EASY ON THE IONS: PHOTON SCATTERING ERRORS FROM
FAR-DETUNED RAMAN BEAMS IN TRAPPED-ION QUBITS
by
ISAM DANIEL MOORE
A DISSERTATION
Presented to the Department of Physics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2023
DISSERTATION APPROVAL PAGE
Student: Isam Daniel Moore
Title: Easy on the Ions: Photon Scattering Errors from Far-Detuned Raman Beams
in Trapped-Ion Qubits
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Physics by:
Dr. Steven Van Enk Chair
Dr. David T. C. Allcock Advisor
Dr. Stephanie Majewski Core Member
Dr. Andrew Marcus Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded June 2023
ii
© 2023 Isam Daniel Moore
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License.
iii
DISSERTATION ABSTRACT
Isam Daniel Moore
Doctor of Philosophy
Department of Physics
June 2023
Title: Easy on the Ions: Photon Scattering Errors from Far-Detuned Raman Beams
in Trapped-Ion Qubits
The viability of quantum computers depends on the development of scalable
platforms with low error rates. Our ”Oregon Ions” group has studied one such
scalable architecture proposal including the limitations placed on logic gate fidelity
by photon scattering. We studied spontaneous Raman scattering-induced errors
in stimulated Raman laser beam-driven logic gates in metastable- and ground-
manifold-encoded qubits. For certain parameter regimes, we found that previous,
simplified models of the process significantly overestimated the gate error rate
due to spontaneous photon scattering. We developed an improved model, which
shows that there is no fundamental lower limit on gate error due to spontaneous
photon scattering for electronic ground state qubits in commonly-used trapped-
ion species when the Raman laser beams are red detuned from the main optical
transition. Additionally, spontaneous photon scattering errors are studied for
qubits encoded in a metastable D5/2 manifold, showing that gate errors below 10
−4
iv
are achievable for all commonly used trapped ions. Furthermore, we extended this
theory from hyperfine to Zeeman qubits and we measured scattering rates from
far-detuned Raman beams in a metastable D5/2 Zeeman qubits in
40Ca+, obtaining
results that matched theoretical expectations. Finally, we present progress towards
implementing a two-qubit Mølmer-Sørensen gate with these Raman beams in
trapped 40Ca+ ions.
This dissertation contains previously published and unpublished material.
v
CURRICULUM VITAE
NAME OF AUTHOR: Isam Daniel Moore
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, Oregon
Wright State University, Dayton, Ohio
DEGREES AWARDED:
Doctor of Physics, 2022, University of Oregon
Bachelor of Science, 2017, Wright State University
AREAS OF SPECIAL INTEREST:
Quantum Computing, Quantum Information, Atomic Physics
PROFESSIONAL EXPERIENCE:
Graduate Research Assistant, University of Oregon, 2019-2023
Graduate Teaching Assistant, University of Oregon, 2017-2019
GRANTS, AWARDS AND HONORS:
Weiser Doctoral Thesis Award, University of Oregon Department of Physics,
2023
PUBLICATIONS:
Moore, I. D., van Enk, S. J. (2021). Self-consistent tomography and
measurement-device independent cryptography. Int. J. Quantum Inf. 19,
2040003
Allcock, D. T. C., Campbell, W. C., Chiaverini, J., Chuang, I. L., Hudson,
E. R., Moore, I. D., Ransford, A., Roman, C., Sage, J. M., Wineland, D. J.,
(2021). omg blueprint for trapped ion quantum computing with metastable
states. APL, 119, 214002.
Moore, I. D., Campbell, W. C., Hudson, E. R., Boguslawski, M. J.,
Wineland, D. J., Allcock, D. T. C. (2023). Photon scattering errors during
stimulated Raman transitions in trapped-ion qubits. PRA, 107, 032413.
vi
Boguslawski, M. J., Wall, Z. J., Vizvary, S. R., Moore, I. D., Bareian, M.,
Allcock, D. T. C., Wineland, D. J., Hudson, E. R., Campbell, W. C. (2023).
Errors in stimulated-Raman-induced logic gates in 133Ba+. arXiv:2212.02608
Quinn, A., Metzner, J., Muldoon, J. E., Moore, I. D., Brudney, S., Das,
S., Allcock, D. T. C., Joglekar, Y. N. (2023). Observing super-quantum
correlations across the exceptional point in a single, two-level trapped ion.
arXiv:2304.12413.
vii
I thank Matt Boguslawski and Zach Wall for help in checking the accuracy
of the calculations. I thank Wes Campbell and Eric Hudson for the extensive
discussions of scattering processes and the Kramers-Heisenberg formula. I thank
Steven van Enk for working with me scientifically and for his advice throughout
graduate school. I also thank Sean Brudney, Gabe Gregory, and Vikram Sandhu
for all their help during graduate school. I thank Dave Wineland, who read this
thesis and provided much helpful guidance during graduate school, and I especially
thank Alex Quinn and Jeremy Metzner who read this thesis and took data for it. I
thank David Allcock for his patient guidance and mentorship to me over the years.
Finally, I thank my friends, family, and Kaylin.
viii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Theory of ion trap quantum computing . . . . . . . . . . . . . . 1
1.2. Transitions Between Energy Levels in Trapped Ions . . . . . . . . 2
1.3. Laser Cooling in Ion Traps . . . . . . . . . . . . . . . . . . . . . 4
1.4. Gates in an Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5. Photon Scattering Errors in Trapped-Ion Quantum Computers . 14
II. OMG ARCHITECTURE . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1. Metastable qubits . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2. omg architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3. Two-Qubit Gates in m Qubits . . . . . . . . . . . . . . . . . . . 23
III. EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . 24
3.1. Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2. Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3. Ion Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4. Trap rf Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5. Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
ix
Chapter Page
3.6. AOM Boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7. ARTIQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.8. Beam Delivery System . . . . . . . . . . . . . . . . . . . . . . . . 41
IV. SCATTERING THEORY . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1. Choice of Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2. Scattering Probability . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3. Power Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 63
V. ZEEMAN QUBITS IN 40CA+ . . . . . . . . . . . . . . . . . . . . . . . 69
5.1. Zeeman Qubits in the D5/2 Manifold . . . . . . . . . . . . . . . . 69
5.2. Scattering Theory for Zeeman Qubits . . . . . . . . . . . . . . . 74
VI. SCATTERING RATE MEASUREMENTS . . . . . . . . . . . . . . . . 78
6.1. 976 nm laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2. Scattering rate experiments . . . . . . . . . . . . . . . . . . . . . 80
VII. TWO-QUBIT GATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.1. Mølmer-Sørensen Gate Setup . . . . . . . . . . . . . . . . . . . . 94
7.2. Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3. Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 101
x
Chapter Page
VIII. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
APPENDICES
A. DERIVATION OF SCATTERING RATE . . . . . . . . . . . . . . . 108
B. DECAY RATE FROM FERMI’S GOLDEN RULE . . . . . . . . . . 115
C. POWER REQUIREMENTS DERIVATION . . . . . . . . . . . . . . 117
D. RECOVERING CLASSICAL LIMITS . . . . . . . . . . . . . . . . . 119
E. QUANTUM CRYPTOGRAPHY PROTOCOL . . . . . . . . . . . . 122
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
xi
LIST OF FIGURES
Figure Page
1.1. Raman transitions schematic . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. The Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1. Alkaline earth ion’s atomic structure . . . . . . . . . . . . . . . . . . . . 20
2.2. omg schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1. Render of our ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2. Photograph of our trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3. Magnetic coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4. Ion camera schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5. Picture of ion camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6. Photograph of laser rack . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7. Laser rack drawer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.8. Inside AOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.9. Blue AOM efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.10. Red AOM efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.11. AOM rack board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.12. AOM board schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.14. Schematic ARTIQ system . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.13. AOM board design schematic . . . . . . . . . . . . . . . . . . . . . . . . 42
3.15. Schematic of trap optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1. g and m qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
xii
Figure Page
4.2. Scattering processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3. Single-qubit gate Raman scattering error . . . . . . . . . . . . . . . . . . 55
4.4. Two-qubit gate Raman scattering error . . . . . . . . . . . . . . . . . . 58
4.5. Rayleigh scattering error ratio . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6. Single qubit gate power requirements . . . . . . . . . . . . . . . . . . . . 67
4.7. Two-qubit gate power requirements . . . . . . . . . . . . . . . . . . . . . 68
5.1. Zeeman m qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2. Laser and rf m qubit operations . . . . . . . . . . . . . . . . . . . . . . . 72
5.3. Polarization impurity scattering . . . . . . . . . . . . . . . . . . . . . . . 74
5.4. Power and detuning requirements for light shift . . . . . . . . . . . . . . 75
5.5. Rayleigh error ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.1. 976 nm laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2. 976 nm optical setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3. Scattering configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.4. mJ = +5/2, σ
− raw measurement results . . . . . . . . . . . . . . . . . 88
6.5. mJ = +5/2, σ
− scattering vs. power results . . . . . . . . . . . . . . . . 89
6.6. mJ = +3/2, σ
− raw measurement results . . . . . . . . . . . . . . . . . 90
6.7. mJ = +3/2, σ
− scattering vs. power results . . . . . . . . . . . . . . . . 91
6.8. mJ = +3/2, π raw measurement results . . . . . . . . . . . . . . . . . . 92
6.9. mJ = +3/2, π scattering vs. power results . . . . . . . . . . . . . . . . . 93
7.1. MS gate schematic geometry . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2. Carrier Rabi flopping, m qubit . . . . . . . . . . . . . . . . . . . . . . . 96
7.3. Sideband Rabi flopping, m qubit . . . . . . . . . . . . . . . . . . . . . . 98
7.4. Phase sensitive vs. phase insensitive geometry . . . . . . . . . . . . . . . 99
xiii
Figure Page
7.5. MS gate schematic state transfer . . . . . . . . . . . . . . . . . . . . . . 100
7.6. Walsh modulation schematic . . . . . . . . . . . . . . . . . . . . . . . . 101
7.7. Walsh modulation preliminary results . . . . . . . . . . . . . . . . . . . 102
7.8. Preliminary Mølmer-Sørensen gate results . . . . . . . . . . . . . . . . . 103
A.1. Scattering derivation frequencies . . . . . . . . . . . . . . . . . . . . . . 109
xiv
LIST OF TABLES
Table Page
1.1. Minimum scattering error during a two-qubit gate in some
commonly-used trapped-ion species. Sourced from Ozeri, 2007. . . . . . 15
4.1. Hyperfine qubit characteristics . . . . . . . . . . . . . . . . . . . . . . . 47
8.1. Scattering characteristics of m and g qubits . . . . . . . . . . . . . . . . 107
xv
CHAPTER I
INTRODUCTION
1.1. Theory of ion trap quantum computing
1.1.1. Earnshaw’s Theorem and the Ion Trap Potential
If asked how to trap a charged particle, you might initially think an
appropriately chosen static electric field will suffice. However, it is not possible
to trap charged particles using static electric fields only. This result is known as
Earnshaw’s Theorem. It is simple to demonstrate: the charged particle needs to be
confined in free space in the presence of an electric field sink, i.e., a point where
the electric force field’s divergence is negative (for a positively-charged particle).
However, this contradicts Gauss’ law. The divergence of an electric force field F
arising from electric potential U in free space is, by Laplace’s equation,
∇ · F = ∇ · (∇U) = ∇2U = 0 (1.1)
where the last step is due to Gauss’ law. This means that we need an
electrodynamic interaction to trap a charged particle. There are many ways to
generate such an interaction, but in our group, we use a linear Paul trap.
1.1.2. Linear Paul Traps and the Mathieu Equations
A linear Paul trap Paul and Steinwedel, 1953 often consists of a pair of
rf electrodes, a pair of ground electrodes, and a pair of dc endcap electrodes
Figure 3.1. The rf electric field null line defines the “trap axis” (for our trap,
1
this is roughly the vector between the endcap electrodes). The rf electrodes (’rod’
electrodes) run parallel to this axis. While the endcap electrodes have a static
potential, two of the rod electrodes have rf signals applied to them with the
other pair being connected to ground. This generates an oscillating quadrupole
electric field and lets us get around Earnshaw’s theorem. Along with the on-axis
confinement provided by the dc endcaps, this creates a ponderomotive confining
potential in 3D.
1.2. Transitions Between Energy Levels in Trapped Ions
A quantum computer requires quantum logic gates. When working with
trapped ions, our gates physically amount to driving transitions between the qubit
states, which are encoded in some energy levels. There are many such transitions,
such as magnetic dipole (M1), electric dipole (E1), and electric quadrupole (E2)
transitions. Although we use other transitions in the lab (such as M1), this work
focuses primarily on E1 transitions. For that reason, in this section, I will discuss
how we can change an ion’s energy by coupling to such E1 transitions.
1.2.1. Stimulated Raman Transitions
Stimulated Raman transitions are a common method for inducing E1
transitions. Here I will explain the mechanism behind Raman transitions, as well
as give a theoretical description of a single-qubit gate and a two-qubit gate that we
have implemented in the lab.
Consider a three level “Λ” system as in Figure 1.1, with one excited state |e⟩
and two lower lying states |1⟩ and |2⟩. We can couple the low-lying states to the
excited state via the electric dipole coupling by applying laser beams to the ion.
2
The strength of this coupling can be characterized by the ‘Rabi frequency’ Ωg. The
Rabi frequency is defined as
E ⟨e| d⃗ · Eϵ̂ |g⟩
Ωg = , (1.2)ℏ
where E is the peak electric field amplitude, ϵ̂ is the polarization direction of the
beam, and g = 1, 2 indexes the lower states.
FIGURE 1.1. Schematic of a stimulated Raman transition between states |1⟩ and
|2⟩ via the excited state |e⟩.
To understand stimulated Raman transitions in this system, we need to
study the Hamiltonian governing their dynamics:
1
H = Ω eiϕ1eik1·r−iδ1t |e⟩ ⟨ | 11 + Ω eiϕ2eik2·r−iδ2t1 2 |e⟩ ⟨2|+ h.c. (1.3)
2 2
Following the discussion in Ballance, 2014, we note that if Ω1 ≪ δ1 and
Ω2 ≪ δ2, then there will be largely no population transfer out of states |1⟩ and
3
|2⟩. If, however, δ = |δ1 − δ2| ≪ δ1, δ2, then we can apply the James-Jerke
approximation James and Jerke, 2007:
Ω1 ( )
H = (|1⟩ ⟨1| − |e⟩ ⟨ | Ω2e ) + (|2⟩ ⟨2| − | Ω1Ω2e⟩ ⟨e|) + eiϕei∆k−iδt |2⟩ ⟨1|+ h.c. ,
4δ1 4δ2 4∆
(1.4)
where ϕ = ϕ1 − ϕ2 and 1 = 1( 1 − 1 ). Final term of Eqn. 1.4 generates a∆ 2 δ1 δ2
coupling between states |1⟩ and |2⟩ through a two-photon emission process. This is
a stimulated Raman transition.
1.3. Laser Cooling in Ion Traps
1.3.1. Doppler Cooling
Simply creating a potential well to confine the ion does not mean it
will stay there. If the ion has sufficient kinetic energy, it can escape the trap.
Therefore to keep the ion trapped, we need to keep it cool. We do this by Doppler
cooling D. J. Wineland, Drullinger and Walls, 1978 and Neuhauser et al., 1978.
Consider a trapped ion which has an electronic transition of frequency
ω0. We can implement Doppler cooling by applying a laser beam tuned red
of this transition, i.e., we apply a laser with frequency ωL = ω0 − ∆, where
∆ is the detuning of the laser beam. Since the ion will be oscillating about its
equilibrium position, the frequency of the laser light experienced by the ion will be
Doppler-shifted. This implies that the transition will be driven more strongly, and
therefore that the ion will preferentially absorb a photon, when the ion is moving
towards the laser beam. In such a case, the ion will lose momentum since it is
traveling opposite the direction of the beam. On average, then, the atom will lose
4
momentum when it absorbs photons. When it later emits this absorbed photon,
it will do so in some random direction. The average change in momentum from
emission is therefore zero. So on net, the laser will cool the ion.
There are limits to this process, of course. You cannot bring the ion to a
standstill. The average emission event does not reduce the ion’s momentum,
but it does change its kinetic energy by ℏ2k2/2m, where k is the transition
wavenumber and m is the ion mass. This may seem contradictory, but it is a
simple consequence of the fact that the velocity averages to zero, but the square
of the velocity does not. So Doppler cooling clearly cannot reduce the ion’s
kinetic energy below the single photon recoil kinetic energy. However, since the
actual equilibrium will be achieved when the heating and cooling rates of the
competing processes are equal, the true limit is somewhat different (and higher):
it is ℏγ/2kB, where γ is the linewidth of the transition and kB is the Boltzmann
constant Letokhov, Minogin and Pavlik, 1977. For certain applications (such as
two qubit gates), it is desirable to be closer to the motional ground state than the
Doppler limit will allow. Fortunately, we are able to beat the Doppler limit by
implementing resolved sideband cooling.
1.3.2. Resolved Sideband Cooling
Resolved sideband cooling is a standard technique in ion trapping Monroe
et al., 1995, Eschner et al., 2003. To understand the mechanism behind this
method, consider a trapped ion with motional frequency ωm. Suppose a laser
beam of frequency ωL is driving an atomic transition of frequency ω0 in this
ion. As a function of ωL, the atomic emission spectrum will have a transition at
ω0, as well as many transitions separated from the strong transition by integer
5
multiples of ωm. The strong transition is referred to as the ‘carrier’ and the weaker
transitions are referred to as the ‘sidebands’. Physically, these transitions are
allowed by emitting a photon and simultaneously absorbing/emitting some number
of motional phonons.
These sideband transitions can be exploited to cool the ion. By setting the
laser frequency to ωL = ω0 − ωm, the ion will become electronically excited most
often when it simultaneously loses a phonon. When coupled with a mechanism for
repumping back to the initial state, this cools the ion down. However, the photons
emitted from this repumping will kick the ion, which causes heating. This issue
can be avoided by operating in the Lamb-Dicke regime.
1.3.3. Lamb-Dicke Regime
The Lamb-Dicke regime is characterized by a weak coupling between the
ion’s atomic and motional states D. Wineland et al., 1998. Mathematically, the
condition to be in the Lamb-Dicke regime is given by
√
⟨Ψ 2 2m| kzz |Ψm⟩ ≪ 1 (1.5)
where |Ψm⟩ is the motional wavefunction, z is the distance away from the ion’s
equilibrium position operator along the axis of interest, and kz is the z-component
of the laser’s wavevector. To better understand the inequality 1.5, we begin by
expressing the operator ẑ as
√
ℏ
ẑ = (a+ a†) (1.6)
2miωm
6
where mi is the ion’s mass, and a and a
† are the ladder operators which,
respectively, lower or raise the photon number of photon number eigenstates |n⟩
(Fock states). If we assume we are in a Fock state |n⟩, then the inequality 1.5 may
be written
√ √ √
⟨ ℏk
2 ℏk2
Ψ | k2z2 |Ψ ⟩ = z ⟨Ψ | (a+ a† 2 zm z m m ) |Ψm⟩ = (2n+ 1) (1.7)√2miωm 2miωm
= η2(2n+ 1) ≪ 1 (1.8)
√
where we have defined the ‘Lamb-Dicke parameter’ η as ℏk2z/2miωm.
Fundamentally, this requirement just says that a small Lamb-Dicke parameter is
needed to be in the Lamb-Dicke regime. The physical meaning of this is that the
kinetic energy change due to a laser photon recoil energy ℏ2k2z/2mi must be much
smaller than the harmonic oscillator energy level separation ℏωm, i.e., the atomic
energy states must be largely decoupled from this motion. This is necessary in
sideband cooling, since after driving a sideband transition, we want spontaneous
emission events (which serve as the repumping process mentioned above) to take
place primarily at the carrier frequency so as to not significantly alter the motional
states and disrupt the cooling process. Satisfying the Lamb-Dicke criterion ensures
this is the case.
1.4. Gates in an Ion Trap
Information processing in a classical computer takes place on bits, which can
take the values 0 or 1. To make a classical computer most useful, it is desirable to
ensure that it is capable of performing any logical operation, i.e., it is able to map
7
any string of bits to any other string of bits. Such a device is called a universal
computer, and the set of logical operations is said to be functionally complete. The
simplest functionally complete set of operators is comprised only of the operator
NAND. The NAND operator is the negation of the logical conjunction of two bits,
and it can be proven that this operator alone can map any string of bits to any
other string of bits. Similarly, in a quantum computer, it is desirable to construct
a set of universal quantum logic gates.
1.4.1. Universal Quantum Logic Gate Sets
Since quantum logic gates can be represented as vectors in an the space of
unitary operators, we need a set of gates that can generate any desired unitary
operation. For a single qubit, you can generate a universal gate set with the the
Pauli matrices,
     0 1 0 −i 1 0 σX = , σY = , σZ =   , (1.9)
1 0 i 0 0 −1
written in the {|0⟩ , |1⟩} basis. You can use these to create operators that rotate
vectors along the “Bloch sphere”. The Bloch sphere is a representation of all
possible single qubit states, with |0⟩ and |1⟩ corresponding to opposite ‘poles’ of
the sphere along the Z-axis, with X and Y corresponding to axes in the equatorial
plane of the sphere. A visualization of the Bloch sphere is given in Fig. 1.2. It is
trivial to see that operators which can map an arbitrary vector to any other vector
on the Bloch sphere would constitute a universal set. The rotation operators do
just that, and are written as
8
−iθ
R (θ) = e σ
−iθ −iθ
X
X 2 , RY (θ) = e
σ
2 Y RZ(θ) = e
σ
2 Z . (1.10)
Here, θ is the angle of the rotation, and X, Y , and Z correspond to the axis of
rotation on the Bloch sphere.
FIGURE 1.2. The Bloch sphere (figure taken from Saad et al., 2021). Poles of the
Z-axis correspond to the states |0⟩ and |1⟩. The angle θ adjusts the population of
|0⟩ and |1⟩ and the angle ϕ adjusts the relative phase.
For the n-qubit case, however, these operators would not constitute a
universal set. The set must be augmented by an entangling, two-qubit gate, such
as the controlled-not (CNOT) gate. The CNOT gate is a two-qubit gate that
9
simply flips the second qubit if the first qubit is |1⟩ but does nothing if the first
qubit is |0⟩. In matrix form, the CNOT gate is
 1 0 0 0  0 1 0 0CNOT =   , (1.11)0 0 0 1
0 0 1 0
written in the {|00⟩ , |01⟩ , |10⟩ , |11⟩} basis. The purpose of the CNOT gate is to
generate entanglement between qubits. You can see that the CNOT gate generates
entanglement by applying it to the state √1 (|0⟩) ⊗ (|0⟩ + |1⟩), which results in the
2
maximally-entangled Bell state √1 (|00⟩+ |11⟩).
2
The rotation operators, plus a phase shift operator, plus CNOT constitute
a universal gate set. This is provable Nielsen and Chuang, 2000, but the proof
is complicated. Intuitively, you can discern that these operators are universal
because they are capable of creating any unitary single-qubit operator and can
also entangle any two qubits together.
In practice, phase shifts, rotation operators, and the CNOT gate are
straightforward enough to implement in ion traps that they are commonly used
to implement general quantum circuits. In our lab, we usually physically perform
these gates using stimulated Raman transitions. We discuss the physics behind
such transitions and their use in implementing our quantum logic gates in the
sections below.
10
1.4.2. Single-Qubit Gate
Perhaps the most commonly-used single-qubit gate is σX gate, i.e., a π
rotation about the X-axis of the Bloch sphere. This is the same as a σY gate,
but with a different phase. For qubits encoded in sublevels of some manifold in
a trapped ion, such gates are usually physically implemented in one of two ways:
stimulated Raman transitions or rf pulses. Stimulated Raman transitions were
discussed in some detail above, but the physics underlying rf-pulse-induced σX
gates is very similar; the only difference is that rf pulses utilize an M1 coupling as
opposed to the E1 coupling used by Raman transitions.
1.4.3. Two-Qubit Gate
Investigation into two-qubit gates in trapped ions began with the Cirac-
Zoller (CZ) gate Cirac and Zoller, 1995). The CZ gate is a means of generating
entanglement between two ions, but it is challenging to implement. Two difficulties
with implementing the CZ gate are the need for individually addressing each
qubit with the laser beams and the first-order dependence of the gate fidelity on
temperature.
In the intervening years since the CZ paper, work has been done to construct
more practical two-qubit gates in trapped ions. Today, the two most common
implementations of a two-qubit gate in trapped-ion quantum computers are the
Mølmer-Sørensen gate (a controlled σx gate) and the light shift gate (a controlled
σz gate). I will discuss each of these gates in turn.
11
1.4.4. The Mølmer-Sørensen Gate
The original discussion of this gate may be found in Mølmer and Sørensen,
1999. The first experimental demonstration was in Sackett et al., 2000. This gate
flips each qubit and applies a phase conditional on their parity.
Physically, an MS gate generates entanglement by coupling the ions internal,
electronic state to the collective motion of the ion crystal. The gate may be driven
by microwaves, quadrupole lasers, or by pairs of Raman beams. Since the latter
case is what we have implemented in the lab, I will discuss it further. We can
implement the MS gate using Raman beams with each detuned by some amount
δ from the resonance of the sideband transitions. This detuning ensures that the
only transitions which flip both ions’ qubit states are driven resonantly. The action
of the gate can be visualized in phase space, where the position and momentum
of the ions form the x and y axes. The joint state of the ion may be represented
as a point in this space. The MS gate acts by steering the ion state around this
space, forming a closed loop at the end of the gate. The area enclosed by this loop
corresponds to the phase that state accumulates, and, for appropriately chosen
parameters, the MS gate will impart phase only to certain joint states of the ion.
We discuss our lab’s progress towards physically implementing this gate in a novel
qubit encoding in Chapter VII
1.4.5. The Light Shift Gate
The light shift (LS) gate Leibfried et al., 2003 Home et al., 2009, like the
MS gate, is a common entanglement-generation mechanism for trapped ions.
Both gates are closely related; they are simply geometric phase gates in different
bases. To physically implement an LS gate, we need to choose a Raman beam
12
polarization such that |0⟩ and |1⟩ couple to it differently. We can then apply a
pair of such beams to generate a standing interference pattern. In this case, the
light shift varies spatially which generates a force. If we adjust the relative phase
of the Raman beams, we can generate a beat note. By tuning this beat frequency
to the motional mode frequency, we can induce a spin-dependent force. Finally, we
can detune the Raman beams from the motional sideband of interest by a small
amount δ, which may be used to control the phase accumulated depending on the
ion pair’s joint state. If we neglect off-resonant terms, the phase accumulated by
each state is approximately Ballance, 2014
|11⟩ : (1 + eiϕm)Ω1
|10⟩ : Ω − eiϕm1 Ω0
(1.12)
|01⟩ : eiϕmΩ1 − Ω0
|00⟩ : −(1 + eiϕm)Ω0,
where ϕm is a the Raman phase difference between the two ions and Ωj is the
single-beam Rabi frequency for the qubit state |j⟩.
For well-chosen experimental parameters, we can set ϕm = π to maximize
the efficiency of this scheme. In this case, ion pairs with the same parity will
experience no force, but opposite-parity ion pairs will experience a force of
±(Ω0 + Ω1). This can be used to implement the logical gate
13
 1 0 0 0 

0 i 0 0
 

Û = (1.13)
0 0 i 0
0 0 0 1
√ √
in the basis {|00⟩ , |01⟩ , |10⟩ , |11⟩}. If given |ψ⟩ = 1/ 2(|0⟩+|1⟩)⊗1/ 2(|0⟩+|1⟩) as
the input state, this gate will yield the maximally entangled state |Ψ⟩ = 1/2(|00⟩+
i |01⟩+ i |10⟩+ |11⟩).
1.5. Photon Scattering Errors in Trapped-Ion Quantum Computers
As discussed above, stimulated Raman transitions are often used to
implement trapped-ion quantum logic gates. In such setups, a pair of laser
beams drives a two-photon stimulated transition between qubit states |0⟩ and |1⟩
through one or more intermediate states. Photon scattering during this process is
unavoidable and reduces the gate fidelity, i.e., how well the actual output state
overlaps with the desired output state. Understanding the fundamental limit
photon scattering places on achievable gate fidelity is potentially crucial for the
viability of trapped-ion quantum computing.
Although photon scattering errors have been previously studied Ozeri, 2007;
Uys et al., 2010; Sawyer and Brown, 2021, our group chose to revisit the topic for
two reasons. First, past work produced a model of scattering that, while suitable
for explaining moderate-detuning gate errors, is inaccurate at larger detunings.
Second, we are interested in characterizing qubits in metastable manifolds (‘m’
qubits), and the previous studies did not examine such qubits. They instead
14
focused solely on qubits with either one or both states encoded in the S1/2 ground
manifold of trapped ions (‘o’ qubits and ‘g’ qubits, respectively).
1.5.1. Past models of Raman Scattering Error
Since 2007, the seminal study of photon scattering errors in trapped-ion
qubits has been Ozeri, 2007. In that work, Raman and Rayleigh scattering errors
were calculated for g qubits in most commonly-used trapped-ion species. While
there was much other useful information in this study, perhaps the most important
message was that there is a fundamental limit on photon scattering suppression
in Raman-beam-driven logic gates in trapped ion species that have low-lying D
manifolds. Put another way, it was said that photon scattering ensures that you
can never reduce the gate error below a certain value. For some species, such as
43Ca+ and 137Ba+, this limit is large enough that it would make error correction
difficult due to the high overhead demands. The limits for two-qubit gates each
species are given in the Table 1.1.
43Ca+ 1.06 ·10−4
87Sr+ 5.0 ·10−5
137Ba+ 1.46 ·10−4
171Yb+ 7 ·10−7
199Hg+ 1 ·10−7
TABLE 1.1. Minimum scattering error during a two-qubit gate in some commonly-
used trapped-ion species. Sourced from Ozeri, 2007.
However, we discovered that these limits do not exist. Ozeri et al. found
such limits because their model assumes that certain effects are negligible,
but these assumptions are valid only at small detunings. The neglected effects
include the detuning dependence of the scattered photon frequency and Lamb-
Dicke parameter, contributions of a second scattering term, interference effects
15
in scattering to the metastable manifolds, and the contribution of the counter-
rotating component of the laser electric field to the Raman transition rate. As
shown below, including such effects changes the limiting behavior of the model,
resulting in no lower bound on gate error, in contrast to the predictions of Ozeri,
et al..
1.5.2. Photon Scattering Errors in Metastable Qubits
The past studies of Raman scattering errors have primarily considered o and
g qubits. However, in a recent proposal Allcock et al., 2021 I co-authored (this
thesis presents a lot of this work, particularly in Chapter II), we showed that
encoding qubits in metastable states of trapped ions (‘m’ qubits) has several
important advantages over g and o type encodings. Since we are interested in
implementing m qubits, it is important for us to characterize the photon scattering
errors for such qubits. Below, we will show that the m qubit scattering error has
largely the same form as in g qubits, though m qubits require longer wavelength
lasers and higher power to reach the same error as g qubits at the same detuning.
16
CHAPTER II
OMG ARCHITECTURE
Typically, ion trap qubits have at least one of their states encoded in the
ground manifold. When both states are so encoded, we call it a g qubit (‘g ’ for
‘ground’); when one state is in the ground manifold, and the other is in some
higher manifold, we call is an o qubit (‘o’ for ‘optical’). In these schemes, you
generally need to load the trap with two ion species: one for your qubit operations,
the other for sympathetic cooling. This is because if you loaded the trap with just
one species and attempted to drive cooling transitions in one of the ions in the
chain, you will also cause scattering in the qubit ions and destroy their coherence.
Loading two species solves this since you can cool on a transition of one species
with a wavelength at which the other species is unlikely to scatter light.
While the multi-species approach has many benefits, it also has several
drawbacks. For example, it requires more lasers, complicated interspecies
operations, and more atomic sources Tan et al., 2015; Bruzewicz et al., 2019;
A. C. Hughes et al., 2020. Additionally, it has multiple downsides due the mass
inequality of the different ion species. First, you have to concern yourself with the
order of the ions in the chain. This is because different ion species ordering can
alter the mode structure. Also, practically speaking, mixed species approaches
require a specific ordering of the ion species in the chain. This is to ensure that
the ion is at the position of maximum intensity for its respective driving lasers.
This is a drawback because enforcing the desired ion order requires relaxing
the ion chain until it assumes the sought-after form. This procedure works well
in a small-scale experiment, but may be be difficult to automate at scale in a
17
proper quantum computer. Second, cooling is less efficient in multi-species chains,
due to their different masses. Third, different ion species will generally have
different sensitivities to stray static fields, which will shift the ions away from
their equilibrium position by different amounts, resulting in a relative position
vector that is no longer aligned with the trap axis. Fourth, gradients in the time-
averaged rf potential and trap anharmonicities can change the inter-ion spacing
since different species will experience a different strength from the ponderomotive
potential. Finally, it is somewhat challenging to transport multi-species ion chains
without substantial motional excitation, though this is not completely infeasible
(see, e.g., Burton et al., 2023).
It would therefore be desirable to implement a scheme that allows ion
trap quantum computing in a single ion species. In collaboration with MIT,
MIT Lincoln Labs, and UCLA, we proposed and implemented such a qubit
scheme: the omg qubit scheme Allcock et al., 2021 (some experimental work
towards implementing these qubits has been conducted by a group at Tsinghua
University Yang et al., 2022).
2.1. Metastable qubits
To begin to understand the usefulness of the omg scheme, we must first
study the m qubit. The m qubit is simply a qubit with both states encoded in
the Zeeman or hyperfine sublevels of a metastable manifold in the trapped ion.
A diagram of the m qubit position in energy space, along with parallel depictions
of o and g qubits, is given in 2.1. For singly-ionized alkaline earth metals, such as
Ca+, we are interested in the metastable D5/2 manifold. In other ion species, other
metastable manifolds are also viable, such as the F7/2 manifold of Yb
+ Ransford
18
et al., 2021. In this work, I will be focusing on D5/2 metastable qubits since they
are what we have studied in our lab.
While they are necessary to implement single-species ion trap quantum
computing, there are a few drawbacks to using metastable qubits. First, they have
a finite lifetime. For example, the D5/2 manifold of Ca
+ lives for about 1.2 seconds.
This does not seriously limit that manifold’s potential for encoding qubits however.
Since most quantum logic gates take on the order of microseconds, potentially
millions of operations can be performed within the lifetime of the D5/2 manifold.
Secondly, preparation and readout are slightly more complicated than in
g qubits. Optical pumping schemes can be used for preparation and readout m
qubits, but more lasers are required to shelve and deshelve population in the D5/2
manifold. However, I performed Python simulations of preparation and readout
for m in the D5/2 manifold on
43Ca+ before we began trapping ions in our lab, and
confirmed that they did not limit the viability of such qubits. I concluded that
greater than 99% preparation and readout fidelity could be achieved, with optical
pumping taking around only 100µs.
2.2. omg architecture
The utility of the omg qubit scheme comes mainly from the separation of
g and m qubits in Hilbert space. This separation lets us divide the various tasks
of quantum computing among the different qubit encodings. Here we will discuss
three modes for omg qubits. We will characterize each mode by an ordered triple,
{q1, q2, q3} with qi = o,m or g) and where q1 corresponds to which encoding is used
for state preparation, q2 which encoding is used for gates, and q3 which encoding
is used for storage. The three modes we will discuss are the {m,m,m} mode, the
19
FIGURE 2.1. The typical structure of alkaline earth ions, including relevant
transitions. The S1/2 ←→ P1/2 transition is used for dissipative operations
(laser cooling, state preparation, and readout). The o qubit transition is electric
dipole (E1) forbidden and typically requires a narrow linewidth laser. Figure taken
from Allcock et al., 2021.
{g,m, g} mode, and the {m, g,m} mode. A diagrammatic explanation of these
modes is given in Figure 2.2.
2.2.1. The {m,m,m} Mode
The main motivation for the {m,m,m} mode is that g qubits are a
natural choice for dissipative operations like laser cooling, state preparation, and
readout. It is therefore natural to consider using m qubits for most of the unitary
20
FIGURE 2.2. Various omg schemes. Figure taken from Allcock et al., 2021.
operations. As shown in Figure 2.2, gates in this mode would be performed by
individually addressing the m qubit ions with Raman laser beams.
Conveniently, because all coherent operations take place in m qubits in this
mode, no coherent transfer between m and o or g qubits is necessary. This, in
turn, means that all requisite operations may be accomplished using only electric-
dipole transitions, and, therefore, the technologically challenging implementation
of narrow linewidth lasers for, e.g., electric-quadrupole transitions is not strictly
necessary. It is also worth noting that this mode has the advantage of allowing for
dissipitative operations such as laser cooling to be performed on g qubits during
single-qubit gates on m qubits in the same ion crystal.
The primary drawback of using this mode is the limited lifetime the
metastable states. However, for many species, including Ca+, the lifetime is long
enough that it will not be a limiting error for gate performance. Since the D5/2
manifold of 40Ca+ has a lifetime of ∼1.2 s, for a gate time of 1-10µs, this would
give an error between 1.2× 10−6 and 1.2× 10−5.
21
2.2.2. The {g,m, g} Mode
The second mode we will consider is characterized in the middle row of
Figure 2.2. Here, coherent transfer between o, m, and g encodings can be achieved
with individually addressable, narrow linewidth laser beams that drive electric-
quadrupole or electric-octopole transitions in the ions. Global beams may then be
used to implement logic gates.
The main upside of this mode is that it exploits the stability of g qubits to
protect stored information during logic gates. One disadvantage, of course, is that
this scheme requires high-fidelity coherent population transfer on quadrupole or
octopole transitions.
2.2.3. The {m, g,m} Mode
The last mode we will consider is described in the final row of Figure 2.2.
This is the first of the three modes we have considered in which gates are
performed on g qubits. Just as in the {g,m, g} mode, this mode requires that we
be able to coherently transfer individual ions from a g to an m qubit, and vice
versa. This capability would be required for gates and helpful for cooling and
readout, although the latter may be accomplished with incoherent methods.
This mode shares a common advantage with the {g,m, g} mode: the storage
qubits are protected from the laser light, so we may apply the laser beams globally
for gates. The difference is that only the qubits need to be converted between m
and g, which is a less stringent requirement than in the {g,m, g} mode. However,
keeping the ions in the metastable manifold by default means that this mode is
more susceptible to the finite lifetime of the m qubit.
22
2.3. Two-Qubit Gates in m Qubits
Two-qubit gates have recently been performed in m qubits for the first time.
One such gate was performed by University of Oxford in m qubits encoded in the
metastable D5/2 manifold of
88Sr+ Bazavan et al., 2023. They implemented it with
the S1/2 ←→ D5/2 quadrupole transition and achieved a gate fidelity of 0.859(5).
Another gate was performed at Innsbruck Roos et al., 2004. Technically, the
gate was not performed in m qubits, but it did generate entangled m qubit states.
The group performed an entangling gate on S1/2, D5/2 o qubits. These qubits were
then converted to m qubits with single qubit operations.
Our group has also made progress towards implementing m qubit two-qubit
gates. Unlike previous implementations, we are applying a two-qubit gate directly
in m qubits using stimulated-Raman transitions. In Chapter VII, I report on our
progress towards implementing such a gate.
23
CHAPTER III
EXPERIMENTAL METHODS
Below, I describe the various components of our experimental apparatus.
3.1. Ion Trap
We use a linear Paul trap in our lab. The schematic of our trap is shown in
Figure 3.1. A view of the physical trap is given in Figure 3.2. 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 vacuums, reaching pressures of order
10−11Torr.
24
FIGURE 3.1. (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.
25
FIGURE 3.2. (a) The assembled rod trap. (b) The rod trap mounted in the
vacuum chamber, as seen through a viewport.
26
3.2. Magnetic Field
When working with trapped ions, it is necessary to apply an external
magnetic field to lift the degeneracy of the atomic sublevels. We accomplish this
by supplying a current to four rectangularly coiled wires, each of different sides
of the spherical octagon chamber in which the trap sits. Additionally, there is
one circularly coiled wire oriented perpendicular to the rectangular coils. One
of the rectangular coil pairs, acting in concert with the Earth’s magnetic field,
primarily determines the magnetic field magnitude and direction, and, therefore,
the ion’s quantization axis. The remaining coils allow us to finely adjust (”shim”)
the magnetic field to align it relative to the incoming laser beams. We supply a
0.5A current with a low-noise source. This setup allows us to generate a 0.498G
magnetic field, which creates a Zeeman splitting in the D5/2 manifold of 2.63MHz.
27
FIGURE 3.3. Coils of copper wire used to generate our trap’s magnetic field
3.3. Ion Detection
We use an imaging system to detect ions and the state of the qubit. The
imaging system is optimized to collect 397 nm photons, which we generate when
we drive the S1/2 ←→ P1/2 transition. The imaging system has a numerical
aperture (NA) of 0.4 and a photomultiplier tube (PMT) efficiency of ∼0.3. The
total measured click efficiency of the system is 1.2%.
28
FIGURE 3.4. Schematic of the ion detection system. Component indicated by ‘a’
is the ion in vacuum, ‘b’ is the aspheric objective, ‘c’ is the adjustable aperture,
‘d’ is the filter holder, ‘e’ is the second stage (re-imaging Lenses and items f-
l), ‘f’ is the slit imaging lens, ‘g’ is the flipper mirror, ‘h’ is the photomultiplier
tube (PMT) imaging lens, ‘i’ is the photomultiplier tube, ‘j’ is the convex camera
imaging lens, ‘k’ is the concave camera imaging lens, and ‘l’ is a CMOS camera
with 3.45 micron resolution.
FIGURE 3.5. The actual ion camera system.
3.4. Trap rf Drive
The rf chain provides the signal needed to provide stable confinement for
the ions as well as reducing noise that contributes to motional decoherence.
The rf chain is comprised of multiple components. One key component is the
29
Squareatron, developed by Jeremy Metzner in collaboration with University
of Oxford Metzner, Allcock and Ballance, 2020. The Squareatron behaves like
a saturated amplifier, taking its input clock signal (which it receives from an
arbitrary wave generator, or AWG) and stripping the harmonics off with a
bandpass filter (BPF) to return stable rf signals. It has an output power of about
10 dBm and can be tuned roughly 1 dBm with a digital to analog converter (DAC)
board. The purpose of the Squareatron is to reduce amplitude modulation noise
to keep the ion’s secular frequency stable. Additionally, the rf chain has a helical
resonator (coupled to a 1 W amplifier to get necessary power) with a step-up of
∼100× and a Q of ∼150. This resonator impedance-matches the amplifier to the
ion trap and provides modest filtering. The Squareatron and amplifier are kept in
an insulating box with a thermostat Thermostat 2022 which helps to stabilize the
ion’s motional frequency.
3.5. Laser Systems
Our 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 a rack
housing the wavemeter. The setup of the room can be seen in Figure 3.6. The
setup of a laser rack drawer is shown in Figure 3.7. 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. Upon emission from the laser, the
beam is redirected 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 off of a diffraction
30
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, which
serves to stabilize the laser frequencies.
31
FIGURE 3.6. The laser racks. Contains our breadboard laser systems used for
ionization, cooling, preparation, and readout.
32
FIGURE 3.7. 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 different AOM racks in a separate room.
33
3.6. AOM Boards
Because our experiments take place on the scale of microseconds, we need a
way to rapidly turn our lasers on and off. Our solution is to use acousto-optical
modulators (AOMs).
AOMs are composed of a piezoelectric material attached to a crystal. When
an rf signal is applied to the piezoelectric material, it causes it to vibrate and
generate a sound wave in the crystal. When laser light passes through the AOM,
the interaction of the light with the vibrating crystal creates discrete, diffracted
beams with frequencies shifted by some integer multiple of the rf frequency. A
picture of the inside of one of our lab’s AOMs is given in Figure 3.8.
FIGURE 3.8. The inside of an AOM. The circuit produces an ac current that
creates a sound wave using a sheet of piezoelectric material attached to the crystal.
The circuit is controlled via the SMA connector on the right side.
34
The optical switch functionality of the AOM comes from the separation
between the diffracted beams. To create our switch, we simply position a slit in
the beam path that allows through the 1st or -1st diffracted beam, but blocks the
0th order. This means that the laser light reaches the ion only when the AOM is
on. Since the rise time of our AOMs is tens of nanoseconds, they provide optical
switches which are more than fast enough for our experiment’s requirements.
Some of our AOM paths are double-pass. This means that the laser beam is sent
through the AOM, the 1st or -1st order beam is picked off, reflected, and sent back
through the AOM once more. This setup has three benefits: in it, the angle of
the beam is constant with changes in AOM frequency, it effectively doubles the
electrical bandwidth, and there is a higher extinction ratio for the beam. This
latter effect is important for some beams, as we want to ensure that we do not
have leakage when the AOMs are off. For example, 854 nm light leakage could lead
to D5/2 deshelving during an m qubit experiment.
Another benefit of using AOMs is that, in conjunction with the SU servo
(see Section 3.7), they allow us to control the optical power of the laser beams at
the ion. This is done by feeding the SU servo the signal from the photodiodes (see
Section 3.8) and having the servo adjust the attenuation of the rf signal to the
AOM to maintain the desired power.
Using AOMs comes at the cost of optical power, however. We choose to
use the first-order diffracted beam out of the AOM, but there is always some
power in the other orders. The first-order diffraction efficiency is controllable by
adjusting the rf power into the AOM. The optimal operating power is different for
Isomet model 1250C-829A ”blue” AOMs is shown in Figures 3.9. The blue AOM
contains a TeO2 crystal and has a center frequency of 260MHz with a 50MHz
35
bandwidth. The optimal operating power for the Isomet model 1205C-1 ”red”
AOMs is shown in Figure 3.10. The red AOM contains a PbMoO4 crystal and has
a center frequency of 200MHz with a 50MHz bandwidth.
It is worth noting that people commonly place AOMs directly on their
optical tables and work with them there. We, however, choose to build our AOM
paths on modular boards housed in rack drawers. This design idea came from the
University of Oxford ion-trapping group. This design has the advantages that
the AOM racks take up less space in the lab (as the drawers may be vertically
stacked in a 19” rack) and are more stable (due to the low beam height), but has
the drawback that the optics’ positions are less reconfigurable. One AOM rack
board (the 393 nm/397 nm board) is shown in Figure 3.11.
FIGURE 3.9. Model 1205C-829A ”blue” AOM first-order diffraction efficiency as a
function of Urukul attenuation.
36
FIGURE 3.10. Model 1205C-1 ”red” AOM first-order diffraction efficiency as a
function of Urukul attenuation.
37
FIGURE 3.11. Example of an AOM rack board in our lab.
We have three AOM boards, and their design is shown in Fig 3.12. More
detailed beam path diagrams are given in Figures 3.13. The top-most board
generates our 850 nm and 854 nm pump beams. We use these beams to pump
population out of the D3/2 and D5/2 manifolds. The board just below that is
designed to control the 854 nm input to the top board as well as the 866 nm
beam. The latter beam is used in Doppler cooling (by driving D3/2 ←→ P1/2
transitions) and for depumping the D3/2 population. The bottom board of the
diagram controls our 397 nm and 393 nm laser beams. The 397 nm beams are for
optical pumping, cooling, and detection on the S1/2 ←→ P1/2 transition, and the
393 nm beam is for optical pumping on the S1/2 ←→ P3/2 transition.
38
FIGURE 3.12. The three AOM boards. The boards are for dissipative operations,
such as cooling, preparation, and readout. The boxes labeled ”1×” and ”2×”
correspond to single- and double-pass AOM filters, respectively.
3.7. ARTIQ
For real-time implementation and remote control of our experiments,
both hardware control and data collection, we utilize the software system
ARTIQ Bourdeauducq et al., 2016 (Advanced Real-Time Infrastructure for
Quantum physics) and the associated hardware from Sinara Kasprowicz, Kulik
et al., 2020; Bourdeauducq et al., 2016. A schematic of the ARTIQ setup is show
in Figure 3.14.
39
a Local ethernet switch
FPGA controller
ARTIQ NAS
ADC Master InuxDB b Kasli
PD Host
PC Grafana
RF DDS Urukul Sampler
AOM SplitterMotional
drive
Amp
DIO MagneticRabi drive
Shutter AOM
DAC
Trap
FIGURE 3.14. Schematic of the ARTIQ setup. (a) Schematic of the connections
between the physical components between the subcomponents of the ARTIQ
system in our lab. (b) Schematic of the SU servo system, and depiction of its
feedback on the optical power measured by the photodiode. Figures made by Alex
Quinn.
In terms of hardware, firstly, the host PC interfaces with an FPGA board
called the ”Kasli,” which, along with the host, also runs the ARTIQ code. The
trap electrodes are controlled by the ”Zotino” board, a digital to analog converter
(DAC). We use the Zotino to apply dc voltages to shim the trap electrodes
and finely adjust the ion’s position. We use ”Urukul” direct digital synthesizer
(DDS) boards Kasprowicz, Harty et al., 2022 as sources of timed rf signals, with
applications to controlling AOMs, driving M1 transitions between Zeeman levels,
and driving ion motion (achieved by applying an rf tone to one of the trap rods).
For reading and generating digital signals, we use DIO boards. This has utility
for reading photon clicks from the photomultiplier tubes and for controlling our
mechanical shutters. Finally, we have the ”Sampler”, which is an analog to digital
40
Laser control
Sinara crate
Shim voltages
Photon
counts
PMT
converter (ADC) board. This is a component of the ”Sampler-Urukul” (SU) servo
system for precise control of laser beam powers on short timescales. The SU servo
uses a pickoff from the laser beams (discussed in the section below) to monitor the
laser power. The SU servo acts as a feedback system, adjusting the rf amplitude on
the Urukul to control adjust the laser power in response to the Sampler reading.
3.8. Beam Delivery System
After the beams are sent from the laser rack room to the trap room, and
there through the AOM boards, they finally reach the beam delivery system.
The primary function of the beam delivery system is to get the laser beams from
the output fibers (usually from the AOM boards) to the ions while ensuring we
maintain the correct beam power, polarizations, and alignment on the ion. After
being passing through the AOM board, the laser beam leaves the fiber via a
collimator and then passes through a PBS to fix the polarization. After this, it
is sent through the photodiode assembly, which picks a fraction of the beam off
to monitor the beam power with the SU servo. After this, the beam is sent to
the optics mounted to the ”tombstone”, a custom-machined component made to
attach to the vacuum hardware. On the tombstone are a pair of 1” fixed mirrors
and a 1” f = 150mm achromatic lens. The beam pointing is controlled by a
piezoelectric-motor-controlled mirror mount, which is controllable by ARTIQ.
After passing through the tombstone optics, the beam is sent to the ion. A
schematic of the relevant beams passing through the trap is shown in Figure 3.15.
The polarization of the beams is controlled by half- and quarter-waveplates
(labeled λ/2 and λ/4 in 3.15).
41
42
AOM 1
PBS 1 DDoouubbllee  
Adjustable Mirrors
PPaassss Fixed Mirrors
Lenses
397 nm light Lens 1
input Lens 2
PBS
AOM
λ/2 Waveplate
PBS 2 λ/4 Waveplate
? +250 NPBS
MHz
NPBS 1
397 nm pump Resonant beamLens 4 Lens 3
output AOM 2
-250 
Lens 6 Red-detuned MHz
beam Lens 5
AOM 3
PBS 3
397/393 nm NPBS 2
pump Lens 8 Lens 7
output
AOM 4
Lens 10 Lens 9
393 nm light
input AOM 5
FIGURE 3.13. Optical paths on the 397/393 nm AOM control board.
C
GTP
4 B/λ
GP
D A
PM
MP π D P6    7  9
PM
MP
GTP
43
D C
λ/4 λ/2 λ/2
+ PD 40 +976 σ Ca
854 π
976 π
A B
LP LP
λ/4 λ/2 λ/2 λ/4
854 σ+/-
PD
  / λ/2
866 σ+/- PD 854 σ+
PD
397 π
PD
393/397 σ+
375/423
FIGURE 3.15. Schematic of beam delivery system. Path shown for each laser beam, as well as the direction they travel
in the trap. Ion shown as yellow circle. Tombstone drawing shown on the right side.
CHAPTER IV
SCATTERING THEORY
Below, after describing our qubit choices and defining our model, we
calculate the photon-scattering-induced errors in single-qubit and two-qubit gates
in m qubits and compare them to g qubits. There are many physical differences
between m and g qubits, such as the dipole coupling between the qubit states and
P1/2 that exists for g qubits but is absent in m qubits, or certain scattering terms
present in m qubits that are not present in g qubits. These physical differences
exert influences of varying magnitudes on the quantitative scattering error, but on
net, they tend to increase the detuning from resonance required for a certain error
in m qubits relative to g qubits. As for the qualitative scattering behavior, we find
that the main difference between the two schemes is the existence of a lower bound
on two-qubit gate errors in m qubits which is absent in g qubits. However, for all
trapped ions considered in this work, this lower bound is sufficiently small (less
than 10−4) that low-overhead error correction is possible for Raman gates in the m
qubit scheme.
After describing the overall behavior of the two different qubit schemes in
the large-detuning model, we discuss the contributions of higher-lying levels to the
scattering rates of the m qubit model. Additionally, for both m and g qubits, we
estimate the contribution to gate error from Rayleigh scattering.
4.1. Choice of Qubit
In what follows, we consider a number of common trapped-ion qubit species:
9Be+, 25Mg+, 43Ca+, 87Sr+, 133Ba+, 135Ba+, 137Ba+, 171Yb+, and 173Yb+. We
44
g qubit m qubit
∆ ∆
P3/2 P3/2
ωf
P1/2 P1/2
|0〉| |1 |
D ω5/2 0 |1〉D| |5/2
D3/2 D3/2
|0〉|
ω0 S1/2
| S1/20 |1〉|
FIGURE 4.1. The g qubits are encoded in hyperfine sublevels of the S1/2 manifold
with gates often performed via Raman transitions; m qubits are encoded in
the hyperfine sublevels of the D5/2 level with gates performed in the same way.
Manifolds that do not participate in qubit operations in the respective schemes are
shown in gray. Figure from Moore, Campbell et al., 2023.
present results for g qubits in all species. For m qubits, we perform calculations
only for the species with a sufficiently long-lived (≳ 1 second lifetime) D5/2
manifold: 43Ca+, 87Sr+, 133Ba+, 135Ba+, and 137Ba+ (see 1 for further discussion).
The attributes of all these ion species are given in Table 4.1.
For our qubit choices, we use hyperfine ‘clock’ qubits for both m and
g qubits, due to their insensitivity to magnetic field noise and corresponding
suppression of Rayleigh dephasing Tan, 2016 (see 2 for the definition of ‘clock
1Yb+ has an F7/2 manifold with a years-long lifetime Allcock et al., 2021, making it a suitable
candidate for m qubits. However, this system is more complex and would require a separate
analysis, so we only present g qubit results for Yb+.
2A clock qubit is defined as having a qubit frequency that does not change with changes in the
B-field (to first order); the B-field magnitude at which this occurs can be called the clock point,
and the B-field direction defines the quantization axis.
45
qubit’.). We give our qubit choices (along with other qubit details) explicitly in
Table 4.1 in terms of F (hyperfine angular momentum quantum number) and mF
(corresponding angular momentum projection quantum number). Since g qubits
are already well-established, we simply follow Ozeri, 2007 in their encoding scheme.
As for m qubits, we choose states that should be relatively easy to prepare and
readout, as well as give consistent behavior across all m qubit species for studying
the scattering errors the states
|0⟩ | 5 5= D5/2, F = I + ,mF = ±|I + − 1|⟩
2 2
and
|1⟩ = | 5 5D5/2, F = I + − 1,mF = ±(I + − 1)⟩
2 2
(depending on the sign of relevant hyperfine splittings), where F is the hyperfine
angular momentum quantum number, mF is the corresponding angular momentum
projection quantum number, and I is the nuclear spin (for further discussion of
this choice, see 3). Details about the qubits in the various species are given in
Table 4.1.
4.2. Scattering Probability
We assume that gates are performed on m and g qubits using stimulated
Raman transitions (Fig. 4.1). These are coherent two-photon processes where
population is transferred between |0⟩ and |1⟩ virtually through higher energy
intermediate states |k⟩. During this process, there is a chance of spontaneous
3These are not necessarily “optimal” qubits. We choose them simply to have consistent
behavior across m qubits for studying the scattering errors.
46
9Be+ 25Mg+ 43Ca+ 87Sr+ 133Ba+ 135Ba+ 137Ba+ 171Yb+ 173Yb+
I 3/2 5/2 7/2 9/2 1/2 3/2 3/2 1/2 5/2
D5/2 5/2τ τD (s) τ
D - - 1.110 0.357 29.856 0.0072
5/2
|0⟩ (m) - - |6,+5⟩ |6,−6⟩ |2,+2⟩ |4,−3⟩ |4,−3⟩ - -
|1⟩ (m) - - |5,+5⟩ |7,−6⟩ |3,+2⟩ |3,−3⟩ |3,−3⟩ - -
|0⟩ (g) |2, 0⟩ |3, 0⟩ |4, 0⟩ |5, 0⟩ |1, 0⟩ |2, 0⟩ |2, 0⟩ |0, 0⟩ |2, 0⟩
|1⟩ (g) |1, 0⟩ |2, 0⟩ |3, 0⟩ |4, 0⟩ |0, 0⟩ |1, 0⟩ |1, 0⟩ |1, 0⟩ |3, 0⟩
Clock point (m, G) - - 2.54 6.49 33.0 1.79 0.0720 - -
ω0/2π (m, GHz) - - 0.025 0.036 0.062 0.012 0.00047 - -
ω0/2π (g, GHz) 1.3 1.8 3.2 5.0 9.9 7.2 8.0 12.6 10.5
d2(ω0/2π)/dB
2 (m, kHz/G2) - - 55.9 36.7 10.6 119 1.72 - -
d2(ω0/2π)/dB
2 (g, kHz/G2) 3.13 2.19 1.21 0.783 0.395 0.545 0.487 0.309 0.373
γP /2π (MHz) 19.4 41.8 23.2 24.0 25.2 25.93/2
D5/2 D5/2α αS α 1 1 0.9350 0.9406 0.7417 0.98751/2
αD
5/2 D5/2αD α - - 0.0063 0.0063 0.02803 0.00173/2
D5/2 5/2α αD α
D - - 0.0587 0.0531 0.2303 0.0108
5/2
ωf/2π (THz) 0.198 2.75 6.68 24.0 57.2 99.8
TABLE 4.1. Characteristics of the qubits and ion species we consider. Throughout
the table, m and g denote values for m or g qubits. I is the nuclear spin; τD5/2
is the D5/2 lifetime; |0⟩ and |1⟩ are the qubit states in the notation |F,mF ⟩; ω0
is the qubit frequency; d2ω0/dB
2 is the second-order B-field-dependence of the
qubit frequency; γP is the decay rate of the P manifold; α is the branching3/2 3/2 M
ratio of P3/2 to manifold M ; and ωf/2π is the fine-structure separation of the P
manifolds. Lifetimes taken from Sahoo et al., 2006; Taylor et al., 1997; γP values3/2
were taken/calculated from Poulsen, Andersen and Skouboe, 1975; Ansbacher, Li
and Pinnington, 1989; Gosselin, Pinnington and Ansbacher, 1988; Safronova, 2010;
Pinnington, Berends and Lumsden, 1995; Pinnington, Rieger and Kernahan, 1997;
αM values taken from Song et al., 2019; H. Zhang et al., 2016; Z. Zhang et al.,
2020; Feldker et al., 2018; ωf values taken from Ozeri, 2007; other values calculated
from atomic parameters.
photon scattering, changing the ion’s state or qubit phase and causing an error.
The scattering probability is therefore important for characterizing the fidelity
achievable by logic gates, as it sets an upper bound.
The rate of ‘Λ and V scattering’ (the processes with upward pointing laser
beam arrows in Fig. 4.2) from qubit states |i⟩ to final state |f⟩ can be calculated
from (see Appendix A for derivation)
47
∑ e2E2 2 ∣∑(jµ ∣Pi ∣ ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂j |i⟩Γf,ΛV = ξℏ2 iγPf +4 ∣ µ µ (ω −∆)
i,j,q Pf P i kPk (4.1)
⟨f | r⃗ · ϵ̂j |k⟩ ⟨k|
)∣ ( )
r⃗ · êq | 2 3i⟩ ∣∣∣ ∆1 + ,µPfµPi (ωki + ωkf +∆) ωPf
where ξi is the occupation probability of qubit state |i⟩ (on average 1 for2
each qubit state in the gates we consider), r⃗ is the position operator of the electron
relative to the atomic core (note that the matrix elements of the electric dipole
operator can be tricky to calculate; see King, 2008 for a simple introduction to
computing them while avoiding pitfalls), γPf is the decay rate from P3/2 to the
manifold containing f , the various ωnm are transition frequencies with n and m
corresponding to states or levels (see 4 for more thorough definition of ωnm), ∆
is the detuning measured from the P3/2 manifold (positive for detuning above
this manifold, and neglecting the energy spread of the manifold’s sublevels), Ej
and ϵ̂j are the electric field and polarization direction of beam j respectively, and
êq is the scattered photon polarization. The parameters µPi (corresponding to
transitions between the P3/2 manifold and the manifold containing state i) and µPf
(corresponding to transitions between the manifolds containing k and f) are the
transition dipole matrix elements of the spin-orbital coupling. Their general form
can be derived by invoking the Wigner-Eckart theorem Brink and Satchler, 1968,
giving
4To be explicit: ωkP is (Ek − EP )/ℏ, where Ek is the mean energy of the manifold containing
|k⟩ and EP is the mean energy of the P3/2 manifold (note this is zero if k corresponds to P3/2);
ωki corresponds to transitions between the manifold containing k and the qubit manifold; ωkf
corresponds to transitions between the manifold containing k and the manifold containing f ; ωPf
corresponds to transitions between the P3/2 manifold and the manifold containing f .
48
∣∣
µul = ∣ √ ∣⟨Ll| |r⃗| |Lu⟩ ∣(2Jl + 1)(2Ll + 1)LlLu1JuJlS∣, (4.2)
where ⟨Lu| |r⃗| |Ll⟩ is the reduced dipole matrix element for transitions between
upper level u and lower level l, S is the spin, L is the orbital angular momentum,
J is the total angular momentum (L + S), and the bracketed term is the Wigner
6j-symbol. The matrix element µul can also be related to the decay rate γul from
the Ju, Lu level to the Jl, Ll level by Steck, 2001
e2ω3
γ ulul = µ
2
3πϵ ℏc3 ul
, (4.3)
0
where ωul is the transition frequency, and e is the charge of the electron. In
m qubits (but not g qubits), ‘ladder scattering’ (the processes with downward
pointing laser beam arrows in Fig. 4.2) can also contribute to the error. The ladder
scattering rate is given by
e2E2µ2 ∑j ∣∣∣∣∑(Pi ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂∗j |i⟩Γf,lad = ξ γℏ2 i Pf4 µPfµPi (ωkP +∆+ 2ω )i,j,q PDk
⟨ | · ∗ | ⟩ ⟨ | · | ⟩ )∣∣∣∣2( )f r⃗ ϵ̂ 3j k k r⃗ êq i 2ωPD +∆+ − 1− .µPfµPi (ωkP ∆+ ωDf ) ωPf
(4.4)
Note that in Eqn. 4.1, the Λ scattering term (first term in the sum within the
modulus) dominates over V scattering; similarly, in Eqn. 4.4, the first term in
the modulus dominates. In both equations, these terms correspond to a two-
photon scattering process where the laser photon is first absorbed or emitted, while
the other weaker terms (what we call the counter-rotating contributions to the
49
g qubit m qubit
∆k{ 〉|k〉| ∆k{ 〉|k〉|
Λ scattering
〉| 〉 Ladderf scattering |i〉|
|i〉| 〉|f〉
V scattering
FIGURE 4.2. All two-photon scattering pathways from state |i⟩ to |f⟩ via excited
state |k⟩ in g qubits and m qubits. Raman laser beam shown in pink, scattered
light shown in red and blue. Processes in which the Raman beam points up in the
diagram are Λ or V scattering events. Processes in which the Raman beam points
down are ladder scattering events (note that ladder scattering events are allowed
only in m qubits). Figure from Moore, Campbell et al., 2023.
scattering rate) correspond to processes where the scattered photon is emitted first
(Fig. 4.2).
We can rewrite Eqns. 4.1 and 4.4 in terms of gul given in Ozeri, 2007 as
eEµul
gul = . (4.5)
2ℏ
Note that in the following calculations, for m qubits, gPi and µPi will correspond
to transitions between the D5/2 and P3/2 manifolds, i.e., u and l in Eqns. 5.2 and
4.5 will correspond to P3/2 and D5/2, respectively. For g qubits, gPi and µPi will
correspond to transitions between S1/2 and P3/2. Assuming that gPi is the same for
both Raman beams, we can now write Eqns. 4.1 and 4.4 as
50
∑ ∣∣∣∣∑(2 ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂j |i⟩Γf,ΛV = gPi ξiγPf +µ
i,j,q Pf
µPi (ωkP −∆)
k ) ( ) (4.6)⟨f | r⃗ · ϵ̂j |k⟩ ⟨k| r⃗ · | ⟩ 2 3êq i ∆
1 +
µPfµPi (ωki + ωkf +∆) ωPf
and
∑ ∣∣∣∑( ⟨f | r⃗ · ê ∗q |k⟩ ⟨k| r⃗ · ϵ̂j |i⟩Γf,lad = g2Pi ξiγPf ∣ µ µ (ω +∆+ 2ω )
i,j,q Pf P i kP PDk (4.7)
⟨f | r⃗ · ϵ̂∗j |k⟩ ⟨k| r⃗ · êq |i⟩
)∣∣∣2( )3− 2ωPD +∆+ 1 ,
µPfµPi (ω ∣kP −∆+ ωDf ) ωPf
where ωDf is the frequency of the transition between D5/2 and the manifold
containing the state f . Eqn. 4.6 gives what we will call the “full model” for g
qubits, and the sum of Eqns. 4.6 and 4.7 gives the corresponding full model for
m qubits. In m qubits, we include contributions to the gate error from some of the
closest higher energy intermediate states, but we neglect such contributions in g
qubits for reasons given in section 4.2.3 below.
We will also define what we call our “simplified model”, in which we neglect
the detuning dependence of the scattered photon frequency (i.e., we assume
(1 + ∆/ω 3Pf ) ≈ 1), neglect the contributions of the higher energy intermediate
manifolds, and assume that only the first term in the squared modulus of Eqn. 4.6
appreciably contributes to the scattering rate in both m qubits and g qubits. This
model results in a simpler version of Eqn. 4.6 D. J. Wineland, 2003,
∑ ∣∣∣∑ ∣2≈ 2 ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂j |i⟩ ∣∣Γf gPi ξiγPf ∣∣ ∣ . (4.8)µ
i,j,q Pf
µPi (ωkP −∆) ∣
k
51
These assumptions have also been made in studies of g qubits (e.g., Ozeri, 2007)
where the 10−4 gate error threshold required detunings on the order of 10THz.
However, as we will show, even at these modest detunings, our model can give
large corrections to this simplified model. Despite this, the simplified model in m
qubits provides an intuitive illustration of the scattering behavior for a large range
of detunings, so we therefore elect to present both models of m qubits throughout
this paper.
4.2.1. Single-qubit gates
We follow Ozeri, 2007 in their choice of a representative single-qubit gate:
a π-rotation around the x-axis of the equivalent Bloch sphere, a σ̂x gate. We will
assume the gates are driven by two laser beams that induce two-photon stimulated
Raman transitions. In the case of m qubits, we assume both beams are purely
π-polarized because we found that they minimize the scattering probability and
power requirements. In g qubits, we assume each beam has equal parts σ+ and σ−
polarization. For this gate, the required time is given by Ozeri, 2007
π
τ1q = | | , (4.9)2 ΩR
where the Rabi frequency ΩR is calculated according to
∑(⟨ )2 1|r⃗ · ϵ̂∗r|k⟩⟨k|r⃗ · ϵ̂b|0⟩ ⟨1|r⃗ · ϵ̂r|k⟩⟨k|r⃗ · ϵ̂∗|0⟩ΩR = gPi + b , (4.10)µ2 2ki(ωkP −∆) µk ki(ωki + ωPi +∆)
where r and b (for red and blue) label the two Raman beams, |0⟩ and |1⟩
are the two qubit states and k indexes the available intermediate states. Since
52
Raman scattering (scattering events for which |i⟩ ≠ |f⟩) is the dominant source of
errors for most species, we will, for the moment, neglect errors caused by Rayleigh
scattering (scattering events for which |i⟩ = |f⟩); Section 4.2.4 below gives a
discussion of these errors. We can now calculate the general Raman scattering
error as
πΓRam
PRam = τ1qΓRam = | | , (4.11)2 ΩR
where ΓRam is given by the sum of the scattering rates to all possible final states
|f⟩ for which |f⟩ ≠ |i⟩. For g qubits, this will be the sum of Eqn. 4.6 over all
relevant final states; for m qubits, it will be the sum of both Eqns. 4.6 and 4.7
over all relevant final states. This scattering error is plotted for g and m qubits in
Fig. 4.3.
In the simplified model of m qubits, we neglect intermediate manifolds aside
from the lowest energy P3/2 manifold, the detuning dependence of the scattered
photon frequency, ladder decay, and the counter-rotating field contribution to
the scattering rate and Rabi frequency (the second terms in the sum of Eqn. 4.6
and Eqn. 4.10). In this case, the sum over k has the same form for all ion species
considered, giving an expression for the m qubit simplified model’s Rabi frequency
for each ion species the form
− 2 g
2
Ω = PiR . (4.12)
15 ∆
and a Raman scattering rate of the form
4 g2 γ ∣∣ γ ∣∣
ΓRam = ρ
Pi = 2ρ ∣ΩR ∣ , (4.13)
15 ∆2 ∆
53
where ρ is the Raman-only fraction of the total scattering rate (the value of ρ can
be inferred from the nearly constant ratio in the low-detuning regime of Fig. 4.5)
and γ is the decay rate of the P3/2 manifold. For the m qubit’s simplified model,
this results in a gate error of
πγ
PRam = ρ | | . (4.14)∆
This simplified model of m qubit scattering gives an especially simple form for
PRam, since only one manifold (P3/2) appreciably contributes to the scattering. The
m qubit’s simplified model behavior is also shown in Fig. 4.3 as the dashed lines.
From Fig. 4.3, we can see that the m qubits and g qubits each, as groups,
have markedly similar behavior because their P3/2 decay rates are all within
∼5MHz (Table 4.1) of each other. As expected, the full model results deviate from
the initial linear regime as the detuning becomes large. In m qubits, the full model
yields a lower scattering probability than the simplified model for red-detuning,
and a higher scattering probability for blue-detuning. The deviations from the m
simplified model can be observed in the lower plot of Fig. 4.3
4.2.2. Two-qubit gates
Consider now a two-qubit Mølmer-Sørensen gate, driven by three Raman
beams Mølmer and Sørensen, 1999; Tan, 2016. We will suppose that the three
beams are comprised of one pair of co-propagating beams of power P and one
beam counter-propagating with this pair with intensity 2P, since this distribution
of power minimizes the scattering error for this beam geometry. For this gate, the
duration is
54
FIGURE 4.3. Raman scattering probability for m qubits and g qubits during
single-qubit σ̂x gate for large detuning. The m qubit ions are labeled by diamonds
and the g qubit ions are labeled by squares. Detuning is measured relative to
P3/2 in m qubits; for g qubits, red detuning is measured relative to P1/2, and blue
detuning is measured relative to P3/2. The lower plot shows zoomed-in regions of
the upper plot with the simplified model behavior shown for m qubits as dashed
lines. Figure from Moore, Campbell et al., 2023.
55
√
√ π Kτ2q = . (4.15)
2 2|ΩR| η
This is the gate time for the two-qubit gate in Ozeri, 2007 but reduced by a factor
√
of 1/ 2 because of the unequal distribution of intensities in this setup. In this
equation, K is the number of loops the |01⟩ and |10⟩ states trace out in phase
space (we set K = 1 for our calculations) and η is the Lamb-Dicke parameter,
which for counter-propagating Raman beams is given Ozeri, 2007 by
1 √ ωL
η = ∆kz0b
(i)
p = 2kLz0√ = 2 z0, (4.16)
2 c
where ∆k is the magnitude of the difference between the two Raman beams’
wavevectors (this difference wavevector being aligned to the mode of interest), kL
and ωL are respectively the wavenumber and frequency of the Raman beams, and
(i) √
bp is the mode participation factor (equal to 1/ 2 here). Note that the simplified
model of Ozeri, 2007 considers perpendicular beams which would increase the
√
Lamb-Dicke parameter by a factor of 2. The root-mean-square spatial spread
of the ground state wavefunction z0 is given by
√
z0 = ℏ/2Mωtrap, (4.17)
where M is the mass of each ion and ωtrap is the frequency of the driven motional
mode.
The scattering probability for this gate is the single-qubit gate scattering
probability scaled by a factor of 4, as well as the extra gate time factor Ozeri,
2007. The factor of 4 comes from two considerations: first, that the two-qubit gate
uses three beams with a total power twice as great as in the total beam power
56
used in the single-qubit gate, and second that there are two ions. Both of these
differences scales the gate error by a factor of 2 to generate an overall increase of 4.
This gives a general form for the two-qubit gate Raman scattering error,
√
π√ΓRam 4 KPR2q = . (4.18)
2 2|ΩR| η
For the simplified model in m qubits, this results in
√
πγ 4 K
PR2q,simp = ρ | | √ . (4.19)∆ 2η
Because η is proportional to the laser frequency, it will in general depend on
detuning. For the simplified model, we neglect this detuning dependence, so the
error probability again exhibits linear behavior. We do, however, include this
dependence in the full model, as the size of the effect is too large to reasonably
ignore. In the full model for both g and m qubits, we can see that the Lamb-Dicke
parameter leads to a further detuning dependence of the form
ωPi
PR2q ∝ | | , (4.20)∆(ωPi +∆)
where ωPi denotes the frequency of the transition between the P3/2 manifold and
the manifold containing the qubit states. The Raman scattering error of a 1-loop
two-qubit Mølmer-Sørensen gate is plotted in Fig. 4.4; the full models of g and m
qubits are shown by the solid curves, and the simplified m qubit model is shown
by the dashed curves.
The qualitative behavior of the sharply-increasing red-detuned m qubit
scattering probability in Fig. 4.4 can be understood by observing that as the
laser frequency approaches zero, the gate time goes to infinity (because the
57
FIGURE 4.4. Raman scattering probability from a 1-loop two-qubit Mølmer-
Sørensen gate on m and g qubits. The m qubit ions are labeled by diamonds and
the g qubit ions are labeled by squares. The simple model for m qubits is shown
in the plot as dashed lines. Detuning is measured relative to P3/2 in m qubits; for
g qubits, red detuning is measured relative to P1/2, and blue detuning is measured
relative to P3/2. Figure from Moore, Campbell et al., 2023.
Lamb-Dicke parameter is approaching zero); however, the frequency of photons
scattered to, e.g., S1/2 approaches a non-zero value (the S1/2 ↔ D5/2 transition
frequency). The gate error, being the product of the scattering rate and gate time,
therefore approaches infinity as the laser frequency goes to zero. This does not
occur in g qubits, because the scattered photon frequency also goes to zero as the
laser frequency approaches zero. Additionally, the scattering rate in g qubits is
generally lower than in m qubits across the detuning range considered. However,
scattering errors in m qubits are largely due to scattering to S1/2, i.e., outside the
qubit manifold. Such errors are easier to detect and correct; indeed, one of our
collaborators recently applied some of the results presented here in an investigation
58
of erasure conversion in quantum error correction with metastable states Kang,
Campbell and Brown, 2022.
The form of the g scattering probability plotted in Figs. 4.3 and 4.4 differs
in several ways from that given in Ozeri, 2007. First, we compute D-manifold
scattering rates directly, whereas Ozeri, 2007 calculates them by multiplying the
total scattering rate by the branching ratio to the D-manifolds. This overestimates
the D-scattering rate, as it neglects interference between the P1/2 and P3/2
manifolds. Second, we include the V scattering process and the counter-rotating
electric field component contribution to the Rabi frequency (i.e., the contribution
to Rabi flopping that is neglected in the rotating-wave approximation). Third,
we include the detuning dependence of the Lamb-Dicke parameter and scattered
photon frequency (while Ozeri, 2007 used a single value of η and ωsc for the entire
detuning range). This latter effect makes the scattering rate proportional to ω3sc,
the cube of the scattered photon frequency (Appendix A). Note that if a detuning
is chosen such that ∆ < −ωPf the contribution to the scattering probability due
to scattering to level f becomes precisely zero. This is because the proportionality
to ω3sc in such a condition renders the f scattering rate negative, which is non-
physical. Another way to see that such scattering is not permitted is to note that
it violates energy conservation.
Finally, it is notable that by neglecting the above effects, the simplified
model of Ozeri, 2007 implied a lower bound on Raman scattering-induced gate
error in g qubits (1.06 × 10−4, 0.50 × 10−4, 1.46 × 10−4, and 0.007 × 10−4
for Ca+, Sr+, Ba+, and Yb+ respectively). This bound appears because the D-
manifold scattering error in that model approaches a non-zero value at large red
detunings. However, in our model, this lower bound does not exist. The Raman
59
scattering error approaches zero at large red detunings. This is as it should be, due
to considerations of energy conservation.
4.2.3. Higher levels
The above models have incorporated scattering probability contributions
from higher levels in m qubits, but not g qubits (a disparity we will justify in
this section). Higher levels contribute to the scattering probability in two ways:
through increasing the overall scattering rate, and increasing or decreasing the
Rabi frequency. Because the Rabi frequency and scattering rate have different
scaling in detuning, the inclusion of higher levels can actually result in a net
decrease in scattering probability for some parameter regimes.
There are two factors which clearly attenuate the magnitude of contributions
from the higher levels to the Raman scattering error: the larger frequency
denominators of the scattering rates and the smaller radial overlap with higher
levels’ wavefunctions. To numerically estimate the contribution of the higher levels,
we calculated it for the P3/2, F5/2, and F7/2 manifolds available in the University of
Delaware database Barakhshan et al., n.d. in Ca+, Sr+, and Ba+.
The contributions from higher levels differ substantially in m qubits and
g qubits. Destructive interference in the Rabi frequency due to higher levels
causes m qubit behavior to change noticeably at large red- and blue-detunings.
Interference effects are largely absent in g qubits, however, since the contributions
of higher P1/2 and P3/2 essentially cancel. The reason for this cancellation is
that Raman scattering flips the electron spin via the spin-orbit coupling L · S
in the excited state. However, the dipole matrix element for this process is equal
in magnitude but opposite in sign for the two fine-structure manifolds P1/2 and
60
P3/2 Cline et al., 1994; Ozeri, 2007. Since we are considering tunings far from the
resonant excitation of the higher levels, these matrix elements largely cancel out
their contributions; deviations from the g qubit model neglecting higher levels do
not exceed more than a few percent until about 1PHz detuning, where the laser
frequency reaches a resonance with a higher P manifold.
Including the higher levels in the m qubit model lowers scattering probability
at the 10−4 error level by a small amount for red detunings. The corrections for
blue detuning, however, can be much larger at the 10−4 error level; inclusion of the
higher levels increases the gate error for blue detuning. These corrections are large
enough that when higher levels are included, Sr+ and Ba+ can no longer get below
the 10−4 error level for blue detuning.
4.2.4. Rayleigh scattering errors
So far, we have neglected Rayleigh scattering-induced errors. Previous
discussions of scattering in the literature (e.g. Ozeri, 2007) have characterized
the infidelity contribution of Rayleigh scattering-induced dephasing as being
proportional to the difference in elastic scattering rates; however, as is shown
in Uys et al., 2010, the dephasing rate due to Rayleigh scattering must be
computed by including interference between the Rayleigh scattering amplitudes.
This can lead to Rayleigh scattering becoming the dominant source of error for
certain parameter regimes. However, since we are considering clock qubits in
both m qubits and g qubits, the Rayleigh-scattering-induced decoherence will be
negligible. Tan, 2016
Rayleigh scattering can however cause errors during two-qubit gates in
two other ways: recoil from the momentum kick during Rayleigh scattering and
61
nonlinearities in the two-qubit gate. Both effects were studied in Ozeri, 2007 and
recoil was found to yield a gate error of the form
⟨|β|2⟩
ϵRay = PE2q , (4.21)
2K
where PE2q is the probability of a elastic Rayleigh scattering event during the
two-qubit gate, β the recoil displacement in phase space due to the scattering
event, and K is the number of loops traced out in phase space (again, K = 1
for our gate). The expected value of |β|2 depends on the polarization choice, but
we can determine an upper bound on the reduction in fidelity by taking the recoil
displacement to equal its maximum value. For our choice of laser beam geometry,
the squared magnitude of the recoil displacement is given by Ozeri, 2007
( )
2 2
| |2 η √1β = + cosθ , (4.22)
2 2
where η is the Lamb-Dicke parameter and θ is the angle of the recoil direction
from the axis along the motional mode to which we are coupling. The maximum
value is at θ = 0, so
( )
⟨|β|2⟩ ≤ η2 3 1+ √ (4.23)
4 2
implying ( )
ϵRay ≤
3 1
PE2qη
2 + √ . (4.24)
8 2 2
The ratio of this upper bound on Rayleigh recoil error to the Raman scattering
error during a two-qubit gate is plotted in Fig. 4.5 for m qubits and g qubits. The
two lightest species, Be+ and Mg+, have the largest Rayleigh scattering errors
62
precisely because they are so light; their low masses increase their sensitivity to
photon recoil. Additionally, they have higher frequency Raman transitions which
makes their Lamb-Dicke parameters larger and therefore further increases their
sensitivity to Rayleigh scattering events. In the other species, for most of the
detuning range, the Rayleigh recoil errors are small. For m qubits, the magnitude
of this contribution to the error is 5 or 6 orders of magnitude lower at the 10−4
error level compared to Raman scattering contribution in the various species of
ions we consider. The g Rayleigh recoil error is roughly 1 to 3 orders of magnitude
smaller than the Raman scattering error for most of the detuning range shown
(again, except for Be+ and Mg+). The m qubits fare better than g qubits with
respect to the Rayleigh recoil error in large part because m qubits have lower
branching ratios to the qubit manifold, which leads to a lower probability of elastic
scattering events. Furthermore, due to their lower frequency transitions, they have
lower Lamb-Dicke parameters, which further suppresses the Rayleigh recoil error.
The infidelity contribution of gate nonlinearities due to recoil momentum
displacement is even smaller. As noted in Ozeri, 2007, the error due to such
nonlinearities is proportional to η4. This is negligible for most g qubits and even
less important in m qubits, due to their smaller Lamb-Dicke parameters. In Be+
and Mg+, this error can still be larger than the Raman scattering error, but it is
small compared to the Rayleigh recoil error.
4.3. Power Requirements
Now we will calculate the power required to achieve a given gate error rate
(recall that we are considering a two beam single-qubit gate and a three-beam two-
qubit gate). We first rewrite Eqn. 4.3,
63
FIGURE 4.5. Ratio of the upper bound on Rayleigh recoil error to Raman
scattering error during a two-qubit gate for m qubits and g qubits. The m qubit
ions are labeled by diamonds and the g qubit ions are labeled by squares. Note
that, for the detunings plotted, the Rayleigh recoil error upper bound exceeds
the Raman scattering error only for g qubits in Be+, Mg+, and Ca+. The steep
declines in the ratio in m qubits at large blue detunings is due to destructive
interference with the F7/2 manifolds in the Rayleigh scattering rate. Figure
from Moore, Campbell et al., 2023.
4ℏω3γ 3/2
ρq = , (4.25)
g2Pi 3πϵ0c
3E2
where ρq is the inelastic fraction of the scattering from P3/2 to the qubit manifold,
ωPi is the frequency of the transition from the qubit manifold to P3/2, and E
is the peak electric field strength. Note that this equation differs from Eqn. 15
of Ozeri, 2007 only by the factor ρq. This is because gPi in Ozeri, 2007 is defined
for S1/2 ↔ P3/2 transitions, and the branching ratio from P3/2 to the lower D levels
was considered small enough that the authors treated the decay rate to S1/2 as the
total decay rate. In m qubits, gPi is defined for transitions from D5/2 ↔ P3/2,
64
but ρq is too small to treat this decay rate as the total. The decay rate to an
individual level must then be the total decay rate weighted by the branching ratio
to that level.
By rewriting g2Pi in Eqn. 4.25 in terms of ΩR and its detuning dependence
r(∆), defined as
∣∣∣∑( ∣⟨1|r⃗ · ϵ̂∗| 2 r|k⟩⟨k| )r⃗ · ϵ̂b|0⟩ ⟨1|r⃗ · ϵ̂r|k⟩⟨k|r⃗ · ϵ̂∗|0⟩ ∣∣r(∆) = Ω bR(∆)|/gPi = ∣∣ + ,µ2 (ω −∆) µ2 ∣ki kP ki(ωki + ωPi +∆) ∣k
(4.26)
we can write the power requirement as a function of detuning for single and
two-qubit gates as (see Appendix C for derivation)
ℏω3 2
P (∆) = Pi
w0 π
1q (4.27)
3c2ρqγ τ1qr(∆)
and √
2ℏω3 w2 π K
P (∆) = √ Pi 02q , (4.28)
3 2c2ρqγ τ2qη(∆)r(∆)
where w0 is the beam waist and η(∆) is the Lamb-Dicke parameter (including
detuning dependence). Because the gate error is also some function of ∆, ϵ(∆)
(Sections 4.2.1 and 4.2.2), we can visualize the power and error requirements by
plotting the set of points (ϵ(∆),P(∆)), shown in Figs. 4.6 and 4.7.
For the simplified model, r(∆) can be directly related to the gate error ϵ,
allowing us to write the power as a function of error as (Appendix C)
5πℏω3 w2 π
P1q(ϵ1q) = ρ
Pi 0 (4.29)
2c2ϵ1q τ1q
65
and
10πℏω3 2
P (ϵ ) = ρ Pi
w0 π K
2q 2q , (4.30)
c2ϵ 22q τ2q η
for single and two-qubit gates, respectively.
As can be seen in Figs. 4.6 and 4.7, the power requirements for g qubits are,
for a large range of detunings, about one order of magnitude smaller than the
required power for m qubits at the same error probability; this is mostly because
of the small branching ratio to D5/2 qubit manifold. However, we note that high
power lasers are readily available at the large red detunings anticipated for m
qubits, so these higher power requirements may not be a limiting factor.
From Fig. 4.7, we can also see the the discrepancies between the simplified
model and the full model reappearing. The full model power curves bend
backwards past the minima of ϵ(∆), entering a regime of diminishing returns
where the gates both take longer and have higher errors. For a fixed gate time,
this farther tuning from resonance means more power is required to drive the gate.
66
FIGURE 4.6. Total power required to achieve a given error rate during a single-
qubit π-rotation gate in the full model (assumes 20 µm beam waist and 1 µs gate
time). The m qubit ions are labeled by diamonds and the g qubit ions are labeled
by squares. Errors for red-detuned g qubits here are calculated from detunings
below P1/2 only. Figure from Moore, Campbell et al., 2023.
67
FIGURE 4.7. Total power required to achieve a given error rate during a two-
qubit Mølmer-Sørensen gate (assumes 20 µm beam waist, 10 µs gate time, and a
5MHz axial trap frequency). The m qubit ions are labeled by diamonds and the
g qubit ions are labeled by squares. Solid lines correspond to full model, dashed
lines correspond to the simplified model. Errors for red-detuned g qubits here are
calculated from detunings below P1/2 only. Figure from Moore, Campbell et al.,
2023.
68
CHAPTER V
ZEEMAN QUBITS IN 40CA+
The analysis of Chapter IV centers around hyperfine qubits in trapped
alkaline earth ions. However, while our lab ultimately plans to trap 43Ca+, we
have so far trapped only 40Ca+, which, having no nuclear spin, has no hyperfine
structure. We chose to trap 40Ca+ first because it has a simpler atomic structure
(downsides will be discussed in the section below). This chapter will give the
details of the Zeeman qubits on which we have performed our experiments and
extends the theory of Chapter IV to them.
5.1. Zeeman Qubits in the D5/2 Manifold
Since 40Ca+ has no nuclear spin, its energy manifolds only possess fine
structure, i.e., they only have Zeeman sublevels. For our experiments, we choose to
encode our qubit as follows: |1⟩ = |D5/2,m = +5/2⟩ and |0⟩ = |D5/2,m = +3/2⟩.
One disadvantage of using this Zeeman qubit is that it is not possible to make a
“clock” qubit with Zeeman levels. The transition frequency of a clock qubit is at
an extremum, such that the derivative of the frequency with respect to magnetic
field is zero. This implies that the frequency is relatively insensitive to magnetic
field noise and Rayleigh scattering. This is possible with hyperfine qubits, but not
in Zeeman qubits, since the qubit frequency is linear in the B-field. However, these
qubits are easy for us to prepare and control, and are sufficient for our experiments
in this work, and so we elect to use them.
It is easy to prepare the qubit in the |0⟩ state via the optical pumping
scheme shown in 5.2. We begin with 397 nm (σ+-polarized) and 866 nm laser beam
69
pulses. This prepares the mJ = +1/2 state in the S1/2 manifold. From there,
we apply 393 nm (σ+-polarized) and 866 nm laser beam pulses to pump out the
population of mJ = +1/2 sublevel of S1/2 via the mJ = +3/2 sublevel of P3/2.
Finally, we apply an 854 nm π-polarized laser beam to depopulate the sublevels
of D5/2, except for the mJ = +5/2, the level we are trying to prepare. We then
repeat this pulse sequence until the desired fidelity is achieved (usually around 7
times). We are able to obtain about 99% preparation fidelity this way.
With the qubit initialized to |0⟩, we are nearly ready to perform our gate.
One problem remains, though: all ∆m = ±1 transitions are degenerate since their
separation frequencies are set by the Zeeman splitting. We therefore need a way to
separate our qubit transition from the other transitions in the D5/2 manifold. We
achieve this by applying a σ+-polarized 854 nm beam. This induces a light shift
on all sublevels aside from the qubit states (this technique was first used in Curtis,
2010; see also Sherman et al., 2013). Importantly, absent polarization impurities, it
also causes no scattering from the qubit states.
After the gate, we perform readout by first shelving the m = +3/2
population in the m = +1/2 state. We do this so that when we deshelve with the
854 nm π-polarized laser beam we avoid back-scattering to the m = +5/2 sublevel.
We can then pump this population out of the D5/2 manifold, and then turn on the
397 nm and 866 nm lasers to check for population in the S1/2 and D3/2 manifolds.
Finally, after this, we can completely depump the D5/2 manifold with the 854 nm
σ−/σ+-polarized laser beams. This is important for resetting the ion into the S1/2
manifold after readout is complete. The complete suite of operations on this qubit
is portrayed in Figure 5.2.
70
FIGURE 5.1. 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.
71
FIGURE 5.2. Laser and rf operations on the 40Ca+ Zeeman qubit in the
D5/2 manifold. The preparation pulse sequence is performed N times, until a
satisfactory preparation fidelity is achieved. 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 depump into the S1/2 manifold
with the 854 nm σ−-polarized laser beam.
72
5.1.1. Light Shifts
As discussed in Section 5.1, we need to energetically separate our qubit
sublevels from the other Zeeman sublevels. We manage this by using an 854 nm
σ+-polarized laser beam. This couples the m = +1/2 sublevel of the D5/2 manifold
to the m = −1/2 sublevel of P3/2 manifold. This generates a light shift on the
m = +1/2 sublevel, lifting the degeneracy as desired. The light shift can be
calculated from
∑
2 | ⟨k| d⃗ · ϵ̂ |D 25/2,m = +1/2⟩ |∆LS = g , (5.1)
∆
k
where k indexes all sublevels of the P3/2 manifold.
It is worth noting that the 976 nm Raman beams are also capable of
generating light shifts. However, the magnitude of the differential light shift from
the 976 nm beams scales with the Rabi frequency, so the different transitions would
never be well-resolved if this were the sole light shift. Therefore, we generally need
the 854 nm light shift beam to make our qubit usable.
While the light shift beam is necessary to isolate the qubit states, it does
introduce one other problem. Any σ− polarization impurity in the 854 nm beam
will cause scattering from the qubit states. However, the σ− polarization impurity
is small, typically around 0.1%. The scattering error from σ− impurities in a
10mW 854 nm beam during a 10µs single-qubit gate is shown in Figure 5.3. From
the figure, it is clear that keeping impurities low is paramount, since we also want
to maintain a large light shift. The relationship between light shift magnitude and
beam power can be seen in Figure 5.4
73
Impurity = 1%
Impurity = 0.1%
1 Impurity = 0.01%10
10 2
10 3
10 4
4 × 102 6 × 102 103 2 × 103 3 × 103
Light shift (kHz)
FIGURE 5.3. Trade-off between scattering error ϵSD from σ
− impurities in a
10mW 854 nm beam during a 10µs single-qubit gate and light shift magnitude for
various σ− fractional impurities.
5.2. Scattering Theory for Zeeman Qubits
There is no essential difference in the derivation of the scattering rate for
Zeeman qubits as compared to hyperfine qubits. The scattering rate will have the
same form as Eqn. 4.6 and Eqn. 4.7, but the matrix elements must be calculated
differently. The transition dipole matrix element between upper and lower states
|u⟩ and |l⟩ is given by
74
SD
20.0
100 kHz light shift
17.5 300 kHz light shift500 kHz light shift
700 kHz light shift
15.0
12.5
10.0
7.5
5.0
2.5
0.0
10 20 30 40 50 60 70 80
Detuning (GHz)
FIGURE 5.4. Power and detuning requirements for various light shifts from the
854 nm σ+-polarized beam.
√
⟨u| dmu−m |l⟩ =(−1)2Ju+Ll−ml (2Jl + 1)(2Ll + 1)l
(5.2)
× Ju1Jlmuml −mu−mlLlLu1JuJlS.
Additionally, when working with Zeeman qubits, one convenient feature of
hyperfine qubits is unavailable to us: the possibility of producing clock qubits.
Because the qubit frequency is determined solely by the Zeeman splitting in
Zeeman qubits, it varies linearly with the B-field, which ensures that there will
be no local minimum at which to generate a clock qubit. Besides increasing
75
Beam power (mW)
dephasing, this also means we cannot use the same argument to neglect Rayleigh-
scattering-induced decoherence as was given in Section 4.2.4. While this will
worsen gate performance, it serves as no obstacle to our goal of measuring the
Raman scattering error and comparing it to the predictions of the theory.
5.2.1. Rayleigh-Scattering-Induced Decoherence
The Rayleigh-scattering-induced decoherence can be calculated as in Uys
et al., 2010,
∑ ∣∣∑( | ⟨k| r⃗ · ϵ̂j | ⟩ | | ⟨ | )0 2 k r⃗ · ϵ̂ 2j |0⟩ |
Γel = g
2
Pi γ ∣Pf ∣ +
∑µ(
2 2
Pi (ωkP −∆) µPi (ωki + ωkf +∆)j k
| ⟨k| )∣ ( )r⃗ · ϵ̂ |1⟩ |2 | ⟨k| r⃗ · ϵ̂ |1⟩ |2 2 3− j j ∣ ∆+ ∣ 1 + .
µ2Pi (ωkP −∆) µ2Pi (ω ∣ki + ωkf +∆) ωPfk
(5.3)
As noted in Section 4.2.4, this decoherence is negligible for clock qubits. Since we
cannot generate a clock qubit in 40Ca+, we cannot neglect the Rayleigh-scattering-
induced decoherence. We can quantify the importance of this decoherence by
plotting its ratio to the Raman scattering decoherence, as in 5.5.
As the figure shows, red-detuning lowers the fractional Rayleigh scattering
error and for our laser system (at -44 THz detuning) will have about 6% more
scattering error on top of ϵSD due to Rayleigh scattering. The error is therefore
not substantially larger.
76
2 × 10 1
10 1
6 × 10 2
4 × 10 2
3 × 10 2
102 101 100 100 101 102
Detuning (THz)
FIGURE 5.5. Ratio of Rayleigh scattering error, ϵRay, to Raman scattering errors
associated with scattering to S1/2 and D3/2, ϵSD.
77
Ray/ SD
CHAPTER VI
SCATTERING RATE MEASUREMENTS
With the scattering rate calculations in hand, we now turn to the
experimental verification of them.
6.1. 976 nm laser setup
As discussed above, although m qubits require larger detunings to achieve
the same gate error as g qubits, the required Raman laser wavelengths for m
qubits in 40Ca+ is in a wavelength range where there are powerful (watts to
kilowatts) lasers available. This enables us to complete the gates in the low error
regime in a reasonable amount of time.
We chose to use a free-space 700 mW 976 nm hybrid external cavity laser
(a HECL; from Innovative Photonics Solutions, model I0976SB0700B) with a
CTL300E-1-1200 Koheron driver. This laser is compact, powerful, and inexpensive,
being about an order of magnitude more powerful than our Toptica lasers and
having two orders of magnitude smaller volume. At 976 nm, this laser is -44THz
detuned from the D5/2 ↔ S1/2 transition. We chose to use this wavelength because
any wavelength longer than 963 nm (see Table 8.1) allows the two-qubit gate to
reach a spontaneous scattering error below 10−4. We choose 976 nm in particular
because it is used to pump erbium-doped fiber amplifiers in telecommunications
and therefore is widely available. Per Figure 4.4, at the -44THz detuning this laser
achieves, we expect a minimum achievable Raman-scattering-induced gate error
of 9 × 10−5. The laser is protected by a optical isolator (Newport, part number
ISO-04-980-MP). The laser can be seen in Figure 6.1.
78
FIGURE 6.1. 976 nm laser.
FIGURE 6.2. 976 nm optical setup.
The laser geometry we use for our scattering experiments is shown in
Figure 6.2. The laser beam is split and then sent through two AOMs. These
are the same AOMs as discussed in Section 3.6. There, it goes into the beam
delivery system and is sent to the ions. Note that, in our scattering measurement
experiments, we adjust the waveplate to divert all power into one beam at a time
and use the undeflected beam out of the AOM in order to maximize available
power at the ion. We keep the AOM on so that we can still use the SU servo (see
Section 3.7) to control the beam power. Because we are using the zeroth order
beam, we have to use a shutter to block the laser beam when we want no 976 nm
light on the ion, and this is much slower than turning off the beam with the AOM,
since the shutter takes a few milliseconds to block the beam.
79
6.2. Scattering rate experiments
The analyses of Chapters IV and V allow us to quantify the scattering rates
in metastable qubits. In this section, I detail our measurements of scattering rates
to the S1/2 and D3/2 manifolds for m qubits encoded in the mJ = +5/2 and
+3/2 Zeeman sublevels of the D5/2 manifold in
40Ca+. The big picture of the
experiment is that we prepare the qubit in one of its two states, then apply one
of the 976 nm Raman beams, either the π- or σ−-polarized beam. We consider σ−
and π-polarized laser beams because they are used to drive Raman gates in our
qubit, and so enables us to infer the Raman scattering error we can expect during
such a gate. Then, in each relevant combination of initial state and laser beam
polarization, we measure the scattering rate (note that we do not consider the
mJ = +5/2 plus π-polarized beam scenario, since, absent polarization impurities,
there will be no scattering in this case).
A more detailed explanation of the experiment is as follows: first, we
calibrate the separation frequency between the mJ = +5/2 and mJ = +3/2
Zeeman sublevels of D5/2, with and without the 976 nm beam applied. This
enables us to measure the light shift of the qubit separation frequency, from which
we infer the laser intensity at the ion. Next, after preparing the ion in either the
mJ = +5/2 or mJ = +3/2 Zeeman sublevel of D5/2, we perform a natural
lifetime measurement (i.e., we measure the lifetime with the 976 nm laser beam
turned off). We do this by preparing the ion in one of the qubit states and wait
for some duration. We then read out the state by first checking for fluorescence
with the 397 nm and 866 nm cooling lasers. If the ion is bright, we infer that we
have scattered or decayed from D5/2. Conditional on the ion being dark in the
previous step, we apply an rf π pulse on the mJ = +3/2 ↔ mJ = +1/2 transition,
80
and subsequently apply an 854 nm π-polarized depump beam, and finally check
for fluorescence using our 397 nm and 866 nm cooling lasers applied. The rf pulse
helps avoid back-scattering to the initial state and the fluorescence tells us if there
was any population in the sublevels of D5/2 (aside from mJ = +5/2). Afterwards,
we also apply an 854 nm beam with both σ+ and σ− polarization to depump the
mJ = +5/2 population and again look for fluorescence with our 397 nm and
866 nm cooling lasers; this checks whether or not the ion has heated excessively.
If the ion has indeed heated excessively, it will appear dark and we reject the
data from this experiment. If instead we detect a bright ion during the 393 nm +
866 nm step, we conclude that the ion was in the mJ = +5/2 state. We count up
how often this happens and then repeat the experiment for various delays between
state preparation and readout. We then fit an exponential curve to this data,
giving us a measure of the natural lifetime τNat of the D5/2 manifold. We then
repeat this experiment, but this time with the 976 nm Raman beam on throughout
the experiment. The data generated by this experiment is another exponentially
decaying lifetime curve, yielding lifetime τRam. We can infer the Raman scattering
rate ΓRam from these two measurements via
1 1
ΓRam = − . (6.1)
τRam τNat
6.2.1. Systematic Errors in Natural Lifetime Measurements
The method describe above of measuring the scattering rate has a knock-
on benefit of checking for certain systematic errors from the measurement, which
allows us to make unbiased measurements of the scattering rate. By systematic
errors, I refer to any effects which lengthen or shorten the lifetime regardless of
81
whether the 976 nm laser is on or off. Because of how we calculate the scattering
rate, such errors will cancel out of the calculation.
An example of such an effect is cooling laser leakage. Early on in our
experiment, we noticed that our natural lifetimes were higher than expected from
the literature. After investigation, we discovered that the 393 nm beam was present
in the trap when it should have been off. This caused scattering from S1/2 to D5/2
and lengthened the measured lifetime. However, because the leakage was present
when the 976 nm laser beam was on and when it was off, this systematic error
canceled out when we calculated the Raman scattering error. After removing the
393 nm leakage, we did not notice any further discrepancies between our measured
natural lifetime and the literature values, as we show below. However, there are
still errors specific to each experimental condition (initial state choice and laser
beam polarization) to consider. In the next section, I will give a detailed discussion
of these three experimental conditions.
6.2.2. Prepare mJ = +5/2 State, Apply 976 nm σ
− Beam
In the first experimental condition, we prepare the mJ = +5/2 state and
apply a σ−-polarized 976 nm laserer beam. The raw data, showing population over
time at various powers, can be seen in Figure 6.4.
To interpret our experimental results correctly, understanding back-scattering
to the D5/2 manifold is paramount. If we scatter back to the mJ = +5/2 state,
this does not hinder our goal of measuring the scattering rate to the S1/2 and D3/2
manifolds (although these Rayleigh scattering events would hurt the qubit during
a gate). The scattering rate remains the same as we expect. However, if the ion
scatters back to the mJ = +3/2 or mJ = +1/2 state, the scattering rate will
82
differ due to the different Clebsch-Gordan couplings of the Zeeman sublevels. The
effect will be small though, since the scattering rate is low enough that we expect,
most probably, zero or one scattering events in each experiment (where the ion
is exposed to the laser for at most one second). The back-scattering contributes
to scattering rate measurement error directly if there are two scattering events,
one to the D5/2 manifold and then one outside of D5/2. The back-scattering can
also contribute if it scatters back to mJ = +3/2 or +1/2 and then fails to scatter
anywhere else. This is because scattering events where the ion first scatters back
to mJ = +5/2 and then to S1/2 or D3/2 do contribute to the scattering rate, but
the rate of these events is different for mJ = +3/2 or +1/2. This effect is, a priori,
likely to be small though, since the total scattering rate is already low and the ion
only scatters to the D5/2 manifold 5% of the time, and the scattering rates from
each of the other states are not significantly different from the scattering rate from
the initial state.
During this experiment, scattering back to the D5/2 manifold can change our
measured scattering. To estimate the size of this effect, suppose the scattering rate
from the initial state to S1/2 and D3/2, ΓSD, is 1Hz. By using the probabilities to
scatter to each state in Figure 6.3 and assuming that at most one back-scattering
event occurs in the experiment (a reasonable assumption given the low branching
ratios), we can calculate the expected scattering rate after accounting for back-
scattering, and find it to be 0.994Hz, a 0.6% difference. Because the scattering
rate is linear in the beam power, this effect will simply decrease the slope of
the linear relationship by 1%. Below, we will verify experimentally that this
effect is negligible by showing that a model which considers only scattering from
83
mJ = +5/2 fits the data very well, and the slope of the fit is 0.7% smaller than the
theory line, in line with our estimate.
P3/2
m = +3/2
m = +1/2
m = -1/2
m = -3/2 m = +5/2 σ-
976 nm σ-
3.91% m = +5/2
1.57% m = +3/2
0.39% m = +1/2
m = -1/2
D5/2
0.63% D3/2
93.5%
S1/2
P3/2
m = +3/2
m = +1/2
m = -1/2
m = -3/2 m = +3/2 σ-
976 nm σ-
m = +5/2
m = +3/2
2.35%
m = +1/2
2.35%
m = -1/2
1.17%
D5/2
0.63%
D3/2
93.5%
S1/2
P3/2
m = +3/2
m = +1/2
m = -1/2 m = +3/2 π
m = -3/2
976 nm π 3.91% m = +5/2
1.57% m = +3/2
0.39% m = +1/2
m = -1/2
D5/2
0.63% D3/2
93.5%
S1/2
FIGURE 6.3. Scattering diagram for the various initial states and beam
polarizations. Possible first-order scattering events indicated by wavy green arrows,
with the dark green arrows indicating the scattering events we actually measure
(i.e., scattering to S1/2 and D3/2). Light red arrows indicate coupling of the other
sublevels (besides the initial state) of D5/2 to P3/2.
84
6.2.3. Prepare mJ = +3/2 State, Apply 976 nm σ
− Beam
In the second experimental condition, we prepare the mJ = +3/2 state and
apply a σ−-polarized 976 nm laserer beam. The raw data, showing population over
time at various powers, can be seen in Figure 6.6.
Back-scattering is also possible for this condition, of course. However, for the
same reasons outlined above, its effect is negligible. We can undertake the same
calculation as in the previous section to confirm this. In this case, we obtain a
scattering rate of 0.986Hz, a 1.4% difference. This is larger that in the mJ = +5/2
case, and we will see in the results below that the theory line is indeed 1.4% higher
than the data fit line (though still within the error bars).
6.2.4. Prepare mJ = +3/2 State, Apply 976 nm π Beam
In the third and final experimental condition, we prepare the mJ = +3/2
state and apply a π-polarized 976 nm laserer beam. The raw data, showing
population over time at various powers, can be seen in Figure 6.8.
Back-scattering plays a role in this condition as well, but in this case it exerts
a strong effect. The reason for this is that the ion can scatter into the mJ = +5/2
state, but once the ion is in this state, it is not susceptible to Raman scattering
from the π-polarized laser beam (there is no mJ = +5/2 level to which it can
couple in the P3/2 manifold). As you can see in Figure 6.3, the ion will scatter
into the mJ = +5/2 state in 3.91% of all scattering events. Making the same
assumptions as above, the scattering rate will be 3.4% lower after accounting for
back-scattering.
Fortunately, this effect is easy to correct for on the experiment side in this
case. To do so, we can just count how often the ion is dark after depumping
85
with the 854 nm π-polarized beam. This serves as a measure of the mJ = +5/2
population. Once we have this measure across all the experiments, we can remove
the back-scattering effect by dividing the population data by 1− ρ+5/2, where ρ+5/2
is the measured population of mJ = +5/2. Note that this option works well only
in this case, since for the other combinations of initial state and laser polarization,
there is no state in which the back-scattering population remains in one sublevel
throughout the experiment.
6.2.5. Experimental Results and Analysis
For each dataset generated, we took a separate natural lifetime measurement.
One advantage of this is it lets us check for any errors that are systematic across
the laser-on and laser-off conditions. We measured a natural lifetime in good
agreement with the literature value of 1168(9)ms (see Kreuter et al., 2005).
Our measured lifetimes were 1165(10)ms, 1154(9)ms, and 1161(13)ms for the
mJ = +5/2 σ
− beam, mJ = +3/2 σ
− beam, and mJ = +3/2 π beam datasets,
respectively.
The scattering rate is linear in the laser intensity applied, and the slope
can be calculated from the Kramers-Heisenberg formula (see Appendix A).
Noting this, we can compare the model we developed in Chapters IV and V the
experiment by varying the laser intensity and comparing the slopes of these theory
lines to the slope of the line fitted to the data. Plots of these lines are given in
figures 6.5, 6.7, and 6.9.
The scattering rate measurements are in good agreement with the theory.
The fitted slope is within the uncertainty range of the theory line’s slope, and vice
versa. The uncertainty ranges of the theory line and the fit line are calculated
86
differently. For the fit line, the uncertainty range spans all lines with slopes within
one standard error of the fitted slope. For the theory line, the uncertainty range
spans all lines with slopes that fall within the uncertainty range of the model
parameters. These parameter uncertainties are dominated by uncertainty in
the P3/2 lifetime (precision of 0.6%, see Meir et al., 2020), so this uncertainty
essentially determines the uncertainty range of the theory line. This corroborates
the scattering rate calculations which went into Chapters IV and V, and therefore
demonstrates low scattering errors are achievable in trapped-ion qubits. Similar
experiments were also performed at UCLA on g qubits in 133Ba+ ions, again
achieving good agreement with the predicted scattering rates (see Boguslawski
et al., 2022).
The next step of this research is to actually implement a two-qubit,
Raman-beam-induced gate using these 976 nm lasers. Our lab’s progress towards
implementing such a gate and increasing its fidelity is the subject of the next
chapter.
87
FIGURE 6.4. Lifetime measurement results for mJ = +5/2 initial state with σ
−-
polarized 976 nm laser beam applied. I fit an exponential decay function 1 − eγt
to the data and infer the scattering rate by taking the difference between the γ fit
parameters of the natural lifetime and laser-incident lifetime datasets.
88
FIGURE 6.5. Scattering rate measurement results for mJ = +5/2 initial state with
σ−-polarized 976 nm laser beam applied.
89
FIGURE 6.6. Lifetime measurement results for mJ = +3/2 initial state with σ
−-
polarized 976 nm laser beam applied. I fit an exponential decay function 1 − eγt
to the data and infer the scattering rate by taking the difference between the γ fit
parameters of the natural lifetime and laser-incident lifetime datasets.
90
FIGURE 6.7. Scattering rate measurement results for mJ = +3/2 initial state with
σ−-polarized 976 nm laser beam applied.
91
FIGURE 6.8. Lifetime measurement results for mJ = +3/2 initial state with
π-polarized 976 nm laser beam applied. Note that all population measurements
shown here were renormalized by subtracting out the measured mJ = +5/2
populations. I fit an exponential decay function 1 − eγt to the data and infer the
scattering rate by taking the difference between the γ fit parameters of the natural
lifetime and laser-incident lifetime datasets.
92
FIGURE 6.9. Scattering rate measurement results for mJ = +3/2 initial state with
π-polarized 976 nm laser beam applied. The data points here were inferred from
raw scattering data with mJ = +5/2 population subtracted out.
93
CHAPTER VII
TWO-QUBIT GATE
As discussed in the introduction, our lab has begun work towards
implementing a two-qubit Mølmer-Sørensen entangling gate in our ion trap.
As discussed in Chapter II, two-qubit gates have previously been used to
generate entanglement in m qubits (see Bazavan et al., 2023 and Roos et al.,
2004). However, we are attempting to implement the first Raman-beam-induced
entangling gate in m qubits. This will serve as a proof-of-principle for m qubit
Raman-driven operations.
7.1. Mølmer-Sørensen Gate Setup
Since we are using the qubit |↓⟩ = |D5/2,mJ = +3/2⟩, |↑⟩ =
|D5/2,mJ = +5/2⟩, we must drive the MS gate using σ−- and π-polarized laser
beams. The beams and magnetic field geometry in our setup are shown in
Figure 7.1.
The Mølmer-Sørensen gate effects entanglement by simultaneously driving
the red- and blue-sideband transitions, and so the detunings of the two σ− beams
are appropriately chosen to drive these transitions. The different detunings of the
σ− beams are generated by applying two rf tones to the AOM for that beam.
The difference between the k⃗ vectors of the π- and σ− beams, ∆k⃗, is at 45◦
with respect to the mode vector of interest (see Figure 7.1) we are interested in
driving. We choose a radial mode because the it is useful for other experiments
we perform and it has a higher frequency than the axial modes, which reduces
the heating rate but lowers the gate speed. However, this has the downside that
94
FIGURE 7.1. Schematic geometry of Mølmer-Sørensen gate physical setup. A
pair of σ−-polarized laser beams travels perpendicular to a π-polarized laser beam,
with all of these beams at 45◦ from the ion chain axis. The radial modes are along
x and y (the difference and sums of these mode vector axes give the axes shown
on the figure). The B field runs parallel to the σ−-polarized laser beams. The
difference between the π-polarized beam’s and each of the σ−-polarized beams’ k
vectors is shown as ∆k⃗.
stabilizing the rf amplitude is more difficult than stabilizing the dc potential (see
Section 3.4). Additionally, while, from the perspective of minimizing scattering
errors, counter-propagating beams would be superior (they would lower the
√
scattering rate by a factor of 2), it comes with several downsides. Because of
the geometry of our trap electrodes and the direction of the B field, a counter-
propagating beam geometry would require us to split the power in the σ− beams
equally between σ+ and σ− polarization, which could waste power. For our MS
gate, the power at the ion is 45mW in the π-polarized beam, with another 45mW
split evenly (22.5mW each) between the two σ−-polarized beams. This uneven
95
distribution of power is optimal from the perspective of minimizing scattering
errors from a three-beam MS setup Moore, Campbell et al., 2023.
Using the pulse sequence discussed in Chapter V, we are able to achieve
a state preparation and readout fidelities of 99.2% and 99.1%, respectively. As
for single-qubit operations, we are able to achieve highly-coherent, stimulated-
Raman-induced Rabi flops, as seen in Figure 7.2. With a Ramsey experiment, we
measure a coherence time of about 1ms. The decay time in the contrast gives us
our coherence time. With spin echoes, we are able to increase the coherence time
to 2ms.
1.0
0.8
0.6
0.4
0.2
0.0
0 10 20 30 40 50
Raman Pulse Duration ( s)
FIGURE 7.2. Carrier transition Rabi flopping at 137(4) kHz Rabi frequency from
976 nm Raman laser beams in the D5/2 Zeeman qubit.
96
Population in mJ = + 3/2
We are also able to drive the sideband transitions quite coherently, albeit less
so than the carrier (because the system is not completely in the motional ground
state). See Figure 7.3. This data is taken after cooling below the Doppler limit, an
important step in implementing the MS gate. Although we could just use resolved
sideband cooling, we instead follow the Doppler cooling with another sub-Doppler
cooling technique, EIT cooling (see Morigi, Eschner and Keitel, 2000 and Lechner
et al., 2016). We use this technique simply because it is faster than resolved
sideband cooling. After this, we cool further with resolved sideband cooling. With
this method, the measured phonon number ⟨n⟩ reaches 0.04(1).
There are two beam geometries to choose from when performing an MS
gate, the phase sensitive and phase insensitive geometries (see A. Hughes, 2021
and Lee et al., 2005). These are shown schematically in Figure 7.4. We do not
have the option to use the phase insensitive geometry, as it requires one of the
beams be σ+-polarized, but this does not couple mJ = +5/2 to any state in P3/2.
We therefore use the phase sensitive geometry instead (the two geometries can be
seen in Figure 7.4). This makes our setup more sensitive to laser-phase-fluctuation-
induced dephasing, although this can be mitigated to some extent (see Lee et al.,
2005 and Section A3 of A. Hughes, 2021).
This setup allows us to generate entanglement because the sidebands couple
the ions’ internal, electronic states to the collective motional state of the ion
crystal. The entanglement generation of this setup can only be appreciated when
looking at the joint state of the two ions, as shown in Figure 7.5.
Figure 7.5 shows why each σ− beam is detuned slightly from the sideband
resonance: this ensures that the intermediate states, |↑↓⟩ and |↓↑⟩ are not
populated. This setup allows population transfer between the states |↑↑⟩ and |↓↓⟩,
97
1.0
0.8
0.6
0.4
0.2
0.0
0 100 200 300 400 500 600
Raman Pulse Duration ( s)
FIGURE 7.3. Sideband transition Rabi flopping at a sideband Rabi frequency of
8.20(2) kHz from 976 nm Raman laser beams in the D5/2 Zeeman qubit. The fit to
this data indicates that the measured phonon number ⟨n⟩ in this run was 0.04(1)
and the Lamb-Dicke parameter is 0.060(2).
and so, for an appropriately chosen gate time, we can map the state |↓↓⟩ to the
Bell state √1 |↑↑⟩+ √1 |↓↓⟩.
2 2
We can increase our gate’s robustness to detuning errors with Walsh
modulation. Walsh modulation involves applying a π phase shift or a π pulse
in the middle of the gate. This flips the trajectory of the ion state in phase
space Hayes et al., 2011. This behavior is shown in Figure 7.6.
The reason this enhances the robustness of the gate is that while the phase
space trajectory may form one incomplete loop on its own, flipping the direction
98
Population in mJ = + 3/2
FIGURE 7.4. Phase insensitive (left) and phase sensitive (right) beam geometries
in a single ion for MS gate.
of the trajectory with Walsh modulation will create two loops which, together, are
more likely to close. We can most clearly see the utility of Walsh modulation by
looking at its direct effects on a single ion. To do so, we measure the population in
the |↑⟩ state in a single ion with and without this π phase shift (these schemes are
called W (0) and W (1), respectively), with results shown in Figure 7.7. We found
that the feature associated with loop closure (found at 1/Tg for W (0) and 2/Tg at
W (1)) is broader for W (1), which shows that,with this setup, we can expect the
MS gate’s performance to be more robust to detuning errors.
99
FIGURE 7.5. Schematic of Mølmer-Sørensen gate’s effect on the joint state of an
ion pair’s energy. The detunings are chosen so that the only resonant coupling is
between the |↑↑⟩ and |↓↓⟩ states.
7.2. Preliminary Results
Early tests of this gate have yielded promising results. In Figure 7.8, I plot
the measured population in each of the possible joint two-ion states as a function
of the detuning from the sideband resonance. Our data shows a population of 80%
in the the |↑↑⟩ and |↓↓⟩ states. However, for an ideal gate, we would expect that
the population would be evenly split between the |↑↑⟩ and |↓↓⟩ states, with no
population in the other state. While our results are consistent with a fidelity of
80%, we will need to perform a parity measurement to confirm this Tan, 2016.
100
FIGURE 7.6. Effect of Walsh modulation on ion state trajectory in phase space.
The first loop would close if not for detuning errors, but the Walsh modulation
flips the trajectory, allowing the ion to make a second loop and close the path.
7.3. Future Directions
There are two primary tasks for the future of the investigation into this
m qubit MS gate. First, we will need to verify entanglement with a parity
measurement. Second, we will need to improve the gate time and fidelity.
To the first point: strictly speaking, population measurements are insufficient
to confirm that we are generating entanglement. To ensure we are truly creating
entangled states, we will need to implement a parity measurement. This will
require that we can control the relative phase of our Raman laser beams, which
is not possible in our current setup. However, we will easily remedy this by
upgrading our ARTIQ crate (Section 3.7) to version 8, which allows us to control
the relative phase of separate rf channels on the DDS boards.
101
FIGURE 7.7. Preliminary results for the calibration experiment with Walsh
modulation. In the W (0) case, shown at the top, the system returns to the
ground state at detuning 1/Tg, while in the W (1) case, shown at the bottom, it
returns at detuning 2/Tg (Tg is 100ms and 200ms for the W (0) and W (1) cases,
respectively). Note the broader feature associated with loop closure in the W (1)
case.
To the second point: in the future, we will explore multiple options for
improving the gate time and fidelity. For the gate time, we completed the MS
gate in 200µs, but the preliminary data was taken with an older setup which
had 45mW of power in each polarization. We estimate that, after accounting
for losses to the intermediate optics, we could achieve a total power of 300mW:
150mW in the π-polarized beam and 75mW in each σ−-polarized beam, with the
102
FIGURE 7.8. Preliminary results for the Mølmer-Sørensen gate with possible
entanglement generation. Population corresponds to the fraction of experiments
when both ions are bright (|↑↑⟩), one ion is bright and the other is dark
(|↑↓⟩/|↓↑⟩), and when both ions are dark (|↓↓⟩). This data was taken without
Walsh modulation. There was 45mW of power in each polarization as discussed
above. The 854 nm beam provided a light shift of 697(1) kHz on the mJ = +1/2
state during the gate, and a differential light shift on the qubit from the 976 nm
Raman beams of 27(1) kHz. The gate time was 200µs.
current setup. We can further reduce the gate time by increasing the intensity
with improved beam focusing. The increased beam power alone will improve the
gate time by a factor of three, so we can expect to achieve a gate time of roughly
60µs with the current setup. Once we have added in Walsh modulation, however,
√
the gate time will be a factor of 2 longer than this.
103
For increasing the gate fidelity, there are many options. First, it is worth
noting that the current experimental setup is, in many ways, superior to the
setup used to take the data seen in Figure 7.8. Since taking the preliminary MS
data, we have improved the qubit and motional coherence. We improved the
qubit coherence with feedforward, which suppresses qubit decoherence from 60Hz
electromagnetic signals from wires in the lab. We improved motional coherence
and frequency stability with temperature stabilization and implementation of a
device which removes amplitude modulation from the rf source 3.4. It is also worth
noting that the heating rate is a common limitation of gate performance, but we
can mitigate this issue by driving the gate faster or using an out-of-phase mode.
Second, the qubit coherence time is only 10 times larger than the gate time, which
is a strong limitation. We could improve this with magnetic shielding, but the
solution we plan to implement is utilizing clock qubits in 43Ca+. This would also
eliminate the need for the 854 nm light shift laser. Additionally, for 43Ca+, the
fidelity limit from Raman scattering with 976 nm Raman beams is 9 × 10−5, so the
gate will likely be limited by the other factors mentioned above, not the Raman
scattering error.
104
CHAPTER VIII
CONCLUSION
In this thesis, we studied scattering errors in trapped-ion qubits. Our
motivation for doing so was the omg scheme, an ion trap quantum computing
architecture that utilizes multiple different qubit types in the same ion species.
Because the omg scheme requires the use of m qubits, and photon scattering
commonly contributes substantially to the errors, we sought to characterize
scattering errors during logic gates in m qubits. In the course of doing so, we
found issues with past models of g qubit scattering errors, and reexamined them
as well.
We therefore constructed a model of two-photon scattering errors during
stimulated Raman transitions in trapped-ion qubits which incorporates all two-
photon scattering processes, as well as the detuning dependencies of all system
parameters involved. We also estimated the contribution to the scattering error
from higher energy levels in m qubits and computed an upper bound on Rayleigh
scattering error in both m and g qubits, finding this latter error to be negligible
in all but the lightest ion species for most of the detuning range considered. We
found that including all the above effects produced non-negligible corrections to
simpler models of the systems, in particular that the more complete model implies
there is no lower bound on Raman scattering-induced infidelity of g qubits as
suggested by past models of such errors Ozeri, 2007. While we originally developed
this theory for hyperfine qubits, it is easily extended to the Zeeman qubits we
actually studied in our experiments.
105
Our experimental study of the scattering rates began from the observation
that the inclusion or exclusion of the various features of the model predicted
different slopes for the linear relationship between the scattering rate and laser
beam intensity. This implied that we could test the accuracy of the calculations
by measuring the scattering rate at various laser intensities, fitting a line to these
points, and comparing the slope of the fit to the slope predicted by the theory.
The experimental results provide strong evidence that the calculation is accurate,
with fit slopes on each data set comfortably within one standard error of the
theory slope. Our collaborators at UCLA conducted a similar experiment on g
qubits in 133Ba+ and obtained results that also verified the theory Boguslawski
et al., 2022.
Finally, the results of the calculations (summarized in Table 8.1) show that,
although m qubits have a lower bound on gate infidelity and the detunings and
powers required for 10−4 error in m qubits are larger than in g qubits, low errors
should still be experimentally achievable in m qubits because high power lasers are
more readily available at the required wavelengths for the ion species considered.
In sum, low Raman scattering errors are achievable in stimulated Raman-driven
gates for both m and g trapped-ion qubits at sufficiently large red detunings.
106
9Be+ 25Mg+ 43Ca+ 87Sr+ 133Ba+ 135Ba+ 137Ba+ 171Yb+ 173Yb+
m - - 0.025 0.036 0.062 0.012 0.00047 - -
ω0/2π (GHz) g 1.3 1.8 3.2 5.0 9.9 7.2 8.0 12.6 10.5
m - - 0.036 0.021 0.028 0.028 0.028 - -
η
g 0.213 0.143 0.077 0.053 0.038 0.038 0.038 0.046 0.047
-40.0 -66.0 -45.3 -45.6 -45.9
m - - - -
∆/2π (THz) (963 nm) (1335 nm) (676 nm) (677 nm) (677 nm)
-1.00 -4.55 -9.05 -13.0 -26.4 -26.6 -26.9 -15.3 -15.4
g
(313 nm) (281 nm) (402 nm) (429 nm) (515 nm) (516 nm) (516 nm) (376 nm) (376 nm)
m - - 4.9 9.1 4.4 4.4 4.5 - -
Power (W)
g 0.067 0.13 0.30 0.37 0.94 0.96 0.98 0.67 0.67
TABLE 8.1. Comparison of g and m qubit gate characteristics. The qubit
frequency for m and g qubits is given in the first two rows. The Lamb-Dicke
parameter η is given for a 5MHz trap frequency and counter-propagating beams at
the P3/2 resonances frequencies (resonance with D5/2 and S1/2 in m and g qubits,
respectively). The detuning ∆ corresponds to the detuning (from P3/2 in m qubits
and from P1/2 in g qubits) required for 10
−4 error, and the corresponding laser
wavelength is given parenthetically below each detuning. Total power requirements
are given for the 10−4 error threshold of the two-qubit Mølmer-Sørensen gate of
Section 4.2.2, driven by 3 Raman beams.
107
APPENDIX A
DERIVATION OF SCATTERING RATE
To derive the general Raman ΛV scattering formula (Eqn. 4.6), begin with
Eqn. 8.7.3 of Loudon, 2000,
∑ πe4ωωscn ∣∣∣∑(⟨f | r⃗ · ϵ̂∗λ |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · ϵ̂∗ )λ |i⟩ ∣∣∣2Γi→f,ΛV = − + δ(ωfi+ωsc−ω).2ϵ2ℏ2V 20 ωk ω ωk + ωscksc,λ k
(A.1)
This equation describes the ΛV scattering rate to state f during virtual
transitions from state i through the manifold containing the states indexed by
k (where all these states are hyperfine sublevels). The transitions are driven by
an n-photon laser beam of frequency ω and polarization ϵ̂, scattering a photon
with frequency ωsc and polarization ϵ̂
∗
λ, where λ indexes the two independent
polarizations in the chosen basis. The rate is calculated by summing contributions
from all scattering modes ksc, λ allowed in the quantization volume V . The
frequencies can be understood by the energy level diagram of Fig. A.1.
Rearranging this equation, we find
∑ ∣ ( )∣2ℏωn πe4ωsc ∣∣∑ ⟨f | r⃗ · ϵ̂∗ |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · ϵ̂∗ |i⟩ ∣∣
Γi→f,ΛV = ∣ λ + λ ∣ δ(ωfi+ωsc−ω).
ϵ 30V 2ϵ0ℏ V ∣ ωk − ω ωk + ωsc ∣
ksc,λ k
(A.2)
108
∆k 〉|k〉|
〉|f〉
ωk
|i〉|
FIGURE A.1. Relevant frequency definitions for Eqn. A.1. The transitions are
driven by a laser beam of frequency ω. Scattered photon frequency is given by
ωsc = ω − ωfi.
Now, if we model the laser field as a classical electric field plane wave with
amplitude E,
E ( )
E (r, t) = ϵ̂ei(k·r−ωt)las + ϵ̂
∗e−i(k·r−ωt) , (A.3)
2
we can calculate the temporal and spatial average of
2
⟨E∗las(r, t) ·
E
Elas(r, t)⟩ = . (A.4)
2
Comparing this to the quantized result in an n photon Fock state, ⟨n| Ê†Ê |n⟩ =
ℏω (n+ 1) allows us to make the replacement, assuming n ≫ 1,
ϵ0V 2
2ℏω
n ≈ E2 (A.5)
ϵ0V
to get
109
∑ ∣∣∣∣∑( )∣∗ ∗ ∣∣∣2E2πe4ωsc ∣ ⟨f | r⃗ · ϵ̂λ |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · ϵ̂λ |i⟩Γi→f,ΛV = − + ∣ δ(ωℏ3 fi+ωsc−ω).4ϵ0 V ωk ω ωk + ωsc
ksc,λ k
(A.6)
Next, we can take the limit of large quantization volume V ,
∑ ∫∫ 2
→ V ωdω scscdΩ , (A.7)
(2π)3 c3
ksc
enabling us to write the spontaneous scattering rate in the form
∑∫∫ ∣E2e4ω3 ∣∑( ∗
Γ = dω dΩ sc ∣ ⟨f | r⃗ · ϵ̂λ |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩i→f,ΛV sc
32π2c3ϵ ℏ3 ∣0 ωk − ω
λ k
⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · ϵ̂∗ )λ |i⟩+ ∣∣∣∣2δ(ωfi + ωsc − ω).ωk + ωsc
(A.8)
Now we wish to perform the integral over dΩ ≡ dϕ dθ sin(θ), the direction
of the scattered photon’s k-vector, k̂sc. For this, it is conceptually helpful to
gather terms into an expression for the vector transition dipole matrix element
for spontaneous scattering,
Ee ∑(≡ ⟨f | )r⃗ |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ |i⟩r⃗sc(ωsc) + (A.9)
2ℏ ωk − ω ωk + ωsc
k
which allows us to write Eq. (A.8) in the form
∑∫∫ e2ω3
Γi→f,ΛV = dωscdΩ
sc |r⃗ ∗ 2
2 3 ℏ sc
· ϵ̂λ| δ(ωfi + ωsc − ω). (A.10)8π c ϵ0
λ
110
Without loss of generality, we will choose a polarization basis such that one
of the basis polarization vectors lies in the plane of r⃗sc and k̂sc; we will call this
vector ϵ̂sc (note that the other basis vector will not contribute to the scattering, as
it will necessarily be perpendicular to r⃗sc). Choosing θ to be the angle between r⃗sc
and k⃗sc, the angle between r⃗
π
sc and ϵ̂sc is − θ, which implies2
( )
|r⃗sc · ϵ̂sc|2 =|r⃗sc|2 cos2 π − θ2
(A.11)
=|r⃗ 2 2sc| sin (θ),
allowing us to carry out the integral over the direction of the spontaneously
emitted photon to get
∫
e2ω3
Γ sc 2i→f,ΛV = dωsc |r⃗sc(ωsc)| δ(ωfi + ωsc − ω)
3πc3ϵ0ℏ
e2(ω − ω 3fi)
= |r⃗sc(ω − ω 23 ℏ fi)|3πc ϵ0
e2(ω − ω 3fi)
= (|r⃗sc(ω − ω 23 ℏ √ fi
)|)
3πc ϵ0
 ∑ 2e2(ω − ω 3fi)= |r⃗sc(ω − ωfi) · ê |2q
3πc3ϵ0ℏ ∑ qe2(ω − ωfi)3
= |r⃗sc(ω − ωfi) · ê 2q|
3πc3ϵ0ℏ q ∑∣∣∣∣∑( )∣∣∣∣2E2e4(ω − ωfi)3 ∣ ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · êq |i⟩= − +12πc3ϵ 30ℏ ωk ω ωk + ω − ωq fi ∣k
(A.12)
where êq is a polarization vector corresponding to π, σ
+, or σ−, with q indexing
these possibilities.
111
Now, if ∆k ≡ ω − ωk is the detuning relative to some level k, we see that
we can write ω − ωfi as ωk − ωfi +∆k. Since we choose to measure the detuning
relative to the P3/2 manifold, we define ∆ = ω − ωPi, where ωPi is the frequency
of the transtion between the manifold containing |i⟩ and P3/2. This allows us to
rewrite ω − ωfi as ωPf + ∆, where ωPf is the frequency of the transition between
the manifold containing state f and P3/2; additionally, we may rewrite ωk − ω as
ωk − ωPi −∆. Rewriting ω − ωfi as such and rearranging gives us
∣ ∣2
E2e4
(
ω3 ∑∣∑Pf ∣∣∣ ⟨
) ( )
f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · êq |i⟩ ∣∣
Γi→f,ΛV = +
12πc3ϵ0ℏ3 ∣∣
∣
( ωk − ωPi −∆ ωk + ωPf +∆ ∣
3
∆
1 +
q∑ k∣∣∑ )∣
ωPf
2
e4ω3 E2Pf ∣ ⟨f |
( )3
r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · êq |i⟩
= +
ℏ ℏ − − ∣∣∣ ∆1 + .3πc3ϵ0 4 2 ωk ωq P i ∆ ωk + ωPf +∆ ∣ ωPfk
(A.13)
We begin the next step by noting Eqn. 4.3,
e2ω3Pf 2 1µPf = ≡ γPf3πϵ0ℏc3 τfl
where γPf is the decay rate from the P3/2 manifold to the manifold containing
state f ; this is given by αfγ, where αf is the branching ratio into the manifold
containing state f and γ is the total spontaneous decay rate for the manifold
containing state l. With this, we can write
112
∣ ( )∣2( )
γ e2E2 ∑∣∣∣∣∑Pf ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · 3ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · ê ∣q |i⟩ ∣ ∆Γi→f,ΛV = µ2Pf 4ℏ2 ∣∣ ( ωk − −
+ ∣ 1 +
ωPi ∆ ωk + ωPf +∆ )∣∣ ωq Pfk 2( )
e2
3
E2 ∑∣∑ ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · êq |i⟩ ∣∣ ∆
= γPf ℏ ∣∣ + ∣ 1 + .4 2 µPf (ωk − ωPi −∆) µPf (ωk + ωPf +∆) ∣ ωq Pfk
(A.14)
Now we divide and multiply by µPi, i.e., the matrix element of Eqn. 5.2
between the P3/2 and the manifold containing i.
∣ (
e2E2 ∑∣∑ ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩
Γi→f,ΛV = γ ∣Pf µPi
4ℏ2 ∣ µPfµPi(ωk − ωPi −∆)q k
⟨f | r⃗ · ϵ̂ |k⟩ ⟨ | | ⟩ )∣∣· ∣2( )3k r⃗ êq i ∆+ 1 +
µPfµPi(ωk + ω ∣Pf +∆) ωPf
e2E2µ2 ∑
Γ Pii→f,ΛV = γPf
4ℏ2 ∣∣
∣∣∑( ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩
µPfµPi(ωq k − ωPi −∆)k
⟨ )∣ ( ) (A.15)f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · êq |i⟩ ∣∣2 3∆+ 1 + .
µ ∣PfµPi(ωk + ωPf +∆) ωPf
Noting the definition of gPi (Eqn. 4.5), this becomes
∑∣∣∣∣∑( ∣22 ∣ ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨
) ( )3
f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · êq |i⟩ ∣∣ ∆
Γi→f,ΛV = γPfgPi + ∣ 1 + .µPfµPi(ωk − ωPi −∆) µPfµq P i(ωk + ωPf +∆) ∣ ωPfk
(A.16)
Finally, we make our result applicable to gates; we do so by averaging over
the undisturbed state |i⟩ during the course of the gate. If we consider a σ̂x gate,
113
then an ion initially in the state |0⟩ will get mapped to |1⟩. The ion’s state |i⟩ then
has a time dependence given by |i(t)⟩ = cos(2πt/τ) |0⟩ + sin(2πt/τ) |1⟩, where τ is
the gate time. Considering this time dependence and averaging Eqn. A.16 over the
gate time gives
∑∣∣∣∣∑( )∣2g2Pi ∣ ⟨f |
( )
r⃗ · 3êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ⟨f | r⃗ · ϵ̂ |k⟩ ⟨k| r⃗ · êq |i⟩ ∣∣ ∆
Γf,ΛV = γPf + ∣ 1 + ,
2 µ
i,q Pf
µPi(ωk − ωPi −∆) µPfµPi(ωk + ωPf +∆) ∣ ωPf
k
(A.17)
where |i⟩ now indexes the two qubit states, |0⟩ and |1⟩.
By the same reasoning, we can get the ladder scattering rate (which
contributes in m qubits but not g qubits),
∣
g2 ∑∣Pi ∣∣∑( ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂∗ |i⟩Γf,lad = γPf 2 µPfµPi (ωi,q kP +∆+ 2ωPD)k
⟨f | r⃗ · ϵ̂∗ | ⟩ ⟨ )∣2( )3k k| r⃗ · êq |i⟩ ∣∣∣ − 2ωPD +∆+ 1 .µPfµPi (ωkP −∆+ ωDf ) ωPf
(A.18)
To generate the full model scattering rate, we sum over all scattering events
i → f except for i → i scattering events, since we are ignoring Rayleigh scattering.
We can get the simplified model’s scattering rate equation by neglecting the
V scattering term in Eqn. A.17 and assuming (1 + ∆/ωPf )
3 ≈ 1
∑∣∣∣∣∑ ∣22≈ gPi ∣ ⟨f | r⃗ · êq |k⟩ ⟨k| r⃗ · ϵ̂ |i⟩ ∣∣Γf γPf 2 µPfµPi (ωkP − ∣∣ . (A.19)∆)i k,q
We can again obtain the Raman scattering rate by summing over all scattering
events except for i → i.
114
APPENDIX B
DECAY RATE FROM FERMI’S GOLDEN RULE
Here we derive Eqn. 4.3. To do so, begin with Fermi’s Golden Rule (Eqn.
1 in J. M. Zhang and Liu, 2016) for the rate of spontaneous decay from excited
atomic state i to a state f ,
2π ∣∣∣ ∣2γi→f = ⟨f | ĤI |i⟩∣∣ ρ(Eω0) (B.1)ℏ
where ρ(Eω0) is the density of states in energy at the energy of a photon with
frequency ω0 (the resonant frequency of the transition between i and f), and ĤI
is the interaction Hamiltonian. The corresponding matrix element is given by (see
§2.2 of Metcalf and Straten, 1999)
√
⟨ | | ⟩ ℏω0f ĤI i = e ⟨f | d⃗ · ϵ̂sc |i⟩
2ϵ0V
where V is the quantization volume, d⃗ is the dipole operator, and ϵ̂sc is the
polarization unit vector of the scattered photon. Putting these together, we get
πe2ω ∣∣ ∣∣20
γi→f = ∣⟨f | d⃗ · ϵ̂sc |i⟩∣ ρ(Eω
ϵ V 0
) (B.2)
0
Now, to find ρ(Eω0), we consider the number of modes N , i.e, the sum of 1 over
all polarizations and wave vectors, which for large quantization volume V (see eqn
1.1.11 in Loudon, 2000) goes as
115
∑∑ ∫ ∫ ∫ ∫ ∫
→ V ω
2 V ω2 V ω2
N = 1 2 dω scscdΩ = 2 dE
sc sc
3 3 ℏ 3 ω
dΩ = dEω
(2π) c (2π) sc c3 π2ℏ sc c3
λ ksc
(B.3)
where the first factor of 2 came from the sum over the two independent
polarizations λ. So
V ω2
ρ(Eω0) =
0 (B.4)
π2ℏc3
which gives
πe2ω V ω2 ∣∣∣ ∣∣∣2 e20 0 ⟨ | · | ⟩ ω3 ∣ ∣2γ = f d⃗ ϵ̂ i = 0i→f sc ∣∣⟨f | d⃗ · ∣ϵ̂ |i⟩∣ (B.5)
ϵ V π2ℏc3 πϵ ℏc3 sc0 0
and, finally, after averaging over the random polarization direction,
e2ω3
γi→f =
0
ℏ ∣∣∣ ∣2⟨f | |d⃗| |i⟩∣∣ (B.6)3πϵ0 c3
116
APPENDIX C
POWER REQUIREMENTS DERIVATION
Here we derive the expressions given in Section 4.3. For a Gaussian laser
beam of power P we consider Steck, n.d.:
E2
4P
= , (C.1)
πw20cϵ0
γ ℏω3 2
= Pi
w0 , (C.2)
g2 2Pi 3c Pαq
where w0 is the laser beam waist and αq is the branching ratio for transitions
between P3/2 and the qubit manifold. Note that in general |Ω 2R| = gPir(∆) for
some r(∆), so we can replace g2Pi with |ΩR|/r(∆). This means that the power can
be written as
ℏω3 w2Pi 0 |ΩR|P = . (C.3)
3c2αqγ r(∆)
For a single-qubit gate, |ΩR| = π/2τ1q where τ1q is the gate time; for a two-qubit
√ √
gate, |ΩR| = π K/2 2τ2qη(∆). This allows us to write the total power required
for each as a function of ∆:
ℏω3 w2 π
P1q(∆) = 2
Pi 0 (C.4)
6c2αqγ τ1qr(∆)
and √
ℏω3 w2 π K
P2q(∆) = 4 √ Pi 0 , (C.5)
6 2c2αqγ τ2qη(∆)r(∆)
117
where the factor of 2 in front of Eqn. C.4 is due to the use of two beams of equal
power, and the factor of 4 in front of Eqn. C.5 is due to the use of two beams of
equal power along with one beam with double the power.
If, as we did in Section 4.2, we neglect the effects of large detuning, we can
write power as a function of the gate error directly. We will start by rewriting
r(∆) = 2/15|∆| in terms of the single-qubit gate error ϵ1q,
πγ 15πγr(∆)
ϵ1q = ρ | = ρ∆| 2
=⇒ r(∆) = 2ϵ1q , (C.6)or for a two-qubit gate,
15ρπγ
√ √
πγ 4√ K 15πγr(∆) 4√ Kϵ2q = ρ | | = ρ∆ 2η √2 2η (C.7)
2ϵ2q 2η
=⇒ r(∆) = √ .
15ρπγ 4 K
Substituting into Eqns. C.4 and C.5, we get
5πℏω3 2
P (ϵ ) = ρ Pi
w0 π
1q 1q (C.8)
2c2ϵ1qαq τ1q
and
10πℏω3 w2 π K
P2q(ϵ2q) = ρ
Pi 0 . (C.9)
c2ϵ2qαq τ 22q η
118
APPENDIX D
RECOVERING CLASSICAL LIMITS
We can test the completeness of our model by seeing if it is able to recover
classical elastic scattering behavior in the limit of large detuning. Considering only
elastic scattering excludes ladder scattering processes (since they are inelastic).
This means we can write the elastic scattering rate as
∣ ( )∣2
4
2 e ω
3 ∣∑ ∣
Γ = E L ∣∣ | ⟨i| | 1 1 ∣r⃗ k⟩ |2 + ∣ . (D.1)
12πc3ϵ ℏ30 ∣ ωki − ωL ωki + ωL ∣
k
For Raman beams with intensity I, we have E2 = 2I/ϵ0c; additionally, we
can replace the scattering rate with the scattering cross-section σ via σ = ΓℏωL/I.
Putting these equations together, we get
28π ω
4
L ∣∣∣∣
∣∑ ( )∣∣2
σ = α | ⟨i| r⃗ |k⟩ |2 1 1 ∣+ ∣ , (D.2)
3 c2 ωki − ωL ωki + ωL ∣
k
where α = e2/4πϵ0ℏc is the fine structure constant.
Now we can calculate the limits. For large blue detuning (ωL ≫ ωki for every
k), we have
( )
1 1 ≈ − ωki− + 2 , (D.3)ω 2ki ωL ωki + ωL ωL
allowing us to rewrite Eqn. D.2 as
∣ ∣2
232π 1
∣
σblue = α ∣∣∣∑ ∣∣ωki| ⟨i| r⃗ |k⟩ |2∣3 c2 ∣ . (D.4)
k
119
Using the Thomas-Reiche-Kuhn sum rule (for the single valence electron only, and
neglecting recoil and the ion’s monopole charge), the sum can be evaluated to
∑
ω | ⟨i| r⃗ |k⟩ |2 ℏki = . (D.5)
2me
k
This gives
232π 1 ℏ2 8πσ 4 2blue = α = α a , (D.6)
3 c2 4m2 3 0e
where a0 is the Bohr radius and α
2a0 is the classical electron radius. So for large
blue detuning, the model recovers the Thomson cross-section of the valence
electron.
For red-detuning, (ωL ≪ ωki for every k), we have
( )
1 1
− + ≈
2
, (D.7)
ωki ωL ωki + ωL ωki
giving
∣ ∣2
28π ω
4
L ∣∣∣∣∑ 2| ⟨i| r⃗ |k⟩ |2 ∣∣σred = α ∣ . (D.8)3 c2 ωki ∣
k
The DC polarizability of the ion can be written as
∑
(0) ≡ 2 ⟨i| r⃗ |k⟩ ⟨k| r⃗ |i⟩+ ⟨k| r⃗ |i⟩ ⟨i| r⃗ |k⟩α e
Ek −
, (D.9)
Ei
k ̸=i
allowing us to rewrite Eqn. (D.8) as
120
8π ω4
σred = α
L
(2 ) ℏ2|α(0)|23 c2e48π ω 4 α(0)L
= ℏ2| |2 (D.10)
3 c 4πϵ0
8π
= k4Lℏ2|α(0)|2.3
We can compare to the classical result by using the Clausius-Mossotti relation,
( )2( ) ( )2( ) ( )2 2
| (0)|2 3ϵ0 ϵr − 1 3ϵ0 n − 1 3ϵ0 4α = = ≈ (n− 1)2, (D.11)
N ϵr + 2 N n2 + 2 N 9
where N is the number density of particles in the material, ϵr is the dielectric
constant, and n is the refractive index. The second equality is true for non-
magnetic media, and the third, approximate equality holds when n ≈ 1. Applying
this to Eqn. (D.10), we find
2k4
σ ≈ Lred |n− 1|2, (D.12)
3πN2
which is the expression for classical Rayleigh scattering (e.g. Jackson, 1999). Note
that if we had ignored the V scattering process, i.e., the 1/(ωki + ωL) term, we
would not have recovered the correct limits.
121
APPENDIX E
QUANTUM CRYPTOGRAPHY PROTOCOL
Below, I detail a quantum cryptography protocol I helped develop with
Steven van Enk Moore and Enk, 2021.
Suppose that Alice and Bob are preaparing and sending separable qubit
states αA and βB, respectively, to a joint measurement device represented by a
measurement operator ξAB. Then the probability of getting a given measurement
is
P = Tr(αA ⊗ βBξAB) (E.1)
If Alice and Bob together would like to be able to tomographically reconstruct the
two-qubit measurement operator ξ, they each need to prepare 4 different (linearly
independent) states of their qubits. Given probabilities of the form (E.1), Alice
and Bob can gather the measured frequencies of detector “clicks” in a 4-by-4 data
matrix whose expectation value should (if there is a unique single-valued operator
ξ) have the form
F kl = Tr{αkA ⊗ βlB ξAB} (E.2)
for k, l = 1 . . . 4. The trace on the right-hand-side of Eq. (E.2) can be calculated by
expanding all operators in the Pauli basis as follows
∑ ∑ ∑
αk = αki σi, β
l = βli σi, ξ = xij σi ⊗ σj. (E.3)
i i i,j
122
Substituting these expansions into the definition of the data matrix and noting
that
Tr{σiσj ⊗ σkσl} = 4δijδkl, (E.4)
yields the equation ∑
F kl = 4 x αk lij i βj. (E.5)
i,j
Eliminating the factor of four by defining S = F/4, we can rewrite the equation for
S as a matrix equation
S = ATXB (E.6)
where X has matrix element xij and A has columns made from vectors of the
Pauli expansion coefficients of Alice’s operators, and B is similarly defined for
Bob. Multiplying both sides of this equation on the left by (AT )−1 [we assume
the inverse exists, i.e., we assume Alice’s 4 operators to be linearly independent]
yields
(AT )−1S = XB. (E.7)
Now we assume that both Alice and Bob have control over their own operators
such that these operators do not vary over the course of the experiment. Next,
suppose that there are two trials, each using a different (not identical) set of
operators αk for Alice but the same set of operators βl for Bob. Then we can
eliminate the unknown matrix X and write the condition on there being a unique
X that does not depend on which operators αk Alice is using, as
(AT1 )
−1S1 = (A
T
2 )
−1S2, (E.8)
123
where the left-hand and right-hand sides of the equation represent trials with
′
different operators αk . One alternative useful way of rewriting this same equation
is
S S−11 2 A
T
2 (A
T )−11 = 1, (E.9)
even though this may fail, namely, if S2 is not invertible. It is helpful to write this
matrix product out in terms of the coefficients:
∑
(S ) (S−11 ik 2 )kl(A
T
2 )lm((A
T
1 )
−1)mj = δij (E.10)
k,l,m
Written this way, it is easy to see how this equation can be used to test for
dependence of ξ on Alice’s operators: suppose, without loss of generality, that ξ
is somehow different for the operator αiA of the first trial. Then every row but the
ith row of S1 is inverted by the remaining three matrices. Therefore, only the ith
row of the left-hand side matrix will differ from the identity. Conversely, since
Alice is able to calculate the left-hand side of (E.10) just from her knowledge
of her operators αk and from the data matrices S1 and S2 she can diagnose,
without needing any knowledge about Bob’s operators βl, with which of her state
preparations the measurement is correlated. She also does not need to reconstruct
the measurement operator ξ.
Note that all we need for this to work is that the trials are not identical.
That is, Alice needs just 5 different states at a minimum [to make two non-
identical sets of 4 states] to be able to run this check.
(By symmetry, Bob could diagnose the presence of correlations between ξ
and his states βl without needing knowledge of Alice’s operators. In this case Bob
would need to prepare at least 5 different states.)
124
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