Hybrid Quantum Nanophotonics

# Hybrid Quantum Nanophotonics – Interfacing Color Center in Nanodiamonds with
$\mathrm{Si}_{3}\mathrm{N}_{4}$-Photonics

Alexander Kubanek*  Anna P. Ovvyan  Lukas Antoniuk  Niklas Lettner  Wolfram H.
P. Pernice
*corresponding authors:
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
##  1 Abstract

This chapter covers recent developments in the field of hybrid quantum
photonics based on color centers in nanodiamonds and
$\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics towards a technology platform with
applications in quantum information processing and quantum information
distribution. The methodological approach can be divided in three main tasks.
First, the fabrication and optimization of
$\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics. Second, the creation,
characterization and control of color centers in nanodiamonds. Third, the
assembly of hybrid quantum photonics by integrating the nanodiamonds into the
photonic structures. One focus will be the efficient interfacing of the color
centers done by optimizing the optical coupling. The chapter describes recent
progress in all three steps and summarizes the established hybrid platform. We
believe, that the hybrid approach provides a promising path to realize quantum
photonic applications, such as quantum networks or quantum repeaters, in the
near future.

##  2 Introduction

Optics and photonics nowadays make a major contribution to leading edge
technologies. As an example, fiber-optic communication system transfer
information over long distances such as transatlantic connection and provide
high bandwidth, low optical loss (from $0.22\,\mathrm{dB}/\mathrm{km}$),
almost no crosstalk and protection against outside electromagnetic waves.
Therefore, effective speed of data transmission with rates up to 11 Tbits/s or
faster became possible winzerFiberopticTransmissionNetworking2018 ;
polettiHighcapacityFibreopticCommunications2013 . Analogous, integrated
photonic circuits (IPC) bogdanovMaterialPlatformsIntegrated2017  and
nanophotonics kinseyExaminingNanophotonicsIntegrated2015  demonstrate strong
potential to provide high-capacity information transmission, processing and
communication on-chip. Operating the technology in the quantum regime
murrayQuantumPhotonicsHybrid2015  could play the key role of an interruptive
technology enabling a plethora of new applications and with the potential to
herald a new era of communication paraisoPhotonicIntegratedQuantum2021 ,
computing and data security.
A field known as integrated quantum photonics enables to establish a
technology based on sub-wavelength light-matter interaction where classic and
quantum effects are conjoined through the utilization of nanophotonic building
blocks. Main components of the integrated quantum photonic circuit include:

  1. 1.

Single photon emitter, ideally producing single photons on-demand;

  2. 2.

Low-loss waveguiding, passive elements such as filters and beamsplitters;

  3. 3.

Reconfigurable active elements such as modulators and switches;

  4. 4.

Cavities to enhance light-matter interaction such as photonic crystal (PhC)
cavities or ring resonators;

  5. 5.

Fast and efficient single-photon detectors.

Integrated photonics enables the development of next-generation devices based
on a technology that can withstand the demands of industrial production.
However, the fabrication and control of quantum systems is at an early stage
of the Technology Readiness Level (TRL). The idea of combining IPCs, which are
rapidly improving due to quickly evolving industry standards, with quantum
optical systems of reduced complexity leverages the advantages of both worlds.
Complex architectures can be realized by merging the strengths of the
different systems. Operating the platform in the quantum regime opens the
field of hybrid quantum photonics bensonAssemblyHybridPhotonic2011 ;
kimHybridIntegrationMethods2020 . Hybrid quantum photonics holds the unique
potential for a scalable technology with quantum properties of high-quality.
Major challenges include the identification of the most promising systems to
be combined and the establishment of a hybridization method that is efficient
and salable for high-throughput production. The hybridization procedure is
called quantum post-processing (QPP) in the following. This chapter focuses on
a relatively new, yet rapidly evolving hybridization method utilizing color
center in nanodiamonds (ND) to post-process silicon nitride
($\mathrm{Si}_{3}\mathrm{N}_{4}$)-photonics.

##  3 Potential Pathways towards Hybrid Quantum Photonics

A number of different strategies to build hybrid quantum devices based on
color center in diamond were explored in the past decade. The methodologies
can be divided in two main concepts. One, the “structure to emitter” approach.
Two, the “emitter to structure” approach.

###  3.1 “Structure to Emitter” Approach

The structure to emitter approach was realized in pioneering experiments by
Barclay et al. barclayHybridPhotonicCrystal2009  and Englund et
al.englundDeterministicCouplingSingle2010 . For example, the coupling between
the mode of a PhC nanocavity and individual negatively-charged nitrogen-
vacancy (NV) centers located close to the surface of bulk diamond, was
realized by in situ lateral scanning of the nanocavity with respect to the
location of the NV center englundDeterministicCouplingSingle2010 . An
alternative realization of the structure to emitter approach builds on the
fabricating of the photonic device around a preselected color center. For
example, lithographic alignment is used to build an integrated platform by
constructing plasmonic waveguides around precharacterized Germanium Vacancy
center (GeV) in NDs in a deterministic fashion
siampourUltrabrightSinglephotonEmission2020a . The platform yields a 10-fold
enhanced optical decay rate and unidirectional emission. Another very
efficient assembly method includes all-silver nanopatch antennas, realized by
silver nanocubes deposited on a silver substrate, where NDs are localized
between in the area of strong field enhancement. For NV centers in NDs a
70-fold enhancement of the fluorescence lifetime and a 90-fold intensity
increase was demonstrated with a count rate of up to 35 million per second
bogdanovUltrabrightRoomTemperatureSubNanosecond2018 . Integration of GeV
centers in NDs into plasmonic modes of dielectric ridges atop colloidal silver
crystals yields a 6-fold Purcell enhancement, $56\,\%$ coupling efficiency and
$33\,\umu\mathrm{m}$ transmission length siampourOnchipExcitationSingle2018 .
A $20\,\umu\mathrm{m}$ single photon propagation length, originating from NV
fluorescence, was realized by building dielectric loaded surface plasmon
polariton waveguides around the NDs using electron-beam lithography yielding
5-fold enhancement in the total decay rate and 58 % coupling efficiency
siampourNanofabricationPlasmonicCircuits2017 . And a chip-integrated cavity-
coupled emission into propagating surface plasmon polariton modes enabled a
42-fold decay rate enhancement
siampourChipintegratedPlasmonicCavityenhanced2017 .

###  3.2 “Emitter to Structure” Approach

The emitter to structure approach is an assembly method complementary to the
structure to emitter approach. In this chapter, we focus on emitter to
structure approaches involving the following steps:

  1. 1.

The quantum emitters are prepared inside small-sized solid state hosts.

  2. 2.

Independently, the photonics devices are optimized with pre-defined
interaction zones.

  3. 3.

In the last step, the emitter is placed in the interaction zone via
nanomanipulation tools, a step which we label as QPP.

In the past years, a number of different hybrid quantum photonic devices were
realized following this procedure. Also nanostructured devices that
concentrate electromagnetic fields in a region much smaller than the
wavelength of light were interfaced with field enhancement of several orders
of magnitude NanophotonicsSurfacePlasmons2006 ;
gramotnevPlasmonicsDiffractionLimit2010 . Other techniques do not rely on
nanomanipulation but instead utilize a charged ND surface to attach larger
quantities of ND to, for example, tapered optical fibers
vorobyovCouplingSingleNV2016 . A few examples of hybrid plasmonics and hybrid
photonics based on color centers in ND and based on the emitter to structure
approach are depicted in the following. However, it is beyond the scope of
this chapter to give a complete overview. NDs containing NV centers were
positioned in the interaction zone of PhC structures by using the pick and
place technique schellScanningProbebasedPickandplace2011 . NV-center in NDs
were also coupled to surface plasmons in a controlled way leading to an
enhanced photon emission rate huckControlledCouplingSingle2011  and further
coupled to quantum plasmonic circuits huckCouplingSingleEmitters2016 . When
coupled to nanopatch antennas the emission rate of single NV center can be
enhanced into an ultrabright regime with a sub-nanosecond emission
bogdanovUltrabrightRoomTemperatureSubNanosecond2018 . NV
andersenHybridPlasmonicBullseye2018  and silicon-vacancy (SiV)
waltrichHighpuritySinglePhotons2021  center in NDs were placed in the center
of bullseye antennas achieving a high-purity single photon source and enabling
access via moderate-NA optics. An essential improvement towards an efficient
integration of color centers in NDs into photonics is the usage of
nanopockets. Placing a ND in a nanopocket within a PhC L3 cavity yields a
robust integration and an up to 10-fold increased Purcell factor as compared
to evanescent coupling could be reached alagappanDiamondNanopocketNew2018 .
Efficient integration of ND into silicon nitride photonics is proposed by
using trenched nanobeam cavities. An optimized ratio between quality factor
and mode volume can be achieved by a redistribution of the cavity modes
originating from the presence of the ND alagappanPurcellEnhancementLight2020 .
Paths towards large-scale integration of individual color centers in NDs were
explored and realized by means of lithographically positioned NV centers into
photonic integrated circuits schrinnerIntegrationDiamondBasedQuantum2020 . As
a result the lithographic positioning yields NDs in $70\,\%$ of the targeted
locations with Purcell-enhanced coupling. Optically detected magnetic
resonances (ODMR) was demonstrated while keeping simultaneous control over
several NV center.
One of the most advanced and optimized procedures to date involves
nanophotonic fabrication, for example of 1D PhC cavities, directly in diamond.
All-diamond nanophotonics was pioneered a decade ago
babinecDiamondNanowireSinglephoton2010  and went through rapid progress since
then to establish efficient spin-photon interfaces
burekHighQualityfactorOptical2014 . Nowadays, the technology builds on well-
developed all-diamond devices to realize quantum applications such as memory-
enhanced quantum communication
bhaskarExperimentalDemonstrationMemoryenhanced2020 ;
sipahigilIntegratedDiamondNanophotonics2016 . In order to realize quantum
computation applications, entanglement needs to be distributed among many
nodes. Therefore, in a second fabrication step all-diamond microchiplets can
be integrated into large-scale non-diamond photonics to realize wafer-scale
devices. In pioneering work, defect free arrays with 128 channels of
negatively-charged GeV- and SiV-center in aluminum nitride photonic integrated
circuits were realized wanLargescaleIntegrationArtificial2020 .

In this review, we focus on the emitter to structure approach and, in
particular, on a process chain where no diamond nanofabrication is required.
Instead, all nanofabrication is performed on classical photonics materials
only, here in particular on $\mathrm{Si}_{3}\mathrm{N}_{4}$. The only
requirement for the diamond host is on the size which needs to be small enough
to ensure efficient optical coupling. The discussed hybrid quantum photonics
is based on color centers in NDs, in particular the SiV center, interfaced
with $\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics. We elaborate on the hybrid
device development on the following points:

  1. 1.

The design, simulation and fabrication of photonics based on
$\mathrm{Si}_{3}\mathrm{N}_{4}$-technology.

  2. 2.

The control of SiV centers in NDs and their integration into
$\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics.

  3. 3.

The performance of the final hybrid quantum device.

The investigated platform holds great potential for future quantum technology
due to the following facts:

  1. 1.

The platform profits from all advantages of a well-established photonics
platform with very low propagation loss thereby enabling integration into
large-scale, highly-functional IPCs.

  2. 2.

The quantum properties of color center in NDs, here in particular the SiV
center, can be optimized separately and enables to improve the optical- and
spin-properties even beyond to what is possible in bulk diamond.

  3. 3.

The possibility of complete on chip control enables a user-friendly, high
bandwidth platform on a small footprint with advantages for scalability.

  4. 4.

The diamond host comprises long-lived nuclear spins that can potentially be
used as quantum storage unit in applications that rely on quantum memories.

##  4 Nanophotonic Device Design of $\mathrm{Si}_{3}\mathrm{N}_{4}$-Photonics
towards Optimal Hybrid Integration

This section recapitulates the essential elements of IPCs optimized for its
use in hybrid quantum photonics.

###  4.1 Single-Mode Planar Optical Waveguide

The key element of all IPCs is the planar waveguide. It confines and guides
light within the photonic circuit and ensures the interconnection between the
photonic devices on the chip. The planar waveguide consists of a core with
high refractive index surrounded by a cladding material of lower refractive
index, thus enabling to confine light to propagate in one direction (Fig. 1).
The $\mathrm{Si}_{3}\mathrm{N}_{4}$-platform is one of the most prominent
platforms for the realization of integrated photonic circuits on-chip. It
provides sufficiently high refractive index contrast from
$\mathrm{Si}_{3}\mathrm{N}_{4}$ to $\mathrm{SiO}_{2}$ enabling tight mode
confinement in the wavequide (Fig. 1 a, b) allowing for photonic circuits with
a small footprints. Furthermore, $\mathrm{Si}_{3}\mathrm{N}_{4}$ provides low-
loss broadband optical transparency covering the entire visible range up to
the near infrared spectrum.

Figure 1: Simulated electric field profile of the transverse electric
fundamental (TE0) mode on $\lambda=738\,\mathrm{nm}$ in a single mode (a)
half-etched rib waveguide (width $450\,\mathrm{nm}$, height of rib
$100\,\mathrm{nm}$) and (b) fully-etched ridge waveguide (width
$450\,\mathrm{nm}$, height $200\,\mathrm{nm}$). (c) $\beta-\omega$ dispersion
diagram for a dielectric planar waveguide.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The guided modes in planar waveguide are the solutions of Maxwell’s equations
for particular boundary conditions of the waveguide and materials expressed as
moving planewaves lefebvrePhotoluminescenceImagingSuspended2006 :

$E_{m}=E_{0m}(y,z)e^{{\rm i}\omega t-\beta_{m}x}$ (1)
$H_{m}=H_{0m}(y,z)e^{{\rm i}\omega t-\beta_{m}x},$ (2)

where $E_{0m}(y,z)$ and $H_{0m}(y,z)$ is the profile of the electric and
magnetic fields, $\beta_{m}=\frac{2\pi}{\lambda}n_{\mathrm{eff},m}$ is the
propagation constant and m being the mode index. The modes are discrete, since
they are trapped in the finite volume of the core of the waveguide. Transverse
electric (TE)-modes are the guided modes of the planar waveguide, where the
only non-vanishing fields are $E_{y}\neq 0,H_{x}\neq 0,H_{z}\neq 0$. The
transverse magnetic (TM)-modes have the non-vanishing field components
$H_{y}\neq 0,E_{x}\neq 0,E_{z}\neq 0$. Therefore, there is only one non-
vanishing component in longitudinal direction for each mode of the planar
waveguide ($H_{x}$ for TE-mode and $E_{x}$ for TM mode). It should be noted,
that transverse electric magnetic modes (TEM) ($H_{x}=0$ and $E_{x}=0$) and
hybrid modes ($E_{y}\neq 0$ and $H_{y}\neq 0$) do not exist in planar
waveguides lefebvrePhotoluminescenceImagingSuspended2006 . A typical
$\beta-\omega$ dispersion diagram is illustrated in Fig.1 c) for three
discrete guided modes. The radiation continuous modes are mostly propagating
in the cladding as not-guided modes. The propagation constant and the
effective refractive index of the guided modes should satisfy the following
inequalities

$\frac{\omega}{c}n_{\mathrm{cut-
off}}<\beta_{m}<\frac{\omega}{c}n_{\mathrm{core}}\;,$ (3) $n_{\mathrm{cut-
off}}<n_{\mathrm{eff,m}}<n_{\mathrm{core}}\;.$ (4)

###  4.2 Apodized Grating Coupler

Integrated on-chip photonic elements are interconnected with optical
waveguides. In order to couple light in and out of the chip grating couplers
are used which convert incident Gaussian-like, free-space modes or modes from
an optical fiber into the waveguide mode and vice versa. The periodic
structure of the coupler, shown in Fig. 2, provides out of plane coupling via
diffraction from every tooth of the device due to the Bragg condition
braggReflectionXraysCrystals1913

$\sin\left(\theta_{in}\right)=\frac{n_{\mathrm{eff}}-m\lambda/\Lambda}{n_{\mathrm{clad}}}\;,$
(5)

where $n_{\mathrm{eff}}$ is the effective refractive index of the mode in the
grating, $n_{\mathrm{clad}}$ is the refractive index of the top cladding of
the grating, $m$ is the diffraction number, $\Lambda$ is the period of the
grating and $\lambda$ is the wavelength of the input light.

Figure 2: Scanning electron microscope (SEM) image of an apodized grating
coupler with a period $\Lambda$ (left panel) and zoom-ins in (middle panel)
and (right panel).
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The coupling efficiency is improved by using the recipe of apodization with a
filling factor $ff=a/\Lambda$. Also, an optimization of the length of the
grating coupler’s taper ensures the matching of the fiber-coupled, input mode
of a Gaussian-like free-space mode bozzolaOptimisingApodizedGrating2015 ;
romero-garciaSiliconNitrideCMOScompatible2013 ;
chenApodizedWaveguideGrating2010  and the waveguide mode. Variation of the
period $a$ of the grating is the tuning mechanism for the diffracted central
wavelength. The measured transmission spectrum of the photonic device, shown
in Fig. 3, is recorded for a waveguide terminated by two apodized grating
couplers, as shown in Fig. 2 a).

Figure 3: (a) Measured Normalized Transmission spectrum of a fabricated half-
etched waveguide terminated by two grating couplers with variation of their
periods $\Lambda$. (b) Measured central wavelength as a function of designed
period $\Lambda$ of the coupler.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The grating couplers act as spectral and polarizing filter, which provide an
alignment-free approach to couple light in and out of the photonic circuit by
converting the incident Gaussian-like free space mode into a guided mode
lomonteEfficientSelfimagingGrating2021 . Furthermore, they are fabricated in
the same electron-beam lithography step as the photonic circuit on-chip and
ensure a compact footprint and easy access to any device on the chip.

###  4.3 Mach-Zehnder Interferometer

Figure 4: Mach-Zehnder interferometer on-chip. (a) Optical micrograph of a
fabricated MZI on $\mathrm{Si}_{3}\mathrm{N}_{4}$ chip with path difference
$\Delta l$. (b) Normalized transmission spectrum of a MZI measured with a
supercontinuum white light source, where the outcoupled light is detected with
a spectrometer. The extinction ratio (ER) at $\lambda=750\,\mathrm{nm}$ equals
$\mathrm{ER}\approx 10\cdot(\log{\left(0.07/0.02\right))}=5.4\,\mathrm{dB}$.
This low value is due to a non-ideal 50/50 splitting ratio as a result of non-
optimized splitter and combiner parameters, and because of the low temporal
coherence of the supercontinuum light, i.e. short coherence length.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The Mach-Zehnder interferometer (MZI) is the fundamental element of amplitude
and phase modulators on chip. In particular, thermo-optical (TO) tunable
filters are used, where a thermally induced phase difference results in the
shift of the interference fringes in the frequency space. In a MZI on chip, as
depicted in Fig. 4 a), the input light is equally split by a Y-splitter into
two beams propagating in two waveguide-arms. The waveguide-arms have a certain
length difference and are then recombined on a Y-combiner. The resulting
interference pattern is shown in Fig. 4 b). The fixed path difference $\Delta
l$ between both arms introduces the corresponding phase difference of
$\Delta\varphi(\lambda)=\frac{2\pi}{\lambda}\cdot\Delta l\cdot
n_{\mathrm{eff}}$, where $n_{\mathrm{eff}}$ is the effective refractive index
of the waveguide mode. Accordingly, the intensity at the output port is
modulated as

$\displaystyle I_{1,\mathrm{MZI}}\left(\lambda\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\cdot I_{0}\cdot\left(e^{-{\rm
i}\left(\varphi_{1}+\Delta\varphi\right)}\cdot
e^{-\frac{\alpha}{2}\left(l+\Delta l\right)}+e^{-{\rm i}\varphi_{1}}\cdot
e^{-\frac{\alpha}{2}l}\right)\cdot\left(\mathrm{h.c.}\right)$ (6)
$\displaystyle=$ $\displaystyle\frac{1}{4}\cdot\
I_{0}\cdot\left(2\cdot\cos{\left(\mathrm{\Delta\varphi}\right)}\cdot
e^{-\alpha\left(l+\frac{\mathrm{\Delta l}}{2}\right)}+e^{-\alpha
l}+e^{-\alpha\left(l+\mathrm{\Delta l}\right)}\right),$ (7)

where $\varphi_{1}$ is the phase of the mode in the shortest arm, $\Delta l$
is the length difference between both arms and
$e^{-\frac{\alpha}{2}\left(l+\Delta l\right)}$ and $e^{-\ \frac{\alpha}{2}l}$
accounts for the propagation loss in the arms respectively. In case of
negligible loss, the intensity at the output port becomes

$I_{1,\mathrm{MZI}}\left(\lambda\right)=\frac{1}{2}\cdot\left(1+\cos{\left(\Delta\varphi\left(\lambda\right)\right)}\right)\;.$
(8)

The free spectral range is determined by

$\Delta\lambda=\frac{\lambda^{2}}{n_{\mathrm{gr}}\cdot\Delta l}\;,$ (9)

where $n_{\mathrm{gr}}$ is the group refractive index of the waveguide mode.
The extinction ratio (ER) is the contrast of the interference fringes and
strongly depends on the optical losses and the splitting ratio of the
Y-splitter (ideally 50/50). It can be derived from (7) as

$\mathrm{ER}=\frac{I_{\mathrm{max}}}{I_{\mathrm{min}}}=\left(\frac{1+e^{-\frac{\alpha}{2}\Delta
l}}{1-e^{-\frac{\alpha}{2}\Delta l}}\right)^{2}\;.$ (10)

MZIs are widely used as on chip ultra-fast modulators
thomsonHighPerformanceMach2013 ; samaniSiliconPhotonicMach2019 ;
lomonteSinglephotonDetectionCryogenic2021 ; chenApodizedWaveguideGrating2010
and biosensors.

###  4.4 Mach-Zehnder Interferometer Thermo-Optical Tunable Filter

The realization of all-integrated, scalable photonic circuits with accessible
quantum emitters requires optical filtration of the excitation light with a
high extinction ratio to distinguish the emitted single photons from the
background signal. The aim is to realize a compact and tunable optical filter
which can be integrated in nanophotonic circuits on-chip. Cascading several
filter stages provides an enhancement of the overall filtration depth. The
tunability of the filtering wavelength could be provided by tuning the
refractive index via the thermo-optical effect densmoreCompactLowPower2009 ;
nedeljkovicMidInfraredThermoOpticModulators2014 ;
chuCompactThermoopticSwitches2005 ; espinolaFastLowpowerThermooptic2003 ;
harrisEfficientCompactLow2014 ; wangSOIThermoopticModulator2003  which can be
controlled with local micro-heaters integrated on chip. Such filters can be
integrated in photonic circuits with an evanescently coupled single photon
emitter, such as the SiV-center.

Figure 5: Optical micrographs of thermo-optical single MZIs equipped with (a)
double-meander and (b) double-spiral shaped microheaters on top of both arms
of the MZI with zoom-ins on (c) meander ($\mathrm{width}=0.5\,\umu\mathrm{m}$)
and on (d) spiral-shaped ($\mathrm{width}=3\,\umu\mathrm{m}$) microheaters,
each with a spiral-folded $\mathrm{Si}_{3}\mathrm{N}_{4}$ waveguide beneath.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The here discussed single TO tunable MZIs are equipped with spiral and meander
shaped microheaters. The design enables to simultaneously obtain maximum and
minimum transmission on the desired wavelengths, in particular the wavelength
of the SiV center fluorescence, at $\lambda=740\,\mathrm{nm}$, and the
excitation light, at $\lambda=532\,\mathrm{nm}$, respectively. MZI photonic
devices on-chip consist of two branches, namely long waveguides which are
recombined on a Y-splitter. A dynamic change of the optical path difference
between both arms leads to amplitude tunability of the mode at the necessary
wavelength of the MZI filter. The dynamic change can be obtained either by
altering the effective refractive index of the propagating mode in one of the
arms or by changing the physical length difference $\Delta l$ between both
arms, whereas the last is fixed in the nanophotonic circuits on-chip. The
result is a thermally induced phase shift $\Delta\varphi$ for the light
travelling through one of the arms of the MZI. The waveguide in one of the
branches should experience a temperature change $\Delta T$ which leads to a
change of the effective refractive index of the propagating mode due to the TO
effect. This can be realized by placing a metal microheaters on top of the MZI
branch. To avoid optical absorption loss due to the presence of the metal in
the vicinity of the waveguide, the nanophotonic structures should be covered
with a cladding layer to isolate the propagating optical mode from the atop
metal microheaters. The TO effect provides a rather weak modification of the
refractive index due to the small thermo-optical coefficient of silicon
nitride, $\frac{dn}{dT}\approx{10}^{-5}K^{-1}$
tuThermalIndependentSiliconNitride2012 , which is approximately ten times
lower than in silicon. Therefore, the branches of the MZI should consist of
long waveguides, several $\mathrm{mm}$, in order to significantly change the
phase difference during the heating and thus increase the shift of the
interference fringes of the MZI. The waveguides are folded in spiral design to
concentrate the heat in the center and thus to increase the efficiency of the
heating as well as to provide a compact device footprint. The developed MZI
filters are composed of two Y-splitters and two long waveguides folded in a
spiral pattern, featuring a small path difference. Two types of devices are
realized where the arms of the MZI waveguides are equipped with spiral or
meander design of microheaters as shown in Fig. 5. The metal heater wire is
aligned with the center of the waveguide to maximize the overlap of the heat
concentration with the guided optical mode.
When applying a voltage to the microheater, the area beneath the waveguide is
locally heated, inducing a phase shift $\varphi_{\mathrm{shift}}$ of the
propagating mode in the MZI arm. The phase difference at the output of the
thermo-optical MZI modulator after heating the longer arm can be expressed as
the following:

$\Delta\varphi=\Delta\varphi_{\mathrm{fixed}}+\varphi_{\mathrm{shift}}=\frac{2\pi}{\lambda}\Delta
l\cdot n_{\mathrm{eff}}+\frac{2\pi}{\lambda}(l_{\mathrm{spiral}}+\Delta
l)\cdot\frac{\mathrm{d}n}{\mathrm{d}T}\Delta T\;.$ (11)

Heating of the shorter arm yields the phase difference

$\Delta\varphi=\Delta\varphi_{\mathrm{fixed}}-\varphi_{\mathrm{shift}}=\frac{2\pi}{\lambda}\Delta
l\cdot
n_{\mathrm{eff}}-\frac{2\pi}{\lambda}(l_{\mathrm{spiral}})\cdot\frac{\mathrm{d}n}{\mathrm{d}T}\Delta
T\;,$ (12)

where $\lambda$ is the desired wavelength in vacuum, $\Delta l$ is the
original fixed length difference between the arms, $l_{\mathrm{spiral}}$ is
the length of the spiral-arm waveguide, $n_{\mathrm{eff}}$ is the effective
refractive index of the guided mode, $\frac{\mathrm{d}n}{\mathrm{d}T}$ is the
thermo-optical coefficient of silicon nitride and $\Delta T$ is the induced
temperature change. The induced temperature difference, which corresponds to
the switching power $P_{\pi}$, which is necessary in order to induce a
$\pi$-phase shift, $\varphi_{\mathrm{shift}}=\pi$, (11, 12) is determined as

$\Delta T_{\pi}=\frac{\lambda}{2\cdot
l_{\mathrm{spiral}}\cdot\frac{\mathrm{d}n}{\mathrm{d}T}}\;.$ (13)

The demonstrated thermo-optical MZI filter, which are equipped with
microheaters on top of both arms, make it possible to double the amplitude of
the possible shift of the interference fringes. The design enables to apply
the required voltage to the microheater on top of both arms separately and
independently. Thus, the interference fringes will be either shifted towards
shorter or longer wavelengths.
An important characteristics of the designed device is the optical propagation
loss, which is determined by measuring the transmission of the photonic
devices containing spiral waveguides with variation in length in the range
$1$–$6.3\,\mathrm{mm}$ and covered with a cladding layer on top equipped with
microheaters. The measured average propagation loss amounts to
$0.49\,\mathrm{dB}/\mathrm{mm}$ at $\lambda=766\,\mathrm{nm}$
ovvyanCascadedMachZehnder2016 .

Figure 6: SEM images of (a) meander ($\mathrm{width}=800\,\mathrm{nm}$) and
(b) spiral ($\mathrm{width}=800\,\mathrm{nm}$) architecture microheaters. (c)
Modulated normalized transmission of a single MZI device equipped with double-
spiral shaped (violet curve) and double-meander shaped (green curve) heaters
with identical $800\,\mathrm{nm}$ heater width. The voltage is applied to the
top microheater of each MZI device. The difference in peak positions are
explained by the different switching powers of the spiral- and meander-shaped
microheater MZIs.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The aim is to optimize the configuration for a filter with minimized switching
power $P_{\pi}$ to obtain a phase shift $\varphi_{\mathrm{shift}}=\pi$. A
typical modulated normalized transmission spectrum for the thermo-optical MZI
devices with spiral-(violet curve) and meander-geometry (green curve) of the
microheaters is shown in Fig. 6. For both measurements, the heater width is
$800\,\,\mathrm{nm}$. Only one arm is heated, while the optical input
wavelength is kept fixed at $\lambda=766\,\mathrm{nm}$. The MZI device with
spiral-shaped microheaters shows more than two times lower switching power
$P_{\pi}=14.83\,\mathrm{mW}$ in comparison to the meander-shaped microheater
at $P_{\pi}=35\,\mathrm{mW}$. By optimizing the geometry of the microheaters
and increasing the active length of the waveguide, the switching power can be
further reduced. The most efficient thermo-optical MZI device achieves a
switching power of $P_{\pi}=12.2\,\mathrm{mW}$ with spiral-shaped heaters with
a width of $200\,\mathrm{nm}$ ovvyanCascadedMachZehnder2016 . This design
achieves an ER of $20\,\mathrm{dB}$. When working at
$\lambda=738\,\mathrm{nm}$ the induced temperature change for the considered
MZI is $\Delta T=20\,\mathrm{K}$.

###  4.5 Cascaded Mach-Zehnder Interferometer as Tunable Filter on Chip

Cascaded TO devices, as shown in Fig. 7, can be employed to further improve
the filtration depth. The maximum ER of such a filter can be obtained when the
interference patterns of both MZIs in cascade are perfectly overlapped. This
can be achieved by applying a scanning voltage to one microheater of one of
the cascaded MZIs to thermally shift and align the interference pattern of
this MZI with respect to the interference pattern of the second MZI.

Figure 7: Optical micrographs of cascaded thermo-optical MZI devices equipped
with (a) double-spiral shaped microheater and (c) double-meander shape.
Comparison of the normalized transmission through a single thermo-optical MZI
device and a cascade of two MZIs supplied with a (b) double-spiral shaped
microheater and (d) double-meander shaped. The length of the spiral waveguide
in both MZIs is $1.78\,\mathrm{mm}$.
Copyright declaration: Firstly published in the PhD Thesis of Dr. Anna P.
Ovvyan ”Nanophotonic circuits for single photon emitters”

The modulated normalized transmission of a single thermo-optical MZI and of a
cascaded thermo-optical MZIs equipped with spiral- (a, b) and meander- (c, d)
design microheaters are shown in Fig. 7. An increase of the ER by
$9.6\,\mathrm{dB}$ and by $4.7\,\mathrm{dB}$ is demonstrated for devices with
a spiral-heater and meander-heater, respectively. The cascaded MZI devices
equipped with spiral microheaters are more efficient in terms of power
consumption as compared to meander microheaters. However, the extinction ratio
of both types of devices are similar and independent of the shape since in
both cases the buffer layer covers the nanophotonic devices and prevents from
absorption of the light.
By simultaneously applying two independent scanning electrical voltages to one
of the microheaters of the first MZI (‘external’ loop) and one of the
microheaters of the second MZI (‘internal’ loop) a corresponding phase shift
in each MZI is induced. Respectively, the interference patterns are overlapped
depending on the induced phase shift. Thus, the modulated intensity at the
output of two cascaded MZIs (assuming 50/50 splitting and without accounting
for propagation loss) is determined as

$\displaystyle I_{2,\mathrm{MZI}}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\cdot(4+4\cos\left(\Delta\varphi_{1}\right)+4\cos\left(\Delta\varphi_{2}\right)$
(15)
$\displaystyle+2\cos\left(\Delta\varphi_{1}+\Delta\varphi_{2}\right)+2\cos\left(\Delta\varphi_{1}-\Delta\varphi_{2}\right))\;,$

where $\Delta\varphi_{1}$ is the phase difference in the first MZI and
$\Delta\varphi_{2}$ is the phase difference in the second MZI.
$\Delta\varphi_{1}$ and $\Delta\varphi_{2}$ can be found by (11, 12) and
depend on which arm was heated.

Figure 8: Analytical modulated intensity at the output of two MZIs as a
function of phase difference $\Delta\varphi_{1}$ and $\Delta\varphi_{2}$ of
the first and second MZI, respectively, according to (15). b) Measured
normalized transmission of two cascaded spiral-arm MZIs equipped with spiral-
shape microheaters ($\mathrm{width}=1\,\umu\mathrm{m}$) by employing a voltage
on top one of the arms of first and second MZIs.
Copyright declaration: Firstly published in the PhD Thesis of Dr. Anna P.
Ovvyan ”Nanophotonic circuits for single photon emitters”

The analytically determined modulated transmission is shown in Fig. 8 a),
whereas the measured normalized transmission at the output of the two thermo-
optical cascaded MZI devices is displayed in b), while two scanning voltages
were applied to one of the microheaters of the first MZI and one of the
microheaters of the second MZI simultaneously. The theoretical and
experimental results are in a good agreement, taking into account that the
dissipated power in the microheater is proportional to $\Delta\varphi$.

### Tuning Performance of Single and Cascaded Thermo-Optical MZI Filters

The developed MZI thermo-optical tunable filters can be integrated in hybrid
nanophotonic circuits on-chip with coupled single photon emitters. The
excitation light is blocked while the emitted fluorescence light is
transmitted, simultaneously.

Figure 9: Modulated insertion loss a thermo-optical spiral-folded arms single
MZI equipped with double-spiral architecture microheater with inner radius of
bend $130\,\umu\mathrm{m}$ and $\mathrm{length}=1.78\,\mathrm{mm}$. Pink curve
corresponds to measured transmission at $738\,\mathrm{nm}$, green curve
corresponds to measurement performed at $532\,\mathrm{nm}$.
Copyright declaration: Firstly published in the PhD Thesis of Dr. Anna P.
Ovvyan ”Nanophotonic circuits for single photon emitters”

In particular, the thermo-optical single and cascaded MZIs shown in Fig. 5 and
7 are intended to be utilized as a tunable filter in the nanophotonic circuit
with coupled SiV color centers. Therefore, the filters should enable to
transmit light at a wavelength around $\lambda=738\,\mathrm{nm}$ while cutting
the excitation light at a wavelength of about $532\,\mathrm{nm}$. The
measurements are performed by delivering both laser lights to the thermo-
optical MZI.
The performance of a single thermo-optical MZI filter with an optimized
architecture (narrow spiral-shape microheater with $200\,\mathrm{nm}$ width)
is shown in Fig. 9. By applying $P=33\,\mathrm{mW}$ to the long arm of a
single MZI yields a $10.6\,\mathrm{dB}$ filtration characteristic. The
transmitted signal at a wavelength of $\lambda=532\,\mathrm{nm}$ (green curve
in Fig. 9) reduces to $-17\,\mathrm{dB}$, whereas the transmission at a
wavelength of $738\,\mathrm{nm}$ (red curve in Fig. 9) yields
$-6.4\,\mathrm{dB}$. The experimentally measured ER of one MZI is higher when
coupling light at $\lambda=738\,\mathrm{nm}$ ($\mathrm{ER}=19.3\,\mathrm{dB}$)
than for coupled light at $532\,\mathrm{nm}$ (Fig. 9) since the waveguide is
designed to be single mode for $\lambda=738\,\mathrm{nm}$. However, this
waveguide is not single mode at $\lambda=532\,\mathrm{nm}$ leading to a
degradation of the interference pattern and a decreasing ER.
The filtration characteristic is improved when utilization the cascaded
thermo-optical MZI as shown in Fig. 7 a) equipped with spiral-shaped
microheaters. By applying two scanning voltages to one of the microheaters of
the first MZI and one of the microheaters of the second MZI shows that the
measured transmission of cascaded MZIs is modulated as a function of the
dissipated power on the microheater of the first and second MZI while coupling
light at $\lambda=738\,\mathrm{nm}$ and $\lambda=532\,\mathrm{nm}$
respectively, as shown in Fig. 10. The maximum ER of the cascaded thermo-
optical MZI device reaches $32.9\,\mathrm{dB}$, $13.6\,\mathrm{dB}$ higher
than with a single MZI filter of the same geometry. The ER at
$\lambda=532\,\mathrm{nm}$ reaches $11.35\,\mathrm{dB}$, which is
$3.55\,\mathrm{dB}$ higher than in the case of a single MZI while coupling
light of the same wavelength. Thus, the maximum filtration characteristics is
obtained at the point of maximum transmission of the light at
$738\,\mathrm{nm}$ and suppression of light at $532\,\mathrm{nm}$. For the
explored cascaded thermo-optical MZI this condition is satisfied when the
applied electrical power on the long arm of the second MZI is
$43\,\mathrm{mW}$ and the electrical power on the long arm of the first MZI is
$40\,\mathrm{mW}$. The obtained filtration depth amounts to $20\,\mathrm{dB}$
which exceeds that of a single MZI by $10\,\mathrm{dB}$
ovvyanCascadedMachZehnder2016 . This design could in the future be used to
separate the zero phonon emission line (ZPL) from the excitation laser for on
chip quantum optical experiments.

Figure 10: Transmission of two cascaded thermo-optical MZIs equipped with
spiral-shaped microheaters as a function of dissipated power by employing a
voltage to the microheaters on top of one of the arms of the first and second
MZIs. a) The contour graph shows transmission at $738\,\mathrm{nm}$,
$\mathrm{ER}=32.9\,\mathrm{dB}$. b) Contour graph displays transmission at
$532\,\mathrm{nm}$, $\mathrm{ER}=11.35\,\mathrm{dB}$. Parameters of MZI: width
of heater $200\,\mathrm{nm}$, minimum radius of spiral-folded waveguide
$130\,\umu\mathrm{m}$.
Copyright declaration: Firstly published in the PhD Thesis of Dr. Anna P.
Ovvyan ”Nanophotonic circuits for single photon emitters”

###  4.6 Photonic Crystal Cavity

In 1887 Lord Rayleigh explored that a periodic structure of dielectric
materials, which is characterized by a periodic dielectric permittivity,
introduces a photonic bandgap rayleighXVIIMaintenanceVibrations1887 ;
lomonteSinglephotonDetectionCryogenic2021 , i.e. a continuous frequency range
within which there are no propagating solutions of Maxwell’s equations for any
wave vector. These modes decay exponentially inside the periodic structure
along the way of propagation. The reason for this decay is that light is
partially reflected at each dielectric layer interface. Therefore, these waves
destructively interfere along the propagation way. In contrast, light in the
wavelength range outside the photonic bandgap propagates through the crystal
without scattering, since the wavevector is conserved. Such a photonic
structure is the basis of Bragg mirrors.

### Origin of Resonance Modes

The 1D version of a PhC is well-known as Bragg reflector (mirror) consisting
of air holes in a semiconductor material. The introduction of a defect in such
a Bragg reflector, for instance by changing the spacing between two central
holes, leads to a line-defect cavity which efficiently traps light in the area
of the defect forming a localized mode. The resonance modes with an energy
within the bandgap can be excited. The modes are discrete, since the modes are
localized in a finite region, i.e. between two Bragg mirrors. The modes are
confined by the bandgap in the longitudinal direction and by total internal
reflection in the two transverse directions. Therefore, the state is localized
only in one dimension of the 1D PhC cavity. The superposition of two plane
waves traveling in opposite directions, which are here extended states with an
exponential decay in the PhC, results in a standing wave. Further, one
considers a TE-like standing cavity mode with $E_{y}$ and $H_{z}$ as the major
field components. The magnetic field distribution of the standing wave is
derived as

$H_{z}=\cos{\left(\omega t-kx\right)}+\cos{\left(\omega
t+kx\right)=2\cdot\cos{\left(\omega t\right)}\cdot\cos{\left(kx\right)}}\;,$
(16)

where $\omega$ is the frequency of the resonance mode,
$k=\frac{2\pi}{\lambda}n_{eff}$ is the wavevector. Taking Bragg’s law into
account braggReflectionXraysCrystals1913  $\lambda=2n_{eff}\cdot a$ and the
wavevector $k=\frac{\pi}{a}$ can be calculated with the period $a$ of the PhC.
Magnetic and electric components of the standing wave are shifted by
$\frac{\pi}{2}$ in space and time. Therefore, the electric field distribution
can be determined as

$\displaystyle E_{y}$ $\displaystyle=$ $\displaystyle\cos{\left(\omega
t+\frac{\pi}{2}-\left(kx+\frac{\pi}{2}\right)\right)}+\cos{\left(\omega
t+\frac{\pi}{2}+\left(kx+\frac{\pi}{2}\right)\right)}$ (17) $\displaystyle=$
$\displaystyle 2\cdot\sin{\left(\omega t\right)}\cdot\sin{\left(kx\right)}\;.$
(18)

The planes of oscillations of $H_{z}$ and $E_{y}$ are orthogonal. The Poynting
vector is determined as

$S_{x}=E_{y}\cdot H_{z}=\sin{\left(2\omega
t\right)}\cdot\sin{\left(2kx\right)}\;.$ (19)

and oscillates with twice the frequency in time and space as compared with the
electric and magnetic field distribution of the standing wave.

### Quality Factor of the Cavity

One of the main figures of merit of a PhC cavity is the quality factor $Q$
which characterizes the number of optical round-trips of light inside the
cavity until the energy of the resonance mode decays by
$\exp\left(-2\pi\right)$. In other words, it indicates the lifetime of the
localized mode inside the cavity joannopoulosPhotonicCrystalsMolding2008 .
There are two main damping mechanisms of the resonance modes for a 1D PhC
cavity:

  1. 1.

The coupling of the resonance mode into the radiative mode which extends in
the air being the decay rate $\frac{1}{Q_{\mathrm{rad}}}$

  2. 2.

The decay into the dielectric waveguide $\frac{1}{Q_{\mathrm{wg}}}$.

The Quality factor is therefore determined as

$\frac{1}{Q}=\frac{1}{Q_{\mathrm{wg}}}+\frac{1}{Q_{\mathrm{rad}}}\;.$ (20)

Increasing the number of segments in each mirror of the PhC cavity increases
$Q_{\mathrm{wg}}$. Thus, ideally the $Q$-value of the 1D PhC cavity with two
perfect mirrors of infinite number of periods, saturates and is limited by
only $Q_{\mathrm{rad}}$. Therefore, the main aim is to suppress the radiation
losses of the resonance mode to increase $Q_{\mathrm{rad}}$. In order to
minimize the radiation losses, i.e. by minimizing the leakage of the resonance
modes into the surrounding cladding, here in air, it is necessary that the
spatial tails of the Fourier harmonics of the resonance modes stay out of the
light cone. This can be obtained by special tapering the envelope function of
the electric field of the resonance modes at the cavity edges
akahaneHighQPhotonicNanocavity2003 . A smooth adiabatic transition between the
Bloch and the guided modes can be achieved by Gaussian attenuation
akahaneHighQPhotonicNanocavity2003  of the electric field pattern of the
resonance mode inside the PhC cavity. This can be achieved by tapering the
holes drilled in the waveguide, by tapering the waveguide width or by changing
the spacing of the holes mukherjeeOnestepIntegrationMetal2011 ;
gongNanobeamPhotonicCrystal2010  while keeping the other parameters constant
quanDeterministicDesignWavelength2011 ; zhouRefractiveIndexSensing2014 ;
zhangUltrahighQTETM2009 ; yangHighQHighsensitivityWidthmodulated2015 .
Interestingly, the sinc-function of the electric field envelope enables the
complete illumination of the Fourier components from the light cone
mccutcheonDesignSiliconNitride2008 .

### Local Density of States and Purcell Enhancement

The spontaneous emission rate is affected by the surrounding environment. For
example, the spontaneous emission rate changes when the emitter is placed in
the mode of a cavity, as described by E.M. Purcell and called the Purcell
effect purcellResonanceAbsorptionNuclear1946 . The PhC cavity can enhance or
inhibit the spontaneous emission rate of the emitter depending on its position
in the cavity field and the detuning of the cavity resonance wavelength, which
needs to match the transition wavelength of the emitter, as proved
experimentally huletInhibitedSpontaneousEmission1985 . Fermi’s Golden rule
diracQuantumTheoryEmission1927  states that the spontaneous emission rate is
proportional to the density of states (DOS). The DOS is determined as the
available number of states for a spontaneously emitted photon at a frequency
$\omega^{(n)}$. The DOS is defined by economouGreenFunctionsQuantum2011 ;
lagendijkResonantMultipleScattering1996

$\mathrm{DOS}(\omega)=\sum_{n}{\delta(\omega-\omega^{\left(n\right)}})\;.$
(21)

In case when the cavity resonance is much larger than the emitter’s optical
transition linewidth the cavity resonance can be treated as a continuum while
the mode density is much larger than its value in free space
harocheExploringQuantumAtoms2006 . Therefore, the spontaneous emission rate
can be enhanced by imposing boundary conditions to the field radiated by the
emitter by means of an optical cavity, as long as the cavity mode can be
considered as a continuum. The local density of states (LDOS) is introduced to
characterize the $\mathrm{DOS}(\omega)$ in a specific position $x$ and in a
certain direction $l$, defined by economouGreenFunctionsQuantum2011 ;
lagendijkResonantMultipleScattering1996

${\mathrm{LDOS}}_{l}(x,\omega)=\sum_{n}{\delta(\omega-\omega^{\left(n\right)}})\cdot\varepsilon(x)\cdot\left|E_{l}^{\left(n\right)}\left(x\right)\right|^{2}\;,$
(22)

where $\varepsilon\left(x\right)$ is the dielectric permittivity of the
material and $E_{l}^{\left(n\right)}(x)$ is the field component of the
standing wave inside the cavity. The $DOS(\omega)$ can be determined as

$\mathrm{DOS}\left(\omega\right)=\sum_{l=1}^{3}\int{{\mathrm{LDOS}}_{l}\left(x,\omega\right)\cdot
d^{3}}x\;.$ (23)

The radiated power of the dipole source is proportional to the overlap
integral between the electric field distribution of the resonance mode
localized in the cavity and the electric field of the dipole-source
tafloveAdvancesFDTDComputational2013 . The LDOS is proportional to this
radiated power and can be determined as tafloveAdvancesFDTDComputational2013

${\mathrm{LDOS}}_{l}\left(x_{0},\omega\right)=\frac{4}{\pi}\varepsilon\left(x_{0}\right)P_{l}\left(x_{0},\omega\right)\;,$
(24)

where $P_{l}\left(x_{0},\omega\right)$ is the power radiated by the linearly
polarized source positioned at $x_{0}$. The resonant cavity mode is
approximated with the complex eigenfrequency $\omega=\omega_{0}-{\rm i}\alpha$
, where $\alpha>0$, thus ensuring the exponential decay of the mode in the PhC
mirror. The $Q$-factor of the mode can be determined as $Q=\omega_{0}/2\alpha$
joannopoulosPhotonicCrystalsMolding2008 . On-resonance LDOS at $\omega^{(n)}$
is derived by joannopoulosPhotonicCrystalsMolding2008  as

${\mathrm{LDOS}}_{\mathrm{res}}\approx\frac{2}{\pi\omega^{\left(n\right)}}\cdot\frac{Q^{\left(n\right)}}{V^{\left(n\right)}}\;,$
(25)

where
$V^{(n)}=\frac{\int{\varepsilon\left|E^{(n)}\right|^{2}}}{\max{{\left(\varepsilon\left|E^{(n)}\right|^{2}\right)}}}$
is referred to as the mode volume purcellResonanceAbsorptionNuclear1946 ;
wardCalculatingPhotonicGreen1998  and $Q^{(n)}/V^{(n)}$ is proportional to the
Purcell factor purcellResonanceAbsorptionNuclear1946 .
The resonance mode of the cavity yields a Lorentzian peak for the total power
radiated by the source as well as an LDOS spectrum as shown in
zhangUltrahighQTETM2009 . The enhancement factor is an average of the LDOS
over the emission linewidth. It should be noted that a lossless resonance mode
leads to a delta-function peak in the LDOS spectrum.
In conclusion, the PhC cavity can enhance the spontaneous emission rate of the
emitter positioned in the cavity mode when the spectral, spatial and
polarization overlap is fulfilled, namely matching the emission pattern of the
source with the resonance mode of the cavity. In this case, the Purcell
enhancement factor $F_{\mathrm{Purcell}}$ can be derived as the ratio of
spontaneous emission rate $\Gamma$ of the emitter coupled to the resonant mode
of the cavity, to the free-space emission rate $\Gamma_{0}$
purcellResonanceAbsorptionNuclear1946 ;
coccioliSmallestPossibleElectromagnetic1998

${F_{\mathrm{Purcell}}}=\frac{\mathrm{\Gamma}}{\mathrm{\Gamma}_{0}}=\frac{3}{4\pi^{2}}\cdot\left(\lambda^{(n)}\right)^{3}\cdot\frac{Q^{\left(n\right)}}{V^{\left(n\right)}}$
(26)

where $\lambda^{(n)}=\lambda^{(n)}_{\mathrm{vac}}/n_{\mathrm{mat}}$ is the
resonance wavelength of the mode containing the host material of the emitter,
here diamond, with the refractive index $n_{\mathrm{mat}}$. Equation (26) is
valid when the linewidth of the resonance mode is broader than the transition
linewidth of the emitter. This condition needs to be fulfilled when the source
is positioned in the maximum of the electric field of the resonance mode and
when the emitters transition is matched spectrally and in polarization
coccioliSmallestPossibleElectromagnetic1998 . A tremendous progress is
obtained in the development of the methods to increase the Purcell
enhancement, which can be achieved either by increasing the quality factor
zhangUltrahighQTETM2009 ; quanDeterministicDesignWavelength2011 ;
zhouRefractiveIndexSensing2014 ; witmerDesignNanobeamPhotonic2016 ;
kuramochiUltrahighQOnedimensionalPhotonic2010  or by reducing the mode volume
miuraUltralowModevolumePhotonic2014 .

###  4.7 Freestanding, Cross-Bar Photonic Crystal Cavities

### Investigation of Modulated Photonic Crystals (Bragg Mirror)

A key challenge for many quantum optics experiments is to increase the light-
matter interaction strength. On-chip PhC cavities are promising candidates to
provide strong Purcell enhancement and high coupling efficiencies. PhC
cavities utilize two non-uniform Bragg gratings, consisting of a number of
periodic segments. In between both Bragg gratings a cavity region is
introduced. In the subsequent experiments quantum emitters will be positioned
in this region yielding an efficient light-matter interface.
The first step in engineering the PhC cavities is the determination of the
Bragg mirror’s parameters, which can efficiently confine light in a small
volume. The aim of computational simulations is to design the Bragg mirror
with a bandgap in the required frequency region, while providing low radiation
loss. The radiation losses can be minimized by modulating the refractive index
of the segments. There are distinguished dielectric types of cavities, where
antinodes of the major electric field component, the $E_{y}$-component, lies
in the dielectric cavity region (in $\mathrm{Si}_{3}\mathrm{N}_{4}$). There
are also air type cavity regions, where the antinodes of the major electric
field component, the $E_{y}$-component, lie in the air (holes). In order to
obtain enhancement of the emitted light, the source should be placed in the
antinode of the major electric field of the corresponding mode. Depending on
the application the device needs to be designed and optimized correspondingly.

#### Choice of the Bragg Mirror Parameters.

The width of the waveguide is chosen to produce a single transverse electric
fundamental mode (TE0) in the wavelength range of interest. The thickness is
pre-determined at $200\,\mathrm{nm}$ by the choice of the conventional silicon
nitride-on-insulator wafer. The period $a$ of the air holes can be estimated
according to Bragg’s law braggReflectionXraysCrystals1913  as
$a=\lambda_{0}/(2n_{\mathrm{eff}})$, where $n_{\mathrm{eff}}$ is the effective
refractive index of the mode. When optimizing for the ZPL of the SiV center at
$\lambda=738\,\mathrm{nm}$ a refractive index of $n_{\mathrm{eff}}=1.546$
needs to be considered. According to Bloch-Floquent theory it is sufficient to
determine the bandgap for a single segment of the 1D periodic PhC, which can
be allocated to the uniform PhC with infinite number of such periods. It
should be noted, that a smooth transition of the resonance mode into the
waveguide mode at the edge mirror-waveguide can be insured by modulation of
the segments. Quadratic interpolation of the filling fraction $ff$
quanDeterministicDesignWavelength2011  of the Bragg mirrors segments reduces
radiation losses. The exploration of the geometric parameters for these device
are performed using the free open-source MPB software
johnsonBlockiterativeFrequencydomainMethods2001 . The aim is to ensure that
all the segments in the mirror cover the necessary bandgap range and the
attenuation constant, the so called mirror strength
quanDeterministicDesignWavelength2011 , of the inner segment is zero, while
for the outer segments this values should be maximized.

### Simulations of the Nanobeam and Cross-Bar PhC with an Internal Cavity

One-dimensional PhC cavities enable the direct coupling of an optical
transition to nanophotonic waveguides and thus provide compatibility to
integrated optical circuits. The light-matter interaction can be strongly
enhanced through the Purcell effect by placing a quantum emitter in the cavity
region.

Figure 11: Schematic of a freestanding cross-bar PhC cavity with an
evanescently coupled emitter (yellow star), placed on top of cavity region in
$xy$ plane (a) and $xz$ plane shown on (b). These sketches are obtained as an
output of the performed simulations using the free open-source software MEEP.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

With the help of a novel cross-bar design (Fig. 11) it is possible to
spatially separate the excitation (waveguide in $y$-direction) and emission
(PhC cavity in $x$-direction). The collection efficiency and emission rate
will be enhanced due to the Purcell effect. Thus, the second step of the 3D
simulations is focused on the determination and optimization of high-$Q$
cavity modes with geometric parameters previously determined via the MPB
simulations, as shown in the previous paragraph by employing the open-source
software MEEP oskooiMeepFlexibleFreesoftware2010 ;
tafloveAdvancesFDTDComputational2013 . The Gaussian pulse dipole source with
an $E_{y}$ electric field is positioned in the center of the cavity with the
pulse centered at $\lambda_{\mathrm{res}}$ and with a bandwidth
$\Delta\lambda$. The electric field distributions of the first three resonance
modes of the cross-bar PhC cavity are shown in Fig. 12. The fundamental
resonant mode, also referred to as the I mode, is characterized by a single
central antinode envelope of the $E_{y}$ pattern. For the second mode, further
called II mode, the envelope consists of two antinode wings. The third mode,
further called III mode, shows an envelope with 3 antinode symmetric around
the center.

Figure 12: Simulated Electric field distribution ($E_{y}$ component) for TE-
like fundamental (a), second-order (b) and third-order resonance mode
profiles, supported in the freestanding cross-bar PhC cavity. The electric
field profiles are superimposed with the dielectric permittivity of the
structure. PhC cavity parameters: cavity length $190\,\mathrm{nm}$,
periodicity $a=300\,\mathrm{nm}$ and number of holes on each side $N=53$.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

As can be seen from the simulations, the quality factor of the I mode is
$Q=10^{3}$ which is 50-times smaller than for the III mode $Q=50\cdot 10^{3}$.
This can be partly attributed to the cross-bar geometry. In Fig. 13 the effect
of the cross-bar design is visualized.

Figure 13: Simulated Electric field distribution ($E_{y}$ component) for the
TE-like I mode (a,d), II mode (b,e) and III mode (c,f) resonance mode profiles
supported in the freestanding nanobeam PhC cavity (a,b,c) and the cross-bar
PhC cavity (d,e,f). The electric field profiles are superimposed with the
dielectric permittivity of the structure. PhC cavity parameters: cavity length
$200\,\mathrm{nm}$, periodicity $a=290\,\mathrm{nm}$ and number of holes on
each side $N=33$.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The $Q$-factor of odd resonance modes of the nanobeam PhC is about two times
higher in comparison with the cross-bar PhC cavity, due to the presence of the
crossed waveguide in the middle of the cavity, where there is an antinode of
the electric field envelope of these modes. In contrast, the cross-bar has not
much influence on the $Q$-factor of even modes, since the antinodes of these
modes are concentrated in the wings of the cavity. Note, that the wavelengths
of the corresponding resonant modes are approximately the same for both types
of PhC cavities.

### Simulation and Experimental Investigation of the PhC Cavity Parameters

The fabricated freestanding, cross-bar PhC cavities are shown in Fig. 14. The
devices are terminated with grating couplers, labeled 1 and 3, to perform
transmission measurements, while the couplers labeled 2 and 4 are utilized for
crosstalk measurements.

Figure 14: Freestanding cross-bar PhC cavity. (a) Optical microscope image of
a PhC cavity device. (b) SEM image of a freestanding PhC cavity with zoom-in
in to the cavity region (c), where the green line (cav.) indicates the
designed cavity length, i.e. the distance between two central segments.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The cavity length, indicated as cav in Fig. 14, is the distance between two
central segments and has the highest influence on the $Q$-factor. The behavior
of the III-order mode is explored since it showed the highest quality factor
according to the simulations. The simulation results of the $Q$-factors for
the freestanding cross-bar PhC cavity in dependence on the cavity length are
shown in Fig. 15. The results are in agreement with measured $Q$ values
fehlerEfficientCouplingEnsemble2019 . The highest measured Q-factor is
$Q=47\cdot 10^{3}$ (Figure 15 b)) at the optimal cavity length of
$190\,\mathrm{nm}$, the simulated value for this device is $Q=51\cdot 10^{3}$.
The resonance wavelength increases with an increase of the cavity length.

Figure 15: Experimentally measured (red dots) and simulated (green curve)
dependency of the (a) $Q$-factor and (b) resonance wavelength of the III-order
TE-like resonance mode on the cavity length for a freestanding cross-bar PhC
cavity with the following parameters: period $a=290\,\mathrm{nm}$, each
modulated Bragg mirror consists $N=53$ segments with quadratic tapered holes.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The measured transmission spectrum of the cross-bar PhC cavity with an
optimized cavity length of $190\,\mathrm{nm}$ is displayed in Fig. 16. The
spectrum shown in a) is measured over a broad wavelength range using a
transmission measurement with a supercontinuum light source. The low
resolution ($1.3\,\mathrm{nm}$ FWHM) and low sensitivity spectrometer does not
enable to detect the first three high $Q$-factor resonance modes. The employed
tunable laser, with a limited tuning range of $765-780\,\mathrm{nm}$, enables
to determine the III- and IV-order resonance modes in the transmitted light.
The cavity resonances III- and IV-order peaks with the highest measured
$Q=47\cdot 10^{3}$ are shown in Fig. 16.

Figure 16: Measured transmission spectrum of freestanding cross-bar PhC
cavities with optimized cavity length $cav=190\,\mathrm{nm}$ (red curve),
$195\,\mathrm{nm}$ (blue curve) by coupling supercontinuum light (a), tunable
laser light in the range $765$–$780\,\mathrm{nm}$ (b) with zoom-in into III-
order modes on (c). The index number of the resonance modes is annotated in
the plots. (d) Measured III-order resonance peak (solid curve) and a
Lorentzian fit (dashed curve), yielding $Q=47\cdot 10^{3}$. Parameters of
measured PhC cavity: period $a=290\,\mathrm{nm}$, $N=53$ quadratic tapered
segments.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

### Tuning Mechanisms for Resonance Wavelength

Controlling the spectral position of the resonance mode is important in order
to achieve the maximum coupling strength. Increasing the periodicity, $a$,
leads to a rise of the resonance wavelength based on Bragg’s law, which is
evident in Fig. 17. The derived tuning coefficient of the resonance wavelength
with the periodicity amounts as $\frac{\delta\lambda}{\delta\Lambda}\approx
0.5$–$2$. Furthermore, an increasing cavity length leads to an increase of the
resonance wavelength. The derived tuning coefficient of the resonance
wavelength with the cavity length amounts as
$\frac{\delta\lambda}{\delta{}cav}\approx 0.02$–$0.05$.
Changing the periodicity and cavity length can be utilized as tuning
mechanisms for the resonance wavelength of the developed cross-bar PhC cavity,
unless a too drastic change of these parameters shifts the bandgap out of the
spectral range of interest.

Figure 17: Experimentally measured resonance wavelength of the I mode as
function of cavity length for freestanding cross-bar PhC devices with
different periods as indicated by the corresponding color and label in the
legend of the plot.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

It should be noted, that the width of the tapered end of the cross-bar and the
number of mirror segments of the PhC cavity also impacts the resonance
wavelength of the mode as experimentally investigated in Fig. 18. The increase
of the width of the crossed waveguides leads to an increase of the effective
refractive index in the middle of the cavity and a shift the resonance towards
longer wavelength would be expected. Surprisingly, it leads to decrease of the
resonance wavelength. This can be explained in the following way: An increase
of the tapering width of the cross-bar leads to a rise of the leakage of the
resonance mode into the crossed waveguide for each cycle of the propagation of
the mode inside the cavity, which leads to an increase of the crosstalk.
Therefore, the penetration of the mode inside the PhC Bragg mirrors is
slightly lower, resulting in a reduction of the effective length of the
cavity, assuming that other parameters are kept constant. The resulting
decrease of the resonance wavelength is illustrated in Fig. 18 a). Indeed,
increasing the width of the crossed waveguide leads to a decrease of the
$Q$-factor due to the higher value of the crosstalk.

Figure 18: (a) Experimentally measured dependency of resonance of the II- and
III-order modes on the number of mirror segments N. Colorful curves correspond
to freestanding PhC devices with different widths of cross-bar indicated in
the legend of the plot. PhC cavity parameters: $a=295\,\mathrm{nm}$,
$cav=200\,\mathrm{nm}$. (b) Resonance wavelength as a function of number of
mirror segments $N$, with the mode index being indicated in the plot. PhC
cavity parameters: $a=290\,\mathrm{nm}$, $cav=200\,\mathrm{nm}$, width of
cross-bar $100\,\mathrm{nm}$.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

Another experimentally investigated trend is illustrated in Fig. 18 b). An
increase of the number of mirror segments elevates the resonance wavelength
for II- and III-order modes because of the increase of the effective length of
PhC cavity, as a result of a deeper penetration of the mode into the Bragg
mirrors from both sides of the cavity. At the same time, for higher order
modes this trend is not always maintained, as indicated in Fig. 18 b). Namely,
higher order modes with lower $Q$-factor experience higher radiation loss.
Starting from a certain $N$, a further increase of the number of segments
yields a shorter penetration depth of the resonance mode into the Bragg
mirror, leading to a reduction of the effective length of the PhC cavity and a
decrease in resonance wavelength. As shown in Fig. 18 b) the effect of
shifting the resonance wavelength is stronger for higher index modes. For III-
and IV-order modes, increasing $N$ still leads to a slight rise of the
resonance wavelength. In contrast, with V-, VI- and VII- modes the resonance
shift towards lower wavelength range starts from $N=40$ mirror segments.

##  5 Color Center in Nanodiamonds

In this section, we describe color centers in diamond hosts with a size small
enough to enable hybrid integration. In case of optical interfacing, the size
needs to be small as compared to the wavelength of the coupled optical
transition. The diamond host could be tailored by nanofabrication towards
specific applications. For example, nanocone structures for directional,
enhanced light emission have been fabricated and are also capable for the
pick-and-place method, here done via tungsten tip
jeonBrightNitrogenVacancyCenters2020 . Therefore, an integration into hybrid
quantum photonics devices is possible. However, in the following we focus on
NDs that do not require any a priori diamond nanostructuring. NDs have a long
research history and a broad application range
alkahtaniFluorescentNanodiamondsPresent ;
qinNanodiamondsSynthesisProperties2021 ;
bradacRoomtemperatureSpontaneousSuperradiance2017 ;
juanCooperativelyEnhancedDipole2017 . Early studies focused on NV centers in
NDs demonstrating, for example, single spin manipulation via ODMR
tislerFluorescenceSpinProperties2009 . However, the spectral instabilities and
uncontrolled fluorescence blinking originating from the close proximity to the
diamond surface as well as from low crystalline diamond quality remained a
long-standing challenge. Also, the deterministic creation of single color
centers per one ND is an ongoing challenge. It was the spectral robustness of
symmetry-protected group-IV color center that enabled to overcome above
mentioned limitations. Even smallest nanodiamonds as small as 1.6 nm,
corresponding to approximately 400 carbon atoms, were shown to be capable to
host SiV center with stable but still blinking fluorescence
vlasovMolecularsizedFluorescentNanodiamonds2014  and single SiV center were
demonstrated in NDs of about 10 nm in size
bolshedvorskiiSingleSiliconVacancy2019 . Narrowband, bright, single photon
emission from SiV centers in NDs was shown in NDs with high crystalline
quality. In early work, the NDs were produced by bead assisted sonic
disintegration of polycrystalline chemical vapor deposition films
neuNarrowbandFluorescentNanodiamonds2011 . Later, the color center were
generated by ion implantation into preselected NDs and high-temperature
annealing was employed to reduce the spectral linewidth
takashimaCreationSiliconVacancy2021 . Spectrally stable single photon emission
even in smallest NDs with a spectral width given by the excited state
lifetime, the so-called Fourier-Transform limit, became possible
jantzenNanodiamondsCarryingSiliconvacancy2016 . The ND-production
neuSinglePhotonEmission2011  and treatment was further improved over the
years. The four-line, finestructure of single SiV center in NDs was observed
for the first time in chemical vapour deposition NDs on
iridiumneuSinglePhotonEmission2011  and later in low-strained, surface-
terminated NDs rogersSingleSiVCenters2019 , as shown in Fig. 19 a). Spectral
diffusion on individual lines was basically absent and the linewidth remained
close to the Fourier-limit as well as the peak position remained at a fixed
frequency, as depicted in Fig. 19 b).
But not only the optical properties in NDs improved over the past years. Also
the spin properties can be enhanced to a level beyond what is possible in bulk
diamond. The spin dephasing time for SiV in bulk diamond is limited by rapid
orbital relaxation which is caused by a resonant phonon
processjahnkeElectronPhononProcesses2015 . In NDs, the orbital relaxation can
be prolonged by locally modifying the phonon density of states in the relevant
frequency range. For NDs of size smaller than about $100\,\mathrm{nm}$ an
extended orbital relaxation rate was recently demonstrated. The orbital
$T_{1}$-time is prolonged by up to one order of magnitude as compared to the
best achievable $T_{1}$-time in bulk diamond
klotzProlongedOrbitalRelaxation2022 , see Fig. 19 c) left panel.
Correspondingly, the spin dephasing time was also prolonged, as inferred from
the dip width in coherent population trapping (CPT) experiments
klotzProlongedOrbitalRelaxation2022 , see Fig. 19 c) right panel. In
conclusion, group-IV color center in NDs and, in particular, the SiV center in
NDs became an attractive quantum system with improved optical and spin
properties at moderate temperatures of a few Kelvin.
At the same time, the NDs can be manipulated and positioned by means of a
atomic force microscope (AFM) on the nanometer-scale without loss of their
optical properties. These are the key properties to enable integration into
hybrid quantum photonics. With respect to external coupling, for example by
means of optical integration, a unique feature is the capability to optimize
all degrees of freedom of the optical coupling term including positioning and
dipole orientationhausslerPreparingSingleSiV2019a . In the next chapter, the
detailed aspects of QPP are discussed.

Figure 19: Color center in nanodiamond. (a) The energy level fine structure of
the SiV center in diamond (upper panel). The finestructure was recently
resolved in NDs (lower panel). (b) The spectral stability is measured in
photoluminescence excitation (PLE) spectroscopy over 10 minutes and reveals
that the central PLE peak position remains within the Fourier-Transform limit,
here 108 MHz. (c) The spin properties of SiV center in NDs can be prolonged as
compared to SiV center in bulk diamond due to a change in the phononic density
of states. The orbital relaxation rate (left panel), disclosed by the orbital
$T_{1}$-time, is measured to be up to 8-times longer than the best possible
value for bulk diamond at similar circumstances. Respectively, the spin
dephasing time $T_{2}^{*}$ is also prolonged in NDs (right panel). $T_{2}^{*}$
is inferred from CPT measurements where the linewidth of the CPT dip
extrapolated to zero power yields $T_{2}^{*}$.
Copyright declaration: (a) Reprinted figure with permission from Häußler, S.,
et al. New J. Phys 21, 103047, 2019. Copyright (2019) author(s), licensed
under https://creativecommons.org/licenses/by/3.0/legalcode. (b) Reprinted
figure with permission from Rogers, L. J., et al. Phys. Rev. Applied, 11,
024073, 2019. Copyright (2019) by the American Physical Society. (c) Reprinted
figure with permission from Klotz, M., et al. Phys. Rev. Lett., 128, 153602,
2022. Copyright (2022) by the American Physical Society.

##  6 Quantum-Postprocessing

The emitter-to-structure approach builds on QPP that needs to be precise and
enable high-throughput processing schrinnerIntegrationDiamondBasedQuantum2020
. In pioneering work, single NDs containing NV center were placed with about
100-nanometer-scale accuracy in the near-field of a
$\mathrm{SiO}_{2}$-microdisk resonator barclayCoherentInterferenceEffects2009
. The nanomanipulation was done by using a fiber taper. In other early work,
NDs were placed on the end facet on an optical fiber positioned within the
core region of the fiber ampem-lassenNanomanipulationDiamondbasedSingle2009
in order to detect the NV fluorescence directly through the
fiberschroderFiberIntegratedDiamondBasedSingle2011 . Nowadays, the most
precise method is established by using AFM-based nanomanipulation capable to
realize assemblies with nanometer precision. Pioneering work was done using
home-build nanomanipulators arranging single-NV-containing NDs
vandersarNanopositioningDiamondNanocrystal2009 . On the one hand, methods
which rely on positioning individual NDs have a lower throughput as compared
to, for example, methods that build on covalent bounding
kianiniaRobustDirectedAssembly2016 . On the other hand, these methods have a
very high precision and enable, in principle, a yield of $100\,\%$. Fig. 20 a)
depicts the assembly of linear chains (left) or 2D-arrangements (right) of
color center containing NDs by means of AFM-nanomanipulation. The positioning
is monitored with high-precision by mapping the topology of the corresponding
ND with an AFM (upper panel) and by mapping the emitter position by
fluorescence detection (lower panel). As a prerequisite, the properties of the
color center in the NDs must persist the QPP hausslerPreparingSingleSiV2019a .
As an example, the rotation of a ND can be monitored with high accuracy by
measuring the change in polarization of the four fine-structure transitions of
a SiV center located inside the ND, as depicted in Fig. 20 b). Other important
properties, such as the optical linewidth, the fine-structure splitting and
the orbital relaxation rate remain unaffected from the nanomanipulation
procedure. Therefore, pre-characterized NDs can be pushed into pre-defined
interaction regions and furthermore optimized for ideal coupling, by means of
lateral and dipole alignment. In pioneering work, a tapered silica fiber is
used to create a small reservoir of NDs close to the target position, the so-
called dip-pen technique andersonNearfieldSurfaceEnhanced2008 . Then, in the
final QPP-step, the NDs are moved into the interaction zone by pushing the
cantilever tip in contact mode against the ND. During the whole QPP-step the
ND is not lifted from the target substrate. With this method, highly accurate
QPP of a planar PhC double-heterostructure cavity was realized
barthControlledCouplingSinglediamond2009 . Similarly, color center containing
NDs were pushed over distances of tens of $\umu$m into the interaction region
of a free-standing PhC, as depicted in Fig. 20 c)
fehlerEfficientCouplingEnsemble2019 ;
fehlerPurcellenhancedEmissionIndividual2020 .
The picking and placing of nanoparticles via AFM was established a decade ago.
The method was used to place NV centers in a ND inside a gallium phosphide PhC
membrane schellScanningProbebasedPickandplace2011 . The pick and place
technique was then further developed enabling high-precision positioning and
rotation of NDs. The high spatial precision of the AFM-based nanomanipulation
enables an extremely high accuracy in assembling hybrid devices. As a recent
example, individual NDs hosting SiV centers were positioned inside the hole of
a PhC cavity fehlerHybridQuantumPhotonics2021 , as summarized in Fig. 20 d).
The technological advancements arising from the high-precision positioning are
discussed in the next section.

Figure 20: The quantum postprocessing. (a) Confocal (upper two panels) and AFM
(lower panel) images of randomly distributed, color centers containing NDs
which are put into regular order of a 1D string (left) and a 2D array (right).
(b) During QPP, properties such as the fine-structure, the optical linewidth
and the spin $T_{1}$-times remain unchanged as demonstrated by the example of
a rotation of an SiV containing ND. The polarization patterns change according
to the rotation. (c) Demonstration of the device postprocessing via the
pushing technique. A precharacterized ND is pushed via the AFM
nanomanipulator, operated in contact mode, along several tens of $\umu$m in
order to reach the interaction zone of the PhC cavity. (d) Demonstration of
the device postprocessing via the pick and place technique. Color centers are
precharacterized in cryogenic, confocal microscopy (left panel) and AFM
microscopy (second left panel). Well-suited NDs are then picked and placed in
the interaction zone of the PhC cavity. Here, the ND is placed with high
precision inside the hole of the PhC cavity.
Copyright declaration: (a) Reprinted figure with permission from Rogers, L.
J., et al. Phys. Rev. Applied, 11, 024073, 2019. Copyright (2019) by the
American Physical Society. (b) Reprinted figure with permission from Häußler,
S., et al. New J. Phys 21, 103047, 2019. Copyright (2019) author(s), licensed
under https://creativecommons.org/licenses/by/3.0/legalcode. (c) Adapted
figure with permission from Fehler et al., Nanophotonics, 9(11), 3655-3662,
2020. Copyright (2020) author(s), licensed under
https://creativecommons.org/licenses/by/4.0/legalcode. (d) Reprinted with
permission from Fehler et al., ACS Photonics 8(9), 2635–2641, 2021. Copyright
(2021) author(s), licensed under https://creativecommons.org/licenses/by-nc-
nd/4.0/legalcode

A remaining bottleneck is the slow throughput of the QPP step. The whole QPP
procedure requires a proper pre-characterization, a highly-accurate
positioning step followed by a post-characterization. A large boost in
efficiency can be expected from incorporating machine learning ideas.
Recently, a 100-fold speedup in single photon emitter classification via
second-order autocorrelation was demonstrated
kudyshevRapidClassificationQuantum2020 .

##  7 Hybrid Quantum Devices Based on
$\mathrm{Si}_{3}\mathrm{N}_{4}$-Photonics Post-Processed with Color Center in
Nanodiamonds

In this last section, we summarize recent developments in hybrid quantum
photonics building on $\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics devices, as
outlined in section 4, and postprocessed with color center in NDs, as outlined
in section 5 and 6. In particular, cross-bar PhC cavities are discussed. We
discriminate two different strategies to establish hybrid quantum photonics
devices. First, optical interfacing achieved by evanescent coupling. This
method was used extensively for different systems in the past. Yet, it is
inherently limited in the achievable coupling strength. Second, the optical
interfacing by embedded coupling. This method puts heavy demands on the QPP-
step but offers ideal performance. Before we discuss both strategies we give a
full characterization of the bare cross-bar PhC cavities.

###  7.1  Full Characterization of a Cross-Bar PhC Cavity for an Evanescently
Coupled Emitter

In this subsection, we discuss the analysis of most important parameters of
the cross-bar PhC cavities which were developed and used for the experimental
realization of hybrid quantum photonic circuits with integrated color centers
in NDs.

### Analysis of the Fundamental Mode of the Cross-Bar PhC Cavity

The cross-bar PhC cavities were developed to enhance the light-matter
interaction, in particular for the SiV emission with a ZPL at
$\lambda=738\,\mathrm{nm}$, and optimized for evanescent coupling to the
fundamental modes as shown in Fig. 21. Further fine tuning of the resonance
wavelength can be obtained by slight variation of the Bragg mirror segments’
period, as discussed in section 4.

Figure 21: (a) SEM image of a freestanding cross-bar PhC cavity. Simulated
Electric field profile ($E_{y}$ component) of the TE-like resonance modes,
superimposed with the cross-bar PhC cavity’s dielectric permittivity for the
I-, II-, III-order resonance modes in part (b–d) respectively. The resonance
wavelength and $Q$-factor are annotated on the plot. PhC cavity parameters:
cavity length $195\,\mathrm{nm}$, periodicity $a=280\,\mathrm{nm}$, quadratic
tapered $N=43$ segments.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The first mode is further explored and its magnetic field ($H_{z}$-component),
Poynting vector $S$ and energy density distributions are determined via extra
cycles of three dimensional Finite Difference Time Domain (3D-FDTD)
simulations. The results are displayed in Fig. 22. However, these simulations
are carried out with a reduced number of mirror segments to decrease
computation time.

Figure 22: Simulation results. (a) Electric field distribution ($E_{y}$
component). (b) Magnetic field distribution ($H_{z}$ component). (c) Poynting
vector distribution. (d) Energy density distribution at the moment of time
when electric energy density component is maximum. (e) Energy density
distribution at the moment of time when magnetic energy density component is
maximum. (f) Analytical Electric field distribution ($E_{y}$ component). All
distributions, superimposed with dielectric permittivity of the structure, of
the fundamental TE-like resonance mode for freestanding cross-bar PhC cavity
with the following parameters: cavity length $195\,\mathrm{nm}$, periodicity
$a=280\,\mathrm{nm}$, quadartic tapered $N=43$ segments.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

Important aspects of the explored electro-magnetic field, Poynting vector and
energy distributions are highlighted for the fundamental resonance mode of the
cross-bar PhC cavity which are evident in Fig. 22.

  * •

The Gaussian envelope $E_{y}=\sin\left(\frac{\pi
x}{a}\right)\cdot\sin{\left(\omega t\right)}\cdot{\exp{\left(-\sigma
x^{2}\right)}}$ of the electric field pattern of the I-order mode is the
result of quadratic tapering of the filling fraction of the segments in each
Bragg mirror from both sides of the cavity. This leads to a suppression of the
mismatch between the cavity and waveguide modes and is ensured by a smooth
transition of the effective refractive index of the latter. This decreases the
radiation loss due to the suppression of the tails of the Fourier harmonics of
the resonance mode inside the light cone.

  * •

The electric $E_{y}$ and magnetic $H_{z}$ field components are reversed, i.e.,
there is an antinode in the $H_{z}$ distribution in the center of cavity as
opposed to the $E_{y}$ pattern. This can be explained by the nature of the
standing wave, where the energy is passed back and forth between the region
where the electric field has a maximum (magnetic field has a node) and the
region where the magnetic field has a maximum (electric field has a node) with
zero energy net stream in both directions, as shown in Fig. 22 d), e). The
standing wave does not transfer energy, since it is the superposition of two
waves traveling in opposite directions, i.e. carry energy in opposite
directions.

  * •

The Poynting vector of the standing wave

$S_{x}=E_{y}\cdot H_{z}=\sin{\left(2\omega
t\right)\cdot\sin{\left(2kx\right)\cdot\exp{\left(-2\sigma x^{2}\right)}}}$
(27)

oscillates at twice the frequency of the corresponding $E_{y}$ and $H_{z}$
field components. It is important that the Poynting vector shows an impact of
both magnetic and electric components at one time, in contrast with the energy
distribution for the standing wave; the time-average of the Poynting vector is
zero.

  * •

The envelope of the energy and the Poynting vector distributions described by
$\exp{\left(-2\sigma x^{2}\right)}$ decays two times faster in comparison with
the electric and magnetic fields.

### The Spatial Map of the Local Density of States Enhancement

According to the Purcell theory, strongly enhanced light-matter interaction
with the emitter is provided when placing the source in the antinode of the
electric field. Furthermore, the emitter’s wavelength and polarization needs
to match the resonance mode of the PhC cavity. Thus, the field pattern is the
first indication where strong interaction with the emitter would occur. The
Purcell enhancement factor approximates the single-resonant LDOS enhancement,
where the latter one is proportional to the overlap integral between the
electric field distribution of the resonance mode of the cavity and the
electric pattern of the source. 3D-FDTD simulations are carried out using the
MEEP software oskooiMeepFlexibleFreesoftware2010 ;
tafloveAdvancesFDTDComputational2013  to compute the LDOS enhancement of an
emitter placed on the cross-bar PhC cavity region. Therefore, the position
($x_{0}$,$y_{0}$,$z_{0}$) of a Gaussian pulse point-dipole – with a major
field component of $E_{y}$ – on the cavity is varied and the corresponding
excitation of the TE-like resonance modes is evaluated. Since the convolution
of the electric field along the $x$-axis is the strongest, mainly the LDOS
spectra dependence on the $x$-axis position of the dipole is considered. The
relative LDOS enhancement spectrum is studied which is equal to the ratio of
the LDOS with the cross-bar PhC cavity compared to the LDOS without any
scattering elements (without PhC cavity). The normalized LDOS spectrum for
different positions of the dipole on top of the cavity region along the
$x$-axis is shown in Fig. 23. Again, these results are evaluated for a PhC
cavity with a reduced number of mirror segments, than in the actual fabricated
device, since the LDOS calculation needs to run for a full period of the
cavity lifetime $2Q/f_{\mathrm{res}}$. The reduction of segments leads to a
decrease of the computation time at a price of a decreased quality-factor,
i.e. linewidth of the resonance peaks.

Figure 23: Simulated LDOS spectrum enhancement of the light emitted from a
dipole placed on top of the freestanding cross-bar PhC cavity, where the
dipole position along the longitudinal direction is varied (a–i). The
positions are annotated in terms of dipole shift from the center along the
$x$-axis in Fig. 11, weighted in units of $a$. The freestanding cross-bar PhC
cavity has the following parameters: cavity length $195\,\mathrm{nm}$,
quadartic tapered $N=18$ segments, periodicity $a=280\,\mathrm{nm}$,.
Analytical Electric field distribution ($E_{y}$ component).
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The spatial map of the simulated LDOS enhancement for odd and even resonance
modes is illustrated in Fig. 24 and allows to sum up the results in a more
conclusive way.

  * •

The centered position of the emitter (position 1 in Fig. 24) inhibits the
coupling into odd modes, since there is a node in the electric field pattern
of the corresponding odd resonance modes (Fig. 23) and at the same time
provides only slight coupling to even resonance modes, due to a weak antinode
in the corresponding electric field distribution. Thus, peaks corresponding to
odd modes are missing in the LDOS enhancement spectrum of Fig.23 a).

  * •

Shifting the emitter away from the center of the cavity along the PhC leads to
an increase in the enhancement for odd modes, while decreasing the coupling to
even ones. The LDOS obtains a maximum for a shift of the emitter of $0.4a$
(position 5 in Fig. 24) with respect to the symmetry plane of the center of
the cavity, since the dipole is successfully coupled to the antinode of the
corresponding odd resonance modes. Coupling into even modes is inhibited
fehlerEfficientCouplingEnsemble2019 .

  * •

The LDOS enhancement factor of the coupling into odd modes is higher in
comparison to even modes and is attributed to a higher $Q$-factor and also to
a lower mode volume of odd modes for this particular PhC cavity.

Figure 24: (a) Simulated LDOS spectrum enhancement of the light emitted from a
dipole placed on top of the freestanding cross-bar PhC cavity, where the
dipole position is varied along the longitudinal direction ($x$-axis in Fig.
11). The positions are indicated in terms of the shift of the dipole from the
center along the $x$-axis in Fig. 11 and are weighted in units of $a$. The
colorful curves correspond to the resonance modes labeled in roman numbers
(I-V) in the plot. (b) SEM image of cavity region of the fabricated and
measured freestanding PhC cavity. The points indicate the position of the
dipole for which the LDOS enhancement spectrum was computed in a) labeled with
the corresponding numbers of the position. The PhC cavity has the following
parameters: cavity length $195\,\mathrm{nm}$, quadratic tapered $N=18$
segments, periodicity $a=280\,\mathrm{nm}$.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

#### Factors Impacting the LDOS Spectra.

The effect of the cross-bar on the LDOS is explored by 3D-FDTD simulations.
The cross-bar mostly influences the fundamental resonance mode, which is
expected from the mode’s electric field distribution. In particular, it
decreases the enhancement (about 5 times) and shifts the resonance wavelength
(about $3\,\mathrm{nm}$) to longer wavelengths as shown in Fig. 25 a). The
influence of the cross-bar on the resonance conditions of the higher order
modes is comparatively small.

Figure 25: (a) Simulated LDOS enhancement of emitted light from a dipole
placed on top of the cavity region of a freestanding cross-bar (red curve) and
freestanding nanobeam (blue curve) PhC cavity with identical parameters. (b)
Simulated LDOS enhancement of emitted light from the dipole placed on top of
cavity (red curve) and inside the center of the cavity (green curve) of a
freestanding cross-bar PhC cavity. (c) Simulated LDOS enhancement of emitted
light from the dipole placed on top of freestanding cross-bar PhC cavity in
the center of the nanoguide (red curve), shifted along $y$-direction on
$0.25w=112.5\,\mathrm{nm}$ (light green curve) and shifted along $y$-direction
on $0.5w-10\,\mathrm{nm}=215\,\mathrm{nm}$ (violet curve), where $w$ is the
width of the waveguide. In all considered simulations of this figure the
dipole was shifted to optimized distance $0.4a$ along the $x$-axis.
Freestanding cross-bar PhC cavity has the following parameters: cavity length
$195\,\mathrm{nm}$, quadratic tapered $N=18$ segments, periodicity
$a=280\,\mathrm{nm}$.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

The theoretically maximum enhancement factor of the spontaneous emission rate
is obtained when the source is embedded inside the PhC cavity leading to an
increase of the enhancement factor of all modes by $\approx 2$ times as shown
in Fig. 25 b). It is important to note, that another consequence of this
optimized position is the suppression of the enhancement of the light coupled
into the radiation modes due to the incomplete bandgap of the 1D PhC. The
effect of the shift of the source from the optimized position along the
$x$-axis is explained in the previous paragraph. While moving the source away
from the center position along the width of the waveguide ($y$-axis) the LDOS
decreases. A shift of $0.25w$ ($=112.5\,\mathrm{nm}$, where $w$ \- width of
waveguide of $w=450\,\mathrm{nm}$) decreases the LDOS by 1.8 times. Placing
the emitter at the edge of the waveguide (shift of the dipole of
$0.5a-10\,\mathrm{nm}=215\,\mathrm{nm}$, violet curve) leads to a decrease of
the LDOS enhancement of about 3 times in comparison with the central position.

#### Considerations Related to the Experimental Coupling of Single Photon
Emitters to the Modes of the PhC Cavity.

In conclusion, the maximal enhancement factor of the spontaneously emitted
light coupled into odd resonance modes of the developed cross-bar PhC cavity
is obtained in case of the $0.4a$ shifted dipole along the longitudinal
direction from the symmetry plane in the center of the cavity (position 5 in
Fig. 24). The fabricated freestanding cross-bar PhC devices consist of
$N=30$–$50$ mirror segments on each side of the cavity, leading to a drastic
increase of the $Q$-factor and the LDOS enhancement factor in comparison with
the computed values for a PhC cavity with $N=15$. It should be noted, that the
LDOS enhancement spectra are computed for the case when the polarization of
the source is perfectly aligned with the field of the TE-like resonance modes
($E_{y}$). However, in reality the discrepancy related with the positioning
and the alignment of the emitter with respect to the polarization of the
resonance modes will decrease observed enhancement factor.

### The Coupling Efficiency of the Emitter into Resonance Modes of the Cross-
Bar PhC Cavity Expressed by the $\beta$-Factor.

A main figure of merit for an emitter coupled to the cavity mode is the
spectrally resolved $\beta$-factor. The coupling efficiency, expressed by the
$\beta$-factor, correlates with the LDOS factor. Thus, it strongly depends on
the spatial position of the emitter in the cavity region. The $\beta$-factor
is the ratio between the amount of spontaneous emitted light (Poynting vector
flux) coupled to the cavity and the total amount of spontaneous emitted light
at this wavelength.

$\beta=\frac{\Gamma_{\mathrm{PhC}}}{\Gamma_{\mathrm{PhC}}+\Gamma_{\mathrm{free}}}$
(28)

where $\Gamma_{\mathrm{PhC}}$ is the emission rate into the cavity mode and
$\Gamma_{\mathrm{free}}$ is the free-space emission of the uncoupled emitter
at the same wavelength. The $\beta$-factor is determined via 3D-FDTD
simulations using the MEEP software, resulting in a wavelength-dependent
$\beta$-factor. The wavelength-dependent coupling efficiency of the emitted
light into the resonance modes of the freestanding cross-bar PhC and further
into the waveguide is illustrated in Fig. 26 for a source placed in the
optimal position (shift of $0.4a$ along the longitudinal direction) on the
cavity . For comparison, the enhancement of the coupling efficiency on
resonance is also computed into the waveguide mode of cross-bar waveguides
with identical geometry but without the PhC (dashed green line in Fig. 26 a)).

Figure 26: (a) Simulated $\beta$-factor transmission coupling efficiency of
emitted light into resonance modes and further longitudinal guided modes of
freestanding cross-bar PhC cavity for the dipole positioned on top of the
cavity region and shifted $0.4a$ away from the symmetry plane along the PhC.
The dashed line shows the $\beta$-factor efficiency of coupling emitted light
into waveguide mode for an identical geometry cross-bar freestanding waveguide
without the PhC. (b) Simulated Transmission spectrum through the considered
cross-bar PhC cavity. (c) Simulated LDOS enhancement spectrum of emitted light
from the considered dipole. Simulated freestanding cross-bar PhC cavity has
the following parameters: cavity length $195\,\mathrm{nm}$, $N=10$ segments,
periodicity $a=280\,\mathrm{nm}$.
Copyright declaration: First published in the PhD Thesis of Dr. Anna P. Ovvyan
“Nanophotonic circuits for single photon emitters” DOI: 10.5445/IR/1000093929

###  7.2 Hybrid Quantum Nanophotonics Established by Evanescent Optical
Coupling

Purcell enhancement of coherent photons can be achieved by resonantly
enhancing the ZPL emission via evanescent coupling of a quantum emitter to
photonic structures. In pioneering work, the ZPL of single NV centers was
enhanced by placing the ND in the interaction region of microdisks
santoriNanophotonicsQuantumOptics2010  and PhC cavities
woltersEnhancementZeroPhonon2010 ; woltersCouplingSingleNitrogenvacancy2012 .
In this subsection, we summarize two recent developments on the evanescent
coupling of color center in NDs to the mode of a high-Q,
$\mathrm{Si}_{3}\mathrm{N}_{4}$ PhC cavity. First, we discuss the efficient
coupling of an ensemble of NV center fehlerEfficientCouplingEnsemble2019 , as
summarized in Fig. 27 a). Second, we discuss the Purcell-enhanced emission of
individual transitions of single SiV center
fehlerPurcellenhancedEmissionIndividual2020 , as summarized in Fig. 27 b).
On the one hand, $\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics is a desired
technology due to excellent photonic parameters such as low-photon loss. On
the other hand, its strong background fluorescence prohibited its use in the
quantum optics regime. In the first experiment
fehlerEfficientCouplingEnsemble2019 , a cross-bar PhC cavity, as introduced in
the previous section, was used to couple an ensemble of NV centers to
individual cavity modes. In this configuration, the background fluorescence
was successfully suppressed by about 20 dB. Based on that breakthrough the
observation of an efficient coupling with a $\beta_{\lambda}$-factor of 0.71
was realized, meaning that more than 2/3 of the photons, which are
spontaneously emitted at a wavelength of $762.1\,\mathrm{nm}$, are directed
into the cavity mode. The overall $\beta$-factor reached 0.14 corresponding to
one out of seven photons, which are spontaneously emitted at any wavelength,
are coupled into the $\mathrm{Si}_{3}\mathrm{N}_{4}$-waveguide. This
corresponds to an overall Purcell-factor $F_{P}=\beta/(1-\beta)$ of 0.16 which
is a reasonably promising number given that it corresponds to the enhancement
averaged over all emitters in the ensemble.
The fact that individual emitters, which have a dipole well aligned with the
cavity mode, can have a much higher Purcell-enhancement was demonstrated in
successive work fehlerPurcellenhancedEmissionIndividual2020 . NDs, now
containing SiV center, were pushed into the interaction zone of the PhC
cavity. Overall, a Purcell enhancement of more than 4 on individual optical
transitions was achieved, in line with four out of five spontaneously emitted
photons guided into the photonic device. Also, the finestructure of individual
SiV center was observed within the mode envelope. To reach this goal the
overlap between the SiV center’s dipole and the cavity electric field was
repeatedly optimized by AFM nanomanipulation. Furthermore, the PhC resonance
condition was tuned to achieve spectral overlap of the emitter ZPL and the
cavity mode. However, the expected lifetime-shortening due to the Purcell
effect to approximately 340 ps was not yet observed.

Figure 27: Hybrid quantum photonics based on evanescently coupled color
centers in NDs. (a) An ND containing an ensemble of NV center is pushed in the
interaction zone of the cavity (upper left panel). The fluorescence coupled to
the cavity mode is measured and its polarization-dependence matches the
polarization properties of the cavity-waveguide system (upper right panel).
The fluorescence signal is strongly enhanced when the system is resonant with
the cavity modes yielding a $\beta_{\lambda}$-factor of 0.71 while the
$\beta$-factor reaches 0.14 averaged over the full ensemble of NV centers. (b)
Resonant coupling of the ZPL of SiV centers enables to resolve the fine-
structure of individual centers. Individual optical transitions are strongly
enhanced reaching Purcell-factors of 4.
Copyright declaration: (a) Reprinted with permission from Fehler et al., ACS
Nano, 13(6) 6891-6898, 2019. Copyright (2019) American Chemical Society, (b):
Reprinted figure with permission from Fehler et al., Nanophotonics, 9(11),
3655-3662, 2020. Copyright (2020) author(s), licensed under
https://creativecommons.org/licenses/by/4.0/legalcode

###  7.3 Hybrid Quantum Nanophotonics Established by Embedded Optical Coupling

The light-matter coupling strength can be further increased by optimizing the
overlap of the cavity field with the position of the quantum emitter. Ideally,
the emitter is embedded directly into the photonics material at the position
of the mode field maximum. With respect to color centers in diamond, this
means that the nanophotonics material needs to be diamond, capable to host the
color centers. Engineering photonic devices out of diamond for that purpose
was pioneered more than a decade ago. The ZPL of NV centers was resonantly
enhanced by fabrication ring resonators in diamond
faraonResonantEnhancementZerophonon2011  and PhC cavities in monocrystalline
diamond faraonCouplingNitrogenVacancyCenters2012 . Integrated nanophotonics
was realized by coupling single NV center to all-diamond ring resonators which
are coupled to all-diamond waveguides that include grating couplers
hausmannIntegratedDiamondNetworks2012 . The research field rapidly evolved
over the following years and is nowadays among the most promising platforms
for the realization of quantum applications such as quantum repeater and
quantum networks bhaskarExperimentalDemonstrationMemoryenhanced2020 .
Recently, suspended PhC cavities in polycrystalline diamond were engineered to
reach 2.5-fold ZPL enhancement of SiV center. Here, the spectral overlap
between the SiV ZPL and the cavity mode was realized without the need of any
post-processing taking advantage of the non-homogeneous thickness of the
diamond layer ondicPhotonicCrystalCavityenhanced2020 . The high precision of
nanomanipulation, as discussed in the previous sections, now makes embedded
coupling also accessible for hybrid quantum systems.
The idea of hybrid embedded coupling is, that instead of placing the ND
outside the photonic device it is integrated into the photonics material. The
photonics material itself does not need to be diamond, in contrast to the all-
diamond devices. Hybrid embedded coupling can, for example, be achieved by
establishing pre-defined nanopockets alagappanDiamondNanopocketNew2018 ;
frochPhotonicNanobeamCavities2020 . Therefore, a very precise nanomanipulation
is required. In the work fehlerHybridQuantumPhotonics2021  the precision of
the QPP was pushed to an extreme by placing a ND, which contains SiV centers,
inside the hole of a PhC cavity by using the AFM-based pick and place
technique. As a result, the corresponding coupling strength got another boost
from an improved Purcell-enhancement and, at the same time, from waveguiding
effects. By employing further optimization steps, such as two-mode coupling
and precise cavity-resonance tuning, the resulting photon flux was increased
by 14-times as compared to free space. For the first time, also the
corresponding lifetime shortening to below 460 ps was directly observed, only
limited by the technical time-resolution of the setup. The results are
summarized in Fig. 28. Therefore, the hybrid quantum photonics platform based
on embedded coupling is among the few spin-based photonics platforms with an
operation bandwidth beyond GHz-rates. The platform is now at a stage to strive
the realization of spin-based quantum applications.

Figure 28: Hybrid quantum photonics based on embedded optical coupling. (a) An
ND containing SiV center is positioned with high precision in the hole of a
PhC cavity. Three cases are compared to demonstrate the advantage of embedded
coupling. 1. The ND in free space. 2. The ND placed inside one hole of a bare
waveguide, studying only the effect of waveguiding. 3. The ND placed in the
hole of a PhC cavity where signal enhancement arises from waveguiding together
with Purcell-enhancement (upper panel). The simulated improvement of the
$\beta_{\lambda}$-factor and the normalized power is plotted as function of
the displacement of the ND from the center of the waveguide (lower panel). (b)
The experimentally determined PL-enhancement is shown as function of the
detuning from two modes. The strongest enhancement of more than 14-times is
reached when the optical transition is in resonance with the composite mode,
comprised of coupling to both, the $2^{nd}$ and $3^{rd}$, mode. (c) Lifetime
shortening for the next order composite mode is plotted as function of
detuning (left panel). Significant lifetime reduction is reached when the
optical transition is in resonance with the cavity mode. The shortest recorded
lifetime reached $460\,\mathrm{ps}$, close to the resolution limit of the
detection system marked as gray shaded area (right panel).
Copyright declaration: (a–c) Reprinted with permission from Fehler et al., ACS
Photonics 8(9), 2635–2641, 2021. Copyright (2021) author(s), licensed under
https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode

##  8 Summary

In this chapter, we summarized the performance of hybrid quantum photonics
based on $\mathrm{Si}_{3}\mathrm{N}_{4}$-devices postprocessed with NDs
containing color centers, in particular NV and SiV-center. We discuss
scalable, low-loss, integrated, $\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics
elements which are necessary for the realization of modern, hybrid quantum
photonics on-chip lombardiPhotostableMoleculesChip2018 ;
elshaariHybridIntegratedQuantum2020 ; khasminskayaFullyIntegratedQuantum2016 ;
wangIntegratedPhotonicQuantum2020 ; pyatkovCavityenhancedLightEmission2016 ;
khasminskayaWaveguideIntegratedLightEmittingCarbon2014 ;
thomsonHighPerformanceMach2013 ; pelucchiPotentialGlobalOutlook2022 ;
wanLargescaleIntegrationArtificial2020 ;
uppuQuantumdotbasedDeterministicPhoton2021 ;
machielseQuantumInterferenceElectromechanically2019  and programmable photonic
circuits feldmannAllopticalSpikingNeurosynaptic2019 ;
feldmannParallelConvolutionalProcessing2021 ;
bogaertsProgrammablePhotonicCircuits2020 ;
dongHighspeedProgrammablePhotonic2022 ; borregaardOneWayQuantumRepeater2020 .
Designed, fabricated, experimentally and computationally fully investigated
1D, freestanding, cross-bar PhC cavities integrated on chip allow us to
simultaneously solve three outstanding challenges, namely:

  1. 1.

Individual, optical transitions of the emitter can efficiently be coupled to
the high-quality, small-volume PhC cavity resonance modes.

  2. 2.

The established pump-probe design enables simultaneous collection of enhanced
fluorescence through the cavity-waveguide channel, while the excitation of the
emitter is realized via crossed waveguides. Thus, both pump and read-out ports
are on-chip and are accessible for additional on-chip photonic technology.

  3. 3.

The cross-bar design ensures spatial separation of the excitation and the
enhanced emission of the coupled emitter in the cavity region, enabling strong
suppression ($\approx-20\,\mathrm{dB}$) of the pump waveguide cross-talk and
correspondingly strong suppression of background fluorescence.

Experimentally efficient integration of NV and SiV centers into the developed
cross-bar PhC cavities and enhancement of the spontaneous emission rate is
demonstrated in the works fehlerEfficientCouplingEnsemble2019 ;
fehlerPurcellenhancedEmissionIndividual2020 ; fehlerHybridQuantumPhotonics2021
. We explore the LDOS enhancement spatial map and determine an optimized
position for the emitter in the cavity region in order to obtain a maximum
enhancement of the emission coupled into certain resonance modes. The insights
were taken into account during the positioning of the NDs into the cavity
interaction region. The simulated coupling efficiency, evaluated coupling
strengths and enhanced spontaneous emission rate of the photon source in
resonance with the PhC and with further waveguide modes of the integrated
photonic device resulted in simulated $\beta$-factors of
$\beta_{\lambda}=75\%$, close to the experimental results of
$\beta_{\lambda}=81\%$ fehlerPurcellenhancedEmissionIndividual2020 .
We elaborate on the potential of the platform in quantum photonics and as
spin-photon interface. Our hybrid device enables coherent optical control with
GHz-bandwidth, access to long-lived electron- and nuclear spin as well as on-
chip control. Ongoing device optimization is required to unfold the full
potential of the platform and includes the following steps:

  * •

Improve the Q-factor of the hybrid photonic devices. In particular, the high-Q
of the bare PhC needs to persist the QPP. For example, when placing a ND in
the interaction zone towards evanescent coupling the Q-factor dropped from
$Q_{\mathrm{bare}}=1000$ for the bare cavity to $Q_{\mathrm{hybrid}}=480$ for
the established hybrid platform fehlerPurcellenhancedEmissionIndividual2020 .
In order to reach higher Q-factors, the scattering induced by the NDs needs to
be reduced. Consequently, much higher Purcell enhancement beyond 500 becomes
feasible.

  * •

Establish color centers in tailored NDs. One path of reducing the scattering
induced by the NDs is to create color centers with good spectral and spin
properties in tailored NDs. Tailoring could mean to use much smaller NDs on
the order of 10 nm or NDs with controlled shape. This challenging task
requires further developments on the material side. Controlling the size and
shape of the ND not only reduces the amount of scattering. It also makes the
influence arising from introducing a particle with different refractive index
into the photonic device more predictable. Therefore, the photonic designs
become more reliable and can be optimized for best performance of the hybrid
device instead of the bare photonic device. For example, system imperfections
such as scattering or a changed refractive index could be taken into account.

  * •

Optimize all degrees of freedom of the coupling term. Optimizing all degrees
of freedom of the coupling term (spatial, spectral and polarization alignment)
unfolds the full potential of the quantum photonic devices. In particular, the
positioning and dipole alignment can, in principle, be fully controlled by
means of nanomanipulation. Therefore, the overall system cooperativity can be
pushed to its theoretical limits.

  * •

Speed-up the QPP towards a high-throughput technology. In order to make the
complete fabrication chain efficient, current bottlenecks in terms of
fabrication time need to be speed up. In particular, the QPP is a slow process
and needs a boost in processing time in order to be compatible with the large-
scale fabrication of $\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics. Ideas to
improve large-scale integration include lithographically positioned color
centers into photonic integrated circuits
schrinnerIntegrationDiamondBasedQuantum2020  or emitter precharacterization
supported by machine learning kudyshevRapidClassificationQuantum2020  as
discussed in this chapter.

We believe that hybrid quantum photonics based on
$\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics and color centers in NDs is a
promising route to establish quantum photonics in a low-loss, high-throughput,
on-chip platform. Essential elements of such integrated photonic circuits
include tunable filters. Cascaded MZI tunable filters equipped with
microheaters on both arms of the MZI to provide a thermo-optical effect are
discussed in this chapter. The filter enables to transmit a desired wavelength
at the interference maximum and, at the same time, blocks the excitation light
at the interference minimum. The developed spiral-type micro-heaters ensure
the lowest switching power of $P_{\pi}=12.2\,\mathrm{mW}$ in comparison with
meander-shaped heaters, which compare favorably with previous reported results
for $\mathrm{Si}_{3}\mathrm{N}_{4}$-platforms. Designed tunable filters are
multifunctional and enable to transmit and block the desired wavelengths in a
wide wavelength range. Therefore, such circuits can be used in a large variety
of future applications in hybrid quantum photonics. We show the transmission
of light at $\lambda=740\,\mathrm{nm}$, according to the SiV fluorescence, and
the corresponding suppression of the excitation light at
$\lambda=532\,\mathrm{nm}$, where cascading two stages of the MZI improves the
filtration depth to obtain $\mathrm{ER}=34.7$–$36.5\,\mathrm{dB}$. The
complete control on-chip will enable to scale the platform to many qubits
interconnected by many photonic channels which are operated with quantum
states of light. At this point the full capabilities of
$\mathrm{Si}_{3}\mathrm{N}_{4}$-photonics can be exploited.
The established coupling and strong Purcell-enhancement of individual
transitions of color centers in NDs is a prerequisite for on-chip quantum
optics experiments. The efficient funneling of single photons into the
photonic waveguides enables in the near future to route those photons through
complex photonic structures. Operating the platform in the quantum regime
enables the realization of emerging quantum optics applications in an
integrated fashion leeIntegratedSinglePhoton2020 . Applications include long-
distance quantum communication and on-chip quantum computing
kokLinearOpticalQuantum2007  as well as quantum photonic networks
ozaydinDeterministicPreparationStates2021 . Furthermore, inherent nuclear
spins within the diamond host residing in close proximity to the color center
could be deployed as long-lived quantum memory. Such storage of quantum
information would prove useful as a photonic memory
kokLinearOpticalQuantum2007  or in applications like quantum repeaters
munroQuantumRepeaters2015  or quantum networks
bhaskarExperimentalDemonstrationMemoryenhanced2020 ;
rufQuantumNetworksBased2021  as well as spin-based quantum technology in
general awschalomQuantumTechnologiesOptically2018 ;
atatureMaterialPlatformsSpinbased2018 .

## Appendix

## Acknowledgements

The summarized projects were funded by the Baden-Württemberg Stiftung in
project Internationale Spitzenforschung, the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) in Project No. 398628099, the BMBF and VDI
in Verbundsproject Q.Link.X and QR.X. AK acknowledges support from IQst.

## References

  * (1) P.J. Winzer, D.T. Neilson, A.R. Chraplyvy, Opt. Express, OE 26(18), 24190 (2018).  DOI 10.1364/OE.26.024190
  * (2) F. Poletti, N.V. Wheeler, M.N. Petrovich, N. Baddela, E. Numkam Fokoua, J.R. Hayes, D.R. Gray, Z. Li, R. Slavík, D.J. Richardson, Nature Photon 7(4), 279 (2013).  DOI 10.1038/nphoton.2013.45
  * (3) S. Bogdanov, M.Y. Shalaginov, A. Boltasseva, V.M. Shalaev, Opt. Mater. Express 7(1), 111 (2017).  DOI 10.1364/OME.7.000111
  * (4) N. Kinsey, M. Ferrera, V.M. Shalaev, A. Boltasseva, J. Opt. Soc. Am. B 32(1), 121 (2015).  DOI 10.1364/JOSAB.32.000121
  * (5) E. Murray, D.J.P. Ellis, T. Meany, F.F. Floether, J.P. Lee, J.P. Griffiths, G.A.C. Jones, I. Farrer, D.A. Ritchie, A.J. Bennett, A.J. Shields, Appl. Phys. Lett. 107(17), 171108 (2015).  DOI 10.1063/1.4935029
  * (6) T.K. Paraïso, T. Roger, D.G. Marangon, I. De Marco, M. Sanzaro, R.I. Woodward, J.F. Dynes, Z. Yuan, A.J. Shields, Nat. Photon. 15(11), 850 (2021).  DOI 10.1038/s41566-021-00873-0
  * (7) O. Benson, Nature 480(7376), 193 (2011).  DOI 10.1038/nature10610
  * (8) J.H. Kim, S. Aghaeimeibodi, J. Carolan, D. Englund, E. Waks, Optica 7(4), 291 (2020).  DOI 10.1364/OPTICA.384118
  * (9) P.E. Barclay, K.M. Fu, C. Santori, R.G. Beausoleil, Opt. Express, OE 17(12), 9588 (2009).  DOI 10.1364/OE.17.009588
  * (10) D. Englund, B. Shields, K. Rivoire, F. Hatami, J. Vučković, H. Park, M.D. Lukin, Nano Lett. 10(10), 3922 (2010).  DOI 10.1021/nl101662v
  * (11) H. Siampour, O. Wang, V.A. Zenin, S. Boroviks, P. Siyushev, Y. Yang, V.A. Davydov, L.F. Kulikova, V.N. Agafonov, A. Kubanek, N.A. Mortensen, F. Jelezko, S.I. Bozhevolnyi, Nanophotonics 9(4), 953 (2020).  DOI 10.1515/nanoph-2020-0036
  * (12) S.I. Bogdanov, M.Y. Shalaginov, A.S. Lagutchev, C.C. Chiang, D. Shah, A.S. Baburin, I.A. Ryzhikov, I.A. Rodionov, A.V. Kildishev, A. Boltasseva, V.M. Shalaev, Nano Lett. 18(8), 4837 (2018).  DOI 10.1021/acs.nanolett.8b01415
  * (13) H. Siampour, S. Kumar, V.A. Davydov, L.F. Kulikova, V.N. Agafonov, S.I. Bozhevolnyi, Light Sci Appl 7(1), 61 (2018).  DOI 10.1038/s41377-018-0062-5
  * (14) H. Siampour, S. Kumar, S.I. Bozhevolnyi, ACS Photonics 4(8), 1879 (2017).  DOI 10.1021/acsphotonics.7b00374
  * (15) H. Siampour, S. Kumar, S.I. Bozhevolnyi, Nanoscale 9(45), 17902 (2017).  DOI 10.1039/C7NR05675C
  * (16) V.M. Shalaev, S. Kawata, _Nanophotonics with surface plasmons_ (Elsevier, 2006).  DOI 10.1016/B978-0-444-52838-4.X5000-0
  * (17) D.K. Gramotnev, S.I. Bozhevolnyi, Nature Photon 4(2), 83 (2010).  DOI 10.1038/nphoton.2009.282
  * (18) V.V. Vorobyov, V.V. Soshenko, S.V. Bolshedvorskii, J. Javadzade, N. Lebedev, A.N. Smolyaninov, V.N. Sorokin, A.V. Akimov, Eur. Phys. J. D 70(12), 269 (2016).  DOI 10.1140/epjd/e2016-70099-3
  * (19) A.W. Schell, G. Kewes, T. Schröder, J. Wolters, T. Aichele, O. Benson, Review of Scientific Instruments 82(7), 073709 (2011).  DOI 10.1063/1.3615629
  * (20) A. Huck, S. Kumar, A. Shakoor, U.L. Andersen, Phys. Rev. Lett. 106, 096801 (2011).  DOI 10.1103/PhysRevLett.106.096801
  * (21) A. Huck, U.L. Andersen, Nanophotonics 5(3), 483 (2016).  DOI 10.1515/nanoph-2015-0153
  * (22) S.K.H. Andersen, S. Bogdanov, O. Makarova, Y. Xuan, M.Y. Shalaginov, A. Boltasseva, S.I. Bozhevolnyi, V.M. Shalaev, ACS Photonics 5(3), 692 (2018).  DOI 10.1021/acsphotonics.7b01194
  * (23) R. Waltrich, B. Lubotzky, H. Abudayyeh, E.S. Steiger, K.G. Fehler, N. Lettner, V.A. Davydov, V.N. Agafonov, R. Rapaport, A. Kubanek, New J. Phys. 23(11), 113022 (2021).  DOI 10.1088/1367-2630/ac33f3
  * (24) G. Alagappan, L.A. Krivitsky, C.E. Png, ACS Omega 3(5), 4733 (2018).  DOI 10.1021/acsomega.8b00139
  * (25) G. Alagappan, L.A. Krivitsky, C.E. Png, J. Opt. 22(2), 025401 (2020).  DOI 10.1088/2040-8986/ab66cd
  * (26) P.P.J. Schrinner, J. Olthaus, D.E. Reiter, C. Schuck, Nano Lett. 20(11), 8170 (2020).  DOI 10.1021/acs.nanolett.0c03262
  * (27) T.M. Babinec, B.J.M. Hausmann, M. Khan, Y. Zhang, J.R. Maze, P.R. Hemmer, M. Lončar, Nature Nanotech 5(3), 195 (2010).  DOI 10.1038/nnano.2010.6
  * (28) M.J. Burek, Y. Chu, M.S.Z. Liddy, P. Patel, J. Rochman, S. Meesala, W. Hong, Q. Quan, M.D. Lukin, M. Lončar, Nat Commun 5(1), 5718 (2014).  DOI 10.1038/ncomms6718
  * (29) M.K. Bhaskar, R. Riedinger, B. Machielse, D.S. Levonian, C.T. Nguyen, E.N. Knall, H. Park, D. Englund, M. Lončar, D.D. Sukachev, M.D. Lukin, Nature 580(7801), 60 (2020).  DOI 10.1038/s41586-020-2103-5
  * (30) A. Sipahigil, R.E. Evans, D.D. Sukachev, M.J. Burek, J. Borregaard, M.K. Bhaskar, C.T. Nguyen, J.L. Pacheco, H.A. Atikian, C. Meuwly, R.M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, M.D. Lukin, Science 354(6314), 847 (2016).  DOI 10.1126/science.aah6875
  * (31) N.H. Wan, T.J. Lu, K.C. Chen, M.P. Walsh, M.E. Trusheim, L. De Santis, E.A. Bersin, I.B. Harris, S.L. Mouradian, I.R. Christen, E.S. Bielejec, D. Englund, Nature 583(7815), 226 (2020).  DOI 10.1038/s41586-020-2441-3
  * (32) J. Lefebvre, D.G. Austing, J. Bond, P. Finnie, Nano Lett. 6(8), 1603 (2006).  DOI 10.1021/nl060530e
  * (33) W.H. Bragg, W.L. Bragg, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 88(605), 428 (1913).  DOI 10.1098/rspa.1913.0040
  * (34) A. Bozzola, L. Carroll, D. Gerace, I. Cristiani, L.C. Andreani, Opt. Express, OE 23(12), 16289 (2015).  DOI 10.1364/OE.23.016289
  * (35) S. Romero-García, F. Merget, F. Zhong, H. Finkelstein, J. Witzens, Opt. Express, OE 21(12), 14036 (2013).  DOI 10.1364/OE.21.014036
  * (36) X. Chen, C. Li, C.K.Y. Fung, S.M.G. Lo, H.K. Tsang, IEEE Photonics Technology Letters 22(15), 1156 (2010).  DOI 10.1109/LPT.2010.2051220
  * (37) E. Lomonte, E. Lomonte, E. Lomonte, F. Lenzini, F. Lenzini, F. Lenzini, F. Lenzini, W.H.P. Pernice, W.H.P. Pernice, W.H.P. Pernice, W.H.P. Pernice, Opt. Express, OE 29(13), 20205 (2021).  DOI 10.1364/OE.428138
  * (38) D.J. Thomson, F.Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J.M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, G.Z. Mashanovich, IEEE Journal of Selected Topics in Quantum Electronics 19(6), 85 (2013).  DOI 10.1109/JSTQE.2013.2264799
  * (39) A. Samani, E. El-Fiky, M. Morsy-Osman, R. Li, D. Patel, T. Hoang, M. Jacques, M. Chagnon, N. Abadía, D.V. Plant, Journal of Lightwave Technology 37(13), 2989 (2019).  DOI 10.1109/JLT.2019.2908655
  * (40) E. Lomonte, M.A. Wolff, F. Beutel, S. Ferrari, C. Schuck, W.H.P. Pernice, F. Lenzini, Nat Commun 12(1), 6847 (2021).  DOI 10.1038/s41467-021-27205-8
  * (41) A. Densmore, S. Janz, R. Ma, J.H. Schmid, D.X. Xu, A. Delâge, J. Lapointe, M. Vachon, P. Cheben, Opt. Express, OE 17(13), 10457 (2009).  DOI 10.1364/OE.17.010457
  * (42) M. Nedeljkovic, S. Stanković, C.J. Mitchell, A.Z. Khokhar, S.A. Reynolds, D.J. Thomson, F.Y. Gardes, C.G. Littlejohns, G.T. Reed, G.Z. Mashanovich, IEEE Photonics Technology Letters 26(13), 1352 (2014).  DOI 10.1109/LPT.2014.2323702
  * (43) T. Chu, H. Yamada, S. Ishida, Y. Arakawa, Opt. Express, OE 13(25), 10109 (2005).  DOI 10.1364/OPEX.13.010109
  * (44) R. Espinola, M. Tsai, J. Yardley, R. Osgood, IEEE Photonics Technology Letters 15(10), 1366 (2003).  DOI 10.1109/LPT.2003.818246
  * (45) N.C. Harris, Y. Ma, J. Mower, T. Baehr-Jones, D. Englund, M. Hochberg, C. Galland, Opt. Express, OE 22(9), 10487 (2014).  DOI 10.1364/OE.22.010487
  * (46) X. Wang, J. Liu, Q. Yan, S. Chen, J. Yu, Chin. Opt. Lett., COL 1(9), 527 (2003)
  * (47) X. Tu, J. Song, T.Y. Liow, M.K. Park, J.Q. Yiying, J.S. Kee, M. Yu, G.Q. Lo, Opt. Express, OE 20(3), 2640 (2012).  DOI 10.1364/OE.20.002640
  * (48) A.P. Ovvyan, N. Gruhler, S. Ferrari, W.H.P. Pernice, J. Opt. 18(6), 064011 (2016).  DOI 10.1088/2040-8978/18/6/064011
  * (49) L. Rayleigh, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24(147), 145 (1887).  DOI 10.1080/14786448708628074
  * (50) J.D. Joannopoulos (ed.), _Photonic Crystals: Molding the Flow of Light_ , 2nd edn. (Princeton University Press, Princeton, 2008).  DOI 10.2307/j.ctvcm4gz9
  * (51) Y. Akahane, T. Asano, B.S. Song, S. Noda, Nature 425(6961), 944 (2003).  DOI 10.1038/nature02063
  * (52) I. Mukherjee, G. Hajisalem, R. Gordon, Opt. Express, OE 19(23), 22462 (2011).  DOI 10.1364/OE.19.022462
  * (53) Y. Gong, B. Ellis, G. Shambat, T. Sarmiento, J.S. Harris, J. Vučković, Opt. Express, OE 18(9), 8781 (2010).  DOI 10.1364/OE.18.008781
  * (54) Q. Quan, M. Loncar, Opt. Express, OE 19(19), 18529 (2011).  DOI 10.1364/OE.19.018529
  * (55) J. Zhou, H. Tian, D. Yang, Q. Liu, L. Huang, Y. Ji, J. Opt. Soc. Am. B, JOSAB 31(8), 1746 (2014).  DOI 10.1364/JOSAB.31.001746
  * (56) Y. Zhang, M.W. McCutcheon, I.B. Burgess, M. Loncar, Opt. Lett., OL 34(17), 2694 (2009).  DOI 10.1364/OL.34.002694
  * (57) D. Yang, H. Tian, Y. Ji, Appl. Opt., AO 54(1), 1 (2015).  DOI 10.1364/AO.54.000001
  * (58) M.W. McCutcheon, M. Lončar, Opt. Express, OE 16(23), 19136 (2008).  DOI 10.1364/OE.16.019136
  * (59) E.M. Purcell, H.C. Torrey, R.V. Pound, Phys. Rev. 69(1-2), 37 (1946).  DOI 10.1103/PhysRev.69.37
  * (60) R.G. Hulet, E.S. Hilfer, D. Kleppner, Phys. Rev. Lett. 55(20), 2137 (1985).  DOI 10.1103/PhysRevLett.55.2137
  * (61) P.A.M. Dirac, N.H.D. Bohr, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 114(767), 243 (1927).  DOI 10.1098/rspa.1927.0039
  * (62) E.N. Economou, _Green’s Functions in Quantum Physics_ (Springer, Berlin; London, 2011).  DOI 10.1007/3-540-28841-4
  * (63) A. Lagendijk, B.A. van Tiggelen, Physics Reports 270(3), 143 (1996).  DOI 10.1016/0370-1573(95)00065-8
  * (64) S. Haroche, J.M. Raimond, _Exploring the Quantum: Atoms, Cavities, and Photons_ (OUP Oxford, 2006).  DOI 10.1093/acprof:oso/9780198509141.001.0001
  * (65) A. Taflove, A. Oskooi, S. Johnson, _Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology_ (2013)
  * (66) A.J. Ward, J.B. Pendry, Phys. Rev. B 58(11), 7252 (1998).  DOI 10.1103/PhysRevB.58.7252
  * (67) R. Coccioli, M. Boroditsky, K.W. Kim, Y. Rahmat-Samii, E. Yablonovitch, IEE Proceedings - Optoelectronics 145(6), 391 (1998).  DOI 10.1049/ip-opt:19982468
  * (68) J.D. Witmer, J.T. Hill, A.H. Safavi-Naeini, Opt. Express, OE 24(6), 5876 (2016).  DOI 10.1364/OE.24.005876
  * (69) E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y.G. Roh, M. Notomi, Opt. Express, OE 18(15), 15859 (2010).  DOI 10.1364/OE.18.015859
  * (70) R. Miura, S. Imamura, R. Ohta, A. Ishii, X. Liu, T. Shimada, S. Iwamoto, Y. Arakawa, Y.K. Kato, Nat Commun 5, 5580 (2014).  DOI 10.1038/ncomms6580
  * (71) S.G. Johnson, J.D. Joannopoulos, Opt. Express, OE 8(3), 173 (2001).  DOI 10.1364/OE.8.000173
  * (72) A.F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson, Computer Physics Communications 181(3), 687 (2010).  DOI 10.1016/j.cpc.2009.11.008
  * (73) K.G. Fehler, A.P. Ovvyan, N. Gruhler, W.H.P. Pernice, A. Kubanek, ACS Nano 13(6), 6891 (2019).  DOI 10.1021/acsnano.9b01668
  * (74) S.W. Jeon, J. Lee, H. Jung, S.W. Han, Y.W. Cho, Y.S. Kim, H.T. Lim, Y. Kim, M. Niethammer, W.C. Lim, J. Song, S. Onoda, T. Ohshima, R. Reuter, A. Denisenko, J. Wrachtrup, S.Y. Lee, ACS Photonics 7(10), 2739 (2020).  DOI 10.1021/acsphotonics.0c00813
  * (75) M.H. Alkahtani, F. Alghannam, L. Jiang, A. Almethen, A.A. Rampersaud, R. Brick, C.L. Gomes, M.O. Scully, P.R. Hemmer, Nanophotonics 7(8), 1423 (2018).  DOI 10.1515/nanoph-2018-0025
  * (76) J.X. Qin, X.G. Yang, C.F. Lv, Y.Z. Li, K.K. Liu, J.H. Zang, X. Yang, L. Dong, C.X. Shan, Materials & Design 210, 110091 (2021).  DOI 10.1016/j.matdes.2021.110091
  * (77) C. Bradac, M.T. Johnsson, M. van Breugel, B.Q. Baragiola, R. Martin, M.L. Juan, G.K. Brennen, T. Volz, Nat Commun 8(1), 1205 (2017).  DOI 10.1038/s41467-017-01397-4
  * (78) M.L. Juan, C. Bradac, B. Besga, M. Johnsson, G. Brennen, G. Molina-Terriza, T. Volz, Nature Phys 13(3), 241 (2017).  DOI 10.1038/nphys3940
  * (79) J. Tisler, G. Balasubramanian, B. Naydenov, R. Kolesov, B. Grotz, R. Reuter, J.P. Boudou, P.A. Curmi, M. Sennour, A. Thorel, M. Börsch, K. Aulenbacher, R. Erdmann, P.R. Hemmer, F. Jelezko, J. Wrachtrup, ACS Nano 3(7), 1959 (2009).  DOI 10.1021/nn9003617
  * (80) I.I. Vlasov, A.A. Shiryaev, T. Rendler, S. Steinert, S.Y. Lee, D. Antonov, M. Vörös, F. Jelezko, A.V. Fisenko, L.F. Semjonova, et al., Nature nanotechnology 9(1), 54 (2014).  DOI 10.1038/nnano.2013.255