# Few-photon transport via a multimode nonlinear cavity: theory and
applications

Yunkai Wang  Holonyak Micro and Nanotechnology Laboratory, University of
Illinois at Urbana-Champaign, Urbana, IL 61801 USA  Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA  Illinois
Quantum Information Science and Technology Center, University of Illinois at
Urbana-Champaign, Urbana, IL 61801 USA  Kejie Fang<EMAIL_ADDRESS>Holonyak Micro and Nanotechnology Laboratory, University of Illinois at
Urbana-Champaign, Urbana, IL 61801 USA  Illinois Quantum Information Science
and Technology Center, University of Illinois at Urbana-Champaign, Urbana, IL
61801 USA  Department of Electrical and Computer Engineering, University of
Illinois at Urbana-Champaign, Urbana, IL 61801 USA

###### Abstract

Few-photon transport via waveguide-coupled local quantum systems has attracted
extensive theoretical and experimental studies. Most of the study has focused
on atomic or atomic-like local quantum systems due to their strong light-
matter interaction useful for quantum applications. Here, we study few-photon
transport via a waveguide-coupled multimode optical cavity with second-order
bulk nonlinearity. We develop a Feynman diagram approach and compute the
scattering matrix of the one- and two-photon transport. Based on the
calculated scattering matrix, we show highly nonclassical photonic effects,
including photon blockade and $\pi-$conditional phase shift, are achievable in
the waveguide-coupled multimode optical cavity system via quantum interference
and linear response engineering. Our results might lead to significant
applications of quantum photonic circuits in all-optical quantum information
processing and quantum network protocols.

##  I Introduction

The architecture of waveguide-coupled local quantum systems enables coupling
flying photons with stationary qubits for light-matter interactions. A prime
example is the waveguide quantum electrodynamics, with several types of local
quantum systems demonstrated successfully, including trapped atoms Tiecke  _et
al._ (2014); Shomroni  _et al._ (2014); Volz  _et al._ (2014), solid state
defects Javadi  _et al._ (2015); Sipahigil  _et al._ (2016), and
superconducting qubits Mirhosseini  _et al._ (2018); Kannan  _et al._ (2020).
This architecture is also relevant to quantum networks where optic fibers link
local quantum nodes for routing, storage, and processing of quantum
information encoded in individual photons Simon (2017). In parallel, a large
body of theoretical work has been carried out to study few-photon transport in
the setting of waveguide-coupled local quantum systems Chang  _et al._ (2007);
Shen and Fan (2007); Liao  _et al._ (2010); Zheng  _et al._ (2010); Rephaeli
and Fan (2012); Xu and Fan (2015). Most of the theoretical and experimental
work, though, has focused on qubit-like quantum entities, which are sought to
provide strong light-matter interactions for quantum information tasks.

Here, we study few-photon transport via a waveguide-coupled multimode optical
cavity with second-order bulk nonlinearity. This is a common device setup used
for cavity-enhanced parametric nonlinear optical processes including second-
harmonic generation and parametric down-conversion Lake  _et al._ (2016); Guo
_et al._ (2017); Luo  _et al._ (2018); Chang  _et al._ (2019); Zhang  _et al._
(2019); Lukin  _et al._ (2020); Lu  _et al._ (2021); Zhao and Fang (2021).
However, its property at the few-photon level has not been fully explored,
because bulk nonlinearity is generally believed to be insubstantial to impact
single photons. We challenge this perspective by investigating the non-
parametric few-photon transport and its physical observations. Besides using a
systematic method from Ref. Xu and Fan (2015) to compute the scattering matrix
($S-$matrix) of few-photon transport, we develop a Feynman diagram approach
which provides further physical insight into the obtained $S-$matrix while
offering mathematical simplicity in its computation. This method is
particularly suited for weak nonlinear systems where only leading order
Feynman diagrams are of interest. Exploiting the result of the $S-$matrix, we
then study observational effects of few-photon transport via the waveguide-
coupled multimode cavity, including photon blockade, conditional phase shift,
and non-classical two-mode correlations. Surprisingly, while these effects are
commonly believed to be associated with strong light-matter interaction, we
find they could become substantial for waveguide-coupled optical cavities with
only weak nonlinearity.

These results reveal a general approach for creation and control of few-photon
correlations via quantum interference and linear response engineering. The
few-photon transport amplitude of waveguide-coupled local quantum systems is a
superposition of the interaction-free, i.e., linear, and interaction-mediated
amplitudes. When the linear transmission coefficient is comparable to the
interaction-mediated amplitude, quantum interference between the two pathways
leads to highly nonclassical correlations between the propagating photons. For
weak nonlinear systems, this might need a tuned linear transmission which
could be achieved, for example, in a one-port waveguide-cavity configuration.
However, this approach is distinct from the post-selection method where
photon-photon correlations are induced by the measurement. Our result might
lead to significant applications in quantum information science using quantum
photonic circuits beyond the parametric regime, including quantum
nondemolition measurement of photons Xia  _et al._ (2016), two-photon quantum
logic gates Langford  _et al._ (2011), and nonlinearity-assisted entanglement
swapping Sangouard  _et al._ (2011).

##  II Scattering matrix of few-photon transport

The system under consideration is a waveguide-coupled trimodal optical cavity
made from $\chi^{(2)}$ materials. A typical realization of such an optical
cavity is a microring resonator (Fig. 1a) which supports traveling-wave
resonances. These resonances couple via the $\chi^{(2)}$ nonlinearity when
frequency-matching and, if necessary, phase-matching conditions are satisfied.
Photons couple in and out of these resonances via the waveguide and meanwhile
dissipate in other loss channels. In the Heisenberg-Langevin framework, this
open quantum system can be modeled by the following effective Hamiltonian
($\hbar=1$)

$H_{\textrm{eff}}=(\omega_{1}-i\frac{\kappa_{1}}{2})a^{\dagger}_{1}a_{1}+(\omega_{2}-i\frac{\kappa_{2}}{2})a^{\dagger}_{2}a_{2}+(\omega_{3}-i\frac{\kappa_{3}}{2})a^{\dagger}_{3}a_{3}+g(a_{1}a_{2}a^{\dagger}_{3}+a^{\dagger}_{1}a^{\dagger}_{2}a_{3}),$
(1)

where $\omega_{j}$ and $\kappa_{j}$ are the frequency and photon loss rate of
the $j$-th resonance, respectively, and $g$ is the trimodal coupling
coefficient. For simplicity, we define a complex frequency
$\alpha_{j}\equiv\omega_{j}-i\frac{\kappa_{j}}{2}$. For the trimodal
interaction to be resonantly enhanced, the three modes need to satisfy the
frequency-matching condition $\omega_{1}+\omega_{2}\approx\omega_{3}$. The
cavity photon loss includes both leakage into the waveguide and intrinsic
losses due to, for example, material absorption and surface scattering, i.e.,
$\kappa_{j}=\kappa_{je}+\kappa_{ji}$, where $\kappa_{je}$ is the cavity-
waveguide coupling rate and $\kappa_{ji}$ is the intrinsic photon loss rate.
Depending on the configuration of the waveguide-cavity coupling, the relation
between the states of the outgoing and incoming photons in the waveguide is
determined by the input-output formalism Walls and Milburn (2007). In this
paper, we specifically consider the case of a cavity side-coupled to a
waveguide unidirectionally—typical for ring resonators—which has the following
input-output relation of the operators,

$a_{\textrm{out},j}(t)=a_{\textrm{in},j}(t)-i\sqrt{\kappa_{je}}a_{j}(t).$ (2)

It turns out such one-port cavity-waveguide configuration is critical for
realizing strong quantum correlations between propagating waveguide photons
coupled via a weak nonlinear optical cavity. However, the computation method
developed here can be straightforwardly generalized to other waveguide-cavity
configurations. For the calculation below, we also assume a linear dispersion
of the waveguide, which is justified given the linewidth of the cavity, and
thus the operation bandwidth of the system, is much smaller than the cavity
frequency Fan  _et al._ (2010).

The goal of this section is to calculate the scattering matrix ($S$-matrix) of
the one- and two-photon transport via the waveguide-coupled trimodal
$\chi^{(2)}$ cavity (Fig. 1b-d). We first use a systematic method from Ref. Xu
and Fan (2015) and apply it to the present case of a multi-mode $\chi^{(2)}$
cavity. We also develop a Feynman diagram approach which provides further
physical insight of the photon transport while lessen the computation
complexity associated with the non-perturbative method of Ref. Xu and Fan
(2015), especially for weak nonlinear systems.

Figure 1: a. A $\chi^{(2)}$ microring cavity with three resonances side-
coupled to a waveguide. Red, green, and blue lines corresponds to $a_{1}$,
$a_{2}$, and $a_{3}$ modes/photons, respectively. b-d. Illustrations of few-
photon transport processes. $\ket{k}$ denotes the single-photon state with
momentum $k$.

###  II.1 One-to-one-photon transport

We first consider the transport of a single photon (Fig. 1b). The amplitude of
transport of an incoming photon at time $t^{\prime}$ to an outgoing photon at
time $t$ is given by the time-domain $S-$matrix

$S_{t;t^{\prime}}\equiv\bra{0}a_{\textrm{out},1}(t)a_{\textrm{in},1}^{\dagger}(t^{\prime})\ket{0},$
(3)

where the photon is assumed to be in resonant with the $a_{1}$ mode. As shown
in Ref. Xu and Fan (2015), the time-domain $S-$matrix is related to the
Green’s function of the local system, i.e., here the nonlinear optical cavity,
by applying Eq. 2 and the quantum causality condition

$\displaystyle\left[a(t),I\left(t^{\prime}\right)\right]=\left[a^{\dagger}(t),I\left(t^{\prime}\right)\right]=0,\quad\text{
for }\quad t\leqslant t^{\prime},$ (4)
$\displaystyle\left[a(t),O\left(t^{\prime}\right)\right]=\left[a^{\dagger}(t),O\left(t^{\prime}\right)\right]=0,\quad\text{
for }\quad t\geqslant t^{\prime},$ (5)

where $I(O)\left(t^{\prime}\right)$ is a shorthand notation for the
input(output) operators that represent either $a_{\text{in(out)
}}\left(t^{\prime}\right)$ or
$a_{\text{in(out)}}^{\dagger}\left(t^{\prime}\right)$. Thus, the single-photon
$S$-matrix becomes

$S_{t;t^{\prime}}=\delta(t-t^{\prime})-\kappa_{1e}G(t;t^{\prime}),$ (6)

where
$G(t;t^{\prime})=\bra{0}\hat{T}[a_{1}(t)a_{1}^{\dagger}(t^{\prime})]\ket{0}$
is the two-point Green’s function, $\hat{T}$ is the time ordering operator and
we have also used
$[a_{\textrm{in(out)}}(t),a_{\textrm{in(out)}}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$
in the derivation.

The Green’s function of the local quantum system is calculated using operators
time-evolved according to the effective Hamiltonian of 1 Xu and Fan (2015).
Thus, we proceed

$\displaystyle G(t;t^{\prime})$ (7) $\displaystyle=$
$\displaystyle\bra{0}e^{iH_{\textrm{eff}}t}a_{1}e^{-iH_{\textrm{eff}}t}e^{iH_{\textrm{eff}}t^{\prime}}a^{\dagger}_{1}e^{-iH_{\textrm{eff}}t^{\prime}}\ket{0}\theta(t-t^{\prime})$
$\displaystyle=$ $\displaystyle
e^{i\alpha_{1}(t^{\prime}-t)}\theta(t-t^{\prime}),$

where $\theta(t-t^{\prime})$ is the Heaviside step function.

The frequency-domain $S$-matrix quantifying the transport amplitude between
output and input photons with fixed frequency or momentum is given by the
Fourier transform of the time-domain $S$-matrix. For the single-photon
transport,

$\displaystyle S_{p_{1};k_{1}}$
$\displaystyle\equiv\mathscr{F}\left\\{S_{t;t^{\prime}}\right\\}$ (8)
$\displaystyle=\int\frac{dt}{\sqrt{2\pi}}e^{ip_{1}t}\int\frac{dt^{\prime}}{\sqrt{2\pi}}e^{ik_{1}t^{\prime}}S_{t;t^{\prime}}$
$\displaystyle=\delta(p_{1}-k_{1})-\kappa_{1e}G(p_{1};k_{1}),$

where $G(p_{1};k_{1})$ is the Fourier transform of the Green’s function of 7.
This definition of Fourier transform leads to
$\bra{k}\ket{k^{\prime}}=\delta(k-k^{\prime})$ given
$\bra{t}\ket{t^{\prime}}=\delta(t-t^{\prime})$, where
$\ket{k}\equiv\mathscr{F}\left\\{\ket{t}\right\\}$. The momentum-space Green’s
function thus is

$G(p_{1};k_{1})=\frac{i}{k_{1}-\alpha_{1}}\delta(p_{1}-k_{1}).$ (9)

As a result, we have

$S_{p_{1};k_{1}}=\left[1-\frac{i\kappa_{1e}}{k_{1}-\alpha_{1}}\right]\delta(p_{1}-k_{1})\equiv
t_{k_{1}}\delta(p_{1}-k_{1}),$ (10)

where $t_{k_{1}}$ is the single-photon transmission coefficient. The
derivation above straightforwardly applies to transport of a single photon in
resonant with other resonances. For the rest of the paper, we use the
subscript $j$ in the momentum to indicate photons in resonant with the $j$-th
resonance.

###  II.2 Two-to-one-photon transport and vice versa

Next, we consider the transport of two $a_{1}$ and $a_{2}$ photons to one
$a_{3}$ photon enabled by the $\chi^{(2)}$ cavity (Fig. 1c). This corresponds
to the sum-of-frequency process in nonlinear optics. Following a similar
procedure, the time-domain $S$-matrix of this process is found to be

$\displaystyle S_{t_{3};t_{1}^{\prime}t_{2}^{\prime}}$
$\displaystyle\equiv\bra{0}a_{\textrm{out},3}(t_{3})a_{\textrm{in},1}^{\dagger}(t_{1}^{\prime})a_{\textrm{in},2}^{\dagger}(t_{2}^{\prime})\ket{0}$
(11)
$\displaystyle=i\sqrt{\kappa_{1e}\kappa_{2e}\kappa_{3e}}\bra{0}\hat{T}[a_{3}(t_{3})a_{1}^{\dagger}(t_{1}^{\prime})a_{2}^{\dagger}(t_{2}^{\prime})]\ket{0}$
$\displaystyle=i\sqrt{\kappa_{1e}\kappa_{2e}\kappa_{3e}}\left(\bra{0}a_{3}(t_{3})a^{\dagger}_{1}(t_{1}^{\prime})a^{\dagger}_{2}(t_{2}^{\prime})\ket{0}\theta(t_{3}-t_{1}^{\prime})\theta(t_{1}^{\prime}-t_{2}^{\prime})+\bra{0}a_{3}(t_{3})a^{\dagger}_{2}(t_{2}^{\prime})a^{\dagger}_{1}(t_{1}^{\prime})\ket{0}\theta(t_{3}-t_{2}^{\prime})\theta(t_{2}^{\prime}-t_{1}^{\prime})\right).$

We introduce $\ket{mnr}$ to denote the state with $m$, $n$, and $r$ photons in
mode $a_{1}$, $a_{2}$, and $a_{3}$, respectively. One key property of the
$\chi^{2}$ trimodal system is that $\\{\ket{100}\\}$, $\\{\ket{010}\\}$, and
$\\{\ket{110},\ket{001}\\}$ form three closed subspaces of the effective
Hamiltonian of 1. Based on this property, we proceed to compute Eq. 11:

$\displaystyle\bra{0}a_{3}(t_{3})a^{\dagger}_{1}(t_{1}^{\prime})a^{\dagger}_{2}(t_{2}^{\prime})\ket{0}$
(12) $\displaystyle=$
$\displaystyle\bra{000}a_{3}(t_{3})\left(\ket{001}\bra{001}+\ket{110}\bra{110}\right)a_{1}^{\dagger}(t^{\prime}_{1})\ket{010}\bra{010}a_{2}^{\dagger}(t^{\prime}_{2})\ket{000}$
$\displaystyle=$
$\displaystyle\left(\bra{001}e^{-iH_{\textrm{eff}}t_{3}}\ket{001}\bra{001}e^{iH_{\textrm{eff}}t_{1}^{\prime}}\ket{110}+\bra{001}e^{-iH_{\textrm{eff}}t_{3}}\ket{110}\bra{110}e^{iH_{\textrm{eff}}t_{1}^{\prime}}\ket{110}\right)e^{-i\alpha_{2}(t_{1}^{\prime}-t_{2}^{\prime})}$
$\displaystyle=$
$\displaystyle\frac{g}{\sqrt{\Delta\alpha^{2}+4g^{2}}}e^{-i\alpha_{2}(t_{1}^{\prime}-t_{2}^{\prime})}(e^{i\lambda_{2}(t_{1}^{\prime}-t_{3})}-e^{i\lambda_{1}(t_{1}^{\prime}-t_{3})}),$

where $\Delta\alpha=\alpha_{1}+\alpha_{2}-\alpha_{3}$ and
$\lambda_{1,2}=\frac{1}{2}(\alpha_{1}+\alpha_{2}+\alpha_{3})\pm\frac{1}{2}\sqrt{\Delta\alpha^{2}+4g^{2}}$
are the eigenvalues of $H_{\textrm{eff}}$ in the subspace spanned by
$\ket{110}$ and $\ket{001}$, i.e.,

$\left[\begin{array}[]{cc}\alpha_{1}+\alpha_{2}&g\\\
g&\alpha_{3}\end{array}\right].$ (13)

The result of
$\bra{0}a_{3}(t_{3})a^{\dagger}_{2}(t_{2}^{\prime})a^{\dagger}_{1}(t_{1}^{\prime})\ket{0}$
is obtained by exchanging the subscripts in Eq. 12. Finally, the momentum-
space $S-$matrix,
$S_{p_{3};k_{1}k_{2}}\equiv\mathscr{F}\left\\{S_{t_{3};t_{1}^{\prime}t_{2}^{\prime}}\right\\}$,
is given by

$S_{p_{3};k_{1}k_{2}}\equiv
i\sqrt{\kappa_{1e}\kappa_{2e}\kappa_{3e}}G(p_{3};k_{1},k_{2})=-ig\frac{\sqrt{\kappa_{1e}\kappa_{2e}\kappa_{3e}}}{\sqrt{2\pi}}\frac{(k_{1}+k_{2}-\alpha_{1}-\alpha_{2})}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})(k_{1}+k_{2}-\lambda_{1})(k_{1}+k_{2}-\lambda_{2})}\delta(p_{3}-k_{1}-k_{2}),$
(14)

where $G(p_{3};k_{1},k_{2})$ is the momentum-space Green’s function.

For the degenerate case, i.e., $a_{1}$ and $a_{2}$ modes are the same mode,
the matrix form of $H_{\textrm{eff}}$ in the subspace spanned by $\ket{110}$
(i.e., $\ket{20}$) and $\ket{001}$ is given by

$\left[\begin{array}[]{cc}2\alpha_{1}&\sqrt{2}g\\\
\sqrt{2}g&\alpha_{3}\end{array}\right].$ (15)

After a similar derivation, we obtain

$S_{p_{3};k_{1}k_{2}}=-2ig\frac{\sqrt{\kappa_{1e}^{2}\kappa_{3e}}}{\sqrt{2\pi}}\frac{(k_{1}+k_{2}-2\alpha_{1})}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{1})(k_{1}+k_{2}-\lambda^{\prime}_{1})(k_{1}+k_{2}-\lambda^{\prime}_{2})}\delta(p_{3}-k_{1}-k_{2}),$
(16)

where
$\lambda^{\prime}_{1,2}=\frac{1}{2}(2\alpha_{1}+\alpha_{3})\pm\frac{1}{2}\sqrt{(2\alpha_{1}-\alpha_{3})^{2}+8g^{2}}$
are the eigenvalues of the matrix of 15. It is worth pointing out that
$\lambda_{1,2}$ ($\lambda^{\prime}_{1,2}$) are functions of the trimodal
coupling coefficient $g$, and thus Eqs. 14 and 16 are non-perturbation result
in terms of $g$, contributed by all virtual processes. These processes
involving creation and annihilation of virtual photons will be revealed using
the Feynman diagram approach in Section III.

The reverse process of one-to-two-photon transport has an $S-$matrix

$S_{t_{1}^{\prime}t_{2}^{\prime};t_{3}}\equiv\bra{0}a_{\textrm{out},1}(t_{1}^{\prime})a_{\textrm{out},2}(t_{2}^{\prime})a^{\dagger}_{\textrm{in},3}(t_{3})\ket{0},$
(17)

whose Fourier transform satisfies

$S_{k_{1}k_{2};p_{3}}\equiv\mathscr{F}\left\\{S_{t_{1}^{\prime}t_{2}^{\prime};t_{3}}\right\\}=S_{p_{3};k_{1}k_{2}}.$
(18)

###  II.3 Two-to-two-photon transport

Last, we consider the process of transport of two $a_{1}$ and $a_{2}$ photons
to two $a_{1}$ and $a_{2}$ photons, without conversion to $a_{3}$ photons
(Fig. 1d). We again start from the time-domain $S$-matrix

$\displaystyle S_{t_{1}t_{2};t^{\prime}_{1}t^{\prime}_{2}}$
$\displaystyle\equiv\bra{0}a_{\textrm{out},1}(t_{1})a_{\textrm{out},2}(t_{2})a_{\textrm{in},1}^{\dagger}(t_{1}^{\prime})a_{\textrm{in},2}^{\dagger}(t_{2}^{\prime})\ket{0}$
(19)
$\displaystyle=\delta(t_{1}-t_{1}^{\prime})\delta(t_{2}-t_{2}^{\prime})-\kappa_{1e}\bra{0}\hat{T}[a_{1}(t_{1})a_{1}^{\dagger}(t_{1}^{\prime})]\ket{0}\delta(t_{2}-t_{2}^{\prime})-\kappa_{2e}\bra{0}\hat{T}[a_{2}(t_{2})a_{2}^{\dagger}(t_{2}^{\prime})]\ket{0}\delta(t_{1}-t_{1}^{\prime})$
$\displaystyle\quad+\kappa_{1e}\kappa_{2e}\bra{0}\hat{T}[a_{1}(t_{1})a_{1}(t_{2})a_{1}^{\dagger}(t_{1}^{\prime})a_{2}^{\dagger}(t_{2}^{\prime})]\ket{0}.$

The second and third terms are the single-photon Green’s function which have
been calculated (see Eq. 7). The last term is new and involves six different
time orderings,

$\displaystyle\bra{0}\hat{T}\left[a_{2}(t_{2})a_{1}(t_{1})a_{1}^{\dagger}(t_{1}^{\prime})a_{2}^{\dagger}(t_{2}^{\prime})\right]\ket{0}=$
$\displaystyle\bra{0}a_{2}(t_{2})a_{1}(t_{1})a_{1}^{\dagger}(t^{\prime}_{1})a_{2}^{\dagger}(t^{\prime}_{2})\ket{0}\theta(t_{2}-t_{1})\theta(t_{1}-t_{1}^{\prime})\theta(t_{1}^{\prime}-t_{2})$
(20)
$\displaystyle+\bra{0}a_{2}(t_{2})a_{1}(t_{1})a_{2}^{\dagger}(t_{2}^{\prime})a_{1}^{\dagger}(t_{1}^{\prime})\ket{0}\theta(t_{2}-t_{1})\theta(t_{1}-t_{2}^{\prime})\theta(t_{2}^{\prime}-t_{1}^{\prime})$
$\displaystyle+\bra{0}a_{1}(t_{1})a_{1}^{\dagger}(t_{1}^{\prime})a_{2}(t_{2})a_{2}^{\dagger}(t_{2}^{\prime})\ket{0}\theta(t_{1}-t_{1}^{\prime})\theta(t_{1}^{\prime}-t_{2})\theta(t_{2}-t_{2}^{\prime})$
$\displaystyle+\bra{0}a_{1}(t_{1})a_{2}(t_{2})a_{2}^{\dagger}(t_{2}^{\prime})a_{1}^{\dagger}(t_{1}^{\prime})\ket{0}\theta(t_{1}-t_{2})\theta(t_{2}-t_{2}^{\prime})\theta(t_{2}^{\prime}-t_{1})$
$\displaystyle+\bra{0}a_{1}(t_{1})a_{2}(t_{2})a_{1}^{\dagger}(t^{\prime}_{1})a_{2}^{\dagger}(t^{\prime}_{2})\ket{0}\theta(t_{1}-t_{2})\theta(t_{2}-t_{1}^{\prime})\theta(t_{1}^{\prime}-t_{2}^{\prime})$
$\displaystyle+\bra{0}a_{2}(t_{2})a_{2}^{\dagger}(t_{2}^{\prime})a_{1}(t_{1})a_{1}^{\dagger}(t_{1}^{\prime})\ket{0}\theta(t_{2}-t_{2}^{\prime})\theta(t_{2}^{\prime}-t_{1})\theta(t_{1}-t_{1}^{\prime}).$

We first calculate
$\bra{0}a_{2}(t_{2})a_{1}(t_{1})a_{1}^{\dagger}(t^{\prime}_{1})a_{2}^{\dagger}(t^{\prime}_{2})\ket{0}$:

$\displaystyle\bra{0}a_{2}(t_{2})a_{1}(t_{1})a_{1}^{\dagger}(t^{\prime}_{1})a_{2}^{\dagger}(t^{\prime}_{2})\ket{0}$
(21) $\displaystyle=$
$\displaystyle\bra{000}a_{2}(t_{2})\ket{010}\bra{010}a_{1}(t_{1})\left(\ket{110}\bra{110}+\ket{001}\bra{001}\right)a_{1}^{\dagger}(t^{\prime}_{1})\ket{010}\bra{010}a_{2}^{\dagger}(t^{\prime}_{2})\ket{000}$
$\displaystyle=$ $\displaystyle
e^{i\alpha_{2}(-t_{2}+t^{\prime}_{2}+t_{1}-t^{\prime}_{1})}\big{(}\bra{110}e^{-iH_{\textrm{eff}}t_{1}}\ket{110}\bra{110}e^{iH_{\textrm{eff}}t^{\prime}_{1}}\ket{110}+\bra{110}e^{-iH_{\textrm{eff}}t_{1}}\ket{001}\bra{001}e^{iH_{\textrm{eff}}t^{\prime}_{1}}\ket{110}\big{)}$
$\displaystyle=$ $\displaystyle
e^{i\alpha_{2}(-t_{2}+t^{\prime}_{2}+t_{1}-t^{\prime}_{1})}\bigg{[}\Big{(}\frac{1}{2}-\frac{1}{2}\frac{\Delta\alpha}{\sqrt{\Delta\alpha^{2}+4g^{2}}}\Big{)}e^{-i\lambda_{1}(t_{1}-t^{\prime}_{1})}+\Big{(}\frac{1}{2}+\frac{1}{2}\frac{\Delta\alpha}{\sqrt{\Delta\alpha^{2}+4g^{2}}}\Big{)}e^{-i\lambda_{2}(t_{1}-t^{\prime}_{1})}\bigg{]}.$

Similarly, the second and third terms in Eq. 20 are found to be

$\bra{0}a_{2}(t_{2})a_{1}(t_{1})a_{2}^{\dagger}(t_{2}^{\prime})a_{1}^{\dagger}(t_{1}^{\prime})\ket{0}=e^{i\alpha_{2}(t_{1}-t_{2})}e^{i\alpha_{1}(t^{\prime}_{1}-t^{\prime}_{2})}\bigg{[}\Big{(}\frac{1}{2}-\frac{1}{2}\frac{\Delta\alpha}{\sqrt{\Delta\alpha^{2}+4g^{2}}}\Big{)}e^{-i\lambda_{1}(t_{1}-t^{\prime}_{2})}+\Big{(}\frac{1}{2}+\frac{1}{2}\frac{\Delta\alpha}{\sqrt{\Delta\alpha^{2}+4g^{2}}}\Big{)}e^{-i\lambda_{2}(t_{1}-t^{\prime}_{2})}\bigg{]},$
(22)
$\bra{0}a_{1}(t_{1})a_{1}^{\dagger}(t_{1}^{\prime})a_{2}(t_{2})a_{2}^{\dagger}(t_{2}^{\prime})\ket{0}=e^{i\alpha_{1}(t^{\prime}_{1}-t_{1})}e^{i\alpha_{2}(t^{\prime}_{2}-t_{2})}.$
(23)

The rest three terms in Eq. 20 are obtained by simply exchanging the indices
$1$ and $2$ in Eqs. 21-23.

Finally, by Fourier transform of Eq. 20, we obtain the momentum-space Green’s
function

$G(p_{1},p_{2};k_{1},k_{2})=-\frac{1}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})}\delta(p_{1}-k_{1})\delta(p_{2}-k_{2})+M(p_{1},p_{2},k_{1},k_{2})\delta(p_{1}+p_{2}-k_{1}-k_{2}),$
(24)

where

$M(p_{1},p_{2},k_{1},k_{2})=-\frac{ig^{2}}{2\pi}\frac{k_{1}+k_{2}-\alpha_{1}-\alpha_{2}}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})(p_{1}-\alpha_{1})(p_{2}-\alpha_{2})(k_{1}+k_{2}-\lambda_{1})(k_{1}+k_{2}-\lambda_{2})},$
(25)

and the momentum-space $S-$matrix

$S_{p_{1}p_{2};k_{1}k_{2}}=t_{k_{1}}t_{k_{2}}\delta(p_{1}-k_{1})\delta(p_{2}-k_{2})+\kappa_{1e}\kappa_{2e}M(p_{1},p_{2},k_{1},k_{2})\delta(p_{1}+p_{2}-k_{1}-k_{2}).$
(26)

For the case of degenerate $a_{1}$ and $a_{2}$, after a similar derivation, we
have

$G(p_{1},p_{2};k_{1},k_{2})=-\frac{1}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{1})}[\delta(p_{1}-k_{1})\delta(p_{2}-k_{2})+\delta(p_{1}-k_{2})\delta(p_{2}-k_{1})]+M(p_{1},p_{2},k_{1},k_{2})\delta(p_{1}+p_{2}-k_{1}-k_{2})$
(27)

and

$S_{p_{1}p_{2};k_{1}k_{2}}=t_{k_{1}}t_{k_{2}}[\delta(p_{1}-k_{1})\delta(p_{2}-k_{2})+\delta(p_{1}-k_{2})\delta(p_{2}-k_{1})]+\kappa_{1e}^{2}M(p_{1},p_{2},k_{1},k_{2})\delta(p_{1}+p_{2}-k_{1}-k_{2}),$
(28)

where

$M(p_{1},p_{2},k_{1},k_{2})=-\frac{2ig^{2}}{\pi}\frac{k_{1}+k_{2}-2\alpha_{1}}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{1})(p_{1}-\alpha_{1})(p_{2}-\alpha_{1})(k_{1}+k_{2}-\lambda^{\prime}_{1})(k_{1}+k_{2}-\lambda^{\prime}_{2})}.$
(29)

Eqs. 26 and 28 show that the $S-$matrix of two-to-two-photon transport
consists of an interaction-free component, which is the product of two single-
photon transmission coefficients, and an interaction-mediated component. The
total transport amplitude is given by the superposition of the two components,
representing the quantum interference between the two pathways. This leads to
profound consequences of the photon-photon interaction via the waveguide-
coupled nonlinear optical cavities and applications in quantum information
science (see Section IV). The interaction-mediated component was also
interpreted as the two-photon bound state previously Shen and Fan (2007).

##  III Feynman diagram approach

Figure 2:  a. Feynman diagram rules for the propagator and vertex. Red, green,
and blue lines corresponds to $a_{1}$, $a_{2}$, and $a_{3}$ mode photons,
respectively. b and c. Feynman diagrams contributing to the Green’s function
of two-to-one-photon transport (b) and two-to-two-photon transport (c). The
part enclosed by the parenthesis is repeated by $n$ times.

In this section, we introduce a perturbation method based on Feynman diagrams
to calculate the momentum-space Green’s function and $S-$matrix. The
perturbation method, suited for weak nonlinear systems with $g/\kappa_{j}<1$,
reveals the relevant physical processes contributing to the few-photon
transport amplitude. It is also mathematically simplified, especially when
only the leading order Feynman diagrams are of interest, compared to the
previous method where careful cancellation of divergence is needed when
performing the Fourier transform of the time-domain Green’s function involving
the Heaviside step function. The Feynman diagram rules here resemble those
developed in quantum filed theory (see, e.g., Ref. Peskin (2018)). In essence,
the $n-$operator Green’s function is calculated using all connected Feynman
diagrams with $n$ external points, which are constructed from basic elements
including “propagator” and “vertex”. The propagator is the two-point free-
particle Green’s function and the vertex corresponds to the bare interaction
term of the Hamiltonian. For the Hamiltonian of 1, the Feynman diagram rules
are given as follows:

  * •

Propagator: $\frac{i}{k-\alpha}$,

  * •

Vertex: $-ig$,

  * •

Impose energy conservation at each vertex: $\frac{1}{\sqrt{2\pi}}\delta(\sum
k-\sum p)$,

  * •

Integrate undetermined momentum: $\int dk\int dp$.

  * •

Multiply the symmetry factor: $m!$ for $m$ propagators of the same mode
connecting two vertices or connecting a vertex with external points.

The Feynman diagrams corresponding to the propagator and vertex are shown in
Fig. 2a. A few differences from the Feynman diagram rules for field operators
in quantum filed theory are worth noting. Here, because the equation of motion
for the cavity mode operator is a linear differential equation, the
propagator, given by Eq. 9, is linearly dependent on the momentum. Further,
because the effective Hamiltonian of the open quantum system already includes
loss terms, i.e., $i\kappa_{j}$, the propagator does not diverge at the real
resonance frequency, avoiding the addition of an infinitesimal imaginary term
in the propagator for performing the momentum integrals.

The three- and four-point momentum-space Green’s functions related to Eqs. 11
and 20 can be calculated using the Feynman diagrams shown in Fig. 2b and c,
respectively. These Feynman diagrams also illustrates the physical processes,
involving creation and annihilation of virtual photons, that contribute to the
photon transport amplitude. After computation of each Feynman diagram
according to the rules above, the three-point Green’s function is found to be

$G(p_{3};k_{1},k_{2})=-\frac{g}{\sqrt{2\pi}}\frac{1}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})(p_{3}-\alpha_{3})}\sum_{n=0}^{\infty}\left(\frac{g^{2}}{(p_{3}-\alpha_{1}-\alpha_{2})(p_{3}-\alpha_{3})}\right)^{n}\delta(p_{3}-k_{1}-k_{2}),$
(30)

where the term of $n=0$ corresponds to the first diagram on the right hand
side in Fig. 2b and each term of $n>0$ corresponds to the diagram with $n$
repeated part in the parenthesis. The four-point Green’s function is found to
be

$\displaystyle G(p_{1},p_{2};k_{1},k_{2})$ (31) $\displaystyle=$
$\displaystyle-\frac{1}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})}\delta(p_{1}-k_{1})\delta(p_{2}-k_{2})$
$\displaystyle-\frac{ig^{2}}{2\pi}\frac{1}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})(p_{1}-\alpha_{1})(p_{2}-\alpha_{2})(k_{1}+k_{2}-\alpha_{3})}\sum_{n=0}^{\infty}\left(\frac{g^{2}}{(k_{1}+k_{2}-\alpha_{1}-\alpha_{2})(k_{1}+k_{2}-\alpha_{3})}\right)^{n}\delta(p_{1}+p_{2}-k_{1}-k_{2}),$

where the first term corresponds to the first diagram on the right hand side
in Fig. 2c, the term of $n=0$ corresponds to the second diagram, and each term
of $n>0$ corresponds to the diagram with $n$ repeated part in the parenthesis.
One can confirm that Eqs. 30 and LABEL:FG22 reproduce Eqs. 14 and 24,
respectively, after performing the summation. As an example of applying the
Feynman diagram rules, we explicitly show the calculation of the $n=1$ diagram
in Fig. 2b. Higher order diagrams are essentially multiple repetition of the
component in the parenthesis of this basic diagram. The formula corresponding
to this diagram, following the Feynman diagram rules from left to right, is
given by

$\displaystyle\frac{i}{k_{1}-\alpha_{1}}\frac{i}{k_{2}-\alpha_{2}}\frac{-ig}{\sqrt{2\pi}}\int
dq_{3}\delta(q_{3}-k_{1}-k_{2})\frac{i}{q_{3}-\alpha_{3}}\frac{-ig}{\sqrt{2\pi}}\int
dq_{1}dq_{2}\delta(q_{1}+q_{2}-q_{3})\frac{i}{q_{1}-\alpha_{1}}\frac{i}{q_{2}-\alpha_{2}}\frac{-ig}{\sqrt{2\pi}}\delta(q_{1}+q_{2}-p_{3})\frac{i}{p_{3}-\alpha_{3}}$
(32) $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2\pi}}\delta(p_{3}-k_{1}-k_{2})\frac{1}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})(p_{3}-\alpha_{3})}\frac{-g^{2}}{2\pi
i}\int
dq_{1}dq_{2}\delta(q_{1}+q_{2}-p_{3})\frac{1}{q_{1}-\alpha_{1}}\frac{1}{q_{2}-\alpha_{2}}\frac{1}{p_{3}-\alpha_{3}}$
$\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2\pi}}\delta(p_{3}-k_{1}-k_{2})\frac{1}{(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})(p_{3}-\alpha_{3})}\frac{g^{2}}{(p_{3}-\alpha_{1}-\alpha_{2})(p_{3}-\alpha_{3})},$

which yields the $n=1$ term in Eq. 30.

We note Eqs. 30 and LABEL:FG22 are power series expansion in terms of
$(g/\kappa)^{2}$, where $\kappa$ is the shorthanded notation for $\kappa_{j}$
or their sum, for input and output photons in resonant with the cavity modes,
i.e., $k_{j},p_{j}\approx\omega_{j}$, and a frequency-matched cavity
($\omega_{1}+\omega_{2}\approx\omega_{3}$). As a result, only leading order
Feynman diagrams are needed for weak nonlinear optical cavities, i.e.,
$(g/\kappa)^{2}<1$, which significantly simplifies the calculation. The
computation here is also divergence free, in contrast to the method used in
the previous Section which encounters divergences during Fourier transform of
the time-domain Green’s function and thus requires careful cancellations Xu
and Fan (2015)

##  IV Observational effects of few-photon transport

In this Section, we discuss several observational effects associated with the
few-photon transport via a waveguide-coupled $\chi^{(2)}$ cavity. Throughout
this Section, we consider input photons in a weak coherent state, which are
commonly used for single-photon level experiments.

###  IV.1 Single-photon down-conversion

We first consider the down-conversion process of $a_{3}$ photons to $a_{1}$
and $a_{2}$ photons. The input weak coherent state of monochromatic $a_{3}$
photons can be written as

$\ket{\psi_{\textrm{in}}}=\ket{0}+\alpha\ket{1_{t}}+\frac{\alpha^{2}}{\sqrt{2}}\ket{1_{t}1_{t}}+O(\alpha^{3}),$
(33)

where $\alpha$ is the amplitude of the coherent state and
$\ket{1_{t}}=\ket{1}e^{-ikt}$ represents a monochromatic single-photon state
with momentum $k$ in the time domain. Note $\ket{1_{t}}\neq\ket{t}$. We are
interested in the second-order correlation function of the output $a_{1}$ and
$a_{2}$ photons:

$\displaystyle g^{(2)}(\tau)$ (34) $\displaystyle=$
$\displaystyle\frac{\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},2}^{\dagger}(t+\tau)a_{\textrm{out},2}(t+\tau)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}}{\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}\bra{\psi_{\textrm{in}}}a_{\textrm{out},2}^{\dagger}(t)a_{\textrm{out},2}(t)\ket{\psi_{\textrm{in}}}},$

which can be calculated using the momentum-space $S$-matrix. To do so, we
first perform a Fourier transform of $\ket{\psi_{\textrm{in}}}$:

$\mathscr{F}\left\\{\ket{\psi_{\textrm{in}}}\right\\}=\int\frac{dt}{\sqrt{2\pi}}e^{ik^{\prime}t}\ket{1}e^{-ikt}=\sqrt{2\pi}\ket{1}\delta(k^{\prime}-k).$
(35)

Thus we define Fourier-transformed monochromatic single-photon state to be
$\sqrt{2\pi}\ket{1_{k}}$. Keeping the leading order term in
$\ket{\psi_{\textrm{in}}}$, we have

$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},2}^{\dagger}(t+\tau)a_{\textrm{out},2}(t+\tau)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
(36) $\displaystyle=$
$\displaystyle|\alpha|^{2}\bra{1_{t}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},2}^{\dagger}(t+\tau)a_{\textrm{out},2}(t+\tau)a_{\textrm{out},1}(t)\ket{1_{t}}$
$\displaystyle=$
$\displaystyle|\alpha|^{2}\bra{1_{t}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},2}^{\dagger}(t+\tau)\ket{0}\bra{0}a_{\textrm{out},2}(t+\tau)a_{\textrm{out},1}(t)\ket{1_{t}}$
$\displaystyle=$ $\displaystyle\frac{|\alpha|^{2}}{2\pi}\int
dp_{1}dp_{2}\bra{1_{k}}a_{\textrm{out},1}^{\dagger}(p_{1})a_{\textrm{out},2}^{\dagger}(p_{2})\ket{0}e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}\bra{0}a_{\textrm{out},2}(\tilde{p}_{2})a_{\textrm{out},1}(\tilde{p}_{1})\ket{1_{k}}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}$
$\displaystyle=$ $\displaystyle\frac{|\alpha|^{2}}{2\pi}\int
dp_{1}dp_{2}S_{p_{1}p_{2};k}^{*}e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}S_{\tilde{p}_{1}\tilde{p}_{2};k}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}$
$\displaystyle=$
$\displaystyle\begin{cases}\frac{|\alpha|^{2}g^{2}\kappa_{1e}\kappa_{2e}\kappa_{3e}}{|(k-\lambda_{1})(k-\lambda_{2})|^{2}}e^{-\kappa_{2}\tau},\tau>0,\\\
\frac{|\alpha|^{2}g^{2}\kappa_{1e}\kappa_{2e}\kappa_{3e}}{|(k-\lambda_{1})(k-\lambda_{2})|^{2}}e^{\kappa_{1}\tau},\tau<0,\end{cases}$

where $S_{p_{1}p_{2};k}$ is given by Eq. 18, and

$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
(37) $\displaystyle=$
$\displaystyle|\alpha|^{2}\bra{1_{t}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{1_{t}}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\int
dp_{1}d\tilde{p}_{1}\bra{1_{k}}a_{\textrm{out},1}^{\dagger}(p_{1})a_{\textrm{out},1}(\tilde{p}_{1})\ket{1_{k}}e^{ip_{1}t-i\tilde{p}_{1}t}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\int
dp_{1}d\tilde{p}_{1}\bra{1_{k}}a_{\textrm{out},1}^{\dagger}(p_{1})\left(\int
dp_{2}\ket{p_{2}}\bra{p_{2}}+\int
dp_{2i}\ket{p_{2i}}\bra{p_{2i}}\right)a_{\textrm{out},1}(\tilde{p}_{1})\ket{1_{k}}e^{ip_{1}t-i\tilde{p}_{1}t}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\left(\int
dp_{1}d\tilde{p}_{1}dp_{2}S_{p_{1}p_{2};k}S_{\tilde{p}_{1}p_{2};k}^{*}+\frac{\kappa_{2i}}{\kappa_{2e}}\int
dp_{1}d\tilde{p}_{1}dp_{2i}S_{p_{1}p_{2i};k}S_{\tilde{p}_{1}p_{2i};k}^{*}\right)e^{ip_{1}t-i\tilde{p}_{1}t}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\frac{\kappa_{2}}{\kappa_{2e}}\int
dp_{1}d\tilde{p}_{1}dp_{2}S_{p_{1}p_{2};k}S_{\tilde{p}_{1}p_{2};k}^{*}e^{ip_{1}t-i\tilde{p}_{1}t}$
$\displaystyle=$
$\displaystyle|\alpha|^{2}\frac{\kappa_{1e}}{\kappa_{1}}\frac{g^{2}(\kappa_{1}+\kappa_{2})\kappa_{3e}}{|(k-\lambda_{1})(k-\lambda_{2})|^{2}},$

where $\ket{p_{2i}}$ represents the state of $a_{2}$ photons in the intrinsic
loss channel. Because the states in the waveguide and the intrinsic loss
channel combined form a complete Hilbert space of the leaked photons from the
cavity, we have the identity

$\int dp_{2}\ket{p_{2}}\bra{p_{2}}+\int dp_{2i}\ket{p_{2i}}\bra{p_{2i}}=I.$
(38)

In other words,
$\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
measures the total $a_{1}$ photon flux, regardless of the place of $a_{2}$
photons, which could either be in the waveguide or the intrinsic loss channel.
Thus, we have to insert Eq. 38 into Eq. LABEL:flux. We have also used the fact
that the $S-$matrix involving $\ket{p_{2i}}$ will be proportional to
$\sqrt{\kappa_{2i}}$, by replacing $\sqrt{\kappa_{2e}}$ in Eq. 14. Exchanging
subscripts 1 and 2 in Eq. LABEL:flux yields the result of
$\bra{\psi_{\textrm{in}}}a_{\textrm{out},2}^{\dagger}(t)a_{\textrm{out},2}(t)\ket{\psi_{\textrm{in}}}$.
The intra-cavity photon-pair generation rate is inferred from Eq. LABEL:flux
to be

$R=|\alpha|^{2}\frac{g^{2}(\kappa_{1}+\kappa_{2})\kappa_{3e}}{|(k-\lambda_{1})(k-\lambda_{2})|^{2}}.$
(39)

Finally, we have

$\displaystyle g^{(2)}(\tau)=$
$\displaystyle\frac{|(k-\lambda_{1})(k-\lambda_{2})|^{2}}{2|\alpha|^{2}g^{2}(\kappa_{1}+\kappa_{2})\kappa_{3e}}\frac{2\kappa_{1}\kappa_{2}}{\kappa_{1}+\kappa_{2}}\times\begin{cases}e^{-\kappa_{2}\tau},\tau>0,\\\
e^{\kappa_{1}\tau},\tau<0,\end{cases}$ (40)

and in the limit $g\ll\kappa_{1,2,3}$,

$\displaystyle g^{(2)}(\tau)\approx$
$\displaystyle\frac{\left((k-\omega_{1}-\omega_{2})^{2}+(\frac{\kappa_{1}+\kappa_{2}}{2})^{2}\right)\left((k-\omega_{3})^{2}+(\frac{\kappa_{3}}{2})^{2}\right)}{2|\alpha|^{2}g^{2}(\kappa_{1}+\kappa_{2})\kappa_{3e}}$
(41)
$\displaystyle\times\frac{2\kappa_{1}\kappa_{2}}{\kappa_{1}+\kappa_{2}}\times\begin{cases}e^{-\kappa_{2}\tau},\tau>0,\\\
e^{\kappa_{1}\tau},\tau<0.\end{cases}$

Interestingly, Eq. 41 is identical to the result of spontaneous parametric
down-conversion derived via a semi-classical approach by assuming non-depleted
classical pumps and using Gaussian moment factoring theorem Clausen  _et al._
(2014); Guo  _et al._ (2017).

###  IV.2 Photon blockade

Next, we consider transport of $a_{1}$ photons via a cavity with degenerate
$a_{1}$ and $a_{2}$ modes. For input of $a_{1}$ photons in this case, there
will be probability of up-conversion to output $a_{3}$ photons; however, we
focus on the process without such up-conversion, which could be separated
apart from the up-converted $a_{3}$ photons because $a_{1}$ and $a_{3}$
photons are disparate in frequency. Assuming monochromatic input $a_{1}$
photons in a weak coherent state given by Eq. 33, we compute the second-order
self-correlation function of the transported $a_{1}$ photons:

$\displaystyle g^{(2)}(\tau)$ (42) $\displaystyle=$
$\displaystyle\frac{\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}^{\dagger}(t+\tau)a_{\textrm{out},1}(t+\tau)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}}{\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}}.$

By keeping the leading order terms in $\ket{\psi_{\textrm{in}}}$, we have

$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
(43) $\displaystyle=$
$\displaystyle|\alpha|^{2}\bra{1_{t}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{1_{t}}$
$\displaystyle=$
$\displaystyle|\alpha|^{2}\bra{1_{t}}a_{\textrm{out},1}^{\dagger}(t)\ket{0}\bra{0}a_{\textrm{out},1}(t)\ket{1_{t}}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\int
dpe^{ipt}\bra{1_{k}}a_{\textrm{out},1}^{\dagger}(p)\ket{0}\int
d\tilde{p}e^{-i\tilde{p}t}\bra{0}a_{\textrm{out},1}(\tilde{p})\ket{1_{k}}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\int dpe^{ipt}S_{p;k}^{*}\int
d\tilde{p}e^{-i\tilde{p}t}S_{\tilde{p};k}$ $\displaystyle=$
$\displaystyle|\alpha|^{2}|t_{k}|^{2}$

and

$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}^{\dagger}(t+\tau)a_{\textrm{out},1}(t+\tau)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
(44) $\displaystyle=$
$\displaystyle\frac{|\alpha|^{4}}{2}\bra{1_{t}1_{t}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}^{\dagger}(t+\tau)a_{\textrm{out},1}(t+\tau)a_{\textrm{out},1}(t)\ket{1_{t}1_{t}}$
$\displaystyle=$ $\displaystyle\frac{|\alpha|^{4}}{2}\int
dp_{1}dp_{2}\bra{1_{k}1_{k}}a_{\textrm{out},1}^{\dagger}(p_{1})a_{\textrm{out},1}^{\dagger}(p_{2})\ket{0}e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}\bra{0}a_{\textrm{out},1}(\tilde{p}_{2})a_{\textrm{out},1}(\tilde{p}_{1})\ket{1_{k}1_{k}}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}$
$\displaystyle=$ $\displaystyle\frac{|\alpha|^{4}}{4}\int
dp_{1}dp_{2}S_{p_{1}p_{2};kk}^{*}e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}S_{\tilde{p}_{1}\tilde{p}_{2};kk}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}$
$\displaystyle=$ $\displaystyle|\alpha|^{4}|t_{k}^{2}+T(k,\tau)|^{2},$

where the $S-$matrix is given by Eq. 28 and

$\displaystyle
T(k,\tau)=-\frac{2g^{2}\kappa_{1e}^{2}}{(2k-\lambda^{\prime}_{1})(2k-\lambda^{\prime}_{2})(k-\alpha_{1})^{2}}e^{-i|\tau|(\alpha_{1}-k)}.$
(45)

Note the additional factor of 1/2 in the third equality of Eq. LABEL:g2num
comes from the definition of Fock state
$\ket{1_{k}1_{k}}=\ket{2_{k}}=\frac{1}{\sqrt{2}}a_{\textrm{in}}^{\dagger
2}(k)\ket{0}$. Finally, we have

$\displaystyle
g^{(2)}(\tau)=\frac{|t_{k}^{2}+T(k,\tau)|^{2}}{|t_{k}^{2}|^{2}}.$ (46)

Figure 3: $g^{(2)}(\tau)$ for various $\kappa_{1e}/\kappa_{1i}$, given
$k=\omega_{1}$, $g/\kappa_{1i}=0.02$, $\kappa_{3i}=2\kappa_{1i}$,
$\kappa_{3e}=0.1\kappa_{1i}$.

This result is remarkable in that even when $|T(k,\tau)|$ is small, which is
the case for practical $\chi^{(2)}$ nonlinear optical cavities with
$g/\kappa\ll 1$, highly nonclassical correlations of transported photons,
i.e., $g^{(2)}(\tau)\neq 1$, could be achieved by making $|t_{k}^{2}|$ close
to $|T(k,\tau)|$. This is due to the quantum interference between the
interaction-free and interaction-mediated amplitudes of the two-photon
transport, as indicated by the $S-$matrix (Eq. 26), especially when the two
amplitudes are compatible. To illustrate this more explicitly, for the one-
port, phase-matched cavity ($2\omega_{1}=\omega_{3}$) and on-resonance input
photons ($k=\omega_{1}$),

$t_{\omega_{1}}^{2}=(\frac{\kappa_{1i}-\kappa_{1e}}{\kappa_{1i}+\kappa_{1e}})^{2}$
(47)

and

$T(\omega_{1},0)=-\frac{8g^{2}\kappa_{1e}^{2}}{\kappa_{1}^{2}(\kappa_{1}\kappa_{3}/2+2g^{2})}.$
(48)

Thus, by making $\kappa_{1i}$ and $\kappa_{1e}$ such that
$t_{\omega_{1}}^{2}\approx-T(\omega_{1},0)$, one obtains $g^{(2)}(0)\approx
0$, which indicates the photon blockade. Remarkably, this can be achieved even
for $g/\kappa_{1,3}\ll 1$, if $\kappa_{1i}\approx\kappa_{1e}$, i.e., close to
the critical coupling between the cavity and waveguide. We note the minus sign
of $T(\omega_{1},0)$ is important in cancellation with $t_{\omega_{1}}^{2}$;
intuitively, this is because during the interaction-mediated process the state
$\ket{20}$ undergoes a Rabi flip with the virtual state $\ket{01}$ and back
again, which yields a Berry phase of $\pi$.

Moreover, the statistical properties of the output $a_{1}$ photons can be
altered, e.g., from anti-bunching to bunching ($g^{(2)}(0)>g^{(2)}(\tau)$), by
controlling $t_{k}^{2}$ via $\kappa_{1e}$ or $\kappa_{1i}$. This is shown in
Fig. 3. For these plots, we have used experimentally achievable device
parameters of a state-of-the-art $\chi^{(2)}$ nonlinear photonic platform with
$g/\kappa_{1i}$ in the range of a few percent Zhao and Fang (2021). The
correlation functions of finite wavepackets can also be calculated using the
$S$-matrix, taking into account of the spectral distribution of the
wavepacket. The result is included in Appendix B, which shows the averaging
effect on the nonclassical correlation due to the finite spectral width of the
wavepacket. However, this is not surprising because the cavity-based mechanism
is expected to work only for wavepackets with spectral width much smaller than
the cavity linewidth.

###  IV.3 Nonlinear phase shift

Figure 4: Nonlinear phase shift for various $\kappa_{1e}/\kappa_{1i}$, given
$g/\kappa_{1i}=0.02$, $\kappa_{3i}=2\kappa_{1i}$,
$\kappa_{3e}=0.1\kappa_{1i}$.

The photon-photon interaction in the $\chi^{(2)}$ cavity also induces a
nonlinear phase shift of the transported photons. To compute the nonlinear
phase shift, we point out the numerator of the second-order correlation
function (Eq. LABEL:g2smdef) actually gives the norm of the time-domain
wavefunction of the transported two photons (see Appendix A). Thus, the
wavefunction of the transported two photons can be written as

$\bra{t,t+\tau}\ket{\psi_{\textrm{out}}}=e^{-ik(2t+\tau)}(t_{k}^{2}+T(k,\tau)).$
(49)

The wavefunction of two photons without the interaction is

$\bra{t,t+\tau}\ket{\psi_{\textrm{out}}^{(0)}}=e^{-ik(2t+\tau)}t_{k}^{2}.$
(50)

The nonlinear phase shift due to the photon-photon interaction thus is

$\displaystyle\varphi_{\textrm{n}}$
$\displaystyle\equiv\textrm{Arg}\left[\bra{t,t+\tau}\ket{\psi_{\textrm{out}}}\right]-\textrm{Arg}\left[\bra{t,t+\tau}\ket{\psi_{\textrm{out}}^{(0)}}\right]$
(51) $\displaystyle=\textrm{Arg}\left[1+\frac{T(k,\tau)}{t_{k}^{2}}\right].$

By making $|t_{k}^{2}|\approx|T(k,\tau)|$, large nonlinear phase shift can be
achieved, which is possible even for the weak coupling regime
$g\ll\kappa_{1}$. Such nonlinear phase shift is a key enabler for quantum
nondemolition measurement of photons Xia  _et al._ (2016) and conditional
quantum logic gates Turchette  _et al._ (1995); Langford  _et al._ (2011).

Fig. 4 shows the calculated nonlinear phase shift (for $\tau=0$) using
experimentally achievable parameters of a state-of-the-art $\chi^{(2)}$
nonlinear photonic platform Zhao and Fang (2021). It is seen that large and
even $\pi-$nonlinear phase shift can be achieved in practical devices.
Actually, $\pi-$nonlinear phase shift is always achieved at $k=\omega_{1}$ as
long as $t_{\omega_{1}}^{2}<-T(\omega_{1},0)$.

###  IV.4 Two-mode nonclassical correlation

In Sections IV.2 and IV.3, we considered a cavity with degenerate $a_{1}$ and
$a_{2}$ modes. Here, we consider a cavity with non-degenerate $a_{1}$ and
$a_{2}$ modes which satisfy the frequency-matching condition
$\omega_{1}+\omega_{2}\approx\omega_{3}$ and show nonclassical two-mode
correlations could be generated. We assume now the input state to be the
product of two weak coherent states in modes $a_{1}$ and $a_{2}$,

$\displaystyle\ket{\psi_{\textrm{in}}}=$
$\displaystyle\ket{\alpha}_{1}\ket{\beta}_{2}$ (52) $\displaystyle=$
$\displaystyle\left(\ket{0}+\alpha\ket{1_{t}}_{1}+\frac{\alpha^{2}}{\sqrt{2}}\ket{1_{t}1_{t}}_{1}+O(\alpha^{3})\right)$
$\displaystyle\times\left(\ket{0}+\beta\ket{1_{t}}_{2}+\frac{\beta^{2}}{\sqrt{2}}\ket{1_{t}1_{t}}_{2}+O(\beta^{3})\right).$

The two-mode and single-mode correlations are, respectively,

$\displaystyle G_{12}(\tau)$ (53) $\displaystyle=$
$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},2}^{\dagger}(t+\tau)a_{\textrm{out},2}(t+\tau)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
$\displaystyle=$ $\displaystyle|\alpha\beta|^{2}\int
dp_{1}dp_{2}\bra{1_{k_{1}}1_{k_{2}}}a_{\textrm{out},1}^{\dagger}(p_{1})a_{\textrm{out},2}^{\dagger}(p_{2})\ket{0}e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}\bra{0}a_{\textrm{out},2}(\tilde{p}_{2})a_{\textrm{out},1}(\tilde{p}_{1})\ket{1_{k_{1}}1_{k_{2}}}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}$
$\displaystyle=$ $\displaystyle|\alpha\beta|^{2}\int
dp_{1}dp_{2}S_{p_{1}p_{2};k_{1}k_{2}}^{*}e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}S_{\tilde{p}_{1}\tilde{p}_{2};k_{1}k_{2}}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}$
$\displaystyle=$
$\displaystyle|\alpha\beta|^{2}|t_{k_{1}}t_{k_{2}}+\widetilde{T}(k_{1},k_{2},\tau)|^{2},$

where the $S-$matrix is given by Eq. 26 and

$\displaystyle\widetilde{T}(k_{1},k_{2},\tau)$ (54)
$\displaystyle=\begin{cases}-\frac{g^{2}\kappa_{1e}\kappa_{2e}}{(k_{1}+k_{2}-\lambda_{1})(k_{1}+k_{2}-\lambda_{2})(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})}e^{-i\tau(\alpha_{2}-k_{2})},\tau>0,\\\
-\frac{g^{2}\kappa_{1e}\kappa_{2e}}{(k_{1}+k_{2}-\lambda_{1})(k_{1}+k_{2}-\lambda_{2})(k_{1}-\alpha_{1})(k_{2}-\alpha_{2})}e^{i\tau(\alpha_{1}-k_{1})},\tau<0,\end{cases}$

and

$\displaystyle G_{11}(\tau)$ (55) $\displaystyle=$
$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}^{\dagger}(t+\tau)a_{\textrm{out},1}(t+\tau)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
$\displaystyle=$ $\displaystyle\frac{|\alpha|^{4}}{2}\int
dp_{1}dp_{2}\bra{1_{k_{1}}1_{k_{1}}}a_{\textrm{out},1}^{\dagger}(p_{1})a_{\textrm{out},1}^{\dagger}(p_{2})\ket{0}e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}\bra{0}a_{\textrm{out},1}(\tilde{p}_{1})a_{\textrm{out},1}(\tilde{p}_{2})\ket{1_{k_{1}}1_{k_{1}}}e^{-i\tilde{p}_{2}t-i\tilde{p}_{1}(t+\tau)}$
$\displaystyle=$ $\displaystyle\frac{|\alpha|^{4}}{4}\int
dp_{1}dp_{2}2t_{k_{1}}^{*}t_{k_{1}}^{*}\delta(p_{1}-k_{1})\delta(p_{2}-k_{1})e^{ip_{1}t+ip_{2}(t+\tau)}$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}2t_{k_{1}}t_{k_{1}}\delta(p_{1}-k_{1})\delta(p_{2}-k_{1})e^{-i\tilde{p}_{2}t-i\tilde{p}_{1}(t+\tau)}$
$\displaystyle=$ $\displaystyle|\alpha|^{4}|t_{k_{1}}|^{4},$

and similarly,

$G_{22}(\tau)=|\beta|^{4}|t_{k_{2}}|^{4}.$ (56)

In Eq. LABEL:G11, we have used the fact that the transport of two $a_{1}$
photons via a non-degenerate cavity is interaction-free because
$2\omega_{1}\neq\omega_{3}$.

To measure the nonclassicality of the two-mode correlation, we define

$\zeta(\tau)\equiv\frac{G_{12}(\tau)}{\sqrt{G_{11}(\tau)G_{22}(\tau)}}=|1+\frac{\widetilde{T}(k_{1},k_{2},\tau)}{t_{k_{1}}t_{k_{2}}}|^{2}.$
(57)

$\zeta(\tau)>1$ leads to violation of the Cauchy-Schwartz inequality and
indicates nonclassical two-mode correlations Mandel and Wolf (1995). Given the
similarity of Eq. 57 and Eq. 64, it is obvious that $\zeta(\tau)>1$ can be
achieved when $|t_{k_{1}}t_{k_{2}}|$ is sufficiently smaller than
$|\widetilde{T}(k_{1},k_{2},\tau)|$.

##  V Conclusion

In summary, we have theoretically studied few-photon transport via a
waveguide-coupled, multimode $\chi^{(2)}$ optical cavity. We used both non-
perturbative method and a new perturbation method based on Feynman diagrams to
compute the $S-$matrix associated with the few-photon transport. The Feynman
diagram approach provides physical insight into the transport process and is
mathematically convenient. The $S-$matrix shows that the two-photon transport
involves quantum interference between linear transmission and interaction-
mediated transport. This effect leads to rather unexpected result that strong
quantum correlations could be realized in weak nonlinear systems by matching
the linear transmission coefficient with the interaction-mediated amplitude.
We numerically showed several pronounced quantum optical effects, including
photon blockade and $\pi-$conditional phase shift, could be achieved in state-
of-the-art nonlinear quantum photonic platforms, which might have a
significant impact on using these systems for quantum information science
applications.

##  Appendix A The wavefunction of transported photons

Here we prove the second-order correlation actually gives the norm of the
output two-photon wavefunction.

$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out}}^{\dagger}(t)a_{\textrm{out}}^{\dagger}(t+\tau)a_{\textrm{out}}(t+\tau)a_{\textrm{out}}(t)\ket{\psi_{\textrm{in}}}$
(58) $\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\int
dp_{1}dp_{2}dp_{3}dp_{4}e^{i(p_{1}t+p_{2}(t+\tau)-p_{3}(t+\tau)-p_{4}t)}\bra{\psi_{\textrm{in}}}\hat{S}^{\dagger}\hat{S}a_{\textrm{out}}^{\dagger}(p_{1})\hat{S}^{\dagger}\hat{S}a_{\textrm{out}}^{\dagger}(p_{2})\hat{S}^{\dagger}\hat{S}a_{\textrm{out}}(p_{3})\hat{S}^{\dagger}\hat{S}a_{\textrm{out}}(p_{4})\hat{S}^{\dagger}\hat{S}\ket{\psi_{\textrm{in}}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\int
dp_{1}dp_{2}dp_{3}dp_{4}e^{i(p_{1}t+p_{2}(t+\tau)-p_{3}(t+\tau)-p_{4}t)}\bra{\psi_{\textrm{out}}}a_{\textrm{in}}^{\dagger}(p_{1})a_{\textrm{in}}^{\dagger}(p_{2})a_{\textrm{in}}(p_{3})a_{\textrm{in}}(p_{4})\ket{\psi_{\textrm{out}}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\int
dp_{1}dp_{2}dp_{3}dp_{4}e^{i(p_{1}t+p_{2}(t+\tau)-p_{3}(t+\tau)-p_{4}t)}\bra{\psi_{\textrm{out}}}a_{\textrm{in}}^{\dagger}(p_{1})a_{\textrm{in}}^{\dagger}(p_{2})\ket{0}\bra{0}a_{\textrm{in}}(p_{3})a_{\textrm{in}}(p_{4})\ket{\psi_{\textrm{out}}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\int
dp_{1}dp_{2}e^{i(p_{1}t+p_{2}(t+\tau))}\bra{p_{1},p_{2}}\ket{\psi_{\textrm{out}}}^{*}\int
dp_{3}dp_{4}e^{-i(p_{3}(t+\tau)+p_{4}t)}\bra{p_{3},p_{4}}\ket{\psi_{\textrm{out}}}$
$\displaystyle=$ $\displaystyle|\bra{t,t+\tau}\ket{\psi_{\textrm{out}}}|^{2}$

where $\ket{t,t+\tau}$ represents the state of two photons at time $t$ and
$t+\tau$, respectively, and thus the end result gives the time-domain
wavefunction of the output two photons. We have used $\hat{S}$ to represent
the operator corresponding to the $S$-matrix defined in the momentum space,
which satisfies the unitary condition $\hat{S}^{\dagger}\hat{S}=\hat{I}$. We
have also assumed the input/output states contain up to two photons, which is
consistent with Eq. 33.

##  Appendix B Correlation functions for photon wavepackets

We consider the input state being a photon wavepacket, i.e.,
$\ket{\psi_{\text{in}}}=\exp(\alpha a^{\dagger}-\alpha^{*}a)\ket{0}$, where
$a=\int dkf(k)a_{k}$ and $f(k)$ is the spectral distribution centered around
$k_{0}$ and satisfying $\int|f(k)|^{2}dk=1$. The first- and second-order
correlation functions for the photon wavepacket are given by

$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
(59) $\displaystyle=$ $\displaystyle|\alpha|^{2}\int
dk_{1}d\tilde{k}_{1}f(k_{1})f(\tilde{k}_{1})\bra{1_{k_{1}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}(t)\ket{1_{\tilde{k}_{1}}}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\int
dk_{1}d\tilde{k}_{1}f(k_{1})f(\tilde{k}_{1})\bra{1_{k_{1}}}a_{\textrm{out},1}^{\dagger}(p_{1})a_{\textrm{out},1}(\tilde{p}_{1})\ket{1_{\tilde{k}_{1}}}e^{i(p_{1}-\tilde{p}_{1})t}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\int
dk_{1}dp_{1}f(k_{1})e^{ip_{1}t}S^{*}_{p_{1}k_{1}}\int
d\tilde{k}_{1}d\tilde{p}_{1}f(\tilde{k}_{1})e^{-i\tilde{p}_{1}t}S_{\tilde{p}_{1}\tilde{k}_{1}}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}\int
dk_{1}f(k_{1})e^{ik_{1}t}t_{k_{1}}^{*}\int
d\tilde{k}_{1}f(\tilde{k}_{1})e^{-i\tilde{k}_{1}t}t_{\tilde{k}_{1}}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}|\widetilde{t}(k_{0},t)|^{2},$

where $\widetilde{t}(k_{0},t)=\int dk_{1}f(k_{1})e^{-ik_{1}t}t_{k_{1}}$, and

$\displaystyle\bra{\psi_{\textrm{in}}}a_{\textrm{out},1}^{\dagger}(t)a_{\textrm{out},1}^{\dagger}(t+\tau)a_{\textrm{out},1}(t+\tau)a_{\textrm{out},1}(t)\ket{\psi_{\textrm{in}}}$
(60) $\displaystyle=$ $\displaystyle\frac{|\alpha|^{4}}{2}\int
dp_{1}dp_{2}dk_{1}dk_{2}\bra{1_{k_{1}}1_{k_{2}}}a_{\textrm{out},1}^{\dagger}(p_{1})a_{\textrm{out},1}^{\dagger}(p_{2})\ket{0}e^{ip_{1}t+ip_{2}(t+\tau)}f(k_{1})f(k_{2})$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}d\tilde{k}_{1}d\tilde{k}_{2}\bra{0}a_{\textrm{out},1}(\tilde{p}_{2})a_{\textrm{out},1}(\tilde{p}_{1})\ket{1_{\tilde{k}_{1}}1_{\tilde{k}_{2}}}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}f(\tilde{k}_{1})f(\tilde{k}_{2})$
$\displaystyle=$ $\displaystyle\frac{|\alpha|^{4}}{4}\int
dp_{1}dp_{2}S_{p_{1}p_{2};k_{1}k_{1}}^{*}e^{ip_{1}t+ip_{2}(t+\tau)}f(k_{1})f(k_{2})$
$\displaystyle\quad\times\int
d\tilde{p}_{1}d\tilde{p}_{2}S_{\tilde{p}_{1}\tilde{p}_{2};\tilde{k}_{1}\tilde{k}_{2}}e^{-i\tilde{p}_{1}t-i\tilde{p}_{2}(t+\tau)}f(\tilde{k}_{1})f(\tilde{k}_{2})$
$\displaystyle=$
$\displaystyle|\alpha|^{4}|\widetilde{t}(k_{0},t)\widetilde{t}(k_{0},t+\tau)+\widetilde{T}(k_{0},t,\tau)|^{2},$

where

$\displaystyle\widetilde{T}(k_{0},t,\tau)=\frac{1}{2}\int
dk_{1}dk_{2}dp_{1}\kappa_{1e}^{2}M(p_{1},k_{1}+k_{2}-p_{,}k_{1},k_{2})e^{-ip_{1}t-i(k_{1}+k_{2}-p_{1})(t+\tau)}f(k_{1})f(k_{2}).$
(61)

To illustrate the effect of finite spectral width of wavepackets, we consider
a spectral distribution of
$f(k)=\sqrt{\frac{2}{\pi\gamma}}\frac{\gamma^{2}}{(k-k_{0})^{2}+\gamma^{2}}$
which leads to

$\displaystyle\widetilde{t}(k_{0},t)=\begin{cases}\sqrt{2\pi\gamma}[e^{-i(k0-i\gamma)t}(1-\frac{i\kappa_{1e}}{k_{0}-i\gamma-\alpha_{1}})-e^{-i\alpha_{1}t}\frac{2\gamma\kappa_{1e}}{(\alpha_{1}-k_{0})^{2}+\gamma^{2}}],\quad
t>0,\\\
\sqrt{2\pi\gamma}e^{-i(k0+i\gamma)t}(1-\frac{i\kappa_{1e}}{k_{0}+i\gamma-\alpha_{1}}),\quad
t<0\end{cases}$ (62)

and

$\displaystyle\widetilde{T}(k_{0},t=0,\tau)=-\frac{4\pi
g^{2}\kappa_{1e}^{2}\gamma}{(k_{0}+i\gamma-\alpha_{1})^{2}(2k_{0}+2i\gamma-\lambda_{1}^{\prime})(2k_{0}+2i\gamma-\lambda_{2}^{\prime})}\times\begin{cases}e^{-i\alpha_{1}\tau},\quad\tau>0,\\\
e^{-2i(k_{0}+i\gamma)\tau+i\alpha_{1}\tau},\quad\tau<0.\end{cases}$ (63)

We assume the wavepacket to be sufficiently long and thus ignore the temporal
wavepacket shape. The normalized second-order correlation function thus is
given by, setting $t=0$,

$\displaystyle
g^{(2)}(\tau)=\frac{|\widetilde{t}(k_{0},0)\widetilde{t}(k_{0},\tau)+\widetilde{T}(k_{0},0,\tau)|^{2}}{|\widetilde{t}(k_{0},0)|^{2}|\widetilde{t}(k_{0},\tau)|^{2}}.$
(64)

We plot $g^{(2)}(0)$ as a function of $\gamma$ in Fig. 5 for a parameter set
which yields $g^{(2)}(0)=0$ for $\gamma\rightarrow 0$, i.e., the continuous-
wave limit. The averaging effect is observed due to the spectral span of the
photon wavepacket, leading to $g^{(2)}(0)$ increasing from 0 to 1. However,
$g^{(2)}(0)$ remains sufficiently small for $\gamma\leq 10^{-2}\kappa_{1}$.

Figure 5:  $g^{(2)}(0)$ as a function of wavepacket width $\gamma$. Other
parameters used in the calculation are $k_{0}=\omega_{1}$,
$g/\kappa_{1i}=0.02$, $\kappa_{1i}/\kappa_{1e}=1.04$,
$\kappa_{3i}=2\kappa_{1i}$, $\kappa_{3e}=0.1\kappa_{1i}$.

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Acknowledgements
This work is supported by US National Science Foundation under Grant No. DMS
18-39177.