# Triangular Quantum Photonic Devices with Integrated Detectors in Silicon
Carbide

Sridhar Majety 111Corresponding author<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, University of California, Davis, CA
95616, USA  Stefan Strohauer  Walter Schottky Institut, School of Natural
Sciences, and MCQST, Technische Universität München, Am Coulombwall 4, 85748
Garching, Germany  Pranta Saha  Department of Electrical and Computer
Engineering, University of California, Davis, CA 95616, USA  Fabian
Wietschorke  Walter Schottky Institut, School of Computation, Information and
Technology, and MCQST, Technische Universität München, Am Coulombwall 4, 85748
Garching, Germany  Jonathan J. Finley  Walter Schottky Institut, School of
Natural Sciences, and MCQST, Technische Universität München, Am Coulombwall 4,
85748 Garching, Germany  Kai Müller  Walter Schottky Institut, School of
Computation, Information and Technology, and MCQST, Technische Universität
München, Am Coulombwall 4, 85748 Garching, Germany  Marina Radulaski
Department of Electrical and Computer Engineering, University of California,
Davis, CA 95616, USA

###### Abstract

Triangular cross-section SiC photonic devices have been studied as an
efficient and scalable route for integration of color centers into quantum
hardware. In this work, we explore efficient collection and detection of color
center emission in a triangular cross-section SiC waveguide by introducing a
photonic crystal mirror on its one side and a superconducting nanowire single
photon detector (SNSPD) on the other. Our modeled triangular cross-section
devices with a randomly positioned emitter have a maximum coupling efficiency
of $89\text{\,}\mathrm{\char 37\relax}$ into the desired optical mode and a
high coupling efficiency (> $75\text{\,}\mathrm{\char 37\relax}$) in more than
half of the configurations. For the first time, NbTiN thin films were
sputtered on 4H-SiC and the electrical and optical properties of the thin
films were measured. We found that the transport properties are similar to the
case of NbTiN on SiO2 substrates, while the extinction coefficient is up to
$50\text{\,}\mathrm{\char 37\relax}$ higher for $1680\text{\,}\mathrm{nm}$
wavelength. Finally, we performed Finite-Difference Time-Domain simulations of
triangular cross-section waveguide integrated with an SNSPD to identify
optimal nanowire geometries for efficient detection of light from TE and TM
polarized modes.

##  1 Introduction

Color centers in silicon carbide (SiC) have been a prominently studied quantum
platform for applications in quantum information processing (QIP). Their
unique combination of spectral homogeneity, emission at near infra-red and
telecommunication wavelengths, long spin coherence, and spin-photon entangling
processes offer key applications in quantum communication, simulation,
computing, and sensing [1, 2, 3, 4, 5, 6, 7].

The optical response of a color center has a sharp peak called the zero phonon
line (ZPL) and a broad phonon side band (PSB). Emission through ZPL, when
there is no dephasing, results in indistinguishable photon creation necessary
for QIP applications. Integration of color centers with photonic devices is an
important step toward their performance improvement and scalability, as shown
in explorations with SiC pillars [8, 9], waveguides [10, 11], and micro- and
nano-resonators [11, 12, 13, 14, 15]. Moreover, recent results on growth and
patterning of superconducting NbN on 3C-SiC [16, 17] for use as the
superconducting nanowire single photon detectors (SNSPDs) indicates that
quantum circuitry can be further miniaturized and optimized.

The challenges in maintaining the pristine quality of color centers have led
photonic integration away from the established nanofabrication processes and
toward alternative approaches that require non-standard sample preparation
[18, 13, 11] or angled etching [19, 20, 10]. The latter, in particular, has
been explored for its potential in wafer-scale processing [21]. Angle etching
produces triangularly shaped devices whose photonic modes have been recently
studied in terms of waveguide propagation [22, 10], photonic crystal band gap
formation [23], photonic crystal cavity resonances [22], and grating coupler
performance [24].

Figure 1: Illustration of SNSPD integrated onto a photonic device with a color
center. Single color center is positioned in the waveguide region such that
the dipole emission reflected by the photonic crystal mirror constructively
interferes with the dipole emission into the waveguide region. The cloud
region in the waveguide represents the segment where the pump light is
filtered (e.g. by a ring oscillator or an inversely designed structure) to
prevent any SNSPD saturation. Single photon detection is performed by the
SNSPDs through absorption of the dipole emission in the waveguide. (Inset) The
cross-section view of the structure, showing the color center (blue) optimally
positioned at the centroid of the triangular waveguide profile.

In this paper, we expand on the studies of triangular SiC photonics by
focusing on methods to optimize collection and detection of color center light
by integration with a photonic crystal mirror (PCM) and an SNSPD, as
illustrated in Figure 1. In particular, we study the case of the nitrogen
vacancy (NV) in 4H-SiC with emission wavelengths in the range
$1176\text{\,}\mathrm{nm}1243\text{\,}\mathrm{nm}$ [25, 26, 27]. When placed
in a symmetric waveguide, at least half of the light is lost in an undesired
direction, but this can be circumvented by the addition of a PCM. PCM has a
periodic variation of refractive index, which results in the formation of
photonic band gaps. Understanding the formation of these band gaps is
necessary for applications in selective reflection of the ZPL and PSB
emissions with the goal of enhancing light collection efficiency [23]. The
other side of the waveguide deals with the efficient detection through hybrid
integration with NbTiN SNSPDs. The SNSPDs are the leading detectors in terms
of quantum efficiency, low dark counts and low timing jitter. Moreover, they
offer an on-chip detection that overcomes the losses involved in the off-chip
detection, thus boosting the success rate of QIP protocols [28]. However,
depending on the type of excitation, the pump laser used for an off-resonant
excitation of the color center can scatter into the detector region and
saturate the SNSPD. There have been successful demonstrations to suppress the
stray pump laser by coating the backside of the chip with an absorbing
material [29] and masking the inactive regions of the chip [30]. To further
suppress the pump laser coupled into the waveguide, spectral filters like ring
resonators [31, 32] and inverse-designed structures [33, 34] can be integrated
in the waveguide region before the detectors, with implementation in the
illustrated cloud segment in Figure 1.

We first find the 1D photonic crystal (PC) conditions for band gap formation
in triangular geometry on the desired wavelength domain (NV in 4H-SiC in this
work) using the Plane Wave Expansion (PWE) and the Finite-Difference Time-
Domain (FDTD) methods. We then utilize the optimal PC design as a PCM in a
triangular waveguide and study the role of color center positioning in
obtaining high emission coupling to the propagating optical mode. Finally, in
a combination of experimental and modeling approaches, we study the triangular
waveguide mode absorption by a NbTiN SNSPD.

##  2 Photonic crystals for efficient propagation of color center emission in
a triangular waveguide

Color centers have an optical dipole-like emission covering a solid angle of
4$\pi$ [10]. This causes $50\text{\,}\mathrm{\char 37\relax}$ higher than in
spin-entangled photon loss in the triangular cross-section waveguide resulting
in low collection efficiency. An in situ mirror, capable of reflecting color
center emission, should theoretically double the collection efficiency. It is
possible to design such a structure artificially by periodic variation of the
dielectric contrast, known as the photonic crystal [35]. For enhanced
collection, we deliberately open up a gap possessing the NV center emission
wavelength in the photonic bands, which is achievable by making a particular
choice of PC parameters [36].

Waveguides support light propagation in a particular direction via total
internal reflection. We implement 1D PC in our triangular cross-section
waveguide to introduce asymmetry for redirecting the color center emission.
However, 1D PCMs have mostly been realized in conventional rectangular
geometry [37, 38, 39], and photonics with triangular geometry has been focused
on making efficient active photonic devices [40, 22, 7]. Photonic band
structures in triangular geometry have not been studied in detail. In this
light, our recent work provides an insightful explanation of the dispersion
relations in SiC triangular geometry [23]. From the band structures obtained
from Plane Wave Expansion (PWE) method, we use Finite-Difference Time Domain
(FDTD) method to analyze the performance of the triangular-cross section 1D
PCM and the dependence of the coupling efficiency to the waveguide mode on the
emitter position with respect to the PCM.

###  2.1 Considerations for integrating color centers into triangular cross-
section devices

Color centers in SiC have multiple orientations due to the presence of
inequivalent lattice sites (h, k). The dipole orientations of the color
centers can be parallel (silicon vacancy in SiC) or at an angle (NV center in
SiC) to the crystal axis (c-axis). In a photonic device fabricated from
commonly available 4H-SiC, grown along the c-plane, this would result in
emissions strongly coupling to the fundamental TM (f-TM) mode [8]. On the
other hand, it has been shown that using 4H-SiC grown along the a-plane, the
color center emission can be predominantly coupled to the fundamental TE
(f-TE) mode [10].

Triangular cross-section waveguides support f-TE/TM modes and other higher
order modes [22]. The fraction of a color center emission coupling to a
supported waveguide mode depends on the position of the emitter within the
triangular cross-section. Highest coupling to a mode can be achieved by
positioning the emitter at the maximum intensity point of that mode. The
maximum intensity point of the f-TE mode is located approximately at the
centroid of the triangle, which is an optimal color center position for
achieving highest coupling efficiency to the f-TE mode [22, 10].

The depth of the centroid from the surface depends on the width and etch angle
of the triangular cross-section waveguide. These dimensions can be decided on
the basis of the implantation capabilities, to ensure that the implanted color
centers achieve the highest coupling efficiency to the f-TE mode. For a given
emission wavelength, a triangular cross-section waveguide has an optimal width
at which it has single mode propagation (f-TE), with high (>
$80\text{\,}\mathrm{\char 37\relax}$) coupling to the f-TE mode [10]. Single
mode propagation in the waveguide is key for applications in QIP [41, 42]. The
optimal width of the triangular cross-section devices increases with the etch
angle and emission wavelength. Hence, triangular cross-section devices with
larger etch angles have more fabrication friendly dimensions. Moreover, they
require shallow implantation depths, which causes less damage to the crystal
and having an emitter closer to the surface is useful for SNSPD absorption.
Advantages like single mode propagation with high coupling efficiency, less
damage to the crystal and better detection in SNSPD outweighs the overall
effect of the color center emission lost into the f-TM or other modes.

###  2.2 Waveguide design with TE and TM band gap

NV center emission in SiC has both TE- and TM-like components [43]. Therefore,
a PCM with polarization-independence is necessary to increase collection
efficiency. For broadband reflectivity, such a mirror must have as large a
complete photonic band gap as possible [44, 45]. Single mode propagation [41,
42] and positioning of emitter at the centroid of the triangular cross-section
[10, 22] ensures that most of the color center emission is coupled into the
waveguide. It was shown that $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ etch
angle has predicted the best results for the complete band gap [23]. But in
this study, an etch angle of $60\text{\,}\mathrm{\SIUnitSymbolDegree}$ was
chosen because the optimal emitter position is closer to the top surface,
offering realistic implantation depths ($70\text{\,}\mathrm{nm}$) and
efficient absorption in the SNSPDs. Moreover, a
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ etch angle supports single mode
propagation for larger dimensions than in the case of
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ [10], resulting in a more
fabrication friendly waveguide design. The width of the waveguide was set to
800 nm, as it supports the f-TE and f-TM modes only.

Figure 2: a) Unit cell of the 1D photonic crystal structure. b) Band structure
for $\alpha$ = $60\text{\,}\mathrm{\SIUnitSymbolDegree}$, $w=1.35a$, and
$r=0.43a$. c) TE/TM photonic band gap and gap size as a function of normalized
hole radii for $w=1.2a,1.35a,1.5a$ respectively. Center point indicates midgap
frequency and error bar indicates gap width for the corresponding TE/TM modes.
The dashed circle shows the TE-TM gap overlap for the band structure shown in
b).

The 1D PC is made of cylindrical air holes in a triangular cross-section SiC
waveguide as shown in Figure 2a. The parameters are as follows: periodicity
along the $y$ axis $a$, waveguide width $w$, hole radius $r$, and etch angle
$\alpha$, which corresponds to half-angle at the apex. The refractive index of
SiC is considered 2.6 for all the simulations. In this study, we focused on
the dispersion relations for $\alpha$ =
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ w.r.t $w$ and $r$. We vary $w$ from
$1.2a$ to $2.25a$ and $r$ from $0.25a$ to $0.45a$. The band structure in
Figure 2b exhibits three properties: i) a wide TE band gap, ii) a narrow TM
band gap, and iii) a partial overlap between TE and TM band gaps, which is the
complete photonic band gap. Figure 2c delineates the changes in the TE/TM gap
$(c/a)$ and the corresponding gap size (gap-midgap ratio) with $r/a$ for $w$
values of 1.2a, 1.35a and 1.5a. Midgap frequency increases with $r/a$ for both
TE and TM band gaps as volumetric fill factor (VFF) and effective refractive
index ($n_{\mathrm{eff}}$) have an inverse relationship with $r/a$ [23]. Gap
size initially increases with $r/a$ as fewer propagating modes are supported,
but declines after the maximum value is reached. Though this trend is observed
for both TE and TM bands, the TM band gap completely vanishes for larger
widths ($w>1.5a$). Moreover, the highest TE gap size
($\sim$$22\text{\,}\mathrm{\char 37\relax}$) is an order magnitude higher
compared to the highest TM gap size ($\sim$$2\text{\,}\mathrm{\char
37\relax}$), which is attributed to the connectivity between high-dielectric
regions [35]. In general, complete band gap mostly occurs for smaller widths
with larger hole radii as depicted in Figure 2c. Nevertheless, the choice of
parameters for designing the reflector is made in such a way that there is a
balance between optimal band gap size and reproducibility of the
nanofabrication.

###  2.3 SiC waveguide design with a PCM

In this section, we use the PC parameters from the PWE method and simulate a
triangular cross-section waveguide with a reflector, using the Finite-
Difference Time-Domain (FDTD) package in Lumerical [22, 10]. This allows for a
comparison of the band gap position estimated from the two different methods,
PWE and FDTD. The FDTD method uses the Fourier transform of the time-domain
signal to record the field data in the frequency domain. To increase the
collection efficiency of the photons from the ZPL and PSB of the NV center in
SiC, the PC parameters were chosen to have a band gap spanning these emission
wavelengths. As mentioned in Section 2.2, the size of the TM band gap (and
thus the complete band gap) is significantly smaller than the size of the TE
band gap, limiting the reflected wavelengths of both TE- and TM-like
components of the NV center emission.

In this study, we focus on two sets of parameters for the PC 1) complete band
gap and 2) NV optimized band gap, for a waveguide with $w$ =
$800\text{\,}\mathrm{nm}$ and $\alpha$ =
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$. In the first scenario, the complete
band gap was achieved using PC parameters of $r$ = $237\text{\,}\mathrm{nm}$
and $a$ = $593\text{\,}\mathrm{nm}$, with a complete band gap from
$1268\text{\,}\mathrm{nm}1274\text{\,}\mathrm{nm}$ (TE band gap:
$1268\text{\,}\mathrm{nm}1579\text{\,}\mathrm{nm}$, TM band gap:
$1249\text{\,}\mathrm{nm}1274\text{\,}\mathrm{nm}$), which is outside the
emission wavelength range of the NV center. In the second scenario, PC
parameters of $r$ = $213\text{\,}\mathrm{nm}$ and $a$ =
$533\text{\,}\mathrm{nm}$ were chosen, resulting in both TE
($1200\text{\,}\mathrm{nm}1494\text{\,}\mathrm{nm}$) and TM
($1170\text{\,}\mathrm{nm}1192\text{\,}\mathrm{nm}$) band gaps individually
overlapping with the emission wavelength range of NV centers, but no complete
band gap.

Figure 3: The reflectivity of photonic crystal mirror as a function of
wavelength for varying number of air holes. Grey regions indicate the band gap
values calculated from the PWE method. a) Reflectivity of the fundamental TE
mode for a NV optimized design. Inset shows the mode profile of the
fundamental TE mode of the waveguide at $1230\text{\,}\mathrm{nm}$. b)
Reflectivity of the fundamental TE mode for a complete band gap design. c)
Reflectivity of the fundamental TM mode for a NV optimized design. Inset shows
the mode profile of the fundamental TM mode of the waveguide at
$1230\text{\,}\mathrm{nm}$. d) Reflectivity of the fundamental TM mode for a
complete band gap design.

As the number of air holes increases, the maximum reflectivity of the PCM
increases and saturates for 20 air holes, as shown in Figure 3a-b. It is
observed that the FDTD calculated wavelength for maximum reflectivity deviates
from the center of the PWE calculated bandgap (grey region) [46]. The
reflectivity curve has a full width half maximum of
$\sim$$200\text{\,}\mathrm{nm}$, large enough to reflect the ZPL and the
entire PSB of the color center emission. At $1230\text{\,}\mathrm{nm}$, the NV
optimized design has $\sim$$85\text{\,}\mathrm{\char 37\relax}$ reflectivity
for the f-TE mode. The reflectivity at a given wavelength depends on two
factors: 1) the effective index of the f-TE mode ($n_{\textnormal{eff}}$) and
2) the fraction of color center emission coupled into the waveguide. While
$n_{\textnormal{eff}}$ decreases monotonically with an increased wavelength,
the coupling efficiency rises to the optimal point (designed at 1230 nm) and
subsequently decreases. The combination of the two effects cause the
reflectivity to gradually increase for the short wavelengths and sharply drop
for the long wavelengths. In the case of the f-TM modes, the reflectivity
values are relatively lower due to the low coupling efficiency to the f-TM
mode ($\sim$$20\text{\,}\mathrm{\char 37\relax}$) and evanescent loss to the
surroundings ($n_{\textnormal{eff}}$ = 1.09).

Figure 4: a) Top view of the waveguide design with reflector having N air
holes. The color center (red dot with red arrows indicating the dipole
orientation) is positioned at a distance $s$ from the edge of the first hole
and its emission propagation indicated by the arrow (brown). b) The variation
of the coupling efficiency of the color center emission to the fundamental TE
mode, in the direction of propagation, as a function of the color center’s
position ($s$) from the photonic crystal.

When a dipole emitter ($1230\text{\,}\mathrm{nm}$) is positioned at the
centroid of the triangular cross-section ($\sim$$70\text{\,}\mathrm{nm}$
depth) of the waveguide ($w$ = $800\text{\,}\mathrm{nm}$, $\alpha$ =
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$), $44\text{\,}\mathrm{\char
37\relax}$ of the $50\text{\,}\mathrm{\char 37\relax}$ light in each direction
of the waveguide is coupled to the f-TE mode ($n_{\textnormal{eff}}$ = 1.574).
In the presence of a PCM, the $50\text{\,}\mathrm{\char 37\relax}$ dipole
emission directed toward the PCM is reflected back towards the emitter,
causing interference with the emission propagating in the waveguide region
(indicated by a brown arrow in Figure 4a). We observe that as the distance
between the emitter and the PCM ($s$) is varied, the total light coupled to
the f-TE mode varies periodically, as shown in Figure 4b. Constructive
(destructive) interference occurs if the additional distance 2$s$ traveled by
the reflected wave results in a phase difference that is an even (odd)
multiple of $\pi$. The periodicity of the coupling efficiency matches with the
calculated value ($\lambda/n_{\textnormal{eff}}$ = $390\text{\,}\mathrm{nm}$)
and the maximum coupling efficiency of $89\text{\,}\mathrm{\char 37\relax}$ is
achieved for values of $s$ that are integer multiples of
$\sim$$390\text{\,}\mathrm{nm}$. There exists a $200\text{\,}\mathrm{nm}$
region with coupling efficiency >$75\text{\,}\mathrm{\char 37\relax}$. Thus,
any randomly positioned emitter in the waveguide has at least
$50\text{\,}\mathrm{\char 37\relax}$ chance of being at a position with high
coupling efficiency.

##  3 Integrated SNSPDs for efficient detection of light in triangular
waveguides

For single-photon detection, SNSPDs [47] provide unrivaled performance metrics
such as high detection efficiency, including at $1550\text{\,}\mathrm{nm}$
wavelength [48], high count rate, low dark-count rate [49], and high temporal
resolution [50]. As such, they have found many applications in quantum
communication and information processing as well as in high energy physics for
charged particle detection [51, 52, 53]. Compared to the top-illumination of
meander SNSPDs and the enhancement of their detection efficiency by use of an
optical cavity with a mirror underneath [54, 55], short waveguide integrated
SNSPDs avoid coupling losses when being interfaced with on-chip devices and
provide smaller recovery times as a result of their lower kinetic inductance
[56, 57]. Due to their superior performance metrics, in this section we study
the potential of integrating SNSPDs onto triangular cross-section SiC
waveguides to detect single-photon emission from color centers embedded in the
waveguide. We start by investigating the fundamental properties of NbTiN films
on SiC substrates and subsequently provide performance estimates for SNSPDs on
SiC. Next, we use FDTD simulations to determine the coupling efficiency from
triangular SiC waveguides to SNSPDs integrated on top of the waveguide and its
dependence on the detector geometry.

###  3.1 Optical and transport properties of NbTiN thin films on SiC

In this section, we investigate the optical and the transport properties of
NbTiN thin films on SiC substrates. For this, we used DC reactive magnetron
sputtering at room temperature with an Ar/N2 atmosphere and a sputtering
target with the stoichiometry Nb_0.7Ti_0.3 to fabricate the NbTiN films
studied in the following.

For simulation of absorption in NbTiN SNSPDs on triangular cross-section SiC
waveguides, the optical constants of the deposited superconductor are
essential, especially because the optical and the transport properties vary
with deposition parameters and the underlying substrates [58]. We used a
variable angle spectroscopic ellipsometer, Model M-2000 of J.A. Woollam Co.,
to measure the optical constants over a wavelength range of
$210\text{\,}\mathrm{nm}1687\text{\,}\mathrm{nm}$. As NbTiN is absorbing at
all measurement wavelengths and one cannot resort to a transparent wavelength
window, it is difficult to decouple and determine the optical constants and
film thickness simultaneously due to correlations between both. However, this
correlation can be overcome by measuring with the interference enhancement
method, where the film is measured on an additional thick transparent
dielectric layer and at multiple angles [59, 60, 61]. As we measured the NbTiN
films on two-sided polished SiC substrates, two additional challenges emerge:
The occurrence of reflections on the back side of the substrate, as well as
the fact that SiC is a birefringent material. However, similar to the
interference enhancement method, the anisotropy helps in decoupling thickness
and optical constants for the NbTiN film [62]. Including backside reflections
of the uniaxial 4H-SiC substrate in the model, we obtain a fitting of the
measured data with a mean square error less than 5, indicating good agreement
between the model and the measured data. The resulting optical constants,
shown in Fig. 5, reveal a smaller extinction coefficient $k$ and a higher
refractive index $n$ for the thicker film and provide the basis for the
subsequent FDTD simulations. Compared to the optical constants of a
$5.5\text{\,}\mathrm{nm}$ thick NbTiN film grown by sputtering on silicon
substrates in the publication of Banerjee [59], we obtain similar results,
although our data shows no features in the range from
$500\text{\,}\mathrm{nm}1000\text{\,}\mathrm{nm}$ and our refractive index is
~$20\text{\,}\mathrm{\char 37\relax}$ and the extinction coefficient
~$50\text{\,}\mathrm{\char 37\relax}$ higher for a wavelength of
$1680\text{\,}\mathrm{nm}$. This agrees with the fact that the growth and
properties of polycrystalline materials such as NbTiN depend on the underlying
substrate.

Figure 5: Extinction coefficient $k$ and refractive index $n$ of
$6\text{\,}\mathrm{nm}$ and $10\text{\,}\mathrm{nm}$ NbTiN films on 4H-SiC
substrates.  Figure 6: Critical temperature and transport measurements of
sputtered NbTiN on 4H-SiC substrates. a) shows an SEM image of the fabricated
cloverleaf structures that were used for performing the magneto-transport
measurements. In graphs b), c), and d), the measured data is shown for films
of nominal thicknesses $6\text{\,}\mathrm{nm}$, $8\text{\,}\mathrm{nm}$, and
$10\text{\,}\mathrm{nm}$.

As fabrication and characterization of SNSPDs is technologically challenging
and time consuming, it is beneficial to estimate the detector performance
directly from properties of the sputtered films—especially when it comes to
new superconducting materials or different substrates. To assess, for example,
possible operating temperatures of the envisioned SNSPDs, it is essential to
know the device’s critical temperature $T_{\mathrm{c}}$ at which it switches
from the superconducting to the normal conducting state. As this temperature
is reduced for nanowires compared to thin films or bulk material [63, 64], the
critical temperature of the film provides an upper limit for reachable SNSPD
critical temperatures. Furthermore, by measuring fundamental film transport
properties such as sheet resistance $R_{\mathrm{sheet}}$, quasiparticle
diffusivity $D$ [65, 66], and Hall coefficient $R_{\mathrm{H}}$, we can
estimate key detector metrics such as depairing current $I_{\mathrm{dep}}$ (an
upper limit for the detector’s switching current) and fall time
$\tau_{\mathrm{fall}}$, and obtain information about disorder in the film (via
the deduced Ioffe-Regel parameter $k_{\mathrm{F}}l$ [67]). The depairing
current density at a temperature $T$ can be calculated according to the
Bardeen model [68, 69, 70, 65] from the superconductor’s energy gap
$\Delta($0\text{\,}\mathrm{K}$)=2.0\,k_{\mathrm{B}}\,T_{\mathrm{c}}$ with the
factor $2.0$ for NbTiN according to [71, 72], the elementary charge $e$, the
resistivity in the normal conducting state $\rho_{\mathrm{nc}}$, and the
quasiparticle diffusivity $D$ by

$j_{\mathrm{dep}}(T)=0.74\cdot\frac{\left(\Delta($0\text{\,}\mathrm{K}$)\right)^{3/2}}{e\rho_{\mathrm{nc}}\sqrt{\hbar
D}}\left(1-\left(\frac{T}{T_{\mathrm{c}}}\right)^{2}\right)^{3/2}\;.$ (1)

A higher switching current of the detector also means a higher amplitude of
the detection pulse, which is beneficial for the signal-to-noise ratio.

Moreover, we can calculate the kinetic inductance from the sheet resistance
$R_{\mathrm{sheet}}$, the energy gap $\Delta$, and the nanowire width $w$ and
length $l$ [65, 73].

$L_{\mathrm{k}}=\frac{\hbar\,R_{\mathrm{sheet}}}{\pi\,\Delta($0\text{\,}\mathrm{K}$)}\frac{l}{w}$
(2)

that can be used to estimate the fall time of a detection pulse [74, 75] by

$\tau_{\mathrm{fall}}=\frac{L_{\mathrm{k}}}{R_{\mathrm{load}}}\;,$ (3)

with $R_{\mathrm{load}}$ denoting the load resistor, which can be either a
shunt resistor or the readout electronics (typically
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$). From the fall time, we can also
estimate the maximum count rate by $1/\tau_{\mathrm{fall}}$. In Fig. 6, we
show the measured transport properties of $6\text{\,}\mathrm{nm}$,
$8\text{\,}\mathrm{nm}$, and $10\text{\,}\mathrm{nm}$ sputtered NbTiN222These
are the nominal thicknesses set during the sputtering process and are used in
conjunction with the measured deposition rate to calculate the necessary
sputtering time. For $6\text{\,}\mathrm{nm}$ and $10\text{\,}\mathrm{nm}$, the
thicknesses have also been fitted within the optical model for the
ellipsometry data and yielded $5.48(20)\text{\,}\mathrm{nm}$ and
$9.62(26)\text{\,}\mathrm{nm}$. on 4H-SiC substrates with one
$8\text{\,}\mathrm{nm}$ film of the same sputtering round on a silicon wafer
with thermally grown SiO2 as reference substrate. From this fundamental data,
we calculate estimates for key detector metrics in Tab. 1. Despite the
difference in refractive index and extinction coefficient found above, a
comparison of the measured and derived performance metrics for NbTiN SNSPDs on
SiC substrates with reference to NbTiN SNSPDs on SiO2 substrate yields similar
values. Also, our measured transport data agree with the literature [76, 67,
77]. However, it seems worthwhile to note that our data for the electron
density is lower than the values found in the literature. Nevertheless, when
comparing the data in [67, 76], the electron density shows a dependence on the
stoichiometry of Nb_xTi_1-xN, and our case with $x=0.7$ fits into the picture
of having a minimum between $x=0.66$ and $x=0.86$.

We conclude that choosing SiC substrates in combination with our sputtering
recipe yields films that have higher absorption while maintaining similar
transport properties. Thus, geometrically identical detectors can be expected
to have a higher detection efficiency with similar electrical performance on
SiC compared to SiO2 substrates. At the same time this can be used to reduce
the optical active area of a detector to enhance its electrical properties
(for example reducing the fall time) while keeping the same detection
efficiency. It is worthwhile to note that the absolute values we obtained for
the fall time of SNSPDs are higher than typically measured fall times that are
in the order of $10\text{\,}\mathrm{ns}$ for $10\times
10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ SNSPDs on SiO2 substrates (measured,
e.g., in [74] as well as in our labs). We speculate that the fabrication
process of detectors modifies the transport properties of the film such that
the kinetic inductivity and thus the fall time are increased compared to the
film-derived ones.

Table 1:  Estimated SNSPD performance metrics from measured film properties. Estimated SNSPD performance metrics from measured film properties. For the derived numbers of the meander SNSPDs we consider $10\times 10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ detectors with $100\text{\,}\mathrm{nm}$ wide wires and a fill factor of $50\text{\,}\mathrm{\char 37\relax}$, while for the double loop SNSPD we assume a $100\text{\,}\mathrm{nm}$ wide and $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long nanowire, similar to the maximum wire length used in the simulations presented in Figures 7 and 8.  |  |  |  | Meander SNSPD | Double loop SNSPD
---|---|---|---|---|---
Substrate |  $d$/nm |  $I_{\mathrm{dep}}($0\text{\,}\mathrm{K}$)$/µA |  $I_{\mathrm{dep}}($4\text{\,}\mathrm{K}$)$/µA |  $L_{\mathrm{k}}$/µH |  $\tau_{\mathrm{fall}}$/ns |  $L_{\mathrm{k}}$/µH |  $\tau_{\mathrm{fall}}$/ns
SiC | $6$ | $14.1$ | $8.9$ | $3.2$ | $63.5$ | $0.254$ | $5.08$
$8$ | $38.1$ | $28.7$ | $1.1$ | $22.0$ | $0.088$ | $1.76$
$10$ | $75.5$ | $60.3$ | $0.6$ | $11.8$ | $0.047$ | $0.95$
SiO2 | $8$ | $40.0$ | $30.4$ | $1.0$ | $20.8$ | $0.083$ | $1.66$

###  3.2 Model of an integrated SNSPD with a triangular cross-section SiC
waveguide

In this section, we look into the absorption efficiency of NbTiN based SNSPDs
integrated on top of a triangular cross-section SiC waveguides based on the
experimental measurements presented in Section 3.1. This provides an upper
limit for the achievable system detection efficiency of such an integrated
detector. The absorption of color center emission (TE- or TM-like) for
different configurations of NbTiN was modeled using the FDTD package in
Lumerical software. The NbTiN was modeled using a complex permittivity model
with ($n$, $k$) values obtained from the ellipsometry measurements discussed
in the Section 3.1. The top and cross-section view of a single strip of NbTiN
integrated on top of a triangular cross-section waveguide are shown in Figure
7a. The single strip has length $l$, width $w$, and thickness $t$, positioned
at the center of the top surface of the waveguide.

Figure 7: Simulated absorption of a single strip NbTiN layer. a) Top and
cross-section view of a single strip NbTiN (yellow) with length $l$, width $w$
and thickness $t$, integrated on a triangular cross-section SiC waveguide
(blue). Red dot indicates the color center and red arrows denotes its dipole
orientation. Absorption of the fundamental TE and fundamental TM modes of the
triangular waveguide as a function of b) width ($l$ =
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t$ = $6\text{\,}\mathrm{nm}$). c)
length ($w$ = $100\text{\,}\mathrm{nm}$, $t$ = $6\text{\,}\mathrm{nm}$). d)
thickness ($w$ = $100\text{\,}\mathrm{nm}$).

When the width of the NbTiN strip ($l$ =
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t$ = $6\text{\,}\mathrm{nm}$) is
increased, the absorption of the f-TE mode remains the same for widths greater
than $100\text{\,}\mathrm{nm}$ and the absorption of the f-TM mode increases
as the width increases, as shown in Figure 7b. It should be noted that the
absorption of the NbTiN strip is sensitive to the polarization of the mode in
the waveguide. The evanescent fields of the f-TE mode are concentrated around
the center of the top surface (inset of Figure 3a), and therefore no changes
in absorption are observed for larger SNSPD widths. In comparison, the
evanescent fields of the f-TM mode are spread across the entire width of the
waveguide (inset of Figure 3c), requiring larger widths of NbTiN for efficient
absorption. As the length ($w$ = $100\text{\,}\mathrm{nm}$, $t$ =
$6\text{\,}\mathrm{nm}$) and thickness ($w$ = $100\text{\,}\mathrm{nm}$) of
the NbTiN strip increases, the absorption of both the f-TE and f-TM mode
increases, as shown in Figure 7c-d. Although larger thicknesses would result
in better absorption, the single photon detection efficiency decreases [78],
which is critical for applications in QIP.

Figure 8: a) Top-view of a single loop NbTiN (yellow) with length $l$, spacing
between adjacent arms $d$, on a triangular cross-section SiC waveguide (blue).
b) Top-view of a double loop NbTiN (yellow) with length L, spacing between two
adjacent arms d, on a triangular cross-section SiC waveguide (blue). Red dot
indicates the color center and red arrows denotes its dipole orientation. c)
Absorption of the fundamental TE and fundamental TM modes of the triangular
waveguide as a function of spacing between adjacent arms, d ($l$ =
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $w$ = $100\text{\,}\mathrm{nm}$,
$t$ = $6\text{\,}\mathrm{nm}$). d) Comparison of absorption of fundamental TE
and fundamental TM modes of the triangular waveguide as a function of length,
for single loop and double loop NbTiN ($w$ = $100\text{\,}\mathrm{nm}$, $d$ =
$100\text{\,}\mathrm{nm}$, $t$ = $6\text{\,}\mathrm{nm}$).

Typical SNSPD based detectors have nanowires fabricated into a meander shape
to increase the optical detection area. To this end, we modeled single loop
and double loop configurations of NbTiN integrated on top of a triangular
cross-section SiC waveguides, as shown in Figures 8a, 8b, respectively. For a
single loop NbTiN ($l$ = $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $w$ =
$100\text{\,}\mathrm{nm}$, $t$ = $6\text{\,}\mathrm{nm}$), as the spacing
between two adjacent arms $d$ increases, the absorption of the f-TE mode
remains constant and the absorption of the f-TM mode increases rapidly, as
shown in Figure 8c. The absorption of the double loop NbTiN ($w$ =
$100\text{\,}\mathrm{nm}$, $d$ = $100\text{\,}\mathrm{nm}$, $t$ =
$6\text{\,}\mathrm{nm}$) is higher than the single loop NbTiN, with a
significant improvement in the absorption of the f-TM mode, as shown in Figure
8d. The polarization sensitivity of the double loop SNSPD at $l$ =
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ almost disappears, thus having
identical detection efficiencies for both f-TE and f-TM modes.

##  4 Discussion

The results presented in this paper demonstrate scalable approaches to
efficiently collect and detect color center emission in triangular cross-
section SiC devices for applications in QIP. Triangular cross-section devices
are promising for chip scale integration of color center photonics. Moreover,
these devices have a rich parameter space (etch angle, width) for optimal
color center integration. At a given emission wavelength, there exists an
optimal width for each triangular cross-section (with an etch angle alpha)
such that the waveguide supports single mode propagation [10], which is
important for applications in QIP. The polarization of the color center
emission depends on the dipole orientation of the color center with respect to
the c-axis. Using 4H-SiC grown along a-plane, most of the color center
emission (silicon vacancy, divacancy, nitrogen vacancy) can be coupled into
the f-TE mode.

Furthermore, the coupling of the color center emission can be improved by
positioning the color center at the maximum intensity point of the f-TE mode,
which is around the centroid of the triangular cross-section. Color centers
are generated in SiC through implantation of high energy projectiles like
electrons, protons, neutrons, ions. The implantation energy and the mass of
the implanted particle determine the implantation damage done to the crystal,
which can only be partially recovered through annealing [79]. Triangular
cross-section devices can be designed depending on the implantation energies
available, while still maintaining the single mode propagation and emitter’s
position at the centroid of the triangular cross-section. The waveguide
simulated in this work ($w$ = $800\text{\,}\mathrm{nm}$, $\alpha$ =
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$) has single mode propagation, with
$\sim$$44\text{\,}\mathrm{\char 37\relax}$ of the $50\text{\,}\mathrm{\char
37\relax}$ dipole emission ($1230\text{\,}\mathrm{nm}$) along one direction of
the waveguide coupled to the f-TE mode.

The collection efficiency ($\sim$$44\text{\,}\mathrm{\char 37\relax}$) on one
end of the designed waveguide can be further improved using a photonic crystal
mirror. The periodic variation of the dielectric constant (air holes) in a PC
results in a photonic band gap for both TE- and TM-like modes in the
waveguide. In a parallel work, a thorough photonic bandgap analysis using PWE
method shows that the PCs can be used for a variety of applications and that
their dimensions can be scaled depending on the color center emission
wavelength [23]. In triangular cross-section PCMs, the TE bandgap is an order
of magnitude larger than the TM bandgap, consequently a relatively much
smaller complete bandgap. Therefore, a more effective approach for using the
PCs as reflectors of c-axis oriented color center emission is to fabricate
them in a-plane SiC substrates [10]. This would ensure that a large portion of
any color center emission is coupled to the f-TE mode, reflecting frequencies
spanning the ZPL and PSB of the color center emission. FDTD analysis shows
that the position of the emitter relative to the PCM determines whether there
is constructive/destructive interference in the waveguide region. We highlight
that using the PCM proposed in this paper, a randomly positioned emitter has a
higher collection efficiency compared to a symmetric waveguide
($\sim$$44\text{\,}\mathrm{\char 37\relax}$) for $75\text{\,}\mathrm{\char
37\relax}$ of cases.

Efficient detection is another important factor for boosting the success rates
of QIP protocols. SNSPDs offer a scalable approach for integrating detectors
onto photonic devices. On-chip detection using SNSPDs can reduce optical
losses, latency, and wiring complexity involved in off-chip detection [28]. As
a basis for SiC waveguide integrated SNSPDs, we investigated, for the first
time, the optical and electrical properties of NbTiN thin films for SNSPD
applications on 4H-SiC substrates. While the absorption of the NbTiN films is
significantly higher on SiC than on SiO2 substrates (up to
$50\text{\,}\mathrm{\char 37\relax}$ for photons of
$1680\text{\,}\mathrm{nm}$), the magneto-electrical properties of the films,
and thus presumably also of the detectors, are similar. Provided that the
fabrication of SNSPDs on SiC waveguides is of similar quality as on SiO2
substrates, we expect that they provide comparable electrical performance and
simultaneously enhanced system detection efficiency due to higher absorption
in the film, making SiC waveguide integrated SNSPDs a promising candidate for
on-chip photon detection in QIP.

##  5 Acknowledgements

This work is supported by the National Science Foundation (CAREER-2047564),
the Bavaria California Technology Center (BaCaTeC) - Internationalization of
the High-Tech-Initiative, the German Federal Ministry of Education and
Research via the funding program Quantum technologies - from basic research to
market (contract numbers 16K1SQ033, 13N15855 and 13N15982) and via the project
MARQUAND (contract number BN105022) and the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy –
EXC-2111 – 390814868.

##  6 Conflict of Interest

The authors have no conflicts to disclose.

##  7 Data Availability Statement

The data that support the findings of this study are available upon reasonable
request from the authors.

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#  $Z_{2}$ Symmetry of $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$
Superstrings

Abbas Ali and Mohsin Ilahi
Physics Department, Aligarh Muslim University,
Aligarh, India

###### Abstract

We discuss the importance of $Z_{2}$ symmetry of large $N=4$ superconformal
algebra for finding its free field realization that is more complete than the
most commonly used one, that is, the Sevrin, Troost and van Proeyen
realization. This in turn has implications for the search of a conformal field
theory dual to superstrings moving in $AdS_{3}\times S^{3}\times S^{3}\times
S^{1}$ background.

##  1 Story of $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ Superstrings

$AdS_{5}$ and $AdS_{3}$ are the two most intensely studied cases of the famous
Maldacena Conjecture about duality between theories with gravity on one side
and conformal field theories without gravity on other [1, 2, 3]. In case of
$AdS_{3}$ three distinct cases have been investigated quite thoroughly. These
are the cases of strings moving on the $AdS_{3}\times
S^{3}\times\mathcal{M}^{4}$ manifolds with
$\mathcal{M}^{4}=T^{4},K3~{}~{}\text{or}~{}~{}S^{3}\times S^{1}$. According to
Maldacena’s original proposal the conformal field theory dual to strings
moving on $AdS_{3}\times S^{3}\times T^{4}$ and $AdS_{3}\times S^{3}\times K3$
are the so called symmetric orifold CFTs $Sym(T^{4})$ and $Sym(K3)$
respectively [4]. More specifically $AdS_{3}/CFT_{2}$ correspondence is
formulated as follows. The 1+1 dimensional CFT describing the Higgs branch of
the D1-D5 system ($Q_{1}$ branes of D1 type and $Q_{5}$ of D5 type aligned in
a specific manner) on $\mathcal{M}^{4}=T^{4}~{}\text{or}~{}K3$ is dual to Type
IIB superstrings moving on the manifold $(AdS_{3}\times
S^{3})_{Q_{1}Q_{5}}\times\mathcal{M}^{4}(Q)$. Here $Q$ specifies the
dependence of $\mathcal{M}^{4}$ on $Q_{1}$ and $Q_{5}$. This immediately leads
to the suggestion that the conformal field theory dual to strings moving on
$AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ should be $Sym(S^{3}\times
S^{1})$ [5].

But there is a problem in this assertion. The problem was very thoroughly
analyzed in Ref.[6] by Gukov, Martinec, Moore and Strominger from a number of
angles. These angles included the analysis of the supergravity solutions,
structure of the moduli space, long strings, singular CFT, intersecting
D-brane configurations, free field realizations (FFRs) of the large N=4
superconformal symmetry, structure of the symmetric product orbifolds, an
index for $Sym(S^{3}\times S^{1})$ and the comparison of the BPS and near BPS
spectra. The end product was that they ruled out all the existing proposals
about the conformal field theory dual to strings on $AdS_{3}\times S^{3}\times
S^{3}\times S^{1}$ geometry at that time.

After that the problem was taken up in Ref.[7]. The technique used there was
of the dynamics of the supersymmetric gauge theory living on the specifically
designed D-brane configuration. Aspects related to the D-brane configurations
and gauge dynamics for $AdS_{3}\times S^{3}\times K3$ and $AdS_{3}\times
S^{3}\times T^{4}$ superstrings had been reviewed in Ref.[8] in detail. The
gauge theory considered in [7] for $AdS_{3}\times S^{3}\times S^{3}\times
S^{1}$ superstrings does give the required central charge but the brane
configuration chosen was such that the geometry resulting from it does not go
over to needed geometry, that is, $AdS_{3}\times S^{3}\times S^{3}\times
S^{1}$. Thus the problem was not resolved.

Next breakthrough came with the Ref.[9] whereby the proposal about
$Sym(S^{3}\times S^{1})$ being the required CFT dual was restored to its
original place in Ref.[10] by revisiting the spectrum analysis of Ref.[11].
But there is a limitation of this renewed assertion. The symmetric product
orbifold $Sym(S^{3}\times S^{1})$ is dual to superstrings moving on
$AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ provided one of the charges
$Q^{\pm}_{5}$ is a multiple of the other. Clearly it is desirable to remove
this limitation.

In this note we approach the problem from the point of view of the free field
realizations. In Ref.[6] this angle too was taken up. The issues the authors
faced are discussed in section 2. What is relevant is the free field
realizations of small, middle and large N=4 superconformal symmetry. The
superconformal symmetry relevant for Type IIB superstrings moving on
$AdS_{3}\times S^{3}\times K3$ is the small N=4 [4] while for Type IIB
superstrings moving on $AdS_{3}\times S^{3}\times T^{4}$ it is the middle N=4
[12]. The superconformal symmetry relevant for the case of Type IIB
superstrings moving on $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ is the
large N=4. In Section 3 we analyze the $Z_{2}$ automorphisms of free field
realization (complete) of large N=4 supercomformal symmetry. In Section 4 we
list the complete free field realization derived in [13] and [14]. We discuss
its implications in Section 5.

##  2 Failure of Tensoring

It is well known, say, for example, from Ref. [5], that the superconformal
symmetry relevant for the $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$
superstrings is the large N=4 one. In view of that it was natural to seek help
of the free field realizations (FFRs) of large N=4 superconformal symmetry to
find the holographic dual to superstrings moving on above background. This was
precisely among various things that were analyzed by Gukov-Martinec-Moore-
Strominger (GMMS) in Ref.[6]. The best free field realization of large N=4
SCFT in 1988 was the one given by Sevrin, Troost and van Proeyen in Ref.[15]
(see also Ref.[16]).

GMMS tried a tensoring of two Sevrin, Troost and van Proeyen free field
realizations to get a complete realization. Their proposal had the following
prescription

$\displaystyle T(z)$ $\displaystyle=$ $\displaystyle
T_{1}(z)+T_{2}(z)+\frac{1}{2}\partial(aU_{1}(z)+bU_{2}(z))\ $ $\displaystyle
G^{a}(z)$ $\displaystyle=$ $\displaystyle
G^{a}_{1}(z)+G^{a}_{2}(z)+\partial(aQ^{a}_{1}(z)+bQ^{a}_{2}(z))$
$\displaystyle A^{\pm i}(z)$ $\displaystyle=$ $\displaystyle A_{1}^{\pm
i}(z)+A_{2}^{\pm i}(z)$ $\displaystyle Q^{a}(z)$ $\displaystyle=$
$\displaystyle Q^{a}_{1}(z)+Q^{a}_{2}(z)$ $\displaystyle U(z)$
$\displaystyle=$ $\displaystyle U_{1}(z)+U_{2}(z).$ (1)

Here the subscripts 1 and 2 are for the two starting FFRs. Values of the
constants $a$ and $b$ were determined to be

$\displaystyle a$ $\displaystyle=$ $\displaystyle
2\frac{k_{1}^{+}k_{2}^{-}-k_{1}^{-}k_{2}^{+}}{k_{1}(k_{1}+k_{2})},$
$\displaystyle b$ $\displaystyle=$ $\displaystyle
2\frac{k_{2}^{+}k_{1}^{-}-k_{2}^{-}k_{1}^{+}}{k_{2}(k_{1}+k_{2})}.$ (2)

Using the prescription $U(z)=\sqrt{\frac{k}{2}}\phi(z)$ for the $U(1)$ current
we get two orthogonal combinations

$U_{+}(z)=\sqrt{\frac{k_{1}}{k_{1}+k_{2}}}\phi_{1}(z)+\sqrt{\frac{k_{2}}{k_{1}+k_{2}}}\phi_{2}(z)$

and

$U_{-}(z)=\sqrt{\frac{k_{2}}{k_{1}+k_{2}}}\phi_{1}(z)-\sqrt{\frac{k_{1}}{k_{1}+k_{2}}}\phi_{2}(z)$
(3)

and in the stress tensor we get

$T(z)=-\frac{1}{2}(\partial\phi_{-}(z))^{2}+\frac{1}{2}Q_{12}(\partial)^{2}\phi_{-}(z)$
(4)

with

$Q_{12}=\sqrt{2}\frac{k_{1}^{+}k_{2}^{-}-k_{2}^{+}k_{1}^{-}}{\sqrt{k_{1}k_{2}(k_{1}+k_{2})}}.$
(5)

From this we get

$Q_{(12)3}\neq Q_{(1(23)}.$ (6)

Hence the tensoring procedure is not associative. Thus the search for a
complete free field realization of the large N=4 superconformal symmetry by
tensoring two copies of Sevrin-Troost-van Proeyen free field realizations was
abandoned. In the next section we list the solution to this two decades old
problem.

##  3 $Z_{2}$ Automorphisms of Free Field Realizations

To overcome the non-associativity encountered by GMMS we shall take help from
the duality and $Z_{2}$ symmetry structure of ADHM instanton sigma models [17,
18, 19, 20]. We believe that the ADHM instanton sigma models can be mapped
onto $AdS_{3}$ superstrings. The orginal ADHM instanton sigma model of Witten
in Ref.[17] is dual to our complementary model of Ref.[18]. This duality gets
promoted to a $Z_{2}$ symmetry in the complete model of Ref.[19]. The complete
model is a concrete system, that is, it is a realization of N=4 supersymmetry.
In the conformal limit it should flow to a system with large N=4
superconformal symmetry. If we have a concrete realization before the
conformal flow the complete free field realization after the flow should also
exist. In this section we examine the $Z_{2}$ symmetry of various free field
realizations of the large N=4 superconformal algebra. This takes us very close
to the problem of finding the complete free field realization.

The simplest example of a large N=4 theory can be realized as a theory of a
free boson, $\phi$, and four Majorana fermions, $\psi_{a}$, $a=0,...,3$.
Particularly, we have in Ref.[21] the following free field realization

$\displaystyle T(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}\psi^{a}\partial\psi_{a}$
$\displaystyle G^{a}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{6}i\epsilon^{abcd}\psi^{b}\psi^{c}\psi^{d}-i\psi^{a}\partial\phi$
$\displaystyle A^{\pm,i}(z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\alpha^{\pm,i}_{ab}\psi^{a}\psi^{b}$ $\displaystyle
Q^{a}(z)$ $\displaystyle=$ $\displaystyle\psi^{a}(z)$ $\displaystyle U(z)$
$\displaystyle=$ $\displaystyle i\partial\phi(z).$ (7)

In Ref.[5] it has been proposed that for the case $k^{+}=k^{-}$ the boundary
SCFT corresponding to Type IIB string theory on $AdS_{3}\times S^{3}\times
S^{3}\times S^{1}$ is a sigma-model based on the symmetric product orbifold of
this $c=3$ theory.

Because of the $k^{+}=k^{-}$ degeneracy the $Z_{2}$ duality under
$k^{+}\leftrightarrow k^{-}$ or $\gamma\leftrightarrow 1-\gamma$ is an
isometry in this case. This means that this free field realization does not
realize the full $Z_{2}$ duality. In turn our quest cannot be answered by the
FFRs in Eq.(3).

Next we take up the Sevrin, Troost and van Proeyen free field realization.
This FFR is realized using a free boson $\phi(z)$, defining $U(1)$ current
$J_{0}(z)=\partial\phi(z)$ such that
$J^{0}(z)J^{0}(w)\sim-\frac{1}{2}(z-w)^{-2}$, four free fermions $\psi^{a}(z)$
with $a=0,\cdots,3$, and three $SU(2)$ Kac-Moody currents $J^{i}(z)$ with
level $k$. The generators of the algebra have the following form in terms of
free fields:

$\displaystyle T(z)$ $\displaystyle=$
$\displaystyle-J^{0}J^{0}-\frac{1}{k+1}J^{a}J^{a}+\psi^{a}\partial\psi^{a},$
$\displaystyle U(z)$ $\displaystyle=$
$\displaystyle\sqrt{k+1}J^{0}(z),~{}~{}~{}~{}~{}Q^{a}(z)=\sqrt{k+1}\psi^{a}(z),$
$\displaystyle A^{-i}(z)$ $\displaystyle=$
$\displaystyle\alpha^{-i}_{ab}\psi^{a}\psi^{b},$ $\displaystyle A^{+i}(z)$
$\displaystyle=$ $\displaystyle J^{i}+\alpha^{+i}_{ab}\psi^{a}\psi^{b},$
$\displaystyle G^{0}(z)$ $\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{0}+\frac{1}{\sqrt{k+1}}J^{i}\psi^{i}+\frac{2}{\sqrt{k+1}}\psi^{1}\psi^{2}\psi^{3}\right],$
$\displaystyle G^{1}(z)$ $\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{1}+\frac{1}{\sqrt{k+1}}(-J^{1}\psi^{0}+J^{2}\psi^{3}-J^{3}\psi^{2})\right.$
(8) $\displaystyle\left.-\frac{2}{\sqrt{k+1}}\psi^{0}\psi^{2}\psi^{3}\right],$

with cyclic expressions for $G^{2}(z)$ and $G^{3}(z)$. It is simple to verify
that the operators given in Eqn.(3) generate the large $N=4$ algebra with
$k^{+}=k+1$ and $k^{-}=1$ [15, 16].

A $Z_{2}$ automorphism gives the following free field realization.

$\displaystyle T(z)$ $\displaystyle=$
$\displaystyle-J^{0}J^{0}-\frac{1}{\kappa+2}J^{a}J^{a}+\psi^{a}\partial\psi^{a},$
$\displaystyle U(z)$ $\displaystyle=$
$\displaystyle\sqrt{k+1}J^{0}(z),~{}~{}~{}~{}~{}Q^{a}(z)=\sqrt{k+1}\psi^{a}(z),$
$\displaystyle A^{-i}(z)$ $\displaystyle=$ $\displaystyle
J^{i}+\alpha^{-i}_{ab}\psi^{a}\psi^{b}$ $\displaystyle A^{+i}(z)$
$\displaystyle=$ $\displaystyle\alpha^{+i}_{ab}\psi^{a}\psi^{b},$
$\displaystyle G^{0}(z)$ $\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{0}+\frac{1}{\sqrt{k+1}}J^{i}\psi^{i}+\frac{2}{\sqrt{k+1}}\psi^{1}\psi^{2}\psi^{3}\right],$
$\displaystyle G^{1}(z)$ $\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{1}+\frac{1}{\sqrt{k+1}}(-J^{1}\psi^{0}+J^{2}\psi^{3}-J^{3}\psi^{2})\right.$
(9) $\displaystyle\left.-\frac{2}{\sqrt{k+1}}\psi^{0}\psi^{2}\psi^{3}\right],$

with cyclic expressions for $G^{2}(z)$ and $G^{3}(z)$.

It is clear from above $Z_{2}$ automorphism that the form of the free field
realization is not maintained. For example the current $J^{i}$ jumps from
$A^{+i}$ to $A^{-i}$. This is a manifestation of the same incompletness of
this FFR that led GMMS to consider the tensoring discussed earlier.

Another sign of the incompleteness of this free field realization is that one
of the $SU(2)$ levels is general the other one is fixed. In the desired or
complete free field realization the values of the levels of two $SU(2)$s
should have general values.

In the $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ superstrings the two
levels are related to the radii of the two spheres. The fixed value of an
$SU(2)$ level implies that one of the radii has a fixed value and hence the
free field realization is not representing general $AdS_{3}\times S^{3}\times
S^{3}\times S^{1}$ superstrings.

We shall now describe the final reason to assert incompletness of the Sevrin-
Troost-van Proeyen free field realization.

While we can take two different and independent singular limits of the large
N=4 superconformal algebra we can take only one singular limit each of above
free field realizations. These are given below for Eqn.(3)

$\displaystyle T(z)$ $\displaystyle=$
$\displaystyle-J^{0}J^{0}-U^{i}U^{i}+\psi^{a}\partial\psi^{a},$ $\displaystyle
U(z)$ $\displaystyle=$ $\displaystyle
J^{0}(z),~{}~{}~{}~{}~{}Q^{a}(z)=\psi^{a}(z),$ $\displaystyle V^{+i}(z)$
$\displaystyle=$ $\displaystyle
U^{i},~{}A^{-i}(z)=\alpha^{-i}_{ab}\psi^{a}\psi^{b},$ $\displaystyle G^{0}(z)$
$\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{0}+U^{i}\psi^{i}+\psi^{1}\psi^{2}\psi^{3}\right],$
$\displaystyle G^{1}(z)$ $\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{1}+(-U^{1}\psi^{0}+U^{2}\psi^{3}-U^{3}\psi^{2})\right.$ (10)
$\displaystyle\left.-\psi^{0}\psi^{2}\psi^{3}\right],$

with cyclic expressions for $G^{2}(z)$ and $G^{3}(z)$.

The other singular limit is given by

$\displaystyle T(z)$ $\displaystyle=$
$\displaystyle-J^{0}J^{0}-U^{i}U^{i}+\psi^{a}\partial\psi^{a},$ $\displaystyle
U(z)$ $\displaystyle=$ $\displaystyle
J^{0}(z),~{}~{}~{}~{}~{}Q^{a}(z)=\psi^{a}(z),$ $\displaystyle A^{+i}(z)$
$\displaystyle=$
$\displaystyle\alpha^{+i}_{ab}\psi^{a}\psi^{b},~{}V^{-i}(z)=U^{i},$
$\displaystyle G^{0}(z)$ $\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{0}+J^{i}\psi^{i}+\psi^{1}\psi^{2}\psi^{3}\right],$
$\displaystyle G^{1}(z)$ $\displaystyle=$ $\displaystyle
2\left[J^{0}\psi^{1}+(-U^{1}\psi^{0}+U^{2}\psi^{3}-U^{3}\psi^{2})\right.$ (11)
$\displaystyle\left.-\psi^{0}\psi^{2}\psi^{3}\right],$

with cyclic expressions for $G^{2}(z)$ and $G^{3}(z)$.

Thus the complete free field realization of large N=4 superconformal symmetry
should be such that we can take two Inönü-Wigner contractions and in each case
still end up in general free field realizations of the middle N=4
superconformal algebra. In other words the Sevrin-Troost-van Proeyen
realization should be augmented in such a way so as to permit two Inönü-Wigner
contractions.

We have duality between Witten’s ADHM sigma model and our complementay model.
We have in above analysis the Sevrin-Troost-van Proeyen realization gets
mapped onto itself under duality. It does not have the $Z_{2}$ symmetry of the
large N=4 superconformal algebra.

The complete free field realization should have a $Z_{2}$ symmetry like the
large N=4 superconformal algebra. The Sevrin-Troost-van Proeyen free field
realization lacks it because $k^{+}=k+1$ and $k^{-}=1$. To have $Z_{2}$
symmetry both $k^{+}$ and $k^{-}$ should have got general values.

Corresponding perspective in case of $AdS_{3}$ superstrings is the following.
These free field realizations correspond to $AdS_{3}\times S_{+}^{3}\times
T^{4}$ and $AdS_{3}\times S_{-}^{3}\times T^{4}$ geometries respectively. The
spheres $S_{+}^{3}$ and $S_{-}^{3}$ in above have fixed radii because the
remaining $SU(2)$ has a fixed level. We require a free field realization where
we can take two singular limits of the same realization. In the resulting
geometries after single singular limits the remaining three spheres will have
general radii.

These are the kind of considerations that lead us to the same quest that led
GMMS to explore tensoring of above two copies of the realization given by
Sevrin, Troost and van Proeyen. The process was not pursued to its logical end
owing to the non-associative character of the tensor product mentioned above.

##  4 Complete Free Field Realization

From the discussion so far it is clear that we need a complete free field
realizations of large N=4 SCFT and that it should exist. In reference [13] we
proposed a strategy to obtain such a realization. This was explicitly
implemented in reference [14]. In Ref.[12] it was used to demonstrate that the
superconformal symmetry relevant for superstrings moving on $AdS_{3}\times
S^{3}\times T^{4}$ geometry is the middle N=4 [22]. After writing [13] and
[14] that we realized that the resulting free field realization already
existed in Ref. [16].

This realization uses two $U(1)$ currents $J^{0}(z)$ and $K^{0}(z)$, two sets
of $SU(2)$ currents $J^{\pm i}(z)$, $i=1,2,3$, and eight free fermions
$\psi^{a}(z),~{}\chi^{a}(z),~{}a=0,\cdots,3$. The expressions for the sixteen
generators $T(z),~{}G^{a}(z),~{}A^{\pm i}(z)$ and $Q^{a}(z)$ are given below:

$\displaystyle T(z)$ $\displaystyle=$
$\displaystyle-J^{0}J^{0}+\sqrt{\frac{\gamma}{6c}}(6-c)\partial
J^{0}-K^{0}K^{0}-\sqrt{\frac{1-\gamma}{6c}}(6-c)\partial K^{0}$
$\displaystyle-$
$\displaystyle\frac{6\gamma}{c}J^{+i}J^{+i}-\frac{6(1-\gamma)}{c}J^{-i}J^{-i}+\psi^{a}\partial\psi^{a}+\chi^{a}\partial\chi^{a},$
$\displaystyle A^{\pm i}(z)$ $\displaystyle=$ $\displaystyle J^{\pm
i}(z)\mp\psi^{0}(z)\psi^{i}(z)\mp\chi^{0}(z)\chi^{i}(z)+\epsilon^{ijk}[\psi^{j}(z)\psi^{k}(z)+\chi^{j}(z)\chi^{k}(z)],$
$\displaystyle Q^{a}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{c}{6\gamma}}\psi^{a}(z)+\sqrt{\frac{c}{6(1-\gamma)}}\chi^{a}(z),$
$\displaystyle U(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{c}{6\gamma}}J^{0}(z)+\sqrt{\frac{c}{6(1-\gamma)}}K^{0}(z),$
$\displaystyle G^{0}(z)$ $\displaystyle=$ $\displaystyle
2[J^{0}\psi^{0}+\sqrt{\frac{\gamma}{6c}}(6-c)\partial\psi^{0}+K^{0}\chi^{0}-\sqrt{\frac{(1-\gamma)}{6c}}(6-c)\partial\chi^{0}$
$\displaystyle+$
$\displaystyle\sqrt{\frac{6\gamma}{c}}(J^{+i}\psi^{i}+2\psi^{1}\psi^{2}\psi^{3})+\sqrt{\frac{6(1-\gamma)}{c}}(J^{-i}\chi^{i}+2\chi^{1}\chi^{2}\chi^{3})],$
$\displaystyle G^{1}(z)$ $\displaystyle=$ $\displaystyle
2[J^{0}\psi^{1}+\sqrt{\frac{\gamma}{6c}}(6-c)\partial\psi^{1}+K^{0}\chi^{1}-\sqrt{\frac{(1-\gamma)}{6c}}(6-c)\partial\chi^{1}$
(12) $\displaystyle+$
$\displaystyle\sqrt{\frac{6\gamma}{c}}(-J^{+1}\psi^{0}+J^{+2}\psi^{3}-J^{+3}\psi^{2}-2\psi^{0}\psi^{2}\psi^{3})$
$\displaystyle+$
$\displaystyle\sqrt{\frac{6(1-\gamma)}{c}}(-J^{-1}\chi^{0}+J^{-2}\chi^{3}-J^{-3}\chi^{2}-2\chi^{0}\chi^{2}\chi^{3})],$

with cyclic expression for $G^{2}(z)$ and $G^{3}(z)$.

Being a complete free field realization of the large N=4 superconformal
symmetry this one has the $\gamma\leftrightarrow 1-\gamma$ duality or $Z_{2}$
automorphism. Its implications and utility in the present case is discussed in
the next section.

##  5 Implications for Holographic Dual to
$AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ Superstrings

The free field realization above is complete in the sense that we can now get
two different singular limit of the same realization and get $AdS_{3}\times
S_{+}^{3}\times T^{4}$ and $AdS_{3}\times S_{-}^{3}\times T^{4}$ geometries
with general radii for $S_{\pm}^{3}$ because remaining $SU(2)$’s have general
values of the levels. The realizations of the middle $N=4$ superconformal
algebra are:

$\displaystyle\hat{T}(z)$ $\displaystyle=$
$\displaystyle-J^{0}J^{0}+\frac{6-\hat{c}}{\sqrt{6\hat{c}}}\partial
J^{0}-K^{0}K^{0}-\frac{6}{\hat{c}}J^{+i}J^{+i}-\frac{6}{\hat{c}}V^{i}V^{i}$
$\displaystyle+\psi^{a}\partial\psi^{a}+\chi^{a}\partial\chi^{a},$
$\displaystyle\hat{U}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\hat{c}}{6}}K^{0}(z),~{}~{}~{}~{}~{}\hat{Q}^{a}(z)=\sqrt{\frac{\hat{c}}{6}}\chi^{a}(z),~{}~{}\hat{U}^{i}(z)=V^{i}(z),$
$\displaystyle\hat{A}^{+1}(z)$ $\displaystyle=$ $\displaystyle
J^{+1}-\psi^{0}\psi^{1}+\psi^{2}\psi^{3}-\chi^{0}\chi^{1}+\chi^{2}\chi^{3},{\rm
cyclic~{}for}~{}\hat{J}^{2,3},$ $\displaystyle\hat{G}^{0}(z)$ $\displaystyle=$
$\displaystyle
2[J^{0}\psi^{0}+\frac{6-\hat{c}}{\sqrt{6\hat{c}}}\partial\psi^{0}+K^{0}\chi^{0}+\sqrt{\frac{6}{\hat{c}}}(J^{+i}\psi^{i}+2\psi^{1}\psi^{2}\psi^{3})+\sqrt{\frac{6}{\hat{c}}}V^{i}\psi^{i}],$
$\displaystyle\hat{G}^{1}(z)$ $\displaystyle=$ $\displaystyle
2[J^{0}\psi^{1}+\frac{6-\hat{c}}{\sqrt{6\hat{c}}}\partial\psi^{1}+K^{0}\chi^{1}+\sqrt{\frac{6}{\hat{c}}}(-J^{+1}\psi^{0}+J^{+2}\psi^{3}$
(13) $\displaystyle-$ $\displaystyle
J^{+3}\psi^{2}-2\psi^{0}\psi^{2}\psi^{3})+\sqrt{\frac{6}{\hat{c}}}(-V^{1}\chi^{0}+V^{2}\chi^{3}-V^{3}\chi^{2})].$

With cyclic expressions for $\hat{G}^{2}(z)$ and $\hat{G}^{3}(z)$.

and

$\displaystyle\bar{T}(z)$ $\displaystyle=$
$\displaystyle-J^{0}J^{0}-K^{0}K^{0}-\frac{6-\bar{c}}{\sqrt{6\bar{c}}}\partial
K^{0}-\frac{6}{\bar{c}}J^{-i}J^{-i}-\frac{6}{\bar{c}}W^{i}W^{i}$
$\displaystyle+\psi^{a}\partial\psi^{a}+\chi^{a}\partial\chi^{a},$
$\displaystyle\bar{U}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\bar{c}}{6}}J^{0}(z),~{}~{}~{}~{}~{}\hat{Q}^{a}(z)=\sqrt{\frac{\bar{c}}{6}}\psi^{a}(z),~{}~{}\bar{U}^{i}(z)=W^{i}(z),$
$\displaystyle\bar{A}^{-1}(z)$ $\displaystyle=$ $\displaystyle
J^{-1}-\psi^{0}\psi^{1}+\psi^{2}\psi^{3}+\chi^{0}\chi^{1}+\chi^{2}\chi^{3},{\rm
cyclic~{}for}~{}\hat{J}^{2,3},$ $\displaystyle\bar{G}^{0}(z)$ $\displaystyle=$
$\displaystyle
2[J^{0}\psi^{0}-\frac{6-\bar{c}}{\sqrt{6\bar{c}}}\partial\chi^{0}+K^{0}\chi^{0}+\sqrt{\frac{6}{\bar{c}}}(J^{-i}\chi^{i}+2\chi^{1}\chi^{2}\chi^{3})+\sqrt{\frac{6}{\bar{c}}}W^{i}\psi^{i}],$
$\displaystyle\bar{G}^{1}(z)$ $\displaystyle=$ $\displaystyle
2[J^{0}\psi^{1}+K^{0}\chi^{1}-\frac{6-\bar{c}}{\sqrt{6\bar{c}}}\partial\chi^{1}+\sqrt{\frac{6}{\bar{c}}}(-W^{1}\psi^{0}+W^{2}\psi^{3}-W^{3}\psi^{2})$
(14) $\displaystyle+$
$\displaystyle\sqrt{\frac{6}{\bar{c}}}(-J^{-1}\chi^{0}+J^{-2}\chi^{3}-J^{-3}\chi^{2}-2\chi^{0}\chi^{2}\chi^{3})].$

With cyclic expression for $\bar{G}^{2}(z)$ and $\bar{G}^{3}(z)$.

On the geometrical side we learn that the $AdS_{3}\times S^{3}\times
S^{3}\times S^{1}$ geometry should go to $AdS_{3}\times S^{3}\times T^{4}$
geometry in the Penrose limit in case of geometry and Inönü-Wigner contraction
of free field realization. Moreover this should happen in two independent ways
to give $AdS_{3}\times S_{+}^{3}\times T^{4}$ and $AdS_{3}\times
S_{-}^{3}\times T^{4}$ geometries.

In our view the complete FFR of the large $N=4$ SCA is the key to address the
problem of finding the CFT that is holographically dual to superstrings moving
on $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ backgrounds. Though the
authors of [6] got distracted by non-associativity in the procedure of
combining two STP realizations but we nevertheless get a complete FFR of the
relevant superconformal symmetry. We shall further present another route to
obtain the complete FFR in [23]. The next task is to take advantage of the
complete FFRs to solve the issue of holographic dual to strings on
$AdS_{3}\times S^{3}\times S^{3}\times S^{1}$.

Acknowledgments: This work was carried out as part of Mohsin Ilahi’s Ph.D.
thesis work.

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