# Maximally efficient biphoton generation by single photon decay in nonlinear
quantum photonic circuits

Mikhail Tokman  Department of Electrical and Electronic Engineering and
Schlesinger Knowledge Center for Compact Accelerators and Radiation Sources,
Ariel University, 40700 Ariel, Israel  Jitendra K. Verma  Department of
Physics and Astronomy, Texas A&M University, College Station, TX, 77843 USA
Jacob Bohreer  Department of Physics and Astronomy, Texas A&M University,
College Station, TX, 77843 USA  Alexey Belyanin<EMAIL_ADDRESS>Department
of Physics and Astronomy, Texas A&M University, College Station, TX, 77843 USA

###### Abstract

We develop a general nonperturbative formalism and propose a specific scheme
for maximally efficient generation of biphoton states by parametric decay of
single photons. We show that the well-known critical coupling concept of
integrated optics can be generalized to the nonlinear coupling of quantized
photon modes to describe the nonperturbative optimal regime of a single-photon
nonlinearity and establish a fundamental upper limit on the nonlinear
generation efficiency of quantum-correlated photons, which approaches unity
for low enough absorption losses.

Introduction.— Future quantum information systems will inevitably rely on
chip-scale integrated photonic circuits for generation and control of quantum
states of light [1, 2, 3, 4, 5, 6, 7, 8, 9]. Nonlinear cavities and waveguides
supporting spontaneous parametric down-conversion (SPDC) would be particularly
attractive for these applications due to their compatibility with mature
semiconductor technology. The bottleneck in any such circuit is an extremely
low efficiency of nonlinear optical interactions between single-photon states
[10], which is usually mitigated by using strong classical laser drive fields.
While this is a viable strategy if one needs just a source of single photon or
biphoton states, more complicated quantum information schemes require multiple
steps (gates) involving nonlinear couplings between single photons. One way to
overcome the low efficiency challenge is to utilize strong coupling of single
photon states with resonant quantum emitters in a nanocavity [11, 12]. Another
approach to single-photon nonlinearities which we are going to propose and
analyze here is to utilize critical coupling [13], or rather its
generalization to nonlinearly coupled systems and quantized fields. We will
show that this approach allows one to achieve efficient generation of
biphotons by parametric decay of single drive photons in low-dissipation
dielectric photonic circuits, even at the optical nonlinearities typical for
standard semiconductor materials. We demonstrate that the nonlinear critical
coupling establishes the fundamental upper limit on the single-photon SPDC,
which approaches unity for low enough absorption losses.

For a specific illustration we consider a standard photonic circuit element
which includes evanescent coupling of quantized waveguide modes to the
quantized modes of a nonlinear ring cavity as in Fig. 1. The usual treatments
based on Heisenberg-Langevin or scattering matrix formalisms [15, 16, 14]
assume a sufficiently weak nonlinearity and some sort of the perturbation
expansion to avoid dealing with nonlinear operator-valued equations. Here, by
extending our recently proposed version of the stochastic Schroedinger
equation (SSE) formalism [17, 18, 19] to the quantum field propagation
problems, we were able to consider strongly nonperturbative nonlinear regimes
and include the coupling of all quantized degrees of freedom to their
dissipative reservoirs. Simple analytical expressions for the biphoton output
fluxes are obtained in the practical limit of low reservoir temperature as
compared to the optical photon energies.

Figure 1: A schematic of a waveguide evanescently coupled to a nonlinear ring
cavity made of a crystal with second order nonlinearity $\chi^{(2)}$. A single
photon propagates as a wave packet of waveguide modes with frequencies
$\omega_{k}$ and couples to the ring cavity mode at frequency $\omega_{d}$
with coupling coefficient $\kappa$. A drive photon decays parametrically into
signal and idler photons $\omega_{1}$ and $\omega_{2}$ which outcouple into
the waveguide with outcoupling coefficients $\kappa_{1,2}$.

Besides the particular application to the nonperturbative SPDC process, our
approach develops the critical coupling theory in the fully quantum domain
including propagation, dispersion, and dissipation, and fluctuation effects on
the same footing, and further generalizes it to include nonlinear optical
coupling. One can apply the same approach to other nonlinear optical processes
such as spontaneous four-wave mixing of single-photon modes. Thus our results
bring together the burgeoning fields of quantum nonlinear optics, integrated
photonics, and telecommunications.

Quantum Dynamics in the single-photon nonlinear coupling.— We begin with
expanding all fields into corresponding spatial modes, i.e., (i) waveguide
modes $\mathbf{E}_{k}(\mathbf{r}_{\mathbf{\bot}})e^{ikz}$ of a waveguide
segment with cross-sectional area S and length L, where
$\mathbf{r}_{\mathbf{\bot}}\mathbf{\bot}\mathbf{z}_{0}$, $\mathbf{z}_{0}$ is
the unit vector along z axis, and (ii) cavity modes
$\mathbf{E}_{N}\left(\mathbf{r}\right)$ of a ring resonator with cross
sectional area $S_{R}$ and radius $R$. The coupling between waveguide and
cavity modes are maximized near the resonance, when $k\approx\frac{N}{R}$ and
$\omega_{k}\approx\omega_{N}$ $(\omega_{k,N}$ are certain eigenfrequencies).
For boundary-value problems one would need to define the waveguide with open
ends and corresponding boundary conditions on the facets. However, since we
want to solve the initial value problem, we can start from a certain field
wave packet at the initial moment of time inside the waveguide (say, near the
boundary) and consider its evolution by expanding it over the waveguide modes
satisfying periodic boundary conditions $k=2\pi n$, $n=0,\pm 1,\pm 2\ldots$
This makes sense until the field goes outside the segment, but we can always
assume its length $L$ to be large enough. All the results will not depend on
the value of $L$.

Proceeding to quantize the fields, one can define the Schroedinger’s operator
of the drive field in the waveguide as

$\hat{\bf{E}}_{wg}({\bf{{r}}}_{\bot},z)=\sum_{k}\big{[}{\hat{c}}_{k}{\bf{E}}_{k}({\bf{r}}_{\bf{\bot}})e^{ikz}+\hat{c}_{k}^{\dagger}{\mathbf{E}}_{k}^{*}(\mathbf{r}_{\bot})e^{-ikz}\big{]},$
(1)

and similarly for the cavity field,

${\hat{\mathbf{E}}}_{R}\mathbf{=}\sum_{N}\big{[}{\hat{c}}_{N}\mathbf{E}_{N}(\mathbf{r})\mathbf{+}{\hat{c}}_{N}^{\dagger}\mathbf{E}_{N}^{\mathbf{*}}(\mathbf{r})\big{]}.$
(2)

Here r is the position vector in the cavity. The normalization given by Eqs.
(S3) and (S4) in the supplemental material (SM) yields standard bosonic
commutation relations for creation and annihilation operators,
${[\hat{c}}_{i}^{\dagger},{\hat{c}}_{j}]=\delta_{ij}$.

The standard perturbative approach would be to solve for the dynamics of the
coupled fields in the Heisenberg’s picture starting from the operator-valued
Maxwell’s equations for the evolution of nonlinearly coupled cavity modes
[20],

$\frac{d\hat{c}_{N}}{dt}+i\omega_{N}{\hat{c}}_{N}=-\frac{i}{\hbar\omega_{N}^{2}}\int_{V}{\mathbf{E}_{N}^{\mathbf{*}}(\mathbf{r}){\ddot{\hat{\mathbf{P}}}}_{ex}(\mathbf{r},t)d^{3}r}$
(3)

where ${\hat{\mathbf{P}}}_{ex}$ is an operator of the external polarization
which include the nonlinear response and any linear coupling from the outside.
Consider three cavity modes, $N=d,1,2$, for which the parametric down-
conversion conditions are satisfied: $\omega_{d}=\omega_{1}+\omega_{2}$ and
$N_{d}=N_{1}+N_{2}$. Then the positive-frequency $\propto e^{-i\omega t}$ part
of the external polarization is

${\hat{\mathbf{P}}}_{ex;d}^{(\mathbf{+})}=\hat{\chi}^{(2)}(\mathbf{r},\omega_{d}=\omega_{1}+\omega_{2})\mathbf{E}_{1}(\mathbf{r})\mathbf{E}_{2}(\mathbf{r}){\hat{c}}_{1}{\hat{c}}_{2}+{\hat{\mathbf{P}}}_{lin}^{(\mathbf{+})},$
(4)
$\hat{\mathbf{P}}_{ex;1}^{(\mathbf{+})}=\hat{\chi}^{(2)}(\mathbf{r},\omega_{1}=\omega_{d}-\omega_{2})\mathbf{E}_{d}(\mathbf{r})\mathbf{E}_{2}^{\mathbf{*}}(\mathbf{r}){\hat{c}}_{d}{\hat{c}}_{2}^{\dagger},$
(5)

and similarly for $\hat{\mathbf{P}}_{ex;2}$, where $\hat{\chi}^{(2)}$ is a
third-rank tensor of the second-order susceptibility, with permutation
properties defined in the SM. The operator
${\hat{\mathbf{P}}}_{lin}^{(\mathbf{+})}$ describes linear excitation of the
cavity mode at the drive frequency by the waveguide field, which can be
parameterized as

$\int_{V}{\mathbf{E}_{d}^{\mathbf{*}}{\hat{\mathbf{P}}}_{lin}^{(\mathbf{+})}d^{3}r}=\sum_{k}{I_{k}\hat{c}}_{k}.$
(6)

Finding the excitation coefficients $I_{k}$ is a boundary-value problem of the
classical electrodynamics. They can be calculated for a specific geometry by
any EM solver or conveniently parameterized for any geometry as shown in Sec.
III of the SM, where the conditions for a single-mode excitation are also
derived. We can assume that $I_{k}$ reaches a maximum at the drive frequency
$\omega_{d}=\omega_{k}$ and decays with increasing detuning
$|\omega_{d}-\omega_{k}|$.

To quantify the SPDC process, we introduce the nonlinear overlap integral of
the cavity modes with the second-order nonlinearity distribution, integrated
over the cavity volume:

$G=\int_{V}{\mathbf{E}_{1}^{\mathbf{*}}(\mathbf{r})\hat{\chi}^{(2)}(\mathbf{r},\omega_{1}=\omega_{d}-\omega_{2})\mathbf{E}_{d}(\mathbf{r})\mathbf{E}_{2}^{\mathbf{*}}(\mathbf{r})d^{3}r}.$
(7)

For numerical estimates we will assume that the ring cavity is made of a
nonlinear material with a constant value of $\chi^{(2)}$ for given field
polarizations. The equations of motion following from Eq. (3) are identical to
the Heisenberg equations of motion with the Hamiltonian

$\displaystyle\hat{H}=$
$\displaystyle\hbar\sum_{i=d,1,2}\omega_{i}({\hat{c}}_{i}^{\dagger}{\hat{c}}_{i}+\frac{1}{2})+\hbar\sum_{k}\omega_{k}({\hat{c}}_{k}^{\dagger}{\hat{c}}_{k}+\frac{1}{2})$
(8)
$\displaystyle-(G{\hat{c}}_{d}{\hat{c}}_{2}^{\dagger}{\hat{c}}_{1}^{\dagger}+h.c.)-\sum_{k}({I_{k}\hat{c}}_{k}{\hat{c}}_{d}^{\dagger}+h.c.),$

see Eqs. (S8)-(S11) in the SM for their explicit form.

The next step in a standard approach would be to couple all fields to their
dissipative reservoirs and solve the resulting Heisenberg-Langevin equations.
However, this approach would lead to operator-valued nonlinear equations [15,
14] and generally work only in the perturbative regime of low SPDC
probability. Instead, we developed the modified version of the SSE formalism
which allows us to find analytic or semi-analytic solutions for the
nonperturbative quantum dynamics with an arbitrarily strong nonlinearity,
determine the maximum nonlinear efficiency, and predict the optimal
parameters.

It is well known from the quantum jumps theory [15, 22, 21, 23, 24, 25, 26,
27] that if the master equation for the dynamics of an open quantum system is
represented in the Lindblad form,
$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]+\hat{\mathcal{L}}(\hat{\rho})$,
the Lindbladian can be always written as
$\hat{\mathcal{L}}(\hat{\rho})=-\frac{i}{\hbar}({\hat{H}}^{(ah)}\hat{\rho}-\hat{\rho}{\hat{H}}^{(ah)\dagger})+\delta\hat{\mathcal{L}}(\hat{\rho})$
where ${\hat{H}}^{(ah)}$ is an anti-Hermitian operator. Then, as we show in
[17, 18, 19], the evolution of the system can be equivalently described by
solving the SSE

$\frac{d}{dt}|\Psi\rangle=-\frac{i}{\hbar}(\hat{H}+{\hat{H}}^{(ah)})|\Psi\rangle-\frac{i}{\hbar}|\mathfrak{R}(t)\rangle$
(9)

where the noise vector satisfies
$\overline{|\mathfrak{R}(t)\rangle\langle\mathfrak{R}(t^{\prime})|}=\hbar^{2}\delta{\hat{\mathcal{L}}(\hat{\rho})}_{\hat{\rho}\Rightarrow\overline{|\Psi\rangle\langle\Psi|}}\delta(t-t^{\prime})$
and the bar means averaging over the reservoir statistics. This approach
provides significant analytical and numerical advantages by reducing the
effective number of degrees of freedom, especially at the single-photon
excitation level. For bosonic modes we have

${\hat{H}}^{(ah)}=\sum_{j}-i\hbar\frac{\mu_{j}}{2}[{\bar{n}}_{\omega_{j}}^{T}{\hat{c}}_{j}{\hat{c}}_{j}^{\dagger}+({\bar{n}}_{\omega_{j}}^{T}+1){\hat{c}}_{j}^{\dagger}{\hat{c}}_{j}],$
(10)
$\delta\hat{\mathcal{L}}(\hat{\rho})=\sum_{j=d,1,2}\left[\mu_{j}{\bar{n}}_{\omega_{j}}^{T}{\hat{c}_{j}}^{\dagger}\hat{\rho}\hat{c}_{j}+(\bar{n}_{\omega_{j}}^{T}+1)\hat{c}_{j}\hat{\rho}{\hat{c}_{j}}^{\dagger}\right],$
(11)

where
${\bar{n}}_{\omega_{j}}^{T}=\left(e^{\frac{\hbar\omega_{j}}{k_{B}T}}-1\right)^{-1}$.
To get analytic results, here we consider the case when the photon frequencies
are much higher than the thermal energy $k_{B}T$, so that we take
${\bar{n}}_{\omega_{j}}^{T}\rightarrow 0$ in Eqs. (10) and (11) and treat the
relaxation constants $\frac{\mu_{1,2}}{2}$ as the total field decay rates of
the signal and idler cavity modes which include the rate of absorption in the
cavity and the rate of cavity field outcoupling into the waveguide:
$\frac{\mu_{1,2}}{2}=\frac{\gamma_{1,2}}{2}+\kappa_{1,2}$. The outcoupling
rate of the drive cavity field back into the waveguide is already included
through the last term in the Hamiltonian (8). There is reciprocity between in-
and outcoupling rates which we both denote as $\kappa$.

For a single-photon drive, the SSE can be solved by the following state vector
ansatz:

$\displaystyle|\Psi\rangle=\sum_{k}C_{k}|1_{k}\rangle\prod_{k^{\prime}\neq
k}|0_{k^{\prime}}\rangle|0_{d}\rangle|0_{1}\rangle|0_{2}\rangle+$
$\displaystyle
C_{d}\prod_{k}{|0_{k}}\rangle|1_{d}\rangle|0_{1}\rangle|0_{2}\rangle+C_{12}\prod_{k}|0_{k}\rangle|0_{d}\rangle|1_{1}\rangle|1_{2}\rangle+$
$\displaystyle
C_{0}\prod_{k}|0_{k}\rangle|0_{d}\rangle|0_{1}\rangle|0_{2}\rangle,$ (12)

which results in a set of linear equations for the probability amplitudes,

$\displaystyle\frac{dC_{k}}{dt}+i\omega_{k}C_{k}=\frac{i}{\hbar}I_{k}^{*}C_{d}$
(13)
$\displaystyle\frac{dC_{d}}{dt}+(i\omega_{d}+\frac{\mu_{d}}{2})C_{d}=\frac{i}{\hbar}G^{*}C_{12}+\frac{i}{\hbar}\sum_{k}{I_{k}C_{k}}$
(14)
$\displaystyle\frac{dC_{12}}{dt}+(i\omega_{1}+i\omega_{2}+\frac{\mu_{1}}{2}+\frac{\mu_{2}}{2})C_{12}=\frac{i}{\hbar}GC_{d}.$
(15)

Note that the noise vector components are neglected in the equations for the
excited state amplitudes due to the assumption of low reservoir temperature.
The noise term must be kept in the equation for $C_{0}$, but the latter
equation can be omitted since $C_{0}$ is not present in other equations and
$|C_{0}|^{2}$ can be obtained from conservation of noise-averaged norm of the
wave vector, $\overline{\langle\Psi|\Psi\rangle}=1$ [17]. Furthermore, we
dropped vacuum terms in the free field Hamiltonian, since they do not affect
the results which depend only on frequency detunings.

Consider the initial condition corresponding to a single-photon wavepacket of
spectral width $\Delta\omega$ in the waveguide and no photons in the cavity,
i.e., $C_{d}(0)=C_{12}(0)=C_{0}(0)=0$, and $C_{k}(0)\neq 0$. The coefficients
$I_{k}$ determine the place of the most effective coupling along _z_ , which
is normally the closest position to the ring cavity. Let all $C_{k}(0)$ are
such that at $t=0$ the field pulse is localized at $z=0$. Although Eqs.
(13)-(15) look like an initial-value problem, in Sec. IV and V of the SM we
show how they include all propagation and linear coupling effects, in
particular the critical coupling regime.

Signal and idler photon fluxes from single-photon SPDC.— Eqs. (13)-(15) are
solved for $C_{12}(t)$ with the help of Laplace transforms and the convolution
theorem in Sec. VI of the SM. The result is

$\left|C_{12}(t)\right|^{2}=\frac{|g|^{2}\left|\Omega_{kd}\right|_{\omega_{k}=\omega_{d}}^{2}}{\left|\kappa_{\Sigma}p_{12}+|g|^{2}\right|^{2}}\left\\{\sum_{k}{\sum_{q}\left[C_{k}(0)C_{q}^{*}(0)\left(e^{-i\left(\omega_{k}-\omega_{q}\right)t}+e^{-i\omega_{k}t}U^{*}(t)+e^{i\omega_{q}t}U(t)\right)\right]}+{\rm
c.c.}\right\\},$ (16)

where

$U(t)=e^{-\left(i\omega_{d}+\frac{\kappa_{\Sigma}+p_{12}}{2}\right)t}[\cosh\left(\theta{t}\right)-\frac{{\kappa_{\Sigma}+p_{12}}}{2\theta}\sinh(\theta{t})],$

$\theta=\sqrt{(\frac{\kappa_{\Sigma}-p_{12}}{2})^{2}-|g|^{2}}$,
$p_{12}=\frac{\mu_{1}}{2}+\frac{\mu_{2}}{2}+\omega_{1}+\omega_{2}-\omega_{d}$,
and $\kappa_{\Sigma}=\frac{\mu_{d}}{2}+\kappa$ is the total decay rate of the
drive field in the cavity which includes its absorption, diffraction, and
outcoupling to the waveguide modes (see Sec. V of the SM). We also introduced
the notations $\frac{i}{\hbar}I_{k}=\Omega_{kd}$ and $\frac{i}{\hbar}G=g$.
Then the photon fluxes outcoupled from the cavity to the waveguide at signal
and idler frequencies $\omega_{1,2}$ are (in photons per second)

$\Pi_{1,2}=2\kappa_{1,2}\left|C_{12}(t)\right|^{2}.$ (17)

The expression (16) is still cumbersome, but it can be easily evaluated
numerically for a given spectrum $C_{k}(0)$ of the incident single-photon
driving pulse, and its meaning is transparent. The two terms containing $U(t)$
describe decaying or overdamped nonlinear Rabi oscillations between the drive
and signal+idler photons, depending on the interplay between the nonlinear
coupling strength and the combination of losses and detuning. Note that the
terms $e^{-i\left(\omega_{k}-\omega_{q}\right)t}$, $e^{-i\omega_{k}t}U^{*}(t)$
and $e^{i\omega_{q}t}U(t)$ are of the same order of magnitude, but the last
two terms describing the Rabi oscillations decay over the timescale no longer
than $\sim\kappa_{\Sigma}^{-1}$, whereas the remaining term
$\propto\sum_{k}{\sum_{q}{C_{k}(0)C_{q}^{*}(0)e^{-i\left(\omega_{k}-\omega_{q}\right)t}}}+$
c.c. describes the signal with a much longer duration
$\sim\frac{2\pi}{\mathrm{\Delta}\omega}$, or even an infinitely long noisy
signal with correlation time $\sim\frac{2\pi}{\mathrm{\Delta}\omega}$.
Therefore, for the drive pulse which is much longer than the timescales of
diffraction and absorption losses in the ring cavity we can keep only this
remaining term.

The unperturbed incident flux of the drive photons in the waveguide is

$\Pi_{wg}=\frac{\upsilon_{g}}{2L}\left[\sum_{k}{\sum_{q}{C_{k}(0)C_{q}^{*}(0)e^{-i\left(\omega_{k}-\omega_{q}\right)t}}}+c.c.\right],$
(18)

where $\upsilon_{g}$ is the group velocity of the drive wavepacket in the
waveguide. Note that in the limit of a large number of modes within the width
of the wavenumber spectrum $\Delta k$, when $\Delta kL\gg 2\pi$, the value of
$\Pi_{wg}$ does not depend on the length of the quantization segment $L$.
Consider the maximum flux in the middle of SPDC bandwidth at exact frequency
matching $\omega_{1}+\omega_{2}-\omega_{d}=0$. Using also
$\left|\Omega_{kd}\right|^{2}\simeq 2\kappa\frac{\upsilon_{g}}{L}$ as shown in
the SM, the resulting signal and idler fluxes are

$\Pi_{1,2}=\frac{8\kappa_{1,2}\kappa|g|^{2}\Pi_{wg}}{\left[\left(\kappa+\frac{\mu_{d}}{2}\right)\left(\kappa_{1}+\kappa_{2}+\frac{\gamma_{1}}{2}+\frac{\gamma_{2}}{2}\right)+|g|^{2}\right]^{2}}$
(19)

This is a simple but very informative expression which gives the universal
dependence of signal and idler fluxes from all relevant parameters, and
includes all possible regimes, such as perturbative (small $|g|$),
nonperturbative (arbitrary $|g|$), loss-dominated, and nonlinear critical
coupling (see below). Moreover, using more general expressions from the SM it
can be generalized to the situations of high reservoir temperatures, broadband
and ultrashort pulses, nonlinear Rabi oscillations, etc. It is also important
that all the input parameters entering the solution of the quantum nonlinear
mixing problem (such as $\kappa$, $g$, $\kappa_{1,2}$, $\mu_{d}$ and
$\gamma_{1,2}$) can be calculated separately, by solving the classical
electrodynamics problem with boundary conditions imposed by the photonic
circuit geometry and with known material parameters. Therefore, the physics of
quantum nonlinear mixing is universal and decoupled from specific experimental
geometry.

Figure 2:  The normalized flux of signal photons $\frac{\Pi_{1}}{\Pi_{wg}}$
(a) as a function of the normalized nonlinear coupling strength $g/g_{max}$
for negligible absorption $\gamma_{1,2}=\mu_{d}=0$ (red line) and for
$\kappa_{1,2}=\kappa$, $\gamma_{1,2}=\mu_{d}=2\kappa$ (blue-dashed line); and
(b) as a function of the normalized linear coupling of the drive field
$(\kappa/\kappa_{max})$ for a fixed $g=4.1\times 10^{8}$ sec-1 and
$\kappa=\kappa_{1,2}=g$, without absorption $\gamma_{1,2}=\mu_{d}=0$ (red
line), and with absorption $\gamma_{1,2}=\mu_{d}=2g$ (blue-dashed line).
Figure 3:  The normalized flux of signal photons $\frac{\Pi_{1}}{\Pi_{wg}}$ as
a function of the normalized linear coupling strengths of the drive and signal
fields for $\gamma_{1,2}=\mu_{d}=g$. Inset: the limit of zero losses, when the
flux depends only on the product $\kappa\kappa_{1}$.

Figures 2 and 3 illustrate some of these dependencies for the signal flux
normalized by the waveguide flux, $\frac{\Pi_{1}}{\Pi_{wg}}$, and assuming
degenerate SPDC, $\omega_{1}=\omega_{2}$, $\gamma_{1}=\gamma_{2}$, and
$\kappa_{1}=\kappa_{2}$ in order to reduce the number of parameters. The
latter assumption can be easily lifted if needed. The maximum possible signal
flux in all figures is of course equal to the incident waveguide flux, i.e.,
$\frac{\Pi_{1}}{\Pi_{wg}}=1$, corresponding to one pair of signal and idler
photons per one photon at the drive frequency in the waveguide. In agreement
with energy conservation, this maximum value is reached when the absorption
losses of all fields are much smaller than linear and nonlinear coupling
frequency scales. Figure 2(a) illustrates the transition from the perturbative
regime $\propto|g|^{2}$ at small $g$ to the nonperturbative regime of large
$g$ as $g=|g|$ changes while keeping other parameters fixed. The maximum flux
is reached at

$g^{2}_{\max}=\left(\kappa+\frac{\mu_{d}}{2}\right)\left(\kappa_{1}+\kappa_{2}+\frac{\gamma_{1}}{2}+\frac{\gamma_{2}}{2}\right).$
(20)

This peak value of the flux is equal to one when absorption losses are
negligible (solid red curve), and decreases quickly with increasing losses as
is clear from Eq. (19) and illustrated by the blue dashed curve.

Figure 2(b) shows the dependence of the signal flux from the linear coupling
strength $\kappa$ between the waveguide and the cavity at the drive frequency.
The peak value of the signal is reached at

$\kappa_{\max}=\frac{\mu_{d}}{2}+\frac{|g|^{2}}{\kappa_{1}+\kappa_{2}+\frac{\gamma_{1}}{2}+\frac{\gamma_{2}}{2}}.$
(21)

Clearly, this expression is the modification of the critical coupling
condition $\kappa=\frac{\mu_{d}}{2}$ derived in [13] for classical fields and
SM for the quantum fields. The significant deviation from standard critical
coupling is possible for large enough values of the nonlinear coupling
strength $g$. To get a specific numerical estimate for the value of $g$, we
consider an InGaP thin-film microring resonator evanescently coupled to a
straight waveguide, similar to the type of a nanophotonic circuit implemented
in [9]. Taking the parameters consistent with their devices: a 5 $\mu$m radius
and $100\times 400$ nm2 of the ring, $\chi^{(2)}=220$ pm/V for InGaP [28],
drive wavelength of 750 nm and signal/idler wavelength of 1500 nm, we obtain
$g\simeq 4.1\times 10^{8}$ s-1. For the best possible performance the
absorption losses have to be smaller than this value, while the coupling has
to be tuned to the critical value (21). The absorption losses for the devices
reported in [9] were not quite there yet, with about 1.5% nonlinearity to loss
ratio; however, there is certainly hope that a ratio close to one will be
achieved soon.

Fig. 3 shows the signal flux when the linear coupling strengths of both the
signal and the drive fields are allowed to vary. The signal flux reaches its
maximum value at the optimal values of the coupling parameters, given by

$\kappa_{\max}^{2}=\frac{\mu_{d}}{2\gamma_{1}}g^{2}+\frac{\mu_{d}^{2}}{4};\;\kappa_{1\max}^{2}=\frac{\gamma_{1}}{2\mu_{d}}g^{2}+\frac{\mu_{d}^{2}}{4}.$
(22)

This is again the nonlinear critical coupling condition. The maximum value of
the normalized flux is given by

$\frac{\Pi_{1\max}}{\Pi_{wg}}=\frac{2}{\left[\sqrt{2+\mu_{d}\gamma_{1}/g^{2}}+\sqrt{\mu_{d}\gamma_{1}/g^{2}}\right]^{2}}.$
(23)

In the limit of negligible losses, the signal flux depends only on the product
of linear coupling parameters,

$\frac{\Pi_{1}}{\Pi_{wg}}=\frac{8\kappa\kappa_{1}g^{2}}{\left(2\kappa\kappa_{1}+g^{2}\right)^{2}},$
(24)

and its maximum value is of course equal to 1; see the inset to Fig. 3.

In conclusion, we developed a general nonperturbative theory of second-order
nonlinear interactions between quantized field modes in dissipative photonic
circuits and proposed a specific realization of maximally efficient SPDC at
the single-photon level. Convenient analytic expressions for the biphoton
fluxes are obtained which provide explicit dependence on all material and
geometric parameters and predict the existence of the nonlinear critical
coupling regime which establishes a fundamental upper limit on the signal and
idler fluxes for a single-photon pump.

This work has been supported in part by the Air Force Office for Scientific
Research Grant No. FA9550-21-1-0272 and National Science Foundation Award No.
1936276. M.T. acknowledges the support by the Center for Integration in
Science of the Ministry of Aliya and Integration, Israel.

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