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Development of an electron paramagnetic resonance system for a single nitrogen-vacancy center in diamond

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DEVELOPMENT OF AN ELECTRON PARAMAGNETIC RESONANCE SYSTEM FOR A SINGLE
NITROGEN-VACANCY CENTER IN DIAMOND

by

Viktor Stepanov

Submitted in Partial Fulfillment of the Requirements

For the Degree of Master of Science in

Chemistry

Dana and David Dornsife
College of Letters, Arts and Sciences

University of Southern California

Accepted by:

Susumu Takahashi, Director of Thesis

Curt Wittig, Committee Member

Stephen E. Bradforth, Committee Member

Peter Z. Qin, Committee Member

ii
© Copyright by Viktor Stepanov, 2013
All Rights Reserved.
iii
ABSTRACT
A nitrogen-vacancy (NV) center is a paramagnetic impurity defect existing in the
diamond. Because of its unique properties including long electron spin coherence time,
photostability and ability to initialize and detect a single NV center, the NV center is a
promising candidate for investigation of fundamental quantum sciences and for
applications to quantum information processing devices as well as for a magnetic sensor
with single spin sensitivity. The spin state of the single NV center spin is initialized by
optical excitation, manipulated by microwave excitation and measured by fluorescence of
the NV center. This thesis describes development of an optically detected electron spin
resonance (ODMR) system which initializes, manipulates and measures the spin state of
a single NV center. I first introduce uniqueness and physical properties of a NV center in
diamond in Chapter 1. The design and performance of the ODMR setup is described in
Chapter 2. Chapter 3 shows continuous-wave and pulsed ODMR experiments performed
by the ODMR system. Finally, in Chapter 4, I discuss the performance of the ODMR
system via analysis of the signal-to-noise ratio on the observed pulsed ODMR data.
iv
TABLE OF CONTENTS
ABSTRACT .......................................................................................................................... iii
LIST OF FIGURES ................................................................................................................. vi
CHAPTER 1: INTRODUCTION OF NITROGEN-VACANCY CENTER IN DIAMOND ......................1
1.1 ELECTRONIC STRUCTURE ....................................................................................1
1.2 SPIN SELECTIVE RELAXATIONS OF NV¯ ..............................................................3
1.3 GROUND STATE SPIN HAMILTONIAN OF NV CENTER ...........................................5
CHAPTER 2: OPTICALLY DETECTED ELECTRON SPIN RESONANCE SETUP .............................7
2.1 SCANNING MECHANISM .......................................................................................9
2.2 FL COLLECTION EFFICIENCY .............................................................................10
2.3 SPATIAL RESOLUTIONS OF NV FL SIGNALS .......................................................15
2.4 NV CENTER TRACKING ......................................................................................16
2.5 ANTIBUNCHING EXPERIMENT ............................................................................17
CHAPTER 3: ODMR MEASUREMENTS OF A SINGLE NV CENTER IN DIAMOND .................21
3.1 CONTINUOUS WAVE ODMR MEASUREMENTS ...................................................21
3.2 RABI OSCILLATIONS MEASUREMENTS ...............................................................23
3.3 SPIN ECHO MEASUREMENTS ..............................................................................25
CHAPTER 4: EXPECTED SNR IN PULSED-ODMR MEASUREMENTS ....................................29
4.1 ANALYSIS OF EXPECTED SNR IN PULSED-ODMR EXPERIMENTS ......................29
v
4.2 COMPARISON BETWEEN THE ANALYSIS AND EXPERIMENTS DATA .....................39
4.3 LIMITING FACTOR OF SNR AND OUTLOOK TO IMPROVEMENTS ..........................39
CONCLUSIONS .....................................................................................................................41
APPENDIX A – ESTIMATE OF THE SPATIAL RESOLUTION OF NV-FL IMAGING .....................42
APPENDIX B – ESTIMATE OF TYPICAL OPTICAL EXCITATION RATE OF NV CENTER AND FL
COLLECTION EFFICIENCY IN OUR SETUP ..............................................................................53

REFERENCES .......................................................................................................................56
vi
LIST OF FIGURES
Figure 1.1 Formation of NV center in diamond ..................................................................1
Figure 1.2 A) Schematics of NV center. B) The energy diagram of NV center C)
Emission spectrum taken from ensemble of NV centers .....................................................2

Figure 1.3 Five level model of NV center ...........................................................................4
Figure 1.4 Magnetic field dependence of energy levels and resonance frequencies of
allowed transitions ...............................................................................................................6

Figure 2.1 ODMR setup developed in our laboratory to study NV center in diamond .......7

Figure 2.2 Illustration of scanning mechanism in optically detected ESR setup.................9
Figure 2.3 Reflection image of the standard grating ..........................................................10
Figure 2.4 GCE as a function of Numerical aperture for a dry objective and oil-
immersion objective ...........................................................................................................11

Figure 2.5 A) Illustration to describe the FL collection efficiency. B) Transmission
coefficient of the FL photon as a function of the incidence angle to the diamond-oil
interface calculated using Fresnel transmission/reflection equations ................................13

Figure 2.6 Summary of NV center FL attenuation in the setup .........................................14

Figure 2.7 A) 3D contour image of NV centers obtained by scanning the laser across the
diamond sample. B) FL profile along white line indicated in A .......................................15

Figure 2.8 Tracking algorithm ...........................................................................................16
Figure 2.9 HBT experiment ...............................................................................................18
Figure 2.10 HBT data ........................................................................................................19

Figure 3.1 Single NV center cw-ODMR ...........................................................................22
Figure 3.2 Rabi oscillation measurement of a single NV center .......................................24
vii
Figure 3.3 SE measurement ...............................................................................................26
Figure 4.1 Five level model of NV center used in simulations .........................................30
Figure 4.2 Dynamics of FL signal in pulsed-ODMR experiment .....................................31
Figure 4.3 Illustration of Rabi signal and contrast definitions...........................................33
Figure 4.4 Calculation results of FL signal and contrast in Rabi experiment....................35
Figure 4.5 SNR
APD
, SNR
TR
and SNR
TOTAL
in pulsed-ODMR experiments estimated for
our setup .............................................................................................................................38

Figure 4.6 SNR
APD
, SNR
TR
and SNR
TOTAL
as a function of laser power ..........................39

Figure A.1 Kirchhoff integral ............................................................................................42
Figure A.2 Intensity distribution of uniform plane wave PSF focused by a thin lens .......45
Figure A.3 Profile of PSF of a focused uniform plane wave .............................................46
Figure A.4 Intensity profiles in the focal plane and along the optical axis are shown for a
uniform plane wave, and different ɤ value of Gaussian laser beam ..................................47

Figure A.5 Illustration of focus shift due to oil-diamond interface ...................................48
Figure A.6 Intensity distribution of laser beam focused through oil-diamond interface ...49
Figure A.7 A) lateral and B) axial intensity distribution for different focused laser beam
penetration depth to the bulk diamond. C) Maximum intensity and shift of focus as
functions of depth ..............................................................................................................50

Figure A.8 FL signal collection and detection ...................................................................51
Figure B.1 Optical and microwave excitation pulse sequence used in time-transient FL
measurements .....................................................................................................................53

Figure B.2 Time-transient FL data and simulated results using the five-level model .......54

1

Chapter 1
Introduction of nitrogen-vacancy center in diamond
A nitrogen-vacancy (NV) center, formed by
a substitutional nitrogen atom next to a vacancy
in a carbon lattice, is an impurity defect center
existing in diamond (Fig.1.1) (1,2). The NV
center possesses remarkable properties,
including long electron spin coherence time
(3,4), superb photostability (5,6,7) and
addressability of single NV center electron spin
that can be polarized by optical excitation (8,9,10), ability to coherently control its spin
state by microwave field (11,12) and to detect a single NV center from spin dependent
fluorescence (FL) emission (13,14,15), that make it suitable for various applications in
quantum technologies, including quantum information processing (16), quantum
cryptography (17) and metrology (18), as well as nanoscale magnetometry (19,20,21,22).
1.1 Electronic structure of NV center
A NV center studied here is a negatively charged NV center (NV¯) (1,23). A NV
center has two unpaired electron forming an S=1 spin system (Fig. 1.2a) (there also exists
neutral NV centers, S=1/2 NV
0

(1,24,25), however NV
0
is not discussed in the remainder

Figure 1.1 Formation of NV center
in diamond.
2

of this thesis). Uniaxial stress measurements (26), electron paramagnetic resonance
(27,28,29) and hole-burning (30,31,32) experiments have confirmed that there exist
dipole allowed transitions between spin triplet ground state
3
A
2
and spin triplet excited
state
3
E in NV centers (Fig.1.2b). Degeneracy between m
s
= 0 and m
s
= ±1in the ground
state is lifted due to the magnetic dipolar interaction between two unpaired electron spins
in NV centers (33). The zero-field splitting of the ground state is 2.87 GHz. Optically
detected magnetic resonance (ODMR) experiments on a single NV center have shown
that, analogously to the ground state
3
A
2
, the excited state
3
E has a S = 1 triplet state with
the zero-field splitting of 1.42GHz (34,35).

Figure 1.2 A) Schematic of a single NV center depicting vacancy (V; white circle),
substitutional nitrogen atom (N; red circle) and three nearest-neighbor carbon atoms to
the vacancy (C; black circle). The NV axis (C
3v
symmetry axis) lies along the Z axis. B)
The energy diagram of the NV center. (Vibrational states are shown as grey lines; the
spin sub-levels are shown as thick solid lines). C) Emission spectrum taken from
ensemble of NV centers.

In addition, NV centers are coupled to vibrational modes of the carbon lattice that
leads to the quasi-continuum of the vibrational states (grey colored states in Fig.1.2b). As
shown in Fig. 1.2c, FL measurement shows that the zero phonon line (ZPL) is at 637nm
(1.945eV). When a NV center is optically excited with a frequency higher than ZPL, the
3

NV center undergoes the transition to the lowest vibrational state due to fast phonon
relaxations followed by a radiative decay primarily to the phonon sidebands according to
the Franck-Condon principle, leading to the extended emission spectrum from 600-
800nm (see Fig.1.2c). Furthermore, the emission in the infrared region is observed with
ZPL at 1042nm (1.190eV) (36), which is attributed to the transition between two singlet
states, namely
1
A to
1
E (37) (Fig.1.2.b). Relative energy levels of the triplet (
3
A
2
,
3
E) and
singlet (
1
A,
1
E) states is not known (37).
1.2 Spin selective relaxations of NV
At room temperature, the ground state spin sub-levels of a single NV center are
mostly equally populated, i.e. no net polarization. However, early experiments have
shown that the spin state of the NV center is polarized into the m
s
= 0 spin state upon
continuous optical excitation, e.g. 70-90 % of the polarization has been demonstrated
(8,10,27,38,39). It is now well established that the optical spin polarization is primarily
due to existence of a spin-dependent non-radiative transition between the excited triplet
state
3
E and intermediate singlet state
1
A (9,10,40). Moreover, spin dependent FL of the
NV center resulting from the spin dependent relaxations enables readout of the single NV
center through measurement of the FL intensity (10,15).
Here I introduce a five-level model (9) to explain the polarization effect and
mechanism of the m
s
= 0 state readout through the FL signal (see Fig.1.3). Upon optical
excitation, the spin sub-levels in the ground state are excited identically to the excited
state
3
E (green solid arrows) following fast vibrational relaxations that bring the NV
center to the lowest vibrational state (black dashed arrows). After the vibrational
4

relaxations, both the NV spin sub-levels decay to the ground state resulting in FL
emission. However, the m
s
= ±1 spin undergoes an additional non-radiative decay from
3
E
state to the intermediate singlet states
1
A and
1
E (the probability of the non-radiative
decay is ~30%), then to the ground m
s
= 0 state. Thus this non-radiative decay process
leads the NV center to initialize into the m
s
= 0 state under continuous optical excitation
In addition, the FL signal of m
s
= 0 state appears higher than of m
s
= ±1 because the
probability to undergo the radiative decay for m
s
= ±1 is lower than that for m
s
= 0,

Figure 1.3 Five level model of a NV center (9). Green solid arrows are optical
transitions upon illumination, black dashed arrows indicate vibrational relaxations in
the excited state
3
E, red arrows stands for radiative decay, solid black arrows – spin
sub-level relaxations through intermediate singlet states, grey lines – vibrational
states in the excited state, black lines – electronic orbitals of NV center.

5

therefore, detection of the FL signals allows us to readout the spin state of the NV center.
In Chapter 4, I will analyze dynamics of the five-level model to understand time-transient
FL signals observed in our experiment.
1.3 Ground state spin Hamiltonian of NV center
The ground state of NV center is
3
A
2
state with total electronic spin S = 1. The
magnetic dipolar interaction between the two unpaired spins results in the zero-field
splitting of D = 2.87 GHz between the m
s
= 0 and m
s
= ±1 levels, therefore the spin
Hamiltonian of NV center electron spin can be written as (hyperfine coupling between
14
N (I=1) and N in the NV center is not considered here):

NV Dipolar Z
H H H 
(1.1)
where
Dipolar
H is the dipolar spin-spin interaction term and
Z
H is the Zeeman term. The
dipolar and Zeeman Hamiltonians are given by

   
2 2 2
1
1
3
Dipolar Z X Y
H D S S S E S S

    


,
(1.2a)

ZB
H g B S   , (1.2b)
where 2.87 D GHz  is the axial term, E ~ 0 because of the axial symmetry of NV center,
2.0028 g  is the isotropic g-factor,
B
 is the Bohr magneton, , BS are magnetic field
vector and NV spin vector operator respectively and
,, X Y Z
S are spin vector operators with
respect to the NV axis (the z axis is along the N-V direction as shown in Fig.1.2a).
Finally, the spin Hamiltonian of the NV center is obtained:
6

 
2
1
1
3
NV Z B
H D S S S g B S 

    



(1.3)
As seen from eqn. 1.3, the energy diagram of NV center depends on the magnetic
field strength as well as the angle between the magnetic field and NV center axis. Figure
1.4 shows simulation of the energy diagram and the resonance frequency of the allowed
transitions (0 ↔ -1 and 0 ↔ 1) as a function of strength of magnetic field applied along
NV center axis.

Figure 1.4 Magnetic field dependence of the energy levels and the resonance frequencies
of the allowed transitions. A) Energy diagram simulated using eqn. 1.3. B) Resonance
frequency simulated using eqn.1.3. The magnetic field is applied along the NV center (z)
axis. 0 1 , 0 1    are the allowed spin transitions.

7

Chapter 2
Optically detected Electron Spin Resonance setup
Figure 2.1 shows an ODMR setup developed in our lab. The setup is based on a
scanning confocal microscopy system and microwave access for electron spin resonance
(ESR) excitation.

Figure 2.1 ODMR setup developed in our laboratory to study NV center in diamond. See
text for details.
8

The setup employs a diode-pump solid-state laser with a single-mode radiation of 100
mW at the wavelength of 532 nm (CrystaLaser CL-532-100-S). The laser out passes
through an acousto-optical modulator (AOM) for pulsed operation. A laser line filter is
used to attenuate the laser emission other than 532 nm, e.g. the fundamental laser
frequency mode (1064 nm). Intensity of the laser to a sample is controlled by series of
neutral density filters (ATTN). After a dichroic mirror, the 532 nm excitation laser is
reflected by a fast steering mirror (FSM) to a pair of lenses (lens1 and lens2 in Fig.2.1)
and to a high numerical aperture oil-immersion objective (NA = 1.4), then the excitation
laser is focused to a diamond sample located at the working distance of the objective. FL
signals emitted by an excited NV center are collected by the same objective. The
collimated FL output from the objective passes through the same lens and FSM, then is
separated by the dichroic mirror and is directed to the detection system. After the dichroic
mirror, a long-pass filter is placed to filter out unwanted background noises. A focusing
lens in the detection system focuses the FL light into an optical fiber connected to an
avalanche photodiode (APD). The diameter of the optical fiber functions as a pin-hole
used in a conventional confocal microscope which determines the spatial resolution along
the z-axis (parallel to the optical axis). In anti-bunching experiment, the beamsplitter is
placed to split FL signals into two APDs, which are connected to the Time-Correlated
Single Photon Counting (TCSPC) module. A sample is mounted on a piezo-electric stage
for a precise sample positioning and tracking. A gold wire (diameter ~25 m  ) is fixed on
a surface of a sample stage for microwave excitation.

9

2.1 Scanning mechanism
Here I describe a scanning mechanism for 2D FL imaging. As shown in Fig. 2.2, the
scanning system is based on a combination of FSM and the two lenses (Lens 1 and 2).
The position of FSM is controlled by a FSM driver (controlled by application of DC
voltage to the driver), which rotates FSM along two orthogonal axes independently with
microradian resolution. For any given FSM orientations, the two-lens setup effectively
directs a laser beam to the back aperture of the objective with a corresponding incident
angle with respect to the optical axis, hence the input excitation with different angles
focuses at different positions defined by the orientation of FSM.

Figure 2.2 Illustration of the XY scanning mechanism in the ODMR setup.

To determine the relationship between the FSM orientation and the focal position of
the excitation, I first made a calibration table using a standard reflective grading. Figure
2.3 shows a measurement of the standard grating. Based on the grating geometry (grating
10

period 3 m  ) and imaging results (almost 15 grating periods were imaged), I found that
the maximum scanning area in our setup is approximately 50 m  by 50 m  .

Figure 2.3 Reflection image of the standard grating. The geometry of the grating is
provided above. Between -9V and 9V were applied for the X and Y axes to image the
maximum area.

2.2 FL collection efficiency
Based on the lifetime of NV centers 12
LIFETIME
ns   (2), the rate of photon emitted
from a NV center is 40×10
3
photons/ms, however 20-30 photons/ms are typically
observed in our measurement. Here I estimate loss of photons in the system by
considering the loss from the microscope objective, optical components, beam splitter
and APD.
11

Figure 2.4 GCE as a function of NA for a dry objective (blue curve) and oil-immersion
objective (red curve).

The collection efficiency of the microscope objective is from a combination of three
components including a geometrical collection efficiency (GCM, 
GEOM
), attenuation due
to diamond-oil interface ( 
D-OIL
) and aberration effect ( 
ABER
). First, GCE is given by a
ratio between the number of photons collected by an objective and the total number of FL
photons emitted by a NV center located at the surface of the diamond (Fig. 2.5).
Collection of the FL photons by an objective is governed by a semi-angle of the
objective, denoted as  , which is defined as sin NA n   , where NA is the objective
numerical aperture, n is the refraction index of a media between the objective and
diamond. Taking into account that FL emission is uniformly distributed over the angle,
GCE can be written as

1 cos 1
11
4 2 2
GEOM
NA
n




 
     





(2.1)
12

where
2
00
sin ' ' dd

   

is the total solid angle the objective, defined by NA. A high
NA objective gives a high value of 
GEOM
(see Fig. 2.4). In our setup we use an oil-
immersion objective (n = 1.518) with NA = 1.4 with collection angle
0
67.25   ,
therefore 30.08%
GEOM
  .
Second, an oil-diamond interface reduces the collection angle due to refraction and
reflection at the diamond-oil interface (see Fig. 2.5a). The transmission of the FL photons
depends on the photon polarization state and the angle of incidence to the surface, as
shown in Fig. 2.5b. The attenuation due to diamond-oil interface (
OIL D 
 ) is written as the
ratio of the number of photons collected within a reduced solid angle ( 2  ) and the
number of photons collected within a solid angle ( 2  ) (the case with no diamond-oil
interface):

   
 
0
0
i i i
D OIL
ii
Td
d


   

  




(2.2)
where  
i
T  and  
i
 are the averaged over photon polarization state transmission
coefficient and the probability of the FL photon emission at
i
 incident angle to the
diamond surface, respectively;        
1
2
i S i P i
T T T     transmission coefficient for
unpolarized light, where  
Si
T  ,  
Pi
T  are Fresnel transmission coefficients for s- and
p-polarizations (defined in the Fig. 2.5b), respectively;  
i
const   for FL source. The
attenuation coefficient due to the interface for our setup is estimated to be 0.284
D OIL



13

based on the eqn. 2.2 (
0
67.25   ,
0
35.68   , 1.518
OIL
n  and 2.4
DIAMOND
n  ; the
integration in the nominator is performed over the full range of the collection angle
  0,
i
  because the total internal reflection angle of the diamond-oil interface is
0
39.23
TIR
  which is higher than the maximal
i
 ). It is interesting to compare the
attenuation coefficient due to the diamond interface in a case of a dry objective (Fig.
2.5a). For a dry objective of NA = 0.92 (
0.92 1.4
30.4% 30.08%
NA NA
GEOM GEOM


   ), the
attenuation coefficient is estimated to be 0.103
D AIR


 , which is mostly three times
smaller than in a case of oil-immersion objective of NA=1.4, making the oil-immersion
objective more preferable for the detection of the FL of a NV center located below the
surface of a diamond crystal.

Figure 2.5 A) Illustration to describe the FL collection efficiency. Two cases, a NV
center is located on the surface (left) and located below the surface (right) are shown. B)
Transmission coefficient of the FL photon as a function of the incidence angle to the
diamond-oil interface calculated using Fresnel transmission/reflection equations.
Transmission coefficients for s-, p- and randomly polarized light are shown by the blue,
red and black lines respectively.

14

Furthermore, diamond-oil interface introduces the aberration effect on the FL
collimation that results in decrease of collimation efficiency, which was estimated to vary
from 1 to 0.2 for a NV center located at 0 to 10 m  below the surface (41), thus I use
0.2
ABER
  . Important to mention, that two attenuation effects (the diamond-oil interface
and the aberrations), are not present when a NV enter is located at a surface, e.g. a case to
study nanodiamonds. In this case, the FL collection efficiencies of a dry objective and
oil-immersion objectives are mostly identical.
In addition, I included FL signal loss due to the beam splitter ( 
BS
= 0.5), optical
components ( 
OPT
= 0.42) and APD quantum efficiency ( 
APD
= 0.7) in the estimate.

Figure 2.6 Summary of NV center FL attenuation in the setup. NV center (black dot)
located in diamond (yellow color) under green laser light illumination (not shown) emits
FL photons in 4  solid angle (red circle), which are collected by the objective and
directed by the optical components into APD. Number of FL photons emitted by a single
NV center and detected by APD is estimated to be 100 photons per millisecond.

Figure 2.4 shows the total estimated collection efficiency
TOTAL
 and the partial
contributions. My estimate shows 
total
= 0.0026 of the total efficiency corresponding to
~100 photons/ms of the detected photons in our system. The estimated value is still 3-5
15

times different from our experimental observation of 20-30 photons/ms. A possible
reason is that laser excitation power of the NV center is smaller than that used in this
estimate. Discussion about the excitation power and the FL intensity is given in Chapter
4.
2.3 Spatial resolutions of NV FL signals
A FL image of NV centers in diamond is shown in Fig. 2.7a. The image was obtained
by measuring the FL signal with scanning the X and Y position of the laser beam. The
integration time of the FL measurement at each pixel is 15ms. The FL profile along the
dashed white line indicated in Fig. 2.7a is plotted in Fig. 2.7b. A high intensity FL peak at
x = 0.5 m corresponds to the NV center located at the focal plane, while the peak of
lower intensity (~5 counts/ms) is the FL of NV center located out of the focal plane.
Moreover, the estimated size of the bright spot agrees well with predicted size of the PSF
(~0.45µm; see Appendix A.1 for details). The FL peak fits well to the Gaussian profile as
expected. The spatial resolution of the system is defined by the full width at half
maximum (FWHM) of the Gaussian peak, 0.17
lateral
r nm  (Fig. 2.7b).

Figure 2.7 A) 3D contour image of NV centers obtained by scanning the laser beam
across the diamond sample. B) FL profile along white line indicated in A.
16

2.4 NV center tracking
It is important to track a single NV center during experiments to maximize the NV-
FL signals continuously. Due to thermal and mechanical instability in the system, the
relative position of a NV center to the microscope objective often drifts during
experiments, consequently the detected FL signal decreases significantly. The magnitude
of the drift is also highly increased upon high-power microwave excitation, and can
exceed a few µm, which is order of magnitude higher than the lateral and axial
resolutions. Therefore, in order to stably perform experiments long enough, we
implemented a tracking algorithm based on a spatial scan of a small volume and
continuously (looped) centering of the NV position.

Figure 2.8 Tracking algorithm. The green dots represent the center of PSF, the red dot
represents the location of a NV center and the black dot is the center of the scanning.
Arrows indicate the direction of the laser scan. We typically use 15ms of the integration
time for the scanning.
17

In the tracking, we first perform a wide range imaging to find the location of a NV
center as shown in Fig. 2.7, then set the central position of the first tracking scan close to
the NV center (black dot in Fig. 2.8). When the tracking is started, the scan begins from
the point called “start” in Fig. 2.8. During the scanning, the laser beam PSF is discretely
shifted in the XY plane with given step sizes (X
step
and Y
step
, Fig. 2.8 shows nine-point
scanning), then, after one XY plane is scanned, PSF is shifted along the Z axis by the step
of Z
step
. Finally, after performing the scan over all XY planes, the center of the next
tracking scan is set to the position giving the maximum FL intensity. Thus, the positions
of the NV center are redefined. In our setup the typical step sizes are X
step
=25nm,
Y
step
=25nm and Z
step
=100nm.
With the tracking, fluctuations of observed FL intensity are now related to small
discrepancy in the locations of between the NV and the center of the scanning and its
drift over the scanning. Using the scan step and the FL profile shown in Fig. 7b, the
fluctuations is estimated ~1.6% for X and Y directions. The fluctuations along the Z
direction is hard to estimate without knowing the fluorescence profile along the Z
direction, however, if one assumes that axial resolution which is estimated by the laser
light PSF analysis (Appendix A), the fluctuation along the Z direction is 2.7%. Thus, the
total fluctuation is 3.5%. Experimentally we typically observe FL fluctuations of 5 – 10%
while continuously tracking the NV center.
2.5 Antibunching experiment
To verify that a single NV center is addressed, we set up for autocorrelation
measurement, which measures the statistics of FL from the NV center. Principle of
18

Hanbury-Brown-Twiss (HBT; sometimes called “start-stop”) experiment is illustrated in
Fig. 2.9.

Figure 2.9 HBT experiment used to identify a single NV center. The experiment
measures the statistics of a single photon source using TCSPC module and two APDs.

FL in the detection system is divided by a beamsplitter into two beams which are
detected by two APDs. Photon detected by the APD results in a TTL pulse output from
the APD. The output of APD1 is connected to the “start” channel of the TCSPC module,
while APD2 output is connected to the channel called “stop”. In this arrangement, TTL
from APD1 starts TCSPC measurement, TTL from APD2 stops the measurement. The
TCSPC module restarts the measurement at each “start” pulse received if no “stop”
signal, and doesn’t initiate the measurement if no “start” signal prior “stop” signal. For
autocorrelation, we use a single event measurement mode in TCSPC which records delay
( ) between the start and stop events and the number of the events for  .
By considering photon number statistics of a light source, signals of the HBT
measurements are characterized by the light intensity fluctuations, which is often
described by the second order correlation function,
19

 
   
   
(2)
I t I t
g
I t I t






(2.3)
where   It means the intensity of the light at time t , ... means time-average over a
long time period. In the case of a classical light source, the second order correlation
function  
(2)
01 g  , where the equality stands for a perfectly coherent light source,
however their photons arrive randomly, while  
(2)
01 g  means that photons tend to be
bunched in time. On the other hand, an antibunched light is described as  
(2)
01 g  . For a
single quantum emitter, the probability of two-photon emission at the same time is zero,
that results in  
(2)
00 g  , whereas in a case of two independent quantum emitters,
 
(2)
1
0
2
g  . Therefore, measurement of  
(2)
1
0
2
g  proves detection of a single NV
center.

Figure 2.10 HBT data. The number of events collected with time delay between two
photons . Time resolution of  , namely time bin width, is software selectable and can be
as small as 1ps for modern TCSPC module.

20

Raw data of HBT experiment is shown Fig. 2.10. An additional time shift (~4 m in
the stop channel) was introduced using a BNC cable corresponding to ~20 ns shift of the
origin as shown in Fig. 2.10. The measurement was performed up to  ~ 2µs to observe
the Poissonian light (  
(2)
1 g  ). We estimated
(2)
0.3 0.5 g  at  = 0 by calculating
the ratio of the number of events at between  = 0 and  = 2µs (non-zero probability at
zero delays is usually regarded as background fluctuations). This confirmed that the
observed FL signal is emitted by a single NV center.
21

Chapter 3
ODMR measurements of a single NV center in diamond
Our ODMR experiment measures ESR signals of a single NV center by detecting FL
signals. A small optical excitation volume in our setup and a low concentration of NV
centers in diamond allow optical address of a single NV center as described in Chapter 2.
The optical excitation is also used to initialize the NV center and to measure the spin state
by means of the FL signal (Chapter 1). Here I discuss applications of continuous-wave
(cw) ODMR and pulsed ODMR experiments to observe ESR signals of a single NV
center.
3.1 Continuous wave ESR measurements
In cw-ODMR measurements, a NV center is continuously excited by optical and
microwave excitation with a constant power, and the FL intensity is measured by APD
while the frequency of the microwave is swept. When the frequency of the microwave is
far from ODMR (01  in the ground state), the NV spin undergoes only optical
cycling under continuous optical excitation resulting in ~80% probability for 0 state.
However, when the excitation is resonant, the population of the NV center is redistributed
between the resonant sub-levels, consequently, decreasing the intensity of FL signal.
22

cw ODMR measurements with zero and finite external magnetic field are shown in
Fig. 3.1. The NV center was continuously tracked during the experiments. Each point on
the measurements was recorded with 10 averaging. In a case of the zero external
magnetic field (Fig. 3.1a), only a single ODMR signal was observed due to the
degeneracy of 1  states. With application of external magnetic field, the degeneracy is
lifted. As the result, ODMR signals for the transitions 01  and 01  were
observed as shown in Fig. 3.1b. The strength of the magnetic field was estimated as 8.7
Gauss based on the frequency shift between the resonances.

Figure 3.1 Single NV center cw-ODMR: a) no external magnetic field and b) external
field are applied. The intensity is normalized by the FL intensity when the microwave
frequency is far from the resonance. , The resonance fields were obtained by fitting the
data with a two-Lorentzian function.

Furthermore, I estimated the noise of cw-ODMR measurements by calculating the
root-mean square deviation of the FL data from the fit. The noise levels of 5.1% and
6.4% were obtained for the case of zero and finite external magnetic field respectively.
Expected noise levels due to the tracking (estimated in Chapter 2) are 11.6% and 17.5%
23

for the zero and finite magnetic field, respectively. Thus, the noise levels on observed
ODMR are below the estimated levels.
3.2 Rabi oscillations measurements
Here I study the transitions of 01  with application of an external magnetic
field where 0 and 1  states can be effectively considered as a two-level system. Then
resonant microwave excitation with various pulse lengths is applied to observe
population oscillations between the two states, known as Rabi oscillations (42).
For the Rabi oscillation experiment, we apply a pulse sequence illustrated in Fig.
3.2a. With application of the first optical pulse ( 2
IN
s   ) and a subsequent dark
interval of
1
6
dark
s   , a NV center spin is polarized into the 0 state. A resonant
microwave excitation pulse with the duration time
MW
 then induces the transition
between the two levels. The population in the 0 state is measured by applying the
second optical (readout) pulse of 300
RO
ns   and by measuring the FL photons by APD
with
RO
 time interval. Fig. 3.2b shows the FL signal as a function of microwave
duration ( 
MW
). The frequency of the microwave excitation was set at resonance of the
01  transition. The measurement was repeated 10
6
times resulting in 300ms
integration time for each microwave duration time
MW
 . The FL signal is normalized by
the total integration time and presented as a photon count rate.
24

Figure 3.2 Rabi oscillation measurement of a single NV center. A) Sequence of optical
and microwave pulses used in the Rabi measurements. B) NV center FL signal as a
function of microwave duration. (FL signal – black squares, fitting to experimental data –
red curve). C) Probability of the m
s
=0 state in the NV center as a function of microwave
duration time.

The maximum value of the FL signal corresponds to the NV at the 0 state, whereas
the minimum FL signal corresponds to the NV at the 1  state. The Rabi measurement
determines the period of π and π/2 pulses. Microwave pulses with an arbitrary duration
prepare the NV into a superposition of 0 and 1  states that is expressed by
    cos / 2 0 sin / 2 1
MW MW
i      , where  is a Rabi frequency that is proportional
to the square-root of microwave power
MW
P . Thus, the population in the 0 state as a
25

function of microwave duration is  
2
cos / 2
MW
  . By fitting the data to cos
2
(  
MW
), I
obtained the Rabi frequency 2 10.5MHz     . The contrast of the Rabi oscillations
determines that FL contrast between 0 and 1  is ~5 counts/ms. From Fig. 3.2c, the
noise level of the Rabi signal is estimated as 13% ( the noise level as much as 20-30% is
also often observed). In Chapter 4, I discuss an estimated signal-to-noise ratio (SNR) in
our Rabi measurement to compare with the experimental data.
3.3 Spin echo measurements
Many pulsed ESR techniques are based on coherent evolution of spin states with
application of microwave excitations; however, noisy local environments around the
target spin often destroy the coherent state. This effect is often characterized by a
coherence time that measures the life time of the spin coherent state. In order to improve
the coherence time, many pulse sequences have been developed. One of the pulse
techniques that cancels out an effect of static and quasi-static environmental noises is
spin echo (SE) technique, which was proposed by Erwin Hahn in 1950 (43).
A pulse sequence of the SE measurement for a single NV center is illustrated in Fig.
3.3a. Similarly to the Rabi experiment, the optical excitation initializes the NV center into
the 0 state, and readout of the NV state is done by application of the second optical
excitation. In the SE measurement, three microwave pulses are applied (π/2-τ-π-τ- π/2),
where τ is the free evolution interval. The first two microwave pulses are same as that in
a conventional SE measurement. The final π/2 pulse converts the echo of the coherence
state into the population difference. The frequency of the microwave excitation is set to
26

the allowed transitions of NV center 01  . Durations of microwave π/2 and π
pulses are determined from Rabi oscillation experiments.

Figure 3.3 SE measurement. A) Pulse sequence used in the SE measurements. B) FL
signal as a function of the free evolution interval (black squares). Red curve is a single
exponential fit to the experimental data. C) Probability to occupy the 0 state as a
function of the free evolution interval, which was obtained by normalizing the SE signal
(black squares - experimental data, red curve – single exponential fit).

During the SE measurements, the spin state of the NV center with application of the
three microwave pulses with the free evolution interval τ is expressed as,
 
 
 
 
2 1 2 1
1
1 0 1 1
22
ii
i
ee
    
    , where  
0
i B i
g b t dt

 

is phase accumulated
27

during i-th interval of free evolution,  
i
bt is a magnetic field at NV center. Upon
application of the second optical pulse, the probability to occupy the 0 state is
measured, which reads as    
21
1
1 cos
2
  . The pulse sequence with a fixed evolution
interval is repeated 10
6
times in order to measure the population in 0 state that can be
expressed as    
21
1
1 cos
2
  , where ... denotes averaging over magnetic fields
due to the environment. In the case of a perfect microwave pulses, the population in 0
is equal to 100% for zero free evolution interval because the NV spin has no time to
experience the noises. The same result is obtained for any given evolution intervals with
static noises at the NV, namely    
12
b t b t const  , because of
21
0   . However,
with fast noises, the accumulated phases during two free precession intervals become
uncorrelated, therefore the population of 0 will be 50%. Here I model the
environmental noises consisting of static components that are canceled out by the SE
sequence and fast components that lead to an exponential decay of SE signal, hence the
signal of the SE experiment is expressed as
2
2
1
1
2
T
e

 
 


(44,45,46), where
2
T is the
decoherence time.
SE data is shown in Fig. 3.3b, where the FL signal is measured as a function of the
free evolution interval. The FL intensity is normalized over the total integration time for
each  and is presented as a FL photon rate in the unit of counts/ms. The SE signal
expressed in units of probability to occupy 0 is also shown in Fig.3.3c. The SE signal
28

was fitted by a single exponential function, as discussed above, resulting in the
decoherence time
2
T of ~2.5µs and the signal amplitude of 2.4 counts/ms that is, as
expected, ~2 times smaller than the Rabi signal (~5 counts/ms). The noise of SE signal
was estimated to be ~15%.
29

Chapter 4
Expected signal-to-noise ratio in pulsed ESR experiments
In Chapter 3, the noise level in our pulsed-ODMR experiments was estimated to be
~13% for the Rabi oscillation measurements and ~15% for the SE experiments. In this
chapter, a quantitative analysis of SNR is considered in order to identify factors that
influence the noise level in our pulsed-ODMR experiments. Improvement of the signal as
well as identification and compression of major noise sources in our setup allows further
enhancement of SNR.
The ODMR setup relies on detection of the NV center FL emission. In order to
estimate SNR, the FL signal is calculated based on the 5 level model of the NV center
introduced in Chapter 1. I also consider two major noise sources contributing to the noise
in the pulsed-ODMR – 1) photocount noise resulting from FL photon number
fluctuations, and quantum efficiency of APD and 2) noise related to the NV center re-
tracking procedure (Chapter2).
4.1 Analysis of expected S/N ratio in pulsed-ESR experiments
In order to calculate expected FL signal in pulsed-ODMR experiments, I model FL
dynamics of the NV center using the five-level system as shown in Fig. 4.1. Taking into
account all possible transitions in the model, the rate equations for all levels are written as
follows
30

   
11 11 33 44 55
1
tP
ex                 , (4.1a)

   
22 22 33 44
1
t
ex              , (4.1b)

   
33 11 22 33
1
t
ex             , (4.1c)

     
44 11 22 44
1
tS
ex               , (4.1d)

55 44 55 t S P
        ,
(4.1e)
where ex is a transition rate from the ground state to the excited state induced by the
optical excitation, ɛ=0.02 is probability of the spin flip excitation, Γ = 77MHz is a
spontaneous decay rate, Γ
S
= 30 MHz is a decay rate to the intermediate singlet state, Γ
P

= 3.3MHz is a polarization rate to the m
s
= 0 ground state,
ii
 is a population in the i-th
state.

Figure 4.1 Five level model of a NV center used in simulations (9). Green solid arrows
are the optical transitions, black dashed arrows indicate vibrational relaxations in the
excited state
3
E, red solid arrows stand for radiative decays, dashed red arrows shows
spin-non conserving radiative decays, solid black arrows are spin sub-level relaxations
through the intermediate singlet states, grey lines are vibrational states in the excited
state, black lines are electronic orbitals of the NV center, “ex” is a excitation rate of the
NV center characterized by a transition dipole of the NV center.
31

In the experiments, a NV center spin is manipulated between |0> <-> |-1>. In the
following treatment, microwave pulses are assumed to be perfect. In our experiments, we
detect the population of m
s
= 0 state. During the Rabi oscillations with
MW
 , the
populations are written as follows

     
22
11 11 22
cos / 2 sin / 2        , (4.2a)

     
22
22 11 22
sin / 2 cos / 2        (4.2b)
where
MW
  (  is Rabi frequency),
11
 and
22
 are populations in the m
s
= 0 and
m
s
= ±1 states after the first dark interval, respectively.

Figure 4.2 Dynamics of FL signal in pulsed-ODMR experiment.
32

Figure 4.2 shows calculated population dynamics in the ground states and transient
FL signal with and without π pulse microwave excitation . In the simulation, I used
similar pulse parameters to our Rabi experiment (the optical excitation rate of
10 ex MHz  ,
11
1/ 3   ,
22
2 / 3   for the initial populations, 2 s of the initialization
pulse, 1µs of the simulation readout pulse instead of 300 ns of the experimental value in
order to display transient evolution of the FL signal for a longer time scale). During the
first optical pulse (the initialization pulse), the population in m
s
= ±1 (state 2 in Fig. 4.1)
is transferred to the m
s
= 0 state (state 1). The FL signal (the sum of populations in state 3
and 4) is increased due to the optical excitation followed by small decrease caused by
decrease of m
s
= ±1 population and later increase due to the polarization effect. When the
optical pulse is over, the population in the ground state is increased on a time scale of the
excited state lifetime, however, the population of m
s
= 0 increases further as remaining
population in the singlet state (state 5) still relaxes into the m
s
= 0 state. During the dark
interval of 6 µs, the populations reach steady values resulting in ~91% spin polarization
in m
s
= 0 (milliseconds of the spin-lattice relaxation time gives no effect here). The effect
of the microwave pulse is seen at 8 µs, where the duration of the microwave pulse and
second dark period in the pulse sequence is neglected in the simulations as it has no effect
in the dynamics although we introduce second dark interval of 5µs in the experiment in
order to separate the microwave pulses and the subsequent optical readout pulse. In a
case of no microwave pulse, the intensity of the FL signal upon the readout pulse is
mostly governed by the population of m
s
= 0 as can be seen in Fig.4.2. When the
microwave π pulse is applied, the populations of the spin states are exchanged (eqn. 4.2)
right before the readout pulse, leading to the overall lower intensity of FL. As seen in
33

Fig.4.2, the resultant transient FL signal within the readout pulse depends on the NV spin
state. Important to note that the width of the optical polarization pulse shows no
importance on the FL signal because even with a small optical excitation power, the
maximum spin polarization will be achieved on the order of 10 repeated pulse sequences
that is negligible comparing to the measurements in our experiments (~ 10
6
times).
However, the population dynamics and the FL signal highly depend on the power and
period of the optical readout. Dependence of the optical readout pulse to SNR is
investigated below.

Figure 4.3 Illustration of Rabi signal and contrast definitions.

For the Rabi experiment, the Rabi signal is given by difference between averaged
excited state populations over the readout pulse interval with and without application of
the microwave π pulse. Thus, the Rabi signal is written as follows

       
0
1
, 0; , 1; ,
RO
RO ex s ex s
RO
Rabi signal rate ex m t ex m t ex dt

  

    

, (4.3)
34

where      
33 44
; , ; , ; ,
ex
t ex t ex t ex        is obtained from the rate equations (4.1)
with the initial condition defined by the first polarization pulse and a microwave  pulse
applied (In eqns 4.2, 0   ( ) corresponds to with (without) microwave π pulse),
RO
 is
the readout pulse duration. With use of Eqn.4.3, the detected FL photons for the total
integration time of
RO
N  is given by

       
0
, 0; , 1; ,
RO
RO ex s ex s
Rabi signal ex N m t ex m t ex dt

         

, (4.4)
where  is a radiative decay rate from the excited state,  is a collection efficiency of the
optical setup, N is the number of pulse sequence applied to NV center for a fixed
experimental parameter. Furthermore, it is convenient to define the contrast in the
measurement as a ratio of the signal to the Rabi signal with no microwave excitation

 
     
 
0
0
0; , 1; ,
,
0; ,
RO
RO
ex s ex s
RO
ex s
m t ex m t ex dt
C ex
m t ex dt





   




. (4.5)
The definition of signal and contrast is illustrated in Figure 4.2, where single Rabi
oscillation period is calculated for pulse sequence used in Fig.4.1 (optical excitation rate
of 10MHz and readout pulse width of 1µs). Dependence of the optical power and readout
width is shown in Fig. 4.4. As expected, the signal highly depends on the optical
excitation power and the readout pulse width. The signal is higher for a strong optical
excitation. In order to increase the signal, a high excitation power and a readout pulse of
100-500ns are preferred. Important to note that the SE signal comparing to the Rabi
35

signal is reduced by a factor of two (Chapter 3). In order to estimate SNR, the noises are
described next.

Figure 4.4 Calculation results of FL signal and contrast in Rabi experiment.

Here I consider two major noises in the system; 1) the photon counting noise
governed by quantum nature of the FL signal, namely photon number fluctuations of a
NV center FL emission, and noise imposed on the FL signal by detector due to
inefficiency of single photon detection by APD and 2) re-tracking noise arises from the
error of NV center position re-tracking procedure, which is used in-between subsequent
measurements in pulsed-ODMR experiments (Chapter 2, Chapter 3).
In general, in order to calculate noise of FL photon emission, exact knowledge of the
NV center emission photon statistics is required. However, the upper bound of the photon
emission noise can be estimated as of Poissonian light because a NV center is a single
photon emitter with a photon number distribution described by sub-Poissonian statistics.
It is well known, that sub-Poissonian distribution is always narrower than Poissonian
photon number fluctuations. Thus, it is safe to describe photon number fluctuations of the
36

NV FL emission
NV
n  as the variance of Poissonian photon source
Poisson
n  that is
related to the average emission photon number n as
Poisson
nn  . In this respect, the
noise of the NV center FL signal is estimated as the noise of Poissonian light

   
0
, 0; ,
RO
NV RO ex s
Noise ex N m t ex dt

     

, (4.6)
For APD, the detected photon number fluctuations are described as
     
22
2
1
APD APD
n n n         (47,48) where n  is the photon number fluctuation,
n is the average photon number
APD
 is the quantum efficiency of APD (in our setup we
use APD of 0.7
APD
  ). Again, if NV center is approximated as Poissonian source
(
NV
nn  ), the noise of photon counts of APD is estimated as

   
0
, 0; ,
RO
APD RO NV ex s
Noise ex Noise N m t ex dt

        

, (4.7)
With the use of Eqns. 4.4 and 4.7, SNR is found

 
     
 
0
0
0; , 1; ,
,
0; ,
RO
RO
ex s ex s
APD RO
ex s
m t ex m t ex dt
N
SNR ex
m t ex dt







   





.
(4.8)
The re-tracking procedure introduces additional fluctuations of FL signals originated
from positioning errors of the NV center at the center of the laser focal, leading to the
laser intensity fluctuations at the NV center. The laser intensity fluctuations were
estimated as 3.5% in Chapter 2. Therefore, FL fluctuations are estimated based on the
37

maximum shift of the FL due to 3.5 % changes of the excitation intensity. The re-tracking
noise is defined as

         
0
, 0; , 0; , 1
RO
TR RO ex s ex s
Noise ex m t ex m t ex dt

        

, (4.9)
where 0.035  . Furthermore, the signal to the re-tracking noise ratio (SNR
TR
) is
expressed as

 
     
       
0
0
0; , 1; ,
,
0; , 0; , 1
RO
RO
ex s ex s
TR RO
ex s ex s
m t ex m t ex dt
SNR ex
m t ex m t ex dt





   

    


. (4.10)
Finally, because the photon count noise and the re-tracking noise are statistically
independent with a low laser power fluctuations (the present case), the total noise is given
by

 
22
,
TOTAL RO APD TR
Noise ex Noise Noise  , (4.11)
thus, the total SNR is

 
22
,
TOTAL RO
APD TR
Signal
SNR ex
Noise Noise
 

,
(4.12)
where Signal ,
APD
Noise and
TR
Noise are given by expressions (4.4), (4.8) and (4.10),
respectively.
Calculation results for SNR
APD
, SNR
TR
and SNR
TOTAL
are shown in Fig. 4.5. The
calculations were performed using Eq. 4.8, Eq. 4.10 and Eqn. 4.12 for SNR
APD
, SNR
TR
and SNR
TOTAL
, respectively, with N = 10
6
and ɣ = 0.0061 that was estimated from time-
transient FL measurements (Appendix B). As seen in Fig. 4.5, >100 of SNR
APD
, SNR
TR

38

and SNR
TOTAL
are expected for an optical excitation rate higher than 50MHz, 250MHz
and 500MHz respectively, and for the readout pulse duration in the range from ~100-
1000ns that have mostly no effect on SNRs. Thus, SNRs in pulsed-ODMR experiments
can be improved effectively by optimizing the optical power and the readout pulse
duration.

Figure 4.5 SNR
APD
, SNR
TR
and SNR
TOTAL
in pulsed-ODMR experiments estimated for
our setup.

39

4.2 Comparison between the analysis and experimental data
Our pulsed-ESR data was taken with 300ns readout pulse and the optical excitation
rate of 5MHz (see Appendix B for details). With this condition, SNR
TOTAL
is calculated
as 8 (see inset in Fig. 4.6) resulting in noise level of 12.5% and 25% for the Rabi and SE
experiments respectively, that is in a good agreement with experimentally observed SNR
- 10-20% for Rabi and 10-30% for SE experiments (Chapter 3).

Figure 4.6. SNR
APD
, SNR
TR
and SNR
TOTAL
as a function of the laser power (The profile
of SNRs along 300
RO
ns   in Fig. 4.5). The readout pulse duration is 300 ns. The inset
shows close-up of SNRs at a low laser power. Laser power is given in units of optical
excitation rate.

4.3. Limiting factor of SNR and outlook to improvements
As seen in Fig. 4.6, SNR
TOTAL
is largely limited by the re-tracking noise for our setup.
The re-tracking errors arise from precision of instrumentation involved in tracking which
40

induces up to 40 nm of the PSF shift in lateral directions and up to 100 nm along the
optical axis.. Therefore, if the PSF shift is confined within a few nm using commercially
available high precision piezo-positioner, the re-tracking noise will be reduced down to
0.01%. If the tracking noise is suppressed, SNR
TOTAL
will be limited by the photocount
noise, therefore, SNR
TOTAL
as high as 100 (total noise signal of 1%) can be readily
achieved by increasing optical excitation rate of NV center to 50MHz (Fig. 4.6 for
SNR
APD
) meaning 10 time increase of laser power in the system. As indicated in Fig. 4.6,
noise signals can also be suppressed to less than 1% with the optical excitation rates of >
550MHz. However, such excitation rate will require the laser power of ~250mW that
would also increase of background fluctuations by 100 times and reduce the spatial
resolution of our ODMR setup.
41

Conclusions
In summary, we developed an ODMR system with ~200 nm of the spatial resolution.
Using the ODMR system, detection of a single NV center was successfully demonstrated
through FL imaging, autocorrelation measurement and cw ODMR spectroscopy. We also
performed pulsed ODMR including Rabi oscillation and SE measurements. 10-20% of
the noise level was observed in the Rabi oscillation.
SNR of the Rabi experiments was analyzed in the thesis. I considered two noise
sources – photocount noise and noise due to re-tracking procedure. I also considered
dependence of the optical excitation power and readout pulse duration on the signal. My
analysis of SNR with parameters for our ODMR setup shows the upper bound of the
noise levels of ~12.5%. This agrees well with our observation of 10-20%. The analysis
shows that the re-tracking dominates the noise in the system, and this noise can be
suppressed by improving a tracking procedure. The analysis also shows that the signal
intensity can be enhanced ~10 times by increasing the excitation laser power ~10 times
and by optimizing the readout pulse duration in the range of 200-800ns. With those
improvements, the noise level in the pulsed ODMR will be suppressed down to ~1%.

42

Appendix A
Estimate of the spatial resolution of NV-FL imaging
In confocal microscope, laser beam is focused by an objective into finite size volume
that is defined by electric field distribution and is usually called as a point-spread
function (PSF; see Fig.A.1a). Dimensions of PSF set the limit on the spatial resolutions
of optical imaging system.

Figure A.1 Kirchhoff integral A) Excitation beam incident onto back aperture of the
objective is focused to a finite volume, which is described by PSF. B) Explanation of
Fresnel-Kirchhoff integral: electric field close to the focal plane of the objective at a
given point (x
f
,y
f
,z
f
) is a superposition of spherical waves traveled from the exit plane
(x,y,0), where electric field distribution is given by objective lens transformation of
electric field at entrance plane (x
0
,y
0
,0).

Electric field distribution of a focused laser beam can be calculated with the use of
Fresnel-Kirchhoff diffraction integral (49):
43

   
 
 
exp
, , , ,0 cos ,
f f f f
ikr
i
E x y z E x y n r dxdy
r 
   
   



,
(A.1)
where   , E x y ,
 
,
f f f
E x y are complex electric field amplitudes at the exit plane and in
the vicinity of the focal plane of an objective, respectively (Fig. A1.b), k - wavenumber
of the laser light, r radius vector from the point   , xy in the exit plane to the
 
, ,z
f f f
xy point, rr  , n - unit vector parallel to the optical axis.
In Fresnel-Kirchhoff diffraction theory each point in the exit plane is considered as a
source of a spherical wave based on Huygens-Fresnel principle. Therefore, Fresnel-
Kirchhoff integral calculates electric field at a given point
 
,,
f f f
x y z as superposition
electric field of all spherical waves traveled over distance r from a plane   , ,0 xy to
 
,,
f f f
x y z point. Electric field of the spherical waves at the exit plane is described by
complex electric field amplitude   , ,0 E x y , which is related to the excitation beam
electric field  
0 0 0
, ,0 E x y at the back aperture of the objective (entrance plane in Fig.
A1.b) as      
0 0 0
, ,0 , ,0 , ,0 E x y t x y E x y  , where   , ,0 t x y is a transmittance of
objective lens. Second term under the integral in eq. A.1
  exp ikr
r

is responsible for the
change of the complex electric field amplitude   , ,0 E x y due to spherical wave
propagation over the distance r , while   cos , nr projects propagation direction of
spherical wave onto an optical axis, hence taking into account only electric field
components that are orthogonal to the optical axis (scalar Fresnel-Kirchhoff diffraction
theory).
44

Transmittance of objective lens -   , ,0 t x y , can be expressed as
      , ,0 , exp[ i , ] t x y P x y x y   , where   , P x y (pupil function) and   , xy  describe
the transformation of electric field phase and amplitude.
In the limit of a thin lens, the objective modifies the wave front of the laser beam
leaving electric field amplitude unaffected by a lens and can be written as follows (49)

 
 
22
0
exp , ,
, ,0 2
0,
ik
x y x y r
t x y f
elsewhere








, (A.2)
where f and
0
r are objective focal distance and radius of objective back aperture,
respectively.
Taking into account cylindrical symmetry of the beam-objective system and adapting
several approximations, namely Fraunhofer approximation (far field approximation) and
paraxial approximation (   cos , 1 nr  ), the Kirchhoff integral (eq. A.1) with the use of
eq. A.2 takes the form:

     
21 2
2
0
00
, , exp
2
f
iu
E v u WF J v d d

      





, (A.3)
where constants were omitted,
0
r
r
  is a normalized modulus of a radius vector in the
exit plane ( r is modulus of a radius vector in the exit plane) (Fig. A1.b),  polar angle
with respect to the optical axis,   , WF  wave front factor that stands for the phase
distribution of excitation laser beam in the entrance plane,
2
f
v r NA


 and
45

2
2
f
u z NA


 are normalized optical coordinates (
f
r - modulus of radius vector in the
focal plane,
f
z is defocus distance from the focal plane),
0
J is zero order Bessel function
of the first kind.
Electric field intensity distributions in the focal plane and along the optical axis for a
uniform plane wave (   ,1 WF   ) can be calculated analytically

 
 
2
1
2
,0
lateral f
Jv
I I v
v




, (A.4a)

 
 
2
sin / 4
0,
/4
axial f
u
I I u
u




. (A.4b)

Figure A.2 Intensity distribution of uniform plane wave PSF focused by a thin lens
(objective) in the focal plane (left) and along the optical axis (right).

Intensity distributions in the focal plane and along optical axis are plotted in Fig. A2,
in a case of uniform plane wave illumination. Intensity distributions in the focal plane
(XY plane in Cartesian coordinate system) and plane orthogonal to the focal plane (XZ
46

plane) are presented in Fig. A3. As expected from geometric optics, the maximum
intensity of the focused light is observed at the focal point of the objective. Lateral
intensity distribution is called the Airy pattern. Size of the focused spot is usually defined
as a distance between two first minimums, therefore spot size of the focused light in the
focal plane and along the optical axis are given in optical units as 7.66
lateral
d  and
25.14
axial
d  (universal values for any given wavelength of excitation light and NA of an
objective), respectively, what gives (with the use of optical coordinates definition) the
well know result for the size of Airy disc - 1.22
Airy
d
NA

 . Moreover, resolutions in each
direction can be defined by adapting the Rayleigh criteria, hence resolution in the focal
plane and along the optical axis are given as 3.83
lateral
res  and 12.57
axial
res 
(converting from the optical units - 0.61
lateral
res
NA

 ,
2
2
axial
res
NA

 ), respectively,
resulting in 0.23
lateral
res m   , 0.54
axial
res m   for our setup ( 532nm   , 1.4 NA  ).

Figure A.3 Profile of PSF of a focused uniform plane wave in the focal plane (left) and
plane orthogonal to the focal plane (right) in the vicinity of the focal point. Results were
obtained using eq. A3.

47

However, the above results are valid in a case of uniform plane wave incent onto
objective. In our case, the excitation beam is derived from the laser, which produces the
single transverse
00
TEM mode. Electric field distribution of
00
TEM mode can be well
approximated by a 2-D Gaussian distribution function; hence it is called as a Gaussian
beam. Taking into account the distribution of the complex electric field amplitude within
the cross-section of a beam produced by the laser and assuming that Gaussian beam is
focused to the back aperture with the use of two lens system (see Fig. 2.2), the wave front
factor takes the form    
22
, exp /
G
WF w     . Intensity distribution of a focused
Gaussian beam is shown in Fig. A.4. The results are presented for different values of 
parameter, which is defined as
0
G
w
r
  , where
G
w is a Gaussian beam waist.

Figure A.4 Intensity profiles in the focal plane and along the optical axis are shown for a
uniform plane wave, and different ɤ value of Gaussian laser beam.

As seen in Fig. A.4, in a case of Gaussian beam, axial and lateral resolution strongly
depends on  , which can be tuned experimentally by choosing different combination of
lenses in front of the objective (Fig. 2.1). It is well known, in the limit of infinite optics,
48

that that the waist of a focused Gaussian beam is decreasing when expanding the beam in
front of the infinite lens. However, due to finite size of the back aperture of the objective,
the Gaussian beam is truncated for 1   resulting in a power loss and intensity
distribution approaches that of the Airy pattern for a large values of  (only slight
deviation from the Airy pattern is observed in a case of 2   ; see Fig. A.4). In order to
transmit most of the laser power with no effect on spatial resolutions, we expand the
Gaussian waist to the size of the back aperture ( 1   case in Fig. A.4).

Figure A.5 Illustration of focus shift due to oil-diamond interface. Black star indicates the
position of a geometric focus.

Moreover, oil-diamond interface introduces spherical aberrations (Fig. A.5). Effects
of a planar interface of mismatched refractive indexes on Gaussian beam focusing were
studied in (50), where authors used the Debye integral formulated by Wolf and extended
49

the Wolf’s theory by including the transformation of electric field at the planar interface
as well as accounting for reflections with the use of Fresnel formulas. Their theory is
limited to consideration of electromagnetic wave far from the exit plane of the objective.

Figure A.6 Intensity distribution of laser beam focused through oil-diamond interface.
Simulation results are shown for different penetration depth 0µm, 10µm, 20µm from left
to right, respectively

Intensity distribution in the vicinity of focal plane was numerically calculated in the
XZ plane using theory described in (50) for the case of Gaussian beam focused by a high
NA objective through oil-diamond interface (Fig. A.6). Zero on Z axis if Fig. A.6 stands
for a Gaussian beam geometric focus defined by NA of the objective in a case of no
diamond (Fig. A.5 black star) and depth (light penetration depth) is defined as a distance
between geometric Gaussian beam focus and diamond surface, hence the diamond
surface coordinates on Z axis in Fig. A.6 are 0µm, -10 µm and -20µm for the focus depth
of 0µm, 10 µm and 20 µm, respectively. Oil-diamond interface introduces focus shift for
different penetration depth, as seen from the shift of PSF maximum intensity position
50

towards positive Z values for larger depth (Fig. A.6 & A.7), as well as spreads out the
PSF along Z axis leaving PSF size in the plane of shifted focus (position of maximum
intensity) mostly unaffected (Fig A.7). The PSF intensity distribution in the plane of
shifted focus and along the optical axis are plotted for different depth values in Fig. A.7a
& A.7b, respectively. Maximum PSF intensity and shift of focus dependences on
penetration depth are presented in Fig. A.7c.

Figure A.7 A) and B) - lateral and axial intensity distribution for different focused laser
beam penetration depth to the bulk diamond, respectively. C) Maximum intensity and
shift of focus as functions of depth.

In a case of no diamond, there is small difference in PSF intensity distribution
calculated by Kirchhoff integral and Debye integral formulated by Wolf. The difference
arises due to paraxial approximation made in scalar Kirchhoff theory that limits the
consideration to objectives with small NA. In contrast, Wolf’s theory describes the
focusing of Gaussian beam by an objective of high NA aperture as in our case (NA=1.4).
However, results derived using scalar Fresnel-Kirchhoff integral have analytical solutions
for the spot size in focal plane and along the optical axis, what can conveniently serve as
a good estimation when designing an optical system because difference between the
predictions of two theories is negligible (Fig. A.7).
51

When the Gaussian beam is focused through oil-diamond interface, the maximum
intensity of PSF at 10µm depth drops down to 20% of the maximum intensity when there
is no diamond surface (Fig. A.7b,c) and the spot size along Z direction is increased
(Fig.A.7b) limiting the optical axis resolution capabilities of the optical setup down to
5
axial
rm   , hence explains our choice to develop the optical setup based on confocal
microscopy technique, which improves the Z direction resolution with the use of
modified detection system that is discussed below.

Figure A.8 FL signal collection and detection. Excitation light pathway and other optical
components are omitted for clarity. PSF of the excitation light is shown in the vicinity of
objective focal point (green area). FL photon pathways are illustrated for a sources
located in focal point, in the defocus plane and in the focal plane away from the optical
axis as solid red line, blue dashed line and red dashed line, respectively. Tube lens may
be considered as an entrance of the detection system.

The aim of confocal microscopy technique is to improve optical axis resolution in
optical imaging system by directing collected FL light to the pin-hole placed after the
dichroic mirror in the detection system (Fig.2.1). The detection principle behind confocal
microscope is illustrated in Fig. A.8. Photons, which are emitted by a FL source located
in the focus of the objective, are collimated and directed to the detection system. With the
52

use of tube lens the FL beam is focused to the pin-hole located at the plane (confocal
plane) that is conjugated to the focal plane. FL intensity is detected by the detector placed
behind the pin-hole. The purpose of a pin-hole is to discriminated the FL signal of single
FL emitters that are located in the defocused plane resulting in improvement of axial
resolution: FL emitted by a defocused source is focused by the tube lens before the pin-
hole; hence the intensity distribution at the pin-hole is broad, which results in high
attenuation of the signal. Attenuation of the later signal highly depends on the choice of
the pin-hole size. Based on the results obtained in (51), the suggested pin-hole radius
is 0.5
p
vM  , where M is objective magnification, what results in the pin-hole diameter
for our setup - 4.35 m
p
d   , which would improve the lateral resolution by factor of 1.4
( 2.74
lateral
res  / 0.16
lateral
res nm  ) and axial resolution by factor of 2
( 6
axial
res  / 0.31
axial
res nm  ). However, consideration in (51) is limited to the case when
light is focused in the homogeneous media; therefore, it is no adequate for our case of oil-
diamond interface.
The choice of a pin-hole diameter in our setup was governed by a general rule of
confocal microscopy in the simplest case, namely pin-hole size should be equal to the
size of the Airy disc magnified by the objective. In this case, the pin-hole size can be
estimated as 33.4µm. We use 50µm diameter optical fiber, which plays the role of a
conventional pin-hole of the same size. In this case the axial resolution is assumed to be
mostly restored down to 0.54 m  . However, in order to obtain the limits of axial
resolution in our optical setup and the best size of a fiber cable, FL PSF at the fiber
(confocal plane) has to be analyzed for the case of oil-diamond interface!
53

Appendix B
Estimate of typical optical excitation rate of NV center and FL collection efficiency
in our setup
In order to predict SNR ratio that encounters only photon noise in our ODESR setup,
the FL collection efficiency of our setup and typical values of optical excitation power at
NV center are estimated based on the results that were obtained in transient FL
measurements, which were performed using pulse sequence that is shown in Fig.B1.
The pulse sequence is mostly identical to the pulse sequence used in Rabi oscillation
experiments except that detection part of the pulse sequence is modified – microwave
pulse width is fixed during the measurement, second optical pulse is applied for a long
time interval (~5µs) and 50
RO
ns   duration APD pulse, which controls the APD
acquisition time, position with respect to the second optical pulse is varied.

Figure B.1 Optical and microwave excitation pulse sequence used in time-transient FL
measurements
54

The transient FL measurements on single NV center are shown in Fig.B.2for the
cases of no microwave applied and microwave durations of 37.5 ns and 75ns. Time-
transient FL signals were modeled by system of 5 rate equations introduced in Chapter 4,
equations 4.2 and taking into account rising time (~40ns) of optical power in the system
and APD readout pulse width of 50ns. Reasonable agreement between simulation results
and experimental data was found for optical excitation power of 5MHz and
0
70   .
Estimated optical excitation rate of NV center is assumed to be a typical value in our
setup and used for SNR estimation that is performed in Chapter 4.

Figure B.2 Time-transient FL data and simulated result using the five-level model.

FL collection efficiency in our setup can be estimated using the relation
count rate population   that relates actual detected photocount rate at APD during the
transient FL measurement to the population in the excited state. In a case of 5MHz
55

optical excitation rate and no microwave applied, maximum NV center population in the
excited state is ~0.06, while maximum photocount rate is 28photons/ms resulting in FL
collection efficiency 0.0061   . This value is used in Chapter 4 to calculate signal to
photocount noise ratio for our ODMR setup.

56

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Abstract (if available)

Abstract A nitrogen‐vacancy (NV) center is a paramagnetic impurity defect existing in the diamond. Because of its unique properties including long electron spin coherence time, photostability and ability to initialize and detect a single NV center, the NV center is a promising candidate for investigation of fundamental quantum sciences and for applications to quantum information processing devices as well as for a magnetic sensor with single spin sensitivity. The spin state of the single NV center spin is initialized by optical excitation, manipulated by microwave excitation and measured by fluorescence of the NV center. This thesis describes development of an optically detected electron spin resonance (ODMR) system which initializes, manipulates and measures the spin state of a single NV center.

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University of Southern California Dissertations and Theses

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Creator Stepanov, Viktor (author)

Core Title Development of an electron paramagnetic resonance system for a single nitrogen-vacancy center in diamond

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Chemistry

Publication Date 07/15/2014

Defense Date 02/24/2014

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

Tag confocal microscopy,magnetic resonance,microscopy,MR,nitrogen‐vacancy center in diamond,NV center,OAI-PMH Harvest,ODMR,optically detected magnetic resonance,quantum mechanics,single spin

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Takahashi, Susumu (committee chair), Bradforth, Stephen E. (committee member), Qin, Peter Z. (committee member), Wittig, Curt (committee member)

Creator Email bonnanno@gmail.com,stepanov@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-442709

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Identifier etd-StepanovVi-2700.pdf (filename),usctheses-c3-442709 (legacy record id)

Legacy Identifier etd-StepanovVi-2700.pdf

Dmrecord 442709

Document Type Thesis

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Rights Stepanov, Viktor

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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