Coherent Control of Hyperfine-coupled Electron and
Nuclear Spins for Quantum Information Processing
by
Jamie Chiaming Yang
Submitted to the Department of Nuclear Science and Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Nuclear Science and Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2008
© Massachusetts Institute of Technology 2008. All rights reserved.
Author ........
..............
Department of N;
...... ...... .....
ience and Engineering
June 2008
...............
Certified by...
David G. Cory
Professor of Nuclear Science and Engineering
, I
Thesis Supervisor
Read by.
Sow-Hsin Chen
Professor of Nuclear Science and Engineering
Thesis Reader
A
Accepted by ........
/
MASSACHUSETTS INST•TTTE
OF TECHNOLOGY
JUL 2 4 2008
LIBRARIES
Jacquelyn C. Yanch
Chair, Department Committee on Graduate Students
Coherent Control of Hyperfine-coupled Electron and Nuclear
Spins for Quantum Information Processing
by
Jamie Chiaming Yang
Submitted to the Department of Nuclear Science and Engineering
on June 2008, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Nuclear Science and Engineering
Abstract
Coupled electron-nuclear spins are promising physical systems for quantum information processing: By combining the long coherence times of the nuclear spins
with the ability to initialize, control, and measure the electron spin state, the favorable properties of each spin species are utilized. This thesis discusses a procedure
to initialize these nuclear spin qubits, and presents a vision of how these systems
could be used as the fundamental processing unit of a quantum computer. The focus of this thesis is on control of a system in which a single electron spin is coupled
to N nuclear spins via resolvable anisotropic hyperfine (AHF) interactions. Highfidelity universal control of this le-Nn system is possible using only excitations on
a single electron spin transition. This electron spin actuator control is implemented
by using optimal control theory to find the modulation sequences that generate
the desired unitary operations. Decoherence and the challenge of making useful
qubits from these systems are also discussed.
Experimental evidence of control using an electron spin actuator was acquired
with a custom-built pulsed electron spin resonance spectrometer. Complex modulation sequences found by the GRadient Ascent Pulse Engineering (GRAPE) algorithm were used to perform electron spin echo envelope modulation (ESEEM)
experiments and simple preparation-quantum operation-readout experiments
on an ensemble of le-1n systems. The data provided evidence that we can generate
any unitary operation on an AHF-coupled le-1n system while sitting on a single
transmitter frequency. The data also guided design of the next iteration of these experiments, which will include an improved spectrometer, bandwidth-constrained
GRAPE, and samples with larger Hilbert spaces.
Thesis Supervisor: David G. Cory
Title: Professor of Nuclear Science and Engineering
Acknowledgments
These years spent at MIT have been instructive and rewarding, and I have many
to thank for this. I would first like to acknowlege David Cory for his guidance,
insight, and support. Throughout my graduate school career, he has fostered good
ideas, asked the necessary questions, and demanded the appropriate rigor, while
being a consistent advocate for his students.
I have benefited greatly from working with various Cory group members: I am
especially grateful to Jonathan Hodges, with whom I collaborated on this project
and from whom I learned a great deal. Yaakov Weinstein and Nicolas Boulant
were very helpful during my introduction to quantum information and NMR as
a summer student. Michael Henry-with whom I started this program, took the
quals, and filled the magnets for several years without incident (that one quench
was entirely my doing)-has been a valuable colleague. In my last months as
a graduate student, I have had the pleasure of transferring this project into the
capable hands of Clarice Aiello and Mohamed Abutaleb. Sekhar Ramanathan has
been a reliable source of knowledge and advice for every experiment conducted in
this lab. It has been a privilege to have spent this time with these and the many
remarkable Cory group members I did not mention by name.
Also, I would like to thank Prof. Kohei Itoh and his graduate student Hiroki
Morishita for hosting me for a summer at Keio University. It was definitely a
worthwhile and educational experience.
Finally, I am grateful for the backing of my family and friends. I will always
value the friendships made here, and am thankful for my friends in California and
New York with whom I enjoyed my vacations from MIT. My deepest gratitude is
reserved for my mother Sheau-shan, my father Fang-chou, and my sister Jackie,
who have always supported me.
Contents
1
Introduction to Electron Spin Actuator Control of a Nuclear Spin Quantum Information Processor
2 Hyperfine-coupled Electron-Nuclear Spin Systems
2.1
11
19
Electron-nuclear spin system Hamiltonian . . . . . . . . . . . . . . . . 19
2.1.1
1 electron- N nuclear spins ....................
2.1.2
1 electron - 1 nuclear spin system . . . . . . . . . . . . . . . . . 21
19
2.2
Polarization of Nuclear Spins .......................
26
2.3
Nuclear Spin Relaxation ..........................
27
2.4
Quantum Information Processing in Electron-Nuclear Spin Systems.
28
2.4.1
le-1n Spin Systems .........................
28
2.4.2
le-Nn Spin Systems ........................
29
3 Achieving Universal Control of Spin Systems
33
3.1
Universality of Electron Spin Actuator Control . . . . . . . . . . . . . 33
3.2
Pulse Engineering: Using Optimal Control to Perform Precise Uni-
3.3
tary Operations ...............................
35
Full System M odel .............................
39
3.3.1
Quality Factor of Resonator . . . . . . . . . . . . . . . . . . . . 39
3.3.2
Inhomogeneity of Magnetic Fields . . . . . . . . . . . . . . . . 40
3.3.3
Crystal Quality ...........................
40
3.3.4
Spin System Hamiltonian .....................
41
4
Spectrometer Design
4.1
4.2
43
Pulsed ESR Spectrometer: Version 1 ...
. .. . . .
4.1.1
Design Specifications .........
. .. . . ..
4.1.2
Probehead .....
4.1.3
Pulse and Receiver Electronics ...
......................
Design Goals
4.2.2
New Probehead...........
4.2.3
Improved Electronics ...........
.
.........
43
. . . . .
. 43
.......
. . . .
Pulsed ESR Spectrometer: Version 2 ...
4.2.1
. . . . ......
. .. . .
. . . . ... .
.. .
48
. . . . . ......
50
. . . . . .
. 50
. .. . . .. .
.
44
.......
.
........
50
.......
53
5
DNP Probe
55
6
Experimental Results
59
6.1
le-1n Spin System: Malonic Acid Radical . . . . . . . . . . . . .... ..
59
6.2
ESEEM .......
61
........................
6.2.1
Hard Pulses .
6.2.2
Shaped Sx pulses .
..
.........
............................
62
.........................
65
6.3
Ramsey Fringe and Refocusing Experiments
. . . . . . . . . . . . . . 69
6.4
Discussion of results .
......
.
....................
73
7 Conclusion
77
A Publications
79
List of Figures
1-1
Quantum information processor with electron spin actuators . . . . .
1-2 Schematic of quantum information processing in a le-Nn spin system
2-1
Energy levels for the le-1n spin system with isotropic hyperfine
coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-2
Local fields seen by nuclear spin . . . . . . . . . . . . . . . . . . . . .
2-3 Energy levels for le-1n spin system with anisotropic hyperfine coup lin g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
Universal control of le-Nn system with electron spin excitations . . .
3-2
Spin actuator control
. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3-3 GRadient Ascent Pulse Engineering (GRAPE) . . . . . .
3-4 Evolution of spin during a j) pulse . . . . . . . . . . .
4-1 Loop-gap resonator and coupling loop . . . . . . . . . .
. . . . . . . 45
4-2
Magnetic field vectors in LGR ...............
. . . . . . . 46
4-3
Block diagram of pulse and receiver electronics . . . . .
. . . . . . . 49
4-4 Magnetic field vectors in BLGR ..............
. . . . . . . 51
4-5 S11 vs. Frequency with varying tuning angle . . . . . .
. . . . . . . 52
4-6
Block diagram of pulse and receiver electronics, version
. . . . . . . 54
5-1
TE011 cavity for DNP experiment ...............
5-2 TE011 magnetic field vectors
. . . . . . . . . . . . . . . ..
5-3 TE011 electric field vectors ...................
5-4 S11 vs. Frequency with varying cavity length . . . . . . . .
6-1
M alonic acid radical
6-2
Simulation of ESEEM with hard pulses: Time-domain data ......
6-3
ESEEM experiment with hard pulses: Time-domain data .......
6-4
ESEEM with hard pulses: Frequency-domain data from simulation
............................
60
63
.63
and experim ent ...............................
64
6-5
ESEEM with shaped GRAPE pulses: Time-domain data ........
67
6-6
ESEEM with shaped GRAPE pulses: Frequency-domain simulation
and experiment data ............................
68
6-7
Energy level illustration of Ramsey fringe experiment .........
70
6-8
Pulse sequence for Ramsey fringe experiment ..............
70
6-9
Pulse sequence for Ramsey fringe refocusing experiment ......
. 72
6-10 Time domain Ramsey fringe signal ......................
6-11 Magnitude spectrum of Ramsey fringe and refocused signal ..... .74
74
Chapter 1
Introduction to Electron Spin
Actuator Control of a Nuclear Spin
Quantum Information Processor
An information processor exploiting entanglement and superposition, uniquely
quantum degrees of freedom, can solve certain problems more efficiently than
classical processors [5]. Liquid-state ensembles of non-interacting molecules containing nuclear spins were recognized to be good testbeds for coherent control
of small (-10 quantum bits or less) quantum systems [18, 28]. Many researchers
have developed techniques for performing high-fidelity quantum operations using liquid-state nuclear magnetic resonance, and used these techniques to explore
quantum information concepts [8, 11, 12, 26, 19, 48]. In the liquid state it is not
known how to access nuclear spins efficiently enough to have a scalable quantum
computer. However, nuclear spins in the solid state, combined with quantum systems such as superconducting circuits, optics, or electron spins, could become an
integral component of a quantum computer.
Both nuclear and electron spins are attractive candidates for qubits because they
are relatively well-isolated from their environment [20, 41, 56]. Nuclear spins in
particular, with magnetic moments several orders of magnitude smaller than that
of electrons, typically have long coherence times. The weak magnetic moment also
means that nuclear spin thermal polarization at reasonable temperatures and fields
is quite small. Initializing nuclear spin qubits thus benefits from methods other
than thermal polarization. For example, dynamic nuclear polarization [1] in solid
state systems using magnetic resonance [62] and optical techniques [82] has been
studied for this purpose. The hyperfine coupling also allows coherent transfer of
information between electron and nuclear spins, opening up new possibilities for
nuclear spin quantum information processing (QIP), including algorithmic cooling [9, 23] using an electron spin. Experimentally, researchers have demonstrated
entanglement between electron and nuclear spins [56, 57, 75] and quantum registers of nuclear spins coupled to an electron spin [20, 58]. The larger magnetic
moment of electrons allows for higher thermal polarization and more efficient state
manipulation. In addition, the transport and optical properties of systems containing electron and nuclear spins provide a variety of options for initialization and
control [86]. Finally, single electron spin detection has been achieved in several systems [29, 36, 45, 72]. In this work we explore a particular combination of electron
and nuclear spins for QIP: Nuclear spins will be used to store information, while
electron spins will initialize qubits, transfer information, and be used for readout.
The key new feature is to exploit the anisotropic hyperfine interaction in selected
electron-nuclear spin systems to achieve universal control with an electron spin
actuator.
Figure 1-1 shows a schematic representation of a possible quantum computer
based on nuclear spin qubits controlled by electron spin actuators. The fundamental processing units of this computer are identical, individually addressable spin
clusters, each containing a single electron and several nuclear spins. By indirectly
manipulating the nuclear qubits with modulations of the electron spin, we simplify
and improve the efficiency of control. The quantum bits are contained in the nuclear spin space of one of the electron spin manifolds, and are separable from the
electron spin. Quantum information is transferred between nuclear and electron
spins using unitary spin actuator operations. Addressability could potentially be
realized with strong magnetic field gradients from nearby microfabricated ferro-
Figure 1-1: Quantum information processor with electron spin
actuators: This
figure shows an array of individually addressable le-Nn spin systems
in which the
electron spin actuator is coupled to several nuclear spins via resolvable
anisotropic
hyperfine interactions. In each cluster a microwave control field drives
gates on the
nuclear spin qubits via the actuator. A spin bus, which transfers
electron spin states
over distance, connects the le-Nn systems to each other, and could
also facilitate
electron spin injection and single-spin readout.
magnets [46] or with voltage gate control of the electron g-factor [84]. A spin bus
transfers the electron spin state between the le-Nn systems, allowing these clusters
to scale to a useful quantum computer. It could also be used to inject polarized
electrons into the le-Nn systems or transfer an electron to a single-spin detection
device. This scheme takes advantage of the primary benefit of nuclear spins-the
long coherence time-and leaves the tasks of control, initialization, measurement,
and information transfer to the electron spin.
We now briefly outline the local spin actuator control needed to process quantum information on these le-Nn systems. The preparation, gate operations, and
measurement steps of a quantum calculation in this le-Nn system are illustrated in
Figure 1-2. We assume high-fidelity gates, an electron T1 longer than the gate times,
and a fully polarized electron spin on which we can make projective measurements.
Nuclear spins in a single electron spin manifold store the quantum information.
We choose the nuclear spin eigenbases in the electron spin-up manifold as the
computational basis:
ITe)®(kY7))
(1.1)
The eigenstates of each nuclear spin qubit are labeled by j e {0,1}. The equilibrium
state consists of a polarized electron and N unpolarized nuclear spins
& ION
Peq = E+ ®
(1.2)
where Ee is the electron polarization operator ITe) (Tel and I is the identity matrix. In
this text S and I are used to label electron spin operators and nuclear spin operators,
respectively. The electron spin ancilla is reset by a projective measurement followed
by a unitary operation Ugi that, conditional on the measurement result, takes the
electron spin to its ground state. Such strong measurements have been performed
using optical techniques [36] and using magnetic resonance force microscopy [72].
A similar procedure was used by Monroe et al. in their demonstration of a CNOT
gate using trapped ions [61]. A series of electron polarization resets and unitary
operations driven by the spin actuator are used to put the nuclear spins into a
state that is useful for quantum computation. First we use the electron to cool
the nuclear spins. The unitary polarization transfer operation UPT followed by an
electron reset puts both the electron spin and the ith nuclear spin in their ground
states:
UprT RESETi(e)
) Ee
Peq
1®(-1) 0 Eni
(N-i).
(1.3)
Eni is the polarization operator a0)ang'i of the ith nuclear spin. UpT, produces a
state pi such that the operations
pi ý-
MopiM
(1.4)
and
UgiMlpiM Ug1
Pi
(1.5)
with measurement operators M0 = ITe) (Tel and M1 = lie) (el, both produce the
(i- 1)0 E n,
same state E 0Dl®
Pi'I = ie
1
®(N-i).
(i-1) 0 E9n g
The operator UPTi generates the state
(N- i)
= (Ee + E e ) 0 1®( i- 1)0 E n 0 ](N-i)
(1.6)
e
If projective measurement yields E', we are in the desired state; if we measure E
we flip the electron spin with operation Ug, to reach the desired state. N repetitions
of this UpT, followed by an electron reset generates the state
Ee
(En)®N.
(1.7)
In electron-nuclear spin systems with isotropic hyperfine couplings, the operators
UpTi = SWAP(e,ni) and Ug, = e- isxR can be used. In our le-Nn spin systems, we
take advantage of the anisotropic hyperfine (AHF) interactions to achieve universal
control with an electron spin actuator, causing the transformation to the nuclear
spin eigenbasis from the nuclear Zeeman basis to differ between the electron spinup and spin-down manifolds. The UPT, and Ug, operators that can be used to
polarize AHF-coupled nuclear spins are presented in Section 2.2.
As will be discussed in Section 2.3, the AHF interaction also subjects nuclear
spins to depolarization due to random electron spin bit flips. Noiseless subsystem
encoding combined with dynamic decoupling can be used to protect against this
noise and create useful nuclear spin qubits. N physical qubits are mapped to M
logical qubits by
ONM.
The desired algorithm is executed by quantum gates on
the logical qubits using engineered microwave pulses on the electron spin actuator.
During the pulses the electron spin and the nuclear spin states are not necessarily
separable, and the qubits need not stay in the protected subspace [12, 26]. Also,
during the algorithm, it could be advantageous to use the resettable electron spin
as a resource. After the algorithm is performed, the logical qubits are mapped back
to the physical nuclear spins by 0'*
Unitary operations transfer the desired
observables to the electron spin for readout.
The electron spin bus that transfers information between the le-Nn systems
could be implemented with spin chains [7, 13, 14], flying qubits [34, 35, 50], or
electrical gates that effect localized wavefunction overlap [41, 58] or mobile electron
transport [77]. The spin bus could also be used to inject spins from a polarized
bath such as a ferromagnet [86]. Measurement of the electron spin state could be
facilitated by transfer to a system that allows single-spin detection.
This thesis focuses on the utilization of anisotropic hyperfine couplings to indirectly and more efficiently control nuclear spin qubits with only electron spin
manipulation. First, the use of electron-nuclear spin systems in QIP is reviewed. A
simple model system is presented, with which an argument for universal control
via an electron spin actuator is made. This work then discusses the algorithm
used to find control sequences and the experimental apparatus to implement these
sequences. Chapter 5 discusses design of a resonator for a related experiment.
Finally, experimental evidence of control using the electron spin actuator is shown.
peq =E,
o(if
(UpTi,-,RESETj(e)
I
E2®(E2)®N
(h
Uphysical
E+ 0 Pfinal
4)
4
U logical
E
fi0pnal
U preparereadout
Figure 1-2: Schematic of quantum information processing in a le-Nn spin system:
The preparation, gate operation, and readout steps of a spin-actuator-controlled
system are shown. Starting from thermal equilibrium, polarization transfer operations and electron spin resets (described in text) are used to polarize the nuclear
spin qubits. The physical qubits are mapped to logical qubits in a protected subspace. The quantum algorithm is performed on the logical qubits. Note that the
algorithm need not be entirely unitary, as the resettable electron spin could be useful as a resource. The result is then mapped back to the physical qubits. Finally,
the desired information is transferred to a physical nuclear qubit and subsequently
swapped to the electron for readout.
Chapter 2
Hyperfine-coupled Electron-Nuclear
Spin Systems
This chapter discusses the use of coupled electron and nuclear spins in solidstate quantum information processors. First, the Hamiltonians and energy-level
diagrams of simple e-n spin systems are presented. These systems have been
studied extensively with magnetic resonance [1, 2, 40, 76, 78, 85]. The implications
of the symmetry of the hyperfine coupling on the relaxation of the nuclear spin
qubits are then discussed. Finally, a few physical electron-nuclear spin systems
are presented. These systems are utilized in QIP implementations and proposals
reviewed in this chapter.
2.1
Electron-nuclear spin system Hamiltonian
2.1.1
1 electron - N nuclear spins
Our study is limited to a single electron coupled to several nuclear spins. Here we
do not address how information is transferred between these clusters, only how
the electron in these systems can be used to achieve universal control within these
systems. The three largest components of the spin Hamiltonian of a single spin-1/2
electron spin coupled to spin-1/2 nuclear spins in a large magnetic field are the
electron Zeeman interaction, the hyperfine interaction, and the nuclear Zeeman
interaction.
Ho = HeZ + Hnz + HHF
(2.1)
The electron Zeeman term describes the interaction of the magnetic moment of
the electron spin with an external magnetic field. In a sufficiently large fielde.g., B0 > 1 Tesla applied to a ge - 2 electron-this interaction is dominant and the
electron spin is quantized along the external field. This term is typically anisotropic,
as given by the tensor ge, which includes effects from the spin-orbit coupling.
Energies are given in frequency units, and fe is the Bohr magneton.
-
ý e
'
->
- BogeS
Hez =
(2.2)
The magnetic moment of the nuclear spin interacts with the external field in a
similar way. The larger mass of the nucleus leads to a nuclear Zeeman term about
3 orders of magnitude smaller than the electron Zeeman term: The ratio of the
Bohr magneton to the nuclear magneton
P11
1800, and
,z(free
electron)
Hz(free proton)
660. The
nuclear g-tensor gni includes the chemical shift, which accounts for shielding by
the local electronic environment of the nucleus. The anisotropy of the interaction
of the electrons with the external fields also causes an anisotropy in the chemical
shift. Both of these effects are typically given in units of parts per million of
the nuclear Zeeman energy, and are thus on the order of one part per billion of
the electron Zeeman energy. Though small, these terms become relevant when
and should be taken into account when
they are comparable to quantum I
quantum operation time'
implementing precise qubit operations.
Hnz = -
h
L BognjIi
(2.3)
i
The hyperfine coupling between the electron and nuclear spins is not dependent on
the external field and can range from tens of kiloHertz to hundreds of megaHertz,
depending on the proximity of the spins. It consists of two components: the Fermi
contact interaction and the dipole-dipole coupling. The Fermi contact interaction
is isotropic, and depends on the electron density at the nucleus. The dipole-dipole
interaction gives rise to anisotropic terms. The total hyperfine interaction is given
below by the tensor A, and riis the vector joining the electron and the ith nuclear
spin:
HHF =
(2.4)
SAIi = HFermi +HDD
i
HFermi
=
23 yofI gepe
gnifn, ITo(ri =
24
0) 2 S Ii
(2.5)
i
_i~fi
li)- S[o
3(S -r-)__(r-.
HDD =
Y
gee_
gninii-a
r5
(2.6)
The Hamiltonian of coupled spin-! electron-spin- 2 nuclear systems also contains
several weaker terms.
Nuclear-nuclear dipole-dipole couplings will likely be
present in any le-Nn spin cluster, but are several orders of magnitude smaller than
the hyperfine couplings. Typically on the order of 10 kHz, n-n dipole interactions
need to be addressed if the calculations approach 100 ps in length. Electron-electron
dipole interactions can be quite strong, but like any dipole-dipole interaction, drop
off as r- 3 .Exchange coupling between electrons requires their orbitals to overlap.
In experiments on ensembles of le-Nn spin systems, the electron spins should be
sparse enough that these couplings are negligible. Systems with grouped electron
spins (S > ½)can have a substantial
zero-field splitting; nuclear spins with I > 21
2
can lead to a quadrupole interaction. In future experimental implementations of
the electron spin actuator, it is likely that these effects will be relevant.
2.1.2
1 electron - 1 nuclear spin system
In a large magnetic field, such that Hez >> HHF, the Hamiltonian can be simplified further. We will first look at the Hamiltonian of a single electron coupled to a single spin-! nucleus with external magnetic field Bo = B02. With a
(b) energy levels
14)= III)
(a) Hamiltonian Components
l
w)
I0
~..........
.......
I)
.**
2 34
11T)
0
We
W23
0e
ITT)
ITT)
i1• 2
I-\__
41,
L=V
I1)=ITJ-
IT)W
)u)
electron
Zeeman
12
nuclear
hyperfine
Zeeman
Figure 2-1: Energy levels for le-1n spin system with isotropic hyperfine coupling:
Part (a) shows the energy splittings due to individual terms of the le-1n isotropic
Hamiltonian. The eigenstates and allowed transitions are shown in (b).
g-tensor in the cartesian lab frame, the electron Zeeman interaction is now Hez =
Bo (gzxSx + gzS, + gzzSz), which can be transformed to Hez =
tive g-factor is geff =
BogeffS. The effec-
+ gy
zx + g z2z and S' is aligned along the new quantization
axis, which is 5 = tan - 1 ••Z.) from the lab frame x-axis and 0 = tan - 1
=
from
the lab frame z-axis. Assuming the nuclear chemical shift anisotropy is negligible, the nuclear Zeeman term is aligned along the z-axis: Hnz = y,,Bolz. Since the
electron Zeeman interaction is the dominant term, the eigenvalue of Sz is a good
quantum number for the le-1n system. The hyperfine terms that do not commute
with Sz do not result in first-order energy shifts of the eigenstates, so we neglect
them here. For a completely isotropic hyperfine term, the total Hamiltonian is now
fully diagonal, with A = Azz:
Ho = CoeSz + anIz + 2mrA Szlz
(2.7)
The resulting energy level structure is shown in Figure 2-1.
Now we consider the anisotropic part of the hyperfine interaction, AzxSzlx +
AzySzly, which can be simplified to a single term by an interaction frame transfor-
mation about the z axis with angle
p = tan - 1 (1). The Hamiltonian is now block
diagonal, with B = /Ax + AzY:
H0 = WeSz + WanIz + 2nA Szlz + 27zB Szlx
(2.8)
This Hamiltonian is diagonalized by the unitary transformation
Ud = e - i(OTEIy+O Ee I y )
(2.9)
where El and El are the electron spin polarization operators
1
Ee = -1 + Sz
2
1
Ee =- 11 - Sz
2
(2.10)
(2.11)
and OT and 01 are the angles of the quantization axes in the electron spin-up and
spin-down manifolds, respectively:
=
2
tan
A -
-B
1= 1
01=2
tan-' A+7
2
(2.12)
( -B)
--'
A+
,
)an
(2.13)
Figure 2-2 illustrates the magnetic field vectors seen by the nuclear spin in
this le-in system and the resultant quantization axis. In this model the hyperfine
coupling is comparable to the nuclear Zeeman term. Part (a) shows the components
of the local field at the nuclear spin. As seen in (b), the nuclear spin is quantized
along the z axis, but the local field magnitude depends on the electron spin state.
As shown in (c), the anisotropic term quantizes the nuclear spin along different axes
in the electron spin up and spin down manifolds. The system has the following
(a)
effective magnetic field vectors at
nuclear spin
(b)
(c)
Isotropic hyperfine coupling
! !
Anisotropic hyperfine coupling
i
-a+ tj,
Figure 2-2: Local fields seen by nuclear spin: In this le-ln system, the nuclear
spin is quantized by the sum of the Zeeman field and the hyperfine field. In
(a) the individual components are shown: the Zeeman field w, aligns with the
z-axis, and the isotropic part a and anisotropic part b of the hyperfine field depend
on the orientation of the electron. Part (b) shows purely isotropic coupling with
magnitude a. If the electron spin is along +2 (-2), the nuclear spin feels field a + wi
(-a + woj). Anisotropic hyperfine coupling (c) tilts the nuclear spin quantization
axis away from the z-axis, mixing the nuclear IT)and 11) states in each electron spin
manifold.
14)=110o
W34
13)= [10
w 23
Ih'\
141=
-,
II•yll
_.12q
I1)=tTa0)
Figure 2-3: Energy levels for le-1n spin system with anisotropic hyperfine coupling:
The nuclear spin eigenstates are now mixtures of the nuclear Zeeman states, and the
mixtures are different in the electron manifolds. The nuclear spin with eigenstates
Iao) and lal) in the the electron spin-up manifold is chosen as the qubit in this system.
The light green arrows (color online) indicate that the forbidden transitions are now
allowed.
eigenvalues, and the energy level diagram is shown in Figure 2-3:
11) = ITao) = IT)0 (sin OT IT)+ cos OT i1))
(2.14)
12) = ITa•) = IT)® (cos OT IT)- sin OT I))
(2.15)
13) = 11 pi) = |1) ®(cos 0 IT)- sin 01 Ii))
(2.16)
14) = 11 Po) = 11) ®(sin 0, IT)+ cos0 1I,))
(2.17)
The following discussions of universal control and the use of the electron as a spin
actuator are based on this model.
2.2
Polarization of Nuclear Spins
We can now discuss the effect of AHF coupling on the nuclear polarization scheme
outlined in Chapter 1. The SWAP operation in the eigenbasis of the electron-ith
nuclear spin system can be used to transfer polarization:
UpTi =
Udi
LISWAP(e, ni)Ud, = UdSWAP(e, ni).
(2.18)
= e-i(O•E iy+OEE-y) is the eigenbasis transformation described in equation 2.9.
Note that the first transformation does nothing since the nuclear spin is in the
identity state. In this le-1n subsystem,
Pi =
(2.19)
UPTiPeqUtTi
= (IT ao)
(Taol + 11 go) (I fo I).
(2.20)
Projective measurement of the electron spin has a probability of 1 of yielding
2
the desired state ITao) (T aol and a probability of 1 of yielding 1 f3o) ( 13o.
If
we measure a spin down electron and apply a lab-frame n-pulse to return the
electron to the ground state, we generate the following unwanted populations and
coherences, with q defined as the difference in quantization axis angles 0T - e,:
1 o0)(3o1
ldfie-isx77 Ut.
Ue
diS~u~i
Ee ®(cos 2 (j)
0ao)
(aol0 + Sin 2 (q) ao)
+ sin(q) cos(q) (lao) (al + Ila) (aol))
(aol
(2.21)
Applying a rotation by -, about ly to this state results in the desired state E' ®
lao) (aol. Thus by using operators UpTL = UdtSWAP(e, ni) and Ug, = eihyle - isx" for the
ith nuclear spin, we can use this process to polarize AHF-coupled nuclear spins.
Another polarization method would be to apply to both manifolds the transformation that diagonalizes the nuclear spin in the electron spin-up manifold. To
do this we rotate the nuclear spin about Iy in both manifolds by the angle - 0 Tafter
26
applying the SWAP operation.
Uri =
e- ' UT Y
UpTI, = USWAP(e, nli)Ur, =
(2.22)
USWAP(e, ni)
(2.23)
This operation puts the nuclear spin in the E' manifold in state lao) and the nuclear
spin in the Ee manifold in a state that transforms to lao) when Ug, = e- is x " is applied:
pi=
(2.24)
UPTipeqUpTi
S
IT ao) (T aol + cos2(q) 1 P3o)(I /ol
+ sin 2 (q)
1.
- sin(q) cos(q) (0 f1o)(4 PiJ + 1 fpi) (4 /o 1)
1io) (, pol
(2.25)
In both of these cases, the UpT, andUg, operations affect only the ith nuclear
spin, leaving the already polarized nuclear spins in their lao) states. We can choose
which procedure is better suited for our spin system; it is likely the latter would
be preferred, since the same Ugi can be used for each spin. Repeated N times, this
procedure produces the desired state ITe) 0 (lao))®N.
2.3
Nuclear Spin Relaxation
Using the le-1n system model, we can explore how nuclear spin relaxation differs
between isotropically and anisotropically coupled systems. Starting with a separable initial state with no electron spin coherence, we look at how the nuclear spin
responds to the electron T1 process of random electron spin flips. In the isotropic
case, as shown in Figure 2-2 (b), the nuclear spin is quantized by a field along
the z axis, but the magnitude of the field depends on whether the electron spin
is up or down. We describe the electron T1 process as a generalized amplitude
damping channel, in which random electron spin flips bring the spin to its equilibrium polarization [64]. These flips subject the nuclear spin to a field along 2 of
randomly varying magnitude, causing the nuclear coherence to acquire a random
phase. Measurement of an ensemble of nuclear spins thus shows a decoherence
rate linked to the electron T1. When the anisotropic part of the hyperfine interaction
is added (Figure 2-2 (c)), the orientation of the nuclear spin quantization axis is also
modulated by electron spin flips. The electron T1 processes in an AHF-coupled
system thus cause variations in both the longitudinal and transverse local fields
experienced by the nuclear spins. This leads to nuclear spin depolarization in
addition to decoherence, linking both the nuclear T1 and T2 to the electron T1.
This simple model helps illuminate issues regarding the creation of viable qubits
from these nuclear spins. In the isotropic case, a simple two-spin decoherence free
subspace (DFS) can protect against the collective 2 noise caused by the electron T1
process [12, 26]. The anisotropy of the hyperfine interaction breaks the symmetry
of the noise incident on the nuclear spins in this DFS. Thus the le-2n system does
not provide us with a qubit resistant to electron T1 processes. In the le-3n system,
a noiseless subspace that protects against all collective noise does exist [83, 33].
Further exploration of techniques to preserve nuclear spin qubits, including other
logical encodings and dynamic decoupling, is needed.
2.4
Quantum Information Processing in Electron-Nuclear
Spin Systems
2.4.1
le-1n Spin Systems
Coherent control of ensembles of hyperfine-coupled spins has been demonstrated
in a few simple systems. Using magnetic resonance techniques, Mehring has shown
entangling operations between an electron and nuclear spin in two systems: a
malonic acid free radical [56] and a nitrogen atom encased in C60 [57, 75]. In both
of these systems, he used electron-nuclear double resonance (ENDOR) to generate
Bell states consisting of one electron qubit and one nuclear qubit. The basis states
of this two qubit system are ITT), ITI), 111), and 1IT) (cf. section 2.1.2 and Figure
2-1). The nuclear spin polarization operators are
1
En = 2 + Iz
1
En = 1I- Iz.
2
(2.26)
(2.27)
Starting with the thermal equilibrium state Po, a pseudopure state is created by a
combination of unitary and nonunitary operations:
cos- 1 (-'))SxE_
S)E+Ix
T2e
3
decay
T 2n decay>
Po = -Sz = -SzE+ - SzE n
1
2
4
-SzEn + "SzE- = -Sz - -Sz
+3
3 32 z
2-(-Sz + Iz- Szlz)
3
(2.28)
(2.29)
(2.30)
This last term, after discarding the identity part, is the pseudo-pure state IT). All
four pseudo-pure basis states can be created using similar sequences. Selective
electron and nuclear spin pulses are used to generate the singlet state:
,T2)EI
it)
_
1 + i]T))
• (T)
2)S - •
-
-(IT)
+
14))
(2.31)
Mehring showed that similar sequences can be used to generate all 4 Bell states. He
also performed this experiment on an 15N atom encased in a C60 buckyball, which
has a spin-3/2 electron coupled to a spin-1/2 nucleus. The Bell states were generated
on a 4-level subsystem. In both of these experiments, it was necessary to use three
control fields-one selective electron spin excitation and two selective nuclear spin
excitations. Using the electron spin actuator, it should be possible to precisely
create these entangled states with only modulated electron spin excitations.
2.4.2
le-Nn Spin Systems
We are also exploring spin systems in which a single paramagnetic center is coupled
to more than one nuclear spin. The same malonic acid radical used in the above
experiments has been shown to have a strongly anisotropic hyperfine coupling
with a
13C
labeled (spin-!) methylene carbon [53]. This sample would allow us
to perform experiments such as the demonstration of a nuclear-nuclear gate using
an electron spin actuator and a study of relaxation in two-spin decoherence free
subspaces. A protected subspace in a le-3n system can potentially be demonstrated
on organic crystal radicals strongly coupled to three protons[39]. However, as we
have experienced in our malonic acid experiments, it is difficult to make highquality single crystals from these organic compounds.
Paramagnetic defects in inorganic crystals could also be viable systems for spin
actuators. Defects and dopants in calcium fluoride crystals that exhibit localized
paramagnetic centers have been studied extensively with ESR and NMR. For example, Mehring studied the spin of a cerium dopant occupying a calcium site that
couples strongly to 9 fluorine nuclear spins [31]. These systems are promising because the electron spin is highly localized and fabrication of high-quality crystals
is relatively straightforward.
Mehring's S-bus experiment used electron-nuclear double resonance (ENDOR)
to perform operations on a single electron coupled to several nuclear spins in a
calcium fluoride crystal [58]. Only the nuclear spin states of one electron manifold were used as qubits. Starting from a thermal equilibrium state in which the
nuclear spins are essentially unpolarized and the electron spins have a Boltzmann
distribution, pulses on the electron spin transitions were used to prepare a nuclear
spin register. NMR pulses were then used to perform gates on the nuclear spins,
followed by electron spin pulses to transfer the desired nuclear information back
to an electron polarization for readout.
The nitrogen-vacancy center in diamond is another promising spin system for
QIP. This center occurs when a substitutional nitrogen and a vacancy occupy neighboring sites in the diamond lattice. This results in a spin triplet ground state with
a long coherence time, even at room temperature. An optical transition of the N-V
center can be used to pump the system into its ground state and to measure the
state of a single center. Lukin's group studied the dynamics of N-V centers coupled
to
13
C nuclear spins [15] and demonstrated that this e-n system could be used as
a quantum register [20]. The Awschalom group studied N-V centers coupled to
electrons of substitutional nitrogen spins, suggesting the possibility of a nitrogen
spin chain between N-V centers [30].
Electron-nuclear spin systems in semiconductors are used in a wide variety of
quantum computer implementations. Kane's silicon-based quantum computer [41]
and hydrogenic quantum computer [77] proposals both utilize phosphorus-doped
silicon. In his silicon-based quantum computer proposal, control of individual
nuclear spin qubits is achieved using nuclear magnetic resonance combined with
the ability to manipulate the hyperfine interaction by using electrical gates to shift
the electron's wavefunction. Interactions between qubits are controlled by using
voltage gates to regulate the overlap of wavefunctions of neighboring electrons. In
Kane's hydrogenic spin quantum computer, which is also silicon-based, the qubit
exists in a subspace of a le-1n spin system. Single qubit operations are performed
with electrical gates and external magnetic fields, and qubit-qubit interactions are
achieved by shuttling electrons from one nuclear spin to another. In both of these
proposals, with isotopic engineering of silicon, it is conceivable that each localized
electron could act as a spin actuator controlling a cluster of spin-1/2
29
Si spins.
Quantum dots-semiconductor structures which spatially confine small numbers of electrons-are also promsing candidates for quantum information processors [49, 66, 44]. Research in quantum dots has not yet demonstrated sufficient
electron localization to be used with this implementation of spin actuator control
of nuclear spins. Single-spin measurements in quantum dots and studies of electrical gate control of electron g-factors [84] and nuclear spin cooling in quantum
dots [16, 81], however, do link to our proposed quantum information processor.
Superconducting circuit qubits are also built on semiconductors, and interact
with substrate spins. Though nuclear spins are typically regarded as sources of
decoherence [65, 71, 21], the possibility of coherent coupling between nuclear spins
and superconducting qubits can be considered. These engineered systems offer
some promise of scalability.
Chapter 3
Achieving Universal Control of Spin
Systems
This chapter gives a simple argument showing that universal control of anisotropic
hyperfine coupled electron-nuclear spin systems can be achieved with only electron spin excitations. We then discuss the use of an optimal control algorithm to
precisely generate unitary operations on the coupled spin system, and the system
model parameters needed for accurate control.
3.1
Universality of Electron Spin Actuator Control
Control of an electron spin in an external magnetic field (Bo = B0o) is achieved
by applying transverse fields (BI = B1x + B1,9) at frequencies resonant with the
electron spin transitions. The Hamiltonian describing the interaction of the spin
with this applied field is
H1 = hgeBi(t) - S
=
(3.1)
-geB1(t) (Sx cos(Comwt + q(t)) + Sy sin (comwt + q(t))
At a fixed transmitter frequency
Wrmw,
(3.2)
we have two degrees of freedom that we can
vary with time: the magnitude of the applied field BI(t) and the phase 0(t). In the
(a)
le-1n
isotropic
hyperfine
(b)
le-1n isotropic
using electronnuclear
double
resonance
control
(c)
1le-1 n
anisotropic
hyperfine
(d)
l e-3n
anisotropic
hyperfine
(.i')
\~-
7,
Yi
Figure 3-1: Universal control of le-Nn system with electron spin excitations. Solid
black edges link nodes which are connected by the operator Sx; dashed red edges
link nodes connected by Ix. This figure shows that in a system with isotropic hyperfine coupling, the states cannot be connected with only electron spin excitations
(a) and require nuclear spin excitations to be fully connected (b). AHF-coupled
spins allow forbidden transitions to be addressed, fully connecting the states (c).
Universal control using only electron spin excitations is possible in systems with
N nuclear spins individually coupled to the electron spin via the AHF interaction
(d).
experiments discussed in this thesis, 0(t) = 0 and spin actuator control is achieved
using only amplitude modulations of Sx:
H1 = -geBi (t)Sx cos (Wmwt)
= o)
(t)Sx cos (wmwt)
(3.3)
In Figure 3-1 we use a simple graphical argument to show that we can achieve
universal control of an AHF-coupled spin system by inducing only electron spin
transitions. The nodes of this graph are the eigenstates of the spin system, as
described in the previous chapter. A solid black edge connects nodes m and n for
which the term (m ISx| n) # 0, and the dashed red edge connects nodes for which
(m
n)
tIx s0.
c
In the isotropic le-1n case (a), an electron spin transition Sx connects state 11) to
14) and 12) to 13). In order to address all four states, at least three transitions need
to be addressed. The isotropically-coupled system requires direct excitation of the
nuclear transitions, as shown in (b), for universal control.
The addition of anisotropic hyperfine coupling (c) allows the Sx operator to
address all 4 electron transitions. These matrix elements indicate how strongly
these terms are addressed, assuming an excitation bandwidth larger than coW,
A,
and B (as defined in Chapter 2):
(1 ISxI 4) = cos(0 T - 0Q)
(3.4)
(1 ISx| 3) = sin(01 - 0Q)
(3.5)
(2 Sxl 3) = cos(A - OT)
(3.6)
(2 |SKi 4) = sin(0Q - 0,)
(3.7)
The "forbidden" transitions are strictly forbidden when OT = 0; the forbidden
and allowed transitions are equal in magnitude when |OT-011 = 71/4. This argument
for universality scales to larger numbers of nuclear spins, provided each nuclear
spin is anisotropically coupled to the electron, as shown in (d) for three nuclear
spins. This requirement hints at an upper limit to the number of nuclear spins
that can be coupled to a single electron spin actuator: Degenerate transitions cause
a loss of addressability. The maximum number of spins is a function of electron
wavefunction, hyperfine strength, linewidth, and the ability to effectively excite
the forbidden transitions.
3.2
Pulse Engineering: Using Optimal Control to Perform Precise Unitary Operations
The use of the electron spin as an actuator allows us to improve the efficiency of
control of nuclear spin qubit states: Because the nutation frequency of a spin is
proportional to its magnetic moment, applying control fields directly to the nu-
14)= [-3o)
13)=
mm
12) = Tot)-
j
I1)= ITO)
Figure 3-2: Spin actuator control: With only a modulated excitation on a single
transition (wavy blue line), we can perform a precise unitary operation (e.g., n)sx,
red arrows) that excites all electron transitions.
clear spins requires longer pulse times and more power than addressing only the
electron spin. Control is also simplified, since we can apply any unitary operation on the le-1n system while sitting on a single transmitter frequency (Figure
3-2). To take advantage of the universality provided by the AHF interaction, we
used an optimal control algorithm to find pulse sequences implementable by our
spectrometer. The Gradient Ascent Pulse Engineering (GRAPE) algorithm, developed by Khaneja [42], finds pulses that implement the desired unitary operations
with high fidelity. This gradient-based pulse optimization scheme represents an
improvement over previous methods with a substantially more efficient method to
calculate derivatives of performance functions.
We seek to apply a desired unitary operation Uw,,ant by applying a set of m
control Hamiltonians Hk to the spin system with natural Hamiltonian H0 . The
control sequence is discretized such that the propagator during time step j is
Uj = e-iAt(Ho+Eki=
uk(j)Hk)
(3.8)
A control sequence of N steps gives the total propagator
Utotal = UNUN-1...U 2 U1.
(3.9)
The control Hamiltonian amplitudes Uk define the modulation of the microwave
excitation-they are the knobs we turn to achieve Uwant. We start with a random
sequence of Uk's and calculate a goodness function P from the sequence. In this
case we use the fidelity squared
(
= (UwantlUtotal)
2
((3.10)
= (UwantIUNUN-l...U 2 U (U1 U2 ...UN-1 UN Uwant).
(3.11)
Using the fact that the inner product is invariant under cyclic permutation,
(UwantlUtotal) = (W U
UwantIUjUj-...U2 U1 .
+2 ..."
(3.12)
We give the bra and ket of equation 3.12 the labels Pj and Xj, respectively:
U ant
+2
Pj =(,
Xj) = ujU j-1... U2U )
(3.13)
(3.14)
Using the matrix exponential formula
d X(t) 10=
1i
e(1-a)x(t)
edtX
dX
eaX(t)dct
(3.15)
we calculate the derivative, to first order, of the propagator Uj with respect to the
amplitude Uk:
=Uj-iAtHkUj
6
Uk
(3.16)
with
-
Hk =
1
-
t
0A
Uj(c)HkUj(-T)dT
Uj(
1 ) = e-iT(Ho+EY=l uk(j)Hk)
(3.17)
(3.18)
For small timesteps, Hk ; Hk and we can calculate the gradient of the goodness
Figure 3-3: GRadient Ascent Pulse Engineering (GRAPE): The pulse is comprised
of N steps of length At. The parameter uk is the amplitude of control Hamiltonian
Hk. Each iteration of the algorithm calculates the goodness of the pulse and a
gradient used to adjust Uk for each timestep in the pulse.
function:
=6-(PjlXj)
6
(iAtHkXjPj) - (PjliAtHkXj) (XjlPj)
Uk
= -2Re ((PliAtHkXj) (X;iP))
(3.19)
A threshold is set for the goodness function, and if it is not met, the control parameters Uk are adjusted according to the calculated gradient:
&I>
uk(j) - Uk(j) + e
6Uk(j)
(3.20)
This sequence, as illustrated in Figure 3-3 from [42], is repeated until a set
of parameters Uk generates a propagator with sufficient fidelity. To illustrate the
evolution of the spin states during application of a GRAPE pulse, Figure 3-4 shows
the evolution on the Bloch sphere of fictitious spins of a le-1n system during
application of a )12 pulse between two eigenstates.
We see that the pulse reliably executes the desired operation, and the evolution
during the pulse is complex: The trajectories of the fictitious spins change drastically and frequently, and can traverse a large range of the Bloch sphere. This
modulation averages out unwanted evolution in the system, as shown with the
strongly modulating pulses used in liquid-state NMR quantum computing studies
[25].
12
z
12
y
(.24 -io724
z
2 Z
z
-4
O 2 12 12
24
24
24
34
34
34
Figure 3-4: Evolution of spin during a )12 n•2
pulse:
puse
Bloch sphere plots of the trajectory
of fictitious spins during application of a ,x pulse.
3.3
Full System Model
3.3.1
Quality Factor of Resonator
Magnetic resonance experiments-especially those in which weak measurements
are performed on an ensemble of spins-typically employ resonators to amplify the
coupling between the spins and the excitation/detection circuitry. At the resonance
frequency, the oscillating magnetic field applied to the spins and the sensitivity of
detection of the precessing spins are enhanced. The quality factor (Q), defined as
2n times the ratio of the power stored in the resonator to the power dissipated in
one oscillation cycle, quantifies the enhancement due to the resonator. To see how
this affects the control of the system with GRAPE pulses, we recall the relation
of the quality factor to the resonator bandwidth: Q = f, where
fo
is the reso-
nance frequency and Af is the bandwidth. The resonator filters the GRAPE pulse,
attenuating modulation frequencies that exceed the bandwidth of the resonator.
Simulations show that this effect significantly degrades the fidelity of the GRAPE
pulses; the experimental results reported in this thesis also indicate this. Subsequently, our GRAPE code has been modified to search for modulation sequences
that account for this filtering, allowing us to find high-fidelity pulses while using
resonators with relatively high Q for pulsed ESR Q > 200 [3].
3.3.2
Inhomogeneity of Magnetic Fields
The simple Hamiltonian presented in Chapter 2 does not account for inhomogeneities in B0 and B1 . Challeges in designing fields for quantum control have
been discussed by Rabitz [10, 68]. In liquid state NMR experiments, open-loop
feedback and pulses robust against B1 inhomogeneity have been demonstrated
[8]. We assume that all the spins in the ensemble experience the same magnetic
fields, but experimentally there are often slight variations across the sample. Spatial static field inhomogeneity can be caused by magnet nonuniformity, sample
shape effects, or the presence of magnetic materials in the probe. Spatial B1 field
variations are largely caused by the resonator. The solenoidal fields of loop-gap
resonators and bridged-loop-gap resonators (cf. Chapter 4) are quite uniform, but
imperfections in fabrication, sample location and size, and interactions between
the sample holder or the sample with the fringing electric fields can lead to B1
inhomogeneity. GRAPE can find pulses robust against these inhomogeneities, but
an accurate model is necessary.
3.3.3
Crystal Quality
High quality single crystals are necessary for these spin ensemble experiments. Ideally, all electron-nuclear spin systems in the ensemble have identical Hamiltonians;
experimentally, the Hamiltonians are often altered by impurities, twinning, and
other crystal defects. Organic crystals, such as the malonic acid used in these experiments, often trap water during crystallization. This can alter the environment
of nearby spins and cause localized heating when oscillating fields are applied.
3.3.4
Spin System Hamiltonian
When searching for pulses, we have considered only the largest components of
the Hamiltonian-Hez, H 1z, and HHF-and considered the weaker terms to be
negligible. The GRAPE code thus finds pulses that perform unitaries on a certain
Hamiltonian, often neglecting the full linewidth of the electron transition. In
malonic acid, for example, the 5 gauss ESR linewidth is caused by interactions with
both the nearby nuclear spins and the more distant matrix protons [47]. Without
using a good model of this, the GRAPE pulse will operate properly only on a subset
of spins.
Chapter 4
Spectrometer Design
Pulsed ESR and ENDOR have been used for spectroscopic applications for more
than 40 years [59, 60]. The available commercial pulsed ESR systems, however,
do not provide the flexibility we need to implement the optimal control pulses.
This chapter discusses the key components of our current and future spectrometer
designs.
4.1
Pulsed ESR Spectrometer: Version 1
4.1.1
Design Specifications
Conducting pulsed ESR experiments at X-band (8-12 GHz) allows us to achieve acceptable polarization and sensitivity without making the implementation of precise
control prohibitively expensive or difficult. More importantly for our experiment,
it makes the nuclear Zeeman term comparable to the hyperfine term, maximizing the magnitude of the forbidden transitions. With the B0 field for an X-band
electron spin transition, the nuclear Zeeman term is on the order of 10 MHz. The
strength of the hyperfine coupling can range from tens of kiloHertz to hundreds
of megaHertz for nearby spins. Using the GRAPE algorithm to find spin actuator
control sequences, we see that a B1 smaller than the nuclear Zeeman and hyperfine
coupling strengths is sufficient. In general, the higher the B1, the more efficient the
control, but this is often costly to achieve. For the malonic acid radical used in our
experiments, we could find sub-microsecond pulses for all desired unitaries with a
B1 of 4 MHz. Another requirement is that our sensitivity needs to be high enough
to measure the signal from a sparse ensemble of spins at cryogenic temperatures.
If the T, is sufficiently brief, we can signal average to improve the signal to noise
ratio (SNR). Reliable averaging requires the absence of low-frequency noise and
drifts in the spectrometer.
4.1.2
Probehead
Resonator
The microwave fields are contained in a simple loop-gap resonator (LGR), introduced by Froncisz and Hyde [27, 54, 55]. The LGR, a lumped-element resonator
with an inductance and capacitance determined by its geometry, has several properties well-suited for pulsed ESR: small volume (large filling factor), uniform magnetic field, and low quality factor (short ringdown time, large bandwidth). For this
spectrometer we designed a two-gap one-loop resonator, a simple hollow conductive cylinder with two slits cut along the length of the cylinder (Figure 4-1). We
can approximate the inductance L as that of a solenoid and the capacitance C of the
gaps as that of parallel plate capacitors in series (r is the radius, Z is the length of
the LGR and the capacitor plates, W is the capacitor plate width, t is the gap size,
and n is the number of gaps):
2
/.o0r
L
2
(4.1)
Z
eWZ
-
C=
(4.2)
nt
The resonant frequency is thus
1
fofo27T -\
C
1_
2
271
nt
eor 2 W
nePOT2W
(4.3)
z
Figure 4-1: Loop-gap resonator and coupling loop: The resonator was fabricated
by cutting two gaps out of oxygen-free high-conductivity copper and mounting it
on the inside of a quartz tube. Inductive coupling to the microwave amplifier and
receiver was provided by a loop formed from the center conductor of a semi-rigid
coaxial cable.
More accurate approximations of the resonant frequency have been reported
[27,54,55]. The maximum microwave field strength at the center of the resonator is
given in [76], where QL is the loaded quality factor of the resonator, P is the incident
power, and Vc is the effective volume of the resonator:
S2QLPYO
B1 = •Vfo
(4.4)
The sensitivity S of the resonator depends on the same parameters:
QL 02
S oc
VT
Vc
(4.5)
At the resonance frequency the fields in the loop-gap resonator are solenoidal.
The current flows along the circumference of the cylinder, and the magnetic field is
directed along the axis of the cylinder. Figure 4-2 shows a vector plot of the magnetic
field at resonance, generated by Ansoft HFSS, a commercial finite element method
H Field[A/m]
5
2599e+02
4.9265e+02
4.5981e*02
,2696e+02
4
3,9412e+02
3.6i2e+02
3 2899e+02
2.9559e+02
2.6275e-02
2.2991e+02
1.9707e+02
1.6922e402
1.3138e+-02
9.,8540e+01
6 5698e+01
3.2855e+01
i.2785e-02
Figure 4-2: Magnetic field vectors in LGR: This vector plot of the magnetic field
along the center axis of the LGR shows the desired mode at resonance. The field is
directed along the z-axis, and is at a maximum at the center of the resonator. The
size of the arrows scale with the magnitude of the field vector at the origin of the
arrow (color online).
(FEM) software package. The size of each arrow is proportional to the magnitude
of the magnetic field vector at the origin of the arrow. The simulation shows the
expected field, a solenoidal field with a maximum at the center of the resonator.
Our LGR was fabricated from oxygen-free high conductivity copper mounted
with cyanoacrylate glue in a quartz tube. With a diameter of 3 mm, gap sizes of
about 0.6 mm, and length 8 mm, the LGR exhibited the desired loaded resonance
of 11 GHz. The resonator is placed inside a 12.75 mm diameter microwave shield
also made from copper tube. The loaded quality factor of the resonator is measured
to be approximately 250, and the maximum B1 field measured was 2.5 Gauss (7
MHz).
Coupling
Inductive coupling to the resonator was accomplished by terminating a semi-rigid
coaxial cable with a loop formed out of the coax's center conductor, as drawn in
Figure 4-1. Coupling via mutual inductance is one of a variety of coupling methods
for loop-gap resonators, and the most commonly used in the literature [70]. Optimal
power transfer to the LGR, as with all transmission line systems, requires that the
impedance of the transmission line matches that of the load (the loop terminationresonator system). While many researchers overcouple their resonators to lower
the Q (and thus the ringdown time), we chose critical coupling to maximize the B1
field and because the ringdown time does not limit this experiment. The critical
coupling condition is met when the resonator is at a certain angular orientation
and distance relative to the loop. The angular orientation is adjusted prior to
closing and pumping down the spectrometer. Using a motion stage and beryliumcopper bellows, we are able to control the distance between the loop and resonator
while under vacuum. This is necessary to optimize matching to the resonator,
as the stainless steel cold finger shrinks by about 3mm when cooled from room
temperature to liquid nitrogen or helium temperatures [22].
Characterization of resonators is performed by measuring the reflected power
from the coaxial cable with either a network analyzer or microwave detector diode.
Access to a network analyzer was limited, so determination of the resonance of the
probe was often performed by systematically varying the matching and using
a diode to measure the reflected power vs. frequency. This was necessary because absorption peaks arising from the copper shield and the transmission line
configuration often obscured the resonance peak from the resonator. Microwave
resonances arising from the box containing the probe were attenuated by inserting
dissipative material into the free space of the box containing the probehead. We
found that the anti-static bags used to package sensitive electronic components
worked well for this purpose.
Sample Holder
The sample was attached with cyanoacrylate glue to the end of a sapphire rod
attached to the end of the cold finger cryostat. Since the sample is in vacuum, its
alignment could not be adjusted after the cryostat was cold. This presented some
challenges in orientation of the sample relative to B0 . To address this, we made a
sample holder that could be transferred at a fixed orientation to a CW-ESR system
where we were able to characterize the sample. The error in rotational alignment
of the crystal is estimated to be +5'.
4.1.3
Pulse and Receiver Electronics
The excitation and detection of the electron spin transitions are implemented with
a standard heterodyne system, as shown in Figure 4-3. On the input side, a singlesideband upconverter first mixes the microwave source signal with a 160 MHz
radiofrequency signal. The signal is subsequently modulated with the GRAPE
pulse shape from a 4 ns resolution arbitrary waveform generator. To implement
phase cycling, this signal is fed into a 6-bit digital phase shifter. Finally the excitation signal is amplified by the 12 W power amplifier and transmitted to the
resonator.
The receiver needs to be sensitive enough to detect the weak signal emitted by
the spin system. To protect the receiver a fast PIN diode switch is left open as
the excitation pulses are delivered to the resonator. After allowing the resonator
to sufficiently ring down, the switch is closed and the signal is passed through
a low-noise preamplifier, mixed down by the microwave carrier signal, and then
amplified again at the intermediate (160 MHz) radiofrequency. At this point the
signal is split into its quadrature components by a 90 degree hybrid, mixed down
to DC, and recorded by a two-channel high-speed digitizer. The choice of detection
phase is implemented by mathematically shifting the phase of the acquired data.
A single computer controls the system timing and data acquisition. PCI card
versions of a 24 channel, 3.3 ns resolution pulse generator and a 10 ns resolution
-
-
'1~'
'5 1 ,Il
~I--"---
-.
ll
is
~II~-
Single-sideband
upconverter
6-bit digital
phase shifter
A
Oh
w~
mixer
-- +
I
w~
fPw7
Loop-gap
T
circulal
re~anmtor
-.
i
PIN
I
diode
switrh
solid-state
source
microwave
--
I
)I
-C
I
111
loop
receiver
front end
pre-- - - - -- - - -
---
1
aamp
,,on'c
mixer
________
A
M"....
luw-pass
filters
Mixers
9
IP hybrid
A
44~1
·s
•q
Figure 4-3: Block diagram of pulse and receiver electronics: The signal paths in
the first version of the spectrometer are shown in this simplified block diagram.
The microwave carrier signal at the frequency of an allowed electron transition is
modulated by the GRAPE pulse mixed in from an arbitrary waveform generator.
The power amplifier delivers this pulse to the resonator, and after ringdown the PIN
diode switch is closed so the signal from the electron spin can enter the receiver. The
signal is amplified with a preamp, mixed down to an intermediate RF frequency,
separated into quadrature components, mixed down again to DC, and digitized.
Phase cycling is implemented with a digital phase shifter.
2-channel digitizer are installed in the computer itself, and the shaped pulses are
transferred to the arbitrary waveform generator (AWG) via GPIB. The triggering of
the power amplifier, the switch protecting the receiver, the AWG, the phase shifter,
and the digitizer are all controlled by the pulse generator. The program to execute
the experiments and acquire the data was written in a combination of LabView and
C++.
4.2
4.2.1
Pulsed ESR Spectrometer: Version 2
Design Goals
To progress further on attaining coherent control of electron-nuclear spin systems,
several modifications to the spectrometer are underway. The results of our experiments highlighted some shortcomings of the first implementation of the spectrometer. One key problem was the inability to cool the sample down to 4K due to
the cold finger setup and thermal leaks between the sample and other parts of the
probehead. Another problem was limited control of the angular orientation of the
sample. Also, the phase noise of our microwave source is rather high compared to
the sources typically used for ESR experiments. The modifications described here
address these issues.
4.2.2
New Probehead
Resonator
The new probehead incorporates a bridged loop-gap resonator (BLGR), introduced
by Pfenninger et al. for its transparency to radiofrequency fields [24, 67]. As shown
in Figure 4-4, this is similar to the loop gap resonator but the metal pieces are
now thinned down to films, and bridges are added to provide capacitance. This
confines electric fields to the space between the layers, reducing the dielectric loss
and heating that occur when electric fields pass through a lossy sample. Also, this
H Field[A/m]
1,6385,5o÷0
4.34911+02
4.0597e+0e2
3,7703e+02
311808e+02
3. 191i4e.2
2.9020e+02
2. 6125e*02
2. 3231e+82
2.0337e+02
1.7443e+02
1.45460+.2
1.1654e+02
8.7598:+e1
5.8655 +01
2.9713e+61
7.694e-81
Figure 4-4: Magnetic field vectors in BLGR: This vector plot shows the desired
mode at resonance. The field is directed along the z-axis, and is at a maximum at
the center of the resonator.
structure allows for tunability by varying the capacitance: In our case we rotate
the bridges relative to the gaps, similar to a tunable BLGR implemented at a lower
frequency by Alfonsetti et al. [4]. A circuit model of the BLGR is found in a paper
by Hirata and Ono [32]. While this model was helpful for quickly approximating
the resonator parameters, we found that finite element method calculations were
needed to more accurately design the resonator and tuning mechanism. These
calculations were performed using Ansoft HFSS.
We fabricated a BLGR with a 4 mm diameter loop with two 0.5 mm gaps and
two 2.0 mm wide bridges at a diameter of 5 mm. The length of the resonator
is 8mm. The resonator is comprised of two quartz pieces held separately (to
enable tuning) by rexolite pieces. Rexolite was chosen for its low microwave loss
and easy machinability. The loop and bridges were initially made using a silver
paint with acrylic binder. However, the quality factor of this BLGR is rather low
(Qunloaded
<
100) and we are exploring alternative methods for making the metal
05 Feb 2008
Ansoft Corporation
XY Plot 1
HFSSDesign2
12:42:24 PM
Y1 --dB(S(LumpPortl,Lum
BLGRangle=5deg
Setup1 :Sweep 1
Y --CI--
dB(S(LumpPortl,Lum
BLG Rangle= 10deg
Setup1 :Sweepi
Y1 -I---
dB(S(LumpPortl,Lum
BLG Rangle= 15deg
Setup i :Sweep i
Y1-7-dB(S(LumpPortl,Lum
BLGRangle=20deg
Setupi :Sweepi
r
dB(S(LumpPortl,Lum
BLGRangle=25deg
Setupi :Sweep 1
Y 1 -0--
dB(S(LumpPortt,Lum
BLGRangle=30deg
Setup :Sweep i
Freq [GHz]
X1= 9.82GHz
Y1= -12.12
10.10GHz
12= -12.51
3= 10.57GHz
13- -9.17
Figure 4-5: S11 vs. frequency with varying tuning angle: The ratio of reflected
signal to incident signal (S11) at the input to the coupling loop is plotted vs. input
frequency for a range of tuning angles. The tuning angle is the difference between
the angular orientation of the bridges and the gaps. The matching, which is
controlled by the distance between loop and the resonator, is not adjusted during
this simulation. FEM calculations show a tuning range of ~750 MHz for this BLGR.
layers, including chemical deposition of silver [80]. The inductive coupling used
with the first version of the spectrometer is unchanged.
Our measurements of this resonator correspond well with the simulations,
showing a resonance close to 10 GHz with a tuning range of more than 500 MHz.
The fields in the BLGR are also solenoidal, as shown in the Ansoft HFSS vector plot
(Figure 4-4). This plot shows a magnetic field directed along the axis of the BLGR,
with its maximum magnitude at the center of the resonator. FEM simulations also
show a large tuning range: Figure 4-5 shows the ratio of reflected to incident signal
at the input port (S11 parameter) vs. frequency for a range of tuning angles. These
dips in S11 show that by varying the relative angle between the bridges and the
gaps, the resonance frequency of the resonator can change by 750 MHz.
Sample Holder
Instead of relying on the thermal conductivity of the sample holder to cool the
sample, we made a sample holder that allows immersion of the sample in cryogen.
This piece was fabricated from Kel-F (PCTFE), which is useful for its low gas
permeability and thermal expansion properties. The walls were made very thin
(0.25 mm) to limit dielectric losses and to allow the sample holder to fit in the
resonator without direct thermal contact.
4.2.3
Improved Electronics
Many of the electronic components of the spectrometer have also been improved
(See Figure 4-6).
We have acquired an X-band bridge with a klystron source,
which has lower phase noise than our original microwave signal generator. A
more powerful microwave amplifier and a two-channel, higher resolution arbitrary
waveform generator will allow us to attain better control with the spin actuator
method. Also, a digitizer with a faster sampling rate will allow us to measure the
echo signal more accurately, and to better analyze the quality of our GRAPE pulses.
MRT"17
•Iltj;ll
" "'li
tI.
I
Y comnponent
WKJ-
QU
6-bit digital
phase shifteir
Single-sideband
upconverter
|
I
M1 -
a .ir;
W
m
Turnahre
Bridged
Loop-gap
Resonator
Quadrature
mixer
1l
)op
mr a
sourceLA
m t--
.z
t.-
com puter__
---------
1I
I
12 HzI
Diitzer2
1
1m
·- ~Q·----·
low-pass
filters
Mixers
-.0'IN"
U(Phybrid
I
'.
Figure 4-6: Block diagram of pulse and receiver electronics, version 2: Improvements are indicated in underlined red text (color online). In addition to the new
resonator, the solid-state microwave source has been replaced by a klystron, an
additional channel has been added to the arbitrary waveform generator, the amplifier power has been increased, and the sampling rate of the digitizer has been
improved.
Chapter 5
DNP Probe
As with the resonator described in Chapter 5, finite element method simulations
were used to design a resonator for dynamic nuclear polarization experiments. Our
goal for this resonator is to maximize the microwave-frequency (66 GHz) magnetic
field, to be able to tune the resonator at low temperature with a variety of samples,
and to be able to move the sample to a nearby RF coil to measure the nuclear
magnetic resonance signal from the sample. A cylindrical TE011 cavity works well
in this situation: The frequency of this cavity can be tuned by moving the ends of
the cylinder; the desired mode does not require current flow between the ends of
the cylinder and the body (in fact, breaking the electrical connection here serves
to filter out some undesired modes); the mode is suited to using a sample holder
along the axis of the cylinder, which is necessary for movement of the sample from
the cylinder into the RF coil. The resonance frequency fo of an ideal, unloaded
TE011 cavity is well known [40]:
(R
2(.82c2
(2R f
2
0) =
2
3832c) + R()
77
L
(5.1)
L is the cavity length, R is the radius, and c is the speed of light in the cavity
space. However, with the introduction of the relatively large sample and sample
holder, a hole on the endplate to allow sample movement, and the coupling iris, it
became expedient to use Ansoft HFSS to design the resonator. The cavity shown
Figure 5-1: TE011 cavity for DNP experiment: This resonator was designed to
permit sample entry from the side, tunability by variation of cavity length, and
matching with a three-stub tuner.
E
Figure 5-2: TE011 magnetic field vectors: This plot shows the magnetic field
along the center axis (2 direction) of the
resonator. This is the desired TE011
mode.
,
Figure 5-3: TE011 electric field vectors:
This plot shows the electric field vectors
of the TE011 mode in the resonator. The
field is entirely in the azimuthal direction.
in Figure 5-1 has a diameter of 5.5 mm, a 1 mm diameter circular iris, a triple-stub
3
tuner, and a length which can vary from 4 mm to 10 mm to accomodate 1 mm
samples with relative permittivities of 1 to 12. The TE011 mode magnetic and
electric fields are defined by the following equations, in cylindrical coordinates
(r, 0, z) with the 2 direction along the axis of the cylinder [40]. Jo is the zeroth-order
Bessel function of the first kind, and Jo is the first derivative of this function. The
parameters kr and kz are the wavevectors in the radial direction and along the z-axis,
respectively, and (krR)oI is the first root of Jo(krr).
(krR)ol
kr
=
R
R
kz
= -
=
3.832
R
(5.3)
L
Hr =
(5.2)
kzHo
Jo(krr) cos(kzz)
(5.4)
Jo(krr) sin(kzz)
(5.5)
Ho Jo(krr) sin(kzz)
(5.6)
2+k2
krHo
Hz =
Ecp
=-
2 +2
Hp = Er = Ez = 0
(5.7)
Shown in Figure 5-2 and Figure 5-3 are vector plots of the TE011 mode magnetic
fields along the z axis and the electric fields in the resonator space, respectively.
These plots show that the modifications made to this cavity do not significantly
distort the vector fields defined in the above equations. Figure 5-4 shows that a
tuning range of 2.3 GHz can be achieved with a change in cavity length from 10
mm to 6 mm. These simulations were performed using a sample with a volume
of -1 mm 3 and a relative permittivity of 4. Simulations also show that the ability
to vary the length of the cavity allows this resonator to also accomodate silicon
samples, which have a relative permittivity of 12.
Ansoft Corporation
iris diam 1.1mm
HFSSDesign1l
05 Feb 2008
03:10:24 AM
Y1 I--dB(S(WavePort 1,WavePor
Plunger_Location= 1.4mm
Setup1 :Sweep 1
Y1 -E---
dB(S(WavePort 1,WavePor
PlungerLocation= 1.6mm
Setupi : Sweepi
Y1 -+--dB(S(WavePortl,WavePor
PlungerLocation= 1.8mm
Setup1 : Sweep i
-10.00
YI -.
--
dB(S(VavePort i,WavePor
PlungerLocation=2r 2mm
Setup I : Sweep 1
-15.00- -
-20.00.--
___
__
----------------____
{--.-.
-25.00--
64.
00
65-b
600
[G9.
67
Freq [GF~z]
K1= 64.91GHz
YI= -8.48
__
. . . 6e
-
30
K2= 67.23GHz
IY2= -7.20
Figure 5-4: S11 vs. Frequency with varying cavity length: The ratio of reflected
voltage to incident voltage (S11) at the input of the waveguide is plotted vs. input
frequency for a range of cavity lengths. Matching using the triple-stub tuner is not
included in this simulation. These FEM calculations show the resonance frequency
varying by ~2 GHz by changing the cavity length from 6mm to 10mm.
Chapter 6
Experimental Results
This chapter presents a series of experiments on a malonic acid single crystal that
demonstrate control achieved using the electron spin actuator. Using GRAPE
pulses to perform electron spin echo envelope modulation (ESEEM) spectroscopy,
we show that we can selectively excite electron and nuclear coherences while applying a relatively weak microwave field at a fixed transmitter frequency. We then
performed a simple preparation-quantum gate-readout experiment, as outlined
in Chapter 1, in which we used GRAPE pulses to generate and refocus a nuclear
spin coherence. The results of these experiments correspond well with the simulations, and provide evidence of efficient and universal control using an electron
spin actuator.
6.1
le-1n Spin System: Malonic Acid Radical
The experiments described in this thesis were conducted on X-ray irradiated single crystal malonic acid, a compound consisting of a methylene group bonded
to two carboxyl groups. Irradiation with X-rays generates several radicals, and
subsequent annealing leaves a single stable radical [73].
The crystal we used
was irradiated with 75 keV x-rays for 2 hours and subsequently annealed for 12
hours at 50 degrees Celsius. This radical, a free electron in the pi orbital of the
methylene carbon, is strongly coupled to the remaining proton from the methy-
CH
Figure 6-1: Malonic acid radical: X-ray irradiation of malonic acid, a molecule
comprised of two carboxyl groups joined by a methylene group, generates a stable
radical. This well-studied paramagnetic center is a free electron localized in the
n7-orbital of the methylene carbon. Its strong coupling to a single proton and weak
coupling to the more distant protons make this malonic acid radical a good testbed
for spin actuator control of a le-1n spin system.
lene group (see Figure 6-1).
This system has been studied by several groups
[6, 17, 47, 51, 52, 63, 73, 74], and the CW-ESR measurements of our crystal correspond well to the reported data. The crystal has two preferred cleavage planes that
intersect along the axis parallel to the direction of the chains of hydrogen-bonded
malonic acid molecules. Since this long crystal axis is nearly parallel to a principal
axis of the hyperfine tensor, it is straightforward to align the crystal in a useful
orientation in the loop-gap resonator. This alignment maximized the volume of
the sample inside the resonator and allowed the sample to be rotated between two
principal hyperfine axes. To maximize the strength of the anisotropic hyperfine
interaction in this plane, we rotated the sample to place the external field direction
halfway between the two principal axes. The error in this angular orientation was
estimated to be
H
±50.
Using a static B0 field of 4240G, our Hamiltonian is:
2n 11.885 GHz Sz - 2n7 18 MHz Iz - 271n 43 MHz Szlz + 271n 14 MHz Szlx
(6.1)
This system has nuclear spin quantization axis angles OT
160 and 0
2.50, and
the following eigenstates and transition frequencies:
11) = ITao)
IT)0 (0.53 I1T)
+ 0.85 1t))
(6.2)
12) = ITa,)
IT)0 (0.85 IT)- 0.53 11))
(6.3)
13) = 11 Mi)
11) 0 (1.0 IT1)
- 0.096 11))
(6.4)
I1)0 (0.096 I1T)
+ 1.0 I1))
(6.5)
14) -= 1 fo)
-
1w141
2n 11.909GHz
(6.6)
1(0121
2n 8MHz
(6.7)
1(341
2n 40MHz
(6.8)
I231
2n 11.861GHz
(6.9)
Our experiments were performed with electron spin excitation and detection at
11.909 GHz, with a maximum B1 field strength of 2.5 Gauss (7 MHz).
6.2
ESEEM
As described in previous chapters, in electron-nuclear spin systems with anisotropic
hyperfine coupling, a microwave pulse can generate coherences on both the allowed and forbidden transitions. In an ESEEM experiment, these coherences are
observed as modulations of the envelope of the electron spin echo and provide
information about the structure of the sample [60, 76]. This has been particularly
useful in spectroscopy of systems with small hyperfine splittings, since a splitting
detectable as a slow modulation of the echo envelope can be obscured in other
methods by broad linewidths. A more quantitative understanding of pulsed electron spin resonance experiments can be achieved with product operator formalism
[37, 69, 76]; in this section I use this formalism to describe the results of two-pulse
ESEEM experiments using hard pulses and using engineered non-selective pulses.
6.2.1
Hard Pulses
In a typical pulse ESR experiment, the maximum power is applied for a short period
of time to generate a specific rotation of the electron spin. Such pulses-called hard
pulses-excite a range of frequencies inversely proportional to their length. A hard
pulse with a weak B1 field can thus only excite a small range of frequencies. (In
NMR these weak pulses are typically called soft pulses, but in this thesis I will call
any pulse in which constant microwave power is applied a hard pulse.)
It is useful to first discuss the spin echo in the absence of an anisotropic hyperfine
interaction. In the case of a weak constant-power pulse, where the hyperfine
term A is sufficiently larger than the applied microwave field B1, the spin echo
experiment observes only electron two-level dynamics entirely in a single nuclear
spin manifold. In our experiments, we addressed and observed the larger electron
spin transition (the I1) - 14) transition). In the isotropic case, in which the nuclear
spin is quantized along the same axis as the electron spin, we observe a spin echo in
the nuclear spin down manifold while the spin in the other manifold is untouched:
)SE
H-n)SxE"_
H0oc
H
po = -Sz = -SzE+ - SzE"
(6.10)
-SzE+ - SyEn
(6.11)
-SzE
+ (Sy cos((Qs - An)T) - Sx sin((7Ds - An)iT) 0 E
(6.12)
-SzE+ + (-Sy cos((i2s - An)T) - Sx sin((Qs - A)•c) 0 E_
(6.13)
-SzE+
_
- SyE_
(6.14)
The detected signal at t = 2c is unmodulated:
(Sx + iSy) = -i/2
(6.15)
If this weak pulse is used to perform ESEEM on an anisotropically coupled
system, the echo envelope modulation reveals information about energy level
Srinvglltinn nf Mannittid, nf Tima fnrmain ýiLn l fnr r-QnF
t^,-ý 1 Qnn.-
07
0.6
0.5
I-
0.4
0.3
0.2
0.1
Time Domain, Centered on Echo Peak (~s)
Figure 6-2: Simulation of ESEEM with hard pulses: Magnitude of the time-domain
echo signal for values of z from 800 ns to 1800 ns (color online).
Figure 6-3: ESEEM experiment with hard pulses: Magnitude of time-domain echo
signal for values of T from 800 ns to 1800 ns (color online).
Magnitude spectrum of indirect domain
simulation
0.18
0.16
0.14
0.12
0.08
0.06
0.04
I2
0.02
jot
60
-_ u-20
-40
"'
0
Freq (MHz)
.L
.
-
--
20
~
I
4
Figure 6-4: ESEEM with hard pulses: Magnitude spectrum of echo peak vs. T data
from simulation and experiment. Both show strong excitation of the 012 coherence
and weak excitation of higher frequency coherences.
splittings within the excitation bandwidth of the pulse. In the experiment, 71/2
and n rotations are generated with 36 ns and 72 ns pulses, respectively, using a B1
of 2.5 Gauss (7 MHz) on resonance with the 11) - 14) transition. Since the pulses are
neither fully non-selective (i.e., able to excite all electron transitions) nor perfectly
selective, the analytical representation of the results of this experiment is rather
involved [37]. Numerical simulation of ESEEM using these pulses show excitation
of coherence on the I1)- 14) and 12) -14)
I1)- 13) and 12) -13)
transitions, and very weak excitation of the
transitions. Figure 6-2 shows the time-domain results from the
simulation. Each row of this plot shows the magnitude of the echo signal obtained
for a specific T. A B0 inhomogeneity (normal distribution with 3 Gauss variance)
was simulated to decohere the terms not contributing to the echo. The delay time
- is varied from 800 ns to 1800 ns. We see in this plot a modulation of 8 MHz,
indicative of the 12) - 14) coherence.
The experimental data (Figure 6-3) show this same modulation and a decay in
the magnitude of the echo peak. From the decay we calculate an electron T2 of
15 ps and a single modulation frequency of 8 MHz. The magnitude spectrum of
the echo envelope data corresponds well with the simulation, as shown in Figure
6-4.
6.2.2
Shaped Sx pulses
By using shaped pulses engineered to generate the unitary Usx = e-isx, we can
perform an ESEEM experiment in which all electron transitions are excited, as if
we were able to implement a short, high-powered pulse.
We first look at the results of an echo on an isotropically coupled le-in system
using pulses generating an electron spin rotation in both nuclear manifolds. Note
that unlike above, where a n7/2 - T we discuss a n/2 -
--S
HoT
-c
- z - echo experiment was described, here
- n7/2 - T - echo experiment.
pO = -Sz
(6.16)
Sy
(6.17)
Sy cos(Qsr) cos(AnT) - Sxlz cos(Osz) sin(Arz)
-S, sin(CsT) cos(AmT) - Sylz sin(QsT) sin(AnT)
Sz cos(Osz) cos(AnT) - Sxlz cos(sz) sin(AnT)
-*
H0-_
(6.18)
Ho
-Sx sin(QsT) cos(Anr) - Szlzj sin(Os-) sin(An7z)
11Sx sin(220s) cos(2An7) - 1-Sxlz
cos(20sz) sin(2A7n)
2
2
1
1
1 Sy(1 - cos(20sT) cos(2Anm)) - -Sylz sin(20•sT) sin(2Anv)
(6.19)
+S, cos(Qsz) cos(An7) - Szlzj sin(Dsz) sin(An7T)
(6.20)
The detected signal at t = 2z is
(S + is)
= -
1 - ei2
sT
cos(2AnT))
(6.21)
When the T2 processes are taken into account, the terms that evolve with Os)
average out to zero. The resultant echo envelope is unmodulated.
Performing this pulse sequence on a le-1n system coupled via an anisotropic
hyperfine interaction reveals information about these couplings in the modulation
of the echoes. The analytical solution can be obtained with a product operator
calculation. Defining the evolution under the internal hamiltonian as UH = re- iH ,
the unitary evolution during this two-pulse ESEEM experiment is:
(6.22)
UESEEM = UHUSxUHUSx.
Using the unitary transformation Ud defined in equation 2.9, we can transform the
density operator between the Cartesian basis and the eigenbasis.
UdUsxU [UdUHUt] UdUSx
d
UESEEM = U [UHUU]
(6.23)
(6.24)
= UUHdUdUSxU UHdUdUSx
The bracketed term is the unitary of the evolution under the diagonalized Hamiltonian Hd through time c. Hd can be defined in terms of the nuclear transitions in
the eigenbasis by
Hd
= WeSz +
W012 EeIz + W34 Ee.Iz
(6.25)
or in terms of the sum and difference frequencies co = C12 + (34 and 6o = W12
- (034
by
(6.26)
Ha = WeSz + W+Iz + wSzlz.
Defining Ud in terms of the sum c and difference q of angles OT and 0Q (equations
2.10 and 2.11) is also useful:
Ud
=
- i(
E
E
i y +O E eI y) .-
(6.27)
S y +r s z y)
- i(
The final density operator from this ESEEM experiment pf =
UESEEM PO
ESEEM
includes terms Sx, Sy, SxIz, SYz, SxIx, Syly, SxIy, and Sylx. The detected echo shows
modulations with the nuclear frequencies 0 12 and w34 and the sum and difference
Figure 6-5: ESEEM with shaped GRAPE pulses: Magnitude of time-domain echo
signal for values of T from 800 ns to 1200 ns (color online)
frequencies 6o and w_.
SiS+
- i cos 2 (q) Sin 2 (j)
i-O2Ps COS(6
(1- 2 sin 2(q) cos2(1))
=
COS2
(Sx+ is,)
+*sina(r/) (cos2(r/) + ei
+
12 T) +
+
COS(a
34
3
))
cos()+)
i2sT) cos(W_ T)
cos 2 () Sin2
(COS(a
(6.28)
Taking into account T2 relaxation, the resulting signal is
(sI + iS
=
i
2
{2 - 2 cos(aW
12T) - 2cos(w 34 T) + cos(0+') + cos(wT_)}.
(6.29)
The same modulations result from the 7/2 - T- n -
-
echo ESEEM experiment
[76], except with twice the amplitude.
This experiment was conducted on a malonic acid crystal using a 400 ns long
GRAPE pulse consisting of 100 four nanosecond steps. The algorithm used the
Hamiltonian presented in Section 2.1.2 and does not account for the cavity quality
factor or field inhomogeneities. Figure 6-5 shows the magnitude of the time-domain
echo signal with z values from 800 ns to 1200 ns. The ESEEM modulations are more
obvious in the fourier transform of the echo signal vs. -, as shown in Figure 6-6.
The data show the expected modulations at w1 2 (8 MHz), a34 (40 MHz), w+ (48
Freq (MHz)
Figure 6-6: ESEEM with shaped GRAPE pulses: Magnitude spectrum of echo peak
vs. T data from simulation and experiment.
MHz), and w (32 MHz); however, the magnitudes of the W12 and c34 peaks are
not equal and an extra peak appears at -(012.
Also shown in Figure 6-6 are the
results of a numerical simulation of this experiment with Q=250 and a gaussian
B0 profile with a 3 Gauss variance. The difference in peak magnitudes is seen
in both simulation and experiment, and can be partially explained by the limited
acquisition bandwidth of the resonator.
6.3
Ramsey Fringe and Refocusing Experiments
Using GRAPE pulses on the le-1n malonic acid radical, this experiment consists of a
simple preparation-nuclear qubit operation-readout sequence. Starting with the
thermal equilibrium state, a unitary operation transfers the electron polarization
to a nuclear polarization. Another unitary operation generates a coherence on
the 11) - 12) transition, which is then allowed to evolve for a time z under the
unperturbed Hamiltonian H0 . A n/2 pulse transfers the coherence back to a nuclear
polarization, which is then converted to an electron spin polarization to be read
out via a spin echo. The resultant echo shows a modulation indicative of the phase
acquired during time T. Figure 6-7 illustrates the operations performed on the
energy levels of the le-1n system.
The evolution of the system during this experiment can be analyzed using
product operator formalism. The notation 0)" represents a 0 rotation about the k
axis in the 2-level system that comprises states m and n. I' is used to indicate the
nuclear spin in the eigenbasis. Noting that the nuclear eigenstates are different in
the electron spin manifolds, I' is defined as follows:
Iz =
(lal) (ai• + l)
(pll) -
Io = lao) (aol + o) (Po
'=
Iai) (aI + 11) (I31
(lao) (aol + po) (3ol)
(6.30)
(6.31)
(6.32)
I
Z
I./-1 I
o/
Figure 6-7: Energy level illustration of Ramsey fringe experiment: (1) Polarization
is transferred from the electron to the nuclear spin. (2) A nuclear coherence is
generated. (3) The coherence is allowed to evolve for time -. (4) The phase
evolution of the nuclear coherence is recorded in the nuclear polarization. (5) The
modulated nuclear polarization is swapped with the electron polarization.
1
3
4
" -v/2)1
2
X
5
-7)2
Echo detect
1,5
3
*1
&,5
0
1,2
0
500
4,5
1000 1500 2000 2500 3000 3500 4000 4500 5000
time (ns)
Figure 6-8: Pulse sequence for Ramsey fringe experiment: The GRAPE pulses used
to generate the Ramsey fringe and the hard pulses used for spin echo readout are
shown.
Using product operator formalism to follow the le-in system evolution in the
eigenbasis, we see that the resulting electron polarization has a modulation of
frequency W012:
77)
24
po = -Sz
I-
(6.33)
-EI' + EeI
(6.35)
(6.34)
7T)12
Hd
e
-E+ (I cos(w 2z)
I sin(w12)) + EIz
-
(6.36)
7 )12
EX0 (I' cos(0
)24
-7-)--S
z
0
1'
12
T)
I x sin(0 12 'r)) + EIe'
+
+ I0 cos(012T)) + SyI 0 sin(a
2 1)
(6.37)
6.38)
The pulse sequence used in this experiment is shown in Figure 6-8. Consecutive unitary operations performed by the spin actuator were combined into single
GRAPE pulses. For example, to execute steps 1 and 2, we used GRAPE to find the
modulation sequence that generates the unitary
U2 U1 = We - il
where
g 12
=
2
e-i-/24 Ud,
(6.39)
E' x , au24 = Sxl - Syl , and Ud is the eigenbasis transformation operator
defined in equation 2.9. We used phase cycling to remove the non-zero-quantum
terms and used a hard pulse spin echo to read out the electron spin polarization.
Pulsing on the I0 electron transition with B1
< W12, (034, (0+, O-,
the echo signal
modulation is a good approximation of the expectation value
(SzoI)
= - cos(0
12
T).
(6.40)
Readout using echo pulses capable of Sx rotations in both nuclear spin manifolds
would exhibit a modulation proportional to
(Sz) = -1 - cos(W12 z).
(6.41)
1
2
)
(/2)
3
4
5
6
1000
2000
Echo detect
-)
24X-r/2)
0
7
3000
time (ns)
4000
5000
6000
Figure 6-9: Pulse sequence for Ramsey fringe refocusing experiment: The GRAPE
pulses used to perform the refocused Ramsey fringe experiment and the spin echo
readout are shown.
The amplitudes of these modulated echo signals depend on the delay of the echo
used for readout, as shown in the sections on ESEEM.
To further demonstrate spin actuator control of the nuclear spin, we applied
a nuclear m pulse to refocus the evolution of the nuclear coherence during -. As
shown in Figure 6-9, at time -/2 we apply a GRAPE pulse that performs the
operation
U4 = Ute-i'x Ud,.
(6.42)
The resulting electron polarization now exhibits no modulation:
7)
Po = -Sz
(6.43)
IZ
(6.44)
24
1)12
-E'I' + E'l
Hd/
I Cos(Wl2T/2) -
-E2 e
-
7T) 1 2 + in)
0 E i' c
(6.45)
I' sin(W12/2)) + El
+ E-I
(6.46)
(6.46)
34
x
Hd-- /2+
> E 0 I cos(w 12 T/2) + I' sin(W127/2)) - EI'
E +I' - E
1'
(6.47)
(6.48)
nT)12
-724
x-
-IX
(6.49)
Sz
(6.50)
The readout echo signal should show no modulation with T.
The time domain data shown in Figure 6-10 compares the data and simulation
of the Ramsey fringe free evolution experiment. As discussed in Section 4.1, the
angular orientation of the crystal has an estimated error of ±5'. We aimed to orient
the crystal at an angle halfway between the principal hyperfine axes, and found
the GRAPE pulses accordingly. The Ramsey fringe data showed an oscillation
frequency 560 kHz lower than the expected value of 7.813 MHz. A rotation of 2.40
from our desired orientation makes the calculated a)12 consistent with the data. We
simulated the experiment using these GRAPE pulses at this angle, and observe
that this simulation data correspond well with the experimental data. We also see
that simulations of the refocusing experiment exhibit the 4 MHz modulation seen
in the data (Figure 6-11).
6.4
Discussion of results
These experiments provide evidence of universal control of a le-in system using
modulations of a microwave field at a fixed frequency. The ESEEM experiments
C\mlldir~n ~~ mhprunr~ hnhuuon · ;tlltoc II~.. INI I)~· n·rl I·nr~ Ci.mll.ttu~ ~~ 63 C1
Figure 6-10: Time domain Ramsey fringe signal: Simulation and experimental data
from the Ramsey fringe experiment are shown. The simulation takes into account
the error in angular orientation of the crystal.
i
FmQuecy OM4)
Figure 6-11: Magnitude spectrum of Ramsey fringe and refocused signal: Data
from the experiments show that the refocusing pulse does remove the Ramsey
fringe modulation, but a significant 4 MHz error signal arises from the bandwidth
of the resonator and the orientation of the crystal. This error is also seen in the
simulation.
using hard pulses and shaped pulses show that we can use GRAPE pulses to gain
access to all the transitions of this AHF-coupled spin system. The Ramsey fringe
and refocusing experiments demonstrate the potential of precise implementation of
arbitrary unitary operations using the electron spin actuator. Though our simulations show good correlation with the data, they do not account for all the observed
dynamics. Also, the data from these and other unreported experiments show decreasing signal-to-noise ratio with increasing length of GRAPE pulse modulation.
The shaped pulses were designed with a Hamiltonian that does not include the
full system model (cf. Section 3.3). We conjecture that the GRAPE pulses are only
able to accurately apply the desired unitary to a subset of the the electron spins
in the crystal, while they average out unwanted evolution in other parts of the
crystal. The planned improvements in the instrumentation, system model, and
sample quality will allow us to address these issues more effectively.
Chapter 7
Conclusion
Here we have introduced the idea of local actuator control for quantum information processing and have shown that it can be used to implement improved
quantum bits. In particular we have experimentally demonstrated such control
via an electron spin actuator with universal control of nuclear spin qubits. In a
le-Nn spin system where each nuclear spin has a resolved anisotropic hyperfine
interaction, any control Hamiltonian that flips the electron spin links all the nuclear
spin states in the system. In this case we can use optimal control techniques to find
modulation sequences to perform any unitary operation on the nuclear spins with
high fidelity. This electron spin actuator improves the efficiency of nuclear qubit
operations compared to direct nuclear spin control, and could act as a fundamental
processing unit in a quantum computer.
Achieving sufficient control to perform a useful quantum computation is a
considerable challenge: A commonly cited error probability required for quantum
computing is 10' [79, 43]. We have not shown such high-fidelity control with the
electron spin actuator, only that such control may be possible using this method.
The focus of our lab will continue to be on achieving precise quantum control of
le-Nn spin systems. In the near future the improvements will include a second
version of the spectrometer, bandwidth-constrained GRAPE, new samples with
more nuclear spins, and a more accurate system model.
One question we might explore with a le-Nn quantum information processors
is the nature of decoherence of nuclear spin qubits in these systems. Also, the
spin actuator could potentially be used to amplify the signal from a single nuclear
spin, analogous to entanglement assisted metrology shown in liquid-state NMR
[11]. The ability to perform precise unitary operations while sitting on a single
transmitter frequency would also have applications for spectroscopy.
Harnessing the substantial resources required for quantum computation is another major challenge. For example, in a proposal for distributed quantum computing with small quantum registers-devices composed of a small number (-5) of
physical two-level systems-it was estimated that with error rates of - 1.7 x 10- 5,
a calculation involving 10' logical qubits and 106 logical operations requires 20
registers per logical qubit [38]. For the le-Nn spin-actuator controlled system to be
usable in a larger quantum information processor, signficant advances are required.
One would need a physical system with regular arrays of le-Nn spin clusters that
can be individually addressed and dynamically coupled and decoupled via a spin
bus. This bus would have to be able to transfer the electron spins efficiently and
with minimal error. Finally, one must be able to polarize and apply a strong measurement to the electron in the le-Nn system without depolarizing the nuclear spin
qubits. Progress in these areas and in the spin actuator control shown in this thesis
could be key steps towards the realization of a quantum computer.
Appendix A
Publications
Jonathan S. Hodges, Jamie C. Yang, Chandrasekhar Ramanathan, and David G.
Cory. Universal Control of Nuclear Spins Via Anisotropic Hyperfine Interactions.
arXiv:0707.2956 [quant-ph], 2007.
G. Rutsch, J. Yang, W. Van Drent, D. Mauri, and J. Li. Vector Magnetometry of
Synthetic Spin Valves. IEEE Transactions on Magnetics, 4010:3712-3714, 2005.
J. Wolfman, D. Mauri, T. Lin, J. Yang, and T. Chen. Low Resistance A12 0 3 magnetic
tunnel junctions optimized through in-situ conductance measurements. Journalof
Applied Physics, 97:123713, 2005.
N. Boulant, K. Edmonds, J. Yang, M. A. Pravia, and D. G. Cory. Experimental
demonstration of an entanglement swapping operation and improved control in
NMR quantum-information processing. Physical Review A, 68:032305, 2003.
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