s O1 O O O Ofigure m[#4]#3

# ‘Sawfish’ Photonic Crystal Cavity for Near-Unity Emitter-to-Fiber
Interfacing in Quantum Network Applications

Julian M. Bopp  Humboldt-Universität zu Berlin, Department of Physics, 12489
Berlin, Germany  Ferdinand-Braun-Institut gGmbH, Leibniz-Institut für
Höchstfrequenztechnik, 12489 Berlin, Germany  Matthias Plock  Zuse Institute
Berlin (ZIB), 14195 Berlin, Germany  Tim Turan  Humboldt-Universität zu
Berlin, Department of Physics, 12489 Berlin, Germany  Gregor Pieplow
Humboldt-Universität zu Berlin, Department of Physics, 12489 Berlin, Germany
Sven Burger  Zuse Institute Berlin (ZIB), 14195 Berlin, Germany  JCMwave GmbH,
14050 Berlin, Germany  Tim Schröder  Humboldt-Universität zu Berlin,
Department of Physics, 12489 Berlin, Germany  Ferdinand-Braun-Institut gGmbH,
Leibniz-Institut für Höchstfrequenztechnik, 12489 Berlin, Germany
Corresponding author: Tim Schröder<EMAIL_ADDRESS>
###### Abstract

Photon loss is one of the key challenges to overcome in complex photonic
quantum applications. Photon collection efficiencies directly impact the
amount of resources required for measurement-based quantum computation and
communication networks. Promising resources include solid-state quantum light
sources, however, efficiently coupling light from a single quantum emitter to
a guided mode remains demanding. In this work, we eliminate photon losses by
maximizing coupling efficiencies in an emitter-to-fiber interface. We develop
a waveguide-integrated ‘Sawfish’ photonic crystal cavity and use finite
element simulations to demonstrate that our system transfers, with
$$97.4$\,\%$ efficiency, the zero-phonon line emission of a negatively-charged
tin vacancy center in diamond adiabatically to a single-mode fiber. A
surrogate model trained by machine learning provides quantitative estimates of
sensitivities to fabrication tolerances. Our corrugation-based design proves
robust under state-of-the-art nanofabrication parameters, maintaining an
emitter-to-fiber coupling efficiency of $$88.6$\,\%$. To demonstrate its
potential in reducing resource requirements, we apply the Sawfish cavity to a
recent one-way quantum repeater protocol.

##  1 Introduction

Photon losses play a critical role in realizing quantum technology
applications [1, 2]. In quantum communication protocols [3] and optical one-
way quantum computation [4], the losses must not exceed a particular
threshold. Losing photons in long-distance quantum communication applications
directly limits information transfer rates as well as maximally bridgeable
communication distances to approximately $100\text{\,}\mathrm{km}\text{/}$
[5]. As opposed to classical communication, the distance cannot be extended
compensating losses by signal amplification due to the no-cloning theorem [6].
Quantum repeaters, however, overcome the no-cloning theorem by exploiting
entanglement [7]. One-way quantum repeaters use multi-photon encoding and
quantum error-correcting codes to protect quantum information from losses and
other errors [8, 9, 10, 11, 12, 13, 14].

The amount of photons $N_{\mathrm{ph}}$ required for such multi-photon
entangled states strongly depends on the employed encoding protocol and the
efficiency of transferring photons from one network node to adjacent nodes.
Recently, Borregaard et al. [3] proposed an efficient one-way quantum repeater
protocol that requires node-to-node (emitter-to-detection) efficiencies of
around $$95$\,\%$ depending on the system error rate $\epsilon_{\mathrm{r}}$
[3]. The emitter-to-detection efficiency includes the photon generation and
fiber coupling on the sender and the photon detection efficiency on the
receiver site. In turn, the photon generation involves an interface between a
stationary atom-like system and photons acting as ‘flying qubits’. A photonic
crystal cavity generally lies at the heart of this interface. It enhances
light-matter interaction by means of the Purcell effect [15] and thus the
emitter’s emission rate [16, 17]. Likewise, the cavity improves photon
collection efficiencies [18]. Although significant effort has been spent
towards controlling and increasing light-matter interaction [1, 18], it
remains an open question how to design interfaces that provide the collection
efficiencies required by next generation quantum repeater protocols, including
the aforementioned proposal by J. Borregaard et al. [3].

In this work, we propose a nanophotonic cavity-to-fiber interface based on a
novel ‘Sawfish’ design (Fig. 1) which reveals near-unity emitter-to-fiber
coupling efficiencies. We benchmark its performance evaluating the cost
function defined by Borregaard et al. [3]

$C=\frac{1}{\Gamma_{\mathrm{tcs}}fp_{\mathrm{trans}}(\eta)}\frac{mL_{\mathrm{att}}}{\tau_{\mathrm{ph}}L}\,,$
(1)

as a function of the emitter-to-fiber collection efficiency $\eta$.
$\Gamma_{\mathrm{tcs}}$ is the tree-cluster state generation rate, $f$ the
secret-bit fraction of the transmitted qubits [19], $p_{\mathrm{trans}}(\eta)$
the transmission probability that depends on $\eta$ and on the tree-cluster
state configuration, $m$ the number of repeater stations, $L_{\mathrm{att}}$
the optical fiber attenuation length, $\tau_{\mathrm{ph}}$ the photon emission
time, and $L$ the total communication distance. $1/C$ can be interpreted as
the secret key rate in units of the photonic qubit emission time per repeater
station and per attenuation length for a given total distance $L$ [3]. The
detection efficiency is implicitly included in $p_{\mathrm{trans}}(\eta)$ (see
supplementary information for details).

With the Purcell factor $F_{\mathrm{P}}$, we define the emitter-to-fiber
coupling efficiency $\eta$ as

$\displaystyle\eta=\beta_{\mathrm{C}}\\!\left(F_{\mathrm{P}}\right)\times\beta_{\mathrm{WG}}\times\beta_{\mathrm{F}}\times
DW\\!\left(F_{\mathrm{P}}\right)\,,$ (2)

where
$\beta_{\mathrm{C}}\\!\left(F_{\mathrm{P}}\right)=F_{\mathrm{P}}/\left(F_{\mathrm{P}}+1\right)$
is the probability to emit a photon into the cavity mode [20, 21],
$\beta_{\mathrm{WG}}$ the cavity-to-waveguide coupling efficiency, and
$\beta_{\mathrm{F}}$ the waveguide-to-fiber coupling efficiency. By
multiplication with the emitter-specific and Purcell factor-dependent [22]
Debye-Waller factor $DW\\!\left(F_{\mathrm{P}}\right)$, we restrict the
emitter-to-fiber efficiency to the ratio of photons emitted into the
respective emitter’s zero-phonon line (ZPL) [23]. Hence, $\eta$ describes the
overall probability that a single photon is emitted into a solid-state
emitter’s ZPL and then successfully transferred to an optical fiber. $\eta$
drastically affects the cost function $C$ and therefore the cluster state size
$N_{\mathrm{ph}}$ that is best suited for distributing a secret key between
two distant parties (insets in Fig. 1). While $C$ is smoothly decreasing,
$N_{\mathrm{ph}}$ decreases stepwise with increasing system efficiency
underlining that photon losses have to be reduced to a particular threshold.
The cost reduction with increasing emitter-to-fiber efficiency becomes
intuitive considering that lower losses result in fewer photons needed for
error correction. This leads to higher state generation rates
$\Gamma_{\mathrm{tcs}}$ as well as potentially fewer repeater stations.

Figure 1:  Overview of the fiber-coupled Sawfish cavity as a building block
for quantum networks. Several participants (blue nodes) communicate over a
quantum network. Repeater stations (orange nodes) bridge long distances
between the participants. Each repeater station is assumed to be based on a
quantum repeater according to a scheme by J. Borregaard et al. [3] with SnV-
centers embedded in Sawfish cavities as sources for quantum communication
resource states. The SnV- is enclosed by two mirror regions in the optical
cavity’s center enabling Purcell enhancement of the $\gamma_{31}$ transition.
In the sketch, the left mirror possesses a higher reflectivity than the right
mirror. The latter transitions into a tapered region to firstly couple part of
the emitted light adiabatically to a waveguide and subsequently to a fiber.
The inset displays the repeater scheme’s cost function $C$ and the tree-
cluster resource state’s required photon number $N_{\mathrm{ph}}$ as functions
of the SnV-to-fiber efficiency (equation (2)) for a reencoding error
$\epsilon_{\mathrm{r}}=$0.1$\,\text{\textperthousand{}}$. Vertical dashed
lines at $\eta_{\mathrm{SnV}}^{\mathrm{real}}=$88.6$\,\%$ correspond to state-
of-the-art cavity quality factors $2Q\approx$17\,000$$ [24] (see subsection
2.5), lines at $\eta_{\mathrm{SnV}}=$93.4$\,\%$ to $2Q\approx$76\,000$$, which
is expected to be within reach in the close future, and lines at
$\eta_{\mathrm{SnV}}^{\mathrm{real}}=$97.4$\,\%$ to an idealized cavity
design.

For the strong dependence $C\\!\left(\eta\right)$, we optimize the Sawfish
cavity and its emitter-to-fiber interface to examine the repeater performance
under ideal and experimentally feasible efficiencies. As a case study, we
consider the negatively-charged tin vacancy center in diamond (SnV-). The SnV-
possesses a ZPL at $484.3\text{\,}\mathrm{THz}\text{/}$ and a comparably large
ground state splitting of $850\text{\,}\mathrm{GHz}\text{/}$ [25] rendering it
a promising optically-active spin emitter [26]. Our spin-photon interface,
however, is applicable to a large variety of solid-state quantum emitters
allowing for its integration with nanophotonic devices based on different
material platforms.

The Sawfish cavity design provides several advantages compared to present
designs. Particularly, it enables for the first time near-unity emitter-to-
fiber coupling efficiencies of up to $$97.4$\,\%$. In contrast to the
multitude of existing photonic crystal cavity geometries which rely on tiny
hole-like features ‘drilled’ into a (suspended) waveguide [24, 27, 16, 17],
the Sawfish cavity is based on large-size corrugation features and a low-loss
adiabatic taper region. For gentle mode confinement [28] leading to high
cavity quality factors $Q$ as well as for adiabatic cavity-to-waveguide
coupling [29, 18], the cavity features have to be precisely adjusted. For the
latter, this is required to convert Bloch modes in the taper region into
waveguide modes. Avoiding mode mismatch-induced scattering losses at cavity-
waveguide interfaces is not straight forward since it necessitates arbitrary
control of the cavity feature sizes [30, 31]. Thus, ‘conventional’ hole-based
geometries suffer from fabrication difficulties in tapering off their hole
diameters adiabatically [30]. In turn, hole-based Bloch waveguides do not
provide arbitrarily smooth transitions into unperturbed waveguide regions. Our
corrugated design overcomes this limit as it relies on open and relatively
large features that can be tapered down adiabatically. We show that tailoring
the photonic potential barrier formed by the Sawfish cavity’s taper region
allows for waveguide-coupled cavities with distinct cooperativities without
reducing their cavity-to-waveguide coupling efficiencies. For instance, tuning
of the cooperativity and the cavity-waveguide coupling is needed to maximize
the state fidelity within the PEPSI scheme used to map photonic polarization-
encoded qubits onto solid-state spins [1]. Furthermore, replacing holes with
corrugation features minimizes undesired surface effects by increasing the
distance between an embedded emitter and the cavity’s surface. As a
consequence, the Sawfish cavity provides fabrication uncertainty-tolerant
near-unity collection of light generated by e.g. a diamond color center
through an adiabatic cavity-to-waveguide followed by an adiabatic waveguide-
to-fiber transition. Even for cavities with realistic and thus relatively high
loss rates, the emitter-to-fiber coupling efficiency hardly decreases.

After optimizing the performance of the emitter-to-fiber coupling, we explore
how nanofabrication tolerances affect the Sawfish cavity. We develop and apply
a surrogate modeling procedure which yields information on how specific
fabrication parameter distributions influence the expected performance. Vice
versa, the procedure reveals parameter distributions minimally required to
make a performance parameter, like the Purcell factor, exceed a certain
threshold. To showcase this modeling technique, we discuss the effect of non-
ideal emitter placement and deviations in the cavity’s thickness, width, and
side wall angles. Besides gauging the robustness of the Sawfish design in
presence of deviations from its ideal configuration, surrogate modeling aids
to anticipate which aspects of the device fabrication require the highest
precision. To achieve comparability with experimentally realized cavity
designs, we consider realistic cavity parameters. In particular, we employ the
highest experimentally achieved diamond photonic crystal cavity quality factor
$Q$ to date reported on [24] to provide a fair benchmark and to outline how
close state-of-the-art emitter-to-fiber interfaces approach the stringent
requirements of one-way quantum repeaters.

Finally, we study the efficacy of the Sawfish cavity loaded with a variety of
color centers such as the negatively-charged nitrogen (NV-) [32] or
negatively-charged group-IV [33] vacancy centers in diamond. Within the frame
of the quantum repeater protocol by J. Borregaard et al., we demonstrate how
perspective improvements in nanofabrication methods give rise to significantly
enhanced cluster state generation rates.

##  2 Results

Firstly, we outline the physical considerations, which inform the design of
the Sawfish cavity. Secondly, a tapered mirror region is added to the cavity
to adiabatically convert the confined cavity mode into a traveling waveguide
mode. This mode is next transferred to a tapered fiber. Finally, we
investigate the cavity design’s robustness under experimental state-of-the-art
conditions such as non-optimal emitter placement and fabrication tolerances.

###  2.1 Cavity design

For constructing a cavity based on corrugation features, a periodic structure
formed by the cavity’s unit cell must result in a photonic band gap. The band
gap has to be large enough to reliably suppress the transmission of light
around its center frequency. We perform eigenmode simulations to determine the
Sawfish unit cell’s dispersion properties and to identify band gaps centered
around the SnV- ZPL. For diamond with a refractive index of $n=$2.41$$ at
$484.3\text{\,}\mathrm{THz}\text{/}$ [34], we find that a unit cell with
lattice constant $a_{i}$ and a sinusoidal corrugation along the $z$ direction
proportional to $\cos^{e}\\!\left(\pi/a_{i}\times z\right)$ with $e=$2$$
(parameters as indicated in Fig. 2a) yields a gap-midgap ratio of only
$$7.8$\,\%$. Raising the exponent $e$ increases the gap-midgap ratio
dramatically due to a higher refractive index contrast between the unit cell’s
edges and its center. For $e=$6$$ with all other parameters unchanged, a gap-
midgap ratio of $$16.1$\,\%$ is obtained. We choose $e=$6$$ since larger
exponents do not affect the band gap significantly. A larger exponent would
instead reduce the distance between an emitter located at the center of one of
the unit cell’s rectangular interfaces and the unit cell’s curved surface. A
small emitter-surface distance potentially promotes undesired interactions
between the emitter’s dipole and surface charges [35, 36].

With the parameters $T=$133\text{\,}\mathrm{nm}\text{/}$$,
$A_{0}=$65\text{\,}\mathrm{nm}\text{/}$$,
$g=$11\text{\,}\mathrm{nm}\text{/}$$, blue solid and dashed lines in Fig. 2b
show the photonic band diagram for a periodic structure built by concatenating
unit cells with lattice constants $a_{0}=$200\text{\,}\mathrm{nm}\text{/}$$
and $a_{4}=$238.5\text{\,}\mathrm{nm}\text{/}$$, respectively. The unit cells
are optimized to maximize the band gap, to permit the transmission of the SnV-
ZPL for the lattice constant $a_{0}$, and to prohibit the transmission for
$a_{4}$. Unit cells with lattice constants $a_{0}$ guide the ZPL emission as a
dielectric mode which is mostly confined in the diamond material. Larger
lattice constants shift the band diagram towards lower frequencies eventually
making the ZPL emission approach the band gap’s center. Unit cells with
lattice constant $a_{4}$ hence act as mirror elements. The cavity’s center
region gently confines a light mode in the vicinity of the SnV- ZPL if $N$
unit cells with increasing lattice constants $a_{0},a_{1},\dots a_{N-1}$ are
arranged on either side around the cavity’s $z=0$ symmetry plane. We reach the
highest quality factor $Q=$1.06\text{\times}{10}^{6}$$ with
$\left[a_{1},\allowbreak{}a_{2},\allowbreak{}a_{3}\right]=\left[$207.0\text{\,}\mathrm{nm}\text{/}$,\allowbreak{}$217.9\text{\,}\mathrm{nm}\text{/}$,\allowbreak{}$228.4\text{\,}\mathrm{nm}\text{/}$\right]$
in addition to $a_{0}$ and $a_{4}$ as mentioned above. $a_{5}$ to $a_{N-1}$
equal $a_{4}$. Fig. 2c visualizes the resulting electric field intensity in
the cavity.

With the cavity’s quality factor $Q$ and its mode volume $V\\!$, the overall
emission rate of an emitter embedded into the cavity is increased by the
Purcell factor $F_{\mathrm{P}}\sim Q/V$ compared to its bulk emission rate
[15]. A high quality factor in combination with a small mode volume therefore
also raises the ZPL emission through an increased Debye-Waller factor
$DW(F_{\mathrm{P}})$ and hence the emitter-to-fiber coupling efficiency
$\eta$. Since each mirror element constituting the cavity contributes to a
photonic potential barrier [37], the quality factor rapidly increases for a
critical amount of mirror elements as soon as tunneling of the cavity mode
through the barrier approaches full suppression (Fig. 2d). In our case,
simulation-based state-of-the-art [38, 24] quality factors above
$1\text{\times}{10}^{6}$ can be reached for $N>20$. The mode volume stays
comparably low at $$0.8$\,\left(\lambda/n\right)^{3}$ over the full range of
investigated cavity period counts with $\lambda$ being the individual
wavelength of the respective cavity’s resonance mode.

As an example, we have chosen the cavity parameters to accommodate the SnV-
ZPL at $484.3\text{\,}\mathrm{THz}\text{/}$. However, the design concept
outlined here can easily be adapted to other defect centers as well as to
completely different material systems. Larger gap widths $g$ might increase
the stability of fabricated devices in trade for slightly narrower band gaps.

Figure 2: Sawfish unit cell and cavity design. a Unit cell of the Sawfish
cavity with its characteristic parameters. $a_{i}$ describes its lattice
constant, $2T$ its thickness, $2A_{0}$ the (untapered) amplitude of a
$\cos^{6}$ function offset by $g$ from the $x=0$ symmetry plane (dashed line)
defining the unit cell’s profile. b Band diagram of a periodic array of
Sawfish unit cells showing TM-like modes. Light modes inside the light cone
(gray-shaded area given by $\nu\geq c\left|k\right|$ with $c$ being the speed
of light) may escape the structure. Other components propagate as Bloch modes
indicated by blue solid and dashed lines for lattice constants
$a_{0}=$200\text{\,}\mathrm{nm}\text{/}$$ and
$a_{4}=$238.5\text{\,}\mathrm{nm}\text{/}$$, respectively. Between the
fundamental Bloch mode (dark blue line) and the next higher air mode (light
blue line), there is a $88.7\text{\,}\mathrm{THz}\text{/}$-wide band gap
(blue-shaded area) for a unit cell lattice constant of $a_{0}$. For this
lattice constant, the SnV- ZPL emission (red line) is guided as the
fundamental mode. Emission is suppressed for a unit cell lattice constant of
$a_{4}$ by the ZPL falling into the band gap (blue-dashed area). The insets
portray the electric field intensity of the modes highlighted with blue
diamonds. c Sawfish cavity geometry. The two symmetric halves of the cavity
consist of $N$ concatenated Sawfish unit cells each with lattice constants
increasing from $a_{0}$ to $a_{4}$. The center plane displays the electric
field intensity. d Convergence behavior of the cavity’s quality factor $Q$
(red) and its mode volume $V$ (blue) depending on the amount of concatenated
unit cells counting from on the cavity’s central $z=0$ symmetry plane.

###  2.2 Cavity to waveguide coupling

Controlled out-coupling of the light mode confined in the cavity requires to
weaken the cavity’s reflectivity at one side of the emitter. In addition to
lowering the reflectivity, we damp the amplitude of the corrugation features
to adiabatically transfer the Bloch mode to a waveguide mode. Hence, an
asymmetric cavity consisting of a weak and a strong mirror needs to be
constructed. The weak mirror consists of $N_{\mathrm{w}}$ unit cells, the
strong mirror of $N_{\mathrm{s}}$ cells, respectively. The unit cells forming
the weak mirror stay at a constant amplitude for $N_{\mathrm{w}}-M$ periods
until they taper down to a waveguide with total width
$2\left(A_{0}+A_{M}+g\right)$ over $M$ periods according to the profile given
by a third-order polynomial as stated in Fig. 3a. $A_{M}$ determines the
waveguide’s width. Barely affecting the cavity-to-waveguide coupling
efficiency, it can be chosen to fit the subsequent fiber coupling
requirements. We find by Bayesian optimization [39] that a profile following a
third-order-polynomial with coefficients $c_{1}=$0.275$$, $c_{2}=$2.243$$ and
$A_{M}=$2.5\text{\,}\mathrm{nm}\text{/}$$ is most efficient for our Sawfish
geometry.

For now, we focus on an ideal system with $N_{\mathrm{s}}=$23$$ and a dipole
perfectly overlapping with the desired TM-like mode. A corresponding untapered
symmetric cavity with $N=23$ possesses a central frequency
$\omega_{\mathrm{c}}=2\pi\times$484.70\text{\,}\mathrm{THz}\text{/}$$ and an
intrinsic scattering loss rate
$\kappa_{\mathrm{s}}=\omega_{\mathrm{c}}/Q_{\mathrm{s}}=2\pi\times$463.8\text{\,}\mathrm{MHz}\text{/}$$
(Fig. 2). By choosing weak mirrors with different amounts of contributing unit
cells $N_{\mathrm{w}}$ and different taper lengths $M$, the Purcell factor
$F_{\mathrm{P}}$ and thus the emitter-cavity cooperativity can be adjusted
over a wide range (Fig. 3b). The waveguide coupling efficiency
$\beta_{\mathrm{WG}}$ defined as the ratio of the Poynting vector integrated
over the waveguide’s end facet to the Poynting vector integrated over the
entire computational domain remains at approximately
$\beta_{\mathrm{WG}}=$99$\,\%$ over the parameter range significant for
cooperativity tuning (Fig. 3c). In turn, the Purcell factor might be
arbitrarily chosen without causing additional scattering losses at the tapered
region.

For each weak mirror period count $N_{\mathrm{w}}$, there is a range of taper
lengths which maximize both, the Purcell factor as well as the waveguide
coupling efficiency. In contrast, the Purcell factor drops if $M$ is chosen
too high for the cavity-to-waveguide loss rate becoming excessive. If $M$ is
chosen too low, $F_{\mathrm{P}}$ increases, but suddenly $\beta_{\mathrm{WG}}$
drops for the cavity-to-waveguide loss rate being reduced to the intrinsic
loss rate $\kappa_{\mathrm{s}}$. Such dependencies emphasize the need to
thoroughly design tapered regions interfacing cavities with waveguides and not
to neglect their influence on the cavities themselves.

To experimentally investigate group-IV defects in diamond like the SnV-, often
highly pure CVD diamond samples are ion-implanted and subsequently annealed
[25, 16, 17]. The respective off-the-shelf diamond substrates are usually cut
along the crystallographic $\left[100\right]$ direction causing dipoles to
enclose an angle of $$54.7$^{\circ}$ with each cartesian axis
($\left[111\right]$ direction) [40, 41]. Consequently, for modeling a
realistic scenario, the effect of rotating the dipole embedded in our Sawfish
design within a plane spanned by the $\left[111\right]$
($\alpha=$54.7$^{\circ}$) and e.g. the $\left[010\right]$
($\alpha=$0$^{\circ}$) direction has to be examined. We observe that the
Purcell factor follows a $F_{\mathrm{P}}\sim
g^{2}\sim\cos^{2}\\!\left(\alpha\right)$ law since the coupling constant $g$
depends on a scalar product between the cavity mode at the dipole’s position
and its orientation. A realistic dipole orientation reduces the Purcell factor
to approximately $$33$\,\%$ of its value obtained for an ideal orientation.
Remarkably, the waveguide coupling efficiency is hardly affected. It is
slightly reduced to $$99.8$\,\%$ of its ideal value (see supplementary
information).

Figure 3: Waveguide-coupled Sawfish cavity design. a The waveguide-coupled
cavity consists of a strong mirror region of $N_{\mathrm{s}}$ unit cells with
amplitude $2A_{0}$ left of the emitter indicated by the small star and a weak
mirror region of $N_{\mathrm{w}}$ unit cells right of the emitter. The
amplitudes $2A_{i}$ ($i\in\left[0,M\right]$) of the unit cells within the weak
mirror region stay constantly at $2A_{0}$ for $N_{\mathrm{w}}-M$ periods
(beginning at the emitter) and taper down to the final amplitude $A_{0}+A_{M}$
according to a third-order polynomial with coefficients $c_{j}$ along the
taper region $M$ as depicted in the inset. The center plane displays the
logarithm of the electric field intensity. b Purcell factor $F_{\mathrm{P}}$
and c waveguide coupling efficiency $\beta_{\mathrm{WG}}$ depending on the
amount of unit cells forming the weak mirror and on the length of the taper
region. $N_{\mathrm{s}}$ is fixed to $23$. The data point highlighted in green
denotes the configuration with the highest waveguide coupling efficiency. For
each point, the [010]-oriented dipole’s emission frequency has been set to the
respective configuration’s resonance.

###  2.3 Waveguide to fiber coupling

Aiming to eventually couple light generated by a cavity-coupled emitter into a
fiber network, we show that the waveguide attached to our Sawfish cavity
allows for efficient fiber coupling. We use a waveguide-fiber interface
according to [42]. The waveguide narrows down to its final width
$w_{0}=$5\text{\,}\mathrm{nm}\text{/}$$ under an angle $\gamma_{\mathrm{WG}}$.
A conical fiber tip with a core refractive index of $n=$1.463$$ tapered down
to a radius $r_{0}=$5\text{\,}\mathrm{nm}\text{/}$$ under an angle
$\gamma_{\mathrm{F}}$ is put on top of the waveguide overlapping it for a
length $v$ (Fig. 4).

A waveguide-to-fiber transmission $\beta_{\mathrm{F}}$ of up to
$\beta_{\mathrm{F}}=$99.4$\,\%$ is reached for
$\gamma_{\mathrm{WG}}=$0.3$^{\circ}$, $\gamma_{\mathrm{F}}=$0.7$^{\circ}$ and
an overlap $v=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$$ (inset in
Fig. 4a). The effective refractive index of the optical mode traveling through
the combined system initially resembles the effective refractive index of the
fundamental waveguide mode, but approaches the one of the fundamental fiber
mode emphasizing successful evanescent coupling [42] (see supplementary
information).

Figure 4: Fiber-coupling of the waveguide attached to the Sawfish cavity. a
Simulated model of the tapered waveguide (gray-shaded) coupled to a tapered
fiber placed on top (blue-shaded) including three cross-sectional views of the
electric field intensity. The inset shows the dependence of the waveguide-to-
fiber transmission $\beta_{\mathrm{F}}$ on the taper angles as defined in b.
The length of the overlapping region $v$ has been optimized for each set of
angles. b Top view of the waveguide-fiber system. A diamond waveguide (red)
with a thickness of $2T$ and an initial total width of
$2\left(A_{0}+A_{M}+g\right)$ as it is attached to the Sawfish cavity is
tapered down under an angle $\gamma_{\mathrm{WG}}$ to a final width $w_{0}$. A
tapered fiber (blue) modeled as a conical frustum with a final radius $r_{0}$
and a taper angle $\gamma_{\mathrm{F}}$ overlaps the tapered waveguide for a
length $v$ measured from the waveguide’s tip. Gray dotted lines denote $z$
coordinates of the planes displaying cross-sectional intensities in a.

###  2.4 Dipole displacement and fabrication tolerances

Until this point, our investigation of the Sawfish cavity was mainly based on
ideal conditions disregarding scattering losses due to e.g. fabrication
tolerances or rough surfaces. To explore how fabrication tolerances impact the
Sawfish cavity’s performance, we perform Monte Carlo sampling on a surrogate
model. Once the surrogate model is trained with computationally expensive
simulation data for a set of fabrication parameters, any uncertainty
distribution of the selected parameters can be evaluated quickly (see
supplementary information). Lacking the need for further computationally
expensive simulations when exchanging the uncertainty distributions renders
the surrogate modeling procedure a powerful tool to efficiently analyze the
impact of fabrication tolerances on any nanostructure. From now on, we
consider realistically $\left[111\right]$-oriented dipoles.

With a dipole displaced by distances $\left[\Delta x,\allowbreak{}\Delta
y,\allowbreak{}\Delta z\right]$ scattering around a zero mean value (ideal
dipole position) with standard deviations of $25\text{\,}\mathrm{nm}\text{/}$
each (Fig. 5a), we obtain
$\overline{F}_{\mathrm{P}}=\left($91.4$\pm$3.2$\right)^{+$18.3$}_{-$26.0$}$
and
$\overline{\beta}_{\mathrm{WG}}=\left($0.9863$\pm$0.0014$\right)^{+$0.0003$}_{-$0.0014$}$
as median values for the Purcell factor and the waveguide coupling efficiency,
respectively. The values’ uncertainties denote combined Monte Carlo and
surrogate uncertainties. Lower and upper bounds indicate lower and upper
standard deviations of the values’ distributions. Fig. 5b and c present the
dependencies $F_{\mathrm{P}}\left(\Delta x,\Delta y,\Delta z\right)$ and
$\beta_{\mathrm{WG}}\left(\Delta x,\Delta y,\Delta z\right)$ as predicted by
the surrogate model. While the Purcell factor only diminishes for high
deviations along the $x$\- and $z$-axis, it is even less affected by dipoles
displaced along the $y$-axis. This effect is understandable considering the
cavity mode profile (inset in Fig. 2b). The electric field spreads further in
$y$-direction compared to the other directions causing the mode overlap to be
less affected by dipole displacements in this direction. In contrast,
$\beta_{\mathrm{WG}}$ is hardly influenced by any displacement.

Likewise, we investigate fabrication geometry parameters $\left[\Delta
T,\allowbreak{}\delta,\allowbreak{}\Delta g\right]$ distributed around their
ideal design values
$\left[$0\text{\,}\mathrm{nm}\text{/}$,\allowbreak{}$90\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$,\allowbreak{}$0\text{\,}\mathrm{nm}\text{/}$\right]$
(Fig. 5d). Rounded corners along the edges highlighted with orange lines in
Fig. 5d as another relevant parameter do not have any influence up to a corner
rounding radius of at least $5\text{\,}\mathrm{nm}\text{/}$. Fig. 5e and f
display predictions of the single fabrication parameters’ impact on
$F_{\mathrm{P}}$ and $\beta_{\mathrm{WG}}$, respectively. In general,
$\beta_{\mathrm{WG}}$ is affected much less than $F_{\mathrm{P}}$ by
fabrication tolerances. However, the Purcell factor strongly depends on the
correct choice of fabrication parameters. This is the case since fabrication
uncertainties shift the unit cells’ band structures away from the cavity
condition rapidly. The system then becomes a pure waveguide supporting Bloch
modes, but no resonant cavity mode, which is expressed in regions with low
Purcell factors but still high waveguide coupling efficiencies. Some
parameters reveal correlations. For instance, a cavity with a lowered side
wall angle works efficiently if the gap is enlarged at the same time. Thus, we
emphasize the need for precise and controllable nanofabrication techniques.

For standard deviations
$\left[$0.8\text{\,}\mathrm{nm}\text{/}$,\allowbreak{}$0.1\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$,\allowbreak{}$0.8\text{\,}\mathrm{nm}\text{/}$\right]$
of the fabrication geometry parameters $\left[\Delta
T,\allowbreak{}\delta,\allowbreak{}\Delta g\right]$, we obtain
$\overline{F}_{\mathrm{P}}=\left($48.5$\pm$26.5$\right)^{+$48.0$}_{-$33.5$}$
and
$\overline{\beta}_{\mathrm{WG}}=\left($0.929$\pm$0.054$\right)^{+$0.057$}_{-$0.141$}$.
$\overline{F}_{\mathrm{P}}$ and $\overline{\beta}_{\mathrm{WG}}$ describe
median values for the ratio of $$95$\,\%$ working Sawfish cavities (see
supplementary information) assuming the given fabrication tolerances and
cavity tuneability within a frequency range between
$483.6\text{\,}\mathrm{THz}\text{/}$ and $485.1\text{\,}\mathrm{THz}\text{/}$.

Figure 5: Influence of dipole displacement and fabrication tolerances. a For
ideal fabrication parameters, the $\left[111\right]$-oriented dipole
(indicated as a star) is displaced along the cartesian axes by distances
$\Delta x$, $\Delta y$, and $\Delta z$ measured from the cavity’s center
point. b, c Colors describe the resulting Purcell factor $F_{\mathrm{P}}$ (b)
and waveguide coupling efficiency $\beta_{\mathrm{WG}}$ (c) for displaced
dipoles. d With an ideally positioned, $\left[111\right]$-oriented dipole,
fabrication geometry parameters are altered. The cavity’s thickness is varied
by $2\Delta T$ and its gap width by $2\Delta g$. Its side wall angle $\delta$
is allowed to deviate from $$90$^{\circ}$. Orange lines denote edges for which
the influence of rounded corners is investigated. e, f Colors describe the
Purcell factors $F_{\mathrm{P}}$ (e) and waveguide coupling efficiencies
$\beta_{\mathrm{WG}}$ (f) for deviations from ideal fabrication geometry
parameters. Predictions by the surrogate model with high uncertainties have
been removed from the displayed planes.

###  2.5 Experimental state-of-the-art conditions

In the previous section, we assessed how fabrication tolerances explicitly
arising from displaced dipoles or deviating fabrication parameters impact the
Sawfish cavity. In this section, we instead investigate the influence of
reduced quality factors without defining the origin of the respective
additional loss channel. Combining both approaches allows to relate desired
quality factors and thus also Purcell factors to minimally required dipole
displacement distributions and fabrication tolerances.

To represent experimental state-of-the-art conditions without modeling every
possible fabrication uncertainty explicitly, we fix the quality factor of an
untapered symmetric Sawfish cavity to $2Q\approx$17\,000$$ by reducing its
length to $N_{\mathrm{s}}=N_{\mathrm{w}}=$11$$ (Fig. 2d). This value is in the
order of magnitude of a currently fabricated photonic crystal cavity [24]. The
factor $2$ takes into account that half of the light leaving a symmetric
cavity will be collected through the weak mirror of a corresponding tapered
asymmetric cavity. The collected part does not contribute to the losses. It is
coupled to the attached waveguide instead. Next, we introduce such a tapered
region to couple the cavity with intrinsic scattering losses and a realistic
$\left[111\right]$-oriented dipole to a waveguide. For the best compatible
taper region as highlighted in Fig. 6, we obtain $F_{\mathrm{P}}=$46.4$$ and
$\beta_{\mathrm{WG}}=$92.9$\,\%$ which agree well with the fabrication
geometry parameters $\left[\Delta T,\allowbreak{}\delta,\allowbreak{}\Delta
g\right]$ distributed as described in subsection 2.4. Hence, surrogate
modeling allows to directly map a desired quality factor to requirements on
fabrication tolerances.

The Purcell factor directly determines $\beta_{\mathrm{C}}$ as well as the
Debye-Waller factor $DW_{\mathrm{SnV}}$. With the emission rate into the ZPL
$\gamma_{31}$ (inset in Fig. 1), the emission rate into the phonon sideband
(PSB) $\gamma_{\mathrm{PSB}}$ and the SnV-’s natural Debye-Waller factor in
bulk $DW_{\mathrm{SnV}}^{0}\approx$60$\,\%$ [41], its Debye-Waller factor in
presence of Purcell enhancement becomes

$DW_{\mathrm{SnV}}\left(F_{\mathrm{P}}\right)=\frac{\gamma_{31}F_{\mathrm{P}}\left(1-DW_{\mathrm{SnV}}^{0}\right)+DW_{\mathrm{SnV}}^{0}\gamma_{\mathrm{PSB}}}{\gamma_{31}F_{\mathrm{P}}\left(1-DW_{\mathrm{SnV}}^{0}\right)+\gamma_{\mathrm{PSB}}}$
(3)

as plotted in Fig. 6a (see supplementary information). With an optimized taper
region (Fig. 6b), we estimate the overall emitter-to-fiber collection
efficiency of a lossy realistic system for the SnV- ZPL to
$\eta_{\mathrm{SnV}}^{\mathrm{real}}=$88.6$\,\%$ neglecting further systemic
optical losses [18]. For ideal conditions (configuration highlighted in Fig.
3b and c), the collection efficiency rises to
$\eta_{\mathrm{SnV}}^{\mathrm{ideal}}=$97.4$\,\%$. Within the frame of the
quantum repeater protocol by J. Borregaard et al. [3], such increases in
$\eta_{\mathrm{SnV}}$ reduce the cluster state size $N_{\mathrm{ph}}$ and
raise the cluster state generation rate $\Gamma_{\mathrm{tcs}}$ by an order of
magnitude as shown in the inset of Fig. 1 and as detailed later in Fig. 7.

Figure 6: Debye-Waller factor and ZPL waveguide coupling efficiency for a
$\bm{\left[111\right]}$-oriented SnV- and realistic losses. a Dependence of
different diamond defect centers’ Debye-Waller factors $DW$ on the Purcell
factor $F_{\mathrm{P}}$. The red cross at $F_{\mathrm{P}}=$46.4$$ highlights
the SnV--based Sawfish cavity configuration with the highest efficiency for
realistic losses (green circle in b). The GeV-’s Debye-Waller factor is not
depicted since it matches the SnV-’s curve. b SnV- ZPL waveguide coupling
efficiency $\beta_{\mathrm{WG}}\times DW_{\mathrm{SnV}}$ for different amounts
of unit cells forming the weak mirror and different lengths of the taper
region. The encircled data point denotes the configuration with the highest
efficiency. The dipole has been adjusted to the cavity’s resonance for each
point.

##  3 Discussion

We introduced and optimized a waveguide-coupled ‘Sawfish cavity’ and estimated
the overall probability of successfully transferring ZPL photons of an
embedded SnV- to an optical fiber. For ideal conditions, a probability of
$\eta_{\mathrm{SnV}}^{\mathrm{ideal}}=$97.4$\,\%$ was obtained which dropped
for state-of-the-art cavity losses [24] only slightly to
$\eta_{\mathrm{SnV}}^{\mathrm{real}}=$88.6$\,\%$. Applying a Monte Carlo
sampling method performed on a surrogate model enabled us to map specific
cavity losses to a set of corresponding fabrication tolerances. Even taking
these scattering losses into account, the overall efficiency
$\eta_{\mathrm{SnV}}^{\mathrm{real}}$ exceeds to date findings in recent work
for hole-based photonic crystal cavities [18]. We highlighted the benefits of
extremely efficient spin-photon interfaces by applying the Sawfish cavity to
the one-way quantum repeater scheme of J. Borregaard et al. [3]. The repeater
nodes’ resource requirements, like the tree-cluster state sizes, reduced
significantly (inset in Fig. 1). Comparing the repeater’s cost function
(equation (1)) for different reencoding errors (Fig. 7a) supports these
findings. In every case, higher overall emitter-to-fiber efficiencies
drastically lower the overall cost. This result further reinforces the
relevance of improving the overall system efficiency in the presence of higher
reencoding errors. Reducing the intrinsic cavity scattering losses from a
state-of-the-art value towards ideal conditions, the tree cluster state
generation rate $\Gamma_{\mathrm{tcs}}$ related to the cost function by $C\sim
1/\Gamma_{\mathrm{tcs}}$ (see supplementary information) rises by over an
order of magnitude (blue curve in Fig. 7b). Even in case of an efficiently
fiber-coupled photon source without Purcell enhancement, this observation
remains almost unchanged (red curve in Fig. 7b). In presence of Purcell
enhancement and intrinsic scattering losses of $2Q\approx$156\,000$$
corresponding to $\eta_{\mathrm{SnV}}=$94.7$\,\%$, the state generation rate
rises from $1.1\text{\,}\mathrm{MHz}\text{/}$ to
$12.5\text{\,}\mathrm{MHz}\text{/}$. Though, already with an intrinsic state-
of-the-art quality factor of $2Q\approx$17\,000$$ [24], the state generation
rate reaches the $\mathrm{MHz}\text{/}$ regime. Expected improvements in
diamond nanofabrication in combination with our improved interface design thus
put the optical requirements of the one-way repeater approach by J. Borregaard
et al. [3] based on diamond vacancy centers embedded in Sawfish cavities
within reach.

Figure 7: Efficiency of the quantum repeater scheme [3] based on a SnV--loaded
Sawfish cavity. a Cost function $C$ (equation (1)) and b tree cluster state
generation rate $\Gamma_{\mathrm{tcs}}$ (see supplementary information) as
functions of the SnV-to-fiber efficiency $\eta_{\mathrm{SnV}}$. The cost
function is shown for different reencoding errors $\epsilon_{\mathrm{r}}$
while the reencoding error is fixed to
$\epsilon_{\mathsf{r}}=$0.1$\,\text{\textperthousand{}}$ for the tree cluster
state generation rate. Red and blue dashed lines serve as guides to the eye.
Vertical dashed lines match the corresponding dashed lines in the insets of
Fig. 1 denoting cavities with different intrinsic scattering losses.

Although, the Sawfish cavity has been optimized for a SnV- center in diamond
to demonstrate its properties and performance, we emphasize that the cavity
design is adaptable to other diamond defect centers as well as to other
material platforms hosting a quantum emitter. Scaling the cavity’s dimensions
adjusts its resonance frequency [43]. Assuming the same cavity-to-waveguide
and waveguide-to-fiber coupling efficiencies but the respective Debye-Waller
factors, we present the expected system efficiencies for different diamond
defect centers in Tab. 1. Clearly, the Sawfish cavity raises the bulk Debye-
Waller factor $DW^{0}$ of every emitter and thus provides high system
efficiencies not only for the SnV-. Particularly, the NV- being susceptible to
surface charge traps [35, 36] profits from an enlarged distance to surfaces as
granted by the Sawfish design.

Having demonstrated advantages of the Sawfish cavity in simulations, its
performance after diamond nanofabrication and as a building block for photonic
integrated circuits demands further investigation. Concerning fabrication
tolerances, predictions of the trained surrogate model necessitate
verification by analyzing batches of fabricated cavities. To this point, it is
not known to what extent rough surfaces possibly caused by diamond etching
affect the Sawfish cavity’s performance.

Optical losses in any spin-photon interface lead to a tremendously reduced
efficiency in quantum information processing applications. This especially
holds if an emitter has to repeatedly mediate entanglement for resource states
involving hundreds of photons [3] or if quantum memories become involved [44].
Whereas there are most efficient photon detectors with $$99.5$\,\%$ detection
efficiency nowadays [45], equally efficient spin-photon interfaces are still
missing. Our system enables a spin-photon interface exceeding the efficiency
of $$83$\,\%$ as needed by the PEPSI scheme [1], even with intrinsic state-of-
the-art scattering losses. Consequently, we deem the Sawfish cavity design to
be critical in interfacing quantum emitters and memories with near-unity
efficiencies as required for scalable quantum networks.

Table 1: Overall emitter-to-fiber efficiencies for different defect centers in diamond assuming the same waveguide $\left(\beta_{\mathrm{WG}}\right)$ and fiber $\left(\beta_{\mathrm{F}}\right)$ coupling efficiencies as found by simulations for the SnV- and either ideal ($F_{\mathrm{P}}=$270.2$$) or realistic state-of-the-art ($F_{\mathrm{P}}=$46.4$$) conditions. $\tau^{0}$ denotes the considered excited state lifetime in bulk diamond. $\tau^{\mathrm{ideal}}$ and $\tau^{\mathrm{real}}$ denote the calculated Purcell-enhanced lifetimes for ideal and realistic conditions, respectively. |  NV- |  SiV- |  GeV- |  SnV-
---|---|---|---|---
$\tau^{0}$ $[$\mathrm{ns}\text{/}$]$ |  $12.2$ [46] |  $1.7$ [47] |  $3.8$ [48] |  $4.5$ [49]
$\tau^{\mathrm{ideal}}$ $[$\mathrm{ns}\text{/}$]$ | $1.3$ | $0.01$ | $0.03$ | $0.04$
$\tau^{\mathrm{real}}$ $[$\mathrm{ns}\text{/}$]$ | $5.1$ | $0.07$ | $0.19$ | $0.23$
$DW^{0}$ $[\%]$ |  $3$ [50] |  $80$ [51] |  $60$ [52] |  $60$ [41]
$DW^{\mathrm{ideal}}$ $[\%]$ | $89.3$ | $99.9$ | $99.6$ | $99.6$
$DW^{\mathrm{real}}$ $[\%]$ | $59.4$ | $99.2$ | $98.0$ | $98.0$
$\eta^{\mathrm{ideal}}$ $[\%]$ | $87.4$ | $97.6$ | $97.4$ | $97.4$
$\eta^{\mathrm{real}}$ $[\%]$ | $53.8$ | $89.7$ | $88.6$ | $88.6$

##  4 Methods

As promising optically-active solid-state spin system, we exemplarily focus on
negatively-charged tin vacancy centers in diamond (SnV-). Their ZPL emission
at $484.3\text{\,}\mathrm{THz}\text{/}$ [25] is coupled to the Sawfish
cavities. Debye-Waller factors are calculated according to a formalism
outlined in [22]. We investigate the system performing full 3d finite element
(FEM) simulations with the software package JCMsuite [53].

Firstly, the geometry parameters of the Sawfish cavity’s central unit cell,
like its lattice constant $a_{0}$, its thickness $2T$, its amplitude $2A_{0}$,
and its gap width $2g$ are chosen to establish a large enough photonic band
gap centered around the SnV- ZPL. Based on eigenmode computations, the higher-
order lattice constants $a_{i}$ are then optimized for large quality factors
using a Bayesian optimizer [54] implemented in JCMsuite. The same optimizer is
employed for designing the cavity-waveguide and waveguide-fiber interfaces.
Investigating electromagnetic energy fluxes originating from a single dipole
emitter placed at the cavity’s center allows to analyze the system’s
efficiency. Furthermore, we account for distributions of cavity design
parameters by performing Monte Carlo sampling on a surrogate model composed of
Gaussian processes and trained by means of machine learning. With the
surrogate model, we gain insights into the cavity’s robustness under displaced
dipoles and fabrication tolerances. Finally, we apply the optimized system
efficiencies to benchmark the potential performance of a one-way quantum
repeater regarding the cost $C$, the required tree-cluster state size
$N_{\mathrm{ph}}$, and the tree cluster state generation rate
$\Gamma_{\mathrm{tcs}}$ following procedures detailed in [3]. For details, see
supplementary information.

#### Data availability

The datasets generated during the current study are available from the
corresponding author on reasonable request.

#### Code availability

JCMsuite, Python, and MATLAB codes to perform and analyze the FEM and quantum
repeater simulations as well as Python codes for Debye-Waller factor
calculations are available from the corresponding author on reasonable
request.

#### Author contributions

J.M.B. designed the Sawfish cavity and its waveguide coupling. M.P. developed
the procedure to analyze the effect of fabrication tolerances. T.T. simulated
the waveguide-to-fiber coupling. G.P. applied the Sawfish cavity to quantum
repeater protocols. T.S. conceived the project. S.B. and T.S. supervised it.
All authors contributed to the manuscript.

#### Acknowledgement

We thank Tommaso Pregnolato for fruitful discussions about realistic state-of-
the-art fabrication scenarios as well as Kurt Busch, Martin Hammerschmidt, Lin
Zschiedrich, and Philipp-Immanuel Schneider for helpful discussions about
simulation methods.

#### Funding

This project is funded by the German Federal Ministry of Education and
Research (BMBF) within the ‘DiNOQuant’ project (No. 13N14921), the ‘QPIS’
project (No. 16KISQ032K), and the ‘siMLopt’ project (No. 05M20ZAA) as well as
by the European Research Council within the ERC Starting Grant ‘QUREP’ (No.
851810) and by the ‘EMPIR’ program (No. 20FUN05 SEQUME) which is co-financed
by the participating states and the European Union’s Horizon 2020 research and
innovation program. We acknowledge further funding from the Einstein
Foundation Berlin within the frame of the ‘Einstein Research Unit on Quantum
Devices’.

#### Competing interests

The authors declare no conflicting interest.

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# Ensemble Synthetic EHR Generation for Increasing Subpopulation Model’s
Performance

Oriel Perets
Department of Industrial Engineering
Ben-Gurion University
<EMAIL_ADDRESS>

Nadav Rappoport
Department of Information Systems Engineering
Ben-Gurion University
<EMAIL_ADDRESS>

###### Abstract

Electronic health records (EHR) often contain different rates of
representation of certain subpopulations (SP). Factors like patient
demographics, clinical condition prevalence, and medical center type
contribute to this underrepresentation. Consequently, when training machine
learning models on such datasets, the models struggle to generalize well and
perform poorly on underrepresented SPs. To address this issue, we propose a
novel ensemble framework that utilizes generative models. Specifically, we
train a GAN-based synthetic data generator for each SP and incorporate
synthetic samples into each SP training set. Ultimately, we train SP-specific
prediction models. To properly evaluate this method, we design an evaluation
pipeline with 2 real-world use case datasets, queried from the MIMIC database.
Our approach shows increased model performance over underrepresented SPs. Our
code and models are given as supplementary and will be made available on a
public repository.

##  1 Introduction

###  1.1 Background

Electronic health records (EHRs) are valuable resources for healthcare
research, providing a wide range of clinical data, including vital signs,
demographics, diagnosis, admissions, and clinical notes [11]. Previous
research shows this data has the potential to be used to develop prediction
and classification models [9] [8], enable the consolidation of risk score and
alerting mechanisms [6], and inform treatment decisions. However, these
records suffer from many challenges [10], including the underrepresentation of
certain SPs, which can limit the generalizability of machine learning models
trained on this data. Factors such as demographic distribution mostly due to
the geographical location of the medical facility, disease prevalence, and
medical facility type (e.g., children’s hospital, nursing home) can all
contribute to the lack of representation in the records [20]. We address this
issue by proposing an ensemble framework based on generative models. Our
framework introduces a new solution for the problem of highly-imbalanced
medical datasets, using synthetic samples added to the training set of each SP
and training an SP-specific prediction model on the augmented data.

Several existing approaches attempt to address the issue of
underrepresentation in EHRs. Some approaches overcome the imbalance by
rebalancing the data or re-weighting specific samples. Other approaches use
synthetic data generation techniques [14]. Under-sampling includes using fewer
samples from the majority population for rebalancing the training set. While
over-sampling does the opposite. One popular approach for over-sampling is the
Synthetic Minority Over-sampling Technique (SMOTE) [2], which creates
synthetic samples for the minority class by interpolating between existing
samples.

###  1.2 Contribution

The contributions of this work are as follows:

  1. 1.

GAN-based ensemble framework. We propose an ensemble framework, for improving
a model’s prediction performance of underrepresented SPs in a dataset. We
first identify poorly performing SP in the dataset, then iteratively generate
and add various amounts of synthetic samples to the training set of each SP
separately. Followed by training an SP-specific prediction model using the
augmented training data (the training set comprised both real and synthetic
samples). We show our proposed framework results in increased model
performance for underrepresented SP. This framework is independent of the
problem settings, including the prediction model type, the task at hand, and
the feature(s) used for the division of samples into subpopulations.

  2. 2.

An evaluation pipeline for synthetic data. We designed and developed a
pipeline for evaluating the effect of adding synthetic samples to the training
set on model performance, the SP sample size, and the amount of synthetic
samples added. We implemented two real-world use cases and datasets to
evaluate our framework. Our pipeline is generic in a way it can be used to
evaluate the ML efficacy of any synthetic data generator.

Our approach has the potential to improve the generalizability of machine
learning models trained on EHRs by addressing the issue of
underrepresentation. Our ensemble framework is a practical solution that can
be applied to any EHR dataset to improve model performance on SPs and ensure
equitable healthcare delivery. We compare our approach with several baselines,
including the baseline models, and several sophisticated rebalancing
techniques.

##  2 Related Work

The problem of imbalanced data distributions is a well-posed problem in many
applications [15]. The specific characteristics of EHR make them more prone to
highly imbalanced data, making their use for training machine learning models
challenging. Several approaches have been proposed in the past to cope with
highly imbalanced data. Here, we refer to ‘class‘ as the feature, or a
combination of features, used for categorizing the samples into SPs as opposed
to the label class.

Preprocessing approaches are data-centered approaches, which alter the
training set itself, prior to model training. These are the simplest methods
for dealing with imbalanced data. Under-sampling techniques remove samples
from the majority class[20]. Mani and Zhang [19] proposed an approach that
uses a KNN classifier to select and remove majority class samples furthest
from the minority class samples. Chawala et al. [2] introduced the Synthetic
Minority Over-Sampling Technique (SMOTE), upon which variations of the
approach have been suggested [16].

Algorithmic approaches do not alter the training data per se, but instead
adjust the learning or decision process to alter the weight of samples from a
given class in the dataset or modify a penalty per class, resulting in a form
of cost-sensitive learning [7]. Barda et al. [1] proposed a custom
recalibration algorithm using PCE and FRAX to asses the calibration between
SPs.

Synthetic Data for Performance Boosting. Synthetic data generation has been
used for addressing several problems in datasets, including privacy [23, 18],
and utility. To the best of our knowledge, methods using synthetic data
neither used an ensemble approach for SP-specific synthetic data generation
nor evaluated the SP-specific prediction model performance. Other methods did
not explore different amounts of synthetic samples and used an augmented
dataset containing both real and synthetic samples to train the prediction
models. These facts make it challenging to exploit synthetic data generation
techniques to address the problem of imbalanced datasets and evaluate the
efficacy of synthetic samples on SP-specific machine learning performance. To
address this, we developed the proposed framework and evaluation pipeline.
Only techniques developed for, or evaluated on machine learning efficacy are
relevant to this study. Che et al. [3] proposed ehrGAN, and SSL-GAN, a
technique involving using a trained ehrGAN in a semi-supervised manner to
boost a CNN-based prediction model performance. Their approach demonstrated
better prediction model performance when training using only synthetic
samples. Torfi et al. [21] proposed CorGAN, which uses CNNs to capture the
correlation between features in the dataset. Their approach demonstrated
similar prediction model performance when trained only on the generated
synthetic samples, to those achieved using real samples. Chin-Cheong et al.
[5] proposed an approach using WGAN for generating high-utility synthetic EHR
to be used in place of real samples for training machine learning models.
Their approach is evaluated for utility using binary classification trained on
the synthetic samples generated and shows similar, but inferior performance to
those trained on the real samples. Xu et al. [22] proposed the Conditional
Tabular GAN (CTGAN), which is used in this research.

Figure 1: Method Workflow: query MIMIC for a dataset, split to SPs, split each
SP to train and test, generate synthetic samples using CTGAN, and evaluate the
model with different amounts (with respect to the SP’s sample size) of
synthetic samples

##  3 Method

In this section, we present the proposed method in detail. The method is
comprised of five steps, namely: (1) identifying under-performing SPs, (2)
splitting the data into train and test sets, (3) training an SP-specific
synthetic data generator, (4) generating and adding synthetic samples to the
training set, and (5) evaluating the performance. A flowchart is presented in
Figure 1.

###  3.1 Datasets and Prediction Tasks

We used two datasets and two corresponding prediction tasks. We used both
MIMIC-III [13], and MIMIC-IV [12] databases to query the following datasets:
(1) 30-day ICU mortality prediction for patients with sepsis-3 [9] and (2)
early mortality prediction for patients with CHF [8]. The datasets were
queried using BigQuery by replicating the inclusion criteria detailed in the
original papers. Data extraction queries are available in the supplementary
material for ease of replication. Minimal preprocessing including dropping
rows with missing values and label encoding was conducted.

###  3.2 Subpopulation Definitions

For each dataset at hand, we define ‘population marker‘ (PM) as the feature(s)
used for the division into SPs. Here, we demonstrate such division using
patients’ ethnicity as the PM. Ethnicity was used for simplicity and
popularity of the demographic feature in medical records and its known bias.
However, our approach treats SP as an abstract concept and can be replicated
using any feature or combination of features in the dataset, e.g., age group,
lab results, vital signs, a combination of demographic features, etc. We
define an SP suitable for applying this approach as one where model
performance over it is systematically lower than the performance on the
overall dataset.

###  3.3 Ensemble-GAN Training

Once patients are categorized to SPs, we (1) split the data into training
(65%) and test (35%), stratified by the outcome of interest to ensure enough
samples from each outcome label are present for evaluating the prediction
model, and ensure sufficient prevalence of the outcome label in the test set.
We then use the training set to (2) train a Conditional Tabular Generative
Adversarial Network (CTGAN) [22] on each of the underrepresented SPs
separately until convergence. The model’s parameters and the training time for
each use case are detailed in the experiments section. (3) Generate synthetic
samples in a range of amounts from 0 to 1000% of the original training set
size and add those samples to the training set of each SP.

###  3.4 Prediction Models

After acquiring the augmented training sets for each SP, we trained two
prediction models, namely Gradient Boosting Classifiers [4]. XGBoost was
chosen for popularity, performance, and ease of use. The model was implemented
using the Sklearn library with the set parameters to minimize variance between
use cases and iterations. One model is trained on the SP-specific augmented
dataset, consisting of both the real samples (65% of the SP samples), and the
synthetic samples (0-1000%, depending on the iteration), and a second model is
trained on 65% of the full dataset, including all SPs, augmented with the SP-
specific synthetic samples – this is done for evaluating the change in
performance caused by the SP-specific synthetic samples over the entire
dataset.

(a) 30-day mortality prediction for Sepsis-3 patients

(b) Early mortality prediction for CHF patients

Figure 2: SP model performance ROAUC, accuracy, and full population with added
SP synthetic samples ROCAUC with respect to the % (of the SP size) synthetic
samples added.

###  3.5 Evaluation Pipeline

Once the models are trained, we evaluate the models using ROCAUC. However, the
evaluation pipeline allows the use of any metric like: ROCAUC, accuracy,
precision, recall, or PRAUC. Then the results are plotted for each % of
synthetic samples added to the training set. Examples of result plots are
presented in Figure 2. Our released code for this evaluation pipeline can be
used for any prediction task and dataset, and any synthetic data generation
technique. The code for this pipeline is available in the supplementary
material.

Table 1: Entire process training and evaluation times, the hardware used, and the CTGAN parameters. Use case | n | Epochs | Batch size | GPU | Time (hrs)
---|---|---|---|---|---
Sepsis-3 | 4,559 | 10 | 50 | Titan RTX | 01:42
CHF | 11,062 | 10 | 50 | Titan RTX | 02:46

(a) 30-day mortality prediction for Sepsis-3 patients

(b) Early mortality prediction for CHF patients

Figure 3: SP sizes per use case ordered by size.

##  4 Experiments

We queried MIMIC-III and MIMIC-IV using Google’s BigQuery interface, for two
use cases:

  1. 1.

Early Mortality Prediction for Patients with CHF [8]. In this dataset, there
are 11,062 samples, where each sample represents a CHF-diagnosed patient in
the first 24 hours in the ICU.

  2. 2.

30-day Mortality Prediction for Patients with Sepsis-3 [9]. In this dataset
there are 4,559 samples, where each sample represents a Sepsis 3-Diagnosed
patient’s first admission to the ICU.

We split the datasets into SPs based on the patient’s reported ethnicity,
namely, ‘Black’, ‘Asian’, ‘Hispanic’, ‘White’, and ‘Other or Unknown’. SP
sizes for each dataset are presented in Figure 3 below. We later split the
datasets to train (65%) and test (35%) stratified by the outcome label, this
was done to ensure sufficient prevalence in the test set of very small SPs, to
enable the evaluation of model performance. We then trained a baseline
prediction XGBoost model using only the real samples in the training set for
each SP, to identify under-performing (ROCAUC lower than the full population),
underrepresented SPs. ROCAUC of the baseline models is presented in Figure 2,
the baseline model is equivalent to the model with 0% synthetic samples added
to the training set. ‘Hispanic’, ‘Asian’, and ‘Black’ were identified as
underperforming SPs for both use cases with respect to the majority population
- ‘White’. We consider ‘Other’ to not be a homogeneous SP, therefore we remove
it from our experiment. Once underperforming SPs have been identified, we
trained a CTGAN model for each of the SPs. Each CTGAN model was trained using
only the training set samples and was not exposed to the test set. The CTGAN
was trained with a discriminator learning rate of $2\cdot 10^{-6}$, a
generator learning rate of $2\cdot 10^{-4}$, and 5 discriminator steps (k=5)
for each generator step. Epochs, batch size, training time, and hardware is
presented in Table 1. The trained CTGAN for each SP is then used to generate
synthetic samples in different amounts ranging from 0% (no synthetic samples
added to the training set), to 1000% (synthetic samples equivalent to 10x the
SP training set sample size were added to the training set), specifically, the
amounts were [0%, 5%, 10%, 15%, 20%, 25%, 30%, 35%, 40%, 45%, 50%, 60%, 70%,
80%, 90%, 100%, 150%, 200%, 500%, 1000%]. We evaluated the performance using
ROCAUC for each SP, and the percentage of synthetic samples added, results are
displayed in Figure 2.

To further evaluate our framework, we compared it with three baselines models,
namely:

  1. 1.

Baseline ensemble learning - implemented as an individual model trained for
each SP, using only the real samples.

  2. 2.

Synthetic Minority Oversampling Technique (SMOTE) - implemented in the Sklearn
’imblearn’ library [17].

  3. 3.

Random Under Sampling (RUS) - implemented in the Sklearn ’imblearn’ library
[17].

For SMOTE, we first split the data into train (65%) and test (35%) sets,
stratified by ethnicity to ensure the proper representation of SP in both
datasets. We then used the training data to perform SMOTE oversampling,
setting the population marker (ethnicity) as the class feature. This resulted
in equal amounts of samples for each SP. We later trained an XGBoost model
using each SP training set and evaluated the performance using each SP’s test
set.

For RUS, we did not initially split the dataset into train and test, this is
done due to the nature of the under-sampling technique. First, we set the
population marker (ethnicity) as the class feature. Then performing RUS with
an "all" sampling strategy (sampling an amount equal to the smallest minority
class from each SP). We then split each SP individually into train (65%) and
test (35%) sets, stratified by the outcome variable, this was done to ensure
the sufficient prevalence of each outcome in the train and test sets. We later
trained an XGBoost model using each SP training set and evaluated the
performance using the SP’s test set.

Table 2: ROCAUC of the different models for each use case and SP. ’n’ corresponds to the test set sample size. ’Ens.’ is the base ensemble model performance, trained on real SP samples and evaluated on SP samples. The Ens. GAN model performance is ROCAUC for the optimal amount of synthetic samples added to the training set.  Use Case | Subpopulation | n | SMOTE | RUS | Ens. | Ens. GAN
---|---|---|---|---|---|---
30 day mortality with Sepsis-3 | White | 1037 | 0.792 | 0.743 | 0.877 | 0.877
Other | 189 | 0.837 | 0.746 | 0.837 | 0.837
Black | 123 | 0.796 | 0.751 | 0.805 | 0.834
Asian | 53 | 0.767 | 0.807 | 0.751 | 0.903
Hispanic | 47 | 0.642 | 0.780 | 0.556 | 0.695
CHF mortality | White | 2543 | 0.728 | 0.794 | 0.723 | 0.725
Other | 386 | 0.692 | 0.777 | 0.718 | 0.723
Black | 495 | 0.682 | 0.837 | 0.739 | 0.774
Asian | 83 | 0.579 | 0.770 | 0.673 | 0.721
Hispanic | 114 | 0.556 | 0.642 | 0.622 | 0.732

Our approach achieved better performance than the baseline model and SMOTE,
and achived higher performance than RUS in two SPs in the Sepsis-3 Mortality
prediction uses case in one SPs in the CHF mortality use case. While
maintaining results for the full population, when adding synthetic samples to
the training set (Table 2).

##  5 Discussion

In this paper, we attempt to address the issue of the underrepresentation of
certain subpopulations (SPs) in electronic health records (EHRs) and their
impact on the performance of ML models over SPs. We apply a SOTA synthetic
data generation technique (CTGAN) [22], together with an ensemble-like
approach to generating and using synthetic samples. Inspired by our
experimental results and based on two real-world use cases from the MIMIC
database, we suggest this ensemble framework improves the model’s performance
for SPs compared to other methods. Consequently increasing the
generalizability of machine learning models in towards SP which we have a very
low amount of samples. This method, although its simplicity, can drastically
increase the predictive power of ML models for those underrepresented SPs. Our
framework consisted of training a GAN-based synthetic data generator, for each
underrepresented SP separately, and incorporating the synthetic samples into
the training set. Followed by training SP-specific prediction models using the
augmented training data. Existing approaches either used synthetic samples to
boost the full model performance or maintain their performance when trained on
synthetic samples compared to real samples (for increased privacy). We
emphasize our approach of adding synthetic samples to the real data to the
best of our knowledge, has not yet been explored. Specifically, synthetic
samples for the purpose of improving SP-specific performance have not been
explored. While many datasets suffer from highly-imbalanced classes and low
representation of certain SPs which ultimately results in decreased
performance and low generalizability.

Furthermore, to evaluate the effectiveness of our method, we designed and
implemented an evaluation pipeline using two real-world use case datasets
queried from the MIMIC database. The evaluation pipeline evaluated both the
efficacy of the synthetic samples, in terms of SP performance, and provided
evidence our proposed framework has merit, compared to other approaches
including SMOTE, RUS, and a vanilla ensemble. Our experimental results not
only suggest better performance over data-balancing techniques but also
preserve model performance over the entire population. More research is needed
to further ensure the generalizability of the framework, and explore different
amounts of synthetic data and other synthetic data generation methods, and
different ways to define SPs. We used a categorical feature of race and
ethnicity, but other data-driven approaches may be explored.

A limitation of our suggested method compared to other methods is an increase
in training time, as it requires training a GAN and generating samples, and
training predictors using larger training sets. However, there is no effect on
the running time during inference.

###  5.1 Impact

Our ensemble framework is a practical solution that can be applied to any EHR
dataset to improve model performance and allow accurate predictions for small
populations. It is independent of the prediction model, the task at hand, and
the population marker used for SP division, making it useful for other
problems.

##  6 Acknowledgements

We gratefully acknowledge the members of our lab for their helpful
conversations and assistance.

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