# Spontaneous Emission in the presence of Quantum Mirrors

Kanu Sinha<EMAIL_ADDRESS>Wyant College of Optical Sciences and Department
of Physics, University of Arizona, Tucson, AZ 85719  Jennifer Parra-Contreras
Wyant College of Optical Sciences and Department of Physics, University of
Arizona, Tucson, AZ 85719  Annyun Das  Wyant College of Optical Sciences and
Department of Physics, University of Arizona, Tucson, AZ 85719  Pablo Solano
<EMAIL_ADDRESS>Departamento de Física, Facultad de Ciencias Físicas y
Matemáticas, Universidad de Concepción, Concepción, Chile

###### Abstract

Arrays of atoms coupled to waveguides can behave as mirrors. We consider an
array of $\Lambda$-type three-level atoms wherein preparing the atoms in one
ground state or another leads to reflection or transmission of the guided
electromagnetic field; a superposition of the two ground states thus
corresponds to a coherent superposition of mirror-like and transparent
boundary conditions. We analyze the spontaneous emission of an excited two-
level atom in the presence of such a quantum mirror, and inside a cavity
formed by quantum mirrors, demonstrating that the resulting dynamics of the
excited atom can exhibit exotic features, e.g., a superposition of Rabi cycle
and exponential decay. Our results pave the way for exploring quantum
electrodynamics (QED) phenomena in a paradigm wherein boundary conditions can
exhibit quantum superpositions and correlations.

Introduction.— The modification of the quantized electromagnetic (EM) field in
the presence of boundaries is a ubiquitous aspect of Quantum Electrodynamics
(QED), and integral to a wide range of phenomena from cavity QED Walther  _et
al._ (2006); Mabuchi and Doherty (2002), to nanophotonics Novotny and Hecht
(2012); D’Amico  _et al._ (2019), and Casimir physics Milonni (1994). While
boundary conditions are typically described in terms of the macroscopic
polarization induced in a medium in the presence of an applied EM field
Jackson (1999), their emergence can be more rigorously established from a
microscopic treatment of media as a continuous distribution of atomic
scatterers Cook and Milonni (1987); Fearn  _et al._ (1996); Feng and Ujihara
(1990).

The remarkable progress in experimental control of atomic systems has enabled
creation of materials at an atom-by-atom scale Chang  _et al._ (2018). For
example, arrays of atoms arranged in a Bragg configuration can exhibit strong
reflection on an incident EM field. Such atomic Bragg mirrors have been
explored theoretically and experimentally with atoms trapped in free-space
optical lattices Deutsch  _et al._ (1995); Weidemüller  _et al._ (1995); Birkl
_et al._ (1995); Schilke  _et al._ (2011); Slama  _et al._ (2005), as well as
near waveguides, enabling enhanced light-matter interactions, and thereby a
strong and coherent reflection of light Chang  _et al._ (2011); Le Kien and
Rauschenbeutel (2014). Such systems are relevant to implementations of various
quantum information processing protocols Chang  _et al._ (2012); González-
Tudela  _et al._ (2015); Paulisch  _et al._ (2016); Asenjo-Garcia  _et al._
(2017); González-Tudela  _et al._ (2017); Shah  _et al._ (2024).

As a striking phenomenon, recent experiments with subwavelength Rydberg atom
arrays and superconducting circuits have demonstrated coherent switching of
the optical response of boundary conditions realized by atomic, or atom-like,
mirrors Srakaew  _et al._ (2023); Mirhosseini  _et al._ (2019). These
experiments bring forth the novel prospect of realizing coherent quantum
superpositions of two disparate boundary conditions on the EM field. The
emergent boundary conditions in the presence of such atomic arrays must
account for the fact that the macroscopic polarization induced in the medium
can exist in a quantum superposition. The consequent QED phenomena in the
presence of such “quantum boundary conditions” are, as yet, largely
unexplored.

In this letter we propose and analyze a system with quantum mirrors, wherein
the emergent optical properties of an atomic array depend on the specific
quantum states of its constituent atoms. Preparing the array of atoms in a
macroscopic quantum superposition of the two states that correspond to
distinct boundary conditions on the EM field, we analyze the spontaneous
emission of a two-level atom near such a quantum mirror (QM), and inside a
cavity formed by two QMs.

Figure 1: Schematic representation of a two-level atom (“Atom”) in front of an
atomic array of $N$ $\Lambda$-type three-level atoms (“Quantum Mirror” (QM))
coupled via a waveguide. The atomic array is arranged in a Bragg configuration
with $|x_{n}-x_{n-1}|=\lambda_{0}/2$, such that for all atoms in the state
$\left|g_{n}\right\rangle$, the array behaves as a mirror. The level
$\left|g^{\prime}_{n}\right\rangle$ being uncoupled from other atomic levels,
the array with all atoms in $\left|g_{n}^{\prime}\right\rangle$ acts
transparently. Figure 2: Preparing an atomic array with $N=100$ atoms
initially in an equal superposition of the two ground states
$\frac{1}{\sqrt{2}}\left(\left|G\right\rangle+\left|G^{\prime}\right\rangle\right)$,
the decay dynamics of the excited two-level atom is plotted for (a)
$x_{1}=3\lambda_{0}/2$ and (b) $x_{1}=5\lambda_{0}/4$. The emitted intensity
of the EM field, for a QM with $N=100$, (c) for the two-level atom-QM
separation $x_{1}=3\lambda_{0}/2$ and state of QM $\left|G\right\rangle$
(field node at $x_{A}$) and (d) $x_{1}=5\lambda_{0}/4$ , state of QM
$\left|G\right\rangle$ (field antinode at $x_{A}$). (e) Intensity for QM in
$\left|G\right\rangle^{\prime}$ (independent of the Atom-QM separation).

Three-level atomic array as a quantum mirror.— Let us consider an array of $N$
three-level atoms coupled to a waveguide, as shown in Fig. 1(a). The ground
level $\left|g_{n}\right\rangle$ (for the $n^{\mathrm{th}}$ atom in the array)
is coupled to the excited level $\left|e_{n}\right\rangle$ with a resonant
transition frequency $\omega_{0}$, while the level
$\left|g_{n}^{\prime}\right\rangle$ is decoupled from other levels. An
initially excited two-level atom, also with a resonant frequency $\omega_{0}$,
is coupled to the waveguide at the position $x_{A}=0$. The interaction
Hamiltonian that describes the interaction between the atoms and the guided
field is given in the interaction picture as SM :

$\displaystyle\tilde{H}_{\mathrm{int}}=$
$\displaystyle\sum_{k}\sum_{n=1}^{N}\hbar
g_{k}\hat{a}_{k}\left[\hat{\sigma}_{n}^{+}e^{ikx_{n}}+\hat{\sigma}^{+}e^{ikx_{A}}\right]e^{-i\left(\omega-\omega_{0}\right)t}+\mathrm{H.c.},$
(1)

where $\hat{\sigma}_{n}^{+}\equiv\left|e_{n}\right\rangle\left\langle
g_{n}\right|$ is the raising operator for the $n^{\mathrm{th}}$ atom in the
array and $\hat{\sigma}^{+}\equiv\left|e\right\rangle\left\langle g\right|$
that for the two-level atom. The bosonic operators $\hat{a}_{k}$ correspond to
the EM field modes in the waveguide. We consider the initial state of the
system to be
$\left|\Psi(0)\right\rangle=\left(c_{A}(0)\left|G\right\rangle+c_{A}^{\prime}(0)\left|G^{\prime}\right\rangle\right)\otimes\left|e\right\rangle_{A}\otimes\left|\left\\{0\right\\}\right\rangle$,
where the states $\left|G\right\rangle\equiv\left|g_{1},g_{2}\dots g_{n}\dots
g_{N}\right\rangle$ and
$\left|G^{\prime}\right\rangle\equiv\left|g^{\prime}_{1},g^{\prime}_{2}\dots
g^{\prime}_{n}\dots g^{\prime}_{N}\right\rangle$ correspond to all the array
atoms being in one ground level or another.

Considering that the interaction Hamiltonian preserves the total number of
excitations in the atoms+field system, the state of the system at a later time
$t$ in the single-excitation subspace is given by:

$\displaystyle\left|\Psi(t)\right\rangle=$
$\displaystyle\left[\left\\{c_{\mathrm{QM}}\hat{\Sigma}^{+}+c_{\mathrm{A}}\hat{\sigma}^{+}+\sum_{k}c_{k}\hat{a}_{k}^{\dagger}\right\\}\left|G\right\rangle\right.$
$\displaystyle\left.+\left\\{c_{\mathrm{A}}^{\prime}\hat{\sigma}^{+}+\sum_{k}c^{\prime}_{k}\hat{a}_{k}^{\dagger}\right\\}\left|G^{\prime}\right\rangle\right]\otimes\left|g\right\rangle\otimes\left|\left\\{0\right\\}\right\rangle,$
(2)

where $c_{A}$ ($c_{k}$ ) and $c_{A}^{\prime}$ $(c_{k}^{\prime})$ represent the
excitation amplitudes for the two-level atom (field mode $k$), for the QM in
state $\left|G\right\rangle$ and $\left|G^{\prime}\right\rangle$,
respectively; $c_{\mathrm{QM}}$ denotes the probability amplitude for there
being an excitation in the QM. We have suppressed the explicit time dependence
of the coefficients for brevity.
$\hat{\Sigma}^{+}=\frac{1}{\sqrt{N}}\sum_{n=1}^{N}(-1)^{n+1}\hat{\sigma}_{n}^{+}$
is the collective raising operator for the array considering that the atoms
are placed at half-wavelength ($\lambda_{0}/2=\pi v/\omega_{0}$) apart from
each other, $v$ being the speed of light in the waveguide. Such a Bragg mirror
configuration, $|x_{n}-x_{n-1}|=\lambda_{0}/2$, is imperative for the array to
behave cooperatively so that it can be described effectively as a single
collective spin. Eliminating the field modes yields the equations of motion
for the atomic excitation amplitudes SM :

$\displaystyle\frac{{d}c_{\mathrm{QM}}}{{d}t}=$
$\displaystyle-\frac{\gamma}{2}\left(Nc_{\mathrm{QM}}+\sqrt{N}e^{ik_{0}x_{1}}c_{\mathrm{A}}\right)$
(3) $\displaystyle\frac{{d}c_{\mathrm{A}}}{{d}t}=$
$\displaystyle-\frac{\gamma}{2}\left(c_{\mathrm{A}}+\sqrt{N}e^{ik_{0}x_{1}}c_{\mathrm{QM}}\right);\frac{{d}c^{\prime}_{\mathrm{A}}}{{d}t}=-\frac{\gamma}{2}c^{\prime}_{\mathrm{A}},$
(4)

where we have made the Born-Markov approximation, assuming a flat spectral
density around the atomic resonance $(g_{k}\approx g_{0})$, and defining
$\gamma\equiv 2\pi g_{0}^{2}$ as the spontaneous emission rate of individual
atoms. We observe that the QM couples to the field with a cooperatively
enhanced coupling $(\sim\sqrt{N}g_{0})$. Solving the above equations of
motion, for the QM in $\left|G^{\prime}\right\rangle$, the decay dynamics of
the two-level atom is given by
$c^{\prime}_{A}(t)=c_{A}^{\prime}(0)e^{-\frac{\gamma t}{2}}$, which
corresponds to that in an open waveguide. For the QM in
$\left|G\right\rangle$, in the limit of $N\gg 1$, the dynamics of the two-
level atom simplifies to:

$\displaystyle c_{\mathrm{A},\infty}(t)\approx
c_{\mathrm{A}}(0)e^{i\frac{\gamma
t}{2}\sin\left(2k_{0}x_{1}\right)}e^{-\frac{\gamma
t}{2}\left(1-\cos\left(2k_{0}x_{1}\right)\right)},$ (5)

which corresponds to the decay of an excited two-level atom in front of a
mirror Cook and Milonni (1987); Milonni and Knight (1973). The accompanying
coefficient for the QM atoms to be excited $(c_{\mathrm{QM}})$ scales as $\sim
1/\sqrt{N}$, vanishing in the large $N$ limit. We note that the appropriate
mirror-like boundary conditions emerge for large $N$, even when the QM atoms
are negligibly excited.

The spontaneous emission dynamics for the two-level atom is illustrated in
Fig.2(a) and (b) as a function of the two-level atom-QM separation, assuming
the initial state of the QM to be
$\frac{1}{\sqrt{2}}\left(\left|G\right\rangle+\left|G^{\prime}\right\rangle\right)$.
For the part of the total state in $\left|G^{\prime}\right\rangle$, the atom
decays exponentially with a rate $\gamma$, as seen from the red dotted curves
in Fig.2(a) and (b). The corresponding emitted intensity is plotted in
Fig.2(e) SM . For the part of the total state in $\left|G\right\rangle$, the
QM provides a mirror-like boundary condition, e.g., for a separation
$x_{1}=p\lambda_{0}/2$ ($p\in\mathbb{N}$) between the two-level atom and the
QM, as the mirror atoms are excited thereby establishing a node of the field
at $x_{A}=0$ (Fig.2(c)), inhibiting the subsequent decay of the two-level atom
(blue solid curves in Fig.2(a)). Likewise, for $x_{1}=(2p+1)\lambda_{0}/4$
there is an antinode of the field at the position of the two-level atom
(Fig.2(d)), leading to a superradiant decay with $2\gamma$ (blue solid curves
in Fig.2(b)) Milonni and Knight (1973).

Quantum erasure.—From the time-evolved state in Eq.(Spontaneous Emission in
the presence of Quantum Mirrors), as well as Fig.2(a) and (b), it is evident
that the decay dynamics of the two-level atom is entangled with the the ground
states of the QM. We now consider the possibility of ‘erasing’ the state
information of the QM by making a projective measurement on the QM atoms in a
basis
$\frac{1}{\sqrt{2}}\left(\left|G\right\rangle+e^{i\phi_{M}}\left|G^{\prime}\right\rangle\right)$111While
a multi-atom projective measurement can be challenging to implement in
practice, we consider such a measurement as an idealized theoretical example.
For example, one could apply a $\pi/2$-pulse to mix $\left|G\right\rangle$ and
$\left|G^{\prime}\right\rangle$, and then post-select on
$\left|G\right\rangle$. Alternatively, in experiments such as Ref. Srakaew
_et al._ (2023), a single qubit controls the whole state of the quantum
mirror, making such projective measurements more accessible.  Scully  _et al._
(1991). Such a measurement gives the post-selected state at the measurement
time $t=t_{M}$:

$\displaystyle\left|\Psi(t_{M})\right\rangle=$
$\displaystyle\frac{1}{2}\left[\left\\{c_{\mathrm{A}}(t_{M})+e^{-i\phi_{M}}c^{\prime}_{\mathrm{A}}(t_{M})\right\\}\hat{\sigma}^{+}+\sum_{k}\left\\{\left(c_{k}(t_{M})\right.\right.\right.$
$\displaystyle\left.\left.\left.+e^{-i\phi_{M}}c^{\prime}_{k}(t_{M})\right)\hat{a}_{k}^{\dagger}\right\\}\right]\left(\left|G\right\rangle+e^{i\phi_{M}}\left|G^{\prime}\right\rangle\right)\left|g\right\rangle\left|\left\\{0\right\\}\right\rangle.$
(6)

Having erased the which-state information for the QM, we see that the
excitation amplitude of the two-level atom is now determined by a coherent
superposition of $c_{A}(t_{M})$ and $c^{\prime}_{A}(t_{M})$ corresponding to
the two different boundary conditions. Starting with an initial state
$\left|\Psi(0)\right\rangle=\frac{1}{\sqrt{2}}\left(\left|G\right\rangle+e^{i\phi_{S}}\left|G^{\prime}\right\rangle\right)\left|e\right\rangle\left|\left\\{0\right\\}\right\rangle$,
in the large $N$ limit, the excitation probability of the two-level atom right
after the erasure becomes
$P_{e}(\phi_{M})\equiv\frac{1}{2}\left|c_{A}\left(t_{M}\right)+e^{-i\phi_{M}}c_{A}^{\prime}\left(t_{M}\right)\right|^{2}$
such that:

$\displaystyle P_{e}(\phi_{M})\approx\frac{e^{-\gamma
t_{M}}}{4}\left[1+e^{\gamma t_{M}\cos(2k_{0}x_{1})}\right.$
$\displaystyle\left.+2e^{\gamma
t_{M}/2\cos(2k_{0}x_{1})}\cos\left(\Delta\phi+\frac{\gamma
t_{M}}{2}\sin\left(2k_{0}x_{1}\right)\right)\right],$ (7)

where $\Delta\phi\equiv\phi_{M}-\phi_{S}$. The first two terms in the above
expression denote the independent excitation probabilities corresponding to
the cases when the QM is in $\left|G^{\prime}\right\rangle$ and
$\left|G\right\rangle$ respectively; the third term arises from the
interference between $\left|G^{\prime}\right\rangle$ and
$\left|G\right\rangle$. If $\left|G\right\rangle$ and
$\left|G^{\prime}\right\rangle$ have different energies, their respective free
Hamiltonians will imprint a dynamical phase $\phi_{S}(t)=\delta t/\hbar$,
where $\delta=\hbar(\omega_{0}-\omega_{0}^{\prime})$ represents the energy
difference between the ground levels. Consequently, $\Delta\phi(t)$ becomes a
function of time and $P_{e}$ will oscillate with $t_{M}$ as in a Ramsey
interferometer of the two metastable states of the quantum mirror. Such
measurements can be experimentally used to distinguish a coherent quantum
superposition of two boundary conditions from a decohered statistical mixture.

Figure 3: (a) Schematic representation of a two level atom in a quantum
cavity. Spontaneous emission dynamics of the two-level atom in the quantum
cavity for (b)$x_{A}=0$, $x_{1}=3\lambda_{0}/2$, (c)$x_{A}=0$,
$x_{1}=5\lambda_{0}/4$ and (d) $x_{A}\approx 0.01\lambda_{0}$,
$x_{1}=3\lambda_{0}/2$. Each QM is composed of $N=100$ atoms.

Spontaneous Emission in a Quantum Cavity.—We now consider the spontaneous
emission from a two-level atom inside a ‘quantum cavity’ formed by two quantum
mirrors, QM1 and QM2, as shown in Fig. 3(a). Each QM is composed of $N$ three-
level atoms positioned at $x_{j}$, with $j=\left\\{1\dots N\right\\}$
$(j=\left\\{-1\dots-N\right\\})$ corresponding to QM1 (QM2). As in the case of
a single QM, the spacing between the adjacent atoms in the array is
$\lambda_{0}/2$, such that each QM can be described in terms of a single
collective spin. Considering that the central two-level atom is initially
excited and the atoms composing the QMs are all initially prepared either in
the state $\left|g_{j}\right\rangle$ or $\left|g^{\prime}_{j}\right\rangle$,
the state of the system at a later time $t$ is given by SM :

$\displaystyle\left|\Psi(t)\right\rangle=\left[\left\\{c_{\mathrm{A},GG}\hat{\sigma}^{+}+c_{\mathrm{QM1}}\hat{\Sigma}_{1}^{+}+c_{\mathrm{QM2}}\hat{\Sigma}_{2}^{+}+\sum_{k}c_{k,GG}\hat{a}_{k}^{\dagger}\right\\}\left|G_{1}G_{2}\right\rangle+\left\\{c_{A,G^{\prime}G^{\prime}}\hat{\sigma}^{+}+\sum_{k}c_{k,G^{\prime}G^{\prime}}\hat{a}_{k}^{\dagger}\right\\}\left|G^{\prime}_{1}G^{\prime}_{2}\right\rangle\right.$
$\displaystyle\left.+\left\\{c_{\mathrm{A},GG^{\prime}}\hat{\sigma}^{+}+c^{\prime}_{\mathrm{QM1}}\hat{\Sigma}_{1}^{+}+\sum_{k}c_{k,GG^{\prime}}\hat{a}_{k}^{\dagger}\right\\}\left|G_{1}G^{\prime}_{2}\right\rangle+\left\\{c_{\mathrm{A},G^{\prime}G}\hat{\sigma}^{+}+c^{\prime}_{\mathrm{QM2}}\hat{\Sigma}_{2}^{+}+\sum_{k}c_{k,G^{\prime}G}\hat{a}_{k}^{\dagger}\right\\}\left|G^{\prime}_{1}G_{2}\right\rangle\right]\left|g\right\rangle\left|\left\\{0\right\\}\right\rangle,$
(8)

where $\left|G_{j}\right\rangle=\left|g_{1},g_{2}\dots g_{N}\right\rangle$ and
$\left|G^{\prime}_{j}\right\rangle=\left|g^{\prime}_{1},g^{\prime}_{2}\dots
g^{\prime}_{N}\right\rangle$ indicates the collective state of QM$j$. The
coefficients
$c_{\mathrm{A},G^{\left\\{\prime\right\\}}G^{\left\\{\prime\right\\}}}$
($c_{k,G^{\left\\{\prime\right\\}}G^{\left\\{\prime\right\\}}}$) denote the
excitation amplitude of the two-level atom (field mode $k$), for the state of
the QMs in
$\left|G_{1}^{\left\\{\prime\right\\}}G_{2}^{\left\\{\prime\right\\}}\right\rangle$.
Similarly the coefficients $c^{\left\\{\prime\right\\}}_{\mathrm{QM1(2)}}$
represent the probability amplitude for the QM1(2) to contain one collective
excitation with the other QM in the state
$\left|G_{2(1)}^{\left\\{\prime\right\\}}\right\rangle$.

While a detailed solution of the system dynamics is presented in the SM SM ,
here we consider three example configurations as illustrated in Fig. 3. In
each case, the system is prepared in an initial state
$\left|\Psi(0)\right\rangle=\frac{1}{\sqrt{2}}\left|e\right\rangle\otimes\left(\left|G_{1}G_{2}\right\rangle+\left|G_{1}^{\prime}G_{2}^{\prime}\right\rangle\right)\otimes\left|\left\\{0\right\\}\right\rangle$.
For the part of the total state when the QMs are in
$\left|G_{1}^{\prime}G_{2}^{\prime}\right\rangle$, the two-level atom ‘sees’
an open waveguide and decays exponentially (red dashed curves in Fig.
3(b)–(d)). For the part of the state wherein the QMs are in the
$\left|G_{1}G_{2}\right\rangle$ subspace, the total system behaves effectively
as a two-level atom coupled to two collective spins corresponding to QM1 and
QM2, each of which exhibits a cooperative coupling enhanced by a factor of
$\sqrt{N}$. The dynamics strongly depends on the length of the cavity formed
by the QMs and the position of the two-level atom therein, as elaborated
below:

  1. 1.

Atom at cavity node ($x_{A}=0$, $x_{1}=p\lambda_{0}/2$, $p\in\mathbb{N}$): In
this case the two-level atom is located at the node of the standing wave modes
arising from the cavity-like boundary conditions, and thus stops decaying once
the boundary conditions are established (solid blue curve in Fig. 3(b)). In
the large $N$ limit, it can be seen that $c_{\mathrm{A},GG,\infty}(t)\approx
c_{\mathrm{A},GG}(0)$ and $c_{\mathrm{QMj},\infty}(t)\propto
1/\sqrt{N}\rightarrow 0$, as indicated by the blue-dashed and green-dashed
curves in Fig. 3(b) SM . A similar condition arises when
$2x_{1}<\lambda_{0}/2$, such that there are no standing wave modes allowed
within the cavity region Milonni and Knight (1973); Hulet  _et al._ (1985).

  2. 2.

Atom at cavity antinode ($x_{A}=0$, $x_{1}=(2p+1)\lambda_{0}/4$,
$p\in\mathbb{N}$): When the two-level atom is located at the antinode of the
standing wave mode in a resonant quantum cavity, it coherently exchanges
excitations with the atoms composing the QMs, as can be seen from Fig. 3(c).
This scenario is akin to the experimentally observed dynamics of
superconducting qubits in cavity QED system Mirhosseini  _et al._ (2019). In
the large $N$ limit, the dynamics of the two-level atom and the QMs is well-
approximated by $c_{\mathrm{A},GG,\infty}(t)\approx
c_{\mathrm{A},GG}(0)e^{-\gamma t/4}\cos\left(\sqrt{\frac{N}{2}}\gamma
t\right)$ and
$c_{\mathrm{QMj,\infty}}\approx\sqrt{2}ic_{\mathrm{A},GG}(0)e^{-\gamma
t/4}\sin\left(\sqrt{\frac{N}{2}}\gamma t\right)$, respectively SM , as seen
from the blue and green dashed curves in Fig. 3(c). Remarkably, in this case
the QMs do not admit an effective passive boundary condition description, with
their excitation dynamics playing an active role in determining the system
behavior even as $N\rightarrow\infty$. However, upon introducing a detuning
between the atomic and cavity resonances, or displacing the two-level atom
from the antinode, such a resonant excitation of the QMs disappears.

  3. 3.

Atom near cavity node ($0<k_{0}x_{A}\ll 1$, $x_{1}=p\lambda_{0}/2$,
$p\in\mathbb{N}$): Being located close to the node of the cavity, while the
small offset from the node allows the two-level atom to still interact with
the cavity modes, the coupling between the two-level atom and the open
waveguide modes is minimized. We observe sustained Rabi oscillations-like
dynamics between the two-level atom and the QM excitations, as depicted by the
blue curve in Fig. 3(d). The dynamics of the two-level atom and those
constituting the QMs is approximated in the large $N$ limit by
$c_{\mathrm{A},GG,\infty}(t)\approx
c_{\mathrm{A},GG}(0)\cos\left(k_{0}x_{A}\sqrt{\frac{N}{2}}\gamma t\right)$ and
$c_{\mathrm{QMj,\infty}}(t)\approx-\frac{i}{\sqrt{2}}\sin\left(k_{0}x_{A}\sqrt{\frac{N}{2}}\gamma
t\right)$ SM  (blue and green dashed curves in Fig. 3(d)). As in case 2, the
resonant excitation of the QM atoms vanishes in the presence of a detuning
between the resonances of the two-level atom and those forming the QMs, or
that of the cavity.

Discussion.— We have demonstrated that an ordered array of waveguide-coupled
$\Lambda$-type three-level atoms, prepared in a superposition of two ground
states, can engender a superposition of two distinct boundary conditions,
e.g., mirror-like and transparent, for the guided EM field modes. We examine
the effect of such a quantum superposition of boundary conditions on the
spontaneous emission of a two-level atom near such a quantum mirror or QM,
demonstrating that the corresponding atomic decay dynamics is entangled with
the state of the QM. We show that making a measurement on the state of the
atoms composing the QM can erase the ‘which-boundary-condition’ information,
resulting in an interference between the system dynamics from the distinct
boundary conditions. We further analyze the spontaneous emission dynamics of a
two-level atom inside a cavity formed by two QMs. Preparing the QMs in a
superposition of the two ground states, the atomic dynamics occurs in a
superposition of having an open waveguide and the atom decaying inside a
cavity. For specific arrangements the atoms constituting the QMs can be
resonantly excited, thereby precluding an effective static boundary condition
description of the system.

Our results are pertinent to ongoing state-of-the-art experiments in waveguide
QED Sheremet  _et al._ (2023), as well as those with free-space atomic arrays
Srakaew  _et al._ (2023). As was recently demonstrated in the experiment by
Srakaew et al, the optical response of a subwavelength 2D array of Rydberg
atoms can be switched from mirror-like to transparent via an ancilla atom.
Preparing such a system in a superposition can potentially result in a quantum
mirror-like boundary condition, which can be tested by measuring the system in
a basis that erases the ‘which-state’ information on the ancilla, thereby
leading to an interference between the two boundary conditions.

From the perspective of applications in quantum information processing,
similar subwavelength 2D arrays of atoms have also been proposed for
generation of photonic Greenberger-Horne-Zeilinger (GHZ) states Bekenstein
_et al._ (2020). Here we start with a GHZ state of atomic arrays coupled to a
waveguide and obtain the subsequent dynamics of the system. Future works will
explore time-reversed renditions of the present dynamics wherein one can
potentially herald an entangled state by starting with an initially
unentangled state and making appropriate measurements on the system.

Finally, quantization of EM field in the presence of boundary conditions that
can be in a quantum superposition or quantum correlated states, is as yet an
open problem. Such a quantization scheme can be studied as an extension of the
previous works on microscopic models of boundary conditions Cook and Milonni
(1987), assuming the induced polarization in a medium to be in a macroscopic
quantum superposition. This will open a new direction of exploring QED
phenomena in the presence of ‘quantum boundary conditions’.

Acknowledgments.— We warmly acknowledge insightful discussions with Elizabeth
A. Goldschmidt, Hakan Türeci, Peter W. Milonni and Pierre Meystre. K.S.
acknowledges support from the National Science Foundation under Grant No.
PHY-2309341, and by the John Templeton Foundation under Award No. 62422. P.S.
is a CIFAR Azrieli Global Scholar in the Quantum Information Science Program.
This research was supported in part by grant NSF PHY-1748958 to the Kavli
Institute for Theoretical Physics (KITP).

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Supplemental Material for “Spontaneous emission in the presence of Quantum
Mirrors”

##  I Single quantum mirror

The total Hamiltonian for a two-level atom near a three-level atom array (QM)
is given by:

$\displaystyle H=H_{A}+H_{F}+H_{\mathrm{int}},$ (S1)

where

$\displaystyle H_{A}=$
$\displaystyle\hbar\omega_{0}\left|e\right\rangle\left\langle
e\right|+\sum_{n=1}^{N}\hbar\omega_{0}\left|e_{n}\right\rangle\left\langle
e_{n}\right|,$ (S2)

is the atomic Hamiltonian, $H_{F}$ is the field Hamiltonian and the
interaction Hamiltonian in the interaction picture is as given in the main
text.

Let us consider the state at time $t$ as:

$\displaystyle\left|\Psi(t)\right\rangle=$
$\displaystyle\left[\sum_{n=1}^{N}\left\\{c_{\mathrm{QM},n}(t)\hat{\sigma}_{n}^{+}+c_{\mathrm{A}}(t)\hat{\sigma}^{+}+\sum_{k}c_{k}(t)\hat{a}_{k}^{\dagger}\right\\}\left|G\right\rangle+\left\\{c_{\mathrm{A}}^{\prime}(t)\hat{\sigma}^{+}+\sum_{k}c^{\prime}_{k}(t)\hat{a}_{k}^{\dagger}\right\\}\left|G^{\prime}\right\rangle\right]\left|g\right\rangle\left|\left\\{0\right\\}\right\rangle.$
(S3)

We obtain the evolution of the atomic and field excitation amplitudes as
follows:

$\displaystyle\frac{{d}c_{\mathrm{QM},n}}{{d}t}=$
$\displaystyle-i\sum_{k}g_{k}c_{k}(t)e^{ikx_{n}}e^{-i\left(\omega-\omega_{0}\right)t}$
(S4) $\displaystyle\frac{{d}c_{\mathrm{A}}}{{d}t}=$
$\displaystyle-i\sum_{k}g_{k}c_{k}(t)e^{-i(\omega-\omega_{0})t};\quad\frac{{d}c^{\prime}_{\mathrm{A}}}{{d}t}=-i\sum_{k}g_{k}c^{\prime}_{k}(t)e^{-i(\omega-\omega_{0})t}$
(S5) $\displaystyle\frac{{d}c_{k}}{{d}t}=$ $\displaystyle-
ig_{k}\left[c_{\mathrm{A}}(t)e^{i(\omega-\omega_{0})t}+\sum_{n=1}^{N}c_{\mathrm{QM},n}(t)e^{-ikx_{n}}e^{i(\omega-\omega_{0})t}\right];\quad\frac{{d}c^{\prime}_{k}}{{d}t}=-ig_{k}c^{\prime}_{\mathrm{A}}(t)e^{i(\omega-\omega_{0})t}$
(S6)

Substituting the field excitation amplitudes in the atomic equations of
motion, yields:

$\displaystyle\frac{{d}c_{\mathrm{QM},n}}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[c_{\mathrm{A}}\left(t\right)e^{i\omega_{0}x_{n}/v}+\sum_{\nu=1}^{N}c_{\mathrm{QM},\nu}\left(t\right)e^{i\omega_{0}\left|x_{n}-x_{\nu}\right|}\right]$
(S7) $\displaystyle\frac{{d}c_{\mathrm{A}}}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[c_{\mathrm{A}}\left(t\right)+\sum_{n=1}^{N}c_{\mathrm{QM},n}\left(t\right)e^{i\omega_{0}x_{n}/v}\right]$
(S8) $\displaystyle\frac{{d}c^{\prime}_{\mathrm{A}}}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}c_{G^{\prime}}^{e,0}(t),$ (S9)

where we have assumed that $g_{k}\approx g_{0}$ for the field modes around the
atomic resonance, and the spontaneous emission rate of an atom into a
waveguide is defined as $\gamma\approx 2\pi\left|g_{0}\right|^{2}$.

Taking Laplace transform of the above equations of motion as
$\tilde{c}_{j}(s)\equiv\int_{0}^{\infty}\mathrm{d}tc_{j}(t)e^{-st}$, we get:

$\displaystyle\left(s+\frac{\gamma}{2}\right)c_{\mathrm{QM},n}(s)+\frac{\gamma}{2}\tilde{c}_{\mathrm{A}}\left(s\right)e^{i\omega_{0}x_{n}/v}+\frac{\gamma}{2}\sum_{\nu\neq
n}\tilde{c}_{\mathrm{QM},\nu}\left(s\right)e^{i\omega_{0}\left|x_{n}-x_{\nu}\right|/v}=$
$\displaystyle c_{\mathrm{QM},n}(0)$ (S10)
$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}_{\mathrm{A}}(s)+\frac{\gamma}{2}\sum_{n=1}^{N}\tilde{c}_{\mathrm{QM},n}\left(s\right)e^{i\omega_{0}x_{n}/v}=$
$\displaystyle c_{\mathrm{A}}(0)$ (S11)
$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}^{\prime}_{\mathrm{A}}(s)=$
$\displaystyle c^{\prime}_{A}(0).$ (S12)

One can define the collective excitation amplitude for the mirror atoms as:

$\displaystyle c_{\mathrm{QM}}(t)\equiv$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{n=1}^{N}c_{\mathrm{QM},n}(t)(-1)^{n-1},$
(S13)

where we have assumed that

$\displaystyle x_{n}=x_{1}+(n-1)\lambda_{0}/2.$ (S14)

The Laplace equations of motion thus simplify to:

$\displaystyle\left(s+\frac{N\gamma}{2}\right)\tilde{c}_{\mathrm{QM}}(s)+\frac{\sqrt{N}\gamma}{2}\tilde{c}_{\mathrm{A}}\left(s\right)e^{ik_{0}x_{1}}=$
$\displaystyle c_{\mathrm{QM}}(0)$ (S15)
$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}_{\mathrm{A}}(s)+\frac{\sqrt{N}\gamma}{2}\tilde{c}_{\mathrm{QM}}\left(s\right)e^{ik_{0}x_{1}}=$
$\displaystyle c_{\mathrm{A}}(0)$ (S16)
$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}^{\prime}_{A}(s)=$
$\displaystyle c^{\prime}_{A}(0).$ (S17)

The equations of motion Eq.(3–4) in the main text correspond to the inverse
Laplace transform of the above.

Assuming that the atoms in the QM are initially unexcited
($c_{\mathrm{QM}}(0)=0$), we obtain:

$\displaystyle\tilde{c}_{\mathrm{A}}(s)=$
$\displaystyle\frac{s+\frac{N\gamma}{2}}{\left(s+\frac{\gamma}{2}\right)\left(s+\frac{N\gamma}{2}\right)-\frac{N\gamma^{2}}{4}e^{2ik_{0}x_{1}}}c_{\mathrm{A}}(0)$
(S18) $\displaystyle\tilde{c}_{\mathrm{QM}}(s)=$
$\displaystyle-\frac{\sqrt{N}\gamma}{2}e^{ik_{0}x_{1}}\frac{1}{\left(s+\frac{\gamma}{2}\right)\left(s+\frac{N\gamma}{2}\right)-\frac{N\gamma^{2}}{4}e^{2ik_{0}x_{1}}}c_{\mathrm{A}}(0)$
(S19)

The poles of the above equation are given by
$\gamma_{\pm}\equiv-\frac{\gamma}{2}\left[\frac{N+1}{2}\pm\sqrt{\frac{(N+1)^{2}}{4}+Ne^{2ik_{0}x_{1}}}\right]$.
This yields the time dynamics of the two-level atom as:

$\displaystyle
c_{\mathrm{A}}(t)=\frac{\gamma_{+}+\frac{N\gamma}{2}}{\gamma_{+}-\gamma_{-}}e^{\gamma_{+}t}-\frac{\gamma_{-}+\frac{N\gamma}{2}}{\gamma_{+}-\gamma_{-}}e^{\gamma_{-}t}$
(S20) $\displaystyle
c_{\mathrm{QM}}(t)=-\frac{\sqrt{N}\gamma}{2\left(\gamma_{+}-\gamma_{-}\right)}e^{ik_{0}x_{1}}\left(e^{\gamma_{+}t}-e^{\gamma_{-}t}\right).$
(S21)

###  I.1 Large $N$ limit

For $N\gg 1$, one obtains from Eq. (S18):

$\displaystyle\tilde{c}_{\mathrm{A}}(s)\approx\frac{c_{\mathrm{A}}(0)}{s+\frac{\gamma}{2}\left(1-e^{2ik_{0}x_{1}}\right)}\implies
c_{\mathrm{A}}(t)\approx e^{-\frac{\gamma
t}{2}\left(1+e^{2ik_{0}x_{1}}\right)},$ (S22)

which corresponds to the decay of an excited atom in front of a mirror Cook
and Milonni (1987). From Eq. (S19) it can be seen that the collective
excitation amplitude for the QM atoms scales as $c_{\mathrm{QM}}\sim
1/\sqrt{N}$, which vanishes in the large $N$ limit.

###  I.2 Field dynamics

The intensity of the field emitted by the atoms as a function of position and
time can be evaluated as
$I\left(x,t\right)=\frac{\epsilon_{0}c}{2}\left\langle\Psi\left(t\right)|\hat{E}^{\dagger}\left(x,t\right)\hat{E}\left(x,t\right)|\Psi\left(t\right)\right\rangle$,
where
$\hat{E}\left(x,t\right)=\sum_{k}\,\mathcal{E}_{k}\hat{a}_{k}e^{i(kx-\omega
t)}$ is the electric field operator at position $x$ and time $t$. More
explicitly, we obtain

$\displaystyle I(x,t)/I_{0}=$
$\displaystyle\left\langle\Psi(t)\right|\left[\sum_{k_{1}}\hat{a}_{k_{1}}^{\dagger}e^{-i(k_{1}x-\omega_{1}t)}\right]\left[\sum_{k_{2}}\hat{a}_{k_{2}}e^{i(k_{2}x-\omega_{2}t)}\right]\left|\Psi(t)\right\rangle$
(S23) $\displaystyle=$
$\displaystyle\sum_{k_{1}}\sum_{k_{2}}\left[\left\langle G\right|\left\langle
g\right|c_{k_{1}}^{\ast}e^{-ik_{1}x}+\left\langle
G^{\prime}\right|\left\langle
g\right|(c^{\prime}_{k_{1}})^{\ast}e^{-ik_{1}x}\right]\left[c_{k_{2}}e^{ik_{2}x}\left|G\right\rangle\left|g\right\rangle+c^{\prime}_{k_{2}}e^{ik_{2}x}\left|G^{\prime}\right\rangle\left|g\right\rangle\right]e^{i(\omega_{1}-\omega_{2})t}$
(S24) $\displaystyle=$ $\displaystyle\left|\sum_{k}c_{k}(t)e^{i(kx-\omega
t)}\right|^{2}+\left|\sum_{k}c^{\prime}_{k}(t)e^{i(kx-\omega t)}\right|^{2}$
(S25)

We substitute the field excitation amplitudes in terms of the atomic
excitation amplitudes using Eq.(S6) and separating the positive and negative
wavevectors $k$ explicitly.:

$\displaystyle I/I_{0}=$ $\displaystyle\left|\int_{0}^{t}\mathrm{d}\tau
e^{-i\omega_{0}\tau}\int_{0}^{\infty}\mathrm{d}\omega
e^{-i\omega(t-\tau)}\left[\left\\{c_{\mathrm{A}}\left(\tau\right)+\sum_{n}c_{\mathrm{QM},n}\left(\tau\right)e^{-ikx_{n}}\right\\}e^{ikx}\right.\right.$
$\displaystyle\left.\left.+\left\\{c_{\mathrm{A}}\left(\tau\right)+\sum_{n}c_{\mathrm{QM},n}\left(\tau\right)e^{ikx_{n}}\right\\}e^{-ikx}\right]\right|^{2}+\left|\int_{0}^{t}\mathrm{d}\tau
e^{-i\omega_{0}\tau}\int_{0}^{\infty}\mathrm{d}\omega
e^{-i\omega(t-\tau)}\left[c^{\prime}_{\mathrm{A}}(\tau)e^{ikx}+c^{\prime}_{\mathrm{A}}(\tau)e^{-ikx}\right]\right|^{2}$
(S26) $\displaystyle=$
$\displaystyle\left|\left[e^{i\omega_{0}x/v}c_{\mathrm{A}}\left(t-x/v\right)\left[\Theta\left(t-x/v\right)-\Theta\left(-x/v\right)\right]+e^{-i\omega_{0}x/v}c_{\mathrm{A}}\left(t+x/v\right)\left[\Theta\left(t+x/v\right)-\Theta\left(x/v\right)\right]\right.\right.$
$\displaystyle\left.\left.+\sum_{n}e^{i\omega_{0}(x-x_{n})/v}c_{\mathrm{QM},n}\left(t-(x-x_{n})/v\right)\left[\Theta\left(t-(x-x_{n})/v\right)-\Theta\left(-(x-x_{n})/v\right)\right]\right.\right.$
$\displaystyle\left.\left.+\sum_{n}e^{-i\omega_{0}(x-x_{n})/v)}c_{\mathrm{QM},n}\left(t+(x-x_{n})/v\right)\left[\Theta\left(t+(x-x_{n})/v\right)-\Theta\left((x-x_{n})/v\right)\right]\right]\right|^{2}$
$\displaystyle+\left|e^{-i\omega_{0}x/v}c^{\prime}_{\mathrm{A}}(t+x/v)\left[\Theta\left(t+x/v\right)-\Theta\left(x/v\right)\right]+e^{i\omega_{0}x/v}c^{\prime}_{A}(t-x/v)\left[\Theta\left(t-x/v\right)-\Theta\left(-x/v\right)\right]\right|^{2}.$
(S27)

We note that the total intensity emitted by the atoms is an incoherent sum of
that emitted from the QM atoms in $\left|G\right\rangle$ and
$\left|G^{\prime}\right\rangle$. In the limit where the QM atoms have a
separation that is small comparable to the coherence length of the emitted
photon $\gamma\left|x_{1}-x_{N}\right|/c\ll 1$, the time delay effects between
the different $c_{\mathrm{QM},n}$ coefficients can be ignored such that
$c_{\mathrm{QM,n}}(t\pm(x-x_{n})/v)\approx c_{\mathrm{QM,n}}(t\pm(x-x_{1})/v)$
Sinha  _et al._ (2020a, b). Thus in the absence of time-delay effects the
above expression for the intensity can be simplified to:

$\displaystyle I/I_{0}\approx$
$\displaystyle\left|\left[e^{i\omega_{0}x/v}c_{\mathrm{A}}\left(t-x/v\right)\left[\Theta\left(t-x/v\right)-\Theta\left(-x/v\right)\right]+e^{-i\omega_{0}x/v}c_{\mathrm{A}}\left(t+x/v\right)\left[\Theta\left(t+x/v\right)-\Theta\left(x/v\right)\right]\right.\right.$
$\displaystyle\left.\left.+e^{i\omega_{0}(x-x_{1})/v}c_{\mathrm{QM}}\left(t-(x-x_{1})/v\right)\left[\Theta\left(t-(x-x_{1})/v\right)-\Theta\left(-(x-x_{1})/v\right)\right]\right.\right.$
$\displaystyle\left.\left.+e^{-i\omega_{0}(x-x_{1})/v}c_{\mathrm{QM}}\left(t+(x-x_{1})/v\right)\left[\Theta\left(t+(x-x_{1})/v\right)-\Theta\left((x-x_{1})/v\right)\right]\right]\right|^{2}$
$\displaystyle+\left|e^{-i\omega_{0}x/v}c^{\prime}_{\mathrm{A}}(t+x/v)\left[\Theta\left(t+x/v\right)-\Theta\left(x/v\right)\right]+e^{i\omega_{0}x/v}c^{\prime}_{A}(t-x/v)\left[\Theta\left(t-x/v\right)-\Theta\left(-x/v\right)\right]\right|^{2}.$
(S28)

##  II Quantum Cavity

In this section we consider an atomic array as a mirror. For the sake of
simplicity the initial state of the total system is assumed to be:

$\displaystyle\left|\Psi(0)\right\rangle=\left|e\right\rangle\otimes\left({c}_{\mathrm{A},GG}(0)\left|G_{1}\right\rangle\left|G_{2}\right\rangle+{c}_{\mathrm{A},GG^{\prime}}(0)\left|G_{1}\right\rangle\left|G^{\prime}_{2}\right\rangle+{c}_{\mathrm{A},G^{\prime}G}(0)\left|G^{\prime}_{1}\right\rangle\left|G_{2}\right\rangle+{c}_{\mathrm{A},G^{\prime}G^{\prime}}(0)\left|G^{\prime}_{1}\right\rangle\left|G^{\prime}_{2}\right\rangle\right)\left|\left\\{0\right\\}\right\rangle$
(S29)

where we consider all of the atoms in QM1 and QM2 to be initially in states
$\left|G_{1,2}\right\rangle=\left|gg...g\right\rangle_{1,2}$ or
$\left|G^{\prime}_{1,2}\right\rangle=\left|g^{\prime}g^{\prime}...g^{\prime}\right\rangle_{1,2}$.

The total Hamiltonian for this system is:

$\displaystyle H=H_{A}+H_{F}+H_{\mathrm{int}}$ (S30)

where

$\displaystyle H_{A}=\hbar\omega_{0}\left|e\right\rangle\left\langle
e\right|+\sum_{m=-1}^{-N}\hbar\omega_{0}\left|e_{m}\right\rangle\left\langle
e_{m}\right|+\sum_{n=1}^{N}\hbar\omega_{0}\left|e_{n}\right\rangle\left\langle
e_{n}\right|.$ (S31)

Here $m$ and $n$ are the index for the mirror atoms in QM1 on the left, and
QM2 on the right.

The interaction Hamiltonian in the interaction picture becomes

$\displaystyle\tilde{H}_{\mathrm{int}}=\sum_{k}\hbar
g_{k}\hat{a}_{k}\left(\sum_{m=-1}^{-N}\hat{\sigma}_{m}^{+}e^{ikx_{m}}+\sum_{n=1}^{N}\hat{\sigma}_{n}^{+}e^{ikx_{n}}+\sum_{k}\hat{\sigma}^{+}e^{ikx_{A}}\right)e^{-i(\omega-\omega_{0})t}+\mathrm{H.c.}$
(S32)

With the above interaction, the atomic state at any time is given by:

$\displaystyle\left|\Psi(t)\right\rangle=$
$\displaystyle\left[\left\\{c_{\mathrm{A},GG}(t)\hat{\sigma}^{+}+\sum_{m=-1}^{-N}c_{e_{m}G}(t)\hat{\sigma}_{m}^{+}+\sum_{n=1}^{N}c_{Ge_{n}}(t)\hat{\sigma}_{n}^{+}+\sum_{k}c_{k,GG}(t)\hat{a}_{k}^{\dagger}\right\\}\left|G_{1}\right\rangle\left|G_{2}\right\rangle\right.$
$\displaystyle\left.+\left\\{c_{\mathrm{A},GG^{\prime}}(t)\hat{\sigma}^{+}+\sum_{m=-1}^{-N}c_{e_{m}G^{\prime}}(t)\hat{\sigma}_{m}^{+}+\sum_{k}c_{k,GG^{\prime}}(t)\hat{a}_{k}^{\dagger}\right\\}\left|G_{1}\right\rangle\left|G^{\prime}_{2}\right\rangle\right.$
$\displaystyle\left.+\left\\{c_{\mathrm{A},G^{\prime}G}(t)\hat{\sigma}^{+}+\sum_{n=1}^{N}c_{G^{\prime}e_{n}}(t)\hat{\sigma}_{n}^{+}+\sum_{k}c_{k,G^{\prime}G}(t)\hat{a}_{k}^{\dagger}\right\\}\left|G^{\prime}_{1}\right\rangle\left|G_{2}\right\rangle\right.$
$\displaystyle\left.+\left\\{c_{\mathrm{A},G^{\prime}G^{\prime}}(t)\hat{\sigma}^{+}+\sum_{k}c_{k,G^{\prime}G^{\prime}}(t)\hat{a}_{k}^{\dagger}\right\\}\left|G^{\prime}_{1}\right\rangle\left|G^{\prime}_{2}\right\rangle\right]\left|g\right\rangle\left|\left\\{0\right\\}\right\rangle.$
(S33)

The equations of motion for the atomic excitation amplitudes are:

$\displaystyle\frac{{d}c_{\mathrm{A},GG}(t)}{{d}t}=$
$\displaystyle-i\sum_{k}g_{k}c_{k,GG}(t)e^{ikx_{A}}e^{-i(\omega-\omega_{0})t};\quad\frac{{d}c_{\mathrm{A},GG^{\prime}}(t)}{{d}t}=-i\sum_{k}g_{k}c_{k,GG^{\prime}}(t)e^{ikx_{A}}e^{-i(\omega-\omega_{0})t}$
(S34) $\displaystyle\frac{{d}c_{\mathrm{A},G^{\prime}G}(t)}{{d}t}=$
$\displaystyle-i\sum_{k}g_{k}c_{k,G^{\prime}G}(t)e^{ikx_{A}}e^{-i(\omega-\omega_{0})t};\quad\frac{{d}c_{\mathrm{A},G^{\prime}G^{\prime}}(t)}{{d}t}=-i\sum_{k}g_{k}c_{k,G^{\prime}G^{\prime}}(t)e^{ikx_{A}}e^{-i(\omega-\omega_{0})t}$
(S35)

Similarly, we have the equations of motion for the excitation amplitudes of
the atoms composing the quantum mirrors:

$\displaystyle\frac{{d}c_{e_{m}G}(t)}{{d}t}=$
$\displaystyle-i\sum_{k}g_{k}c_{k,GG}(t)e^{ikx_{m}}e^{-i(\omega-\omega_{0})t};\quad\frac{{d}c_{e_{m}G^{\prime}}(t)}{{d}t}=-i\sum_{k}g_{k}c_{k,GG^{\prime}}(t)e^{ikx_{m}}e^{-i(\omega-\omega_{0})t}$
(S36) $\displaystyle\frac{{d}c_{Ge_{n}}(t)}{{d}t}=$
$\displaystyle-i\sum_{k}g_{k}c_{k,GG}(t)e^{ikx_{n}}e^{-i(\omega-\omega_{0})t};\quad\frac{{d}c_{G^{\prime}e_{n}}(t)}{{d}t}=-i\sum_{k}g_{k}c_{k,G^{\prime}G}(t)e^{ikx_{n}}e^{-i(\omega-\omega_{0})t}$
(S37)

For the field excitation amplitudes:

$\displaystyle\frac{{d}c_{k,GG}(t)}{{d}t}=$
$\displaystyle-i\left[g_{k}c_{\mathrm{A},GG}(t)e^{-ikx_{A}}+\sum_{m=-1}^{-N}g_{k}c_{e_{m}G}(t)e^{-ikx_{m}}+\sum_{n=1}^{N}g_{k}c_{Ge_{n}}(t)e^{-ikx_{n}}\right]e^{i(\omega-\omega_{0})t}$
(S38) $\displaystyle\frac{{d}c_{k,GG^{\prime}}(t)}{{d}t}=$
$\displaystyle-i\left[g_{k}c_{\mathrm{A},GG^{\prime}}(t)e^{-ikx_{A}}+\sum_{m=-1}^{-N}g_{k}c_{e_{m}G^{\prime}}(t)e^{-ikx_{m}}\right]e^{i(\omega-\omega_{0})t}$
(S39) $\displaystyle\frac{{d}c_{k,G^{\prime}G}(t)}{{d}t}=$
$\displaystyle-i\left[g_{k}c_{\mathrm{A},G^{\prime}G}(t)e^{-ikx_{A}}+\sum_{n=1}^{N}g_{k}c_{G^{\prime}e_{n}}(t)e^{-ikx_{m}}\right]e^{i(\omega-\omega_{0})t}$
(S40) $\displaystyle\frac{{d}c_{k,G^{\prime}G^{\prime}}(t)}{{d}t}=$
$\displaystyle-
ig_{k}c_{\mathrm{A},G^{\prime}G^{\prime}}(t)e^{-ikx_{A}}e^{i(\omega-\omega_{0})t}$
(S41)

Substituting the field amplitudes in the atomic equations of motion, and
making the Born-Markov approximation one obtains:

$\displaystyle\frac{{d}c_{\mathrm{A},GG}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[c_{\mathrm{A},GG}(t)+\sum_{m=-1}^{-N}c_{e_{m}G}(t)e^{-i\omega_{1}(x_{m}-x_{A})/v}+\sum_{n=1}^{N}c_{Ge_{n}}(t)e^{i\omega_{2}(x_{n}-x_{A})/v}\right]$
(S42) $\displaystyle\frac{{d}c_{\mathrm{A},GG^{\prime}}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[c_{\mathrm{A},GG^{\prime}}(t)+\sum_{m=-1}^{-N}c_{e_{m}G^{\prime}}(t)e^{-i\omega_{1}(x_{m}-x_{A})/v}\right]$
(S43) $\displaystyle\frac{{d}c_{\mathrm{A},G^{\prime}G}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[c_{\mathrm{A},G^{\prime}G}(t)+\sum_{n=1}^{N}c_{G^{\prime}e_{n}}(t)e^{i\omega_{2}(x_{n}-x_{A})/v}\right]$
(S44)
$\displaystyle\frac{{d}c_{\mathrm{A},G^{\prime}G^{\prime}}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}c_{\mathrm{A},G^{\prime}G^{\prime}}(t)$ (S45)
$\displaystyle\frac{{d}c_{e_{m}G}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[\sum_{\mu=-1}^{-N}c_{e_{\mu}G}(t)e^{i\omega_{1}\left|x_{m}-x_{\mu}\right|/v}+c_{\mathrm{A},GG}(t)e^{-i\omega_{A}(x_{m}-x_{A})/v}+\sum_{n=1}^{N}c_{Ge_{n}}(t)e^{i\omega_{2}(x_{n}-x_{m})/v}\right]$
(S46) $\displaystyle\frac{{d}c_{e_{m}G^{\prime}}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[\sum_{\mu=-1}^{-N}c_{e_{\mu}G^{\prime}}(t)e^{i\omega_{1}\left|x_{m}-x_{\mu}\right|/v}+c_{\mathrm{A},GG^{\prime}}(t)e^{-i\omega_{A}(x_{m}-x_{A})/v}\right]$
(S47) $\displaystyle\frac{{d}c_{Ge_{n}}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[c_{\mathrm{A},GG}(t)e^{i\omega_{A}(x_{n}-x_{A})/v}+\sum_{m=-1}^{-N}c_{e_{m}G}(t)e^{i\omega_{1}(x_{n}-x_{m})/v}+\sum_{\nu=1}^{N}c_{Ge_{\nu}}(t)e^{i\omega_{2}\left|x_{n}-x_{\nu}\right|/v}\right]$
(S48) $\displaystyle\frac{{d}c_{G^{\prime}e_{n}}(t)}{{d}t}\approx$
$\displaystyle-\frac{\gamma}{2}\left[c_{\mathrm{A},G^{\prime}G}(t)e^{i\omega_{A}(x_{n}-x_{A})/v}+\sum_{\nu=1}^{N}c_{G^{\prime}e_{\nu}}(t)e^{i\omega_{2}\left|x_{n}-x_{\nu}\right|/v}\right].$
(S49)

We have assumed the spontaneous emission rate to be $\gamma\approx
2\pi|g\left(\omega_{0}\right)|^{2}$ for a sufficiently flat spectral density
of the field. We now define the collective excitation amplitudes for the QMs:

$\displaystyle c_{\mathrm{QM1}}(t)\equiv$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{m=1}^{N}c_{e_{m}G}(t)(-1)^{m-1};\quad
c^{\prime}_{\mathrm{QM1}}(t)\equiv\frac{1}{\sqrt{N}}\sum_{m=1}^{N}c_{e_{m}G^{\prime}}(t)(-1)^{m-1}$
(S50) $\displaystyle c_{\mathrm{QM2}}(t)\equiv$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{n=1}^{N}c_{Ge_{n}}(t)(-1)^{n-1};\quad
c^{\prime}_{\mathrm{QM2}}(t)\equiv\frac{1}{\sqrt{N}}\sum_{n=1}^{N}c_{G^{\prime}e_{n}}(t)(-1)^{n-1}$
(S51)

Note that the position of the $n^{\mathrm{th}}$ atom in mirror 2 is

$\displaystyle x_{n}=$ $\displaystyle x_{1}+(n-1)\lambda_{0}/2$ (S52)
$\displaystyle\implies e^{ik_{0}x_{n}}=$ $\displaystyle
e^{ik_{0}x_{1}}e^{i\pi(n-1)}=e^{ik_{0}x_{1}}(-1)^{(n-1)}$ (S53)

Similarly, the position of the $m^{\mathrm{th}}$ atom in mirror 1 is:

$\displaystyle x_{m}=$ $\displaystyle x_{-1}-(m-1)\lambda_{0}/2$ (S54)
$\displaystyle\implies e^{-ik_{0}x_{m}}=$ $\displaystyle
e^{-ik_{0}x_{-1}}e^{i\pi(m-1)}=e^{ik_{0}x_{1}}(-1)^{(m-1)},$ (S55)

where we have assumed, without loss of generality, that $x_{-1}=-x_{1}$.

Taking the Laplace transform of the atomic equations of motion
(Eq.(S42)–(S49)) in terms of the collective mirror excitation amplitudes, we
obtain:

$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}_{\mathrm{A},GG}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}+x_{A})}\tilde{c}_{\mathrm{QM1}}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}-x_{A})}\tilde{c}_{\mathrm{QM2}}(s)=c_{\mathrm{A},GG}(0)$
(S56)
$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}_{\mathrm{A},GG^{\prime}}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}+x_{A})}\tilde{c}^{\prime}_{\mathrm{QM1}}(s)=c_{\mathrm{A},GG^{\prime}}(0)$
(S57)
$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}_{\mathrm{A},G^{\prime}G}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}-x_{A})}\tilde{c}^{\prime}_{\mathrm{QM2}}(s)=c_{\mathrm{A},G^{\prime}G}(0)$
(S58)
$\displaystyle\left(s+\frac{\gamma}{2}\right)\tilde{c}_{\mathrm{A},G^{\prime}G^{\prime}}(s)=c_{\mathrm{A},G^{\prime}G^{\prime}}(0)$
(S59)
$\displaystyle\left(s+\frac{N\gamma}{2}\right)\tilde{c}_{\mathrm{QM1}}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}+x_{A})}\tilde{c}_{\mathrm{A},GG}(s)+\frac{N\gamma}{2}e^{2ik_{0}x_{1}}\tilde{c}_{\mathrm{QM2}}(s)=c_{\mathrm{QM1}}(0)$
(S60)
$\displaystyle\left(s+\frac{N\gamma}{2}\right)\tilde{c}^{\prime}_{\mathrm{QM1}}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}+x_{A})}\tilde{c}_{\mathrm{A},GG^{\prime}}(s)=c^{\prime}_{\mathrm{QM1}}(0)$
(S61)
$\displaystyle\left(s+\frac{N\gamma}{2}\right)\tilde{c}_{\mathrm{QM2}}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}-x_{A})}\tilde{c}_{\mathrm{A},GG}(s)+\frac{N\gamma}{2}e^{2ik_{0}x_{1}}\tilde{c}_{\mathrm{QM1}}(s)=c_{\mathrm{QM2}}(0)$
(S62)
$\displaystyle\left(s+\frac{N\gamma}{2}\right)\tilde{c}^{\prime}_{\mathrm{QM2}}(s)+\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}-x_{A})}\tilde{c}_{\mathrm{A},G^{\prime}G}(s)=c^{\prime}_{\mathrm{QM2}}(0),$
(S63)

Each QM acts as an effective collective spin with a cooperative coupling of
$\sqrt{N}\gamma$ to the waveguide. Eq. (S59) can be immediately solved to
obtain an exponential decay for
$c_{\mathrm{A},G^{\prime}G^{\prime}}(t)=c_{\mathrm{A},G^{\prime}G^{\prime}}(0)e^{-\gamma
t/2}$.

The remaining equations can be expressed in matrix notation as:

$\displaystyle\bar{\bar{M}}(\tilde{s})\underbrace{\left(\begin{array}[]{c}\tilde{c}_{\mathrm{A},GG}(\tilde{s})\\\
\tilde{c}_{\mathrm{A},GG^{\prime}}(\tilde{s})\\\
\tilde{c}_{\mathrm{A},G^{\prime}G}(\tilde{s})\\\
\tilde{c}_{\mathrm{QM1}}(\tilde{s})\\\
\tilde{c}^{\prime}_{\mathrm{QM1}}(\tilde{s})\\\
\tilde{c}_{\mathrm{QM2}}(\tilde{s})\\\
\tilde{c}^{\prime}_{\mathrm{QM2}}(\tilde{s})\end{array}\right)}_{\equiv\tilde{{\mathbf{C}}}(\tilde{s})}=\frac{1}{\gamma}\underbrace{\left(\begin{array}[]{c}c_{\mathrm{A},GG}(0)\\\
c_{\mathrm{A},GG^{\prime}}(0)\\\ c_{\mathrm{A},G^{\prime}G}(0)\\\
c_{\mathrm{QM1}}(0)\\\ c^{\prime}_{\mathrm{QM1}}(0)\\\ c_{\mathrm{QM2}}(0)\\\
c^{\prime}_{\mathrm{QM2}}(0)\end{array}\right)}_{\equiv{\mathbf{C}}(0)},$
(S78)

where $\tilde{s}\equiv s/\gamma$ and the matrix $\bar{\bar{M}}(\tilde{s})$ is
defined as:

$\displaystyle\bar{\bar{M}}(\tilde{s})=$
$\displaystyle\left(\begin{array}[]{ccccccc}\tilde{s}+\frac{1}{2}&0&0&\frac{\sqrt{N}}{2}e^{ik_{0}(x_{1}+x_{A})}&0&\frac{\sqrt{N}}{2}e^{ik_{0}(x_{1}-x_{A})}&0\\\
0&\tilde{s}+\frac{1}{2}&0&0&\frac{\sqrt{N}\gamma}{2}e^{ik_{0}(x_{1}+x_{A})}&0&0\\\
0&0&\tilde{s}+\frac{1}{2}&0&0&0&\frac{\sqrt{N}}{2}e^{ik_{0}(x_{1}-x_{A})}\\\
\frac{\sqrt{N}}{2}e^{ik_{A}(x_{1}+x_{A})}&0&0&\tilde{s}+\frac{N}{2}&0&\frac{N}{2}e^{2ik_{0}x_{1}}&0\\\
0&\frac{\sqrt{N}}{2}e^{ik_{A}(x_{1}+x_{A})}&0&0&\tilde{s}+\frac{N}{2}&0\\\
\frac{\sqrt{N}}{2}e^{ik_{A}(x_{1}-x_{A})}&0&0&\frac{N}{2}e^{2ik_{0}x_{1}}&0&\tilde{s}+\frac{N}{2}&0\\\
0&0&\frac{\sqrt{N}}{2}e^{ik_{A}(x_{1}-x_{A})}&0&0&0&\tilde{s}+\frac{N}{2}\end{array}\right)$
(S86)

The solution to the atomic dynamics can be thus numerically obtained in terms
of the inverse Laplace
${\mathbf{C}}(t)=\mathcal{L}^{-1}\left[\frac{1}{\gamma}\bar{\bar{M}}^{-1}(\tilde{s}){\mathbf{C}}(0)\right]$.

###  II.1 Large $N$ limit

Let us consider the dynamics in the subspace where both QMs are in the ground
state $\left|G\right\rangle$ (Eqs.(S56), (S60), (S62)). Using the initial
conditions $c_{\mathrm{QM1}}(0)=c_{\mathrm{QM2}}(0)=0$ in Eq.(S60) and (S62),
yields:

$\displaystyle\tilde{c}_{\mathrm{QM1}}(s)=-\frac{\sqrt{N}\gamma}{2}e^{ik_{0}x_{1}}\frac{\left[e^{ik_{0}x_{A}}\left(s+\frac{N\gamma}{2}\right)-e^{-ik_{0}(x_{A}-2x_{1})}\frac{N\gamma}{2}\right]}{\left(s+\frac{N\gamma}{2}\right)^{2}-\left(\frac{N\gamma}{2}e^{2ik_{0}x_{1}}\right)^{2}}\tilde{c}_{\mathrm{A},GG}(s)$
(S87)
$\displaystyle\tilde{c}_{\mathrm{QM2}}(s)=-\frac{\sqrt{N}\gamma}{2}e^{ik_{0}x_{1}}\frac{\left[e^{-ik_{0}x_{A}}\left(s+\frac{N\gamma}{2}\right)-e^{ik_{0}(x_{A}+2x_{1})}\frac{N\gamma}{2}\right]}{\left(s+\frac{N\gamma}{2}\right)^{2}-\left(\frac{N\gamma}{2}e^{2ik_{0}x_{1}}\right)^{2}}\tilde{c}_{\mathrm{A},GG}(s)$
(S88)

Substituting the above in Eq.(S56), we obtain:

$\displaystyle\tilde{c}_{\mathrm{A},GG}(s)=c_{\mathrm{A},GG}(0)\left[\frac{\left(s+\frac{N\gamma}{2}\right)^{2}-\left(\frac{N\gamma}{2}e^{2ik_{0}x_{1}}\right)^{2}}{\left(s+\frac{\gamma}{2}\right)\left\\{\left(s+\frac{N\gamma}{2}\right)^{2}-\left(\frac{N\gamma}{2}e^{2ik_{0}x_{1}}\right)^{2}\right\\}-\frac{N\gamma^{2}}{2}e^{2ik_{0}x_{1}}\left\\{\cos\left(2k_{0}x_{A}\right)\left(s+\frac{N\gamma}{2}\right)-\frac{N\gamma}{2}e^{2ik_{0}x_{1}}\right\\}}\right].$
(S89)

We now consider the large $N$ limit for the QMs for following cases as
discussed in the main text:

  1. 1.

Atom at cavity node $(x_{A}=0,x_{1}=p\lambda_{0}/2,p\in\mathbb{N})$: One can
simplify Eq.(S89) in this case as follows

$\displaystyle\tilde{c}_{\mathrm{A},GG}(s)=c_{\mathrm{A},GG}(0)\left[\frac{{s+N\gamma}}{\left(s+\frac{\gamma}{2}\right)\left(s+N\gamma\right)-\frac{N\gamma^{2}}{2}}\right],$
(S90)

such that in the large $N$ limit this can be approximated as:

$\displaystyle\tilde{c}_{\mathrm{A},GG}(s)\approx\frac{1}{s}c_{\mathrm{A},GG}(0)\implies
c_{\mathrm{A},GG}(t)=c_{\mathrm{A},GG}(0).$ (S91)

Substituting Eq.(S90) in Eq.(S87) and (S88), we get
$\tilde{c}_{\mathrm{QM1}}(s)=\tilde{c}_{\mathrm{QM2}}(s)=-\frac{\sqrt{N}\gamma}{2}\frac{1}{s+N\gamma}\tilde{c}_{\mathrm{A},GG}(s)$,
such that in the large $N$ limit both
$c_{\mathrm{QM1}}(t),c_{\mathrm{QM2}}(t)\propto 1/\sqrt{N}\rightarrow 0$.

  2. 2.

Atom at cavity antinode $(x_{A}=0,x_{1}=p\lambda_{0}/4,p\in\mathbb{N})$: In
this case Eq.(S89) becomes:

$\displaystyle\tilde{c}_{\mathrm{A},GG}(s)=c_{\mathrm{A},GG}(0)\frac{s}{s^{2}+\frac{\gamma}{2}s+\frac{N\gamma^{2}}{2}},$
(S92)

which can be approximated in the large $N$ limit as:

$\displaystyle\tilde{c}_{\mathrm{A},GG}(s)\approx\frac{c_{\mathrm{A},GG}(0)}{2}\left[\frac{1}{s+\frac{\gamma}{4}+i\sqrt{\frac{N}{2}}\gamma}+\frac{1}{s+\frac{\gamma}{4}-i\sqrt{\frac{N}{2}}\gamma}\right]$
(S93) $\displaystyle\implies$ $\displaystyle c_{\mathrm{A},GG}(t)\approx
c_{\mathrm{A},GG}(0)e^{-\gamma t/4}\cos\left(\sqrt{\frac{N}{2}}\gamma
t\right).$ (S94)

Substituting Eq.(S92) in Eq.(S87) and (S88) yields:

$\displaystyle\tilde{c}_{\mathrm{QM1}}(s)=\tilde{c}_{\mathrm{QM2}}(s)=-\frac{i\sqrt{N}\gamma}{2\left(s^{2}+\frac{\gamma}{2}s+\frac{N\gamma^{2}}{2}\right)}c_{\mathrm{A},GG}(0)$
(S95)

Taking the large $N$ limit,

$\displaystyle\tilde{c}_{\mathrm{QM1}}(s)=\tilde{c}_{\mathrm{QM2}}(s)\approx\frac{c_{\mathrm{A},GG}(0)}{\sqrt{2}}\left[\frac{1}{s+\frac{\gamma}{4}+i\sqrt{\frac{N}{2}}\gamma}-\frac{1}{s+\frac{\gamma}{4}-i\sqrt{\frac{N}{2}}\gamma}\right]$
(S96) $\displaystyle\implies$ $\displaystyle
c_{\mathrm{QM1}}(t)=c_{\mathrm{QM2}}(t)\approx\sqrt{2}ic_{\mathrm{A},GG}(0)e^{-\gamma
t/4}\sin\left(\sqrt{\frac{N}{2}}\gamma t\right)$ (S97)

  3. 3.

Atom near cavity node $(0<k_{0}x_{A}\ll
1,x_{1}=p\lambda_{0}/2,p\in\mathbb{N})$: We define $\phi_{A}\equiv
k_{0}x_{A}$, such that $\cos\left(2k_{0}x_{A}\right)\approx 1-2\phi_{A}^{2}$,
thus Eq.(S89) becomes:

$\displaystyle\tilde{c}_{\mathrm{A},GG}(s)\approx$ $\displaystyle
c_{\mathrm{A},GG}(0)\left[\frac{s\left(s+N\gamma\right)}{s\left(s+\frac{\gamma}{2}\right)\left(s+N\gamma\right)-\frac{N\gamma^{2}}{2}\left\\{s-2\phi_{A}^{2}\left(s+\frac{N\gamma}{2}\right)\right\\}}\right]$
(S98)

Taking the large $N$ limit:

$\displaystyle\tilde{c}_{\mathrm{A},GG}(s)\approx$ $\displaystyle
c_{\mathrm{A},GG}(0)\left[\frac{s}{s^{2}+\phi_{A}^{2}\frac{N\gamma^{2}}{2}}\right]\implies
c_{\mathrm{A},GG}(t)\approx
c_{\mathrm{A},GG}(0)\cos\left(\phi_{A}\sqrt{\frac{N}{2}}\gamma t\right)$ (S99)

To obtain the dynamics of the atoms constituting the QMs, we substitute
Eq.(S98) in Eqs.(S87) and (S88):

$\displaystyle\tilde{c}_{\mathrm{QM1}}(s)\approx-\frac{\sqrt{N}\gamma}{2}\frac{e^{ik_{0}x_{A}}\left(s+\frac{N\gamma}{2}\right)-e^{-ik_{0}x_{A}}\frac{N\gamma}{2}}{s\left(s+\frac{\gamma}{2}\right)\left(s+N\gamma\right)-\frac{N\gamma^{2}}{2}\left\\{s-2\phi_{A}^{2}\left(s+\frac{N\gamma}{2}\right)\right\\}}c_{\mathrm{A},GG}(0)$
(S100)
$\displaystyle\tilde{c}_{\mathrm{QM2}}(s)\approx-\frac{\sqrt{N}\gamma}{2}\frac{e^{-ik_{0}x_{A}}\left(s+\frac{N\gamma}{2}\right)-e^{ik_{0}x_{A}}\frac{N\gamma}{2}}{s\left(s+\frac{\gamma}{2}\right)\left(s+N\gamma\right)-\frac{N\gamma^{2}}{2}\left\\{s-2\phi_{A}^{2}\left(s+\frac{N\gamma}{2}\right)\right\\}}c_{\mathrm{A},GG}(0)$
(S101)

Taking the large $N$ limit of the above, we get:

$\displaystyle\tilde{c}_{\mathrm{QM1}}(s)\approx-\tilde{c}_{\mathrm{QM2}}(s)\approx-\frac{\sqrt{N}\gamma}{2}\frac{i\phi_{A}}{s^{2}+\frac{N\gamma^{2}}{2}\phi_{A}^{2}}c_{\mathrm{A},GG}(0)$
(S102) $\displaystyle\implies$ $\displaystyle c_{\mathrm{QM1}}(t)\approx-
c_{\mathrm{QM2}}(t)\approx-\frac{i}{\sqrt{2}}\sin\left(\phi_{A}\sqrt{\frac{N}{2}}\gamma
t\right).$ (S103)