Laser-free trapped-ion quantum logic with a radiofrequency

magnetic field gradient

by

R. Srinivas

B.Sc., National University of Singapore, 2013

M.S., University of Colorado, Boulder, 2016

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2020
This thesis entitled:

Laser-free trapped-ion quantum logic with a radiofrequency magnetic field gradient

written by R. Srinivas

has been approved for the Department of Physics

Dr. Andrew C. Wilson

Dr. David J. Wineland

Date

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

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

mentioned discipline.
iii

Srinivas, R. (Ph.D., Physics)

Laser-free trapped-ion quantum logic with a radiofrequency magnetic field gradient

Thesis directed by Dr. Andrew C. Wilson and Dr. David J. Wineland

Coupling the internal (spin) states of trapped ions to their shared motion is essential for

applications in metrology, quantum simulation, and quantum information processing. Spin-motion

coupling requires a state-dependent force and is typically performed with laser-based interactions.

However, laser-based interactions can be limited by photon scattering, which is the leading error

in the highest fidelity two-qubit gates demonstrated thus far. Laser-free methods, which are not

limited by photon scattering, have been proposed and demonstrated using either static magnetic

field gradients, or magnetic field gradients close to the qubit frequency at gigahertz frequencies.

We develop a new laser-free method of spin-motion coupling that instead uses a radiofrequency

magnetic field gradient. We extend this technique to novel two-qubit entangling gates that are

intrinsically insensitive to qubit frequency errors, achieving a symmetric Bell-state fidelity of 0.999(1),

competitive with the highest-fidelity laser-based gates. The insensitivity to qubit frequency errors

enables laser-free individual addressing which we employ to create anti-symmetric Bell states with

fidelity 0.998(1). These techniques are demonstrated in a surface-electrode trap with integrated

microwave and rf circuitry.
Dedication

To my sister, a constant inspiration.

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v

Acknowledgements

It is somewhat strange to have only one author on this thesis when so many have contributed

to the work.

First, I would like to thank my advisor Dave Wineland. Aside from being an inspiring

physicist, he is one of the nicest people I have ever interacted with. I have always appreciated his

open door to all the questions I have had over the years, profound or (mostly) not. His presence

was greatly missed when he left.

I’ve had the privilege of having not one, but two advisors. Andrew Wilson has been a great

source of advice throughout my PhD, for problems physics related and not. He has an incredibly

calm approach to dealing with the trickiest problems, and always remained affable regardless of the

situation. These are qualities I seek to emulate. He also owes me a cake which I mention here for

posterity.

Dietrich “Didi” Leibfried has been the victim of my many physics questions over the years.

He has a knack of distilling the most important physics from any problem and has inspired me to

be more clear and precise in all that I do. I hope to convince him one day that Beach House is a

good band.

Daniel Slichter and David Allcock have also been great mentors, and friends. They were

postdocs at the start of my PhD and did much of the heavy lifting initially in setting up and

planning the experiment. Everything I know (which isn’t much) about rf and microwaves is from
vi

them. David Allcock is a master ground-loop sniffer and great problem solver in the lab. I’ve learnt

a lot of debugging skills from him, which is basically to put everything on the scope. Daniel Slichter

has a great eye for detail and is a master of electronics. Much of the hardware on our experiment

only works due to his careful designs. He has also been a great source of advice for both physics

and non-physics related issues and I will miss being able to rely on him. Both David and Daniel

have plied me with alcohol over the years which is hard to separate from Stockholm Syndrome.

Along with Shaun Burd, Daniel, David and I made up the original “μ-crew” on the “magtrap”.

Shaun is very much the laser guru on our experiment, and has been a great friend and companion,

often there in the late hours when data was taken. He is a great person to bounce ideas off, and it’s

been great having him down in the trenches so to speak.

Hannah Knaack has been a breath of fresh air since she joined the experiment. Her enthusiasm

has been a welcome change from the cynicism that Shaun and I had accumulated by then. Her

sense of humour will be sorely missed. I wish we had had a longer overlap so she could pronounce

words properly; we have worked on iodine but not aluminium.

Tyler Sutherland has been a great person for us to collaborate with; he’s been the source of

so many good ideas that we could implement in our setup. Hard to believe he is a professor now,

y’all best pay attention to what he gets up to next!

Our experiment was embedded in the larger “Quantum 1” lab. I have fond memories of

spending time, both in and out of the lab, with fellow graduate students Susanna Todaro and Katie

McCormick, and postdocs Jonas Keller and Robert J ̈ordens. Justin Niedermeyer, Felix Knollmann,

and Laurent “Loz” Stephenson joined the lab more recently, it’s been great working with them for

the brief amount of time that it’s been.

It’s been my privilege to be part of the Ion Storage Group, where I’ve had the opportunity to

interact not only with great physicists, but people as well. John Bollinger has done a great job of

taking over as group leader and has always provided a listening ear. Ting Rei was a great person to

bug with random questions and helped me get settled in the group. In no particular order, I’ve had

enlightening discussions with Lin Yiheng, John Gaebler, Ryan Bowler, Yong Wan, Daniel Kienzler,
vii

Dan Cole, Stephen Erickson, Hou Panyu, Jenny Wu, Aaron Hankin, Ethan Clements, Kevin Boyce,

Cui Kaifeng, Dave Hume, Sam Brewer, Jwo-Sy Chen, May Kim, Jose Valencia, Dave Leibrandt, Jim

Bergquist, James Chin-wen Chou, Alejandra Collopy, Julian Schmidt, Christoph Kurz, Shon Cook,

Kevin Gilmore, Joe Britton, Elena Jordan, Matt Affolter, Justin Bohnet, Bryan Sawyer, Shlomi

Kotler, Dustin Hite, Dave Pappas and Kyle McKay. Apologies to anyone I might have missed out

on inadvertently. Amongst these group, Aaron Hankin, Elena Jordan, and Alejandra Collopy had a

desk next to mine but subsequently abandoned me for greener pastures with more natural light.

Being at NIST, we benefited greatly from our theory collaborators Scott Glancy, Emanuel

Knill, and Alex Kwiatkowski who have helped us navigate the murky waters of fidelity analysis.

Theodore “Ted” Stauffer provided the lifeblood of our experiment, liquid helium. He kept

up with our increasing helium consumption and odd schedules, without which the experiments

described in this thesis would not be possible.

I’ve been somewhat sheltered from administrative tasks throughout my studies. At NIST,

I am grateful to Anne Reidy, Kelsey Light, Kelsey Rodriguez, Petrina Potts, Kerry Yannacito,

Camille Mandarino, Shannon Kelly amongst others for all the work behind the scenes. At PREP,

Tiffany Mason, JoAnne Stein, and Lisa Valencia have helped sort out all of my travel and helped to

keep food on the table. Jeanne Nijhowne at CU has saved me multiple times from missing deadlines

and making sure everything was in order.

Outside of work, I’ve been lucky to have a fantastic group of friends to keep me sane. In

particular, I’d like to thank Kevin Gilmore and Gargi Datta, who aside from being a constant

source of support, put me up for the last few months of my PhD and very generously allowed me

to take their cat Koshka. In Boulder, I’ve had the pleasure of knowing (in no particular order)

Caroline Whiteman, Nicole Simmons, Dan Hackett, Sven and Kat D ̈onges, Judith Olson, Trevor

Doner, Abhinav Prem, Robert “Bob” Hueftle and Anita Wagner, Marion Boulet, Pete Siegfried,

Ng Kia Boon, Zach Ulibarri, Tim Large, Nigel Arruda and Katy Wilkin, Bryan Changala, Joseph

Isaac Samaniego, and Stijn DeSmet. Special thanks also to Alejandra Collopy and Andrew Spott

for taking me in and letting me play with their wonderful pets Daphne, Apollo, and Artemis.
viii

I’ve had the benefit of many great teachers and mentors throughout my life, Bjørn Hessmo

and David DeMille in particular are to blame for setting me down the path of experimental atomic

physics.

Leaving Boulder is a little easier with another project to look forward to. For that, I thank

David Lucas and Chris Ballance for providing the next step. I’ve also had some great discussions

with Tom Harty at Oxford over the years.

My parents and sister have been a never-ending source of support throughout all my life. My

nephew and nieces have been a great source of entertainment and help keep things in perspective.

In particular, my niece Mallika was born when I started, and is now a fully functional human being

who can even play the piano. In the mean time we’ve messed around with some ions...

I would like to thank Hannah Knaack, Laurent Stephenson, Felix Knollman, Dave Wineland,

Andrew Wilson, David Allcock, and Daniel Slichter for a close reading of the thesis. Future readers

of the thesis should also be indebted to them, this document would have been much worse otherwise!
Contents

Chapter

1 Introduction 1

1.1 Quantum entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Quantum computing with trapped ions . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Trapped ion quantum logic 5

2.1 Linear rf Paul Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Trapped ion hyperfine qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Trapped ion spin-motion coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Sideband transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Position-dependent energy shifts . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Laser-based spin-motion coupling . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 Laser-free spin-motion coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.5 Near-qubit-frequency oscillating magnetic-field gradient . . . . . . . . . . . . 14

2.4 Entangling gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Geometric phase gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

ix
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2.4.2 Mølmer-Sørensen gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.3 ˆσzσˆz gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Phase gate errors and suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Motional errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 Reducing motional errors: multi-loop sequences . . . . . . . . . . . . . . . . . 22

2.5.3 Qubit errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.4 Gate duration errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Non-classical states of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Apparatus 28

3.1 Surface electrode trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Cryogenic operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 High-field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Compensation coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5.1 Beam orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5.2 BD, BDD, and RD beamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5.3 VECSEL system for Raman beams . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.4 Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Trapping rf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7 Ion loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.8 Clock distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.9 Microwave generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.9.1 DDS Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.9.2 Microwave carrier drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.9.3 Microwave sideband drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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3.9.4 Combining microwave drives . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.10 RF generation and delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.11 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.12 Experimental control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Qubit control 50

4.1 25Mg+ qubit at intermediate magnetic field . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Field-sensitive “stretch” qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 State preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 State readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.3 Rabi spectroscopy and Rabi flopping . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Field-insensitive “clock” qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 Checking clock state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.2 State preparation and readout . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.3 Rabi flopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.4 Qubit coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.1 Micromotion compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.2 x shim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.3 Detection beam positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4.4 Single qubit rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Laser-based spin-motion coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5.1 Ground state cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5.2 Heating rate measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.3 Photon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6 Microwave spin-motion coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6.1 Nulling the microwave magnetic field . . . . . . . . . . . . . . . . . . . . . . . 73
xii

4.6.2 Microwave sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.7 Motional frequency tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.7.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7.3 Seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.7.4 Example operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.7.5 Laser-induced charging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7.6 Future improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.8 Qubit frequency tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.9 Ion transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Spin-motion coupling with a radiofrequency magnetic field gradient 88

5.1 Spin-motion coupling with an oscillating magnetic-field gradient . . . . . . . . . . . 88

5.2 Residual oscillating magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3 Experimental implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 Spectroscopy with an oscillating magnetic field gradient . . . . . . . . . . . . . . . . 96

5.5 Measuring and nulling the oscillating magnetic field . . . . . . . . . . . . . . . . . . 98

5.6 Sideband Rabi frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.6.1 ac Zeeman shift from microwave fields . . . . . . . . . . . . . . . . . . . . . . 100

5.6.2 Characterizing the sideband Rabi frequency . . . . . . . . . . . . . . . . . . . 101

5.7 Sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.8 Pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.9 Comparison to static gradient scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.10 Comparison to near-qubit-frequency oscillating gradient scheme . . . . . . . . . . . . 106

5.11 Oscillating electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.11.1 Measuring electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.11.2 Effect of electric field on ac Zeeman shift . . . . . . . . . . . . . . . . . . . . 108
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5.11.3 Shift on motional frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.11.4 Reducing the effect of the electric field . . . . . . . . . . . . . . . . . . . . . . 109

5.12 Higher-order interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.12.1 Oscillating gradient interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.12.2 Sideband interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.12.3 Single sideband interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Entangling gates with a radiofrequency oscillating magnetic field gradient 114

6.1 Bichromatic interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.1.1 Intrinsic dynamical decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.1.2 Pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.1.3 Physical implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.1.4 ˆσzσˆz interaction without microwave fields . . . . . . . . . . . . . . . . . . . . 127

6.1.5 Rabi frequency imbalance of microwave fields . . . . . . . . . . . . . . . . . . 127

6.2 Experimental implementation and calibration of ˆσzσˆz gates . . . . . . . . . . . . . . 128

6.2.1 Calibrating microwave Rabi frequencies . . . . . . . . . . . . . . . . . . . . . 129

6.2.2 Calibration of mode orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2.3 Measuring ac Zeeman shifts from magnetic field at ωg . . . . . . . . . . . . . 133

6.2.4 Tuning motional mode frequencies . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2.5 Cooling with a radiofrequency gradient and a bichromatic microwave field . . 135

6.2.6 Gate pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2.7 Calibrating π/2 pulse phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2.8 Gate duration and detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2.9 Stability of calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3 Gate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.4.1 Comparison to other laser-free gates . . . . . . . . . . . . . . . . . . . . . . . 144
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6.4.2 Insensitivity to qubit frequency offsets . . . . . . . . . . . . . . . . . . . . . . 146

6.4.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4.4 Temperature insensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.5 Gates at Doppler temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.6 Pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4.7 Qubit decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.4.8 Maximizing argument of J2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.4.9 Gates with fewer loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.4.10 Comparison to laser-based gates . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.5 Mølmer-Sørensen gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.5.1 Errors from dynamical decoupling field . . . . . . . . . . . . . . . . . . . . . . 155

7 Single-ion addressing with radiofrequency magnetic field gradient 156

7.1 Overview of laser-free single-ion addressing . . . . . . . . . . . . . . . . . . . . . . . 157

7.2 ac Zeeman shift from oscillating magnetic field on two ions . . . . . . . . . . . . . . 158

7.3 Generating anti-symmetric Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3.1 Effective single-ion rotation with Ramsey sequence . . . . . . . . . . . . . . . 161

7.3.2 Singlet data and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.4 Rabi spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.4.1 Generating singlet states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.4.2 Comparison of Ramsey and Rabi individual addressing sequences . . . . . . . 171

8 Fidelity analysis 173

8.1 Measuring fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.1.1 Triplet state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.1.2 Singlet state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8.2 Leakage correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.2.1 Triplet state fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
xv

8.2.2 Singlet state fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.3 Simulating data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.3.1 Mapping populations to counts . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.3.2 State preparation and readout errors . . . . . . . . . . . . . . . . . . . . . . . 186

8.3.3 Characterizing leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.3.4 Characterizing other parameters . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.3.5 Asymmetry in ion counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.4 Analyzing simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.4.1 Triplet state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.4.2 Singlet state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9 Conclusion 203

9.1 Improvements to entangling gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.2 Randomized benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.3 New trap geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.4 Mixed-species entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Bibliography 206

Appendix

A Baker-Campbell-Hausdorff Theorem 216

A.1 Common interaction picture transformations . . . . . . . . . . . . . . . . . . . . . . 217

A.1.1 Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

A.1.2 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

A.1.3 Modified BCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
xvi

B Single qubit rotations 220

B.1 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

B.2 Parity flopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

C Displacements in phase space 224

C.1 Rotating frame transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

C.1.1 Transformation of displacement operator . . . . . . . . . . . . . . . . . . . . . 225

C.2 Static gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

C.3 Oscillating gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

C.3.1 Rotating frame at ωg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

C.4 Summary of displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

C.5 Pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

D Derivations of Hamiltonians 233

D.1 Magnus Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

D.2 Geometric phase gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

D.3 Static gradient with detuned microwaves . . . . . . . . . . . . . . . . . . . . . . . . . 235

D.4 Oscillating gradient with carrier microwaves . . . . . . . . . . . . . . . . . . . . . . . 238

D.5 Comparison of displacements between static and oscillating gradients . . . . . . . . . 239

D.6 Derivation of spin-flip Rabi frequency from electric field sideband . . . . . . . . . . . 240

D.6.1 Comparison to classical derivation . . . . . . . . . . . . . . . . . . . . . . . . 243
Tables

Table

4.1 Microwave transition frequencies, first and second order magnetic field sensitivities

for 25Mg+ at 212.8 G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 π pulse durations for single-qubit rotations for state preparation and readout. . . . . 70

4.3 Heating rates for different motional modes. . . . . . . . . . . . . . . . . . . . . . . . 72

6.1 Comparison of laser-free entangling gates. . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2 Estimated error budget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3 Comparison of estimated errors with different gate sequences . . . . . . . . . . . . . 153

6.4 Comparison with laser gates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xvii
Figures

Figure

1.1 Schematic of Stern Gerlach experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Paul trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Sideband transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Geometric phase gates in phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Detuning errors and Walsh sequences for the mode of motion used in the gate. . . . 22

2.5 Simulation of detuning errors with Walsh modulation. . . . . . . . . . . . . . . . . . 23

2.6 Simulations of multi-loop sequences with incoherent motional errors. . . . . . . . . . 24

2.7 Simulations of Bell-state infidelity for a fixed detuning error versus temperature for

different Walsh sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Simulations of gate duration errors with Walsh modulation. . . . . . . . . . . . . . . 27

3.1 Photograph of the “magtrap.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 False color SEM of “magtrap”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Nulling magnetic fields at the ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Pillbox and base flange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Vacuum system on optical table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xviii
xix

3.6 Cryogenic operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7 High-field coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Beam orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.9 Beamlines for blue doppler (BD), blue doppler detuned (BDD), and red doppler (RD)

laser beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.10 Synchronization of devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.11 DDS synchronization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.12 Microwave carrier drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.13 Microwave red sideband drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.14 Combining carrier and sideband microwave drives. . . . . . . . . . . . . . . . . . . . 47

3.15 Generating high power MHz currents. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.16 Part of microwave and rf setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.17 Image of two ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Energy level diagram for 25Mg+ at 212.8 G. . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Calibration experiment for currents in compensation coils. . . . . . . . . . . . . . . . 55

4.3 Calibration of repump duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Histogram of two ion photon counts . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 Rabi spectroscopy of “stretch” qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6 Rabi flopping on the “stretch” qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.7 Plot of the shift of the “clock” qubit frequency relative to the calculated fieldindependent frequency versus the applied magnetic field. . . . . . . . . . . . . . . . . 60

4.8 Calibration of delay duration in readout of “clock” qubit. . . . . . . . . . . . . . . . 61

4.9 Rabi flopping on “clock” qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.10 Ramsey experiment on “clock” qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.11 Calibration of in-plane micromotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.12 Calibration of out-of-plane micromotion with microwave gradient. . . . . . . . . . . . 66
xx

4.13 Calibration of compensation voltages along trap axis. . . . . . . . . . . . . . . . . . . 67

4.14 Calibration of the in-plane position of detection beam. . . . . . . . . . . . . . . . . . 68

4.15 Calibration of the out-of-plane position of detection beam. . . . . . . . . . . . . . . . 69

4.16 Calibration of π-pulse times using microwaves for single-qubit rotations. . . . . . . . 70

4.17 Verification of ground-state cooling of the out-of-phase axial mode with Raman beams. 71

4.18 Nulling magnetic field using multiple electrodes. . . . . . . . . . . . . . . . . . . . . 74

4.19 Blue sideband transition using microwave magnetic field gradient. . . . . . . . . . . . 75

4.20 Overview of motional frequency tracking. . . . . . . . . . . . . . . . . . . . . . . . . 78

4.21 Sample calibration scans of microwave sidebands for drift tracking. . . . . . . . . . . 80

4.22 Example motional frequency tracking data. . . . . . . . . . . . . . . . . . . . . . . . 81

4.23 Histogram of difference between predicted and measured motional frequency. . . . . 82

4.24 Effect of charging from Raman beams on a radial frequency. . . . . . . . . . . . . . . 83

4.25 Example qubit frequency tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.26 Transport of a single ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.27 Compensation voltages during ion transport. . . . . . . . . . . . . . . . . . . . . . . 87

5.1 Schematic of spin-dependent displacements with gradients . . . . . . . . . . . . . . . 91

5.2 Surface-electrode trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Sideband transitions with applied microwaves and an oscillating gradient . . . . . . . 97

5.4 Full microwave spectroscopy with an oscillating gradient . . . . . . . . . . . . . . . . 97

5.5 Example Rabi flopping for J0 transition. . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6 Spin-flip transition corresponding to J0 . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7 Spin-flip transitions corresponding to an oscillating magnetic field . . . . . . . . . . . 99

5.8 Example data for determining sideband Rabi frequency . . . . . . . . . . . . . . . . 102

5.9 Sideband characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.10 Sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.11 Pulse shaping of microwave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
xxi

5.12 Microwave spectroscopy with and without pulse shaping . . . . . . . . . . . . . . . . 105

5.13 Dependence of ac Zeeman shift on radial frequency . . . . . . . . . . . . . . . . . . . 109

6.1 Relative strengths of different interactions in the bichromatic interaction picture . . 119

6.2 Spectrum of applied fields for two-qubit entangling gate. . . . . . . . . . . . . . . . . 125

6.3 Spectrum of frequency modulation of the spin-dependent force term. . . . . . . . . . 126

6.4 Broadening of Rabi spectroscopy by intrinsic dynamical decoupling. . . . . . . . . . 131

6.5 Calibration of microwave field Rabi frequencies. . . . . . . . . . . . . . . . . . . . . . 132

6.6 Qubit coherence with and without intrinsic dynamical decoupling. . . . . . . . . . . 133

6.7 Radial mode frequency as a function of bias voltage on RF electrodes. . . . . . . . . 134

6.8 Sideband cooling with bichromatic microwave field and radiofrequency magnetic field

gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.9 Gate pulse sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.10 Scan of relative phase of the final π/2 pulse in gate sequence to the first . . . . . . . 139

6.11 Simulated gate detuning scans at different gate durations. . . . . . . . . . . . . . . . 140

6.12 Gate detuning scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.13 Raw population and parity data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.14 Raw reference data for fidelity analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.15 Parity scan of triplet state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.16 Gate insensitivity to qubit frequency shifts. . . . . . . . . . . . . . . . . . . . . . . . 147

6.17 Temperature dependence of gate motional errors. . . . . . . . . . . . . . . . . . . . . 149

6.18 Effect of gate pulse shaping sequence on qubit. . . . . . . . . . . . . . . . . . . . . . 151

6.19 Parity scan from Mølmer-Sørensen gate with radiofrequency gradient. . . . . . . . . 154

7.1 Ion crystal and electrode configuration for gates and single-ion addressing . . . . . . 158

7.2 Pulse sequence for spin-echo Ramsey sequence. . . . . . . . . . . . . . . . . . . . . . 159

7.3 Ramsey experiment with two ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.4 Calibration of single arm duration in Ramsey sequence for making a singlet state. . . 165
xxii

7.5 QuTiP simulation of the sensitivity of of the singlet fidelity to the Ramsey sequence

duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.6 Calibration of phase in Ramsey sequence for making a singlet state. . . . . . . . . . 166

7.7 QuTiP simulation of the sensitivity of singlet fidelity to the Ramsey sequence phase. 167

7.8 Raw data for preparing a singlet state. . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.9 Parity analysis for singlet and triplet state. . . . . . . . . . . . . . . . . . . . . . . . 168

7.10 Microwave Rabi spectroscopy of two ions with a differential ac Zeeman shift . . . . . 169

7.11 Duration scan of the single-ion addressing pulse. . . . . . . . . . . . . . . . . . . . . 170

7.12 Calibrating the phase of the single-ion addressing pulse. . . . . . . . . . . . . . . . . 171

7.13 Raw data for generating singlet state with Rabi spectroscopy. . . . . . . . . . . . . . 172

8.1 Comparison of count histograms between experimental and simulated data. . . . . . 187

8.2 Leakage estimation with varying leakage. . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.3 Leakage estimation with more data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.4 Estimation of other parameters in simulation. . . . . . . . . . . . . . . . . . . . . . . 190

8.5 Effect of count asymmetry on triplet and singlet fidelity. . . . . . . . . . . . . . . . . 191

8.6 Triplet fidelity estimate with varying leakages. . . . . . . . . . . . . . . . . . . . . . . 192

8.7 Analysis of simulated triplet data with varying fidelities. . . . . . . . . . . . . . . . . 193

8.8 Histograms of simulated triplet fidelities. . . . . . . . . . . . . . . . . . . . . . . . . . 194

8.9 Analysis of triplet fidelities with more data. . . . . . . . . . . . . . . . . . . . . . . . 195

8.10 Analysis of the standard deviation of simulated triplet data with more data. . . . . . 195

8.11 Simulated triplet data with detuning errors. . . . . . . . . . . . . . . . . . . . . . . . 196

8.12 Singlet fidelity estimate with varying leakages. . . . . . . . . . . . . . . . . . . . . . 197

8.13 Analysis of simulated singlet data with varying fidelities. . . . . . . . . . . . . . . . . 198

8.14 Histograms of simulated singlet fidelities. . . . . . . . . . . . . . . . . . . . . . . . . 199

8.15 Analysis of simulated singlet fidelities with more data. . . . . . . . . . . . . . . . . . 200
xxiii

8.16 Analysis of standard error of the mean of simulated singlet data analysis with more

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

8.17 Analysis of simulated singlet data with an error in the phase of the Ramsey sequence.201

8.18 Analysis of simulated singlet data with an error in the duration of the Ramsey sequence.202
You can try the best you can

The best you can is good enough

Radiohead

1

Introduction

Magnetic field gradients have a long history in atomic physics. The Stern-Gerlach experiment in 1922 using a 1000 T/m gradient1 was one of the earliest demonstrations of quantum

behaviour [Gerlach and Stern, 1922]. In this experiment, a beam of silver atoms was sent through

an inhomogeneous magnetic field, and their resulting trajectories were recorded on a screen upon

exiting the magnetic field. According to classical theory, the angular momentum of these atoms

would have a random orientation with respect to the magnetic field gradient and there would be a

broad central peak of undeflected atoms. Instead, two distinct peaks2 were observed, corresponding

to the internal spin states of the unpaired outermost electron. These spin states are quantized, and

experienced a spin-dependent force due to the magnetic field gradient, resulting in two possible

trajectories as shown in Fig. 1.1.

Other examples of magnetic field gradients in atomic physics include the continuous SternGerlach effect [Dehmelt, 1986] for precision electron g-factor measurements [Van Dyck et al., 1987],

and magnetic trapping of neutral atoms [Pritchard, 1983, Raab et al., 1987] which was an important

1

In our experiment, we only generate gradients of about 100 T/m, about an order of magnitude lower. Progress!

2 Somewhat serendipitously due to the reaction of cigar smoke with silver, producing silver sulphide. See

Ref. [Friedrich and Herschbach, 2003] for an excellent overview of this experiment. No cigars were used for this thesis.

1
2

step towards the subsequent observation of Bose-Einstein condensates [Anderson et al., 1995, Davis

et al., 1995]. The focus of this thesis is the use of magnetic field gradients with trapped ions to

demonstrate another aspect of quantum behaviour, entanglement.

silver atoms 

|↑

|↓

∇



B N

S

Figure 1.1: Schematic of Stern Gerlach experiment. A beam of silver atoms enters a region with a

magnetic field gradient. The atoms experience a force that depends on their angular momentum.

Only two deflected trajectories were observed; the angular momentum or spin of these atoms is

quantized and can take one of two values, |↑i or |↓i corresponding to the mJ = ±1/2 states of the

silver atom.

1.1 Quantum entanglement

Entangled states of matter exhibit behaviour that cannot be explained classically. An example

of an entangled state is

|ψi =

1

√

2

(|↓↓i + |↑↑i), (1.1)

where |↑i and |↓i can correspond, for example, to the two spin states previously discussed. A striking

feature of this state is that it cannot be decomposed into a product state of individual spins; a

measurement on one of the spins will immediately determine the spin-state of the whole system,
3

regardless of how far each spin might be separated from the other. Einstein described this feature

as a “spooky action at a distance” [Bell and Bell, 2004], and along with Podolsky and Rosen argued

that the quantum description of nature was incomplete [Einstein et al., 1935]. In response, John

Stewart Bell pointed out that local realism is inherently incompatible with quantum mechanics, and

formulated inequalities that could be verified experimentally with entangled states [Bell, 1964]. These

experiments, known as Bell tests, were soon carried out using entangled states of photons [Freedman

and Clauser, 1972, Aspect et al., 1982] and subsequently loophole free tests were performed [Hensen

et al., 2015, Shalm et al., 2015, Giustina et al., 2015].

1.2 Quantum computing with trapped ions

Aside from verifying fundamental aspects of quantum theory, the ability to make entangled

states also enables quantum computing. Quantum computers hold the promise of solving many

problems intractable to classical computers, with applications ranging from cryptography [Shor,

1994] to nitrogen fixation [Reiher et al., 2017]. Trapped ion systems are one of the leading candidates

towards that goal; trapped ions have excellent coherence properties [Bollinger et al., 1991, Fisk

et al., 1995, Wang et al., 2017] and have been used to demonstrate some of the highest-fidelity

single [Harty et al., 2014] and two-qubit entangling operations [Gaebler et al., 2016, Ballance et al.,

2016] across any platform. However, there are many challenges still in scaling these systems towards

a practical device [Bruzewicz et al., 2019].

One such challenge is improving the fidelity of fundamental operations such as two-qubit gates

that generate entanglemenet. These gates are typically performed with laser-based interactions,

which have led to the highest two-qubit gate fidelities of 0.9992(4) and 0.999(1) in Ref. [Gaebler

et al., 2016] and Ref. [Ballance et al., 2016] respectively. However, both of these gates were limited

by photon scattering, intrinsic to the laser-based interaction used.

Alternative laser-free methods that eliminate photon scattering have been proposed and

demonstrated using magnetic field gradients to perform these entangling operations. These laser-free

methods can employ either static magnetic field gradients [Mintert and Wunderlich, 2001, Khromova
4

et al., 2012] or magnetic field gradients oscillating close to the qubit frequency at gigahertz

frequencies [Wineland et al., 1998, Ospelkaus et al., 2008, Ospelkaus et al., 2011]. In addition

to eliminating photon scattering, these methods enable simpler phase and amplitude control at

microwave frequencies(∼GHz), as opposed to the ∼ 1000 THz radiation used in laser-based gates.

While there are no fundamental limits to the fidelity of laser-free operations, technical challenges

have limited the demonstrations thus far.

1.3 Thesis outline

In this thesis, we develop a new technique that addresses some of the challenges of previous

laser-free methods. Instead of a static or gigahertz magnetic field gradient, we use a radiofrequency

magnetic field gradient to generate a spin-dependent force. We give an overview of the trapped

ion physics in Chapter 2, while Chapter 3 describes the apparatus we built, including the surfaceelectrode trap with integrated microwave and rf circuitry used to perform all the experiments in

this thesis. Chapter 4 outlines the various calibrations we perform, as well as provides a description

of the relevant atomic structure of our choice of trapped ion, 25Mg+. We first use a radiofrequency

gradient to couple the internal state of a single ion to its motion in Chapter 5. Chapter 6 describes

how we extend this technique to two-qubit entangling gates, while Chapter 7 explains how we

achieve laser-free single-ion addressing as well. Finally, we discuss how we perform the fidelity

analysis for the entangled states we created in the preceding chapters in Chapter 8.
Seasons change and our love went cold

Feed the flame ’cause we can’t let go

Run away, but we’re running in circles

Post Malone

2

Trapped ion quantum logic

In this chapter, we discuss the basics of trapping ions, how to couple their internal states (here

denoted as spin states to indicate they are states of different angular momentum) to their motion,

and how to perform entangling gates. We also include some numerical simulations of different error

mechanisms in entangling gates.

2.1 Linear rf Paul Traps

There are several trapping configurations that can be used for charged particles. We focus

on the linear rf Paul trap, which is used in the experiments described in this thesis. We mainly

follow Ref. [Wineland et al., 1998] and Ref. [Foot, 2007]. Earnshaw’s theorem states that a charged

particle cannot be in stable equilibrium when acted upon by only a static electric field. This theorem

follows from Maxwell’s equations where ∇ · ~ E~ = 0 for a region with no charge density, i.e., all the

electric field lines do not point to one location. One way around this constraint is to use oscillating

potentials instead.

We analyze the simple four-rod configuration in Fig. 2.1, where a potential V0 cos ΩT t is

applied to two diagonally opposite electrodes a distance R from the trapping location. The applied

5
6

V0 cos ΩT t

R

x

y

Figure 2.1: Paul trap. An oscillating potential V0 cos ΩT t is applied to the two electrodes in dark

grey, while the other two electrodes are grounded. The ion (red) is trapped at the center, a distance

R from each of the electrodes.

potential is oscillating with frequency ΩT and amplitude V0. For the potential Φ in the (x-y) plane,

the boundary conditions are

Φ = V0 cos ΩT t, x = ±R, y = 0,

Φ = 0, x = 0, y = ±R, (2.1)

which are satisfied by

Φ = Φ0 +

V0 cos(ΩT t)

2



1 +

x

2 − y2

R2



. (2.2)

Here, we make the approximation that we have a quadratic dependence close to the axis. The

electric field along the x direction is

Ex = −

∂Φ

∂x (2.3)

= −

V0

R2

cos(ΩT t)x, (2.4)
7

which gives rise to the equation of motion

M

d

2x

dt

2

= −

eV0

R2

cos(ΩT t)x, (2.5)

where we assume the ion has a single elementary charge e and mass M. Here, a classical treatment

is sufficient. We make a substitution τ = ΩT t/2, which leads to

d

2x

dτ

2

= −

4eV0

Ω2

TMR2

cos(2τ )x. (2.6)

This equation is a form of the Mathieu equation

d

2x

dτ

2

+ (ax − 2qx cos 2τ )x = 0. (2.7)

In our simplified case, ax = 0 and qx = (2eV0)/(Ω2

TMR2

)  1. In practice, ax will be non-zero if an

additional static quadrupole potential is applied to the electrodes (see Ref. [Wineland et al., 1998]).

The solution is

x = x0 cos 

qxτ

√

2

+ θ0

 

1 +

qx

2

cos 2τ



(2.8)

= x0 cos 

qxΩT

2

√

2

t + θ0

 

1 +

qx

2

cos ΩT t



(2.9)

where x0 and θ0 are determined by initial conditions. The term on the left is usually referred to as

the “secular” motion with frequency

ωx =

qxΩT

2

√

2

=

eV0

√

2ΩTMR2

. (2.10)

The faster oscillating term at ΩT is referred to as micromotion1. It has the effect of modulating the

ion’s wavepacket position at ΩT , but with a small enough amplitude that we can ignore it for the

1 Distinct from the “excess” micromotion we compensate for when a stray static electric field pushes the ion off the

rf null [Berkeland et al., 1998].
8

key results described below. We get an identical frequency along the y-direction; this degeneracy is

broken by a static potential. Additional end caps with static potentials can provide confinement

along the z axis.

2.2 Trapped ion hyperfine qubits

The hyperfine states within a trapped ion are useful as qubits due to their long coherence

times; up to tens of seconds [Langer et al., 2005, Harty et al., 2014] or even minutes [Bollinger et al.,

1991, Fisk et al., 1995, Wang et al., 2017] have been demonstrated. The qubit can be described by

the Hamiltonian

Hˆ

qubit =

~ω0

2

σˆz, (2.11)

where ω0 is the qubit frequency. For trapped ion hyperfine qubits, ω0/2π is typically in the

GHz range. As these qubits are within the S manifold of the ion, electric dipole transitions are

forbidden. Instead, single-qubit rotations can be performed using either two-photon stimulated

Raman transitions via an intermediate electronic state, or magnetic dipole transitions using an

oscillating magnetic field at ω0 [Wineland et al., 1998]. Both of these methods have been used

for high-fidelity single-qubit rotations using lasers [Ballance et al., 2016, Gaebler et al., 2016]

and microwaves [Brown et al., 2011, Harty et al., 2014]. In particular, Ref. [Harty et al., 2014]

demonstrated single-qubit-rotation gate errors at the 1 × 10−6level. Ref. [Bruzewicz et al., 2019]

contains an overview of other types of trapped-ion qubits that can be used such as optical frequency

qubits.

2.3 Trapped ion spin-motion coupling

Aside from the internal spin states, the ions also have motional states. To a very good

approximation [Wineland et al., 1998], the ions’ modes of motion can be described by a quantum

harmonic oscillator with Hamiltonian
9

Hˆmotion = ~ωraˆ

†

a, ˆ (2.12)

where ωr is the frequency of the motional mode. For a single ion, the position operator rˆ for a single

mode is

rˆ = r0(ˆa + ˆa

†

), (2.13)

with r0 =

p

~/(2mωr) describing the ion’s ground state extent. The creation and annihilation

operators are aˆ

† and aˆ respectively. For the experiments in this thesis using 25Mg+ and motional

frequencies which are typically ωr/2π ≈ 6 MHz, r0 ≈ 6 nm. The motional states are usually described

in the Fock state basis |ni.

Coupling the ions’ internal states to their motion, or spin-motion coupling, is the workhorse

for many applications in trapped ions, from elementary logic gates [Monroe et al., 1995] to quantum

logic spectroscopy [Schmidt et al., 2005] that has enabled some of the world’s most precise clocks,

for example Ref. [Brewer et al., 2019]. These demonstrations all involved only sideband transitions

(described below), using the motion as an information bus between ion spin states [Cirac and Zoller,

1995]. Here, we discuss how these sideband transitions are generated from a field gradient and how

they can be implemented with and without laser beams.

2.3.1 Sideband transitions

First, let us look at the Hamiltonian that corresponds to the potential energy of an ion, given

by

Hˆ = Vˆ (r), (2.14)

where the potential is dependent on position. Expanding to first order,

Hˆ ≈ Vˆ (r)







r=raverage

+

∂V

∂r







r=raverage

r0(ˆa + ˆa

†

), (2.15)
10

where raverage is the average position of the ion. We see that if there is a gradient in the potential, we

now have terms that couple to the motion. To see sideband transitions, let us analyze a Hamiltonian

describing a field with frequency ω that can drive spin-flip transitions in an ion.

Hˆ ≈ 2~Ωf cos (ωt)ˆσx + 2~Ωg cos (ωt)ˆσx(ˆa + ˆa

†

), (2.16)

where Ωf corresponds to the coupling of the internal states of the ion to the field, while Ωg is the

coupling of the internal states and the ion motion to the gradient following Eq. 2.15. Transforming

to the interaction picture with respect to Hˆ

0 =

~ω0

2

σˆz +~ωraˆ

†aˆ (the ion qubit and motional energies),

we obtain (following Appendix. A.1)

Hˆ

I ≈ ~Ωf (ˆσ+e

−iδt + ˆσ−eiδt) + ~Ωg(ˆσ+e−iδt + ˆσ−eiδt)(ˆae−iωrt + ˆa†

e

iωrt

), (2.17)

where δ = ω − ω0. The spin-raising and lowering operators are σˆ+ = |↑i h↓| and σˆ− = |↓i h↑|

respectively, where |↑i and |↓i are the internal spin states of the trapped ion. When δ = 0, we

implement |↓i ↔ |↑i spin-flip transitions (ignoring fast-rotating terms). Similarly, when δ = ωr, we

implement the blue sideband transition with the interaction

Hˆ

bsb = ~Ωg(ˆσ+aˆ

† + ˆσ−aˆ). (2.18)

This interaction drives the |↓i |ni ↔ |↑i |n + 1i transition with ∆n = 1. If instead δ = −ωr, we

implement the red sideband transition

Hˆ

rsb = ~Ωg(ˆσ+aˆ + ˆσ−aˆ

†

), (2.19)

which drives the |↓i |ni ↔ |↑i |n − 1i transition with ∆n = −1. These transitions are summarized

in Fig. 2.2.
11

hω ̄ r

n=0

n=1

n=2

∆

n = 

0

∆

n = 

1

∆

n= -1

hω ̄ 0

Figure 2.2: Schematic of sideband transitions. The ion has spin states denoted by |↓i and |↑i with

energy ~ω0. The ion also has motional (Fock) states denoted by n with energies ~ωr. A carrier

transition that flips the ion spin but leaves the motional state unchanged (∆n = 0) is denoted by

the green arrow. A blue (red) sideband transition corresponds to adding (subtracting) a motional

quantum when starting in |↓i, such that ∆n = 1 (∆n = −1).

2.3.2 Position-dependent energy shifts

Not all spin-motion coupling results in sideband transitions. If instead we have an interaction

that modulates the qubit frequency at frequency ω, the Hamiltonian becomes

Hˆ ≈ 2~Ωf cos (ωt)ˆσz + 2~Ωg cos (ωt)ˆσz(ˆa + ˆa

†

), (2.20)

which is identical to Eq. 2.16, except that σˆx has been replaced by σˆz. Again, going into the

interaction picture with respect to Hˆ

0,

Hˆ

I ≈ 2~Ωf cos ωtσˆz + 2~Ωg cos ωtσˆz(ˆae−iωrt + ˆa

†

e

iωrt

). (2.21)

The first term corresponds to an oscillation of the qubit frequency at frequency ω. The second term,

while arising from the gradient, does not drive sideband transitions, but instead spin-dependent
12

displacements that are the basis for the entangling gates described in Sec. 2.4.3. However, an

additional field can be used to drive sideband transitions as we discuss in Chapter 5.

2.3.3 Laser-based spin-motion coupling

Laser-based spin-motion coupling typically involves driving electric dipole transitions according

to the Hamiltonian

Hˆ

e = −~μd · E. ~ (2.22)

The ion’s electric dipole operator is ~μd and E~ is the electric field. In this case, our qubit states will

correspond to different electronic levels in the atom (for example S and P states). For an electric

field E~ = Exxˆ cos(kr − ωt + φ), where k is the wavevector 2π/λ and ω its frequency,

Hˆ

e = ~Ωe(ˆσ+ + ˆσ−)



e

i(kr−ωt+φ) + e−i(kr−ωt+φ)



. (2.23)

The Rabi frequency is Ωe = −μdxEx/4. Expanding to first order following Eq. 2.15,

Hˆ

e ≈~Ωe(ˆσ+ + ˆσ−)



e

i(−ωt+φ

0

) + e−i(−ωt+φ

0

)



field

+~Ωekr0(ˆσ+ + ˆσ−)(ˆa + ˆa

†

)



e

i(−ωt+φ

0

) + e−i(−ωt+φ

0

)



, gradient (2.24)

where φ

0 = kraverage + φ. The gradient term has a Rabi frequency that is smalelr than the field

term by a factor η = kr0, where η is the Lamb-Dicke parameter that describes the ratio of the

wavelength to the ion’s ground-state wavefunction extent. Transforming into the interaction picture

with respect to Hˆ

0, we obtain

Hˆ

I =~Ωe(ˆσ+e

−iδt + ˆσ−eiδt) field

+~Ωeη(ˆσ+e

−iδt + ˆσ−eiδt)(ˆae−iωrt + ˆa†

e

iωrt

), gradient (2.25)
13

where for simplicity we have dropped the complex phase corresponding to e

iφ0

. We now have all

the ingredients to drive spin-flip or sideband transitions by appropriate choice of δ as shown in

Eq. 2.17. Note that here we assume the interaction of the dipole moment with just one field. A

similar derivation follows for two-photon transitions except two fields would have to be considered,

and instead of k we have ∆k, the difference between the k-vectors of the two fields [Wineland et al.,

1998].

2.3.4 Laser-free spin-motion coupling

Instead of using the ion’s electric dipole moment, we can also use its state-dependent magnetic

dipole moments coupled to a magnetic field gradient. Laser-free spin-motion coupling has been

proposed and demonstrated using static magnetic field gradients [Mintert and Wunderlich, 2001,

Johanning et al., 2009] and oscillating magnetic field gradients close to the qubit frequency [Ospelkaus

et al., 2008, Ospelkaus et al., 2011]. We first look at how these gradients can be generated in the

near-field relative to current-carrying electrodes, and give an overview of the different types of

gradients that can be used.

2.3.4.1 Near-field magnetic field gradients

As we have seen in this chapter, we need field gradients to couple the ions’ spin to their motion.

For a plane wave, this gradient is proportional to ∂/∂r exp (ikr) ∝ 1/λ. Thus, for laser-based

methods, this gradient can be large as the wavelength is small, ∼ 300 nm. For microwave or rf

transitions, this gradient would be negligible as the wavelengths are now ∼ 100 mm, 6 orders of

magnitude higher. However, as pointed out in Ref. [Wineland et al., 1998]2, near-field gradients can

also be used. The interaction describing an ion’s magnetic moment in a magnetic field is

2 One of the reasons we call ref. [Wineland et al., 1998] the “Bible”. Every time you think you have a good idea

you check the “Bible” and Dave thought about it 20 years ago.
14

HˆB = −~μ · B~

≈ −~μ · B~







r=raverage

− ~μ ·

∂B

∂r







r=raverage

r, ˆ

(2.26)

where ~μ is the ion magnetic moment, and B~ is the magnetic field. We have expanded the magnetic

potential following Eq. 2.15. In future chapters, the first term in the second line of Eq. 2.26 is what

we will refer to as the field component, and the second the gradient.

For simplicity, we consider a magnetic field generated by an infinitely long current-carrying

wire and its gradient in the radial direction

B =

μ0I

2πd,

∂B

∂d = −

μ0I

2πd2

,

(2.27)

where d is the distance from the wire and I can be time-dependent. The gradient is independent of

the wavelength and dependent only on the ion-electrode distance (with the near-field approximation

of d  λ). Thus, by placing the ion very close to a current-carrying electrode, we can generate a

magnetic field gradient that is large enough to couple the ion’s spin to its motion.

2.3.5 Near-qubit-frequency oscillating magnetic-field gradient

Following Eq. 2.26, for a magnetic field that couples the qubit states, B~ = Bxxˆ cos ωt, the

Hamiltonian is

Hˆ

I = −~μ · B~

≈~ΩB(ˆσ+e

−iδt + ˆσ−eiδt) field

+~Ωg(ˆσ+e

−iδt + ˆσ−eiδt)(ˆae−iωrt + ˆa†

e

iωrt

), gradient (2.28)

where we have transformed into the interaction picture with respect to Hˆ

0. The Rabi frequencies

are
15

ΩB =

Bx

2~

μ↓↑, (2.29)

corresponding to spin-flip transitions3 with δ = 0, and μ↓↑ = h↓| μx |↑i. The Rabi frequency of the

gradient is

Ωg =

∂B

∂r r0

2~

μ↓↑, (2.30)

which drives sideband transitions when δ = ±ωr. The matrix element μ↓↑ describes the coupling of

the ion spin states via the magnetic moment and r0 is the ground state extent4.

2.3.5.1 Static magnetic field gradients

Instead of an oscillating magnetic field gradient, a static magnetic field gradient can also be

used. The magnetic field gradient can be from permanent magnets [Lake et al., 2015, Weidt et al.,

2016, Khromova et al., 2012] or a wire with a dc current [Welzel et al., 2019]. This gradient is

typically used in conjunction with additional oscillating fields close to the qubit frequency to drive

sideband transitions as we discuss in Sec. 5.1.

2.3.5.2 Radiofrequency magnetic field gradient

Finally, we demonstrate a new laser-free method of spin-motion coupling that uses a radiofrequency magnetic field oscillating close to the ion motional frequency, in addition to oscillating fields

close to the qubit frequency; this technique is discussed in more detail in Chapter 5.

2.4 Entangling gates

Trapped ion qubits have weak direct spin-spin interactions due to the Coloumb repulsion

that keeps the ion-ion spacing large. However, as pointed out in Ref. [Cirac and Zoller, 1995], the

3 This is equivalent to Ωx

in Ref.[Ospelkaus et al., 2008].

4 This is equivalent to Ωx

j,n in Ref.[Ospelkaus et al., 2008].
16

spin-motion coupling techniques discussed in the previous section enable strong spin-spin interactions

via the shared motion of the ions. By using tightly focused beams on ions in their ground state of

motion, entangled states can be created using only sideband interactions on each ion. Experimental

demonstrations soon followed using a single ion entangled with its motional state [Monroe et al.,

1995], and later with multiple ions [Schmidt-Kaler et al., 2003]. More recently, this technique has

been used to entangle atomic and molecular ion qubits [Lin et al., 2020].

However, there are two big disadvantages to the Cirac-Zoller technique. First, it requires

cooling to the ground state of motion. Any motional occupation outside of n = 0 would result

in a gate error. Second, individual addressing of ions is necessary. Both of these problems are

circumvented with geometric phase gates.

2.4.1 Geometric phase gates

Geometric phase gates use spin-dependent forces to drive closed spin-dependent trajectories in

phase space. As the ions return to their original position in phase space, their motion is disentangled

at the end of the sequence independent of the initial state of motion, eliminating the need for ground

state cooling. Broadly, there are two classes of phase gates. First, are the σˆxσˆx gates proposed

in Ref. [Mølmer and Sørensen, 1999, Sørensen and Mølmer, 1999], which was first demonstrated

in Ref. [Sackett et al., 2000]. Second, are the σˆzσˆz gates which were proposed in Ref. [Milburn

et al., 2000] and inspired the demonstration of a high fidelity phase gate5in Ref. [Leibfried et al.,

2003]. While the two interactions broadly generate the same dynamics [Lee et al., 2005], the origins

of spin-dependent forces are physically distinct. We discuss their specific implementations after a

general overview of phase gates. The gate interaction for two ions is

Hˆ

g = ~Ωg (ˆσi1 ± σˆi2)



aeˆ

i∆t + ˆa†

e

−i∆t



, (2.31)

where Ωg is the Rabi frequency corresponding to the gradient producing the spin-dependent force

and ∆ is the gate detuning. The spin operators are (ˆσi1 + ˆσi2) for the center-of-mass mode and

5 Also known as the Didi gate.
17

(ˆσi1 − σˆi2) for the out-of-phase mode. The spin operator σˆi can have i = x, y, z while 1 and 2 specify

the ion index. Following Appendix D.2, the time evolution operator for this interaction is

Uˆ

g = Dˆ(α(t))e

−iφ(t)

, (2.32)

where Dˆ is the displacement operator. The displacement is

α(t) = Ωg

∆

(ˆσi1 ± σˆi2) (e

−i∆t − 1), (2.33)

and the phase is

φ(t) =

Ω

2

g

∆2

(ˆσi1 ± σˆi2)

2

(∆t − sin (∆t)). (2.34)

This propagator describes a spin-dependent trajectory in phase space, with a spin-dependent

geometric phase. To generate spin-spin entanglement while disentangling the spin states from

the motion, we require that the displacement α = 0 at the end of the gate sequence. We fulfill

this requirement by choosing the gate duration and detuning such that ∆ = 2Kπ/tg, where K is

an integer corresponding to the number of loops in the phase-space trajectory. The phase space

dynamics are shown in Fig. 2.3.

To generate the maximally entangled state, we need φ(t) = π/2. As the eigenvalues of

(ˆσi1 ± σˆi2) are 2 and 0, solving for ∆ and tg we obtain

∆ = ±4

√

KΩg, (2.35)

tg =

2π

√

K

4Ωg

. (2.36)

Note that the positive and negative values of ∆ correspond to different maximally entangled states.

For the specific case of the spin operators in Eq. 2.31 being (σˆz1 − σˆz2), the acquired phases for

each of the different spin states (∆ positive) are
18

x

p

x

a) b) p

|↑↑

|↑↓

|↓↓ |↓↑

Figure 2.3: Geometric phase gates in phase space for (σˆz1 − σˆz2) in Eq. 2.31. The center of the

ions’ motional wavefunction is indicated by the blue dot. a) Two of the four two-ion spin states

undergo a circular trajectory in phase space. Returning to the origin, these spin states acquire a

geometric phase proportional to the area enclosed in phase space and are disentangled from the

ion motion. b) The other two spin states do not experience a spin-dependent force and stay at the

origin. Setting the acquired geometric phase to π/2 enables the generation of entangled states. This

example shows the initial position of the ion at the origin in phase space but, in principle the ion

could be initialized anywhere.

|↓↓i → |↓↓i,

|↑↑i → |↑↑i,

|↓↑i → −i|↓↑i,

|↑↓i → −i|↑↓i.

(2.37)

This spin-dependent phase, along with global π/2





x

pulses before and after the phase-space

displacement, produces a maximally entangled state,

|↓↓i → 1

√

2

(|↓↓i + i|↑↑i). (2.38)

2.4.2 Mølmer-Sørensen gates

Having gone through the general theory, we now discuss specific implementations of the

geometric phase gates. For the Mølmer-Sørensen gate, the interaction is generated by simultaneous
19

application of red and blue sideband interactions that are symmetrically detuned from the qubit

resonance. In other words, if the blue sideband is detuned by ∆, the red sideband is detuned by

−∆. The Hamiltonian in the interaction picture is

Hˆms = Hˆ

bsb + Hˆrsb

= ~Ωg



(ˆσ+1 + ˆσ+2)ˆa

†

e

−i∆t + (ˆσ−1 + ˆσ−2)ˆaei∆t BSB

+ (ˆσ+1 + ˆσ+2)ˆaei∆t + (ˆσ−1 + ˆσ−2)ˆa

†

e

−i∆t



RSB

= ~Ωg (ˆσx1 ± σˆx2)



aeˆ

i∆t + ˆa†

e

−i∆t



. (2.39)

While our spin operators correspond to σˆx here, adjusting the phase of the BSB and RSB interactions

can result in σˆφσˆφ interactions, where σˆφ = cos φσˆx + sin φσˆy. This technique has resulted in both

high-fidelity laser-based gates [Gaebler et al., 2016] and laser-free gates [Harty et al., 2016, Weidt

et al., 2016, Zarantonello et al., 2019].

2.4.3 σˆzσˆz gates

The ˆσzσˆz interaction is

Hˆ

g = ~Ωg (ˆσz1 ± σˆz2)



aeˆ

i∆t + ˆa†

e

−i∆t



. (2.40)

The physical basis of this force is the gradient of an oscillating energy shift at the ion, for laser-based

gates from the gradient of a spin-dependent ac Stark shift [Leibfried et al., 2003]. This technique

has been used to demonstrate high-fidelity gates such as Ref. [Ballance et al., 2016, Sch ̈afer et al.,

2018]. There has also been a laser-free demonstration of this interaction using static magnetic field

gradients [Khromova et al., 2012]. This interaction is the basis for the entangling gates in Chapter 6.

One disadvantage of the σˆzσˆz interaction is that states that have a spin-dependent oscillating

ac Stark shift are typically static magnetic field-sensitive as well, in contrast with the MS gates that

work with any qubit states. However, there has been a recent demonstration that used an auxiliary
20

state to generate this interaction on a field-insensitive qubit [Baldwin et al., 2020a]. In contrast to

MS gates however, only two laser beams are required with a difference frequency close to the motion

at MHz frequencies. For MS gates using Raman transitions, two pairs of laser beams are required,

one for each sideband, each with a difference frequency close to the qubit at GHz frequencies. The

use of ac Stark shifts also enables the generation of a spin-dependent force on qubits of a different

species using the same pair of beams [Ballance et al., 2015, Hughes et al., 2020].

2.5 Phase gate errors and suppression

With an understanding of how phase gates work, we discuss the different error mechanisms in

phase gates, with a focus on the motional errors as those cause the dominant errors in the gates we

later demonstrate.

2.5.1 Motional errors

While the ions’ spin is ideally disentangled from the motion at the end of the gate, the spins

are entangled with the motion during their trajectories. Thus, any error due to the motion would

affect the gate as well. We can broadly divide these errors into incoherent and coherent errors.

Incoherent errors include heating or motional dephasing, while coherent errors include fixed detuning

and gate duration errors. Following Ref. [Sepiol, 2016], we model incoherent errors using a Lindblad

master equation

ρ ̇ = −

i

~

[H, ρ ˆ ] + 1

2

X

k



2Cˆ

kρCˆ†

k − (ρCˆ†kCˆ

k + Cˆ†

kCˆ

kρ)



, (2.41)

where Cˆ

k =

√γn

Aˆ

k is the collapse operator. The rate of decoherence is γk and Aˆ

k is the operator that

governs the decoherence mechanism and is typically either Pauli spin, or creation and annihilation

operators. Here, we have used the notation in QuTiP [Johansson et al., 2013], which we use later to

model these errors. For our simulations, Hˆ will correspond to the gate interaction.
21

2.5.1.1 Heating

Heating refers to incoherent heating of the ion motion from for example, coupling to a thermal

bath. This heating causes an increase in the ions’ motional state occupation as a function of

time [Brownnutt et al., 2015], characterized by n ̄ ̇ , where n ̄ is the average motional occupancy.

Following Ref. [Harty et al., 2014], the master equation for this process is,

ρ ̇ = −

i

~

[H, ρ ˆ ] + n ̄ ̇

2



2ˆa

†

ρaˆ − (ρaˆaˆ

† + ˆaaˆ†

ρ) + 2ˆaρaˆ

† − (ρaˆ†

aˆ + ˆa

†

aρˆ ),



(2.42)

which corresponds to Eq. 2.41 with two collapse operators: √

n ̄ ̇ aˆ

† and √

n ̄ ̇ aˆ.

2.5.1.2 Motional dephasing

Fast frequency fluctuations in the ion motion [Harty, 2013] can cause dephasing, but not

necessarily heating. For the axial mode, this could result from fluctuations in the dc voltages that

provide the axial confinement. Amplitude fluctuations in the trapping rf voltage will cause the

same effect for the radial modes. This dephasing can be measured by a Ramsey experiment on the

motional states. For a Fock state superposition √

1

2

(|ni + |n + ∆ni), the Ramsey contrast decays

with time constant (∆n)

2

τc. Using this definition, the master equation is

ρ ̇ = −

i

~

[H, ρ ˆ ] + 1

τc



2ˆa

†

aρˆ aˆ

†

aˆ − (ρ(ˆa

†

aˆ)

2 + (ˆa†

aˆ)

2

ρ)



, (2.43)

which corresponds to Eq. 2.41 with a single collapse operator p

(2/τc)ˆa

†aˆ in Eq. 2.41.

2.5.1.3 Detuning errors

Detuning errors refer to miscalibrations or changes of the motional frequency which result

in an error in the gate detuning ∆. This error will result in incomplete loop closure and residual

spin-motion entanglement that will reduce the gate fidelity. We require all of the motional wave

functions of the different spin states to overlap at the end of the gate sequence; if there is no overlap
22

there is no interference between the spin states that gives rise to entanglement. An example of a

detuning error is shown in Fig. 2.4a).

x

p

x

a) b) p

Figure 2.4: Detuning errors and Walsh sequences for the mode of motion used in the gate. a) A

detuning error results in an incomplete loop closure for one or more of the spin states. b) This

error can be mitigated by a two-loop sequence, flipping the sign of the spin-dependent force in

between the loops. This results in a trajectory that ends closer to the starting point. This sequence

corresponds to Walsh 1 modulation.

2.5.2 Reducing motional errors: multi-loop sequences

An effective method of reducing motional errors is to employ multi-loop sequences6. As

pointed out in Ref. [Hayes et al., 2012], changing the sign of the spin-dependent force during a gate

sequence, or Walsh modulation, reduces the effect of detuning errors as shown in Fig. 2.4b). A

Walsh 1 sequence corresponds to one phase change, Walsh 3 to two, and Walsh 7 to five. Walsh 1,

3, and 7 sequences also have a minimum of 2, 4, and 8 loops respectively.

We perform a simulation in QuTiP to investigate the effect of Walsh modulation on static

detuning errors as shown in Fig. 2.5. For static errors, using Walsh 1 modulation reduces the

infidelity by about an order of magnitude. Higher Walsh sequences reduce this error further but

less significantly. Note that higher Walsh sequences might help for time-varying detuning errors.

Multi-loop sequences can also reduce the effects of incoherent motional errors. We plot a

simulation with both heating and dephasing in Fig. 2.6. We model both errors following Eq. 2.42 and

6 Unlike humans on a roller coaster, ions have less motion sickness the more loops they do.
23

2 4 6 8

Number of loops (K)

10

3

10

2

Bell-state infidelity

No Walsh

Walsh 1

Walsh 3

Walsh 7

Figure 2.5: Simulation of detuning errors with Walsh modulation. Using a gate Rabi frequency of

Ωg/2π = 1.25 kHz, we simulate a detuning error of 200 Hz. We plot the Bell-state fidelity versus

the number of loops for different Walsh sequences: no Walsh (1-8 loops), Walsh 1 (2, 4, 6, 8

loops), Walsh 3 (4, 8 loops), Walsh 7 (8 loops). Using Walsh 1 modulation as opposed to no

Walsh modulation reduces the error by an order of magnitude. Using higher Walsh modulation and

increasing the number of loops reduces the error further, but less significantly. For all simulations,

we use Ωg/2π = 1.25 kHz and adjust the gate duration and detuning according to the number of

loops (see equations 2.35 and 2.36).

Eq. 2.43 respectively. We find that increasing the number of loops reduces the infidelity, independent

of the Walsh modulation. The reduction in this error also roughly follows 1/

√

K scaling. Purely

geometrically, these errors would be larger the larger the ions’ excursion in phase space. The phase

space trajectory has a radius that is proportional to 1/

√

K; the more loops there are, the smaller

each loop has to be to acquire the same geometric phase.

2.5.2.1 Temperature dependence of motional errors

While geometric phase gates are nominally insensitive to the initial temperature of the mode,

higher temperatures make the gate more sensitive to other errors. We only expand Eq. 2.25 to first

order with respect to the Lamb-Dicke parameter η. Higher order terms in the gradient expansion

(for example ˆa

2

) will add error terms that scale more quickly with  ̄n.

We plot the temperature dependence of the gate fidelity versus the average motional occupation
24

10

3

6 × 10

4

2 × 10

3

Bell-state infidelity

a) n = 20/s

1/ K

No Walsh

Walsh 1

Walsh 3

Walsh 7

2 4 6 8

Number of loops (K)

10

3

3 × 10

4

4 × 10

4

6 × 10

4

Bell-state infidelity

b) 1/ c = 10/s

1/ K

No Walsh

Walsh 1

Walsh 3

Walsh 7

Figure 2.6: Simulations of multi-loop sequences for two types of incoherent motional errors: a)

heating rate n ̄ ̇ = 20/s, b) motional dephasing with 1/τc = 10/s. For each error, we plot the Bell-state

fidelity versus the number of loops for different Walsh sequences: no Walsh (1-8 loops), Walsh 1 (2,

4, 6, 8 loops), Walsh 3 (4, 8 loops), Walsh 7 (8 loops). For all simulations, we use Ωg/2π = 1.25 kHz

and adjust the gate duration and detuning according to the number of loops (see equations 2.35

and 2.36). For both errors, we find that increasing the number of loops reduces the error. We find

no effect from the Walsh modulation. We additional plot a dashed line showing 1/

√

K scaling for

each of the infidelities.

number in Fig. 2.7. For the gate sequence without Walsh modulation, the error increases by almost

an order of magnitude going from an n ̄ of 0 to 4. Walsh modulation significantly reduces this

temperature dependence. A more detailed analysis of the sensitivity of different errors to temperature

for a Walsh 7 sequence is shown in Fig. 6.4.4. Note that this temperature dependence is more

problematic for laser-based interactions.
25

0 1 2 3 4

n

10

3

10

2

10

1

Bell-state infidelity

No Walsh

Walsh 1

Walsh 3

Walsh 7

Figure 2.7: Simulations of Bell-state infidelity for a fixed detuning error versus temperature for

different Walsh sequences. The detuning error is 200 Hz, with Ωg/2π = 1.25 kHz for all gates. Walsh

modulation significantly reduces the temperature dependence of this error.

2.5.3 Qubit errors

Aside from motional errors, there are also qubit errors that will affect the gate fidelity.

2.5.3.1 Photon scattering

The highest-fidelity entangling gates are the laser-based gates reported in Ref. [Gaebler et al.,

2016] and Ref. [Ballance et al., 2016], with fidelities of 0.9992(4) and 0.999(1) respectively. They had

photon scattering errors of 5.7 × 10−4 and 4 × 10−4respectively. This photon scattering comprises

both Raman and Rayleigh scattering. Raman scattering is inelastic, destroying the coherence of

the qubit, as opposed to Rayleigh scattering which is elastic [Ozeri et al., 2007]. Raman scattering

can be reduced by increasing the detuning of the Raman transition further, but at the cost of

increasing laser power to maintain the same gate Rabi frequency. Rayleigh scattering on the other

hand approaches an asymptotic value.
26

2.5.3.2 Qubit decoherence

Lastly, the qubit can also suffer from decoherence during the gate sequence. This effect is

particularly problematic for field-sensitive qubits. Any changes in the magnetic field can cause shifts

in the energy levels of the qubit during the gate. Fortunately for σˆzσˆz gates, qubit frequency shifts

commute with the gate and can be echoed out.

For MS gates on the other hand, this error does not commute. Even though they can be

performed on field sensitive qubits, fluctuations in ac Zeeman or ac Stark shifts can still cause qubit

decoherence. This decoherence can be mitigated by an additional dynamical decoupling [Harty

et al., 2016]. We discuss dynamical decoupling in more detail in Sec. 6.1.1.

2.5.4 Gate duration errors

Gate duration errors are closely related to detuning errors; they can both cause incomplete

closure of the phase-space loop, but gate duration errors can further cause the wrong geometric phase

to be acquired. These errors are also mitigated by the Walsh modulation discussed in Sec. 2.5.2. For

a fixed gate duration error of 5% we plot the gate infidelity versus detuning in Fig. 2.8. Higher-order

Walsh sequences reduce the sensitivity to gate duration errors.

2.6 Non-classical states of motion

While the focus of this chapter has been on spin-motion coupling to generate entanglement

between ion spin states via the motion, the ion motional states are also active areas of research in

themselves. Non-classical states of motion such as squeezed states [Meekhof et al., 1996, Kienzler

et al., 2015, Burd et al., 2019], and Fock-state superpositions [Leibfried et al., 2002, McCormick

et al., 2019] have been generated. The motional states of trapped ions can also be used as a qubit

in the context of continuous variable quantum computation [Fl ̈uhmann et al., 2019].
27

0.15 0.10 0.05 0.00 0.05 0.10 0.15

/ g1

10

5

10

4

10

3

10

2

Bell-state infidelity

No Walsh

Walsh 1

Walsh 3

Walsh 7

Figure 2.8: Simulations of gate duration errors with Walsh modulation. We have a fixed 5% gate

duration error for the different gate sequences, and plot the fidelity versus detuning (∆ + δ∆),

normalized to the one loop detuning ∆g1.
I don’t need no money, fortune, or fame (ooh hey hey hey)

I’ve got all the riches baby one man can claim (oh yes I do)

I guess you’d say

What can make me feel this way?

My girl (my girl, my girl)

Talkin’ ’bout my girl (my girl)

The Temptations 3

Apparatus

An experimental thesis would not exist without an apparatus. In this chapter, we discuss

key features that enable the experiments described subsequently. The trap fabrication and vacuum

assembly was done by Daniel Slichter and David Allcock, while Robert J ̈ordens set up many of

the beamlines used in the experiment. Shaun Burd developed a new laser system using vertical

external-cavity surface-emitting lasers (VECSELs) that initially were used only for sideband cooling,

but were later used for Doppler cooling and detection as well [Burd et al., 2016]. Though this

apparatus was redesigned and rebuilt from the ground up in a new lab, it is very much the second

generation of the apparatus used in Ref. [Ospelkaus et al., 2008] and Ref. [Warring et al., 2013a],

which contain many technical details relevant to this setup.

3.1 Surface electrode trap

At the heart of this apparatus is the surface electrode trap for near-field magnetic-field

gradients, or the “magtrap” as shown in Fig. 3.1. This trap was fabricated at the NIST clean room

by Daniel Slichter and David Allcock. The trap has a c-cut sapphire substrate and the electrodes

28
29

are electroplated gold, with a thickness of 8 μm. A thermal epoxy1is used to attach it a pedestal

underneath. Ribbon bonds are used to connect the trap to an alumina printed circuit board.

Figure 3.1: Photograph of the “magtrap”, the surface electrode trap2used in this thesis. The trap

is electroplated gold on a sapphire substrate. A thermal epoxy is used to attach it a pedestal

underneath. Ribbon bonds connect it to an alumina printed circuit board.

To more clearly see the electrode structure, we display a false color image from a scanning

electron microscope (SEM) in Fig. 3.2. Conceptually, it is identical to the trap described in Sec. 2.1,

but with all the electrodes in a single plane. The red electrodes are used for the oscillating rf voltage

needed to generate a trapping potential. They are connected to the rf resonator on one end and

open on the other. We can also apply a dc bias voltage to these electrodes to tilt the ion modes in

the radial (y-z) plane. We have 9 pairs of dc electrodes along the trap axis (x) that provide the

axial confinement. These electrodes are also used to apply potentials (see Sec. 4.4.1) to compensate

for stray electric fields. Additionally, changing the voltages applied to these electrodes enables

transport of the ions along the trap axis. As discussed in Sec. 4.9, most of the experiments in this

1 EPO-TEK H21D.

2 Even Admiral Ackbar would agree.
30

thesis were conducted where the ions are loaded as shown in Fig. 3.2.

Figure 3.2: False color SEM image of the “magtrap”. We apply rf voltages to the red electrodes

to generate the trapping potential. Nine pairs of dc electrodes provide confinement along the trap

axis (x). The MHz and GHz currents used to the generate the magnetic fields and magnetic field

gradients are applied to the electrodes in blue, here labeled 1, 2, and 3.

The most important electrodes for the physics in this thesis are shown in blue. We apply

large MHz and GHz currents (∼ 1 A rms) to these electrodes in order to generate magnetic fields

and magnetic field gradients. These electrodes are shorted to ground at one end of the chip. By

design, the magnetic field gradients are only along the radial (y-z) plane. We have three electrodes

for these currents, as did Ref. [Ospelkaus et al., 2011] and Ref. [Allcock et al., 2013], to null the

magnetic field at the ion. By adjusting the phase and amplitudes of the currents applied to all

three electrodes, we are able to null the magnetic field at the ion while keeping a magnetic field

gradient (see Fig. 3.3). At least for the spin-motion coupling scheme described in Sec. 2.3.5, it

is advantageous to null the magnetic field while keeping the gradient. Nulling the magnetic field

eliminates off-resonant transitions as well as fluctuations in the ac Zeeman shift, improving gate

fidelities. Alternative trap structures can be designed to null the magnetic field using only one
31

electrode such as in Ref. [Hahn et al., 2019]. However, one disadvantage is that it can be challenging

to align the rf null to the null of the magnetic field. In contrast, having three electrodes ensures

that there are always enough degrees of freedom to null the magnetic field at the ion.

I3 I2 I1

B



2

B



1

B



3 ion

Figure 3.3: Schematic of nulling the magnetic field at the ion. If the current in electrode 2

is π out of phase, and the relative amplitudes adjusted, the magnetic field at the ion is nulled

(B~

1 + B~2 + B~3 = 0). Crucially, the sum of the gradients is not zero (∇~ B1 + ∇~ B2 + ∇~ B3 6= 0). The

direction of the magnetic field gradient from each electrode is along their respective dotted lines.

One consideration for our electrode geometry is the choice of qubit. The magnetic fields that

we generate will mostly be linearly polarized. Thus, using a qubit whose transition is only driven by

σ

+ polarization results in only using at most half of the linear polarization generated (one must

also factor in the direction of the quantization axis). Our apparatus was originally designed to use

a “clock” qubit whose transition is driven by a π-polarized magnetic field. This qubit is discussed

in more detail in Sec. 4.3. Of course, the experiments we ended up performing took a slight detour

from this plan3.

3 Electrodes 1-3 were grounded intentionally to give us the flexibility to pursue different types of gradients.
32

3.2 Vacuum system

Figure 3.4 shows the gold-plated copper pillbox encasing the trap. The trap as well as the

alumina PCB are mounted in this pillbox, which is attached to the base flange of our vacuum

system. Our MHz and GHz currents are applied to electrodes 1, 2, and 3 in Fig. 3.2 via the SMA

ports and cables shown in Fig. 3.4b) and c) respectively. A copper braid provides the thermal link

to the flow cryostat. There are additional resistors and capacitors not indicated on the PCB that

provide in-vacuum filtering.

We show the assembled vacuum system on the optical table in Fig. 3.5. The vacuum system

itself is obscured by other components, such as as the four argon ion guns attached to the top flange.

The guns were intended for in situ ion milling, which has been shown to reduce trap heating rates

by two orders of magnitude at room temperature [Hite et al., 2012, Daniilidis et al., 2014]. However,

more recent results from Ref. [Sedlacek et al., 2018] suggest that ion milling might in fact make

heating rates worse at cryogenic temperatures. Thus, as we had a fortunately low heating rate to

begin with, and ion milling carries the risk of creating electrical shorts between electrodes, we have

eschewed ion milling thus far; we have these argon ion guns but have not dared to use them4. The

other components shown in Fig. 3.5 include the flow cryostat and helium recovery line, the high

field coils, the camera, and imaging system which we discuss in subsequent sections.

3.3 Cryogenic operation

Cryogenic operation offers a number of advantages for ion trapping. First, heating rates are

typically orders of magnitudes lower [Deslauriers et al., 2006, Chiaverini and Sage, 2014, Brownnutt

et al., 2015, Sedlacek et al., 2018]. Second, condensation of residual gasses (cryopumping) in our

vacuum system significantly reduces the pressure, increasing the ion lifetime. At room temperature,

we have a pressure of ≈ 2×10−10 Torr and a single ion lifetime of about 3 min. Operating at closer

to 15 K, the pressure drops to below 5×10−11 Torr, and our single ion life time is more than 8 hours.

4 Not Chekhov’s argon ion guns. One of the guns is also shorted.
33

Figure 3.4: a) Pillbox disassembled. On the left we have the lid that has the imaging port with

a gold mesh. On the right, we have the trap pedestal and alumina PCB. The alumina PCB has

resistors and capacitors for filtering not shown. The high power MHz and GHz currents come in via

SMA ports that are connected to electrodes 1, 2, and 3 in Fig. 3.2. We have an on-chip platinum

resistance temperature detector (RTD) to measure the temperature close to the trap. The atomic

flux from the magnesium oven comes through the neutral atom port. b) and c) show the assembled

pillbox attached to the base flange. b) Our connections to the dc electrodes come in via micro-d

connectors. Also shown are the rf feedthrough and the copper braid that provides the thermal link

to the flow cryostat. c) We indicate the stainless steel SMA cables used for MHz and GHz currents

applied to electrodes 1, 2, and 3.
34

Figure 3.5: Vacuum system on optical table. The trap, within its vacuum system, is at the center

of this image. Its top flange has four argon ion guns for in situ ion milling. We have large high-field

coils for generating our quantization magnetic field, and smaller compensation coils to compensate

for stray fields as well as fine adjustment of the magnetic field. An additional compensation coil

underneath the vacuum system is not shown. We point out the imaging system, camera, and

photomultiplier tube (PMT) used to collect and count photons from the ions. Lastly, we also show

the flow cryostat and helium recovery line used for cryogenic operation.

Our two ion lifetime becomes ∼ 1 hour, making the entangling gate experiments in later chapters

feasible.

The key components of our cryogenic operation are shown in Fig. 3.6 including the Janis

ST-400 flow cryostat. When running experiments, we use liquid helium to cool the trap to ≈ 15 K.

Helium flows to the trap via a transfer line from a connected dewar (≈ 80 L) at a pressure of about

10 psi. This pressure is reached naturally by inserting the transfer line into the dewar. A secondary

valve ensures that excess pressure is released into the recovery line rather than into the lab. At this

temperature, we consume ≈ 1 L per hour. When not running experiments, we usually warm up

to about 50 K to reduce the helium consumption to ≈ 0.3 L per hour. We usually consume about
35

Figure 3.6: Cryogenic operation. a) The flow cryostat makes thermal contact with the copper

pillbox via a thermal link. b) Helium dewar connected to the trap. Liquid helium flows via a transfer

line to the trap. Warm helium gas goes into the yellow recovery line into the larger NIST helium

recovery system. We have a flow regulator for the helium gas flow, controlling the temperature of

the trap. A temperature controller enables monitoring of the temperature both at the flow cryostat

and at the trap.

100-150 L per week5. Approximately 90% of this helium is recovered at NIST6. Cycling between

5 Liquid helium costs more than a graduate student.

6

In some sense, we have a very large closed-cycle system.
36

50 K and 15 K, our trap position is maintained within a few μm and our laser beams typically need

minimal realignment.

When not running experiments, for example during a global pandemic7, we switch over to

liquid nitrogen instead. We maintain our trap temperature at about 80 K, consuming approximately

3 L of liquid nitrogen per day. We use liquid nitrogen to minimize costs. We refrain from warming up

to room temperature to avoid thermal expansions or contractions of various in-vacuum components.

3.4 High-field coils

For our experiments, we require a fairly large static magnetic field of 212.8 G (see Sec. 4.3)

at the trap. We use foil-wound copper coils from Stangenes Industries as shown in Fig. 3.7. We

supply about ≈ 60 A, with a voltage drop of about 15 V across the two coils connected in series.

At this current, these coils require water cooling to stay close to room temperature. We have an

interlock that shuts off our power supply if either the water flow is turned off, or if a temperature

sensor on the coils reaches 50◦C. We measured the magnetic field along the axis of the coils and it is

fairly uniform over a region of about 2 mm. Based on Ref. [Merkel et al., 2019], we actively stabilize

the current in our trap and also feedforward on the 60 Hz noise in the power supply8. With this

magnetic field stabilization, we are able to extend our coherence time (T

∗

2

) on our field sensitive

“stretch” qubit from ∼ 100 μs to ∼ ms.

3.4.1 Compensation coils

Aside from the high field coils, we have smaller compensation coils as shown in Fig. 3.5. These

coils help to compensate for stray fields at the ion, as well as offer fine adjustment the magnetic field

and thus the beam polarization at the ion (see Sec. 4.2.1.1). These coils have axes orthogonal to

the high-field coils. We typically apply a current of 0-2 A to each of the coils, generating magnetic

fields of 0-2 G at the ion. At these smaller currents, we have neither current stabilization nor water

7 Our trap has stayed cold thanks to heroic efforts from ACW.

8 Mainly done by SCB and DTCA.
37

Figure 3.7: Coils to generate magnetic field of 212.8 G. These coils were designed by Stangenes

Industries. We indicate where the water cooling and power supply are connected. The height of

20.8 cm refers to the coil diameter.

cooling.

3.5 Laser system

As much as we would like a completely laser-free experiment, we need lasers for photoionization,

state preparation (including repumping), and detection. Additionally, we have a pair of Raman

beams for sideband cooling. Here, we discuss key details of our laser system. For the rest of this

section, we use the following nomenclature9for the different laser beams:

(1) Blue doppler (BD) - The laser beam we use for detection close to resonance. We also use it

for the final stage of Doppler cooling (≈ 10 MHz red detuned).

(2) Blue doppler detuned (BDD) - Laser beam that is ≈ 370 MHz red detuned from resonance

for initialization of the state and preliminary cooling.

9 Largely historical.
38

(3) Red doppler (RD) - Laser beam used for repumping (see Fig. 4.1). This beam is ≈ 1302 MHz

red detuned from resonance (but resonant with a different transition, see Fig. 4.1).

(4) Raman beams - The two beams that we use to drive stimulated Raman transitions with σ

+

and π beam polarizations respectively. The two-photon transitions are driven through a

virtual state ∼ 50 GHz blue detuned from the P3/2transition.

On resonance refers to the 2S1/2 ↔2 P3/2transition in Fig. 4.1.

3.5.1 Beam orientations

Figure. 3.8 shows the orientation of the beams relative to the trap axis (see Fig. 3.2) and the

quantization axis set by the static magnetic field B0. Our Doppler cooling beams are tilted 22.5◦

from the y-axis, ensuring cooling of all three motional modes for a single ion. We have two Raman

beams, whose net momentum ∆~k also has components along all three modes.

3.5.2 BD, BDD, and RD beamlines

We ultimately need UV light at 280 nm for our transitions. However, to generate the BD,

BDD, and RD laser beams, we first start in the infrared. For the experiments described in this

thesis, the infrared source was first a fiber-laser system10 that was later replaced by a VECSEL

system (see Sec. 3.5.3). From the 1118 nm (≈ 700 mW) light, we double to the green (≈ 559 nm)

using a periodically poled lithium niobate (PPLN) waveguide11 for second harmonic generation

(SHG). At this point, we have approximately 200 mW of power in the green, ≈ 5 mW of which we

use to lock the laser12 to an iodine absorption line as an absolute frequency reference. The rest of

the green light is sent to a cavity with a BBO (β-BaB2O2) crystal for SHG of UV light. This cavity

was designed by Christian Ospelkaus based on Ref. [Wilson et al., 2011]. We obtain ≈ 5 mW of UV

light out of the doubler.

10 Menlo Systems orange one.

11 NTT electronics.

12 Mainly done by SCB.
39

x

y

PI

Raman

π

Raman

σ+

BD, BDD, RD

σ+

22.5°

B0

Figure 3.8: Beam orientations. Our BD, BDD, and RD beams (see main text) are all σ

+ polarized,

reaching the trap from the same fiber. They are aligned 22.5◦to the y-axis in Fig. 3.2, enabling

cooling of all three modes of a single ion. We have additional Raman beams with π and σ

+

polarizations that we use for resolved sideband cooling. The static magnetic field B0 from the

high-field coils (Sec. 3.4) defines the quantization axis. We have a photoionization (PI) beam that

we use for ionizing neutral magnesium.

This UV light is now used to generate the BD, BDD, and RD beams as shown in Fig. 3.9.

This figure is a simplified schematic that does not include all the optical components required. All

three laser beams are coupled into the same fiber that takes the light closer to the trap. We use a

solarization-resistant, hydrogen-cured fiber as described in Ref. [Colombe et al., 2014]. We typically

get ∼ 10 μW of UV at the trap. We stabilize the intensity of the BD laser beam for detection by

picking off part of the light and using a digital servo [Leibrandt and Heidecker, 2015]. The beam

waist (1/e2intensity) for these lasers is ≈ 14 μm.

3.5.3 VECSEL system for Raman beams

We use a vertical external-cavity surface-emitting laser (VECSEL) system for our Raman

beams as described in Ref. [Burd et al., 2016]. This system was designed and built by Shaun Burd.
40

UV Doubler

BD1

BDD

330

MHz

261

MHz

232

MHz

RD 205

MHz

BD2

trap

+892

MHz

+522

MHz

BD

BDD

RD

-410

MHz

Figure 3.9: Beamlines for blue doppler (BD), blue doppler detuned (BDD), and red doppler (RD)

laser beams. The BD beam goes does double pass through the BD1 acousto optic modulator (AOM)

(2 × 330 MHz shift) and then a single pass at the BD2 (232 MHz shift) AOM. The BDD only goes

through a double pass at the BDD AOM (2 × 261 MHz shift). Lastly, the RD goes through one

double pass (−2 × 205 MHz shift). Both the RD and BDD beams only go through to the fiber when

the BD2 AOM is off. The shifts for each of the three beams are shown in the inset. All three beams

go through the same fiber to the trap.

Starting at 1118 nm, we follow a similar process to double the light twice to obtain UV light at

≈ 280 nm. However, each of the laser beams shown in Fig. 3.8 used for the π and σ

+ is coupled to

separate fibers. Each beam has amplitude stabilization as well.

3.5.4 Photoionization

To generate magnesium ions, we employ a two-photon photoionization process. As described

in Ref. [Madsen et al., 2000], two 285 nm photons can ionize neutral magnesium, the first photon

driving the 3s

2 1S0 ↔ 3s3p1P1 transition, and the second exciting an electron out to the continuum.
41

This light is generated using a Raman-fiber-amplifier-based system13 that first generates 1140 nm

light that is doubled to 570 nm. The green light is then sent to a cavity containing a BBO crystal14

that generates UV light at 285 nm. The UV light is sent to the trap via a fiber, and we obtain

≈ 1 mW of UV light at the trap, which is more than sufficient for loading. We do not lock the laser

frequency, instead monitoring it on a wavemeter, and adjusting it accordingly. In the future, we

plan on switching to a VECSEL-based system and using a home-built cavity doubler15.

3.6 Trapping rf

We need an oscillating potential to trap ions. Our voltage source is an AnaPico APSIN3000

waveform generator set to 68.579 MHz. We stabilize its amplitude following Ref. [Harty, 2013] before

it is amplified and sent to a resonator. We use a λ/2 resonator made from a Heliax cable designed

by Daniel Slichter. This resonator has a quality factor of about 90, enabling amplitudes of ≈ 30 V

at the trap rf electrodes shown in Fig. 3.2. We can also couple in DC voltages to tilt the ion radial

modes, or apply oscillating potentials to perform parametric modulation of the radial motional

frequencies [Burd et al., 2019].

3.7 Ion loading

To load ions, we resistively heat a hollow stainless steel tube filled with neutral magnesium.

The tube has an aperture drilled in the sidewall for a flux of neutral magnesium atoms to escape.

We run about 5.3 A through this wire for one minute, generating a flux of neutral magnesium atoms.

We then pulse the PI on while keeping the oven running, periodically checking the counts on the

PMT to see if we have loaded. We continue this until the desired number of ions is loaded. We

have observed that running the oven continuously for too long a period inhibits loading. We suspect

that when the oven is too warm, residual gases in the vacuum that had been cryopumped start to

evaporate, increasing the pressure at the trap.

13 MPB VRFA-SF series.

14 LAS GmbH WaveTrain

15 Someday.
42

Loading multiple ions can occasionally be challenging. We have found that reducing the

axial confinement and/or the radial confinement can help. We reduce the axial confinement by

multiplying all the voltages on the dc electrodes by a factor (less than 1, excluding shim voltages to

compensate for stray fields). We reduce the radial confinement by changing the frequency of the

trap rf voltage 16 , thus moving it off resonance and reducing the voltage amplitude at the trap. We

typically move it by ∼ 100 kHz until ions have been loaded and then change it back to its original

frequency.

3.8 Clock distribution

Many devices in our apparatus require a reference signal. We have:

(1) APSIN3000 - Trap rf generator.

(2) PDQ17 - Arbitrary wave form generator based on Ref. [Bowler et al., 2013]. We use these

PDQs to:

(a) Generate rf currents.

(b) Generate baseband voltages for IQ modulators for pulse shaping microwaves.

(3) Microwave DDS’ - AD9914 direct digital synthesizers (DDS’) used for microwave sources.

(4) Crate DDS’ - DDS’ in experimental control “crate” that control acousto-optic modulator

(AOM) frequencies, for example.

(5) Field-programmable gate array (FPGA) that controls our experimental sequences.

It would be desirable if the relative phase between all of these reference signals were stable, as drifts

in that relative phase would change the relative phase of the fields at the ions and have physical

effects.

16 This technique can also be used to get rid of excess ions, a procedure we call the Knaack maneuver.

17 Pretty darn quick.
43

We stabilize these phases using a 10 MHz signal either from a hydrogen maser18 or a rubidium

frequency standard19 . A schematic for our phase stabilization is shown in Fig. 3.10. We send

the 10 MHz signal to our APSIN3000 that generates our trap rf , as well as to a 100 MHz crystal

oscillator20 . This 100 MHz signal is now sent to the various PDQs, as well as a 24× frequency

multiplier21 . The resulting 2.4 GHz signal is now used to clock both the crate and microwave DDS’.

The internal clock of one of the crate DDS’ at 100 MHz is used as a reference to our experimental

control FPGA. This DDS is also used to synchronize the internal clocks of the microwave DDS’(see

Sec. 3.9.1).

10 MHz

APSIN3000

68.579 MHz

crystal oscillator

100 MHz

rf PDQ

5 MHz

microwave

 PDQs

x24

2.4 GHz

microwave

DDS’

crate DDS’

÷ 24

100 MHz

FPGA

÷ 384

6.25 MHz

Figure 3.10: Synchronization of devices. Most of the essential devices in the apparatus are

synchronised to a 10 MHz signal, see main text for details.

18 One of the many benefits of working at NIST.

19 SRS FS 725.

20 Wenzel 100 MHz-SC.

21 Wenzel IFM-5R-100-24-13-13.
44

3.9 Microwave generation

We now describe the generation of the various microwave currents for our experiment. We

first explain how the different DDS’ are synchronized and then each of the chains22 for the carrier

and sideband drives.

3.9.1 DDS Synchronization

Our microwave sources are AD9914 DDS’. Clocked at 2.4 GHz, we use them to generate

microwave tones from 1.3-2.2 GHz for our various transitions. We also use it to generate MHz

tones for driving the ion motion or squeezing as shown in Ref. [Burd et al., 2019]. As we use more

than one DDS, it is important to sychronize their internal clocks (at 100 MHz) to both each other

and the experiment FPGA so all phases are completely deterministic. At every power cycle, the

internal clocks of these DDS’s are randomly in one of 16 different phases. We synchronize these

DDS’ following the AN-1254 application note, where we use the “SYNC OUT” signal at 6.25 MHz23

from the crate DDS that also references the FPGA (see Fig. 3.10). Figure 3.11 shows the internal

clocks of these DDS’s before and after the synchronization.

3.9.2 Microwave carrier drive

The microwave components used to generate fields that drive spin-flip transitions are shown

in Fig. 3.12. We need to generate tones at 1.3-2.2 GHz; this frequency range is within the second

Nyquist zone (using the first Nyquist image) that has a steep power dependence [Analog Devices,

2009]. To minimize the power variation across frequencies, we use a frequency doubler to operate

within the first Nyquist zone (using fundamental frequency). Figure 3.12 shows the chain for

electrode 1. We have additional switches not shown that let us apply the tone from this chain to

electrodes 2 and 3 as well. For electrode 2, we have an additional amplifier that lets us drive faster

spin-flip transitions. In contrast, we have an attenuator for electrode 3. This configuration enables

22 Much of this was put together by DTCA and DHS.

23 2.4 GHz/384
45

Figure 3.11: DDS synchronization. We synchronize the four DDS’ we use for generating our

microwave tones to a crate DDS. We monitor their internal clocks at 100 MHz to check their

synchronization. a) Their internal clocks are not synchronized and are at one of 16 random phases

after a power cycle. b) Their internal clocks are now synchronized.

a wide range of amplitudes to be used within an experimental sequence. We also have additional

chains with IQ modulators for electrode 1 that enable pulse shaping of carrier fields used in later

chapters.

3.9.3 Microwave sideband drive

The chain for the microwave sideband drive for a single electrode is shown in Fig. 3.13. In

contrast to the carrier drive, we only need to generate frequencies within a narrow range of about 10

MHz either side of the “clock” qubit frequency at 1.69 GHz. Thus, we omit the frequency doubler

and just operate within the second Nyquist zone (using the first image). We have an IQ modulator
46

Carrier

DDS

ZRL-1150LN

x2

FK-3000+ UTE

CT-2006-CT

ZASWA2-50DR+

UTE

CT-2348-0

Figure 3.12: Microwave chain for carrier drive, generating microwave fields that drive spin flips at

the ion. The unlabeled components are bandpass filters that remove the Nyquist image as well as

harmonics from the amplifier and doubler.

that enables pulse shaping, and for adjusting the phase and amplitude of the microwave tone for

each electrode. This functionality is essential for the nulling described in Sec. 3.1. The sideband

transitions require much more power to drive than the spin flip transitions, hence the higher power

amplifier used in this chain.

RSB

DDS

ZRL-2400-LN+ Nova

0165IAS

I

Q

PDQ

PDQ

Digital

attenuator

Nova

0165IAS

ZASWA2-50DR+

ZHL-30W

-252-S+

UTE

CT-2348-0

Figure 3.13: Microwave chain for red sideband drive for a single electrode, generating microwave

gradients that drive sideband transitions. The unlabeled components are bandpass filters that remove

the Nyquist image as well as a narrowband cavity filter after the higher power ZHL-30W-252-S+

amplifier. We have a power splitter (ZC4PD-18-S+) for the chain for the two other electrodes (not

shown).

3.9.4 Combining microwave drives

We combine the carrier and sideband drives for each electrode as shown in Fig. 3.14. We use

a combination of 3 dB hybrids and directional couplers to combine the different microwave currents

for a single electrode.
47

3 dB

10 dB 20 dB

Carrier

interlock

RSB

BSB

monitor

trap 

electrode

diplexer

rf drive

Figure 3.14: Combining carrier and sideband microwave drives. We first use a 3 dB hybrid, then 10

and 20 dB directional couplers. The unused port of the 3 dB hybrid is used is used to monitor the

output power and interlock the power amplifiers if the average power delivered to the trap becomes

too large.

rf

PDQ

SLP-5+

ZASWA2-50DR+

SLP-5+ ZABP 16+ coupler

trap 

electrode

microwave

drive

ZHL-5W-1+ diplexer

Figure 3.15: Chain to generate high power MHz currents for a single electrode. The coupler enables

monitoring of the signals on the scope. The diplexer provides a resonant step up of about 3 in the

current for a 5 MHz tone.

3.10 RF generation and delivery

The components required to generate higher power MHz currents for one electrode are shown

in Fig. 3.15. We modify one of the PDQs described in Ref. [Bowler et al., 2013] to generate the

MHz tones for the radiofrequency gradient. We use a PDQ instead of a DDS as the PDQ very easily

allows incorporation of pulse shaping (we need the IQ modulators to do this for the microwave

sideband drive). Right before the trap, we use a resonant diplexer24 that allows a factor of 3

enhancement in the current at the trap electrodes for a 5 MHz tone. Before the diplexer, we have

24 Designed by DHS.
48

about 2W at 5 MHz for each electrode25.

Figure 3.16: Part of microwave and rf setup. a) We show the microwave PDQs, IQ modulators,

switches, amplifiers, and filters from Fig. 3.13. b) The rf PDQ and high power amplifiers26in

Fig. 3.15.

3.11 Imaging

We use custom designed “Slichter optics” for our imaging system, and are able to obtain a

magnification of ≈ 44. We are able to collect about 0.5% of the photons on our PMT27 resulting

in about 50 counts per 200 μs detection for a single ion. An image of two ions on the camera28 is

shown in Fig. 3.17.

3.12 Experimental control

If the surface electrode trap is the heart, the FPGA in our control crate29 is very much the

brain. We use a Xilinx Kintex-7 KC705 evaluation board, that is connected to 40 40 digital I/O

25 HMK has led efforts to increase this to about 5W.

26 Blossom, Bubbles, and Buttercup for electrodes 1, 2, and 3 respectively. Originally to be named after Daenerys’

dragons but recent events forced a reevaluation.

27 Hamamatsu R7600P-203. 34% quantum efficiency at 280 nm.

28 Andor DV887ECS-UVB.

29 Hardware designed by DHS.
49

Figure 3.17: Image of two ions on the camera. The ions are approximately 3 μm apart.

(DIO) channels and 12 DDS’ via its FPGA mezzanine card (FMC) connectors. We use the DIO

channels to trigger rf switches that control our AOMs, and thus the lasers at the ions. We also use

rf switches to control which microwave/rf current is pulsed on which electrode. The DIO channels

are also used as inputs, such as to record PMT counts. The DDS’ in the crate are used primarily

to generate the frequencies for the AOMs in our laser beamlines. In addition, we have 4 serial

peripheral interface (SPI) channels, which we use for example to program our microwave DDS’.

To program the desired pulse sequences to the FPGA, we use the Advanced Real-Time

Infrastructure for Quantum physics (ARTIQ) control system [Bourdeauducq et al., 2016]. Our

experiments are Python files that are compiled and sent to the core device, a soft-core CPU on

the FPGA, which executes the code. Note that only a subset of Python can be executed on the

core device. One big advantage of using Python, aside from its accessibility and readability, is its

modularity that easily allows many experiments to share the same code. An important example of

this modularity is the drift tracking code used in Sec. 4.7.

For timing, the FPGA has an internal clock at 125 MHz that we bypass using the 100 MHz

clock from one of the DDS’ (see Fig. 3.10). The timing resolution is 1/8th the clock period which in

our case is 1.25 ns. It is important to be aware of this subdivision, as when using other devices such

as the rf PDQs or DDS’ whose timing resolution is 10 ns (100 MHz clock), we need to ensure that

experiments start at the same subdivision for every repetition of the experiment so other devices

have the same phase every time.
Every breath you take and every move you make,

Every bond you break, every step you take,

I’ll be watching you

The Police

4

Qubit control

In this chapter we introduce the main protagonist1 of this thesis, 25Mg+. We describe its

energy levels at our operating static magnetic field of 212.8 G that produces a field-independent

“clock” qubit, its microwave transitions, and our qubit preparation and readout sequences. We

discuss the various calibrations that we perform, including compensating for stray fields that push

the ion off the rf null. We also describe how we couple the ion to its motion, using microwave

gradients as well as laser-based interactions, for ground-state cooling. We explain how we track both

the ions’ motional and qubit frequencies within an experimental sequence; the motional frequency

tracking especially is an essential component of all remaining experiments described in this thesis.

Finally, we have a brief discussion of our even briefer foray into ion transport.

4.1 25Mg+ qubit at intermediate magnetic field

The hyperfine Hamiltonian for 25Mg+ in its electronic ground state is

1 Sometimes antagonist.

50
51

“stretch”

|3, −1

|3, −3

|3, 3

|3, 2

|3, 1

|3, 0

|3, −2

|2, 2

|2, 1

|2, 0

|2, −1

|2, −2

2

S1/2

2

P3/2

|4, 4

cooling/detection

280 nm, σ+

F = 2

F = 3

“clock”

“bright”

“dark”

1.686 GHz

|↑

|↓

B0 = 212.8 G

1.326 GHz

repump

|mJ = 3/2, mI = 3/2

Figure 4.1: Energy level diagram for 25Mg+ at a static magnetic field of 212.8 G. We use two

sets of qubit states in this thesis. First, the field-sensitive “stretch” qubit transition corresponds

to |F = 3, mF = 3i ↔ |2, 2i. Second the field-insensitive “clock” qubit transition is |3, 1i ↔ |2, 1i.

The grey dotted lines indicate additional transitions for shelving to the |2, −1i state for readout.

We have an additional repump laser that couples the |2, 2i state to the 2P3/2|mJ = 3/2, mI = 3/2i

state. Note that |mJ = 3/2, mI = 3/2i is an approximate designation; the hyperfine interaction in

the P state is much weaker so mJ and mI are “better” quantum numbers. This repump sends

population in the |2, 2i state primarily to the |3, 2i state.

Hˆ

hfs = hAI · J, (4.1)

where the hyperfine constant is A = −596.254376 MHz [Itano and Wineland, 1981] and h is Planck’s

constant. J and I represent the electron and nuclear angular momentum operators respectively.

The total angular momentum of the atom is
52

F = I + J. (4.2)

Using this result in Eq. 4.1,

Hˆ

hfs =

hA

2

(F

2 − I2 − J2

), (4.3)

which has eigenstates |IJFmF i, where mF is the projection of F along a quantization axis [Foot,

2007] defined by a static magnetic field B. For small values of this magnetic field, all states with the

same F are degenerate and their energies are

Ehfs '

hA

2

(F(F + 1) − I(I + 1) − J(J + 1)). (4.4)

For 25Mg+, I = 5/2 and J = 1/2 in the ground state. The hyperfine splitting between F = 3 and

F = 2 at zero magnetic field is ∆Ehfs/h = 3A = −1.78876 GHz. Following Ref. [Woodgate, 1970],

the Hamiltonian for an atom with both hyperfine and Zeeman interactions is

Hˆ = hAI · J − (μJ + μI ) · B, (4.5)

The magnetic moments are

μJ = −gjμBJ,

μI = g

0

IμBI.

(4.6)

Its nuclear g factor to electronic g factor ratio is g

0

I = gI/gJ = 9.299484 × 10−5

. We operate at

an intermediate magnetic field of 212.8 G (21.28 mT), which breaks the degeneracy between the F

states, where none of the F, I, or J quantum numbers are good (mF = mI + mJ still good). We use

the Breit-Rabi [Breit and Rabi, 1931] formula2to calculate the energy shifts to our |F, mF i states

2 Applies only to J = 1/2.
53

∆E(F, mF ) = −

∆Ehfs

2(2I + 1) − g

0

IμBBmF ±

∆Ehfs

2

r

1 +

4mF xB

2I + 1

+ x

2B2, (4.7)

where x =

(gj+g

0

I

)μB

∆Ehfs

. In Eq. 4.7, + corresponds to F = I + 1/2 and − to F = I − 1/2. We use this

result to calculate the energy levels of the different states shown in Table. 4.1. The magnetic field of

212.8 G was chosen such that the frequency of the |3, 1i ↔ |2, 1i transition is first-order insensitive

to a change in the quantization axis magnetic field.

Transition (|F, mF i ↔ |F, mF i) Frequency f (GHz) df /dB (MHz/G) d

2

f /dB2(kHz/G2)

|3, 3i ↔ |2, 2i 1.326456 -1.973 1.120

|3, 2i ↔ |2, 2i 1.460516 -1.144 2.240

|3, 1i ↔ |2, 2i 1.573543 -0.572 2.283

|3, 2i ↔ |2, 1i 1.573432 -0.572 2.283

|3, 1i ↔ |2, 1i 1.686459 0 2.327

|3, 0i ↔ |2, 1i 1.786044 -0.443 2.100

|3, 0i ↔ |2, −1i 1.975445 1.252 1.570

Table 4.1: Microwave transition frequencies, first and second order magnetic field sensitivities for

25Mg+ at 212.8 G. Our field-sensitive “stretch” qubit (|3, 3i ↔ |2, 2i) and field-insensitive “clock”

qubit (|3, 1i ↔ |2, 1i) are in bold.

4.2 Field-sensitive “stretch” qubit

We first discuss preparation and readout of our “stretch” qubit on the field-dependent

|F = 3, mF = 3i ↔ |F = 2, mF = 2i transition. We call this qubit the “stretch” state qubit as it

involves the |F = 3, mF = 3i state, which has the highest mI and mj values. We also describe

single-qubit rotations using microwave fields. This qubit is used primarily in the experiments

described in this thesis as we require a field-sensitive qubit.

4.2.1 State preparation

We prepare the |3, 3i state via optical pumping using the BDD and BD laser described in

Sec. 3.5. Using σ

+ polarized light as shown in Fig. 4.1, we should end up in the |3, 3i state starting
54

from any state in the 2S1/2 manifold. If this beam is not completely σ

+ polarized or aligned along

the quantization axis, we will get imperfect state preparation and leftover population primarily in

the |2, 2i state. After first adjusting the polarization to maximize the counts as a coarse alignment,

we use two additional compensation coils providing fields nominally orthgonal to the quantization

axis for finer adjustment.

4.2.1.1 Calibrating compensation fields for beam polarization

We calibrate the currents in the compensation coils (see Sec. 3.4.1) using the following

experiment.

(1) Prepare ion nominally in the |3, 3i state.

(2) Shelve the |3, 3i state to the “dark” |2, −1i state with microwave pulses. Leftover population

in the |2, 2i state is now in the |3, 3i.

(3) Turn on a microwave field on the |3, 3i ↔ |2, 2i transition for variable duration t.

(4) Shelve the |2, 2i state to the |2, 1i state.

(5) Detect population in the |3, 3i state.

Thus, by looking at the contrast of the Rabi flopping, we have a measure of the population

left in the |2, 2i state after optical pumping. We adjust the currents in the compensation coils to

minimize this contrast. An example scan is shown in Fig. 4.2.

4.2.1.2 Additional state-preparation sequence

Additionally, we perform the following sequence to transfer any residual population in the

|3, 2i or |2, 2i state to the |3, 3i state.

(1) |3, 3i → |2, 2i (also transfers population from |2, 2i → |3, 3i).

(2) |2, 2i → |3, 2i (population originally in |3, 3i now in |3, 2i).
55

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Duration ( s)

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Counts

Figure 4.2: Calibration experiment for currents in compensation coils. If the compensation coils

are not calibrated, we will have imperfect state preparation and leftover population in the |2, 2i

state. We pulse a microwave field on resonance with the |3, 3i ↔ |2, 2i transition to measure this

leftover population. We adjust the currents in the compensation coils to minimize the amplitude of

this Rabi flopping.

(3) Pulse repump laser and then drive |3, 3i → |2, 2i. We repeat this step 5 times. At this

point, the population should be in the |3, 2i state.

(4) |3, 2i → |2, 2i.

(5) |2, 2i → |3, 3i.

The population transfers are performed with microwave π pulses.

4.2.1.3 Repump calibration

Our repump laser sends population in the |2, 2i state primarily to the |3, 2i state, and

secondarily to the |3, 3i state (see Fig. 4.1). This repump is used in the state preparation sequence

described in the previous section as well as in ground-state cooling. To calibrate the pulse duration

for the repump, we prepare the |2, 2i state and measure how much of the population is removed

from this state as a function of the pulse duration. Experimental data is shown in Fig. 4.3.
56

0 2 4 6 8 10

Repump duration ( s)

10

15

20

25

30

35

40

45

Counts

Figure 4.3: Calibration of repump duration. Starting in the |2, 2i state which is dark, we pulse the

repump laser for a variable duration and subsequently transfer the population in the |3, 2i state to

the “bright” |3, 3i state and then detect. After 5 μs, most of the state in the |2, 2i state is pumped

out.

4.2.2 State readout

To read out the state, we first transfer population from the |2, 2i state to the |2, −1i state via

a sequence of microwave π pulses. We then detect the fluorescence of the |3, 3i state by driving a

cycling transition to the 2P3/2 manifold as shown in Fig. 4.1. The microwave shelving sequence is

(1) |2, 2i → |3, 1i

(2) |3, 1i → |2, 1i

(3) |2, 1i → |3, 0i

(4) |3, 0i → |2, −1i

This sequence is shown in Fig. 4.1. This gives us an increased separation between our “bright” |3, 3i

state and the “dark” |2, −1i which is ≈ 300 MHz further away from the |2, 2i state. Aside from the

detuning, this state also requires more scattered photons to “repump” into the bright state [Langer,
57

2006]. We apply our detection light for 200 μs. The count histograms for two ions are shown in

Fig. 4.4. We typically see about 56 counts (including background counts of about 5) for the one ion

bright state and and about 107 counts for the two ion bright states. A more detailed discussion of

state preparation and readout errors can be found in Sec. 8.3.2.

For most of the data in this thesis, we use Poissonians to model the counts for the zero ion,

one ion, and two ion bright states. This model is then used to extract ion populations from a given

set of counts using maximum likelihood estimation3.

0 25 50 75 100 125 150

Counts

0

500

1000

1500

2000

2500

3000

3500

Frequency

Dark

Pi/2 pulse

Bright

Figure 4.4: Histogram of two ion photon counts. We measure the photon counts with ions in the

dark state (blue), after a π/2 pulse applied to the “stretch” qubit (orange), and in the bright state

(green). There are about 19,000 measurements per histogram.

4.2.3 Rabi spectroscopy and Rabi flopping

We perform Rabi spectroscopy by pulsing a microwave field for about 60 μs with a variable

detuning from the transition frequency at 1.326 GHz as shown in Fig. 4.5. For these data we apply

a current to electrode 3, which generates a lower Rabi frequency of about ≈ 4 kHz. This transition

frequency has an approximate linear dependence with the applied magnetic field. We use the

3 Primarily done by DHS.
58

difference frequency between the transition we measure and our calculated value as a measure of

our magnetic field4. Typically, we do not adjust the current to our magnetic field coils unless this

difference is larger than 50 kHz. We instead use this transition frequency to track magnetic fields

and adjust our microwave frequencies as discussed in more detail in Sec. 4.8.

40 20 0 20 40

Detuning (kHz)

0.0

0.2

0.4

0.6

0.8

1.0

Population

Figure 4.5: Rabi spectroscopy of “stretch” qubit. Starting in the |3, 3i state, we pulse a microwave

field with a variable detuning for about 60 μs from the |3, 3i ↔ |2, 2i transition and measure the

population in the |3, 3i state.

We also show Rabi flopping on this transition in Fig. 4.6. Here, instead of applying the

current to electrode 3, we apply it to electrode 2 which can produce higher Rabi frequencies (due to

an additional amplifier in the chain) of about 370 kHz. We use this electrode for fast population

transfer during state preparation and readout.

4.3 Field-insensitive “clock” qubit

Our magnetic field produces a field-insensitive “clock” qubit using the |3, 1i and |2, 1i states.

While we do not use this qubit for the new spin-motion coupling technique and entangling gates

that we discuss in Chapters 5 and 6, which require a magnetic field sensitive qubit, we do use it for

4 Note that there will be an offset due to the ac Zeeman shift from an oscillating magnetic field at ωrf.
59

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Duration ( s)

0.0

0.2

0.4

0.6

0.8

1.0

Population

Figure 4.6: Rabi flopping on the “stretch” qubit. Starting in the |3, 3i state, we pulse a microwave

field driving the |3, 3i ↔ |2, 2i transition and measure the population in the |3, 3i state.

our microwave sidebands and for some other measurements.

4.3.1 Checking clock state

We check that we have set our magnetic field to the correct value by measuring the frequency

of the clock transition as a function of the applied magnetic field. The data is shown in Fig. 4.7. For

each of the points, we change the current in our power supply for the high field coils (see Sec. 3.4).

For these measurements, the current is varied from 58.89–61.05 A. We then measure the frequency of

the “stretch” qubit using Rabi spectroscopy. We use the difference frequency between its measured

value and its calculated value at 212.8 G to determine the magnetic field strength. We then perform

Rabi spectroscopy on the “clock” qubit to measure its frequency. The minimum is not at 0 due to

an ac Zeeman shift from an oscillating magnetic field at ωrf. This field causes an ac Zeeman shift

of ≈ 2.5 kHz on the qubit frequency. The magnetic field oscillating at ωrf could be more carefully

characterized using the methods outlined in Sec. 5.5.
60

209 210 211 212 213 214 215 216

Calculated magnetic field (G)

5

10

15

20

25

30

35

Clock qubit shift (kHz)

Figure 4.7: Plot of the shift of the “clock” qubit frequency relative to the calculated field-independent

frequency versus the applied magnetic field. We vary the strength of the applied magnetic field by

changing the current in the magnetic field coils. The value of the magnetic field is calculated using

the frequency of the “stretch” qubit. The minimum shift is not 0 due to an additional ac Zeeman

shift from a magnetic field at ωrf.

4.3.2 State preparation and readout

To prepare the |3, 1i state, we first follow the steps in Sec. 4.2.1 to prepare the |3, 3i state.

We then transfer from |3, 3i to |3, 1i using two microwave π pulses that drive the |3, 3i → |2, 2i and

|2, 2i → |3, 1i transitions sequentially.

While state preparation is straightforward, state readout is not. With an arbitrary superposition of |3, 1i and |2, 1i, one needs to be careful transferring populations out of these states. For

example, the |3, 1i ↔ |2, 2i transition frequency is ≈ 100 kHz away from the |3, 2i → |2, 1i transition

frequency. Thus, a π pulse from |3, 1i → |2, 2i can off-resonantly drive population from |2, 1i → |3, 2i.

This can be mitigated by using a Rabi frequency that is small compared to the difference frequency

or by employing pulse-shaping. We use an alternative approach using a composite pulse sequence as

follows

(1) π/2 pulse on |3, 1i → |2, 2i transition.
61

(2) Wait for a duration t.

(3) π/2 pulse on |3, 1i → |2, 2i transition.

This sequence will transfer the population from the |3, 1i state to the |2, 2i state. By adjusting the

delay duration t, the part of the amplitude in the |2, 1i state that was excited by the initial π/2

pulse to the |3, 2i state will be returned to |2, 1i. The delay duration t is calibrated experimentally

by preparing the state in |2, 1i and minimizing the transfer of population to the |3, 2i state. An

example calibration is shown in Fig. 4.8.

6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00

Delay duration ( s)

3

4

5

6

7

8

Counts

Figure 4.8: Calibration of delay duration in readout of “clock” qubit. We prepare the ion in the

|2, 1i state and then perform a Ramsey sequence on the |3, 1i ↔ |2, 2i transition that off-resonantly

drives the |2, 1i ↔ |3, 2i transition. We then measure the population transfer to the |3, 2i state. By

calibrating the delay duration between the two π/2 pulses in the Ramsey sequence, we can transfer

population from |3, 1i → |2, 2i while maintaining the original population of the |2, 1i state.

4.3.3 Rabi flopping

We show Rabi flopping on the |3, 1i ↔ |2, 1i transition in Fig. 4.9.
62

0.0 0.5 1.0 1.5 2.0

Pulse puration ( s)

0.0

0.2

0.4

0.6

0.8

1.0

Population in |3,1>

Figure 4.9: Rabi flopping on “clock” qubit. We start in the |3, 1i state and pulse a microwave field

on resonance with the |3, 1i ↔ |2, 1i for a variable duration. We measure the population in the

|3, 1i state at the end of the sequence.

4.3.4 Qubit coherence

To measure the coherence of the “clock” qubit, we perform a Ramsey experiment. Starting in

the |3, 1i state we use the following sequence.

(1) π/2 pulse on |3, 1i ↔ |2, 1i transition with phase φ = 0.

(2) Wait time t.

(3) π/2 pulse on |3, 1i ↔ |2, 1i transition with variable phase.

As the qubit decoheres, the contrast in this measurement will decrease. Out to wait times of 100 ms,

we see almost no degradation of the qubit contrast on this field-insensitive transition. A plot of the

data is shown in Fig. 4.10.

4.4 Calibrations

In this section we outline the calibrations that are performed regularly:
63

0 1 2 3 4 5 6

/2 phase (radians)

0.0

0.2

0.4

0.6

0.8

1.0

Population

Figure 4.10: Ramsey experiment on “clock” qubit. We start in the |3, 1i state and perform a π/2

pulse on the “clock” transition. We then wait for 100 ms before a second π/2 pulse with a variable

phase. The contrast of this scan is a measure of the qubit coherence. We see almost no loss in

contrast with wait times out to 100 ms.

(1) Compensating for ion micromotion in (y) and out of the trap electrode plane (z).

(2) Compensating for stray electric fields along the trap axis (x).

(3) Beam positions on the ions.

(4) Calibrating single-qubit rotations.

The directions of the different axes follow Fig. 3.2. The ion micromotion compensation is

typically recalibrated every time ions are loaded due to charging effects from the photoionization

beam. The single-qubit rotations are calibrated once a day and the other calibrations less often.

Compensating for micromotion first is important to ensure the beams positions are calibrated

correctly.
64

4.4.1 Micromotion compensation

If there are unwanted stray electric fields at the position of the ion, it will be pushed off the rf

null and experience micromotion: coherent motion driven by the electric field oscillating at the rf

trapping frequency ωrf. We measure the in-plane micromotion using the ion fluorescence and the

out of plane micromotion using magnetic field gradients. We compensate for this micromotion by

changing the potentials on the dc electrodes to move the ion towards the rf null. The axes follow

those in Fig. 3.2.

4.4.1.1 In-plane micromotion

As described in Ref. [Berkeland et al., 1998], micromotion will cause a modulation of the ion

fluorescence on resonance due to a Doppler-shift-induced frequency modulation. Thus, maximizing

the ion fluorescence allows us to minimize the micromotion. We use simulations to generate a set of

voltages5 on the dc electrodes that provide an electric field at the ion in the plane of the trap along

the y direction. We scan the ion fluorescence as a function of this compensating electric field. Data

are shown in Fig. 4.11.

4.4.1.2 Out-of-plane micromotion

Our laser beams are orthogonal to the motion of the ion out of the plane of the trap. Ion

fluorescence is not affected by micromotion in this direction and we need to use another method.

We use the technique described in Ref. [Warring et al., 2013a], using a microwave magnetic field

gradient. If the ion is not at the rf null, the electric field at the trapping rf frequency ωrf will drive

ion motion at that frequency. If we have a microwave field gradient that is detuned from the qubit

frequency ω0 by ωrf, we can drive micromotion sideband transitions that flip the ion spin. The Rabi

frequency of this transition is

5 Thanks DHS!
65

50 100 150 200 250 300 350

Electric field (V/m)

46

48

50

52

54

56

58

60

Counts

Figure 4.11: Calibration of in-plane micromotion. We measure the ion fluorescence as a function

of the electric field along the yˆ direction. When the electric field we apply compensates for stray

electric fields at the ion and minimizes the in-plane micromotion, we see maximum fluorescence.

Ωmm =

B0μB

4~

rmm, (4.8)

where B0is the magnitude of the magnetic field gradient (π-component) we apply along the direction

of the micromotion, μB is the Bohr magneton6and rmm is the amplitude of the ion micromotion

at ωrf. Thus, by adjusting the shim voltages to (ideally) null Ωmm, we can compensate the ion

position out of the plane of the trap. We note that the exact detuning of the field gradient from

ω0 is shifted slightly from ωrf due to the ac Zeeman shift from the detuned microwave field (see

Sec. 4.3.1). Rabi oscillations at the micromotion sideband are shown in Fig. 4.12 for three different

values of the shim voltage. While in principle we could use a microwave gradient detuned from

any microwave transition, we use the field-insensitive “clock” qubit for its longer coherence time,

which allows sensing of smaller Ωmm and thus better micromotion compensation. Additionally, a

field-sensitive transition would also be sensitive to an oscillating magnetic field at ωrf that can also

cause spin-flip transitions (see Sec. 5.2).

6 The astute reader might wonder where the matrix element went. It has been factored in, but for the “clock”
66

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Pulse duration (ms)

10

20

30

40

50

60

Counts

0 V/m (Calibrated)

50 V/m

100 V/m

Figure 4.12: Calibration of out-of-plane micromotion with microwave gradient. We pulse a

microwave gradient detuned from the “clock” qubit frequency ω0 by the trap rf frequency ωrf. We

perform this experiment for three different values of the compensation voltages applied: at the

calibrated value (blue), 50 V/m off (orange) and 100 V/m off the calibrated value (green points).

When the ion is displaced from the rf null, the Rabi frequency of this transition increases.

4.4.2 x shim

Stray electric fields along the trap axis will push the ion off the center of the trap but not

cause micromotion since the rf electric field is zero along the rf null line, parallel to the trap axis.

However, it is still desirable to have the ion at the nominal trap center. Our shim voltages are

calculated at this point, and it should minimize anharmonicities, improving the ion lifetime. To

compensate for stray fields along this direction, we measure the ion position as a function of the

axial confinement. For a harmonic oscillator,

F = −kx, (4.9)

where here, F is the force from a stray field, k is the spring constant, and x is the displacement of

the ion from its equilibrium position without any stray fields. By changing the axial confinement

transition, that corresponds to ≈ 1.001μB [Warring et al., 2013a].
67

strength, we change the value of the spring constant. The displacement x = −F/k. Thus, if the

ion position x is constant for different values of k, we have F = 0. We measure the ion position by

looking at the peak of its fluorescence versus beam position as shown as shown in Sec. 4.4.3. We

change the axial confinement by changing all of our dc voltages (without compensation) by the

same scale factor. The two scaling factors for the data in Fig. 4.13 are 0.36 and 0.76.

300 280 260 240 220 200 180 160

Electric field (V/m)

4

2

0

2

Difference in position at 

lower and higher confinements

(

m

)

Figure 4.13: Calibration of compensation voltages along trap axis. We measure the ion position

at two values of the axial confinement. We plot the difference in position at those two values as a

function of the electric field applied along the trap axis. When the ion is well-compensated, the

difference will be 0.

4.4.3 Detection beam positions

Our beams for detection, Doppler cooling, repumping, and optical pumping follow the same

path to the ion after the same fiber. Thus, aligning one of the beams on the ion aligns all of them.

The horizontal alignment is controlled using a motorized actuator whose position we scan. Using

the ion, we optimize the beam position by maximizing its fluorescence as shown in Fig. 4.14.

The vertical (out-of-plane) alignment is controlled via a mirror mount with a piezoelectric

adjuster. Changing the applied voltage changes the vertical position of the beam. To distinguish

between ion fluorescence and scatter from the surface, we measure the counts with the ion in its
68

4 2 0 2 4 6

Horizontal position of laser ( m)

40

45

50

55

60

Counts

Figure 4.14: Calibration of in-plane position of detection beam. We measure the ion fluorescence

as a function of the beam position which is varied using a motorized actuator. We use a quadratic

fit to determine the position at the maximum.

bright state and in its dark state. We pick the point with the highest difference between bright and

dark state counts as shown in Fig. 4.15.

4.4.4 Single qubit rotations

We use our microwave fields to perform single-qubit rotations. For example, we perform π/2

pulses for Ramsey sequences, two-qubit gates, and π pulses for spin-echos, state initialization, and

shelving sequence for state readout. To perform these single-qubit rotations accurately, we need to

calibrate the duration of the pulses. A coarse calibration is done using Rabi oscillations as shown in

Fig. 4.6. However, the π-pulse duration extracted from Rabi oscillations does not take into account

the rise and fall times of our microwave switches that also induce some rotation.

To more precisely calibrate the duration of π or π/2 pulses, we perform a sequence of pulses

with a short delay in between. As shown in Fig. 4.16, we measure the qubit population as we

perform a sequence of 200 π pulses in a row with a 5 μs gap between pulses on the “stretch” qubit.

This gap also mitigates duty cycle effects to a certain extent. Starting in |↑i, we perform an even
69

20 25 30 35 40 45 50

Actuator voltage (V)

0

10

20

30

40

50

Counts

Difference in bright and dark counts

Dark counts

Figure 4.15: Calibration of the out-of-plane position of the detection beam. We measure the ion

fluorescence as a function of the beam position which is varied by changing the actuator voltage.

We measure both the bright and dark state counts and plot the difference as well as the dark state

counts as a function of voltage. We fit a quadratic to the difference and set the voltage to the

maximum.

number of π pulses so the ion should ideally be returned to its initial dark population. If the pulse

duration is too long or too short, we get an over or under rotation that we measure with longer

sequences. Based on this measurement, we can apply a correction t to the π pulse duration

t =

tπ

No. of pulses for π rotation error

. (4.10)

For the data in Fig. 4.16, about 118 pulses resulted in a π rotation error, or a complete transfer

from dark to bright. From a π pulse duration tπ of about 0.673 μs, this corresponds to a timing

error of about 6 ns. As we cannot distinguish between an under or over rotation, we experimentally

try both adding and subtracting this correction and choose the time that results in a reduction

of the error. The initial coarse calibration of this π pulse duration was done via Rabi flopping;

we attribute the 6 ns discrepancy to the rise and fall times of the switches. We perform a similar

calibration for the π/2 pulses, which are not simply half the π time, again due to the switches.
70

0 25 50 75 100 125 150 175 200

No. of pulses

0.0

0.2

0.4

0.6

0.8

1.0

Population

Coarse

calibration

Fine

calibration

Figure 4.16: Calibration of π-pulse times using microwaves for single-qubit rotations on the “stretch”

qubit. We plot the population versus the number of π pulses applied. Starting in the dark |2, 2i

state, we apply an even number of π pulses. If the timing of the π pulse is incorrect, we will see

coherent excitation to the |3, 3i state which we measure. Based on the number of pulses that transfer

|2, 2i → |3, 3i, we can determine the timing offset. Based on the data in the blue points which

produced an error, we applied a 6 ns correction to the π time as shown in the orange points.

Transition (|F, mF i ↔ |F, mF i) π pulse duration (μs)

|3, 3i ↔ |2, 2i 0.673

|3, 2i ↔ |2, 2i 0.299

|3, 1i ↔ |2, 2i 2.329

|3, 1i ↔ |2, 1i 0.366

|3, 0i ↔ |2, 1i 2.628

|3, 0i ↔ |2, −1i 4.254

Table 4.2: π pulse durations for single-qubit rotations for state preparation and readout. These

transitions are driven by a current applied to electrode 2.

The π times for the transitions used in the state preparation and shelving sequence are listed

in Table. 4.2. Each of these π pulse durations are calibrated using the method outlined previously.
71

4.5 Laser-based spin-motion coupling

As described in Sec. 3.5.3, we use a VECSEL based system for implementing stimulated

Raman transitions. We drive coherent two-photon |3, 3i ↔ |2, 2i transitions via the intermediate P

states. As the effective k vector for this transition has an overlap with all three motional modes

of a single ion, we can use these lasers for 3D ground-state cooling [Burd et al., 2016]. For the

experiments described in this thesis, we use this transition to cool the out-of-phase axial mode and

sometimes the radial modes.

To align each beam on the ions, we maximize the ac Stark shift on the ions due to the Raman

beams, as measured using a microwave Ramsey experiment. With two ions, the beams might need

slight readjustment from the one ion calibration to ensure they are equally illuminating both ions.

4.5.1 Ground state cooling

We use the Raman beams to cool the out-of-phase axial mode (≈ 3.8 MHz) as shown in

Fig. 4.17. Cooling this mode is important for the gates we perform in chapter 6.

0.10 0.05 0.00 0.05 0.10

Detuning from sideband transition (MHz)

50

60

70

80

90

100

110

Counts

Red sideband

Blue sideband

Figure 4.17: Verification of ground-state cooling of the out-of-phase axial mode with Raman beams.

We scan across the sideband transitions after cooling the mode, exciting the blue sideband but not

the red. This asymmetry indicates that the mode is close to its ground state (Eq. 4.11).
72

4.5.2 Heating rate measurements

After ground-state cooling, we can add in a variable wait duration before measuring the

sideband asymmetry as shown in the previous section. Following Ref. [Leibfried et al., 2003] for a

single ion, the mean thermal occupation  ̄n is

n ̄ =

R

1 − R

, (4.11)

where R is the ratio of the red to blue sideband excitation probabilities.

R =

Prsb

Pbsb

. (4.12)

We note that this relation holds for a thermal state regardless of the duration for which each of the

blue or red sideband pulse is applied. Typically, this is adjusted to the π time of the blue sideband

transition from the ground state to maximize the contrast. Thus, by measuring the sideband

asymmetry as a function of wait duration after ground state cooling, we have a measure of n ̄ as

a function of time and can determine the heating rate. The heating rates of the different modes

measured in this setup are shown in Table. 4.3.

Mode Frequency (MHz) Heating rate (quanta/s)

Axial center-of-mass 2.4 60

Radial center-of-mass 7.3 18

Radial out-of-phase 7.2 <2

Table 4.3: Heating rates for different motional modes.

4.5.3 Photon scattering

The Raman beams cause photon scattering (Raman and Rayleigh), sometimes causing

transitions out of the “stretch” qubit manifold. When using the Raman beams for ground-state

cooling in other experiments, such as our two-qubit gates, we find that we need to add an extra π
73

pulse on the |3, 1i ↔ |2, 2i transition before performing the state preparation sequence in Sec. 4.2.1.2.

This extra pulse helps recover any population lost from the ground state cooling to the |3, 1i state

and return it to the |3, 3i state.

4.6 Microwave spin-motion coupling

Similar to Ref. [Ospelkaus et al., 2011], we can drive sideband transitions using a magnetic

field gradient oscillating close to the “clock” qubit frequency as described in Sec. 2.3.5.

4.6.1 Nulling the microwave magnetic field

As described in Ref. [Ospelkaus et al., 2008], we first null the microwave magnetic field at

the ion’s position. When we apply a current to the electrode, we generate a magnetic field B and

a magnetic field gradient. While we need a magnetic field gradient to drive sideband transitions,

nulling the magnetic field B is desirable as it minimizes the ac Zeeman shift at the ions which is

proportional to B2. A static ac Zeeman shift can be calibrated out, but fluctuations in this ac

Zeeman shift will cause qubit decoherence, even though the qubit itself is insensitive to fluctuations

in the quantization magnetic field. These fluctuations could be due to Rabi frequency fluctuations

of the microwave fields. Operating at the B field null reduces the sensitivity as

∆ac ∝ B

2

,

δ∆ac ∝ BδB.

(4.13)

Nulling the microwave magnetic field also minimises off-resonant transitions, for example for the

|3, 1i ↔ |2, 2i transition7.

In our trap, nulling the magnetic field involves applying currents of a given frequency to

multiple electrodes, and adjusting the phases and amplitudes of each current to cancel the magnetic

field at the ion. Other traps instead incorporate an electrode geometry with a meander to null the

7 Off-resonant transitions can also be mitigated by pulse shaping.
74

magnetic field automatically [Hahn et al., 2019], though successfully overlapping the B field null and

the rf null requires both complex finite element analysis simulations and accurate microfabrication.

To null the magnetic field, we first apply currents at the qubit frequency ω0 to both electrodes

2 and 3 (see Fig. 3.2). On resonance, the magnetic field will drive carrier transitions. We adjust

the relative phase and amplitudes of the currents to minimize the Rabi frequency Ωμ of this

transition. As Ωμ ∝ B, this measurement is first order sensitive to the magnetic field. Note that this

measurement is only sensitive to B field polarizations that drive the qubit transition, which in this

case requires π polarization. The effect of this nulling is shown in Fig. 4.18. With currents applied

to both electrodes 2 and 3, the Rabi flopping seen with just electrode 3 is reduced considerably.

However, going to longer pulse durations we see some flopping likely due to duty cycle effects.

0 20 40 60 80 100

Duration ( s)

5

10

15

20

Counts

Only electrode 3

Electrode 2 and 3

Figure 4.18: The magnetic field oscillating at the qubit frequency ω0 is nulled by applying currents

to multiple electrodes. With a current applied only to electrode 3, we see fast Rabi flopping as seen

by the blue points. By applying a current to electrode 2 as well, with an amplitude of 0.77 and

phase of 0.92π relative to electrode 3, we are able to minimize the Rabi flopping. We start to see

flopping at longer timescales most probably due to thermal duty cycle effects that shift the relative

phases and amplitudes slightly. The counts are lower in this plot as a 100 μs detection time was

used.

Compared to using a current at ω0 to calibrate the nulling, there might be slight variations in
75

the phase and amplitudes required when the microwave current is applied at ω0 ± ωr for sideband

transitions. This variation can be verified by measuring the ac Zeeman shift on the “clock” qubit

that varies as B2. The ac Zeeman shifts on other transitions such as |3, 1i ↔ |2, 2i offer measures of

the other polarization components of the B field [Warring et al., 2013b].

4.6.2 Microwave sidebands

After nulling the magnetic field, we then drive sideband transitions by applying a current

at ω0 ± ωr where +(-) corresponds to the blue(red) sideband. An example sideband transition is

shown in Fig. 4.19. These sideband transitions are used for the motional frequency tracking in the

next section.

6.868 6.869 6.870 6.871

Detuning from qubit frequency (MHz)

0.0

0.2

0.4

0.6

0.8

1.0

Population

Figure 4.19: Blue sideband transition using a microwave magnetic field gradient oscillating at

ω0 + ωr. The population excited from the |↓i (|3, 1i) state is shown a a function of detuning from

the qubit frequency. Starting in the ground state of motion, we are able to completely flip the spin

while adding a motional quantum. The sideband was pulsed on for 770 μs.
76

4.7 Motional frequency tracking

Most quantum information experiments with trapped ions involve coupling their internal

spin-states to their motion. In order to do this, one most know the motional frequency to some

precision. Unfortunately, for our system this frequency drifts quite substantially. These drifts are

primarily due to the proximity of our ions to the surface; being close to the surface increases the

strength of our gradients and thus the spin-motion coupling, but also makes us more sensitive to

surface effects such as charging which can change the motional frequencies. This charging is caused

by the 280 and 285 nm light that we use with 25Mg+. In this section, we describe how we keep track

of the drifts in the motional frequencies as we conduct our experiments. This drift tracking is used

for practically every experiment that involves the ions’ motion in this apparatus.

4.7.1 Overview

We created a drift tracking (DT) module that could be easily imported into different experiments. The DT is interleaved in between repetitions of a particular experiment; the number of

repetitions is a parameter that can be set. When the DT is run, it records the time at which it is

running, measures the motional frequency and then outputs that frequency.

We keep track of all frequencies measured as a function of time and perform a simple linear

fit to predict what the frequency will be in the future. Setting the frequency to the most recent

estimate, without including linear extrapolation of the current drift trend, produced less accurate

predictions of the motional frequency. Empirically, fitting the 5 most recent points works well; too

few points and the DT becomes vulnerable to fast fluctuations, too many and the DT is unable to

adapt to changes.

An overview of each sequence of the DT procedure is shown in Fig. 4.208. The steps are

(1) Initialize hardware: We set the parameters for the drift tracking, namely the attenuation

values for the digital attenuators controlling the strength of the sidebands we apply. In

8 HMK points out that no Boulder thesis is complete without a 420 reference.
77

some experiments, where IQ baseband voltages are also modified from their values for the

microwave sideband used for drift tracking, the IQ parameters will also need to be set.

(2) Based on previous measurements, we calculate the predicted motional frequency. Subsequent

measurements use this calculated value.

(3) We now begin measurements of the motional frequency, starting with preparing the ion in

the |↓i state of the “clock” qubit.

(4) We now pulse the microwave blue sideband on either side of the center frequency, determined

in step 2, where there is a steeper response. We perform these measurements in an interleaved

fashion, alternating between negative and positive detunings. The detuning values used

here are based on the calibration of the sideband that is done beforehand.

(5) We perform state readout, and perform thresholding of the measured counts to determine

the qubit populations.

(6) We then reset the hardware so the subsequent experimental measurements can be done.

Steps 3-5 are repeated, typically for a total of 100 repetitions per negative and positive

detuning.

(7) We perform analysis of the measured qubit populations. This step is done on the host

computer, not the FPGA. Here, we calculate the ratio of the populations measured on either

side of the detuning, and based on the ratio, the actual motional frequency. We now include

the most recent measurement of the motional frequency into our linear fit. We save the new

fit parameters which are used for subsequent calculations of the motional frequency.

4.7.2 Calibration

To calibrate the drift tracking parameters, we run a sideband scan similar to that shown in

Fig. 4.19. However, instead of a ground- state cooled ion, we typically perform our drift tracking
78

record time

Drift 

tracking

block

1. initialize hardware

2. calculate frequency

measurements

3.state prep

4.negative

detuning

4.positive

detuning

5. detect

7. analyze

frequency

6. reset hardware

or

x200

Figure 4.20: Overview of motional frequency tracking. See main text for details.

with Doppler cooled ions. While ground-state cooled ions would offer increased contrast, the duty

cycle of the experiment would be reduced due to the ground state cooling. Furthermore, if the

ground state cooling is performed with lasers, there might be additional charging as well.

Sample calibration scans for three different pulse durations of 400, 800 and 1600 μs are shown

in Fig. 4.21a. In applying the microwave sidebands for these different durations, the only parameter

that is changed is the attenuation on the digital attenuator in 6 dB steps. Renulling of the magnetic
79

field is not done. In addition, a separate experiment is performed to measure the ac Zeeman shift

from the residual magnetic field as described in Sec. 4.6.1. This shift would generate a systematic

shift to any measured motional frequencies. We typically measure this once, and only recalibrate

this shift if there are hardware changes to the microwave lines.

From the counts measured in the calibration experiment, we determine the ion populations as

shown in Fig. 4.21b. Simple thresholding is used to determine the populations. The data shown

in Fig. 4.21b uses a threshold of 20 and 72 for the zero ion bright and two ion bright populations

respectively. Any count below 20 is classified as population 0, between 20 and 72 population 1, and

above 72 as two.

From the population points, we fit a resonance line to determine the slope at the population

point halfway in between the minimum and maximum and its corresponding detuning. For the data

shown, this point occurs at a population of 1.6. These parameters are used for subsequent tracking.

4.7.3 Seeding

We typically interleave measurements of the motional frequency with the experiment we want

to run. However, to ensure the drift tracking “locks”, we usually seed the drift tracking. This

seeding involves running the drift tracking continuously without any experimental measurements to

determine the parameters for the linear fit before the experiment starts. This typically involves

measuring the frequency about 20 times. The first five measurements have no feedforward; subsequent

points use the linear fit to predict future points. This seeding is usually used in conjunction with

“dummy” points of the experiments which allows the drift tracking to adapt to the duty cycle of the

experiment and keep lock.

4.7.4 Example operation

As an example operation, we show the drift tracking data from one of the two-qubit entangling

gates discussed in Sec. 8.1.1 in Fig. 4.22. The population data from measurements made on either

side of the predicted motional frequency are shown in Fig. 4.22a. The drift tracking is “locked”
80

60

70

80

90

100

110

120

Counts

a)

400 s

800 s

1600 s

7.259 7.260 7.261 7.262 7.263 7.264 7.265

Detuning from qubit frequency (MHz)

1.0

1.2

1.4

1.6

1.8

2.0

Threshold populations

b)

400 s

800 s

1600 s

Figure 4.21: Sample calibration scans of microwave sidebands for drift tracking for three different

sideband pulse durations of 400, 800 and 1600 μs. a) Plot of raw counts versus detuning. b) Plot of

qubit populations derived from thresholds and resulting fits. The parameters extracted from the fit

are used for subsequent tracking.

as the populations are close to 1.5. If the drift tracking is “unlocked” both of these population

measurements would be closer to 2, i.e. the sideband is far off resonance and causes no spin flips.

The drift of the motional mode over the course of normal operation is shown in Fig. 4.22b. Here we

see a drift of about 1.2 kHz over 6 min. Larger drifts of about 10 kHz have also been observed.

The precision of our drift tracking, where we plot the difference between the predicted and

measured value, is shown in Fig. 4.23. From the histogram, we extract a mean of -2.4 Hz and a
81

1.2

1.4

1.6

1.8

Populations

a)

Negative detuning

Positive detuning

0 100 200 300 400

Time (s)

6.90425

6.90450

6.90475

6.90500

6.90525

6.90550

Motional frequency (MHz)

b) Figure 4.22: Example motional frequency tracking data from a gate experiment. a) Population

measurements versus time. These populations are measured on either side of the predicted central

peak. The red points correspond to a negative detuning while the blue points to a positive detuning.

b) Motional frequency versus time. We plot the motional frequency versus time. Here we see about

a 1.25 kHz drift over the course of a typical entangling gate experiment. Each point of the drift

tracking precedes 200 repetitions of the entangling gate experiment.

standard deviation of 67 Hz. This measurement was taken using a sideband probe duration of 1.6 ms.

Longer times can be used, but there is a trade off between the length of the probe and how reliably

the drift tracking stay locked, as the time intervals between frequency updates become too long.

Empirically, 1.6 ms is about as long a probe duration as can be used in practice. For coarser scans

where the precision is not as important we use 400 μs probes.
82

100 0 100

Predicted frequency - measured frequency (Hz)

0.000

0.002

0.004

0.006

0.008

Probabililty density

= -2.4 Hz,

= 66.9 Hz

Figure 4.23: Histogram of difference between predicted and measured motional frequency from

data in Fig. 4.22. We plot the probability density of measuring a particular frequency difference in

red. The mean is close to 0 and the standard deviation of about 67 Hz.

4.7.5 Laser-induced charging

When we first discovered drifts in the motional frequency when performing gate experiments,

we were somewhat perplexed. We eventually tracked it down to duty cycle effects from the laser

beams and implemented several iterations of the drift tracking before it reached the version described

in this thesis. With the drift tracking, we examined the laser-induced charging more concretely as

shown in Fig. 4.24.

With a single ion, we measure the motional frequency as a function of time. However, after

some number of measurements, we pulse our Raman beams on but detuned from any transition

during our drift tracking before measuring the frequency. We see large fluctuations in the motional

frequency on the order of 100 Hz that are correlated with the application of the Raman beams.

These fluctuations also have fast time scales, causing spikes over a few seconds. When running the

drift tracking, we usually try to keep the duty cycle of the laser (Doppler cooling and or Raman)

consistent with the main experiment. This can be controlled by changing the amount of Doppler
83

cooling before the measurement is performed during the drift tracking.

Figure 4.24: Effect of charging from Raman beams on a radial frequency. We measure the motional

frequency on a radial mode versus time. The darker regions indicate when Raman beams were also

pulsed during the drift tracking sequence. We see sharp increases when the Raman beams are first

applied that decreases when they are turned off. We attribute these fluctuations to charging of

contaminants on the surface of our electrodes from the laser.

4.7.6 Future improvements

25Mg+ is not as light as 9Be+ and uses 280 nm for cooling and detection as opposed to 313 nm,

which should cause less charging. Other experiments in our group that use 9Be+ close to the surface

see weaker effects from charging. Thus, one way to improve our drift tracking would be to reduce

the drifts in the first place by using a different ion. Fortunately, this apparatus also has Ca ovens;

Ca+ ions will hopefully enable mixed-species operations that minimize the use of Mg lasers.

Additionally, we typically require a scan of the motional frequency to initialize the drift

tracking i.e. a starting point before the seeding occurs. This step could be eliminated by a

continuous monitoring of the motional frequency in between experiments.

Furthermore, more advanced algorithms to predict the motional frequency could also be used.
84

In particular, we find that the drift tracking has consistent behaviour for identical experimental

sequences outside of the drift tracking. If motional frequency drifts from previous runs of the

experiments could also be incorporated in the tracking, it might help to track the drifts in subsequent

runs of the experiments more precisely. Since all the drift tracking data is stored for every

experimental run, different algorithms could be tested with existing data.

4.8 Qubit frequency tracking

Aside from motional frequency drifts, we also employ a similar procedure for tracking the

magnetic-field-sensitive “stretch” qubit frequency. The qubit frequency would drift due to changes

in our nominally static quantization field. An example of qubit frequency tracking is shown in

Fig. 4.25. Instead of using Rabi spectroscopy on the sideband transition, we instead use the spin-flip

transition as shown in Fig. 4.5. We increase the π-pulse duration of this transition to about 1 ms by

reducing the microwave amplitude in order to increase the precision. We are usually able to predict

the qubit frequency to about 50 Hz, which corresponds to a magnetic field shift of 2.3×10−5 G

(2.3 nT). These fluctuations are small. In fact, while running these experiments we often saw qubit

frequency shifts when routine tasks such as opening cupboards or drawers close to the experiment.

We also observed shifts due to cryogen dewars being transported outside the lab.9.

4.9 Ion transport

As shown in Fig. 3.2, there are multiple trap zones that could be used for experiments. The

advantage of performing experiments away from the load zone would be to minimize the effects of

charging from the higher power 285 nm beams used for loading. However, for the majority of the

experiments in the thesis, we only performed experiments in the load zone. One does not simply

transport ions into the experiment zone10.

9 Even if we don’t end up with a quantum computer with our setup, at least we have a very expensive magnetic

field sensor.

10 Even Boromir would agree walking into Mordor is easier.
85

0 50 100 150

Time (s)

8.0

8.2

8.4

8.6

8.8

9.0

Qubit frequency (kHz)

a) +1.32646e6

100 50 0 50 100

Predicted frequency - measured frequency (Hz)

0.000

0.002

0.004

0.006

0.008

0.010

Probabililty density

b)

= -1.8 Hz,

= 46.2 Hz

Figure 4.25: Example qubit frequency tracking. a) Plot of qubit frequency versus time. We see

drifts on the kHz level over a period of about 3 minutes. b) Histogram of the difference between the

predicted and measured frequencies. The mean μ is -1.8 Hz and the standard deviation σ is about

46 Hz.

Ion transport involves changing the potentials11 of the DC electrodes, moving the minimum

of the axial confining potential along the trap axis. We only attempted transport that was well into

11 Transport potentials generated by DHS. This is just a statement of fact, not assigning blame for our transport

woes.
86

the adiabatic regime on the millisecond timescale. We were initially successful in this and were able

to transport up to two ions reliably. Figure 4.26 shows one of our more successful attempts moving

a single ion three electrodes over or by a distance corresponding to≈ 165 μm from its initial position.

165 μm

Figure 4.26: Transport of a single ion from the load zone to the experiment zone. The bright dot

indicates the ion position, which has moved about 165 μm.

Unfortunately, that reliability was short-lived. We eventually started losing ions during the

transport. Instead of moving the ion all 165 μm at once, we broke the transport into smaller steps.

We noticed that the ion was getting severely decompensated during the transport, as evidenced

by the shim voltages required. We adjusted the shim voltages at smaller intervals as shown in

Fig. 4.27. Large changes in the z-shim, on the order of kV/m were required. We attribute this large

spatial variation to charging of the trap surface. Debugging transport is somewhat of a vicious cycle

as if the ion is lost, reloading will make the transport even harder due to the charging from the

photoionization beam.

We suspect our transport became more unreliable due to a degradation of our UV fiber,

resulting in a loss of power for Doppler cooling. Reducing the amount of cooling meant the ion was

more likely to leave the trap if it were uncompensated. However, changing the fiber did not fix

this issue and we eventually stopped working on transport altogether to focus on experiments in
87

0 20 40 60 80 100

Distance from load zone ( m)

2000

1000

0

1000

2000

3000

Shim value (V/m)

x

y

z

Figure 4.27: We plot the compensation voltages required during the ion transport as a function of

its position from the load zone (0 μm is at the load zone). We found that there were large changes

in the required z-shim.

the load zone. This provides a good juncture to terminate the transport discussion and focus on

experiments in the subsequent chapters.
All we hear is radio ga ga,

Radio blah blah,

Radio, what’s new?

Someone still loves you

Queen 5

Spin-motion coupling with a radiofrequency

magnetic field gradient

In this chapter, we discuss a new method of laser-free spin-motion coupling that uses an

oscillating magnetic-field gradient close to the motional frequency, at radio frequency. This technique

was developed during this Ph.D. and addresses some of the challenges present in previous laser-free

methods. We discuss the physics underlying this mechanism, demonstrate sidebands using this

technique, and use those sidebands to cool a motional mode close to its ground state. The results

presented in this chapter were published in [Srinivas et al., 2019]. We note that similar physics was

observed using an intensity gradient of a running optical lattice [Ding et al., 2014], which inspired

this work.

5.1 Spin-motion coupling with an oscillating magnetic-field gradient

In this section, we derive the sideband interaction that couples the spin to the motion using

an oscillating gradient. For simplicity, we first assume the ion experiences only a gradient oscillating

at ωg and a microwave field close to its qubit frequency. We discuss the effects of an additional

88
89

oscillating magnetic field at ωg in section 5.2. The Hamiltonian can be written as

Hˆ (t) =~ω0

2

σˆz + ~ωraˆ

†

aˆ

+ 2~Ωg cos(ωgt)ˆσz(ˆa + ˆa

†

)

+ 2~Ωμ cos (ω0 + δ)t



σˆx.

(5.1)

The first two terms correspond to the ion’s qubit and motional energies, with frequencies ω0

and ωr respectively. The next term corresponds to the spin-dependent force arising from a magnetic

field gradient with a Rabi frequency Ωg that is oscillating at ωg. The last term corresponds to the

microwave field with spin-flip Rabi frequency Ωμ oscillating close to the qubit frequency ω0 with a

detuning δ. The Rabi frequency from the gradient1 Ωg is defined as

Ωg ≡

r0(ˆr · ∇Bg)

4

dω0

dBz







Bz=|B~0|

. (5.2)

A quantization axis zˆ is defined by a static magnetic field B~

0. The sensitivity of the qubit

frequency ω0 to changes in an additional magnetic field Bz along this quantization axis is described

by dω0/dBz. This sensitivity in general depends on the magnitude of B~

0 as well. The gradient

of the magnetic field oscillating at ωg is described by ∇Bg, where Bg is the magnetic field along

the quantization axis oscillating at ωg. The ground-state extent of the ion’s wave function for the

motional mode along the rˆ direction is characterized by r0 =

p

~/2Mωr for ion mass M. This gives

an implicit ω

−1/2

r dependence to Ωg. This dependence is present in all types of spin-motion coupling,

including laser-based methods.

The microwave Rabi frequency2 Ωμ is

Ωμ =

Bx

2~

h↓| μx |↑i, (5.3)

1 This is equivalent to Ωz

j,n in Ref. [Ospelkaus et al., 2008].

2 This corresponds to Ωx

in Ref. [Ospelkaus et al., 2008].
90

where Bx is the component of the oscillating microwave magnetic field perpendicular to the

quantization axis and μx is the component of the ion’s magnetic moment along the same direction.

The dynamics of this Hamiltonian in the lab frame can be difficult to analyze. To elucidate

the physical mechanism, let us first look at the static gradient case by setting ωg = 0. Going

into the interaction picture with respect to the Hamiltonian corresponding to the qubit frequency,

H0 =

~ω0

2

σˆz, our interaction picture Hamiltonian is

Hˆ

static(t) = 2~Ωgσˆz(ˆa + ˆa

†

) spin-dependent displacement

+~ωraˆ

†

aˆ motional frequency

+~Ωμ



σˆ+e

−iδt + ˆσ−eiδt

. microwave term (5.4)

The first term corresponds to a spin-dependent displacement, the second term to the motional

frequency, and the last term to the microwave fields which drive spin flips, where σˆ+ = |↑i h↓| and

σˆ− = |↓i h↑|.

In the absence of the spin-dependent displacement, the microwave field can only drive spin-flip

transitions3. Sideband transitions where ∆n 6= 0 are forbidden as the different |ni Fock states

are orthogonal. However, if |↑i is displaced relative to |↓i, spatial overlap between the motional

wavefunctions for different spin states is modified, enabling sideband transitions [F ̈orster et al.,

2009]. In order to drive these transitions, the microwave field would have to be detuned by the

motional frequency with δ = ±ωr. This process is illustrated in Fig. 5.1(a). This effect can also be

viewed as similar to Franck-Condon factors that appear in molecular spectroscopy [F ̈orster et al.,

2009, Hu et al., 2011].

The dynamics with an oscillating gradient in Eq. 5.1 are similar to the static gradient case

when going into a frame of reference oscillating at the qubit frequency ω0 and the oscillating gradient

frequency ωg. This transformation is accomplished by going into an interaction picture with respect

to H0 =

~ω0

2

σˆz + ~ωgaˆ

†aˆ. We obtain the interaction picture Hamiltonian Hωg

(t),

3 We refer to the microwave field. A microwave gradient detuned by ±ωr can also drive sideband transitions but

this is a different mechanism altogether. See Eq. 2.26.
91

≈ ∆xg

∆x

⇒

hω ̄ r h ̄(ωr−ωg)

∆xg cos ωgt Interaction frame at ωg

n=0

n=1

n=2

∆n

= 

0

∆n

= 

1

∆n

= 

-1

∆n

= 

0

∆n

= 

1

∆n= 

-1

(a) (b)

Figure 5.1: Schematic of a qubit coupled to a harmonic oscillator with a spin-dependent displacement

from (a) a static gradient or (b) an oscillating gradient4. An additional microwave field drives

|↓i ↔ |↑i transitions. (a) For a static gradient, detuning the microwave field by ± ωr drives sideband

transitions with ∆n = ±1. (b) An oscillating gradient at ωg is formally equivalent to a static gradient

in the interaction frame oscillating at ωg, ignoring fast-oscillating terms. The sideband transitions

now occur at detunings ±(ωr − ωg). A more detailed discussion of the different displacements can

be found in appendix C.

Hˆ

ωg

(t) =~Ωgσˆz



(ˆa + ˆa

†

) spin-dependent displacement

+(ˆae−2iωgt + ˆa

†

e

2iωgt

)



counter-rotating terms

+~(ωr − ωg)ˆa

†

aˆ modifed “motional frequency”

+~Ωμ



σˆ+e

−iδt + ˆσ−eiδt

. microwave term (5.5)

In this interaction frame, we effectively get a static spin-dependent displacement. However, instead

of the microwave field driving sideband transitions at detunings of ±ωr, they now occur at detunings

of δ = ±(ωr − ωg). We have an additional counter-rotating term shown in the second line which

can be ignored if ωg  Ωg.

To obtain the sideband transitions, we transform Eq. 5.5 into a second interaction picture

with respect to the term ~(ωr − ωg)aˆ

†aˆ. One can also go straight into the interaction picture with

respect to H0 =

~ω0

2

σˆz + ~ωraˆ

†aˆ from Eq. 5.1. The order in the text is chosen for pedagogical

4 This figure is significantly out of scale. The ion’s ground state extent is ∼ 5.7 nm, while the displacements

are ∼ 30 pm. Artistic liberty was exercised to make the displacement more than a pixel. More detail on how the

displacement can be estimated can be found in appendix C.
92

reasons. We obtain the interaction Hamiltonian HI (t),

Hˆ

I (t) =~Ωgσˆz

h

(ˆae−i(ωr−ωg)t + ˆa

†

e

i(ωr−ωg)t

)

+(ˆae−i(ωr+ωg)t + ˆa

†

e

i(ωr+ωg)t

)

i

+~Ωμ



σˆ+e

−iδt + ˆσ−eiδt

.

(5.6)

In order to transfer into the interaction picture with respect to the gradient term (corresponding

to Ωg), we derive the propagator using the first term of the Magnus expansion. This step is required

as the commutator of Hˆ

I (t) with itself at different times is not zero, i.e. [HˆI (t), Hˆ

I (t

0

)] 6= 0. The

second term in the Magnus expansion is just a global phase, which is ignored. Terms that are higher

order in the Magnus expansion are thus vanishing. A detailed derivation is in appendix D. The

propagator Uˆ†

g (t) is

Uˆ†

g

(t) = exp h Ωg

ωr − ωg

σˆz



− aˆ(e

−i(ωr−ωg)t − 1)

+ ˆa

†

(e

i(ωr−ωg)t − 1)

+

Ωg

ωr + ωg

σˆz



− aˆ(e

−i(ωr+ωg)t − 1)

+ ˆa

†

(e

i(ωr+ωg)t − 1)

i

.

(5.7)

Using this propagator, we derive the interaction Hamiltonian with respect to the gradient

term Hˆ

I

0(t),

Hˆ 0

I

(t) =Uˆ†

g

(t)~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt)Uˆ

g(t). (5.8)

Making use of the Baker-Campbell-Hausdorff theorem (appendix A), keeping only the lowest

order terms, we obtain Hˆ

I

0(t),
93

Hˆ 0

I

(t) '~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt)

+ 2~ΩgΩμ(ˆσ+e

−iδt − σˆ−eiδt)

×

n 1

ωr − ωg



aˆ

†

(e

i(ωr−ωg)t − 1) − aˆ(e−i(ωr−ωg)t − 1)

+

1

ωr + ωg



aˆ

†

(e

i(ωr+ωg)t − 1) − aˆ(e−i(ωr+ωg)t − 1)o

.

(5.9)

We obtain sideband transitions when δ = ±(ωr − ωg), where (+) corresponds to blue sideband

transitions and (-) corresponds to red sideband transitions. We obtain the Hamiltonian Hˆ

sb,

Hˆ

sb = ±

2~ΩgΩμ

ωr − ωg

(ˆσ±aˆ

† + ˆσ∓aˆ), (5.10)

where the sideband Rabi frequency Ωsb is

Ωsb =

2ΩgΩμ

ωr − ωg

. (5.11)

For Eqs. 5.10 and 5.11, we have neglected fast-rotating terms corresponding to a detuned carrier

field and the additional sideband at ωr + ωg, which are detuned by δ and 2ωg, respectively. Errors

caused by these terms are coherent, and can be mitigated, by for example ramping the gradient and

microwaves on and off adiabatically. Note that the additional sidebands for δ = ±(ωr + ωg) will

have a sideband Rabi frequency that is proportional to 1/(ωr + ωg), and thus will be significantly

weaker than the sidebands at δ = ±(ωr − ωg) when ωg is close to ωr.

5.2 Residual oscillating magnetic field

To generate the gradient for the sideband interactions, we apply a current oscillating at ωg to

our trap electrodes. However, the current will not only give us an oscillating magnetic-field gradient,

but also an oscillating magnetic field at the ion. This oscillating magnetic field will modulate the

frequency of our field-sensitive qubit which gives rise to some interesting effects. To simplify the

analysis, we ignore the effect of the gradient and the motion. Our Hamiltonian is then
94

Hˆ

0(t) =~ω0

2

σˆz + 2~Ωz cos(ωgt)ˆσz

+2~Ωμ cos (ω0 + δ)t



σˆx.

(5.12)

The term with Ωz corresponds to the magnetic field oscillating at ωg, where Ωz is

Ωz ≡

Bg

4

dω0

dBz







Bz=|B~0|

. (5.13)

The terms in Ωz follow the same definitions as Eq. 5.2, with Bg the magnitude of the magnetic

field along the quantization axis oscillating at ωg. Effectively, this term causes a modulation of

the qubit frequency with frequency ωg and amplitude Ωz. To see this frequency modulation more

clearly, we go into the interaction picture with respect to the first two terms of Eq. 5.12, deriving

the propagator Uˆ

I (t),

Uˆ†

I

(t) = exp  i

~

Z t

0

h

~ω0

2

σˆz + 2~Ωz cos(ωgt

0

)ˆσz

i

dt0



= exp iω0t

2

σˆz +

2iΩz

ωg

sin (ωgt)ˆσz



.

(5.14)

We thus obtain the interaction Hamiltonian for the last term in Eq. 5.12,

Hˆ

I (t) =~Ωμ

 

e

−iδtσˆ+

X∞

m=−∞

Jm



4Ωz

ωg



e

imωgt

!

+ H.c, (5.15)

where we use the Jacobi-Anger expansion,

e

iz sin θ =

X∞

m=−∞

Jm(z)e

imθ

. (5.16)

When δ = mωg, we implement spin-flip transitions described by the Hamiltonian Hˆm,

Hˆm = ~ΩμJm



4Ωz

ωg



(ˆσ+ + ˆσ−), (5.17)
95

where we have dropped the fast-rotating terms. This applies for any oscillating magnetic field,

including those generated by rf trapping potentials [Meir et al., 2018]. The Rabi frequency Ωm of

these transitions is

Ωm = ΩμJm



4Ωz

ωg



. (5.18)

Thus, the oscillating magnetic field would modify the microwave term from the previous

section and would need to be considered for the sideband transitions as well. Modifying Eq. 5.10,

Hˆ

sb,m = ±

2~ΩgΩμ

ωr − ωg

Jm



4Ωz

ωg



(ˆσ±aˆ

† + ˆσ∓aˆ). (5.19)

The largest sideband Rabi frequency corresponds to the largest value of the Bessel function

Jm



4Ωz

ωg



. This value occurs for J0(0), which is achieved when Bg = 0, i.e. the oscillating magnetic

field along the quantization is nulled. We explain this nulling process in detail in section 5.5.

5.3 Experimental implementation

For these experiments, we use a field-sensitive “stretch qubit” with a transition frequency

of ω0/2π = 1.326 GHz as described in section 4.2.1. The magnetic field sensitivity of this transition is (dω0/dBz)/2π = −19.7 MHz/mT. The qubit preparation and readout is described in

Sec. 4.2. For these experiments with a single ion, the motional frequencies are (ωa, ωr1, ωr2)/2π ≈

(3.2, 6.2, 7.6) MHz, where ~a is along the axis of the trap, and ~r1, ~r2 lie in the radial plane (Fig.5.2).

By design, the magnetic field gradient is only in the radial plane and thus the spin-motion coupling

is only for the two radial modes.

To realize spin-motion coupling, we apply simultaneous currents (which are ramped on and

off over 10 μs) to the trap electrodes at two frequencies, ωg and ω0 + δ. We apply up to 0.5(1) A rms

per electrode at ωg/2π = 5 MHz, corresponding to 6(1) mW of dissipation in the trap electrodes5;

5 These electrodes are shorted to ground at their far end, so most of the power is reflected. See Sec. 3.1 for details.
96

dissipation from the drive at ω0 + δ is  1 mW. Currents oscillating at ωg are applied to electrodes

1, 2 and 3 as shown in Fig. 5.2. The microwave currents close to the qubit frequency are applied to

either electrode 1 or 2 for the experiments described here.

25

Mg+

1 rf 2 3 rf

30 μm

dc

Top view

View 

along 

a

r1

r2

22.5°

B



0

dc

r1

r2

a

Figure 5.2: Schematic of the surface electrode trap with the relevant details for the experiments

discussed in this chapter. Oscillating currents in electrodes 1, 2, and 3 generate magnetic fields

and magnetic field gradients at the ion. A static magnetic field B~

0 parallel to the surface of the

trap defines the quantization axis zˆ. From the electrode geometry, the relevant gradients oscillating

at ωg are only along the radial modes ~r1 and ~r2. Thus, sideband transitions are only observed for

those modes using the technique described in this chapter.

5.4 Spectroscopy with an oscillating magnetic field gradient

We first perform microwave spectroscopy with the oscillating gradient applied. The microwave

pulse is applied for 500 μs and the detuning δ is varied. The ion is initialised in |↓i to its Doppler

temperature (n ̄ ≈ 2) so both motion-adding blue sideband (∆n = 1) and motion-subtracting

red sideband (∆n = −1) transitions could be observed. A reduced spectrum with the strongest

sidebands for each of the radial modes is shown in Fig. 5.3. For δ = 0, the spin-flip transition is

driven resonantly. The blue (red) sideband transitions occur when δ = +[ωri − ωg] (δ = −[ωri − ωg]),

where ωri is the frequency of the radial mode ~r1 or ~r2.

However, the full spectroscopy shows additional spin-flip transitions, as shown in Fig.5.4.

We see spin-flip transitions at δ = mωg, as the magnetic field Bg = 0 as described in Eq. 5.17. 6
97

4 2 0 2 4

Microwave detuning (MHz)

0.0

0.2

0.4

0.6

0.8

1.0

Population

( r1 g) r1 g

( r2 g) r2 g

Figure 5.3: Sideband transitions with microwaves and an oscillating magnetic field gradient. In

addition to a magnetic field gradient oscillating at ωg, an additional microwave pulse with detuning

δ is applied. Blue (red) sideband transitions with ∆n = 1 (∆n = −1) occur when δ = +[ωri − ωg]

(δ = −[ωri − ωg]).6

14 12 10

0.0

0.2

0.4

0.6

0.8

Population

3 g

( r2 + g)

( r1 + g)

2 g

10 12 14

3 g

r2 + g

r1 + g

2 g

4 2 0 2 4

/2 (MHz)

0

( r1 g) r1 g

( r2 g)

(2 g r2) 2 g r2

r2 g

g g

Figure 5.4: Microwave spectroscopy with an applied magnetic field gradient oscillating at ωg/2π =

5 MHz. The microwave field has a detuning δ which is varied. Spin-flip transitions that leave the

motional state unchanged (∆n = 0), denoted by the green-dotted lines, occur when δ = ±mωg. Blue

sideband transitions with ∆n = 1 (blue dashed lines) occur when δ = +(ωri ± ωg). Red sideband

transitions with ∆n = −1 (red dash-dotted lines) occur when δ = −(ωri ± ωg).

In addition to the blue (red) transitions at δ = +(−)[ωri − ωg], we see sideband transitions at

δ = +(−)[ωri+ωg]. We also observe weaker higher-order transitions due to the frequency modulation

of the qubit frequency as described in Eq. 5.19.

6 The eagle-eyed reader might observe that these transitions are slightly shifted from the exact values of δ =

±[ωri − ωg]). This shift is due to the ac Zeeman shift from the detuned microwave fields as described in Sec. 5.6.1.
98

5.5 Measuring and nulling the oscillating magnetic field

We characterize the effect of the oscillating magnetic field by measuring the Rabi frequency

of the spin-flip transitions Ωm as a function of Bg following Eq. 5.17. We apply a current oscillating

at ωg to only electrode 2, in addition to a microwave field on resonance with Rabi frequency

Ωμ/2π = 375 kHz. This microwave field is also generated by a current applied to electrode 2, and

this Rabi freuqency is measured without any applied current at ωg (Bg = 0). We vary the amplitude

of the current at ωg and measure the Rabi frequency of the spin-flip transitions as a function of this

amplitude. We perform this first for the J0 transition.

For each amplitude of the oscillating magnetic field Bg, we measure the Rabi frequency of

the J0 transition. An example measurement is shown in Fig. 5.5 for Bg = 0. From these data, we

extract a Rabi frequency. We repeat this process for different values of Bg as shown in Fig. 5.6.

While increasing the value of Bg, the frequency of the spin-flip transition shifts slightly due to an

ac Zeeman shift from Bg. This shift must be accounted for to keep the spin-flip transitions on

resonance. We scan the microwave detuning δ to calibrate this shift. A fit of the Rabi frequencies as

a function of Bg to the J0 Bessel function gives a scaling factor from the amplitude of the applied

current oscillating at ωg to the amplitude of the oscillating magnetic field Bg at the ion.

0 1 2 3 4 5

Time ( s)

0.0

0.2

0.4

0.6

0.8

1.0

P

Figure 5.5: Example Rabi flopping data for the J0 transition. For these data the amplitude of Bg

was set to 0 and Ω0/2π was 375 kHz. The ion was initialized to |↓i and the microwave field drives

the |↓i ↔ |↑i transition.

We repeat the measurements for the m = {1, 2, 3, 4, 5} transitions, changing the detuning of
99

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

Bg (mT)

0

50

100

150

200

250

300

350

400

0

/

2

(

k

H

z

)

Figure 5.6: Spin-flip transition Rabi frequency Ω0 as a function of magnetic field Bg oscillating

at ωg. For these data, the microwave field is nominally on resonance after taking into account the

additional ac Zeeman shift (δ = 0).

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

Bg (mT)

0

50

100

150

200

250

300

350

400

m

/

2

(

k

H

z

)

= 0

= g

= 2 g

= 3 g

= 4 g

= 5 g

m = 0

m = 1

m = 2

m = 3

m = 4

m = 5

m = 0

m = 1

m = 2

m = 3

m = 4

m = 5

Figure 5.7: Plot of the spin-flip Rabi frequency Ωm versus the amplitude of the oscillating magnetic

field Bg. Spin-flip transitions occur when the detuning of the microwave field δ = mωg. The theory

curves indicated by the dashed lines have no free parameters. Error bars are smaller than the data

points.

the microwave field to δ = mωg accordingly. As before, there is a slight offset to the exact detuning,

with an additional contribution to the ac Zeeman shift from the detuned microwave field which is

accounted for. The results are shown in Fig. 5.7. Note that for the m > 0 transitions, the theory

lines have no free parameters. We use the same scaling factor that we determined from the J0

transition. We see that the data seem to fit Eq. 5.17 well7.

In order to maximize the sideband Rabi frequency Ωsb, we set Bg = 0 and thus Ωz to zero

7 Would calculating Bessel functions be a more practical application of near-term quantum computers than

factoring? Certainly for our system.
100

as shown in Eq. 5.19. We set Bg = 0 by minimizing the value of Ω1 by adjusting the phases and

amplitudes of the currents oscillating at ωg for all three electrodes 1, 2, and 3. As Bg → 0, Ω1 is

proportional to Bg

8

. At J0(0), the sideband Rabi frequency is insensitive to variations in Bg to

first order. We note that nulling Bg does not mean that all components of the oscillating magnetic

field are also nulled. However, only the component of the magnetic field along the quantization field

will create this frequency modulation effect; other perpendicular components produce an ac Zeeman

shift of less than 100 kHz for the experiments in this chapter.

5.6 Sideband Rabi frequency

In addition to the magnitude of Bg, the sideband Rabi frequency also depends on the Rabi

frequency of the microwave field Ωμ. Setting J0 = 0 in Eq. 5.19 to achieve the largest sideband Rabi

frequency, we obtain our initial expression for the sideband Rabi frequency in Eq. 5.11 where

Ωsb =

2ΩgΩμ

ωr − ωg

.

At first glance, it may seem that we can make this Rabi frequency arbitrarily large by

increasing the microwave Rabi frequency Ωμ, which would be easy to do in our system without large

currents. However, an important effect that needs to be considered is the ac Zeeman shift from this

microwave field.

5.6.1 ac Zeeman shift from microwave fields

To drive the blue (red) sideband transition, a microwave field detuning of δ = +(−)[ωr − ωg]

is required. For ωr − ωg > 0, a blue (red) detuned microwave field will generate a negative (positive)

ac Zeeman shift that will need to be accounted for. This shift will push the absolute value of the

detuning from the unshifted qubit frequency closer to 0 as Ωμ is increased (Eq.5.21). This effect

ultimately limits the value of Ωμ that can be used. The ac Zeeman shift ∆ac from a detuned field δ

is

8

J1(x) ∝ x, x → 0
101

∆ac = δ ∓

q

δ

2 + 4Ω2

μ

, (5.20)

where - (+) corresponds to a blue (red) detuned field respectively [Foot, 2007]. This shift would

modify the actual detuning required to δ = ±(ωr − ωg) + ∆ac. Solving for δ we obtain,

δ = ±

q

(ωr − ωg)

2 − 4Ω2

μ

. (5.21)

Thus, 2Ωμ < |ωr − ωg| and Ωsb ≤ Ωg when using an oscillating gradient.9. As 2Ωμ approaches

ωr − ωg, the microwave field is shifted closer to resonance, driving spin-flip transitions.

5.6.2 Characterizing the sideband Rabi frequency

We experimentally verify Eq. 5.11, keeping Ωg constant but varying the microwave Rabi

frequency Ωμ. For each value of Ωμ, we first measure the ac Zeeman shift by scanning the detuning

of the microwave field from the qubit frequency ω0 and looking for the sideband transition. We

measure Ωsb for the r1 mode by first cooling to its ground state, then driving the blue sideband

transition. Example data is shown in Fig. 5.8. For these experiments ωr1 − ωg/2π ≈ 1.2 MHz.

We repeat these measurements for different values of Ωμ, and a plot of both the sideband

Rabi frequency Ωsb and the microwave detuning δ to drive the blue sideband transition is shown in

Fig. 5.9. We see a linear variation in Ωsb, and as 2Ωμ → (ωr1 − ωg), δ → 0. From a linear fit of the

data, we determine that Ωg/2π =1.383(6) kHz. We can use this Rabi frequency to determine the

magnetic field gradient. Following Eq. 5.2.

rˆ · ∇Bg =

4Ωg

r0



dω0

dBz







Bz=|B~0|

−1

. (5.22)

For these experiments, r0 ≈ 5.7 nm, and the frequency sensitivity of the transition is ≈ 19.7 GHz/T,

yielding a peak magnetic field gradient10of 49.4(2) T/m along the mode ~r1.

9 There has to be some conservation principle here, but generating sidebands is hard and being able to crank up

the microwave Rabi frequency would have been too easy a way out.

10 For the gate experiments we perform later, we increase this gradient to ≈ 120 T/m!
102

a)

b)

Figure 5.8: Example time and frequency scans on the blue sideband transition for two microwave

Rabi frequencies a) Ωμ/2π ≈ 0.12 kHz and b) Ωμ/2π ≈ 0.57 kHz. On the left, we show Rabi flopping

on the sideband transition and on the right a detuning scan over the transition. The detuning scan

is used to determine the frequency of the ac Zeeman shift and the time scan is used to determine

the sideband Rabi frequency Ωsb. An exponentially decaying sine function is fit to the time scan

data to account for qubit decoherence due to fluctuating magnetic fields. As Ωμ is increased, the

sideband Rabi frequency Ωsb is increased.

5.7 Sideband cooling

Finally, we use the resolved sideband at δ ≈ −(ωr1 − ωg) to cool the r1 mode, as shown in

Fig. 5.10. Starting from a Doppler-cooled mean phonon occupation of n ̄ ≈ 2, we use a sequence of

twelve 150 μs red sideband pulses with interleaved optical repumping to reach n ̄ = 0.09(7). The total

cooling duration is ≈ 2.5 ms, more than an order of magnitude faster than previous demonstrations

of microwave cooling using the static gradient scheme [Weidt et al., 2015, Sriarunothai et al., 2018].

This speed-up is partly from using higher motional frequencies, which results in a lower initial

thermal occupation after Doppler cooling. For the data in this section, we use only one microwave

field in addition to the gradient to perform this cooling. We extend this technique to two microwave
103

0.00 0.25 0.50 0.75 1.00

2 /( r1 g)

0.0

0.2

0.4

0.6

0.8

1.0

s

b

/

g

0.0

0.2

0.4

0.6

0.8

1.0

/

(

r

1

g

)

Figure 5.9: We plot the normalized blue sideband Rabi frequency Ωsb/Ωg (black squares, left axis)

and the normalized microwave detuning δ/(ωr1 − ωg) from Eq. (5.21) (gray circles, right axis) as

a function of 2Ωμ/(ωr1 − ωg). Error bars are smaller than the data points. The red dashed line

denotes the limit on Ωμ described in the text. The black line is a linear fit to the data; the gray line

is a theoretical plot following Eq. 5.21.

fields in Sec. 6.2.5.

6 4 2 0 2 4 6

Sideband detuning (kHz)

0.0

0.2

0.4

0.6

0.8

Population

r1 RSB

r1 BSB

Figure 5.10: Populations in |↑i after blue (BSB, circles) and red (RSB, squares) sideband analysis

pulses on a ground-state-cooled ion initialised in |↓i versus the detuning of the microwaves from the

r1 sideband transition. Both the cooling and the analysis pulses are performed using the oscillating

gradient sidebands. Lines are fits giving a final Fock state population of  ̄n = 0.09(7).
104

5.8 Pulse shaping

For the data in this section, we operate in a limit where our microwave Rabi frequency Ωμ is

of the same order of magnitude as its detuning δ. Using a square pulse for the microwave would

thus result in significant off-resonant excitation. To minimize this effect, we employ pulse-shaping

of our microwave fields.

Specifically, we use a Blackman envelope [Harris, 1978] to adiabatically ramp the microwave

pulse on and off over 10 μs. This pulse shaping allows us to operate with 2Ωμ/(ωr − ωg) = 0.9,

and Ωsb close to its maximum value of Ωg as shown in the previous section. We program an

approximation of the Blackman envelope to our IQ modulator, which controls the amplitude of the

microwave field. An example waveform is shown in Fig. 5.11.

0 5 10 15 20 25 30

Time ( s)

0.0

0.2

0.4

0.6

0.8

I (V)

Figure 5.11: Plot of voltage versus time for the I port (I) of the IQ modulator. The voltage I is

directly proportional to the microwave field amplitude at the ion. The microwave field is ramped on

(blue and orange segments) and off (red and purple segments) over 10 μs. The field has a constant

amplitude for a duration of 10 μs (green segment).

We plot the spectroscopy data from Fig. 5.3, but this time include data without pulse shaping

in Fig. 5.12. We see that without the pulse shaping, there is significant off-resonant excitation.

This excitation is purely an effect on the spin transition, and for the parameters of the experiment

obscures the sideband transitions.
105

4 2 0 2 4

Microwave detuning (MHz)

0.0

0.2

0.4

0.6

0.8

1.0

Population

( r1 g) r1 g

( r2 g) r2 g

No pulse shaping

Pulse shaping

Figure 5.12: Identical plot to Fig. 5.3, except the data without pulse shaping are included. Pulse

shaping significantly reduces the off-resonant excitation of the spin, allowing us to resolve the

sideband transitions.

Section 6.1.2 contains a more detailed discussion on pulse shaping, especially for a bichromatic

microwave field.

5.9 Comparison to static gradient scheme

The sideband Rabi frequency using an oscillating gradient is Ωsb =

2ΩgΩμ

ωr−ωg

from Eq. 5.11. For

a static gradient, the Ωsb is instead,

Ωsb =

4ΩgΩμ

ωr

. (5.23)

The microwave field detuning required to drive this sideband transition would be δ = ωr, without

considering the effects of ac Zeeman shifts. Following the discussion in Sec. 5.6.1, 2Ωμ ≤ ωr, and

the maximum value of the sideband Rabi frequency using a static gradient is 2Ωg. The maximum

value using an oscillating magnetic field gradient is Ωg, as half of the gradient strength is lost as
106

a counter-rotating term that does not participate in the interaction(Eq. 5.5). However, Ωsb = Ωg

is obtained for a lower microwave Rabi frequency Ωμ using an oscillating gradient as opposed to

a static gradient. Using an oscillating gradient, 2Ωμ = ωr − ωg as opposed to 2Ωμ = ωr/2 for the

static gradient. For surface traps especially, this is the difference between 2Ωμ/2π ≈ 1 MHz or

2Ωμ/2π ≈ 3 MHz for the oscillating and static gradients respectively, an order of magnitude in the

microwave power. In principle, ωr − ωg could be made smaller by setting ωg closer to ωr, but this is

limited by the electric field at ωg as discussed in sec 5.11.

5.10 Comparison to near-qubit-frequency oscillating gradient scheme

Compared to the near-qubit-frequency oscillating gradient scheme that requires gradients at

GHz frequencies, it is much easier to generate large currents at radio frequencies. The maximum

current that can be applied will likely be limited by Joule heating in the trap electrodes. This

heating is significantly lower for a current at megahertz frequencies than at gigahertz frequencies

due to the larger skin depth. From [Pozar, 2009], the skin depth δs for a conductor is defined as

δs =

r

2

ωμσ

, (5.24)

where ω is the angular frequency of the current, μ the magnetic permeability and σ the conductivity.

For gold, μ = μ0 = 4π × 10−7 N/A2 and σ = 4.098 × 107 S/m. For, ω/2π = 1.7 GHz, the skin depth

δs ≈ 2 μm. For, ω/2π = 5 MHz, the skin depth δs ≈ 35 μm. For our 8 μm thick electrodes, operating

at 5 MHz is effectively dc as far as heat dissipation goes. Furthermore, the magnitude of induced

return currents in neighboring electrodes [Warring et al., 2013a] is reduced at lower frequencies,

yielding a larger gradient at the ion for a given applied current.

An additional challenge is that one near-field gradient, and hence one large GHz oscillating

current, is needed per sideband. Thus, for a typical Mølmer-Sørensen like interaction, two such

gradients, from two large GHz currents, would be required.
107

5.11 Oscillating electric field

The finite impedance of the current-carrying electrodes results in an oscillating potential on

these electrodes at ωg. The resulting gradient in this potential at the ions gives rise to an electric

field that drives the ion’s motion. As for a classical oscillator, the amplitude of this motion is

∝ (ω

2

r − ω

2

g

)

−1

. As ωg → ωr, the amplitude of this motion gets larger, causing the ion to sample

increasingly anharmonic regions of the trap, and may even cause the ion to leave the trap altogether.

These effects will set a practical limit on how close ωg can be set to ωr.

5.11.1 Measuring electric field

In order to measure the electric field, we use a technique described in [Warring et al., 2013a].

This technique is similar to what we use for compensating excess micromotion out of the plane of the

trap as described in Sec. 4.4.1.2. An oscillating electric field at ωg will cause ion motion oscillating

at the same frequency. This motion enables a microwave magnetic field gradient to cause spin-flip

transitions when detuned from the qubit frequency by ±ωg. Following the discussion in Sec. 5.2, we

note that an oscillating magnetic field Bg at ωg can also give rise to spin flips when a microwave

magnetic field with detuning ωg from the qubit frequency is applied. To isolate the contribution

to the Rabi frequency due to electric fields at ωg, we perform the measurement using our “clock”

qubit, the hyperfine transition (|F = 3, mF = 1i ↔ |F = 2, mF = 1i) which is first-order insensitive

to magnetic field fluctuations at our applied magnetic field of |B~

0| = 21.3 mT. This yields Ωz ≈ 0

even if there is an oscillating magnetic field Bg at ωg at the ion, and as a result will not cause spin

flips. The Rabi frequency of the spin flips due to the electric field is given by,

Ωspin-flip =

2ΩμsbΩeωr

(ω2

r − ω2g

)

, (5.25)

where Ωe = qEr0/2~ and Ωμsb is the Rabi frequency of the sideband from the oscillating gradient

close to the qubit frequency. The amplitude of the oscillating electric field at the ion is E and q is

the elementary charge (Appendix D.6). Using the same parameters for the phases and amplitudes
108

when the magnetic field at ωg is nulled at the ion, Ωspin-flip/2π ≈ 3.7 kHz. The Rabi frequency

corresponding to the microwave magnetic field gradient Ωμsb/2π ≈ 0.5 kHz. Using these values, we

estimate the electric field amplitude to be ≈ 10 V/m at the ion.

5.11.2 Effect of electric field on ac Zeeman shift

The electric field oscillating at ωg also has an effect on the ac Zeeman shift due to the magnetic

field oscillating at ωg. While we null the magnetic field to a large extent, the electric field would

cause the ion to oscillate around the null point with amplitude re, where re ∝ (ω

2

r −ω

2

g

)

−1

. This will

cause the ion to sample a magnetic field whose maximum value is proportional to this amplitude,

B ≈ ∇Bre. Since the ac Zeeman shift ∆ac is proportional to B2,

∆ac ∝ r

2

e

,

∴ ∆ac ∝



1

ω2

r − ω2g

2

.

(5.26)

We measured the ac Zeeman shift as a function of the radial mode frequency ωr and use this

simple model to fit the data, which is shown in Fig. 5.13. We see good agreement between the data

and the fit. However, this model only accounts for the coupling of the electric field to one mode of

motion. Other modes which have an overlap with the electric field would have additional shifts with

a similar scaling as Eq. 5.26, except that ωr is replaced by the particular mode frequency.

5.11.3 Shift on motional frequency

Finally, the electric field also gives rise to a pseudopotential that modifies the curvature of the

electric fields at the ion, and thus the motional frequencies. For these experiments, the shift12 on

ωr was about 4 kHz. We measure this shift by comparing the motional frequency with and without

11 We were puzzled for a long time that we couldn’t null the ac Zeeman shift closer to 0. However, once we figured

out this effect, the ac Zeeman shifts were not really problematic for any experiments we wanted to as long as we could

calibrate it so we did not investigate this further.

12 This shift increases to about 15 kHz in later experiments as we increased the current at ωg.
109

5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6

Radial frequency (MHz)

100

200

300

400

500

AC Zeeman shift (kHz)

Figure 5.13: Plot of the ac Zeeman shift due to the electric field11at ωg versus the radial mode

ωr. The data are shown by the blue points and the blue line is a fit following Eq. 5.26. For these

experiments, ωg/2π = 5 MHz.

the application of currents oscillating at ωg. A similar effect was seen due to microwave electric

fields in [Warring et al., 2013a], with shifts of about 3 kHz on the motional frequencies.

5.11.4 Reducing the effect of the electric field

The effects from these electric fields can be reduced in a number of ways. First, with multiple

ions, differential motional modes which are not excited by a uniform electric field can be used.

In addition, one can compensate these oscillating electric fields by applying electric potentials

oscillating at the same frequency to the electrodes. One can also use trap designs which place the

current-carrying electrodes beneath a conducting layer [Welzel et al., 2019] which shields electric

fields more strongly than magnetic fields.
110

5.12 Higher-order interactions

As seen in Sec. 5.1, a spin-dependent displacement can cause a non-zero overlap between

otherwise orthogonal motional wavefunctions for the different spin states. For small displacements,

this allows sideband interactions that change the Fock state by ∆n = ±1 using an additional

microwave field. To allow higher order transitions, for example ∆n = ±2, a larger displacement

would be required. Using a static gradient, this would simply correspond to a larger gradient.

However, using an oscillating gradient, we can exploit the additional degree of freedom of being

able to set ωg relative to ωr to obtain a larger displacement without increasing the magnitude of

the field gradient.

Analogous to a classical harmonic oscillator, driving it close to resonance allows larger

displacements without a larger force. One application which we explore in this section is the

generation of higher-order interactions in a and a

† by tuning the oscillation frequency of the gradient

close to the oscillation frequency of the harmonic oscillator, i.e. setting ωg close to ωr.

5.12.1 Oscillating gradient interaction

Since we will be operating in a regime where ωg → ωr, we will ignore the term ∝ 1/(ωr + ωg) in

Eq. 5.7 and the propagator simplifies to

U

†

g

(t) ≈ exp h Ωg

ωr − ωg

σz



− aˆ(e

−i(ωr−ωg)t − 1)

+ ˆa

†

(e

i(ωr−ωg)t − 1)

i

.

(5.27)

We now use the BCH theorem to transform the microwave term into the interaction picture with

respect to the gradient term.

e

ACBe−AC = B + [AC, B] + 1

2

[AC, [AC, B]]... (5.28)

where A =

Ωg

ωr−ωg

σz, C =



−a(e

−i(ωr−ωg)t − 1) + a†

(e

i(ωr−ωg)t − 1)

and B = ~Ωμ(σ+e

−iδt +σ−eiδt).

To evaluate the commutator we make use of,
111

[σz, σ+] = 2σ+

[σz, σ−] = −2σ−,

and

[B, C] = 0

[A, C] = 0.

Using the result from appendix A.1.3,

Hg = e

ACBe−AC

= ~Ωμ



σ+e

−iδt exp( 2Ωg

ωr − ωg



− a(e

−i(ωr−ωg)t − 1) + a†

(e

i(ωr−ωg)t − 1)

)

+ σ−e

iδt exp( −2Ωg

ωr − ωg



− a(e

−i(ωr−ωg)t − 1) + a†

(e

i(ωr−ωg)t − 1)

)



= ~Ωμ



σ+e

−iδtD(α) + σ−eiδtD(−α)



,

(5.29)

where the displacement operator D(α) is

D(α) = exp (αa† − α

∗

a), (5.30)

with α

α =

2Ωg

ωr − ωg

(e

i(ωr−ωg)t − 1). (5.31)

5.12.2 Sideband interactions

We can expand the displacement operator to obtain sideband interactions. Keeping up to the

second order terms gives
112

D(α) = exp (αa† − α

∗

a)

≈ 1 + (αa† − α

∗

a) + 1

2

(α

2

a

†

2

− |α|

2

(a

†

a + aa†) + α

∗2

a

2

).

5.12.3 Single sideband interactions

Just keeping the first order terms, we obtain the first sideband interaction where,

Hsb = ~Ωμ(σ+e

−iδt − σ−eiδt)(αa† − α∗

a)

=

2~ΩμΩg

ωr − ωg

(σ+e

−iδt − σ−eiδt)(a†

(e

i(ωr−ωg)t − 1) − a(e−i(ωr−ωg)t − 1))

(5.32)

When δ = ωr − ωg we obtain the blue sideband interaction and when δ = −(ωr − ωg) we obtain the

red sideband interaction as described by Eq. 5.10.

As discussed in Sec. 5.6.1, the detuned microwave causes an ac Zeeman shift. Thus, the Rabi

frequency of the microwave can only be as large as the detuning for the single sideband due to the

ac Zeeman shift (2Ωμ < ωr − ωg). Thus, the maximum sideband Rabi frequency is Ωg.

5.12.3.1 Second sideband interaction

The second order terms give us the second sideband interaction. Dropping the a

†a and aa†

terms that are at different detunings,

Hssb =

~Ωμ

2

(σ+e

−iδt + σ−eiδt)(α2

a

†

2

+ α

∗2

a

2

)

≈

~Ωμ

2

 2Ωg

ωr − ωg

2

(σ+e

−iδt + σ−eiδt)(a†

2

(e

2i(ωr−ωg)t − 2ei(ωr−ωg)t + 1)

+ a

2

(e

−2i(ωr−ωg)t − 2e−i(ωr−ωg)t + 1)).

When δ = 2(ωr − ωg), we get the second blue sideband and when δ = −2(ωr − ωg) we get the second

red sideband. Similarly, because of the ac Zeeman shift, the maximum Rabi frequency of this second

sideband is Ω2

g/(ωr − ωg).
113

These interactions can be enhanced simply by tuning ωg closer to ωr i.e they can exist even

without a second order spatial derivative of the magnetic field; Ωg only depends on the first order

derivative. Of course, one must be careful of the electric field oscillating at ωg; as ωg → ωr, the

electric field also gets closer to resonance. The frequency difference ωr − ωg ∼ Ωg for this interaction

to have non-negligible strength.
There are two colours in my head

Radiohead

6

Entangling gates with a radiofrequency

oscillating magnetic field gradient

In the previous chapter, we discussed how one can use a radiofrequency magnetic field gradient

with one microwave field to couple an ion’s spin to its motion. In this chapter, we extend that

technique to implementing two-qubit entangling gates that use two microwave fields symmetrically

detuned from the qubit frequency. We analyze the dynamics of these fields in the “bichromatic”

interaction picture: the interaction picture with respect to the two microwave fields. We show that

we can choose between σˆyσˆy Mølmer-Sørensen-type gates and σˆzσˆz gates simply by changing the

detuning of the microwave fields. In addition, by appropriately choosing the Rabi frequency of

the microwave fields, we can make the gate less sensitive to qubit frequency fluctuations without

additional fields, known as intrinsic dynamical decoupling (IDD). These theoretical results were

published in [Sutherland et al., 2019].

Next, we experimentally demonstrate two-qubit entangling gates using a radiofrequency

magnetic field gradient. We implement both σˆyσˆy and σˆzσˆz gates, the latter being the focus of

this chapter. By tuning the microwave fields to the IDD point, we show an improvement in our

114
115

qubit coherence by more than an order of magnitude. We also measure the insensitivity of the

two-qubit gate to qubit frequency offsets, which enables the single-ion addressing presented in the

next chapter.

6.1 Bichromatic interaction picture

We first consider a general Hamiltonian for laser-free gates comprising an interaction with a

bichromatic microwave field whose components are symmetrically detuned from the qubit frequency

in addition to a magnetic field gradient that couples the ions’ spin to one mode of their motion.

The Hamiltonian is

Hˆ

lab(t) = ~ω0

2

Sˆ

z + ~ωraˆ

†

aˆ + 2~ΩμSˆ

i

n

cos([ω0 + δ]t) + cos([ω0 − δ]t)

o

+ 2~Ωgf(t)Sˆ

j

n

aˆ + ˆa

†

o

.

(6.1)

This Hamiltonian is for an N-ion string1, where the spin operators are Sˆ

i ≡

PN

n

σˆi,n, with i ∈ {x, y}

and j ∈ {x, y, z}. We assume for now that we are using the center-of-mass mode, so all ions

participate equally in the motion2. The Ωμ term describes two microwave fields with the same Rabi

frequency, one with a detuning +δ and one with a detuning −δ from the qubit frequency ω0. The

Ωg term represents a field gradient that couples the ions’ internal states to their motion, which is

characterized by creation and annihilation operators aˆ and aˆ

† and frequency ωr. The function f(t)

describes the time dependence of the field gradient, which we take to be either static or oscillating

sinusoidally.

We transform from Eq. 6.1 into an interaction picture with respect to Hˆ

0 = ~ω0Sˆz/2 +~ωraˆ

†aˆ,

i.e. the qubit and motional frequency. Making use of the results in Sec. A.1 including the rotating

wave approximation, we obtain the Hamiltonian,

1

In other sections, n is used to identify the motional Fock state |ni. However, in this chapter we use it to denote

the number of ions to maintain consistency with the paper in which these results were published [Sutherland et al.,

2019].

2 For other modes of motion, we would need to add coefficients bn to the spin operator for the gradient term such

that Sˆi ≡

P

n

bnσˆi,n. For the simple case of two identical ions, on the center of mass modes, b1 = b2 = 1. On the

out-of-phase modes, b1 = 1 = −b2.
116

Hˆ (t) = Hˆ

μ(t) + Hˆg(t)

= 2~ΩμSˆ

i cos(δt) + 2~Ωgf(t)Sˆj

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

. (6.2)

We identify two main components of the Hamiltonian, Hˆ

μ(t) and Hˆg(t), which we refer to as

the microwave field term and the gradient term, respectively. To more explicitly derive Hˆ

μ, let us

first analyze the case of Sˆ

i = Sˆx,

2~ΩμSˆ

x

n

cos([ω0 + δ]t) + cos([ω0 − δ]t)

o

→ ~Ωμ

nX

n

σˆ+,ne

−iδt +

X

n

σˆ−,ne

iδt blue-detuned microwave field

+

X

n

σˆ+,ne

iδt +

X

n

σˆ−,ne

−iδto

red-detuned microwave field,

= 2~ΩμSˆ

x cos(δt), (6.3)

where we make use of e

iδt + e−iδt = 2 cos(δt) as we transform into rotating frame of the qubit. We

dropped the fast-rotating terms through the rotating wave approximation. A similar result can be

shown for Sˆ

i = Sˆy and is left as an exercise for the reader3

. We assume that all ions are identical

with the same qubit frequencies, and can all be addressed with a single pair of microwave fields.

We can generalize this formalism to the case of multiple qubit frequencies, either for different ion

species, or for ions of the same species with different qubit frequencies as discussed for example in

[Khromova et al., 2012], by using multiple pairs of microwave fields. Note that the result in Eq. 6.2

is valid unconditionally for Sˆ

j = Sˆz. This result also holds for specific cases j ∈ {x, y}, such as the

gradient term comprising two gradients oscillating with frequencies symmetrically detuned from the

qubit, i.e. f(t) = cos([ω0 + δ

0

]t) + cos([ω0 − δ

0

]t).

We transform Eq. 6.2 into the bichromatic interaction picture, the interaction picture with

respect to Hˆ

μ. In performing this step, we make use of the formalism developed in [Roos, 2008] for

laser-driven gates. We first derive the propagator from Hμ,

3

I’ve always wanted to say that.
117

Uˆ

μ(t) = exp n

−

i

~

Z t

0

dt0Hμ(t

0

)

o

= exp n− 2iΩμSˆ

i

Z t

0

dt0cos(δt0)

o

= exp n− iF(t)Sˆ

i

o

, (6.4)

with F(t) ≡

2Ωμ sin(δt)

δ

. The interaction picture Hamiltonian is then,

Hˆ

I (t) = Uˆ†

μ

(t)Hˆ

g(t)Uˆμ(t)

= 2~Ωgf(t)

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

e

iF(t)Sˆ

iSˆ

je

−iF(t)Sˆ

i

,

(6.5)

where the terms corresponding to the motion commute with the terms corresponding to the qubit.

All the interesting dynamics are in the qubit term with only spin operators, which we can simplify

by using e

ibσˆi = ˆI cos(b) + iσˆi sin(b), where b is scalar quantity. First, using the commutativity of

the different ion operators,

e

iF(t)Sˆ

iSˆ

je

−iF(t)Sˆ

i =

X

n

e

iF(t)ˆσi,n σˆj,ne−iF(t)ˆσi,n . (6.6)

Now focusing on the single-ion operators and dropping the n subscript for simplicity,

e

iF(t)ˆσiσˆje−iF(t)ˆσi

=

n

ˆI cos(F(t)) + iσˆi sin(F(t))o

σˆj

n

ˆI cos(F(t)) − iσˆi sin(F(t))o

=ˆσj cos2(F(t)) − iσˆjσˆi cos(F(t)) sin(F(t)) + iσˆiσˆj cos(F(t)) sin(F(t)) + ˆσiσˆjσˆi sin2(F(t))

=ˆσj cos2(F(t)) + i[ˆσi, σˆj ] cos(F(t)) sin(F(t)) + ([ˆσi, σˆj ] + ˆσjσˆi)ˆσi sin2(F(t))

=ˆσj



cos2(F(t)) + sin2(F(t))+ [ˆσi

, σˆj ] sin(F(t)) (i cos(F(t)) + ˆσi sin(F(t)))

=ˆσj + i[ˆσi, σˆj ] sin(F(t)) (cos(F(t)) − iσˆi sin(F(t)))

=ˆσj + i[ˆσi, σˆj ] sin(F(t))e

−iF(t)ˆσi

,

(6.7)

where we have also made use of ˆσ

2

i = ˆI. Using this result in Eq. 6.5 gives
118

Hˆ

I (t) = 2~Ωgf(t)

n

aeˆ

−iωrt + ˆa†

e

iωrt

onSˆ

j + i[Sˆi

, Sˆ

j ] sin(F(t))e

−iF(t)Sˆ

i

o

.

If i = j, then Eq. 6.7= ˆσj , and Hˆ

I (t) = Hˆ

g(t). However, if i 6= j, Eq. 6.7 becomes

σˆj + i[ˆσi, σˆj ] sin(F(t))e

−iF(t)ˆσi

,

=ˆσj − 2ijkσˆk sin(F(t))e

−iF(t)ˆσi

,

=ˆσj − 2ijkσˆk sin(F(t))(cos(F(t)) − isin(F(t))ˆσi),

=ˆσj − 2ijkσˆk sin(F(t)) cos(F(t)) + 2iijkσˆkσˆi sin2(F(t)),

=ˆσj − ijkσˆk sin(2F(t)) − 2ˆσj sin2(F(t)),

=ˆσj cos(2F(t)) − ijkσˆk sin(2F(t)),

(6.8)

where we understand that σˆkσˆi = iσˆj and ijk is the Levi-Civita symbol. We can use the Jacobi-Anger

expansion [Abramowitz and Stegun, 1972], to simplify this further.

cos(z sin θ) = J0(z) + 2X∞

n=1

J2n(z) cos(2nθ),

sin(z sin θ) = 2X∞

n=0

J2n+1(z) cos((2n + 1)θ),

where Jn is the n

th Bessel function4

. Finally, using z =

4Ωμ

δ

, θ = δt, and putting back the summation

Sˆ

i =

PN

n

σˆi,n in Eq. 6.5 we obtain

Hˆ

I (t) = 2~Ωgf(t)

n

aeˆ

−iωrt + ˆa†

e

iωrt

onSˆ

j

h

J0

4Ωμ

δ



+ 2X∞

n=1

J2n

4Ωμ

δ



cos(2nδt)

i

−2ijkSˆ

k

X∞

n=1

J2n−1

4Ωμ

δ



sin([2n − 1]δt)

o

, (6.9)

This is the main result from [Sutherland et al., 2019]. One interpretation of this result is that the

Hˆ

μ term effectively provides a frequency modulation of the gradient term Hˆg. When i 6= j, we

4 Not to be confused with the n in Sˆi =

PN

n

σˆi,n which refers to the ion number. We stick with n here to be

consistent with Ref. [Sutherland et al., 2019].
119

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

4 /

0.0

0.2

0.4

0.6

0.8

1.0

| J

n

(

4

/

)|

Intrinsic dynamical

decoupling (IDD)

J0

J1

J2

Figure 6.1: Plot of |Jn(4Ωμ/δ)| versus 4Ωμ/δ. The interactions described in Eq. 6.9 are proportional

to Bessel functions Jn; J0, J1, and J2 are plotted. Each of those interactions correspond to a

different condition on δ and different spin operators. Any qubit frequency error would be scaled

by the J0 function as described in Sec. 6.1.1. By choosing 4Ωμ/δ such that J0 = 0, we perform

intrinsic dynamical decoupling (IDD). The σˆzσˆz gate that we perform experimentally has a gate

Rabi frequency that is proportional to J2.

obtain an infinite series of resonances which can be used to implement different spin-spin interactions

between ions, with a suitable choice of parameters. Even resonances (2n) produce an interaction

corresponding to the spin term Sˆ

j in Hˆg. Odd resonances (n = 2n − 1) correspond to [Sˆi

, Sˆ

j ], the

commutator of the spin-terms in Hˆ

g and Hˆμ. The strengths of these interactions correspond to the

Bessel functions Jn as shown in Fig. 6.1. We can choose which interaction to implement by choosing

the value of δ, the detuning of the bichromatic fields. We assume that the Rabi frequency of the

gradient term Ωg  δ, such that when we pick one resonance we can ignore the others5. The other

interactions will have excitations proportional to 

Ωg

δ

2

. We analyze three different implementations

Hˆ

g in Sec. 6.1.3; a static magnetic field gradient, a magnetic field gradient oscillating at near-qubit

frequencies, and finally what we implement in this thesis, a radiofrequency magnetic field gradient

near the motional frequency.

5 For our experiments, Ωg/2π ∼kHz and δ/2π ∼MHz.
120

6.1.1 Intrinsic dynamical decoupling

Aside from elucidating the dynamics of the spin-dependent force term Hˆ

g in the presence

of large bichromatic microwave fields, the bichromatic interaction picture also reveals important

dynamics of qubit frequency errors. This analysis shows that we can perform dynamical decoupling

of the qubit without any additional fields. Dynamical decoupling is useful for error suppression [Viola

and Lloyd, 1998, Viola et al., 1999, Uhrig, 2007] and has been used in a number of different ion

trapping experiments such as [Biercuk et al., 2009, Timoney et al., 2011, Tan et al., 2013] and more

specifically in laser-free gates such as demonstrated in Ref. [Weidt et al., 2016, Harty et al., 2016].

In contrast to those implementations, we do not require additional fields for dynamical decoupling.

To illustrate this intrinsic dynamical decoupling (IDD), we add an error term to our Hamiltonian in Eq. 6.9

Hˆ

z =

~

2

Sˆ

z, (6.10)

where  is the magnitude of the qubit frequency shift, which we assume is the same for all the ions.

This qubit frequency shift can arise from a shift in the quantization axis magnetic field, changes in

the ac Zeeman shift from the microwave fields, or just an error in calibrating the qubit frequency.

This shift can be static or time-varying (slowly compared to 1/δ). This error term transforms into

the bichromatic interaction picture as

Hˆ

I,z =

~ε

2

n

Sˆ

z

h

J0



4Ωμ

δ



+ 2P∞

n=1 J2n



4Ωμ

δ



cos(2nδt)

i

+2ikzSˆ

k

P∞

n=1 J2n−1



4Ωμ

δ



sin([2n − 1]δt)

o

. (6.11)

As long as  does not have frequency components near nδ, most of the terms in Eq. 6.11 will be

varying quickly with time and can be ignored. As n gets larger, Jn also in general gets smaller.

Thus, the leading error term, especially if  is static, is proportional to J0



4Ωμ

δ



. Thus, by choosing

4Ωμ/δ ≈ 2.405 such that J0



4Ωμ

δ



= 0, we are only left with the higher order terms that oscillate
121

quickly. At this point, the J1 and J2 values are approximately 0.519 and 0.432, as opposed to their

peak values of 0.582 and 0.486 respectively. Thus, choosing the IDD point will come at a slight cost

to the interaction strength. We investigate IDD experimentally in Sec. 6.2.1.1.

6.1.2 Pulse shaping

In this chapter, we have explained how analyzing our Hamiltonians in the bichromatic

interaction picture has simplified the dynamics. Here, we show how we can employ pulse shaping

of the microwave fields to move smoothly in and out of the interaction picture, such that the

dynamics in the original interaction picture (with respect to the qubit and motion) are identical to

the dynamics in the bichromatic interaction picture.

As shown in Eq. 6.2, our Hamiltonian is composed of two parts,

Hˆ (t) = Hˆ

μ(t) + Hˆg(t), (6.12)

where Hˆ

μ(t) is our bichromatic microwave term and Hˆg(t) is the gradient term. We assume Hˆμ

commutes with itself at all times so its propagator can be determined simply as

Uˆ(t) = exp n−

i

~

Z t

0

dt0Hˆ

μ(t

0

)

o

. (6.13)

The Hamiltonian in the bichromatic interaction picture Hˆ

I (t) is

Hˆ

I (t) = Uˆ†

(t)Hˆ

g(t)Uˆ(t). (6.14)

In this interaction frame, our initial state |ψ(t)i → |φ(t)i = Uˆ†

|ψ(t)i, whose time evolution is

governed by,

i~ |φ ̇(t)i = Hˆ

I (t)|φ(t)i. (6.15)
122

Solving this equation, we obtain the time evolution operator Tˆ

I (t) such that |φ(t)i = TˆI (t)|φ(0)i.

The final state in our original frame is

|ψ(tf )i = Uˆ(tf )|φ(tf )i

= Uˆ(tf )Tˆ

I (tf )|φ(0)i

= Uˆ(tf )Tˆ

I (tf )Uˆ†

(0)|ψ(0)i.

(6.16)

Uˆ†(0) = ˆI, so if Uˆ(tf ) → ˆI, we find that

|ψ(tf )i → Tˆ

I (tf )|ψ(0)i, (6.17)

and the state in the bichromatic interaction picture and the ion frame (with respect to qubit and

motion) are identical. We achieve this condition of Uˆ(tf ) → ˆI by pulse shaping our microwave fields,

turning the microwave fields on and off smoothly in a time τ that is long compared to 2π/δ. We

add time dependence to Ωμ in Hˆ

μ(t) such that

Hˆ

μ(t) → 2Ωμg(t) cos(δt)Sˆi

. (6.18)

We impose the following boundary conditions on g(t)

g(0) = g(tf ) = 0,

g(τ ≤ t ≤ tf − τ ) = 1,

(6.19)

i.e. our microwaves are off at the start and end of the gate sequence and between the ramps they

have a constant amplitude. We now evaluate Uˆ(tf ),

Uˆ(tf ) = exp n− i

Z tf

0

dt

0

2Ωμg(t

0

) cos(δt0)Sˆ

i

o

,

= exp n2iΩμ

δ

 Z τ

0

dt0g ̇(t

0

) sin(δt0) + Z tf

tf −τ

dt0g ̇(t

0

) sin(δt0)



Sˆ

i

o

,

(6.20)
123

where we have integrated by parts and made use of the boundary conditions on g(t). If g ̇(t) is

small compared to δ, then the value of the integrals will be roughly proportional to 2π/(τ δ), so for

τ  2π/δ, these integrals will vanish and Uˆ(tf ) → ˆI. This condition can also be fulfilled by picking

ramp times such that δτ = 2nπ. We note that the pulse shaping will change the optimal gate time

slightly as there are some gate dynamics during the ramps. In practice, we do not fulfill τ  2π/δ

to reduce the ramp times, which is discussed in more detail in Sec. 6.4.6.

6.1.3 Physical implementations

Using the formalism that we have discussed so far, we describe three different types of laser-free

entangling gates using three different types of magnetic field gradients: a static magnetic field

gradient, a magnetic field gradient oscillating at near-qubit frequencies, and a magnetic field gradient

oscillating at radio frequencies close to the motional frequency. The latter method was discussed

in the previous chapter and is used for the gates we perform in this chapter. Each of these three

implementations will have different spin operators Sˆ

i and Sˆj and time dependencies of the gradient

f(t).

6.1.3.1 Static magnetic field gradient

We study a static magnetic field gradient with a pair of microwave fields symmetrically

detuned from the qubit frequency as described in [Lake et al., 2015]. Equation 6.2, with Sˆ

i = Sˆx,

Sˆ

j = Sˆz and f(t) = 1, becomes

Hˆ (t) = 2~ΩμSˆ

x cos(δt) + 2~ΩgSˆz

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

. (6.21)

In the bichromatic interaction picture, this Hamiltonian transforms as,
124

Hˆ

I (t) = 2~Ωg

n

aeˆ

−iωrt + ˆa†

e

iωrt

onSˆ

z

h

J0

4Ωμ

δ



+ 2X∞

n=1

J2n

4Ωμ

δ



cos(2nδt)

i

+2Sˆ

y

X∞

n=1

J2n−1

4Ωμ

δ



sin([2n − 1]δt)

o

. (6.22)

We obtain a ˆσyσˆy interaction for (2n − 1)δ ≈ ωr and a ˆσzσˆz interaction for 2nδ ≈ ωr.

6.1.3.2 Magnetic field gradient oscillating at near-qubit frequencies

We next study a magnetic field gradient oscillating close to the qubit frequency [Ospelkaus

et al., 2011, Harty et al., 2016]. We consider the case that the microwave fields are not nulled

and are oscillating at the same frequencies and have the same spin-operator as the gradient. This

corresponds to Sˆ

i = Sˆx, Sˆj = Sˆx, and f(t) = cos(δt) in Eq. 6.2. The interaction Hamiltonian for

this system is,

Hˆ (t) = 2~Ωμ cos(δt)Sˆ

x + 2~Ωg cos(δt)Sˆx

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

. (6.23)

We note that the microwave term commutes with the gradient term, giving the bichromatic

interaction picture Hamiltonian

Hˆ

I (t) = 2~Ωg cos(δt)Sˆ

x

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

. (6.24)

This Hamiltonian results in a σˆxσˆx interaction. While previous implementations nulled the oscillating

magnetic fields at the ion, we show that those fields can be used for intrinsic dynamical decoupling.

6.1.3.3 Magnetic field gradient oscillating at near-motional frequencies

We extend the technique developed in the previous chapter to two microwave fields in addition

to the oscillating magnetic field gradient close to the motional frequency at radio frequency. The
125

2π × MHz 2π × GHz

ωr

ωg

ω0

δ

Figure 6.2: Spectrum of applied fields for two-qubit entangling gates. We have a strong radiofrequency magnetic field gradient oscillating at ωg close to the motional mode frequency ωr. At the

gigahertz end of the spectrum, we have two microwave fields that are symmetrically detuned by δ

from the qubit frequency ω0.

spectrum of fields used is shown in Fig. 6.2. With Sˆ

i = Sˆx, Sˆj = Sˆz, and f(t) = cos(ωgt), Eq. 6.2

becomes

Hˆ (t) = 2~Ωμ cos(δt)Sˆ

x + 2~Ωg cos(ωgt)Sˆz

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

, (6.25)

where we follow the same definitions for Ωμ and Ωg as in Sec. 5.1. Note that this reduces to Eq. 6.21

for ωg = 0. In the bichromatic interaction picture, our Hamiltonian is

Hˆ

I (t) = 2~Ωg cos(ωgt)

n

aeˆ

−iωrt + ˆa†

e

iωrt

onSˆ

z

h

J0

4Ωμ

δ



+2X∞

n=1

J2n

4Ωμ

δ



cos(2nδt)

i

+ 2Sˆ

y

X∞

n=1

J2n−1

4Ωμ

δ



sin([2n − 1]δt)

o

.

(6.26)

Compared to the static gradient case, the resonant interactions occur when nδ = |ωr ± ωg|

as opposed to nδ = ωr. For smaller values of δ, lower microwave Rabi frequencies Ωμ are required

to achieve the same argument of the Bessel functions 4Ωμ/δ. We can view the microwave term

in Eq. 6.25 as providing a frequency modulation to the gradient term δ as illustrated in Fig. 6.3.

Because the term providing the frequency modulation does not commute with the gradient term,
126

3 2 1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

| J

n|

g 2 g + 2

z z z z

g 3 g + 3

y y y y

g

z z

g g +

y y y y

r

Figure 6.3: Spectrum of frequency modulation of the spin-dependent force term. The microwave

term in Eq. 6.25 provides a frequency modulation at the microwave detuning δ of the gradient term

which provides the spin-dependent force. Even terms correspond to an effective σˆzσˆz interaction

and odd terms correspond to an effective σˆyσˆy interaction. When one of the frequency modulation

terms is close to the motional frequency ωr, we can drive an entangling interaction. We operate at

the IDD point where J0



4Ωμ

δ



= 0.

odd resonances ((2n + 1)δ ≈ |ωr ± ωg|) implement a σˆyσˆy interaction while even resonances

(2nδ ≈ |ωr ± ωg|) implement a ˆσzσˆz interaction.

For our gate demonstrations, we use the J2 resonance where 2δ ≈ ωr − ωg which corresponds

to

Hˆ

zz = ~ΩgJ2



4Ωμ

δ



(ˆσz1 − σˆz2)

n

aeˆ

i∆t + ˆa†

e

−i∆t

o

, (6.27)

where the gate Rabi frequency (following Eq. 2.31) is ΩgJ2



4Ωμ

δ



and ∆ = 2δ − (ωr − ωg). We

have also replaced Sˆ

z with the spin operators for each of the two ions for the out-of-phase radial

mode. This gate is the focus of the rest of the chapter. However, we also briefly discuss the

Mølmer-Sørensen-type ˆσyσˆy interaction in Sec. 6.5 which occurs when δ ≈ ωr − ωg, and

Hˆms = i~ΩgJ1



4Ωμ

δ



(ˆσy1 + ˆσy2)

n

−aeˆ

i∆t + ˆa†

e

i∆t

o

, (6.28)

where the gate Rabi frequency is ΩgJ1



4Ωμ

δ



and ∆ = δ − (ωr − ωg) and we have replaced Sˆ

y with

the individual spin operators for the center of mass mode on which the gate was performed.
127

6.1.4 σˆzσˆz interaction without microwave fields

From Eq. 6.25, we note that we could drive a σˆzσˆz interaction directly by removing the

microwave fields and just having our gradient frequency ωg close to the motional frequency ωr. In

this case, our Hamiltonian would be,

Hˆ (t) = 2~Ωg cos(ωgt)Sˆ

z

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

. (6.29)

This type of gate has been proposed in [Leibfried et al., 2007, Ospelkaus et al., 2008]. However, this

gate is challenging to implement as there is a spin-independent electric field that will be oscillating

at ωg. This electric field can excite the ion motion and cause other adverse affects as described in

Sec. 5.11. This gate has been implemented in our apparatus [Burd, 2020], with a highest Bell-state

fidelity of approximately 0.8.

6.1.5 Rabi frequency imbalance of microwave fields

Thus far, we have assumed that our bichromatic field comprises two microwave fields with

equal amplitude. In this section, we analyze the effect of an amplitude imbalance between the two

fields. For this analysis, we ignore the gradient term. From Eq. 6.3, our Hamiltonian is

2~ΩμSˆ

x cos([ω0 + δ]t) + 2(~Ωμ + ~)Sˆx cos([ω0 − δ]t)

→ ~Ωμ

nX

n

σˆ+,ne

−iδt +

X

n

σˆ−,ne

iδt blue-detuned microwave field

+

X

n

σˆ+,ne

iδt +

X

n

σˆ−,ne

−iδto

red-detuned microwave field

+~

X

n

σˆ+,ne

iδt +

X

n

σˆ−,ne

−iδt

red-detuned field with amplitude 

= 2~ΩμSˆ

x cos(δt) + ~



Sˆ

x cos(δt) − Sˆy sin(δt)



,

where the red-detuned microwave field has a slightly different Rabi frequency Ωμ + . We now
128

transform to the bichromatic interaction picture with respect to the Ωμ term and obtain

Hˆ

I,(t) = ~Sˆ

x cos(δt)

+~

n

Sˆ

y sin(δt)

h

J0

4Ωμ

δ



+ 2X∞

n=1

J2n

4Ωμ

δ



cos(2nδt)

i

+2Sˆ

z sin(δt)

X∞

n=1

J2n−1

4Ωμ

δ



sin([2n − 1]δt)

o

.

(6.30)

The terms on the first and second lines have fast oscillations at δ and can be ignored. However, the

term on the last line will produce stationary terms corresponding to σˆz (Sˆ

z). If one is performing a

σˆzσˆz gate corresponding to the even resonances in Eq. 6.26, this term will commute with the gate

interaction and can be treated as a static shift that can be echoed out. However, if one is performing

a σˆyσˆy gate corresponding to the odd resonances, this error term will no longer commute and will

produce an error. This gate error will be small if Ωg  . As Ωg/2π ∼kHz, we would require 

to be much smaller than that. This requirement can be challenging to achieve experimentally as

Ωμ/2π ∼MHz. Thus, both fields would need to be balanced at the 10−6level; practically anything

smaller than 10−3 would be difficult. We note that an additional dynamical decoupling field could

be added to reduce sensitivity to this term, but that has its own challenges as discussed in Sec. 6.5.

6.2 Experimental implementation and calibration of σˆzσˆz gates

We follow a similar implementation as described in Sec. 5.3. We perform a σˆzσˆz gate

corresponding to the J2 resonance in Eq. 6.26. We use an out-of-phase radial mode6 with frequency

ωr/2π ≈ 6.9 MHz. The frequency of our gradient is ωg/2π = 5 MHz. For the σˆzσˆz gate, the detuning

of the microwave fields is δ ≈ (ωr − ωg)/2, with δ/2π ≈ 0.95 MHz. For these gate experiments, we

use currents of ≈ 1.2 A rms per electrode, producing a peak magnetic field gradient of ≈ 120 T/m at

the ion. This corresponds to a gradient Rabi frequency of Ωg/2π ≈ 1.25 kHz. In the next sections,

we describe the calibration experiments performed and then discuss the results of our experiments.

As we use the out-of-phase (rocking) radial mode, the calibrations that require spin-motion

6 We initially performed the gate on the center of mass mode, but the highest fidelity entanglement was obtained

with the out-of-phase rocking mode. We discuss these other variations including two and four loop gates in Sec. 6.4.9.
129

coupling need to be done with two ions. Initially when the gate was performed on the center of

mass mode, many of the gate dynamics could be investigated with a single ion. Because the two-ion

lifetime is much shorter than the single-ion lifetime, we separate the calibrations into those that can

be done with a single ion and those that require two.

For the single-ion calibrations, we perform the following steps:

• Calibrate the microwave π and π/2 pulses used in the gate as described in Sec. 4.4.4.

• Null the oscillating magnetic field as discussed in Sec. 5.5.

• Following the discussion in Sec. 6.1.1, we tune the Rabi frequency of the microwave fields to

be at the intrinsic dynamical decoupling (IDD) point.

• Tune the bias voltage to maximize the gradient along the mode.

• Measure the ac Zeeman shift from the magnetic field orthogonal to the quantization axis

oscillating at ωg.

With two ions:

• We describe how we choose our motional frequencies to avoid resonances at ωg.

• We calibrate the cooling of the motional mode using the radiofrequency gradient as described

in Sec. 5.7, except we perform this cooling with two microwave fields instead of one.

• We calibrate the gate pulse sequence and the phase of the final π/2 pulse to take into

account any asymmetries in our gate ramps.

• We calibrate the gate time and detuning to optimize the fidelity of our gate.

6.2.1 Calibrating microwave Rabi frequencies

We want to set the Rabi frequency of the microwave fields such that J0



4Ωμ

δ



= 0. In this

case, the detuning δ/2π ≈ 0.95 MHz, which requires Ωμ/2π = 0.571 MHz. We perform a coarse
130

calibration by measuring this Rabi frequency on resonance for each of the microwave lines. A finer

calibration is performed using an additional field on resonance. This additional field has a much

weaker Rabi frequency Ωc/2π ≈4.2 kHz. The Hamiltonian of these three fields is

Hˆ (t) =2~(Ωμ + ~)Sˆ

x cos([ω0 + δ + ∆]t) blue-detuned field

+2~ΩμSˆ

x cos([ω0 − δ + ∆]t) red-detuned field

+2~ΩcSˆ

x cos([ω0 + ∆]t). weaker carrier field

We adjust the phase of the weaker carrier field to be the same as the beatnote of the blue and

red detuned fields. We choose the pulse duration such that the weaker field drives a spin flip on

resonance. We add a frequency offset ∆ to all three fields. This offset can be seen as a qubit

frequency error as described in Sec. 6.1.1. If the red and blue detuned microwave fields are balanced

and at the IDD point, we should see a broadening of the resonance driven by the weaker field as

shown in Fig. 6.4.

Following the discussion in Sec. 6.1.5, if the two fields have an imbalance , an additional

term also weakly drives spin flips and we have an additional shift that is not removed by the IDD.

This imbalance results in an asymmetry in the scan that we can use to calibrate the Rabi frequency

more precisely. The data for this calibration is shown in Fig. 6.5.

6.2.1.1 Intrinsic dynamical decoupling

Our qubit is field-sensitive; therefor the qubit will have frequency fluctuations due to changing

(nominally static) magnetic fields or changing ac Zeeman shifts from the various oscillating magnetic

fields. In order to perform a high-fidelity gate, we need to suppress the effects of these fluctuations.

We investigate the effect of intrinsic dynamical decoupling (IDD) on the qubit coherence by

performing a spin-echo Ramsey experiment with and without IDD. The results are presented in

Fig. 6.6. We find that the IDD extends our qubit coherence by more than order of magnitude, and

for gate durations shorter than 1 ms, the effects of qubit decoherence are small.
131

600 400 200 0 200 400 600

Qubit frequency offset (kHz)

0.0

0.2

0.4

0.6

0.8

1.0

Population

IDD

No IDD

Figure 6.4: Rabi spectroscopy with and without intrinsic dynamical decoupling (IDD). With the

bichromatic field off (no IDD), we have only one field with Rabi frequency Ωc/2π ≈ 4.2 kHz which

is pulsed for the duration of a π pulse that flips the ion spin from |↓i → |↑i. As we sweep the

frequency of this field, we see a narrow resonance shown by the points in green. With the IDD on

the other hand (purple points), we see a broadening of this line as the IDD reduces the sensitivity

to qubit frequency offsets.

6.2.2 Calibration of mode orientation

We tune the bias voltage on our RF electrodes as described in Sec. 3.6. As we change the

bias, we change the orientation of the radial modes, and thus their overlap with the magnetic field

gradient. As we maximize the overlap of the gradient with the mode, we maximize the gate speed

using that mode. Increasing the gate speed typically increases the fidelity and reduces sensitivity

to many different errors. As shown in [Warring et al., 2013a] using a similar trap geometry, this

gradient is optimized at the maximum frequency of the low-frequency radial mode. We measure this

frequency as a function of bias voltage and plot this variation in Fig. 6.7. We set the bias voltage to

sit at the maximum frequency.

We note that it is experimentally challenging to verify that this mode orientation has the

largest gradient. Verification would require careful calibrations of the gate times and detunings, and

improvements in the gate speed would be small. For these experiments we simply set the bias as
132

a)

b)

c)

Figure 6.5: Plot of population in |↑i versus qubit frequency offset for three cases of the bichromatic

field Rabi frequences. For each of the three plots, the Rabi frequency of the blue-detuned microwave

field is varied while the Rabi frequency of the red-detuned microwave field is kept constant at

the ideal value of Ωμ. An additional carrier pulse transfers |↓i → |↑i on resonance. For each

qubit frequency offset, the frequency of all three fields is shifted by that amount. a) The blue

detuned microwave has a Rabi frequency of 0.996 Ωμ. b) The blue detuned microwave has a Rabi

frequency of ≈1.006 Ωμ. c) The blue detuned microwave has a Rabi frequency of Ωμ. When the

Rabi frequencies of the blue and red detuned fields are balanced, we see no asymmetry in the scan.

described and optimized the gate parameters there. Simulations could also be performed to verify

the mode orientation.
133

10

1 100 101 102

Total spin-echo duration (ms)

0.2

0.4

0.6

0.8

1.0

Spin-echo Ramsey contrast

With IDD

Without IDD

Figure 6.6: Qubit coherence with and without intrinsic dynamical decoupling (IDD). We perform

a spin-echo Ramsey experiment with and without IDD and plot the contrast as a function of the

interrogation time. For each point, we scan the phase of the second π/2 pulse in the spin-echo

Ramsey sequence and measure the contrast of the resulting oscillation. We find that with IDD (blue

points) we extend the qubit coherence by more than an order of magnitude compared to the bare

qubit (orange points).

6.2.3 Measuring ac Zeeman shifts from magnetic field at ωg

The qubit frequency experiences an ac Zeeman shift from the magnetic fields oscillating at ωg.

While we null the magnetic field at ωg along the quantization axis, the orthogonal components of the

field are not nulled and give rise to a significant ac Zeeman shift on the qubit frequency. We measure

this for both the |F = 3, mF = 3i ↔ |F = 2, mF = 2i and |F = 3, mF = 2i ↔ |F = 2, mF = 2i

transition. The former is the qubit transition for our gate and the shift is ≈ −400 kHz. The

latter corresponds to a additional transition required for our repump sequence for cooling while the

gradient at ωg is on and the shift is ≈ −240 kHz.

6.2.4 Tuning motional mode frequencies

It is important to be aware of the frequencies of all six of the two-ion modes. Crucially, we

want to avoid any mode frequencies, or the difference frequency of any of the modes at ωg. An
134

0.34 0.36 0.38 0.40 0.42 0.44 0.46

Bias voltage (V)

6.935

6.940

6.945

6.950

6.955

6.960

6.965

6.970

Radial frequency (MHz)

Figure 6.7: Plot of radial frequency versus bias voltage. This radial frequency is the lower radial

frequency for a single ion. The higher radial frequency is at ≈7.4 MHz. As we change the bias

voltage, we change the orientation of the radial modes and their overlap with the gradient. We set

the bias to the maximum of the curve, corresponding to a bias voltage of 0.405 V, to maximize the

overlap with the gradient.

electric field at ωg would excite those modes as described in Sec. 5.11. Furthermore, there is also

the oscillating magnetic field at ωg to consider. If the magnetic field along the quantization axis is

not nulled completely, the Hamiltonian for the residual magnetic field appears in the bichromatic

interaction picture as

Hˆ

I,z =

~Ωz

2

cos(ωgt)

n

Sˆ

z

h

J0



4Ωμ

δ



+ 2P∞

n=1 J2n



4Ωμ

δ



cos(2nδt)

i

−2Sˆ

y

P∞

n=1 J2n−1



4Ωμ

δ



sin([2n − 1]δt)

o

, (6.31)

where we follow a similar derivation as Sec. 6.1.1 and Ωz is defined as in Sec. 5.5. If nδ ≈ ωg, some

of the higher order terms in Eq. 6.31 will no longer be fast-oscillating and cannot be ignored. These

terms could excite additional resonances that cause gate errors. These resonances can be avoided

by choosing ωr and ωg such that nδ for a particular gate is far from resonance. Experimentally, ωg

is fixed for us due to hardware considerations and we adjust ωr by adjusting the bias as described
135

in Sec. 3.6 or the scale for the axial confinement. The Raman beams are used to find the difference

frequencies between our radial and axial modes as our microwave and rf gradients do not couple to

the axial modes (see Sec. 3.1).

6.2.5 Cooling with a radiofrequency gradient and a bichromatic microwave field

We have shown in Sec. 5.7 that a motional mode can be cooled with a radiofrequency gradient

and a single microwave field. Here, we extend this to two microwave fields, carefully tuning the

blue sideband off resonance so it causes negligible heating. The reasons are two-fold. First, as we

increase the Rabi frequency of the microwave fields, we increase the ac Zeeman shift. This limits the

microwave Rabi frequencies that can be used as seen in Sec. 5.6.1. By including a second microwave

field with the opposite sign for its detuning δ, we cancel the ac Zeeman shifts from each of the fields

to first order. Second, we perform the cooling sequence before the gate. Thus, by turning on all the

fields that are required for the gate during the cooling, we warm up all the relevant hardware and

reduce duty cycle or thermal effects during the gate itself.

For a single microwave field, δred = −(ωr − ωg) to drive the red sideband, ignoring the

effects of ac Zeeman shifts. To cool it with two fields, we have a second blue-detuned field at

δblue = ωr − ωg + ∆offset, where ∆offset is an offset from resonance. Without ∆offset, we would drive

a spin-dependent force corresponding to the σˆyσˆy interaction in Eq. 6.26. These values for δ are

different from the detuning of the gate fields at δ = ±

ωr−ωg

2

. We keep the Rabi frequency of the

microwave fields at the same values that are used for the gate sequence. The cooling sequence could

be optimized further by adjusting this Rabi frequency to maximize the sideband interactions, but

we eschew this step as this would be slow in our experiment.

Factoring in the ac Zeeman shifts, the detunings for both the fields are

δred = −(ωr − ωg) + ∆ac, (6.32)

δblue = ωr − ωg + ∆ac + ∆offset, (6.33)
136

where ∆ac is the ac Zeeman shift from all three fields: the two microwave fields and the magnetic

field at ωg. With all three fields, ∆ac/2π ≈ −390 kHz as opposed to -400 kHz without the microwave

fields. This shift was measured with ∆offset/2π = 100 kHz. We pick this value to be large compared

to the sideband Rabi frequency, so it does not cause additional heating by driving motion-adding

transitions.

For our cooling sequence, we use thirty 30 μs red sideband pulses interleaved with repump

sequences to cool the two ion mode used for the gate close to ground state. If the gate operation

is performed on the center-of-mass mode that is present with a single ion, the cooling sequence

as well as the shifts can be calibrated with a single ion. For the two-ion out-of-phase mode, we

perform these calibrations with both ions. Data showing cooling with the bichromatic field is shown

in Fig. 6.8. We find that with cooling, ∆ac is shifted slightly, most likely due to thermal effects from

heat dissipation in the trap electrodes when all the fields are on for the cooling sequence.

We note that for the out-of-phase radial mode, we also cool the out-of-phase axial mode. The

two modes have a significant coupling that can affect the coherence of the mode we use for the

gate [Home et al., 2011]. As we do not have a magnetic field gradient along the axial mode, we

perform cooling of this mode with our Raman beams before cooling the out-of-phase radial mode as

described above.

6.2.6 Gate pulse sequence

The gate pulse sequence that we use corresponds to an 8-loop Walsh 7 sequence [Hayes et al.,

2012] (see Sec. 2.5.2). We ramp the gradient and the bichromatic fields up and down 8 times. We

turn the gradient at ωg on first using a 5 μs Blackman envelope, and then we turn on the bichromatic

fields also with a Blackman envelope over 5 μs. We ramp the gradient at ωg first, as we also have an

ac Zeeman shift from the magnetic fields at ωg that shifts the qubit frequency. Our bichromatic

fields are calibrated to this shifted value. This pulse sequence is shown in Fig. 6.9. Aside from the

gradient and the bichromatic fields, we have an additional field on resonance with the qubit at ω0 to

perform the initial and final π/2 pulses as well as the π pulses during the gate sequence to provide
137

0.0

0.2

0.4

0.6

0.8

1.0

Population

a)

P

P +

P

40 20 0 20 40

Sideband detuning (kHz)

0.0

0.2

0.4

0.6

0.8

1.0

Population

b)

P

P +

P

Figure 6.8: Red sideband transition with a bichromatic microwave field and radiofrequency magnetic

field gradient. We scan the detuning from the red sideband transition on the out-of-phase rocking

mode, with the two fields and the gradient pulsed for 30 μs. a) Without cooling, we see excitation

of the ion populations from their initial internal states of |↓↓i. b) After cooling, we see that the

red sideband transition is significantly suppressed, indicating that the ions are cooled close to the

ground state of this mode. We use a cooling sequence of thirty 30 μs red sideband pulses.

the Walsh 7 modulation. We alternate the phase of the π pulses to correct for any π pulse errors

that cause under or over rotations [Levitt, 1996].

We shorten the ramp times compared to those in Sec. 5.8 to reduce the time taken just for the

ramps during our gate sequence. During the ramps, the qubit is also unprotected and vulnerable to

decoherence as the intrinsic dynamical decoupling only occurs when the bichromatic fields are on at
138

π/2x π/2xπ

y

π

x

π

y

π

x

π

y

ωg

ω0

ω− δ

ω0 + δ

Figure 6.9: Gate pulse sequence for our Walsh 7 σˆzσˆz gate. The field gradient at ωg and the

bichromatic microwave fields at ω0 ± δ, where δ ≈ ωr − ωg, are ramped up and down 8 times for

our 8-loop gate, with ω0 being the qubit frequency, and ωr the motional mode frequency. We have

an additional field on resonance with the qubit for our initial and final π/2 pulses and for the 5

π pulses during the gate sequence to produce the Walsh 7 modulation. The total gate duration

including the π/2 pulses is 740 μs.

their calibrated amplitude.

6.2.7 Calibrating π/2 pulse phase

Our gate implementation is in theory robust to fluctuations in the qubit frequency. Aside

from the intrinsic dynamical decoupling, qubit frequency fluctuations commute with the gate and

are echoed out by the π pulses in the gate sequence. Thus, the initial and final π/2 pulses should

have the same phase. In practice, however, we find that we have to make a small adjustment to

the phase of the final π/2 pulse. We believe this adjustment is due to thermal effects that result in

a time variation in the ac Zeeman shift from the magnetic field at ωg. This time variation would

result in the ac Zeeman shift during the first ramp not being identical to subsequent ramps. Our

pulse sequence should remove up to quadratic variations of this shift, with the phase adjustment

taking into account the higher-order variation. Having calibrated the gate time and detuning, we

scan the phase of the second π/2 pulse and set the phase to be at the minimum of the one-ion

bright population. This scan is shown in Fig. 6.10. We find the correction required to be on the

order of 0.002×2π radians or 0.8◦.
139

0.10 0.05 0.00 0.05 0.10

Relative phase of second /2 pulse/(2 )

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Population

P

P +

P

Figure 6.10: Scan of relative phase of the final π/2 pulse in the gate sequence to the first. We fit

a quadratic to the one-ion bright data (green line) and determine the phase correction from the

minimum of the fit. For this particular scan, the minimum was at a phase of 0.002×2π radians.

6.2.8 Gate duration and detuning

Finally, we discuss the gate duration and detuning calibrations. In principle, both these

parameters are determined completely by the gate Rabi frequency. However, for our system,

optimizing the gate determines our gate Rabi frequency. For a coarse calibration, we perform gate

detuning scans at different gate durations and try to match the scans to numerical simulations

performed using QuTiP [Johansson et al., 2013]. Simulations are shown in Fig. 6.11 with experimental

data in Fig. 6.12. We perform detuning scans as opposed to duration scans to keep the duty cycle

of gate fields and gradients constant and to minimize thermal effects.

A finer calibration is then performed by varying the gate duration and detuning in smaller

steps and measuring the fidelity at each point. We then adjust both the gate detuning and duration

to maximize the gate fidelity. We find the optimal gate duration to be 740 μs. Following Sec. 2.5.4,

we are unable to resolve timing errors at the 5% level with the Walsh 7 seqeuence. We find that the

optimal gate detuning has day-to-day variations at the 100 Hz level.
140 0.0

0.2

0.4

0.6

0.8

Population

a) P

P +

P

0.0

0.2

0.4

0.6

0.8

1.0

Population

b) P

P +

P

4 5 6 7 8

/(2 = 1)

0.0

0.2

0.4

0.6

0.8

Population

c) P

P +

P

Figure 6.11: We perform simulations of gate detuning scans at three different durations: a) t = 0.9tg,

b) t = tg and c) t = 1.1tg, where tg is the duration required to generate the maximally entangled

state |Φi = 1/

√

2 (|↓↓i + |↑↑i) and ∆ is the gate detuning (see Eq. 6.27). Depending on the duration

of the interaction, the populations of P↑↑ and P↓↓ (red and blue lines) cross at different points above

the null of the one ion bright population P↑↓+↓↑ (green line). This feature can be used to calibrate

the gate time close to its optimal value. Experimental data is shown in Fig. 6.12.

6.2.9 Stability of calibrations

The bulk of the data for the gate results was taken over a two month period from December

2019 to January 2020. During this time, the microwave parameters for the intrinsic dynamical

decoupling (IDD), the phases for the nulling of the magnetic field at ωg, the bichromatic cooling

calibrations, and the gate duration were not adjusted from mid-December7 onwards. However, we
141

8 10 12 14 16

Gate detuning /2 (kHz)

0.0

0.2

0.4

0.6

0.8

Population

P

P +

P

Figure 6.12: Gate detuning scan at total gate duration of 740 μs. We see similar features to the

simulated data in Fig. 6.11. We use this scan as a coarse calibration of the gate time and detuning.

For finer calibration of those parameters, we measure the gate fidelity at different values of the gate

time and detuning.

typically calibrated the π and π/2 times used in the gate sequence daily. With each pair of ions that

were loaded, we performed the micromotion compensation and adjusted the motional frequencies

such that δ = (ωr − ωg)/2 was close to the frequency where the IDD was calibrated. We also

adjusted the gate detuning at the 100 Hz level to optimize the fidelity.

6.3 Gate data

While the fidelity analysis is detailed in Chapter. 8, we explain what raw data is collected.

We measure the population and parity after a gate sequence. Raw data is shown in Fig. 6.13. In

taking this data, we include 20 dummy points which are not used in the analysis at the start of

the experiments in order for both the motional frequency tracking and thermal effects to stabilize.

We then measure the populations after an additional parity π/2 pulse with a variable phase. For

7 This included several power outages and large temperature swings (> 5 K) in the lab. This stability offers

the most compelling argument for laser-free gates; path length fluctuations are much easier to ignore with 60 m

wavelengths (5 MHz gradient) as opposed to wavelengths ∼ 300 nm.
142

this measurement, we use a precalibrated phase offset to take more points at the peaks of the

parity amplitude. In total, we measure the populations for 52 parity phases from 0 to 2π. In

addition, within the same run we also measure the populations after the gate sequence without any

parity pulse. We make 40 population measurements. Both the parity and population measurements

are done with 200 repetitions each. These data were taken with interleaved measurements of the

motional frequency in between every parity or population point as described in Sec. 4.7. More

repetitions could be done per point, but that measurement would likely have to be interleaved

with the motional frequency tracking before the end of all the repetitions. The drift tracking for

these data is shown in Fig. 4.22. 200 repetitions were chosen for these data for simplicity and also

ion-lifetime considerations.

0.0 0.2 0.4 0.6 0.8 1.0

/2 analysis phase/2

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 6.13: Raw population and parity data. We measure both the populations and parity within

a single run of the experiment, where each point is averaged from 200 repetitions. For horizontal

axis points below 0 or above 2π, there is no parity pulse and we simply measure the populations at

the end of the gate sequence. The first twenty points of the experiment are “dummy” points that

are not used for subsequent analysis. These data are used to generate the parity scan in Fig. 6.15.

In addition to the population and parity measurements, we also measure three different

reference states to characterize our state preparation and measurement (SPAM). These states are
143

both ions bright, |↓↓i, both ions dark |↑↑i, following the application of a π pulse, and the ions in

an equal superposition of |↓↓i, |↑↑i, |↑↓i, |↓↑i after a π/2 pulse. The first measurement lets us

characterize the two-ion bright state, the second the two-ion dark, and the third a measure of the

one-ion bright state. Ideally, one would be able to prepare the one-ion bright state as well but that

would require high-fidelity single-ion addressing. For the reference data, we take 500 repetitions per

point, which corresponds to about 18,500 measurements per state. We use the exact same state

preparation as we do for the gate sequence. Example data is shown in Fig. 6.14.

We note that each set of gate and reference data is taken with a single set of two ions. We

perform the calibrations as described earlier, but want to take into account the changing conditions

of the trap due to charging from our photoionization beams.

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 6.14: Raw reference data for fidelity analysis. From the initial state of |↓↓i, we alternate

between doing nothing, applying a π pulse, and applying a π/2 pulse and then measuring the ion

populations. These measurements would correspond to P↓↓ ≈ 1, P↑↑ ≈ 1 and finally P↓↓ = P↑↑ ≈ 0.25,

P↑↓+↓↑ ≈ 0.5. These measurements provide the reference data for the fidelity analysis as described

in the text. Each point is an average of 500 measurements.

6.4 Results

Using our Walsh 7 σˆzσˆz gate sequence, we obtain a Bell-state fidelity of 0.999(1). A scan of

the parity of our entangled state is shown in Fig. 6.15. This result is the highest laser-free Bell-state
144

fidelity to date, and is competitive with the highest-fidelity two-qubit gates across any platform.

0 1 2 3 4 5 6

/2 analysis phase (radians)

1.0

0.5

0.0

0.5

1.0

Parity

Figure 6.15: Parity scan of the entangled state produced at the end of the gate sequence. For

these data, we perform the entangling operation and measure the ion populations after applying

an additional π/2 pulse with a variable phase. The parity is P = P↑↑ + P↓↓ − P↓↑+↑↓. The raw ion

populations are shown in Fig. 6.13. These data are not corrected for SPAM errors.

6.4.1 Comparison to other laser-free gates

We compare our result to other laser-free trapped-ion entangling gates in table 6.1. Our result

is the first gate using a radio-frequency magnetic field gradient, and we have managed to improve

the Bell-state fidelity while simultaneously reducing the gate duration compared to other results

with a similar fidelity. In comparison to schemes that use gradients close to the qubit at gigahertz

frequencies, we also only require one field gradient. The entangling gate in [Weidt et al., 2016]

required four microwave fields per ion, as opposed to two fields for our scheme. This reduction is

due to our intrinsic dynamical decoupling scheme that does not require extra fields.

We note that our technique has also enabled generation of larger magnetic field gradients at

the ion. However, this does not translate proportionately to a larger gate Rabi frequency as one

must also take into account the factor of the J2 function.
Gradientfrequency

No. offield gradients

No. ofmicrowavefields

Gate interaction B-field gradient

(T/m)

Gate Rabifrequency

(kHz)

Gate duration (μs) Bell-state infidelity

(10−3)

Year 

1.69 GHz 2 0 σˆxσˆx 35.3 0.9 400 240(30) 2011 

Static 1 0 σˆzσˆz 19 0.02 8000 360(50) 2012 

3.2 GHz 2 1 σˆxσˆx 7 0.154 3250 3(1) 2016 

Static 1 8 σˆxσˆx 24 0.093 2700 15(12) 2016 

1.1 GHz 2 0 σˆxσˆx 19 0.59 808 18(12) 2019 1.1 GHz 2 0 σˆxσˆx 19 0.59 2938 3(1) 2019 

5 MHz 1 2 σˆzσˆz 120 1.25 740 1(1) 2020 

Table 6.1: Comparison of laser-free entangling gates. Gate Rabi frequencies are listed following our convention wh

duration is t = 2π/(4Ωgate) (see Eq. 2.35). Ramp times are included in the gate duration for this work but not factoestimate. We consider only the field gradients and fields required to generate the spin-spin interaction.

8 NIST 2011 - [Ospelkaus et al., 2011], Siegen 2012 - [Khromova et al., 2012], Oxford 2016 - [Harty et al., 2016], Sussex 2016 - [Wei

- [Hahn et al., 2019] PTB 2- [Zarantonello et al., 2019]
146

6.4.2 Insensitivity to qubit frequency offsets

Our gate is insensitive to qubit frequency offsets in two ways. First, we employ the intrinsic

dynamical decoupling described in Sec. 6.1.1. Second, qubit frequency offsets which are proportional

commute with the σˆzσˆz interaction that we generate, and thus can be echoed out. We investigate

this experimentally and the data are shown in Fig. 6.16. For these data, we deliberately add a

frequency offset to our blue and red detuned microwave fields for the gate operation. Explicitly, the

detunings of the fields are

δred = −(ωr − ωg)/2 + ∆ac + ∆offset,

δblue = (ωr − ωg)/2 + ∆ac + ∆offset,

(6.34)

where ∆ac is the ac Zeeman shift from the field at ωg and ∆offset is an offset that we add. This

offset mimics a qubit frequency miscalibration with the gradient and bichromatic microwave fields

on. We note that the frequency of the carrier for the π and π/2 pulses was not modified, nor did we

change any other gate parameters.

From the data, we can see that the gate fidelity is > 0.99 over a range of ∆offset/2π from -200

to 200 kHz. This range is much larger than the gate Rabi frequency which is ∼kHz. For an σˆxσˆx,

this error would not commute with the gate interaction and would severely impact the fidelity. We

exploit this insensitivity to qubit frequency offsets to perform single-ion addressing in chapter 7.

6.4.3 Error analysis

We think our gate fidelity is primarily limited by motional errors. These errors come mainly

from motional dephasing and motional frequency fluctuations as described in Sec. 2.5. We perform

simulations in QuTiP with our gate parameters to estimate the errors in Table 6.2.

For the motional frequency error, we extract the errors from our motional frequency tracking

data (see Sec. 4.7 and Fig. 4.23) and use that in our simulation.

The heating rate on our rocking mode is expected to be an order of magnitude lower than
147

400 200 0 200 400

Qubit frequency offset (kHz)

10

4

10

3

10

2

10

1

Bell-state infidelity

Figure 6.16: Plot of Bell-state fidelity versus qubit frequency offset during the gate. The qubit

frequency offset is added to both the red- and blue-detuned microwave fields. For a region of

±200 kHz, the Bell-state fidelity is still > 0.99, demonstrating the robustness of this gate to such

shifts. This region is much larger than the gate Rabi frequency Ωg/2π ≈ 1 kHz. This robustness is

a consequence of both the intrinsic dynamical decoupling as well as performing a σˆzσˆz gate that

commutes with frequency errors which are proportional to ˆσz.

the heating on the center-of-mass mode, which is about 40 quanta/s. A heating rate of a few

quanta/s is challenging to measure, especially with our relatively short two-ion lifetimes. Based on

measurements we have made, we place an upper bound on this heating rate at 4 quanta/s.

We believe motional dephasing is our leading error source. However, this is also a challenging

measurement. From separate experiments using squeezed states [Burd, 2020], we estimate this

dephasing rate to be 5-10/s, which corresponds to a motional coherence time of 100-200 ms.

Error mechanism Error (10−4)

Motional dephasing 5-10

Motional frequency fluctuations < 5

Heating < 1

Total 5-10

Table 6.2: Estimated error budget.
148

6.4.4 Temperature insensitivity

While we believe that the infidelity of our highest-fidelity gate is dominated by motional

errors, our gate implementation is relatively robust to the initial temperature of the gate mode. This

dependence was examined using numerical simulations as shown in Fig. 6.17, following the model for

gate errors discussed in Sec. 2.5. We assume a gate Rabi frequency Ωg/2π = 1.25 kHz, and calculate

the error due to static detuning errors, motional dephasing, and motional heating at different

thermal occupations n ̄ from 0 to 4. For a multi-loop sequence, the temperature dependence of these

errors gets smaller and smaller, and for a Walsh 7 sequence, the only error that is significantly

temperature dependent is from motional dephasing up to our Doppler temperature with  ̄n ≈ 2.

6.4.5 Gates at Doppler temperature

The highest fidelity gate was performed on the out-of-phase radial mode with ground state

cooling on both that mode and the out-of-phase axial mode. Without ground-state cooling, we find

that the increased temperature mainly makes us more sensitive to the motional dephasing error. If

we could eliminate ground state cooling from the state preparation, that would increase the duty

cycle of our experiments substantially. Ground state cooling typically takes several milliseconds,

especially for multiple modes of motion.

We perform the gate with ions at the Doppler temperature with n ̄ ≈ 2. For this experiment,

we perform the ground state cooling of the motional mode with the radiofrequency gradient and the

microwaves as before but subsequently pulse the Doppler cooling laser to bring the ions back to their

Doppler temperature. This step is done to keep the duty cycle of the high-power radiofrequency

gradient roughly constant. We verify that the ions are at the Doppler temperature with n ̄ ≈ 2 in a

separate experiment. Note that we perform this gate on the center of mass mode, as at the Doppler

temperature, the out-of-phase radial mode has increased dephasing due to the out-of-phase axial

stretch mode. We obtain a fidelity of 0.992(2) at the Doppler temperature.

The fidelity of laser gates would suffer more significantly at the Doppler temperature due
149

0 50 100 150 200 250 300

Gate detuning error (Hz)

10

5

10

4

10

3

Bell-state infidelity

a)

n=0.0

n=0.1

n=0.5

n=1.0

n=2.0

n=4.0

0 10 20 30 40 50

Motional dephasing (s

1

)

10

4

10

3

10

2

Bell-state infidelity

b)

n=0.0

n=0.1

n=0.5

n=1.0

n=2.0

n=4.0

0 20 40 60 80 100

Motional heating (quanta s

1

)

10

4

10

3

Bell-state infidelity

c)

n=0.0

n=0.1

n=0.5

n=1.0

n=2.0

n=4.0

Figure 6.17: We analyze the temperature dependence of gate errors from three sources of motiondependent error: a) static detuning errors, b) motional dephasing, and c) motional heating. With

our multi-loop Walsh 7 sequence, the only error that exhibits significant temperature dependence is

from motional dephasing, up to our Doppler temperature with  ̄n ≈ 2.

to the higher-order terms in aˆ and aˆ

† present in the Hamiltonian from the Lamb-Dicke expansion.
150

These higher-order terms are negligible for our system as the second-order spatial derivatives of our

magnetic field are relatively small.

6.4.6 Pulse shaping

We ramp our gradient and microwave fields on and off over 5 μs using a Blackman envelope.

These ramps make up 160 μs of our 740 μs gate sequence, as we ramp the fields and gradient on and

off eight times. Thus, there is a large incentive to keep these ramps as short as possible. However,

we operate in a regime where our ramp times are not long compared to 2π/δ, which for our gate is

≈ 1 μs. Our short ramps result in qubit errors that are dependent on the gate detuning. For some

detunings, we can operate with ramp times that are an integer multiple of 2π/δ as discussed in

Sec. 6.1.2. We investigate this effect with a single ion, performing the gate sequence and measuring

how much population is lost. The data are shown in Fig. 6.18, where we see that the ion sometimes

makes transitions to the bright state (|↑i) of the ion at specific gate detunings. These excitations

will cause a gate error.

Ideally, we would not operate at those specific problematic gate detunings. However, our

motional frequency ωr changes as we run the gate experiments. This change is as much as a few kHz

over the duration of the gate experiments, due to charging of the surface from our lasers as described

in Sec. 4.7. We track this frequency and change δ ≈ ωr − ωg accordingly to keep the gate detuning

∆ = 2δ − (ωr − ωg) constant. However, as δ changes, we suspect that we can sometimes hit one

of these pulse shaping resonances. Keeping the gate duration and detuning constant and running

multiple gate sequences one after another, we see variation in the gate fidelity at the 0.001–0.005

level as the motional frequency changes.

These effects could be mitigated by keeping δ constant, which can be achieved by either

keeping ωr fixed or changing ωg instead. Changing ωg might affect the magnitude of the fields

and gradient at the ion as our diplexer has a fixed bandwidth. Phases and amplitudes for nulling

might also be affected by changing ωg. Keeping ωr fixed would involve feeding back on the motional

frequency instead of just tracking it. In the future, this problem can be mitigated by using a different
151

20 15 10 5 0 5 10 15 20

/2 (kHz)

0.00

0.01

0.02

0.03

0.04

0.05

Population

Figure 6.18: Effect of gate pulse shaping sequence on qubit. We perform the gate sequence on a

single ion starting in |↓i and measure its population in |↑i. Ideally, the ion should return to |↓i after

the gate sequence. However, we see excitations of the |↑i state at specific gate detunings due to

off-resonant carrier excitation from the fast microwave ramps.

ion with friendlier wavelengths, reducing charging.

6.4.7 Qubit decoherence

Our gate sequence should be robust to fluctuations in the qubit frequency; we have both

spin-echo pulses as well as intrinsic dynamical decoupling. To measure the qubit decoherence due to

such fluctuations, we perform the gate sequence with a single ion. With a single ion, the out-of-phase

mode is absent and the nearest motional mode is ≈ 100 kHz away from where the gate fields are

calibrated, eliminating spin-motion coupling. Thus, we can isolate the effect of the gate sequence

purely on the ion spin.

Starting in the |↓i state, the ion’s final population should be in the |↑i state following the

gate sequence. Any error due to decoherence would result in ion population in the |↓i state. With a

scan of 5 × 105 points, we measure a |↓i population of 0.0069(4) with the gate fields. As a control,

we make a measurement of 0.0076(4) with just a π pulse. The difference in populations between
152

between these two measurements is consistent with zero, showing that we see no effect from qubit

decoherence within the statistics of the measurement.

6.4.8 Maximizing argument of J2

As shown in Eq. 6.27, our gate Rabi frequency is ΩgJ2



4Ωμ

δ



. We tune 4Ωμ

δ = 2.4048 to operate

at the intrinsic dynamical decoupling (IDD) point such that J0



4Ωμ

δ



= 0 and J2



4Ωμ

δ



≈ 0.432

using Ωμ/2π = 0.571 MHz for δ ≈ 0.95 MHz.

We also attempted gates at J2



4Ωμ

δ



≈ 0.486 by adjusting the microwave Rabi frequency Ωμ.

Here, 4Ωμ

δ = 3.0542 with Ωμ/2π = 0.723 MHz. We reduced the gate duration from 740 μs to 650 μs,

with a fidelity of ≈0.997(2). We believe the fidelity reduction is due to operating away from the

IDD point, where slow qubit frequency fluctuations are still echoed out by our π pulses but we

become more sensitive to faster fluctuations. At this Rabi frequency, J0



4Ωμ

δ



= −0.278, which

still reduces our sensitivity to qubit frequency fluctuations but not quite as much as with IDD.

6.4.9 Gates with fewer loops

Our highest fidelity gate used an eight-loop Walsh 7 sequence that has a duration of 740 μs.

We also tried two-loop Walsh 1 and four-loop Walsh 3 sequences. The results are summarized in the

table below. For the motional frequency fluctations, we assume a 100 Hz static shift. The heating

on the centre-of-mass (COM) mode is 40 quanta/s, compared to <4 quanta/s on the out-of-phase

(OOP) rocking mode. We assume a motional dephasing of 10/s (or motional coherence time of

100 ms).

While our errors are broadly consistent with the measured fidelities for the Walsh 7 sequence,

both on the COM and OOP modes, they are not consistent with our Walsh 1 or Walsh 3 sequences.

This inconsistency suggests that our Walsh 1 or Walsh 3 sequences were not as well-calibrated.

There could be additional timing errors unaccounted for. In addition, we have assumed a static shift

for our motional frequency fluctuations, but the motional frequency drifts might be time-varying

during the gate sequence. This time variation would make the Walsh 7 sequence more robust
153

Walsh sequence 1 3 7 3 7

Loops 2 4 8 4 8

Mode COM COM COM OOP OOP

Duration (μs) ≈ 300 ≈ 500 ≈ 800 ≈ 500 740

Fidelity 0.987 0.993 0.997 0.993 0.999

Errors

Motional dephasing 8 × 10−4 5 × 10−4 4 × 10−4 5 × 10−4 4 × 10−4

Heating 2.9 × 10−3 2 × 10−3 1.4 × 10−3 1 × 10−4 1 × 10−5

Motional frequency fluctuations 5 × 10−4 2 × 10−4 1 × 10−4 2 × 10−4 1 × 10−4

Table 6.3: Comparison of estimated errors with different gate sequences.

compared to other sequences.

6.4.10 Comparison to laser-based gates

We also compare our gate to laser-based gates in table 6.4. While our gate is significantly

slower, we achieve fidelities competitive with some of the highest fidelity laser-based gates. Our

gates also not limited by photon scattering errors.

Duration (μs) Bell-state infidelity (10−3) Photon scattering error (10−4) Year Ref9

30 0.8(4) 5.7 2016 NIST

100 1(1) 4 2016 Oxford

1.6 2(1) 6 2018 Oxford

740 1(1) 0 2020 This work

Table 6.4: Comparison with laser-based gates.

6.5 Mølmer-Sørensen gates

When we first attempted entangling gates with a radiofrequency magnetic field gradient, we

tried Mølmer-Sørensen (MS) type gates. In the discussion in the previous chapter, a single microwave

field was used with the gradient to generate a sideband interaction. The natural extension of this

was to add a second microwave field to produce an MS type interaction as described in Eq. 6.28,

9 NIST 2016 [Gaebler et al., 2016], Oxford 1 2016 [Ballance et al., 2016], Oxford 2018 [Sch ̈afer et al., 2018]
154

Hˆms = i~ΩgJ1



4Ωμ

δ



(ˆσy1 + ˆσy2)

n

−aeˆ

i∆t + ˆa†

e

i∆t

o

.

At the time, we were unaware of intrinsic dynamical decoupling and instead added an extra

field as described in [Harty et al., 2016] to give us dynamical decoupling from qubit frequency shifts

during the gate sequence. We obtained a fidelity of ≈ 0.95 with a gate time of about 1 ms. The

parity scan from this gate sequence is shown in Fig. 6.19.

The main challenge for this gate is that since qubit frequency errors (∝ σˆz) did not commute

with the gate interaction (∝ σˆy), we needed to calibrate qubit frequency shifts very precisely. We

needed to calibrate these shifts to the level of our gate Rabi frequency ∼ 1 kHz, even though

the extra dynamical decoupling field afforded us some insensitivity to this. We were constantly

challenged with keeping track of small changes in the magnetic field as well as calibrating ion crystal

rotations so both ions saw the same ac Zeeman shift from the magnetic field at ωg. All of these

challenges were mitigated by performing the σˆzσˆz gate instead. In addition, the extra dynamical

decoupling field potentially added error terms to our interaction as discussed in the next section.

0 1 2 3 4 5 6

/2 analysis phase/2

1.0

0.5

0.0

0.5

1.0

Parity

Figure 6.19: Parity scan from Mølmer-Sørensen gate with radiofrequency gate. The parity amplitude

is ≈ 0.95.
155

6.5.1 Errors from dynamical decoupling field

The full Hamiltonian including the extra dynamical decoupling field is, similar to Eq. 6.25

Hˆ (t) =2~Ωμ cos(δt)Sˆ

x + 2~Ωg cos(ωgt)Sˆz

n

aeˆ

−iωrt + ˆa†

e

iωrt

o

+ 2~ΩddSˆ

y,

(6.35)

where the last term corresponds to the extra field that we have added with Rabi frequency Ωdd.

Here, we have chosen this field to be proportional to Sˆ

y, so it commutes with the MS interaction.

However, in order for it to commute with the MS interaction, which is proportional to [Sˆ

x, Sˆz], it

cannot commute with the bichromatic microwave term which is proportional to Sˆ

x. Thus, analyzing

this term in the bichromatic interaction picture,

Hˆ

I,(t) = ~Ωddn

Sˆ

y

h

J0

4Ωμ

δ



+ 2X∞

n=1

J2n

4Ωμ

δ



cos(2nδt)

i

+2Sˆ

z

X∞

n=1

J2n−1

4Ωμ

δ



sin([2n − 1]δt)

o

.

(6.36)

First, the strength of our dynamical decoupling field is modified by a factor of J0



4Ωμ

δ



, and

we have additional error terms that might reduce the gate fidelity.
Just beat it, beat it

Beat it, beat it, beat it

Michael Jackson

7

Single-ion addressing with radiofrequency

magnetic field gradient

In this chapter1, we demonstrate single-ion addressing using the magnetic field gradient

oscillating at ωg. If the two-ion crystal is twisted relative to the trap axis (and thus the currenty

carrying electrodes), the two ions will see slightly different magnetic fields, and thus slightly different

ac Zeeman shifts. The qubit transitions for the two ions become separated in frequency, and we use

this frequency shift for individual addressing, either via a Ramsey sequence or Rabi spectroscopy.

While using a spatially varying ac Zeeman shift for individual addressing has been shown in [Warring

et al., 2013b], we integrate this single-ion addressing with our gate sequence in order to create

antisymmetric Bell states. This integration is made possible by our gate’s insensitivity to qubit

frequency offsets as shown in Sec. 6.4.2, where we can create high-fidelity entangled states even

though our two qubits have slightly different frequencies during the gate operation.

1 Alternative chapter title from HMK - Hot singlets in your area!

156
157

7.1 Overview of laser-free single-ion addressing

Laser-based single-ion addressing typically involves focusing beams to spot sizes smaller than

the ion separation. Single-ion addressing was first demonstrated using tightly focused lasers [Na ́egerl

et al., 1999], which has more recently been extended to using ac Stark shifts to shift the frequency [Blatt and Roos, 2012]. The ions’ position in the beam can also be adjusted such that

each ion sees a different optical phase2[Rowe et al., 2001]. Such techniques are not possible with

microwave radiation whose wavelength is typically much larger than the ion spacing.

Instead, laser-free individual addressing methods have been demonstrated using static magnetic

field gradients. Using a static magnetic field gradient along the trap axis, for example [Johanning

et al., 2009], ions in a string are naturally separated in frequency space and can be addressed

individually. This technique has been extended to an eight-ion string to demonstrate single-ion

microwave addressing with cross-talk errors at the 10−5level [Piltz et al., 2014]. One challenge

with a permanent static gradient, as in the experiments above, is that is that while all the ions are

individually addressable, they are also at different frequencies. Each ion will require its own fields

and phase tracking. For example, a two-qubit gate with dynamical decoupling required four fields

per ion [Weidt et al., 2016]. This gradient can instead be generated from a dc current that can be

turned on and off such as in [Wang et al., 2009] for addressing.

Spatial addressing can also be achieved using novel trap designs [Aude Craik et al., 2014].

Using a microfabricated surface trap with integrated microwave electrodes [Aude Craik et al., 2017],

microwave fields that drive spin-flip transitions in one zone coherently cancel in other zones. For

ions in zones 960 μm apart, the calculated crosstalk error was 1 × 10−6.

An oscillating magnetic field gradient can also be used for single-ion addressing. In [Warring

et al., 2013a], four methods were demonstrated using gigahertz magnetic fields and gradients: driving

the qubits on resonance, using the micromotion sideband, and using differential ac Zeeman shifts

with either Ramsey or Rabi spectroscopy. For each of these methods, the ions were switched between

2 Also known as the “tricky-trick”.
158

two spatial configurations over 80 μs such that one of the ions was along the RF and magnetic field

null, while the other was displaced from the null. The same configuration does not work for both

the gate operation and the individual addressing.

In our experiment, we use a similar method to that described in [Warring et al., 2013a].

However, since our gate is insensitive to qubit frequency shifts, we are able to keep the ion

configuration constant between gate and individual addressing operations. Performing this single-ion

addressing after a gate sequence, we are able to create an antisymmetric Bell state.

7.2 ac Zeeman shift from oscillating magnetic field on two ions

∇B



∇B



1

2

3

a) b)

1

2

3

Figure 7.1: Ion crystal and electrode configuration for entangling gates and single-ion addressing.

Our ion crystal is twisted relative to the current carrying electrodes, where the dotted line indicates

the ion’s original position along the axis. We use the same ion crystal configuration for both

entangling gates and single-ion addressing. a) For entangling gates, we apply a current oscillating

at ωg to all three electrodes 1, 2, and 3. b) For single-ion addressing, a current is applied only to

electrode 1.

If the two-ion crystal is not well-aligned to the electrode generating the magnetic field, the

two ions will see a slightly different magnetic field and thus a different ac Zeeman shift (see Fig. 7.1).

We first measure this shift using a spin-echo Ramsey sequence. The pulse sequence we use is

(1) π/2 pulse around φ axis.
159

(2) Turn on differential shift for some duration t (excluding ramp durations).

(3) π pulse around φ axis. This is our spin-echo pulse.

(4) Wait for duration t (excluding ramp durations) with no fields on.

(5) π/2 pulse around φ axis.

This pulse sequence is shown in Fig. 7.2. We explicitly label the phase φ. For just this spin-echo

sequence, only the relative phase between the initial and final π/2 pulses is consequential. However,

if the Ramsey sequence follows another sequence, for example an entangling gate, this phase will be

important. This sequence is used for both measuring the differential shift as well as implementing

an effective single-qubit rotation in Sec. 7.3.1. The results of this experiment are shown in Fig. 7.3.

Measuring the fluorescence of the ions, we see the beating of two sinusoidal oscillations, with

frequencies corresponding to the ac Zeeman shift for each ion. We see a slow envelope corresponding

to the difference frequency between the two ions and faster oscillations corresponding to their mean

frequency. From the data, we extract a difference frequency of ≈ 19.5 kHz. For this experiment,

the current at ωg was applied only to electrode 1 (see Fig. 7.1) to generate the differential shift.

We find that applying the current to only one electrode creates larger differential shifts, enabling

faster single-ion addressing. We note that the same ion crystal configuration that was used for these

experiments was also used for the highest-fidelity gates described in the previous chapter.

π/2φ π/2φ πφ

ωg

ω0

t

Figure 7.2: Pulse sequence for spin-echo Ramsey sequence. This pulse sequence is used to both

measure the differential ac Zeeman shift, as well as to perform an effective single-qubit rotation. We

apply a current at ωg only to electrode 1 (see Fig. 7.1). A field on resonance at ω0 is used for the π

and π/2 pulses around the same axis φ. The single arm duration t does not include the duration of

the ramps to turn on and off the current.
160

0 20 40 60 80 100

Duration ( s)

20

40

60

80

100

Counts

Figure 7.3: We perform a Ramsey experiment to measure the differential ac Zeeman shift between

two ions from a magnetic field oscillating at ωg. The solid line is a fit to the data assuming the sum

of two sines. From the fit, the two ions have a difference frequency of ≈ 19.5 kHz. The absolute ac

Zeeman shift of each ion is ≈ 2.5 MHz. This value is much larger than its fitted values of ≈ 400 kHz

due to aliasing. However, the difference frequency is more important as it imparts a differential

phase shift that we use for single-ion addressing.

7.3 Generating anti-symmetric Bell states

The four Bell states for a two-qubit system are

|Φ

+i =

1

√

2

(|↑↑i + |↓↓i),

|Φ

−i =

1

√

2

(|↑↑i − |↓↓i),

|Ψ

+i =

1

√

2

(|↓↑i + |↑↓i),

|Ψ

−i =

1

√

2

(|↓↑i − |↑↓i).

(7.1)

Three of these states, |Φ

+i, |Φ−i, and |Ψ+i are symmetric under an exchange of qubits, while |Ψ−i

is antisymmetric. We can transform between the symmetric states using global rotations. However,

in order to transform a symmetric Bell state to an antisymmetric one, such as |Φ

+i → |Ψ−i, we

need a non-global interaction. We generate this interaction via the differential ac Zeeman shift
161

described previously. We also discuss the use of Rabi spectroscopy for this purpose in Sec. 7.4.

7.3.1 Effective single-ion rotation with Ramsey sequence

We use the same sequence in the previous section to generate ion-dependent phase shifts that

help us create an effective single-ion rotation. Analyzing the single ion case first, let us assume

that there is a qubit frequency shift ∆i from the magnetic field at ωg which adds new terms to the

Hamiltonian,

Hˆ

zi =

~∆i

2

σˆzi, (7.2)

where i is the ion index. Its corresponding propagator is,

Uˆ

zi(t) = exp −

i

~

Hˆ

zit

= exp −i

∆it

2

σˆzi

,

= cos

∆it

2

ˆI − isin

∆it

2

σˆz

=





cos ∆it

2 − isin ∆it2

0

0 cos ∆it

2 + isin ∆it2



 .

(7.3)

We now evaluate the overall propagator on the single ion assuming a spin-echo sequence as

shown in Fig. 7.2. Making use of the results in appendix B.1, the overall time evolution operator is

Uˆ

i = Uˆ π

2

Uˆ

πUˆziUˆ π

2

=

1

2





1 −ie−iφ

−ieiφ 1









0 −ie−iφ

−ieiφ 0









e

−i∆it/2 0

0 e

i∆it/2









1 −ie−iφ

−ieiφ 1





=





− cos ∆it

2

e

−iφ sin ∆it

2

−e

iφ sin ∆it

2 − cos ∆it2



 ,
162

where we use the basis |↓i =





0

1





and |↑i =





1

0



. The Bell-state we create in Chap. 6 is

|Φi = √

1

2

(|↓↓i + i|↑↑i). For simplicity, we apply the propagator to the symmetric Bell state

|Φ

+i = √

1

2

(|↓↓i + |↑↑i),

Uˆ

1Uˆ2 |Φ

+i =

1

√

2

 

cos

∆1t

2

cos

∆2t

2

+ e

−2iφ sin

∆1t

2

sin

∆2t

2



|↑↑i

+



cos

∆1t

2

cos

∆2t

2

+ e

2iφ sin

∆1t

2

sin

∆2t

2



|↓↓i

+



−e

−iφ sin

∆1t

2

cos

∆2t

2

+ e

iφ cos

∆1t

2

sin

∆2t

2



|↑↓i

+



−e

−iφ cos

∆1t

2

sin

∆2t

2

+ e

iφ sin

∆1t

2

cos

∆2t

2



|↓↑i !.

(7.4)

For the case of φ = 0,

Uˆ

1Uˆ

2 |Φ

+i =

1

√

2



cos

∆1 − ∆2

2

t|↑↑i + cos

∆1 − ∆2

2

t|↓↓i

− sin

∆1 − ∆2

2

t|↑↓i + sin

∆1 − ∆2

2

t|↓↑i .

(7.5)

When t = π/(∆1 − ∆2),

Uˆ

1Uˆ

2 |Φ

+i =

1

√

2

(|↓↑i − |↑↓i)

= |Ψ

−i,

(7.6)

which gives us the singlet state. Note that we can also achieve the singlet state using φ = π. An

alternative, perhaps more intuitive way of analyzing this is to first look at the effect of a π/2 pulse

on |Φ

+i. Following appendix B.2,

Uˆ π

2

1Uˆ π

2

2 |Φ

+i =

1

2

√

2



(1 − e

i2φ

)|↓↓i − 2i cos φ(|↓↑i + |↑↓i) + (1 − e

−i2φ

|↑↑i).



(7.7)

When φ = 0 or φ = π,
163

Uˆ π

2

1Uˆ π

2

2 |Φ

+i =

−i

√

2

(|↓↑i + |↑↓i), (7.8)

which is |Ψ+i = √

1

2

(|↓↑i + |↑↓i) up to a global phase. At this point, if we apply the differential shift,

Uˆ

z1Uˆz2 |Ψ

+i =

1

√

2



e

−i(∆1−∆2)t/2

|↓↑i + e

i(∆1−∆2)t/2

|↑↓i

=

1

√

2

e

−i(∆1−∆2)t/2



|↓↑i + e

i(∆1−∆2)t/ |↑↓i

,

(7.9)

which will result in the singlet state when t = π/(∆1 − ∆2). The larger the differential shift we

can create, the faster we can implement this operation as well. The singlet state is invariant under

global rotations and thus the additional π and π/2 pulses would leave it unaffected and could have

been omitted entirely. However, we use them to cancel out any common shifts, and this sequence is

more suited for general individual-qubit rotations rather than to only produce the singlet state. In

order to generate the singlet state, we need to calibrate the phase φ relative to the phase of the

entangled state we make, as well as the duration of the differential shift.

7.3.1.1 Calibration of Ramsey sequence duration and phase

Our calibration experiment for the duration and phase of the Ramsey sequence is

(1) Perform gate to generate symmetric entangled state.

(2) Spin-echo Ramsey sequence in Sec. 7.2 with phase φ.

(3) Additional π/2 pulse with phase 6= φ to analyze parity.

In order to calibrate the duration and phase of the Ramsey sequence, we first make our

entangled state with our gate operation. We then apply the Ramsey sequence after the gate

operation with some single-arm duration t and phase φ. For the calibration experiments, we apply

an additional π/2 on both ions with a fixed phase at the end before measuring the population. This

global π/2 pulse helps us distinguish between |Ψ+i and |Ψ−i. |Ψ+i will always be rotated to some
164

combination of |Φ

+i and |Φ−i and will produce populations in |↓↓i and |↑↑i. The antisymmetric

singlet state |Ψ−i on the other hand, will be invariant under the π/2 rotation and will only have

populations in |↓↑i and |↑↓i. We note that this final π/2 pulse cannot have the same phase as the

spin-echo sequence in step two3; if it does, we will produce some combination of |Ψ+i and |Ψ−i

which we cannot distinguish from measuring the ion populations.

First keeping the phase of the π/2 pulses in the Ramsey sequence fixed close to its optimal

value, we scan the duration in each arm as shown in Fig. 7.4. The optimal duration is t = π/(∆1−∆2).

For (∆1 − ∆2)/2π ≈ 20 kHz, this is roughly 25 μs. Experimentally, we find this duration t is slightly

lower than the measured value of the differential shift, due to the pulse shaping. We ramp the field

up and down over 5 μs, giving us a shift in that duration which is not accounted for in the x-axis of

the plot. Taking the ramps into account, assuming that on average the shift is only on for half the

duration each 5 μs ramp, the total duration is roughly the 25 μ s we expect.

We perform a numerical simulation in QuTiP to determine how precisely we need to calibrate

this duration. This sensitivity is shown in Fig. 7.5. To keep the error below 1 × 10−3, we need to

calibrate the duration at the ±500 ns level. This sensitivity could be reduced by using a composite

pulse sequence. We perform a similar calibration for the phase of the Ramsey sequence with respect

to the entangled state we produce at the end of the gate operation. We keep the duration constant

at close to its optimal value and instead scan the phase as shown in Fig. 7.6. Similarly, we perform

an analysis of the sensitivity to this phase using a numerical simulation (see Fig. 7.7). To keep the

error in the singlet state below 1 × 10−3, we need to calibrate the phase to better than 0.03 radians4

.

7.3.2 Singlet data and results

Similar to Sec. 6.3, the fidelity analysis is detailed in Chapter 8. Here we describe the data

that was collected. To measure the fidelity of the singlet state we produce, we measure both the

3 Our pulse sequence will be Uˆ π

2

Uˆ π

2

UˆπUˆziUˆ π

2

= Uˆ2πUˆziUˆ π

2

. This sequence will leave us the state shown in Eq. 7.9

which is constantly alternating between |Ψ

+i and |Ψ−i.

4 Or 0.005 of 2π.
165

10 15 20 25 30

Single arm duration in Ramsey sequence ( s)

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 7.4: Calibration of single arm duration in Ramsey sequence for making a singlet state. The

phase of the Ramsey sequence is close to its optimal value. As we scan the duration, we rotate

between |Ψ+i and |Ψ−i. The additional π/2 pulse turns |Ψ+i into a combination of |↓↓i and |↑↑i,

|Ψ−i is unchanged. We use a quadratic fit to the single-ion population (green line) to determine the

optimal duration for the Ramsey sequence.

populations and parity after we produce the singlet state. To measure the population, we simply

perform our entangling gate followed by the addressing sequence. To measure the parity, we add a

π/2 pulse at the end with a variable phase. This pulse helps us distinguish between the singlet and

any other Bell state, as the other Bell states will produce a parity oscillation, or in the case of |Ψ+i,

will always be rotated to a superposition of |↓↓i and |↑↑i. These rotations are shown in more detail

in Appendix B.2.

Raw data are shown in Fig. 7.8. Similar to Sec. 8.1.1, we have 20 dummy points at the start

that allow the motional frequency tracking and thermal effects to stabilize. We make measurements

of the populations with and without an additional π/2 pulse with a variable phase. For this singlet

state, we cycle through 6 different parity phases 7 times, each with 200 repetitions. We also make

40 population measurements without the additional π/2 pulse, also with 200 repetitions. In Fig. 7.9

we show the parity analysis of both the triplet and singlet states we produce. The triplet state
166

23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0

Ramsey single-arm duration ( s)

10

5

10

4

10

3

10

2

Singlet infidelity

Figure 7.5: QuTiP simulation of the sensitivity of of the singlet fidelity to the Ramsey sequence

duration. Starting with a perfect triplet state |Φ

+i, we plot the error in making a singlet state

versus the duration of the Ramsey sequence. We assume the phase is calibrated at its optimal value

and the two ion shifts are ∆1/2π = 2500 kHz and ∆2/2π = 2520 kHz. The difference frequency

(∆2 − ∆1)/2π = 20 kHz which requires 25 μs for the single-arm duration.

0.25 0.30 0.35 0.40 0.45 0.50

Ramsey sequence phase/2

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 7.6: Calibration of phase in Ramsey sequence for making a singlet state. The time of the

Ramsey sequence is close to its optimal value. At the optimal phase value, we should be producing

|Ψ−i. We use a quadratic fit to the single-ion population (green line) to determine the optimal

phase.
167

0.010 0.005 0.000 0.005 0.010

Ramsey sequence phase/2

10

6

10

5

10

4

10

3

Singlet infidelity

Figure 7.7: QuTiP simulation of the sensitivity of singlet fidelity to the Ramsey sequence

phase.Starting with a perfect triplet state |Φ

+i, we plot the error in making a singlet state versus

the phase of the Ramsey sequence. We assume the duration is calibrated at its optimal value and

the two ion shifts are ∆1/2π = 2500 kHz and ∆2/2π = 2520 kHz.

corresponding to |Φi exhibits a parity oscillation from +1 to -1, while the singlet state |Ψ−i has

a fixed parity of close to -1. Based on the fidelity analysis discussed in Chap. 8, we are able to

produce singlets with fidelity 0.998(1), corrected for SPAM errors.

7.4 Rabi spectroscopy

Initially, the single-qubit rotations were implemented using Rabi spectroscopy. In this

implementation, the differential ac Zeeman shift separated the ions’ qubit frequencies, and they

could be individually addressed using an additional microwave field that drives spin-flip transitions.

When this field was tuned to a particular ion’s qubit frequency, spin flips would be driven only on

that ion.

To perform this rotation, we first turn on the gradient at ωg, which also produces a magnetic

field at ωg. If the two ions are not aligned to the electrodes, they each see a different ac Zeeman

shift from the magnetic field ∆i. We then turn on an additional microwave field. The Hamiltonian
168

0.0 0.2 0.4 0.6 0.8 1.0

/2 analysis phase/2

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 7.8: Raw data for preparing a singlet state. All points are averaged over 200 repetitions. For

phases < 0 and > 2π, we only measure the populations without an additional π/2 pulse (x-axis

irrelevant). To measure the parity, we have 7 measurements of 6 different π/2 pulse phases, each

with 200 repetitions each. The points for phases < 0 are ”dummy points” which are not used for

further analysis.

0.0 0.2 0.4 0.6 0.8 1.0

/2 analysis phase/2

1.0

0.5

0.0

0.5

1.0

Parity

Figure 7.9: Parity analysis of both the triplet and singlet states we produce. The parity is

P = P↑↑ + P↓↓ − P↓↑+↑↓. The triplet state corresponding to |Φi = √

1

2



|↓↓i + e

iθ |↑↑i

exhibits a

parity oscillation from +1 to -1. The singlet state |Ψ−i = √

1

2

(|↓↑i − |↑↓i) on the other hand has a

fixed parity of -1. These data are not corrected for SPAM errors.
169

for the two ions in the interaction picture is

Hˆ =

~∆1

2

σz1 +

~∆2

2

σz2 differential ac Zeeman shift

+~Ωc



(ˆσ+1 + ˆσ+2)e

−i(δct+φ) + (ˆσ−1 + ˆσ−2)ei(δct+φ)



, microwave field (7.10)

where Ωc and δc are the Rabi frequency and detuning of the additional field respectively. For these

experiments, Ωc/2π ≈ 4 kHz. After applying the differential ac Zeeman shift, we pulse the additional

microwave field long enough to perform a π pulse and scan its detuning δc as shown in Fig. 7.10.

For δc = ∆i, we ideally perform a π pulse on the resonant ion. The differential Rabi frequency

(∆1 − ∆2)/2π ≈ 20 kHz is not large compared to the microwave Rabi frequency Ωc. Thus, we see

some off-resonant excitation of the second ion.

100 90 80 70 60 50 40

Detuning of carrier pulse (kHz)

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 7.10: Rabi spectroscopy of two ions with a differential ac Zeeman shift from the magnetic

field oscillating at ωg. We scan the detuning of an additional microwave field in the presence of the

differential shift. As the detuning of the field is equal to the shifted qubit frequencies of each of the

ions, we see excitations of the one-ion-bright population (green points) from their initial |↑↑i state.

As the differential shift is not large compared to the Rabi frequency of the microwave field, we see

some off-resonant excitation of the second ion.

Setting the detuning of the microwave field to one of the resonances, we also scan the duration
170

as shown in Fig. 7.11. We see oscillations in the one-ion-bright population as we selectively excite

only a single ion. Again, since our microwave Rabi frequency is not small compared to the ion

frequency splitting, we see some off-resonant excitation of the other ion.

0 20 40 60 80 100 120 140

Duration of carrier pulse ( s)

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 7.11: Duration scan of the single-ion addressing pulse. We fix the detuning of the microwave

field to one of the resonances in Fig. 7.10 and scan the duration. Initializing the ions to |↑↑i, we

drive the |↑↑i ↔ |↑↓i transition. As our microwave field Rabi frequency is not small compared to

the frequency splitting of the two ions, we observe off-resonant excitation of the second ion, giving

rise to some population in the |↓↓i state.

7.4.1 Generating singlet states

Similar to Sec. 7.3.1, we use this Rabi pulse to transform our entangled state after the gate

sequence |Φ

+i, to the singlet state |Ψ−i. Ideally, we implement

Uˆ

π1

ˆI2 |Φ

+i =

−i

√

2



e

iφ |↑↓i + e−iφ |↓↑i

, (7.11)

performing a spin flip on one ion and leaving the other unchanged. When φ = π/2 or 3π/2, we

create the singlet state, and when φ = 0 or π we create the triplet |Ψ+i state.

To calibrate this phase, we perform our single-ion pulse after a gate operation that creates
171

the state |Φ

+i, and then add an additional π/2 pulse to distinguish between |Ψ+i and |Ψ−i. The

state |Ψ+i will always be rotated to a state of even parity and |Ψ−i will be unchanged as shown in

appendix B.2. Figure 7.12 shows this behaviour as a function of the single-ion pulse phase.

0.0 0.2 0.4 0.6 0.8 1.0

Phase of carrier pulse/2

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 7.12: Scan of the phase of the single-ion pulse after a gate sequence. An additional π/2

pulse is used to distinguish between |Ψ+i and |Ψ−i. The state |Ψ+i corresponds to the minima of

the one-ion population (green points) and |Ψ−i to the peaks.

Finally, we use this calibrated phase to create the singlet state and analyze its fidelity following

the same procedure described in Sec. 8.1.2. The raw data is shown in Fig. 7.13.

7.4.2 Comparison of Ramsey and Rabi individual addressing sequences

We have demonstrated single-ion addressing after a gate sequence to create a singlet state

using both Ramsey and Rabi sequences. We achieve better singlet fidelity using the Ramsey sequence

as shown in Fig. 7.8. We also prefer the Ramsey sequence as our addressing duration can be as

fast as the inverse of the differential ac Zeeman shift, 2π/(∆1 − ∆2), for the two ions. In contrast,

the Rabi sequence requires the Rabi frequency for the additional pulse Ωc  |∆1 − ∆2|, increasing

the duration for the pulse in order to avoid off-resonant excitations. While we use a square pulse

for this additional pulse, pulse-shaping could also be used to avoid off-resonant excitation of the
172

0.0 0.2 0.4 0.6 0.8 1.0

/2 analysis phase/2

0.0

0.2

0.4

0.6

0.8

1.0

Population

P

P +

P

Figure 7.13: Raw data for generating singlet state with Rabi spectroscopy. For phases < 0 and

> 2π, we only measure the population of the singlet state and omit the addtional π/2 pulse. For

phases between 0 and 2π, we have an additional π/2 pulse with that phase. Ideally, the singlet state

should produce a constant one-ion population of 1. However, we see that we have an imperfect

singlet and have a slight oscillation in the ion populations.

non-resonant ion while minimizing the duration penalty.

In a larger string of ions however, the Rabi sequence might provide a more straightforward

way of individual addressing. The two-ion case that we have provides a simple implementation of

the Ramsey sequence, with more ions we would likely need a longer sequence as well as additional

global rotations.
Why’d you have to go and make things so complicated?

Avril Lavigne

8

Fidelity analysis

We have described how we have created entangled states in the previous chapters, but how

well can we make them? We now explain how we determine the fidelity of the entangled states we

have created. We discuss how we correct for state preparation and measurement (SPAM) errors,

including leakage out of the qubit manifold before laser-free operations. We validate our analysis

with simulated data. We want to ensure that we are estimating our true fidelity accurately, and

that if there is a bias it is towards a lower fidelity; we want a number we can trust.

Unless otherwise stated, all error bars in the figures in this chapter correspond to the standard

error of the mean, σ/√n where σ is the standard deviation and n is the number of measurements.

In addition, for most of the simulated data, we try to replicate our experimental data as closely

as possible, especially with regards to the number of measurements made as described in Sections

6.3, and 7.3.2. In some cases, we generate simulated data with more statistics than experimental

data1. We describe this increase by a “data factor”, where if the “data factor” is 10, we have 2000

repetitions per point rather than the 200 in the experiment.

1

If only it were that easy to take that much more data in real life.

173
174

8.1 Measuring fidelity

We want to measure how well we can implement a two-qubit gate. One could use process

tomography to measure the final state fidelities for all possible input states, but that requires

preparing input states better than you can entangle them. This requires robust individual addressing,

which is what we are trying to demonstrate in the first place.

More sophisticated methods such as randomized benchmarking [Emerson et al., 2005, Knill

et al., 2008, Gaebler et al., 2012] involve doing multiple random sequences of single-qubit and

two-qubit gates to extract an average error per gate. Randomized benchmarking of two-qubit

gates requires the single-qubit gate error, including individual addressing, to be much smaller than

the two-qubit gate error. Symmetric subspace benchmarking [Baldwin et al., 2020b] removes the

requirement for individual addressing. However, our limited two-ion lifetime and possible duty cycle

effects in our system could potentially make benchmarking challenging, i.e. not all gates in a longer

sequence would be the same.

We have developed a new technique for entangling gates, and the fidelity of making the

singlet state is our measure of the individual addressing error. Thus, we decided to keep our fidelity

measurement as simple as possible and only measure the fidelity of producing a target Bell state

from a specific starting state. We only perform one entangling gate. As pointed out in Ref. [Ballance,

2017], this is a “cheat”. We assume all input states would have the same operation fidelity. We also

have the challenge of extracting a two-qubit gate error lower than our SPAM error. This chapter

explains how we address that challenge.

8.1.1 Triplet state

We measure the fidelity of producing a target Bell state. The target state of our gate sequence,

starting from |↓↓i, is

|Φi =

1

√

2



|↓↓i + e

iθ |↑↑i

. (8.1)
175

The fidelity F of producing that target state is

F = | hψ|Φi |2 =

1

2

hψ|(|↓↓i h↓↓| + |↑↑i h↑↑|)|ψi Populations

+

1

2

hψ|(e

−iθ |↓↓i h↑↑| + eiθ |↑↑i h↓↓|)|ψi, Coherences (8.2)

where |ψi is the state that we produce at the end of our gate sequence. The first line in Eq. 8.2

corresponds to the populations, or the diagonal elements of our density matrix, and the second line

corresponds to the coherences, the off-diagonal elements. We can measure the populations naturally

by looking at the fluorescence of the ions. However, measuring the coherences is harder. Instead of a

direct measurement, we use an additional π/2 pulse with a variable phase φ to map the off-diagonal

elements to the populations [Sackett et al., 2000]. We define the parity as

P = P↑↑ + P↓↓ − P↓↑ − P↑↓, (8.3)

where P↑↑ for example is the population in the |↑↑i state. Following Appendix B.2, the parity will

be a function of the analysis phase φ such that

P(φ) = A cos(2φ), (8.4)

where the magnitude of A is equal to the magnitude of the coherences in Eq. 8.2. Thus, our fidelity

is

F = | hψ|Φi |2 =

1

2

(P↑↑ + P↓↓) Populations

+

1

2

|A|. Coherences (8.5)

We note that we need both population and parity measurements to distinguish our state from

a mixed state, as well as to ensure that we produce an entangled state with a fixed phase θ.
176

8.1.2 Singlet state

Our measurement of the singlet state follows a similar procedure, where we measure its

populations and its parity. The singlet state is

|Ψ

−i =

1

√

2

(|↓↑i − |↑↓i). (8.6)

The parity of the singlet state as a function of the π/2 phase is

P(φ) = −1, (8.7)

for the ideal singlet. For non-ideal singlets, we analyze a couple of cases. If our state is a combination

of |Ψ−i and |Ψ+i, the parity of the state would be less negative than -1 as the parity of |Ψ+i after

applying a π/2 pulse is +1. If our state contains any population in one of the other triplet states,

we will have a parity signal that oscillates as cos φ. We could in theory fit this oscillation, however

in practice it is difficult to fit a low amplitude sinusoid. We instead take the average value of our

parity across all phases. For any residual triplet, the average value of this oscillation will be zero.

We make a measurement of the parity with six different π/2 pulse phases equally spaced between 0

and 2π. These measurements also average to zero as

nX=5

n=0

cos(2nπ

3

+ φ0) =

nX=2

n=0



cos(2nπ

3

+ φ0) + cos((2n + 1)π

3

+ φ0)



= 2 cos(π

3

)

nX=2

n=0

cos 

(4n + 1)π

3

+ φ0



= 0.

(8.8)

If there is any non-singlet population, the average parity measured is less negative. Overall, our

singlet fidelity is
177

F = | hψ|Ψ

−i |2 =

1

2

(P↓↑ + P↑↓) Populations

−

1

2

A, Coherences (8.9)

where A is the average parity measured across all π/2 phases. For the ideal singlet, F = 1. For

|Ψ+i for example, A = 1 and F = 0.

8.2 Leakage correction

The previous analysis assumes perfect SPAM. However, neither our state preparation nor

measurement is perfect. We need to account for repumping, depumping, and leakage. Repumping

occurs when a nominally dark ion is optically pumped by the detection beam to the fluorescing

state, and starts to scatter photons. Depumping refers to a nominally bright state that is optically

pumped to a dark state due to imperfect polarization of the detection beam. Leakage occurs when

our ion population is outside the qubit manifold, for example due a scattering event before the gate

sequence. When preparing the ion in the qubit state |↓i, there is a small probability that the ion is

instead in an auxiliary state |ai that does not participate in the gate dynamics. There are several

such states within the hyperfine manifold. This state is also a dark state that does not fluoresce. To

be clear, these measurement and leakage errors do not occur during the gate sequence, only before

(leakage) or after during measurement (repumping and depumping)2.

Our raw data includes both reference and gate data as described in Sections 6.3 and 7.3.2.

From the reference data, we are able to extract depumping, repumping, leakage, as well as ion

fluorescence rates. This process is discussed in more detail in Sec. 8.3.2. Using these extracted

parameters, we are able to use maximum likelihood estimation to extract the ion populations from

the count histograms3in the gate data, largely removing any measurement error. Our analysis

leaves us only leakage to correct. The initial state of each ion ρ0,1 is

2 And at some level these are all caused by our lasers. Even with our laser-free gate there is no escape.

3 Count histogram analysis mainly done by DHS.
178

ρ0,1 = (1 − )ρ↓ + ρa, (8.10)

where  is the probability of producing the leaked state, ρ↓ = |↓i h↓|, and ρa = |ai ha|. For two ions,

the initial state is thus,

ρ0,2 = (1 − )

2

ρ↓↓ + (1 − )(ρ↓a + ρa↓) + 

2

ρaa. (8.11)

We now analyze the effect of this leaked state on both the triplet and singlet fidelity analysis.

Specifically, we need to determine how our gate sequence and parity analysis affect populations with

one leaked state (ρ↓a and ρa↓) or two leaked states (ρaa).

8.2.1 Triplet state fidelity

We assume the leaked state |ai and |↑i are dark, while |↓i is bright. Let the probabilities of two

ions bright, one ion bright, and zero ions bright be P2, P1, and P0 respectively4. These probabilities

correspond to the different ion states P(|ψi) as

P2 = P(|↓↓i),

P1 = P(|↑↓i) + P(|↓↑i) + P(|a ↓i) + P(|↓ ai),

P0 = P(|↑↑i) + P(|↑ ai) + P(|a ↑i) + P(|aai).

(8.12)

For an entangling gate with fidelity F,

ρ↓↓ → F ρ↓↓+↑↑ +

(1 − F)

2

(ρ↓↑ + ρ↑↓), (8.13)

where ρ↓↓+↑↑ is the density matrix corresponding to the triplet state5

√

1

2

(|↓↓i + |↑↑i. We assume

that any error in the entangling operation results in an incoherent mixture of |↓↑i and |↑↓i. This

4 These are Pff , P1f , and Pdd in Ref. [Ballance, 2017].

5 Not the exact entangled state we create ( √1

2

(|↓↓i + i|↑↑i), but we assume this for simplicity here.
179

assumption holds if the leading error is motional dephasing. If instead the leading error were

coherent, such as a timing or detuning error, this would have to be modified.

8.2.1.1 Gate flips spin of single ion

We now analyze a couple of different cases depending on the effect of the gate sequence on a

single-ion spin state. At the end of our entangling operation, our gate operation transforms the

single-ion state |↓i → |↑i due to the sequence of π/2 and π pulses (see Fig. 6.9). Thus, our state ρg

is

ρg = F(1 − )

2

ρ↓↓+↑↑ +

(1 − F)

2

(1 − )

2

(ρ↓↑ + ρ↑↓) + (1 − )(ρ↑a + ρa↑) + 

2

ρaa. (8.14)

Measuring the fluorescence of this state,

P2 =

F(1 − )

2

2

,

P1 = (1 − F)(1 − )

2

,

P0 =

F(1 − )

2

2

+ 2(1 − ) + 

2

.

(8.15)

Analysing each component of the fidelity measurement, our population would be

1 − P1 = 1 − (1 − F)(1 − )

2

= F + (1 − F)2 − 

2

(1 − F)

≈ F,

(8.16)

for (1 − F),   1. For the parity measurement, our entangled state alternates between |ψei =

√

1

2

(|↓↓i + |↑↑i) and |ψoi = √

1

2

(|↑↓i + |↓↑i) depending on the parity phase, and is in general a

superposition of the two even and odd states. The parity pulse also transforms the following states

as
180

|↓↑i → 1

2

(|↓i + |↑i)(|↓i − |↑i)

=

1

2

(|↓↓i + |↑↓i − |↓↑i − |↑↑i),

|↑↓i → 1

2

(|↓↓i − |↑↓i + |↓↑i − |↑↑i),

|↑ ai → 1

√

2

(|↓ ai − |↑ ai),

|a ↑i → 1

√

2

(|a ↓i − |a ↑i),

|aai → |aai

(8.17)

We assume a particular phase for these transformations, but as the error state is an incoherent

mixture of |↓↑i, |↑↓i and the leaked states, this phase is not consequential. When we produce the

even parity state |ψei with fidelity F, the two ion populations after a parity pulse are

P2 =

F(1 − )

2

2

+

(1 − F)(1 − )

2

4

,

P1 =

(1 − F)(1 − )

2

2

+ (1 − ),

P0 =

F(1 − )

2

2

+ (1 − ) + 

2 +

(1 − F)(1 − )

2

4

.

(8.18)

With this state, the parity P is

P = 1 − 2P1

= 1 −



(1 − F)(1 − 2 + 

2

) + 2 − 2

2



≈ 1 − (1 − F) − 2

= F − 2,

(8.19)

assuming (1 − F)  1. When we produce the odd parity state |ψoi, the two ion populations after a

parity pulse are
181

P2 =

(1 − F)(1 − )

2

4

,

P1 = F(1 − )

2 +

(1 − F)(1 − )

2

2

+ (1 − ),

P0 = (1 − ) + 

2 +

(1 − F)(1 − )

2

4

.

(8.20)

The parity is,

P = 1 − 2P1

= 1 −



2F(1 − 2 + 

2

) + (1 − F)(1 − 2 + 

2

) + 2 − 2

2



≈ 1 − (2F − 4F  + (1 − F) + 2)

≈ −(F − 2).

(8.21)

Thus, the parity oscillation should have an amplitude of F −2 and be symmetric about 0. Combining

the parity measurement with our population, our measured fidelity Fm is

Fm =

1

2



1 − (1 − F)(1 − )

2 + F(1 − )2



=

1

2



1 + (2F − 1)(1 − )

2



=

1

2



2F − 2(2F − 1) + 

2

(2F − 1)

= F − (2F − 1) + 

2

(2F − 1)

≈ F − ,

(8.22)

Thus, to obtain the true fidelity F, we would need to add  to our measured fidelity.

8.2.1.2 Gate does not flip spin of single ion

We now assume the gate operation leaves the single ion state unchanged such that |↑i → |↑i. This

is not the case for our experiment but we include it for completeness. Our density matrix ρg is

ρg = F(1 − )

2

ρ↓↓+↑↑ +

(1 − F)

2

(1 − )

2

(ρ↓↑ + ρ↑↓) + (1 − )(ρ↓a + ρa↓) + 

2

ρaa. (8.23)
182

Measuring the fluorescence of this state,

P2 =

F(1 − )

2

2

,

P1 = (1 − F)(1 − )

2 + 2(1 − ),

P0 =

F(1 − )

2

2

+ 

2

.

(8.24)

Thus, for a fidelity measurement, our population, (1 − P1) would be ≈ (F − 2), and the parity

amplitude would be at most 1 − 2 + 

2

, giving a fidelity of approximately F − 2. The analysis

of the parity is unchanged as in the previous section. Thus, 2 would have to be added to the

measured fidelity Fm instead of just  in the previous case. While not applicable to our experiment,

this analysis shows the importance of being aware of what the gate sequence does to a single spin,

as it will affect the leakage correction. We note that an additional π pulse to half the population

measurements would average6 out this effect to F −  .

8.2.2 Singlet state fidelity

We now turn to the fidelity analysis of the singlet state. Following Eq. 8.14, the density

matrix at the end of the gate sequence is

ρg = F(1 − )

2

ρ↓↓+↑↑ +

(1 − F)

2

(1 − )

2

(ρ↓↑ + ρ↑↓) + (1 − )(ρ↑a + ρa↑) + 

2

ρaa.

With this state, we now perform a single-qubit rotation on only one of the ions, flipping its spin.

We assume this addressing pulse is perfect for now, producing the singlet state from the triplet

state. The state including gate errors and leakage is

ρs = F(1 − )

2

ρ↓↑−↑↓ +

(1 − F)

2

(1 − )

2

(ρ↓↓ + ρ↑↑) + (1 − )(ρ↑a + ρa↓) + 

2

ρaa.

Measuring the populations at this point,

6 Pointed out to us by our theory collaborator Alex Kwiatkowski. Unfortunately, too late to take more data.
183

P2 =

(1 − F)

2

(1 − )

2

P1 = F(1 − )

2 + (1 − ),

P0 =

(1 − F)

2

(1 − )

2 + (1 − ) + 2

.

(8.25)

Thus, the single ion population would be ≈ (F − ) keeping terms first order in  and assuming

1 − F  1.

Including a parity pulse, the singlet state should be unchanged but the other spin states

should be transformed into an equal superposition of |↓i and |↑i. Now, the probability of measuring

one ion bright is

P1 = F(1 − )

2 + (1 − ) + (1 − F)

2

(1 − )

2

= (F +

1 − F

2

)(1 − )

2 + (1 − )

=

1 + F

2

(1 − )

2 + (1 − )

≈

1 + F

2

− ,

(8.26)

again assuming (1 − F),  << 1. the parity P

P = 1 − 2P1

= 1 − (1 + F)(1 − )

2 − 2(1 − )

≈ −F + 2

(8.27)

Subtracting the parity from the one ion populations and then dividing by two for Fm,
184

Fm =

1

2



F(1 − )

2 + (1 − ) + (1 + F)(1 − )2 − 1 + 2(1 − )



=

1

2



(1 + 2F)(1 − )

2 + 3(1 − ) − 1



=

1

2



(1 + 2F)(1 − 2 + 

2

) + 3(1 − ) − 1



=

1

2



(1 + 2F) − 2(1 + 2F) + (1 + 2F)

2 + 3(1 − ) − 1



≈

1

2

(2F +  − 4F )

= F(1 − 2) + 

2

≈

1

2

(2F − 3)

= F −

3

2

.

(8.28)

8.3 Simulating data

In order to validate our analysis, we create simulated data where we can simulate various

gate errors as well as leakage errors. The goal is to have simulated data with controllable fidelities

that we can test our analysis on. We create these data using QuTiP [Johansson et al., 2013]. Our

simulated entangling gate does not contain the full Hamiltonian described in Eq. 6.1, only the

interaction described by Eq. 6.27. Our simulation includes the π pulses required for the Walsh 7

modulation (Sec. 6.2.6) as well as errors such as heating and motional dephasing (Sec. 2.5). For

generating singlet states, we have an additional sequence after the entangling gate that follows our

experimental implementation in the previous chapter closely.

The free parameters in our simulation are the output state after the gate sequence, the leakage,

depumping, and repumping rates. We also set the mean zero-ion, one-ion, and two-ion bright

counts c0, c1 and c2 respectively which we later use to sample from a Poisson distribution. Each

set of simulated gate data has its own set of simulated reference data, in order to replicate our

experimental data as closely as possible. To determine the fidelity of a set of gate data, we analyze

the corresponding reference data to characterize our SPAM errors, including leakage. These values

are subsequently used to determine the fidelity from the simulated gate data.
185

8.3.1 Mapping populations to counts

Our analysis will ultimately use the ion counts to extract a population. Our simulation first

creates a density matrix corresponding to a particular state. To account for projection noise, we

make a “measurement” for every repetition of the simulation. We generate a random number from

0 to 1, and based on the value, assign an outcome following the original density matrix. To simplify

the “measurement”, we group populations that are identical after swapping the two ions as our

experiment cannot distinguish between the state of each ion. For example, ρ↓↑ and ρ↑↓ are grouped

together. We then map the “measured” populations to two ions bright, one ion bright, and zero

ions bright following Eq. 8.12.

To factor in depumping and repumping, we assign a probability that repumping and depumping

occurs during detection. We generate a random number from 0 to 1 to determine if a repump

or depump event occurs. If it does occur, a new random number between 0 and 1 is used to

determine when it occurs and the counts are modified accordingly. As an example, let us assume

the ions are initially in |↑↑i, the dark state, and it stays in the dark state for a fraction f of the

detection duration, before it is repumped to a bright state (instantaneously). The counts for this

“measurement” would be the sum of a sample of two Poissonians with λ = f c0 and λ = (1 − f)c1

respectively.

In our simulation, we make the following assumptions:

(1) There is only one depump or repump event during each “measurement”.

(a) The zero-ion-bright state only repumps once to the one-ion-bright state.

(b) The one-ion-bright state only repumps or depumps once to the two-ion-bright and

zero-ion-bright state respectively.

(c) The two-ion-bright state only depumps once to the one-ion-bright state.

(2) The leaked state behaves identically to the dark state and has the same probability of

repumping. Since the probability of having a leaked event and a repumping event is low,
186

we treat the leaked state identically to simplify the simulation.

8.3.2 State preparation and readout errors

An example of simulated and experimental data is shown in Fig. 8.1. The parameters for the

simulated data are obtained from fitting the experimental data. We obtain repump and depump

probabilities of 0.0192 and 0.0169 respectively. The ion counts c0, c1, and c2 are 5, 56, and 107

respectively, and the leakage was  = 0.0035. We generate count histograms for three different ion

states: the bright state corresponding to the starting state in Eq. 8.11 (Fig. 8.1c), the starting

state after a π/2 pulse (Fig. 8.1b), and the dark state corresponding to the starting state after a

π pulse (Fig. 8.1a). For each of those three states, we generate 200,000 measurements which are

renormalized to the experimental data that has about 18,500 measurements per state (see Sec. 6.3),

such that the total number of measurements plotted for both the simulated and experimental data

is identical.

Figure 8.1 shows good agreement between our model (green lines) and the data (blue lines).

For reference, we also plot the results from a purely Poissonian model (orange lines) where there

is no repumping, or depumping, or leakage. This model works fairly well but fails to capture the

behaviour of the tails that occur a couple of orders of magnitudes below the peaks.

8.3.3 Characterizing leakage

We vary the leakage in the simulation from 0 to 0.005 per qubit and check that our analysis is

able to track this as shown in Fig. 8.2. We find good agreement between the simulated and inferred

values, with larger uncertainties as the leakage increases. We then check that the leakage estimate

improves as we increase the amount of data in the simulation. The estimated value converges to its

set value of 0.0035 per qubit as shown in Fig. 8.3.
187 10

1

10

0

10

1

10

2

10

3

a) Dark 

Data

Only Poisson

Poisson 

with depumping, 

repumping,

and leakage

10

1

10

0

10

1

10

2

10

3

b) /2 pulse

Data

Only Poisson

Poisson 

with depumping, 

repumping,

and leakage

0 25 50 75 100 125 150

10

1

10

0

10

1

10

2

10

3

c) Bright 

Data

Only Poisson

Poisson 

with depumping, 

repumping,

and leakage Figure 8.1: Comparison of count histograms between experimental and simulated data for three

different states. a) Dark state after applying a π pulse to the initial state in Eq. 8.11. b) State after

π/2 pulse to the initial state. c) The initial bright state. For the simulated data, we generate 200,000

“measurements” that are renormalised to the experimental value of about 18,500 measurements.

This rescaling causes the counts can go below 1 for the simulated data. We see good agreement

between the experimental data (blue) and our model that incorporates depumping, repumping, and

leakage (green). For reference we also include data from a purely Poissonian model.
188

0.000 0.001 0.002 0.003 0.004 0.005

Simulated leakage

0.00025

0.00020

0.00015

0.00010

0.00005

0.00000

0.00005

Inferred leakage -

simulated leakage

Figure 8.2: Analysis of leakage estimation with varying leakages. For each simulated leakage, we

generate 100 sets of simulated data. We plot the mean of the inferred leakages minus the simulated

leakage versus the simulated leakage.

10

0

10

1

10

2

Data factor

0.000150

0.000125

0.000100

0.000075

0.000050

0.000025

0.000000

Inferred leakage -

simulated leakage

Figure 8.3: Analysis of leakage estimation with more data. For a fixed simulated leakage of

 = 0.0035 per qubit, we generate 100 sets of simulated data. We plot the mean of the inferred

leakages minus the simulated leakage. As we increase the amount of data in the simulation, we find

that the our estimates get both more accurate and more precise.
189

8.3.4 Characterizing other parameters

We perform a similar analysis with the other parameters required to map a given population

to counts: the repump rate, depump rate, zero ion, and one-ion-bright counts. The results are shown

in Fig. 8.4. Similar to Fig. 8.3, the more data we provide the better we are able to extract the true

value of the parameter from the count histograms. These data show our analysis and simulations

are behaving consistently.

8.3.5 Asymmetry in ion counts

A slightly less significant effect we examine is an asymmetry in the ion counts. If the ions

are unequally illuminated by the detection beam, they will have slightly different fluorescence rates

and the one-ion-bright histogram would have a central peak that is the sum of two Poissonians

with different means instead. For a fixed triplet and singlet fidelity of 0.999, we examine this effect

and find that it is negligible for the estimated fidelity. Moreover, this effect only seems to bias us

negatively, predicting a lower fidelity than the actual value. In our experimental data we see little

count asymmetry. Thus, we do not factor in count asymmetry into our simulations.

8.4 Analyzing simulated data

Confident that our analysis is characterizing our simulated reference data and determining the

leakage (and other parameters) appropriately, we now check our analysis with simulated gate data

with varying fidelities. The goal is to establish if there are any systematic biases in our analysis and

to sanity check the leakage correction. For the simulations in this section, the mean counts are set

to 5, 57 and 109 for the zero-ion-bright, one-ion-bright and two-ion-bright histograms respectively7.

There was no asymmetry between the the ion counts for these simulations. The repump and depump

rates are 0.0192 and 0.0169 respectively. When simulating data with varying fidelities, we are

fortunately able to measure the true fidelity of the underlying state with QuTiP, which is what

we will refer to as simulated fidelity. This is in contrast to the fidelities we extracted from count
190

0.0004

0.0003

0.0002

0.0001

0.0000

Inferred repump rate -

simulated repump rate

a) Repump rate = 0.0192

0.0003

0.0002

0.0001

0.0000

0.0001

0.0002

Inferred depump rate -

simulated depump rate

b) Depump rate = 0.0167

0.001

0.000

0.001

0.002

0.003

0.004

Inferred dark counts -

simulated dark counts

c) Zero ion bright counts = 5

10

0 101 102

Data factor

0.006

0.004

0.002

0.000

0.002

Inferred bright counts -

simulated bright counts

d) One ion bright counts = 57

Figure 8.4: Estimation of other parameters in simulation as we increase the amount of data. The

parameters are: a) Repump rate, b) Depump rate, c) Zero-ion-bright counts, and d) One-ion-brightcounts. For each data factor, we generate 100 sets of simulated data. Similar to the leakage in

Fig. 8.3, the more data we have, the the more the analysis converges to the set simulation values.
191

5 4 3 2 1 0 1 2 3 4 5

Count asymmetry

0.00150

0.00125

0.00100

0.00075

0.00050

0.00025

0.00000

0.00025

Inferred fidelity -

simulated fidelity

Singlets

Triplets

Figure 8.5: Effect of count asymmetry on triplet and singlet fidelity. For a simulated triplet and

singlet fidelity of 0.999, we find that count asymmetry has little effect on the analysis, and if at

all it biases the fidelity estimate negatively. The count asymmetry has a stronger effect on the

singlet state. The singlet state more heavily involves measuring one-ion-bright counts which are

more significantly affected by count asymmetry.

histograms, as is the case for the fidelity analysis on experimental data. Every set of simulated gate

data also has its own simulated reference data.

8.4.1 Triplet state

We first characterize our analysis for the triplet state. Following Eq. 8.5, we need to determine

both the parity and the populations. The parity A is determined from the parity oscillation with

maximum likelihood estimation and the populations are extracted from the count histograms of the

corresponding points. We first look at varying leakages and then different types of errors.

8.4.1.1 Leakage correction

We check our leakage correction by setting the triplet fidelity to 0.999 and varying the leakages

from 0 to 0.005 per qubit. The results are plotted in Fig. 8.6. From the analysis, the leakage

7 We spent a long time trying to debug our analysis when the two ion counts were at 107 instead of 109. It is very

important that c2 = 2c1 − c0 where c0, c1 and c2 are the mean zero, one and two ion counts respectively.
192

correction seems to track with increasing leakages.

0.000 0.001 0.002 0.003 0.004 0.005

Simulated leakage

0.00150

0.00125

0.00100

0.00075

0.00050

0.00025

0.00000

0.00025

Inferred fidelity -

simulated fidelity

Figure 8.6: Analysis of triplet fidelity with varying leakage. For each simulated leakage, we generate

100 sets of simulated data with a fixed underlying fidelity of 0.999. We plot the mean of the inferred

fidelities from the count histograms minus the simulated fidelity.

8.4.1.2 Motional dephasing

We analyze the effect of motional dephasing on the estimated fidelity for fidelities from 0.99

to 1, We add motional dephasing to our gate model until the required fidelity is reached, i.e. lower

fidelities need more motional dephasing. We then make 1000 sets of simulated data per simulated

fidelity, with each data set having the same statistics as the experiment described in Sec. 6.3.

This analysis is plotted in Fig. 8.7. We see that our analysis has a negative bias that infers a

lower fidelity than the true fidelity. This bias gets worse as we get closer to a true underlying fidelity

of 1. This bias is likely due to our populations and parity estimates being constrained in the region

[0, 1]. As the fidelity gets closer to 1, we are closer to this boundary. We plot the histograms of the

inferred fidelities for simulated fidelities of 0.998, 0.999 and 1 in Fig. 8.8. Note that our analysis

method does allow fidelity estimates above 1, due to the leakage correction.

We verify that the bias in the fidelity estimation is reduced with increasing statistics in

Fig. 8.9. Indeed, the more statistics we have, the closer the inferred fidelity is to the simulated
193

0.990 0.992 0.994 0.996 0.998 1.000

Simulated fidelities

0.0010

0.0008

0.0006

0.0004

0.0002

0.0000

Inferred fidelities

- simulated fidelities

Figure 8.7: Analysis of simulated triplet data with varying fidelities. For each simulated fidelity, we

generate 1000 sets of simulated data. We plot the mean of the inferred fidelities from the count

histograms minus the simulated fidelities. We see a negative bias across all fidelities that gets more

negative the closer we get to a fidelity of 1.

fidelity. We check that the spread in our fidelity estimates are also well-behaved in Fig. 8.10. As

one would expect, the spread decreases as we have more data and the standard error of the mean

follows 1/

√

n scaling where n is the number of measurements.

8.4.1.3 Detuning errors

Finally, aside from motional dephasing errors, we also check that our fidelity analysis works

well with our next leading error. This error arises from miscalibrations of the motional frequency

that will cause an error in the detuning of the gate. Figure 8.11 shows this analysis. We are able to

estimate the true fidelity as a function of detuning well, usually with a negative bias close to the

optimal detuning.

8.4.2 Singlet state

We perform a similar set of analyses for the singlet states, following equations 8.9 and 8.28 to

estimate the fidelity. We determine the parity A and populations directly from the count histogram
194

0.990 0.992 0.994 0.996 0.998 1.000 1.002 1.004

0

50

100

150

200

250

a) Simulated triplet fidelity:0.9980

Simulated

fidelity

=0.9976

=0.0019

0.992 0.994 0.996 0.998 1.000 1.002 1.004

0

50

100

150

200

250

b) Simulated triplet fidelity:0.9990

Simulated

fidelity

=0.9985

=0.0018

0.992 0.994 0.996 0.998 1.000 1.002 1.004

0

50

100

150

200

250

c) Simulated triplet fidelity:1.0000

Simulated

fidelity

=0.9992

=0.0017

Figure 8.8: Histograms of inferred fidelities from analysis for underlying triplet fidelities of 0.998,

0.999, and 1. For each simulated fidelity, we generate 1000 sets of simulated data. The mean and

standard deviation are indicated by the red line and red shaded region. As the fidelity approaches 1,

we see a larger negative bias between the mean of the analysed fidelities and the simulated fidelity.
195

10

0

10

1

10

2

Data factor

0.0010

0.0008

0.0006

0.0004

0.0002

0.0000

0.0002

Inferred fidelity -

simulated fidelity

Figure 8.9: Analysis of simulated triplet data with more data compared to the experiment. For

each data factor, we generate 100 sets of simulated data. We plot the mean of the inferred fidelities

minus the simulated fidelities.

10

0

10

1

10

2

Data factor

10

5

10

4

Standard error of the mean

Figure 8.10: Analysis of standard deviation of simulated triplet data analysis with more data. As we

generate more data, the standard deviation decreases linearly on a log-log scale as one would expect.

The green dashed line corresponds to a 1/

√

n scaling where n is the number of measurements which

will be proportional to the data factor.

measurements with and without the additional π/2 pulse respectively (see Sec. 7.3.2).
196

0.94

0.95

0.96

0.97

0.98

0.99

1.00

Fidelities

a)

Simulated

Count histograms

0.10 0.05 0.00 0.05 0.10

Fractional detuning error

0.0010

0.0005

0.0000

0.0005

Inferred fidelities -

simulated fidelities

b)

Figure 8.11: Simulated triplet data with detuning errors. a) We plot the fidelity as a function of

the fractional detuning error. The insensitivity we see close to zero is due to the Walsh 7 sequence

we employ (see sec. 6.2.6). b) We plot the mean of the inferred fidelities minus the simulated fidelity.

We find that our estimated fidelity tracks the simulated fidelity well.
197

8.4.2.1 Leakage correction

We first verify our leakage correction. For a fixed fidelity of 0.999, we vary the leakage from 0

to 0.005 per qubit. The results are shown in Fig. 8.12.

0.000 0.001 0.002 0.003 0.004 0.005

Simulated leakage

0.0008

0.0007

0.0006

0.0005

0.0004

0.0003

0.0002

0.0001

0.0000

Inferred fidelity -

simulated fidelity

Figure 8.12: Analysis of singlet fidelity with varying leakage. For each simulated leakage, we

generate 100 sets of simulated data with a fixed underlying fidelity of 0.999. We plot the mean of

the inferred fidelities minus the simulated fidelity.

8.4.2.2 Entangling gate errors

We first look at singlet states with varying fidelities. Here, the error is only from motional

dephasing during the entangling gate; the individual addressing is perfect. For the singlets, we also

verify our fidelity estimations using the populations from the simulations before mapping them to

counts8. The analysis is shown in Fig. 8.13. Similar to the triplets in Fig. 8.7, we see an overall

negative bias that gets worse as we get closer to 1. For the fidelities extracted from the populations,

the bias is fairly independent of the underlying fidelity. Histograms for simulated fidelities of 0.998,

0.999 and 1 are shown in Fig. 8.14.

To check if this bias is due to the limited statistics used in the experiment, we generated

8

In principle we could have done this for the triplets as well. Unfortunately, at the time of writing, our code was

only setup to do the analysis from the count histograms.
198

0.990 0.992 0.994 0.996 0.998 1.000

Simulated fidelities

0.0006

0.0004

0.0002

0.0000

0.0002

0.0004

Inferred fidelities

- simulated fidelities

Count histograms

Simulated populations

Figure 8.13: Analysis of simulated singlet data with varying fidelities. For each simulated fidelity, we

generate 1000 sets of simulated data. We plot the mean of the inferred fidelities minus the simulated

fidelities. We plot the fidelities from the count histograms as well as the simulated populations

(which includes projection noise). Our measurements are negatively biased, with the bias from

the simulated populations being fairly uniform across all fidelities. The fidelities from the count

histograms have a worse bias the closer the true underlying fidelity is to 1.

simulated data with more statistics. The simulated fidelity was set to 0.999. Figure 8.15 shows

that as we have more simulated data the bias decreases for fidelities derived from both the count

histograms and the simulated populations.

8.4.2.3 Individual addressing errors

Next, we investigate the effect of individual addressing errors, from incorrectly calibrating

either the phase or the duration of the Ramsey sequence discussed in Sec. 7.3.1.1. The results

are plotted in Figures 8.17 and 8.18 respectively. For both cases, the estimated fidelity tracks the

simulated fidelity closely. Near the optimal phase and duration, we have a negative bias.
199

0.994 0.996 0.998 1.000 1.002

0

50

100

150

200

250

a) Simulated singlet fidelity:0.9980

Simulated

fidelity

=0.9977

=0.0014

0.994 0.996 0.998 1.000 1.002

0

50

100

150

200

250

b) Simulated singlet fidelity:0.9990

Simulated

fidelity

=0.9986

=0.0014

0.995 0.996 0.997 0.998 0.999 1.000 1.001 1.002

0

50

100

150

200

250

c) Simulated singlet fidelity:1.0000

Simulated

fidelity

=0.9994

=0.0012

Figure 8.14: Histograms of inferred fidelities from analysing the count histograms for underlying

singlet fidelities of 0.998, 0.999 and 1. For each simulated fidelity, we generate 1000 sets of simulated

data. The mean and standard deviation are indicated by the red line and red shaded region. As the

fidelity approaches 1, we see a larger negative bias between the mean of the analysed fidelities and

the simulated fidelity.
200

10

0

10

1

10

2

Data factor

0.0010

0.0008

0.0006

0.0004

0.0002

0.0000

0.0002

Inferred fidelity -

simulated fidelity

Count histograms

Simulated populations

Figure 8.15: Analysis of simulated singlet data with more data compared to the experiment. We

plot the mean of the inferred fidelities minus the simulated fidelities. As we increase the amount of

data, our inferred fidelity is less biased for both the fidelities from the count histograms (blue) and

simulated populations (orange).

10

0

10

1

10

2

Data factor

10

5

10

4

Standard error of the mean

Count histograms

Simulated populations

Figure 8.16: We plot the standard error of the mean as a function of the amount of simulated data

for both the analysis from the count histograms (blue) and the simulated populations (orange). The

green dashed line corresponds to 1/

√

n scaling where n is the number of measurements, which is

proportional to the data factor.
201

0.992

0.994

0.996

0.998

Fidelities

a)

Simulated

Count histograms

0.10 0.05 0.00 0.05 0.10

Fractional Ramsey phase error

0.0008

0.0006

0.0004

0.0002

0.0000

0.0002

Inferred fidelities -

simulated fidelities

b)

Figure 8.17: Analysis of simulated singlet data with an error in the phase of the Ramsey sequence.

For each error value, we simulate 100 sets of data. a) We plot the fidelity as a function of the

error, with the orange line indicating the simulated fidelities while the blue points are the fidelities

extracted from the data points. b) We plot the difference between the inferred and simulated

fidelities.
202

0.975

0.980

0.985

0.990

0.995

1.000

Fidelities

a)

Simulated

Count histograms

0.10 0.05 0.00 0.05 0.10

Fractional Ramsey duration error

0.0006

0.0004

0.0002

0.0000

0.0002

0.0004

Inferred fidelities -

simulated fidelities

b)

Figure 8.18: Analysis of simulated singlet data with an error in the duration of the Ramsey sequence.

For each error value, we simulated 100 sets of data. a) We plot the fidelity as a function of the error,

with the orange line indicating the simulated fidelities while the blue points correspond to fidelities

extracted from the count histograms. b) We plot the difference between the inferred and simulated

fidelities.
Every year is getting shorter never seem to find the time.

Plans that either come to naught or half a page of scribbled lines.

Hanging on in quiet desperation is the English way,

The time is gone, the song is over,

Thought I’d something more to say.

Pink Floyd 9

Conclusion

We have demonstrated a new technique for laser-free spin-motion coupling in trapped ions that

uses a radiofrequency magnetic field gradient. We extended this technique to two-qubit entangling

gates, generating symmetric and antisymmetric Bell states with fidelities of 0.999(1) and 0.998(1)

respectively. The symmetric Bell-state fidelity is competitive with the highest fidelity across any

platform. The antisymmetric Bell-state fidelity is as far as we know, the highest that has been

demonstrated, integrating both high-fidelity entanglement and individual addressing, both without

lasers. Where do we go from here?

9.1 Improvements to entangling gate

Our gate duration is currently 740 μs. Preliminary experiments have shown that we can

reduce that duration to approximately 560 μs while maintaining a fidelity of ≈ 0.999 with current

hardware. To reduce the duration further, we can increase the current used to generate the magnetic

field gradient. We currently use about 1.2 A, but we estimate that our trap can handle close to 3 A

safely. If we can reduce the gate duration without any other adverse effects, that should improve

the fidelity as well. Motional errors decrease as the gate duration is decreased.

203
204

Aside from increasing the current, improvements could be made to the pulse shaping to reduce

the ramp times for turning our radiofrequency gradient and microwave fields on and off. These

ramps comprise a significant portion of the gate duration at 160 μs.

9.2 Randomized benchmarking

Aside from reducing the error in our gates, we want to characterize the error more carefully.

For our entangling gate, we currently perform only one gate and measure the Bell-state fidelity from

one input state. Furthermore, our analysis is complicated by the state preparation and measurement

error that is larger than our gate error.

Randomized benchmarking [Knill et al., 2008] of our two-qubit operations, where a series of

two-qubit and single-qubit operations are performed, would enable more sensitive characterization

of our errors, assuming we do not have duty cycle related effects with longer sequences. The

individual addressing we demonstrate makes benchmarking viable in our system. Benchmarking

also provides a more practical demonstration for our gates; multiple gates are required for any

useful computation. Currently, laser-free randomized benchmarking in trapped ions has only been

performed for single-qubit operations [Harty et al., 2014, Piltz et al., 2014].

9.3 New trap geometries

The techniques described in this thesis could enable new trap geometries. The use of a

current at megahertz instead of gigahertz frequencies vastly simplifies the current handling in surface

electrode traps. Larger electrodes could be used for larger currents, and thus larger gradients at the

ion for stronger spin-motion coupling. Instead of a single-layer trap that we use, a multi-layer trap

with buried electrodes would enable larger currents while reducing the effect of the electric field.

9.4 Mixed-species entanglement

Finally, the new technique we have developed also simplifies mixed-species entangling gates.

Using field-sensitive states on both Mg+ and Ca+, for which we have an oven in our system, would
205

enable a single gradient to couple the spins of both species to their shared motion. Compared to our

current implementation, just two additional fields symmetrically detuned from the Ca+ qubit would

be required to entangle an Mg+ and Ca+ ions. To date, no mixed species entangling operations have

been performed without lasers. Mixed-species entanglement would enable an “all-electronic” qubit,

where the state preparation and logical operations could be performed on the Mg+ qubit without

lasers. Their internal states could then be mapped to the Ca+ qubit for readout, which has lasers

at longer wavelengths (≈ 400 nm), that will hopefully cause less charging than lasers at 280 nm.

High-fidelity entanglement and mixed species operations should enable a host of experiments to be

carried out. We have built a versatile system and it will be exciting to see what else it is capable of!
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A

Baker-Campbell-Hausdorff Theorem

The Baker-Campbell-Hausdorff (BCH) theorem helps us evaluate expressions in the form of

e

Aˆ

Beˆ −Aˆ= Bˆ + [A, ˆ Bˆ] + 1

2

[A, ˆ [A, ˆ Bˆ]]... (A.1)

where Aˆ and Bˆ are operators. One usually encounters this when trying to take some Hamiltonian

into the interaction picture with another term. For example, taking a Hamiltonian Hˆ into the

interaction picture with respect to Hˆ

0, we first find the time evolution operator for Hˆ0, Uˆ0(t)

Uˆ

0(t) = exp 

−

i

~

Z t

0

Hˆ

0dt



. (A.2)

To obtain the interaction picture Hamiltonian Hˆ

I (t),

Hˆ

I (t) = Uˆ†

0

(t)HˆUˆ

0(t), (A.3)

which is in the form of eq. A.1, with Aˆ =

i

~

R t

0 Hˆ

0dt and Bˆ = Hˆ .

216
217

If [A, ˆ Bˆ] = sBˆ where s is some constant,

e

Aˆ

Beˆ −Aˆ= e

sB. ˆ (A.4)

A.1 Common interaction picture transformations

We now go through common interaction picture transformations that were encountered in the

main text. We list them here to avoid repetition and to focus on the physics in the main text while

having the details here.

A.1.1 Qubit

The qubit energy is usually described by

Hˆ

qubit =

~ω0

2

σˆz, (A.5)

where ω0 is the qubit frequency. The propagator corresponding to this interaction is

U

†

qubit = exp(iω0tσz). (A.6)

For another operator Bˆ, going into the interaction picture with respect to Hˆ

qubit is

Bˆ

I = Uˆ†

qubitBUˆ

qubit. (A.7)

If Bˆ ∝ σˆz, Bˆ

I = Bˆ. If Bˆ ∝ σˆx,y, we understand that

σˆx = ˆσ+ + ˆσ−,

σˆy = −iσˆ+ + iσˆ−,

[ˆσz, σˆ+] = 2ˆσ+,

[ˆσz, σˆ−] = −2ˆσ−.

(A.8)
218

Thus, for the example of some field oscillating at ω that couples our two-level system such that

Hˆ = 2~Ω cos ωtσˆx, (A.9)

Our interaction Hamiltonian Hˆ

I with respect to the qubit is

Hˆ

I = 2~Ω cos ωtUˆ†

qubitσˆxUˆ

qubit

= 2~Ω cos ωt σˆ+e

iω0t + ˆσ−e−iω0t



= ~Ω(e

iωt + e−iωt)



σˆ+e

iω0t + ˆσ−e−iω0t



= ~Ω



σˆ+e

−iδt + ˆσ−eiδt

,

(A.10)

where δ = ω − ω0 and we have made the rotating wave approximation that ω, ω0  |ω − ω0|.

A.1.2 Motion

The ions’ motional energy is described by

Hˆmotion = ~ωraˆ

†

a, ˆ (A.11)

where ωr is the motional mode frequency. The propagator corresponding to this interaction is,

U

†

motion = exp(iωraˆ

†

at ˆ ). (A.12)

We usually transform the creation and annihilation operators aˆ and aˆ

† with respect to this

term. We use the commutator relations,

[ˆa

†

a, ˆ aˆ] = [ˆa

†

, aˆ]ˆa = −a, ˆ

[ˆa

†

a, ˆ aˆ

†

] = ˆa

†

[ˆa, aˆ

†

] = ˆa

†

,

(A.13)
219

where we understand that [a, ˆ aˆ

†

] = 1 and [AˆB, ˆ Cˆ] = Aˆ[B, ˆ Cˆ] + [A, ˆ Cˆ]Bˆ. Using these results, we

obtain,

Uˆ†

motionaˆUˆmotion = ˆae−iωrt

,

Uˆ†

motionaˆ

†Uˆmotion = ˆa†

e

iωrt

.

(A.14)

A.1.3 Modified BCH

Equation A.1 can be modified to

e

AˆCˆ

Beˆ −AˆCˆ= Bˆ + [AˆC, ˆ Bˆ] + 1

2

[AˆC, ˆ [AˆC, ˆ Bˆ]]... (A.15)

We assume that [A, ˆ Cˆ] = 0, [C, ˆ Bˆ] = 0, and [A, ˆ Bˆ] = sBˆ where s is some constant.

[AˆC, ˆ Bˆ] = Aˆ[C, ˆ Bˆ] + [A, ˆ Bˆ]Cˆ

= sBˆC, ˆ

and

[AˆC, ˆ [AˆC, ˆ Bˆ]] = [AˆC, s ˆ BˆCˆ]

= [A, s ˆ BˆCˆ]Cˆ

= s

2BˆCˆ2

.

Thus,

e

AˆCˆ

Beˆ −AˆCˆ= Bˆ + sBˆCˆ +

1

2

s

2BˆCˆ2

...

= Beˆ sCˆ.

(A.16)
B

Single qubit rotations

Here we describe some of the single-qubit operations presented in the main text in more detail.

B.1 Rotation matrices

First, looking at the Hamiltonian for a resonant field with a variable phase on a single spin,

Hˆ = ~Ω(ˆσ+e

−iφ + ˆσ−eiφ)

= ~Ω(ˆσx cos φ + ˆσy sin φ).

The time evolution operator is

220
221

Uˆ(t) = exp −

i

~

Ht ˆ

= exp (−iΩt(ˆσx cos φ + ˆσy sin φ)),

= cos Ωt

ˆI − isin Ωt(ˆσx cos φ + ˆσy sin φ),

where exp (−iAωt ˆ ) = cos (ωt)

ˆI − iAˆ sin (ωt) for Aˆ2 = ˆI.

For a π

2

pulse,

Uˆ π

2

= Uˆ(t =

π

4Ω) = 1

√

2

(

ˆI − iσˆx cos φ − iσˆy sin φ)

For a π pulse,

Uˆ

π = Uˆ(t =

π

2Ω) = −iσˆx cos φ − iσˆy sin φ.

For qubit frequency shifts,

Hˆ = ~

∆

2

σˆz.

The corresponding time evolution operator is

Uˆ

z(t) = cos ∆

2

t

ˆI − isin

∆

2

tσˆz.

B.2 Parity flopping

Here we derive the effect of a parity π/2 pulse on two-qubit entangled states. First analysing

the effect of the propagator Uˆ

1 for a π/2 pulse on a single ion,
222

Uˆ

1 |↓i =

1

√

2

(|↓i − ie−iφ |↑i),

Uˆ

1 |↑i =

1

√

2

(−ieiφ |↓i + |↑i).

Evaluating the propogators on the two-ion states,

Uˆ

1Uˆ

2 |↓↓i =

1

√

2

(|↓i − ie−iφ |↑i) ⊗

1

√

2

(|↓i − ie−iφ |↑i)

=

1

2

(|↓↓i − ie−iφ(|↑↓i + |↓↑i) − e

−i2φ

|↑↑i),

Uˆ

1Uˆ

2 |↓↑i =

1

√

2

(|↓i − ie−iφ |↑i) ⊗

1

√

2

(−ieiφ |↓i + |↑i)

=

1

2

(−ieiφ |↓↓i + |↓↑i − |↑↓i − ie−iφ |↑↑i),

Uˆ

1Uˆ

2 |↑↓i =

1

√

2

(−ieiφ |↓i + |↑i) ⊗

1

√

2

(|↓i − ie−iφ |↑i)

=

1

2

(−ieiφ |↓↓i − |↓↑i + |↑↓i − ie−iφ |↑↑i),

Uˆ

1Uˆ2 |↑↑i =

1

√

2

(−ieiφ |↓i + |↑i) ⊗

1

√

2

(−ieiφ |↓i + |↑i)

=

1

2

(−e

i2φ

|↓↓i − ieiφ(|↓↑i + |↑↓i) + |↑↑i).

Applying the propagators to the Bell states,

Uˆ

1Uˆ

2(

1

√

2

(|↓↑i + |↑↓i)) = −i

√

2

(e

iφ |↓↓i + e−iφ |↑↑i).

And the parity, P(φ) := P↑↑ + P↓↓ − P↓↑+↑↓, is always 1 regardless of phase.
223

For the singlet state,

Uˆ

1Uˆ

2(

1

√

2

(|↓↑i − |↑↓i)) = 1

√

2

(|↓↑i − |↑↓i),

P(φ) = −1.

For the other triplet states,

Uˆ

1Uˆ2(

1

√

2

(|↓↓i + |↑↑i)) = 1

2

√

2



(1 − e

i2φ

)|↓↓i − 2i cos φ(|↓↑i + |↑↓i) + (1 − e

−i2φ

|↑↑i)



P(φ) = 1

8

(2 − 2 cos 2φ + 2 − 2 cos 2φ − 8 cos2 φ)

=

1

8

(4 − 4 cos 2φ − 4 − 4 cos 2φ)

= − cos 2φ,

Uˆ

1Uˆ

2(

1

√

2

(|↓↓i − |↑↑i)) = 1

2

√

2



(1 + e

i2φ

)|↓↓i − 2 sin φ(|↓↑i + |↑↓i) − (1 + e

−i2φ

|↑↑i)



P(φ) = 1

8

(2 + 2 cos 2φ + 2 + 2 cos 2φ − 8 sin2 φ)

=

1

8

(4 + 4 cos 2φ − 4 + 4 cos 2φ)

= cos 2φ.

These triplet states have parity oscillations that are π periodic in φ.
C

Displacements in phase space

This section outlines how we calculate displacements using static and oscillating gradients.

The goal is derive the expressions for the displacements quantum mechanically and show that they

agree with classical intuitions.

C.1 Rotating frame transformations

We first quickly recap the theory required to transfer into and out of different rotating frames.

These are specifically for rotating frames at the ion’s motion. We also equivalently call this moving

into the interaction frame with respect to some Hamiltonian. For example, moving into the rotating

frame with respect to the ion motion, the Hamiltonian describing its motion is Hˆm = ~ωraˆ

†aˆ where

ωr is the ion’s motional frequency. The propagator for this Hamiltonian is

Uˆ†

rot(t) = exp (iωraˆ

†

at ˆ ). (C.1)

For a general operator Aˆ, it transforms at Aˆ → Uˆ†

rot(t)AˆUˆ

rot(t). For the creation and

annihilation operators aˆ, aˆ

†

, we make use of the Baker-Campbell-Hausdorff (BCH) formula to

224
225

perform this transformation. For simpler commutation relations where [A, ˆ Bˆ] = sBˆ, e

AˆBeˆ −Aˆ = esBˆ.

Making use of the commutators [ˆa

†a, ˆ aˆ] = −aˆ and [ˆa†a, ˆ aˆ†

] = ˆa

†

,

aˆ → e

−iωrt

aˆ

aˆ

† → eiωrt

aˆ

†

.

(C.2)

C.1.1 Transformation of displacement operator

Making use of those transformations for aˆ and aˆ

†

, we can evaluate the transformation on

a displacement operator, which is what the propagators we are interested in will look like. The

displacement operator Dˆ(α) is

Dˆ(α) = exp (αaˆ

† − α∗

aˆ), (C.3)

where α is the displacement in phase space. Making use of BCH again, where e

Aˆ

e

Bˆ = eAˆ+Bˆ+[A, ˆ Bˆ]/2

,

we can rewrite Dˆ(α) as

Dˆ(α) = e

(αaˆ

†−α∗aˆ)

= e

(αaˆ

†−α∗aˆ)

e

|α|

2/2

e

−|α|

2/2

= e

−|α|

2/2

e

αaˆ

†

e

−α

∗aˆ

.

(C.4)

Now if we want to transform this displacement operator out of a specific rotating frame, we

can undo the previous transformation. In some rotating frame, the displacement is

Dˆ(αrot) = Uˆ†

rotDˆ(α)Uˆ

rot. (C.5)

Transforming out of this rotating frame,
226

Dˆ(α) = Uˆ

rotDˆ(αrot)Uˆ†

rot

= e

−|αrot|

2/2Uˆ

rote

αrotaˆ

†

e

−α

∗

rotaˆUˆ†

rot

= e

−|αrot|

2/2Uˆ

rote

αrotaˆ

†

Uˆ†

rotUˆ

rote

−α

∗

rotaˆU

†

rot

= e

−|αrot|

2/2

e

αrotaˆ

†e−iωrt

e

−α

∗

rotaeˆ

iωrt

= Dˆ(αe−iωrt),

(C.6)

where we made use of e

Aˆ

e

Bˆ

e

−Aˆ = Beˆ s

for [A, ˆ Bˆ] = sBˆ. So now we have an easy way of transforming

displacement operators from one frame to another, which is useful in transforming the displacement

operators that we’ve evaluated before in a frame rotating at the ion’s motion.

C.2 Static gradient

Let us start first with the static gradient case. Our propagator for the gradient, which is a

displacement operator, in the rotating frame of the ion is

Uˆ†

g

(t) = exp 2Ωg

ωr

σˆz(−aeˆ

−iωrt + ˆa†

e

iωrt

)

1

= Dˆ



σˆz

2Ωg

ωr

e

iωrt



= Dˆ(ˆσzαrot),

(C.7)

where Ωg is

Ωg =

r0B0

z

4~

(μz↑↑ − μz↓↓). (C.8)

This is the term describing the gradient spin-motion coupling strength; this term is proportional to

the ground state extent, the B field gradient, and the matrix element. The displacement in this

rotating frame at the ion’s motion is

αrot =

2Ωg

ωr

e

iωrt

. (C.9)
227

We want to find the change in the position, rˆ = r0(aˆ + aˆ

†

), where r0 is the ground state extent

of the motional mode with frequency ωr.

rˆrot = Dˆ(αrot)r0(ˆa + ˆa

†

)Dˆ†(αrot)

= r0(ˆa − αrot + ˆa

† − α∗

rot)

= r0(ˆa + ˆa

† − 2Re(αrot)).

(C.10)

To obtain the spin-dependent displacement ∆xrot, we need to multiply 2r0Re(αrot) by a factor of 2

as |↑i is displaced one direction, and |↓i the other direction.

∆xrot = 4r0Re(αrot)

=

8r0Ωg

ωr

cos ωrt

=

8r

2

0B0

z

4~ωr

(μz↑↑ − μz↓↓) cos ωrt

=

8~

2mωr

B0

z

4~ωr

(μz↑↑ − μz↓↓) cos ωrt

=

B0

z

(μz↑↑ − μz↓↓)

mω2

r

cos ωrt

=

FB

mω2

r

cos ωrt,

(C.11)

where we define the magnetic force FB = B0

z

(μz↑↑ − μz↓↓). Note that this is in the rotating frame of

the ion’s motion and we have a displacement that is oscillating at the motional frequency. Moving

to the lab frame, we make use of Eq. C.6 to obtain the displacement

α = αrote

−iωrt

=

2Ωg

ωr

.

(C.12)

And following a similar process as before we obtain the displacement in the lab frame,

1

I’ve dropped the constant terms from a(e

−iωrt − 1) as that comes about from the pulse shaping. See that section

for more details.
228

∆x =

FB

mω2

r

. (C.13)

This displacement is in complete agreement with a classical harmonic oscillator that is

displaced by a constant force (which is turned on adiabatically).

C.3 Oscillating gradient

We now determine the displacement of an oscillating gradient. Again, our starting point is

the displacement operator that we obtain having gone into the rotating frame of the ion.

Uˆ†

g

(t) = exp h Ωg

ωr − ωg

σˆz(−aeˆ

−i(ωr−ωg)t + ˆa†

e

i(ωr−ωg)t

)+

Ωg

ωr + ωg

σˆz(−aeˆ

−i(ωr+ωg)t + ˆa†

e

i(ωr+ωg)t

)

i

= Dˆ



σˆz

h Ωg

ωr − ωg

e

i(ωr−ωg)t +

Ωg

ωr + ωg

e

i(ωr+ωg)t

i

= Dˆ(ˆσzαrot),

(C.14)

where the displacement in the rotating frame is

αrot =

Ωg

ωr − ωg

e

i(ωr−ωg)t +

Ωg

ωr + ωg

e

i(ωr+ωg)t

. (C.15)

Moving to the lab frame,

α = αrote

−iωrt =

Ωg

ωr − ωg

e

−iωgt +

Ωg

ωr + ωg

e

iωgt

. (C.16)

Now calculating the displacement ∆x,
229

∆x = 4r0Re(α)

= 4r0

 Ωg

ωr − ωg

+

Ωg

ωr + ωg



cos ωgt

= 4r0

2Ωgωr

ω2

r − ω2g

cos ωgt

=

8r

2

0B0

z

(μz↑↑ − μz↓↓)

4~

ωr

ω2

r − ω2g

cos ωgt

=

8FB

4~

~

2mωr

ωr

ω2

r − ω2g

cos ωgt

=

FB

m(ω2

r − ω2g

)

cos ωgt

(C.17)

In the lab frame, we have a displacement that’s oscillating at ωg. The amplitude of this

displacement also increases as ωg → ωr. This displacement is in agreement with a classical harmonic

oscillator driven non-resonantly.

C.3.1 Rotating frame at ωg

We finally analyze the displacement in the rotating frame at ωg. We follow a similar procedure

to transform the displacement into this frame. We first look at the displacement in phase space,

αωg = αeiωgt

=

h Ωg

ωr − ωg

e

−iωgt +

Ωg

ωr + ωg

e

iωgt

i

e

iωgt

=

Ωg

ωr − ωg

+

Ωg

ωr + ωg

e

i2ωgt

(C.18)

Now to calculate the displacement of the spins,
230

∆xωg = 4r0Re(αωg

)

= 4r0

Ωg

ωr − ωg

+

Ωg

ωr + ωg

cos 2ωgt



=

4r

2

0FB

4~

1

ωr − ωg

+

1

ωr + ωg

cos 2ωgt



=

FB

2mωr

1

ωr − ωg

+

1

ωr + ωg

cos 2ωgt



≈

FB

2mωr(ωr − ωg)

,

(C.19)

where in the last line we have dropped the fast oscillating displacement. Comparing this displacement

in the rotating frame at ωg to amplitude of the displacement in the lab frame,

∆xωg

∆x

≈

FB

2mωr(ωr − ωg)

. FB

m(ω2

r − ω2g

)

=

ωr + ωg

2ωr

,

(C.20)

Which is a number that is close to but slightly smaller than 1. Thus, as far as the figure is concerned,

the displacement in this frame should be roughly the same amplitude as the displacement in the lab

frame.

C.4 Summary of displacements

The displacements we need for the figure are summarised below.

Static gradient Oscillating gradient

Rotating frame Lab Lab ωg

Displacement ∆x

FB

mω2

r

FB

m(ω2

r−ω2

g

)

cos ωgt

FB

2mωr(ωr−ωg)

C.5 Pulse shaping

We finally look at the effect of pulse shaping on the gradient propagator. For simplicity, we

just do this for the static gradient case. From the previous set of notes, the propagator associated
231

with this gradient is,

Uˆ†

g

(t) = i2Ωgσˆz

Z t

0

(ˆae−iωrt1 + ˆa

†

e

iωrt1

)dt1

=

2Ωg

ωr

σˆz

h

−aeˆ

−iωrt1 + ˆa

†

e

iωrt1

it

0

=

2Ωg

ωr

σˆz(−aˆ(e

−iωrt − 1) + ˆa†

(e

iωrt − 1)).

(C.21)

There is an extra term that corresponds to the force suddenly being turned on at t = 0. Now

let us assume that the force is instead ramped on slowly over time τ with the amplitude function

F(t), where F(0) = 0, F(τ ) = F(t) = 1. Evaluating the propagator here,

Uˆ†

g

(t) = i2Ωgσˆz

Z t

0

F(t1)(ˆae−iωrt1 + ˆa

†

e

iωrt1

)dt1

= i2Ωgσˆz

Z t

τ

F(t1)(ˆae−iωrt1 + ˆa

†

e

iωrt1

)dt1

+ i2Ωgσˆz

Z τ

0

F(t1)(ˆae−iωrt1 + ˆa

†

e

iωrt1

)dt1

=

2Ωg

ωr

σˆz

h

F(t1)(−aeˆ

−iωrt1 + ˆa†

e

iωrt1

)

it

τ

+

2Ωg

ωr

σˆz

h

F(t1)(−aeˆ

−iωrt1 + ˆa†

e

iωrt1

)

iτ

0

−

2Ωg

ωr

Z τ

0

F

0

(t1)(−aeˆ

−iωrt1 + ˆa†

e

iωrt1

)dt1

=

2Ωg

ωr

σˆz

h

F(t)(−aeˆ

−iωrt + ˆa†

e

iωrt

)

i

−

2Ωg

ωr

Z τ

0

F

0

(t1)(−aeˆ

−iωrt1 + ˆa†

e

iωrt1

)dt1,

(C.22)

where we have made use of the boundary conditions and that the fact that F

0

(t) = 0 for t > τ .

If we assume a linear ramp for F(t), so F

0

(t) = 1/τ for τ ≥ t ≥ 0, the last term would scale as

(1/ωr)/τ . Thus, if we ramp the force on slowly compared to 1/ωr, the last term can be ignored.

This would be analogous to a classical harmonic oscillator with a static force. If harmonic

oscillator is initially at no displacement and the force is turned on adiabatically, it will eventually

have a fixed but static displacement depending on the magnitude of the force. If instead the force

was turned on suddenly, it would be oscillating at its natural frequency with amplitude corresponding
232

to that force.
D

Derivations of Hamiltonians

Here we collate the lengthier derivations for equations in the main text.

D.1 Magnus Expansion

We use the Magnus expansion to evalute the time evolution operator for Hamiltonians that

do not commute with itself at different times. Let Aˆ(t) = −

i

~Hˆ

g(t). The time-evolution operator

corresponding to Hˆ

g is,

Uˆ

e(t) = expX∞

k=1

Ωˆ

k(t),

where,

Ωˆ

1(t) = Z t

0

Aˆ(t1)dt1

Ωˆ

2(t) = 1

2

Z t

0

dt1

Z t1

0

dt2[Aˆ(t1), Aˆ(t2)]

Ωˆ

3(t) = 1

6

Z t

0

dt1

Z t1

0

dt2

Z t2

0

dt3

n

[Aˆ(t1), [Aˆ(t2), Aˆ(t3)]] + [[Aˆ(t3), [Aˆ(t2), Aˆ(t1)]]o.

233
234

D.2 Geometric phase gates

We derive the propogator of the gate interaction (Eq. 2.31)

Hˆ

g = ~Ωg (ˆσi1 ± σˆi2)

n

aeˆ

i∆t + ˆa†

e

−i∆t

o

,

We determine the Magnus expansion with

Aˆ = −iΩg (ˆσi1 ± σˆi2)

n

aeˆ

i∆t + ˆa†

e

−i∆t

o

. (D.1)

The first term is

Ωˆ

1(t) = −iΩg (ˆσi1 ± σˆi2)

Z t

0

dt1

n

aeˆ

i∆t1 + ˆa†

e

−i∆t1

o

= −iΩg (ˆσi1 ± σˆi2)



aeˆ

i∆t1

i∆

+

aˆ

†

e

−i∆t1

−i∆

t

0

=

Ωg

∆

(ˆσi1 ± σˆi2)

n

−aˆ(e

i∆t − 1) + ˆa†

(e

−i∆t − 1)o

= Dˆ



Ωg

∆

(ˆσi1 ± σˆi2) (e

−i∆t − 1)

.

(D.2)

The second term is

Ωˆ

2(t) = −

1

2

Ω

2

g

(ˆσi1 ± σˆi2)

2

Z t

0

Z t1

0

dt1dt2

h

aeˆ

i∆t1 + ˆa†

e

−i∆t1

, aeˆ

i∆t2 + ˆa†

e

−i∆t2

i

= −

1

2

Ω

2

g

(ˆσi1 ± σˆi2)

2

Z t

0

Z t1

0

dt1dt2(e

i∆(t1−t2) − e−i∆(t1−t2)

)

= −iΩ

2

g

(ˆσi1 ± σˆi2)

2

Z t

0

Z t1

0

dt1dt2 sin ∆(t1 − t2)

= −i

Ω

2

g

∆

(ˆσi1 ± σˆi2)

2

Z t

0

dt1[cos ∆(t1 − t2)]t1

0

= −i

Ω

2

g

∆

(ˆσi1 ± σˆi2)

2

Z t

0

dt1(1 − cos(∆t1))

= −i

Ω

2

g

∆2

(ˆσi1 ± σˆi2)

2

[∆t1 − sin (∆t1)]t

0

= −i

Ω

2

g

∆2

(ˆσi1 ± σˆi2)

2

(∆t − sin (∆t)).

(D.3)
235

Further terms vanish. The first term corresponds to a spin-dependent displacement, the second to a

spin-dependent phase.

D.3 Static gradient with detuned microwaves

First, looking at the case with just a static gradient with detuned microwaves, the interaction

Hamiltonian,

Hˆ

I (t) = Hˆ

g(t) + Hˆμ(t),

Hˆ

g(t) = 2~Ωgσˆz(ˆae−iωrt + ˆa

†

e

iωrt

),

Hˆ

μ(t) = ~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt),

(D.4)

where Ωg describes the gradient term,

Ωg =

brr0B0

z

4~

(μz↑↑ − μz↓↓).

μz↑↑ = h↑| μz |↑i and μz↓↓ = h↓| μz |↓i describe the matrix elements of the ion’s magnetic moment

along the quantization axis, z. B0

z

is the gradient of the B field along the quantization axis in the

direction of the motional mode. This term is Ωz

z

in Ref. [Ospelkaus et al., 2008]. The Rabi frequency

of the microwave field on resonance is described by Ωμ. The frequency of the motional mode of

interest is ωr. The detuning of the microwave field from the qubit frequency is δ.

We want to go into the interaction picture with respect to the gradient term Hˆ

g(t). However,

once you include the aˆ and aˆ

†

terms, the commutator at different times is no longer 0. Fortunately,

it’s a relatively straightforward application of the Magnus expansion in Appendix D.1.

Ωˆ

3 and higher vanish and Aˆ(t) = −i2Ωgσˆz(aeˆ

−iωrt + aˆ†

e

iωrt

). So now evaluating the nonvanishing terms,
236

Ωˆ

1(t) = Z t

0

Aˆ(t1)dt1

= −i2Ωgσˆz

Z t

0

(ˆae−iωrt1 + ˆa

†

e

iωrt1

)dt1

=

2Ωg

ωr

σˆz

h

aeˆ

−iωrt1 − aˆ

†

e

iωrt1

it

0

=

2Ωg

ωr

σˆz(ˆa(e

−iωrt − 1) − aˆ†

(e

iωrt − 1)).

To evaluate Ωˆ

2(t), we first need to look at the commutator [Aˆ(t1), Aˆ(t2)],

[Aˆ(t1), Aˆ(t2)] = −4Ω2

g

[ˆae−iωrt1 + ˆa

†

e

iωrt1

, aeˆ

−iωrt2 + ˆa†

e

iωrt2

]

= −4Ω2

g

(e

−iωr(t1−t2) − eiωr(t1−t2)

)

= −8iΩ

2

g

sin (ωr(t1 − t2)).

This term is independent of the spin/motional state, so would commute with all the terms and can

be ignored. Higher order commutators (Ωˆ

3 onwards would also vanish as this term would commute

with Aˆ(t). Thus,

Uˆ†

e

(t) = exp 2Ωg

ωr

σˆz(−aˆ(e

−iωrt − 1) + ˆa†

(e

iωrt − 1)) (D.5)

Now to get the new interaction Hamiltonian Hˆ 0

I

,

Hˆ 0

I

(t) =Uˆ

e(t)

†

~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt)Uˆ

e(t).

We make use of BCH here,

e

Aˆ

Beˆ −Aˆ= Bˆ + [A, ˆ Bˆ] + 1

2

[A, ˆ [A, ˆ Bˆ]]...
237

where Aˆ =

2Ωg

ωr

σˆz(−aˆ(e

−iωrt − 1) + ˆa†

(e

iωrt − 1)) and Bˆ = ~Ωμ(ˆσ+e−iδt + ˆσ−eiδt).

The relevant commutators are,

[ˆσz, σˆ+] = 2ˆσ+

[ˆσz, σˆ−] = −2ˆσ−.

Thus,

[A, ˆ Bˆ] = 4~ΩgΩμ

ωr

(ˆσ+e

−iδt − σˆ−eiδt)(−aˆ(e−iωrt − 1) + ˆa†

(e

iωrt − 1)).

The higher order terms will go as ( Ω0

g

ωr

)

n

so we can probably ignore them. This gives us the

interaction Hamiltonian Hˆ 0

I

,

Hˆ 0

I

(t) =~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt)+

4~ΩgΩμ

ωr

(ˆσ+e

−iδt − σˆ−eiδt)(−aˆ(e−iωrt − 1) + ˆa†

(e

iωrt − 1)).

(D.6)

When δ = ωr we get the BSB interaction,

Hˆ

bsb =

4~ΩgΩμ

ωr

(ˆσ+aˆ

† + ˆσ−aˆ),

dropping the fast rotating terms. Similarly when δ = −ωr we get the RSB interaction which gives

us,

Hˆ

rsb = −

4~ΩgΩμ

ωr

(ˆσ+aˆ + ˆσ−aˆ

†

).
238

D.4 Oscillating gradient with carrier microwaves

We now derive this for the oscillating gradient case, following a similar procedure as before.

Hˆ

I (t) =2~Ωgσˆz cos ωgt(ˆae−iωrt + ˆa

†

e

iωrt

)

+ ~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt)

=~Ωgσˆz(e

iωgt + e−iωgt

)(ˆae−iωrt + ˆa

†

e

iωrt

)

+ ~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt)

=~Ωgσˆz(ˆae−i(ωr−ωg)t + ˆa

†

e

i(ωr−ωg)t

) negative frequency component

+ ~Ωgσˆz(ˆae−i(ωr+ωg)t + ˆa

†

e

i(ωr+ωg)t

) positive frequency component

+ ~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt).

Here, the negative and positive frequency components of the oscillating gradient have explicitly

been separated. We can now going through the same process as before to obtain the propagator,

U

†

e

(t) = exp h Ωg

ωr − ωg

σˆz(−aˆ(e

−i(ωr−ωg)t − 1) + ˆa†

(e

i(ωr−ωg)t − 1))+

Ωg

ωr + ωg

σˆz(−aˆ(e

−i(ωr+ωg)t − 1) + ˆa†

(e

i(ωr+ωg)t − 1))i

.

(D.7)

This propagator corresponds to two displacements with different magnitudes oscillating at

different frequencies. This is elaborated on more in section D.5. Now to obtain the interaction

Hamiltonian,

Hˆ 0

I

(t) =~Ωμ(ˆσ+e

−iδt + ˆσ−eiδt)+

2~ΩgΩμ

ωr − ωg

(ˆσ+e

−iδt − σˆ−eiδt)(ˆa†

(e

i(ωr−ωg)t − 1) − aˆ(e−i(ωr−ωg)t − 1))+

2~ΩgΩμ

ωr + ωg

(ˆσ+e

−iδt − σˆ−eiδt)(ˆa†

(e

i(ωr+ωg)t − 1) − aˆ(e−i(ωr+ωg)t − 1)).

(D.8)

Which more or less gives us the dynamics as before. The only differences are that ωr → ωr±ωg,

and that we lose a factor of two in the gradient term. That could be understood by the fact that
239

with the oscillating gradient, the sideband gets split into two frequency components. However, you

can probably more than make up for that by making ωr − ωg smaller in the denominator.

The BSB is now at δ = ωr ± ωg and the RSB with δ = −(ωr ± ωg). In practice, you would

probably always want to use the ωr − ωg sideband as that will be much stronger. Of course, one is

also closer to the carrier so would have to employ pulse shaping to get rid of off-resonant effects.

D.5 Comparison of displacements between static and oscillating gradients

We want to look at the expressions we’ve derived in terms of displacements. Looking at the

propagator for the static gradient,

Uˆ

static = exp

2Ωg

ωr

σˆz(−aˆ

†

(e

iωrt − 1) + ˆa(e−iωrt − 1)).

We can rewrite this as a displacement operator Dˆ(α),

Dˆ(ˆσzα) = exp (ˆσz(αaˆ

† − α∗

aˆ)),

where for the static gradient,

αstatic = −

2Ωg

ωr

(e

iωrt − 1) (D.9)

Essentially, we get a spin-dependent displacement on the ion, and this displacement is

oscillating in time. For the oscillating gradient case, the two displacements α−, α+ corresponding

to negative or positive frequency components are,

α− = −

Ωg

ωr − ωg

(e

i(ωr−ωg)t − 1) (D.10)

α+ = −

Ωg

ωr + ωg

(e

i(ωr+ωg)t − 1). (D.11)
240

Summarising,

Static gradient Oscillating gradient

Rotating frame ωr ωr − ωg ωr + ωg

Displacement proportional to 2Ωg/ωr Ωg/(ωr − ωg) Ωg/(ωr + ωg)

D.6 Derivation of spin-flip Rabi frequency from electric field sideband

We are using the magnetic field gradient to measure the E field from the oscillating field

at ωg/2π =5 MHz. This procedure follows Ref. [Warring et al., 2013a]. Let’s first start with the

Hamiltonian from this E field.

Hˆ

e(t) = 2~Ωe cos ωgt(ˆa

†

e

iωrt + ˆae−iωrt

)

= ~Ωe(e

iωgt + e−iωgt

)(ˆa

†

e

iωrt + ˆae−iωrt

)

= ~Ωe



(ˆa

†

e

iδ−t + ˆae−iδ−t

) + (ˆa

†

e

iδ+t + ˆae−iδ+t

)



,

where ωg is the angular frequency of this electric field, ωr is the frequency of the motional mode,

δ± = ωr ± ωg, and Ωe is the Rabi frequency associated with this E field. This expression is after

going into the interaction picture with respect to the ion’s motion (~ωra

†a) and thus the creation

and annihilation operators pick up the time-dependent terms.

Ωe =

qEr0

2~

.

q is the charge of the ion, E is the peak amplitude of the electric field at the ion and r0 the ground

state extent of the ion along that particular mode of motion.

We are using the oscillating gradient close to a GHz to measure this. The Hamiltonian of the

microwave gradient is,
241

Hˆ

μw = ~Ωsb(ˆσ+e

−iδt + ˆσ−eiδt)(ˆa†

e

iωrt + ˆae−iωrt

),

where Ωsb is the ground state sideband Rabi frequency, and δ is the detuning of the microwave

gradient from the qubit. Again, this is after going into the interaction picture with respect to the

qubit ( ~ω0

2

σˆz) and the motion ~ωr(a

†a).

Now, we want to go into the interaction picture with respect to the E field term. We have to

evaluate the time evolution operator from this Hamiltonian, Hˆ

e. However, what’s a little tricky is

that the commutator at different times for this Hamiltonian is not 0. One solution is to use the

Magnus expansion as described in Appendix D.1.

For this case Ωˆ

3 onwards vanishes. Now evaluating these quantities for the expression we

have,

Ωˆ

1(t) = −i

Z t

0

dt1Ωe



(ˆa

†

e

iδ−t1 + ˆae−iδ−t1

) + (ˆa

†

e

iδ+t1 + ˆae−iδ+t1

)



= −iΩe



aˆ

†

e

iδ−t1 − aeˆ−iδ−t1

iδ−

+

aˆ

†

e

iδ+t1 − aeˆ

−iδ+t1

iδ+

t

0

= Ωe



−aˆ

†

(e

iδ−t − 1) + ˆa(e−iδ−t − 1)

δ−

+

−aˆ

†

(e

iδ+t − 1) + ˆa(e−iδ+t − 1)

δ+



.

Now for Ωˆ

2, let’s first evaluate the commutator,

[Aˆ(t1), Aˆ(t2)] = −Ω

2

e

[(ˆa

†

e

iδ−t1 + ˆae−iδ−t1

) + (ˆa

†

e

iδ+t1 + ˆae−iδ+t1

),

(ˆa

†

e

iδ−t2 + ˆae−iδ−t2

) + (ˆa

†

e

iδ+t2 + ˆae−iδ+t2

)]

= −Ω

2

e

(−e

iδ−(t1−t2) + e−iδ−(t1−t2) − ei(δ−t1−δ+t2) + e−i(δ−t1−δ+t2)

− e

iδ+(t1−t2) + e−iδ+(t1−t2) − ei(δ+t1−δ−t2) + e−i(δ+t1−δ−t2)

),

using [a, ˆ aˆ

†

] = 1. One can see that all the operators have been removed from this expression and we

are left with scalar, time-dependent terms. Since this is spin/motion independent, it would just

contribute a global phase that we can just ignore.
242

Thus, back to our original problem of the propagator of the electric field

Uˆ

e(t) = exp Ωˆ1(t)

= exp Ωe



−aˆ

†

(e

iδ−t − 1) + ˆa(e−iδ−t − 1)

δ−

+

−aˆ

†

(e

iδ+t − 1) + ˆa(e−iδ+t − 1)

δ+

,

and

Uˆ†

e

(t) = exp Ωe



aˆ

†

(e

iδ−t − 1) − aˆ(e−iδ−t − 1)

δ−

+

aˆ

†

(e

iδ+t − 1) − aˆ(e−iδ+t − 1)

δ+

.

Now we wish to evaluate the interaction Hamiltonian Hˆ

I (t) = Uˆ†

e (t)Hˆ

μwUˆe(t). We can make

use of BCH,

[Ωˆ

1(t)

†

, Hμw] = " Ωe



aˆ

†

(e

iδ−t − 1) − aˆ(e−iδ−t − 1)

δ−

+

aˆ

†

(e

iδ+t − 1) − aˆ(e−iδ+t − 1)

δ+

 ,

~Ωsb(ˆσ+e

−iδt + ˆσ−eiδt)(ˆa†

e

iωrt + ˆae−iωrt

)

#

=~ΩsbΩe(ˆσ+e

−iδt + ˆσ−eiδt)

×



−e

−iωrt

(e

iδ−t − 1) − eiωrt

(e

−iδ−t − 1)

δ−

+

−e

−iωrt

(e

iδ+t − 1) − eiωrt

(e

−iδ+t − 1)

δ+



=~ΩsbΩe(ˆσ+e

−iδt + ˆσ−eiδt)



−e

−iωgt + e−iωrt − eiωgt + eiωrt



×(

1

ωr − ωg

+

1

ωr + ωg

)

=~ΩsbΩe(ˆσ+e

−iδt + ˆσ−eiδt)



−e

−iωgt + e−iωrt − eiωgt + eiωrt



(

2ωr

ω2

r − ω2g

)

=

2~ΩsbΩeωr

(ω2

r − ω2g

)

(ˆσ+e

−iδt + ˆσ−eiδt)



−e

−iωgt + e−iωrt − eiωgt + eiωrt



.

When δ = ωg, we thus get spin flips whose Rabi frequency is,

Ωspin-flip =

2ΩsbΩeωr

(ω2

r − ω2g

)

. (D.12)

From that we should be able to evaluate the magnitude of the E field.
243

D.6.1 Comparison to classical derivation

From Ref. [Ospelkaus et al., 2008],

Ωsb =

B0r0

2~

μB,

where B0is the gradient of the magnetic field and μB the magnetic moment. r0 =

q ~

2mωr

. Plugging

those values in,

Ωspin-flip =

B0r0

2~

μB

qEr0

2~

2ωr

ω2

r − ω2g

=

B0μB

2~

qE

~

ωr

ω2

r − ω2g

~

2mωr

=

B0

2

μB

2~

qE

m

1

ω2

r − ω2g

=

B0

2

μB

2~

re,

(D.13)

where re is the displacement of the ion by the E field. For a classical harmonic oscillator with

frequency ω0 that’s driven at some non-resonant frequency ω without damping, the amplitude of

the displacement is F/m

ω

2

0−ω2

. For our case, F = qE and now this agrees with equation 4 in Ulrich’s

paper exactly now.

Note: We are not considering other modes of motion which might come in to play, but to a

smaller extent.