was long enough to

minimize coherent effects between the transfer into and out of the computational manifold. The

coherence times of these transfer transitions are typically ∼ 1 ms or less.

Errors caused by spontaneous photon scattering [Ozeri 07] from the Raman beams were estimated to be ∼ 6×10−3 for the 25Mg+ ion and ∼ 1×10−3 for the 9Be+ ion. This error was deduced

by performing numerical master equation simulations including the Lindblad operators describing

the spontaneous scattering of photons (see Sec. 4.4.2.1), with the scattering rate calculated with

the Kramer-Heisenberg equation [Cline 94] shown in Eq. 4.29. We also experimentally measured

the Raman scattering rate by applying the laser pulses used for the entangling gate but with the

MS interaction set far-detuned from both the red and blue sideband excitations such that they do

not induce any coherent excitation. The scattering rate is determined from loss of population in the

qubit manifold; these measurements agree with the calculations made using the Kramer-Heisenberg

equation.

We measured the heating of the in-phase mode [Turchette 00a] due to ambient electric field

noise to be approximately 160 quanta/s, which contributes an error of approximately 4 × 10−3

to the created Bell state. This error was estimated using a master equation simulation with the

Lindblad operators describing the heating (see Sec. 4.4.4).

Other known error sources include imperfect single-qubit pulses, off-resonant coupling to

spectator hyperfine states and the other motional modes, mode frequency fluctuations, qubit decoherence due to magnetic field fluctuations, laser intensity fluctuations, optical phase fluctuations,

and calibration errors. Each of these sources contributes error on the order of 10−3 or less. Details

of the evaluation of these errors are discussed in Sec. 4.4.
113

GG

�

9Be+

25Mg+

|↑⟩

|↑⟩

μWv

RR(

ππ

2 , 0)

μWv

RR(

ππ

2 , 0)

μWv

RR(

ππ

2 , 0)

μWv

RR(

ππ

2 , 0)

Creating Bell State: 1

2 (|↑↑⟩ + |↓↓⟩)

Measurement 

Setting: αα11 or αα22

D

D

Measurement 

Setting: ββ11 or ββ22

Bell test

RRzz(

ππ

2 )

Figure 5.6: Pulse sequence for the CHSH-type Bell inequality experiment on the mixed-species

Bell state. The black dashed box denotes the “black box” which prepares the two particles of

different species. The measurement settings αi, βj are applied (with microwave fields) before

making fluorescence measurements. The laser pulse sequence for the implementation of Gb is shown

in Fig. 5.2. The rotations R(θ, φ) and Rz(ξ) are defined in Eq. 2.13 and Eq. D.5, respectively.

Rz(ξ) rotation is accomplished by adjusting the phase of the DDS which is used to produce the

microwave fields.

Errors due to motional heating can be significantly reduced if the out-of-phase mode is used

instead of the in-phase mode for the state-dependent force because the heating rate of the outof-phase mode was measured to be ∼ 30 quanta/s. We used the in-phase mode because for the

25Mg+ ion, it has a larger normal mode amplitude (0.265) compared to the out-of-phase mode

(0.072). This results in less spontaneous emission error for a given strength of the state-dependent

force. Error caused by spontaneous emission can be reduced by increasing Raman detuning at the

cost of higher laser intensity to keep the Rabi rate constant [Ozeri 07]. When this experiment was

performed, the available optical power in each of the 25Mg+ Raman beams was only ∼ 1 mW,

compared to ∼ 50 mW in each of the Raman beams for the 9Be+ ion.

5.3.2 Mixed-Species Bell Inequality Test

Using the 9Be+ - 25Mg+ Bell state, we performed a Clauser-Horne-Shimony-Holt (CHSH)

type Bell-inequality test [Clauser 69]. The pulse sequence is shown in Fig. 5.6. The source, which

can be treated as a “black box”, ideally creates a Bell state |Φ i = 1

+ (|↑↑i + |↓↓i) and is regarded 2

as the starting point of the following Bell test experiment.

Two measurement setting choices on each species, labeled as α1, α2 for the 9Be+ ion, and
114

β1, β2 for the 25Mg+ ion, were applied to the prepared state before making qubit state measurements. This is equivalent to choosing the measurement basis for the state that is present before

the measurement settings are applied. Joint measurement outcomes of |↑↑i and |↓↓i were recorded

as correlated events while |↑↓i and |↓↑i were recorded as anticorrelated events. Each of the four

measurement setting combinations were performed with multiple repetitions, then iterated with

different measurement setting combinations. For each measurement setting combinations, we computed the correlation of these outcomes

Ncorrelated(αi, βj ) − Nanticorrelated(αi, βj ) q(αi, βj ) = , (5.9) Ncorrelated(αi, βj ) + Nanticorrelated(αi, βj )

for i, j = 1, 2. For a local realistic theory to be valid, the sum of correlation BCHSH (at expectation)

must satisfy [Clauser 69, Rowe 01]

BCHSH = |q(α1, β1) + q(α2, β1)| + |q(α1, β2) − q(α2, β2)|

≤ 2. (5.10)

The measurement settings are applied with microwave-induced spin rotations R(π , αi or βj ) 2

which are controlled by classical parameters. Ideally, the rotations for the 9Be+ ion can be described

as:

1 �  |↑i → √ |↑i − e−iαi |↓i , 2

1 �  |↓i → √ |↓i − eiαi |↑i , (5.11) 2

and analogously for the 25Mg+ ion with αi replaced by βj . They are implemented by setting

the durations and phases of the microwave pulses transmitted from two microwave antennas (one

for each species, see Sec. 2.1.4 and Sec. 2.2.3 for more details). The microwave-frequency electric

potentials applied to the antennas are produced by a FPGA-controlled DDS. The classical variables

are the phases of the oscillating fields that implement a particular setting αi and βj .

The measurement settings αi, βj and the correlation outcomes q(αi, βj ) are given in Table

5.1. We determined the sum of correlation BCHSH to be 2.70(2). This inequality, measured on an
115

Measurement Setting on 9Be+

Measurement Setting on 25Mg+

= −π α1 8

= −π β1 8

= −π α1 8

3π β2 = 8

3π α2 = 8

= −π β1 8

3π α2 = 8

3π β2 = 8

q(αi, βj ) −0.635(9) −0.721(8) −0.711(8) 0.638(9)

Table 5.1: Correlation values for the four measurement setting combinations in the CHSH Bell

inequality experiment performed on a mixed-species entangled state.

entangled system consisting of different species, agrees with the predictions of quantum mechanics while eliminating the detection loophole [Pearle 70, Clauser 74] but not the locality loophole

[Bell 85]. The closely related chained Bell inequality experiments are detailed in Chap. 8.

5.4 Mixed-Species Controlled-NOT Gates

As the Gb gate combined with single qubit rotations make up a universal gate set [Barenco 95,

Bremner 02, Zhang 03], we can use them to construct other unitary quantum gate operations.

The controlled-NOT gate is one of the most common entangling logic gates used in the

literature [Nielsen 00]. A controlled-NOT gate “flips” the state of the “target” qubit depending

on the state of the “control” qubit. Here, we define the CNOT gate as the controlled-NOT gate

where the first qubit serves as the control and the second qubit acts as the target. The unitary

matrix of this operation is given in Eq. 4.9 with the basis states defined in Eq. 5.8. Similarly,

the controlled-NOT gate with the control and target qubit roles inverted can be expressed by the

matrix

⎛ ⎞ ⎜

1 0 0 0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0 0 0 1

⎟ CNOTinvert = ⎜ ⎟ ⎜ ⎟ . (5.12)

⎜⎜

0 0 1 0 ⎟⎟ ⎝ ⎠ 0 1 0 0

These two CNOT gates can be implemented with the pulse sequences including Gb gates and singlequbit rotations as shown in Fig. 5.7, where the 9Be+ ion acts as the control of the CNOT gate and

the target of CNOTinvert gate. Figure 5.7 provides detailed parameters which were not discussed

in Ref. [Tan 15] for brevity. The construction of CNOT presented here and the construction of
116

GG�

9Be+

25Mg+

MS

Interaction

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 )

μWv

RR(

ππ

2 ,

ππ

2 )

μWv

RR(

ππ

2 ,- ππ2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 ) RRzz(− ππ

2 )

RRzz(

ππ

2 )

GG�

9Be+

25Mg+

MS

Interaction

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 )

μWv

RR(

ππ

2 ,

ππ

2 )

μWv

RR(

ππ

2 ,- ππ2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 ) RRzz(

ππ

2 )

RRzz(− ππ

2 )

(a) CCCCCCCC Gate

(b) CCCCCCCCiiiiiiiiiiii Gate

Figure 5.7: (a) Pulse sequence for the CNOT gate (Eq. 4.9) with 9Be+ qubit as the “control” and

25Mg+ qubit as the “target”. (b) Pulse sequence for the CNOTinvert gate (Eq. 5.12), where the

two qubits change roles, i.e. the 25Mg+ qubit as the “control” and the 9Be+ qubit as the “target”.

The rotations R(θ, φ) and Rz(ξ) are defined in Eq. 2.13 and Eq. D.5, respectively. Here, the Rz(ξ)

rotations are accomplished by adjusting the phase of the DDS used to induce microwave transitions

 (see footnote 2), which can be absorbed into the microwave-induced R(π , −π ) pulses. 2 2
117

controlled-NOT gate in Ref. [Tan 15] are related with single-qubit Rz(ξ) rotations. Here, Rz(ξ)

rotations are accomplished by shifting the phase of the DDS which is used to induce microwave

transitions. In our experiment, the Rz rotations (in Fig. 5.7) are effectively absorbed into the

R(π

2 , −π ) rotation during our calibration procedures. Here, the phases of the R(θ, φ) rotations must 2

also take into account the residual σˆz rotations which can be caused by inaccurate bookkeeping or

miscalibrations of pulse durations or frequencies for the implementation of the CNOT gate with

the laser-induced Gb gate2 . The actions of these calibrations are verified by confirming that the

controlled-NOT gates processes the four states (given in Eq. 5.8) correctly according to the truth

table in Eq. 4.8.

5.4.1 Quantum Non-Demolition Measurement

Quantum logic spectroscopy (QLS) is a process where one ion is used to readout the spin state

of another ion. Traditionally, this task is accomplished by implementing the proposal presented

in Ref. [Schmidt 05]. The working principle of the QLS procedure is very similar to the CiracZoller gate (see Sec. 4.2) in the sense that it relies on the blockade provided by the motional

ground state to conditionally transfer information from one ion to another ion, where both ions are

trapped in the same potential well. The QLS procedure first transfers the population information

from a “spectroscopy” ion to the shared motional mode, then this information is transferred to

a “logic” ion., which is then detected. The QLS method is useful to readout certain ion species

where a direct fluorescence measurement is not readily applicable. This might be due to a lack

of cycling transitions (e.g. molecular ions [Wolf 16]), or hard-to-access laser wavelength required

for fluorescence detection (e.g. Al+ ions [Rosenband 08]). For example, QLS has been successfully

applied to a quantum logic clock [Chou 10].

Figure 5.8.(a) shows the pulse sequence used in a typical QLS experiment. After cooling a

2 furthermore, due to insufficient numbers of DDSs available to us, we use a single DDS to drive microwave

transitions on both the 9Be+ (qubit frequency ∼ 1.2 GHz) and 25Mg+ (qubit frequency ∼ 1.8 GHz) ions. As the

phases of the qubits’ superposition states are defined by the applied microwave field, changing the microwave-DDS

frequency gives phase offsets and effectively applies Rz(ξ) rotations to the qubits. However, these Rz(ξ) rotations can

be reversed and the phases of the qubits can be recovered by changing the phase of the microwave field appropriately

at a later time in the sequence
118

b.) Phase-gate-assisted quantum logic spectroscopy

9Be+

25Mg+

D

CCCCCCCC Gate

|↑⟩

GG� μWv

RR(

ππ

2 ,- ππ2

RRzz ) (− ππ

2 )

RRzz(

ππ

2)

μWv

RR(

ππ

2 ,

ππ

2 )

9Be+

Motion |nn = 0⟩

D 25Mg+ |↑⟩

a.) Original recipe of quantum logic spectroscopy

|↓⟩|0⟩ ↔ |↑⟩|1⟩

transition

|↓⟩|0⟩ ↔ |↑⟩|1⟩

transition

αα|↑⟩ + ββ|↓⟩

αα|↑⟩ + ββ|↓⟩

Figure 5.8: Pulse sequences of performing Rabi flopping on the 9Be+ (spectroscopy) ion as detected

with 25 the Mg+ (logic) ion using (a) conventional quantum logic spectroscopy [Schmidt 05], and

(b) the controlled-NOT gate based procedure. The CNOT gate implemented here is defined by the

matrix in Eq. 4.9, with the pulse sequence of the Gb gate shown in Fig. 5.2. Ground state cooling

which was applied prior to the qubit state preparations, are not shown in this figure.
119

shared motional mode to near the ground state, a π pulse on the |↓i|n = 0i ↔ |↑i|1i (where n is

the Fock state quantum number) transition is applied to the spectroscopy ion; this is followed by

applying an equivalent π pulse to the logic ion, which is initially prepared in the |↑i state. As a

demonstration here, the 9Be+ ion mimics the spectroscopy ion and the 25Mg+ ion mimics the logic

ion. We consider the actions of the pulse sequence (Fig. 5.8.(a)) with the input state of either the

|↑iBe|↑iMg state or the |↓iBe|↑iMg state, when the motional mode is cooled to the ground state:

where −→ and −→ indicate applying the |↓i|n = 0i ↔ |↑i|1i transitions to the spectroscopy (9Be+)

|↑iBe|↑iMg|0i S

−→ |↑iBe|↑iMg|0i L−→ |↑iBe|↑iMg|0i, (5.13)

|↓iBe|↑iMg|0i S

−→ |↑iBe|↑iMg|1i L−→ |↑iBe|↓iMg|0i, (5.14)

S L

ion and the logic (25Mg+) ion, respectively. If the spectroscopy ion is prepared in a superposition

state, this pulse sequence implements the following action:

S (α|↑iBe + β|↓iBe)|↑iMg|0i −→ |↑iBe|↑iMg (α|0i + β|1i)

  L

−→ |↑iBe α|↑iMg + β|↓iMg |0i, (5.15)

indicating that the state of the 9Be+ ion is mapped to the 25Mg+ ion.

Analogously, as a controlled-NOT gate is also a conditional operation (i.e., the state of the

target qubit remains unchanged if the control qubit is in the |↑i state and the target qubit’s

state is flipped if the control qubit is in the |↓i state), a controlled-NOT gate can also be used

to execute quantum logic spectroscopy. Moreover, because our implementations of the CNOT and

CNOTinvert gates inherit the robustness against motional excitation from the MS gate [Sørensen 99,

Sørensen 00, Milburn 00, Solano 99], it offers certain advantages compared to the conventional QLS

method where the performance is strongly tied to performance of the ground state cooling applied

prior to the motion-subtracting mapping pulses.

As an illustration, we compared the CNOT gate with the method used in conventional QLS

[Schmidt 05] procedure in a series of quantum logic readout experiments. The pulse sequences

for the comparison experiments are shown in Fig. 5.8. Both procedures were calibrated with the

motional mode initialized close to the motional ground state.
120

0 10 20 30 40 50 0.0

0.2

0.4

0.6

0.8

1.0

Time (μs)

P

(↑〉Mg

)

Mapping with QLS

Mapping with CNOT

Figure 5.9: Rabi flopping of the Be ion detected on the Mg ion with the motional modes

cooled to near the ground state (n ̄ ' 0.05). P(|↑iMg) is the probability of finding the 25Mg+ qubit

in the |↑i state. In this case, both the QLS and our controlled-NOT gate mapping procedures

performed approximately equally. Error bars correspond to standard error of the mean.

 9 + 25 + 
121

0 10 20 30 40 50 0.0

0.2

0.4

0.6

0.8

1.0

Time (μs)

P

(↑〉Mg

)

Mapping with QLS

Mapping with CNOT Figure 5.10: Rabi flopping of the 9Be+ ion detected on the 25Mg+ ion with the motional modes

cooled to the Doppler temperature of n ̄ ' 4. The controlled-NOT mapping technique, which

makes use of the mixed-species gate, performed better than the original QLS procedure due to the

relative low sensitivity to motional excitations [Sørensen 99, Sørensen 00]. Error bars correspond

to standard error of the mean.
122

Figure 5.9 and Fig. 5.10 show the Rabi flopping of the 9Be+ qubit as detected on the 25Mg+

ion, which was initially prepared in the |↑i state. For the ions’ motional modes cooled near the

ground state ( ̄n ' 0.05) with Raman sideband cooling applied to the 9Be+ ion (see Sec. 2.1.5),

the performances of both procedures are similar in terms of the population contrast. For the ions’

motional modes cooled to the Doppler temperature ( ̄n ' 4), the contrast of the conventional QLS

method (red dots) was reduced significantly compared to transfer with a controlled-NOT gate (blue

squares).

A typical QLS sequence consists of a ground state cooling sequence followed by transfer pulses

that map the information from the spectroscopy ion to the logic ion before reading out on the

logic ion (Fig. 5.8.(a)). This procedure is usually performed with multiple repetitions to improve

statistics. Compared to the conventional QLS, since the population contrast of our geometricphase-gate controlled-NOT procedure is less sensitive to thermal excitation of the motional modes,

the ground state cooling sequence could be removed or substantially shortened. This enables an

increase of spectroscopy probe duty cycle as well as relaxing the requirement for preparing the

motional state into the ground state. This procedure might potentially be useful for the quantum

logic readout of certain ion species with (i) relatively short trapping lifetimes, e.g. highly-charged

or certain molecular ions, or (ii) relatively short decay lifetimes compared to the duration required

for ground state cooling.

Although the technical requirements for the implementation of a geometric-phase gate are

more demanding, e.g., the need for coherent phase control of the laser phases described in Sec.

5.2.3, the ingredients required, i.e. blue and red-sideband excitations, are readily available in most

cold, trapped-ion experiments.

5.5 SWAP Gate for Quantum Information Mapping

The 9Be+ qubit’s phase information is not accessible on the 25Mg+ with the CNOT mapping

procedure described in the previous section. This might not be of concern for the application of

quantum non-demolition measurement with quantum logic using the controlled-NOT gates con-
123

SWAP

9Be+

25Mg+ D μWv

RR(

ππ

2 , φφ)

|↑⟩

|↑⟩

μWv

RR(

ππ

2 , 0) CCCCCCCCiiiiiiiiiiii

Gate

CCCCCCCC

Gate

CCCCCCCC

Gate

Figure 5.11: The pulse sequence for a Ramsey experiment with a SWAP gate. We used three

controlled-NOT gates to construct a SWAP gate. The first and third controlled-NOT gates were

implemented with the CNOT gate (Eq. 4.9) with the 25Mg+ ion as the target, with pulse sequence

indicated in Fig. 5.7.(a). The second controlled-NOT gate was implemented as depicted in Fig.

5.7.(b), with the 9Be+ ion as the target of the CNOTinvert gate. The overall operation inside the

red dashed box implements the SWAP gate as shown in Eq. 5.16.

structed with geometric phase gates, because in spectroscopy applications usually only the population information of the spectroscopy ion is of interest. However, in a broader context of general

quantum information applications, it is desirable to transfer both the population and phase information between qubits of different species. To preserve this phase information, we constructed

a SWAP gate that interchanges the quantum state of the two qubits [Nielsen 00], which can be

described by the following unitary matrix

⎛ ⎞ ⎜

1 0 0 0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0 0 1 0

⎟ SWAP = ⎜ ⎟ ⎜ ⎟ . (5.16)

⎜ ⎟ ⎜

0 1 0 0

⎟ ⎝ ⎠ 0 0 0 1

With the 9Be+ and 25Mg+ qubits prepared to the αB|↑iBe + βB|↓i |↑i |↓i Be and αM Mg + βM Mg, the

SWAP gate implements the following operation:

    SWAP (αB|↑iBe + βB|↓i |↑i |↓i −→ |↑i |↓i Be) αM Mg + βM Mg (αM Be + βM Be) αB|↑iMg + βB|↓iMg .

(5.17)

Our SWAP gate is constructed with three controlled-NOT gates. Figure 5.11 shows the pulse

sequence of a Ramsey-type experiment where the first Ramsey (microwave-induced) π/2 pulse was

applied to the 9Be+ ion to create an equal superposition in the |↑i and |↓i states, and the second
124

(microwave-induced) π/2 pulse with a variable phase φ was then applied to the 25Mg+ ion after

implementing the SWAP gate. Two Ramsey fringes detected on the 25Mg+ ion as a function of φ

are shown in Fig. 5.12. Data shown as blue squares corresponds to the case where the ions’ axial

motional modes are initialized to near the ground state ( ̄n ' 0.05), and data shown as red dots

corresponds to Doppler cooling with n ̄ ' 4. The contrast with Doppler cooling is reduced because

the Lamb-Dicke limit [Wineland 98] is not rigorously satisfied, which can reduce the fidelity of the

MS gate (see Sec. 4.4.6), which in turn affects the performance of the SWAP gate.

5.6 Mixed-Species Cirac-Zoller Gate

We also implemented a mixed-species Cirac-Zoller gate [Cirac 95] and used it to create a

Bell state (see Sec. 4.2 for a description of the Cirac-Zoller gate). The pulse sequence of our

implementation is shown in Fig. 5.13. First, we cooled the axial motional modes to near their

ground states ( ̄n ' 0.05). Then, with both qubits initialized to the |↑i state followed by a π/2

pulse induced by a microwave field on the 9Be+ qubit, a controlled-NOT gate with 9Be+ as the

“control” and 25Mg+ as the “target” implemented using the Cirac-Zoller protocol created a Bell

1 state √ (|↑↑i − i|↓↓i). 2

The Cirac-Zoller protocol requires a red-sideband 2π pulse to be applied on a transition

between one of the qubit states and an auxiliary state, |Auxi. Here, we chose the |F = 2, mF = 2i

state of the 25Mg+ ion as the |Auxi state, and the red-sideband 2π pulse was driven on the

|3, 1i ↔ |Auxi transition (see Fig. 2.6 for the energy level diagrams of the 25Mg+ ion).

Figure 5.14 shows the coherent parity flopping of the mixed-species Bell state created by our

implementation of the Cirac-Zoller gate. The data was obtained by applying microwave-induced

R(π/2, φ) pulses to each of the qubits after generating the Bell state with the pulse sequence

shown in Fig. 5.13. We estimated the Bell state fidelity to be ∼ 0.95. We did not investigate

the properties and imperfections of this protocol in detail since its performance is sensitive to

the motional excitation, making it less favorable compared to the mixed-species Mølmer-Sørensen

protocol (see Sec. 4.2).
125

1.0

0 1 2 3 4 5 6 0.0

0.2

0.4

0.6

0.8

φ

P

(↑〉Mg

)

Doppler temperature (n ≃ 4)

Near ground state (n ≃ 0.05)

Figure 5.12: The Ramsey fringes measured on the 25Mg+ ion after applying the SWAP gate as

depicted in Fig. 5.11. This shows that the phase information is preserved between the two species

and transferred from the 9Be+ ion to the 25Mg+ ion. Error bars correspond to standard error of

the mean.

Controlled-NOT Gate

μμWv

RR(

ππ

22 , 00)

μμWv

RR(

ππ

22 , 00)

BSB

RR(ππ, 0)

BSB

RR(ππ, 0)

RSB to Aux

RR(22 , 0)

Controlled-Phase Gate

9Be+

25Mg+

|↑⟩

|↑⟩

Motion

μμWv

RR(

ππ

22 , 00)

|nn = 0⟩

Figure 5.13: Pulse sequence for creating a Bell state √

1 (|↑↑i − i|↓↓i) using a combination of single- 2

qubit pulses and the Cirac-Zoller gate [Cirac 95]. For high fidelity, this protocol requires the

motional mode to be prepared in its ground state. The notations “BSB” and “RSB” denote blue

and red-sideband transitions, respectively. The controlled-phase gate and the controlled-NOT gate

are defined in Eq. 4.6 and Eq. 4.9, respectively. The definition of the R(θ, φ) rotation is in Eq.

2.13. Except for the “RSB to Aux” pulse, all other pulses are driven between the qubit |↑i and |↓i

states. The pulse sequence corresponding to ground state cooling of the motional modes (applied

prior to the operations shown) are not shown in this figure.
126

Figure 5.14: Parity oscillation of the Bell state created by the Cirac-Zoller gate [Cirac 95]. The

analysis pulses were microwave-induced π/2 pulses with variable phase φ. Error bars correspond

to standard error of the mean.
127

A previous implementation of a two-ion Cirac-Zoller gate employed two 40Ca+ ions cotrapped

in the same potential well [Schmidt-Kaler 03], and suffered considerable single-qubit addressing

error (∼ 3 %), whereas here addressing error is approximately zero.

5.7 Summary and Conclusion

In summary, by using state-dependent forces induced by two different wavelength laser beams

(near 313 nm and 280 nm), we demonstrated a deterministic mixed-species entangling gate with a

9Be+ and a 25Mg+ ion. The highest achieved mixed-species Bell state fidelity created by this gate

was 0.979(1). Together with single-qubit gates, we demonstrated the universality of this entangling

gate by constructing controlled-NOT gates and a SWAP gate. These two logic gates enabled us to

transfer information between qubits of different species, and their robustness against thermal excitation was compared with the original recipe of the quantum logic spectroscopy [Schmidt 05]. This

entangling gate can be added to the elementary quantum computing toolkit, which is potentially

useful in realizing a large-scale quantum information processor [Monroe 13] or a quantum network

[Moehring 07] using the distinctive advantages of different ion species.

Although the demonstration here used hyperfine qubits, our entangling gate technique is

general enough that it can be applicable to qubits based on optical transitions, or a combination

of hyperfine and optical qubits. This would also make this gate useful for readout in quantum

logic clocks [Chou 10, Schulte 16]. Furthermore, as our gate scheme employs an effective spin-spin

interaction, this technique can be potentially generalized for the study of quantum many-body

physics, as well as quantum simulation.

A similar and complimentary work has also been carried out at the University of Oxford

trapped-ion group with two isotopes of calcium ions, i.e. 40Ca+ and 43Ca+ [Ballance 15]. In that

work, the isotope shift between the qubits is small enough that one set of laser beams (near 397 nm)

is sufficient to implement the two-isotope entangling gate. A gate fidelity of 0.998(6) was achieved

using the σˆzσˆz gate (see Sec. 4.3.2 and Ref. [Leibfried 03] for a description of the σˆzσˆz gate).
Chapter 6

9Be+ High Fidelity Universal Gate Set

6.1 Introduction

It is well known that certain single-qubit gates and two-qubit entangling gates make up a universal gate set and combinations of them are sufficient to realize other quantum gates [Barenco 95,

Bremner 02, Zhang 03]. Therefore, the problem is reduced to achieving high fidelity for these two

operations. It is generally agreed that individual logic gate errors must be reduced below a certain

threshold, often taken to be around 10−4, to achieve fault-tolerance [Preskill 98, Knill 10, Ladd 10]

without excessive overhead in the number of physical qubits required to implement a logical qubit.

This level has been achieved in some experiments for all elementary operations including state

preparation and readout [Myerson 08], with the exception of two-qubit entangling gates, emphasizing the importance of improving multi-qubit gate fidelities.

For trapped-ion systems, various atomic species including 9Be+, 25Mg+, 40Ca+, 43Ca+, 88Sr+,

137Ba+ 111Cd+ 171Yb+ , , and have been investigated as qubit candidates systems. As various

ions differ in mass, electronic, and hyperfine structure, they each have technical advantages and

disadvantages. The 9Be+ ion is the lightest ion currently considered for quantum information

processing (QIP), and as such, has several potential advantages. The relatively light mass yields

deeper traps and higher motional frequencies for given applied potentials, and facilitates fast ion

transport [Bowler 12, Walther 12]. A light mass also yields stronger laser-induced spin-motion

coupling (proportional to the Lamb-Dicke parameter), which can yield less spontaneous emission

error for a given laser intensity [Ozeri 07]. However, a disadvantage of 9Be+ ion qubits compared
129

to some heavier ions such as 40Ca+ and 43Ca+ [Benhelm 08, Ballance 15] has been the difficulty of

producing and controlling the ultraviolet (313 nm) light required to drive 9Be+ stimulated-Raman

transitions.

In this chapter, we describe an experimental implementation of a high-fidelity universal gate

set with 9Be+ ions. By taking advantage of recent technological developments as described in Chap.

3, we demonstrate laser-induced single-qubit computational gate errors of 3.8(1)×10−5 and realize

a deterministic two-qubit gate to produce the Bell state |Φ 1

+i = √ (|↑↑i + |↓↓i) with high fidelity. 2

Partial state tomography analyzed with a maximum likelihood (ML) algorithm is used to evaluate

the fidelity of the created Bell state. Together with characterizing the effects of known error sources

with numerical simulations and calibration measurements, we deduce an entangling gate infidelity

or error of  = 8(4) × 10−4.

The organization of this chapter is the following: we briefly describe the setup relevant to

these experiments, followed by results of the two-qubit gate in Sec. 6.3. In Sec. 6.4 details of

the two-qubit gate’s error sources are characterized and discussed. Validity of using the created

Bell state fidelity as a representation of gate fidelity is investigated in Sec. 6.5. The partial state

tomography used for the evaluation of the created Bell state and certain imperfections using this

method are discussed in Sec. 6.6. Lastly, in Sec. 6.7, we describe the experiment and the results

characterizing single-qubit gates using a randomized benchmarking technique [Knill 08].

6.2 Experimental Setup

In this experiment, 9Be+ ions are confined at the E zone depicted in Fig. 3.1. For a single

9Be+ ion confined in this zone, the axial z harmonic mode frequency is ωz ' 2π × 3.58 MHz1 ,

while the transverse (radial) mode frequencies are ωx ' 2π × 11.2 MHz, and ωy ' 2π × 12.5 MHz.

Two trapped 9Be+ ions are separated by ' 3.94 μm and are aligned along the axial z direction.

Their z motion can be described by two normal modes, the center-of-mass (C) and stretch (S)

√ modes with frequencies ωC = ωz and ωS = 3ωz respectively. The motion of the ith ion is written

1 angular frequency
130

zi = zi,C0(ˆa + ˆa†) + zi,S0(ˆb + ˆb†) where a, ˆ aˆ† and ˆb, ˆb† are the lowering and raising operators for the

√ p √ p C and S modes and z1,C0 = z2,C0 = z0/ 2, z1,S0 = −z2,S0 = z0/ 2 3, where z0 = ~/(2mωz).

The two-qubit entangling gate is realized by the Mølmer-Sørensen (MS) interaction [Sørensen 99,

Sørensen 00, Milburn 00, Solano 99] on the stretch mode (see also Sec. 4.3.4). The MS protocol

requires simultaneous excitation of a blue-sideband transition with a detuning of δ and a redsideband transition with a detuning of −δ on the selected motional mode. Following Eq. 4.20, the

Hamiltonian is

X  

H = ~ ˆ ˆ η Ωσˆ+ be−i(δt+φj,r) + b†

ei(δt−φj,b) S j + h.c., (6.1)

j=1,2

where Ω is the resonant carrier transition Rabi rate, ηS = |Δk|z1,S0 ' 0.19 is the Lamb-Dicke

parameter of the stretch mode, σˆ+

j is the spin raising operator for the jth ion, and φj,b(r) is the

phase of the blue (red) sideband interaction on the jth ion. Starting in the |↑↑i state and setting

ηSΩ = δ/4, this interaction produces the entangled state

 P  1 −i( 1 (φ +φ |Φi = √

)+π/2) |↑↑i + e j=1,2 2 j,b j,r |↓↓i (6.2) 2

after a duration tgate = 2π/δ.

The details of the laser beam setup is given in Sec. 3.4.3. Figure 6.1 shows the relevant

stimulated-Raman laser beams used in the experiments described in this chapter. The MølmerSørensen gate is implemented with three laser beams labeled by k1, k2a, and k2b. Each of the

beams in path 2 excite one of the sidebands when paired with the k1 beam. Note that we implement

the above entangling operation on the second micromotion sideband to maximize the spin-motion

coupling strength and state-dependent forces with the ions subjected to micromotion (see Sec.

3.1). This induces higher errors caused by spontaneous emission and will be discussed later in Sec.

6.4. Similar to our implementation of the mixed-species entangling gate described in Chap. 5, the

entanglement created by this laser setup depends on the relative optical phases between paths 1

and 2, causing the phase factor φj,b + φj,r in Eq. 6.2 to fluctuate. This laser phase sensitivity can

be removed by using the same technique described in Sec. 5.2.3. The polarization of the k1 beam
131

ωω − 2ωωRRRR − ωω0,Be + ωωSS + δδ

ωω

ωω − 2ωωRRRR − ωω0,BBBB − ωωSS − δδ

kk2222

kk2222

kk11

9Be+

kk11 is turned off for 

ωω single-qubit gates : Frequency of beam kk11

ωωRRRR: Radio frequency for trapping of ions

ωω0: 9Be+ qubit frequency, 2ππ × 1.207 GHz

ωωSS: Motional stretch mode frequency

δδ: Detuning for sideband transitions

B field: 119.446 G 

Axial direction, zẑ

Figure 6.1: Laser beam geometry for stimulated-Raman transitions. Each of the k2a and k2b

beams when paired with the k1 beam give a wave vector difference Δk aligned along the axial z

direction, and separately excite the blue and red sideband transitions on the axial stretch mode. All

three beams are applied to implement the Mølmer-Sørensen interaction on the second micromotion

sideband. Copropagating beams k2a and k2b are used to implement single-qubit gates described

in Sec. 6.7. Details of laser beam setup is given in Sec. 3.4.3.
132

is adjusted such that the intensity ratio of σ+ to σ− components is approximately 8 : 2, with the

σ+ component used for driving the MS gate and the σ− component used for sideband cooling on

the |2, 2i → |1, 1i transition [Monroe 95b] (see also Sec. 2.1.5). The k2a and k2b beams are set to

be π-polarized to maximize the entangling gate Rabi rate for a given amount of laser intensity.

The MS protocol requires the blue and red sideband excitations to have equal Rabi rates.

Imbalanced Rabi rates cause errors as discussed in Sec. 4.4.8. Rabi rate imbalance between the

blue and red sideband excitations is minimized by adjusting the laser power in beams k2a and k2b

(Fig. 6.1) differentially while keeping the total laser power in path 2 fixed. During calibration of the

blue sideband’s Rabi rate, the beam in path 2 that is responsible for the red sideband excitation is

shifted off resonant from any transition. Similarly when the Rabi rate of the red sideband transition

is calibrated. This strategy is used to ensure that the transition frequency shifted by the AC Stark

effect does not change during the calibration.

6.3 Experimental Results

Following the initial Doppler cooling, the ions are sideband cooled with a series of |F = 2, mF = 2i|ni →

|1, 1i|n − 1i transitions (n is the Fock state quantum number), each followed by repumping [Monroe 95b].

This cooling yields mean mode occupation numbers n ̄C ' 0.01 and n ̄S ' 0.006 and the ions being

pumped to the |2, 2i state at the end of the cooling sequence. The cooling transitions are driven

on the positive second micromotion sideband as described in Sec. 2.1.5.

Mapping from the |2, 2i state to the |1, 1i = |↑i state is accomplished by a microwavedriven composite π pulse [Levitt 86] composed of a sequence of R(θ, φ) pulses R(π, 0), R(π, π/3),

R(π, π/6), R(π, π/3), R(π, 0), where the definition of R(θ, φ) is given in Eq. 2.13. After gate

operations, population in the computational qubit is transferred to the measurement qubit before

applying the state-dependent fluorescence detection (see Fig. 2.1). Firstly, population in the

|↑i state is transferred back to the |2, 2i state with the same composite pulse sequence. Then,

population in the |2, 0i = |↓i state is transferred or “shelved” to the |1, −1i using a microwave π

pulse, followed by another microwave π pulse which transfers remaining |↓i population to the |1, 0i
133

state.

We 1 use the gate to ideally prepare the Bell state |Φ+i = √ (|↑↑i + |↓↓i)2 . To evaluate 2

the gate’s performance, we perform partial state tomography analyzed with a maximum likelihood

(ML) algorithm to deduce the fidelity of the experimentally prepared state. The details of the

analysis method are discussed in Sec. 6.6. From the Bell-state fidelity as determined by the ML

method, we can estimate the MS gate fidelity (see Sec. 6.5). To first order, the inferred Bellstate fidelity does not include errors due to imperfect |2, 2i state preparation and measurement.

This is advantageous because this enables us to investigate the gate error at a level smaller than

the imperfect initialization into the |2, 2i state. Here, we estimate the impurity of the |2, 2i state

after the initial optical pumping to be approximately 2 × 10−3 (details in Sec. 6.6.2), whereas

an entangling gate error as small as 8 × 10−4 can be determined. Furthermore, this ML method

allows an arbitrary photon-count distribution to be analyzed, removing the need for making certain

assumptions, which are detailed in the later section of Sec. 6.6.

In a first experiment, we vary the laser beam power and determine the error of the Bell state

as a function of the gate duration for a fixed Raman detuning of Δ ' −2π × 730 GHz (Fig. 6.2).

The various curves in the figures show the expected errors due to spontaneous emission, errors due

to motional mode frequency fluctuations, and errors in the composite microwave pulse used for

|2, 2i ↔ |1, 1i = |↑i state transfer. We also determine the error of the created Bell state by varying

the Raman detuning, Δ. The results are shown in Fig. 6.3 where we keep a fixed gate duration of

tgate ' 30 μs while adjusting the laser beam powers for each Raman detuning. The minimum error

obtained is 8(4) × 10−4 for Δ ' −2π × 900 GHz, which yields a ML-Bell-state fidelity of 0.9992(4).

An important contribution to the ML-Bell-state error is due to the imperfect transfers from the

|2, 2i state to the qubit |↑i state (for both qubits) before the application of the gate, and the reverse

procedure that transfers |↑i population back to the |2, 2i state before detection. The total fidelity

of these transfer pulses, limited by magnetic field fluctuations and the quality of the microwave

2 here, we drop the phase term shown in Eq. 6.2; the phase of the laser pulse used for analyzing the coherence of

the created Bell state is stable relative to the phase term shown in Eq. 6.2, can therefore be effectively dropped
134

40 60 80 100 120 140 160 180

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

tgate (μs)

Error

105 Hz fluctuation

95 Hz fluctuation

85 Hz fluctuation

Qubit state preparation error

Bell state error (ML algorithm)

Figure 6.2: ML-Bell-state error (red dots) as a function of gate duration tgate for a constant Raman

beam detuning Δ ' −2π × 730 GHz. The black line shows the separately determined error and

uncertainty (gray shade) due to the microwave pulses used for |2, 2i ↔ |↑i state transfer. The

three dashed lines show the sum of the expected gate errors including photon scattering and mode

frequency fluctuations (which are slow compared to gate durations shown) for three different r.m.s.

magnitudes of mode frequency fluctuations. Errors due to these errors are discussed below. The

gate error increases quadratically with increasing tgate due to such frequency fluctuations; however,

for tgate ' 30 μs the error due to frequency fluctuations is approximately 1 × 10−4.
135

pulses, is investigated with separate experiments analyzed with the same ML algorithm. We find

transfer = 4(3) × 10−4. This is averaged over multiple data evaluations across multiple days; the

uncertainty is the standard deviation of these data. While this error does not in principle affect

the gate performance, we conservatively do not subtract it from our gate fidelity estimate due to

its relatively large uncertainty. The uncertainties of the inferred errors presented here are deduced

by parametric bootstrap resampling [Efron 93] with 500 resamples.

In the next section, we describe the characterization of individual errors sources through calibration measurements and numerical simulations. Together with investigating the average fidelity

with numerical simulations in Sec. 6.5, we deduce that the fidelity of the ML-Bell-state is a good

representation of the average gate fidelity.

It would be advantageous to evaluate the gate performance with full process tomography

or randomized benchmarking to confirm our assessment. We did not perform randomized benchmarking because ion motional excitation gives additional errors. This excitation occurs during ion

separation (to provide individual ion addressing) and because of anomalous heating [Turchette 00a]

during the required long sequences of gates. These problems can eventually be solved as in our previous demonstration where the gate fidelity was measured by interleaved randomized benchmarking

[Gaebler 12] or by process tomography [Navon 14]. In these two experiments, the gate error was

consistent with the measured two-qubit state fidelity [Horodecki 99, Nielson 02, Gaebler 16].

Our ML algorithm assumes that the reference histograms are perfect representations of the

qubit measurement. Experimental issues that can affect the validity of this assumption are investigated and discussed in Sec. 6.6.1. We determine a lower bound of 0.999 on the purity of the |2, 2i

state for one ion prepared by optical pumping, as detailed in Sec. 6.6.2. With this, we put a lower

bound of 0.997 on the overall Bell state fidelity.

6.4 Error Sources

Table 6.1 lists the individually evaluated errors for the highest entangling gate fidelity through

calibration measurements and numerical simulations.
136

Figure 6.3: ML-Bell-state error (red dots), plotted as a function of −2π/Δ for a constant gate

duration of approximately 30 μs, where Δ is detuning of the Raman laser beams relative to the

2S1/2 ↔ 2P1/2 electronic transitions. The simulated contributions to the Bell state error from

Raman and Rayleigh scattering (for details see Sec. 4.4.2) are shown with the blue and purple

dashed lines respectively. For large |Δ|, the Raman scattering error approaches zero, however,

the Rayleigh scattering error remains approximately constant at 1.7 × 10−4. The black line is the

sum of the Raman and Rayleigh scattering errors, and the composite microwave pulses used for

qubit state preparation and detection (uncertainty indicated by the gray band). Error bars for the

measured Bell state fidelity are determined from parametric bootstrap resampling [Efron 93] of the

data and represent a 1-σ statistical confidence interval.
137

Errors ×10−4

Spontaneous emission (Raman)

Spontaneous emission (Rayleigh)

Motional mode frequency fluctuation

Rabi rate fluctuation

Laser coherence

Qubit coherence

Stretch-mode heating

Finite Lamb-Dicke approximation

Off-resonant coupling

4.0

1.7

1

1

0.2

<0.1

0.3

0.2

<0.1

|2, 2i ⇔ |↑i two-way transfer 4

Table 6.1: Individually evaluated errors for the entangling gate at a Raman detuning of Δ '

−2π×900 GHz, and a gate duration of approximately 30 μs. Off-resonant coupling includes coupling

of the qubit states to other hyperfine states and their sidebands, which can includes motional mode

sidebands and micromotion sidebands. The last (transfer) error reduces the ML-Bell-state fidelity

but should minimally affect the gate fidelity.
138

6.4.1 Spontaneous Emission Induced by Raman Laser Beams

Spontaneous emission error is caused by randomly scattered photons when exciting stimulatedRaman transitions. It can be separated into Raman and Rayleigh scattering. Raman scattering

processes are inelastic and project an ion’s internal state to one of the other hyperfine states (see

Fig. 2.1 for energy level diagram for the 9Be+ ion), destroying coherence. Rayleigh scattering

processes are elastic and do not necessarily cause spin decoherence [Ozeri 07]; however, momentum

kicks from photon recoil cause uncontrolled displacements of the motional state, which can result

in phase errors in the final target states of the two-qubit gates. See Sec. 4.4.2 for more details on

spontaneous scattering of photon induced by Raman laser beams. The Kramer-Heisenberg equation (Eq. 4.29) is used to calculate Raman scattering probabilities, and verified with measurements

made on ions. The dynamics of the Mølmer-Sørensen interaction are studied with a numerical

master equation simulation (Sec. 4.4.1) including the Lindblad operators shown in Sec. 4.4.2.1 to

investigate the effect of Raman scattering on gate performances. The recoil error is calculated with

Eq. 4.38.

Raman scattering can be reduced by increasing the Raman detuning, |Δ|, at the cost of higher

laser intensity to maintain the same gate Rabi rate. However, Rayleigh scattering error reaches an

asymptotic values as |Δ| is increased (see Fig. 6.3 and Fig. 4.5). This error is proportional to the

Lamb-Dicke parameter (Eq. 4.38) and thus could be reduced by increasing the trap frequency (Eq.

2.7); it can also be reduced by using multiple loops in phase space [Ozeri 07, Hayes 12] (Eq. 4.38).

However, these methods reduce the gate Rabi rate and thus increase Raman scattering error. In

our experiment, eliminating the axial micromotion would allow us to increase Δ by a factor of ξ '

2 which would lower the Raman scattering error by a factor of 2ξ, and the Rayleigh scattering error

by a factor of ξ while maintaining the same gate duration.

Spontaneous Raman scattering can result in leakage of population from the qubit manifold

(Sec. 4.4.2.1). The resulting states will predominantly be detected as dark and falsely associated

with the qubit |↓i state. This creates a systematic bias that overestimates the actual Bell state
139

fidelity. Through simulations with the Lindblad operators described in Eq. 4.33, we found that

such a bias is approximately 4 × 10−5 for the Bell state fidelity created at a Raman detuning of

−2π × 900 GHz and approximately 1.5 × 10−3 for −2π × 90 GHz Raman detuning.

6.4.2 Motional Mode Frequency Fluctuations

Motional mode frequency fluctuations also cause errors. For the stretch mode, the sources

of frequency fluctuations (which are slow compared to the gate durations shown in Fig. 6.2) are

(i) fluctuations in the DC potentials applied to electrodes for trapping, (ii) fluctuating electric-field

gradients from uncontrolled charging of electrode surfaces, and (iii) non-linear coupling to radial

“rocking” modes [Roos 08, Nie 09]. By measuring the lineshape for exciting the motional state

of a single ion with injected RF “tickle” potentials on the trap electrodes at frequencies near the

mode frequencies, we estimate the first two sources contribute fluctuations of approximately 50 Hz.

Mode frequency fluctuations can also be caused by electrode charging induced by UV beam light

scattering off the trap surfaces [Harlander 10] so this effect may becomes more pronounced when

higher laser power is used.

For (iii), to a good approximation, the shift of the stretch mode frequency, δωS, from excitation of the rocking modes is given by δωS = χ(nx + ny + 1) where χ is a non-linear coupling parameter, nx and ny are Fock state occupation numbers of the two radial rocking modes

[Roos 08, Nie 09]. For our parameters χ ' 45 Hz. Our Raman laser beam geometry did not allow

direct measurement of the n ̄x and n ̄y radial mode excitations. Therefore, the final temperature

is estimated from the (thermal) Doppler cooling limit (Eq. 2.2), taking into account heating due

to photon recoil during sideband cooling of the axial modes. From this, we estimate the stretch

mode frequency fluctuations from experiment to experiment to be approximately 100 Hz r.m.s. As

these fluctuations are dependent on the occupation numbers of the radial modes, the error can be

suppressed by cooling the radial modes to near the ground state. Figure 6.2 shows three master

equation simulations (combined with Monte Carlo simulation) curves for different total values of

the r.m.s. frequency fluctuation of the motional mode. The simulation also includes the effect of
140

Raman scattering (Lindblad operators shown in Sec. 4.4.2.1) with the fluctuations in motional

mode frequency described by the Lindblad operator shown in Eq. 4.39. These curves follow the

trend of the data and are consistent with our known sources. The error due to these fluctuations is

approximately 1 × 10−4 for the shortest gate durations. Moreover, the trends in Fig. 6.2 indicate

that gate performance could be further improved with higher gate speed. Our gate speed is limited

by the available laser intensity for the given Raman laser beams detuning, Δ.

6.4.3 Rabi Rate Fluctuations

Errors are also caused by fluctuations in the sideband transitions’ Rabi rates, causing fluctuations in the state-dependent forces (see also Sec. 4.4.7). Sources are (i) fluctuations of the ions’

micro-motion amplitude along the axial direction, (ii) fluctuations in the laser beam intensities at

the ion locations, and (iii) fluctuations in the Debye-Waller factor associated with the center-ofmass (C) mode [Wineland 98]. In our experiments, the latter contributes the largest error (for

errors caused by Rabi rate fluctuations).

Given a thermal distribution of the C mode and following Eq. 4.47 and Eq. 4.52, we can

estimate the r.m.s. Rabi rate fluctuation and the resulting entangling gate error. With n ̄C '

0.01 at the beginning of the MS interaction, we find h δΩi ' 6 × 10−3 and we deduce an error of Ω

approximately 1 × 10−4 to |Φ+i. Because this mode experiences anomalous heating during the

application of the gate, the actual error contribution increases with the gate duration. The heating

rate for the COM mode is measured to be ∼ 80 quanta/s. For our 30 μs gate duration, this implies

a change of Δ ̄nC ' 0.001 averaged over the duration of the gate. Therefore, the error caused by the

changes of the Debye-Waller factor from heating can be neglected for our fastest gate durations.

Because the two-qubit-gate Raman transitions are driven on the second micro-motion sideband, the Rabi rates are proportional to the second-order Bessel function J2(|Δk|zμm) ' 0.48,

where |Δk|zμm = 2.9 is the modulation index due to the micromotion-induced Doppler shift and

is proportional to the applied RF voltage VRF . For the conditions of the experiment, J2(|Δk|zμm)

is near a maximum such that the Rabi rate is relatively insensitive to fluctuations in VRF . Our
141

measurements show that the radial mode frequencies can drift by up to 10 kHz over the course of

several experiments; this would imply a relative drift in VRF of ∼ 1 × 10−3 and a corresponding

change in the Rabi rate of 3 × 10−4, which contributes an error that is negligible compared to the

other errors. Relationship between gate error and Rabi fluctuation is given in Eq. 4.47.

Laser intensity fluctuations can be assumed to be comparable to the fluctuations measured

from the single-qubit benchmarking experiments (see Sec. 6.7), which are estimated to be ∼

1×10−3. This makes the laser intensity fluctuation’s contribution of Rabi rate fluctuations negligible

compared to that of the fluctuating Debye-Waller factors. Laser intensity fluctuations also cause

fluctuations in the AC Stark shifts, which we measure to be ∼ 1 kHz at a Raman detuning of

−2π × 900 GHz and induce a negligible error.

6.4.4 Smaller Error Sources

Smaller sources of error are (i) laser beam phase fluctuations between beam paths 1 and 2

during each experiment (Sec. 4.4.9), (ii) individual qubit decoherence (Sec. 4.4.10), (iii) heating of

the axial stretch mode [Turchette 00a] (Sec. 4.4.4), (iv) imperfect Lamb-Dicke approximation (Sec.

4.4.6), (v) off-resonant coupling to spectator transitions (Sec. 4.4.5), and (vi) Rabi rate imbalance

between the two spin-motion excitations (Sec. 4.4.8). Each of these sources contributes ∼ 10−5

error to the entangling gate.

Sources of transition frequency and phase fluctuations include fluctuations in the laser beam

phases φj,b and φj,r, and fluctuations in the qubit frequency. Fluctuations due to relative length

changes between paths 1 and 2 are measured by recombining the two beams after they exit the

UV fibers (see Fig. 6.1 and Fig. 3.8), detecting with a fast photo-diode, and measuring the phase

of the beat note using the AOM RF sources as a reference. We measured a phase drift of ∼ π

after ∼ 1 s, which is likely due to temperature drift of the optical elements in the setup. We also

observed small phase oscillations with frequencies of a few hundred Hertz, which can be attributed

to acoustic vibrations in the laboratory. With this, we estimate an error of ∼ 2 × 10−5 to the

gate. The measured coherence time of the qubit from Ramsey experiments is approximately 1.5 s
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(Fig. 2.2), which implies an r.m.s. qubit transition frequency error of 1 Hz, giving negligible error

compared to other sources.

The heating rate of the axial stretch mode is measured to be less than 1 quantum per second

and contributes an error of less than 3 × 10−5 to |Φ+i, estimated with master equation simulation

with Lindblad operators given in Eq. 4.40 and Eq. 4.41. The MS interaction is robust against

finite thermal excitation in the Lamb-Dicke limit, η  1 [Sørensen 99, Sørensen 00]. However, due

to the small mass of 9Be+ ions, this condition is not rigorously satisfied and the sensitivity to finite

motional excitation must be considered. Because of this effect, the realistic MS interaction applied

to the ions deviates from the desired Hamiltonian given in Eq. 6.1. The error due to this is given

by Eq. 4.46, which contributes an error of less than 2 × 10−5 for our parameters. Even within the

Lamb-Dicke limit, finite thermal excitation increases the sensitivity of error due to motional mode

frequency fluctuations [Hayes 12]. For our parameters, this error is negligible.

Off-resonant coupling to spectator transitions is suppressed by employing laser pulse shaping

(see Sec. 3.8). The rise and fall durations of the gate pulse are adjusted such that the Fourier

component at the frequencies of spectator transitions is sufficiently small. Spectator transitions

include the carrier and center-of-mass sideband transitions as well as other atomic transitions

that can be coupled by micromotion sidebands (the Zeeman splittings between atomic states are

comparable to ωRF ). If a square pulse is used instead of a shaped pulse, we estimate an error of

1 × 10−4 for a gate duration of 30 μs [Sørensen 00]. Rabi rate imbalance between the blue and red

sideband excitations is inferred by: first, performing extended Rabi flopping experiments on each

sideband transition with their respective laser beams (see Fig. 6.1). The Rabi rate(ΩR) of each

transition is then deduced by fitting the Rabi flopping data to the function e−t/τAcos(ΩRt+φ0

)+B

with ΩR represents the Rabi rate. Finally, the error caused by Rabi rate imbalance effect is

estimated by performing numerical Schr ̈odinger equation simulation. The measured Rabi rate

ratio of the two sideband excitations is typically ∼ 0.99 or better, this contributes ∼ 10−5 or lower

error to the entangling gate (see also Sec. 4.4.8).



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6.5 Average Fidelity

The performance of our entangling gate is evaluated by using the gate to create a Bell state.

Here, we investigate the performance of the gate over all input states by employing numerical

simulation with known experimental imperfections. Our simulation is performed by solving master

equations (Eq. 4.28), and combining with Monte Carlo simulations for certain error mechanisms.

Different methods are chosen for different error mechanisms (see Sec. 4.4.1).

One metric to quantify the gate’s performance is the “average gate fidelity”, Favg [Horodecki 99,

Nielson 02, Gaebler 16]:

6 3 1 Favg = S+ + S− − , (6.3) 5 5 5

with

XXh i 1 † S+ = (hU1| ⊗ hU2|) Gˆ ρnoisy(U1, U2)Gˆideal (|U1i ⊗ |U2i) , (6.4) ideal 36

U1 U2

1 XXh�
   �   i † S  ˆ − = U1 ⊗ U2 G ρnoisy(U1, U2)Gˆideal U1 ⊗ U2 , (6.5) ideal 36

U1 U2

  where |Uii is an eigenstate of the Pauli operators σˆx, ˆσy or σˆz for the ith qubit, and Ui is the

state orthogonal to |Uii. We fix a consistent phase for these eigenstates throughout. The operator

Gˆideal is the ideal entangling operation, ρnoisy(U1, U2) represents the resultant density matrix of the

imperfect entangling operation with the input states of |U1i ⊗ |U2i. With 36 different input states,

our simulations including known imperfections yield the summands in Eq. 6.4 and Eq. 6.5. We

found Favg lies within the uncertainty of the inferred Bell state fidelity measurement. With this,

we deduce that our the Bell state fidelity inferred with our ML analysis is a good representation of

the average gate fidelity.

6.6 State Detection and Tomography

A partial state tomography analyzed with a maximum likelihood (ML) algorithm is used to

evaluate the fidelity of the Bell state created by the entangling gate. This ML method has been

developed in collaboration with members of a NIST theory group led by E. Knill, with A. Keith
144

as lead. Other theory contributors include: S. Glancy and K. Coakley. The details of the partial

state tomography and ML method are one of the main focus of A. Keith’s master’s research, and

will be published elsewhere.

We will briefly discuss certain issues about inferring information about the prepared twoqubit quantum states from the photon count distributions and the maximum-likelihood method we

used to circumvent some of these problems. We also study certain issues associated with using the

ML method for the fidelity determination of the experimentally created Bell state.

The detection procedure is the standard state-dependent fluorescence technique where the

qubit up state of each ion is transferred to the bright state of the measurement qubit, and the qubit

down state is transferred (“shelved”) to the detection dark state before the detection laser beam is

applied (see Sec. 2.1.7 for details on fluorescence detection on the 9Be+ ions). Because both ions

are illuminated with a detection laser beam (beam waist of ∼30 μm), which is much larger than

the ion separation (∼ 4 μm), the photon counts come from both ions.

The recorded joint photon-count histograms collected by illuminating both ions simultaneously are drawn from mixtures of three possible count distributions qj (c) (c = 0, 1, 2, ...., C indicates

the photon counts) corresponding to the distinguishable ion subspaces spanned by (i) |↑↑i (q1(c)),

(ii) |↑↓i or |↓↑i (q2(c)), and (iii) |↓↓i (q3(c)) states. Because of the finite efficiency of our photon collection apparatus and some optical pumping during detection, the three count distributions

overlap, particularly those of subspaces (ii) and (iii) as shown in Fig. 6.4. Therefore an exact

determination of the subspace to which the ions are projected cannot be determined in a single

experiment.

Nevertheless, we can infer the ions’ density matrix statistically from repetitions of the experiment provided that the count distributions for each projected subspace are known. These

distributions can be inferred from reference experiments by fitting to a parametrized model of the

distributions. A simple class of such models is given by mixtures of Poissonians with different

means. The uncertainty requirements of our experiments and effects such as optical pumping during photon collection imply that we cannot use such models unless they have an excessively large
145

(a)

(b)

Figure 6.4: The photon-counts histogram for two 9Be+ ions prepared into an equal superposition

of |↑i and |↓i states of each ion before making joint fluorescence measurements. (a) Data is plotted

in a linear scale. (b) Same data plotted in a log scale. Count distributions corresponding to the

|↑↓i or the |↓↑i states (on average 30 photon counts) and the |↑↑i state (on average 60 photon

counts) overlap significantly. These overlapping distributions cause errors in the determination of

the quantum state using a single detection.
146

number of parameters, in which case overfitting becomes an issue. Figure 6.5 shows the typical

photon-count distributions for both ions prepared in the bright state (|2, 2i) or the dark state

(|1, −1i) of the measurement qubit. To obtain Fig. 6.5.(a), we apply optical pumping to prepare

the ion in the |2, 2i state before applying detection laser; for Fig. 6.5.(b), we first apply optical

pumping to the |2, 2i state, then transfer the population to the |1, −1i state and the |1, 0i state

(the dark state of the measurement qubit) with a series of π pulses driven with microwave field

(see Sec. 2.1.7 for more details). A simple inspection of these histograms indicates the difficulty of

using simple fitting models to accurately extract quantum state information.

Our maximum likelihood (ML) analysis avoids these issues by statistically inferring states

without requiring a model for the ideal count distributions. It requires sets of reference and data

histograms. The data histograms involve observations of an identically prepared state ρ modified

by analysis pulses. It infers a representative density matrix ρˆ. Because the different observations

are not “informationally complete”, ρˆ is not intended to match ρ precisely but the measurements

are designed so that the extracted fidelities of interest do match to within a statistical uncertainty.

In our experiment, we obtain four reference histograms ri(c) (i = 1, 2, 3, 4). These histograms

are associated with known populations in the measurement qubit. Each reference histogram is

obtained by observing known ion states prepared as follows: For r1(c), the state is prepared by

optical pumping both ions to the |2, 2i state, Fig. 6.5(a) shows a typical r1(c) histogram. For

r2(c), this optical pumping is followed by implementing the transfer |2, 2i → |↑i with a composite

microwave pulse, followed by the transfers |↑i → |↓i and shelving into one of the states |1, −1i or

|1, 0i with microwave π pulses as described above. A typical r2(c) histogram is shown in Fig. 6.5(b).

For r3(c), the optical pumping is followed by the microwave-driven spin-echo sequence consisting

of R(π/2, 0), R(π, 0), R(π/2, π/2) pulses on the |2, 2i ↔ |↑i transition, followed by transferring the

population in the |↑i state to the |1, −1i or |1, 0i state as was done for r2(c). The histogram r4(c)

is obtained like r3(c) but with the phase of the third pulse set to 3π . The change in phase does not 2

change the state when the initial state and pulses are as designed. Fig. 6.4 shows a typical r3(c)

or r4(c) histogram. Data histograms hk(c) are obtained directly from the prepared state ρ or by
147

(a) (b)

Figure 6.5: Typical photon-counts histograms for two 9Be+ ions prepared in the (a.) |2, 2i state and

(b.) the “shelved” states of |1, −1i or |1, 0i state. Both distributions exhibit non-standard photoncount distribution (e.g., the Poissonian distribution). For (a), this is caused by depumping process

due to imperfect detection laser beam polarization. For (b), finite frequency separation between

the atomic states allows population to leak from the dark state to the bright state (via optical

pumping); in which case the ions start to fluoresce. See Fig. 2.1 for 9Be+ energy level structure;

see also Fig. 6.6 for the detection behavior of each of the Zeeman states in the 9Be+ ion’s electronic

ground state. These issues render simple fitting models, e.g. to Poissonians, incapable of extracting

accurate quantum state information.
148

applying analysis pulses on the prepared state. The analysis pulses are global R(π/2, nπ/4) pulses,

for n = 0, 1, ..., 7. These pulses are applied using the laser beams from paths 1 and 2 (Fig. 6.1) to

maintain laser beam phase stability relative to the two-qubit gate. The rotation R(θ, φ) is given in

Eq. 2.13.

To determine ρˆ, we maximize the logarithm of the likelihood of the observed histograms

with respect to the unknown qj (c) and ρ to be determined. Given these unknowns, reference

P histogram ri(c) is constructed from the distribution j aijqj (c), where the aij are “populations”

determined from the ideal prepared states. For example, the r1(c) and r2(c) reference histograms

ideally correspond to count distributions q1(c) and q2(c), respectively; ideally, r3(c) and r4(c) each

corresponds to the mixture of count distribution 0.25q1(c)+ 0.5q2(c)+ 0.25q3(c). Similarly, the data

P histograms hk(c) are sampled from the distribution bkjqj (c), where the populations bkj are a j

linear function of ρ. Given these distributions, the log likelihood is given by

⎡ ⎛ ⎞⎤ ⎡ ⎛ ⎞⎤ X X 

4,C 3

X X 

9,C 3

log (Prob(r, h|q, a, b)) = ⎣ ri(c)log ⎝ aijqj (c)⎠⎦ + ⎣ hk(c)log ⎝ bkjqj (c)⎠⎦ i=1,c j=1 k,c j

+const. (6.6)

To maximize the log likelihood, we take advantage of the separate convexity of the optimization problem in the qj (c) and ρ and alternate between optimizing with respect to the qj (c) and

ρ. We used a generic optimization method for the first and the “RρR” algorithm [Hradil 04] for

the second to keep ρ physical during the optimization. The quality of the model fit can be determined by a bootstrap likelihood-ratio test [Boos 03]. We found that our data’s log-likelihood-ratio

is within two standard deviations of the mean bootstrapped log-likelihood-ratio.

A modification of the ML analysis is required to reduce the number of parameters needed

for the qj (c) and the complexity of the algorithms. For this, we bin the counts into seven bins of

consecutive counts [Lin 15, Lin 16]. The binning is done separately by setting aside a random 10 %

of each reference histogram and using this as a training set to determine a binning that maximizes

state information quantified by a mutual-information-based heuristic. The heuristic is designed to

characterize how well we can infer which of the four training set reference histograms a random
149

count histogram is sampled from.

6.6.1 Assumptions with the ML Method

The ML analysis assumes that the reference histograms are sampled from count distributions

corresponding to states with known populations. The actual populations deviate by small amounts

from this assumption, because here, the references are designed to yield histograms associated with

known populations in the measurement qubit, instead of the computational qubit. We considered

two systematic effects, which can lead to such deviations: (i) optical pumping that ideally prepares

the |2, 2i state may have fidelity as low as 0.999 (see Sec. 6.6.2), and (ii) imperfections in the transfer

pulses between the computation qubit manifold (the |↑i and |↓i states) and the measurement qubit

manifold (the |2, 2i state as the bright state and the |1, −1i and |1, 0i states as the dark state). The

latter is dominated by the transfer pulses between the |2, 2i and |↑i states, which has a measured

fidelity of 0.9996(3). Errors in the transfer between the |↓i and the |1, −1i and |1, 0i states have

less effect, since to a high degree all three of these states are detected as dark (see Fig. 6.6).

Similarly, to first order, the population of the prepared state corresponds to the collected reference

histograms are all in the measurement qubit manifold; contribution from the small populations

outside the measurement qubit manifold are observationally equivalent to population in |↓i since

they are dark to a high degree (see Fig. 6.6). If these populations are the same in all experiments,

they are equivalent to a background contribution. With this in mind, we inspect the systematic

effects from (i) imperfect optical pumping and (ii) imperfect transfers in more detail.

Consider the effect of imperfect optical pumping of population to the |2, 2i state. In Sec.

6.6.2, we estimate an upper bound of 10−3 population (per ion) in states other than the |2, 2i

state after optical pumping. This population is distributed over the other Zeeman states in a way

that depends on details of the optical pumping process. With one exception considered below,

population in states other than |2, 2i is nominally detected as dark in all reference histograms

(Fig. 6.6). The ML algorithm infers qj (c) as if all populations were in the measurement qubit

manifold. Provided the population in other states remains dark in all experiments, it is treated as
150

a background and effective subtracted with our ML method. The effect is that the ML algorithm

infers the renormalized density matrix on the qubit rather than the actual one with the trace

reduced by the population outside the qubit manifold. To correct for the effect of imperfect optical

pumping, we subtracted 2 from the ML-inferred Bell state fidelity, where  = 10−3 is the population

(per ion) outside of the computational qubit manifold during the Bell state preparation and outside

the measurement qubit manifold when detection laser is applied. This is consistent with the effect

on Bell state fidelity determined by performing numerical simulations taking imperfect optical

pumping preparation of the |2, 2i state into account (see next section). The exception to this

model is that a fraction of the population outside the |2, 2i state after optical pumping is in the

qubit manifold, i.e. the |↑i = |1, 1i state and the |↓i = |2, 0i state. Two situations are considered:

(a) In the case where non-|2, 2i state is in the |↑i state, reference histogram r3(c) and r4(c) are

affected differently compared to the situation above. This is because in this case, the population

stays inside the measurement qubit manifold. With respect to the background interpretation, this

is equivalent to having the |2, 2i population in references r3(c) and r4(c) exceed 0.5 (for each ion)

by ξ ≤ /2. To determine such effect on the ML inferred fidelity, a sensitivity analysis on simulated

data is performed by varying the ML assumed populations for these references according to the

parameter ξ. The change in fidelity is found to be small compared to our uncertainties. (b) Non-

|2, 2i population in the |↓i state after optical pumping enters the qubit state at the beginning of Bell

state preparation and in this context does not behave as a dark state independent of the analysis

pulses. However, simulations of the optical pumping process show that the non-|2, 2i population in

the |↓i state is less than 10−4, which is small compared to our uncertainties. This optical pumping

simulation is performed as follows: first, we include an impurity (π component) in the polarization

of the laser beam responsible for the optical pumping (ideally pure σ+ polarized). The population

in the |↓i = |2, 0i state is inferred by a laser beam polarization which gives 0.999 (for one ion)

optical pumping fidelity (see next section). Our simulation closely follows that described in Ref.

[Langer 06], and is further discussed in the next section.

Now we consider the second effect, i.e. imperfect transfer pulses between the computational
151

qubit and the measurement qubit. This is dominated by the transfer of |2, 2i to the |↑i computational qubit state before the application of the gate. The errors of the transfer pulse result in

population remaining in the |2, 2i state, outside the computational qubit manifold and unaffected

by the MS interaction and the following analysis pulses. To first order, the transfer pulse after the

gate will exchange the population in the |2, 2i state and the |↑i = |1, 1i state. The |1, 1i state will

be nominally detected with high probability as dark (see Fig. 6.6), independent of the analysis

pulses. While this effect is equivalent to a leakage error on the Bell state being analyzed, it is not

accounted for by the ML analysis. Transfer error after the application of the gate results in extra

dark population that depends on the final population in the |1, 1istate, which in turn can depend

on the analysis pulses. Thus, both effects are inconsistent with the analysis pulse model that is

assumed by the ML analysis. Such inconsistencies, if significant, are expected to show up in the

bootstrapped likelihood-ratio test, but did not. To confirm this, a second sensitivity analysis is

performed on simulated data, where we modified the ML model to include an extra dark state outside the qubit manifold and included the expected transfer pulse effects by modifying the analysis

pulses with pulses coupling the qubit to the extra state. The effect on the inferred error of this

modification was also found to be small compared to the uncertainties of the Bell state fidelity

inferred using the ML method.

These two sensitivity analysis were performed by A. Keith, who is leading the development

of the ML analysis with details to be published elsewhere.

6.6.2 Imperfect Optical Pumping and Lower Bound of Bell State Fidelity

The maximum-likelihood method, to first order, does not include the errors due to the imperfect preparation and measurement of the |2, 2i state. This error is to be included when reporting

the lower bound of the overall Bell state fidelity. This is essentially the problem of quantifying the

purity of the |2, 2i state prepared by optical pumping.

There are eight Zeeman states in the 2S1/2 electronic ground state and sixteen Zeeman states

in the 2P3/2 electronic excited state of the 9Be+ ion (see Fig. 2.1 for energy level diagram of the
152

9Be+ ion; the Zeeman states in the 2P3/2 levels are not shown except for the 2P3/2|3, 3i state).

The laser beam for the state-dependent fluorescence measurement is set to be σ+ polarized, and

on resonance with the 2S1/2|2, 2i ↔ 2P3/2|3, 3i transitions. Due to finite frequency separations

between different Zeeman states, starting with a non-2S1/2|2, 2i state, the detection laser beam can

off-resonantly excite electronic transitions between Zeeman/hyperfine states in the 2S1/2 and 2P3/2

levels, and optically pump the population to the |2, 2i state. The ion starts emitting fluorescence

photons after it is pumped to the |2, 2i state under the illumination of the detection laser beam.

Each Zeeman state responds differently and uniquely as shown in Fig. 6.6, which gives the simulated

histogram corresponding to each of the Zeeman/hyperfine states when the detection laser is applied.

The simulation used here is an analytic method which closely following that described in Ref.

[Langer 06] (see also Ref. [Acton 06]). Our simulation includes twenty four energy levels (eight

2S1/2 states and sixteen 2P3/2 states); the energy splittings and matrix elements between different

levels are calculated at the magnetic field of B = 119.446 G taking into account both the hyperfine

and the Zeeman interactions.

The lower bound of the |2, 2i state purity can be deduced by deriving an upper bound on the

error  of preparing the |2, 2i state after applying optical pumping. The population is dominantly in

the |2, 2i state but we do not know precisely which of the remaining Zeeman states are populated.

We write the density matrix of a single ion for this situation as

X 

7

ρ = (1 − )|2, 2ih2, 2| + i|ΨiihΨi|, (6.7)

i=1

where  = P7i and the |Ψii represent the hyperfine/Zeeman states excluding the |2, 2i state. One i

strategy for setting an upper bound on  is to choose a cut-off count β and compare the small-count

P “tail” probabilities t = c<β h(c).

Let tb be the tail probability of h|2,2i. Because state preparation and detection are not

perfect, we have tb ≥ tb, where t

 ̄  ̄ b is the tail probability of perfectly prepared |2, 2i states. The

tail probabilities ti for |Ψii are large, as verified by experimentally preparing each |Ψii state and

measuring its count distribution. From this we can set a lower bound on ti such that ti > l. With
153

10 20 30 40

20

40

60

80

100

Photon counts

Number of events

|2,2〉

10 20 30 40

0.1

1

10

100

1000

Photon counts

Number of events

|2,1〉

10 20 30 40

0.1

1

10

100

1000

Photon counts

Number of events

|2,0〉

10 20 30 40

0.1

1

10

100

1000

Photon counts

Number of events

|2,-1〉

10 20 30 40

0.1

1

10

100

1000

Photon counts

Number of events

|2,-2〉

10 20 30 40

0.1

1

10

100

1000

Photon counts

Number of events

|1,1〉

10 20 30 40

0.1

1

10

100

1000

Photon counts

Number of events

|1,0〉

10 20 30 40

0.1

1

10

100

1000

Photon counts

Number of events

|1,-1〉

Figure 6.6: Simulated detection histograms for a 9Be+ ion initialized to each of the eight Zeeman

states in the 2S1/2 electronic ground state of the 9Be+ ion (see Fig. 2.1). Each histogram is obtained

with 1, 000 detection events. Except for the |2, 2i state, all other figures are shown in log scale.

Each state responds differently and uniquely when the detection beam is applied. Besides the |2, 2i

state, all other Zeeman states are detected as dark to a high degree. Furthermore, the F = 1

manifold states (|F = 1, mF = 0, ±1i) have smaller fluorescence compared to the F = 2 manifold

states. The bright state of the measurement qubit is the |2, 2i state (which corresponds to the qubit

|↑i state), and the dark state is the |1, −1i and the |1, 0i states (which corresponds to the qubit

|↓i state). See Sec. 2.1.7 for details on the fluorescence detection of the 9Be+ ions. The simulation

here uses the saturation parameter, s = I/Isat = 0.5 and assumes that the detection laser beam is

pure σ+ with its frequency tuned on resonance with the 2S1/2|2, 2i ↔ 2P3/2|3, 3i transition. The

detection efficiency (which experimentally relates to the sum of the quantum efficiency of the PMT

and the collection efficiency of the imaging system) is set so that the |2, 2i state fluorescence counts

equal to 30 on average in our simulation.
154

this, we can write

X 

7

tb = (1 − )t

 ̄b + iti (6.8)

i

≥ (1 − )t

 ̄b + l (6.9)

tb ≥  (l − t

 ̄b) + t ̄b, (6.10)

or

tb − t

 ̄b

 ≤ (6.11) l − t

 ̄b

tb ≤ . (6.12) l − t

 ̄b

For our parameters of l = 0.8 and t

 ̄b = 0, we estimate an upper bound on  of 1 × 10−3. We also

numerically simulate the effect of imperfect optical pumping and find that the Bell state error scales

linearly as a function of . We also numerically simulate the effect of imperfect optical pumping

and find that the Bell state error scales linearly as a function of . This is consistent with the lower

bound on overall Bell state fidelity inferred in the previous section by considering the effect on the

ML analysis (see Sec. 6.6.1).

6.7 9Be+ Single-Qubit Gate

Single-qubit gates are driven with co-propagating beams k2a and k2b shown in Fig. 6.1

with their frequency difference set to be the qubit frequency of ω0 ' 2π × 1.207 GHz. With this

copropagating beam geometry, single-qubit gates are negligibly affected by ion micromotion since

Δk of these two beams projected along any mode of motion corresponds to a microwave wavelength,

which is much larger than the size of motional wavepacket. Here, the Raman detuning, Δ from the

2S1/2 ↔2 P1/2 transition frequency is set to be ∼ −2π × 730 GHz.

We employ a randomized benchmarking technique [Knill 08] to evaluate the performance

of the single-qubit gates. Compared to quantum process tomography or other methods for the

characterization of gate performance, randomized benchmarking is more efficient and provides

separate determination of the state preparation and measurement error (SPAM) and the error per
155

computational gate (gate). Randomized benchmarking also offers scalability advantage as it allows

gate to be determined with a number of measurements which scales polynomially with the number

of qubits. Furthermore, using long sequences of random gates make the procedure sensitive to

errors similar to that in arbitrary, lengthy QIP computations.

Randomized benchmarking techniques have been used in platforms other than trapped-ion

systems such as neutral atoms [Olmschenk 10], liquid NMR [Ryan 09], and superconducting qubits

[Chow 09]. Using randomized benchmarking, the lowest single-qubit gate error achievable in any

platform was measured to be 1.0(3) × 10−6 [Harty 14].

In our implementation of the single-qubit randomized benchmarking, multiple sequences with

random gates are pre-generated for a given set of sequence lengths. Here, sequence lengths of 1, 3,

8, 21, 55, 144, 233, 377, 610, and 987 are implemented. Each set of sequence lengths consists of 100

unique sequences. Each sequence consists of a different number of π and π/2 pulses around the x,

y, and z axes of the Bloch sphere, as well as identity gates. The π pulses are performed with two

sequential π/2 pulses, each with a duration ∼ 2 μs. Rotations about the z axis are accomplished

by shifting the phase of the direct digital synthesizer that is keeping track of the qubit’s phase; the

identity gate is implemented with a 1 μs wait time. Each “computational gate” is taken from the

Clifford group [Gottesman 98], and consists of a random Pauli gate (π pulse) and a random Clifford

gate (π/2 pulse). With the qubit initialized to a pure state, and in the absence of gate errors, the

outcome of each sequence is deterministic and the ion will be found in either the |↑i or |↓i state.

Sequences are randomly chosen from the pre-generated pool to be implemented during run

time, with each sequence repeated 100 times to collect statistics. The fidelity of each sequence is

recorded and the decay of fidelity is determined as a function of the number of random computational gates for a given sequence length. The results are shown in Fig. 6.7. Here, state detection

and measurement for a single 9Be+ ion is relatively straightforward. As shown in Fig. 2.5, the

photon-count distribution for the bright and dark states are well separated so we determine the

ion’s state by setting a threshold of 12 counts (count ≤ 12 is recorded as dark).

The reason for implementing computational gates with the Clifford group is because the noise
156

0 200 400 600 800 1000

0.96

0.97

0.98

0.99

1.00

Sequence length

Fidelity

Raman detuning Δ: -2π x 730 GHz

Error per computational gate: 3.8(1) x 10-5

Figure 6.7: Average fidelity for single-qubit-gate randomized benchmarking sequences, plotted as

a function of sequence length. We determine the average error per computational gate to be

gate = 3.8(1)×10−5 and state preparation and measurement error to be  = 2.0(3)×10−3 SPAM for

these data sets. Error bars show the standard error of the mean for each point.
157

of the Clifford group is intrinsically depolarized, i.e. the average fidelity of an operation applied to a

pure state is identical to the operation fidelity averaged over different pure states [Knill 08]. Under

this depolarizing assumption, randomized benchmarking yields an average fidelity as a function of

the number of computational gates. The average fidelity decreases to the asymptotic values of 1/2

with sufficiently large number of computational gates. The average fidelity with a sequence length

of l is [Knill 08]

1 1 F = + (1 − SPAM)(1 − 2gate)

l

. (6.13) 2 2

By fitting this expression to the data on Fig. 6.7, we deduce an error per computational gate of

gate = 3.8(1) × 10−5, and SPAM = 2.0(3) × 10−3. For Δ ' −2π× 730 GHz used here, spontaneous

emission error [Ozeri 07] is calculated to be 2.5 × 10−5. This value is estimated using the KramerHeisenberg equation (Eq. 4.29), and is verified with a separate measurement made on the ion. The

remaining error is dominated by Rabi rate fluctuations of approximately 1 × 10−3 due to imperfect

laser power stabilization. Fluctuations in the Rabi rate are inferred from separate experiments

that calibrate the Rabi rates of the laser-induced qubit transition; the associated error is inferred

from a Schr ̈odinger equation (Eq. 4.27) or a master equation (Eq. 4.28) simulation combined

with a Monte-Carlo simulation that includes the measured power fluctuations. These calibration

experiments are interleaved with the benchmarking experiments.

6.8 Summary

In summary, we demonstrated high-fidelity single-qubit and two-qubit laser-induced gates

with trapped 9Be+ ions. The single-qubit gates fidelity of 0.999962(1) (corresponding to an error

of  = 3.8(1) × 10−5) exceeds some threshold estimates for fault-tolerant error correction with

reasonable overhead. A partial state tomography is developed to analyze our two-qubit gate data.

The highest achieved two-qubit gate fidelity is measured to be 0.9992(4) (corresponding to an error

of  = 8(4)×10−4). The sources of error have been identified and strategies to further suppress them

were discussed. This result is a factor of ∼ 50 improvement compared to the previous best effort
158

of implementing a high-fidelity two-qubit gate with 9Be+ ions in our laboratory [Tan 13] (see also

Chap. 7). The improvement was made by taking advantage of various technological developments

[Wilson 11, Bowler 13, Colombe 14, Leibrandt 15] and using an ion trap [Blakestad 11] which has

a lower anomalous heating [Turchette 00a] due to uncontrolled electric field noise.

Similar laser-driven single-qubit and two-qubit gate has also been demonstrated with 43Ca+

ions by the trapped-ion group at Oxford [Ballance 16]. Single-qubit computational gate with an

error of 6.6(3)×10−5 were evaluated with randomized benchmarking techniques on a magnetic-field

insensitive transition. Two-qubit gates were implemented with the σˆzσˆz gate (see Sec. 4.3.2 and

Ref. [Leibfried 03]) with an error of 1(1) × 10−3. Along with our two-qubit gate presented here,

these results appear to be the highest multi-qubit gate fidelities reported to date in any physical

platform.

However, despite the improvement, two-qubit gates remain the only elementary QIP operation which has yet to be demonstrated with error lower than the practical fault-tolerant threshold

of ∼ 10−4. In this context, we plot the history of two-qubit gate errors achieved with (i) laser driven

interactions applied on trapped-ion qubits, (ii) microwave-driven interactions applied on trappedion qubits, and (iii) superconducting circuits in Fig. 6.8. Although the fidelity of the two-qubit

entangling gate provides a benchmark for the level of quantum control across different platforms, it

provides little information about the scalability of different systems. We are not aware of a simple

metric for a system’s scalability for the realization of a practical QIP device.
159

This work

9Be+ ̶ 25Mg+ gate

see Chap. 5

Dynamical decoupling gate

see Chap. 7

Figure 6.8: Two-qubit gate errors (or the errors of the created Bell state) reported in the literature over time. This figure is not comprehensive but shows the best two-qubit gate results, as

well as two other type of two-qubit gate results presented in this thesis. Gate implementations

in three platforms are shown, i.e. (i) laser-driven gates on ion qubits [Turchette 98, Sackett 00,

Rowe 01, Leibfried 03, Benhelm 08, Tan 13, Tan 15, Ballance 16, Gaebler 16], (ii) microwavedriven gates on ion qubits [Ospelkaus 11, Weidt 16, Harty 16], and (iii) superconducting qubits

[Steffen 06, DiCarlo 09, Chow 12, Barends 14]. The highest fidelity achieved with ion qubits using a microwave-based implementation is 0.997(1) [Harty 16] and 0.9944(5) with superconducting

qubits [Barends 14]. Data in this figures are of courtesy from Prof. David Lucas (University of

Oxford) who collected and consolidated the data.
Chapter 7

Dynamical Decoupling Two-Qubit Geometric Phase Gate

7.1 Introduction

Based on extensions of the spin-echo technique [Hahn 50], dynamical decoupling was first

proposed and studied theoretically in Ref. [Viola 98] by driving a control Hamiltonian that continuously induces oscillation between the states of a two-level system. Since then, various proposals

and implementations using dynamical decoupling schemes to suppress decoherence errors affecting

the quantum systems of interests have been reported. The general theory of dynamical decoupling

was formulated in Ref. [Viola 99] and reviewed in a more recent paper [Lidar 14].

Most dynamical decoupling schemes have been implemented for the protection of quantum

memories against environmental noise [Du 09, de Lange 10, Ryan 10, Szwer 11] such as external

magnetic field fluctuations. For quantum information processing, it is also desirable to have dynamical decoupling built into quantum gates or have quantum gates directly applicable to quantum

memories which are protected by dynamical decoupling techniques. Such operations must be designed to provide protection against environmental noise, but not from the control fields required to

apply the coherent operations. Single-qubit gates featuring protection with dynamical decoupling

have been demonstrated in Ref. [Levitt 86, Souza 12, Zhang 14]. Demonstrations of multi-qubit

gates with dynamical decoupling built-in include Ref. [van der Sar 12, Liu 13, Zhang 15] with

solid-state-based qubits and with trapped-ion qubits [Tan 13, Piltz 13]. Furthermore, dynamical

decoupling has also been used to lengthen the lifetime of an entangled state [Gustavsson 12].

In this chapter, we describe an experimental implementation of an entangling gate based on
161

a spin-dependent force featuring continuous dynamical decoupling using two trapped-ion qubits.

The decoupling is provided by a control field that continuously induces oscillation between the

logical states of the qubit. With this, the qubit becomes less sensitive to dephasing errors which

can be caused by external magnetic field fluctuations. Furthermore, fluctuations in the AC Stark

shifts induced by the laser beams that are used for driving coherent operations through stimulatedRaman process are also suppressed. This technique is analogous to the bang-bang control used in

classical platforms (see for example Ref. [Takahashi 70, Macki 82]).

The entangling gate scheme described and implemented here was proposed in the theoretical

paper by Bermedez et al. [Bermudez 12]. The proposal considers the weak effective spin-spin

coupling regime, where the motional mode that serves as the mediator for the spin-dependent

force is only virtually excited. Here, we modify the original proposal to be applicable in the

strong effective spin-spin coupling regime in order to maximize gate speed for a given laser-induced

transition Rabi rate. This enables us to minimize gate errors due to spontaneous scattering of

photons caused by stimulated-Raman laser beams (see Sec. 4.4.2). Furthermore, a faster gate

speed suppresses other error sources such as motional heating and other thermal-related errors.

Following the publication of our work [Tan 13], a theoretical study for the gate scheme in the

strong coupling regime was presented in Ref. [Lemmer 13], providing a complementary framework

for the results presented in our experimental demonstration.

7.2 Experiment Setup and Implementation

This experiment was carried out in a legacy experimental setup with a different trappedion apparatus located in a different laboratory. However, most of the crucial elements for the

manipulations of trapped ions used here are very similar to that described in other parts of this

thesis. A detailed description of this legacy experimental apparatus is available in Ref. [Jost 10].

Similar to the X-junction trap [Blakestad 11] (see also Sec. 3.1), the ion trap used here is also

a linear Paul trap [Paul 90]. As shown in Fig. 7.1, this trap consists of eight pairs of control

electrodes for shuttling of ions but does not feature a junction for arbitrary reordering of an ion
162

E L

400 μμm

Figure 7.1: The electrodes of the ion trap used for the experiment described in this chapter. The

ions are loaded in zone L and transported to zone E for the entangling gate to be implemented.

See Ref. [Jost 10] for details of the design and construction of this trap.

crystal. The ions were first loaded in zone L and then transported to zone E, where entangling

gate operations were carried out.

For this ion trap, the axial micromotion experienced by the ions is substantially smaller than

in the X-junction trap. Therefore, we do not apply the differential voltage on the electrodes for

detection (described in Sec. 2.1.1). Furthermore, the spin-motion stimulated-Raman transitions

were driven on the micromotion carrier. The motional mode heating [Turchette 00a] rate of this

trap is higher compared to the case of the X-junction trap. For a single 9Be+ ion, we measure the

heating rate to be ' 0.5 quanta/ms at an axial trapping frequency of ' 2π × 2.7 MHz1 . Heating

rate affects the performance of the entangling gate and will be discussed in Sec. 7.3 (see also Sec.

4.4.4).

The laser beam setup used for the experiment here suffered certain drawbacks compared to

that described in the rest of this thesis. Firstly, here the laser beam profiles and pointing fluctuations were considerably worse due to the unavailability of the ultraviolet (UV) fibers [Colombe 14].

Secondly, the laser power stabilization for the laser beams was not as robust as that used in Chap.

6 and we did not have access to apparatus with sufficient servo bandwidth to employ dynamic

shaping of laser intensity profiles. Furthermore, due to the limited available laser power when this

experiment was carried out, the Raman detuning used here induced a higher spontaneous emission

1 angular frequency
163

rate compared to that used in Chap. 6. Higher photon scattering rate causes higher gate error,

see details in Sec. 4.4.2. The laser beam setup used for the experiment here suffered certain drawbacks compared to that described in the rest of this thesis. Firstly, here the laser beam profiles

and pointing fluctuations were considerably worse due to the unavailability of the ultraviolet (UV)

fibers [Colombe 14]. Secondly, the laser power stabilization for the laser beams was not as robust

as that used in Chap. 6 and we did not have access to apparatus with sufficient servo bandwidth

to employ dynamic shaping of laser intensity profiles. Furthermore, due to the limited available

laser power when this experiment was carried out, the Raman detuning used here induced a higher

spontaneous emission rate compared to that used in Chap. 6. Higher photon scattering rate causes

higher gate error, see details in Sec. 4.4.2.

Two 9Be+ ions are confined along the axis of a linear Paul trap and have an axial center-ofmass (COM) mode (angular) frequency of 2π × 2.6 MHz and a stretch mode frequency of 2π × 4.5

MHz. For a single 9Be+ ion, the radial secular frequencies are set to be 2π × 12.5 MHz and

2π × 11.8 MHz. Here, we used the hyperfine state of |F = 2, mF = 1i = |↓i and |1, 0i = |↑i as

our qubit. Although this choice of hyperfine states for the qubit is different than that used in

other experiments described in this thesis, the properties of these qubits are very similar. The

qubit frequency is ω0 ' 2π × 1.207 GHz, which is first-order insensitive to changes in the applied

magnetic field of B ' 0.011964 mT (119.64 G). The direction of the applied magnetic field is aligned

45◦ with respect to the trap axis. Table A.2 lists the magnetic-field insensitive transitions in the

electronic ground state of the 9Be+ ion.

As described in Sec. 4.3.5, the Bermudez protocol requires a resonant carrier spin-flip excitation and a single spin-motion sideband excitation with a detuning of δ from a selected motional

mode. Here, the detuned spin-motion sideband excitation is driven through a stimulated-Raman

process by a pair laser beams labeled by kCo1 and k90 in Fig. 7.2. These two beams are aligned

√ such that their vector difference Δkz = 2 2π/λ aligned along the axial direction. In this configuration, only the motional mode along the trap axis interacts with the laser beams. We set the

beat-note frequency of these two laser beams to be blue detuned by a frequency δ from the stretch
164

k90

kCo1

ωL − (ω0+ων + δ) ωL

90∘

Δk ω0

B = 11.964 mT

Helical antenna

Ions

Figure 7.2: Laser beam setup to drive the two-qubit gate with the Bermudez protocol in the case

of a microwave-induced carrier excitation. Two lasers beams labeled as kCo1 and k90 are used to

excite the detuned spin-motion sideband transitions. Their wave vector difference Δk is aligned

such that only the axial motional mode can be coupled.
165

mode blue sideband transition. In our setup, the carrier excitation can be driven by (i) a microwave

field, or (ii) a laser field. We investigated both methods in two different sets of experiments. The

microwave fields are delivered with a helical antenna located outside the vacuum system, with the

frequency set and impedance matched to the source at ω0 ' 2π × 1.207 GHz. We achieve a carrier

π-pulse duration of approximately 11 μs. The laser field to drive the carrier excitation includes an

additional beam kCo2 that paired is with the beam kCo1 (see Fig. 7.3). The relative frequency

difference of these two beams is set to be ω0, and the carrier π-transition has a duration of approximately 5 μs. As these two beams are copropagating, the Rabi frequency is not affected by

the ion’s motion [Wineland 98]. These two beams are generated inside a single AOM (with center

frequency ∼ 600 MHz) analogous to that described in Sec. 3.4.3. The Raman detuning is set to

be ' 260 GHz (' 160 GHz ) red detuned from the 2S1/2 ↔ 2P1/2 transitions for the microwave

(laser)-induced-carrier gate. In the rotating-wave approximation, and after transforming into the

interaction frame of both the qubit and the motion, the Hamiltonian of our implementation the

Bermudez protocol in the Lamb-Dicke limit is

X  

σ+ iφC σ+ † −i(δt+φj

0

H = ) ~ ΩC ˆj e + iΩj ˆ aˆ e + h.c., (7.1) j

j=1,2

where ΩC denotes the Rabi rate of the carrier excitation that provides the dynamical decoupling,

and Ωj = Ω0,jηξj , where Ω0,j is the Rabi rate of the resonant laser-driven carrier excitation, η is

the Lamb-Dicke parameter for the selected normal mode with ξj as the mode amplitude of the jth

ion. The respective phases for the carrier and sideband excitation are φC and φ0

j = ΔkX0,j + Δφ,

where Δk and Δφ are the difference in wave vectors and phases of the optical fields driving the

blue sideband transition, respectively. The equilibrium position of the the jth ion is denoted by

X0,j .

Since the interesting case is when ΩC  |Ωj |, we go to the interaction frame where the states

are dressed by the carrier excitation. With this condition, the eigenvalues of the dressed system

under the influence of a certain mechanism that shifts the qubit frequency by ω0 are given by Eq.
166

Δk

B = 11.964 mT

Ions

k90 kCo2 kCo1

ωL − (ω0+ων + δ)

ωL + ω0

ωL

90∘

Figure 7.3: The laser beam setup where carrier excitation is induced by stimulated-Raman process.

This setup is very similar to that depicted in Fig. 7.2. Here, the microwave antenna is turned off

and an additional beam labeled as kCo2 is adjusted such that its frequency is shifted ω0 relative to

beam kCo1. Together, these two beams excite carrier transitions.
167

4.25:

   ! 2 1 02 ω 1 ω02

± ΩC + + ΩC + · · · . (7.2) 2 ΩC 2 Ω2

C

Therefore, the influence of ω0 becomes a second order effect and decreases as ΩC increases. Now we

consider the effects of the sideband terms in the dressed-state picture. In the |+i, |−i basis where

1 |↑i = √ (|+i + |−i), 2

1 |↓i = √ (|+i − |−i), (7.3) 2

the Hamiltonian described in Eq. 7.1 becomes

P     H = ~ Ω 0

i j + −i(δt+φ0 ) i(δt+φ ) | i h e j j +| − |−i j j |−ij j aˆ† − aeˆ j

2

P    

~ Ωj −2iΩC t 2iΩC t 0 ) i(δt+φ0

+i |−i | e

i ) h+ − |+ − (δt+φ i |−i †

j j je aˆ e j + aeˆ j j j . (7.4) 2

The first term in the expression above is identical to Eq. 4.16, which describes a Hamiltonian of a

geometric phase gate with the spin-dependent force acting on the rotated basis. The second term

induces off-resonant transitions between the dressed states |+i and |−i and can be dropped when

ΩC  |Ωj |. We set φC = 0 for simplicity; this does not affect the discussion that follows. The

entangled states produced by this protocol are insensitive to optical path length changes between

the non-copropagating laser beams (kCo1 and k90 in Fig. 7.2 and Fig. 7.3) that occur on a time

scale that is long compared to the gate duration [Tan 13].

However, similar to the situation in Sec. 5.2.1, the geometric phase accumulated after applying the gate depends on the basis states. Applying Eq. 5.5 to the case here by writing φ = φ0 M,j j ,

after setting Ω1 = Ω2 = Ω, the geometric phase accumulated by the different parity basis states

after a duration of 2π/δ is

8πΩ2   2 Δk(X0,1 − X 0,2) φ++,−− = cos , δ2 4

  8πΩ2

2 Δk(X0,1 − X0,2) φ+−,−+ = sin . (7.5) δ2 4

With this, it becomes obvious that the geometric phases applied to the basis states depend upon

the distance between the two ions, i.e. X0,1 − X0,2. Here, we adjust the strength of the harmonic
168

confinement to make the ions’ separation distance equal to πp/Δk where p is an integer. By doing

so, the entangling gate speed can be optimized for a given Ω and δ by setting the geometric phases

for the different parity qubit state to differ by π/2.

In the case where the condition of ΩC  Ω is not rigorously satisfied, the evolution of the

populations oscillates and the final state depends on the finite carrier Rabi frequency. Figure 7.4

shows a simulation result of the interaction Hamiltonian described in Eq. 7.1 with ΩC = 15Ωj=1,2.

In our experimental implementation, we remove the fast oscillation dependence by performing

spin-echo type sequences.

For the case where the carrier excitations are driven by the microwave field, we applied a

π-pulse with a π -phase shift relative to the carrier in the middle of the gate sequence, as depicted in 2

Fig. 7.5. This phase shift operation is accomplished by reprogramming the direct-digital synthesizer

which is used for the generation of the microwave field (see Sec.3.6). The application of this pulse

makes the ions undergo a multiple-loop trajectory in motional phase space which can further

suppresses errors in the gate detuning δ and gate duration that can lead to residual spin-motion

entanglement at the end of the gate operation [Hayes 12].

The spin-echo sequence for the case where the laser is used to induce carrier transition is

shown in Fig. 7.6. The carrier excitation is continuously applied but a π-phase shift is implemented

halfway through the gate. This phase shift is accomplished by changing the phase of the RF signal

used for the generation of the kCo2 laser beam (see Fig. 7.3). In contrast to the microwave-carrier

case, this spin-echo sequence induces a single-loop phase-space trajectory. It corrects only for

errors in the carrier Rabi frequency but does not suppress errors that lead to residual spin-motion

entanglement.

Each experiment repetition begins with Doppler cooling and optical pumping of the ion’s

population to the |F = 2, mF = 2i state, followed by Raman sideband cooling [Monroe 95b] of the

axial COM and stretch modes to n ̄ of ' 0.2 and ' 0.05, respectively. This is followed by applying

a composite pulse sequence [Levitt 86] induced by copropagating beams kCo1 and kCo2 (Fig.

7.3), consisting three resonant pulses R(π/2, 0), R(π, π/2), and R(π/2, 0) (the rotation operation
169

Figure 7.4: Simulated population evolution of Hamiltonian in Eq. 7.1 with both qubits initialized

to the |↑i state. Although a maximally entangled state can be created at t = tgate, due to the

finite ratio between the carrier transition Rabi rate ΩC and the sideband transition Rabi rate

Ω = Ω1 = Ω2, the populations undergo oscillations at the carrier Rabi frequency, making the target

state highly sensitive to the interaction time. Here, the simulation parameter is ΩC = 15Ω.

ttgggggggg

2 ,φφ

Ωttgggggggg

2 ,φφ′

ttππ,φφ +

ππ

2

Microwave

carrier excitation

Detuned 

sideband excitation

ttgggggggg

2 ,φφ

Ωttgggggggg

2 ,φφ′

Figure 7.5: Pulse timing sequence for the microwave-induced-carrier gate. A π rotation with a π/2

phase with respect to the previous pulse refocuses the fast spin population oscillations induce by

the carrier excitation (see Fig. 7.4). This pulse can suppress miscalibrations and errors in the gate

duration and detuning, δ [Hayes 12].
170

Laser-driven

carrier excitation

Detuned 

sideband excitation

ttgggggggg

2 ,φφ

ttgggggggg

2 ,φφ + ππ

ttgggggggg,φφ′

Figure 7.6: Pulse timing sequence for the laser-induced-carrier gate. The phase of the carrier

excitation is shifted by π during the second half of the gate.

R(θ, φ) is given in Eq. 2.13) that transfers each each ion from the |2, 2i state into the |2, 1i = |↓i

state. With both ions initialized to the |↓↓i state, the entangling gate ideally creates the Bell state

|Φ i = √

1

+ (|↓↓i + |↑↑i). 2

The standard state-dependent resonance fluorescence technique is used to measure the ions’

state at the end of the gate operation. First, the |↓i states are transferred to the |2, 2i state by using

the same composite pulse sequence as for the state initialization, and the |↑i state is transferred

to the |1, −1i state by use of a single “shelving” π-pulse (see Fig. 2.1 for 9Be+ ion’s energy level

diagram). This is followed by turning on the detection laser beam for 250 μs. Ions in the |2, 2i

state fluoresce “bright” and register an average of ∼ 30 photons per ion on a photomultiplier tube.

Ions in the |1, −1i state fluoresce “dark” and register an average background counts of 3 photons.

Detection counts yield three possible outcomes: two ions bright (|↓↓i), one ion bright (|↑↓i and

|↓↑i), and zero ions bright (|↑↑i). The probabilities of those outcomes, P2, P1, and P0 respectively,

are determined by fitting a triple Poissonian function to the histogram of counts obtained in each

experiment.

The population evolution as a function of the gate interaction duration with carrier excitation

driven by laser beams is shown in Fig. 7.7, with the state |Φ+i (in the ideal case) created at

tgate = 2π ' 105 μs. The coherence of the created Bell state is investigated by applying an δ

“analysis” carrier π/2-pulse to the ions. Figure 7.8 shows the parity (P2 + P0 − P1) plotted as

a function of the phase φ of the analysis pulse for the case of the microwave-induced-carrier gate

with a gate duration of tgate = 4π = 250 μs. The resulting oscillation of the parity is fitted to the δ
171

Figure 7.7: Evolution of the populations of |↓↓i (blue points), |↑↑i (red) and anti-aligned spin

states (green) as a function of the duration of simultaneous application of laser-induced carrier and

detuned sideband excitations. The phase of the carrier is shifted by π at half of the interrogation

time for each point (see Fig. 7.6). The gate time for this case is approximately 105 μs, at which

point the Bell state |Φ i = √

1

+ (|↓↓i + |↑↑i) (in the ideal case) is created. The solid lines show the 2

results of numerical master equation simulation that include contributions from (i) spontaneous

scattering of photon induced by the Raman laser beams, and (ii) state preparation and detection

error.
172

function Acos(2φ + φ0). The performance of the gate was characterized by measuring the created

Bell state’s fidelity, which is determined by the contrast of the parity oscillation (A) and the total

qubit population in the |↑↑i and |↓↓i (P2 + P0). The Bell state fidelity, F, is (P0 + P2 + A)/2

[Sackett 00]. For the microwave-induced-carrier gate we find P0 + P2 = 0.988(4) and A = 0.960(8),

which gives F = 0.974(4). For the laser-induced-carrier gate we determine P0 + P2 = 0.961(1) and

A = 0.930(8), which gives F = 0.946(4).

7.3 Error Sources

The sources of error which lead to imperfect creation of the Bell state were investigated with

numerical simulations and calibration measurements. Certain sources of error in the ∼ 10−4 level

or lower are not described here. Table 7.1 shows the individually evaluated error contributions

for both the microwave-induced carrier and the laser-induced-carrier gates. For the laser-inducedcarrier gate, the error caused by spontaneous scattering of photons is dominated by the laser beams

that are used to drive the carrier excitation (the kCo1 and kCo2 beams in Fig. 7.3). Furthermore,

due to a limited amount of power available from the laser source, the Raman detuning was restricted

to ' 160 GHz to allocate sufficient power for the laser beams responsible for the carrier excitation.

For the microwave-induced-carrier gate, we were able to increase the Raman detuning to ' 260

GHz and thus reduce the spontaneous emission error compared to the laser-induced-carrier gate.

We calculated the spontaneous scattering rate using the Kramer-Heisenberg equation as shown

in Eq. 4.29 and then deduced the associated gate error by numerically simulating the master

equation containing the out-of-manifold scattering and bit-flip scattering Lindblad operators (see

Sec. 4.4.2.1). We also performed the following experiments to estimate the Raman scattering rate;

first, the ions were prepared in their qubit states, then the Raman laser beams were turned on

with their frequency difference set far-off resonant from any transition, so their primary effect was

to induce photon scattering. Finally the scattering rates were determined by measuring how much

population that remained in the qubit states. The measurements agrees with the calculation using

the Kramer-Heisenberg equation. Errors due to Rayleigh scattering (Sec. 4.4.2.2) were estimated
173

0

Π

2

Π

3 Π

2

2 Π

-1

-0.5

0

0.5

1

Phase Φ of analysis pulse

Parity

Figure 7.8: The coherence of the state produced by the microwave-induced-carrier gate is investigated by applying an analysis π/2-pulse with a variable phase. The contrast of the parity oscillation

is determined by fitting A cos(2φ + φ0) + B to the data points.
174

Infidelities/Errors Microwave

(×10−3)

Laser

(×10−3)

Spontaneous emission

State preparation and detection

Carrier drive infidelities

Debye-Waller factor

Fast oscillation term

Imperfect sideband drive

Motional heating of strectch mode

2.8

9.1

1.3

∼ 10

∼ 3

∼ 1

< 1

19

17

16

∼ 6

< 1

∼ 1

< 1

Table 7.1: Individually evaluated errors that contribute to the imperfect creation of the Bell state

|Φ+i state with carrier excitation driven by (i) microwave field and (ii) laser field. See Sec. 4.4 for

detail discussions on error sources in the implementation of two-qubit gates and their respective

evaluation strategies.

to be negligible in the gate implementation here.

The errors for state preparation and measurement include imperfect optical pumping that

initialized the ions to the |2, 2i state, and errors in the transfers of population into and out of the

qubit manifold. The improved state preparation and detection for the microwave-induced-carrier

gate was achieved by a more careful calibration of the optical pumping laser beams’ polarization and

alignment with respect to the applied magnetic field. By using a microwave field for the qubit-state

transfers and better composite pulse sequences (e.g. see Eq. 2.12 and Eq. 2.14), state preparation

and detection errors could be further suppressed to the level demonstrated in Chap. 6.

Errors due to fluctuations of the carrier Rabi frequency that are slow compared to the gate

duration are highly suppressed by the spin-echo techniques (Fig. 7.5 and Fig. 7.6). However,

fluctuations on the time scale of the gate duration cause error. This error source was characterized

by performing the gate sequences with only the carrier drive applied to the ions and measuring the

probability the ions end in |↓↓i state for the laser-induced carrier or the |↑↑i state for the microwaveinduced carrier. The higher error for the laser-induced-carrier can be attributed to laser intensity

and beam-pointing fluctuations. We estimated the error caused by fluctuations in the laser-induced

sideband excitation due to laser intensity and beam-pointing fluctuation to be approximately 10−3

for both cases. This error was determined by performing a Monte Carlo simulation of the sideband
175

excitation incorporating measured laser-intensity and beam-pointing fluctuations. Errors related to

laser intensity and beam-pointing fluctuations could be suppressed by better laser beam control as

demonstrated in Chap. 6, e.g. UV fibers [Colombe 14] and better laser power stabilization including

the dynamic shaping of laser power profile (see Sec. 3.8). A double-dressed state technique has

been proposed to suppressed errors due to imperfect carrier excitation [Cohen 15].

As the spin-dependent force was applied on the axial stretch mode, the error due to motional dephasing associated with the anomalous heating [Turchette 00a] of this mode is negligible.

However, due to (i) the finite temperature ( ̄n ' 0.2) of the COM mode at the beginning of the

gate implementation and (ii) the heating of the COM mode (∼ 1 quanta/ms), the performance of

the gate was affected through a fluctuating Debye-Waller factor (see Sec. 4.4.7). This error was

estimated with Eq. 4.47 and Eq. 4.51. The microwave-carrier gate suffered more from this error

mechanism due to its longer gate duration, as compared to the laser-carrier gate.

As shown in Eq. 7.4, the Hamiltonian describing the Bermudez gate contains two terms.

The first represents the spin-dependent force and the second is a fast oscillation term which can

be neglected if the Rabi rate of the carrier excitation is sufficiently high, i.e. ΩC  Ω. In our

implementation this condition was not rigorously satisfied and caused errors in the gate. This error

is determined by performing numerical simulations with experimental values as input parameters,

assuming they are noise free. Figure 7.9 shows the master equation simulation results of Bell state

fidelity as a function of ΩC for different levels of motional dephasing rate as described by Eq. 4.39. Ω

Errors associated with the imperfections of carrier excitation and the fast-oscillating term

can be suppressed by better microwave field delivery that increases the Rabi rate and fidelity of

microwave-driven transitions. Furthermore, a higher carrier Rabi rate also allows the gate speed

to be increased, reducing the effect of error mechanisms associated with motional heating and

thermal-related issues.

Other sources of error include qubit dephasing, motional frequency fluctuations, off-resonance

coupling to spectator transitions, optical phase fluctuations, imperfect global rotations and calibra-
176

Figure 7.9: Master equation simulations are used to study the Bell state fidelity as a function of

carrier Rabi rate (ΩC) to sideband Rabi rate (Ω) ratio. The Hamiltonian described in Eq. 7.1

and the motional dephasing Lindblad term described by Eq. 4.39 are included. Four different

motional dephasing rates, Γ, are considered and color coded in this figure. Two groups of the gate

implementations are shown: (i) solid lines (marked with the asterisk symbol (∗) in the legend)

indicate a spin-echo sequence using a π-pulse rotation with π/2 phase shift as depicted in Fig. 7.5,

and (ii) dashed lines represent spin-echo sequence implemented by shifting the phase of the carrier

drive at the middle of the gate, as shown in Fig. 7.6. Each simulation begins with the ions in

the |↑↑i state and with the interactions turned on for a gate time of tgate. The duration of the

single-qubit gate for the π-pulse rotation used in spin-echo sequence (i) is not included in tgate to

provide a straightforward comparison. For (i), Ω is set to be δ and the fidelity of the created Bell 2

1 state is evaluated at tgate = δ . For (ii), the spin-echo sequence produces a two-loop gate. In this

δ 2 case, Ω is set to be √ and tgate = δ . A total of 11 Fock states are used in the simulation. 2 2
177

tion errors. These sources are estimated to contribute errors on the order of a few 10−4 or lower to

the gate, and they were not studied extensively here (see Sec. 4.4 for discussions regarding these

error sources).

7.4 Summary

Inspired by the proposal presented in Ref. [Bermudez 12], which considered a dynamicaldecoupling entangling gate operating in the weak effective spin-spin coupling regime, we modified

the proposal and demonstrated two gate schemes which operate in the strong coupling regime to

maximize gate speed for a given laser-induced transition Rabi rate. By creating a Bell state and

evaluating its fidelity, the performance of our gate implementations were investigated. The highest

Bell state fidelity achieved was measured to be 0.974(4). The error sources that limit the creation

of a perfect Bell state were studied and we determined that these errors were dominated by various

technical issues at the time these gate schemes were implemented. Technical improvements that

were made to allow the implementations of the high-fidelity universal gate set (described in Chap.

6) would also benefit the gate scheme described here.

A similar scheme has also been investigated in another trapped-ion group. In Ref. [Harty 16]

a Bell state with a fidelity of 0.997(1) was created by employing a modified scheme which is

referred to as the dynamical-decoupling Mølmer-Sørensen gate. The gate scheme was implemented

with two detuned motional sideband excitations (similar to that of the Mølmer-Sørensen protocol

described in Sec. 4.3.4), together with a carrier excitation that provided the dynamical decoupling.

This gate scheme alleviates the requirement that the Rabi rate of the carrier excitation be much

larger than the sideband excitations, while still providing protection against qubit dephasing. The

gate implementation was accomplished by an all-microwave scheme, and the built-in dynamical

decoupling reduced the gate error due to qubit dephasing caused by fluctuating AC Zeeman shifts

induced by fluctuating microwave fields.
Chapter 8

Chained Bell Inequality Experiment

In this chapter, we describe chained Bell inequality (CBI) [Pearle 70] experiments performed

with pairs of trapped ions. Ion pairs used here include a pair of 9Be+ ions and a 9Be+ - 25Mg+

pair. The experiments were implemented with the techniques developed for a trapped-ion quantum information processing (QIP) using a quantum charge coupled device architecture outlined in

Ref. [Wineland 98] and Ref. [Kielpinski 02]. These include (i) magnetic field insenstive (“clock”)

state qubits which offer long coherence time, (ii) robust state preparation and measurement, (iii)

shuttling of ions for their individual addressing and measurements, (vi) generation of mixed-species

entanglement (described in Chap. 5), (iv) high-fidelity deterministic generation of Bell states,

and (v) single-qubit gates, (the latter two are described in Chap. 6). High performance of these

techniques is essential for the realization of the experiments described here.

The detection efficiency in our experiment is near 100 %, and the measurement outcomes of

every trial in each experiment were recorded. This avoid making the “fair sampling” assumption,

such that our experiments successfully close the “detection loophole” [Pearle 70, Clauser 74].

This chapter is organized in the following fashion: we will first use the better known Clauser,

Horne, Shimony, and Holt (CHSH)-type [Clauser 69] experiment to provide background and then

introduce the chained Bell inequality experiment, including a brief summary on loopholes in Bell

inequality experiments and previous CBI experiments. After a brief description of the experiment

setup, results are presented in Sec. 8.3. The lowest chained Bell inequality parameter determined

from our measurements is 0.296(12). This value is significantly lower than 0.586, the minimum
179

fraction derived from a perfect CHSH-type experiment. Lastly in Sec. 8.4, we briefly discuss

one consequence of our Bell inequality results in the context of quantum certification under the

“self-testing” framework [Mayers 04, McKague 12].

8.1 Background and Motivation

The incompatibility between quantum physics and local-realistic theory was first presented by

Einstein, Podolsky, and Rosen [Einstein 35] and formulated by John Bell with a mathematical inequality [Bell 64]. This was followed by proposals for this conflict to be investigated in the so-called

“Bell inequality experiments”. The most notable one is that presented by Clauser, Horne, Shimony,

and Holt (CHSH) [Clauser 69]. With each particle subjected to two measurement settings, a CHSH

experiment employs two particles to be jointly measured with four different measurement setting

combinations. In such experiments, the correlation outcomes must satisfy the CHSH inequality if

local realism is valid, the CHSH inequality can be written as [Rowe 01]

BCHSH = |q(α1, β1) + q(α2, β1)| + |q(α1, β2) − q(α2, β2)|

≤ 2, (8.1)

with

Ncorrelated(αk, βl) − Nanticorrelated(αk, βl) q(αk, βl) = , (8.2) Ncorrelated(αk, βl) + Nanticorrelated(αk, βl)

where ak and bl with k, l = 1, 2 are the measurement setting choices applied on system a and b,

respectively.

The first experimental confirmation of this violation by a quantum system was performed

by Freedman and Clauser [Freedman 72], and refined by Aspect, Grangier and Roger [Aspect 82].

Along with many other Bell inequality experiments performed with various systems since these

pioneering works, the experimental results provide evidence that quantum physics contradicts the

predictions of local realistic theories. However, all of these experiments are subjected to certain

“loopholes”, which in principle, could allow a local system to show violation of the Bell inequality.

A major milestone was reached with three loophole-free CHSH-type Bell inequality experiments
180

[Hensen 15, Shalm 15, Giustina 15]. However, although these experiments reject with high confidence the null hypothesis of “nature is local”, they are limited in the extent to which their data

may assert the hypothesis that their data exhibits maximum nonlocality. Here, we are using the

convention that “local” is referred to the distribution of outcomes which obeys Bell’s inequalities,

and “nonlocal” is referred to the distribution that violates Bell inequalities.

Elitzur, Popescu, and Rohrlich [Elitzur 92] described a model of the distribution of outcomes

measured from a quantum state as a mixture of a local-realistic distribution, which obeys Bell’s

inequalities, and another distribution that does not. That is, a probability distribution P for the

outcomes of a experiment can be written as

P = plocalP L + (1 − plocal)P NL, (8.3)

where P L represents a local joint probability distribution and P NL represents a non-local distribution, with plocal as the weight of the local component bound by 0 ≤ plocal ≤ 1. For an ideal

CHSH Bell inequality experiment, the lowest attainable upper bound on the local content plocal in

the distribution created by a maximally entangled state is ∼ 0.586 [Barrett 06, Christensen 15]. In

principle, this bound can be lowered to zero by using a chained Bell inequality (CBI) experiment

[Pearle 70].

As indicated in Fig. 8.1.(a), CBI experiments are a generalization of a CHSH-type experiment. During each trial, a source that may be treated as a “black box” emits two systems labeled

a and b, respectively. The experimentalist records the measurement outcomes after choosing a

pair of measurements to perform separately on a and b. We use the symbols ak, bl to denote the

respective measurement settings and akbl for the pair. There is a hierarchy in which the Nth CBI

experiment involves 2N different settings. The N = 2 CBI experiment is equivalent to the CHSH

Bell inequality experiment. The settings for general N are chosen from the set

Z = {a1b1, a1b2, a2b2, a2b3, . . . , aN−1bN , aN bN , aN b1}. (8.4)

Each local measurement has a binary outcome which we call B for “bright” or D for “dark”. The

outcome of the trial is recorded as c(x, y) = 1 if x = y or 0 if x 6= y, where x is the outcome
181

Source

Measurement 

setting: aakk

Measurement 

setting: bbll

Bell test

Measurement:

x = B or D

Measurement:

y = B or D

aa

bb

c(x,y) = 0 or 1

(a)

(b)

aa11

aa22

aa33

⋮

aaNN−11

aaNN

bb11

bb22

bb33

⋮

bbNN−11

bbNN

Figure 8.1: (a) Illustration of a Bell inequality experiment. A source emits two systems a and b, here

two 9Be+ ions. After choosing measurement settings ak and bl, the experiment implements Hilbertspace rotation operations (which are controlled with classical variables) on the ions respectively.

Then a standard fluorescence based measurement in a fixed basis is applied to each ion. This is

equivalent to choosing the measurement basis for the state that is present before the measurement

settings are applied. Each system’s measurement outcome is labeled B for “bright” or D for “dark”,

corresponding to the observation of fluorescence or not. From the joint measurement we record “c

= 1” if the outcomes are the same and “c = 0” if they are not. (b) “Chaining” of the measurement

settings for the Nth CBI experiment. The measurement settings can be visualized as a chain where

akbk and ak+1bk+1 are linked by akbk+1, and the chain is closed by the settings aN b1. The CHSH

Bell inequality experiment corresponds to the special case of N = 2.
182

from system a and y is the outcome from system b. The probability to obtain c(x, y) = 1 may

depend on the choices ak and bl, so we define that probability to be the correlation C(ak, bl) =

P(BB|akbl) + P(DD|akbl), where P(xy|akbl) is the probability that system a yields measurement

outcome x and system b yields measurement outcome y when the measurement settings pair is akbl.

We define the chained Bell parameter to be:

IN = C(a1, b1) + C(a1, b2) + C(a2, b2) + C(aN , bN ) + (1 − C(aN , b1)). (8.5)

If the experiment is governed by a local hidden variable model, then the chained Bell inequality

IN ≥ 1 (8.6)

must be satisfied [Pearle 70]. Note that IN can be estimated using only the record of the settings

akbl and outcomes c(x, y), without knowledge of the mechanism of the source, thus it is referred to

as a “black box”. It was shown in Ref. [Barrett 06] that the chained Bell parameter IN is always

an upper bound on plocal. Furthermore, IN is shown in Ref. [Bierhorst 16] to be a least upper

bound for plocal under the assumption that the distributions are non-signaling, in the sense that

each party’s measurement outcomes do not depend on the remote party’s setting choice. In the

limit of N → ∞ and with perfect experimental conditions, CBI experiments could be used to show

that plocal → 0, demonstrating complete departure from local realism.

8.1.1 “Loopholes” and Previous Experiments

Similar to a CHSH-type experiment, a CBI experiment may be subject to “loopholes” that,

in principle, allow local systems to show violation of the inequality. Loopholes can also be viewed

as certain interpretations of the local-realistic theory. These loopholes arise when one must rely on

various supporting assumptions that are made in the design and execution of the experiments, but

which cannot be absolutely verified. Loopholes in Bell inequality experiments are reviewed in the

recent articles of Ref. [Brunner 14] and Ref. [ ̊A. Larsson 14].

For example, if the setting choice for a can be communicated to b (or vice-versa), the “locality

loophole” is opened. Ensuring space-like separation between the choices and remote measurement
183

events closes this loophole [Bell 85]. The “detection loophole” [Pearle 70, Clauser 74] is opened by

making the fair-sampling assumption, which says that a subset of the data can be used to represent

the entire data set. This assumption is often used when some trials fail to produce outcomes due

to inefficient detectors. High efficiency detectors are required to close the detection loophole and

observe violation of the inequality [Cabello 09]. The minimum detection efficiency required to close

the detection loophole for the Nth CBI experiment is given by [Cabello 09]

2

ηmin(N) = N , (8.7) cos( π ) + 1 N−1 2N

assuming that the measurement efficiencies on a and b are equal and that a maximally entangled

state is measured. This emphasizes the important of high detection efficiencny in large N CBI

experiments. If the analysis of the data assumes that the outcomes of the trials are independent

and identically distributed (i.i.d.), the “memory loophole” is opened [Barrett 02]. For example, one

way to determine IN is by running each of the CBI setting pairs akbl for a total number of Mk,l

trials respectively and calculating

PMk,l

i=1 c(xi, yi) C(ak, bl) = , (8.8) Mk,l

(where i indexes the trials) to estimate each C(al, bk) term in Eq. 8.5. This analysis requires the

i.i.d. assumption for standard error estimates to be valid. This loophole can also be associated with

certain memory effects which can exists in experimental apparatus that produce data erroneously

due to the time sequential nature of the execution of experiments. The memory loophole can

be closed by applying appropriate analysis techniques to an experiment that uses randomized

measurement settings from trial to trial [Gill 03].

Previous experiments on the CBIs (N ≥ 3) employed entangled photons pairs [Pomarico 11,

Aolita 12, Stuart 12, Christensen 15]. The lowest yet reported upper bound on plocal is approximately 0.126 [Christensen 15]. However, to our knowledge all previous CBI experiments with N ≥ 3

suffer from the locality, detection, and memory loopholes.

There exist five previous CHSH-type Bell inequality experiments with trapped ions [Rowe 01,

Matsukevich 08, Pironio 10, Ballance 15, Tan 15]; all of them closed the detection loophole. The
184

first detection-loophole-closing Bell inequality experiment was performed with two 9Be+ ions by

Rowe et al. [Rowe 01]. In this case the ions were always confined in one potential well during applications of measurement settings and measurements. In Ref. [Matsukevich 08] and Ref.

[Pironio 10], two 171Yb+ ions were separately confined in two traps separated by a distance of ∼ 1

m. Entanglement was created by a heralded scheme with photonic links, where each of the two

remotely located ions was first entangled with the polarization states of an emitted photon, then

the entanglement of the two ions was accomplished via interference and joint measurement of the

two photons. Two different isotopes of calcium ions, i.e. 43Ca+ and 40Ca+ were subjected to a Bell

inequality test in Ref. [Ballance 15]. Although the Bell state was created by a high-fidelity deterministic entangling gate, this experiment suffered measurement errors on the 40Ca+ ion. Lastly, a

CHSH Bell inequality experiment was also carried out on a pair of 9Be+ and 25Mg+ ions, described

in Sec. 5.3.2 and in Ref. [Tan 15]. However, none of the trapped-ion experiments, including the

experiments described here, are able to close the locality loophole. In Table 8.1 we list all CHSH

Bell inequality experiments (across all platforms) that close the detection loopholes.

8.2 Experiment Setup

Five trapping zones were used in this experiment, they are labeled as EL, E, S, E0, ER in

Fig. 8.2. The ions can be confined together in a single harmonic well, or separately confined in

different locations along the trap axis. Time varying potentials are applied to the control electrodes

to deterministically separate ions and transport them between different locations [Blakestad 11,

Bowler 12]. Two 9Be+ ions are initially confined in zone S to interact with laser beams for the

generation of entanglement. Trapping parameters in S are set to be similar to that when ions

are confined in E, where manipulation of ions in the experiments described in Chap. 5 and 6

were performed. The two states of the ions are encoded in the “clock” qubit states described in

Sec. 2.1, i.e. |F = 2, mF = 0i = |↓i and |1, 1i = |↑i. This clock qubit is realized by applying an

external magnetic field of ∼ 119.446 G, which makes the transition frequency first-order insensitive

to magnetic-field fluctuations [Langer 05]. We measured a qubit coherence time of approximately
185

S ER EL E E’ 

200 μm

bb aa

Figure 8.2: Layout of the relevant segmented trap electrodes. Each CBI experiment begins with

one ion located in zone E and the other in zone E0

. The blue dots, which indicate the ions, are

overlaid on a photograph showing the trap electrodes (gold). By transporting the ions in and out

of zone S, we individually implement settings and measure each ion sequentially. The ions are

separated by at least ∼ 340 μm when settings akbl are applied, a distance much larger than the

laser beams waist of ∼ 25 μm.
186

1.5 s (see Sec. 2.1). Such a long-lived memory storage duration is essential when taking into account

the duration for each trial, as detailed later.

To deterministically create a Bell state, we employ a pulse sequence as depicted in Fig. 8.3.

This sequence consists of a two-qubit entangling gate and multiple global rotations induced by

laser beams, and is equivalent to that described in Sec. 5.2.3. The entangling gate is implemented

using the Mølmer-Sørensen (MS) protocol [Sørensen 99, Sørensen 00] as described in Chap. 6.

The global pulses shown inside the dashed box in Fig. 8.3 are induced by the same set of (noncopropagating) laser beams (all three beams shown in Fig. 6.1) that drive the MS interaction.

By embedding the MS interaction in such a Ramsey sequence, the overall pulse sequence in the

dashed box produces entanglement which has negligible sensitivity to slow phase drift between laser

beams [Lee 05, Tan 15], and implements the operations |↑↑i → |↑↑i, |↑↓i → i|↑↓i, |↓↑i → i|↓↑i,

and |↓↓i → |↓↓i. Global rotations shown labeled as “Co Carrier” in Fig. 8.3 were induced by the

copropagating laser beams used for the single-qubit rotations as described in Sec. 6.7. All laser

beams are focused and directed at zone S (Fig. 8.2), with beam waists of ∼ 25 μm. With both

ions initialized to the |↑↑i state, the overall pulse sequence in Fig. 8.3 creates the entangled state

1 1 |Φ+i = √ (|↑↑i + |↓↓i). From this state, we can effectively create the |Φ−i = √ (|↑↑i − |↓↓i) state 2 2

by appropriately shifting the phase of the pulses that implement the measurement settings bl. This

is equivalent to a π/2-rotation around the z-axis of the Bloch sphere.

By measuring the population and the coherence of the created Bell state [Sackett 00], we

determine the fidelity of the overall sequence in Fig. 8.3 to be ' 0.99 (this evaluation of the Bellstate fidelity is similar to that used and described in Chap. 5 and Chap. 7). The Bell state fidelity

is lower than that achieved in Chap. 6 due to a higher error dominated by spontaneous emission of

photons induced by the Raman laser beams [Ozeri 07]. Here, different laser parameters, including

Raman detuning, laser intensities, and polarizations compared to those used in Chap. 6 are chosen

to allow different operations (e.g. two-qubit gate, global rotation, and single-qubit rotation) to

be implemented with the same laser beam lines. Furthermore, in Chap. 6 the entangling gate

was applied with ions confined in zone E (Fig. 8.2), but here entanglement is created in S. We
187

9Be+

(ion aa) Mølmer-Sørensen

(MS) Interaction

90 Carrier 

RR(

ππ

2 , − ππ2 )

90 Carrier 

RR(

ππ

2 , − ππ2 )

90 Carrier

RR(

ππ

2 ,

ππ

2 )

90 Carrier 

RR(

ππ

2 ,

ππ

2 )

|↑⟩

|↑⟩

9Be+

(ion bb)

Co Carrier 

RR(

ππ

2 , 0)

Co Carrier 

RR(

ππ

2 , 0)

Co Carrier

RR(

ππ

2 , 0)

Co Carrier 

RR(

ππ

2 , 0) RRzz(

ππ

2 )

Figure 8.3: Pulse sequence for generating the entangled state |Φ 1

+i = √ (|↑↑i + |↓↓i). The notation 2

R(θ, φ) represents the rotation with angle θ about an axis in the x-y plane of the Bloch sphere,

and φ is the azimuthal angle of the rotation axis. Operation Rz(ξ) is the rotation with an angle

ξ around the z axis of the Bloch sphere. The angle θ is adjusted by varying the length of the

laser pulse, and φ and ξ are adjusted with the phases of the RF signal driving the AOMs that

controls the laser beams. Laser pulses in the dashed box all use the same set of laser beams, which

makes the sensitivity of the created state to slow phase drifts between the two Raman beam paths

negligible [Lee 05, Tan 15]. The two laser beams used to drive the stimulated-Raman transitions for

the pulses outside of the dashed box (labeled as “Co Carrier”) are copropagating (the laser beams

labeled as k2a and k2b in Fig. 6.1) which eliminates phase drifts due to path length differences

in the beams [Gaebler 16] (see also Sec. 6.7). The Rz rotation is implemented by shifting the

phase of the direct digital synthesizer controlling the laser pulses that implement the subsequent

measurement settings bl.
188

measure a factor of ∼ 2 higher axial motional heating [Turchette 00a] rate (∼ 80 quanta/s for the

center-of-mass mode) at S than in E. All these factors reduce the fidelity of the entangling gate

that creates the Bell states (see Sec. 4.4).

  After the states Φ+/− are created at zone S (Fig. 8.2), the ions are separated and placed

in two potential wells located in zone E and E0

, separated by ∼ 340 μm. These processes represent

the source in Fig. 8.1 and prepare the two ions a and b for the measurement of IN described below.

To apply the settings ak and bl to the ions individually, the ion in E is first transported to

EL while the ion in E0 is simultaneously transported to S (see Fig. 8.4). We then illuminate the

ion with Raman-transition-inducing laser beams to zone S to implement the measurement setting

ak without disturbing ion b. Subsequently, time-varying voltages are applied to implement the

simultaneous well transportation operations S → ER and EL → S. With ion b located in zone

S, the laser beams implement the measurement setting bl. The Raman laser beams used for the

applications of measurement settings are the copropagating beams labeled as “Co Carrier ” in Fig.

8.3. Ideally, these operation can be described as the following rotations:

1 �  |↑i → √ |↑i − e−irk |↓i , r r r 2

1 �  |↓i → √ |↓i − eirk |↑i , (8.9) r r r 2

where r = a or b to represent each of the ions, and the angles rk = ak or bk are

(2k − 1)π

ak = , (8.10) 2N

(l − 1)π bl = − , (8.11) N

which are chosen from Eq. 8.4. These angles minimize the expected value of IN if the produced

entangled state is ideal [Braunstein 90]. These rotation operations are implemented by setting the

amplitude and phase of the Raman laser beams with an acousto-optic modulator (AOM) (see Sec.

3.4.3). The radio-frequency electric field driving the AOM is produced by a field-programmable

gate array (FPGA)-controlled direct digital synthesizer (see Fig. 3.11). The classical variable is

the phase of the oscillating field that implements a particular setting ak or bl. The experiment
189

sequence is depicted in Fig. 8.4.

After applying the measurement settings, the ions are recombined into zone S for “shelving”

pulses to be applied on both ions simultaneously. Then, similar transport procedures separate

and move the ions into zone S sequentially to be individually measured with the standard statedependent fluorescence technique. When the detection laser is turned on, we detect on average 30

photon counts per ion for ion in the |↑i state on a photomultiplier tube and about 2 counts when

the ion is in the |↓i state. See Sec. 2.1.6 and Sec. 2.1.7 for details on the shelving and fluorescence

detection of the 9Be+ ions. The photon collection apparatus is positioned to image zone S with a

field of view of approximately 50 μm in this experiment (see Sec. 3.3). We label a measurement

outcome “dark” (D) if six or fewer photons are observed and “bright” (B) if more than 6 are

observed. Figure 8.5 shows the typical detection photon histograms of the two 9Be+ ions. With

this, we obtain the four possible joint-measurement fluorescence outcomes BB, BD, DB, DD, for

each experiment. These outcomes correspond to the states |↑↑i, |↑↓i, |↓↑i, and |↓↓i.

The durations taken for each separation are ∼ 300 μs, and simultaneous transport takes ∼

250 μs. With the detection duration of ∼ 330 μs and the durations taken by other operations, the

total duration shown in Fig. 8.4 takes ∼ 3 ms. Due to this extended duration, it becomes important

for the ion’s state memory storage time to be long lived. Using states associated with magneticfield sensitive transitions to encode the ions’ would not be ideal in this experiment, because the

memory storage times of a magnetic-field sensitive transitions in a 9Be+ ion are typically a few ms,

which is comparable to (or shorter than) the duration for the pulse sequence shown in Fig. 8.4.

Magnetic-field sensitivities of hyperfine transitions in the electronic ground state of a 9Be+ ion are

shown in Fig. A.1.

When the state |Φ+i is prepared, we compute an estimate I

b

N of IN as shown in Eq. 8.5 with

Eq. 8.8 used to estimate the C(ak, bl) terms. For the state |Φ−i we instead use anticorrelations and

compute

�  I

bA

N = A(a1, b1) + A(a1, b2) + A(a2, b2) + A(aN , bN ) + 1 − A(aN , b1) , (8.12)
190

9Be+ |↑⟩

9Be+

Source

Bell state

Generation |↑⟩

Separate

ELS aall

Recombine

Shelving

SER

D

Laser 

pulse

bbkk

Separate

D

SER

Bell test

Measurement

Laser 

pulse

EL E S E’ ER

EL E S E’ ER EL E S E’ ER EL E S E’ ER EL E S E’ ER

ELS

Measurement 

setting applications

Figure 8.4: Experimental sequence for one trial. The notation ELS refers to transport to place

the ion b in zone EL and ion a in zone S. Similarly for the operation SER. The entangled state is

generated as shown in Fig. 8.3 with the ions located in zone S. Time-varying potentials are applied

to control electrodes for the separation, shuttling and recombination of the ions [Blakestad 11,

Bowler 12]. The total duration of the entire sequence shown here is approximately 3 ms. The

initial optical pumping, Doppler cooling and ground-state cooling which are applied prior to the

Bell state generation are not shown here. The total duration of one trial is approximately 8 ms.
191

Figure 8.5: Typical detection photon histograms that were obtained when we apply the statedependent fluorescence detection on each ion sequentially. During the state measurement of ion

a (b) at zone S, ion b (a) is located in zone EL (ER) such that it does not interact with the

detection laser beam. We choose the threshold of 6 for the differentiation of the “bright” and

“dark” measurement outcomes.
192

where A(ak, bl) = 1−C(ak, bl). The measured I

bA

N is equivalent to I

b

N for the purpose of quantifying

plocal

8.3 Experiment Results

We performed the experiment for the CBI parameter N ranging from 2 to 15. Figure 8.6

shows the experimentally obtained CBI parameter I

b

N as a function of N. The data points in Fig.

8.6 were obtained with multiple sequential trials having the same settings, then iterated across

different choices of settings. The error bars are calculated under the assumption that the settings

qP and outcomes are i.i.d. The error bars indicate the propagated standard errors 2

j , with j

p j = χj (1 − χj )/(Mj − 1) where Mj is the number of trials (here Mj ∼ 2, 000) and χj is the

averaged correlated or anticorrelated outcome for the jth setting pair.

Two different data sets are collected using two 9Be+ ions. The first data set was obtained with

the Bell state |Φ−i undergoing trials and by computing the CBI parameters according to Eq. 8.12.

The second data set was carried out with |Φ+i and the CBI parameters were calculated according

to Eq. 8.5. These two data sets were collected ∼ 6 months apart. Due to miscalibrations and our

inability to completely reproduce the experimental conditions, there exists noticable differences

between the two data sets and finer features within each data set. Furthermore, the very high

demands of fidelity and the high sensitivities to errors of the elementary operations required for the

realization of these experiments can also contribute to these observed behaviors. The I

b2 experiment

took a total of ∼ 5 minutes, the one for I

b15 ∼ 20 minutes. The lowest value of I

b

N is obtained for

the N = 9 data run, where I

b9 = 0.296(12).

Using the same apparatus, we also performed the CBI experiment on an entangled pair of

9Be+ and 25Mg+ ions. The computed I

b

N values are shown as orange dots in Fig. 8.6. The

generation of the mixed-species entangled state is described in Chap. 5 and Ref. [Tan 15]. In

this experiment, the ions remain confined in a single zone throughout the entire sequence. The

rotations implementing the settings choices are applied with microwave fields tuned to each ion. The

frequencies are ' 1.2 GHz for the 9Be+ ion and ' 1.8 GHz for the 25Mg+ ion. The determinations
193

2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.3

0.4

0.5

0.6

0.7

N

⋀

I

N or

⋀

I

N



CHSH test

Limit of CHSH test

Be+-Mg+ pair

Two Be+ (|Φ-〉 state)

Two Be+ (|Φ+〉 state)

Figure 8.6: Experimentally measured values I

b

N and I

bA

N as a function of N. Data represented by

black and blue dots are obtained with two 9Be+ ions, with black (blue) dots corresponding to tests

on |Φ+i (|Φ−i). These two data sets were obtained approximately six months apart. The difference

between them and the finer features within each data set are probably due to miscalibrations and

our inability to reproduce exact experimental conditions. Orange dots are data from test on |Φ+i

prepared on a 9Be+-

25Mg+ pair. The dashed line indicates the lowest upper bound on the local

content attainable in a perfect CHSH-type experiment.
194

of the final states of the two ions are made with detection lasers at ∼ 313 nm for the 9Be+ and at

∼ 280 nm for the 25Mg+ ion. Due to higher background photon counts when the CBI experiments

were performed on the 9Be+-

25Mg+ pair, bright thresholds of 11 and 12 were used for 9Be+ and

25Mg+, respectively. See Chap. 2 for detailed energy level structures, and descriptions of detection,

and manipulation of the 9Be+ and 25Mg+ ions.

8.4 Self-Testing of Bell States

It was shown in Ref. [Popescu 92] that if the CHSH inequality (Eq. 8.1) is violated at

√ the maximum value of 2 2 by a bipartite system, the system must be in a maximally entangled

state. This is a remarkable result because it enables the validation of the quantum entanglement

using only the observed correlations. In other words, violation of Bell inequality can serve as an

entanglement witness. Such a characterizations with minimal assumptions on the quantum devices

is formalized under the self-testing framework [Mayers 04, McKague 12]. With this framework, an

experimentalist is able to quantify the quality of the entangled state given an amount of violation

of a Bell inequality [Bardyn 09, Yang 14, Bancal 15, Kaniewski 16]. Furthermore, because the

quantum device is treated as a black box, such characterization can offer a device-independent

assessment of the physical system under inspection. Thus, self-tested certification can be used as

a benchmark for QIP devices across different platforms.

Here, we use the method provided by Ref. [Kaniewski 16] to infer a self-tested Bell-state

fidelity. Our lowest measurement of I

b2 corresponds to a CHSH inequality parameter (sum of

correlations) of BCHSH = 2.80(2), which is calculated using the following equation:

BCHSH = 2 (1 − I2) + 2. (8.13)

We infer a self-tested Bell-state fidelity lower bound (at the 95 % confidence level) of FL ∼ 0.958.

This is calculated with [Kaniewski 16]

  1 BCHSH − βS FL = 1 + √ , (8.14) 2 2 2 − βS
195

√ where βS = (16+14 2)/17. There also exist other methods to infer a self-testing Bell-state fidelity,

see for example Ref. [Bardyn 09] and Ref. [Bancal 15].

Our self-tested fidelity appears to be the highest for a deterministically created Bell state.

Several previous experiments have reported violation of the CHSH inequality (the CBI with N = 2)

while closing the detection loophole [Rowe 01, Matsukevich 08, Pironio 10, Ballance 15, Tan 15,

Ansmann 09, Hofmann 12, Pfaff 13, Vlastakis 15, Dehollain 16], and [Hensen 16] closed all loopholes. Other experiments [Shalm 15, Giustina 15] have closed the detection loophole, but they

reported violation of other Bell inequalities, for which self-tested fidelity bounds are not available.

Table 8.1 summarizes the BCHSH parameters determined in several previous detection-loophole

closing experiments and ours, as well as the resulting self-testing singlet-fidelity lower-bound according to Eq. 8.14 at the 50 % and 95 % confidence levels. For the 50 % confidence fidelity lower

bound we use the point estimates of BCHSH given in Table 8.1, and for the 95 % confidence lower

bound we replace BCHSH in the equation above with a 95 % confidence lower bound on BCHSH

assuming that each estimate of BCHSH is normally distributed with standard deviation given by

the uncertainties in the table.

8.5 Summary

In summary, we used the quantum information processing (QIP) techniques based on trappedions described in this thesis to perform a chained Bell inequality (CBI) experiment. The lowest CBI

parameter determined from our measurements is 0.296(12) in a N = 9 CBI experiment. This value is

significantly lower than 0.586, the minimum fraction derived from a perfect Clauser-Horne-ShimonyHolt (CHSH)-type experiment (the CBI with N = 2). Furthermore, for the conventional case of

the CHSH inequality, the fidelity of the Bell state is certified to be 0.958 at the 95 % confidence

level under a self-testing protocol [Kaniewski 16]. This result is the highest for a deterministically

created Bell state, and verifies that the basic elements of trapped-ion QIP techniques employed here

work well in combination, demonstrating the versatility and promising capabilities of QIP based

on trapped-ions.
196

Reference System BCHSH F50%

L F95%L

[Rowe 01] Two 9Be+ 2.25(3) 0.600 0.566

[Matsukevich 08] Two 171Yb+ 2.54(2) 0.800 0.778

[Ansmann 09] Phase qubits 2.0732(3) 0.477 0.477

[Pironio 10] Two 171Yb+ 2.414(58) 0.713 0.647

[Hofmann 12] Two Rb0 2.19(9) 0.558 0.456

[Pfaff 13] One NV 2.30(5) 0.634 0.577

[Vlastakis 15] Trans. & cavity 2.30(4) 0.634 0.589

[Ballance 15] 40Ca+ & 43Ca+ 2.228(15) 0.584 0.567

[Tan 15] 9Be+ & 25Mg+ 2.70(2) 0.911 0.888

see also Sec. 5.3.2

[Dehollain 16] One 31P in Si 2.70(9) 0.911 0.809

[Hensen 16] Two NV 2.38(14) 0.690 0.530

This work Two 9Be+ 2.80(2) 0.980 0.958

Table 8.1: Results from CHSH experiments without the fair-sampling assumption. The table shows

each experiment’s measured CHSH parameter BCHSH with one standard deviation uncertainty from

the references and self-testing fidelity lower bounds at the 50 % (F50%) L and 95 % (F95%) confidence L

levels determined according to Eq. 8.14.
Chapter 9

Outlook

This dissertation work focuses on implementing two-qubit entangling gates in the context of

scaling trapped-ion quantum information processing (QIP) in the quantum charge coupled device

(QCCD) architecture [Wineland 98, Kielpinski 02]. That is, the entangling gates were implemented

inside ion traps which are designed for scaling and some of the experiments utilize all of the basic

features required for scaling. Furthermore, the use of multiple ion species is incorporated which

is also regarded as an important utility for the realization of a practical trapped-ion based QIP

device.

While improvements on gate performance have been made, and incorporation of most trappedion QIP elementary operations in one experiment which can be used as benchmarks for these operations has been demonstrated, it still remains a tremendous challenge for an operational large-scale

trapped-ion quantum information processor to be realized. Based on the experience and lessons

learned from these experiments, here we briefly summarize potential areas of improvement for

future experiments.

9.1 Better Trap design

9.1.1 Axial Micromotion

In our experiment a major source of error for two-qubit entangling gates is the spontaneous

scattering of photons induced by the laser beams used for exciting stimulated-Raman transitions

(Sec. 4.4.2). In the X-junction trap, due to axial micromotion caused by the particular trap design
198

and imperfections in the trap construction (see Sec. 3.1 and Ref. [Blakestad 10]), the Raman

scattering rate for a 9Be+ ion is increased by a factor of ∼ 4 compared the case where micromotion

is absent. Therefore future trap designs must be carefully simulated to suppress this effect and

fabrication techniques need to be further developed.

9.1.2 Segmented-Stray-Fields Compensating Electrodes

Segmented electrodes are useful for the compensation of stray electric fields, as depicted in

Fig. 9.1. The X-junction trap has only one bias electrode that provides a micromotion compensating

electric field along the x axis (see Fig. 9.1). While it is possible to null excess RF micromotion at any

given location along the z-axis with only the diagonal control electrodes and the bias electrode, it is

a non-trivial problem to simultaneously null excess RF micromotion at different locations along the

z direction with this configuration. Simultaneous minimization of excess micromotion at multiple

locations along the axis is important for storing ions at different locations simultaneously and for

shuttling operations. Such designs have been pursued in a linear Paul trap [Pyka 14, Kienzler 15]

and might be employed for future work in scalable traps.

9.2 Cooling of Trap Apparatus

A major limiting factor to increasing the number of trapped ions is due to collisions of ions

with background gas in the vacuum chamber. Such collisions can cause unacceptable heating and

ion reordering, and ion loss through exothermic chemical reactions [Wineland 98]. This can be

mitigated by employing cryogenic pumping [Wilson 14, Niedermayr 14, Bruzewicz 15, Alonso 16].

Most gases which are commonly found in the background of a high vacuum system condense

on material surfaces at temperatures ≤ 20 K; therefore operating at these temperatures should

substantially reduce the background gas collisions rates. Another advantage of cooling is that the

heating rate of motional modes [Turchette 00a] due to uncontrolled noisy electric fields could also

be suppressed [Labaziewicz 08, Bruzewicz 15, Brownnutt 15]. Low heating rates are important to

suppress errors in two-qubit entangling gates as discussed in Chap. 4.
199

Shim electrode

RF 

electrode

Control 

electrode

RF 

electrode

Control 

electrode

Shim electrode

Bias electrode

ion

�xx

�yy

(not to scale) zẑis pointing into the page

Figure 9.1: Schematic showing the cross section of a potential linear Paul trap with multiple shim

electrodes. Our X-junction trap (see Sec. 3.1) consists of two pairs of RF electrodes and two

pairs of control electrodes (inside the grey dashed box), as well as a single bias electrode. The

control electrodes are segmented along the z direction. In the proposed design, in addition to the

electrodes inside the dashed box, two segmented “shim” electrodes that run parallel to the RF and

control electrodes along the z axis will also be included. This design enables excess micromotion at

different locations along the z-axis to be better compensated simultaneously. In addition to being

important for the implementations of motional mode cooling and fluorescence detection, better

compensation of excess micromotion is particularly crucial for implementing two-qubit entangling

gates and shuttling of ions. Shuttling operations include transporting, separating, and recombining

ions.

 
200

Drawbacks of operating a trapped-ion apparatus at the cryogenic temperatures include (i)

restrictions on the heat load allowable, (ii) restrictions on laser beams access due to cryogenic

apparatus and heat shields, and (iii) for liquid Helium bath cryostats, the cost of Helium.

9.3 Rapid Loading of Ions

A fast loading mechanism for ions can be advantageous. This becomes increasingly important

as the required number of ions increases for more complicated experiments. Currently, loading uses

resistively-heated ovens in combination with photoionization laser beams. The ion loading rate is

typically one ion per minute limited by the need to control the number of ions loaded; furthermore,

the heat generated by the ovens can increase background pressure.

Two approaches for the rapid loading of ions have been investigated in several groups: (i) integration of a neutral atom magneto-optical trap (MOT) with an ion trap [Cetina 07, Sage 12], and

(ii) ablation loading of ions [Hashimoto 06, Leibrandt 07, Hendricks 07, Sheridan 11, Zimmermann 12,

Hankin 16, Chen 16, Todaro 16].

The ablation loading technique usually employs a pulsed laser beam that strikes a target

and leads to a rapid ejection of a mixture of atoms, molecules, ions, and electrons. If needed, a

photoionization laser beam can be added to create the desired ions. In addition to providing a fast

loading rate, this technique also minimizes the heat load compared to the conventional resistivelyheated oven. This makes the ablation technique particularly attractive for a cryogenic ion trap.

In addition, no wire connections and electrical feedthroughs are required for the ablation target.

Furthermore, for multiple species applications, different mixtures of the desired atom species can

be infused into one ablation target [Guggemos 15, Hankin 16].

Another approach for a rapid loading of ions is to incorporate an “ion reservoir” in a QCCD

trap; a reserve of ions is maintained in a designated trap zone and individual ions can be transported

into the experimental region “on demand”. This strategy can be combined with any loading

technique.
201

Operation Duration (typical values)

Single-qubit gate

Two-qubit gate

Qubit state preparation

Transport of one ion

Separation of two ions

Sideband cooling

Doppler cooling

“Shelving”

Measurement

10 μs

30 μs

50 μs

250 μs

300 μs

2000 μs

600 μs

200 μs

300 μs

Table 9.1: Typical durations required for elementary operations using the X-Junction trap. The

actual durations of these operations vary, as they depend on motional mode structure, transition

excitation methods, and ion location in the trap array. “Shelving” refers to the pulses that transfer

ions’ states from the computation qubit to the measurement qubit before making fluorescence

measurement (see Sec. 2.1.7 and Sec. 2.2.6).

9.4 Improved Microwave Delivery

One strategy to improve the delivery of microwave fields to the ions is to integrate a microwave

antenna with an ion trap [Ospelkaus 08, Harty 14, Weidt 16]. With an antenna placed in close

proximity with the trapped ions, Rabi rates and fidelities of magnetic-dipole transitions in ions

can be significantly increased for a given microwave power, or with reduced power, the temporal

instability of the microwaves would be improved due to a reduced thermal effect. Benefits include:

(i) performance of the mapping operation between the computational qubit and the measurement

qubit could be improved (see Sec. 2.1.6 and Sec. 2.2.5), (ii) in the implementation of the dynamical

decoupling gate described in Chap. 7, the error source associated with the infidelity of microwaveinduced transition could be suppressed (see Sec. 7.3), and (iii) with higher fidelity microwave

transitions, one could explore more complicated dynamical decoupling schemes [Cohen 15].

9.5 Switchable Noise Filtering for Control Electrodes

As shown in Table 9.1, durations required for shuttling operations are currently a factor of

∼ 10 greater than the durations taken by quantum gates. One reason for this is because a series of

low-pass filters are incorporated to reduce externally-injected electric potential noise on the control
202

electrodes (see Sec. 3.1), which is important to suppress errors of two-qubit gates. We want to be

able to keep this feature but also enable high speed ion shuttling [Bowler 12] which is prevented if

we include strong low-pass filtering.

A potential approach is to implement different filtering stages that can be chosen with high

speed switches. Arranged in a parallel configuration, one stage provides heavy filtering, the other

provides minimal or no filtering. The latter is chosen during shuttling, the heavy filter is then

switched in when a static potential well is applied for the application of quantum gates. Highspeed switching of trapping potential has been investigated in Ref. [Alonso 13].

9.6 Electronically-Induced-Transparency Cooling with an Atomic Tripod

System

Another possibility to speed up the overall processing is to employ more efficient scheme

for the cooling of motional modes. The typical duration of a Raman-sideband cooling sequence

is on the order of a few ms, and is currently the longest-duration operation (see Table 9.1) in an

experimental sequence.

This could be mitigated by employing an electronically-induced-transparency (EIT) cooling

scheme [Morigi 00, Morigi 03]. Compared to cooling with Raman-sideband transitions, EIT cooling

has been demonstrated to offer (i) speed up of the cooling processes, (ii) reduction in the required

laser intensities, and (iii) simplifications of experimental implementations. In addition, it can also

provide broad cooling bandwidth such that multiple motional modes are cooled simultaneously

[Roos 00, Lin 13b, Lechner 16].

Previous implementations of EIT cooling relied on close approximations to three-level “Λ”

atomic systems (one excited level which is connected to two lower energy levels), including one

demonstrated with 24Mg+ ions [Lin 13b] in the legacy system where the experiments described

in Chap. 7 were performed. However, there exists no simple Λ systems in the ions used in this

thesis. It was suggested in Ref. [Evers 04] that a tripod-EIT cooling scheme (one excited level and

three lower energy levels) could offer better cooling efficiency compared to a Λ system. Here, using
203

an additional ground state to engineer a double-EIT profile that can offer better suppression of

both the carrier and the blue-sideband excitations, improved cooling should be possible. Reference

[Lu 15] proposes a similar tripod-EIT cooling scheme that employs two laser beams paired with a

microwave-induced transition. These possibilities should be investigated for extensions of the work

presented here.
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Appendix A

Details of 9Be+

The relevant properties of the neutral 9Be atom and the 9Be+ ion are summarized in Table

A.1.

The 9Be+ ion has a nuclear spin of 3/2, and the interaction between the nuclear and electron spin gives rise to hyperfine structure. Also, Zeeman sublevels are split in energy when an

external magnetic field is applied. In the intermediate field regime where the hyperfine and Zeeman interaction strengths are comparable, the energy levels exhibit curvature as a function of the

applied external magnetic field due to the competition between these two interactions. At certain magnetic-field strengths, some transition frequencies become insensitive to small changes in

the applied magnetic field. Such transitions are sometimes referred to as “clock” transitions or

field-insensitive transitions.

The eigenenergies and eigenstates that take both the hyperfine and Zeeman interactions into

account can be determined by solving the Breit-Rabi formula (see Ref. [Langer 06, Jost 10] for

details in the case of 9Be+). Table A.2 summarizes the magnetic-field independent points and their

respective transitions available in a 9Be+ ion.

The detailed energy level diagram in the 2S1/2 electronic ground state of the 9Be+ ion at a

field B = 0.0119446 T (119.446 G) is shown in Fig. A.1.
222

9Be+ ion properties

Nuclear spin

Electron g-factor

Nuclear to electron g-factor ratio

3/2

-2.00226206(42)

0.0002134779853(2)

[Wineland 85]

[Wineland 85]

Hyperfine constant of the 2S1/2 state

Magnetic dipole (A)

Hyperfine constant of the 2P1/2 state

Magnetic dipole (A)

Hyperfine constant of the 2P3/2 state

Magnetic dipole (A)

Electric quadrupole (B)

-625.008837.048(10) MHz

-118.6(3.6) MHz

|A| ≤ 0.6 MHz

Not available

[Wineland 85]

[Bollinger 85]

[Poulsen 75]

P1/2 and P3/2 levels natural decay rate 2π × 19.4(5) MHz [Poulsen 75]

Fine structure splitting of

P1/2 and P3/2 levels at B = 0 197.150(64) MHz [Bollinger 85]

2S1/2 state hyperfine splitting at B = 0

2P1/2 state hyperfine splitting at B = 0

1,250.018 MHz

237.2 MHz

Calculated value

Calculated value

2S1/2 ↔ 2P1/2 transition frequency

(line center of hyperfine structure)

2S1/2 ↔ 2PP/2 transition frequency

(line center of hyperfine structure)

957,199.6525(1200) GHz

957,396.8020(1349) GHz

[Bollinger 85]

[Bollinger 85]

Neutral Be atom properties

2s2 ↔ 2s2p transition wavelength

2s2p to continuum excitation wavelength

234.9329 nm

306.4919 nm

[Kramida 97]

[Kramida 97]

Table A.1: Relevant properties of the 9Be atom and 9Be+ ion which are important for this thesis.

The references from which the properties are extracted are shown in the last column. See also Ref.

[Shiga 11] for the 9Be+ ion’s ground state hyperfine constant that takes the diamagnetic correction

into account. The hyperfine energy level splittings at a field B = 0 is calculated by solving the

Breit-Rabi equation at a vanishing field.
223

Transition B-field

(T)

Transition

frequency (MHz)

d2f

dB2

(MHz/mT2)

|F = 2, mF = 0i ↔ |1, 1i

|2, 1i ↔ |1, 0i

|2, 1i ↔ |1, 1i

|1, 1i ↔ |1, 0i

|2, 1i ↔ |2, 0i

0.0119446

0.0119643

0.0223073

0.160182

0.174719

1,207.496

1,207.353

1,082.547

322.552

324.548

0.6097

0.6097

0.7252

1.059 × 10−3

0.6247 × 10−3

Table A.2: Magnetic-field insensitive transitions and their respective magnetic-field strengths in

the 2S1/2 electronic ground state for the 9Be+ ion in the intermediate field regime. The transition

frequencies and the second-order magnetic-field sensitivities are also given.
224

86.547 MHz

76.248 MHz

102.799 MHz

86.690 MHz

76.390 MHz

69.073 MHz

Qubit transition1207.4958 MHz

⟩ | ↑

⟩ | ↓

1018.007 MHz

1120.806 MHz

1515.754 MHz

1.039 MHz/G

1.764 MHz/G

726 kHz/G

1207.353 MHz

1370.290 MHz

B = 119.446 G 9Be+

⟩ |FF = 22, mmFF − 22

⟩ |22, −11

⟩ |22, 00

⟩ |22, 11

⟩ |22, 22

⟩ |11, 00

⟩ |11, 11

⟩ |11, −11

2SS1/

2.331 MHz/G

2 1.291 MHz/G

566 kHz/G

2nd order qubit field sensitivity: 6.097kHz/G2 the of adjacent total frequencythe eentransitions t

w

an

et

where

b

relev

transition

some

strength

of

differences

the erehw

frequency

sensitivities

field magnetic

ers.bumn

T,

The

magnetic-field

wlo

tum

0.0119446

.

at

s

quan

fluctuation

from

do

e

go

of

olv

The

field

magnetic-field

ev

oth

b

old.

magnetic

are

b

they states

F

in

m

wn

a

axis

sho

the

to

at

are

y

b

ion

tization

+ Be 9

insensitive

the

transitions

designated

quan the

certain

are

of

first-order

along

state

is

of

states

jection

frequencies

ground

The

o pr

t.

transition

electronic

prin

its

i1 1

transition

ter

and

, |

ligh

F

↔

The

i

in

tum

0

=

the

wn

A.1:

F

m

sho

and

momen

,

Figure

2 =

also

els,

F|

levareangular
Appendix B

Details of 25Mg+

Table B.1 lists the relevant properties of the neutral magnesium atom and the 25Mg+ ion’s

properties. Table B.2 shows the magnetic-field insensitive points and their respective transitions for

the 25Mg+ ion’s 2S1/2 electronic ground state. For the work presented in this thesis, the magnetic

field is set at a insensitive point for the 9Be+ ion, i.e. ∼ 119.446 G. The detailed energy levels

diagram in the 2S1/2 electronic ground state of the 25Mg+ at B = 119.446 G is shown in Fig. B.1.
226

25Mg+ ion properties

Nuclear spin

Nuclear to electron g-factor ratio

5/2

9.299484(75) × 10−5 [Itano 81]

Hyperfine constant of the 2S1/2 state

Magnetic dipole (A)

Hyperfine constant of the 2P1/2 state

−596.254376(54) MHz [Itano 81]

Magnetic dipole (A) −101.70 MHz (theory) [Sur 05]

Hyperfine constant of the 2P3/2 state

−102.16(16) MHz (experiment) [Nguyen 09]

Magnetic dipole (A) -18.89 MHz [Sur 05]

Electric Quadrupole (B) 22.91 MHz [Sur 05]

P1/2 and P3/2 levels natural decay rate 2π × 41.3(3) MHz [Clos 14]

2S1/2 state hyperfine splitting at B = 0

2P1/2 state hyperfine splitting at B = 0

1,788.763 MHz

305.1 MHz

Calculated value

Calculated value

2S1/2 ↔ 2P1/2 transition frequency

(line center of hyperfine structure)

1,069,339.957(5) GHz [Clos 14]

2S1/2 ↔ 2PP/2 transition frequency

(line center of hyperfine structure)

1,072,084.547(5) GHz [Clos 14]

Isotope shift between 25Mg+ and 24Mg+

2S1/2 ↔ 2P1/2 1,620(19) MHz [Batteiger 09]

2S1/2 ↔ 2P3/2

Isotope shift between 26Mg+ and 24Mg+

1,621(19) MHz [Batteiger 09]

2S1/2 ↔ 2P1/2 3,084.905(93) MHz [Batteiger 09]

2S1/2 ↔ 2P3/2 3,087.560(87) MHz [Batteiger 09]

Neutral Mg atom properties

3s2 ↔ 3s3p transition wavelength

3s3p to continuum excitation wavelength

285.296 nm

375.647 nm

[Martin 80]

[Martin 80]

Table B.1: Relevant properties of the 25Mg+ ion and neutral magnesium. The references from

which the properties are extracted are shown in the last column. See also Ref. [Drullinger 80] and

Ref. [Ansbacher 89] for precision spectroscopy measurements in Mg+ ions. The hyperfine energy

level splittings at a field B = 0 is calculated by solving the Breit-Rabi equation at a vanishing field.
227

Transition B-field

(T)

Transition

frequency (MHz)

d2f

dB2

(MHz/mT2)

|F = 3, mF = 0i ↔ |2, 1i

|3, 1i ↔ |2, 0i

|3, 1i ↔ |2, 1i

|3, 1i ↔ |2, 2i

|3, 2i ↔ |2, 1i

|3, 2i ↔ |2, 2i

|2, 2i ↔ |2, 1i

|3, 2i ↔ |3, 1i

|2, 1i ↔ |2, 0i

|3, 1i ↔ |3, 0i

0.0109464

0.0109584

0.0212784

0.0331568

0.0331673

0.0425569

0.122563

0.123202

0.340915

0.438352

1,763.031

1,762.974

1,686.462

1,539.102

1,538.929

1,333.265

346.187

346.827

301.558

303.524

0.4330

0.4330

0.3902

0.4979

0.4979

0.5889

8.293 × 10−3

8.035 × 10−3

0.1066 × 10−3

0.02140 × 10−3

Table B.2: The magnetic-field insensitive transitions and their respective magnetic-field strengths in

the 2S1/2 electronic ground state for the 25Mg+ ion in the intermediate field regime. The transition

frequencies and the second-order magnetic-field sensitivities are also given.
228

48.11 MHz

50.51 MHz

65.69 MHz

60.65 MHz

56.62 MHz

53.30 MHz

50.45 MHz

53.24 MHz

60.59 MHz

56.56 MHz

Qubit transition1763.184 MHz

Magnetic-field independent point for 9Be+ ⟩ |22, 00 ↔ ⟩ |11, 11 transition

2075.24 MHz

42.71 kHz/G 2.16 MHz/G

1519.69 MHz

1.52 MHz/G

1585.38 MHz

1646.03 MHz

973 kHz/G

643 kHz/G

⟩ | ↑

⟩ | ↓

⟩ |FF = 33, mmFF = −33

⟩ |33, −11

⟩ |33, 00

⟩ |33, −22

⟩ |33, 11

⟩ |33, 33

⟩ |33, 22

⟩ |22, 00

⟩ |22, 11

⟩ |22, 22

⟩ |22, −11

⟩ |22, −22 25Mg+

2SS1/2

B = 119.446 G

1.73 MHz/G

1976.80 MHz

935 kHz/G

1873.11 MHz

2.47 MHz/G

i1, 1|↔

G.

i0=

464 .

F

m

109

, 2

'

=

B

F|

at

ion

+ Be 9 the

fluctuations

for toinp e

magnetic-field

magnetic-field-insensitiv

to einsensitiv ret-ord s

the

fir is

at ion

+ Mg

transition

25

i0

the

, 2|

of

↔

state

i1=

ground

F m,

3

=

electronic

F| + Mg 25

The

The

B.1: Figure

transition.
Appendix C

Atom-light Interaction Hamiltonian

In this appendix, we give the Hamiltonians describing an atom’s interaction with monochromatic light fields and the Hamiltonians governing the spin-motion transitions driven by stimulatedRaman processes. We will start with the simplest case of a two-level system.

C.1 Two-Level Atoms - Light Interaction

The Hamiltonian of a two-level system interacting with a monochromatic electromagnetic

(EM) field (Fig. C.1) can be written as

H = H0 + V (t), (C.1)

where H0 = ~ω1|1ih1|+~ω2|2ih2| is the system’s bare Hamiltonian, with |1i and |2i representing the

unperturbed states with energies ~ω1 and ~ω2, respectively. Their energy difference is ω0 = ω2−ω1.

detuning Δ

|1 ⟩

|2 ⟩

ωω0 ωω

Figure C.1: A two-level system with a detuned interaction.
230

V (t) describes the interaction between the atom dipole operator, d, with an externally applied

monochromatic EM field, i.e.

V (t) = −d · EσˆEcos(ωt − φ), (C.2)

where E, ω, and φ are the amplitude, frequency, and the phase the of applied EM field, respectively.

The electric field’s polarization vector σˆE is expressed in terms of linear (πˆ), left (σˆ−) and right

  + − − (σˆ ) circular polarizations such that σˆE =  σˆ + 0πˆ + + σˆ+ with − 2 | | + 0  

2 + |+ 2

| = 1.

Transforming into the interaction picture of the unperturbed system by writing

iH t 0 −iH t 0

H = e ~ I V (t)e ~ , (C.3)

and after making the rotating-wave approximation, we get

H −iΔt

I = ~Ω|2ih1|e + h.c., (C.4)

where Ω = −d·EσˆE and Δ = ω − (ω2 − ω1) are the resonant transition Rabi rate and the detuning 2~

of the electromagnetic field from the resonant transition frequency, respectively. For simplicity, we

have set φ = 0. In our case, Eq. C.4 is applicable to a transition induced by laser fields between

an electronic ground state and an excited state. It can also describe magnetic dipole transitions

driven by microwave fields.

C.2 Stimulated-Raman Transition

The simplest case of a stimulated-Raman transition can be seen as a three-level system

interacting with two monochromatic light fields (Fig. C.2). For simplicity, we assume that the

laser field with frequency ωa only couples the |1i and |3i states, and does not couple the |2i and

|3i states, similarly for the laser field (with frequency ωb) coupling the |2i and |3i states. Following

Eq. C.4, the interaction Hamiltonian in this case is

H = ~Ω |3ih1|e−iΔ1t + ~Ω |3ih2|e−iΔ2t

I 1 2 + h.c., (C.5)

where |3i is the state in which both the |1i and |2i states are coupled to with two light fields that

are coherent with each other. The two light fields are detuned by Δ1 and Δ2 from each of the
231

|3 ⟩

Δ1

|1 ⟩

|2 ⟩

Δ2

ωωaa

ωωbb

Figure C.2: A three-level system with two detuned interactions. For simplicity, we assume that the

light field labeled as ωa does not couple the |2i and |3i states, and the light field labeled as ωb does

not couple the |1i and |3i states.
232

transitions, respectively. To relate to our qubit system, we will write |1i = |↓i and |2i = |↑i. Thus,

H −iΔ1t −iΔ2t

I = ~Ω1|3ih↓|e + ~Ω2|3ih↑|e + h.c.. (C.6)

This equation can be transformed into an effective Hamiltonian written as

~Ω2

1 ~Ω2

H 2

ef f = − (|3ih3| − |↓ih↓|) − (|3ih3| − |↑ih↑|) + Δ1 Δ2

  Ω1Ω2 i(Δ2−Δ1)t −i(Δ2−Δ1)t ~ |↑ih↓|e + |↓ih↑|e , (C.7) 2Δ ̄

where Δ ̄ = Δ1Δ2 . There are multiple approaches to arrive at Eq. C.7, see for example, Ref. Δ1+Δ2

[Wineland 98] with a Schrodinger’s equation approach and Ref. [James 07] which give a relatively

compact formula for the derivation of the effective Hamiltonian describing the time-averaged dynamics of the system.

The first and second term of Eq. C.7 represent the AC Stark shifts induced by the laser fields,

and the third term represents the effective Hamiltonian coupling the qubit |↑i and |↓i states via

the stimulated-Raman transition. The latter term can be separated out from the AC Stark shifts

terms when we are only interested in the dynamics between the qubit states. A quick inspection

and comparison between this term and Eq. C.4 indicates that stimulated-Raman transition can be

viewed as an effective coherent Rabi oscillation on the |↑i ↔ |↓i transition described by

i(Δ2−Δ1)t σ H + Raman = ~ΩRamanˆ e + h.c., (C.8)

with an effective Rabi rate of

Ω1Ω2 ΩRaman = , (C.9) 2Δ ̄

where σˆ+ = |↑ih↓| and its conjugate is σˆ− = |↓ih↑|. Equation C.8 can be rewriten as

i(Δω−ω0)t σ H + Raman = ~ΩRamanˆ e + h.c., (C.10)

where Δω = ωb − ωa, is the frequency difference of the two EM fields depicted in Fig. C.2. A

spin-flip oscillation can be driven by setting Δω = ω0.

One important effect that has not yet been considered here is the spontaneous emission

of photon from the electronic excited state when driving the stimulated-Raman transition. The
233

spontaneous photon emission rate is approximately proportional to Δ

Γ

R ×Ωef f where Γ is the excited

state’s decay rate and ΔR = Δ1 or Δ2. In most cases, ΔR  Γ such that photon scattering process

is suppressed compared to the coherent interaction described by Eq. C.8. Details of spontaneous

emissions are described in Sec. 4.4.2.

C.2.1 Rabi Rate of Stimulated-Raman Transitions

The descriptions above provide a simple framework for stimulated-Raman transitions without

detailing the coupling strength. The Rabi rate depends on the laser beam electric fields parameter

as well as the electronic structure and properties of the addressed atom. If there are j possible

excited states that give coupling between the |↑i and |↓i states, the Rabi rate of stimulated-Raman

transition driven by two laser fields which are copropagating is [Wineland 98]

P P σp1 σ X

p2

E1E2 p1 h↑|dE · ˆE,1|ji p2 hj|dE · ˆE,2|↓i ΩRaman = , (C.11) 4~2 Δj j

where p1 and p2 index the polarization components for each of the laser beams, respectively. The

atom’s electric dipole operator is denoted by dE.

C.3 Spin-Motion Stimulated-Raman Transition

The descriptions above deal only with the interaction between the internal states (spin) of an

atom with the electromagnetic fields. Now, adding the motional degree of freedom, the unperturbed

system Hamiltonian for a single trapped atom can be written as

X

3

ω0 † H0 = ~ (|↑ih↑| − |↓ih↓|) + ~ων,iaˆiaˆi, (C.12) 2

i

where ω0 is the qubit frequency. For one ion, there are three motional modes, each with a harmonic

oscillator frequency of ων,i, and aˆi represents the harmonic oscillator annihilation operator for the

ith mode.

Consider the two-photon stimulated-Raman transition described in Sec. C.2. Taking into account the propagation direction of the two laser beams, the interaction between the dipole operator
234

and the applied EM waves is

X 

2

V (t) = − dE · EiσˆE,icos(ki · zˆ − ωit + φi), (C.13)

i

where ki, ωi and φi are the wave vector, the laser frequency, and the laser beam phase of the ith

beam at the location of the ion. The operator of the ion’s motion is given by

  zˆ = z0 aˆ + ˆa† , (C.14)

p where z0 = ~/(2mων) is the zero-point amplitude of the ion’s motional wavefunction, with m as

the mass of the ion. The effective Hamiltonian associated with this dipole-light interaction takes

the form of Eq. C.10:

i[Δk·zˆ−(Δω−ω0)t+(φ2−φ1)] σ H + I = ~Ωˆ e + h.c., (C.15)

where Ω is the Rabi rate which can be computed with Eq. C.11.

In this thesis, we use a pair of laser beams with their wave vector difference Δk = k2 − k1

aligned along the axial (z) direction. In this configuration, we only have to consider one motional

mode for the interaction. For a 9Be+ ion in a trap with ωz = 2π × 3.6 MHz, we have z0 ' 12.5 nm.

With Eq, C.14, Eq. C.15 can be rewritten as

σ+ i[η(aˆ+ˆa†

H )−(Δω−ω0)t+(φ2−φ1)] I = ~Ωˆ e + h.c., (C.16)

where η = |Δk|z0 is the Lamb-Dicke parameter. In the limit where η  1, we can expand the first

term in the exponent:

i(η(ˆa+ˆa †

)

2 e

†)) = 1 + iη(ˆa + ˆa†

) − η2(ˆa + ˆa + ... (C.17)

such that Eq. C.16 becomes

h i σ+ iω0t †

) H 2 i[−Δωt+(φ2−φ1)] I = ~Ωˆ e 1 + iη(ˆa + ˆa†) − η2(ˆa + ˆa + ... e + h.c.. (C.18)

Now, we will go into the interaction frame (Eq. C.3) of the harmonic oscillator describing the ion’s

motion, then

h i σ+ iω0t † iωzt

) − η2(ˆ † −iωzt) H 2

I = ~Ωˆ e 1 + iη(ˆae−iωzt + ˆa e ae−iωzt + ˆa e + ...

i[−Δωt+(φ2−φ1)] ×e + h.c.. (C.19)
235

In most cases, the laser frequency difference Δω = ω1 − ω2 is set near the qubit frequency ω0

or ω0 ± mωz, where m = 1, 2, ... such that most non-resonant terms in Eq. C.19 can be neglected.

For example, by setting Δω = ω0, and dropping high frequency terms, we have

HCarrier = ~Ωˆσ+ei(φ2−φ1) + h.c., (C.20)

which is analogous to Eq. C.10 and describes a carrier excitation that drives |↓, ni ↔ |↑, ni spin-flip

oscillations. Likewise, by setting Δω = ω0 − ωz, the Hamiltonian after dropping fast oscillating

terms is

H i(φ2−φ1) RSB = ~ηΩˆσ+aeˆ + h.c., (C.21)

which induces a spin-flip |↓, ni ↔ |↑, n − 1i while removing a quanta from the motional harmonic

oscillator. This excitation is a red-sideband transition. Analogously, a blue-sideband transition is

described by the Hamiltonian

H † i(φ2−φ1) BSB = ~ηΩˆσ+aˆ e + h.c., (C.22)

after setting Δω = ω0 + ωz and neglecting fast oscillating terms.

It is straightforward to generalize these Hamiltonians to the case where multiple ions are

interacting with the laser light fields. For example, for two ions trapped in a harmonic well, there

are two normal modes along the axial direction. In this case, we can write the position operator zˆj

of the jth ion in terms of normal mode coordinates

X zˆj = z0,kξk,j (ˆak + ˆa†

), (C.23) k

k=1,2

p where z0,k = ~/(2mωk) with ωk, ξk,j and aˆk are the normal mode frequency, mode amplitude (of

the jth ion) and the annihilation operator for the kth motional mode.

Following Eq. C.16, the interaction Hamiltonian describing two ions interacting with light

fields arranged such that their Δk aligned along the axial direction is [Wineland 98]

" # X X   σ+ † HI = ~ Ωj ˆj Exp i ηkξk,j aˆk + ˆa − i(Δω − ω0)t + i(φ2,j − φ1,j ) + h.c., (C.24) k

j=1,2 k

where Ωj is the resonant Rabi rate of the laser-induced carrier transition for the jth ion, it can be

computed with Eq. C.11.
Appendix D

Matrix Representations of State Vectors and Qubit Operations

D.1 Qubit State Vectors

For a two-level atom, the qubit |↑i and |↓i states are defined as the following matrices:

⎛ ⎞ ⎛ ⎞ ⎜

1

⎟ ⎜

0

⎟ |↑i = ⎝ ⎠ , |↓i = ⎝ ⎠ . (D.1)

0 1

The state vectors for the compound system of two two-level atoms are the results from tensor

products of individual atom’s state vectors, for example

⎛ ⎞ ⎜

1

⎛ ⎞ ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟

⎜

1

⎟ ⎜

1

⎟ ⎜

0

⎟ |↑↑i = ⎝ ⎠ ⊗ ⎝ ⎠ = ⎜ ⎟ ⎜ ⎟ . (D.2)

0 0 ⎜ ⎟ ⎜

0

⎟ ⎝ ⎠ 0

Likewise,

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜

0

⎟ ⎜

0

⎟ ⎜

0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜

1

⎟ ⎜

0

⎟ ⎜

0

⎟ |↑↓i = ⎜ ⎟ ⎜ ⎟ , |↓↑i = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ , |↓↓i = ⎜ ⎟ . (D.3)

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0

⎟ ⎜

1

⎟ ⎜

0

⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 1
237

D.2 Qubit Rotations

Rotation about an axis in the x-y plane of the Bloch sphere of a two-level atom is represented

by the following matrix:

⎛ ⎞ ⎜

cos(θ/2) −ie−iφsin (θ/2)

⎟ R(θ, φ) = ⎝ ⎠ , (D.4)

−ieiφsin (θ/2) cos (θ/2)

where θ is the rotation angle and φ is the azimuthal angle of the rotation axis.

The rotation around z axis of the Bloch sphere with an angle of ξ is defined as

⎛ ⎞ ⎜

e−iξ/2 0

⎟ Rz(ξ) = ⎝ ⎠ . (D.5)

0 eiξ/2

Similar to the construction of a compound system from smaller systems, the operator representing a global rotation on a two-qubit compound system can be constructed using the tensor

product:

Rtwo(θ, φ) = R(θ, φ) ⊗ R(θ, φ). (D.6)

Likewise, the (two-qubit) operator representing a rotation applied on the first qubit without influencing the second qubit is

⎛ ⎞ ⎜

1 0

⎟ R(θ, φ) ⊗ ⎝ ⎠ . (D.7)

0 1

D.3 Fock State Vectors

Each motional degree of freedom is described by a simple harmonic oscillator. The Fock basis

state |ni can be represented by the following matrices:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜

1

⎟ ⎜

0

⎟ ⎜

0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜

0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ |0i = ⎜ ⎟ , |1i = ⎜ ⎟ , |2i = ⎜ ⎟ ⎜

0

⎟ ⎜

0

⎟ ⎜

1

⎟ , etc. (D.8)

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎜ . ⎟ ⎜ ⎟ ⎟ ⎜ . ⎟ ⎜ . . . ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0
238

D.4 Creation and Annihilation Operators

The simple harmonic oscillator annihilation operator, aˆ, and creation operator, aˆ†, can be

represented by the following matrices

⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 

⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 

,

0 1 0 0 · · · 0 · · ·

√ 0 0 2 0 · · · 0 · · ·

√ 0 0 0 3 · · · 0 · · ·

. . 0 0 0 0 . . . . · · ·

. . .

. . . .. .. √ . . . . . n · · ·

0 0 0 0 · · · 0 · · ·

. . . . . . . . . . . . . . . . . . . . .

aˆ = (D.9)

⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 

⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 

0 0 0 0 · · · 0 · · ·

1 0 0 0 · · · 0 · · ·

√ 0 2 0 0 · · · 0 · · ·

√ 0 0 3 0 · · · 0 · · ·

. . . . . . . . . . . . . . . . . . · · ·

√ 0 0 0 0 n 0 · · ·

. . . . . . . . . . . . . . . . . . . . .

† aˆ = . (D.10)

when applying aˆ or aˆ† to a Fock state |ni, they give

√ aˆ|ni = n|n − 1i, (D.11)

√ ˆ

a†|ni = n + 1|n + 1i. (D.12)

D.5 Atom and Motion Compound System

Similar to that in Eq. D.2, the state vector of the compound system consisting of a two-level

atom and one motional mode can be constructed with the tensor product. For example, an atom
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in the |↑i state prepared in the ground state of motion is

⎛ ⎞ ⎜

1

⎟ ⎜ ⎟

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜

0

⎟

⎜

1

⎟ ⎜ ⎟ |↑i|n = 0i = ⎝ ⎠ ⊗ ⎜ ⎟ ⎜

0

⎟ . (D.13)

0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜

0

⎟ ⎝ ⎠ .

.

.

Similarly, operators acting on the compound system can be constructed with tensor products. It

is straightforward to generalize state vectors and operators to a systems of higher dimension.

D.6 Summary of Qubit Operators

Here we write down some of the common qubit operators which are frequently used in this

thesis.

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜

0 1

⎟ ⎜

0 −i

⎟ ⎜

1 0

⎟ σˆx = ⎝ ⎠ , σˆy = ⎝ ⎠ , σˆz = ⎝ ⎠ . (D.14)

1 0 i 0 0 1

⎛ ⎞ + σˆx + iσˆ 0

y ⎜

1

⎟ σˆ = |↑ih↓| = = ⎝ ⎠ , (D.15) 2 0 0

⎛ ⎞ − σˆx − iσˆy ⎜

0 0

⎟ σˆ = |↓ih↑| = = ⎝ ⎠ . (D.16) 2 1 0

D.7 Three-Level Atom

For some cases, it is desirable to include an auxiliary atomic state to study the dynamics of

the system such as (i) the implementation of the Cirac-Zoller protocol [Cirac 95] (see also Sec. 4.2

and Sec. 5.6), (ii) the effect of out-of-manifold Raman spontaneous scattering processes (see Sec.

4.4.2.1), and (iii) imperfect qubit state preparation and measurement (see for example, Sec. 6.6.2).

We label the auxiliary state as |Ai.
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In this case, the state vectors are modified such that

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜

1

⎟ ⎜

0

⎟ ⎜

0

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ |↑i = ⎜ 0 ⎟ , |↓i = ⎜ 1 ⎟ , |Ai = ⎜ ⎟ ⎜

0 ⎟ . (D.17)

⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 1

Here, we use the superscript 0 to differentiate these state vectors from those of Eq. D.1.

D.8 Qubit Operators in a Three-Level Atom

The R(θφ) rotation (Eq. D.4) between the qubit |↑i and |↓i states becomes

⎛ ⎞ ⎜

cos(θ/2) −iφe−iφsin (θ/2) 0

⎟ ⎜ ⎟

R0 ⎜ ⎟ (θ, φ) = ⎜ cos

⎜ −iφeiφsin (θ/2) (θ/2) 0 ⎟ . (D.18)

⎟ ⎝ ⎠ 0 0 1

Similarly for the Rz rotation (Eq. D.5) between the |↑i and |↓i states:

⎛ ⎞ ⎜

e−iξ/2 0 0

⎟ ⎜ ⎟ ⎜ ⎟ R0

z(ξ) = ⎜ 0 eiξ/2 0 ⎟ . (D.19)

⎜ ⎟ ⎝ ⎠ 0 0 1
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Other commonly used operators are:

⎛ ⎞ ⎜

0 1 0

⎟ ⎜ ⎟

σˆ+0 = 0 0 |↑i h↓| = ⎜⎜

0 0 0 ⎟ , (D.20)

⎟ ⎝ ⎠ 0 0 0

⎛ ⎞ ⎜

0 0 0

⎟ ⎜ ⎟ −0 0 0 σˆ = |↓i h↑| = ⎜ 0 0 ⎟ ⎜

1 , (D.21)

⎟ ⎝ ⎠ 0 0 0

⎛ ⎞ ⎜

0 0 1

⎟ ⎜ ⎟ + 0 σˆ = |↑i hA| ↑A = ⎜ 0 0 ⎟ ⎜

0 , (D.22)

⎟ ⎝ ⎠ 0 0 0

⎛ ⎞ ⎜

0 0 0

⎟ ⎜ ⎟

σˆ− 0 = |↓i h | ↓A A = ⎜ 0 0

⎜

1 ⎟ , (D.23)

⎟ ⎝ ⎠ 0 0 0

⎛ ⎞ ⎜

0 0 0

⎟ ⎜ ⎟ 0 σˆ+ = |Aih↑| A↑ = ⎜ 0 (D.24)

⎜

0 0 ⎟ , ⎟ ⎝ ⎠ 1 0 0

⎛ ⎞ ⎜

0 0 0

⎟ ⎜ ⎟ 0 σˆ− = |Aih↑| A↓ = ⎜ 0 0 0 ⎟ . (D.25)

⎜ ⎟ ⎝ ⎠ 0 1 0

With these, it is straightforward to generalize and construct other operators, as needed.