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Development of high-frequency and high-field optically detected magnetic resonance

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Development of high-frequency and high-field optically detected magnetic resonance

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Content DEVELOPMENT OF HIGH-FREQUENCY AND HIGH-FIELD OPTICALLY
DETECTED MAGNETIC RESONANCE
by
Benjamin Fortman
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMISTRY)
May 2022
Copyright 2022 Benjamin Fortman
Dedicated to Ashna, the love of my life,
and to my teachers: past, present, and future.
ii
Acknowledgements
I would like to thank my advisor, Dr. Susumu Takahashi, for his guidance, mentoring,
and constructive, critical feedback throughout my Ph.D. His assistance, experimental
patience, and belief in me have made this work possible.
I would like to express my gratitude to my parents, Jeff and Martha, and my family
for their support, words of encouragement, and belief in me.
I would also like to acknowledge the former graduate students, as it is their work
that laid the foundation for mine. I thank Dr. Franklin Cho for his excellent work
building and constructing the HF-EPR spectrometer that I spent so much time working
on. I thank Dr. Viktor Stepanov for his work on realizing optically detected magnetic
resonance at high fields and his advice at the beginning of my Ph.D. I thank Dr. Rana
Akiel for her assistance during my time as a teaching assistant and work with diamond
surface chemistry. I thank Dr. Chathuranga Abeywardana for his assistance with many
experiments, guidance with experimental troubleshooting, and helpful advice. I greatly
thank Dr. Zaili Peng for her helpful advice, critical feedback, experimental assistance
with the HF-EPR spectrometer, and for her many textbook recommendations that guided
me through my degree.
I thank my current lab-mates, Laura Mugica-Sanchez, Michael Coumans, Ana Gur-
genidze, and Yuhang Ren, for their assistance and critical insights. I wish them the very
iii
best as they work to finish their degrees. Best wishes also go to the excellent undergrad-
uates whom I have had the pleasure of working with: Meera Mistry, Armand Tadjali,
Junior Pena, Yuhang Ren, Yuxiao Hang, Noah Tischler, Cooper Selco, Zihan Zhao, and
Kyle Shi. I also wish Wendy Marquina, my mentee for the Young Researchers Program,
the best of luck in her studies.
I would also like to thank my qualifying commitee members, Dr. Alexander Ben-
derskii, Dr. Stephan Haas, Dr. Andrey Vilesov, and Dr. Daniel Lidar for their guidance
and support. Special thanks goes to Dr. Andrey Vilesov and Dr. Daniel Lidar for being
on my dissertation commitee. Lastly, I would like to thank all the chemistry depart-
ment staff who have assisted me during my time at USC, especially Michele Dea, Allan
Kershaw, Magnolia Benitez, Thuc Do, and Joseph Lim. Special thanks also goes to
the USC machine shop for their excellent work. I also thank the many customer ser-
vice support technicians who answered when I called for assistance, especially Arctic
Chill service support center technician Scott Stegall for his assistance troubleshooting
our chiller system.
I thank financial support from the Chemical Measurement and Imaging program
in the National Science Foundation (NSF), (CHE-2004252 (with partial co-funding
from the Quantum Information Science program in the Division of Physics) and CHE-
1611134) and the NSF Condensed Matter Physics program (DMR-1508661).
iv
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables viii
List of Figures ix
Abbreviations xi
List of Physical Constants xiii
List of Units xiv
List of Symbols xv
Abstract xviii
Chapter 1:Introduction 1
1.1 Overview and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 High Field ESR . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Single Spin ESR . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Nanoscale ESR and NMR at High Field . . . . . . . . . . . . . . . . . 6
Chapter 2:Quantum Sensing using Electron Spin Resonance (ESR) Spec-
troscopy 9
2.1 Static Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Electron Zeeman Interaction . . . . . . . . . . . . . . . . . . . 11
2.1.2 Zero-field splitting . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Hyperfine interaction . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.4 Nuclear Zeeman Interaction . . . . . . . . . . . . . . . . . . . 14
2.1.5 Nuclear Quadrupole Interaction . . . . . . . . . . . . . . . . . 15
2.2 Continuous Wave (cw) ESR spectroscopy . . . . . . . . . . . . . . . . 15
v
2.2.1 cw-ESR: Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 cw-ESR: Spectral analysis . . . . . . . . . . . . . . . . . . . . 20
2.3 Pulsed ESR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Free Induction Decay (FID) . . . . . . . . . . . . . . . . . . . 26
2.3.3 Spin Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.4 Stimulated Echo . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.5 Inversion Recovery . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.6 Double Electron Electron Resonance (DEER) . . . . . . . . . . 36
2.3.7 Double Resonance Electron-Nuclear Spectroscopy . . . . . . . 39
2.4 Optically Detected Magnetic Resonance (ODMR) with the Nitrogen Va-
cancy (NV) Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.1 NV Center in Diamond: Overview . . . . . . . . . . . . . . . . 45
2.4.2 NV Optical Properties . . . . . . . . . . . . . . . . . . . . . . 46
2.4.3 Pulsed ODMR . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.4 NV-ESR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.4.5 NV-NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter 3:Instrumentation 69
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 HF ESR Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 High Field ODMR Spectrometer . . . . . . . . . . . . . . . . . . . . . 72
3.3.1 Laser Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.2 Optical Setup and Sample Positioning . . . . . . . . . . . . . . 73
3.3.3 Fluorescence Detection and Autocorrelation . . . . . . . . . . . 74
3.3.4 Electron Nuclear Double Resonance (ENDOR) setup . . . . . . 78
3.4 Low-field ODMR Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.1 Optical Setup and Sample Positioning . . . . . . . . . . . . . . 80
3.4.2 Microwave Excitation System . . . . . . . . . . . . . . . . . . 80
3.5 Implementation of Shaped Pulses . . . . . . . . . . . . . . . . . . . . . 81
3.5.1 I/Q modulation: a review . . . . . . . . . . . . . . . . . . . . . 81
3.5.2 Shaped Pulses for Low Frequency ODMR . . . . . . . . . . . . 82
3.5.3 Shaped Pulses for High Frequency ODMR . . . . . . . . . . . . 84
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter 4:Understanding Linewidth of ESR Spectrum Detected by a Single
NV Center in Diamond 89
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.1 Diamond sample . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.2 115 GHz ESR spectroscopy . . . . . . . . . . . . . . . . . . . 91
vi
4.2.3 ODMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Experimental Results and Discussion . . . . . . . . . . . . . . . . . . . 92
4.3.1 cw-ESR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 ODMR: NV Identification and Field Calibration . . . . . . . . . 93
4.3.3 NV-ESR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 5:Demonstration of NV-detected ESR spectroscopy at 115 GHz and
4.2 Tesla 103
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Experiment and discussion . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.1 Ensemble NV-ESR . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.2 Single NV-ESR . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter 6:Electron-electron double resonance detected NMR spectroscopy
using ensemble NV centers at 230 GHz and 8.3 Tesla 114
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter 7:Conclusion 131
Chapter A: Spin Dynamics of Chirp Pulses in a Two Level System 134
A.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Chapter B:Contribution of Dipolar Field Fluctuations to the Lineshape 140
B.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Chapter C: Experimental Considerations 146
C.1 Experimental Noise analysis . . . . . . . . . . . . . . . . . . . . . . . 146
C.1.1 Experimental Example for High-field NV-based ESR . . . . . . 147
C.2 NV-ESR based on Dynamical Decoupling . . . . . . . . . . . . . . . . 151
C.3 HF ENDOR Pulse Broadening Dependence at 115 GHz . . . . . . . . . 153
Bibliography 154
G.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
vii
List of Tables
6.1 State identification and energy values determined from Eq. 6.1 . . . . . 124
6.2 Simulated transition energies calculated from Table 6.1 . . . . . . . . . 125
viii
List of Figures
2.1 Magnetization from Bloch Equations . . . . . . . . . . . . . . . . . . . 19
2.2 P1 centers Energy diagram and Experimental spectrum . . . . . . . . . 23
2.3 Rabi Oscillations Pulse Sequence . . . . . . . . . . . . . . . . . . . . . 25
2.4 Free induction decay (FID) Pulse Sequence. . . . . . . . . . . . . . . . 27
2.5 Spin Echo (SE) Decay Pulse Sequence . . . . . . . . . . . . . . . . . . 31
2.6 Stimulated Echo Decay (STE) Pulse Sequence . . . . . . . . . . . . . . 33
2.7 Double Electron-Electron Resonance (DEER) Pulse Sequence . . . . . 37
2.8 Sequences for Pulsed Electron-Nuclear DOuble Resonance (ENDOR) . 41
2.9 Sequence for Pulsed Electron-electron Double resonance detected NMR
(EDNMR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.10 Energy level diagram of the NV center . . . . . . . . . . . . . . . . . . 47
2.11 Initialization and fluorescence readout of the NV center . . . . . . . . . 48
2.12 Experimental autocorrelation data . . . . . . . . . . . . . . . . . . . . 49
2.13 Continous Wave Optically detected magnetic resonance (cw-ODMR)
measured from an ensemble NV sample . . . . . . . . . . . . . . . . . 53
2.14 Rabi Oscillations from a single NV center . . . . . . . . . . . . . . . . 55
2.15 Pulsed ODMR (pODMR) measurements from single NV centers . . . . 56
2.16 Free induction decay (FID) measurement from a single NV center . . . 57
2.17 Spin echo (SE) measurements from a single NV center . . . . . . . . . 59
2.18 Spin echo (STE) measurements from ensemble NV centers . . . . . . . 60
2.19 T
1
inversion recovery (T1IR) measurements from a single NV center . . 62
2.20 NV-ESR from a single NV at 33.4 mT . . . . . . . . . . . . . . . . . . 63
2.21 EDNMR from an ensemble of NV centers at 230 GHz and 8.305 T . . . 65
2.22 Mims-ENDOR at 115 GHz (4.197T) from an ensemble NV system . . . 66
3.1 Overview of HF ODMR system . . . . . . . . . . . . . . . . . . . . . 70
3.2 Overview of fluorescence detection system for single photon counting . 75
3.3 Overview of fluorescence detection system using the Thorlabs rapid re-
sponse avalanche photo detector (RAPD) . . . . . . . . . . . . . . . . . 77
3.4 Overview of low field ODMR setup . . . . . . . . . . . . . . . . . . . 79
3.5 Implementation of frequency swept chirp pulses on the low field ODMR
setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
ix
3.6 Implementation of frequency swept chirp pulses on the low field ODMR
setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.7 Implementation of frequency swept chirp pulses at 115 GHz on a single
NV center at 4.197 T . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1 cw-ESR spectrum of P1 centers taken at 115 GHz at room temperature . 93
4.2 ODMR experiment to identify a NV center . . . . . . . . . . . . . . . . 94
4.3 Pulsed ODMR data collected from NV1 at 33.4 mT . . . . . . . . . . . 95
4.4 NV-ESR of NV1 with = 5:7s . . . . . . . . . . . . . . . . . . . . . 97
4.5 Dependence of DEER pulse length on NV-ESR linewidth . . . . . . . . 100
4.6 Dependence of DEER pulse length on NV-ESR linewidth for NVs 2-5 . 102
5.1 Overview of HF ODMR system . . . . . . . . . . . . . . . . . . . . . 106
5.2 Characterization of an ensemble NV system at 115 GHz and 4.2 Tesla . 108
5.3 HF NV-ESR using an ensemble NV system . . . . . . . . . . . . . . . 109
5.4 115 GHz NV-ESR spectroscopy of P1 centers using a single NV . . . . 110
6.1 EDNMR Experimental Setup at 230 GHz . . . . . . . . . . . . . . . . 117
6.2 Ensemble ODMR at 230 GHz . . . . . . . . . . . . . . . . . . . . . . 120
6.3 NV detected EDNMR at high field. . . . . . . . . . . . . . . . . . . . . 123
6.4 NV detected EDNMR of at 8.3 Tesla . . . . . . . . . . . . . . . . . . . 126
A.1 Numerical Simulations of a Landau-Zener Transition . . . . . . . . . . 138
C.1 Experimental Noise Analysis for HF NV-ESR . . . . . . . . . . . . . . 148
C.2 NV-ESR using Dynamical Decoupling . . . . . . . . . . . . . . . . . . 152
C.3 Variation of Pulse Length for ENDOR at 115 GHz . . . . . . . . . . . . 153
x
Abbreviations
AC Alternating Current
AOM Acousto-optic modulator
APD Avalanche photo-diode
AWG Arbitrary waveform generator
cw-ESR continuous wave ESR
DAQ Digital to analog converter
DEER Double electron electron resonance
EDNMR Electron-electron double resonance detected NMR
ENDOR Electron-nuclear double resonance
ESEEM Electron spin echo envelope modulation
ESR Electron Spin Resonance
EPR Electron Paramagnetic Spin Resonance
FID Free Induction
FL fluorescence
FWHM full width at half maximum
HF High Field
HTA High turning angle
MRI Magnetic Resonance Imaging
MW Microwave
NMR Nuclear Magnetic Resonance
NV Nitrogen-Vacancy
xi
NV-ESR NV-detected ESR
NV-NMR NV-detected NMR
ODMR Optically Detected Magnetic Resonance
P1 substitutional nitrogen
RAPD Rapid response Avalanche photo-diode
RF Radio-frequency
R.o. Readout
SE Spin Echo
STE Stimulated Echo
T1-IR T
1
-Inversion recovery
TLS Two-level-system
TTL Transistor-transistor logic
VISA Virtual instrument software application
xii
List of Physical Constants

e
9.2740096810
24
kgm
2
s
2
T
1
or JT
1
s Bohr magneton

n
5.05078369910
27
kgm
2
s
2
T
1
or JT
1
s Nuclear magneton
e 1.60217663410
19
AS Elementary charge
h 6.6260695710
34
kgm
2
s
1
or J s Plank constant
k
B
1.3806610
23
kgm
2
s
2
K
1
or JK
1
s Boltzmann constant

0
4 10
7
NA
2
or TMA
1
s Permeability of free space
xiii
List of Units
dB Decibel
eV electron V olt (1:602 10
19
J)
fW femtowatt (1 10
15
W)
mW Milliwatt (1 10
3
W)
kW Kilowatt (1 10
3
W)
kHz Kilohertz (1 10
3
Hz)
MHz Megahertz (1 10
6
Hz)
GHz Gigahertz (1 10
9
Hz)
THz Terahertz (1 10
12
Hz)
T Tesla
xiv
List of Symbols
A Secular hyperfine interaction (Eq. 2.58)
A Hyperfine interaction tensor (componentsA
x
,A
y
, andA
z
; Eq. 2.9)
a
iso
isotropic hyperfine interaction; Eq. 2.10)
Alpha state
B Pseudosecular hyperfine interaction (Eq. 2.58)
~
B
0
Static magnetic field vector (componentsB
x
,B
y
, andB
z
and magnitudeB
0
; Eq. 2.3)
~
B
1
electromagnetic field vector (magnitudeB
1;max
, frequency!, and phase; Eq. 2.17)
Beta state
b
i
magnetic field of the i-th spin
D Electron zero-field interaction tensor (componentsD
x
,D
y
, andD
z
; Eq. 2.7)
Variance in the magnetic field due to the Ornstein-Uhlenbeck process (Eq. 2.38
N
v
Spin population difference (Eq. 2.24)
! Resonance offset (Eq. 2.19)
!
1=2
Half width at half maximum (Eq. B.13)
E Electron strain interaction (Eq. 2.7)
E
n
Eigenenergy of state n (Eq. 2.24)
g Electron g-tensor (with componentsg
x
,g
y
, andg
z
; Eq. 2.3)
g
2
() Second order correlation function (Eq. 2.65)

1
(
2
) Characteristic decay time of state 1 (state 2) (Eq. 2.66)

e
Gyromagnetic ratio for an electron (Eq. 2.4)

n
Gyromagnetic ratio for a nuclei (Eq. 2.14)

1
Amplitude damping relaxation operator (Eq. 2.22)

2
Phase damping relaxation operator (Eq. 2.22)
g
n
Nuclear g -factor (Eq. 2.13)
h Depth of an EDNMR hole (Eq. 2.64)
xv
H Rotating frame Hamiltonian (Eq. 2.19)
H
0
Static spin Hamiltonian (Eq. 2.1)
H
1
Transverse spin Hamiltonian (Eq. 2.27)
H
Dip
Dipolar interaction Hamiltonian (Eq. 2.11)
H
EZ
Electron zeeman interaction Hamiltonian (Eq. 2.3)
H
HF
Electron-nuclei hyperfine interaction Hamiltonian (Eq. 2.8)
H
NQ
Nuclear quadrupole interaction Hamiltonian (Eq. 2.15)
H
NZ
Nuclear zeeman interaction Hamiltonian (Eq. 2.13)
H
P 1
Hamiltonian for the substitutional nitrogen center (Eq. 2.13)
H
ZFS
Electron zero-field interaction Hamiltonian (Eq. 2.5)
I Nuclear spin angular momentum quantum value with nuclear spin numberm
I
^
I Nuclear spin angular momentum vector operator (components
^
I
x
,
^
I
y
, and
^
I
z
; Eq. 2.13)
I Identity matrix
I
a
Intensity of allowed transition (Eq. 2.63)
I
f
Intensity of forbidden transition (Eq. 2.63)
k Sweep rate (Eq. A.1)
k
12
Driving rate of excitation from state 1 to state 2 (Eq. 2.65)
L Lindbladian (Eq. 2.22)
L(; !) Intrinsic lineshape. (Eq. 2.57)
M Bulk magnetitization (componentsM
x
,M
y
, andM
z
; Eq. 2.24)
n
B
Concentration of B spins
n
H
Size of the state space for a Hamiltonian (Eq. 2.2)
N
s
Population of state S (Eq. 1.1)

Rabi Frequency (Eq. 2.17)
!
0
Electron precession frequency
!
c
Angular frequency of the carrier wave (Eq. 3.1)
!
s
Precessional frequency of state S (Eq. 1.1)
P Nuclear quadrupole Tensor (componentsP
x
,P
y
, andP
z
Eq. 2.16)
P
ab
Probability of a state transition (Eq. 2.27)
j i Wavefunction of a pure state (Eq. 2.20)
j (0)i Wavefunction at the initial time
q Electric field gradient (Eq. 2.16)
Q Nuclear quadrupole moment (Eq. 2.16)
r Vector between two spins, with magneitude r.
^
R Rotation operator (Eq. 2.30)
xvi
Density matrix (Eq. 2.21)
S Electron spin angular momentum quantum value with spin numberm
s
^
S Electron spin angular momentum vector operator (components
^
S
x
,
^
S
y
, and
^
S
z
; Eq. 2.3)

sin
2
2

L
Pulse dependent lineshape function (Eq. 2.57)
t Time
T Temperature (Eq. 1.1)
T Anisotropic Hyperfine Interaction Tensor (Eq. 2.9)
T
1
Spin-lattice relaxation (Eq. 2.22)
T
2
Spin-spin relaxation (Eq. 2.22)
T

2
Intrinsic spin dephasing (Eq. 2.22)
Pulse delay time

D
Storage delay time (Eq. 2.47c)
t
c
Correlation time due to the Ornstein-Uhlenbeck process. (Eq. 2.38)
Angle betweenA
z
andB
z
t
p
Pulse time
U Unitary transformation (Eq. 2.19)
V V olume integral (Eq. B.3)
Relative intensity of bunching and antibunching (Eq. 2.66)
Nuclear quadrupole assymetry parameter (Eq. 2.16)

s
Selectivity parameter (Eq. 2.62)

mix
Mixing parameter (Eq. 2.63)
xvii
Abstract
Magnetic resonance spectroscopies, including nuclear magnetic resonance (NMR) and
electron spin resonance (ESR) are invaluable spectroscopic techniques for characteri-
zation of molecular structures. Magnetic resonance at high magnetic fields has many
advantages. NMR at high magnetic fields is highly advantageous because of its high
spectral resolution and improved sensitivity, enabling the resolution of closely related
chemical shifts and offering new insights into the study of complex molecules. ESR
spectroscopy at high magnetic fields improves spin polarization, increases control over
spin dynamics, improves insight into molecular motion, and offers high spectral resolu-
tion into closely related spin systems. The increased resolution enables investigation of
complex systems with similar g values.
The nitrogen-vacancy (NV) center, due to its unique properties, has enabled widespread
study of nanoscale NMR and ESR at low magnetic fields. However, there have been few
studies of NV-detected NMR and ESR at high magnetic fields due to the technical chal-
lenges involved. In addition, conventional NV-detected NMR based on the detection of
alternating current (AC) magnetic field sensing is not applicable at high magnetic fields
and therefore requires the development of alternate techniques.
This thesis is focused on the development of NV-based ESR and NMR spectro-
scopies at high magnetic fields. The organization is as follows: Chapter 1 provides an
xviii
introduction and motivation for high-field ESR and NMR and relevant technical chal-
lenges involved that make the extension to high-field NV-based magnetic resonance
difficult. Chapter 2 provides an overview of pulsed ESR spin techniques and their appli-
cations for optically detected magnetic resonance using NV centers in diamond. Chapter
3 overviews the experimental apparatus and hardware required for the implementation
of high-field optically detected magnetic resonance. Chapter 4 demonstrates a tech-
nique for NV-detected ESR (NV-ESR) with high-spectral resolution and a method of
extracting an intrinsic linewidth representative of ESR spectral properties. Chapter 5
demonstrates high-field NV-ESR at a field strength of 4.2 Tesla using both single and
ensemble NV-centers. Chirp pulses are used to increase the excitation bandwidth and
improve population transfer, enabling NV-ESR for single NV centers. Chapter 6 in-
troduces a technique applicable for NV-detected NMR (NV-NMR) at high-fields and
utilizes this technique for the first performance of NV-NMR at 8.3 Tesla, a field strength
comparable to research grade NMR.
xix
Chapter 1: Introduction
1.1 Overview and Motivation
Quantum mechanics, since its development in the early 20
th
century, has been funda-
mental to our understanding of the world. The observation of quantized spin angular mo-
mentum, through the Stern-Gerlach experiment, changed our understanding of atomic
particles.
1
Since the first in 1932, eight nobel prizes have been awarded for research in
the field of quantum mechanics and this field continues to be at the forefront of modern
day research.
The Stern-Gerlach experiment provided the first evidence for quantized angular mo-
mentum. Since then, the observation of nuclear magnetic resonance (NMR), has gone
on to revolutionize our understanding of spin and advanced the fields of chemistry and
physics.
2, 3
NMR builds upon the principle that particles with quantized spin-angular
momentum precess in a magnetic field. This precessional frequency is a remarkably
sensitive probe of the particles’ local environment, with different nuclear isotopes pro-
ducing a characteristic frequency (gyromagnetic ratio) dependent on the magnetic field.
The precessional frequency also produces spectral signatures indicative of molecular
connectivity and its behavior depends on the environment. These properties have made
NMR a standard technique for many applications. Three applications where NMR has
1
a large impact are: (1) characterization of molecules during synthetic research in inor-
ganic and organic chemistry, (2) chemical research into biomolecular proteins, where
protein structure can be correlated to biological function and, (3) magnetic resonance
imaging (MRI), where NMR can non-invasively provide millimeter scale imaging for
tracking real-time mental processing.
4
NMR’s capacity to offer insight into molecular
environments, chemical bonding and chemical structure at room-temperature make it
a powerful spectroscopic technique. For example, NMR is a standard tool for quality
control of heparin, an anti-coagulant used for the prevention of heart attacks.
5
NMR is
also used to develop new treatments for malaria, a devastating disease responsible for
hundreds of thousands of deaths each year.
6
The related field of electron spin resonance (ESR), also known as electron para-
magnetic resonance (EPR), has developed complementarily to NMR. ESR addresses
the magnetic field response of unpaired electron spins and was demonstrated in 1944.
7
While unpaired electron spins remain less common, they exhibit a stronger magnetic
field response than nuclei, owing to their significantly larger charge to mass ratio. ESR
excels in applications where unpaired electron spins can act as probes of their local
environment. ESR is a natural choice for the measurement of reaction and catalytic
intermediates as electrons are fundamental in chemical reactions. For example, rapid
freeze-quench ESR allows direct measurement of intermediates as a reaction progresses
and has been utilized for the identification of reaction mechanisms in a a variety of pho-
tochemical reactions.
8–10
The development of site-directed spin-labels, stable radicals
that can be controllably integrated into biological structures, has provided tremendous
insight into large biomolecules. The implementation of spin-labels into biomolecular
proteins and DNA/RNA allows insight into the molecule’s structure and conformational
2
dynamics in its aqueous environment, a significant advantage for proteins that are in-
trinsically disordered and cannot be crystallized for X-ray crystollagraphic measure-
ments.
11–13
ESR is widely applicable for the identification of paramagnetic defects and
unpaired radical spins in semiconductors, insulators, conducting metals, and radicals in
solution.
14–18
In semiconductors, the presence of a chemically inert, periodic lattice sta-
bilizes point defects consisting of incomplete bonds and partially unfilled orbitals. ESR
is therefore sensitive to only the point defects and enables measurement of the local en-
vironment surrounding each defect. For radicals in solution, ESR enables measurement
of the rotational correlation times of the spin-radicals. This can provide insight into
local bond rigidity and molecular motion.
12
1.1.1 High Field ESR
While low field ESR has been widely developed, the development of high-field ESR
techniques remain a challenge. The motivation for high-field ESR is analagous to that
in NMR: high magnetic fields greatly increase the spectral resolution and improve sensi-
tivity. The increase in field strength increases the frequency difference between closely
related chemical species and enables resolution of small chemical shifts. High field
NMR offers new insights into molecules with many similar nuclei, low gyromagnetic
ratios, and low natural abundance, such as for
17
O NMR in pharmaceutical compounds
and biomacromolecules.
19, 20
Commercial NMR magnets operating at 28.2 T (proton
Larmor frequency of 1.2 GHz) have recently become available, with hybrid magnets at
fields of 35.2 T (corresponding to 1.5 GHz proton NMR) being available in user facil-
ities.
21, 22
Pulsed ESR spectroscopy at higher frequencies and magnetic fields becomes
more powerful for finer spectral resolution, enabling clear spectral separation of systems
3
with similar g values.
23, 24
This is advantageous in the investigation of complex and het-
erogeneous spin systems.
11, 25
A high frequency of Larmor precession is also less sensi-
tive to motional narrowing, enabling the ESR investigation of structures for molecules
in motion.
12, 26
In addition, a high Larmor frequency provides greater spin polarization:
improving sensitivity
23, 24
and providing control of spin dynamics.
27, 28
These advantages
have driven the development of ESR instrumentation at higher frequencies. Commer-
cial pulsed ESR instrumentation is available at X-band (9:4 GHz;0.3T), Q-band (35
GHz;1.25T), W-band (95 GHz;3.3T) and recently at Millimeter wave (263 GHz;
9.4T). The main limitation in the development of pulsed HF ESR is the development of
high-power, high-frequency microwave sources operating in the mm-wave to terahertz
regime. The development of these sources is a technical challenge under active research,
but the fundamental limitation is the reduced skin depth of high frequency alternating
current in conductors. The skin depth is proportional to the frequency of the wave lead-
ing to significant losses in the millimeter wave regime. This reduced skin-depth makes
the fabrication of electrical components, such as amplifiers, mixers, and transfer lines,
extremely lossy, requiring advanced design options such as quasioptical propagation.
These design challenges limit the options for pulsed HF ESR frequency sources. A few
sources are: (1) Frequency multiplier chains from an intermediate frequency of 9-10
GHz to the frequency of interest. These offer modest power (mW) and high-tunability
and can be implemented in a quasioptical MW bridge. This is used in the HF optically
detected magnetic resonance (ODMR) spectrometer utilized within this work.
29, 30
(2)
Gyrotron and Klystron sources. These consist of a vacuum tube that uses the motion of
electrons around a magnetic field to generate high power ( kW), and high frequency
( GHz) radiation, but are generally not tunable over a wide frequency band and not
easily pulsed. Commercial sources are available, but their usage is largely limited to
4
dynamic nuclear polarization. (3) Free electron lasers. These consist of an accelerated
beam of electrons that is passed through an undulator to generate high frequency (GHz-
THz), high power (kW) radiation. These sources depend on a large user facility and
are not readily available. However, high-field pulsed ESR was demonstrated with a free
electron laser source.
31
The development of high frequency sources remains an area of
active research.
1.1.2 Single Spin ESR
Both ESR and NMR require large numbers of spins for the detection of signals. ESR has
a slightly higher sensitivity, due to the larger gyromagnetic ratio of the unpaired electron,
so will be the focus of this section. Analagous logic applies to NMR. There are two
primary reasons for low sensitivity: (1) detection efficiency and (2) small population
difference. The detection efficiency is directly related to the energies of the photons
involved. For example, a single photon in the mid-infrared region has an energy of1eV
while an X-band (9 GHz) photon has an energy of 410
5
eV . Therefore, single photon
detectors are not available in the MW or RF regime. In typical ESR measurements, the
MW intensity is monitored as a function of field. When the MW is in resonance with
an ESR transition, the MW detector observes a slight reduction in MW intensity that
is detected as an ESR signal. Due to the low energy of these photons, many photons
are required to overcome thermal noise and register as a measurable signal. The second
reason is related to the relative spin-state population. The population of an energy level
is determined by the Boltzmann distribution:
N
s
=
exp(
~!s
k
B
T
)
P
S
exp(
~!s
k
B
T
)
; (1.1)
5
whereN
s
is the population of state s,!
s
is the precessional frequency of state s,~ is the
reduced Planck constant,k
B
is the Boltzmann constant, andT is the temperature.As the
signal intensity is proportional to the population difference (I/ N), large numbers of
spins are required to generate a measurable signal. For example, at X-band and 300K,
the polarization is only 0:1%. For conventional ESR > 10
10
spins are required to
generate a detectable signal. A large ensemble of spins is not ideal, as the presence of
ensemble averaging obscures the structure, function, and dynamics that take place on
the level of a single molecule. Therefore, it is highly desirable to reduce the sample size
to the single spin, or small spin ensemble level. Single spin sensitivity in a magnetic
resonance technique potentially eliminates ensemble averaging in heterogeneous and
complex systems and has great promise to directly probe fundamental interactions and
biochemical function.
Advanced pulsed techniques, such as double electron-electron resonance (DEER)
and electron-nuclear double resonance (ENDOR), are routinely employed in conven-
tional magnetic resonance to measure the dipolar coupling between pairs of electron
and nuclear spins.
26, 32
If the spin-state of a single spin can be directly read-out (without
detecting photons), magnetic resonance of single and small spin ensembles should be
possible. This single spin can act as a sensor to read-out the magnetic resonance spec-
trum of nearby dipolar coupled electron and nuclear spins. Ideally, the sensor would
offer single-spin sensitivity, be operable on the nanoscale, have high physical and chem-
ical stability, function at high magnetic fields, and be operational at room temperatures.
1.2 Nanoscale ESR and NMR at High Field
The NV center in diamond fulfills all these required characteristics. The two-atom defect
in diamond consists of a substitutional nitrogen atom and an adjacent vacancy site. The
6
NV center has an atomic-scale footprint, spin-dependent fluorescence enabling optical
readout of the spin state, long coherence times, and high sensitivity to local magnetic
fields.
33–36
Additionally, the defect exists in the diamond lattice, a chemically inert ma-
terial well known for its high durability. While the existence of the NV center was
known for some time, the observation of coherent oscillations from a single defect led
to the rapid utilization of the NV center for quantum sensing.
37
Shortly after, quan-
tum sensing was demonstrated with single and ensemble NV centers.
34, 38–40
Since then,
many sensing methodologies and applications have been demonstrated using the NV
center. The NV center’s unique properties have enabled both nanoscale NMR and ESR
with sensitivity down to the level of a single spin.
41–47
NV-detected NMR (NV-NMR)
is now widely used at low magnetic fields (< 0:1 T), such as for NV depth estimation,
liquid state NMR, two-dimensional NMR, hyperpolarized NMR, nanodiamond based
NMR, and even for selective spin manipulation in a 10-qubit quantum register.
48–52
NV-
detected ESR (NV-ESR) has been used to investigate biological molecules at the single
molecule level.
53, 54
The current status of NV-based sensing has been summarized in
a number of review articles.
55–58
However, a featureless “g = 2” signal is often ob-
served, causing spectral overlap with target NV-ESR signals, precluding spin identifica-
tion.
54, 59–61
NV-ESR at high-fields can be used to resolve this signal. However, only a
few investigations of NV centers have been performed at high magnetic fields,.
62–64
This thesis will describe general methods development for NV-ESR and NV-NMR
at high magnetic fields. Chapter 2 will provide an overview of the spin-physics for EPR
and NV properties required for the performance of NV-detected magnetic resonance.
Chapter 3 will overview the experimental apparatus and necessary equipment for the
performance of high-field ODMR. Chapter 4 will discuss the nature of the NV-ESR
7
line-shape and establish a procedure to obtain a high resolution spectrum represent-
ing intrinsic properties of the sample. Chapter 5 discusses the necessary techniques to
achieve the first demonstration of NV-ESR at a high magnetic field (4.2 T). Chapter 6
demonstrates the first application of NV-NMR at a magnetic field strength comparable
to commercial research grade NMR (8.3T), which corresponds to a proton larmor fre-
quency of 350 MHz. The thesis is briefly summarized in Chapter 7, before a few useful
results are presented in the Appendix.
8
Chapter 2: Quantum Sensing using
Electron Spin Resonance (ESR)
Spectroscopy
ESR spectroscopy is a well developed field that centers on the interaction of unpaired
electron spin(s) with an applied magnetic field. As discussed in Chapter 1, the primary
focus of this work is based upon the NV center in diamond. This defect center is an
S = 1 spin system and exhibits spin physics that well described in pulsed ESR. There-
fore, a background of ESR spectroscopy will be reviewed before describing the unique
properties of the NV center. The fundamentals of ESR will be reviewed here, but the
interested reader is encouraged to refer to the variety of excellent textbooks on the sub-
ject.
26, 32, 65, 66
In general, the field of ESR is complementary to the field of NMR and
the excellent textbooks discussing the spin dynamics of NMR are directly applicable to
ESR.
67–69
Optically detected magnetic resonance (ODMR) and quantum sensing tech-
niques exhibit spin dynamics that are well described by a foundation of pulsed ESR.
Therefore, after an overview of pulsed ESR, experimental examples will be presented
demonstrating the pulsed ESR techniques using ODMR.
9
2.1 Static Spin Hamiltonian
The static spin Hamiltonian of a general spin system will first be described in the labo-
ratory frame.
26, 65
This was first derived for a general spin system (H
0
) with an effective
electron spinS coupled tom nuclei with spinI.
70
The static spin Hamiltonian may be
written as a sum of its respective components:
H
0
=H
EZ
+H
ZFS
+H
HF
+H
NZ
+H
NQ
; (2.1)
where the first term (H
EZ
) describes the electron Zeeman interaction, the second term
(H
ZFS
) denotes the zero field splitting, the third term (H
HF
) denotes the hyperfine in-
teraction, the fourth term (H
NZ
) denotes the nuclear zeeman interaction, and the fifth
term (H
NQ
) denotes the nuclear quadrupole interaction. The strength of each term in
the Hamiltonian depends on the spin system and its environment.
The state space of the spin system depends on the number of electron and nuclear
spins and their respective angular momentum. For a general spin system, the size of the
state space can be expressed as:
n
H
=
n
Y
k=1
(2S
k
+ 1)
m
Y
k=1
(2I
k
+ 1); (2.2)
wheren is the number of electron spins with angular momentumS andm is the number
of nuclear spins with angular momentumI. As Eq. 2.2, implies, the overall size of the
state space increases quickly when n;m > 1. A useful tool for quickly constructing
basis states is the tensor product. The basis of a spin system can be expanded,j i =
j
S
i
j
I
1
i:::
j
I
N
i, where multi-spin operators can be constructed from single spin
operators.
10
2.1.1 Electron Zeeman Interaction
The electron Zeeman interaction describes the interaction between an electron spin sys-
tem and the magnetic field. This interaction may be written as:
H
EZ
=
e
=~
~
B
|
0
g
^
S; (2.3)
where
e
is the Bohr magneton, ~ is the reduced planck constant,
~
B
0
|
is the transpose
of a vector describing the applied magnetic field with elementsB
x
,B
y
, andB
z
, g is a
tensor with three principal axesg
x
,g
y
, andg
z
, and
^
S is a vector containing spin operators
^
S
x
,
^
S
y
, and
^
S
z
. In some cases, it may be helpful to define the gyromagnetic ratio for an
electron,
e
. This is defined for an isotropicg as,

e
=
e
g=~: (2.4)
The values of theg matrix depend upon the degree of spin-orbit interaction of the un-
paired electron spin. Theg value of a free electron is known to an extremely high degree
of precision, currently 2.00231930436256(35).
71
When unpaired electrons are located
within molecular orbitals they experience an admixture of orbital angular momentum
with spin angular momentum. This causes a deviation of the value of g from that of
the free electron and can be used to probe the electron’s surrounding environment. The
orientation of the three principalg axes is chosen to reflect that of the molecular frame.
The relationship between the values describes the symmetry of the spin system rela-
tive to the molecular frame. For an isotropic g-value, the interaction of the electron is
equivalent in all directions relative to the molecular frame and the three g values are
11
equivalent,g
x
=g
y
=g
z
. This is the case for a completely unpaired electron, or a free-
radical with little orbital mixture, such as in organic molecules. G-value anisotropy can
exist in multiple forms, depending on the localization of the electron within the molec-
ular orbital. For spin-systems with two symmetrical axes, the g value can exhibit axial
symmetry. In this case, the interaction along two axes of the molecular frame are equiv-
alent, while one is different. Therefore, these have two unique g values and are defined
in terms of parallel and perpendicular components, ie, g
x
= g
y
= g
?
g
z
= g
k
. Some
spin-molecular systems exhibit orthorhombic symmetry where the interaction is unique
along each cartesian axis and therefore contain three unique g values,g
x
6=g
y
6=g
z
.
2.1.2 Zero-field splitting
For systems with more than one unpaired electronS 1, there is interaction between
the electrons termed zero-field splitting. The coupling between two unpaired electron
spins may be written as:
H
ZFS
=

0
4
g
2
e

2
e
"
^
S
T
1

^
S
2
r
3

3(
^
S
T
1
r)(
^
S
T
2
r)
r
5
#
; (2.5)
where
^
S
1
and
^
S
2
are the vectors of spin operators for the first and second electron,
respectively, andr is the vector between them. Here, the g values for both electrons are
assumed to be equal to the free electron and isotropic. By expanding the dot products
and rearranging, this can be represented as:
H
ZFS
=D
x
^
S
2
x
+D
y
^
S
2
y
+D
z
^
S
2
z
; (2.6)
12
or equivalently,
H
ZFS
=D[
^
S
2
z
1=3S(S + 1)] +E(
^
S
2
x

^
S
2
y
); (2.7)
whereD = 3D
z
=2 andE = (D
x
D
y
)=2. For axially symmetric systems,D6= 0 and
E = 0. TheD term is often termed the zero-field splitting term as it results in splitting
of the spin states independently of the applied magnetic field.
2.1.3 Hyperfine interaction
The hyperfine interaction between an electron spin and a nuclear spin may be defined
as:
H
HF
=
^
S
|
A
^
I; (2.8)
where the tensor A can be written as a sum of isotropic and anisotropic components.
This gives:
A =a
iso
I +T; (2.9)
wherea
iso
is the isotropic hyperfine interaction,I is an identity matrix, andT is a tensor
that contains anisotropic components. The overlap of the electron wavefunction at the
nucleus, also known as the Fermi contact interaction, determines the isotropic hyperfine
coupling,
a
iso
=
2
0
3~
g
e
g
n

n
j
0
(0)j
2
; (2.10)
whereg
n
is the nuclear g value,
n
is the nuclear magneton, andj
0
(0)j
2
represents the
electron spin density at the nucleus.
13
Dipolar interaction between the electron and nuclei is represented by the tensorT.
This dipolar interaction may be written as:
H
Dip
=

0
4
g
e
g
n

n
"
^
S
T

^
I
r
3

3(
^
S
T
r)(
^
I
T
r)
r
5
#
; (2.11)
wherer represents the vector between the electron and nuclear spin. Upon rearranging,
it can be found that,
H
Dip
=
^
S
|
T
^
I; (2.12)
where the tensorT describes the electron-nuclear dipolar interaction.
2.1.4 Nuclear Zeeman Interaction
The nuclear Zeeman interaction describes the interaction of a magnetic nuclei with a
magnetic field. It may be written as:
H
NZ
=
n
g
n
=~
~
B
|
0
^
I; (2.13)
where
^
I is the nuclear spin operator. The value ofg
n
is generally regarded to be isotropic
and, along with
^
I, is determined by the identity of the nuclear spin. It is standard from
the field of NMR to discuss nuclei in terms of their gyromagnetic ratio,
n
. This is
defined as,

n
=
n
g
n
=~; (2.14)
wheren depends on the identity of the nuclear spin. Tables of
n
for magnetically active
nuclear isotopes are readily accessible.
67
14
2.1.5 Nuclear Quadrupole Interaction
The nuclear quadrupole interaction is defined for nuclei withI 1 using a nuclear elec-
trical quadrupole moment, Q. This term represents the interaction of a non-spherical
charge distribution with an electric field gradient and is described by the following
Hamiltonian:
H
NQ
=
^
I
|
P
^
I; (2.15)
whereP is the nuclear quadrupolar tensor. This can be written as:
H
NQ
=
e
2
qQ
4I(2I 1)~
h
3
^
I
2
z
I(I + 1)
2
) +(
^
I
2
x

^
I
2
y
)
i
; (2.16)
where e is the charge of an electron, q is the electric field gradient and = (P
x

P
y
)=P
z
is an asymmetry parameter. The largest principal value ofP is given byP
z
=
e
2
qQ=4I(2I 1)~ with the labeling convention P
z
P
y
P
x
. Since the principal
axes of the nuclear quadrupolar term sum to zero (P
x
+P
y
+P
z
= 0), the asymmetry
parameter is bounded with 0 1. Where = 0 whenP
x
= P
y
and = 1 when
P
x
=P
y
= 2P
z
.
67
2.2 Continuous Wave (cw) ESR spectroscopy
The field of ESR began with the successful detection of electron spin resonance from
a few systems of hyrdated salts in 1944.
65
Since its initial demonstration, the field of
ESR has grown rapidly. Significant improvements in instrumentation have resulted in
the rapid growth of the field with a wide variety of applications, such as for character-
ization of defects in solid-state samples, determination of intermolecular distances in
biomolecules through spin-labeling, for the investigation of transition metal ions, and
15
for the characterization of systems with conducting electrons. Continuous wave ESR
is now a standard spectroscopic technique with commercially available instrumentation
at a variety of field strengths available. Depending on the experiment, different field
strengths can be used to isolate field dependent spin parameters, such as the Zeeman
interaction, from non-field dependent spin parameters, such as the hyperfine interaction.
As discussed in Chapter 1, higher field strengths offer improved g-factor resolution and
spin-polarization, but remains under-development.
In the most common application of cw-ESR, a spin system is placed within a mag-
netic field, B
0
, while a fixed electromagnetic field, B
1
, is applied with frequency !.
In typical experiments, the intensity of the reflectedB
1
is monitored while the applied
magnetic field is swept to avoid frequency related artifacts.
2.2.1 cw-ESR: Model
To better understand cw-ESR it is useful to consider anS = 1=2 spin system with an
isotropic g value as a two-level system (TLS). Application of a static magnetic fieldB
0
results in splitting of the spin states due to the electron Zeeman interaction described in
Eq. 2.3. This interaction produces two eigenstates,ji andji, with energiesE
;
=
!
0
=2, where!
0

e
B
0
. The z axis of the spin system is taken to be collinear withB
0
.
An electromagnetic field (B
1
) is applied with an oscillating magnetic field orthogonal to
B
0
. Taking the rotating wave approximation, this has the form:
B
1
(t) =
[cos(!t +)
^
S
x
+ sin(!t +)
^
S
y
]; (2.17)
where

e
B
1;max
=2 is the Rabi frequency,B
1;max
is the maximum amplitude of the
electromagnetic radiation,! is the frequency of the electromagnetic radiation,t is time,
16
and is the phase of the electromagnetic radiation. At this point, the Hamiltonian for
the spin system is:
H
0
(t) =!
0
^
S
z
+
[cos(!t +)
^
S
x
+ sin(!t +)
^
S
y
]: (2.18)
It is useful to define a resonance offset, ! =!!
0
, and transform the Hamiltonian to
a frame rotating a frequency of! using the unitary transformation,U = exp(i!
^
S
z
(t).
The Hamiltonian in the rotating frame (H =U
1
H
0
(t)U) can be written:
H = !
^
S
z
+
(
^
S
x
cos +
^
S
y
sin); (2.19)
where the time dependence of the Hamiltonian has been removed by transforming to the
rotating frame. It is standard convention to set = 0 for an “x-pulse” and ==2 for
a “y-pulse”. The evolution of the spin-state att is given by the Schr¨ odinger equation:
i~
dj i
dt
=
^
Hj i; (2.20)
where ~ is the reduced Planck’s constant, andj i is a vector representing the wave-
function of a pure state. It will be necessary to include mixed states in our discussion,
therefore we will consider density matrices given by, =j ih j. The Schr¨ odinger
equation can be extended to describe the dynamics of a density operator. The evolution
of the spin state, including relaxation, is described using the Lindblad equation:
d
dt
=
i
~
[H;] +L; (2.21)
where is the density matrix of the system andL is the Lindbladian.
72
The Lindbladian
describes the effect of relaxation on the quantum system, resulting in mixed states. The
17
Lindbladian may be written to account for amplitude and phase damping. In the basis
of and states, this takes the form:
L =
2
6
6
4

1
=2[
4

1

4eq

1eq
]
2

2

2

3

1
=2[
4

1

4eq

1eq
]
3
7
7
5
; (2.22)
where
1
= 1=T
1
represents the influence of amplitude damping,
2
= 1=T
2
represents
the influence of phase damping,
n
represents the populations of the density matrix (ie,
= [

1

2

3

4
]), and
neq
are the populations of the density matrix at thermal equilibrium.
The cw-ESR measurement is performed by measuring the system in a steady-state
where the spin-relaxation processes are slow relative to the measurement time. There-
fore, cw-ESR signals can be calculated with Eq. 2.21 and settingd=dt = 0 and =
0
,
the density matrix of the system at thermal equilibrium (
0
= (I +~!
0
=k
B
T
^
S
z
)=2),I
is the identity matrix,k
B
is the Boltzmann constant,T is the temperature.
67
The steady
state solutions are expressed as:

1
= 1=2 P
eq
1 !
2
T
2
2
2(1 +

2
T
2
T
1
+ !
2
T
2
2
)
(2.23a)

4
= 1=2 + P
eq
1 !
2
T
2
2
2(1 +

2
T
2
T
1
+ !
2
T
2
2
)
(2.23b)

2
=

3
= P
eq
(i
)(1 !
2
T
2
2
)
2(
2
+i!)(1 +

2
T
2
T
1
+ !
2
T
2
2
)
: (2.23c)
18
Frequency Offset [MHz]
-10 -5 0 5 10
0.0
0.5
-0.5
-1.0
1.0
Mx
My
Mz
Intensity [a.u.]
Figure 2.1: Magnetization from Bloch Equations. The Magnetization was cal-
culated from Eq. 2.24 usingM
0
z
=1,
= 0:1 MHz,T
2
= 100s, andT
1
= 1
ms. The frequency offset (!) is plotted along the x-axis The intensities ofM
x
andM
y
have been normalized to 1 for clarity.
By considering the expectation values from the density matrix, one can calculate the
bulk magnetization. The “Bloch equations” describing the steady state magnetization
are found by evaluating Eq. 2.23 withh
^
S
n
i =g
B
Tr(
^
S
n
) as:
M
x
=M
0
z

!T
2
2
1 + !
2
T
2
2
+

2
T
1
T
2
(2.24a)
M
y
= +M
0
z

T
2
1 + !
2
T
2
2
+

2
T
1
T
2
(2.24b)
M
z
=M
0
z
1 + !
2
T
2
2
1 + !
2
T
2
2
+

2
T
1
T
2
; (2.24c)
whereM
0
z
is the static magnetization (M
0
z
= 1=2~
e
N
v
), and N
v
is the difference
in spin population per unit volume (N
v
=
eq
=m
3
).
65
Figure 2.1 shows a plot of
Eqs.2.24. From Eq. 2.24 it is evident that the strength of the respective signals depends
on the relative relaxation values (T
1
/T
2
) and the strength of the applied excitation field
(
). During a cw-ESR experiment, the signalsM
x
,M
y
, andM
z
are experimental ob-
servables. Phase sensitive detection, as realized through a lock-in amplifier, is typically
19
used to detect bothM
x
andM
y
simultaneously. As the sweep approaches resonance, the
bulk magnetization M
z
is reduced to zero while the magnetization M
x
first increases,
then decreases after resonance, corresponding to a dispersion signal. The absorption
signal (M
y
) is at a maximum while on resonance. It is convention to display ESR sig-
nals in dispersion mode, as this offers higher sensitivity to small changes in lineshape.
However, as can be seen in Fig. 2.1, both lineshapes are representative of the same spin
dynamics.
2.2.2 cw-ESR: Spectral analysis
Spectral analysis of ESR signals can provide detailed information on spin species. As
the focus of this thesis is on defects in diamond, here I discuss spectral analysis of a
single substitutional nitrogen impurity in diamond (P1 center). The P1 center consists
of a single unpaired electron spin (S = 1=2) that is coupled to a nitrogen spin in the
diamond lattice. Nitrogen has two stable, magnetically active isotopes,
14
N (I = 1)
with a natural abundance of 99.6% and
15
N (I = 1=2) with a natural abundance of
0.4%. Due to the relevant natural abundances, the P1 center consisting of a
14
N nuclei
with aS = 1=2,I = 1 spin system is most common and will be considered below. The
P1 center has the following Hamiltonian:
H
P 1
=g
e
=~B
0
^
S
z
+A
k
^
S
z
^
I
z
+A
?
(
^
S
x
^
I
x
+
^
S
y
^
I
y
); (2.25)
where the tensors were expanded using the isotropic g value (g = 2:0024) and axial
symmetry of the hyperfine interaction (A
k
= 114 MHz, A
?
= 82 MHz).
16, 27
Due to
this symmetry, a (111) cut diamond possesses P1 centers in two distinct groups: axially
aligned defects aligned along the [111] orientation and non-axially aligned defects with
20
[111], [111], and [111] orientations. According to Eq. 2.2, there will be 6 eigenstates for
the axial orientation. For a sufficiently strong (g
e
=~B
0
A
k;?
) magnetic field aligned
along the quantization axis, the eigenenergies of the states may be written as:
E
1
(jm
s
=1=2;m
I
= +1i) =!
0
=2A
k
=2 (2.26a)
E
2
(jm
s
=1=2;m
I
= 0i) =!
0
=2 (2.26b)
E
3
(jm
s
=1=2;m
I
=1i) =!
0
=2 +A
k
=2 (2.26c)
E
4
(jm
s
= +1=2;m
I
=1i) = +!
0
=2A
k
=2 (2.26d)
E
5
(jm
s
= +1=2;m
I
= 0i) = +!
0
=2 (2.26e)
E
6
(jm
s
= +1=2;m
I
= +1i) = +!
0
=2 +A
k
=2; (2.26f)
where!
0
=g
e
=~B
0
,m
s
(m
I
) was used to denote the spin (nuclear) quantum number,
and the state energy (E
n
) was written in order of increasing energy. The transition
probability between two eigenstates,jai andjbi, is described as:
P
ab
=jhajH
1
jbij
2
; (2.27)
whereH
1
= g
e
=~B
1
^
S, andB
1
was described in Eq. 2.17. During a typical cw-ESR
experiment, a fixed frequency B
1
is applied and the magnetic field is slowly varied.
When the energy difference between two states nears the frequency of the appliedB
1
there are three possible scenarios predicted by Eq. 2.27: (1) an allowed ESR transition
whereP 1, (2) a forbidden ESR transition where 1 P6= 0, and (3) no transition
whereP = 0. Allowed ESR transitions involve a change of the spin quantum number
by m
s
=1, while the nuclear quantum number remains the same and are therefore
much greater in intensity than forbidden transitions. Forbidden transitions are all other
21
spin quantum transitions where m
s
6= 1 such as a double quantum transition where the
electron and nuclei simultaneously flip (m
s
=1 & m
I
=1). These transitions
will be discussed in more detail in section 2.3.7.
As shown in Fig. 2.2(a), there are three allowed ESR transitions where m
s
=1:
E
1
$E
6
=!
0
+A
k
(2.28a)
E
2
$E
5
=!
0
(2.28b)
E
3
$E
4
=!
0
A
k
; (2.28c)
where a transition between the energy states is denoted by an arrow. The transitions for
the nonaxial transitions can be found in a similar manner, giving an additional set of
three lines where the spacing between the lines is smaller asA
?
T
1
.
34
2.3.5 Inversion Recovery
Inversion recovery is an experiment designed to measure the spin-lattice relaxation time,
also known as T
1
. This time corresponds to the amount of time required to relax to
thermal equilibrium, given by the Boltzmann distribution. Therefore, the evolution may
be written as:
j
IR
i =
^
U
1
(t)
^
R

j (0)i; (2.49)
where the operators are defined as in Eq. 2.41. In this case, the initial statej (0)i
depends on how the system was initialized and is typically the thermal equilibrium,ji,
orji states. The evolution of theji state can be written as:
j
IR
i =
^
U
1
(t)
^
R

ji (2.50a)
j
IR
i =
^
U
1
(t)iji (2.50b)
The final magnetization is then read out by measuring the population of the final
state,P

=jhj
IR
ij
2
. The population of the state then decays according to:
M
z
=M
0

1 exp(
t
T
1
)

; (2.51)
where M
0
is the static magnetization. The rate of the decay depends on the local en-
vironment. Two common sources ofT
1
decoherence are flip-flop interaction with local
paramagnetic spins or coupling to lattice phonons. In diamond samples, the coupling to
spin baths and lattice phonons can be controlled by varying the temperature, resulting
in drastic improvements in the T
1
relaxation time
27, 85
In addition, the T
1
decoherence
35
rate can be selectively tuned forT
1
-based magnetometry by controlling the the relative
splitting of energy levels between nearby spins.
86–88
2.3.6 Double Electron Electron Resonance (DEER)
Double electron-electron resonance, also known as DEER, is a form of dipolar spec-
troscopy that builds upon the methods discussed previously. In DEER, two frequencies
of radiation are employed: one to drive the probe spin (A spin) and another to drive the
target spin (B spin). The model introduced here is a variation of the model previously
introduced.
89
The B spin is dipolar coupled to the A spin. Therefore the A spin experi-
ences a magnetic fieldb that depends on the spin state of the B spin. The dipolar field
that the A spin experiences can be written as:
m
s
b =m
s

0

B
~
4
(1 3 cos
2
)
r
3
; (2.52)
where
a
(
b
) is the gyromagnetic ratio of the A (B) spin,r is the distance between the
spins, is the angle between the A spin’s quantization axis and the B spin, andm
s
is the
magnetic quantum number of the B spin.
The pulse sequence is the same as that discussed in section 2.3.3 with an additional
pulse applied to the B spins at the same time as the pulse is applied to the A spin as
seen in Fig. 2.7(a). The evolution of the spin state can then be expressed using Eq. 2.42
withb(t
0
) =b
s
(t
0
)+b, separating the field contribution of the B spin from the magnetic
36
MW1
p
2
MW2
y
z
x
b
a
y
z
x
b
a
y
z
x
b
a
y
z
x
b
a
(b)
(a)
p
t t
p
D µ g
a
dbt
Figure 2.7: Double Electron-Electron Resonance (DEER) Pulse Sequence. (a)
Pulse sequence used for measurement of DEER. The pulse sequence is a SE
sequence with an additional -pulse applied at the frequency of B spins. (b)
Bloch sphere representation of the pulse sequence. The A spin accumulates a
phase during the first evolution period. Application of a-pulse to the B spin
results in a change in the local magnetic field at the A spin. As shown in Eq.
2.55, spins do not rephase at 2 under the DEER condition
field due to bath spins (b
s
(t
0
)). The evolution of the spin-state under the DEER sequence
may then be written as:
j
DEER
i =
1
p
2
[exp

i
a
=2(
Z
2t
t
b
s
(t
0
)dt
0
b=2) +i
a
=2(
Z
t
0
b
s
(t
0
)dt
0
+b=2)

ji
+i exp

+i
a
=2(
Z
2t
t
b
s
(t
0
)dt
0
b=2)i
a
=2(
Z
t
0
b
s
(t
0
)dt
0
+b=2)

ji];
(2.53)
where the B spin was assumed to havem
s
=1=2 and the sign ofb is reversed as the
additional-pulse flips the B spin, reversing its phase accumulation. By assuming that
b
s
(t
0
) is constant during the experimental time scale the spin state may be written as:
j
DEER
i =
1
p
2
[exp (+i
a
b))ji +i exp (i
a
b)ji]; (2.54)
37
where the time period between the time points (t
2
t
1
) was set to be. Furthermore,
the expectation valuesh
^
S
x
i andh
^
S
y
i may be found:
h
^
S
x
i = 1=2 sin(
a
b2) (2.55a)
h
^
S
y
i = 1=2 cos(
a
b2); (2.55b)
where trigonometric substitutions were used to simplify the exponentials. From Eq.
2.55, it is clear that in the case of a single A and single B spin the expectation values
will oscillate proportional to the field and evolution period. As done previously, it is
useful to consider a model where many B spins interact with a single A spin.
89
The
DEER intensity of an ensemble of B spins may be written as,
I
DEER
= exp

2
0

2
e
g
a
g
b

9
p
3~
n
B

sin
2

2

L

; (2.56)
where thehsin
2
2
i
L
term represents the effective population inversion of the DEER
pulse. This is given as:

sin
2

2

L
=
Z
+1
1

2
(!)
2
+

2
sin
2

p
(!)
2
+

2
t
p
2

L(; !)d; (2.57)
where! is the frequency of the applied MW pulse (MW2 for DEER),t
p
is the applied
pulse length, andL(; !) is an intrinsic ESR lineshape with linewidth given by !.
90
The intrinsic ESR lineshape represents the broadening of individual line positions and
can be represented by a normalized sum of Lorentzian or Gaussian functions depending
on the relevant broadening mechanisms.
38
2.3.7 Double Resonance Electron-Nuclear Spectroscopy
The interaction between an electron spin and its corresponding nuclei will be reviewed
in the following section. There are two main types of double resonance ESR techniques
used to measure nuclear spins: Electron-Nuclear Double Resonance (ENDOR) and
Electron-electron double resonance detected NMR (EDNMR). Both techniques build
upon the previously discussed spin dynamics and can be well described using a sys-
tem of two coupled spins. Therefore, the Hamiltonian of a coupled two spin system
(S = 1=2;I = 1=2) will be briefly considered. The Hamiltonian can be written in the
lab frame as,
^
H
EN
=
e
B
0
^
S
z

n
B
0
^
I
z
+A
^
S
z
^
I
z
+B(
^
S
x
^
I
x
+
^
S
y
^
I
y
); (2.58)
where Eqs. 2.3, 2.8, and 2.13 were used. In addition, the hyperfine interaction was sep-
arated into axial (A =A
k
cos
2
+A
?
sin
2
) and nonaxial (B = (A
k
A
?
) cos sin)
components. The angle is the angle betweenA
k
andB
0
. In the high field, weak cou-
pling limit (ie,
e
B
0

n
B
0
> A;B), there are four possible states.The four states
have six possible transitions with energies:
!

=E
1
$E
2
=

(!
I
+A=2)
2
+ (B=2)
2

1=2
(2.59a)
!

=E
3
$E
4
=

(!
I
A=2)
2
+ (B=2)
2

1=2
(2.59b)
!
13
=E
1
$E
3
=!
0
+ 1=2(!

!

) (2.59c)
!
24
=E
2
$E
4
=!
0
1=2(!

!

) (2.59d)
!
14
=E
1
$E
4
=!
0
+ 1=2(!

+!

) (2.59e)
!
23
=E
2
$E
3
=!
0
1=2(!

+!

); (2.59f)
39
where, !
0
=
e
B
0
and !
I
=
n
B
0
are the electron and nuclear larmor frequencies,
respectively, andE
n
is used to denote the energy of the n-th state.
26
The allowed NMR
transitions (m
s
= 0; m
I
=1) are!

and!

, the allowed ESR transitions (m
s
=
1; m
I
= 0) are !
13
and !
24
, the forbidden double quantum transitions (m
s
=
1; m
I
=1) are!
14
and!
23
.
In an ENDOR measurement, an RF pulse is applied in sequence with MW pulses
driving the allowed NMR transitions. The RF pulse manipulates the spin state of the nu-
clear spins. In an EDNMR measurement, suitable MW pulses are applied to selectively
drive the nominally forbidden double quantum transitions, enabling NMR spectroscopy
of the nuclear spins. Both techniques will be discussed in the following sections.
Pulsed ENDOR
There are two main types of pulsed ENDOR: Mims and Davies Endor, which are shown
in Fig.2.8(a) and (c), respectively.
Both sequences utilize an RF pulse to drive an allowed NMR transition and change
the population of nuclear spin states, which are detected at the end of the pulse sequence.
Mims ENDOR is based upon the stimulated echo discussed in Sec. 2.3.4 and was first
introduced in 1965.
91
As seen in Fig.2.8(a), the sequence applies an RF pulse during
the central period
D
of a STE sequence. The frequency of the RF pulse is swept, when
the pulse is in resonance with an NMR transition, it flips the nuclear spin state and
changes the intensity of the STE. To better describe the phenomenon, it is convenient to
setB = 0, which simplifies the transitions in Eq. 2.59. As discussed in Sec. 2.1.3 in the
case of weak hyperfine interaction, the nuclear spins are coupled largely through dipolar
interaction to the electron spin. Therefore, the interaction of the nuclear spin causes a
shift in the resonance frequency experienced at the electron spin, which may be written
40
D µ At
MW
(a) Mims Endor
y
z
x
b
a
(b)
p
2
p
2
p
2
t t
t
D
RF
y
z
x
b
a
y
z
x
b
a
z
x
b
a
z
x
b
a
x
z
x
b
a
p
D µ AT
p
MW
(c) Davies Endor
Detection
(d)
T
p
p
2
p
t t
RF
p
w
0
w
0
Dw
I
w
0
Dw
I
Figure 2.8: Sequences for Pulsed Electron-Nuclear DOuble Resonance (EN-
DOR). (a) Pulse sequence used for Mims ENDOR. The pulse sequence is a
STE sequence with an additional -pulse applied at the frequency of nuclear
spins during the delay period
D
. (b) Bloch sphere representation of the pulse
sequence. The A spin accumulates a phase during the first evolution period.
Application of a-pulse to the nuclear spins changes the local magnetic field
at the A spin. The difference in the local magnetic field causes the spin-state to
rephase at a different point on the Bloch sphere, reducing the echo intensity as
shown in Eq. 2.61. (C) Pulse sequence used for Davies ENDOR. The pulse se-
quence consists of a long, selective MW pulse followed by an RF pulse to drive
nuclear transitions. A SE sequence is employed for detection of the nuclear
spin polarization. (d) Representation of the electron spin resonance distribution
centered at!
0
. The first, selective MW pulse burns a hole in the resonance spec-
trum. Application of an RF pulse moves resonant nuclear spins with resonant
nuclear frequency !
I
back into the hole burnt in the spectrum, increasing the
echo intensity.
b
N
= (E
1
E
3
) (E
2
E
4
) =A. The evolution of the spin state is written using Eq.
2.47a and making the substitution thatb(t
0
) = b
s
(t
0
) +b
N
. The evolution of the spin
state may be written as:
j
mims
i =
^
U
2
(
D
)
1
p
2
[exp (+i
n
b
N
))jii exp (i
n
b
N
)ji]; (2.60)
41
where the RF pulse applied during
D
is assumed to be a- pulse on the coupled nuclear
spins. The nuclear spin flip induces a phase shift of the spin state, as shown in Fig.
2.8(b). Evaluating the expectation value it is found that:
h
^
S
y
i = 1=4(1 + cos(A)) (2.61)
The pulse sequence for Davies ENDOR seen in Fig.2.8(c), where an initial MW
pulse is applied to burn a hole in an ESR spectrum.
92
The RF pulse is applied afterwards
to shift the relative polarization of nuclear spins. If the applied RF pulse is in resonance
with an NMR transition, the population of the spin states in the hole changes, as seen in
Fig.2.8(d). The population is then read out using a detection sequence. The sensitivity
of Davies ENDOR depends strongly on the length of the MW pulse used to burn a hole
in the ESR spectrum and the hyperfine interaction of the nuclear spins. This relationship
can be expressed as:
I
Davies
=I
max
p
2
s

2
s
+ 1=2
; (2.62)
where
s
=t

A=(2) is a selectivity parameter that depends on the pulse length (t

) and
the hyperfine interaction. I
max
is the maximum ENDOR intensity when
s
= 1=
p
2.
93
The amplitude oft

is adjusted to ensure that the hole burning pulse produces a rota-
tion. While in general, the hyperfine resolution increases with longer pulse lengths, the
number of contributing spins decreases due to the reduced excitation pulse bandwidth.
Therefore the pulse intensity and length must be chosen carefully to maximize signal
resolution while ensuring sufficient signal intensity.
42
DµI
f
MW1 (n
0
)
(a)
Detection
(b) n
1
< n
0
p
2
p
t t
MW2 (n
1
)
HTA
w
0
w
0
+w
I
w
0
-w
I
MW1
MW2
(d) n
1
= w
0
+ w
I
w
0
w
0
+w
I
w
0
-w
I
MW1
MW2
(c) n
1
= w
0
w
0
w
0
+w
I
w
0
-w
I
MW1
MW2
Figure 2.9: Sequence for Pulsed Electron-electron Double resonance detected
NMR (EDNMR). (a) Pulse sequence used for EDNMR. The pulse sequence
consists of a long high-turning angle (HTA) pulse applied with MW2 at
1
,
followed by a SE sequence using MW1 at
0
for detection. (b) Representation
of the electron spin resonance distribution centered at !
0
. Double quantum
transitions are offset by the nuclear larmor frequency (!
I
). When
1
is far from
resonance, the intensity of the echo remains unchanged. (c) When
1
= !
0
at the central blind spot, the ESR transition is saturated, resulting in very low
echo intensity. (d)When
1
=!
0
!
I
, the HTA pulse drives a double quantum
transition, resulting in a slight reduction in the echo intensity and detection of
EDNMR.
Electron-electron double resonance detected NMR (EDNMR)
EDNMR is a form of hyperfine spectroscopy that utilizes two microwave frequencies,
MW1 (
0
) and MW2 (
1
) to selectively drive forbidden transitions.
94
EDNMR transi-
tion frequencies can be explained using Eq. 2.58, but require state mixing to partially
allow the forbidden transition intensity. Therefore, either nuclear quadrupolar mixing
or an angle of misalignment between the hyperfine axis and the static magnetic field is
required (ie,B6= 0).
As shown in 2.9(a), EDNMR measurements vary the frequency (
1
) of a high turning
angle (HTA) MW2 pulse, while MW1 applies a detection pulse sequence, such as spin
43
echo, at
0
to measure the spin polarization of an ESR transition.
94
The intensity of
the echo is monitored, while the frequency of
1
is swept. When the HTA pulse is not
on resonance, the echo intensity does not change (Fig. 2.9(b)). As
1
approaches the
central allowed transition (Fig. 2.9(c)) there is significant population transfer leading
to a highly intense change and the so-called “central blind spot”. Since the central
blind spot highly distorts EDNMR signal, the measurement is typically performed at
a magnetic field sufficiently high to shift the nuclear transitions outside of the central
blind spot. When
1
matches the frequency of forbidden transitions (
1
=!
0
!
I
; for
this example !
I
!

;!

) forbidden transitions are induced from hyperfine coupled
nuclei as shown in Fig. 2.9(d). The intensity of the transitions may be written as:
I
f
=
j!
2
I
1=4(!

+!

)
2
j
!

!

= sin
2

mix
(2.63a)
I
a
=
j!
2
I
1=4(!

!

)
2
j
!

!

= cos
2

mix
; (2.63b)
where
mix
= 1=2(

),
;
= tan
1
(B=(A 2!
I
)), and I
f
(I
a
) is the in-
tensity of the forbidden (allowed) transitions. The transition intensity directly depends
on the relative strength of the off-axial hyperfine splitting B. This is comparable to
other ESR spectroscopies that utilize forbidden transitions such as electron-spin echo
envelope modulation (ESEEM), where the modulation depth is k = sin
2
(2).
32
The
EDNMR signal corresponds to the depth of the holes (h) in Fig. 2.9(d), and is given by:
h = 1I
a
cos

t
HTA
p
I
f

I
f
cos

t
HTA
p
I
a

(2.64a)
h 1I
a
cos

t
HTA
p
I
f

1
4
(
t
HTA
)
2
I
f
; (2.64b)
44
where t
HTA
is the length of the high turning angle pulse.
94
Equation 2.64a accounts
for possible overlap of forbidden and allowed transitions of different spin packets. By
assuming a very small forbidden transition probabilityI
f
, such that
t
HTA
p
I
f
1,
eq. 2.64b can be obtained. Application of a long HTA pulse improves the likelihood of
population transfer, but the total length of the HTA pulse must be short relative toT
1
in
order to maximize the observable signal.
2.4 Optically Detected Magnetic Resonance (ODMR)
with the Nitrogen Vacancy (NV) Center
The NV center exhibits a number of useful properties, such as stable fluorescence, a
nanoscale footprint, and high sensitivity to magnetic fields that make it an excellent can-
didate for quantum sensing. Since the initial observation of coherent oscillations from
a single NV center, the field has exploded with many implementations of NV centers
for quantum sensing, as single photon sources, and as biochemical probes of their local
magnetic environments. The properties of the NV center will be briefly described here,
but have been thoroughly summarized in a number of useful review articles.
55–58
One
of the NV’s most useful properties is its capability for optically detected magnetic res-
onance (ODMR). ODMR is performed by manipulating the spin state of the NV center
and draws strongly on the spin-physics discussed in the preceding section.
2.4.1 NV Center in Diamond: Overview
The NV center is an optically active defect center in diamond consisting of a substi-
tutional nitrogen next to an adjacent vacancy site. While the existence of NV defect
centers has been known through ESR for some time, the first detection of a single NV
45
center and observation of coherent Rabi oscilliations were fundamental in its develop-
ment for quantum sensing.
16, 33, 37
The properties of the NV center are related to its
host material, the diamond lattice. Diamond is an excellent host material for a quantum
sensor due to its wide bandgap, high thermal stability, and chemical stability. The NV
center has been shown to have excellent sensitivity to magnetic fields (> 1fT), is opera-
tional at room temperature, and has an atomic-scale footprint; making it an ideal sensor
for nanoscale measurements.
34, 35, 38, 95, 96
2.4.2 NV Optical Properties
The NV center is known to exist in several charges states: NV

, NV
0
, and NV
+
.
97
However, only the negatively charged NV center, NV

, exhibits optical polarization.
The NV center consists of six electrons: two donated by the substitutional nitrogen
atom, three from the carbon atom dangling bonds, and one absorbed from the diamond
lattice. These electrons occupy a set of four molecular orbitals as shown in Fig.2.10. The
lowest of these, the electronic ground state
3
A
2
, is a spin triplet split into sublevels by a
zero-field interactionD, of 2:87 GHz. The
3
A
2
state is coupled to an excited triplet state,
3
E, that has a weaker zero-field interaction of 1.42 GHz. The transition between the
3
A
2
ground state to the
3
E state is spin state preserving with a zero-phonon line of 638 nm,
and is typically excited using above-bandgap (532 nm) light into the phonon sideband.
55
Excitation of the
3
E state rapidly decays with photon emission back into the
3
A
2
with
an average fluorescence lifetime of 10-20 ns.
98–101
A non-radiative transition also exists
between the
3
E and
1
A
1
states mediated by spin-orbit coupling. Intersystem crossing
occurs with a much higher probability from the m
s
=1 states than the m
s
= 0 of
3
E and does not conserve spin angular momentum.
102
The
1
A
1
state is often termed the
metastable state as it takes significantly longer ( 300 ns) before a radiative transition
46
1
E
1
A
1
3
E
532 nm 600-850 nm
3
A
2
D
|M
s
= 0ñ
|M
s
= ±1ñ
|M
s
= 0ñ
|M
s
= ±1ñ
Figure 2.10: Energy level diagram of the NV center. Excitation with above
bandgap (532 nm) light optically polarizes the NV in thejm
s
= 0i state due
to the coupling between the excited state and the metastable state. The ground
state zero field splitting, D = 2:87 GHz.
(zero-phonon line 1042 nm) to the
1
E state. Intersystem crossing from the
1
E state to
the
3
A
2
state preferentially populates them
s
= 0 level of the ground state, completing
the optical cycle.
103
Initialization and readout
The spin-state selectivity of the intersystem crossing allows for optical initialization of
them
s
= 0 state of up to 87% at room temperature.
104
The preferential intersystem
crossing of them
s
=1 states results in a lower rate of fluorescence from them
s
=1
states relative to them
s
= 0 state. This allows clear identification of the spin state as
seen in Fig. 2.11. Therefore, a laser pulse serves to initialize the spin state and readout
fluorescence indicative of the spin state, with the readout window calibrated to the first
300 ns of the laser pulse. For all ODMR sequences, a long laser pulse is used at the
beginning of the sequence and a short laser pulse is used at the end of the sequence
to induce fluorescence representative of the population difference. Conventional ESR
detection is based on either the absorption or emission of photons and therefore detects
47
Initialization R.o.
Laser
FL
MW
p
(a)
(b)
|m
s
=0
|m
s
= -1
Readout (R.o)
Initialization
Figure 2.11: Initialization and fluorescence readout of the NV center. (a) Pulse
sequence used for calibration of initialization and readout windows. A long
laser pulse was applied to initialize the spin state before application of a MW
-pulse to change the spin state. Fluorescence (FL) readout was performed by
integrating the first 300 ns of the trace. (b) Experimentally measured time
dependent FL traces. When a long laser pulse is applied tojm
s
= 0i, the FL in-
creases rapidly before stabilizing. When a laser pulse is applied tojm
s
=1i,
the FL is initially lower, before stabilizing. Using a time dependent FL trace, the
initialization and readout (R.o.) pulse lengths can be calibrated. Data courtesy
of Dr. Chathuranga Abeywardana.
in the transverse plane of the Bloch sphere. ODMR spectroscopy utilizes many similar
sequences as pulsed ESR. However, as ODMR spectroscopy reads out the population
difference between spin states, it is necessary to apply a final=2-pulse to convert the
spin-states into a population difference.
Autocorrelation
Autocorrelation, also known as anti-bunching, measurements can be used to determine
if detected fluorescence is from a single quantum emitter and provide insight into the
48
(b) (a)
NV B NV A
t [ns]
g
2
(t)
-50 0 50
0
1
2
Exp Fit
Error 0.5
t [ns]
-50 0 50
Figure 2.12: Experimental autocorrelation data. (a) Autocorrelation data for
NV-A. The experimental data is shown as the blue line, the solid red line is a
fit to Eq. 2.66 and the dashed red lines represent the 95% confidence intervals.
A dashed line drawn at 0.5 indicates the threshold for a single quantum emitter.
NV-A has ag
(2)
(0) = 0:28 0:08 identifying it as a single NV . (b) Autocorre-
lation data for NV-B. NV-B has ag
(2)
(0) = 0:65 0:05, and is therefore not a
single NV .
photodynamics of the NV center. The autocorrelation function (g
(2)
()) was obtained
by correlating the arrival times () of photons detected using a Hanbury Brown-Twiss
interferometer as discussed in detail in Section 3 and Fig. 3.2.
105
The second order
correlation function is defined as:
g
2
()
hI(t)I(t +)i
hI(t)ihI(t +)i
; (2.65)
whereI(t) (I(t +)) is the intensity of light measured at timet (t +). The values for
I(t) andI(t +) are measured after the beamsplitter using separate photodetectors. For
a single NV center at zero time delay ( = 0), an emitted photon may only be detected at
one photodetector. Therefore single NV centers exhibit antibunching withg
(2)
(0)< 0:5.
Autocorrelation data for two NVs, NV-A and NV-B, is shown in Fig. 2.12.
The data was analyzed using a previously introduced model of the photodynamics
of NV centers.
100
The autocorrelation data exhibits a characteristic growth rate,
1
,
49
that is dependent on the rate of fluorescence and strength of laser excitation, k
12
. NV
centers also exhibit photon bunching due to the existence of a metastable state with a
long lifetime relative to the excited state. Shelving of the electron from the excited state
to the metastable state acts to “trap” the electron and causeg
(2)
()> 1:0 with increased
driving rate,k
12
. At longer times, the population on the metastable state decays with a
characteristic rate,
2
, until the photons are completely uncorrelated at long delay times
(ie, g
(2)
() 1:0) to which the experimentally observed autocorrelation function is
normalized. The intensity of antibunching and bunching signals is dependent on the
state population and is therefore dependent on the strength of laser excitation relative
to the decay rates. The relative intensity of the corresponding signals is represented by
. This is distinguished from incoherent background photons through the usage of an
additional parameter, , and is the ratio of NV fluorescence (S) to total fluorescence
(S +B) (ie, =S=(S +B)). The autocorrelation function is then described using the
following model:
g
(2)
() = 1 +
2
(e
jj
1
+ ( 1)e
jj
2
); (2.66)
where the parameters were introduced previously.
100
The FL lifetime of the excited
state is given by
FL
= 1=(
1
k
12
). For the AC measurements described here, the
laser power was set below 1 mW so thatk
12
is smaller than observed error values and

FL
1=(
1
).
100
Furthermore, the lifetime of the metastable state is given by
MS
=
=
2
. Due to the sub-poissionian statistics of light, the observed signal was subject to
shot noise. Therefore,g
(2)
() was measured continuously for each NV until sufficient
photons were accumulated for satisfactory signal-to-noise.
As seen in Fig. 2.12(a), NV-A had ag
(2)
(0) = 0:28 0:08, with uncertainty calcu-
lated from the 95% confidence intervals of the fit, proving that the observed FL signal
50
is from a single photon emitter. By fitting with Eq. 2.66, values of
FL
= 10:7 0:3
ns,
MS
= 344 8 ns, and = 0:85 0:01. As seen in Fig. 2.12(b), NV-B had a,
g
(2)
(0) = 0:65 0:05 therefore, the experiment shows that the observed FL signal is
not from a single photon emitter. NV-B does exhibit similar FL lifetimes to NV-B with
values of
FL
= 8:8 0:3 ns,
MS
= 253 10 ns, and = 0:59 0:01 extracted from
the fit to Eq. 2.66.
2.4.3 Pulsed ODMR
Having discussed the optical properties of the NV center, the NV centers spin-physics
as probed through ODMR will be discussed. The spin physics of pulsed ODMR builds
strongly on the material discussed in Sec. 2.3, with the primary difference being the
detection mechanism. In conventional pulsed ESR, the absorption or emission of MW
photons is detected. In comparison, pulsed ODMR is detected by inducing FL from
the NV center that is representative of the spin population. Therefore, an additional
=2-pulse is required to convert spin-states in the transverse plane into a measurable
population difference. This has minor implications as discussed in the following sec-
tions.
NV: Hamiltonian
The Hamiltonian of the NV center is briefly reintroduced. This Hamiltonian may be
written in the lab frame as:
H
NV
=g
NV

e
=~
~
B
^
S +D
^
S
z
2
+A
?
(
^
S
x
^
I
x
+
^
S
y
^
I
y
) +
^
S
z
A
k
^
I
z
; (2.67)
51
where the electron Zeeman interaction (g
NV
= 2:0028), zero field splitting (D = 2:87
GHz), and hyperfine interaction with the internal nitrogen nuclear spin (H
N
) were in-
cluded. The internal nitrogen spin can be either
14
N (I = 1) or
15
N (I = 1=2), which
are 99.6% and 0.4% naturally abundant, respectively. When the hyperfine splitting can
be resolved, the number and splitting of lines can be used to identify the nucleus of
interest. For example,
14
N will result in three hyperfine lines with A
k
=2:1 MHz,
A
?
=2:7 MHz, and an additional quadrupolar term,P =5:0 MHz, while
15
N will
result in two hyperfine lines withA
k
= +3:0 MHz,A
?
= 3:7 MHz.
106
When fabricat-
ing shallow NV centers, this can be useful to verify if the detected NV was fabricated
or naturally occurring.
88, 107, 108
For this reason, NV fabrication is generally performed
using
15
N.
cw-ODMR and Zeeman splitting
Application of a static magnetic field B
0
lifts the degeneracy of the m
s
= 1 sub-
levels according to Eq. 2.67. The degree of splitting is proportional to the magnitude
of B
0
along the NV spin axis. The resonances can be observed by applying contin-
uous laser excitation, sweeping the frequency of an applied microwave (MW) field
and monitoring the fluorescence intensity. A reduction in fluorescence is seen when
the microwave frequency is in resonance with the splitting between either the lower
(jm
s
= 0i$jm
s
=1i) or the upper (jm
s
= 0i$jm
s
= +1i) transitions, as seen
in Fig. 2.13. The relative peak positions can be used to determine the strength of the
magnetic field and offset angle () from the NV axis. This is done by using a lorentzian
function (L(!; !;!
c
) = 1=
!
!
2
+(!!c)
2
), where ! is the width of the resonance dis-
tribution, and!
c
is the resonance position determined by diagonalization of Eq. 2.67.
A fitting function can then be constructed in software such as MATLAB with three
52
Frequency [MHz]
1800 1850 3850 3900 3950
95
100
Contrast [%]
Sim.
Fit
Figure 2.13: Continous Wave Optically detected magnetic resonance (cw-
ODMR) measured from an ensemble NV sample. Clear dips were resolved
for the lower (jm
s
= 0i$jm
s
=1i) and upper (jm
s
= 0i$jm
s
= +1i)
transitions of an NV ensemble. By fitting with Eq. 2.67 and a lorentzian res-
onance frequency distribution, it was determined thatB
0
= 36:71 0:05 mT,
= 4:3 0:6
o
, and ! = 8:0 0:8 MHz.
input parameters, B
0
, misalignment from NV-axis (), and !. This allows for the
NV-alignment field and angle to be determined precisely as shown in 2.13. In cer-
tain samples, application of high resolution pulsed techniques can be used to limit MW
broadening and resolve the hyperfine splitting of the internal nitrogen spin. Extension of
the fitting routine to account for the hyperfine interaction then allows for a more precise
determination of field strength and angle.
Spin Dynamics of the NV center
Application of pulse sequences to probe the spin dynamics can provide more detailed
information about the local environment and weaker couplings. These measurements
utilize the spin dynamics discussed for pulsed ESR in Sect.2.3. Experimental examples
of Rabi oscillations, Pulsed ODMR FID, SE, STE, and T1-IR will be discussed as these
are utilized for NV-ESR and NV-NMR as discussed in the following sections. For the
53
following discussion, the NV-center will be treated as a TLS. This can be done if a
sufficiently strong static magnetic fieldB
0
is applied such that one of the transitions can
be selectively addressed. Either transition can be used, but the lower (m
s
= 0$m
s
=
1) transition is typically more convenient from an experimental standpoint.
Rabi Oscillations
Rabi Oscillations are observed by initializing the NV intojm
s
= 0i with a laser pulse
before applying a variable length MW pulse on resonance with the lower transition.
Three sequential experiments were performed as shown in Fig. 2.14(a), by observing
the amount of fluorescence with no MW pulse (Max.), the amount of fluorescence with a
fixed MW pulse applied (Sig.), and the amount of fluorescence observed when a MW-
pulse is applied (Min.). From these measurements the population of thejm
s
= 0i state
was determined. As seen in Fig. 2.14(b), the spin state oscillates betweenjm
s
= 0i and
jm
s
=1i with the Rabi frequency as described in Eq. 2.32a. Data is presented for a
single NV , where
= 12:5 MHz, resulting in a -pulse of 40 ns and a =2-pulse of
20 ns. These pulse lengths were then used for measurements of the spin dynamics and
calibrated separately for each NV sample.
Pulsed ODMR
The measurement of pulsed ODMR will be discussed next. In order to resolve the hyper-
fine coupling to the internal nitrogen spin, it is necessary to reduce power broadening.
A pulsed ODMR measurement with a single, low power pulse is ideal for this as the ob-
served fluorescence contrast will be maximized if the length and power of the pulse
are correctly calibrated. If the amplifier is operating in the linear regime, the required
power for a longerpulse may be calculated.
54
t
p
[ms]
Exp. Fit
P|m
s
=0> (a.u.)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4
(b)
(a)
Laser
MW
Init. R.o.
t
p
Init. R.o.
p
Init. R.o.
N
Max. Sig. Min.
Figure 2.14: Rabi Oscillations from a single NV center. (a) Pulse sequence
used for the measurement of Rabi oscillations. An initial 5 s laser pulse is
used to initialize the laser pulse before a final 300ns laser pulse is used to con-
vert the population difference into a population difference. Three sequential
experiments are performed: (Max.) where no MW pulse is applied between
initialization and readout, (Sig.) where a microwave pulse of length t
p
is ap-
plied. (Min.) where a microwave pulse corresponding to a-pulse is applied.
Each experiment was then repeated N times, where N was typically 10
5
-10
7
.
(b) Experimentally measured Rabi oscillations. The fluorescence intensity was
converted to Pjm
s
= 0i using Pjm
s
= 0i =
MaxSig
MaxMin
, where Max was the
maximum FL observed, Sig was the FL observed after application of a MW
pulse, and Min was the minimum amount of FL observed (ie,Pjm
s
= 0i = 0).
Once the length of apulse is known, pulsed ODMR can be performed as shown
in Fig. 2.15. The pulse sequence is similar to that used in the measurement of Rabi
oscillations, with a fixed pulse length and the power of the pulse adjusted to perform a
-rotation. Pulsed ODMR data from a single NV in a type-1b diamond is shown in Fig.
2.15(b). By reducing the power of the MW pulse, power broadening is reduced, result-
ing in three clearly resolved peaks at 1950:7, 1953:1, and 1955:1 MHz. The presence of
three peaks and the hyperfine splitting is in agreement with the presence of
14
N, which
is expected from naturally occurring NVs due to its higher relative abundance. Pulsed
55
(a)
(b) (c)
Init. R.o.
Laser
14
N
15
N
MW
t
p
2095 1950 1955 2100 2105
Frequency [MHz] Frequency [MHz]
Contrast [a.u.]
Contrast [a.u.]
Figure 2.15: Pulsed ODMR (pODMR) measurements from single NV centers.
(a) Pulse sequence used for the measurement of pODMR. Laser pulses were
used to initialize and readout the NV center’s spin state. A fixed microwave
pulse of 1s was applied with the power adjusted to perform a rotation. (b)
Experimentally measured pODMR from the lower (m
s
= 0 $ m
s
= 1)
transition of a single NV at 32:7 mT in a type-1b diamond. Three peaks were
resolved with hyperfine splitting in agreement with
14
N. The single NV was
NV4 from the following reference.
109
(c) Experimentally measured pODMR
from the lower transition of a single NV at 27:5 mT that was fabricated in an
electric grade diamond. Two peaks were resolved with hyperfine splitting of 3.0
MHz in agreement with
15
N. The single NV was fabricated using previously
described methods.
107, 108
ODMR data from a single NV fabricated using shallow ion implantation of
15
N in an
electric grade diamond is shown in Fig. 2.15(c). The NV was fabricated using previ-
ously described methods.
107, 108
Peaks are clearly resolved at 2098:0 and 2101:1 MHz.
The presence of two peaks and splitting between them is in excellent agreement with
the presence of
15
N, indicating that the NV was fabricated using the shallow energy ion
implantation process. While not discussed in detail here, the relative intensity of the
peaks measured in pODMR provides direct insight into the polarization of the nitrogen
spin. This can be utilized as a quantum memory for single-shot readout and to probe the
spin-dependent photodynamics of the polarization process.
110–112
56
(a)
(b)
Init. R.o.
Laser
MW
p
2
p
2
t
t [ms]
0 1 2 3 4
0.5
1.0
Exp.
Fit
P|m
s
=0 [a.u.]
Figure 2.16: Free induction decay (FID) measurement from a single NV center.
(a) Pulse sequence used for the measurement of the FID. Laser pulses were used
to initialize and readout the NV center’s spin state. The MW sequence consisted
of a =2-pulse followed by a delay period and a final =2-pulse to project
the spin state into a population difference. (b) Experimentally measured FID
from the lower (m
s
= 0$ m
s
=1) transition of a single NV at 26:0 mT.
The hyperfine splitting of the NV was previously measured in 2.15(c). In this
experiment,=2-pulses of 20 ns were used. Fitting with Eq. 2.40 was used to
determineT

2
= 1:2 0:2s.
Free induction decay (FID)
The free induction decay (FID) experiment will be discussed next. As discussed in Sec.
2.3.2, the FID measures the behavior of a spin state in the transverse plane of the Bloch
sphere following an initial=2-pulse. The pulse sequence for ODMR measurement of
the FID is shown in Fig. 2.16(a). This consists of initialization of the spin state before an
initial=2-pulse followed by a period and a final=2-pulse to transfer the coherence
back into a population difference before the population is read out by a final laser pulse.
The presence of hyperfine coupling results in a static shift of the resonance frequencies.
The FID may be calculated by taking into accountA
z
= 3:03 in Eq. 2.40 MHz for the
15
N hyperfine coupling constant.
106
57
The experiment was performed at a MW1 frequency of 2100 MHz and a magnetic
field of 26:0 mT. As shown in Fig. 2.16, Eq. 2.40 is in excellent agreement with the
observed data. From this fit,T

2
was determined to be 1.2 0.2s.
Spin Echo (SE)
Next the SE measurement will be described as introduced in Sec. 2.3.3. The pulse
sequence for measurement of SE is shown in Fig. 2.17(a). Within the SE measurement,
a=2-pulse places the NV center in a superposition after initialization. The spin state
evolves for a period during before application of a-pulse. After application of the
-pulse, the spin state rephases during the second period. An additional =2-pulse
is applied to convert the coherence to a population difference that is measured using a
readout pulse. During the measurement the maximum intensity of the echo is monitored
and both periods are varied simultaneously. As the evolution time 2 approachesT
2
,
the spin state is less efficiently refocused. Once 2 T
2
, the spin state is mixed at
the end of the sequence in equal populations ofm
s
= 0 andm
s
=1. The coherence
decay of a SE measurement may be written as:
SE(t) =
1
2
[exp((t=T
2
)
3
) + 1]; (2.68)
for NV centers in diamond.
47, 74, 113
Experimental measurement of SE for a single NV in diamond is shown in Fig.
2.17(b). At short 2 times the spin state is efficiently refocused. The periodic revivals
in echo intensity are indicative of an electron spin echo envelope modulation (ESEEM)
effect due to coupled
13
C and
14
N spins.
40
At longer time periods the echo intensity
decays to the mixed state (Pjm
s
= 0i = 0:5) according to Eq.2.68 with a decay time of
45s.
58
(a)
(b)
Init. R.o.
Laser
MW
p
2
p
p
2
t t
2t [ms]
0 50 100 150 200 250
0.5
1.0
0.0
P|m
s
=0 [a.u.]
Exp.
Sim.
Figure 2.17: Spin echo (SE) measurements from a single NV center. (a) Pulse
sequence used for the measurement of SE. Laser pulses were used to initialize
and readout the NV center’s spin state. The MW sequence consisted of a=2-
pulse followed by a delay period, a second-pulse, an additional delay period
, before a final=2-pulse projects the spin state into a population difference.
(b) Experimentally measured SE from the lower (m
s
= 0$ m
s
=1) tran-
sition of a single NV at 32:7 mT. The NV was previously measured in 2.15(b).
In this experiment, =2-pulses of 24 ns and -pulses of 48 ns were used. A
simulation of Eq. 2.68 shows that aT
2
of 45s is in good agreement with the
experimentally observed coherence decay.
Stimulated Echo (STE)
Next the STE measurement will be described as introduced in Sec. 2.3.4. The pulse
sequence for a STE experiment is shown in Fig. 2.18(a). In the STE pulse sequence,
the first MW pulse places the NV center in a coherent superposition of spin states after
initialization intom
s
= 0 using a laser pulse. The coherent superposition accumulates a
phase during the first
1
period that is transferred into the longitudinal plane by a second
=2-pulse. A longer delay
D
then stores the spin state in the transverse plane before
another=2-pulse tips the spin state back into the transverse plane. After a time period

1
=
2
, the spin state rephases. A final=2-pulse converts the rephased echo back into
a population difference that is measured using an additional laser pulse.
59
(a)
Init. R.o.
Laser
MW
p
2
p
2
t
1
t
D p
2
p
2
t
2
(b)
t
2
[ms]
9 10 11
Contrast [a.u.]
(c)
t
1
+t
2
[ms]
0 200 400 600
Contrast [a.u.]
Exp.
Sim.
Figure 2.18: Spin echo (STE) measurements from ensemble NV centers. (a)
Pulse sequence used for the measurement of STE. Laser pulses of 100 and 15
s were used to initialize and readout the NV center’s spin state, respectively.
The MW sequence consisted of a=2-pulse followed by a delay period, a sec-
ond=2-pulse, a storage delay period
D
, a third=2-pulse, and an additional
delay period before a final=2-pulse projects the spin state into a population
difference. (b) Experimentally measured echo intensity STE from the lower
(m
s
= 0$m
s
=1) transition of ensemble NV centers at 4:196 T. In this ex-
periment,=2-pulses of 0.6s were used. The first delay period was fixed at
10s and the storage delay
D
was set at 100s while the second delay period

2
was swept. An echo was observed with a maximum at time 10s with a full
width at half maximum of 0:84s. (c) Experimentally measured echo intensity
STE for several
1
parameters. In each case,
1
and
D
were fixed, while
2
was
varied. A simulation of Eq. 2.68 shows that aT
2
of 350s is in good agreement
with the experimentally observed coherence decay.
In practice, the echo amplitude can be measured by varying
1
and
2
simultaneously.
However, for some samples with very longT
2
, this can be problematic as laser intensity
fluctuations and/or mechanical motion may prevent an accurate measurement of the FL
after all coherence has decayed (
1
+
2
T
2
). An alternative measurement, that
can also be utilized for the measurement of SE, is therefore discussed. An alternative
measurement of T
2
can be performed by performing several sequential measurements
with fixed
1
and
D
periods while varying
2
. In this way, the echo refocusing can
60
be observed at a period where
1
=
2
in addition to a baseline fluorescence intensity.
Therefore, drifts in baseline fluorescence intensity can be corrected and the coherence
timeT
2
may be obtained. The presented method is used to describe experimental results
for a dNV (element six) sample containing NV centers with long coherence times. As
shown in Fig.2.18(b) variation of
2
results in echo rephasing when
1
=
2
with a
clearly resolved baseline. After repeating the measurement for several
1
times, the
echo intensity is observed to decrease as seen in Fig. 2.18(c). Using Eq. 2.68, aT
2
of
350s is determined in excellent agreement with the observed data.
T
1
-Inversion Recovery (T1-IR)
Next the measurement of the spin-lattice relaxation time (T
1
) will be described as in-
troduced in Sec. 2.3.5. For the measurement of inversion recovery, two sequential
measurements are performed. The pulse sequence is shown in Fig. 2.19(a) For both
measurements, the spin state is initialized in them
s
= 0 state before an evolution period
T . After the evolution periodT , the spin state is read out using an additional laser pulse.
For one measurement (Sig1) the spin state is left in them
s
= 0 state during the evolution
period, while for the other measurement (Sig2) a-pulse is applied to convert the spin
state to m
s
=1. Both Sig1 and Sig2 relax to the mixed state during the period T .
Therefore, by taking the difference between the spin states and fitting to an exponential,
theT
1
time can be extracted.
Experimental results for a single NV center are shown in Fig. 2.19(b). Sig1 is seen
to have a slight downward trend while Sig2 has a slight growth. As seen in Fig. 2.19(c),
the difference exhibits a clear exponential decay and is well fit by a single exponential
decay. From the fit, a T
1
relaxation time of 300 100 s was extracted. In practice,
extraction of theT
1
relaxation time for single NV centers is often a lengthy measurement
61
(b)
T [ms]
Contrast [%]
D Contrast [%]
(c)
T [ms]
0 200 400 600 800 0 200
100
95
90
0
5
400 600 800
Exp.
Sim.
Sig1
Sig2
T T
(a)
Laser
MW
Init. R.o. Init. R.o.
p
Sig1 Sig2
Figure 2.19: T
1
inversion recovery (T1IR) measurements from a single NV
center. (a) Pulse sequence used for the measurement ofT
1
. Laser pulses of were
used to initialize and readout the NV center’s spin state. The MW sequence
consisted of two sequential measurements: Sig1 without a -pulse and Sig2
with a-pulse before an evolution periodT . AfterT , the spin state was readout
using a laser pulse. (b) Experimentally measured T1IR curves for Sig1 and
Sig2. echo intensity STE from the lower (m
s
= 0$ m
s
=1) transition of
ensemble NV centers at 32:7 mT. In this experiment, a -pulse of 48 ns was
used. (c) Experimentally measured change in contrast (Sig1-Sig2). A fit to a
single exponential decay shows that aT
1
of 300 100s is in good agreement
with the experimentally observed fluorescence decay.
due to the large time scales (ms) and required experimental repetitions ( 10
4
10
5
).
However, the described measurement allows for estimation of the T
1
relaxation time
even with limited experimental measurement time.
2.4.4 NV-ESR
The previously discussed spin dynamics set a foundation for the discussion of NV-ESR.
NV-ESR will be briefly introduced, but will be discussed in detail in Chapters 4 and 5
for single and ensemble NV centers at low and high magnetic fields. The measurement
62
(a)
(b)
Init. R.o.
Laser
MW1
p
2
p
p
2
t t
MW2
p
MW2 Frequency [MHz]
800 900 1000 1100
0.5
1.0
0.0
P|m
s
=0 [a.u.]
Exp.
Ref.
Figure 2.20: NV-ESR from a single NV at 33.4 mT. (a) Pulse sequence used for
the measurement of NV-ESR. After initialization, a SE sequence was applied
using MW1. A MW2 pulse was applied during the second evolution period,
before the final=2-pulse and laser readout pulse. The MW2 pulse was swept in
frequency. 24 ns 40 ns 5 us tau 56ns p1 (b) Experimentally measured NV-ESR
spectrum from P1 centers in diamond (Sig.) alongside a reference experiment
with no MW2 pulse (Ref.). In this experiment, a=2-pulse of 24 ns, a MW1
-pulse of 40 ns, a MW2-pulse of 56 ns, and a period of 5s was used.
of NV-ESR is based upon a double electron-electron resonance technique that was intro-
duced within Sec. 2.3.6. During a SE-based DEER measurement, an additional pulse is
applied to invert the spin of a target B spin and therefore change the phase accumulated
at the A spin. An example of an NV-ESR sequence is shown in Fig. 2.20(a). To perform
NV-ESR, the coherence time is first measured by applying the SE sequence and varying
the interpulse spacing as shown in Fig. 2.17(b). Once a suitable time was selected,
NV-ESR is performed by varying the frequency of the MW2 pulse with a fixed de-
lay. The results of the experiment alongside a reference experiment are shown in Fig.
63
2.20(b). The reference experiment is performed without a MW2 pulse. Clear reductions
in Fl intensity are observed in the experiment at 825, 855, 945, 1025, and 1055 MHz
in excellent agreement with the five expected peaks from the substitutional nitrogen
(P1) center. DEER measurements may also be performed using dynamical decoupling
sequences as shown in appendix C.
2.4.5 NV-NMR
There are three main hyperfine spectroscopic methods, ESEEM, EDNMR, and ENDOR.
The first of these has become a routine measurement in NMR, but is generally limited to
low-field applications. ESEEM is generally performed at low fields for several reasons:
(1) the intensity of the ESEEM signal depends on state mixing, which decreases as the
electron Zeeman interaction dominates over the hyperfine coupling resulting in a 1=B
0
field dependence.
26, 32
(2) ESEEM requires that the interpulse spacing is modulated at
the nuclear larmor frequency, requiring shorter periods as the field strength increases.
Technical limitations make the generation of high-frequency pulses difficult, resulting
in low MW power or difficulty in pulse control.
28
NV-based NMR has become a standard technique with many demonstrations in the
literature.
48–51, 114
The other two techniques, EDNMR and ENDOR are designed for
high-field ESR and will be discussed in the following sections.
NV: EDNMR
EDNMR was introduced in Sect. 2.3.7 and was performed on an ensemble of NV cen-
ters. The pulse sequence used to perform EDNMR is described in Fig. 2.21(a) and
consists of an initialization pulse, a HTA pulse applied using MW2, a-pulse applied
using MW1 on the NV transition at frequency
0
, and then a readout pulse to induce
64
(b) (c)
(a)
Init.
High Turning Angle Pulse
R.o.
Laser
MW1
MW2
p
MW2 Frequency [GHz]
229.92 229.94 229.96 229.98
MW2 offset Frequency [MHz]
Min.
-40 -20 20 0 40
100.0
99.5
99.0
Contrast [%]
101
100
99
EDNMR Int. [%]
100ms
300ms
500ms
1000ms
n
0
14
N
14
N
Max.
Figure 2.21: EDNMR from an ensemble of NV centers at 230 GHz and 8.305
T. (a) Pulse sequence used for the measurement of EDNMR. After initialization
a long High turning angle pulse (HTA), was applied using MW2. MW1 applied
a-pulse at the NV’s center transition. (b) EDNMR spectra collected by vary-
ing the frequency of the HTA pulse, using a HTA pulse of 500s. References
used to normalize the data (Max, Min, and
0
) are indicated with arrows. (c)
Normalized EDNMR data showing the influence of varying the pulse length on
the intensity of
14
N signals.
laser fluorescence that is integrated. In the present example,
0
= 229:9528 GHz. Dur-
ing the EDNMR measurement, the frequency of the MW1 pulse is fixed while the fre-
quency of the HTA pulse is varied. As the frequency of HTA pulse is changed, there
is weak excitation of EDNMR transitions before a large reduction in fluorescence due
to the central blind spot. As shown in Fig. 2.21(b), the central blind spot occurs at
the central resonance frequency
0
and allows determination of the minimum contrast.
The maximum contrast is determined from an area where there are no EDNMR tran-
sitions. The data is normalized to the intensity of the central blind spot according to
I
EDNMR
: = (MaxSig)=(MaxMin), whereSig is the experimentally observed
contrast. The normalized data for EDNMR with several different HTA pulse lengths is
shown in Fig. 2.21(c). Clear signals from
14
N are resolved at 30 MHz. As expected
65
(b)
Frequency [MHz]
44.88 44.92 44.96
99.76
99.75
99.74 Contrast [%]
T
pulse
MW1
RF
(a)
Init. R.o.
Laser
p
2
p
2
t t
D p
2
p
2
t
Figure 2.22: Mims-ENDOR at 115 GHz (4.197T) from an ensemble NV sys-
tem. (a) Pulse sequence used for the measurement of NV-ENDOR. A STE se-
quence was applied with an additional RF pulse applied during the delay period

D
. Laser pulses were used to initialize and readout the spin-state. (b) ENDOR
data from
13
C using a of 10s and a 400s RF pulse,T
pulse
.The diamond is
the same as used in Fig.2.18.
from Eq. 2.64b, a long HTA pulse is observed to increase the EDNMR signal intensity
until the signal intensity begins to saturate.
NV: ENDOR
NV-based Mims ENDOR was performed on an ensemble of NV centers at 4.197 T
and 115 GHz. The Mims ENDOR measurement was introduced in sect. 2.3.7. The
pulse sequence used for NV-ENDOR is described in Fig. 2.22(a) and consists of a STE
sequence with a delay and
D
. During the storage delay of the STE sequence, an
additional RF pulse is applied. The RF pulse is significantly longer than the pulses used
to drive the NV spin due to the weaker gyromagnetic ratio of nuclear spins. The pulse
length is selected to perform a-rotation on the target nuclear spins, but also influences
the relative EDNMR contrast and width as shown in appendix C. As shown in Fig.
2.22(b), by sweeping the frequency of the RF pulse, a clear reduction in fluorescence
66
intensity is observed at 44.9322 MHz which is in agreement with the expected line
position for
13
C at a field of 4.197 T The peak shape agrees well with a lorentzian
lineshape with a width of only 9:4 0:3 kHz.
2.5 Summary
Within this chapter, the foundations for pulsed ESR and ODMR were described. Starting
from a static Hamiltonian, fundamental interactions between spins and with a magnetic
field were described. These interactions formed the basis for cw-ESR, a technique that
remains in wide usage today due to its high sensitivity to interactions at the atomic level.
A brief example of how the spectral analysis can be performed to correlate the atomic
structure with the static Hamiltonian was given using the subsitutional nitrogen (P1)
center in diamond. Having described cw-ESR, the discussion was extended to pulsed-
ESR methods. These methods provide more precise control over the spin dynamics and
allow for the extraction of weaker couplings and interactions that may not be resolved
within cw-ESR. Rabi oscillations describe the interaction of an electromagnetic field
with a spin-system on resonance and enable calibration of pulse lengths that are used in
successive experiments. The measurement of the FID allows direct correlation between
the minimum observable linewidth and the spin relaxation rateT

2
. An extension of the
time domain FID is the SE measurement that introduces an additional pulse to refo-
cus the dephasing that contributes to the FID. This measurement enables measurement
ofT
2
, sometimes called the phase memory or coherence time. Dynamical decoupling
techniques build upon the SE technique, where more advanced pulse timing and phase
control can be utilized to drastically extend the coherence time. The STE technique was
presented as a variation of the SE sequence where an additional delayT
d
was incorpo-
rated that stores the spin states in the longitudinal plane where they are less sensitive to
67
phase relaxation, instead relaxing according to T
1
, which is generally longer than T
2
.
The lattice-relaxation timeT
1
can be measured using aT
1
-inversion recovery sequence.
These measurements of spin-relaxation formed a foundation for the measurement of
weak spin interaction, which is done for electron and nuclear spins using DEER and
ENDOR techniques.
After providing an overview of spin dynamics as related to pulsed ESR, a model-
system, the NV-center, was introduced to provide experimental examples of the previ-
ously described measurements. Before discussing the spin-dynamics, the optical prop-
erties of the NV-center were briefly introduced. Identification of single-NV centers via
autocorrelation was also described, which can be correlated directly to the photodynam-
ics. Pulsed ODMR was next described, first beginning with the zeeman interaction as
measured using cw-ODMR, before describing experimental examples of the previously
described pulsed ESR techniques using both single and ensemble NV-centers. Specific
examples of NV-ESR and NV-NMR using DEER, EDNMR, and ENDOR were then
described.
The presented techniques represent a brief introduction into the spin-dynamics that
can be measured using the NV center. The following chapters provide examples of the
previously described techniques for several applications.
68
Chapter 3: Instrumentation
This chapter concerns instrumentation used for performing ODMR experiments on NV
centers in diamond. The high field ODMR spectrometer is built upon an high field
ESR spectrometer that was described previously.
30, 115
The original design and addition
of optical components to this spectrometer was described in a previous publication.
62
This section will describe the overview, operation, and modifications of the ODMR
spectrometer.
3.1 Overview
Both low field and high field ODMR spectrometers were designed in a modular fashion
for a highly flexible and adaptable system that enables complementary experiments and
facilitates optimization and troubleshooting. The high field ODMR spectrometer was
designed on top of the existing high field ESR spectrometer for in-situ operation of high
field ESR experiments.
The overall experimental diagram is shown in Fig. 3.1. All experimental protocols
including instrumental control, instrument timing, and data integration is performed
using a central computer. The instrumental control is programmed in LabVIEW with
instrument interfacing conducted primarily through VISA (Virtual Instrument Software
Application). The central computer is equipped with a pulse generation board (PB-500)
69
B
0
Power
combiner
Corrugated
waveguide
2. Quasioptics
1. HF source
3. Detection
system
Steering
Mirror
Lenses
Dichroic
Mirror
Photo
diode
Oscilloscope
Freq.
Muitiplier
MW2
Rectangular
pulses
MW1
Chirped
pulses
Laser filters
4. ODMR system
AOM
532 nm
laser
APD2
APD1
Diamond
NV
Wire
5. Sample
Stage
6. MW/RF
System
Figure 3.1: Overview of HF ODMR system. The HF ODMR system consists
of five components: (1) A HF microwave component, (2) a quasioptical MW
propagation system, (3) a HF ESR detection system, (4) an ODMR system.
The NV detection system utilizes a photodiode for ensemble experiments and
avalanche photodiodes for single NV experiments. (5) A diamond sample is
mounted upon a sample stage within a variable field magnet. (6) A high power
amplifier and MW/RF source is coupled to the sample stage using cables. A
thin copper wire (40 AWG) is placed on top of the diamond sample for delivery
of MW/RF radiation to the diamond sample. HF MW excitation propagates
through free space from the waveguide to the top of the sample stage. Optical
access is obtained through the bottom of the sample stage.
for orchestration of experimental timing using TTL pulses. A detailed description of
each experimental component follows.
70
3.2 HF ESR Spectrometer
As seen in Fig.3.1 (1), the 115/230 GHz ESR system employs a high-power solid-state
source consisting of an 8-10 GHz synthesizer, solid-state positive-intrinsic-negative
(pin) diode switch, microwave amplifiers, and frequency multipliers. The output power
of the source system is 100 mW at 230 GHz and 480 mW at 115 GHz. Two synthe-
sizers are employed for DEER experiments. An IQ mixer (Miteq) controlled by an
arbitrary wave generator (AWG; Keysight) has recently been implemented in MW1 for
pulse shaping of high-frequency microwaves. The frequency range of the microwave
source is 107-120 GHz and 215-240 GHz. The output power of the HF microwave is
700 mW at 115 GHz and 100 mW at 230 GHz. HF microwaves are propagated to a
sample using a home-built quasioptical bridge and a corrugated waveguide (Thomas-
Keating), as seen in Fig.3.1 (2). As demonstrated previously, quasioptics are suitable
for a high-frequency ESR spectrometer because of their capacity for low-loss and broad-
band propagation.
29, 116
A sample is mounted at the end of the corrugated waveguide and
positioned at the field center of a room temperature bore within a superconducting mag-
net system. As seen in Fig.3.1 (3), ESR signals are isolated from the excitation using
induction mode operation.
116
For ESR detection, a superheterodyne detection system is
employed in which 115/230 GHz is down-converted into the intermediate frequency of
3 GHz then down-converted again to in-phase and quadrature components of dc signals.
Details of the system have been described elsewhere.
30, 115
No microwave resonator is
employed for implementation of wide bandwidth DEER techniques. The magnetic field
at the sample is adjustable between 0 to 12.1 Tesla.
71
3.3 High Field ODMR Spectrometer
The HF ODMR system is built upon the existing HF ESR spectrometer that was de-
scribed previously, enabling in-situ experiments of both ESR and ODMR.
30, 115
A con-
focal microscope system for ODMR has been built on an optical table below the magnet.
3.3.1 Laser Excitation
Laser excitation is directed via single mode fiber to an optical table below the supercon-
ducting magnet. An overview of the laser excitation system is provided in Fig. 3.1 (4).
Laser excitation is provided by a 100 mw diode-pumped 532-nm laser (Crystalaser). The
laser excitation is focused with a lens into an acousto-optic modulator (AOM; Isomet-
1250C) which is driven by a radiofrequency signal to generate a controllable diffraction
pattern. The radiofrequency signals were generated by a signal generator (Rohde &
Schwarz) before being passed through a high gain-low noise amplifier (Minicircuits)
and into the AOM. Typical diffraction efficiency, defined as the ratio between the input
laser power and output laser power, was1:7 for the zeroth and first order diffraction
spots, respectively. The isolation (ratio between the “on” and “off” output powers for
each diffraction spot), is significantly improved by utilizing the first order diffraction
spot: the zeroth order spot offers9 dB of isolation while the first order offers nearly
30 dB. Therefore, the first order diffraction spot is selected with a pinhole and col-
limated using a second lens. The first order beam then passes through a low pass and
narrow band filter (Omega Optics, Thorlabs) (532 nm3 nm) to remove the residual
pump laser light (1064 nm). After filtering, the laser is then directed into a single mode
optical fiber (Thorlabs) using a high precision manual stage (Thorlabs) and microscope
objective (Olympus; 10X). Due to the small mode diameter of the single mode fiber,
72
careful positioning and a good match between the numerical aperture of the optical fiber
and the microscope objective is important for high coupling efficiency. In the present
setup, a coupling efficiency of50% was achievable, resulting in30 mW of output
laser power after the single mode fiber.
3.3.2 Optical Setup and Sample Positioning
As seen in Fig. 3.1 (4), the output of the laser-coupled single mode fiber is directed to
a fiber coupling stage (Thorlabs) where it is then coupled into free-space. The laser is
directed through a series of mirrors to a dichroic mirror (Omega Optics), fast-steering
mirror (Newport), a system of tube lenses (Thorlabs), and then a microscope objective
(Zeiss). The microscope objective focuses laser excitation to the surface of the diamond.
The setup is configured for operation with both oil and dry objectives. The diamond is
mounted upon a coverslip and z- positioner (Attocube) that controls the relative sample
position between the waveguide and microscope objective and is seen in Fig. 3.1 (5).
The translational positioning is controlled using the fast steering mirror and the system
of tube lenses. By changing the input angle of the laser excitation into the fast steering
mirror, the relative angle of the laser excitation into the microscope objective can be
changed, thus changing the output angle after the objective and allowing for scanning
across the surface of the diamond sample. As the laser excitation induces fluorescence,
spatial imaging can be performed by changing the angle of the laser excitation. Fluores-
cence collected by the microscope objective is directed back through the tube lens and
separated from the laser excitation using the dichroic mirror and an additional longpass
filter (Omega Optics). The fluorescence is then directed either into a photodiode (Thor-
labs) or coupled into a single mode fiber using a microscope objective for direction to a
fluorescence detection/autocorrelation setup.
73
3.3.3 Fluorescence Detection and Autocorrelation
The fluorescence detection setup is shown in Fig. 3.1 (4) and will be explained in more
detail within this section. Fluorescence detection and integration is performed in two
ways depending on the intensity of the fluorescence.
Single photon counting
For single NVs and small NV ensembles, low amounts of fluorescence are typically
observed (fW;<40 kCts/s). In these cases, the fluorescence is collected from free space
using a fiber coupling setup and then directed to a single photon counting setup using a
single mode optical fiber (Thorlabs). The experimental setup is shown in Fig. 3.2(a). For
pulsed ODMR measurements, all available fluorescence is directed into a single photon
counter (Excelitas). The photon counter produces an output TTL pulse for each detected
photon. These TTL pulses are then counted using a fast digital to analog converter
(National Instruments), as seen in figure. During pulsed ODMR measurements, the
digital analog converter counts the photons produced during each readout period and
integrates these each time the experiment is performed ( 10
4
10
6
per data point).
This averaging is necessary to achieve sufficient photon statistics and overcome shot-
noise. The typical readout window was chosen to maximize fluorescence contrast, with
a length of 300 ns, as discussed in Sec. 2.4.2.
Autocorrelation
For single-NV measurements it is necessary to confirm that the fluorescence emitted is
from a single photon source. A Hanbury-Brown-Twiss Interferometer was implemented
to perform autocorrelation (also known as anti-bunching) measurements and verify the
presence of single-photon statistics.
105
The autocorrelation setup is shown in Fig. 3.2(b)
74
(a)
(b)
APD2
APD1
TCSPC
FL
Single
mode fiber
Multi
mode fiber
Beamsplitter
Lens
t
D
~5m Delay
Line
APD1
DAQ
FL
Readout Window
Figure 3.2: Overview of fluorescence detection system for single photon count-
ing. (a) Detected fluorescence is coupled to an optical fiber that is directed to an
APD. The APD produces TTL pulses when photons are detected. TTL pulses
are integrated using a fast DAQ during the readout window. (b) For autocor-
relation measurements, the detected fluorescence is coupled to a single mode
fiber that is then directed to an autocorrelation setup. Within the autocorrelation
setup, the single mode fiber is recoupled into free space before being directed
through a lens, 50:50 beamsplitter, and into multi-mode optical fibers. The out-
put of each optical fiber is directed to a separate APD, with an extra 5m of cable
placed after APD2. The BNC delay output of each APD is coupled to a time
correlated single photon counting (TCSPC) module. The extra cable delays the
arrival of TTL pulses from APD2 by
D
. Within (b), the TTL pulse driver and
DAQ are omitted for clarity.
and was realized by collimating the output of the single-mode fiber using a microscope
objective (Newport). The collimated fluorescence was then focused through a 50:50
beamsplitter (Thorlabs) using an anti-reflection (850-1150 nm) coated lens into two
separate multi-mode fiber optical cables. Each fiber optical cable is then directed into a
separate single photon counter (APD1 and APD2). The output of APD1 is directed into
a TTL pulse driver (Spincore) that duplicates the TTL output into two channels; one
75
of these is directed to the fast DAQ for continued photon measurement and the other
is directed to the input of a a time correlated single photon counting device (TCSPC;
PicoQuant). The output of APD2 is then directed through a long delay line (5m) into
the other channel of the TCSPC for measurement of the autocorrelation function (g
2
()).
Photon collection is continued until sufficient photons are accumulated. The long delay
line shifts the arrival time of the TTL pulses so that if two photons pass through the
beamsplitter and are counted at the same time by APD1 and APD2, the voltage pulse
from APD2 will arrive after a delay period
D
. The length of
D
depends on the speed
of light and the length of the delay cable. The fluorescence intensity is not attenuated
before either beam splitter in order to maximize the available fluorescence counts.
Measurement of High Fluorescence Intensity
For samples with a high concentration of NV centers (ie, ppm), the fluorescence inten-
sity is in the nW-W range and will saturate the single photon counting module. There-
fore, a rapid response avalanche photo diode (RAPD; Thorlabs) is used. The RAPD has
a higher dynamic range than the Excelitas single photon counter and outputs a current
proportional to the input light intensity. The output current must be integrated. As shown
in Fig. 3.3, this is done in one of two ways: (a) using a fast oscilloscope (Tektronix) to
average the fluorescence trace over multiple measurements. The averaged waveform is
then transferred to a computer where the relevant portions (Sig. and Ref.) are numer-
ically integrated. (b) using an analog boxcar integrator (Stanford Research Systems)
operated in toggling mode. When triggered, the boxcar integrator integrates the signal
within a gated window. By operating in toggling mode, the boxcar integrator alternates
polarity of the integrated signal. Therefore, the boxcar integrator needs to be triggered
twice, once to integrate the fluorescence corresponding to a signal of interest and once
76
(a)
(b)
FL
Boxcar
Integrator
DAQ
Sig.
+ -
Ref.
FL
Fast Oscilloscope
Sig.
Ref.
Figure 3.3: Overview of fluorescence detection system using the Thorlabs rapid
response avalanche photo detector (RAPD). Fluorescence was directed to the
RAPD and then converted to an electrical signal. (a) Using a fast oscilloscope,
the averaged output of the RAPD was recorded. This output was then trans-
ferred to a computer where numerical integration of the signal (Sig.) and ref-
erence (Ref.) portions of the waveform was performed. (b) Using a boxcar
integrator operated in toggling mode, the boxcar was triggered twice: once dur-
ing the signal (Sig.) portion of the waveform and once during the reference
(Ref.) portion of the waveform. The boxcar integrator recorded the difference
between these waveforms as a constant voltage. This was measured using a
digital to analog convertor (DAQ).
to integrate the fluorescence from a reference signal. By operating in this mode, the
analog output of the boxcar integrator is proportional to the fluorescence contrast. The
output of the boxcar integrator, a constant voltage proportional to the integrated signal,
is then converted to a digital value using a digital to analog converter. Typical integra-
tion windows were 15 s for the RAPD, owing to the responsivity rate ( MHz) and
the larger detection volume. Usage of the boxcar integrator can be advantageous, as
it can be operated with very high repetition rates, which can be difficult to work with
using digital scopes due to the limited available memory and sampling rates. However,
boxcar integrators can also be sensitive to small changes in the power supply input and
fluctuations in the room temperature, resulting in baseline-drift over long measurement
times.
77
3.3.4 Electron Nuclear Double Resonance (ENDOR) setup
The capabilities of the HFODMR/HFESR system were recently extended by implement-
ing an Electron Nuclear Double Resonance (ENDOR) setup. This setup is shown in Fig.
3.1(6) and is based on a MW source (SG386) that is capable of broadband MW genera-
tion (DC-8 GHz). This source is coupled to a pin switch (Minicircuits) for high on-off
isolation that is then coupled into a high power amplifier (Minicircuits). The amplifier
is selected based upon the desired frequency range and required output gain. Currently,
two amplifiers are used: one in the range of 10-500 MHz and another in the range of
2-4 GHz. The output of the amplifier is directed through SMA cables positioned within
the waveguide down to the sample stage. Once at the sample stage, the SMA cable is
coupled to a high gauge (44 AWG) insulated copper wire placed across the surface of
the diamond near the focal spot of the microscope objective. The output of the wire is
connected to a second SMA cable that is routed through the waveguide to a high-power
attenuator and terminated by a 50 ohm resistor.
The addition of an additional MW source to the existing HFODMR/HFESR spec-
trometer significantly extends the capabilities. With a suitable amplifier, the MW source
enables NV ESR measurements from 0 to 0.4 T in addition to those already enabled up
to 8.6 T. This allows for more advanced measurements that require simultaneous ma-
nipulation of electron and nuclear spins, such as dynamic nuclear polarization (DNP)
and triple hyperfine correlation spectroscopy, possible.
32
3.4 Low-field ODMR Setup
For the single-NV HF ODMR experiment, a conventional confocal microscope setup,
routinely used for NV ODMR experiments, is employed. Due to the modular nature of
78
Laser
Dichroic
Mirror
Filters
B
0
Sample
x,y,z
Piezo
MW2
MW
Single
Mode Fiber
FL Detection
Pin
AOM
APD1
APD2
Figure 3.4: Overview of low field ODMR setup. The system is modular and
based on the laser excitation and fluorescence detection as shown in Fig. 3.1.
The output of the laser excitation is directed through a dichroic mirror to a
fixed microscope objective. The sample is mounted on an adjustable X,Y ,Z
piezo stage. Imaging is performed by moving the sample position relative to the
microscope objective. Fluorescence (FL) is collected using APDs and directed
into an autocorrelation setup. Two microwave sources are employed for double
resonance measurements. These are directed through a pin switch and a power
combiner before passing through a high power amplifier and directed to a 20
m gold wire placed on the sample surface. The microwave circuit is terminated
using a high power attenuator and a 50 ohm resistor. A magnetic field is applied
using a fixed permanent magnet opposite the microscope objective.
the optical setup, the experiment uses similar components to the high field ODMR setup
with a few differences as shown in Fig. 3.4. These have been described in several prior
publications.
107, 109
79
3.4.1 Optical Setup and Sample Positioning
As shown in Fig. 3.4, the laser-coupled single mode fiber is directed to a fiber coupling
stage where it is coupled into free-space. The laser passes through several mirrors for
alignment before passing to a dichroic mirror and into a microscope objective. The
sample is mounted on a sample stage that is placed within a clamp mounted on a sub-
micrometer XYZ positioning stage with manual and piezo drives (Thorlabs). The focal
point of the laser directed through the microscope objective is fixed while the position
of the sample can be adjusted in and out of focus for three dimensional control and
imaging of the sample. Fluorescence collected at the sample stage is directed back
through the dichroic mirror, through a longpass filter, and into either a photodiode or
coupled directly into a single mode fiber for detection using an autocorrelation setup.
A static magnetic field can be applied by mounting a permanent magnet opposite the
microscope objective. The strength of the field is proportional to the distance between
the permanent magnet and the NV center (/r
3
).
3.4.2 Microwave Excitation System
The low-field ODMR system has a microwave (MW) system based upon two sources, a
SG386 (Stanford Research Systems) and a SynthNV (Windfreak Technologies, LLC).
The SG386 is controlled by the central computer and operated in I/Q modulation mode
and triggered by separate BNC outputs from the PB-500. The output of the SG386 is
directed through a pin switch (Minicircuits) for improved isolation and then coupled
to a MW combiner (Minicircuits). The synthNV output is directed directly to a pin
switch (Minicircuits) before being coupled into the MW combiner. The sum of the MW
combiner passes into a high-gain amplifier (Minicircuits) operating in the range of either
80
0.8 to 2 GHz or 2 to 4 GHz. The output of the amplifer is directed into an SMA port
on the copper plated sample stage before being directed to a 20m gold wire applied
to the surface of the diamond sample. The gold wire acts as an antenna, directing the
high power MW excitation to the surface of the diamond. Development of a wideband
coplanar waveguide with flat frequency response can improve the MW circuit. The gold
wire is then coupled back to the copper plated sample stage and SMA port where a high
power attenuator is connected and terminated with a 50 ohm-resistor.
3.5 Implementation of Shaped Pulses
Shaped pulses, generated by amplitude and frequency modulation, offer an additional
level of control compared to traditional single-frequency pulses. The spin dynamics for
this system is described in appendix A. Shaped pulses were implemented on both low-
field and high-field ODMR setups. As I/Q modulation is used for the implementation of
shaped pulses, a brief review of the principles will be discussed first before describing
the implementation on both low and high field ODMR measurements.
3.5.1 I/Q modulation: a review
An I/Q modulator uses two channels of input,I andQ, to allow for amplitude and phase
modulation of a carrier wave. I/Q modulation is widely used, as it offers a highly flexible
output where the phase () and amplitude (A) of the output wave can be continuously
modulated depending on the relative inputs ofI andQ. Output of an ideal I/Q mixer
can be written as:
A cos(!
c
t +) =I cos(!
c
t)Q sin(!
c
t); (3.1)
81
whereA is the amplitude of the output wave,!
c
is the angular frequency of the carrier
wave,t is time, is a phase shift,I is the input of the I channel (I = A
I
cos()),Q is
the input of the Q channel (Q = A
Q
sin()), andA
I
(A
Q
) is the amplitude of the I (Q)
channel.
The amplitude, frequency, and phase of the output wave can be controlled by ma-
nipulating the inputI andQ functions. For example, to implement an X and Y pulse
in a pulse sequence the phase of the Y pulse must by 90
o
offset from the X pulse.
This can be easily accomplished using eq.3.1, by setting = 0 during the X pulse and
==2 during the Y pulse. More interestingly, I/Q modulation can be used to control
the amplitude and frequency of a carrier wave. The instantaneous amplitude is given by
A =
q
A
2
I
+A
2
Q
whereA
I
andA
Q
can be varied according to the desired function. The
frequency of the output wave can be controlled by continuously varying the phase such
that,(t) =
R
t
0
!(t)dt, where!(t) is the offset frequency at time pointt. The frequency
can either be a constant offset, where(t) = !t, or swept according to some modula-
tion function. In the case of a linear frequency sweep (ie, chirp pulse), the frequency
is varied from !
a
to !
b
during a time period, t
swp
. The modulation function is then,
!(t) =
!a!
b
tswp
t = kt (k is the sweep rate in units of Hz/s), resulting in a phase offset,
(t) = kt
2
=2. More advanced pulse shapes, such as WURST, hyperbolic secant, and
lorentzian pulses can be implemented with a suitable choice of amplitude and frequency
modulation functions.
117, 118
3.5.2 Shaped Pulses for Low Frequency ODMR
The implementation of shaped pulses is straightforward for the low-frequency ODMR
system. In this case, a two-channel arbitrary waveform generator (AWG; Keysight) is
connected to the input I/Q modulation ports of the SG386 and preloaded with the I/Q
82
(b)
(c)
(a)
MW1
Laser Init RO
Pulse Time [ns]
0 400 800
0
30
20
10
0.5
0.0
-0.5
4
8
52ns p pulse 800ns Chirp
I
Q
MW Frequency [MHz]
2650 2700 2750
W [MHz] w [MHz] Amp. [V]
100
90
80
Contrast [%]
Figure 3.5: Implementation of frequency swept chirp pulses on the low field
ODMR setup. (a) Pulse parameters. The amplitude of the pulse was ramped
up to the maximum Rabi frequency (
) of 9.6 MHz during a 800 ns pulse. The
frequency (!) was swept from 15 to 25 MHz. The output I/Q waveforms are
shown with a maximum amplitude of0:5 V . (b) Pulse sequence used for the
measurement. After intialization, either a chirp pulse or a-pulse was applied.
(c) Experimental implementation of chirp pulses. The carrier frequency was
varied during the measurement. The chirp pulse can be seen with a broader
linewidth and offset relative to the monochromatic-pulse (52 ns).
waveforms required to produce the desired output waveform. At the appropriate time in
the sequence, the AWG is triggered using a separate TTL pulse from the pulse generator
(PB-500). The output pulse is then passed through an open pin switch and to the high
power amplifier. The timing between the AWG output and pin switch is adjusted to
account for a slight delay (170 ns) after the AWG is triggered before the pulse outputs.
An example of a frequency swept pulse implemented using the low-field ODMR
setup is shown in Fig. 3.5. The amplitude of the pulse was ramped up, held at the
maximum amplitude, and then ramped down. The frequency of the pulse was offset and
swept from 15-25 MHz, with a Rabi frequency of 9:6 MHz based on the- pulse length.
These parameters, as well as the output I/Q waveforms are seen in Fig. 3.5(a) and show a
clear frequency oscillation. The pulse sequence is shown in Fig. 3.5(b), where the pulse
was applied between the laser initialization and readout pulses. The experimental data is
83
shown in Fig. 3.5(c). In the experiment, the carrier frequency was incremented to sweep
over the visible resonance with the AWG triggered at each frequency in the sequence.
Implementation of a frequency swept pulse resulted in a wider bandwidth than ODMR
performed with a conventionalpulse. In addition, the chirp pulse produced similar
optical contrast, indicating full inversion of the spin state.
3.5.3 Shaped Pulses for High Frequency ODMR
The frequency and amplitude modulation capability is implemented using a single-
sideband I/Q mixer (Miteq; DC-500 MHz) that aims to modulate microwaves before
the frequency multiplier chain at a frequency of 8-10 GHz output of the MW synthe-
sizer before the frequency multiplication chain. The implementation of shaped pulses
is more complicated for the high-frequency ODMR system due to the presence of the
frequency multiplication chain, which has a non-linear amplitude response, as shown
previously.
119
An AWG is used to inputI andQ signals into the I/Q mixer with a circuit
designed for full dynamic range with 0 to 200 mV peak-to-peak amplitude. A two-
channel AWG is connected to the input I/Q modulation ports and preloaded with the I/Q
waveforms required to produce the desired output waveform. At the appropriate time
in the sequence, the AWG is triggered using a separate TTL pulse from the PB-500.
The I/Q outputs from the AWG then play out, modulating the carrier frequency, and the
pulse is then passed through an open pin switch to the frequency multiplication chain.
The timing between the AWG output and pin switch is adjusted to account for a slight
delay after the AWG is triggered before the pulse outputs.
The pulse shaping characteristics of the HFESR/ODMR system were explored us-
ing the 115 GHz configuration. The output pulse power can be measured using the
84

(a) (b) (c)
Time [ns]
100 200 300 400
Input Power [%]
0 5 10 15 20
Input Power [mV]
0 10 20 30 40
50
100
0
Power [%]
50
100
0
Output Power [%]
Time [ns]
0 500 1000 1500
50
100
0
Power [%]
Input
Output
Input
Output
Figure 3.6: Implementation of amplitude control using the AWG on the HF
ESR setup at 115 GHz. The output voltage was recorded using the HF ESR
receiver system and a power detector.
30
(b) Input of an “ideal” Gaussian pulse,
as shown in blue produces a severely distorted output pulse shown in red. (a)
Variation of input voltage against output voltage. The output powere is seen
to exhibit a nonlinear amplitude response, increasing from 0 to 100% over an
input scale of 5 to 10%, which corresponds to an input voltage range from 10
to 20 mV . (c) Correcting for the amplitude response by predistorting the input
using (b) to account for the nonlinear response function produces a nearly ideal
output pulse. Figures and data courtesy of Cooper Selco
detection system shown in Fig. 3.1, with a power detector (Aglient) placed in the detec-
tion system after the subharmonically pumped mixer. The power detector converts the
downconverted signal from the superheterodyne detection system to an output voltage
representative of the pulse power. The frequency multiplication chain has a non-linear
amplitude response, which severely distorts an input Gaussian pulse as seen in Fig.
3.6(a). The nonlinear amplitude response of the frequency modulation chain is seen in
Fig. 3.6(b), where a slight variation of input power (5%-10%) results in a large change
in output power (0%-100%). However, by measuring the non-linear amplitude response,
the input pulse can be predistorted as seen in Fig. 3.6(c) to produce the expected ampli-
tude response. Remaining imperfections can be corrected using a feedback control loop
to produce the desired pulse shape.
Frequency modulation is also achievable using the high frequency ODMR system
and accounting for the multiplication chain (12 for 115 GHz;24 for 230 GHz).
85
1ms Chirp
10ms Chirp
0.6ms p
MW Frequency [GHz]
114.78 114.79 114.80 114.81
(b)
100
90
80
Contrast [%]
(a)
MW
Laser
Init RO
Figure 3.7: Implementation of frequency swept chirp pulses at 115 GHz on
a single NV center at 4.197 T. (a) Pulse sequence used at 115 GHz. The the
carrier frequency was swept while applying the chirp pulse. The spin state was
initialized before and readout after application of the microwave pulse. The
amplitude of the pulse was fixed at the maximum Rabi frequency of 0.8 MHz,
and the frequency was swept from 1 to 11 MHz (after the x12 multiplication
chain). (b) Experimental data showing the influence of chirp pulses of different
lengths. For comparison, the influence of apulse (600 ns) is shown.
Current implementation of slow frequency shifts (< 1 MHz before multiplication) are
successfully implemented by the IQ mixer/AWG as shown in Fig. 3.7. Experimental
measurement of HFODMR using chirp pulses is performed using the pulse sequence
shown in Fig. 3.7(a). The spin state is first initialized and later readout after applying a
MW pulse. During this experiment, the sweep rate of the pulse is varied by fixing the
sweep range (111MHz) and varying the length of the pulse. As discussed in appendix
A, a slower sweep rate is more efficient at transferring population, provided the length
of the pulse is short relative to T
2
. Experimental results are presented in Fig. 3.7(b).
As expected, a longer pulse (slower sweep rate) results in improved population inver-
sion. Surprisingly, it is also revealed that the overall contrast increases drastically using
frequency shaped pulses. Both chirp pulses are more effective at inverting population
than the rectangularpulse as seen in Fig. 3.7(b). This is due to the improved MW
86
excitation bandwidth relative to the intrinsic linewidth determined by the NV center’s
hyperfine coupling to nitrogen. Due to the low Rabi frequency (0.8 MHz) in the present
case, the usage of hard pulses results in a significant reduction in population inversion
and corresponding signal contrast. In this case, the usage of frequency swept pulses
overcomes this limitation. This is possible provided the spin-relaxation times are suffi-
ciently long. With larger frequency shifts, additional filtering will be required to reduce
the harmonics generated by imperfect frequency cancellation from the I/Q mixer. If
not removed prior to multiplication, these harmonics will be amplified by the frequency
multiplication chain producing spurious frequency spikes and distorting the pulse shape.
3.6 Summary
In this chapter, the fundamentals for a high field ODMR/ESR spectrometer were dis-
cussed. First an overview of the HF ESR spectrometer was provided, as the HF ODMR
spectrometer utilizes this instrumentation as a base. The ODMR spectrometer is built in
a modular fashion so that the laser excitation system and fluorescence detection system
can be used for both low and high field ODMR. A discussion of fluorescence measure-
ments followed with examples of the autocorrelation setup and the procedure for mea-
surement of both high and low fluorescence intensity. The recent addition of an RF/MW
to the system for ENDOR enhances the capabilities of the present system, making low
field (0-0.4T) and high field (> 8.3T ODMR of the same NVs straightforward. The
low-field ODMR system was briefly described, as many components are shared with the
high-field ODMR system. The main difference between the low and high-field ODMR
setups were the sample positioning and MW delivery system. Next the application of
shaped pulses for ODMR was discussed. A brief review of the principles of I/Q modula-
tion was provided before a discussion of the implementation of frequency swept pulses
87
on both low field and high field ODMR setups. Pulse shaping on the high field ODMR
setup is more challenging due to the properties of the nonlinear frequency multiplication
chain (nonlinear amplitude response and sensitivity to undesirable harmonics), but is a
promising route to increase sensitivity and enhance control of the spin dynamics.
88
Chapter 4: Understanding Linewidth
of ESR Spectrum Detected by a Single
NV Center in Diamond
Materials presented in this chapter can also be found in the article titled Understanding
the Linewidth of the ESR Spectrum Detected by a Single NV Center in Diamond by
Benjamin Fortman and Susumu Takahashi in the Journal of Physical Chemistry A 123,
6350-6355 (2019).
4.1 Introduction
ESR spectral analysis, in which the position, intensity, and line-shape of the ESR spec-
trum is carefully analyzed to extract spin parameters including g-values, hyperfine and
spin-spin couplings, zero-field splittings and rotational correlation times of systems, is
widely and routinely used for characterizations and investigations in science and engi-
neering fields. Examples include identification of paramagnetic defect contents in semi-
conductors,
14–16
investigations of structures and conformational dynamics of biological
molecules,
11–13
and characterizations of photochemical reactions.
8–10
89
The nitrogen-vacancy (NV) center is of significant interest in quantum sensing due
to its unique properties, including long-lived quantum coherence of the NV centers’
spin states and high sensitivity to external magnetic fields.
27, 33, 35, 38, 95, 96
Using a sin-
gle NV center, nanoscale ESR detection of several types of spins in solid state and
biological systems has been demonstrated.
41, 42, 45–47, 53, 59, 61, 120
For applications of NV-
detected ESR (denoted NV-ESR) spectral analysis, it is critical to understand the nature
of NV-ESR line-shape and to establish a procedure to obtain a high resolution spec-
trum representing intrinsic properties of the sample. This chapter establishes a method
for obtaining high-resolution NV-ESR spectra and implements lineshape analysis for
extraction of the intrinsic linewidth using NV-ESR.
Within this chapter, the nature of an NV-ESR spectrum of single substitutional ni-
trogen defects in diamond (called P1 centers) is investigated. The NV-ESR spectrum is
obtained using a double electron-electron resonance (DEER) pulse sequence, which uti-
lizes pulses at two distinct microwave (MW) frequencies to coherently control the NV
center and target spins. By studying the spectral line width as a function of the DEER
pulse length, we identify a significant contribution of the DEER excitation bandwidth
to the observed NV-ESR linewidth at short pulse lengths. At long pulse lengths, we ob-
serve that the ESR linewidth is limited by inhomogeneous broadening of the detected P1
ESR frequency (T

2
-limit), representing intrinsic spin dynamics of P1 spins. Moreover,
by employing a long DEER pulse, we observed that the ESR linewidth is as narrow as
0.3 MHz and, with the improvement of the spectral resolution, we clearly resolve a small
splitting (2 MHz) in P1 ESR that originates from the anisotropic hyperfine coupling and
four different orientations of the P1 spins.
90
4.2 Materials and Methods
4.2.1 Diamond sample
A single crystal (2.0 2.0 0.3 mm
3
) of (111)-cut high pressure high temperature
type-Ib diamond (purchased from Sumitomo electric industries) was used in this study.
4.2.2 115 GHz ESR spectroscopy
The 230 GHz/115 GHz ESR system employs a high-power solid-state source consists of
an 8-10 GHz synthesizer, pin switch, microwave amplifiers, and frequency multipliers.
The output power of the source system is 100 mW at 230 GHz and 700 mW at 115 GHz.
The 230/115 GHz excitation is propagated using a quasioptical bridge and a corrugated
waveguide and couples to a sample located at the center of a 12.1 T cryogenic-free
superconducting magnet. ESR signals are isolated from the excitation using induction
mode operation.
116
For ESR detection, we employ a superheterodyne detection system
in which 115 GHz is down-converted into the intermediate frequency (IF) of 3 GHz then
down-converted again to in-phase and quadrature components of dc signals. Details of
the system have been described elsewhere.
30, 115
In the present experiment, the magnetic
field modulation strength was adjusted to maximize the intensity of ESR signals with-
out distorting the lineshape (typical modulation amplitude of 0.02 mT with modulation
frequency of 20 kHz).
4.2.3 ODMR spectroscopy
The ODMR system is based on a homebuilt confocal microscope system. A 100-
mW 532-nm laser (Crystalaser) is passed through an acousto-optic modulator (Isomet
91
1250C) before being directed through a low-pass filter (Omega) and into a single mode
fiber (Thorlabs). The output of the fiber is directed through a dichroic mirror and up
through a microscope objective (Zeiss 100X) to the sample stage. Fluorescence (FL)
is detected by an avalanche photodiode (Excelitas) through a high-pass filter (Omega)
and another single mode fiber. The autocorrelation measurement is performed with
a Hanbury Brown-Twiss interferometer.
105
For ODMR, microwave (MW) excitation
is directed from the sources (Stanford Research Systems SG386 and Rohde-Schwarz
SML03) through a power combiner, and a high gain amplifier to the sample stage. A 20
m gold wire is placed on the surface of the diamond for MW excitation and coherent
control of the NV centers.
4.3 Experimental Results and Discussion
4.3.1 cw-ESR
We first perform 115 GHz continuous wave ESR (cw-ESR) spectroscopy to identify
impurity contents within the diamond sample. Figure 4.1 shows cw-ESR data of the di-
amond sample with application of an external magnetic field along the [111] direction.
We observe five pronounced ESR signals from P1 centers (S = 1=2,I = 1,A
x;y
= 82
MHz andA
z
= 114 MHz).
16
The intensity of 115 GHz wave excitation was reduced
to avoid the saturation of the ESR signal and the intensity of magnetic field modulation
was carefully adjusted to maximize the signal-to-noise ratio of the ESR signal without
distortion. The P1 spectrum consists of five ESR signals due to the four possible ori-
entations of P1 and the anisotropic hyperfine coupling. Namely, the signals at 4.104,
4.108 and 4.113 T correspond to the ESR of P1 centers oriented along the [111] direc-
tion while the signals at 4.105, 4.108 and 4.112 T are from the other three orientations,
92
Intensity [a.u.]
115 GHz
4.104 4.108 4.112
B
0
[T]
4.1122 4.1127
0.15 mT
Figure 4.1: cw-ESR spectrum of P1 centers taken at 115 GHz at room temper-
ature. The inset graph shows the spectrum of them
I
=1 ESR signal. The
ESR linewidth is 0:15 0:02 mT. A field modulation of 0.01 mT at 20 kHz
and a field sweep rate of 0:01 mT/s were used. Reprinted with permission from
Ref.
109
Copyright 2019 American Chemical Society.
[

111], [1

11], and [11

1]. As shown in the inset of Fig. 4.1, the linewidth of the observed
ESR is 0:15 0:02 mT for the signal at 4.113 T.
4.3.2 ODMR: NV Identification and Field Calibration
We next measure P1 ESR using a single NV center in diamond. For NV-ESR measure-
ments, we employ a homebuilt ODMR system as shown in Fig. 4.2(a). Figure 4.2(b)
shows a FL image of the diamond sample as well as an isolated FL peak for the present
experiment (denoted as NV1). An autocorrelation measurement, as shown in Fig. 4.2(c),
shows a dip in the signal att = 0 that proves the FL emission is from a single quantum
emitter. cw-ODMR measurements, where the FL intensity is monitored while sweep-
ing the microwave frequency, are then performed on the isolated FL spot, as shown in
Fig. 4.2(d). The observed ODMR signals correspond to the m
S
=1 and m
S
= 0,
93
Laser
Dichroic
Mirror
Filters
B
0
Sample
x,y,z
Piezo
MW2
MW
Single
Mode Fiber
FL Detection
Pin
(a)
[µm]
[µm]
0 4
0
4
Cts/ms
NV 1
1.0
0.9
0.8
1.95 3.80
Frequency [GHz]
Exp.
Fit
Contrast [a.u.]
t [ns]
g
2
(t)
0 200
0
2
(b)
(c)
(d)
|0
>
↔|-1
>
|0
>
↔|+1
>
5 18 30 40
AOM
APD1
APD2
Figure 4.2: ODMR experiment to identify a NV center. (a) Diagram of the
experimental setup. (b) Spatial FL image with NV1 indicated in the solid red
circle. (c) Autocorrelation measurement of NV1. The dotted red line drawn at
0.5 indicates the threshold for single quantum emitters. (d) cw-ODMR signals
from the lower (j0i$j1i) and upper (j0i$j+1i) transitions. The sig-
nal is normalized by the FL intensity without MW excitation. Reprinted with
permission from Ref.
109
Copyright 2019 American Chemical Society.
and them
S
= +1 andm
S
= 0 transitions of the NV center (S = 1, g = 2:0028 and
D = 2:87 GHz). Therefore, we determine this spot to be a single NV center. From the
observed ODMR frequencies we determined the applied magnetic field to be 33.4 mT
with a polar angle of 6:1 0:1 degrees from the [111] axis.
94
1.0
0.0
0.5
t

[ns]
0 50 100
Init. R.o. Laser
MW
t
Exp.
Fit
(a)
T
2
≈ 40 μs
MW
π
π
2
π
2
t t
Exp.
Sim.
P|m
s
=0> [a.u.]
1.0
0.5
2t [μs]
0 30 60
(b)
P|m
s
=0> [a.u.]
Figure 4.3: Pulsed ODMR data collected from NV1 at 33.4 mT. (a) Measure-
ment of Rabi oscillations. The Rabi oscillations show a pulse time of 40
ns. The pulse sequence is shown in the inset. The data is normalized to re-
flect the probability of the NV center being in them
S
= 0 state (Pjm
S
= 0i).
47
For all pulsed ODMR presented, a 5 s laser pulse is used to initialize the
spin state while a 300 ns laser pulse is used for readout. Microwave pulses
(shown as blue rectangles) are applied to drive thej0i $ j1i transition.
Each pulse sequence is repeated 10
4
-10
6
times for an unweighted averaging
of each data point. (b) Spin echo measurement. The spin echo data shows a
spin decoherence time (T
2
) of 40s for NV1. Data is shown in agreement with
I(t) = exp[(t=T
2
)
3
].
47, 74, 113
Reprinted with permission from Ref.
109
Copy-
right 2019 American Chemical Society.
4.3.3 NV-ESR
Next, we perform the NV-ESR experiment. We first conduct Rabi oscillation and spin
echo (SE) measurements to determine pulse lengths and the spin coherence time (T
2
) for
NV-ESR. As shown in the inset of Fig. 4.3(a), the Rabi measurement is performed by
first initializing the spin state intoj0i with a long laser pulse before applying a variable
length of MW pulses. The final spin state is then read out using a short laser pulse to
induce FL from the NV center (see the inset of Fig. 4.3(a)).T
2
is measured using a Hahn
spin echo sequence (see the inset of Fig. 4.3(b)). We determined theT
2
of NV1 to be 40
s.
95
After the Rabi and SE experiments, we perform NV-ESR using a DEER technique,
as shown in Fig. 4.4(a). NV-ESR is performed by measuring the change in a coherent
state of the NV center as a function of the frequency of the DEER pulse. The coher-
ent state change is induced by a shift of the magnetic dipole field of target spins due
to the population inversion of target spins induced by the DEER pulse. For this mea-
surement, a of 5.7s was chosen to reduce decoherence of the NV center. As shown
in Fig. 4.4(a), the resulting spectrum exhibits five peaks, in agreement with P1 ESR. In
the measurement, the MW intensity is adjusted to ensure the DEER pulse length per-
forms a rotation of the P1 center spins. Figure 4.4(b) shows Rabi oscillations of P1
centers measured by NV-ESR with differentT values (see the sequence in Fig. 4.4(b)).
To explain the results, we consider the following NV-ESR model which describes the
spin dynamics of an ensemble of two-level systems.
89
Using this model, the intensity of
NV-ESR is given by,
I
NVESR
= exp

2
0

2
B
g
NV
g
B
T
9
p
3~
n

sin
2

2

L

(4.1)
where
0
is the vacuum permeability,
B
is the Bohr magneton,g
NV
is theg-value of
the NV center,g
B
is the g value of target spins,T is the time for phase to accumulate
after application of the DEER pulse, ~ is the reduced Planck constant, and n is the
concentration of target spins. Thehsin
2
2
i
L
term represents the effective population
inversion of the DEER pulse given as,
90

sin
2

2

L
=
Z
+1
1

2
(!)
2
+

2
sin
2

p
(!)
2
+

2
t
p
2

L(; !)d
where
is the Rabi frequency of the target spins, ! is the frequency of MW2, t
p
is
the applied pulse length, andL(; !) is an intrinsic ESR line of P1 spins where !
96
(a)
P|m
s
=0> [a.u.]
1.0
0.5
Frequency [MHz]
800 900 1000 1100
Sim.
(b)
t
p
[ns]
P|m
s
=0> [a.u.]
0 100 200 300
0.4
0.6
0.8
0.4
0.6
0.8
Sig. SE Sim.
π
π/2
MW
t t
p\2
MW2
T
t
p
Sig.
Ref.
p\2 p
MW
t t
p\2 p\2 p
MW2
p
Figure 4.4: NV-ESR of NV1 with = 5:7s. (a) NV-ESR spectrum obtained
for NV1 with a 56-ns MW2-pulse. SE intensity at the same is shown as a
reference. The DEER pulse length was chosen to maximize signal for the axial
P1 orientation. ESR frequencies calculated from the P1 spin Hamiltonian are
shown in the stick spectrum. (b) Rabi oscillations of P1 centers measured by
NV-ESR. The NV-ESR signal is plotted againstt
p
(solid black). The distance
to the end of the sequence is indicated byT and was 5.7 (2.0)s for the upper
(lower) data. SE data (Solid blue) is shown as a reference. The simulation using
Eq. 4.1 is shown in red. The pulse sequence used for the NV-ESR Rabi exper-
iment is shown to the right. Reprinted with permission from Ref.
109
Copyright
2019 American Chemical Society.
represents the linewidth. Therefore, Eq. 4.2 includes the effects of the MW excitation
and the ESR line on the NV-ESR signal. The P1 Rabi data were simulated with Eq. 4.1
by fixing T while allowing n to vary. As shown in Fig. 4.4(b), the simulations were
found to be in good agreement with the experiments. We found that NV-ESR intensity
depends on the value ofT . As shown in Fig. 4.4(b), NV-ESR withT = 2:0s exhibits
a high intensity contrast between and=2 pulses.
We seek to determine the origin of the observed linewidth in Fig. 4.4(a) by extract-
ing the contribution from the MW excitation bandwidth. The contribution is studied by
analyzing the NV-ESR linewidth as a function of the DEER pulse length (t
p
). In the
experiment, the MW power is adjusted to maintain a-pulse for all pulse lengths. As
shown in Fig. 4.5(a), the spectrum narrows and the shape of the spectrum changes as
the pulse length of the DEER pulse increases. In order to characterize the linewidth,
97
we fit each spectrum to a sum of two Lorentzians with resonance positions for all ori-
entations of P1 centers (i.e.,m
I
= 0 transitions for the [111] orientation and the other
three orientations at 943 and 945 MHz, respectively). The extracted full widths at half
maximum (FWHM) are summarized in Fig. 4.5(b) where the linewidths strongly de-
pend ont
p
below at
p
of 0.4s. To explain the dependence of the pulse length on the
linewidth, we analyze contributions to the linewidth by fitting FWHM calculated using
Eq. 4.1 with the experimental FWHM where ! (Lorentzian linewidth) is a fit parame-
ter. As shown in Fig. 4.5(b), we found excellent agreement with the observed linewidths
with ! = 1:6 MHz with 90% confidence bounds of (1:4; 1:8) MHz. In the figure, we
also show partial contributions of the MW excitation and ! where the contribution
of the MW excitation bandwidth is obtained by numerical calculation of FWHM using
L(; !) = () in Eq. 4.1. This analysis verified that the MW excitation bandwidth
is a major contribution of the NV-ESR linewidth when t
p
is shorter than0.4s. As
shown in the inset of Fig. 4.5(b), the observed NV-ESR spectrum is well-explained by
the simulation using Eq. 4.1 with ! = 1:6 MHz. Moreover, the obtained high spectral
resolution NV-ESR spectrum allows clear identification of them
I
= 0 P1 ESR signals
for the [111] orientation and the non-[111] orientations which are separated by only 2
MHz. This small splitting was not well resolved in a previous experiment performed
at a similar magnetic field.
47
The splitting is due to the contribution of the anisotropic
hyperfine interaction comparable to the Zeeman energy at the low magnetic field which
sets the ESR frequency from the non-[111] orientations to 945 MHz at 33.4 mT while
the [111] orientation remains at 943 MHz. We next confirm the nature of the intrinsic
linewidth ! = 1:6 MHz by comparing with the spin dephasing time (T

2
). T

2
re-
laxation time originates from an inhomogeneous distribution of ESR frequencies and
represents the linewidth in many conventional ESR experiments. Given the and=2
98
pulse lengths with T = 2 s as shown in Fig. 4.4(b), we perform a DEER Ramsey
experiment to measure T

2
. To confirm the observed signal, a concurrent experiment
varying the position of the-pulse (Ref.) was performed with the sequence shown in
Fig. 4.5(c). We observed exponential behavior from the Ramsey measurement, as shown
in Fig. 4.5(c). The observed signal was then analyzed by fitting the data with Eq. 4.1
wherehsin
2
2
i
L
= exp(t=T

2
). We observed aT

2
= 118 34 ns from the analysis for
NV1. The value ofT

2
corresponds to a FWHM of 2:7 1:0 MHz, a value in reasonable
agreement with the ! extracted from frequency measurements. Furthermore, from the
analysis of the NV-ESR intensity at 943 MHz, the detected magnetic dipole field (B
Dip
)
is420 nT.
47
This strength of the magnetic field corresponds to an axially aligned single
spin at a distance of16 nm.
Moreover, we investigate NV-ESR spectroscopy with other single NV centers (NV2-
5). As summarized in Fig. 4.6, NV2-5 also exhibit a strong pulse length dependence
similar to what we observed in NV1. For NV2, ! of 0.9 (0.9, 1.0) MHz was ob-
served in agreement with the measuredT

2
of 240 125 ns, as shown in Fig. 4.6(a). The
linewidth observed for NV3 (! = 1:0 MHz) was similar to that of NV2 (Fig. 4.6(b)).
The linewidth observed for NV4 and NV5 (! = 0:3 MHz) was similar in magnitude,
but significantly smaller than NV1-3 (Fig. 4.6(c) and (d)). For NV4, we measure the
m
I
= 0 ESR transition and resolve a very narrow linewidth that allowed for clear reso-
lution of the two peaks originated from the [111] and the other orientations of P1 spins.
In NV5, the NV-ESR linewidth with a pulse length of 2s was only 0.3 (0.2, 0.4) MHz.
Overall, the results from NV1-5 provide clear examples of nanoscale ESR investigation
of the inhomogeneity in ESR signals. This is shown by variation in ESR linewidths as
measured by different NVs located within the same diamond crystal. Moreover, the ob-
served linewidths of NV-ESR are much smaller than that in HF ESR (see Fig. 4.1). This
99
(c)
t [ns]
0 200 400 600
Sig.
Ref.
Fit
P|m
s
=0> [a.u.]
0.4
0.6
0.8
T
2
*
=118 ± 34 ns
(b)
Exp.
Exc. BW
Int. LW
BW+LW
Linewidth [MHz]
5
10
15
20
25
Pulse Length [μs]
0.0 0.5 1.0
Freq. [MHz]
940 950
P|m
s
=0> [a.u.]
1.0
0.8
Sig.
Ref.
Sim.
1 μs
0.8 μs
0.4 μs
0.1 μs
0.05 μs
Intensity [a.u.]
(a)
Frequency [MHz]
915 930 945 960 975
MW
t t
p\2
MW2
t
T
p\2 p
p\2 p\2
p
Sig.
Ref.
Figure 4.5: Dependence of DEER pulse length on NV-ESR linewidth. (a) NV-
ESR spectra taken using various-pulse lengths;-pulse lengths are indicated
in the legend. (b) The NV-ESR linewidth as a function of the pulse length. The
red solid line is the result of a nonlinear least squares regression using Eq. 4.1
and ! = 1:6 (90 % confidence bounds of (1:4; 1:8)) MHz. Fitting was done
with weights 1=
2
. The blue and green dashed lines show partial contributions.
The blue dashed line is the MW excitation bandwidth, while the green dashed
line shows !. The inset graph shows the spectrum taken using a-pulse of
1 s. The spectrum was normalized for the probability of the NVj0i state.
The simulated spectrum based on a linewidth of 1:6 MHz is shown in red. (c)
Ramsey measurement using NV-ESR to measureT

2
. Pulses were applied 2s
before the end of the sequence. The pulse sequence used for NV-ESR Ramsey
is shown to the right. The spacing between the pulses (t) was varied in the
NV-ESR Ramsey measurement (Sig.). A reference experiment was performed
concurrently with the position of a single-pulse being varied byt. NV center
(P1 center) pulse times were 40 (64) and 24 (32) ns, for the and=2 pulses
respectively. The fit is shown in red. Reprinted with permission from Ref.
109
Copyright 2019 American Chemical Society.
is most likely because of the significant difference in the sample size between the two
experiments. Conventional ESR obtains signal from all spins within the millimeter scale
sample while the sample volume in NV-ESR is confined within several to a-few-tens of
nanometers from the NV center. This significantly smaller size of the sample volume
100
limits the number of detected P1 spins. In the present case, P1 ESR in the nanometer-
scaled sample volume has significantly smaller inhomogeneity compared with conven-
tional ESR. As shown by the previous conventional ESR investigations, there are two
major contributions to the P1 ESR linewidth; hyperfine couplings to
13
C nuclear spin
baths and magnetic dipole couplings to P1 spin baths.
16, 89
When the P1 concentration
is low, the ESR linewidth as narrow as0.3 MHz is broadened by the hyperfine cou-
plings to the
13
C nuclear spin baths. On the other hand, when the P1 concentration is
high, the linewidth is broader due to the coupling of the P1 spin baths and depends on
the P1 concentration. Therefore the present result strongly suggests that the variation of
observed P1 linewidths is due to inhomogeneity of P1 densities or spatial configurations
of P1 spin baths within the detected nanoscale volume. Furthermore, for NV 4 and 5,
the contribution from the P1 spin bath is negligible on the linewidths (0.3 MHz) while
the hyperfine coupling to
13
C spin baths is the major contribution.
4.4 Conclusion
Within this chapter we investigated the nature of the NV-ESR linewidth by studying P1
ESR. We found that the spectral resolution depends strongly on the length of the DEER
pulse. This was particularly evident when pulse lengths are shorter than 0.4s. Upon
using long pulse lengths, the minimum resolved linewidth was found to be limited by
inhomogeneous broadening of P1 ESR (T

2
-limit). This linewidth was found to vary be-
tween NV centers, indicating spatial inhomogeneity of local magnetic fields surrounding
each NV center. Since NV-ESR is useful for investigation of many spin systems with
single spin sensitivity, the ability to perform high-resolution NV-ESR is critical. The
present work provides important context into the improvement of spectral resolution. In
particular, we demonstrated resolution of a small ESR splitting (2 MHz) by improving
101
5
10
15
Width [MHz]
Pulse Length [μs]
0.0 0.5 1.0 1.5
NV 4
Width [MHz]
5
10
15
Pulse Length [μs]
0.0 0.5 1.0
NV 5
NV 3
Pulse Length [μs]
0.0 0.5 1.0 1.5 2.0
5
10
Width [MHz]
(a) (b)
(c)
NV 2
Width [MHz]
10
20
30
Pulse Length [μs]
0.0 0.5 1.0
Exp.
Exc. BW
Int. LW
BW+LW
Freq. [MHz]
944 948
P|m
s
=0> [a.u.]
0.7
0.6
Sig.
Ref.
Sim.
0.8
P|m
s
=0> [a.u.]
0.7
0.6
Freq. [MHz]
948 952
P|m
s
=0> [a.u.]
0.9
0.8
Freq. [MHz]
948 952 944
Freq. [MHz]
936 940 944
P|m
s
=0> [a.u.]
1.0
0.8
(d)
Figure 4.6: Dependence of DEER pulse length on NV-ESR linewidth for NVs
2-5.!
0
was set to the resonance position(s) for linewidth extraction as discussed
in the main text. (a) Result of NV2. = 19:5 s and B
0
= 37:7 mT. The
red solid line shows the fit result and the blue and green dashed lines show
partial contributions from the MW excitation and !, respectively. From the
fit, ! = 0:9(0:9; 1:0) MHz was obtained. The inset graph shows the spectrum
taken using a-pulse of 1s (green) with the fitted spectrum shown in red. (b)
Result of NV3. = 15s andB
0
= 37:8 mT. From the fit, ! = 1:0(0:5; 1:5)
MHz was obtained. The inset graph shows the spectrum taken using a-pulse
of 1s. (c) Result of NV4. = 8:81s andB
0
= 32:7 mT. Linewidth data
was extracted using a sum of two equal-width Lorentzians with!
1
and!
2
set
to be 941 and 943 MHz respectively. From the fit, ! = 0:3(0:0; 0:6) MHz
was obtained. The inset graph shows the spectrum taken using a-pulse of 1.6
s with the simulated spectrum shown in red. (d) Result of NV5. = 15s
andB
0
= 37:8 mT. From the fit, ! = 0:3(0:2; 0:4) MHz was extracted. The
inset graph shows the spectrum taken using a-pulse of 2s. Reprinted with
permission from Ref.
109
Copyright 2019 American Chemical Society.
the spectral resolution and identified dominant coupling between P1 and surrounding
electron and nuclear spins. Furthermore, the present technique will be applicable for
various NV-ESR investigations including identification of multiple types of spins and
study of spin dynamics.
102
Chapter 5: Demonstration of
NV-detected ESR spectroscopy at 115
GHz and 4.2 Tesla
Materials presented in this chapter can also be found in the article titled Demonstra-
tion of NV-detected ESR spectroscopy at 115 GHz and 4.2 Tesla by Benjamin Fortman,
Junior Pena, Karoly Holczer, and Susumu Takahashi in Applied Physics Letters 116,
174004 (2020)
5.1 Introduction
NV-ESR offers the capability to detect single or small numbers of electron
spins
41–43, 45–47
and to investigate biological molecules at the single molecule level.
53, 54
Development of an ESR technique with single spin sensitivity potentially eliminates
ensemble averaging in heterogeneous and complex systems, and has great promise to
directly probe fundamental interactions and biochemical function. The g-factor is ex-
tremely useful for the identification of spin species. However, a featureless “g = 2”
signal is often observed with NV-ESR, causing spectral overlap with target ESR signals,
which may prevent spin identification.
54, 59–61
103
Similar to NMR spectroscopy, pulsed ESR spectroscopy at higher frequencies and
magnetic fields becomes more powerful for finer spectral resolution, enabling clear spec-
tral separation of systems with similar g values.
23, 24
This is advantageous in the inves-
tigation of complex and heterogeneous spin systems.
11, 25
A high frequency of Larmor
precession is also less sensitive to motional narrowing, enabling the ESR investigation of
structures for molecules in motion.
12, 26
In addition, a high Larmor frequency provides
greater spin polarization: improving sensitivity
23, 24
and providing control of spin dy-
namics.
27, 121
On the other hand, pulsed HF ESR often has the disadvantage of long pulse
times due to low HF microwave power. The low microwave power limits the excitation
bandwidth, and consequently the sensitivity of pulse ESR measurements. NV-detected
ESR (indicated as NV-ESR) will overcome this limitation and improve the sensitivity of
HF ESR drastically. However, only a few investigations of NV centers have been per-
formed at high magnetic fields,
62–64
and NV-ESR has not been demonstrated at a high
magnetic field.
In this work, we demonstrate NV-ESR at a Larmor frequency of 115 GHz, corre-
sponding to a magnetic field of4.2 Tesla. The HF NV-ESR experiment is performed
with both an ensemble and a single NV system. Within the ensemble experiment, we
start the characterization of NV centers using optically detected magnetic resonance
(ODMR), a measurement of Rabi oscillations, and a spin echo measurement to deter-
mine a spin decoherence time (T
2
). Then, we utilize a double electron-electron reso-
nance (DEER) sequence to perform NV-ESR spectroscopy of single-substitutional ni-
trogen (P1) centers in diamond. We find that the observed NV-ESR spectrum is in
excellent agreement with the spectrum of P1 centers. In the single NV-ESR experiment,
we start the identification and the characterization of a single NV center. For high fi-
delity coherent control, we apply chirp pulses which improve population inversion and
104
optical contrast. We then implement a DEER sequence with chirp pulses. The observed
NV-ESR signal is in agreement with P1 centers. This work provides a clear demon-
stration of HF NV-detected ESR, and provides a foundation for the study of external
spins with high spectral resolution NV-ESR. Furthermore, the presented experimental
strategies are applicable to NV-ESR at higher magnetic fields.
5.2 Experiment and discussion
Figure 5.1 shows an overview of a home-built HF ODMR system used in the experi-
ment. The HF ODMR system consists of a HF microwave source (Virginia Diode, Inc.),
quasioptics, a 12.1 Tesla cryogenic-free superconducting magnet (Cryogenics), and a
confocal microscope system for ODMR. The HF ODMR system is built upon the ex-
isting HF ESR spectrometer that was described previously.
30, 115
Therefore the system
enables in-situ experiments of both ESR and ODMR. As seen in Fig. 1, the HF source
contains two microwave synthesizers (MW1 and MW2), a power combiner, and a fre-
quency multiplier chain. An IQ mixer (Miteq) controlled by an arbitrary wave generator
(AWG; Keysight) has recently been implemented in MW1 for pulse shaping of high-
frequency microwaves. Two synthesizers are employed for DEER experiments. The
frequency range of the microwave source is 107-120 GHz and 215-240 GHz. In this
experiment, we use a frequency range of 107-120 GHz where the output power of the
HF microwave is 480 mW at 115 GHz. HF microwaves are propagated to a sample us-
ing a home-built quasioptical bridge and a corrugated waveguide (Thomas-Keating). As
demonstrated previously, quasioptics are suitable for a high-frequency ESR spectrom-
eter because of their capacity for low-loss and broadband propagation.
29, 116
A sample
is mounted at the end of the corrugated waveguide and positioned at the field center
of a room temperature bore within a superconducting magnet system. No microwave
105
B
0
Power
combiner
Corrugated
waveguide
2. Quasioptics
1. HF source
3. Detection
system
Steering
Mirror
Lenses
Dichroic
Mirror
Photo
diode
Laser filters
4. ODMR system
Oscillo-
scope
AOM
532 nm
laser
APD2
APD1
Freq.
Multiplier
MW2
Rectangular
pulses
MW1
Chirped
pulses
Diamond NV
5. Sample Stage
Diamond Diamond Diamond Diamond Diamond
MW
Figure 5.1: Overview of HF ODMR system. The HF ODMR system consists
of five components. (1) A HF microwave component, (2) a quasioptical MW
propagation system, (3) a HF ESR detection system, (4) an ODMR system.
The NV detection system utilizes a photodiode for ensemble experiments and
avalanche photodiodes for single NV experiments. (5) A diamond sample is
mounted upon a sample stage within a variable field magnet. HF MW excitation
propagates through free space from the waveguide to the top of the sample stage.
Optical access is obtained through the bottom of the sample stage. Reproduced
from Ref.,
122
with the permission of AIP Publishing.
resonator is employed for implementation of wide bandwidth DEER techniques. The
magnetic field at the sample is adjustable between 0 to 12.1 Tesla. For the single-NV
HF ODMR experiment, we employ a conventional confocal microscope setup routinely
used for NV ODMR experiments.
107, 109
The details of the single-NV ODMR system
have been described previously.
62
For the ensemble HF ODMR experiment, we direct
the fluorescence (FL) to a photodiode (Thorlabs), implemented before the coupling stage
106
to the single mode fiber. This is connected to a fast oscilloscope (Le Croy) for measure-
ment of the time-domain FL signal. The detection volume in the ensemble experiment
is in the range of a few m
3
. Results presented here have been obtained using two
samples, both are 2.0 x 2.0 x 0.3 mm
3
size, (111)-cut high pressure, high temperature
type-Ib diamonds from Sumitomo Electric Industries. The crystal used for the single
NV experiment had been previously shown to contain single NV centers with reason-
ably long spin-relaxation times and coupling to P1 centers.
109
The other crystal, used
for the ensemble experiment, was subjected to successive irradiations with high energy
(4 MeV) electron beam and annealing processes (at 1000
o
C) in order to increase the
NV center density. Exposure to a total fluence of 1:2 10
18
e

/cm
2
resulted in approx-
imately 8% NV/N ratio as determined from the X-band (9 GHz) ESR spectrum of the
sample (see the supplementary material).
122
Characterization with ensemble HF ESR
measurements reveals strong ESR signals from both P1 and NV centers, indicating that
both NV and P1 concentrations are more than 1 ppm.
5.2.1 Ensemble NV-ESR
We first perform ODMR of ensemble NV centers. This experiment was performed by
monitoring the FL intensity, while the frequency of a 500 ns MW pulse was varied.
As seen in Fig. 5.2(a), a reduction in FL was observed at 114:78 GHz and 120:51
GHz, which correspond to the lower (jm
s
= 0i$jm
s
=1i) and upper (jm
s
= 0i$
jm
s
= +1i) transitions of a [111] oriented NV with a polar angle of 1:8 degrees. By
fixing the frequency at 114:78 GHz and varying the length of the pulse, Rabi oscillations
were observed as seen in Fig. 5.2(b). From this measurement,=2 and pulse lengths
of 204 and 420 ns, respectively, were extracted. Extracted pulse times were used in a
spin echo experiment. A spin echo relaxation time of 2:40:3s was observed, as seen
107
(c)
0.0 5.0 10.0
2t [ms]
T
2
= 2.4 ± 0.3 ms
(b)
MW1
p
2
p
2
p t t
(a)
Contrast [%]
MW1 Frequency [GHz]
114.76 114.78 120.50 120.52
Exp
Fit
MW1 t
p
Laser
Init RO
0.5
1.0
0.0
0.2
0.4
0.0
Contrast [%]
0.0 1.0 2.0 3.0
4.0
t
p
[ms]
MW1 t
p
0.5
1.0
0.0
Contrast [%]
Figure 5.2: Characterization of an ensemble NV system at 115 GHz and 4.2
Tesla. (a) Pulsed ODMR signals of NV centers. A pulse length (t
p
) of 500 ns
was used. To confirm the observed signal, sequential measurements were made
with (Sig) and without (Ref) application of MW irradiation. The contrast in
percentage (Exp = (Ref-Sig)/Ref) was plotted for the analysis. A Lorentzian fit
is shown in red. Laser pulses with durations of 5 and 10s were used for initial-
ization (Init) and readout (RO), respectively. (b) Rabi oscillations measurement.
The data was taken by varying the length oft
p
. From the data,=2 and pulse
lengths of 212 and 402 ns were obtained. (c) Spin echo decay data obtained
by varying the inter-pulse delay time . T
2
was determined to be 2:4 0:3
s when fit to a single exponential decay. The pulse sequences were shown in
the inset. All ensemble experiments were repeated 10
3
times for averaging.
Reproduced from Ref.
122
with the permission of AIP Publishing.
in Fig. 5.2(c). We next perform NV-ESR using the ensemble NV system with a DEER
sequence. In the DEER sequence, a separate MW pulse (MW2) is applied during the
spin echo sequence as shown in the inset of Fig. 5.3. When the frequency of the pulse
matches the ESR frequency of target spins, the target spins flip, and then the dipolar
field from the target spins experienced by the NV center changes. This change results
in a reduction of the refocused echo intensity. In this manner, an ESR signal of weakly
dipolar coupled spins located in the nanometer scale region surrounding the NV center
can be detected.
41, 47
As shown in Fig. 5.3, application of this pulse sequence reveals five
108
117.60 117.50 117.70
MW2 Frequency [GHz]
Exp Sim
Contrast [%]
0.2
0.3
0.1
MW1
p
2
p
2
p
t t
MW2
p
Figure 5.3: HF NV-ESR using an ensemble NV system. A simulation of the
spectrum based upon DEER spin dynamics is shown in the red dashed line.
89
For the simulation,B
0
= 4:1963 Tesla with a polar angle of 1:8 degrees from
the from the NV center’s [111] axis. Due to the small polar angle, sensitivity
in azimuthal angle variation was below the observed linewidth. The pulse se-
quence is shown in the inset. Each experimental scan is shown in a different
color. In this measurement, = 1 s and
MW 2
= 560 ns were used. The
simulation is offset for clarity. Reproduced from Ref.
122
with the permission of
AIP Publishing.
distinct reductions in FL intensity at 117:49, 117:52, 117:61, 117:69, and 117:72 GHz.
These dips are in excellent agreement with the spectrum simulated from the P1 center’s
Hamiltonian (S = 1=2,I = 1,g = 2:0024,A
?
= 82 MHz,A
k
= 114 MHz).
16, 89
5.2.2 Single NV-ESR
Having resolved the spectrum of P1 centers from the ensemble system, we next discuss
the demonstration of NV-ESR using a single NV center. For this we begin by taking a FL
image of the diamond, as is shown in Fig. 5.4(a). From the observed FL image, a well
isolated FL spot is selected (denoted as NV1). An autocorrelation measurement per-
formed on the FL spot, as shown in the upper inset of Fig. 5.4(a), confirms the observed
FL is from a single quantum emitter. As can be seen in the lower inset of Fig. 5.4(a),
upon increasing the magnetic field we observe the level anti-crossing of both the ground
109
0
10
Contrast [%]
Offset Frequency [MHz]
20
-20 -10 0 10 20
1 ms
10 ms
Sim
p pulse
1.0
0.5
0.0
P|M
s
= 0>
(c)
MW1
(d)
0.5
0.6
0.7
P|M
s
= 0>
MW2 Frequency [GHz]
117.52117.54 117.62117.64 117.72117.74
Sig
Ref
Sim
t t
MW1
MW2
[mm]
[mm]
0
0
10
15
NV1
3
3 1
1
5 30
Cts/ms 0
200 0
1
2
t [ns]
g
2
(t)
100 50
Field [mT]
FL Cts.
(a)
0
10
20
0.0 0.5 1.0 1.5
Contrast [%]
P|M
s
= 0>
t
p
[ms]
1.0
0.5
0.0
Exp
Sim
(b)
MW1 tp
Figure 5.4: 115 GHz NV-ESR spectroscopy of P1 centers using a single NV .
(a) FL image of NV1. The top inset figure shows an autocorrelation measure-
ment on NV1. The dashed red line in the top inset indicates the threshold for a
single quantum emitter. The bottom inset shows the FL from NV1 as a function
of magnetic field. Dips at 50 and 100 mT correspond to the level anti-crossing
of the excited and ground states of the NV center, respectively.
62
(b) Rabi data
taken at 4:197 Tesla corresponding to them
S
= 0$ m
S
=1 transition of
NV1. The normalized percentage of FL contrast (dashed blue line) is plotted
against the pulse length. The simulation is shown in red. For all single NV
measurements, the signal was averaged for (10
5
10
6
) scans with a 5s initial-
ization and 300 ns readout pulse. (c) Pulsed ODMR data taken using a MW1
pulse. These pulses were implemented experimentally by varying the position
of the carrier frequency and sweeping from 1 to 11 MHz (after the multiplica-
tion chain). The data is plotted with respect to the center resonance frequency.
The simulations are shown in red for the 1 s and 10 s pulse lengths. The
effect of a monochromatic pulse is shown for comparison. (d) NV-ESR spec-
trum using NV1 taken with a MW1 chirp pulse length of 1s, = 4s, and

MW 2
= 1s. The NV-ESR was measured by varying the frequency of MW2
(Sig). Clear reductions in FL intensity were observed at ESR frequencies of P1
centers compared to a reference with no MW2 pulse (Ref). The simulation is
shown for P1 centers.
and excited state, thereby confirming the FL spot as a single NV center.
62, 86
After the
detection of a single NV center, the superconducting magnet was set in a persistent field
mode at 4.2 Tesla. The magnetic field strength was then measured via an in-situ ESR
110
measurement of P1 centers. Next we perform pulsed ODMR on the NV center and re-
solve clear reductions in FL intensity at both 114.80 and 120.53 GHz, corresponding
to the lower and upper transitions (data not shown). From these ODMR signals, we
determined the NV center to be aligned in the magnetic field with a polar angle of 1.6
degrees from the [111] axis. We next perform a measurement of the Rabi oscillations for
NV1 at 114.80 GHz. The FL intensity against pulse length is shown within Fig. 5.4(b)
and shows clear Rabi oscillations. Analysis of the signal was then performed by simu-
lating dynamics of a two-level system using the Liouville-von Neumann equation with
a resonance-frequency distribution due to the hyperfine coupling of
14
N (2:2 MHz). As
shown in Fig. 5.2(b), the experiment and the simulation show excellent agreement. The
analysis also reveals that the limited excitation bandwidth results in incomplete popula-
tion inversion of the NV center spin state. The estimated population inversion is only
40%.
The small population inversion of a rectangular pulse is a significant challenge for
the NV-ESR experiment because of errors in the preparation and readout of the quantum
coherent state used in the DEER sequence. The result can be poor signal-to-noise in the
measurement. To overcome this, we employ a pulse shaping technique. The recent
development of high temporal resolution (sub-ns) arbitrary waveform generators has
triggered significant progress in various pulse shaping techniques for ESR. Here we
focus on chirp pulses, a class of pulses that have been used to demonstrate broadband
control over wide frequency ranges. Chirped pulses are now routinely used in ESR
spectroscopy at X- and Q-( 34 GHz) bands.
123–125
In addition, it has recently been
demonstrated at a frequency of 200 GHz.
119
Chirp pulses offer wider spectral excitation,
an ability to correct pulse imperfections, and generally higher fidelity than rectangular
pulses.
126
The effectiveness of chirp pulses is due to the principle of adiabatic passage,
111
whereby population transfer is achieved by sweeping an applied electromagnetic field
through resonance at a sufficiently slow rate.
68, 127, 128
Here, we utilize a linear frequency-swept chirp pulse at 115 GHz. Figure 5.4(c)
shows the efficiency of chirp pulses with two different frequency-swept rates; 10 MHz-
swept in a duration of 1s and 10s. A clear increase in contrast is observed via the
application of chirp pulses, indicating an improvement in both population transfer and
excitation bandwidth compared with the rectangular pulse.T
2
was also determined to be
20s from a spin echo measurement (see the supplementary material). Furthermore,
we use the Liouville-von Neumann equation to calculate the observed behavior as a
function of frequency based on the chosen pulse length and sweep width. As seen in
Fig. 5.4(c), the observed result is in good agreement with the simulation, confirming
that the usage of chirp pulses increases population transfer for the NV center to nearly
100%.
We next utilize chirp pulses to perform NV-ESR with the NV center by applying
three pulses with a fixed delay between them. Similarly to the previous work,
109
a long
MW2 pulse (
MW 2
= 1 s) was utilized for high spectral resolution NV-ESR. The
result of this experiment is shown in Fig. 5.4(d). Clear reductions in intensity are ob-
served compared to a sequence without the MW2 pulse. The obtained spectrum agrees
very well with the three resolved peaks and the spectral splitting of the P1 center.
16, 89
In
addition, the2 MHz linewidth of the observed peaks is of similar width to high resolu-
tion, inhomogeneously broadened, NV-ESR signals resolved at low magnetic fields.
109
5.3 Conclusion
In summary, we have demonstrated NV-ESR of P1 centers at a Larmor frequency of
115 GHz and the corresponding magnetic field of 4.2 Tesla. We have also shown that
112
the application of chirp pulses improves excitation bandwidth and population inversion
of the NV center spin-state at high magnetic fields. The high magnetic field achieved
in this measurement represents a step towards high-resolution NV-based spectroscopy.
In the future, the presented HF ODMR system and technique can be further extended
for the operation at a ESR Larmor frequency of 230 GHz (corresponding to 8.2 Tesla)
for higher spectral resolution. HF NV-ESR offers insight into complex radical spin
systems, with high spectral resolution and the capability of the detection of nanoscale
heterogeneity of external spins, spectral separation from unwanted ESR signals such
as diamond surface spins, and applications for the study of spins within complex in-
tracellular environments. Furthermore, the present demonstration sets the basis for HF
NV-detected NMR spectroscopy enabling high resolution NMR of a small number of
molecules in spatially and temporally heterogeneous environments. HF NV-detected
NMR spectroscopy will be useful for a variety of investigations including structures and
dynamics of biomacromolecules and chemical environments of solid state surfaces and
interfaces.
113
Chapter 6: Electron-electron double
resonance detected NMR spectroscopy
using ensemble NV centers at 230 GHz
and 8.3 Tesla
Materials presented in this chapter can also be found in the article titled Electron-
electron double resonance detected NMR spectroscopy using ensemble NV centers at
230 GHz and 8.3 Tesla by Benjamin Fortman, Laura Mugica-Sanchez, Noah Tischler,
Cooper Selco, Yuxiao Hang, Karoly Holczer, and Susumu Takahashi in Journal of Ap-
plied Physics 130, 083901 (2021).
6.1 Introduction
NMR spectroscopy is routinely used in chemical synthesis for structural analysis of
small molecules. NV-detected NMR is now widely used at low magnetic fields (< 0:1
T), such as for NV depth estimation, liquid state NMR, two-dimensional NMR, hyper-
polarized NMR, nanodiamond based NMR, and even for selective spin manipulation in
a 10-qubit quantum register.
48–52
114
NMR at high magnetic fields greatly increases the spectral resolution and improves
sensitivity. The increase in field strength increases the frequency difference between
closely related chemical species and enables resolution of small chemical shifts. High
field NMR offers new insights into molecules with many similar nuclei, low gyromag-
netic ratios, and low natural abundance, such as for
17
O NMR in pharmaceutical com-
pounds and biomacromolecules.
19, 20
Commercial NMR magnets operating at 28.2 T
(proton Larmor frequency of 1.2 GHz) have recently become available, with hybrid
magnets at fields of 35.2 T (corresponding to 1.5 GHz proton NMR) being available
in user facilities.
21, 22
Implementation of NV-detected NMR at a high magnetic field is
highly desirable. However, there have only been a handful of studies on NV-based sens-
ing at high magnetic fields due to technological challenges involved with combining a
NV ODMR system with a high magnetic field ESR system.
62–64, 122
In this paper, we discuss the implementation of NV-detected NMR at a high mag-
netic field. NV-detected NMR can be achieved by hyperfine spectroscopic techniques.
There are three primary pulsed ESR hyperfine spectroscopic techniques: electron spin
echo envelope modulation (ESEEM), electron-nuclear double resonance (ENDOR), and
electron-electron double resonance detected NMR (EDNMR).
26, 91, 94
Most NV-detected
NMR spectroscopy performed at a low magnetic field is based on ESEEM where the
hyperfine coupling between the NV center and nuclear spins mixes the spin-state and re-
sults in periodic revivals of the echo intensity.
40
This technique functions very efficiently
only when the energies of the hyperfine coupling and the nuclear Larmor frequency are
comparable. Therefore, it works well at a low magnetic field, but becomes unfeasible
at a high magnetic field.
26
In ENDOR and EDNMR techniques, the population differ-
ence of an ESR transition is monitored via a detection scheme while either pulsed RF
(ENDOR) or off resonance MW (EDNMR) radiation is applied to drive polarization
115
transfer. Nuclear identification and determination of hyperfine coupling is performed
based on the frequency of polarization transfer. EDNMR uses a high turning angle
(HTA) pulse to drive population transfer on forbidden transitions. At higher fields, the
Zeeman interaction more completely dominates over the hyperfine interaction, reduc-
ing state mixing and consequently, transition probability. Therefore, higher magnetic
fields require stronger or longer HTA pulses to induce polarization transfer. Both ED-
NMR and ENDOR are promising for NV-based sensing, as they are applicable to single
and ensemble NV systems and are limited by the longitudinal relaxation time, T
1
, in-
stead of the transverse relaxation time, T
2
. The T
1
relaxation time for NV ensembles
has been shown to extend dramatically (up to minutes) at low temperature.
27, 85
More
recently, EDNMR has emerged as a promising technique due to its higher sensitivity
and resiliency against RF related artifacts.
129
EDNMR has an additional advantage over
ENDOR in that it does not require an additional RF power amplifier or tuned RF circuit
and can thus be readily implemented over a large frequency range for the detection of
nuclei with a wide range of gyromagnetic ratios.
Within this work, we demonstrate optically detected magnetic resonance (ODMR)
on the NV center at the highest field and frequency to date, 8.3 T, corresponding to
the NV’s Larmor frequency of 230 GHz (proton Larmor frequency of 350 MHz). We
successfully implement EDNMR using ensemble NV centers and detect
13
C nuclear
bath spins in the diamond crystal. Since the EDNMR technique is limited byT
1
, notT
2
,
NV-detected NMR based on EDNMR can take advantage of the NV center’s long T
1
to perform measurements with a long HTA pulse. With development of suitable pulse
capabilities, the described NV-detected NMR technique will be advantageous for the
development of NV-detected NMR at higher fields and frequencies where the microwave
power is often limited.
31, 122, 130
116
Quasioptics
Dichroic
Mirror
Photodiode
Lenses
Corrugated
Waveguide
Fast Steering
Mirror
Tx
12.1 T
Magnet
Microscope
Objective
Sample
x24
B
0
Signal
Integrator
Laser
AOM
Central Computer
PB-500 & DAQ
PIN MW1
PIN MW2
Figure 6.1: Overview of the experimental setup. The transmission (Tx) setup
consists of two independently controllable frequency sources (MW1 and MW2)
that pass through PIN switches to a frequency multiplication chain. High
frequency MW excitation is propagated through quasioptics and a corrugated
waveguide to the sample stage within a 12.1 Tesla variable field magnet. Pulsed
laser excitation is directed through an acousto-optic modulator (AOM) and an
optical fiber to a system of lenses, a fast steering mirror, and the sample stage.
At the sample stage, a microscope objective directs laser intensity and collects
sample fluorescence. The fluorescence is redirected through a dichroic mirror
to a photodiode where it is integrated using either gated boxcar integrators or
a fast oscilloscope. The MW components, laser, and boxcar integrators are all
controlled through a central computer equipped with a fast TTL logic board and
digital to analog converter (DAQ). The magnetic field (B
0
) is aligned with the
optical axis. Reproduced from Ref.
131
with the permission of AIP Publishing.
6.2 Methods and Materials
A home-built, high field (HF) ODMR spectrometer operating in the band of 215-240
GHz was used. An overview of the experimental setup is shown in Fig. 6.1. The di-
amond sample was mounted at the center of a variable field 12.1 T superconducting
magnet (Cryogenic Limited). Microwave (MW) excitation was produced by a solid
state source (Virginia Diodes) and directed through quasioptics to the sample stage. The
117
output power of both channels from the source was 115 mW at 230 GHz. Laser ex-
citation was produced from a solid-state single mode laser (Crystalaser) and directed
through an acousto-optic modulator (Isomet), single mode fiber (Thorlabs), and micro-
scope objective (Zeiss100X, NA=0.8) before reaching the sample stage. The excitation
beam position was controlled using a fast steering mirror (Newport) and a system of
lenses below the microscope objective. Fluorescence (FL) collected at the objective was
directed back through a dichroic mirror and fluorescence filters (Omega Optics) before
being detected using a photodiode (Thorlabs 130A2). The typical excitation spot size
was a few m
2
. Typical laser excitation of 4 mW at the sample stage resulted in
1-2W of detected FL. The output of the photodiode was directed to a signal integra-
tor. Integration was performed using either a pair of analog boxcar integrators (Stanford
Research Systems SR250) or a fast digitizing oscilloscope (Tektronix MSO64B). The
analog output of the boxcar integrators was digitized using a fast DAQ (National Instru-
ments PCIe-6321). Gate timing was controlled using a gated TTL logic board (Spin-
Core Technologies PB-500). Additional details of the HF-ESR/ODMR spectrometer
have been described previously.
30, 62, 115, 122
For this study, two samples were used. Sam-
ple 1 was a 2.0 2.0 0.3 mm
3
size, (111)-cut high pressure, high temperature type
Ib diamond from Sumitomo Electric Industries. Sample 2 was a hexagonal 4.4 3.9
0.5 mm
3
size, (111)-cut high pressure high temperature type-Ib diamond obtained from
Element Six. Both diamonds had previously been subjected to high energy (4 MeV)
electron beam irradiation and were exposed to a total fluence of 1:2 10
18
e

=cm
2
fol-
lowed by an annealing process at 1000
o
C. This treatment produced a NV concentration
greater than 1 ppm.
122
118
6.3 Discussion
We begin by performing pulsed ODMR on ensemble NV centers. For pulsed ODMR,
the relative FL intensity was monitored while a MW pulse was varied in frequency.
As seen in Fig. 6.2(a), clear reductions in FL intensity were resolved at 229:953 GHz
and 235:687 GHz, corresponding to the lower (jm
S
= 0i$jm
S
=1i) and upper
(jm
S
= 0i$jm
S
= +1i) transitions of a [111] oriented NV with a polar offset angle
of 1:50 0:02 degrees. Next, Rabi oscillations were recorded by fixing the frequency of
MW1 at 229:953 GHz (jm
S
= 0i$jm
S
=1i transition) and varying the pulse length
as seen in Fig. 6.2(b). From these measurements, damped oscillations and a pulse
length of 1:9 s was observed. Next, the NV ensemble’s spin-lattice relaxation time,
T
1
, was recorded. For this measurement, the duration between the laser initialization
and readout pulse () was varied (see Fig. 6.2 (c). Two sequential measurements were
performed by varying the spacing between initialization and readout with and without a
MW pulse before normalization. AT
1
time of 3:9 0:2 ms was found by fitting to a
single exponential decay.
Next we perform EDNMR using the NV center. As shown in Fig. 6.3(a), EDNMR
is a form of high field hyperfine spectroscopy that utilizes two microwave frequencies,
MW1 (
0
) and MW2 (
1
). EDNMR measurements vary the frequency (
1
) of a HTA
MW2 pulse, while MW1 applies a detection pulse sequence, such as Hahn echo, at

0
to measure the spin polarization of an ESR transition.
94
As the frequency of
1
is
swept, the frequency shifts on resonance with transitions below the central transition
(
1
<
0
) due to weakly coupled hyperfine nuclei, as seen in Fig. 6.3(b). These tran-
sitions are generally forbidden as they involve a flip of both the electron and nuclear
spin (m
S
= 1; m
I
= 1). The forbidden transitions become weakly allowed with
119
0
t [ms]
T
1
= 3.9 ± 0.2 ms
0.5
0.0
1.0
Relative
Contrast [a.u.]
MW1
Laser
Init RO
p
t
5 10
Exp.
Fit
0 2 4 6
MW1
t
p
t
p
[ms]
0.2
0.4
0.0
Contrast [%]
8
(b)
(a)
(c)
Contrast [%]
229.94 229.96 235.68
Exp.
Fit MW1
t
p
Laser
Init RO
0.1
0.2
0.0
0.3
0.4
MW1 Frequency [GHz]
235.70
Figure 6.2: Ensemble ODMR at 230 GHz. (a) Pulsed ODMR data. For all
ODMR measurements, laser pulses of 20s and 15s were used for initializa-
tion (Init) and readout (RO), respectively. After initialization, a MW1 pulse (t
p
)
of 1:9s was applied and varied in frequency. Clear reductions in FL intensity
were resolved at 229:953 GHz and 235:687 GHz, corresponding to the lower
(jm
S
= 0i$jm
S
=1i) and upper (jm
S
= 0i$jm
S
= +1i) transitions of
the NV center. The magnetic field was found to be 8:306 T with a polar angle
of 1:50 0:02

. Fitting was performed using nonlinear least squares regres-
sion and the NV center Hamiltonian (S = 1,D = 2870 MHz,g = 2:0028).
27
(b) Measurement of Rabi oscillations. The frequency of MW1 was set at the
lower resonance and the pulse length was varied. From the observed oscilla-
tions, a pulse length of 1:9s was found. (c) Measurement ofT
1
relaxation.
Measurements were performed with (Sig1) and without (Sig2) a pulse. The
difference (Sig2-Sig1) was normalized and then fit to a single exponential de-
cay.
85
Data was collected using (a) 10 scans, (b) 18 scans, and (c) Sig1 and Sig2
were measured sequentially with 5 scans each. Reproduced from Ref.
131
with
the permission of AIP Publishing.
120
partial state mixing, leading to polarization transfer and a reduction in the ESR signal
intensity. This change is detected as an EDNMR signal. Application of a long HTA
pulse improves the likelihood of population transfer, but the total length of the HTA
pulse must be short relative toT
1
in order to maximize the observable contrast. As
1
approaches the central allowed transition (m
S
= 1; m
I
= 0) there is significant pop-
ulation transfer leading to a highly intense change and the so-called ”central blind spot”.
Since the central blind spot highly distorts EDNMR signal, in practice the measurement
is performed at a frequency range outside of the central blind spot. After passing the
central blind spot,
1
then induces forbidden transitions from hyperfine coupled nuclei
with a positive frequency offset (
1
>
0
) relative to the central transition. For NV
detected EDNMR, the spin population can be directly detected via optical spin state
readout, eliminating the need for an echo detection sequence. EDNMR with the NV
center has an advantage over conventional EDNMR, as optical initialization of the NV
center ensures high spin polarization and improves EDNMR sensitivity. The usage of
optical initialization shortens the measurement time by eliminating the need for long
cycle delays between subsequent experiments (typicallyT
1
).
As shown in Fig. 6.3(a), we perform the experiment by applying an initialization
laser pulse, MW2 HTA pulse at frequency
1
, MW1 pulse at frequency
0
, and laser
readout pulse. During the experiment
1
is varied while
0
is fixed at the lower NV
resonance. When the HTA pulse drives a transition, the population of thejm
S
= 0i
spin state is reduced before the MW1 pulse transfers the population to thejms =1i
state. Therefore, when the HTA pulse is in resonance with a transition, an increase in
the FL intensity is observed. For the present experiment, a HTA pulse length of 500s
was chosen. In principle, longer length pulses, up toT
1
, can be applied. Figure 6.3(c)
shows the result of the experiment and we observe signals at88,64,30, +28, and
121
+65 MHz. The strong change in the FL intensity at 0 MHz corresponds to the central
blind spot. The signals at64 and +65 MHz give the hyperfine coupling constant of
129 MHz, consistent with nearest neighbor
13
C hyperfine interaction (126-130 MHz)
splitting the allowed ESR transition.
132–134
The reduced intensity relative to the central
transition corresponds to the low natural abundance of
13
C (1.1%) and low probability
of nearest neighbor locality.
Next we discuss signals at88 MHz. In order to understand the signals we discuss
the following Hamiltonian:
H
NV
=
B
g
NV
~
B
0

~
S +D
~
S
z
2
+H
N
+H
C
; (6.1)
where D = 2:87 GHz, g
NV
= 2:0028, and
~
S is the electronic spin operator.
27
H
N
andH
C
represent the Hamiltonians of hyperfine coupled nitrogen in the NV center and
surrounding
13
C bath spins. The nuclear spin Hamiltonians may be written as:
H
N
=
14
N
~
B
0

~
I
1
+
~
S
~
A14
N

~
I
1
+PI
2
1z
; (6.2a)
H
C
=
13
C
~
B
0

~
I
2
+
~
SA13
C

~
I
2
; (6.2b)
where
nuc
represents the gyromagnetic ratios (3:077 and 10:708 MHz/T for
14
N and
13
C, respectively),
~
I
1
(
~
I
2
) is the
14
N (
13
C) nuclear spin operator,
~
A
nuc
is the hyperfine
interaction (
14
N: A
?
=2:14 MHz,A
k
=2:70 MHz), and P represents the nuclear
quadrupole interaction (5:0 MHz).
106
We focus our study on weakly coupled
13
C
nuclear bath spins. Using Eq. 6.1, we determine all eigenvalues based on the observed
magnetic field. The observed states and energies are listed in Table 6.1. From Table 6.1,
it is seen that the m
S
= 0 states are not evenly spaced around zero. This spacing is
induced by partial field misalignment and nuclear quadrupole interaction that mixes the
122
MW1 (n
0
)
p
MW2 (n
1
) HTA
Laser Init RO
m
s
= 0, m
I
=+
m
s
= 0, m
I
=-
m
s
= -1, m
I
=-
m
s
= -1, m
I
=+
ESR (n
0
)
Forbidden
ESR (n
0
)
n
1
> n
0
n
1
< n
0
-100
Frequency Offset [MHz]
-50 0 50 100
Exp.
(a) (c)
(b)
n
1
< n
0
n
1
> n
0
13
C
13
C
14
N
14
N
1
0
2
3
EDNMR Intensity [%]
Figure 6.3: NV detected EDNMR at high field. (a) Pulse sequence used in
the NV-detected EDNMR experiment. In the experiment, a HTA pulse was
applied with MW2 at frequency
1
before a pulse was applied with MW1 at
frequency
0
. The frequency of
0
was set to match the lower transition. The
application of a pulse increases the sensitivity by isolating the FL of [111]
oriented NV centers from non axial orientations. (b) Energy level diagram.
Nuclei coupled via weak hyperfine interaction are represented bym
I
= + and
m
I
=. During the experiment, the frequency of the HTA pulse is swept from
below (
1
<
0
) to above (
1
>
0
) the central ESR resonance. Population
is transferred when the HTA pulse is in resonance with the difference between
coupled states, resulting in an increase in the observed FL. Due to the length
of the HTA pulse and state mixing induced by the hyperfine interaction, this
occurs for both allowed and forbidden transitions. The intensity of the central
blind spot is due to the allowed transitions. (c) Experimental spectra. The data
are shown with reference to the MW frequency offset (
1

0
) and normalized to
the intensity of the central blind spot. In the present case,
0
= 229:9528 GHz.
A 500 s HTA pulse and 1.9 s pulse were used. The length of the HTA
pulse was chosen to minimize the influence of T
1
relaxation after population
transfer. EDNMR signals due to forbidden transitions involving
14
N and
13
C
are indicated. Grey stars are used to indicate peaks due to allowed transitions
from nearest neighbor
13
C lattice sites. For (c), data was collected using 20
scans over a period of 11 hours. Reproduced from Ref.
131
with the permission
of AIP Publishing.
states and results in twelve non degenerate energy levels. We next calculate allowed
transitions (m
S
= 1, m
I
= 0) and double quantum transitions (m
S
= 1, m
I
=
1) involving a simultaneous electron and nuclear spin flip. We tabulate the allowed
transitions and double quantum transitions involving
13
C and
14
N spin flips in Table 6.2.
As seen in Table 6.2, the allowed ESR transitions are spaced by the axial hyperfine
123
Table 6.1: State identification and energy values determined from Eq. 6.1 based
on a magnetic field of 8.306 Tesla with an offset angle of 1.5 degrees and
A13
C
= 1 kHz. The nuclear magnetic spin value of
14
N (
13
C) is shown as
m
I1
(m
I2
).
Statejm
S
;m
I1
;m
I2
i Energy [MHz]
-1, +1, +1/2 -230023.7
-1, 0, +1/2 -229995.3
-1, -1, +1/2 -229976.9
-1, +1, -1/2 -229934.8
-1, 0, -1/2 -229906.3
-1, -1, -1/2 -229887.9
0, +1, +1/2 -73.1
0, 0, +1/2 -42.5
0, -1, +1/2 -21.9
0, +1, -1/2 15.9
0, 0, -1/2 46.4
0, -1, -1/2 67.0
coupling to
14
N, contributing to the central blind spot. As shown in the inset of Fig.
6.4(a), the signals at30 and +28 MHz are in excellent agreement with the predicted
peak positions for
14
N predicted in Table 6.2. The proximity to the central spot makes
identification of the peaks at18 and 20 MHz difficult, but a dip at18 MHz is in
agreement with the expected peak position. The polarity inversion of the signals is
under further investigation. The observed signals are not symmetric due to the nuclear
quadrupole interaction. The main graph of Fig. 6.4(a) shows the signals at88 MHz
in excellent agreement with double quantum transitions for
13
C bath spins and are well
spaced from the central blind spot. We next repeat the measurements on additional
locations to confirm the observed signals. We adjust the mirror to position 2, 50
m from position 1, and repeat EDNMR measurements. We also include data from
a separate experimental run as position 3. For position 3, the sample was removed
124
Table 6.2: Simulated transition energies calculated from Table 6.1. The states
involved in the transition are listed in the left and central columns, while the
calculated difference is shown in the right column. For clarity, the transition
relative to the central transition (
sim:

obs:
) was tabulated (
obs:
= 229:9528
GHz). Allowed transitions (m
S
= 1, m
I
= 0) are shown in the top panel.
The middle and bottom panel show double quantum transitions (m
S
= 1,
m
I
= 1) involving a simultaneous electron and nuclear spin flip. The middle
panel shows transitions involving
14
N and the bottom panel shows transitions
involving
13
C.
j0;m
I1
;m
I2
i j1;m
I1
;m
I2
i E [MHz]
+1, +1/2 +1, +1/2 -2.1
+1, -1/2 +1, -1/2
0, +1/2 0, +1/2 0.0
0, -1/2 0, -1/2
-1, +1/2 -1, +1/2 2.1
-1, -1/2 -1, -1/2
+1, +1/2 0, +1/2 -30.6
+1, -1/2 0, -1/2
0, +1/2 -1, +1/2 -18.4
0, -1/2 -1, -1/2
-1, +1/2 0, +1/2 20.6
-1, -1/2 0, -1/2
0, +1/2 +1, +1/2 28.4
0, -1/2 +1, -1/2
+1, +1/2 +1, -1/2 -91.1
0, +1/2 0, -1/2 -88.9
-1, +1/2 -1, -1/2 -86.8
+1, -1/2 +1, +1/2 86.8
0, -1/2 0, +1/2 88.9
-1, -1/2 -1, +1/2 91.1
from the setup and replaced, resulting in a different sample location. Fig. 6.4(a) shows
13
C EDNMR signals at88 MHz for all positions in excellent agreement with the
simulation.
125
(a)
Sample 1
Sample 2
(b)
-40 -20 0 20 40
0
1
Freq. Off. [MHz]
EDNMR [%]
-40 -20 0 20 40
0
1
Freq. Off. [MHz]
EDNMR [%]
Frequency Offset [MHz]
80 100 -80 -100
Pos.3 Pos.2
Pos.1 Sim.
13
C
14
N
Pos.3
Pos.2
Pos.1
13
C
14
N
EDNMR Intensity [%]
EDNMR Intensity [%]
-90 90
Frequency Offset [MHz]
-100 -90 -80 90 80 100
0
1
2
3
4
5
0
1
2
4
5
6
3
Figure 6.4: NV detected EDNMR at 8.3 Tesla (a) NV detected EDNMR from
sample 1. EDNMR detection of
13
C is shown in the main graph and EDNMR
detection of
14
N is shown in the inset. Data is offset for clarity. The pre-
sented data is from three areas: positions 1 and 2 were spaced 50m apart,
position 3 was taken after removing and replacing the sample. The data for
position 3 was integrated with boxcar integrators, all other measurements were
integrated using the fast oscilloscope. For position 3, variations in the exper-
imental setup resulted in slightly different parameters: the magnetic field was
8.298 T with a polar offset angle of 1:9 0:1
o
. Rabi oscillations showed a
pulse length of 1:6 s. The change in magnetic field resulted in a small
( 0:1 MHz) shift in the transition frequencies. The stick spectrum shows
double quantum transitions from Table 6.2. A simulation based upon
13
C cou-
pling to the NV center is shown in red. The red line shows a simulation of
L(!; !;!
i
) = A=
P
!
i
!=(!
2
+ 4(!!
i
)
2
) where A is an amplitude
and the sum runs over the resonance positions (!
i
). A nitrogen spin concentra-
tion of 70 ppm (! = 3:2 MHz) was used in agreement with sample properties.
Nonlinear regression ofL(!; !;!
i
) was used to determine ! from the ex-
perimental data (fits not shown). ! was measured to be 2:3 0:3, 2:9 0:4,
and 3:7 0:8 MHz for positions 1, 2, and 3 respectively. (b) NV detected ED-
NMR from sample 2. EDNMR detection of
13
C is shown in the main graph and
EDNMR detection of
14
N is shown in the inset. Data is offset for clarity. The
presented data is from three areas: position 1, 2 and 3 were spaced 50m
apart from each other. The stick spectrum shows the position of double quan-
tum transitions. For sample 2, ! was measured to be 2:7 0:3, 2:9 0:3, and
2:5 0:3 MHz for positions 1, 2, and 3 respectively. Reproduced from Ref.
131
with the permission of AIP Publishing.
We next investigate the linewidth of the
13
C signals in more detail. We plot the sig-
nals related to double quantum transitions in Fig. 6.4 and show the transitions as a stick
spectrum. The calculated three transition frequencies are ranged by 4:3 MHz, which
126
is comparable to the observed linewidth. In general, the EDNMR linewidth is depen-
dent on a variety of factors, including both intrinsic properties, such as spin relaxation
times, and experimental parameters, such as the HTA pulse length and intensity.
129
In
the present case, the observed linewidth was observed to be constant when HTA pulse
lengths from 300 1000 s were used, suggesting that the linewidths are broadened
by internal dynamics. Therefore, we focus our discussion on magnetic dipole coupling
from surrounding spins which can contribute to the observed linewidth. In general, the
magnetic field at an “A” spin fluctuates due to the interaction with random spin flips of
dipolar-coupled “B” spins. When the concentration of “B” spins is sufficiently dilute,
this interaction broadens the linewidth of the “A” spin by inducing a distribution of Lar-
mor frequencies. In this case, the Larmor frequency fluctuations (!), at an “A” spin
from the j-th dipolar coupled “B” spins may be written as:
!
j
=
a
b
j
=

0

a

b
~
4
(1 3 cos
2

j
)m
j
r
3
j
; (6.3)
where
a
(
b
) is the gyromagnetic ratio of the “A” (“B”) spin,
0
is the permeability
of free space, and ~ is the reduced planck constant. The spin state of the j-th spin is
given bym
j
(m
j
=1=2 for anS = 1=2 spin) with
j
representing the angle between
the vector joining the spins, r
j
, and the applied magnetic field. Now by considering
that “B” spins are randomly distributed and the populations of the up- and down-states
of “B” spins are equal, we can average !
j
by considering the probability of finding
a spin at the j-th position and integrating over possible angles and spin states.
68, 73, 135
127
The integral gives the full-width at the half-maximum of the Lorentzian function as a
linewidth (!), which may be written as:
! =
X
j
!
j
=
2
0
~
9
p
3

a

b
n; (6.4)
where n is the concentration of “B” spins in units of spins per cubic meter. In the case
of the present EDNMR study, “A” spin is the NV center and “B” spins are surrounding
paramagnetic spins such as P1 centers and
13
C nuclear spins. The concentration of
nitrogen in the present sample was estimated to be 70 ppm from a 230 GHz pulsed
ESR measurement of the P1 center’sT
2
(T
2
= 1:07 0:01s; data not shown).
89
Using
70 ppm for the concentration of P1 centers, we obtained ! = 3:2 MHz. As shown
in Fig. 6.4, the simulated peaks with the three resonance frequencies and ! gives
excellent agreement with the observed data. Furthermore, the observed linewidth is in
excellent agreement with the linewidth of the lower NV resonance (Fig. 6.2 (a)) and
with previous work on type-Ib diamonds.
89
The use of high purity, isotopically purified
diamonds with low concentrations of paramagnetic spins can be used to further improve
the spectral resolution and is the subject of current work. For example, we note that
Eq. 6.4 predicts ! 0:2 MHz from dipolar broadening due to natural abundance
13
C.
We next discuss measurements on sample 2. All measurements previously discussed
were repeated on sample 2. From measurement of both the lower and upper ODMR
transitions, the magnetic field was determined to be 8.306 T with a polar offset angle of
1:88 0:03
o
. Rabi oscillations showed a pulse length of 1:6s and theT
1
relaxation
time was measured as 3:8 0:3 ms (data not shown). As seen in Fig. 6.4(b), EDNMR
was measured at three different locations on sample 2, with EDNMR signals from
13
C
resolved at88 MHz in each location. The observed signals are in excellent agreement
with the expected peak positions. The slight variation in the observed height and width
128
from sample 1 indicates small sample to sample variation. The inset shows EDNMR
signals resolved from
14
N. Clear signals are resolved at31 and +28 MHz in excellent
agreement with the simulated peak positions and sample 1.
In summary, we have demonstrated pulsed ODMR on an ensemble system of NV
centers at 8.3 Tesla and 230 GHz. Ensemble NV centers were utilized to perform pulsed
EDNMR with optical readout of the spin population. EDNMR signals were resolved
from
13
C bath spins with the linewidth limited by the concentration of paramagnetic im-
purities. This work provides a clear demonstration of NV center detected EDNMR, and
establishes groundwork for the implementation of NV-detected NMR at higher magnetic
fields, with shallow NV centers, and for the study of nuclei with a variety of gyromag-
netic ratios. EDNMR can resolve spins whose gyromagnetic ratios shift the resonance
from the central blind spot. Nuclei with large gyromagnetic ratios, such as
1
H and
19
F , are excellent candidates for future research. Signals from bath
13
C spins were
resolved in this work. From previous measurements, it is known that weakly coupled
13
C hyperfine interaction is on the order of 10 100 kHz.
52, 136
Based on a dipolar
calculation, a hyperfine coupling of more than 10 kHz is expected for surface protons
within 8 nm of NVs. With the fabrication of NVs with T
1
times of a few ms, stable
photoluminescence, and a depth of at least 8 nm,
114, 137, 138
NV-NMR of protons at the
diamond surface will be detectable with the presented EDNMR technique. Chemical
functionalization techniques can be used to bring spins of interest within close proxim-
ity of shallow NV centers.
17, 139
Furthermore, the described technique is limited by the
comparative length of the HTA pulse relative toT
1
relaxation. AsT
1
can be extended
up to several seconds at cryogenic temperatures, this technique can utilize a long HTA
pulse to perform measurements at higher fields and frequencies where microwave power
is often limited.
27, 85
With the development of suitable pulsing techniques, this method
129
will enable measurements in higher magnetic fields, such as those in the National High
Magnetic Field Laboratory.
140, 141
130
Chapter 7: Conclusion
The work presented within this thesis greatly extends the capabilities for NV-based ESR
and NMR measurements at high magnetic fields. While the technique continues to
develop and the spectral resolution continues to improve, there remain a number of
challenges which will briefly be summarized below.
Chapter 2 introduced fundamental concepts used within the field of ESR and ex-
tended these concepts to the NV-center for the application of ODMR. ESR was shown
to provide direct insight into Hamiltonian parameters which have direct dependence on
nanoscale interactions and fundamental constants. ESR remains a growing field, but the
insight enabled by ESR is invaluable for spin system characterization under a variety of
samples. Pulsed ESR provides for an additional level of control over the spin-system
compared to continuous wave ESR measurements. Fundamental pulsed ESR measure-
ments, such as Rabi oscillations, measurement of the FID, SE, STE, and Inversion re-
covery were described and connected to lattice parameters. These pulsed measurements
provided a foundation for more advanced double resonance ESR measurements: DEER,
ENDOR, and ELDOR-detected NMR. Double resonance measurements are fundamen-
tal to the performance of NV-ESR. The description of NV-ESR followed with a brief
overview of the optical properties including spin-state initialization and readout that are
important to consider. Examples of the previously described ESR measurements were
131
then provided for the NV-center for a variety of samples. The detection of internal elec-
tron and nuclear spins was demonstrated for P1 centers,
13
C, and
14
N. Pulsed Endor
enabled resolution of a
13
C line with a width of only 9 kHz, a tremendous advance in
terms of spectral resolution.
Chapter 3 discussed the experimental hardware required for the performance of these
measurements. The optical components for HF ODMR were implemented on a custom-
built HF ESR spectrometer.
30, 62
Improvements to the system, such as the implementa-
tion of a setup for the measurement of ensemble samples with high fluorescence, the
implementation of an RF/MW system for ENDOR measurements, and the implemen-
tation of shaped pulses greatly extended the versatility of the system. The HF ODMR
system was designed in a modular setup for versatility in experimental setup. This de-
sign is invaluable as it allows one to switch between setups in a few short hours and
design measurements for optimization that minimize instrumental downtime.
Chapter 4 described the development of a high resolution NV-ESR technique. This
technique was implemented on single NV-centers in diamond and allowed for spectral
separation and clear resolution of closely related peaks. Variation of the pulse length
and power was used to clearly resolve the intrinsic linewidth of the sample which was
shown to be in agreement withT

2
.
Chapter 5 detailed the first experimental example of NV-ESR at high fields. This
represented an important experimental step, as the previous demonstration of high-field
NV-based measurements did not resolve NV-ESR.
62
Importantly, this was done for both
single and ensemble NV centers. These measurements identified how the limited MW
strength available at high fields placed a severe limitation on the performance of NV-
ESR. By either increasing the total number of NV-centers (as in ensemble NV-ESR)
132
or by implementing frequency swept chirp pulses, improved sensitivity of NV-ESR is
realized.
Chapter 6 detailed the demonstration of NV-NMR at the highest magnetic field to
date. NV-NMR at high fields is arguably more impactful than NV-EPR at high fields
due to the close proximity of nuclear spins that cannot be separated without high mag-
netic fields. The complexity of biomolecular environments, where many identical nuclei
differ only in slight shifts in their chemical environment adds further promise and com-
plexity to the challenge. To date, the performance of NV-NMR has been nearly exclu-
sively done at low-fields (< 0:1T) using methods that are not applicable for NV-NMR
at high fields. This work demonstrates that weakly coupled
13
C spins can be routinely
detected using EDNMR at a field of 8:3 T, a field strength comparable to commercial,
research-grade NMR. Moreover, the technique demonstrates that detection of external
nuclear spins at high field is possible with proper sample design.
133
Appendix A: Spin Dynamics of Chirp
Pulses in a Two Level System
Frequency swept pulses, also known as chirped or adiabatic pulses, offer an addi-
tional level of control over the spin-system compared with conventional monochromatic
pulses. These pulses fall within a larger class of shaped pulses where careful control of
the amplitude and frequency can be used to improve population inversion and overall
fidelity of spin-control. A chirped pulse is a frequency swept pulse, where the frequency
is varied between two set frequencies, !
a
and !
b
, during a pulse time, t
p
. While it is
most common to vary the frequency linearly, for a linear chirp pulse, this is not required
and there are a variety of frequency and amplitude modulation patterns that have been
explored in the literature.
118, 126, 127, 142
These pulses become “adiabatic” pulses when
they fulfill the adiabatic condition. This may be expressed as:
k

2
1
kT
2

; (A.1)
where k = (!
b
!
a
)=t
p
is the sweep rate (here expressed in MHz/s) and
is the
Rabi frequency (MHz).
68, 123, 127
Equation A.1 requires that d!=dt be slow relative to

, which in NMR terminology means that the effective magnetization will complete
many cycles around the magnetic field vectorB
1
and appear to “follow”B
1
as it moves
through the spin transition, thereby adiabatically inverting the spin state. On the other
134
hand, the sweep rate must be fast relative to the transverse relaxationT
2
, as the slower
sweep rates will result in more time spent in the transverse plane. This field has been
well-explored in the NMR literature, as frequency selective pulses and echo sequences
are in regular use for slice selction within magnetic resonance imaging (MRI).
143, 144
Due to the recent development of fast sampling rate (> 1GSa/s) arbitrary waveform
generators (AWGs), there has been a large interest in these pulses within EPR, primarily
at X-band,
123–125, 128, 145, 146
with recent extensions to high frequency EPR.
119
Within this section, the Hamiltonian of a two level system (TLS) under chirped
excitation will be introduced, before being utilized to derive the Landau-Zener equation
which describes the probability of a non-adiabatic transition. Numerical simulations
exploring the influence of pulse parameters on the results will also be shown.
A.1 Derivation
Beginning from the Hamiltonian in the rotating frame (Eq. 2.19), the Hamiltonian of a
TLS is given by:
H
LZ
(t)=~ =

k(t
t
p
2
)

^
S
z
+

^
S
x
; (A.2)
where the sweep width k was symmetric (ie, !
a
= !
b
) during the pulse time t
p
.
Equation A.2, has diagonal and off-diagonal terms, therefore:
H
LZ
(t) =
2
6
6
4
E
1
(t) 0
0 E
2
(t)
3
7
7
5
+
2
6
6
4
0 ~
=2
~
=2 0
3
7
7
5
(A.3a)
=H
0
+H
1
; (A.3b)
135
whereE
1
(t) = ~k=2(tt
p
=2) andE
2
(t) =~k=2(tt
p
=2) are the first and second
eigenvalues ofH
0
.H
0
will be considered the static Hamiltonian andH
1
will be analyzed
as a perturbation.H
0
has two wavefunctions which may be found using the Schr¨ odinger
equation (Eq. 2.20)

1
(t) = exp(i=~
Z
t
0
k=2(t
0
t
p
=2)dt
0
) (A.4a)
= exp

ik
4~
t
2

exp

ik
4~
t
p
t

(A.4b)

2
(t) = exp

ik
4~
t
2

exp

ik
4~
t
p
t

; (A.4c)
where
1
(t) and
2
(t) are the wavefunctions for eigenvaluesE
1
(t) andE
2
(t), respec-
tively. Next a general wavefunction in a TLS will be considered, (t) =

C
1
(t)
1
(t)
C
2
(t)
2
(t)

,
evolving under the Schr¨ odinger equation (Eq. 2.20). Therefore:
i~
d
dt
0
B
B
@
C
1
(t)
1
(t)
C
2
(t)
2
(t)
1
C
C
A
=
2
6
6
4
E
1
(t) ~
=2
~
=2 E
2
(t)
3
7
7
5
0
B
B
@
C
1
(t)
1
(t)
C
2
(t)
2
(t)
1
C
C
A
: (A.5)
At this point it is convenient to drop the subscript for time (t) when solving for the
constants:
d
dt
C
1

1
=ik=2(tt
p
=2)C
1

1
i
=2C
2

2
(A.6a)
_
C
1

1
ik=2(tt
p
=2)C
1

1
=ik=2(tt
p
=2)C
1

1
i
=2C
2

2
(A.6b)
_
C
1

1
=i
=2C
2

2
; (A.6c)
_
C
1
=i
=2C
2
exp

ik
2
t
2

exp

ik
2
t
p
t

; (A.6d)
136
where the product rule was used to expand the left hand side, and the dot indicates the
derivative. Multiplication by the adjoint was used to simplify Eq. A.6c. Similarly, it
can be found that
_
C
2

2
=i
=2C
1

1
and
_
C
2
=i
=2C
1
exp

ik
2
t
2

exp

ik
2
t
p
t

.
Differentiation and rearranging of
_
C
2
yields:

C
2
+ik(tt
p
=2)
_
C
2
(
=2)
2
= 0; (A.7)
where

C
2
is the second derivative with respect to time. This equation may be solved
using contour integration.
147
This yields:
P
LZ
= 1 exp

2
2jkj

; (A.8)
whereP
LZ
is the probability of an adiabatic transition. Notably, this equation provides
an analytical solution for the left hand side of the adiabatic condition (Eq. A.1). When
k is large relative to
, the adiabatic condition is fulfilled and an adiabatic transition is
likely. When
is large relative tok, the adiabatic condition is not fulfilled.
A.2 Numerical Simulations
Further insight into the spin-dynamics of a two-level system can be obtained through
numerical simulations of Eq. A.2 Using the Liouville-von Neuman Equation (Eq. 2.21).
For the following simulations, the Hamiltonian was assumed to be constant over a time
interval (1ns) small relative to the sweep width and time propagated using an ordinary
differential equation solver in MATLAB. The population of each element of the density
matrix ( = [

1

2

3

4
]) was calculated at each time step. In each simulation, the initial
137
0 5 10
(a) (b) (c)
a [MHz/ms] T
p
[ms]
10
1
10
0
10
2
10
3
10
4
0.5
1.0
0.0
P|b
0.5
1.0
0.0
P|b
0.5
1.0
0.0
P|b
Simulation
Analytical
k = 0.2
k = 2
k = 20
-100 0 50 -50 100
Offset Frequency

[MHz]
k = 2
k = 20
p-pulse
Figure A.1: Numerical Simulations of a Landau-Zener Transition. For all sim-
ulationst
p
= 10s,
= 1 MHz, and the sweep width was varied symmetri-
cally in order to shiftk. (a) Simulation of Pji by varying the sweep rate. Eq.
A.8 is plotted as the analytical solution. The simulation for each sweep rate is
shown as a blue square. (b) Time domain traces fork = 0:2, 2, and 20 MHz/s.
Dashed arrows are drawn from (a). (b) Frequency domain traces fork = 2 and
20 MHz/s and a rectangular-pulse plotted against offset frequency relative
to the center of the sweep. Sincet
p
is fixed, a largerk corresponds to a wider
sweep width.
state was the state (
0
=jihj) and the population of the state was calculated
(Pji =
4
). Therefore, Pji = 0, unless an adiabatic transition occurred.
A comparison of numerical simulations to Eq. A.8 are shown in Fig. A.1(a). As
expected, there is excellent agreement with the analytical solution. The deviation from
the analytical solution observed at k < 1 is due to the finite pulse length. As seen in
Fig. A.1(b), for k = 0:2, damped Rabi oscillations are observed, which are expected
as the sweep width reduces relative to
. In the limit of k = 0, Rabi oscillations are
recovered. For k = 2 and 20, small oscillations in Pji are seen before a transition
at the center of the sweep. Frequency swept pulses are also utilized due to their wide
excitation bandwidth. A clear example of this is shown in Fig. A.1(c). The excitation
bandwidth of a rectangular, -pulse is shown compared to the excitation profiles for
k = 2, and 20. Compared to the-pulse, both chirp pulses induce a population change
over a much broader frequency range.
138
A.3 Applications
Experimental examples of chirp pulses are provided in Fig. 3.7 and Fig. 5.4. For NV
centers, the wider excitation bandwidth of chirp pulses results in improved population
inversion and increased contrast when the maximum Rabi frequency is smaller than the
hyperfine coupling to the internal nitrogen spin.
While the discussion here focused on linearly swept pulses, a variety of shapes are
known in the literature with improved shapes under development. These include Sinc,
Sech/Tan, and WURST pulse shapes.
117, 118, 126, 127, 142, 148
More advanced pulse shaping
techniques such as a shortcut to adiabaticity can also be implemented.
149, 150
The extension of chirp pulses to chirp echo sequences is promising, but not straight-
forward due to a nonlinear phase shift induced during the pulse.
142
The most widely
applied technique to correct for this phase shift is the B¨ ohlen-Bodenhausen echo se-
quence where the amplitude and lengths of the pulses are carefully controlled.
151, 152
More advanced techniques chirp echo sequences have been developed, including usage
of composite pulses and sequence design using optimal control theory.
142, 153, 154
Based
on a suggestion by Steffan Glaser, one promising route for a phase-corrected chirp echo
sequence is to implement a series of three chirp pulses with, an echo period, the same
pulse length, and same sweep width as done previously.
122
Ideally, the amplitude of
the pulses would be carefully controlled so that the first and last pulses perform =2
rotations and the central pulse performs a -rotation. The application of chirp echo
sequences is an actively growing area of research with several examples of dynamical-
decoupling sequences using chirp pulses recently demonstrated for NV centers.
155, 156
139
Appendix B: Contribution of Dipolar
Field Fluctuations to the Lineshape
B.1 Derivation
For homogeneously broadened lines the observed linewidth is broadened by dipolar
interaction with neighboring spins. The following discussion focuses on a two-spin
model of A spins and B spins. There is dipolar interaction between the spins and it is
assumed that the B spins are sufficiently far from resonance that they are not excited
upon excitation of the A spins. The discussion here is general and applicable for both
electron and nuclear spins. In the present case, both A and B spins areS (I) = 1=2. The
presented derivation follows that previously presented in the literature.
68, 135
In general, the magnetic field at an A spin fluctuates due to the interaction with
random spin flips of dipolar coupled B spins. This interaction broadens the linewidth of
the A spin by inducing a distribution of larmor frequencies. The system can be treated
in the quasistatic regime when the timescale of the fluctuations is slow relative to the
measurement time.
73
In this case, the larmor frequency fluctuations (!), at an A spin
from j dipolar coupled B spins may be written as:
! =
a
b =

0

a

b
~
4
X
j
(1 3 cos
2

j
)m
j
r
3
j
(B.1)
140
where
a
(
b
) is the gyromagnetic ratio of the A (B) spin,
0
is the permeability of free
space, and~ is the reduced planck constant.
157
The sum runs to the j-th dipolar coupled
B spin, with
j
representing the angle between the vector joining the spins,r
j
, and the
applied magnetic fieldB
0
. The spin state of the j-th spin is given bym
j
(m
j
=1=2
for anS = 1=2 spin).
The goal is to find the intensity distributionI(!) for all A spins in the sample. To
do this it is necessary to integrate over spatial variables to include all possible configu-
rations fromn B spins. The bounds of this integration are confined to the configurations
which give rise to a shift in the linewidth (!2 !d!) The probability of finding
the j-th configuration (p
j
) is given by the product:
p
j
=
n
Y
j
(dV
j
)=V; (B.2)
Where dV
j
= 2r
2
j
dr
j
d(cos(
j
)) and V is the total volume. Therefore, the desired
function is:
I(!)d(!) =
n
Y
j
(dV
j
)=Vd(!)(!); (B.3)
where the function is expressed in integral form and expresses the likelihood of ob-
serving a particular frequency !:
(!) = 1=(2)
Z
+1
1
d exp
"
i

!

0

a

b
~
4
X
j
(1 3 cos
2

j
)m
j
r
3
j
!#
:
(B.4)
141
Since Eq. B.3 contains a product over spatial variables, the problem to be solved be-
comes:
I(!)d(!) =d(!)=(2)
Z
V
Z
V
:::
Z
V
n
Y
j
(dV
j
)=V
Z
+1
1
d exp
"
i

!

0

a

b
~
4
X
j
(1 3 cos
2

j
)m
j
r
3
j
!#
(B.5)
At this point it is necessary to simplify the integral. The high temperature limit can
be imposed, meaning that m
j
is equally likely to be in the spin state1=2 for a S =
1/2 spin. This can be removed from the integration by expanding the integration region
r2 [0; +1] to r2 [1; +1] and doubling the volume (V ! 2V ). Each definite
integral is now similar, so the product of integrals may be simplified and the indices can
be dropped. Therefore, the equation becomes:
I(!) =1=(2)
Z
+1
1
d
Z
2V
(dV )=2V exp

i

!

0

a

b
~
4
(1 3 cos
2
)
2r
3

N
;
(B.6)
whereN is an integer denoting the number of B spins. Utilizing the definition ofdV ,
the integrals may be expanded:
I(!) =1=(2)
Z
+1
1
d exp [i(!)]

=V
Z
+1
1
r
2
dr
Z

0
sin()d() exp

i

0

a

b
~
4
(1 3 cos
2
)
2r
3

N
(B.7)
142
In order to solve the integrals, some rearranging is required. To do so, the second
line of Eq. B.7 will be considered:

=V
Z
+1
1
r
2
dr
Z

0
sin()d() exp

i

0

a

b
~
4
(1 3 cos
2
)
2r
3

N
(B.8a)

=V
Z
+1
1
r
2
dr
Z

0
sin()d() exp[K]

N
(B.8b)

1
V
[VV +V exp[K]

N
(B.8c)

1
V (1 exp[K])
V

N
(B.8d)

1
V
0
V

N
; (B.8e)
where the constant K was used to represent the functions inside the exponent (K =
h
i

0

a
b
~
4
(13cos
2
)
2r
3
i
), the integral of dV was used, and:
V
0
=
Z
+1
1
r
2
dr
Z

0
sin()d()

1 exp

i

0

a

b
~
4
(1 3 cos
2
)
2r
3

: (B.9)
Plugging back in, Eq. B.7, becomes:
I(!) = 1=(2)
Z
+1
1
d exp [i(!)]

1
V
0
V

N
(B.10a)
= 1=(2)
Z
+1
1
d exp[i(!)] exp[n
B
V
0
]; (B.10b)
where the limit was taken asN!1,V !1. To further simplify, the definition of
B spin density, n
B
= N=V (number of B spins per cubic meter), and the exponential
function were used.
143
The last step is to solve forV
0
. Starting from Eq. B.9,
V
0
=
Z

0
sin()d()
Z
+1
1

1 exp

ib
u
2

du=3 (B.11a)
=
Z

0
sin()d()jbj=3 (B.11b)
=

0

A

B
~
8 3
Z

0
sin()d()

3 cos
2
() 1

(B.11c)
=

0
~
9
p
3

A

B
; (B.11d)
whereb =

0

A

B
~
8
(3 cos
2
() 1) and the substitutionu = 1=r
3
was used to solve the
integration. Therefore, the final integral takes the form:
I(!) = 1=(2)
Z
+1
1
d exp[i(!)] exp

n
B

0
~
9
p
3

A

B

; (B.12)
Which, upon closer inspection is the fourier transform of a lorentzian function:
I(!) = 1=()
!
1=2
(!)
2
+ (!
1=2
)
2
(B.13)
where ! was used to replace !, and !
1=2
is the half-width at half-maximum
(!
1=2
=

0
~
9
p
3

A

B
n
B
). Equivalently, this can be expressed using the full-width at
half maximum (FWHM; !)
I(!) = 1=()
!
(!)
2
+ 4(!)
2
; (B.14)
where the 4 preserves normalization and:
! =
2
0
~
9
p
3

A

B
n
B
; (B.15)
144
which is the equation used within the main text.
145
Appendix C: Experimental
Considerations
C.1 Experimental Noise analysis
Photon-based measurements are limited in sensitivity by the number of photons col-
lected. This principle is more generally referred to as shot noise and is defined as:
SNR =
N
p
N
=C
p
N; (C.1)
where SNR is the signal-to-noise ratio, N is the number of collected photons, and C
is a proportionality constant. It is discussed in detail in a number of articles and refer-
ences.
158–160
The experimentalist working with photon based devices, such as the NV-
center, wants to improve data quality. There are two main options after optimization of
their experimental setup:
1. Increase the intensity (and/or contrast) of the detected fluorescence.
2. Signal-average for longer.
Considerable effort has been devoted to the first item in the above list. A non-
exhaustive list includes: (1) improving photon collection efficiency with a higher NA
146
objective or fabrication of microstructures into the surface of the diamond, (2) optimiz-
ing the laser intensity/polarization to maximize fluorescence and observable contrast, (3)
increasing the number of NV-centers within the sample, and (4) implementing contrast
improvement methods such as spin-to-charge conversion.
58, 161–163
This section will be focused on analysis of signal-averaging with regards to exper-
imental noise. Signal averaging for longer will generally improve the signal-to-noise
ratio, but experimental variables may limit the efficacy of long signal averaging periods.
This will help to answer the question of “How long should I let it run?” The experiment
should be run long enough to resolve the signals of interest, but this can be complicated
by experimental noise sources. A few common noise sources in NV-based measure-
ments include fluctuations of laser intensity, variations in sample positioning due to
thermal expansion and cooling, mechanical vibrations, and background light intensity.
The following analysis was used extensively for Chapter 4 and Chapter 5 to resolve
small signals.
C.1.1 Experimental Example for High-field NV-based ESR
The following example is based on data presented in Fig. 5.4 (d). This experiment as
performed on a single NV at high-field to resolve the coupling of nearby P1 centers.
The pulse sequence is shown in Fig. C.1(a) and consists of four separate sequences:
Max. is used to track the maximum amount of fluorescence from the NV-center, Sig.
is the sequence of interest where the MW2 pulse is stepped incrementally in frequency,
Min. is a reference sequence used to determine the maximum amount of fluorescence
contrast, and Norm. is another reference sequence used to isolate the influence of the
MW2 pulse. In this experiment, Max, Sig., Min. and Norm are run forN repetitions
(typically, 1e
3
< N < 1e
6
), before stepping the MW2 frequency and repeating the
147
(a)
Laser
MW1
Init. R.o. N
M
Max. Sig. Min. Norm
MW2
p
Init. R.o. Init. R.o. Init. R.o.
7e+3
8e+3
9e+3
Fl. [Cts.]
(b)
MW2 Frequency [GHz]
117.620 117.625 117.630 117.635
Max.
Sig.
Min.
Norm
Prob. [a.u.]
M = 10
Fl. [Cts.]
3.2e+5
3.6e+5
(c)
MW2 Frequency [GHz]
117.620 117.625 117.630 117.635
Max.
Sig.
Min.
Norm
Prob. [a.u.]
M = 41
m [Cts.]
s [Cts.]
0
0.0
300
3.5e+5
(d)
Number of Scans (M)
0 20 40
Max. Min.
Signal to Noise (SNR)
10
20
(e)
Number of Counts
1e+3 1e+5
SNR
Shot Noise
s
s
m s
m
s
m
m
Figure C.1: Experimental Noise Analysis for HF NV-ESR. The data presented
is the same as Fig. 5.4 (d). (a) Pulse sequence used for the measurement of NV-
ESR. Max, Min, and Norm. act as references while the Frequency of MW2 is
varied. The experiment is repeatedN times while the frequency is incremented
forM scans. (b) Experimental data shown forM = 10. The histograms show
the average () and standard deviation () for Max and Min. (c) Experimental
data shown forM = 41. (d) Plot of and against number of scans (M). (e)
SNR analysis using Eqs. C.2 and C.1 withC = 1=15.
N repetitions. The fluorescence counts from these N repetitions is recorded by the
computer. The MW2 frequency is then incremented in a fixed range (between 117.620
and 117.636 GHz in this example), before the frequency stepping is repeated over the
frequency range for M scans. After the M-th scan is completed, the total number of
148
fluorescence counts is integrated separately for each channel. For the present measure-
ment, the time taken to step the frequency ( ms) is significantly longer than the length
of each sequence (s). The ratios betweenN andM determine the relative timescale
of the signal integration. For stable experiments where the sample positioning does not
change significantly over several hours it is helpful to use a largerN value and shorter
M. In the present experiment,N = 8e
4
was found to give good results. In practice, the
rate of averaging should be optimized with the total number of detected photons approx-
imately equal to the product of N and M. The NV-based ESR measurement is given
in Fig. C.1(b) after 10 scans (M = 10). The data is displayed with 95% confidence
intervals calculated from the standard deviation of the collected photons for each data
point. Max. and Min. lines are cleanly separated, however there is still significant over-
lap between Sig. and Norm. No parameter is varied for either Max. or Min. sequences
during the measurement. Therefore, in the absence of experimental fluctuations, they
should yield consistent amounts of fluorescence for each frequency point. However,
this is not what is observed. To better understand the relative noise, a histogram of the
observed values was plotted on the right panel of Fig. C.1(b) for Max. and Min. This
shows that fluctuations in the experimental setup reduce the observed standard deviation
(
Max

Min
) to100 counts when averaged across all frequency points. When the
number of scans is increased (M = 41), the data quality improves as seen in Fig. C.1(c).
The standard deviation for each data point is similar to that in Fig. C.1(b), but appears
smaller relative to the larger number of collected photons ( 3:5e
5
). In this case, the
NV-ESR signal at 117:627 GHz is cleanly resolved. As can be seen in the histogram,
there still exists a significant distribution of counts (
Max

Min
300). However,
the average accumulated counts (
Max
and
Min
) increased faster than the noise due to
experimental fluctuations (
Max
and
Min
). This is not always observed and depends on
149
many experimental parameters, including laser and positioning stability. Therefore, it is
important to monitor during an experiment to ensure that signal-averaging is improving
the quality of the data (particularly if averaging for long time periods, such as several
days).
Both and can be monitored during the course of an experiment as seen in Fig.
C.1(d) for Max. and Min. The difference between
Max
and
Min
grows linearly with
an increase in the number of experimental scans. However,
Max
and
Min
also increase
with the number of experimental scans. Therefore, a useful metric to evaluate the signal
to noise is:
SNR
NV
=

Max

Min
p

2
Max
+
2
Min
(C.2)
A plot of Eq. C.2 and Eq. C.1 against the average number of counts (
Max
+
Min
=2) is
shown in Fig. C.1(e). The graph shows good agreement between the experimental and
ideal SNR withC = 1=15. If significant deviation from the expected
p
N dependence
is observed, this indicates that experimental fluctuations are limiting the observed SNR.
In this case, adjustments must be made to improve the experimental setup or increase
the signal contrast.
150
C.2 NV-ESR based on Dynamical Decoupling
The previously discussed spin dynamics set a foundation for the discussion of NV-ESR.
The measurement of NV-ESR is based upon a double electron-electron resonance tech-
nique as discussed within Sec. 2.4.4. For a DEER measurement using a dynamical
decoupling sequence, multiple pulses are applied in order to maximize the accumulated
phase shift. An example of a DEER sequence using the XY8-2 dynamical decoupling
sequence is shown in Fig. C.2(a). The XY8-N sequence is based on the application of
eight pulses with alternating phase in the orderXYXYYXYX,
where X denotes a pulse with 0
o
phase offset and Y denotes a 90
o
phase offset.
78
The
usage of this sequence results in an extension of the coherence time compared to a con-
ventional spin echo technique.
48, 113
To perform NV-ESR, the coherence time is first
measured by applying the dynamical decoupling sequence and varying the interpulse
spacing as shown in Fig. C.2(b). For this example, two sequential experiments were
performed: one with the last pulse as a =2-pulse, where the spin state was directed
back tom
s
= 0, and one with the last pulse as a 3=2-pulse where the spin state was
directed to m
s
=1. Application of these pulse sequences ensured that the experi-
mentally observed decay was due to the observed spin dynamics.
48, 51
Normalization of
the fluorescence decay (Signal = (Exp:1Exp:2)=(Exp:1 +Exp:2) resulted in an
exponential decay as shown in Fig. C.2(c) with a coherence time,T
2
= 9:4 0:7s,
which is extended from the Hahn echoT
2
of 1:3 0:6s. The dip in the exponential
decay time at 5s is due to the detection of the 4th harmonic of
13
C signal.
164
Once the
coherence decay is measured, a time of 216 ns (total evolution time 7s) is cho-
sen to maximize the evolution time and the observed signal intensity. Next an NV-ESR
experiment is performed by fixing the time and varying the frequency of the MW2
151
(a)
N
(b) (c)
(d)
(e)
Init. R.o.
Laser
MW1
MW2
p
2
x
p
x
p
y
p
x
p
y
p
y
p
x
p
y
p
x
p
2
x
3p
2
x
t t 2t 2t 2t 2t 2t 2t 2t
or
p p p p p p p p
Evolution time [ms]
0 5 10 15 20
Frequency [MHz]
800 850 900 950 1000 1050
Frequency [MHz]
800 850 900 950 1000 1050
Evolution time [ms]
0 5 10 15 20
100
98
96
94
92
Contrast [%]
98
96
94
Contrast [%]
3
2
1
0
-1
Signal [D%]
2
1
0
Signal [D%]
Exp.1
Exp.2
Exp.1
Exp.2
Exp.
Fit
Figure C.2: NV-ESR from a sample of ensemble NVs. All measurements were
performed on the lower transition.(a) Pulse sequence used for the measurement
of NV-ESR. An XY-8-2 dynamical decoupling sequence was applied to the NV
center. The final pulse was a=2-pulse for Exp.1 and a 3=2-pulse for Exp.2.
MW2 pulses were applied only during the NV-ESR measurements as shown in
(d) and (e). (b) Experimentally measured coherence decay from variation of.
In this experiment,=2-pulses of 12 ns,-pulses of 22 ns, and 3=2-pulses of
34 ns were used. (c) Normalized coherence decay. An exponential decay fit
shows thatT
2
= 9:4 0:7s. (c) Observed NV-ESR utilizing a MW2-pulse
of 32 ns. The final pulse was a=2-pulse for Exp.1 and a 3=2-pulse for Exp.2.
(d) Normalized NV-ESR showing ESR signals.
pulses. The result of this experiment is shown in Fig. C.2(d). A clear signal is observed
at 970 MHz. By normalizing the data as described above, the signal to noise ratio is
152
slightly improved, resolving a broad signal at 920 MHz in addition to the signal at 970
MHz.
C.3 HF ENDOR Pulse Broadening Dependence at 115
GHz
As presented in Sec. 2.4.5,NV-based Mims ENDOR was performed on an ensemble of
NV centers at 4.197 T and 115 GHz. The pulse sequence is shown in Fig. C.3(a). A
strong dependence on the pulse length was observed as seen in Fig C.3(b), where the
line width starts at10 kHz and then reduces to 4 kHz.
(b)
Frequency [MHz]
44.85 44.90 45.00 44.95
99.80
99.75
Contrast [%]
200ms
400ms
50ms
100ms
T
pulse
MW1
RF
(a)
Init. R.o.
Laser
p
2
p
2
t t
D p
2
p
2
t
Figure C.3: ENDOR at 115 GHz (4.197T) from an ensemble NV system. (a)
Pulse sequence used for the measurement of NV-ENDOR. A STE sequence was
applied with an additional RF pulse applied during the delay period
D
. Laser
pulses were used to initialize and readout the spin-state. (b) ENDOR data from
13
C using a of 10s and varying the length of the RF pulse,T
pulse
. The data
is offset vertically for clarity. The diamond is the same as used in Fig.2.18.
153
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Abstract (if available)

Abstract Magnetic resonance spectroscopies, including nuclear magnetic resonance (NMR) and electron spin resonance (ESR) are invaluable spectroscopic techniques for characterization of molecular structures. Magnetic resonance at high magnetic fields has many advantages. NMR at high magnetic fields is highly advantageous because of its high spectral resolution and improved sensitivity, enabling the resolution of closely related chemical shifts and offering new insights into the study of complex molecules. ESR spectroscopy at high magnetic fields improves spin polarization, increases control over spin dynamics, improves insight into molecular motion, and offers high spectral resolution into closely related spin systems. The increased resolution enables investigation of complex systems with similar g values.
The nitrogen-vacancy (NV) center, due to its unique properties, has enabled widespread study of nanoscale NMR and ESR at low magnetic fields. However, there have been few studies of NV-detected NMR and ESR at high magnetic fields due to the technical challenges involved. In addition, conventional NV-detected NMR based on the detection of alternating current (AC) magnetic field sensing is not applicable at high magnetic fields and therefore requires the development of alternate techniques.
This thesis is focused on the development of NV-based ESR and NMR spectroscopies at high magnetic fields. The organization is as follows: Chapter 1 provides an introduction and motivation for high-field ESR and NMR and relevant technical challenges involved that make the extension to high-field NV-based magnetic resonance difficult. Chapter 2 provides an overview of pulsed ESR spin techniques and their applications for optically detected magnetic resonance using NV centers in diamond. Chapter 3 overviews the experimental apparatus and hardware required for the implementation of high-field optically detected magnetic resonance. Chapter 4 demonstrates a technique for NV-detected ESR (NV-ESR) with high-spectral resolution and a method of extracting an intrinsic linewidth representative of ESR spectral properties. Chapter 5 demonstrates high-field NV-ESR at a field strength of 4.2 Tesla using both single and ensemble NV-centers. Chirp pulses are used to increase the excitation bandwidth and improve population transfer, enabling NV-ESR for single NV centers. Chapter 6 introduces a technique applicable for NV-detected NMR (NV-NMR) at high-fields and utilizes this technique for the first performance of NV-NMR at 8.3 Tesla, a field strength comparable to research grade NMR.

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Creator Fortman, Benjamin Michael (author)

Core Title Development of high-frequency and high-field optically detected magnetic resonance

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Publication Date 03/21/2022

Defense Date 03/09/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag electron paramagnetic resonance,electron spin resonance,EPR,ESR,high field,nitrogen-vacancy center,NMR,nuclear magnetic resonance,NV center,OAI-PMH Harvest,ODMR,optically detected magnetic resonance,quantum sensing

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Takahashi, Susumu (committee chair), Lidar, Daniel (committee member), Vilesov, Andrey (committee member)

Creator Email bfortman@email.wm.edu,bfortman@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC110816579

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