Control and measurement of a single-ion quantum

harmonic oscillator

by

Katherine Casey McCormick

B.S., Astrophysics, University of California, Santa Cruz, 2012

B.A., Mathematics, University of California, Santa Cruz, 2012

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

2019
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This thesis entitled:

Control and measurement of a single-ion quantum harmonic oscillator

written by Katherine Casey McCormick

has been approved for the Department of Physics

Dietrich Leibfried

David J. Wineland

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iii

McCormick, Katherine Casey (Ph.D., Physics)

Control and measurement of a single-ion quantum harmonic oscillator

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

A trapped-ion harmonic oscillator represents a rich and well-controlled platform for

tests of quantum physics, with applications to quantum simulations, quantum sensing, and

quantum information processing. Owing to its net charge, the ion’s motion is affected by

changes in the electric potential. Consequently, electric field and potential curvature noise

will limit the coherence of the ion’s harmonic oscillation. While in some respects, this serves

as a limitation in the performance of the trapped ion system (for example, in the fidelity of

two-qubit gates or quantum simulations with motional quanta), it also opens an opportunity

to use the ion as a precise field sensor and to use engineered quantum states of motion to

enhance the sensitivity to these fields beyond the standard quantum limit (SQL). This thesis

describes the implementation of such applications on a single beryllium ion confined in a Paul

trap.

In the thesis, I present the theory of an ideal harmonic oscillator and the implementation of this system using the motion of a single trapped ion, including producing special

quantum states of motion; these states include the energy eigenstates of the harmonic oscillator, called number states, up to n = 100, superpositions of number states of the form

√

1

2 (|0+ |n), with n up to 18, and coherently displaced number states with an average occupation  ̄n up to approximately 300. I describe how we use these quantum states of motion

for precise sensing of fluctuations of the harmonic oscillator frequency. The sensitivity of

the √

1

2 (|0+ |n) state ideally follows the 1/n Heisenberg scaling for frequency sensitivity.

Additionally, I describe investigations of the spectrum of motional frequency noise using a

series of coherent displacements of the motion of the ion, with features similar to Ramsey and

dynamical decoupling sequences for two-level systems. While these displaced states don’t
iv

improve sensitivity over the SQL, they can be simply and rapidly implemented with trapped

ions. I discuss extensions to all of these experiments, including to multiple ions in a new

trap designed to confine 3 or 4 ions in individually tunable potential wells.
Dedication

For Mom and Dad.
vi

Acknowledgements

These 7 years have been a time of great personal growth. I owe a lot of this to my

experiment, where my tenacity and gumption was tested many times. Whenever the cryostat

broke in a new and interesting way, I was grateful to have fellow graduate students Susanna,

Raghu, and Shaun in the lab to offer suggestions and sympathy, postdocs David A. (now a

professor at University of Oregon) and Daniel (now a staff scientist) as well as my co-advisor,

Dave W. to give sage advice and encouragement, and Andrew, who, despite being my de

facto advisor, got his hands dirty and helped with a lot of the tedious tasks and never gave

up on me. Later additions to the lab whom I am grateful to are Hannah, who has taken

on an ambitious project, but always greets her problems with humor, and Jonas, who saved

me from the lab, taking over many of my half-finished projects, so that I could turn to my

computer to write 2 papers, a proposal, and this thesis, all in succession. I know that I leave

this experiment in very thoughtful and capable hands and I’m excited to see what comes out

of it with Jonas at the helm. Finally, I would like to express endless gratitude to my advisor,

Didi. He gave me support, encouragement, and pressure when I needed it to persevere and

never settle for less than excellence. An additional thanks goes especially to Dave and Didi,

who have had to read a lot of my writing in the past year or so. Their comments on my

work have greatly improved my writing, and I am grateful to them for the many hours spent

on this.

Outside of the lab, I found camaraderie in a few different circles. The dinner club nights

and “chool” bus bouldering trips with Mike, Matt, Molly, Steve, Tom, Dan, and company
vii

provided well deserved diversion through the early years of grad school. Later, being a

member of the CU women’s ultimate team, Colorado Kali, was one of the most rewarding

things I have participated in. I am inspired by the dedication, grit, and weirdness of that

group of women. Finally, I thoroughly enjoyed living in our little bus commune-turned

conventional house with Dani, Britt, Erica, Nicole, and Buck over the last few years.

Thank you, Mom and Dad, for the lifetime of unwavering love and support, and Ryan

and Matthew for the jokes, games, and conversations about science, politics, climbing, and

everything in between. Dad and Matthew, I will cherish the memories made during our

annual 14er hikes and hope that this tradition continues in the PNW and wherever else life

takes us.

And finally, Dan. These last 2 years with you have been wonderful. Thank you for the

snacks, back-rubs, encouragement, and tolerance of the stressball I’ve been during the final

stretch of graduate school.
viii

Contents

Chapter

1 Introduction 1

1.1 The harmonic oscillator as a test bed for quantum physics .......... 1

1.2 Quantum-enhanced measurements . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Application to trapped-ion quantum computing ................ 4

1.4 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Quantum harmonic oscillators 7

2.1 Wave mechanics representation ......................... 8

2.2 Phase-space representation and Wigner functions ............... 8

2.2.1 Time evolution of Wigner functions and the rotating frame . . . . . . 11

2.3 Number states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Number-state superpositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Displaced number states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Decoherence in quantum oscillators . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 Quantum vs. classical measurements . . . . . . . . . . . . . . . . . . . . . . 19

3 Beyond the 1-d oscillator model: trapped-ion harmonic oscillators 21

3.1 Producing the harmonic potential – RF Paul traps . . . . . . . . . . . . . . 22

3.2 Beryllium electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ix

3.3 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Doppler cooling in 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Stimulated Raman transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.1 Lamb-Dicke Approximation . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.2 Ground state cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.3 Raman interactions outside the Lamb-Dicke regime . . . . . . . . . . 42

3.5 Microwave transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Apparatus 46

4.1 Surface electrode traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.1 Double-well trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.2 Triangle trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Laser system, microwaves, and resonant electric field excitation of motion . . 52

4.2.1 Doppler cooling and detection laser . . . . . . . . . . . . . . . . . . . 53

4.2.2 Repump laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.3 Raman laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.4 Frequency doubling cavity with BBO . . . . . . . . . . . . . . . . . . 60

4.2.5 Spatial filtering of Raman beams . . . . . . . . . . . . . . . . . . . . 63

4.2.6 Photoionization laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.7 Microwave drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.8 Excitation of motion with resonant electric fields . . . . . . . . . . . 65

4.3 UHV copper vacuum enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Radio-frequency trap drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.7 Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.8 Experimental Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
x

4.8.1 Loading ions into a trap . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.8.2 Doppler and ground state cooling . . . . . . . . . . . . . . . . . . . . 76

4.8.3 Micromotion compensation . . . . . . . . . . . . . . . . . . . . . . . . 77

4.8.4 State-selective fluorescence detection . . . . . . . . . . . . . . . . . . 79

4.8.5 Motional state analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Number states 83

5.1 Producing pure number states . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Time scales of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Exploring error sources and resulting populations through simulations . . . . 88

5.4 Potential application: force-sensing with number states . . . . . . . . . . . . 88

6 Mode-frequency sensing with number state superpositions 92

6.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 Number-state superposition creation . . . . . . . . . . . . . . . . . . . . . . 94

6.3 Interferometer characterization . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4 Mode-frequency tracking with number-state interferometers . . . . . . . . . . 97

6.5 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.5.1 Time scales of experiments . . . . . . . . . . . . . . . . . . . . . . . . 100

6.5.2 Sources of decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5.3 Auto-balanced frequency tracking experiments . . . . . . . . . . . . . 102

6.5.4 Anharmonic contributions . . . . . . . . . . . . . . . . . . . . . . . . 103

6.6 Extensions to work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7 Mode-frequency sensing with coherent displacements 106

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.2 States and ion fluorescence signals from coherent displacements . . . . . . . 109

7.3 Noise sensing with motion-echo sequences . . . . . . . . . . . . . . . . . . . 113
xi

7.3.1 Basic principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3.2 Effects of oscillator frequency fluctuations . . . . . . . . . . . . . . . 116

7.3.3 Motion-echo sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3.4 Response to a monochromatic modulation . . . . . . . . . . . . . . . 119

7.4 Experimental implementation and results . . . . . . . . . . . . . . . . . . . . 121

7.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.4.2 Displaced number states . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.4.3 Motion-echo experiments . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8 Future directions: Coupled harmonic oscillators and experiments with multiple ions 132

8.1 Previous attempt with triangle trap . . . . . . . . . . . . . . . . . . . . . . . 132

8.1.1 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.2 Design changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.3 Possible experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9 Conclusion 138

Bibliography 139

Appendix

A Precision limit of “classical” measurements on “classical” states 147
xii

Tables

Table

5.1 Duration of pulse sequences to produce number states . . . . . . . . . . . . . 87

6.1 Duration of pulse sequences to produce number-state superpositions . . . . . 101
Figures

Figure

2.1 Wavefunctions and energies of the first few harmonic oscillator eigenstates. . 9

2.2 Wigner functions of first few harmonic oscillator number states, from n = 0−5. 14

2.3 Wigner functions of superpositions of number states of the form |0+|n, with

n = 1 − 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Wigner functions and corresponding number state populations of coherent

states with α = 0, 1, and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Schematic of a frictionless ball sitting on a saddle potential under influence of

gravity depicting concept of a confining pseudopotential. . . . . . . . . . . . 23

3.2 Schematic of a canonical linear RF Paul. . . . . . . . . . . . . . . . . . . . . 24

3.3 Energy level structure of singly ionized 9Be. . . . . . . . . . . . . . . . . . . 28

3.4 Schematic sketch depicting velocity-selective absorption of Doppler cooling

light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Schematic of a two-photon stimulated Raman transition . . . . . . . . . . . 36

3.6 Spatial wavefunction extent compared with Raman laser wavelength . . . . . 40

3.7 Rabi frequencies versus number state n, without the Lamb-Dicke approximation 44

4.1 Potential and pseudopotential contours of a three-dimensional RF Paul trap

compared to surface electrode traps . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Microscope images of “double-well” surface electrode trap . . . . . . . . . . . 48
xiv

4.3 Macroscopic trap with 3-dimensional RF confinement compared with a surfaceelectrode trap with 3-dimensional RF confinement . . . . . . . . . . . . . . . 49

4.4 Microscope images of the triangle trap . . . . . . . . . . . . . . . . . . . . . 50

4.5 Scaled-up 2-d-array traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.6 Beryllium energy levels, with laser beams and microwaves represented as colored arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7 Saturation measurement of detection beam . . . . . . . . . . . . . . . . . . . 55

4.8 Blue Doppler beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.9 Red Doppler beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.10 Raman beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.11 Raman double-pass AOM photo . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.12 Effect of spatially filtering of Raman beam . . . . . . . . . . . . . . . . . . . 64

4.13 Schematic of electrical configuration for microwave, tickle, and RF monitor

connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.14 Comparison of the old and new pillboxes . . . . . . . . . . . . . . . . . . . . 69

4.15 Photograph of bath cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.16 Comparison of the new and old imaging systems . . . . . . . . . . . . . . . . 72

4.17 Inventor drawing of new imaging system and mounting configuration . . . . 73

4.18 Comparison of bright histogram to an ideal Poissonian histogram of same

average value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 Generating number states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 Sideband flopping on number states . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Simulation of n = 100 number state creation . . . . . . . . . . . . . . . . . 89

5.4 Simulation of force-sensing protocol for number states up to n=100 . . . . . 91

6.1 Generating superpositions of number states . . . . . . . . . . . . . . . . . . . 95

6.2 Interference and sensitivity of different number-state superpositions . . . . . 98
xv

6.3 Oscillator frequency tracking using number-state interferometers . . . . . . . 101

6.4 Schematic illustrating the auto-balanced feedback loop applied to local oscillator104

7.1 Schematic phase-space sketch of the displacements motion-echo sequence . . 115

7.2 Spin-flip probability Pπ

↓ of ion after tickle excitation on versus detuning . . . 123

7.3 Response of ion to tickle excitation versus detuning . . . . . . . . . . . . . . 125

7.4 Response of ion to tickle excitation versus coherent drive duration . . . . . . 127

7.5 Characterization of motion-echo experiments with tones of 10 kHz and 100 kHz129

7.6 Noise sensing with motion-echo experiments . . . . . . . . . . . . . . . . . . 130

8.1 Images of ions loaded in the triangle trap . . . . . . . . . . . . . . . . . . . . 133

8.2 DC and RF electrodes of triangle trap used in previous experiments . . . . . 135

8.3 Triangle trap electrode redesigns . . . . . . . . . . . . . . . . . . . . . . . . . 136
Chapter 1

Introduction

In this thesis, I will describe a series of experiments that use quantum states of a single

trapped ion to characterize its center-of-mass motion and fields that affect it. From my

perspective, there are several motivations for this work: 1) it allows us to test quantum mechanics (QM) in new regimes and develop techniques that can be applied to other systems,

for example, macroscopic systems, which will provide additional tests of QM. 2) It enables

us to utilize QM to our benefit, using quantum coherence to obtain better measurement

precision than we would otherwise be able to achieve. 3) It provides a way forward for

higher-fidelity quantum computations with trapped ions that rely on the motional coherence

of the ions.

1.1 The harmonic oscillator as a test bed for quantum physics

Because of its strange and counterintuitive predictions, quantum mechanics has a long

history of inspiring spirited arguments among prominent physicists. Albert Einstein famously

proclaimed, “God does not play dice,” as he had deep discomfort with the probabilistic nature of quantum mechanics, introducing the EPR paradox to argue that quantum mechanics

is not a complete description [Einstein et al., 1935; Einstein, 1936]. Erwin Schr ̈odinger, too,

did not like the implications of quantum mechanics, and through correspondence with Ein-
2

stein, he developed the “cat paradox” which has since come to be known as “Schr ̈odinger’s

cat” [Schr ̈odinger, 1935]. The premise of the paradox is this: a cat is in an isolated box with

a radioactive particle. When the particle decays, it triggers a mechanism that kills the cat.

At a time equal to the half-life of the particle, an observer opens the box to find the cat

either dead if the particle decayed or alive if the particle did not. Quantum mechanics says

that before the observer opens the box, the cat is in a superposition of both dead and alive,

entangled with the particle’s state.

Schr ̈odinger’s cat paradox represents a description of events which is at odds with human

intuition; this has unsettled many scientists, opening opportunity to resolve this apparent

paradox by refining our intuition of quantum mechanics and exploring the boundary between

the quantum and classical descriptions of nature. This has inspired interest in observing

quantum mechanics on mesoscopic and macroscopic scales, and continues to be an intriguing area of study from a philosophical and foundational perspective. Researchers continue to

design experiments to push the boundaries of quantum mechanics by producing larger and

larger quantum states. For example, researchers have prepared macroscopic superpositions

of states in a superconducting circuit [Wilhelm et al., 2001], entangled the vibrational modes

of two particles consisting of 1016 atoms, separated by 15 cm [Lee et al., 2011], and observed

wave-particle duality in compounds with masses of approximately 104 times the mass of a

proton [Eibenberger et al., 2013].

The harmonic oscillator, a thoroughly studied, well understood, and ubiquitous physical

system, could provide the ideal platform for such tests of quantum mechanics. The textbook

example of a harmonic oscillator is a mass on a spring which oscillates due to the restoring

force provided by the spring. Other examples include: a pendulum swinging back and forth

under gravity, an LC electrical circuit, light in an optical cavity, microfabricated cantilevers

and membranes, and a single ion in a harmonic trap. In principle, every harmonic oscillator
3

is a quantum harmonic oscillator, but the quantum mechanics that governs the dynamics

of the system is not typically apparent until we go to the very small and very cold. Recent

experiments have studied quantum mechanical behavior of higher energy states [McCormick

et al., 2019a; Joshi et al., 2019] or of larger, more massive oscillators [Lee et al., 2011; Chu

et al., 2018], at scales where the behavior is usually well described by classical mechanics.

Such experiments serve to explore this boundary between quantum and classical physics and

demonstrate a growing level of quantum control over relatively large harmonic oscillators.

1.2 Quantum-enhanced measurements

With improved control over harmonic oscillator systems, researcher have been able to

push the limits of systems exhibiting hallmarks of quantum mechanics farther than ever

before. But as the states become more non-classical, they also become more susceptible to

perturbations from the environment. This increased coupling to the environment is often

stated as one of the reasons that macroscopic quantum superpositions are typically not observed [Zurek, 1991]. While this poses limitations on the level of quantum control that we

can achieve, it also opens an opportunity to use non-classical states as more sensitive probes

of certain external perturbations. Therefore, efforts toward improved quantum control are

not only of philosophical interest—more non-classical states could be used towards precision

measurements, potentially contributing to technological advances and to answering fundamental questions about the nature of the universe. While the idea of using non-classical

states to improve the performance of sensors is not new [Helstrom, 1976; Caves et al., 1980;

Wineland et al., 1992; Bollinger et al., 1996], we have now achieved the level of measurement sensitivity where we benefit from using quantum techniques to enhance the sensitivity.

LIGO, for example, has recently added the use of squeezed light in one arm of the interferometer for a factor of ∼ 3 dB enhancement in their sensitivity to gravitational waves, which
4

promises about an 8-fold increase in detected events [Aasi, 2013]. The use of entangled

many-atom states for enhanced performance in atomic clocks and atom interferometers is

also on the horizon [Wineland et al., 1998; Polzik and Ye, 2016; Engelsen et al., 2016].

Likewise, the non-classical states of motion of a single trapped ion can be used for enhanced

measurement sensitivity. Since ions have a net charge, they are particularly susceptible to

electric fields. While this susceptibility affects the coherence of the quantum motion of the

ion and limits the quantum states that we can produce, it also means that the more susceptible states are better probes of this electric field noise. Developing quantum-enhanced

techniques for precise electric field and curvature measurement with a trapped-ion mechanical oscillator could contribute to discoveries in fundamental physics, surface science, and, as

I will detail next, improved fidelity in quantum computations.

1.3 Application to trapped-ion quantum computing

In most trapped-ion quantum computing proposals, gates are achieved by coupling the

ions’ motion to their internal electronic and hyperfine states via lasers or microwave interactions. To generate a two-qubit gate, most proposals also rely on the fact that ions interact

with one another via the Coulomb force, so oscillation of one ion is felt by the other [Cirac

and Zoller, 1995]. Two-qubit gate fidelities achieved are among the highest for current implementations of qubit gates, but one of the main limitations is dephasing and heating of the

harmonic oscillator motion providing the coupling between the qubits [Gaebler et al., 2016;

Ballance et al., 2016].

Some of these errors can be attributed to electric field noise. The dominating mechanism

that produces electric field noise on trap electrodes is not known and an active area of investi-
5

gation. Of particular interest is the electric field noise near the harmonic oscillator frequency

which heats the harmonic oscillator—this is known as anomolous heating because the source

is unclear and cannot be attributed to Johnson or technical noise [Brownnutt et al., 2015].

The experiments I will discuss in this thesis contribute to further study of the motional coherence of ions, helping to answer questions about anomolous heating and electric field noise.

Unlike other studies on anomalous heating, these methods also provide a way to measure

noise components at frequencies away from the harmonic oscillator frequency. This allows us

to follow the oscillator frequency as it changes over time, minimizing errors associated with

undetected deviations between the oscillation frequency and, for example, frequencies of a

laser driving a two-qubit gate operation. This could help identify and alleviate the added

susceptibility to fluctuating stray fields when using traps with very small ion-to-electrode

distances, which offer significant advantages in terms of scalability and speed of operation

in high-fidelity quantum computing.

1.4 Thesis structure

This thesis is structured as follows. I will first briefly cover quantum harmonic oscillators, including introducing different representations of quantum harmonic oscillator states

of particular interest. Then I will discuss the implementation of a trapped-ion harmonic oscillator, including parts of the system that go beyond the simple one-dimensional harmonic

oscillator model and how we are able to isolate a single mode of oscillation in the system

to perform experiments which can be described by a simple 1-d model. Following a chapter

describing the experimental apparatus, I will discuss experiments involving three different

classes of states of a quantum harmonic oscillator: number (or Fock) states, superpositions

of number states of the form |0+ |n(for simplicity, normalization is dropped), and coher-
6

ently displaced number states. Finally, I will discuss how these experiments performed with

one ion can be extended to multiple ions, and in particular our plans with three ions in a

triangular configuration.
Chapter 2

Quantum harmonic oscillators

“Everything is a harmonic oscillator or a two-level system.”

- Professor Thomas Degrand, University of Colorado, 2012.

Harmonic oscillators are among the most ubiquitous physical systems. Harmonic motion

is one of the first topics a first-year physics student encounters, and it is revisited time and

time again in different contexts. An ion trapped in a confining potential is one of the simplest

examples of a harmonic oscillator system. An object follows harmonic motion in a restoring

potential, governed by spring constant k, of the form V (x) = 1

2 kx2 so that the Hamiltonian

that describes the system is

H = 1

2m

pˆ

2 +

1

2

mω2

xˆ2, (2.1)

where m is the mass of the object, ω =

k

m is the frequency of oscillation, and ˆx and ˆp

are the position and momentum operators. Replacing ˆx and ˆp with the raising and lowering

operators, ˆa and ˆa†

, where ˆx =



2mω (ˆa† + ˆa) and ˆp = i

mω

2 (ˆa† − aˆ), we have

H = ω(

1

2 + ˆn) (2.2)

where ˆn = ˆa†

aˆ. Eigenenergies En = ω(1/2+n), with n ≥ 0 an integer, are equidistant from

one another, like rungs in a ladder (hence why ˆa and ˆa† are sometimes referred to as “ladder

operators”). The eigenstates are called number states because of this linear dependence of

the energy on the number n.
8

2.1 Wave mechanics representation

The eigenstates projected into position space, ψ(x) = x|ψcan be determined by

finding the solutions to the Schr ̈odinger equation in position space, Hˆ |ψ(x)= E|ψ(x),

which becomes a second-order differential equation. The solutions are [Sakurai, 1993]:

ψn(x) = 1

√

2nn!

mω

π

1

4

e− mωx2

2Hn

mω

x



(2.3)

where Hn(z) is a Hermite polynomial:

Hn(z)=(−1)nez2 dn

dzn



e−z2



. (2.4)

The wavefunctions for the first few eigenstates are shown in Fig. 2.1. As we can see from

this figure, as n increases, the wavefunction becomes more dispersed and has n nodes in the

amplitude, behaving more and more nonclassically. These are eigenstates, so the observables

are stationary. There is no time evolution of the state, other than an overall phase factor

einωt, which cannot be observed directly. However, the frequency difference between such

phase factors can be observed in interferometric experiments, as described in detail in Sec.

2.4 below and later in Ch. 6.

2.2 Phase-space representation and Wigner functions

It is sometimes convenient to express the behavior of a mechanical system in phase

space—where the position is plotted on the x-axis and the momentum is plotted on the

y-axis. This representation is particularly convenient for harmonic potentials, where, in
9

0.04 0.02 0.00 0.02 0.04

x ( m)

Energy

wavefunction amplitude

Figure 2.1: Wavefunctions and energies of the first few harmonic oscillator eigenstates of a

9Be+ ion, where ω = 2π × 7 MHz. The wavefunction amplitudes (not normalized) versus

position for n = 0 − 5 are plotted, with offsets equal to the energy of the state. The black

parabola represents the harmonic confining potential.
10

classical mechanics, the evolution of a particle undergoes simple harmonic motion:

x(t) = A sin(ωt + φ) (2.5)

p(t) = m

dx(t)

dt = mAω cos(ωt + φ). (2.6)

If we define dimensionless variables  ̃x and  ̃p to be

x ̃ = 1

2

2mω

x (2.7)

p ̃ = 1

2

 2

mω

p, (2.8)

then when plotted in phase space, the harmonic motion traces circles, with the radius

α = mω/(2)A proportional to the amplitude of the motion.1

In quantum mechanics, we can visualize a representation of the quantum state in phase

space by defining a quasiprobability distribution, or Wigner function [Wigner, 1932]:

W(x, p) = 1

π

 +∞

−∞

ψ∗(x + y)ψ(x − y)e2ipy/dy. (2.9)

If ψ is a pure state, then integrating the Wigner function over momentum p gives  +∞

−∞ dpW(x, p) =

|ψ(x)|

2, which is the probability distribution of the state in position space and is positive for

all x. Likewise,  +∞

−∞ dxW(x, p) = |Φ(p)|

2. Note that the Wigner function can be negative

in parts, hence why it is called a quasiprobability distribution.

We can visualize certain features and dynamics of quantum states of a harmonic oscillator in phase space using this Wigner function representation, replacing x and p with their

respective unitless counterparts,  ̃x and  ̃p, defined above. There exist other phase-space representations of quantum states (e.g. Glauber-Sudarshan P, Husimi Q, etc.), but the Wigner

1 Although we are discussing a classical harmonic oscillator and of course, is a quantum unit, we use

this scaling factor in anticipation of the transition to our discussion of quantum harmonic oscillators. In

classical mechanics, the unitless phase-space representation can be accomplished using any constant with

units of action in place of .
11

function is most prevalent and is sufficient in establishing the desired physical intuition for

the quantum states of the harmonic oscillator systems.

2.2.1 Time evolution of Wigner functions and the rotating frame

The interaction picture [Sakurai, 1993] is commonly used in atomic physics when the

Schr ̈odinger picture Hamiltonian can be written as

Hˆ (t) = Hˆ0 + VˆI (t) (2.10)

where Hˆ0 is a time-independent, unperturbed Hamiltonian, and VˆI (t) is a time-dependent

interaction term. In the interaction picture, we define an interaction Hamiltonian:

Hˆint = eiHˆ0t/

VˆI (t)e−iHˆ0t/(2.11)

and transform the original, Schr ̈odinger-picture wavefunction |ΨS(t)to |Ψ(  ̃ t)= eiH0t/|ΨS(t)

so that the transformed states evolve according to

d |Ψ(  ̃ t)

dt = Hˆint |Ψ(  ̃ t). (2.12)

So, the state |Ψ(  ̃ t)evolves only according to the interaction term. (We can drop the tilde

notation when it is clear which frame is used.) If there is no interaction, then the state in the

interaction picture coincides with the state in the Heisenberg picture and is constant. An

observable OˆS that is time-independent in the Sch ̈odinger picture becomes time-dependent,

and evolves according to OˆI = eiH0t/OˆSe−iH0t/.

In our case, where we have a harmonic oscillator and we visualize the time evolution via

the rotation of Wigner functions in phase space, moving to the interaction picture amounts

to moving to a frame that is rotating with frequency ω0 = H0/in phase space, so the ladder
12

operators in the interaction picture become:

aˆ† → aˆ†

eiω0t (2.13)

aˆ → aeˆ −iω0t

. (2.14)

An unperturbed harmonic oscillator with no interactions with its environment or externally

applied driving field (VˆI (t) = 0) will not move in this frame. However, any deviations of the

actual oscillator frequency ω from the value ω0 that defines the rotating frame will cause the

Wigner function of the state to rotate about the origin at a rate δω = ω − ω0. Throughout

the remainder of this thesis we will use this interaction picture to describe the evolution of

the quantum states.

In the next few sections, we will discuss a few special types of quantum states of motion,

noting important features in their energy and phase-space representations.

2.3 Number states

As introduced above, number states are eigenstates of the harmonic oscillator Hamiltonian, and hence are rather boring in the energy basis. When we examine the Wigner

functions (see Fig. 2.2) of this class of states, however, two important features of number

states become apparent. First, the radial dependence of the Wigner function becomes more

finely structured as we use increasingly high values of n. (This becomes intuitively clear

when considering that integration of the Wigner functions along the momentum axis will

yield the square modulus of the position wavefunctions (see Fig. 2.1 for the wavefunctions

themselves).) Second, the Wigner functions have continuous azimuthal symmetry in phase

space. These two features will become especially important when we discuss the proposed

use of these states for quantum-enhanced force sensing later. The first feature gives us the
13

quantum enhancement, and the second allows us to use this quantum enhancement regardless of the phase of the displacement.

2.4 Number-state superpositions

Because of the quantum coherent interference between the different constituents of the

superposition, the Wigner functions of states of the form |0+ |n(omitting normalization)

exhibit features that are not present in a single number state alone. Chiefly, the continuous rotational symmetry in the single number state case instead becomes a discrete, n-fold

symmetry (see Fig 2.3). We will discuss later how this can be used for quantum-enhanced

measurements of the harmonic oscillator frequency.

2.5 Coherent states

The study of coherent states is a fascinating and rich area from both a physical and a

mathematical perspective. We can derive many of their interesting qualities starting from

the formal, mathematical definition of a coherent state: it is defined to be an eigenstate of

the lowering operator ˆa:

aˆ|α= α|α. (2.15)

The form of |αthat satisfies this relationship is

|α= e− |α|

2

2

	∞

n=0

αn

√

n!

|n. (2.16)

This is a Poissonian distribution over number states |n, with an average value α|nˆ|α=

n ̄ = |α|

2. This can be expressed in terms of the ground state wavefunction. Since (ˆa†

)n|0=
14

0.6

0.0

0.6

-3

0

3

|0 |1 |2

-3 0 3

-3

0

3

|3

-3 0 3

|4

-3 0 3

|5

x

p

Figure 2.2: Wigner functions of first few harmonic oscillator number states, from n = 0 − 5.

Salient features of number states are: azimuthal symmetry in phase space, and fine structure

in the radial dependence of the Wigner function that increases in complexity with n. Both

of these features are key to the metrological usefulness of these states and will be discussed

in later chapters.

-3

0

3

|0 + |1 |0 + |2 |0 + |3

-3 0 3

-3

0

3

|0 + |4

-3 0 3

|0 + |5

-3 0 3

|0 + |6

x

p

0.6

0.0

0.6

Figure 2.3: Wigner functions of superpositions of number states of the form |0+ |n, with

n = 1 − 6. While the single number states are azimuthally symmetric, the interference

between the two components of these superpositions causes n-fold azimuthal symmetry,

making the superposition states sensitive to rotations about the origin with a sensitivity

that increases linearly with n. This will be discussed in more detail in later chapters.
15

√

n!|n, from Eq. 2.16 we obtain:

|α= e− |α|

2

2

	∞

n=0

(αaˆ†)n

n! |0= e− |α|

2

2 eαaˆ†

|0= eαaˆ†−α∗aˆ|0(2.17)

where the last step is completed by applying the Baker-Haussdorf Lemma [Sakurai, 1993].

The operator applied to the ground state is called the displacement operator Dˆ(α) = eαaˆ†−α∗aˆ

because it displaces the ground state from the origin by an amount α in phase space (see

Figure 2.4). For real α, the displacement is proportional to the momentum operator, enacting a translation along x, while purely imaginary α is proportional to the position operator

and produces a momentum kick.

Let us now look at the dynamics of a displaced ground state, or coherent state. We

find the expectation values xˆand pˆ(calculated in the interaction picture):

α|xˆ|α=

 

2mω0

α|(ˆae−iω0t + ˆa†eiω0t)|α=

 2

mω0

|α| cos(ω0t + φ) (2.18)

α|pˆ|α= −i

mω0

2 α|(ˆaeiω0t − aˆ†

e−iω0t)|α=

mω0

2 |α|sin(ω0t + φ), (2.19)

with φ = Arg[α]. In scaled units the relation between α and the expectation values of

position and momentum is particularly simple:

α|x ̃|α= Re[α],α|p ̃|α= Im[α]. (2.20)

The position and momentum expectation values obey the classical equations of motion of

an object undergoing simple harmonic motion, as in Eqs. 2.5 and 2.6, where A =

2

mω0 |α|.

The uncertainties in both quadratures, Δx and Δp, are:

Δx = |xˆ2 − xˆ2| =

 

2mω0

(2.21)

Δp = |pˆ2 − pˆ2| =

mω0

2 . (2.22)

Notably, the time- and α-dependence drops out, leaving a state that, regardless of its amplitude of motion, satisfies the minimum uncertainty allowed by Heisenberg:

ΔxΔp ≥



2

, Δ ̃xΔ ̃p ≥

1

4

. (2.23)
16

0.0

0.6

-3 0 3

-3

0

3

Im( )

= 0

-3 0 3

Re( )

= 1

-3 0 3

= 2

0 10

0

0.25

0.5

0.75

1.0

Probability

0 10 0 10

Number state n

Figure 2.4: Wigner functions and corresponding number state populations of coherent states

with α = 0, 1, and 2. From the Wigner functions, one can see that the coherent states

correspond to a ground state Wigner function that is displaced by amount α in phase space.

The population follows a Poissonian distribution with the average  ̄n = |α|

2.
17

So, the dynamics of a coherent state can be described in phase space by a circular, minimumuncertainty, Gaussian Wigner function travelling in circles in phase-space. In pictorial representations, the Wigner function is often replaced by a disk with a diameter corresponding

to the width of the Gaussian.

To summarize, the key features of a coherent state are:

• it is an eigenstate of the lowering operator: satisfies ˆa |α= α |α

• it is composed of a Poissonian distribution over number states with average value

n ̄ = |α|

2

• it obeys classical equations of motion

• it has time-independent position and momentum uncertainties, with the minimum

uncertainty allowed by the Heisenberg uncertainty principle, with time independent

position and momentum uncertainties which are shared equally.

Because they are classically behaving states, they serve as a baseline when defining quantum

advantages. A harmonic oscillator state is said to provide a quantum advantage if no coherent state exists that performs equally well for the task at hand. By definition, one cannot

derive a quantum advantage from using these states for metrology, but as we will see in later

chapters, the ease and speed with which these states can be produced can give a practical

advantage in the lab over more exotic quantum states.
18

2.6 Displaced number states

In the previous section, we saw that a coherent state is generated by applying the

displacement operator Dˆ(αd) to the ground state. What if instead we applied the displacement operator to a higher number state |Ψ(0)= |n? This would result in a more involved

probability distribution over number states m [Carruthers and Nieto, 1965]:

P(n) m =  ̄n|n−m|e−n ̄ n<!

n>!

(L|n−m| n< ( ̄n))2

, (2.24)

where  ̄n = |αd|

2, n< (n>) is the lesser (greater) of the integers n and m and La

n(x) is a

generalized Laguerre polynomial.

The application of operators to quantum states of motion helps us expand our toolbox

of quantum motional control and can be useful for future applications to other quantumenhanced motional state metrology protocols.

2.7 Decoherence in quantum oscillators

There are two main types of motional decoherence in harmonic oscillators: one type

either adds or subtracts energy to the oscillator, driving transitions between oscillator levels,

while the other does not drive transitions between levels, but causes the phase evolution of

the various energy levels to decohere. The former is driven by noise components at the oscillator frequency [Carruthers and Nieto, 1965] and is sometimes called dissipation, damping,

or heating, and the latter is produced by noise at frequencies that differ from the oscillator

frequency and is usually referred to as dephasing. When we think about how the different

types of decoherence affect the Wigner functions of states in phase space, we note that,

since the energy is proportional to the square of the magnitude of the displacement from the

origin, |α|

2, then the first type of decoherence, the dissipation, would act radially in phase

space. By the same reasoning, dephasing must not have any radial component, since the
19

average energy of the state does not change, so dephasing causes only rotations around the

origin in phase space.

2.8 Quantum vs. classical measurements

With the phase-space and Wigner functions introduced, we can now look at the difference between classical and quantum measurements more rigorously. As we have shown above,

a coherent state of a quantum harmonic oscillator exhibits time-dependent expectation values consistent with classical mechanics, so if we compare the phase-space representation of

these states with those of the other quantum states introduced in this chapter, we can determine situations in which other quantum states would be more metrologically useful than

a classically behaving state.

A fundamental limit to how precisely we can measure an observable is given by the Heisenberg uncertainty principle. For a coherent, or classically behaving, state, the uncertainty

disk is shared equally in both quadratures of phase space. If we have quantum control of

the harmonic oscillator, then we can make a quantum state to “bend” the rules a bit. A

classic example of this is squeezing, where the uncertainty in one of the quadratures of α is

“squeezed” to less than /2 at the cost of a greater uncertainty in the other quadrature.

This allows us to measure either the position or the momentum of an oscillator with precision

above the limits set by a coherent state, but not both [Caves et al., 1980].

In this spirit, we can also generate states of motion that allow us to measure other observables with higher precision than a classical experiment would allow. If we want to be

maximally sensitive to the two different types of decoherence described previously (dissipation and dephasing), then we want to use states whose uncertainty is lowest in the direction
20

we would like to measure. If we want to measure dissipation, then we want a state that

varies strongly in the radial direction, but is relatively insensitive to rotations. If we want to

measure dephasing, the opposite is true: we want a state that is very sensitive to rotations

but not as sensitive to radial displacements. These can both be achieved with number states

|nand superpositions of the form |0+ |n, respectively (note the radial dependence of the

Wigner functions in Fig. 2.2 and rotational dependence of the Wigner functions in Fig. 2.3).

Comparing these Wigner functions to that of a coherent state (Fig. 2.4), we see that there is

much finer structure in the more quantum-like states. This “sub-Planck” structure [Zurek,

2001] is one way of understanding the advantage gained when we use quantum states of

motion.
Chapter 3

Beyond the 1-d oscillator model: trapped-ion harmonic oscillators

“Everything is a harmonic oscillator or a two-level system.”

- Professor Thomas Degrand, University of Colorado, 2012.

In truth, as you will see, the trapped-ion system is a bit more complicated than either

a harmonic oscillator or a two-level system. However, nearly all of the physics demonstrated

in the experiments presented in later chapters can be distilled to a single harmonic oscillator

coupled to a single two-level system. In this chapter, I will describe the physics that goes

into trapping, cooling, and controlling the electronic and motional states of the ion in order

to model the system in this simple way.

There are a few main elements of our system that complicate the simple 1-d oscillator model

of the ion that we introduced in the previous chapter. First, the ion is in 3-d space, so it has

3 spatial motional degrees of freedom. Second, the way we generate some of the harmonic

potential is by applying time-varying electric fields that oscillate at frequency ΩT  ω. This

causes the ion to perform oscillations at this higher frequency, called micromotion, on top

of the slower, secular frequency ω of oscillation due to the time-averaged harmonic pseudopotential. Finally, our ability to control the motion of and learn information about the

motional states of the ion is derived from coupling electronic and hyperfine states to the

motion, so understanding the electronic states of our ion, in this case a beryllium ion, is
22

crucial to understanding the experiments.

3.1 Producing the harmonic potential – RF Paul traps

Owing to its nonzero net charge, an ion can be confined with certain configurations of

electric fields. While static electric fields alone will not confine an ion in all three spatial

dimensions simultaneously, we can set up a time-varying potential made up of an RF electric

field that does confine the ion in 3-d. A simple way of demonstrating this concept is by imagining a frictionless ball sitting on the equilibrium position of a saddle-point potential under

the influence of gravity (see Fig. 3.1). This potential is stable (confining) in one direction,

but unstable (anti-confining) in the other direction. With any small displacement from the

equilibrium position in the anti-confining direction, the ball will start to roll down the hill.

But if we switch the polarity of the potential before the ball rolls a significant amount, then

the previously anti-confining direction will now be confining and vice-versa, restoring the

ball to the equilibrium position. If ΩT is fast compared to the ball changing direction to roll

downhill, then when averaged over one period 2π/ΩT , the time-varying surface produces a

confining pseudopotential.

In the same way, we a can set up a time-varying saddle potential for an ion using RF

electric fields, which is the concept behind a Paul trap [Drees and Paul, 1964]. A linear

Paul trap is composed of four linear rods, set up parallel and often equidistant to the z-axis

(usually labelled the axial direction, while x and y are called the radial directions), on the

lines (x, y, z)=(±R, 0, z) and (0, ±R, z), with RF applied to the two rods intersecting the

x-axis and the other two rods with static voltages applied (see Fig. 3.2). At a given time t,
23

a) b)

Figure 3.1: Schematic of a frictionless ball sitting on a saddle potential under influence of

gravity depicting concept of a confining pseudopotential. a) At a given point in time, the

potential is anti-confining in one direction, so the ball will start to roll downhill. b) If the

polarity of the potential is switched fast enough, the direction the ball was originally rolling

down is now confining and the ball will be forced towards its equilibrium position. If this

polarity-switching is repeated periodically in time at a high enough rate, the flapping saddle

produces a time-averaged confining pseudopotential that keeps the ball near the equilibrium

position.
24

Figure 3.2: Schematic of a canonical linear RF Paul trap. It is composed of four linear rods,

set up parallel and equidistant to the z-axis on the lines (x, y, z)=(±R, 0, z) and (0, ±R, z).

An RF potential V0 cos(ΩT t) is applied to the two rods intersecting the x-axis and static

voltages U0 are applied to the four endcaps on the other two rods.
25

the potential that an ion positioned near the z-axis will feel is:

Φ 	

V0 cos(ΩT t)

2



1 +

x2 − y2

R2



, (3.1)

where V0 and ΩT are, respectively, the amplitude and frequency of the RF potential applied

to the the rods, and R = κRFR, with κRF a geometric factor that depends on the shape,

spacing, and orientation of the electrodes and is of order unity. In the z-direction, the

potential is determined by a static electric field produced by the potential U0 applied equally

to all the endcap electrodes shown in Figure 3.2. The static potential produced from these

electrodes is

Φs = κDCU0




z2 − 1

2

(x2 + y2

)



(3.2)

where κDC is a geometrical factor similar to κRF . In the axial direction, the potential is

static and harmonic near the center of the trap, so that an ion obeys the standard harmonic

oscillator equations of motion, with frequency of oscillation ωz = 2κqU0/m. The x- and

y-dependent oscillating term from the RF potential means that the full, non-time-averaged

motion in these directions is given by the Mathieu equations:

d2x

dζ2 +




ax + 2qx cos(2ζ)



x = 0 (3.3)

d2y

dζ2 +




ay + 2qy cos(2ζ)



y = 0 (3.4)

where the following substitutions were made:

ζ = ΩT t

2 , (3.5)

ax = 4q

mΩ2

T

(κDC

U0

z2

0

), (3.6)

ay = − 4q

mΩ2

T

(κDC

U0

z2

0

), (3.7)

qx = −qy = 2qV0

Ω2

TmR2 . (3.8)

General solutions to these equations can be found using the Floquet Theorem [Leibfried
26

et al., 2003; McLachlan, 1947]:

xi(ζ) = Aeiβxi ζ 	

+∞

n=−∞

C2nei2nζ + Be−iβiζ 	

+∞

n=−∞

C2ne−i2nζ (3.9)

for xi ∈ {x, y}, and where βxi is a real-valued characteristic exponent and the coefficients C2n

are functions of ax and qx. Determining these coefficients based on boundary conditions is not

straightforward and not especially illuminating, but in the special case where |ax|, |qx|

2  1,

we can approximate the solution to lowest order in axi and qxi :

xi(t) 	 2AC0 cos 

βxi

ΩT

2

t


1 − qxi

2

cos ΩT t



(3.10)

where βxi 	 axi + q2

xi

/2. Under these conditions, the trajectory of the ion in each radial

direction is described by two different types of motion: first, the secular motion, which

describes the harmonic motion from the ion moving in the pseudopotential created by the

RF field. This frequency is ωi = βxi

ΩT

2 ≈ qV0/(

√2ΩTmR2). The second term describes the

ion oscillating at the RF frequency ΩT . This is known as micromotion, and is multiplied

by a factor of qxi

2 compared with the secular motion. Given |qxi |  1, this term can often

be ignored, leaving us with an approximately harmonic 3-dimensional system defined by

DC confinement in the axial direction, and RF pseudopotential confinement in the radial

directions:

Φ = 1

2

m(ωxx2 + ωyy2 + ωzz2

). (3.11)

Two key features worth noting are:

• The radial secular frequencies of the ion depend inversely on the RF frequency ΩT ,

mass m and the square of the distance R2.

• The pseudopotential scales inversely with mass, so the strength of the radial confinement depends on the mass of the ion being confined.
27

Note that the approximation in Eq. 3.10 was performed at the location of the RF pseudopotential minimum. If the ion is significantly off of the minimum, micromotion will become

much more of a concern. To minimize this effect, which typically appears in real traps due

to stray fields or geometric imperfections of the electrode structures, we apply small correction voltages to overlap the static and RF potential minima. This is known as micromotion

compensation and is an important calibration step before experiments are performed (see

Sec. 4.8.3) .

All of the experiments presented in this thesis were performed using the axial motional

mode, whose potential ideally is time-independent and only determined by static voltages.

In this section we have shown that even the ion’s radial confinement is harmonic under the

right conditions, so all of the experiments should also be possible to implement on a radial

mode of the ion.

3.2 Beryllium electronic structure

In order to cool the ion and generate non-classical states of motion, we use the ion’s

electronic structure to address certain internal transitions which are modified by or selectively allowed depending on the motional state of the ion. Singly-ionized 9Be has a number of

features that are advantageous in demonstrating motional state control. First, it has a single

valence electron, so its electronic structure resembles that of alkali atoms (see Figure 3.2).

The 12S1/2 ↔ 22P3/2 transition is dipole allowed, with a natural linewidth of Γ = 2π × 19.4

MHz. This transition has a very high photon scattering rate because of the short excitedstate lifetime, which makes it convenient for fluorescence detection and Doppler cooling.

A second feature that we exploit throughout this thesis is the non-zero nuclear spin, I = 3/2.
28

197.2 GHz

313 nm

1.25 GHz

270 MHz

Figure 3.3: Energy level structure of singly ionized 9Be. The nuclear spin is I = 3/2, leading

to hyperfine structure. A weak magnetic field Zeeman splits the degeneracy of the mF

sublevels. This figure is adapted from reference [Jost, 2001].
29

This leads to hyperfine sublevels in the ground state that have extremely long lifetimes. Having a long-lived, effective two-level system which can be coupled to the motion of the ion is

essential to all of the experiments in this thesis.

We will now examine this electronic structure, noting shifts and splittings in energy levels due to coupling between different excitations. The three energy levels we are concerned

with are (in spectroscopic notation, n2S+1LJ ) 12S1/2, 22P1/2 and 22P3/2. Here, S is the quantum number denoting the spin of the electronic state, L is the orbital angular momentum,

and J = L + S is the total electronic angular momentum. The coupling of the electron’s

spin to the orbital angular momentum, given by the Hamiltonian [Griffiths, 2005]:

Hso =

 e2

8π0

 1

m2c2r3S · L (3.12)

which causes the 197.2 GHz splitting between the 2P1/2 and 2P3/2 levels. The electron’s spin

also couples to the nuclear spin, producing the hyperfine splitting.

Hhf = hAJ · I (3.13)

where A is the hyperfine constant. In singly-ionized 9Be, the nuclear spin is I = 3/2. The

hyperfine splitting in the ground state is 1.25 GHz, in the P1/2 state the hyperfine splitting

is 237 MHz, and in the P3/2 state the splitting is less than 1 MHz [Langer, 2006].

In the presence of a weak external field in the z-direction, the different hyperfine sublevels

|F,mF with mF describing the magnitude of the z-projection of the angular momentum

quantum number F, will experience Zeeman splitting EZ = −gF μBmFB [Griffiths, 2005],

where μB = e

2m is the Bohr magneton and gF is the g-factor. For the ground state of beryllium, gF is −1/2 and +1/2 for F = 2 and F = 1, respectively [Langer, 2006]. We apply a

field of 14.3 Gauss, so the low-field Zeeman approximation is appropriate, and the splitting

in the ground state between neighboring Zeeman sublevels is approximately 10 MHz.
30

Finally, while we will not go into the mathematical details here, it is worth noting that

the RF fields that confine the ion also cause a small amount of mixing between mF sublevels in the trap we used for our experiments. This means that transitions that otherwise

would not be allowed are weakly allowed. We will see the effect of this when looking at

state-selective fluorescence later.

3.3 Doppler Cooling

To have full control of the motion of a trapped ion, we must first be able to cool the

ion to its ground state. Doppler cooling is the first step in this process. First proposed

in 1975 [Wineland and Dehmelt, 1975; H ̈ansch and Schawlow, 1975] and demonstrated

in [Wineland et al., 1978; Neuhauser et al., 1978], the basic idea behind this method of

cooling is to use the momentum of a photon to damp the motion of the ion. If we imagine

the ion oscillating back and forth in one dimension, with laser light at frequency ωL incident on the ion from the right (see Figure 3.4), then, in order to damp the motion rather

than amplify it, we would like the ion to preferentially absorb light when it is moving in

the direction opposite to the photon’s momentum. We can do this by choosing a laser

frequency close to, but lower than, a resonant transition in the ion. So when the ion is

moving to the right in Fig. 3.4, the frequency of the photons is Doppler shifted closer to

resonance in its reference frame, which increases the probability of absorbing the photon and

experiencing a momentum kick against the direction of motion of the ion. When the ion

moves in the opposite direction, the probability of absorption is lowered, so on average a

moving ion experiences more momentum kicks that slow it down than those that speed it up.

Each absorption event is accompanied by an emission event that ejects a photon in an
31

a) b)

Figure 3.4: Schematic sketch depicting velocity-selective absorption of Doppler cooling light.

If the light is red-detuned from an electronic transition, and a) the ion is moving with velocity

v in the direction opposite the laser k-vector, then in the frame at rest with ion, the photons

will be Doppler-shifted toward resonance, increasing the probability of absorption and the

ion receiving a momentum kick in the direction opposite the ion’s motion. b) If the ion is

moving in the same direction as the k-vector, then the laser will be Doppler shifted further

from resonance and the ion will be less likely to absorb a photon.
32

approximately random direction—this causes the ion to heat by an amount (k)2

2m from the

recoil energy of the emitted photon, where k is the wavevector of the emitted photon and m

is the mass of the atom. Because of this heating mechanism, there is a limit to the temperature that can be achieved by Doppler cooling. Following [Leibfried et al., 2003], we will now

derive the minimum temperature that can be achieved with Doppler cooling in the case of

an ion trapped in a one-dimensional harmonic potential:

V (z) = 1

2

mω2

z z2 (3.14)

where m and ωz are the ion’s mass and frequency of oscillation, respectively. The ion’s

velocity will obey simple harmonic motion:

v(t) = v0cos(ωzt). (3.15)

Obviously, this is a very simplistic model—a more realistic model of a trapped ion would

include all 3 modes of oscillation, the existence of micromotion, and other electronic energy

levels that off-resonantly couple to the Doppler cooling light field. Nevertheless, this simplistic approach allows us to examine the physics which limits Doppler cooling, motivating

the need for another method of laser cooling to reach the ground state (see next section).

We assume the weak binding limit, ωz  Γ, where Γ is the linewidth of the atomic transition.

This means that a photon is absorbed and reemitted much faster than a period of mechanical oscillation. In our experiments, we use the 2S1/2 ↔ 2P3/2 transition, so Γ = 2π × 19.4

MHz. The frequency of oscillation in the axial direction is approximately 7-8 MHz, so this

weak binding limit should be valid. In this limit, we can model the radiation pressure as a

continuous force that depends on the velocity of the ion. This force is equal to the momentum transferred to the ion with each absorption event times the rate of absorption-emission

cycles:

Fabs = kΓρee (3.16)
33

where ρee = e|ρ|eis the probability of being in the excited state (with ρ the density matrix

of the electronic states of the ion and |edenoting the excited 2P3/2 state). With each absorption event, a photon is reemitted, but since the direction of the reemission is symmetric

in z, the force the atom feels from emission is averaged to zero.

When the ion is illuminated with a laser beam traveling in the +z direction, with saturation parameter s = 2|Ω|

2/Γ2, where Ω is the on-resonance Rabi frequency, this probability

is:

ρee = s/2

1 + s + (2δef f /Γ)2 (3.17)

where δef f is effective detuning from resonance, taking into account the Doppler shift of the

light field from the reference frame of the moving ion:

δef f ≈ Δ − kvz (3.18)

and Δ is the detuning in the lab frame, assuming vz  c. If we also assume the velocity is

low enough that the effect from Doppler broadening is much less that the linewidth Γ, then

we can linearize the force in vz:

F ≈ F0(1 + κvz) (3.19)

where

F0 = kΓ s/2

1 + s + (2Δ/Γ)2 (3.20)

is the velocity-independent, average radiation pressure that displaces the ion’s equilibrium

position by a small amount, and

κ = 8kΔ/Γ2

1 + s + (2Δ/Γ)2 (3.21)

is a “friction coefficient” which leads to damping when Δ < 0. The cooling rate, averaged

over many oscillation periods is

E ̇

c = Favz≈ F0(vz+ κv2

z ) = F0κv2z (3.22)
34

since vz= 0 for a confined particle. This treatment neglects that the laser beam also acts

as a heating source, particularly as vz → 0. When v = 0, every absorption or emission event

will cause the ion to heat from the photon recoil. Even when vz is non-zero, but small, the

relatively broad linewidth Γ of most transitions that are used in Doppler cooling means that

some absorbed photons will actually contribute to heating the ion (the ion is traveling in

the same direction as the photon-momentum kick) rather than cooling. To get a sense for

the magnitude of these heating events, let us look at the heating rates of the absorption and

emission when vz = 0. The change in the ion’s energy from each absorption-emission cycle

is:

E ̇

h = 1

2m

dp2

dt = E ̇

abs + E ̇

em = E ̇abs(1 + ξ) = 1 + ξ

2m

2

k2Γρee(v = 0) (3.23)

where the factor of ξ is inserted to take into account the fact that while the photon momentum kick from an absorption event will always be purely along ˆz since the laser beam

is pointed in the −z direction, the photon can be reemitted into free space, so some of the

emitted photon’s recoil momentum will not contribute to heating in the z direction.

The ion will arrive at a steady-state energy when the cooling (Eq. 3.22) and heating (Eq.

3.23) rates are balanced:

mv2

z = kBT = Γ

8

(1 + ξ)




(1 + s) Γ

2Δ +

2Δ

Γ



(3.24)

For a low saturation parameter, the lowest temperature is achieved when Δ = Γ/2, which,

for Γ = 2π × 19.4 MHz and ζ = 2/5 (for dipole radiation where the dipole is oriented along

z), leads to T ≈1 mK.
35

3.3.1 Doppler cooling in 9Be+

We made many assumptions in our treatment of Doppler cooling above, many of which

do not perfectly hold in beryllium. One of the main assumptions that is broken is that the

system can be isolated to only two electronic energy levels. As is apparent from Sec. 3.3

above, there are many hyperfine sublevels in the ground state that the ion can fall into upon

the reemission of the photon. By making the polarization of the Doppler beam σ−, the ion

nearly always stays in the 2S1/2 |2, −2↔ 2P3/2 |3, −3cycling transition, but occasionally,

due to polarization imperfections or the RF-induced mixing of the mF sublevels mentioned in

Sec. 3.2, the ion will scatter into one of the other hyperfine states. This makes the situation

significantly more complicated and I will not cover the technical details here, but by applying other laser beams resonant with the 2S1/2 |2, −1↔ 2P1/2 and 2S1/2 |1, −1↔ 2P1/2

transitions, we can “repump” population that has fallen outside of the cycling transition

back to 2S1/2 |2, −2.

3.4 Stimulated Raman transitions

To cool from the Doppler temperature to the ground state of motion, we need a transition with a narrow linewidth so that nearly every scattered photon contributes to cooling—

this is known as the resolved sideband limit. In this limit, we can tune the frequency of the

laser to ωL = ω↓↑ − ωz, where ω↓↑ is the frequency of the transition between the ground and

excited states, and ωz is the ion’s oscillation frequency, so that each absorbed photon transfers the ion to the excited state and removes a quantum of motional energy. In 9Be+, the

narrow linewidth transition we use is a coherent two-photon transition between the hyperfine

levels |F,mF = |2, −2and |1, −1of the 2S1/2 ground state, via a virtual excitation to the

2P1/2 level. There are three relevant electronic energy levels: |↓= |2, −2, |↑= |1, −1, and
36

e

Figure 3.5: Schematic of a two-photon stimulated Raman transition from |↑to |↓, via a

third level, |e. The frequencies of the two laser beams are depicted by the length of the two

purple arrows.
37

|e= 2P1/2, as shown in Fig. 3.5. We perform a Raman transition |↓↔ |↑by applying

two laser beams with frequencies ω1 = ωe − ω↓ − δ − Δ and ω2 = ωe − ω↑ − Δ (also shown

in Fig. 3.5).

The Hamiltonian that describes this interaction is given by:

HI = −μ · E = −μ · (ˆ1E1ei(

k1·z−ω1t+φ1) + ˆ2E2ei(k2·z−ω2t+φ2)

) + H.C. (3.25)

where μ is the electric dipole moment corresponding to the transition and E1,2, ˆ1,2, 

k1,2,

and φ1,2 are, respectively, the amplitude, polarization, k-vector, and phase of the electric

field from the laser with frequency ω1,2. Transforming to the interaction picture, defined by

H0 = (ω↓ |↓↓| + ω↑ |↑↑| + ωe |ee|, this becomes

Hint = g1ei

k1·z+φ1 eiΔt |↓e| + g2eik2·z+φ2 eiΔt |↑e| + H.C. (3.26)

where

g1 = −E1

2

↓ |μ · 1|e(3.27)

g2 = −E2

2

↑ |μ · 2|e. (3.28)

If the detuning Δ of the laser beams from the excited state |eis large enough, then we

can assume that |eis negligibly populated and our relevant system reduces to the two-level

subsystem of |↓and |↑. This is known as adiabatic elimination—we will not go through

the derivation here, as it can be found in Ref. [Wineland et al., 1998; Wineland, 2003]. This

gives rise to an effective coupling between |↓and |↑:

Hint,ef f = −




|g1|

2

Δ |↓↓|+g1g∗

2

Δ e−iδt |↓↑| e−i(Δk·z+δφ)

+g2g∗

1

Δ eiδt |↑↓| ei(Δk·z+δφ)

+|g2|

2

Δ |↑↑| 

(3.29)

where Δk = k1 − 

k2, δ = ω↓↑ − (ω1 − ω2), and δφ = φ1 − φ2. This equation captures two

phenomena—the first and last terms describes an energy shift of the two levels due to the

off-resonant coupling between |eand either |↓or |↑, or an AC Stark shift. Because the
38

values g1 and g2 depend on the intensity and polarization of laser light used, we can adjust

the relative intensities and polarizations between the beams such that the difference in Stark

shifts between |↑and |↓can be minimized. We can absorb these terms into H0, leaving only

the off-diagonal elements, which describe coupling between |↓and |↑, with Rabi frequency

Ω0 = −g1g∗

2/Δ.

Since this is now effectively a two-level system, we can write the Hamiltonian in terms

of the Pauli matrices:

σˆz =

⎛

⎜⎝

1 0

0 −1

⎞

⎟⎠ , σˆx =

⎛

⎜⎝

0 1

1 0

⎞

⎟⎠ , σˆy =

⎛

⎜⎝

0 −i

i 0

⎞

⎟⎠ . (3.30)

So Eq. 3.29 can be written as:

Hint = Ω0(ˆσ−e−i(δt+Δk·z+δφ) + ˆσ+e−i(δt+Δk·z+δφ)

) (3.31)

where ˆσ± = 1/2(ˆσx ± iσˆy). We also want to include the motional energy in the Hamiltonian

that defines the interaction frame. In Ch. 2, we used the variable ω0 for this frequency.

However, for the purposes of deriving the Raman interactions which couple to the motion,

we will assume a perfect harmonic oscillator with no mode frequency noise, so we define

ω0 = ωz. The new unperturbed, time-independent part of the Hamiltonian which will define

our interaction frame is now (rescaling the electronic part by ω↑↓

2 and neglecting the ωz2

harmonic oscillator zero-point energy)

H0 = H(e) + H(m) = 

ω↓↑

2

σˆz + ωzaˆ†

aˆ (3.32)

In the new interaction frame, the time-independent position operator z is replaced with the

time-dependent Heisenberg position operator z(t) = z0(ˆae−iωzt+ˆa†

eiωzt), where z0 =



2mωz .

Eq. 3.31 becomes:

Hint = (/2)Ω0σ+ exp[iη(ˆae−iωzt + ˆa†

eiωzt)]ei(δφ−δt) + H.C. (3.33)

where δ = ω − ω↓↑ and η = Δk · z0 is the Lamb-Dicke parameter, which is the ratio of

the extent of the ion’s ground state spatial wavefunction to the effective wavelength of the
39

Raman beams 2π/Δk. When the detuning δ = mωz with m an integer, then the Raman

coupling is resonant with a transition between |↓|nand |↑|n + m. When m = 0, no

motional energy is exchanged and we call this a carrier transition. When m > 0 (m < 0), we

call this a blue (red) sideband since it is blue (red) detuned from the carrier. We often refer

to the magnitude |m| as the order of the sideband, denoting the number of quanta that are

added or subtracted with a single π-pulse.

3.4.1 Lamb-Dicke Approximation

If the effective wavelength of the Raman beams 2π/Δk is much larger than the extent

of the ion’s spatial wavefunction, then the ion experiences only a small part of the laser beam

intensity pattern (see Fig. 3.6 a) and b)) and we can apply the Lamb-Dicke approximation,

expanding the exponentials that describe the spatial intensity pattern of the light in Eq.

3.33 to lowest order:

HLD

int = (/2)Ω0σ+[1 + iη(ˆae−iωzt + ˆa†

eiωzt)]ei(φ−δt) + H.C. (3.34)

So the Rabi frequency of a transition between number states |n − 1and |nwill be proportional to √n in the Lamb-Dicke regime.

3.4.2 Ground state cooling

After Doppler cooling to ∼1 mK (typically corresponding to  ̄n ≈ 0.2 − 5), we cool to

close to the ground state by Raman sideband cooling. By using these extremely narrow1

Raman transitions, we are deep in the resolved sideband regime, so nearly every scat1 Since both states coupled by the transition are stable ground states, the spectral width of the transitions

is determined by the Fourier transform of the envelope of the laser pulses driving the Raman transitions. In
40

1

0

1

wavefunction amplitude

0.2 0.1 0.0 0.1 0.2

1

0

1

0.2 0.1 0.0 0.1 0.2

x ( m)

Figure 3.6: Spatial wavefunction (shown in colors) extent in a ω0 = 2π × 7 MHz oscillator

mode for 9Be+ compared with Raman laser effective wavelength 2π/Δk, with λ1 	 λ2 313

nm and a 90 degree angle between k1 and k2 for number states a) n = 0, b) n = 2, c)

n = 10, and d) n = 40. a) and b) For small spatial wavefunction extent compared with

the wavelength, the laser intensity pattern can be approximated to be linear across the ion’s

wavefunction (Lamb-Dicke approximation). c) and d) As the number state increases, the

wavefunction becomes more extended, so the ion sees more of the sinusoidal intensity pattern

of the Raman field, and this approximation no longer holds.
41

tered photon contributes to cooling, suppressing many of the heating mechanisms that were

present in Doppler cooling. We sideband cool by applying an approximate π-pulse on the

red sideband (RSB) |↓|n↔ |↑|n − 1, then using repump beam |↑→ 2P1/2 |2, −2. The

ion will decay to the ground state after scattering an average of 3 repump photons, at which

point the sideband π-pulse and repump procedure can be repeated. The optimal number

and duration of pulses depend on the  ̄n of the Doppler cooling limit prior to sideband cooling

and the magnitude and behavior of the heating mechanisms. I will now qualitatively discuss

the main heating sources that will ultimately limit the achievable temperature.

First, while the repump lasers are necessary for performing multiple sideband cooling cycles,

they also contribute to heating the ion. This energy will be removed in the following RSB

pulse with very high probability, except for the final repump pulse, which is not followed by

a RSB pulse. For this final repump step, the probability of the repump interacting with the

ion is very low, since the majority of the population will be in |↓|n = 0. If, however, the

ion is found to be in |↑, then this final repump step will impart  ̄n = β k2

2mωz ≈ 0.05 quanta

of energy on the ion, assuming ωz = 2π × 7 MHz and β is a factor of order 1 depending

on the k-vector of the repump beam and directionality of the spontaneous emission event

relative to the mode being cooled, and the average number of repump photons scattered

before the ion decays to the desired |↓state. Components of the directions of the absorbed

and spontaneously emitted photons that are not along the mode direction will contribute to

heating the other modes.

Second, there is some probability of off-resonantly exciting the carrier or blue sideband

(BSB), but this is detuned by δ = ωz and δ = 2ωz, respectively, so is suppressed by sinc2( δ

Ω),

this work, we mostly used square pulses which have a Fourier transform proportional to sin(T(ω −ω0))/(ω −

ω0), where ω0 is the frequency of the carrier oscillations under the envelope and T the pulse duration.
42

where Ω is the Rabi frequency of the respective off-resonant transition.2 We want to maintain a low Rabi frequency relative to the oscillation frequency to keep the probability of

exciting the off-resonant carrier and BSB low. It is also worth considering the Raman intensity, which dictates the π-time tπ of the Raman interactions. The off-resonant excitation

probability will be enhanced if the mode frequency falls on a peak of one of the side-lobes

of the sinc2-lineshape, which occur at (k + 1/2)/tπ for k an integer, while it can be zero if

the mode-frequency co-incides with a zero of the lineshape.

Finally, the ion suffers heating from the electric field noise from the surface of the trap

chip, so the slower the cooling process, the higher the final temperature will be because of

this heating rate. Finding the Raman sideband Rabi frequency that achieves a compromise

between avoiding too much heating during the cooling process and higher BSB excitation

probability is necessary for achieving the lowest possible  ̄n.

3.4.3 Raman interactions outside the Lamb-Dicke regime

Even if we are operating with a fairly low Lamb-Dicke parameter, as n increases, the

spatial wavefunction will spread out more and more, sampling larger parts of the laser beam

intensity profile, and the Lamb-Dicke approximation will become less and less valid (see Fig.

3.6). In this case, we can no longer approximate the exponential function by its leading

linear tem as in Eq. 3.34, and we must keep higher-order terms in the expansion. Without

the Lamb-Dicke approximation, the 2-level interaction Hamiltonian is given by Eq. 3.33. A

2 Eq. 3.34 implies that the blue sideband Rabi frequency is reduced relative to the carrier by a factor

η < 1, so typically off-resonant carrier excitation is the dominant process. For this to contribute to heating,

the ion has to scatter on a blue sideband during repumping, which hapens with probability 3η2

r , where

ηr < 1 is the Lamb-Dicke parameter of the repump laser. After an off-resonant transition on the BSB,

the ion has picked up a quantum already and is most likely to be repumped on the carrier (probability

(1 − η2

r )3 	 1 − 3η2r ).
43

resonant transition between |↓|nand |↑|n + mcan be driven when the detuning is an

integer multiple of the oscillation frequency, δ = mωz. We can find the Rabi frequencies for

a transition at a given starting number state n by expanding the exponential and reordering

according to the “order” of the transition, or the change in n, of each term [Wineland et al.,

1998; Leibfried et al., 2003]. The Rabi frequency for the mth order transition from level n is

Ωn,n+m = Ω0|n + m|eiη(ˆa+ˆa†)

|n| = Ω0e−η2/2η|m|



n<!

n>!

L|m| n< (η2

), (3.35)

where n< (n>) is the lesser (greater) of n + m and n, and Lα

n(x) is a generalized Laguerre

polynomial. Changes in the Rabi frequency with n are plotted for different Lamb-Dicke

parameters in Fig. 3.7.

3.5 Microwave transitions

A final tool in our ion-control toolbox is the use of microwaves to couple between pairs

of the sublevels in the ground state hyperfine manifolds. It is important to be able to initialize the desired internal state before any subsequent steps involving controlling the motion

of the ion, because whether a sideband interactionadds or subtracts a quantum of motion

depends on the initial internal state of the ion. In principle, we could use Raman carrier

transitions for initialization, but the polarization of the Raman beams was chosen to maximally drive the |2, −2↔|1, −1transition and to minimize AC Stark shifts, so only certain

transitions can be driven. For example, since one beam has only σ+ and σ−-polarized light

and the other has only π-polarized light, only transitions where ΔmF = ±1 can be driven

with the beams. If, however, we deliver microwaves to the ion with an electrode whose Bfield has components both along and perpendicular to the quantization axis, then ΔmF = 0

transitions can be driven in addition to the ΔmF = ±1 transitions. Another reason that

microwaves are preferred over Raman beams for carrier transitions is that the Raman beams
44

Figure 3.7: Rabi frequencies versus number state n, without the Lamb-Dicke approximation.

a) Rabi frequency on the first blue sideband (Δn = 1) vs. n for different Lamb-Dicke

parameters, compared with Lamb-Dicke approximation, the ∼ 1/

(n + 1) line (black). b)

Rabi frequency using the experimental value of η ≈ 0.26 vs. n for different order sideband

transitions Δn = 1 − 4.
45

are not infinitely detuned from the intermediate level, so the Raman transitions will suffer

from spontaneous emission from this level, leading to imperfections in the Raman operations.

Microwaves do not have this drawback.

We do not use the microwaves for sideband transitions. In order to be able to do this,

we would need a sizable oscillating magnetic field gradient that would provide the coupling

between the ion’s motion and its internal energy levels, which can be difficult to achieve.

While this is not an area of investigation for this thesis, this approach is being pursued by

several groups [Ospelkaus et al., 2011; Harty et al., 2016] including our own [Srinivas et al.,

2019], and is a promising avenue for many ion trap experiments in the future.
Chapter 4

Apparatus

Most of my work done in the Ion Storage Group falls into one of two categories: first,

experiments involving the motion of a single trapped ion in a linear surface electrode trap,

and second, updates made to the apparatus to accomodate a new generation of surface electrode traps for two-dimensional arrays of ions. Because of this, in this chapter, I will be

describing older versions of the setup that were used in the experiments I performed and

updates I and others have made in anticipation of working with future traps with larger

numbers of electrodes.

4.1 Surface electrode traps

In Section 3.1, I used the canonical linear RF Paul trap geometry to describe how RF

fields generate the pseudopotential that confines the ions. However, if we would like to develop a scalable architecture for trapped-ion quantum computing, then fabricating ion traps

on a single, two-dimensional, microfabricated surface would be advantageous. Among other

advantages, this allows us to have precise individual control of each ion’s trapping potential

while still maintaining a close ion-to-ion distance, and hence sufficient Coulomb coupling,

between ions. Microfabricated surface-electrode traps allow us to make very small electrodes

to give us this fine control of individual ions [Chiaverini et al., 2005; Seidelin et al., 2006].
47

a) b) c)

Figure 4.1: Potential contours (blue lines and background shading) and pseudopotential

contours (red lines) of a a) three-dimensional RF Paul trap compared to surface electrode

traps with b) a “four-wire” design and c) a “five-wire” design. For b) and c), the top panels

are top-down views of the RF (red) and ground (black) electrodes, and the bottom panels

are side-views.

The basic idea of the three-dimensional linear RF trap and the linear surface electrode trap

is the same: the RF provides confinement in two dimensions, creating a one-dimensional,

RF-null line along the axis of the trap. The ion is confined axially by DC electrodes. Applying the concept from the 3D geometry to a surface is a kind of reverse Origami. If you

imagine “unfolding” the three dimensions of the macroscopic linear Paul trap onto a single

plane, then you would recover the geometry of a surface electrode trap. Fig. 4.1 shows two

typical designs for the RF electrode geometry of a surface electrode trap: the “four-wire”

design (Fig. 4.1b) and the “five-wire” design (Fig. 4.1b). One advantage of the four-wire

design is that the antisymmetry of the potentials in transverse direction rotates the radial

normal modes of oscillation of the ion to each be at 45◦ from the trap plane, which makes it

possible to cool all three modes of the ion with a laser beam parallel to the surface.
48

a) b)

RF

DC1

DC2

DC3

DC4

DC5

DC6

DC7

DC8

DC9

DC10

DC11

DC12

Figure 4.2: Microscope images of “double-well” surface electrode trap, with trapping RF

potentials applied to shaded red electrodes and the other electrodes held at RF-ground. a)

zoomed out and b) zoomed in to trapping region, where the red circle is where the ion is

trapped 40 μm above the surface. DC electrodes DC1-DC12 are labeled.

4.1.1 Double-well trap

The trap used in the experiments presented in this thesis is a four-wire, surfaceelectrode trap designed by Kenton Brown in ca. 2010 and used in various experiments involving the coupling of two ions in two separate wells [Brown et al., 2011; Wilson et al.,

2014]. Figure 4.2 shows microscope images of this trap, with its 8 μm thick gold electrodes

electroplated on crystalline quartz and 5 μm-wide gaps between electrodes.

The trap has 12 DC electrodes, which are wire bonded to a “filterboard,” which serves

two purposes: first, it provides a breakout of the DC lines, allowing connectors to be more

easily attached, and second, it contains the 3.5 nF capacitors to ground and 10 kΩ resistors

that make up the RC low-pass filters, which reduce the amount of externally injected technical and Johnson noise that the ion experiences.
49

a) b)

Figure 4.3: a) macroscopic, 3D needle trap, from the University of Washington ion trapping

group [Shu et al., 2011]. b) The needle trap geometry unfolded onto a plane: the “detector”

trap, a surface electrode trap with 3D RF confinement from Daniel Slichter of the NIST Ion

Storage Group [Slichter et al., 2017].

4.1.2 Triangle trap

Up until now, I have only discussed linear RF traps, but there exists another canonical

macroscopic trap design that produces a zero-dimensional RF-null point, rather than a onedimensional RF-null line. One example of this type of trap is shown in Fig. 4.3a). The same

concept of “unfolding” the 3D structure onto a 2D plane can be applied to this situation

(see Fig. 4.3).

This concept applies to the triangle trap, except that its RF electrode shape was designed so that there are instead three RF null points at the corners of an equilateral triangle

40 μm above the trap surface in which the ions are confined (in fact, there are four RF nulls,

but three were intentionally designed) (see Fig. 4.4). Shaping the RF electrode in such

a way as to tile many pseudopotential minima in a two-dimensional plane is scalable, and

could potentially serve as an alternative to the “quantum CCD” architecture [Mielenz et al.,

2016] (see Fig. 4.5). Rather than being transported to different zones as in the quantum

CCD idea [Wineland et al., 1998; Kielpinski et al., 2002], this “microtrap array” architecture would forego the transport of ions and allow the coupling between ions to be tuned
50

a) b) c)

240 um

1 mm

12 mm

Figure 4.4: Microscope images of the triangle trap. a) Entire chip + interposer, b) zoomed

in on both traps, and c) zoomed in on 30 μm triangle, with red dots indicating ion locations.

by changing the relative normal modes of oscillation between sites through the application

voltages on of DC electrodes. When ions’ oscillation frequencies are very different, they

will oscillate independently, but when they are tuned on resonance with one another, the

ions’ motion can be highly coupled, allowing phonons (quanta of motion) to be exchanged

between sites [Brown et al., 2011; Wilson et al., 2014].

The triangle trap we use is a multi-layer trap fabricated by Sandia National Labs (Fig.

4.4). To have full control of the ion’s motion at each site, many DC electrodes are needed.

Three are needed at each site to cancel any stray fields, and five each are needed to tune the

field curvature (hence, the normal modes of oscillation) that the ion experiences. To avoid

applying very high voltages to achieve this control, two more electrodes are added to make

the problem underconstrained. There are two triangle traps on each chip, so this means that

we need at least 60 electrodes on a single chip, with some of the electrodes being as small

as ∼ 5 μm2. This requires more than 100 classified fabrication steps to create the elaborate

electrode geometries (see Fig. 4.4) and complicated array of vias through the various layers

to connect each electrode to the bond pads on the edges of the trap chip (Fig. 4.4). The

two traps on the chip are both triangles, one with 30 μm distances between trapping sites

and one with 20 μm distances. The trap chip is diced into bow-tie-like shape to minimize
51

a)

b)

c)

Figure 4.5: Scaled-up 2d-array traps. Conceptual design of how to scale to larger microtrap

arrays. Designs for the rf electrode shape (gray region) for traps with a) 3 sites, b) 18 sites,

and c) 69 sites, where each site is denoted by a red dot. Figure from [Mielenz et al., 2016].
52

scatter of the laser beams off of the trap surface.

The trap chip is bonded onto a silicon interposer, also fabricated at Sandia, which has

integrated 1.5 nF trench capacitors to ground on each DC line to low-pass filter the DC

lines. This eliminates the need to put capacitors on the filterboard as was done previously

for the double-well trap, but the relatively low break-down voltage of the trench capacitors

of 20 V limits the range of applicable potentials.

4.2 Laser system, microwaves, and resonant electric field excitation of

motion

In order to load, cool, and control the quantum state of beryllium ions, we need lasers

for photoionization, Doppler cooling, Raman transitions, and repumping (see Fig. 4.6). In

addition, we use microwaves applied to electrode DC1 of the trap depicted in Fig. 4.2.

With the exception of the photoionization laser, the lasers we use are all produced with

the same basic scheme: two near-infrared lasers at approximately 1050 nm and 1550 nm

are summed (sum-frequency generation, or SFG) together in a single-pass periodically poled

lithium niobate (PPLN) non-linear crystal to produce red light at approximately 626 nm,

then this light is frequency doubled (second-harmonic generation, or SHG) in a resonant

bow-tie cavity with a Brewster’s angle-cut β-barium borate (BBO) crystal centered at the

tighter of the two cavity mode waists (approximately 200 μm and 20 μm) [Wilson et al.,

2011]. The resulting 313 nm light is passed through various AOMs to achieve the desired

frequency and ability to switch beams on and off, then fiber-coupled into a hydrogen-loaded

and cured solid-core, photonic crystal fiber [Colombe et al., 2014]. Following the output of

the fiber, the beams are collimated and the polarization suitably rotated, then each beam is

focused with a 15 cm focal length lens into the UHV chamber at the trapping site of the ion.
53

In the next few subsections, I will describe the beam paths of the lasers, then discuss certain

aspects of the setup in more detail. The SFG and SHG are covered in detail in [Wilson et al.,

2011].

4.2.1 Doppler cooling and detection laser

The Blue Doppler (BD) beam can be tuned to resonance for detection or be reddetuned by 10 MHz for Doppler cooling, and is set to approximately half the saturation

intensity (∼ 0.1 W/cm2, see Fig. 4.7). The Blue Doppler Detuned (BDD) is generated from

the same source, but red-detuned by approximately 500 MHz so that it provides far-detuned

cooling light for highly energetic ions (for example when the ion is first loaded into the trap)

and repumping from other ground state hyperfine levels, including those in the F = 1 manifold. To this end, the BDD is detuned such that it is roughly halfway between the F=2 and

F=1 manifolds to 2P3/2 transitions and its intensity on the ions is set to approximately 10

times the saturation intensity.

To produce the BD/BDD light, we start with an NKT Photonics erbium-doped fiber laser

that is sent through a fiber amplifier to produce 3 W of 1549.408 nm light. This light is

combined with 2 W of light from a Koheras Boostik ytterbium-doped fiber seed laser +

amplifier at 1051.132 nm. This laser produces 6 W in total but is split between the BD, RD,

and Raman beamlines. After summing to 626.267 nm light, some of the light is picked off

by a waveplate + polarizing beamsplitter and sent through a -290 MHz, double-pass offset

AOM, then delivered to the iodine lock setup with a short optical fiber. See Fig. 4.8 for a

schematic. In the iodine lock, we use Doppler-free saturated absorption spectroscopy with

pump and probe beams counter-propagating in a 10 cm long iodine cell to resolve the individual hyperfine lines of the iodine molecules. We modulate the frequency of the 90 MHz
54

197.2 GHz

1.25 GHz

BD

BDD

Raman 1

Raman 2

RD 1

RD 2

MW

313.133 nm 313.198 nm

~7 MHz

Tickle

= 80 GHz

Figure 4.6: Beryllium energy levels, with laser beams and microwaves represented as colored

arrows. The Blue Doppler Detuned (BDD) beam (blue arrow labeled “BDD”) is used for a

first stage of Doppler cooling, and the Blue Doppler (BD) beam (blue arrow labeled “BD”) is

used for Doppler cooling and fluorescence detection. The two Raman beams (purple arrows

labeled “Raman 1” and “Raman 2”) are used for resolved sideband cooling to the ground

state and motional state preparation. The two Red Doppler (RD) beams (red arrows labeled

“RD 1” and “RD 2”) are used as repumps in case of a scattering event leaving the state in a

different hyperfine state than |2, −2. Microwaves (green arrow labeled “MW”) are delivered

to the trap by running a current through a DC electrode. The microwave frequency can be

set to transfer populations between any two states with Δm = 0, ±1 in the ground state

hyperfine manifold. Finally, the electric field tickle tone (labeled “Tickle”) is set to the the

harmonic oscillator frequency, which coherently excites motion in the ion.
55

Figure 4.7: Saturation measurement of ion 2S1/2 |2, −2↔ 2P3/2 |3, −3cycling transition

versus intensity of detection beam. Red points are data and black line is a fit to the data.

The power is measured at each point and intensity is calculated based on an estimated 20

μm beam waist at the location of the ion. Based on this measurement, we choose to set

our detection beam intensity to ∼0.1 W/cm2, corresponding to ∼15 counts when the ion is

bright.
56

pump AOM to be able to detect the saturation signal (determined by measurement of the

power of the probe beam by a photodiode) with a lock-in amplifier.

The red light is then doubled in a bow-tie cavity, followed by a 220 MHz noise-eater AOM

which stabilizes the power of the beam by monitoring the photodiode signal from a small

amount of the beam which is picked off by a glass plate and feeding back to noise eater AOM

drive. The beam power after the noise eater is held constant by dumping a variable amount

of power (roughly 5-10%) into the first order diffracted beam of the noise eater AOM (see

Fig. 4.8). A 50/50 beamsplitter splits the BD and BDD light, which are double-passed

through +220 MHz and -80 MHz AOMs, respectively. The same beamsplitter recombines

the beams, which are then fiber-coupled into a 0.5 m long UV fiber to be focused on the ion.

4.2.2 Repump laser

The Red Doppler (RD) beam line contains two different frequencies: one is resonant

with the |↑↔ 2P1/2 transition, and the other is resonant with the 2S1/2|2, −1↔ 2P1/2

transition. These are used to repump the ion when it scatters into an undesired ground state

hyperfine level and after transitions to hyperfine states other than |2, −2during sideband

cooling and other internal state manipulation.

The 626.385 nm red light of the RD is produced from summing an NP photonics “The

Rock” fiber laser + amplifier, which produces 3 W of 1550.192 nm light, and 2 W of the

shared 1050 nm laser. Some of the light is then picked off by a waveplate + polarizing beamsplitter and sent through a -204 MHz offset AOM, then delivered to the iodine lock (Fig. 4.9).

Following the doubling and noise-eater stage which is set up similar to the BD (see above),
57

Figure 4.8: Blue Doppler beamline. Two NIR fiber lasers at approximately 1050 nm and

1550 nm are summed to generate 626 nm light, then doubled to produce 313 nm light. The

beam is split into BD and BDD beamlines and double-passed through separate AOMs at

+221 MHz and -80 MHz, respectively, then recombined and delivered through a fiber to the

trap.
58

the light is split by a 50/50 beamsplitter, where one half is double-passed through a +221

MHz AOM and the other is double-passed through a -415 MHz AOM. They are recombined

on the same 50/50 beam splitter, combined with the BD on a subsequent 50/50 beamsplitter, and delivered through the UV fiber to the trap along with the BD. See Fig. 4.9 for a

schematic.

4.2.3 Raman laser

The two Raman beams are detuned by approximately + 80 GHz from the 2S1/2 ↔ 2P1/2

transition and by approximately 1.3 GHz from each other to induce transitions between the

|2, −2= |↓and |1, −1= |↑hyperfine levels of the ground state 2S1/2. We also need to set

the polarization of the beams so that angular momentum is conserved. One of our beams

is π-polarized and the other is σ+ + σ− polarized. Only the σ− photons drive transitions,

but it turns out that using both σ+ and σ− with roughly equal intensity largely cancels the

AC Stark shifts difference of the levels participating in the transition from the off-resonant

coupling to the upper P-levels (see Sec. 3.4).

The 626 nm red light is produced from summing 3 W of an NKT fiber laser and amplifier at 1550.032 nm with 2 W of the shared 1050 nm light with a PPLN crystal in the same

way as for the BD and RD. Since fluctuations of the detuning from the 2P1/2 level only

changes the Raman Rabi frequency to second order, the absolute detuning is not so critical,

so we don’t lock the wavelength to an iodine feature like we do for the BD and RD.

After the 626 nm light is doubled to 313 nm and split into two beams, the relative phase

stability between the two beams is very important. Because of this, we want to eliminate any

path-length differences that vary in time. This would chiefly be caused by vibrating mirrors
59

Doubler

626 nm -> 313 nm

AOM

-415 MHz

SFG

1050 nm + 1550 

nm -> 626 nm

RD Beamline

Noise Eater

Iodine Lock

AOM

-75 MHz

photodiode 

feedback signal to trap

50/50 

beamsplitter

pick-off 

mirror

AOM

221 MHz

AOM

-204 MHz

AOM

220 MHz

Figure 4.9: Red Doppler beamline. Two NIR fiber lasers at approximately 1050 nm and

1550 nm are summed to generate 626 nm light, then doubled to produce 313 nm light. The

beam is split into RD1 and RD2 beamlines and double-passed through separate AOMs at

+221 MHz and -415 MHz, respectively, then recombined and delivered through a fiber to

the trap.
60

or other optics and air currents in the beam path. To mitigate these effects, we periscope

the beam height from the typical 4-inch height down to 2 inches, so that we can use shorter

posts with reduced mechanical vibration. The beam passes through a half-waveplate and a

Glan laser prism to split the light into beams R1 and R2, rather than the 50/50 beam splitter

used for the BD and RD, which only allows a maximum of 25% of the light to be delivered to

the trap following the recombination (however, the remaining beams can be sent to another

trap setup in the same laboratory). The R1 (R2) beam is then double-passed through a

+320 (-320) MHz AOM, tunable over approximately ±30 MHz, with RF derived from the

same source, for a total frequency difference between the two beams of 320 × 4=1.28 GHz.

The beams are then recombined on the Glan laser prism, and coupled into a UV fiber to be

delivered to the trap. In order to be able to have the maximum amount of light available

to deliver to the trap, we spatially separate the incoming and outgoing beams in the double

passes by a corner cube retro-reflector. This vertically offsets the outgoing beam, which we

can then pick off with a d-shaped mirror after the two beams are recombined in the Glan

laser prism. The entire double-pass setup is boxed up to eliminate air currents as much as

possible. See Figs. 4.10 and 4.11 for a diagram and a picture, respectively, of the setup.

4.2.4 Frequency doubling cavity with BBO

As mentioned, much of the detail about the SHG cavities for producing the 313 nm

light is covered in [Wilson et al., 2011], but we have learned of some quirks connected with

using BBO crystals at relatively high powers and relatively short wavelength that we note

here. First, since BBO is hygroscopic, it is necessary to gently purge the crystal with dry

oxygen. Keeping the crystal mount housing at a slightly positive pressure also helps mitigate the optical tweezer effect, where dust particles are tweezed to the the part of the beam

with the highest intensity. The beam waist is inside the crystal, so the dust particles get
61

Doubler

626 nm -> 313 nm

AOM

+320 MHz

SFG

1050 nm + 1550 

nm -> 626 nm

Raman Beamline

Noise Eater

HWP

D-shaped 

mirror

Glan laser 

prism

HWP

photodiode 

feedback signal to trap

pick-off 

mirror

AOM

-320 MHz

AOM

220 MHz

Figure 4.10: Raman beamline. Two NIR fiber lasers at approximately 1050 nm and 1550

nm are summed to generate 626 nm light, then doubled to produce 313 nm light. The beam

is split into two beamlines and double-passed through separate AOMs at ±320 MHz whose

frequency is controlled by the same DDS, then recombined and delivered through a fiber.
62

	






 









 !"#

Figure 4.11: Photo of Raman double-pass AOM setup. The lid of the box protecting against

air currents was removed to take this picture.
63

deposited on the surface of the crystal and sometimes are burned on, damaging the crystal.

We suspect that oil in the oxygen line and the decomposition of certain types of tubing used

to deliver the oxygen have contributed to additional surface contamination and damage of

some of our crystals, so it is worth putting reasonable thought and care into this aspect of

the doubler. We have an oil filter right after the oxygen cylinder regulator, and have replaced

the problematic tubing with one that does not decompose to the same extent.

4.2.5 Spatial filtering of Raman beams

Because the Raman beams deliver relatively high power (∼ 500 μW) in a beam waist

of approximately 15 μm at the position of the ion and the ion is only 40 μm away from

the trap surface, a non-negligible amount of the Raman light in the wings of the beam can

scatter off of the electrode surfaces. This seems to contribute significantly to trap charging

and the ion motional decoherence, so we try to minimize this scatter as much as possible.

Following the output of the fiber, we send the two Raman beams through a spatial filter,

where we focus them through a 25 μm diameter pinhole, cutting off approximately 20%

of the light. The beam is then recollimated with a 15 cm lens. Fig. 4.12 shows the effect of the spatial filtering on the focused beam shape of the ion. Note that the side lobes

are eliminated and the beam looks much closer to the ideal Gaussian profile after the filtering.

4.2.6 Photoionization laser

Beryllium is ionized with two photons at 235 nm—the first photon excites one of the

valence electrons to a the first excited P-state in neutral beryllium, and the second excites

the electron to the continuum, ionizing the atom. The photoionization light that we use
64

a) b)

Figure 4.12: Raman beam shape a) before and b) after spatial filtering. Note that camera

exposure time is set so that the center of the beam is significantly oversaturated so that we

can see the weaker outer part of the beam.
65

is produced from a Coherent Verdi laser that pumps an M-squared CW Ti:Sapph at 940

nm, which is then doubled twice. The first doubling stage to 470 nm is achieved with a

commercial M-squared doubler and the second doubling stage to 235 nm is a bow-tie cavity

with BBO, adapted from the design of the 626 nm to 313 nm doublers. The 235 nm ionization light is then sent through a short (20 cm) piece of UV fiber to be spatially filtered and

delivered to the trap.

4.2.7 Microwave drive

Two hyperfine sublevels can be connected via the resonant magnetic field from produced from a microwave current applied to DC electrode 1 on the trap (see Fig. 4.2). The

tone is derived from a DDS with a 1 GHz clock. To avoid aliasing, we low-pass the output, limiting the frequency of the DDS signals to less than 500 MHz. To reach the desired

frequency range of 1.22-1.28 GHz to connect the Zeeman sublevels between the ground hyperfine states, we use Minicircuits components to frequency quadrupole and further filter

this DDS tone. After the quadrupling stage, the tone is connected to the high-pass port of

a triplexer which is capacitively coupled to DC1 by a 3.5 nF capacitor (see Fig. 4.13).

4.2.8 Excitation of motion with resonant electric fields

Near-resonant excitation of ion motion with a “tickle” is a very useful technique to

find the secular frequencies of the ion without needing Raman sideband transitions. This is

especially useful when first testing a new trap or a trap under new conditions. We apply an

oscillating voltage to a trap electrode which generates a time-dependent field at the location
66

triplexer

low-pass 0-60 MHz

high-pass 270 MHz -

band-pass 60-270 

MHz

tickle

RF monitor

microwave

to trap circulator

 power detector

3.5 nF

DC1

3.5 nF

B A

C (50 Ohm)

10 kOhm

DC voltage 

applied

Figure 4.13: Schematic of electrical configuration for microwave, tickle, and RF monitor

connections. The three are connected through a triplexer with high-pass (270 MHz - ),

low-pass (0 - 60 MHz), and band-pass (60 - 270 MHz) ports, which are sent to DC1 of the

double-well trap through a 3.5 nF capacitor. The DC voltages are applied to the electrode

from the top-right in the schematic, through a low-pass filter composed of a 10 kΩ resistor

and a 3.5 nF capacitor to ground.
67

of the ion, E (t) = E0 cos (ωtict), that creates a driving force:

F = qE0 cos (ωtict) (4.1)

on the ion. If ωtic is resonant with one of the secular frequencies of the ion and E0 has

a component along the mode direction, this acts as a driven harmonic oscillator, exciting

motion in the ion, which we then detect by observing a change in the fluorescence level. For

higher precision, we can detect the departure of the ion from the ground state, as will be

discussed in Ch. 7.

The resonant electric field is produced from a DDS at 1-20 MHz frequency, corresponding to the ion oscillation mode frequencies, and connected to the low-pass port of a triplexer

which is capacitively coupled to DC1 in the double well trap by the 3.5 nF capacitor (see

Fig. 4.13).

4.3 UHV copper vacuum enclosure

The experiments were performed in a copper UHV vacuum enclosure (the “pillbox”)

that was designed and assembled by Kenton Brown. While it was successful in housing a

trap that produced many results (e.g., [Brown et al., 2011; Wilson et al., 2014]), it was not

ideal for quickly switching between traps or for traps with a large number of DC electrodes.

Following the lead of Daniel Slichter in the “cryo 2” experiment (another cryogenic ion trap

apparatus in our lab), I also designed a new pillbox for our setup, nicknamed “cryo 1.”

Nearly all of the trap and oven mounting parts are identical to the cryo 2 pillbox, which

should allow us to not only easily switch between traps in cryo 1, but also swap traps between cryo 1 and cryo 2. A side-by-side comparison between the old and new pillboxes is

shown in Fig. 4.14. The new base plate is made of copper for good thermal conductivity
68

from the cryostat cold plate to the trap, but has a stainless steel ring brazed into the copper

in which all of the electrical feedthroughs are welded (see Fig. 4.14b). There is also a 3/8

inch stainless steel tube welded into the ring with an L-shaped Swagelock connector on the

end. In the other side of the connector, we can insert a 3/8 inch copper tube through which

we pump out the pillbox, then subsequently pinch off the copper tube to create a vacuum

tight seal. The lid is secured to the baseplate with an indium wire seal and has 9 viewports,

8 for laser access and one on the top of the lid for imaging, which are also sealed with indium.

Following the design and machining of the individual parts, postdoc Jonas Keller has taken

over the assembly and testing of the new pillbox. He has been able to achieve 10−7 Torr base

pressure at room temperature (without baking), with no detectable leaks, an improvement

over the aging older pillbox. Installation of a re-designed triangle trap (see Ch. 8) into this

new system is imminent.

4.4 Cryostat

The UHV pillbox is mounted to the cold head of a ∼ 1 m tall bath cryostat, which

holds approximately 40 L each of liquid helium and liquid nitrogen, with both spaces isolated

from each other and the surrounding room-temperature air by a 10−6 Torr vacuum space.

Because of the need to thermally isolate the coldhead from the environment, all of the dewar

spaces are attached at the top of the cryostat, constituting a 1 meter-long pendulum. Any

vibrations of the cryostat will cause the trap, and hence the ion, to move relative to the laser

beams. The most detrimental effect of this will be errors in the Raman beam operations

from unstable optical phases and Rabi frequencies. To mitigate this as much as possible, we

have added kevlar braided straps that wrap around the bottom of the coldhead in-vacuum

to reduce pendulum vibrations. This provides the mechanical support needed with minimal
69

a) b)

Figure 4.14: A comparison of the old and new pillboxes: a) The old pillbox. Note the

handwiring of ∼100 connections, and the walls on all sides, which made taking traps in and

out tricky and risky. b) The new pillbox. Commercial connectors and made-to-order flex

PCB are used to break out DC wiring, and all components are mounted to and plugged in

on the baseplate, allowing more maneuverability in assembly. Electrical feedthroughs and

pump-out tube are welded into the stainless steel ring, which is brazed into the copper.
70

thermal conductivity between the 4 K space and room temperature. We also use four 2-inch

diameter posts attached to the bottom of the outside of the cryostat to rigidly register the

bottom of the cryostat to the optical table.

4.5 Imaging system

The imaging system used in experiments described in this thesis is an in-vacuum

Schwarzschild objective, designed by previous postdoc Christian Ospelkaus, made of two

mirrors optically contacted to two glass pillars and crossbar. It has fairly high magnification

(adjustable between 40 and 50) with low aberrations, imaging both 313 nm and 235 nm light

with the same focal distance. However, it also has a very small depth of field, which means

that we have to be able to move it relative to the object we want to image in vacuum. We

achieve this by using an Attocube actuator, which uses a stack of piezo-electric transducers

that slip-stick against another piece, moving like an inchworm for a total travel of approximately 5 mm. This usually worked well, but after having the Attocube sieze in vacuum at

4 K repeatedly, we decided to design a new imaging system that did not have any moving

parts in-vacuum.

In fact, the new imaging system is completely outside the vacuum enclosure. It uses two f =

60 mm, aspheric lenses made by Asphericon as a one-to-one relay to produce a primary image

approximately 180 mm from the object plane. A Thorlabs high-NA 40x microscope objective

then produces a secondary image either on the Andor CCD camera or the Hamamatsu PMT

(a flipper mirror that can be moved into the beam paths allows us to choose between the two).
71

Figure 4.15: Photograph of bath cryostat, with copper pillbox attached (outer vacuum can

removed). Susanna Todaro (left) and I (right) are pictured for scale. We are normal-height

humans.
72

crossbar

spacers

primary mirror secondary mirror

Attocube

positioning 

stage

a) b)

f = 60 mm 

aspherics Thorlabs 

LMU-40X-UVB

 imaged on 

camera/PMT

Figure 4.16: a) Photograph of Schwarzschild objective, designed by Christian Ospelkaus. The

two mirrors are connected to each other through optically contacted glass spacers that need

to be matched very precisely (order a few μm) in height to produce aberration-free images.

The secondary mirror is held by a cross bar that requires similar tolerances as the spacers.)

This is mounted onto the Attocube positioning stage. b) Ray diagram of the new, out-ofvacuum imaging system, which has replaced the Schwarzschild objective. Two f =60-mm

aspheric lenses (Asphericon AFL50-60-S-X) produce an unmagnified primary image, which

then is magnified by a 40x microscope objective from Thorlabs (LMU-40X-UVB).
73

Figure 4.17: Inventor drawing of new imaging system and mounting configuration, designed

by Jonas Keller.
74

4.6 Radio-frequency trap drive

The ions are confined by a pseudopotential created from an RF electric field oscillating

at approximately 125 MHz. Because the amplitude of the RF that is applied to the trap

in part determines the confinement of the ion, any amplitude noise can translate into ion

mode-frequency noise, which can cause motional decoherence. So it is very important to

have a stable RF source. To get a low-amplitude noise RF signal at the trap, we start with

a Holzworth low phase noise synthesizer as our clock, which triggers an amplitude-stable

RF source (developed by David Lucas’ group at Oxford and David Allcock ) that generates

a square wave supplied by a very stable 5 V DC voltage source. After band-pass filtering

which eliminates the higher harmonics of the square wave, we are left with a phase- and

amplitude-stable sinusoidal RF signal.

Another place where care must be taken to avoid amplitude noise is in the RF resonator.

This can introduce amplitude noise in two ways. First, if the quality factor of the resonator

is so high that some of the input frequency excursions lie on the slope of the resonator

transmission profile, then the source frequency noise gets translated into amplitude noise.

For this reason, we use a resonator with a modest quality-factor of ∼ 100, which, at 125

MHz RF frequency, corresonds to 1.25 MHz bandwidth, which is much larger than the RF

frequency noise. Second, if mechanical vibrations cause the resonator to change its shape,

hence changing its capacitive coupling, this can also affect the amplitude stability. We have

addressed this by replacing our previous resonator consisting of a spring-like wound cylindrical coil, which was very susceptible to mechanical vibrations, with an inductive resonator

on a printed circuit board (designed by Daniel Slichter).

We have the ability to monitor the power of the RF signal on the trap by measuring the

pickup on a nearby DC electrode, DC1 (see Fig. 4.2). This DC is connected to the triplexer,
75

so we monitor the signal using the third and final port of the triplexer, which is band-passed

from 60 MHz to 270 MHz, followed by a circulator and diode-based power detector (see Fig.

4.13).

4.7 Magnetic field coils

The quantization axis is defined by a 14.3 Gauss field at 45◦ from the axial direction

of the trap in a plane parallel to the trap electrode surface. The field is produced with

water-cooled coils in a Helmholtz configuration, with 9.69 A of current produced from an

HP 6628A DC power supply. The power supply has relatively low noise, and with this setup

we passively achieve 10−4 relative magnetic field stability. Although was not used in the

experiments presented here, we also have the option of stabilizing the current. Since we are

only stabilizing the current and not the field at the location of the ion, if the coils are not

rigidly mounted or there are other fluctuating magnetic fields not derived from the bias field

coils, we will not necessarily be able to keep the field at the ion stable to the level that is

given by the current stability.

4.8 Experimental Primitives

I will briefly describe, in roughly chronological order, a few experimental procedures

that we perform in either preparation of or as steps for detection of the experiments described

in later chapters. These involve loading the ion, controlling and measuring the motion of

the ion, and state detection primitives.
76

4.8.1 Loading ions into a trap

We load beryllium with a resistive “oven,” which consists of 50 μm diameter beryllium

wire wrapped around 100 μm diameter tungsten wire. The tungsten wire itself is then wound

into a spiral of roughly 2 mm diameter. We run a current (usually ∼ 1.2 A) through the oven,

which heats the beryllium to approximately 1300 ◦C, sublimating the metal and creating a

flux of neutral beryllium atoms passing through the trapping region with an average velocity

of 1700 m/s. While the oven emits atoms, we also apply our photoionization (PI) beam and

Doppler cooling (BD/BDD) beams focused on the trapping region. If a neutral beryllium

atom passes through this region, the PI can ionize the atom through a two-photon process,

with one photon bringing the atom from the 1S ground state to the 1P0 first excited state,

and a second photon exciting the electron to the continuum. Generally, the potential depth

in surface electrode traps are around 20-30 meV, so we expect that we are only trapping a

velocity class at the lower-end tail of the velocity distribution. This question, among other

questions relating to ion production and loading efficiency, is the subject of study of Hannah

Knaack’s current project.

4.8.2 Doppler and ground state cooling

We first Doppler cool the ion, using the BD and BDD beams. The BDD serves two

purposes: it is responsible for initially cooling the ion when it is first loaded and very

hot. Second, it acts as a repumper if the ion scatters into the wrong hyperfine sublevel,

making the BD now blue-detuned. The BD is detuned by −Γ/2 ≈ -10 MHz from the

2S1/2 |2, −2↔ 2P3/2 |3, −3cycling transition and cools the ion to near the Doppler temperature of ∼1 mK.
77

Following the Doppler cooling stage, in most experiments, we ground state cool using pulsed

Raman sideband transitions interspersed with repumping pulses from the RD. Using the predicted  ̄n based on our Doppler temperature and mode frequency, we apply a certain number

of red sideband pulses (usually 10-20 on the axial mode and 5-10 on the radial modes, since

they are higher frequency so have lower  ̄n), with each sideband pulse followed by an RD

pulse to repump to the |2, −2state. After this, we typically achieve  ̄n ≈ 0.05 on the axial

mode and  ̄n ≈ 0.01 on the radial modes. Sometimes, since the initial occupations in the radial modes are already so low after Doppler cooling, we do not need to sideband cool them.

The trade-off is between suffering more charging of the trap surface due to the increased

number of Raman pulses for sideband cooling and having slightly smaller Debye-Waller factors [Wineland et al., 1998] due to the increased occupation in the radial modes.

4.8.3 Micromotion compensation

In order to have an ion whose radial motion is well described by a harmonic oscillator

at the secular frequency and that scatters near-resonant photons at the optimal rate, we

must compensate stray electric fields at the position of the ion. We apply voltages on various DC electrodes to create an electric field that is equal and opposite to the stray field at

the postion of the ion, so that the position of zero static field overlaps with a position where

the RF field is minimal and micromotion is negligible. We use a few different techniques to

accomplish this.

First, to find the correct position perpendicular to the electrode plane, we scan the voltage applied to the “mesh,” which is an electrically floating gold screen parallel to and ∼ 2

mm above the trap surface (so that the ion is sandwiched between the trap chip and the

mesh). Applying a voltage to the mesh creates a nearly homogeneous electric field (similar
78

to a parallel plate capacitor) that is perpendicular to the trap surface and pushes the ion

vertically, which, for the double well trap, primarily affects the micromotion in the plane

of the trap chip that has a component along the wave-vector of the BD beam. For an ion

with micromotion due to the RF field, the micromotion velocity varies with time according

to v = v0 cos (ΩT t), where ΩT is the RF frequency. In the ion’s oscillating frame, the the

first-order Doppler shift can be expressed as a phase-modulation of the laser field [Keller

et al., 2015]:

E (t) = (E0ei(ωLt+φ(t))) = (E0e

i(ωLt+

k·v0 ΩT cos(ΩT t))), (4.2)

thereby modulating the fluorescence from the near-resonant BD beam that has frequency

ωL and k-vector 

k. If the trap drive frequency is larger than the linewidth of the transition, ΩT > Γ, this modulation causes the fluorescence to follow a zero-order Bessel-function

pattern with modulation index k · v0/ΩT as we scan the mesh voltage [Riehle, 2004]. To

minimize micromotion in the direction parallel to the BD, we apply a voltage on the mesh

corresponding to the maximum of the zero-order Bessel function.

Second, we find the correct axial position. This is not stricly compensating micromotion,

since the ion is confined by DC voltages in the axial direction, but there exist stray electric

fields in this direction that cause the ion’s equilibrium position to differ from the intended,

simulated position. We find that the ion is better behaved if we compensate in this direction

also. Axial compensation is accomplished by looking at the position of the ion on the camera

and changing the “scale,” which is a common scaling factor applied to the linear combination

of voltages to the different DC electrodes that generates the harmonic confinement in the

axial direction (added compensation voltages are unaffected by this scale parameter). If the

ion moves after the scale has been changed, we know that the stray fields have not been

entirely canceled. Then we apply an axial shim voltage to push the ion until the stray field

and compensation field cancel each other. In this case, the ion does not move when the
79

harmonic potential is scaled.

Finally, we need to compensate the stray field in the remaining direction. The direction

of micromotion induced by a stray field in this direction is perpendicular to the electrode

plane and BD beam, so we can’t use the fluorescence to compensate. Instead, we use the

fact that the electrode that we use for microwaves produces a magnetic field gradient along

the direction of this micromotion. We drive a current in this electrode at a frequency that is

detuned by the RF frequency ΩT from the |F,mf = |2, 0↔|1, 0“clock” transition. Due

to the micromotion, the magnetic field gradient is translated into an amplitude modulation

of the microwave field. An amplitude-modulated oscillating field can be decomposed into the

sum and the difference frequency of its carrier oscillation and the frequency of the amplitude

modulation. One of these components is on resonance with the clock transition and can

drive it with a strength that is proportional to the micromotion amplitude. By adjusting a

shim to maximize the π-time of this transition, we can compensate in this final direction.

Some amount of iteration among all three of these techniques is necessary to compensate

simultaneously in all directions, probably because the compensation fields are not perfectly

orthogonal.

4.8.4 State-selective fluorescence detection

Regardless of the experiment we are performing, the information we receive is the number of PMT counts detected when observing ion fluorescence for a given detection period.

In nearly all cases, we want to determine whether the ion is “bright,” in the |↓state, or

“dark,” in the |↑state or another hyperfine state not resonantly excited by the BD beam.

Since there is a non-zero level of background counts (mostly due to stray light from the BD

beam), the nominally dark state has an average of 0.3 counts in a 400 μs detection window,
80

and the bright state typically has an average of 12-15 counts, with both obeying approximately Poissonian statistics. We set a detection threshold of 3 counts, labeling everything

below 3 to be dark and everything with 3 counts and above to be bright. We suffer a slightly

elevated error compared to ideal Poissonian distributions centered at 0.3 and 15 counts that

have very little overlap. Our bright state histogram is not exactly a Poissonian because

the bright state can be depumped into other hyperfine levels during the detection period.

The deviation of an example bright state histogram from an ideal Poissonian distribution

is shown in Fig. 4.18. To some extent this can be mitigated by making sure we have perfectly σ−-polarized light, but the presence of off-resonant magnetic fields produced by the

RF trap drive mixes the mF sublevels, causing some amount of depumping to be unavoidable.

Another possible source of error in the state detection is off-resonantly driving a transition from |↑into the cycling transition. For this reason we apply two microwave π-pulses,

|↑→ |2, −1and |2, −1→|1, 1, to shelve |↑to |F = 1, mF = 1which scatters fewer

photons on average.

4.8.5 Motional state analysis

All of the experiments I will describe in the remainder of this thesis involve the motion of an ion, so we need a way of detecting the motion. We can do this by transferring information about the motion to the hyperfine levels of the ion, then reading out via

state-selective fluorescence as described above. If the hyperfine state is |↓and the motional state is restricted to being in only two number states, usually the two lowest, where

|ψ= |↓(α |n = 0+ β |n = 1), then we simply need to apply a RSB π-pulse on |n = 1,

which will flip the spin of the part of the state in |n = 1and remove a quantum of motion,

but leave the |n = 0part untouched, so |ψ= (α |↓+ β |↑)|n = 0. We can then detect
81

Figure 4.18: Comparison of bright histogram to an ideal Poissonian histogram of same

average value. The tail of the histogram is due to depumping from the bright state to darker

states during the 400 μs detection period.
82

in the usual way, by applying the BD resonant laser and observing fluorescence. The probability of observing fluorescence counts above the treshold will be approximately equal to |α|

2.

If the motional state is not restricted to two number states, then more complicated techniques are required. Instead of applying a RSB π-pulse, if we scan the duration of the RSB

pulse (or BSB or carrier, which work equally well) and observe the fluorescence of the ion

as a function of this pulse duration, we obtain a flopping trace, where each part of the

motional superposition state is Rabi flopping simultaneously, but with different Rabi frequencies. By determining the Fourier components in the flopping curve, we can determine

the relative populations of the various number states that are occupied [Meekhof et al., 1996].

However, the flopping method does not give us all of the relative phase information between different components of the motional state. In order to know everything about the

state, we need to perform state reconstruction. One method involves applying a known and

variable displacement before obtaining the flopping trace, then repeating the flopping experiments described above many times for different displacement values to obtain the Wigner

function of the state [Leibfried et al., 1996]. This is a time-consuming analysis tool that is

only performed in rare cases, and in fact is not used in any of the experiments presented

here, but I mention it for the sake of completeness.
Chapter 5

Number states

In developing control of the quantum state of a trapped-ion oscillator, a good starting

point is to make individual number states. We can use this experiment as a benchmark to

measure against other harmonic oscillator systems, including previous trapped-ion experiments [Meekhof et al., 1996], photons [Hofheinz et al., 2008], and mesoscopic mechanical

oscillators [Chu et al., 2018]. This is also a metrologically useful class of states for measuring

forces that displace the state in phase space [Ziesel et al., 2013; Wolf et al., 2018]. Below I

will discuss how we generate the states, our experimental results, and sources of error. I will

also propose a protocol for force sensing with high number states outside the Lamb-Dicke

regime.

5.1 Producing pure number states

Starting from |↓|0, the generation of higher number states can be accomplished by

first applying a microwave π-pulse to transform the initial state to |↑|0, followed by a series

of alternating red (|↑|k→ |↓|k + 1, RSB) and blue (|↓|k→ |↑|k + 1, BSB) sideband

π-pulses at frequencies ω↓↑ − ω and ω↓↑ + ω respectively, where ω is the harmonic oscillator

frequency. Here, each pulse flips the spin of the internal state and adds a quantum of motion [Meekhof et al., 1996] (Figure 5.1 a and b). In general, we can use sidebands of order l
84

a

BSB

RSB

b

    ...   Detect Prepare 

into   RSB flop 

Figure 5.1: Generating number states. a) Relevant energy levels and transitions for motional

state creation. Blue sideband (BSB) pulses transfer population between |↓|kand |↑|k+1,

while red sideband (RSB) pulses transfer population between |↑|kand |↓|k + 1. b) Pulse

sequence for generating a harmonic oscillator number state. Alternating RSB and BSB πpulses are applied, with each pulse adding one quantum of motion (or more quanta on higher

order sidebands, see text) and flipping the spin of the state. To analyze the resulting state,

a RSB pulse is applied for a variable duration (labeled “RSB flop”) and the final spin state

is detected via state-selective fluorescence.
85

at frequencies ω↓↑ − lω and ω↓↑ + lω to add l quanta of motion with a single π-pulse. While

the Rabi frequencies of higher order sidebands on a mode with Lamb-Dicke parameter η < 1

are suppressed by ηl for small n, for higher number states, the Rabi frequencies can be much

larger than that of the first-order sideband [Leibfried et al., 2003] (see Fig. 5.2 d).

We demonstrate control over the motional state of the ion by preparing it in different (approximately pure) number states and Rabi-flopping on RSBs to determine the contrast, decay,

and n-dependent Rabi frequency [Meekhof et al., 1996; Leibfried et al., 2003] (see Fig. 5.2 a).

With the use of only first-order sidebands to create |↑|n = 40, we are able to achieve RSB

flopping (|↑|n = 40↔ |↓|n = 41) with greater than 70 % contrast. If we make use of up to

fourth-order sidebands to create the motional state, we observe approximately 50 % contrast

when we flop |n = 100on the fourth-order RSB (Fig. 5.2 b). To verify that the population

participating in the fourth-sideband flopping is in the desired number state, we also flop

on the second and third-order sideband (Fig. 5.2 c), which have a distinctly different Rabi

frequencies, and compare the Rabi frequencies of second to fourth order flopping to theory

(colored symbols in Fig. 5.2 d). For the second sideband, we measured a Rabi frequency (in

units of Ω01) of 0.2183 ± 0.008 for n = 100, which agrees with theory within one standard

deviation. The calculated Rabi-frequencies in n = 99 and n = 101 are more than three

standard deviations away from this measured value. Similar comparisons for other values of

n further establish confidence that intermediate states are prepared as desired and we can

indeed transfer approximately 50 % of the population to n = 100.

5.2 Time scales of experiments

The state preparation consists of Doppler cooling (∼ 120 μs) followed by ground state

cooling (∼ 110 μs) and a microwave carrier π-pulse |↓→ |↑(∼ 5 μs). At the end of each
86

0 20 40 60 80 100 120

Time ( s)

0.0

0.2

0.4

0.6

0.8

1.0

Population in |

n = 40 to 41

0 20 40 60 80 100 120

Time ( s)

0.0

0.2

0.4

0.6

0.8

1.0

Population in |

n = 100 to 104

0 20 40 60 80 100 120

Time ( s)

0.0

0.2

0.4

0.6

0.8

1.0

Population in |

n = 100 to 103

0 20 40 60 80 100 120

Number state n

0.0

0.5

1.0

1.5

2.0

2.5

3.0

n, n0/ 0, 1

Sideband Rabi frequencies vs n

n = 1

n = 2

n = 3

n = 4

a

C

b

d

Figure 5.2: Sideband flopping on number states. a) RSB flopping on the 1st-order sideband

of an |↑|n = 40state prepared using 1st-order RSB and BSB pulses. The curve shows

the probability of measuring |↓as a function of pulse duration during 1st RSB flopping

to |↓|n = 41. Each point represents an average over 200 experiments. The error bars

represent one standard deviation of the mean in all figures. Solid black lines show theory fits

to the data with the Rabi frequency, the initial contrast and an exponential decay constant

as fit parameters. b) RSB flopping on the 4th-order sideband of |↑|n = 100. The curve

shows the probability of measuring |↓as a function of pulse duration during 4th-order RSB

flopping to |↓|n = 104. Each point represents an average over 500 experiments. c) Same

as b on the 3rd-order sideband of |↑|n = 100↔ |↓|n = 103. d) First to 4th-order

sideband Rabi frequencies. All 1st-order data points (blue triangles) are fit to determine the

Lamb-Dicke parameter η = 0.2632(2). Curves for higher-orders are plotted for the same η.

Measured Rabi-frequencies for higher-order sidebands (colored symbols) are consistent with

theory (solid colored lines) for all orders. The π-pulse duration from |n = 0to |n = 1

is approximately 13 μs. Pulse durations required to produce higher number states can be

calculated based on this and the calculated Rabi frequencies for higher number states (see

Sec. 5.2).
87

n T(|n) (μs)

2 22

4 38

6 51

8 63

12 86

16 109

20 133

40 196

80 293

100 335

Table 5.1: Duration of pulse sequences to produce number states, with a base Rabi frequency

Ω0,1 = π/(13 μs). Beyond n = 40, we make use of higher-order sidebands, which allows us

to use transitions with higher Rabi frequencies and skip “rungs” as we move the ion up the

number-state ladder.

experiment, a 400 μs detection pulse on the cycling transition is applied. The Raman red

sideband |↑|0→ |↓|1π-time, which sets the Rabi-frequency Ω0,1, is approximately 13 μs,

so the duration of individual experiments can be calculated by using the measured Ω0,1 as a

“base” unit to calculate the sideband π-times for higher n according to (see also Eq. 7.10):

Ωn,n+s/Ω0,1 = η|s|−1



n<!

n>!

L|s|

n< (η2

) (5.1)

where L|s|

n is the generalized Laguerre polynomial and n> (n<) is the greater (lesser) of n+s

and n (these Laguerre functions are plotted in Fig. 5.2d). We can then sum the durations

of the individual sideband pulses used. Table 5.1 lists durations of the sequences used to

generate the number states. It is possible to substantially decrease these durations, as long

as the sidebands are still resolved.
88

5.3 Exploring error sources and resulting populations through simulations

We believe that the decay in contrast of the high-n flopping traces is largely due to

Raman beam pointing fluctuations on the ion, causing the intensity (and hence, the Rabi frequency) to fluctuate. This not only will cause decay in contrast even for a perfectly prepared

number state (since we are averaging over 100s of flopping experiments with slightly different

Rabi frequencies), but it will also contribute significantly to imperfect state preparation.

We think that, while there is some dispersion around the nominal n value from heating,

most of our initial loss in flopping contrast and subsequent decay is due to small amounts of

population left in each number state as we move the ion up the number state ladder.

Figure 5.3a shows the resulting 3rd sideband flopping from a simulation with a random

error (σ = 3%) in each preparation pulse and including heating (rate of 17 quanta/s determined in independent experiments). The final state has 44(2)% population in the target n

= 100 level (see Fig. 5.3b) in the histogram obtained by averaging over 30 runs of the simulation. The flopping trace produced by this distribution looks very similar to the experiment

(Fig. 5.2c).

5.4 Potential application: force-sensing with number states

Number states up to n = 3 have recently been demonstrated to exhibit quantumenhanced sensitivity to displacements [Wolf et al., 2018]. Notably, this enhancement is in all

directions, regardless of the phase of the displacement, unlike squeezed states, for example.1

We believe that with our ability to produce very high number states, we can potentially

1 If one has two harmonic oscillators, it is possible to measure displacements in both phase space quadratures by using two-mode squeezed states. With only one harmonic oscillator, though, Fock states are the
89

1.0

0.0

0.4

0.6

0.8

0.2

0 50 60 70 10 20 30 40 0.0

0.4

0.2

0.1

0.3

0 20 100 40 60 80

a) b) Population Population

Time (us) Number state n

Figure 5.3: Simulation of n = 100 number state creation with random error (σ = 3%) per

sideband pulse and 17 quanta/s heating rate. a) 3rd-order sideband flopping on resulting

state, and b) histogram of population in number states up to n = 110.

demonstrate an even higher sensitivity to forces than this previous work from [Wolf et al.,

2018], which demonstrated a 1.3 dB enhancement over the standard quantum limit with

an n = 1 number state. A typical method for detecting displacements from the ground

state is to apply a RSB π-pulse to an ion in |↓|m, where |mis the motional state. The

RSB transfers information about the motion into the spin, which allows us to read out via

state-selective fluorescence (see Ch. 4.8.5). The main feature of the ground state that allows

this method to work is that the RSB does not couple to |↓|n = 0. There are other values

of n which do not couple, or couple very weakly, to certain sideband interactions; these are

the points of zero Rabi frequency, plotted for our Lamb-Dicke parameter in Fig 5.2d above.

If we produce a number state |nand then the state is displaced by a small amount , the

resulting state following this displacement is:

|Ψ= Dˆ()|↓|n≈ |↓C(

√

n + 1 |n + 1− 

∗√n |n − 1+ |n), (5.2)

where Dˆ() = eaˆ†−∗aˆ is the displacement operator and C = √

1

||2(2n+1)+1

is a normalization

factor. If the Rabi frequency Ωn,n = 0, then following an approximate sideband π-pulse

on the |↓|n + 1↔ |↑|n + 1and the |↓|n − 1↔ |↑|n − 1transitions, then the

probability of measuring the state in |↑is:

P(|↑) ≈ ||

2

(2n + 1) (5.3)

optimal state for the phase-insensitive measurement.
90

for   1. The use of the number state enhances the signal by a factor of order n over the

displaced ground state experiment. Fig. 5.4 shows an example of the signal vs. displacement

for different number states, compared with the ground state. In this simulation, the LambDicke parameter is adjusted from η ≈ 0.22 (for n = 70) to η ≈ 0.30 (for n = 40) so that

the target number state coincides with a zero of either the 1st- or 2nd-order sideband Rabi

frequency (switching from 1st- to 2nd-order for n = 60 and higher to maintain reasonable

values of η). Adjusting η can be achieved by adjusting the strength of the ion’s confinement

to change the extent of the ground-state wavefunction or by changing the k-vector difference

of the Raman beams. The values in Fig. 5.4 correspond to a range of mode frequencies

from approximately 5 MHz to 9.3 MHz, which is easily achieved in our apparatus. These

initial simulations hold promise for potentially measuring displacements 25 times smaller

than we could with the ground state. Similar performance with a squeezed state of motion

would require nearly 30 dB of squeezing, and would only be sensitive to displacements along

directions with reduced uncertainty in the squeezed state.

For small displacements, this method is equivalent to measuring the overlap of the state

prior to displacement with the final, displaced state, as in Ref. [Wolf et al., 2018]. In this

reference, they also show that a perfect overlap measurement saturates the Quantum CramerRao bound. Our proposal is an alternative scheme for measuring the overlap, which might

be more practical for higher number states.
91

Figure 5.4: a) Simulation of force-sensing protocol for number states up to n=100. b)

Zoomed in, with dashed lines tangent to curves at point of maximal slope.
Chapter 6

Mode-frequency sensing with number state superpositions

As mentioned in Ch. 2, superpositions of the form |0+|nare great tools for measuring

the harmonic oscillator frequency. In fact, given the constraint that we cannot use number

states higher than |n, the |0+|nstate allows us to measure the oscillation frequency with

the best sensitivity possible, at the Heisenberg limit. Below I will discuss how we use these

states to measure the oscillation frequency, how we generate the states, and our experimental

results involving characterizing the states and measuring the ion oscillation frequency over

time. Finally, I will discuss possible extensions of this work.

6.1 Principle of operation

Enhanced interferometric sensitivity of superpositions of eigenstates depends on the

difference in energy between the two states—in the experiments here where the first state

of the superposition is the ground state this energy difference scales linearly with n. We

consider a harmonic oscillator with a frequency of oscillation ω0 + δω(t), where δω(t) is

a small, time-dependent correction due to noise and drift relative to the frequency ω0 of

an ideal reference oscillator, which we will call a local oscillator in keeping with common

terminology. We implement a Ramsey-type experiment, where the first effective π/2-pulse

creates the state |Ψn= √

1

2 (|0+ |n), where n denotes the Fock state number (in the

remainder of this chapter we will omit normalization). In a frame rotating at the local
93

oscillator frequency, |0and |nwill acquire a relative phase between the two states that is

proportional to n and equal to the integral of the fluctuations δω(t) over time T. After a

duration T the state is

|ΨnT = |0+ einφ |n,

φ = −

 T

0

δω(t)dt. (6.1)

Subsequently, a second effective π/2-pulse synchronous with the local oscillator recombines

the number-state superposition to the ground state if nφ = (2m + 1)π or to |nif nφ =

2mπ with m an integer. For general φ, the final state (up to a global phase) is |Ψnf =

cos(nφ/2)|n− isin(nφ/2)|0, with a probability of being in the ground state given by

P0 = 1/2[1 − cos(nφ)]. (6.2)

To characterize the harmonic oscillator in interferometric measurements, we want to determine small deviations of φ around some nominal value with maximal sensitivity. This occurs

when the slope |∂P0/∂φ| = |n/2 sin(nφ)| = n/2, namely when nφ 	 mπ/2, with m an odd

integer (m = ±1 in the tracking experiments described below). We want to minimize

δφ = ΔP0

|∂P0/∂φ|

, (6.3)

where ΔP0 = P2

0 −P02 is the standard deviation of a population measurement that

can discriminate between |0and |n. To a good approximation, the measurement is projection noise limited [Itano et al., 1993], which implies ΔP0 = P0(1 − P0). For nφ 	 mπ/2,

ΔP0 = 1/2 and δφ = 1/n, which is the Heisenberg limit and can only be reached with nonclassical oscillator states. In fact, the state |0+ |nsatisfies the Margolus-Levitin bound

for the maximal rate of evolution [Margolus and Levitin, 1998], implying that no other combination of states with quantum numbers between 0 and n can produce interference fringes

with higher sensitivity to motional frequency changes. The phase uncertainty of classically

behaving interferometers, which we define as those that use coherent oscillator states of the
94

same average excitation number  ̄n = n/2 and measurements of the oscillator energy (equivalent to the mean occupation number), will only reduce as δφclass = 1/n (see Appendix A).

In practice, the effective π/2-pulses will not have perfect fidelity and there will be added

noise above the fundamental projection noise. Such imperfections reduce the contrast C

(0 ≤ C ≤ 1) from the ideal value C = 1, which can be incorporated as P0 = C/2[1−cos(nφ)].

In our experiments C decreases as the complexity of preparing superpositions increases with

larger n. Additionally, since a single experiment only gives us one bit of information (the

ion is found in either |0or |n), we need to perform multiple experiments to accumulate

statistics to determine a phase shift. If the mode-frequency noise is not stable over the time

period required to acquire statistics, then the contrast of our interferometer is reduced. This

becomes more of a problem with the higher order interferometers, since the susceptibility

to mode-frequency noise increases with n. This limits the measurable gains in sensitivity to

n ≤ 12 in our specific experimental setting (see below).

6.2 Number-state superposition creation

The motional superposition |↑(|0+ |2) is straightforward to prepare by replacing

the first RSB pulse by a π/2-pulse |↑|0→ |↑|0+ |↓|1followed by a BSB π-pulse,

which transforms |↓|1→ |↑|2while not affecting |↑|0(see Fig. 1 a). To realize the

effective π/2-pulse |0→|0+ |nwhen n > 2, after the initial RSB π/2-pulse the |↑|0

component is “shelved” with a microwave π-pulse to |aux|0that is unaltered by subsequent pairs of BSB and RSB π-pulses that promote the |↓|kcomponent to |↓|k + 2.

The preparation is finished by a final microwave π-pulse |aux|0→ |↑|0and a BSB pulse

|↓|n − 1+ |↑|0→ |↑(|0+ |n) that promotes the |↓|n − 1component to |↑|nwhile

leaving the |↑|0component unaltered.
95

a

b

BSB

RSB MW

... 

2         ...      ...   ... Detect 

2     Prepare 

into 

""

2

""

2 ""

Figure 6.1: Generating superpositions of number states. a) Relevant energy levels and

transitions for motional state creation. Blue sideband (BSB) pulses transfer population

between |↓|kand |↑|k + 1, while red sideband (RSB) pulses transfer population between

|↑|kand |↓|k + 1. The blue sideband does not couple to |↑|0(crossed out, faded blue

arrow) because there is no energy level below the ground state. Transitions between the

states |↑and |auxare driven by a microwave field (MW, indicated in green), which does

not change k. b) Pulse sequences and trap frequencies for number-state interferometers. To

characterize the interferometers, the first effective π/2-pulse (labeled “π/2”) creates |0+|n,

followed by a free-precession period during which the mode frequency is deliberately increased

by Δω. An effective π-pulse (“π”) swaps the phase of the superposition, |0+ eiφ|n→

eiφ|0+ |n. After another free-precession period with the mode frequency reduced by Δω,

a final effective π/2-pulse closes the interferometer. For the composition of effective pulses

see text.
96

6.3 Interferometer characterization

We characterize the enhanced sensitivity of each interferometer by inserting an effective

“spin-echo” type π-pulse between wait periods of duration T and purposely induce trap frequency changes Δω with equal magnitude and opposite sign before and after the echo pulse

(Fig. 6.1 c). We use this echo technique because trap frequency fluctuations or drifts that

alter the mode frequency by approximately the same amount in both arms are suppressed.

For the first wait period this ideally results in an order-dependent phase φ = −n Δω T.

The echo pulse is composed of the following steps: first the pulses of the effective π/2-

pulse are applied in reverse order to ideally give |↑(|0+ eiφ |n) → |↑|0+ eiφ |↓|1.

Second, a RSB π-pulse results in eiφ |↑|0+ |↓|1, which is then walked up the numberstate ladder as described for the first effective π/2-pulse. Ideally the effective echo πpulse accomplishes |0+ eiφ |n→ eiφ |0+ |n. In this way, the induced trap frequency

change −Δω during the second wait period constructively adds to the interferometer phase

eiφ |0+ |n→ eiφ |0+ e−iφ |n, which is transformed to −isin(φ)|↓|1+ cos(φ)|↑)|0

by the final effective π/2-pulse, so the induced interferometer phase φ can be read out by

measuring the probability to find |↓.

We find that as we increase n in the superposition for T = 100 μs, the phase accumulation

increases linearly with n as expected, but, due to accumulated imperfect state preparation

steps, the contrast of the interference fringes (and therefore the fringe slope) is reduced (see

Fig. 6.3 a), reducing the signal in Eq.(6.3) of the higher-order interferometers. Given this effect, we observe the highest sensitivity with the |0+|12superposition state, which achieves

a 8.5(2) dB (7.1(4) times) improvement over a perfect |0+ |1interferometer (Fig. 6.3 b).

The n = 12 interferometer also performs 3.2(2) dB better than an ideal classical interferom-
97

eter (see Appendix A). To similarly increase sensitivity with ideal squeezed states [Loudon,

2000], we would require approximately 6 dB of squeezing below the vacuum noise for the

ideal interferometer.

6.4 Mode-frequency tracking with number-state interferometers

We can use this enhanced sensitivity to precisely track motional mode frequency

changes over time. We perform two Ramsey-type experiments with the phase of the final effective π/2-pulses equal to ±π/2 so that when the pulses are resonant with the mode

frequency, the resulting |↓population from each Ramsey experiment is ideally 1/2. A

difference between the populations for the +π/2 and −π/2 cases provides an error signal

that we feed back to the local oscillator to follow the fringe pattern as the mode frequency

drifts due to changes in stray electric fields and the sources providing the electrode potentials.

This procedure is complicated by the fact that the sideband transition frequencies are shifted

by the AC Stark effect from the Raman beams. These result in phases beyond those described in Eq. (1) that shift the interferometer fringes. To mitigate this effect, as well as

to subtract out non-zero phase accumulation during the creation of the superposition state,

we use auto-balanced Ramsey spectroscopy [Sanner et al., 2017]. Instead of using two Ramsey experiments that provide the error signal used to feed back to the pulse frequency, we

interleave four Ramsey experiments with two different Ramsey times, Tshort and Tlong (typically 20 μs and 100 μs, respectively). The phase between the two π/2-pulses is adjusted

to compensate for systematic phases according to the error signal from the short Ramsey

experiments, and the frequency of the local oscillator is adjusted according to that from the

long Ramsey experiments. These phase and frequency adjustments are applied equally to

both the short- and long-pulse Ramsey experiments. This suppresses all contributions to
98

2 1 0 1 2

0.0

0.5

1.0

0.2 0.1 0.0 0.1 0.2

/2 (kHz)

0.0

0.5

1.0

Population in |

100 101

Number of phonons

10 1

100

|0 + |2

|0 + |4

|0 + |8

|0 + |12

|0 + |16

|0 + |18

a b

1 1

Heisenberg limit

Figure 6.2: Interference and sensitivity of different number-state superpositions. a) Interference fringes for number state interferometers with n= 2, 4, 8 and 12. Each data point is

averaged over 250 experiments and uses a wait time of 100 μs before and after the effective

π-pulse. The fringe spacing is reduced as 1/n as expected for Heisenberg scaling. At the

same time, the fringe contrast is reduced with higher n due to the larger number of imperfect

pulses and the higher susceptibility to mode-frequency changes that are not stable over all

250 experiments for each data point. This reduces the fringe slopes for n > 12 below the

maximal slope reached for n = 12. b) Experimentally determined noise-to-signal ratio δφ as

defined in Eq.(6.3) as a function of order n (colored dots) together with the theoretical lines

for a perfect classical interferometer at 1/

√n and the 1/n Heisenberg limit valid for ideal

number-state interferometers.
99

phase accumulation other than the phase difference accumulated due to the different freeprecession times, which is unperturbed by laser beam couplings [Sanner et al., 2017]. This

procedure is described in more detail in Sec. 6.5.3 below.

When tracking the mode frequency in this way we can record the frequency error versus

time for different-n interferometers. We can then determine the overlapping Allan deviation [Howe et al., 1981] as a function of averaging interval and compare it for the different

interferometers (Fig. 6.4 a). This data was taken while interleaving experiments with the

|0+ |2, |0+ |4, |0+ |6and |0+ |8interferometers to allow for a direct comparison

of their sensitivity under the same noise and drift conditions. For long averaging periods,

trap frequency drifts dominate the uncertainty. As expected, this gives the same asymptotic

long-time slope of the Allan deviation for all interferometers. Importantly, the increased

sensitivity of higher-n interferometers reduces the time interval required to average down

to a certain level for n ≤ 8. For the n = 8 interferometer, we observe the minimum Allan

standard deviation at approximately 23 s of averaging in this interleaved comparison. By

running only the n = 8 interferometer sequences, we increase the measurement duty cycle

which accelerates the rate with which the Allan deviation approaches its minimum. Under

these conditions, the minimal fractional frequency Allan deviation of 2.6(2)×10−6 (∼ 19 Hz

at 7.2 MHz) is reached at approximately 4 seconds of averaging (red triangles in Fig. 6.4 b).

To further increase the measurement rate, we record the population differences determined

in all four Ramsey experiments comprising the auto-balance sequence without feeding back

on the local oscillator frequency. This eliminates the latency due to computer control of the

frequency tracking. As long as the populations of four Ramsey experiments uniquely determine the frequency change, we can run a series of n = 8 interferometer experiments, each

taking 4 ms, without feedback to shorten the time to reach the minimum of 2.9(4) ×10−6 to

approximately 0.5 s before the uncompensated mode frequency drift produces an increasing

Allan deviation (Fig. 6.4 b, blue circles). While the minimum value of the Allan deviation
100

is not lower when taking data in this fashion as compared to when tracking the drift, this

experiment gives an idea of how quickly we could average down to the level of a few parts in

10−6 if tracking latency is minimized. We suspect that the Raman beams contribute to charging the trap and increasing mode-frequency instabilities (see Sec. 6.5.2), so we attribute the

difference in drift rate between Fig: 6.4a) and b) to the difference in Raman beam duty cycle.

6.5 Experimental details

6.5.1 Time scales of experiments

As with experiments described in Ch. 5, the state preparation consists of Doppler

cooling (∼ 120 μs) followed by ground state cooling (∼ 110 μs) and a microwave carrier

π-pulse |↓→ |↑(∼ 5 μs). At the end of each experiment, a 400 μs detection pulse on the

cycling transition is applied. The Raman red sideband |↑|0→ |↓|1π-time, which sets the

Rabi-frequency Ω0,1, is approximately 13 μs, so the duration of individual experiments can

be calculated by using Ω0,1 as a “base” unit to calculate the sideband π-times for higher n

according to Eq. 7.10. We can then sum the durations of the individual sideband pulses and

of the ∼ 5 μs microwave |↑→ |auxπ-pulses used. Table 6.1 lists durations of the sequences

used to generate number state superpositions. It is possible to substantially decrease these

durations, as long as the sidebands are still resolved.

The sidebands in our interferometers rely on the same Raman laser coupling that is used in

most two-qubit gates, so the time scale for a single BSB or RSB pulse is on the same order as

a typical gate time. This limits the usefulness for improving gate fidelity, but interferometric

tracking would certainly help with longer-term drifts and reduce the duration required to

measure the trap frequency with a certain precision.
101

1 5 10 50 100

5

10

20

averaging time(s)

fractional Allan deviation(10- 6

)

, with feedback

, no feedback

0.05 0.10 0.50 1 5 10 2

4

6

8

10

averaging time(s)

fractional Allan deviation(10- 6

)

a b

Figure 6.3: Oscillator frequency tracking using number-state interferometers. a) Interleaved

comparison of the Allan standard deviation of tracked fractional trap frequencies vs. averaging time found with n = 2, n = 4, n = 6 and n = 8 interferometers. The repetition rate

of a single run, comprised of long (100 μs Ramsey time) and short (20 μs Ramsey time)

auto-balance sequences on both sides of the fringe respectively, was approximately 7/s. The

n = 8 interferometer produces the lowest fractional frequency Allan deviation. Trap frequency drifts begin to dominate the Allan deviation at 10’s of seconds. b) Fractional modefrequency uncertainty vs. averaging time for two series of n = 8-only interferometer runs to

maximize measurement duty cycle. We are able to achieve a minimal fractional frequency

Allan deviation of 2.6(2) ×10−6 at approximately 4 seconds of averaging time with feedback

activated (red triangles) and an experiment rate of approximately 43/s as defined above.

The Allan standard deviation for averaging times up to 1 s without feedback activated is

shown by the blue circles. The minimum is reached after 0.5 s with the experiment rate

increased to approximately 250/s.

n T(|0+ |n) (μs)

2 16

4 41

6 54

8 66

12 89

16 112

Table 6.1: Duration of pulse sequences to produce number-state superpositions, with a base

Rabi frequency Ω0,1 = π/(13 μs).
102

6.5.2 Sources of decoherence

At the 7.2 MHz axial mode frequency, the heating rate is relatively low, approximately

17 quanta/s. Consequently, the dominant source of decoherence in these experiments is dephasing due to fluctuations in the mode frequency. These fluctuations probably arise from

technical noise from the voltage source, uncontrolled charging from stray light scattering

off the dielectric material between the trap electrodes and amplitude instabilities in the rf

source that generates the pseudo-potential. We also attribute longer time-scale drift, on the

order of minutes to hours and a magnitude of ∼ 1 − 10 Hz/s to uncontrolled charging and

discharging. This charging affects mode-frequency tracking experiments and leads to unpredictable deviations from the ideal white-noise 1/

√τ behavior during experimental runs to

determine Allan-variances. A more detailed investigation of this noise and ways to further

suppress it is currently in progress in our laboratory.

6.5.3 Auto-balanced frequency tracking experiments

The auto-balanced sequence (see Fig. 6.4) for mode-frequency tracking is comprised

of four interleaved Ramsey experiments, two each with Ramsey times tshort and tlong, where

the phase of the second effective π/2-pulse is +π/2 relative to the first π/2-pulse for one of

the experiments and −π/2 for the other. For a given Ramsey time, the two experiments

with +π/2 and −π/2 relative phase interrogate the fringe close to its largest positive and

negative slope respectively. If the long Ramsey experiments are on exact resonance, both

should result in the same average population in |↓, so the signal difference is zero on average.

If the Ramsey experiment is off resonance, the signal difference provides a non-zero error
103

signal plong which is used to calculate the offset δω between the assumed mode frequency

ωa and the actual mode frequency. The offset δω is fed back to the local oscillator (LO),

which updates its frequency to ω = ωa + δω. This in turn updates the frequency of the

BSB and RSB pulses to ωLO = ω↓↑ + ωa + δω and ωLO = ω↓↑ − ωa − δω, respectively, where

ω↓↑ is the energy difference between |↑and |↓, as described in the main text. The short

Ramsey experiments provide an error signal pshort, as described above, which is now used

to compute a phase offset δφ that is added to the relative phase between the two effective π-pulses, φLO = ±π/2 + δφ. Feedback on this phase reduces “frequency pulling” due

to non-zero phase accumulation during the pulses, which can be caused by slowly drifting,

pulse-synchronous systematic errors such as AC-Stark shifts. For a more detailed description

of the auto-balanced Ramsey technique, see Ref. [Sanner et al., 2017].

6.5.4 Anharmonic contributions

We do not expect the anharmonicity of the trapping potential to be a significant limitation or source of systematic error. Previous calculations [Home et al., 2011] for a similar

trap and estimates based on a numerical simulation of our current trap predict an anharmonic component of a few parts in 10−7 per quantum on the axial mode frequency. With a

maximal superposition state of |0+ |8used in the mode-tracking experiments, this would

cause an offset of ∼ 1 × 10−6, which is within the minimum measurement uncertainty.

6.6 Extensions to work

In a natural extension of the work presented here, it should be possible to observe

such mode-frequency noise during free precession by refocusing with one or more effective

π-pulses. This would allow us to filter the response of the ion to certain spectral components
104

,

,

LO

Figure 6.4: Schematic illustrating the auto-balanced feedback loop applied to local oscillator

(LO), a frequency source used as a reference to compare to the ion’s oscillation frequency.

The LO controls the phases and frequencies of the BSB and RSB laser-pulses (see Fig.

6.1) during mode-frequency tracking experiments. The difference between the populations

measured after a pair of Ramsey experiments with long wait times provides an error signal,

plong, used to feed back on the LO frequency, ωLO. Similarly, a second pair of Ramsey

experiments with short wait times provides and error signal, pshort, used to feed back on an

additional LO phase offset φLO between the first and second effective π/2-pulses. The long

and short Ramsey experiments are interleaved, with φLO and ωLO applied equally to both.

For more details on auto-balanced Ramsey experiments, see Ref. [Sanner et al., 2017].
105

of the motional mode-frequency noise, providing a quantum lock-in analyzer in analogy to

characterizations of magnetic field noise with a trapped ion [Kotler et al., 2011]. By using

number-state superpositions, we can transfer the quantum advantage demonstrated in the

experiment reported here to achieve “quantum gain” in such lock-in measurements. These

quantum-enhanced spectrum analysis experiments have been led by Jonas Keller and will

be written up soon.

More generally, we expect that the techniques demonstrated here can be applied to characterize other harmonic oscillators in the quantum regime with increased precision and on time

scales that were previously inaccessible. Such capabilities could support quantum metrology

and improve the prospects of fault-tolerant quantum information processing, where some of

the most advanced experimental platforms are limited by harmonic oscillator coherence.
Chapter 7

Mode-frequency sensing with coherent displacements

The third set of experiments based on generating a “quantum” state of motion I will

discuss is with coherent displacements [McCormick et al., 2019b]. As discussed previously,

coherent states are classically behaving quantum states, so we will not gain quantum enhancement from this method, unless we perhaps coherently displace non-classicial states of

motion in a hybrid approach. The coherent displacements are produced by tickling (see Sec.

4.2.8) the ion, which is relatively simple to implement in the lab. Because of the ease of

implementation and short operation time (typically ∼ 1 μs), we still are able to characterize

the ion’s motion with a sensitivity on the order of that of the best superposition state we

used by operating at higher average occupation numbers. In this chapter, I will discuss the

quantum theory of coherent displacements and experiments we performed involving coherent displacements of number states, including spectral analysis of the mode frequency noise

using a series of coherent displacements with alternating signs.

7.1 Introduction

The manipulation of quantum states of a harmonic oscillator is a theme of current

interest across a wide range of experimental platforms. Often the methods developed on one

platform, in our case a single, harmonically bound atomic ion, can be adapted to many other
107

platforms, after suitable modifications of the original procedures. An important example is

resolved sideband-cooling of micro-fabricated oscillators, theoretically described in [WilsonRae et al., 2007; Marquardt et al., 2007], that bears strong analogies to the methods that

were first developed for single-ion mechanical oscillator systems [Neuhauser et al., 1978;

Wineland and Itano, 1979; Diedrich et al., 1989]. In this chapter, ground-state cooling is

combined with another basic idea that is widely applicable across harmonic oscillator platforms, namely coherent displacements that can be conveniently implemented with a classical

force that is near resonant with the harmonic oscillator frequency [Carruthers and Nieto,

1965]. The added ingredient used here is a suitable two-level system, in our case two internal electronic states of a single ion, that can be coupled to the harmonic oscillator motion and

read out with a projective measurement to gain information about the state of the harmonic

oscillator. A superconducting qubit is just one example of an effective two-level system that

has been coupled to a co-located micro-fabricated harmonic oscillator [Chu et al., 2018].

Therefore, the methods described in this chapter might also be adaptable and useful to the

rapidly growing community that studies harmonic oscillator systems in the quantum regime.

Exciting the harmonic oscillator motion of trapped charged particles with a weak oscillating electric field has long been used to determine motional frequencies and subsequently

the charge-to-mass ratio in various ion trap based devices, for example in ion-trap mass

spectrometers [Paul and Steinwedel, 1953; Myers, 2013] (see also Sec. 4.2.8). The response

of the ions can be detected by counting resonantly ejected particles, by resonance absorption of the driving field [Dehmelt and Major, 1962], or through image currents in the trap

electrodes [Dehmelt and Walls, 1968]. For certain atomic ions, it is possible to detect the

ion motion through changes of scattered light that depends on velocity via the Doppler effect [Wineland et al., 1983] (see also Sec. 4.2.8). An important practical advantage of a

tickle is that it only interacts with the charge of the ion, therefore it is immune to magnetic

field or AC-Stark shifts that may restrict how well the harmonic oscillator frequency can be
108

determined spectroscopically, for example by resolving the motional sidebands of internal

transitions of ions [Leibfried et al., 2003]. Many other harmonic oscillator systems have

analogous mechanisms available, for example excitation of a micro-fabricated resonator by

driving it with a piezo-electric element or with a capacitively coupled electric circuit.

The tickle method can be further refined with atomic ions that are cooled close to the

ground state of their motion and can be coupled to a two-level system through resolved

sideband transitions [Wineland and Itano, 1979; Leibfried et al., 2003]. Near the ground

state of motion ( ̄n  1), the probability of driving a “red sideband” transition, where the

internal state change of the ion is accompanied by reducing the number of quanta in the

motional state |n→|n − 1, is strongly suppressed; this can be used to determine the

average harmonic oscillator occupation number  ̄n [Diedrich et al., 1989; Monroe et al., 1995;

Leibfried et al., 2003]. Starting near the ground state, a resonant tickle can add quanta of

motion such that the red sideband can be driven again, as discussed qualitatively in [Home

et al., 2011]. For weak excitation,  ̄n ≤ 1, we observe responses close to the Fourier limit of

the tickle pulse, as I will describe in more detail and have experimentally demonstrated.

If the tickle excitation acts longer or with a larger strength, an ion in the ground state

can be displaced to coherent states with an average harmonic oscillator occupation number  ̄n  1. The Rabi frequency of sideband transitions depends non-linearly on  ̄n, which

leads to collapse and revival of internal state changes that are one of the hallmarks of the

Jaynes-Cummings model [Jaynes and Cummings, 1963; von F ̈orster, 1975; Meekhof et al.,

1996]. Here we examine the probability of changing the internal state theoretically and

experimentally, as a function of tickle detuning relative to the frequency of a harmonic oscillator motional mode of the ion. When probing the red sideband transition after displacing

to  ̄n  1, we observe rich sets of features with steep and narrow side lobes around the

resonance center. Such nonlinear responses can in principle be used to find the frequency of
109

the motion with better signal-to-noise ratio than what the Fourier limit implies for smaller

coherent states where  ̄n ≤ 1 and the response of the ion is essentially linear.

A sequence of coherent displacements alternating with free evolution of the motion, inspired by spin-echos [Hahn, 1950] and dynamical decoupling [Ernst et al., 1987; Alvarez and  ́

Suter, 2011], can be used to obtain a frequency-filtered response of the ion. We implement

and characterize such sequences by observing and modeling the ion response to deliberately

applied, monochromatic modulations of the trapping potential curvature. Similar sequences

can then be used without applied modulations. In this case, the response of the ion can be

attributed to harmonic oscillator frequency noise that is intrinsic to our system, allowing

us to characterize noise on the trap potential in a frequency range of 500 Hz to 400 kHz, a

wide frequency range that has not been studied in detail in previous work [Talukdar et al.,

2016]. With this method, several narrow band technical noise components (spurs) in our

setup were identified through the direct response of the ion. The noise was traced back to

digital-to-analog converters (DACs) used in our setup and was eliminated by replacing the

DACs with analog power supplies.

7.2 States and ion fluorescence signals from coherent displacements

We consider a single ion with charge q and mass m confined in a harmonic trapping

potential with minimum position at r0, such that the motion of the ion can be described

by three normal harmonic oscillator modes with approximate frequencies ωz ≤ ωy ≤ ωx.

By using a coordinate system where the axis directions coincide with the normal mode

directions, we can write the ion position as r = r0 + δr. The interaction of the ion with an
110

additional uniform electric field E can be described as

HE = q(E · δr). (7.1)

For the normal mode in the z-direction we introduce ladder operators ˆa and ˆa† to write a

time independent Hamiltonian as

H0 = ωzaˆ†

a. ˆ (7.2)

We have suppressed the ground state energy since it is a constant term that does not change

the dynamics. The real oscillator we observe in the experiment has frequency noise and is

therefore not always sufficiently described by H0. We will consider the change in dynamics

due to frequency noise in more detail in section 3. We replace δz by its equivalent quantum

mechanical operator δzˆ = z0(ˆa† + ˆa) with z0 = /(2mωz) the ground state extent of the

oscillator. For the normal mode in this direction, and in the interaction picture relative to

H0, the interaction with an oscillating electric field Ez(t) = E0 cos(ωt + φ) becomes

HI = Ωz(ˆa†

eiωzt + ˆae−iωzt)(ei(ωt+φ) + e−i(ωt+φ))

= Ωz(ˆa†

e−i(δt+φ) + ˆaei(δt+φ) + ˆa†ei(σt+φ) + ˆae−i(σt+φ)), (7.3)

with the coupling Ωz = qE0z0/(2) and δ = ω − ωz, σ = ωz + ω. If the oscillating field is

close to resonance with the normal mode, |δ|  σ, the faster-rotating terms containing σ

can be neglected to a good approximation and the interaction takes the form of a coherent

drive detuned by δ

HI 	 Ωz(ˆa†

e−i(δt+φ) + ˆaei(δt+φ)). (7.4)

We can formally integrate the equation of motion for HI [Carruthers and Nieto, 1965;

Glauber, 1963; Leibfried et al., 2003] to connect an initial state |Ψ(0)at t = 0 when the

electric field is switched on to the coherently displaced state after evolution for duration t,

|Ψ(t)

|Ψ(t)= Dˆ(α(t))eiΦ(t) |Ψ(0), (7.5)
111

where

Dˆ(α) = exp(αaˆ† − α∗

aˆ),

α(t) = −iΩze−iφ  t

0

e−iδτdτ = −Ωze−iφ 1 − e−iδt

δ ,

Φ(t) = Im 
 t

0

α(τ ){∂τα∗(τ )}dτ=

Ωz

δ

2

[sin(δt) − δt]. (7.6)

The phase Φ(t) can play an important role, for example, in two-qubit gates [Leibfried et al.,

2003; Lee et al., 2005] or interferometric experiments that combine internal degrees of freedom of the ion with motional states [Monroe et al., 1996; Hempel et al., 2013]. Here, we will

be interested only in the average occupation number  ̄n = Ψ(t)|aˆ†

aˆ|Ψ(t), which does not

depend on Φ(t). If the initial state is the harmonic oscillator ground state, |Ψ(0)= |0the

average occupation is

n ̄(t) = |α(t)|

2 = 2 Ωz

δ

2

[1 − cos(δt)]. (7.7)

On resonance (δ = 0) the coherent state amplitude grows linearly in t as α(t) = eiφΩzt and

the energy of the oscillator quadratically as  ̄n(t)=Ω2

zt

2. For a coherent state (displaced n = 0

ground state), the probability distribution over number states |mis a Poisson distribution

with average  ̄n

P(0)

m = n ̄me−n ̄

m! . (7.8)

An initial number state with |Ψ(0)= |n, displaced by Dˆ(αd), results in a more involved

probability distribution [Carruthers and Nieto, 1965]

P(n) m =  ̄n|n−m|

e−n ̄ n<!

n>!

(L|n−m| n< ( ̄n))2

, (7.9)

where  ̄n = |αd|

2, n< (n>) is the lesser (greater) of the integers n and m and La

n(x) is a

generalized Laguerre polynomial.

In our experiments, the harmonic oscillator motion is coupled by laser fields to a two-level

system with states labelled |↓and |↑with energy difference E↑ − E↓ = ω↓↑ > 0. The
112

internal state is initialized in |↓by optical pumping. After the state of motion is prepared, the state |↓|Ψ(t)can be driven on a red sideband, resulting in population transfer

|↓|m↔ |↑|m − 1for all m > 0 while the state |↓|0is unaffected [Leibfried et al., 2003].

The Rabi frequencies depend on m > 0 as

Ωm,m−1 = Ω0e−η2/2

η

 1

m

L1

m−1(η2

), (7.10)

where Ω0 is the Rabi frequency of a carrier transition |↓↔ |↑of an atom at rest, η = kzz0

is the Lamb-Dicke parameter, with kz the component of the effective wavevector along the

direction of oscillation. After driving the red sideband of state |↓|Ψ(t)(where the state

prior to displacement is |Ψ(0)= |n) for duration τ , the probability of having flipped the

internal state to |↑is

P↑(τ ) = 1

2



1 − P(n)

0 − 	∞

m=1

P(n) m cos(2Ωm.m−1τ )



. (7.11)

We set the red sideband drive duration to be equivalent to a resonant π-pulse on the |↓|1↔

|↑|0transition, which implies 2Ω1,0τ = π. For an arbitrary displaced number state the

probability of the ion to be in |↑for this pulse duration becomes

Pπ

↑ = 1

2



1 − P(n)

0 − 	∞

m=1

P(n) m cos(π

Ωm.m−1

Ω1,0

)



. (7.12)

This probability is not a monotonic function of  ̄n and exhibits maxima and minima as the

displacement changes. Experimental observations of this behavior for displaced number

states and comparisons to the predictions of Eq. (7.12) will be discussed in section 7.4.2.

When the detuning in Eq. (7.7) is δ = 0, the coherent drive displaces |Ψ(0)along circular

trajectories in phase space that can turn back onto themselves. For δ t = mπ with m a

non-zero integer, |α(t)| will reach a maximum of 2Ωz/δ for m odd and return to zero for m

even. The non-monotonic behavior of Pπ

↑ with respect to  ̄n creates feature-rich lineshapes

when this probability is probed as a function of the displacement detuning δ relative to the

harmonic oscillator frequency.
113

7.3 Noise sensing with motion-echo sequences

The motion displacements discussed above enable sensitive tests of the ion’s motional

coherence. Electric field amplitude noise that overlaps with the motional frequencies heats

the ion out of the ground state, and is observed in all traps at a level that often exceeds resistive heating by orders of magnitude. This “anomalous heating,” is well documented [Turchette et al., 2000; Brownnutt et al., 2015; Talukdar et al., 2016], but the

sources are not well understood. On much longer time scales than the ion-oscillation period,

motional frequencies are known to drift over minutes and hours due to various causes, for

example slow changes in stray electric fields and drifts of the sources that provide the potentials applied to the trap electrodes (see also Sec. 6.4). Much less work has been done

to characterize noise in the frequency range in between the harmonic oscillator frequency

and slow drift [Talukdar et al., 2016]. The high Q of most trapped-ion systems implies that

this lower frequency noise contributes negligibly to heating, but does cause dephasing of the

ion’s motional state that can be detected with coherent displacements. Here, we construct

sequences of coherent displacements that alternate with periods of free evolution and suppress the sensitivity to slow drifts of the harmonic oscillator frequency. This allows us to

isolate harmonic oscillator frequency noise in a specific frequency band, in analogy to an

AC-coupled electronic spectrum analyzer. This method is suitable for detecting noise at

frequencies in a range of 500 Hz to 400 kHz in the experiments described here. Measurements at lower frequencies are in part limited by the heating rate of the system, while the

highest noise frequencies we can characterize are determined by the duration of coherent

displacements, which in our system can be implemented on time scales of order a few μs.
114

7.3.1 Basic principle

The sequences of coherent displacements discussed here are closely related to spin-echo

experiments and dynamical decoupling in two-level systems [Hahn, 1950; Ernst et al., 1987;

Alvarez and Suter, 2011]. In analogy to the classic  ́

(π/2-pulse)-τa-(π-pulse)-τa-(π/2-pulse)

spin-echo sequence [Hahn, 1950] with τa the duration of a free-precession period, the ideal

“motion-echo”pulse sequence consists of

Dˆ(Ωzτd/2)-τa-Dˆ(−Ωzτd)-τa-Dˆ(Ωzτd/2),

where τd characterizes the duration of the displacement and the minus sign in the argument

of the second displacement indicates that the phase φ of the displacement drive has changed

by π relative to the other displacement operations. To simplify this initial discussion, we

assume that all displacements are instantaneous and not affected by fluctuations in the oscillator frequency. This condition is similar to the “hard-pulse” limit for spin-echo sequences.

If the frequency of the oscillating electric field in Eq. (7.1), which we call the “local oscillator frequency” in this context, is on resonance with the harmonic oscillator frequency,

the displacements in the sequence add up to zero, so any initial state is displaced back onto

itself at the end of the sequence (see Fig. 7.1 (a)). In analogy to a spin-echo sequence, if the

local oscillator differs from the harmonic oscillator frequency by a small, constant detuning

δ  2π/τd, the sequence will still result in a final state that is very close to the initial state

(see Fig. 7.1 (b)). However, if the detuning changes sign between free-precession periods (see

Fig. 7.1 (c)), the final state will not return to the initial position and in general information

about the oscillator frequency fluctuations can be gained from the final displacement. This

basic echo sequence can be expanded by including additional blocks of the form

τa-Dˆ(Ωzτd)-τa-Dˆ(−Ωzτd)

after the first displacement Dˆ(−Ωzτd) in analogy to dynamical decoupling sequences in twolevel systems. Ideally, this increases the number of free-precession sampling windows which
115

1

3

5

1

3

5

1

3

5

a) b) c)

4

2

4

2

4

2

Figure 7.1: Schematic phase-space sketch of the displacements in the simplest motion-echo

sequence. Here, all displacements are assumed to act instantaneously (hard-pulse limit), such

that the effect of oscillator detuning during displacements can be neglected. (a) Without

fluctuations of the oscillator frequency, the ground-state minimum uncertainty disk (green)

is coherently displaced by Ωzτd/2 (step 1), then remains stationary during a free-precession

period (step 2), it is then displaced symmetrically through the origin by −Ωzτd (step 3),

followed by another free-precession period (step 4). The final displacement by Ωzτd/2 (step

5) returns the state to the origin. (b) With a small, constant detuning, the state drifts

perpendicular to the direction of the first displacement in step 2. However, it drifts an equal

amount in the opposite direction during step 4, to therefore return to the ground state after

step 5. This immunity to constant detuning can be thought of as a harmonic oscillator

analogy to a spin-echo sequence in a spin-1/2 system. (c) If the detuning changes sign

between steps 2 and 4, the state does not return to the origin. In all three cases (a)-(c), the

final state reflects the sum of additional displacements during the operation of the sequence

that are caused by time-dependent changes in harmonic oscillator detuning.
116

leads to a longer sampling time and narrower filter bandwidth of the extended sequence,

while still producing no net displacement from the initial state if the harmonic oscillator is

stable, even if the local oscillator is slightly detuned from the harmonic oscillator resonance.

7.3.2 Effects of oscillator frequency fluctuations

If the local oscillator frequency is not on resonance with the harmonic oscillator frequency, or if the detuning is not constant in time, a realistic coherent drive (not assuming the

hard-pulse limit) will not always displace the state of motion along a straight line. To model

this situation, we define ω0 to be the time-independent frequency at which the interaction

frame with respect to H0 rotates and set this to be equal to the local oscillator frequency

and ω0 = ω − δ(t). The differential equation describing the coherent displacement α(t) as a

function of time becomes

α ̇(t) = αiδ(t) − iΩze−iφ, (7.13)

where δ(t) is the instantaneous detuning between the harmonic oscillator and the local

oscillator at time t. In the special case where δ does not depend on time and α(0) = 0, the

solution is consistent with α(t) from Eq. (7.6) after taking the change of reference frame into

account. If there is noise on the trap frequency, δ(t) will fluctuate randomly as a function

of t. A general solution of Eq. (7.13), at time t0 + τ as it evolves from the initial state α(t0)

at time t0, can be formally written as

α(t0, τ ) = exp[iI1(t0, τ )][α(t0) − iΩze−iφI2(t0, τ )], (7.14)

with

I1(t0, τ ) =  t0+τ

t0

δ(τ1)dτ1,

I2(t0, τ ) =  t0+τ

t0

exp 
−i

 τ2

t0

δ(τ1)dτ1



dτ2. (7.15)
117

This form is useful for numerical calculations and can be explicitly solved for special cases of

δ(t). Motion-echo sequences are most useful if the accumulation of phase during τ is small,

I1(t0, τ )  2π. In such cases, we can expand the exponential functions in Eqs. (7.14) and

(7.15) to linear order and make the approximation

α(t0, τ ) 	 {α(t0) − iΩzτe−iφ} + i(α(t0) − iΩzτe−iφ)I1(t0, τ )

−Ωze−iφI3(t0, τ ),

I3(t0, τ ) =  t0+τ

t0

 τ2

t0

δ(τ1)dτ1



dτ2. (7.16)

The different terms in Eq. (7.16) have straightforward interpretations: the term in curly

braces characterizes the displaced coherent state for no detuning, δ(t) = 0. Finite detuning

rotates this state around the origin in phase space and to lowest order this effect is captured

by the term proportional to I1(t0, τ ). The final term reflects the effect of the detuning during

displacement, which results in a correction proportional to I3(t0, τ ). For a free-precession

period, Ωz = 0; during displacement, Ωz = 0.

In this linear approximation, it is straightforward to keep track of the displacements and

the corrections from δ(t) = 0 when periods of driving and free precession are concatenated.

Because corrections on earlier corrections are higher order than linear, the correction from

each period only acts on the zero-order displacement of any previous period. This implies

that the zero-order terms in curly brackets and the corrections can be summed up separately

for a sequence. In this way, we can calculate the total zero-order displacement αn and first

order correction Δαn of a sequence with n steps starting at time t = 0 in state |α(0). For

the k-th step starting at tk, the displacement drive Rabi frequency is Ωz,k, the drive duration

τk, and the phase φk. In the linear approximation with α(0) = α0 the sums are

αn = α0 − i

	n

k=1

Ωz,kτke−iφk , (7.17)

Δαn = i

	n

k=1

αk−1 − iΩz,kτke−iφk

I1(tk, τk) − Ωz,ke−iφk I3(tk, τk)



. (7.18)
118

7.3.3 Motion-echo sequences

We restrict ourselves to sequences with N steps acting on an initial ground state,

α0 = 0, where the sum over all unperturbed displacements of a sequence is αN = 0. In

this way, the actual final state is equal to |ΔαN and directly reflects the effects of non-zero

detuning. Moreover, we can construct the displacements in such a way, that a constant

detuning δ = 0, results in ΔαN = 0. This mimics the feature of spin-echo sequences that

small constant detunings have no effect on their final spin state. The motion-echo sequences

preserve this feature, if the linear approximation is valid, even when taking the effect of

the detuning of the displacement operations into account. For constant δ, the integrals

I1(tk, τk) = δτk and I3(tk, τk)=1/2 δτ 2

k are independent of tk and the total displacement

simplifies to

ΔαN = iδ	

N

k=1

τk(αk−1 + i/2 Ωz,ke−iφk τk). (7.19)

For the motion-echo sequences, Ωz,k = Ωz is the same for all displacements and the coherent

state parameter before each of the Dˆ(±Ωzτd) operations is ∓Ωzτd/2. In this case, the second

contribution in the (...) braces is ±1/2Ωzτd, equal and opposite to the initial state parameter,

so all displacement terms in the sum Eq. (7.19) are equal to zero individually, except for the

first and last displacement which is Dˆ(Ωzτd/2). However, since α0 = 0, these terms sum to

iδτd/2(1/2Ωzτd/2 − Ωzτd/2+1/2Ωzτd/2) = 0, which leaves only the free-precession terms to

be considered. All sequences contain an even number 2na of free-precession periods (na > 0,

integer), with half of them contributing iδτaΩzτd/2 each and the other half −iδτaΩzτd/2, so,

as previously noted (see Fig. 7.1) these terms also sum to zero and ΔαN = 0 for a constant

detuning.
119

7.3.4 Response to a monochromatic modulation

Next, we can determine the response to a monochromatic modulation at frequency ωn

of the form δn(t) = An cos(ωnt + φn). On the one hand, the harmonic oscillator frequency

can be deliberately modulated in this way, which enables us to compare the response of

the motion-echo sequence to the theoretical expectation. On the other hand, some of the

frequency noise acting on the oscillator can be characterized as a noise spectrum consisting of

a sum of such modulation terms with distinct frequencies ωn,j , generally varying amplitudes

An,j and random phases φn,j . In addition, the harmonic oscillator may be affected by noise

with a continuous spectrum, but we will restrict ourselves to discrete, narrow-band noise

spurs here. The noise spectrum can be characterized with motion-echo sequences, if the

response to a monochromatic modulation at ωn allows for determination of that frequency

within a band that depends on the resolution of the sequence. The amplitude of the response

ΔαN is proportional to the noise amplitude and is zero when averaged over the random noise

phase φn (denoted by ...), but because the final occupation  ̄nfin is proportional to |αN |

2,

after integrating over φn, we get an average final occupation

n ̄fin= 1

2π

 2π

0

|ΔαN |

2

dφn. (7.20)

This is proportional to the noise power (proportional to A2

n) inside the filter bandwidth of

the motion-echo.

For the monochromatic modulation, the integrals I1 and I3 have analytic solutions:

I1(t0, τ ) = An

ωn

[sin(ωn(t0 + τ ) + φn) − sin(ωnt0 + φn)]

I3(t0, τ ) = An

ω2

n

[cos(ωnt0 + φn) − cos(ωn(t0 + τ ) + φn) −

−ωnτ sin(ωnt0 + φn)]. (7.21)

The integrals depend on t0 and τ , therefore the sum over a motion-echo sequence is nonzero in general. Inserting the integrals into Eq. (7.17) and summing over the motion-
120

echo sequences is tedious but straightforward, and yields closed expressions for the final

displacement ΔαN and the corresponding average occupation number of the motion  ̄nfin =

|ΔαN |

2. Taking the average over the random phase φn yields

n ̄fin= 8A2

nΩ2z

ω4

n

sin2[ωnτd/4]{cos[ωnτd/4] − cos[ωn(τa + 3τd/4)]}2 ×

×

sin2[naωn(τd + τa)]

sin2[ωn(τd + τa)] . (7.22)

If the free-evolution time τa is varied in the motion-echo sequence, the expression in the

upper line produces an envelope that is oscillating at frequency ωn with phase shifts proportional to τd.

1 The first main peak appears when ωn(τd + τa) 	 π and the spacing

between adjacent main peaks is exactly Δτa = 2π/ωn, which allows for determination of ωn

from this interference pattern. The width of the narrow main peaks can be characterized

by the distance δτa of the two zeros of the response closest to a peak, which are spaced by

δτa = 2π/(naωn). It is possible to resolve a pair of main peaks produced by modulations at

ωn and ωn + δωn respectively, as separate maxima if |δωn| ≥ π/[na(τa + τd)]. If a continuous

noise power spectral density a2

n(ωn) is sampled in this way, δωn determines the bandwidth

of the sample filter that relates the noise power density to the actual noise power detected

in this band.

To have n ̄finapproximately represented by the ion state population Pπ

↑ , the average mode

occupation should be kept below n ̄fin≤ 1, which is possible by reasonable choices for the

displacement Ωzτd and the number of free-precession periods 2na. Choosing either the size

of the displacement or the number of displacements to be too large has the same effect as

over-driving the mixer in an electronic spectrum analyzer, which leads to a response that is

not linear in the input signal, resulting in a distorted output.

1 The expression in the lower line is equivalent to the intensity far-field pattern of a transmission grating

with na slits [Born and Wolf, 1980] and describes a na-times sharper response that produces a more narrowly

peaked interference pattern with nearly symmetric side lobes.
121

7.4 Experimental implementation and results

7.4.1 Experimental setup

Experiments are performed with the linear surface-electrode trap described in Sec.

4.1.1. The coherent displacements are performed on the axial mode, with frequency ωz 	

2π × 8 MHz. Direct “carrier”-transitions between the states |↓|nand |↑|nare driven by

microwave fields induced by a ω0 	 2π×1.281 GHz current through electrode DC1 (Fig. 4.2).

The ion is prepared in |↓|0with a fidelity exceeding 0.99 by Doppler laser cooling, followed

by ground-state cooling and optical pumping. Sideband transitions |↓|n↔ |↑|n ± 1are

implemented with stimulated Raman transitions driven by two counter-propagating laser

beams. This allows us to prepare nearly pure number states of the motion as described in

more detail in [McCormick et al., 2019a] and Ch. 5. We implement the tickle as described

in Sec. 4.2.8 and distinguish measurements of the |↑and |↓states with state-dependent

fluorescence (see Sec. 4.8.4).

In the experiments detailed below, the signal indicates the deviation of the final motional

state from |n = 0. Population in the ground state of motion is discriminated from that in

excited states of motion by performing the RSB pulse theoretically described in Sec. 4.8.5

and Eq. (7.12), connecting population in |↓|n > 0to |↑|n − 1while leaving population

in |↓|n = 0unchanged. For average excitation  ̄n ≤ 1/2 the probability Pπ

↑ of changing

the internal state is approximately equal to  ̄n. A subsequent microwave carrier π-pulse exchanges population in |↓and |↑followed by detection. The |↑state has a low average

count rate, which minimizes shot noise in the photomultiplier signal. This is helpful when

determining small deviations from |n = 0with high signal-to-noise ratio.
122

7.4.2 Displaced number states

As briefly described in [Home et al., 2011], tickling an ion that has been cooled to near

the motional ground state to determine the ion oscillation frequency is a practical calibration

tool. The experiment is performed as follows: the ion is cooled to near the ground state

and prepared in |↓, then a tickle tone with a fixed amplitude and detuning δ is applied for

a fixed duration τd = 13 μs. The resulting coherent state is characterized by applying a

RSB π-pulse for the |↓|1→ |↑|0transition followed by a microwave carrier π-pulse, then

detection of the internal state via state-selective fluorescence as described above.

The symbols with error bars (1-σ statistical error, from shot noise in the photo-multiplier

counts averaged over 600 experiments per detuning value) in Fig. 7.2 show the measured

Pπ

↓ as a function of tickle detuning for low on-resonance occupation ( ̄n = 0.61(1)). The line

shape is well described by Eqs. (7.7) and (7.12). The solid line is a fit to these equations

with  ̄n = 0.61(1) as the only free parameter after subtracting an offset of 0.05(1) due to stray

light background and imperfect ground state cooling that was determined independently. In

this case, Pπ

↓ is roughly linear in  ̄n and reflects the sinc2-shape of the Fourier transform of

the square-envelope tickle pulse. Keeping the excitation small gives us the practical advantage that only one prominent peak in Pπ

↓ versus detuning exists, making fitting to find the

resonance frequency straightforward. However, the precision with which we can determine

the frequency of oscillation is Fourier-limited by the pulse duration.

With the development of a theoretical understanding of line shapes for larger displacements,

where Pπ

↓ is non-linear in  ̄n, we have found that we can determine the resonant frequency

with a precision that increases approximately linearly with the size of the excitation |α| in

the range of 0 < |α| < 17, which implies that we can improve on the Fourier limit of the
123

-2 -1 1 2

0.1

0.2

0.3

0.4

0.5

detuning 

Figure 7.2: Spin-flip probability Pπ

↓ (See Sec. 4.2) of ion after 13 μs tickle excitation on |↓

versus detuning from ion oscillation frequency. The average occupation  ̄n of the ion motion

in response to tickle excitation is mapped onto the spin state by applying a RSB pulse,

which connects levels |↓|nto |↑|n − 1for n > 0, while leaving population in |↓|n = 0

unchanged. A subsequent microwave carrier π-pulse exchanges populations in |↑and |↓to

reduce measurement projection noise. The solid line is a fit using Eq. (7.7), free parameter  ̄n,

and an experimentally determined vertical offset of 0.05(1) added to account for background

counts and imperfect ground state cooling. The fit yields an on resonance average occupation

of  ̄n = 0.61(1).
124

tickle pulse. We expect that the linear trend continues beyond |α| = 17, but anharmonicity

and trap stability will limit the useful range of |α|.

We have measured Pπ

↓ (given by Eq. 7.12 following the microwave carrier π-pulse to exchange population between |↓and |↑) versus detuning of the tickle frequency for various

displacement amplitudes up to |α| ≈ 17, corresponding to a coherent state with an average

occupation of  ̄n ≈ 300. Fig. 7.3 shows four such cases with  ̄n of 3.22(3) (Fig. 7.3a), 10.4(1)

(Fig. 7.3b), 98.4(7) (Fig. 7.3c) and 299(1) (Fig. 7.3d). The lines are fits with free parameters Ωz and harmonic oscillator resonance frequency ω0. An experimentally determined

vertical offset of 0.05(1) is added to the fit function to account for background counts, as in

the evaluation of the data presented in Fig. 7.2.

The steep slopes of some of the line-shape features imply a stronger response to small

changes in the detuning, as compared to cases where  ̄n ≤ 1 (dashed lines in Fig. 7.3). Moreover, the response is symmetric around δ = 0, so these steeper slopes can be exploited to

find δ = 0 without a detailed understanding of the line shapes beyond this symmetry. This

enables line-center determination with a signal-to-noise ratio beyond the Fourier limit of the

linear response ( ̄n ≤ 1).

To validate the generalization of our theory to displaced number states with quantum numbers n ≥ 1 (see Eq. 7.9), we produce number states up to n = 6, as described in Ch. 5

and [McCormick et al., 2019a] and scan the amplitude of the coherent displacement while the

tickle frequency is resonant with the ion’s oscillation (see also [Ziesel et al., 2013] for earlier

experiments on displaced number states with n up to 3). As the amplitude of the coherent

state increases, the Rabi frequency of the RSB interaction varies with n as predicted by Eqs.

(7.10) and (7.12), producing the non-monotonic response of the ion’s fluorescence shown in

Fig. 7.4 together with fits based on Eq. (7.12), with the tickle strength Ωz and contrast of
125

detuning

- 3 - 2 - 1 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

n  3.2

- 3 - 2 - 1 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

n  10

- 1.5 - 1.0 - 0.5 0.5 1.0 1.5

0.2

0.4

0.6

0.8

n  98

- 0.5 0.5

0.2

0.4

0.6

0.8

1.0

n  300

a)

c) d)

b)

Figure 7.3: Response of ion to tickle excitation versus detuning from the ion oscillation

frequency for maximal excitations on resonance of a)  ̄n = 3.22(3), b)  ̄n = 10.4(1), c)  ̄n =

98.4(7) and d)  ̄n = 299(1). As the average occupation of the coherent state increases, the

line shape becomes more sharply featured. Solid lines are fits with free parameters Ωz and

ωz. We determine the maximum excitation  ̄n of each experiment from the fitted values of

Ωz. Dashed lines represent a Fourier-limited response resulting from a weaker excitation

( ̄n = 0.64) for comparison.
126

the final state readout as free parameters. We perform this experiment by preparing the ion

in pure number states n = 0 (Fig. 7.4 a)), n = 2 (Fig. 7.4 b)), n = 4 (Fig. 7.4 c)) and n = 6

((Fig. 7.4 d)) and applying a resonant tickle tone with fixed Ωz for durations ranging from

0.05 to 12 μs, resulting in coherent displacements up to |αd| ≈ 20. Displaced number states

are useful to quantum-enhanced sensing of forces with high number states [Wolf et al., 2018].

7.4.3 Motion-echo experiments

We first perform the motion-echo experiments with a 10 kHz or 100 kHz tone applied

to one of the trap electrodes. The tone modulates the potential curvature of the trap at the

position of the ion and therefore the ion’s oscillation frequency. The purpose of this is twofold: To experimentally characterize the response of the ion and compare the results against

theory for a known perturbation and to explore the range of noise that is detectable with

this method in our setup. With the tone applied continuously and the phase φn changing

randomly from experiment to experiment (to mimic the uncontrolled phase of noise), we

perform a series of motion-echo sequences with various numbers of free-precession periods

2na, scanning the wait time τa. We find that Pπ

↓ depends on the relationship between τa

and the frequency of the applied tone in the expected manner according to Eq. (7.22). This

can be seen by comparing the measured points in Fig. 7.5 to the solid lines that show fits

with An as a free parameter. All fitted values of An are consistent with each other to within

2 times the standard deviation (see caption of Fig. 7.5) and indicate a relative modulation

depth of An/ω0 	 1.4 × 10−4. Similar experiments were performed with applied tones from

500 Hz to 400 kHz and while the results qualitatively agreed with theory, attenuation and

distortion of the tones through various filters with uncharacterized parasitic capacitance and

resistive loss at 4 K in our experimental setup prevented us from comparing quantitatively

to the theory.
127

displacement amplitude

a)

c) d)

b)

2 4 6 8 10 12

0.2

0.4

0.6

0.8

2 4 6 8 10 12

0.2

0.4

0.6

0.8

2 4 6 8 10 12

0.2

0.4

0.6

0.8

2 4 6 8 10 12

0.2

0.4

0.6

0.8

Figure 7.4: Response of ion to tickle excitation versus coherent drive duration for initial

number states a) n = 0, b) n = 2, c) n = 4 and d) n = 6. As the displacement of the

ion’s motion increases, the Rabi frequency of the RSB interaction varies non-monotonically.

Solid lines are produced with theory in Eq. (7.12) using fit parameters of the tickle strength

Ωz and contrast of the final state readout, and an experimentally determined vertical offset

of 0.05(1) to account for background counts and imperfect initial state preparation. Fitted

values of Ωz and the contrast of the final state are, respectively, a) 2π× 0.2300(2) MHz and

0.884(4), b) 2π× 0.2453(3) MHz and 0.899(5), c) 2π× 0.2516(6) MHz and 0.886(6) and d)

2π× 0.2558(8) MHz and 0.894(5).
128

Finally, we perform motion-echo experiments without a purposely applied tone, to sense

and characterize intrinsic frequency noise in our setup. With na = 10 (20 free-precession periods), τd = 4 μs, and coherent displacements Ωzτd/2=3.44(2), we find several peaks in the

time scan corresponding to a single narrow-band noise spur at ωn 	 2π× 260 kHz having an

amplitude An = 2π × 2.4(2) kHz (see Fig. 7.6a), which corresponds to a relative modulation

depth An/ω0 	 3 × 10−4. In this run, the electrode potentials are supplied from digital to

analog converters (DACs), so the spurs are likely caused by cross-talk of digital circuitry in

the DACs to the outputs. After switching the electrode potential sources to low-noise analog power supplies, we observed a nearly uniform noise floor without the spurs (see Fig. 7.6b).

In principle the sensitivity can be increased arbitrarily by choosing a larger value of the

coherent drive strength (which is proportional to Ωz, see Eq. (7.22)). However, there will

be technical limitations that will depend largely on the setup. If noise on the coherent drive

begins to dominate the outcome of an experiment, then the harmonic oscillator noise can

no longer be accurately deduced. Even without any noise in the harmonic oscillator or the

drive, anharmonicities will eventually deform the wave-packet and reduce the overlap of the

final state with the ground state.

7.5 Discussion

Coherent displacements are a universal concept that applies to all harmonic oscillators

and, because they correspond to a classical, near-resonant force on the oscillator, they can

often be implemented in a simple way in concrete experimental settings. Here, we characterized the responses of a single, harmonically trapped atomic ion to an electric field that

oscillates close to resonance with the ion’s mechanical motion. The ion response was then
129

10 kHz 100 kHz

2na = 8 2na = 12

2na = 6

2na = 4

2na = 2 2na = 4

2na = 6

2na = 8







 





 















































 





  





 





 







 

 







 































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0.06

0.10

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0 5 10 15 20

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0 5 10 15 20

0.10

0.20

0.30

arm time/s

Figure 7.5: Characterization of motion-echo experiments with tones of 10 kHz and 100 kHz

for various numbers of free-precession periods 2na (see top of plots). Lines are fits based on

Eqs. (7.22) and (7.12) with free parameters An. For the experiments with the 10 kHz tone

applied, fitted values of An were 2π× 1.3(4) kHz, 1.5(1) kHz, 1.4(1) kHz and 1.2(1) kHz for

2na = 2, 4, 6 and 8, respectively. Likewise for the 100 kHz tone, fitted values of An were

2π× 1.0(4) kHz, 1.1(3) kHz, 1.1(2) kHz and 1.1(1) kHz for 2na = 4, 6, 8 and 12, respectively.
130





















































































 





































































































 







































0 5 10 15 20

0.05

0.10

0.15

0.20

a)

b)































































































































































































































































0 5 10 15 20

0.05

0.10

0.15

0.20

Figure 7.6: Noise sensing with motion-echo experiments. The two sub-plots compare noise

on electrode potentials delivered from two different sources: a) Digital-to-Analog converters

(DACs) and b) linear bench power supply. Experiments were performed with na = 10 (20

free-precession periods) and coherent displacements of Ωzτd/2=3.44(2). The solid black

line in a) is a fit to the data taken using the DACs based on Eq. (7.22) with free parameters

An and ωn, and a vertical offset of 0.07 to account for background counts and imperfect state

preparation. From this fit, we determine that the DACs introduce noise at ωn/(2π) 	 260

kHz with An = 2π × 2.4(2) kHz.
131

characterized by coupling the ion harmonic oscillator to an internal two-level system of the

ion.

By applying a sequence of coherent displacements, alternating with free evolution of the

state of motion, we obtained a frequency-filtered response of the harmonic oscillator. In this

way, fluctuations of the harmonic oscillator frequency in a certain frequency band can be

isolated and sensitively detected by the ion itself. This technique can be used as a complementary method in investigating the behavior of surface charges and answering questions

about the sources of anomalous heating.

We anticipate that the basic concepts exhibited in the experiments described in this chapter

can be transferred to other harmonic oscillator systems and foresee various extensions and

refinements of the current trapped-ion work. For example, it should be possible to find

interesting modifications of the coherent displacement sequences by utilizing different displacement patterns in phase space or by displacing non-classical quantum states [McCormick

et al., 2019a; Wolf et al., 2018].
Chapter 8

Future directions: Coupled harmonic oscillators and experiments with multiple

ions

One main motivation for studying the motion of a single trapped ion is for a better

understanding of limitations of motional coherence in applications to quantum information

processing. To this end, our next step is to apply the techniques discussed here with one ion

to systems with two or three. The triangle trap has now replaced the double-well trap in

our setup, which should allow us to trap three or more ions in separate, individually tunable

wells in the future. An interesting first set of experiments would involve manipulating the

coupling between the quantum motion in the different wells.

8.1 Previous attempt with triangle trap

Much of chapter 4 was spent describing updates to the apparatus in order to install the

new triangle trap. We actually had previously installed an earlier version of the triangle trap

in the setup for a brief period, had loaded ions in this trap, and were beginning preliminary

characterization experiments. We were able to load in all three sites as well as a spurious

trap in the center (see Fig. 8.1).
133

a) b)

Figure 8.1: Images of a) 3 and b) 4 ions loaded in the triangle trap. The ion in the center

in b) is loaded in a spurious trap that is located ∼10 μm out of the plane of the other ions,

which is why it is slightly defocused.
134

8.1.1 Challenges

A major problem in controlling the ions in this triangle trap was that the DC electrode voltages necessary to compensate micromotion were much too large, particularly in

the direction in the yi directions (defined in Fig. 8.2). This is mainly due to the electrode

geometry—there are very few electrodes near the ion that aren’t symmetric about xi, which

means they will produce fields mainly in the xi and zi directions, but not the yi directions

(see Fig. 8.3).

Shortly after this discovery, the RF-electrode in the trap developed a short to ground

and we were no longer able to trap ions. We suspect that a spike in the RF power shorted

the RF electrode by some breakthrough mechanism. After this event, the resistance of the

RF electrode to ground, which should nominally be infinite, was less than 10 Ω. Nevertheless, this short experience with the triangle trap inspired electrode geometry design changes

that we think will make the new triangle trap more successful.

8.2 Design changes

There were a few main design changes made to the next generation triangle traps.

First, the 20 μm triangle was abandoned—based on simulations, the ability to compensate

stray fields in this was going to be even worse than in the 30 μm triangle, which was already

marginal, given the upper limit of 20 V that can be applied to the DC electrodes. So, we now

have two 30 μm traps, each with a slightly altered electrode geometry. One has the same

number of electrodes as the previous design, but the shape of the electrodes underneath each

ion is altered to provide more field in the yi-direction for micromotion compensation (see Fig.

8.3). The alternative design mitigates this problem by changing the electrode geometry, and
135

rf

cu0

cu1

cu6

cl0

cl1 cu5 cu2

cl2 al1

cu3 al2 cu4

al0

bu4 au5 au6

au3

au4 bu3 au2

bu2 au1

bu5 bl2 bu1

bu6 au0

bl1

bu0

bl0

y

z

^x ^

^

c c

c

y

z x

^

^a ^

a

a

y z

x

^

^

^

b

b b

Figure 8.2: DC and RF electrodes of triangle trap used in previous experiments. Note the

lack of electrodes near the trapping sites (black dots) which would produce a field in the yi

directions. This was intended to reduce the number of gaps between electrodes close to the

ions but ended up as a major design flaw that hindered our ability to cancel stray fields in

this direction.
136

a) c) y b)

z x

^

^ ^

y

z x

^

^a ^

a

a a

a

a

cu4 al2

au5 au6

au3

au4 au2

au1

al1

Figure 8.3: Triangle trap electrode redesigns. a) Original electrode geometry, zoomed in on

one trapping site, b) new electrode geometry to provide more control over larger stray fields

in the y-direction, c) second triangle electrode geometry, with more electrodes to control up

to four ions, including one in the central spurious trap. Ion locations are represented by

black dots.

also adds more electrodes to the center to allow us to tune the trapping parameters of the

center spurious well, effectively becoming a “diamond” trap with four tunable trapping sites

(see Fig. 8.3 c). Finally, a microwave antenna that provides a magnetic field with the proper

polarization to drive the microwave transitions between hyperfine states (described in Sec.

4.2.7) is integrated on the chip. This allows the ion to experience higher microwave fields

and gradients, since it is much closer to the ions than the previous triangle trap microwave

line, which was ∼ 3 mm above from the ions on a Faraday cage that surrounds the trap.

8.3 Possible experiments

A natural starting point for experiments in the triangle trap would be to perform extensions of the work described here for multiple ions. With the use of non-classical states

of motion extended to multiple ion sites, we could introduce new degrees of freedom and

complexity, like phonon tunneling and effective beamsplitter operations. We could think

about doing Sagnac interferometry with the phonons traveling around the triangle as an
137

alternative to the proposed matter-wave trapped-ion Sagnac interferometer [Campbell and

Hamilton, 2017].

We could also begin to explore certain bosonic many-body models and lattice gauge theories, with the phonons acting as the bosonic quasiparticles of interest. The dynamics of the

phonons and interactions between neighboring sites could be controlled either by tuning the

relative oscillation frequencies of the sites, or using photon-assisted tunneling (PAT), where

the energy gap between neighboring sites is provided by a photon from a strong microwave

or RF drive [Bermudez et al., 2012]. PAT also introduces a new degree of freedom: the

phase of the driving field is imprinted on the tunneled phonon wavefunction, which allows

for interesting and controllable phase differences of the phonons as they traverse the triangle

with different path orientations, which mimic the phases encountered when encircling field

lines that are crucial for simulating lattice gauge theories.
Chapter 9

Conclusion

By exploring the boundaries of quantum mechanics, we can refine our notions of this

strange, often counter-intuitive theory, devise new tests of its validity, and facilitate advancements in quantum metrology and quantum information processing. The harmonic oscillator,

an elementary and ubiquitous physical system with a simple correspondence between classical and quantum descriptions, is an ideal platform for such experiments. In this thesis, I

have presented a series of experiments that extend quantum control of a harmonic oscillator

over prior art. I used this control to determine the oscillation frequency with sensitivity

better than the standard quantum limit, approaching the Heisenberg limit. I also presented

experiments based on coherent displacements of quantum states, using these displacements

to devise a spectrum analyzer for sensitive detection of mode-frequency noise over a wide

range of frequencies. I discussed potential extensions to all of these experiments, including

applying these techniques to multiple ions, for example in the new triangle trap. These

methods highlight a growing level of quantum control in trapped-ion systems, which could

benefit quantum metrology and quantum information processing and open new opportunities

for quantum simulations with ions and phonons as well as mesoscopic mechanical oscillators.
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Appendix A

Precision limit of “classical” measurements on “classical” states

There is fairly general agreement in the literature that any “classical” limit to measurement precision on the frequency of a harmonic oscillator will scale as proportional to

√n ̄, where  ̄n is the average occupation number [Braun et al., 2018]. Despite this, there is

no general agreement on the pre-factor to that scaling, which is necessary to fully define

a “standard quantum limit.” [Braun et al., 2018] The limit we establish here assumes an

observer with classical resources (defined below) that are perfectly implemented.

For comparisons of our experiments to a well-defined classical reference experiment, we

extend the notions of classical experiments with light fields given by Roy Glauber [Glauber,

2006] to a general harmonic oscillator. For light fields, Glauber restricted classical sources

to coherent light. In this spirit, we broaden the possible physical implementations to any

system that can be described as a harmonic oscillator, but limit the admissible operations to

coherent displacements. Harmonic oscillator quantum observables are expressed with ladder

operators ˆa and ˆa† and a coherent state is an eigenstate of the annihilation operator with

aˆ|α= α|α. The oscillator’s average number of quanta is then given by the expectation

value  ̄n of the number operator  ̄n = α|aˆ†

aˆ|α= |α|

2. Glauber restricted classical measurements to measuring intensities for light fields (for example the intensities arising on a screen

due to the interference of the two light fields from a double slit arrangement). We generalize

light field intensity to number expectation values  ̄n as the permitted classical measurements
148

in a harmonic oscillator. The attainable signal-to-noise of such measurements will be limited by shot-noise, given by the standard deviation of a Poisson distribution with mean  ̄n,

ΔPn ̄ = √n ̄, for an ideal measurement on a coherent state with unit quantum efficiency and

no excess noise. To compare to number-state superpositions of the form |Ψn= |0+|n, we

require that the classical interferometer uses no more energy than the competing numberstate interferometer,  ̄n = Ψn|aˆ†

aˆ|Ψn≤ n. This definition of equal resources is somewhat

arbitrary, for example one could also argue for the same maximal energy. However, the

coherent states have no well defined maximal energy and it is always possible to rescale from

our definition to another definition of equal resources. The scaling factor would likely be of

order unity in most cases and irrespective of its value, the ideal non-classical interferometer will eventually outperform the classical counterpart due to its more favorable scaling in  ̄n.

With the definitions above, we can devise an interferometer experiment that has the salient

features of a Ramsey experiment, but is based on classical states and measurements. A

Ramsey experiment consists of two excitations with known relative phase, separated by

“free precession” for duration T. An example of a classical-like Ramsey experiment could

be to send a short rf-pulse from a reference oscillator with known frequency and phase into

a near-resonant circuit. The pulse will ring up the circuit which then evolves freely for T.

A second pulse is the sent to the circuit and depending on its phase relative to the first

pulse and the phase the excitation has picked up in the resonant circuit during T, the two

pulses interfere constructively to further build up the field in the circuit or interfere destructively, diminishing the circuit excitation. For a general harmonic oscillator, this can

conveniently be described in phase-space in a frame oscillating at the frequency of the reference oscillator. Starting in the ground state, the first excitation creates a coherent state

|α1, where we can choose phase space coordinates that make α1 ≥ 0 real without losing generality. During a free-precession time T the state picks up a phase φT which transforms it to

|α1T = |α1 exp(iφT ). In analogy to a Ramsey experiment, the second excitation is chosen
149

to have the same magnitude with phase φ relative to the first excitation, α2 = α1 exp(iφ).

This transforms the state (up to a global phase that is of no consequence to this experiment)

into |α= |α1[1+ exp(i(φT −φ))]. Following Glauber’s definition of classical measurements,

we measure the average occupation  ̄n:

n ̄ = 4α2

1




1

2 +

1

2 cos(φ − φT )



(A.1)

For a fair comparison to a number-state superposition |0+|n, we would like to restrict our

resources to  ̄n ≤ n for all possible φ and φT , so we choose the maximum value 4α2

1 = n. Under

this restriction, we recover a classical version of the Ramsey fringes described by Eq. (6.2),

where we measure  ̄n = 0 when φ − φT = (2m + 1)π and  ̄n = n when φ − φT = 2mπ, with

m integer. We want to minimize the noise-to-signal ratio Eq.(6.3) restricted to coherent

excitations and a small free-precession phase |φT |	 0  π. The noise-to-signal ratio to

minimize is

δφc = ΔPn ̄

|∂Pn ̄/∂φT |











φT =0

=

n(1/2+1/2 cos(φ))

n/2 sin(φ) =

1

n

1+1/ cos2(φ)

2 . (A.2)

The minimum occurs when φ = π, which describes two equal and opposite displacements

that put the harmonic oscillator back to the vacuum state if the free-precession phase φT = 0.

Noise and signal both vanish for this interferometer in such a way that their ratio stays finite

at δφc = 1/

√n.

This is the best one could possibly do in the case of no excess noise in the system. In

practice, when some added noise is present, we would choose a value of φ that maximizes the

signal, as we did for the case of the number state interferometer. In this case, as was also

true for the number state interferometer, the point of maximal signal occurs for φ = π/2, so

the practical noise-to-signal-ratio yields:

δφp =

n(1/2+1/2 cos(φ))

n/2 sin(φ)











φ=π/2

=

2

n

. (A.3)
150

This interferometer has no practical drawbacks, optimally suppresses additional noise and

leads to an increase of the noise-to-signal over the optimal value by √2. This implies that

the advantage of a number-state interferometer may increase by √2 over the ideal limit in

a realistic setting, where excess noise (such as current noise or dark counts of a detector)

is almost inevitable. This also is arguably a more direct comparison to the |0+ |ninterferometer experiment, where we choose the same relative phase between the two pulses of

φ = π/2 to maximize the signal. Despite the case to be made for using the more practical

limit in our comparisons, we strictly use the more stringent ideal classical interferometer in

all our sensitivity comparisons of the number-state interferometers in the main text.

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