M a s t e r’ s T h e s i s
Coupling of Two Nitrogen-Vacancy
Ensembles through a Cavity Bus
presented by
Peter Clemens Strassmann
under the supervision of
Dr. Dmitry Krimer1
Prof. Dr. Stefan Rotter1
Prof. Dr. Sebastian Huber2
conducted at the
Institute for Theoretical Physics
of the
Vienna University of Technology
submitted in February 2014
II
1 Institute
2 Institute
for Theoretical Physics, TU Wien, A-1040 Vienna, Austria.
for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerland.
Abstract
This thesis presents a theoretical framework to describe the steady-state transmission through a microwave resonator coupled to two nitrogen-vacancy ensembles.
Two different approaches are discussed and directly compared to experiments
that are presently being carried out in the group of J. Majer. The first approach
relies on a simple scattering model based on a Green’s function technique which
provides a qualitiative description of the setup. The second approach, based on a
quantum optical model, is more accurate as it incorporates quantum mechanical
effects like dephasing in the spin ensembles. We find good agreement between
this quantum optical model and the experiments, which, however, could still
be improved by taking into account the effects of adjacent nitrogen-vacancy
ensembles.
III
IV
ABSTRACT
Contents
Abstract
III
1 Introduction
1
2 Description of the system
2.1 Microwave resonator . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Nitrogen-vacancy centers . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Properties of a single nitrogen-vacancy center . . . . . . .
2.2.2 Nitrogen-vacancy centers forming an ensemble . . . . . .
2.2.3 Spectral distribution of the nitrogen-vacancy center ensemble
2.3 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . .
3
4
4
5
8
11
12
3 Green’s function formalism
15
3.1 Green’s function in general . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Application of the formalism . . . . . . . . . . . . . . . . 17
3.2 Simulations compared to the experiments . . . . . . . . . . . . . 18
4 Quantum optical model
4.1 Tavis-Cummings Hamiltonian and equations of motion . . . . . .
4.2 Steady-state transmission through the cavity . . . . . . . . . . .
4.2.1 Transmission through the cavity coupled to a single ensemble
4.2.2 Dimensionless transmission . . . . . . . . . . . . . . . . .
4.2.3 Transmission through the cavity coupled to several ensembles
4.3 Simulations compared to the experiment . . . . . . . . . . . . . .
23
23
25
25
26
26
27
5 Summary and outlook
35
Acknowledgment
37
V
VI
CONTENTS
Chapter 1
Introduction
In the last decade, so-called “hybrid” quantum systems attracted much interest
since these allow the manipulation of qubits by coupling them to a mode of a
resonator for the coherent processing of quantum information. Several implementations of such setups have been proposed, based on the concepts of nuclear
magnetic resonance [GC97], circuit quantum electrodynamics (QED) [WSB+ 04],
cavity QED [RBH01], and trapped ions [WMI+ 98].
Circuit QED with its superconducting circuits is especially promising for
quantum information processing (QIP) as the circuits and complex composite
structures are fabricated on chips, resulting in advantageous properties in terms
of scalability and hardware integration [DS13]. In contrast to natural atoms,
these systems also provide a high customizability. Several different kinds of two
level systems like charge (Cooper pair box [BVJ+ 98]), flux [MOL+ 99], or transmon [KYG+ 07] qubits have been studied. In all these two level systems, however,
the short coherence time is a severe limiting factor. Currently a lot of effort is
spent on finding an alternative memory with longer coherence time [DS13]. Two
pioneers of the field recently stated that the next development is to gather a
“logical memory with longer lifetime than the physical qubits” [DS13]. Once such
a memory is found, one will have to demonstrate the communication between
such memories, e.g. via a cavity bus [MCG+ 07].
Several systems with longer coherence times are studied to provide a storage
for quantum information. Such systems could be ultracold atoms [HMSR10],
solid-state systems such as molecular ensembles [RDD+ 06] or nitrogen-vacancy
centers (NV-centers) [KOB+ 10]. NV-centers are promising candidates for the
storage of qubits in quantum information processing as their magnetic spin
states provide very long coherence times [SRA+ 12, AKN+ 11]. The current
limit is about 50 s [AKN+ 11] with the theoretical limit being in the range of
100 s [BNT+ 09]. It was recently shown that one can strongly couple an ensemble
of NV-centers to a cavity mode [SRA+ 12].
Distant superconducting qubits have been shown to communicate via a cavity bus [MCG+ 07]. An implementation of the complex teleportation protocol
based on the principle of entanglement of qubits via a cavity bus recently succeeded [SSO+ 13]. After such complex QIP protocols can be implemented with
high fidelity, the need for memory heightens because the number of calculation
steps is limited by the decoherence of the quantum states, i.e. by the qubit
implementation. In this context, we are interested in ensembles of NV-centers as
1
2
CHAPTER 1. INTRODUCTION
a memory. We take large ensembles as their size determines the coupling to the
cavity bus. To understand the interaction between multiple ensembles with such
a cavity is still a topic of current research. The most recent experiments, carried
out by the group of J. Majer, test nitrogen-vacancy ensembles (NV-ensembles)
in two diamond crystals for their interaction via such a cavity bus [Pet13]. The
experiments successfully monitor the strong coupling for each of the ensembles
with the cavity bus. An important step towards a complete understanding of
the coupled system is a theoretical framework that accurately describes the
experimental data. In this thesis, we give preliminary results for the physical
parameters that describe this coupled system similar to what was done before
for one diamond crystal. The present work extends the use of the last model
by embedding the magnetic field dependency. For an improved description the
influence of different resonance frequencies is encountered too.
There are several theoretical approaches available to describe such a complex
system. On the one hand, a simple input output formalism strongly simplifies
the problem. On the other hand, a quantum optical model describes the system
more adequately in view of the fact that it includes effects like dephasing.
One method of applying a simple input-output formalism uses Green’s functions and the Dyson equation. It describes the system components solely based
on the resonance position and width of the collective NV ensemble. Another
advantage of using Green’s functions is that these provide a relation to the
transmission through the system [Dat97].
The steady-state transmission through the quantum system can be written
in terms of the quantum mechanical field operators. Additionally, this procedure includes the broadening of the ensemble, which, following previous analysis,
we consider to be distributed as a q-Gaussian [SRA+ 12]. This continuous distribution can be fully incorporated in our theoretical model which we solve
numerically.
The present study starts by introducing the system and its components in
chapter 2, i.e. the resonator in section 2.1 and the NV-ensemble in section 2.2.
At the end of this chapter the experiments of transmission through this system
are described. Chapter 3 is dedicated to the Green’s function formalism as a
first approach to describe the measured data. With the understanding obtained
by this toy model, a more in depth quantum optical approach is set up which is
described in chapter 4. Chapter 5 summarizes the results and gives an outlook
of future research.
Chapter 2
Description of the system
The system studied in the present thesis consists of the following components:
A superconducting microwave waveguide resonator interacts with the nitrogenvacancy ensembles (NV-ensembles) of two carbon diamond crystals positioned
directly on top of the resonator. The interaction is a magnetic coupling between
the electromagnetic field inside the resonator and the magnetic dipole moment
of each NV-center. This chapter explains these components of the composite
systems. Section 2.1 is dedicated to the resonator. The properties and concepts of
the NV-centers are introduced in section 2.2. Section 2.3 covers the transmission
through this system and the associated measurements.
Figure 2.1: The system setup we consider consists of a resonator and two NVensembles of two diamond crystals.
In order to probe and characterize the system, we study coherent microwave
transmission through the resonator. The microwave signal is sent through the
waveguide which is coupled in and out at two capacitive ends of the resonator.
The resonator is part of a superconducting transmission line fabricated on a chip.
The magnetic field interacts with the magnetic dipole moment of the NV-centers
in the two diamond crystals. Hence the crystals are positioned at the anti-nodes
of the magnetic field in the resonator to maximize the coupling strength. An
illustration of this composite system is provided in Fig. 2.1. The resonator and a
large ensemble of NV-centers are strongly coupled via the electromagnetic field.
3
4
2.1
CHAPTER 2. DESCRIPTION OF THE SYSTEM
Microwave resonator
The coplanar waveguide resonator features resonant waves with corresponding
discrete eigenfrequencies. The nodes of these resonant waves’ magnetic field are
located at the two ends of the resonator. Fig. 2.2 shows the resonator carrying
a standing wave with a wave length of λ, the second lowest resonance mode. At
the two anti-nodes of the magnetic field, this wave provides strong coupling to
the corresponding NV-ensemble due to the high magnetic field strength. The
Q-factor of the second lowest mode is about Q ≈ 6.32 · 104 , which is determined
ν
by the formula Q = ∆ν
from the experimental transmission pattern of the bare
resonator.
Figure 2.2: The superconducting microwave coplanar waveguide resonator,
viewed from the top, has a length that coincides with the wave length λ. The
superconducting material is drawn in gray, i.e. the top and bottom parts indicate
ground whereas in between is the waveguide with the resonator. The resonator is
separated from the rest of the waveguide on each side by an in and out-coupling
capacity. The resonant magnetic field inside the resonator is drawn in red.
One should mention the resonator’s length, l ≈ 4.31 cm chosen such that
the second lowest mode is close to resonance with the NV-center’s transition
frequency. The wave length λ = l corresponds to a frequency of ωcav ≈ 2.91 GHz
for the bare resonator, i.e. without crystals lying on top of it. The frequency
is decreased by the carbon’s dielectric influence. Waves in this frequency range
are also called microwaves. The NV-center’s transition frequency can be tuned
into resonance with this cavity mode. Resonance is achieved by applying an
appropriate magnetic field to the NV-center as shown in section 2.2.1.
An additional way to increase the coupling strength is to increase the spatial overlap between the crystal and the resonator because the magnetic field
decreases with distance. This is designed by forming curves of the resonators
parts with high magnetic field amplitude on the chip as shown in Fig. 2.5.
2.2
Nitrogen-vacancy centers
The nitrogen-vacancy ensembles (NV-ensembles) are realized in a diamond crystal that contains a large number of specific color defects i.e. the NV-centers as
shown in Fig. 2.3. The first section explains the properties of a single nitrogenvacancy center, mainly its Hamiltonian and the resulting transition frequencies
between the eigenstates. The properties of ensembles formed by NV-centers are
2.2. NITROGEN-VACANCY CENTERS
5
discussed in 2.2.2 followed by section 2.2.3, which provides a description of the
spectral distribution of the NV-centers in the ensemble.
Figure 2.3: This figure shows a single NV-center substituted into a unit cell for
the diamond crystal. Diamond has an face-centered cubic Bravais lattice with
a two atomic basis where the atoms are separated by a quarter of the diagonal
through the unit cell. The nitrogen atom is drawn in red and the vacancy
is indicated by a transparent sphere. The vacancy can substitute any of the
carbon atoms (black) on the four adjacent lattice sites of the nitrogen atom.
The boundary of the unit cell is indicated by yellow edges.
2.2.1
Properties of a single nitrogen-vacancy center
A nitrogen-vacancy center (NV-center) is a color defect in single crystalline
diamond that consists of a substitutional nitrogen atom and an adjacent carbon
vacancy. The symmetry axis of the NV-center, which connects the nitrogen
atom and the vacancy (the two defect positions), is called the NV-axis in the
present work. There are four possible NV-axes corresponding to the four nearest
neighboring sites of the nitrogen atom, where a vacancy can be placed. There are
neutral and negatively charged NV-centers, both of which are stable color defects.
The neutral NV-center features five electrons due to dangling bonds adjacent to
the vacancy. The negatively charged NV-centers capture an additional electron
from the bulk [JW06]. Only the single negatively charged NV-centers are relevant
in the present context because these provide a spin-1 ground state with zero-field
splitting in contrast to the neutral NV-centers with a spin- 12 ground state. The
zero-field splitting lifts the degeneracy of the energy levels up such that a small
magnetic field is sufficient for manipulation of the transition frequencies which
are discussed in the following subsection. The negatively charged NV-centers
also exhibit optically detected magnetic resonance [GDT+ 97, SCLD14]. For
simplicity, we will leave away the term “negatively charged” in the following.
The property we are most concerned about is the NV-center’s magnetic
moment, which is effectively described by a spin-one state of the electron configuration. The transition frequencies of the spin-one state are crucial later on for
the coupling to the microwave resonator’s magnetic field and are determined by
the Hamiltonian, which is explained in the following subsection.
6
CHAPTER 2. DESCRIPTION OF THE SYSTEM
Hamiltonian of a single nitrogen-vacancy center
The lowest part of the electronic spectrum of such a single NV-center can be welldescribed by a spin S = 1 state. We take an external homogeneous magnetic field
into account, which shifts the transition frequencies between the eigenenergies
of the Hamiltonian, which we are interested in.
The Hamiltonian for a single NV-center can be written as
HSpin = HEZI + HZFS ,
(2.1)
where the first and second term describe the Electron Zeeman Interaction and
Zero-Field Splitting, respectively.1 The Hamiltonian that describes the electron
Zeeman interaction is given as
HEZI = µe B · Ŝ
(2.2)
with the magnetic moment of the electron µe = ge µB , the external magnetic
field vector B and the spin operator in vector form Ŝ. The Hamiltonian of the
zero-field interaction is given as
HZFS = h DŜz2 + E(Ŝx2 − Ŝy2 )
(2.3)
with the longitudinal zero-field splitting D and transversal zero-field splitting
E [HW04]2 . The value of D is temperature dependent whereas a finite value E
corresponds to breaking the NV-center’s C3v symmetry by stress in the crystal
(or an applied electrical field) [Ams12]. In the spin 1 basis |−1i , |0i , |+1i, the
Hamiltonian has the explicit form


µe
√
D − µe Bz
(Bx − iBy )
E
2
µ
µe
√
0
(Bx − iBy )
(2.4)
HNV =  √e2 (Bx + iBy )
.
2
µe
√
(B
+
iB
)
D
+
µ
B
E
x
y
e
z
2
The eigenvalues of the Hamiltonian at vanishing magnetic field depend on
the longitudinal and transversal zero-field splitting only. The eigenenergies to
the corresponding eigenstate are 0 Hz at |0i, D + E at √12 (|+1i + |−1i), and
D − E at √12 (|+1i − |−1i), which are depicted in Fig. 2.4 with the connecting
wiggled lines describing the transition frequencies. A finite zero-field splitting
lifts the degeneracy of the three eigenenergies.
1 The interaction with the nuclear spin (e.g. via hyperfine interaction) is neglected at this
point. The nuclear magnetic moment has a small contribution as the large majority of all
carbon and nitrogen atoms have vanishing nuclear magnetic moments. Hence this contribution
is ignored at this point.
2 Eq. (2.3) is derived from the formula


Dx
0
0
Dy
0  Ŝ.
HZFS = hŜT  0
0
0
Dz
We redefine the zero-field splitting tensor’s components. The longitudinal D = 32 Dz and
transversal component E = 12 (Dx − Dy ) by using the normalization Dx + Dy + Dz = 0.
Hence the Hamiltonian adapts to
1
HZFS = h D Ŝz2 − Ŝ2 + E(Ŝx2 − Ŝy2 ) .
3
The term − 31 Ŝ2 is ignored for simplicity as it only affects the energy offset. It originates from
the separation in longitudinal and transversal parts.
7
2.2. NITROGEN-VACANCY CENTERS
Energy
h̄(D + E)
E
h̄D
E
h̄(D − E)
√1
2
(|1i + |−1i)
√1
2
(|1i − |−1i)
D
0J
|0i
Figure 2.4: The eigenenergies and eigenstates at vanishing magnetic field B = 0.
The components of the external magnetic field in Eq. (2.4) are written in the
frame of reference of the NV-center’s orientation. In the following, the magnetic
field will be transformed into the frame of reference of the lab. The orientation of
the crystals on the chip deviates only in the horizontal plane and is accomplished
by a rotation of the magnetic field in spherical coordinates. The magnetic field’s
z-component is given by the direction of the NV-axis. The four possible NV-axes,
in the crystal’s reference frame coordinates, are
 
 
 
 
1
−1
−1
1
1
1
1
1
zA = √ 1 , zB = √ −1 , zC = √  1  , zD = √ −1 ,
3 1
3
3 −1
3 −1
1
The x- and y-component are defined by the convention xA k (−1, −1, 2) and
yA k (1, −1, 0) [Ams12] and the others are symmetrically analogous. Hence
the corresponding unitary transformation matrices from the crystal’s to the
NV-center’s frame of reference are




−1 −1
2
1
2
√
√1
√
1 √
1
3 − 3 0 ,
3
0 ,
UA = √
UB = √ −√3
6 √2 √2 √2
6 − 2 −√ 2 √ 2




1
−1
−2
−1 √1
−2
√
√
√
1
1
0 ,
0 .
UC = √ −√3 −√ 3
UD = √ √3
√
√3
√
6 − 2
6
2 − 2
2 − 2 − 2
The magnetic field is


sin(θB ) cos(φB − φcrystal )
B = B  sin(θB ) sin(φB − φcrystal ) 
cos(θB )
in the crystal’s reference frame. The angles θB and φB determine the orientation
of the magnetic field in the lab frame as in Fig. 2.5 and the angle φcrystal uniquely
determines the orientation of the crystals within the lab frame as the crystals
have no vertical deviation angle. We assume that the magnetic field amplitude
lies in the horizontal plane too, i.e. θB = 90◦ .
8
2.2.2
CHAPTER 2. DESCRIPTION OF THE SYSTEM
Nitrogen-vacancy centers forming an ensemble
φcrystal,2
φcrystal,1
The setup, as the top view shows in Fig. 2.5, contains two differently oriented
crystals (in the lab frame, see last section) containing large ensembles of NVcenters. The first paragraph clarifies what is ment by “large” ensembles whereas
the second paragraph gives insight to the chosen crystal orientation. Afterward
the different ensemble orientations are discussed corresponding to the single
NV-center’s orientation dependent on the magnetic field angle and strength.
x
B
φB
y
Figure 2.5: This figure shows the design of the resonator and the crystals positioned on top of it. The orientation of the external magnetic field and the
NV-centers stay in this plane.
The diamond crystals in the experimental setup contain about N = 1012
NV-centers involved in the coupling (which is smaller than the total number in
the crystal) in order to enhance the coupling strength as described by Amsüss
et al. [AKN+ 11]. Ensembles of NV-centers feature an effective coupling to the
cavity that scales with the square-root of the number of NV-centers [HMSR10]
as inherent in the quantum optical model in chapter 4. This increase in the
coupling strength is crucial for entering the strong coupling regime and motivates
why we are dealing with a whole ensemble of NV-centers.
The horizontal orientations of the diamond crystals φcrystal,1 = 15◦ and
φcrystal,2 = −9◦ , as in Fig. 2.5, differ from each other such as to lift the degeneracy
between both spin ensembles. In fact the exact angles are picked at random. If
the two crystals (with all spin ensembles) were aligned with each other, both
crystals
√ would appear as a single crystal with enhanced coupling strength of a
factor 2. This enhancement is due to the fact that the coupling strength scales
with the square root of the number of NV-centers with the same orientation (as
mentioned before and explained in section 4.2.3).
Fig. 2.6 depicts the transition frequencies as a function of their angular
dependence for zero-field splitting with longitudinal component D = 2.8807 GHz
and transversal component E = 10 MHz. The transition frequencies of different
ensembles cross each other (called degeneracy) at different angles, e.g. at a
magnetic field with a horizontal angle of φB = 48◦ . At this angle both crystals
can be viewed as a single crystal with twice the number of spins. Hence the
9
2.2. NITROGEN-VACANCY CENTERS
Spin Frequency ωs [GHz]
3.1
8
7
6
5
4
3
2
1
3
2.9
2.8
2.7
0
20
40
60
80
100
120
140
Horizontal Angle φB [degrees]
160
180
Figure 2.6: The eigenenergies of both crystals are shown in this figure at constant magnetic field B 6= 0 as a function of the horizontal angle of the magnetic
field φB . The transition frequencies between these eigenenergies of the corresponding crystal are shifted exactly by the amount of the crystals’ orientation,
i.e. φcrystal,1 = 15◦ and φcrystal,2 = −9◦ . In this graphic the magnetic field
strength is assigned to a fix value of 6.1 mT. The transition frequencies show
degeneracies, e.g. at an angle of 48◦ . The dotted vertical lines at 48◦ and 66◦
describe the angles corresponding to Fig. 2.7a and Fig. 2.7b, respectively.
√
coupling strength is enhanced by a factor of 2.3 In turn, the field is far from
degeneracy at a horizontal angle of φB = 66◦ . The angles φB = 48◦ and
66◦ are chosen according to the data provided by the experiments. Each of
the four pairs that form an avoided crossing correspond to the same NV-axis’
direction. We will refer to those NV-centers with NV-axes pointing in the
same direction as a “subensemble”. Fig. 2.6 additionally shows the avoided
crossing due to the degeneracy of the spin transition frequencies of the NVcenters that point in the same direction. The coupling strength is enhanced at
the degeneracy of different subensembles’ transition frequencies but these do not
display an avoided crossing. There are further degeneracies between transition
frequencies of different subensembles at angles of φB = 3◦ , 93◦ , and 138◦ . The
180◦ -periodicity is due to the symmetry of the square of the sine and cosine
function. The subensembles with an orthogonal orientation in the xy-plane are
therefore shifted about the angle φB = 90◦ .
The transition frequencies of the NV-centers are shown in Fig. 2.7 for the
same zero-field splitting parameters as in Fig. 2.6 but dependent on variable
magnetic field strength. Fig. 2.7a and Fig. 2.7b cut through Fig. 2.6 at the two
dotted lines at 48◦ and 66◦ , respectively. As indicated in Fig. 2.6, always two
subensembles are degenerate with each other at an angle of 48◦ .4 The red (1)
3 Consistently
the symmetry appears in the Hamiltonian too. For the exemplar case of the
degeneracy at 48◦ , the equality cos(45◦ ± ∆φ) = sin(45◦ ∓ ∆φ) for ∆φ = φB − 45◦ − φcrystal
is applied. Hence the field component By vanishes and the others Bx , Bz stay the same in
the NV-center’s reference system.
4 Each of the depicted lines in Fig. 2.6 and Fig. 2.7b is still degenerate by two subensembles
10
Transition frequency ωs [GHz]
CHAPTER 2. DESCRIPTION OF THE SYSTEM
3.2
7,8
5,6
3,4
1,2
3.1
3
2.9
2.8
2.7
2.6
0
3
6
9
Magnetic field B [mT]
12
15
Transition frequency ωs [GHz]
(a) The magnetic field is pointing in the direction of φB = 48◦ .
3.2
8
7
6
5
4
3
2
1
3.1
3
2.9
2.8
2.7
2.6
0
3
6
9
Magnetic field B [mT]
12
15
(b) The magnetic field is pointing in the direction of φB = 66◦ .
Figure 2.7: The transition frequency of the NV spin states ωs is shown as
a function of the magnetic field strength at horizontal angles φB = 48◦ and
φB = 66◦ in Fig. 2.7a and 2.7b, respectively, (i.e. θB = 90◦ ). The different colors
correspond to different spin ensembles.
11
2.2. NITROGEN-VACANCY CENTERS
and violet (8) lines describe the behavior of the degenerate NV states M = −1
and M = +1, respectively, along the [+1, +1, +1] and [−1, −1, +1] direction
of the crystal with the orientation angle φcrystal = 15◦ . The magenta (2) and
purple (7) lines describe the behavior of the degenerate NV states M = −1 and
M = +1, respectively, along the [+1, +1, +1] and [−1, −1, +1] direction of the
crystal with the orientation angle φcrystal = −9◦ . The cyan (3) and brown (6)
lines describe the behavior of the degenerate NV states M = −1 and M = +1,
respectively, along the [+1, −1, −1] and [−1, +1, −1] direction of the crystal with
the orientation angle φcrystal = −9◦ . The blue (4) and olive (5) lines describe the
behavior of the degenerate NV states M = −1 and M = +1, respectively, along
the [+1, −1, −1] and [−1, +1, −1] direction of the crystal with the orientation
angle φcrystal = 15◦ .
2.2.3
Spectral distribution of the nitrogen-vacancy center
ensemble
The NV-centers in the ensemble have inhomogeneously broadened transition
frequencies around the quantum mechanically predicted spin transition frequency
ωs . The density of states ρ(ω) for a single ensemble is assumed to be q-Gaussian
distributed with q = 1.389 following the study of Sandner et al., who found this
distribution to fit most accurately [SRA+ 12]. This inhomogeneous broadening
appears due to electrical strain in the diamond crystal [Ams12].
The q-Gaussian distribution is defined by
1
ρ(ω) =
Cq ∆
1 − (1 − q)
ω − ωs
∆
1
2 ! 1−q
(2.5)
for ∆ > 0 and q ∈ [1, 3[, where the variable ∆ relates to the Full Width Half
Maximum of the q-Gaussian distribution
s
1 − 2q−1
FWHM = 2∆
.
(2.6)
1−q
The normalization constant is
√
3−q
 √πΓ( 2(q−1) ) , 1 < q < 3,
1
)
Cq = √q−1Γ( q−1
 π,
q = 1.
The Gaussian and Lorentz distributions are limiting cases of the q-Gaussian
distribution for the values q = 1 and q = 2, respectively. Fig. 2.8 depicts the
different distributions with the same value for ∆ = 1, for which, however, the
FWHM is different, consistent with Eq. (2.6). The higher populated wings of the
Lorentzian distribution are clearly visible in the comparison shown in Fig. 2.8.
at the angle θB = 90◦ , where the z-components are still the same.
12
CHAPTER 2. DESCRIPTION OF THE SYSTEM
Relative density ρ(ω)
0.4
0.3
0.2
0.1
0
−9
−6
−3
0
Frequency ω
3
6
9
Figure 2.8: Comparison of a Gaussian, Lorentzian and q-Gaussian (with
q = 1.389) distribution shown in normalized units using the colors blue, green,
and red, respectively. The integral over the density function is normalized to one.
For simplicity the frequency is dimensionless and the central transition frequency
is set ωs = 0. All distributions have the same dimensionless parameter ∆ = 1.
2.3
Experimental data for transmission through
the coupled quantum system
The transmission through the quantum system described in the previous sections
provides a convenient tool to study the state of the system and its physical behavior. The corresponding steady-state transmission measurements are currently
carried out at the Atominstitut in the group of J. Majer [Pet13]. This section
is dedicated to exploring and explaining the transmission characteristics of a
coherent steady-state microwave signal referred to as a “probe” in the following.
The experiments provide data sets for some magnetic field angles. Each
data set contains values of the complex transmission at variable magnetic field
strengths and probe frequencies. The data sets A and C were measured at a
horizontal angle of φB = 48◦ whereas the data set B contains the values for a
horizontal angle of φB = 66◦ . The zero magnetic field data of A and B were
measured at the same temperature of about 50 mK. During the measurement,
the cooling process is stopped and the temperature increases. In the data sets
A and C, the temperature increases with increasing magnetic field about 15 mK
per 15 mT. In contrast to the data set C where the temperature increases with
decreasing magnetic field at considerably higher temperature. The temperature
in C is almost 90 mK at zero magnetic field. Fig. 2.9 compares the data sets A,
B, and C at zero magnetic field in order to check their consistency. This is an
independent measure of how good the data from the measurement satisfy the
consistency for conclusions of working with the same parameters for different
data sets. It is immediately clear that this condition is not satisfied for the
data set C. The other two data sets show higher consistency. From this we
conclude that both data sets might be described by the same parameter set in
the following chapters.
13
−40
−50
−60
−70
2.760
2.758
2.756
2.754
2.752
2.750
2.748
2.746
2.744
2.742
−80
2.740
Relative Transmission T (dB)
2.3. EXPERIMENTAL DATA
Probe Frequency ωp [GHz]
Figure 2.9: Data sets (A in blue, and B in magenta, C in red) with the same
experimental setup compared at zero magnetic field.
In this section the three dimensional diagrams depict the transmission in
a logarithmic scale in order to stress the characteristics of the data, which are
mentioned in the following paragraphs. Nevertheless in the following chapters,
the transmission is scaled linearly. The reason for this difference is that on a
logarithmic scale the error for comparing the theory and experiment would grow
exponentially towards lower transmission. Additionally we are interested in the
transmission at and close to resonance only.
Generally speaking, the resonator’s transmission peaks at the cavity resonance frequency, i.e. ωp = ωcav , see Fig. 2.10. Note, however, that this resonance
frequency is shifted and has an avoided crossing at a finite magnetic field strength.
The avoided crossing appears due to the resonance between the resonator cavity
and the spin transition frequency. The strong coupling reduces the transmission
by a large amount at resonance between the cavity and NV-center’s frequencies
ωcav = ωs , which can be seen in the experimental data in Fig. 2.10.
Note that the transmission has an asymmetry due to the influence of other
cavity resonance frequencies. This is motivated by the fact that the transmission
pattern at the wings of the transmission resonance is independent of the magnetic
field strength and direction except at the avoided crossing of course. Far from
the avoided crossing in the experimental data in Fig. 2.10, the minimum at a
frequency of about 2.763 GHz and the additional local maximum at a frequency of
about 2.72 GHz can be understood as the interaction of different cavity resonance
frequencies. It is a well-known problem in the experiment.
At resonant transmission ωprobe = ωcav , the value of the maximum transmission at a magnetic field above the avoided crossing is not reached in the
experiment. This reduction of the transmission in the experiment is expected
to be due to an interaction with nuclear magnetic moments of the NV-center’s
neighboring 13 C or its 14 N atom.
14
CHAPTER 2. DESCRIPTION OF THE SYSTEM
Probe Frequency ωp [GHz]
2.78
2.76
2.74
2.72
0
3
−90
6
9
Magnetic field B [mT]
−80
−70
−60
12
−50
Relative Transmission T (dB)
15
−40
(a) Experimental data set A at φB = 48◦ .
Probe Frequency ωp [GHz]
2.78
2.76
2.74
2.72
0
3
−90
6
9
Magnetic field B [mT]
−80
−70
−60
12
−50
Relative Transmission T (dB)
15
−40
(b) Experimental data set B at φB = 66◦ .
Figure 2.10: The figures show the data for the measured field- and probe
frequency-dependent transmission from the experiments at the Atominstitut
in logarithmic scale.
Chapter 3
Green’s function formalism
Finding the transmission through the coupled quantum system, described in
chapter 2, requires the solution of a complex scattering problem. In principle,
such a problem is very hard to solve as the resonator couples to a large number
of NV-centers. In this chapter, the problem will be simplified drastically and it
will be shown that this simplification is reasonable for modeling the experimental
results. The resonator and all ensembles are described each by a single resonance
term with a well-defined resonance frequency and a resonance width. The goal is
to find an expression for the transmission through the system in such resonance
terms. We will employ a formalism that relates the transmission to the Green’s
functions. Green’s functions have the advantage that the coupling between the
resonance terms is well-described and that they are directly proportional to
the transmission. We will thus consider first the Green’s function for a single
resonance and shall then investigate the coupling of such resonances to obtain
an expression for the transmission. These steps are explained in section 3.1.
Section 3.2 discusses the simulations achieved by applying these steps.
3.1
Green’s function in general
The scattering properties of an electromagnetic wave entering the system through
one lead attached to the microwave cavity are described by the Helmholtz equation,
(∇2 + k 2 )G(x, x0 , k) = −δ(x − x0 ).
(3.1)
Note that this equation is formally equivalent to the stationary Schrödinger
equation,
(Ĥ0 − E)G(x, x0 , E) = −δ(x − x0 ),
(3.2)
as used, e.g., for the description of electron scattering problems. In the present
work, it is equivalent to deal with the Helmholtz equation or the Schrödinger
equation as we are only interested in the stationary solution.
In the following we will work with the Green’s function which we will compose
of concatenated single-site Green’s functions (see Fig. 3.1),
Ĝsingle-site =
1
(E − Ĥsite’s system + ih̄κsite’s system )−1 ,
∆x
15
(3.3)
16
CHAPTER 3. GREEN’S FUNCTION FORMALISM
as realized on a tight-binding chain of sites connected to their nearest neighbors.
In addition, we will add three specific single-site Green’s functions to the system
studied: One for the cavity and two for the NV-ensembles.
NV-ensemble
Left lead
Right lead
Resonator
Figure 3.1: The schematic sketches the composed discrete system in a reduced
way to illustrate the setup of Green’s functions.
The Green’s function of semi-infinite chains for the input- and output-leads
of the cavity is given by
Ĝsi =
eiθ
,
V̄
(3.4)
where the complex phase is
θ = arccos(E/(2V ) + 1).
(3.5)
1
The discrete integrated hopping matrix element is V̄ = V ∆x = − 2∆x
for
1
V = − 2∆x2 with the mesh step-size ∆x.
The Green’s function for a combined system Ĝtot can be composed based on
the Green’s functions of many unconnected sites Ĝ0 using the Dyson equation
Ĝtot =
1 − Ĝ0 V̂tot
−1
Ĝ0 ,
(3.6)
where V̂tot stands for the coupling between the sites [Rot99] 1 .
1
This equation can be derived from the general definition of a Green’s function
Ĝtot := lim
ε→0+
= lim
ε→0+
E + iε − Ĥ
−1
−1
E + iε − (Ĥ0 + V̂tot )
−1
= E − (Ĥ0 + V̂tot )
−1
= (E − Ĥ0 )(1 − (E − Ĥ0 )−1 V̂tot )
−1 −1
= 1 − (E − Ĥ0 )−1 V̂tot )
E − Ĥ0
−1
= 1 − Ĝ0 V̂tot
Ĝ0 .
If the term E − Ĥ0 is not invertible, the limit can not be taken at the intermediate step. This
is the case at resonant energies without loss.
17
3.1. GREEN’S FUNCTION IN GENERAL
3.1.1
Application of the formalism
For the studied setup, the matrices for the unconnected sites’ Green’s function
Ĝ0 and the connecting integrated hopping matrix V̂tot are of the following form2 ,


Ĝsi
0
0
0
 0 Ĝcav
0
0 
,
Ĝ0 = 
(3.7)
 0
0
ĜNV 0 
0

0
0
 V̂left
V̂tot = 
 0
V̂Fano
V̂left
0
V̂int
V̂right
0
0
V̂int
0
0
Ĝsi

V̂Fano
V̂right 
.
0 
0
(3.8)
A reduced scheme of the system is illustrated in figure Fig. 3.1.
In all the following Green’s functions, the energy E = h̄ωp is a variable scan
parameter for the figures in the section 3.2 corresponding to the probe frequency.
The Green’s function of the microwave cavity is given by the single-site Green’s
function in equation (3.3),
Ĝcav =
1
(E − Hcav + ih̄κcav )−1 ,
∆x
(3.9)
where the resonance frequency of the cavity is h̄1 Ĥcav = 2.750 GHz and the decay
rate is κcav = 170 kHz. The Green’s function of the semi-infinite leads is given by
Eq. (3.4), where the mesh step-size is chosen ∆x = √ω1cav . The Green’s function
of the NV-centers is of the same form as the cavity’s Green’s function. But the
resonance frequency Hh̄NV is calculated from the transition frequency between
the eigenstates of a single NV-center. The Hamiltonian is given in Eq. (2.4) in
section 2.2.1.
The Hamiltonian from Eq. (2.4), i.e. the resonant energy level transitions,
will be incorporated into an effective Green’s function that describes the NVensemble in the next section. In fact this Hamiltonian describes only single
spin dynamics, which will be compensated by taking a larger decay rate of
the NV-centers to account for the inhomogeneous broadening of the entire NVspin ensemble. As will be shown in a subsequent chapter, the inhomogeneous
broadening can also be described more rigorously in a more advanced quantum
optical model.
The coupling between the subsystems is described by the parameters V̂int .
The quantum optical model provides such a parameter in chapter 4.
The complex transmission for scattering through the cavity from the left to
the right lead [Dat97] is given as follows
t = ih̄
sin(θ)
Ĝtot (1, 5).3
∆x
(3.10)
2 In fact the matrices are higher dimensional in order to describe the coupling of the cavity
with each sub-ensemble separately as they couple independently of each other.
3 The complex reflection coefficient at the left lead is
r = ih̄
sin(θ)
Ĝtot (1, 1) − 1.
∆x
18
CHAPTER 3. GREEN’S FUNCTION FORMALISM
The prefactor to the Green’s function is in the limit ∆x → 0 equal to the wavevector k = lim∆x→0 sin(θ)
∆x , which is linear with respect to the velocity, i.e. the
prefactor normalizes the entry of the scattering matrix, which describes the flux
amplitudes of incoming and outgoing waves. The complex phase θ is defined in
equation (3.5).
3.2
Simulations compared to the experiments
As explained in chapter 2, the experimental setup consists of a microwave cavity
in a transmission line which couples to two NV-ensembles on top of it. Each of
the two NV-ensembles has an individual horizontal orientation angle of φ1 = 15◦
and φ2 = −9◦ with respect to the x-axis of the lab frame. The Hamiltonian of
the NV-center has a degeneracy if the magnetic field is at an angle of φB = 48◦ .
The relative transmission in figures Fig. 3.3, Fig. 3.2, Fig. 3.5, and Fig. 3.4
shows qualitatively and quantitatively good agreement between theory and experiment. This agreement is obtained by fitting a set of parameters for the model to
both experimental data sets, which are introduced in section 2.3. The parameters
of the cavity and coupling to the leads are fitted at zero magnetic field whereas
the other parameters are fitted at B = 6.14 mT where the degenerate ensembles,
i.e. at an angle of φB = 48◦ , cross the cavity’s resonance frequency. At the
crossing, we can determine the coupling between the cavity and NV-ensembles,
the resonance frequencies of the NV-ensembles, and their loss. We choose the
cavity’s resonance frequency to be the same as the NV-ensembles’ resonance
frequency at a magnetic field strength of about 6.14 mT. An avoided crossing
in the relative transmission is obtained for this set of parameters, see Fig. 3.2b.
The influence of the further interacting cavity modes is taken into account by
adding a Fano term to the transmission. A Fano term is an additional constant
transmission with a relative phase. This enables us to model the minimum in
the transmission, which was mentioned in section 2.3.
From Fig. 3.3 and Fig. 3.5, we can conclude that we have reached qualitatively
good consistency between our theory and the experimental data over both a
large range of magnetic field and a wide range of probe frequency around the
cavity resonance frequency. Nevertheless there is a big discrepancy between
the theoretical toy model and the experiment at the avoided crossing. The
Green’s function formalism that we use is a purely classical approach, which
does not describe quantum mechanical effects apart from wave interference
inherent already in a classical wave description. It is therefore necessary to
consult a more appropriate description – the quantum optical model, which will
be discussed in chapter 4. At resonant transmission ωp = ωcav , the transmission
predicted by the Green’s function formalism achieves higher values at magnetic
field strengths above the avoided crossing than below, which is unexpected.
At the angle of φB = 66◦ , two avoided crossings between the NV-ensembles
and the cavity resonance appear. This is because the degeneracy of the NVsubensembles is lifted at this angle as already mentioned in Fig. 2.6.
19
3.2. SIMULATIONS COMPARED TO THE EXPERIMENTS
·10−4
2
2.78
2.77
2.76
2.75
2.74
2.73
0
2.72
1
2.71
Relative Transmission T
3
Probe Frequency ωp [GHz]
(a) No external magnetic field is applied.
·10−6
2
2.78
2.77
2.76
2.75
2.74
2.73
0
2.72
1
2.71
Relative Transmission T
3
Probe Frequency ωp [GHz]
(b) An external magnetic field is applied such that the transition frequency of
the NV-ensemble is at resonance with the cavity frequency.
Figure 3.2: The blue curves show the experimental data of the relative transmission at different magnetic field strength at φB = 48◦ , whereas the red curve
shows the theoretical description with fit parameters adapted to the experimental
data.
20
CHAPTER 3. GREEN’S FUNCTION FORMALISM
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
1
1.5
2
2.5
3
Relative Transmission T
12
15
3.5
4
·10
(a) Experimental data set A.
−4
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
1
1.5
2
2.5
3
Relative Transmission T
12
15
3.5
4
−4
·10
(b) Results of the Green’s function method with the parameters being fitted to
the according experiment.
Figure 3.3: These images show the relative transmission as a function of the
applied magnetic field strength and of the probe frequency. The magnetic field
angles are kept fixed at the degenerate case at φB = 48◦ and θB = 90◦ .
21
3.2. SIMULATIONS COMPARED TO THE EXPERIMENTS
·10−4
2
2.78
2.77
2.76
2.75
2.74
2.73
0
2.72
1
2.71
Relative Transmission T
3
Probe Frequency ωp [GHz]
(a) No external magnetic field is applied.
·10−6
1.5
1
2.78
2.77
2.76
2.75
2.74
2.73
0
2.72
0.5
2.71
Relative Transmission T
2
Probe Frequency ωp [GHz]
(b) An external magnetic field is applied such that the transition frequency of
the NV-ensemble is at resonance with the cavity frequency.
Figure 3.4: The blue curves show the experimental data of the relative transmission at different magnetic field strength at φB = 66◦ , whereas the red curve
shows the theoretical description with fit parameters adapted to the experimental
data.
22
CHAPTER 3. GREEN’S FUNCTION FORMALISM
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
12
1
1.5
2
Relative Transmission T
15
2.5
·10−4
(a) Experimental data set B.
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
1
1.5
2
Relative Transmission T
12
15
2.5
·10−4
(b) Results of the Green’s function method with the parameters being fitted to
the according experiment.
Figure 3.5: These diagrams show the relative transmission as a function of the
applied magnetic field strength and of the probe frequency. The magnetic field
angles are kept fixed at the degenerate case at φB = 66◦ and θB = 90◦ .
Chapter 4
Quantum optical model
In chapter 3 we have already seen that the main features and the resonant
splitting can be captured rather well within a classical scattering formalism
using the Green’s function approach. However, for a more quantitative analysis,
a specific shape of the inhomogeneous distribution of the spin density, which is
unavoidable in real experiment and stands for the dephasing processes in our
system, has to be taken into account. These facts require that we go beyond this
simple model treating the problem in the framework of a full quantum optical
approach. This approach will be introduced in section 4.1 with a short derivation
of the Hamiltonian and the corresponding equations of motion. In section 4.2 we
derive an expression for the complex transmission of the cavity, which is related
to the steady-state solution of the equations of motion. The derivation follows
the same structure as in previous work [Krib, Har13]. A comparison with the
experiment is provided in section 4.3.
4.1
Tavis-Cummings Hamiltonian and equations
of motion
The Hamiltonian for the composite system of a single cavity mode and an
ensemble of spins with an external coherent probe field is given by
H = HCav + HEns + HInt + HProbe
(4.1)
with
HCav = h̄ωcav a† a,
HEns =
h̄
2
HInt = ih̄
N
X
(4.2)
ωk σkz ,
(4.3)
k=1
N
X
gk σk− a† + gk∗ σk+ a ,
(4.4)
k=1
HProbe = ih̄ ηa† e−iωp t − η ∗ ae+iωp t
[Kria].
(4.5)
The frequencies ωcav , ωp , and ωk are the cavity resonance frequency, coherent
probe frequency, and k-th spin transition frequency, respectively. The operators
23
24
CHAPTER 4. QUANTUM OPTICAL MODEL
σk+ , σk− , σkz are the Pauli spin-operators for the k-th NV-center. The electromagnetic field mode in the cavity is quantized in terms of the field operators a
following the second quantization formalism. The magnetic field inside the cavity
interacts with the magnetic moment of the NV-centers via dipole interaction
in the first order. The interaction part of the Tavis-Cummings Hamiltonian is
written in the well-known form, where the counter-rotating terms are neglected
due to the rotating wave approximation. This approximation is well fulfilled
since the cavity frequency is much larger as compared to the collective coupling
strength of the spin ensemble to a cavity (see section 4.2.1 for more details).
The coupling strength for a single NV-center is assumed to be of the order of
gk ≈ 10 Hz. At last, the letter η stands for the amplitude of a coherent microwave
pumping field.
The Heisenberg equation for an arbitrary operator A(t) in the Heisenberg
picture is given by
i
∂t A = [H, A] .
(4.6)
h̄
The
in
is the commutator.
Using the commutation
relations
bracket
this equation
a, a† = 1, σj+ , σk− = δj,k σkz , σj− , σkz = 2δj,k σk− , σj+ , σkz = −2δj,k σk+ and
introducing the corresponding loss terms leads to the following equations of
motion,
i
[H, a] − κa,
h̄
i
∂t a† = [H, a] − κa† ,
h̄
i
[H, a] −
h̄
i
∂t σk+ = [H, a] −
h̄
i
∂t σkz = [H, a] .
h̄
∂t σk− =
∂t a =
γ −
σ ,
2 k
γ +
σ ,
2 k
The decay rates of the cavity mode and single NV-spin excitation are κ and γ,
respectively, which characterize the loss of the quantum system.
In the frame rotating with ωp , the equations acquire the following form,
∂t ã = −(κ + i∆c )ã +
N
X
k=1
∂t ㆠ= −(κ − i∆c )ㆠ+
gk σ̃k− − η,
N
X
k=1
gk σ̃k+ − η ∗ ,
γ
+ i∆k )σ̃k− + gk ãσkz ,
2
γ
∂t σ̃k+ = −( − i∆k )σ̃k+ + gk ㆠσkz ,
2
∂t σkz = −2gk (ㆠσ̃k− + ãσ̃k+ ).
∂t σ̃k− = −(
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
The operators ã = aeiωp t , ㆠ= a† e−iωp t , σ̃k− = σk− eiωp t , σ̃k+ = σk+ e−iωp t are redefined in the rotating frame. The variables ∆c = ωcav − ωp and ∆k = ωk − ωp
denote the detuning of the probe frequency with respect to the cavity’s resonance
frequency and to the k-th spin, respectively.
In order to decouple the system of operator equations, we take the expectation values of the operators and only the operators are time dependent in the
Heisenberg picture. The approximation hσkz i ≈ −1 is valid if the number of the
4.2. STEADY-STATE TRANSMISSION THROUGH THE CAVITY
25
excited spins is small compared to the ensemble size. This assumption is well
fulfilled for the experimental results discussed in this work which are performed
at low temperatures and at low intensities of a probe signal.1 The factorization
of expectation values hãσkz i = hãihσkz i, hㆠσkz i = hㆠihσkz i for independent eigenstates is applied. Therefore, Eq. (4.11) is decoupled from the rest and plays no
role here. Finally, we end up with the folloiwng set of equations
∂t hãi = −(κ + i∆c )hãi +
N
X
k=1
gk hσ̃k− i − η,
(4.12)
γ
+ i∆k )hσ̃k− i − gk hãi.
(4.13)
2
The equations for the Hermitian conjugated operators contain no additional
information and are therefore dismissed.
∂t hσ̃k− i = −(
4.2
Steady-state transmission through the cavity
In the steady-state the temporal derivative vanishes and the equations of motion
ˆ =+
(κ + i∆c )hãi
N
X
k=1
ˆ − i + η,
gk hσ̃
k
γ
ˆ − i = −gk hãi,
ˆ
( + i∆k )hσ̃
k
2
can be decoupled and solved independently
ˆ−i =
hσ̃
k
ˆ =
hãi
igk
∆k −
γ
2
ˆ =−
hãi
∆c − iκ −
iη
PN
gk η
PN
∆k −
∆c − iκ − k=1
γ
2
2
gk
∆k −i γ2
2
gk
k=1 ∆k −i γ2
The complex transmission through the cavity is defined by
√
2i κout κin
hĉt i
t=
=
,
2
PN
gk
hĉin i
∆c − iκ − k=1 ∆k −i
γ
(4.14)
2
where the operators of the injected
and the transmitted wave are for linear
√
η
response ĉin = √2κ
and ĉt = 2κout a, respectively.2
in
4.2.1
Transmission through the cavity coupled to a single
ensemble
In the next step a continuous density of states for the spin ensemble (with its
coupling gk for each individual spin to the cavity) is introduced as the continuous
limit of the finite sum over single spins
ρ(ω) =
N
1 X 2
gk δ(ω − ωk ).
Ω2
k=1
1 The
2 The
experiments are performed at a temperature
√of the order of 50 mK.
operator of the reflected wave is ĉr = ĉin + 2κin a.
26
CHAPTER 4. QUANTUM OPTICAL MODEL
R
The density of states3 is normalized according to R ρ(ω) dω = 1 so that the
qP
N
2
collective coupling Ω =
k=1 gk . The continuous limit is well justified thanks
to the large number of spins coupled to the cavity (typically of the order of 1012
[SRA+ 12]).
The formula Eq. (4.14) for the complex transmission is finally rewritten in
the continuous limit as
√
2i κout κin
,
(4.15)
t(ωp ) =
R
ρ(ω)
ωcav − ωp − iκ − Ω2 R ω−ω
γ dω
−i
p
2
which was derived previously by [Krib].
4.2.2
Dimensionless transmission
For the purpose of simplicity and to get rid of the large numbers which are undesirable during numerical simulations, we introduce the following dimensionless
quantities:
SI units [Hz]
ωcav
ωp
ωs
γ
κ
κin
κout
Ω
∆
Relative units
1
ωp
ω̃p = ωcav
ωs
ω̃s = ωcav
γ
γ̃ = ωcav
κ
κ̃ = ωcav
in
κ̃in = ωκcav
κout
κ̃out = ωcav
Ω̃ = ωΩ
cav
˜ = ∆
∆
ωcav
ρ(ω)
ρ̃(ω̃) =
1
˜
Cq ωcav ∆
1 − (1 − q)
ω̃−ω̃s
˜
∆
1
2 1−q
With this transformation the complex transmission is given by
√
2i κ̃out κ̃in
t(ω̃p ) =
.
R
ρ̃(ω)
dω
1 − ω̃p − iκ̃ − Ω̃2 R ω−ω̃
−i γ̃
p
4.2.3
(4.16)
2
Transmission through the cavity coupled to several
ensembles
The formula for the complex transmission coefficient can be generalized to n
non-interacting spin ensembles with different transition frequencies ωsj similar
to the previous section for a single ensemble. The j-th ensemble has a number
of Nj spins contributing to the finite sum
N
X
k=1
gk2 δ(ω
− ωk ) =
Nj
n X
X
j=1 kj =1
gk2j δ(ω − ωkj ).
3 The ensemble of a large number of NV-centers can be described by a continuous distribution,
which we model by the q-Gaussian distribution explained in section 2.2.3.
4.3. SIMULATIONS COMPARED TO THE EXPERIMENT
27
The continuous densities of states for each spin ensemble is given by
ρj (ω) =
Nj
1 X 2
gkj δ(ω − ωkj ),
Ω2j
kj =1
with collective coupling
v
u N
j
uX
u
Ωj = t
gk2j .
kj =1
R
The densities of states are normalized as in section 4.2.1, R ρj (ω) dω = 1.
The formula for the complex transmission in Eq. (4.14) then reads
√
2i κout κin
t(ωp ) =
.
(4.17)
R
Pn
ρ (ω)
ωcav − ωp − iκ − j=1 Ω2j R ω−ωj p −i γ dω
2
Example of merging two spin ensembles Let us assume, for simplicity,
two spin ensembles with the mean frequencies ωs1 , ωs2 . The ensembles are tuned
by an external magnetic field. If it is oriented such that both mean frequencies
are the same, then the degeneracy takes place so that the response of the system
is such as if we deal with a single ensemble only. If additionally the number of
spins coupled to a cavity are the same for each ensemble, then their coupling
strengths are
√equal, Ω1 = Ω2 . Hence, the common collective coupling is given
by ΩTot = 2Ω1 in this specific case. The following equality supports this
statement,
Z
Z
Z
ρ1 (ω)
ρ2 (ω)
ρ1 (ω)
2
2
Ω21
dω
+
Ω
dω
=
2Ω
γ
γ
γ dω.
2
1
R ω − ωp − i 2
R ω − ωp − i 2
R ω − ωp − i 2
(4.18)
For n different degenerate non-interacting
ensembles
or
spins,
the
common
col√
lective coupling scales with n if each ensemble couples with the same coupling
strength to the cavity mode (as mentioned in section 2.2.2).
4.3
Simulations compared to the experiment
This section describes the results of the quantum optical model. The Green’s
function formalism discussed in chapter 3 already shows qualitatively good agreement with the experimental data.
Some further assumptions are made in order to simplify the model. The
subensembles with transition frequencies that do not cross the cavity resonance
frequency have a negligible small dispersive influence to the resonance in the
transmission4 and are, therefore, ignored in this section. Hence only two ensembles are required in order to describe the avoided crossings in the case of a
horizontal oriented magnetic field as mentioned in section 2.2.2. The ensembles
are modeled with distributed transition frequencies according to Eq. (2.5). We
further assume the coupling strengths of the cavity to each of the ensembles to
4 The dispersive shift of the transmission is proportional to the square of the coupling
strength divided by the detuning.
28
CHAPTER 4. QUANTUM OPTICAL MODEL
have the same value Ω1 = Ω2 = Ω since the size of two diamonds is approximately the same in the experiment. From discussions with our experimental
colleagues we conclude that these values might vary at most 10% from each
other.
The physical parameters under the above assumptions are determined by
analyzing two different cases. The case of zero magnetic field at first determines
the cavity parameters. The resonant case, where the cavity frequency and the
ensemble’s central transition frequency are equal ωcav = ωs , is analyzed in order
to complete the set of parameters. A unique set of parameters is chosen to
describe both data sets. The values of the physical parameters are summarized
in table 4.1.
Physical parameters
Cavity resonance frequency ωcav
Cavity decay rate κ
Longitudinal zero field splitting D
Transversal zero field splitting E
q-Gaussian parameter q
Broadening of the q-Gaussian γq
Collective coupling to each ensemble Ω
Values
2π · 2.748 475 GHz
2π · 0.275 MHz
2.8807 GHz
10 MHz
1.389
2π · 7.75 MHz
2π · 8.5 MHz
Table 4.1: Physical parameters describing the transmission through the present
quantum system.
In the first case, the parameters of the cavity are found to accurately describe
the experimental data at zero magnetic field. It is remarkable that the cavity
resonance frequency ωcav can be determined up to a precision of a few kHz. In
fact the resonance is so narrow that different data sets might have recognizably
different values of the cavity resonance frequency at zero magnetic field. In
Fig. 4.3a, we notice a dispersive shift of the cavity resonance frequency. The
finite difference between the theoretical and experimental curve at no magnetic
field is due to imperfections in the measurement setup, as the model applies
same parameter set and, therefore, the same cavity resonance frequency for both
comparisons. In Fig. 2.9 the same dispersive shift between different data sets
is already observed. These limitations also affect the quality of the theoretical
description of the system in Fig. 4.1b. Conclusively, the cavity frequency should
be initialized at each measurement anew but we do not in order to show the
quality of common parameters for both data sets. The decay rate of the cavity
κ is determined by the broadening of the transmission resonance and has no
remarkable difference for small thermal differences.
The second case is to find the remaining parameters at resonance between the
cavity frequency and the ensemble’s central transition frequency. The latter is
determined by the zero-field splitting parameters D, which was slightly adapted
in order to achieve an improved description5 , and E (introduced in section
2.2.1). The broadening around this central frequency is modeled by a q-Gaussian
distribution. The broadening of the distribution γq can be fitted to achieve the
correct depth of the minimal transmission between the two maxima surrounding
5 Only the longitudinal parameter has a significant influence at a field strength of B =
6.14 mT.
4.3. SIMULATIONS COMPARED TO THE EXPERIMENT
29
the avoided crossing6 . The last parameter that remains is the coupling strength
of the ensemble to the cavity, which determines the splitting of these two maxima
in transmission.
When comparing our theoretical predictions, following from the above model,
with the expeirmental data in Fig. 4.1b and Fig. 4.3b, we find that the model
still does not fit very well to the data because of the interaction with further
cavity resonances at the wings of the transmission resonance (see discussion in
section 2.3). One simple and intuitive way to overcome this discrepancy is to
renormalize the data to the expected shape of the curve at e.g. zero magnetic
field. The renormalization takes the difference between the experimental data
and the quantum optical fit at vanishing magnetic field at the wings weighted
by a Fermi-Dirac distribution in order to neglect the finite difference close to the
resonance. The cyan curves in Fig. 4.1b and Fig. 4.3b represent the renormalized
data from the experiments.
Conclusion With the final renormalization, the correspondence between theory and experiment can be compared. The following paragraphs are dedicated
to evaluate the quality and predictive values of the quantum optical model. The
first paragraph covers the consistency of the model by comparing the simulations of one with two ensembles. The second paragraph presents the comparison
between the presented quantum optical approach and the Green’s functions
method.
As shown in the previous section by Eq. (4.18), the transmission at φB = 48◦
should behave in the same way for one and two resonant ensembles that do
not interact with each other. The simulations including one or two resonant
ensembles give the same results for the transmission function in the degenerate
case (horizontal angle φB = 48◦ ) as expected (not shown).
The quantum mechanical description is able to correctly account for the
interaction between the cavity and the NV-ensemble. It also includes the qGaussian distribution of the spin transition frequencies. For these reasons, the
description of the avoided crossing in Fig. 4.1b and Fig. 4.3b shows qualitatively
good agreement with the experiment. At an angle of φB = 48◦ the agreement
is very good even on the quantitative level. The qualitative degradation of the
model at φB = 66◦ is mainly due to the fact that the resonance frequency is not
fitted to the data set B but to the data set A. The data sets are described in
section 2.3. The shift of the cavity frequency ωcav here also implies a decrease
of the left and an increase of the right maximum in Fig. 4.3b. From both figures
can be concluded that the assumption Ω1 = Ω2 is fulfilled rather well.
By comparing Fig. 4.1b with Fig. 3.2b, we recognize an improvement of
describing the change in transmission for the avoided crossing at resonance
between the cavity and the ensembles. An improvement is also visible in the three
dimensional plots for the maximal transmission at strong field, e.g. B = 15 mT.
The degeneracy between the ensembles of both crystals is lifted for a magnetic
field at a generic horizontal angle. At an angle of φB = 66◦ each of the crystals
can be addressed uniquely depending on the magnetic field strength. This is
an important fact for quantum information processing as only one ensemble
interacts with the cavity at a time.
6 The shape of the q-Gaussian q = 1.389 is assumed to be the same as in the publication of
Sandner et al. [SRA+ 12] since the setup with the system components is similar.
30
CHAPTER 4. QUANTUM OPTICAL MODEL
·10−4
2
2.78
2.77
2.76
2.75
2.74
2.73
0
2.72
1
2.71
Relative Transmission T
3
Probe Frequency ωp [GHz]
(a) Transmission at no magnetic field.
2
2.78
2.77
2.76
2.75
2.74
2.73
0
2.72
1
2.71
Relative Transmission T
3
·10−6
Probe Frequency ωp [GHz]
(b) Transmission at a magnetic field with an angle of φB = 48◦ and a strength
of B = 6.14 mT.
Figure 4.1: The transmission is depicted as a function of the probe frequency
ωp . The formula for the transmission given in equation Eq. (4.16) is drawn in
red. The experimental data (data set A) is drawn in blue whereas in Fig. 4.1b
also the renormalized data is colored in cyan. The magnetic field is applied
at an angle of φB = 48◦ with a strength B = 6.14 mT in Fig. 4.1b such that
the degenerate spin ensembles’ central frequency is at resonance with the cavity
frequency.
31
4.3. SIMULATIONS COMPARED TO THE EXPERIMENT
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
1
1.5
2
2.5
3
Relative Transmission T
12
15
3.5
4
·10
(a) Experimental data set A.
−4
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
1
1.5
2
2.5
3
Relative Transmission T
12
15
3.5
4
−4
·10
(b) Results of the quantum optical model with the parameters being fitted to
the according experiment.
Figure 4.2: These diagrams show the relative transmission as a function of the
applied magnetic field strength and of the probe frequency. The magnetic field
angles are kept fixed at the degenerate case at φB = 48◦ and θB = 90◦ .
32
CHAPTER 4. QUANTUM OPTICAL MODEL
·10−4
2
2.78
2.77
2.76
2.75
2.73
2.71
0
2.74
1
2.72
Relative Transmission T
3
Probe Frequency ωp [GHz]
(a) Transmission at no magnetic field.
·10−6
1.5
1
2.78
2.77
2.76
2.75
2.74
2.73
0
2.72
0.5
2.71
Relative Transmission T
2
Probe Frequency ωp [GHz]
(b) Transmission at a magnetic field with an angle of φB = 66◦ and a strength
of B = 6.14 mT.
Figure 4.3: The transmission is depicted as a function of the probe frequency ωp .
The formula for the transmission given in equation Eq. (4.16) is drawn in red.
The experimental data (data set B) is drawn in blue whereas the renormalized
data is colored in cyan. The magnetic field is applied at an angle of 66◦ chosen
at the same strength as in Fig. 4.1b.
33
4.3. SIMULATIONS COMPARED TO THE EXPERIMENT
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
12
1
1.5
2
Relative Transmission T
15
2.5
·10−4
(a) Experimental data set B.
Probe Frequency ωp [GHz]
2.78
2.77
2.76
2.75
2.74
2.73
2.72
2.71
0
0
3
0.5
6
9
Magnetic field B [mT]
1
1.5
2
Relative Transmission T
12
15
2.5
·10−4
(b) Results of the quantum optical model with the parameters being fitted to
the according experiment.
Figure 4.4: These diagrams show the relative transmission as a function of the
applied magnetic field strength and of the probe frequency. The magnetic field
angles are kept fixed at the degenerate case at φB = 66◦ and θB = 90◦ .
34
CHAPTER 4. QUANTUM OPTICAL MODEL
Chapter 5
Summary and outlook
In this thesis the coupling of a resonator with two nitrogen-vacancy ensembles (NV-ensembles) was studied, which is an interesting topic in the context
of quantum information processing. NV-ensembles are expected to have the
desired properties as a memory for quantum information in circuit quantum
electrodynamics.
The system is probed by a transmission of a steady-state microwave signal
through the resonator. The resonator has one mode, we are concerned about,
but the transmission suffers from the influence of different resonance frequencies.
There are two simple approaches used to work around this issue. The Fano
term and an other simple renormalization of the transmission to the expected
behavior at zero magnetic field.
The NV-center’s spin transition frequencies depend on the external magnetic
field’s direction and strength. An ensemble of such NV-centers provides the
advantage of an enhanced coupling strength. A unique set of system parameters
like the coupling strength is found by applying the methods mentioned below to
describe the experimental data for the transmission measurements of the group
of J. Majer. The experiments provide data for different transmission frequencies
through the cavity and different magnetic field applied to the NV-ensembles.
The crucial point is to fit the data with few the parameters by keeping the model
simple.
In order to address this problem, two different approaches were applied. One
being very simple but not very quantitative and the other being more complex,
however, yielding better quantitative agreement. The first approach involving a
simple Green’s functions technique on a discretized grid already shows the results
of strong coupling between the resonator and the individual NV-ensembles. The
second and more elaborate approach is based on a quantum optical model, which
in addition provides a quantitatively good description of the avoided crossing.
The key is here that the contribution of dephasing, which is modeled as a qGaussian distribution, can be taken into account in this method. By applying
these approaches, it is possible to describe the system with its full magnetic
field dependence. This means the magnetic field enters the model as a system
parameter.
This thesis provides the fundamental concepts for the description of the
steady-state transmission. This work enables one to further explore the magnetic field dependence of the transmission. In a next step, one could relax the
35
36
CHAPTER 5. SUMMARY AND OUTLOOK
assumptions of equal coupling strength and adapt different coupling strengths for
the subensembles to improve the quantitative fit. A further interesting extension
of this work would be to include additional subensembles that do not cross the
cavity’s resonance frequency or include the effect of nuclear spins. In the long
run, it would also be interesting to study the dynamic aspects of transmission
through the system of two coupled NV-ensembles.
Acknowledgment
First of all, I would like to thank all those who helped me to complete this
thesis, especially all members of the Rotter Group who supported me with
their knowledge. My deepest gratitude goes to Dr. Dmitry Krimer, who was a
major help with his broad knowledge, even about Fortran77. I am very grateful
to Prof. Stefan Rotter for the possibility of supervising my thesis with great
enthusiasm. Special thanks go to Dr. Johannes Majer for his patience to explain
me all experimental details, for his good ideas and advices, and the Swiss-German
conversations. My gratitude goes also to Prof. Sebastian Huber, who supported
me with his help during the four months of my thesis and beyond. It was a
beautiful but short time in Austria, which I enjoyed not only as a research
experience at TU Wien but also as a cultural experience.
37
38
ACKNOWLEDGMENT
Bibliography
[AKN+ 11] R. Amsüss, C. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner,
S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer and J. Majer, Cavity QED with Magnetically Coupled Collective Spin States, Phys. Rev. Lett. 107, 060502 (Aug 2011).
[Ams12]
R. Amsüss, Strong coupling of an NV spin ensemble to a superconducting resonator, Dissertation, Technische Universität Wien,
2012.
[BNT+ 09] G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham,
R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler,
V. Jacques, P. R. Hemmer, F. Jelezko and J. Wrachtrup, Ultralong spin coherence time in isotopically engineered diamond, Nature
Materials 8(5), 383–387 (May 2009).
[BVJ+ 98] V. Bouchiat, D. Vion, P. Joyez, D. Esteve and M. H. Devoret,
Quantum coherence with a single Cooper pair, Physica Scripta
1998(T76), 165–170 (Mar 1998).
[Dat97]
S. Datta, Electronic transport in mesoscopic systems, Cambridge
University Press, Cambridge, UK, 1997.
[DS13]
M. H. Devoret and R. J. Schoelkopf, Superconducting Circuits for
Quantum Information: An Outlook, Science 339(6124), 1169–1174
(Mar 2013).
[GC97]
N. A. Gershenfeld and I. L. Chuang, Bulk Spin-Resonance Quantum
Computation, Science 275(5298), 350–356 (1997).
[GDT+ 97] A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup and
C. v. Borczyskowski, Scanning Confocal Optical Microscopy and
Magnetic Resonance on Single Defect Centers, Science 276(5321),
2012–2014 (1997).
[Har13]
B. Hartl, Cavity protection in Cavity QED with an inhomogeneous
ensemble of emitters strongly coupled to a single cavity mode, Bachelorarbeit, Technische Universität Wien, 2013.
[HMSR10] K. Henschel, J. Majer, J. Schmiedmayer and H. Ritsch, Cavity QED
with an ultracold ensemble on a chip: Prospects for strong magnetic
coupling at finite temperatures, Phys. Rev. A 82, 033810 (Sep
2010).
39
40
BIBLIOGRAPHY
[HW04]
H. Haken and C. Wolf, Molecular Physics and Elements of Quantum
Chemistry, Springer, Berlin Heidelberg New York, 2004.
[JW06]
F. Jelezko and J. Wrachtrup, Single defect centres in diamond: A
review, physica status solidi (a) 203(13), 3207–3225 (2006).
[KOB+ 10] Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng,
A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F.
Barthe, P. Bergonzo and D. Esteve, Strong Coupling of a Spin
Ensemble to a Superconducting Resonator, Phys. Rev. Lett. 105,
140502 (Sep 2010).
[Kria]
D. Krimer, Equations derivations from Hamiltonian, personal communication.
[Krib]
D. Krimer, Solutions of the volterra integral equation for the different
values of the probe frequency ωp , personal communication.
[KYG+ 07] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster,
J. Majer, A. Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf,
Charge-insensitive qubit design derived from the Cooper pair box,
Phys. Rev. A 76, 042319 (Oct 2007).
[MCG+ 07] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson,
J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff,
A. Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf,
Coupling superconducting qubits via a cavity bus, Nature 449(7161),
443–447 (Sep 2007).
[MOL+ 99] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der
Wal and S. Lloyd, Josephson Persistent-Current Qubit, Science
285(5430), 1036–1039 (1999).
[Pet13]
N. Peterschofsky, Coupling NV Ensembles via a Superconducting
Resonator, Bachelorarbeit, Technische Universität Wien, 2013.
[RBH01]
J. M. Raimond, M. Brune and S. Haroche, Manipulating quantum
entanglement with atoms and photons in a cavity, Rev. Mod. Phys.
73, 565–582 (Aug 2001).
[RDD+ 06] P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkopf
and P. Zoller, Hybrid Quantum Processors: Molecular Ensembles
as Quantum Memory for Solid State Circuits, Phys. Rev. Lett. 97,
033003 (Jul 2006).
[Rot99]
S. Rotter, A Modular Recursive Green’s Function Method for Quantum Transport, Diplomarbeit, Technische Universität Wien, 1999.
[SCLD14] R. Schirhagl, K. Chang, M. Loretz and C. L. Degen, NitrogenVacancy Centers in Diamond: Nanoscale Sensors for Physics and
Biology, Annual Review of Physical Chemistry 65(1), null (2014).
BIBLIOGRAPHY
41
[SRA+ 12] K. Sandner, H. Ritsch, R. Amsüss, C. Koller, T. Nöbauer, S. Putz,
J. Schmiedmayer and J. Majer, Strong magnetic coupling of an
inhomogeneous nitrogen-vacancy ensemble to a cavity, Phys. Rev. A
85, 053806 (May 2012).
[SSO+ 13]
L. Steffen, Y. Salathe, M. Oppliger, P. Kurpiers, M. Baur, C. Lang,
C. Eichler, G. Puebla-Hellmann, A. Fedorov and A. Wallraff, Deterministic quantum teleportation with feed-forward in a solid state
system, Nature 500(7462), 319–322 (aug 2013).
[WMI+ 98] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King
and D. M. Meekhof, Experimental Issues in Coherent QuantumState Manipulation of Trapped Atomic Ions, Journal of Research of
the National Institute of Standards and Technology 103(3), 259–328
(1998).
[WSB+ 04] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang,
J. Majer, S. Kumar, S. M. Girvin and R. J. Schoelkopf, Strong
coupling of a single photon to a superconducting qubit using circuit
quantum electrodynamics, Nature 431(7005), 162–167 (Sep 2004).