#  Engineering bound state in continuum via giant atom in photonic waveguide

Xiaojun Zhang  Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun 130024, China  Mingjie Zhu  Center for Quantum
Sciences and School of Physics, Northeast Normal University, Changchun 130024,
China  Zhihai Wang<EMAIL_ADDRESS>Center for Quantum Sciences and
School of Physics, Northeast Normal University, Changchun 130024, China

###### Abstract

The bound state in the continuum (BIC) in photonic system has been widely used
in the field of lasing and sensing. We here find the controllable BIC in an
artificial giant atom-dressed one-dimensional photonic waveguide. The giant
atom couples to the waveguide via two distant sites. We find that the energy
and the photonic distribution in the BIC can be controlled on demand by tuning
the frequency and the size of the giant atom as well as its coupling phase
with the waveguide. More interestingly, we predict the quantum beats in the
atomic and photonic dynamical evolution, which is induced by the oscillation
between the BIC and bound state outside the continuum (BOC). These findings
provide an approach to manipulate the waveguide system via the bound states,
and can be applied in the quantum information processing.

##  I introduction

The bound state in the continuum (BIC) is a kind of peculiar states, which are
remain localized even though it resides in the radiation continuum BIC1 ; BIC2
; BIC3 . With the assistance of the BIC in photonic system, the light-matter
interaction can be significantly enhanced bicre1 ; bicre2 . Therefore, the BIC
is nowadays used to generate low threshold laser laser1 ; laser2 ; laser3 ,
enhance the nonlinear responds nonlinear1 ; nonlinear2 , realize the quantum
sensing sense1 ; sense2  and high efficient wave guiding waveguiding1 ;
waveguiding2 ; waveguiding3 ; waveguiding4 , etc.

The waveguide provides the channel for photons travelling in the confined
structure. Meanwhile, the waveguide also serves as a data bus, to induce the
coherent interaction and incoherent collective dissipation between quantum
emitters. In Ref. bicAN , the authors revealed that BIC can avoid the adverse
effects of environment on quantum information processing, and proposed a
feasible experimental scheme to realize BIC.

In the traditional waveguide community, the atom or artificial atom was
usually considered as point like dipole. However, in 2014, the coherent
coupling between the superconducting transmon qubit and the surface acoustic
waves S1  break down the dipole approximation. In this setup, the size of the
transmon is in the same order of the acoustic wavelength, and one has to
consider the effects of non-local coupling via multiple points, leading to
giant atom quantum optics. Subsequently, the giant atom setup was realized by
coupling the transmon or magnon spin ensemble to the curved transmission line
and proposed in the synthesis dimension domain sq1 ; sq2 ; sq3 ; mq . The
interference and retardation effect due to the non-local coupling in the giant
atom system has evoked lots of attentions theoretically to demonstrate some
exotic phenomena, such as non-Markovian oscillation NM1 ; NM2 , decoherence-
free interaction AF2018 ; AC2020 , retardation effect chengre ; LG2017 , Lamb
shift Lambpra14  and chiral photonic population XW2021 ; XW2022 ; AS2021 . The
combination of the the topological photons and the giant atom also induces
some interesting effects, for example the chiral zero mode chengre ; tp2 , the
vacuum like dressed states vds , the wide band photonic reflection tpg .

Recently, people have found that the BIC can exist in the giant atom-waveguide
coupling system, where the waveguide is characterized by the linear or cosine
type dispersion relation BIC6 ; BIC7 ; ZZ . The underlying physics under the
BIC generation is such giant atom is decoupled from the waveguide due to the
destructive interference effect. Once the condition of the BIC is fulfilled,
we can obtain the persistent oscillation without dissipation. Moreover, a
controllable magic cavity QED system is also simulated under the BIC physics
ZZ . Since a controllability quantum system is necessary in quantum
information processing, how to control the BIC in the giant atom scenario is
deserved to investigated in detail.

In this paper, we promote a scheme to realize the BIC manipulation by coupling
a giant atom to the coupled resonator waveguide via two sites non-locally. We
analytically obtain the photonic distribution for the BIC. Moreover, we find
that both of the BIC and bound states out of continuum (BOC) participate the
dynamics of the system, and the BIC-BOC oscillation leads to the exotic
quantum beat, which does not exist in the tradition small atom setup.

##  II model

Figure 1: Sketch of the waveguide QED setup, where a giant atom is coupled to
a coupled resonator waveguide via the $0$th and the $N$th sites.

As schematically shown in Fig. 1, a single giant atom is coupled with the CRW
via the $0th$ and $N$th sites. The Hamiltonian of the structure in real space
reads $H=H_{A}+H_{c}+H_{I}$ where $(\hbar=1)$

$\displaystyle H_{A}$ $\displaystyle=$ $\displaystyle\Omega|e\rangle\langle
e|,$ (1) $\displaystyle H_{c}$ $\displaystyle=$
$\displaystyle\omega_{c}\sum_{j}a_{j}^{\dagger}a_{j}-\xi\sum_{j}(a_{j+1}^{\dagger}a_{j}+a_{j}^{\dagger}a_{j+1}),$
(2) $\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle
g[(a_{0}^{\dagger}+a_{N}^{\dagger}e^{i\phi})\sigma_{-}+{\rm H.c.}].$ (3)

Here, $H_{A}$ and $H_{c}$ are the Hamiltonian of the GA and the CRW
respectively and $H_{I}$ represents their interaction. Here we have neglected
the counter-rotating terms and performed the dipole approximation at each
coupling point between the GA and the CRW. $\Omega$ is the transition
frequency of the GA between the ground state $|g\rangle$ and the excited state
$|e\rangle$. $\omega_{c}$ is the intrinsic frequency of each resonator in the
CRW. $a_{j}$ is the annihilation operator of the $j$th resonator in the CRW,
$\sigma_{+}=|e\rangle\langle g|$ is the raising operator of the GA. $\xi$ is
the hopping strength between the nearest resonators in CRW. $g$ is the real
coupling strength between the GA and the connected resonator. We have assumed
that the coupling phase in the left leg is zero while that for the right leg
is $\phi$.

We consider that the CRW is composed by infinite resonators so that the
waveguide satisfies the translation invariance. By the Fourier transformation
$a^{\dagger}_{k}=\sum_{j}a^{\dagger}_{j}\exp(-ikj)/\sqrt{N_{c}}$ with
$N_{c}\rightarrow\infty$ being the length of the waveguide, the Hamiltonian of
CRW $H_{c}$ is expressed as $H_{c}=\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k}$
where the dispersion relation satisfies $\omega_{k}=\omega_{c}-2\xi\cos k$. It
describes that the single photon energy band is centered at $\omega_{c}$ and
posses a width of $4\xi$. In term of the operator $a_{k}$, the interaction
Hamiltonian becomes

$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{N_{c}}}\sum_{k}[a_{k}^{\dagger}\sigma_{-}(1+e^{i(kN+\phi)})+{\rm
H.c.}],$ (4)

$N_{c}\rightarrow\infty$ is the length of CRW. It is obvious that the coupling
amplitude between the GA and its resonate mode in the CRW is

$G(\phi)=\frac{g}{\sqrt{N_{c}}}(1+e^{i(KN+\phi)})$ (5)

which is $\phi$ dependent and $K$ satisfies $\Omega=\omega_{c}-2\xi\cos K$.

##  III Modulation of the BIC

In Figs. 2(a) and (c), we numerically plot the energy spectrum of system in
the single excitation subspace for different atomic resonant frequency and
coupling phase $\phi$. As expected, in both of the parameter conditions, we
find a pair of bound states which locate above and below the continual band
respectively. In what follows, we name them as bound states out of continuum
(BOC). These BOCs originate from the broken of the translational symmetry of
the waveguide induced by the GA. Not only in the GA setup, the BOCs also exist
in the CRW system which couples to the traditional small atom(s) Peter16pra ;
ZLpra08 ; Qiao2021 .

In the continual band, we also observe a bound state when the condition
$|G(\phi)|^{2}=0$ is satisfied BIC7  , this is the so called BIC ZZ . The
underlying physics is that the GA is decoupled from the resonant mode in the
waveguide. However, the rest off resonant modes contributes and the BIC is
still the atom-photon dressed state.

To further investigate the character of the BIC, we resort to the single
photon wave function ansatz of

$\left|\phi\right\rangle=\alpha\sigma_{+}|G\rangle+\sum_{k}\beta_{k}a_{k}^{\dagger}\left|G\right\rangle,$
(6)

where $|G\rangle$ represents that both of the GA and the waveguide are in
their ground state. According to the Schrödinger equation
$H|\phi\rangle=E|\phi\rangle$, we can get

$\displaystyle E$ $\displaystyle=$
$\displaystyle\Omega+\frac{g^{2}}{\pi}\int_{-\pi}^{\pi}\frac{1+\cos(kN+\phi)}{E-\omega_{c}+2\xi\cos
k}dk,$ (7) $\displaystyle\frac{\beta_{k}}{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{[g+ge^{i(kN+\phi)}]}{E-\omega_{c}+2\xi\cos
k}$ (8)

and the photon distribution in the real space reads

$\displaystyle{\beta_{j}}=\int_{-\pi}^{\pi}\beta_{k}e^{-ikj}dk.$ (9)

In Ref. ZZ , we have shown the BIC in the system when the giant atom
($N=4m+2,m=0,1,2\cdots$) is resonant with the bare resonator in the waveguide
($\Omega=\omega_{c}$) by taking the coupling phase to be zero ($\phi=0$).
Moving to the non-resonant case, the energy spectra in Fig. 2 (a) and (c)
illustrate that the BIC (blue line) locate at the atomic frequency $\Omega$
when the coupling phase satisfies $|G(\phi)|^{2}=0$. As proven analytically in
the appendix, $E=\Omega$ is always a solution of Eq. (7) in the condition of
$|G(\phi)|^{2}=0$ for arbitrary $\omega_{c}-2\xi<\Omega<\omega_{c}+2\xi$,
which agrees with the results in Fig. 2 (a) and (c). Furthermore, we can also
obtain the photonic distribution for the BIC in Figs. 2 (b) and (d). Here, the
solid bar represents the numerical results which comes from the direct
diagonalization of the Hamiltonian and the empty circles is the analytical
results which is expressed as (see Appendix for detail)

$\frac{\beta_{j}}{\alpha}=\begin{array}[]{l}\left\\{\begin{matrix}0&,&j<0\&j>N\\\
\frac{2g\sin Kj}{\sqrt{4\xi^{2}-\Delta^{2}}}&,&0\leq j\leq
N\end{matrix}\right.\end{array}$ (10)

where $\Delta=\Omega-\omega_{c}$ is the detuning between the giant atom and
the bare frequency of the resonator in the waveguide.

Figure 2: (a,c) The energy diagram of the giant atom-CRW coupled system where
the thick blue line in the continuum is the BIC. (b,d) The photonic
distribution in the BIC. We have set $g=0.1\xi,\omega_{c}=0$. The other
parameters are set as $\Omega=-\xi,N=6,\phi=\pi$ in (a,b) and
$\Omega=-\sqrt{2}\xi,N=12,\phi=0$ in (c,d).

Explicitly, we here find some exotic characters of the single photon BIC as
shown in Figs. 2 (b) and (d). For $\Omega/\xi=-1,\phi=\pi,N=6$, we observe a
dimerised character in Fig. 2 (b). That is, the photon is bounded in the
resonators dimmer by dimmer inside the regime covered by the giant atom and
the photonic intensity is homogenous in each resonator of the dimmer. This can
be explained by the fact that the photon in the $Kth$ mode propagating to the
left interferes with that propagating to the right which is revealed by Eq.
(10). Since $\Omega/\xi=-1$ and $K=\pi/3$, there are photons at $j=1,2,4$ and
$5$ but no photons at $j=3$. On the contrary, for the parameter of
$\Omega/\xi=-\sqrt{2},\phi=\pi,N=12$, we will reach a trimerised BIC as shown
in Fig. 2 (d). It shows that in each of the trimmer, the photonic intensity in
the central resonator is two times larger than those in the side ones. In this
case $K=\pi/4$, so there is no photon distribution between the two legs of the
giant atom at $j=4m\,(m=0,1,2,3)$ as is demonstrated in Eq. (10). Therefore,
the frequency and the photonic distribution of the BIC can be modulated by the
size and the frequency of the giant atom as well as its coupling phase to the
waveguide.

Besides the above mentioned BIC, in Figs. 2 (a) and (c), we can also observe a
pair of BOCs, locating below and above the continuum asymmetrically, which is
different from that in the resonant case ($\Omega=\omega_{c}$). Together with
the BIC, the system exhibit an exotic dynamical behavior for both of the atom
and the photonic evolution.

##  IV dynamics

In this section, we will investigate the dynamics of the system for both of
the atomic and photonic counterpart under the parameter of
$\Omega=-\xi,N=6,\phi=\pi$, in which case the energy spectrum is given in Fig.
2(a). We set the initial state as $|\psi(0)\rangle=\sigma_{+}|G\rangle$, in
which the giant atom is in its excited state while all of the resonators are
in their ground states. In Fig. 3, we numerically illustrate the dynamical
evolution of the system, which is governed by
$|\psi(t)\rangle=\exp(-iHt)|\psi(0)\rangle$. In Fig. 3(a), we show the atomic
excitation $P_{e}=|\langle\psi(t)|e\rangle|^{2}$, we can observe the slow
variable envelope and the fast oscillation in each of the envelope. The
dynamics of the photonic counterpart
$\beta_{j}=\langle\psi(t)|a_{j}^{\dagger}|G\rangle$ is shown in Fig. 3(b). We
can observe that the photon emitted by the giant atom is nearly completely
trapped inside the covered regime. We further detailed plot the evolution of
the photonic distribution in the resonator $j=0,1,3$ in Fig. 3(c). We observe
that the photon can never be excited in the $3$th resonator, that is
$|\beta_{3}|=0$ during the time evolution. For $j=1$, we find the similar
quantum beats phenomenon with the atomic dynamics and the same result for
$j=2$ is not shown in the figure. For $j=0$, we can observe a single high
oscillation frequency.

Figure 3: The dynamical evolution of the system. (a) The dynamics of the
atomic population in its excited states. (b,c) The dynamics of the photonic
distribution. (d) The frequency spectrum of the atomic and photonic dynamical
evolution which us obtained by the numerical Fourier transformation from the
time domain to the frequency domain. The parameters are set as
$\omega_{c}=0,\Omega=-\xi,g=1.1\xi,N=6,\phi=\pi$.

To understand the above dynamical evolution, we write the initial state as

$\displaystyle|\psi(0)\rangle\simeq
c_{U}|\phi_{U}\rangle+c_{I}|\phi_{I}\rangle+c_{L}|\phi_{L}\rangle+\sum_{k}c_{k}|\phi_{k}\rangle,$
(11)

where $|\phi_{U}\rangle$ and $|\phi_{L}\rangle$ are the upper and lower BOC
with eigen energy $E_{U}$ and $E_{L}$ respectively, and $|\phi_{I}\rangle$ is
the BIC with eigenenergy $E_{I}$. $|\phi_{k}\rangle$ is the $k$th propagating
photonic mode in the waveguide, and the C-numbers $c_{m}\,(m=U,I,L,k)$ denote
the corresponding amplitudes. Extracting from the exact numerical results in
Fig. 2(a), the eigen energy of the system is illustrated in the cartoon figure
in the upper panel Fig. 3(d). Here, the continual band, which locates in the
regime of $[-2\xi,2\xi]$ by setting $\omega_{c}=0$, is demonstrated by the
blue bar, and the BIC and the BOCs are represented by the dark red bar. Since
the propagating modes will not contribute to the atomic dynamics when the
evolution time is long enough. Therefore, in the long time limit, we will have

$\displaystyle P_{e}(t)$ $\displaystyle=$
$\displaystyle|e^{-iE_{L}t}c_{L}^{2}+e^{-iE_{I}t}c_{I}^{2}+e^{-iE_{U}t}c_{U}^{2}|^{2}$
(12) $\displaystyle=$
$\displaystyle|e^{i\delta_{L}t}c_{L}^{2}+c_{I}^{2}+e^{-i\delta_{U}t}c_{U}^{2}|^{2},$
$\displaystyle|\beta_{j}(t)|^{2}$ $\displaystyle=$
$\displaystyle|e^{-i\delta_{U}t}c_{U}d_{U,j}+e^{i\delta_{L}t}c_{L}d_{L,j}+c_{I}d_{I,j}|^{2}.$
(13)

where $\delta_{L}=E_{I}-E_{L}$ and $\delta_{L}=E_{U}-E_{I}$ are the detunings
between of BOCs and the BIC, and
$d_{\alpha,j}=\langle\phi_{\alpha}|a_{j}^{\dagger}|G\rangle\,(\alpha=U,I,L)$
denotes the photonic excitation amplitude on the $j$th resonator of the
waveguide when the system is in the bound state $|\phi_{\alpha}\rangle$.

To further investigate the dynamical property of the system, we apply the
numerical fast Fourier transformation technique to obtain the frequency
spectrum for $P_{e}$ and $|\beta_{j}|\,(j=0,1)$ in the lower panels of Fig.
3(d). We can observe three significant peaks for for $P_{e}$ and
$|\beta_{1}|$, which locate at the frequency $\delta_{L}$, $\delta_{U}$ and
$\delta_{U}+\delta_{L}$, agreeing the analytical expression in Eq. (12). It
implies that the dynamics of the giant atom is induced by the oscillations
between the single BIC and the two BOCs (which contributes the peaks located
at $\delta_{U}$ and $\delta_{L}$) and between the two BOCs (which contributes
the peaks located at $\delta_{U}+\delta_{L}$). On the contrary, we only find a
single peak which locates at the frequency of $\delta_{U}+\delta_{L}$ in the
results of $|\beta_{0}|$. This can also be explained by the analytical
expression in Eq. (13), in which $d_{I,0}=0$ [see Fig. 2(b)], and there only
exists a single non-zero frequency component at $\delta_{U}+\delta_{L}$. It
further indicates that the oscillation between the two BOCs contributes to the
photonic dynamics in the zeroth cavity.

##  V Conclusions

In this paper, we have investigate the possibility of control the photonic
distribution in the BIC in a giant atom waveguide QED system. The BIC-BOC
oscillation leads to the quantum beats dynamics. The giant atom can be
realized by the superconducting transmon qubit, and the coupled resonator
waveguide up to tens of sites 22prx ; s2023  and has also been fabricated in
superconducting circuits. In these experimental realizations, the parameters
can be achieved in the regime of $g\ll\xi\approx 100-200$ MHz, and therefore
our predicted dynamical behavior can be observed even the giant atom suffers
the spontaneous emission with the the life time of $T_{1}=10{\rm\mu s}$
matter2020 .

Beyond the specific setting considered in this work, our investigate can be
develop to the system consist of more giant atoms or one giant atom which
couples to the waveguide via more than one sites with the on-demand coupling
strengths and phases. In these more complicated setups, the BIC and BOC is
deserved to be investigated and is hopefully to design the needed dynamical
process for quantum information processing.

###### Acknowledgements.

This work is supported by the funding from Jilin Province (Grant Nos.
20230101357JC and 20220502002GH) and the National Science Foundation of China
(Grant No. 12375010).

##  Appendix A Bound state in the continuum

In Eqs. (7) and (10), we have given the transcendental equations for the
energies of the bound states and the wave function of the BIC. Here, we give a
detailed derivation by transforming to momentum space.

We begin the Hamiltonian in Eq. (4), and the wave function in Eq. (6), the
Schödinger equation $H|\phi\rangle=E|\phi\rangle$ yields

$\displaystyle\alpha(E-\Omega)=\frac{g}{\sqrt{N_{c}}}\sum_{k}\beta_{k}[1+e^{-i(kN+\phi)}],$
$\displaystyle\alpha(E-\omega_{c}+2\xi\cos
k)\beta_{k}=\frac{g}{\sqrt{N_{c}}}[1+e^{i(kN+\phi)}].$ (14)

By eliminating $\alpha$ and $\beta_{k}$, the equation satisfied by $E$ can be
obtained.

$\displaystyle E$ $\displaystyle=$
$\displaystyle\Omega+\frac{g^{2}}{\pi}\int_{-\pi}^{\pi}\frac{1+\cos(kN+\phi)}{E-\omega_{c}+2\xi\cos
k}dk,$ (15)

which is Eq. (7) in the main text.

Next, we will prove that $E=\Omega$ is a solution of the above transcendental
equation, that is to prove

$\displaystyle
M=\int_{-\pi}^{\pi}\frac{1+\cos(kN+\phi)}{\Omega-\omega_{c}+2\xi\cos k}dk$
(16)

is always zero. To this end, we rewrite $M$ as

$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\frac{1+\cos(kN+\phi)}{\Omega-\omega_{c}+2\xi\cos
k}dk$ (17) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{-\pi}^{\pi}\frac{2+e^{i(kN+\phi)}+e^{-i(kN+\phi)}}{\Delta+\xi(e^{ik}+e^{-ik})}dk$
$\displaystyle=$ $\displaystyle\frac{1}{2\xi
i}\oint_{|z|<1}\frac{2z^{N}+z^{2N}e^{i\phi}+e^{-i\phi}}{(z-z_{1})(z-z_{2})z^{N}}dz,$

where $\Delta=\Omega-\omega_{c}$ and $z_{1,2}=-(\Delta/2\xi)\pm
i\sqrt{1-(\Delta/2\xi)^{2}}=e^{\pm iK}$ which satisfies $|z_{1,2}|=1$.
Applying the residue theorem to solve the above integral, only the $N$-order
singular points $(z=0)$ is needed to be considered. Therefore,

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{\pi}{\xi}\frac{1}{(N-1)!}\frac{d^{N-1}}{dz^{N-1}}[\frac{2z^{N}+z^{2N}e^{i\phi}+e^{-i\phi}}{(z-z_{1})(z-z_{2})}]|_{z=0}$
(18) $\displaystyle=$ $\displaystyle\frac{\pi
e^{-i\phi}}{2i\xi\sqrt{1-(\Delta/2\xi)^{2}}}(z_{2}^{-N}-z_{1}^{-N})$
$\displaystyle=$
$\displaystyle\frac{\pi}{2i\xi\sqrt{1-(\Delta/2\xi)^{2}}}(e^{-i(KN+\phi)}-e^{i(KN-\phi)})$
$\displaystyle=$ $\displaystyle 0,$

where we have used the condition $G(K)=G(-K)=0$ for the BIC.

From Eq. (14), we will have

$\displaystyle\frac{\beta_{k}}{\alpha}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{N_{c}}}\frac{1+e^{i(kN+\phi)}}{E-\omega_{c}+2\xi\cos
k}.$ (19)

Going back into the real space, we will have

$\displaystyle\frac{\beta_{j}}{\alpha}$ $\displaystyle=$
$\displaystyle\frac{g}{2\pi}\int_{-\pi}^{\pi}\frac{[1+e^{i(kN+\phi)}]e^{-ikj}}{\Omega-\omega_{c}+2\xi\cos
k}dk$ (20) $\displaystyle=$
$\displaystyle\frac{g}{2i\pi\xi}\oint_{|z|<1}\frac{1+z^{N}e^{i\phi}}{(z-z_{1})(z-z_{2})z^{j}}dz.$

When $j<0$, there is no singular point in the integral domain, so

$\displaystyle\frac{\beta_{j}}{\alpha}=\frac{g}{2i\pi\xi}\oint_{|z|<1}\frac{1+z^{N}e^{i\phi}}{(z-z_{1})(z-z_{2})z^{j}}dz=0.$
(21)

When $0\leq j\leq N$,

$\oint_{|z|<1}\frac{z^{N}e^{i\phi}}{(z-z_{1})(z-z_{2})z^{j}}dz=0.$ (22)

Therefore, we will achieve

$\displaystyle\frac{\beta_{j}}{\alpha}$ $\displaystyle=$
$\displaystyle\frac{g}{2i\pi\xi}\oint_{|z|<1}\frac{1}{(z-z_{1})(z-z_{2})z^{j}}dz$
(23) $\displaystyle=$
$\displaystyle\frac{g}{\xi(j-1)!}\frac{d^{j-1}}{dz^{j-1}}[\frac{1}{(z-z_{1})(z-z_{2})}]|_{z=0}$
$\displaystyle=$
$\displaystyle\frac{g}{2i\xi\sqrt{1-(\Delta/2\xi)^{2}}}(z_{1}^{j}-z_{2}^{j})$
$\displaystyle=$
$\displaystyle\frac{g}{2i\xi\sqrt{1-(\Delta/2\xi)^{2}}}(e^{iKj}-e^{-iKj})$
$\displaystyle=$
$\displaystyle\frac{g\sin(Kj)}{\xi\sqrt{1-(\Delta/2\xi)^{2}}}.$

When $j>N$

$\displaystyle\frac{\beta_{j}}{\alpha}$ $\displaystyle=$
$\displaystyle\frac{g}{\xi(j-1)!}\frac{d^{j-1}}{dz^{j-1}}[\frac{1}{(z-z_{1})(z-z_{2})}]|_{z=0}$
(24)
$\displaystyle+\frac{g}{\xi(j-N-1)!}\frac{d^{j-N-1}}{dz^{j-N-1}}[\frac{e^{i\phi}}{(z-z_{1})(z-z_{2})}]|_{z=0}$
$\displaystyle=$
$\displaystyle\frac{g}{2i\xi\sqrt{1-(\Delta/2\xi)^{2}}}(z_{2}^{-j}-z_{1}^{-j}+z_{2}^{-(j-N)}-z_{1}^{-(j-N)})$
$\displaystyle=$
$\displaystyle\frac{ge^{iKj}}{2i\xi\sqrt{1-(\Delta/2\xi)^{2}}}(1+e^{i(KN+\phi)})$
$\displaystyle-\frac{ge^{-iKj}}{2i\xi\sqrt{1-(\Delta/2\xi)^{2}}}(1+e^{i(-KN+\phi)})$
$\displaystyle=$ $\displaystyle 0.$

Up to now, we obtain the photonic distribution in BIC which is given in Eq.
(10) in the main text.

## References

  * (1) F. H. Stillinger, and D. R. Herrick, Bound states in the continuum, Phys. Rev. A 11, 446 (1975).
  * (2) D. C. Marinica, A. G. Borisov, and S. V. Shabanov, Bound States in the Continuum in Photonics, Phys. Rev. Lett. 100, 183902 (2008).
  * (3) M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Surface Bound States in the Continuum, Phys. Rev. Lett. 108, 070401 (2012).
  * (4) M. Kang, T. Liu, C. T. Chan, and M. Xiao, Applications of bound states in the continuum in photonics, Nat. Rev. Phys. 5, 659 (2023).
  * (5) S. I. Azzam, and A. V. Kildishev, Photonic bound states in the continuum: from basics to applications, Advanced Optical Materials 9, 2001469 (2021).
  * (6) Y. Yu, A. Sakanas, A. R. Zali, E. Semenova, K. Yvind, and J. Mørk, Ultra-coherent Fano laser based on a bound state in the continuum, Nat. Photon. 15, 758 (2021).
  * (7) A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, Lasing action from photonic bound states in continuum, Nature 541, 196 (2017).
  * (8) M. Hwang, H. Lee, K. Kim, K. Jeong, S. Kwon, K. Koshelev, Y. Kivshar, and H. Park, Ultralow-threshold laser using super-bound states in the continuum, Nat. Commun. 12, 4135 (2021).
  * (9) L. Carletti, K. Koshelev, C. D. Angelis, and Y. Kivshar, Giant nonlinear response at the nanoscale driven by bound states in the continuum, Phys. Rev. Lett. 121, 033903 (2018).
  * (10) Z. Liu, Y. Xu, Y. Lin, J. Xiang, T. Feng, Q. Cao, J. Li, S. Lan, and J. Liu, High-Q quasibound states in the continuum for nonlinear metasurfaces, Phys. Rev. Lett. 123, 253901 (2019).
  * (11) S. Romano, G. Zito, S. Torino, G. Calafiore, E. Penzo, G. Coppola, S. Cabrini, I. Rendina, and V. Mocella, Label-free sensing of ultralow-weight molecules with all-dielectric metasurfaces supporting bound states in the continuum, Photon. Res. 6, 726 (2018).
  * (12) S. Romano, G. Zito, S. N. Lara Yépez, S. Cabrini, E. Penzo, G. Coppola, I. Rendina, and V. Mocellaark, Tuning the exponential sensitivity of a bound-state-in-continuum ptical sensor, Opt. Express 27, 18776 (2019).
  * (13) Z. Yu, Y. Tong, H. K. Tsang, and X. Sun, High-dimensional communication on etchlesslithium niobate platform with photonic bound states in the continuum, Nat. Commun. 11, 2602 (2020).
  * (14) Z. Yu, and X, Sun, Acousto-optic modulation of photonic bound state in the continuum, Light Sci. Appl. 9, 1 (2020).
  * (15) I. Benea-Chelmus, S. Mason, M. L. Meretska, D. L. Elder, D. Kazakov, A. Shams-Ansari, L. R. Dalton, and F. Capasso, Gigahertz free-space electro-optic modulators based on Mie resonances, Nat. Commun. 13, 3170 (2022).
  * (16) H. Zhang, S. Liu, Z. Guo, S. Hu, Y. Chen, Y. Li, Y. Li, and H. Chen, Topological bound state in the continuum induced unidirectional acoustic perfect absorption, Sci. China-Phys. Mech. Astron. 66, 284311 (2023).
  * (17) C. Chen, C.-J. Yang, and J.-H. An, Exact decoherence-free state of two distant quantum systems in a non-Markovian environment, Phys. Rev. A 93, 062122 (2016).
  * (18) M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekström, G. Johansson, and P. Delsing, Propagating phonons coupled to an artificial atom, Science 346, 207 (2014).
  * (19) G. Andersson, B. Suri, L. Guo, T. Aref, and P. Delsing, Non-exponential decay of a giant artificial atom, Nat. Phys. 15, 1123 (2019).
  * (20) B. Kannan, M. J. Ruckriegel, D. L. Campbell, A. F. Kockum, J. Braumüller, D. K. Kim, M. Kjaergaard, P. Krantz, A. Melville, B. M. Niedzielski, A. Vepsäläinen, R. Winik, J. L. Yoder, F. Nori, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Waveguide quantum electrodynamics with superconducting artificial giant atoms, Nature 583, 775 (2020).
  * (21) A. M. Vadiraj, A. Ask, T. G. McConkey, I. Nsanzineza, C. W. Sandbo Chang, A. F. Kockum, and C. M. Wilson, Engineering the level structure of a giant artificial atom in waveguide quantum electrodynamics, Phys. Rev. A, 103, 023710 (2021).
  * (22) Z.-Q. Wang, Y.-P. Wang, J. Yao, R.-C. Shen, W.-J. Wu, J. Qian, J. Li, S.-Y. Zhu, and J. Q. You, Giant spin ensembles in waveguide magnonics, Nat. Commun. 13, 7580 (2022).
  * (23) L. Guo, A. F. Kockum, F. Marquardt, and G. Johansson, Oscillating bound states for a giant atom, Phys. Rev. Research 2, 043014 (2020)
  * (24) C. A. González-Gutiérrez, J. Román-Roche, and D. Zueco, Distant emitters in ultrastrong waveguide QED: Ground-state properties and non-Markovian dynamics, Phys. Rev. A 104, 053701 (2021).
  * (25) A. F. Kockum, G. Johansson and F. Nori, Decoherence-free interaction between giant atoms in waveguide quantum electrodynamics, Phys. Rev. Lett. 120, 140404 (2018).
  * (26) A. Carollo, D. Cilluffo and F. Ciccarello, Mechanism of decoherence-free coupling between giant atoms, Phys. Rev. Research 2, 043184 (2020).
  * (27) W. Cheng, Z. Wang, and Y.-X. Liu, Topology and retardation effect of a giant atom in a topological waveguide, Phys. Rev. A 106, 033522 (2022).
  * (28) L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, Giant acoustic atom: A single quantum system with a deterministic time delay, Phys. Rev. A 95, 053821 (2017).
  * (29) A. F. Kockum, P. Delsing, and G. Johansson, Designing frequency-dependent relaxation rates and Lamb shifts for a giant artificial atom, Phys. Rev. A 90, 013837 (2014).
  * (30) X. Wang, T. Liu, A. F. Kockum, H.-R. Li, and F. Nori, Tunable chiral bound states with giant atoms, Phys. Rev. Lett. 126, 043602 (2021).
  * (31) X. Wang, Z.-M. Gao, J.-Q. Li, H.-B. Zhu, and H.-R. Li, Unconventional quantum electrodynamics with a Hofstadter-ladder waveguide, Phys. Rev. A 106, 043703 (2022).
  * (32) A. Soro, and A. F. Kockum, Chiral quantum optics with giant atoms, Phys. Rev. A 105, 023712 (2022).
  * (33) D. Wang, C. Zhao, Y. Yan, J. Yang, Z. Wang, and L. Zhou, Topology-dependent giant-atom interaction in a topological waveguide QED system, Phys. Rev. A 93, 062122 (2024).
  * (34) L. Leonforte, A. Carollo, and F. Ciccarello, Vacancy like Dressed States in Topological Waveguide QED, Phys. Rev. Lett. 126, 063601 (2021).
  * (35) W. Zhao, T. Tian, and Z. Wang, Single-photon scattering and bound states in a one-dimensional waveguide with topological giant atom, arXiv: 2401. 02104 (2024).
  * (36) Q. Qiu, Y. Wu, and X. Lü, Collective Radiance of Giant Atoms in Non-Markovian Regime, Sci. China Phys. Mech. Astron. 66, 224212 (2023).
  * (37) K. H. Lim, W. Mok, and L. Kwek, Oscillating bound states in non-Markovian photonic lattices, Phys. Rev. A 107, 023716 (2023).
  * (38) X. Zhang, C. Liu, Z. Gong, and Z. Wang, Quantum interference and controllable magic cavity QED via a giant atom in a coupled resonator waveguide, Phys. Rev. A 108, 013704 (2023).
  * (39) G. Calajó, F. Ciccarello, D. Chang, and P. Rabl, Atom-field dressed states in slow-light waveguide QED, Phys. Rev. A 93, 033833 (2016).
  * (40) L. Zhou, H. Dong, Y. Liu, C. P. Sun, and F. Nori, Quantum supercavity with atomic mirrors, Phys. Rev. A 78, 063827 (2008).
  * (41) L. Qiao, Y.-J. Song, and C.-P. Sun, Quantum phase transition and interference trapping of populations in a coupled-resonator waveguide, Phys. Rev. A 100, 013825 (2021).
  * (42) M. Scigliuzzo, G. Calajò, F. Ciccarello, D. P. Lozano, A. Bengtsson, P. Scarlino, A. Wallraff, D. Chang, P. Delsing, and S. Gasparinetti, Controlling Atom-Photon Bound States in an Array of Josephson-Junction Resonators, Phys. Rev. X 12, 031036 (2022).
  * (43) X. Zhang, E. Kim, D. K. Mark, S. Choi, and O. Painter, A superconducting quantum simulator based on a photonic-bandgap metamaterial, Science 379, 278 (2023).
  * (44) M. Kjaergaard, M.E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Annu. Rev. Condens. Matter Phys. 11, 369 (2020).