# Correlated two-photon scattering in a one-dimensional waveguide coupled to
two- or three-level giant atoms

Wenju Gu<EMAIL_ADDRESS>School of Physics and Optoelectronic
Engineering, Yangtze University, Jingzhou 434023, China  He Huang  School of
Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023,
China  Zhen Yi  School of Physics and Optoelectronic Engineering, Yangtze
University, Jingzhou 434023, China  Lei Chen  School of Physics and
Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China  Lihui
Sun  School of Physics and Optoelectronic Engineering, Yangtze University,
Jingzhou 434023, China  Huatang Tan<EMAIL_ADDRESS>Department of
Physics, Huazhong Normal University, Wuhan 430079, China

###### Abstract

We investigate the two-photon scattering processes in a one-dimensional
waveguide coupled to either a two-level or three-level giant atom. By
manipulating the accumulated phase shift between the two coupling points, we
are able to effectively modify the characteristics of these scattering
processes. Utilizing the Lippmann-Schwinger formalism, we obtain the exact
two-photon interacting scattering wavefunctions of these two systems.
Additionally, analytical expressions for the incoherent power spectra and
second-order correlations are derived. The incoherent spectrum, which is
defined by the correlation of the bound state, provides valuable insights into
photon-photon correlations. It serves as a useful indicator of the degree of
photon-photon correlation between scattered photons. Furthermore, the second-
order correlation function gives a direct measure of the photon-photon
correlation. For photons scattered by the two-level giant atom, manipulating
the accumulated phase shift allows for improvement of the photon-photon
correlation and adjustment of the evolution of the second-order correlation.
In the case of the three-level giant atom, the photon-photon correlation can
be substantially increased. The photon-photon interaction and correlation
distance of the scattered photons can be further enhanced by tuning the
accumulated phase shift. Moreover, the statistical properties can be adjusted
by the control field.

††preprint: APS/123-QED

##  I Introduction

Exploiting atom-photon interactions plays a critical role in the emerging
field of waveguide quantum electrodynamics (QED) [1, 2, 3, 4], which opens up
novel opportunities for both fundamental physics [5, 6, 7] and quantum
information processing [8, 9, 10]. The conventional atom-photon interaction
has been studied within the small atom regime, where the interaction is
localized due to the presence of atoms in a region significantly smaller than
the wavelength of photons. However, recent investigations have focused on a
new paradigm known as “giant atoms”, which exhibits nonlocal couplings. This
has been achieved by coupling artificial atoms to propagating fields (e.g.,
surface acoustic waves) with wavelengths smaller than atomic sizes [11], or
through meandering waveguides at separated points [12, 13]. The introduction
of nonlocal interactions enables the occurrence of various remarkable
phenomena that are not achievable in the case of small atoms. These include
frequency-dependent Lamb shifts and relaxation rates [14], waveguide-mediated
decoherence-free subspaces [15, 16], nonexponential decay [11, 17, 18, 19],
oscillating bound states [20], and the enhanced spontaneous sudden birth of
entanglement [21].

In waveguide QED systems, the presence of a one-dimensional (1D) continuum of
modes allows for a broader range of field bandwidths and facilitates strong
coupling to local emitters by reducing the mode volume [1]. The study of few-
photon scattering is well-established in the strong-coupling regime, where the
interaction between light and the emitter dominates over loss and dephasing
effects. Various theoretical techniques have been developed to investigate
few-photon scattering in the context of small atoms. These techniques include
the real-space Bethe-ansatz method [22, 23, 24, 25, 26], the Lehmann-Symanzik-
Zimmermann reduction [27, 28], the wave-packet evolution approach [29, 30],
the Lippmann-Schwinger (LS) formalism [31, 32, 33], input-output formalism
[34, 35], as well as the Dyson series summation [36].

The phenomenon of photon scattering in waveguide QED systems involving giant
atoms has gained significant attention due to the unique interference effects
[37, 38, 39, 40, 41, 42, 43]. It is worth noting that the majority of studies
in this area primarily focus on single-photon interactions, with relatively
limited exploration of few-photon scattering within the giant atom regime. The
presence of nonlinearity in atomic systems would induce photon-photon
correlations, and such photon-photon correlations can, in turn, influence the
transmission and reflection of photons through phenomena like induced
tunneling or blocking of photons [44, 45, 46, 47]. Therefore, gaining a clear
understanding of the nature of photon-photon correlations is crucial to
comprehending the functioning of various quantum devices, such as a switchable
mirror [48] or a single-photon router [49]. In the context of few-photon
scattering, the wavefunctions typically exhibit a common structure: the two-
particle plane wave with momenta of photons rearranged and the bound state.
The plane-wave component arises from the coherent scattering, while the bound
state component originates from the incoherent scattering. The bound state
decays exponentially as the distance between the two photons increases, and is
associated with the two-particle irreducible T-matrix in scattering theory
[50]. Recently, the eigenstates of the scattering matrix have been
successfully obtained for the even mode in a two-level giant system using a
reasonable wavefunction hypothesis [51].

In this paper, we employ the LS formalism to analyze the two-photon scattering
processes involving both two- and three-level giant atoms coupled to a 1D
waveguide. By utilizing this approach, we are able to obtain the analytical
two-photon interacting scattering wavefunctions for these systems.
Additionally, the incoherent power spectrum is derived from the correlation of
the bound state, with the total flux serving as an indicator of photon-photon
correlation. The second-order correlation function provides a direct measure
of photon-photon correlation. Through our analysis, we find that the
accumulated phase shift can be utilized to enhance the photon-photon
correlation and control the evolution of the second-order correlation for
photons scattered by the two-level giant atom. In comparison to the two-level
giant atom, the photon-photon correlation can be substantially enhanced for
the system of the three-level giant atom. Moreover, we find that tuning the
accumulated phase shift between the two coupling points allows for further
improvement of photon-photon interaction and an increase in the decay distance
of the second-order correlation. These properties can be explained by the
poles of these systems. Furthermore, the statistics property of the scattering
photons in the three-level giant atom system can be controlled by adjusting
the control laser at the resonance condition. In the presence of the control
field, the incident photons pass by the system coherently, thereby maintaining
unchanged statistics. However, in the absence of the control field, the system
effectively behaves as a two-level model, resulting in bunched transmitted
photons and antibunched reflected photons.

The paper is organized as follows. In Sec. II, we analyze the two-photon
scattering process in the system of the two-level giant atom coupled to a 1D
waveguide. We investigate various aspects, including the derivation of the
two-photon interacting scattering eigenstate, the incoherent power spectrum,
and the second-order correlation function. In Sec. III, we extend our analysis
to the case of the three-level giant atom, where we examine similar quantities
such as the two-photon interacting scattering eigenstate, the incoherent power
spectrum, and the second-order correlation function. The conclusions drawn
from our study are given in Sec. IV. Additionally, a summary of technical
details relevant to the calculation of the incoherent power spectrum is
presented in Appendix.

##  II The system of a two-level giant atom coupled to a 1D waveguide

Figure 1: Schematic illustration of a two-level giant atom side coupled to a
one-dimensional (1D) waveguide. The atomic transition between the ground state
$\left|{g}\right\rangle$ and the excited state $\left|{e}\right\rangle$ is
coupled to waveguide modes positioned at $x=-d/2$ and $d/2$. The coupling
strength is denoted as $V$.

We begin by considering the correlated two-photon dynamics in a system
consisting of a two-level giant atom side coupled to a 1D waveguide, as shown
in Fig. 1. The configuration can be feasibly realized experimentally by
establishing a coupling between a superconducting qubit and either meandering
microwave transmission lines [13, 12] or propagating surface acoustic waves
(SAWs) [11, 52]. In the microwave transmission line setup, the size of the
qubit is significantly smaller than the wavelength, thereby enabling us to
treat it as a point-like object at each coupling point. In the SAWs system,
the interdigital transducer (IDT) forms the two transmon islands and couples
with the SAWs propagating on the substrate. The entire IDT consists of two
local IDTs at opposite ends, which are sufficiently distant from each other.
Each IDT consists of $N$ pairs of fingers, arranged such that the distance
between neighboring finger pairs matches the corresponding phonon wavelength,
allowing for a constant coupling. Moreover, the distance between the centers
of the two local IDTs can be made to be considerably larger than the SAW
wavelengths, permitting us to approximate the interaction as point couplings.
For convenience, let us assume that the atomic transition
$\left|{g}\right\rangle\leftrightarrow\left|{e}\right\rangle$ interacts with
the waveguide field at the coupling points of $x=-d/2$ and $d/2$,
respectively. While the scattering process involving single photons has been
extensively studied in previous works [43, 53], our focus in this paper lies
on the investigation of two-photon dynamics in order to explore photon-photon
interactions. In real space, the Hamiltonian governing the system can be
written as ($\hbar=1$ hereafter)

$\displaystyle\hat{H}=$
$\displaystyle\left(\omega_{0}-i\gamma_{e}/2\right)\hat{\sigma}_{ee}$
$\displaystyle-i\nu_{g}\int
dx\left[\hat{a}^{\dagger}_{R}(x)\partial_{x}\hat{a}_{R}(x)-\hat{a}^{\dagger}_{L}(x)\partial_{x}\hat{a}_{L}(x)\right]$
$\displaystyle+\frac{V}{2}\sum_{\alpha=R,L}\int
dxM(x)\left[\hat{a}^{\dagger}_{\alpha}(x)\hat{\sigma}^{-}+\hat{\sigma}^{+}\hat{a}_{\alpha}(x)\right].$
(1)

Here $M(x)=\delta(x+d/2)+\delta(x-d/2)$ denotes the coupling points. The
operators $\hat{a}^{\dagger}_{R}(x)$ and $\hat{a}^{\dagger}_{L}(x)$ are the
creation operators for the right-moving and left-moving modes in real space,
and $\nu_{g}$ is the group velocity. We will take $\nu_{g}=1$ for simplicity.
For the two-level giant atom, $\omega_{0}$ is the atomic transition frequency,
and $\gamma_{e}$ is the spontaneous emission rate of the excited state to
modes other than the waveguide continuum. The atomic transition
$\left|{g}\right\rangle\leftrightarrow\left|{e}\right\rangle$ couples to the
waveguide modes at $x=-d/2$ and $x=d/2$ with an equal strength
$V=\sqrt{2\Gamma}$, where $\Gamma$ denotes the atomic spontaneous decay rate
to the waveguide continuum. In the strong coupling limit, the value of Purcell
factor $P$ is large, i.e., $P=\Gamma/\gamma_{e}\gg 1$. For instance,
$\Gamma/2\pi=2\text{MHz}$ and $\gamma_{e}/2\pi=0.03\text{MHz}$ are
demonstrated in the experimental setup [12]. This indicates that the
spontaneous decay is mainly to the waveguide compared with all the other
modes. In this case, the atomic spontaneous dissipation rate $\gamma_{e}$ can
be neglected.

The total excitation number is conserved for the Jaynes-Cummings model of
interaction between the atom and waveguide field. Thus, in the single-
excitation subspace, the eigenstate of the system can be written in the form

$\displaystyle\left|{\phi_{1}(k)}\right\rangle_{\alpha}=$ $\displaystyle\int
dx\left[\phi_{R}^{\alpha}(k,x)\hat{a}^{\dagger}_{R}(x)+\phi_{L}^{\alpha}(k,x)\hat{a}^{\dagger}_{L}(x)\right]\left|{0,g}\right\rangle$
$\displaystyle+u_{e}^{\alpha}(k)\left|{0,e}\right\rangle,$ (2)

where $\alpha=\\{R,L\\}$, and $\phi_{R/L}^{\alpha}(k,x)$ denote the
probability amplitudes of creating the right-moving and left-moving photons in
real space for the incident photon with the wavevector $k$ in the
$\alpha$-direction. Here $u_{e}^{\alpha}(k)$ is the excitation amplitude of
the atom, and $\left|{0,g}\right\rangle$ denotes the vacuum state of the
system. The probability amplitudes can be determined by the Schrödinger
equation
$\hat{H}\left|{\phi_{1}(k)}\right\rangle_{\alpha}=k\left|{\phi_{1}(k)}\right\rangle_{\alpha}$,
which gives

$\displaystyle\left(\omega_{0}-k\right)u_{e}^{\alpha}(k)+\sqrt{\frac{\Gamma}{2}}\sum_{\alpha^{\prime}=R,L}\int
dxM(x)\phi_{\alpha^{\prime}}^{\alpha}(k,x)$ $\displaystyle=0,$
$\displaystyle(-i\partial_{x}-k)\phi_{R}^{\alpha}(k,x)+\sqrt{\frac{\Gamma}{2}}M(x)u_{e}^{\alpha}(k)$
$\displaystyle=0,$
$\displaystyle(i\partial_{x}-k)\phi_{L}^{\alpha}(k,x)+\sqrt{\frac{\Gamma}{2}}M(x)u_{e}^{\alpha}(k)$
$\displaystyle=0.$ (3)

The solution of the wavefunction takes the following form

$\displaystyle\phi_{R}^{R}(k,x)$
$\displaystyle=\frac{e^{ikx}}{\sqrt{2\pi}}\big{\\{}\theta(-d/2-x)+t_{1}(k)\big{[}\theta(x+d/2)$
$\displaystyle-\theta(x-d/2)\big{]}+t_{2}(k)\theta(x-d/2)\big{\\}},$
$\displaystyle\phi_{L}^{R}(k,x)$
$\displaystyle=\frac{e^{-ikx}}{\sqrt{2\pi}}\big{\\{}r_{1}(k)\theta(-d/2-x)$
$\displaystyle+r_{2}(k)\left[\theta(x+d/2)-\theta(x-d/2)\right]\big{\\}},$
$\displaystyle\phi_{R}^{L}(k,x)$
$\displaystyle=\frac{e^{ikx}}{\sqrt{2\pi}}\big{\\{}r_{1}(k)\theta(x-d/2)$
$\displaystyle+r_{2}(k)\left[\theta(x+d/2)-\theta(x-d/2)\right]\big{\\}},$
$\displaystyle\phi_{L}^{L}(k,x)$
$\displaystyle=\frac{e^{-ikx}}{\sqrt{2\pi}}\big{\\{}\theta(x-d/2)+t_{1}(k)\big{[}\theta(x+d/2)$
$\displaystyle-\theta(x-d/2)\big{]}+t_{2}(k)\theta(-d/2-x)\big{\\}},$ (4)

where $\theta(x)$ is the Heaviside step function. Here, $t_{2}(k)$ and
$r_{1}(k)$ denote the single-photon transmission and reflection amplitudes in
the regions where $x>d/2$ and $x<-d/2$, respectively. Additionally, $t_{1}(k)$
and $r_{2}(k)$ are the probability amplitudes for transmission and reflection
between two coupling points $-d/2<x<d/2$. These amplitudes are calculated as

$\displaystyle t_{1}(k)$
$\displaystyle=\frac{\omega_{0}-k-i\Gamma/2\left(1+e^{ikd}\right)}{\omega_{0}-k-i\Gamma\left(1+e^{ikd}\right)},$
$\displaystyle t_{2}(k)$ $\displaystyle=\frac{\omega_{0}-k+\Gamma\sin
kd}{\omega_{0}-k-i\Gamma\left(1+e^{ikd}\right)},$ $\displaystyle r_{1}(k)$
$\displaystyle=\frac{i\Gamma\left(1+\cos
kd\right)}{\omega_{0}-k-i\Gamma\left(1+e^{ikd}\right)},$ $\displaystyle
r_{2}(k)$
$\displaystyle=\frac{i\Gamma/2\left(1+e^{ikd}\right)}{\omega_{0}-k-i\Gamma\left(1+e^{ikd}\right)},$
$\displaystyle u_{e}^{\alpha}(k)$
$\displaystyle=\frac{-\sqrt{\Gamma/\pi}\cos(kd/2)}{\omega_{0}-k-i\Gamma\left(1+e^{ikd}\right)}.$
(5)

In order to study the two-photon scattering process, we employ the LS equation
to obtain the eigenstate for two-photon scattering [31, 32]. To account for
the nonlinear effect of the two-level atom, we use a bosonic representation
which includes an on-site interaction [54, 31],

$\displaystyle H=H_{0}+V,\hskip
5.0ptV=\frac{U}{2}\hat{d}^{\dagger}\hat{d}(\hat{d}^{\dagger}\hat{d}-1).$ (6)

The atomic operators $\hat{\sigma}^{\pm}$ in $\hat{H}_{0}$ are replaced by the
bosonic creation and annihilation operators $\hat{d}^{\dagger}$ and $\hat{d}$,
respectively. To map the atomic ground and excited states to zero- and one-
boson states, $U\rightarrow\infty$ should be taken in the end to eliminate
occupations greater than $1$. In this bosonic representation, when $U=0$, the
Hamiltonian corresponds to a non-interacting Hamiltonian that can be easily
solved. The non-interacting two-photon eigenstate is given by

$\displaystyle\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=\frac{1}{\sqrt{2}}\left|{\phi_{1}(k_{1})}\right\rangle_{\alpha_{1}}\otimes\left|{\phi_{1}(k_{2})}\right\rangle_{\alpha_{2}}.$
(7)

The LS equation establishes a relationship between the fully interacting two-
photon eigenstates
$\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$ and
$\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$ as

$\displaystyle\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=$
$\displaystyle\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$
$\displaystyle+G^{R}(E)V\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}},$
(8)

where $G^{R}(E)=1/(E-H_{0}+i0^{+})$ is the retarded Green’s function, and
$E=k_{1}+k_{2}$ is the total energy of two incident photons. The two-particle
identity operator in real space can be expressed as

$\displaystyle I_{2}=$ $\displaystyle
I_{2}^{x}\otimes\left|{0}\right\rangle\left\langle{0}\right|+I_{1}^{x}\otimes\left|{d}\right\rangle\left\langle{d}\right|+I_{0}^{x}\otimes\left|{dd}\right\rangle\left\langle{dd}\right|,$
$\displaystyle I_{n}^{x}=$ $\displaystyle\sum_{\alpha_{1}\cdots\alpha_{n}}\int
dx_{1}\cdots dx_{n}\left|{x_{1}\cdots
x_{n}}\right\rangle_{\alpha_{1}\cdots\alpha_{n}}\left\langle{x_{1}\cdots
x_{n}}\right|,$ (9)

where $\left|{0}\right\rangle$ is the ground state, and we denote the single-
excitation state
$\left|{d}\right\rangle=\hat{d}^{\dagger}\left|{0}\right\rangle$ and the two-
excitation state $\left|{dd}\right\rangle=\hat{d}^{\dagger
2}\left|{0}\right\rangle/\sqrt{2}$. By inserting the identity operator into
Eq. (8), we have

$\displaystyle\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$
$\displaystyle=\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}+G^{R}(E)VI_{2}\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$
$\displaystyle=\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}+UG^{R}(E)\left|{dd}\right\rangle\langle{dd}\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}.$
(10)

To determine the two-photon interacting eigenstate, we project Eq. (10) onto
$\left\langle{dd}\right|$:

$\displaystyle\langle{dd}\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=$
$\displaystyle\left(1-UG_{dd}\right)^{-1}$
$\displaystyle\times\langle{dd}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}},$
(11)

where $G_{dd}=\left\langle{dd}\right|G^{R}(E)\left|{dd}\right\rangle$. Next,
we project Eq. (10) onto a two-photon basis state
${}_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\left\langle{x_{1}x_{2}}\right|$
and taking the limit $U\rightarrow\infty$, the two-photon interacting
eigenstate eventually becomes

${}_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\langle
x_{1}x_{2}\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\langle
x_{1}x_{2}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$
$\displaystyle-
G_{xd}^{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}(x_{1},x_{2})G_{dd}^{-1}\langle
dd\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}},$ (12)

where
$G_{xd}^{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}(x_{1},x_{2})=_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\left\langle{x_{1}x_{2}}\right|G^{R}(E)\left|{dd}\right\rangle$,
and $x_{1}$ and $x_{2}$ refer to the positions of the photons. The first term
denotes the plane-wave state, while the second term contains all the
nonlinearity and is commonly referred to as the two-photon bound state. In
order to solve the Green’s functions, we utilize the two-photon non-
interacting scattering eigenstates to establish a two-particle identity in
momentum space

$\displaystyle I_{2}^{\prime}=\sum_{\alpha_{1},\alpha_{2}}\int
dk_{1}dk_{2}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}\left\langle{\phi_{2}(k_{1},k_{2})}\right|.$
(13)

The Green’s functions can be finally derived in the following form

$\displaystyle G_{dd}$ $\displaystyle=\sum_{\alpha_{1},\alpha_{2}}\int
dk_{1}dk_{2}\frac{\langle
dd\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}\left\langle{\phi_{2}(k_{1},k_{2})}\right|dd\rangle}{E-k_{1}-k_{2}+i0^{+}},$
$\displaystyle G_{xd}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})$
$\displaystyle=\sum_{\alpha_{1}^{\prime},\alpha_{2}^{\prime}}\int
dk_{1}dk_{2}\frac{{}_{\alpha_{1}\alpha_{2}}\langle
x_{1}x_{2}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\left\langle{\phi_{2}(k_{1},k_{2})}\right|dd\rangle}{E-k_{1}-k_{2}+i0^{+}},$
(14)

with the elements

$\displaystyle\langle
dd\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$
$\displaystyle=u_{e}^{\alpha_{1}}(k_{1})u_{e}^{\alpha_{2}}(k_{2}),$
${}_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\langle
x_{1}x_{2}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$
$\displaystyle=\frac{1}{2}\Big{[}\phi_{\alpha_{1}^{\prime}}^{\alpha_{1}}(k_{1},x_{1})\phi_{\alpha_{2}^{\prime}}^{\alpha_{2}}(k_{2},x_{2})$
$\displaystyle+\phi_{\alpha_{1}^{\prime}}^{\alpha_{2}}(k_{2},x_{1})\phi_{\alpha_{2}^{\prime}}^{\alpha_{1}}(k_{1},x_{2})\Big{]}.$
(15)

When the propagating time of photons between the coupling points is shorter
than the atomic lifetime $1/\Gamma$, the Markovian approximation can be
applied [55]. In this case, the wavevector $k$ in the phase factors can be
replaced by a constant $k_{0}=\omega_{0}/\nu_{g}$, and the accumulated phase
shift between the coupling points is denoted as $\vartheta=k_{0}d$. The phase
shift can be adjusted by changing the distance between these points. Then, by
performing double integrals with the use of the standard contour integral
techniques, the Green’s functions become

$\displaystyle G_{dd}$
$\displaystyle=\frac{1}{E-2\omega_{0}+i2\Gamma^{\prime}},$ $\displaystyle
G_{xd}^{RR}(x_{1},x_{2})$
$\displaystyle=\frac{-\Gamma(1+\cos\vartheta)}{E-2\omega_{0}+i2\Gamma^{\prime}}e^{iEx_{c}}e^{i(E/2-\omega_{0})\left|{x}\right|-\Gamma^{\prime}\left|{x}\right|},$
(16)

where $\Gamma^{\prime}=\Gamma(1+e^{i\vartheta})$. During the contour
integration, the conditions $x_{1}>d/2$, $x_{2}>d/2$, and $x=x_{2}-x_{1}$,
$x_{c}=(x_{1}+x_{2})/2$ are used. It can be proven that
$G_{xd}^{LL}(-x_{1},-x_{2})=G_{xd}^{RR}(x_{1},x_{2})=G_{xd}^{RL}(x_{1},-x_{2})=G_{xd}^{LR}(-x_{1},x_{2})$
because of the parity symmetry. Substituting these expressions into Eq. (12),
the two-photon interacting eigenstate becomes

$\displaystyle\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{RR}=$
$\displaystyle\int
dx_{1}dx_{2}\Bigg{[}\frac{f_{RR}(x_{1},x_{2})}{\sqrt{2}}\hat{a}^{\dagger}_{R}(x_{1})\hat{a}^{\dagger}_{R}(x_{2})$
$\displaystyle+\frac{f_{LL}(x_{1},x_{2})}{\sqrt{2}}\hat{a}^{\dagger}_{L}(x_{1})\hat{a}^{\dagger}_{L}(x_{2})$
$\displaystyle+f_{RL}(x_{1},x_{2})\hat{a}^{\dagger}_{R}(x_{1})\hat{a}^{\dagger}_{L}(x_{2})\Bigg{]}\left|{0}\right\rangle.$
(17)

The coefficients are

$\displaystyle f_{RR}(x_{1},x_{2})=$
$\displaystyle\frac{e^{iEx_{c}}}{\sqrt{2}\pi}\left[t_{2}(k_{1})t_{2}(k_{2})\cos\Delta_{1}x+B_{k_{1}k_{2}}(x)\right],$
$\displaystyle f_{LL}(x_{1},x_{2})=$
$\displaystyle\frac{e^{-iEx_{c}}}{\sqrt{2}\pi}\left[r_{1}(k_{1})r_{1}(k_{2})\cos\Delta_{1}x+B_{k_{1}k_{2}}(x)\right],$
$\displaystyle f_{RL}(x_{1},x_{2})=$
$\displaystyle\frac{e^{iEx/2}}{2\pi}\Big{[}t_{2}(k_{1})r_{1}(k_{2})e^{2i\Delta_{1}x_{c}}$
$\displaystyle+r_{1}(k_{1})t_{2}(k_{2})e^{-2i\Delta_{1}x_{c}}+2B_{k_{1}k_{2}}(x_{c})\Big{]},$
$\displaystyle B_{k_{1}k_{2}}(x)=$
$\displaystyle\frac{\Gamma^{2}(1+\cos\vartheta)^{2}e^{i(E/2-\omega_{0})|x|-\Gamma^{\prime}|x|}}{\left(E/2-\omega_{0}+i\Gamma^{\prime}\right)^{2}-\Delta_{1}^{2}},$
(18)

where $\Delta_{1}=(k_{1}-k_{2})/2$ corresponds to half of the energy
difference between two incident photons. For the case of $\vartheta=0$, the
interacting eigenstate can be simplified to the same form as that of the small
two-level atom scenario with a single coupling point in Ref. [56] by replacing
$\Gamma$ with $\Gamma/4$. The factor $4$ arises from the constructive
interference between two coupling points.

For the input of the two-particle plane wave, the scattering wavefunction
exhibits a common constituent: the two-particle plane wave with rearranged
momenta of the photons and the bound state. The plane wave originates from the
coherent scattering, while the bound state originates from the incoherent
scattering. As the distance between the two photons increases, the bound state
decays exponentially. The bound state can also be referred to as the two-
particle irreducible T-matrix in scattering theory [50].

###  II.1 Incoherent power spectrum

The two-photon interacting eigenstate is composed of two components: the plane
wave arising from coherent scattering, and the bound state arising from
photon-photon interactions. To investigate their impacts on scattering
processes, we first consider the power spectrum or resonance fluorescence,
which is the Fourier transform of the first-order correlation function,

$\displaystyle S_{\alpha}(\omega)=\int dte^{-i\omega
t}\left\langle{\psi_{2}}\right|\hat{a}^{\dagger}_{\alpha}(x_{0})\hat{a}_{\alpha}(x_{0}+t)\left|{\psi_{2}}\right\rangle,$
(19)

where $x_{0}$ is the position of the detector located far away from the
scattering region. Here $S_{\alpha}(\omega)$ accounts for the spectral
decomposition of the photons in the two-photon interacting wavefunction
$\left|{\psi_{2}}\right\rangle$. The detailed derivation of the power spectrum
is presented in Appendix A. In general, the power spectrum consists of the
coherent and incoherent parts, i.e.,
$S_{\alpha}(\omega)=S_{\alpha}^{\text{coh}}(\omega)+S_{\alpha}^{\text{incoh}}(\omega)$.
The coherent scattering component contributes a $\delta$-function, while the
incoherent scattering component is determined by the correlation of the bound
state in the wavefunction, as shown in Eq. (54).

To explore the physical implications of the incoherent power spectrum, we
consider the case where $\alpha=R$, specifically focusing on the two-photon
transmission state. In this case, the incoherent power spectrum can be
expressed in the form

$\displaystyle
S_{R}^{\text{incoh}}(\omega)=\frac{4}{\pi^{2}}\left|{\Upsilon(\omega)}\right|^{2},$
(20)

where

$\displaystyle\Upsilon(\omega)=$
$\displaystyle\frac{\Gamma^{2}(1+\cos\vartheta)^{2}}{(E/2-\omega_{0}+i\Gamma^{\prime})}$
$\displaystyle\times\frac{1}{(E-\omega_{0}-\omega+i\Gamma^{\prime})(\omega-\omega_{0}+i\Gamma^{\prime})}.$
(21)

This expression is provided in Appendix A. When transforming the transmitted
state from real space to frequency space, it can be represented as
$t_{2}(\omega_{1})t_{2}(\omega_{2})\hat{a}_{R}^{\dagger}(\omega_{1})\hat{a}_{R}^{\dagger}(\omega_{2})\left|{0}\right\rangle+\frac{i}{\pi}\int
d\omega\Upsilon(\omega)\hat{a}_{R}^{\dagger}(E-\omega)\hat{a}^{\dagger}_{R}(\omega)\left|{0}\right\rangle$.
The first term describes the independent propagation of the two photons, while
the second term represents the final state of the two photons after undergoing
inelastic scattering. According to the principle of energy conservation, the
two scattered photons are always generated in pairs with frequencies of
opposite signs. The coefficient $\Upsilon(\omega)$ quantifies the generation
of these photon pairs. Therefore, the incoherent power spectrum serves as a
direct measure of the production of photon pairs with a frequency of $\omega$.

Furthermore, the total incoherent power spectrum is defined by

$\displaystyle F(k)=$ $\displaystyle\sum_{\alpha}\int d\omega
S_{\alpha}^{\text{incoherent}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{4}{\pi}\int dxB_{k_{1}k_{2}}^{*}(x)B_{k_{1}k_{2}}(x).$
(22)

It serves as a quantitative measure for the overall strength of correlations,
and also provides a direct measure of the bound-state term. Under the
assumption of a narrow bandwidth of incident photons, where the spectral width
of the wave packet is significantly smaller than $\Gamma$, the wave packet
approaches to a $\delta$ function. This implies that the incident photons have
the equal frequency $k_{1}=k_{2}=k$. In this case, the expression for the
total incoherent power spectrum can be simplified as

$\displaystyle
F(k)=\frac{4\Gamma^{3}(1+\cos\vartheta)^{3}}{\pi\left|{k-\omega_{0}+i\Gamma^{\prime}}\right|^{4}}=\frac{4}{\pi}\frac{\left|{B_{k_{1}k_{2}}(0)}\right|^{2}}{\Gamma(1+\cos\vartheta)},$
(23)

which is consistent with the single coupling point case (by replacing $\Gamma$
with $\Gamma/4$) when $\vartheta=0$ [57].

Figure 2: The total incoherent power spectrum $F(k)$ as a function of the
incident frequency $k$ with different values of the accumulated phase shift
$\vartheta$. The other parameter is $\omega_{0}=100\Gamma$.

The total incoherent power spectrum $F(k)$ as a function of the incident
frequency $k$ is shown in Fig. 2. A large value of $F(k)$ indicates strong
correlation effects, since the incoherent scattering originates from the
correlation of the bound state. Therefore, the peak value of $F(k)$
corresponds to the strongest correlation, and $k_{\text{peak}}$ denotes the
optimal incident frequency to obtain photon-photon correlations. It also shows
that the peak value of $F(k)$ and $k_{\text{peak}}$ vary with the phase shift
$\vartheta$. For example, when $\vartheta=0.85\pi$, it can be observed that
the peak value of $F(k)$ increases by approximately an order of magnitude
compared to its value at $\vartheta=0$. This implies a significant enhancement
of photon-photon correlations. Physically, the position and width of the peak
can be explained by the pole of the system, which corresponds to the zero of
the denominator in the single-photon transmission or reflection amplitude
$t(k)$ or $r(k)$. We denote the pole as $z=\tilde{\omega}-i\tilde{\Gamma}$,
where the real part $\tilde{\omega}=\omega_{0}+\Gamma\sin\theta$ represents
the eigenfrequency and $\tilde{\Gamma}=\Gamma(1+\cos\theta)$ denotes the
effective decay rate. The position of the peak corresponds to the
eigenfrequency $\tilde{\omega}$, while its width is determined by
$\tilde{\Gamma}$. Moreover, the peak value is given by
$F_{\text{peak}}=4/\pi\tilde{\Gamma}$. In comparison to the single coupling
point of a small atom, where the peak value is $8/\pi\Gamma$, it can be
surpassed in the giant atom with $\vartheta\subset(2\pi/3,5\pi/3)$. This
indicates that the smaller effective decay rate can enhance photon-photon
correlations, and it is adjustable through the change of the phase shift
$\vartheta$ in the two-level giant atom system.

###  II.2 Second-order correlation function

Next, we utilize the second-order correlation function to demonstrate the
spatial interaction between photons [58]. The second-order correlation
function of the transmitted and reflected fields ($x_{1}>d/2$, $x_{2}>d/2$ and
$x=x_{2}-x_{1}$) are defined as follows:

$\displaystyle G_{\alpha}^{(2)}(x)=$
$\displaystyle\left\langle{\psi_{2}}\right|\hat{a}^{\dagger}_{\alpha}(x_{1})\hat{a}^{\dagger}_{\alpha}(x_{2})\hat{a}_{\alpha}(x_{2})\hat{a}_{\alpha}(x_{1})\left|{\psi_{2}}\right\rangle$
$\displaystyle=2\left|{f_{\alpha\alpha}(x_{1},x_{2})}\right|^{2}.$ (24)

This correlation function represents the probability of detecting a photon at
$x_{2}$ after detecting the first one at $x_{1}$. The expression is directly
proportional to the rate at which two photons are transmitted or reflected,
and is determined by the interference between the plane-wave term and the
bound-state term. In order to briefly illustrate the effect of the bound
state, we examine the difference between the probability of two-photon
detection and the independent single-photon detection when $x=0$, under the
condition that $k_{1}=k_{2}=k$, denoted as
$\chi_{R}=2\pi^{2}\left|{f_{RR}(0)}\right|^{2}-\left|{t_{2}(k)}\right|^{4}$
for the transmitted field and
$\chi_{L}=2\pi^{2}\left|{f_{LL}(0)}\right|^{2}-\left|{r_{1}(k)}\right|^{4}$
for the reflected field. If $\chi_{R}>0$, it indicates that the bound state
enhances the transmission of two photons, resulting in a phenomenon known as
photon-induced tunneling, which serves as a signature of photon bunching.
Conversely, if $\chi_{R}<0$, it implies that the bound state can suppress the
transmission of two photons, leading to photon blockade [44]. In Fig. 3, it is
shown that $\chi_{R}>0$, indicating that the transmitted photons are bunched,
while $\chi_{L}<0$, suggesting that reflected photons are antibunched.
Therefore, the statistical properties of photons are determined by the
interference between the plane-wave and the bound-state terms.

Figure 3: The difference between the probability of two-photon detection and
the independent single-photon detection when $x=0$, denoted as
$\chi_{R}=2\pi^{2}\left|{f_{RR}(0)}\right|^{2}-\left|{t_{2}(k)}\right|^{4}$
for the transmitted field and
$\chi_{L}=2\pi^{2}\left|{f_{LL}(0)}\right|^{2}-\left|{r_{1}(k)}\right|^{4}$
for the reflected field, as functions of the incident frequency $k$ and the
accumulated phase shift $\vartheta$. The other parameter is
$\omega_{0}=100\Gamma$.

It is also useful to introduce the normalized second-order correlation
function [32, 59, 60]

$\displaystyle
g_{\alpha}^{(2)}(x)=\frac{G^{(2)}_{\alpha}(x)}{\left|{{}_{\alpha}\langle
x_{1}\left|{\phi_{1}(k_{1})}\right\rangle_{R}}\right|^{2}\left|{{}_{\alpha}\langle
x_{2}\left|{\phi_{1}(k_{2})}\right\rangle_{R}}\right|^{2}}.$ (25)

This function is normalized by the single-photon transmission and reflection
probability. Furthermore, for incident photons with equal frequency, i.e.,
$k_{1}=k_{2}=k$, and by substituting the explicit expressions, the normalized
second-order correlation functions can be simplified as

$\displaystyle g^{(2)}_{R}(x)$
$\displaystyle=\left|{1+\frac{\Gamma^{2}(1+\cos\vartheta)^{2}e^{i(k-\omega_{0})\left|{x}\right|-\Gamma^{\prime}\left|{x}\right|}}{(k-\omega_{0}-\Gamma\sin\vartheta)^{2}}}\right|^{2},$
$\displaystyle g^{(2)}_{L}(x)$
$\displaystyle=\left|{1-e^{i(k-\omega_{0})\left|{x}\right|-\Gamma^{\prime}\left|{x}\right|}}\right|^{2}.$
(26)

These correlation functions are depicted in Fig. 4. Here we choose the
frequency of the input field to be resonant with the atomic transition
frequency, i.e., $k=\omega_{0}$. As a result, the second-order correlation
function is primarily determined by $\Gamma^{\prime}$. When $\vartheta=0$, the
behavior of $g^{(2)}_{R/L}(x)$ resembles that of the single coupling point,
which has been extensively investigated in theory as well as experiment with
microwave photons [49]. The transmitted photons exhibit bunching behavior,
while the reflected photons display anti-bunching. The correlations quickly
reach the value of $1$ with little structure. Therefore, the initial value
$g^{(2)}_{R/L}(0)$ makes a good prediction of the overall correlation nature.
It should be noted here that at resonance, i.e., $k=\omega_{0}$, the single-
photon transmission rate becomes zero, leading to a divergence in the
normalized second-order correlation function of $g^{(2)}_{R}(x)$. In physics,
$g^{(2)}_{L}(0)=0$ of the reflected field is due to that the two-level atom
can only absorb and emit one photon at a time [49].

Figure 4: The normalized second-order correlation functions, denoted as
$g^{(2)}_{R}(x)$ for transmitted photons (a) and $g^{(2)}_{L}(x)$ for
reflected photons (b), as a function of $x$ with different values of
$\vartheta$. The other parameters are $\omega_{0}=100\Gamma$ and
$k=\omega_{0}$.

However, when $\vartheta$ takes values of $0.5\pi$, $0.75\pi$, and $0.85\pi$,
$g^{(2)}_{R}(x)$ exhibits oscillations between bunching and antibunching.
Moreover, it takes a longer distance to reach the value of $1$. This behavior
indicates that the photons become periodically organized in time and space. In
contrast to the behavior observed in the case of $\vartheta=0$, where the
initial correlation can predict whether the system generates bunching or
antibunching of photons, the oscillatory feature presents a challenge in using
the initial correlation as a reliable predictor. These features can be
explained by the pole of the system, which appears in the exponential factor
of Eq. (26). The oscillation arises from beating between the eigenfrequency
and the incident frequency, and its duration is determined by the effective
decay rate. Therefore, for $\vartheta=0$, the effective decay rate reaches the
maximum value $2\Gamma$ and $g^{(2)}_{R/L}(x)$ rapidly reaches $1$. For
$\vartheta=0.5\pi$, the effective decay rate becomes equal to $\Gamma$,
resulting in an extended oscillation period. For $\vartheta=0.75\pi$ and
$0.85\pi$, the effective decay rates further decrease, leading to even longer-
lasting oscillations.

##  III The system of a three-level giant atom coupled to a 1D waveguide

Figure 5: Schematic illustration of a three-level giant atom side coupled to a
1D waveguide. The transition between the ground state $\left|{g}\right\rangle$
and the excited state $\left|{e}\right\rangle$ is coupled to waveguide modes
positioned at $x=-d/2$ and $d/2$ with strength $V$. Additionally, the
transition between the excited state $\left|{e}\right\rangle$ and the
metastable state $\left|{s}\right\rangle$ is driven by a laser field of
frequency $\omega_{c}$ with the Rabi frequency $\Omega$, and the detuning
between the transition frequency and the laser frequency is $\Delta$.

We extend our investigation to explore the dynamics of correlated two-photon
processes in a system consisting of a 1D waveguide side coupled to a three-
level $\Lambda$-type giant atom, as shown in Fig. 5. The configuration of a
giant transmon coupled to a transmission line has been demonstrated in Ref.
[13], and its characteristic phenomenon, i.e., electromagnetically induced
transparency (EIT), has been observed. For the single coupling point, it
demonstrates that the second-order correlation of scattered photons can be
tuned by adjusting the Rabi frequency of the control field [61]. Now, our
focus is specifically on investigating how the accumulated phase shift affects
non-classical properties of scattered photons. The system under consideration
is described by the Hamiltonian ($\hbar=1$ and the group velocity $\nu_{g}=1$)

$\displaystyle\hat{H}$ $\displaystyle=-i\int
dx\left[\hat{a}^{\dagger}_{R}(x)\partial_{x}\hat{a}_{R}(x)-\hat{a}^{\dagger}_{L}(x)\partial_{x}\hat{a}_{L}(x)\right]$
$\displaystyle+(\omega_{0}-i\gamma_{e}/2)\hat{\sigma}_{ee}+(\omega_{0}-\Delta)\hat{\sigma}_{ss}+\frac{\Omega}{2}(\hat{\sigma}_{es}+\hat{\sigma}_{se})$
$\displaystyle+\frac{V}{2}\sum_{\alpha=R,L}\int
dxM(x)\left[\hat{a}^{\dagger}_{\alpha}(x)\hat{\sigma}_{ge}+\hat{\sigma}_{eg}\hat{a}_{\alpha}(x)\right],$
(27)

where $M(x)=\delta(x+d/2)+\delta(x-d/2)$ represents the positions of the
coupling points, where $d$ is the separation between them. The parameter
$\omega_{0}$ denotes the transition frequency of the atomic levels
$\left|{g}\right\rangle\rightarrow\left|{e}\right\rangle$, and $\Delta$
represents the detuning between the transition frequency of
$\left|{s}\right\rangle\rightarrow\left|{e}\right\rangle$ and the laser
driving frequency $\omega_{c}$. It is worth noting that we consider the strong
atom-waveguide coupling regime, where the interaction strength
$\Gamma=V^{2}/2$ is significantly larger than the atomic spontaneous emission
rate $\gamma_{e}$. The regime is realizable in experimental setups, and the
atomic spontaneous dissipation rate $\gamma_{e}$ can be neglected. Within the
single-excitation subspace, the eigenstate of the system can be written in the
form

$\displaystyle\left|{\phi_{1}(k)}\right\rangle_{\alpha}=$ $\displaystyle\int
dx\left[\phi_{R}^{\alpha}(k,x)\hat{a}^{\dagger}_{R}(x)+\phi_{L}^{\alpha}(k,x)\hat{a}_{L}^{\dagger}(x)\right]\left|{0,g}\right\rangle$
$\displaystyle+u_{e}^{\alpha}(k)\left|{0,e}\right\rangle+u_{s}^{\alpha}(k)\left|{0,s}\right\rangle.$
(28)

Here $\phi^{\alpha}_{R/L}(k,x)$ indicate the probability amplitudes of
creating the right-moving and left-moving photon in real space for the
$\alpha$-direction incoming photon with the wavevector $k$, respectively;
$u_{e}^{\alpha}(k)$ is the excitation amplitude of the atomic level
$\left|{e}\right\rangle$; $u_{s}^{\alpha}(k)$ is the excitation amplitude of
the atomic level $\left|{s}\right\rangle$. These probability amplitudes can be
determined by the Schrödinger equation
$\hat{H}\left|{\phi_{1}(k)}\right\rangle_{\alpha}=k\left|{\phi_{1}(k)}\right\rangle_{\alpha}$,
which obey

$\displaystyle(-i\partial_{x}-k)\phi_{R}^{\alpha}(k,x)+\sqrt{\frac{\Gamma}{2}}M(x)u_{e}^{\alpha}(k)=0,$
$\displaystyle(i\partial_{x}-k)\phi_{L}^{\alpha}(k,x)+\sqrt{\frac{\Gamma}{2}}M(x)u_{e}^{\alpha}(k)=0,$
$\displaystyle(\omega_{0}-k)u_{e}^{\alpha}(k)+\frac{\Omega}{2}u_{s}^{\alpha}(k)$
$\displaystyle+\sqrt{\frac{\Gamma}{2}}\sum_{\alpha^{\prime}=R,L}\int
dxM(x)\phi_{\alpha^{\prime}}^{\alpha}(k,x)=0,$
$\displaystyle(\omega_{0}-\Delta-k)u_{s}^{\alpha}(k)+\frac{\Omega}{2}u_{e}^{\alpha}(k)=0.$
(29)

The solutions of these probability amplitudes take the following form

$\displaystyle\phi_{R}^{R}(k,x)=$
$\displaystyle\frac{e^{ikx}}{\sqrt{2\pi}}\big{\\{}\theta(-d/2-x)+t_{1}(k)\big{[}\theta(x+d/2)$
$\displaystyle-\theta(x-d/2)\big{]}+t_{2}(k)\theta(x-d/2)\big{\\}},$
$\displaystyle\phi_{L}^{R}(k,x)=$
$\displaystyle\frac{e^{-ikx}}{\sqrt{2\pi}}\big{\\{}r_{1}(k)\theta(-d/2-x)$
$\displaystyle+r_{2}(k)\left[\theta(x+d/2)-\theta(x-d/2)\right]\big{\\}},$
$\displaystyle\phi_{R}^{L}(k,x)=$
$\displaystyle\frac{e^{ikx}}{\sqrt{2\pi}}\big{\\{}r_{1}(k)\theta(x-d/2)$
$\displaystyle+r_{2}(k)\left[\theta(x+d/2)-\theta(x-d/2)\right]\big{\\}},$
$\displaystyle\phi_{L}^{L}(k,x)=$
$\displaystyle\frac{e^{-ikx}}{\sqrt{2\pi}}\big{\\{}\theta(x-d/2)+t_{1}(k)\big{[}\theta(x+d/2)$
$\displaystyle-\theta(x-d/2)\big{]}+t_{2}(k)\theta(-d/2-x)\big{\\}},$ (30)

where the coefficients are

$\displaystyle t_{1}(k)$
$\displaystyle=\frac{(\omega_{0}-\Delta-k)\left[\omega_{0}-k-i\frac{\Gamma}{2}\left(1+e^{i\vartheta}\right)\right]-\frac{\Omega^{2}}{4}}{(\omega_{0}-\Delta-k)\left[\omega_{0}-k-i\Gamma\left(1+e^{i\vartheta}\right)\right]-\frac{\Omega^{2}}{4}},$
$\displaystyle t_{2}(k)$
$\displaystyle=\frac{(\omega_{0}-\Delta-k)\left(\omega_{0}-k+\Gamma\sin\vartheta\right)-\frac{\Omega^{2}}{4}}{(\omega_{0}-\Delta-k)\left[\omega_{0}-k-i\Gamma\left(1+e^{i\vartheta}\right)\right]-\frac{\Omega^{2}}{4}},$
$\displaystyle r_{1}(k)$
$\displaystyle=\frac{i\Gamma(\omega_{0}-\Delta-k)(1+\cos\vartheta)}{(\omega_{0}-\Delta-k)\left[\omega_{0}-k-i\Gamma\left(1+e^{i\vartheta}\right)\right]-\frac{\Omega^{2}}{4}},$
$\displaystyle r_{2}(k)$
$\displaystyle=\frac{i\Gamma/2(\omega_{0}-\Delta-k)(1+e^{i\vartheta})}{(\omega_{0}-\Delta-k)\left[\omega_{0}-k-i\Gamma\left(1+e^{i\vartheta}\right)\right]-\frac{\Omega^{2}}{4}},$
$\displaystyle u_{e}^{\alpha}(k)$
$\displaystyle=\frac{-\sqrt{\Gamma/\pi}(\omega_{0}-\Delta-k)\cos\frac{\vartheta}{2}}{(\omega_{0}-\Delta-k)\left[\omega_{0}-k-i\Gamma\left(1+e^{i\vartheta}\right)\right]-\frac{\Omega^{2}}{4}},$
$\displaystyle u_{s}^{\alpha}(k)$
$\displaystyle=\frac{\Omega/2\sqrt{\Gamma/\pi}\cos\frac{\vartheta}{2}}{(\omega_{0}-\Delta-k)\left[\omega_{0}-k-i\Gamma\left(1+e^{i\vartheta}\right)\right]-\frac{\Omega^{2}}{4}}.$
(31)

We have made the assumption that the wavevector $k$ in the accumulated phase
shift is a constant $k_{0}$ under the Markovian approximation, and replaced
$k_{0}d$ with $\vartheta$.

In order to construct the wavefunction for two-photon scattering, the LS
equation is employed by replacing the atomic operators with bosonic operators.
To ensure compliance with level statistics, an additional on-site repulsion
denoted as $U$ needs to be introduced and assumed to be infinite in the end.
For a three-level atom, apart from the repulsion for each upper level, an
additional term must be included to fully eliminate the double occupancy. Here
the repulsion operator $\tilde{V}$ is

$\displaystyle\tilde{V}=\frac{U}{2}\left(\hat{b}_{e}^{\dagger}\hat{b}_{e}^{\dagger}\hat{b}_{e}\hat{b}_{e}+\hat{b}_{s}^{\dagger}\hat{b}_{s}^{\dagger}\hat{b}_{s}\hat{b}_{s}+2\hat{b}_{e}^{\dagger}\hat{b}_{e}\hat{b}_{s}^{\dagger}\hat{b}_{s}\right).$
(32)

The coefficient of the last term is chosen for convenience, and any
coefficient would be canceled out after taking $U\rightarrow\infty$ [62]. By
introducing an appropriate on-site interaction $\tilde{V}$, it becomes
possible to calculate the two-photon wavefunction
$\left|{\psi_{2}}\right\rangle$ following the preceding steps. As a result, in
the limit $U\rightarrow\infty$, the two-photon interacting eigenstate in the
coordinate representation takes the form

${}_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\langle
x_{1}x_{2}\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\langle
x_{1}x_{2}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}$
$\displaystyle-\sum_{i,j=1}^{3}G^{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}_{i}(x_{1},x_{2})(G^{-1})_{ij}\langle
j\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}.$ (33)

Here, for simplicity, we denote
$\left|{1}\right\rangle=\left|{d_{e}d_{e}}\right\rangle$,
$\left|{2}\right\rangle=\left|{d_{e}d_{s}}\right\rangle$ and
$\left|{3}\right\rangle=\left|{d_{s}d_{s}}\right\rangle$. The elements of the
Green’s functions are defined as follows:

$\displaystyle
G_{i}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})=_{\alpha_{1}\alpha_{2}}\langle
x_{1}x_{2}|G^{R}(E)\left|{i}\right\rangle$
$\displaystyle=\sum_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\int
dk_{1}dk_{2}\frac{{}_{\alpha_{1}\alpha_{2}}\langle
x_{1}x_{2}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\left\langle{\phi_{2}(k_{1},k_{2})}\right|i\rangle}{E-k_{1}-k_{2}+i0^{+}},$
$\displaystyle G_{ij}=\left\langle{i}\right|G^{R}(E)\left|{j}\right\rangle$
$\displaystyle=\sum_{\alpha_{1}\alpha_{2}}\int dk_{1}dk_{2}\frac{\langle
i\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}\langle\phi_{2}(k_{1},k_{2})\left|{j}\right\rangle}{E-k_{1}-k_{2}+i0^{+}},$
$\displaystyle G^{-1}=\left(\begin{array}[]{ccc}G_{11}&G_{12}&G_{13}\\\
G_{21}&G_{22}&G_{23}\\\ G_{31}&G_{32}&G_{33}\\\ \end{array}\right)^{-1},$ (37)

with the use of relations

$\displaystyle\langle
1\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=$
$\displaystyle u_{e}^{\alpha_{1}}(k_{1})u_{e}^{\alpha_{2}}(k_{2}),$
$\displaystyle\langle
2\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=$
$\displaystyle\frac{1}{2}\big{[}u_{e}^{\alpha_{1}}(k_{1})u_{s}^{\alpha_{2}}(k_{2})$
$\displaystyle+u_{e}^{\alpha_{1}}(k_{2})u_{s}^{\alpha_{2}}(k_{1})\big{]},$
$\displaystyle\langle
3\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=$
$\displaystyle u_{s}^{\alpha_{1}}(k_{1})u_{s}^{\alpha_{2}}(k_{2}),$
${}_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\langle
x_{1}x_{2}\left|{\phi_{2}(k_{1},k_{2})}\right\rangle_{\alpha_{1}\alpha_{2}}=$
$\displaystyle\frac{1}{2}\Big{[}\phi_{\alpha^{\prime}_{1}}^{\alpha_{1}}(k_{1},x_{1})\phi_{\alpha^{\prime}_{2}}^{\alpha_{2}}(k_{2},x_{2})$
$\displaystyle+\phi_{\alpha^{\prime}_{1}}^{\alpha_{2}}(k_{2},x_{1})\phi_{\alpha^{\prime}_{2}}^{\alpha_{1}}(k_{1},x_{2})\Big{]}.$
(38)

By employing the standard contour integral techniques to evaluate the double
integral, we can obtain

$\displaystyle G_{11}$
$\displaystyle=\frac{16\Gamma^{2}\cos^{4}\frac{\vartheta}{2}}{(\lambda_{1}-\lambda_{2})^{2}}\left[\frac{(\lambda_{1}-\omega_{0}+\Delta)^{4}}{f_{1}(\lambda_{1},\lambda_{2})}-\frac{(\lambda_{1}-\omega_{0}+\Delta)^{2}(\lambda_{2}-\omega_{0}+\Delta)^{2}}{f_{2}(\lambda_{1},\lambda_{2})}+\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{22}$
$\displaystyle=\frac{4\Gamma^{2}\Omega^{2}\cos^{4}\frac{\vartheta}{2}}{(\lambda_{1}-\lambda_{2})^{2}}\left[\frac{(\lambda_{1}-\omega_{0}+\Delta)^{2}}{f_{1}(\lambda_{1},\lambda_{2})}-\frac{(\lambda_{1}+\lambda_{2}-2\omega_{0}+2\Delta)^{2}}{4f_{2}(\lambda_{1},\lambda_{2})}+\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{33}$
$\displaystyle=\frac{\Omega^{4}\Gamma^{2}\cos^{4}\frac{\vartheta}{2}}{(\lambda_{1}-\lambda_{2})^{2}}\left[\frac{1}{f_{1}(\lambda_{1},\lambda_{2})}-\frac{1}{f_{2}(\lambda_{1},\lambda_{2})}+\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{12}$
$\displaystyle=\frac{8\Omega\Gamma^{2}\cos^{4}\frac{\vartheta}{2}}{(\lambda_{1}-\lambda_{2})^{2}}\left[\frac{(\lambda_{1}-\omega_{0}+\Delta)^{3}}{f_{1}(\lambda_{1},\lambda_{2})}-\frac{(\lambda_{1}-\omega_{0}+\Delta)^{2}(\lambda_{2}-\omega_{0}+\Delta)}{f_{2}(\lambda_{1},\lambda_{2})}+\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{13}$
$\displaystyle=\frac{4\Omega^{2}\Gamma^{2}\cos^{4}\frac{\vartheta}{2}}{(\lambda_{1}-\lambda_{2})^{2}}\left[\frac{(\lambda_{1}-\omega_{0}+\Delta)^{2}}{f_{1}(\lambda_{1},\lambda_{2})}-\frac{(\lambda_{1}-\omega_{0}+\Delta)(\lambda_{2}-\omega_{0}+\Delta)}{f_{2}(\lambda_{1},\lambda_{2})}+\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{23}$
$\displaystyle=\frac{2\Omega^{3}\Gamma^{2}\cos^{4}\frac{\vartheta}{2}}{(\lambda_{1}-\lambda_{2})^{2}}\left[\frac{(\lambda_{1}-\omega_{0}+\Delta)}{f_{1}(\lambda_{1},\lambda_{2})}-\frac{(\lambda_{1}-\omega_{0}+\Delta)}{f_{2}(\lambda_{1},\lambda_{2})}+\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{21}$ $\displaystyle=G_{12},\hskip 5.0ptG_{31}=G_{13},\hskip
5.0pt,G_{32}=G_{23},$ $\displaystyle G_{1}^{RR}(x_{1},x_{2})$
$\displaystyle=\frac{2\Gamma\cos^{2}\frac{\vartheta}{2}}{\lambda_{1}-\lambda_{2}}\left[\frac{(\lambda_{1}-\omega_{0}+\Delta)(\lambda_{1}+\omega_{0}-\Delta-E)}{(2\lambda_{1}-E)(\lambda_{1}+\lambda_{2}-E)}e^{iEx_{2}-i\lambda_{1}t}-\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{2}^{RR}(x_{1},x_{2})$
$\displaystyle=\frac{\Omega\Gamma\cos^{2}\frac{\vartheta}{2}}{2(\lambda_{1}-\lambda_{2})}\left[\frac{2\omega_{0}-2\Delta-E}{(2\lambda_{1}-E)(\lambda_{1}+\lambda_{2}-E)}e^{iEx_{2}-i\lambda_{1}t}-\lambda_{1}\leftrightarrow\lambda_{2}\right],$
$\displaystyle G_{3}^{RR}(x_{1},x_{2})$
$\displaystyle=-\frac{\Omega^{2}\Gamma\cos^{2}\frac{\vartheta}{2}}{2(\lambda_{1}-\lambda_{2})}\left[\frac{1}{(2\lambda_{1}-E)(\lambda_{1}+\lambda_{2}-E)}e^{iEx_{2}-i\lambda_{1}t}-\lambda_{1}\leftrightarrow\lambda_{2}\right].$
(39)

The expressions for $\lambda_{1}$ and $\lambda_{2}$ are given as

$\displaystyle\lambda_{1}=$
$\displaystyle\frac{1}{2}\Big{[}2\omega_{0}-\Delta-i\Gamma^{\prime}+\sqrt{(-\Delta+i\Gamma^{\prime})^{2}+\Omega^{2}}\Big{]},$
$\displaystyle\lambda_{2}=$
$\displaystyle\frac{1}{2}\Big{[}2\omega_{0}-\Delta-i\Gamma^{\prime}-\sqrt{(-\Delta+i\Gamma^{\prime})^{2}+\Omega^{2}}\Big{]},$
(40)

with $\Gamma^{\prime}=\Gamma(1+e^{i\vartheta})$. Furthermore, the functions
$f_{1}(\lambda_{1},\lambda_{2})$ and $f_{2}(\lambda_{1},\lambda_{2})$ are
defined as

$\displaystyle f_{1}(\lambda_{1},\lambda_{2})=$
$\displaystyle(2\lambda_{1}-E)(\lambda_{1}-\lambda_{1}^{*})^{2}(\lambda_{1}-\lambda_{2}^{*})^{2},$
$\displaystyle f_{2}(\lambda_{1},\lambda_{2})=$
$\displaystyle(E-\lambda_{1}-\lambda_{2})(\lambda_{1}-\lambda_{1}^{*})(\lambda_{2}-\lambda_{2}^{*})\left|{\lambda_{1}-\lambda_{2}^{*}}\right|^{2}.$
(41)

During the contour integration, we utilize the conditions $x_{1}>d/2$ and
$x_{2}>d/2$, and define $x=x_{1}-x_{2}$ and $x_{c}=(x_{1}+x_{2})/2$. It can
also be demonstrated that
$G_{i}^{LL}(-x_{1},-x_{2})=G_{i}^{RL}(x_{1},-x_{2})=G_{i}^{LR}(-x_{1},x_{2})=G_{i}^{RR}(x_{1},x_{2})$
for $i=1,2,3$, owing to the parity symmetry. Finally, the expression for the
two-photon interacting eigenstate becomes

$\displaystyle\left|{\psi_{2}(k_{1},k_{2})}\right\rangle_{RR}=$
$\displaystyle\int
dx_{1}dx_{2}\Bigg{[}\frac{f_{RR}^{(\Lambda)}(x_{1},x_{2})}{\sqrt{2}}\hat{a}_{R}^{\dagger}(x_{1})\hat{a}_{R}^{\dagger}(x_{2})$
$\displaystyle+\frac{f_{LL}^{(\Lambda)}(x_{1},x_{2})}{\sqrt{2}}\hat{a}^{\dagger}_{L}(x_{1})\hat{a}_{L}^{\dagger}(x_{2})$
$\displaystyle+f_{RL}^{(\Lambda)}(x_{1},x_{2})\hat{a}_{R}^{\dagger}(x_{1})\hat{a}_{L}^{\dagger}(x_{2})\Bigg{]}\left|{0}\right\rangle.$
(42)

The coefficients can be written as the sum of the plane-wave and bound-state
terms

$\displaystyle f_{RR}^{(\Lambda)}(x_{1},x_{2})=$
$\displaystyle\frac{e^{iEx_{c}}}{\sqrt{2}\pi}\left[t_{2}(k_{1})t_{2}(k_{2})\cos\Delta_{1}x+B^{(2)}_{k_{1}k_{2}}(x)\right],$
$\displaystyle f_{LL}^{(\Lambda)}(x_{1},x_{2})=$
$\displaystyle\frac{e^{-iEx_{c}}}{\sqrt{2}\pi}\left[r_{1}(k_{1})r_{1}(k_{2})\cos\Delta_{1}x+B^{(2)}_{k_{1}k_{2}}(x)\right],$
$\displaystyle f_{RL}^{(\Lambda)}(x_{1},x_{2})=$
$\displaystyle\frac{e^{iEx/2}}{2\pi}\Big{[}t_{2}(k_{1})r_{1}(k_{2})e^{2i\Delta_{1}x_{c}}$
$\displaystyle+r_{1}(k_{1})t_{2}(k_{2})e^{-2i\Delta_{1}x_{c}}+2B_{k_{1}k_{2}}^{(2)}(x_{c})\Big{]}.$
(43)

The bound-state term $B^{(2)}_{k_{1}k_{2}}(x)$ at the resonance condition,
i.e., $\Delta=0$, can be expressed as

$\displaystyle
B_{k_{1}k_{2}}^{(2)}(x)=A_{+}e^{i(E/2-\gamma_{+})\left|{x}\right|}+A_{-}e^{i(E/2-\gamma_{-})\left|{x}\right|}.$
(44)

Here, $\gamma_{\pm}$ are the values of $\lambda_{1,2}$ in Eq. (40) with
$\Delta=0$, i.e.,

$\displaystyle\gamma_{\pm}=\omega_{0}-i\frac{\Gamma^{\prime}}{2}\pm\frac{\sqrt{\Omega^{2}-{\Gamma^{\prime}}^{2}}}{2},$
(45)

and the coefficients $A_{\pm}$ are

$\displaystyle A_{\pm}=$
$\displaystyle\frac{\Gamma^{2}(1+\cos\vartheta)^{2}}{2\sqrt{\Omega^{2}-{\Gamma^{\prime}}^{2}}}\Big{[}\pm(k_{1}+k_{2}-2\omega_{0})\frac{\Omega^{2}}{4}$
$\displaystyle+(k_{1}-\omega_{0})(k_{2}-\omega_{0})(\sqrt{\Omega^{2}-{\Gamma^{\prime}}^{2}}\mp
i\Gamma^{\prime})\Big{]}$
$\displaystyle/\Big{[}(k_{1}-\gamma_{+})(k_{1}-\gamma_{-})(k_{2}-\gamma_{+})(k_{2}-\gamma_{-})\Big{]}.$
(46)

If $\Omega=0$, these coefficients can be reduced to the same form as those of
the two-level giant atom derived above. In the following discussion, we also
consider the narrow bandwidth limit for incident photons, where the
frequencies of the incident photons are equal, i.e., $k_{1}=k_{2}=k$.

###  III.1 Incoherent power spectrum

According to the previous discussion on the two-level giant atom system, the
total incoherent power spectrum can be utilized as a metric for the photon-
photon correlation because it originates from the correlated bound state. In
the case of a three-level giant atom, the exact derivation of the total
incoherent power spectrum $F(k)$ is provided in Appendix A, which is

$\displaystyle
F(k)=\frac{8i}{\pi}\sum_{m,n=+,-}\frac{A_{m}^{*}A_{n}}{\gamma_{m}^{*}-\gamma_{n}}$
(47)

The results are numerically shown in Fig. 6. Notably, when the system reaches
the perfect electromagnetically-induced transparency (EIT) window, i.e.,
$k=\omega_{0}$, the value of $F(k)$ becomes zero. This indicates that all
photons are scattered coherently, and no bound state is formed in the
wavefunction. This observation aligns with the expression of the bound-state
term in Eq. (44), which becomes zero when $k=\omega_{0}$. Thus, it is
consistent with the notion that photon-photon correlation and incoherent
scattering are related. Additionally, $F(k=\omega_{0})=0$ at the perfect
transparency remains unaffected by the phase shift $\vartheta$, and even the
number of identical atoms coupled to the waveguide [62]. This phenomenon is
commonly referred to as fluorescence quenching [63, 35].

Figure 6: The total incoherent power spectrum $F(k)$ of the three-level giant
atom as a function of the incident frequency $k$ with different values of
$\vartheta$. The other parameters are $\omega_{0}=100\Gamma$ and
$\Omega=\Gamma/2$.

A significant level of photon-photon correlation is indicated by a large value
of $F(k)$. Figure 6 illustrates that the peak value of $F(k)$ for the three-
level giant atom is considerably higher than that of the two-level giant atom.
This suggests that the three-level giant atom generates photon-photon
correlation more efficiently. Furthermore, when $\vartheta=0$, the shape of
$F(k)$ exhibits symmetry and resembles that of the single-point coupling [62].
However, for other phase shifts such as $\vartheta=0.5\pi$, $0.75\pi$ and
$0.85\pi$, the magnitude of incoherent scattering increases significantly, and
the line shape changes from symmetry to asymmetry. This phenomenon can be
explained by the presence of an asymmetric pole structure. The poles referred
to here are $\gamma_{\pm}$ in Eq. (45). When $\vartheta=0$, the real parts of
the pole are symmetric with respect to $\omega_{0}$. However, for
$\vartheta=0.5\pi$, $0.75\pi$ and $0.85\pi$, they become asymmetric.
Additionally, the height of the peak is inversely proportional to its width,
which is determined by the imaginary part of the pole. For example, when
$\vartheta=0.5\pi$, the two poles are $\omega_{0}+1.03-0.97i$ and
$\omega_{0}-0.032-0.03i$. As a result, the main peak is located at
$\omega_{0}-0.032$ while the other peak is negligible due to its large
imaginary part. Similarly, for $\vartheta=0.75\pi$, the two poles are
$\omega_{0}+0.78-0.27i$ and $\omega_{0}-0.072-0.025i$, leading to the main
peak being located at $\omega_{0}-0.072$. The height of the peak is higher due
to the smaller imaginary part. At the same time, the other peak is located at
$\omega_{0}+0.78$ with a smaller height due to its slightly larger imaginary
part. Also, for $\vartheta=0.85\pi$, the two poles are
$\omega_{0}+0.56-0.091i$ and $\omega_{0}-0.11-0.017i$, leading to the main
peak being located at $\omega_{0}-0.11$ with a higher height. Simultaneously,
the other peak located at $\omega_{0}+0.56$ also becomes evident due to its
small imaginary part.

When the Rabi frequency increases, the transparency window becomes wider.
Within the transparency window, the scattered photons are not confined to the
bound state, as shown in Fig. 7. The peak values of $F(k)$ can be observed
around $\omega_{0}\pm\frac{\Omega}{2}$, but their amplitudes decrease.
Therefore, a strong Rabi frequency can hinder the photon-photon correlation
and result in simpler and less structured effects. This characteristic may be
advantageous in certain circumstances [61]. The structure can also be
explained by the poles $\gamma_{\pm}$, which approach
$\omega_{0}\pm\frac{\Omega}{2}-i\frac{\Gamma^{\prime}}{2}$. The amplitudes
become smaller due to the larger values of the imaginary parts.

Figure 7: The total incoherent power spectrum $F(k)$ of the three-level giant
atom as a function of the incident frequency $k$ with different values of
$\vartheta$. The other parameters are $\omega_{0}=100\Gamma$ and
$\Omega=5\Gamma$.

###  III.2 Second-order correlation function

Likewise, to briefly illustrate the effect of the bound state on the
transmitted and reflected field ($x_{1}>d/2$, $x_{2}>d/2$, and
$x=x_{2}-x_{1}$) in the system of the three-level giant atom, we first examine
the difference between the probability of the two-photon detection and the
single-photon detection when $x=0$. Under the condition of $k_{1}=k_{2}=k$, we
denote
$\chi_{R}=2\pi^{2}\left|{f_{RR}^{(\Lambda)}(0)}\right|^{2}-\left|{t_{2}(k)}\right|^{4}$
for the transmitted field and
$\chi_{L}=2\pi^{2}\left|{f_{LL}^{(\Lambda)}(0)}\right|^{2}-\left|{r_{1}(k)}\right|^{4}$
for the reflected field. If $\chi_{R}>0$, it indicates that the bound state
enhances the transmission of two photons, resulting in a phenomenon known as
photon-induced tunneling, which serves as a signature of photon bunching.
Conversely, if $\chi_{R}<0$, it implies that the bound state can suppress the
transmission of two photons, leading to photon blockade. In Fig. 8, it shows
that $\chi_{R}>0$, indicating that the transmitted photons are bunched, while
$\chi_{L}<0$, suggesting that the reflected photons are antibunched.
Therefore, the statistical properties of photons are determined by the
interference between the plane-wave and bound-state terms.

Figure 8: The difference between the probability of the two-photon detection
and the single-photon detection when $x=0$, denoted as
$\chi_{R}=2\pi^{2}\left|{f_{RR}^{(\Lambda)}(0)}\right|^{2}-\left|{t_{2}(k)}\right|^{4}$
for the transmitted field (a) and
$\chi_{L}=2\pi^{2}\left|{f_{LL}^{(\Lambda)}(0)}\right|^{2}-\left|{r_{1}(k)}\right|^{4}$
for the reflected field (b), as functions of the incident frequency $k$ and
the accumulated phase shift $\vartheta$. The other parameters are
$\omega_{0}=100\Gamma$ and $\Omega=\Gamma/2$.

Similarly, the normalized second-order correlation functions are utilized to
investigate the photon-photon correlation of the transmitted and reflected
field in the system. Upon performing calculations, the normalized correlation
functions can be expressed in the form

$\displaystyle g_{R}^{(2)}(x)=$
$\displaystyle\Bigg{|}1+\frac{A_{+}}{t_{2}(k_{1})t_{2}(k_{2})}e^{i(E/2-\gamma_{+})\left|{x}\right|}$
$\displaystyle+\frac{A_{-}}{t_{2}(k_{1})t_{2}(k_{2})}e^{i(E/2-\gamma_{-})\left|{x}\right|}\Bigg{|}^{2},$
$\displaystyle g_{L}^{(2)}(x)=$
$\displaystyle\Bigg{|}1+\frac{A_{+}}{r_{1}(k_{1})r_{1}(k_{2})}e^{i(E/2-\gamma_{+})\left|{x}\right|}$
$\displaystyle+\frac{A_{-}}{r_{1}(k_{1})r_{1}(k_{2})}e^{i(E/2-\gamma_{-})\left|{x}\right|}\Bigg{|}^{2}.$
(48)

From the definition, it can be proven that at the perfect EIT window, where
$k=\omega_{0}$, we have $g^{(2)}_{R/L}(x)=1$ due to the bound term in Eq. (44)
being equal to zero. This implies that all of the photons are scattered
coherently, without any structured correlation effects, which is consistent
with the existence of a zero value for $F(k)$. It suggests that the
statistical properties of the scattering photons can be adjusted by the
control field in the three-level giant atom under the resonance condition. In
the presence of the control field, the incident photons pass by the system
coherently, thereby maintaining unchanged statistics. However, in the absence
of the control field, the system effectively behaves as a two-level model,
resulting in bunched transmitted photons and antibunched reflected photons.
Actually, the generation of strong photon interactions indicated by the peak
value of $F(k)$ appears slightly away from the EIT condition [61].
Furthermore, it can be verified that when $x=0$, we have $g_{L}^{(2)}(0)=0$,
indicating that the reflected photons are initially antibunched. As for the
transmitted photons, the second-order correlation function at $x=0$ is given
by

$\displaystyle g_{R}^{(2)}(0)=$
$\displaystyle\Bigg{|}1+\Gamma^{2}(1+\cos\vartheta)^{2}$
$\displaystyle\times\prod_{i=1}^{2}\frac{(k_{i}-\omega_{0})}{(\omega_{0}-k_{i})(\omega_{0}-k_{i}+\Gamma\sin\vartheta)-\frac{\Omega^{2}}{4}}\Bigg{|}^{2}.$
(49)

It can be seen that when the incident photons have equal frequency, the
transmitted photons are initially bunched. The analysis is consistent with the
results presented in Fig. 8 and similar to that of the single coupling point
[62].

Figure 9: The normalized second-order correlation functions, denoted as
$g^{(2)}_{R}(x)$ for transmitted photons (a) and $g^{(2)}_{L}(x)$ for
reflected photons (b), as a function of $x$ with different values of
$\vartheta$. The other parameters are $\Omega=\Gamma/2$ and
$\omega_{0}-k=\Omega/2$.

To illustrate the spatial correlation between photons, we have chosen
$\omega_{0}-k=\Omega/2$ for the numerical plot of $g^{(2)}_{R/L}(x)$ in Fig.
9. It should be noted that the initial value cannot be used to predict the
overall photon-photon correlation due to the complex nature of
$g_{R/L}^{(2)}(x)$ resulting from the beating between two eigenfrequencies and
the incident frequency. For the given parameters in Fig. 9, considering that
$\left|{\text{Im}[\gamma_{+}]}\right|\gg\left|{\text{Im}[\gamma_{-}]}\right|$,
the behavior of $g_{R/L}^{(2)}(x)$ at small values of $x$ is primarily
determined by the term of $A_{+}$. It can be easily verified that
$g_{R}^{(2)}(x)$ rapidly decreases within the scale of
$\pi/(\text{Re}[\gamma_{+}]-k)$, while $g_{L}^{(2)}(x)$ quickly reaches a
local peak at the same scale. On the other hand, for large values of $x$, the
contribution of the term $A_{+}$ becomes negligible due to its fast decay.
Instead, it is dominated by the beating of the term $A_{-}$ with the incident
frequency. In addition to the enhanced photon-photon correlation compared to
the two-level atom, the correlation lasts for a longer distance. The long
decay distance can be characterized by the smaller value of the imaginary part
of $\gamma_{-}$. When $\vartheta=0$, the decay time is on the scale of
$8\Gamma/\Omega^{2}$, which is longer than that of two-level atom when
employing the weak Rabi frequency $\Omega$. It should be noted here that the
single-photon transmission rate becomes zero for $\vartheta=0$, resulting in a
divergence in the normalized second-order correlation function of the
transmitted field. Therefore, its evolution is not depicted in Fig. 9.
Moreover, for $\vartheta=0.5\pi$, $0.75\pi$, and $0.85\pi$, the imaginary
parts become even smaller, indicating the possibility of achieving
significantly long-distance correlations.

##  IV Conclusions

In conclusion, we have employed the LS method to investigate the two-photon
scattering processes involving two- and three-level giant atoms coupled to a
1D waveguide. We focus on effects of the accumulated phase shift acquired by
photons as they travel between coupling points. The multiple coupling points
of the giant atom give rise to interference effects that are absent in small
atoms. Based on the analytical results for the total incoherent power spectra
and second-order correlation functions of scattered photons, we have found
that the accumulated phase shift can be utilized to enhance the strength and
distance of photon-photon interactions. In the case of a two-level giant atom,
photon-photon interactions are enhanced, and the evolution of the second-order
correlation displays an oscillation between bunching and anti-bunching.
Comparatively, the system of a three-level giant atom exhibits significantly
increased photon-photon correlation, surpassing that of the two-level
counterpart. The photon-photon interaction and the correlation distance can be
further enhanced by tuning the accumulated phase shift between the two
coupling points. These characteristics can be explained by analyzing the poles
of the system.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(under Grants No.1150403, 61505014,12174140.)

##  Appendix A Calculation of the incoherent power spectrum

###  A.1 The two-level giant atom

The power spectrum or resonance fluorescence is defined as the Fourier
transform of the first-order coherence function

$\displaystyle S_{\alpha}(\omega)=\int dte^{-i\omega
t}\left\langle{\psi_{2}}\right|\hat{a}^{\dagger}_{\alpha}(x_{0})\hat{a}_{\alpha}(x_{0}+t)\left|{\psi_{2}}\right\rangle,$
(50)

where $x_{0}$ is the position of a detector located far away from the
scattering region. For example, considering $\alpha=R$, the first-order
coherence in $\left|{\psi_{2}}\right\rangle$ is

$\displaystyle\left\langle{\psi_{2}}\right|\hat{a}_{R}^{\dagger}(x_{0})\hat{a}_{R}(x_{0}+t)\left|{\psi_{2}}\right\rangle$
$\displaystyle=2\int
dx^{\prime}f_{RR}^{*}(x_{0},x^{\prime})f_{RR}(x_{0}+t,x^{\prime}).$ (51)

Under the condition of the narrow bandwidth for incident photons, they have
equal frequency, i.e., $k_{1}=k_{2}=k=E/2$, and

$\displaystyle f_{RR}^{*}(x_{0},x^{\prime})f_{RR}(x_{0}+t,x^{\prime})=$
$\displaystyle\frac{e^{iEt/2}}{2\pi^{2}}\Big{[}\left|{t_{2}(E/2)}\right|^{4}+t_{2}^{2}(E/2)B_{k_{1}k_{2}}^{*}(x)$
$\displaystyle+t_{2}^{*2}(E/2)B_{k_{1}k_{2}}(x-t)+B_{k_{1}k_{2}}^{*}(x)B_{k_{1}k_{2}}(x-t)\Big{]},$
(52)

with $x=x^{\prime}-x_{0}$. By substituting Eqs. (51) and (52) into Eq. (19),
the power spectrum can be divided into two parts,

$\displaystyle
S_{R}(\omega)=S_{R}^{\text{coh}}(\omega)+S_{R}^{\text{incoh}}(\omega),$ (53)

where the coherent part contains terms proportional to
$\delta(0)\delta(\omega-E/2)$ due to the use of delta-normalized plane waves.
On the other hand, the incoherent part is related to the correlation of the
bound-state term

$\displaystyle S_{R}^{\text{incoh}}(\omega)=\frac{1}{\pi^{2}}\iint
dtdxe^{i(E/2-\omega)t}B_{k_{1}k_{2}}^{*}(x)B_{k_{1}k_{2}}(x-t).$ (54)

After the integration, the incoherent power spectrum becomes

$\displaystyle
S_{R}^{\text{incoh}}(\omega)=\frac{4}{\pi^{2}}\left|{\Upsilon(\omega)}\right|^{2},$
(55)

where

$\displaystyle\Upsilon(\omega)=$
$\displaystyle\frac{\Gamma^{2}(1+\cos\vartheta)^{2}}{(E/2-\omega_{0}+i\Gamma^{\prime})}$
$\displaystyle\times\frac{1}{(E-\omega_{0}-\omega+i\Gamma^{\prime})(\omega-\omega_{0}+i\Gamma^{\prime})}.$
(56)

The total incoherent power spectrum is obtained by summing the right- and
left-moving incoherent power spectra,

$\displaystyle
S^{\text{incoh}}(\omega)=S_{R}^{\text{incoh}}(\omega)+S_{L}^{\text{incoh}}(\omega).$
(57)

Here it can be proven that
$S_{R}^{\text{incoh}}(\omega)=S_{L}^{\text{incoh}}(\omega)$. The total
incoherent power can be used to measure the overall strength of photon-photon
correlations to show the non-classical effects. It also provides a direct
measure of the bound-state term. The definition of the total incoherent power
spectrum is

$\displaystyle F(k)=\int d\omega S^{\text{incoh}}(\omega)=\frac{4}{\pi}\int
dxB_{k_{1}k_{2}}^{*}(x)B_{k_{1}k_{2}}(x).$ (58)

Via integrating over $\omega$, it becomes

$\displaystyle
F(k)=\frac{4\Gamma^{3}(1+\cos\vartheta)^{3}}{\pi\left|{k-\omega_{0}+i\Gamma^{\prime}}\right|^{4}}.$
(59)

###  A.2 The three-level giant atom

The incoherent power spectrum in the system of the three-level giant atom can
be calculated by following the same procedure as that of the two-level giant
atom, which yields

$\displaystyle
S_{R}^{\text{incoh}}(\omega)=\frac{1}{\pi^{2}}\left|{\sum_{m=+,-}\frac{A_{m}(E-2\gamma_{m})}{(E-\omega-\gamma_{m})(\omega-\gamma_{m})}}\right|^{2},$
(60)

where $\gamma_{\pm}=\pm\frac{1}{2}\sqrt{\Omega^{2}-\Gamma^{\prime
2}}+\frac{i}{2}\Gamma^{\prime}$. By performing integration over the frequency
$\omega$, the total incoherent power spectrum $F(k)=2\int d\omega
S_{R}^{\text{incoh}}(\omega)$ becomes

$\displaystyle
F(k)=\frac{8i}{\pi}\sum_{m,n=+,-}\frac{A_{m}^{*}A_{n}}{\gamma_{m}^{*}-\gamma_{n}}.$
(61)

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