The Sound of Ions: Using Trapped Atomic Ion Motion for Quantum Computation
and Sensing
by
JEREMY M. METZNER
A dissertation accepted and approved in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in Physics
Dissertation Committee:
Daniel Steck Chair
David Allcock Advisor & Core member
Tien-Tien Yu Core Member
George Nazin Institutional Representative
University of Oregon
Spring 2024
© 2024 Jeremy M. Metzner
All rights reserved.
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DISSERTATION ABSTRACT
Jeremy M. Metzner
Doctor of Philosophy in Physics
Title: The Sound of Ions: Using Trapped Atomic Ion Motion for Quantum
Computation and Sensing
Encoding qubits in the electronic states of atoms has enabled the ability to
perform computations, sense the environment, and gain a deeper understanding of
other physical systems through quantum simulation. In the trapped ion platform,
each of these applications is made possible, or enhanced by coupling the qubit to
the motion of the trapped ions. The motion can be used to do more than just
mediate interactions with qubits and is itself a resource for computation, sensing
and simulation. The work presented here focuses on methods for manipulation
and entanglement of trapped ion motional states in a spin-independent way while
retaining the spin to enable measurement of the quantum state of the motion.
We have shown the ability to use a set of spin-independent operations including
displacement, beam splitter, squeezing and two-mode squeezing, to sense the phase
of the oscillator states with precision exceeding the standard quantum limit (SQL)
by up to 5.9 dB and approaching the ultimate Cramér-Rao bound. With qubits
still generally required for utilizing the motional states for quantum information
experiments, qubit measurement makes it difficult to preserve any motional state
during state detection. We have developed the technique of encoding both states in
a metastable manifold which could enable methods to preserve particular motional
states of longer-chains during state detection. We present preliminary results
demonstrating this preservation of motional states using a mixed atomic-species
trap at Lincoln Lab. The metastable encoding also provides the unique ability to
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engineer and explore non-Hermitian qubit Hamiltonians, where we have shown the
ability to break the quantum speed limit. This dissertation includes co-authored
material that is pending publication.
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ACKNOWLEDGEMENTS
I would like to start by thanking my advisor David Allcock for allowing
me to be a part of growing a new lab, for sharing his organizational skills and his
great depth of knowledge of the trapped ion field. I really appreciate all the time
and effort spent reviewing my work and making sure it was presentable. I also
thank Dave Wineland for being a co-advisor, always providing feedback on work
and always taking an interest what was going on in the lab. A special thanks to
my starting lab mates Alex Quinn and Daniel Moore for sharing the load of all
of those packages and always providing stimulating conversations, especially in
the early days trapped in B61 together. I appreciate all of the members of the
lab that I have worked with, Sean Brudney, Gabe Gregory, Evan Ritchie, Oliver
Miller and Vikram Sandhu for being a part of the process and always being an ear
for my complaints. Thanks also to outside collaborators Yogesh Joglekar, Shaun
Burd and Colin Bruzewicz for sharing their knowledge. Finally I want to thank
my family and friends. I want to thank Claire Albrecht and Matt Ball for helping
me to survive my first year of classes, being my community in the department and
always keeping me from getting ahead of myself. A special shoutout to my wife
Sahlea Tubbeh, who wasn’t my wife the whole time I was working on my PhD, but
was my rock and source of happiness the whole time. Thanks to my parents and
in-laws for always taking the time to come visit and share in the magic of Oregon
and for always making an effort to understand what I work on.
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To my wife Sahlea. . .
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TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1. Trapped ion harmonic motion . . . . . . . . . . . . . . . . . . 16
1.2. Harmonic oscillators and qubits . . . . . . . . . . . . . . . . . 17
1.3. Effects of qubit measurements . . . . . . . . . . . . . . . . . . 18
1.4. Interferometry with trapped ions . . . . . . . . . . . . . . . . 20
1.5. Thesis layout . . . . . . . . . . . . . . . . . . . . . . . . . 22
II. QUANTUM HARMONIC OSCILLATORS . . . . . . . . . . . . . 24
2.1. Fock basis ladder . . . . . . . . . . . . . . . . . . . . . . . 24
2.2. Wigner function representation . . . . . . . . . . . . . . . . . 27
2.3. Oscillator modes with ion chains . . . . . . . . . . . . . . . . 28
2.3.1. Motional modes . . . . . . . . . . . . . . . . . . . . . 29
2.3.2. Mixed mass crystals . . . . . . . . . . . . . . . . . . . 30
2.4. Driven harmonic oscillators and parametric modulation . . . . . . 32
2.4.1. Multi-mode operators . . . . . . . . . . . . . . . . . . 34
2.5. Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.1. Displacement and coherent states . . . . . . . . . . . . . 37
2.5.2. Squeezing and single mode squeezed states . . . . . . . . . 38
2.5.3. Phase operator . . . . . . . . . . . . . . . . . . . . . 40
2.5.4. Beam splitter . . . . . . . . . . . . . . . . . . . . . . 42
2.5.5. Two-mode squeezing and two mode squeezed vacuum . . . . 44
2.5.6. Thermal states . . . . . . . . . . . . . . . . . . . . . 46
2.6. Non-Gaussian states . . . . . . . . . . . . . . . . . . . . . . 47
7
Chapter Page
2.6.1. Fock states . . . . . . . . . . . . . . . . . . . . . . . 48
2.6.2. Cross-Kerr interaction . . . . . . . . . . . . . . . . . . 48
2.6.3. Tri-squeezing . . . . . . . . . . . . . . . . . . . . . . 53
III. ATOMIC SPIN QUBITS IN OSCILLATORS . . . . . . . . . . . . 55
3.1. Calcium-40 . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2. Qubit encodings . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1. Ground-state qubits . . . . . . . . . . . . . . . . . . . 56
3.2.2. Optical qubits . . . . . . . . . . . . . . . . . . . . . 57
3.2.3. Metastable qubits . . . . . . . . . . . . . . . . . . . . 58
3.3. Zeeman qubits in the D5/2 Manifold . . . . . . . . . . . . . . . 58
3.4. State preparation . . . . . . . . . . . . . . . . . . . . . . . 59
3.5. Single qubit rotations . . . . . . . . . . . . . . . . . . . . . 61
3.5.1. RF magnetic fields . . . . . . . . . . . . . . . . . . . 61
3.5.2. Raman transitions . . . . . . . . . . . . . . . . . . . 63
3.6. Spin-motion coupling . . . . . . . . . . . . . . . . . . . . . 65
3.6.1. Lamb-Dicke regime . . . . . . . . . . . . . . . . . . . 65
3.6.2. Carrier . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6.3. Red sideband (RSB) . . . . . . . . . . . . . . . . . . 66
3.6.4. Blue sideband (BSB) . . . . . . . . . . . . . . . . . . 67
3.7. State detection . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7.1. Mode thermalization . . . . . . . . . . . . . . . . . . 69
IV. EXPERIMENTAL SETUP . . . . . . . . . . . . . . . . . . . . 71
4.1. Rod trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2. RF electronics . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1. Helical resonator . . . . . . . . . . . . . . . . . . . . 74
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Chapter Page
4.2.2. Squareatron amplitude stabilization . . . . . . . . . . . . 75
4.2.3. Filtering AM Noise (Squareatron 5000) . . . . . . . . . . 76
4.2.4. Tuning board . . . . . . . . . . . . . . . . . . . . . . 76
4.2.5. Amplifier . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.6. Temperature stabilization . . . . . . . . . . . . . . . . 78
4.3. Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1. Laser rack . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.2. Photoionization . . . . . . . . . . . . . . . . . . . . . 82
4.3.3. Locking lasers to a wavemeter . . . . . . . . . . . . . . 85
4.3.4. Raman laser system . . . . . . . . . . . . . . . . . . . 88
4.4. Motional state cooling . . . . . . . . . . . . . . . . . . . . . 88
4.4.1. Doppler cooling . . . . . . . . . . . . . . . . . . . . . 89
4.4.2. EIT cooling . . . . . . . . . . . . . . . . . . . . . . 91
4.4.3. Resolved sideband cooling . . . . . . . . . . . . . . . . 94
4.5. Electronic drives . . . . . . . . . . . . . . . . . . . . . . . 96
4.5.1. Resonant drive . . . . . . . . . . . . . . . . . . . . . 96
4.5.2. Parametric drive resonator circuit . . . . . . . . . . . . . 96
4.5.3. Drive filtering . . . . . . . . . . . . . . . . . . . . . 97
4.5.4. Pulse shaping . . . . . . . . . . . . . . . . . . . . . 97
V. NON-HERMITIAN HAMILTONIANS . . . . . . . . . . . . . . . 99
5.1. Unitary dynamics and quantum speed limit . . . . . . . . . . . . 99
5.2. Lindblad model and master equation solver . . . . . . . . . . . . 100
5.3. Post-selection and non-Hermitian Hamiltonians . . . . . . . . . . 102
5.4. Finding the exceptional point . . . . . . . . . . . . . . . . . . 106
5.5. Legget-garg inequality . . . . . . . . . . . . . . . . . . . . . 107
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Chapter Page
5.6. Violation of quantum speed limit . . . . . . . . . . . . . . . . 109
VI. LASER-FREE CONTROL OF TRAPPED-ION
QUANTUM HARMONIC OSCILLATORS . . . . . . . . . . . . . 114
6.1. State characterization . . . . . . . . . . . . . . . . . . . . . 114
6.2. Coherent states . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3. Thermal states . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4. Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . 117
6.5. Two-mode squeezed states . . . . . . . . . . . . . . . . . . . 118
6.6. Beam splitter . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.7. Off-resonant motional coupling . . . . . . . . . . . . . . . . . 120
VII. PROTECTED MODE INVESTIGATION FOR MID-
CIRCUIT MEASUREMENT . . . . . . . . . . . . . . . . . . . . 122
7.1. Mode structure and protected modes . . . . . . . . . . . . . . . 122
7.2. Surface electrode trap with mixed species ion chain . . . . . . . . 124
7.3. Heating rates . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.3.1. Sideband thermometry . . . . . . . . . . . . . . . . . 126
7.4. Motional Fock state Ramsey interferometry . . . . . . . . . . . . 130
7.5. Perturbations to protection . . . . . . . . . . . . . . . . . . . 131
7.5.1. Radiation pressure . . . . . . . . . . . . . . . . . . . 132
7.5.2. Stray electric fields . . . . . . . . . . . . . . . . . . . 133
VIII.SU(2) AND SU(1,1) INTERFEROMETRY FOR
PRECISION PHASE MEASUREMENT . . . . . . . . . . . . . . . 135
8.1. Verification of two-mode squeezed state with beam splitter . . . . . 138
8.2. Cramer-Rao bounds . . . . . . . . . . . . . . . . . . . . . . 139
8.3. Theoretical bounds . . . . . . . . . . . . . . . . . . . . . . 144
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Chapter Page
IX. CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . 149
9.1. Phase insensitive displacement amplification with two-
mode squeezed states . . . . . . . . . . . . . . . . . . . . . 149
9.1.1. Potential experimental implementation . . . . . . . . . . 151
9.2. Phase distributions . . . . . . . . . . . . . . . . . . . . . . 152
9.3. Correlation measurements below the CH bound . . . . . . . . . . 154
9.4. Continuous variable quantum information processing . . . . . . . . 155
APPENDIX: SQUAREATRON 5000 AND TUNING BOARD . . . . . . . 157
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 162
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LIST OF FIGURES
Figure Page
1.1. Qubit and motion visualization . . . . . . . . . . . . . . . . . . . 19
1.2. Interferometer geometries . . . . . . . . . . . . . . . . . . . . . 22
2.1. Mixed mass motional modes . . . . . . . . . . . . . . . . . . . . 32
2.2. single mode Gaussian operations . . . . . . . . . . . . . . . . . . 42
2.3. Action of a beamsplitter . . . . . . . . . . . . . . . . . . . . . . 44
2.4. Two mode squeezed state visualization . . . . . . . . . . . . . . . 46
2.5. Fock state Wigner functions . . . . . . . . . . . . . . . . . . . . 48
2.6. Kerr interaction resonances . . . . . . . . . . . . . . . . . . . . 51
2.7. Wigner functions of two modes after Kerr interaction . . . . . . . . . 53
2.8. Wigner function of tri-squeezed state . . . . . . . . . . . . . . . . 54
3.1. 40Ca+ level structure . . . . . . . . . . . . . . . . . . . . . . . 56
3.2. Zeeman m qubit . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3. Laser and RF m qubit operations . . . . . . . . . . . . . . . . . 61
3.4. Raman transition theory . . . . . . . . . . . . . . . . . . . . . 64
3.5. State detection count histogram . . . . . . . . . . . . . . . . . . 69
4.1. Render of our ion trap . . . . . . . . . . . . . . . . . . . . . . 72
4.2. Photograph of our trap . . . . . . . . . . . . . . . . . . . . . . 73
4.3. Squareatron AM noise measurement . . . . . . . . . . . . . . . . 77
4.4. RF chain temperature stabilization . . . . . . . . . . . . . . . . . 79
4.5. Photograph of laser rack . . . . . . . . . . . . . . . . . . . . . 81
4.6. Laser rack drawer . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7. Rendering of photoionization board . . . . . . . . . . . . . . . . . 84
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Figure Page
4.8. Image of the photoionization board . . . . . . . . . . . . . . . . . 84
4.9. OSA board image . . . . . . . . . . . . . . . . . . . . . . . . 87
4.10. Raman beam delivery optics . . . . . . . . . . . . . . . . . . . . 89
4.12. EIT schematic and results . . . . . . . . . . . . . . . . . . . . . 94
4.13. Parametric modulation setup . . . . . . . . . . . . . . . . . . . 98
5.1. non-Hermitian Hamiltonian in 40Ca+ . . . . . . . . . . . . . . . . 103
5.2. non-Hermitian and Lindblad model comparison . . . . . . . . . . . 105
5.3. K3 measurement results . . . . . . . . . . . . . . . . . . . . . . 108
5.4. Branching ratio effects on maximum K3 . . . . . . . . . . . . . . . 110
5.5. Breaking the quantum speed limit . . . . . . . . . . . . . . . . . 111
6.1. Displaced state qubit probabilities . . . . . . . . . . . . . . . . . 116
6.2. Squeezed state qubit probabilities . . . . . . . . . . . . . . . . . 118
6.3. Two mode squeezed state qubit probabilities . . . . . . . . . . . . . 119
6.4. Beamsplitter calibration . . . . . . . . . . . . . . . . . . . . . . 120
6.5. Off resonant motional effects on qubit probabilities . . . . . . . . . . 121
7.1. motional mode spectroscopy . . . . . . . . . . . . . . . . . . . . 124
7.2. Lincoln standard surface electrode trap . . . . . . . . . . . . . . . 125
7.3. Multi ion sideband probabilities . . . . . . . . . . . . . . . . . . 129
7.4. two ion motional interferometry pulse sequence . . . . . . . . . . . . 131
7.5. Motional interferometry phase fringes . . . . . . . . . . . . . . . . 132
8.1. Motional interferometer experimental setup . . . . . . . . . . . . . 137
8.2. Two mode squeezed state verification . . . . . . . . . . . . . . . . 139
8.3. Phase fringes of different motional state interferometers . . . . . . . . 141
8.4. Interferometer sensitivities . . . . . . . . . . . . . . . . . . . . . 147
9.1. Single mode displacement amplification . . . . . . . . . . . . . . . 151
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Figure Page
9.2. amplification phase sensitivity . . . . . . . . . . . . . . . . . . . 152
A.1. Squareatron 5000 electronic schematic . . . . . . . . . . . . . . . 158
A.2. Squareatron tuning board schematic . . . . . . . . . . . . . . . . 159
A.3. Squareatron and tuning board PCB render . . . . . . . . . . . . . 160
A.4. Squareatron and tuning board image . . . . . . . . . . . . . . . . 161
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LIST OF TABLES
Table Page
1. two ion sideband qubit probabilities . . . . . . . . . . . . . . . . . 127
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CHAPTER I
INTRODUCTION
1.1 Trapped ion harmonic motion
Understanding the nature of harmonic oscillators is an essential part of
understanding basic physics given that many physicals systems can be modeled
as harmonic oscillators (Cao & Voth, 1995) including the electromagnetic
field (Bennett, Barlow, & Beige, 2015). Systems near a potential energy minimum
can often be approximated by harmonic oscillations with an appropriate Taylor
expansion about that minimum. This means many physical systems can be
modeled as a harmonic oscillator. In the quantum regime, where we work
near minimum energy states the harmonic approximation works very well for
many systems and quantum harmonic oscillators are a part of many different
platforms (J. Aasi et al. (Ligo Scientific Collaboration), 2013; Oliver et al.,
2005; Regal, Teufel, & Lehnert, 2008). Independent of the platform, quantum
harmonic oscillators are being used for sensing of weak signals (quantum
metrology) (C. Caves, 1981; McCormick et al., 2019), mediating interactions with
other systems (J. I. Cirac & Zoller, 1995; Schmidt-Kaler et al., 2003) and even for
processing of quantum information (Flühmann et al., 2019; Lloyd & Braunstein,
1999). Further development of the techniques used to control these systems will
likely lead to a deeper understanding of quantum mechanics and provide avenues
for new applications of these technologies.
In this thesis I present work that extends the current capabilities of the
control and readout of trapped ion oscillators. Trapped ions represent some of
the most ideal harmonic oscillators given that they can be trapped in deep and
highly harmonic potentials. Trapping an ion in vacuum provides exquisite isolation
from the environment which enables ion oscillations with long quantum coherence
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times. Significant work has been done to demonstrate the ability to cool and
coherently control the ion motion (Meekhof, Monroe, King, Itano, & Wineland,
1996; Wineland, Drullinger, & Walls, 1978). The dynamics of these oscillator
states can be controlled using laser-based methods, as well as with static and
oscillating magnetic field gradients, where these techniques also effect the atomic
internal electronic states (Leibfried et al., 2002). We develop techniques that
involve modulation of trapping electric field gradient, which enables manipulation
of these oscillator states (Heinzen & Wineland, 1990) in a way that doesn’t effect
the qubit. These techniques allow for the generation of different types of classical
and quantum states that are useful for metrology and computation. Parametric
modulation of the trapping potential is one of these techniques which allows
for the ability to couple two distinct oscillator states of a single trapped-ion
and generate entangled states of motion (Leibfried et al., 2002). Generation of
entangled motional states and or qubits can provide a resource for metrology at
the Heisenberg limit and is a necessary component in quantum computation. The
techniques and interactions presented here are not uniquely possible with trapped-
ions but leveraging some of the strengths of this platform can enable a more direct
path to utilizing quantum harmonic oscillators.
1.2 Harmonic oscillators and qubits
Trapping ions in three dimensions means three harmonic oscillators per
ion. With each ion comes a single qubit which can be coupled by utilizing the
collective motion of any of the oscillator modes. The collective motion gives rise
to discrete normal modes (James, 1998), which like single ion oscillators can be
manipulated and used for metrology and information processing. The state of
these oscillators are difficult to detect directly. The common method of detecting
the motional states is to couple them to the ion’s internal electronic states using
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laser fields (J. I. Cirac & Zoller, 1995; Vogel & Filho, 1995). This technique is
possible when a interaction field is applied with an effective wavelength that
is not significantly greater than the size of the motional wavepacket (Meekhof
et al., 1996). This can be done with laser fields or with strong magnetic field
gradients (Mintert & Wunderlich, 2001; Ospelkaus et al., 2011). The simplest
method for this coupling is to use a field that couples two levels of the ion’s
electronic structure and is detuned by the motional frequency (J. Cirac, Parkins,
Blatt, & Zoller, 1996; Meekhof et al., 1996). This requires a suitable choice of a
two-level system chosen from a vast range of different energy levels. The initial
(uncoupled) full state (of the qubit and the oscillator )can∑be written as∞
ψ = cos (θ/2) |↓⟩+ sin (θ/2)eiϕ |↑⟩ ⊗ cn |n⟩ , (1.1)
n=0
where θ and ϕ are arbitrary real parameters, |↓⟩ , |↑⟩ represent the two different
energy levels of the qubit, and cn are the amplitude of each oscillator Fock state
|n⟩. The qubits can be visualized with the Bloch sphere representation shown in
Fig. 1.1, with the equally spaced levels of the Harmonic oscillator shown. There are
several options when it comes to choosing two levels to encode a qubit in trapped-
ions with frequency scales ranging from MHz to THz and different advantages to
each encoding. The ability to take advantage of different encodings could help to
scale quantum systems without incorporating additional physical systems (Allcock
et al., 2021).
1.3 Effects of qubit measurements
Building a quantum computer or a quantum sensor requires incorporating
quantum systems into larger devices which follow the rules of quantum mechanics
but provide the ability the measure a quantum state and give a classical
outcome (Nelsen & Chuang, 2001). This scheme requires the ability to change
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Figure 1.1. a) Bloch sphere representation of a qubit, where qubit populations are
measured in the Z basis and angles θ and ϕ give the phase relationship between
qubit states. b) Harmonic potential that confines two trapped ions represented with
the circles and internal states. Energy eigenstates of the harmonic potential are
evenly spaced levels.
from quantum coherences to classical measurement results by collapsing the
wavefunction into a discrete output. This interface between quantum and classical,
to our knowledge, means updating the description of the quantum state in a
non-unitary way (Bassi, Lochan, Satin, Singh, & Ulbricht, 2013). The choice of
when and where this update happens is critical for preserving a certain desired
quantum state. In practice this is done by combining coherent control of qubit
states and the ability to detect a subset of the full quantum system. The non-
unitary measurement often leads to disturbances of quantum states which should
be preserved and updated depending on the outcome of the measurement. There
are techniques that enable the ability to preserve the quantum state stored in
the electronic states of ions, by decoupling those states from the state detection
process. The preservation of the quantum state of certain oscillators is more
difficult. When using lasers to detect the state of a qubit, in the event of absorption
and emission of photons, the position and momentum of the ion gets kicked
19
which drives the quantum state of the oscillators towards a classical mixed
state (Wineland et al., 1978). Consequences of the geometrical structure of the
normal modes of ion chains means detection of certain ions internal states can be
done without perturbing every oscillator state of the ions. Understanding the limits
of this geometrical protection will help to enable classical feedback schemes that
could assist in the error correction schemes involving bosonic codes.
1.4 Interferometry with trapped ions
Some early physics experiments involved interfering two classical sources
of light to make measurements about the universe, such as the existence of the
ether or the speed of light (Kennard, 1922; Michelson & Morley, 1887). The field
of light being described as a harmonic oscillator implies that harmonic oscillators,
like trapped ions, can be used for interferometry. In modern times, interferometers
are used to measure many physical properties like length, in order to detect
gravitational wave detection (J. Aasi et al. (Ligo Scientific Collaboration), 2013),
but can also be used to measure properties of the laser used, like wavelength and
linewidth (Monchalin et al., 1981). We will use these same concepts to measure the
phase of trapped ion harmonic oscillators. Using narrow band lasers enables sub-
wavelength resolution of the parameter that is being measured. Using quantum
resources can lead to improved sensitivities that can get below the shot-noise
limit of the light source called the standard quantum limit (SQL) (Demkowicz-
Dobrzański, Jarzyna, & Ko lodyński, 2015). The basic strategy is to allow to one
or more modes of electromagnetic fields, or oscillator wave packets, to interfere
and to measure the interference pattern seen by some detector. The phase of the
interacting fields can be varied experimentally, or by some physical phenomenon,
which can be detected as a shift in the interference pattern. The interference
pattern and the sensitivity to phase shifts depends on the physical implementation
20
of the interferometer. The earliest interferometer is the Michelson interferometer,
used to verify the lack of an ether (Michelson & Morley, 1887). More recently,
the Mach-Zehnder interferometer was developed which has two outputs which can
be measured (Jakeman, Oliver, & Pike, 1975); the Michelson and Mach-Zehnder
interferometer geometries are shown in Fig. 1.2 . The signal detected from these
interferometers gets stronger when using stronger fields, however the scaling of the
sensitivity, which depends on the signal to noise ratio, is set by the measurement
technique being used. Quantum resources enable the ability to measure phases at
the fundamental Heisenberg limit by reducing noise below the SQL. Entanglement
of quantum systems can help in reaching the fundamental limits in sensitivity,
which involves using multiple modes that are highly correlated, but reducing the
quantum fluctuations in a single mode can also improve sensitivity. Having the
ability to implement a variety of interaction elements and input states enables
the ability to employ different interferometers for detection of different phases e.g.
single mode, common mode, difference mode. These techniques, in combination
with displacement detection are important for progress towards fully characterizing
the state of the modes of interest, or characterizing operations that transform the
state.
21
a) b)
Figure 1.2. a) Michelson interferometer with a single beam splitter element and
a single accessible output. b) the Mach-Zehnder interferometer which enables
two inputs and the ability to measure two outputs with only one additional beam
splitter element.
1.5 Thesis layout
The topics in this thesis cover development of techniques for control of
trapped-ion mechanical oscillator states using laser-free methods as well as methods
for preservation of oscillator states during readout. Metastable qubits provide the
toolbox for measurement in this work and will be discussed in detail along with the
experimental studies that are possible using these states.
Chapter 2 reviews the theory of quantum harmonic oscillators with a focus
of parametric modulation to couple two independent modes of motion.
Chapter 3 describes the possible encoding schemes for qubits for atoms
with a similar structure to Ca+. This includes techniques for coupling these two-
level systems and performing state detection of the spin states or spin-motion
coupling for detection of motional states.
Chapter 4 covers the experimental systems that are used throughout this
work. This chapter covers a range of systems starting from how to produce ions
from neutral atoms to how the state of the ion is detected.
22
Chapter 5 is a presentation of work to demonstrate non-Hermitian
dynamics and the quantum limits that they can exceeded. This work is made
possible with the use of metastable qubits which enables a dissipitive channel to
engineer this interactions. This chapter includes work pending publication co-
authored by A. Quinn, S. Brudney, I.D. Moore, J.E. Muldoon, S. Das, D.T.C
Allcock and J.N. Joglekar.
Chapter 6 focuses on the mechanisms for generating and calibrating
experiments using the motional state of the trapped ion. This chapter connects
heavily to the theories in Chapter 2 to bring those motional operations and
characterization to the experiment.
Chapter 7 discusses a method for preserving a particular motional state
during detection of the internal qubit state. This work relies on the mode geometry
of longer ion chains and is an exploration of the techniques to determine the
protection that this particular mode has.
Chapter 8 is a presentation of SU(2) and SU(1,1) interferometers using
the motional states of trapped ions. The limit of the sensitivity is discussed and
presented alongside the experimental results. This chapter includes work pending
publication that is co-authored by A. Quinn, S. Brudney, I.D. Moore, S.C. Burd,
D.J Wineland and D.T.C Allcock.
Chapter 9 is the outlook for future work with the motional states of
trapped ions. This includes discussion of potential experiments using two-mode
squeezed states for quantum amplification and how to measure their correlations.
An outlook for potential implementations of continuous variable quantum
computing is included.
23
CHAPTER II
QUANTUM HARMONIC OSCILLATORS
Following from the realization that many physical systems can be modeled
as harmonic oscillators, it is necessary to discuss the mathematical details of this
model. Mechanical harmonic oscillators are defined beginning with a system that
is subject to a restoring force that is proportional to its displacement x from
equilibrium. This force is given by Hooke’s law F = −kx with spring constant k
1
which corresponds to a potential V (x) = kx2. When considering the kinetic and
2
potential energy of this system we have the full Hamiltonian for a one-dimensional
harmonic oscillator
1 1
H = p2 + mω2x2, √ (2.1)2m 2
k
where we introduce the oscillator frequency defined as ω = with oscillator
m
mass m and momentum p. This Hamiltonian describes the total energy of the
system and leads to oscillations of the position and momentum given by
x(t) = A sin (ωt+ ϕ), (2.2)
p(t) = Amω cos (ωt+ ϕ), (2.3)
where A, ϕ are the amplitude and phase of the oscillations and depend on the
initial conditions. This classical description is very useful for analysis of classical
systems, however a quantum treatment is necessary to get a deeper understanding
these systems.
2.1 Fock basis ladder
Before we can describe the ladder of energy states available to the quantum
harmonic oscillator we must first make the transformation to working with
operators x̂r and p̂r (Gerry & Knight, 2005) instead of canonical variables of
24
position and momentum x, p. The full Hamiltonian is now written as
p̂2r 1Ĥ = + mω2x̂2r. (2.4)2m 2
These canonical operators must satisfy the commutation relation [x̂r, p̂r] = iℏ,
where ℏ is Planck’s constant. We can now rewrite the position and momentum
operators in terms of dimensionless constants and use quanta creation and
annihilation operators â†, a giving√
≡ mω 1x̂ x̂r = (↠+ â) (2.5)
2ℏ 2
and √
√ p̂ ≡
1 i
p̂ = (â†r + â). (2.6)
2mωℏ 2
ℏ
We will define x0 = which is the extent of the ground state wavefunction.
2m
Using these transformations we can now re-write the Hamiltonian in terms of its
second-quantization form
Ĥ = ℏω(â†â+ 1/2). (2.7)
At this point, it is useful to define the commutation relation for creation and
annihilation operators as [â†, â] = 1. From this commutation relation we can define
the number operator N̂ = â†â. This operator is connected to the Fock states where
the eigenstates of N̂ are defined to be Fock states |n⟩ where n ∈ {0, 1, 2, 3, ...}. The
associated eigenvalues are given by (n + 1/2)ℏω and we make the observation of
ℏω
the ground state energy is , which is an energy offset and has no effect on the
2
dynamics. Gaining or losing n quanta of energy is accomplished with successive
application of the creation (raises n) or annihilation (lowers n) operators onto
the state |n⟩. For a ground state to exist we terminate the ladder at |0⟩ with any
further application of the lowering operator resulting in zero. These relations are
written out completely as
25
N̂ |n⟩ = n |n⟩ (2.8)
â |0⟩ = 0 (2.9)
† √â |n⟩ = n+ 1 |n+ 1⟩ (2.10)
√
â |n⟩ = n |n− 1⟩ . (2.11)
Now that the Fock state has been defined we can also talk about the density matrix
representation as this is useful to discuss certain states of the harmonic oscillator.
We will stick to pure states where we can define the density matrix as
ρ = |ψ⟩ ⟨ψ| . (2.12)
T∑he Fock state ladder forms an orthonormal basis with ⟨n|m⟩ = δn,m and
n |n⟩ ⟨n| = 1. The density operator representation of the oscillator is then written
as ∑∞
ρ = bmn |m⟩ ⟨n| and bmn = ⟨m|ρ|n⟩ . (2.13)
n,m=0
Before moving on to other representations of quantum harmonic oscillators it is
worth mentioning consequences of multi-mode systems given that this is a focus of
the work presented here. These relations are summed up by the fact that operators
can only operate on a Hilbert space that contain the states that the operators
belong to,
[âi, â
†
j] = δij (2.14)
[âi, â ] = [â
†
j i , â
†
j] = 0. (2.15)
26
Finally, the creation and annihilation operators in a multi-dimensional space change
the eigenvalues of the number operat∑or, ∑
N = n = â†i i âi. (2.16)
i i
2.2 Wigner function representation
Moving away from the number basis we can begin to speak in more classical
analogs of position and momentum. The position |x⟩ and momentum |p⟩ basis
plays a key role in discussion of phase space and the oscillator state |ψ⟩ can be
written in this basis as ∫ ∞
|ψ⟩ = dxψ(x) |x⟩ , (2.17)
−∞
where ψ(x) is a complex wavefunction that gives weight to each position eigenstate
and can be written as ψ(x) = ⟨x|ψ⟩. The position and momentum representation
are connected by a Fourier transform g∫iven by∞
ϕ(p) = √1 dxe−2ixpψ(x). (2.18)
π −∞
From this relation we can begin to see naturally how the Heisenberg uncertainty
principal arises. If we were to consider a particle that has a position that is
perfectly well defined given by a delta function δ(x) then when we take the Fourier
transform we get ∫
1 ∞F 1(δ(x)) = dxδ(x)eiωx = , (2.19)
2π −∞ 2π
which is a constant function spread across all momentum space evenly. With this in
mind, we define the Heisenberg uncertainty relation for position and momentum as
1 2
∆x∆p ≥ |⟨[x̂, p̂]⟩|2 ℏ= , (2.20)
4 4
27
where ∆x = ⟨x̂ − ⟨x̂⟩2⟩ is the variance of the position. We will now define the
Wigner function (Wigner, 1932) f∫or an〈arbitrar∣∣y ∣∣density〉matrix ρ̂1 ∞ 1 1
W (x, p) = x+ q∣∣ρ̂∣∣x− q eipq/ℏ∣ 〉 dq, (2.21)2πℏ −∞ 2 2
where ∣x± 1q are eigenstates of the position operator. It is important to point out
2
that this is a quasi-probability distribution (Gerry & Knight, 2005) function and
not the only one that can be used to represent the state, but is the one used in this
thesis. If we assume a pure state∫defined above then we get
1 ∞ 1 1
W (x, p) = ψ∗(x− q)ψ(x+ q)eipq/ℏdq. (2.22)
2πℏ −∞ 2 2
Without writing out all the details it is important to note that by integrating over
position or m∫omentum we get∞ ∫ ∞
W (x, p)dp = |ψ(x)|2 and W (x, p)dx = |ϕ(p)|2 (2.23)
−∞ −∞
which are the probability density for position x and momentum p. Working
backwards we can note that like a probability density the Wigner function is real
valued, but in contrast the Winger function can take on negative values which is
a signature of quantum states and is why it is referred to as a quasi-probability
distribution.
2.3 Oscillator modes with ion chains
Generally ion traps are operated with more than one ion in order to perform
2-qubit gates, as well as allow for sympathetic cooling (Kielpinski et al., 2000).
For a system of N ions, the potential can be written as a sum of the quadratic
potentials generated by the trap and the Coulomb potential due to the charge of
the ions.
∑ [NM ∑3 Q2 ∑ ∑ ] 13 −2
V = ω2i x
2 2
2 ni
+ (xni − xmi) (2.24)
8πϵ
n=1 i=1 0 n,m=1 i=1
m ̸=n
28
The secular frequencies of the harmonic motion are set by the strength of the
confinement. With weak axial confinement, the ions will create a linear crystal.
When this confinement strength is increased the ion crystal undergoes a phase
transition to other structural phases (Yan et al., 2016). We will remain in the linear
crystal regime for the work in our lab and for the purposes of this analysis, under
the assumption that
√
ω1 = ω2 = ω3/ α, (2.25)
where α ≪ 1 for linear crystals, and 1,2,3 are x, y, z respectively.
2.3.1 Motional modes. For N ions there are 3N motional modes of
the ion crystal. The motion is analyzed by expanding the potential to second order
about the equilibrium positions denoted by x̄ni (Marquet, Schmidt-Kaler, & James,
2003), where n indexes the ion and i the dimension. These positions are determined
by solving the equations ∣
∂V ∣∣∣ = 0. (2.26)∂xni 0
It is more convenient to work with the dimensionless equilibrium position u = x̄ni/l
that is normalized to a natural length(scale in thi)s problem.
Q2
l = (2.27)
4πϵ0Mω23
This distance is approximately equivalent to the cubed distance between ions in a
two ion crystal. The displacements of the ions from their equilibrium position are
written as
qni(t) = xni(t)− x̄ni. (2.28)
29
Expanding about these equilibrium∑positions the Lagrangian is approximated asN ∑3M 1 ∑N ∑3 ∣∂2 ∣
[ L =
2
] q̇ni −
V
V ∣0 −[ qmiqnj2 2 ∂xmi∂x ∣∑ ∑ n=1 i=1 ∑ ∑m,n=1 i,j=1 ∑ nj 0 ]N N 2 N N
≈ M Mq̇2 2 2 2n3 − ω3 Amnqm3qn3 + q̇2 2 ni
− ω3 Bmnqmiqni ,
n=1 m,n=1 i=1 n=1 m,n=1
(2.29)
where V0 has no impact on the motion and is ignored and the tensors Amn and Bmn
are given by  ∑
N 1
1 + 2 if m = np=1 |u − u |3m pp ̸=m
Amn =  (2.30) −2 if m ≠ n|u 3m −(up| )
1 1
Bmn = δmn − Amn. (2.31)
α 2
The eigensystem of Amn and Bmn describe the axial and radial motion respectively.
The eigenvectors are the same for both, meaning all 3 dimensions have the same
mode structure. However, the oscillation frequencies of the radial modes are
different.
2.3.2 Mixed mass crystals. In the section above it was assumed
that all ions in the crystal have the same mass and charge. Having different species
of ions allows each species to have a different role given its different electronic
structure. The normal modes are still eigenvectors of the Amn, but we must now
re-normalize to account for the mixed mass (Kielpinski et al., 2000). This analysis
makes the assumption of an odd number of ions and that the center ion has a
different mass, which will be labeled by the index nc with mass m, all others having
mass M. This treatment is useful for the work presented in Chapter VII. The
30
Lagrangian is no[w
m M ∑written as ] [ ]N ∑N 2 N N2 2 2 M ∑ m ∑ ∑L = q̇ + q̇ 2 2 2n 3 n3 − ω3 Amnqm3qn3 + q̇n + q̇ − ω B q q .2 c 2 2 2 ci ni 3 mn mi ni
n=1 m,n=1 i=1 n=1 m,n=1
n̸=nc n̸=nc
(2.32)
√
The mass dependence is removed by using the transformation Qni = qni mn where
µ = m/M∑. This produc∑es the re-normalized Lagr[angian ]N 2 N 21 ω ′ 1 ∑ ∑N ∑N
L = Q̇2n3 − 3 A 2
′
mnQm3Qn3 + Q̇ni − ω23 BmnQmiQni2 2 2
n=1 m,n=1 i=1 n=1 m,n=1
(2.33)
where 
Amn m,n ̸= nc
′
Amn =  √A / µ m or n = n ,m ̸= n (2.34) mn cAmn/µ m = n = nc
′
and the same modification is true for Bmn. One notable consequence of having
as mixed mass ion chain is that there is no longer a rigid body mode where each
ion has the same amplitude oscillation; this is shown in Fig.2.1. This enables the
possibility of these modes, normally called center of mass, exchanging energy with
other modes, which is not possible with equal mass ions.
31
Figure 2.1. Normal modes and their oscillation frequencies for 3 ion chains. The
normal mode vectors give the amplitude of oscillation described by qni. Results for
40Ca+/88Sr+/40Ca+ chains and 88Sr+/40Ca+/88Sr+ chains are also shown, which are
of interest for experiments presented here.
2.4 Driven harmonic oscillators and parametric modulation
Now that the oscillator states can be described mathematically we can
work out how different oscillator states are generated. Specifically we can consider
what types of operations we can generate using an ion trap. The geometry of
macroscopic ion traps generally enable the ability to easily generate linear and
quadratic potentials. Starting with linear potentials we can see what types of
quantum operators we generate. Assuming a potential of the form V̂ (t) =
x̂
F sin (ωdt+ ϕ) the full Hamiltonian governing these dynamics is written as
x0
Ĥ = ℏω(â†a+ 1/2)− F (â+ â†) sin (ωdt+ ϕ), (2.35)
where we have transformed to working with the creation and annihilation
operators. If we move into an interaction picture where we are rotating in phase
32
with the oscillator we get
Ĥ = −Fx (âeiωt + â†e−iωtI 0 ) sin (ωdt+ ϕ). (2.36)
We can see from this transformation, we get terms that rotate at different rates,
ωd + ω and ωd − ω. In the case that the coupling rate, defined by gd = Fx0/ℏ is
much slower than the mode frequency gd ≪ ω then the fast rotating terms ω + ωd
can be ignored, called the rotating wave approximation (RWA). The interaction
Hamiltonian is then reduced to
ℏgd
Ĥ −iδt+iϕ † iδt−iϕI = − (âe − â e ). (2.37)
2i
The interaction detuning is defined as δ = ωd − ω and we can see the effect on the
state when we tune into resonance with the oscillator ωd = ω. In this case we get a
time independent Hamiltoni(an and )the unita(ry evolution operato)r can be written asgdt
Û = exp iĤ t/ℏ = exp − (âeiϕI I − â†eiϕ) , (2.38)
2i
which is the definition of the displacement operator (Carruthers & Nieto, 1965)
gdt
D̂(α = eiϕ) where the amplitude depends linearly on the duration the drive is
2
applied for. This operator allows for linear translations anywhere in phase space
while maintaining minimum uncertainty and generates states with a common
classical analog (Alonso et al., 2016). This technique is used widely in the trapped
ion field in order to do things like calibrate motional frequencies. Next we will look
at higher order operations that can generate states with more of a quantum nature.
In order to get higher order operations we need to implement higher order
potentials at the location of the ion. Harmonic potentials, as described above, are
how the ions are confined in space. We can modulate this potential to implement a
Hamiltonian with x̂2 given by
Ĥ = ℏ 2ω(â†â+ 1/2)− ℏg sin (ωst− θ)(↠+ â2 + 2â†â+ 1). (2.39)
33
In the same manner as for linear potentials, we move into a interaction frame
rotating at ω and drop fast rotating terms at (2ω + ωs) and ωs which reduces ĤI
to
ℏg 2
ĤI = − (â2e−2iωt+ωst−iθ − ↠e2iωt−iωst+iθ). (2.40)
2i
Time dependence of this Hamiltonian is removed by the choice of modulation
frequency to be ω = 2ωs,
iℏg 2
ĤI = (â
2e−iθ − ↠eiθ). (2.41)
2
In this rotating frame we can write the unitary evolution operator when applied for
a duration t as ( ) (gt 2 )
ÛI = exp iĤIt/ℏ = exp (â2eiθ − ↠eiθ) . (2.42)
2
This unitary is defined as the squeezing operator Ŝ(ξ) and the squeezing parameter
ξ is given by ξ = gteiθ (Heinzen & Wineland, 1990). The states generated by these
operators will be described in further detail in the next section, but first we will
take a look at the types of operators than can be generated involving more than
one mode of oscillation.
2.4.1 Multi-mode operators. When considering multiple orthogonal
modes of oscillation of a single trapped ion we can generalize the oscillator
Hamiltonian to be ∑
Ĥ0 = ℏωi(â†i âi + 1/2). (2.43)
i∈{x,y,z}
In the same manner as above we can apply an oscillating potential that will now be
generalized to be U(r, t) = U(r) cos (ωt+ ϕ). The multi-mode interactions can be
seen by expanding this poten∑tial to second order∂U 1 ∑ ∂2U
U(r+ δr) ≃ U(r0) + |r=r0δri + |r=r0δriδrj, (2.44)∂i 2 ∂i∂j
i∈{x,y,z} i∈{x,y,z}
34
where we can drop terms besides second order terms using the same rotating wave
approximation, reducing the potential to ∑ ∂2U
U(r+ δr) ≃ 2−δ(ia,ib) |r=r0δriδrj. (2.45)∂i∂j
i∈{x,y,z}
This curvature expression has several terms where the non-cross (x2, y2, z2) terms
give rise to the single mode interactions. In order to couple two different modes
of oscillation we need a terms that looks like x̂ŷ where x̂ and ŷ are the position
operators for the two orthogonal modes of oscillation. The interaction Hamiltonian
is then written as
ĤI = −ℏg sin (ωtmt− ϕ)(x̂ŷ). (2.46)
If we define the x mode of oscillation to have frequency ωx with annihilation
operator â and the y mode to have frequency ωy with annihilation operator b̂, we
can transform into the rotating frame. Making a transformation into a frame at ωx
and ωy is possible given that â and b̂ commute, given by
ei/ℏĤ0tâe−i/ℏĤ0t = âe−iωxt
(2.47)
ei/ℏĤ0tb̂e−i/ℏĤ0t = b̂e−iωyt.
The full interaction Ham[iltonian is then expanded to be ]
Ĥ = −ℏg sin (ω † † it(ωx+ωy) −it(ωx+ωy) † it(ωx−ωy) † −it(ωx−ωy)I tmt− ϕ) â b̂ e + âb̂e + â b̂e + âb̂ e
(2.48)
and we now have access to different resonant interactions. We will first let ωtm =
ωx − ωy and can drop terms that oscillate at 2ωx,y using the RWA. This interaction
is typically called the beam splitter interaction which enables coherent exchange of
energy between modes (Gorman, Schindler, Selvarajan, Daniilidis, & Häffner, 2014)
and is written as
Ĥ = ℏg(âb̂†eiϕBS + â†b̂e−iϕ). (2.49)
35
The other choice of modulation frequency we can make is to set ωtm = ωx + ωy and
ignore the same fast oscillating terms. This interaction is generally called two-mode
squeezing and generates highly correlated oscillator states and is expressed as
Ĥ = ℏg(â†b̂†eiϕ + âb̂e−iϕTMS ). (2.50)
All of the interactions described in this section can be used for implementations
of quantum computation and quantum metrology in a variety of platforms, but
require different methods to produce. In the following section these states that
are generated from these interactions will be described in more detail along with
potential use cases.
2.5 Gaussian states
In order to talk about the states generated with the Hamiltonians given
above, it is necessary to work again in the language of phase space and the Wigner
function. In order to do this we must define the most relevant quantities of the
Wigner function, which are the statistical moments of the quantum state. The first
moment is sometimes called the displacement vector, but more simply described as
the mean value
x̄ := ⟨x̂⟩ = Tr(x̂ρ), (2.51)
where again the quantum state is described by the density operator ρ. The second
moment is called the covariance matrix V with the elements defined by
1
Vij := ⟨{∆x̂i,∆x̂j}⟩, (2.52)
2
with ∆x̂i := x̂i − ⟨x̂i⟩ and {, } is the anti-commutator. The variance in the
quadrature operators x̂ is then just the diagonal elements of this matrix
Vii = V (x̂i)
(2.53)
V (x̂ ) = ⟨x̂2i i ⟩ − ⟨x̂i⟩2.
36
It is also useful to note that through this formalism the uncertainty relation
ℏ
appears as V (x̂)V (p̂) ≥ . For Gaussian states alone the quantum state can
2
be completely characterized by these two moments ρ̂ = ρ̂(x̄, V ) and the Wigner
function can take the general for{m }
exp [−(1/2)(x− x̄)TV −1(x− x̄)]
W (x) = √ , (2.54)
(2π)N detV
which is that of a Guassian function. This is another place where the Wigner
negativity is an important aspect of the quantum state as a state is only Gaussian
if and only if its Wigner function is non-negative. With Gaussian states roughly
mathematically defined we can again take a look at the operators that we can
apply to these states. Each of the above operators is unitary and transforms
the state according to ρ̂ → Uρ̂U † or just |ψ⟩ → U |ψ⟩ for the pure states we
are considering. All of these operators are considered Gaussian operators if they
transform a Gaussian state into another Gaussian state and each one of these
operators and the final state is described below.
2.5.1 Displacement and coherent states. The displacement
operator has already been defined above but we will discuss in more detail
the effect of this operator on the vacuum state of an oscillator in terms of the
displacement parameter α. The magnitude and phase of α is set by the driving
field coupling, duration and phase. More generally we can define α = (x + ip)/2
which can help to clarify the action in phase space. Application of this operator
makes the transformation |α⟩ = D̂(α) |0⟩ which maintains the same variance as
the vacuum state (V = I) but shifts the mean value (x̄ = dα) which is clear when
looking at the Wigner function of the coherent state,
2 2
W (α, β) = e−2|α−β| . (2.55)
π
37
Coherent states are also the eigenstates of the annihilation operator â |α⟩ = α |α⟩
and when expanded in the number(basis can)be∑written as∞ n
|α⟩ 1= exp − | |2 √αα |n⟩ . (2.56)
2
n=0 n!
When making a measurement of the phonon number of a coherent state α, like
photon counting a laser field, the probabilities are given by a Poisson distribution
2n
Pn∑= |⟨n|α⟩|
2 2 |α|= e−|α| , (2.57)
n!
where the mean value ⟨n̂⟩ = n nPn = |α|2 which is equal to the variance.
The number distribution and Wigner function can both be visualized in 2.2a).
Coherent states are used widely in trapped ions (Gorman et al., 2014; Mihalcea,
2023), appearing in the entanglement of spin states (Mølmer & Sørensen, 1999),
but they will also be important for performing classical interferometry experiments
that use classical laser fields.
2.5.2 Squeezing and single mode squeezed states. We can now
discuss in more detail the effect of the squeezing operator that has been defined
above. In optics, a single photon in a pump field at frequency 2ω is spit into 2
photons at frequency ω via a non-linear medium. The same effect happens with
trapped ions where the driving field at 2ω generates phonon pairs of the motion,
but without the inefficiencies of using a non-linear medium. Since this operator is
second order in the annihilation and creation operators the state that is generated
is a superposition of the even Fock states |2n⟩ and that state can be written in the
number basis as
√ 1 ∑∞ √(2n)!|ξ⟩ = −einθ tanh(r)n |2n⟩ . (2.58)
cosh(r) 2nn!n=0
The even state populations are expressed as
(tanh r)2n (2n)!
P2n = . (2.59)
cosh r (2nn!)2
38
The effects of squeezing are more obvious in phase space where the Wigner function
is given as ( )
2
W (r, θ, α) = exp − 2e2r Re[e−iθ/2α]2 − 2e−2r Im[e−iθ/2α]2 (2.60)
π
which is a two-dimensional Gaussian with different variances which can
be seen in 2.2b). In order to get the mean phonon number (expectation of
the number operator) it is easiest to first see how the squeezing operator
transforms the annihilation and creation operator, given by the Boguliubov
transformations (Bogoljubov, 1958):
Ŝ†(ξ)âŜ(ξ) = â cosh r − â†eiθ sinh r (2.61)
Ŝ†(ξ)â†Ŝ(ξ) = ↠cosh r − âe−iθ sinh r (2.62)
Using these transformations the expectation value of the number operator evaluates
simply to ⟨n̂⟩ = ⟨0|Ŝ†(ξ)â†âŜ(ξ)|0⟩ = sinh2 r. Unlike the displacement operator,
squeezed states are fixed at the origin with expectation values of position and
momentum remaining zero ( ⟨ξ|x̂|ξ⟩ = ⟨ξ|p̂|ξ⟩ = 0). Since the mean value of
these Gaussian distributions remain unchanged it is more interesting to look at
the quadrature variances. At a phase θ = 0 the quadrature operators q̂ = (x̂, p̂)T
are transformed under the mapping q̂ →S(r)q̂ wheree−r 0
S(r) :=   , (2.63)
0 er
where the covariance matrix is then given by V = S(r)S(r)T = S(2r). The effect is
to shrink one quadrature variance below the quantum shot noise limit and stretch
the other all while maintaining the Heisenberg uncertainty relation. Having a
mechanism to shrink one quadrature variance below the shot noise limit is a key
ingredient in metrology experiments that achieve sensitivity below the standard
39
quantum limit that is achievable with coherent states. (C. Caves, 1981) The phase
of the the squeezing operator sets the quadrature that is squeezed and can be used
for amplification protocols at certain phases.
2.5.3 Phase operator. Before moving onto the effects of two-mode
operations it is worth discussing the phase operator and its effect on oscillator
states. The phase of a quantum state is what metrology experiments may aim to
sense and given the need understand this phase it is worth discussing. The phase
of a quantum state is generally tracked with a local oscillator in the experimental
control system, which can be provided by a laser or some other oscillator. Phase
can naturally be added in optical experiments by passing a field through a medium
or increasing optical path length but looks different with trapped ions. Since the
ions are fixed in space, instead of increasing an optical path length, additional
time delays can be added, as well as changing the trapping potential to shift
the oscillator frequency, or moving the local oscillator off resonant with a frame
rotating with the oscillator for some fixed time. With a digitally programmable
oscillator, this can be done by advancing the phase programmed to the device. No
matter the method, the effect is to implement a free propagation Hamiltonian of
Ĥ = 2θâ†a. This looks just like the bare oscillator Hamiltonian and is made up
of the number operator that preserves the energy of the state. The unitary phase
operator is given as ( )
R̂(θ) = exp −iθâ†â (2.64)
and transforms the annihilation operator by another Boguliubov transformation by
â → e−iθâ. The mapping for the quadratures is givenby q̂ → R(θ)q̂ where cos θ sin θR(θ) :=  (2.65)
− sin θ cos θ
40
which is a simple rotation matrix with angle θ. In phase space, for a coherent state,
this looks like a rotation about the origin with angle θ as shown in 2.2a). Later we
will implement experiments to detect these sort of phase rotations with sensitivities
beyond the SQL.
41
a)
b)
c)
Figure 2.2. a) number distribution and Wigner function of a coherent state
with α=2. The displacement operator translates the Gaussian centered at 0 by
magnitude |α| with phase rotations shown by the state subtending the angle θ.
Variance of the state is shown by the black circle b) number distribution and
Wigner function of a squeezed state with r=1. Population of only the even Fock
states can be seen clear and variance of the squeezed state being below that of a
coherent state only in one quadrature. c) number distribution and Wigner function
of a thermal state with n̄=1, where the increased variance of the the SQL (coherent
state) can be seen clearly
2.5.4 Beam splitter. The two-mode unitary is a key part of quantum
optics and is the basis for most interferometers. The beam splitter unitary operator
42
is given as [ ]
B̂(θ) = exp θ(â†b̂− âb̂†) , (2.66)
where θ sets the transmissivity of the beam splitter τ = cos2 θ which is fixed to
between 0 and 1. In the trapped ion platform we have the ability to tune this value
to whatever we want by tuning the coupling rate or the interaction time, where
τ = 1/2 is usually the default in optics, shown in 2.3. The annihilation operators
are again transformed under a Boguliubov transformation by
â 
 
 √ √τ 1− τ→ â√ √ . (2.67)
b̂ − 1− τ τ b̂
It can be seen from the form of the transformation matrix there is a similarity to
rotation matrices. In the most simplified description this view is a consequence of
the beam splitter belonging the the SU(2) group which will be discussed more in
detail later. The mapping for the quadrature operators q̂ := (x̂a, p̂a, x̂b, p̂
T
b) is given
by q̂ → B(τ)q̂ where
 √ √τI 1− τI
B(τ) :=  √ √  , (2.68)
− 1− τI τI
where I is the identity matrix.
43
a) b)
Beam 
Splitter
Figure 2.3. a) action of the beam splitter when τ=1/2 on the input state |α = 2, 0⟩.
Total energy (phonon) number is conserved in this process. b) covariance matrix
for the multi-mode state shown that the two modes are left unentangled after the
beam splitter operation.
2.5.5 Two-mode squeezing and two mode squeezed vacuum. In
a similar fashion to the single mode squeezing where a pump photon at 2ω is split
into two photons, now those photons are generated into two different modes with
the pump field at ωa + ωb. As shown above, this can be done with a driving field
that has some cross curvature terms that implement a bilinear terms â†b̂† in the
Hamiltonian. The corresponding Gaus[sian unitary is ]given by
Ŝ2 = exp r(âb̂− â†b̂†)/2 , (2.69)
where r described the amount of two-mode squeezing and is defined as r = 2gt.
For the quadrature operators q̂ := (x̂a, p̂a, x̂b, p̂b)
T the mapping for the two-mode
squeezer is given by q̂ → S2(r)q̂ where cosh rI sinh rZS2(r) :=  , (2.70)
sinh rZ cosh rI
44
 1 0 where Z = . The output state when applied to vacuum is sometimes
0 −1
called an Einstein-Podolsky-Rosen (EPR) st∑ate where√ ∞
|TMSV ⟩ = 1− λ2 (−λ)n |na, nb⟩ (2.71)
n=0
and |na, nb⟩ now describes the number state of two modes with λ = tanh r. The
question of quadrature variance becomes more interesting with this state with the
covariance matrix given by
 √

νI ν2 − 1Z
VTMSV = √  , (2.72)
ν2 − 1Z νI
where ν = cosh r describes the quadrature variance. The reduced variance is not
seen in the quadratures of a single mode but it can be checked that
V (x̂−) = V (p̂ ) = e
−2r
+ (2.73)
√ √
where x̂− := (x̂a − x̂b)/ 2 and p̂+ := (p̂a + p̂b)/ 2 are new EPR quadrature
variables. Working in this new basis enables squeezing to be more clearly seen and
is visualized in 2.4 The consequences of these new quadrature variances means
that for r > 0 the correlations between these two modes beat the quantum
shot noise limit. This entangled Gaussian state is another common state used in
quantum optics and the elements used to generate this state also play a role in
interferometry.
Before moving on, it is worth discussing the phonon number probabilities
and what it means to look at a single mode of this entangled state. The joint
phonon probability distribution for the two-mode squeezed vacuum only has
diagonal elements
P (n1, n2) = P (n, n)δn1,nδn,n2 (2.74)
45
and those elements are given by
(tanh r)2n
P (n, n) = 2 . (2.75)cosh r
If a measurement of only a single mode is made, this is equivalent to tracing over
a single mode, the result is Trb[ρ̂] = ρ̂
th
a (n̄), which is a thermal state with average
phonon number of n̄ = sinh2 r (C. M. Caves, Zhu, Milburn, & Schleich, 1991). This
means that the mean phonon number of single mode is related to the amount of
two-mode squeezing, which will be important later.
a) b)
Figure 2.4. a) Wigner function representation of a two-mode squeezed state in the
EPR-variable basis. The squeezing is seen in the mode correlations and not single
mode quadratures. b) correlation matrix for a two-mode squeezed state with off-
diagonal elements giving proof of mode correlations.
2.5.6 Thermal states. Given the presence of thermal states when
making measurements of two-mode squeezed states, it is relevant to include some
discussion of these states. Thermal states are not only present when working
with two-mode squeezed states but are common in trapped ions and exist as a
consequence of the ions coupling to the environment or after some dissipative
operation. These Gaussian states are not pure states and the mixed state density
matrix is given by ∑∞
ρ̂th = (1− e−ℏωβ) e−βℏωn |n⟩ ⟨n| , (2.76)
n=0
46
where β = 1/kBT with kB being the Boltzmann constant and temperature
T . Measurements of the average phonon number (n̄) can be made by using the
motional sidebands (Bergquist, Itano, & Wineland, 1987). The temperature relates
to n̄ by,
ℏω 1
T = n̄ , (2.77)kB ln[ ]
1 + n̄
which allows the density matrix to be∑re-(written)in terms of n̄ to be∞ n
th 1 n̄ρ̂ = |n⟩ ⟨n| . (2.78)
1 + n̄ 1 + n̄
n=0
The phonon number distribution is then gi(ven by)n
1 n̄
Pn = . (2.79)
1 + n̄ 1 + n̄
The Gaussian nature of these states can again be seen by looking at the Wigner
distribution given by
2 1 [ ]
W (n̄, α) = exp −2|α|2/(1 + 2n̄) . (2.80)
π 1 + 2n̄
Both quadrature variances are equal, giving a circular state like a coherent state,
however the variances grow above those of a coherent state with increasing n̄ which
can be seen in 2.2c).
2.6 Non-Gaussian states
Gaussian states are exceedingly common in quantum optics and are a key
ingredient in continuous variable quantum computing (CVQC) and are a resource
for quantum metrology. Despite this, non-Gaussian states are necessary to complete
a universal set of operations for CVQC (Lloyd & Braunstein, 1999) and can be
advantageous in some quantum metrology experiments (Wolf et al., 2019). All
of these states require higher order interactions, being at least third order in the
annihilation and creation operators, or a coupling to a discrete variable system like
a qubit. The potentials necessary to implement these higher order interaction are
47
generally weak without a more electrodes (Hou et al., 2022). A few non-Gaussian
states will be mentioned here for their potential relevance.
2.6.1 Fock states. Fock states, sometimes called number states, have
already been mentioned heavily since it is natural to work in a basis where states
are described as a superposition of Fock states (Gerry & Knight, 2005). Shown
in the number probability distributions in 2.2, Gaussian states are given by a
superposition of Fock states, but single Fock states can be generated and are non-
Gaussian. The Wigner function for these single Fock states is given by
(−1)m 2 2 2 2
W (x, p) = e−(x +p )Lm(2(p +x )), (2.81)
π
where Lm(x) denotes the m-th degree Laguerre polynomial. Figure 2.5 highlights
the non-Gaussian nature of these state through the Wigner negativity.
0.1 0.3 0.1
0.0 0.2 0.0
0.1 0.1 0.1
0.2 0.0 0.2
0.3 0.1 0.3
2 2 2
1 1 1
2 1 0 2 1 0 2 00 1 p 0 1 p 1 0 1 p
x 1 2 2 x 1 2 2 x 1 2 2
Figure 2.5. Wigner functions for the n = 1, 2, 3 states which highlight the Wigner
negativity and the spherical symmetry of the states
2.6.2 Cross-Kerr interaction. With a single ion there are
approximately no anharmonic terms in the potential experienced by an ion. When
considering the problem of scaling the capabilities of quantum systems using
the motional states of trapped ions, it is necessary to start working with more
ions. There is more physics to consider when working with more than one ion. In
Chapter VII there is discussion of how the geometries of the modes can be used
48
advantageously when detecting the internal state of the ion. In this section we
will explore the interactions that arise from considering the additional Coulomb
interaction present with multiple ions.
The Kerr interaction is a non-linear interaction that appears when
considering multiple ions. This Hamiltonian is of the form ĤKerr ∼ ξn̂an̂b where ξ
is the coupling strength between mode a and mode b with number operators n̂a and
n̂b (the number operator being given by n̂
†
a = a a). This interaction can be found
by expanding the potential, from eq. 2.24, to fourth order, producing an additional
term
−Mω
2 ∑
3 Cm,n,pqp3(2qm3qn3 − 3qm1qn1 − 3qm2qn2). (2.82)
2l
m,n,p
This term describes coupling between different ions. The coupling strength is given
by the tensor Cmnp which is∑written asN (uq − um)(δmn − δqn)(δmp − δqp)
Cmnp = . (2.83)
(um − u )4q=1 q
q≠ m
This complete Hamiltonian can be re-written in terms of the normal modes using
the transformation ∑3N
Xn = Amnqm, (2.84)
i=1
where Xn is the normal mode vector, qm is the position vector, and Amn is the
matrix of eigenvectors. Using this transformation the Hamiltonian can be written
as H = H0 + HI , where H0 describes the uncoupled collective oscillations, and HI
describes the perturbation that can couple modes (Marquet et al., 2003). This term
is written as
2 ∑NMω
HI =
3 DijkZk(2ZiZj − 3XiXj − 3YiYj), (2.85)
2l
i,j,k=1
49
where i, j, k now index the mode. The mode-mode coupling coefficients Dαβγ are
defined as ∑N
(α) (β) (γ)Dαβγ = Cνσλbν bσ bλ , (2.86)
ν,σ,λ=1
(p)
where the bn are the eigenvectors of Amn. Quantization of this Hamiltonian
involves replacing the position operators with their ladder operator representation.
Ignoring terms that have zero matrix elements for population transfer between
some initial and final state, the Hamiltonian can be written in the interaction
representation ∑asN [ ]
Ĥi ≈ −
D
ℏ ijk (−) (+)3ϵ ω √ 2â (b̂†b̂ + ĉ†ĉ )eiδ ωzt + â (b̂†b̂† + ĉ†ĉ†)eiδ ωzt3 k j j k + h.a,
4 γ i i i j i jiγjµk
i,j,k=1
(2.87)
where ϵ characterizes the strength of the non-linear interaction, and γi,µk are the
radial and axial eigenvalues and is given[by ]
√1 ℏω3ϵ = . (2.88)
4 2 α2 2fscMc
√ √
The mode coupling resonance condition is given by δ(±) = γi ± γj −
√
µk.
This condition puts a constraint on what α can be for resonant mode coupling.
Resonances can only occur for
16µk
α = . (2.89)
4µ2k + µ
2
i + µ
2
j − 8µk + 4µkµi + 4µkµj − 2µiµj
There is also a bound placed on what α can be. When α is too large, the matrix
Bmn is no longer positive definite, which is the result of unstable transverse
oscillation modes. This is the boundary for when a linear crystal transforms into
a ”zig-zag” type structure. There is also a minimum value of α for which resonant
50
Figure 2.6. (a) shows the phase boundary for trapped ions. Above the critical
value of α, the crystal is no longer linear. Each of the green x’s represent a possible
resonant mode coupling from the Kerr interaction. There are no resonances below
the minimum value of α. (b) shows the non-dispersive regime where coherent
exchange of phonons occur between coupled modes. This simulation is of a 3-ion
crystal with the radial mode prepared in the n = 1 or n = 2 Fock state and the
axial mode prepared in a n̄ = 0.01 thermal state.
mode coupling is no longer possible. This minimum value is given by 16µ N if N is oddµN(9µ + 2µ − 8) + µ2N N−1 N−1
αmin =

(2.90)
 4 if N is even
3µN − 2
The constraint on α limits the modes that can resonantly couple; Fig.2.6.
shows the boundaries for this coupling. There are also two different types of
resonances given in this problem. δ(−) is associated with the creation of a radial
phonon and simultaneous annihilation of both a axial and a radial phonon and is
referred to as a resonance of the first kind. δ(+) is the detuning from resonances of
the second kind which creates two radial phonons in different modes by annihilating
an axial phonon, with the resonance condition conserving energy. The coupling for
this type of resonance tends to be exceedingly weak (Marquet et al., 2003).
51
There are several applications for the Kerr interaction. It could be used as
a non-Gaussian operation, but other possible uses could include non-destructively
reading out the motional modes of the ions S. Ding, Maslennikov, Hablützel, Loh,
and Matsukevich (2017b). There are also different regimes of this interaction, near
resonance with δ ∼ 0 and δ ≫ ξ. In the first regime there is coherent exchange
of phonons between coupled modes. The second is the dispersive regime S. Ding,
Maslennikov, Hablützel, Loh, and Matsukevich (2017a); S. Ding et al. (2017b);
Home, Hanneke, Jost, Leibfried, and Wineland (2011); Nie, Roos, and James (2009)
where there is no coherent exchange of phonons, and instead there is a frequency
shift of the coupled modes relative to the occupation number of that coupled mode.
I built out a set of simulations using the QuTiP python package Johansson, Nation,
and Nori (2013) to fully characterize this interaction and understand the quantum
mechanical evolution of states and extended the theory to include mixed-species ion
chains. Evolution Under this Hamiltonian is shown in Fig. 2.7, which highlights the
Wigner negativity associated with non-Gaussian states.
52
Wigner function
1.0
4
0.8
2
0.6
0
0.4
2
0.2
4
0.0
0 2 4 6 8 10 4 2 0 2 4
Fock number x1
Wigner function
1.0
4
0.8
2
0.6
0
0.4
2
0.2
4
0.0
0 2 4 6 8 10 4 2 0 2 4
Fock number x2
Figure 2.7. a) Number probability distribution and Wigner function for the axial
mode highlighting the non-Gaussian nature of the state of some characteristic
interaction time with the radial mode b)radial mode number probability
distribution and Wigner function shown that only even Fock states are populated
and more clearly showing Wigner negativity.
2.6.3 Tri-squeezing. The final non-Gaussian state worth mentioning
is the tri-squeezed state. Generation of this state could be pursued with methods
similar to parametric modulation. In this case, a cubic potential would need to be
applied where additional trap electrodes have been used to implement this type
of potential for mode coupling (Hou et al., 2022). Modulation of this potential at
3ω leads to only population of the |3n⟩ states and has a 60o rotational symmetry
53
Occupation probability Occupation probability
p2 p1
in phase space. Only a little exploration of this state has been done previously,
and some using laser field interactions (Sutherland & Srinivas, 2021) have been
shown Băzăvan et al. (2024), but knowledge of applications is underdeveloped.
Wigner function
1.0
4
0.8
2
0.6
0
0.4
2
0.2
4
0.0
0 10 20 30 4 2 0 2 4
Fock number x
Figure 2.8. Number probability distribution and Wigner function for the tri-
squeezed state showing the 3-fold rotation symmetry
54
Occupation probability
p
CHAPTER III
ATOMIC SPIN QUBITS IN OSCILLATORS
Any charged particle can generally be trapped in a linear Paul trap with the
proper choice of trapping parameters. For small excitations, a particle trapped in
this potential can generally be treated as a quantum harmonic oscillator. Alkaline
earth elements are most commonly trapped in ion traps, due to their simple
electronic structure, with a single valence electron for singly ionized species. This
structure makes state discrimination straightforward. Choice of atomic isotope
is usually taken from the stable isotopes (longer lived radioactive isotopes can
be used) that exist with reasonable abundances. The choice of isotope is also
important for encoding qubits as many isotopes have nuclear spin that gives rise
to hyperfine structure. All of these considerations dictate the experimental setups
that are necessary to prepare, control and readout the chosen ion.
3.1 Calcium-40
The ion work described in this thesis has been done entirely with 40Ca+.
This isotope has 20 protons and 20 neutrons which give it zero nuclear spin
meaning no hyperfine structure has to be considered. This species has transitions
between different orbitals that are at wavelengths where semiconductor diode lasers
are available. Calcium ions have a ground state S1/2 manifold, short-lived P1/2 and
P3/2 manifolds and metastable D3/2 and D5/2 manifolds. Transition wavelengths
and Zeeman sub-levels are shown in Fig. 3.1
55
 +3/2
+1/2
 - 1/2
 - 3/2
P3/2
+1/2
P  -1/2 1/2  +5/2
 +3/2
+1/2
 - 1/2
 - 3/2
 - 5/2
D5/2
  +3/2
+1/2
 - 1/2
 - 3/2
m
32
 n D
7 3/2
mJ= +1/2

S mJ= -1/21/2
Figure 3.1. Atomic structure of 40Ca+ showing different orbitals with their Zeeman
sub-levels, which are non-degenerate when a magnetic field is applied. Also shown
are the wavelengths of the transitions. OMG labels the different qubits that can be
used that make use of different states.
3.2 Qubit encodings
Given the large number of electronic states that exist within atoms, the
choice of qubit encoding is an important decision. The very short lifetimes of
certain states makes them unsuitable for qubit levels. The common encodings
used in atoms, with a similar structure to a calcium ion, fall into what is called the
‘OMG’ architecture (Allcock et al., 2021). How the different encodings work and
what the letters in this acronym stand for are explained in the following sections.
3.2.1 Ground-state qubits. The G in OMG stands for ground
state qubit. Ground state qubits are generally split by MHz-GHz depending on
56
393 nm
397 nm
729 nm
 nm
854
50 n
m m
8  n866
if these are Zeeman qubits or hyperfine qubits. These qubit states can be coupled
with microwaves or with two-photon stimulated-Raman transitions. In order for
the state of the qubit to be determined, these qubits require “shelving” of one of
the states to another level that does not participate in the cycling transition. A
cycling transition uses a closed number of electronic levels and enables continuous
absorption and stimulated emission in order to collect photons. This enables the
ability to discriminate which qubit state is “bright” or“dark”. This type of shelving
naturally requires the use of another metastable state that does not decay for the
duration of the state readout. The ability to drive this transition coherently leads
to another natural qubit encoding.
3.2.2 Optical qubits. Optical qubits, or O qubits, generally consist of
one qubit state being in the ground state manifold with the other being encoded
in a metastable manifold, separated by the energy of an optical photon. The
transition for this qubit in 40Ca+ is an electric quadrupole transition with a narrow
linewidth, given the relatively long life of the metastable state, and can be directly
coupled with a single optical wavelength photon. The laser used for coupling these
qubit states generally needs to be linewidth narrowed by locking to a high finesse
cavity. Readout of optical qubits does not require any extra shelving steps as the
“bright” and“dark” states are naturally distinguishable. Having one state, with
optical qubits, or two states, with ground state qubits, that are naturally coupled
to the cycling transition means that operations that involve fluorescence couple
directly to qubit states. This is a technical difficulty when using multiple ions of
a single species as all qubits are projected when performing state detection. Many
quantum computing implementations us two atomic species with spectrally resolved
transitions (Home, 2013) to address this problem. In order to get around this
57
limitation, it is possible to encode both qubit states into a manifold where both
qubit states are not affected by beams that drive fluorescence.
3.2.3 Metastable qubits. Qubit states encoded into metastable
states, M qubits, require either Raman or microwave couplings, with qubit
splittings also in the MHz-GHz range, similar to G qubits. This encoding scheme
enables the ability to preserve quantum states when attempting to drive the
cycling transition, but has some other technical difficulties. Zeeman qubit energy
levels are made non-degenerate with the application of a magnetic field, but qubit
splittings are naturally degenerate. This is not an issue in the ground state with
only two Zeeman sublevels, but with more Zeeman sublevels in the metastable
manifold this would be a problem. With degenerate energy splittings it would be
impossible to only drive a two level transitions in the metastable manifold. The
highest or lowest two energy states can be isolated with the application of an AC
Stark shift that couples only to the other four levels (Sherman et al., 2013). State
discrimination also requires moving population from one qubit state out of the
metastable state, called deshelving. We have done significant work to implement
qubits in metastable states with the ability to perform single qubit operations and
perform state detection. The specific scheme for how this encoding works is shown
in Fig. 3.2
3.3 Zeeman qubits in the D5/2 Manifold
∣∣ For the work pr〉esented here,∣∣our qubit is enco〉ded as follows:|1⟩ = D5/2,m = +5/2 and |0⟩ = D5/2,m = +3/2 . All the energy levels in
isotopes without nuclear spin are linearly dependent on the magnetic field, for
the magnetic fields used in this work. This dependence of the qubit states on the
magnetic field makes any qubit encoding first order sensitive to fluctuations in the
magnetic field. This makes working with Zeeman qubits more difficult given the
58
shorter coherence times without effective magnetic shielding in our setup. However,
these qubits are straightforward for us to prepare and control, as compared to
43Ca+ which has 48 D5/2 states, and are sufficient for our experiments in this
work, which makes this qubit an ideal candidate for initial exploration of using
metastable qubits.
Figure 3.2. The D5/2 Zeeman qubit. The qubit is isolated with 854 nm light shift
beam and the gates are driven with the 976 nm Raman laser beams, or an RF
magnetic field (not shown).
3.4 State preparation
Coherent qubit operations are the basis for performing quantum information
experiments. In atomic systems we do not have a simple two-level system, requiring
that we first must address the problem of population outside the qubit manifold.
59
Optical pumping is the method used independent of encoding in O,M, or G. O and
G have the same preparation scheme as they both involve the same ground state
level for one of the qubit states. This process can happen with a single round of
optical pumping.
Optical pumping requires using a transition where the population ends in
a fixed final state and cannot leave it on the timescale of the experiments. The
cycling transition in 40Ca+ involves applying 397 nm laser radiation on the S1/2 →
P1/2 transition. Unfortunately, without another laser the population would not be
entirely in a single state because 6% of the population will decay from the P1/2 into
the D3/2 manifold. This can be addressed with 866 nm laser radiation to re-pump
population from D +3/2 → P1/2. Choosing the 397 nm light to be σ polarized and
the 866 nm light to have σ+/− components, leaves all population in S1/2 mj = +1/2.
This is the final step needed to get the O and G qubits prepared.
There are a few more rounds of optical pumping necessary to prepare
the metastable qubit without using a 729 nm laser to drive the quadrupole
transition (Sherman et al., 2013). After preparing mj = +1/2 in S1/2, 393 nm
radiation polarized with σ+ will drive population to highest energy Zeeman level
in P3/2, mj = +3/2. Population will only decay into the D5/2 manifold ≃5% of
the time, spreading into mj = +5/2,+3/2,+1/2 depending on the polarizations
chosen. All population can be pumped to the mj = +5/2 by using 854 nm π
polarized radiation to re-pump the other states. Given the probabilistic nature
of this process, repeating the whole sequence multiple times results in high fidelity
state preparation, this sequence is shown in Fig. 3.3 for the qubit of choice in this
work.
60
Figure 3.3. [Figure made by Alex Quinn]Laser and RF operations on the 40Ca+
Zeeman qubit in the D5/2 manifold. The preparation pulse sequence is performed
N times, until preparation fidelity has saturated and further rounds of pumping
don’t significantly change the fidelity. The qubit operations (using the 976 nm
Raman beams) may then be performed while the 854 nm light shift beam is on. To
read out the qubit, we first shelve the m = +3/2 population in the m = +1/2
state and then deshelve this population with an 854 nm π-polarized laser beam.
We then check for population in the S1/2 and D3/2 manifolds with the 397 nm and
866 nm laser beams. Finally, we can pump into the S1/2 manifold with the 854 nm
σ−-polarized laser beam.
3.5 Single qubit rotations
The first step in realizing quantum gates is performing single qubit
rotations. This primitive is the one of the requirements for being able to perform
universal computations and is a stepping stone to performing two qubit gates.
These qubit operations require the ability to couple two energy states of a quantum
system in order to transfer population from one state to the other. The method
used to couple these states has a large impact of the performance of the quantum
computer. The three common methods have yielded near perfect single qubit
operations (Ballance, Harty, Linke, Sepiol, & Lucas, 2016; Srinivas et al., 2021),
but can have different potential benefits or drawbacks. What follows is a discussion
of two methods used in the experiments described here.
3.5.1 RF magnetic fields. Using RF radiation to couple qubits
states is a very attractive approach given that this method is straight forward
61
to implement in the trapped ion platform. Additionally, this method does not
suffer from issues arising in steering the optical beams towards trapped ions
and controlling the laser frequency, phase and amplitude noise. Addressing
individual ions with this technique is non-trivial, but some techniques has been
demonstrated (Srinivas et al., 2023). In order to use RF radiation we use a trap
electrode that runs along the trap axis. A current is passed through this electrode
which generates a dynamic magnetic field B(x, t) = cos(ωbt)B(x) with a gradient
of the amplitude perpendicular to the trap axis. The ion and the magnetic field
are coupled via the matrix element µx of the magnetic dipole moment. Writing the
magnetic field with position x about the equilibrium position of the ion xj plus a
small displacement ∆x we have B(x) = Bj + ∆xB
′. We can write the Hamiltonian
corresponding to the qubit’s harmonic motiona nd coupling to the magnetic field as
ℏω0
H = σz + ℏωna†a+ σx cos(ωbt)µxBj + σx cos(ωbt)µxB′[q(a† +√a)], (3.1)2
with Pauli operators σz and σx, mode angular frequency ωn and q = ℏ/(2mωn)
being the ion’s spatial extent in the ground state. Using the RWA this Hamiltonian
is re-written as
H = ℏσ (Ωe−iδt + Ω ae−i(δ+ωn)t+ n ) + h.c. (3.2)
with the detuning δ = ωb − ω0 and the Rabi frequencies Ω = Bjµx/(2ℏ) and
Ω = B′n µxq/(2ℏ) (Ospelkaus et al., 2011). When the RF frequency ωb is brought on
resonance with the qubit frequency the first term in the above equation dominates
the dynamics. This enables arbitrary single qubit rotations by simply tuning the
field strength and/or interaction time. The second term corresponds the motional
sidebands, which gives a method for a coupling between the internal states and the
oscillator states given by σ+a. This coupling is only possible with a magnetic field
gradient B′ and requires a large magnetic field gradient to achieve a strong coupling
62
which is not accessible in our setup. The effects of spin-motion coupling will be
discussed in later sections.
3.5.2 Raman transitions. Despite the advantages of using RF
radiation for coupling two levels systems, the long wavelength of this radiation
means a strong magnetic field gradient is required to enable spin-motion coupling.
This coupling is the method for generating spin-spin entanglement and requires
more complicated trap engineering as compared to laser methods. Laser beams
that drive stimulated-Raman transitions can be used instead, which have shown
some of the highest fidelity one and two qubit gates (Ballance et al., 2016). Besides
the technical challenges of using laser beams, it is also important to understand
the off-resonant spontaneous Raman scattering from using these beams (Ozeri et
al., 2007). This effect is suppressed at large detunings and we will explore below
how this coupling works to better understand the limitations. This type of two
level coupling requires an third excited state to exist that can be adiabatically
eliminated in the description by proper choice of laser parameters. The existence
of the excited state is what gives rise to the possibility of stimulated-Raman
transitions. The most common arrangements of these levels is the Λ scheme, where
the two lower energy states (ground states) have a much smaller splitting than the
detuning from the excited state. The Hamiltonian for this coupling can be written
as
H = ∆1 |2⟩ ⟨2|+∆2 |3⟩ ⟨3|+ Ω12(σ1eiδt + σ†e−iδt1 ) + Ω
†
32(σ2 + σ2), (3.3)
where we can choose |1⟩ to be at zero energy. |2⟩ is the excited state and |3⟩ is the
other “ground” state. ∆2,3 are detunings of the lasers with respect to the dipole
transition frequency to the excited state. This energy structure and laser tuning
63
can be seen in Fig. 3.4. There is a two-photon resonance when δ = 0 or
ωL1 −
E1 − E2
ωL2 = (3.4)ℏ
where ωL1,2 is the laser frequency of the field coupling |1⟩,|3⟩ to |2⟩. The single
photon Rabi frequencies are Ω12,32 which drive transitions with operator σ1,2 =
|1, 3⟩ ⟨2|.
Population is prevented from getting into |2⟩ with successful adiabatically
elimination of this state. When the detuning from the intermediate excited level
is sufficiently large this state can be ignored and the problem is reduced to an
effective two-level system. This is qualitatively argued by thinking about the
time-energy uncertainty principle such that when the atom transitions to this
“virtual” state at energy ∆, it remains in this state for a time ∼ 1/∆. In the
event that ∆ ≫ Ω12,Ω3,2, δ this time is much shorter than the evolution time for
|1⟩ → |3⟩. This makes the dynamics dominated by the slow dynamics such that the
probability amplitude in |2⟩ adiabatically follows that of states |1⟩ and |3⟩.
∆ ∆21 δ
ω32
ω12
Figure 3.4. A three level Λ type structure with states |1⟩ , |2⟩ , |3⟩. Transition
frequencies between |1⟩ , |3⟩ and |2⟩ are ω12, ω32 with coupling lasers detuned by
∆1,∆2 and relative detuning δ
.
64
3.6 Spin-motion coupling
So far the motion of the ion hasn’t been thoroughly addressed when
considering coupling of a two-level system. Ignoring the motion is reasonable to
do when treating the ion like a point particle with a fixed location given by its
equilibrium position. In reality there ion position is described by a wavefunction
with a non-zero spatial extent such that the ion feels a position dependent force.
Even with this consideration it is possible to ignore this force if we assume the the
wavelength of the applied field is much larger that the wavefunction extent of the
motional state, such that the ion sees no effective gradient. This consideration of
the ion motional wavefunction and the wavelength of the field factor into what is
called the Lamb-Dicke regime (Meekhof et al., 1996).
3.6.1 Lamb-Dicke regime. In one dimension, if we consider the
position of the ion x, we can write the coupling of a field ion about the equilibrium
position as
kx = kxeq + η(a+ a
†) (3.5)
where
η ≡ kx0 √ (3.6)
with k being the wave vector of the field and x0 = ℏ/2mωx being the spread
of the ground state wavefunction of the ion with mass m. The energy transferred
to the ion from a photon of the field absorbed by the atom is given by ℏω 2xη =
ℏ2k2/2m. The Lamb-Dicke parameter η therefore describes how strongly the field
couples to a mode of the ion. Operation within the Lamb-Dicke regime is satisfied
as long as ⟨ψ|k(x− xeq)|ψ⟩ ≪ 1 where |ψ⟩ is the motional state wavefunction. This
criteria is to just re-state that the amplitude of the ion’s motion is small compared
65
to the wavelength. This criteria is essential for the validity of the Hamiltonians we
use to describe the spin-motion coupling.
When the internal levels are coupled by an electric field the Hamiltonian is
written as
H = −µ · E(x, t) = ℏΩ(σ + σ )(ei(kx−ωt+ϕ) + e−i(kx−ωt+ϕ)d + − ) (3.7)
with electric dipole moment µd. When moving into the interaction representation
we get ( )
HI = −ℏΩσ+ exp i[η(ae−iωxt + a†eiωxt)− δt+ ϕ] + h.c. (3.8)
with δ ≡ ω − ω0. Now, considering the Lamb-Dicke regime we can make the
approximation that ( )
exp iη(ae−iωxt + a†eiωxt) ≈ 1 + η(ae−iωxt + a†eiωxt). (3.9)
With this approximation we get effectively three different terms we can identify,
each with their own unique effects.
3.6.2 Carrier. When δ = 0, the first order term dominates and the
other terms can be ignored making the RWA. This term simplifies to
H iϕ −iϕcarrier = ℏΩ(σ+e + σ−e ) (3.10)
which drives population between qubit states without affecting the motional state..
3.6.3 Red sideband (RSB). When δ = −ωx the carrier term retains
a time dependence which will now be ignored with the same RWA when ωx ≫ Ω.
There are second order terms that no longer have time dependence that are given
by
HRSB = ηℏΩ(σ aeiϕ+ + σ−a†e−iϕ). (3.11)
When starting in a lower internal energy state, this Hamiltonian leads to a flip
of the qubit state to the higher energy state while subtracting a phonon from the
66
motion. This transition occurs with Rabi frequency
√
Ωn→n−1 = Ω nη (3.12)
which is lower than the carrier Rabi frequency when η ≪ 1.
3.6.4 Blue sideband (BSB). The last resonant condition for second
order terms is when δ = ωx. The resulting Hamiltonian is now
HBSB = ηℏΩ(σ+a†eiϕ + σ ae−iϕ− ), (3.13)
which now when starting in the lower energy state, drives a state flip and adds a
phonon to the motion. The Rabi frequency for this transition is given by
√
Ωn→n+1 = Ω n+ 1η. (3.14)
These Hamiltonians are known more commonly as the Jaynes-Cummings and anti-
Jaynes-Cummings Hamiltonians and are very important for engineering many
interactions in trapped-ions and other platforms (Shore & Knight, 1993). In our
work, this interactions will be the foundation of how to determine the motional
state, but they can be used to create spin-spin entanglement and be used for a host
of other interesting physics with longer ion chains.
3.7 State detection
State readout essentially entails turning all qubit encodings into optical
qubits. This is required because it enables a state-dependent fluorescence. In order
to perform state detection with metastable qubits one of the qubit states must first
be deshelved. The |↓⟩ qubit state is dehselved using a pulse of 854 nm laser which is
π polarized meaning the laser does not couple to the |↑⟩ state. Given ∼ 5% of the
population will return the the D5/2 mj = +5/2,+3/2,+1/2 states after decaying
from the P3/2 manifold, shelving the |↓⟩ population in the mj = +1/2 is required
to prevent a readout error. This sequence is shown in Fig. 3.3. After the |↓⟩ state is
deshelved the cycling transition with 397 and 866 nm lasers, will make population
67
in this state fluoresce, making this the “bright” state and leaving any population
in the metastable manifold “dark”. The ground state is “bright” because we can
collect and count photons at 397 nm.
In order to assign a state based on the number of photons that are counted,
a threshold is assigned (Myerson et al., 2008). The bright state will scatter photons
that are collected with some efficiency η at a rate of RB, but detectors also have a
dark rate detection rate of RD. These fixed rates provide Poissonian statistics given
by
λne−λ
P (n) = , (3.15)
n!
where n is the number of photons and λ = Rtb, the product of the detection rate R
and the detection time bin tb.
There are inherently errors in this method of state detection. The average
readout error is ϵ = (ϵB + ϵD)/2, where ϵB is the fraction of events where an ion
prepared in the bright state was measured to be in the dark state (likewise for ϵD).
A statistical treatment is used to determine the threshold for how many photons
there need to be for the ion to be considered bright. This threshold is chosen based
on the intersection of the bright and dark Poissonian distributions.
tb(Rb −Rd)
nthreshold = (3.16)Rb
ln ( )
Rd
This intersection point shows how longer detection times can help to reduce
readout errors, but will ultimately be limited by the lifetime of the qubit. Larger
bright count rates and longer detection time bins helps separate the means of the
Poissonian distributions.
This method can be extended to more than 1 ion but does not provide
the ability to distinguish which ion is bright, only the number of bright ions. The
thresholds are decided again with the intersection of Poissonian distributions given
68
by the count rate for the number of bright ions, and can be seen in Fig. 3.5 with
a 2ms readout time. The overlap is not equivalent for each distribution and can
become significant with a larger number of ions as the distributions spread more
and more. For long ion chains this method will lead to significant readout errors
and other techniques are required. For all the work in this thesis this method
provides readout fidelity >99% for 1 ion.
0.25
0.20
0.15
0.10
0.05
0.00
0 50 100 150 200 250 300
Photon counts
Figure 3.5. Histogram of counts recorded in 100 shots across 81 iterations of an
arbitrary experiment. The red curves show the theoretical Poissonian distributions
for a given count rate and detection period of 1 ms. Black vertical dash lines show
the intersection points between the 0/1 bright state and the 1/2 bright state.
3.7.1 Mode thermalization. The cycling transition, which used
to discriminate between bright and dark states, means that millions of photons
are scattered from the ion every second. This is a very powerful technique for
determining the internal state of the ion, but there are consequences for the
motional states of the ion. Motional states, that have been prepared into some non-
thermal state, are driven towards a thermal state by scattering of photons when a
laser is tuned to the red sideband. We have already seen that the recoil energy is
69
Count probability
the energy given to the oscillator from the photon and depends on the mass and
frequency of the ion and illuminating field.
This effect of photon scattering makes it impossible to preserve motional
states and perform readout of qubit states. We will consider in Ch. VII how this
problem can be overcome with longer ion chains for particular geometries. In
Ch. IV we will explore how this same effect of photon scattering can be used to
cool motional states that start in a relatively large thermal state.
70
CHAPTER IV
EXPERIMENTAL SETUP
Many experimental systems are used in order to trap, cool, manipulate and
detect the state of ions. An easy to follow discussion of how an ion trap works
can be found in in (Foot, 2005) and will not be exhaustively covered here. This
chapter will provide descriptions and applications of experimental systems that
enable quantum information processing.
4.1 Rod trap
Ions, in this work, are trapped using a room temperature linear Paul trap in
our lab with a schematic shown in Figure 4.1. A view of the physical trap is given
in Figure 4.2. The ion-electrode separation ‘r0’ is 0.75mm, the needle-to-needle
spacing ‘a’ is ∼3mm, and the rods have a diameter ‘2re’ of 0.5mm. In order to
minimize the rate of background gas collisions, we keep the trap under ultra-high
vacuum, reaching pressures of order 10−11Torr.
71
Figure 4.1. (a) Schematic view of our linear rod trap along the trap axis. The
gold circles are cross-sections of the different rods in the trap. In the figure, re is
the electrode radius and r0 is the trap axis to electrode surface distance or trap
radius. Two opposing inner rods are connected to the oscillating voltage Vrf at
frequency ΩT with offset Ur and the other two inner rods are held at ground. (b)
Schematic side view of the linear rod trap where a is the tip-to-tip distance between
the needles and the yellow dot is an oversized ion giving its approximate location in
the trap. Following the axes in figure (a), the upward direction in (b) is the (x + y)
axis and the out-of-the-page direction is the (x− y) axis.
72
Figure 4.2. (a) The assembled rod trap. (b) The rod trap mounted in the vacuum
chamber, as seen through a viewport.
73
4.2 RF electronics
This section discusses how the trapping potential is generated and the
requirements for stable trapping, without getting into the full theory of how an
ion trap works. The potential to trap an ion of mass m and charge e can be written
as
e2V 2η2
ψ = 0 (x2 + y2). (4.1)
4mr4Ω2rf
Ωrf is the RF angular frequency of the trapping potential, r is the ion electrode
separation, V0 is the amplitude of the RF voltage and η is an efficiency factor that
depends on the geometry of the trap. This true potential provides a ponderomotive
pseudopotential with an oscillation secular frequency of
eV0η
ωs = √ (4.2)
2mr2Ωrf
The equations of motion of the ion is given by the Mathieu equations and have
stability criteria that depend on several of these parameters. For common masses
of atoms used in an ion trap, the trapping potential must be an AC potential,
in radio frequency band, for stable trapping. Depending on the mass of the
ion, the geometry of trap, the trap frequency and desired secular frequency,
necessary trapping voltages for stable trapping can vary from V to kV. Ion traps
can trap room temperature atoms with trap depths of order 10 eV which enable
days long ion lifetimes. Increasing the trapping voltage also increases the mode
frequency which can enable more efficient cooling of motional modes. Working with
excessively high voltages should be avoided to prevent electrical breakdown and
experimentally intrusive temperatures due to power dissipated in the trap.
4.2.1 Helical resonator. The helical resonator used in these
experiments is a quarter wave resonator, meaning one end of the coil is ground
and the other end is at the voltage maximum. This resonator has an antenna
74
coil on the input that is is inductively coupled to the main coil, which allows for
impedance matching with a high ratio of output to input voltage. A large voltage
step up and narrow bandwidth of the resonator are desired to lower the input
power requirements and reduce noise coupling to the trap.
The length and diameter of the resonator and main coil length are all
parameters that can effect the resonant frequency and the quality factor of the
resonator. After testing, a 6” diameter resonator provided a suitable quality
factor and resonant frequency. The design follows a lumped circuit element
model (Siverns, Simkins, Weidt, & Hensinger, 2012), but manual adjustment of
antenna and resonator coils is necessary to get the best possible quality resonator.
The resonator was characterized using a tracking generator coupled to the
resonator and looking for the resonant drop in reflected power on the spectrum
analyzer. The quality factor is given by the full width at half max (-3 dB) of the
resonance. The voltage step up is measured as a ratio of the input to output
voltage levels. Capacitive loads, like a scope probe, changes the impedance, so a
calibrated capacitive divider was used to reduce the capacitive load to a negligible
level. We have achieved a quality factor as high as 150 and a step up of nearly 80
with a resonant frequency of 14.13MHz. With a 1W amplifier you have ∼10 V
amplitude in a 50Ω system. With a voltage step up of 80 this gives approx 0.8 kV,
and a ∼1.8MHz radial secular frequency.
4.2.2 Squareatron amplitude stabilization. Running an RF ion
trap for coherent control of ions, with long motional coherence times, is not as
simple as connecting a waveform generator to the helical resonator when using
radial modes. While this method works when using axial modes that are not
coupled to this potential, waveform generators alone can introduce significant noise,
which leads to motional decoherence. The goal is to achieve ultra-low noise RF
75
signals, which provide the trapping strength and enable the motional coherence
time we need. Amplitude modulation (AM) noise leads to fluctuations of the
trapping frequencies, and causes motional decoherence. The motional coherence
time τc, assuming white noise, is related the the AM noise power spectral density
NAM0 (Harty, 2013) by
AM
τ−1
1 2N0
c = ωr . (4.3)2 Pc
Pc is the power in the RF carrier and ωr is the motional frequency of the mode of
interest.
4.2.3 Filtering AM Noise (Squareatron 5000). The Squareatron
5000 is a circuit that filters the AM noise of a clock source and produces an
ultra-low noise RF tone (10s of MHz) and was originally designed at Oxford
University (Harty, 2019). Modifications were made to the original design to
improve performance and enable greater flexibility. The circuit diagrams are
included in Appendix A. The board is designed to use a highly saturated amplifier
to “clip” the input signal, producing a constant amplitude square-wave. This
element is combined with a pass band filter to strip the harmonics and output the
original frequency sine wave with the AM noise greatly reduced.
The basic design of the circuit involves using a low phase noise
logic converter (LTC6957) (“LTC6957 Low Phase Noise, Dual Output
Buffer/Driver/Logic Converter”, 2017) on the input and a low drift precision
reference to have a stable power supply. This combination of devices generates a
low noise output with no way to tune the output amplitude.
4.2.4 Tuning board. Having the ability to tune the amplitude of
the output directly allows for tuning of the ion trap secular frequencies. This
will be critical for CVQC in as mode frequency tuning is required for operations
76
Figure 4.3. (a) The setup for measuring AM noise is shown (Enrico, 2006). A
calibration source at frequency νs is used calibrate the power spectral density
(PSD) to be in terms of the RF carrier power at frequency ν0. A power splitter
splits the signal into Pa and Pb and two diode detectors are used to pick out
the AM noise from the carrier. The two channel setup allows for a correlated
measurement and gives a noise level below the noise floor of each detector. (b)
shows the measured AM noise power densities typical lab devices. The right
hand ordinate gives the motional coherence time for a 3MHz mode that would
be obtained with white amplitude noise at the level given by the left hand ordinate.
like coupling modes with the Kerr interaction, amongst other operations. To
achieve this goal, I have designed and tested a circuit board that attaches to the
Squareatron (Metzner, 2020). A 16-bit digital to analog converter (DAC) has a
reference supplied by the stable power reference from the Squareatron and can be
programmed to adjust the power rail of the logic converter. The circuit diagram
is shown in Appendix A. This additional circuit was tested and has shown that
it does not inject noise into the Squareatron. We would like to have coherence
times that are longer than the heating rate for the mode of interest. This would
require coherence times of at least 100 ms for our heating rates, which are shown in
section 4.4.3. Fig. 4.3 shows the motional coherence times above this threshold and
orders of magnitude better than typical lab devices.
4.2.5 Amplifier. The necessary power to trap an ion is well above
what the Squareatron can output. The helical resonator, described in section 4.2.1,
77
provides a significant voltage step up, however, an amplifier is still necessary
after the Squareatron to get to the power levels we need. The Minicircuits ZX60-
100VH+, 36 dB gain amplifier is used to get to the desired powers. The amp is
connected to the Squareatron with attenuators to avoid compression and output
over 30 dBm of power. To increase the secular frequencies, which might be desirable
for faster operations involving the motion, even more power is necessary, which
likely also requires an increase in the trap RF frequency for stability. To achieve
this, we have designed a 4 Watt amplifier that has two four-way splitters and
four amplifiers. The splitters divide and combine 6 dB of power coherently before
and after the amplifiers; this allows us to run at a higher output power without
compression. This parallel amplifier has 1 dB compression at 34 dBm out power
and saturates at 38 dBm. The AM noise test with both of these amplifiers show the
noise levels are not not noticeably above the levels without an amplifier. Note that
the 4 amp configuration divides the additive AM noise from the amp by 4 (Scott,
2015).
4.2.6 Temperature stabilization. After assembly of the Squareatron
and amplifier was completed, the combined temperature coefficient was measured
to be -1300 ppm/k, which was determined by measuring the change in the output
power as a function of the temperature (controlled with a hotplate). Power
fluctuations are very important to stabilize due the the direction conversion from
RF power to trap frequency. Fast fluctuations can lead to motional decoherence
and slow drifts need to be calibrated constantly. We have measured a secular
frequency response of -3Hz/mK, which matches closely to the measure temperature
coefficient. This means we require we require few mK temperature stability to the
frequencies from drifting significantly on the time scale of the experiment. Fig.
4.12a) shows a housing that was constructed that hosts a Peltier TEC connected
78
to a thermostat. Both the Squareatron and amplifier are in thermal contact a
bulk plate of aluminum and the whole box is filled with foam insulation. The
Thermostat (Thermostat , 2022) runs a PID loop that controls the TEC in order
to reach a target temperature. PID loop parameters are automatically tuned to
minimize temperature oscillations. Fig. 4.12b) shows the temperature stability that
was achieved after the thermostat is used.
a) b)
Figure 4.4. a) combined mounting of Squareatron and amplifier in thermal contact
with an aluminium plate. The aluminum plate has a heat sync mounted on top
of a TEC to help stabilize the temperature. The whole mounting is placed inside
and insulated box to prevent airflow from causing temperature fluctuations.b) time
series data of the trap secular frequencies with and without the active temperature
stabilization showing temperature stabilization reducing fluctuations to <100Hz
over hours.
4.3 Laser systems
There are a broad range of lasers required to go from neutral Calcium to
ground state cooled and coherently manipulated ions. A large fraction of the
physical space in the lab can be dominated by laser heads and beam delivery
optics. To minimize the space required for this, we have built rack mounted
optical setups that contain all the lasers and optics to deliver laser beams to
several different traps. This setup also enables active locking each laser to a fixed
79
wavelength and monitoring of the lasing mode. Each of the laser systems used in
the experiment are detailed in the following sections.
4.3.1 Laser rack. All the lab’s laser systems are stored in one room
with three racks: a rack containing drawers which house our Toptica Littrow ECDL
lasers (Model ‘DL Pro’), a rack containing Toptica laser controllers (Model ‘DLC
Pro’) and photoionization laser board, and a rack housing the wavemeter and
laser diagnostic equipment. These rack mounted drawers can be seen in Figure 4.5,
with the standard laser setup inside the rack drawer shown in Figure 4.6. In each
drawer, a Toptica laser in mounted to an optical breadboard. The Toptica laser
parameters are set by the controllers mounted in the adjacent rack. The beam
is redirected, from the laser head, by a periscope to be level with half-inch optics
mounts. In order to prevent amplified stimulated emission a few nanometers away
from the lasing mode from reaching the optical fiber, the beam is then reflected
from a diffraction grating before being sent down a path of polarizing beam
splitters, where half-waveplates allow us to adjust how much power goes down
each arm. These separated beams are then coupled into fibers to be delivered to
the AOM boards in the trap rooms. One of the beams on each board is sent to
the wavemeter and optical spectrum analyzer(OSA) on the OSA board, where the
wavemeter is used to stabilize the laser frequencies.
80
Figure 4.5. The laser racks. Contains our breadboard laser systems used for
ionization, cooling, preparation, and readout.
81
Figure 4.6. The inside of a laser rack drawer. The beam from a Toptica laser is
picked off by multiple polarizing beam splitters and is coupled into multiple fibers
to be sent to the trap and the OSA board.
4.3.2 Photoionization. Photoionization is used in our lab instead
of electron bombardment in part because photo-ionization allows for isotope
selectivity. Photo-ionization also reduces the charging of insulating parts of the trap
which causes drifting electric fields. Photo-ionization has been shown to be ∼ 104
82
times more efficient (Lucas et al., 2004) than electron bombardment, leaving fewer
neutral atoms that are deposited onto the trap. Neutral atoms are loaded into
an oven, made from a steel tube, as granules of calcium. About 4 amps is passed
through the oven heating, vaporizing, and ejecting the neutral atoms from a hole in
the side. The atoms are ejected into the interaction region where the lasers are used
for ionization. Two lasers at 423 nm and 375 nm are necessary. The 423 nm laser
will excite the neutral to a high lying 4s4p1P1 level at which point the 375 nm laser
will take it to the continuum.
An optical bread board that has the 375 nm laser mounted on it and the
423 nm laser brought on from a optical fiber is used to combine these two laser
beams for delivery to the trapping setup. Four different traps can be run by using
beam splitters to pick off a quarter of the 375 nm laser power, shown in Fig. 4.7 and
Fig. 4.8. The 423 and 375 nm lasers are brought together using a dichroic mirror
and coupled into a fiber that is delivered to the trap. Shutters are also mounted in
front of each fiber coupler so the lasers can be shut off once the required number of
ions are loaded into the trap.
83
Figure 4.7. A 3D rendering of the photoionization board showing the 4 possible
trap connections and the optics to combine the 375 nm and 423 nm light.
Figure 4.8. Image of the photionization board mounted in a drawer with two
output collimators and shutters installed.
84
4.3.3 Locking lasers to a wavemeter. Analysis of the lasing mode
and wavelength of each laser is made possibly by fiber coupling the individual lasers
to two different fiber switches. The fiber switches are designed to operate in the
range of the UV and near-infrared wavelengths that the lasers lie within. The
output of the fiber switches are coupled onto the OSA board. This board has two
different scanning Fabry-Pérot interferometers (ThorLabs SA200) for each of the
primary wavelength ranges used in our lab.
An additional fiber coupler couples the light to a wavemeter, via a photonic
crystal fiber (PCF) that is single mode for all wavelengths, to monitor and correct
drift in frequency of the lasers using the Oxford wavemeter analysis and display
software (WAnD) (Oxford, 2019). An error signal is generated, for a proportional-
integral-derivative (PID) loop to stabilize, by recording the actual wavelength
and taking the difference with a given reference wavelength. The laser grating
has a piezo attached which can be fed back on to stabilize the wavelength. The
optimal gain for each laser was found by ramping the gain and monitoring the
laser frequency variance over a 1 minute window. The wavemeter is temperature
stabilized and has proven to drift by less than 100 MHz per year allowing for the
wavemeter to operate without regular calibration with a frequency reference like
the ion.
ECDLs can commonly start lasing on multiple modes, which can be
prevented by monitoring the lasing mode and tuning the laser when necessary. The
laser mode is monitored with the scanning Fabry-Pérot cavities, which is driven
by a piezo electric crystal with a photodiode to collect the transmitted light. The
piezo is used to sweep the length of the cavity which causes light to be transmitted
only when the cavity length is resonant with the laser mode. To operate the OSA
board electronics including a piezo driver and a photodiode amplifier are necessary.
85
These electronics and cavity have been characterized to compare the Finesse to the
cavity specifications, as well as the signal to noise ratio (SNR) of the photodiode
to compare it to the amplifier design spec. The cavity has a calculated Finesse
between 210-270 for both cavities which matches well to the specification. The
photodiode amplifier output is connected to a digital multi-meter (DMM) (Keithley
DMM6500). The DMM is triggered off of the AWG that generates the ramping
voltage which scans the cavity piezo. The measured voltages as displayed along
with the reference detuning on the WAnD software Oxford (2019) The completed
configuration can be seen in Fig. 4.9.
86
OSA OSA
To wavemeter
Collimator
Collimator
Figure 4.9. A picture of the OSA board mounted in a drawer showing the input
collimators, the Fabry-Perot cavities and the output collimator to the wavemeter.
87
4.3.4 Raman laser system. The laser system that enables
metastable coherent operations is different from the other laser systems presented
here. The wavelength of this laser is 976 nm which means it is 122 nm detuned from
D5/2 ↔ P3/2 resonance in calcium. This large detuning reduces the spontaneous
Raman scattering rate, but means a significant amount of power is required to
get two qubit gate operations on a timescale faster than decoherence. While two
photons from a single beam is sufficient to drive Raman transitions, as long as the
polarization has π/σ− components, we built an orthogonal geometry, see Fig. ??, to
have a non-zero ∆k that is in the radial plane. This geometry allows for coupling
to both of the radial modes, but not the axial mode.
The 976 nm laser setup for this work was using a free space (IPS
I0976SB0750B) direct-diode laser. This laser is a volume-holographic (VHG)
hybrid external cavity laser (HECL) mounted in a butterfly package. This package
is mounted on a Koheron industrial laser controller which manages temperature
and laser current. The free space laser is directed into a Newport ISO-04-980-MP
optical isolator to prevent optical feedback. The beam is then split into two paths
with a waveplate and a beamsplitter. Each separate path is focused through an
AOM to have separate switching and frequency control of the individual beams.
There is an optical pickoff on one path which sends light to the OSA board for laser
monitoring. Both beams are then collimated into fibers which are delivered onto
raised platforms on either side of the trap. Piezo mirrors and waveplates provide
the spatial and polarization degrees of freedom to get the desired polarizations
aligned onto the ion.
4.4 Motional state cooling
In order to be able to perform operations with ions with high fidelity, it is in
general necessary to first cool the ions to their motional ground state. This process,
88
Figure 4.10. Optical setup for Raman beam delivery showing the laser head
(HECL) the optical isolator (OI) a shutter, to prevent beam leakage onto the ion
(not used in work presented here), waveplate (WP) and polarizing beam splitter
(PBS) for relative power control in the two arms, acousto-optic modulators (AOM)
for frequency control, and fiber couplers to get the light to the ions.
in trapped ions, generally makes up the majority of the experimental duty cycle
and consists of multiple stages. This section is devoted to discussing the theory and
implementation behind the stages of cooling that are used in these experiments.
4.4.1 Doppler cooling. Doppler cooling is a critical part of trapping
ions, even when not specifically worried about operations on the motional states.
This type of cooling is very fast and enables the very long lifetimes of trapped ions
by preventing heating out of the trap.
The Hamiltonian considered here is the same as Eq. 3.7, which is a simple
Dipole coupling Hamiltonian where the motion has to be taken into account due to
the non-zero wavepacket extent. The excited state in this case has a spontaneous
decay rate Γ, which can be considered in the full Lindblad master equation
discussed in section 5.2. The spontaneous emission has a mechanical effect that
does not effect the ion’s momentum on average due to the spatial symmetry of the
emission (Eschner, Morigi, Schmidt-Kaler, & Blatt, 2003). We have already seen
that this emission does come with a recoil energy Espont = ℏωRα, where ωR is the
emitted photon angular frequency and α describes the average component of the
recoil energy on the motional axis (Wineland & Itano, 1979). Spontaneous emission
has a dipole pattern which does not effect all motional degrees of freedom equally.
89
The spontaneous emission spectrum can be seen by sweeping the laser
detuning across the atomic resonance. In theory this couples a given state |g, n⟩
to several excited states |e,m⟩ with each of their own Rabi frequencies (Eschner
et al., 2003). The additional resonances, which are the motional sidebands, are
separated by the trap frequency which means that the spectral resolution of the
resonances depend on the relation between the line width Γ and the trap frequency
ωx. For Doppler cooling, the 397 nm line is ∼ 21.6MHz wide making the sideband
transitions impossible to resolve (the weak confinement limit). We will consider
only transitions on the first red and blue sideband as the transition rates of higher
order transitions are greatly suppressed in the Lamb-Dicke regime.
In order to consider cooling in this scattering process, we must consider the
rates at which transitions are driven on the red and blue sideband. The rate to go
from |g, n⟩ → |g, n+ 1⟩ is given by R+ = (n + 1)A+ and the rate for |g, n⟩ →
|g, n− 1⟩ is R− = nA−, where
Ω2
A± = η
2[cos2(θ)W (∆∓ ωx) + αW (∆)] (4.4)
Γ
with W (∆) = 1/(4∆2/Γ2 + 1), which is found in Eschner et al. (2003). R± have two
terms that describe either absorption on the sideband and emission on the carrier
or vice versa with their own rates and relative probability. In order to see the
dynamics of the vibrationa∑l state we can look at the rate of change of the average
vibrational number ⟨n⟩ = ∞n=0 n ⟨n|ρ|n⟩
d ⟨n⟩ = −(A− − A+) ⟨n⟩+ A+, (4.5)
dt
found in Eschner et al. (2003).This equation has a steady state solution as long
as A− > A+, which is to say as long as the scattering rate on the red sideband is
greater than that on the blue sideband. This relation is always true when ∆ < 0
(when the laser is tuned to a lower frequency than the atomic resonance). The
90
solution to this equation is given as
⟨n⟩ (t) = ⟨n⟩ (0) exp[−(A− − A+)t] + n̄{1− exp[−(A− − A+)t]}, (4.6)
where ⟨n⟩ (t) is the vibrational quantum number at time t and n̄ = A+/(A− − A+)
is the steady-state value. In the weak confinement limit, we get Doppler cooling
and find the maximum ratio of A−/A+ is obtained at ∆ = −Γ/2 which gives the
smallest steady-state temperature (n̄) (Wineland et al., 1978). With the ideal
two-level system, this steady state temperature is given as n̄ =≃ Γ/2ωx. This
mathematical description of motional heating and cooling will be important in
Chapter VII, where heating rates are measured. Eq. 4.6 can be fit to heating rate
data to extract the Lamb-Dicke parameter.
In the experiment, Doppler cooling is a little bit more complicated. As
mentioned in section 3.6, there is a lower lying D manifold that must be re-pumped
with 866 nm light. There are also multiple Zeeman sub-levels that population is
spread amongst. The natural linewidth is also not observed due to the Doppler
broadening from finite temperatures of the ion (Wineland & Itano, 1979). When
the ion is first ionized, it is very hot and has a linewidth significantly broader than
the natural linewidth. In order to still cool hot ions efficiently a far red detuned
397 nm beam is used. The 397, 866 nm beams are brought in 45◦ to the trap axis
which enables cooling of all modes, seen in ??. Radial mode temperatures, have a
Doppler limit of n̄ ≃ 10 at 1.8 MHz. In order to get bellow the Doppler limit other
cooling techniques are required.
4.4.2 EIT cooling. Electromagnetically Induced Transparency
(EIT) cooling is another method of atomic cooling that can reach near ground
state temperatures (Eschner et al., 2002). This method of cooling is unique from
the method above, in that some slightly different physics is used to alter the
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spontaneous emission spectrum to suppress scattering on the blue sideband, which
leads to heating.
EIT cooling is similar to a Raman transition and involves a three-level
system and involves cancellation of absorption on one transition induced by
simultaneous coherent driving of another transition. This type of dark resonance
is straight forward to realize in trapped ion systems given the large number of
available states (Lechner et al., 2016). The effect arises from an interference
of the two pathways to the excited level. This method suppresses the carrier
transition on |g, n⟩ → |e, n⟩ and enhances absorption on the cooling transition
|g, n⟩ → |e, n− 1⟩ (Eschner et al., 2002). The method for EIT cooling consists
of a ground state |g⟩ that is coupled to a metastable state |r⟩ with an intense
“pump” laser with Rabi frequency Ωr and detuning ∆r = ωr − ωre, where ωre
is the bare atomic transition wavelength. The third state is an excited state |e⟩
which is coupled to |r⟩ with a “probe”(cooling) laser with Rabi frequency Ωg and
detuning ∆g = ωg−ωge. The spectrum for this excited state transition is a Fano-like
profile (Marangos, 1998) with a zero at ∆g = ∆r and is an asymmetric profile when
δr ̸= 0. This profile encapsulates the cooling nature of this setup. When considering
the motion this zero of the Fano-like profile corresponds to the carrier transition
|g, n⟩ → |e, n⟩. When ∆r > 0 with a suitable Rabi frequency the spectrum can be
altered to have |g, n⟩ → |e, n− 1⟩ transition on the maximum of the profile and
the |g, n⟩ → |e, n+ 1⟩ transition lying near zero excitation probability (Eschner et
al., 2002). The strong coupling laser also induces an AC stark shift that shifts the
metastable state by √
δ = ( ∆2r + Ω
2
r − |∆r|)/2. (4.7)
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The carrier transition is eliminated and red sideband absorption is maximized when
∆g = ∆r and δ ≃ ωx. (4.8)
Making use of these dipole transition means that the achievable scattering rates
can be significant and lead to rapid cooling. This types of cooling also couples to
all modes of motion that have overlap with the k vectors of the lasers used. This
means this method is very effective at cooling all modes quickly when compared to
resolved sideband cooling, described below.
In the experiment, |g⟩ |e⟩ are the two Zeeman sub-levels of the S1/2 manifold
and |r⟩ being a Zeeman sublevel in the P1/2 manifold. The pump is a σ+ beam
with the probe beam being π polarized. The π polarized beam couples both
ground state manifolds to both metastable Zeeman states, a large Rabi asymmetry
Ωr ≫ Ωg helps suppress other transitions being driven. Finding optimal beam
tunings can be difficult to determine experimentally. In order to tune up the
experiment a basic type of thermometry can be performed as a function of the
pump and probe detuning. At a set detuning, the metastable qubit can be prepared
and a red sideband driven. In the event that that ion is in the ground state, the
red sideband does nothing. This enables the detuning to be scanned to find which
detuning minimizes the bright state probability as shown in Fig. 4.12. The pump
and probe beam are detuned 150MHz from the atomic resonance, with ideal
cooling when the beams are detuned approximately 4MHz from each other. The
radial modes are cooled from the start mean occupations number of n̄ ≃ 20 to
n̄ ≃ 1 in 1ms with 2.7 uW of power in the pump beam and 0.15 uW in the probe.
93
 m = +1/2 
m = -1/2
P1/2
π
 σ+

S1/2 m = +1/2
m = -1/2
Figure 4.12. a) Level structure of the 40Ca+ transitions used in the experiment for
EIT cooling. Cooling is implemented using the Zeeman sublevels of the S1/2 ↔ P1/2
dipole transition. All polarizations used in the experiment are shown. Grey
transitions do not contribute to EIT cooling. b) Qubit bright state probability
after applying a red sideband as a function of the relative EIT beam detuning.
Probabilities near zero indicate where cooling to ground state is maximized for one
of the radial modes.
4.4.3 Resolved sideband cooling. Resolved sideband cooling
works in a manner similar to the Doppler cooling. When For Γ < ωx (strong
confinement limit) the motional sidebands are well resolved. A laser can be tuned
to the red motional sideband at ∆ = −ωx. We perform sideband cooling after
the metastable qubit has been prepared. The linewidth of the qubit transition can
then be arbitrarily controlled depending on the pulse times and the Rabi frequency.
A π pulse on this transition will subtract a single vibration quanta and flip the
qubit state. In order to dissipate energy from the system a dissipative reset of the
original qubit state is required. With this procedure, very small values of n̄ can be
achieved. For Γ ≪ ωx, n̄ ≃ (Γ/ωx)2 ≪ 1 (Eschner et al., 2003), which shows with
increased mode frequencies it is possible to get even cooler. In our experiment, the
Rabi frequency and pulse times used limit the linewidth Γ of the transition to be
94
∼1 kHz, which means for radial mode frequencies of 1.8MHz, final mean occupation
number of n̄ ≃ 3x10−4 is in principle possible.
The use of metastable qubits makes this procedure slightly more
complicated. The state preparation scheme laid out in section 3.4 means that on
average there is ∼20 photons scattered to prepare the qubit. This means that
preparing the qubit from the ground state of motion will add some vibrational
quanta. The Lamb-Dicke parameter for this scatter is ∼ η = 0.01 and the energy
added will be ∼ η2β, where β is the number of photons scattered. This will limit
the temperature to be n̄ ∼ 0.1.
The full procedure procedure of resolved-sideband cooling involves preparing
the qubit, a π pulse on the red sideband and then using the 854 nm laser to
deshelve population in one qubit state and reset the qubit. This is repeated several
times to prepare a motional state near the ground state. It is possible to get
slightly closer to the ground state by doing a post-selection procedure to project
closer into the ground state. In this case, after the 854 deshelving pulse is used,
most of the population left in the initial qubit state is associated with the ground
state. This means that only running experiments where the qubit is measured
to be in the initial state, after cooling, will give a slightly lower initial average
vibrational quanta. This same process can be repeated several times to get closer
to the ground state. It is important to note that this procedure is much slower than
Doppler cooling. It takes 30µs to prepare the qubit, around 50µs to perform the
red sideband pulse and another 30µs to deshelve the qubit. This whole procedure
is repeated 20 before diminishing returns. This full sequence can take several
milliseconds. In order to speed up the cooling rate EIT cooling is used to bridge
the gap between the Doppler limit and the ground state.
95
A combination of these cooling techniques has enabled us to achieve mode
temperatures of n̄ < 0.15 in 7-8 ms, including Doppler, EIT and resolved sideband
cooling, which corresponds to over 90% in the ground state. Cooling times are
longer than when only cooling a single mode. We have also measured the heating
rate using the sideband ratio technique shown in Bergquist et al. (1987). Heating
rates for the radial modes are n¯̇ = 3quanta/sec.
4.5 Electronic drives
In order to apply the oscillating forces described in Chapter 2, the ion
trap needs to be connected electronically to a source that provides the necessary
potentials. The electronics drives in this section only couple to the radial modes.
RF filters are also implemented to reduce noise that can lead to decoherence of
these motional modes.
4.5.1 Resonant drive. The resonant drive involves applying an
oscillating force at the frequency of the harmonic oscillator. This force (known as
the “tickle”) is implemented by applying an oscillating potential to one of the the
DC electrodes of the ion trap, shown in Fig. 8.1. The potential is provided using a
Urukul direct digital synthesizer (DDS) Kasprowicz et al. (2022) controlled by an
FPGA and ARTIQ software and gateware Kasprowicz et al. (2020). This DDS
enables precision control of the amplitude, frequency, and phase of the applied
potential. This generates a force F (t) = qE0 sin(ωt+ ϕ) where E0 is the component
of the electric field along the motional mode of interest. This also means that the
tickle drive does not couple to all radial modes equally. This force is a generator
of the displacement operator and with a arbitrary amplitude and duration can
generate displacements of arbitrary size.
4.5.2 Parametric drive resonator circuit. In order to generate
second order motional operations, we modulate the trapping potential using the
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RF electrodes of the trap. Applying this potential to the RF electrodes ensures it
is aligned with the RF null. In order to apply this potential to the RF electrodes,
it is necessary to connect a DDS to the ground of the helical resonator. In order
to prevent opening the trapping RF to the DDS, parallel resonant tank circuits
are used, shown in Fig. 4.13. These tank circuits are resonant with twice the
mode frequency at 3.6MHz and the difference of the single ion mode frequencies
at 30 kHz. The circuits are off resonant at the trap RF frequency which ensures
preventing high impedance to the DDS. Tuning the DDS frequency means that any
of two-mode squeezing, single mode squeezing or beam splitter Hamiltonians can be
realized. Coupling rates of these Hamiltonians are not expected to be the same due
to different coupling coefficients and the mode geometry.
4.5.3 Drive filtering. The parametric drive circuit relies on strong
modulation of the potential curvature with any noise in the driving electronics
causing excitation of the ion secular motion. In order to prevent this, this signal
source needs to be filtered. In order to filter out frequencies near the secular
frequency without filtering the beam splitter frequency, the squeezing and beam
splitter signal are provided from different sources and combined after the filters.
A custom high-pass elliptical filter on a custom PCB was built to filter out low
frequency noise from the DDS with a cutoff around 3 MHz.
4.5.4 Pulse shaping. Another significant concern performing these
operations is off-resonant driving from features being narrowly split. The radial
modes are commonly only split of 10s of kHz, which means resonant interactions
are only split by 10s of kHz. Filtering is not effective for frequencies so close to
the carrier. When using a DDS, by default the pulse have a ∼ns ramp time, giving
almost ideal square pulses. The spectral decomposition of a square pulse is a a
Sinc2 function with significant power in side lobes. This can result in off resonant
97
Trapping RF Squareatron amp
ion trap
helical resonator
 beam splitter RF
tank circuits and 
 squeeze RF lters
Figure 4.13. Diagram of the trapping RF connected to the Squareatron, amplifier
and helical resonator. Beam splitter and squeezing RF are connected to an
elliptical filter and tank circuits. The helical resonator is connected to the ion
trap.
driving of single mode and two mode squeezing simultaneously. These effects can
be mitigated by shaping the pulse. In order to shape the pulses, an RF multiplier
board (ADL5932-EVALZ) is used. This board takes the RF input from the DDS
and multiplies it with a DC ramp to remove the spurious spectral components.
The DC ramp is provided by an AWG (Rigol DG1022Z) that is triggered by a
TTL pulse at the beginning of every experimental sequence. A trapezoidal window
is used in the experiment with a rise time of 15µs. This rise time is sufficient to
reduce power, in the frequency components that are >100 kHz from the carrier, to
levels where off-resonant driving was not observed.
98
CHAPTER V
NON-HERMITIAN HAMILTONIANS
Quantum information experiments generally involve two separate but
equally important processes, the coherent unitary dynamics used for tranformation
of the quantum state and the dissipative dynamics that enable measurements
of this state. Combining unitary dynamics with dissipative operations enables
exploring a entirely different type of Hamiltonian. These non-Hermitian
Hamiltonians are relatively unexplored and could provide benefits for quantum
information processing. The work presented in this chapter explores the dynamics
of a non-Hermitian Hamiltonian and was done together with Alex Quinn who ran
the experiments and is co-first author on work pending publication (Quinn et al.,
2023).
5.1 Unitary dynamics and quantum speed limit
Before discussing how to implement non-Hermitian Hamiltonians and what
that means, it is important to discuss, in more detail, unitary evolution and the
corresponding limits. We have already seen that given any state |η⟩ in a Hilbert
space H, there is some other state |ψ⟩ in H such that under unitary evolution,
|η⟩ = U |ψ⟩. This mapping of a Hilbert space onto itself is true if
UU † = I, (5.1)
and in addition the inner product is preserved under unitary evolution,
(U |w⟩)†U |ϕ⟩ = ⟨ω|U †U |ϕ⟩ . (5.2)
This preservation of the inner product will also only be true when
U †U = I. (5.3)
These two equalities, eq. 5.1 and eq. 5.3, show that U † is equivalent to the inverse
U−1 of U . In order to tie this back to Hamiltonians and time evolution we use the
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density operator representation of Schrödinger’s equation
i
ρ̇(t) = − [H, ρ(t)], (5.4)
ℏ
known as the Von Neumann equation. In this representation, H is the Hamiltonian
governing the system and can be time dependent. The solution to this equation
is in general given by the same unitary operators described above. This unitary
operator U(t; t0) propagates the state of the system from some initial state at t0 to
some final state ρ(t) at some time t,
ρ(t) = U(t; t0)ρ(t )U
†
0 (t; t0). (5.5)
In many of the cases we have seen in Chapter II, if H is time-independent the
solution is given by U(τ) = exp[−iH(τ)/ℏ], for all t, t0 with τ = t− t0.
After looking at the operators that evolve some given state to another, it
is reasonable to ask, what is the fastest rate achievable to go from the initial state
to the final state? When considering the minimum time needed for any state of a
system to evolve into an orthogonal state τ⊥, it has been shown that with a fixed
average energy E it is always true that
ℏ
τ⊥ ≥ . (5.6)
4E
π
This can be put in terms of a Rabi frequency Ω as τSQL = . This is the
2Ω
minimum transit time for a given state to reach an antipodal state. This is
equivalent to doing a full rotation on the Bloch sphere.
5.2 Lindblad model and master equation solver
Real physical system can only be treated with the description above in the
limits of perfectly isolated systems. In reality we have to deal with open quantum
systems, where the system is interacting with the environment. Being able to
perfectly describe the environment and the systems coupling to the environment
can be very difficult and makes this problem significantly more complicated and
100
often requires numerical techniques. The full Hamiltonian is instead written as
H = HS +HE +Hint, (5.7)
where HS is the Hamiltonian of just the system of interest, HE is the environment
Hamiltonian, and Hint describes the interaction between the two. Under these
conditions, if the solution to U(τ) is known, the dynamics of the system of interest
can still be drawn from a larger state ρ by tracing over the environment’s degrees of
freedom
ρS(t) = TrE[ρ(t)]. (5.8)
In order to describe the evolution of the system, the Von Neumann equation can be
extended to become the Lindblad m∑aster(equation, { })i 1
ρ̇(t) = − [H, ρ(t)] + γk Lkρ(t)L†k − L
†
kLk, ρ(t) , (5.9)ℏ 2
k
where ρ and H is the system density operator and Hamiltonian respectively. Lk are
Lindblad operators which represent some non-unitary process like decoherence at
some rate γk. The operator {., .}, in eq. 5.9 is the anti-commutator of the operands.
The Lindblad operators are generators of incoherent transitions as opposed to
coherent transitions given by the Hamiltonian. When working with discrete energy
states it is relevant to look at incoherent couplings, sometimes given by natural
decay from some excitepd state |e⟩ to some ground state |g⟩. This process is
mediated by the Lindblad operator
L↓ = |g⟩⟨e| , (5.10)
with |g⟩ = L †↓ |e⟩, but L↓ = |e⟩⟨g| ̸= L↓. The Lindblad master equation is used
to approximate the evolution of the density operator with a weak coupling to
a memory-less environment. This Lindblad model will be the starting point for
101
realizing non-Hermitian Hamiltonians. A incoherent (dissipative) channel will be a
necessary but a non-sufficient condition for realizing this type of Hamiltonian.
5.3 Post-selection and non-Hermitian Hamiltonians
Experimental realization of a incoherent channel, which can be combined
with coherent control, requires considered selection of qubit encodings. The
use of metastable qubits enables an almost ideal realization of a non-Hermitian
Hamiltonian. With most qubit encodings it is possible to use coupling to the
short lived P manifolds for dissipation, however population generally returns to
the qubit states, which scrambles the coherent qubit dynamics. Population can
be coherently coupled to the S1/2 ground state which enables dissipation, but this
limits the achievable effective dissipation rate given that the coherent transfer rate
must be sufficiently slow to prevent population from coming back to the qubit.
This effectively limits the possible parameter regime to Ω > γ.
With these considerations in mind, using a single 40Ca+ ion with states
|↑⟩ ≡ |m = +5/2⟩ and |↓⟩ ≡ |m = +3/2⟩ within the metastable D5/2 manifold
as the two-level system is ideal (Fig. 5.1a) (Sherman et al., 2013). Through post-
selection, we realize the non-Hermitian Hamiltonian (ℏ = 1)
H(γ) = Jσx + iγσz (5.11)
as follows. The Hermitian Rabi drive Jσx ≡ Ω(|↑⟩ ⟨↓| + |↓⟩ ⟨↑|) is implemented by
using resonant radio frequency pulses at the qubit frequency. Additionally, the |↓⟩
state, called the lossy-state, is coupled to the auxiliary, short-lived P3/2 state |A⟩
using π-polarized light with pulse-strength ΩA, where population then primarily
(93.5%) decays to the S1/2 ground-state |g⟩ with decay rate γg. The dissipative
dynamics of the four levels {|↑⟩ , |↓⟩ , |A⟩ , |g⟩} are described by a Lindblad equation
with two Hermitian drives Ωσx and ΩA(|↑⟩ ⟨A| + |A⟩ ⟨↑|), and three spontaneous-
102
a c
Post-selec�on
b
d e
Figure 5.1. (figure from Quinn et al. (2023))Two-level non-Hermitian trapped
ion. a. Four levels of a 40Ca+ ion {|↑⟩ , |↓⟩ , |A⟩ , |g⟩} have a density matrix
ρ4(t) that is governed by Lindblad equation with two coherent drives and three
spontaneous-emission dissipators, most dominant of which, to the |g⟩ level, is
shown. b. Post-selection, i.e. eliminating the short-lived |A⟩ level and conditioning
the ion on not being in the |g⟩ level, leads to a non-Hermitian qubit within the
{|↑⟩ , |↓⟩} manifold; both Rabi drive J and the anti-Hermitian term γ in Eq.(5.11)
can be independently varied. c. Population-transfer difference ∆P (t) measured
at short times t ≲ 15µs shows the transition from the PT -symmetric phase
(∆P > 0) to a PT -broken phase (∆P < 0), with the EP at γEP = J . d. The
|↓⟩-level probability p↓(t) measured over 200 µs shows three skewed oscillations
that are increasingly affected by the backflow to the qubit manifold as γ increases
(γ/J = 0.18 dark green, γ/J = 0.37 medium, γ/J = 0.73 light blue-green) e. |∆|
extracted from parabolic fits to ∆P (t) shows the expected transition at the EP.
(Data: symbols, theory: solid lines, best-fit: broken lines; error bars in c-d are s.d.
from 400 shots.)
103
-broken
-symm
√
emission dissipators, the most dominant being γg |g⟩⟨A|. When γg ≫ JA,
adiabatically eliminating the auxiliary level and conditioning the ion on not being
in ground state generates the anti-Hermitian potential iγσz = iγ(|↑⟩ ⟨↑| − |↓⟩ ⟨↓|)
with γ = J2A/γg ≪ γg (Fig. 5.1b).
In order to model this system, it is simpler and has proven sufficient to
work with a three level system with |↓⟩ , |↑⟩ , |e⟩, with a qubit coupling rate Ω
and Lindblad operator L↓,e = |e⟩⟨↓| at rate γ. In order to capture some of the
experimental imperfections, it is necessary to take into account the ∼5% of the
population that decays back into the D5/2 manifold from the P3/2. There are two
different Lindblad operators that can be used to capture this effect, Le,↓ = |↓⟩⟨e| =
L†↓,e and Le,↑ = |↑⟩⟨e|. This processes will have rates that are proportional to γ and
constrained by the specific Zeeman branching ratio, γ↓ = 0.039γ and γ↑ = 0.015γ.
There is some population that can go to lower Zeeman sublevels, but including a
fourth state was found to have a negligible effect.
Using Qutip (Johansson et al., 2013), the Lindblad master equation can
be used to include a bare qubit coupling with Lindblad operators. This can be
compared directly to using the non-Hermitian Hamiltonian given in eq. ?? seen
in Fig. 5.2 In order to simulate post selection, the norm needs to be re-defined. The
final state after some time evolution is given by
|ψ⟩ = a |↓⟩+ b |↑⟩+ c |e⟩ , (5.12)
and the norm will be redefined from N = |a|2+ |b|2+ |c|2 to N = |a|2+ |b|2. This re-
normalization is used to just capture the probabilities of the population remaining
within the qubit manifold.
104
= 0.04 = 0.57 = 1.03 = 1.81
Non-Hermitian Hamiltonian
Lindblad model
Experimental data
= 0.18 = 0.37 = 0.73
Figure 5.2. (figure from Quinn et al. (2023))A comparison of non-Hermitian
Hamiltonian versus a Lindblad model across a range of different values of γ/J . The
first two rows show results starting in different intital states. The third row gives
longer time dynamics, highlighting the strong deviation from the non-Hermitian
model at long times, especially at larger values of γ/J
105
5.4 Finding the exceptional point
One of the significant features of this type of non-Hermitian Hamiltonian
H(γ) is the parity-time (PT ) symmetry with P = σx and complex conjugation
as the T -o√perator (Naghiloo, Abbasi, Joglekar, & Murch, 2019). Its eigenvalues
±∆ = ± J2 − γ2 change from real (PT -symmetric phase) to imaginary (PT -
broken phase) at the exceptional-point (EP) degeneracy γ = γEP = J ; both
J, γ are experimentally determined and can be independently controlled. Post-
selection preserves the state-norm in the qubit manifold, and leads to a nonlinear
Schrödinger-like equation for the qubit (Brody & Graefe, 2012; Varma, Muldoon,
Paul, Joglekar, & Das, 2023) with solution given by
| G(t) |ψ(0)⟩ψ(t)⟩ = √ , (5.13)
⟨ψ(0)|G†(t)G(t) |ψ(0)⟩
where G(t) = cos(∆t)12 − iH sin(∆t)/∆ is the non-unitary time-evolution operator.
Traditionally, the qubit level probabilities p↓(t) and p↑(t) = 1 − p↓(t) are measured
at times 0 ≤ t ≲ T∆ ≡ 2π/|∆| to characterize the transition across the EP. In our
setup, the small fraction (5.87%) of the population from the auxiliary state decays
back into the D5/2 manifold means at times t ∼ T∆ this backflow creates deviations
from Eq.(5.11) as the effective description. This approach this becomes ill-suited to
observe the transition (Fig. 5.1d).
Instead, we use the sign of the population-transfer difference ∆P (t) ≡
Pγ(t) − PJ(t) to differentiate the PT -symmetric phase (∆P > 0) from
the PT -broken phase (∆P < 0) without determining the eigenvalue gap
|2∆| (L. Ding et al., 2021). Here Pγ(t) ≡ | ⟨↓|G(t) |↑⟩ |2 = (J2/∆2) sin2(∆t)
and P (t) ≡ | ⟨−|G(t) |+⟩ |2 = (γ2/∆2J ) sin2(∆t) denote population transfers
√
to respective antipodal states in the two bases, and |±⟩ ≡ (|↑⟩ ± |↓⟩)/ 2.
Thus ∆P (t) = sin2(∆t), obtained at even short times |∆|t ≪ 1, changes sign
106
at the EP and allows determination of PT -symmetry breaking transition. The
population transfers Pγ(t), PJ(t) are related to the level-transition probabilities
p↓, p− and the exponentially decaying successful-post-selection fractions F↑, F+ —
four experimentally measured quantities — as follows:
P 2γtγ(t) = e F↑(t)p↓(t); PJ(t) = e
2γtF+(t)p−(t). (5.14)
Figure 5.1c shows that the measured ∆P (t) changes from positive to negative as γ
traverses across the EP at γEP = J . Detecting this transition through a single time-
instance data in a non-Hermitian qubit is an embodiment of quantum advantage
over classical PT -symmetric dimers that require data over time t ∼ T∆.
5.5 Legget-garg inequality
The general protocol for measuring the LG parameter K3 is schematically
shown in Fig. 5.3a. We use Q = σz as the dichotomous observable with eigenvalues
±1 and corresponding one-dimensional projectors |↑⟩ ⟨↑| and |↓⟩ ⟨↓|. With the
judicious choice of lossy-state as the initial state and equally-spaced times t1 = 0,
t2 = t, t3 = 2t, the LG parameter becomes
K3(t, γ) = C(t) + F (t)− C(2t), (5.15)
where, unlike the unitary case, C21 = C(t) and C32 = F (t) are distinct functions.
Starting from one, K3(t) reaches the Lüder bound of 1.5 at Jt = π/6 ≈ 0.523
for a Hermitian qubit. At small γ, expansion around this point gives K3 ≈
√
KL3 + 7 3γ/(8J) thereby exceeding the Lüder bound. Near the EP, similar analysis
predicts that maxK3 occurs at Jt ≈ 0.35 and approaches its algebraic maximum
in a vanishingly small window at short times t ∝ γ−1 ln(2γ/J) deep in the PT -
broken region γ/J ≫ 1. Optimizing K3 over the space of {|ψ(0)⟩ , Q} significantly
broadens the window where K3 supersedes the Lüder bound. However, in that case,
107
a b
c
d
g e
f
Run
Figure 5.3. (figure from Quinn et al. (2023))Super-quantum correlations in
K3(t). a. General protocol for determining K3(t) comprises three, pairwise two-
time projective measurements of a dichotomous observable. b. Measured K3(t, γ)
exceeds the Lüder bound of 1.5 over a wide range of γ/J . Different shades of
green (K3 ≤ 1.5) and red (K3 > 1.5) represent different γ/J values. c. With
judicious choice of the lossy-state as the initial state, the maxK3 occurs at time
tmax(γ) shorter than the unitary-limit value Jt = π/6. Thus, coherent, non-
unitary dynamics generate stronger correlations faster. d., e. Time-series for
K3(t) below the EP (γ = 0.78J) and above the EP (γ = 2.04J) show that the
location of maxK3 shifts to lower time-values as γ is increased. f. For a given
(t, γ/J), the measured K3 remains stable over time across experiments carried out
over five hours, thereby showing the robustness of super-quantum correlations in
coherent, non-unitary dynamics. A red dotted line shows the average K3 = 1.768(8)
across these runs. In each run, the Lüder bound, shown by the red dashed line, is
exceeded by at least 10.5 standard deviations. g. K3(t) in the unitary case spans
the range from 1.5 to -3 (Emary, Lambert, & Nori, 2013). (data: symbols; theory:
lines; each point denotes the average of 400 shots.)
108
these stronger correlations take longer time than Jt = π/6 to develop (Varma et al.,
2023).
Figure 5.3b shows the experimentally measured K3(t, γ) time-series with
γ/J values ranging across the EP with K3 > 1.5 marked in varying shades of red
that indicate increasing γ. Correspondingly, varying shades of green indicate K3
values that remain below the Lüder bound. As γ increases, the time at which K3 is
maximum shortens, to less than half its unitary-limit value at γ/J ≈ 2 (Fig. 5.3c).
A close-up time-series across the EP shows this trend clearly (Fig. 5.3d,e). Our
observed values reach a maximum K3 = 1.768(8), clearly superseding the Lüder
bound of 1.5. The coherent, non-unitary dynamics of a PT -symmetric qubit, thus,
spawn temporal correlations that are stronger than the Lüder bound, in a time that
is shorter that one dictated by the unified quantum speed limit (Levitin & Toffoli,
2009; Margolus & Levitin, 1998).
Accessing parameter region where K3 can approach its algebraic maximum
is challenging for two reasons. First, increasing γ by ramping up the JA drive
suppresses the successful post-selection fraction; second, at a fixed value of γ,
reducing J slows down the dynamics, amplifies deviations from the non-Hermitian,
PT -symmetric qubit model, and suppresses maxK3 relative to predictions from
such a model. The effects of the branching ratio in the backflow can be seen in
Fig. 5.4 where the maximum achievable value of K3 is shown as a function of the
branching ratio.
5.6 Violation of quantum speed limit
The excesses above 1.5 in K3(t) arise primarily from the temporal variation
of the norm of G(t) in Eq.(5.13). For a unitary qubit, the Lüder bound emerges
from a constant speed-of-evolution on the Bloch sphere (Campaioli, Pollock, &
Modi, 2019; Levitin & Toffoli, 2009; Margolus & Levitin, 1998). Being equal for
109
2.0
1.9
1.8
1.7
1.6
1.5
b0.000 0.00ranc0h.005 0.25ing 0r.010 0.50atio 0i.015 0.75nto n0.020 1.00on-di0s.025 1.25s /Jipat0i.030
1.50
ve st0 1.75a.035te 0.040 2.00
Figure 5.4. Surface contour of the Maximum value of K3 possible across different
values of γ/J taking into account the branching ratio into the non-dissipative state.
The black line highlights the true experimental value, which prevents reaching
higher values of K3.
110
Maximum K3
a b d
c
Figure 5.5. (figure from Quinn et al. (2023))Moving faster than quantum-
speed-limit on the Bloch sphere. a. In unitary case, the transit times to and
from an antipodal state are equal, bounded below by unified quantum speed limit
τQSL = π/(2J); coherent dynamics of PT -symmetric Hamiltonian break this
reciprocity. b. Starting from the lossy state, measured p↓(t) shows a faster transit
to the non-lossy state |↑⟩ as γ is increased; theory lines are obscured by the data
symbols. c. In contrast, starting in the non-lossy state, measured p↓(t) shows a
slower return from the |↑⟩ state. The incomplete transfer to the lossy-state|↓⟩ at
γ = 0.57J also reflects the backflow to the qubit manifold. d. Measured transit
time T↓→↑ shows a 50% reduction relative to its minimum mandated value τQSL;
the error in measured T↑→↓(γ) is due to the backflow and the incomplete Rabi flop
(Supplementary Information). Data: symbols; theory: lines.
111
Non-Hermi�an, Unitary
-broken
-symmetric
antipodal states, for a two-level system, it results in the unified lower bound τQSL =
π/2J for the minimum transit-time required to reach an antipodal state (Fig. 5.5a).
The coherent, non-unitary dynamics generated by G(t) break this reciprocity, and
the new, non-reciprocal transit times satisfy T↓→↑(γ) ≤ τQSL ≤ T↑→↓(γ). As γ
is increased, T↓→↑ is suppressed from τQSL to 1/J at the EP, to γ
−1 ln(4γ/J) for
γ ≫ J . The return time satisfies
π
T↑→↓(γ) + T↓→↑(γ) = (5.16)
∆(γ)
when γ < J , and diverges at the EP. Past the EP, a complete transit from the non-
lossy-state |↑⟩ to the lossy-state |↓⟩ cannot occur.
Measured occupations p↓(t) show that the antipodal transit time is
shortened when starting in the |↓⟩ state (Fig. 5.5b) and increased when starting
in the |↑⟩ state (Fig. 5.5c) as γ/J increases. The extracted transit-time T↓→↑(γ)
monotonically decreases showing a 50% reduction across the EP, while T↑→↓
diverges as the EP is approached (Fig. 5.5d). This non-reciprocal transit is a
signature that a non-Hermitian Hamiltonian commingles salient features of unitary
and dissipative dynamics.
By implementing a two-level non-Hermitian Hamiltonian in a single,
dissipative ion, we have demonstrated stronger temporal correlations that supersede
all known models comprising unitary or dissipative, linear quantum maps Budroni
and Emary (2014); Budroni, Moroder, Kleinmann, and Gühne (2013). These
correlations are generated faster than those permitted by standard quantum speed
limits Levitin and Toffoli (2009); Margolus and Levitin (1998). While making clear
that non-Hermiticity is necessary for superseding the Lüder bound, we have laid to
rest implicit assumptions regarding the special role played by the EP in maximizing
the LG parameter Karthik, Akshata Shenoy, and Devi (2021); Lu et al. (2024);
112
Naikoo, Kumari, Banerjee, and Pan (2021); Wu et al. (2023) (Supplementary
Information).
These dual features of the coherent, non-unitary dynamics generated by PT -
symmetric, non-Hermitian Hamiltonians – stronger correlations, faster – may pave
the way for new models of quantum correlations and entanglement generation.
113
CHAPTER VI
LASER-FREE CONTROL OF TRAPPED-ION QUANTUM HARMONIC
OSCILLATORS
Chapter II has laid out the theoretical and experimental toolbox for
operators that generate a range of classical and non-classical states of trapped
ion motion without using laser fields. With that foundation, we can discuss the
method of analyzing the states that these operators generate in order to calibrate
experiments and perform useful measurements. The results of these calibration are
critical for a full understanding of the experiments laid out in chapter VIII.
6.1 State characterization
The primary method for extracting information from an ion is to use qubits.
This allows access to a single bit of information that when averaged over many
experimental runs, gives insight into processes that affect the state of the qubit.
This means that in order to gain information about the motional states that have
been prepared, the sideband interaction are critical for mapping information about
the motional state onto the qubit state. We have already seen that using the BSB
interaction leads to transitions between |n, ↓⟩ and |n+ 1, ↑⟩ when starting from
a Fock state. This means that probability of measuring the spin state |↓⟩ after
applying the BSB Hamiltonian for some time t i(s given by√ )
|⟨↓| ⟩|2 2 n+ 1ΩtP↓(t) = ψ(t) = cos . (6.1)
2
When the ion is in some superposition of Fock states the probability to measure |↓⟩
must include the probabilities of a∑ll Fock sta(tes and is given as∞ √ )
2 n+ 1ΩtP↓(t) = Pn cos , (6.2)
2
n=0
where Pn is the probability of finding the Harmonic oscillator is some Fock state
|n⟩ before the application of the BSB Hamiltonian. In order to take into account
114
spin and motional decoherence, full numerical simulations of the master equation
can be done with phenomenological decay factors taken into account. Under
consideration of motional decoherence alone, caused by white frequency noise, the
qubit probabilities can be re-w(ritten
1 ∑as∞ (− √ ))P↓(t) = 1 + P γntne cos n+ 1Ωt , (6.3)
2
√n=0
where the decay constant γn = γ0 n+ 1. This means motional states that occupy
larger Fock states will decay more rapidly. This generally puts a constraint on the
size of the states that can be generated with phase coherence in the experiment.
The RSB Hamiltonian is more useful when trying to make measurements of
the ground state population. When applying this Hamiltonian with Rabi frequency
Ω for time t the qubit probabilit(ies ar∑e given as∞ (√ ))1
P −γnt↓(t) = 1 + Pne cos nΩt . (6.4)
2
n=0 √
From this, it is clear that the Rabi frequency Ω n is zero when in the ground
state. This makes sense as you cannot subtract a phonon from the vacuum state.
This type of measurement will be best when we consider time reversal protocols
that should return the motional state to the ground state as this Hamiltonian can
be applied to determine if return to the ground state was successful.
6.2 Coherent states
Generation of coherent state without the use of lasers has already been
described in section 2.5.1 using an oscillating electric field at the mode frequency
for a duration t. The BSB analysis can be seen in Fig. 6.1 which shows a collapse
and revival of the contrast which is expected for this type of Jaynes-Cummings
interaction (Berman & Ooi, 2014) with coherent states. The coupling rate of the
motional state and the qubit state is set by the Rabi frequency and Lamb-Dicke
parameter is a fit parameter in Fig. 6.1a). Fitting the theory to the data also
115
enables the coherent state amplitude α to be extracted. In order to determine the
coherent state amplitude α as a function of drive duration at a fixed amplitude,
a linear fit is applied. The slope of this line is used to determine the coupling
rate of the displacement Hamiltonian. The linear relationship between drive
duration and coherent state amplitude is shown in Fig. 6.1b). This calibration is
important to determine coherent state amplitude that sets the sensitivity for some
interferometers in Ch. VIII.
 
BSB pulse duration (µs) RF Drive duration (µs)
Figure 6.1. (a) Fits to time series data produced by BSB Rabi flopping with the
motion in a coherent state. Data reproduces the characteristic collapse and revival
of a coherent state. (b) Calibration data for determining the coupling rate of the
displacement operations. Data is taken by fitting time series data like in a) for
varying durations of the displacement operation. The slope of the linear fit gives
the coupling rate.
6.3 Thermal states
After the photon scattering events from cooling sequences, the ion’s motion
is theoretically in a thermal state. The density matrix, that describes this state,
does not have off-diagonal elements (coherences) which is a fully mixed state.
Sideband analysis still enables determination of the Fock state populations. When
using this type of analysis for thermometry after cooling, it is generally more useful
to use the RSB. As the mean phonon number is reduced, the |↓⟩ state population
116
P↓
α
tends to 1. This enables scanning and finding cooling parameters that maximize
the ground state population.
As has been shown in section 2.5.6, thermal states are also a product
of tracing over a single mode of a two-mode squeezed state. Using the BSB
Hamiltonian, the qubit populations are fit to using numerical simulations to
determine the mean phonon number n̄. Thermal state fitting can be seen in
Fig. 6.3a). The relationship between n̄ and the squeeze amplitude rtms is n̄ =
2 sinh(rtms). This relationship then gives the size of the two-mode squeezed state
when measuring the thermal state.
6.4 Squeezed states
Section 2.5.2 showed how squeezed states can be generated with a
parametric drive at twice the mode frequency, applied for a duration tr. Squeezed
vacuum or squeezed coherent states are generated by applying this interaction
on either the ground state, or a coherent state. The analysis in that section also
showed how this interaction should only populate the even Fock states. Using the
BSB analysis enables measuring the squeezed state amplitude r. The parametric
coupling strength can also be measured with a method similar to that used for
coherent states. Since r = gtr, the slope of a plot of r vs tr can be fit to give
the value of g. The calibration for the experiment involves keeping tr fixed and
varying g with the maximum coupling rate given by the amount of squeezing at full
amplitude for the given squeezing time, which is shown in Fig. 6.2.
117
 
BSB pulse duration (µs) Drive amplitude (full scale)
Figure 6.2. (a) Fits to time series data produced by blue sideband Rabi flopping
with the motion in a squeezed. (b) Calibration data for determining the coupling
rate of the displacement operations. Data is taken by fitting time series data like in
a) after varying the amplitude of the driving field with a fixed pulse duration. The
maximum coupling rate is taken as the the coupling rate when the DDS is at full
amplitude.
6.5 Two-mode squeezed states
Given that making a qubit measurement after using a sideband interaction
to a single mode is equivalent to a partial trace of a single mode, this state is
difficult to measure. Measuring the thermal state can be used to calibrate the
experiment so long as there is additional evidence that an actual two-mode
squeezed state has actually been generated. Given the commonality of thermal
states of motion after cooling, further evidence is necessary. The thermal state
measurement was used for calibration once other experiments have been used to
verify this operation and is shown in Fig. 6.3. The beam splitter can be used to
effectively measure the mode in another basis and can be seen in section 8.1. The
other method we employed to verify the coherence of the state, was to implement a
time reversal protocol to bring the motion back to the ground state and is shown in
section 8.2.
118
P↓
rsqueeze
 
ηa,bΩt Drive amplitude (full scale)
Figure 6.3. (a) Fits to time series data produced by blue sideband Rabi flopping
on either radial mode with the motion in a two mode squeezed state. Tracing over
a single mode when using sideband flopping produces a thermal state. The x-axis
is rescaled with the relative Lamb-Dicke parameter to show overlap in time scales
of the oscillations (b) Calibration data for determining the coupling rate of the two
mode squeeze operation. Data is taken by fitting time series data like in a) after
varying the amplitude of the driving field. The maximum coupling rate is taken as
given coupling rate at full amplitude of the DDS.
6.6 Beam splitter
To produce the beam splitter operation described by the Hamiltonian in
section 2.5.4, we digitally trigger the drive AWG to output a pulse consisting of an
integer number of cycles (30 cycles in the runs in Fig. 6.4) at a fixed amplitude,
starting at a phase of zero. We cannot continuously adjust the length of these
pulses and therefore must calibrate the pulse amplitude to produce a 50/50 beam
splitter pulse. Specifically, we select an amplitude that maximizes the fringe
contrast of our SU(2) interferometer (i.e. that allows a pair of 50/50 beam splitter
pulses to fully transfer a displacement from one mode to another). The starting
phase of each pulse is fixed which means to adjust the relative phase between beam
splitter pulses in our SU(2) experiments we adjust the delay time tdelay between
pulses, giving a phase offset of (ωa − ωb)tdelay, as shown in Fig. 6.4.
119
P↓
rTMS
mode a: initial ~3.4
0.8 mode b: initial ~0
0.6
0.4
0.2
0.0
60 70 80 90 100 110 120 130
delay time ( s)
Figure 6.4. Starting with a coherent state in the low-frequency mode, the SU(2)
interferometer is performed with a variable delay between beam splitter pulses,
allowing coherent transfer of the coherent state between modes. The amplitude of
the 50/50 beam splitter pulses used can be calibrated by maximizing contrast on
mode a fringes. Delay time is time passed between beam splitter pulses.
6.7 Off-resonant motional coupling
Due to the narrow frequency splitting of our motional modes, the sideband
interaction results in off-resonant driving of our second mode which is separated
in frequency from the first mode by only 33 kHz for the results shown in Ch. VIII.
The resulting interaction Hamiltonian is now more accurately written as
Ĥ = ℏΩ/2(σ̂+â†eiδ1t + σ̂−âe−iδ1t + σ̂+b̂†eiδ2t + σ̂−b̂e−iδ2t) (6.5)
where δ1 − δ2 ≈ 2π × 33 kHz. There is no known analytical solution for the
qubit probabilities under evolution of this Hamiltonian therefore cannot be used.
In order to get a model free fit, master equation simulations were performed using
QuTiP (Johansson et al., 2013) where each Fock state probability in the initial
state is a fit parameter while preserving the trace. Estimates of fit uncertainties
with this method proved the be a challenge due to weak dependence on the off-
resonant mode. Fitting was performed starting with an ideal state and holding
120
P
all Fock states except one constant. The single Fock state was iterated through
all Fock states for both mode a and b which may produce an under estimate of
the uncertainty. Data where either mode a or b was driven resonantly was fit
independently. Figure 6.5 shows the results of the fit Fock state fits with the model
state fit to the probabilities for both modes independently and overlaid.
(a)
(b)
Figure 6.5. (a) Fits to the output of two-mode squeezing under interaction
described above. Blue bars indicate mode a and red bars are mode b with
associated error bars. Black line is a thermal state model fit to the Fock state
fits. (b) Fock state fits to output of the beam splitter after two-mode squeezing.
Blue data is mode a and red is mode b with associated error bars. Black line is a
squeezed state model fit to the Fock state fits.
121
CHAPTER VII
PROTECTED MODE INVESTIGATION FOR MID-CIRCUIT MEASUREMENT
In order to run quantum algorithms, there are many instances where it is
important to be able to make measurements of a subset of the full systems. This
type of measurement can enable things like error correction that can employ a
classical feedback mechanism (Ahn, Wiseman, & Milburn, 2003). This problem
has been addressed when using spin qubits in different traps and having individual
addressing, which is possible by using tightly focused laser beams (Nägerl et al.,
1999). This can also be done with multiple ion species (Wright et al., 2016) or the
OMG scheme (Allcock et al., 2021), where not all qubits have a state that couples
to the detection laser.
This problem of mid-circuit measurement is more complicated when dealing
with the motion. With a single ion, the ion is involved in all modes of motion,
which means that photon scattering will heat and decohere every mode of motion.
Chapter VI showed that with more ions, more interesting mode structure arises.
By symmetry, when there is an odd number of ions in a chain, the center ion does
not participate in the motion of of the symmetric modes. This chapter covers the
details of these “protected” modes and the experimental setup that was used to
test the limits of the protection offered ,from photon scattering, when using the ion
that doesn’t participate in the motion.
7.1 Mode structure and protected modes
Section 2.3 covers how when considering the Coulomb coupling present with
multiple ions, a collective motional structure is present with N linearly independent
modes for each principle axis of the trap. The mode structure for 3 ions is also
characterized in section 2.3. With a odd number of ions a consequence of the
symmetry means that the center ion does not participate in the motion. This
122
participation is characterized by the vector describing the components of motion
of the ion j in the motional mode α. This is important when considering the Lamb-
Dicke parameter for each ion in the c√hain given by
ℏ
ηj,α = k · e’ j,α, (7.1)
2mjωα
where e’ j,α is this mode-ion vector and k is the wavevector of the laser. With the
center ion in the stretch mode having a zero Lamb-Dicke parameter, it is decoupled
from this mode and should not show up in spectrscopy of the motional modes
performed on the center ion, which can be seen in Fig. 7.1. For this experimental
work, it is important to have the individual addressing of the qubits, in order to
be able to investigate the effects of probing only the center ion in the 3 ion chain.
This work is made possible by using a mixed species ion chain of 40Ca+ and 88Sr+
trapped ions in a surface electrode trap at Lincoln lab. The mode structure of this
mixed species chain can be seen in Fig. 2.1.
123
Figure 7.1. Spectroscopy of calcium and strontium ions showing the motional
sidebands detuned clearly resolved from the carrier transition. The protected
stretch mode is seen in the spectroscopy of the outer strontium ions, but is not
present when performing spectroscopy with the center calcium ion.
7.2 Surface electrode trap with mixed species ion chain
The experimental setup at Lincoln Lab was used in order to experimentally
verify that this protected mode can be preserved during ion detection, given
the mixed species capabilities with this setup. All data was collected by Colin
124
Bruzewicz and I developed experimental sequences and performed data analysis.
More complete details of the experimental setup can be seen in Bruzewicz,
McConnell, Stuart, Sage, and Chiaverini (2019). This setup consists of a cryogenic
surface-electrode trap with a strong magnetic shield to prevent decoherence of
the Zeeman qubits used for this work. This 2-dimensional surface-electrode trap
consists of a 2µm-thick, patterned aluminum layer on a sapphire substrate and can
be seen in Fig. 7.2. Using segmented electrodes enables to ability to control the
position and twisting of the crystal which affect the mode structure of a chain of
ions. The protected mode structure only exists under the assumption that the ion
chain lies along the null of the RF potential.
Figure 7.2. [Figure made by Colin Bruzewicz] Schematic of the surface electrode
trap used for this experimental work. Trapping RF electrodes are labelled and
run along the length of the trap axis. All other electrodes are DC electrodes with
individual connections to enable arbitrary control of the ion location.
125
7.3 Heating rates
A first order check on whether the protected mode is preserved during ion
detection is to determine if this mode leaves the ground state after state detection
of the center ion. In order to determine this, a heating rate is measured. All
modes of motion have stochastic anomalous heating (Deslauriers et al., 2006)
√ √
that can be modeled with Lindblad operators L̂1 = ṅâ and L̂2 = ṅâ†
where ṅ is the anomalous heating rate and â(â†) are the annihilation (creation)
operators for the mode of interest. This type of heating (decoherence) arises
from the oscillator coupling to a bath of an infinite number of oscillators at
thermal equilibrium (Turchette et al., 2000). This heating rate can be measured
by performing thermometry of the ion as a function of time. In addition to the
anomalous heating, when considering photon scattering, can experience excitations
on the red or blue sideband sideband during fluorescence. Excitations on the red
sideband are suppressed when near the ground state, which means carrier and blue
sideband excitations dominate. As a consequence, the motion is quickly driven the
the Doppler limit where an equilibrium is reached. If the heating rate measured
as a function of the duration of the state detection is not significantly different
than the anomalous heating rate, it can be assumed that mode heating from state
detection is not the primary cause of decoherence.
7.3.1 Sideband thermometry. For a single ion, measuring the mean
motional occupation n̄ is fairly straight forward using the red and blue sideband.
Equation 2.79 showed the populations for a thermal state and eqs. 6.4 and 6.3 gave
the qubit probabilities on the red and blue sideband. Taking a ratio of the red and
blue sideband qubit probabilities P rsb↑ (t) and P
bsb
↑ (t) gives (Bergquist et al., 1987)
r = P rsb
n̄
↑ (t)/P
bsb
↑ (t) = . (7.2)n+¯ 1
126
Populations ( red sideband ) ( blue sideband2 )2
|⟨↓↓|Ψ⟩|2 − n n+ 11 [1− cos(grΩt/2)] 1− [1− cos(gbΩt/2)]
2n− 1 2n+ 3
|⟨↓↑|Ψ⟩|2 n n+ 1, |⟨↑↓|Ψ⟩|2 sin2(g 2
− r
Ωt/2) sin (gbΩt/2)
2(2n 1) 2(2n+ 3)
|⟨↑↑| ⟩|2 n(n− 1) − 2 (n+ 1)(n+ 2)Ψ [1 cos(grΩt/2)] [1− cos(gbΩt/2)]2
(2n− 1)2 (2n+ 3)2
Table 1. Popu√lations of the four different p√ossible qubits states of a two ion systemsafter resonant red or blue sideband transitions on the state |Ψ, n⟩ with pulse length
t.gr = ηm 2(2n− 1) and gb = ηm 2(2n+ 1), where ηm is the Lamb-Dicke
parameter of either the center of mass mode ηc or the stretch mode ηs
This ratio of the excitations on the two sidebands is a simple algebraic equation
and conveniently does not depend on the sideband pulse duration or Rabi
frequency. This ratio tends to 1 for n̄ ≫ 1, which can also be near the Doppler
limit, and tends to n̄ for n̄ ≪ 1. A simple rearrangement of this expression gives
the mean occupation as a function of the sideband ratio
r
n̄ = . (7.3)
1− r
For the longer chain of ions considered here, the problem of sideband thermometry
is not as straightforward. While there are three ions in the chain, sidebands are
only driven on two of them, so this can be considered a two ion problem. For
two ions, the sideband probabilities are shown in table 1, taken from (Webster,
2005), for each of the measurement outcomes of the state |Ψ⟩. For two ions, there
is not a time and Rabi frequency independent ratio of the sidebands. Despite this
limitation, a joint fit of all the qubit probabilities can be performed to extract the
mean motional occupation. Fig. 7.3 shows the red and blue sideband probabilities
for the three possible measurement outcomes. A fit to the data is used to get the
peak height. The peak height for all three outcome probabilities is used to perform
a least squares fit and return the mean motional occupation number as well as the
pulse duration and Rabi frequency.
127
The heating rate above the anamolous heating rate depends on the intensity
and detuning from atomic resonance of the detection laser. The anamalous heating
rate is ∼1 quanta/sec with heating rates from photon scattering varying from as
low as 2 quanta/sec to as high as 90 quanta/sec. The Lamb-Dicke parameter has
been estimated to be as low as 10−4 which means of order 108 photons would need
to be scattered to get to a thermal state with n̄ = 1. As is, this leads to limited
heated during a detection cycle and could be reduced further by mitigating effects
that break the symmetry of the mode.
128
Figure 7.3. Fits to the red and blue sideband transitions on the two Sr+ ions. The
ratio of the sideband peak heights are used in a joint fit to extract the average
motional quanta in order to determine a heating rate.
129
7.4 Motional Fock state Ramsey interferometry
Heating of a motional mode is only one mechanism of decoherence.
Determining the heating rate of the protected mode provides only a lower bound
on the possible coherence time. In order to determine the full coherence time, the
phase relation between Fock states should be measured. For single ions, a motional
Ramsey type experiment is straightforward to implement. Starting with a carrier
π/2 pulse on a pure qubit state and the ground state of motion gives the state
|Ψ⟩ = √1 (|↓⟩ + |↑⟩) ⊗ |n = 0⟩. With a π pulse on the blue sideband the spin
2
superposition can be mapped onto the motion, |Ψ⟩ = |↑⟩ ⊗ √1 (|0⟩ + |1⟩). A
2
time reversal of this procedure enables to ability measure for how long the phase
relationship between the |0⟩ and |1⟩ Fock states can be tracked.
For two ions, it is not possible to perform the π pulse on the blue sideband
that gives the above result. A more involved sequence is necessary to remove the
spin superposition while giving a motional superposition. Fig. 7.4 shows a pulse
sequence involving a blue and red sideband pulse which produces approximately the
final state |Ψ⟩ = |↓↓⟩ ⊗ (0.33 |0⟩ + 0.93 |4⟩). The same time reversal scheme can be
used here to measure the motional coherence. It is also important to point out that
using a higher Fock state in this experiment means that the expected decoherence
rate should be 4x larger than when using a |0⟩,|1⟩ superposition. The contrast of
this signal reduced for this sequence, given the lack of a equal superposition, which
means sensitivity is not increased 4x for this interferometer. The phase of the final
pulse can be varied to provide fringes which can be used to determine the coherence
time. As longer delay times are used there is an expected decay in the contrast of
the fringes and this decay can be fit to give a photon scattering-free coherence time.
When state detection is performed on the center ion during this Ramsey sequence,
any decoherence caused by this detection beam will lead to a more rapid decay
130
in the contrast. Preliminary results show that there is negligible loss in contrast
after 400 µs of photon scattering, consistent with the heating rate of ∼4 quanta/sec
heating rate. Results can be seen in Fig. 8.3.
Similar results have been shown in Hou et al. (2022), where the beam
splitter interaction has been used to coherently swap the state of non-protected
modes with the protected modes. This ability has been used to preserve the desired
state in the protected mode during state detection.
Figure 7.4. [Figure made by Colin Bruzewicz] Pulse sequence used to determine the
motional coherence of the stretch mode. Combining red and blue sideband pulses
enables the ability to prepare a motional superposition of Fock states while leaving
the spin in a single spin state. Quotations are around the π to indicate that these
pulses do no achieve exact population swapping.
7.5 Perturbations to protection
In order for the Lamb-Dicke parameter of the center ion to remain zero,
the ion chain must be perfectly aligned with the RF null of the trap. There are
a number of different ways that the ion chain can become offset from the RF
131
Figure 7.5. Phase scans of the final sideband pulse using the motional
interferometer sequence shown in 7.4. The first panel shows fringes after a 400µs
delay without the use of the 397 nm fluorescing beam. The second panel shows
similar fringe contrast after the same amount of delay time but with the fluorescing
beam on during the delay.
null. The detection field itself can also exert a force, called radiation pressure,
on the ions and cause a breaking in the symmetry. Stray electric fields are the
main contributor to an offset and can lead to a significant departure from the ideal
mode geometry. These types of offset can generally be detected and compensated
for by detecting excess micromotion and leads to additional motional sidebands
when performing spectroscopy. Both of these effects have been considered and
investigated in this section.
7.5.1 Radiation pressure. State detection is performed using a near
resonant laser on a broad dipole allowed transition. This laser exerts a force on the
ion from the laser field which time averages to
F = ℏkΓρee, (7.4)
132
where k is the wavevector of the laser, Γ is the natural linewidth of the dipole
transition and ρee = ⟨e|ρ̂|e⟩ is the steady-state population of the excited state.
This probability depends on the tuning of the laser and is given by
s/2
ρee = (7.5)
1 + s+ (2∆/Γ)2
where s = 2|Ω|2/Γ2 is the saturation parameter, Ω is the Rabi frequency and
∆ is the laser detuning. For typical laser detunings used for Doppler cooling
∆ = −Γ/2 and s = 1, this force is of order 10−20 N and can lead to displacements
of the ion from equilibrium of order nm. This shift leads to changes in the mode
structure and mode frequency, but from models would only lead to a Lamb-Dicke
parameter of the protected mode of less than order 10−3. This small of a Lamb-
Dicke parameter would mean that heating induced from this effect would be below
the anomalous heating rate and therefore not easy to detect.
7.5.2 Stray electric fields. The presence of unwanted electric
fields at the location of the ions is a certainty. It is possible through different
techniques to measure the effect of these fields (Berkeland, Miller, Bergquist, Itano,
& Wineland, 1998) and use trap electrodes to compensate for them. In order to
consider the effects of these fields it is worth briefly discussing the potential that
the ion is subject to. The potentials can be separated into static and dynamic
potentials:
V
ϕst(r) = (χ x
2 + χ y2 + χ z2) +E · r
r2
x y z
0 (7.6)
URF
ϕ (r) = (κ x2 + κ y2RF x y + κzz
2) cos(ΩRF t),
r20
where r ≡ (x, y, z), r0 is the distance from the ion to the nearest electrode and
χx,y,z, κx,y,z are dimensionless shape parameters dependent on trap geometry. E
is a static electric field that is added to account the effects of stray fields. There
are constraints on the shape parameters to satisfy Laplace’s equation (Griffiths,
133
1981) and considering the oscillating part of the potential in equation 7.6, the
ponderomotive part of the potential is given as
qU2
Φ RF 2 2 2pond(r ,m) = (κxx + κyy + κzz ) (7.7)
mΩ2 4RF r0
which from the nature of this potential is mass dependent and gives rise to secular
oscillations at a frequency √
2q2U2
ω = RF
κx,y,z 2qV χx,y,z
x,y,z + . (7.8)
m2Ω2 r4RF 0 mr
4
0
The mass dependence of this potential is very important for the mixed species ion
chain considered here. Any uncompensated electric fields will cause displacement of
the ion chain from the RF null, where the displacement of the ions will depend on
the depth of the trapping potential and the mass of the ions. In a chain consisting
of strontium and calcium ions, the strontium ions have more than double the
mass and will be effected more significantly by uncompensated electric fields.
Unequal displacements of the ions cause a sort of zig-zag structure in the ion chain
and break the symmetry that decouples the center ion from participating in any
mode. Besides careful calibration to ensure stray fields have been calibrated, this
is another potential benefit to using the OMG architecture where all ions are the
same mass and uncompensated stray fields won’t break the chain symmetry.
134
CHAPTER VIII
SU(2) AND SU(1,1) INTERFEROMETRY FOR PRECISION PHASE
MEASUREMENT
Given that state motional state preparation, control and readout has been
covered in chapters IV, VI, VII, we can discuss methods for using trapped ion
motion for quantum sensing. In the following, we demonstrate different classes of
interferometers using the motional modes of a single trapped atomic ion. With a
demonstration of these interferometers, we show the ability to concatenate several
operations on the oscillator states of two modes in a phase coherent manner.
Performing these operations with digitally programmable frequency sources for
relative phase control simplifies significant calibration challenges when developing
more complicated circuits.
Yurke et al. (Yurke, McCall, & Klauder, 1986) proposed schemes to
reach the Heisenberg limit (Pezzé & Smerzi, 2009) using different classes of
interferometers. Here, we work within the framework of the Cramér-Rao
bound (Agarwal & Davidovich, 2022; Zwierz, Pérez-Delgado, & Kok, 2010) which
sets the ultimate limit for the resolution of a sensing scheme. This bound is
described by a parameter variance, instead of the Heisenberg limit which is set
directly by the number of resources used, i.e. photon or phonon number (Zwierz et
al., 2010), making it more suitable to the limits of the experiments reported here.
The required interactions for the interferometers (described below) are colloquially
called displacing, beam splitting, single-mode squeezing, and two-mode squeezing.
We have already shown the ability to perform these interactions and characterize
the states that are produced. Performance of these interferometers can, in practice,
be limited only by the motional heating of the ion resulting from fluctuating patch
potentials on the trap electrodes (Deslauriers et al., 2006; Hite et al., 2013). Here,
135
we demonstrate an SU(2) interferometer (Mach-Zender), which has previously
been demonstrated in (S. Ding et al., 2017b; Gorman et al., 2014; Leibfried et al.,
2002), (coherent input states are used here in contrast to those works), as well as
single-mode and two-mode SU(1,1)1 interferometers. Each of the interferometers
described here is sensitive to different phase shifts, which are single-mode phase ϕa
(single-mode SU(1,1)), difference phase ϕa − ϕb (SU(2)), or sum phase ϕa + ϕb (two-
mode SU(1,1)), where the a, b labels refer to different modes. We show performance
close to the Cramér-Rao bounds for these interferometers.
To demonstrate these different interferometers we use the experimental
setup shown in Chapter IV. Experiments are performed using the two ‘radial’
modes of the ion (normal to the trap axis) with frequencies ωa ≈ 2π × 1.80MHz
and ωb ≈ 2π × 1.83MHz, and oscillator energy eigenstates denoted by |na, nb⟩. The
mode splitting can be adjusted by changing a DC potential on the RF electrodes
which also rotates the mode angle and can be used to vary the achievable coupling
rate of the interactions described below. We use qubit states |↑⟩ ≡ |mL = +5/2⟩
and |↓⟩ ≡ |mL = +3/2⟩ within the metastable D5/2 manifold which has a lifetime
of ≃ 1.1 seconds (Kreuter et al., 2005), where mL is the total angular momentum
projection along the direction of the quantization magnetic field of 0.498G. The
qubit transition frequency is ω0 ≈ 2π × 2.63MHz. This is a magnetic field sensitive
qubit with a coherence time of ≃ 1 ms, while stable trapping potentials provide
motional coherence times over 20ms. In each experiment the qubit is first prepared
in the |↑⟩ |0, 0⟩ with the methods shown in Chapter IV . Mode temperatures
correspond to n̄a,b < 0.15. Motional state analysis is performed using the techniques
described in Ch. VI. The necessary interactions are generated by applying the same
1The Lie-group labelling is used to refer to the groups that contain the operators that generate
these states.
136
pair of Raman beams that drive the qubit, but detuned above (red sideband) or
below (blue sideband), the qubit frequency. The red sideband is again used to
verify preparation or return to the motional ground state. The blue sideband
interaction is used to fully characterize the generated state populations (Meekhof
et al., 1996). Generation of the motional states is accomplished by applying
a) (b)
B
Δk
976 σ -
976 
Ω πtrap
Figure 8.1. (figure from Metzner et al. (2023)) (a) View of trap along the trap
axis showing the electrical connections:pink—RF quadrupole electrodes, trap and
parametric drive, purple—displacement drive and DC electrode, yellow— other DC
electrodes and relevant control electrodes. The green box represents the circuitry
necessary to combine the drives. Mode a and mode b label the two different ‘radial’
mode vectors of a single ion with the angles from the horizontal shown. (b) A 3D
rendering of the trap showing direction of the 976 nm laser used for motional state
preparation and state characterization, and cyan ball representing the 40Ca+ ion
time-varying potentials to the trap electrodes (see Fig. 8.1) producing a unitary
operation described in Ch. II. The displacement operation is produced by applying
an oscillating potential, at the mode frequency, to a single trap electrode (purple
electrode Fig. 8.1). Single-mode squeezing, two-mode squeezing and the beam
splitter are accomplished by applying an oscillating potential at twice the mode
frequency (2ωa), at the sum of the mode frequencies (ωa + ωb) or at the difference
frequency (ωa − ωb) respectively (pink electrodes Fig. 8.1). These modulated
137
potentials yields a gene[ralized squeezing interaction Hamiltonian,g ]
Ĥ = iℏ âiâj exp(iδt) exp(−iθ)− â†i â
†
j exp(−iδt) exp(iθ) (8.1)2
where g is the parametric coupling strength, which has a different value for
each operation, δ is the detuning from the interaction resonance (either single
mode squeezing (mode a or b) or two-mode squeezing), and θ is the phase of
the parametric modulation. With the proper choice of detuning this implements
the unitary squeezing operator with squeezing parameter magnitude given by
rSMS = gt and the two mode squeezing operator with rTMS = 2gt. Here,
we used a single-mode displacement coupling of g = 2π × 1.37 kHz which
gives a displacement parameter α = gt. The maximum single mode squeezing
coupling is g = 2π × 3.99 kHz and a maximum two-mode squeezing coupling of
g = 2π × 1.15 kHz. Maximum coupling refers to a coupling rate calculated from
fits to the states with the maximum voltage we used and the relevant pulse time for
each operation. The beam splitter coupling is g = 2π × 0.66 kHz calculated using
calibration data shown in Chapter VI.
8.1 Verification of two-mode squeezed state with beam splitter
In order to verify the proper performance of the beam splitter operation,
we use a calibration experiment described in section 6.6. Verification of the two-
mode squeezed state is more complicated given we are only able to readout a single
mode in a single shot of the experiment. Using the blue sideband readout method,
described above, is equivalent to tracing over a single mode and a thermal state is
measured (C. M. Caves et al., 1991). This is analogous to measuring a single qubit
from a Bell state which produces a random qubit state. Additionally, we can use
a beam splitter operation after the two-mode squeezer to disentangle the modes
and produce single-mode squeezed states (Fig. 8.2). Performing both of these
138
measurements verifies we have produced the target two-mode squeezed state and
enables the use of these states for interferometric measurements.
(a)
T M S    B S
B S B B S B
(b) (c)
Figure 8.2. (figure from Metzner et al. (2023))(a) General quantum circuit diagram
for characterizing the two-mode squeezed state with blue sideband readout
performed at one of two instances, to verify we are producing the expected output
(b) Measurement after the two-mode squeezed state and before beam splitter
produces a thermal state. (c) applying the beam splitter operation after the two-
mode squeezer and then characterizing the state produces two single-mode squeezed
states with squeezing parameter r equal to that of the two-mode squeezed state.
8.2 Cramer-Rao bounds
Implementing time reversal protocols is a method for generating these
types of interferometers and is a powerful tool for sensing applications (Colombo
et al., 2022). These protocols also enable us to verify that we are coherently
139
generating the desired states by time-reversing back to the initial state. The
SU(2) interferometer (Fig. 8.3c) is implemented by first displacing mode a in our
experiment, which generates the desired initial coherent state. The beam splitter
is then used to generate a superposition of coherent and vacuum states. The
phase of the final beam-splitter is controlled by using a variable delay, such that
the accumulated phase is (ωa − ωb)tdelay. The final beam splitter will completely
transfer the phonons to mode b with the correct choice of relative phase, and the
red sideband readout will verify we have returned to the ground state (Fig. 8.3c).
We use a maximum displacement of α ≃ |6| to maintain a similar maximum n̄ in
the single and two-mode squeezed states.
The single-mode (Fig. 8.3a) and two-mode (Fig. 8.3b) SU(1,1)
interferometers are implemented by using the parametric drive tuned to either twice
the frequency of mode a or the sum frequency of mode a and b respectively. To
avoid increased decoherence, from long pulse durations, these pulses are run at a
fixed duration with the drive voltage being varied to control the size of the state. A
second squeezing pulse is then applied (Fig. 8.3b) with a varied phase controlled by
the driving electronics. We are limited in the amount of two-mode squeezing we can
generate due to our increased sensitivity to common-mode phase fluctuations that
this interferometer is sensitive to.
140
(a)
SMS SMS(φ)
RSB
(b)
TMS TMS(φ)
RSB
(c)
D(α)
BS BS(φ)
RSB
Figure 8.3. (figure from Metzner et al. (2023)) (a) Single-mode SU(1,1)
interferometer output for different amounts of squeezing is shown by varying the
relative phase of the second squeezing pulse shown in the circuit diagram (inset).
(b) repeat of (a) with two-mode squeezing and two-mode SU(1,1) shown in the
interferometer circuit inset.(c) SU(2) interferometer output shown for various sizes
of initial displacement where relative phase is controlled with a variable delay
shown in the SU(2) circuit inset.
141
In order to determine the sensitivities to small phase shifts that are
achievable with these different experiments, we calculate the quantum Fisher
information for a pure state (Agarwal & Davidovich, 2022)
F(ϕ) = 4(∆Ĝ)2 (8.2)
where (∆Ĝ)2 is the variance in the initial state of the probe (coherent state,
squeezed state, two-mode squeezed state) and Ĝ = i[dÛ †/dϕ]Û . For phase sensing
†
Û = eiϕa a and therefore Ĝ = a†a and the Fisher information is proportional to
variance in the phonon number of the probe states. The phase sensitivity is limited
by the Cramér-Rao bound (Agarwal & Davidovich, 2022) given by
1
∆ϕ ≥ √ . (8.3)
F(ϕ)
This means for the SU(2) interferometer we are limited at
1 1
∆ϕ = = √ (8.4)
α ⟨N⟩d
where ⟨N⟩d is the expected mean phonon number for the displacement generated
and this limit corresponds to the expected standard quantum limit. The limit for
the single-mode SU(1,1) interferometer is given by (Monras, 2006)
√ 1∆ϕ = (8.5)
8 ⟨N⟩SMS (⟨N⟩SMS + 1)
where ⟨N⟩SMS is the expected mean phonon number for the input squeezed state.
The two-mode SU(1,1) limit is given by (Agarwal & Davidovich, 2022)
1
∆ϕ = √ (8.6)
⟨N⟩TMS (2 + ⟨N⟩TMS)
142
where ⟨N⟩TMS is the expected mean phonon number for the input two-mode
squeeze state.
The sensitivities achieved are calculated from the experimental data using
the Fisher information (Agarwal & D∑avidovich[, 2022) ]21 dPj(ϕ)
F (ϕ) = (8.7)
P (ϕ) dϕ
j j
where Pj(ϕ) is the probability of getting the experimental results j if the value of
the parameter is ϕ. In our experiment with only two possible outcomes, spin-up or
spin-down, this simplifies to [ ]2
1 ∂P↓
F (ϕ) = . (8.8)
P↓(1− P↓) ∂ϕ
The variance and slope of the signal are expressed analytically and are fit to the
phase signals that we generate (Fig. 8.3). These theoretical curves are derived,
in section 8.3, without any adverse effects and represent the ideal experimental
output (Fig. 8.4). Systematic offsets in the phase fringes, seen in the single-mode
squeezing phase scans, from π are due to an additional pseudopotential (Leefer
et al., 2017) when the parametric drive is applied. This additional potential
accounts for a maximum shift of 10 kHz of the mode frequencies. In the single
mode SU(1,1) experiment the interaction frequency is calibrated at the maximum
coupling meaning as the coupling is turned down, the mode frequencies decrease
and larger phase offsets are produced. This phase offset is calibrated away in the
two-mode phase scans by calibrating the mode frequencies at every parametric
drive voltage. The maximum phase sensitivity is extracted and plotted against
the prediction from the quantum Fisher information (Fig. 8.4). The results we
achieve deviate from the ideal limit due to motional and qubit decoherence as
well as motional heating. Initial mode temperatures also affects these results and
contribute to vertical offsets of the curves in Fig. 8.3. The ideal experimental
143
output deviates from the CR bound due to the choice of red side-band readout
times, where optimal pulse time varies with n̄. Red side-band pulse times were not
optimized across all n̄ due to the constraints on pulse times due to off-resonant
driving of other modes and increased pulse length causing further decoherence.
8.3 Theoretical bounds
To obtain an analytical expression for the output of these interferometers
we must determine the action of the red sideband on the final state and determine
the expectation values for the final qubit projection operator. We calculate this
below for the two-mode SU(1,1) case. The single-mode case can be treated in a
similar manner. We start by representing the two-mode squeezed state in the Fock
basis (Kurochkin, Prasad, & Lvovsky, 2014)
∞
| ⟩ 1
∑
TM√S = tanh
n r |n, n⟩ =
c∑osh(r√n=0 )∞ n (8.9)1 λ |n, n⟩
1 + λ 1 + λ
n=0
where λ = sinh r. We need to determine |out⟩ = TMS(ϕ)TMS |0, 0⟩. For phase
other than ϕ = π the output is still a two-mode squeezed state with a different two-
mode squeeze parameter r, which we can express as a function of the phase.
r(ϕ) = sinh−1(sinh r0 cosϕ/2), (8.10)
where r0 is the input two-mode squeeze state magnitude. The red sideband
Hamiltonian is written as (Meekhof et al., 1996)
Ω
HRSB = ℏη (σ+a+ σ−a†). (8.11)
2
144
Ω is the qubit carrier Rabi frequency, η is the Lamb-Dicke
parameter (Meekhof et al., 1996) for the given mode, and σ+ and σ− are the single
qubit raising and lowering operators. The unitary operator for the Hamiltonian can
be written as
iβHRSBt
U = e ℏ (8.12)
where β = ηΩt/2. Expanding this unitary into an infinite series gives
∑∞ (iβ)j|n, n⟩ → (σ+a+ σ−a†)j |↓, n, n⟩ . (8.13)
j!
j=0
Only terms having equal numbers of raising and lower operators for spin and
motion will have a final contribution to the projection, so we can re-write this state
∑∞
|↓ ⟩ (iβ)
j
URSB , n, n = (σ
+aσ−a†)j |↓, n, n⟩ , (8.14)
j!
j=0
where {j} is only even integers. This leaves the final spin state unaffected and only
the number operator a†a remains, leaving us
(iβ)2j
U jRSB |↓, n, n⟩ = n |↓, n, n⟩ =
(2j)!
(8.15)
(iβ)2j√ 2j √
n |↓, n, n⟩ = cos (β n) |↓, n, n⟩ .
(2j)!
The expectation value of the spin projector is therefore
∑⟨↓, n, n||↓ ⊗⟩ ⟨↓(⊗||↓, n,)n⟩ =∞ n1 √ t λ (8.16)
cos ( nηΩ )
1 + λ 2 1 + λ
n=0
145
〈 〉
For the sake of simplicity we will call this expectation value X̂ and we define the
variance 〈 〉 〈 〉2
〈 〉 〈∆X̂〉 2 = X̂2 − X̂ . (8.17)
For spin projection X̂2 = X̂ . We also need an expression for the derivative of
the expectation value as a function of the phase;
〈 〉
d X̂ dλ 1
(〈 〉 =∑dϕ dϕ 1 + λ∞ √ ( )n−11 Ωt λ
X̂ + cos2 ( nη )n (8.18)
1 + λ ( ))2 1 + λn=0
− λ1
1 + λ
dλ
where = −2 cosh2 r sinϕ sinh2 r. The same process is carried out and shown for
dϕ
the coherent state in the SU(2) interferometer as these results differ significantly
from the two-mode SU(1,1). A coherent state is expressed in the Fock basis as
∑∞ 2n2
|α⟩ α= e−|α| /2 |n⟩ . (8.19)
n!
n=0
The red sideband produces a similar output,
〈 〉
⟨↓, n||↓ ⊗∑⟩ ⟨↓ ⊗||↓, n⟩ = X̂ =∞ α2n √ (8.20)
e−|α|
2
cos2 ( nηΩt/2)
n!
n=0
with α a function of the phase, α = α0 sinϕ/2. The variance can again be expressed
in terms of this expectation value as in eq. 8.17 and the derivative of ⟨X̂⟩ is given
as
146
〈 〉
d X̂ ∞ 2n
|α|2
∑ α √
= e cos2 ( nηΩt/2)(−α2n+1 + nα2n−1) (8.21)
dϕ n!
n=0
/ ideal
/ ideal
/ ideal
Figure 8.4. (figure from Metzner et al. (2023))Ideal experimental phase sensitivities
(δϕ) for the SU(2) (dashed green line), single mode SU(1,1) (dashed orange
line) and two-mode SU(1,1) dashed red line) are shown and can be compared to
experimental results shown as dots with error bars and Cramér-Rao (CR) bounds
in solid lines.
We have demonstrated a wide class of trapped-ion motional state
interferometers using a ‘laser-free’ parametric drive achieving single and two-
mode SU(1,1) interferometer sensitivities of 5.9(2) dB and 4.5(2) dB below the
SQL, respectively and a SU(2) interferometer sensitivity within 0.67(5) dB of the
SQL. These results illustrate the measurement of phase shifts near the Cramér-
Rao bound for common-mode, relative, and single-mode phase fluctuations.
Enhanced sensitivity for measuring these motional phase fluctuations could
enable characterization and mitigation of motional frequency noise which can
benefit experiments involving motional states. Beside providing a framework for
implementing quantum-enhanced sensing protocols, our methods provide the
147
flexibility to concatenate multiple motional operations into circuits for potential
applications in CVQC.
148
CHAPTER IX
CONCLUSION AND OUTLOOK
Chapter VIII is one example of a method for using two-mode squeezed
states for quantum sensing. There are other uses for one and two-mode squeezed
states in quantum sensing of displacements (Cardoso, Rossatto, Fernandes, Higgins,
& Villas-Boas, 2021) and general cases where reducing quantum fluctuations is
necessary to improve signal to noise (Lawrie, Lett, Marino, & Pooser, 2019). These
methods, combined with time reversal protocols provide amplification of a signal for
detection (Colombo et al., 2022). In addition to making use of two-mode squeezed
states, if is important to be able to measure properties of these states, for example,
a direct witness of the mode entanglement. This chapter explores some possible
experiments that could be performed in our setup to further demonstrate the
ability to characterize and make use two-mode squeezed states. It is also important
to consider how this work may fit into the larger context of continuous variable
quantum computing.
9.1 Phase insensitive displacement amplification with two-mode
squeezed states
Previous theoretical and experimental work has shown single mode squeezed
states can be used to amplify displacements (Burd et al., 2019; Yurke et al., 1986).
By first squeezing the motional ground state, which reduces quantum fluctuations
along one quadrature, displacements along this quadrature can be amplified by
performing time reversal of the squeezing operation. The state is returned to a
minimum uncertainty coherent state with a larger amplitude, αf = Gαi with gain
G. This amplification is given by
D̂(α †f ) = Ŝ (ξ)D̂(αi)Ŝ(ξ), (9.1)
149
where D̂(α) is the displacement operator with amplitude |α| and Ŝ(ξ) is the single
mode squeeze operator with complex squeeze parameter ξ = reiθ. The issues
with this scheme is that the final displacement depends on the phase relationship
between the squeeze and the displacement, α = α cosh(r)+α∗eiθf i i sinh(r), which can
be seen in 9.1. Only when the displacement is along the squeezed axis do you get
maximum amplification, G = er. If the phase of the displacement is not known this
leads to the maximal gain being washed out over the many experimental iterations.
Two-mode squeezed states, in theory, can out perform single mode squeezed states
when the displacement phase is unknown. This scheme is written as
(a) (b) (a)
D̂a(αf )D̂b(αf ) = T̂
†(ξ)D̂a(αi )T̂ (ξ), (9.2)
where now we consider two modes with annihilation (creation) operators â, b̂(â†, b̂†)
of the two modes. The displacement operators D̂a,b(α) act independ(ently on eith)er
mode and the two-mode squeezing operator is given by T̂ (ξ) = exp ξ∗âb̂− ξâ†b̂† .
Independent of the squeeze and displacement phase relationship the final
(a) (a) (b) (a)
displacements are αf = cosh(r)αi and αf = sinh(r)αi . Given that the
displacement and squeezing phase are not a part of this expression, the gain
becomes independent of these phases. This means that the displacement phase
does not need to be known and a gain of G = cosh(r) is fixed. This cosh(r) gain
matches what has been shown in Burd et al. (2024) using single mode squeezing,
however their protocol requires equal amplitude displacements in each pulse period
and would be unsuitable for transient displacements.
Chapter II showed how the beam splitter can be used to effectively rotate
the basis and work in the EPR variable basis. This result means that the beam
splitter can be used at the end of the above amplification protocol to retrieve full
gain. G = er. The effect of the beam splitter is to transfer the small displacement
150
in the second mode back to the originally displaced mode. In order to properly
make this transformation, another phase is introduced into the protocol. This
means that given an unknown displacement phase, the addition of the beam splitter
makes this a phase sensitive amplification and suffers the same loss in gain.
The addition of a second mode in this scheme does provide access to a
second bit of information. This second bit can be conditioned to potentially
increase the gain in the event that the displacement phase is not known. This
increase in gain is achieved by post selecting on the secondary mode being in the
ground state. The secondary mode is only in the ground state if the action of the
beam splitter gives the ideal EPR transformation.
  
α = iα0 α = e-riα0
r
α = α α = e α0 0
Figure 9.1. a) single mode quadrature squeezing which shows variance reduced
below the SQL (black dashed lines) for one quadrature, with variance increased
in the other. b) displacement of squeezed state along two different trajectories
with a π/2 phase difference. c) reversal of the squeezing showing amplification of
displacement when displaced along the squeezed axis and loss when displaced along
anti-squeezed axis.
9.1.1 Potential experimental implementation. Most of the
elements for a demonstration of this technique have been performed and shown
in Chapter VI and VIII. The last remaining question is how to measure the gain.
In Burd et al. (2019), small displacements were used which meant that the final
state overlapped significantly with the ground state. This makes a single RSB pulse
151
 
Figure 9.2. a) output amplification of a displacement on mode a, when using two
mode squeezing and measuring either mode a or mode b. b) phase sensitivity
comparison of the two amplifications schemes, with a fixed rsms = rtms = 1, with
varied gain dependent on the displacement phase for single mode squeezing and no
sensitivity to the phase when using two mode squeezing, but with gain below the
maximum of the single mode squeezing scheme.
ineffective for measuring the gain, which is why a more complicated readout scheme
was used in that work. For this demonstration, if larger displacements are used, a
single RSB pulse is sufficient to measure the gain. With a fixed RSB pulse duration
is used, the qubit probability can be measured as a function of the size of the initial
displacement. The slope of this signal can be used to retrieve the gain.
When adding in the beam splitter, there are more optimal beam splitter
fractions that can be used. Post-selection is done by performing a RSB pulse on the
secondary mode and rejecting shots where the qubit is flipped.
9.2 Phase distributions
Measuring correlations in two-mode squeezed states is not straight forward
when it is difficult to directly measure the motional states. Mapping the motion to
a qubit also does not provide an obvious solution to how to measure some joint
property of two modes. One potential solution has been laid out theoretically
in Selvadoray and Sanjay Kumar (1997). Given the types of phase profiles that
152
are shown in section chapter VIII, there may be an avenue for generating full two-
dimensional phase profiles as a method for measuring mode correlations. The phase
distribution of a two-mode state |ψ⟩ is
1
P (θ1, θ2) = |⟨θ1, θ2|ψ⟩|2, (9.3)
(2π)2
where θ1 and θ2 are the phases of the individual modes and −π ≤ θ1, θ2 ≤ π.
The two-mode phase state |θ1, θ2⟩ is defined to be the eigenstate of the two-
mode Susskind-Glogower (Susskind & Glogower, 1964) phase operator ê =
(1 + â†â)−1/2â(1 + b̂†b̂)−1/2b̂, giv∑ing∞
|θ1, θ2⟩ = exp[i(n1θ1 + n2θ2)] |n1, n2⟩ . (9.4)
n1,n2=0
This phase state is non-normalizable and non-orthogonal, but does form a complete
set and the phase distributio∫n is nor∫malized:π π
dθ1 dθ2P (θ1, θ2) = 1. (9.5)
−π −π
Measuring the two-mode squeezed state overlap with this phase state is not exactly
realizable, however the phase one-dimensional phase distributions that are shown
in chapter VIII may be a close enough approximation to be able to measure
correlations. The correlations between the phases of the two modes is defined by
C12 = ⟨θ1θ2⟩ − ⟨θ1⟩ ⟨θ2⟩ . (9.6)
We have already seen that the marginal Wigner distribution corresponding to
one mode obtained by tracing over the other mode is a thermal state. The phase
distribution of this single mode state state has the property that P (θ1) = P (−θ1),
meaning that ⟨θ1⟩ = 0 and by the same argument, ⟨θ2⟩ = 0. Hence the correlation
C12 is given entirely by ∫ π ∫ π
C12 = dθ1 dθ2θ1θ2P (θ1, θ2). (9.7)
−π −π
153
This means that if this phase distribution can be generated, the mode correlations
can be measured by integrating over the distribution. Work still needs to be done
to determine how well the procedure in chapter VIII is actually approximating a
phase distribution. It will also be necessary to individually vary the phase of the
individual modes instead of just the sum-mode phase.
9.3 Correlation measurements below the CH bound
Phase distributions is only one possible avenue to measuring two-mode
squeezed state correlations. Another method based on work from Banaszek and
Wódkiewicz (1999), is to test nonlocality in phase space. This type of experiment
can be performed with a phonon counting type experiment which leads directly
to a measurement that is described by the Wigner function. This function is
given by joint phonon count correlations. The setup for this experiment will be
to take a correlated source of the ion, like the two-mode squeezed state, make
conditional displacements on either, or both modes, and then be able to to make
joint measurements of the modes. The measurement that can be used to do this,
which is very adaptable to trapped ions, is to use events when non phonons are
registered. This is very similar to a Bell type test with a strict analogy to two-
particle coincidence experiments. The adjustable polarizers used in those photon
experiments are replaced with coherent displacements of α and β.
The joint probability of no-count events happening simultaneously with both
detections is
Qab(α, β) = ⟨Ψ|Q̂a(α)⊗ Q̂b(β)|Ψ⟩ , (9.8)
where Q̂(ab) are projection operators of a single mode,
Q̂(α) = D̂(α) |0⟩⟨0| D̂†(α), (9.9)
154
with displacement operator D̂(α). The full measurement is performed for two
setting of the coherent displacement for each mode: zero or α for mode a and zero
or β for mode b. This full set of measurements enables calculating the Clauser-
Horne combination (Clauser & Horne, 1974):
CH = Qab(0, 0) +Qab(α, 0) +Qab(0, β)−Qab(α, β)−Qa(0)−Qb(0), (9.10)
which for local theories in bounded by the inequality −1 ≤ CH ≤ 0. With arbitrary
control of the size of the displacements and relative phase of the displacements a
range of values can be achieved. For equal amplitude displacements |α| = |β| = A
and a phase difference β = e2iϕα the values are
CH = −1 +Ae−A − 2Ae−2A sin2 ϕ, (9.11)
which violates the inequality when the displacements have opposite phases β = −α.
Testing this theory with trapped ions would not pose any significant
difficulties. Every piece of this scheme as been demonstrated besides the detection
scheme. There are several avenues to explore in terms of addressing this detection
scheme. One possible avenue is to use two ions and implement different qubit
encodings on each ion with the OMG architecture.
9.4 Continuous variable quantum information processing
The ability to process information using quantum system is not constrained
to only using two discrete energy states (qubits). This paradigm can be scaled to
include many discrete states (qudits) and can even be scaled to an infinite number
of equally spaced states, or continuous energy spectra (continuous variable). While
schemes employing qubits have been widely developed, significantly less effort has
been given to realizing other methods of quantum information processing. Many of
the elements necessary of a universal set of CV operations have been demonstrated
here. A source of non-Gaussian states is necessary for universal computation. This
155
is already possible with motional sidebands for generating Fock states. Sidebands
can also be used to engineer higher order Hamiltonians (Sutherland & Srinivas,
2021). A non-Gaussian operation could be provided by the native Coulomb
interaction, but tends to be very slow. Higher order parametric modulation is
another avenue for providing this non-linear interaction, but would require at least
qubic terms in the trapping potential.
Harmonic oscillators provide the possibility of a CV quantum information
processor, but it is also possible to encode qubits into these systems. Bosonic codes
exist which can be implements in trapped ions and provide advantages for error
correction over existing schemes. One of these bosonic codes is the Gottesman,
Kitaev and Preskill (GKP) code which is a grid state making use of a superposition
of squeezed and displaced states. A truncated approximation to this infinite energy
state has been generated in the trapped ion platform at ETHZ (Flühmann et
al., 2019). The state engineering has been performed using laser based methods,
but performance could theoretically be improved with parametric modulation.
Current work is focused on generating two GKP state gates. This is especially
relevant for the techniques demonstrated here. With the capabilities of current ion
traps, CVQC is a paradigm within reach and effort needs to start shifting towards
demonstrations of quantum gates for use in quantum algorithms.
156
APPENDIX
SQUAREATRON 5000 AND TUNING BOARD
This appendix includes the design schematics and PCB layout of the
Squareatron 500 and the additional Tuning board. The Squareatron design is
modified from the original Oxford design (Harty, 2019) to include pads for the
tuning board to connect to; the update design is shown in Metzner (2020). The
tuning board is powered by the Squareatron and uses the 3.3V reference for the
DAC. The tuning board has a RJ45 connector mounted to allow for programming
from a field-programmable gate array (FPGA) used in the lab.
157
Figure A.1. Schematic diagram of the Squareatron 5000 showing all of the
components used
158
Figure A.2. Schematic diagram of the tuning board that attaches to the
Squareatron
159
Figure A.3. A rendering the the Squareatron and tuning board PCBs
160
Figure A.4. Image of the Squareatron and tuning board connected and mounted in
a box
161
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