# Deterministic Shaping of Quantum Light Statistics

Garrett D. Compton  Department of Physics and Astronomy, Webster Hall,
Washington State University, Pullman, WA 99164-2814<EMAIL_ADDRESS>Mark G. Kuzyk  Department of Physics and Astronomy, Webster Hall, Washington
State University, Pullman, WA 99164-2814

(March 8, 2024)

###### Abstract

We propose a theoretical method for the deterministic shaping of quantum light
via photon number state selective interactions. Nonclassical states of light
are an essential resource for high precision optical techniques that rely on
photon correlations and noise reshaping. Notable techniques include quantum
enhanced interferometry, ghost imaging, and generating fault tolerant codes
for continuous variable optical quantum computing. We show that a class of
nonlinear-optical resonators can transform many-photon wavefunctions to
produce structured states of light with nonclassical noise statistics. The
devices, based on parametric down conversion, utilize the Kerr effect to tune
photon number dependent frequency matching, inducing photon number selective
interactions. With a high amplitude coherent pump, the number selective
interaction shapes the noise of a two-mode squeezed cavity state with minimal
dephasing, illustrated with simulations. We specify the requisite material
properties to build the device and highlight the remaining material degrees of
freedom which offer flexible material design.

##  1 Introduction

Systems that utilize quantum information resources generally take advantage of
coherent macroscopic superposition and/or entanglement. The increased
information density and sensitivity to measurements of such quantum states,
compared to their classical counter parts, is central to their application in
modern quantum information technologies like quantum key distribution[1, 2],
quantum computing [3, 4, 5], and metrology at the fundamental noise limit[6,
7, 8]. Efforts toward engineering the spatio-temporal profile of quantum
states of light have already found great success[9, 10, 11]. This article
focuses instead on shaping the noise statistics of quantum states of light for
such applications. Both macroscopic superposition and entanglement are
generically found in states supported by nonlinear systems [12, 13]. However,
optical nonlinearities are typically weak for closed systems. Much attention
has therefore been paid to developing methods for generating quantum states of
light which utilize wavefunction collapse as a source of nonlinear evolution,
where a particular measurement outcome heralds the desired quantum state [14,
15, 16, 17, 18]. Though hugely successful, heralded state preparation is
inherently non-deterministic, making sequential preparations low probability
events.

The approach presented here overcomes low heralding success rates using the
paradigm of deterministic generation of quantum states. Examples in the
literature can be found where the probabilistic nature of measurement is
circumvented[19, 20, 21, 22, 23, 24, 25, 26, 20, 27, 28, 21, 29, 30, 31].

One example utilizes the quantum Zeno effect [19, 20, 21]. By strongly
coupling a specially engineered system to an environment, the state of the
system is continuously projected to a subspace of states that are effectively
uncoupled from the environment, called a decoherence free subspace (DFS).
External driving fields may then evolve states deterministically and
coherently within the DFS.

With the advent of epsilon-near-zero materials [22] and polariton enhanced
scattering, commonly facilitated by electronically induced transparency, [23,
24, 25, 26], large effective nonlinearities become increasingly realistic as
well. When realized in a high-Q optical resonator, strong optical
nonlinearities with relatively long evolution time and low loss facilitate
deterministic quantum optical state engineering. Trapped atom and other
quantum emitter systems show great promise for this application [20, 27, 28,
21, 29, 30, 31].

Such systems accomplish universal control of an oscillator state by relying on
an array of emitters or by coupling to a single two level emitter. With an
array of quantum emitters, the quantum information content of a state of
interest is first written onto the array, typically in a decoherence free
subspace, then mapped to super-radiant light-matter states which couple to an
optical waveguide [20, 27, 28, 21]. Alternatively, arbitrary state
transformations may be realized in a cavity with a single quantum emitter by
repeatedly addressing the emitter with a control beam to introduce a Berry
phase to cavity field Fock states and applying coherent displacement
operations to the cavity field in sequences prescribed within [29, 30, 31].
Notably, Krastanov et al. [30, 31] utilize a dispersive coupling between the
emitter and a microwave cavity field to realize a selective number-dependent
arbitrary phase (SNAP) gate which selectively rotates the phase of each Fock
state of the cavity field.

This work introduces an alternative technique for precision shaping of quantum
noise statistics without the need for precise emitter engineering and
addressing. We consider a material with first, second, and third order
electric susceptibilities at optical frequencies 111It is straightforward to
generalize our approach for nonperturbative constitutive relations. and show
that when second and third order nonlinearities are comparable in strength,
the optical Kerr effect alters the frequency matching condition for optical
parametric oscillation such that the interaction between field states is
strongly photon number dependent. Our analysis separates the phase rotating
interactions from the mode mixing interactions, providing us a deeper
intuition for the dynamics we observe. Specifically, we first solve the
combined first and third order system exactly and study the second order mode
mixing in the interaction picture. The general dynamical characteristics are
illustrated and we utilize the effect to demonstrate a procedure for
deterministic generation of approximate Fock states, among other interesting
states. General material constraints are highlighted and developed into
figures of merit to guide experimental realizations of this method.

##  2 Background

We begin with a brief discussion of electromagnetic field quantization. Dirac
recognized that Hamiltonian mechanics has simple rules for canonical
quantization, that Lagrangian mechanics is well suited for relativistic gauge
invariant classical field theories, and that they both poorly handle what the
other does well. So, to formulate relativistic quantum theory, he generalized
Hamiltonian mechanics to handle systems with constraints[32]. His mechanics
encodes symmetries that are simple to write into a Lagrangian, into
constraints on a Hamiltonian, recovering Hamilton’s equations of motion with a
generalized Poisson bracket called the Dirac bracket. Canonical quantization
of Dirac’s bracket reveals, in part, Dirac’s relativistic theory of quantum
mechanics. Born and Infeld adopted Dirac’s procedure to show that Maxwell’s
equations in the Coulomb gauge assume a Hamiltonian formulation [33, 34] where
the Hamiltonian is the classical field energy

$H=\int_{V}dv\;\int_{-\infty}^{t}dt\;\frac{E\cdot\partial_{t^{\prime}}D}{2}+\frac{H\cdot\partial_{t^{\prime}}B}{2}$
(1)

and the fields obey a version of the canonical Dirac bracket relations,

$\begin{split}&\\{B^{j}(r),B^{k}(r^{\prime})\\}=\\{D^{j}(r),D^{k}(r^{\prime})\\}=0\\\
&\\{D^{j}(r),B^{k}(r^{\prime})\\}=\epsilon^{jlk}\frac{\partial}{\partial
x^{l}}\delta(r-r^{\prime})\equiv\delta^{tr}(r-r^{\prime})\;,\end{split}$ (2)

with the second equation defining the transverse delta function
$\delta^{tr}(r-r^{\prime})$. 222The transverse delta function captures the
constraints on the field enforced by the Coulomb gauge. In particular, note
that in a medium without free charges or free currents, Maxwell’s equations
ensure $D$ and $B$ are always transverse to the direction of radiation
propagation, unlike $E$ and $H$. Furthermore, since both $D$ and $B$ are
divergenceless, so must be their commutator. The transverse delta function is
consistent with these two properties. It has no divergence and acts as an
ordinary delta function on transverse fields.According to Dirac’s canonical
quantization, the field amplitudes are promoted to operators, and the Dirac
brackets become commutators

$\\{D^{j}(r),B^{k}(r^{\prime})\\}=\delta^{tr}(r-r^{\prime})\rightarrow[D^{j}(r),B^{k}(r^{\prime})]=i\hbar\delta^{tr}(r-r^{\prime})\;.$
(3)

Quesada et al. [35] later showed that, to correctly recover Maxwell’s
equations in nonlinear dielectric media from the Born-Infeld Dirac brackets,
it is imperative to consider the displacement field and magnetic flux density
to be the fundamental dynamical variables. Here ’fundamental’ indicates that
$D$ and $B$ are Hermitian linear combinations of Bose operators, i.e. they are
Boson fields, whereas the electric and magnetic fields $E$ and $H$ are
generally higher polynomials of Bose operators333For a rigorous discussion see
[35]. Quesada provides the following intuitive argument for why neither $E$
nor $H$ can be fundamental: Suppose the Hamiltonian is order $N+1$ in Bose
operators. If $E$ is fundamental, it is of order 1. If $B$ is order 1, then
$\partial_{t}B\sim[B,H]\sim\nabla\times E$ is order N. Thus $E$ is order N, a
contradiction. A similar argument holds for $H$. No contradiction occurs with
$D$ and $B$ as fundamental. . For modern derivations and perspectives of
quantum field theory in linear dielectric media, see [36, 37, 38, 39].

For a finite linear cavity, the eigenstates of the system are enforced by the
dielectric structure and boundary conditions of the cavity. Let us restrict
ourselves to one effective spatial dimension, scalar susceptibilities, and a
single polarization of the field modes without loss of generality444Our
analysis depends only on the algebraic structure of the Hilbert space and the
dynamics generated by the Hamiltonian, so is easily generalized to richer mode
structures.. This amounts to a single parameter family of eigenstates.
Consider a uniform cavity of length $L$ with periodic boundary conditions, as
is the case for a microring resonator [40]. The allowed wavenumbers, $k$,
within the cavity are integer multiples of $\frac{2\pi}{L}$. The fields $D$
and $B$ are physical, therefore real valued. Each may then be expanded with
respect to the basis of eigenmodes in the following form,

$\begin{split}&D(z)=\sum_{k}D_{k}(z)=\frac{1}{\sqrt{AL}}\sum_{k}\alpha_{k}e^{ikz}+\alpha_{k}^{*}e^{-ikz}\\\
&B(z)=\sum_{k}B_{k}(z)=\frac{1}{\sqrt{AL}}\sum_{k}\beta_{k}e^{ikz}+\beta_{k}^{*}e^{-ikz}\;,\end{split}$
(4)

where $A$ represents the transverse cross sectional area of the modes in the
cavity555If the refractive index is not isotropic, $A$ defines an effective
cross sectional area, discussed in [40]..

The field amplitudes of a single photon, $\zeta_{k}$ – defined later, are
determined such that the energy of a photon with well defined momentum $\hbar
k$ agrees with the Einstein energy of a photon, $\hbar\omega(k)$. Suppose a
dispersive permitivity such that the Fourier components of the linear electric
field and displacement field are related by
$\tilde{E}(k)=\frac{1}{\epsilon(k)}\tilde{D}(k)$. If the medium is dispersive,
each frequency mode solves a different eigenvalue equation, so normalization
requires special care [37]. We rescale the amplitude by the single photon
amplitude, $\alpha_{k}\rightarrow\zeta_{k}a_{k}$, and promote the amplitude
variable to a Bose operator obeying the canonical commutation relations

$[a_{k},a_{k^{\prime}}]=0,\quad[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}\;.$
(5)

The displacement field operator becomes

$D(z)=\sum_{k}i\sqrt{\frac{\hbar\omega(k)v_{g,k}n_{k}^{3}}{2ALc}}\Big{(}a_{k}e^{ikz}-a^{\dagger}e^{-ikr}\Big{)}\equiv\sum_{k}i\zeta_{k}\Big{(}a_{k}e^{ikz}-a^{\dagger}e^{-ikr}\Big{)}\;,$
(6)

where we have adopted a common quantum optics phase convention of initial
phase $\pi/2$, $v_{g,k}$ is the group velocity, and $n_{k}$ the refractive
index of the mode with wave-vector $k$. The magnetic flux field operator
obtains a similar form[37].

Having properly dealt with linear dispersion, the Hamiltonian for linear
quantum optics takes the form of an infinite set of harmonic oscillators

$H^{(1)}=\sum_{k}\hbar\omega(k)a_{k}^{\dagger}a_{k}\;.$ (7)

###  2.1 Nonlinear Quantum Optics

We introduce nonlinear optics assuming that the dispersion of nonlinear
susceptibilities is negligible for frequency bands we are interested in. The
electric field is related to the displacement field and polarization field,
$P$, via [41]

$E=D-4\pi P=\frac{1}{\epsilon}D-4\pi P^{nl}\;.$ (8)

Recall from Equation (1) the electric field contribution to the field energy,
$\frac{E\cdot D}{2}$. The energy term $\frac{1}{\epsilon}D\cdot D$ is handled
in the linear part of the Hamiltonian upon quantization. The remaining
interaction $-P^{nl}\cdot D$ relays the effect of the nonlinear polarization
$P^{nl}$.

We assume that the polarization at a point in space depends on the
displacement field at that point, i.e. the response function is local.
Expanding the nonlinear polarization in a power series of the displacement
field amplitudes up to third order, the part of the Hamiltonian associated
with classical nonlinear dynamics is

$\begin{split}H^{(nl)}&=\frac{-P^{nl}\cdot D}{2}=\\\
&-2\pi\sum_{k_{0},k_{1},k_{2}}\int
dr^{3}\;\Gamma^{(2)}(\vec{r};\omega(k_{0}),\omega(k_{1}),\omega(k_{2})))D_{k_{0}}(\vec{r},t)D_{k_{1}}(\vec{r},t)D_{k_{2}}(\vec{r},t)\\\
&-2\pi\sum_{k_{0},k_{1},k_{2},k_{3}}\int
dr^{3}\;\Gamma^{(3)}(\vec{r};\omega(k_{0}),\omega(k_{1}),\omega(k_{2}),\omega(k_{3})))\\\
&\times
D_{k_{0}}(\vec{r},t)D_{k_{1}}(\vec{r},t)D_{k_{2}}(\vec{r},t)D_{k_{3}}(\vec{r},t)\equiv
H^{(2)}+H^{(3)},\end{split}$ (9)

where $\Gamma^{(2)}$ is the second order displacement field susceptibility
that is responsible for optical parametric oscillation (OPO) – often referred
to as parametric down conversion. $\Gamma^{(3)}$ is the third order
displacement field susceptibility responsible for four effects: the two
optical Kerr effects called self phase modulation (SPM) and cross phase
modulation (XPM), a mode mixing process called four wave mixing (FWM), and a
static field induced OPO. Both are defined by the power series expansion of
the polarization field in terms of displacement field [40].

The ordinary nonlinear susceptibilities are defined analogously for the
electric field [42], namely

$P=\Gamma^{(1)}D+\Gamma^{(2)}D^{\otimes 2}+\Gamma^{(3)}D^{\otimes
3}+...=\chi^{(1)}E+\chi^{(2)}E^{\otimes 2}+\chi^{(3)}E^{\otimes 3}+...\;$ (10)

Substituting the relation $D=E+4\pi P(E)$ into Equation (10) and collecting
terms with common powers of electric field, ignoring tensor structure and
dispersion, yields the relations

$\
\begin{split}\Gamma^{(1)}=\frac{\chi^{(1)}}{\epsilon},\quad\Gamma^{(2)}=\frac{\chi^{(2)}}{\epsilon^{3}},\quad\Gamma^{(3)}=\frac{\chi^{(3)}}{\epsilon^{4}}-8\pi\frac{(\chi^{(2)})^{2}}{\epsilon^{5}}\;.\end{split}$
(11)

##  3 Approach

We aim to show how strong Kerr nonlinearities affect OPO. The essential result
is that photon number dependent frequency shifts from the Kerr effect generate
a photon number dependent detuning in OPO. The number dependent detuning is
exploited for coherent deterministic number selective shaping of noise
statistics in select modes. To make this effect clear, we constrain our view
to the subspace of three optical modes engaging in OPO, called signal, idler,
and sum666In general, the process we discuss can occur over the spectrum of
cavity modes. This will be addressed in future papers.. We first solve the
system exactly under the action of the sum of Hamiltonian terms that preserve
photon number in each mode. Evolution under this Hamiltonian gives nontrivial
structure to the time dependent displacement field operator; it is scaled by a
diagonal unitary operator with a frequency that is a function of the photon
number operators. OPO is analyzed in the interaction picture, exhibiting
photon number selective behavior. The nature of this behavior is discussed,
and explored numerically in Section 4.1.

###  3.1 Eigenstates of Kerr Hamiltonian

The Hamiltonian $H^{(3)}$ from Equation (9) is responsible for two distinct
photon number density preserving processes, SPM and XPM, and a mode mixing
process, FWM. FWM is a rich effect, leading to optical frequency combs and
soliton propagation in micro-resonators [43]. However, our aim is to study an
entirely different phenomenon in the same environment, so we do not wish to
handle these complications presently. We will suppose the dispersion relation
of our system is specified such that no FWM is phase matched for the active
modes.

Unlike FWM, SPM and XPM are always phase matched. These effects, together with
the linear Hamiltonian, dominate the phase dynamics of the system. The
corresponding Hamiltonian is diagonal in the Fock basis and, equivalently,
conserves photon number for each mode. Our present objective is to solve for
the time evolution of the field operator under photon number density
preserving Hamiltonian $H_{0}\equiv H^{(1)}+H^{(3)}$ such that
$H=H_{0}+H^{(2)}$, where $H^{(1)}$, $H^{(2)}$, $H^{(3)}$ are defined in
equations (7) and (9). We later show how this time evolution affects
$H^{(2)}$, producing a photon number selective interaction under the right
conditions. We will consider the dynamics of three modes interacting in the
cavity; a sum, signal, and idler mode with frequencies
$\omega_{\Sigma},\omega_{s},\omega_{i}$ respectively.

We assume permutation invariance of all susceptibilities777Permutation
invariance typically does not hold near resonance, where we expect our system
to reside given the constraints derived later. However, accounting for
variability in $\Gamma^{(3)}$ simply amounts to substituting the average
$\bar{\Gamma}^{(3)}$ in place of $\Gamma^{(3)}$ and does not change the
results. and abbreviate the frequency dependence of Kerr susceptibilities
$\Gamma^{(3)}(-\omega_{1},\omega_{1},-\omega_{2},\omega_{2}))=\Gamma^{(3)}_{1,2}=\Gamma^{(3)}_{2,1}$.
Furthermore, we uphold the rotating wave approximation888The rotating wave
approximation is synonymous with the secular approximation. and assume
spatially constant third order susceptibility. After ordering all permutations
of the creation and annihilation operators to be simple powers of the mode
number operators $N_{j}=a_{j}^{\dagger}a_{j}$, the Hamiltonian $H_{0}$
pertaining to the sum, signal, and idler modes is

$\begin{split}H_{0}&=\Big{(}\hbar\omega_{\Sigma}-12\pi
L\zeta_{\Sigma}^{2}(2\zeta_{\Sigma}^{2}\Gamma^{(3)}_{\Sigma,\Sigma}+\zeta_{s}^{2}\Gamma^{(3)}_{\Sigma,s}+\zeta_{i}^{2}\Gamma^{(3)}_{\Sigma,i})\Big{)}N_{\Sigma}\\\
&+\Big{(}\hbar\omega_{s}-12\pi
L\zeta_{s}^{2}(2\zeta_{s}^{2}\Gamma^{(3)}_{s,s}+\zeta_{i}^{2}\Gamma^{(3)}_{s,i}+\zeta_{\Sigma}^{2}\Gamma^{(3)}_{s,\Sigma})\Big{)}N_{s}\\\
&+\Big{(}\hbar\omega_{i}-12\pi
L\zeta_{i}^{2}(2\zeta_{i}^{2}\Gamma^{(3)}_{i,i}+\zeta_{\Sigma}^{2}\Gamma^{(3)}_{i,\Sigma}+\zeta_{s}^{2}\Gamma^{(3)}_{i,s})\Big{)}N_{i}\\\
&-24\pi
L\Big{(}\zeta_{\Sigma}^{4}\Gamma^{(3)}_{\Sigma,\Sigma}N_{\Sigma}^{2}+\zeta_{s}^{4}\Gamma^{(3)}_{s,s}N_{s}^{2}+\zeta_{i}^{4}\Gamma^{(3)}_{i,i}N_{i}^{2}\\\
&\quad\quad\quad\quad+\zeta_{\Sigma}^{2}\zeta_{s}^{2}\Gamma^{(3)}_{\Sigma,s}N_{\Sigma}N_{s}+\zeta_{\Sigma}^{2}\zeta_{i}^{2}\Gamma^{(3)}_{\Sigma,i}N_{\Sigma}N_{i}+\zeta_{i}^{2}\zeta_{s}^{2}\Gamma^{(3)}_{i,s}N_{i}N_{s}\Big{)}\end{split}$
(12)

Heisenberg’s equation of motion, $\dot{a}=\frac{1}{i\hbar}[a,H_{0}]$, tells us
the time evolution for the field annihilation operators
$a_{\Sigma},a_{s},a_{i}$.

To condense our notation, let us define the Kerr frequencies

$g_{ij}\equiv\frac{24\pi
AL}{\hbar}\zeta_{i}^{2}\zeta_{j}^{2}\Gamma^{(3)}_{ij}$ (13)

where $g_{jj}$ are the frequencies of self phase modulation and $g_{i,j\neq
i}$ are the frequencies of cross phase modulation. These frequencies are
proportional to the classical Kerr frequency shifts $\delta\omega$ in
accordance with the relation $g\langle N\rangle=\delta\omega I$ where $I$ is
the classical intensity and $\langle N\rangle$ is the mean photon number. The
time evolution of each field annihilation operator under $H_{0}$ is

$\begin{split}a_{\Sigma}(t)&=\exp\Big{\\{}-i\Big{(}\omega_{\Sigma}-(g_{\Sigma\Sigma}+\frac{g_{\Sigma
s}}{2}+\frac{g_{\Sigma i}}{2})-g_{\Sigma\Sigma}(2N_{\Sigma}+1)+g_{\Sigma
s}N_{s}+g_{\Sigma i}N_{i})\Big{)}t\Big{\\}}a_{\Sigma}(0)\\\
a_{s}(t)&=\exp\Big{\\{}-i\Big{(}\omega_{s}-(g_{ss}+\frac{g_{s\Sigma}}{2}+\frac{g_{si}}{2})-g_{ss}(2N_{s}+1)+g_{s\Sigma}N_{\Sigma}+g_{si}N_{i})\Big{)}t\Big{\\}}a_{s}(0)\\\
a_{i}(t)&=\exp\Big{\\{}-i\Big{(}\omega_{i}-(g_{ii}+\frac{g_{i\Sigma}}{2}+\frac{g_{is}}{2})-g_{ii}(2N_{i}+1)+g_{i\Sigma}N_{\Sigma}+g_{is}N_{s})\Big{)}t\Big{\\}}a_{i}(0)\;.\end{split}$
(14)

Note that the exponentials above do not commute with the $t=0$ field
annihilation operators. To further simplify our notation we call $a(0)=a$, and
use $\xi$ to represent the exponential operators on the right hand side of
Equation (14) such that

$a_{\Sigma}(t)=\xi_{\Sigma}a_{\Sigma},\quad a_{s}(t)=\xi_{s}a_{s},\quad
a_{i}(t)=\xi_{i}a_{i}$ (15)

Substituting these operators into Equation (6) yields the time dependent
displacement field operator for the nonlinear field described by $H_{0}$,

$D_{k}(z,t)=i\zeta_{k}\Big{(}\xi_{k}(t)a_{k}e^{ikz}-e^{-ikz}a_{k}^{\dagger}\xi_{k}^{*}(t)\Big{)}\;.$
(16)

The time dependent displacement field operator is the essential result of this
subsection and the starting point for our analysis in the following subsection
3.2. Let us take a brief detour to understand how the $\xi$ operators affect
optical pumping and derive our first constraint.

####  3.1.1 Pump photon blockade

The $\xi$ operators neatly quantify the photon blockade effect, reviewed in
[26], which affects optical pumping of the resonator. To see this, consider a
cavity pumped by a near monochromatic beam that is near resonance to a cavity
mode with wavenumber $k_{j}$. As the photon number in the cavity increases
from $N_{j}$ to $N_{j}+1$, the frequency shifts by $g_{jj}(2N_{j}+1)$. If the
frequency of the next excitation shifts out of the pump line width, the cavity
will no longer be excited, i.e. additional photons are blockaded from entering
by the photons already in the cavity. Consequently, obtaining a high cavity
intensity requires a pump beam with a wide linewidth in the presence of very
strong Kerr nonlinearities. However, a wide pump line width yields unwanted
excitations, so it is best for the Kerr nonlinearities involving the pump
frequency to be small.

If small Kerr frequencies at the pump frequency are challenging to achieve, we
consider the following constraint: Self phase modulation by a positive
susceptibility red shifts the frequency by $-g_{jj}N_{j}^{2}$. If
$\Gamma^{(3)}_{jj}>0$, the largest photon number achievable in the cavity
pumped by a beam with a spectral edge at some fraction $f$ of $\omega_{j}$ is

$N_{MAX}=\sqrt{\frac{(1-f)\omega_{j}}{g_{jj}}}$ (17)

When considering optical parametric oscillation in the next sections, we drive
the sum frequency cavity mode with an external pump. It is most important that
the external pump spectrum does not overlap with the signal and idler modes to
avoid unwanted excitation. Therefore it is necessary that $f>1/2$ and
preferable that $f>3/4$. We discuss later in Section 4.2, and simply note now,
that a large coherent state for the sum mode minimizes dephasing and enhances
OPO efficiency. Because unwanted excitations and high pump intensity are
coincident, it is best for the Kerr nonlinearities involving the pump
frequency to be negative, if not small.

###  3.2 Number Selective Optical Parametric Oscillation

The previous section solved the Kerr Hamiltonian exactly and introduced the
operators $\xi$ to simplify our analysis of optical parametric oscillation.
With an exact time evolution for the Kerr photons in hand, we construct the
Hamiltonian for number selective optical parametric oscillation (NSOPO). In
the interaction picture, the OPO Hamiltonian evolves via the photon number
preserving propagator, $U_{0}=\exp\\{-iH_{0}t/\hbar\\}$. From Equation (9) we
obtain the time dependent OPO Hamiltonian in the interaction picture,

$H_{I}=U_{0}^{\dagger}H^{(2)}U_{0}=-2\pi\sum_{k_{0},k_{1},k_{2}}\int
dr^{3}\;\Gamma^{(2)}(\vec{r};\omega(k_{0}),\omega(k_{1}),\omega(k_{2})))D_{k_{0}}(\vec{r},t)D_{k_{1}}(\vec{r},t)D_{k_{2}}(\vec{r},t)\;.$
(18)

We assume the susceptibility $\Gamma^{(2)}$ is spatially local, spatially
constant, and frequency permutation invariant999As is the case for
$\Gamma^{(3)}$, permutation invariance is not valid for the strong
nonlinearities required of NSOPO. However, accounting for variability simply
amounts to substituting the average $\bar{\Gamma}^{(2)}$ in place of
$\Gamma^{(2)}$ and does not change the result.. The time dependent
displacement field operators, Equation (16), are substituted into Equation
(18). Terms of the form $a^{\dagger}_{\Sigma}a^{\dagger}_{i}a^{\dagger}_{s}$
and $a_{\Sigma}a_{s}a_{i}$ are neglected by assuming the rotating wave
approximation for the three modes, but the time dependence from the Kerr
induced frequency mismatch is preserved. All possible permutations of the
three relevant mode contributions to the field operators are reordered to a
common tractable form, accounting for the noncommutativity of $a$ and $\xi$.
The simplified interaction Hamiltonian is thus

$H_{I}=-i12\pi
AL\zeta_{\Sigma}\zeta_{s}\zeta_{i}\left(\Gamma^{(2)}_{\Sigma,s,i}a^{\dagger}_{\Sigma}a_{s}a_{i}\xi_{\Sigma}^{*}\xi_{s}\xi_{i}e^{-i(2g_{ss}+2g_{ii}+g_{si}-g_{\Sigma
s}-g_{\Sigma i})t}-h.c.\right)$ (19)

where h.c. denotes hermitian conjugate.

Suppose the sum frequency mode is a high amplitude coherent state
$|\alpha_{\Sigma}\rangle$ such that
$\bar{N}_{\Sigma}=|\alpha_{\Sigma}|^{2}\gg\bar{N}_{s},\bar{N}_{i}$. Then we
can reasonably replace the operator $a_{\Sigma}$ with the c-number
$\alpha_{\Sigma}$ without worrying much about dephasing effects. For example,
if shaping of signal and idler modes is done in a range of $m$ photons, the
purity of the manipulated quantum state is on the order of
$e^{-|\frac{m}{2\alpha_{\Sigma}}|^{2}}$101010See Section 4.2.

To simplify notation we define the NSOPO complex squeezing parameter $\Xi$ and
OPO resonance detuning frequency $\Delta(N_{\Sigma},N_{s},N_{i})$ such that

$\begin{split}&\Xi e^{i\Delta t}\equiv\Big{(}\frac{12\pi
AL}{\hbar}\zeta_{\Sigma}\zeta_{s}\zeta_{i}\Gamma^{(2)}_{\Sigma,s,i}\alpha_{\Sigma}^{*}\Big{)}\Big{(}\xi_{\Sigma}^{*}\xi_{s}\xi_{i}e^{-i(2g_{ss}+2g_{ii}+g_{si}-g_{\Sigma
s}-g_{\Sigma i})t}\Big{)}\end{split}$ (20)

where

$\Xi\equiv\frac{12\pi
AL}{\hbar}\zeta_{\Sigma}\zeta_{s}\zeta_{i}\Gamma^{(2)}_{\Sigma,s,i}\alpha_{\Sigma}^{*}\;,$
(21)

$\begin{split}\Delta\equiv\delta+G_{\Sigma}N_{\Sigma}+G_{S}(N_{s}+1)+G_{i}(N_{i}+1)\;,\end{split}$
(22)

and

$\begin{split}&\delta\equiv\omega_{\Sigma}-\omega_{s}-\omega_{i}+2(g_{ss}+g_{ii}-g_{\Sigma\Sigma})\\\
&G_{\Sigma}\equiv g_{\Sigma s}+g_{\Sigma i}-2g_{\Sigma\Sigma}\\\ &G_{s}\equiv
2g_{ss}+g_{is}-g_{\Sigma s}\\\ &G_{i}\equiv 2g_{ii}+g_{si}-g_{\Sigma
i}\end{split}$ (23)

are the set $\\{G\\}$ of detuning weights for each photon number, as well as a
static detuning $\delta$.

The interaction Hamiltonian is now expressed compactly as

$H_{I}=-i\hbar\Big{(}a_{s}a_{i}\Xi e^{i\Delta t}-e^{-i\Delta
t}\Xi^{*}a_{i}^{\dagger}a_{s}^{\dagger}\Big{)}$ (24)

Without Kerr nonlinearities, non-degenerate parametric down conversion
generates 2-mode squeezing between the signal and idler modes [44]. With Kerr
nonlinearities, the squeezing parameter hosts a detuning oscillation with
photon number dependent frequency, making two mode squeezing amplitude
dependent. Let us explore precisely how this feature affects the interaction.

The OPO detuning $\Delta(N_{\Sigma},N_{s},N_{i})$ governs how number states
experience 2-mode squeezing. A simple, and very useful case to study is when a
large-$\alpha_{\Sigma}$ coherent state $|\alpha_{\Sigma}\rangle$ is injected
into the nonlinear cavity in the presence of a 2-mode squeezed state of signal
and idler frequencies. Experimentally this would also require a method for
coherently injecting a 2-mode squeezed state into the nonlinear cavity
initially in the vacuum state. Acknowledging the technological
challenge111111This may be realized, for example, with an acousto-optic
modulator. , we will assume it is possible and investigate its
consequences121212This example also captures the behavior of NSOPO via
degenerate parametric down conversion and second harmonic generation. In these
cases, the fundamental mode and second harmonic are entirely uncorrelated and
coupled into the cavity without a need to preserve correlations. We choose to
study NSOPO on a two mode squeezed state rather than a single mode state to
highlight how noise shaping is identically present in the entangled signal and
idler modes..

Assume an initial 2-mode squeezed state – a state which is a superposition of
all equal populations of signal and idler photons. Evident from the
interaction operator $a_{\Sigma}^{\dagger}a_{s}a_{i}$, the difference in
signal and idler photon numbers is a conserved quantity, so the state remains
a superposition of equal photon numbers for all times much less than the
cavity decay time. When $G_{\Sigma}=0$, the OPO detuning $\Delta$ is
independent of the sum frequency photon number and NSOPO is coherently driven
by the sum frequency beam. Each OPO interaction event will decrement or
increment the signal and idler photon numbers by one, transporting probability
amplitude along the space of states. Adjacent states see a difference in
detuning of

$\gamma\equiv G_{s}+G_{i}=2(g_{ss}+g_{ii}+g_{is}-g_{\Sigma\Sigma})$ (25)

Suppose there is a photon number, $N_{s}=N_{i}=n^{\prime}$ such that the
detuning is minimal. We call $n^{\prime}$ the shape center. Then the OPO
Hamiltonian in Dirac notation takes the form

$H_{I}=-i\hbar\sum_{m=0}^{\infty}(m+1)\Big{(}\Xi
e^{i(\gamma(n^{\prime}-m)+\Delta(n^{\prime}))t}|m\rangle\langle
m+1|-\Xi^{*}e^{-i(\gamma(n^{\prime}-m)+\Delta(n^{\prime}))t}|m+1\rangle\langle
m|\Big{)}.$ (26)

This Hamiltonian generates particularly interesting evolution when $\Xi$ and
$\gamma$ are of similar magnitude, however challenging to describe
analytically. We explore this behavior in the following section.

##  4 Results and Discussions

In the previous section we developed a theoretical method for directly
targeting photon number states to coherently shape the quantum statistics of
an optical field in a cavity. In the following we numerically simulate the
model and illustrate how the parameters can be controlled to generate quantum
states with useful structure. The constraints established here and in Section
3 are elaborated on and brought together to determine what regions of
parameter space are viable for experimental realization.

###  4.1 Simulations

The time dependent Schrödinger equation corresponding to the interaction
Hamiltonian, Equation (26), is numerically evolved with a Crank-Nicolson
method, which preserves unitarity. Figure 1 reveals the characteristic
behavior of NSOPO. It shows that probability amplitude tends to coalesce
around number states separated by a spacing

$s=\frac{2\pi}{\gamma t}$ (27)

where states see constructive interference from two mode squeezing. These
peaks rapidly fall off in amplitude when near the shape center, but fall off
more slowly when away from the shape center maintaining a wide tail in their
height distribution. The peak heights increase monotonically with $|\Xi|$
until reaching a maximum, after which the peak begins to wind itself and
develops a complicated structure.

The peak nearest the shape center deviates from periodic spacing, consequently
shifting the pattern for states with higher photon number. The location of the
central peak depends on the phase of the NSOPO squeezing parameter $\Xi$. For
$\Xi=-i$, the peak is centered on the shape center. For $\Xi=i$, there is a
deep trough at the shape center with nearest peaks equidistant to the left and
right. For $\Xi=1(-1)$, the nearest peak and trough are to the right and left
(left and right) of the shape center. The plots in Figure 1 underlay the
function

$\frac{1}{N}\Big{[}\text{sinc}\Big{(}\gamma(n^{\prime}-m)\tau+arg(\Xi)+\pi/2\Big{)}(m+1)\tau+1\Big{]}$
(28)

which approximately replicates the height and location of each peak, breaking
accuracy at the central peak. This function is obtained by integrating the
Hamiltonian over time $\tau$ and acts as a more tractable guideline for
engineering states with shaping.

Figure 1: General characteristics of NSOPO. In each plot the state is
initially a uniform real valued distribution. The plots differ only in the
phase of the NSOPO complex squeezing parameters $\Xi$. For all plots, the
gouge center $n^{\prime}=60$, the Kerr detuning $\gamma=\pi$, and the
evolution time $\tau=0.08$. Orange dots are spaced by the predicted peak
spacing, $s=\frac{2\pi}{\gamma\tau}$, and aligned with the shaping center. The
green curve shows a first order approximation of shaping that follows the peak
positions and magnitude closely. The blue curve is the shaped photon
probability distribution resulting from an in initial uniform distribution.

These shaping principles can be applied in several ways. Setting
$arg(\Xi)=-\pi/2$ such that the shape center is aligned with the peak of a
state and the peak spacing $s$ to be greater than the width of the state, the
photon number probability will coalesce into an approximate Fock state. Plot
(a) in Figure 2 illustrates this process, generating a state with signal
photon number probabilities $P(n=52)\approx 45\%$, $P(n=54,53,51,50)\approx
11\%$. Alternatively, if the shape center is set far from the concentration of
initial state probabilities, clean periodic oscillations in photon number
probabilities, with period $s$, are introduced into the state. Plots (b-d) in
Figure 2 illustrate this process. (b) shows precision shaping of a coherent
state with $s=2$. (c) is the same as (b) but acting on a two-mode squeezed
vacuum. (d) shows shaping with $s=6$.

Figure 2: Examples of photon probability shaping offered by NSOPO. (a) An
approximate Fock state of $n=52$ accomplished by placing the shaping center on
the peak of an initial coherent state with amplitude $\alpha_{s,i}=7$. (b) A
precision shaping of the same coherent state. Parameter $\gamma t$ is chosen
such that peak spacing $s$ is two photons. (c) A two mode squeezed vacuum
state with $s=2$. (d) A two mode squeezed vacuum with $s=6$.

In tandem with photon number probabilities, NSOPO offers control over states
in optical phase space. The following demonstrates an interesting case of this
control; generating a family of displaced Schrödinger cats with a single
operation. Recall that, in the Schrödinger picture, the modes experience self
phase modulation and cross phase modulation via the Kerr effect. Introducing
the condition $G_{\Sigma}=0$ into Equation (12), we see that the term of the
Hamiltonian

$\begin{split}H^{Kerr}=-\hbar\big{(}g_{ss}N_{s}^{2}+g_{ii}N_{i}^{2}+g_{is}N_{i}N_{s}\big{)}\end{split}\;,$
(29)

generates non-linear dynamics for the signal and idler modes. Noting that $N$
is an integer, we see that states of a single mode are periodic in time $t$
when each Kerr frequency satisfies $gt=2\pi m$, with $m$ an integer. For
simple rational values of $m$, the evolution maps coherent states to
generalized coherent states [12].

We extend this argument to the two mode squeezed state, recalling
$N_{i}=N_{s}$. We choose a preferred final state by selecting $m$ to satisfy

$(g_{ss}+g_{ii}+g_{si})t=2\pi m=(\gamma/2+g_{\Sigma\Sigma})t\;.$ (30)

Equation (27) provides a similar constraint regarding NSOPO peak spacing. If
we suppose $g_{\Sigma\Sigma}=0$, then $(g_{ss}+g_{ii}+g_{si})=\gamma/2$ making
Equation (30) and Equation (27) the same constraint with $2m=1/s$. Setting
$s=4$ we obtain a four-phase Schrödinger cat state. Figure 3 shows the action
of NSOPO on a coherent state, (a), that is split into a generalized four-phase
Schrödinger cat state, (b). Depending on the phase $arg(\Xi)$, NSOPO restricts
the amplitude of the cat to two of four phases, shown in (c-f). The axes in
each plot are independent phase space quadratures of the displacement field.
Not shown in Figure 3, shifting the shape center by one rotates the action of
NSOPO among the operations shown in (c-f). Namely, under
$n^{\prime}\rightarrow n^{\prime}+1$, the plots map to each other according to
$c,d,e,f\rightarrow f,c,d,e$. This behavior alludes efficient quantum logic
gates for macroscopic quantum states, though further investigation is needed
to understand the range of this capability.

Figure 3: (a) Wigner function of an initial coherent state of amplitude
$\alpha=7$. (b) The coherent state subject to Kerr rotation without any NSOPO.
(c-f) The state shaped with NSOPO complex squeezing parameter $\pm 2,\pm 2i$.
In all plots, blue regions are positive valued and red regions are negative.
Every state is shaped with parameters $\gamma=\pi/2,\tau=1,n^{\prime}=8$. As
the phase of $\Xi$ is varied by increments of $\pi/2$, number selective
shaping changes predictably.

Figure 4 shows the effect of incoherent loss on the generation of an
approximate Fock state, similar to that shown in Figure 2 (a). In this
instance, an initial product state $|\psi\rangle$ of a signal/idler two-mode
coherent state with amplitude $\alpha_{s,i}=2$ and a large coherent pump state
is evolved according to the local Lindblad master equation

$\begin{split}\frac{d\rho(t)}{dt}&=\frac{1}{i\hbar}[H^{(1)}+H^{(2)}+H^{(3)},\rho(t)]\\\
&+\frac{\Gamma_{cav}}{2}\Big{(}2a_{s}\rho(t)a^{\dagger}_{s}-\rho(t)a^{\dagger}_{s}a-a_{s}^{\dagger}a_{s}\rho(t)+2a_{i}\rho(t)a^{\dagger}_{i}-\rho(t)a^{\dagger}_{i}a-a_{i}^{\dagger}a_{i}\rho(t)\Big{)}\end{split}$
(31)

where $\rho(0)=|\psi\rangle\langle\psi|$ and $\Gamma_{cav}$ represents the
incoherent decay rate of the cavity at the signal and idler frequencies.

To characterize the viability of the system for generating useful and pure
quantum states, we introduce a parameter that is similar in spirit to the
cooperativity [45]. Cooperativity is defined for light-matter systems with
specific quantum models of the material and quantifies the ratio of the light-
matter interaction rate to the loss rate of the system. Our theory is derived
from a constitutive relation, Equation (LABEL:eq:PExpandDandE), so does not
offer the detail required to provide the exact cooperativity. However the
strength to loss ratio $\gamma/\Gamma_{cav}$ of the adjacent state detuning
$\gamma$, i.e. the Kerr frequency of the signal-idler state, and the
incoherent decay rate of the cavity at the signal and idler frequencies
$\Gamma_{cav}$, serves a similar role.

Plot (a) shows the growth of quantum infidelity $1-F$ with respect to
reciprocal NSOPO peak spacing $\gamma t=\frac{2\pi}{s}$. Each curve represents
a different value of the strength to loss ratio $\gamma/\Gamma_{cav}$. For
$\gamma t<1$, infidelity scales linearly with $\gamma t$. For $|\Xi|t=.3$,
$n^{\prime}=4$, and $\gamma t=0.9$, we find the numerical result

$\Gamma_{cav}/\gamma\approx\frac{1-F}{3}\;.$ (32)

This inverse proportionality between strength to loss ratio and infidelity is
substantial. Interestingly, generating approximate Fock states of higher
photon number require a smaller $\gamma t$ such that the spacing is larger,
thus high Fock state generation demands less of the strength to loss ratio for
the same quantum fidelity.

Figure 4: Both plots present data from a simulation generating an approximate
N=5 Fock state in the presence of incoherent loss. The state is evolved from
an initial two mode squeezed coherent state of amplitude $\alpha=2$. The
Hamiltonian parameters are $|\Xi|\tau=.3$, $\gamma\tau=0.9$ and
$n^{\prime}=4$. Plot (a) shows the growth of quantum infidelity $1-F$ with
respect to reciprocal NSOPO peak spacing $\gamma t=\frac{2\pi}{s}$. Each curve
represents a different value of the strength to loss ratio
$\gamma/\Gamma_{cav}$ where $\gamma$ is the the adjacent state detuning and
$\Gamma_{cav}$ is the decay rate of the cavity at the signal and idler
frequencies. Plot (b) shows the effect of loss on corresponding photon number
probabilities.

###  4.2 Pump Dephasing

A key challenge to overcome for preserving the quantum coherence of noise
shaped states of light is to mitigate the dephasing effects of the pump beam.
The two criteria to satisfy are a strong pump beam, quantified below, and a
vanishing detuning weight for the pump frequency, $G_{\Sigma}=0$.
Specifically, we demand that the detuning weights satisfy

$G_{\Sigma}N_{\Sigma}\approx|2g_{\Sigma\Sigma}-g_{\Sigma s}-g_{\Sigma
i}||\alpha_{\Sigma}|^{2}\ll G_{s}N_{s}+G_{i}N_{i}=\gamma N_{s}\;.$ (33)

A vanishing pump detuning weight ensures that the desired OPO detuning
$\Delta$ is independent of pump photon number, rendering NSOPO not number
selective for the pump field. This condition is important because a pump
photon number selective process implies that far fewer states in the entire
Hilbert space contribute to OPO, dramatically decreasing the efficiency of the
process. Furthermore, if the interaction were number selective for the pump,
the state of the pump field would vary dramatically for different possible
interaction histories, resulting in strong dephasing.

When NSOPO is not number selective for the pump, the OPO interaction is simply
a displacement operation, which minimally disturbs the structure of the pump
field. To illustrate, consider the state $|\alpha_{\Sigma},n_{s},n_{i}\rangle$
time evolved under the OPO interaction $H_{I}$ with $G_{\Sigma}=0$. The result
is a superposition of products of shifted coherent states in the sum frequency
and incremented number states in the signal and idler. The coherent states are
at most shifted by a displacement corresponding to the maximum photon number
change of the signal and idler. For large amplitude coherent states, shifts in
the coherent states on the scale of shifts in the signal and idler yield
minimal deviations in the overlap between the coherent pump states.

To obtain a generic measure of state purity, we note that the histories of
Fock state amplitudes are carried along with NSOPO peaks. The pump beam
coherent states that are entangled with either peak acquire dephasing
contributions from pump beam coherent states with mean shifted by the interval
of photon numbers the NSOPO peak has traveled over its evolution. This
entanglement between the pump, signal, and idler contributes a dephasing in
the signal and idler density matrix. The purity of the signal/idler density
matrix, $Tr(\rho_{s/i}^{2})$, is well approximated by the squared overlap of
the shifted and unshifted pump beam states. Thus, for a pump beam in a
coherent state $|\alpha_{\sigma}\rangle$, signal/idler in initial two mode
squeezed coherent state $|\alpha_{s,i}\rangle$, and spacing $s$, the purity
measure is

$\gamma_{purity}=\exp\Big{\\{}-\Big{|}\frac{\alpha_{s,i}}{4s\alpha_{\Sigma}}\Big{|}^{2}\Big{\\}}\;.$
(34)

###  4.3 Summary of Constraints and Figures of Merit

Five conditions must be met in order to realize coherent NSOPO. The following
subsections summarize each, roughly ordered from simple to difficult to
satisfy.

  1. 1.

In the presence of strong Kerr nonlinearities, exciting a cavity pump field to
high intensity requires a broad external pump linewidth. However, a broad
external pump will excite superfluous modes in the system. Suppose the pump
beam is a broad spectrum coherent state $|\alpha_{\Sigma}\rangle$. To mitigate
unintended interactions from modes above the signal and idler frequency, we
restrict the magnitude of the sum mode self phase modulation frequency by
Equation (17) to obtain

$g_{\Sigma\Sigma}<\frac{\omega_{\Sigma}}{4|\alpha_{\Sigma}|^{4}}\;.$ (35)

Thus, it is essential for $g_{\Sigma\Sigma}$ to be small or negative.

  2. 2.

The purity measure Equation (34) indicates the constraint

$\frac{\alpha_{s,i}}{4\alpha_{\Sigma}}\ll s\;.$ (36)

The precision of noise shaping is limited by the relative uncertainty of the
signal/idler photon number noise w.r.t. the pump photon number noise and is
generally the easy to control.

  3. 3.

Numerical simulations place useful values of $\frac{\Xi}{\gamma}$ in the range
$0.1$ to $10$. For an order of magnitude estimate of this constraint, consider
a case where $g_{ss}=g_{ii}=g_{is}$ and $g_{\Sigma\Sigma}=0$. Using the
definitions of $\gamma$, $\Xi$,and $g$ from equations (25), (21), (13), (6),

$\frac{\Xi}{\gamma}=\frac{\Xi}{6g}\approx\frac{\zeta_{\Sigma}\Gamma^{(2)}\alpha_{\Sigma}^{*}}{12\zeta_{s}\zeta_{i}\Gamma^{(3)}}\;.$
(37)

The quantity $\zeta$ is the single photon displacement field amplitude in
Equation (6). Careful use of Equation (11) relates the displacement field
susceptibilities to the more commonly measured electric field
susceptibilities. Expanding the ratio with the definitions of each frequency
yields

$\begin{split}\frac{\Xi}{\gamma}&\approx\sqrt{\frac{LA}{18\hbar\omega}}\sqrt{\frac{cv_{g,\Sigma}n_{\Sigma}^{3}}{v_{g,s}v_{g,i}n_{s}^{3}n_{i}^{3}}}\frac{\Gamma^{(2)}\alpha_{\Sigma}^{*}}{\Gamma^{(3)}}\\\
&\approx\sqrt{\frac{LA}{18\hbar\omega}}\sqrt{\frac{cv_{g,\Sigma}}{v_{g,s}v_{g,i}}}\sqrt{\frac{1}{n_{\Sigma}n_{s}n_{i}}}\frac{\chi^{(2)}\alpha_{\Sigma}^{*}}{\chi^{(3)}}\in[0.1,10]\end{split}$
(38)

This criterion is straightforward to satisfy in real materials. For however
large the magnitude of $\chi^{(3)}$ is, the strength of OPO can be increased
with the intensity of the pump beam and the quantization volume.

  4. 4.

As discussed in Subsection 4.2 above, shaping maintains quantum coherence if
the pump field detuning weight vanishes, $G_{\Sigma}=0$. Equation (33)
provides a qualitative expression of the requisite smallness of $G_{\Sigma}$.
The upshot is that greater pump power requires higher precision in the
proximity of $G_{\Sigma}$ to zero. This condition is easiest to satisfy if
Kerr frequencies related to the pump are independently small, which is
challenging if the other Kerr frequencies are to be incredibly large. Appendix
Appendix discusses a class of systems that can satisfy exactly this scenario
by generating nonlinearities on resonance with the signal and idler, and off
resonance with the pump.

  5. 5.

To apply the theory developed in this paper, all shaping should occur on a
time scale $\tau$ shorter than that defined by the acceptable infidelity
$1-F_{acc}$ and desired spacing according to

$\frac{\Gamma}{\gamma}\frac{2\pi}{s}=\frac{\tau}{\tau_{cav}}<\frac{1-F_{acc}}{s}\;.$
(39)

where $\tau_{cav}=1/\Gamma_{cav}$ is the cavity decay time. This constrains
the lower bound on the Kerr frequencies $g_{ss},g_{ii},g_{si}$. Let us assume
$g_{ss}=g_{ii}=g_{si}=g$ and $g_{\Sigma,\Sigma}=0$ for simplicity. Restating
the numerical result for the high fidelity regime Equation (32), but
substituting in the cavity lifetime $\tau_{cav}=\frac{2Q}{\omega}$ we obtain

$\frac{3}{1-F}\approx\frac{\gamma}{\Gamma_{cav}}=\frac{12gQ}{\omega}\;.$

Substituting $\zeta$ from Equation (6) into the definition of the Kerr
frequencies $g$ from Equation (13), and relating $\Gamma^{(3)}$ to
$\chi^{(3)}$ with Equation (11), we find the strength to loss ratio is

$\frac{\gamma}{\Gamma_{cav}}=72\pi
Q\frac{\hbar\omega}{V}\frac{v_{g}^{2}}{c^{2}}\frac{\chi^{(3)}}{n^{2}}$ (40)

Finally, substituting in the classical form of the nonlinear refractive index
$n_{2}=\frac{24\pi^{2}\chi^{(3)}}{cn^{2}}$ we obtain a figure of merit
$\mathcal{F}$ for NSOPO that is proportional to the strength to loss ratio,

$\frac{1}{1-F}\approx\frac{Q\hbar\omega v_{g}^{2}n_{2}}{\pi
Vc}\equiv\mathcal{F}$ (41)

This figure of merit commands our attention; it is the most challenging
constraint to practically satisfy among this list. Candidate systems will have
large $\mathcal{F}$ at the signal and idler frequencies and small
$\mathcal{F}$ at the sum frequency. In Appendix Appendix we show that
compliance is likely attainable by addressing the state of the art for
achieving strong nonlinearities with minimal loss and controllable group
velocity and showing that these systems perform well within the boundary of
the constraints.

###  4.4 Control Parameters

The shape center $n^{\prime}$ is controlled by modulating the linear
refractive index of the material, therefore the static detuning parameter
$\delta$, in time. To understand the extent of modulation we may ask, what are
the possible rates of change for the shape center with respect to refractive
index shifts? Recall the expression for $\delta$, (23), contains the frequency
difference $\omega_{\Sigma}-\omega{s}-\omega_{i}$. In the simplest case, the
frequencies are related to the fixed wavenumbers of each mode by

$\omega=\frac{kc}{n}\;.$ (42)

Where $\frac{d\omega}{dn}$ is maximal, so is $\frac{d\delta}{dn}$. Therefore
it is desirable for the refractive index of each mode to be near zero. This
conclusion is agreeable with the figures of merit derived above. One can
change the refractive index of the material by, for example, introducing a
static field, or by utilizing acoustic oscillations in the cavity material.

The NSOPO squeezing parameter, (21), is primarily controlled by modulating the
amplitude and phase of the pump beam. In practice one should be aware that
changing the shape center affects the strength of NSOPO in three ways.

  1. 1.

Equation (26) shows that OPO strength is scaled by $(m+1)$, where $m$ is the
signal photon number of the state being acting upon. Thus the OPO strength is
scaled by the shape center.

  2. 2.

In some materials, e.g. organic materials, changing a static field to modulate
the refractive index will reorient molecules affecting $\Gamma^{(2)}$.

  3. 3.

Static fields will also introduce a $\Gamma^{(3)}$ contribution to OPO
strength $\Xi$.

Ultimately, determining the parameters for preparing a particular state will
be a matter of empirical tuning guided by rough numerical estimates.

A final note on state control: if the static field can be modulated on a time
scale faster than the state preparation time $\tau$, more complicated shaping
of the field noise statistics can be accomplished, beyond fixed shaping
parameters.

##  5 Conclusion

We have proposed a novel method for the deterministic manipulation of quantum
states of light. The underlying theoretical tools required for its
implementation are developed and the physical constraints for its practical
implementation described. The method utilizes strong Kerr nonlinearities to
introduce photon number dependence into the frequency matching condition for
optical parametric oscillation, rendering the OPO interaction photon number
selective. Number selective interactions expand the capabilities of current CV
quantum optical state control and preparation techniques by introducing
targeted deterministic manipulations to subspaces of a light field’s Fock
space.

Simulations indicate the generation of highly nonclassical quantum states of
light. We establish five necessary conditions to realize NSOPO in practice,
which will aid the search for candidate materials and optical configurations.
The constraints are consistent with a common figure of merit, $\mathcal{F}$,
expressed in Equation (41). The best materials for NSOPO will have large
$\mathcal{F}$ at the signal and idler frequencies, and small $\mathcal{F}$ at
the sum frequency. The constraints can likely be realized in specialized
systems that are driven into electronically induced transparency and
spontaneously generated coherence.

Future research will aim to realize the requisite constraints in detailed
models of highly resonant cavity light coupled with matter systems, with
special attention paid to dark resonances. From specific models, the approach
derived in this paper will be adapted to accommodate Hamiltonians beyond the
simple cubic and quartic interactions treated above. These new models will be
used to guide experimental design and to search for novel sculpted states.
Future research will also investigate generation of entanglement between two
uncorrelated signal and idler states, and study how noise shaping can be
generalized to pulses of light, in particular to solitons in optical
resonators with attention paid to mitigating the dephasing effects of quantum
soliton propagation.

Acknowledgments: We thank Ned and Nancy Wogman for the Wogman fellowship, the
WSU CAS Graduate Research Scholarship, and the donors to the Innovative
Physics Fund and the Department of Physics for support.

## Appendix

The present theoretical paper describes how one might achieve deterministic
shaping of a Fock state. This appendix gives a plausibility argument that the
required figure of merit is achievable. However, meeting the constraints
discussed in section 4.3 may require new materials and device architectures.
Here we individually consider the measured properties of known material
systems to estimate a reasonable range of the figure of merit.

EIT is an example of a process that exhibits the requisite properties for
NSOPO. Schmidt and Imamoglu [26] showed that a four level system can be
configured to obtain a giant cross phase modulation strength. With the
additional presence of spontaneously generated coherence (SGC), which can be
accomplished in a five level system [46], the EIT window can be shifted such
that the Kerr nonlinearity is maximal when absorption is near zero.
Furthermore, the five level system proposed offers direct control over the
group velocity, which ranges from highly sub- to super-luminal and with photon
number dependent behavior [47].

Obtaining giant nonlinearities in near resonant systems with EIT gives us two
other desirable features for NSOPO. The peaks of the linear and nonlinear
susceptibilities are sufficiently sharp near resonance to make FWM minimal due
to a rapidly increasing phase mismatch and rapidly decaying interaction
strength. In addition, off resonant nonlinearities will be comparatively weak.
Such a system can be tuned so that the sum frequency is off resonance with the
material while the signal and idler are on resonance. Then Constraint (33) is
satisfied by virtue of small Kerr nonlinearities at the pump frequency and FWM
can be reliably neglected for all modes.

Bang and Doai, [46, 47], report values of self and/or cross phase modulating
nonlinear index in a gas of 85Rb with density $10^{8}$ atoms/cm3 on the order
of

$n_{2}=\frac{24\pi^{2}\chi^{(3)}}{cn^{2}}\approx 5\times 10^{-13}\frac{s\cdot
cm^{2}}{erg}$ (43)

In principle, the atom density can be made orders of magnitude greater in a
solid state device. With modest concentrations of the active molecule on the
order of $10^{18}$ molecules/cm3 [48], the third order susceptibility for non-
interacting molecules is enhanced in proportion to the molecular density,
increasing the nonlinear index to

$n_{2}\approx 5\times 10^{-3}\frac{s\cdot cm^{2}}{erg}\;.$ (44)

Suppose the signal and idler wavelengths are on the order of $1\mu m$, the
resonator cross-sectional area is $1\mu m\times 1\mu m$, the group velocity of
the signal and idler are both $c$. Then the figure of merit $\mathcal{F}$ is
to within an order of magnitude given by

$\mathcal{F}=\frac{Q}{L}10^{3}\;.$ (45)

For a length on the order of $10\lambda$ and cavity quality factor on the
order $10^{4}$ [49, 50], the infidelity $\mathcal{F}^{-1}$ is $10^{-10}$. This
scenario is the upper boundary of feasible materials of this type, so we do
not expect to reach it in practice, but it offers several orders of magnitude
of leeway for trade-offs in designing materials and devices that shape photon
statistics.

While this rough calculation shows the plausibility of deterministic shaping
given known materials properties, an important question remains as to the
plausibility of making a solid state waveguide device. Realizing the benefits
of EIT and SGC in a solid state waveguide device would require a material that
in analogy is made with atoms that do not strongly interact with each other. A
possible material system is a dye-doped polymer, which can be fabricated into
waveguides, and whose linear and nonlinear-optical properties can be
controlled with atom/molecule doping.

One way to incorporate the active atom/molecule into a solid is to use a
polymer, sol-gel host, or other host – which would lead to higher dopant
concentration, thus a larger nonlinearity [51]. Such materials can be made
into thin films and fibers [52] and photolithographic techniques can be used
to make waveguide resonators [53]. Interactions between the host material and
the atom/molecule can be minimized by using molecular pincers – called
chelates – to isolate them. Chelated rare-earth atoms are commonly used in
amplifiers [54]. The ability to get higher densities of active molecules to
increase nonlinearity, the flexibility of choosing molecules for their
spectral properties, the shielding from the environment to mitigate de-phasing
through chelation, and the reduction of inhomogeneous broadening by cooling
the material all provide the means for optimization.

Each of these approaches presents challenges, but designing a device is far
beyond the scope of this paper. However, the numbers suggest that making a
device for deterministic shaping of quantum light statistics is possible and
our work defines the material requirements that will guide future experimental
efforts.

## References

  * [1] Nicolas Sangouard, Christoph Simon, Nicolas Gisin, Julien Laurat, Rosa Tualle-Brouri, and Philippe Grangier.  Quantum repeaters with entangled coherent states.  J. Opt. Soc. Am. B, 27(6):A137, June 2010.
  * [2] J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen, and A. S. Sørensen.  Hybrid long-distance entanglement distribution protocol.  Phys. Rev. Lett., 105:160501, Oct 2010.
  * [3] Atsushi Sakaguchi, Shunya Konno, Fumiya Hanamura, Warit Asavanant, Kan Takase, Hisashi Ogawa, Petr Marek, Radim Filip, Jun-Ichi Yoshikawa, Elanor Huntington, Hidehiro Yonezawa, and Akira Furusawa.  Nonlinear feedforward enabling quantum computation.  Nat. Commun., 14(1):3817, July 2023.
  * [4] Daniel Gottesman, Alexei Kitaev, and John Preskill.  Encoding a qubit in an oscillator.  Phys. Rev. A, 64(1), June 2001.
  * [5] D. A. Lidar, I. L. Chuang, and K. B. Whaley.  Decoherence-free subspaces for quantum computation.  Phys. Rev. Lett., 81:2594–2597, Sep 1998.
  * [6] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino.  Photonic quantum metrology.  AVS Quantum Sci., 2(2):024703, June 2020.
  * [7] M. Oszmaniec, R. Augusiak, C. Gogolin, J. Kołodyński, A. Acín, and M. Lewenstein.  Random bosonic states for robust quantum metrology.  Phys. Rev. X, 6:041044, Dec 2016.
  * [8] Manuel Unternährer, Bänz Bessire, Leonardo Gasparini, Matteo Perenzoni, and André Stefanov.  Super-resolution quantum imaging at the heisenberg limit.  Optica, 5(9):1150–1154, Sep 2018.
  * [9] Genko S Vasilev, Daniel Ljunggren, and Axel Kuhn.  Single photons made-to-measure.  New Journal of Physics, 12(6):063024, jun 2010.
  * [10] O. Morin, M. Körber, S. Langenfeld, and G. Rempe.  Deterministic shaping and reshaping of single-photon temporal wave functions.  Phys. Rev. Lett., 123:133602, Sep 2019.
  * [11] Yiwen Huang, Zhantong Qi, Yilin Yang, Yuanhua Li, Yiwei Sun, Yongzhi Tang, Fengchao Ni, Lanting Li, Yuanlin Zheng, and Xianfeng Chen.  Frequency-insensitive spatiotemporal shaping of single photon in multiuser quantum network.  Npj Quantum Inf., 9(1), September 2023.
  * [12] B Yurke and D Stoler.  Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion.  Phys. Rev. Lett., 57(1):13–16, July 1986.
  * [13] Xanda C. Kolesnikow, Raditya Weda Bomantara, Andrew C. Doherty, and Arne L. Grimsmo.  Gottesman-kitaev-preskill state preparation using periodic driving.  2023\.
  * [14] Warit Asavanant, Kota Nakashima, Yu Shiozawa, Jun-Ichi Yoshikawa, and Akira Furusawa.  Generation of highly pure schrödinger’s cat states and real-time quadrature measurements via optical filtering.  Opt. Express, 25(26):32227, December 2017.
  * [15] Kan Takase, Jun-Ichi Yoshikawa, Warit Asavanant, Mamoru Endo, and Akira Furusawa.  Generation of optical schrödinger cat states by generalized photon subtraction.  Phys. Rev. A (Coll. Park.), 103(1), January 2021.
  * [16] Yi-Ru Chen, Hsien-Yi Hsieh, Jingyu Ning, Hsun-Chung Wu, Hua Li Chen, Zi-Hao Shi, Popo Yang, Ole Steuernagel, Chien-Ming Wu, and Ray-Kuang Lee.  Generation of heralded optical ‘schroedinger cat’ states by photon-addition.  2023\.
  * [17] Kosuke Fukui, Mamoru Endo, Warit Asavanant, Atsushi Sakaguchi, Jun-Ichi Yoshikawa, and Akira Furusawa.  Generating the Gottesman-Kitaev-Preskill qubit using a cross-kerr interaction between squeezed light and fock states in optics.  Phys. Rev. A (Coll. Park.), 105(2), February 2022.
  * [18] A Ben Hayun, O Reinhardt, J Nemirovsky, A Karnieli, N Rivera, and I Kaminer.  Shaping quantum photonic states using free electrons.  Sci. Adv., 7(11):eabe4270, March 2021.
  * [19] A Beige, D Braun, B Tregenna, and P L Knight.  Quantum computing using dissipation to remain in a decoherence-free subspace.  Phys. Rev. Lett., 85(8):1762–1765, August 2000.
  * [20] A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac.  Deterministic generation of arbitrary photonic states assisted by dissipation.  Phys. Rev. Lett., 115:163603, Oct 2015.
  * [21] Frank Verstraete, Michael M Wolf, and J Ignacio Cirac.  Quantum computation and quantum-state engineering driven by dissipation.  Nat. Phys., 5(9):633–636, September 2009.
  * [22] Larissa Vertchenko, Maxim Nikitin, and Andrei Lavrinenko.  Near-zero-index platform in photonics: tutorial.  J. Opt. Soc. Am. B, 40(6):1467, June 2023.
  * [23] A Camacho-Guardian, M Bastarrachea-Magnani, T Pohl, and G M Bruun.  Strong photon interactions from weakly interacting particles.  Phys. Rev. B., 106(8), August 2022.
  * [24] Raphael F Ribeiro, Jorge A Campos-Gonzalez-Angulo, Noel C Giebink, Wei Xiong, and Joel Yuen-Zhou.  Enhanced optical nonlinearities under collective strong light-matter coupling.  Phys. Rev. A (Coll. Park.), 103(6), June 2021.
  * [25] Xin-Xin Hu, Chang-Ling Zhao, Zhu-Bo Wang, Yan-Lei Zhang, Xu-Bo Zou, Chun-Hua Dong, Hong X. Tang, Guang-Can Guo, and Chang-Ling Zou.  Cavity-enhanced optical controlling based on three-wave mixing in cavity-atom ensemble system.  Opt. Express, 27(5):6660–6671, Mar 2019.
  * [26] A Imamoḡlu, H Schmidt, G Woods, and M Deutsch.  Strongly interacting photons in a nonlinear cavity.  Phys. Rev. Lett., 79(8):1467–1470, August 1997.
  * [27] D. Porras and J. I. Cirac.  Collective generation of quantum states of light by entangled atoms.  Phys. Rev. A, 78:053816, Nov 2008.
  * [28] M Perarnau-Llobet, A González-Tudela, and J I Cirac.  Multimode fock states with large photon number: effective descriptions and applications in quantum metrology.  Quantum Science and Technology, 5(2):025003, feb 2020.
  * [29] C. K. Law and J. H. Eberly.  Arbitrary control of a quantum electromagnetic field.  Phys. Rev. Lett., 76:1055–1058, Feb 1996.
  * [30] Stefan Krastanov, Victor V. Albert, Chao Shen, Chang-Ling Zou, Reinier W. Heeres, Brian Vlastakis, Robert J. Schoelkopf, and Liang Jiang.  Universal control of an oscillator with dispersive coupling to a qubit.  Phys. Rev. A, 92:040303, Oct 2015.
  * [31] Reinier W. Heeres, Brian Vlastakis, Eric Holland, Stefan Krastanov, Victor V. Albert, Luigi Frunzio, Liang Jiang, and Robert J. Schoelkopf.  Cavity state manipulation using photon-number selective phase gates.  Phys. Rev. Lett., 115:137002, Sep 2015.
  * [32] P. A. M. Dirac.  Generalized hamiltonian dynamics.  Canadian Journal of Mathematics, 2:129–148, 1950.
  * [33] M Born and L Infeld.  On the quantization of the new field equations I.  Proc. R. Soc. Lond., 147(862):522–546, December 1934.
  * [34] M Born and L Infeld.  Foundations of the new field theory.  Proc. R. Soc. Lond. A Math. Phys. Sci., 144(852):425–451, March 1934.
  * [35] Nicolás Quesada and J E Sipe.  Why you should not use the electric field to quantize in nonlinear optics.  Opt. Lett., 42(17):3443, September 2017.
  * [36] Navin A. R. Bhat and J. E. Sipe.  Hamiltonian treatment of the electromagnetic field in dispersive and absorptive structured media.  Phys. Rev. A, 73:063808, Jun 2006.
  * [37] Michael G Raymer.  Quantum theory of light in a dispersive structured linear dielectric: a macroscopic hamiltonian tutorial treatment.  J. Mod. Opt., 67(3):196–212, February 2020.
  * [38] Mark Hillery and Leonard D Mlodinow.  Quantization of electrodynamics in nonlinear dielectric media.  Phys. Rev. A Gen. Phys., 30(4):1860–1865, October 1984.
  * [39] Peter D Drummond and Mark Hillery.  The quantum theory of nonlinear optics.  Cambridge University Press, Cambridge, England, May 2014.
  * [40] Nicolas Quesada, Lukas Helt, Matteo Menotti, Marco Liscidini, and John Sipe.  Beyond photon pairs: Nonlinear quantum photonics in the high-gain regime.  Adv. Opt. Photonics, April 2022.
  * [41] J D Jackson.  Electrodynamics, Classical.  Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, April 2003.
  * [42] Robert W Boyd.  Nonlinear optics.  Academic Press, San Diego, CA, 4 edition, March 2020.
  * [43] Alessia Pasquazi, Marco Peccianti, Luca Razzari, David J Moss, Stéphane Coen, Miro Erkintalo, Yanne K Chembo, Tobias Hansson, Stefan Wabnitz, Pascal Del’Haye, Xiaoxiao Xue, Andrew M Weiner, and Roberto Morandotti.  Micro-combs: A novel generation of optical sources.  Phys. Rep., 729:1–81, January 2018.
  * [44] Mohosin Alam, Swapan Mandal, and Mohamed Ridza Wahiddin.  Squeezing, mixed mode squeezing, amplitude squared squeezing and principal squeezing in a non-degenerate parametric oscillator.  Optik, 157:1035–1052, 2018.
  * [45] Ronen M. Kroeze, Brendan P. Marsh, Kuan-Yu Lin, Jonathan Keeling, and Benjamin L. Lev.  High cooperativity using a confocal-cavity–qed microscope.  PRX Quantum, 4:020326, May 2023.
  * [46] Nguyen Huy Bang and Le Van Doai.  Colossal kerr nonlinearity without absorption in a five-level atomic medium.  Sci. Rep., 14(1), January 2024.
  * [47] Le Van Doai.  The effect of giant kerr nonlinearity on group velocity in a six-level inverted-y atomic system.  Phys. Scr., 95(3):035104, March 2020.
  * [48] K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling.  Electro‐optic phase modulation and optical second‐harmonic generation in corona‐poled polymer films.  Applied Physics Letters, 53(19):1800–1802, 11 1988.
  * [49] Xiaoyang Cheng, Jianxun Hong, Andrew M. Spring, and Shiyoshi Yokoyama.  Fabrication of a high-q factor ring resonator using lscvd deposited si3n4 film.  Opt. Mater. Express, 7(7):2182–2187, Jul 2017.
  * [50] Tao Ling, Sung-Liang Chen, and L. Jay Guo.  Fabrication and characterization of high q polymer micro-ring resonator and its application as a sensitive ultrasonic detector.  Opt. Express, 19(2):861–869, Jan 2011.
  * [51] Carl W Dirk and M G Kuzyk.  Squarylium dye-doped polymer systems as quadratic electrooptic materials.  Chem. Mater., 2(1):4–6, January 1990.
  * [52] D J Welker, J Tostenrude, D W Garvey, B K Canfield, and M G Kuzyk.  Fabrication and characterization of single-mode electro-optic polymer optical fiber.  Opt. Lett., 23(23):1826–1828, December 1998.
  * [53] J Zheng, M Lu, C J Wagner, B T Cunningham, and J G Eden.  Spectral characteristics of single and coupled microresonator lasers comprising a replica-molded bragg grating and dye-doped polymer.  J. Opt. Soc. Am. B, 29(2):209, February 2012.
  * [54] C Koeppen, S Yamada, G Jiang, A F Garito, and Larry R Dalton.  Rare-earth organic complexes for amplification in polymer optical fibers and waveguides.  J. Opt. Soc. Am. B, 14(1):155, January 1997.


# GIST-AiTeR Speaker Diarization System for VoxCeleb Speaker Recognition
Challenge (VoxSRC) 2023

###### Abstract

This report describes the submission system by the GIST-AiTeR team for the
VoxCeleb Speaker Recognition Challenge 2023 (VoxSRC-23) Track 4. Our
submission system focuses on implementing diverse speaker diarization (SD)
techniques, including ResNet293 and MFA-Conformer with different combinations
of segment and hop length. Then, those models are combined into an ensemble
model. The ResNet293 and MFA-Conformer models exhibited the diarization error
rates (DERs) of 3.65% and 3.83% on VAL46, respectively. The submitted ensemble
model provided a DER of 3.50% on VAL46, and consequently, it achieved a DER of
4.88% on the VoxSRC-23 test set.

Index Terms: VoxSRC-23, Speaker Diarization, Variational Bayes, Multiscale
Embedding, Ensemble Model

##  1 Introduction

Speaker diarization (SD) is a task of determining “who spoke when,” and it
plays a vital role in many applications involving multi-speaker audio data.
Traditionally, SD is framed as a clustering problem where each segment of
speech is assigned a particular speaker label. A typical clustering-based SD
system is modular in nature and comprises components for voice activity
detection (VAD), speaker embedding extraction, and speaker clustering [1].
Once speaker labels are associated with each segment, segments sharing the
same label are grouped together. By improving speaker embedding, traditional
SD systems have achieved remarkable accomplishments through substantial
reduction of speaker ambiguity. Recent efforts to refine this process have
been concentrated on exploring methodologies such as overlapped speech
detection (OSD) [2], end-to-end neural SD (EEND) [3], and target-speaker voice
activity detection (TS-VAD) [4] to mitigate missed speaker errors.

In this report, we propose a clustering-based SD system specifically tailored
for the Diarization Task of the VoxCeleb Speaker Recognition Challenge 2023
(VoxSRC-23). The proposed approach integrates several processing stages such
as VAD, speaker embedding extraction, clustering, and OSD.

The remainder of this report is organized as follows: Section 2 delves into
the proposed system architecture, outlining four distinctive systems developed
by different combinations of SD modules with various hyperparameter settings.
Then, it also describes the construction of fusion systems to enhance
diarization efficiency. Section 3 presents and discusses the experimental
results. Finally, Section 4 concludes this report.

##  2 GIST-AiTeR SD System

The following subsections will explain each SD module in detail. In accordance
with [5], the provided challenge dataset is divided into two distinct subsets:
a development set (DEV402) and a validation set (VAL46) that are composed of
the initial 402 recordings and the remaining 46 recordings, respectively.

Table 1: Comparison of Precision and F1-score of different OSD modules on
VAL46

OSD Module |  Precision (%) |  F1-score (%)
---|---|---
GIST-AiTeR (VoxSRC-22) [6] | 88.81 | 52.45
ResNet+LSTM with Multitask detection | 90.16 | 53.81

###  2.1 VAD and OSD

VAD identifies speech among audio segments. In this work, we utilized the
identical approach as the one used in our VoxSRC-22 submission system [6].
Specifically, we constructed an ensemble method with ResNet+LSTM and
SincNet+LSTM modules, averaging their posterior predictions.

For OSD, we utilized the same ResNet+LSTM model structure and training dataset
as those used in [6]. In addition, multitask detection in [7] was employed for
activating speech, overlapping speech, and speaker changes. In particular, we
used weighted binary cross entropy (BCE) loss to train for OSD owing to the
data imbalance problem. This imbalance occurs because the number of overlapped
speech frames is significantly smaller than that of single speech or silence
frames. The weighted BCE loss was also used for speaker change detection,
effectively addressing imbalance problem. To adjust the hyperparameters in our
model, we utilized Pyannote’s hyperparameter optimization [8]. In particular,
we set the $\beta$ value to 0.1 in the $F_{\beta}$ score calculation, thereby
increasing the importance of precision in the optimization process. This
adjustment ensures accurate identification of overlap regions while
prioritizing precision.

Table 1 compares the precision and F1-score of different OSD modules on VAL46.
As shown in the table, the ResNet+LSTM model with the multitask detection
approach improved both precision and F1-score compared to our submission
system from the previous year. Consequently, we selected this model as the OSD
module for our proposed SD system.

Table 2: Performance comparison of our different versions of SD systems

Method | Time-scale | VAL46 | VoxSRC-23
---|---|---|---
|  | DER | JER | DER | JER
GIST-AiTeR [6] | - | 3.56 | 27.63 | 5.67 | 33.51
MFA-Conformer | 1.0 / 0.5 | 3.98 | 25.89 | - | -
2.0 / 1.0 | 3.83 | 24.33 | - | -
3.0 / 1.5 | 4.11 | 26.02 | - | -
ResNet293 | 1.5 / 0.25 | 3.65 | 24.97 | - | -
DOVER-Lap |  | 3.50 | 25.23 | 4.88 | 29.69

###  2.2 Speaker Embedding

In our speaker embedding process, we employed two models: MFA-Conformer,
trained with a multiscale approach and consistent with the process described
in the previous year’s implementation [6], and ResNet293 provided by Wespeaker
[9]. While MFA-Conformer was employed solely in the previous year, we
introduced ResNet293 as an innovative component to complement MFA-Conformer.
The ensemble of the MFA-Conformer and ResNet293 forms the core of the proposed
SD system, providing a robust solution to extract speaker embedding.

###  2.3 Clustering

The dimension of speaker embedding vectors obtained from both MFA-Conformer
and ResNet293 was first reduced into 128 using the linear discriminant
analysis (LDA). Subsequently, two probabilistic LDA (PLDA) models were
formulated by utilizing the 128-dimensional embedding vectors, derived from
both VoxCeleb1 and VoxCeleb2 and the VoxConverse DEV402 set. Then, these two
PLDA models were interpolated into one with a factor of 0.9.

Next, agglomerative hierarchical clustering (AHC) was applied to a given
utterance, where PLDA scores were employed to allocate a cluster index to
individual frame. A threshold for AHC clustering was determined with a two-
component Gaussian mixture model using a shared variance over all PLDA scores
from the utterance.

After that, the variational Bayesian hidden Markov model (VB-HMM) [10] was
applied to reassign the cluster index to each frame, considering time
dependencies. This approach allows more freedom to find optimal results and
converge to the appropriate number of speaker models.

##  3 Experimental Results

Table 2 compares the SD performance of different versions of our system,
including our previous model from VoxSRC-22 [6], MFA-Conformer, ResNet293, and
the final ensemble DOVER-Lap model. The performance was evaluated on both
VAL46 and VoxSRC-23 test set. In this report, the performance of MFA-Conformer
was evaluated across three different combinations of segment length and hop
length, such as (1 s / 0.5 s), (2 s / 1 s), and (3 s / 1.5 s), with the hop
length being half of the segment length. However, the ResNet293 embedding was
evaluated using one pair of (1.5 s / 0.25 s). As shown in the table, different
combinations of segment and hop lengths for MFA-Conformer led to varied
performance results on VAL46, where a segment/hop length of (2.0 / 1.0)
yielded the best diarization error rate (DER) of 3.83%. On the other hand, the
ResNet293 speaker embedding lowered DER by 0.18% than MFA-Conformer, which was
the best among all the single models evaluated in this work.

Next, we ensembled various models according to different speaker embedding
techniques alongside varying segment and hop lengths, through DOVER-Lap [11],
which culminated in a significantly reduced DER of 3.50% on VAL46. This
strategy of incorporating diverse modeling parameters played an important role
in the success of the ensemble system, leading to its selection for the
challenge. Consequently, the system obtained from this ensemble approach
achieved a DER of 4.88% on the challenge test set.

##  4 Conclusion

In this report, we explored various SD techniques and evaluated their
performance on VAL46 and VoxSRC-23 test set. The ResNet293 and MFA-Conformer
models exhibited varied performance depending on segment and hop lengths,
resulting in the best DERs of 3.65% and 3.83%, respectively. By integrating
all the individual models according to different combinations of segment and
hop lengths through DOVER-Lap, we achieved a significantly reduced DER of
3.50% on VAL46. Notably, this ensemble approach led to a DER of 4.88% on the
challenge test set, ranked fourth place in the VoxSRC-23.

##  5 Acknowledgements

This work was supported in part by Institute of Information & Communications
Technology Planning & Evaluation funded by the Korea Government (MSIT)
(2022-0-00963) and by Culture, Sports, and Tourism R&D Program through the
Korea Creative Content Agency grant funded by the Ministry of Culture, Sports,
and Tourism in 2023 (R2022060001, Contribution Rate: 50%).

## References

  * [1] S. H. Shum, N. Dehak, R. Dehak, and J. R. Glass, “Unsupervised methods for speaker diarization: An integrated and iterative approach,” _IEEE/ACM Trans. Audio Speech Lang. Process._ , vol. 21, no. 10, pp. 2015–2028, 2013.
  * [2] K. Boakye, B. Trueba-Hornero, O. Vinyals, and G. Friedland, “Overlapped speech detection for improved speaker diarization in multiparty meetings,” in _Proc. ICASSP_ , 2008, pp. 4353–4356.
  * [3] Y. Fujita, S. Watanabe, S. Horiguchi, Y. Xue, and K. Nagamatsu, “End-to-end neural diarization: Reformulating speaker diarization as simple multi-label classification,” _arXiv preprint, arXiv:2003.02966_ , 2020.
  * [4] I. Medennikov _et al._ , “Target-speaker voice activity detection: A novel approach for multi-speaker diarization in a dinner party scenario,” in _Proc. Interspeech_ , 2020, pp. 274–278.
  * [5] W. Wang, D. Cai, Q. Lin, L. Yang, J. Wang, J. Wang, and M. Li, “The DKU-DukeECE-Lenovo system for the diarization task of the 2021 VoxCeleb speaker recognition challenge,” _arXiv preprint, arXiv:2109.02002_ , 2021\.
  * [6] D. Park, Y. Yu, K. W. Park, J. W. Kim, and H. K. Kim, “GIST-AiTeR system for the diarization task of the 2022 VoxCeleb speaker recognition challenge,” _arXiv preprint, arXiv:2209.10357_ , 2022.
  * [7] M. Kunešová and Z. Zajíc, “Multitask detection of speaker changes, overlapping speech and voice activity using wav2vec 2.0,” in _Proc. ICASSP_ , 2023, pp. 1–5.
  * [8] H. Bredin, “pyannote.audio 2.1 speaker diarization pipeline: principle, benchmark, and recipe,” in _Proc. Interspeech_ , 2023, pp. 1983–1987.
  * [9] H. Wang _et al._ , “Wespeaker: A research and production oriented speaker embedding learning toolkit,” in _Proc. ICASSP_ , 2023, pp. 1–5.
  * [10] F. Landini _et al._ , “Bayesian HMM clustering of x-vector sequences (VBx) in speaker diarization: theory, implementation and analysis on standard tasks,” _Computer Speech & Language_, vol. 71, p. 101254, 2022\.
  * [11] D. Raj, L. P. Garcia-Perera, Z. Huang, S. Watanabe, D. Povey, A. Stolcke, and S. Khudanpur, “DOVER-Lap: A method for combining overlap-aware diarization outputs,” in _Proc. IEEE SLT_ , 2021, pp. 881–888.