Download Quantum Nonlinear Optics in Lossy Coupled-Cavities in Photonic Crystal Slabs

Quantum Nonlinear Optics in Lossy
Coupled-Cavities in Photonic Crystal Slabs
by
Mohsen Kamandar Dezfouli
A thesis submitted to the
Department of Physics, Engineering Physics and Astronomy
in conformity with the requirements for
the degree of Doctorate of Philosophy
Queen’s University
Kingston, Ontario, Canada
May 2015
c Mohsen Kamandar Dezfouli, 2015
Copyright Abstract
A general formalism is developed that can be used to obtain photon dynamics in
coupled-cavity system in leaky photonic crystal slabs. This is accomplished using a
non-Hermitian projection operator, where the coupled-cavity modes, known as quasimodes, are used as a basis. Because of this, intrinsic features of these quasimodes
such as the leakage and the non-orthogonality are included in a self-consistent manner. The projection technique can be used to represent the Hamiltonian of a typical
system in the basis of the quasimodes. In addition, the corresponding quantum Master equation and adjoint quantum Master equation are provided. By employing these,
the time dependence of the density matrix and Heisenberg operators can be obtained.
In particular, a multimode Jaynes-Cummings Hamiltonian is obtained for photonic crystal slabs interacting with multiple quantum dots. As a proof of principle, a
simple system with two quasimodes is considered, where the mode non-orthogonality
affects the photon dynamics in a non-trivial manner. It is shown that, while the number of photons in each quasimode decays off, it also oscillates due to the quasimode
non-orthogonality.
Using the same projection technique, the problem of nonlinear photon pair generation via spontaneous four-wave mixing in photonic crystal slabs is discussed. The
main objective is to examine the effect of loss on pair generation in systems such as
i
photonic molecules and coupled-resonator optical waveguides. Several conclusion are
made. In addition to the overall loss rates of the pump, signal and idler photons, the
loss difference between signal and idler channels plays an important role in minimizing the number of unpaired photon in the system. Also, there is a trade-off between
source brightness and higher order generation depending on the losses in the system.
This is important, because both the number of unpaired photons and the number of
multiple photon pairs degrade device performance. Moreover, when slow light devices
are considered, the probability of finding photon pairs at particular locations is affected both by the dispersive behavior of the waveguide and the lossy behavior. This
is important as it opens up different possible design strategies that one might want
to use in lossy systems.
ii
Acknowledgments
I would like to thank my supervisor, Marc Dignam, who is not only a patient, caring,
supportive and well organized supervisor but is also an excellent teacher. I have
always admired the value Mark places on education and will always remember and
appreciate the great qualities he has both as a supervisor and as a teacher.
I am grateful to all the members of our group with whom I really enjoyed discussing
both physics and other topics of interest. I learned a lot from each of them and am
reminded of how much someone can touch another’s life without knowing it.
I would also like to thank Mike Steel and John Sipe for useful discussions that
gave strength to my research on the nonlinear pair generation. The approach they
both take to solving physics problems was inspiring to me.
I owe my deepest gratitude to my parents and my brothers who have always loved
me and supported me in every stage of my life. They have done so much for me and
I am not sure if I will ever be able to do the same for them.
Last but definitely not least, my best friend who is also the beautiful love of my
life, Golnaz. The part of my life I have spent with her most certainly stands out as
the best. Being with her, I have learned how small things can make a big difference.
Being with her, I have learned how much two people can care for one another. Being
with her, I have learned to love. My sincerest thanks!
iii
Statement of Originality
I developed the treatment of quantum nonlinear optics in photonic crystal slab structures presented in this thesis. This includes both of the mathematical formulation
and the numerical implementations for the pair generation as well as the pair detection problems. That also includes FDTD calculation of the single-defect quasimodes
and TB calculation of the coupled-cavity quasimodes in photonic crystal slabs. The
formulation of the pair generation was published as: [M. Kamandar Dezfouli, M. M.
Dignam, M. J. Steel, and J. E. Sipe, Physical Review A 90, 043832 (2014)]. The
manuscript on the pair detection problem is under preparation.
With regards to the quantum optics part of the thesis, I started working on the
multi-photon projection technique (chapter 3) when I first joined the group. This
work was started by David Fussell and Sean Doutre, two former members of Marc
Dignam’s group here at Queen’s. However, all the results presented in this section of
the thesis were independently derived by me. This was published as: [M. M. Dignam
and M. Kamandar Dezfouli, Physical Review A 85, 013809 (2012)].
I also did considerable work on coupled-cavity designs for vertical-cavity surfaceemitting lasers which is not presented in this thesis. This was published as: [M.
Kamandar Dezfouli, M. M. Dignam, AIP Advances 4, 123003 (2014)].
iv
Contents
Abstract
i
Acknowledgments
iii
Statement of Originality
iv
Contents
v
List of Tables
viii
List of Figures
ix
Glossary
xi
Chapter 1:
Introduction
1.1 Cavity quantum electrodynamics . . . . .
1.2 Nonlinear optics using microcavities . . . .
1.3 Features of coupled-cavity PCS structures
1.4 Thesis overview . . . . . . . . . . . . . . .
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Chapter 2:
Photonic Crystal Slabs and Quasimodes
2.1 Waves in periodic dielectric medium . . . . . . . . . .
2.2 Photonic crystal slabs . . . . . . . . . . . . . . . . .
2.3 Single-cavity quasimode . . . . . . . . . . . . . . . .
2.4 Coupled-cavity quasimodes . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 3:
Quasimode Projection Technique
3.1 The basis of true modes . . . . . . . . . . . . . . . . .
3.2 The basis of quasimodes . . . . . . . . . . . . . . . . .
3.3 Quasimode photon-number states . . . . . . . . . . . .
3.4 Non-Hermitian projection and the system Hamiltonian
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3.5
3.6
3.7
3.8
3.4.1 Quasimode ladder operators . . . . . . . . . . . . . . . .
3.4.2 Projected free-field Hamiltonian . . . . . . . . . . . . . .
Hermitian projection and system observables . . . . . . . . . . .
3.5.1 Ladder operators for restricted QMs . . . . . . . . . . .
3.5.2 Restricted Hamiltonian . . . . . . . . . . . . . . . . . . .
3.5.3 Restricted operators and restricted density matrix . . . .
Master equation for projected density matrix . . . . . . . . . . .
A new simplified representation . . . . . . . . . . . . . . . . . .
3.7.1 Adjoint Master equation for Heisenberg operators . . . .
3.7.2 Expectation value of observables using a simplified basis
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4:
Quantum Optics in the Quasimode Representation
4.1 Projected Jaynes-Cumming Hamiltonian . . . . . . . . . . . . . .
4.2 Photon dynamics in a PCS with two coupled cavities . . . . . . .
4.2.1 Obtaining the quasimode basis . . . . . . . . . . . . . . .
4.2.2 Photon number expectation . . . . . . . . . . . . . . . . .
4.2.3 One-photon evolution . . . . . . . . . . . . . . . . . . . . .
4.2.4 Two-photon evolution . . . . . . . . . . . . . . . . . . . .
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 5:
5.1
5.2
5.3
5.4
5.5
5.6
Nonlinear pair-generation in leaky coupled-cavity
tems
The linear Hamiltonian . . . . . . . . . . . . . . . . . . . . . . .
The nonlinear SFWM Hamiltonian . . . . . . . . . . . . . . . .
Heisenberg evolution of number operators . . . . . . . . . . . .
Lossy dynamics of photon-pair generation . . . . . . . . . . . .
5.4.1 Lossless pump limit . . . . . . . . . . . . . . . . . . . . .
Effect of multiple pair generation . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6:
Pair Detection in CROW Structures
6.1 Tight-Binding model for CROW structures . . . . .
6.1.1 CROW dispersion relation . . . . . . . . . .
6.1.2 CROW nonlinear overlap function . . . . . .
6.2 Heisenberg field operators and correlation functions
6.3 Pair detection probabilities in lossy CROWs . . . .
6.4 Nonlinear SFWM in the square-lattice CROW . . .
6.4.1 Pulse propagation through the CROW . . .
6.4.2 Pair generation in square-lattice CROW . .
6.4.3 Pair detection in square-lattice CROW . . .
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6.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Chapter 7:
Conclusion and Future Work
Bibliography
148
152
Appendix A: Commutation relation between projected ladder operators
165
Appendix B: Coupled set of Master equations for two-photon dynamics
168
Appendix C: Electric field operator in QM representation
174
Appendix D: Second order ansatz for number operator
178
Appendix E: First and second order ansatzen for product of number
operators
182
Appendix F: Higher order contributions to Sk1 ,k2 ,k3 ,k4
185
Appendix G: Evaluating the phase-matching sum for Sk1 ,k2 ,k3 ,k4
190
Appendix H: Heisenberg evolution of the ladder operators
192
vii
List of Tables
6.1
List of the first four Ãm and B̃m overlap integrals . . . . . . . . . . . 121
viii
List of Figures
1.1
Examples of CC structures in triangular PCSs . . . . . . . . . . . . .
3
1.2
Visualize of a quantum emitter inside an optical cavity . . . . . . . .
5
1.3
Schematic of a nonlinear optical process . . . . . . . . . . . . . . . .
10
1.4
Total internal reflection . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1
Multilayer dielectric slabs as a 1D photonic crystal . . . . . . . . . .
16
2.2
Two-band model dispersion for a 1D stack . . . . . . . . . . . . . . .
19
2.3
Schematic of the confinement mechanisms in a square lattice PCS . .
21
2.4
The band structure for a square-lattice PCS . . . . . . . . . . . . . .
23
2.5
Visualization of a PCS cavity inside a computational volume. . . . . .
24
2.6
FDTD calculated mode for a single defect . . . . . . . . . . . . . . .
26
3.1
Schematic of a PCS and the surrounding environment. . . . . . . . .
33
3.2
Schematic of two coupled defects in a PCS. . . . . . . . . . . . . . . .
35
4.1
Two defects are asymmetrically coupled in a PCS . . . . . . . . . . .
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4.2
FDTD calculated real electric field . . . . . . . . . . . . . . . . . . .
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4.3
Real part of the complex TB-calculated electric field . . . . . . . . . .
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4.4
The total photon number as a function of time . . . . . . . . . . . . .
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4.5
The total photon number as a function of time . . . . . . . . . . . . .
88
ix
5.1
Visualization of three coupled cavities in a PCS with non-zero χ(3) . . 103
5.2
Two-photon expectation value for three different cases
5.3
Comparison between the two-photon and the one-photon-only . . . . 107
5.4
Photon dynamics when the pump mode is lossless . . . . . . . . . . . 108
5.5
N1 (t) as a function of time . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6
Plot of dimensionless noise figure R (t) |G|2 /γp2
5.7
Plot of dimensionless R (t) |G|2 /γ̄ 2 . . . . . . . . . . . . . . . . . . . 113
6.1
Visualization of a CROW structure in a square-lattice PCS. . . . . . 118
6.2
Plot of frequency dispersion, quality factor, . . . . . . . . . . . . . . . 122
6.3
Plot of G(2) for the linear dispersion given in Eq. (6.30) . . . . . . . . 131
6.4
Spatial dispersion of the pump pulse due to propagating . . . . . . . 137
6.5
Comparison between Nsi (t) and N1 (t) for the square-lattice CROW . 139
6.6
Energy conservation mainly derives the nonlinear process . . . . . . . 141
6.7
Pair detection probability as a function of pump wavevector . . . . . 144
6.8
Detection probability for different cases of pump and signal separation. 146
x
. . . . . . . . 104
. . . . . . . . . . . . 112
List of Abbreviations
CAR Coincidence to Accidental Ratio
CC
Coupled Cavity
CROW Coupled Optical Resonator Waveguide
FDTD Finite Difference Time Domain
JC
Jaynes-Cummings
NLO Nonlinear Optics
PBG Photonic Band Gap
PCS Photonic Crystal Slab
PWE Plain Wave Expansion
QD
Quantum Dot
QED Quantum Electrodynamics
QM
Quasimode
SFWM Spontaneous Four Wave Mixing
xi
1
Chapter 1
Introduction
Photonics, and in particular integrated photonics, is an active area of research with
applications in biosensing, quantum communication and quantum information sciences. The key in many of these applications is to have a large degree of control over
light propagation in dielectric media. For example, light confinement in structured
dielectrics enables sensitivities to tiny refractive index changes of the environment
due to the presence of foreign bodies such as chemicals, molecules and viruses, which
can be of great use in sensing, particularly biosensing [1]. Another application is to
use such photonic structures to capture, manipulate and release light on nano-second
time scales [2]. This enables the use of photons robustness to environmental noises as
well as their fast travel times in quantum communication and quantum information
processing [3, 4].
Optical microcavities [5] can be used as key elements in controlling and manipulating light in many integrated applications [6–11]. They can confine light over small
regions of space for long periods of times and thus can enhance the light-matter interaction in many systems. Among all the different types of optical platforms [5],
2
photonic crystal slabs (PCSs) [12] offer optical cavities with both high quality factors (Q) and small mode volumes (V ) in an integrable fashion. The combination
of the high Q and the small V is of great importance in many applications in integrated photonics. There are other optical microcavities that can perform better
than PCS microcavities, but only in certain aspects. For example, whispering gallery
microtoroids and microspheres can support modes with very large Q’s of about 109 ,
but coupling in and out remains challenging, particularly in an integrated manner.
Also, microring resonators are integrable devices that offer better fabrication surface
smoothness compared to PCSs, but the large mode volume of the confined mode is
limiting for many applications. Indeed, the large mode volume is a general feature
of many whispering gallery modes such as in microtoroids and microspheres. In addition, by using PCSs, it is possible to design optical modes with different spatial
patterns [13–18], which brings extra flexibility compared to other platforms available.
Single cavities in PCSs have been used in many different devices for cavity quantum electrodynamics (QED) [10, 11, 21–29] and nonlinear optics (NLO) [9, 30, 31].
However, in certain applications, multiple resonances are needed to perform the
desired function. For example, biexciton states in QDs can emit two photons at
slightly different frequencies that can be used for the preparation of entangled photons. Coupled-cavity structures in PCSs, such as the one shown in Fig. 1.1a, offer
resonances that are close in frequency and have modal overlap in space. Therefore,
when coupled to QDs, they can be used to capture the both of the emitted photons
from biexcitonic emission. Also, in nonlinear optical devices, usually several fields of
different frequencies such as the pump field and the generated field, are interacting
over the nonlinear region. Coupled cavities (CC) in PCSs can be used to provide
3
(a) Two coupled cavities in a triangular PCS used for the all-optical coherent control of
vacuum Rabi oscillations [19]. Each cavity is formed by filling three holes in a row while
fine adjustments on neighboring holes might be done to optimize the device.
(b) A coupled optical resonator waveguide in a triangular PCS. Each individual cavity is
formed by filling one hole. A similar structure with a different design for the individual
cavity has been used to generate entangled photons [20].
Figure 1.1: Examples of CC structures in triangular PCSs made by perforating air
holes in dielectric slabs such as Si or GaAs. Such structures can be used
for integrated QED as well as integrated NLO.
multiple resonances [19, 20, 32–36]. In particular, coupled-resonator optical waveguide (CROW) structures [20, 37] can transmit light in a narrow range of frequencies
centered about the resonance frequency of the individual cavity used to build the
CROW. These structures also offer low group velocities for the propagating light that
results in pulse compression inside CROW. Therefore an enhanced nonlinear interaction over a range of resonant frequencies of the structure is expected. An example of
1.1. CAVITY QUANTUM ELECTRODYNAMICS
4
such a structure in PCSs is shown in Fig. 1.1b.
To design and fully exploit PCS-CC devices, further understanding of photon
dynamics in these systems is required. The majority of the theoretical modeling of
these devices has been done using finite-difference time-domain (FDTD) calculations,
which do not generally give insight into the detailed dynamics in photonic circuits.
Instead, a general formalism capable of incorporating mutual interactions between
different units on a PCS circuit is desired that outputs predictions of the operation of
the device. To this end, two inherent features of PCS-CC systems must be addressed:
1) The “scattering” loss due to only partial internal reflection of the confined light at
the slab surface [12] and 2) The non-orthogonality of the CC modes due to potential
asymmetries in designed circuits [38, 39].
The objective of this thesis is to develop a general mathematical formalism that
can be used to study leaky PCS-CC devices for use in both cavity-QED and NLO.
In the remainder of this introduction, a basic description of cavity-QED and NLO
is presented. In addition, a more detailed discussion of scattering loss and mode
non-orthogonality in PCSs is given along with the first steps towards a theoretical
treatment of systems with such characteristics.
1.1
Cavity quantum electrodynamics
Traditional cavity-QED [40, 41] is the study of the quantized modes of an electromagnetic field inside a cavity interacting with quantum emitters such as atoms, as
visualized in Fig. 1.2. Due to the imposed boundary conditions at the mirror surfaces,
solutions to Maxwell’s equation are different than solutions in free space. Therefore,
the modes accessible to an emitter placed in between mirrors are different than in
1.1. CAVITY QUANTUM ELECTRODYNAMICS
5
emitter
cavity
γ
g
Figure 1.2: Visualize of a quantum emitter inside an optical cavity. The important
parameters are cavity loss rate, γ, and coupling between the field inside
cavity and the emitter, g. In this schematic, any loss associated with the
emitter is neglected.
free space and this results in a modified behavior of the quantum emitter.
The Jaynes-Cumming (JC) Hamiltonian [42, 43] is a standard approach when
studying cavity-QED. In this model, the quantum emitters are considered to be simply
two-level systems described by the ground state |gi and the excited state |ei, with
corresponding frequencies ωg and ωe . Assuming that the cavity supports a single
mode and only one emitter is placed inside it, the JC Hamiltonian describing the full
coupled behavior, including the interaction, is given by
H ≡ H emitter + H f ield + H int
=
1
~ ω eg σ z + ~ ω a† a + g σ + a + a† σ − .
2
(1.1)
(1.2)
Here, ωeg = ωe − ωg is the transition frequency of the emitter, ω is the resonant
frequency of the single-mode cavity and g is the coupling constant between the field
and the cavity mode. Note that the Hamiltonian of Eq. (1.2) is Hermitian and the
cavity mode is considered to be lossless. In reality though, the cavity mode is leaky
with some known loss rate γ for the stored field amplitude. As we will discuss shortly,
1.1. CAVITY QUANTUM ELECTRODYNAMICS
6
the cavity leakage plays an important role in the coupled cavity-emitter dynamics and
must be addressed appropriately.
In the JC Hamiltonian of Eq. (1.2), the field quantization can be done in an
standard manner where the cavity mode is described by a set of annihilation and
creation operators, a and a† , that satisfy the bosonic commutation relation
a, a† = 1.
(1.3)
The σ z , σ + and σ − are the emitter-associated operators defined as
σ + ≡ |ei hg|
(1.4)
σ − ≡ |gi he|
(1.5)
σ z ≡ |ei he| − |gi hg| ,
(1.6)
where, σ+ takes the emitter to the excited state from the ground state, σ− does the
reverse and σz represents the population inversion of the emitter. It is easy to see
that the three above-defined operators follow the same algebra as the Pauli matrices
for a spin 1/2 particle:
σ+, σ− = σz
(1.7)
σ + , σ z = 2σ +
(1.8)
σ − , σ z = −2σ − .
(1.9)
1.1. CAVITY QUANTUM ELECTRODYNAMICS
7
The Hamiltonian in Eq. (1.2) can be extended to multimode systems interacting with
a number of emitters in an straightforward manner as long as a well-defined basis of
orthogonal field modes is available. However, as will be discussed in this thesis, extra
caution must be taken when dealing with PCS systems as the modes of CC systems
in these platforms can be non-orthogonal.
The exact behavior of an emitter in the presence of the cavity depends on how
the coupling g compares to the cavity loss, γ. In general, two situations are possible:
1) In the weak coupling regime, where g γ, energy transfer between the cavity
mode and the emitter is slower than the leakage. In other words, photons emitted
from QD are gone before they can be reabsorbed. However, the presence of the cavity
still modifies the spontaneous emission behavior of the emitter as quantified by the
so-called Purcell factor, Fp , given by
3
Fp = 2
4π
λ0
n
3
Q
,
V
(1.10)
where, λ0 is the free-space wavelength, n is the refractive index of the medium, and
Q and V are the quality factor and the mode volume of the cavity, respectively. The
larger the Purcell factor is for a given mode, the larger the corresponding spontaneous
emission rate will be. It is clear that, in order to achieve large Purcell factors, a large
Q and small V are desired, which both can be achieved at the same time in PCS
cavities.
2) In the strong coupling regime, where g γ, energy transfer between the
two parties is faster than leakage. In other words, the field inside cavity and the
emitter exchange excitations many times before photons leak outside. The wellknown phenomenon of Rabi splitting where energy levels of the coupled system are
1.2. NONLINEAR OPTICS USING MICROCAVITIES
8
shifted in frequency with respect to both the bare cavity and the bare emitter occur
in the strong coupling regime.
Self-assembled QDs coupled to semiconductor optical microcavities [5], such as
PCS cavities, can serve as integrated alternatives to lab-size cavity-QED setups while
all the physics remains unchanged. As mentioned earlier, large Q and small V are
easily achievable in PCS structures, and therefore a large enhancement of Fp is expected in the weak coupling regime. In fact, controlling the spontaneous emission rate
of a QD that is carefully (at the right location with respect to the modal shape of
the cavity mode) coupled to the localized mode of a defect cavity in a PCS, has been
successfully performed [23, 25, 44]. This is important for making on-demand single
photon sources, where exciting the QD can produce a single photon whenever needed.
In the strong coupling regime, successful demonstration of Rabi splitting in coupled
cavity-QD systems has been achieved [24,45]. In addition, bringing PCS cavities into
strong coupling with QDs [28, 29, 46, 47] has enabled many integrated devices such as
entangled photon sources [26], optical nonlinearities at the single-photon level [10,27]
and quantum logic gates [11].
1.2
Nonlinear optics using microcavities
Nonlinear optics is the study of the nonlinear response of a material when an external,
intense electromagnetic fields is applied. In a standard treatment, the electromagnetic
field induces a time-dependent polarization that can re-radiate the electromagnetic
energy either at same frequency as the incident field or at some other frequencies
(see schematic in Fig. 1.3). At low intensities, the induced polarization is linearly
dependent on the applied filed amplitude. Neglecting spatial and temporal dispersion,
1.2. NONLINEAR OPTICS USING MICROCAVITIES
9
as well as anisotropy, we have
PL (r, t) = 0 χ(1) E (r, t) ,
(1.11)
where PL (r, t) is the induced linear polarization, E (r, t) is the external electric field
and χ(1) is the linear susceptibility of the medium. Re-radiation from this linear
polarization occurs at the same frequency as the field.
When the intensity is increased, higher order responses can be generated, which
have a nonlinear dependence on the incident field, such that
PN L (r, t) = 0 χ(2) E2 (r, t) + 0 χ(3) E3 (r, t) + · · · ,
(1.12)
where again spatial dispersion, temporal dispersion and anisotropy effects are neglected. Here, PN L (r, t) is the induced nonlinear polarization, χ(2) is the second
order nonlinear susceptibility tensor and χ(3) is the third order nonlinear susceptibility tensor. Higher order nonlinearities can be constructed in the same manner.
Re-radiation due to the nonlinearly induced polarization can occur either at initial
applied frequency or at some other frequencies as long as energy is conserved. Considering a single-frequency pump field, the well-known phenomena of second and third
harmonic generation occurs through the second and third order nonlinearities, respectively [48]. Using two pump fields at two different frequencies can lead to phenomena
such as sum frequency generation, difference frequency generation, and four-wave
mixing [48].
The nonlinear optical susceptibilities of different materials are usually small, which
means that one requires strong pump intensities to generate appreciable nonlinear
1.3. FEATURES OF COUPLED-CAVITY PCS STRUCTURES
ω
6=
1
10
ωp
Strong Pump Laser (ωp )
ωp
Nonlinear Medium
ω2
6=
ωp
Figure 1.3: Schematic of a nonlinear optical process where a strong pump laser generates light at frequencies different than the pump frequency. The pump
laser is also present at the output, but possibly depleted.
fields. However, in integrated devices such as PCS cavities, the strong confinement of
the light increases the intensities, which enhances the effective nonlinear interaction
between the pump and the medium and therefore increases the device efficiency.
This has been exploited in a variety of different nonlinear experiments performed
in PCSs, including optical bistability [30, 49, 50], optical switching [9] and four-wave
mixing [36, 51].
1.3
Features of coupled-cavity PCS structures
As mentioned earlier, scattering loss and mode non-orthogonality are two features
of PCS-CC systems that must be addressed in any study of these systems. Here, I
discuss these issues in more detail and present the tools and approaches taken in this
thesis to include them in our formulation of these systems.
PCS cavities are known to be inherently leaky due to the finite thickness of the
1.3. FEATURES OF COUPLED-CAVITY PCS STRUCTURES
k1 =
11
2πn1
λ0
n1
2πn2
k2 =
λ0
n2 > n1
Figure 1.4: Total internal reflection at the interface between two dielectrics with different refractive indices. There are always wavevectors in high-index
medium that won’t couple to any wavevector in low-index medium.
slab material in which the periodic structure is embedded. This is referred to as
“scattering” loss, as it is simply due to partial transmission of the confined light
at the slab surface. As sketched in Fig. 1.4, when propagating from a high index
medium into a low index medium, the tangential wavevector along the boundary must
be conserved. For a given wavelength, there will be wavevector components in the
high index medium that provide larger tangential wavevectors at the surface than
the maximum allowed wavevector in low index medium. These components undergo
total reflection upon incidence on the boundary and will always remain in the higher
index region. On the other hand, there will be always components of the light that
scatter into low index region and propagate there, freely. These are the wavevectors
for which the tangential components are matched on both sides of the surface. The
escaped light carries energy away and results in leaky microcavities in PCSs.
Scattering loss can either limit or at least degrade the performance of cavitybased devices. For example, consider a four-wave mixing experiment in a nonlinear
1.3. FEATURES OF COUPLED-CAVITY PCS STRUCTURES
12
CROW [20, 37], where the system is pumped at some frequency to generate a pair
of photons at neighboring frequencies, below and above pump frequency. Generated
photon pairs can be used for heralded single photon sources, where detecting one
photon (idler) heralds the existence of the other photon (signal). But, in the presence
of the loss, as is the case in PCSs, delivery of a signal photon at the output port
of the device is not certain, even when the idler photon is being detected. This
can result in faulty operation of the next unit on an integrated circuit, designed for
quantum information processing for example. Therefore, it is necessary to include
the scattering loss in any study of these systems.
The standard quantum mechanical treatment of leaky systems with a number of
optical modes can be done using the so-called “Master” equation [52]
i
dρS (t)
= − [H (t) , ρS (t)] + L (ρS (t)) ,
dt
~
(1.13)
where, ρS (t) is the system density matrix, H (t) is a Hermitian Hamiltonian responsible for the unitary evolution of the system and L is the Lindblad super-operator
that includes leakage information associated with different optical channels of the
system. In this thesis, a different strategy is taken that uses leaky modes of PCSCC structures as a basis onto which the system Hamiltonian is projected. As will
be shown, this leads to a non-Hermitian projected Hamiltonian for the system that
includes leakage right from the start. However, the projected Hamiltonian does not
give accurate evolution of the density matrix and system observables, as the probability is not conserved in the system. To overcome this issue, appropriate Lindblad
terms, in generalization to the standard Lindblad super-operator, are found that when
added to the dynamical equation for the density matrix, conservation of probability
1.4. THESIS OVERVIEW
13
is guaranteed.
In addition to loss, when individual cavities are coupled in a PCS, depending
on the degree of symmetry held by the full structure, the full CC modes of the
system can form a non-orthogonal basis [38, 39]. In the past, non-orthogonal leaky
modes of the PCS-CC structures have been used to calculate the optical Green tensor
and to formulate the quantum Hamiltonian for systems with only one photon, in our
group [53]. However, a general quantum mechanical Master equation with as many as
CC modes and photons, was needed to be developed. Indeed, the projection technique
discussed in the previous paragraph, has been done such that the non-orthogonality
of the basis is included as well. Therefore, we offer a self-consistent mathematical
formalism that can be used in any study of PCS-CC structures when many QDs and
photons are present.
Although, PCS platforms are the focus of attention in this thesis, from a theoretical point of view, the developed formalism can be adopted to study other leaky
optical systems, possibly with non-orthogonal modes, in an straightforward manner.
1.4
Thesis overview
The outline of this thesis is as follows. In chapter 2, the basic discussion of PCS structures, including the photonic band gap (PBG), localized defect modes and numerical
calculation methods for obtaining such modes is presented. In chapter 3, PCS-CC
modes are used to project the system dynamics via a non-Hermitian projection operator. In chapter 4, the quantum Master equation represented in the basis of PCS
modes is used to study photon dynamics in coupled cavity-QD systems governed by
the JC Hamiltonian. In chapter 5, the adjoint quantum Master equation represented
1.4. THESIS OVERVIEW
14
in PCS modes is used to study quantum NLO in CC systems. In particular, the
pair generation problem via spontaneous four wave mixing (SFWM) is considered. In
chapter 6, to complete the work done on pair generation, pair detection probabilities
are calculated for CROW structures, where give information on finding generated
pairs at specified locations in the system. Finally, conclusions and future directions
are presented in chapter 7.
15
Chapter 2
Photonic Crystal Slabs and
Quasimodes
Index difference guiding of light is a common effect used to control light in geometries
such as optical fibers and silicon waveguides. However, as a serious limitation, one
needs to avoid sharp bends in order to keep the optical loss low in such systems.
Photonic Crystals (PCs), are structured dielectrics, where the periodic changes in dielectric index of refractive forbids light propagation over a range of frequencies in any
direction within the periodic medium. This is a fundamentally different mechanism
than index difference guiding that has been used to control light flow in arbitrary
geometries [12].
In this chapter, a stack of dielectric layers is used to demonstrate how the periodic
change in refractive index of the medium leads to a PBG. In addition, it is shown
how a 2D PBG can be employed to obtain a full 3D control over light propagation in
PCS structures, particularly to make defect cavities for optical devices.
2.1. WAVES IN PERIODIC DIELECTRIC MEDIUM
l1
1
16
l2
1
1
1
1
z
2
2
2
2
Figure 2.1: Multilayer dielectric slabs as a 1D photonic crystal, where periodic modulation of the index can open up a PBG at the edge of the Brillouin zone.
This is essentially the same as the nearly free electron model in solid-state
physics used to calculate electronic bands in a metal.
2.1
Waves in periodic dielectric medium
Consider a periodic stack of alternating layers of two different dielectric materials
as shown in Fig. 2.1. Each layer extends to infinity in the transverse direction (xy
plane) but has a finite length of either l1 or l2 along propagation direction, z. Due to
the continuous symmetry of the layers in transverse direction, solutions to Maxwell’s
equations do not depend on x or y. Therefore, Maxwell’s wave equation reduces to
ω 2
d2 Ek (z)
+
(z)
Ek (z) = 0.
dz 2
c
(2.1)
Here, Ek (z) is the solution corresponding to the kth wavevector in the periodic multilayer structure that is assumed to be polarized along x, ω is the associated real
frequency, (z) is the periodic dielectric function, and c is the speed of the light in
vacuum.
The periodic symmetry of the layered structure suggests that one Fourier expands
2.1. WAVES IN PERIODIC DIELECTRIC MEDIUM
17
dielectric function, (z), as well as the solution, Ek (z), as:
(z) =
X
εm exp
m
Ek (z) =
X
m
2iπmz
l
2πm
Em exp i k +
z ,
l
(2.2)
(2.3)
where, εm and Em are the expansion coefficients for the dielectric function and the
electric field, respectively, and l = l1 +l2 is the period of the stack structure. Inserting
Eqs. (2.2) and (2.3) into Eq. (2.1) leads to
−
X
m
2
2πm
2πm
Em k +
exp i k +
z
l
l
X ω 2
2iπm1 z
2πm2
+
εm1 Em2 exp
exp i k +
z = 0.
c
l
l
m ,m
1
(2.4)
2
Now, consider the case where only the E0 and E−1 are significant. This is valid
when we are interested in the band behavior near the edge of the Brillouin zone at
k = π/l. Indeed, E0 and E−1 are associated with forward-going and backward-going
waves in the periodic system that interact the most strongly at k = π/l, simply
because they are degenerate (have the same frequency). It is then straightforward to
see that only the following two equations are relevant:
ω 2 ω 2
2
=
ε1 E−1
E0 k − ε0
c
c
(
)
2
ω 2
ω 2
2π
E−1
k−
− ε0
=
ε−1 E0 .
l
c
c
(2.5)
(2.6)
This is effectively a two-band model for the dispersion of the multilayer dielectric
2.1. WAVES IN PERIODIC DIELECTRIC MEDIUM
18
system. To obtain the stack dispersion, the determinant of the coefficients must go
to zero
ω 2
ω 2
2
k − ε0
−
ε1
c
c
2
ω 2
ω 2
2π
−
ε
k
−
−
ε
−1
0
c
l
c
= 0,
(2.7)
which leads to a quadratic equation in both ω 2 and k 2
(
ε20 − |ε1 |2 ω 4 − ε0 c2
2 )
2
2π
2π
2
2
4 2
k + k−
ω +c k k−
= 0.
l
l
(2.8)
This leads to the dispersion relation for the 1D periodic dielectric stack:


v
2 ) 
u

2

2π
u


2π 
2 2


2
k2 + k −
u


4 ε0 − |ε1 | k k −


l
u
l
u
1 ± u1 −
"
2 #2  .

2 ε20 − |ε1 |2
u


2π


t


ε20 k 2 + k −




l
(
ε 0 c2
ω2 ≈
(2.9)
To visualize the above-derived dispersion, the Fourier components, ε0 and ε1 , of
the periodic dielectric stack must be determined. The stack shown in Fig. 2.1 can be
mathematically represented by
(z) =



1



n (l1 + l2 ) ≤ z ≤ (n + 1) l1 + nl2
(2.10)




 2 (n + 1) l1 + nl2 ≤ z ≤ (n + 1) (l1 + l2 ) .
Here, n is an integer that labels layers of the 1D periodic system. The zeroth and the
2.1. WAVES IN PERIODIC DIELECTRIC MEDIUM
19
frequency (2πc/l)
0.4
0.3
0.2
0.1
0
0
0.25
0.5
0
0.25
0.5
0
0.25
0.5
wavevector (2π/l)
Figure 2.2: Two-band model dispersion for a 1D stack of alternating layers of dielectrics for three different cases: On the left, 1 = 2 = 11.56 results in
a continuous band. In the middle, a 30% change in dielectric constant,
1 = 11.56 and 2 = 0.71 , opens a small gap at the edge of the Brillouin
zone. On the right, a wide gap is obtained as a result of 70% difference in
the dielectric constant of the adjacent layers, 1 = 11.56 and 2 = 0.31 .
first order Fourier expansion coefficients are then
ˆ
ε1
l
l1 1 + l2 2
(2.11)
l1 + l2
0
ˆ l
2iπz
1 − 2
πl1
iπl1
=
exp −
(z) dz =
sin
exp −
. (2.12)
l
π
l
l
0
ε0 =
(z) dz =
2.2. PHOTONIC CRYSTAL SLABS
20
Assuming that l1 = l2 , these reduce to
ε0 =
1 + 2
2
(2.13)
ε1 =
1 − 2
.
iπ
(2.14)
Therefore, ε0 is the average dielectric constant of the system, while ε1 is proportional
to the index difference between two adjacent layers.
In Fig. 2.2, the dispersion (2.9) for three different cases is plotted as a function
of k over the first Brillouin zone. On the left, there is no index difference between
layers, 1 = 2 = 11.56, which is equivalent to having a homogeneous 1D system
with continuous symmetry and linear dispersion everywhere (including the edge of
the Brillouin zone). In the middle, an index difference of 1 − 2 = 0.31 between
π
layers opens up a band gap at at k = . On the right, increasing the index difference
l
to 1 − 2 = 0.71 makes the gap even wider as seen from the plots. Therefore, a
periodic index modulation of the medium results in frequency gap where no wave can
propagate inside the stack. This is similar to the electronic band gap that electrons
experience in the periodic potential of a crystal.
2.2
Photonic crystal slabs
The idea of a three-dimensional PBG was first discussed by E. Yablonovitch [54] and
S. John [55] in 1987. In 1991, it was experimentally verified by E. Yablonovitch et
al. [56] in a face-centered-cubic lattice for microwaves. Since then, several complex,
multilayer lithographic procedures have been used to demonstrate a complete 3D PBG
at optical wavelengths [57–59], using complicated multilayer fabrication procedures.
2.2. PHOTONIC CRYSTAL SLABS
21
y
z
x
Figure 2.3: Schematic of the confinement mechanisms in a square lattice PCS, where
two crystals are made on both sides of a rectangular slab and the light
is confined in the middle. On the top panel, a top view of the slab is
shown, where reflection of electromagnetic waves off the periodic region
is indicated, assuming that the frequency of the impinging waves sits
inside the PBG. In the bottom panel, a side view of the PCS is shown at
the location of the dashed red line, where reflection of the electromagnetic
waves off the interface between dielectric and air from top and bottom of
the slab are shown.
Quasi two-dimensional PBG structures can be fabricated using mature nanofabrication techniques to produce a periodic arrangement of air-holes in slab dielectrics such as Si and GaAs, with much better fabrication qualities than 3D photonic
crystals. As depicted in Fig. 2.3, in such slabs, light propagation is prohibited within
the slab due to 2D PBG and is limited perpendicular to slab due to total internal
reflection and thereby 3D confinement of the light can be achieved.
Since a complete PBG that forbids light propagation in all directions does not
exist in PCSs, the band structure analysis is slightly different than in 3D PCs. In
2.3. SINGLE-CAVITY QUASIMODE
22
Fig. 2.4, an example of a photonic band structure for a square PCS is plotted. The
specifics are given on the figure caption. The gray solid area on the plot represents
the region above the light cone where leaky modes of the PCS are located. All the
modes below the light cone are indeed guided modes of the slab and experience no
scattering loss. The solid gold region represents the effective PBG between the first
and the second bands where no wave is allowed to propagate. If one removes the
overlayed light cone region, no real gap exist when considering with both guided and
leaky modes.
2.3
Single-cavity quasimode
Defects can be made in PCSs by removing one or a few of the lattice sites, by say, filling
some holes with slab material. Defects can support localized modes with wavelengthsized volumes and sharp resonances in frequency. To obtain modes and the corresponding frequencies of such modes, Maxwell’s equations in 3D must be solved in a
non-magnetic medium:
∇ × E (r, t) = −
∇ × H (r, t) =
1 ∂B (r, t)
c
∂t
1 ∂D (r, t)
c
∂t
(2.15)
(2.16)
∇ · D (r, t) = 0
(2.17)
∇ · B (r, t) = 0,
(2.18)
where, D = (r) E and H = µ−1
0 B.
Given the complex geometry of a PCS, analytic solutions to Maxwell’s equations
2.3. SINGLE-CAVITY QUASIMODE
23
0.7
leaky modes region
frequency (ωc/2πd)
0.6
0.5
0.4
0.3
0.2
0.1
0
Γ
X
M
Γ
wavevector (k)
Figure 2.4: The band structure for a square-lattice PCS calculated using the MIT
Photonic Bands (MPB) package. Here, Γ = (0, 0), X = (0, 1) π/d and
K = (1, 1) π/d are the locations of the high-symmetry corners of the
reduced Brillouin zone, where d is the lattice period. For this calculation,
the hole radius and the slab thickness were r = 0.4 d and h = 0.7 d,
respectively. The slab refractive index was also n = 3.4. The gray solid
fill represents the light cone where the modes below are the guided modes
of the slab and the modes above are the leaky modes. Here, only TE
modes, the modes where the electric field is mainly within the slab and
magnetic field is mainly in transverse direction z, are plotted. TE modes
hold even symmetry with respect to the slab middle along z axis. The
gold solid fill represent the PBG, which ranges from ωd/2πc = 0.3079 to
ωd/2πc = 0.3408.
2.3. SINGLE-CAVITY QUASIMODE
24
Figure 2.5: Visualization of a PCS cavity inside a computational volume. The computational volume is large enough to avoid any unphysical effects due
to space truncation. Also, PML boundary conditions are applied on all
outer surfaces of the box.
are not available. Numerical methods, such as a plane-wave expansion (PWE) [60]
and finite-difference time-domain (FDTD) [61] need to be employed instead. For the
purpose of this thesis, FDTD was used to obtain the defect modes and related information such as the resonant frequency and the quality factor. Normally, a computational
volume, Vc , that contains the PCS defect structure and some of the surrounding volume, is considered. The computational volume is chosen large enough to eliminate
any numerical errors in the calculated mode due to space truncation. On the all of
outer surfaces of this computational volume a Perfectly Matched Layer (PML) [62]
is applied that simulates the system as if it was not truncated spatially. This eliminates all of the non-physical back-reflections off the boundaries of the computational
volume (see Fig. 2.5).
2.3. SINGLE-CAVITY QUASIMODE
25
As an example, the FDTD-calculated mode, M̃ (r), for a single-hole defect in a
square-lattice PCS is plotted in Fig. 2.6. As shown, the designed cavity is single
mode and the mode is strongly localized about the defect region, both in the plane
of slab and in the transverse direction. A complex frequency, ω̃ = ω − iγ, is obtained
from the FDTD calculation, the imaginary part of which describes the cavity leakage.
More details are given in the caption.
The electric field associated with the mode inside the computational volume can
be represented as
∗
E (r, t) = M̃ (r) e−iω̃t + M̃∗ (r) eiω̃ t .
(2.19)
Outside Vc , no accurate information on the mode behavior is available to us. Indeed,
it has been shown that these modes diverge at large distances from the slab and
therefore calculating quantities such as the mode volume requires care [63, 64]. This
can be important for example for integrated cavity-QED applications in which an
accurate value for the mode volume can be important in the estimation of quantities
such as Purcell factor. In general, this seems to be a serious issue when low-Q modes,
such as the ones in plasmonic platforms, are concerned. However, when dealing with
the high-Q modes, the strong confinement of the field to the cavity region generally
makes corrections to standard definitions due to such considerations very small. Such
localized modes provide the dominant response of the system and can be used to
study optical behavior of PCS in a narrow range of frequency inside the computational
region. Therefore, it is appropriate for them to be called PCS “quasimodes” (QMs),
as opposed to the system true modes that form a complete well defined basis over the
entire space.
26
y
2
1
0
3.1
3.2
3.3
wavelength (µm)
z
monitor value (a.u.)
2.4. COUPLED-CAVITY QUASIMODES
x
Figure 2.6: FDTD calculated mode for a single defect within a PCS of thickness
h = 0.7 d and hole radius r = 0.4 d, where d is the lattice period. On the
left, the cavity response is plotted in frequency domain, indicating singlemode behavior. On the right, the Ex component of the mode is plotted,
first from the top at z = 0 and then form the side at y = 0. The resonant
complex frequency obtained from FDTD is ω̃d/2πc = 0.3120−i7.2×10−6 ,
corresponding to a quality factor of Q u 20, 000. Note that the real part
of the frequency sits inside the PBG calculated using MPB and presented
in Fig. 2.4.
2.4
Coupled-cavity quasimodes
Obtaining QMs for PCS-CC systems requires solving Maxwell’s equations, Eq. (2.15)
- Eq. (2.18), using the refractive index function of the full structure. As far as
FDTD implementation is concerned, this does not necessarily cause any further complications. However, CCs can often support multiple resonances that are close in
frequency and overlap in space, which could make FDTD impractical for identifying
2.4. COUPLED-CAVITY QUASIMODES
27
system modes in a reasonable computational time. It could become even worse and
one may not be able to find them at all if the system symmetry is low. In addition, the
computational domain for CCs is normally large compared to that for single cavities
and thus adds to the need for much greater computational resources and time.
FDTD can be still used when either there are only few coupled cavities (see
Fig. (1.1a)) or when the CC structure has sufficient symmetry (see Fig. (1.1b)). When
FDTD is not practical, tight-binding (TB) calculations can be used to obtain QMs
of the PCS-CC, once the QMs of the individual defects are obtained using FDTD
calculations. Note that, in many cases, identical individual cavities are used to build
the CC slab structure, and therefore only one FDTD calculation, for an individual
defect, must be performed.
We now discuss the TB approach to CC systems. We start with the Helmholtz
equation for the individual defects,
f q (r) −
∇×∇×M
e 2q
Ω
q (r) M̃q (r) = 0,
c2
(2.20)
where Ω̃q is the complex frequency of qth QM corresponding to the qth single defect
and q (r) is the dielectric function for the PCS with only qth cavity present. The QMs
e m (r), satisfy the same equation but using the full dielectric
of the entire PCS-CC, N
function, (r),
2
em
e m (r) − ω
e m (r) = 0,
(r) N
∇×∇×N
c2
(2.21)
where ω
em is the complex frequency of the mth QM of the PCS-CC system. The TB
2.4. COUPLED-CAVITY QUASIMODES
28
expansion of full QMs in terms of the single-defect QMs can be written as
e m (r) =
N
X
f q (r) ,
vemq M
(2.22)
q
where vemq are the unknown expansion coefficients. Substituting this expansion into
Eq. (2.21) and then using the Eq. (2.20) leads to
X
2
f q (r) = ω
e 2 q (r) M
vemq Ω
em
(r)
q
X
f q (r) .
vemq M
(2.23)
q
q
f ∗ (r) and integrating over
Now, multiplying both sides of the above equation with M
p
all space, it is easy to see that
X
2
epq Ω
e 2 vemq = ω
A
em
q
q
X
epq vemq ,
B
(2.24)
q
e and the coupling matrix, B
e , defined as
with the overlap matrix, A,
ˆ
epq =
A
f ∗ (r) · M
f q (r)
d3 rq (r) M
p
(2.25)
f ∗ (r) · M
f q (r) .
d3 r (r) M
p
(2.26)
ˆ
epq =
B
The overlap matrix uses only the single-defect dielectric function whereas the coupling matrix has contributions from all of the individual defects in the coupled-defect
structure.
Eq. (2.24) can be written in the following form
2 e
eΩ
ev
em = ω
em ,
A
em
Bv
(2.27)
2.5. SUMMARY
29
e is a diagonal matrix, whose elements are Ω̃2 and v
em is a vector, whose
where Ω
q
elements are vemq . This is a generalized non-Hermitian eigenvalue problem that can
be solved numerically to obtain the complex frequencies of the CC modes as well as
the TB expansion coefficients of Eq. (2.22). As mentioned earlier, in many cases all of
the single defects are exactly the same. This means that they support the same QM
with the same resonance frequency. In this case, the generalized eigenvalue problem
of Eq. (2.27) simplifies to
2 e
ev
e0 A
em = ω
em .
Ω
em
Bv
(2.28)
e a different coupling matrix, K
e can be
Rather than using the coupling matrix B,
defined that incorporates the difference dielectric function, δq = (r) − q (r), such
that
ˆ
e pq =
K
f ∗ (r) · M
f q (r) .
d3 rδq (r) M
p
(2.29)
This coupling matrix is easier to calculate as it is zero everywhere inside computational volume except on the qth defect, inside slab material. Using this, the following
eigenvalue problem needs to be solved to obtain the coupled-defect QMs of the system:
2
ev
e +K
e v
e0 A
em = ω
em .
Ω
em
A
2.5
(2.30)
Summary
In this chapter, the fundamental concept of the PBG that is crucial for light manipulation in photonic crystals was discussed. In particular, it was mentioned how a
two dimensional PBG can be combined with total internal reflection in a dielectric
slab to achieve effective three dimensional confinement of the light in PCS structures,
2.5. SUMMARY
30
such as defect microcavities. In addition, the standard techniques that one can use
to obtain quasimodes of PCS microcavities was reviewed, both for single-cavity and
coupled-cavity designs. Because defect modes are leaky, any comprehensive study of
quantum optics and nonlinear quantum optics in these platforms must take scattering
loss into consideration. In the next chapter, we seek such solutions where the mode
loss is incorporated in a self-consistent manner. As will be seen, this can be done by
projecting the dynamics in the frequency range of interest onto the quasimodes.
31
Chapter 3
Quasimode Projection Technique
As discussed in the previous chapter, QMs can be used to study the dominant optical
response of PCS devices in the frequency range of interest [21, 38, 39, 53, 65]. In
this chapter, the elements of a non-standard, non-Hermitian projection technique
are presented, where the system QMs are used as the basis. The technique will be
used to project the free-field Hamiltonian, which contains the leakage information of
the PCS-CC structures. Any interaction Hamiltonian, both a field-QD interaction
and a nonlinear light-matter interaction, can be added in a straightforward manner.
Indeed, the quantum Master equation obtained for the projected non-Hermitian freefield Hamiltonian preserves its form as long as only Hermitian terms are added to the
system Hamiltonian, as there is no change in the loss characteristics of the system.
The chapter elaborates the technical details of the work presented in [66].
3.1. THE BASIS OF TRUE MODES
3.1
32
The basis of true modes
Consider the solutions, f̃µ (r), to the Helmholtz equation
∇ × ∇ × f̃µ (r) −
ωµ2
(r) f̃µ (r) = 0,
c2
(3.1)
inside of the very large volume, V , surrounding the PCS of interest (see Fig. 3.1),
where (r) is the structure dielectric function, ωµ is the real frequency of the µth
mode, c is the speed of the light in vacuum and the field goes to zero at the boundary. Alternatively, the boundary condition might be chosen to be periodic in some
situations. These solutions are considered to be “true modes” of the system. Far
away from the PCS, true modes are close to plane waves in free space, whereas close
to the PCS, due to the presence of the defects, sharp resonances may appear. The
true modes form an orthogonal basis over V such that
ˆ
d3 r (r) f̃µ∗ (r) · f̃ν (r) = δ ⊥ (µ − ν) ,
(3.2)
V
where the transverse delta function, δ ⊥ , indicates that only transverse solutions that
h
i
satisfy ∇ · (r) f̃µ (r) = 0 are considered. True modes also form a complete basis
such that
ˆ
dµ f̃µ∗ (r) · f̃µ (r0 ) = δ ⊥ (r − r0 ) .
(3.3)
They thus form a well-defined basis with which PCS structures can be studied, in
theory. Unfortunately, the true modes form a continuum, which generally makes them
impractical for computational purposes, unless extensive symmetries are present, for
example in defect-free structures where they form the Bloch modes [60].
3.2. THE BASIS OF QUASIMODES
33
Vc
V
PCS
entire space
Figure 3.1: Schematic of a PCS and the surrounding environment. The term “entire
space” is used to denote that the region surrounding the PCS structure is
indeed very large that terminates in either a hard boundary or a periodic
boundary, far from the structure. This is the volume within which the
true modes are defined.
3.2
The basis of quasimodes
In contrast to true modes, QMs do not form a complete basis, but can be expanded
in terms of true modes as
ˆ
e m (r) =
N
dµ c̃mµ f̃µ (r) ,
(3.4)
e m (r) is the mth QM of the system and
where, N
ˆ
e m (r) ,
d3 r (r) f̃µ∗ (r) · N
c̃mµ =
V
(3.5)
3.2. THE BASIS OF QUASIMODES
34
are the expansion coefficients. Since true modes are defined over the entire space,
integration in Eq. (3.5) is over the entire space volume, V . In practice, FDTD calculates the QMs only inside of computational volume, Vc . Therefore, one can define
e c (r), such that
restricted QMs, N
m
e c (r) =
N
m



e m (r) r ∈ Vc
N


0
(3.6)
elsewhere.
Similar to the full QMs, restricted QMs can be also expressed in terms of true modes
as
ˆ
e c (r) =
N
m
where
dµ q̃mµ f̃µ (r) ,
(3.7)
ˆ
e c (r) ,
d3 r (r) f̃µ∗ (r) · N
m
q̃mµ =
(3.8)
V
are the new expansion coefficients for the restricted QMs. Mathematically, integration
in Eq. (3.8) runs over the entire space as well, however, due to the specific definition
of the restricted QMs one only needs to perform integrals over the computational
volume. As shown later, defining two different types of QMs is necessary to obtain
the appropriate non-Hermitian Hamiltonian for lossy PCS structures.
As discussed briefly in the introduction chapter, QMs for PCS-CC structures are
in general non-orthogonal. Even for the simple two-defect structure shown in Fig.
3.2, an asymmetry introduced in the defect locations results in a non-zero overlap
integral between different QMs, where the overlap integral between QMs is defined
as
ˆ
emn ≡
O
e c∗ (r) · N
e n (r) .
d3 r (r) N
m
(3.9)
3.3. QUASIMODE PHOTON-NUMBER STATES
35
Figure 3.2: Schematic of two coupled defects in a PCS. On the left, defect are placed
symmetrically; this makes the QMs of the total structure orthogonal. On
the right, one of the defect is shifted to the right and as a result, the QMs
are not orthogonal anymore.
It is easy to see that
ˆ
emn =
O
ˆ
dµ
ˆ
dν
∗
q̃mµ
c̃nν
d3 r (r)f̃µ∗ (r) · f̃ν (r)
(3.10)
V
ˆ
=
ˆ
dµ
∗
dν q̃mµ
c̃nν δ (µ − ν) .
(3.11)
ˆ
=
∗
dµ q̃mµ
c̃nµ .
(3.12)
emm = 1.
Without any loss of generality, QMs can be normalized such that O
3.3
Quasimode photon-number states
Similar to the standard quantization of plane-waves in free space [43], quantized true
modes [67, 68] can be described using photon creation and annihilation operators, a†µ
and aµ , that satisfy the bosonic commutation relation,
h
i
aµ , a†µ0 = δµµ0 .
(3.13)
3.3. QUASIMODE PHOTON-NUMBER STATES
36
Accordingly, the one-photon state for the µth true mode is defined through the action
of an annihilation operator on vacuum
|fµ i = a†µ |0i ,
(3.14)
where subsequent operations of creation and annihilation operators, simply add and
remove photons, respectively.
Now, using Eqs. (3.4) and (3.7), the one-photon state for the mth unrestricted
QM can be constructed as
ˆ
dµ c̃mµ a†µ |0i .
|φm i =
(3.15)
Similarly, the adjoint one-photon state for the mth restricted QM is
ˆ
∗
dµ q̃mµ
h0| aµ .
hχm | =
(3.16)
As a direct consequence of the above definitions, we find
ˆ
hχm |φn i =
ˆ
dµ
ˆ
=
∗
dν q̃mµ
c̃nν 0 aµ a†ν 0
(3.17)
∗
dν q̃mµ
c̃nν δµν
(3.18)
ˆ
dµ
ˆ
=
∗
dµ q̃mµ
c̃nµ
emn ,
=O
which is the overlap between the corresponding QMs.
(3.19)
(3.20)
3.3. QUASIMODE PHOTON-NUMBER STATES
37
The generalization of one-photon number states are the M -photon QM number
states with photons in states m1 , m2 , ..., mM . These can be defined as
ˆ
dµ1 . . . dµM c̃m1 µ1 . . . c̃mM µM a†µ1 . . . a†µM |0i ,
|φm1 . . . φmM i = Am1 ... mM
(3.21)
where Am1 ... mM is a normalization constant. Similarly, the restricted M -photon QM
number state with photons in states m1 , m2 , . . . , mM can be defined as
ˆ
hχm1 . . . χmM | = Am1 ... mM
dµ1 . . . dµM q̃m1 µ1 . . . q̃mM µM aµ1 . . . aµM h0| .
(3.22)
Note that we have defined these states such that there may be more than one photon
in a given QM, in which case some of the mi will be the same. It is straightforward
to see that the inner product of these two generalized photon number states is
hχm1 . . . χmM |φm01 . . . φm0M i = Am1 ... mM Am01 ... m0M
X
Õm1 m01 . . . ÕmM m0M ,
(3.23)
perms
where the sum is over the M ! permutations of the indices. Therefore, the proper
normalization constant is given by
Am1 ... mM = qP
1
.
(3.24)
perms Õm1 m1 . . . ÕmM mM
For example, for M = 3, this gives
2 2 2
n
o−1/2
.
= 1 + Õm1 m2 + Õm2 m3 + Õm3 m1 + 2Re Õm1 m2 Õm2 m3 Õm3 m1
Am1 m2 m3
(3.25)
In Eq. (3.24), if there is only one mode, then all of the mi are the same and we obtain
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
38
the usual result
1
Am... m = √ .
M!
3.4
(3.26)
Non-Hermitian projection and the system Hamiltonian
The next step is to project the system Hamiltonian onto the basis of the QMs. In
the language of linear algebra, if we have a basis of orthogonal vectors, the image of
a general vector |ψi onto one specific element of a basis |ψi i is obtained using the
projection operator
p̂i = |ψi i hψi | .
(3.27)
To ensure that multiple operations of projector operator leave the state vector unchanged, p̂i needs to be idempotent, i.e.,
p̂2i = p̂i .
(3.28)
The generalization of this for an orthogonal basis of vectors is the full projector
operator that projects onto a subspace of the entire basis space rather than one
particular element. This can be defined as
p̂ =
X
|ψi i hψi | ,
(3.29)
i
which is idempotent as a direct result of the orthogonality of the basis vectors.
In contrast, QMs form a non-orthogonal basis. Thus, constructing the corresponding projector operator is not as straightforward. However, regardless of QMs
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
39
non-orthogonality, if an operator Pb can be constructed such that
Pb 2 = Pb,
(3.30)
then the projection onto the basis of QMs can be performed.
To this end, consider the situation where we have total of N QMs. Let us define
the following projection operator that incorporates both restricted and unrestricted
QMs:
TbM ≡
N
X
N
X
{mi }=1 {m0 }=1
Pem1 m01 . . . PemM m0M
|φm1 . . . φmM ihχm01 . . . χm0M |,
M !Am1 ... mM Am01 ... m0M
(3.31)
i
where {mi } and {m0i } are the shorthand for all mi and m0i from 1 to N with i running
from 1 to M . In this expression, P̃ is the inverse of Õ such that
X
P̃ml Õln =
l
X
Õml P̃ln = δnm .
(3.32)
l
Note that, TbM is not Hermitian and projects a general photon state, |ψi, onto Mphoton states in the basis of unrestricted QMs through the overlap matrix of the QM
states with the restricted QM states.
Using the projector TbM that projects onto the QM photon number states with total
number of M photons, the full projector operator that projects onto every possible
photon number state, can be constructed as
Pb =
∞
X
M =0
TbM .
(3.33)
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
40
Here, the vacuum projector, Tb0 , needs to be defined as
Tb0 ≡ |0ih0|.
(3.34)
The projector (3.33) is the extension to a multi-photon system of the projector that
was employed in [53] for single-photon states. Much like TbM , the Pb is non-Hermitian.
The importance of this will be discussed in the remainder of this chapter.
Using Eqs. (3.31) and (3.23), it is straightforward to see that
TbM2 =
N
X
N
X
N
X
N
X
|φm1 . . . φmM ihχm000
. . . χm000
|
1
M
(3.35)
{mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
×
=
P̃m1 m01 . . . P̃mM m0M
. . . P̃m00M m000
P̃m001 m000
1
M
hχm01 . . . χm0M |φm001 . . . φm00M i
000
M !Am1 ... mM Am01 ... m0M M !Am001 ... m00M Am000
1 ... mM
N
X
N
X
N
X
N
X
|φm1 . . . φmM ihχm000
. . . χm000
|
1
M
(3.36)
{mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
i
. . . P̃m00M m000
P̃m001 m000
1 X
1
M
P̃m1 m01 Õm01 m001 . . . P̃mM m0M Õm0M m00M .
000 M !
M !Am1 ... mM Am000
...
m
1
M
perms
e is the inverse of the O,
e we have
Since P
N
X
N
X
X
P̃m1 m01 Õm0 m00 . . . P̃mM m0M Õm0
1
{ }=1 {
m0i
m00
i
1
00
M mM
= M !.
(3.37)
}=1 perms
Therefore, it is found that
TbM2 = TbM .
(3.38)
Because different terms in the full projector expansion associated with different M
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
41
are independent, an idempotent TbM guarantees an idempotent Pb, which proves Pb
is a projection operator. In the following sections, this projection operator will be
used to obtain QM ladder operators, the effective system Hamiltonian, as well as the
corresponding dynamical equations for the projected density matrix and the projected
observables of the system.
3.4.1
Quasimode ladder operators
The projected creation operator for the `th QM can be defined as
ˆ
β`†
≡ Pb
=
dµ c̃`µ a†µ Pb
∞ ˆ
X
(3.39)
dµ c̃`µ TbM +1 a†µ TbM ,
(3.40)
M =0
where in the second line the definition of Pb is used along with the fact that, due
to the action of the creation operator, only photon number states with one photon
difference can contribute. Using definition of TbM in Eq. (3.31) we find
β`†
=
∞
N
X
X
N
X
N
X
N
X
ˆ
dµ c̃`µ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
i
P̃m1 m01 ...P̃mM +1 m0M +1
(M + 1)!Am1 ...mM +1 Am01 ...m0M +1
P̃m001 m000
...P̃m00M m000
1
M
000
M !Am001 ...m00M Am000
...m
1
M
× |φm1 ...φmM +1 ihχm01 ...χm0M +1 |a†µ |φm001 ...φm00M ihχm000
...χm000
|,
1
M
(3.41)
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
42
which can be further expanded using Eqs. 3.21 and 3.22:
β`†
=
∞
N
X
X
N
X
N
X
N
X
ˆ
dµ c̃`µ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
P̃m1 m01 ...P̃mM +1 m0M +1 P̃m001 m000
...P̃m00M m000
1
M
|φm1 ...φmM +1 ihχm000
...χm000
|
1
M
000
(M + 1)!Am1 ...mM +1 M !Am000
1 ...mM
ˆ
ˆ
∗
∗
× dµ1 ...dµM +1 q̃m01 µ1 ...q̃m0M +1 µM +1 dν1 ...dνM c̃m001 ν1 ...c̃m00M νM
×
× h0| aµ1 ...aµM aµM +1 a†µ a†ν1 ...a†νM |0i .
(3.42)
The vacuum expectation of the product of ladder operators can be evaluated in a
standard manner to obtain
β`†
=
N
∞
X
X
N
X
N
X
N
X
ˆ
dµ c̃`µ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
P̃m1 m01 ...P̃mM +1 m0M +1 P̃m001 m000
...P̃m00M m000
1
M
|φm1 ...φmM +1 ihχm000
...χm000
|
1
M
000
000
(M + 1)!Am1 ...mM +1 M !Am1 ...mM
ˆ
ˆ
∗
∗
× dµ1 ...dµM +1 q̃m01 µ1 ...q̃m0M +1 µM +1 dν1 ...dνM c̃m001 ν1 ...c̃m00M νM
×
×
X
perms(µ,µi )
δ (µ1 − ν1 ) ...δ (µM − νM ) δ (µ − µM +1 ) .
(3.43)
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
43
It is now straightforward to see that
β`† =
∞
N
X
X
N
X
N
X
N
X
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
P̃m1 m01 ...P̃mM +1 m0M +1 P̃m001 m000
...P̃m00M m000
1
M
|φm1 ...φmM +1 ihχm000
...χm000
|
1
M
000
(M + 1)!Am1 ...mM +1 M !Am000
1 ...mM
ˆ
X
∗
∗
∗
× dµ1 ...dµM dµ
q̃m
c̃`µ (3.44)
0 µ c̃m00 µ1 ...q̃m0 µ c̃m00 µ q̃m0
1
M M
1 1
M M
M +1 µ
perms(µ,µi )
×
=
∞
N
X
X
N
X
N
X
N
X
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
P̃m1 m01 ...P̃mM +1 m0M +1 P̃m001 m000
...P̃m00M m000
1
M
|φm1 ...φmM +1 ihχm000
...χm000
|
1
M
000
(M + 1)!Am1 ...mM +1 M !Am000
...m
1
M
X
×
perms
=
i
∞
N
X
X
Õm01 m001 ...Õm0M m00M Õm0M +1 l
(3.45)
(m0i )
N
X
N
X
N
X
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
×
i
i
...P̃m00M m000
P̃m001 m000
1
1
M
|φm1 ...φmM +1 ihχm000
...χm000
|
1
M
000
(M + 1)! M !Am1 ...mM +1 Am000
...m
1
M
X
perms(mi )
δm1 m001 ...δmM m00M δmM +1 l .
(3.46)
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
44
This can be further simplified into the final expression for the projected creation
operator for the `th QM:
β`† =
∞
N
X
X
N
X
M =0 {mi }=1 {m0 }=1
P̃m1 m01 . . . P̃mM m0M
|φl φm1 . . . φmM ihχm01 . . . χm0M |. (3.47)
M !Alm1 ... mM Am01 ... m0M
i
Similarly, the projected annihilation operator for the `th QM is
α` =
∞
N
X
X
N
X
M =0 {mi }=1 {m0 }=1
i
P̃m1 m01 . . . P̃mM m0M
|φm1 . . . φmM ihχ` χm01 . . . χm0M |. (3.48)
M !A`m01 ... m0M Am1 ... mM
Since, Pb is not Hermitian,
β`† 6= (α` )† ,
(3.49)
which is a non-standard feature of working with the QMs. In addition, because
Pb projects onto the basis of unrestricted QMs, projected creation and annihilation
operators create and destroy a photon in the `th unrestricted QM, respectively. It
is also useful to note that QM ladder operators satisfy the following commutation
relations (see Appendix A)
†
enm Pb,
αn , βm
=O
(3.50)
[αn , αm ] = 0,
(3.51)
†
= 0.
βn† , βm
(3.52)
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
3.4.2
45
Projected free-field Hamiltonian
Now we discuss the projection of the free-field Hamiltonian, H, onto the basis of
QMs, where H is initially represented in the basis of the system true modes as
ˆ
H=
dµ ~ωµ a†µ aµ .
(3.53)
Interaction Hamiltonians of different types will be considered in later chapters. The
procedure is more or less the same as was used in the derivation of the projected
ladder operators, apart from some extra details that we will see shortly. The effective
projected free-field Hamiltonian is defined as
H̃ P ≡ PbH Pb
=
∞ ˆ
X
(3.54)
dµ~ωµ TbM a†µ aµ TbM ,
(3.55)
M =0
where in the second line, we have used the fact that, due to the nature of number
operator a†µ aµ only projectors of the same photon number M matter. Instead of a†µ aµ
it is easier to evaluate aµ a†µ − 1. The extra term corresponds to a constant energy
term which can be dropped from the Hamiltonian and therefore we effectively have
P
H̃ =
N
∞
X
X
N
X
N
X
N
X
ˆ
dµ~ωµ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
i
P̃m1 m01 ...P̃mM m0M
P̃m001 m000
...P̃m00M m000
1
M
000
M !Am1 ...mM Am01 ...m0M M !Am001 ...m00M Am000
1 ...mM
× |φm1 ...φmM ihχm01 ...χm0M |aµ a†µ |φm001 ...φm00M ihχm000
...χm000
|
1
M
(3.56)
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
=
∞
N
X
X
N
X
N
X
N
X
46
ˆ
dµ~ωµ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M !Am1 ...mM
M !Am000
1 ...mM
ˆ
ˆ
∗
∗
× dµ1 ...dµM q̃m01 µ1 ...q̃m0M µM dν1 ...dνM c̃m001 ν1 ...c̃m00M νM
×
× h0| aµ1 ...aµM aµ a†µ a†ν1 ...a†νM |0i
=
∞
N
X
X
N
X
N
X
N
X
(3.57)
ˆ
dµ~ωµ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M !Am1 ...mM
M !Am000
...m
1
M
ˆ
ˆ
∗
∗
× dµ1 ...dµM q̃m01 µ1 ...q̃m0M µM dν1 ...dνM c̃m001 ν1 ...c̃m00M νM
×
X
×
δ (µ1 − µ) δ (µ2 − ν1 ) ...δ (µM − νM −1 ) δ (µ − νM )
(3.58)
perms(µ,µi )
=
∞
N
X
X
N
X
N
X
N
X
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M !Am1 ...mM
M !Am000
1 ...mM
ˆ
ˆ
X ˆ
∗
∗
∗
×
dµ2 q̃m02 µ2 c̃m001 µ2 ... dµM q̃m0M µM c̃m00M −1 µM dµ~ωµ q̃m
0 µ c̃m00 µ .
M
1
perms(m0i )
×
(3.59)
The obvious difference in this derivation compared to the earlier derivation of the
projected creation operator is that the integration over µ is not as straightforward as
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
47
before, because it involves the frequencies. Indeed, to proceed further here requires
using a very important property of QMs. Let us assume that the positive-frequency
electric field associated with the mth QM can be written as
−iω̃m t
e
E(+)
,
m (r, t) = Nm (r) e
(3.60)
for r inside the computational volume, Vc . This defines the temporal behavior of the
QMs inside Vc which contains both oscillation and decay. This temporal behaviour
can and has been verified for a variety of structures using FDTD simulations where
the initial state is one of the QMs [69]. Following [53], multiplying this equation with
e c∗ (r) and integrating over the entire space leads to
N
n
ˆ
ˆ
e c∗ (r) · N
e m (r) e−iω̃m t
dr3 (r) N
n
3
e c∗ (r) · E(+) (r, t) =
dr (r) N
n
m
V
(3.61)
V
= Õnm e−iω̃m t .
(3.62)
The left-hand side on the other hand, can be further simplified by expanding fields
in terms of the system true modes as follows:
ˆ
e c∗ (r) · E(+) (r, t)
dr3 (r) N
n
m
LHS =
(3.63)
V
ˆ
ˆ
3
=
dr (r)
ˆ
∗ ∗
dµq̃nµ
f̃µ
(r) ·
dνc̃mν f̃ν (r) e−iων t
(3.64)
dr3 (r) f̃µ∗ (r) · f̃ν (r)
(3.65)
V
ˆ
ˆ
∗
dµq̃nµ
=
ˆ
−iων t
dνc̃mν e
V
ˆ
=
∗
dµq̃nµ
c̃mµ e−iωµ t .
(3.66)
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
48
Therefore, we arrive at the following useful equation:
ˆ
Õnm e
−iω̃m t
∗
c̃mµ e−iωµ t .
dµq̃nµ
=
(3.67)
By differentiating both sides of this equation with respect to time and evaluating at
time t = 0, we obtain
ˆ
ω̃m Õnm =
∗
dµ ωµ q̃nµ
c̃mµ ,
(3.68)
which relates the frequency of the mth QM to the frequencies of the true mode components. This is exactly what we needed to be able to project the system Hamiltonian
onto the basis of QMs. Now, it is clear that both restricted and unrestricted QMs
were needed. Indeed, without doing this, the non-Hermitian nature of the system
Hamiltonian in the basis of the leaky QMs would not have been captured. If one only
defines unrestricted QMs, then integrations over the entire space cannot be performed
as there is simply no information available outside Vc . The presence of the restricted
QMs forces any integral contributions outside Vc to go to zero. On other hand, because the true modes are defined over the entire space, originally the unrestricted
QMs are needed to be defined to be able to use the useful properties of true modes
such as completeness and orthogonality. Moreover, note that the definition of the
QM field in Eq. (3.60) is only valid for the unrestricted QMs not for the restricted
QMs, as the FDTD-calculated modes are calculated using PML boundary conditions
3.4. NON-HERMITIAN PROJECTION AND THE SYSTEM
HAMILTONIAN
49
rather than hard boundary conditions. Using Eq. (3.68), we find
P
H̃ =
∞
N
X
X
N
X
N
X
N
X
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M !Am1 ...mM
M !Am000
1 ...mM
X
×
perms
=
i
∞
N
X
X
Õm02 m001 ...Õm0M m00M −1 ~ω̃m00M Õm01 m00M
(3.69)
(m01 )
N
X
N
X
M =0 {mi }=1 {m00 }=1 {m000 }=1
i
×
i
...P̃m00M m000
P̃m001 m000
1
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M ! M !Am1 ...mM Am000
...m
1
M
X
×
δm2 m001 ...δmM m00M −1 ~ω̃m00M δm1 m00M
(3.70)
perms(m1 )
=
∞
N
X
X
N
X
M =0 {mi }=1 {m000 }=1
i
× ~ω̃m1
...P̃mM m000
P̃m1 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|,
1
M
000
M !Am1 ...mM Am000
1 ...mM
(3.71)
which, using expressions for projected ladder operators, can be written as
P
H̃ =
N
X
†
~ω̃m P̃mn βm
αn .
(3.72)
mn=1
Note that the projected Hamiltonian is non-Hermitian due to the action of nonHermitian projection operator, Pb, which was constructed to provide such non-standard
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
50
projection in first place.
3.5
Hermitian projection and system observables
Observables represent physical quantities and must be Hermitian operators with realvalued expectation values. Projecting observables using the non-Hermitian projector,
Pb, results in non-Hermitian operators, which are not appropriate. On the other hand,
all the measurement must be performed in the computational volume where we have
accurate information about the system. Therefore, for observables, there is no need
to incorporate two different QM basis, restricted and unrestricted. This was only
required when we wanted to obtain information on photon leakage from within the
computational volume.
The projection operators for physical observables can be defined using only restricted QMs as
b≡
Q
∞
N
X
X
N
X
M =0 {mi }=1 {m0 }=1
P̃m1 m01 . . . P̃mM m0M
|χm1 . . . χmM ihχm01 . . . χm0M |.
M !Am1 ... mM Am01 ... m0M
(3.73)
i
This is identical to Pb in construction, except that it is Hermitian, because only
emn and hχm |χn i =
restricted QMs have been used. Using the fact that hχm |φn i = O
emn , it is easy to prove that
O
b2 = Q
b
Q
(3.74)
b = Pb
PbQ
(3.75)
bPb = Q.
b
Q
(3.76)
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
51
These will be useful for proving some of the later results presented in this thesis.
3.5.1
Ladder operators for restricted QMs
Similar to unrestricted creation operator, the restricted creation operator can be
defined as
ˆ
b†m
b
=Q
b
dµ q̃mµ a†µ Q.
(3.77)
b
Unlike the Pb-projected creation operator, the Q-projected
creation operator creates
photons in restricted QMs. Using properties in Eqs. (3.74), (3.75) and (3.76), it is
easy to show that
†
b m
Qβ
= b†m
(3.78)
b m = bm .
Qα
(3.79)
b
Therefore, the Q-projected
creation and annihilation commutator reduces to
bO
enm .
bn , b†m = Q
(3.80)
b replaces the nonThis is similar to Eq. (3.50) except that the restricted projector Q
Hermitian projector Pb and creation and annihilation operators are Hermitian conjugate of each other.
3.5.2
Restricted Hamiltonian
Now that the connection between restricted and unrestricted creation and annihilation operators is obtained, it is straightforward to project the previously obtained
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
52
non-Hermitian Hamiltonian onto the basis of restricted QMs. Indeed, the final Hamiltonian for the system needs to be defined as
bH̃ P Q.
b
H̃ Q ≡ Q
(3.81)
†
Since by its definition, βm
has projector operators of type Pb on both sides, using
Eq. (3.75), it is easy to see that
†
† b
βm
= βm
Q.
(3.82)
Thus we obtain
Q
H̃ =
N
X
†
b m
b
~ω̃m P̃mn Qβ
αn Q
(3.83)
† b
b m
b
~ω̃m P̃mn Qβ
Qαn Q
(3.84)
b
~ω̃m P̃mn b†m bn Q,
(3.85)
mn=1
=
N
X
mn=1
=
N
X
mn=1
where we have used Eqs. (3.78) and (3.79) in going from the second line to the
b onto the restricted QM space is
third line. Now, the operation of the projector Q
equivalent to the operation of unitary operator 1. Therefore, if one only operates
within this space, the projected Hamiltonian is effectively
H̃ Q =
N
X
mn=1
~ω̃m P̃mn b†m bn .
(3.86)
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
3.5.3
53
Restricted operators and restricted density matrix
In general, for a physical observable A, the corresponding restricted operator can be
defined as:
b Q.
b
AQ ≡ QA
(3.87)
The expectation value for the operator AQ is given by
b Qi
b
hAQ i ≡ T rhρ QA
= T rhρQ AQ i,
(3.88)
where
b Q,
b
ρQ ≡ Qρ
(3.89)
is the restricted, projected density matrix for which we must obtain the dynamical
equation of motion. Note that the density matrix of the system before projection
evolves according to the total Hermitian Hamiltonian of the system, H, such that
ρ (t) = e−iHt/~ ρ (0) eiHt/~ .
(3.90)
If the system is initially confined to the subspace spanned by the unrestricted basis
states (as it is in the cases of interest), then
ρ (0) = Pbρ (0) Pb† .
(3.91)
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
54
Therefore, it is easy to see that
b −iHt/~ Pbρ (0) Pb† eiHt/~ Q
b
ρQ (t) = Qe
bPbe−iHt/~ Pbρ (0) Pb† eiHt/~ Pb† Q.
b
=Q
(3.92)
(3.93)
To proceed further, it is critical to be able to prove the following for any power of the
Hamiltonian:
l
PbH l Pb = H̃ P .
(3.94)
To do so, we start with the following:
ˆ
l
l
PbH Pb = ~
=
dµ1 dµ2 ωµ1 ...ωµs P̂ a†µ1 aµ1 ...a†µl aµl P̂
∞
N
X
X
N
X
N
X
N
X
(3.95)
ˆ
~
l
dτ1 ...dτl ωτ1 ...ωτl
(3.96)
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M !Am1 ...mM
M !Am000
...m
1
M
ˆ
ˆ
∗
∗
× dµ1 ...dµM q̃m
dν1 ...dνM c̃m001 ν1 ...c̃m00M νM
0 µ ...q̃m0 µ
1 1
M M
×
× h0| aµ1 ...aµM a†τ1 aτ1 ...a†τl aτl a†ν1 ...a†νM |0i .
However, using the bosonic commutation relation between ladder operators, it is
straightforward to see that
†
aτ aτ , a†ν = δ (τ − ν) a†ν ,
(3.97)
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
55
which in successive applications leads to
M
X
a†τ aτ , a†ν1 ...a†νM =
δ (τ − νn ) a†ν1 ...a†νM .
(3.98)
n=1
This can be used to obtain
PbH l Pb =
∞
N
X
X
N
X
N
X
N
X
ˆ
l
dτ1 ...dτl ωτ1 ...ωτl
~
(3.99)
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M !Am1 ...mM
M !Am000
...m
1
M
ˆ
ˆ
∗
∗
× dµ1 ...dµM q̃m01 µ1 ...q̃m0M µM dν1 ...dνM c̃m001 ν1 ...c̃m00M νM
×
×
M
X
δ (τ1 − νn1 ) ...δ (τl − νnl )
n1 ,...,nl =1
× h0| aµ1 ...aµM a†ν1 ...a†νM |0i
=
∞
N
X
X
N
X
N
X
N
X
ˆ
~
l
dτ1 ...dτl ωτ1 ...ωτl
(3.100)
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M
|φm1 ...φmM ihχm000
...χm000
|
1
M
000
M !Am1 ...mM
M !Am000
1 ...mM
ˆ
ˆ
∗
∗
× dµ1 ...dµM q̃m01 µ1 ...q̃m0M µM dν1 ...dνM c̃m001 ν1 ...c̃m00M νM
×
×
M
X
δ (τ1 − νn1 ) ...δ (τl − νnl )
n1 ,...,nl =1
×
X
perms(µi )
δ (µ1 − ν1 ) ...δ (µM − νM ) .
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
56
After some work, we obtain
PbH l Pb = ~l
∞
X
N
X
M
X
ω̃mn1 ...ω̃mnl
(3.101)
M =1 {mi }{m0 }=1 n1 ,...nl =1
i
×
P̃m1 m01 ...P̃mM m0M
|φm1 ...φmM ihχm01 ...χm0M |.
M !Am1 ...mM Am01 ...m0M
l
It is straightforward to show that H̃ P is given exactly by same expression and
therefore the desired result for the photon portion of the Hamiltonian is obtained.
When there are other terms such as photon-QD interaction terms, the proofs for
the other terms in the Hamiltonian follow in a similar fashion. However, for the
interaction term, there are two approximations that need to be made. First, the QDs
must have negligible coupling to any modes other than the QMs in the calculation.
This will be the case when the QD couples strongly to one or more of the QMs and
when the transition frequency of the QD is well within the photonic bandgap of the
PCS. The second approximation is that we assume that the continuum contribution
to the Lamb shift can be neglected. This is generally the case and one can simply
correct for this by adjusting the transition frequencies of the QDs. These points are
discussed in more detail for the one-photon effective Hamiltonian in Ref. [53].
Using the newly confirmed result given in Eq. (3.94), it can be seen that
b
bPbe−iH̃ P t/~ Pbρ (0) Pb† eiH̃ P † t/~ Pb† Q.
ρQ (t) = Q
(3.102)
3.5. HERMITIAN PROJECTION AND SYSTEM OBSERVABLES
57
Thus
o
dρQ
d n b b −iH̃ P t/~ b
P†
b
=
QP e
P ρ (0) Pb† eiH̃ t/~ Pb† Q
dt
dt
o
−i b n P b
†
† P†
b
b
b
b
=
Q H̃ P ρ (t) P − P ρ (t) P H̃
Q
~
o
−i n b P
b − Qρ
b (t) H̃ P † Q
b .
QH̃ ρ (t) Q
=
~
(3.103)
Also, it is easily shown that
bH̃ P = H̃ Q Q.
b
Q
(3.104)
Therefore, we arrive at
−i n Q b b b b Q† o
dρQ
=
H̃ QρQ − QρQH̃
dt
~
i
i
= − H̃ Q ρQ + ρQ H̃ Q† .
~
~
(3.105)
Note that H̃ Q is not Hermitian and therefore this equation is different than the
standard Heisenberg equation of motion for the density matrix of a Hermitian system.
The projected density matrix obtained this way does not conserve total probability
3.6. MASTER EQUATION FOR PROJECTED DENSITY MATRIX 58
in the system. In fact
Tr
dρQ
dt
i i = − Tr H̃ Q ρQ + Tr ρQ H̃ Q†
~
~
i i = − Tr H̃ Q ρQ + Tr H̃ Q† ρQ
~
~
i i h Q
Q†
ρQ
= − Tr H̃ − H̃
~
6= 0.
(3.106)
(3.107)
(3.108)
(3.109)
For a Hermitian Hamiltonian the trace becomes zero and the probability is conserved.
But, the projected non-Hermitian Hamiltonian results in a non-conserving probability
projected density matrix. As presented in the next section, to address this, extra
terms must be added to the dynamical equation Eq. (3.105) such that the trace
becomes time-independent.
3.6
Master equation for projected density matrix
For open quantum systems, where the system of interest is allowed to interact with
the rest of the world (known as the reservoir), the evolution of the system density
matrix, ρS (t), is given by the Lindblad equation [52]
X †
dρS
i
1 †
1
†
= − [H, ρS ] +
Γk Ak ρS Ak − Ak Ak ρS − ρS Ak Ak ,
dt
~
2
2
k
(3.110)
where Ak are the Lindblad operators of the open system, Γk is the loss rate for kth
channel and H is a Hermitian Hamiltonian describing the unitary part of the system
evolution. Indeed, the Lindblad terms are placed such that the leakage from inside the
3.6. MASTER EQUATION FOR PROJECTED DENSITY MATRIX 59
system into the surrounding environment is accounted for and therefore the density
matrix obtained from solving this equation is norm-conserved.
On the other hand, the QM-projected Hamiltonian, H Q , contains information
on both the unitary and the non-unitary evolution of the system in the restricted
volume of the entire space. Thus, it is convenient to break down H Q into Hermitian
and non-Hermitian pieces
Q
Q
H̃ Q ≡ Hher
+ Hnon
,
(3.111)
where
Q
Hher
N
X
~
[ω̃m + ω̃n∗ ] P̃mn b†m bn
≡
2
mn=1
(3.112)
Q
Hnon
N
X
~
[ω̃m − ω̃n∗ ] P̃mn b†m bn .
≡
2
mn=1
(3.113)
e is Hermitian, indeed. Now, if we define the Hermitian matrix
Note that the matrix P
M as
Mmn ≡
1
[(iω̃m ) + (iω̃n )∗ ] P̃mn ,
2
(3.114)
the non-Hermitian part of the effective Hamiltonian can be re-written as
Q
Hnon
≡ −i~
N
X
Mmn b†m bn .
(3.115)
mn=1
Because M is Hermitian, another Hermitian matrix S can be found such that
M ≡ SS,
(3.116)
3.6. MASTER EQUATION FOR PROJECTED DENSITY MATRIX 60
therefore
Q
Hnon
N X
N
X
≡ −i~
Smp Spn b†m bn
(3.117)
mn=1 p=1
= −i~
N
X
L†p Lp ,
(3.118)
p=1
where
Lp ≡
X
Spm bn .
(3.119)
m
Now, Eq. (3.105) can be re-written as
N
N
X
dρQ
i h Q Qi X †
b
ρQ L†p Lp + L,
Lp Lp ρQ −
= − Hher
,ρ −
dt
~
p=1
p=1
(3.120)
b has been added such that the Master equawhere the unknown Lindblad operator L
tion produces a probability conserving density matrix. Comparing Eq. (3.110) and
Eq. (3.120) it is easy to see that
b≡2
L
N
X
Lp ρQ L†p .
(3.121)
p=1
=2
N X
N
X
Smp Spn bm ρQ (t) b†n
(3.122)
p=1 mn=1
=2
N
X
Mmn bm ρQ (t) b†n
(3.123)
mn=1
=i
N X
mn=1
Q
ω̃m P̃mn bm ρ
(t) b†n
−
ω̃n∗ P̃mn bm ρQ
(t) b†n
.
(3.124)
3.6. MASTER EQUATION FOR PROJECTED DENSITY MATRIX 61
Thus, the projected master equation that should be used to obtain the dynamics in
the leaky coupled-defect systems is
N X
dρQ
i
i
= − H̃ Q ρQ + ρQ H̃ Q† + i
ω̃m P̃mn bm ρQ (t) b†n − ω̃n∗ P̃mn bm ρQ (t) b†n .
dt
~
~
mn=1
(3.125)
It should be noted that the determination of the norm-conserving term of the Master
equation (3.125) does not depend upon the exact form of the Hermitian Hamiltonian
for the unitary evolution. Therefore, adding an arbitrary Hermitian Hamiltonian to
the system does not require any further modification of the existing result as far
as the scattering loss treatment is concerned. Indeed, as presented in the following
chapters of this thesis, extra Hermitian Hamiltonian pieces, for example to account
for photon-QD interaction, can be added and still same form of Master equation can
be used to obtain the correct evolution for the system.
Note that, if an orthogonal basis labeled by m exists and the Hermitian part of
the Hamiltonian is given by
Hher =
X
~ωm b†m bm ,
(3.126)
m
then Eq. (3.125) reduces to the standard Master equation of the Lindblad form [52]
N
X
i
1 †
1 Q
dρQ
Q
Q
†
Q
†
= − Hher , ρ (t) + i
Γm bm ρ (t) bm − bm bm ρ (t) − ρ (t) bm bm ,
dt
~
2
2
m=1
(3.127)
where the dissipation rate is Γm = 2γm . Indeed, γm in our model represent the
amplitude loss rate than the intensity loss rate.
3.7. A NEW SIMPLIFIED REPRESENTATION
3.7
62
A new simplified representation
The Hamiltonian of Eq. (3.86) can be represented in a simplified fashion if we introb
duce a new set of operators, cm , using the previously defined Q-projected
annihilation
operators. We define them to be
cm ≡
N
X
Pemn bn ,
(3.128)
emn cn .
O
(3.129)
n=1
where the inverse relation is
bm =
N
X
n=1
With this definition, it is straightforward to see that
b nm ,
cn , b†m = Qδ
(3.130)
which is similar to the normal commutation relation for bosonic operators. The
corresponding Hamiltonian in this new representation can be written as
Q
H̃ =
N
X
~ω̃m b†m cm .
(3.131)
m=1
The corresponding simplified Master equation then becomes
N
X
i Q Q i Q Q†
dρQ
∗
= − H̃ ρ + ρ H̃ + i
ω̃m cm ρQ (t) b†m − ω̃m
bm ρQ (t) c†m .
dt
~
~
m=1
(3.132)
3.7. A NEW SIMPLIFIED REPRESENTATION
3.7.1
63
Adjoint Master equation for Heisenberg operators
The introduction of the creation operator, c†m , does more than just simplify the Hamiltonian and the Master equation. This can be seen by looking at the time-dependence
of the new restricted operators.
The time dependence of operators is given by the adjoint Master equation [52]
N
X
i
i
dAH
∗ †
= − AH H̃ Q + H̃ Q† AH + i
ω̃m b†m AH cm − ω̃m
c m A H bm .
dt
~
~
m=1
(3.133)
The connection between the Master equation for the density matrix and the adjoint
Master equation for the time-dependent operators is a generalization of the standard
Schrödinger and Heisenberg pictures in quantum mechanics. Using this, it can be
seen that
N
X
dcn
i
i Q†
Q
∗ †
= − cn H̃ + H̃ cn + i
ω̃m b†m cn cm − ω̃m
c m c n bm
dt
~
~
m=1
= −iω̃n cn ,
(3.134)
(3.135)
e prowhere we have used the commutation relation of Eq. (3.130). Note that the Q
jection effectively reduces to 1 in the restricted space. Therefore
cn (t) = cn e−iω̃n t ,
(3.136)
where cn is the Schrödinger operator while cn (t) is the Heisenberg, time-dependent
operator. Thus, we see that it is the cn (t) operators rather than the bn (t) operators
that have the simple decaying-harmonic time evolution. The evolution of the bn (t)
3.7. A NEW SIMPLIFIED REPRESENTATION
64
operator is more complicated, but is easily found using the relation
bm (t) ≡
N
X
enm cn (t) .
O
(3.137)
n=1
Note that, the difference between the cm and bm operators are all results from the
non-orthogonality properties of the QMs and as soon as one makes the QM basis
orthogonal, bm and cm become equivalent.
3.7.2
Expectation value of observables using a simplified basis
The commutation relation of Eq. (3.130) suggests that both b†m and cm be used to
define a basis to represent the density matrix in the QM basis. Let us define the
M -photon number state as
|B; M ; n1 , n2 ...nN i ≡ √
n1 nN
1
b†1
... b†N
|0i,
n1 !...nN !
(3.138)
and define the conjugate M -photon number state as
hC; M ; n1 , n2 ...nN | ≡ √
1
h0| (cN )n1 ... (c1 )nN .
n1 !...nN !
(3.139)
Obviously, in these definitions, the total number of photons has to equal M , in other
words
N
X
i=1
ni = M.
(3.140)
3.7. A NEW SIMPLIFIED REPRESENTATION
65
With these definitions, we see from the commutation relations that these states are
orthonormal such that


 1 if all n0 = ni
i
0
0
0
.
hC; M ; n1 , n2 ...nN |B; M ; n1 , n2 ...nN i =

 0 otherwise
(3.141)
Thus these two sets of vectors form a biorthogonal basis. Also, we have the generalization of the standard results:
ci |B; M ; n1 , n2 ...nN i =
b†i |B; M ; n1 , n2 ...nN i =
√
ni |B; M − 1; n1 , n2 ... (ni − 1) , ...nN i
√
ni + 1 |B; M + 1; n1 , n2 ... (ni + 1) , ...nN i
and
√
hC; M ; n1 , n2 ...nN | b†i = hC; M − 1; n1 , n2 ... (ni − 1) , ...nN | ni
√
hC; M ; n1 , n2 ...nN | ci = hC; M + 1; n1 , n2 ... (ni + 1) , ...nN | ni + 1.
Note that with these definitions, the state |B; M ; n1 , n2 ...nN i (|C; M ; n1 , n2 ...nN i) is
Q†
an eigenstate of H Q (Hef
f ). It can be shown that the restricted projector can be
written in terms of these states as
b≡
Q
∞ X
X
M =0 {ni }
|B; M ; n1 , n2 ...nN i hC; M ; n1 , n2 ...nN | ,
(3.142)
3.7. A NEW SIMPLIFIED REPRESENTATION
66
or equivalently as
b=
Q
∞ X
X
|C; M ; n1 , n2 ...nN i hB; M ; n1 , n2 ...nN | .
(3.143)
M =0 {ni }
This is the standard form of a biorthogonal projector.
With this expression for the projector, we see that the expectation value of a
restricted operator AQ is
hAQ i = Tr ρQ (t) AQ
=
∞ X
∞
N
X
X
N
X
hC; M ; n1 , n2 ..., nN | ρQ (t) |B; M 0 ; n01 , n02 ..., n0N i
M =0 M 0 =0 {ni }=1 {n0 }=1
i
× hC; M 0 ; n01 , n02 ..., n0N | AQ |B; M ; n1 , n2 ..., nN i
=
∞ X
∞
N
X
X
N
X
(3.144)
hB; M ; n1 , n2 ..., nN | ρQ (t) |C; M 0 ; n01 , n02 ..., n0N i
M =0 M 0 =0 {ni }=1 {n0 }=1
i
× hB; M 0 ; n01 , n02 ..., n0N | AQ |C; M ; n1 , n2 ..., nN i .
(3.145)
Therefore, to evaluate expectation values, we need to have all of the matrix elements of
ρQ between our two different different restricted basis states. In using this expression,
it is important to note that
hC; M ; n1 , n2 ...nN | AQ |B; M ; n1 , n2 ...nN i =
6 hB; M ; n1 , n2 ...nN | AQ |C; M ; n1 , n2 ...nN i ,
(3.146)
even when AQ is Hermitian, and so we must be careful with our matrix elements.
3.8. SUMMARY
3.8
67
Summary
In this chapter, the details of a general formalism were presented that can be used to
obtain photon dynamics (as well as quantum dot dynamics, if present) in CC systems
in PCSs. This was done using a non-standard, non-Hermitian projection operator
that includes the scattering loss and the mode non-orthogonality in a self-consistent
manner. The non-Hermitian projection operator was then used to project the system
Hamiltonian for the electromagnetic field onto the basis of QMs. In addition, a
generalized Master equation was derived by solving which the evolution of the system
density matrix can be obtained. Moreover, a generalized adjoint Master equation
was provided that can be used to obtain time dependence of different Heisenberg
operators.
In the remainder of this thesis, the developed formalism will be used to study both
loss and mode non-orthogonality in example CC systems. In particular, a quantum
optical study is presented in chapter 4 where a multimode JC Hamiltonian in the
presence of multiple QDs is projected onto the leaky non-orthogonal basis of the
QMs. Also, in chapters 5 and 6, the projection formalism will be used to study the
effect of loss on pair generation in CC systems via the nonlinear process of SFWM.
68
Chapter 4
Quantum Optics in the Quasimode
Representation
What are the consequences of the non-standard QM features on the quantum optical
behavior of a given PCS circuit designed for cavity-QED applications? To model
quantum optics in PCSs when there are potentially a number of cavities and a number
of QDs present, the non-Hermitian projection technique presented in the previous
chapter will be used to transform the JC Hamiltonian from the basis of the true
modes to the basis of the QMs. A simple PCS with two asymmetrically coupled
cavities will be used to show how the formalism works and what the effect of QM
non-orthogonality is on photon dynamics. More complex structures can be treated
in the same manner.
4.1. PROJECTED JAYNES-CUMMING HAMILTONIAN
4.1
69
Projected Jaynes-Cumming Hamiltonian
Consider a PCS with N QMs, potentially coupled to ND QDs. In order to apply the
QM projection technique, one can write down the JC Hamiltonian, Eq. (1.2), in the
basis of true modes as
H ≡ Hdots + Hf ield + Hint
=
ND
X
1
j=1
2
(4.1)
ˆ
~ωjeg σjz
+
dµ ~ωµ a†µ aµ
+
ND ˆ
X
∗ † −
dµ g̃jµ σj+ aµ + g̃jµ
aµ σj ,
(4.2)
j=1
where, ωjeg is the resonance frequency of the jth QD and
r
g̃jµ ≡ −i
~ωµ
d̃j · f̃µ (rj ),
20
(4.3)
is the complex coupling constant between the jth QD and the µth true mode. Here,
d̃j = d˜j d̂j is the dipole-transition matrix element of the jth QD, and rj is its position.
As before, the projection needs to be done in two steps: first the non-Hermitian
b In
projection is done using Pb and then the Hermitian projection is applied using Q.
doing this, dealing with the QDs Hamiltonian is trivial, as the associated operators
commute with field operators. Therefore one easily finds
P
Hdots
≡ PbHdots Pb
=
X
j
~ωjeg σjz Pb.
(4.4)
(4.5)
4.1. PROJECTED JAYNES-CUMMING HAMILTONIAN
70
The free-field Hamiltonian has been already projected onto the basis of QMs:
N
X
H̃fPield =
†
~ω̃m P̃mn βm
αn .
(4.6)
m,n=1
Therefore, only the interaction Hamiltonian presents any difficulties. This is not difficult as the creation and annihilation operators have already been projected. Indeed
P
Hint
≡ PbHint Pb
=
ND ˆ
X
(4.7)
∗ b † b −
dµ g̃jµ σj+ Pbaµ Pb + g̃jµ
P aµ P σj
(4.8)
j=1
=
NQ
ND X
X
c∗ † −
P̃mn g̃jm σj+ αn + g̃jn
βm σj ,
(4.9)
j=1 mn=1
where
ˆ
g̃jm ≡
dµ g̃j,µe
cmµ
(4.10)
∗
∗
dµ g̃j,µ
qemµ
.
(4.11)
ˆ
c∗
g̃jm
≡
Using the expression for the QM expansion in the basis of true modes, Eq. (3.4),
the time-dependent QM can be constructed as
ˆ
−iω̃m t
e m (r) e
N
=
dµ c̃mµ f̃µ (r) e−iωµ t ,
(4.12)
4.1. PROJECTED JAYNES-CUMMING HAMILTONIAN
71
which can be differentiated and then evaluated at t = 0 to obtain
ˆ
e m (r) =
ω̃m N
dµ c̃mµ ωµ f̃µ (r) .
(4.13)
Consecutive derivations can be performed to obtain exactly same equation for any
power of frequency:
ˆ
k e
ω̃m
Nm
dµ c̃mµ ωµk f̃µ (r) .
(r) =
(4.14)
Thus, this can be generalized to any analytic function, h (ωµ ) that can be expanded
in terms of powers of ωµ such that
ˆ
e m (r) =
h (ω̃m ) N
dµ c̃mµ h (ωµ ) f̃µ (r) .
(4.15)
This is very useful, as it can be used to find expressions for the coupling constant in
terms of QM field amplitudes rather than the true-mode field amplitudes. Thus, we
obtain
ˆ
g̃jm =
dµ g̃j,µe
cmµ
ˆ
= −id̃j ·
r
= −i
r
dµ
(4.16)
~ωµ
e
cmµ f̃µ (raj )
20
~ω̃m
e m (raj ) .
d̃j · N
20
(4.17)
(4.18)
Similarly
r
c∗
g̃j,m
=i
~ωm ∗ e c∗
d̃ · Nm (raj ) .
20 j
(4.19)
c∗
∗
Note that for raj inside Vc , g̃j,m
= g̃j,m
. The full Pb projected non-Hermitian JC
4.1. PROJECTED JAYNES-CUMMING HAMILTONIAN
72
Hamiltonian of the photon-QD system is thus
P
H̃ef
f =
ND
X
1
j=1
2
~ωjeg σjz Pb +
N
X
†
~ω̃m P̃mn βm
αn +
m,n=1
ND X
N
X
∗
† −
P̃mn g̃jm σj+ αn + g̃jn
βm
σj .
j=1 m,n=1
(4.20)
b projector on H̃ef f to obtain the final Hamiltonian
Now it is time to apply the Q
Q
b P b
H̃ef
f ≡ QH̃ef f Q,
(4.21)
that is used in the Master equation. Again, the only non-trivial part is the interaction
Hamiltonian that as a matter of fact has been projected in chapter 2. It is easy to
see that
b †Q
b = b† Q
b
Qβ
m
m
(4.22)
b nQ
b = bn Q.
b
Qα
(4.23)
Therefore, the final effective JC Hamiltonian in the basis of the restricted quasimodes
is
Q
H̃ef
f
=
ND
X
1
j
2
~ωjeg σjz
+
N
X
m,n=1
~ω̃m P̃mn b†m bn
+
ND X
N
X
c∗ † −
P̃mn g̃jm σj+ bn + g̃jn
bm σj ,
j=1 m,n=1
(4.24)
b operation on the restricted space is equivalent
where, we have used the fact that Q
b
to the identity operator 1. This Q-projected
JC Hamiltonian is still non-Hermitian
due to the previous operation of non-Hermitian projector Pb. However, if all QDs are
inside of the computation volume (as we assumed from the start) then the QD term
and the interaction terms are all Hermitian. In the limit that frequencies become
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
73
real and the non-orthogonality of the QM vanishes, this projected non-Hermitian
Hamiltonian reduces to the standard multi-mode JC Hamiltonian.
Using definitions for cm operators, the Hamiltonian (4.24) can be further simplified
to
Q
H̃ef
f
=
ND
X
1
j
2
~ωjeg σjz
+
N
X
m=1
~ω̃m b†m cm
+
ND X
N
X
c∗ † −
g̃jm σj+ cm + g̃jm
cm σj .
(4.25)
j=1 m=1
Therefore, the Master equation of Eq. (4.26) can be used for the projected nonHermitian JC density matrix:
N
X
dρQ
i Q Q i Q Q†
∗
= − H̃ef f ρ + ρ H̃ef f + i
ω̃m cm ρQ (t) b†m − ω̃m
bm ρQ (t) c†m .
dt
~
~
m=1
(4.26)
Note that, because the QD Hamiltonian and the interaction Hamiltonian are Hermitian, the Lindblad term found for the density matrix using the projected free-field
Hamiltonian is the same for the density matrix using the full projected JC Hamiltonian.
4.2
Photon dynamics in a PCS with two coupled cavities
To demonstrate how the generalized quantum Master equation can be used to study
PCSs, we consider a simple two-defect slab structure. As visualized in Fig. 4.1, the
PCS consist of square array of air-holes of radius r = 0.35 d, where d is the lattice
period, in a dielectric slab with refractive index of n = 3.4. The PCS structure is
9 d wide and 10 d long and the slab has a finite thickness of h = 0.6 d. Asymmetric
placement of the defects ensures the non-orthogonality of the QMs. For simplicity,
we assume that there are no QDs present and we focus on the effects of the leakiness
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
74
PML
PML
PML
PML
Figure 4.1: Two defects are asymmetrically coupled in a PCS to create nonorthogonal QMs. The left defect support a higher Q-factor compare to
the defect on the right, closer to the slab edge. The underlying slab structure supports a PBG ranging from ωd/2πc = 0.2896 to ωd/2πc = 0.3095,
which was calculated using MPB package. The real parts of the resonant
frequency of the two defects are inside this gap.
and the non-orthogonality of the QMs on the photon dynamics.
4.2.1
Obtaining the quasimode basis
Separate FDTD calculations were first performed to obtain the single-defect QMs.
The structure was placed at the center of a computational box of size 13 d × 14 d × 3 d.
The simulation domain was ended inside slab, within the plane of the slab, to avoid
any complications due to reflections off the slab edge in xy plane. A PML layer was
applied on all the outer surfaces of the computational box to simulate the outgoing
boundary condition. A grid of 20 points per lattice period was used to obtain enough
accurate mode profiles for post-FDTD numeric, when calculating the overlap and the
coupling matrices discussed in chapter 2. These was then used to solve the eigenvalue
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
75
problem of Eq. (2.27) to obtain the QMs for the full CC structure.
Each individual cavity supports a relatively low-Q quadrupole mode. The FDTD
f L (r), and rightcalculated values for the complex frequencies of the left-defect QM, M
f R (r), are
defect QM, M
Ω̃L d
= 0.297 − i8.24 × 10−5
2πc
(4.27)
Ω̃R d
= 0.298 − i1.01 × 10−3 ,
2πc
(4.28)
respectively. These correspond to Q-factors of QL = 1805 for the left defect and
QR = 147 for the right defect. The output from the FDTD for each individual cavity
is a real-valued field amplitude at all points inside Vc that is plotted in Fig. 4.2. Shown
is the cut through the xy plane at z = 0 for the x-component of the electric field.
Obviously, the field inside the right cavity leaks more outside of the periodic air-hole
region.
To proceed further, complex QM amplitudes associated with different cavities
must be constructed. To do so, we note that the real-valued electric field for each
cavity-mode can be represented in terms of the corresponding QM as
f L,R (r) e−iΩ̃L,R t + M
f ∗ (r) eiΩ̃∗L,R t .
EL,R (r, t) = M
L,R
(4.29)
Therefore, if one outputs the real-valued FDTD-calculated electric field at t and t + τ
in time, the corresponding complex-valued QM can be constructed as
∗
f L,R (r) =
M
EL,R (r, t + τ ) − EL,R (r, t) eiΩ̃L,R τ
h
i h
i .
2 sin Re Ω̃L,R τ exp 2 Im Ω̃L,R τ
(4.30)
76
y-axis
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
x-axis
x-axis
Figure 4.2: FDTD calculated real electric field along the x-axis of the PCS for the left
defect (on the left panel) and for the right defect (on the right panel). As
seen, because the right defect is placed close to the edge of the periodic
air-holes in slab dielectric, the corresponding modes is more leaky and
some of the field penetrates more outside the holes.
The obtained QMs for the individual defects can be used in the TB formalism
discussed in chapter 2, to obtain the complex frequencies of the CC modes,
ω̃1 d
= 0.29785 − i6.34 × 10−4
2πc
(4.31)
ω̃2 d
= 0.29719 − i4.59 × 10−4 ,
2πc
(4.32)
as well as the TB expansion coefficients given by Eq. (2.22). The real part of the
TB-obtained QMs for the two CCs are plotted in Fig. 4.3. Indeed, these two QMs
are approximately the symmetric and the anti-symmetric superpositions of the two
individual-defect QMs with respect to a vertical line half way between them. From
the map, it can be seen that, in contrast to the single-defect QMs, the coupled-defect
77
y-axis
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
x-axis
x-axis
Figure 4.3: Real part of the complex TB-calculated electric field along the x-axis of
the PCS for the first QM (on the left panel) and for the second QM (on
the right panel). Compare to the single defects, the two coupled-defect
QMs suffer almost the same way from loss as they both have components
from the low-Q defect and the high-Q defect.
QMs have appreciable amplitudes over both the left and the right defects.
Finally, we need to calculate the overlap matrix between the two coupled-defect
QMs. Integrating over the entire computational volume leads to


Õ = 

0.02 − i0.80 
,
0.02 + i0.80
1.0
1.0
(4.33)
with a very pronounced non-orthogonality of the coupled-defect QMs, which results
from the close proximity of the defects and the structure asymmetry.
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
4.2.2
78
Photon number expectation
Now that all the elements that go into the QM projected quantum Master equation
are available, photon dynamics for different initial conditions can be evaluated. The
natural quantity to take a look at is the number of photons in the system at different times. In the same manner as the free-field Hamiltonian, the projected number
operator is given by
ˆ
b
NQ = Q
=
N
X
b
dµa†µ aµ Q
b†m cm ,
(4.34)
(4.35)
m=1
which can be used to obtain the photon number expectation
hNQ i (t) = Tr ρQ (t) NQ .
(4.36)
Since, the restricted number operator has a diagonal representation in the biorthogonal basis, using Eq. (3.144), the photon number expectation reduces to
hNQ i =
∞
N
X
X
hC; M ; n1 , n2 ...nN | ρQ |B; M ; n1 , n2 ...nN i
M =0 {ni }=1
× hC; M ; n1 , n2 ...nN | NQ |B; M ; n1 , n2 ...nN i .
(4.37)
This means that only diagonal elements of the density matrix contribute to the photon
number expectation. Note that this is the same conclusion arrived at if one is working
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
79
in an orthogonal standard basis with standard inner product definitions. However,
the simple result here only arises due to the definition of the bi-orthogonal basis.
Indeed, the non-orthogonality elements still exist in our calculations, but the new
representation simplifies the calculation to some extent.
4.2.3
One-photon evolution
When there is only one photon present in the system, the calculation of the projected
density matrix involves the following three B-type photon number states
|B; 0; 0, 0i
(4.38)
|B; 1; 1, 0i
(4.39)
|B; 1; 0, 1i ,
(4.40)
with the corresponding C-type conjugate states
hC; 0; 0, 0|
(4.41)
hC; 1; 1, 0|
(4.42)
hC; 1; 0, 1| .
(4.43)
The photon number expectation in this basis is
Q
hNQ i = ρQ
1,0;1,0 + ρ0,1;0,1 ,
(4.44)
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
80
where
Q
B; 1; 1, 0
ρQ
1,0;1,0 ≡ C; 1; 1, 0 ρ
(4.45)
Q
B; 1; 0, 1 .
ρQ
0,1;0,1 ≡ C; 1; 0, 1 ρ
(4.46)
It is very important to note that these diagonal elements do not represent photon
number expectations for the particular states they are labeled with, as in an orthogonal inner product space. But, summation of all diagonal element leads to total photon
number expectation in the system.
The time dependence of the diagonal elements of the density matrix now needs
to be determined to obtain the time dependence of the photon number expectation.
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
81
This is done using the generalized quantum Master equation, Eq. (4.26), to obtain
dρQ
M,{ni }
dt
∞
N
i X X
Q
0
0
0
0
=−
hC; M ; n1 , n2 ..., nN | H̃ef
f |B; M ; n1 , n2 ..., nN i
~ M 0 =0 0
{ni }=1
× hC; M 0 ; n01 , n02 ..., n0N | ρQ |B; M ; n1 , n2 ..., nN i
∞
N
i X X
hC; M ; n1 , n2 ..., nN | ρQ |B; M 0 ; n01 , n02 ..., n0N i
+
~ M 0 =0 0
{ni }=1
Q†
× hC; M 0 ; n01 , n02 ..., n0N | H̃ef
f |B; M ; n1 , n2 ..., nN i
+i
N X
∞
∞
X
X
N
X
N
X
ω̃m hC; M ; n1 , n2 ..., nN | cm |B; M 0 ; n01 , n02 ..., n0N i
m=1 M 0 =0 M 00 =0 {n0 }=1 {n00 }=1
i
i
× hC; M 0 ; n01 , n02 ..., n0N | ρQ |B; M 00 ; n001 , n002 ..., n00N i
× hC; M 00 ; n001 , n002 ..., n00N | b†m |B; M ; n1 , n2 ..., nN i
−i
N X
∞
∞
X
X
N
X
N
X
∗
ω̃m
hC; M ; n1 , n2 ..., nN | bm |B; M 0 ; n01 , n02 ..., n0N i
m=1 M 0 =0 M 00 =0 {n0 }=1 {n00 }=1
i
i
× hC; M 0 ; n01 , n02 ..., n0N | ρQ |B; M 00 ; n001 , n002 ..., n00N i
× hC; M 00 ; n001 , n002 ..., n00N | c†m |B; M ; n1 , n2 ..., nN i .
(4.47)
The expectation values in the first and the third lines are easy to evaluate as we know
how cm and b†m operate on the elements of the biorthogonal basis. However, this is
not the case in the second and fourth lines. However, the bm and c†m operators can
be expanded in terms cm and b†m to allow simple evolution of these terms. Indeed, for
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
82
the diagonal elements of the density matrix, we find
dρQ
1,0;1,0
Q
∗
∗
= −iω̃1 ρQ
+
i
ω̃
P̃
Õ
+
ω̃
P̃
Õ
1,0;1,0
1 11 11
2 12 21 ρ1,0;1,0
dt
∗
∗
+ i ω̃1 P̃21 Õ11 + ω̃2 P̃22 Õ21 ρQ
1,0;0,1
(4.48)
dρQ
0,1;0,1
Q
∗
∗
= −iω̃2 ρQ
+
i
ω̃
P̃
Õ
+
ω̃
P̃
Õ
22
1 11 12
2 12 22 ρ0,1;1,0
dt
∗
∗
+ i ω̃1 P̃21 Õ12 + ω̃2 P̃22 Õ22 ρQ
0,1;0,1 .
(4.49)
and
These are coupled to the dynamics of the some of the off-diagonal elements of the
projected density matrix. Therefore the following off-diagonal components must be
evaluated as well
dρQ
1,0;0,1
Q
∗
∗
= −iω̃1 ρ12 + i ω̃1 P̃11 Õ12 + ω̃2 P̃12 Õ22 ρQ
1,0;1,0
dt
+ i ω̃1∗ P̃21 Õ12 + ω̃2∗ P̃22 Õ22 ρQ
1,0;0,1
(4.50)
dρQ
0,1;1,0
∗
∗
P̃
Õ
+
ω̃
P̃
Õ
ρQ
= −iω̃2 ρQ
+
i
ω̃
12
21
11
11
21
0,1;1,0
2
1
dt
+ i ω̃1∗ P̃21 Õ11 + ω̃2∗ P̃22 Õ21 ρQ
0,1;0,1 .
(4.51)
and
This gives a set of four ordinary first-order differential equations that can be solved
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
83
for an arbitrary initial density matrix to obtain the one-photon evolution in our
coupled-cavity system with two non-orthogonal QMs.
For the initial condition, we first consider two situations where the one photon is
initially in one of the two QMs (details of the pumping of these states is not considered
here), i.e.
1 ψ (0) = |B; 1; 1, 0i
(4.52)
2 ψ (0) = |B; 1; 0, 1i .
(4.53)
Note that in constructing the corresponding density matrices, the initial conditions
are:
ρ1 (0) ≡ |B; 1; 1, 0i hB; 1; 1, 0|
(4.54)
ρ2 (0) ≡ |B; 1; 0, 1i hB; 1; 0, 1| .
(4.55)
The log-scale plot of the photon number expectation is shown in Fig. (4.4). Plain
decays are observed in agreement with expectations as the slopes of the lines equal
twice the imaginary part of the QM complex frequencies.
We now, consider a situation where the initial photon is launched in either the right
or the left individual defect mode. Since the coupled-defect QMs can be expressed in
terms of the individual-defect QMs,
e m (r) =
N
X
q
f q (r) ,
vemq M
(4.56)
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
84
the inverse transformation can be used to represent the one-photon state in one of
the single cavities in terms of the coupled-cavity QMs. Therefore, such initial state
can be written as
3 ψ (0) = uL,1 |B; 1; 1, 0i + uL,2 |B; 1; 0, 1i
(4.57)
4 ψ (0) = uR,1 |B; 1; 1, 0i + uR,2 |B; 1; 0, 1i ,
(4.58)
where
2
X
ũpm ṽmq = δpq .
(4.59)
m=1
When the PCS is launched in |ψ 3 (0)i state, the photon is initially away from the edge
of the PCS, where if the PCS is launched in |ψ 4 (0)i state, the photon starts off closer
to the edge of the slab. As plotted on the log scale in Fig. (4.4), in the former case,
the photon initially leaks out slowly, but after a time τ = π/ω12 where ω12 ≡ ω1 − ω2 ,
it will beat over to the right single-cavity mode, where it will experience a higher rate
of loss. Conversely, if the photon is initially in the right cavity mode, then it will leak
rapidly out of the system at early times but the decay rate will slow once in is in the
left single cavity mode.
Even though we have not included QDs in here, the strong dependence of the
dynamics on the initial conditions is directly related to the strong dependence of the
rate of spontaneous emission on the location of a QD if there was one. Different
locations of the QD lead to effectively different initial conditions for the emitted
photon. This points to the possibility of designing a coupled cavity system to control
the emission and evolution of photons, by choosing the location of the QD to give the
desired initial superposition of QM states.
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
100
First Quasimode
Second Quasimode
Left cavity mode
Right cavity mode
10−1
Photon Expectation
85
10−2
10−3
10−4
10−5
10−6
0
1
2
3
4
5
γ2 t
Figure 4.4: The total photon number as a function of time for the structure of Fig.
4.1 for different one-photon initial states: First QM, |ψ 1 (0)i = |B; 1; 1, 0i
(yellow dot); Second QM, |ψ 2 (0)i = |B; 1; 0, 1i (green short-dash) ; Left
cavity mode |ψ 3 (0)i = uL,1 |B; 1; 1, 0i + uL,2 |B; 1; 0, 1i (red long-dash);
Right cavity mode |ψ 4 (0)i = uR,1 |B; 1; 1, 0i + uR,2 |B; 1; 0, 1i (blue solid).
4.2.4
Two-photon evolution
Let us now consider the situation where there are two photons present at time t = 0,
in the system. As before, the photon number dynamics can be obtained using the
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
86
Master equation 4.26. The basis of our calculation in this case are
|B; 2; 2, 0i
(4.60)
|B; 2; 1, 1i
(4.61)
|B; 2; 0, 2i
(4.62)
|B; 1; 1, 0i
(4.63)
|B; 1; 0, 1i
(4.64)
|B; 0; 0, 0i ,
(4.65)
and the corresponding conjugate C-type states. The photon number expectation in
this case is given by
Q
Q
Q
Q
hNQ i = 2ρQ
2,0;2,0 + 2ρ1,1;1,1 + 2ρ0,2;0,2 + ρ1,0;1,0 + ρ0,1;0,1 .
(4.66)
Similar to the procedure for the one-photon dynamics, some of the diagonal elements
in this representation are coupled to some other off-diagonal members of the projected
Q
Q
density matrix. For example, calculating ρQ
2,0;2,0 involves calculating ρ2,0;1,1 and ρ0,2;2,0
through
dρQ
2,0;2,0
∗
∗
P̃
Õ
+
ω̃
P̃
Õ
ρQ
= −2iω̃1 ρQ
+
2i
ω̃
12
21
11
11
2,0;2,0
2,0;2,0
1
2
dt
√ + 2i ω̃1∗ P̃21 Õ11 + ω̃2∗ P̃22 Õ21 ρQ
2,0;1,1 ,
(4.67)
4.2. PHOTON DYNAMICS IN A PCS WITH TWO COUPLED
CAVITIES
87
dρQ
2,0;1,1
Q
∗
∗
+
i
ω̃
= −2iω̃1 ρQ
P̃
Õ
+
ω̃
P̃
Õ
2,0;1,1
1 11 11
2 12 21 ρ2,0;1,1
dt
√ ∗
∗
+ 2i ω̃1 P̃11 Õ12 + ω̃2 P̃12 Õ22 ρQ
2,0;2,0
+
√ ∗
2i ω̃1 P̃21 Õ11 + ω̃2∗ P̃22 Õ21 ρQ
2,0;0,2
+ i ω̃1∗ P̃21 Õ12 + ω̃2∗ P̃22 Õ22 ρQ
2,0;1,1
(4.68)
and
√ ∗
dρQ
2,0;0,2
Q
∗
= −2iω̃1 ρQ
+
2i
ω̃
P̃
Õ
+
ω̃
P̃
Õ
2,0;0,2
1 11 12
2 12 22 ρ2,0;1,1
dt
∗
∗
+ 2i ω̃1 P̃21 Õ12 + ω̃2 P̃22 Õ22 ρQ
2,0;0,2 .
(4.69)
Other relevant Master equations that must be used to obtain the two-photon dynamics
are given in Appendix B.
Now, we consider the following three different scenarios for the initial state of the
system
1 ψ (0) = |B; 2; 2, 0i
(4.70)
2 ψ (0) = |B; 2; 0, 2i
(4.71)
3 ψ (0) = |B; 2; 1, 1i .
(4.72)
Plotted in Fig. 4.5, are the corresponding dynamics as a function of time on a
log scale. In general, similar conclusions to the one-photon case are valid in same
situations. However, the most interesting situation arises when the two photons
4.3. SUMMARY
88
Photon Expectation
100
|B; 2, 2, 0i
|B; 2, 0, 2i
|B; 2, 1, 1i
10−1
10−2
10−3
10−4
10−5
10−6
0
1
2
3
4
5
γ2 t
Figure 4.5: The total photon number as a function of time for the structure of Fig. 2
for different two-photon initial states: |ψ 1 (0)i = |B; 2; 2, 0i (green shortdash); |ψ 2 (0)i = |B; 2; 0, 2i (red long-dash) ; |ψ 3 (0)i = |B; 2; 1, 1i (blue
solid).
start out in different QMs, namely |ψ 3 (0)i, that is plotted as blue solid in Fig. 4.5.
One might have expected that the photons would have simply decayed with the two
different decay rates, but instead the non-orthogonality introduces oscillations into
the dynamics. This is a non-trivial effect that must be taken into consideration in
any study of CC structures with non-orthogonal QMs in PCSs.
4.3
Summary
In this chapter, the effective JC Hamiltonian for multimode CC systems interacting
with multiple QDs was obtained in the basis of the QMs. As an example, two cavities
4.3. SUMMARY
89
placed asymmetrically within a square-lattice PCS were considered, where the nonzero spatial overlap between the system QMs demonstrated the non-orthogonality of
the QMs. Using the Master equation derived in chapter 3, the evolution of the density
matrix in the basis of the QMs was obtained. From that, the number of photons in
the system was monitored as a function of the time where the non-trivial oscillation
behavior associated with the QMs non-orthogonality was observed. Our formalism
can be used to study photon dynamics in complex PCS circuits, where there may
be a benefit in having non-orthogonal QMs in order to perform certain tasks in an
optical network. In addition, in situations where the QMs non-orthogonality is a
given in the design of the system, our formalism is capable of estimating the degree
of modification to the photon behavior that one must expect.
90
Chapter 5
Nonlinear pair-generation in leaky
coupled-cavity systems
Photon pair-generation through nonlinear processes is of great importance in quantum photonics, particularly as a means to obtain heralded single photon sources [70]
and entangled photons sources [20] for on-chip integrated devices. This is true for
systems both with cubic [51, 71–79] and quadratic nonlinearities [80–84]. In particular, the cubic nonlinear process of spontaneous four wave mixing (SFWM), where
the system is pumped at one frequency and signal and idler photons are detected
at neighboring frequencies, is increasingly being studied experimentally in nanophotonic structures [85–90]. To date, however the effects of loss in a quantum mechanical
picture have been treated only in optical fibers, silicon waveguides and silicon wire
waveguides [86, 87, 89, 90].
In this chapter, we apply our QM projection technique to find the effective full
Hamiltonian of the SFWM in CC systems. The obtained Hamiltonian is then used
to calculate perturbative but analytical expressions for the time-evolution of different
5.1. THE LINEAR HAMILTONIAN
91
operators of interest. This is done using the Heisenberg picture and the adjoint Master
equation, rather than the Schrödinger picture and the regular Master equation. The
techniques discussed in this chapter are based on the work presented in [91].
For the purpose of this chapter, we assume the existence of an orthogonal QM
basis. The orthogonality could be the consequence of underlying symmetry of the
structure of desired interest, such as a CROW in a PCS, or due to lack of overlap
between the modes. Although generalization to a continuum of modes is straightforward, for simplicity we continue to work with a discrete basis. It should be also
noted that effects such as two-photon absorptions that can occur at high intensities
are ignored in this chapter and in chapter 6.
5.1
The linear Hamiltonian
The effective linear Hamiltonian of the free field was obtained already in Chapter 3.
However, when QMs are orthogonal to each other, the overlap matrix reduces to an
identity matrix which in turn reduces the QM projected free-field Hamiltonian to
H̃LQ =
X
~ω̃m b†m bm ,
(5.1)
m
where now, the ladder operators satisfy the standard bosonic commutation relation
bm , b†n = δmn .
(5.2)
5.1. THE LINEAR HAMILTONIAN
92
The corresponding adjoint master equation for a given Heisenberg operator, ÂH (t),
can be also written as
N X
bH (t)
i b
dA
i Q† b
Q
∗ † b
bH (t) bm − ω̃m
− AH (t) H̃L + H̃L AH (t) + i
ω̃m b†m A
bm AH (t) bm .
dt
~
~
m=1
(5.3)
It should be remembered that operators without explicit time dependence are Schrödinger
operators. As we have seen before, the ladder operators obey the simple dynamical
equations
dbn (t)
= −iω̃n bn (t)
dt
(5.4)
db†n (t)
= iω̃n∗ b†n (t) .
dt
(5.5)
and
This can be generalized to any normally-ordered product of ladder operators. To
show this, it is convenient to first transform the Master equation (5.3) into the much
more useful form:
N X
dÂH (t)
i b
i Q† b
Q
† b
∗ † b
=− A
H̃
+
H̃
A
+
i
ω̃
b
A
b
−
ω̃
b
A
b
H L
H
m m H m
m m H m
dt
~
~ L
m=1
N
N
i X ∗ †
i X
†
b
bH
~ω̃m AH bm bm +
~ω̃ b bm A
=−
~ m=1
~ m=1 m m
(5.6)
(5.7)
N
i X
bH bm − ~ω̃ ∗ b† A
bH bm
+
~ω̃m b†m A
m m
~ m=1
=−
N
N
X
i X
∗ †
bH bm + i
bH b† − b† A
b
b
~ω̃m A
~ω̃
b
b
A
−
A
b
m H
H m ,
m
m
~ m=1
~ m=1 m m
(5.8)
5.1. THE LINEAR HAMILTONIAN
93
which can be recast as
dÂH (t)
= L0 ÂH (t)
dt
(5.9)
h
i
i
iX
iX ∗ † h
†
~ω̃m ÂH (t) , bm bm +
~ω̃m bm bm , ÂH (t) .
≡−
~ m
~ m
(5.10)
This equation yields simple Heisenberg dynamics for any normally-ordered product of ladder operators under the time evolution of the system linear Hamiltonian,
with loss included systematically. For example, for the simplest product operator,
b†n1 bn2 (t), using the ansatz b†n1 bn2 (t) ≡ fn1 ,n2 (t) b†n1 bn2 , it is straightforward to
see that Eq. (5.9) leads to
d b†n1 bn2 (t)
ifn ,n (t) X
=− 1 2
~ω̃m b†n1 bn2 , b†m bm
dt
~
m
+
=−
(5.11)
ifn1 ,n2 (t) X
~ω̃m b†n1 bn2 , b†m bm
~
m
+
=−
ifn1 ,n2 (t) X ∗ † ~ω̃m bm bm , b†n1 bn2
~
m
ifn1 ,n2 (t) X ∗ † ~ω̃m bm bm , b†n1 bn2
~
m
(5.12)
ifn1 ,n2 (t) X
~ω̃m b†n1 δn2 m bm
~
m
+
ifn1 ,n2 (t) X ∗ †
~ω̃m bm δn1 m bn2
~
m
= ifn1 ,n2 (t) ω̃n∗ 1 − ω̃n2 b†n1 bn2 ,
(5.13)
(5.14)
5.2. THE NONLINEAR SFWM HAMILTONIAN
94
which is equivalent to
d b†n1 bn2 (t)
= i ω̃n∗ 1 − ω̃n2 b†n1 bn2 (t) .
dt
(5.15)
The parentheses around the two operators indicates that this pair of operators is to be
considered as one Heisenberg operator. We use this notation because in the presence
of nonlinearity and loss, it can be shown that in general for two Heisenberg operators
 and B̂
ÂB̂ (t) 6= Â (t) B̂ (t) .
(5.16)
Note that when the nonlinearity is not present the operator dynamics is factorable;
for example from Eq. (5.15) it can be shown that
b†n bn (t) = b†n (t) bn (t) .
(5.17)
Eq. (5.15) can be generalized to any normally ordered product of ladder operators.
For example,
d b†n1 b†n2 b†n3 bn4 bn5 (t)
= i ω̃n∗ 1 + ω̃n∗ 2 + ω̃n∗ 3 − ω̃n4 − ω̃n5 b†n1 b†n2 b†n3 bn4 bn5 (t) .
dt
(5.18)
This is quite powerful, as any given operator can be written as sums of products of
normally-ordered ladder operators.
5.2
The nonlinear SFWM Hamiltonian
We now find the QM representation of the nonlinear portion of the system Hamiltonian. Following [92], the Hamiltonian responsible for the nonlinear process of SFWM
5.2. THE NONLINEAR SFWM HAMILTONIAN
95
is
HN L
o
=−
4
ˆ
(3)
d3 r χijkl (r) (r) Ei (r) Ej (r) Ek (r) El (r) ,
(3)
where, subscripts i, j, k, l label the components of the electric field and χijkl is the
third-order nonlinear susceptibility tensor.
Although direct projection of the nonlinear Hamiltonian can be done using similar
steps to those taken to project ladder operators and the free-field Hamiltonian in
Chapter 3, it is easier to use a straight expansion of the field operator in terms of
QMs, instead. Indeed, the real advantage of working in QM basis is that they provide
a more practical basis than the true modes for the study of optical systems, over the
frequency range of interest. As shown in Appendix C, the positive-frequency part of
the electric field can be expanded in terms of an orthogonal basis of QMs as
(+)
ẼQ
(r) =
N
X
r
m=1
~ω̃m
Ñm (r) bm .
20
(5.19)
Including only the photon-number-conserving terms associated with two-photon processes, the projected nonlinear Hamiltonian is given by
HNQL
N
N
N
N
o X X X X
=−
4 m =1 m =1 m =1 m =1
1
2
3
4
r
~ω̃m1
20
r
~ω̃m2
20
r
~ω̃m3
20
r
~ω̃m4
20
ˆ
×
(3)
∗
d3 r χijkl (r) (r) Ñm
(r) Ñm2 j (r) Ñm3 k (r) Ñm4 l (r) b†m1 b†m2 bm3 bm4 + h.c.,
1i
(5.20)
where, m1 and m2 label signal and idler modes in the SFWM process, while m3 and
m4 label the pump modes. Now, assuming that the imaginary parts of the complex
5.3. HEISENBERG EVOLUTION OF NUMBER OPERATORS
96
frequencies (only) under the square roots are small, and defining
Sm1 ,m2 ,m3 ,m4
√
~ ωm1 ωm2 ωm3 ωm4
≡
16 o
ˆ
(3)
∗
∗
× d3 r χijkl (r) (r) Ñm
(r) Ñm
(r) Ñm3 k (r) Ñm4 l (r) , (5.21)
1i
2j
the projected nonlinear Hamiltonian can be written as
HNQL
=−
N
X
N
X
~ Sm1 ,m2 ,m3 ,m4 b†m1 b†m2 bm3 bm4 + h.c..
(5.22)
m1 ,m2 =1 m3 ,m4 =1
The quantity Sm1 ,m2 ,m3 ,m4 represents the strength of the effective nonlinear interaction
between different QMs and is negligible unless the QMs overlap significantly over the
nonlinear region.
The full adjoint master equation, including the nonlinear interaction in the system,
becomes
ih
i
dÂH (t)
= L0 ÂH (t) −
ÂH (t) , HNQL .
dt
~
(5.23)
In general, analytic solutions to this equation are not available. However, as shown in
the next section, analytic expressions for the time evolution of the operators of interest
can be obtained using perturbation theory applied to the nonlinear Hamiltonian.
5.3
Heisenberg evolution of number operators
Signal and idler photon statistics are of primary interest in a typical pair-generation
experiment. In order to detect signal and idler photons unambiguously, we assume
that their frequencies are well separated from the pump frequency and from each
5.3. HEISENBERG EVOLUTION OF NUMBER OPERATORS
97
other. Although photon generation near the pump band can take place, it is unobservable in practice due to the need to filter the pump. It is therefore necessary to be
able to calculate the number of generated photons of each type in our system, as a
function of time. The goal in this section is to calculate the Heisenberg time evolution
for the photon-number operator as a perturbation expansion in Sm1 ,m2 ,m3 ,m4 of the
form
(0)
(1)
(2)
b†n bn (t) = b†n bn
(t) + b†n bn
(t) + b†n bn
(t) + · · · ,
(5.24)
when the real part of the signal mode frequency, ωn , is far from the pump-band
modes.
From Eq. (5.23), the adjoint master equation for the photon-number operator is
i
d b†n bn (t)
ih † Q
†
= L0 bn bn (t) −
bn bn (t) , HN L .
dt
~
(5.25)
In the zeroth order approximation in the nonlinear Hamiltonian, the evolution is
simply
(0)
(0)
d b†n bn
(t)
= L0 b†n bn
(t) ,
dt
(5.26)
which has the elementary solution,
b†n bn
(0)
(t) = b†n bn e−2γn t ,
(5.27)
when we set n1 = n2 = n in Eq. (5.15) and then integrate over time. Thus photons in
the signal and idler QMs would decay exponentially with a rate 2γn if there were no
nonlinear interaction. Assuming that the pump field is in a lossy multimode coherent
5.3. HEISENBERG EVOLUTION OF NUMBER OPERATORS
98
state [88], we can express it as
|Φpump i =
Y
†
eαφp bp −α
∗ φ∗ b
p p
|vaci .
(5.28)
p
Then for any mode p with a frequency within the pump band, we have
bp |Φpump i = αφp |Φpump i ,
(5.29)
where α is the overall coherent state amplitude, φp is the pump amplitude for the pth
P
mode, and p |φp |2 = 1. That is, the state of the pump field can be considered as a
product of lossy coherent states in different modes with amplitudes αφp . One would
usually take the pump field to be nonzero only for a few QMs that are close to each
other in frequency. As the signal and idler modes are in vacuum initially, the total
initial state of the system factors as
|Φi i ≡ |Φpump i ⊗ |vacisi .
(5.30)
where |vacisi is the vacuum state for the signal and idler modes. Therefore, the
zeroth order solution in Eq. (5.27) makes no contribution to the expectation value of
the photon-number operator
D
b†n bn
(0)
E
(t) = e−2γn t vac b†n bn vac = 0.
(5.31)
The nonlinear Hamiltonian appears in the first order equation,
(1)
i
(1)
d b†n bn
(t)
i h † (0)
= L0 b†n bn
(t) −
bn bn
(t) , HNQL .
dt
~
(5.32)
5.3. HEISENBERG EVOLUTION OF NUMBER OPERATORS
99
Using expressions for the zeroth order number operator (5.27) and the projected
nonlinear Hamiltonian, we obtain
b†β bβ
(0)
(t) , HNQL
= −e−2γn t
N
X
N
X
~ Sm1 ,m2 ,m3 ,m4 b†n bn , b†m1 b†m2 bm3 bm4
m1 ,m2 =1 m3 ,m4 =1
−2γn t
+e
N
X
N
X
† †
∗
~ Sm
bm4 bm3 bm2 bm1 , b†n bn .
1 ,m2 ,m3 ,m4
m1 ,m2 =1 m3 ,m4 =1
(5.33)
Since, n 6= m3 , m4 (note that m3 and m4 correspond to pump modes), it is straightforward to see that
†
bn bn , b†m1 b†m2 bm3 bm4 = b†n bn , b†m1 b†m2 bm3 bm4
(5.34)
= b†n b†m2 bm3 bm4 δnm1 + b†n b†m1 bm3 bm4 δnm2
(5.35)
= b†m1 b†m2 bm3 bm4 δnm1 + b†m2 b†m1 bm3 bm4 δnm2
(5.36)
= b†m1 b†m2 bm3 bm4 (δnm1 + δnm2 ) .
(5.37)
Therefore
(
)
(1)
X X
d b†n bn
(t)
(1)
= L0 b†n bn
(t)+ 2ie−2γn t
Sn,m2 ,m3 ,m4 b†n b†m2 bm3 bm4 + h.c. ,
dt
m m ,m
2
3
4
(5.38)
where we have used the even symmetry of Sm1 ,m2 ,m3 ,m4 under m1 and m2 permutation.
5.3. HEISENBERG EVOLUTION OF NUMBER OPERATORS
100
To solve Eq. (5.38), we try the ansatz
b†n bn
(1)
(t) =
X X
† †
h(1)
n,m2 ,m3 ,m4 (t) bn bm2 bm3 bm4 + h.c.,
(5.39)
m2 m3 ,m4
(1)
where hn,m2 ,m3 ,m4 (t) is a c-number function. With this choice, the action of the signal
and idler operators on the vacuum guarantees zero for the first order expectation value
of the number operator
b†n bn
(1)
(t) = 0.
(5.40)
However, we need the first order solution to (5.39) for use in the second order approximation.
Inserting (5.39) into (5.38), and using the result for the time dependence of normally ordered (unperturbed) operators obtained from Eq. (5.9) we find
(1)
dhn,m2 ,m3 ,m4 (t)
∗
−2γn t
= i ω̃n∗ + ω̃m
− ω̃m3 − ω̃m4 h(1)
Sn,m2 ,m3 ,m4 .
n,m2 ,m3 ,m4 (t) + 2ie
2
dt
(5.41)
(1)
The fact that we have obtained an ordinary differential equation for hn,m2 ,m3 ,m4 (t)
indicates that the ansatz of Eq. (5.39) provides a solution. Integrating this leads to
h(1)
n,m2 ,m3 ,m4
∗ +ω̃ ∗ −ω̃
i(ω̃n
m3 −ω̃m4 )t
m2
ˆ
(t) = 2i Sn,m2 ,m3 ,m4 e
t
dt0 e−2γn t e−i(ω̃n +ω̃m2 −ω̃m3 −ω̃m4 )t .
0
∗
∗
0
0
(5.42)
Following a similar procedure (see Appendix D) we find the second order expectation
5.3. HEISENBERG EVOLUTION OF NUMBER OPERATORS
101
for the number operator:
D
b†n bn
(2)
E
(t) = 8|α|4 Re

X X X

ˆ
t
×e−2γn t
∗
φ φ∗ 0 φ∗ 0
Sn,m2 ,m3 ,m4 Sn,m
0 φ
0
2 ,m3 ,m4 m3 m4 m3 m4
m2 m3 ,m4 m03 ,m04
∗ +ω̃ ∗ −ω̃ ∗ −ω̃ ∗
−i
ω̃
t0
0
0
n
m
2
0
m3
m4
ˆ
)
t
dt00 ei(
∗ +ω̃ ∗ −ω̃
ω̃n
m3 −ω̃m4
m2
dt e
)
t00
,
t−t0
0
(5.43)
which is the first non-zero contribution. Because the photon number expectation
obtained here is second order in the nonlinear interaction, Sm1 ,m2 ,m3 ,m4 , it has a
square dependence on the pump field intensity (forth power dependence on the pump
field amplitude) for different pump modes.
In addition to counting signal and idler photons in the system, it is also important
to determine the number of signal-idler pairs in the system. Mathematically, the
expectation value of the product of two number operators associated with signal and
idler photons, b†n1 bn2 b†n2 bn2 (t), is required when n1 6= n2 . Similar to the procedure
used for the number operator (see Appendix E), the first non-zero contribution for
the product of two photon-number operators is also second order in the perturbation
and is given by
D

X X
E
(2)
b†n1 bn1 b†n2 bn2
(t) = 8|α|4 Re
Sn1 ,n2 ,m3 ,m4 Sn∗1 ,n2 ,m03 ,m04 φm3 φm4 φ∗m03 φ∗m04

0
0
m3 ,m4 m3 ,m4
×e−2(γn1 +γn2 )t
ˆ
t
∗ +ω̃ ∗ −ω̃ ∗ −ω̃ ∗
−i
ω̃
t0
0
0
n
n
1
2
0
m3
m4
ˆ
t
dt e
0
)
dt00 ei(ω̃n1 +ω̃n2 −ω̃m3 −ω̃m4 )
t00
,
t−t0
(5.44)
5.4. LOSSY DYNAMICS OF PHOTON-PAIR GENERATION
102
where the n1 and n2 modes are both assumed to sit far away in frequency from the
pump modes.
In this subsection, we have obtained approximate but analytic expressions for the
averaged signal, idler and signal-idler paired photons in a lossy CC system.
5.4
Lossy dynamics of photon-pair generation
To demonstrate the impact of loss on the pair generation process, we consider a
simple CC model in which there are only three QMs, one for each of the signal,
idler and pump photons. For example, these could be the resonant frequencies of the
three coupled cavities in the so-called photonic molecule structures in photonic crystal
slabs, as visualized in Fig. 5.1. For simplicity, we also assume that the signal and
idler frequencies are in resonance with the pump frequency such that ωs + ωi = 2ωp .
This is a restriction imposed upon the real part of the frequencies only as we wish
to investigate the effect of QM loss on the pair generation process under resonant
conditions, where pair generation can be maximal. In addition, we assume that at
t = 0, the pump photons are already loaded in the pump mode, perhaps using a
waveguide or through fiber coupling. This loading process could be included in our
model explicitly, but here we wish to focus on the physics of the generation process
and so simply assume the initial state to be one with the pump mode in a coherent
state.
The very first quantity of interest in our lossy model is the average number of
photon pair generated for different system loss rate situations. For the simple three
5.4. LOSSY DYNAMICS OF PHOTON-PAIR GENERATION
103
PCS with non-zero χ(3)
signal (ωs )
pump (ωp )
idler (ωi )
Figure 5.1: Visualization of three coupled cavities in a PCS with non-zero χ(3) . Such
a structure can support three modes close in frequency and overlapping
in space suitable for nonlinear pair-generation experiments via SFWM.
CC system, using Eq. (5.44), we define the scaled two-photon expectation to be
1
Nsi (t) ≡
8|G|2
ˆ
=e
−4γp t
t
b†s bs b†i bi
(2)
(t)
0 −2(γ̄−γp )t0
ˆ
0
=
8 (γ̄ − γp )2
00
dt00 e−2(γ̄−γp )t
dt e
e−4γ̄t 1 − e−2(γ̄−γp )t
t0
(5.45)
(5.46)
0
2
.
(5.47)
Here G ≡ Sms ,mi ,p,p |αφp |2 is simply a constant representing the coupling between the
three modes in the presence of the nonlinearity and γ̄ = (γs + γi ) /2 is the average loss
in the signal and idler modes. Note that this function only depends on the sum of the
losses in the signal and idler modes and not on each separately. As expected, at t = 0
there are no pairs as this is our initial condition. The same is true when t → ∞, since
all of the photons have leaked out of the system. The two-photon expectation value
5.4. LOSSY DYNAMICS OF PHOTON-PAIR GENERATION
0.035
104
γ̄ = 2γp
γ̄ = γp
γ̄ = γp /2
0.03
γp2 Nsi (t)
0.025
0.02
0.015
0.01
0.005
0
0
1
2
γp t
3
4
Figure 5.2: Two-photon expectation value for three different cases: green-dashed,
when γ̄ = γp /2; red-dotted, when γ̄ = γp ; blue-solid, when γ̄ = 2γp .
(normalized to the effective nonlinear interaction strength) is shown as a function
of time in Fig. 5.2. The results are plotted for three different cases with different
ratios of the average loss in the signal and idler, γ̄, to the loss in the pump mode,
γp . The general trend in all three cases is similar: there is a rise at early times due
to the nonlinear generation followed by an exponential decay due to photon leakage.
However, for a given pump loss, as γ̄ increases the two-photon expectation value in
general decreases.
As discussed in the introduction of the thesis, photon loss can also result in false
heraldings where the signal photon might not be delivered to the output port of
the device even when the heralding idler detector has already clicked. Therefore,
to understand how scattering loss affects pair dynamics within the CC system, we
5.4. LOSSY DYNAMICS OF PHOTON-PAIR GENERATION
105
consider the expectation value of the number of signal plus idler photons in the
system that are not in pairs, due to leaking out either a signal or an idler photon.
Quantitatively, this can be calculated as the following
N1 (t) ≡ Ns (t) + Ni (t) − 2Nsi (t) ,
(5.48)
where
Ns (t) ≡
1 D † (2) E
bs bs
(t)
8|G|2
ˆ
t
−4γp t
0 −2(γ̄−γp )t0
(5.49)
ˆ
t0
00
dt00 e−(γs −γi −2γp )t
dt e
=e
(5.50)
0
0
e−4γp t
=
γs − γi − 2γp
1 − e(4γp −2γs )t 1 − e−(2γ̄−2γp )t
+
4γp − 2γs
2γ̄ − 2γp
,
(5.51)
and
1
Ni (t) ≡
8|G|2
ˆ
=e
−4γp t
t
b†i bi
(2)
(t)
0 −2(γ̄−2γp )t0
(5.52)
ˆ
e−4γp t
=
γi − γs − 2γp
00
dt00 e−(γi −γs −2γp )t
dt e
0
t0
(5.53)
0
1 − e(4γp −2γi )t 1 − e−(2γ̄−2γp )t
+
4γp − 2γi
2γ̄ − 2γp
,
(5.54)
are the scaled signal and idler averaged photon numbers. In Eq. (5.48), signal and idler
photons are counted separately, and the photons that are still in pairs are subtracted.
Thus, this represents the one-photon-only expectation.
5.4. LOSSY DYNAMICS OF PHOTON-PAIR GENERATION
106
Depicted in Fig. 5.3 are the two-photon and one-photon-only expectations as a
function of time, when γ̄ = γp . As seen, both paired and unpaired expectations have
the same general behavior as a function of time. The rise time for the one-photononly result is longer than that for the two-photon result. This occurs because the
signal and idler photons must be generated first in order for them to decay into the
one-photon state. Also, the one-photon-only curve has a larger maximum, so that at
later times it is more likely for the system to be detected in a one-photon state when
loss is present. At large times, the two-photon expectation decays approximately
as exp (−4γp t) whereas the unpaired photons decay approximately as exp (−2γp t).
Therefore, the one-photon-only state of the system decays slower which again confirms
the larger probability of the system carrying unpaired photons.
Note that if we assume that signal and idler modes are lossless, we can immediately
conclude that Ns (t) = Ni (t) = Nsi (t) irrespective of the loss in the pump, so that
N1 (t) = 0, in agreement with expectations. In such a case, one never finds only one
photon in the system as the generated photon pairs do not leak out.
5.4.1
Lossless pump limit
Let us now consider the situation where the pump mode is lossless (γp = 0) but the
signal and idler modes are not. In Fig. 5.4 we plot the photon-pair expectation along
with the one-photon-only expectation when only the pump is lossless. We see that
at large times, the expectation values tend to steady state values given by
Nsi (∞) =
1
,
8γ̄ 2
(5.55)
5.4. LOSSY DYNAMICS OF PHOTON-PAIR GENERATION
Photon Expectation
0.025
107
γp2 N1 (t)
γp2 Nsi (t)
0.02
0.015
0.01
0.005
0
0
1
2
γp t
3
4
Figure 5.3: Comparison between the two-photon expectation and one-photon-only
expectation when γ̄ = γp . Two-photon expectation (red-dashed) is compared to the only one-photon expectation (blue-solid).
and
N1 (∞) =
1
1
− 2.
2γs γi 4γ̄
(5.56)
In this case, because the pump photons last forever in the system, they can constantly
convert into signal and idler photons, reaching a steady state at longer times. This
would be the situation that would arise if we imagine that the photons in the pump
mode are coupled in from outside at a rate such that the new photons added to the
mode exactly balance those lost due to scattering losses. Since in the derivation of
Eqs. (5.43) and (5.44), no use of the undepleted pump approximation has been made,
one might expect the photon expectation of the signal and idler modes to drop at
5.4. LOSSY DYNAMICS OF PHOTON-PAIR GENERATION
Photon Expectation
0.25
108
γ̄ 2 N1 (t)
γ̄ 2 Nsi (t)
0.2
0.15
0.1
0.05
0
0
1
2
γ̄ t
3
4
Figure 5.4: Photon dynamics when the pump mode is lossless but γs = γi . Twophoton expectation (red-dashed) is compared to the one-photon-only expectation (blue-solid).
long times. However, note that these results are at the second order in perturbation
theory and it can be shown that pump depletion does not appear to this order.
Note that the analytic expressions of Eqs. (5.55) and (5.56) suggest that one
should keep the photon decay rates low if one wishes to obtain a high steady state
pair generation rate. However, as we will see later, there is trade-off between the
signal brightness and multiple pair generation that depends on the loss constants.
In Fig. 5.5, the one-photon-only expectation value is shown for different values
of the difference between signal and idler photon loss rates, again for the case where
γp = 0. As seen, increasing loss-rate difference results in higher values for the onephoton-only expectation, especially at longer times after generation. This means that
5.5. EFFECT OF MULTIPLE PAIR GENERATION
0.4
109
γs − γi = 0
γs − γi = γ̄/2
γs − γi = γ̄
0.35
γ̄ 2 N1 (t)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
γ̄ t
3
4
Figure 5.5: N1 (t) as a function of time in three different cases: blue-solid, when γs −
γi = 0; red-dotted, when γs − γi = γ̄/2; green-dashed, when γs − γi = γ̄.
any loss-rate difference between signal and idler photons increases the probability of
a false signal heralding in the system.
5.5
Effect of multiple pair generation
In addition to loss, multiple pair generation [93] is a key consideration in the design
of a heralded single-photon source. Multiple pair generation can result in more than
one signal photon being delivered to a quantum circuit or apparatus, even when the
heralding detector fires only once. Since typically only one photon per qubit mode
should be delivered to the next unit on an integrated chip, the presence of multiple
photons due to higher order pair generation would likely result in a faulty operation.
5.5. EFFECT OF MULTIPLE PAIR GENERATION
110
Therefore, one of the main challenges in obtaining a reliable heralded single photon
source is to keep the multiple pair generation as low as possible.
In practice, the noise figure commonly known as the “coincidence to accidental
ratio” (CAR) in the experimental literature, is a measure of how often real single
photons are delivered in a given heralded single photon source. Although dark counts
contribute to such noise figure for low pump power, when the pump power is high the
effect of higher order pair generation is usually the main contributor to the observed
CAR. Therefore, it is important to examine the effect of leakage loss on multiple pair
generation in CC systems.
We shall now discuss the lossy dynamics of multiple pairs using our Heisenberg
approach, by examining the probability of finding multiple pairs as well as single pairs
in the system. The probability of finding only one pair of signal and idler photons in
the Heisenberg picture is
p11 (t) = Tr (ρ0 P11 (t)) ,
(5.57)
where ρ0 is the initial state of the system given in Eq. (5.30) and P11 (t) is the
Heisenberg projection operator onto the |1s 1i i state corresponding to one photon in
each one of the signal and idler modes. This projector is given by
P11 (t) = (bs bi )† (t) |vaci hvac| (bs bi ) (t) .
(5.58)
Thus, the probability of finding a single pair in the system is
p11 (t) = |hvac |(bs bi ) (t)| vaci|2 .
(5.59)
5.5. EFFECT OF MULTIPLE PAIR GENERATION
111
In the same manner, the probability of finding two pairs due to second order generation is
p22 (t) = |hvac |(bs bs bi bi ) (t)| vaci|2 .
(5.60)
We note that these quantities are distinct from detection probabilities as they represent the probabilities of the photons being anywhere within the system, rather than
at a particular position where there is a detector. Calculation of pair detection probability is discussed in the next chapter of this thesis. Using our perturbative treatment
of the adjoint master equation and following procedures similar to those presented
earlier for photon number expectations, the first non-zero approximations (first order
for p11 and second order for p22 ) are found to be
2 −4γp t
p11 (t) = |G| e
1 − e−2(γ̄−γp )t
γ̄ − γp
2
,
(5.61)
and
4 −8γp t
p22 (t) = 4 |G| e
1 − e−2(γ̄−γp )t
γ̄ − γp
4
.
(5.62)
Note that the second order contribution to p11 (t) is in fact zero. These results are
new and give compact, analytic expressions for the effect of scattering loss. Also, it
can be observed that both these expressions only depend on the average loss in the
signal and idler modes.
As a noise figure, we evaluate the ratio between the one-pair and two-pair probabilities
R (t) ≡
p11 (t)
.
p22 (t)
(5.63)
It is easy to see that R is proportional to the inverse of the pump intensity. This is
5.5. EFFECT OF MULTIPLE PAIR GENERATION
107
112
γ̄ = 2γp
γ̄ = γp
γ̄ = 2γp /2
106
R(t) |G|2 /γp2
105
104
103
102
101
100
0
1
2
γp t
3
4
Figure 5.6: (a). Plot of dimensionless noise figure R (t) |G|2 /γp2 in three different
cases: green-dashed, for γ̄ = 2γp ; red-dotted, for γ̄ = γp ; blue-solid, for
γ̄ = γp /2; (b) and (c) show p11 and p22 as a function of time for the same
conditions.
analogous to what is observed in the experimental CAR at high powers and reflects
the familiar trade-off between signal brightness and CAR in a typical experiment.
Figure 5.6 shows the noise figure R for different values of the average loss in the
signal and idler modes. As can be seen, the noise figure is larger when the pump loss
is smaller than the signal and idler average loss. This means that it is more likely
that multiple pairs will be generated if the pump mode stays longer in the system.
In the case of a lossless pump, R reaches the steady state value of
R (∞) =
γ̄ 2
,
4 |G|2
(5.64)
5.5. EFFECT OF MULTIPLE PAIR GENERATION
105
lossy pump
lossless pump
104
R(t) |G|2 /γ̄ 2
113
103
102
101
100
10−1
0
1
2
γ̄ t
3
4
Figure 5.7: (a). Plot of dimensionless R (t) |G|2 /γ̄ 2 in two different cases: red-dashed,
when γp = 0; blue-solid, when γ̄ = γp ; (b) and (c) show p11 and p22 as a
function of time for the same conditions.
as t → ∞. This can be seen in Fig. 5.7, which shows the noise figure for both a
lossless and lossy pump. It is important to note that the steady state value of R
increases with increasing loss in the signal and idler modes. Therefore, although it
is necessary to keep signal and idler loss rates low if one wants a bright pair signal,
there is a trade-off as lower losses result in higher rates of multiple pair generation.
This is because the decay rate of multiple pairs is higher than that of a single pair.
To be more concrete, in the lossless pump regime, the signal brightness steady state
value is directly proportional to the fundamental factor |G|2 /γ̄ 2 , the exact same factor
that the multiple-pair ratio is inversely proportional to. Therefore, from device-design
point of view, if one interested in a certain operational ratio R for a device, regardless
5.5. EFFECT OF MULTIPLE PAIR GENERATION
114
of the loss rates in the system, there is always a pump field amplitude that can achieve
this. At first glance, it seems that the loss can be easily neglected from any study in
this regard and the common sense must be followed that the lesser the loss the better
the brightness will be. However, as can be seen form Fig. 5.6, with smaller loss rates,
it needs a longer time to obtain the same operational R for a higher loss rate. This is
fine, as long as the multiple-pair issue is concerned. However, in addition, when loss
is present, there are chances that the idler photons will be detected even when the
signal photons are already lost, or vice-versa. Using Eqs. (5.55) and (5.56), it is easy
to see that, the ratio between unpaired photons and paired photons in the system in
steady state is
4 (γs + ∆/2)
N1 (∞)
=
− 2,
Nst (∞)
γs
(5.65)
where we have expressed everything in terms of signal-loss, γs , and signal and idler
loss-difference, ∆. For a fixed ∆, this ratio starts at infinity when γs → 0, and
approaches the value of 2 when γs → ∞. This means that for higher loss rates, the
quality of the pairs generated is higher. Of course, the price we have to pay is to have
a less bright source. To overcome the brightness issue however, multiplexing using
a number of duplicated devices has been proposed [70]. Therefore, in a practical
situation, it seems that one needs to optimize a multiplexed single photon source
device with aim of finding a range of desired CAR factors and signal brightnesses
with respect to the loss rates present in the system [94, 95].
5.6. SUMMARY
5.6
115
Summary
In this chapter, the non-Hermitian projection technique was employed to obtain the
full effective Hamiltonian of the system including the nonlinear interaction in the basis of the QMs. Although, the particular nonlinear process of SFWM was considered,
this method can be adopted to any desired nonlinear processes in an straightforward
manner. In contrast to the Schrödinger picture in the previous chapter, the adjoint
Master equation was used to obtain time dependence of relevant Heisenberg operators in the system, such as the photon number operators. Using the Heisenberg
picture allows straightforward generalization of the calculation method to detection
probabilities that will be presented in the next chapter.
Analysis of the photon number expectations were carried out in a three-mode
photonic molecule model where the resonance frequencies of the system matched the
desired frequencies of pump, signal and idler in a typical nonlinear pair generation
experiment. As expected, the presence of the loss forces the nonlinearly generated
photon pair state to move to an unpaired state after some time, depending on its
magnitude, specially when unbalanced loss rates for signal and idler photons are
considered. In addition, a trade-off between the loss low operation and high CAR
photon source was verified. This study will be generalized to pair generation in
CROW structures where the photon detection probabilities at particular location are
of interest, in the next chapter.
116
Chapter 6
Pair Detection in CROW
Structures
In the previous chapter, pair generation in CC structures was studied in the presence
of scattering loss. In particular, the photon number expectations and the probability
of finding signal-idler pairs within the entire system were calculated. However, there
are situations where the probability of finding photons of interest, signal and idler,
at specific locations in the CC system is desired. In particular, in coupled resonator
optical waveguides (CROWs), usually the pump laser enters the waveguide from one
end and the output light that includes the generated light is collected at the other
end.
CROW structures are slow light devices that can enhance the nonlinear interaction between the light and the medium considerably. Standard slow light devices
such as line-defect waveguides in PCSs benefit from the increased interaction time
for the light propagating through. In comparison, because CROW structures are
6.1. TIGHT-BINDING MODEL FOR CROW STRUCTURES
117
built from cavities with strong modal confinement for the captured light, an additional enhancement factor of the nonlinear interaction is offered. Indeed, PCS-CROW
structures have been reported to be the first to offer effective nonlinearity exceeding
10, 000 (W m)−1 [51] and they have been already used to generate entangled photon
pairs [20].
In this chapter, we further extend our Heisenberg formulation of the photon pairgeneration problem to obtain analytic expressions for photon-detection probabilities
in CROW systems. To this end, we evaluate the second order coherence function by
calculating the time-evolution of the field operator in the presence of the nonlinearity
and loss. The obtained expression are then used to calculate pair generation and
detection first for simple lossy dispersion models and then for a real lossy CROW in
a square-lattice PCS.
6.1
Tight-Binding model for CROW structures
Consider a CROW structure such as the one shown in Fig. 6.1 that is formed by
coupling a large number of cavities in a row. Transmission occurs when the light
tunnels from one cavity to another. For a CROW structure extended to infinity,
the periodicity suggests the traveling Bloch modes, Ñk (r), as a basis for study of
the system, where k is the propagation wavevector. These Bloch modes form an
orthogonal basis and can be expanded in terms of individual cavity quasimodes,
M̃p (r), as
Ñk (r) = Ak
N
X
p=−N
M̃p (r) eipkD ,
(6.1)
6.1. TIGHT-BINDING MODEL FOR CROW STRUCTURES
d
x=0
118
D
ωs
ωp
ωi
nonlinear region with χ(3)
Figure 6.1: Visualization of a CROW structure in a square-lattice PCS. The base
cavity used to build this waveguide is the cavity shown in Fig. 2.6. The
exact same parameter for the PCS platform are used here. The spacing
between adjacent cavities is only one lattice period d.
where p is an integer labeling different cavities that runs from 1 to the total number
of defects, 2N + 1, and D is the period of the CROW structure. In the example
shown in Fig. 6.1, we have D = 2d, where d is the PCS period in the direction of the
propagation. Also, Ak is the normalization factor which the particular expression for
depends on the order of TB model one uses. For example, if only the overlap between
modes sitting on the same site is considered, the normalization factor is
Ak u √
1
.
2N + 1
(6.2)
For ease of derivation, we set our reference point at the center of the CROW structure
and assume that there is always an odd number of defects present. The periodic
symmetry of the CROW structure can be used to obtain a compact expression for
the dispersion relation of the waveguide as well as for the nonlinear overlap function
between different traveling modes, as discussed below.
6.1. TIGHT-BINDING MODEL FOR CROW STRUCTURES
6.1.1
119
CROW dispersion relation
Assuming that all of the individual cavities are single mode (perhaps in the narrow
frequency range of interest) and support the same mode, M̃ (r), with the complex
frequency ω̃0 , using Eq. (6.1) the TB dispersion of Eq. (2.23) reduces to
ω̃02
N
X
eipkD p (r) M̃ (r − pDx̂) = ω̃k2 (r)
p=−N
N
X
eipkD M̃ (r − pDx̂) ,
(6.3)
p=−N
where, M̃ (r − pDx̂) is the mode sitting on the pth cavity of the CROW. Multiplying
both sides of this equation by M̃∗ (r) and integrating over all space, leads to
ω̃02
N
X
ipkD
e
p=−N
Ãp =
ω̃k2
N
X
eipkD B̃p ,
(6.4)
p=−N
where
ˆ
Ãp ≡
d3 rp (r) M̃∗ (r) · M̃ (r − pDx̂)
(6.5)
d3 r (r) M̃∗ (r) · M̃ (r − pDx̂) .
(6.6)
ˆ
B̃p ≡
From Eq. (6.4) the TB dispersion for the CROW is found to be
v
u PN
u p=−N eipkD Ãp
.
ω̃k = ω̃0 t PN
ipkD B̃
p
p=−N e
(6.7)
6.1. TIGHT-BINDING MODEL FOR CROW STRUCTURES
120
Using the fact that
Ãp = Ã−p
(6.8)
B̃p = B̃−p ,
(6.9)
it is straightforward to see that
v
P
u
u Ã0 + p6=0 eipkD Ãp
t
ω̃k = ω̃0
P
B̃0 + p6=0 eipkD B̃p
(6.10)
v
P
u
ipkD + e−ipkD ) Ã
u Ã0 + N
p
p=1 (e
t
= ω̃0
PN
B̃0 + p=1 (eipkD + e−ipkD ) B̃p
(6.11)
v
P
u
u Ã0 + 2 N
p=1 cos (pkD) Ãp
t
= ω̃0
.
PN
B̃0 + 2 p=1 cos (pkD) B̃p
(6.12)
For example, in the limit that Ã0 u 1, B̃0 u 1 and the only other non-zero overlap
integral is Ã1 , via a Taylor expansion of the square root function we obtain
ω̃k u ω̃0
1 + Ã1 cos (kD) ,
(6.13)
which is similar to the TB expression for the dispersion used in Ref. [96].
The above-presented TB model can be used to obtain the frequency dispersion of
the CROW structure shown in Fig. 6.1. Using the FDTD mode of the base cavity
that is shown in Fig. 2.6, the first four elements of the coupling integrals Ãm and
B̃m are calculated and listed in Table 6.1. As can be seen, the B̃m integrals are real,
which is due their specific and the structure symmetry. Also, because the mode is
6.1. TIGHT-BINDING MODEL FOR CROW STRUCTURES
m
Ãm
B̃m
0
1.0
1.0024
1
7.1 × 10−2 + i1.9 × 10−5
8.0 × 10−2
2
1.8 × 10−2 − i4.8 × 10−6
2.0 × 10−2
3
2.1 × 10−3 + i8.4 × 10−6
2.8 × 10−3
4 −3.5 × 10−4 − i2.3 × 10−6
121
−1.7 × 10−4
Table 6.1: List of the first four Ãm and B̃m overlap integrals calculated for the
quadrupole modes of the single-defect cavity in square-lattice PCS.
confined to the cavity region in space, the magnitude of overlap integrals drops off
when the spacing between cavities is increased along the propagation direction.
Using the values listed in Table 6.1, the TB dispersion for the square-lattice
CROW of Fig. 6.1 is plotted in Fig. 6.2a. In addition, the quality factor, the group
velocity and the group index over the entire band are plotted in Fig. 6.2b, Fig. 6.2c
and Fig. 6.2d, respectively. The red lines in the top panel of plots represent the values
for the resonance frequency and the quality factor of a single defect. Note that, there
are oscillations in the Q, while the actual value changes by about a factor of 10 from
one side of the band to the other side. This means that traveling modes of the CROW
that are closely located in frequency can experience significantly different loss rates,
which in turn can affect the pair generation and therefore the detection probabilities
for the signal and idler photons depending on where the operating frequency is.
6.1. TIGHT-BINDING MODEL FOR CROW STRUCTURES
(a)
(b)
120
90
ω0
0.312
Q (10 3 )
ωd/2πc
0.314
0.310
60
Q0
30
0.308
0
0
0.2
0.4 0.6
kD/π
0.8
1
0
0.2
(c)
0.040
160
0.030
120
0.020
40
0.000
0
0.2
0.4 0.6
kD/π
0.8
1
0.8
1
80
0.010
0
0.4 0.6
kD/π
(d)
ng
vg /c
122
0.8
1
0
0.2
0.4 0.6
kD/π
Figure 6.2: Plot of frequency dispersion, quality factor, group velocity and group
index for the square-lattice CROW visualized in Fig. (6.1). The overlap
integrals of Table (6.1) were used in the TB dispersion of Eq. (6.12)
to obtain the data. On the top panel, red lines represent the resonant
frequency and the quality factor for the based cavity.
6.1. TIGHT-BINDING MODEL FOR CROW STRUCTURES
6.1.2
123
CROW nonlinear overlap function
The nonlinear overlap integral of Eq. (5.21) can be also simplified when the traveling
modes are expanded in terms of individual cavity modes. If we make the assump(3)
(3)
(3)
tion that χxxxx = χyyyy = χzzzz and the other elements of the nonlinear tensor are
negligible, then it can be easily seen that
Sk1 ,k2 ,k3 ,k4
~ωp2 |Ak |4
=
16o
N
X
exp {−i [p1 k1 + p2 k2 − p3 k3 − p4 k4 ] D}
p1 ,p2 ,p3 ,p4 =−N
ˆ
×
(3)
d3 r χiiii (r) (r) M̃i∗ (r − p1 Dx̂)M̃i∗ (r − p2 Dx̂)M̃i (r − p3 Dx̂)M̃i (r − p4 Dx̂),
(6.14)
where i sums over all of the three Cartesian components and the real part of the
frequencies under the square root are approximated by the real part of the complex
frequency for the pump mode. Two additional assumption will be made at this point.
1) Only when all of the modes in the integral are on the same site is the integrand
non-negligible. This is justified as the modes are usually tightly confined to the
cavity regions and the overlap falls off rapidly as they are moved apart. However,
this approximation can easily be relaxed if necessary and the details are given in
Appendix F. 2) The length of the nonlinear region (shown in purple in Fig. 6.1) is
taken to be different (shorter) than the full length of the CROW. This gives us the
ability to model pump propagation before entering the nonlinear region. Indeed, in
our formulation of the problem, coupling in and out from CROW is not considered
and instead it is assumed that the pulse is traveling down the linear part of the
CROW at time t = 0 toward the nonlinear region. Using these two assumptions, the
6.2. HEISENBERG FIELD OPERATORS AND CORRELATION
FUNCTIONS
124
nonlinear overlap function can be further simplified to
Sk1 ,k2 ,k3 ,k4
~ωp2 |Ak |4
u
16o
ˆ
M
4 X
(3)
exp {−i [k1 + k2 − k3 − k4 ] pD} .
d3 r χiiii (r) (r) M̃i (r)
p=−M
(6.15)
Note that the sum over p runs over only the number of cavities inside the nonlinear
region, 2Nnl + 1. Now, as shown in Appendix G, the sum in Eq. (6.15) can be easily
evaluated to obtain
Sk1 ,k2 ,k3 ,k4
~ωp2 |Ak |4
u
16 o
ˆ
d
3
(3)
r χiiii
4 sin (∆k (2N + 1) D/2)
nl
(r) (r) M̃i (r)
, (6.16)
sin (∆k D/2)
where ∆k ≡ k1 +k2 −k3 −k4 and the integration runs over the entire nonlinear region.
Assuming that the nonlinear susceptibility is constant within the slab material, as it
is expected to be, we find
(3)
Sk1 ,k2 ,k3 ,k4
~ωp2 χxxxx |Ak |4 sin (∆k (2Nnl + 1) D/2)
,
u
16 o Vef f
sin (∆k D/2)
(6.17)
where we define the effective nonlinear mode volume to be
Vef f ≡ ˆ
slab
1
4 4 4 .
d3 r (r) M̃x (r) + M̃y (r) + M̃z (r)
(6.18)
Here, the integration runs over only the slab material inside the nonlinear region.
6.2
Heisenberg field operators and correlation functions
In a standard experiment, to measure the number of generated photons at a given
frequency, a filtering unit is used to spectrally decompose the light collected from one
6.2. HEISENBERG FIELD OPERATORS AND CORRELATION
FUNCTIONS
125
end of the CROW. The filtered light is then sent to broad-band detectors for photon
counting. The total process is indeed a narrow-band detection scheme where at the
end, the number of detector clicks represents number of photons generated at the
specified frequency. Below, it is shown how the second order coherence function can
be used to evaluate such detection probabilities.
Let us first evaluate the following second order coherence function
D
E
(−)
(−)
(+)
(+)
G(2) (r, t1 , t2 , t01 , t02 ) ≡ EQ (r, t01 ) EQ (r, t02 ) EQ (r, t2 ) EQ (r, t1 ) ,
(6.19)
where different arrival times at different detectors are allowed for different photons.
This is necessary if one wants to consider detectors that filter photons that arrive
at the detectors. Also, the time arguments for the positive and negative frequency
parts of the field associated with each detector are also different, which can be used
to appropriately include the detection time period in the model.
The positive-frequency part of the field operator at the location r and time t can
be expanded in terms of CROW traveling modes as
(+)
EQ
(r, t) = i
X
r
k
~ω̃k
Ñk (r) bk (t) ,
20
(6.20)
where all the time dependence resides in the Heisenberg lowering operator bn (t). Note
that, the sum over k excludes the pump range of frequencies, as they are filtered before
detection, in real experiments. Recall that, the Heisenberg annihilation operator has
the following zeroth order time dependence
(0)
bk (t) = e−iω̃k t bk .
(6.21)
6.2. HEISENBERG FIELD OPERATORS AND CORRELATION
FUNCTIONS
126
As shown in Appendix H, a similar to that employed in chapter 5 can be used to obtain
the following first order approximation to the Heisenberg annihilation operator:
(1)
bk (t) =
X
hk,k2 ,k3 ,k4 (t) b†k2 bk3 bk4 ,
(6.22)
k2 ,k3 ,k4
where
ˆ
t
−iω̃k t
hk,k2 ,k3 ,k4 (t) = 2i Sk,k2 ,k3 ,k4 e
dt0 ei(ω̃k +ω̃k2 −ω̃k3 −ω̃k4 )t .
∗
0
(6.23)
0
Here, k2 runs over both signal and idler frequencies bands, while k3 and k4 run over
only the pump frequency band. Note that, b†k is the Hermitian conjugate of bk ,
therefore, an expression for it does not need to be obtained independently.
The second order coherence function of Eq. (6.19) can be recast as
(2)
G
(r, t1 , t2 , t01 , t02 )
=
~ω0
20
2 X X
Ñ∗k10 (r) Ñ∗k20 (r) Ñk2 (r) Ñk1 (r)
k1 ,k2 k10 ,k20
D
E
× b†k0 (t01 ) b†k0 (t02 ) bk2 (t2 ) bk1 (t1 ) ,
1
2
(6.24)
where the complex frequency under the square roots in Eq. (6.20) are approximated
by the real part of the resonance frequency of the individual cavity, ω0 .
Now we calculate the coherence function of Eq. (6.24). In a nonlinear SFWM
process, there are initially no photons present in either the signal or the idler range of
frequencies. Therefore, because all the sums run over signal and idler modes, which
are in the vacuum state in the beginning, trivially there is no zeroth order contribution to the expectation value of the product of ladder operators in Eq. (6.24).
Considering the first order approximation for the expectation value, only one of the
6.2. HEISENBERG FIELD OPERATORS AND CORRELATION
FUNCTIONS
127
ladder operators involved can be of the first order, which in turns forces the expectation value to go to zero as well. However, the first non-zero expectation is a second
order approximation that is indeed first order in two of the ladder operators (the
annihilation operator on the most right-side and the creation operator on the most
left-side). Using time evolution for ladder operators from Eqs. (6.21) and (6.24) (and
their Hermitian conjugate expressions), we obtain
(2)
G
(r, t1 , t2 , t01 , t02 )
=
~ω0
20
2 X X
Ñk2 (r) Ñk1 (r) φk3 φk4 e−iω̃k2 t2 hk1 ,k2 ,k3 ,k4 (t1 )
k1 ,k2 k3 ,k4
×
XX
iω̃ ∗0 t02 ∗
k2
hk10 ,k20 ,k30 ,k40
Ñ∗k10 (r) Ñ∗k20 (r) φ∗k30 φ∗k40 e
(t01 ) .
k10 ,k20 k30 ,k40
(6.25)
This can be simplified by expanding the Bloch traveling QMs, Ñk (r), in terms of
individual cavity QMs, M̃ (r − pDx̂), as done in Eq. (6.1). We set r = pobs Dx̂,
where pobs represent the location of interest where the light collection and detection
occurs. Then assuming that only cavity QMs sitting at the same site contribute
significantly to the second order coherence function, we obtain
(2)
G
pobs , t1 , t2 , t01 , t02
=
~ω0
20
2 4
M̃obs ×
XX
φk3 φk4 e
(6.26)
i(k1 +k2 )p
obs D e−iω̃k2 t2 hk1 ,k2 ,k3 ,k4 (t1 )
k1 ,k2 k3 ,k4
×
XX
∗ 0
−i k0 +k0 p
D iω̃ 0 t
φ∗k30 φ∗k40 e ( 1 2 ) obs e k2 2 h∗k10 ,k20 ,k30 ,k40 (t01 ) ,
k10 ,k20 k30 ,k40
where, M̃obs is the averaged QM amplitude inside the observation segment of the
6.2. HEISENBERG FIELD OPERATORS AND CORRELATION
FUNCTIONS
128
CROW. We employ this averaging because in practice, with some out-coupling mechanism, light at one end of the CROW is gathered and directed toward the filtering
unit and then to the detectors.
Mathematically, in the frequency domain, the second order coherence function,
G(2) (r, ωs , ωi ), can be used to evaluate the probability of detecting photon pairs at
desired frequencies. It is defined as
D
E
(−)
(−)
(+)
(+)
G(2) (r, ωs , ωi ) ≡ EQ (r, ωs ) EQ (r, ωi ) EQ (r, ωi ) EQ (r, ωs ) ,
where
ˆ
(+)
EQ
t+∆T /2
0
(+)
dt0 eiωt EQ (r, t0 ) ,
(r, ω) ≡
(6.27)
(6.28)
t−∆T /2
is the Fourier transform of the electric field operator of the light collected from the
device. Here, t is the observation time and ∆T is the detection time interval, where
for simplicity both have been omitted from the arguments of the E+ (r, ω).
Using the previously obtained result for G(2) (r, t1 , t2 , t01 , t02 ), it is straightforward
to show that G(2) (r, ωs , ωi ) is given by
G(2) pobs , ωs , ωi = 4
2 4
~ω0 i(k +k )p
D
X X
φk3 φk4 Sk1 ,k2 ,k3 ,k4 e 1 2 obs
M̃obs 20
k1 ,k2 k3 ,k4
ˆ
t+∆T /2
×
i(ωi −ω̃k2 )t2
ˆ
t+∆T /2
dt2 e
t−∆T /2
dt1 e
t−∆T /2
i(ωs −ω̃k1 )t1
ˆ
t1
dt e
0
2
.
0 i(ω̃k1 +ω̃k∗2 −ω̃k3 −ω̃k4 )t0 (6.29)
This is an analytic expression that provides the probability of finding signal-idler
pairs at the observation location, pobs Dx̂, along the CROW.
6.3. PAIR DETECTION PROBABILITIES IN LOSSY CROWS
6.3
129
Pair detection probabilities in lossy CROWs
Before we study real CROW dispersions such as the one shown in Fig. 6.2, it is important to examine our formalism in a simple model where only the nonlinear interaction
and the scattering loss are present and dynamics associated with phase matching, frequency matching and group velocity dispersion are ignored. In particular, we consider
the simple linear dispersion,
π ,
ω (k) = ω0 + vg k −
2D
(6.30)
where the phase matching is naturally satisfied everywhere across the band. Here, ω0
is the resonant frequency of the individual cavities used to build the CROW and vg
is the constant group velocity. Although the actual value of these quantities are not
crucial to the conclusions that will be made in this section, the resonance frequency for
the individual cavity is set to ω0 d/2πc = 0.3, the group velocity of the corresponding
CROW is set vg = c/40 and the CROW period is chosen to be D = 2d, similar to
the values that one can have in a real PCS-CROW structure such as the one shown
in Fig. 6.1 and Fig. 6.2.
For the initial excitation, we consider a Gaussian pulse centered at kp in wavevector
and placed spatially at t = 0 at xp ,
(k − kp )2
φk ≡ φ0 exp −
2∆2
where φo is the maximum amplitude and
√
!
exp (−ikxp ) ,
(6.31)
2 ln 2 ∆ is the full-width at half-maximum.
Because phase matching is naturally satisfied regardless of the pump wavevector, we
6.3. PAIR DETECTION PROBABILITIES IN LOSSY CROWS
130
set kp = 0.5 π/D in this section. The pulse width is given by ∆ = 0.02 π/D which is
not too critical in the case of the linear dispersion. However, as will be discussed in
more detail, for real CROW structures caution must be taken in choosing values for
this parameter. Assuming that the origin is at the middle of the CROW, the pump
pulse is launched at xp = −100 D, that is the left end of the 200 D long CROW.
The quantities t and ∆T in Eq. 6.29 must be also determined in order to be able
to calculate the detection map. It is natural for the observation point to be set at
the exit end of the waveguide, xobs = 100 D. Therefore, the observation time t in
calculations corresponding to the time taken for the pump to travel from the launch
point to the observation point. In addition, ∆T in calculations is set such that all
the contribution from the one pulse propagating through the CROW is accounted
for. Although depending on the spatial extent of the pump this can change, but to
be on the safe side, the chosen time corresponds to the time taken by the pulse to
propagate the full length of the CROW. These two time constants are kept the same
for all the other calculations presented in the remaining of this chapter.
For the scattering loss, two scenarios are considered: 1) The case where the scattering loss is flat everywhere in the band, which means that every traveling mode
of the waveguide suffers an equal amount of loss. 2) The case where the scattering
loss is also allowed to have a linear dependence on the wavevector, starting with the
minimum loss at one end of the band and reaching the maximum loss at the other
end. Eq. (6.29) is used to calculate the probability of detecting photon pairs in these
two situation (see Fig. 6.3). On the left, a constant loss rate of γd/2πc = 10−5
corresponding to a quality factor of Q = 15, 000 is assumed for all the modes. A
uniform detection occurs which means that the probability of detecting signal-idler
6.3. PAIR DETECTION PROBABILITIES IN LOSSY CROWS
1
1
(a)
(b)
0.8
0.8
0.6
0.6
ki D/π
ki D/π
131
0.4
0.2
0.4
0.2
0
0
0
0.2
0.4 0.6
ks D/π
0.8
1
0
0.2
0.4 0.6
ks D/π
0.8
1
Figure 6.3: Plot of G(2) for the linear dispersion given in Eq. (6.30) where yellow
represents the maximum value. (a) Plot of G(2) when the loss is kept
constant everywhere. (b) Plot of G(2) where the scattering loss in linearly
dependent on the wavevector. Details of the parameters used in these
calculations are given in the text.
pairs at different places on the band is not affected by the loss, except by an overall
uniform reduction. On the right, the loss is linearly increased form γd/2πc = 10−6 to
γd/2πc = 10−4 . The corresponding quality factor varies between Q = 150, 000 and
Q = 1, 500. As can be seen from the color map, the detection probability across the
band is not uniform anymore. The probability of detecting signal-idler pairs is lower
at one end of the band compared to the other end.
The fact that the detection probability is not symmetric under a permutation of ωs
and ωi is not consistent with our physical expectation of the system. The fundamental
reason behind this inconsistency is discussed below, where the appropriate solution
that leads to physical detection probabilities is derived.
In a standard formulation of the two-photon detection problem, the probability
6.3. PAIR DETECTION PROBABILITIES IN LOSSY CROWS
132
amplitude, Tf i , of the two detectors making the transition from the ground state to
the excited state can be written as [97]
2 ˆ t
ˆ t1
E
D i
(1)
(2)
(2)
(1)
dt2 Φf Hdet (t1 ) Hdet (t2 ) + Hdet (t1 ) Hdet (t2 ) Φi ,
dt1
Tf i (t, t0 ) = −
~
t0
t0
(6.32)
where |Φi i and |Φf i are the initial and final states of the combined field and detector,
t − t0 is the detection time window and
(j)
Hdet ≡ −dj · E(+) (t) ,
(6.33)
is the detector Hamiltonian for j = 1, 2. Here, dj is the dipole moment of the
jth detector and the E(+) is the positive-frequency electric field. Note that only
the detector Hamiltonians are present in this expression because we are using the
interaction picture. If the detectors are located in a passive linear medium, the
Hamiltonians of the two different detectors commute at all times,
h
i
(1)
(2)
Hdet (t1 ) , Hdet (t2 ) = 0.
(6.34)
As a result, the probability amplitude for detecting photon pair reduces to
2 ˆ t
ˆ t
E
D i
(1)
(2)
dt1
dt2 Φf Hdet (t2 ) Hdet (t1 ) Φi ,
Tf i (t, t0 ) = −
~
t0
t0
(6.35)
where now, both time integrations run from t0 to t. Because the final state of the
detector and the field are not concerned, one needs to sum over all the possible final
6.3. PAIR DETECTION PROBABILITIES IN LOSSY CROWS
133
states to obtain the detection probability, Pf i , as
Pf i (t, t0 ) =
2 2 ˆ
d1k d2k ~4
ˆ
t
t0
ˆ
t
t
dt01
dt2
dt1
t0
ˆ
t
t0
dt02
t0
E
(−) 0
(+)
(+)
(−) 0
× Φi E1k (t1 ) E2k (t2 ) E2k (t2 ) E1k (t1 ) Φi ,
D
(6.36)
where djk and Ejk are the dipole moment and the electric field components along the
polarization axis of the optical filters that are aligned such that maximum detector
clicks are obtained. Note that one requires to integrate a time dependent second
order coherence function with four different time arguments.
However, when the nonlinear interaction is present, the situation is quite different.
As mentioned earlier, field operators at different times do not commute. Therefore,
Eq. (6.34) is not valid anymore and the two different terms in the original expression of Eq. (6.32) for the probability amplitude of the transition must be separately
considered. In our notation, there must be four different contributions to the second
order coherence function in the Heisenberg representation; these are given by
D
E
(−)
(−)
(+)
(+)
G(2) (r, ωs , ωi , ωi , ωs ) ≡ EQ (r, ωs ) EQ (r, ωi ) EQ (r, ωi ) EQ (r, ωs )
E
(r, ωi )
(6.38)
D
E
(−)
(−)
(+)
(+)
G(2) (r, ωi , ωs , ωi , ωs ) ≡ EQ (r, ωi ) EQ (r, ωs ) EQ (r, ωi ) EQ (r, ωs )
(6.39)
D
E
(−)
(−)
(+)
(+)
G(2) (r, ωi , ωs , ωs , ωi ) ≡ EQ (r, ωi ) EQ (r, ωs ) EQ (r, ωs ) EQ (r, ωi ) .
(6.40)
(2)
G
(r, ωs , ωi , ωs , ωi ) ≡
D
(6.37)
(−)
EQ
(−)
(r, ωs ) EQ
(+)
(r, ωi ) EQ
(+)
(r, ωs ) EQ
All of these must be added to obtain the properly symmetrized second order coherence
function, which takes the final form
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
pobs , ωs , ωi = 4
G(2)
s
(ˆ
t+∆T /2
×
~ω0
20
2 4 X X
i(k +k )p
D
φk3 φk4 Sk1 ,k2 ,k3 ,k4 e 1 2 obs
M̃obs k1 ,k2 k3 ,k4
ˆ
i(ωi −ω̃k2 )t2
t+∆T /2
dt2 e
dt1 e
t−∆T /2
ˆ
t+∆T /2
+
t−∆T /2
134
dt2 ei(ωs −ω̃k2 )t2
ˆ
i(ωs −ω̃k1 )t1
t+∆T /2
dt1 ei(ωi −ω̃k1 )t1
t−∆T /2
dt0 ei(ω̃k1 +ω̃k2 −ω̃k3 −ω̃k4 )t
∗
0
ω̃k1 +ω̃k∗ −ω̃k3 −ω̃k4
2
)2
) .
0
t−∆T /2
ˆ
t1
ˆ
t1
dt0 ei(
0
t0
(6.41)
This is clearly symmetric under the exchange of the signal and idler frequencies.
6.4
Nonlinear SFWM in the square-lattice CROW
In this section, we study nonlinear pair-generation for the real CROW structure of
Fig. 6.1. The cavity used to build this CROW is the one shown in Fig. 2.6. Both pair
generation and pair detection will be discussed in detail in what follows.
6.4.1
Pulse propagation through the CROW
As mentioned earlier, the Gaussian pulse of Eq. (6.31) is considered for the initial
excitation of the system. The parameter ∆ that sets the pump width in wavevector,
and therefore in frequency and space, must be decided upon carefully. On the one
hand, the duration of typical laser pulses that might be used in real experiments
must be considered. In addition, one wants a narrow enough pump pulse for signal
and idler extraction purposes. Finally, the degree of spatial distortion that the pump
experiences as it propagates down the CROW structure depends on the spectral width
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
135
as well. We take into account all of these consideration in deciding upon a proper
value for the parameter ∆.
Pico-second pulses have typically been used to conduct nonlinear SFWM experiments in PCS structures [76]. Considering a Gaussian pulse that is 4 ps long in time,
the width of the pulse in (angular) frequency space is given by
∆ω =
1
= 250 GHz.
∆t
(6.42)
Assuming that one desires to operate at λ = 1.5 µm, the corresponding frequency is
ω/2π = 200 THz. Therefore, the ratio between the pulse width and the operating
frequency is approximately
∆ω
u 2 × 10−4 .
ω
(6.43)
Now, one has to determine the ratio of the pulse width to the transmission bandwidth
of the CROW system. For the square-lattice CROW structure of interest, the transmission bandwidth approximately ranges from ω1 d/2πc = 0.308 to ω2 d/2πc = 0.314,
while the resonance frequency of the base cavity is ω0 d/2πc = 0.311. Therefore, the
ratio of CROW’s bandwidth, ∆ω BW , to the resonance frequency ω0 is
∆ω BW
u 0.02.
ω0
(6.44)
In an active version of this CROW system, where ω0 is the operating frequency,
Eqs. (6.43) and (6.44) can be used to estimate the ratio between the pulse width,
∆ω, and CROW’s bandwidth in frequency space, ∆ω BW , to be
∆ω
u 0.01,
∆ω BW
(6.45)
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
136
which can be used to find an estimation on the pulse width, ∆, in wavevector space.
Because a nonlinear dispersion relation is involved, the actual value depends on the
operating frequency but can be estimated to lie between ∆ = 0.01 π/D and ∆ =
0.03 π/D.
Let us now examine the pulse distortion due to propagation for different values
of ∆. Plotted in Fig. 6.4 is the spatial envelope of a Gaussian pulse before (on the
left) and after (on the right) the pulse has propagated the full length of a 200 D
long CROW. As seen, for ∆ = 0.1 π/D, the pulse is both broadened and distorted.
In this case, a relatively wide range of frequency components are included in the
pulse envelope and in particular, distortion is expected as each frequency components
travels with a different speed. For ∆ = 0.05 π/D the pulse has been somewhat
broadened, but the envelope is mostly preserved. Only a small asymmetry in the
envelope is observed in this case. Finally, for ∆ = 0.02 π/D, some broadening has
occurred while the Gaussian envelope is completely preserved. Fortunately, this value
of ∆ falls within the range of values that was estimated based on the time extent of
4 ps for the pulse. Also, note that pump pulses with ∆ < 0.02 π/D are expected to
even suffer less broadening from propagation.
6.4.2
Pair generation in square-lattice CROW
Now, consider a 200 D long version of the square-lattice CROW that is pumped
with a Gaussian pulse located at xp = −100 D at the beginning of the time, t = 0.
The expressions for the photon number expectation values obtained in the previous
chapter, Eqs. (5.44) and (5.48), can be adopted to monitor the number of paired and
unpaired photons in the system as a function of time, as the pump travels through.
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
before
after
∆ = 0.10 π/D
∆ = 0.10 π/D
amp. (a.u.)
1
0.4
0.8
0.6
0.2
0.4
0.2
0
0
-100
-50
0
50
100
-100
amp. (a.u.)
∆ = 0.05 π/D
-50
0
50
100
∆ = 0.05 π/D
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-100
-50
0
50
100
-100
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-100
-50
0
x (D)
50
-50
0
50
100
∆ = 0.02 π/D
∆ = 0.02 π/D
amp. (a.u.)
137
100
-100
-50
0
x (D)
50
100
Figure 6.4: Spatial dispersion of the pump pulse due to propagating a full CROW
length. Three different pulse width are considered as labeled in the figure. In general, the wider the pulse is in the wavevector space the larger
dispersion experiences due to propagation.
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
138
The other relevant parameters for the calculations of this subsection are the following:
kp = 0.60 π/D
(6.46)
ks = 0.65 π/D
(6.47)
ki = 0.55 π/D
(6.48)
∆ = 0.01 π/D.
(6.49)
The particular kp is of interest for reasons that will be presented shortly. Using the
TB expression for the nonlinear overlap function, Sk1 ,k2 ,k3 ,k4 , (as well as expression in
Eq. (6.31) for the Gaussian pump), Eqs. (5.44) and (5.48) can be evaluated to obtain
the desired photon number expectations. Two different situations are considered:
either only half of the CROW is nonlinear or the entire CROW is nonlinear. The
calculated photon expectations are plotted in Fig. 6.5. In general, conclusions similar
to the previously-discussed results of chapter 5 can be made in both cases. Due to
the action of scattering loss, the expectation value of the unpaired photons rises some
time after the signal-idler pairs have been generated. When the CROW includes a
linear portion, see Fig. 6.5a, the generation only starts when the pump enters the
nonlinear region of the CROW, whereas in the other case, see Fig. 6.5b, generation
starts right from the beginning.
At later times, the number of the unpaired photons grows while the number of
the paired photons drops. However, when the entire CROW is nonlinear, the width
associated with expectation peaks is wider, as there is more time for the pulse to
interact with the χ(3) material and generate more pairs. It is obvious that a number
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
0.05
paired
unpaired
139
(a)
Photon Exp. (a.u.)
0.04
0.03
0.02
0.01
0
0
0.12
50
100
vg t/D
150
paired
unpaired
200
(b)
Photon Exp. (a.u.)
0.1
0.08
0.06
0.04
0.02
0
0
50
100
vg t/D
150
200
Figure 6.5: Comparison between Nsi (t) and N1 (t) for the 200 D long square-lattice
CROW with χ(3) nonlinearity, in two different cases. On the top where
the length of the nonlinear region is only 20 D, and on the bottom where
the entire CROW is made from nonlinear material.
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
140
of unpaired photons are present in the system which is not desired, as it degrades
the performance of the device either as a single photon source or as a entangled
photons source. This is expected to be a performance characteristic of similar CROW
structures that support traveling modes with quality factors of the order 104 to 105 .
6.4.3
Pair detection in square-lattice CROW
We now turn to photon detection probabilities for the CROW structure of Fig. (6.1).
In general, frequency matching and phase matching are known to be the critical
conditions for nonlinear optical processes to occur [48]. This can be examined by
plotting a color map that represents the detection probability as a function of both
signal and idler wavevectors, using Eq. (6.41)
For the square-lattice CROW of interest, the exact same parameters as before are
used for the excitation and length of the CROW, but we first set the length of the
nonlinear region to only 20 D to minimize the effect of phase mismatch. Plotted in
Fig. 6.6a, is the frequency mismatch, ωs +ωi −2ωp , as the signal and idler wavevectors
are scanned across the CROW’s band and the minimum is shown in black. Note that
this requires no complicated calculation and can be done simply using the dispersion
(2)
data plotted in Fig. 6.2. In Fig. 6.6b, the corresponding Gs is plotted in the same
fashion where the bright curve in yellow represents the maximum detection probability. As can be seen, the photon pair generation is determined primarily by the energy
(2)
conservation, as expected. If one re-plots the exact same Gs on frequency axes rather
than wavevector axes as has been done in Fig. 6.6b, the bright curve appears as a
straight line, which is the confirmation of the energy conservation condition given by
2ωp = ωs + ωi .
(6.50)
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
1
1
(b)
0.8
0.8
0.6
0.6
ki D/π
ki D/π
(a)
0.4
0.2
0.4
0.2
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
ks D/π
0.4
0.6
0.8
1
ks D/π
(c)
(d)
0.313
ωi d/2πc
0.313
ωi d/2πc
141
0.312
0.311
0.312
0.311
0.311
0.312
ωs d/2πc
0.313
0.311
0.312
0.313
ωs d/2πc
Figure 6.6: Energy conservation mainly derives the nonlinear process of pair generation. In addition, depending on the nonlinear length of the CROW
phase matching features can appear. (a). Plot of frequency mismatch
|ωs + ωi − 2ωp | for the square-lattice CROW shown in Fig. 6.1. (b). Plot
(2)
of Gs (r, ωs , ωi ) for the same CROW where only 20 D of the CROW is
nonlinear. (c). Same data as in (b) plotted on frequency axes rather than
wavevector. Here (and in (d) as well), only sub-data is shown for visu(2)
alization purposes). (d). Plot of Gs (r, ωs , ωi ) on frequency axes where
half of the CROW is nonlinear.
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
142
In addition, a similar calculation was performed when the length of the nonlinear region was extended from 20 D to 100 D, namely half of the full CROW (see Fig. (6.6)).
Then, the traveling pump interacts with the nonlinear medium over a longer distance
and and now due to the phase matching the extent of the bright region in the detection
map is smaller.
In order to understand the effect of loss, it is important to see how the pair detection probability changes when the pump wavevector is scanned through the CROW’s
band. In practice, one avoids both ends of the dispersion, as the group velocities
there are extremely slow. Operating at these regions is not desirable because, due to
the fabrication imperfections, huge scattering losses will be experienced [98] by the
pump as well as by the signal and idler. In addition, when operating too close to
the band edges, signal and idler extraction becomes difficult due to potential overlap with the pump modes. Here, the pair detection probability is given in arbitrary
units, but one can use characteristics of the detection units in a given experiment
such as the filter functions and the detectors dipole moments to obtain the numbers
of pairs detected, which will be done in future works. The main focus in here is the
physical understanding of the pair detection in lossy CROW systems which remains
unchanged regardless of the choice of units for the pair detection probability.
It should be noted that the fixed ∆ for the pump in these simulations implies
slightly different time durations for the excitation pulses. Although the precise durations may be important in arriving at a detailed device design, these changes in the
pulse durations do not significantly affect the overall conclusions presented below.
(2)
Plotted in Fig. 6.7a is Gs where the pump wavevector ranges from kp = 0.2π/D
to kp = 0.8π/D and the entire CROW is considered to be nonlinear. In particular, the
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
143
spacing between signal and pump wavevectors was set to |ks − kp | = 0.03 π/D while
the idler frequency was forced on resonance such that 2ωp = ωs + ωi . The spacing
between pump and signal is chosen not so close that it would cause them to overlap
in frequency, while at the same time not so far apart as that it would cause a large
phase mismatch and features not associated with loss would appear. Basically, we
have tried to stay within the region about the pump where the local linear behavior
of the dispersion is dominant. More simulations were performed where signal and
pump frequencies were further apart and those result will be discussed shortly.
In Fig. 6.7a, the maximum detection probability occurs at kp u 0.6 π/D which
roughly coincides with the local maximum in the Q plot. This is due to reduced loss
for the pump pulse within the χ(3) medium compared to the adjacent modes with
lower Q’s. To confirm this, we have performed a similar calculation where all the
parameters are kept the same as in the previous calculation but the TB dispersion is
replaced with a simple linear dispersion. This enforces zero group velocity dispersion
everywhere across the band and eliminates any dispersive pump behavior. Thus, the
detection probability is expected to be mostly affected by the dependence of the Q
of the modes on ks , ki and kp .
As plotted in Fig. 6.7b, with the simple linear dispersion, the trend of the detection
probability follows the exact same trend of the Q plot of Fig. (6.2), which confirms
that detection probability is correlated with the Q of the pump mode. However, it is
obvious that one can easily be operating somewhere in the band that has a higher detection rate compared to the local peak about kp u 0.6 π/D. Compared to Fig. 6.7a,
this suggests that perhaps an interplay between the leaky and the dispersive characteristics of the CROW plays a role in the overall dynamics of the pair generation,
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
144
20
(a)
(2)
Gs (a.u.)
16
12
8
4
0
0.2
0.3
0.4
0.5
kp D/π
0.6
0.7
0.8
30
(b)
25
15
(2)
Gs (a.u.)
20
10
5
0
0.2
0.3
0.4
0.5
kp D/π
0.6
0.7
0.8
Figure 6.7: Pair detection probability as a function of pump wavevector for the
CROW structure shown in Fig. 6.1 for the resonant condition 2ωp =
ωs +ωi . On the top, the real dispersion of square CROW are used whereas
for the bottom only the Q’s are adopted.
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
145
because for the actual dispersion, the waveguide modes around kp = 0.6 π/D have
lower group velocities than the waveguide modes around kp = 0.35π/D. Therefore,
the former benefits from increased interaction time with the nonlinear medium and
therefore the effective pair generation rate for the pump mode can be enhanced.
Depicted in Fig. 6.8, are the results of calculations performed, where the spacing
between signal and pump wavevectors were increased from ks − kp = 0.03 π/D to
ks − kp = 0.07 π/D. Simulations were also performed for two different lengths of the
nonlinear region: 20 D and 200 D. Comparing these results, several conclusions can
be made. First of all, increasing the spacing between signal and pump modes brings in
additional dynamics that are associated with the phase matching, as they only appear
when the entire CROW is considered nonlinear. In addition, depending on the actual
value of the difference, the dominant response of the system is determined either by
the high-Q features of the CROW or by the fastest group velocity features. Note that
in some of the plots in Fig. 6.8, a second peak appears that matches the location of
the fast group velocity around kp u 0.3 π/D, where a minimum phase mismatch can
be achieved. Finally, the actual relative ratio of the detection probability for the two
peaks in these plots can vary, again depending on the separation between signal and
the pump. Therefore, a clear signature that both dispersion and loss are responsible
for the overall dynamics of the system is present.
These results are very important, as one must decide the operation condition,
depending on requirements of the experiment. For example, one may decide to increase the spacing between the signal and the pump modes to benefit from higher
detection probabilities associated with low group index regime as well as ease of signal extraction. The price one pays is to perhaps suffer more pulse distortion due to
(2)
Gs (a.u.)
6.4. NONLINEAR SFWM IN THE SQUARE-LATTICE CROW
Nnl = 10 D
Nnl = 100 D
ks − kp = 0.03 π/D
ks − kp = 0.03 π/D
0.25
20
0.2
15
0.15
10
0.1
0.05
5
0
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.2 0.3 0.4 0.5 0.6 0.7 0.8
ks − kp = 0.05 π/D
ks − kp = 0.05 π/D
(2)
Gs (a.u.)
0.25
6
0.2
4
0.15
0.1
2
0.05
0
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.2 0.3 0.4 0.5 0.6 0.7 0.8
ks − kp = 0.07 π/D
ks − kp = 0.07 π/D
6
0.25
(2)
Gs (a.u.)
146
0.2
4
0.15
0.1
2
0.05
0
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
kp D/π
0.2 0.3 0.4 0.5 0.6 0.7 0.8
kp D/π
Figure 6.8: Detection probability for different cases of pump and signal separation for
the resonant condition 2ωp = ωs + ωi . Every calculation was performed
twice, for Nnl = 10 D on the left and for Nnl = 100 D on the right.
6.5. SUMMARY
147
propagation, as the center of the pulse always travels the fastest. One also has to
deal with phase mismatch contributions as the dispersion is not linear everywhere.
On the other hand, if one decides to choose a small separation between pump and
signal modes to benefits from low phase mismatch and pump distortion, the price
paid is the difficulty in signal extraction. Therefore, a careful consideration of lossy
and dispersive behavior of the desired CROW structure would be beneficial.
6.5
Summary
In this chapter, pair detection via the nonlinear process of SFWM was studied in
CROW structures. Due to the presence of the nonlinear interaction, a standard second
order correlation function calculation was shown to lead to an asymmetric result for
the detection probability when ωs and ωi were switched. It was found that there are
other contributions to the total detection probability that must be included in our
case to obtain the appropriate detection map. In addition, the detection probability
across the entire band of the square-lattice CROW was examined, where the detection
maximums took place either where the phase mismatch was the lowest or the quality
factor had a local maximum. This confirmed that both the dispersive behavior and
the lossy behavior of the CROW must be taken into account in a comprehensive study
of these systems.
148
Chapter 7
Conclusion and Future Work
The primary goal of this thesis was to develop a formalism that includes intrinsic
properties of CC quasimodes in photonic crystal slab structures and can be used to
study both quantum optics and quantum nonlinear optics in such systems. Specifically, scattering loss and non-orthogonality of these modes was incorporated as both
can affect the photon dynamics considerably. The approach taken was to use a nonstandard, non-Hermitian projection operator based on the system quasimodes that
contains all the necessary information about intrinsic properties of these modes.
Detailed methods were developed that show how the projection operator can be
used to obtain the effective Hamiltonian in different CC systems. In particular, an
effective representation of the multimode JC Hamiltonian for the CCs interacting with
multiple QDs was obtained. This Hamiltonian can be used in the generalized quantum
Master equation also derived in the thesis to obtain photon and QD dynamics for
complex PCS circuits, in the Schrödinger picture. A simple two mode system was
used to show how non-orthogonality of the QMs can affect the photon dynamics in a
non-trivial manner.
149
On the quantum NLO side, an effective representation of the Hamiltonian for
the nonlinear process of SFWM was obtained in the basis of QMs. The SFWM
Hamiltonian, along with the adjoint quantum Master equation, which has been also
represented in the basis of the QMs, can be used to obtain time dependence of any
operator in the Heisenberg picture. This approach was used to model the pair generation experiment in both photonic molecules and CROW structures in PCSs. The
focus of our studies in these cases was the effect of lossy characteristics of the QMs
rather than the non-orthogonality. In these systems loss enters in a non-trivial manner, where not only the overall loss rates in the system are important, but also the
loss difference between signal and idler channels are found to be playing role in minimizing the number of unpaired photons in the system; a feature that to the best of
my knowledge has not been reported in the literature before. Moreover, the important interplay between scattering loss and higher order photon pair generations was
studied. The trade-off between having a bright source and having a high-quality pair
device was discussed. Again to best of my knowledge, such considerations have been
studied for pair generation experiments in optical fibers but not in CROW structures,
particularly in PCSs.
In the particular case of the CROW structure, analytic expressions for the detection probabilities were obtained that can be used to calculate the probability of
finding photons of a given energy at any point in the system. This is in contrast to the
photon expectation that estimates the overall photon numbers in the entire system.
It was found that the standard second order coherence function calculation is not
sufficient for systems with nonlinear interactions and therefore additional considerations must be taken to obtain a symmetric detection map of the system. Therefore, I
150
suggest using the calculation scheme presented in this thesis to obtain the pair detection probabilities when working with systems with nonlinear interactions. This was
used to study the pair detection probability for a square-lattice CROW structures as
a function of pump wavelength. The detection probability was found to be related to
the quality factor of the propagating pump mode. In addition, the interplay between
the lossy characteristics and the slow light characteristics of the square-lattice CROW
were studied where it was found that in any design of such system, one must both
include both the lossy and the dispersive behavior into consideration. A message that
has not been conveyed in the related literature so far.
The formalism developed in this thesis is quite general in at least two different
aspects. 1) Adopting the current formalism for other Hamiltonians of interest, both
in quantum optics and quantum nonlinear optics, should be straightforward. Indeed,
the specific choice of Hamiltonian here comes form experimental motivation not from
the desire to make a simple case study for proof of principle. 2) Although developed
for CC structures in PCSs, the formalism can be used to study any optical system
where a set of leaky and possibly non-orthogonal modes are present. This can includes
microring resonator as well as line-defect waveguides in PCSs.
However, now that the fundamentals are known and the proof that our approach
works is available, further steps could be taken to generalize this approach to a wider
range of applications and systems. For example, for all of the structures of interest in
this thesis, the coupling mechanism was not considered. Normally in PCSs coupling
in and out from the device is done in an integrated fashion on the same chip as the
device, such as a line-defect waveguide or a grating. Additional considerations are
needed to develop mechanism for input/output coupling in a consistent manner with
151
the current formalism. One might picture this by adding appropriate terms to the
system Hamiltonian, or equivalently to the dynamical equation of motion.
Moreover, pair generation experiments have been also performed both in linedefect PCS waveguides and in silicon waveguides [76, 89]. Adopting the current formalism to those cases is straightforward in the sense that a set of QMs can be made
available. However, finding a compact expression for the nonlinear overlap function
as was done in chapter 6 was made possible by expanding the traveling Bloch modes
of the CROW in terms of the localized modes of the single cavities that are not available for a line-defect waveguide. One solution might be to employ localized Wannier
functions in a similar way to treat the pair generation in line-defect waveguide. This
would also allow the methods of this thesis to be applied to systems consisting of
defect cavities coupled to line-defect waveguides. Such system show great potential.
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152
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165
Appendix A
Commutation relation between
projected ladder operators
In this appendix, the commutation relation between projected annihilation and creation operator is derived. Derivation of other commutation relation can be done in
the same manner. This was stated in chapter 3 where the QM projection formalism
was presented.
We first expand the commutation relation
†
†
†
αn , βm
= αn βm
− βm
αn ,
(A.1)
in order to evaluate each term separately. Using expressions obtained for the projected
ladder operators in the text, it is easy top see that
166
†
αn βm
=
∞
N
X
X
N
X
N
X
N
X
(A.2)
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
i
i
×
P̃m001 m000
...P̃m00M m000
P̃m1 m01 ...P̃mM m0M
1
M
000
M !Am1 ...mM Anm01 ...m0M M !Amm001 ...m00M +1 Am000
1 ...mM
× |φm1 ...φmM ihχn χm01 ...χm0M |φm φm001 ...φm00M ihχm000
...χm000
|
1
M
=
∞
N
X
X
N
X
M =0 {mi }=1 {
m0i
×
N
X
N
X
}=1 {
m00
i
}=1 {
m000
i
(A.3)
}=1
P̃m1 m01 ...P̃mM m0M P̃m001 m000
...P̃m00M m000
1
M
000
M !Am1 ...mM
M !Am000
...m
1
M
× |φm1 ...φmM ihχm000
...χm000
|
1
M
X
perms
=
∞
N
X
X
N
X
Õnm
M =0 {mi }=1 {m000 }=1
Õnm Õm01 m001 ...Õm0M m00M
(m00
i)
P̃m1 m000
...P̃mM m000
1
M
|φm1 ...φmM ihχm000
...χm000
|,
1
M
000
M !Am1 ...mM Am1 ...m000
M
i
(A.4)
where on the second step we have used the fact that the product of P̃ and Õ matrices
is identity matrix. Comparing the obtained expression to the definition of the nonHermitian projector Pb we have
†
α n βm
= Õnm Pb.
(A.5)
167
Using a similar procedure, we obtain
†
βm
αn =
∞
N
X
X
N
X
M =0 {mi }=1 {m000 }=1
P̃m1 m000
...P̃mM m000
1
M
|φm φm1 ...φmM ihχn χm000
...χm000
|.
1
M
000
M !Amm1 ...mM Anm000
...m
1
M
i
(A.6)
This will give zero when applied on states with only M photons. Therefore, we have
†
αn , βm
= Õnm Pb.
(A.7)
Note that a given operator, including the commutator of other operators, is only
defined by its action on state of the system.
168
Appendix B
Coupled set of Master equations
for two-photon dynamics
In this appendix, the full set of coupled Master equations are listed that are used to
obtain the photon dynamics in the two-defect PCS in chapter 4, where initially there
are two photons present in the system.
Q
Q
The diagonal ρQ
1,1;1,1 is coupled to ρ1,1;0,2 and ρ1,1;2,0 as the following:
dρQ
1,1;1,1
Q
∗
∗
= −i (ω̃1 + ω̃2 ) ρQ
+
i
Õ
P̃
ω̃
+
Õ
P̃
ω̃
11
11
21
12
1,1;1,1
1
2 ρ1,1;1,1
dt
√ + 2i Õ11 P̃21 ω̃1∗ + Õ21 P̃22 ω̃2∗ ρQ
1,1;0,2
√ ∗
∗
+ 2i Õ12 P̃11 ω̃1 + Õ22 P̃12 ω̃2 ρQ
1,1;2,0
+ i Õ12 P̃21 ω̃1∗ + Õ22 P̃22 ω̃2∗ ρQ
1,1;1,1 ,
(B.1)
169
and
√ dρQ
1,1;0,2
Q
∗
∗
= −i (ω̃1 + ω̃2 ) ρQ
+
2i
+
Õ
P̃
ω̃
Õ
P̃
ω̃
22 12 2 ρ1,1;1,1
12 11 1
1,1;0,2
dt
∗
∗
+ 2i Õ12 P̃21 ω̃1 + Õ22 P̃22 ω̃2 ρQ
1,1;0,2 ,
(B.2)
and
dρQ
1,1;2,0
Q
∗
∗
= −i (ω̃1 + ω̃2 ) ρQ
+
2i
Õ
P̃
ω̃
+
Õ
P̃
ω̃
21 12 2 ρ1,1;2,0
11 11 1
1,1;2,0
dt
√ ∗
∗
+ 2i Õ11 P̃21 ω̃1 + Õ21 P̃22 ω̃2 ρQ
1,1;1,1 .
(B.3)
Q
Q
The diagonal ρQ
0,2;0,2 is coupled to ρ0,2;1,1 and ρ0,2;2,0 as the following:
√ dρQ
0,2;0,2
Q
∗
∗
= −iω̃2 ρQ
+
2i
Õ
P̃
ω̃
+
Õ
P̃
ω̃
12 11 1
22 12 2 ρ0,2;1,1
0,2;0,2
dt
∗
∗
+ 2i Õ12 P̃21 ω̃1 + Õ22 P̃22 ω̃2 ρQ
0,2;0,2 ,
(B.4)
and
dρQ
0,2;1,1
Q
∗
∗
= −2iω̃2 ρQ
+
i
Õ
P̃
ω̃
+
Õ
P̃
ω̃
11 11 1
21 12 2 ρ0,2;1,1
0,2;1,1
dt
√ ∗
∗
+ 2i Õ11 P̃21 ω̃1 + Õ21 P̃22 ω̃2 ρQ
1,1;1,1
+
√ 2i Õ12 P̃11 ω̃1∗ + Õ22 P̃12 ω̃2∗ ρQ
0,2;2,0
+ i Õ12 P̃21 ω̃1∗ + Õ22 P̃22 ω̃2∗ ρQ
0,2;1,1 ,
(B.5)
170
and
dρQ
0,2;2,0
Q
∗
∗
= −2iω̃2 ρQ
+
2i
+
Õ
P̃
ω̃
Õ
P̃
ω̃
21 12 2 ρ0,2;2,0
11 11 1
0,2;2,0
dt
√ ∗
∗
+ 2i Õ11 P̃21 ω̃1 + Õ21 P̃22 ω̃2 ρQ
0,2;1,1 .
(B.6)
Q
Q
Q
Q
Q
Q
The diagonal ρQ
1,0;1,0 is coupled to ρ1,0;0,1 , ρ2,0;2,0 , ρ2,0;0,2 , ρ2,0;1,1 , ρ1,1;1,1 , ρ1,1;0,2 and
ρQ
1,1;2,0 as the following (note that some coupled equations are already obtained):
dρQ
1,0;1,0
Q
∗
∗
= −iω̃1 ρQ
+
i
Õ
P̃
ω̃
+
Õ
P̃
ω̃
11
11
21
12
1,0;1,0
1
2 ρ1,0;1,0
dt
+ i Õ11 P̃21 ω̃1∗ + Õ21 P̃22 ω̃2∗ ρQ
1,0;0,1
+ 2iω̃1 ρQ
2,0;2,0
+ iω̃2 ρQ
1,1;1,1
+ 2i Õ11 P̃11 ω̃1∗ + Õ21 P̃12 ω̃2∗ ρQ
2,0;2,0
√ ∗
∗
+ 2i Õ11 P̃21 ω̃1 + Õ21 P̃22 ω̃2 ρQ
2,0;1,1
+
√ 2i Õ12 P̃11 ω̃1∗ + Õ22 P̃12 ω̃2∗ ρQ
1,1;2,0
+ i Õ12 P̃21 ω̃1∗ + Õ22 P̃22 ω̃2∗ ρQ
1,1;1,1 ,
(B.7)
171
and
dρQ
1,0;0,1
Q
∗
∗
= −iω̃1 ρQ
+
i
+
Õ
P̃
ω̃
Õ
P̃
ω̃
22 22 2 ρ1,0;0,1
12 21 1
1,0;0,1
dt
∗
∗
+ i Õ12 P̃11 ω̃1 + Õ22 P̃12 ω̃2 ρQ
1,0;1,0
+
+
√
√
(B.8)
2iω̃1 ρQ
2,0;1,1
2iω̃2 ρQ
1,1;0,2
√ ∗
∗
+ 2i Õ11 P̃11 ω̃1 + Õ21 P̃12 ω̃2 ρQ
2,0;1,1
+ 2i Õ11 P̃21 ω̃1∗ + Õ21 P̃22 ω̃2∗ ρQ
2,0;0,2
+ i Õ12 P̃11 ω̃1∗ + Õ22 P̃12 ω̃2∗ ρQ
1,1;1,1
√ ∗
∗
+ 2i Õ12 P̃21 ω̃1 + Õ22 P̃22 ω̃2 ρQ
1,1;0,2 .
Q
Q
Q
Q
Q
Q
The diagonal ρQ
0,1;0,1 is coupled to ρ0,1;1,0 , ρ0,2;0,2 , ρ0,2;2,0 , ρ0,2;1,1 , ρ1,1;1,1 , ρ1,1;0,2 and
172
ρQ
1,1;2,0 as the following (note that some coupled equations are already obtained):
dρQ
0,1;0,1
Q
∗
∗
= −iω̃2 ρ0,1;0,1 + i Õ12 P̃11 ω̃1 + Õ22 P̃12 ω̃2 ρQ
0,1;1,0
dt
+ i Õ12 P̃21 ω̃1∗ + Õ22 P̃22 ω̃2∗ ρQ
0,1;0,1
+ iω̃1 ρQ
1,1;1,1
+ 2iω̃2 ρQ
0,2;0,2
+ i Õ11 P̃11 ω̃1∗ + Õ21 P̃12 ω̃2∗ ρQ
1,1;1,1
+
√ 2i Õ11 P̃21 ω̃1∗ + Õ21 P̃22 ω̃2∗ ρQ
1,1;0,2
√ ∗
∗
+ 2i Õ12 P̃11 ω̃1 + Õ22 P̃12 ω̃2 ρQ
0,2;1,1
+ 2i Õ12 P̃21 ω̃1∗ + Õ22 P̃22 ω̃2∗ ρQ
0,2;0,2 ,
(B.9)
173
and
dρQ
0,1;1,0
Q
∗
∗
= −iω̃2 ρQ
+
i
+
Õ
P̃
ω̃
Õ
P̃
ω̃
21 12 2 ρ0,1;1,0
11 11 1
0,1;1,0
dt
∗
∗
+ i Õ11 P̃21 ω̃1 + Õ21 P̃22 ω̃2 ρQ
0,1;0,1
+
+
√
√
2iω̃1 ρQ
1,1;2,0
2iω̃2 ρQ
0,2;1,1
√ ∗
∗
+ 2i Õ11 P̃11 ω̃1 + Õ21 P̃12 ω̃2 ρQ
1,1;2,0
+ i Õ11 P̃21 ω̃1∗ + Õ21 P̃22 ω̃2∗ ρQ
1,1;1,1
+ 2i Õ12 P̃11 ω̃1∗ + Õ22 P̃12 ω̃2∗ ρQ
0,2;2,0
√ ∗
∗
+ 2i Õ12 P̃21 ω̃1 + Õ22 P̃22 ω̃2 ρQ
0,2;1,1 .
(B.10)
174
Appendix C
Electric field operator in QM
representation
In this appendix, the QM representation of the positive-frequency electric field operator is derived, which in particular was used in chapter 5.
We start by the positive-frequency part of the electric field represented in the basis
of the system true modes
ˆ
Ẽ
(+)
(r) = i
r
dµ
~ωµ
aµ f̃µ (r) .
20
(C.1)
The projected field operator onto the basis of the non-orthogonal QMs can be constructed as the following:
(+)
(+)
ẼQ (r) = PeẼQ (r) Pe
=
∞ ˆ
X
M =0
r
dµ
(C.2)
~ωµ
f̃µ (r) TbM aµ TbM +1 .
20
(C.3)
175
Similar to the procedure taken in chapter 3, it is straightforward to see that
(+)
ẼQ
(r) =
∞
N
X
X
N
X
N
X
N
X
ˆ
r
dµ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
~ωµ
f̃µ (r)
20
i
P̃m001 m000
...P̃m00M +1 m000
P̃m1 m01 ...P̃mM m0M
1
M +1
000
M !Am1 ...mM Am01 ...m0M (M + 1)!Am001 ...m00M +1 Am000
1 ...mM +1
× |φm1 ...φmM ihχm01 ...χm0M |aµ |φm001 ...φm00M +1 ihχm000
...χm000
|
1
M +1
=
∞
N
X
X
N
X
N
X
N
X
ˆ
r
dµ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
~ωµ
f̃µ (r)
20
i
...P̃m00M +1 m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M +1
|φm1 ...φmM ihχm000
...χm000
|
1
M +1
000
M !Am1 ...mM (M + 1)!Am000
...m
1
M +1
ˆ
ˆ
∗
∗
dµ1 ...dµM q̃m
0 µ ...q̃m0 µ
1 1
M M
×
dν1 ...dνM +1 c̃m001 ν1 ...c̃m00M +1 νM +1
× h0| aµ1 ...aµM aµ a†ν1 ...a†νM a†νM +1 |0i
=
∞
N
X
X
N
X
N
X
N
X
ˆ
r
dµ
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
×
(C.5)
~ωµ
f̃µ (r)
20
i
...P̃m00M +1 m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M +1
|φm1 ...φmM ihχm000
...χm000
|
1
M +1
000
M !Am1 ...mM (M + 1)!Am000
...m
1
M +1
ˆ
×
(C.4)
ˆ
∗
∗
dµ1 ...dµM q̃m
0 µ ...q̃m0 µ
1 1
M M
X
perms(µ,µi )
dν1 ...dνM +1 c̃m001 ν1 ...c̃m00M +1 νM +1
δ (µ1 − ν1 ) ...δ (µM − νM ) δ (µ − νM +1 ) .
(C.6)
176
The delta functions involved can be used to simplify the νi integrals and obtain
(+)
ẼQ (r) =
∞
N
X
X
N
X
N
X
N
X
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
i
...P̃m00M +1 m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M +1
|φm1 ...φmM ihχm000
...χm000
|
1
M +1
000
M !Am1 ...mM (M + 1)!Am000
1 ...mM +1
ˆ
ˆ
×
X
∗
dµ1 q̃m
0 µ c̃m00 µ1 ...
1
1 1
perms(m0i )
r
ˆ
~ωµ
× dµ
f̃µ (r) c̃m00M +1 µ .
20
∗
dµM q̃m
0 µ c̃m00 µ
M M
M M
(C.7)
Integration over µ can be done using the very useful identity of Eq. 4.15 and thus we
find
(+)
ẼQ (r) = i
∞
N
X
X
N
X
N
X
N
X
M =0 {mi }=1 {m0 }=1 {m00 }=1 {m000 }=1
i
×
i
i
...P̃m00M +1 m000
P̃m1 m01 ...P̃mM m0M P̃m001 m000
1
M +1
|φm1 ...φmM ihχm000
...χm000
|
1
M +1
000
M !Am1 ...mM (M + 1)!Am000
...m
1
M +1
s
×
X
perms
(m00
i)
Õm01 m001 ...Õm0M m00M
~ω̃m00M +1
20
Ñm00M +1 (r) .
(C.8)
177
Now, it is easy to see that
(+)
ẼQ (r) = i
∞
N
X
X
N
X
N
X
M =0 {mi }=1 {m00 }=1 {m000 }=1
i
×
i
P̃m001 m000
...P̃m00M +1 m000
1
1
M +1
|φm1 ...φmM ihχm000
...χm000
|
1
M +1
000
M ! (M + 1)!Am1 ...mM Am000
1 ...mM +1
s
×
X
perms
=i
N
∞
X
X
δm1 m001 ...δmM m00M
~ω̃m00M +1
20
(m00
i)
N
X
N
X
s
~ω̃m00M +1
20
000
M =0 {mi }=1 m00
M +1 =1 {m }=1
Ñm00M +1 (r)
(C.9)
Ñm00M +1 (r)
i
×
...P̃mM m000
P̃m00M +1 m000
P̃m1 m000
1
M
M +1
000
M !Am1 ...mM Am000
1 ...mM +1
|φm1 ...φmM ihχm000
...χm000
|.
1
M +1
(C.10)
Using expression for the projected annihilation operator, Eq. (3.48), the projected
field operator can be written in the following form
(+)
ẼQ
(r) = i
N
X
m,n=1
r
~ω̃m
Ñm (r) P̃mn αn .
20
178
Appendix D
Second order ansatz for number
operator
In this appendix, the derivation of the second order ansatz that led to a non-zero
expectation value for the photon number operator in chapter 5, is derived.
The second order approximation to the photon number expectation of Eq. (5.43)
is governed by
(2)
i
(2)
d b†n bn
(t)
i h † (1)
†
= L 0 bn bn
(t) −
bn bn
(t) , HN L ,
dt
~
(D.1)
179
where can be expanded out as
(2)
(2)
d b†n bn
(t)
= L0 b†n bn
(t)
dt
+i
X X X X
(D.2)
h(1)
n,m2 ,m3 ,m4 (t) Sm01 ,m02 ,m03 ,m04
m2 m3 ,m4 m01 ,m02 m03 ,m04
h
i
× b†n b†m2 bm3 bm4 , b†m0 b†m0 bm03 bm04 + h.c.
1
+i
X X X X
2
∗
h(1)
n,m2 ,m3 ,m4 (t) Sm01 ,m02 ,m03 ,m04
m2 m3 ,m4 m01 ,m02 m03 ,m04
h
i
× b†n b†m2 bm3 bm4 , b†m0 b†m0 bm02 bm01 + h.c..
4
3
Consider the commutator on the second line of Eq. (D.2) that involves only raising
operators corresponding to signal and idler indices. Therefore, no matter what this
commutator simplifies to, its expectation value on the initial state (5.30) is zero. The
second commutator on the other hand, simplifies to
180
h
i
b†n b†m2 bm3 bm4 , b†m0 b†m0 bm02 bm01 = b†n b†m2 b†m0 bm3 bm02 bm01 δm4 m04 + b†n b†m2 b†m0 bm3 bm02 bm01 δm4 m03
4
4
3
3
+ b†n b†m2 b†m0 bm4 bm02 bm01 δm3 m04 + b†n b†m2 b†m0 bm4 bm02 bm01 δm3 m03
3
4
+ b†n b†m2 bm02 bm01 δm4 m04 δm3 m03 + b†n b†m2 bm02 bm01 δm4 m03 δm3 m04
− b†m0 b†m0 b†m2 bm02 bm3 bm4 δm01 n − b†m0 b†m0 b†n bm02 bm3 bm4 δm01 m2
4
3
4
3
− b†m0 b†m0 b†m2 bm01 bm3 bm4 δm02 n − b†m0 b†m0 b†n bm01 bm3 bm4 δm02 m2
4
3
4
3
− b†m0 b†m0 bm3 bm4 δm01 n δm02 m2 − b†m0 b†m0 bm3 bm4 δm01 m2 δm02 n .
4
3
4
3
(D.3)
From all these, only the two terms that have four ladder operators and are negative
(1)
in sign have a non-zero expectation value. Moreover, Sm1 ,m2 ,m3 ,m4 and hm1 ,m2 ,m3 ,m4 (t)
in Eq. (D.2) are invariant under index exchange m1 ↔ m2 (m3 ↔ m4 ). Therefore,
we obtain
*
(2) + D
(2) E
d b†n bn
(t)
= L0 b†n bn
(t)
dt
(D.4)



 X X X
D
E
†
†
∗
− 2i
h(1)
(t)
S
b
b
b
b
+
h.c.
.
0
0
0
0
m
m
3
4
n,m2 ,m3 ,m4
m4 m3

 m m ,m 0 0 n,m2 ,m3 ,m4
2
3
4
m3 ,m4
Suggested by this equation, the second order ansatz that we employ for the number
181
operator corresponding to terms with only non-zero expectations is
b†n bn
(2)
X X
(t) =
hm3 ,m4 ,m0 ,m0 (t) b†m0 b†m0 bm3 bm4 + h.c.
(2)
3
m3 ,m4 m03 ,m04
4
4
3
(D.5)
Note that, all other terms which we have not represented in our ansatz evolve independently from each other and from terms we have shown above, thus will have no
indirect contribution to the final expectation value. Substituting (D.5) into (D.1) and
using simple dynamics of normally ordered operators under L0 presented in Eq. (5.9),
we obtain the differential equation
(2)
dhm3 ,m4 ,m0 ,m0 (t)
3
4
dt
(2)
∗
∗
= i ω̃m
hm3 ,m4 ,m0 ,m0 (t)
0 + ω̃m0 − ω̃m3 − ω̃m4
3
4
3
− 2i
X
4
(D.6)
∗
h(1)
n,m2 ,m3 ,m4 (t) Sn,m2 ,m03 ,m04 ,
m1 ,m2
(2)
for hm3 ,m4 ,m0 ,m0 (t). Integrating this, we obtain
3
4
(2)
hm3 ,m4 ,m0 ,m0 (t) = −2i
3
X
4
∗
Sn,m
0
0
2 ,m3 ,m4
(D.7)
m2
i ω̃ ∗ 0 +ω̃ ∗ 0 −ω̃m3 −ω̃m4 t
×e
m3
ˆ
t
−i ω̃ ∗ 0 +ω̃ ∗ 0 −ω̃m3 −ω̃m4 t0
0
dt0 h(1)
n,m2 ,m3 ,m4 (t ) e
m4
m3
m4
.
0
This can be then used to derive the photon number expectation given by Eq. (5.43).
182
Appendix E
First and second order ansatzen for
product of number operators
In this appendix, the derivation of the both first order and second order ansatzen
that led to a non-zero expectation value for the product of photon number operators
in chapter 5, is derived.
For the product of number operators, the zeroth order evolution is simply
b†n1 bn1 b†n2 bn2
(0)
(t) = e−2γn1 t e−2γn2 t b†n1 bn1 b†n2 bn2 .
(E.1)
This has zero expectation when operating on the vacuum, thus we continue with the
first order perturbation
d b†n1 bn1 b†n2 bn2
dt
(1)
(t)
= L0 b†n1 bn1 b†n2 bn2
(1)
(t) −
i
(0)
ih †
bn1 bn1 b†n2 bn2
(t) , HN L .
~
(E.2)
183
Unfortunately, this commutator does not yield as simple an expression as Eq. (5.38)
for the first order perturbation. However, it is easy to see that, if normally ordered,
every term in this commutator has zero expectation value when operating on vacuum
for the signal and idler modes. On the other hand, some of the terms will eventually
contribute to the expectation value for the product operator through the second order perturbation. Only considering such terms, the proper ansatz for the first order
perturbation is
b†n1 bn1 b†n2 bn2
(1)
(t) =
X
gn(1)
(t) b†n1 b†n2 bm3 bm4 + h.c.
1 ,n2 ,m3 ,m4
(E.3)
m3 ,m4
(1)
Substituting Eq. (E.3) into Eq. (E.2) gives a differential equation for the gn1 ,n2 ,m3 ,m4 (t)
which we integrate to obtain
gn(1)
(t) = 2iSn1 ,n2 ,m3 ,m4 ei(ω̃n1 +ω̃n2 −ω̃m3 −ω̃m4 )t
1 ,n2 ,m3 ,m4
∗
ˆ
×
t
∗
(E.4)
dt0 e−2γn1 t e−2γn2 t e−i(ω̃n1 +ω̃n2 −ω̃m3 −ω̃m4 )t .
0
0
∗
∗
0
(E.5)
0
This expression is then subjected to one final iteration of the dynamical master
equation to derive the second order evolution of the product of number operators. It
can be seen that the adjoint master equation for the expectation value of the product
operator is
184
*
d b†n1 bn1 b†n2 bn2
dt
−


(2)
+
D
(2) E
(t)
= L0 b†n1 bn1 b†n2 bn2
(E.6)
E
D
X X
2i

(t)
†
†
∗
h(1)
n1 ,n2 ,m3 ,m4 (t) Sn1 ,n2 ,m03 ,m04 bm0 bm0 bm3 bm4 + h.c.
4
m3 ,m4 m03 ,m04
3


.

This means that, the second order ansatz for the product operator has to be set
to
b†n1 bn1 b†n2 bn2
(2)
X X
(t) =
gm3 ,m4 ,m0 ,m0 (t) b†m0 b†m0 bm3 bm4 + h.c.
(2)
3
m3 ,m4 m03 ,m04
4
4
3
(2)
The corresponding gm3 ,m4 ,m0 ,m0 (t) is
3
4
(2)
gm3 ,m4 ,m0 ,m0 (t) = −2i Sn∗1 ,n2 ,m03 ,m04
3
(E.7)
4
×e
i ω̃ ∗ 0 +ω̃ ∗ 0 −ω̃m3 −ω̃m4 t
m3
ˆ
−i ω̃ ∗ 0 +ω̃ ∗ 0 −ω̃m3 −ω̃m4 t0
t
dt0 gn(1)
(t0 ) e
1 ,n2 ,m3 ,m4
m4
m3
m4
.
0
(E.8)
(1)
(2)
The given gn1 ,n2 ,m3 ,m4 (t) and gm3 ,m4 ,m0 ,m0 (t) found here, may then be used to
3
4
obtain the expectation value of the product of number operators given by Eq. (5.44).
185
Appendix F
Higher order contributions to
Sk1,k2,k3,k4
In chapter 6, a compact expression for the nonlinear overlap function was obtained
when only contributions form modes sitting on the same site were considered. In this
appendix, we generalize this to include higher order contributions when either one or
two of the four modes under the nonlinear overlap integral are shifted.
Consider the case where only one of the modes under the overlap integral in
Eq. (6.14) is shifted away by one CROW period, D. There are four possible ways to
186
do so as there are four modes involved. Therefore
(1)
Sk1 ,k2 ,k3 ,k4
N
~ωp2 X
=
exp {−i [k1 + k2 − k3 − k4 ] pD − ik1 D}
16o p=−N
ˆ
(3)
x)M̃i∗ (r)M̃i (r)M̃i (r)
d3 r χiiii (r) (r) M̃i∗ (r − Db
×
N
~ωp2 X
+
exp {−i [k1 + k2 − k3 − k4 ] pD − ik2 D}
16o p=−N
ˆ
(3)
d3 r χiiii (r) (r) M̃i∗ (r)M̃i∗ (r − Db
x)M̃i (r)M̃i (r)
×
N
~ωp2 X
exp {−i [k1 + k2 − k3 − k4 ] pD + ik3 D}
+
16o p=−N
ˆ
(3)
d3 r χiiii (r) (r) M̃i∗ (r)M̃i∗ (r)M̃i (r − Db
x)M̃i (r)
×
N
~ωp2 X
exp {−i [k1 + k2 − k3 − k4 ] pD + ik4 D}
+
16o p=−N
ˆ
×
(3)
d3 r χiiii (r) (r) M̃i∗ (r)M̃i∗ (r)M̃i (r)M̃i (r − Db
x),
(F.1)
where as before, overlap integrals associated with different ps are approximated to be
the same and the real part of the frequencies under the square root are approximated
with the real part of complex frequency for the pump mode. This expression can be
187
recast in the following compact form
(1)
Sk1 ,k2 ,k3 ,k4 =
(3)
M
X
~ωp2 χxxxx
C̃1 exp {−i (k1 + k2 ) D}
exp {−i [k1 + k2 − k3 − k4 ] pD}
16o
p=−M
(3)
M
X
~ωp2 χxxxx ∗
+
C̃1 exp {i (k3 + k4 ) D}
exp {−i [k1 + k2 − k3 − k4 ] pD} .
16o
p=−M
(F.2)
where
ˆ
d3 r (r) M̃x∗ (r − Db
x)M̃x∗ (r)M̃x (r)M̃x (r)
C̃1 ≡
ˆ
d3 r (r) M̃y∗ (r − Db
x)M̃y∗ (r)M̃y (r)M̃y (r)
+
ˆ
+
d3 r (r) M̃z∗ (r − Db
x)M̃z∗ (r)M̃z (r)M̃z (r).
(F.3)
The sum over p has been also evaluated in Appendix (G), thus we obtain
(1)
Sk1 ,k2 ,k3 ,k4
(3)
o
~ωp2 χxxxx n
−i(k1 +k2 )D
∗ i(k3 +k4 )D sin (∆k (2M + 1) D/2)
=
C̃1 e
+ C̃1 e
.
16o
sin (∆k D/2)
(F.4)
A similar procedure can be taken for the case where two of modes are shifted one
188
CROW period to obtain:
(2)
Sk1 ,k2 ,k3 ,k4 =
N
~ωp2 X
exp {−i [k1 + k2 − k3 − k4 ] pD − i (k1 + k2 ) D}
16o p=−N
ˆ
(3)
x)M̃i (r)M̃i (r)
x)M̃i∗ (r − Db
d3 r χiiii (r) (r) M̃i∗ (r − Db
×
N
~ωp2 X
+
exp {−i [k1 + k2 − k3 − k4 ] pD − i (k1 − k3 ) D}
16o p=−N
ˆ
(3)
d3 r χiiii (r) (r) M̃i∗ (r − Db
x)M̃i∗ (r)M̃i (r − Db
x)M̃i (r)
×
N
~ωp2 X
exp {−i [k1 + k2 − k3 − k4 ] pD − i (k1 − k4 ) D}
+
16o p=−N
ˆ
(3)
d3 r χiiii (r) (r) M̃i∗ (r − Db
x)M̃i∗ (r)M̃i (r)M̃i (r − Db
x)
×
+
N
~ωp2 X
exp {−i [k1 + k2 − k3 − k4 ] pD − i (k2 − k3 ) D}
16o p=−N
ˆ
(3)
d3 r χiiii (r) (r) M̃i∗ (r)M̃i∗ (r − Db
x)M̃i (r − Db
x)M̃i (r)
×
N
~ωp2 X
exp {−i [k1 + k2 − k3 − k4 ] pD − i (k2 − k4 ) D}
+
16o p=−N
ˆ
(3)
d3 r χiiii (r) (r) M̃i∗ (r)M̃i∗ (r − Db
x)M̃i (r)M̃i (r − Db
x)
×
N
~ωp2 X
exp {−i [k1 + k2 − k3 − k4 ] pD + i (k3 + k4 ) D}
+
16o p=−N
ˆ
×
(3)
d3 r χiiii (r) (r) M̃i∗ (r)M̃i∗ (r)M̃i (r − Db
x)M̃i (r − Db
x). (F.5)
189
This can be recast into the following form:
(3)
(2)
Sk1 ,k2 ,k3 ,k4
~ωp2 χxxxx sin (∆k (2M + 1) D/2)
=
16o
sin (∆k D/2)
(F.6)
n
× C̃2 e−i(k1 +k2 )D + C̃2∗ ei(k3 +k4 )D
−i(k1 −k3 )D
+C̃3 e
+e
−i(k1 −k4 )D
+e
−i(k2 −k3 )D
−i(k2 −k4 )D
+e
o
.
where we have defined
ˆ
d3 r (r) M̃x∗ (r − Db
x)M̃x∗ (r − Db
x)M̃x (r)M̃x (r)
C̃2 ≡
ˆ
d3 r (r) M̃y∗ (r − Db
x)M̃y∗ (r − Db
x)M̃y (r)M̃y (r)
+
ˆ
+
d3 r (r) M̃z∗ (r − Db
x)M̃z∗ (r − Db
x)M̃z (r)M̃z (r),
(F.7)
and
ˆ
d3 r (r) M̃x∗ (r − Db
x)M̃x∗ (r − pDb
x)M̃x (r − Db
x)M̃x (r)
C̃3 ≡
ˆ
d3 r (r) M̃y∗ (r − Db
x)M̃y∗ (r − pDb
x)M̃y (r − Db
x)M̃y (r)
+
ˆ
+
d3 r (r) M̃z∗ (r − Db
x)M̃z∗ (r − pDb
x)M̃z (r − Db
x)M̃z (r).
(F.8)
Note that C̃2 is indeed a real number and its complex conjugate is not present in the
expression.
190
Appendix G
Evaluating the phase-matching
sum for Sk1,k2,k3,k4
In this appendix, the derivation used to evaluate the phase-matching sum of Eq. (6.15),
is presented.
We start by rearranging the elements of the following sum:
M
X
exp (−iβp)
= 1 + eiβ + e−iβ + · · · + eiβM + e−iβM
(G.1)
= 1 + eiβ + · · · + eiβM + e−iβ + · · · + e−iβM
(G.2)
eiβ 1 − eiβM
e−iβ 1 − e−iβM
=1+
+
,
1 − eiβ
1 − e−iβ
(G.3)
p=−M
where β ≡ ∆k D and we have used geometric series identities on the last step. Further
191
simplification can be obtained using simple algebra:
M
X
eiβ eiβM/2 e−iβM/2 − eiβM/2
e−iβ e−iβM/2 eiβM/2 − e−iβM/2
exp (−iβx) = 1 +
+
iβ/2 (e−iβ/2 − eiβ/2 )
e
e−iβ/2 (eiβ/2 − e−iβ/2 )
p=−M
=1+
eiβ/2 eiβM/2 sin (βM/2) e−iβ/2 e−iβM/2 sin (βM/2)
+
sin (β/2)
sin (β/2)
=1+
sin (βM/2) iβ/2 iβM/2
e e
+ e−iβ/2 e−iβM/2
sin (β/2)
=1+
2 sin (βM/2) cos (β (M + 1) /2)
sin (β/2)
=1+
sin (βM/2 + β (M + 1) /2) + sin (βM/2 − β (M + 1) /2)
sin (β/2)
=1+
sin (β (2M + 1) /2) + sin (−β/2)
sin (β/2)
=1+
sin (β (2M + 1) /2)
−1
sin (β/2)
=
sin (β (2M + 1) /2)
.
sin (β/2)
(G.4)
192
Appendix H
Heisenberg evolution of the ladder
operators
In this appendix, the Heisenberg time-dependent ladders operators are obtained to
first order in the nonlinear perturbation. This was used in chapter 6, where detection
probabilities were calculated.
In our new notation, the full projected system Hamiltonian is
(
Q
H̃ef
f =
X
m
~ω̃k b†k bk −
)
X
~ Sk1 ,k2 ,k3 ,k4 b†k1 b†k2 bk3 bk4 + h.c. ,
(H.1)
k1 ,k2 ,k3 ,k4
where, k1 and k2 label signal and idler modes in the SFWM process, while k3 and
k4 label the pump modes. Also, the corresponding adjoint AMster equation for the
Heisenberg operator, Â (t), can be recast as
ih
i
dÂH (t)
Q
= L0 ÂH (t) −
ÂH (t) , H̃ef
f ,
dt
~
(H.2)
193
where
i
h
i
iX
i X ∗ †h
L0 ÂH (t) ≡
~ω̃k b†k , ÂH (t) bk +
~ω̃k bk bk , ÂH (t) .
~ k
~ k
(H.3)
We start by obtaining the time-dependent ladder operators of the system. The
zeroth order evolution for the lowering operator, bn (t), is given by
(0)
dbk (t)
(0)
= L0 bk (t)
dt
h
i
iX ∗ †
iX
~ω̃k0 b†k0 , bk bk0 +
~ω̃k0 bk0 [bk0 , bk ]
=
~ k0
~ k0
= −iω̃k bk .
(H.4)
(H.5)
(H.6)
This has the trivial solution of
(0)
bk (t) = e−iω̃k t bk .
(H.7)
The first order approximation is given by
(1)
ih
i
dbk (t)
(1)
(0)
= L0 bk (t) −
bk (t) , HN L
dt
~
h
i
X
(1)
† †
−iω̃k t
= L0 bk (t) + i e
Sk1 ,k2 ,k3 ,k4 bk , bk1 bk2 bk3 bk4 .
(H.8)
(H.9)
k1 ,k2 ,k3 ,k4
It is straightforward to see that
h
i
bk , b†k1 b†k2 bk3 bk4 = b†k2 bk3 bk4 δkk1 + b†k1 bk3 bk4 δkk2 .
(H.10)
194
Therefore, using the underlying symmetry of the S under k1 ↔ k2 permutation, we
have
(1)
X
dbk (t)
(1)
= L0 bk (t) + 2i e−iω̃k t
Sk,k2 ,k3 ,k4 b†k2 bk3 bk4 .
dt
k ,k ,k
2
3
(H.11)
4
If we now use the ansatz
X
(1)
bk (t) ≡
hk,k2 ,k3 ,k4 (t) b†k2 bk3 bk4 ,
(H.12)
= i ω̃k∗2 − ω̃k3 − ω̃k4 b†k2 bk3 bk4 .
(H.13)
k2 ,k3 ,k4
Since
L0
b†k2 bk3 bk4
Therefore
dhk,k2 ,k3 ,k4 (t)
= i ω̃k∗2 − ω̃k3 − ω̃k4 hk,k2 ,k3 ,k4 (t) + 2i e−iω̃k t Sk,k2 ,k3 ,k4 .
dt
(H.14)
Integrating this equation leads to
i(ω̃k∗ −ω̃k3 −ω̃k4 )t
hk,k2 ,k3 ,k4 (t) = 2i Sk,k2 ,k3 ,k4 e
ˆ
t
2
dt0 e−i(ω̃k +ω̃k2 −ω̃k3 −ω̃k4 )t ,
∗
0
(H.15)
0
which can be recast as
ˆ
−iω̃k t
hk,k2 ,k3 ,k4 (t) = 2i Sk,k2 ,k3 ,k4 e
t
dt0 ei(ω̃k +ω̃k2 −ω̃k3 −ω̃k4 )t .
∗
0
(H.16)
0
Therefore, the ansatz (H.12) is indeed the first order approximation to the lowering
195
operator. The raising operator can be written as
b†k
(1)
(t) =
X
k2 ,k3 ,k4
h∗k,k2 ,k3 ,k4 (t) b†k4 b†k3 bk2 .
(H.17)