# Single-photon scattering in giant-atom waveguide systems with chiral
coupling

Shu-Yu Li  Fujian Key Laboratory of Quantum Information and Quantum Optics &
Department of Physics, Fuzhou University, Fuzhou 350116, People’s Republic of
China  Ze-Quan Zhang  Fujian Key Laboratory of Quantum Information and Quantum
Optics & Department of Physics, Fuzhou University, Fuzhou 350116, People’s
Republic of China  Lei Du  School of Physics and Center for Quantum Sciences,
Northeast Normal University, Changchun 130024, China  Yong Li  Center for
Theoretical Physics & School of Physics and Optoelectronic Engineering, Hainan
University, Haikou 570228, China  Huaizhi Wu  Fujian Key Laboratory of Quantum
Information and Quantum Optics & Department of Physics, Fuzhou University,
Fuzhou 350116, People’s Republic of China

###### Abstract

We study single-photon scattering spectra of a giant atom chirally coupled to
a one-dimensional waveguide at multiple connection points, and examine
chirality induced effects in the scattering spectra by engineering the
chirality of the coupling strengths. We show that the transmission spectra
typically possess an anti-Lorentzian lineshape with a nonzero minimum, but
when the chirality satisfies some specific conditions independent of the
number of coupling points, the transmission spectrum of an incident photon can
undergo a transition from complete transmission to total reflection at
multiple frequency “windows”, the width of which can be flexibly tuned in situ
by engineering the coupling strengths of a certain disordered coupling point.
Moreover, we show that a perfect nonreciprocal photon scattering can be
achieved due to the interplay between internal atomic spontaneous emission and
the chirally external decay to the waveguide, in contrast to that induced by
the non-Markovian retardation effect. We also consider the non-Markovian
retardation effect on the scattering spectra, which allows for a photonic band
gap even with only two chiral coupling points. The giant-atom-waveguide system
with chiral coupling is a promising candidate for realizing single-photon
routers with multiple channels.

##  I INTRODUCTION

Waveguide quantum electrodynamics (QED) (Roy et al., 2017; Gu et al., 2017),
which studies the interaction between atoms (or other quantum emitters) and
free propagating photons in a one-dimensional (1D) waveguide, has been
experimentally demonstrated in many state-of-the-art architectures, such as
trapped (natural) atoms coupled with optical fibers (Vetsch et al., 2010;
Corzo et al., 2016; Sørensen et al., 2016) or photonic crystal waveguides
(Goban et al., 2014, 2015; Yan and Wei, 2015) and superconducting qubits
coupled with transmission lines (Astafiev et al., 2010; Hoi et al., 2012,
2013; van Loo et al., 2013; Liu and Houck, 2017). Typically, the (natural)
atoms are orders of magnitude smaller than optical (microwave) wavelengths of
the continuous bosonic modes in the 1D waveguide, therefore, they can be
viewed as point-like emitters to justify the dipole approximation. Waveguide
QED systems with natural ‘small’ atoms can potentially be used to construct
quantum network (Kimble, 2008; Duan and Monroe, 2010; Wehner et al., 2018) and
simulate quantum many-body physics (Bello et al., 2023; Wang et al., 2021;
Mahmoodian et al., 2020). On the other hand, extending the small atom platform
to artificial ’giant’ atomic systems has attracted significant recent
attention, in part because it represents a breakdown of the dipole
approximation where the scale of atoms becomes comparable to the wavelength of
the light they interact with. The artificial giant atoms have been well
designed and recently demonstrated with superconducting qubits coupled to
short-wavelength surface acoustic waves (Gustafsson et al., 2014; Manenti et
al., 2017; Guo et al., 2017; Andersson et al., 2019, 2020; Wang et al., 2022)
or a microwave-waveguide at multiple discrete points (Frisk Kockum et al.,
2014; Kannan et al., 2020; Vadiraj et al., 2021). The paradigmatic systems
show striking interference effects which depend on both the distance between
different coupling points and the photonic frequency, and open the door to
several novel phenomena, including frequency-dependent Lamb shift and
relaxation rate (Frisk Kockum et al., 2014; Qiu et al., 2023), decoherence-
free atomic states (Kockum et al., 2018; Carollo et al., 2020; Kannan et al.,
2020; Du et al., 2023), non-Markovian decay dynamics (Guo et al., 2020;
Cilluffo et al., 2020; Guo et al., 2017; Andersson et al., 2019; Lim et al.,
2023), and chiral light-matter interactions (Soro and Kockum, 2022; Wang et
al., 2021; Du et al., 2021, 2022; Wang and Li, 2022; Zhou et al., 2023).

Waveguide QED system with giant atoms has emerged as a new promising platform
for engineering transport of photons and single-photon routing (Du and Li,
2021; Du et al., 2021; Chen et al., 2022; Cai and Jia, 2021; Feng and Jia,
2021; Peng and Jia, 2023; Zhao et al., 2022; Yin et al., 2022; Zhou et al.,
2023; Ask et al., ; Soro and Kockum, 2022; Qiu et al., 2023). The system
enables strong tunable atom-waveguide coupling and the engineering of time
delay, manifesting multiple-point interference and non-Markovian retardation
effects in the photon scattering spectra (Du et al., 2021; Chen et al., 2022;
Cai and Jia, 2021; Yin et al., 2022; Zhou et al., 2023). Herein, the photon
scattering spectra can exhibit interesting features, such as
electromagnetically induced transparency (Feng and Jia, 2021; Peng and Jia,
2023; Ask et al., ), atomic decay induced nonreciprocity (Du et al., 2021;
Chen et al., 2022; Zhou et al., 2023), and photonic band gap (Cai and Jia,
2021; Peng and Jia, 2023), and these features can be applied to probe
collective radiance and topological states (Feng and Jia, 2021; Peng and Jia,
2023; Soro and Kockum, 2022) with a chain of two-level giant atoms in both the
Markovian and non-Markovian regimes (Qiu et al., 2023). Moreover, an
experimental setup with chiral interfaces between giant atoms and waveguides
has recently become a reality based on technological progress, e.g., by
coupling transmon qubits (Koch et al., 2007; Joshi et al., 2023) to meandering
transmission lines with circulators (Sathyamoorthy et al., 2014; Sliwa et al.,
2015; Chapman et al., 2017; Müller et al., 2018). In these setups, the
coupling between waveguide modes and giant atoms depends on the propagation
direction of the light. Despite the chirality of their coupling, one can still
observe perfect collective radiance and decoherence-free dark states (Soro and
Kockum, 2022) inaccessible with small atoms, and furthermore non-Markovianity
induced nonreciprocity (Du et al., 2021; Zhou et al., 2023) and photon
frequency conversion [Du et al., 2021], whereas the previous inspiring studies
about single-photon routing are limited to special cases, e.g., a (multilevel)
giant atom with two asymmetric coupling points (Du and Li, 2021; Du et al.,
2021). Thus, it is unclear what can bring about by engineering the chirality
with more than two coupling points.

In this paper, we study single-photon scattering spectra and its
nonreciprocity in a waveguide chirally coupled to a giant atom with multiple
coupling points. We focus on chirality induced effects in the scattering
spectra by considering three chiral coupling regimes with coupling points
being equally spaced: (1) bidirectional even coupling (BEC), (2)
unidirectional un-even coupling (UUEC), and (3) bidirectional un-even coupling
(BUEC), respectively. Here, the Lamb shifts and the effective decay rates
depend not only on the distance (or phase delay) between the coupling points,
but also on the propagation direction of light. In the Markovian regime, an
incident photon, which must be completely transmitted in the BEC and the UUEC
regime, can be totally reflected in the BUEC regime by engineering the chiral
condition of the coupling strengths (independent of the number of coupling
points). It is remarkable that the transmission spectra can exhibit multiple
total reflection "windows" with the width being flexibly controlled, and
moreover their locations can be independent of the coupling strengths of the
disordered connection point. Furthermore, by taking atomic spontaneous decay
into account, we find that when the chirality in atom-waveguide couplings is
canceled out by the internal dissipation, counter-intuitively, it can lead to
a perfect nonreciprocal photon scattering (with transmission probability of
unity in the forward direction and zero in the backward) inaccessible by non-
chiral waveguide QED systems. Note that the nonreciprocity is not induced by
the non-Markovian retardation effect (Zhou et al., 2023), but the chirality of
the multiple-point coupling. Finally, we consider the non-Markovian
retardation effect on the scattering spectra, which show a photonic band gap
in the intermediate non-Markovian regime even with only two chiral coupling
points, and exhibit an oscillatory pattern between total reflection and full
transmission in the deep non-Markovian regime. The giant-atom-waveguide system
with chiral coupling makes it a promising platform for realizing single-photon
routers with multiple frequency channels.

The paper is organized as follows: Section II introduces the theoretical
model, and calculates the transmission and reflection amplitudes of an
incident photon by using a real-space scattering method, where the giant
atom’s Lamb shifts and effective decay rates are derived and discussed.
Section III presents the scattering spectra for three chiral coupling
conditions (i.e. the BEC, the UUEC, and the BUEC conditions) in the Markovian
regime, and discusses intriguing features induced by the chiral coupling.
Section IV shows that nonreciprocal photon scattering can be realized due to
the cooperative effect of chiral coupling and atomic spontaneous decay.
Section V further discusses the scattering spectra in the non-Markovian
regime, with a conclusion given by Section VI.

##  II MODEL AND METHOD

Figure 1: Schematics of a giant-atom system. A two-level giant atom (with
excited state $|e\rangle$ and ground state $|g\rangle$) couples to the one-
dimensional waveguide at multiple points (labeled by the coordinates $z_{j}$),
where $g_{L_{j}}$($g_{R_{j}}$) are coupling strengths at the $j$th coupling
point for left-going (right-going) waveguide modes, $\phi_{ij}$ are distance-
dependent phases for the photon propagating between the $i$th and the $j$th
coupling points, and $\gamma$ is the spontaneous decay rate of the atom to the
bath environment.

As schematically shown in Fig.1, the system consists of a two-level giant atom
chirally coupled to a 1D waveguide at $N$ discrete points (labeled $j$,
coordinates $z_{j}$). The coupling strengths between the giant atom and
waveguide modes depend on the propagation direction of the light. Under the
rotating-wave approximation, the total Hamiltonian of the system in real space
can be written as ($\hbar=1$ hereafter) (Gonzalez-Ballestero et al., 2016)

$H=H_{a}+H_{w}+H_{int},$ (1)

with

$\displaystyle H_{a}=$ $\displaystyle(\omega_{e}-i\gamma)|e\rangle\langle e|,$
$\displaystyle H_{w}=$
$\displaystyle\int_{-\infty}^{+\infty}dza_{L}^{\dagger}\left(z\right)\left(\omega_{0}+iv_{g}\frac{\partial}{\partial
z}\right)a_{L}\left(z\right)$
$\displaystyle+a_{R}^{\dagger}\left(z\right)\left(\omega_{0}-iv_{g}\frac{\partial}{\partial
z}\right)a_{R}\left(z\right),$ $\displaystyle H_{int}=$
$\displaystyle\int_{-\infty}^{+\infty}dz\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}\delta(z-z_{j})\left[g_{L_{j}}a_{L}^{\dagger}\left(z\right)e^{ik_{0}z}\right.$
$\displaystyle+\left.g_{R_{j}}a_{R}^{\dagger}\left(z\right)e^{-ik_{0}z}\right]|g\rangle\langle
e|+\text{H.c.},$

where $H_{a}$ is the bare atomic Hamiltonian, with $\omega_{e}$ being the
transition frequency between the ground state $|g\rangle$ and the excited
state $|e\rangle$, and $\gamma$ being the spontaneous emission rate induced by
the non-waveguide modes in the environment. $H_{w}$ is the bare waveguide
Hamiltonian, with $a_{L}^{\dagger}\left(z\right)$
$[a_{R}^{\dagger}\left(z\right)]$ and $a_{L}(z)$ [$a_{R}(z)$] the creation and
annihilation operators of the left-propagating (right-propagating) modes at
position $z$. $\omega_{0}$ is the central frequency around which a linear
dispersion relation under consideration is given by
$\omega(k)=\omega_{0}+\left(k-k_{0}\right)v_{g}$ with $k$ the wave vector of
the incident photon, $k_{0}$ the wave vector corresponding to $\omega_{0}$,
and $v_{g}$ the group velocity in the vicinity of $\omega_{0}$ (Shen and Fan,
2009). $H_{int}$ is the interaction Hamiltonian with $g_{L_{j}}$ and
$g_{R_{j}}$ the coupling strengths for the atom interacting with a left-going
and right-going photon at the position $z=z_{j}$, respectively, where in real
space, the giant atom behaves as a potential $\delta\left(z-z_{j}\right)$ at
each coupling point. Note that the atom-waveguide coupling at multiple points
enable us to observe system dynamics in both the Markovian and non-Markovian
regimes (Soro and Kockum, 2022; Zhou et al., 2023), which strongly depends on
the accumulated phases $\phi_{ij}=k|z_{i}-z_{j}|$ [or
$\phi_{ij}=(\omega-\omega_{0})\tau_{ij}+k_{0}|z_{i}-z_{j}|$] of photons
propagating between any two of the $N$ coupling points with
$\tau_{ij}=\left|z_{i}-z_{j}\right|/v_{g}$.

We consider the single-photon scattering problem by engineering chirality of
the atom-waveguide couplings, where the system is constrained to the single
excitation subspace, and then the eigenstate of Hamiltonian (1) is given by

$\displaystyle\left|\psi\right\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dz\left[c_{gl}\left(z\right)a_{L}^{\dagger}\left(z\right)+c_{gr}\left(z\right)a_{R}^{\dagger}\left(z\right)\right]|g,0\rangle$
(2) $\displaystyle+c_{e0}|e,0\rangle,$

where $c_{gl}\left(z\right)$ [$c_{gr}\left(z\right)$] is the probability
amplitude of the states $a_{L}^{\dagger}\left(z\right)|g,0\rangle$
[$a_{R}^{\dagger}\left(z\right)|g,0\rangle$], describing a left-propagating
(right-propagating) photon at position $z$ and the atom in $|g\rangle$, and
$c_{e0}$ is the probability amplitude of the atom in the excited state
$|e\rangle$, finding no photon in the waveguide. Inserting Eqs. (1) and (2)
into the stationary Schrödinger equation $H|\psi\rangle=E|\psi\rangle$, we
find that the probability amplitudes obey the following relations:

$\displaystyle Ec_{gl}\left(z\right)=$
$\displaystyle(\omega_{0}+iv_{g}\frac{\partial}{\partial
z})c_{gl}\left(z\right)$
$\displaystyle+\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}\delta(z-z_{j})g_{L_{j}}e^{ik_{0}z}c_{e0},$
$\displaystyle Ec_{gr}\left(z\right)=$
$\displaystyle(\omega_{0}-iv_{g}\frac{\partial}{\partial
z})c_{gr}\left(z\right)$
$\displaystyle+\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}\delta(z-z_{j})g_{R_{j}}e^{-ik_{0}z}c_{e0},$
$\displaystyle(\Delta+i\gamma)c_{e0}=$ $\displaystyle\int
dz\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}\delta(z-z_{j})[g_{L_{j}}e^{-ik_{0}z}c_{gl}\left(z\right)$
$\displaystyle+g_{R_{j}}e^{ik_{0}z}c_{gr}\left(z\right)],$ (3)

where $\Delta=\omega(k)-\omega_{e}$ is the detuning between the incident
photons and the atomic transition $|g\rangle\leftrightarrow|e\rangle$. Now
suppose a photon is incident from the left port and the atom is initially in
$|g\rangle$, the probability amplitudes, due to the $\delta$-function
potential effect of the atom at the coupling point, can be formed as

$\displaystyle c_{gr}\left(z\right)=$ $\displaystyle
e^{i(k-k_{0})z}[\Theta\left(z_{1}-z\right)$
$\displaystyle+\stackrel{{\scriptstyle[}}{{j}}=1]{N-1}{\sum}t_{j}\Theta\left(z-z_{j}\right)\Theta\left(z_{j+1}-z\right)$
$\displaystyle+t_{N}\Theta\left(z-z_{N}\right)],$ $\displaystyle
c_{gl}\left(z\right)=$ $\displaystyle e^{-i(k-k_{0})z}[r_{1}\Theta(z_{1}-z)$
$\displaystyle+\stackrel{{\scriptstyle[}}{{j}}=2]{N}{\sum}r_{j}\Theta(z-z_{j-1})\Theta(z_{j}-z)],$
(4)

where $t_{j}$ ($r_{j}$) is the transmission (reflection) amplitude for the
$j$th coupling point, $t_{N}$ ($r_{1}$) is the transmission (reflection)
amplitude for the last (first) coupling point, and $\Theta(z-z_{j})$ are the
Heaviside step function. While for the right-incident case, the probability
amplitudes alternatively take the form

$\displaystyle c_{gr}\left(z\right)=$ $\displaystyle
e^{i(k-k_{0})z}[\stackrel{{\scriptstyle[}}{{j}}=1]{N-1}{\sum}\tilde{r}_{j}\Theta(z-z_{j})\Theta(z_{j+1}-z)$
$\displaystyle+\tilde{r}_{N}\Theta(z-z_{N})],$ $\displaystyle
c_{gl}\left(z\right)=$ $\displaystyle
e^{-i(k-k_{0})z}[\tilde{t}_{1}\Theta(z_{1}-z)$
$\displaystyle+\stackrel{{\scriptstyle[}}{{j}}=2]{N}{\sum}\tilde{t}_{j}\Theta(z-z_{j-1})\Theta(z_{j}-z)$
$\displaystyle+\Theta\left(z-z_{N}\right)],$ (5)

with $\tilde{t}_{j}$, $\tilde{r}_{j}$ being the corresponding probability
amplitudes as in the right-incident case.

Substituting Eqs. (4) and (5) into (3), one readily obtains the transmission
and reflection amplitudes for a left-incident photon:

$\displaystyle t_{N}$
$\displaystyle=\frac{\left(\Delta-\Delta_{LS}\right)+i\left(\gamma+\Gamma{}_{x}-\Gamma{}_{y}\right)}{\left(\Delta-\Delta_{LS}\right)+i\left(\gamma+\Gamma{}_{x}+\Gamma{}_{y}\right)},$
$\displaystyle r_{1}$
$\displaystyle=\frac{-i\Gamma_{xy}}{\left(\Delta-\Delta_{LS}\right)+i\left(\gamma+\Gamma{}_{x}+\Gamma{}_{y}\right)},$
(6)

and for a right-incident photon:

$\displaystyle\tilde{t}_{1}$
$\displaystyle=\frac{\left(\Delta-\Delta_{LS}\right)+i\left(\gamma-\Gamma{}_{x}+\Gamma{}_{y}\right)}{\left(\Delta-\Delta_{LS}\right)+i\left(\gamma+\Gamma{}_{x}+\Gamma{}_{y}\right)},$
$\displaystyle\tilde{r}_{N}$
$\displaystyle=\frac{i\Gamma_{xy}^{*}}{\left(\Delta-\Delta_{LS}\right)+i\left(\gamma+\Gamma{}_{x}+\Gamma{}_{y}\right)},$
(7)

where $\Delta_{LS}\equiv\Delta_{x}+\Delta_{y}$ is the overall Lamb shift
contributed by both the left- and right-propagating directions waveguide modes
from interference between connection points, with

$\Delta_{x}=\frac{1}{2}\stackrel{{\scriptstyle[}}{{i}},j]{N}{\sum}x_{i}x_{j}\text{sin}\phi_{ij},\text{
}\Delta_{y}=\frac{1}{2}\stackrel{{\scriptstyle[}}{{i}},j]{N}{\sum}y_{i}y_{j}\text{sin}\phi_{ij},$
(8)

and $(x_{j},y_{j})\equiv(g_{Lj},g_{Rj})/\sqrt{v_{g}}$; correspondingly,
$\Gamma_{x}$ and $\Gamma_{y}$ are direction-dependent relaxation rates given
by (Cai and Jia, 2021; Frisk Kockum et al., 2014)

$\Gamma_{x}=\frac{1}{2}\stackrel{{\scriptstyle[}}{{\\{}}i,j\\}=1]{N}{\sum}x_{i}x_{j}\text{cos}\phi_{ij}=\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}\left(\frac{x_{i}^{2}}{2}+\stackrel{{\scriptstyle[}}{{j}}=i+1]{N}{\sum}x_{i}x_{j}\text{cos}\phi_{ij}\right),$
(9)
$\Gamma_{y}=\frac{1}{2}\stackrel{{\scriptstyle[}}{{\\{}}i,j\\}=1]{N}{\sum}y_{i}y_{j}\text{cos}\phi_{ij}=\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}\left(\frac{y_{i}^{2}}{2}+\stackrel{{\scriptstyle[}}{{j}}=i+1]{N}{\sum}y_{i}y_{j}\text{cos}\phi_{ij}\right),$
(10)

and

$\Gamma_{xy}=\left(\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}e^{ikz_{i}}x_{i}\right)\left(\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}e^{ikz_{j}}y_{j}\right).$
(11)

It follows that the transmission and reflection probabilities are defined by
$T_{l}=\left|t_{N}\right|^{2}$ ($T_{r}=\left|\tilde{t}_{1}\right|^{2}$) and
$R_{l}=\left|t_{1}\right|^{2}$ ($R_{r}=\left|\tilde{t}_{N}\right|^{2}$),
respectively.

Revisiting the small-atom limit (Frisk Kockum et al., 2014), where the atom
interacts with the left-going (right-going) waveguide modes by a single
connection point (e.g., at position $z_{j}$), the atomic relaxation rates into
the continuum modes can be simply derived by Fermi’s golden rule and are given
by $x_{j}^{2}/2$ and $y_{j}^{2}/2$, and the atomic ground state is not shifted
by the atom-waveguide coupling (i.e., $\Delta_{x(y)}=0$) (Frisk Kockum et al.,
2014). In the case of a giant-atom with $N$ connection points, when the
couplings with left-going and right-going photons are uniformly symmetric,
i.e., $x_{i}=x_{j}=y_{i}=y_{j}$ (for all $i\neq j$), one finds
$\Delta_{x}=\Delta_{y}$ and $\Gamma_{x}=\Gamma_{y}$ regardless of the details
of phases $\phi_{ij}$, then the transmission and reflection probabilities read

$T_{l}=T_{r}=\frac{\left(\Delta-2\Delta_{x}\right)^{2}+\gamma^{2}}{\left(\Delta-2\Delta_{x}\right)^{2}+\left(\gamma+2\Gamma{}_{x}\right)^{2}},$
(12)
$R_{l}=R_{r}=\frac{4\Gamma_{x}^{2}}{\left(\Delta-2\Delta_{x}\right)^{2}+\left(\gamma+2\Gamma{}_{x}\right)^{2}}.$
(13)

For $\gamma=0$, the reflection probabilities have the standard Lorentzian
lineshapes centered at $\Delta=2\Delta_{x}$ with the full-width at half-
maximum (FWHM) $4\Gamma{}_{x}$ and the transmission probabilities have the
anti-Lorentzian lineshapes. There exists a singular point with
$\Gamma{}_{x}=0$ for which the giant atom decouples from the waveguide,
leading to $T_{l}=T_{r}=1$. Moreover, Eqs. (12) and (13) imply that
nonreciprocal photon scattering can not happen whether $\gamma=0$ or
$\gamma\neq 0$.

To clearly see chirality induced features, we now consider the $N$ coupling
points being equally spaced and firstly focus on the effect of even or uneven
chiral coupling in the Markovian regime, where the propagating time of photons
travel between the leftmost and the rightmost coupling points
$\tau_{1N}=\left(z_{N}-z_{1}\right)/v_{g}$ is short compared to the
characteristic relaxation time $\sim\tilde{\Gamma}^{-1}$ of the giant atom
with
$\tilde{\Gamma}=\frac{1}{2}(\sum_{i=1}^{N}x_{i})^{2}+\frac{1}{2}(\sum_{i=1}^{N}y_{i})^{2}+\gamma$,
corresponding to the decay rate for $\tau_{1N}\rightarrow 0$. However, when
$\tau_{1N}\sim 1/\tilde{\Gamma}$, the giant atom-waveguide interaction will
enter the non-Markovian regime (Guo et al., 2017; Frisk Kockum et al., 2014;
Cai and Jia, 2021; Frisk Kockum, 2021), i.e., the time evolution of the system
can depend on what the system state was at an earlier time, and will be
discussed later in Sec. V. It should be emphasized that the transmission and
reflection amplitudes in Eqs. (6)-(7) are valid in both the Markovian and the
non-Markovian regimes. Moreover, given that the bandwidth $\Delta$ is of the
order of $\tilde{\Gamma}$, we can examine the Markovian physics based on Eqs.
(8)-(10) under the condition of
$\tilde{\Gamma}\tau_{1N}\sim|\Delta|\tau_{1N}\ll 1$, and neglect the
contribution of $|\Delta|\tau_{1N}$ to the $\phi_{ij}$ dependent effect. In
the next section, we first replace the phases $\phi_{ij}$ by
$\tilde{\phi}_{ij}=(\omega_{e}-\omega_{0})\tau_{ij}+k_{0}|z_{i}-z_{j}|$, and
the transmission and reflection amplitudes are rephrased in terms of the phase
difference between adjacent coupling points $\tilde{\phi}_{12}$, i.e.,
$\phi_{ij}=|j-i|(\Delta\tau_{12}+\tilde{\phi}_{12})\rightarrow\tilde{\phi}_{ij}=\left|j-i\right|\tilde{\phi}_{12}$.
We take $\tilde{\phi}_{12}$ modulo $2\pi$ [denoted by
mod$(\tilde{\phi}_{12},2\pi)$ in the following] into account for its
effectiveness, and study photon scattering under different chiral coupling
regimes.

Figure 2: Transmission probabilities $T_{l(r)}$ in the Markovian regime with
chiral coupling. (a) $T_{l(r)}$ given by Eq. (24) and Eq. (25) as functions of
$\Delta$ by considering $x_{2}/x_{1}=\\{1,2,3\\}$ with $N=2$ coupling points.
The dashed lines correspond to the single-photon transmission spectrum under
the condition of $x_{1}+x_{2}=y_{1}+y_{2}$,
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, and the solid lines correspond to
$|x_{1}-x_{2}|=|y_{1}-y_{2}|$, $\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$. For
$\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$, $T_{l(r)}$ remains unity under the
even coupling condition $x_{2}=x_{1}$, while for $x_{2}\neq x_{1}$, $T_{l(r)}$
become the anti-Lorentzian lineshape with a narrow half width
$2(x_{1}-x_{2})^{2}$ and undergo a sudden jump from unity to zero at resonance
$\Delta=0$. (b)-(d) $T_{l(r)}$ versus the coupling strengths $x_{i}/x$ and
$y_{i}/x$ at the the $i$th coupling point with (b) $N=2$,
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, $\Delta=\Delta_{LS}=0$, (c) $N=5$,
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, $\Delta=\Delta_{LS}=0$, and (d) $N\geq
2$, $\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$,
$\Delta=\Delta_{LS}$, respectively. Here, we consider the $N$ coupling points
are equally spaced, and set $(x_{j},y_{j})/x=(1,2)$ for $j\neq i$. The white
dot-dashed lines indicate $T_{l(r)}=1$ under the condition of unidirectional
uneven coupling $x_{i}=x$ or $y_{i}=y$. The white dashed lines correspond to
$T_{l(r)}=0$ under the condition of $\Gamma{}_{x}=\Gamma{}_{y}$ with respect
to the case of bi-directional uneven coupling, except the point of
intersection between the lines. The white dots indicate the three different
coupling regimes discussed later.

##  III photon scattering with chiral coupling: The Markovian regime

By considering the chiral coupling with $\gamma=0$ and assuming $x_{i}$
($y_{i}$) being positive real values, one typically has
$\Gamma_{x}\neq\Gamma_{y}$, i.e.,
$\frac{1}{2}\sum_{\\{i,j\\}=1}^{N}(x_{i}x_{j}-y_{i}y_{j})\text{cos}\phi_{ij}\neq
0$. The chirality can be divided into three different regimes: (1)
bidirectional even coupling (BEC) $x_{i}=x_{j},y_{i}=y_{j},$ but $x_{i}\neq
y_{i}$; (2) unidirectional un-even coupling (UUEC) $x_{i}=x_{j},y_{i}\neq
y_{j}$, or $x_{i}\neq x_{j},y_{i}=y_{j}$; (3) bidirectional un-even coupling
(BUEC) $x_{i}\neq x_{j},y_{i}\neq y_{j}$. Remarkably, we find that
$\Gamma_{x}=\Gamma_{y}$ can be achieved by appropriately tuning the uneven
coupling strengths distributed at different coupling points. As a result, it
offers a flexible way to control the transmission and reflection of a single
incident photon, and allows for in situ photon routing at multiple frequency
windows. This will be discussed in detail later.

###  III.1 Bidirectional even coupling

We first consider the bidirectional even coupling regime for a giant atom with
$N\geq 2$ coupling points, where the coupling strengths of the $N$ coupling
points are identical for the same propagation direction, and are set to
$x_{i}=x$ and $y_{i}=y$. Due to the multiple-point interference effect, the
effective Lamb shift in total takes the form

$\Delta_{LS}=\frac{1}{2}\left(x^{2}+y^{2}\right)\frac{N\text{sin}\tilde{\phi}_{12}-\text{sin}N\tilde{\phi}_{12}}{1-\text{cos}\tilde{\phi}_{12}},$
(14)

and the sum (difference) of the two effective decay rates is given by

$\displaystyle\Gamma_{x}\pm\Gamma_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(x^{2}\pm
y^{2}\right)\frac{\text{sin}^{2}\left(\frac{1}{2}N\tilde{\phi}_{12}\right)}{\text{sin}^{2}\left(\frac{1}{2}\tilde{\phi}_{12}\right)},$
(15)

which shows that the photon interferes with itself multiple times at the two
(left and right) propagation directions independently.

(1) When the propagating phase satisfies
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, the Lamb shift vanishes regardless of
the coupling directions, but the effective decay rates are $N^{2}$-enhanced
and are given by

$\Gamma_{x}\pm\Gamma_{y}=\frac{\left(\sum_{i=1}^{N}x_{i}\right)^{2}\pm\left(\sum_{i=1}^{N}y_{i}\right)^{2}}{2}=\frac{1}{2}N^{2}\left(x^{2}\pm
y^{2}\right).$ (16)

The transmission probabilities then read

$\displaystyle T_{l}=T_{r}$ $\displaystyle=$
$\displaystyle\frac{4\Delta^{2}+N^{4}\left(x^{2}-y^{2}\right)^{2}}{4\Delta^{2}+N^{4}\left(x^{2}+y^{2}\right)^{2}}.$
(17)

In contrast to the uniformly symmetric coupling case ($x=y$), the chirality
leads to a nonvanishing transmission $T_{l(r)}\neq 0$ at resonance $\Delta=0$,
i.e.,

$T_{l(r)}=\left(\frac{\Gamma_{x}-\Gamma_{y}}{\Gamma_{x}+\Gamma_{y}}\right)^{2}=\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)^{2},$
(18)

corresponding to $T_{l(r)}$ in the large $N\rightarrow\infty$ limit.

(2) For $\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$
($m^{\text{$\prime$}}=1,2,...,N-1$), the Lamb shift can be nonvanishing and is
given by

$\Delta_{LS}=\frac{N}{2}\left(x^{2}+y^{2}\right)\text{cot}\left(\frac{m^{\text{$\prime$}}\pi}{N}\right),$
(19)

but the effective decay rates vanish ($\Gamma_{x}=\Gamma_{y}=0$) due to the
destructive interference effect among the coupling points, which is analogous
to the case with uniformly symmetric coupling, then a photon incident from the
left or the right will be completely transmitted, independent of the frequency
detuning between photon and atomic transition.

###  III.2 Unidirectional un-even and bidirectional un-even coupling

The results for the BEC regime can be understood intuitively, but those for
the UUEC and the BUEC regimes are not self-evident. To obtain instructive
insight, we first consider the simplest model with only $N=2$ coupling points
in the Markovian limit, where the Lamb shifts are $\Delta_{x}=\Delta_{y}=0$
both for $\text{mod}(\tilde{\phi}_{12},2\pi)=0$ and
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$, but the sum
and difference of the two effective decay rates $\Gamma_{x}\pm\Gamma_{y}$
strongly depend on the chirality of the coupling strengths. For
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, the effective decay rates are simply
given by

$\Gamma_{x}\pm\Gamma_{y}=\frac{(x_{1}+x_{2})^{2}\pm(y_{1}+y_{2})^{2}}{2},$
(20)

and correspondingly the transmission probabilities are

$\displaystyle T_{l(r)}$ $\displaystyle=$
$\displaystyle\frac{4\Delta^{2}+\left[\left(x_{1}+x_{2}\right)^{2}-\left(y_{1}+y_{2}\right)^{2}\right]^{2}}{4\Delta^{2}+\left[\left(x_{1}+x_{2}\right)^{2}+\left(y_{1}+y_{2}\right)^{2}\right]^{2}};$
(21)

for $\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$, the effective decay rates are
alternatively given by

$\displaystyle\Gamma_{x}\pm\Gamma_{y}$ $\displaystyle=$
$\displaystyle\frac{(x_{1}-x_{2})^{2}\pm(y_{1}-y_{2})^{2}}{2},$ (22)

and the transmission probabilities read

$T_{l(r)}=\frac{4\Delta^{2}+\left[\left(x_{1}-x_{2}\right)^{2}-\left(y_{1}-y_{2}\right)^{2}\right]^{2}}{4\Delta^{2}+\left[\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}\right]^{2}}.$
(23)

Note that in the UUEC regime, i.e., $x_{1}=x_{2}$ and $y_{1}\neq y_{2}$ (or
$y_{1}=y_{2}$ and $x_{1}\neq x_{2}$), Eq. (21) reduces to

$T_{l(r)}=\frac{\Delta^{2}}{\Delta^{2}+\left(x_{1}+x_{2}\right)^{4}},$ (24)

if the coupling strengths satisfy $x_{1}+x_{2}=y_{1}+y_{2}$, which possesses
the anti-Lorentzian line shape centered at $\Delta=0$ with the FWHM
$2(x_{1}+x_{2})^{2}$ and the minimum $T_{l(r)}(\Delta=0)=0$, in contrast to
$0<T_{l(r)}<1$ of the BEC regime, see Fig. 2(a); on the other hand, the
transmission probabilities [Eq. (23)] for
$\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$ are constant unity (i.e.,
$T_{l(r)}\equiv 1$) regardless of the specific value of the detuning $\Delta$,
exhibiting the same feature as in the BEC regime. In the BUEC regime, the
overall profile of $T_{l(r)}$($\Delta$) for
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$ is below unity and
$T_{l(r)}(\Delta=0)=0$ again for $x_{1}+x_{2}=y_{1}+y_{2}$; while for
$\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$, it is remarkable that, under the
condition of $|x_{1}-x_{2}|=|y_{1}-y_{2}|$ (corresponding to
$\Gamma_{x}=\Gamma_{y}\neq 0$), $T_{l(r)}$($\Delta$) [Eq. (23)] reduce to

$T_{l(r)}=\frac{\Delta^{2}}{\Delta^{2}+\left(x_{1}-x_{2}\right)^{4}},$ (25)

which have the minimum $T_{l(r)}(\Delta=0)=0$ at resonance, and possess the
FWHM $2(x_{1}-x_{2})^{2}$ that can be infinitely narrow for
$x_{2}/x_{1}\rightarrow 1$, in contrast to $T_{l(r)}(\Delta)\equiv 1$ for both
the BEC and the UUEC regimes. In other words, an incident photon can be
totally reflected or fully transmitted for the "destructive" interference
phases $\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$ by modulating the chirality of
the coupling strengths, and moreover, the width of the reflection window can
be flexibly tuned [as shown in Fig. 2(a)], offering the potential application
for sensing and optical switch (Cai and Jia, 2021; Peng and Jia, 2023; Gu et
al., 2017).

For $N>2$, we consider the simplified model where the coupling strengths at
the $i$th coupling point $x_{i}$ ($y_{i}$) for the left-propagating (right-
propagating) photons are uniquely different from that [assumed to be identical
to $x$ ($y$)] of all the other ($N-1$) coupling points. A brief calculation
shows that the Lamb shift is given by

$\displaystyle\Delta_{LS}=$
$\displaystyle\frac{x^{2}+y^{2}}{2}\left(NS_{1}-S_{N}\right)-\frac{x(x-x_{i})+y(y-y_{i})}{2}$
$\displaystyle\text{$\times$}\left(2S_{1}+S_{i-1}-S_{i}+S_{N-i}-S_{N-i+1}\right),$
(26)

with
$S_{j}(\tilde{\phi}_{12})=\text{sin}(j\tilde{\phi}_{12})/(1-\text{cos}\tilde{\phi}_{12})$,
$j\in\mathbf{N}$, and the effective decay rates are

$\displaystyle\Gamma_{x}\pm\Gamma_{y}=$ $\displaystyle\frac{x^{2}\pm
y^{2}}{2}C_{N}+\frac{\left(x-x_{i}\right)^{2}\pm\left(y-y_{i}\right)^{2}}{2}$
$\displaystyle+\frac{x\left(x-x_{i}\right)\pm y\left(y-y_{i}\right)}{2}$
$\displaystyle\times\left(C_{i-1}-C_{i}+C_{N-i}-C_{N-i+1}\right),$ (27)

with
$C_{j}(\tilde{\phi}_{12})=\text{sin}^{2}\left(\frac{1}{2}j\tilde{\phi}_{12}\right)/\text{sin}^{2}\left(\frac{1}{2}\tilde{\phi}_{12}\right)$.
Note that both the Lamb shifts and the decay rates strongly depend on the
differences (or chirality) of the coupling strengths $\left(x-x_{i}\right)$
and $\left(y-y_{i}\right)$, and moreover, are related to the specific location
of the disorder (i.e., the $i$th coupling point).

(1) Again, the Lamb shifts vanish for $\text{mod}(\tilde{\phi}_{12},2\pi)=0$
due to $S_{j}(\tilde{\phi}_{12})=0$, but the decay rates in Eq. (27) become
dependent on the number of the coupling points according to

$\Gamma_{x}\pm\Gamma_{y}=\frac{\left[\left(N-1\right)x+x_{i}\right]^{2}\pm\left[\left(N-1\right)y+y_{i}\right]^{2}}{2}.$
(28)

It follows that the transmission probabilities are

$T_{l(r)}=\frac{4\Delta^{2}+\left\\{\left[\left(N-1\right)x+x_{i}\right]^{2}-\left[\left(N-1\right)y+y_{i}\right]^{2}\right\\}^{2}}{4\Delta^{2}+\left\\{\left[\left(N-1\right)x+x_{i}\right]^{2}+\left[\left(N-1\right)y+y_{i}\right]^{2}\right\\}^{2}},$
(29)

and thus, the incident photon is totally reflected at $\Delta=0$ if the
chirality of the coupling strengths satisfies

$\left(N-1\right)x+x_{i}=\left(N-1\right)y+y_{i},$ (30)

which reduces to $x+x_{i}=y+y_{i}$ for $N=2$ as discussed before. In Fig. 2(b)
[Fig. 2(c)], we show $T_{l(r)}[\Delta=0,\text{mod}(\tilde{\phi}_{12},2\pi)=0]$
as functions of $x_{i}$ and $y_{i}$ (in units of $x$) with $y/x=2$ and $N=2$
($N=5$), and indicate $T_{l(r)}=0$ (or $R_{l(r)}=1$) corresponding to the
condition [Eq. (30)] by white dashed lines. Around the resonance,
$T_{l(r)}(\Delta)$ under the condition of Eq. (30) possess the anti-Lorentzian
line shape with the FWHM $2\left[\left(N-1\right)x+x_{i}\right]^{2}$ scaling
as $N^{2}$.

Figure 3: Transmission probabilities $T_{l(r)}$ versus $\Delta$ and
$\text{mod}(\tilde{\phi}_{12},2\pi)$ for $N=2$, 5, 6, respectively. We
consider the rightmost (i.e., $N$th) coupling point which has the coupling
strengths $(x_{N},y_{N})/x$ different to that of the others with
$(x_{j},y_{j})/x=(1,2)$ ($j\neq N$). As indicated in Fig. 2(d), we set
$(x_{N},y_{N})/x=\\{(1,2),(3,2),(3,4)\\}$, which (from top to bottom)
correspond to the BEC [(a)-(c)], the UUEC [(d)-(f)], and the BUEC [(g)-(i)]
regime, respectively. The white solid lines are used to label the solutions
for $\Delta=\Delta_{LS}(\tilde{\phi}_{12})$, and the horizontal gray dashed
lines label the phases
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$
($m^{\text{$\prime$}}=1,2,...,N-1$). Panels (j)-(l) depict transmission
spectrum in the BUEC regime by fixing
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$, where
specially for $N$ being an even number, one can observe $T_{l(r)}=0$ at the
resonance $\Delta=0$ with $N-2$ dips symmetrically around it.

(2) For $\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$, there
exist the mathematical identities
$C_{j}(\tilde{\phi}_{12})=C_{N-j}(\tilde{\phi}_{12})$ and
$S_{j}\left(\tilde{\phi}_{12}\right)=-S_{N-j}\left(\tilde{\phi}_{12}\right)$
($j=0,1,...,N$), from which we readily find the Lamb shifts being nonvanishing
and $N$-dependent

$\displaystyle\Delta_{LS}=$ $\displaystyle
NS_{1}\frac{x^{2}+y^{2}}{2}-\left(S_{1}+S_{i-1}-S_{i}\right)$
$\displaystyle\times\left[x(x-x_{i})+y(y-y_{i})\right],$ (31)

and more interestingly, the $N$-independent effective decay rates

$\Gamma_{x}\pm\Gamma_{y}=\frac{1}{2}\left[(x-x_{i})^{2}\pm(y-y_{i})^{2}\right].$
(32)

In analogy to the case of $N=2$, $\Gamma_{x}\pm\Gamma_{y}$ are simply
determined by the chirality of the coupling strengths $\left(x-x_{i}\right)$
and $\left(y-y_{i}\right)$. As a result, an incident photon is perfectly
transmitted [$T_{l(r)}(\Delta)\equiv 1$] if $x=x_{i}$ or $y=y_{i}$, regardless
of the specific value of the detuning $\Delta$. In the BUEC regime, when the
coupling strengths of the disordered coupling point fulfill the condition

$|x-x_{i}|=|y-y_{i}|,$ (33)

the transmission probabilities reduce to

$T_{l(r)}=\frac{\left(\Delta-\Delta_{LS}\right)^{2}}{\left(\Delta-\Delta_{LS}\right)^{2}+\left(x-x_{i}\right)^{4}},$
(34)

which again possess the anti-Lorentzian line shape of the tunable width
$2(x-x_{i})^{2}$ but with the center ($T_{l(r)}=0$) shifted to
$\Delta=\Delta_{LS}$. In Fig. 2(d), we have shown the density plot of
$T_{l(r)}(\Delta=\Delta_{LS})$ as functions of the chiral coupling
$(x_{j},y_{j})/x$, and indicated the UUEC regime by horizontal and vertical
dot-dashed lines and the BUEC regime with $|x-x_{i}|=|y-y_{i}|$ by white
dashed line. The intersection of the white dashed line with the coordinate
$(x_{j},y_{j})/x=(1,1)$ corresponds to the BEC regime with $T_{l(r)}=1$.
Furthermore, if we take $i=N$, i.e., the coupling strength of the leftmost (or
rightmost) is different from that of the others, Eq. (31) then reduces to
$\Delta_{LS}=\frac{N}{2}\left(x^{2}+y^{2}\right)\text{cot}(m^{\text{$\prime$}}\pi/N)$,
which is exactly the same to that of the BEC regime, see Eq. (19). Note that
$\Delta_{LS}$ is now independent of the coupling strengths {$x_{N}$, $y_{N}$}
at the $N$th coupling point, therefore, by engineering the chirality of the
disordered coupling point, the transmission probabilities $T_{l(r)}(\Delta)$
can be tuned from zero to unity in situ at
$\Delta=\frac{N}{2}\left(x^{2}+y^{2}\right)\text{cot}(m^{\text{$\prime$}}\pi/N)$
with $m^{\text{$\prime$}}=1,2,...,N-1$, and the width of the transmission
window given by $2(x-x_{N})^{2}$ can be flexibly controlled.

In Fig. 3, we plot the transmission probabilities $T_{l(r)}$ as functions of
the detuning $\Delta$ and the two-point phase delay $\tilde{\phi}_{12}$ modulo
$2\pi$ [i.e., $\text{mod}(\tilde{\phi}_{12},2\pi)$] with $y/x=2$ and
$N=\\{2,5,6$}. We consider the set of coupling strengths
$(x_{N},y_{N})/x=\\{(1,2),(3,2),(3,4)\\}$ for the disordered (or the $N$th)
coupling point, corresponding to the BEC, the UUEC, and the BUEC regime,
respectively. We first look at the case of $N=2$ shown in Fig. 3(a), (d) and
(g), for $\text{mod}(\tilde{\phi}_{12},2\pi)=0$, the incident photon is
partially reflected (i.e., $0<T_{l(r)}<1$) in the BEC regime with
$\text{(}x_{N},y_{N})/x=(1,2)$, and nevertheless is totally reflected (i.e
$T_{l(r)}=0$) in the UUEC regime with $\text{(}x_{N},y_{N})/x=(3,2)$
(satisfying $x+x_{i}=y+y_{i}$). In both cases, the incident photon will be
completely transmitted for $\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$, which are
indicated by the gray dashed line. As a result, the two red regions
(corresponding to high transmission probabilities), that are divided by the
white solid line with regard to the condition
$\Delta=\Delta_{LS}(\tilde{\phi}_{12})$, connect to each other around
$\Delta=0$. In contrast, in the BUEC regime with
$\text{(}x_{N},y_{N})/x=(3,4)$ (satisfying $|x-x_{i}|=|y-y_{i}|$), there
appears a blue channel that separates the two high transmission regions and
has the minimum $T_{l(r)}=0$ precisely at $\Delta=0$ with
$\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$. Extending the results to $N=5$
($N=6)$ and comparing them of the three regimes, we then find $N-1$ “total
reflection windows” at
$\Delta\sim\Delta_{LS}=\frac{N}{2}\left(x^{2}+y^{2}\right)\text{cot}(\tilde{\phi}_{12}/2)$
with $\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$
($m^{\text{$\prime$}}=1,2,...,N-1$) in the BUEC regime, and the opened
channels connect the $N$ disjunct blue regions for the BEC and the UUEC
regimes. Moreover, the transmission spectrum along
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$ (indicated by
the horizontal dashed lines) possess the anti-Lorentzian lineshape centered at
$\Delta=\Delta_{LS}$ with tunable width [see Figs. 3(j), (k) and (l)]. It is
worth noting that $\Delta_{LS}$ is independent of the specific values of the
coupling strengths $(x_{i},y_{i})$ for $i=N$, thus, by setting
$\Delta=\Delta_{LS}(\tilde{\phi}_{12}=\frac{2m^{\prime}\pi}{N})$, the setup
can potentially act as a multi-channel photon router, with the frequencies of
incident photons $\omega_{e}+\Delta_{LS}(\tilde{\phi}_{12})$ and the number of
channels $N-1$ under the condition of
$\text{mod}(\tilde{\phi}_{12},2\pi)\in(0,2\pi)$.

##  IV Nonreciprocal photon scattering with atomic spontaneous emission

Figure 4:  Nonreciprocal transport of a single photon by including the
spontaneous emission decay $\gamma/x^{2}=0.2$. The contrast ratio $I$ versus
$x_{i}/x$ and $y_{i}/x$ for (a) $N=5$, $\text{mod}(\tilde{\phi}_{12},2\pi)=pi$
and (d) $N\geq 2$,
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$, with the
corresponding transmission probability $T_{l}$ ($T_{r}$) for a left-incident
(right-incident) photon being shown in (b) [(c)] and (e) [(f)]. The white
dashed lines in panels (a), (d) indicate the optimal nonreciprocity condition
with $I=1$. The white hyperbola dashed lines in (b) [(c)] and (e) [(f)] are
used to label the transmission probabilities with $T_{l(r)}=0$, corresponding
to $\gamma+\Gamma_{x}=\Gamma_{y}$ or $\gamma+\Gamma_{y}=\Gamma_{x}$. Other
parameters are the same as in Figs. 2 (c)-(d).

We now take the atomic spontaneous decay into account, i.e., $\gamma\neq 0$.
Previously, we mentioned that nonreciprocal photon scattering (i.e.,
$T_{l}\neq T_{r}$) can not occur for the giant atom with uniformly symmetric
coupling. Moreover, it is remarkable that only reciprocal photon transport can
be observed with $\gamma=0$ despite the even or uneven chiral coupling.
However, we will see that, by engineering the chirality of the coupling
strengths, nonreciprocal photon transfer can be realized, due to the joint
effect of the chirality (i.e., $\Gamma_{x}\neq\Gamma_{y}$) and the atomic
spontaneous decay $\gamma\neq 0$, and while the reflections are reciprocal
$R_{l}=R_{r}$, see Eqs. (6) and (7). In other words, the atomic spontaneous
decay of the giant atom due to its coupling to the thermal environment and the
external decay due to the interfaced waveguide can cooperatively lead to
nonreciprocal photon transport.

As an example, for $\gamma=\Gamma_{x}-\Gamma_{y}$ and $\Delta=\Delta_{LS}$, a
photon incident from the left will be partially transmitted with the
probability

$T_{l}=\left(1-\frac{\Gamma_{y}}{\Gamma_{x}}\right)^{2},$ (35)

while a photon incident from the right will be completely isolated, i.e.,
$\text{ $T_{r}$}=0$. For a generic situation, the nonreciprocity in
transmission can be described by a contrast ratio defined as

$I=\left|\frac{T_{l}-T_{r}}{T_{l}+T_{r}}\right|=\frac{2\gamma\left|\Gamma_{x}-\Gamma_{y}\right|}{\left(\Delta-\Delta_{LS}\right)^{2}+\gamma^{2}+\left(\Gamma_{x}-\Gamma_{y}\right)^{2}}.$
(36)

Thus, the nonreciprocity can only be observed when both the conditions
$\gamma\neq 0$ and $\Gamma_{x}\neq\Gamma_{y}$ are simultaneously satisfied. By
setting $\Delta=\Delta_{LS}$, one achieves the optimal nonreciprocity $I=1$
with $\gamma=|\Gamma_{x}-\Gamma_{y}|,$ where the spontaneous decay rate
$\gamma$ cancels out the difference between the effective decay rates in the
forward and backward direction. For $\Delta\neq\Delta_{LS}$, the contrast
ratio is always less than one.

In Figs. 4(a) and 4(d), we plot the contrast ratio $I$ as functions of the
coupling strengths $(x_{i},y_{i})/x$ for $y/x=2$, $\gamma/x^{2}=0.2$, (a)-(c)
$N=5$, and (d)-(f) $N\geq 2$. Correspondingly, we show the forward (backward)
transmission probabilities $T_{l}(\Delta=\Delta_{LS})$
[$T_{r}(\Delta=\Delta_{LS})$] for a left-incident (right-incident) photon in
Figs. 4(b)-4(c) with $\text{mod}(\tilde{\phi}_{12},2\pi)=0$ and Figs.
4(e)-4(f) with $\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$.
As discussed above, the optimal nonreciprocity ($I=1$) condition for
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$ is

$\gamma=\pm\frac{\left[\left(N-1\right)x+x_{i}\right]^{2}-\left[\left(N-1\right)y+y_{i}\right]^{2}}{2},$
(37)

and alternatively for
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$
($m^{\text{$\prime$}}=1,2,...,N-1$) is

$\gamma=\pm\frac{\left(x_{i}-x\right)^{2}-\left(y_{i}-y\right)^{2}}{2},$ (38)

corresponding to the hyperbola branches indicated by the white dashed lines in
Fig. 4(a) and Fig. 4(d). Note that for $\text{mod}(\tilde{\phi}_{12},2\pi)=0$,
one can only observe the tail of hyperbola branches since we consider the
coupling strengths being positive values; when physical parameters (coupling
strengths) are extended to the complex plane, the full hyperbola branches will
be clearly seen in the lower left quadrant. However, the optimal
nonreciprocity is achieved at the expense of low transmission probabilities
$T_{l(r)}$ [shown in Fig. 4(b) and Fig. 4(c)]. For
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$, we previously
show in the UUEC regime $T_{l}=T_{r}\equiv 1$ along $y_{i}/y=1$ ($x_{i}/x=1$)
for $\gamma=0$, in contrast to that, when $\gamma\neq 0$, there emerges an
avoided crossing at the BEC point $(x_{i},y_{i})/x=(1,2)$ with a gap $\sim
2\sqrt{2\gamma}$ of the coupling strength $y_{i}/y$ ($x_{i}/x$) for $T_{l}$
($T_{r}$) in Fig. 4(e) [Fig. 4(f)]. Here, the optimal nonreciprocity [Eq.
(38)] corresponds to the total reflection of the incident photon $T_{l}=0$ or
$T_{r}=0$, as indicated by the white dashed lines in Fig. 4(e) and Fig. 4(f).
In particular, by considering the UUEC regime (i.e., $\Gamma_{x}=0$ or
$\Gamma_{y}=0$), we obtain the optimal nonreciprocity $I=1$ with $T_{l}=0$
($T_{r}=1)$ or $T_{l}=1$ ($T_{r}=0)$ for a left-incident (right-incident)
photon, since the chiral coupling $y_{i}$ ($x_{i}$) at the $i$th coupling
point is counteracted by the spontaneous decay according to
$y_{i}=y\pm\sqrt{2\gamma}$ ($x_{i}=x\pm\sqrt{2\gamma}$). Extending to a
generic picture including the BUEC regime, one finally observes $I=1$ on the
hyperbola branches in Fig. 4(d).

##  V The Non-Markovian regime

So far, we consider only the Markovian regime, namely, the time for light to
travel between the leftmost and the rightmost coupling points is much less
than the characteristic relaxation time $\tau_{1N}\ll\tilde{\Gamma}^{-1}$. For
the giant atom with $N$ equally spaced coupling points, where $\tilde{\Gamma}$
is on the same order of $N^{2}x^{2}$ (with regard to the uniformly symmetric
coupling) (Guo et al., 2017; Cai and Jia, 2021; Ask et al., 2019), the system
in the non-Markovian regime implies $\tau_{1N}=(N-1)\tau_{12}$ being
comparable to or even larger than $\tilde{\Gamma}^{-1}$. Moreover, by
considering the bandwidth of the order $|\Delta|\sim\tilde{\Gamma}$ or
$|\Delta|/x^{2}\sim N^{2}$, the effect of $\Delta\tau_{12}$ on the propagating
phase $\phi_{12}=\Delta\tau_{12}+\tilde{\phi}_{12}$ (and therefore the non-
Markovian retardation effect) can not be ignored (Du et al., 2021; Zhou et
al., 2023).

In Fig. 5, we show transmission spectra for $N=2$ coupling points as functions
of $\Delta/x^{2}$ for $x^{2}\tau_{12}=0.25,1,2.5$, respectively. We consider
the set of coupling strengths with
$(x_{1},y_{1},x_{2},y_{2})/x=(1,0.5,1,0.5)$,
$(x_{1},y_{1},x_{2},y_{2})/x=(1,0.5,1,1.5)$, and
$(x_{1},y_{1},x_{2},y_{2})/x=(1,0.5,0.5,1)$, according to the specific chiral
conditions Eqs. (30) and (33). For $x^{2}\tau_{12}=0.25$, the weak non-
Markovian effect starts to influence the transmission spectra. For
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, all of the scattering spectrum have a
single minimum at $\Delta=0$ but with a different width for different chiral
coupling strengths. For $\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$, the
transmission probabilities in the UUEC [with
$(x_{1},y_{1},x_{2},y_{2})/x=(1,0.5,1,1.5)$] and the BEC [with
$(x_{1},y_{1},x_{2},y_{2})/x=(1,0.5,1,0.5)$] regime decline weakly from
$T_{l(r)}(\Delta=0)=1$ as $\Delta$ increases, while in the BUEC regime [with
$(x_{1},y_{1},x_{2},y_{2})/x=(1,0.5,0.5,1)$], $T_{l(r)}(\Delta)$ quickly
transits from zero to almost one for being out of resonance [see Fig. 5(b)].
For $x^{2}\tau_{12}=1$, the system enters the intermediate non-Markovian
regime. For $\text{mod}(\tilde{\phi}_{12},2\pi)=0$, there emerges a frequency
window with the transmission probabilities approximately unchanged, and in
particular, $T_{l(r)}$ remain zero in the frequency window for the BUEC
regime, which is referred to as a photonic band gap with
$|\Delta|/x^{2}\propto N^{2}$ (Cai and Jia, 2021); for
$\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$, $T_{l(r)}$ start to oscillate with
additional minima. In the deep non-Markovian regime with $x^{2}\tau_{12}=2.5$
and $\text{mod}(\tilde{\phi}_{12},2\pi)=0$, the flat band structure splits and
exhibits side valleys, and only in the case with respect to the BUEC regime,
we observe $T_{l(r)}=0$ at the two side valleys close to $\Delta=0$. As
$\Delta$ increases, $T_{l(r)}(\Delta)$ in the BEC and the BUEC regimes then
oscillate between zero and one both for
$\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$ and
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$.

Figure 5: Transmission spectra in the non-Markovian regime with $N=2$ and
$\gamma=0$, where (a), (c), (e) $\text{mod}(\tilde{\phi}_{12},2\pi)=0$ (the
left panels), and (b), (d), (f) $\text{mod}(\tilde{\phi}_{12},2\pi)=\pi$ (the
right panels). The photon propagation time is set as
$x^{2}\tau_{12}=\\{0.25,1,2.5\\}$, corresponding to the weak, intermediate,
and deep non-Markovian regimes, respectively. Coupling strengths for the two
coupling points are
$(x_{1},y_{1},x_{2},y_{2})/x=\\{(1,0.5,1,0.5),(1,0.5,1,1.5),(1,0.5,0.5,1)\\}$,
which correspond to the BEC (blue), the UUEC (green), and the BUEC (red)
regime, respectively. Figure 6: Transmission spectrum in the non-Markovian
regime with $N=5$ and $\gamma=0$, where (a), (c), (e)
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, and (b), (d), (f)
$\text{mod}(\tilde{\phi}_{12},2\pi)=0.8\pi$ [i.e.,
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$ with
$m^{\text{$\prime$}}=2$ without loss of generality]. We set the coupling
strengths of the $N$th coupling point as
$(x_{N},y_{N})/x=\\{(1,0.6),(1,2.6),(0,1.6)\\}$ and that of the others as
$(x_{j},y_{j})/x=(1,0.6)$ ($j\neq N$), which correspond to the BEC (blue), the
UUEC (green), and the BUEC (red) regime, respectively. Moreover, the photon
propagation time is set as $x^{2}\tau_{12}=\\{0.01,0.04,0.1\\}$, corresponding
to the weak, intermediate, and deep non-Markovian regimes, respectively.

In Fig. 6, we show the transmission spectra as functions of $\Delta/x^{2}$
with $N=5$ for $x^{2}\tau_{12}=0.01,0.04,\text{ and }0.1$, respectively. We
consider the propagating time $\tau_{1N}\sim\left(N^{2}x^{2}\right)^{-1}$ from
the first to the last coupling point, and the frequency bandwidth
$|\Delta|\sim N^{2}x^{2}$. The disordered coupling point locates at the $N$th
coupling point with $(x_{N},y_{N})/x=(0,0.6)$ and $(x_{N},y_{N})/x=(1,2.6)$,
and the coupling strengths for the other coupling points are
$(x_{j},y_{j})/x=(1,0.6)$, corresponding to the BUEC and the UUEC regimes
respectively. Moreover, the coupling strengths $(x_{N},y_{N})/x=(0,0.6)$
satisfy both the conditions $x_{N}-y_{N}=\left(1-N\right)\left(x-y\right)$ and
$|x_{N}-x|=|y-y_{N}$|, while the coupling strengths $(x_{N},y_{N})/x=(1,2.6)$
satisfy $x_{N}-y_{N}=\left(1-N\right)\left(x-y\right)$ only. For
$\text{mod}(\tilde{\phi}_{12},2\pi)=0$, the transmission spectra in the three
regimes $x^{2}\tau_{12}=\\{0.01,0.04,0.1\\}$ all show similar features to that
of $N=2$. In the intermediate and deep non-Markovian regimes, the transmission
probabilities saturate to unity after a few oscillations as $\Delta$
increases, see Figs. 6(a), 6(c) and 6(e). For
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$, the lineshape
of the scattering spectra becomes asymmetric due to an odd number of coupling
points, see Figs. 6(b), 6(d) and 6(f), where we take $m^{\text{$\prime$}}=2$
[i.e., $\text{mod}(\tilde{\phi}_{12},2\pi)=0.8\pi$] as an example. In
particular, since the Lamb shift $\Delta_{LS}$ is nonvanishing for
$\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$, the time delay
$\sim\Delta\tau_{12}$ will lead to a shift of the dip (with respect to
$T_{l(r)}\approx 0$) for the BUEC regime [i.e., $(x_{N},y_{N})/x=(0,0.6)$],
while in comparison, the transmission probabilities remain unity at resonance
[i.e., $T_{l(r)}(\Delta=0)\equiv 1$] for the UUEC [$(x_{N},y_{N})/x=(1,2.6)$]
and the BEC [$(x_{N},y_{N})/x=(1,0.6)$] regimes. Remarkably, since there may
not exist an appropriate detuning $\Delta$ fulfilling both
$\Gamma_{x}=\Gamma_{y}$ and $\Delta=\Delta_{LS}$, the transmission probability
for the BEC regime suffers from a sudden jump to
$T_{l(r)}(\Delta_{LS})=\left[(\Gamma_{x}-\Gamma_{y})/(\Gamma_{x}+\Gamma_{y})\right]^{2}$
as $\Delta$ increases [see blue curve in Figs. 6(b), 6(d), and 6(f)]. In
general for $\text{mod}(\tilde{\phi}_{12},2\pi)=2m^{\text{$\prime$}}\pi/N$,
$T_{l(r)}(\Delta)$ gradually decline in a oscillatory pattern as $\Delta$
increases in the UUEC [$(x_{N},y_{N})/x=(1,2.6)$] and the BEC
[$(x_{N},y_{N})/x=(1,0.6)$] regimes, and moreover $T_{l(r)}(\Delta)$ is
nonvanishing even at $\Delta=\Delta_{LS}$ for the BUEC regime
$\\{x_{N},y_{N}\\}/x=(0,0.6)$. These phenomena become more obvious as the
system enters the intermediate and deep non-Markovian regimes.

##  VI CONCLUSION

In conclusion, we have studied the photon scattering spectra by a giant atom,
which is chirally coupled to a waveguide at multiple equally spaced points.
The chirality is categorized in terms of the evenness of coupling strengths in
the left and right propagation directions, and is divided into three regimes,
i.e., the BEC, the UUEC, and the BUEC regimes. In the Markovian limit and for
the two-point propagating phase being $\text{mod}(\tilde{\phi}_{12},2\pi)=0$,
the transmission spectra possess the anti-Lorentzian line shape centered at
$\Delta=\Delta_{LS}$ for all three regimes, where the incident photon can be
totally reflected when the chiral coupling satisfies certain conditions
related to the number of coupling points; for the two-point propagating phase
satisfying mod($\tilde{\phi}_{12},2\pi)=2m^{\prime}\pi/N$, the incident photon
is fully transmitted at $\Delta=\Delta_{LS}$ in the BEC and the UUEC regimes,
but can be totally reflected in the BUEC regime by simply modulating the
coupling strengths at any one of the coupling points, where, interestingly, a
controllable single-photon routing can be implemented in situ at a multiple of
frequency “windows”. The photon transport can become nonreciprocal if the
atomic spontaneous decay is taken into account, and the nonreciprocity
represented by the contrast ratio can be varied from zero to one by
engineering the chirality in atom-waveguide couplings, differing from the non-
Markovianity induced effect in chiral waveguide QED systems (Du et al., 2021;
Zhou et al., 2023). The non-Markovian retardation effect manifested by the
scattering spectra reflects itself mainly in the photonic band gap and the
oscillatory behavior between total reflection and full transmission,
especially in the intermediate and the deep non-Markovian regimes,
respectively. The giant-atom-waveguide system with chiral coupling thus offers
flexible ways for single-photon routing.

###### Acknowledgements.

H.W. acknowledges support from the National Natural Science Foundation of
China (NSFC) under Grants No. 11774058 and No. 12174058. Y.L. acknowledges
support from the NSFC under Grants No. 12074030 and No. 12274107.

## References

  * Roy et al. (2017) D. Roy, C. M. Wilson, and O. Firstenberg, Rev. Mod. Phys. 89, 021001 (2017).
  * Gu et al. (2017) X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, Phy. Rep. 718-719, 1 (2017).
  * Vetsch et al. (2010) E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, Phys. Rev. Lett. 104, 203603 (2010).
  * Corzo et al. (2016) N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, Phys. Rev. Lett. 117, 133603 (2016).
  * Sørensen et al. (2016) H. L. Sørensen, J.-B. Béguin, K. W. Kluge, I. Iakoupov, A. S. Sørensen, J. H. Müller, E. S. Polzik, and J. Appel, Phys. Rev. Lett. 117, 133604 (2016).
  * Goban et al. (2014) A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, et al., Nat. Commun. 5, 3808 (2014).
  * Goban et al. (2015) A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, Phys. Rev. Lett. 115, 063601 (2015).
  * Yan and Wei (2015) C.-H. Yan and L.-F. Wei, Opt. Express 23, 10374 (2015).
  * Astafiev et al. (2010) O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, Science 327, 840 (2010).
  * Hoi et al. (2012) I.-C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P. Delsing, and C. M. Wilson, Phys. Rev. Lett. 108, 263601 (2012).
  * Hoi et al. (2013) I.-C. Hoi, C. Wilson, G. Johansson, J. Lindkvist, B. Peropadre, T. Palomaki, and P. Delsing, New J. Phys. 15, 025011 (2013).
  * van Loo et al. (2013) A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, Science 342, 1494 (2013).
  * Liu and Houck (2017) Y. Liu and A. A. Houck, Nat. Phys. 13, 48 (2017).
  * Kimble (2008) H. J. Kimble, Nature 453, 1023 (2008).
  * Duan and Monroe (2010) L.-M. Duan and C. Monroe, Rev. Mod. Phys. 82, 1209 (2010).
  * Wehner et al. (2018) S. Wehner, D. Elkouss, and R. Hanson, Science 362, eaam9288 (2018).
  * Bello et al. (2023) M. Bello, G. Platero, J. I. Cirac, and A. González-Tudela, Sci. Adv. 5, eaaw0297 (2023).
  * Wang et al. (2021) X. Wang, T. Liu, A. F. Kockum, H.-R. Li, and F. Nori, Phys. Rev. Lett. 126, 043602 (2021).
  * Mahmoodian et al. (2020) S. Mahmoodian, G. Calajó, D. E. Chang, K. Hammerer, and A. S. Sørensen, Phys. Rev. X 10, 031011 (2020).
  * Gustafsson et al. (2014) M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekström, G. Johansson, and P. Delsing, Science 346, 207 (2014).
  * Manenti et al. (2017) R. Manenti, A. F. Kockum, A. Patterson, T. Behrle, J. Rahamim, G. Tancredi, F. Nori, and P. J. Leek, Nat. Commun. 8, 975 (2017).
  * Guo et al. (2017) L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, Phys. Rev. A 95, 053821 (2017).
  * Andersson et al. (2019) G. Andersson, B. Suri, L. Guo, T. Aref, and P. Delsing, Nat. Phys. 15, 1123 (2019).
  * Andersson et al. (2020) G. Andersson, M. K. Ekström, and P. Delsing, Phys. Rev. Lett. 124, 240402 (2020).
  * Wang et al. (2022) Z.-Q. Wang, Y.-P. Wang, J. Yao, R.-C. Shen, W.-J. Wu, J. Qian, J. Li, S.-Y. Zhu, and J. Q. You, Nat. Commun. 13, 7580 (2022).
  * Frisk Kockum et al. (2014) A. Frisk Kockum, P. Delsing, and G. Johansson, Phys. Rev. A 90, 013837 (2014).
  * Kannan et al. (2020) B. Kannan, M. J. Ruckriegel, D. L. Campbell, A. Frisk Kockum, J. Braumüller, D. K. Kim, M. Kjaergaard, P. Krantz, A. Melville, B. M. Niedzielski, et al., Nature 583, 775 (2020).
  * Vadiraj et al. (2021) A. M. Vadiraj, A. Ask, T. G. McConkey, I. Nsanzineza, C. W. S. Chang, A. F. Kockum, and C. M. Wilson, Phys. Rev. A 103, 023710 (2021).
  * Qiu et al. (2023) Q.-Y. Qiu, Y. Wu, and X.-Y. Lü, Sci. China Phys. Mech. Astron. 66, 224212 (2023).
  * Kockum et al. (2018) A. F. Kockum, G. Johansson, and F. Nori, Phys. Rev. Lett. 120, 140404 (2018).
  * Carollo et al. (2020) A. Carollo, D. Cilluffo, and F. Ciccarello, Phys. Rev. Res. 2, 043184 (2020).
  * Du et al. (2023) L. Du, L. Guo, and Y. Li, Phys. Rev. A 107, 023705 (2023).
  * Guo et al. (2020) S. Guo, Y. Wang, T. Purdy, and J. Taylor, Phys. Rev. A 102, 033706 (2020).
  * Cilluffo et al. (2020) D. Cilluffo, A. Carollo, S. Lorenzo, J. A. Gross, G. M. Palma, and F. Ciccarello, Phys. Rev. Res. 2, 043070 (2020).
  * Lim et al. (2023) K. H. Lim, W.-K. Mok, and L.-C. Kwek, Phys. Rev. A 107, 023716 (2023).
  * Soro and Kockum (2022) A. Soro and A. F. Kockum, Phys. Rev. A 105, 023712 (2022).
  * Du et al. (2021) L. Du, Y.-T. Chen, and Y. Li, Phys. Rev. Res. 3, 043226 (2021).
  * Du et al. (2022) L. Du, Y. Zhang, J.-H. Wu, A. F. Kockum, and Y. Li, Phys. Rev. Lett. 128, 223602 (2022).
  * Wang and Li (2022) X. Wang and H.-R. Li, Quantum Sci. Technol. 7, 035007 (2022).
  * Zhou et al. (2023) J. Zhou, X.-L. Yin, and J.-Q. Liao, Phys. Rev. A 107, 063703 (2023).
  * Du and Li (2021) L. Du and Y. Li, Phys. Rev. A 104, 023712 (2021).
  * Chen et al. (2022) Y.-T. Chen, L. Du, L. Guo, Z. Wang, Y. Zhang, Y. Li, and J.-H. Wu, Commun. Phys. 5, 215 (2022).
  * Cai and Jia (2021) Q. Y. Cai and W. Z. Jia, Phys. Rev. A 104, 033710 (2021).
  * Feng and Jia (2021) S. L. Feng and W. Z. Jia, Phys. Rev. A 104, 063712 (2021).
  * Peng and Jia (2023) Y. P. Peng and W. Z. Jia, Phys. Rev. A 108, 043709 (2023).
  * Zhao et al. (2022) W. Zhao, Y. Zhang, and Z. Wang, Front. Phys. 17, 42506 (2022).
  * Yin et al. (2022) X.-L. Yin, Y.-H. Liu, J.-F. Huang, and J.-Q. Liao, Phys. Rev. A 106, 013715 (2022).
  * (48) A. Ask, Y.-L. L. Fang, and A. F. Kockum, eprint arXiv:2011.15077.
  * Koch et al. (2007) J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007).
  * Joshi et al. (2023) C. Joshi, F. Yang, and M. Mirhosseini, Phys. Rev. X 13, 021039 (2023).
  * Sathyamoorthy et al. (2014) S. R. Sathyamoorthy, L. Tornberg, A. F. Kockum, B. Q. Baragiola, J. Combes, C. M. Wilson, T. M. Stace, and G. Johansson, Phys. Rev. Lett. 112, 093601 (2014).
  * Sliwa et al. (2015) K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Phys. Rev. X 5, 041020 (2015).
  * Chapman et al. (2017) B. J. Chapman, E. I. Rosenthal, J. Kerckhoff, B. A. Moores, L. R. Vale, J. A. B. Mates, G. C. Hilton, K. Lalumière, A. Blais, and K. W. Lehnert, Phys. Rev. X 7, 041043 (2017).
  * Müller et al. (2018) C. Müller, S. Guan, N. Vogt, J. H. Cole, and T. M. Stace, Phys. Rev. Lett. 120, 213602 (2018).
  * Gonzalez-Ballestero et al. (2016) C. Gonzalez-Ballestero, E. Moreno, F. J. Garcia-Vidal, and A. Gonzalez-Tudela, Phys. Rev. A 94, 063817 (2016).
  * Shen and Fan (2009) J.-T. Shen and S. Fan, Phys. Rev. A 79, 023837 (2009).
  * Frisk Kockum (2021) A. Frisk Kockum, in _International Symposium on Mathematics, Quantum Theory, and Cryptography_ , edited by T. Takagi, M. Wakayama, K. Tanaka, N. Kunihiro, K. Kimoto, and Y. Ikematsu (Springer Singapore, Singapore, 2021), pp. 125–146.
  * Ask et al. (2019) A. Ask, M. Ekström, P. Delsing, and G. Johansson, Phys. Rev. A 99, 013840 (2019).