# Non-Markovian Collective Emission of Giant emitters in the Zeno Regime

QingYang Qiu  School of Physics and Institute for Quantum Science and
Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P.
R. China  Wuhan institute of quantum technology, Wuhan, 430074, China  Xin-You
Lü<EMAIL_ADDRESS>School of Physics and Institute for Quantum Science
and Engineering, Huazhong University of Science and Technology, Wuhan, 430074,
P. R. China  Wuhan institute of quantum technology, Wuhan, 430074, China

###### Abstract

We explore the collective Zeno dynamics of giant artificial atoms that are
coupled, via multiple coupling points, to a common photonic or acoustic
reservoir. In this regime, the establishment of atomic cooperativity and the
revivification of exponential decay, are highly intertwined, which is utterly
beyond the non-Markovian regime with only retarded backaction. We reveal that
giant atoms build up their collective emission smoothly from the decay rate of
zero to that predicted by Markovian approximation, and show great disparity
between different waveguide QED setups. As a comparison, the step-like growth
of instantaneous decay rates in the retardation-only picture has also been
shown. All of these theoretical pictures predict the same collective behavior
in the long time limit. From a phenomenological standpoint, we observe that
the atomic superradiance exhabits significant directional property. In
addition, the subradiant photons feature prolonged oscillation in the early
stage of collective radiance, where the energy is exchanged remarkably between
giant emitters and the field. Our results might be probed in state-of-art
waveguide QED experiments, and fundamentally broaden the fields of collective
emission in systems with giant atoms.

††preprint: APS/123-QED

##  I INTRODUCTION

Spontaneous emission from an unstable system, as one of the earliest quantum
field theory phenomena, was first predicted by Einstein and subsequently
extended to multi-atomic ensembles in $1953$ which is known as Dicke
superradiance Dicke (1954). These significant contributions have profoundly
solidified the concept of exponential decay law. However, it is not the whole
physical picture because the realistic spontaneous emission is generally
exposed to a non-Markovian environment. Indeed, non-Markovianity can have a
wealth of physical origins Hu  _et al._ (1992, 1993); Tufarelli  _et al._
(2013); Vasile  _et al._ (2014); Gröblacher  _et al._ (2015); Breuer  _et al._
(2016); de Vega and Alonso (2017) with a hallmark of strong informational
back-flow. Light-matter interactions within such a non-Markovian bath open new
routes for unprecedented phenomena, such as genuinely non-Markovian steady
states Ask and Johansson (2022); Cilluffo  _et al._ (2024), collective
superradiant burst Sinha  _et al._ (2020); Solano  _et al._ (2023) and
engineered Floquet bound state Bai and An (2023). In particular, the
retardation induced non-Markovianity has been demonstrated to endow collective
radiation with a common intuition: Each emitter initially undergoes
independent exponential decay, altering its decay behavior upon interaction
with delayed photons emitted by other emitters Sinha  _et al._ (2020).

Recently, Zeno regime of spontaneous emission comes to the fore in the
community of waveguide quantum electrodynamics (QED), revealing how point-like
atoms cooperatively modify their decay behavior in the building of collective
emission Zhang (2023). In this context, atoms complete their significant
transition from non-exponential to exponential decay by exchanging virtual
photons in a very short time. This time scale evaluates the duration of non-
exponential decay, i.e., the Zeno time $\tau_{Z}$, during which the atomic
survival probability is quadratic in time Wilkinson  _et al._ (1997); Crespi
_et al._ (2019); Giacosa and Pagliara (2021); Liu  _et al._ (2023).

Featuring exotic self-interference effects and the nature of non-local
couplings Frisk Kockum  _et al._ (2014); Guo  _et al._ (2017); Frisk Kockum
(2020), “giant” artificial atoms (GAs) have spurred a rapidly growing interest
in the emerging area of waveguide QED. It has been manipulated to interact
with surface acoustic waves (SAWs) Andersson  _et al._ (2015) or meandering
coplanar waveguide (CPW) Kannan  _et al._ (2020); Vadiraj  _et al._ (2021);
Joshi  _et al._ (2023) via multiple connecting points as sketched in Fig.1.
Leveraging the state-of-the-art quantum device, such a novel designation has
recently been extended to “giant” ferromagnetic spin ensemble Wang  _et al._
(2022, 2023). Of particular interest is the exploration of novel non-Markovian
effects with GAs, and considerable efforts have been made to manifest a rich
variety of intriguing outcomes, including tunable localization-delocalization
quantum phase transition Terradas-Briansó  _et al._ (2022), oscillating bound
states Guo  _et al._ (2020); Lim  _et al._ (2023); Xu and Guo (2024), enhanced
Dicke superradiance Qiu  _et al._ (2023) and so on Yin  _et al._ (2022); Du
_et al._ (2022); Cheng  _et al._ (2022); Guo  _et al._ (2024); Li and Shen
(2024). However, it seems that the mechanism of non-negligible time-delayed
feedback underlies all of these efforts. Thus, the scenario of collective
emission beyond the retardation, i.e., GAs go Zeno region, is still
unexplored.

Figure 1: Schematic diagram of theoretical model and experimental
implementation for a linear chain of $N$ GAs. (a) and (b) respectively show
$N$ two-legged separate and braided superconducting qubits are coupled to a 1D
waveguide at multiple connecting points with the light blue dots denote the
coupling points. The experimental setup in separate case is depicted
accordingly in (c), i.e., $N$ transmon qubits (superconducting quantum
interference devices marked by black circles) are embedded in a surface
acoustic waves transmission line. Each qubit interacts with the propagating
phonons via piezoelectric effect utilizing the large interdigital transducers
from its two islands. For braided case, panel (d) exhibits $N$ X-mon qubits
coupled capacitively to a meandering microwave coplanar waveguide with the
central waveguide (dark green line) is terminated to $50\Omega$.

In this work, we have investigated the collective radiative dynamics of a
linear chain of $N$ GAs coupling to a 1D waveguide, and show that it exhibits
prominent non-Markovianity beyond retardation. To incorporate the relaxation
dynamics into Zeno regime, the array of GAs is designed to guarantee
$\tau_{0}\ll\tau_{Z}$, where $\tau_{0}$ is the time taken by a single photon
or phonon traveling between adjacent coupling points. On this occasion, the
GAs build up their cooperativity obeying the combined action of time-delayed
feedback and memory effect of the electromagnetic field. As a consequence, we
find that the non-Markovianity can be enhanced by increasing the coupling
points or modifying the waveguide QED setups. In contrast to the previous
achievements obtained outside the Zeno region, the full development of
collective emission shows a dominating signature: Each emitter grows their
instantaneous decay rate smoothly from zero to that predicted by Markovian
theory, and predictions from different waveguide QED setups are distinguished
by the building speed of emission and the magnitude of the memory effect. We
also put forward step-like growth of decay rates in the retardation-only
picture for the purpose of comparison. In an anticipated manner, these
predictions can eventually coalesce with each other at long enough time. To
get further insight into the collective Zeno physics, we have also studied the
dynamics of the field emitted from the array of GAs, during the establishment
of collective radiation. Interestingly, we find that the chiral radiation
occurs when the superradiant development of GAs is asymmetric. Moreover, a
remarkable oscillation can be captured in the early stage of subradiance,
where the atoms are periodically emitting and absorbing photons.

##  II MODEL HAMILTONIAN AND ZENO TIME OF WAVEGUIDE QED

We consider $N$ GAs interacting with an open 1D waveguide through multiple
coupling points and apply a strict two-level approximation with the ground
state $\ket{g}$ and the excited state $\ket{e}$. Figs. 1 (a) and 1 (b)
illustrate the theoretical model of two-legged separate (all coupling points
of each atom are outside those of another atom) and braided (part of the
coupling points of each atom is inside those of another atom) GAs,
respectively. The waveguide QED setups with giant atoms have at least two
different experimental platforms associated with SAWs or superconducting
transmission line, as sketched respectively in Figs. 1 (c) and 1 (d).

The total Hamiltonian for the GAs+bath system reads
$H_{{\rm{tot}}}=H_{A}+H_{B}+H_{{\rm{int}}}$, where
$H_{A}=\omega_{0}\sum\limits_{n=1}^{N}\sigma_{n}^{\dagger}\sigma_{n}$ is the
free Hamiltonian for the GAs of resonance frequency $\omega_{0}$, with the
creation operator of $n^{{\rm{th}}}$ atom is
$\sigma^{\dagger}_{n}=\ket{e}_{n}\bra{g}$;
$H_{B}=\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}\omega_{k}a_{k}^{\dagger}a_{k}$
is the free Hamiltonian for the 1D continuum of bosonic modes where
$a_{k}^{\dagger}$ is the generation operator of the waveguide mode with wave
number $k$, satisfying bosonic commutation relation
$[a_{k},a_{k^{\prime}}^{\dagger}]=2\pi\delta(k-k^{\prime})$. Here we adopt a
linear dispersion relation $\omega_{k}=|k|v_{g}$ with $v_{g}$ is the group
velocity of the field. To isolate the non-Markovian effect stemming from the
band edges of photonic or phononic bath Vats and John (1998); Ferreira  _et
al._ (2021); Burgess and Florescu (2022), the resonant wave number
$k_{0}=\omega_{0}/v_{g}$ is assumed to be far enough away from the cutoff of
the wave number $\Lambda$. The nonlocal interactions between $M$-legged GAs
and bosonic field are described by ($\hbar=1$)

$\displaystyle\\!\\!H_{{\rm{int}}}\\!=\\!-i\sum_{n=1}^{N}\sum_{m=1}^{M}\\!\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}g_{k}(\sigma_{n}\\!+\\!\sigma^{\dagger}_{n})a_{k}e^{ikx^{n}_{m}}\\!+\\!{\rm{H.c.}},$
(1)

where $x^{n}_{m}$ denotes the $m^{{\rm{th}}}$ coupling point of the
$n^{{\rm{th}}}$ giant atom ($x^{n}_{m_{1}}<x^{n}_{m_{2}}$ if $m_{1}<m_{2}$).
In the context of waveguide QED, there are many general choices for the
specific form of coupling strength $g_{k}$, among which constant and linear
spectral density are frequently considered. The above two types of coupling
formalism are referred to as “const-wQED” and “lin-wQED” corresponding to
$g_{k}=\sqrt{\Gamma_{0}v_{g}/2}$ and $g_{k}=\sqrt{\Gamma_{0}v_{g}|k|/2k_{0}}$,
respectively, with $\Gamma_{0}$ is the relaxation rate at single coupling
point.

Figure 2: The instantaneous decay rate $\Gamma_{{\rm{ins}}}$ (in units of
$\Gamma_{0}$) of a single giant atom as a function of scaled time
$\omega_{0}t$. Time evolved decay rates are depicted by (a) gradually
increasing the atomic coupling points with fixed
$\Gamma_{0}/\omega_{0}=10^{-4}$ or (b) changing the ratio of
$\Gamma_{0}/\omega_{0}$. In panel (a), we distinguish different coupling
points by coloring: $M$=2 (blue), $M$=3 (red), $M$=4 (green), and the
numerical results obtained from const-wQED (solid lines) or lin-wQED (dotted
lines) have been marked. Also, we utilize gradient color to distinguish the
data under different ratios in panel (b), with the predictions of const-wQED
(solid lines) or lin-wQED (dashed lines) have also been marked. Other
parameters are $\Lambda/\omega_{0}=10^{4}$ and $d=0.1\pi/\omega_{0}$.

Commonly, the Hamiltonian $H_{{\rm{tot}}}$ can not be solved exactly and the
rotating-wave approximation (RWA) is usually applied. Instead, the
counterrotating terms in Eq.(1) can not be neglected in our case, since the
time scale we focused on is extremely short. In order to analysis the impact
of the full Hamiltonian 1 on the short-time collective dynamics, we employ a
polaron-like unitary transform Drummond (1987); Stokes and Nazir (2022); Yan
(2023) on $H_{{\rm{tot}}}$, i.e., $H_{{\rm{eff}}}=e^{iU}H_{{\rm{tot}}}e^{-iU}$
with

$\displaystyle\\!\\!U\\!=\\!\sum_{n=1}^{N}\sum_{m=1}^{M}\\!\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}\frac{g_{k}}{\omega_{0}+\omega_{k}}\\!(\sigma_{n}\\!+\\!\sigma^{\dagger}_{n})a_{k}e^{ikx^{n}_{m}}\\!+\\!{\rm{H.c.}}.$
(2)

The transformed Hamiltonian is given by
$H_{{\rm{eff}}}=H_{A}+H_{B}+H_{{\rm{JC}}}+\mathcal{O}(g^{2}_{k})$ with

$\displaystyle
H_{{\rm{JC}}}=-i\sum_{n=1}^{N}\sum_{m=1}^{M}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}g^{{\rm{JC}}}_{k}\sigma_{n}^{\dagger}a_{k}e^{ikx^{n}_{m}}+\\!{\rm{H.c.}},$
(3)

where $g^{{\rm{JC}}}_{k}=\frac{2\omega_{0}g_{k}}{\omega_{0}+\omega_{k}}$ is
the scaled coupling of Jaynes- Cummings (JC) type interaction;
$\mathcal{O}(g^{2}_{k})$ contains terms of order higher than $g_{k}$ and thus
can be safely dropped in the limit of $g_{k}\ll 1$.

We now proceed by determining the duration of nonexponential decay for a
single excited giant atom with $M$ legs, which can be well characterized by
the Zeno time $\tau_{Z}$. After some standard algebra, we obtain the Zeno time
$\tau_{Z}^{-2}=\frac{2\Gamma_{0}\omega_{0}}{\pi}M\ln(\Lambda/k_{0})$ for the
lin-wQED; and the one for the const-wQED has the form of (see Appendix. A for
more details)

$\displaystyle\tau_{Z}^{-2}\\!=\\!\frac{2\Gamma_{0}\omega_{0}}{\pi}[M^{2}+2\sum_{n=1}^{M-1}n\varphi(\chi_{n}(1+\Lambda)\\!-\\!\chi_{n}(1))],$
(4)

where $\chi_{n}(x)\\!\equiv\\!(M-n)[\sin(n\varphi){{\rm{Ci}}(n\varphi
x)}\\!-\\!\cos(n\varphi){{\rm{Si}}(n\varphi x)}]$ and $\varphi\equiv k_{0}d$
have been introduced for simplicity, with $d$ the minimal distance between
adjacent coupling points. Note that the distance between two coupling points
from a single giant atom is $d$ for separate two-legged GAs and $3d$ for
braided two-legged GAs, as we can see from Figs.1(a)-(b). The integral
functions ${\rm{Ci}}(x)$ and ${\rm{Si}}(x)$ are defined as follows:

$\displaystyle{\rm{Csi}}(x)\equiv{\rm{Ci}}(x)+i{\rm{Si}}(x)\equiv\int_{x}^{\infty}dz\frac{e^{iz}}{z}.$
(5)

We note that the results of Zeno time for the two mentioned waveguide-QED
setups are cutoff-dependent, arising from the self-interference effect of an
individual giant atom, which is different from a single “small” atom. The
quantity $\tau_{Z}^{-2}$ shows different behaviors by increasing cutoff wave
number $\Lambda$: divergence for lin-wQED and convergence for const-wQED,
hence a more pronounced memory effect of the radiation field is expected for
the latter. Importantly, the collective radiation can completely established
during the Zeno time under the assumption of weak atom-field couplings. More
specifically, $\tau_{Z}\gg d/v_{g}$ is guaranteed by
$\omega_{0}\gg\ln(\Lambda/k_{0})$ for lin-wQED and $\omega_{0}\gg 1$ for
const-wQED. We determine the Zeno time for $N$ GAs in Appendix.A for
completeness.

Figure 3: The instantaneous decay rate $\Gamma_{{\rm{ins}}}$ (in units of
$\Gamma_{0}$) of (a) $N=10$ separate GAs (background-color: pink) or (b)
braided GAs (background-color: pale green), as a function of scaled time
$\omega_{0}t$. The panels labeled by “Total” and “No.$n$” represent the decay
rate of the whole GAs or $n^{{\rm th}}$ giant atom, respectively. The
numerical predictions obtained from const-wQED (blue), lin-wQED (red) or
retardation (dark green) have also been marked. The minimal distance $d$
between adjacent coupling points is taken as $d=0.1\pi/k_{0}$ in panel (a) and
$d=0.25\pi/k_{0}$ in panel (b). Other parameters are$M=2$ and
$\Gamma_{0}/\omega_{0}=10^{-6}$.

##  III EQUATIONS OF MOTION AND THEIR SOLUTIONS

In this section, we study the dynamical process of spontaneous emission from
$N$ GAs into waveguide, according to the excitations-conservative Hamiltonian
$H_{{\rm{eff}}}$. Assuming that the dynamics of total GAs plus field system is
limited in the single excitation subspace, the state at $t>0$ is

$\displaystyle|\psi(t)\rangle=\sum_{n=1}^{N}c_{n}(t)\sigma_{n}^{\dagger}|G\rangle+\int_{-\Lambda}^{\Lambda}\alpha_{k}(t)a_{k}^{\dagger}|G\rangle,$
(6)

where $c_{n}(t)$ and $\alpha_{k}(t)$ are the amplitudes of probability to find
an excitation populated in $n^{{\rm{th}}}$ giant atom or in propagating mode
with wave number $k$, respectively, at time $t$. $|G\rangle$ is the ground
state of the total system, where all of the GAs are in their lower state
$\ket{g}$ while the waveguide modes are empty. Utilizing the
schr$\ddot{o}$dinger equation $i|\dot{\psi}(t)\rangle=H_{{\rm
eff}}|\psi(t)\rangle$, the atomic equation of motion (EOM) in the interaction
picture is given by

$\displaystyle\dot{c}_{n}(t)=$
$\displaystyle-\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}\int_{0}^{t}d\tau|g_{k}^{{\rm
JC}}|^{2}c_{n^{\prime}}(\tau)$ $\displaystyle\times
e^{i(\omega_{0}-\omega_{k})(t-\tau)}e^{ik(x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}})},$
(7)

with a mathematical form tracing back the whole historical dynamics of the
system. In general, Eq.(III) has no simple closed form of the analytical
description. Here, we present an exact solution in terms of convolution
integral:

$\displaystyle
c_{n}^{{\rm\frac{const}{lin}}}(t)=c_{n}(0)+\frac{2\Gamma_{0}i}{\pi}\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{t}d\tau
c^{{\rm\frac{const}{lin}}}_{n^{\prime}}(\tau)$
$\displaystyle\times[\mathcal{K}_{1}(\varphi_{mm^{\prime}}^{nn^{\prime}},\phi)-\mathcal{K}_{2}(\varphi_{mm^{\prime}}^{nn^{\prime}},\phi)\mp
2\mathcal{K}_{3}(\varphi_{mm^{\prime}}^{nn^{\prime}},\phi)],$ (8)

where
$\varphi_{mm^{\prime}}^{nn^{\prime}}=k_{0}|x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}}|$
is the acquired field phase and $\phi=\omega_{0}(t-\tau)$, the expressions for
the kernels $\mathcal{K}_{1},\mathcal{K}_{2}$ and $\mathcal{K}_{3}$ are given
in Appendix.B. Note that $c_{n}^{{\rm const}}(t)/c_{n}^{{\rm lin}}(t)$ denotes
the dynamical solution for const/lin-wQED (the initial state $c_{n}(t=0)$ does
not depend on the specific choices of waveguide setups).

The non-Markovian dynamics in the regime of non-negligible time delays, can
also be derived under several assumptions in Appendix.B, and the resulting
atomic EOM induced merely by retardation is

$\displaystyle\dot{c}_{n}(t)\\!=\\!$
$\displaystyle-\frac{M}{2}\Gamma_{0}c_{n}(t)\\!\\!-\\!\\!\sum_{m\neq
m^{\prime}}\frac{\Gamma_{0}}{2}\beta_{n}(\tau_{mm^{\prime}}^{nn},\varphi_{mm^{\prime}}^{nn})\Theta(t\\!-\\!\tau_{mm^{\prime}}^{nn})$
$\displaystyle\\!\\!-\\!\sum_{n^{\prime}\neq
n}\sum_{m,m^{\prime}=1}^{M}\\!\frac{\Gamma_{0}}{2}\beta_{n^{\prime}}(\tau_{mm^{\prime}}^{nn^{\prime}},\varphi_{mm^{\prime}}^{nn^{\prime}})\Theta(t\\!-\\!\tau_{mm^{\prime}}^{nn^{\prime}}),$
(9)

where $\beta_{n}(\tau,\varphi)\equiv c_{n}(t-\tau)e^{i\varphi}$ and
$\tau_{mm^{\prime}}^{nn^{\prime}}=|x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}}|/v_{g}$
is the time taken by a single photon or phonon traveling between the $m^{{\rm
th}}$ coupling point of $n^{{\rm th}}$ atom and the $m^{\prime{\rm th}}$
coupling point of $n^{\prime{\rm th}}$ atom.

The first term on the right-hand side of Eq. (III) describes the spontaneous
emission processes due to the Markovian dynamics. The second term arising from
the nature of nonlocal coupling of a single $M$-legged giant atom, and the
remainder describes the atomic relaxation dynamics mediated by the delayed
photons released from other GAs. In contrast to Eq.(III), the time evolution
given by Eq.(III) predicts that the atomic dynamics response at time $t$
depends only on the state of the system at some specific delay times (this
prediction will be labeled as “retard” in the following). Up to now, the
collective emission phenomena of $N$ GAs with arbitrary configurations would
be readily simulated based on the Eq. (III) and Eq. (III).

Figure 4: Time evolution of the change on the scaled atomic population $\Delta
P_{e}\omega_{0}/\Gamma_{0}$ for atom 1 in the array of separate GAs (a) and
braided GAs (b). Subradiant instantaneous decay rates (in units of
$\Gamma_{0}$) for separate GAs (c) and braided GAs (d) with fixed
$d=0.1\pi/k_{0}$. Other parameters, legend and the notations of background-
coloring are the same as Fig.3.

##  IV THE BUILDING OF NON-MARKOVIAN COOPERATIVITY: BEYOND RETARDATION

In the Zeno regime, the interference properties of the field would be modified
drastically due to the joint action of retardation and memory effect of the
field. The rich interference between emission from the connected coupling
points may enhance or suppress the decay of collective GAs. This phenomenon is
famous as super/subradiance Ressayre and Tallet (1977); Sheremet  _et al._
(2023); Tiranov  _et al._ (2023), respectively. Generally speaking, the
collective dynamics is sensitive to the specific configurations of GAs,
initial states, and the acquired propagating phase between neighboring
coupling points. The nonexponential decay of a linear chain of $N$ GAs can be
characterized quantitatively by instantaneous decay rate

$\displaystyle\Gamma_{{\rm ins}}=-\frac{d}{dt}\ln P_{e}(t),$ (10)

where $P_{e}(t)=\sum\limits_{n}^{N}|c_{n}(t)|^{2}$ for total decay rate of
GAs, and $P_{e}(t)=|c_{n}(t)|^{2}$ for individual decay.

###  IV.1 Single giant atom: spontaneous emission

We begin by investigating the spontaneous emission of a single giant atom with
$M$ coupling points. As shown in Fig.2(a), we plot $\Gamma_{{\rm ins}}$ (in
units of $\Gamma_{0}$) versus scaled time $\omega_{0}t$ by increasing the
number of coupling points from $M=2$ to $M=4$. We show that the decay rates
for both of the waveguide QED setups are growing gradually from zero over time
until they reach the critical “radiation burst”. This emerging burst, for
instance, is roughly about $5.46\Gamma_{0}$ for $M=2$. After which the decay
rates will exhibit oscillating behavior for a while, and eventually be
stabilized on the Markovian decay rate

$\displaystyle\Gamma_{{{\rm
Mar}},M}=\Gamma_{0}\frac{\sin^{2}M\varphi/2}{\sin^{2}\varphi/2}$ (11)

at long time limit. The non-Markovianity of const- wQED is more obvious then
that of lin-wQED, and can be enhanced by adding coupling points according to
the anticipation from their Zeno time, i.e., Eq.(4). Moreover, the influence
of ratios $\Gamma_{0}/\omega_{0}$ on the instantaneous decay rates has also
been plotted in Fig.2(b), implying that the establishment of emission is not
sensitive to this ratio.

###  IV.2 Many giant atoms: collective emission

We proceed by considering $N$ GAs bathed in a common radiation field where the
interference behaves constructive. The considered initial state of the system
inherits the analogue of timed-Dicke state with the concept originated from
the “small” atoms Scully  _et al._ (2006). For illustration purpose, we
briefly describe the preparation mechanism of the initial state.

The interaction between incident single photon with wave number $k$ and GAs is

$\displaystyle\\!\\!V(t)\\!\\!=\\!\\!-i\sum_{n=1}^{N}\sum_{m=1}^{M}g^{{\rm
JC}}_{k}\sigma^{\dagger}_{n}a_{k}e^{ikx^{n}_{m}\\!-i(\omega_{k}\\!-\\!\omega_{0})t}\\!+\\!{\rm{H.c.}},$
(12)

and the corresponding unitary time evolution operator is
$\mathcal{U}(t)=\mathcal{T}e^{-i\int_{0}^{t}V(t^{\prime})t^{\prime}}$ where
$\mathcal{T}$ the time ordering operator. When a single resonant photon is
incident from the waveguide, the state of the GAs+field system at time $t$
will be evolved into

$\displaystyle\mathcal{U}(t)a_{k_{0}}^{\dagger}\ket{G}\\!\approx\\!a_{k_{0}}^{\dagger}\ket{G}\\!+\\!g^{{\rm
JC}}_{k_{0}}t\sum_{n=1}^{N}\sum_{m=1}^{M}\sigma^{\dagger}_{n}e^{ik_{0}x^{n}_{m}}\ket{G},$
(13)

where $g^{{\rm JC}}_{k_{0}}t$ is assumed to be very small. Therefore, if we
fail to probe the photon in the output channel, then we conclude that the
total system has been prepared in

$\displaystyle\ket{\Psi_{k_{0}}}=(\frac{\Gamma_{0}}{\Gamma_{{{\rm
Mar}},M}N})^{1/2}\sum_{n=1}^{N}\sum_{m=1}^{M}\sigma^{\dagger}_{n}e^{ik_{0}x^{n}_{m}}\ket{G}.$
(14)

The dynamical descriptions in Sec. III combined with the specific initial
state $\ket{\Psi_{k_{0}}}$ enable the whole observation for the building of
collective emission. As shown in Fig.3, we plot the instantaneous decay rates
$\Gamma_{{{\rm ins}}}$ for $N=10$ separate GAs (see Fig.3 (a), background-
color: pink) and braided GAs (see Fig.3 (b), background-color: pale green),
and the spacing $d$ is taken as $0.1\pi/k_{0}$ and $0.25\pi/k_{0}$,
respectively. Here, the curves of $\Gamma_{{{\rm ins}}}$ for total GAs and
individual giant atom have been labeled as “Total” and “No.$n$” with $n$
enumerates the emitters, respectively.

The predictions of the total $\Gamma_{{{\rm ins}}}$ from the const-wQED (blue)
and lin-wQED (red) setups feature a continuous growth, while the one obtained
from only retardation (dark green) gives the steplike increase. These three
curves are inconsistent with each other in the early stage of time evolution,
but eventually tend to be consistent roughly at $t>10/\omega_{0}$ for separate
GAs and longer time for braided GAs.

Figure 5: Bosonic field density of emissions from two separate GAs that are
prepared initially in (a)-(b) $\ket{\Psi_{k_{0}}}$ and (d)-(e)
$\ket{\Psi_{{\rm sub}}}$, respectively. Panels labeled by “retaed”/“const” are
plotted according to Eq.(16)/Eq.(17). (c) and (f) are the corresponding
instantaneous decay rates. We have distinguish the numerical results for atom
$1$ (solid lines) and atom $2$ (dotted lines) in (c), and the blue/dark green
curves are predictions from const-wQED/retardation. Furthermore, the vertical
dashed lines plot the positions of coupling points $x/d=x^{n}_{m}/d$ where
$(x^{1}_{1},x^{1}_{2},x^{2}_{1},x^{2}_{2})=(-1.5d,-0.5d,0.5d,1.5d)$; the
transverse dashed lines plot the minimal delay time $\omega_{0}t=k_{0}d$. The
spacing $d=0.2\pi/k_{0}$ for (a)-(c) and $d=0.5\pi/k_{0}$ for (d)-(f) are
used. Other parameters are $M=2,\Gamma_{0}/\omega_{0}=10^{-6}$ and
$\Lambda/\omega_{0}=10^{4}$.

For the purpose of further insights of collective behavior in the Zeno regime,
we select five GAs, No.$1$, $2$, $5$, $8$, $10$, and plot their individual
decay rate $\Gamma_{{{\rm ins}}}$. We find that the GAs in atomic array could
exhibit a wide variety of decay behaviors. Firstly, the stabilized decay rate
of individual GAs tends to increase as they are positioned closer to the
right, and the scenario will be the opposite when
$\ket{\Psi_{k_{0}}}\rightarrow\ket{\Psi_{-k_{0}}}$. Secondly, the three theory
frameworks reveal distinct emission dynamics. On the one hand, taking atom 1
(No.$1$) in separate GAs array as an example, the blue curve shows that the
atom quickly enters a long time absorption dynamics after experiencing a short
time radiation; the red curve shows that the atom persistently emits
excitations into the waveguide while the dark green curve implying two
radiation-absorption cycles. Nevertheless, all three curves would be dominated
by absorption in the case of braided GAs. It can also be clearly captured by
the change on atomic population $\Delta P_{e}(t)=P_{e}(t)-P_{e}(0)$, as shown
in Figs.4(a)-(b). On the other hand, taking atom 10 (No.$10$) as an another
example, the dark green curve shows a stable step-like upward trend for the
separated GAs. Instead, it shows an oscillating zigzag upward trend for
braided GAs, with the upper and lower boundaries of the oscillation are close
to blue and red curves periodically. Thus, the received delay signal enable
the decay rate that is predicted by retardation-only picture to exchange
faithfully between const-wQED and lin-wQED. Last but not least, the above
three curves agree better at long time for the middle atom (i.e., No.$5$) in
atomic array.

We are now in a position to study the subradiant emission of $N$ GAs where the
interference between their emissions is destructive. In the Markovian
approximation, the effective Hamiltonian for atomic excitations is available
by tracing out the waveguide modes Asenjo-Garcia  _et al._ (2017); Ke  _et
al._ (2019); Ilin  _et al._ (2023), and has the form of

$\displaystyle\\!\\!\mathcal{H}_{{\rm
eff}}\\!=\\!\sum_{n,n^{\prime}=1}^{N}\\!(\omega_{0}\delta_{n,n^{\prime}}\\!-\\!i\Gamma_{0}\\!\\!\sum_{m,m^{\prime}=1}^{N}e^{ik_{0}|x^{n}_{m}\\!-\\!x^{n^{\prime}}_{m^{\prime}}|})\sigma_{n}^{\dagger}\sigma_{n^{\prime}}.$
(15)

Numerical diagonalization of $\mathcal{H}_{{\rm eff}}$ in the one-excitation
manifold gives $N$ eigenstates. After that the subradiant state $|\Psi_{{\rm
sub}}\rangle$ might be found by searching the state with minimal decay rate.

We plot the subradiant decay rates $\Gamma_{{{\rm ins}}}$ for $N=10$ separate
GAs and braided GAs in Figs.4 (c) and (d), respectively. It shows that the
subradiant emission is established by undergoing several radiation-absorption
cycles. The instantaneous decay rates $\Gamma_{{{\rm ins}}}$ oscillate around
$0$ with different amplitudes for three predictions. More concretely, the dark
green curve predicted from retardation-only picture has largest amplitude.
Later discussion shows the opposite case indicating that the const-wQED
possesses more strong non-Markovianity in the building of subradiance [see
Fig. 5(f)]. This pronounced revival of atomic survival probability gives rise
to intriguing phonomenon of oscillating bound states.

##  V CHIRAL EMISSION AND OSCILLATING BOUND STATES

Figure 6: Bosonic field density of emissions from a single giant atom that is
prepared initially in its excited state. Upper raw: intensity distributions
for a $3$-legged giant atom with the numerical predictions are predicted from
(a) retardation-only or (b) cons-wQED. (c) plots the corresponding
instantaneous decay rates of spontaneous emission. The panels in lower raw are
the same as that in upper raw but $M=5$, and the distance between adjacent
coupling points is $d=0.6641\pi/k_{0}$ for (a)-(c) and $d=0.7985\pi/k_{0}$ for
(d)-(f). Other parameters are $\Gamma_{0}/\omega_{0}=10^{-6}$ and
$\Lambda/\omega_{0}=10^{6}$.

Until now, we have studied the non-Markovian collective phenomena in a linear
chain of $N$ GAs, with the attention is mainly focused on atomic dynamics. The
asymmetric superradiant behavior in Fig.3 and the significant memory effect of
subradiance predicted by const-wQED, inspire us to explore the dynamics of
bosonic field. Thus, a more complete dynamical description of atom-field
quantum system is expected.

An important figure of merit to quantify the dynamics of bosonic field in this
regard is its bosonic field density Sinha  _et al._ (2020); Qiu  _et al._
(2023) $I(x,t)\propto\bra{\psi(t)}E^{+}(x,t)E(x,t)\ket{\psi(t)}$, where
$E(x,t)$ is the electric field operator. The distribution function $I(x,t)$
describes the probability density at position $x$ and time $t$ to find a
single photon for all possible wave vectors $k$. After some algebra presented
in Appendix. C, we find the exact formula of bosonic field density in the
retardation-only picture:

$\displaystyle\\!\\!\\!\\!\frac{I(x,t)}{\mathcal{I}_{{\rm
retard}}}\\!\\!=\\!\\!\big{|}\sum_{n=1}^{N}\\!\sum_{m=1}^{M}\\!\sum_{p=\pm}\\!\beta_{n}(t_{s}^{p},\\!\varphi_{s}^{p})[\Theta(t\\!-\\!t_{s}^{p})\\!-\\!\Theta(-t_{s}^{p})]\big{|}^{2},$
(16)

where $t_{s}^{\pm}\equiv\pm\frac{x-x^{n}_{m}}{v_{g}}$ and
$\varphi_{s}^{p}=k_{0}t_{s}^{p}$. Similarly, the distribution function
$I(x,t)$ for const-wQED is given by

$\displaystyle\frac{I(x,t)}{\mathcal{I}_{{\rm
const}}}=\big{|}\sum_{n=1}^{N}\sum_{m=1}^{M}\int_{0}^{t}d\tau c^{{{\rm
const}}}_{n}(\tau)\mathcal{S}_{nm}(t-\tau)\big{|}^{2},$ (17)

with the kernel

$\displaystyle\mathcal{S}_{nm}(t-\tau)=\mathcal{F}_{nm}(\phi,1+\Lambda)-\mathcal{F}_{nm}(\phi,1),$
(18)

where
$\mathcal{F}_{nm}(\phi,x)\equiv\sum\limits_{\zeta=\pm}e^{i(\phi+\phi^{\zeta})}{\rm
Csi}^{*}(x\phi^{\zeta})$ and $\phi^{\pm}=\phi\pm k_{0}(x-x^{n}_{m})$ have been
introduced for simplicity. Here, $\mathcal{I}_{{\rm retard}}$ and
$\mathcal{I}_{{\rm const}}$ are the normalization constants. One may note that
summation terms in Eqs.(16) describe the unidirectional emission from all
possible coupling points. These emissions collide each other and modify the
interference properties of the electromagnetic field, which can be shown from
their rich interference patterns.

We plot the bosonic field density $I(x,t)$ for the fields emitted by two
separate GAs when atoms are initially prepared in $\ket{\Psi_{k_{0}}}$, as
shown in Figs.5(a) (retard) and (b) (const-wQED). The corresponding
instantaneous decay rates for atom $1$ (solid lines) and atom $2$ (dotted
lines) are plotted in Fig.5(c). In this case, the atom $2$ decays faster than
atom $1$ leading to an asymmetric distribution of emitted fields. Thus, the
interference properties of emissions are strongly modified and eventually
exhibit obvious chirality. As a consequence, most of the energy is carried
away by the right-moving waveguide modes. Noticed that the opposite occurs by
replacing $\ket{\Psi_{k_{0}}}\rightarrow\ket{\Psi_{-k_{0}}}$.

We proceed by exploring the intriguing interference behaviors when the GAs
build their subradiance. The pronounced revival of atomic excitations in Fig.
5(f) at early time implying that the energy is alternately stored between GAs
and the fields. We plot the bosonic field density $I(x,t)$ for two separate
GAs in Figs.5 (d) and (e) according respectively to the Eqs. (16) and (17),
when atoms are prepared initially in $|\Psi_{{\rm sub}}\rangle$. The
corresponding decay rates for total GAs are shown in panel (f) for comparison
purpose. As anticipated, the field intensity predicted by const-wQED is
enhanced/weakened during the time window of atomic radiation/absorption
stages. Such “oscillating” behavior of bound state would vanish when the
system dynamics enters completely the regime of exponential decay.

This oscillating behavior for bound states can also arise in the scenario of
single giant atom when dark state is formed. In order to illustrate it
intuitively, we first plot the intensity distributions of emissions emitted
from a $M=3$-legged giant atom according to Eqs. (16) and (17), as shown
respectively in Figs.6(a) and (b). We find the atom periodically emits and
absorbs photon or phonon in const-wQED which can also be witnessed in
Fig.6(c). The similar case occurs for a $5$-legged giant atom [Figs.6(d)-(f)]
with richer interference patterns, where photons or phonons bouncing back and
forth between certain coupling points. Furthermore, we stress that the
discovered oscillating and long-lived bound states in Guo  _et al._ (2020) are
the consequence of coexistence of bound modes in the picture of retardation-
only. In our case, the similar phenomenon appears, but the mechanism based on
the combination of both non-negligible delay feedback and memory effect of the
field and only exists in the early stage of spontaneous emission.

##  VI CONCLUSION AND OUTLOOKS

In summary, we have studied the spontaneous radiation of a linear chain of $N$
$M$-legged GAs that are coupled to a 1D waveguide, with arbitrary atomic
configurations. We illustrate the collective emission by comparing three
completely different theoretical formalisms which give rise to abundant
dynamical performance. These predictions enable us to show how GAs build up
their collective emission. In particular, the time evolution of atomic
population features significant environment memory effect with a highly
relaxed temporal resolution. Moreover, we have also investigated the dynamics
of the emitted field, characterized by their intensity distribution, to give a
more complete perspective for the radiation physics. Interestingly, when
prepared in certain superradiant states, the system allows directional
emission where the most of energy is either carried away by right or left
propagating modes. We also predict a significant oscillating phenomenon in the
development of atomic subradiance. Possible applications of our results might
be efficient protection of quantum information in the quantum networks with
GAs via dissipation engineering Verstraete  _et al._ (2009); Harrington  _et
al._ (2022), or fundamentally useful for theoretical predictions of Zeno
physics Lizuain  _et al._ (2010); Mouloudakis and Lambropoulos (2022); Sun
_et al._ (2023).

As for future works, it is instructive to explore the many-body signatures of
collective emission Masson  _et al._ (2020); Cardenas-Lopez  _et al._ (2023);
Liedl  _et al._ (2024); Masson  _et al._ (2024) with GAs in the Zeno regime.
In this case, the system dynamics is no longer limited to the single
excitation space and interesting multi-particle correlations might emerge. One
can also extend the coupling mechanism to that with higher dimensional
photonic or phononic bath González-Tudela  _et al._ (2019); Vega  _et al._
(2023). And interesting physics may be explored by combining the present
results with platforms pertaining to topological Bello  _et al._ (2019) or
chiral quantum optics Lodahl  _et al._ (2017).

## ACKNOWLEDGMENTS

This work is supported by the National Key Research and Development Program of
China (Grant No. 2021YFA1400700), the National Natural Science Founda- tion of
China (Grant No. 12125402) and the Innovation Program for Quantum Science and
Technology (No. 2021ZD0301500). The computation is completed in the HPC
Platform of Huazhong University of Science and Technology.

##  APPENDIX A CALCULATIONS OF THE ZENO TIME

In this section, we give the derivations of the Zeno time $\tau_{Z}$ for a
single giant atom and $\tau_{Z,N}$ for $N$ GAs, with arbitrary coupling
points. The definition of Zeno time is given by the short-time expansion of
the atomic survival probability

$\displaystyle\left|\bra{\psi_{0}}e^{-iH_{{\rm
JC}}t}\ket{\psi_{0}}\right|^{2}\\!=\\!1-t^{2}/\tau^{2}_{Z}\\!+\\!\cdots,$ (19)

where $\ket{\psi_{0}}$ is the initial state of the emitters+field system.

### 1\. Zeno time for a single giant atom

We begin by considering a $M$-legged giant atom initially prepared in
$\ket{\psi_{0}}=\sigma^{\dagger}\ket{G}$, and then the Zeno time can be
obtained easily according to Eq.(19)

$\displaystyle\tau_{Z}^{-2}=$ $\displaystyle\bra{\psi_{0}}H_{{\rm
JC}}^{2}\ket{\psi_{0}}-\bra{\psi_{0}}H_{{\rm JC}}\ket{\psi_{0}}^{2}$
$\displaystyle=\sum_{m,m^{\prime}=1}^{M}\int_{0}^{\Lambda}\frac{dk}{\pi}(g_{k}^{{\rm
JC}})^{2}\cos[k(x^{1}_{m}-x^{1}_{m^{\prime}})].$ (20)

We then substitute the coupling strength $g_{k}^{{\rm JC}}$ for const-wQED and
lin-wQED into Eq.(A), with the obtained results are denoted respectively as
$\tau_{Z_{{\rm const}}}$ and $\tau_{Z_{{\rm lin}}}$. For the former, we have

$\displaystyle\tau_{Z_{{\rm const}}}^{-2}$
$\displaystyle=\sum_{m,m^{\prime}=1}^{M}\int_{0}^{\Lambda}\frac{dk}{\pi}\frac{4\Gamma_{0}v_{g}/2}{(1+k/k_{0})^{2}}\cos(k|x^{1}_{m}-x^{1}_{m^{\prime}}|)$
$\displaystyle=\frac{2\Gamma_{0}\omega_{0}}{\pi}\big{\\{}M^{2}+2\sum_{n=1}^{M-1}n\varphi(M-n)\times$
$\displaystyle[\sin(n\varphi){\rm{Ci}}(n\varphi
z)-\cos(n\varphi){\rm{Si}}(n\varphi z)]|_{1}^{1+\Lambda}\big{\\}},$ (21)

and for the latter, we have

$\displaystyle\tau_{Z_{{\rm lin}}}^{-2}=$
$\displaystyle\sum_{m,m^{\prime}=1}^{M}\int_{0}^{\Lambda}\frac{dk}{\pi}\frac{2\Gamma_{0}v_{g}k/k_{0}}{(1+k/k_{0})^{2}}\cos(k|x^{1}_{m}-x^{1}_{m^{\prime}}|)$
$\displaystyle=$ $\displaystyle-\tau_{Z_{{\rm
const}}}^{-2}+\frac{2\Gamma_{0}\omega_{0}}{\pi}\big{\\{}M\ln(\Lambda+1)+2\sum_{n=1}^{M-1}(M-n)$
$\displaystyle\times[\cos(n\varphi){\rm Ci}(nz\varphi)+\sin(n\varphi){\rm
Si}(nz\varphi)]\big{|}_{1}^{\Lambda+1}\big{\\}}$ $\displaystyle\approx$
$\displaystyle\frac{2\Gamma_{0}\omega_{0}}{\pi}M\ln(\Lambda+1),$ (22)

where $\varphi=k_{0}d$ and a short hand replacement
$\Lambda/k_{0}\rightarrow\Lambda$ has been made.

### 2\. Zeno time for many giant atoms

Now we consider giant atoms prepared in a so-called timed-Dicke state

$\displaystyle|\psi_{q}\rangle=\mathcal{N}\sum_{n=1}^{N}\sum_{m=1}^{M}e^{iqx^{n}_{m}}\sigma_{n}^{\dagger}|G\rangle,$
(23)

where $\mathcal{N}=\frac{1}{\sqrt{N}}\frac{\sin(\varphi/2)}{\sin(M\varphi/2)}$
is the normalization coefficient. Thus, the Zeno time $\tau_{Z,N}$ of state
$\ket{\psi_{q}}$ can be found by inserting Eq.(23) into Eq.(19):

$\displaystyle\tau_{Z,N}^{-2}=$ $\displaystyle\bra{\psi_{q}}H_{{\rm
JC}}^{2}\ket{\psi_{q}}-\bra{\psi_{q}}H_{{\rm JC}}\ket{\psi_{q}}^{2}$
$\displaystyle=$
$\displaystyle[N\mathcal{N}^{2}\sum_{mm^{\prime}}e^{iq(x^{1}_{m}-x^{1}_{m^{\prime}})}\\!-\\!(N\mathcal{N}^{2}\sum_{mm^{\prime}}e^{iq(x^{1}_{m}-x^{1}_{m^{\prime}})})^{2}]\omega_{0}^{2}$
$\displaystyle+\mathcal{N}^{2}\sum_{nn^{\prime}}^{N}\sum_{mm^{\prime}}^{M}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}(g_{k}^{{\rm
JC}})^{2}e^{i(k-q)(x^{n^{\prime}}_{m^{\prime}}-x^{n}_{m})}$ $\displaystyle=$
$\displaystyle
F^{2}_{q}\big{\\{}(1-F^{2}_{q})\omega_{0}^{2}+\tau_{Z}^{-2}+\\!\frac{1}{N}\sum_{n\neq
n^{\prime}}^{N}\sum_{mm^{\prime}}^{M}\sum_{jj^{\prime}}^{M}$ $\displaystyle
e^{-iq(x^{n^{\prime}}_{m^{\prime}}\\!-\\!x^{n}_{m})}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}(g_{k}^{{\rm
JC}})^{2}e^{ik(x^{n^{\prime}}_{j^{\prime}}\\!-\\!x^{n}_{j})}\big{\\}},$ (24)

where $F_{q}=\sin^{2}\frac{M}{2}q\varphi/\sin^{2}\frac{1}{2}q\varphi$ and
$\tau_{Z}$ is the Zeno time of a single giant atom, i.e., Eq.(A). Noticed that
we have assumed that each giant atom in atomic array is identical, no matter
what the configuration of GAs is.

##  APPENDIX B TIME EVOLUTION OF $N$ GIANT ATOMS

We are now in a position to discuss the time evolution of the system composed
of $N$ GAs coupling to a 1D waveguide. In the interaction picture, the system
Hamiltonian is

$\displaystyle\\!H_{{\rm JC}}^{{\rm
int}}\\!\\!=\\!\\!-i\sum_{n=1}^{N}\sum_{m=1}^{M}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}g_{k}^{{\rm
JC}}\sigma_{n}^{\dagger}a_{k}\\!e^{i(\omega_{0}\\!-\\!\omega_{k})t}\\!e^{ikx^{n}_{m}}\\!+\\!{{\rm
H.c.}}.$ (25)

Given the wave function ansatz in the single excitation space

$\displaystyle\ket{\psi(t)}=\sum_{n=1}^{N}c_{n}(t)\sigma_{n}^{\dagger}\ket{G}+\int_{-\Lambda}^{\Lambda}\alpha_{k}(t)a_{k}^{\dagger}\ket{G},$
(26)

we can immediately obtain the coupled differential equations of the emitters-
field system by applying the schr$\ddot{o}$dinger’s equation
$i\ket{\dot{\psi}(t)}=H_{{\rm JC}}^{{\rm int}}\ket{\psi(t)}$, which have the
form of

$\displaystyle\dot{c}_{n}(t)$
$\displaystyle=-\sum_{m=1}^{M}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}g_{k}^{{\rm
JC}}\alpha_{k}(t)e^{i(\omega_{0}-\omega_{k})t}e^{ikx^{n}_{m}},$ (27)
$\displaystyle\dot{\alpha}_{k}(t)$
$\displaystyle=\sum_{n=1}^{N}\sum_{m=1}^{M}g_{k}^{*{\rm
JC}}c_{n}(t)e^{-i(\omega_{0}-\omega_{k})t}e^{-ikx^{n}_{m}}.$ (28)

The probability amplitudes of bosonic modes can be solved formally as follows

$\displaystyle\\!\\!\\!\alpha_{k}(t)\\!=\\!\sum_{n=1}^{N}\\!\sum_{m=1}^{M}\\!g_{k}^{*{\rm
JC}}\int_{0}^{t}d\tau
c_{n}(\tau)\\!e^{-i(\omega_{0}\\!-\\!\omega_{k})\tau}\\!e^{-ikx^{n}_{m}}.$
(29)

Therefore, the atomic equations of motion can be derived by inserting Eq.(29)
into Eq.(27):

$\displaystyle\dot{c}_{n}(t)=$
$\displaystyle-\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}\int_{0}^{t}d\tau\big{|}g_{k}^{{\rm
JC}}\big{|}^{2}c_{n^{\prime}}(\tau)$ $\displaystyle\times
e^{i(\omega_{0}-\omega_{k})(t-\tau)}e^{ik(x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}})}.$
(30)

In the following, we will solve numerically the Eq.(B) by considering
different waveguide-QED setups.

### 1\. Solutions of equations of motion

The above integro-differential equation Eq.(B) can be transformed to an
integral equation

$\displaystyle c_{n}(t)=$ $\displaystyle
c_{n}(0)-\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{t}dt^{\prime}\int_{0}^{\Lambda/k_{0}}\frac{dz}{\pi}\int_{0}^{t^{\prime}}d\tau
k_{0}|g^{{\rm
JC}}_{k_{0}z}|^{2}c_{n^{\prime}}(\tau)e^{i\omega_{0}(1-z)(t^{\prime}-\tau)}\cos(z\varphi^{nn^{\prime}}_{mm^{\prime}})$
$\displaystyle=$ $\displaystyle
c_{n}(0)+\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{t}d\tau
c_{n^{\prime}}(\tau)\int_{0}^{\Lambda}dz\frac{\cos(z\varphi_{nn^{\prime},mm^{\prime}})|g^{{\rm
JC}}_{k_{0}z}|^{2}k_{0}(1-e^{i\omega_{0}(1-z)(t-\tau)})}{\pi
i(1-z)\omega_{0}},$ (31)

where we have defined $\varphi^{nn^{\prime}}_{mm^{\prime}}\equiv
k_{0}|x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}}|$ for simplicity. We then proceed
by plugging the coupling strength of const-wQED
$g_{k}=\sqrt{\Gamma_{0}v_{g}/2}$ into Eq.(B), with the obtained solution is
given by

$\displaystyle c^{{\rm const}}_{n}(t)=$ $\displaystyle
c_{n}(0)+\frac{2\Gamma_{0}i}{\pi}\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{t}d\tau
c^{{\rm
const}}_{n^{\prime}}(\tau)\int_{0}^{\Lambda}dz\frac{\cos(x\varphi^{nn^{\prime}}_{mm^{\prime}})}{(1+z)^{2}}\frac{1-e^{i\omega_{0}(1-z)(t-\tau)}}{z-1}$
$\displaystyle=$ $\displaystyle
c_{n}(0)+\frac{2\Gamma_{0}i}{\pi}\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{t}d\tau
c^{{\rm
const}}_{n^{\prime}}(\tau)[\mathcal{K}_{1}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)-\mathcal{K}_{2}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)-2\mathcal{K}_{3}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)],$
(32)

where $\phi=\omega_{0}(t-\tau)$, and the kernels in Eq.(B) are given by

$\displaystyle\mathcal{K}_{1}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)=$
$\displaystyle\frac{1}{4}[\cos(\varphi^{nn^{\prime}}_{mm^{\prime}}){\rm
Ci}(z\varphi^{nn^{\prime}}_{mm^{\prime}})-\sin(\varphi^{nn^{\prime}}_{mm^{\prime}}){\rm
Si}(z\varphi^{nn^{\prime}}_{mm^{\prime}})\\!-\\!\frac{e^{i\varphi^{nn^{\prime}}_{mm^{\prime}}}}{2}{\rm
Csi}(z\varphi^{nn^{\prime}-}_{mm^{\prime}})\\!-\\!\frac{e^{-i\varphi^{nn^{\prime}}_{mm^{\prime}}}}{2}{\rm
Csi^{*}}(z\varphi^{nn^{\prime}+}_{mm^{\prime}})]\big{|}_{-1}^{\Lambda-1}$
$\displaystyle\mathcal{K}_{2}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)=$
$\displaystyle\frac{1}{4}[\cos(\varphi^{nn^{\prime}}_{mm^{\prime}}){\rm
Ci}(z\varphi^{nn^{\prime}}_{mm^{\prime}})+\sin(\varphi^{nn^{\prime}}_{mm^{\prime}}){\rm
Si}(z\varphi^{nn^{\prime}}_{mm^{\prime}})-\frac{e^{i(\phi-\varphi^{nn^{\prime}-}_{mm^{\prime}})}}{2}{\rm
Csi}(z\varphi^{nn^{\prime}-}_{mm^{\prime}})-\frac{e^{i(\phi+\varphi^{nn^{\prime}+}_{mm^{\prime}})}}{2}{\rm
Csi^{*}}(z\varphi^{nn^{\prime}+}_{mm^{\prime}})]\big{|}_{1}^{\Lambda+1}$
$\displaystyle\mathcal{K}_{3}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)=$
$\displaystyle\frac{1}{4}\frac{\cos(\Lambda\varphi^{nn^{\prime}}_{mm^{\prime}})}{\Lambda+1}[e^{-i(\Lambda-1)\phi}-1]+1-e^{i\phi}-\big{\\{}\varphi^{nn^{\prime}}_{mm^{\prime}}[\cos(\varphi^{nn^{\prime}}_{mm^{\prime}}){\rm
Si}(z\varphi^{nn^{\prime}}_{mm^{\prime}})-\sin(\varphi^{nn^{\prime}}_{mm^{\prime}}){\rm
Ci}(z\varphi^{nn^{\prime}}_{mm^{\prime}})]$
$\displaystyle+\frac{i}{2}\varphi^{nn^{\prime}-}_{mm^{\prime}}e^{i(2\phi-\varphi^{nn^{\prime}}_{mm^{\prime}})}{\rm
Csi}(z\varphi^{nn^{\prime}-}_{mm^{\prime}})-\frac{i}{2}\varphi^{nn^{\prime}+}_{mm^{\prime}}e^{i(2\phi+\varphi^{nn^{\prime}}_{mm^{\prime}})}{\rm
Csi}(z\varphi^{nn^{\prime}+}_{mm^{\prime}})\big{\\}}\big{|}_{1}^{\Lambda+1},$
(33)

where
$\varphi^{nn^{\prime}\pm}_{mm^{\prime}}\equiv\varphi^{nn^{\prime}}_{mm^{\prime}}\pm\phi$
is introduced. Following the similar procedures, we can easily obtain the time
evolution of atomic population for lin-wQED, which is given by

$\displaystyle c^{{\rm
lin}}_{n}(t)=c_{n}(0)+\frac{2\Gamma_{0}i}{\pi}\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{t}d\tau
c^{{\rm
lin}}_{n^{\prime}}(\tau)[\mathcal{K}_{1}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)-\mathcal{K}_{2}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)+2\mathcal{K}_{3}(\varphi^{nn^{\prime}}_{mm^{\prime}},\phi)].$
(34)

As the dynamics controlled by Eq.(25) conserves the total number of
excitations, thus the normalization of atomic probability amplitudes obtained
in Eq.(B) and Eq.(34) should be done by making a transformation
$c_{n}(t)\rightarrow c_{n}(t)/\sqrt{|c_{n}(t)|^{2}+N_{B}(t)}$. Here,
$N_{B}(t)=\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}|\alpha_{k}(t)|^{2}$ is the
number of excitations in the 1D waveguide. After some algebra, the expression
of $N_{B}(t)$ has the form of

$\displaystyle
N_{B}(t)\\!=\\!\sum_{n,n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{t}d\tau\int_{0}^{t}d\tau^{\prime}c_{n}(\tau)c_{n^{\prime}}(\tau^{\prime})B_{nn^{\prime};mm^{\prime}}(\tau\\!-\\!\tau^{\prime}),$
(35)

where the kernel $B_{nn^{\prime};mm^{\prime}}$ is given by

$\displaystyle B^{{\rm
const}}_{nn^{\prime};mm^{\prime}}(\tau\\!-\\!\tau^{\prime})=\frac{2\Gamma_{0}\omega_{0}}{\pi}e^{-i\phi}\big{[}-\frac{\cos(z\varphi^{nn^{\prime}}_{mm^{\prime}})}{1+z}e^{i\phi
z}\big{|}_{0}^{\Lambda}+\sum_{s=\pm}\frac{i}{2}\phi_{s}e^{-i\phi_{s}}{\rm
Csi}(z\phi_{s})\big{|}_{1}^{\Lambda+1}\big{]}$ (36)

for const-wQED, and

$\displaystyle B^{{\rm
lin}}_{nn^{\prime};mm^{\prime}}(\tau\\!-\\!\tau^{\prime})=\frac{\Gamma_{0}\omega_{0}}{\pi}e^{-i\phi}\sum_{s=\pm}e^{-i\phi_{s}}{\rm
Csi}(z\phi_{s})\big{|}_{1}^{\Lambda+1}-B_{nn^{\prime};mm^{\prime}}^{{\rm
const}}(\tau-\tau^{\prime})$ (37)

for lin-wQED, where $\phi_{\pm}=\phi\pm\varphi^{nn^{\prime}}_{mm^{\prime}}$ is
introduced. Up to now, the exact solutions for atomic time evolution are found
for both the waveguide setups, as shown in Eq.(III) in the main text.

### 2\. Non-Markovian dynamics in the retardation-only picture

In order to obtain the equations of motion considering only the retardation,
one can rewrite the Eq.(B) in the limit of $\Lambda\rightarrow\infty$ as
follows

$\displaystyle\dot{c}_{n}(t)=$
$\displaystyle-\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{-\infty}^{\infty}\frac{dk}{2\pi}\int_{0}^{t}d\tau|g_{k}^{{\rm
JC}}|^{2}c_{n^{\prime}}(\tau)e^{i(\omega_{0}-\omega_{k})(t-\tau)}e^{ik(x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}})}$
$\displaystyle=$
$\displaystyle-\sum_{n^{\prime}=1}^{N}\sum_{m,m^{\prime}=1}^{M}\int_{0}^{\infty}\frac{d\omega_{k}}{2\pi
v_{g}}|g_{k}^{{\rm JC}}|^{2}\int_{0}^{t}d\tau
c_{n^{\prime}}(\tau)\big{[}e^{i\omega_{0}(t-\tau)}e^{i\omega_{k}\frac{(x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}})}{v_{g}}-(t-\tau)}+e^{i\omega_{0}(t-\tau)}e^{i\omega_{k}\frac{(-x^{n}_{m}+x^{n^{\prime}}_{m^{\prime}})}{v_{g}}-(t-\tau)}\big{]}$
$\displaystyle\approx$ $\displaystyle-\frac{|g_{k_{0}}^{{\rm
JC}}|^{2}}{v_{g}}[2M\int_{0}^{t}d\tau c_{n}(\tau)\delta(t-\tau)+\sum_{m\neq
m^{\prime}}\int_{0}^{t}d\tau
c_{n}(\tau)\delta(t-\frac{|x^{n}_{m}-x^{n}_{m^{\prime}}|}{v_{g}}-\tau)e^{ik_{0}|x^{n}_{nm}-x^{n}_{m^{\prime}}|}]$
$\displaystyle-\sum_{n^{\prime}\neq
n}^{N}\sum_{m,m^{\prime}=1}^{M}\frac{\Gamma_{0}}{2}c_{n^{\prime}}(t-\frac{|x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}}|}{v_{g}})e^{ik_{0}|x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}}|}\Theta(t-\frac{|x^{n}_{m}-x^{n^{\prime}}_{m^{\prime}}|}{v_{g}})$
$\displaystyle\\!\\!=\\!$
$\displaystyle-\frac{M}{2}\Gamma_{0}c_{n}(t)\\!-\\!\sum_{m\neq
m^{\prime}}\frac{\Gamma_{0}}{2}c_{n}(t\\!-\\!\frac{\varphi^{nn}_{mm^{\prime}}}{\omega_{0}})e^{i\varphi^{nn}_{mm^{\prime}}}\Theta(t\\!-\\!\frac{\varphi^{nn}_{mm^{\prime}}}{\omega_{0}})\\!-\\!\sum_{n^{\prime}\neq
n}^{N}\sum_{m,m^{\prime}=1}^{M}\frac{\Gamma_{0}}{2}c_{n^{\prime}}(t\\!-\\!\frac{\varphi^{nn^{\prime}}_{mm^{\prime}}}{\omega_{0}})e^{i\varphi^{nn^{\prime}}_{mm^{\prime}}}\Theta(t\\!-\\!\frac{\varphi^{nn^{\prime}}_{mm^{\prime}}}{\omega_{0}})$
$\displaystyle=$
$\displaystyle-\frac{M}{2}\Gamma_{0}c_{n}(t)\\!-\\!\sum_{m\neq
m^{\prime}}\frac{\Gamma_{0}}{2}\beta_{n}(\tau_{mm^{\prime}}^{nn},\varphi_{mm^{\prime}}^{nn})\Theta(t\\!-\\!\tau_{mm^{\prime}}^{nn})\\!-\\!\sum_{n^{\prime}\neq
n}\sum_{m,m^{\prime}=1}^{M}\frac{\Gamma_{0}}{2}\beta_{n^{\prime}}(\tau_{mm^{\prime}}^{nn^{\prime}},\varphi_{mm^{\prime}}^{nn^{\prime}})\Theta(t\\!-\\!\tau_{mm^{\prime}}^{nn^{\prime}}),$
(38)

which is exactly the Eq.(III) in the main text, and $\Theta(\bullet)$ is the
Heaviside step function. Such a simplified equation of motion for GAs
indicates that the time evolution of atom $n$ at time $t$ is decided only by
the historical dynamics at certain instants
$\tau_{mm^{\prime}}^{nn^{\prime}}$. As a consequence, it shows great
discrepancy from that predicted from const-wQED and lin-wQED.

##  APPENDIX C DETAILED CALCULATIONS OF FIELD INTENSITY DISTRIBUTION

We now turn to focus on the dynamical properties of the bosonic field emitted
from the coupled GAs, which can be characterized by its field intensity
distribution

$\displaystyle
I(x,t)=\frac{\epsilon_{0}v_{g}}{2}\bra{\psi(t)}E^{\dagger}(x,t)E(x,t)\ket{\psi(t)},$
(39)

where $E(x,t)$ is the electric field operator at position $x$ and time $t$
with a general form

$\displaystyle\\!\\!E(x,t)\\!=\\!\int_{0}^{\Lambda}dk\mathcal{E}_{k}[e^{ikx}a_{R}(k)\\!+\\!e^{-ikx}a_{L}(k)]e^{-i\omega_{k}t}.$
(40)

Here, $a_{R/L}(k)$ is the annihilation operator of the right-/left-moving mode
with frequency $\omega_{k}$ and $\mathcal{E}_{k}$ is assumed to be a constant
$\mathcal{E}_{k_{0}}$ for all $k$. Combined with the Eq.(29) and Eq.(40), the
field density distribution $I(x,t)$ for const-wQED can be calculated as
follows

$\displaystyle I(x,t)=$
$\displaystyle\frac{\epsilon_{0}v_{g}|\mathcal{E}_{k_{0}}\big{|}^{2}}{2}|\langle
G|\int_{0}^{\Lambda}dk[e^{ikx}a_{R}(k)+e^{-ikx}a_{L}(k)]$ $\displaystyle\times
e^{-i\omega_{k}t}|\psi(t)\rangle\big{|}^{2}$ $\displaystyle=$ $\displaystyle
8\Gamma_{0}v_{g}I_{0}\big{|}\sum_{n=1}^{N}\sum_{m=1}^{M}\int_{0}^{t}d\tau\int_{0}^{\Lambda}dk\frac{1}{1+k/k_{0}}c_{n}(\tau)$
$\displaystyle\cos[k(x-x_{m}^{n})]e^{-i(\omega_{k}-\omega_{0})(t-\tau)}\big{|}^{2}$
$\displaystyle=$ $\displaystyle
2\Gamma_{0}v_{g}I_{0}\big{|}\sum_{n=1}^{N}\sum_{m=1}^{M}\int_{0}^{t}d\tau
c_{n}(\tau)\int_{1}^{1+\Lambda/k_{0}}dz\frac{k_{0}}{z}$
$\displaystyle\times[e^{i(r_{nm}-\phi)z}e^{i(2\phi-
r_{nm})}\\!+\\!e^{-i(r_{nm}\\!+\\!\phi)z}e^{i(2\phi\\!+\\!r_{nm})}]\big{|}^{2}$
$\displaystyle\\!\\!\\!=$ $\displaystyle
2\Gamma_{0}v_{g}I_{0}|\sum_{n=1}^{N}\sum_{m=1}^{M}\int_{0}^{t}d\tau
c_{n}(\tau)\\!S_{nm}(t\\!-\\!\tau)|^{2}\\!,\\!\\!$ (41)

which is exactly the Eq.(17) in the main text, where
$I_{0}=\frac{\epsilon_{0}v_{g}|\mathcal{E}_{k_{0}}|^{2}}{2},r_{nm}=k_{0}(x-x_{m}^{n})$
and the atomic probability amplitude $c_{n}(\tau)$ is given by Eq.(B). The
kernel $S_{nm}(t-\tau)$ in Eq.(C) reads

$\displaystyle\mathcal{S}_{nm}(t-\tau)=\mathcal{F}_{nm}(\phi,1+\Lambda)-\mathcal{F}_{nm}(\phi,1).$
(42)

here, we have introduced the notation
$\mathcal{F}_{nm}(\phi,x)\equiv\sum\limits_{\zeta=\pm}e^{i(\phi+\phi_{\zeta})}{\rm
Csi}^{*}(x\phi_{\zeta})$ for simplicity.

We proceed by calculating the field intensity of the emitted field in
retardation-only picture. In the limit of large enough
$\Lambda\rightarrow\infty$, we have

$\displaystyle I(x,t)=$
$\displaystyle\frac{\epsilon_{0}v_{g}|\mathcal{E}_{k_{0}}\big{|}^{2}}{2}|\langle
G|\int_{0}^{\infty}dk[e^{ikx}a_{R}(k)+e^{-ikx}a_{L}(k)]\times
e^{-i\omega_{k}t}|\psi(t)\rangle\big{|}^{2}$ $\displaystyle=$
$\displaystyle\frac{I_{0}\Gamma_{0}}{2v_{g}}\Big{|}\int_{-\infty}^{\infty}dz\int_{0}^{t}d\tau\sum_{n=1}^{N}\sum_{m=1}^{M}c_{n}(\tau)[e^{iz(\frac{x-x_{m}^{n}}{v_{g}}+\tau-t)}\times
e^{i\omega_{0}(\frac{x-x_{m}^{n}}{v_{g}}-t)}+e^{-iz(\frac{x-x_{m}^{n}}{v_{g}}-\tau+t)}e^{-i\omega_{0}(\frac{x-x_{m}^{n}}{v_{g}}+t)}]\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\frac{2\pi^{2}I_{0}\Gamma_{0}}{v_{g}}\Big{|}\sum_{n=1}^{N}\sum_{m=1}^{M}\big{\\{}c_{n}(t\\!-\\!x_{m}^{n}/v_{g}\\!+\\!x/v_{g})e^{i\omega_{0}(x_{m}^{n}\\!-\\!x)/v_{g}}[\Theta(-\frac{x_{m}^{n}\\!-\\!x}{v_{g}})\\!-\\!\Theta(t\\!-\\!\frac{x_{m}^{n}\\!-\\!x}{v_{g}})]$
$\displaystyle+c_{n}(t\\!+\\!x_{m}^{n}/v_{g}\\!-\\!x/v_{g})e^{-i\omega_{0}(x_{m}^{n}-x)/v_{g}}[\Theta(\frac{x_{m}^{n}-x}{v_{g}})-\Theta(t+\frac{x_{m}^{n}-x}{v_{g}})]\big{\\}}\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\frac{2\pi^{2}I_{0}\Gamma_{0}}{v_{g}}|\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{p=\pm}c_{n}(t-p\frac{x-x_{m}^{n}}{v_{g}})e^{i\omega_{0}p\frac{x-x_{m}^{n}}{v_{g}}}\times[\Theta(t-p\frac{x-x_{m}^{n}}{v_{g}})-\Theta(-p\frac{x-x_{m}^{n}}{v_{g}})]|^{2},$
(43)

which is exactly the Eq.(16) in the main text. Therefore, the calculations of
the field intensity distribution for const-wQED and the one with only
retardation are completed.

## References

  * Dicke (1954) R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99 (1954).
  * Hu  _et al._ (1992) B. L. Hu, J. P. Paz, and Y. Zhang, “Quantum brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise,” Phys. Rev. D 45, 2843 (1992).
  * Hu  _et al._ (1993) B. L. Hu, J. P. Paz, and Y. Zhang, “Quantum brownian motion in a general environment. ii. nonlinear coupling and perturbative approach,” Phys. Rev. D 47, 1576 (1993).
  * Tufarelli  _et al._ (2013) T. Tufarelli, F. Ciccarello, and M. S. Kim, “Dynamics of spontaneous emission in a single-end photonic waveguide,” Phys. Rev. A 87, 013820 (2013).
  * Vasile  _et al._ (2014) R. Vasile, F. Galve, and R. Zambrini, “Spectral origin of non-markovian open-system dynamics: A finite harmonic model without approximations,” Phys. Rev. A 89, 022109 (2014).
  * Gröblacher  _et al._ (2015) S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, “Observation of non-markovian micromechanical brownian motion,” Nat. Commun. 6, 7606 (2015).
  * Breuer  _et al._ (2016) H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, “Colloquium: Non-markovian dynamics in open quantum systems,” Rev. Mod. Phys. 88, 021002 (2016).
  * de Vega and Alonso (2017) I. de Vega and D. Alonso, “Dynamics of non-markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
  * Ask and Johansson (2022) A. Ask and G. Johansson, “Non-markovian steady states of a driven two-level system,” Phys. Rev. Lett. 128, 083603 (2022).
  * Cilluffo  _et al._ (2024) D. Cilluffo, L. Ferialdi, G. M. Palma, G. Calajò, and F. Ciccarello, “Multimode-cavity picture of non-markovian waveguide qed,” (2024), arXiv:2403.07110 [quant-ph] .
  * Sinha  _et al._ (2020) K. Sinha, P. Meystre, E. A. Goldschmidt, F. K. Fatemi, S. L. Rolston, and P. Solano, “Non-markovian collective emission from macroscopically separated emitters,” Phys. Rev. Lett. 124, 043603 (2020).
  * Solano  _et al._ (2023) P. Solano, P. Barberis-Blostein, and K. Sinha, “Dissimilar collective decay and directional emission from two quantum emitters,” Phys. Rev. A 107, 023723 (2023).
  * Bai and An (2023) S.-Y. Bai and J.-H. An, “Floquet engineering to overcome no-go theorem of noisy quantum metrology,” Phys. Rev. Lett. 131, 050801 (2023).
  * Zhang (2023) Y.-X. Zhang, “Zeno regime of collective emission: Non-markovianity beyond retardation,” Phys. Rev. Lett. 131, 193603 (2023).
  * Wilkinson  _et al._ (1997) S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Niu, B. Sundaram, and M. G. Raizen, “Experimental evidence for non-exponential decay in quantum tunnelling,” Nature 387, 575 (1997).
  * Crespi  _et al._ (2019) A. Crespi, F. V. Pepe, P. Facchi, F. Sciarrino, P. Mataloni, H. Nakazato, S. Pascazio, and R. Osellame, “Experimental investigation of quantum decay at short, intermediate, and long times via integrated photonics,” Phys. Rev. Lett. 122, 130401 (2019).
  * Giacosa and Pagliara (2021) F. Giacosa and G. Pagliara, “Leggett-garg inequalities and decays of unstable systems,” Phys. Rev. A 104, 052225 (2021).
  * Liu  _et al._ (2023) Q. Liu, W. Liu, K. Ziegler, and F. Chen, “Engineering of zeno dynamics in integrated photonics,” Phys. Rev. Lett. 130, 103801 (2023).
  * Frisk Kockum  _et al._ (2014) A. Frisk Kockum, P. Delsing, and G. Johansson, “Designing frequency-dependent relaxation rates and lamb shifts for a giant artificial atom,” Phys. Rev. A 90, 013837 (2014).
  * Guo  _et al._ (2017) L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, “Giant acoustic atom: A single quantum system with a deterministic time delay,” Phys. Rev. A 95, 053821 (2017).
  * Frisk Kockum (2020) A. Frisk Kockum, “Quantum optics with giant atoms—the first five years,” in  _Mathematics for Industry_ (Springer Singapore, 2020) p. 125–146.
  * Andersson  _et al._ (2015) G. Andersson, B. Suri, L. Guo, T. Aref, and P. Delsing, “Non-exponential decay of a giant artificial atom,” Nat. Phys. 15, 1123–1127 (2015).
  * 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, A. Vepsäläinen, R. Winik, J. L. Yoder, F. Nori, T. P. Orlando, S. Gustavsson, and W. D. Oliver, “Waveguide quantum electrodynamics with superconducting artificial giant atoms,” Nature 583, 775–779 (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, “Engineering the level structure of a giant artificial atom in waveguide quantum electrodynamics,” Phys. Rev. A 103, 023710 (2021).
  * Joshi  _et al._ (2023) C. Joshi, F. Yang, and M. Mirhosseini, “Resonance fluorescence of a chiral artificial atom,” Phys. Rev. X 13, 021039 (2023).
  * 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, “Non-exponential decay of a giant artificial atom,” Nat Commun 13, 7580 (2022).
  * Wang  _et al._ (2023) X. Wang, Q.-Y. Qiu, K.-W. Huang, and H. Xiong, “Nonreciprocal single-photon scattering in giant-spin-ensemble–waveguide magnonics,” Phys. Rev. A 108, 063715 (2023).
  * Terradas-Briansó  _et al._ (2022) S. Terradas-Briansó, C. A. González-Gutiérrez, F. Nori, L. Martín-Moreno, and D. Zueco, “Ultrastrong waveguide qed with giant atoms,” Phys. Rev. A 106, 063717 (2022).
  * Guo  _et al._ (2020) L. Guo, A. F. Kockum, F. Marquardt, and G. Johansson, “Oscillating bound states for a giant atom,” Phys. Rev. Res. 2, 043014 (2020).
  * Lim  _et al._ (2023) K. H. Lim, W.-K. Mok, and L.-C. Kwek, “Oscillating bound states in non-markovian photonic lattices,” Phys. Rev. A 107, 023716 (2023).
  * Xu and Guo (2024) L. Xu and L. Guo, “Catch and release of propagating bosonic field with non-markovian giant atom,” New J. Phys. 26, 013025 (2024).
  * Qiu  _et al._ (2023) Q.-Y. Qiu, Y. Wu, and X.-Y. Lü, “Collective radiance of giant atoms in non-markovian regime,” Sci. China Phys. Mech. Astron. 66, 7580 (2023).
  * Yin  _et al._ (2022) X.-L. Yin, W.-B. Luo, and J.-Q. Liao, “Non-markovian disentanglement dynamics in double-giant-atom waveguide-qed systems,” Phys. Rev. A 106, 063703 (2022).
  * Du  _et al._ (2022) L. Du, Y.-T. Chen, Y. Zhang, and Y. Li, “Giant atoms with time-dependent couplings,” Phys. Rev. Res. 4, 023198 (2022).
  * Cheng  _et al._ (2022) W. Cheng, Z. Wang, and Y.-x. Liu, “Topology and retardation effect of a giant atom in a topological waveguide,” Phys. Rev. A 106, 033522 (2022).
  * Guo  _et al._ (2024) A.-L. Guo, L.-T. Zhu, G.-C. Guo, Z.-R. Lin, C.-F. Li, and T. Tu, “Phonon superradiance with time delays from collective giant atoms,” Phys. Rev. A 109, 033711 (2024).
  * Li and Shen (2024) Z. Y. Li and H. Z. Shen, “Non-markovian dynamics with a giant atom coupled to a semi-infinite photonic waveguide,” Phys. Rev. A 109, 023712 (2024).
  * Vats and John (1998) N. Vats and S. John, “Non-markovian quantum fluctuations and superradiance near a photonic band edge,” Phys. Rev. A 58, 4168 (1998).
  * Ferreira  _et al._ (2021) V. S. Ferreira, J. Banker, A. Sipahigil, M. H. Matheny, A. J. Keller, E. Kim, M. Mirhosseini, and O. Painter, “Collapse and revival of an artificial atom coupled to a structured photonic reservoir,” Phys. Rev. X 11, 041043 (2021).
  * Burgess and Florescu (2022) A. Burgess and M. Florescu, “Non-markovian dynamics of a single excitation within many-body dissipative systems,” Phys. Rev. A 105, 062207 (2022).
  * Drummond (1987) P. D. Drummond, “Unifying the p$\cdot$a and d$\cdot$e interactions in photodetector theory,” Phys. Rev. A 35, 4253 (1987).
  * Stokes and Nazir (2022) A. Stokes and A. Nazir, “Implications of gauge freedom for nonrelativistic quantum electrodynamics,” Rev. Mod. Phys. 94, 045003 (2022).
  * Yan (2023) Y. Yan, “Spontaneous emission spectrum from a v-type artificial atom in a strong-coupling regime: Dark lines and line narrowing,” Phys. Rev. A 108, 043712 (2023).
  * Ressayre and Tallet (1977) E. Ressayre and A. Tallet, “Quantum theory for superradiance,” Phys. Rev. A 15, 2410 (1977).
  * Sheremet  _et al._ (2023) A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakinskiy, and A. N. Poddubny, “Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations,” Rev. Mod. Phys. 95, 015002 (2023).
  * Tiranov  _et al._ (2023) A. Tiranov, V. Angelopoulou, C. J. van Diepen, B. Schrinski, O. A. D. Sandberg, Y. Wang, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, A. S. Sørensen, and P. Lodahl, “Collective super- and subradiant dynamics between distant optical quantum emitters,” Science 379, 389 (2023).
  * Scully  _et al._ (2006) M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of $n$ atoms: Timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
  * Asenjo-Garcia  _et al._ (2017) A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, “Exponential improvement in photon storage fidelities using subradiance and “selective radiance” in atomic arrays,” Phys. Rev. X 7, 031024 (2017).
  * Ke  _et al._ (2019) Y. Ke, A. V. Poshakinskiy, C. Lee, Y. S. Kivshar, and A. N. Poddubny, “Inelastic scattering of photon pairs in qubit arrays with subradiant states,” Phys. Rev. Lett. 123, 253601 (2019).
  * Ilin  _et al._ (2023) D. Ilin, A. V. Poshakinskiy, A. N. Poddubny, and I. Iorsh, “Frequency combs with parity-protected cross-correlations and entanglement from dynamically modulated qubit arrays,” Phys. Rev. Lett. 130, 023601 (2023).
  * Verstraete  _et al._ (2009) F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, “Quantum computation and quantum-state engineering driven by dissipation,” Nature Phys 5, 633 (2009).
  * Harrington  _et al._ (2022) P. M. Harrington, E. J. Mueller, and K. W. Murch, “Engineered dissipation for quantum information science,” Nat. Rev. Phys. 4, 660 (2022).
  * Lizuain  _et al._ (2010) I. Lizuain, J. Casanova, J. J. García-Ripoll, J. G. Muga, and E. Solano, “Zeno physics in ultrastrong-coupling circuit qed,” Phys. Rev. A 81, 062131 (2010).
  * Mouloudakis and Lambropoulos (2022) G. Mouloudakis and P. Lambropoulos, “Coalescence of non-markovian dissipation, quantum zeno effect, and non-hermitian physics in a simple realistic quantum system,” Phys. Rev. A 106, 053709 (2022).
  * Sun  _et al._ (2023) Y. Sun, T. Shi, Z. Liu, Z. Zhang, L. Xiao, S. Jia, and Y. Hu, “Fractional quantum zeno effect emerging from non-hermitian physics,” Phys. Rev. X 13, 031009 (2023).
  * Masson  _et al._ (2020) S. J. Masson, I. Ferrier-Barbut, L. A. Orozco, A. Browaeys, and A. Asenjo-Garcia, “Many-body signatures of collective decay in atomic chains,” Phys. Rev. Lett. 125, 263601 (2020).
  * Cardenas-Lopez  _et al._ (2023) S. Cardenas-Lopez, S. J. Masson, Z. Zager, and A. Asenjo-Garcia, “Many-body superradiance and dynamical mirror symmetry breaking in waveguide qed,” Phys. Rev. Lett. 131, 033605 (2023).
  * Liedl  _et al._ (2024) C. Liedl, F. Tebbenjohanns, C. Bach, S. Pucher, A. Rauschenbeutel, and P. Schneeweiss, “Observation of superradiant bursts in a cascaded quantum system,” Phys. Rev. X 14, 011020 (2024).
  * Masson  _et al._ (2024) S. J. Masson, J. P. Covey, S. Will, and A. Asenjo-Garcia, “Dicke superradiance in ordered arrays of multilevel atoms,” PRX Quantum 5, 010344 (2024).
  * González-Tudela  _et al._ (2019) A. González-Tudela, C. S. Muñoz, and J. I. Cirac, “Engineering and harnessing giant atoms in high-dimensional baths: A proposal for implementation with cold atoms,” Phys. Rev. Lett. 122, 203603 (2019).
  * Vega  _et al._ (2023) C. Vega, D. Porras, and A. González-Tudela, “Topological multimode waveguide qed,” Phys. Rev. Res. 5, 023031 (2023).
  * Bello  _et al._ (2019) M. Bello, G. Platero, J. I. Cirac, and A. González-Tudela, “Unconventional quantum optics in topological waveguide qed,” Sci. Adv. 5, eaaw0297 (2019).
  * Lodahl  _et al._ (2017) P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473 (2017).