# Engineering photonic band gaps with a waveguide-QED structure containing an
atom-polymer array

M. S. Wang  School of Physical Science and Technology, Southwest Jiaotong
University, Chengdu 610031, China  W. Z. Jia<EMAIL_ADDRESS>School of
Physical Science and Technology, Southwest Jiaotong University, Chengdu
610031, China

###### Abstract

We investigate the generation and engineering of photonic band gaps in
waveguide quantum electrodynamics systems containing periodically arranged
atom-polymers. We first consider the configuration of a dimer array coupled to
a waveguide. The results show that if the intra- and inter-cell phase delays
are properly designed, the center and the width of the band gaps, as well as
the dispersion relation of the passbands can be modified by adjusting the
intra-cell coupling strength. These manipulations provide ways to control the
propagating modes in the waveguide, leading to some interesting effects such
as slowing or even stopping a single-photon pulse. Finally, we take the case
of the tetramer chain as an example to show that, in the case of a larger
number of atoms in each unit cell, tunable multi-gap structures and more
sophisticated band-gap engineering can be realized. Our proposal provides
efficient ways for photonic band-gap engineering in micro- and nano-quantum
systems, which may facilitate the manipulation of photon transport in future
quantum networks.

††preprint: APS/123-QED

##  I Introduction

Waveguide quantum electrodynamics (wQED) systems are realized by strongly
coupling quantum emitters (e.g., atoms, artificial atoms, cavities, and so on)
to a one-dimensional (1D) waveguide and are ideal platform for the
manipulation of strong emitter-photon interactions Roy  _et al._ (2017); Gu
_et al._ (2017); Sheremet  _et al._ (2023). The high coupling efficiency
between emitters and photonic modes makes wQED setups optimal for single-
photon level quantum devices Shen and Fan (2005a); Chang  _et al._ (2006);
Shen and Fan (2007); Zhou  _et al._ (2008); Astafiev  _et al._ (2010); Shi
_et al._ (2015). In a 1D wQED setup containing multiple emitters, the coherent
and dissipative interactions between them can be mediated by their common
electromagnetic environment, giving rise to a large number of intriguing
phenomena, including superradiant and subradiant phenomena Dicke (1954);
Vetter  _et al._ (2016); van Loo  _et al._ (2013); Zhang and Mølmer (2019); Ke
_et al._ (2019); Wang  _et al._ (2020), long-range quantum entanglement
between atoms Zheng and Baranger (2013); Gonzalez-Ballestero  _et al._ (2014);
Facchi  _et al._ (2016); Mirza and Schotland (2016), cavity QED with atomic
mirrors Chang  _et al._ (2012); Mirhosseini  _et al._ (2019); Nie  _et al._
(2023), decoherence-free interactions in giant artificial atoms Kockum  _et
al._ (2018), topologically protected scattering spectra Nie  _et al._ (2021),
multiple Fano interferences Tsoi and Law (2008); Liao  _et al._ (2015); Cheng
_et al._ (2017); Ruostekoski and Javanainen (2017); Mukhopadhyay and Agarwal
(2019), control-field-free atomic coherence effects Mukhopadhyay and Agarwal
(2020); Ask  _et al._ ; Jia and Cai (2022); Feng and Jia (2021), and so on.

In recent years, considerable attention has been devoted to photonic band-gap
materials in micro- and nano-quantum structures. One potential avenue for
achieving this objective is through the use of a conventional photonic crystal
waveguide (PCW) Bendickson  _et al._ (1996); Hood  _et al._ (2016), in which
the periodicity of the dielectric structure is employed to generate a set of
Bloch bands. Moreover, in the microwave domain, PCW has a series of circuit
versions, including, for example, coplanar waveguide with periodically
modulated impedance Liu and Houck (2017); Sundaresan  _et al._ (2019),
resonator array waveguide Kim  _et al._ (2021); Ferreira  _et al._ (2021);
Scigliuzzo  _et al._ (2022); Zhang  _et al._ (2023), and circuit-crystal
metamaterials with Josephson junction arrays Rakhmanov  _et al._ (2008);
Hutter  _et al._ (2011); Zueco  _et al._ (2012). Another method to generate
photonic band gaps is based on wQED structures with periodically arranged
emitters (either an atom array or a cavity array) side coupled to a 1D linear
waveguide Xu  _et al._ (2000); Shen and Fan (2005b); Yanik  _et al._ (2004);
Shen  _et al._ (2007); Yi  _et al._ (2010); Witthaut and Sørensen (2010); Fang
and Baranger (2015); Mirza  _et al._ (2017); Mirhosseini  _et al._ (2018); He
_et al._ (2021); Greenberg  _et al._ (2021); Tang  _et al._ (2022); Peng and
Jia (2023); Berndsen and Mirza (2023); Berndsen  _et al._ (2024). Achieving
highly tunable band-gap modulation in periodic structures that have already
been prepared is of significant importance. This type of band-gap engineering
can facilitate the manipulation of photon transport and the control of light-
matter interactions in a variety of ways. In the case of band-gap structures
produced by wQED containing multiple atoms, existing schemes for band-gap
modulation are mainly based on the adjustment of the atomic frequency Shen
_et al._ (2007); He  _et al._ (2021); Greenberg  _et al._ (2021) and the
application of an external control field Witthaut and Sørensen (2010).

In micro- and nano-quantum systems, such as superconducting quantum circuits,
the tunability of the coupling strength between artificial atoms is a
prominent feature Hime  _et al._ (2006); Niskanen  _et al._ (2007); Baust  _et
al._ (2015); Zhu and Jia (2019). It seems reasonable to posit that the tunable
coupling between atoms can be harnessed to engineer the band-gap structure
formed by wQED systems containing an atomic array. In this paper we present a
proposal to achieve this goal, as shown schematically in Fig. 1. In our
proposal, a periodic atomic array is coupled to a 1D waveguide, with each unit
cell containing at least two atoms with tunable couplings between the nearest
neighboring atoms, forming a so called polymer. The transfer matrix of the
system is derived using the real-space formalism, and then the corresponding
scattering coefficients are expressed analytically in terms of Chebyshev
polynomials. Using these results, we further analyze the band-gap structures
in the scattering spectra and demonstrate the potential for band-gap
engineering through the use of interatomic couplings. Specifically, for the
case where each cell is an atom-dimer (i.e., a cell with two coupled atoms),
band-gap structures with highly tunable center frequencies and widths can be
realized by properly adjusting the intra-cell coupling strength and designing
the interatomic phase delays. Moreover, these kinds of modulations can also be
used to engineer the propagating modes in the passband. In particular, if a
band gap is closed at a specific frequency, the Bloch modes in the vicinity of
this frequency will become slow modes with a linear dispersion relation.
Furthermore, if the width of a passband is dynamically compressed by
modulating the coupling strength, the formation of a flat band can enable the
setup to slow down or even stop a single-photon pulse at will. Finally, the
results for a dimer array can be generalized to the case of a polymer chain
with a greater number of atoms in each unit cell. This enables the realization
of a tunable multi-band-gap structure and the implementation of more
sophisticated band-gap engineering.

The remainder of this paper is organized as follows. In Sec. II, we present a
theoretical model that includes the system Hamiltonian and corresponding
single-photon transport equations. Furthermore, using the transfer matrix
approach, we derive the expressions of the single-photon scattering
coefficients. In Sec. III, we present some general discussions on the band-gap
structure produced by a wQED system containing a polymerized atomic chain. In
Sec. IV, we analyze tunable band-gap structures and propagation-mode control
in a wQED system coupled by an atom-dimer chain. In Sec. V, we further examine
band-gap engineering in a wQED system containing an atom-tetramer chain.
Finally, further discussions and conclusions are given in Sec. VI.

##  II Theoretical description of single-photon scattering

###  II.1 Hamiltonian and single-photon transport equations

Figure 1: Sketches of (a) an atom-dimer chain ($N=2$) coupled to a 1D
waveguide, (b) an atom-tetramer chain ($N=4$) coupled to a 1D waveguide.

In this study, we concentrate on the wQED structures comprising $M$
periodically spaced polymers, each containing $N$ two-level atoms with
identical transition frequency $\omega_{\mathrm{a}}$, coupled to photonic
modes in a 1D waveguide with linear dispersion, as schematically shown in Fig.
1. The atoms are located at position $x_{m,s}$ ($m=1,2,\cdots,M$ labels the
cells and $s=1,2,\cdots,N$ denotes the atoms in each cell). The lattice
constant of the polymer array is $L=x_{m,s}-x_{m-1,s}$. The distance between
neighboring atoms in a cell is $a=x_{m,s}-x_{m,s-1}$. The strength between the
$s$th atom and the $(s+1)$th atom in a cell is $J_{s}$. The coupling strength
between each atom and the waveguide is $g$. Under the rotating-wave
approximation, the Hamiltonian of the system in the real space can be written
as $(\hbar=1)$:

$\begin{split}\hat{H}=&\sum_{\rho}\int\mathrm{d}x{\hat{a}_{\rho}^{\dagger}(x)\left(-i\tau_{\rho}v_{\mathrm{g}}\frac{\partial}{\partial
x}\right)\hat{a}_{\rho}\left(x\right)}\\\
&+\sum_{m=1}^{M}\sum_{s=1}^{N}\omega_{\mathrm{a}}\sigma_{m,s}^{+}\sigma_{m,s}^{-}+\sum_{m=1}^{M}\sum_{s=1}^{N-1}\left(J_{s}\sigma_{m,s}^{+}\sigma_{m,s+1}^{-}+\mathrm{H.c.}\right)\\\
&+\sum_{\rho}\sum_{m=1}^{M}\sum_{s=1}^{N}g\int\mathrm{d}x\delta(x-x_{m,s})\left[\hat{a}_{\rho}^{\dagger}\left(x\right)\sigma_{m,s}^{-}+\mathrm{H.c.}\right],\end{split}$
(1)

where $\rho=\mathrm{R},\mathrm{L}$ and $\tau_{\mathrm{R},\mathrm{L}}=\pm 1$.
$\hat{a}_{\mathrm{R}}^{\dagger}(x)$ [$\hat{a}_{\mathrm{R}}(x)$] and
$\hat{a}_{\mathrm{L}}^{\dagger}(x)$ [$\hat{a}_{\mathrm{L}}(x)$] are the
bosonic creation (annihilation) operators of the right- and left-going wave at
position $x$. $v_{\mathrm{g}}$ is the group velocity of photons in the
waveguide. $\sigma_{m,s}^{+}$ ($\sigma_{m,s}^{-}$) is the raising (lowering)
operator of the $s$th atom in the $m$th cell.

In the single-excitation manifold, the scattering eigenstate can be written as

$|\Psi\rangle=\sum_{\rho}\int\mathrm{d}x\Phi_{\rho}\left(x\right)\hat{a}_{\rho}^{\dagger}\left(x\right)\left|\emptyset\right\rangle+\sum_{m=1}^{M}\sum_{s=1}^{N}f_{m,s}\sigma_{m,s}^{+}\left|\emptyset\right\rangle,$
(2)

where $|\emptyset\rangle$ is the vacuum state with no photons propagating in
the waveguide and the atoms occupying their ground states. $\Phi_{\rho}(x)$ is
the single-photon wave function in the $\rho$ mode. $f_{m,s}$ is the
excitation amplitude of the $s$th atom in the $m$th cell.

We assume that a single photon with energy $\omega=v_{\mathrm{g}}k$ is
initially incident from the left, where $k$ is the wave vector of the photon.
The corresponding right(left)-going wave function $\Phi_{\mathrm{R}}(x)$
[$\Phi_{\mathrm{L}}(x)$] takes the following Ansatz:

$\displaystyle\Phi_{\mathrm{R}}\left(x\right)$ $\displaystyle=$ $\displaystyle
e^{ikx}\Big{\\{}\theta\left(x_{1,1}-x\right)$
$\displaystyle+\sum_{m=1}^{M}\sum_{s=1}^{N}t_{m,s}\left[\theta\left(x-x_{m,s}\right)-\theta\left(x-x_{m,s+1}\right)\right]\Big{\\}},$
$\Phi_{\mathrm{L}}\left(x\right)=e^{-ikx}\sum_{m=1}^{M}\sum_{s=1}^{N}r_{m,s}\left[\theta\left(x_{m,s}-x\right)-\theta\left(x_{m,s-1}-x\right)\right].$
(3b)

Here, we define $x_{1,0}=-\infty$ and $x_{M,N+1}=+\infty$. $t_{m,s}$
($r_{m,s}$) is the transmission (reflection) amplitude of the $s$th atom in
the $m$th cell. And the transmission (reflection) amplitude of the last
(first) atom $t_{M,N}\equiv t$ ($r_{1,1}\equiv r$) is defined as the
transmission (reflection) amplitude of the atom array. $\theta(x)$ denotes the
Heaviside step function.

From the eigenequation
$\hat{H}\left|\Psi\right\rangle=\omega\left|\Psi\right\rangle$, we can obtain
the following set of equations:

$t_{m,s}-t_{m,s-1}+i\sqrt{\frac{\Gamma}{2}}\tilde{f}_{m,s}e^{-i\phi_{m,s}}=0,$
(4a)
$r_{m,s+1}-r_{m,s}-i\sqrt{\frac{\Gamma}{2}}\tilde{f}_{m,s}e^{i\phi_{m,s}}=0,$
(4b) $\displaystyle
J_{s-1}\tilde{f}_{m,s-1}-\Delta\tilde{f}_{m,s}+J_{s}\tilde{f}_{m,s+1}$
$\displaystyle+\frac{1}{2}\sqrt{\frac{\Gamma}{2}}\left[\left(t_{m,s}+t_{m,s-1}\right)e^{i\phi_{m,s}}+\left(r_{m,s+1}+r_{m,s}\right)e^{-i\phi_{m,s}}\right]=0.$
(4c)

Note that in the above equations the inter-cell connection relations
$t_{m,0}=t_{m-1,N},r_{m,N+1}=r_{m+1,1}$ and the boundary conditions
$t_{1,0}=1,r_{1,1}=r,t_{M,N}=t,r_{M,N+1}=0$ are satisfied.
$\tilde{f}_{m,s}=f_{m,s}/\sqrt{v_{\mathrm{g}}}$ is the atomic excitation
amplitude scaled by $\sqrt{v_{\mathrm{g}}}$. $\Gamma=2g^{2}/v_{\mathrm{g}}$ is
the decay rate of the atoms into the guided modes.
$\Delta=\omega-\omega_{\mathrm{a}}$ is the detuning between the photon and the
atom. The phase factor corresponding to position $x_{m,s}$ is defined as
$\phi_{m,s}=kx_{m,s}$.

###  II.2 Transfer matrix and scattering coefficients

From Eqs. (4a) and (4b), after iteration, we can obtain

$t_{m,s}=t_{m-1,N}-i\sqrt{\frac{\Gamma}{2}}\sum_{j=1}^{s}e^{-i\phi_{m,j}}\tilde{f}_{m,j},$
(5a)
$r_{m,s}=r_{m+1,1}-i\sqrt{\frac{\Gamma}{2}}\sum_{j=s}^{N}e^{i\phi_{m,j}}\tilde{f}_{m,j}.$
(5b)

From now on, we rewrite the phase factor of the $s$th atom in the $m$th cell
as $\phi_{m,s}=\beta_{m}+\phi_{s}$. $\beta_{m}=(m-1)\beta$ is the reference
phase of the $m$th cell (without loss of generality, we have chosen
$\beta_{1}=0$), where $\beta=\phi_{m,s}-\phi_{m-1,s}=kL$ is the phase delay
between neighboring cells. $\phi_{s}=[s-(N+1)/2]\alpha$ is the relative phase
of the $s$th atom in each cell with respect to the reference phase, where
$\alpha=\phi_{m,s}-\phi_{m,s-1}=ka$ is the phase delay between neighboring
atoms in a unit cell. Note that the phase delays

$\alpha=ka=\left(1+\frac{\Delta}{\omega_{\mathrm{a}}}\right)\alpha_{0},~{}~{}\beta=kL=\left(1+\frac{\Delta}{\omega_{\mathrm{a}}}\right)\beta_{0}$
(6)

is detuning dependent, where
$\alpha_{0}=\omega_{\mathrm{a}}a/{v_{\mathrm{g}}}$ and
$\beta_{0}=\omega_{\mathrm{a}}L/{v_{\mathrm{g}}}$ are detuning-independent
phase delays defined in terms of the frequency of the resonant photons.

Using the above definitions and substituting Eqs. (5a) and (5b) into Eq. (4c),
we have

$\displaystyle\left(\Delta\mathbf{I}-\mathbf{H_{c}}\right)\tilde{\mathbf{f}}_{m}=e^{i\beta_{m}}{\mathbf{V}}t_{m-1,N}+e^{-i\beta_{m}}{\mathbf{V}}^{*}r_{m+1,1},$
(7)

where $\mathbf{I}$ is the $N$-dimensional identity matrix. $\mathbf{H_{c}}$ is
the effective non-Hermitian Hamiltonian of each cell, with elements

$\left(\mathcal{H}_{\mathrm{c}}\right)_{ss^{\prime}}=J_{s}\delta_{s,s^{\prime}-1}+J_{s^{\prime}}\delta_{s-1,s^{\prime}}-i\frac{\Gamma}{2}e^{i\left|\phi_{s}-\phi_{s^{\prime}}\right|}.$
(8)

$\mathbf{V}$ and $\tilde{\mathbf{f}}_{m}$ take the form

$\mathbf{V}=\sqrt{\frac{\Gamma}{2}}\left(e^{i\phi_{1}},e^{i\phi_{2}},\cdots,e^{i\phi_{N}}\right)^{\top},$
(9a)
$\tilde{\mathbf{f}}_{m}=\begin{pmatrix}\tilde{f}_{m,1},&\tilde{f}_{m,2},&\cdots,&\tilde{f}_{m,N}\end{pmatrix}^{\top}.$
(9b)

According to Eqs (5a) and (5b), we can read the relations satisfied by the
transmission (reflection) amplitudes of neighboring cells:

$t_{m,N}=t_{m-1,N}-ie^{-i\beta_{m}}{\mathbf{V}}^{\dagger}\tilde{\mathbf{f}}_{m},$
(10a)
$r_{m,1}=r_{m+1,1}-ie^{i\beta_{m}}{\mathbf{V}}^{\top}\tilde{\mathbf{f}}_{m}.$
(10b)

On the other hand, Eq. (7) allows us to express the vector
$\tilde{\mathbf{f}}_{m}$, which is composed of the excitation amplitudes of
the atoms in the $m$th cell, in terms of the corresponding input amplitudes
$t_{m-1,N},r_{m+1,1}$. Substituting this result into Eqs. (10a) and (10b), one
can obtain the connection relation between the input and the output amplitudes
of the $m$th unit cell

$\binom{t_{m,N}}{r_{m,1}}=\mathscr{B}^{-1}_{m}\mathcal{S}\mathscr{B}_{m}\binom{t_{m-1,N}}{r_{m+1,1}},$
(11)

where $\mathscr{B}_{m}=\mathrm{diag}(e^{i\beta_{m}},e^{-i\beta_{m}})$ and the
scattering matrix $\mathcal{S}$ takes the form

$\mathcal{S}=\begin{pmatrix}{1-i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathbf{c}})^{-1}\mathbf{V}}&{-i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathbf{c}})^{-1}\mathbf{V}^{*}}\\\
{-i\mathbf{V}^{\top}(\Delta\mathbf{I}-\mathbf{H}_{\mathbf{c}})^{-1}\mathbf{V}}&{1-i\mathbf{V}^{\top}(\Delta\mathbf{I}-\mathbf{H}_{\mathbf{c}})^{-1}\mathbf{V}^{*}}\end{pmatrix}.$
(12)

Alternatively, the connection relation can be expressed as

$\binom{t_{m-1,N}}{r_{m,1}}=\mathscr{B}^{-1}_{m}\mathcal{T}\mathscr{B}_{m}\binom{t_{m,N}}{r_{m+1,1}}.$
(13)

Here

$\mathcal{T}=\begin{pmatrix}\frac{1}{1-i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V}}&\frac{i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V}^{*}}{1-i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V}}\\\
\frac{-i\mathbf{V}^{\top}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V}}{1-i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V}}&\frac{\mathrm{Det}[\mathcal{S}]}{1-i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V}}\end{pmatrix}$
(14)

represents the transfer matrix, which establishes a relationship between the
light field on either side of the $m$th unit cell. As indicated by Eq. (13),
the connection relation for a single-polymer setup (i.e., the case of $M=1$)
with the reference phase set to zero acquires the following form

$\left(\begin{array}[]{c}1\\\
\tilde{r}\end{array}\right)=\mathcal{T}\left(\begin{array}[]{c}\tilde{t}\\\
0\end{array}\right),$ (15)

where $\tilde{t}$ and $\tilde{r}$ are the corresponding transmission and
reflection amplitudes. Furthermore, by inserting Eq. (14) into Eq. (15), it
can be observed that

$\tilde{t}=1-i\mathbf{V}^{\dagger}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V},$
(16a)
$\tilde{r}=-i\mathbf{V}^{\top}(\Delta\mathbf{I}-\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{V}.$
(16b)

In addition, the atomic chain under consideration satisfies time reversal
symmetry. Consequently, the transfer matrix can also be expressed in terms of
$\tilde{t}$ and $\tilde{r}$ as follows Xu  _et al._ (2000)

$\mathcal{T}=\begin{pmatrix}\frac{1}{\tilde{t}}&\frac{\tilde{r}^{*}}{\tilde{t}^{*}}\\\
\frac{\tilde{r}}{\tilde{t}}&\frac{1}{\tilde{t}^{*}}\end{pmatrix}.$ (17)

Using Eq. (13) iteratively $M$ times in succession and noticing the conditions
$t_{0,N}=1$, $r_{1,1}=r$, $t_{M,N}=t$, $r_{M+1,1}=0$, and $\beta_{1}=0$, we
finally obtain the following connection relation between the reflection and
transmission amplitudes for the entire atomic chain:

$\left(\begin{array}[]{c}1\\\
r\end{array}\right)=\tilde{\mathcal{T}}^{M}\left(\begin{array}[]{c}te^{iM\beta}\\\
0\end{array}\right),$ (18)

with

$\tilde{\mathcal{T}}=\mathcal{T}\mathcal{T}_{\beta},~{}~{}\mathcal{T}_{\beta}=\begin{pmatrix}e^{-i\beta}&0\\\
0&e^{i\beta}\end{pmatrix}.$ (19)

Thus the transmission and reflection amplitudes for the polymer chain read

$t=\frac{e^{-iM\beta}}{[\tilde{\mathcal{T}}^{M}]_{11}},~{}~{}r=\frac{[\tilde{\mathcal{T}}^{M}]_{21}}{[\tilde{\mathcal{T}}^{M}]_{11}},$
(20)

Notice that the determinant of the matrix $\tilde{\mathcal{T}}$ is $1$. Based
on Abeles’s theorem Abelès, Florin (1950), we can write the matrix
$\tilde{\mathcal{T}}^{M}$ in terms of Chebyshev polynomials of the second
kind:

$\tilde{\mathcal{T}}^{M}=U_{M-1}(y)\tilde{\mathcal{T}}-U_{M-2}(y)\mathcal{I},$
(21)

with

$y=\frac{1}{2}\mathrm{Tr}\left(\tilde{\mathcal{T}}\right).$ (22)

$\mathcal{I}$ is the two-dimensional identity matrix. According to Eqs. (20)
and (21), one can obtain the transmittance and the reflectance of the polymer
chain (see Appendix A for details)

$T=\left|t\right|^{2}=\frac{1}{1+\zeta U_{M-1}^{2}(y)},$ (23a)
$R=\left|r\right|^{2}=\frac{\zeta U_{M-1}^{2}(y)}{1+\zeta U_{M-1}^{2}(y)},$
(23b)

where
$\zeta\equiv|\tilde{\mathcal{T}}_{12}|^{2}=|\tilde{\mathcal{T}}_{21}|^{2}$. It
should be pointed out that the relation $T+R=1$ is satisfied as a consequence
of the conservation of photon number. Therefore, in what follows, we will
focus on the reflectance $R$ alone when analyzing the band-gap structures.

Moreover, we will focus on the regime where the phase-accumulated effects for
detuned photons can be ignored, i.e., one can approximate the detuning-
dependent phase delays defined in Eq. (6) as $\alpha\simeq\alpha_{0}$ and
$\beta\simeq\beta_{0}$. Or equivalently, we can say that the Markovian
approximation can be performed, i.e., the time delays between the coupling
points can be neglected. In Appendix B, we present the Markovian conditions
for our system, which are easily satisfied under typical parameters of wQED
experiments Sheremet  _et al._ (2023). In what follows, we will use this
approximation in our analysis to keep the physics transparent, while in the
full numerical calculations we still make the phase delay $\alpha$ and $\beta$
depend on the detuning in order to obtain accurate results.

Figure 2: Reflection spectra for a dimer chain with (a) $M=1$, (b) $M=3$ and
(c) $M=10$. The band gap is represented by the shaded region. Other parameters
are set as $\alpha_{0}=\pi/2$, $\beta_{0}=\pi$, $J=2\Gamma$, and
$\omega_{\mathrm{a}}=10^{4}\Gamma$. Figure 3: (a) The reflection coefficient
$R$ and $y$ as functions of $\Delta$ for an atom-dimer chain with $6$ cells.
The blue dashed lines are used to mark $y=\pm 1$. The parameters are set as
$\alpha_{0}=\pi/2$, $\beta_{0}=\pi$, $J=\Gamma/2$, and
$\omega_{\mathrm{a}}=10^{4}\Gamma$. (b) The dispersion curve for a dimer chain
with the same parameters used in panel (a). In panels (a) and (b), the band
gap of width $W$ is represented by the shaded region (with $|y|>1$). (c) The
curves show the decomposed components contributed by the collective
excitations corresponding to the spectra shown in panel (a). $w$ represents
the distance between the two narrowest peaks, which are located at the
innermost and correspond to the two most subradiant states. (d), (e) Details
of the curves in the blue dashed boxes in panel (c).
$\tilde{\Gamma}_{\mathrm{ms}}$ denotes the width of the narrowest peaks. (f)
The blue circles illustrate the width $\tilde{\Gamma}_{\mathrm{ms}}$ as a
function of $M$. (g) The red squares show the distance $w$ as a function of
$M$. As the value of $M$ increases, the value of $w$ approaches $W$. Other
parameters used in panels (f) and (g) are the same as those used in panel (a).

##  III Formation of band-gap structure

In this section, we present some general discussions on the band-gap structure
produced by a polymerized atomic chain. The existence of band gaps is a common
feature of periodic structures. For our system, it is not surprising that
band-gap structures can form as the number of cells $M$ increases. To
illustrate this point, we present in Figs. 2(a)-2(c) the reflection spectra
for an atom-dimer array for varying values of $M$. As $M$ increases, the
spectrum with two total reflection points for small $M$ evolves into a band-
gap structure for relatively large $M$. Moreover, it can be observed from Eq.
(23b) that the solutions to the equation for $\Delta$

$y(\Delta)=\cos\left(\frac{l\pi}{M}\right)\equiv y_{l}$ (24)

are the zero points of the function $R(\Delta)$, since $y_{l}$
($l=1,2,\cdots,M-1$) is a root of the the Chebyshev polynomial of the second
kind $U_{M-1}(y)$ Greenberg  _et al._ (2021); Peng and Jia (2023).
Importantly, these zero reflection points outside the band gap indicate the
frequencies of the propagating modes that are permitted by the passbands.

In order to gain insight into the formation of the band gap, we consider a
periodic chain comprising an adequate number of cells and present the
dispersion relation based on Bloch’s theorem (see Appendix C for details):

$\cos\left(qL\right)=y\left(\Delta\right).$ (25)

Here $q$ represents the Bloch wave number, which is used to label the
propagating modes that are permitted by the wQED system with atomic chain.
Clearly, Eq. (25) tells us that for frequencies satisfying the inequality

$|y(\Delta)|>1,$ (26)

no propagating mode exists, resulting in band-gap structures with $R=1$ in the
reflection spectrum, as shown by the shaded region in Fig. 3(a). The
corresponding dispersion relation, which can only be defined in the pass
bands, is shown in Fig. 3(b).

On the other hand, the formation of a spectrum with band-gap structures can be
understood in terms of interference effects between different scattering
channels corresponding to the excitations of the collective modes. To this
end, we decompose the reflection spectrum into the superpositions of several
Lorentzian-type amplitudes contributed by the collective excitations (see
Appendix D for details):

$r=\sum_{j=1}^{N^{\prime}}\mathcal{L}_{j},~{}~{}\mathcal{L}_{j}=\frac{d_{j}}{\Delta-\tilde{\Delta}_{j}+i\frac{\tilde{\Gamma}_{j}}{2}}.$
(27)

Here $N^{\prime}=MN$ is the total number of atoms, $\tilde{\Delta}_{j}$
represents the detuning between the $j$th collective mode and the atoms,
$\tilde{\Gamma}_{j}$ denotes the effective decay of the $j$th collective mode,
and $d_{j}$ determines the weight of each Lorentzian component. In Figs.
3(c)-3(e), we illustrate this decomposition by taking the case of a dimer
array (with $\alpha_{0}=\pi/2,~{}\beta_{0}=\pi$) as an example. It can can be
seen that the Lorentzian-type amplitudes $\mathcal{L}_{j}$ are symmetrically
distributed around $\Delta=0$. The band-gap region with near total reflection
is mainly attributed to the excitation of the most superradiant states [see
Fig. 3(c)]. While the two narrowest excitation amplitudes corresponding to the
most subradiant states are situated at the innermost [see Figs. 3(d) and
3(e)], and the corresponding width, denoted as $\tilde{\Gamma}_{\mathrm{ms}}$,
decreases sharply with increasing number of cells $M$ [see Fig. 3(f)]. The
Fano-type destructive interferences between these two most subradiant
excitations and the superradiant ones result in the formation of a pair of
innermost reflection minima and the corresponding steep band-gap walls [see
Figs. 3(c)-3(e)]. This phenomenon becomes more remarkable as the parameter $M$
increases. Consequently, as the value of $M$ increases to approximately $M>5$,
the distance $w$ between the centers of the two innermost amplitudes
approaches the width of the band gap $W$ [which is obtained from the condition
(26)], as shown in Fig. 3(g). This result indicates that the condition (26)
can even be used to calculate the gap width of a chain containing a relatively
small number of cells.

##  IV Tunable photonic band gap with a dimer chain

Let us first consider the simplest case of $N=2$, namely a dimer array coupled
to a waveguide, as shown in Fig. 1(a). As demonstrated in the preceding
sections, once the expressions for the quantities $\zeta$ and $y$ [from which
we can determine the scattering coefficients (23a) and (23b), as well as the
dispersion relation (25)] have been derived, a more comprehensive examination
of the scattering spectrum and the band-gap structure is possible.
Specifically, in the Markovian regime with $\alpha\simeq\alpha_{0}$ and
$\beta\simeq\beta_{0}$, the quantities $\zeta$ and $y$ for a dimer array take
the form

$\zeta=\frac{\Gamma^{2}\left(2\Delta\cos\alpha_{0}+2J+\Gamma\sin\alpha_{0}\right)^{2}}{4\left[\Delta^{2}-J(J+\Gamma\sin\alpha_{0})\right]^{2}},$
(28a) $\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{2}\Gamma^{2}\sin(\beta_{0}-\alpha_{0})\sin\alpha_{0}+\Gamma(J\cos\alpha_{0}+\Delta)\sin\beta_{0}}{\Delta^{2}-J(J+\Gamma\sin\alpha_{0})}$
(28b) $\displaystyle+\cos\beta_{0},$

which are strongly influenced by the intra-cell phase difference $\alpha_{0}$
and the inter-cell phase differences $\beta_{0}$. Here we focus on the cases
that the intra- and inter-cell phase delays satisfy either the Bragg or the
anti-Bragg condition. The following analysis will demonstrate that, with the
exception of the case where both the intra- and inter-cell phase delays
satisfy the Bragg condition (see Sec. IV.1), spectra with highly tunable band-
gap structures can be generated in the remaining cases by varying the intra-
cell coupling strength $J$ (see Secs. IV.2-IV.4).

Figure 4: Reflection spectra for a dimer chain with (a) $\alpha_{0}=\pi$ and
(b) $\alpha_{0}=2\pi$. Other parameters are set as $\beta_{0}=4\pi$,
$J=2\Gamma$, and $\omega_{\mathrm{a}}=10^{5}\Gamma$.

###  IV.1 $\alpha_{0}=n\pi$ and $\beta_{0}=m\pi$: spectrum with Lorentzian
line shape

We begin with the assumption that both the intra- and inter-cell phase
differences satisfy the Bragg condition, with $\alpha_{0}=n\pi$
($n\in\mathbb{N}^{+}$) and $\beta_{0}=m\pi$ ($m\in\mathbb{N}^{+}$, $m>n$).
Accordingly, Eqs. (28a) and (28b) can be simplified as

$\zeta=\frac{\Gamma^{2}}{\left[\Delta-(-1)^{n}J\right]^{2}},$ (29a)
$y=(-1)^{m}.$ (29b)

Plugging these expressions into Eq. (23b) and using the identity
$U^{2}_{M-1}[(-1)^{m}]=M^{2}$, the reflectance can be further written as

$R=\frac{M^{2}\Gamma^{2}}{\left[\Delta-(-1)^{n}J\right]^{2}+M^{2}\Gamma^{2}}.$
(30)

Obviously, similar to an uncoupled atom array (with $J=0$) under Bragg
condition, the reflection spectrum shows the well-known phenomenon of
superradiance, with a line width $2M\Gamma$ (scaled as the number of atoms
$2M$). However, the center frequency is shifted to $\Delta=(-1)^{n}J$ instead
of at $\Delta=0$. These properties are shown in Figs. 4(a) and 4(b), where we
take the case of $\alpha_{0}=\pi$ ($n=1$) and $\alpha_{0}=2\pi$ ($n=2$) as
examples.

Figure 5: (a) Reflectance for a dimer chain as a function of $\Delta$ and $J$.
The parameters are set as $M=15$, $\alpha_{0}=\pi/2$, $\beta_{0}=3\pi$, and
$\omega_{\mathrm{a}}=10^{4}\Gamma$, respectively. (b) The width $W$ of the
band gap as a function of $J$ corresponding to the case shown in panel (a).
(c) Reflectance for a dimer chain as a function of $\Delta$ and $J$ with
intra-cell phase delay $\alpha_{0}=3\pi/2$. Other parameters are the same as
those used in panel (a). (d) The width $W$ as a function of $J$ corresponding
to the case shown in panel (c). The dashed lines in panels (c) and (d) are
employed to indicate the value of $J$ corresponding to $W=0$. Figure 6:
Dispersion relations for a dimer chain for various values of $J$ in the
vicinity of $J=\Gamma/2$. The parameters are the same as those used in Fig.
5(c).

###  IV.2 $\alpha_{0}=(n-1/2)\pi$ and $\beta_{0}=m\pi$: band gap with tunable
width

We proceed to assume that $\alpha_{0}=(n-1/2)\pi$ ($n\in\mathbb{N}^{+}$) and
$\beta_{0}=m\pi$ ($m\in\mathbb{N}^{+}$, $m\geq n$), i.e., the
intra(inter)-cell phase difference satisfies the anti-Bragg (Bragg) condition.
Under these conditions, Eqs. (28a) and (28b) can be simplified as

$\zeta=\frac{\Gamma^{2}\left[2J-(-1)^{n}\Gamma\right]^{2}}{4\left\\{\Delta^{2}-J\left[J-(-1)^{n}\Gamma\right]\right\\}^{2}},$
(31a)
$y=(-1)^{m}\left(1-\frac{\frac{1}{2}\Gamma^{2}}{\Delta^{2}-J\left[J-(-1)^{n}\Gamma\right]}\right).$
(31b)

After substituting Eq. (31b) into the band-gap condition (26), we find that
except for the case of $n\in\mathbb{E}^{+},J=\Gamma/2$ (where the gap is
closed), the region of the band gap is

$|\Delta|<|J\pm\Gamma/2|.$ (32)

Namely, we can obtain a single band gap centered about $\Delta=0$, with width

$W=\left|2J\pm\Gamma\right|.$ (33)

Here “$+$” (“$-$”) corresponds to $n\in\mathbb{O}^{+}$ ($n\in\mathbb{E}^{+}$).
It should be emphasized that the value of $m$ do not influence the solution
set of the inequality (26), thus the range and width of the band gap are only
dependent on the parity of $n$. A similar conclusion can be drawn with regard
to the situations discussed in the following subsections (Sec. IV.3 and IV.4).
Importantly, the results obtained in Eqs. (32) and (33) indicate that the
width of the band gap can be manipulated by changing the coupling strength $J$
between the atoms within a cell, which is a significant advantage over a chain
of non-interacting atoms.

Without loss of generality, we will consider the cases of $\alpha_{0}=\pi/2$
($n=1$) and $\alpha_{0}=3\pi/2$ ($n=2$) as illustrative examples to
demonstrate the modulation effect of $J$ on the width of the band gap. To this
end, we plot the reflection spectra and corresponding gap width under these
phase delays for varying $J$ in Fig. 5. Specifically, when $\alpha_{0}=\pi/2$
and $\beta_{0}=3\pi$, the bandwidth increases linearly with $J$. Particularly,
if $J\gg\Gamma$, we can obtain a wide band gap with width $W\simeq 2J$ [see
Figs. 5(a) and 5(b)]. On the other hand, when $\alpha_{0}=3\pi/2$ and
$\beta_{0}=3\pi$, the bandwidth $W$ first decreases linearly, reaching a
minimum value of zero at $J=\Gamma/2$, and then increases linearly with $J$ to
a wide band gap [see Figs. 5(c) and 5(d)]. Additionally, the dispersion
relations for this case around $J=\Gamma/2$ are plotted in Figs. 6(a)-6(c). It
is shown that at $J=\Gamma/2$ the gap is closed, so the atom-coupled
waveguide, like a bare waveguide, allows photons of all frequencies [see Fig.
6(b)]. But its dispersion relation has been greatly modified compared to a
bare waveguide. In particular, around the resonance point $\Delta=0$, one can
obtain an effective linear dispersion
$|\Delta|\simeq\tilde{v}_{\mathrm{g}}|q|$, with $\tilde{v}_{\mathrm{g}}=\Gamma
L/4$ [see the dashed lines in Fig. 6(b)]. Moreover, the ratio between the
group velocities of the perturbed and the bare waveguides can be estimated as
$\tilde{v}_{\mathrm{g}}/v_{\mathrm{g}}\sim\Gamma/\omega_{\mathrm{a}}\ll 1$,
which means that under these parameters, the atom-coupled waveguide can be
considered as a linear slow-light waveguide.

###  IV.3 $\alpha_{0}=n\pi$ and $\beta_{0}=(m-1/2)\pi$: band gap with tunable
center frequency

Figure 7: Reflectance for a dimer chain as functions of $\Delta$ and $J$, with
(a) $\alpha_{0}=\pi$, (b) $\alpha_{0}=2\pi$. The red dashed lines indicate the
position of the center of the band gap. Other parameters are set as $M=15$,
$\beta_{0}=5\pi/2$, and $\omega_{a}=10^{4}\Gamma$.

We subsequently discuss the case of $\alpha_{0}=n\pi$ ($n\in\mathbb{N}^{+}$)
and $\beta_{0}=(m-1/2)\pi$ ($m\in\mathbb{N}^{+}$, $m>n$), i.e., the
intra(inter)-cell phase difference satisfies the Bragg (anti-Bragg) condition.
Under these conditions, Eqs. (28a) and (28b) can be simplified as

$\zeta=\frac{\Gamma^{2}}{\left[\Delta-(-1)^{n}J\right]^{2}},$ (34a)
$y=\frac{(-1)^{m+1}\Gamma}{\Delta-(-1)^{n}J}.$ (34b)

Similar to Sec. IV.2, with the help of the condition (26), we can find that
the region of the band gap is

$\mp J-\Gamma<\Delta<\mp J+\Gamma,$ (35)

indicating that the center of the band gap is located at $\Delta=\mp J$ (where
“$-$” corresponds to $n\in\mathbb{O}^{+}$ and “$+$” corresponds to
$n\in\mathbb{E}^{+}$), and the corresponding width is

$W=2\Gamma.$ (36)

This result shows that the center of the band gap can be tuned by changing the
intra-cell coupling strength $J$. As shown in Figs. 7(a) (with
$\alpha_{0}=\pi$ and $\beta_{0}=5\pi/2$) and 7(b) (with $\alpha_{0}=2\pi$ and
$\beta_{0}=5\pi/2$), due to the existence of the intra-cell coupling, the
center of the band gap is red-shifted or blue-shifted by $J$ in comparison
with a non-interacting atomic chain [see the red dashed lines in Figs. 7(a)
and 7(b)], while the width of the band gap remains unchanged.

Figure 8: (a) Reflectance for a dimer chain as a function of $\Delta$ and $J$.
The parameters are set as $M=15$, $\alpha_{0}=\pi/2$, $\beta_{0}=5\pi/2$, and
$\omega_{\mathrm{a}}=10^{4}\Gamma$, respectively. (b) The width $W$ of the
band gap and the width $W^{\prime}$ of the center passband as functions of $J$
corresponding to the case shown in panel (a). (c) Reflectance for a dimer
chain as a function of $\Delta$ and $J$, with intra-cell phase delay
$\alpha_{0}=3\pi/2$. Other parameters are the same as those used in panel (a).
(d) The widths $W$ and $W^{\prime}$ as functions of $J$ corresponding to the
case shown in panel (c). In panels (c) and (d), the dashed lines are employed
to indicate the value of $J$ corresponding to the disappearance of gaps,
whereas the dash-dotted lines are used to indicate the disappearance of a
center passband.

###  IV.4 $\alpha_{0}=(n-1/2)\pi$ and $\beta_{0}=(m-1/2)\pi$: tunable double
band gaps and center passband

####  IV.4.1 Tunable band-gap structures

Finally, we discuss the case of $\alpha_{0}=(n-1/2)\pi$ ($n\in\mathbb{N}^{+}$)
and $\beta_{0}=(m-1/2)\pi$ ($m\in\mathbb{N}^{+}$, $m>n$), i.e., both the
intra- and inter-cell phase differences satisfy the anti-Bragg condition. For
this case, Eqs. (28a) and (28b) can be simplified as

$\zeta=\frac{\Gamma^{2}\left[2J-(-1)^{n}\Gamma\right]^{2}}{4\left\\{\Delta^{2}-J\left[J-(-1)^{n}\Gamma\right]\right\\}^{2}},$
(37a)
$y=\frac{(-1)^{m+1}\Gamma\Delta}{\Delta^{2}-J\left[J-(-1)^{n}\Gamma\right]}.$
(37b)

For $n\in\mathbb{O}^{+}$, different from the previous cases, there are always
two regions satisfying the condition (26), with

$J<|\Delta|<J+\Gamma.$ (38)

These regions with no propagating modes will form two band gaps centered at
$\Delta=\pm(J+\Gamma/2)$ with equal width

$W=\Gamma.$ (39)

We refer to them as the right and left band gaps, respectively [see Fig.
8(a)]. And the region $-J<\Delta<J$ between the two gaps forms a passband
whose width is proportional to $J$

$W^{\prime}=2J.$ (40)

In a word, we can obtain a symmetrical spectral structure with a passband
sandwiched between two band gaps of equal width. In addition, the centers of
the gaps, and thus the width of the passband, show a linear relationship with
$J$, as shown in Fig. 8(a). To show these more clearly, the widths $W$ and
$W^{\prime}$ as functions of $J$ are given in Fig. 8(b).

If $n\in\mathbb{E}^{+}$, as in the case of $n\in\mathbb{O}^{+}$, two band-gap
regions symmetric about $\Delta=0$ appear in the reflection spectra for
different $J$, except for the cases of $J=\Gamma/2$ (where the gaps are
closed) and $J=\Gamma$ (where the two gaps merge into one). But a richer
modulation of the band-gap structure can be obtained by varying the intra-cell
coupling strength $J$ [see Fig. 8(c)]. According to the condition (26), we can
obtain the regions of the gaps, the width $W$ of each gap, and the width
$W^{\prime}$ of the passband between the gaps, as summarized in Table 1.
Correspondingly, the widths $W$ and $W^{\prime}$ as functions of $J$ are
plotted in Fig. 8(d). Specifically, in the region $0<J<\Gamma$, the two band
gaps are centered at $\Delta=\pm\Gamma/2$ and the width of each gap first
decreases linearly (with $W=\Gamma-2J$) to zero at $J=\Gamma/2$ [where the
gaps vanish, see the dashed lines in Figs. 8(c) and 8(d)] and then increases
linearly (with $W=2J-\Gamma$). Finally the two gaps merge into a single band
of width $2\Gamma$ at $\Delta=\Gamma$ [see the dash-dotted lines in Figs. 8(c)
and 8(d)]. Accordingly, the width of the center passband first increases
linearly (with $W^{\prime}=2J$) to a local maximum when $J\to\Gamma/2$ (note
that at $J=\Gamma/2$, the center passband cannot be defined because the gaps
are closed) and then decreases linearly [with $W^{\prime}=2(\Gamma-J)$] to
zero at $\Delta=\Gamma$. In the region $J>\Gamma$, the gap splits again into
two, centered at $\Delta=\pm(J-\Gamma/2)$ and of equal width $J$. And the
width of the corresponding center passband increases linearly [with
$W^{\prime}=2(J-\Gamma)$].

Table 1: Summary of the regions of the gaps, the width $W$ of the two gaps, and the width $W^{\prime}$ of the center passband between the gaps for different ranges of $J$ for the case of $n\in\mathbb{E}^{+}$ discussed in Sec. IV.4. | Range of $J$ | Gap regions | $W$ | $W^{\prime}$
---|---|---|---|---
| $0<J<\Gamma/2$ | $J<|\Delta|<\Gamma-J$ | $\Gamma-2J$ | $2J$
| $\Gamma/2<J<\Gamma$ | $\Gamma-J<|\Delta|<J$ | $2J-\Gamma$ | $2(\Gamma-J)$
| $J>\Gamma$ | $J-\Gamma<|\Delta|<J$ | $\Gamma$ | $2(J-\Gamma)$

####  IV.4.2 Center passband engineering

Figure 9: The reflection spectrum and the function $y(\Delta)$ in the region
of the central passband. The intra-cell coupling strength is set as
$J=3\Gamma/2$, and the remaining parameters are the same as those utilized in
Fig. 8(c). The horizontal dashed lines are given by $y=y_{l}=\cos(l\pi/M)$
(from top to bottom: $l=1,2,\cdots,M-1$), and the values of $\Delta$ indicated
by the vertical dashed lines are the solutions to the equation
$y(\Delta)=y_{l}$, which is consistent with the points of $R=0$. These
solutions correspond to the frequencies of the propagating modes in the
passband. The band gaps are represented by the shaded region (with
$|y(\Delta)|>1$). Figure 10: Dispersion relations for a dimer chain for
various values of $J$ in the vicinity of $J=\Gamma$. The parameters are the
same as those used in Fig. 8(c).

Compared to the previous subsections, the most significant feature of the
situation now discussed is the possibility of realizing a highly engineered
center passband. As we have discussed earlier, its width is easily tunable by
the intra-cell coupling strength $J$. Here we will further discuss its other
properties. As discussed in Sec. III, the solutions to Eq. (24) can give the
zero reflection points in the reflection spectrum. For present case, as shown
in Fig. 9, the center branch of $y(\Delta)$ can be used to fix the $M-1$ zero
reflection points within the passband, which correspond to the propagating
modes permitted by this band. According to Eqs. (24) and (25), one can find
that the wave numbers of these propagating modes are $q=\pm l\pi/(ML)$
($l=1,2,\cdots,M-1$).

If the frequencies of photons are concentrated within the center passband, the
atom-perturbed system can be regarded as a nonlinear waveguide with finite
bandwidth, with the group velocity being considerably less than that of the
propagating photons in a bare waveguide. By changing $J$, the band width and
thus the dispersion relation of the center passband can also be easily
engineered. To illustrate, we consider the case of $n\in\mathbb{E}^{+}$ and
start with a coupling strength that falls within the range of
$\Gamma/2<J<\Gamma$ [see Fig. 10(a)]. Subsequently, the value of $J$ is
increased until it reaches $J\simeq\Gamma$, during which the bandwidth of the
center passband is compressed and ultimately reduces to zero. Consequently,
the center passband becomes a flat band, as illustrated in Fig. 10(b). It is
evident that, in this situation, the density of the traveling modes becomes
exceedingly high, while the group velocity for photons in this band is close
to zero. As a result, it is possible to slow or even stop a single-photon
pulse at will in this setup by dynamically controlling $J$. If the coupling
strength $J$ is precisely equal to $\Gamma$, the center passband will
disappear, as the two gaps will merge into a single gap of width $2\Gamma$. As
the value of $J$ is increased to satisfy $J>\Gamma$, the group velocity of the
center passband undergoes a change in sign, as illustrated in Fig. 10(c). This
indicates that the sign of the group velocity (i.e., the direction of wave-
packet propagation) for the passband can also be controlled by selecting an
appropriate $J$. We note that similar modulations can also be achieved in a
wQED structure containing a resonator array Yanik  _et al._ (2004) or a chain
of superconducting qubits Shen  _et al._ (2007) through the appropriate
adjustment of the detuning between the emitters in a cell.

Similar to the second case discussed in Sec. IV.2, when $n\in\mathbb{E}^{+}$
and $J=\Gamma/2$ [where the gaps are closed, indicated by the dashed lines in
Figs. 8(c) and 8(d)], the atom-coupled waveguide can be utilized as a linear
waveguide when the modes in the vicinity of $\Delta=\pm\Gamma/2$ are of
interest. When $\alpha_{0}=3\pi/2$, $\beta_{0}=5\pi/2$ [the parameters used in
Fig. 8(c)], the corresponding dispersion relations in the linear regions can
be approximated as $|\Delta-\Gamma/2|=\tilde{v}_{\mathrm{g}}|q|$ and
$|\Delta+\Gamma/2|=\tilde{v}_{\mathrm{g}}|q\pm\pi/L|$, respectively. Here
$\tilde{v}_{\mathrm{g}}=\Gamma L/2$ is the group velocity at
$\Delta=\pm\Gamma/2$ of the perturbed waveguide. We also have
$\tilde{v}_{\mathrm{g}}/v_{\mathrm{g}}\sim\Gamma/\omega_{\mathrm{a}}\ll 1$,
which indicates the slow-light effect.

Figure 11: (a) Reflectance for a tetramer chain as a function of $\Delta$ and
$J$ for fixed $\eta=5$. (b) The details in the box in panel (a). (c) The
widths $W_{c}$, $W_{s}$, and $W_{p}$ as functions of $J$ for fixed $\eta=5$.
(d) Reflectance for a tetramer chain as a function of $\Delta$ and $\eta$ for
fixed $J=\Gamma/2$. (e) The details in the box in panel (d). (f) The widths
$W_{c}$, $W_{s}$, and $W_{p}$ as functions of $\eta$ for fixed $J=\Gamma/2$.
The gray dashed lines in (a)-(f) are used to mark the positions of
$f(\tilde{J},\eta)=0$, at which the three gaps converge to form a single one.
Other parameters are set as $M=15$, $\alpha_{0}=\pi/2$, $\beta_{0}=2\pi$ and
$\omega_{\mathrm{a}}=10^{4}\Gamma$.

##  V Tunable photonic band gaps with a polymer chain

The discussions in last section can be generalized to the case of a polymer
chain, wherein each unit cell contains a greater number of atoms than two.
Naturally, as the number of atoms in a cell increases, multi-band-gap
structure can be realized. Furthermore, the design of intra-cell interactions
enables the implementation of more sophisticated band-gap engineering. In this
section, we utilize the case of a tetramer ($N=4$) array coupled to a
waveguide, as shown schematically in Fig. 1(b), as an example to discuss these
properties. The coupling strengths between neighboring atoms in a cell are set
as $J_{1}=J_{3}=J$ and $J_{2}=\eta J$, where the dimensionless parameter
$\eta$ is introduced to enhance the tunability of the system. The phase
factors are set as $\alpha_{0}=\pi/2$ and $\beta_{0}=2\pi$. Correspondingly,
the function $y(\Delta)$ under the Markovian approximation takes the form

$y(\Delta)=\frac{\Delta^{4}-\mathcal{A}\Delta^{2}+\mathcal{B}}{\Delta^{4}-\mathcal{C}\Delta^{2}+\mathcal{D}},$
(41)

where

$\displaystyle\mathcal{A}=2\Gamma^{2}+\left(\eta+2\right)\Gamma
J+\left(\eta^{2}+2\right)J^{2},$
$\displaystyle\mathcal{B}=\left(\Gamma^{2}+J^{2}\right)\left[\left(\eta+2\right)\Gamma
J+J^{2}\right]+\frac{1}{2}\Gamma^{2}\left[\Gamma^{2}+\left(\eta+2\right)^{2}J^{2}\right],$
$\displaystyle\mathcal{C}=\mathcal{A}-2\Gamma^{2},~{}~{}\mathcal{D}=\left[\left(\Gamma+J\right)^{2}+\eta\Gamma
J\right]J^{2}.$

Substituting Eq. (41) into the band-gap condition (26), we can obtain the
band-gap regions for different values of $J$ and $\eta$. To classify the
solutions to the inequality (26), one can define the following discriminant
function

$f\left(\tilde{J},\eta\right)=3\tilde{J}^{2}\eta^{2}+2\left(4\tilde{J}^{2}+\tilde{J}\right)\eta-1,$
(42)

with $\tilde{J}=J/\Gamma$. For $f(\tilde{J},\eta)\neq 0$, the reflection
spectra indicate that a triple-band-gap structure, comprising a center gap and
two side gaps of equal width, is distributed symmetrically about $\Delta=0$,
as shown in Figs. 11(a) (for fixed $\eta=5$ and varying $J$) and 11(d) (for
fixed $J=\Gamma/2$ and varying $\eta$). The details of these spectra around
the regions satisfying $f(\tilde{J},\eta)=0$ are provided in Figs. 11(b) and
11(e). Furthermore, both the centers and the widths of the two side gaps and
the two passbands between the gaps are highly tunable by varying the values of
$J$ and $\eta$. The regions of the gaps, the width $W_{c}$ of the center gap,
the width $W_{s}$ of the two side gaps, and the width $W_{p}$ of the two
passbands between the gaps are summarized in Table 2. The quantities $\delta$
and $\tilde{\delta}_{1,2}$ in the table take the form

$\delta=\frac{1}{2}\left[\Gamma+\left(\eta+2\right)J\right],$ (43a)
$\tilde{\delta}_{1,2}=\frac{1}{2}\left[\pm\eta
J+\sqrt{\left(\Gamma+2J\right)^{2}+\left(\Gamma+\eta J\right)^{2}}\right].$
(43b)

It can be demonstrated that the relations
$0<\tilde{\delta}_{2}<\tilde{\delta}_{1}$, $0<\delta<\tilde{\delta}_{1}$, and

$\left.\delta-\tilde{\delta}_{2}\left\\{\begin{array}[]{ll}>0,&f\big{(}\tilde{J},\eta\big{)}>0\\\\[8.61108pt]
=0,&f\big{(}\tilde{J},\eta\big{)}=0\\\\[8.61108pt]
<0,&f\big{(}\tilde{J},\eta\big{)}<0\end{array}\right.\right.$

are satisfied. In accordance with the reflection spectra illustrated in Figs.
11(a) and 11(d), the widths $W_{c}$, $W_{s}$, and $W_{p}$ are plotted as
functions of $J$ (for fixed $\eta$) and of $\eta$ (for fixed $J$) in Figs.
11(c) and 11(f), respectively. These results show that a tetramer chain can
provide greater flexibility in band-gap engineering.

Table 2: Summary of the regions of the gaps, the width $W_{c}$ of the center gap, the width $W_{s}$ of the two side gaps, and the width $W_{p}$ of the two passbands between the gaps for the setup discussed in Sec. V under condition $f\big{(}\tilde{J},\eta\big{)}<0$ or $f\big{(}\tilde{J},\eta\big{)}>0$. | Condition | Gap regions | $W_{c}$ | $W_{s}$ | $W_{p}$
---|---|---|---|---|---
| $f\big{(}\tilde{J},\eta\big{)}<0$ | $|\Delta|<\delta\cup\tilde{\delta}_{2}<|\Delta|<\tilde{\delta}_{1}$ | $2\delta$ | $\tilde{\delta}_{1}-\tilde{\delta}_{2}$ | $\tilde{\delta}_{2}-\delta$
| $f\big{(}\tilde{J},\eta\big{)}>0$ | $|\Delta|<\tilde{\delta}_{2}\cup\delta<|\Delta|<\tilde{\delta}_{1}$ | $2\tilde{\delta}_{2}$ | $\tilde{\delta}_{1}-\delta$ | $\delta-\tilde{\delta}_{2}$

In the case of $f(\tilde{J},\eta)=0$, the three band gaps merge into a single
one centered about $\Delta=0$, with width $2\tilde{\delta}_{1}$, as
illustrated by the dashed lines in Figs. 11(a)-11(f). Importantly, if we
select an appropriate path in the parameter space spanned by $(\eta,J)$ [i.e.,
from the area satisfying $|f(\tilde{J},\eta)|>0$ to a point on the curve
$f(\tilde{J},\eta)=0$], the two passbands between the gaps can be easily
compressed and ultimately becomes a flat band when $f(\tilde{J},\eta)\to 0$.
This property is illustrated in Figs. 12(a)-12(c), in which $\eta$ is held
constant while $J$ is allowed to vary. Furthermore, the values of $\eta$ and
$J$ employed in these figures have been selected to satisfy the conditions
$f(\tilde{J},\eta)<0$, $f(\tilde{J},\eta)\simeq 0$, and $f(\tilde{J},\eta)>0$,
respectively. Thus, like the case discussed in Sec. IV.4.2, it is possible to
slow or even stop a single-photon pulse by modulating the relevant parameters.
However, this time we are able to control the propagation of photon pulses in
two distinct passbands simultaneously. Finally, it can be seen from Figs.
12(a) and 12(c) that the group velocities of the two passbands between the
gaps can also be designed by selecting suitable values of the parameters
$\eta$ and $J$.

Figure 12: Dispersion relations for a tetramer chain, with fixed $\eta=5$ and
varying $J$: (a) $J=0.02\Gamma$ [satisfy $f(\tilde{J},\eta)<0$], (b)
$J=0.059\Gamma$ [satisfy $f(\tilde{J},\eta)\simeq 0$] and (c) $J=0.1\Gamma$
[satisfy $f(\tilde{J},\eta)>0$]. Other parameters are set as
$\alpha_{0}=\pi/2$, $\beta_{0}=2\pi$, and $\omega_{\mathrm{a}}=10^{4}\Gamma$.

##  VI Conclusions

In this paper, we have analyzed the generation and engineering of photonic
band gaps in wQED systems containing multiple atom-polymers. We first consider
the configuration of a dimer array coupled to a waveguide with the intra- and
inter-cell phase delays satisfying either the Bragg or anti-Bragg condition.
It was found that, with the exception of both the intra- and inter-cell phase
delays satisfying the Bragg condition, the photonic band-gap structures can be
observed in the spectra for the remaining three cases. More importantly, the
band-gap structures in these cases can be manipulated in a variety of ways
(e.g., the center and the width of the gaps, as well as the dispersion
relation of the passbands) by adjusting the intra-cell coupling strength.
Furthermore, modulation of the dispersion relation and bandwidth provides an
effective way to manipulate the propagating modes in the waveguide, leading to
some interesting phenomena such as slowing or even stopping a single-photon
pulse. Finally, the results for a dimer array can be generalized to the case
of a greater number of atoms in each unit cell. This enables the realization
of a tunable multi-gap structure and the implementation of more sophisticated
band-gap engineering.

In our proposal, the key point is the use of tunable couplings between atoms
to realize band-gap engineering. This can be easily achieved in micro- and
nano-quantum systems such as superconducting quantum circuits Gu  _et al._
(2017), where superconducting qubits or superconducting quantum interference
devices (SQUIDs) are usually used as tunable coupling elements Hime  _et al._
(2006); Niskanen  _et al._ (2007); Baust  _et al._ (2015); Zhu and Jia (2019).
Thus, in summary, we present an experimentally feasible proposal to produce
tunable band-gap structures in 1D wQED setups, which may provide novel ways to
control single-photon propagation. Our proposal may have potential
applications in long-distance quantum networking and quantum communication.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(NSFC) under Grants No. 61871333.

##  Appendix A  Derivation of Eqs. (23a) and (23b)

According to Eq. (21), one can obtain

$\left[\tilde{\mathcal{T}}^{M}\right]_{11}=\tilde{\mathcal{T}}_{11}U_{M-1}(y)-U_{M-2}(y),$
(44a)
$\left[\tilde{\mathcal{T}}^{M}\right]_{21}=\tilde{\mathcal{T}}_{21}U_{M-1}(y).$
(44b)

Substituting these results into Eq. (20) and taking the square of the modulus
yields the transmission and reflection coefficients

$T=|t|^{2}=\frac{1}{\left|\tilde{\mathcal{T}}_{11}U_{M-1}(y)-U_{M-2}(y)\right|^{2}},$
(45a)
$R=|r|^{2}=\frac{\left|\tilde{\mathcal{T}}_{21}\right|^{2}U_{M-1}^{2}(y)}{\left|\tilde{\mathcal{T}}_{11}U_{M-1}(y)-U_{M-2}(y)\right|^{2}}.$
(45b)

Note that the elements of the transfer matrix $\tilde{\mathcal{T}}$ satisfy
the relation $\mathrm{Det}(\tilde{\mathcal{T}})=1$,
$\tilde{\mathcal{T}}_{11}^{*}=\tilde{\mathcal{T}}_{22}$ and
$\tilde{\mathcal{T}}_{12}^{*}=\tilde{\mathcal{T}}_{21}$ [see Eq. (17)]. Thus
we have $|\tilde{\mathcal{T}}_{11}|^{2}-|\tilde{\mathcal{T}}_{21}|^{2}=1$ and
$\tilde{\mathcal{T}}_{11}+\tilde{\mathcal{T}}_{11}^{*}=\tilde{\mathcal{T}}_{11}+\tilde{\mathcal{T}}_{22}=\mathrm{Tr}(\tilde{\mathcal{T}})=2y$.
Using these results and the identity

$U^{2}_{M-1}(y)+U^{2}_{M-2}(y)-2yU_{M-1}(y)U_{M-2}(y)=1,$ (46)

the dominator of $T$ and $R$ can be further simplified as

$\displaystyle\left|\tilde{\mathcal{T}}_{11}U_{M-1}(y)-U_{M-2}(y)\right|^{2}$
(47) $\displaystyle=$
$\displaystyle\left|\tilde{\mathcal{T}}_{11}\right|^{2}U^{2}_{M-1}(y)+U^{2}_{M-2}(y)-(\tilde{\mathcal{T}}_{11}+\tilde{\mathcal{T}}_{11}^{*})U_{M-1}(y)U_{M-2}(y)$
$\displaystyle=$
$\displaystyle\left(1+\left|\tilde{\mathcal{T}}_{21}\right|^{2}\right)U^{2}_{M-1}(y)+U^{2}_{M-2}(y)-2yU_{M-1}(y)U_{M-2}(y)$
$\displaystyle=$ $\displaystyle
1+\left|\tilde{\mathcal{T}}_{21}\right|^{2}U^{2}_{M-1}(y).$

Finally, by defining the quantity
$\zeta=|\tilde{\mathcal{T}}_{21}|^{2}=|\tilde{\mathcal{T}}_{12}|^{2}$, one can
obtain the transmittance and reflectance described by Eqs. (23a) and (23b).

##  Appendix B The Markovian conditions

In this paper, we focus on the regime in which the detuning-dependent phase
delays defined in Eq. (6) can be approximated as $\alpha\simeq\alpha_{0}$ and
$\beta\simeq\beta_{0}$. Here we will present the condition that allows us to
make this approximation. The accumulated phase factor between two ends of the
atomic chain under consideration in this paper is
$(M-1)\beta=(1+\Delta/\omega_{\mathrm{a}})(M-1)\beta_{0}$. Obviously, for a
not so small $M$, if the term caused by the detuning effects
$(M-1)\beta_{0}\Delta/\omega_{\mathrm{a}}\simeq
M\beta_{0}\Delta/\omega_{\mathrm{a}}\ll 1$, it can be safely neglected. Note
that for the band-gap cases (the cases in Secs. IV.2-IV.4 and V), the detuning
range we are interested in is approximately $\Delta\sim\Gamma$, and the order
of magnitude of $\beta_{0}$ we care about is $\beta_{0}\sim 1$, so we end up
with

$M\ll\frac{\omega_{\mathrm{a}}}{\Gamma},$ (48)

which is easily satisfied under typical parameters for wQED systems Sheremet
_et al._ (2023). And for the case discussed in Sec. IV.1), we have $\Delta\sim
M\Gamma$, the Markovian condition becomes

$M\ll\sqrt{\frac{\omega_{\mathrm{a}}}{\Gamma}}.$ (49)

Furthermore, under these conditions the detuning-dependent term corresponding
to any other pair of coupling points can also be ignored, since the distance
between them is smaller than the length of the atomic chain. In a word, once
the condition (48) (valid for the cases discussed in Secs. IV.2-IV.4 and V) or
(49) (valid for the case discussed in Sec. IV.1) is satisfied, we can safely
let $\alpha\simeq\alpha_{0}$ and $\beta\simeq\beta_{0}$.

##  Appendix C Derivation of dispersion relation

Consider an infinite periotic polymer chain, we write the photonic wave
functions to the immediate left and right of the $m$th unit cell as

$u^{(k)}_{m}\left(x\right)=t_{m-1,N}e^{ikx}+r_{m,1}e^{-ikx},$ (50a)
$u^{(k)}_{m+1}\left(x\right)=t_{m,N}e^{ikx}+r_{m+1,1}e^{-ikx}.$ (50b)

According to Bloch’s theorem, we have
$u_{m+1,k}\left(x+L\right)=u_{m,k}\left(x\right)e^{iqL}$, where $q$ is the
Bloch wave number, which leads to the following relations

$t_{m,N}=t_{m-1,N}e^{i\left(qL-\beta\right)},~{}~{}r_{m+1,1}=r_{m,1}e^{i\left(qL+\beta\right)}.$
(51)

Substituting Eqs. (17) and (51) into Eq. (13) yields the following linear
equations for $t_{m-1,N}$ and $r_{m,1}$:

$\left(\mathscr{B}^{-1}_{m}\mathcal{T}\mathscr{B}_{m-1}-e^{-iqL}\mathcal{I}\right)\begin{pmatrix}t_{m-1,N}\\\
r_{m,1}\end{pmatrix}=0.$ (52)

The non-zero solutions for $t_{m-1,N}$ and $r_{m,1}$ requires the determinant
of the coefficient matrix to be zero, resulting in the following dispersion
relation

$\cos\left(qL\right)=\mathrm{Re}\left(\frac{1}{\tilde{t}}e^{-i\beta}\right)=\frac{1}{2}\mathrm{Tr}\left(\tilde{\mathcal{T}}\right)=y\left(\Delta\right).$
(53)

##  Appendix D Expressions of scattering amplitudes in terms of collective
modes

By starting from the transport equations (4a)-(4c) and performing some
algebra, we can obtain the scattering amplitudes of the atomic chain Peng and
Jia (2023)

$t=1-i\mathbf{W}^{\dagger}(\Delta\mathbb{I}-\mathbf{H})^{-1}\mathbf{W},$ (54a)
$r=-i\mathbf{W}^{\top}(\Delta\mathbb{I}-\mathbf{H})^{-1}\mathbf{W}.$ (54b)

Here $\mathbb{I}$ is the $N^{\prime}$-dimensional identity matrix.
$N^{\prime}=MN$ is the total number of atoms. $\mathbf{W}$ takes the form

$\mathbf{W}=\sqrt{\frac{\Gamma}{2}}\left(e^{i\phi_{1}},e^{i\phi_{2}},\cdots,e^{i\phi_{N^{\prime}}}\right)^{\mathrm{\top}}.$
(55)

$\mathbf{H}$ is the effective non-Hermitian Hamilton matrix of the atomic
array, with elements

$\mathcal{H}_{nn^{\prime}}=\mathcal{J}_{n}\delta_{n,n^{\prime}-1}+\mathcal{J}_{n^{\prime}}\delta_{n-1,n^{\prime}}-i\frac{\Gamma}{2}e^{i\left|\phi_{n}-\phi_{n^{\prime}}\right|},$
(56)

with $n,n^{\prime}=1,2,\cdots,N^{\prime}$. Here the coupling strength
$\mathcal{J}_{n}$ between neighboring atoms is defined as
$\mathcal{J}_{mN+s}=J_{s}$ and $\mathcal{J}_{mN}=0$ ($m=0,1,2,\cdots,M-1$ and
$s=1,2,\cdots,N-1$). The first two terms of Eq. (56) describe the direct atom-
atom coupling, and the last term summarizes the atom-waveguide decay (for
$n=n^{\prime}$) as well as the coherent and dissipative atom-atom interactions
mediated by the waveguide modes (for $n\neq n^{\prime}$).

To better understand the physical properties of the scattering process, we
rewrite the scattering amplitudes in terms of collective modes of the atomic
chain:

$t=1-i\sum_{j=1}^{N^{\prime}}\frac{\left(\mathbf{W}^{\dagger}\mathbf{U}_{j}^{\mathscr{R}}\right)\left(\mathbf{U}_{j}^{\mathscr{L}^{\dagger}}\mathbf{W}\right)}{\Delta-\lambda_{j}},$
(57a)
$r=-i\sum_{j=1}^{N^{\prime}}\frac{\left(\mathbf{W}^{\top}{\mathbf{U}^{\mathscr{R}}_{j}}\right)\left({\mathbf{U}^{\mathscr{L}}_{j}}^{\dagger}\mathbf{W}\right)}{\Delta-\lambda_{j}},$
(57b)

where $\mathbf{U}^{\mathscr{R}}_{j}$ and $\mathbf{U}^{\mathscr{L}}_{j}$ are
the right and left eigenvectors of the non-Hermitian Hamilton matrix
$\mathbf{H}$, and $\lambda_{j}$ and $\lambda^{*}_{j}$ are the corresponding
complex eigenvalues, satisfying
$\mathbf{H}\mathbf{U}^{\mathscr{R}}_{j}=\lambda_{j}\mathbf{U}^{\mathscr{R}}_{j}$,
$\mathbf{H}^{{\dagger}}\mathbf{U}^{\mathscr{L}}_{j}=\lambda^{*}_{j}\mathbf{U}^{\mathscr{L}}_{j}$,
and
${\mathbf{U}^{\mathscr{L}}_{j}}^{\dagger}\mathbf{U}^{\mathscr{R}}_{j^{\prime}}={\mathbf{U}^{\mathscr{R}}_{j}}^{\dagger}\mathbf{U}^{\mathscr{L}}_{j^{\prime}}=\delta_{jj^{\prime}}$
Brody (2013).

In the Markovian regime we are interested in, the phase accumulation effect
caused by the detuned photons can be ignored, thus the quantities
$\mathbf{W}$, the non-Hermitian Hamiltonian $\mathbf{H}$, and the
corresponding eigenvalue $\lambda_{j}$ and eigenvector
$\mathbf{U}_{j}^{\mathscr{L,R}}$ are independent of the detuning $\Delta$ of
photons. The transmission and reflection amplitudes [Eqs. (57a) and (57b)] can
be further written as

$t=1+\sum_{j=1}^{N^{\prime}}{\frac{c_{j}}{\Delta-\tilde{\Delta}_{j}+i\frac{\tilde{\Gamma}_{j}}{2}}},$
(58a)
$r=\sum_{j=1}^{N^{\prime}}{\frac{d_{j}}{\Delta-\tilde{\Delta}_{j}+i\frac{\tilde{\Gamma}_{j}}{2}}}.$
(58b)

These results show superpositions of several Lorentzian-type amplitudes
contributed by the collective excitations.
$\tilde{\Delta}_{j}=\mathrm{Re}(\lambda_{j})|_{\Delta=0}$ is the detunning
between the $j$th collective mode and the atoms, and
$\tilde{\Gamma}_{j}=-2\mathrm{Im}(\lambda_{j})|_{\Delta=0}$ is the effective
decay of the $j$th collective mode.
$c_{j}=-i[(\mathbf{W}^{\dagger}\mathbf{U}_{j}^{\mathscr{R}})(\mathbf{U}_{j}^{\mathscr{L}^{\dagger}}\mathbf{W})]|_{\Delta=0}$
and
$d_{j}=-i[(\mathbf{W}^{\top}\mathbf{U}_{j}^{\mathscr{R}})(\mathbf{U}_{j}^{\mathscr{L}\dagger}\mathbf{W})]|_{\Delta=0}$
determine the weight of each Lorentzian component.

## References

  * Roy  _et al._ (2017) D. Roy, C. M. Wilson, and O. Firstenberg, Rev. Mod. Phys. 89, 021001 (2017).
  * Gu  _et al._ (2017) X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, Phys. Rep. 718-719, 1 (2017).
  * Sheremet  _et al._ (2023) A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakinskiy, and A. N. Poddubny, Rev. Mod. Phys. 95, 015002 (2023).
  * Shen and Fan (2005a) J.-T. Shen and S. Fan, Phys. Rev. Lett. 95, 213001 (2005a).
  * Chang  _et al._ (2006) D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, Phys. Rev. Lett. 97, 053002 (2006).
  * Shen and Fan (2007) J.-T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007).
  * Zhou  _et al._ (2008) L. Zhou, Z. R. Gong, Y.-x. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).
  * Astafiev  _et al._ (2010) O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, Science 327, 840 (2010).
  * Shi  _et al._ (2015) T. Shi, D. E. Chang, and J. I. Cirac, Phys. Rev. A 92, 053834 (2015).
  * Dicke (1954) R. H. Dicke, Phys. Rev. 93, 99 (1954).
  * Vetter  _et al._ (2016) P. A. Vetter, L. Wang, D.-W. Wang, and M. O. Scully, Physica Scripta 91, 023007 (2016).
  * van Loo  _et al._ (2013) A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, Science 342, 1494 (2013).
  * Zhang and Mølmer (2019) Y.-X. Zhang and K. Mølmer, Phys. Rev. Lett. 122, 203605 (2019).
  * Ke  _et al._ (2019) Y. Ke, A. V. Poshakinskiy, C. Lee, Y. S. Kivshar, and A. N. Poddubny, Phys. Rev. Lett. 123, 253601 (2019).
  * Wang  _et al._ (2020) Z. Wang, H. Li, W. Feng, X. Song, C. Song, W. Liu, Q. Guo, X. Zhang, H. Dong, D. Zheng, H. Wang, and D.-W. Wang, Phys. Rev. Lett. 124, 013601 (2020).
  * Zheng and Baranger (2013) H. Zheng and H. U. Baranger, Phys. Rev. Lett. 110, 113601 (2013).
  * Gonzalez-Ballestero  _et al._ (2014) C. Gonzalez-Ballestero, E. Moreno, and F. J. Garcia-Vidal, Phys. Rev. A 89, 042328 (2014).
  * Facchi  _et al._ (2016) P. Facchi, M. S. Kim, S. Pascazio, F. V. Pepe, D. Pomarico, and T. Tufarelli, Phys. Rev. A 94, 043839 (2016).
  * Mirza and Schotland (2016) I. M. Mirza and J. C. Schotland, Phys. Rev. A 94, 012302 (2016).
  * Chang  _et al._ (2012) D. E. Chang, L. Jiang, A. V. Gorshkov, and H. J. Kimble, New J. Phys. 14, 063003 (2012).
  * Mirhosseini  _et al._ (2019) M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. Dieterle, A. J. Keller, A. Asenjo-Garcia, D. E. Chang, and O. Painter, Nature (London) 569, 692 (2019).
  * Nie  _et al._ (2023) W. Nie, T. Shi, Y.-x. Liu, and F. Nori, Phys. Rev. Lett. 131, 103602 (2023).
  * Kockum  _et al._ (2018) A. F. Kockum, G. Johansson, and F. Nori, Phys. Rev. Lett. 120, 140404 (2018).
  * Nie  _et al._ (2021) W. Nie, T. Shi, F. Nori, and Y.-x. Liu, Phys. Rev. Appl. 15, 044041 (2021).
  * Tsoi and Law (2008) T. S. Tsoi and C. K. Law, Phys. Rev. A 78, 063832 (2008).
  * Liao  _et al._ (2015) Z. Liao, X. Zeng, S.-Y. Zhu, and M. S. Zubairy, Phys. Rev. A 92, 023806 (2015).
  * Cheng  _et al._ (2017) M.-T. Cheng, J. Xu, and G. S. Agarwal, Phys. Rev. A 95, 053807 (2017).
  * Ruostekoski and Javanainen (2017) J. Ruostekoski and J. Javanainen, Phys. Rev. A 96, 033857 (2017).
  * Mukhopadhyay and Agarwal (2019) D. Mukhopadhyay and G. S. Agarwal, Phys. Rev. A 100, 013812 (2019).
  * Mukhopadhyay and Agarwal (2020) D. Mukhopadhyay and G. S. Agarwal, Phys. Rev. A 101, 063814 (2020).
  * (31) A. Ask, Y.-L. L. Fang, and A. F. Kockum, arXiv:2011.15077 .
  * Jia and Cai (2022) W. Z. Jia and Q. Y. Cai, Eur. Phys. J. Plus 137, 1082 (2022).
  * Feng and Jia (2021) S. L. Feng and W. Z. Jia, Phys. Rev. A 104, 063712 (2021).
  * Bendickson  _et al._ (1996) J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107 (1996).
  * Hood  _et al._ (2016) J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, Proc. Natl. Acad. Sci. 113, 10507 (2016).
  * Liu and Houck (2017) Y. Liu and A. A. Houck, Nat. Phys. 13, 48 (2017).
  * Sundaresan  _et al._ (2019) N. M. Sundaresan, R. Lundgren, G. Zhu, A. V. Gorshkov, and A. A. Houck, Phys. Rev. X 9, 011021 (2019).
  * Kim  _et al._ (2021) E. Kim, X. Zhang, V. S. Ferreira, J. Banker, J. K. Iverson, A. Sipahigil, M. Bello, A. González-Tudela, M. Mirhosseini, and O. Painter, Phys. Rev. X 11, 011015 (2021).
  * Ferreira  _et al._ (2021) V. S. Ferreira, J. Banker, A. Sipahigil, M. H. Matheny, A. J. Keller, E. Kim, M. Mirhosseini, and O. Painter, Phys. Rev. X 11, 041043 (2021).
  * Scigliuzzo  _et al._ (2022) M. Scigliuzzo, G. Calajò, F. Ciccarello, D. Perez Lozano, A. Bengtsson, P. Scarlino, A. Wallraff, D. Chang, P. Delsing, and S. Gasparinetti, Phys. Rev. X 12, 031036 (2022).
  * Zhang  _et al._ (2023) X. Zhang, E. Kim, D. K. Mark, S. Choi, and O. Painter, Science 379, 278 (2023).
  * Rakhmanov  _et al._ (2008) A. L. Rakhmanov, A. M. Zagoskin, S. Savel’ev, and F. Nori, Phys. Rev. B 77, 144507 (2008).
  * Hutter  _et al._ (2011) C. Hutter, E. A. Tholén, K. Stannigel, J. Lidmar, and D. B. Haviland, Phys. Rev. B 83, 014511 (2011).
  * Zueco  _et al._ (2012) D. Zueco, J. J. Mazo, E. Solano, and J. J. García-Ripoll, Phys. Rev. B 86, 024503 (2012).
  * Xu  _et al._ (2000) Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000).
  * Shen and Fan (2005b) J.-T. Shen and S. Fan, Opt. Lett. 30, 2001 (2005b).
  * Yanik  _et al._ (2004) M. F. Yanik, W. Suh, Z. Wang, and S. Fan, Phys. Rev. Lett. 93, 233903 (2004).
  * Shen  _et al._ (2007) J.-T. Shen, M. L. Povinelli, S. Sandhu, and S. Fan, Phys. Rev. B 75, 035320 (2007).
  * Yi  _et al._ (2010) H. Yi, D. S. Citrin, and Z. Zhou, Opt. Express 18, 2967 (2010).
  * Witthaut and Sørensen (2010) D. Witthaut and A. S. Sørensen, New J. Phys. 12, 043052 (2010).
  * Fang and Baranger (2015) Y.-L. L. Fang and H. U. Baranger, Phys. Rev. A 91, 053845 (2015).
  * Mirza  _et al._ (2017) I. M. Mirza, J. G. Hoskins, and J. C. Schotland, Phys. Rev. A 96, 053804 (2017).
  * Mirhosseini  _et al._ (2018) M. Mirhosseini, E. Kim, V. S. Ferreira, M. Kalaee, A. Sipahigil, A. J. Keller, and O. Painter, Nat. Commun. 9, 3706 (2018).
  * He  _et al._ (2021) S. He, Q. He, and L. F. Wei, Opt. Express 29, 43148 (2021).
  * Greenberg  _et al._ (2021) Y. S. Greenberg, A. A. Shtygashev, and A. G. Moiseev, Phys. Rev. A 103, 023508 (2021).
  * Tang  _et al._ (2022) J.-S. Tang, W. Nie, L. Tang, M. Chen, X. Su, Y. Lu, F. Nori, and K. Xia, Phys. Rev. Lett. 128, 203602 (2022).
  * Peng and Jia (2023) Y. P. Peng and W. Z. Jia, Phys. Rev. A 108, 043709 (2023).
  * Berndsen and Mirza (2023) T. Berndsen and I. M. Mirza, Phys. Rev. A 108, 063702 (2023).
  * Berndsen  _et al._ (2024) T. Berndsen, N. Amgain, and I. Mirza, J. Opt. Soc. Am. B 41, C9 (2024).
  * Hime  _et al._ (2006) T. Hime, P. A. Reichardt, B. L. T. Plourde, T. L. Robertson, C.-E. Wu, A. V. Ustinov, and J. Clarke, Science 314, 1427 (2006).
  * Niskanen  _et al._ (2007) A. O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, S. Lloyd, and J. S. Tsai, Science 316, 723 (2007).
  * Baust  _et al._ (2015) A. Baust, E. Hoffmann, M. Haeberlein, M. J. Schwarz, P. Eder, J. Goetz, F. Wulschner, E. Xie, L. Zhong, F. Quijandría, B. Peropadre, D. Zueco, J.-J. García Ripoll, E. Solano, K. Fedorov, E. P. Menzel, F. Deppe, A. Marx, and R. Gross, Phys. Rev. B 91, 014515 (2015).
  * Zhu and Jia (2019) Y. T. Zhu and W. Z. Jia, Phys. Rev. A 99, 063815 (2019).
  * Abelès, Florin (1950) Abelès, Florin, Ann. Phys. 12, 596 (1950).
  * Brody (2013) D. C. Brody, J. Phys. A-Math. Theor. 47, 035305 (2013).


f_{i_{r-1}}\otimes f_{j_{1}}\otimes...\otimes f_{j_{s}})\big{]}.$

Notice that the index $s$ – appearing in the sum sums over $i$ and $j$ – is a
dummy index and after renaming the other indices we get

$\displaystyle=\sum_{s=1}^{n}\sum_{r=1}^{n-s+1}g_{r}\big{[}m^{\prime}_{s}\otimes\mathrm{id}^{\otimes
r-1}\sum_{i_{1}+...+i_{r+s-1}=n}(f_{i_{1}}\otimes...\otimes
f_{i_{r+s-1}})+...$ $\displaystyle\qquad\qquad\qquad+\mathrm{id}^{\otimes
r-1}\otimes
m_{s}^{\prime}\sum_{i_{1}+...+i_{r+s-1}=n}(f_{i_{1}}\otimes...\otimes
f_{i_{r+s-1}})\big{]}.$

Now set $r^{\prime}:=r+s-1$

$\sum_{s=1}^{n}\sum_{r^{\prime}=s}^{n}g_{r^{\prime}-s+1}\big{[}m^{\prime}_{s}\otimes\mathrm{id}^{\otimes
r^{\prime}-s}+...+\mathrm{id}^{\otimes r^{\prime}-s}\otimes
m_{s}^{\prime}\big{]}\sum_{i_{1}+...+i_{r^{\prime}}=n}(f_{i_{1}}\otimes...\otimes
f_{i_{r^{\prime}}}).$

Rename the index $r^{\prime}$ to $r$ and then interchange the first two sums:

$\displaystyle\quad\sum_{s=1}^{n}\sum_{r=s}^{n}g_{r-s+1}{m^{\prime}}_{s}^{r}\sum_{i_{1}+...+i_{r}=n}(f_{i_{1}}\otimes...\otimes
f_{i_{r}})$
$\displaystyle=\sum_{r=1}^{n}\sum_{s=1}^{r}g_{r-s+1}{m^{\prime}}_{s}^{r}\sum_{i_{1}+...+i_{r}=n}(f_{i_{1}}\otimes...\otimes
f_{i_{r}})$
$\displaystyle=\sum_{r=1}^{n}\sum_{s=1}^{r}\sum_{j_{1}+...+j_{s}=r}m_{s}^{\prime\prime}(g_{j_{1}}\otimes...\otimes
g_{j_{s}})\sum_{i_{1}+...+i_{r}=n}(f_{i_{1}}\otimes...\otimes f_{i_{r}})$
$\displaystyle=\sum_{s=1}^{n}m_{s}^{\prime\prime}\sum_{r=s}^{n}\sum_{j_{1}+...+j_{s}=r}\sum_{i_{1}+...+i_{r}=n}(g_{j_{1}}\otimes...\otimes
g_{j_{s}})(f_{i_{1}}\otimes...\otimes f_{i_{r}})$

It remains to show that

$\displaystyle\sum_{r=s}^{n}\sum_{j_{1}+...+j_{s}=r}\sum_{i_{1}+...+i_{r}=n}(g_{j_{1}}\otimes...\otimes
g_{j_{s}})(f_{i_{1}}\otimes...\otimes f_{i_{r}})$
$\displaystyle=\sum_{i_{1}+...+i_{s}=n}(g\circ
f)_{i_{1}}\otimes...\otimes(g\circ f)_{i_{s}}$ (54)

for all $n$ and all $1\leq s\leq n$. The proof is by induction: The induction
start is $n=s=1$ where both sides of equation (A) are equal to $g_{1}\circ
f_{1}$. Fix $n\geq 1$ and assume (A) is true for all $1\leq m\leq n$ and all
$1\leq s\leq m$. First observe that for $n+1$ and $s=1$ both sides of equation
(A) are equal to $(g\circ f)_{n}$. Now let $2\leq s\leq n+1$ and start with
the right-hand side of equation (A):

$\displaystyle\sum_{i_{1}+...+i_{s}=n+1}(g\circ
f)_{i_{1}}\otimes...\otimes(g\circ f)_{i_{s}}$
$\displaystyle=\sum_{i_{1}=1}^{n+2-s}(g\circ
f)_{i_{1}}\otimes\sum_{i_{2}+...+i_{s}=n+1-i_{1}}(g\circ
f)_{i_{2}}\otimes...\otimes(g\circ f)_{i_{s}}$
$\displaystyle\overset{\mathrm{i.a.}}{=}\sum_{i_{1}=1}^{n+2-s}(g\circ
f)_{i_{1}}\otimes\sum_{r=s-1}^{n+1-i_{1}}\sum_{j_{2}+...+j_{s}=r}\sum_{k_{1}+...+k_{r}=n+1-i_{1}}(g_{j_{2}}\otimes...\otimes
g_{j_{s}})(f_{k_{1}}\otimes...\otimes f_{k_{r}})$
$\displaystyle\overset{k:=i_{1}}{=}\sum_{k=1}^{n+2-s}\sum_{r=s-1}^{n+1-k}\sum_{l=1}^{k}\sum_{i_{1}+...+i_{l}=k}\sum_{j_{2}+...+j_{s}=r}\sum_{k_{1}+...+k_{r}=n+1-k}$
$\displaystyle\qquad\qquad\qquad\qquad(g_{l}\otimes g_{j_{2}}\otimes...\otimes
g_{j_{s}})(f_{1_{1}}\otimes...\otimes f_{i_{l}}\otimes
f_{k_{1}}\otimes...\otimes f_{k_{r}}).$

Now set $r^{\prime}:=n+1-r$

$\displaystyle\quad\sum_{k=1}^{n+2-s}\sum_{r^{\prime}=k}^{n+2-s}\sum_{l=1}^{k}\sum_{i_{1}+...+i_{l}=k}\sum_{j_{2}+...+j_{s}=n+1-r^{\prime}}\sum_{k_{1}+...+k_{n+1-r^{\prime}}=n+1-k}$
$\displaystyle\qquad\qquad\qquad\qquad(g_{l}\otimes g_{j_{2}}\otimes...\otimes
g_{j_{s}})(f_{1_{1}}\otimes...\otimes f_{i_{l}}\otimes
f_{k_{1}}\otimes...\otimes f_{k_{n+1-r^{\prime}}}).$

Now interchange the sums as follows

$\sum_{k=1}^{n+2-s}\sum_{r^{\prime}=k}^{n+2-s}\sum_{l=1}^{k}=\sum_{r^{\prime}=1}^{n+2-s}\sum_{k=1}^{r^{\prime}}\sum_{l=1}^{k}=\sum_{r^{\prime}=1}^{n+2-s}\sum_{l=1}^{r^{\prime}}\sum_{k=l}^{r^{\prime}}.$

The main equation becomes

$\displaystyle\sum_{r^{\prime}=1}^{n+2-s}\sum_{l=1}^{r^{\prime}}\sum_{k=l}^{r^{\prime}}\sum_{i_{1}+...+i_{l}=k}\sum_{j_{2}+...+j_{s}=n+1-r^{\prime}}\sum_{k_{1}+...+k_{n+1-r^{\prime}}=n+1-k}$
$\displaystyle\qquad\qquad\qquad\qquad(g_{l}\otimes g_{j_{2}}\otimes...\otimes
g_{j_{s}})(f_{1_{1}}\otimes...\otimes f_{i_{l}}\otimes
f_{k_{1}}\otimes...\otimes f_{k_{n+1-r^{\prime}}})$
$\displaystyle=\sum_{r^{\prime}=1}^{n+2-s}\sum_{l=1}^{r^{\prime}}\sum_{j_{2}+...+j_{s}=n+1-r^{\prime}}(g_{l}\otimes
g_{j_{2}}\otimes...\otimes g_{j_{s}})$
$\displaystyle\quad\sum_{k=l}^{r^{\prime}}\sum_{i_{1}+...+i_{l}=k}\sum_{k_{1}+...+k_{n+1-r^{\prime}}=n+1-k}(f_{1_{1}}\otimes...\otimes
f_{i_{l}}\otimes f_{k_{1}}\otimes...\otimes f_{k_{n+1-r^{\prime}}})$
$\displaystyle=\sum_{r^{\prime}=1}^{n+2-s}\sum_{l=1}^{r^{\prime}}\sum_{j_{2}+...+j_{s}=n+1-r^{\prime}}(g_{l}\otimes
g_{j_{2}}\otimes...\otimes
g_{j_{s}})\sum_{i_{1}+...+i_{n+1-r^{\prime}+l}=n+1}(f_{i_{1}}\otimes...\otimes
f_{i_{n+1-r^{\prime}+l}})$

Where the last equality uses Lemma A.1. Shift the index $l$ in the following
way $r:=r^{\prime}-l+1$

$\displaystyle=\sum_{r^{\prime}=1}^{n+2-s}\sum_{r=1}^{r^{\prime}}\sum_{j_{2}+...+j_{s}=n+1-r^{\prime}}(g_{r^{\prime}-r+1}\otimes
g_{j_{2}}\otimes...\otimes
g_{j_{s}})\sum_{i_{1}+...+i_{n+2-r}=n+1}(f_{i_{1}}\otimes...\otimes
f_{i_{n+2-r}})$

Then after interchanging the sums

$\sum_{r^{\prime}=1}^{n+2-s}\sum_{r=1}^{r^{\prime}}=\sum_{r=1}^{n+2-s}\sum_{r^{\prime}=r}^{n+2-s}$

shift the index $r^{\prime}$ as follows $j_{1}:=r^{\prime}-r+1$

$\displaystyle\sum_{r=1}^{n+2-s}\sum_{j_{1}=1}^{n+3-s-r}\sum_{j_{2}+...+j_{s}=n+2-r-j_{1}}(g_{j_{1}}\otimes...\otimes
g_{j_{s}})\sum_{i_{1}+...+i_{n+2-r}=n+1}(f_{i_{1}}\otimes...\otimes
f_{i_{n+2-r}})$
$\displaystyle=\sum_{r=1}^{n+2-s}\sum_{j_{1}+...+j_{s}=n+2-r}(g_{j_{1}}\otimes...\otimes
g_{j_{s}})\sum_{i_{1}+...+i_{n+2-r}=n+1}(f_{i_{1}}\otimes...\otimes
f_{i_{n+2-r}}).$

Finally shift the index $r$ as follows $r^{\prime}:=n+2-r$ we obtain

$\sum_{r=s}^{n+1}\sum_{j_{1}+...+j_{s}=r^{\prime}}(g_{j_{1}}\otimes...\otimes
g_{j_{s}})\sum_{i_{1}+...+i_{r^{\prime}}=n+1}(f_{i_{1}}\otimes...\otimes
f_{i_{r^{\prime}}}).$

This completes the induction step.

## References

  * [Ada78] John Frank Adams  “Infinite loop spaces”  Princeton University Press, 1978
  * [Ber14] Alexander Berglund  “Homological perturbation theory for algebras over operads”  In _Algebraic & Geometric Topology_ 14.5 Mathematical Sciences Publishers, 2014, pp. 2511–2548
  * [BM03] Clemens Berger and Ieke Moerdijk  “Axiomatic homotopy theory for operads”  In _Commentarii Mathematici Helvetici_ 78.4 Springer, 2003, pp. 805–831
  * [Bro67] Ronald Brown  “The twisted Eilenberg-Zilber Theorem”  In _Simposio di Topologia (Messina, 1964)_ , 1967, pp. 33–37
  * [BV06] John Michael Boardman and Rainer M. Vogt  “Homotopy invariant algebraic structures on topological spaces”  Springer, 2006
  * [BV68] John Michael Boardman and Rainer M. Vogt  “Homotopy-everything $H$-spaces”  In _Bulletin of the American mathematical society_ 74.6 American Mathematical Society, 1968, pp. 1117–1122
  * [Del+75] Pierre Deligne, Phillip Griffiths, John W. Morgan and Dennis Sullivan  “Real homotopy theory of Kähler manifolds”  In _Inventiones mathematicae_ 29.3 Springer, 1975, pp. 245–274
  * [DL59] Albrecht Dold and Richard Lashof  “Principal quasifibrations and fibre homotopy equivalence of bundles”  In _Illinois Journal of Mathematics_ 3.2 University of Illinois at Urbana-Champaign, 1959, pp. 285–305
  * [EL53] Samuel Eilenberg and Saunders Mac Lane  “On the groups H ($\pi$, n), I”  In _Annals of Mathematics_ JSTOR, 1953, pp. 55–106
  * [FHT12] Yves Félix, Stephen Halperin and Jean-Claude Thomas  “Rational homotopy theory”  Springer, 2012
  * [Fuk+10] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono  “Lagrangian intersection Floer theory: anomaly and obstruction, Part II”  American Mathematical Soc., 2010
  * [Fuk93] Kenji Fukaya  “Morse homotopy, $A_{\infty}$ Category and Floer homologies”  In _Proceeding of Garc Workshop on Geometry and Topology_ , 1993  Seoul National Univ
  * [GH78] Phillip Griffiths and Joseph Harris  “Principles of algebraic geometry”  Wiley Online Library, 1978
  * [GJ+90] Ezra Getzler and John D.S. Jones  “$A_{\infty}$-algebras and the cyclic bar complex”  In _Illinois J. Math_ 34.2 Citeseer, 1990, pp. 256–283
  * [GLS89] V.K.A.M. Gugenheim, Larry A. Lambe and James D. Stasheff  “Perturbation theory in differential homological algebra. I”  In _Illinois J. Math_ 33.4 Citeseer, 1989, pp. 566–582
  * [GLS91] V.K.A.M. Gugenheim, Larry A. Lambe and James D. Stasheff  “Perturbation theory in differential homological algebra II”  In _Illinois Journal of Mathematics_ 35.3 Duke University Press, 1991, pp. 357–373
  * [GM81] Phillip Griffiths and John W. Morgan  “Rational homotopy theory and differential forms”  Springer, 1981
  * [Hic19] Jeff Hicks  “Some Practical Constructions with filtered $A_{\infty}$ algebras”, 2019
  * [Hic19a] Jeff Hicks  “Wall-crossing from Lagrangian Cobordisms”, 2019  arXiv:1911.09979 [math.SG]
  * [Hin03] Vladimir Hinich  “Tamarkins proof of Kontsevich formality theorem”  In _Forum Mathematicum_ 15(4) Walter de Gruyter GmbH & Co. KG Berlin, Germany, 2003, pp. 591–614
  * [HK91] Johannes Huebschmann and Tornike Kadeishvili  “Small models for chain algebras”  In _Mathematische Zeitschrift_ 207.1 Springer, 1991, pp. 245–280
  * [Hue83] Johannes Huebschmann  “The homotopy type of $F\psi^{q}$. The complex and symplectic cases”  In _Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983)_ 55, 1983, pp. 487–518
  * [Hue89] Johannes Huebschmann  “The mod-p cohomology rings of metacyclic groups”  In _Journal of Pure and Applied Algebra_ 60.1 Elsevier, 1989, pp. 53–103
  * [JY09] Mark W. Johnson and Donald Yau  “On homotopy invariance for algebras over colored PROPs”  In _Journal of Homotopy and Related Structures_ 4.1, 2009, pp. 275–315
  * [Kad05] Tornike Kadeishvili  “On the homology theory of fibre spaces”, 2005  arXiv:math/0504437
  * [Kad08] Tornike Kadeishvili  “Cohomology $C_{\infty}$-algebra and Rational Homotopy Type”, 2008  arXiv:0811.1655 [math.AT]
  * [Kad80] Tornike Kadeishvili  “On the homology theory of fibre spaces”  In _Russian Mathematical Surveys_ 35, 1980, pp. 231–238
  * [Kan88] Richard M. Kane  “The homology of Hopf spaces”  North-Holland, 1988
  * [Kel01] Bernhard Keller  “Introduction to $A$-infinity algebras and modules”  In _Homology, homotopy and applications_ 3.1 International Press of Boston, 2001, pp. 1–35
  * [Kel02] Bernhard Keller  “Addendum to:“Introduction to $A$-infinity algebras and modules”[HHA, v. 3 (2001) No. 1, pp. 1-35]”  In _Homology, Homotopy and Applications_ 4.1 International Press of Boston, 2002, pp. 25–28
  * [Kel06] Bernhard Keller  “A-infinity algebras, modules and functor categories”  In _Contemporary Mathematics_ 406 Providence, RI: American Mathematical Society, 2006, pp. 67–94
  * [Kon95] Maxim Kontsevich  “Homological algebra of mirror symmetry”  In _Proceedings of the international congress of mathematicians_ , 1995, pp. 120–139  Springer
  * [KS00] Maxim Kontsevich and Yan Soibelman  “Homological mirror symmetry and torus fibrations”, 2000  arXiv:math/0011041
  * [Laa04] Pepijn Philippe Immanuel Laan  “Operads: Hopf algebras and coloured Koszul duality”, 2004
  * [LV12] Jean-Louis Loday and Bruno Vallette  “Algebraic operads”  Springer, 2012
  * [Man10] Marco Manetti  “A relative version of the ordinary perturbation lemma”, 2010  arXiv:1002.0683 [math.KT]
  * [Mar04] Martin Markl  “Homotopy algebras are homotopy algebras”  In _Forum Mathematicum_ 16 Walter de Gruyter GmbH & Co. KG Berlin, Germany, 2004, pp. 129–160
  * [Mar04a] Martin Markl  “Transferring $A_{\infty}$(strongly homotopy associative) structures”, 2004  arXiv:math/0401007
  * [Mar92] Martin Markl  “A cohomology theory for A (m)-algebras and applications”  In _Journal of pure and applied algebra_ 83.2 Elsevier, 1992, pp. 141–175
  * [Mar99] Martin Markl  “Simplex, associahedron, and cyclohedron”  In _Contemporary Mathematics_ 227 American Mathematical Society, 1999, pp. 235–266
  * [May72] J. May  “The geometry of iterated loop spaces” 271, Lecture Notes in Mathematics  Springer-Verlag, 1972
  * [Mer98] Sergei A. Merkulov  “Strongly homotopy algebras of a Kähler manifold”, 1998  arXiv:math/9809172
  * [Mil56] John Milnor  “Construction of universal bundles, II”  In _Annals of Mathematics_ JSTOR, 1956, pp. 430–436
  * [Mil59] John Milnor  “On spaces having the homotopy type of a CW-complex”  In _Transactions of the American Mathematical Society_ 90.2 JSTOR, 1959, pp. 272–280
  * [MSS02] Martin Markl, Steven Shnider and James D. Stasheff  “Operads in algebra, topology and physics”  American Mathematical Soc., 2002
  * [Pet20] Dan Petersen  “A closer look at Kadeishvili’s theorem”, 2020  arXiv:1905.10877 [math.QA]
  * [Pro86] Alain Prouté  “Algèbres différentielles fortement homotopiquement associatives…”, 1986
  * [Qui69] Daniel Quillen  “Rational homotopy theory”  In _Annals of Mathematics_ JSTOR, 1969, pp. 205–295
  * [Sei02] Paul Seidel  “Fukaya categories and deformations”, 2002  arXiv:math/0206155
  * [Smi80] Vladimir A. Smirnov  “Homology of fibre spaces”  In _Russian Mathematical Surveys_ 35.3 IOP Publishing, 1980, pp. 294–298
  * [Sta06] James D. Stasheff  “H-spaces from a homotopy point of view”  Springer, 2006
  * [Sta63] James D. Stasheff  “Homotopy associativity of H-spaces. I”  In _Transactions of the American Mathematical Society_ 108.2, 1963, pp. 275–292
  * [Sta63a] James D. Stasheff  “Homotopy associativity of H-spaces. II”  In _Transactions of the American Mathematical Society_ 108.2, 1963, pp. 293–312
  * [Sta92] James D. Stasheff  “Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras”  In _Quantum groups_ Springer, 1992, pp. 120–137
  * [Sug57] Masahiro Sugawara  “A condition that a space is group-like”  In _Mathematical Journal of Okayama University_ 7.2 Department of Mathematics, Faculty of Science, Okayama University, 1957
  * [Sug57a] Masahiro Sugawara  “On a condition that a space is an H-space”  In _Mathematical Journal of Okayama University_ 6.2 Department of Mathematics, Faculty of Science, Okayama University, 1957
  * [Sul77] Dennis Sullivan  “Infinitesimal computations in topology”  In _Publications Mathématiques de l’Institut des Hautes Études Scientifiques_ 47.1 Springer, 1977, pp. 269–331
  * [Swe69] Moss E. Sweedler  “Hopf algebras”  Benjamin, New York, 1969
  * [Zho19] Jiawei Zhou  “On the Construction of Minimal Model for Some A-infinity Algebras”, 2019