# Chiral SQUID-metamaterial waveguide for circuit-QED

Xin Wang<EMAIL_ADDRESS>Institute of Theoretical Physics, School of
Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China,
Ya-Fen Lin  Institute of Theoretical Physics, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, People’s Republic of China,  Jia-Qi Li
Institute of Theoretical Physics, School of Physics, Xi’an Jiaotong
University, Xi’an 710049, People’s Republic of China,  Wen-Xiao Liu
Department of Physics and Electronics, North China University of
Water Resources and Electric Power, Zhengzhou 450046, People’s Republic of
China  Hong-Rong Li  Institute of Theoretical Physics, School of Physics,
Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China,

###### Abstract

Superconducting metamaterials, which are designed and fabricated with
structured fundamental circuit elements, have motivated recent developments of
exploring unconventional quantum phenomena in circuit quantum electrodynamics
(circuit-QED). We propose a method to engineer 1D Josephson metamaterial as a
chiral waveguide by considering a programmed spatiotemporal modulation on its
effective impedance. The modulation currents are in the form of travelling
waves which phase velocities are much slower than the propagation speed of
microwave photons. Due to the Brillouin-scattering process, non-trivial
spectrum regimes where photons can propagate unidirectionally emerge.
Considering superconducting qubits coupling with this metamaterial waveguide,
we analyze both Markovian and non-Markovian quantum dynamics, and find that
superconducting qubits can dissipate photons unidirectionally. Moreover, we
show that our proposal can be extended a cascaded quantum network with
multiple nodes, where chiral photon transport between remote qubits can be
realized. Our work might open the possibilities to exploit SQUID metamaterials
for realizing unidirectional photon transport in circuit-QED platforms.

##  I introduction

The interaction between matter and quantized electromagnetic fields has been
the central topic of quantum optics for more than half a century Lamb and
Retherford (1947); Scully and Zubairy (1997); Cohen-Tannoudji  _et al._
(1998); Walls and Milburn (2007); Clerk  _et al._ (2010). The impressive
progresses in quantum electrodynamics (QED) and quantum technologies provide
solid foundations for quantum information science Clerk  _et al._ (2010);
Buluta  _et al._ (2011); Xiang  _et al._ (2013); Underwood  _et al._ (2012);
Reiserer and Rempe (2015); Chang  _et al._ (2018). In past two decades,
superconducting quantum circuits (SQCs) with Josephson junctions have been
developed as a well-performed artificial platform for microwave photonics
Nakamura  _et al._ (1999); Blais  _et al._ (2004); Koch  _et al._ (2007);
Clarke and Wilhelm (2008); You and Nori (2011); Gu  _et al._ (2017); et al.
(2019a); Clerk  _et al._ (2020); Blais  _et al._ (2021); et al. (2021). The
versatility of SQCs stems from the flexibilities in both fabrication and
controlling processes, which allows to realize many exotic quantum phenomena
such as dynamical Casimir effects and ultrastrong light–matter couplings
Johansson  _et al._ (2009, 2010); et al. (2011); Lähteenmäki  _et al._ (2013);
et al. (2011); Beaudoin  _et al._ (2011); Kockum  _et al._ (2019); Bourassa
_et al._ (2009); et al. (2010).

In circuit-QED the boson modes of microwave photons are routinely supported by
conventional circuit elements such as LC resonators, transmission-line
resonators and 1D open coplanar-waveguides et al. (2007, 2008); Bourassa  _et
al._ (2012); Clem (2013); Doerner  _et al._ (2018). Recently exploring
intriguing QED phenomena with metamaterial structures in SQCs have attracted a
lot of interests Leib  _et al._ (2012); Liu and Houck (2017); et al. (2017);
Kollár  _et al._ (2019); Ma  _et al._ (2019); Carusotto  _et al._ (2020);
Mazhorin  _et al._ (2022). Compared with standard circuit-QED elements,
superconducting metamaterials can be designed and fabricated in various kinds
of spatial structures, which might have exotic band spectrum and nontrivial
vacuum modes Mirhosseini  _et al._ (2018); Scigliuzzo  _et al._ (2021). For
example, by engineering the hopping rates among microwave resonators or SQC
qubits, the photonic metamaterials analogue to topological lattices and
strongly correlated matters were realized Roushan  _et al._ (2017); Kim  _et
al._ (2021). When the inductors and capacitors in the usual discrete model of
a 1D transmission line are interchanged, the left-handed metamaterials can
also be fabricated, which might work as a waveguide with an effective negative
index of refraction Caloz  _et al._ (2004); Egger and Wilhelm (2013); et al.
(2019b); Messinger  _et al._ (2019); Indrajeet  _et al._ (2020).

The Josephson-chain metamaterials, which inherit the advantages of Josephson
junctions such as nonlinearity and tunability, have been widely employed for
different quantum engineering tasks Masluk  _et al._ (2012); Altimiras  _et
al._ (2013); Weissl (2014); et al. (2015); Krupko and Nguyen (2018); Cosmic
_et al._ (2018); Planat  _et al._ (2020); Esposito  _et al._ (2022); Sinha
_et al._ (2022). For example, many-body quantum optics in ultra-strong
coupling regimes was successfully demonstrated in a Josephson chain platform
by exploiting its high characteristic impedance et al. (2019c). Moreover,
given that each node junction is replaced with a superconducting quantum
interference device (SQUID), the SQUID chain is possible to be engineered as a
tunable high-impedance ohmic reservoir by applying position-dependent magnetic
flux Rastelli and Pop (2018). When the SQUID chain are biased with a space-
time varying flux, analogue Hawking radiation can be reproduced Nation  _et
al._ (2009, 2012); Tian  _et al._ (2017); Tian and Du (2019); Lang and
Schützhold (2019); Blencowe and Wang (2020). To create an event horizon, the
group velocity of the modulation signals should be comparable to the
propagation velocity.

In contrast to proposals which are analogs of creating cosmological particles
Nation  _et al._ (2009); Lang and Schützhold (2019), in this study we focus on
the parameter regime where the velocity of the modulation signals is much
slower than the photon’s propagation speed, which was rarely considered in
previous studies. The scenario is akin to realizing nonreciprocal sound
propagation in an elastic metamaterial via spatiotemporally modulating the
stiffness Swinteck  _et al._ (2015); Trainiti and Ruzzene (2016); Yi  _et al._
(2017); Croënne  _et al._ (2017); Riva  _et al._ (2019); Karkar  _et al._
(2019); Chen  _et al._ (2019); Attarzadeh  _et al._ (2020), where the
Brillouin-scattering process will lead to chiral acoustic wave transport. In
this work we first derive the dispersion relation of the metamaterial
waveguide by employing the generalized Bloch-Floquet expansion formula. We
find that there are three non-trivial dispersion regimes, which respectively
correspond to left/right chiral emission and band gaps. Then we show that the
metamaterial waveguide can be engineered as a chiral quantum channel when
coupling it to superconducting qubits. The chiral transport is similar to the
proposal of realizing nonreciprocal photonic transport via acoustic modulation
Calajó  _et al._ (2019). However, the modulating signal in our proposal is the
electromagnetic current bias (rather than acoustic waves), which can be tuned
much faster and freely. Our proposal can be employed for demonstrating chiral
quantum optics widely discussed in Refs. Cirac  _et al._ (1997); Petersen  _et
al._ (2014); Lodahl  _et al._ (2017); Zhang  _et al._ (2021); Guimond  _et
al._ (2020); Gheeraert  _et al._ (2020); Solano and Sinha (2021); Kannan  _et
al._ (2022); Wang  _et al._ (2022); Wang and Li (2022), and provide a novel
method to realize nonreciprocal transport of microwave photons without using
the bulky ferrite circulator Hogan (1953); Caloz  _et al._ (2018). Very recent
studies with circuit-QED setups showed that chiral emission from qubit pairs
(coupled to a common waveguide) can be realized via linear interference
process Guimond  _et al._ (2020); Gheeraert  _et al._ (2020); Solano and Sinha
(2021); Kannan  _et al._ (2022). However, the chiral transport is restricted
at the single-photon level. Our proposal does not have such limitations and
can maintain the chiral behaviour even in the multi-excitation regime Pichler
_et al._ (2015); Mahmoodian  _et al._ (2018); Prasad  _et al._ (2020);
Mahmoodian  _et al._ (2020); Kusmierek  _et al._ (2022). In the last part, we
show that our proposal can be extended as a chiral quantum network where
remote nodes are mediated with no information back flow.

##  II Model

Figure 1: The proposed setup where a SQUID-metamaterial waveguide couples to a
transmon qubit. The artificial waveguide is composed by a SQUID array, which
impedance is modulated by a programmed current bias in each unit structure.
The current signal $I(j,t)$ changing with both SQUID’s positions $j$ and time
$t$ will spatiotemporally modulated the flux $\Phi_{j}$ through each SQUID via
mutual inductance $M_{j}$. The Josephson (ground) capacitance of each SQUID is
denoted as $C_{J}$ ($C_{0}$). The node flux $\phi_{j}$ at position $j$ follows
the Kirchhoff current relation in Eq. (2). A transmon qubit, which is in the
form of a split junction with Josephson capacitance $C_{q}$, interacts with
the waveguide at position $x_{0}=0$ via a coupling capacitance $C_{J}^{g}$.

As shown in Fig 1, we consider that the Josephson-metamaterial waveguide is
composed by a SQUID array which allows microwave photons propagating along it
Weissl (2014); et al. (2015); Krupko and Nguyen (2018). Each node is connected
to the ground with a capacitance $C_{g}$. Two neighbor sites are separated
with distance $d_{0}$. The SQUID can be viewed as a nonlinear Josephson
inductance $L_{j}$ in parallel with a capacitance $C_{J}$. A series of current
biases which can be programmed with external signals will produce site-
dependent flux $\Phi_{j}$ via mutual inductance $M_{j}$. The relation between
$L_{j}$ and $\Phi_{j}$ is given by Wallquist  _et al._ (2006); Sandberg and et
al. (2008); Eichler and Wallraff (2014); Pogorzalek and Fedorov (2017);
Eichler and Petta (2018); Wang  _et al._ (2021)

$L_{j}=\frac{L_{0}}{\cos\left|\frac{\pi\Phi_{j}}{\Phi_{0}}\right|},\quad
L_{0}=\frac{\Phi_{0}^{2}}{8\pi^{2}E_{s0}},$ (1)

where $E_{s0}$ is the Josephson energy of one junction. Defining the node flux
as $\phi_{j}$, we obtain the following Kirchhoff current equation for the
SQUID chain Weissl (2014); et al. (2015)

$\displaystyle\frac{\phi_{j+1}-\phi_{j}}{L_{j+1}}-\frac{\phi_{j}-\phi_{j-1}}{L_{j}}=$
$\displaystyle
C_{g}\ddot{\phi}_{j}+C_{J}\left(\ddot{\phi}_{j}-\ddot{\phi}_{j-1}\right)-C_{J}\left(\ddot{\phi}_{j+1}-\ddot{\phi}_{j}\right).$
(2)

We assume that the current biases are composed of a static part and a time-
dependent part, respectively. That is,

$\Phi_{j}(t)=M_{j}[I_{0}+\delta If(j,t)],$ (3)

where $I_{0}$ ($\delta I$) is the amplitude of the DC (AC) part satisfying
$\delta If(j,t)\ll I_{0}$. We define $G_{j}(t)$ as

$\displaystyle
G_{j}(t)=\frac{1}{L_{j}(t)}=\frac{1}{L_{0}}\left[\alpha_{0}+\delta\alpha
f(j,t)\right],$ (4)

where

$\displaystyle\alpha_{0}=\cos\left(\frac{\pi M_{j}I_{0}}{\Phi_{0}}\right),$
(5) $\displaystyle\delta\alpha=-\sin\left(\frac{\pi
M_{j}I_{0}}{\Phi_{0}}\right)\frac{\pi M_{j}\delta I}{\Phi_{0}}.$ (6)

As discussed in Appendix A, given that $\phi_{j}$ varies slowly in the length
scale $d_{0}$ (i.e., long wavelength limit), the difference Equation (2) can
be written in a quasi-continuous form. Defining $c_{g}=C_{g}/d_{0}$
($c_{J}=C_{J}/d_{0}$) as the ground (Josephson) capacitance per unit length,
the wave function in the quasi-continuous limit becomes Wang  _et al._ (2021)

$c_{g}\frac{\partial^{2}\phi(x,t)}{\partial
t^{2}}=c_{J}d_{0}^{2}\frac{\partial^{4}\phi(x,t)}{\partial t^{2}\partial
x^{2}}+\frac{\partial}{\partial
x}\left[g(x,t)\frac{\partial\phi(x,t)}{\partial x}\right],$ (7)

where $g(x,t)$ is expressed as

$g(x,t)=\frac{1}{l_{J}}\left[\alpha_{0}+\delta\alpha f(x,t)\right],\quad
l_{J}=\frac{L_{0}}{d_{0}},$ (8)

with $l_{J}$ being the Josephson inductance per unit length. The
spatiotemporal modulation signal is encoded in $f(x,t)$. When $C_{J}=0$, Eq.
(7) is simplified as a transmission-line equation in which the impedance is
modulated by a space-time-varying wave. A non-zero Josephson capacitance
$c_{J}$ will entangle both spatial and temporal differentials together, which
will produce a nonlinear dispersion for the SQUID waveguide. In Appendix A and
following discussions, we discuss its effects and give the conditions where
$c_{J}$ can be neglected.

$d_{0}$ |  $C_{g}$ |  $C_{J}$ |  $L_{0}$ |  $\alpha$ |  $\delta\alpha/\alpha$ |  $k_{d}/(2\pi)$ |  $v_{0}$ |  $v_{d}$
---|---|---|---|---|---|---|---|---
$1~{}\mu\text{m}$ |  0.2 fF |  100 fF |  0.2 nH |  0.3 |  $0.15$ |  $0.25\times 10^{4}\text{m}^{-1}$ |  $2.7\times 10^{6}~{}\text{m/s}$ |  $(0.05\sim 0.08)v_{0}$
Table 1: The parameters adopted for calculating the spectrum of the modulated
SQUID-metamaterial waveguide.

The eigen-wave functions of the field are obtained by adopting a generalized
Bloch-Floquet expansion Slater (1958); Trainiti and Ruzzene (2016); Calajó
_et al._ (2019)

$\phi_{lk}(x,t)=e^{i(\widetilde{\omega}_{l}(k)t-kx)}\sum_{n=-\infty}^{n=\infty}u_{ln}(k)e^{in(\Omega_{d}t-k_{d}x)},$
(9)

where $\widetilde{\omega}_{l}$ is the quasi-energy of the $l$th band, and $k$
is the wave number. For convenience, we define $v_{0}=1/\sqrt{l_{J}c_{g}}$ as
the propagation velocity of the static SQUID metamaterial waveguide without
modulation. Now we consider the bias current $g(x,t)$ varying periodically in
both space and time, i.e.,

$g(x,t)=g(x+\lambda_{d},t+T_{d}).$

The simplest modulation signal is a travelling wave, and in this work we adopt
$f(x,t)$ as

$f(x,t)=\cos{(\Omega_{d}t-k_{d}x)}.$ (10)

The phase velocity is $v_{d}=\Omega_{d}/k_{d}$ with $k_{d}=2\pi/\lambda_{d}$
($\Omega_{d}=2\pi/T_{d}$) being the wave number (angular frequency). By
substituting Eq. (9) into Eq. (7), the dispersion relation is obtained by
solving a quadratic eigenvalue problem Trainiti and Ruzzene (2016). Detailed
derivations can be found in Appendix A.

According to the experiments in Refs. et al. (2015); Krupko and Nguyen (2018);
et al. (2019c), we list the parameters adopted for our numerical simulation in
Table 1. Although we can employ the generalized Floquet form in Eq. (9) to
derive the spectrum, it does not mean that all kinds of modulation waves can
produce stable eigenmodes in the SQUID transmission line. As discussed in Ref.
Cassedy (1967), when the modulation velocity $v_{d}$ is faster than $v_{0}$,
the eigenfrequencies become complex, indicating that the oscillations are
time-growing and unstable. The high-speed modulation signal can be employed
for conversing vacuum fluctuations into photons, which is analog to the
Hawking effect in a gravitational system Nation  _et al._ (2009); Blencowe and
Wang (2020).

Figure 2: (a) Dispersion relation between quasi-energy $\omega_{l}(k)$ and
wave number $k$ for $C_{J}=500C_{g}$ and $C_{J}=0$, respectively. The enlarged
plot of the band gap [the area inside the box in (a)] is depicted in (b). The
dispersion relations for different Floquet orders $\\{l,n\\}$ [corresponding
to time-dependent part in Eq. (9)] are shown. The distribution ratio of each
Floquet order is mapped with colors. The nontrivial spectra can be divided
into three parts. In the red (blue) area, the microwave photons propagate into
the right (left) direction; In the grey area, the microwave photon will be
localized due to no propagating modes.

In this study, we focus on the parameter regimes with $v_{d}\ll v_{0}$. The
first Brillouin zone (BZ) is limited within $k\in(-0.5k_{d},0.5k_{d}]$. There
will be Brillouin-scattering processes between modes $k$ and $k+k_{d}$ (which
are wave numbers for the eigenmodes of the static waveguide) due to the
conservation of momentum. Additionally, the conservation of energy also
requires $\omega_{l}(k)\pm\Omega_{d}\simeq\omega_{l}(k+k_{d})$. Since
$\Omega_{d}$ is much smaller than $\omega_{l}$ at $k=k_{d}/2$, the modes
around $k\simeq\pm 0.5k_{d}$ will interact with each other. The interactions
between modes produces two band gaps, which is similar to the appearance of an
anti-crossing point in a two-coupled-mode system.

We assume that the waveguide is long enough ($L\rightarrow\infty$) to support
the microwave photons propagating without reflection. By quantizing the field,
the root-mean-square voltage operator $\hat{V}(x,t)$ is written as Gu  _et
al._ (2017)

$\hat{V}(x,t)\simeq-i\sum_{lk}\sqrt{\frac{\hbar\widetilde{\omega}_{l}(k)}{2C_{t}}}\left[\phi_{lk}^{*}(x,t)a_{lk}-\phi_{lk}(x,t)a^{\dagger}_{lk}\right],$
(11)

where $C_{t}=Lc_{g}$ is the total capacitance of the waveguide, and $a_{lk}$
($a_{lk}^{\dagger}$) is the annihilation (creation) operator for mode $k$ in
the $l$th quasi-energy band. We numerically plot $\omega_{l}(k)$ changing with
wave number $k$ in Fig. 2(a) by adopting parameters in Table 1. We find that
an energy gap emerges between the two lowest bands in the first BZ
$(-0.5k_{d},0.5k_{d}]$. As discussed in Appendix A, under the following
condition

$\sqrt{\frac{c_{J}}{c_{g}}}\ll\frac{1}{\left(k+nk_{d}\right)d_{0}},$ (12)

the nonlinear effect led by $c_{J}$ can be neglected. By adopting the
parameters in Table 1, one finds that $k_{d}d_{0}=0.25\times 10^{-2}\ll 1$.
According to Eq. (12), even when $C_{J}=500C_{g}$, the nonlinear effect of
Josephson capacitance is not apparent, and the dispersion relation is quite
close to that with $C_{J}=0$.

Equation (9) shows that the time-dependent part of $\phi_{lk}(x,t)$ depends on
both band index $l$ and Floquet order $n$, and the distribution ratio of the
$n$th Floquet order is $|u_{ln}(k)|^{2}$ (note that
$\sum_{n}|u_{ln}(k)|^{2}=1$) for a certain mode $lk$. Since the modulation
signal $g(x,t)$ can be viewed as a perturbation, $|u_{ln}(k)|$ decreases
quickly with $n$ according to our numerical calculations. For the parameters
in Table 1, it is accurate enough to consider $n=0,\pm 1$, and the
corresponding dispersion relation is plotted in Fig. 2(b) with
$|u_{ln}(k)|^{2}$ being mapped with colors.

The modulation signal $g(x,t)$ propagates unidirectionally (rather than a
standing wave carrying opposite momentums $\pm k_{d}$), which will open two
_asymmetric_ energy gaps located around $k=\pm 0.5k_{d}$ (see Fig. 2).
Consequently, the unconventional spectrum regime in Fig. 2 can be divided into
three parts. The gray area between two quasi-energy bands, where
$|u_{ln}(k)|\simeq 0$ are valid for all Floquet orders $\\{l,n\\}$, is the
conventional band gap with no propagating mode in the waveguide. In red (blue)
area where point A (B) is located, the dispersion relation is asymmetric and
the Floquet order $\\{l=1,n=0\\}$ ($\\{l=2,n=-1\\}$) possesses the highest
distribution ratio $|u_{ln}(k)|$. In these two regimes microwave photons
propagate unidirectionally, which will be discussed below by considering a
superconducting qubit interacting with this metamaterial waveguide.

##  III Chiral emission of superconducting qubits

###  III.1 Interaction Hamiltonian of metamaterial waveguide coupling to
superconducting qubits

As depicted in Fig. 1(a), we first consider the simplest QED setup where a
transmon qubit interacts with the metamaterial waveguide via a coupling
capacitance $C^{g}_{J}$ at $x_{0}$. The transom qubit is composed by two
identical junctions which form a SQUID loop, and its Hamiltonian is written as
Blais  _et al._ (2004); Koch  _et al._ (2007)

$H_{\text{q}}=4E_{C}(\hat{n}-n_{g})^{2}-2E^{q}_{J}\cos\\!\left(\frac{\pi\Phi_{\text{q}}}{\Phi_{0}}\right)\cos{\hat{\phi}},$
(13)

where $\hat{n}$ ($\hat{\phi}$) is the charge (phase) operator of the transmon,
$C_{\Sigma}=C^{q}_{J}+C_{q}$ with $C_{q}$ being the Josephson capacitance, and
$E^{q}_{J}$ ($E_{C}=e^{2}/(2C_{\Sigma})$) is the Josephson (charging) energy
of the junction. The interaction Hamiltonian between the transmon and the
waveguide is written as Koch  _et al._ (2007)

$H_{\text{int}}=2e\frac{C^{q}_{J}}{C_{\Sigma}}\hat{V}(x,t)\hat{n}.$ (14)

In the limit $E_{C}\ll E_{J}^{q}$, the transmon can be viewed as a Duffing
nonlinear oscillator. Given that only the two lowest energy levels are
considered, $H_{\text{q}}$ can be approximately written as Koch  _et al._
(2007)

$H_{\text{q}}=\frac{1}{2}\omega_{q}\sigma_{z},\qquad\omega_{q}=4\sqrt{E_{C}E^{q}_{J}}-E_{C},$
(15)

where the charge operator is

$\hat{n}=-i\sqrt[4]{\frac{E_{J}^{q}}{4E_{C}}}\frac{\sigma_{-}-\sigma_{+}}{\sqrt{2}}.$
(16)

In the rotating frame of
$\omega_{q}\sigma_{z}/2+\sum_{lk}\hbar\widetilde{\omega}_{l}(k)a^{\dagger}_{lk}a_{lk}$
and by adopting the rotating-wave approximation, we rewrite $H_{\text{int}}$
as

$\displaystyle H_{\text{int}}$ $\displaystyle=$ $\displaystyle\hbar
g_{0}\sum_{lk}e^{i\omega_{q}t}\sigma_{+}a_{lk}\phi^{*}_{lk}(x_{c},t)+\text{H.c.}$
(17) $\displaystyle=$
$\displaystyle\hbar\sum_{lk}g_{lk}\sigma_{+}a_{lk}e^{i\omega_{q}t}$
$\displaystyle\times$
$\displaystyle\left[e^{-i\widetilde{\omega}_{l}(k)t}\sum_{n=-\infty}^{n=\infty}u^{*}_{ln}(k)e^{-in\Omega_{d}t}\right]+\text{H.c.},$

where the coupling position is set at $x_{0}=0$ without loss of generality,
and $g_{lk}$ is the coupling strength which is derived as

$g_{lk}=\sqrt{2}\frac{eC^{q}_{J}}{\hbar
C_{\Sigma}}\sqrt[4]{\frac{E_{J}^{q}}{4E_{C}}}\sqrt{\frac{\hbar\widetilde{\omega}_{l}(k)}{2C_{t}}}.$
(18)

Here we take the transmon as an example. As discussed in experimental work in
Refs. Rastelli and Pop (2018); et al. (2019c); Wang  _et al._ (2021), the
SQUID-metamaterial waveguide can also interact with a flux or charge qubit,
and all these circuit-QED setups can be employed to demonstrate the
unconventional emission behaviors discussed in this work.

###  III.2 Chiral emission in Markovian regime

Figure 3: (a) The time-dependent evolution $|c_{e}(t)|^{2}$ for different
transmon frequencies, and the field distributions are shown in (b-e),
respectively. The corresponding relations are (b) bidirectional emission
$\rightarrow\omega_{q}=2.70~{}\text{GHz}$, (c) right emission
$\rightarrow\omega_{q}=3.02~{}\text{GHz}$, (d) trapped bound state
$\rightarrow\omega_{q}=3.34~{}\text{GHz}$ and (e) left emission
$\rightarrow\omega_{q}=3.64~{}\text{GHz}$. The system’s parameters are adopted
from Table 1 with $v_{d}=0.05v_{0}$.

Similar to the standard spontaneous emission process, the intensity of
radiation field should be narrowly centered around the atomic transition
frequency $\omega_{q}$. Therefore, by replacing $\widetilde{\omega}_{l}(k)$ in
Eq. (18) with $\omega_{q}$, $g_{lk}$ becomes mode-independent, i.e.,
$g_{lk}\simeq g_{0}$. We assume that a single excitation is initially in the
transmon qubit, while the waveguide is in vacuum state $|0\rangle$. Due to the
conservation of excitation number for the Hamiltonian in Eq. (17), the
system’s state at time $t$ can be expressed in the single-excitation subspace
as

$|\Psi(t)\rangle=c_{e}(t)|e,0\rangle+\sum_{l,k}c_{lk}(t)|g,1_{lk}\rangle,$
(19)

where $|1_{lk}\rangle$ represents a single photon being excited in mode
$a_{lk}$. The evolution governed by $H_{\text{int}}$ can be derived from the
following coupled differential equations

$\displaystyle\frac{\partial c_{e}(t)}{\partial t}$ $\displaystyle=$
$\displaystyle-
ig_{0}\sum_{lkn}c_{lk}u_{ln}(k)e^{[i\Delta_{l}(k)-in\Omega_{d}]t},$ (20)
$\displaystyle\frac{\partial c_{lk}(t)}{\partial t}$ $\displaystyle=$
$\displaystyle-
ig_{0}c_{e}(t)e^{-i\Delta_{l}(k)t}\sum_{n}u_{ln}^{*}(k)e^{in\Omega_{d}t},$
(21)

where $\Delta_{l}(k)=\omega_{q}-\widetilde{\omega}_{l}(k)$ is the frequency
detuning between the qubit and the mode $a_{lk}$. By substituting the
integration form of Eq. (21) into Eq. (20), we obtain

$\displaystyle\frac{\partial c_{e}(t)}{\partial t}$ $\displaystyle=$
$\displaystyle-
g_{0}^{2}\int_{0}^{t}dt^{\prime}G(t,t^{\prime})c_{e}(t^{\prime}),$ (22)
$\displaystyle G(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{lknn^{\prime}}e^{-i(n-n^{\prime})\Omega_{d}t}u_{ln}(k)u_{ln^{\prime}}^{*}(k)e^{i[\Delta_{l}(k)-n\Omega_{d}](t-t^{\prime})},$
(23)

where $G(t,t^{\prime})$ is the time-delay correlation function. Given that
$n\neq n^{\prime}$, $e^{-i(n-n^{\prime})\Omega_{d}t}$ are fast oscillating
terms, and their contributions to the evolution will be significantly
suppressed when the decaying time scale is much longer than the oscillating
period $\Omega_{d}^{-1}$. Under these conditions, we can only keep the
resonant term $n=n^{\prime}$, which is similar to the rotating-wave
approximation Calajó  _et al._ (2019). Finally we obtain

$\displaystyle G(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{lkn}|u_{ln}(k)|^{2}e^{i[\Delta_{l}(k)-n\Omega_{d}](t-t^{\prime})}$
(24) $\displaystyle=$ $\displaystyle\sum_{ln}\frac{L}{2\pi}\int
dk|u_{ln}(k)|^{2}e^{i[\Delta_{l}(k)-n\Omega_{d}](t-t^{\prime})}.$

In the emission spectrum, the intensity of the field will center at the modes
satisfying the resonant condition

$\Delta_{l}(k)-n\Omega_{d}=0,$ (25)

which solutions of $k$ are denoted as $k_{ln}$. For example, in Fig. 2(b) for
the frequency at the dashed position in the red regime, we mark the resonant
positions (green dot $A$) for $\\{l=1,n=0\\}$. Around each $k_{ln}$, one can
approximately derive the dispersion relation as

$\Delta_{l}(k)=\Delta_{l}(k_{ln})+v_{ln}(k-k_{ln}),$ (26)

where

$v_{ln}=\frac{\partial\Delta_{l}(k_{ln})}{\partial k}\Big{|}_{k_{ln}}$

is the group velocity. When the qubit transition frequency is far away from
the band edges, $G(t,t^{\prime})$ can be written as

$\displaystyle G(t,t^{\prime})$ $\displaystyle\simeq$
$\displaystyle\sum_{ln}\frac{L}{2\pi}|u_{ln}(k_{ln})|^{2}\int_{-\infty}^{\infty}d(\delta
k)e^{-iv_{ln}\delta k(t-t^{\prime})}$ (27) $\displaystyle=$
$\displaystyle\sum_{ln}\frac{L|u_{ln}(k_{ln})|^{2}}{|v_{ln}|}\delta(t-t^{\prime}).$

The delta function in Eq. (27) is valid given that the bandwidth of the
waveguide’s spectrum is approximately infinite compared with the interaction
strength. By substituting Eq. (27) into Eq. (23), we obtain

$\displaystyle\frac{\partial c_{e}(t)}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{T}}{2}c_{e}(t),\quad\Gamma_{T}=(\Gamma_{R}+\Gamma_{L}),$
(28) $\displaystyle\Gamma_{R(L)}$ $\displaystyle=$
$\displaystyle\Gamma_{0}\sum_{ln}|u_{ln}(k_{ln})|^{2}\Theta(\pm v_{ln}),$ (29)

where $\Theta(x)$ is the Heaviside step function, $\Gamma_{R(L)}$ corresponds
to the decay rate to the right (left) hand of the waveguide, and $\Gamma_{0}$
is the characteristic decay rate for the qubit which is derived as

$\Gamma_{0}=\frac{g_{0}^{2}L}{v_{0}}=\frac{1}{\hbar
v_{J}}\left(\frac{eC^{q}_{J}}{C_{\Sigma}}\right)^{2}\sqrt[2]{\frac{E_{J}^{q}}{4E_{C}}}\frac{\omega_{q}}{2c_{g}}.$
(30)

In the single-excitation subspace expressed in Eq. (19), the photonic wave
function in real space is Scully and Zubairy (1997)

$\displaystyle\psi_{\gamma}(x,t)=\sum_{lkn}c_{lk}(t)e^{ikx}u^{*}_{ln}(k)e^{-in(\Omega_{d}t-k_{d}x)}.$
(31)

The photonic energy decaying into the right (left) hand side is expressed as

$\Phi_{R/L}=\left|\int_{0}^{\pm\infty}|\psi_{\gamma}(x^{\prime},t)|^{2}dx^{\prime}\right|,\quad
t\rightarrow\infty.$ (32)

By neglecting the local decoherence and dephasing of the transmon, the chiral
factor is defined as Lodahl  _et al._ (2017)

$\beta_{\pm}=\frac{\Gamma_{R(L)}}{\Gamma_{R}+\Gamma_{L}}=\frac{\Phi_{R(L)}}{\Phi_{R}+\Phi_{L}}.$
(33)

As indicated in Eq. (29), the directional emission rates depend on both
distributions ratios $|u_{ln}(k_{ln})|$ and group velocities $v_{ln}$. For
example, given that the qubit frequency lies in the red regime of Fig. 2(b),
the Floquet order $\\{l=1,n=0\\}$ has the largest $|u_{ln}(k_{ln})|$ (see
point A), and the corresponding $v_{ln}$ is positive. Consequently, the photon
will be chirally emitted to the right direction. In the blue regime (point B),
the emission chirality will be reversed. Below we will numerically discuss the
emission behaviors in different frequency regimes.

###  III.3 Numerical discussions

The chiral spontaneous emission process can well be described by the master
equation, which however discards much information due to a lot of
approximations. For example, master equation cannot describe the directional
field distribution and Non-Markovian dynamics led by band-edge effects. To
avoid those problems, we numerically simulate the unitary evolution governed
by Hamiltonian in Eq. (17) by discretising the modes of the modulated
waveguide in the momentum space. The photonic field in the waveguide is
recovered from Eq. (31). The details about numerical methods are presented in
Appendix B.

By changing the transmon’s frequency, we plot the evolutions of
$|c_{e}(t)|^{2}$ in Fig. 3(a). When $\omega_{q}$ is in the conventional
dispersion regime [$\omega_{q}/(2\pi)=2.70~{}\text{GHz}$], the transmon
exponentially decays its energy, and the photonic field is symmetrically
emitted into both left and right side [see Fig. 3(b)]. In the right chiral
regime with $\omega_{q}/(2\pi)=3.02~{}\text{GHz}$ [see red horizontal line in
Fig. 2(b)], the solutions for Eq. (25) correspond to the intersection points
with the dispersion curves of different Floquet orders. The intersection point
A with the Floquet order $\\{l=1,n=0\\}$ has the largest distribution ratio
$|u_{ln}(k)|\simeq 1$, while the other $|u_{ln}(k)|$ are of extremely low
amplitudes. The group velocity for point A is positive. Therefore, the
transmon will chirally emit photons into the right part of the waveguide, as
indicated by Eq. (29). Similarly, when transmon frequency is in the blue
regime, most of the emitted photonic field will distribute on its left hand
side. The chiral field distributions changing with time are shown in Fig. 3(c,
e), respectively. Moreover, the chiral factor is about $\beta_{\pm}\simeq
0.95$, indicating that this SQUID-metamaterial waveguide can be implemented as
a well-performance directional quantum bus.

Figure 4: (a) The chiral factor $\beta_{+}$ changes with $\omega_{q}$ by
setting modulation group velocities as $v_{d}=0.05v_{g}$ and
$v_{d}=0.08v_{g}$. (b) The corresponding dispersion relation of the quasi-
energy $\omega_{l}(k)$ of the waveguide.

When the qubit frequency lies within the band gap [the gray regime in Fig.
2(b)], the distribution ratios for all Floquet orders are around zero, i.e.,
$|u_{ln}(k)|\simeq 0$, indicating that there is no resonant mode which can
lead to exponential decay of the transmon qubit. In this scenario, the
transmon only decays its partial energy into the waveguide [see Fig. 3(a) for
$\omega_{q}/(2\pi)=3.34~{}\text{GHz}$], and the field is localized around the
coupling position without propagating outside, as shown Fig. 3(d)]. This
unconventional evolution can be understood from Fig. 2(b): in the gray regime,
the coefficients of all the Floquet orders are around zero, i.e.,
$|u_{ln}(k_{ln})|\simeq 0$. Due to no resonant mode, only the modes which are
of large detuning will contribute significantly to the dynamics. Those modes
are of large density due to band edge effects, and the field will be localized
in the form of bound state. The scenario is akin to an emitter being prevented
from spontaneous emission when it is trapped by the band gap of a photonic
crystal waveguide Goban and et al. (2014); González-Tudela and et al. (2015);
Douglas  _et al._ (2016).

In this proposal the modulation signal is programmable. Therefore, the
dispersion relation of the waveguide can be tailed freely, which enables us to
control the qubit dynamics on demand. For example, given that the modulation
phase velocity $v_{d}$ is switched oppositely, the chiral direction with a
certain qubit frequency is also reversed. In Fig. 4(a), we plot the chiral
factor $\beta_{+}$ changing with transmon frequency by setting modulation
velocity as $v_{d}=0.05v_{g}$ and $v_{d}=0.08v_{g}$, respectively. When
$v_{d}=0.05v_{g}$, the quasi-energy band $l=1$ and $l=2$ are separated by a
finite band gap [see gray area in Fig. 2(b)]. The gap leads to the trapped
bound state when the qubit frequency lies within this regime [see Fig. 3(d)].
As discussed in Ref. Trainiti and Ruzzene (2016), when increasing the
modulation velocity, the gap disappears, which can be found from the
dispersion relation for $v_{d}=0.08v_{g}$ depicted in Fig. 4(b). The chirality
will be smoothly switched from left to right without any gaps when increasing
the qubit frequency. With a larger $v_{d}$ the chirality is enhanced and the
directional bandwidth becomes wider [see Fig. 4(a)], which is due to that the
Brillouin-scattering process emerges between the modes with a large energy
difference. However, as discussed in Refs. Trainiti and Ruzzene (2016);
Cassedy (1967), when the modulation velocity $v_{d}$ is comparable to $v_{0}$,
the waveguide’s eigenfrequencies become complex, indicating the field is time-
growing and unstable. Due to this, the modulation velocity $v_{d}$ should be
much smaller than $v_{0}$ to avoid the unstable phenomena emerging in the
whole setup.

##  IV Chiral photon flow between two quantum nodes

Figure 5: By mediating remote nodes, the metamaterial waveguide can work as a
common quantum bus in a chiral quantum network. The metamaterial waveguide is
placed in a tube which is below the superconducting temperature (see
experiments in in Ref. Magnard  _et al._ (2020)). To suppress the thermal
microwave noise, each node is placed in a dilution refrigerator which
temperature is around $T\sim 10~{}\text{mk}$.

By considering multiple nodes interacting with the same metamaterial
waveguide, our proposal in Fig. 1 can be extended as a chiral quantum network.
The schematic cascaded network is depicted in Fig. 5, where nodes are placed
in separated dilution refrigerators with temperature $\sim 10~{}\text{mK}$ to
suppress the thermal noise. As discussed in Ref. Magnard  _et al._ (2020), the
metamaterial waveguide can be inside a multi-sleeve tube which is below the
superconducting critical temperature. This experimental method allows to
connect transmons located in different cryogenic refrigerators. Given that the
transmons’ frequencies are identical, the interaction Hamiltonian can be
written as

$H_{\text{int},2}=\hbar
g_{0}\sum_{i}\sum_{lk}e^{i\omega_{q}t}\sigma_{i}^{+}a_{lk}\phi^{*}_{lk}(x_{i},t)+\text{H.c.},$
(34)

where $x_{i}$ is the coupling position of the $i$th node. As discussed in
Appendix C, we can derive the cascaded master equation for multiple nodes by
tracing over the waveguide’s freedoms. Taking that two transmons chirally
decay/absorb the right propagating photons for example, the evolution is
governed by

$\displaystyle\dot{\rho}_{s}(t)$ $\displaystyle=$ $\displaystyle-
iH_{\text{eff}}\rho_{s}+i\rho_{s}H^{\dagger}_{\text{eff}}+\mathcal{L}\rho_{s}\mathcal{L}^{\dagger},$
$\displaystyle H_{\text{eff}}$ $\displaystyle=$
$\displaystyle-i\sum_{i}\frac{\Gamma_{R}}{2}\sigma_{i}^{+}\sigma_{i}^{-}-i\sum_{i>j}\Gamma_{R}\sigma_{i}^{+}\sigma_{j}^{-}.$
(35)

where $\rho_{s}$ is the reduced density matrix operator for two transmons,
$\Gamma_{R}$ is the decay rate to the right side, and
$\mathcal{L}=\sqrt{\Gamma_{R}}(\sigma_{1}^{-}+\sigma_{2}^{-})$ is the
collective jump operator. The last term in $H_{\text{eff}}$ is unique to the
cascaded quantum system, and describes the irreversible process that a photon
emitted by transmon $j$ will be reabsorbed by transmon $i$, while the
information back flow is prevented Lodahl  _et al._ (2017). As discussed in
Appendix C, when deriving the cascaded master Eq. (35), we assume that the
propagating time $\tau_{ij}=(x_{i}-x_{j})/v_{ln}$ between two nodes is much
smaller than the decaying time scale $\Gamma_{R}^{-1}$. Therefore, we can
adopt the Markovian approximation $t-\tau\simeq t$, and the evolution becomes
independent of time delay. This approximation is valid only when the
separation distance $x_{ij}$ is much shorter than the wavepacket’s size.

Figure 6: (a) The directional photon flow from transmon 1 ($x=0$) to transmon
2 ($x=125k_{d}$). Due to the time-delay effect, transmon 2 is excited with a
retardation time $t=x_{12}/v_{0}$. (b) The field distribution along the
waveguide changes with $t$. At $x=125k_{d}$, the emitted photon is scattered
by transmon 2. Due to directional transport, the scattered field only
propagates into the right direction. The adopted parameters are the same with
those in Fig. 3(c).

In cascaded master equation (35), the evolution information, such as the field
distribution and time-decay effects, has been discarded due to taking a trace
over waveguide’s freedoms. To proceed beyond those limitations, our simulation
is based on the unitary evolution governed by the Hamiltonian in (34), which
can well describe the time-delay effects Wang and Li (2022). Numerical details
can be found in Appendix B. In Fig. 6(a), by assuming a single excitation
initially in transmon 1, we plot the evolution of two transmons with a
separation distance $x_{2}-x_{1}=125\lambda_{d}$. At the frequency adopted in
Fig. 3(c), the group velocity is around $v_{ln}\simeq 0.83v_{0}$, which is
slower than $v_{0}$ due to the nonlinear dispersion relation around the band
edge. Consequently the wave front arriving at transmon 2 is calculated as
$\tau_{ij}=125\lambda_{d}/0.83v_{0}\simeq 0.022\text{ms}$ [see Fig. 6(a)]. The
field distribution $|\psi_{\gamma}(x,t)|^{2}$ is depicted in Fig. 6(b), where
one finds that the photon emitted by transmon 1 propagates unidirectionally,
and the field absorbed by transmon 2 (inside the box) is aslo re-emitted to
the right side. Due to no photonic energy back flow, transmon $1$ will not be
re-excited. Those numerical results show that our proposal can behave as a
well-performed cascaded quantum network.

##  V Summary and outlooks

In this work, we propose how to employ the Josephson array as chiral
metamaterial waveguide for circuit-QED. The waveguide is in the form of SQUID
chain which impedance is modulated with bias currents. When the bias signals
are programmed as travelling waves, the symmetry between the left and right
propagating modes is broken due to the Brillouin-scattering process. We also
discuss the quantum optical phenomena by considering superconducting qubits
coupling to this metamaterial waveguide. By applying the optimized modulating
parameters, the qubit will emit photons unidirectionally, and chiral factor
can approach 1. Last we extend our proposal as a multi-node quantum network,
and demonstrate that the chiral transport between remote nodes can be
realized. Compared with routing microwave photons unidirectionally with the
bulky ferrite circulators, our proposal does not require strong magnetic
field, and the direction can be freely tuned by the programmed bias signals.

Note that we only focus on slow travelling waves to modulate the SQUID-chain’s
impedance. Exploring other modulating parameters or forms, such as standing-
wave modulations and pulses with different shapes, might allow us to observe
more intriguing QED phenomena in this metamaterial platform. As discussed in
experimental studies in Refs. et al. (2019c); Planat  _et al._ (2020), the
SQUID number in a single metamaterial waveguide can be around $10^{3}\simeq
10^{4}$, indicating that our proposal is within the capability of current
technology. We hope that our work can inspire more efforts being devoted to
exploiting SQUID metamaterials for controlling microwave photons in SQC
setups.

##  VI Acknowledgments

The quantum dynamical simulations are based on open source code QuTiP
Johansson  _et al._ (2012, 2013). X.W. is supported by the National Natural
Science Foundation of China (NSFC) (Grant No. 12174303 and No. 11804270), and
China Postdoctoral Science Foundation No. 2018M631136. W.X.L. was supported by
the Natural Science Foundation of Henan Province (No. 222300420233). H.R.L. is
supported by the National Natural Science Foundation of China (NSFC) (Grant
No.11774284).

## APPENDICES

##  Appendix A Deriving the dispersion relation of the modulated SQUID-
metamaterial waveguide

We start from the left side of Eq. (2), i.e., the difference terms related to
time-dependent inductance $L_{j}(t)$. To derive the corresponding quasi-
continuous differential form, we first rewrite it as

$\frac{\phi_{j}-\phi_{j-1}}{L_{j}(t)}-\frac{\phi_{j+1}-\phi_{j}}{L_{j+1}(t)}=\frac{\phi_{j}-\phi_{j-1}}{L_{j}(t)}-\frac{\phi_{j+1}-\phi_{j}}{L_{j}(t)}+\frac{\phi_{j+1}-\phi_{j}}{L_{j}(t)}-\frac{\phi_{j+1}-\phi_{j}}{L_{j+1}(t)}.$
(A1)

In this work we assume that each SQUID’s size is much smaller than wavelengths
of both modulation wave and microwave photons. Therefore, we replace
$\phi_{j}(t)\rightarrow\phi(x,t)$ and $f(j,t)\rightarrow f(x,t)$. Note that
$G_{j}(t)$ becomes

$G(x,t)=\frac{1}{L_{0}}\left[\alpha_{0}+\delta\alpha f(x,t)\right].$ (A2)

By replacing $x=id_{0}$, Eq. (A1) is changed as

$\frac{\phi_{j}-\phi_{j-1}}{L_{j}(t)}-\frac{\phi_{j+1}-\phi_{j}}{L_{j}(t)}+\frac{\phi_{j+1}-\phi_{j}}{L_{j}(t)}-\frac{\phi_{j+1}-\phi_{j}}{L_{j+1}(t)}=-G(x,t)\frac{\partial^{2}\phi(x,t)}{\partial
x^{2}}d_{0}^{2}-\frac{\partial G(x,t)}{\partial
x}\frac{\partial\phi(x,t)}{\partial x}d_{0}^{2}.$ (A3)

Similarly, the left side in Eq. (2) which contains capacitance terms can also
be written in quasi-continuous differential form. Finally, the nonlinear wave
function of the modulated SQUID waveguide is derived as

$C_{g}\frac{\partial^{2}\phi(x,t)}{\partial
t^{2}}-C_{J}\frac{\partial^{4}\phi(x,t)}{\partial t^{2}\partial
x^{2}}d_{0}^{2}=G(x,t)\frac{\partial^{2}\phi(x,t)}{\partial
x^{2}}d_{0}^{2}+\frac{\partial G(x,t)}{\partial
x}\frac{\partial\phi(x,t)}{\partial x}d_{0}^{2},$ (A4)

which is equivalent to the form in Eq. (7).

Given that the modulation is of the travelling wave form, one can decompose
the wave function in Eq. (2) as a matrix form by employing the orthogonal
relations. The dispersion relation is obtained by solving the following
quadratic eigenvalue problem

$\left[\omega_{l}^{2}(k)\hat{M}_{2}+\omega_{l}(k)\hat{M}_{1}+\hat{M}_{0}\right]\hat{U}(k)=0,$
(A5)

where $M_{1,2}$ are diagonal matrices and expressed as

$\displaystyle\hat{M}_{2}=\text{diag}\left[...,-c_{J}(d_{0})^{2}\left(k+nk_{d}\right)^{2}-c_{g},...\right],$
(A6)
$\displaystyle\hat{M}_{1}=\text{diag}\left[...,-2n\Omega_{d}[c_{J}(d_{0})^{2}\left(k+nk_{d}\right)^{2}+c_{g}],...\right].$
(A7)

The matrix $\hat{M}_{0}$ is

$\displaystyle\hat{M}_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\ddots&\ddots&\ddots&0&0&0&\vdots\\\
...&T_{n-1,n-2}&T_{n-1,n-1}&T_{n-1,n}&0&0&...\\\
...&0&T_{n,n-1}&T_{n,n}&T_{n,n+1}&0&...\\\
...&0&0&T_{n+1,n}&T_{n+1,n+1}&T_{n+1,n+2}&0\\\
\vdots&0&0&0&\ddots&\ddots&\ddots\end{matrix}\right),$ (A8)

where

$\displaystyle
T_{n,n}=-\left(n\Omega_{d}\right)^{2}\left[c_{J}(d_{0})^{2}\left(k+nk_{d}\right)^{2}+c_{g}\right]+\frac{\alpha_{0}}{l_{0}}\left(k+nk_{d}\right)^{2},$
(A9) $\displaystyle T_{n,n\pm
1}=\frac{\delta\alpha}{2l_{0}}\left\\{\left(k+\left(n\pm
1\right)k_{d}\right)^{2}+\left(k+\left(n\pm
1\right)k_{d}\right)k_{d}\right\\}.$ (A10)

We find that the Josephson capacitance $c_{J}$ only appears in the diagonal
terms of $\hat{M}_{0,1,2}$, from which on can easily find that $c_{J}$ can be
neglected under the following condition

$c_{J}(d_{0})^{2}\left(k+nk_{d}\right)^{2}\ll c_{g}.$ (A11)

Further discussions of nonlinear dispersion led by $c_{J}$ can be found in
Sec. II of the main text.

##  Appendix B Numerical methods for simulating spontaneous emission and non-
Markovian dynamics

Although the master equation can well describe the spontaneous emission of
emitters, the Born-Markovian approximation is not valid when the interaction
strength is comparable to the band width of baths. Moreover, the information
of emitted photons will be traced off when deriving the master equation. To
avoid those, our simulation is based on the unitary evolution governed by the
original Hamiltonian in Eqs. (17, 34). In the single-excitation subspace and
taking for the case of two transmons for example, the system’s state is
written as

$|\psi(t)\rangle=\sum_{lk}c_{lk}(t)|g,g,1_{lk}\rangle+c_{e1}(t)|e,g,0\rangle+c_{e2}(t)|g,e,0\rangle,$

where $|1_{lk}\rangle$ represents the single excitation being in $lk$th mode,
and $|0\rangle$ corresponds to the waveguide in its vacuum. Since the
Hamiltonian is expressed in momentum space, we first need to calculate
waveguide’s eigen-frequencies and wavefunctions by employing the method
presented in Appendix A. Next we discretise modes in the first BZ
$k\in(-0.5k_{m},0.5k_{m}]$ with a large number $N$, which is equal to
considering a waveguide with a length $L=N\lambda_{m}$. A similar method can
be found in Ref Wang and Li (2022). In the single-excitation subspace, the
Hamiltonian in Eq. (17) [Eq. (34)] can be mapped into a matrix with dimension
$N+1$ ($N+2$) when the system contains a single emitter (two emitters). Taking
two emitters ($Q=2$) for example, the corresponding matrix is written as

$\displaystyle
H_{\text{int}}=\left[\begin{matrix}w_{lk_{1}}&0&...&0&g_{0}\phi_{lk_{1}}(x_{1},t)&g_{0}\phi_{lk_{1}}(x_{2},t)\\\
0&w_{lk_{2}}&\ddots&...&g_{0}\phi_{lk_{2}}(x_{1},t)&g_{0}\phi_{lk_{2}}(x_{2},t)\\\
\vdots&\ddots&...&0&\vdots&\vdots\\\
0&...&0&w_{lk_{N}}&g_{0}\phi_{lk_{N}}(x_{1},t)&g_{0}\phi_{lk_{N}}(x_{2},t)\\\
g_{0}\phi^{*}_{lk_{1}}(x_{1},t)&g_{0}\phi^{*}_{lk_{2}}(x_{1},t)&...&g_{0}\phi^{*}_{lk_{N}}(x_{1},t)&w_{qa}&0\\\
g_{0}\phi^{*}_{lk_{1}}(x_{2},t)&g_{0}\phi^{*}_{lk_{2}}(x_{2},t)&...&g_{0}\phi^{*}_{lk_{N}}(x_{2},t)&0&w_{qb}\\\
\end{matrix}\right],$ (B1)

where $\phi_{lk_{i}}(x_{j},t)$ is numerically obtained via methods in Appendix
A. After obtaining the matrix form in Eq. (B1), we can numerically solve the
evolution governed by the Schrödinger equation. At certain time $t=t_{i}$, the
amplitude $c_{kl}(t)$ of each mode is recorded, and the field distribution can
be recovered by substituting them into Eq. (31). Employing this method, we
obtained both transmon’s and waveguide’s evolution shown in the main text.

##  Appendix C Cascaded master equation for multi-nodes system

In this part we will derive the cascaded master equation for multiple emitters
mediated by the metamaterial waveguide. By expanding the Schrödinger equation
to the second order, the evolution of the system is expressed as Scully and
Zubairy (1997)

$\displaystyle\dot{\rho}_{s}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\mathrm{Tr}_{R}{\left[H_{\mathrm{int},2}(t),\rho(t)\right]}-$
(C2)
$\displaystyle\\!\frac{1}{\hbar^{2}}\mathrm{Tr}_{R}\int_{t_{0}}^{t}{\left[H_{\mathrm{int},2}(t),\left[H_{\mathrm{int},2}(t^{\prime}),\rho(t^{\prime})\right]\right]}dt^{\prime},$

where $\rho$ ($\rho_{s}$) is the density matrix operator of the whole system
(the emitters), and $\text{Tr}_{R}$ represents taking a trace over the
waveguide’s freedoms. We assume that the waveguide is always approximately in
its vacuum state. Therefore, the correlation relations for the waveguide’s
modes satisfy $\langle a_{lk}\rangle=\langle a^{\dagger}_{lk}\rangle=\langle
a^{\dagger}_{lk}a_{lk}\rangle=0$ and $\langle
a_{lk}a^{\dagger}_{lk}\rangle=1$. By substituting those relations into Eq.
(C2), we obtain

$\displaystyle\dot{\rho}_{s}(t)$ $\displaystyle=$ $\displaystyle-
g_{0}^{2}{\frac{L}{2\pi}}\sum_{i,j}\int_{t_{0}}^{t}dt^{\prime}A\left(x_{i},x_{j},t,t^{\prime}\right)$
(C3)
$\displaystyle\big{[}\sigma_{i}^{+}\sigma_{j}^{-}\rho_{s}(t^{\prime})-\sigma_{j}^{-}\rho_{s}(t^{\prime})\sigma_{i}^{+}\big{]}+\mathrm{H}.\mathrm{c}.,$

where the correlation function $A\left(x_{i},x_{j},t,t^{\prime}\right)$ is
defined as Calajó  _et al._ (2019)

$\displaystyle
A\left(x_{i},x_{j},t,t^{\prime}\right)=e^{i\omega_{q}\left(t-t^{\prime}\right)}\sum_{lk}\phi_{lk}^{*}(x_{i},t)\phi_{lk}(x_{j},t)$
(C4) $\displaystyle=$
$\displaystyle\sum_{ln}|u_{ln}(k_{ln})|^{2}\int_{-\infty}^{\infty}{d}(\delta
k)e^{-i\delta kv_{ln}\left[(t-t^{\prime})-\frac{x_{ij}}{v_{ln}}\right]}$
$\displaystyle=$
$\displaystyle\sum_{ln}\frac{2\pi}{|v_{ln}|}|u_{ln}(k_{ln})|^{2}\delta\left[(t-t^{\prime})-\frac{x_{ij}}{v_{ln}}\right],$

with $x_{ij}=x_{i}-x_{j}$ being the distance between two emitters. We have
neglected the phase terms
$e^{i\left(k_{ln}+k_{d}\right)\left(x_{i}-x_{j}\right)}$ in
$A\left(x_{i},x_{j},t,t^{\prime}\right)$. Note that the integral regime is
within $t^{\prime}\in[t_{0},t]$. Therefore, the $\delta$-funtion will produce
non-zero value in Eq. (C3) only under the condition $t-x_{ij}/v_{ln}\leq t$.
The delay-time $\tau_{ij}=x_{ij}/v_{ln}$ ($i\neq j$) corresponds to the
propagating time between two separated emitters. When $x_{i}>x_{j}$
($x_{i}<x_{j}$), the non-zero time-delay correlation is mediated by the right
(left) propagating modes with $v_{ln}>0$ ($v_{ln}<0$). Given that $\tau_{ij}$
is much smaller than the time scale of spontaneous emission, we can replace
$t-\tau\rightarrow t$, and Eq. (C3) is simplified as

$\displaystyle\dot{\rho}_{s}(t)$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{\Gamma_{R}+\Gamma_{L}}{2}\sum_{i}\big{[}\sigma_{i}^{+}\sigma_{i}^{-}\rho_{s}(t)-\sigma_{i}^{-}\rho_{s}(t)\sigma_{i}^{+}\big{]}$
(C5)
$\displaystyle-\sum_{i>j}\Gamma_{R}\big{[}\sigma_{i}^{+}\sigma_{j}^{-}\rho_{s}(t)-\sigma_{j}^{-}\rho_{s}(t)\sigma_{i}^{+}\big{]}+$
$\displaystyle\\!\sum_{i<j}\Gamma_{L}\big{[}\sigma_{i}^{+}\sigma_{j}^{-}\rho_{s}(t)-\sigma_{j}^{-}\rho_{s}(t)\sigma_{i}^{+}\big{]}\Big{\\}}\\!+\\!\mathrm{H}.\mathrm{c}.,$

where $\Gamma_{R(L)}$ is the decay rate into right/left side, which are
expressed in Eq. (29). In Eq. (C5) we have employed the properties of
$\delta$-function

$\displaystyle\int_{t_{0}}^{t}\delta\left(t-t^{\prime}-\frac{x_{ij}}{v_{ln}}\right)\rho_{s}(t^{\prime})dt^{\prime}=\left\\{\begin{array}[]{lr}\frac{1}{2}\rho_{s}(t)\quad
x_{ij}=0,\\\ \rho_{s}(t)\quad x_{ij}/v_{ln}=0^{+},\\\ 0\quad
x_{ij}/v_{ln}<0.\end{array}\right.$

When the chiral factor of the whole system approaches $\beta_{\pm}\simeq 1$,
we obtain the cascaded master equation in the main text.

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