# Few-photon transport in Fano-resonance waveguide geometries

Kristoffer B. Joanesarson  Department of Photonics Engineering, DTU Fotonik,
Technical University of Denmark, Building 343, 2800 Kongens Lyngby, Denmark
Jake Iles-Smith  Department of Photonics Engineering, DTU Fotonik, Technical
University of Denmark, Building 343, 2800 Kongens Lyngby, Denmark  Mikkel
Heuck  Department of Photonics Engineering, DTU Fotonik, Technical University
of Denmark, Building 343, 2800 Kongens Lyngby, Denmark  Jesper Mørk
<EMAIL_ADDRESS>Department of Photonics Engineering, DTU Fotonik,
Technical University of Denmark, Building 343, 2800 Kongens Lyngby, Denmark

###### Abstract

We present a theoretical study of Fano interference effects in few-photon
transport. Under appropriate conditions, a local defect in an optical
waveguide induces a highly asymmetric transmission lineshape, characteristic
of Fano interference. For a two-level emitter placed adjacent to such a
defect, here modeled as a partially transmitting element, we find an
analytical expression for the full time evolution of single-photon wavepackets
and the emitter excitation probability. We show how the partially transmitting
element affects the emitter lifetime and shifts the spectral position of the
effective system resonances. Using input-output formalism, we determine the
single and two-photon $S$-matrices for both a two-level emitter and a cavity-
emitter system coupled to a waveguide with a partially transmitting element.
We show how the Fano interference effect can be exploited for the
implementation of a Hong-Ou-Mandel switch in analogy with a tunable linear or
nonlinear beam splitter.

††preprint: APS/123-QED

##  I Introduction

A long standing goal for quantum optics is the deterministic control of light
at the single-photon level Kimble (2008). Optical waveguides coupled to
(artificial) atoms are examples of promising candidates for guiding and
manipulating the photonic qubits with the potential for scalable on-chip
quantum information processing Lodahl  _et al._ (2015). To efficiently
manipulate the optical quantum states one can exploit that strong light-matter
coupling can mediate photon-photon interactions and give rise to highly non-
linear properties Chang  _et al._ (2014); Roy  _et al._ (2017). Examples of
waveguide geometries with demonstrated strong light-matter interactions
include nanofibres coupled to atoms Mitsch  _et al._ (2014), superconducting
waveguides coupled to tunable plasmon qubits Mirhosseini  _et al._ (2019),
plasmonic nanowires Akimov  _et al._ (2007), and photonic crystal waveguides
coupled to semiconductor quantum dots Arcari  _et al._ (2014); Javadi  _et
al._ (2015).

Within few-photon scattering experiments and theoretical studies, the
waveguide geometry is often modelled as an infinite or semi-infinite
collection of harmonic oscillators coupled to one or more discrete-energy-
level quantum systems such as a single two-level emitter (TLE) Shen and Fan
(2007); Nysteen  _et al._ (2015a), a Jaynes-Cummings (JC) system (i.e. a TLE
coupled to a cavity) Shi  _et al._ (2011); Shen and Fan (2009a); Rephaeli and
Fan (2012), multi-level emitters Witthaut and Sørensen (2010); Zheng  _et al._
(2013); Das  _et al._ (2018a, b), multiple emitters Rephaeli  _et al._ (2011);
Cheng  _et al._ (2017), whispering gallery mode cavities Shen and Fan (2009b),
and a Kerr nonlinear cavity Liao and Law (2010); Xu and Li (2014). Common for
these studies is that the transport properties for the waveguide geometries
are almost always assumed to depend only on the specific type of discrete-
energy-level quantum system and not on the waveguide properties. In this work
we investigate the few-photon transport properties in waveguides having a
local defect placed in the mirror-symmetry line of a quantum system Heuck  _et
al._ (2013), as sketched in Fig. 1(a). In Fig. 1(a) an initially right
propagating optical pulse is partially transmitted and reflected after
interacting with a scattering system, $S$, and a partially transmitting
element (PTE). $S$ is coupled to the waveguide with a rate $\Gamma$, the PTE
is characterized by a strength $V$, and dissipation is included through
coupling to non-guided modes with leakage rate $\gamma$. In Fig. 1(b) and 1(c)
we sketch two limiting cases where the PTE is absent and with maximal
strength, corresponding to a side-coupled system and a blocked waveguide,
respectively. In this work we will analyse three specific quantum systems,
represented by $\mathcal{S}$ in Fig. 1(d): i. a TLE, ii. a cavity, and iii. a
coupled emitter-cavity system.

Figure 1: Waveguide geometries. (a) A Fano-resonance waveguide geometry
coupled to a quantum system $\mathcal{S}$ with coupling rate $\Gamma$ and
coupled to non-guided modes with coupling rate $\gamma$. The waveguide has a
PTE in the mirror-symmetry line with strength $V$. (b) and (c) show the
limiting cases of a quantum system side coupled to a waveguide, and coupled to
a blocked waveguide, respectively. (d) The specific quantum systems
$\mathcal{S}$ analyzed in this work. We consider i. a TLE with energy
separation $\hbar\omega_{\mathrm{e}}$, ii. a cavity with resonance frequency
$\omega_{\mathrm{c}}$, and iii. a combined cavity-emitter system with coupling
coefficient $g$.

For the waveguide geometry in Fig. 1(a), with either of the three
aforementioned quantum systems, an incident photon far detuned from the system
resonance(s) will only interact with the PTE, which we assume has a flat
spectral response, i.e. the transmission and reflection coefficients are
frequency independent. In the two limiting cases for the PTE strength, i.e.
the waveguide geometries in Fig. 1(b) and (c), the photons will therefore
completely transmit and reflect, respectively. On the other hand, for an
incident photon close to being resonant with the quantum system, interaction-
mediated transport can result in (partial) reflection for the situation in
Fig. 1(b), and (partial) transmission for the situation in Fig. 1(c) Shen and
Fan (2009a). In classical optics, experiments demonstrate that engineering the
direct waveguide path, e.g. with a local hole impurity in a photonic crystal
waveguide (similar to that depicted in Fig. 1(a)), can be used to exploit
linear and nonlinear scattering effects more efficiently, with potential
applications for switching, sensing, and pulse shaping Heuck  _et al._ (2013);
Yu  _et al._ (2014); Bekele  _et al._ (2019). This improvement is due to Fano
interference, which is characterized by a highly asymmetric transmission
lineshape originating from combining a discrete mode system with a continuum
of modes Suh  _et al._ (2003); Fan  _et al._ (2003); Limonov  _et al._ (2017).
Here, we analyze the quantum optical scattering properties of Fano-resonance
waveguide geometries, from now on referred to as Fano-waveguide geometries,
and determine to what extent two-photon nonlinear effects can be enhanced.

The Fano interference effect for one and two-photon scattering in a waveguide
coupled to a TLE was studied in the recent work by Xu and Fan Xu and Fan
(2016). We extend this analysis of Fano-resonance waveguide geometries in the
following ways: (i) We determine analytical expressions for the dynamical
evolution of single-photon wavepackets in a waveguide with a PTE coupled to a
TLE. (ii) We determine the single and two-photon $S$-matrix for a JC system
coupled to a waveguide with a PTE. (iii) We investigate two-photon scattering
for envelopes with finite spectral widths and how the Fano interference
affects few-photon switching.

The outline of this paper is as follows. In Sec. II, we introduce the
waveguide Hamiltonian, emphasizing the incorporation of the partially
transmitting element. In Sec. III we solve the dynamical single-excitation
transport properties for the system sketched in Fig. 1(a), where $S$
represents a TLE. In Sec. IV we determine the single and two-photon scattering
matrices for both a TLE and a cavity-emitter coupled Fano-resonance waveguide
geometry. We account here for dissipation via coupling to non-guided modes. In
Sec. V we use our scattering results to analyze few-photon switching
properties. Specifically, we investigate a tunable Houng-Ou-Mandel
interferometer and how the Fano interference effect may be exploited. Finally,
in Sec. VI, we summarize our findings.

##  II Waveguide with a partially transmitting element

The temporal evolution of photons in the waveguide geometry sketched in Fig.
1(b), in the absence of coupling to a system, is described by the Hamiltonian
Xu and Fan (2016); Fan  _et al._ (2010)

$\displaystyle\begin{split}H_{\mathrm{wg},0}&=\sum_{\mu=R,L}\int_{-\infty}^{\infty}\differential{k}\,v_{g}\hbar
k\,b_{\mu}^{\dagger}(k)b_{\mu}(k),\end{split}$ (1)

where $v_{g}$ is the group velocity and $b_{\mu}^{\dagger}$($b_{\mu}$) is the
bosonic creation (annihilation) operator satisfying the commutator relation
$\commutator{b_{\mu}(k)}{b_{\nu}^{\dagger}(k^{\prime})}=\delta_{\mu\nu}\delta(k-k^{\prime})$.
The subscript $\mu$ refers to independent waveguide channels corresponding to
right (R) and left (L) propagating modes (and later dissipative modes). We
have assumed operation in a regime, where a linear approximation to the
dispersion relation is valid, and chosen a reference frame where the
wavenumbers are centred around zero. From here on, the absence of integration
limits indicate integration from $-\infty$ to $\infty$. Furthermore, we will
work in normalized units with $\hbar=v_{g}=1$, such that energies,
wavenumbers, and frequencies are measured in the same units.

To describe the propagation of photons in the waveguide geometry sketched in
Fig. 1(a) with $\Gamma=0$, we add to the Hamiltonian in Eq. (1) a term
corresponding to a local change in the optical density of states. We model the
combined waveguide and PTE using the Hamiltonian
$H_{\mathrm{wg}}=H_{\mathrm{wg},0}+H_{\mathrm{PTE}}$, with

$\displaystyle
H_{\mathrm{PTE}}=V\int\frac{\differential{k}}{\sqrt{2\pi}}\int\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}b_{R}^{\dagger}(k)b_{L}(k^{\prime})+\mathrm{h.c.},$
(2)

as used in Xu and Fan (2016). Using input-output formalism Collett and
Gardiner (1984); Gardiner and Collett (1985); Gardiner  _et al._ (1987) we
determine the scattering properties for the waveguide with a PTE. We start by
defining the single-photon scattering eigenstates Fan  _et al._ (2010)

$\displaystyle\ket{k^{+}}_{\nu}=b^{\dagger}_{\mathrm{in},\nu}(k)\ket{\varnothing},\qquad\ket{p^{-}}_{\mu}=b^{\dagger}_{\mathrm{out},\mu}(p)\ket{\varnothing},$
(3)

where $|\varnothing\rangle$ is the vacuum state and
$b^{\dagger}_{\mathrm{in(out)},\nu}(k)$ is the Fourier transformed input
(output) field operator for the mode $\nu$ defined as

$\displaystyle b_{\text{in(out)},\nu}(t)$
$\displaystyle=\int\frac{\differential{k}}{\sqrt{2\pi}}\,b_{\nu}(k,t_{\mathrm{in(out)}})e^{ik\quantity(t_{\mathrm{in(out)}}-t)},$
(4)

where $b_{\nu}(k,t)=e^{iHt}b_{\nu}(k)e^{-iHt}$ is in the Heisenberg picture.
Photons are incident in the waveguide at large negative times,
$t_{\mathrm{in}}\rightarrow-\infty$, and exit the waveguide at
$t_{\mathrm{out}}\rightarrow\infty$. Assuming $V\in\mathbb{R}$, the input-
output relation in vector form is

$\displaystyle\mathbf{b}_{\text{out}}(t)$
$\displaystyle=\mathbf{C}\mathbf{b}_{\text{in}}(t),$ (5)

with

as shown by Xu and Fan Xu and Fan (2016). We interpret the dimensionless
parameter $V$ as the PTE strength, as it determines the proportion of
reflected and transmitted light characterized by the frequency independent
transmission and reflection coefficients $|t_{B}|^{2}$ and $|r_{B}|^{2}$,
respectively. We note that $|t_{B}|^{2}+|r_{B}|^{2}=1$ as required by the
loss-less PTE. Furthermore, we note that both coefficients are uniquely
confined in the interval $[0,\,1]$ for $V\in[0,\,2]$. $V=0$ and $V=2$
corresponds to having a waveguide geometry like that of Fig. 1(b) and that of
Fig. 1(c), respectively. $V=2/\quantity(1+\sqrt{2})$ gives a balanced PTE i.e.
$|t_{B}|^{2}=|r_{B}|^{2}=1/2$.

##  III A Fano-resonance waveguide system with a two-level emitter – exact
dynamics

In this section we consider a TLE symmetrically and frequency-independently
coupled to a linearly dispersive waveguide with a PTE in the mirror-symmetry
line. This particular geometry corresponds to the illustration in Fig. 1(a)
with the TLE in place of $\mathcal{S}$ (System i. in Fig. 1(d)). The
Hamiltonian is

$\displaystyle\begin{split}H&=\omega_{\mathrm{e}}\sigma_{+}\sigma_{-}+\sum_{\mu=L,R}\int\differential{k}\,kb_{\mu}^{\dagger}(k)b_{\mu}(k)\\\
&\quad+V\int\frac{\differential{k}}{\sqrt{2\pi}}\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}\,\quantity[b_{R}^{\dagger}(k)b_{L}(k^{\prime})+\mathrm{h.c.}]\\\
&\quad+\int\differential{k}\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\quantity(\sigma_{-}\sum_{\mu=R,L}b_{\mu}^{\dagger}(k)+\mathrm{h.c.}).\end{split}$
(6)

The first term describes the bare waveguide with a PTE, as introduced in Sec.
II. The second term governs the free evolution of the TLE, where
$\sigma_{+}$($\sigma_{-}$) is the raising (lowering) operator. The last term
accounts for the emitter-waveguide coupling in the dipole and rotating wave
approximation Scully and Zubairy (1997). Because $[H,\hat{N}]=0$, where
$\hat{N}=\sigma_{+}\sigma_{-}+\sum_{\mu=L,R}\int\differential{k}\,b_{\mu}^{\dagger}(k)b_{\mu}(k)$
is the excitation number operator, we can solve the dynamics in the single-
excitation manifold by considering the single-excitation state vector
expansion

$\displaystyle\ket{\psi(t)}=\chi(t)\sigma_{+}\ket{\varnothing}+\sum_{\mu=L,R}\int\differential{k}\xi_{\mu}(t,k)b_{\mu}^{\dagger}(k)\ket{\varnothing}.$
(7)

The coefficient $\chi(t)$ describes the dynamical emitter excitation with
$|\chi(t)|^{2}$ being the probability of finding the TLE in its excited state.
The complex envelope shape at time $t$ for the part of the photon traveling to
the right(left) is given by $\xi_{R(L)}(t,k)$. From the Schrödinger equation,
we find a set of coupled differential equations for the state vector
coefficients

$\displaystyle i\dot{\chi}(t)$
$\displaystyle=\omega_{e}\chi(t)+\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\int\differential{k}\,\quantity[\xi_{R}(t,k)+\xi_{L}(t,k)],$
(8a) $\displaystyle i\dot{\xi}_{R}(t,k)$
$\displaystyle=k\xi_{R}(t,k)+\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\chi(t)+\frac{V}{2\pi}\int\differential{k^{\prime}}\,\xi_{L}(t,k^{\prime}),$
(8b) $\displaystyle i\dot{\xi}_{L}(t,k)$
$\displaystyle=k\xi_{L}(t,k)+\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\chi(t)+\frac{V}{2\pi}\int\differential{k^{\prime}}\,\xi_{R}(t,k^{\prime}).$
(8c)

Without a PTE, i.e. $V=0$, the solution is well-known Chen  _et al._ (2011);
Liao  _et al._ (2015). For a non-zero $V$ and with known initial conditions,
$\chi(t_{i})$, $\xi_{R}(t_{i},k)$, and $\xi_{L}(t_{i},k)$, an analytical
solution is also possible and given in its exact form in App. A. In the case
$V\neq 0$, the emitter excitation coefficient is

$\displaystyle\chi(t)$
$\displaystyle=\chi(t_{i})e^{-\quantity[\frac{\tilde{\Gamma}}{2}+i\omega_{\mathrm{e}}]\quantity(t-t_{i})}-i\frac{\tilde{\Gamma}}{2\sqrt{\pi\Gamma}}\int\differential{k^{\prime}}\,\xi(t_{i},k^{\prime})e^{-ik^{\prime}\quantity(t-t_{i})}\frac{1-e^{-\quantity[\frac{\tilde{\Gamma}}{2}+i\quantity(\omega_{\mathrm{e}}-k^{\prime})]\quantity(t-t_{i})}}{\frac{\tilde{\Gamma}}{2}+i\quantity(\omega_{\mathrm{e}}-k^{\prime})},$
(9)

with
$\xi(t_{i},k^{\prime})=\xi_{R}(t_{i},k^{\prime})+\xi_{L}(t_{i},k^{\prime})$.
We have also defined here an effective complex coupling rate which depends on
the PTE stength:

$\displaystyle\tilde{\Gamma}=\frac{\Gamma}{1+\frac{iV}{2}}.$ (10)

###  III.1 Single-photon emission by an emitter relaxation process

Here we study the emitter relaxation process for an initially excited TLE in a
waveguide with a PTE, corresponding to the geometry sketched in Fig. 1(a) with
system i. in place of $\mathcal{S}$. With $\chi(t_{i})=1$ and
$\xi_{R}(t_{i},k)=\xi_{L}(t_{i},k)=0$, Eq. (9) reduces to

$\displaystyle\chi(t)=e^{-\quantity[\frac{\tilde{\Gamma}}{2}+i\omega_{\mathrm{e}}]\quantity(t-t_{i})}.$
(11)

The effect of the PTE is to increase the emitter lifetime by a factor of
$\mathrm{Re}\\{\quantity(1+iV/2)^{-1}\\}^{-1}=1+(V/2)^{2}$. In the long-time
limit, $t_{f}\longrightarrow\infty$, we find for the photon envelopes

$\displaystyle\quantity|\xi_{R}(t_{f},k)|^{2}=\quantity|\xi_{L}(t_{f},k)|^{2}=\frac{\frac{\Gamma}{4\pi}}{1+\quantity(\frac{V}{2})^{2}}\quantity|\frac{1}{\quantity(k-\omega_{\mathrm{e}})+i\frac{\tilde{\Gamma}}{2}}|^{2}.$
(12)

A photon generated by the relaxation process will thus have a Lorentzian
spectrum even if we introduce a PTE. We see, however, that the center
frequency is shifted by
$\mathrm{Im}\\{\tilde{\Gamma}/2\\}=-\Gamma\frac{V}{2}(1+(V/2)^{2})^{-1}$. Both
the frequency shift and altered emitter lifetime are characteristic of a Lamb
shift and the Purcell effect induced by a change in the local density of
states Rybin  _et al._ (2016); Lodahl  _et al._ (2015), here caused by the PTE
in the waveguide.

###  III.2 Single-photon scattering dynamics

We can use the analytical expressions for the state vector coefficients to
calculate how a single photon scatters off the TLE as well as the probability
of finding the emitter in the excited state as a function of time. In the
following, we take the input state to be an initially right-propagating single
photon, entering from the left side of the waveguide, with a Gaussian spectral
envelope

$\displaystyle\xi_{R}(k,t_{i})=\quantity(\frac{1}{2\pi\sigma^{2}})^{1/4}e^{-\quantity(\frac{k-k_{c}}{2\sigma})^{2}}e^{-it_{i}\quantity(k-k_{c})}.$
(13)

The spectral width, $\sigma$, and the center frequency $k_{c}$ uniquely
characterize the spectral shape. We assume that the initial time, $t_{i}$, is
chosen such that the real-space envelope is far away from the emitter.

In the absence of a PTE ($V=0$), the emitter excitation probability,
$|\chi(t)|^{2}$, can reach a maximum of $0.4$ when the envelope is centered
around $\omega_{\mathrm{e}}$ and has a width of $\sigma_{0}\approx 0.73\Gamma$
Chen  _et al._ (2011). If a PTE is introduced, i.e. $V>0$, this maximum value
is reached by changing the envelope parameters according to

$\displaystyle k_{c}$
$\displaystyle=\omega_{\mathrm{e}}+\frac{\Gamma}{2}\times\mathrm{Im}\quantity{\frac{1}{1+\frac{iV}{2}}}\equiv\tilde{\omega}_{\mathrm{e}},$
(14) $\displaystyle\sigma$
$\displaystyle=\sigma_{0}\times\mathrm{Re}\quantity{\frac{1}{1+\frac{iV}{2}}}.$
(15)

This translation and rescaling, respectively, is again due to a change in the
local density of states from introducing a PTE as discussed in Sec. III.1, and
is easily verified by insertion into Eq. (9). From here on, we will refer to
$\tilde{\omega}_{\mathrm{e}}$ as the effective emitter resonance frequency. In
Fig. 2, we plot the envelope dynamics as well as the emitter excitation
probability and scattering probabilities for the above given envelope
parameters using three different values of $V$, corresponding to an unblocked
waveguide ($V=0$), a balanced PTE ($V=\sqrt{2}/\quantity(1+\sqrt{2})$), and a
blocked waveguide ($V=2$). The initial condition is
$\chi(t_{i})=\xi_{L}(t_{i},k)=0$, and $\xi_{R}(t_{i},k)$ is given by Eq. (13).
Fig. 2(a) shows how the photon propagates in the waveguide. When the
wavepacket reaches the emitter and the PTE at $x=0$, it partially transmits
and reflects. In row 1 of Fig. 2(a), frequencies far-off-resonance are
transmitted as they do not interact with the emitter. This is also evident
from row 1 of Fig. 2(b), which shows the asymptotic behavior of the scattered
state. In row 2 of Fig. 2(a) the incident photon is scattered into a left and
a right propagating part with equal probability. The scattered state of Fig.
2(a) row 3 is qualitatively similar to row 1 of Fig. 2(a) but with the key
difference that the reflected and transmitted envelope shapes have
interchanged. This is of course due to the fact that frequencies far-off-
resonance are now reflected, whereas frequencies in the vicinity of
$\tilde{\omega}_{\mathrm{e}}$ are transmitted via the interaction with the
emitter. The asymptotic output envelopes as well as the input envelope are
plotted in Fig. 2(b), clearly showing which frequencies are reflected and
which are transmitted. For the waveguide without the PTE (row 1) and with a
fully blocking PTE (row 3), the output spectrum is symmetric around
$\tilde{\omega}_{\mathrm{e}}$. For the balanced PTE (row 2), the transmitted
and reflected wavepackets are anti-symmetric around
$\tilde{\omega}_{\mathrm{e}}$. The asymptotic scattering behavior will be
analyzed in depth in Sec. IV. In fig. 2(c) we plot the probability of
measuring the emitter in its excited state as well as measuring a right or a
left propagating photon versus time. We see that for all three PTE strengths a
maximum excitation probability of $0.4$ occurs for $t\approx 1/\Gamma$, when
the envelope center is close to the emitter position. In row 2 of Fig. 2(c),
we see that the asymptotic output photons are equal parts reflected and
transmitted. This is in fact always true for the balanced PTE in case of a
symmetric envelope shape and a center frequency of
$\tilde{\omega}_{\mathrm{e}}$.

Figure 2: Single-photon scattering in a PTE-waveguide system coupled to a TLE.
The input state is an initially right-propagating single photon, entering from
the left side of the waveguide, with a Gaussian spectral envelope with
$k_{c}=\omega_{\mathrm{e}}+\Gamma\times\mathrm{Im}\\{\quantity(1+iV/2)^{-1}\\}/2$
and $\sigma=\sigma_{0}\times\mathrm{Re}\\{\quantity(1+iV/2)^{-1}\\}$. Column
(a): Single-photon propagating in the waveguide with $V=0$ (row 1),
$V=2/\quantity(1+\sqrt{2})$ (row 2), and $V=2$ (row 3). The color scale shows
the field amplitude vs. position and time. Column (b): Spectral envelope in
the asymptotic limit for the transmitted (red) and reflected (blue) part of
the wavepacket. Column (c): Emitter excitation probability $|\chi|^{2}$
(black) as well as the probabilities for finding a photon propagating in the
right (red) and left (blue) direction versus time.

##  IV Few-photon scattering matrices

There are many ways of determining the scattering properties for waveguide
systems considered in this work. One approach is by determining the full
dynamics using a wavefunction approach, as we did in Sec. III, and calculating
the asymptotic behavior. Such an approach can also be used to determine the
evolution of a two-photon wavepacket in a waveguide coupled to a JC system
Stolyarov (2019). If only the asymptotic wavepackets are of interest, however,
determining the scattering matrix – or $S$-matrix – can offer a computational
advantage. The $S$-matrix can be found in a number of ways by using e.g. the
Lippmann-Schwinger formalism Shen and Fan (2007), path-integral techniques Shi
_et al._ (2015), Dyson series Hurst and Kok (2018), or via the input-output
relations using the method introduced by Fan et al. in Fan  _et al._ (2010).
In this section we will use the latter as this method is both common and
transparent.

###  IV.1 Single-photon $S$-matrix

The single-photon $S$-matrix, which describes the scattering process of a
single-photon input, is defined by

$\displaystyle
S_{p;k}^{\mu;\nu}={}_{\mu}\innerproduct{p^{-}}{k^{+}}_{\nu}=\matrixelement{\varnothing}{b_{\mathrm{out},\mu}(p)}{k^{+}}_{\nu},$
(16)

where $\ket{p^{-}}$ and $\ket{k^{+}}$ are the scattering eigenstates defined
in Sec. II. A single-photon state thus scatters according to

$\displaystyle\ket{\psi_{\mathrm{in}}^{(1)}}$
$\displaystyle=\sum_{\nu}\int\differential{k}\,\xi_{\mathrm{in},\nu}(k)b_{\nu}^{\dagger}(k)\ket{\varnothing}$
$\displaystyle\stackrel{{\scriptstyle S}}{{\longrightarrow}}$
$\displaystyle\,\ket{\psi_{\mathrm{out}}^{(1)}}=\sum_{\mu}\int\differential{p}\,\xi_{\mathrm{out},\mu}(p)b_{\mu}^{\dagger}(p)\ket{\varnothing},$
(17)

where $\xi_{\mathrm{in},\nu}$($\xi_{\mathrm{out},\mu}$) is the spectral
single-photon input (output) envelope. From the definition of the $S$-matrix,
the single-photon output envelopes are determined from

$\displaystyle\xi_{\mathrm{out},\mu}(p)=\sum_{\nu}\int\differential{k}\,S_{p;k}^{\mu;\nu}\xi_{\mathrm{in},\nu}(k).$
(18)

In order to determine $S_{p;k}^{\mu;\nu}$ we first choose a specific waveguide
geometry and determine a set of input-output relations.

###  IV.2 Input-output relations

The relation between incoming and outgoing fields can be described in terms of
a set of input-output relations Gardiner and Collett (1985). In determining
the input-output relations we assume a time-independent Hamiltonian similar to
that of Eq. (6) but for an initially unspecified system and system-waveguide
Hamiltonian term. Furthermore, we introduce coupling to additional,
unspecified, waveguide channels, which account for dissipation into a
reservoir Rephaeli and Fan (2013). The Hamiltonian then reads

$\displaystyle\begin{split}H&=\sum_{\mu}\int\differential{k}\,kb_{\mu}^{\dagger}(k)b_{\mu}(k)+H_{\mathrm{s}}+H_{\mathrm{s-wg}}\\\
&\quad+\sum_{\mu\neq\nu}V_{\mu,\nu}\int\frac{\differential{k}}{\sqrt{2\pi}}\int\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}b_{\mu}^{\dagger}(k)b_{\nu}(k^{\prime}).\end{split}$
(19)

The first and last terms are similar to the waveguide and PTE contributions in
Eq. (1) and Eq. (2), respectively. The second and third term correspond to the
relevant system and its coupling to the waveguide, respectively. We use the
Heisenberg equation of motion, $\dot{c}=-i\commutator{c}{H}$, where $c$ is an
arbitrary operator. For the waveguide operators we use that $c=b_{\mu}(k)$ and
$[b_{\mu},H_{\mathrm{s}}]=0$:

$\displaystyle\begin{split}\dot{b}_{\mu}(t,k)&=-i\commutator{b_{\mu}(t,k)}{H_{\mathrm{s-wg}}}\\\
&\quad-
ikb_{\mu}(t,k)-i\sum_{\nu/\mu}V_{\mu,\nu}\int\frac{\differential{k^{\prime}}}{2\pi}b_{\nu}(t,k^{\prime}).\end{split}$
(20)

This, as well as the input-output field operators as defined in Eq. (4), is
sufficient to determine the full set of input-output relations and in turn the
single-photon $S$-matrix. We illustrate this with two paradigmatic examples;
first the case of a TLE and then a JC system coupled to a Fano-waveguide.

###  IV.3 Single-photon $S$-matrix – TLE

The single-photon $S$-matrix for a Fano-waveguide geometry coupled to a TLE in
the mirror-symmetry line was first calculated in Xu and Fan (2016). We start
by summarizing the results of Xu and Fan (2016) before extending the analysis
to include both a TLE and a cavity. We use the Hamiltonian

$\displaystyle\begin{split}H&=\omega_{\mathrm{e}}\sigma_{+}\sigma_{-}+\sum_{\mu}\int\differential{k}\,kb_{\mu}^{\dagger}(k)b_{\mu}(k)\\\
&\quad+\sum_{\mu\neq\nu}V_{\mu,\nu}\int\frac{\differential{k}}{\sqrt{2\pi}}\int\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}b_{\mu}^{\dagger}(k)b_{\nu}(k^{\prime})\\\
&\quad+\sum_{\mu}\int\differential{k}\quantity(\frac{\kappa_{\mu}}{\sqrt{2\pi}}\sigma_{+}b_{\mu}(k)+\mathrm{h.c.}),\end{split}$
(21)

which is similar to Eq. (6), except here we have accounted for an arbitrary
number of waveguide modes. The total lifetime of the emitter is
$1/\sum_{\mu}|\kappa_{\mu}|^{2}$. The relevant input-output equations are (see
App. B)

$\displaystyle\dot{\sigma}_{-}(t)$
$\displaystyle=A\sigma_{-}(t)+\mathbf{B}\mathbf{b}_{\mathrm{in}}(t)+\hat{f}(t),$
(22a) $\displaystyle\mathbf{b}_{\mathrm{out}}(t)$
$\displaystyle=\mathbf{C}\mathbf{b}_{\mathrm{in}}(t)+\mathbf{D}\sigma_{-}(t),$
(22b)

with

$\displaystyle A$
$\displaystyle=-i\omega_{\mathrm{e}}-\frac{1}{2}\bm{\kappa}^{\dagger}\mathbf{G}^{-1}\bm{\kappa},$
(23a) $\displaystyle\mathbf{B}$
$\displaystyle=-i\bm{\kappa}^{\dagger}\mathbf{G}^{-1},$ (23b)
$\displaystyle\mathbf{C}$ $\displaystyle=\mathbf{G}^{*}\mathbf{G}^{-1},$ (23c)
$\displaystyle\mathbf{D}$ $\displaystyle=-i\mathbf{G}^{-1}\bm{\kappa},$ (23d)
$\displaystyle\hat{f}(t)$
$\displaystyle=2i\bm{\kappa}^{\dagger}\mathbf{G}^{-1}\sigma_{+}(t)\sigma_{-}(t)\mathbf{b}_{\mathrm{in}}(t),$
(23e) $\displaystyle\mathbf{G}$
$\displaystyle=\quantity(\mathds{1}+\frac{i}{2}\mathbf{V}).$ (23f)

Inserting the input-output relations, Eq. (22a)-(22b), in Eq. (16), we find

$\displaystyle
S_{p;k}^{\mu;\nu}=C_{\mu;\nu}\delta(p-k)+D_{\mu}\mathcal{G}^{\nu}_{p;k},$ (24)

with

$\displaystyle\mathcal{G}^{\nu}_{p;k}=\matrixelement{\varnothing}{\tilde{\sigma}_{-}(p)}{k^{+}}_{\nu}=i\quantity(p-iA)^{-1}B_{\nu}\,\delta(p-k).$
(25)

Eq. (24) is found by Fourier transforming Eq. (22a) and using
$\langle\varnothing|\hat{f}=0$. Eq. (23e) is derived using
$[\sigma_{-},\sigma_{+}]=\mathds{1}-2\sigma_{+}\sigma_{-}$ but would have
yielded the same result had we assumed
$[\sigma_{-},\sigma_{+}]\approx\mathds{1}$. This is sometimes called the low-
excitation approximation Rephaeli and Fan (2012) but in fact gives the exact
result regardless, for single-photon scattering. As we will soon see, the low-
excitation approximation is also exact for the single-photon case if we
replace the TLE with a JC system.

In matrix form, Eq. (24) has the simple expression

$\displaystyle\mathbf{S}_{p;k}=\quantity[\mathbf{C}+i\frac{\mathbf{D}\mathbf{B}}{p-iA}]\delta(p-k).$
(26)

Although we have accounted for an arbitrary number of waveguide modes, we take
the PTE to affect only the transmission between two modes, here named right
($R$) and left ($L$), i.e.

$\displaystyle
V_{\mu,\nu}=\begin{cases}V&\text{if}\,\quantity(\mu,\nu)=\quantity(R,L)\,\text{or}\,\quantity(\mu,\nu)=\quantity(L,R)\\\
0&\text{otherwise}.\end{cases}$ (27)

Physically, this corresponds to a waveguide geometry where scattering into
dissipative channels is only possible via the system and not via the PTE. In
the special case where the TLE couples symmetrically to two waveguide channels
($R$ and $L$), with $\kappa_{R}=\kappa_{L}=\sqrt{\Gamma/2}$, and also couples
to a dissipation channel ($D$), with $\kappa_{D}=\sqrt{\gamma}$, we find

where the transmission and reflection coefficients are

$\displaystyle t(p)$
$\displaystyle=t_{B}-\frac{i\frac{\tilde{\Gamma}^{2}}{2\Gamma}}{\quantity(p-\omega_{\mathrm{e}})+i\quantity(\tilde{\Gamma}+\gamma)/2},$
(28a) $\displaystyle r(p)$
$\displaystyle=r_{B}-\frac{i\frac{\tilde{\Gamma}^{2}}{2\Gamma}}{\quantity(p-\omega_{\mathrm{e}})+i\quantity(\tilde{\Gamma}+\gamma)/2},$
(28b)

respectively. Here, $\tilde{\Gamma}=\Gamma/\quantity(1+iV/2)$, as in Sec. III,
which reduces to the form found in Rephaeli and Fan (2013) in the absence of a
PTE.

In Fig. 3(a) we plot the absolute squared transmission coefficient for the
Fano-waveguide coupled to a TLE. We have chosen to illustrate three different
PTE strengths both without ($\gamma=0\,\Gamma$) and with
($\gamma=0.1\,\Gamma$) dissipation. Without a PTE, row 1 of Fig. 3(a), the
transmission spectrum is a simple Lorentzian lineshape. The dip in
transmission is the result of destructive interference between the photon path
of direct transmission and the transmission path via the emitter Roulet  _et
al._ (2016). A single incoming quasi-monochromatic photon resonantly tuned to
a lossless emitter will thus reflect. Even when we account for dissipation,
the dip is quite pronounced. For a balanced PTE, row 2 of Fig. 3(a), the
transmission spectrum has a characteristic asymmetric Fano lineshape. Light
incident from one end of the waveguide is transmitted due to partial
transmission via the PTE and via the emitter. When these two paths add
constructively, an increase in transmission is seen and vice versa;
destructive interference results in a decrease in transmission. This gives a
finite spectral distance between the maximum and minimum value in the
spectrum. For a fully blocking PTE, row 3 of Fig. 3(a), the lineshape is again
Lorentzian, but flipped upside-down relative to row 1. When we account for
dissipation, the contribution from loss is more pronounced when high
transmission coincides with the effective emitter resonance frequency. This is
a consequence of light transmitted via the emitter being more susceptible to
loss as only the emitter is assumed to couple to a dissipation channel.

Figure 3: Absolute squared transmission coefficient, $|t(k)|^{2}$, for Fano-
waveguide geometries with (a) a TLE and (b)-(c) a JC system with
$g=0.5\,\Gamma$ and $g=1\,\Gamma$, respectively. The cavity energy is chosen
such that
$\omega_{\mathrm{c}}=\omega_{\mathrm{e}}-\mathrm{Im}\\{\tilde{\Gamma}\\}/2$.
The PTE strengths are $V=0$ (row 1), $V=2/\quantity(1+\sqrt{2})$ (row 2), and
$V=2$ (row 3). In column (a), the dissipation rates are $\gamma=0$ (full
black) and $\gamma=0.1\,\Gamma$ (dashed blue). See Sec. V for an explanation
of the red and green regions. In column (b) and (c), the dissipation rates are
$\quantity(\gamma_{\mathrm{c}},\gamma_{\mathrm{e}})=\quantity(0,0)$ (full
black),
$\quantity(\gamma_{\mathrm{c}},\gamma_{\mathrm{e}})=\quantity(0.1\,\Gamma,0)$
(dashed blue), and
$\quantity(\gamma_{\mathrm{c}},\gamma_{\mathrm{e}})=\quantity(0.1\,\Gamma,0.1\,\Gamma)$
(dashed dotted orange).

###  IV.4 Single-photon $S$-matrix – coupled cavity-emitter system

We now determine the single-photon $S$-matrix for the Fano-waveguide geometry
coupled to a cavity-emitter system, as sketched in Fig. 1(a), where
$\mathcal{S}$ is replaced by quantum system iii. of Fig. 1(d). In the rotating
wave approximation, the Hamiltonian is

$\displaystyle\begin{split}H&=\omega_{\mathrm{c}}a^{\dagger}a+\omega_{\mathrm{e}}\sigma_{+}\sigma_{-}+g\quantity(\sigma_{-}a^{\dagger}+\sigma_{+}a)\\\
&\quad+\sum_{\mu}\int\differential{k}\,kb_{\mu}^{\dagger}(k)b_{\mu}(k)\\\
&\quad+\sum_{\mu\neq\nu}V_{\mu,\nu}\int\frac{\differential{k}}{\sqrt{2\pi}}\int\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}b_{\mu}^{\dagger}(k)b_{\nu}(k^{\prime})\\\
&\quad+\sum_{\mu}\int\differential{k}\quantity(\frac{\kappa_{\mathrm{c},\mu}a^{\dagger}+\kappa_{\mathrm{e},\mu}\sigma_{+}}{\sqrt{2\pi}}b_{\mu}(k)+\mathrm{h.c.}).\end{split}$
(29)

Here, $a$($a^{\dagger}$) is the annihilation(creation) operator for the cavity
and satisfy $\commutator{a}{a^{\dagger}}=1$. $g$ is the emitter-cavity
coupling strength which we take to be real. Compared to Eq. (21), we have
introduced an extra direction-dependent coupling coefficient for the cavity,
$\kappa_{\mathrm{c},\mu}$. We define a system operator vector,
$\hat{\mathbf{c}}=\matrixquantity(a&\sigma_{-})^{T}$. The relevant input-
output equations are (see App. B)

(30a)

with

(30aea) (30aeb) (30aec) (30aed) (30aee)

Aside from an extra added dimension from having two independent system
operators, these expressions look very similar to that of just the TLE, with
$\hat{f}$ again satisfying $\bra{\varnothing}\hat{f}=0$. The single-photon
$S$-matrix for this system is then found in a similar manner as for the case
of the TLE. We find

$\displaystyle\mathbf{S}^{(1)}_{p;k}=\quantity[\mathbf{C}+i\mathbf{D}\quantity(\mathds{1}p-i\mathbf{A})^{-1}\mathbf{B}]\delta(p-k).$
(30aeaf)

As before, we determine the transmission and reflection coefficients in a
specific case. We assume that the cavity couples symmetrically to two
waveguide channels ($R$ and $L$), with
$\kappa_{\mathrm{c},R}=\kappa_{\mathrm{c},L}=\sqrt{\Gamma/2}$, as well as to a
dissipation channel ($D_{\mathrm{c}}$), with
$\kappa_{\mathrm{c},D_{\mathrm{c}}}=\sqrt{\gamma_{\mathrm{c}}}$. Furthermore,
we assume that the emitter couples only to the dissipation channel
$D_{\mathrm{e}}$ with strength
$\kappa_{\mathrm{e},D_{\mathrm{e}}}=\sqrt{\gamma_{\mathrm{e}}}$, i.e.
$\kappa_{\mathrm{e},R}=\kappa_{\mathrm{e},L}=0$. The system now corresponds to
a lossy JC system coupled to a Fano-waveguide. We find for the transmission
and reflection coefficients:

$\displaystyle t(p)$
$\displaystyle=t_{B}-\frac{i\quantity(p-\omega_{\mathrm{e}}+i\frac{\gamma_{\mathrm{e}}}{2})\frac{\tilde{\Gamma}^{2}}{2\Gamma}}{\quantity[p-\omega_{\mathrm{c}}+i\frac{\tilde{\Gamma}+\gamma_{\mathrm{c}}}{2}]\quantity[p-\omega_{\mathrm{e}}+i\frac{\gamma_{\mathrm{e}}}{2}]-g^{2}},$
(30aeaga) $\displaystyle r(p)$
$\displaystyle=r_{B}-\frac{i\quantity(p-\omega_{\mathrm{e}}+i\frac{\gamma_{\mathrm{e}}}{2})\frac{\tilde{\Gamma}^{2}}{2\Gamma}}{\quantity[p-\omega_{\mathrm{c}}+i\frac{\tilde{\Gamma}+\gamma_{\mathrm{c}}}{2}]\quantity[p-\omega_{\mathrm{e}}+i\frac{\gamma_{\mathrm{e}}}{2}]-g^{2}}.$
(30aeagb)

In the limit $g=0$, we see that the scattering coefficients reduce to the case
of the TLE with $\omega_{\mathrm{c}}\rightarrow\omega_{\mathrm{e}}$ and
$\gamma_{\mathrm{c}}\rightarrow\gamma_{\mathrm{e}}$ as expected due to the
low-excitation approximation. Furthermore, we note that the scattering
coefficients are in agreement with Shen and Fan (2009a) for $V=0$.

In Fig. 3(b) and (c) we plot the absolute squared transmission coefficient for
three different sets of dissipation rates using $g=0.5\,\Gamma$ and
$g=1\,\Gamma$, respectively. We choose the cavity frequency to be
$\omega_{\mathrm{c}}=\omega_{\mathrm{e}}-\mathrm{Im}\\{\tilde{\Gamma}\\}/2$.
As in the example of the TLE, row 1, 2, and 3 corresponds to having no PTE, a
balanced PTE, and a fully blocking PTE in the waveguide, respectively. Without
the PTE (row 1) the two dips in the transmission spectrum occur at the
eigenfrequencies of the JC system, i.e. $k=\omega_{\mathrm{e}}\pm g$.
Qualitatively, column (b) and (c) are very similar. It is clear, however, that
as $g$ increases the lineshape will approach two copies of the TLE spectrum
(Fig. 3(a) separated by $2g$. As expected, the effect of dissipation is again
largest when a transmission peak coincides with one of the system
eigenfrequencies. When the photon can escape via the emitter as well as via
the cavity, the loss increases.

###  IV.5 Two-photon S-matrices

The two-photon $S$-matrix is calculated in a similar way as Eq. (16), except
now with an additional photon before and after scattering. We denote the input
directions $\nu_{1}$ and $\nu_{2}$, and the input frequencies $k_{1}$ and
$k_{2}$. The output directions are $\mu_{1}$ and $\mu_{2}$, and the output
frequencies $p_{1}$ and $p_{2}$. The general expression for the two-photon
$S$-matrix is:

$\displaystyle S$
${}_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}={}_{\mu_{1}\nu_{2}}\innerproduct{p_{1}p_{2}^{-}}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}}$
$\displaystyle=S_{p_{1};k_{1}}^{\mu_{1};\nu_{1}}S_{p_{2};k_{2}}^{\mu_{2};\nu_{2}}+S_{p_{1};k_{2}}^{\mu_{1};\nu_{2}}S_{p_{2};k_{1}}^{\mu_{2};\nu_{1}}+iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}.$
(30aeah)

The first two terms are the contributions from linear interactions. The third
term is called the bound state term and accounts for the nonlinear
interactions. For systems with a unique ground state and symmetrical waveguide
coupling, the $S$-matrix can be shown to have the following form Xu  _et al._
(2013); Xu and Fan (2017)

$\displaystyle
iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}=\mathcal{M}_{p_{1},p_{2},k_{1},k_{2}}\delta(p_{1}+p_{2}-k_{1}-k_{2}),$
(30aeai)

The delta function ensures energy conservation in the system. For a Fano-
waveguide symmetrically coupled to a lossless TLE, we find (see App. B.3)

$\displaystyle\mathcal{M}_{p_{1},p_{2},k_{1},k_{2}}=\frac{1}{\pi}\frac{\tilde{\Gamma}}{\sqrt{2\Gamma}}\mathcal{G}(p_{1})\mathcal{G}(p_{2})\quantity[\mathcal{G}(k_{1})+\mathcal{G}(k_{2})],$
(30aeaj)

with

$\displaystyle\mathcal{G}(k)=\frac{\frac{\tilde{\Gamma}}{\sqrt{2\Gamma}}}{k-\omega_{\mathrm{e}}+i\frac{\tilde{\Gamma}}{2}}.$
(30aeak)

Replacing the TLE for a JC system, we find (see App. B.4)

$\displaystyle\mathcal{M}_{p_{1},p_{2},k_{1},k_{2}}$
$\displaystyle=\frac{-2ig}{\sqrt{2\pi}}\mathcal{G}_{\mathrm{e}}(p_{1})\mathcal{G}_{\mathrm{e}}(p_{2})\mathcal{G}_{\mathrm{e-c}}(p_{1}+p_{2},k_{1},k_{2}),$
(30aeal)

with

$\displaystyle\mathcal{G}_{\mathrm{e-c}}(p_{1}+p_{2},k_{1},k_{2})$
$\displaystyle=-\frac{\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}}{\sqrt{2\pi}}\frac{\quantity[p_{1}+p_{2}-2\omega_{\mathrm{c}}+i\tilde{\Gamma}]\quantity(\mathcal{G}_{\mathrm{e}}(k_{1})+\mathcal{G}_{\mathrm{e}}(k_{2}))+2g\quantity(\mathcal{G}_{\mathrm{c}}(k_{1})+\mathcal{G}_{\mathrm{c}}(k_{2}))}{\quantity[p_{1}+p_{2}-\quantity(\omega_{\mathrm{c}}+\omega_{\mathrm{e}})+i\frac{\tilde{\Gamma}}{2}]\quantity[p_{1}+p_{2}-2\omega_{\mathrm{c}}+i\tilde{\Gamma}]-2g^{2}},$
(30aeam)

and

$\displaystyle\matrixquantity(\mathcal{G}_{\mathrm{c}}(k)\\\
\mathcal{G}_{\mathrm{e}}(k))=\frac{1}{\quantity[k-\omega_{\mathrm{c}}+i\frac{\tilde{\Gamma}}{2}]\quantity[k-\omega_{\mathrm{e}}]-g^{2}}\matrixquantity(\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}\quantity(k-\omega_{\mathrm{e}})\\\
\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}g).$ (30aean)

Going beyond two-photon scattering can be achieved by an expansion of the
expression in Eq. (30aeah), see eg. Xu and Fan (2015). The caveat, however, is
the fast increase in the number of terms one has to consider. We therefore
limit ourselves to the analysis of single and two-photon scattering properties
in the next section.

##  V Hong-Ou-Mandel switch

It has recently been experimentally demonstrated that the Fano effect can be
used to make a tunable quantum optical filter in a waveguide Foster  _et al._
(2019). Recent theoretical results have shown that the asymmetry of the Fano
resonance can be used to suppress electron-phonon interactions in quantum dots
Denning  _et al._ (2019). In this section, we investigate how the Fano effect
may be used to implement a tunable Hong-Ou-Mandel (HOM) switch. As was
suggested by Roulet et al. Roulet  _et al._ (2016), with circulators in
optical waveguides and the ability to efficiently turn the HOM interference on
and off, a configurable integrated on-chip circuit can be realized. As an
example, they analyzed the scattering behavior of two initially counter-
propagating photons in a waveguide coupled to a TLE. In this section we extend
the analysis to include a PTE element in the waveguide, and investigate how
the Fano interference effect may be exploited. We will assume that the two
incident photons are identical, with Gaussian envelopes, i.e.

$\displaystyle\ket{LR}=\int\differential{k_{L}}\xi(k_{L})b_{L}^{\dagger}(k_{L})\int\differential{k_{R}}\xi(k_{R})b_{R}^{\dagger}(k_{R})\ket{\varnothing},$
(30aeao)

where $\xi(k)$ is the same spectral envelope as in Eq. (13). We define an
ideal HOM switch as having the following properties

$\displaystyle\ket{LR}\longrightarrow\begin{cases}\ket{LR}&\text{switch-
OFF},\\\ \frac{1}{\sqrt{2}}\quantity[\ket{LL}+\ket{RR}]&\text{switch-
ON},\end{cases}$ (30aeap)

where

$\displaystyle\ket{LL(RR)}=\frac{1}{\sqrt{2}}\quantity[\int\differential{k}\xi(k)b_{L(R)}^{\dagger}(k)]^{2}\ket{\varnothing}.$
(30aeaq)

In the switch-OFF configuration, the output state, just as the input,
represents two photons propagating in different channels, whereas in the ON
configuration, the output is a superposition of two photons propagating in the
same channel. The latter is equivalent to a beam splitter transformation,
except here the output arms are mapped onto the input arms.

###  V.1 Quasi monochromatic input photons

We first consider quasi monochromatic photons, i.e. the limit where
$\sigma\rightarrow 0$ (See Eq. (13)). Because we have assumed the photons to
be quasi monochromatic, correlations arising from the nonlinear term
$\mathcal{M}_{p_{1},p_{2},k_{1},k_{2}}$ (Eq. (30aeaj)) in the scattering
matrix effectively vanishes Nysteen  _et al._ (2015a). In other words, the
scattering event is linear, and the resulting output state is found by
considering the two photons independently. This means, the switch is in the
OFF configuration, i.e. the output state equals the input state $|LR\rangle$,
when the transmission coefficient is either 0 or 1, as this corresponds to
both photons being perfectly reflected or transmitted, respectively.
Conversely, the ON-configuration is reached when the absolute squared
transmission coefficient is $1/2$. In this case, each photon scatters into an
equal superposition of being transmitted and reflected, but the terms
corresponding to a counter-propagating output cancel. From the spectra in Fig.
3(a) we can therefore easily identify the ON and OFF configurations indicated
by the green and red regions, respectively. For $V=0$ (row 1) the OFF
configuration is reached when the two photons are either far detuned from the
emitter, or when the two photons both are resonant with the emitter. In the
former case, each photon will transmit without interaction and in the latter
case, the two photons will both reflect as if there were a mirror. For the
Lorentzian spectrum in row 1, the ON configuration is reached when the photon-
emitter detuning is $\omega=\pm\Gamma/2$.

An unaddressed caveat for the protocol by Roulet et al. Roulet  _et al._
(2016) is that in order to realize the ON-configuration in the absence of a
PTE ($V=0$), the emitter must be tuned to the energy where the transmission
lineshape is the steepest, making the protocol highly susceptible to errors.
In case of a balanced PTE, the ON-configuration is reached when the emitter-
photon detuning is large compared to the waveguide coupling rate, $\Gamma$, or
when $k_{c}-\omega_{\mathrm{e}}=\tilde{\omega}_{\mathrm{e}}$ (see row 2 of
Fig. 3(a)). The OFF-configuration is reached for
$k_{c}-\omega_{\mathrm{e}}=\tilde{\omega}_{\mathrm{e}}\pm(2+\sqrt{2})\Gamma/8$.
Both the ON and OFF-configurations can thus be realized at frequencies where
the derivative of the transmission coefficient is (arbitrarily close to) 0,
thus suggesting that introducing a balanced PTE in the waveguide results in a
HOM switch that is more robust with respect to small perturbations in both the
emitter and photon energy. The price of achieving this, however, is that it
requires an emitter detuning of several linewidths. For a PTE strength of
$V=2$, the transmission spectrum is again Lorentzian, with ON-configurations
at points where the slope is steep (row 3 of Fig. 3(a)).

The analysis of the TLE can be carried over to the JC system: The OFF-
configuration is reached for a transmission coefficient of zero or unity,
whereas the ON-configuration is reached when the absolute squared transmission
coefficient equals $1/2$. For the balanced PTE, this again means that both the
ON-and OFF-configurations can be reached at points where the slope is zero
(see Fig. 3(b) and (c)).

###  V.2 Finite-width envelopes

As a final investigation, we consider the scattering of counter-propagating
photons with finite spectral width in Fano-resonance waveguides. As has been
shown for the case of a TLE in a waveguide with $V=0$, strong scattering-
induced photon correlations are possible for envelope widths comparable to the
waveguide coupling rate Nysteen  _et al._ (2015b). In this connection, it was
also suggested how the TLE can be used as a beam splitter for photons with
envelopes of finite width. For Gaussian envelopes centered around
$\omega_{\mathrm{e}}$, the scattering properties depend highly on the envelope
width $\sigma$ (see Fig. 4(a)). For spectrally narrow or broad envelopes, two
counter propagating photons will continue to be counter-propagating post
scattering, i.e. the OFF-configuration is achieved. The reason for the former
was explained in the previous subsection. The reason for the latter can most
easily be explained by considering the wide distribution in frequencies; the
photon is practically everywhere far-detuned from the emitter and thus does
not interact with it. For $\sigma=0.43\,\Gamma$, the probability of a HOM-like
interference is maximized. For this width the trade-off between having too
narrow and too broad spectral envelopes is minimized. The ON-configuration is
reached with an error probability (the probability of scattering into a
counter-propagating output state, $|LR\rangle$)), of $0.11$.

Figure 4: Scattering probabilities for two counter-propagating photons on a
TLE versus spectral width. The probability of counter-propagating output
photons are shown in green and the probability for co-propagating output
photons are shown in red. The full line shows the result for scattering off a
TLE, whereas the dashed line shows the result for scattering off a cavity,
i.e. without the bound state contribution in the $S$-matrix. The envelopes are
Gaussian with $k_{c}=\tilde{\omega}_{\mathrm{e}}$. The PTE strengths are (a)
$V=0$, (b) $V=2/(1+\sqrt{2})$ and (c) $V=2$.

In Fig. 4(b), we plot the scattering probabilities versus envelope width in
case of a balanced PTE, again for envelopes centered around
$\tilde{\omega_{\mathrm{e}}}$. In this case, the ON-configuration is reached
for spectrally narrow or broad envelopes because only the PTE contributes to
the scattering. The OFF configuration is maximized for $\sigma=0.36\,\Gamma$,
also with an error probability (the probability of detecting two co-
propagating photons) of $0.11$. As seen from Fig. 4(c), a PTE strength of
$V=2$ gives qualitatively the same results as having no PTE. We note that a
center frequency of $\tilde{\omega}_{\mathrm{e}}$ was chosen as this maximizes
the switching contrast (not shown here).

The dashed lines in Fig. 4 correspond to scattering off a cavity (system ii.
in Fig. 1(d)) in place of the TLE. This corresponds to setting the bound state
term in Eq. (30aeai) equal to zero, as can easily be derived using
$[a,a^{\dagger}]=\mathds{1}$ in place of
$[\sigma_{-},\sigma_{+}]=\mathds{1}-2\sigma_{+}\sigma_{-}$ for the two-photon
$S$-matrix derivation (see Sec. IV.3). Because there are no nonlinear
scattering effects for this system, the minimum error probability is $0.5$ in
all three cases.

Finally, we plot in Fig. 5 the two-photon input and output spectra for the
envelope widths that minimize the error probabilities, defined above. Ideally,
the input spectrum (fig. 5(a)) would be identical to the desired output
spectrum (row 1 and row 3 of Fig. 5(b) for $V=0$ and $V=2$, respectively, and
row 2 of Fig. 5(c) for $V=2/(1+\sqrt{2})$). We see, however, that the output
spectra accumulates correlations in the spectra due to wave mixing arising
from the nonlinear term in Eq. (30aeaj), resulting in elongated envelope
densities. We see that the Fano resonance effect does not alter the induced
correlations.

Figure 5: Two-photon envelope densities (color scale) vs. frequencies. The
input state (column (a)) is taken to be a counter-propagating two-photon state
traveling towards a TLE in a waveguide with a PTE. The input photons have
identical Gaussian envelopes with the center frequency
$k_{c}=\tilde{\omega}_{\mathrm{e}}$. In row 1 the PTE strength is $V=0$ and
the spectral width is $\sigma=0.43\Gamma$. In row 2 the PTE strength is
$V=2/(1+\sqrt{2})$ and the spectral width is $\sigma=0.36\Gamma$. In row 3 the
PTE strength is $V=2$ and the spectral width is $\sigma=0.21\Gamma$. Column
(b) shows the co-propagating output part ($LL$ or $RR$) and column (c) 3 shows
the counter-propagating output spectrum ($LR$).

##  VI Summary

In this work we have analyzed the effect of Fano interference on few-photon
scattering in waveguide geometries containing a partially transmitting
element. In Sec. I we briefly motivated the study of such waveguide
geometries, and in Sec. II we described the model Hamiltonian for a waveguide
with a partially transmitting element as introduced in Xu and Fan (2016). The
work presented in this paper contributes to the study of Fano waveguide
geometries in a quantum optical context by analyzing dynamical single-photon
scattering, scattering of photons with finite-width envelopes, and providing
analytical expressions for single and two-photon $S$-matrices for more
complicated waveguide geometries than studied previously. Below we give a
brief summary of our results presented in Sec. III to Sec. V.

In Sec. III we found analytical expressions for the single-excitation dynamics
for a two-level emitter in a waveguide with a partially transmitting element
in the mirror-symmetry line. We showed that the emitter lifetime increases
with the strength of the partially transmitting element, and that the emitted
photons are redshifted. Furthermore, we determined the conditions for which a
single-photon with Gaussian envelope maximizes the emitter excitation
probability. In the limit of a vanishing partially transmitting element, we
reproduce the results of e.g. Chen  _et al._ (2011) and Liao  _et al._ (2015).

In Sec. IV we established a way of calculating the single-photon $S$-matrix
for a general Fano-waveguide geometry including the effect of dissipation. We
also determined the two-photon S-matrix for a two-level emitter and a Jaynes-
Cummings system. In case of a TLE, we showed that the results agree with that
of Xu and Fan Xu and Fan (2016). In case of a Jaynes-Cummings system, we
obtain the same $S$-matrix coefficients as Shen and Fan (2009a) in the limit
of a vanishing partially transmitting element.

In Sec. V we discussed the switching properties of a Fano-waveguide geometry
and compared the results to that of a waveguide without a partially
transmitting element, as studied by Roulet et al. in Roulet  _et al._ (2016).
We showed that for quasi-monochromatic photons, the Fano interference effect
can be exploited to generate tunable Hong-Ou-Mandel interference between
counter-propagating photons in a way that is robust against small changes in
the emitter energy. For finite width-envelopes, we investigated the Fano
interference effect on the induced spectral correlations and found no
advantage compared to the case without a Fano interference.

##  Appendix A Single-excitation dynamics in a Fano-waveguide geometry coupled
to a TLE

The Hamiltonian for a TLE coupled to a linearly dispersive waveguide with a
PTE is given by (see Eq. (6))

$\displaystyle\begin{split}H&=\omega_{\mathrm{e}}\sigma_{+}\sigma_{-}+\int\differential{k}\,k\quantity(b_{R}^{\dagger}(k)b_{R}(k)+b_{L}^{\dagger}(k)b_{L}(k))+\int\differential{k}\,\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\quantity(\sigma_{-}\quantity[b_{R}^{\dagger}(k)+b_{L}^{\dagger}(k)]+\mathrm{h.c.})\\\
&\qquad+V\int\frac{\differential{k}}{\sqrt{2\pi}}\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}\,\quantity[b_{R}^{\dagger}(k)b_{L}(k^{\prime})+\mathrm{h.c.}].\end{split}$
(30aear)

The state-vector expansion in the single-excitation manifold is

$\displaystyle\ket{\psi(t)}=\chi(t)\sigma_{+}\ket{\varnothing}+\int\differential{k}\,\xi_{R}(k,t)b_{R}^{\dagger}(k)\ket{\varnothing}+\int\differential{k}\,\xi_{L}(k,t)b_{L}^{\dagger}(k)\ket{\varnothing}.$
(30aeas)

Inserting in the Schrödinger equation, we find the following equations of
motion:

$\displaystyle i\partial_{t}\chi(t)$
$\displaystyle=\omega_{e}\chi(t)+\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\int\differential{k}\,\quantity[\xi_{R}(k,t)+\xi_{L}(k,t)],$
(30aeata) $\displaystyle i\partial_{t}\xi_{R}(k,t)$
$\displaystyle=k\xi_{R}(k,t)+\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\chi(t)+\frac{V}{2\pi}\int\differential{k^{\prime}}\,\xi_{L}(k^{\prime},t),$
(30aeatb) $\displaystyle i\partial_{t}\xi_{L}(k,t)$
$\displaystyle=k\xi_{L}(k,t)+\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\chi(t)+\frac{V}{2\pi}\int\differential{k^{\prime}}\,\xi_{R}(k^{\prime},t).$
(30aeatc)

We define $\xi(k,t)=\xi_{R}(k,t)+\xi_{L}(k,t)$ and make the variable change
$\chi(t)=\chi^{\prime}(t)e^{-i\omega_{\mathrm{e}}t}$ and
$\xi(k,t)=\xi^{\prime}(k,t)e^{-ikt}$:

$\displaystyle i\partial_{t}\chi^{\prime}(t)$
$\displaystyle=\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}e^{i\omega_{\mathrm{e}}t}\int\differential{k}\,\xi^{\prime}(k,t)e^{-ikt},$
(30aeaua) $\displaystyle i\partial_{t}\xi^{\prime}(k,t)$
$\displaystyle=\frac{\sqrt{\Gamma}}{\sqrt{\pi}}\chi^{\prime}(t)e^{-i\quantity(\omega_{\mathrm{e}}-k)t}+\frac{V}{2\pi}\int\differential{k^{\prime}}\,\xi^{\prime}(k^{\prime},t)e^{i\quantity(k-k^{\prime})t}.$
(30aeaub)

We formally integrate Eq. (30aeaub) from $t_{i}$ to $t$ and then multiply by
$-ie^{-ikt}$:

$\displaystyle\begin{split}\xi^{\prime}(k,t)e^{-ikt}&=\xi^{\prime}(k,t_{i})e^{-ikt}-i\frac{\sqrt{\Gamma}}{\sqrt{\pi}}\int_{t_{i}}^{t}\differential{t^{\prime}}\,\chi^{\prime}(t^{\prime})e^{-ik\quantity(t-t^{\prime})}e^{-i\omega_{\mathrm{e}}t^{\prime}}-i\frac{V}{2\pi}\int_{t_{i}}^{t}\differential{t^{\prime}}\,\int\differential{k^{\prime}}\,\xi^{\prime}(k^{\prime},t^{\prime})e^{-ik^{\prime}t^{\prime}}e^{ik\quantity(t^{\prime}-t)}.\end{split}$
(30aeav)

We integrate over all $k$ and find

$\displaystyle\int\differential{k}\,\xi^{\prime}(k,t)e^{-ikt}=\quantity(1+\frac{iV}{2})^{-1}\int\differential{k}\,\xi^{\prime}(k,t_{i})e^{-ikt}-i\quantity(1+\frac{iV}{2})^{-1}\sqrt{\pi\Gamma}\chi^{\prime}(t)e^{-i\omega_{\mathrm{e}}t}.$
(30aeaw)

We insert above expression in Eq. (30aeaua)

$\displaystyle\partial_{t}\chi^{\prime}(t)$
$\displaystyle=-i\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\quantity(1+\frac{iV}{2})^{-1}\int\differential{k}\,\xi^{\prime}(k,t_{i})e^{-i\quantity(k-\omega_{\mathrm{e}})t}-\frac{\Gamma}{2}\quantity(1+\frac{iV}{2})^{-1}\chi^{\prime}(t).$
(30aeax)

After formal integration we find

$\displaystyle\chi(t)$
$\displaystyle=\chi(t_{i})e^{-\quantity[\frac{\tilde{\Gamma}}{2}+i\omega_{\mathrm{e}}]\quantity(t-t_{i})}-i\frac{\tilde{\Gamma}}{2\sqrt{\pi\Gamma}}\int\differential{k^{\prime}}\,\xi(t_{i},k^{\prime})e^{-ik^{\prime}\quantity(t-t_{i})}\frac{1-e^{-\quantity[\frac{\tilde{\Gamma}}{2}+i\quantity(\omega_{\mathrm{e}}-k^{\prime})]\quantity(t-t_{i})}}{\frac{\tilde{\Gamma}}{2}+i\quantity(\omega_{\mathrm{e}}-k^{\prime})},$
(30aeay)

where we have defined $\tilde{\Gamma}=\Gamma/\quantity(1+\frac{iV_{RL}}{2})$.
Next, we solve for $\xi(k,t)$. Although this is not technically a photon
envelope coefficient it is a helpful variable. From combining Eq. (30aeaua)
and Eq. (30aeaub) we find

$\displaystyle\partial_{t}\xi^{\prime}(k,t)$
$\displaystyle=-ie^{ikt}\quantity[\frac{\sqrt{\Gamma}}{\sqrt{\pi}}\quantity(1-\frac{V}{\Gamma}\omega_{\mathrm{e}})\chi(t)+i\frac{V}{\sqrt{\pi\Gamma}}\partial_{t}\chi(t)].$
(30aeaz)

We solve by formal integration

$\displaystyle\begin{split}\xi(t,k)&=\xi(t_{i},k)e^{-ik\quantity(t-t_{i})}-i\frac{\sqrt{\Gamma}}{\sqrt{\pi}}\quantity(1-\frac{V}{\Gamma}\omega_{\mathrm{e}})\int_{t_{i}}^{t}\differential{t^{\prime}}\,e^{ik\quantity(t^{\prime}-t)}\chi(t^{\prime})+\frac{V}{\sqrt{\pi\Gamma}}\int_{t_{i}}^{t}\differential{t^{\prime}}\,e^{ik\quantity(t^{\prime}-t)}\partial_{t^{\prime}}\chi(t^{\prime}).\end{split}$
(30aeba)

To solve for $\xi_{R}(k,t)$ we use that

$\displaystyle\xi_{L}^{\prime}(t,k)$
$\displaystyle=\xi_{L}^{\prime}(t_{i},k)-i\frac{\sqrt{\Gamma/2}}{\sqrt{2\pi}}\int_{t_{i}}^{t}\differential{t^{\prime}}\,\chi^{\prime}(t^{\prime})e^{-i\quantity(\omega_{\mathrm{e}}-k)t^{\prime}}-i\,\frac{V}{2\pi}\int_{t_{i}}^{t}\differential{t^{\prime}}\int\differential{k^{\prime}}\,\xi_{R}^{\prime}(t^{\prime},k^{\prime})e^{-i\quantity(k^{\prime}-k)t^{\prime}},$
(30aebb)

and insert in Eq. (30aeatb). With this we find

$\displaystyle\begin{split}\xi_{R}(t,k)&=\xi_{R}(t_{i},k)e^{-ik\quantity(t-t_{i})}-i\sqrt{\frac{\Gamma/2}{2\pi}}\frac{1-iV}{1-\frac{iV}{2}}\int_{t_{i}}^{t}\differential{t^{\prime}}\,e^{ik(t^{\prime}-t)}\chi(t^{\prime})\\\
&\quad-\frac{i}{2\pi}\frac{V}{1-i\frac{V}{2}}\int_{t_{i}}^{t}\differential{t^{\prime}}\,e^{ik(t^{\prime}-t)}\int\differential{k^{\prime}}\,\xi_{L}(t_{i},k^{\prime})e^{-ik^{\prime}\quantity(t^{\prime}-t_{i})}-\frac{1}{\pi}\frac{\quantity(V/2)^{2}}{1-i\frac{V}{2}}\int_{t_{i}}^{t}\differential{t^{\prime}}\,e^{ik(t^{\prime}-t)}\int\differential{k^{\prime}}\,\xi(t^{\prime},k^{\prime}).\end{split}$
(30aebc)

To summarize, the dynamical evolution of the excitation probability is
calculated from Eq. (30aeay). The dynamical evolution of the right-propagating
envelope is calculated using Eq. (30aebc), where $\xi$ is given in Eq.
(30aeba). To calculate the left-propagating envelope we simply use
$\xi_{L}(t,k)=\xi(t,k)-\xi_{R}(t,k)$. From this, all the state vector
coefficients in Eq. (30aeas) are known provided we know the initial conditions
$\chi(t_{i})$, $\xi_{R}(t_{i},k)$, and $\xi_{L}(t_{i},k)$ for some initial
time $t_{i}$.

##  Appendix B Single and two-photon $S$-matrix calculations

We start from the Hamiltonian in Eq. (29) for a cavity-emitter system. We
repeat the relevant terms here for easy reference:

$\displaystyle\begin{split}H&=\sum_{\mu}\int\differential{k}\,kb_{\mu}^{\dagger}(k)b_{\mu}(k)+\sum_{\mu\neq\nu}V_{\mu,\nu}\int\frac{\differential{k}}{\sqrt{2\pi}}\int\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}b_{\mu}^{\dagger}(k)b_{\nu}(k^{\prime})+H_{\mathrm{s}}+H_{\mathrm{s-wg}},\end{split}$
(30aebda) $\displaystyle H_{\mathrm{s}}$
$\displaystyle=k_{\mathrm{c}}a^{\dagger}a+\omega_{\mathrm{e}}\sigma_{+}\sigma_{-}+g\sigma_{-}a^{\dagger}+g^{*}\sigma_{+}a,$
(30aebdb) $\displaystyle H_{\mathrm{s-wg}}$
$\displaystyle=\sum_{\mu}\int\differential{k}\frac{\kappa_{\mathrm{c},\mu}a^{\dagger}+\kappa_{\mathrm{e},\mu}\sigma_{+}}{\sqrt{2\pi}}b_{\mu}(k)+\mathrm{h.c.}.$
(30aebdc)

In the limit $g=0$, and $\kappa_{\mathrm{c},\mu}=0$ for all $\mu$, this
reduces to the Hamiltonian in Eq. (21).

###  B.1 Input-output relations

From the Heisenberg equation of motion we find for the waveguide operators

$\displaystyle\partial_{t}b_{\mu}(k,t)$
$\displaystyle=-i\commutator{b_{\mu}(k)}{H}=-ikb_{\mu}(k,t)-i\frac{\kappa_{\mu,\mathrm{c}}}{\sqrt{2\pi}}a(t)-i\frac{\kappa_{\mu,\mathrm{e}}}{\sqrt{2\pi}}\sigma_{-}(t)-i\frac{\sum_{\nu\setminus\mu}V_{\mu\nu}\Phi_{\nu}(t)}{\sqrt{2\pi}},$
(30aebe)

where

$\displaystyle\Phi_{\nu}(t)=\frac{1}{\sqrt{2\pi}}\int\differential{k}\,b_{\nu}(k,t).$
(30aebf)

We rewrite Eq. (30aebe)

$\displaystyle\partial_{t}\quantity{e^{ikt}b_{\mu}(k,t)}$
$\displaystyle=-i\frac{\kappa_{\mu,\mathrm{c}}}{\sqrt{2\pi}}a(t)e^{ikt}-i\frac{\kappa_{\mu,\mathrm{e}}}{\sqrt{2\pi}}\sigma_{-}(t)e^{ikt}-i\frac{\sum_{\nu\setminus\mu}V_{\mu\nu}\Phi_{\nu}(t)}{\sqrt{2\pi}}e^{ikt}.$
(30aebg)

We formally integrate from time $t_{\mathrm{in}}$ to $t$ and multiply through
by $e^{-ikt}$

$\displaystyle\begin{split}b_{\mu}(k,t)-e^{ik\quantity(t_{\mathrm{in}}-t)}b_{\mu}(k,t_{\mathrm{in}})&=-i\int_{t_{\mathrm{in}}}^{t}\differential{t^{\prime}}\,\frac{\kappa_{\mu,\mathrm{c}}}{\sqrt{2\pi}}a(t^{\prime})e^{ik\quantity(t^{\prime}-t)}-i\int_{t_{\mathrm{in}}}^{t}\differential{t^{\prime}}\,\frac{\kappa_{\mu,\mathrm{e}}}{\sqrt{2\pi}}\sigma_{-}(t^{\prime})e^{ik\quantity(t^{\prime}-t)}\\\
&\quad-i\frac{\sum_{\nu\setminus\mu}V_{\mu\nu}}{\sqrt{2\pi}}\int_{t_{\mathrm{in}}}^{t}\differential{t^{\prime}}\,\Phi_{\nu}(t^{\prime})e^{ik\quantity(t^{\prime}-t)}.\end{split}$
(30aebh)

We integrate over all $k$ and divide through by $\sqrt{2\pi}$

$\displaystyle\begin{split}\Phi_{\mu}(t)-b_{\mathrm{in},\mu}(t)&=-i\frac{\kappa_{\mu,\mathrm{c}}}{2}a(t)-i\frac{\kappa_{\mu,\mathrm{e}}}{2}\sigma_{-}(t)-i\frac{\sum_{\nu\setminus\mu}V_{\mu\nu}\Phi_{\nu}(t)}{2}\end{split}$
(30aebi)

The factor of $1/2$ comes from integrating over ”half” the delta function. In
matrix form, this becomes

We note, that had we formally integrated from $t$ to $t_{\mathrm{out}}$
instead, we would have found

$\displaystyle\quantity(\mathds{1}-\frac{i}{2}\mathbf{V})\bm{\Phi}$
$\displaystyle=\mathbf{b}_{\mathrm{out}}(t)+\frac{i}{2}\bm{\kappa}\mathbf{c}.$
(30aebj)

From Eq. (LABEL:eq:app_inOut_phi_in) and Eq. (30aebj) we find the first input-
output relation

$\displaystyle\mathbf{b}_{\mathrm{out}}=\mathbf{C}\mathbf{b}_{\mathrm{out}}+\mathbf{D}\mathbf{c},$
(30aebk)

with

$\displaystyle\mathbf{C}=\quantity(\mathds{1}-\frac{i}{2}\mathbf{V})\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1},$
(30aebl)

and

$\displaystyle\mathbf{D}=-i\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\bm{\kappa}.$
(30aebm)

To find the second input-output relation, we use Heisenberg equation of motion
on the two system operators, $a$ and $\sigma_{-}$. This gives the following
equations:

and

$\displaystyle\begin{split}\partial_{t}\sigma_{-}&=-i\omega_{\mathrm{e}}\sigma_{-}-ig^{*}a-i\matrixquantity(0\hfil&1)\bm{\kappa}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\mathbf{b}_{\mathrm{in}}-\frac{1}{2}\matrixquantity(0&1)\bm{\kappa}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\bm{\kappa}\mathbf{c}\\\
&\qquad+2ig^{*}\sigma_{+}\sigma_{-}a+2i\sigma_{+}\sigma_{-}\matrixquantity(0\hfil&1)\bm{\kappa}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\mathbf{b}_{\mathrm{in}}+\sigma_{+}\sigma_{-}\matrixquantity(0&1)\bm{\kappa}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\bm{\kappa}\mathbf{c}\end{split}$
(30aebn)

In matrix form this becomes

$\displaystyle\dot{\mathbf{c}}(t)=\mathbf{A}\mathbf{c}(t)+\mathbf{B}\mathbf{b}_{\mathrm{in}}(t)+\matrixquantity(0\\\
\hat{f}(t)),\qquad\mathbf{c}(t)=\matrixquantity(a(t)\\\ \sigma_{-}(t)),$
(30aebo)

with

$\displaystyle\mathbf{B}$
$\displaystyle=-i\bm{\kappa}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1},$
(30aebp)

and

###  B.2 $S$-matrix calculations

We first Fourier transform our input-output equation in Eq. (30aebo) and
rearrange terms:

$\displaystyle\mathbf{c}(k)=i\quantity(k\mathds{1}-i\mathbf{A})^{-1}\mathbf{B}\mathbf{b}_{\mathrm{in}}(k)+i\quantity(k\mathds{1}-i\mathbf{A})^{-1}\matrixquantity(0\\\
\tilde{f}(k)),$ (30aebq)

where

$\displaystyle\begin{split}\tilde{f}(k)&=\quantity(2ig^{*}+\matrixquantity(0\hfil&1)\bm{\kappa}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\bm{\kappa}\matrixquantity(1\\\
0))\int\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}\sigma_{+}(k^{\prime})\sigma_{-}a(k+k^{\prime})\\\
&\quad+2i\matrixquantity(0\hfil&1)\bm{\kappa}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\int\frac{\differential{k^{\prime}}\differential{k^{\prime\prime}}}{2\pi}\sigma_{+}(k^{\prime})\sigma_{-}(k^{\prime\prime})\mathbf{b}_{\mathrm{in}}(k+k^{\prime}-k^{\prime\prime}).\end{split}$
(30aebr)

Using $\langle\varnothing|\hat{f}=0$, the single-photon $S$-matrix (Eq. (16))
can be expressed in terms of $A,\,B,\,C,$ and $D$ (see Eq. (30aeaf)):

$\displaystyle\mathbf{S}^{(1)}_{p;k}=\quantity[\mathbf{C}+i\mathbf{D}\quantity(\mathds{1}p-i\mathbf{A})^{-1}\mathbf{B}]\delta(p-k).$
(30aebs)

The two-photon $S$-matrix is not so straight forward. We start from the
definition and insert the input-output relations

$\displaystyle S_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}$
$\displaystyle={}_{\mu_{1}\mu_{2}}\innerproduct{p_{1}p_{2}^{-}}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}}=\matrixelement{\varnothing}{b_{\mathrm{out},\mu_{1}}(p_{1})b_{\mathrm{out},\mu_{2}}(p_{2})}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}}$
(30aebt)
$\displaystyle=\sum_{\nu^{\prime}}\int\differential{k^{\prime}}\matrixelement{\varnothing}{b_{\mathrm{out},\mu_{1}}(p_{1})}{k^{\prime+}}_{\nu^{\prime}}\,{}_{\nu^{\prime}}\matrixelement{k^{\prime+}}{b_{\mathrm{out},\mu_{2}}(p_{2})}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}}$
(30aebu)
$\displaystyle=S_{p_{1};k_{1}}^{\mu_{1};\nu_{1}}S_{p_{2};k_{2}}^{\mu_{2};\nu_{2}}+S_{p_{1};k_{2}}^{\mu_{1};\nu_{2}}S_{p_{2};k_{1}}^{\mu_{2};\nu_{1}}+iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}},$
(30aebv)

with

$\displaystyle
iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}=\sum_{\nu^{\prime}}S^{\mu_{1};\nu^{\prime}}(p_{1})\,{}_{\nu^{\prime}}\matrixelement{p_{1}^{-}}{i\frac{iD_{\mathrm{c},\mu_{2}}A_{\mathrm{ce}}\hat{f}(p_{2})+D_{\mathrm{e},\mu_{2}}\quantity(p_{2}-iA_{\mathrm{cc}})\hat{f}(p_{2})}{\quantity(p_{2}-iA_{\mathrm{cc}})\quantity(p_{2}-iA_{\mathrm{ee}})}}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}}.$
(30aebw)

So far, these calculations have been completely general. In the two
subsections below, we determine the matrix coefficients and $S$-matrices in
two special cases, first in case of a TLE coupled to a waveguide with a PTE,
and second in case of a JC system coupled to a waveguide with a PTE.

###  B.3 $S$-matrix for a TLE

For a TLE only, i.e. $g=0$ and $\kappa_{\mathrm{c},\mu}=0$ for all $\mu$, Eq.
(30aebr) simplifies to

$\displaystyle\begin{split}\tilde{f}(k)&=2i\bm{\kappa}_{\mathrm{e}}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}\int\frac{\differential{k^{\prime}}\differential{k^{\prime\prime}}}{2\pi}\sigma_{+}(k^{\prime})\sigma_{-}(k^{\prime\prime})\mathbf{b}_{\mathrm{in}}(k+k^{\prime}-k^{\prime\prime}).\end{split}$
(30aebx)

The term for the bound state contribution (Eq. (30aebw)) becomes

$\displaystyle
iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}=\sum_{\nu^{\prime}}S^{\mu_{1};\nu^{\prime}}(p_{1})\,{}_{\nu^{\prime}}\matrixelement{p_{1}^{-}}{\frac{iD_{\mathrm{e},\mu_{2}}\hat{f}(p_{2})}{\quantity(p_{2}-iA_{\mathrm{ee}})}}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}}.$
(30aeby)

We insert $\tilde{f}(k)$ and find

$\displaystyle
iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}=\frac{-i}{\pi}D_{\mathrm{e},\mu_{2}}\sum_{\nu^{\prime}}S^{\mu_{1};\nu^{\prime}}(p_{1})\mathcal{G}_{\nu^{\prime}}^{*}(p_{1})\mathcal{G}_{\nu_{2}}(p_{2})\quantity[\mathcal{G}_{\nu_{1}}(k_{1})+\mathcal{G}_{\nu_{1}}(k_{2})]\delta(p_{1}+p_{2}-k_{1}-k_{2}),$
(30aebz)

where we have defined

$\displaystyle\mathcal{G}_{\nu}(k)$
$\displaystyle=\frac{-i\quantity{\bm{\kappa}_{\mathrm{e}}^{\dagger}\quantity(\mathds{1}+\frac{i}{2}\mathbf{V})^{-1}}_{\nu}}{k-iA_{\mathrm{ee}}}$
(30aeca)

For a Fano-waveguide symmetrically coupled to a lossless TLE, the matrices and
matrix elements are

(30aecba) (30aecbb) (30aecbc)

With this, Eq. (30aebz) simplifies to

$\displaystyle
iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}=\frac{1}{\pi}\frac{\tilde{\Gamma}}{\sqrt{2\Gamma}}\mathcal{G}(p_{1})\mathcal{G}(p_{2})\quantity[\mathcal{G}(k_{1})+\mathcal{G}(k_{2})]\delta(p_{1}+p_{2}-k_{1}-k_{2}),$
(30aecbcc)

with

$\displaystyle\mathcal{G}(k)=\frac{\frac{\tilde{\Gamma}}{\sqrt{2\Gamma}}}{k-\omega_{\mathrm{e}}+i\frac{\tilde{\Gamma}}{2}}.$
(30aecbcd)

###  B.4 $S$-matrix for a JC system

Here, we consider the specific case of a Fano-waveguide symmetrically coupled
to a lossless JC system. In this case, the emitter only couples to the
waveguide via the cavity and not directly. We therefore take
$D_{\mathrm{e},\mu}$ to be zero for all $\mu$ and the term for the bound state
contribution (Eq. (30aebw)) becomes

$\displaystyle
iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}=\sum_{\nu^{\prime}}S^{\mu_{1};\nu^{\prime}}(p_{1})\,{}_{\nu^{\prime}}\matrixelement{p_{1}^{-}}{i\frac{iD_{\mathrm{c},\mu_{2}}A_{\mathrm{ce}}\hat{f}(p_{2})}{\quantity(p_{2}-iA_{\mathrm{cc}})\quantity(p_{2}-iA_{\mathrm{ee}})}}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}},$
(30aecbce)

where

$\displaystyle\begin{split}\tilde{f}(k)&=2ig^{*}\int\frac{\differential{k^{\prime}}}{\sqrt{2\pi}}\sigma_{+}(k^{\prime})\sigma_{-}a(k+k^{\prime}).\end{split}$
(30aecbcf)

Inserting this and assuming $g=g^{*}$, we find

$\displaystyle
iT_{p_{1}p_{2};k_{1}k_{2}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}=\frac{2ig}{\sqrt{2\pi}}\sum_{\nu^{\prime}}S^{\mu_{1};\nu^{\prime}}(p_{1})\mathcal{G}_{\mathrm{e}}^{*}(p_{1})\mathcal{G}_{\mathrm{e}}(p_{2})\matrixelement{\varnothing}{\sigma_{-}a(p_{1}+p_{2})}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}},$
(30aecbcg)

where we have defined

$\displaystyle\mathcal{G}_{\mathrm{e}}(k)$
$\displaystyle=\frac{\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}g}{\quantity[k-\omega_{\mathrm{c}}+i\frac{\tilde{\Gamma}}{2}]\quantity[k-\omega_{\mathrm{e}}]-g^{2}}.$
(30aecbch)

We also define

$\displaystyle\mathcal{G}_{\mathrm{c}}(k)=\frac{\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}\quantity(k-\omega_{\mathrm{e}})}{\quantity[k-\omega_{\mathrm{c}}+i\frac{\tilde{\Gamma}}{2}]\quantity[k-\omega_{\mathrm{e}}]-g^{2}}.$
(30aecbci)

Note that we here consider $\sigma_{-}a(k)$ to be a single combined operator.
In order to determine the matrix element
$\langle\varnothing|\sigma_{-}a(p_{1}+p_{2})|k_{1}k_{2}^{+}\rangle_{\nu_{1}\nu_{2}}$,
we need a closed set of equations extending beyond the input-output equations.
We find

$\displaystyle\partial_{t}{\sigma_{-}a(t)}=\quantity(-i\omega_{\mathrm{e}}\sigma_{-}a(t)-igaa(t)+f(t)a(t))+\quantity(\quantity(-i\omega_{\mathrm{c}}-\frac{\tilde{\Gamma}}{2})\sigma_{-}a(t)-i\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}\sigma_{-}(t)\sum_{\mu=L,R}b_{\mathrm{in},\mu}(t)).$
(30aecbcj)

We see that an equation of motion for the combined operator $aa$ is also
needed:

$\displaystyle\partial_{t}a^{2}(t)=2\quantity(-i\omega_{\mathrm{c}}-\frac{\tilde{\Gamma}}{2})aa(t)-2ig\sigma_{-}a(t)-2i\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}a(t)\sum_{\mu=L,R}b_{\mathrm{in},\mu}(t).$
(30aecbck)

After Fourier transforming and rearranging terms, we find in matrix-vector
form

We again use that $\langle\varnothing|\hat{f}=0$ (and therefore also
$\langle\varnothing|\tilde{fa}=0$) and find

$\displaystyle\matrixelement{\varnothing}{\sigma_{-}a(p)}{k_{1}k_{2}^{+}}_{\nu_{1}\nu_{2}}$
$\displaystyle=-\frac{\frac{\tilde{\Gamma}/2}{\sqrt{\Gamma/2}}}{\sqrt{2\pi}}\frac{\quantity[p-2\omega_{\mathrm{c}}+i\tilde{\Gamma}]\quantity(\mathcal{G}_{\mathrm{e}}(k_{1})+\mathcal{G}_{\mathrm{e}}(k_{2}))+2g\quantity(\mathcal{G}_{\mathrm{c}}(k_{1})+\mathcal{G}_{\mathrm{c}}(k_{2}))}{\quantity[p-\quantity(\omega_{\mathrm{c}}+\omega_{\mathrm{e}}+i\tilde{\Gamma}/2)]\quantity[p-2\omega_{\mathrm{c}}+i\tilde{\Gamma}]-2g^{2}}\delta(p-k_{1}-k_{2}).$
(30aecbcl)

If we insert this into Eq. (30aecbcg) and simplify, we obtain the expression
in Eq. (30aeal).

## Acknowledgments

K.B.J, J.I.-S., M.H., and J.M. acknowledge funding from the Danish Council for
Independent Research (DFF-4181-00416). M.H. acknowledges support from the
Villum Foundation. J.M. acknowledges support from the European Research
Counsil (ERC) under the European Union Horizon 2020 Research and Innovation
Programme (Grant no. 834410 Fano). Furthermore, the authors acknowledge
helpful discussions with Yi Yu.

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