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Single Electron Spin Qubits in Silicon
Quantum Dots
David Zajac
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Jason R. Petta
September 2018
c© Copyright by David Zajac, 2018.
All rights reserved.
Abstract
Electron spins in quantum dots form ideal two-level systems for implementing
quantum computation in the solid state. While spin states can have extremely long
quantum coherence times, addressing single spins and coupling large arrays of spins
have been formidable experimental challenges. Research over the past several decades
has resulted in a variety of creative approaches to address these problems, yielded new
insights into the physics of spins in semiconductors, and demonstrated many of the
basic criteria for quantum computation.
This thesis presents a systematic study of the physics and quantum control of spins
in few-electron Si/SiGe quantum dots. We present novel designs for quantum dot de-
vices that yield improved control of single electron wavefunctions. We demonstrate
full control of single electron spin states by placing a quantum dot in the vicinity of a
strong magnetic field gradient produced by a micron-scale ferromagnet, and quantify
the control fidelity using randomized benchmarking. Utilizing the exchange interac-
tion between neighboring spins in combination with arbitrary single-spin rotations,
we present one of the first demonstrations of all the criteria for universal quantum
computation (initialization, readout, and a universal set of gates) with electron spins
in a single device. Finally, we take the first steps towards controlling a large array of
quantum dots by deterministically shuttling single electrons through an array of nine
quantum dots.
iii
Acknowledgements
Firstly I would like to thank my advisor Jason Petta for all of the guidance he has
provided over the course of my PhD. Throughout the years working in his lab I’ve
learned an incredibly diverse set of skills including sweating copper pipes, designing
printed circuit boards, fabricating nanoscale electronics, installing dilution refrigera-
tors, manipulating microwave signals, and many more. Above all I’ve learned from
Jason how to approach problems with intense focus and how to avoid getting dis-
tracted from the big picture goal. I’ve learned by example how to give a good, clear
presentation, how to get in the lab and make things happen, and the genius that is
the Sawzall. I’m truly grateful for my experience in working in his lab and have no
doubt that it will serve me well in the future.
Throughout my time in Petta Lab I’ve been privileged to work alongside a great
cohort of peers. Xiao Mi has demonstrated a truly impressive work ethic and an
uncanny ability to plow through tough problems. George Stehlik has the ability to
lift the spirits of those around him with a childish sense of wonder and playfulness,
and I will miss racing him to the coffee machine after lunch each day. Ke Wang
really helped me get started in the lab and passed on indispensable knowledge of
the cleanroom that was key to my success within the group. Peace always impressed
me with his efficiency and ingenuity in the lab. For someone so quiet in lab, I
was amazed by Yinyu’s ability to relentlessly churn out new results. Tom Hazard
always lightened the mood around the lab, and has been great to have as a climbing
partner and friend. Stefan Putz always kept the lunchtime discussions interesting
with intriguing trivia. Felix Borjans and Adam Mills have both impressed me with
their programming skills, and I have no doubt the lab will be in good hands with
them alongside the experimental expertise of Anthony Sigillito.
I also could not have done the work in this thesis without the support of our
collaborators. Jake Taylor and Guido Burkard provided invaluable theory support
iv
while exploring the exchange interaction between neighboring spins. Max Russ and
Michael Gullans have both been extremely patient in explaining new theory concepts
to me. Also, none of this could have been possible without financial support from
agencies that fund our research. The work in this thesis was supported by US Depart-
ment of Defense under contract H98230-15-C0453, the Army Research Office through
Grant No. W911NF-15-1-0149, the Gordon and Betty Moore Foundations EPiQS Ini-
tiative through grant GBMF4535, the National Science Foundation through Grants
No. DMR-1409556 and No. DMR-1420541, and the Spanish Ministry of Economy
and Competiveness through Grant MAT2017-86717-P. Also, the devices these ex-
periments were performed on were fabricated in the Princeton University Quantum
Device Nanofabrication Laboratory.
I’ve also had the support of numerous friends and family members over the span
of my PhD. I wish I could thank you all individually, but know that even if you’re not
mentioned here I appreciate everything that you’ve done for me over this significant
period of my life. Throughout my time at Princeton I could always count on Zach
Sethna to get me out of the lab to play softball, get in a quick game of pool between
study sessions, or host a night of drinks at the Z-bar. As for Josh Hardenbrook, I’ll
always remember our Monday night fantasy league, you forcing me to go to the gym
at unreasonable hours, and your intense drive for life.
Life at Princeton would not have been what it was without KL by my side. You’ve
been so supportive throughout this entire process and have helped me grow as a
person. I’m excited to move to NYC with you and start a new chapter in our lives
which I’m sure will be every bit as special as our time together in Princeton. Robin
and Michelle, you’ve treated me like family and given me a home away from home,
and I can’t say how grateful I am for that. The number of ways you’ve helped me
over the years is immeasurable, but if it could be measured I have a feeling it would
be in six-packs and racks of ribs. Dick and Mo, you’ve also become family to me, and
v
taught me some of the best jokes I know. Mattia I’ve felt like a brother with you,
sharing apartments over the past few years. Princeton doesn’t feel the same without
you, and I hope you’ll move back to the east coast once you get fed up with the San
Fransisco weather. Leslie, I can always count on you to make me smile. Spending
time with you always reminds me of what’s most important in life.
Of course the people who have most shaped who I am throughout my life is my
family. Tommy, I couldn’t have asked for a better brother in life. Thank you for
supporting me from all the way across the country. Whether it’s airport phone calls,
late night gaming, or family vacations, it’s always great to spend time with you. Aunt
Noreen, you’ve also been a big part of my life for as long as I can remember. As a
kid I remember always being excited to spend the day with Aunt Noreen and that’s
still true to this day.
Lastly, no one has supported me more than my parents. Dad, thanks for inspiring
my passion for building things with my own hands. From constructing skateboard
ramps to pool decks, watching and learning from you has shaped the way I approach
problems and came in handy during my research. Mom, probably no one has impacted
who I am more than you. You’ve always encouraged me to pursue my interests and
supported me in my pursuit. Whether it was driving me to hockey practice or signing
me up for guitar lessons, you’ve always been there for me. Thanks Mom and Dad for
making all of this possible. I couldn’t have done it without you.
vi
The work described in this dissertation has been published in the following articles
and presented at the following conferences:
Appl. Phys. Lett. 106, 223507 (2015).
Phys. Rev. Appl. 6, 054013 (2016).
Science 359, 439 (2018).
Nature 555, 599 (2018).
American Physical Society Meeting, March 2015, San Antonio, Texas.
Silicon Quantum Electronics Workshop, August 2015, Takamatsu, Japan.
MRS Fall Meeting, November 2016, Boston, Massachusetts.
American Physical Society Meeting, March 2017, New Orleans, Louisiana.
Silicon Quantum Electronics Workshop, August 2017, Hillsboro, Oregon.
Spin Qubit 3, November 2017, Sydney, Australia.
American Physical Society Meeting, March 2018, Los Angeles, California.
vii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Introduction 1
1.1 Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Experimental Quantum Bits (Qubits) . . . . . . . . . . . . . . . . . . 2
1.3 Introduction to Quantum Dots . . . . . . . . . . . . . . . . . . . . . 5
1.4 Electron Spins as Qubits . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Quantum Dot Device Physics 14
2.1 Ultra-Coherent Quantum Materials . . . . . . . . . . . . . . . . . . . 14
2.2 Quantum Well Heterostructures . . . . . . . . . . . . . . . . . . . . . 16
2.3 Full Device Architecture . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Gate Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Single Quantum Dots 23
3.1 Model of a Single Quantum Dot . . . . . . . . . . . . . . . . . . . . . 23
3.2 Coulomb Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Coulomb Diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
viii
3.4 Lever Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Charge Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Charging Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Electron Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.8 Orbital Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Magnetospectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.10 Measurements of Real-Time Tunneling Events . . . . . . . . . . . . . 39
4 Coupling Quantum Dots 41
4.1 Constant Interaction Model . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Double Quantum Dot Charge States . . . . . . . . . . . . . . . . . . 43
4.3 Finite Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Interdot Tunnel Coupling . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Dipole Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Single Electron Spin Qubits 50
5.1 Initialization and Readout . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Spin Lifetime (T1) Measurements . . . . . . . . . . . . . . . . . . . . 53
5.3 Micromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Electron Spin Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.5 Power Dependence of Rabi Frequency . . . . . . . . . . . . . . . . . . 59
5.6 Spin Coherence Time (T2) Measurements . . . . . . . . . . . . . . . . 61
5.7 Single Qubit Randomized Benchmarking . . . . . . . . . . . . . . . . 64
5.8 State Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 Two Qubit Gates and Entanglement 70
6.1 The Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 AC Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3 Conditional Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . 76
ix
6.4 DC Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.5 State Tomography and Entanglement . . . . . . . . . . . . . . . . . . 82
7 Large Scale Architectures 87
7.1 Device Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Controlling Large Arrays . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3 Long-Distance Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 92
8 Conclusions and Outlook 96
Bibliography 98
x
List of Tables
2.1 Isotopic Composition of Ga, As, and Si . . . . . . . . . . . . . . . . . 15
7.1 Characterization Measurements of a Nine Dot Array . . . . . . . . . 89
xi
List of Figures
1.1 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Qubit Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Quantum Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Nanowire Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Planar Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Loss-DiVincenzo Proposal . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Dephasing Due to Nuclear Spins . . . . . . . . . . . . . . . . . . . . . 16
2.2 Si/SiGe Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Quantum Well Transmission Electron Microscope Image . . . . . . . 17
2.4 Packaged Quantum Dot Device . . . . . . . . . . . . . . . . . . . . . 18
2.5 Quantum Dot Device Structure . . . . . . . . . . . . . . . . . . . . . 19
2.6 Depletion-Mode Quantum Dots . . . . . . . . . . . . . . . . . . . . . 21
2.7 Multilayer Gate Designs . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 Model of a Single Quantum Dot . . . . . . . . . . . . . . . . . . . . . 24
3.2 Chemical Potentials of a Quantum Dot . . . . . . . . . . . . . . . . . 26
3.3 Coulomb Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Coulomb Diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Charge Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Electron Temperature from Charge Sensing . . . . . . . . . . . . . . . 33
xii
3.7 Electron Temperature from Transport . . . . . . . . . . . . . . . . . . 33
3.8 Lever Arm from Temperature . . . . . . . . . . . . . . . . . . . . . . 35
3.9 Pulsed Gate Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 36
3.10 Magnetospectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.11 Single Electron Tunneling . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 Doulbe Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Finite Bias Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Double Quantum Dot Orbitals . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Tunnel Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Capacitively Coupled Double Quantum Dots . . . . . . . . . . . . . . 49
5.1 Elzerman Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Spin Readout Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Readout Window vs Magnetic Field . . . . . . . . . . . . . . . . . . . 53
5.4 T1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 Two Qubit Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6 Micromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.7 Two Qubit Control Sequence . . . . . . . . . . . . . . . . . . . . . . . 57
5.8 Adiabatic Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.9 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.10 Rabi Frequency vs Microwave Power . . . . . . . . . . . . . . . . . . 61
5.11 Ramsey Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.12 Hahn Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.13 Clifford Randomized Benchmarking . . . . . . . . . . . . . . . . . . . 64
5.14 Single Qubit Gate Fidelities . . . . . . . . . . . . . . . . . . . . . . . 65
5.15 Quantum State Tomography . . . . . . . . . . . . . . . . . . . . . . . 67
5.16 Symmetric Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
xiii
6.1 The Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Detuning Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Resonant AC Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Tilt vs Barrier Control . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5 Effect of Exchange on Energy Levels . . . . . . . . . . . . . . . . . . 77
6.6 Frequency Domain Exchange Spectroscopy . . . . . . . . . . . . . . . 78
6.7 Calibrating a CNOT . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.8 Conditional Phase Oscillations . . . . . . . . . . . . . . . . . . . . . . 80
6.9 Time Domain Exchange Spectroscopy . . . . . . . . . . . . . . . . . . 81
6.10 Raw Bell State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.11 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . 84
6.12 CPhase Generated Bell States . . . . . . . . . . . . . . . . . . . . . . 85
7.1 Large Scale Linear Array Architecture . . . . . . . . . . . . . . . . . 88
7.2 Charge Pumping Schematic . . . . . . . . . . . . . . . . . . . . . . . 90
7.3 Chemical Potential Space . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4 Charge Pumping Trajectory . . . . . . . . . . . . . . . . . . . . . . . 92
7.5 1e, 2e and 3e Charge Pumping . . . . . . . . . . . . . . . . . . . . . . 93
7.6 Long Range Spin-Cavity Coupling . . . . . . . . . . . . . . . . . . . . 94
xiv
Chapter 1
Introduction
“It’s not a Turing machine, but a machine of a different kind.”
-Richard Feynman
1.1 Quantum Computing
Quantum computers were first proposed as a way to simulate physics beyond what
is possible with modern, transistor-based computers [1]. The intuition behind this
idea came from the fact that many of the general phenomena in quantum field theory
have counterparts in condensed matter theory. For instance, a spin wave in a lattice
of spins behaves like a boson despite the fact that it arises from a system consisting
solely of fermions. A natural question to ask is whether or not it’s possible to engineer
a machine consisting of simple quantum systems, such as electron spins, that can then
be used to make predictions about the behavior of other physical systems that are
too complex to simulate classically. This is the basic concept of a quantum computer,
and the idea of building one with electron spins is the main subject of this thesis.
At the time this idea was first proposed very little was known about the theo-
retical power of a quantum computer. As with classical computer science the first
steps towards quantifying the computational power of a quantum computer were
1
made by giving a physical model of such a machine. In 1985 the idea of a Universal
Quantum Computer, the quantum analog of a Universal Turing Machine, was put
forth by David Deutsch [2]. In addition to providing a physical model for quantum
computation, Deutsch, in collaboration with Richard Jozsa, provided one of the first
examples of a problem where quantum computers appear to be able to solve a well-
defined mathematical problem exponentially faster than any corresponding algorithm
a classical computer [3]. Although the problem solved by their algorithm is somewhat
contrived, it is nonetheless a clearly stated mathematical problem.
While quantum algorithms and simulations provide some hint to the potential
utility of quantum machines, research into quantum computing has yielded important
technological advances that may have applications of their own outside the realm
of computation. For instance, research in superconducting qubits has led to the
development of electrical amplifiers whose performance is limited by the fundamental
laws of physics [4,5]. These amplifiers have led to the first non-demolition detection of
traveling single photons [6,7]. Other important developments include the development
of the first single-photon controlled transistor [8], high fidelity current standards for
metrology [9], the creation of novel entangled states involving thousands of particles
[10], and even advancements in fundamental tests of the laws of physics [11].
1.2 Experimental Quantum Bits (Qubits)
The basic building block of Deutsch’s Universal Quantum Computer is the qubit. A
qubit is any two-level quantum system which can be initialized into a well-defined
state, manipulated into any arbitrary superposition of the two basis states, and whose
final state can be accurately measured [12]. The mathematical description of a qubit
can be summarized as having a wavefunction described by
2
|ψ〉 = α|0〉+ β|1〉, (1.1)
subject to the standard normalization constraint,
|α|2 + |β|2 = 1, (1.2)
and whose measurement outcomes are given by the Copenhagen interpretation with
probabilities
P0 = |α|2,
and P1 = |β|2.(1.3)
Figure 1.1: Bloch sphere representation of a two level system. The cardinal directionscorrespond to the eigenstates of the Pauli x, y and z operators. An arbitrary superposition|ψ〉 is represented by a vector along the surface of the Bloch sphere.
While this is an accurate description of a two level system, it’s often move conve-
nient to represent the state of a qubit graphically as a vector in space. This can be
done by defining
α = cos(θ2
),
and β = eiφ sin(θ2
),
(1.4)
3
where θ and φ represent the polar and azimuthal angles of a vector restricted to lying
on a three dimensional sphere of unit radius. This sphere is known as the Bloch
sphere and the vector representing the state is known as the Bloch vector.
It is important to note that the x, y, and z coordinates of the Bloch vector do
not typically represent the Cartesian coordinates of a laboratory. For most of the
examples in this thesis, they will correspond the coordinates of a rotating reference
frame, but in general they have no correspondence with the three dimensional space we
live in. Instead they may represent the phase or amplitude of an electrical oscillator,
or different superpositions of two orbital states in an atom.
Figure 1.2: Modern Qubit Platforms: (a) Typical trapped ion setup. Figure from [13]. (b)Five qubit superconducting chip from the Martinis group [14]. (c) Nine quantum dot devicein Si/SiGe. (d) Atomic structure of nitrogen vacancy center in diamond. Figure from [15].Braiding scheme for topological qubits. Figure from [16].
Some of the leading platforms for the development of qubits are shown in Fig.
1.2. The two most advanced platforms are trapped ion qubits and superconducting
qubits, shown in Fig. 1.2 (a) and (b), where systems containing tens of qubits are
currently being studied. So far trapped ion qubits have boasted the highest single
4
qubit fidelities [17], and have a superior connectivity compared to superconducting
qubits [18]. However, industrial efforts in quantum computing have favored solid state
qubits such as superconducting qubits and quantum dot qubits (Fig. 1.2 (c)) due to
the ability to leverage standard fabrication techniques in their production. Nitrogen
vacancy centers in diamond (Fig. 1.2 (d)) have the rare ability to operate at room
temperature but pose major difficulties to efforts in scaling up due to their random
placement in the lattice [19, 20]. Another more speculative approach to building a
large scale quantum computer based on topologically protected quantum states has
also emerged as a new field of qubit research (Fig. 1.2 (e)). Topological qubits have
been suggested to have extremely long lifetimes and resilience to environmental noise,
however experimental efforts in this field are still in their early stages [21–23].
1.3 Introduction to Quantum Dots
Figure 1.3: Electronic density of states g(E) in lower dimensions. (a) The density of statesfor bulk objects displaying a
√E dependence on energy. (b) A quantum well or 2D density
of states is constant as a function of energy until an excited state is reached in the confineddimension. (c) Density of states in a quantum wire showing Van Hove singularities whenexcited states are reached in the confined dimensions. (d) Discrete density of states of aquantum dot.
5
The term quantum dot is a somewhat whimsical name for the most extreme ex-
ample of low-dimensional materials. The simplest understanding of a semiconductor
comes from treating the conduction electrons as free particles with a modified effective
mass confined within a three-dimensional (3D) box. Through solving Schrodinger’s
equation we can solve for the states accessible to the electrons and the energies of
those states. For a typical macroscopic piece of semiconductor the number of states
that may be occupied by electrons is so large that it doesn’t make sense to talk
about individual orbital states, but rather the number of states per unit energy or
the density of states. In 3D we can calculate the density of states as follows: The
wavefunctions allowed in a 3D box are
ψ(x, y, z) = A sin(kxx) sin(kyy) sin(kzz), (1.5)
subject to the constraint,
kx,y,z =nπ
L, n = 1, 2, 3... (1.6)
so that the wavefunction will be zero at the limits of the box (with sides length
L).
Since the energy of a particle is related to k by
E(k) =~2k2
2m∗, (1.7)
where ~ is the reduced Planck’s constant and m∗ is the effective mass of the particle,
we can find the number of states up to energy E by finding the number of states
within a sphere of radius k in k-space. The number of states within the sphere is
given by
N = 2× 1
8×(L
π
)3
× 4πk3
3, (1.8)
6
where the factor of 2 is for spin degeneracy. Now we can calculate the density of
states, normalized per unit volume, to be
g(E) =dN
dE=dN
dk
dk
dE=
√2
π2~3m∗3/2
√E. (1.9)
If we were to shrink one of the dimensions of the box, we would see the energies
associated with that degree of freedom increase, and at some point the energy splitting
associated with that degree of freedom would become so great that electrons would
no longer become excited along that axis of motion. At this point the material may
be treated as effectively a two-dimensional (2D) material, since the third degree of
freedom no longer plays a role in the dynamics of the electrons. Following a similar
analysis as above, the resulting 2D density of states is
g(E) =m∗
π~2(1.10)
2D materials can be achieved in a number of ways in the lab. For instance by
exfoliating a piece of graphite it is possible to isolate a single atomic layer of graphite
(graphene), or by interfacing two dissimilar semiconductors it is possible to reduce the
band gap of the structure via strain or differences in electron affinity in the vicinity
of the interface resulting in what is known as a quantum well [24,25]. Quantum wells
are used as the platform for constructing many quantum dot devices as is the case
for the devices presented in this thesis.
Continuing down the path of lower dimensions we could shrink the box along a
second dimension. The resulting structure, known as a quantum wire, would only
allow electrons to be excited along one motional degree of freedom. The simplest
experimental realization of a quantum wire is a semiconductor nanowire of sufficiently
small radius [26]. The resulting density of states in 1D is
7
g(E) =1
π~
√m∗
2E. (1.11)
Finally if all dimensions of our box have reached a small enough size scale, the
electrons trapped inside will no longer be able to move like free particles along any
direction. By extension, we have moved from a quantum wire down to a “quantum
dot” [27]. In 0D the states are now well separated from each other and the resulting
‘density of states’ for such a structure is a series of delta functions
g(E) = 2∑i
δ(E − Ei). (1.12)
The simplest realization of a quantum dot is semiconductor nanocrystal. These
nanocrystals have well defined emission spectra determined by their size, and have
found application in television displays as a result [28]. However, nanocrystals can
be difficult to work with, and in trying to make devices it is often more convenient
to form a quantum dot by starting with a quantum wire or quantum well.
While Earnshaw’s theorem tells us that it is not possible to confine a charged
particle with DC electric fields in 3 dimensions [29], it turns out that it is possible
in lower dimensions. First, let’s see a simple example making use of a semiconductor
nanowire. Figure 1.4 (a) shows schematically how two neighboring quantum dots
can be formed by electrostatically gating a quantum wire. The device consists of an
InAs nanowire suspended above 5 metallic electrodes, commonly referred to as gate
electrodes or simply gates. The wire is contacted by metal on either side to form
a source (S) and drain (D) for conduction band electrons. By placing alternating
positive and negative voltages on the gate electrodes, an electrostatic potential in
the shape of a double well is formed along the length of the wire. From both the
radial confinement of the wire combined with electrostatic potential from the gates,
electrons are confined to two small regions of the wire: two quantum dots. An SEM
8
Figure 1.4: Nanowire double quantum dot device. (a) Schematic illustration of the devicestructure. An InAs nanowire is suspended on two pedestals above a series of gate electrodes.Alternating positive and negative biases applied to the gates create a double well potentialdefining the double quantum dot. (b) Scanning electron micrograph of a real suspendednanowire quantum dot device. Figure from [30].
image of a real suspended nanowire device fabricated by Thomas Hartke in our lab
is shown in Fig. 1.4 (b) [30].
Although gated nanowires are a nice pedagogical example of a quantum dot de-
vice, nanowires, similar to nanocrystals, are also difficult to work with in real world
fabrication processes due to the difficulty of placing them in precise locations. The
most readily fabricated quantum dot devices are instead planar or lateral, gate-defined
quantum dots. These are formed by gating 2D materials rather than 1D materials. A
schematic illustration of a quantum dot device making use of a 2D interface is shown
in Fig. 1.5 (a) [31]. In this example a two-dimensional electron gas (2DEG) is formed
by a thin layer of AlGaAs grown on top of a GaAs wafer, with the electrons accumu-
lating on the GaAs side. The interface between the AlGaAs and GaAs is typically
less than 200 nm from the surface of the wafer.
On the surface of the wafer are again metallic gate electrodes, only this time with
a more complicated geometry than in the case of the nanowire device. The 2DEG
is insulated from surface electrodes except at specific locations on the wafer where
contact into the 2DEG has been deliberately made. These special contacts, called
ohmic contacts, play a very different role from the gate electrodes, allowing electrons
to flow into and out of the 2DEG through the surface of the wafer. Meanwhile the
9
Figure 1.5: (a) Schematic of a planar double quantum dot device. Surface electrodes definedepleted regions in the 2DEG below. Small puddles of charge formed by the depleted regiondefine the quantum dots. Scanning electron micrographs of single (a) and double quantumdot (b) gate geometries show the intended positions of the formed dots. Figure from [31].
gate electrodes are used to selectively deplete regions of the 2DEG by application of
a negative voltage. The gate geometries are chosen to selectively deplete the 2DEG
in the shape of a corral, trapping single electrons inside the corrals. Gate geometries
for a single quantum dot and double quantum dot (DQD) are shown in Fig. 1.5 (b)
and (c) respectively.
1.4 Electron Spins as Qubits
The electron spin has long been used as the quintessential example of a two level
system in introductory textbooks on quantum mechanics [32, 33], but the idea of
actually trapping and manipulating single electrons in the laboratory has only recently
become feasible. Early quantum dot devices proved that it was possible to trap small
puddles of electrons in solid state devices, and to control the addition of electrons one
at a time into the device [34–36]. Even before the fist observation of a one-electron
gate-defined quantum dot [37], it became clear that quantum dots could provide a
platform for manipulating single electrons in the solid state. In 1998 the first full
proposal of using electron spins in quantum dots to build a quantum computer was
put forth by David Loss and David DiVincenzo [38].
10
Figure 1.6: Figure 1 from the Loss-DiVincenzo proposal [38]. Qubit levels are defined bythe spin-up and spin-down states of a single electron in a magnetic field. Two qubit controlis achieved by modulating a tunnel barrier between two neighboring dots. Single qubitcontrol is achieved through selective coupling to a local ferromagnet, and spin readout isperformed using a spin valve and a sensitive electrometer.
The Loss-DiVincenzo proposal laid out in detail how quantum dots could be op-
erated as qubits. The essential features of their proposal are summarized by Fig. 1
of their 1998 paper, shown here in Fig. 1.6. The circular objects sketched out in
their figure are intended to represent the wavefunctions of single electrons confined
in lateral gate-defined quantum dots. State initialization is achieved by cooling the
electrons in the presence of a uniform magnetic field, while single qubit manipula-
tion is achieved by locally controlling the interaction of a single quantum dot with
a ferromagnet. Two qubit gates are executed by raising and lowering the tunnel
barrier between neighboring electrons via a gate voltage. By allowing tunneling be-
tween neighboring electrons they are subject to a Heisenberg exchange interaction
J(t) leading to swapping of angular momentum between the pair. Finally, the spin
state can be read out by allowing it to tunnel through a spin valve to a neighboring
reservoir. If the presence of the electron is detected on the reservoir using a sensitive
electrometer, then we know it was a spin up electron.
While their proposal gave a clear blueprint for building a quantum computer with
quantum dots, none of the basic elements of their proposal had been experimentally
11
demonstrated at the time of writing. In fact, it would be another seven years be-
fore the first observation of single spin dynamics in gate-defined quantum dots [39].
Nonetheless, the individual components of their proposal have now been demonstrated
by various groups [39–41], and in chapters 5 and 6 we will see how all the components
of the Loss-DiVincenzo architecture can be realized in a single solid-state quantum
dot device.
1.5 Thesis Overview
This thesis will demonstrate how electrons in quantum dots can be used to meet
the requirements of a Universal Quantum Computer. Chapter 2 will introduce the
physical structure of quantum dot devices, explaining both the geometric construc-
tion of a quantum dot device as well as materials considerations. We will briefly
review the evolution of quantum dot devices from a historical perspective and see
how improvements in device design have made possible the experiments presented
throughout the following chapters. Chapters 3 and 4 will cover the essential physics
of trapping single charges in quantum dots. Chapter 3 will introduce the basic the-
oretical model of a quantum dot, and show how measurements can be performed to
probe basic properties of a single quantum dot such as charging energy, orbital ener-
gies, and valley physics. We will also see how one quantum dot can be used to gain
information about another through a technique known as charge sensing, and use it
to explore the dynamics of single electron tunneling events. Chapter 4 will expand on
Chapter 3 by introducing coupling between quantum dots. We will see how the en-
ergy levels of one quantum dot can be effected by neighboring quantum dots through
tunnel barriers, capacitive coupling, and dipolar coupling. In Chapters 5 and 6 the
emphasis will move from charge to spin, and we will see how these two degrees of
freedom can interact. Chapter 5 will focus on single electron spins. We will first see
12
how electron spin states can be initialized and readout allowing us to probe one of
the most critical parameters of a qubit, its lifetime T2. From here we will see how to
control the evolution of the wavefunction to create arbitrary superposition states and
assess the fidelity of our control. In Chapter 6 we will explore how coupling between
quantum dots can be used to generate entangled states, and make use of correlations
measurements between two spins to quantify entanglement. Finally in Chapter 7 we
will outline some initial efforts towards building and controlling a large scale quan-
tum computer based on quantum dots, as well as some ideas for future generations
of quantum processors.
13
Chapter 2
Quantum Dot Device Physics
“...the foundations of transistor electronics were created by
making errors and following hunches that failed to give what
was expected.”
-William Shockley
2.1 Ultra-Coherent Quantum Materials
This chapter will outline the physical structure of quantum dot devices in greater de-
tail, layout the basic measurement setup for a quantum dot device, and explain some
of the considerations that go into choosing a semiconductor material and designing a
gate structure, and the starting point for our device structure is choosing a material
for the quantum well heterostructure. While the devices presented in this thesis are
made exclusively on Si/SiGe quantum wells it is instructive to compare them with the
material system which was utilized for most of the early quantum dot experiments-
the GaAs/AlGaAs quantum well.
The GaAs/AlGaAs quantum well has served as a platform for many ground-
breaking experiments in low dimensional physics, including the first observation of
the fractional quantum hall effect [42]. It has also been the primary material system
for the development of high electron mobility transistors [43] and high energy efficient
14
Isotopes of Ga, As, and SiIsotope Natural
Abundance (%)Nuclear Spin
69Ga 60.1 3/271Ga 39.9 3/275As 100 3/228Si 92.2 029Si 4.7 1/230Si 3.1 0
Table 2.1: All stable isotopes of Ga and As carry 3/2 nuclear spin, leading to dephasing inGaAs quantum dot devices. However, 95.3 % of natural Si has zero nuclear spin. Si canalso be isotopically purified to grow pure 28Si quantum wells to further enhance coherencetimes in Si-based quantum dots.
solar cells [44]. Due to the nearly perfect lattice matching of GaAs and AlAs, an un-
derstanding of GaAs/AlGaAs interfaces was developed much earlier than in Si/SiGe,
and high quality GaAs/AlGaAs 2DEGs are more readily available to experimentalists.
Due to the relative ease of growing high quality GaAs/AlGaAs heterostructures,
nearly all early quantum dots were fabricated on this material. While GaAs still of-
fers the best platform for studying charge physics in quantum dots, it unfortunately
cannot provide a clean environment for spins. The reason for this can be seen in Table
2.1. Unfortunately nature provides no stable isotopes of Ga or As with zero nuclear
spin. Therefore any electron residing in a GaAs quantum well will experience hyper-
fine interactions with ∼1 million nuclear spins as depicted schematically in Fig. 2.1
(a). Since these nuclear spins randomly fluctuate between consecutive measurements,
they create an effective magnetic field seen by the electron spin that is randomly
oriented with magnitude on the order of a mT. As a result the spin decoherence time
in GaAs is only ∼10 ns as seen in the data of Fig. 2.1 (b). By comparison ∼95 %
of naturally abundant Si has zero nuclear spin. The abundance of nuclear spin free
isotopes also makes it possible to grow isotopically purified Si quantum wells enabling
decoherence times as long as 28 ms [45].
15
Figure 2.1: (a) An electron spin in GaAs experiences hyperfine interactions with the ran-domly oriented nuclei in the substrate leading to an effective fluctuating magnetic field Bnuc.Figure from [46]. (b) This fluctuating hyperfine field leads to dephasing times of ∼10 ns.Data from [39].
2.2 Quantum Well Heterostructures
In our devices we confine the electrons in one direction by making use of a quantum
well heterostructure, and the other two through an electric potential. The quantum
wells we use are made from a heterostructure consisting of Si and SiGe. A typical
Si/SiGe quantum well structure is shown in Fig. 2.2. From the bottom up the
structure starts with a Si0.7Ge0.3 substrate (this is usually referred to as a ‘virtual
substrate’ because it’s typically grown on top of a Si wafer, but we need not go into
those details for the purpose of this thesis). Then a thin layer (∼10 nm) of pure Si is
deposited, before a spacer layer of more Si0.7Ge0.3. An additional few nanometers of
pure Si are grown on top to cap the wafer and prevent oxidation of the SiGe.
Even though Si has a larger band gap than SiGe, the 2DEG counterintuitively
forms in the Si layer of this structure. The reason for this surprising result comes
from lattice mismatch. The difference in lattice constants between Si and SiGe are
∼ 1%, but when the Si layer is sufficiently thin it will adhere to the lattice constant
of the SiGe on either side of it. The resulting strain on the Si layer reduces its band
gap creating a quantum well with a square potential profile as shown in Fig. 2.2. A
16
Figure 2.2: Typical Si/SiGe undoped quantum well structure. Strain in the Si layer pullsdown the conduction band of the Si below that of the SiGe in a square profile that dipsbelow the Fermi level EF set by voltages applied to the surface of the wafer.
Figure 2.3: Schematic and cross sectional transmission electron micrograph of a quantumwell heterstructure used by our group. The SiGe spacer layer is 34nm thick and the Siquantum well is ∼7 nm thick.
17
cross-sectional transmission electron microscope (TEM) image of a real quantum well
heterostructure used in our device fabrication is shown in Fig. 2.3.
2.3 Full Device Architecture
Figure 2.4: (a) Optical image of a quantum dot chip in a printed circuit board. The chipcontains four devices, one of which is wirebonded to the circuit board, ready to be measured.(b) Scanning electron micrograph of the microscopic structure of the device, showing thefanout of the metal electrodes from the active region of the device towards the macroscopicpads of metal used for wirebonding.
Figure 2.4 shows both the macroscopic and microscopic structure of a real quan-
tum dot device sitting in a printed circuit board ready to be measured. One of the
four devices on this 6x6 mm piece of Si/SiGe is connected to the circuit board for
testing via wirebonds. While the device shown in Fig. 2.4 (a) is ∼ 4 mm x 1 mm
in area, the region of the device which we call the quantum dot is typically less than
1 µm2. The main reason for the large footprint of the device is the need to connect
tens of macroscopic wires (diameter of 25 µm) between the device and the printed
circuit board which houses it during testing. For this reason the real device design
consists mostly of sprawling metallic electrodes that fan out from the microscopic,
18
Figure 2.5: Schematic cross section of a standard quantum dot device. Ohmic contactsare made to doped regions of the wafer to allow voltage control of the Fermi level EF inthe quantum well. Gate electrodes on the surface form the confinement potential of thequantum dot V (x), and are insulated from the doped regions of the wafer by Al2O3.
active region of the device to a large array of macroscopic pads used for wirebonding
as seen in Fig. 2.4 (b).
Rather than dissecting the 5 or 6 orders of magnitude of device structure, let’s look
at the main features of the device schematically. The basic structure of a quantum dot
device are shown in the schematic cross section of Fig. 2.5. Ohmic contact is made to
the 2DEG by heavily implanting selected regions of the wafer with phosphorous, and
then evaporating Ti/Au electrodes on top of the implanted regions. These contacts
allow us to set the Fermi level EF inside the quantum well, as well as to measure
current or conductance through the quantum dot. Gate electrodes (typically Al) are
then evaporated on the surface of the device to allow us to create a confinement
potential V (x) in the quantum dot below. Individual gates are dedicated to tuning
the density of electrons in the reservoirs of the dots, as well as for tuning the tunnel
barriers on either side of the dot, and the depth of the quantum dot confinement
19
potential. These electrodes are insulated from the wafer with Al2O3 in regions where
they overlap with the ohmic implants.
2.4 Gate Designs
The gate structure in the vicinity of the quantum dots is most critical in defining
overall device performance. Many of the early quantum dot devices utilized a single
layer of gate electrodes which formed a corral around the quantum dot, defining
the dot by its perimeter, as shown in Fig. 2.6 (a)-(c). The gates in these devices act
primarily to deplete electrons in selective regions of the 2DEG. These early depletion-
mode devices made it possible to see single electron charging effects with an unknown
number of electrons trapped on the dot, but were not capable of emptying the dot
completely of free electrons.
In an effort to reach the last electron transition in these devices gate dimensions
were eventually shrunk to near the limits of standard ebeam lithography techniques.
Although the shape of the electrodes changed slightly to accommodate fabrication
demands, the structures of these few electron quantum dots still maintained their
corral-like structure. Figure 2.6 (d)-(f) show standard few-electron single, double, and
triple quantum dot depletion-mode devices. While this single-layer, depletion-mode
gate structure has been the workhorse design for many of the first demonstrations of
single spin phenomena in quantum dots, it offers limited control over the electronic
wavefunction in the dot. The reason for this is that the design does not allow any
way of changing the depth of the quantum dot confinement potential without also
changing its shape. Because of this it is often not possible to reach a desired electron
number while also maintaining a desired tunnel coupling in devices like these [49].
In order to have increased control over the confinement potential our group has
moved to a multilayer gate structure [50–52]. Using multiple layers of gate metal sep-
20
Figure 2.6: (a)-(c) Early quantum dot devices had very obvious corral shapes, but werenormally not capable of reaching the last electron transition due their size. SEM imagesfrom [34–36]. Few electron quantum dot devices maintained the corral shape, but had muchsmaller dimensions. Designs from some of the first single (d), double (e), and triple (f) fewelectron quantum dot devices. SEM images from [37,47,48].
arated by only a few nanometers of insulating oxide, we’re able to create confinement
potentials that would not be possible in single layer devices. Our devices consist of
three layers of overlapping aluminum gates, electrically isolated from each other by
aluminum’s robust, native oxide. A typical gate geometry for making two neighboring
single quantum dots is shown in Fig. 2.7 (a)-(c) at each stage of fabrication. The first
gate layer consists of large screening gates which act as depletion gates. Since they
will screen electric fields from subsequent gate layers, they will only allow consecutive
gate layers to affect the 2DEG in the 1D transport channels defined by this layer.
We prevent electron accumulation outside of the transport channel by maintaining a
low voltage on the layer 1 gates. Layer 2 consists solely of accumulation gates. The
large paddles on either side of each transport channel accumulate large reservoirs of
electrons while the small pin shaped gates accumulate a small puddle of charge inside
21
of the transport channel forming a quantum dot. The final layer fills the gaps between
each dot and its reservoir, allowing us to tune the tunnel barriers between the dots
and their reservoirs. Extending to double or triple quantum dot devices is simple in
this geometry, only requiring additional gates to fall into the 1D transport channels
defined in the first gate layer as shown in Figs. 2.7 (d) and (e).
Figure 2.7: Multilayer gate designs. Typical geometries for each layer of a device are shownin (a)-(c). Aluminum’s native oxide is used to separate the three layers electrically. Double(d) and triple (e) quantum dot devices follow naturally from adding more gates to each 1dtransport channel.
22
Chapter 3
Single Quantum Dots
“An electron is no more (and no less) hypothetical than a star.
Nowadays we count electrons one by one in a Geiger counter,
as we count the stars one by one on a photographic plate.”
-Sir Arthur Eddington
3.1 Model of a Single Quantum Dot
While real quantum devices are rather complicated, many of the basic phenomena of
quantum dots can be captured by a simple capacitance model [31,53]. In this model
our device consists of a small conductor representing the dot with a fixed capacitance
to our gate electrodes. Tunnel barriers from the dot to neighboring source and drain
reservoirs are represented by capacitors in series with tunable resistors that allow
charge to move on and off the dot. A schematic representation of a real confinement
potential for a quantum dot is shown in Fig. 3.1 (a), and the corresponding capacitor
model is shown in Fig. 3.1 (b). An SEM image of a typical quantum dot device is
shown in Fig. 3.1 (c), and a cross sectional TEM image through the active region of
the device is shown in Fig. 3.1 (d) with the capacitor model overlayed on top.
With this simple model, we can easily write down the energy of the quantum dot
as it is simply the energy of a capacitor plus any energy associated with the quantum
23
Figure 3.1: (a) A real quantum dot consists of a confinement potential filled with electronsbeing controlled by nearby gate voltages. However, a simple capacitance based model can beconstructed as in (b), where the dot is represented by a capacitor with parallel capacitancesand resistances to the source and drain. A cross section of the device shown in (c) isdisplayed in (d) with the capacitance model overlayed.
confinement of the electrons on the dot [54]. For a quantum dot with N electrons the
energy can be expressed as
U(N) =Q2
2C+
N∑n=1
En(B), (3.1)
where C is the total capacitance of the dot, Q is the total charge on the dot, and
En(B) is the single-particle energy level occupied by the nth electron which depends
on the magnetic field B. We can relate the charge on the quantum dot Q to the
voltages and capacitance in the device through the definition of capacitance
24
Q = CS(VS − Vdot) + Cg(Vg − Vdot) + CD(VD − Vdot)
=⇒ Q+ CSVS + CgVg + CDVD = CVdot,(3.2)
where Vdot is the voltage of the quantum dot and the total capacitance is just the
sum of the capacitances to the dot C = CS + Cg + CD. Using this relation we can
express the energy of the dot in terms of gate voltages and electron number
U(N) =(−eN + CSVS + CgVg + CDVD)2
2C+
N∑n=1
En(B), (3.3)
where e is the charge of an electron.
From the energy of the quantum dot we can define a more useful quantity: the
electrochemical potential of the dot µ(N). This is the energy required to add the N th
electron to the dot, and is defined as
µ(N) = U(N)− U(N − 1)
= (N − 1
2)EC −
ECe
(CSVS + CgVg + CDVD) + EN ,(3.4)
where we define the charging energy EC = e2/C. Unlike the energy of the dot, the
chemical potential of each state in the dot shifts linearly with respect to gate voltage.
This allows us to label the states of the dot as a ladder of chemical potentials that
can be shifted up and down together by tuning gate voltages as depicted in Fig. 3.2.
3.2 Coulomb Blockade
The chemical potential picture for a quantum dot allows us to quickly see the mech-
anism behind one of the most basic quantum dot phenomenon: Coulomb blockade.
This phenomenon is illustrated schematically in Fig. 3.3. On either side of our quan-
tum dot are two reservoirs of electrons with nearly continuous spectra, filled to some
25
Figure 3.2: A quantum dot can be thought of as a ladder of chemical potentials correspond-ing to different charge states of the dot. The black walls on either side of the dot representtunnel barriers separating it from continuous reservoirs on either side with chemical poten-tials µL and µR.
Fermi level with a Fermi-Dirac distribution of occupied states. States well above the
Fermi level will have an occupation of zero, since there are no electrons available with
high enough energy to fill them. States well below the Fermi sea will become occupied
with electrons which will become ‘stuck’ on the dot, since all the states in the Fermi
sea at that energy are already occupied.
If there are no chemical potentials in the dot near the Fermi level then we see
that electrons can neither move on or off the dot, and therefore current cannot flow
through the dot. When the dot is in this state it is said to be in Coulomb blockade.
If however we were to change the voltages on the gate electrodes, then eventually one
of the chemical potentials in the dot would cross the Fermi level. At this point there
are both filled and empty states on either side of the dot, and so if we put a small
voltage across the dot we would see a corresponding current through the dot. This
resonant tunneling condition gives rise to abrupt peaks, known as Coulomb blockade
peaks, in the current or conductance through the dot as seen in Fig. 3.3 (b) and (c).
26
Figure 3.3: (a) Electron density in the quantum well for two neighboring single quantumdots. Electrons can only flow through each dot when the chemical potential of a state inthe dot is resonant with the Fermi level of the reservoirs as shown in (b). (c) The currentI through the dot therefore exhibits Coulomb blockade peaks as a function of gate voltageVg.
3.3 Coulomb Diamonds
The condition for current to flow through the dot not only depends on the gate
voltages, but also the chemical potentials of the reservoirs. The reservoirs of the dot
are connected to the ohmic contacts of the device, and therefore the chemical potential
at the Fermi level is simply the voltage on the corresponding ohmic multiplied by the
charge of an electron. So, if we have a grounded drain contact and apply 1 mV to
the source contact we have a source-drain chemical potential bias of 1 meV across
the dot as depicted in Fig. 3.4 (a).
Now the condition for current to flow is that a chemical potential in the dot has
to lie between the Fermi levels of the two reservoirs. When this condition is met,
27
Figure 3.4: (a) By applying a voltage VSD across the dot we create an offset in the chem-ical potentials on either side of the dot. (b) When the chemical potential of a state inthe dot is between that of the two reservoirs we see current through the dot, leading toCoulomb diamonds. The configuration of chemical potentials at various points in the di-amond and slopes of the transitions are shown in (c). (d) Coulomb diamonds also oftenexhibit additional resonance conditions when excited states fall into the bias window.
the state has filled states on one side and empty states on the other side allowing
electrons to move from the high energy reservoir to the lower energy reservoir. If we
plot the current through the dot as a function of both gate voltage and source-drain
bias, we see that current is forbidden in diamond shaped regions known as Coulomb
diamonds as shown in Fig. 3.4 (b).
The slopes of the current region boundary can now be understood in terms of our
capacitance model. If the drain is grounded (VD = 0) then the shift in the chemical
potential of the dot comes only from changes in Vg and VS. If the source is changed
by an amount VSD then the gate must be changed by a corresponding amount of
VSDCS/Cg to cancel its effect and keep the chemical potential of the dot aligned with
28
the drain. This results in the slope eCg/CS of the positive edge of the Coulomb
diamond seen in Fig. 3.4 (c). By a similar argument the negatively sloped edge of
the diamond must have slope eCg(C − CS). It is also worth pointing out that the
presence of excited states in the bias window lead to extra resonances with the same
slope seen inside the diamond as in Fig. 3.4 (d).
3.4 Lever Arm
If we’re using a well calibrated voltage source to apply a source-drain bias across the
dot, then we know chemical potentials of the reservoirs very accurately. Up until now
however, we have not known how the voltages applied to our gates relate to actual
energy shifts in the dot. Using our Coulomb diamond measurements we can now
quantify this relationship in a straightforward way. Since we know that the borders
of the Coulomb diamond correspond to when the chemical potential of the dot is lined
up with the each of the reservoirs, we can relate the voltage range on the gate where
we see current to an actual energy shift of eVSD of the chemical potential on the dot.
This ratio is usually given in units of meV/mVg, and is known as the ‘lever arm’ of
the gate, typically called α.
The lever arm is extremely important for measuring energies in quantum dots, as
we most commonly gain information about the dot by changing gate voltages rather
than source-drain bias. For example say we would like to know how much energy
it takes to add an electron to a dot. We simply multiply the voltage between two
consecutive Coulomb blockade peaks by the lever arm. This quantity, known as the
addition energy contains important information about both the charging energy and
the single-particle states in our quantum dot as seen in the equation
Eadd(N) = µ(N)− µ(N − 1) = EC + ∆E, (3.5)
29
where ∆E is the difference in the energy of the single-particle states occupied by
consecutive electrons. While the Coulomb diamond offers perhaps the most intuitive
measurement of the lever arm, there are many methods to extract the lever, arm some
of which are more convenient than others depending on the device configuration. We
will outline a few more methods in the following sections.
3.5 Charge Sensing
For many applications of quantum dots to quantum information we need to need to
know exactly how many electrons are on the dot. So how can we tell if our dot has
just one electron or hundreds? Fortunately we can deduce this information by making
use of a second quantum dot through a technique known as charge sensing.
Figure 3.5: Charge sensing in the DQD device shown in (a). We tune the DQD to haveone electron in each transport channel as shown in (b). Then we plot the derivative of thecharge sensing quantum dot’s conductance dgs/dVL as a function of the gate voltages in thedot being sensed. The resulting plot shown in (c) is the charge stability diagram of the dot.The black lines denote boundaries between stable charge configurations.
As shown in section 3.2, the current through a quantum dot is extremely sensitive
to gate voltage. This is simply a statement that the chemical potential of the dot
is very sensitive to local electric fields, and therefore we can use the current through
one quantum dot to sense changes in the electric field coming from another nearby
quantum dot. Figure 3.5 shows how this technique works. We will refer to the
30
quantum dot through which we measure current as the charge sensor or charge sensing
quantum dot from here on. First we tune the chemical potential of the sensor dot so
that an electron transition is near resonant with its source drain reservoir and measure
the current or conductance through it. While we are monitoring the conductance
through the sensor quantum dot, we tune the gate voltages of a neighboring quantum
dot and look for changes in the measured conductance. The resulting plot is known
as a charge stability diagram of the dot being sensed. A typical single dot stability
diagram is shown in Fig. 3.5 (c). The absence of charge transitions at low voltages in
Fig. 3.5 (c) indicates that we have reached the last electron transition in the device.
3.6 Charging Energy
Now that we have a way of measuring the absolute electron number and the lever
arm we can measure one of the most basic parameters in our model for the quantum
dot: the self-capacitance of the dot. We know from equation 3.5 that the addition
energy of the dot is the charging energy plus any extra energy associated with the
single-particle levels occupied by our electrons. We also know that at zero magnetic
field the lowest two single particle states will be degenerate due to the spin degree of
freedom of the electron. Therefore the first addition energy of the dot will simply be
EC . Using our charge sensing data we can multiply the first addition voltage in our
quantum dot by the lever arm to arrive at the charging energy.
For the data shown in Fig. 3.5 (c) we extract a charging energy of 6 meV, which
is typical for the quantum dots presented in this thesis. From the charging energy we
can determine that the total capacitance of our quantum dot is C = EC/e2 = 24.3
aF. From this capacitance we can also estimate the size of the electron wavefunction.
Since we know that the wavefunction of the dot will be approximately shaped like a
disk, we can use the classical formula for a disk to estimate its radius r = C/8εrε0 ≈ 29
31
nm, where εr = 11.7 for Si and ε0 = 8.85 × 10−11 F/m is the electric permittivity of
free space. This number is consistent with the expected radius of ∼30 nm from the
COMSOL simulation shown in Fig. 3.5 (b).
3.7 Electron Temperature
Another extremely important property of our devices that can be nicely measured
using charge sensing is the actual temperature of the electrons in the 2DEG Te. While
the temperature of the mixing chamber of the dilution refrigerator can reach as low
as TMC ∼10 mK, the electrons in the device are often much hotter due to their
direct electrical connection to higher temperature stages in the fridge and incomplete
thermalization with the lattice. In order to measure the actual electron temperature
in the device we can make use of the charge sensor to measure the occupation of a
single particle state as a function of energy. Since the charge sensor signal is directly
proportional to the charge occupation of the dot we simply measure the charge sensor
current near an electron transition in the dot as shown in Fig. 3.6. Since the state
is occupied by a fermion, we can fit the resulting curve to a Fermi-Dirac distribution
using our lever arm α to convert gate voltage Vg into an energy
f(Vg) =1
e(αVg−EF )/kTe + 1(3.6)
where EF is the Fermi level of the reservoir and k is Bolzmann’s constant.
We could have extracted this information directly from transport through the
quantum dot as well, although its much harder to see how, and in practice this
measurement seems to be more susceptible to error. Transport through the quantum
dot is of course also related to the Fermi-Dirac distributions of the reservoirs, but
deriving the relation between current through the dot and temperature is a subtle
theoretical problem. For details I point the reader to the discussion by Beenaker
32
Figure 3.6: Charge sensor current I plotted as a function of gate voltage Vg near a chargetransition. The step in current is fit to a Fermi-Dirac distribution yielding an electrontemperature of Te = 85 mK.
Figure 3.7: Conductance g through a quantum dot at the N=0 to 1 charge transition. Thedata are fit using equation 3.7 to extract an electron temperature of Te =40 mK.
[55]. In practice using the conductance through the dot to measure the electron
temperature can be done as follows.
33
First, we must ensure that the conductance through either barrier of the quantum
dot is significantly less than a conductance quantum gl,r e2/h. Actually this
requirement also applies to our charge sensing technique, but in practice this is much
easier to achieve in the charge sensing scenario since the charge sensor signal persists
even at tunneling rates that are far too slow to produce a measurable current through
the dot (even a tunneling rate of 1 GHz only produces ∼160 pA of current). The
reason for this condition on the tunneling rates can be understood from a simple
argument based on classical circuits and the time-energy uncertainty relation. If we
represent the dot as a capacitor C connected to the reservoir by a resistance R, then
we know that the charging time of the dot is τ ∼ RC. If use this time as the lifetime
of a state on the dot, then the minimum uncertainty in the energy of the state must
satisfy τ = RC > h/∆E. In order to distinguish consecutive charge transitions
which are separated by the charging energy, we need for R > h/e2, as implied by our
condition gl,r e2/h.
Next we measure the conductance over a blockade peak with zero source-drain
bias, ensuring that the signal we use to measure the conductance is smaller in ampli-
tude than the expected electron temperature (i.e. for an electron temperature of 100
mK, our voltage signal should not exceed V ∼ kTe/e = 8.6µV ). Finally we fit the
conductance through the dot to
g ∝ cosh−2(αe(V − V0)
2kTe
), (3.7)
if the spacing of single-particle energy levels is much larger than the temperature
∆E kTe or using the equation
g ∝ cosh−2(αe(V − V0)
2.5kTe
), (3.8)
34
if the spacing of single-particle energy levels is much smaller than the temperature
∆E kTe, and transport occurs through many single-particle states. In both equa-
tions α is the lever arm, k is Boltzmann’s constant, and V0 is the voltage where the
blockade peak occurs. Transport data from the N=0 to 1 charge transition of a quan-
tum dot are shown in Fig. 3.7, fit to equation 3.7. Since the width of this coulomb
blockade peak is proportional to temperature, we can also extract the lever arm if we
know what the temperature is. This can be done by intentionally heating the mixing
chamber and plotting the full width half maximum of the blockade peak vs TMC as
is done in Fig. 3.8. The resulting slope is a reflection of the conversion between gate
voltage and energy.
Figure 3.8: The full width half maximum and electron temperature Te of a Coulomb block-ade peak is plotted as a function of the temperature of the mixing chamber plate of thedilution refrigerator TMC. By fitting the slope of the data we extract a lever arm of α =0.13.
35
3.8 Orbital Energy
In this section we will see how to measure the orbital exited states of the dot that
result from its quantum confinement. The technique used here was first demonstrated
by Elzerman et al. [56]. By applying a square wave to the plunger gate of the dot we
can modulate the chemical potential of the dot between two levels. When these two
levels are on either side of the Fermi level of the reservoir our charge sensor will tell
us the average occupation of the dot. For small amplitude square waves, we will only
be loading an electron into and out of the ground state orbital as shown in Fig. 3.9
(a). However as we increase the amplitude of the square wave, we eventually pulse
the chemical potential so far below the Fermi sea that we can now load into either
the ground state or the first excited state thereby changing the loading rate of the
dot. When this happens we see a change in the average response of the charge sensor
as shown in Fig. 3.9 (b).
Figure 3.9: By modulating the gate voltage with a square wave, we force the chemicalpotential to shift between high and low levels. (a) When the square wave amplitude issmall, loading and unloading of the dot occurs only through the ground state. (b) When theamplitude is larger we eventually begin loading through both the ground state and excitedstate, increasing the effective loading rate and changing our time averaged occupation ofthe dot. (c) Charge sensor response vs pulse amplitude VP shows a clear second transitionabove the ground state at voltage VOrb indicating the energy of the first orbital excitedstate.
36
From the data in Fig. 3.9 (c) we extract an orbital excited state energy of ∼2.4
meV using our lever arm of 0.11 eV/mVg. We can do a back of the envelope calculation
to check this value with our expected size of the dot. From our capacitance value
we estimate that the wavefunction of the dot is a disk with a radius of 29 nm. For
a 2D square box of the same area (πr2 = L2) we would expect an orbital energy
of Eorb = 3~2π2/2m∗L2 = 2.2 meV where ~ is the reduced Planck’s constant and
m∗ = 0.19me is the effective mass and me is the free electron mass. This rough
calculation agrees well with our measured value.
3.9 Magnetospectroscopy
In silicon there is an additional degree of freedom associated with the crystal structure,
known as the valley degree of freedom. Unlike GaAs, Si is an indirect band gap
semiconductor, with a sixfold degenerate conduction band minimum [57]. These
minima are known as valleys. In a strained Si quantum well the four valleys associated
with the in plane directions are split off from the out of plane valleys, with the out
of plane valleys forming the ground state. The remaining splitting of these two out
of plane valleys is a very poorly understood quantity that is expected to depend on
the atomistic details of the quantum well as well as the electric field at the quantum
well [58, 59]. In practice the splitting of the out of plane valleys can vary wildly in
quantum dot devices [60], but is generally less than ∼ 100 µeV in Si/SiGe devices.
This small energy scale makes it difficult to see valley splitting in the pulsed gate
spectroscopy methods of the previous section.
A more sensitive method for discerning the low lying energy levels of a quantum
dot is magnetospectroscopy. This technique was first demonstrated in GaAs devices
[61, 62] to observe spin filling in quantum dots, but has also been used to measure
valley splittings in Si quantum dots [63, 64]. The technique works by measuring the
37
Figure 3.10: Magnetospectroscopy data of the first 4 transitions in a quantum dot. Theshifts of the charge transitions with applied magnetic field can be interpreted as a reflectionof adding the Nth electron to the energy level diagrams shown beneath the data. From thelocation of the kinks in the charge transitions we extract a valley splitting of ∆v = 60µeV.
spin configuration of the first few electrons loaded into the quantum dot. According
to Eqn. 3.5, the energy, and therefore the gate voltage, at which a new electron
can be added to the device depends both on the capacitance of the dot and on the
energies of the single particle states. Since we know that spin down electrons shift
down in energy with magnetic field, we also expect the voltage of the first electron
transition to shift down with magnetic field as shown in Fig. 3.10 (a). In the absence
of any low lying excited states we would expect the voltage of the second electron
transition to increase with applied magnetic field B. However, as seen in Fig. 3.10
(b) the transition only increases up to ∼ 0.5 T, after which it turns downward. This
indicates that the Zeeman splitting of the ground state has exceeded the first excited
state of the dot as shown in the energy level diagrams in Fig. 3.10. The third electron
again shows this energy level crossing (Fig. 3.10 (c)), but the fourth electron is forced
to occupy a spin-up state at all magnetic fields (Fig. 3.10 (d)). From the magnetic
38
field value where the second and third electron transitions change direction we can
extract the valley splitting of the dot ∆v = 60 µeV.
3.10 Measurements of Real-Time Tunneling Events
Up until now, we have only looked at the time averaged occupation of the quantum
dot. Now we will turn to the actual dynamics of the quantum dot occupation and
see their relation to the Fermi-Dirac distribution of statistical mechanics. Figure 3.11
(a) shows the time domain response of the charge sensor over a small voltage range
near a quantum dot charge transition. Several time traces of the current are plotted
at different chemical potentials in Fig. 3.11 (b).
As we can see, the current through the charge sensor actually fluctuates between
two discrete values when the dot is near the Fermi level. In earlier measurements we
simply assumed that the current signal from the charge sensor was proportional to the
charge occupation on the dot, but now that we can resolve single electron tunneling
events we can assign absolute probabilities to the dots occupation. Using a simple
threshold current to distinguish the two charge states, we plot the directly measured
charge occupation as a function of gate voltage in Fig. 3.11 (c). The data are fit to
Fermi-Dirac distribution yielding an electron temperature of Te = 120 mK. As we will
see in the following sections, this ability to resolve single electron tunneling events on
timescales shorter than the spin relaxation time of the electron will form the basis of
qubit readout.
39
Figure 3.11: (a) Charge sensor current as a function of time and gate voltage near the 0 to1 charge transition. (b) Time traces at the voltages marked by white dashed lines in (a).(c) Dot occupation 〈N〉 as a function of gate voltage, fit to a Fermi-Dirac distribution.
40
Chapter 4
Coupling Quantum Dots
“Fundamentally, improvements in control are really
improvements in communicating information within an
organization or mechanism.”
-John Von Neumann
4.1 Constant Interaction Model
We now generalize our capacitance model to include an arbitrary number of gates
and dots following the discussion from [54]. Consider a network of N separate voltage
nodes, with capacitive coupling between nodes. The charge on the jth node can be
written in terms of the charges on all the capacitors connecting it to the other nodes
in the system
Qj =N∑k=1
qjk =N∑k=1
cjk(Vj − Vk), (4.1)
where Vj is the voltage on the jth node. We can express this relation in matrix form
as
~Q = C~V , (4.2)
41
if we define the diagonal elements of the capacitance matrix C to be
Cjj =N∑k 6=j
cjk, (4.3)
and the off-diagonal elements to simply be the capacitances from equation 4.1 with
a minus sign
Cjk = Ckj = −cjk. (4.4)
This simple matrix equation allows us to succinctly write down the energy of the
network:
U =1
2~V · C~V (4.5)
However, in order to use this equation we need to know the voltage on every node
in the network as well as the full capacitance matrix. In general we can only measure
the capacitances of the dots and not their voltages. Likewise, we have control over the
voltage on the gate electrodes, but generally don’t know their capacitances. In order
to make equation 4.5 more useful we need to separate the gate electrode nodes from
the dot nodes. By dividing the nodes based on whether they are “voltage nodes” or
“charge nodes”, we can rewrite equation 4.2 in terms of a block matrix
~Qc
~Qv
=
Ccc Ccv
Cvc Cvv
~Vc~Vv
, (4.6)
where ~Qc and ~Vc are the charges and voltages on the charge nodes and ~Qv and ~Vv are
the charges and voltage on the voltage nodes. We can now solve this equation for the
voltages on the charge nodes
42
~Vc = C−1cc
(~Qc − Ccv~Vv
). (4.7)
The resulting equation allows us to determine the voltage on the dots only requir-
ing knowledge of the charge on the dots, the capacitances between the dots, and the
capacitances between the dots and the gates, all of which can be readily measured.
We will discuss in more detail how to measure the capacitance matrix of an array
of quantum dots in section 7.2, and use it to design control sequences for up to 9
quantum dots.
4.2 Double Quantum Dot Charge States
Now that we have seen the abstract representation of N quantum dots we will see how
to how to experimentally characterize multiple dots starting with the simplest case of
just two dots: the double dot. If there were no capacitive coupling between the two
dots, then the charge on each quantum dot would simply be determined by its plunger
gate voltage as in the stability diagram of Fig. 4.1 (a). As the capacitive interaction
between dots is increased the the square charge regions in the charge stability diagram
become honeycomb shaped as in Fig. 4.1 (b). The presence of an electron on one dot
shifts the charge transitions of the second dot, giving rise to new lines in the stability
diagram that represent interdot charge transitions. If the coupling between the dots
dominates the total capacitance of each dot, then the dots have effectively merged
into a single quantum dot as shown in Fig. 4.1 (c).
An important condition is met at the edges of the interdot charge transitions.
These points are referred to as triple points, because three three charge states are
degenerate at these points. Due to the degeneracy of charge states, current can pass
through the double dot only at these triple points. Analogously the Coulomb blockade
43
Figure 4.1: DQD stability diagrams shown for varying mutual capacitances. (a) Completelyuncoupled dots result in a simple checkerboard pattern of stable charge configurations. (b)For moderately coupled quantum dots we see a honeycomb lattice of stable charge config-urations. The charge transitions with positive slope represent interdot charge transitions.(c) When the coupling between dots becomes exceedingly large, they merge to form a sin-gle dot. (d) The edges of the interdot charge transitions form ’triple points’ where threedifferent charge states are degenerate in energy and electrons can pass through the device.
peaks of a single dot could be thought of as double points, where the N and N+1
charge states are degenerate.
44
4.3 Finite Bias
As in the case of a single dot, if we now add a source-drain bias then current flow is
possible over a wider range of gate voltages. After the electron hops onto the first dot,
it needs to move to the second dot which is generally at a different chemical potential.
Therefore the electron generally passes through the double dot via inelastic tunneling
processes. Because of the very low temperature, the electron cannot generally move
from a lower energy dot to a higher energy dot, but it can easily reach lower energy
states by emitting a phonon into the lattice. The requirement that the second dot
have a lower potential than the first gives rise to triangular regions of current in gate
voltage space known as finite bias triangles.
Figure 4.2: (a) Schematic illustration of finite bias triangles in a DQD. Chemical potentialconfigurations of the DQD are shown schematically for the vertices of the triangles. (b)Current plotted as a function of left and right gate voltage for a typical DQD device witha source-drain bias of 500 µeV applied across the dots.
Finite bias triangles and the corresponding alignment of chemical potential at
various points in the triangles are shown in Fig. 4.2 (a). Experimentally measured
finite bias triangles are plotted in Fig. 4.2 (b).
45
4.4 Interdot Tunnel Coupling
So far we have seen how classical electrostatic interactions between the dots alters
their energy levels. Now we will see the impact of quantum mechanics on the orbital
energies of coupled quantum dots. As the potential barrier between dots is decreased
we not only increase the capacitance between dots, but the orbital states of the two
dots begin to overlap. Although the proper quantum mechanical treatment of this
problem is to solve Schrodinger’s equation in the double well potential created by our
electrodes, the main features of this problem including level repulsion, and bonding
and antibonding states can be captured by a simple matrix model for the charge
states.
Figure 4.3: Energy level diagram of a coupled two level system. The red dashed linesrepresent the energy levels if no coupling was present. The gap of the avoided crossing istwice the tunnel coupling tc from equation 5.3.
Moving along an axis in voltage space that is perpendicular to the interdot charge
transition, the energies of the two charge states move in opposite directions. In the
basis of the left and right charge states |L〉, |R〉 the Hamiltonian of these two charges
states with no coupling is simply
46
H =
ε/2 0
0 −ε/2
, (4.8)
where the detuning ε is the difference between the energy of the |L〉 and |R〉 states.
The tunnel barrier between the two orbitals can be represented by an off-diagonal
element called the tunnel coupling tc
H =
ε/2 tc
tc −ε/2
. (4.9)
The resulting energy level diagram from this hamiltonian is plotted in Fig. 4.3. At
zero detuning the states are fully hybridized superpositions of the left and right dot
occupation with an energy splitting of 2tc.
Figure 4.4: (a) Double quantum dot charge stability diagram showing the detuning axisε in voltage space. (b) Charge sensor signal near zero detuning for varying values of themiddle barrier voltage VMB1 , and the corresponding extracted tunnel coupling.
This hybridization can be seen directly in the charge sensor response near an
interdot charge transition. In Fig. 4.4 (b) we plot the charge sensor response along
the detuning axis defined in Fig. 4.4 (a) for varying tunnel coupling. As the tunnel
coupling is increased the charge sensor response changes more gradually between that
of the (1,0) region to that of the (0,1) region due to the larger hybridization of the
47
two charge states. The tunnel couplings are extracted by fitting to the equation of
Dicarlo et al. [65]
P(0,1) =1
2
[1 +
ε
Ωtanh
(Ω
2kTe
)], (4.10)
where Te ∼ 40 mK is the electron temperature and Ω =√ε2 + 4t2c is the energy
difference of the hybridized charge states.
4.5 Dipole Coupling
We have now seen how direct capacitive coupling between neighboring electrons can
shift each others chemical potentials and how tunnel coupling can shift the energies
of neighboring electrons. In this section we will see how two neighboring double
quantum dots can affect each other through dipole coupling. This coupling has been
used successfully in GaAs singlet-triplet qubits to demonstrate high fidelity two-qubit
gates [66, 67]. The stability diagrams of two neighboring double quantum dots are
shown in Fig. 4.5 (a) and (b). We operate each double quantum dot with one
electron. In order to see the dipole interaction between the two DQDs we sweep each
of them across their (1,0)-(0,1) interdot charge transitions. The resulting stability
diagram is shown in Fig. 4.5 (c), where the charge configuration in each region is
shown schematically. The change in charge configuration of the right DQD produces
a shift in the location of the interdot charge transition of the left DQD by an amount
∆εL = 0.77 mV. Using the lever arm this corresponds to an energy shift of 200 µeV
or a 50 GHz two-qubit operation speed if each DQD were operated as a qubit.
48
Figure 4.5: Charge stability diagrams of two neighboring DQDs are plotted in (a),(b). (c)The capacitive interaction between the two DQDs is extracted by measuring the quadruple-dot stability diagram as a function of εL and εR. The interdot charge transition of the leftDQD shifts by an amount ∆εL = 0.77 mV when εR is swept across the (0,1) to (1,0) interdotcharge transition.
49
Chapter 5
Single Electron Spin Qubits
“I think that a particle must have a separate reality
independent of the measurements. That is an electron has spin,
location and so forth even when it is not being measured. I like
to think that the moon is there even if I am not looking at it.”
-Albert Einstein
5.1 Initialization and Readout
In this chapter we begin to see how electrons in quantum dots can be utilized as
spin qubits for quantum computing applications, and the first step towards full qubit
control being able to distinguish the two spin states of the dot. To do this we make
use of a technique that’s commonly referred to as Elzerman readout, which was first
demonstrated in 2004 in a GaAs quantum dot [40]. The steps of the experiment are
illustrated in Fig. 5.1. First we pulse the chemical potential of the dot well below
the Fermi level as shown in Fig. 5.1 (a). This causes electrons from the reservoir to
randomly load into either spin state on the dot, which we have now split by applying
an external magnetic field to the sample. Next we adjust the chemical potential of the
dot such that the two spin states straddle the Fermi level of the reservoir as shown in
Fig. 5.1 (b). In this arrangement spin up electrons will be able to tunnel off the dot
50
Figure 5.1: Spin readout. (a) We randomly load a spin state by pulsing the chemicalpotential of the dot far below the Fermi level. (b) We then align the spin states on eitherside of the Fermi sea, so that only spin up electrons can tunnel out. (c) With a spin downstate remaining on the dot, we pulse the chemical potential high to unload the electronbefore repeating the experiment.
leaving it open to be replaced with a spin down state while spin down electrons will
remain stuck on the dot without enough energy to tunnel out. Finally, we pulse the
chemical potentials of both levels well above the Fermi level to empty the dot and
restart the experiment as shown in Fig. 5.1 (c).
If the charge sensor were only coupled to electric fields from the dot itself, the
signal from the charge sensor during the experiment would look like the upper plot in
Fig. 5.2 (a), only changing value when electrons moved on or off the dot. However,
real charge sensors often have direct capacitive coupling to many gates in the device.
Because of this the real signal from the charge sensor will also shift during each step
of the experiment due to the voltage pulses applied to the dot whose spin is being
measured, as seen in the lower plot of Fig. 5.2 (a). If we average the signal from
the charge sensor over many cycles of the experiment we see the signal shown in the
red trace of Fig. 5.2 (b). The small bump at the beginning of the read step is the
signature of spin up electrons on our charge sensor.
51
Figure 5.2: (a) If our device had no cross-capacitances our charge sensor signal would resem-ble the upper plot during our readout experiment, only changing when the dot occupationchanges. However, the real signal from the device also exhibits offsets at each stage of theexperiment as shown in the lower plot. (b) Time averaged current during the experimentwith (blue) and without (red) spin-down initialization. (c) Single shot traces during theread phase of the experiment clearly showing single electron hopping events from spin upelectrons.
The Elzerman method for readout the electron spin also suggests a way of initial-
izing a spin state. Notice that at the end of read phase the dot is holding a spin down
electron regardless of which spin state was on it before the start of the read phase.
Therefore we can initialize an empty dot by going to the same chemical potential
configuration used for readout and waiting longer than the characteristic tunneling
time of the barrier. This initialization was used in a four step measurement (we added
an initialization step to the beginning of the experiment) to produce the blue curve in
Fig. 5.2 (b). The absence of the spin bump indicates the success of our initialization
step.
If the signal-to-noise ratio of our measurement is high enough we can even readout
the spin state in single-shot measurements. A few example single-shot measurements
are shown in Fig. 5.2 (c). Similar to our measurements from section 3.10 we can use
a simple threshold value to distinguish spin up signals from spin down, allowing us
to assign absolute probabilities to our spin measurements.
52
Figure 5.3: (a) Probability of hopping during the read phase of our experiment as a functionof read level for varying magnetic fields. (b) Extracted read window width as a function ofmagnetic field whose slope is given by the lever arm.
Our spin measurements only work in a very small range of chemical potentials.
If we set the chemical potential of the dot too high spin down electrons are likely to
hop out due to thermal excitation yielding a false spin up reading. However, if we
put the chemical potential too low spin up electrons will not have empty states in
the reservoir to hop into leading to false spin down readings. We can easily see the
working range of the measurement by executing our simple three step measurement
for varying read levels. The resulting probability of seeing a tunneling event is shown
in Fig. 5.3 (a) for different values of magnetic field. The width of this readout window
as a function of magnetic field is plotted in Fig. 5.3 (b) and gives us yet another way
to measure the lever arm, since we know how the external magnetic field affects the
Zeeman splitting of the spin states.
5.2 Spin Lifetime (T1) Measurements
We are now poised to make the first characterization measurement of our qubit.
By holding the electron for a varying time and then reading out the spin state, we
53
can measure the relaxation time of the spin state T1. Decay curves of the spin up
probability are shown for a few different magnetic field values in Figs. 5.4 (a-c).
By fitting the decay curve at each value of field, we can see the T1 dependence on
magnetic field B which is plotted in Fig. 5.4 (d). We observe an approximately power
law dependence of the relaxation rate on magnetic field Γ = 1/T1 ∝ B4.7.
Figure 5.4: T1 decay curves taken at B = 2 T (a), 1.5 T (b) and 1 T (c) showing thatT1 gets longer at lower magnetic field. (d) The spin relaxation rate 1/T1 as a function ofmagnetic field, showing a ∼ B4.7 dependence at high fields.
54
5.3 Micromagnets
In order to manipulate the spin state, we need a way of manipulating the magnetic
field seen by the dot. Currently there are two predominant ways of achieving this.
The first is to directly apply an oscillating magnetic field to the spin by running an AC
current through a nearby stripline. However, striplines only offer modest magnetic
field strengths (typical Rabi frequencies less than 1 MHz) [45], and provide no way
of selectively addressing a single spin in a large array. The second method we adopt
in the following experiments is to place a small ferromagnet (micromagnet) near the
electrons, and then to oscillate the position of the electron in the fringing field of
the magnet using AC electric fields so that the electron experiences an effective AC
magnetic field [68–71]. By cleverly choosing the geometry of the magnet, we can also
design it so that each electron has a different total Zeeman splitting allowing us to
address the spins separately.
Figure 5.5: Two qubit device consisting of a double quantum dot and a single dot chargesensor. (a) False-color SEM image of the device showing the primary control gates L, M, R,and S. (b) Schematic cross section of the device showing where the electrons are trapped.
Using micromagnets to drive spin transitions has been studied extensively by the
group of Siego Tarucha, and we will base our magnet geometry off of their simulation
55
Figure 5.6: (a) Geometry of the Co micromagnet layer. (b) Schematic cross section of thedevice. The fringing field of the micromagnet produces a strong dBy/dz gradient for spindriving.
studies [72]. The geometry of our micromagnet is shown in Fig. 5.6. We fabricate the
micromagnet using cobalt, a strong ferromagnet with a high saturation magnetization
of ∼1.8 T that is easy to lift off. In order to understand the driving mechanism for
our spins, let’s consider the fringing field of the micromagnet when viewing a cross
section of the device as is shown in Fig. 5.6. Starting from the piece of Co on the
left of the image the field of the micromagnet arches down into the the plane of the
quantum well and then back up into the Co on the right side of the image. Therefore
if we can move the electron along the z-direction, the component of the fringing field
which changes most drastically is the out of plane field. We use the S-gate to push
and pull the electron in the z-direction by changing the voltage applied to it creating
a changing magnetic field through the gradient dBy/dz.
56
5.4 Electron Spin Resonance
The double quantum dot stability diagram of the device is shown in Fig. 5.7 (a).
Overlayed on top of the device is the path through voltage space that is traversed
through the course of each single-shot experiment. The basic step being executed at
each point in voltage space is also shown schematically in Fig. 5.7 (c). In order to
readout the spins separately, we read them out sequentially. A typical readout signal
is shown in Fig. 5.7 (b).
Figure 5.7: (a) Charge stability diagram of the DQD with the voltage trajectory overlayedon the data. (b) A typical readout trace showing two spin-up detections. (c) We start inthe (0,0) charge region with both dots empty. During steps B and C we initialize both dotswith spin down electrons. In the middle of the (1,1) region we manipulate the spins (D),before returning to the points marked E and G to readout the spin states.
The magnetic field changes generated by shifting the position of the electron are
still far too small to change the quantization axis of the spin significantly. Therefore
a simple DC change in the S-gate voltage will not produce a spin rotation. Instead,
we make use of the magnetic resonance techniques used in standard electron-spin
57
Figure 5.8: (a) By applying a 40 MHz frequency chirp we achieve adiabatic inversion of thespin, widening the effective linewidth to ∼ 2 mT. (b) Using a fixed frequency drive we cansee that the true linewidth is 1 mT.
resonance (ESR) and nuclear magnetic resonance (NMR) experiments where a small
oscillating magnetic field applied perpendicular to a large static magnetic field is used
to induce Rabi oscillations. We can do this by applying an AC (microwave frequency)
voltage to the S-gate.
In order to see the effect of our microwave drive, we must first find the resonance
frequency of the spin which is set by the total magnetic field at the position of the
electron. However, when we start the experiment it is often difficult to predict the
total magnetic field accurately due to the unknown field added by the micromagnet.
Typically we can estimate the magnetic field to within ∼50 mT (1.4 GHz), but the
linewidth of the spin state is only ∼36 µT (1 MHz) making the task of finding the
spin resonance condition very time consuming. However, we can make use of a clever
pulse technique known as an adiabatic inversion pulse to speed up our search [73].
The adiabatic inversion pulse has the ability to address spins within a large range
of frequencies (∼40 MHz in our experiment), and induce a nearly perfect spin flip
without needing knowledge of the exact driving strength of the perpendicular field. It
works by applying a long microwave drive (typically 500 µs - 1 ms) whose frequency
is swept linearly in time. When the frequency of the microwave drive is far detuned
58
from the resonance frequency of the spin the effective magnetic field in the rotating
frame is along the z axis of the Bloch sphere with sign determined by the sign of the
frequency detuning. When the drive is on resonance however the effective magnetic
field in the rotating frame is along the x or y axis of the Bloch sphere depending of
the choice of rotating frame. Therefore if we sweep the frequency of the microwave
drive through resonance, we rotate the effective magnetic field acting on the spin by
180 degrees. We can view this as taking the system through an avoided crossing of
size gµB1 where g is the Lande’ g-factor, µ is the Bohr magneton, and B1 is the
driving field acting on the spin. If we traverse this avoided crossing adiabatically the
spin will track with the instantaneous eigenstates of the Hamiltonian and rotate the
spin by 180 degrees. The effective linewidth of the of the spin under this pulse is seen
to be greater than 1 mT in Fig. 5.8 (a). After we find the approximate resonance
frequency we can make a finer search using a simple fixed-frequency microwave pulse
with a square envelope, seeing that the actual linewidth is much less than 1 mT as
in Fig. 5.8 (b).
From here we can verify the expected Rabi flopping dependence on drive time and
frequency near the resonance condition as seen in Fig. 5.9. We observe the largest
amplitude oscillations when driving directly on resonance as in Fig. 5.9 (c) with a
characteristic decay time TRabi2 =15 µs. We can now measure the Rabi oscillation
quality factor Q = nπ/TRabi2 ∼120, implying a π-pulse fidelity of 99.2 %.
5.5 Power Dependence of Rabi Frequency
The Rabi frequency is expected to scale linearly with the magnitude of the oscillating
magnetic field, but we don’t have direct control over the magnetic field. We only con-
trol it indirectly through the position of the electron which is also indirectly controlled
through the voltage amplitude of our microwave drive. By measuring the dependence
59
Figure 5.9: Spin up probability of the left PL↑ (a) and right PR↑ (b) spins plotted as a functionof drive frequency and drive time revealing Rabi oscillations. (c) Direct on resonance Rabioscillations with a decay time of ∼15 µs.
of Rabi frequency on microwave drive power, which is shown in Fig. 5.10, we can
make some simple deductions about the relation between electric field, magnetic field
and position of the electron.
First we see that at low drive powers, the frequency is linearly dependent on drive
amplitude. In fact the roll-off seen at high powers has been verified to come from
saturation of the amplifier in our microwave signal generator and not a property of
the device. This suggests that the electric field is directly proportional to position of
the electron, and that our magnetic field gradient is approximately constant within
the range of drive amplitudes accessible in our experiment. From Rabi frequency we
60
Figure 5.10: Rabi oscillations of the left (a) and right (c) spins as a function of appliedmicrowave power. The extracted Rabi frequencies as a function of microwave amplitude forthe left (b) and right (d) spins show a linear dependence on drive amplitude up to the pointwhere the microwave amplifier begins to saturate.
can deduce a perpendicular magnetic field of ∼0.57 mT. Based on our simulations of
the micromagnet this suggests a displacement of the electron ∼0.3 nm.
5.6 Spin Coherence Time (T2) Measurements
In this section we will measure one of the most important properties of our qubit
T2. While T1 tells us how long a simple spin up state will last, the qubit often looses
its information long before this process takes effect. The decoherence time T2 tells
us exactly when that occurs. The measurement is known as a Ramsey experiment,
and simply consists of two microwave bursts rather than one as in the case of a Rabi
61
Figure 5.11: Ramsey fringes. The spin up probabilities of the left (a) and right (b) spinsplotted as a function of drive frequency and free-evolution time for a Ramsey experiment.The oscillation frequency of the fringes correspond to the frequency detuning our our drivefrom the resonance frequency of the spin.
experiment. Based on our Rabi oscillations we time the pulses to be π/2 rotations.
The first π/2 rotation puts our qubit into a superposition and then at some later
time τ we execute a second π/2 rotation to project state back onto the z-axis. If no
decoherence has occurred we end with a spin up state. However, if we have lost track
of the true state of the qubit due to a shift in the resonance frequency of the spin, we
end up seeing equal probabilities of spin up and spin down.
The results of this experiment are shown in Fig. 5.11 as a function of our free
evolution time τ and our driving frequency. Each drive frequency defines a new
rotating frame for our experiment. When the rotating frame perfectly matches the
Larmor frequency of the spin, we see a simple decay from P↑ ∼ 1 to P↑ ∼ 0.5. If our
frequency is different from the Larmor frequency then the qubit state rotates around
the Bloch sphere at the detuning frequency during its free evolution giving rise to
oscillations in the spin up return probability known as Ramsey fringes. By fitting the
decay we extract a decoherence time of TRamsey2 ∼ 1.3µs.
Notice that our TRamsey2 time is significantly shorter than our TRabi
2 time, suggesting
that perhaps we can extend the lifetime of our qubit through a series of rotations.
62
Figure 5.12: The spin up return probability for a Hahn echo pulse sequence is plotted asa function of total free-evolution time. The resulting decay indicates a coherence time ofT echo2 =80 µs.
This is true in the special case when the source of dephasing is quasi-static noise.
Quasi-static means that although the parameters of our qubit may fluctuate in time,
they are constant on the time scale of a single measurement which happens to be
the case for many spin qubits. In our case the dominant source of noise in our qubit
the randomly fluctuating 29Si nuclei in the quantum well, and the dynamics of the
nuclear spins occur on timescales significantly longer than 1 µs.
The most simple way to decouple our spin from low frequency noise is through a
Hahn echo pulse sequence. In this pulse sequence we split the free evolution time of
our Ramsey sequence into two parts of equal length and do a full π rotation of the
qubit in between. By flipping the qubit halfway through its evolution we ensure that
any phase accumulated during the first half of its evolution due to a static error term
will ‘unwind’ during the second half of its evolution. The resulting decay curve we
get from a Hahn echo sequence is shown in Fig. 5.12. Remarkably we increase the
decoherence time of the spin from 1.3 µs to TEcho2 = 80µs by including just a single
echo pulse.
63
5.7 Single Qubit Randomized Benchmarking
The results of our T2 measurements show an interesting behavior. If we do nothing
to the spin but prepare a superposition and read it out, we get a decoherence time
of TRamsey2 = 1.3 µs. If we do just a single rotation we can make it live as long
as TEcho2 = 80 µs, but if we continually manipulate the state the decoherence time
becomes shorter (TRabi2 = 15 µs). This brings us to perhaps the most important single
qubit parameter of all: the control fidelity, also called the gate fidelity or sometimes
simply the single qubit fidelity. In the end we don’t just want to create a superposition
and preserve it. We want to manipulate it to process information. Therefore we need
a measurement which doesn’t simply tell us how long the qubit will live, but how
many operations we can make on the qubit before losing all information. We need a
way of benchmarking the performance of our qubit.
Figure 5.13: Clifford randomized benchmarking protocol. K random sequences ofN Cliffordrotations are measured R times to extract a spin up return probability as a function of N .The resulting probability is then fit to extract the average Clifford fidelity FC.
64
Figure 5.14: Randomized benchmarking data for the two qubits from Fig. 5.5. By fittingthe spin up return probabilities as a function of N we extract Clifford fidelities of 99.3%and 99.7% for the left and right qubits respectively.
The prevailing method for characterizing gate errors is known as Clifford random-
ized benchmarking [74, 75]. Formally the Clifford group Cn on n qubits is defined
as
Cn = U : UPU † ∈ Pn∀P ∈ Pn, (5.1)
where U is the set of all unitary operators and Pn is the set of all Pauli operators
on n qubits. For a single qubit we can think of C1 as the set of rotations which
permute the ±x, ±y, and ±z directions of the Bloch sphere. The reason for char-
acterizing our qubits on this set of rotations is that randomly sampling this group
is a computationally efficient task, and they form a universal gate set for quantum
computation [76].
The randomized benchmarking protocol proceeds as follows: We construct K
random sequences of N rotations from the Clifford group that brings the initial state
(| ↓〉) to a well defined final state | ↑〉. For each series we measure the spin-up
return probability P↑ by repeating the sequence R times. We plot the average return
probability P↑ of all K sequences as a function of N , and fit the resulting decay curve
65
to extract an average Clifford fidelity FC . The randomized benchmarking protocol is
summarized schematically in Fig. 5.13.
The resulting decay curves for two of our qubits are shown in Fig. 5.14. The data
are fit to the equation
P↑(N) = P0(2FC − 1)N + Pinf , (5.2)
to extract average Clifford fidelities of 99.3 % for the left spin and 99.7 % for the
right.
5.8 State Tomography
At this point we have fully characterized our single qubit gates. However, before we
turn our attention to two-qubit gates I would like to show how we can effectively
measure our qubit along an arbitrary axis to fully characterize the state of a qubit
at a given point in time. This technique is known as quantum state tomography and
amounts to measuring the density matrix of our qubit.
In the Schrodinger picture of quantum mechanics, our experiment offers no way
of measuring the x or y component of the spin. However, if we consider it from the
Heisenberg picture then we can measure different observables by simply rotating the
spin before reading out on our fixed z-axis in the lab. In order to measure the x-
component of the spin for instance we simply apply a -π/2 rotation about the y-axis
before executing Elzerman readout. To measure the y-component we apply a +π/2
rotation about the x-axis. Using these operations to constitute measurements of the
σx or σy operators, we can fully reconstruct the density matrix using the following
equation:
ρ =1
2(I + 〈X〉σx + 〈Y 〉σy + 〈Z〉σz) . (5.3)
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Figure 5.15: State tomography of a spin up state. (a) Experimental data extracted bymeasuring the X, Y, and Z components of the spin. (b) Ideal spin up state. The extractedstate fidelity is 90 %.
The measured density matrix of a spin-up state is plotted in Fig. 5.15 (a), while
the ideal spin-up density matrix is plotted in Fig. 5.15 (b). Notice that our real
denisty matrix contains errors in the form of significant population of the | ↓〉 state as
well as off-diagonal matrix elements. The population of the | ↓〉 state is to be expected
due to the less than unity visibility of our readout. However, the off-diagonal terms
arise from a peculiarity of our readout technique.
One of the characteristics of Elzerman readout is that the fidelity of reading the
| ↑〉 state is often different than the fidelity of reading the | ↓〉 state. This is because
charge sensor bandwidth affects the | ↑〉 signal differently than the | ↓〉 signal. The
67
Figure 5.16: (a) Asymmetric readout. Spin up state measured by projecting X and Y witha single rotation resulting in asymmetric readout errors, yielding a state fidelity of 90%.(b) Symmetric readout. The X, Y, and Z components of the spin are measured along bothpossible directions with an equal number of measurements, resulting in a symmetric readouterrors yielding a state fidelity of 93.5%.
| ↓〉 signal corresponds to simply observing no hopping events, meaning that it is
essentially a low frequency signal. The | ↓〉 signal however is a square pulse in current,
containing very high frequency components. Therefore as we reduce the bandwidth
of the charge sensor measurement we filter out signals representing | ↑〉 measurements
reducing the fidelity of measuring spin-up F↑. Meanwhile we actually increase the
fidelity of spin-down measurements F↓ because we are less likely to falsely identify
high frequency noise as a spin-up signal. The asymmetry of our readout can be seen
68
in the Rabi oscillations of Fig. 5.9 by the fact that the mean of the oscillations is less
than 0.5.
There are two ways we can deal with the asymmetry of our charge sensor: we can
either try to post process the data taking into account the F↑ and F↓ fidelities or we
can try to measure the Pauli operators in a way that is insensitive to the bias of the
detector. Here I will explain how to alter our measurement protocol to eliminate bias
in the detector, leaving the discussion of post-processing to the next chapter.
Consider the task of measuring σx. What we have done so far is to rotate the
state with a -π/2 rotation about the y-axis before executing readout. In this case
a spin-up measurement counts as a +X measurement and spin-down counts as -X
measurement. We could have equally rotated the spin with a +π/2 rotation about
the y-axis before executing readout, in which case a spin-up measurement would be
counted as -X measurement and spin-down would be counted as a +X measurement.
The first scenario biases our detector for measuring -X, while the second biases our
detector towards +X. Instead of choosing either method, we can perform an equal
number of both measurements and combine the results. When measuring the Pauli
operators in this way we do not alter the visiblity of our spin-state, but we remove
any bias from the measurement. The results of measuring the density matrix this
way are shown in Fig. 5.16. We can see that the off-diagonal errors in our density
matrix are largely removed, yielding a higher target state fidelity.
69
Chapter 6
Two Qubit Gates and
Entanglement
“I would not call that ‘one’ but rather ‘the’ characteristic trait
of quantum mechanics, the one that enforces its entire
departure from classical lines of thought. By the interaction the
two representatives (or ψ-functions) have become entangled.”
-Erwin Schrodinger
6.1 The Exchange Interaction
In this chapter we’ll combine the single qubit control demonstrated in the last chapter
with the exchange interaction to demonstrate the criteria for universal quantum com-
putation with spins. The exchange interaction between two particles arises from the
symmetrization requirement of identical Fermions. The requirement is that the wave-
function of two identical Fermions must be antisymmetric under particle exchange,
or in other words the wavefunction must pick up a minus sign when you swap the
identity of two particles:
ψFermion(1, 2) = −ψFermion(2, 1). (6.1)
70
This can be achieved by placing the two particles in an antisymmetric combination
of single-particle states:
ψFermion(1, 2) =1√2
(|α〉1|β〉2 − |β〉1|α〉2) =
− 1√2
(|α〉2|β〉1 − |β〉2|α〉1) = −ψFermion(2, 1).
(6.2)
When the two electrons are well separated from each other this symmetrization
requirement has no impact on their energies. However, when we consider the case
when they occupy the same dot this results in an effective interaction between the spin
states. For example, say that the orbital states of the dot are labeled |g〉, |e〉, .. and
consider two electrons in the ground state orbital with a triplet spin configuration:
ψ(1, 2) = |g〉1|g〉2 ⊗1√2
(| ↓〉1| ↑〉2 + | ↑〉1| ↓〉2) . (6.3)
Swapping the indices on this wavefunction leaves it unchanged. Therefore this
cannot be a valid state for our two electrons. However, if the electrons were both in
the ground state orbital with a spin singlet configuration, for example:
ψ(1, 2) = |g〉1|g〉2 ⊗1√2
(| ↓〉1| ↑〉2 − | ↑〉1| ↓〉2) , (6.4)
then swapping indices does result in an overall minus sign in the wavefunction. Or if
we put the electrons in an antisymmetric combination of ground and excited orbital
states with a spin triplet configuration
ψ(1, 2) =1√2
(|g〉1|e〉2 − |e〉1|g〉2)⊗1√2
(| ↓〉1| ↑〉2 + | ↑〉1| ↓〉2) , (6.5)
we again arrive at a valid wavefunction.
71
The need for the triplet spin configuration to be associated with a higher energy
orbital state can be captured by a simple spin Hamiltonian of the form:
H = J ~S1 · ~S2, (6.6)
where the constant J represents the strength of the exchange interaction. Since this
term arises from the overlap between the wavefunctions we can tune it by changing
the tunnel coupling in our device.
The idea of using exchange to produce a two-qubit gate with quantum dots was
first proposed by Loss and DiVincenzo [38,77]. Their idea was to turn on and off the
tunnel coupling between spins to control the duration of the exchange interaction.
Since the singlet and triplet states are the eigenstates of the exchange interaction,
a | ↑↓〉 state will evolve into a | ↓↑〉 and back under the effect of exchange. If the
exchange pulse is timed properly the resulting evolution takes | ↓↑〉 to 1√2(| ↓↑〉+| ↑↓〉),
an operation known as a√
SWAP which has been shown to be a universal two-qubit
gate [12].
An experimental√
SWAP was first demonstrated in 2005 [39]. This experiment
which set the standard for exchange based control in quantum dots made use of the
detuning of the DQD, or ’tilt’ of the double well potential, to both initialize a spin
singlet and control the exchange between the two spins as shown in Fig. 6.1 (a).
The resulting exchange oscillations are shown in Fig. 6.1 (b). This type of tilt based
control has been utilized in a number of experiments on singlet-triplet qubits [79–81]
as well as for demonstrating exchange-only qubits in triple quantum dots [78,82,83],
as shown in Fig. 6.1 (c).
The basic idea of these experiments is to pulse towards the (1,1)-(2,0) transition in
the stability diagram as shown in Fig. 6.2 (a) to observe oscillations between the | ↑↓〉
and | ↓↑〉 states, which we now attempt in our device. Since exchange acts between
antiparallel spin states we first flip the left spin, and then execute our exchange pulse
72
Figure 6.1: Conventional exchange oscillations in quantum dot qubits. (a) Single tripletoscillations in a DQD where exchange is controlled by the tilt the double well confinementpotential. Figure from [39]. (b) Exchange oscillations in an exchange only triple dot qubit,again controlled by the detuning axis of the triple dot. Figure from [78].
to tilt the double well potential as shown schematically in Fig. 6.2 (b). The pulse is
a simple 10 ns square pulse whose amplitude we vary. The resulting oscillations are
shown in Fig. 6.2 (c). While we do observe anticorrelated oscillations, the amplitude
is small and they decohere significantly as we pulse closer to zero detuning. The reason
for the partial oscillations can be understood in terms of a Bloch sphere defined in
the | ↑↓〉, | ↓↑〉 basis. In this basis J and ∆Bz are orthogonal. Therefore in order to
see full oscillations between | ↑↓〉 and | ↓↑〉, we require that J ∆Bz which is ∼
200 MHz. So the partial oscillations we observe in Fig. 6.2 are about a tilted axis
which is a combination of J and ∆Bz. By the time J ∆Bz, the Hamiltonian is so
sensitive to voltage fluctuations that the oscillations are no longer coherent.
6.2 AC Exchange
While directly turning on the exchange interaction does not yield high fidelity S-T
oscillations in the high magnetic field gradient regime, there is still a way to utilize
73
Figure 6.2: (a) After initializing both electrons in a spin-down state, the left spin is flippedbefore applying a detuning pulse to the DQD to turn on exchange. (b) Charge stabilitydiagram of the DQD showing the trajectory of the detuning pulse in voltage space. (c)The spin up probabilities of the left (red) and right (blue) electrons for a 10 ns exchangepulse to a value of detuning ε. (d) In the singlet-triplet Bloch sphere the exchange term Jis overwhelmed by the large field gradient ∆Bz resulting in the partial oscillation shown in(c).
the tilt method to execute full amplitude coherent oscillations. The idea of using
tilt based exchange in a regime where ∆Bz J was first demonstrated in GaAs
DQDs [67] where the nuclear spins have been polarized to generate a large ∆Bz. The
basic idea is to oscillate the exchange term resonant with the ∆Bz splitting of the
| ↑↓〉, | ↓↑〉 states. We modify our previous experiment as shown in Fig. 6.3 (a) and
(b) by adding a sign wave at the plateau of our square pulse at the frequency given
by ∆Bz. The resulting oscillations, shown in Fig. 6.3 (c) and (d), correspond to full
amplitude SWAP oscillations.
Still the oscillations decohere fairly quickly due to the high sensitivity of the
exchange interaction to gate voltage in the vicinity of the (1,1)-(2,0) transition. In
74
Figure 6.3: Full amplitude SWAP oscillations can be attained by adding a sinusoidal oscil-lation to the detuning after pulsing toward the (1,1)-(2,0) transition as in (a). (b) Againwe initialize in | ↓↓〉 and flip the left spin before turning on the exchange interaction. Nowthe exchange term forms a small oscillating perpendicular component on the singlet-tripletBloch sphere. (c) When the exchange frequency ω matches the field gradient ∆Bz we seefull amplitude anti-correlated oscillations of the two spins.
fact several experiments have demonstrated that the sensitivity of exchange to charge
noise can be greatly reduced by operating in the middle of the (1,1) region near the
so-called symmetric operating point [84,85]. In order to do this in our device we make
use of control over the tunneling barrier rather than the detuning. We execute this
type of control by sitting in the middle of the (1,1) region at the point shown in Fig.
6.4 (a). The oscillations generated by barrier control shown in Figs. 6.4 (d),(e) have
a significantly higher quality factor than those achieved with tilt control (Figs. 6.4
(b),(c)).
75
Figure 6.4: Comparison of tilt and barrier control of the exchange interaction. (a) Tiltcontrol of the exchange interaction requires operating near the (1,1)-(2,0) charge transitionwhere it is very sensitive to the plunger gate voltages VL and VR. Barrier control however,can be performed in the middle of the (1,1) charge region at a sweet spot where exchangeis first order insensitive to plunger gate voltage fluctuations. Barrier control exchangeoscillations (d),(e) clearly show a higher quality factor than the oscillations achieved withtilt control (b),(c).
6.3 Conditional Rotations
While we can realize full amplitude SWAP operations using AC exchange, there are
simpler ways to make use of the exchange interaction when we have a large magnetic
field gradient present. One of the simplest two qubit gates we can realize in this new
regime is a conditional rotation. A conditional rotation (CROT) can most easily be
understood by looking at how the exchange interaction modifies our energy levels.
As we can see in Fig. 6.5, the exchange term lowers the energy of the antiparallel
spin states with respect to the parallel spin states. The result of this is that the spin
resonance condition of one spin is dependent on the state of the other spin. So rather
than having a single resonant frequency for the left spin, independent of the right, we
end up with two resonant frequencies for the left spin fL|ψR〉=|↑〉 and fL
|ψR〉=|↓〉 for the
two different states of the right spin.
76
Figure 6.5: When exchange is turned off (left) resonance frequencies of the | ↓↓〉 - | ↑↓〉 and| ↓↑〉 - | ↑↑〉 transitions are identical indicating that the left spin has a single spin resonancefrequency fLJ=0. However, when exchange is turned on (right) the | ↓↓〉 - | ↑↓〉 frequencybecomes fL|ψR〉=|↓〉 and the | ↓↑〉 - | ↑↑〉 frequency becomes fL|ψR〉=|↑〉 so that the resonancefrequency of left spin is dependent on the state of the right spin. Similarly the frequencyof the right spin becomes dependent on the state of the left.
The first thing we notice is that these state dependent frequencies offer a con-
venient way of measuring the exchange interaction since their splitting is exactly J .
In order to do this, we first rotate the right spin by applying a microwave drive of
duration τR as shown in Fig. 6.6 (b). Next we apply a long, low power probe tone to
the left spin for a time much longer than its decoherence time τL T2. This probe
tone will leave the spin in a mixed state if its frequency fLp is near the resonance
condition of the left spin. The resulting spin up probabilities of both spins are plot-
ted in Fig. 6.6 (c),(d). The right spin exhibits simple Rabi oscillations as a function
of τR while the left spin’s resonance frequency is found to oscillate back and forth
at the same rate as the Rabi oscillations of the right spin. The separation between
these two resonance conditions is J . We plot the frequency spectrum of the left spin
averaged over τR for a few different values of the middle barrier voltage in Fig. 6.6
77
Figure 6.6: Measuring the conditional frequencies of the left spin (a). (b) We start bydriving the right spin with a drive of length τR and then probe the left spin with a long, lowpower drive at frequency fLP that is much longer than the coherence time T2. The resultingspin up probability of the left spin (c) shows switching between two distinct resonanceconditions, correlated with the Rabi oscillations of the right spin (d). (e) Measuring thespectrum of the left spin for varying voltages on the middle barrier gate allows us to extractthe strength of the exchange couping J as a function of the middle barrier voltage VM asshown in (f).
78
Figure 6.7: Calibrating a CNOT gate. (a) While exchange is off we drive the spins at fLJ=0
and fRJ=0 to prepare different input states to our conditional rotation. While exchange isturned on we drive the left spin at fL|ψR〉=|↑〉 to execute a left spin rotation conditioned on
the state of the right spin. (b) After preparing an input state, we simultaneously apply asquare pulse to VM and a microwave drive at fL|ψR〉=|↑〉 for time τP to execute the conditionalrotation. The resulting spin up probabilities as a function of τP for the four logical inputstates. At a drive time τP = 130 ns we realize a CNOT gate.
(e). Finally we can now plot J as a function of VM in Fig. 6.6 (f), where we have
effectively mapped the exchange interaction onto a voltage that we can control.
Now that we have mapped out J as a function of gate voltage, we can use time
domain measurements of the left spin to calibrate a CNOT gate. The basic experiment
for a conditional rotation is outlined in Fig. 6.7. We use the voltage VM to to turn
on and off the exchange interaction between the two spins. While the VM is low, each
spin has an independent frequency as shown in Fig. 6.7 (a). We then use a square
pulse on VM to shift the frequencies by J/2, and apply a microwave drive at fL|ψR=|↑〉〉
for time τP as shown in Fig. 6.7 (b). The effect of this square pulse and microwave
drive are plotted in Fig. 6.7 (c) for the four logical basis state inputs. We see from the
lower two panels, that nothing happens to either spin when the right spin is pointing
79
down. However, when the right spin is prepared in the spin up state we get full
amplitude oscillations in the left spin. At τP = 130 ns this operation corresponds to
a CNOT gate, another universal two-qubit gate.
6.4 DC Exchange
Figure 6.8: (a) We can directly measure the conditional phases picked up during an exchangepulse by preparing the control qubit (left spin) in either the spin up or spin down state andplacing the exchange pulse in one of the free evolution periods of a Hahn echo sequenceapplied to the target qubit (right spin). (b) The resulting spin up return probability ofthe target qubit PR↑ is plotted as a function of the exchange time τJ . The difference inaccumulated phase when the control qubit is spin up (blue trace) versus spin down (redtrace) is the conditional phase from the exchange interaction.
We have now seen how to execute a SWAP operation and a CNOT gate using the
exchange interaction. In this section we will see perhaps the most natural two-qubit
80
gate that can be executed in our system, the CPHASE gate. As we have seen in the
last section the frequency of each qubit depends on the state of the other. Therefore
the phase evolution of each qubit is dependent on the state of the other spin. This can
be seen a bit more explicitly by performing a simple echo experiment on one qubit
where we place the exchange pulse on one half of the free evolution as shown in Fig.
6.8 (a). In this case our J gate is only the square pulse applied to VM without any
microwave drive.
Figure 6.9: By performing the experiment of Fig. 6.8 for varying pulse amplitudes on themiddle barrier voltage VM , we can map the strength of the exchange interaction J as afunction of VM .
By performing the echo on the right spin for both input states of the left spin, we
can directly see the phase shift induced by the exchange pulse. The resulting spin
projections of the right spin are plotted in Fig. 6.8 (b). The difference in conditional
frequencies ∆fR = fR|ψL=|↑〉〉 − f
R|ψL=|↓〉〉 leads to the difference in frequency of the red
and blue sine waves in Fig. 6.8 (b). By performing these time domain measurements
for varying VM we can again map the strength of J as a function of VM as shown in
Fig. 6.9. If we stop the exchange interaction at a point where the difference in the
two conditional phases is 180 degrees, we realize a CPHASE gate.
81
6.5 State Tomography and Entanglement
Now that we have calibrated a few different two qubit gates, we can see how two qubit
interactions lead to entanglement between qubits via quantum state tomography.
First we start with the CNOT gate based on conditional rotations from section 6.3.
When the control qubit of a CNOT gate is placed into a superposition, the resulting
two-qubit wavefunction is a superposition of | ↓↓〉 and | ↑↑〉, a completely nonclassical
correlated state. We prepare and readout this state as shown in Fig. 6.10 (a). State
tomography for two qubits now requires all possible combinations of I, X, Y, and
Z measurements on each qubit leading to 15 measurements in total (II need not be
measured). The resulting probabilities from these 15 measurements can be used to
reconstruct the full two qubit density matrix through
ρ =1
4(II + 〈IX〉Iσx + 〈IY 〉Iσy + 〈IZ〉Iσz
+ 〈XI〉σxI + 〈XX〉σxσx + 〈XY 〉σxσy + 〈XZ〉σxσz
+ 〈Y I〉σyI + 〈Y X〉σyσx + 〈Y Y 〉σyσy + 〈Y Z〉σyσz
+ 〈ZI〉σzI + 〈ZX〉σzσx + 〈ZY 〉σzσy + 〈ZZ〉σzσz).
(6.7)
The raw data from these measurements are shown in Fig. 6.10 (b) for the state
|ψtarget〉 =1√2
(| ↓↓〉 − i| ↑↑〉) . (6.8)
The off-diagonal terms in the imaginary part of the density matrix represent entangle-
ment correlations. Similar to section 5.8 the dominant errors are seen on the diagonal
of the real part of the density matrix as expected from our limited readout visibility.
The overall state fidelity from this measurement is F = 〈ψtarget|ρ|ψtarget〉 = 56% due
to the large readout errors.
82
Figure 6.10: (a) A Bell state is prepared by placing the control qubit (right spin) into asuperposition state before applying a CNOT gate to the target qubit (left spin). (b) Byappending the appropriate post rotations and measuring all two-qubit correlations we canreconstruct the full two-qubit density matrix. We extract a state fidelity F = 56 % fromthe raw data.
Based on our readout fidelities we can try to estimate what the density matrix
would look like if we had perfect readout [71]. For each measurement axis we can
imagine that the real probabilities of the four logical basis states are the set P =
P|↓↓〉, P|↓↑〉, P|↑↓〉, P|↑↑〉, but that they are transformed to the measured probabilities
PM = PM|↓↓〉, P
M|↓↑〉, P
M|↑↓〉, P
M|↑↑〉 through the limited readout visibility. For each spin
we can define the transformation through a fidelity matrix
Fi =
Fi,↓ 1− Fi,↑
1− Fi,↓ Fi,↑
, (6.9)
where Fi,↑ and Fi,↓ are the fidelity of accurately identifying a spin up or spin down
state on dot i. We can extract the elements of the fidelity matrix directly from the
83
Figure 6.11: By transforming the data from Fig. 6.10 we can effectively remove the effectof readout errors on our state tomography. The raw data (a) are transformed using thereadout fidelity matrices of equation 6.9 and maximum likelihood estimation (MLE) is usedto arrive at the density matrix in (b). The removal of readout errors brings the state fidelityfrom F = 56 % for the raw data to F = 78 % for the transformed data.
amplitude and offset of the single spin Rabi oscillations. Now we can relate the full
measured set of probabilities to the actual spin probabilities through
PM = (FL ⊗ FR)P. (6.10)
Solving this equation for P we can calculate what the spin probabilities would be
without readout errors. This process however can lead to unphysical density matrices,
therefore after we construct the new density matrix we perform maximum likelihood
84
Figure 6.12: (a) Gate sequence for generating Bell states from a CPHASE or CZ gate. Thesign of the z rotations after the CZ gate determine which of the four Bell states is created.The two qubit density matrices of the four Bell states are shown in (b)-(e) with an averagestate fidelity of 83 %.
estimation to ensure that the density matrix is physical. The result is shown in Fig.
6.11. This process brings our final state fidelity to 78 %.
85
Lastly, we can also make use of the CPHASE gate to produce entangled states. A
CPHASE (or CZ) gate is also a universal two-qubit gate that we can use to produce
entanglement. The gate sequence for making the four Bell states is shown in Fig.
6.12 (a). By changing the sign of the two z rotations in this pulse sequence we
can separately realize each Bell state. The four Bell states produced with this gate
sequence are shown in Fig. 6.12 (b)-(e). The fidelities realized with this gate (82-84
%) are slightly higher than that of the resonantly driven CNOT gate (78 %), but
both fidelities are likely still limited by errors related to readout and their imperfect
removal from the density matrix. A more robust method for ascertaining gate fidelities
is two-qubit randomized benchmarking. Further experiments are needed to properly
compare the different entangling gates presented in this chapter.
86
Chapter 7
Large Scale Architectures
“We can only see a short distance ahead, but we can see plenty
there that needs to be done.”-Alan Turing
7.1 Device Architecture
In this chapter we will discuss possible approaches to building a full large-scale quan-
tum processor using quantum dots, and demonstrate some basic control experiments
in a 9 qubit device. While a 2D grid of qubits would be ideal for implementing many
popular coding schemes, fabrication constraints combined with the need for charge
sensors make 2D architectures difficult to realize under realistic assumptions about
fabrication capabilities. Instead we will focus on laying a path toward large 1D arrays.
The simplest scheme for scaling up is to establish a gate structure unit cell which
can be repeated and linked together with other unit cells. We take our unit cell
to consists of three logical qubits per charge sensor as seen in Fig. 7.1 (a). As
a proof-of-concept for our device we have fabricated and fully characterized a nine
quantum dot array and fully characterized the charge characteristics of each dot. A
scanning electron microscope (SEM) image of the device built from three unit cells
is shown in Fig. 7.1 (b). We can measure all three charge sensors simultaneously by
87
Figure 7.1: (a) Architecture for an arbitrarily sized linear array of quantum dots withrepeating units of 3 computational dots per charge senor quantum dot. (b) Prototypedevice of 9 quantum dots and 3 charge sensors. The three charge sensors are measuredsimultaneously by an alternating series of signal generators and amplifiers along the sensordot chain using AC lock-in techniques. (c) Simulated charge density in the 2DEG when thedevice is tuned near few electron occupancy.
connecting alternating signal generators and current amplifiers to the ohmic contacts
of the charge sensors and using standard AC lock-in techniques to differentiate the
signal coming from each charge sensor. A simulation of the resulting charge density
in the 2DEG for this device is shown in Fig. 7.1 (c). The basic characterization
measurements for all nine dots in the array are summarized in Table 7.1.
88
7.2 Controlling Large Arrays
In smaller quantum dot devices consisting of only two or three dots, we can easily
tune the quantum dots to a desired number of electrons by hand. However, in larger
arrays we will need to develop more sophisticated techniques for designing control
pulses. As a demonstration of the general principles of controlling large arrays, we
present a basic charge pumping experiment in the nine dot arrray.
The basic concept of this experiment is shown in Fig. 7.2. The device and a
schematic illustration of the confinement potential of the nine dots are shown in Fig.
7.2 (a) and (b). We load a single electron into dot 1, and pump it through the array
by lowering the chemical potential of each dot in turn as shown in Fig. 7.2 (c).
One of the key concepts in controlling the array is using measurements of the
capacitance matrix to establish orthogonal control over the chemical potentials of
each dot. Normally when we change voltages in the device, they effect the chemical
potentials of several dots, leading to the slopes of the charge stability diagrams that
we see in gate-voltage space as shown in Fig. 7.3 (a). However, by fitting the stability
diagrams, we can extract the capacitance matrix from the slopes of the charge tran-
sitions and locations of the triple points in voltage space. Using this knowledge, we
can then design ‘virtual gates’ which are linear combinations of gate voltages which
Dot α (meV/mV) Ec (meV) Eorb (meV)
1 0.14 6.6 2.72 0.13 6.1 2.63 0.11 5.6 2.14 0.14 7.3 3.35 0.14 7.2 3.36 0.14 7.1 3.07 0.14 7.7 3.58 0.14 7.1 3.49 0.13 7.2 3.4
Table 7.1: Lever-arm conversion between gate voltage and energy α, charging energy Ec,and orbital excited state energy Eorb for each of the nine dots in the linear array.
89
Figure 7.2: Nine dot charge pumping experiment. Dots are pumped through the ninedot device shown in (a) by manipulating the confinement potential formed beneath thegates shown in (b). (c) Schematic illustration of the confinement potential throughout thepumping sequence.
allow us to tune the chemical potential of each dot separately as shown in the stability
diagram of Fig. 7.3 (b).
In practice we measure the capacitance matrix through pairwise DQD stability
diagrams as we form the array. We start by forming dots 1 and 2 and extracting a
capacitance matrix for only those two dots. Next, we add the third dot and measure
the capacitance matrix between dots 2 and 3, and append it to the 1-2 capacitance
matrix. In general when we add the N th dot, we only need to update matrix elements
Cij where i, j ∈ N,N − 1. We continue measuring pairwise capacitance matrices
until the full nine dot array is formed. The ability to neglect non-nearest neighbor
coupling is afforded to us by the small cross capacitances of our multilayer gate
architecture.
Using the experimentally measured capacitance matrix we can now directly cal-
culate the trajectory in voltage space that will shuttle a single charge through the
90
(a) (b)
GS1 (e2/h)0.1 0.2
575
590
600 615VP2 (mV)
VP
1 (m
V)
(0,0)
(1,0) (1,1)
(0,1) 695
0.30.20.1GS1 (e2/h)
600 625
720
u1
(mV
)
u2 (mV)
(0,0)
(1,0) (1,1)
(0,1)
Voltage Space Mode Space
u1
uN
…𝛿𝝁 = R11e
1 0
0 1
…
…
… …
R1N
…
……
VP1
VPN
…
R11
RN1 RNN…
𝛿𝝁 = e
(c) (d)
Figure 7.3: (a) The relation between chemical potentials of the dots and gate voltages~V can be represented as a matrix R with many off diagonal terms representing cross-capacitances that lead to the sloped transitions seen in regular charge stability diagrams.(b) By measuring the capacitance matrix we can define virtual gate voltages ~u we canestablish orthogonal control over the chemical potentials, which is reflected by the verticaland horizontal charge transitions seen in the stability diagrams taken in virtual gate space.
array. For simplicity we show the trajectory through a four dot array in Fig. 7.4.
The chemical potential of each dot is plotted as a function of time in Fig. 7.4 (d),
and the corresponding trajectory in voltage space is plotted in Fig. 7.4 (a)-(c).
Charge pumping data for the full nine dot array are presented in Fig. 7.5. The
potential configuration of the dots is shown schematically in Fig. 7.5 (a). Since our
capacitance matrix has only nearest neighbor coupling terms, we can begin pumping
another electron as soon as the first electron gets to at least dot 3. This allows us to
easily pump two or three electrons at a time. The current resulting from the pump
can be directly measured in the drain of the nine dot array. If we plot the current as
a function of two of our virtual gate parameters, we see a rectangular region where
91
Figure 7.4: Charge pumping sequence for a quadruple dot. (a)-(c) Pumping trajectorymapped onto the pairwise stability diagrams of the four dots measured in virtual gate space.(d) Chemical potentials of each dot as a function time for a simple pumping trajectorydesigned in virtual gate space using the experimentally measured capacitance matrix. (e)Schematic illustration of the chemical potential configuration of the four dots throughoutthe pumping process.
the pump works due to the orthogonality of the virtual gates as seen in the color-plot
of Fig. 7.5 (c). The current within this rectangular region is constant and does not
depend on the exact value of the virtual gates as shown in the 1D trace of Fig. 7.5
(c). The current generated by the pump is plotted as a function of pump frequency
in Fig. 7.5 (b) for 1e, 2e and 3e charge pumping.
7.3 Long-Distance Coupling
While in principle any arbitrary quantum algorithm can be implemented in a 1D
array, having the ability to couple distant qubits lifts many constraints on fabrication
and can possibly improve the performance of algorithms by increasing connectivity
92
(a)
(b) (c)
1 electron pumping 2 electron pumping 3 electron pumping
740
710
965930
80I (pA)
u1 (m
V)
u9 (mV)
3e pumping
f = 18.4 MHz-10
-5
0
5
10
I (p
A)
20151050f (MHz)
+3ef+2ef+1ef
-3ef-2ef-1ef
0 8I (pA)
Figure 7.5: Charge pumping in the full nine dot array. (a) Electrons are shuttled bysequentially lowering the chemical potential of each dot to push the electron through thearray. Once an electron has moved to the third dot of the array we may begin loading thenext electron, allowing us to pump either 1, 2, or 3 electrons at a time. (b) We plot thecurrent generated from the pump as a function of the pumping frequency for 1e, 2e, and3e pumping in both directions through the device. (c) We directly measure the currentgenerated by the pump and plot it as a function of the offsets of two of the virtual gates.We see a rectangular region of current indicating the region of chemical potential spacewhere the pump works properly.
between qubits. Due to its importance for potential large scale architectures, we
briefly review recent progress in coupling distant spins via a microwave photon.
The idea of coupling distant solid state qubits with photons was first developed
in superconducting qubits. The field is now generally referred to as circuit quantum
electrodynamics (cQED). As early as 2007 two qubit coupling between two distant
transmons was achieved with transmon qubits coupled to a coplanar waveguide res-
onator [86]. Cavity coupling has also served as a method for qubit readout, achieving
fast, high-fidelity readout for transmon qubits [87,88]. The first steps toward realizing
cQED experiments with quantum dots were taken by attempting to couple the charge
degree of freedom of the electron to a cavity [89]. Our group first realized strong cav-
ity coupling to a single electron in a quantum dot device in 2016, by placing a single
electron in a DQD and tuning it to near an interdot charge transition [90]. In this
93
Figure 7.6: (a) Optical image of a cQED device for strong spin-photon coupling. Two DQDswith Co micromagnets (b) are place on either side of a superconducting coplanar waveguideresonator. The effective spin orbit interaction from the fringing field of the micromagnetallows the spin degree of freedom to couple to the electric field of the cavity. (c) When thespin is tuned through resonance with the cavity by applying an external magnetic field Ban avoided crossing is clearly visible in the transmission spectra of the cavity.
tuning configuration the electron has an enhanced dipole moment due to its ability
to hop between two neighboring quantum dots.
In order to utilize our charge-photon coupling to generate a spin-photon interac-
tion, we induce an artificial spin-orbit interaction using a Co micromagnet. The device
is shown in Figs. 7.6 (a) and (b). Two double quantum dots are placed on either side
of the cavity, and micromagnets above each dot give a large magnetic field gradient
across the double quantum dot. To further enhance the coupling strength the width
of the center pin of the resonator was also shrunk to increase the impedance of the
resonator resulting in a larger electric field at each double dot. Through the strong
94
charge-photon coupling combined with the spin-charge hybridization from the micro-
magnet, the first strong coupling of a single spin in a quantum dot to a microwave
cavity was realized by Mi et al. just this year [91].
In this device, both DQDs have shown strong coupling to the cavity. However, in
the current device one of the plunger gates from each device share a DC voltage from
their DC connection to the cavity center pin. This constraint makes it difficult to
tune both quantum dots into resonance with the cavity at the same time. Nonetheless
it is easy to imagine alternative coupling schemes which would allow both quantum
dots to be individually tuned opening the door to two-qubit gates between electrons
separated by millimeters. This capability will surely play a key role making quantum
integrated circuits containing tens or hundreds of quantum dot qubits.
95
Chapter 8
Conclusions and Outlook
The main focus of this thesis has been integrating the work of many decades of
prior research into a cohesive single effort to build devices capable of performing
quantum computation with single electron spins in Si. Throughout the early chapters
of the thesis we learned how subtle changes to device designs can produce significant
improvements to device performance. The intuition behind these modifications could
not have come without the collaborative efforts of countless researchers before me, and
it is my hope that the studies presented here will help to inspire future experiments
towards multiqubit devices.
One of the most critical areas in the field right now is the need to better understand
two-qubit interactions in quantum dots. Through the work presented in chapter 6
we have attempted to scout more of the terrain of two-qubit coupling in quantum
dots. While more characterization is needed to quantify gate fidelities and determine
sources of error, I think the early results are promising that spin-qubits could become
one of the leading qubit technologies in the coming years. I have no doubt that two-
qubit fidelity measurements will become routine within the research community over
the next few years. With the control techniques outlined in chapter 7 I think it is
likely that processors containing tens of spin qubits will be realized in the coming
96
decade. While it is difficult to see if any major roadblocks lie ahead, I think it is clear
that spin based qubits will enable exciting new physics experiments and help to push
the limits of our understanding of nature.
97
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