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††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Single-photon transport in a whispering-gallery mode microresonator
directionally coupled with a two-level quantum emitter

Jiangshan Tang (唐江山)  College of Engineering and Applied Sciences, National
Laboratory of Solid State Microstructures, and Collaborative Innovation Center
of Advanced Microstructures, Nanjing University, Nanjing 210023, China  School
of Physics, Nanjing University, Nanjing 210023, China  Lei Tang (唐磊)  College
of Engineering and Applied Sciences, National Laboratory of Solid State
Microstructures, and Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210023, China  Keyu Xia (夏可宇)
<EMAIL_ADDRESS>College of Engineering and Applied Sciences, National
Laboratory of Solid State Microstructures, and Collaborative Innovation Center
of Advanced Microstructures, Nanjing University, Nanjing 210023, China  School
of Physics, Nanjing University, Nanjing 210023, China

###### Abstract

We investigate the single-photon transport problem in the system of a
Whispering-Gallery mode microresonator directionally coupled with a two-level
quantum emitter (QE). This QE-microresonator coupling system can usually be
studied by cavity quantum electrodynamics and the single-photon transport
methods. However, we find that if we treat a two-level QE as a single-photon
phase-amplitude modulator, we can also deal with such systems using the
transfer matrix method. Further, in theory, we prove that these three methods
are equivalent. The corresponding relations of respective parameters among
these approaches are precisely deduced. Our work can be extended to a
multiple-resonator system interacting with two-level QEs in a chiral way.
Therefore, the transfer matrix method may provide a convenient and intuitive
form for exploring more complex chiral QE-resonator interaction systems.

##  I Introduction

The interaction of light and matter at the single-quantum level is the basis
of essential physics of many phenomena and applications [1], which has been
extensively explored in various quantum systems, such as the single-mode
waveguide coupling to quantum emitters (QEs) [2, 3, 4, 5, 6, 7], the Fabry-
Perot cavity [8, 9, 10, 11, 12] and the Whispering-Gallery mode (WGM)
microresonator [13, 14, 15, 16, 17]. In recent years, an emerging field of
research called “chiral quantum optics” [18], in which the light-matter
coupling is direction-dependent, has received extensive attention in the field
of quantum nonreciprocity [19] and exhibited chiral interactions of light and
QEs [20, 21, 22, 23, 24, 25, 26].

To realize the chiral light-matter interaction, an external magnetic field is
usually required to induce the magneto-optical effect [27] or make the energy
of the QE undergo a Zeeman splitting [28]. It greatly limits the
miniaturization and integration of single-photon devices. Recently, an all-
optical approach, based on valley-selective response in transition metal
dichalcogenides, has been described to overcome the limitations of magnetic
materials [29]. Towards on-chip chiral single-photon interfaces, non-magnetic
schemes have been proposed based on a WGM microresonator chirally coupled with
a two-level QE [20, 30, 19, 18, 23].

Theoretically, the single-photon transport problem in the system of a WGM
microresonator coupled to a waveguide can be solved by methods such as the
cavity quantum electrodynamics (CQED) [31, 32, 33, 34, 35, 36], and the
single-photon transport (SPT) [37, 38, 4, 15, 39], and the transport matrix
(TM) [40]. Further, the WGM microresonator system containing a two-level QE
has also been discussed under the framework of CQED and SPT theory [41, 13, 4,
15], even extending to the chiral interactions [20, 19, 18, 23]. However, how
to deal with the chiral interaction of a WGM microresonator with a two-level
QE using the TM method is still not available. And the inner link among these
three methods also remain to be revealed.

In this paper, we demonstrate that the CQED, the SPT, and the TM methods are
equivalent in dealing with the single-photon transport problem in a chiral QE-
microresonator system. In the TM method, the effect of a two-level QE can be
regarded as a single-photon phase-amplitude modulator. By introducing a
nonlinear coefficient into the transfer relation, we can use the TM method to
solve the single-photon transmission. Furthermore, we present the
corresponding relations for the parameters among the three methods.

##  II System and model

Figure 1: Schematic of a chiral QE-microresonator system. A two-level QE is
coupled to a WGM microresonator in a chiral way to form the QE-microresonator
system. A waveguide is side coupled to the microresonator as input and output
ports. A scatterer on the microresonator is considered to introduce
backscattering. The arrows represent the propagating direction of a single
photon for an input to the port 1 (green) or port 2 (red).

Schematic diagram of the system studied in this paper is shown in Fig. 1. A
WGM microresonator, which can be made with various material platforms, such as
silicon oxynitride [42, 43], ploymers [44], or silicon on insulator [45, 46,
47], simultaneously couples to a two-level QE and a waveguide to form a QE-
microresonator system. A microresonator, with a radius $R$, supports two
optical WGMs propagating in either clockwise (CW) or counterclockwise (CCW)
direction. The evanescent fields of the WGMs are almost perfectly circular
polarized with its polarization locked to the propagation direction and thus
possesse optical chirality. In practice, backscattering usually needs to be
considered due to the surface roughness of microresonator. In this paper, we
treat the backscattering as a scatterer [48], as shown in the black mark in
Fig. 1.

In our system, a two-level QE is chirally coupled to the microresonator. The
chiral interaction between the QE and the evanescent field of WGM can be
achieved by initializing QE in a specific spin ground state or shifting the
transition energy with a polarization-selective optical Stark effect [20, 30,
19, 18, 10, 23, 7]. Here, we assume that the QE only interacts with the CCW
WGM, because the evanescent field of the CCW mode is $\sigma^{+}$ polarized
and only the $\sigma^{+}$-polarized transition of the prepared QE is allowed.
Note that the QE-microresonator coupling strength is dependent on the
propagating direction of light in our chiral system. In the forward case, the
incident light from port $1$ excites the CCW mode in the microresonator, and
it strongly couples with the QE. In the backward (port-$2$ incident) case, the
CW mode is decoupled from the QE and thus the coupling strength $g$ is
negligible (i.e., $g\approx 0$). One can use a precisely positioned atom [19,
49], a quantum dot (QD) [50, 22, 28, 51], or a nanopillar covered by
monolayers [52, 53, 54] to construct the two-level QE.

Below, we first provide the CQED, the SPT, and the TM methods to solve the
response of the system. Then we show these three methods are equivalent if we
treat the two-level QE as a single-photon phase-amplitude modulator. We only
discuss the forward case ($g\neq 0$) in detail, and the backward case
corresponds to the system without a QE ($g=0$).

###  II.1 Cavity Quantum Electrodynamics Method

In this section, we discuss the CQED method to solve our model. For a coupled
atom-microresonator system, it has been analyzed in Ref. [13]. Here, we
discuss the chiral coupling using the same approach. We consider a two-level
QE with transition frequency $\omega_{\text{qe}}$ is coupled to the CCW mode
and decoupled to the opposite mode. The two degenerate WGMs, with same
resonant frequency $\Omega$ and dissipation $\kappa_{\text{tol}}$, are assumed
to be coupled with each other in a strength $h$ due to the scatterer. Here, we
divide the dissipation $\kappa_{\text{tol}}$ into two parts, the intrinsic
decay rate of $\kappa_{\text{in}}$ and the external loss of
$\kappa_{\text{ex}}$, satisfying
$\kappa_{\text{tol}}=\kappa_{\text{in}}+\kappa_{\text{ex}}$. A weak coherent
field of frequency $\omega$ with a amplitude $\alpha_{\text{in}}$ drives the
CCW mode $a$. To a good single-photon approximation, $\alpha_{\text{in}}\ll
1$. In a frame rotating at the frequency $\omega$, the Hamiltonian of our
system can be obtained [55]

$\displaystyle H=$
$\displaystyle-\Delta_{1}a^{\dagger}a-\Delta_{2}\sigma^{+}\sigma^{-}-\Delta_{1}b^{\dagger}b+i\sqrt{2\kappa_{\text{ex}}}\alpha_{\text{in}}\left(a^{\dagger}-a\right)$
(1)
$\displaystyle+g\left(a^{\dagger}\sigma^{-}+\sigma^{+}a\right)+h\left(a^{\dagger}b+b^{\dagger}a\right)\;,$

where $g$ represents the coupling strength between the CCW mode and the QE.
$\Delta_{1}=\omega-\Omega$ and $\Delta_{2}=\omega-\omega_{\text{qe}}$ are the
detunings. $b$ is the annihilation operator of the CW mode. $\sigma^{\pm}$ are
the raising and lowering operators describing the two-level QE. It is worth
noting that if we consider the coupling of two microresonators instead of the
scatterer, the Hamiltonian has the same form as Eq. (1). In this case, $h$
describes the coupling strength between the two microresonators. Introducing
the dissipation of the QE, $\gamma$, the evolution of the system can be found
by solving the master equation,

$\displaystyle\dot{\rho}=$
$\displaystyle-\mathrm{i}[H,\rho]+\kappa_{\text{tol}}\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right)$ (2)
$\displaystyle+\kappa_{\text{tol}}\left(2b\rho
b^{\dagger}-b^{\dagger}b\rho-\rho b^{\dagger}b\right)$
$\displaystyle+\gamma\left(2\sigma^{-}\rho\sigma^{+}-\sigma^{+}\sigma^{-}\rho-\rho\sigma^{+}\sigma^{-}\right)\;,$

where $\rho$ is the density operator. From Eq. (2), we can derive the
equations of motion,

$\displaystyle\dot{a}=i\tilde{\Delta}_{1}a+\alpha_{\text{in}}\sqrt{2\kappa_{\text{ex}}}-ig\sigma^{-}-ihb\;,$
(3a)
$\displaystyle\dot{\sigma}^{-}=i\tilde{\Delta}_{2}\sigma^{-}+ig\sigma_{z}a\;,$
(3b) $\displaystyle\dot{b}=i\tilde{\Delta}_{1}b-iha\;,$ (3c)

and have the steady-state solution

$a=\frac{i\alpha_{\text{in}}\sqrt{2\kappa_{\text{ex}}}\tilde{\Delta}_{1}\tilde{\Delta}_{2}}{\tilde{\Delta}_{1}\left(\tilde{\Delta}_{1}\tilde{\Delta}_{2}+\sigma_{z}g^{2}\right)-\tilde{\Delta}_{2}h^{2}}\;,$
(4)

where, $\sigma_{z}=\sigma^{+}\sigma^{-}-\sigma^{-}\sigma^{+}$,
$\tilde{\Delta}_{1}=\Delta_{1}+i\kappa_{\text{tol}}$ and
$\tilde{\Delta}_{2}=\Delta_{2}+i\gamma$. For simplicity, we have omitted the
notation $\langle\cdots\rangle$. According to the input-output relation,
$a_{\text{out}}=\alpha_{in}-\sqrt{2\kappa_{\text{ex}}}a$, we can get the
transmission amplitude in the port 2:

$t_{\omega}=\frac{\tilde{\Delta}_{1}\left[\left(\Delta_{1}+i\kappa_{\text{in}}-i\kappa_{\text{ex}}\right)\tilde{\Delta}_{2}+\sigma_{z}g^{2}\right]-\tilde{\Delta}_{2}h^{2}}{\tilde{\Delta}_{1}\left(\tilde{\Delta}_{1}\tilde{\Delta}_{2}+\sigma_{z}g^{2}\right)-\tilde{\Delta}_{2}h^{2}}$
(5)

The transmission of port 2 can be obtained from $T=|t_{\omega}|^{2}$.

Moreover, the fully quantum dynamics of the system can be solved by a
numerical solution to the master Eq. (2) using truncated space of photon
number for the WGMs.

###  II.2 Single-Photon Transport Method

Hereafter, we consider only a single photon in our system. Based on the
single-photon transport theory [37, 4, 15], our previous work has given a
transmission amplitude for the system interesting in this paper, see Eq. (15)
in Ref. [23]. Since the QE is decoupled to the CW mode in our system, the form
of the transmission amplitude reads:

$t_{\omega}=\frac{\tilde{\Delta}_{1}\left[\left(\Delta_{1}+i\kappa_{\text{in}}-i\kappa_{\text{ex}}\right)\tilde{\Delta}_{2}-g^{2}\right]-\tilde{\Delta}_{2}h^{2}}{\tilde{\Delta}_{1}\left(\tilde{\Delta}_{1}\tilde{\Delta}_{2}-g^{2}\right)-\tilde{\Delta}_{2}h^{2}}\;.$
(6)

If we consider $\sigma_{z}=-1$ in the CQED method, that is, the weak probe
field approximation [55], we can find that the Eq. (5) and the Eq. (6) are
equivalent. In the Sec. II.4, we will verify that the TM method is consistent
with the SPT method.

###  II.3 Transport Matrix Model

Next, we study the chiral QE-microresonator system using the TM method. Under
the notation in Fig. 1, the coupling relation between the waveguide and the
microresonator can be written as

$\left\\{\begin{array}[]{ l }{a_{1}=t^{*}b_{1}-\kappa^{*}a_{0}}\\\
{b_{0}=ta_{0}+\kappa
b_{1}}\end{array}\;,\quad\left\\{\begin{array}[]{l}c_{1}=t^{*}d_{1}-\kappa^{*}c_{0}\\\
d_{0}=tc_{0}+\kappa d_{1}\end{array}\right.\right.\;,$ (7)

where $t$ and $\kappa$ are the transmission and the coupling coefficients, and
$|t|^{2}+|\kappa|^{2}=1$ for lossless coupling. Written in a matrix form:

$\left(\begin{array}[]{l}a_{0}\\\ b_{0}\\\ c_{0}\\\
d_{0}\end{array}\right)=\frac{1}{\kappa^{*}}\left(\begin{array}[]{cccc}-1&t^{*}&0&0\\\
-t&1&0&0\\\ 0&0&-1&t^{*}\\\
0&0&-t&1\end{array}\right)\left(\begin{array}[]{l}a_{1}\\\ b_{1}\\\ c_{1}\\\
d_{1}\end{array}\right)\equiv M_{\text{cpl}}\left(\begin{array}[]{l}a_{1}\\\
b_{1}\\\ c_{1}\\\ d_{1}\end{array}\right).$ (8)

The size of the QE and the scatterer is much smaller than the structure of
microresonator, so theoretically they can be treated as particles. Without
loss of generality, we assume that the QE and the scatterer are in the same
position, see Fig. 1. Thus, the coupling points of the waveguide, the QE and
the microresonator divide the microresonator into two parts with lengths
$L_{1}$ and $L_{2}$, satisfying $L_{1}+L_{2}=2\pi R$. The field component
notations are shown in Fig. 1. When a single photon propagates around the
microresonator, it will accumulate propagation phases $\theta_{j}=\beta
L_{j}$, and may attenuate with loss $\alpha_{j}(L_{j})$ ($j=1,2$) [40]. We
take $\theta=\theta_{1}+\theta_{2}$ and $\alpha=\alpha_{1}\alpha_{2}$. The
factor $\beta$ is the propagation constant in the microresonator as given by
$\beta=n_{\text{eff}}\omega/c$, where $n_{\text{eff}}$ is the effective
refractive index and $\omega$ is the frequency. Thus, we have the transfer
relation

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\leavevmode\nobreak\ \left(\begin{array}[]{l}a_{1}\\\ b_{1}\\\ c_{1}\\\
d_{1}\end{array}\right)=M_{\text{pro}}\left(\begin{array}[]{l}a_{1}^{\prime}\\\
b_{1}^{\prime}\\\ c_{1}^{\prime}\\\ d_{1}^{\prime}\end{array}\right)\;,$ (9i)
$\displaystyle
M_{\text{pro}}=\left(\begin{array}[]{cccc}\alpha_{1}^{-1}e^{-i\theta_{1}}&0&0&0\\\
0&\alpha_{2}e^{i\theta_{2}}&0&0\\\ 0&0&\alpha_{2}^{-1}e^{-i\theta_{2}}&0\\\
0&0&0&\alpha_{1}e^{i\theta_{1}}\end{array}\right)\left(\begin{array}[]{c}a_{1}^{\prime}\\\
b_{1}^{\prime}\\\ c_{1}^{\prime}\\\ d_{1}^{\prime}\end{array}\right)\;.$ (9r)

We refer to $M_{\text{cpl}}$ and $M_{\text{pro}}$ as coupling and propagation
matrices. Combining Eqs. (8) and (9), we obtain the transfer matrix as

$\left(\begin{array}[]{l}a_{0}\\\ b_{0}\\\ c_{0}\\\
d_{0}\end{array}\right)=M_{\text{cpl}}M_{\text{pro}}\left(\begin{array}[]{l}a_{1}^{\prime}\\\
b_{1}^{\prime}\\\ c_{1}^{\prime}\\\ d_{1}^{\prime}\end{array}\right)\;.$ (10)

We consider a single input of port 1 ($c_{0}=0$). It excites the CCW-direction
WGM. In the following, we will discuss four cases:
1\. No two-level QE and No scatterer

We first consider the case without two-level QEs and scatterers, the transfer
relation of the field amplitudes in the microresonator can be directly
obtained, $a_{1}^{{}^{\prime}}$=$b_{1}^{{}^{\prime}}$ and
$c_{1}^{{}^{\prime}}$=$d_{1}^{{}^{\prime}}$. In the absence of scatterers, the
CCW and CW modes are decoupled. We can get the transmission amplitude in the
port 2 [40],

$t_{\omega}=\frac{b_{0}}{a_{0}}=\frac{-t+\alpha e^{i\theta}}{-1+\alpha
t^{*}e^{i\theta}}\;.$ (11)

2\. No two-level QE and Consider scatterer

In this case, we consider the effect of the scatterer in the microresonator.
The relation between the amplitudes can be written as
$b_{1}^{{}^{\prime}}=t_{s}a_{1}^{{}^{\prime}}+r_{s}c_{1}^{{}^{\prime}}$ and
$d_{1}^{{}^{\prime}}=t_{s}c_{1}^{{}^{\prime}}+r_{s}a_{1}^{{}^{\prime}}$.
$t_{s}$/$r_{s}$ are the transmission/reflection coefficients introduced by the
scatterer, and they satisfy $|t_{s}|^{2}+|r_{s}|^{2}=1$ when the dissipation
of the scatterer is neglected. The two WGMs are coupled to each other in this
case. We assume the scatterer is weak, thus we can write $t_{s}/r_{s}$ in the
following form [48]:

$t_{s}=\text{cos}\epsilon\approx 1-\frac{\epsilon^{2}}{2},\quad
r_{s}=i\text{sin}\epsilon\approx i\epsilon\;.$ (12)

Then we have the transmission amplitude in the port 2,

$t_{\omega}=\frac{b_{0}}{a_{0}}=\frac{-t+\alpha
e^{i\theta}\frac{t_{s}-t^{*}\alpha e^{i\theta}}{1-t_{s}t^{*}\alpha
e^{i\theta}}}{-1+\alpha t^{*}e^{i\theta}\frac{t_{s}-t^{*}\alpha
e^{i\theta}}{1-t_{s}t^{*}\alpha e^{i\theta}}}\;.$ (13)

3\. Single two-level QE and No scatterer

Here, we study the effect of a two-level QE directionally coupled to a
microresonator. Because the QE is in a specific spin ground state or the
polarization-selective energy level transition, the coupling of the QE and the
evanescent field on the microresonator is direction-dependent. The reflection
of single-photon propagation will vanish due to such chiral QE-light
interaction [20, 23]. In this case, the single photon will not excite the CW
mode, leading to the decoupling between the CCW and CW modes. We assume the
single photon through the two-level QE with a transmission coefficient
$t_{\text{qe}}$, i.e., $b_{1}^{{}^{\prime}}=t_{\text{qe}}a_{1}^{{}^{\prime}}$
and $d_{1}^{{}^{\prime}}=c_{1}^{{}^{\prime}}$, such that

$t_{\omega}=\frac{b_{0}}{a_{0}}=\frac{-t+\alpha
e^{i\theta}t_{\text{qe}}}{-1+\alpha t^{*}e^{i\theta}t_{\text{qe}}}\;.$ (14)

The specific form of $t_{\text{qe}}$ will be discussed below.

4\. Single two-level QE and Consider the scatterer

Combining with the above discussions, we can get the transfer relation of the
field amplitudes, with considering both a two-level QE directionally coupled
to the microresonator and a scatterer,
$b_{1}^{{}^{\prime}}=t_{\text{qe}}\left(t_{s}a_{1}^{{}^{\prime}}+r_{s}c_{1}^{{}^{\prime}}\right)$
and $d_{1}^{{}^{\prime}}=t_{s}c_{1}^{{}^{\prime}}+r_{s}a_{1}^{{}^{\prime}}$.
The transmission amplitude can be calculated as

$t_{\omega}=\frac{b_{0}}{a_{0}}=\frac{-t+\alpha
e^{i\theta}t_{\text{qe}}\frac{t_{s}-t^{*}\alpha
e^{i\theta}}{1-t_{s}t^{*}\alpha e^{i\theta}}}{-1+\alpha
t^{*}e^{i\theta}t_{\text{qe}}\frac{t_{s}-t^{*}\alpha
e^{i\theta}}{1-t_{s}t^{*}\alpha e^{i\theta}}}\;.$ (15)

###  II.4 Single-Photon Phase-amplitude Modulator

We define the round-trip time of microresonator, $\tau_{\text{rt}}=2\pi
Rn_{\text{eff}}/c$, that a photon needs to make a round trip in the
microresonator of length $2\pi R$. It is the inverse of the free spectral
range $\mathcal{F}$, i.e., $\tau_{\text{rt}}=1/\mathcal{F}$ [56]. Since
$\mathcal{F}\gg 1$ for a microresonator, $\tau_{\text{rt}}$ is a small amount.
We have
$\text{exp}(i\theta)=\text{exp}\left[i\left(\omega-\Omega\right)\tau_{\text{rt}}\right]\approx
1+i\Delta_{1}\tau_{\text{rt}}$. For a single photon having travelled a round
trip in the microresonator, we can get $a_{1}(\tau_{\text{rt}})=\alpha
t^{*}a_{1}(0)$ from the transfer relation. The circulating power meets
$|a_{1}(\tau_{\text{rt}})|^{2}=\alpha^{2}t^{2}|a_{1}(0)|^{2}$. On the other
hand, we have
$|a_{1}(\tau_{\text{rt}})|^{2}=\text{exp}(-2\kappa_{\text{tol}}\tau_{\text{rt}})|a_{1}(0)|^{2}$,
from the dissipation of the microresonator. Hence, we obtain

$\displaystyle\alpha=e^{-\kappa_{\text{in}}\tau_{\text{rt}}}\approx
1-\kappa_{\text{in}}\tau_{\text{rt}}\;,$ (16a) $\displaystyle
t=e^{-\kappa_{\text{ex}}\tau_{\text{rt}}}\approx
1-\kappa_{\text{ex}}\tau_{\text{rt}}\;.$ (16b)

Because the size of the two-level QE is much smaller than the bend structure
of the microresonator, the interaction between the evanescent field and the QE
can be equivalent to that of a waveguide directionally coupled with a two-
level QE [20], with a transmission coefficient

$t_{\text{qe}}=\frac{\omega-\omega_{\text{qe}}+i\left(\gamma-\Gamma\right)}{\omega-\omega_{\text{qe}}+i\left(\gamma+\Gamma\right)}.$
(17)

$\Gamma$ is the decay rate from the QE into the microresonator. Therefore,
substituting the Eqs. (12), (16) and (17) into Eq. (15), and ignoring the
second-order small quantity, we have

$\displaystyle t_{\omega}=\frac{-t+\alpha
e^{i\theta}t_{\text{qe}}\frac{t_{s}-t^{*}\alpha
e^{i\theta}}{1-t_{s}t^{*}\alpha e^{i\theta}}}{-1+\alpha
t^{*}e^{i\theta}t_{\text{qe}}\frac{t_{s}-t^{*}\alpha
e^{i\theta}}{1-t_{s}t^{*}\alpha e^{i\theta}}}$ (18) $\displaystyle\approx$
$\displaystyle\frac{\kappa_{\text{ex}}\tau_{\text{rt}}-1+\left(1-\kappa_{\text{in}}\tau_{\text{rt}}+i\Delta_{1}\tau_{\text{rt}}\right)\left[\left(1-\frac{2i\Gamma}{\tilde{\Delta}_{2}+i\Gamma}\right)\left(1+\frac{\varepsilon^{2}}{i\tilde{\Delta}_{1}\tau_{\text{rt}}+\varepsilon^{2}/2}\right)\right]}{-1+\left(1+i\tilde{\Delta}_{1}\tau_{\text{rt}}\right)\left[\left(1-\frac{2i\Gamma}{\tilde{\Delta}_{2}+i\Gamma}\right)\left(1+\frac{\varepsilon^{2}}{i\tilde{\Delta}_{1}\tau_{\text{rt}}+\varepsilon^{2}/2}\right)\right]}$
$\displaystyle\approx$
$\displaystyle\frac{\tilde{\Delta}_{1}\left[\left(\Delta_{1}+i\kappa_{\text{in}}-i\kappa_{\text{ex}}\right)\tilde{\Delta}_{2}-\Gamma\left(2/\tau_{\text{rt}}-\kappa_{\text{tol}}\right)\right]-\tilde{\Delta}_{2}\frac{\varepsilon^{2}}{\tau_{\text{rt}}^{2}}}{\tilde{\Delta}_{1}\left[\tilde{\Delta}_{1}\tilde{\Delta}_{2}-\Gamma\left(2/\tau_{\text{rt}}-\kappa_{\text{tol}}\right)\right]-\tilde{\Delta}_{2}\frac{\varepsilon^{2}}{\tau_{\text{rt}}^{2}}}$
$\displaystyle=$
$\displaystyle\frac{\tilde{\Delta}_{1}\left[\left(\Delta_{1}+i\kappa_{\text{in}}-i\kappa_{\text{ex}}\right)\tilde{\Delta}_{2}-\Gamma\left(2\mathcal{F}-\kappa_{\text{tol}}\right)\right]-\tilde{\Delta}_{2}\left(\varepsilon\times\mathcal{F}\right)^{2}}{\tilde{\Delta}_{1}\left[\tilde{\Delta}_{1}\tilde{\Delta}_{2}-\Gamma\left(2\mathcal{F}-\kappa_{\text{tol}}\right)\right]-\tilde{\Delta}_{2}\left(\varepsilon\times\mathcal{F}\right)^{2}}.$

Comparing Eq. (18) with Eq. (6), we can find that if we take

$\Gamma\left(2\mathcal{F}-\kappa_{\text{tol}}\right)=g^{2},\quad\varepsilon\times\mathcal{F}=h\;,$
(19)

then the TM method and the SPT method are consistent. This also proves that
the assumption of Eq. (17) is valid.

Therefore, the two-level QE can be treated as a single-photon phase-amplitude
modulator, that causes a change in propagation phase and an amplitude
modulation when the single photon passes through it. We divide Eq. (17) into
two parts:
$t_{\text{qe}}=\exp\left(i\varphi_{\text{pha}}\right)\exp\left(-\varphi_{\text{dis}}\right)$,
where
$\exp\left(i\varphi_{\text{pha}}\right)=\text{arg}\left(t_{\text{qe}}\right)$
represents the change of the phase and
$\exp\left(-\varphi_{\text{dis}}\right)=|t_{\text{qe}}|$ describes the
attenuation of the amplitude. It is worth noting that the additional
propagation phase introduced by the two-level QE can be equivalent to a change
of the effective resonance frequency of the microresonator. When a single
photon travels around the microresonator, in the absence of the QE, we have
$\theta=2\pi R\beta=2\pi m\omega/\Omega$, where $m=\Omega n_{\text{eff}}R/c$
is the modal number. But if we consider the chiral QE-microresonator
interaction, the additional propagation phase $\varphi_{\text{pha}}$ leads to
$\theta+\varphi_{\text{pha}}=2\pi m\omega/\Omega_{\text{eff}}$. The effective
resonance frequency of the microresonator is

$\Omega_{\text{eff}}\approx\Omega\left(1-\frac{\varphi_{\text{pha}}\Omega}{2\pi
m\omega}\right)\;,$ (20)

and $\Omega_{\text{eff}}\approx\Omega\left[1-\varphi_{\text{pha}}/2\pi
m\right]$ for $\Omega/\omega\approx 1$.

In general, by equating a two-level QE directionally coupled with a
microresonator to a single-photon phase-amplitude modulator, we can use the TM
method to solve the single-photon transport problem in such chiral QE-
microresonator systems. This only needs to be multiplied by a transmission
coefficient $t_{\text{qe}}$ in the transfer relation. Further, this approach
can be extended to the system in which multiple QEs are coupled to
microresonators in a chiral way.

Figure 2: The transmission spectra of a waveguide coupled with a
microresonator. The blue solid, the red dashed, and the green dotted curves
are calculated by the TM method, the CQED method and the SPT method,
respectively. The setting in the following figures is the same. (a) In absence
of backscattering. (b) and (c) In presence of the backscattering with
strengths $h=\kappa_{\text{in}}$ and $h=10\kappa_{\text{in}}$, respectively.
See the Sec. III for other parameters.
Figure 3: The transmission spectra for a chiral QE-microresonator system
without considering the backscattering. $\Gamma=0.1\gamma$, $\Gamma=\gamma$
and $\Gamma=100\gamma$ in (a-c), respectively.

##  III Results

Below we numerically study our system to prove the consistency of the three
methods. For the TM method and the SPT method, we solve the Eqs. (15) and (6)
directly, whereas for the CQED method, we perform a full quantum dynamics
simulation using Eq. (2). We set a prepared QD as the two-level QE for
coupling to a silicon-based microresonator in a chiral way. The experimentally
available parameters are chosen as [50, 57, 47]: $R=\text{10.5
}\leavevmode\nobreak\ \mathrm{\SIUnitSymbolMicro}\mathrm{m}$,
$n_{\text{eff}}=1.5$, $\mathcal{F}/2\pi=3\leavevmode\nobreak\
\mathrm{T}\mathrm{Hz}$, $\alpha_{\text{in}}=0.1$, and
$\gamma/2\pi=6\leavevmode\nobreak\ \mathrm{M}\mathrm{Hz}$. The conversion
relationships between the parameters of the three methods are given by Eqs.
(16) and (19). We take $t=\alpha=0.99$, thus satisfying the critical coupling
condition,
$\kappa_{\text{ex}}/2\pi=\kappa_{\text{in}}/2\pi=30\leavevmode\nobreak\
\mathrm{G}\mathrm{Hz}$. The frequency of the QD is resonant with the
microresonator, i.e., $\omega_{\text{qe}}=\Omega$. We first consider the case
without a two-level QD, corresponding to $\Gamma=0\leavevmode\nobreak\ (g=0)$,
shown in Fig. 2. When the strength $h=0$, the deep of transmission appears at
the resonance point [see Fig. 2(a)]. As the strength $h$ increases, the
transmission spectrum gradually splits [see Fig. 2(b-c)]. The calculation
results of the three methods are exactly the same.

Then we consider the chiral coupling of a two-level QD. By modeling the two-
level QD chirally coupled to the microresonator as a single-photon phase-
amplitude modulator, we can use the TM method to solve such problems. Figure 3
shows the transmission spectra without scatterers. The presence of the two-
level QD causes the transmission spectrum to split [23]. We can find that the
transmission spectra calculated by the three methods are consistent regardless
of whether it is under weak coupling, $\Gamma=0.1\gamma$ and $\Gamma=\gamma$
($g/\kappa_{\text{tol}}=0.03$ and $g/\kappa_{\text{tol}}=0.1$ ), or strong
coupling, $\Gamma=100\gamma$ ($g/\kappa_{\text{tol}}=1$). The results taking
into account the effect of backscattering are shown in Fig. 4. We consider the
case of strong coupling, $\Gamma=100\gamma$. Whether it is in the case of weak
backscatter [see Fig. 4(a)] or strong backscatter [see Fig. 4(b)], the
calculation results are consistent. Therefore, our numerical results further
confirm the above theoretical analyses and prove the correctness of the
parameter relationships of these three methods.

Figure 4: The transmission spectra for a chiral QE-microresonator system.
$\Gamma=100\gamma$, $h=\kappa_{\text{in}}$ in (a) and $\Gamma=100\gamma$,
$h=10\kappa_{\text{in}}$ in (b).

##  IV Conclusion

We demonstrate that a two-level QE can be treated as a single-photon phase-
amplitude modulator in a chiral QE-microresonator system. Based on this, we
can solve the single-photon transport problem by the method of TM. Theoretical
analyses and numerical results confirm that the TM method is consistent with
CQED and SPT methods. The conversion for the parameters of these three methods
is explicitly derived. Without loss of generality, the TM method can be
extended to solve the single-photon transport problem of any number of two-
level QEs chirally coupled to multiple microresonators.

## Acknowledgements

This work was supported by the National Key R&D Program of China (Grants No.
2019YFA0308700, No. 2017YFA0303703, No. 2017YFA0303701), the National Natural
Science Foundation of China (Grant Nos. 11874212, 11890704,
61671279,11574145,11690031), the Fundamental Research Funds for the Central
Universities (021314380095) and Program for Innovative Talents and
Entrepreneurs in Jiangsu (Grant No. JSSCTD202138).

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