Structured Emission and Entanglement Dynamics of a Giant Atom in a Photonic Creutz Ladder

Abstract

We explore the spontaneous emission dynamics of a giant atom coupled to a photonic Creutz ladder, focusing on how flat-band frustration and synthetic gauge fields shape atom–photon interactions. The Creutz ladder exhibits perfectly flat bands, Aharonov–Bohm caging, and topological features arising from its nontrivial hopping structure. By embedding the giant atom at multiple spatially separated sites, we reveal interference-driven emission control and the formation of nonradiative bound states. Using both spectral and time-domain analyses, we uncover strong non-Markovian dynamics characterized by persistent oscillations, long-lived entanglement, and recoherence cycles. The emergence of bound-state poles in the spectral function is accompanied by spatially localized photonic profiles and directionally asymmetric emission, even in the absence of band dispersion. Calculations of von Neumann entropy and atomic purity confirm the formation of coherence-preserving dressed states in the flat-band regime. Furthermore, the spacetime structure of the emitted field displays robust zig-zag interference patterns and synthetic chirality, underscoring the role of geometry and topology in photon transport. Our results demonstrate how flat-band photonic lattices can be leveraged to engineer tunable atom–photon entanglement, suppress radiative losses, and create structured decoherence-free subspaces for quantum information applications.

1. Introduction

The interplay between quantum emitters and engineered low-dimensional photonic systems has emerged as a central theme in contemporary quantum optics, with waveguide quantum electrodynamics (QED) providing the theoretical backbone for analyzing such interactions [1,2,3]. These systems offer unparalleled opportunities for tailoring light–matter coupling with high precision, which is vital for realizing robust platforms for quantum communication and information processing. Recent technological advancements have enabled the realization of controllable photonic reservoirs—including one-dimensional waveguides and discrete photonic lattices—capable of interfacing coherently with quantum emitters. Experimental implementations span various platforms, including photonic crystal structures [4], superconducting circuit QED architectures [5,6,7], and synthetic atomic arrays [8,9], each contributing novel functionalities to quantum photonics.

A particularly innovative concept to emerge in this context is that of the giant atom (GA). A GA refers to a quantum emitter whose coupling points to the surrounding photonic reservoir are separated by distances comparable to, or exceeding, the relevant photonic wavelength [10,11]. This spatial separation allows the emitted or absorbed field to accumulate a finite phase between coupling points, leading to interference between spatially distributed emission channels and enabling nonlocal effects absent in conventional (small) atoms, which couple locally at a single point. Such nonlocal coupling can strongly modify the emission spectrum, produce frequency-dependent Lamb shifts and emission rates [12,13], and enable decoherence-free interactions mediated by structured continua [14,15,16,17,18,19]. It can also give rise to bound or quasi-localized states with oscillatory features [20,21,22,23,24], thereby making giant atoms a versatile platform for exploring non-Markovian dynamics and structured light–matter interactions in waveguide QED.

Various experimental strategies have successfully implemented GAs. One prominent direction uses superconducting qubits coupled to either surface acoustic waves [11,25,26,27,28,29,30,31,32,33,34,35] or microwave transmission lines and waveguides [13,15,36], offering detailed control over both the geometry and strength of the atom–field interaction. Complementary theoretical proposals have further diversified possible implementations, including designs based on synthetic frequency lattices [37] and photonic integrated circuits [38]. More recently, the GA paradigm has been generalized to encompass extended quantum systems such as giant molecules [39,40,41,42,43,44] and collective spin ensembles [45].

Although numerous studies have focused on GAs embedded in continuum waveguides [12,13,14,15,17,19,20,21,23,36,42,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61], where the spectral density remains approximately flat, increasing attention is now being paid to structured photonic environments such as tight-binding lattices. These engineered reservoirs exhibit complex dispersion relations and nontrivial spectral features [18,24,38,42,52,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85], enabling phenomena such as strong non-Markovian effects, bound states in the continuum, and synthetic gauge interactions.

Recent developments have highlighted the Creutz ladder [86] as a paradigmatic platform for exploring flat-band physics, synthetic gauge fields, and topological effects in low-dimensional photonic systems. Theoretically, the Creutz ladder hosts topologically nontrivial phases and features perfectly flat energy bands under specific flux configurations, which result in Aharonov–Bohm (AB) caging and disorder-free localization [87,88]. Circuit QED architectures have demonstrated how artificial gauge potentials can be engineered in such systems using dynamically modulated transmon arrays, enabling chiral edge states and nonreciprocal photon transport [89]. In cold atom setups, the Creutz ladder has been realized via resonant lattice shaking between orbital states, allowing the control of inter-leg couplings and the observation of topological charge pumping [90]. Experimental work with photonic waveguide arrays has directly observed AB cages, confirming the collapse of energy bands and the emergence of compact localized states [87]. Extensions to multilayer geometries reveal rich entanglement and topological transitions driven by the pairing and delocalization of compact Wannier modes [91]. These features have also enabled studies of many-body localization and ergodicity breaking in flat-band regimes without disorder [92]. Recent work has pushed further into higher-order topological regimes, where the Creutz ladder supports robust corner and edge modes protected by symmetry constraints [93]. Parametric cavity simulations have added a new layer of control, enabling programmable synthetic lattices for probing nontrivial topological dynamics [94]. Meanwhile, non-Hermitian generalizations have enabled tunable AB caging and extended localization control through anti-$\mathcal{PT}$-symmetric imaginary couplings [95]. Importantly, in the context of waveguide QED, the Creutz ladder supports robust, vacancy-like dressed bound states where localized photons orbit the emitter with topologically protected features [96]. These advancements position the Creutz ladder as an ideal setting for exploring exotic atom–photon interactions in structured quantum environments.

Building on these developments, and extending the theoretical framework introduced in our recent studies on giant atoms in synthetic-gauge-field lattices [84,85], the present work investigates the dynamics of a quantum emitter—specifically, a giant atom—embedded in a photonic Creutz ladder. In Refs. [84,85], we analyzed single and two giant atoms in a one-dimensional coupled-resonator waveguide threaded by a synthetic gauge phase, focusing on chiral emission control, decoherence-free interactions, and gauge-phase-tunable bound states. Here, we turn to the Creutz ladder, whose flat-band structure, topological winding, and Aharonov–Bohm caging give rise to qualitatively different transport and localization properties. By leveraging these features, we explore how spatially extended atom–field coupling can produce non-Markovian emission dynamics, robust bound states, and localized photon profiles. In contrast to previous studies that treated either the emitter or the lattice in isolation, our model highlights the interplay between the internal structure of the giant atom and the synthetic magnetic flux threading the ladder. The rest of this manuscript is organized as follows: In Section 2, we introduce the system Hamiltonian and outline the theoretical framework for describing the time evolution. Section 3 presents numerical simulations of population dynamics, emission heatmaps, and spectral properties, while identifying key features linked to flat-band effects and nonlocal coupling. Finally, Section 4 summarizes the main findings and discusses future directions for structured light–matter interactions in synthetic quantum lattices.

2. Theory

2.1. Creutz Ladder Model

The Creutz ladder is a quasi-one-dimensional tight-binding lattice that exhibits a rich topological structure and flat energy bands, depending on its parameters. It consists of two parallel chains (legs) connected by vertical rungs and cross-diagonal hoppings (see Figure 1), as well as purely imaginary intra-leg hoppings that emulate a synthetic magnetic flux. Each unit cell contains two sites: $a_n$, corresponding to the upper leg, and $b_n$, corresponding to the lower leg, where $n\in \mathbb{Z}$ is the index of the unit cell.

The real-space tight-binding Hamiltonian of the Creutz ladder is given by

$$\begin{array}{c}\hfill H=\sum _n\left. [-t\left. (a_n^{†}{b}_n+a_{n+1}^{†}{b}_n\right)-i\gamma \left(a_{n+1}^{†}{a}_n-b_{n+1}^{†}{b}_n\right)+\mathrm{H}.\mathrm{c}.\right],\end{array}$$

(1)

where $t\in \mathbb{R}$ denotes the hopping amplitude along both the vertical (rung) and diagonal directions, and $\gamma \in \mathbb{R}$ denotes the strength of the imaginary hopping along each leg. The operators $a_n^{†}$ and $b_n^{†}$ create a photon at site $a_n$ and $b_n$, respectively.

Applying the Fourier transform

$$a_n={\displaystyle \frac{1}{\sqrt{N}}}\sum _k{a}_k{e}^{ikn},b_n={\displaystyle \frac{1}{\sqrt{N}}}\sum _k{b}_k{e}^{ikn}.$$

(2)

The Hamiltonian in momentum space becomes

$$\begin{array}{c}\hfill H\left(k\right)=-2tcosk{\sigma }_x-2\gamma sink{\sigma }_z,\end{array}$$

(3)

where ${\sigma }_x$ and ${\sigma }_z$ are Pauli matrices acting on the sublattice space spanned by $(a_k,b_k)$. In compact form, the Bloch Hamiltonian can be written as $H\left(k\right)=\overrightarrow{d}\left(k\right)·\overrightarrow{\sigma }$, with the vector $\overrightarrow{d}\left(k\right)=\left(-2tcosk,0,-2\gamma sink\right)$.

The corresponding frequency spectrum consists of two symmetric bands given by

$$\begin{array}{c}\hfill {\omega }_{\pm }\left(k\right)=\pm \sqrt{{(2tcosk)}^2+{(2\gamma sink)}^2}.\end{array}$$

(4)

For the special case $\gamma =t$, the bands become completely flat with frequency ${\omega }_{\pm }\left(k\right)=\pm 2t$, independent of momentum. This flat-band regime is a hallmark of the Creutz ladder and is often associated with localization phenomena and enhanced interaction effects.

In Figure 2, we plot the dispersion relation of Equation (3) with $t=1$ and $k\in [-\pi ,\pi ]$. The choice $\gamma /t=0.5$ represents a generic dispersive regime, while $\gamma /t=1$ corresponds to the analytic flat-band limit ${\omega }_{\pm }=\pm 2t$ of the Creutz ladder.

The Creutz ladder possesses a variety of symmetries. First, it exhibits chiral (or sublattice) symmetry which is generated by the operator $\mathrm{\Gamma }={\sigma }_y$, satisfying $\mathrm{\Gamma }H\left(k\right){\mathrm{\Gamma }}^{-1}=-H\left(k\right)$, or equivalently $\left\{H\right(k),\mathrm{\Gamma }\}=0$. This guarantees that the energy eigenvalues appear in symmetric $\pm E$ pairs. Time-reversal symmetry (TRS) is broken when $\gamma \ne 0$ due to the purely imaginary hoppings, but is restored in the special case $\gamma =0$.

Topologically, the Creutz ladder supports nontrivial winding numbers. The topological invariant is defined by the winding of the vector $\overrightarrow{d}\left(k\right)$ around the origin in the $(d_x,d_z)$ plane

$$\nu ={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }dk{\displaystyle \frac{d}{dk}}arg\left[d_x\left(k\right)+i{d}_z\left(k\right)\right].$$

(5)

This winding number takes integer values $\nu =\pm 1$, depending on the sign of $\gamma$, and characterizes the topological phase of the system [88,97]. For open boundary conditions, the nontrivial topology manifests as zero-energy edge states that are protected by chiral symmetry [88,97]. These properties make the Creutz ladder an ideal platform for exploring flat-band topology and nontrivial edge dynamics in synthetic photonic systems.

2.2. Giant Atom in a Creutz Ladder

We now introduce a two-level GA with energy spacing ${\omega }_0$, coupled to P sites of the Creutz ladder. The atomic Hamiltonian remains

$$H_{\mathrm{atom}}=\Delta {\sigma }^{†}\sigma ,$$

(6)

with detuning $\Delta ={\omega }_0-{\omega }_c$, and raising/lowering operators ${\sigma }^{†}$, $\sigma$.

For simplicity, we assume that the GA couples only to sublattice a at positions ${\left\{n_p\right\}}_{p=1}^P$, though generalizations to both legs are possible. The interaction Hamiltonian is

$$H_{\mathrm{int}}=g\sum _{p=1}^P\left({\sigma }^{†}{a}_{n_p}+a_{n_p}^{†}\sigma \right).$$

(7)

The full system Hamiltonian becomes

$$H=H_{\mathrm{atom}}+H_{\mathrm{lat}}+H_{\mathrm{int}}.$$

(8)

Working in the single-excitation subspace, we expand the total wavefunction as

$$|\psi \left(t\right)\rangle =C_e\left(t\right)|e\rangle \otimes |\mathrm{vac}\rangle +\sum _n\left[A_n\left(t\right)|g\rangle \otimes |1_{a_n}\rangle +B_n\left(t\right)|g\rangle \otimes |1_{b_n}\rangle \right],$$

(9)

where $|1_{a_n}\rangle =a_n^{†}|\mathrm{vac}\rangle$, and $|1_{b_n}\rangle =b_n^{†}|\mathrm{vac}\rangle$. The amplitudes $A_n\left(t\right)$, $B_n\left(t\right)$, and $C_e\left(t\right)$ evolve via the Schrödinger equation:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{C}_e\left(t\right)& =-i\Delta C_e\left(t\right)-ig\sum _{p=1}^P{A}_{n_p}\left(t\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{A}_n\left(t\right)& =-ig{C}_e\left(t\right)\sum _{p=1}^P{\delta }_{n,n_p}-it\left[B_n\left(t\right)+B_{n-1}\left(t\right)\right]-\gamma \left[A_{n+1}\left(t\right)-A_{n-1}\left(t\right)\right],\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{B}_n\left(t\right)& =-it\left[A_n\left(t\right)+A_{n+1}\left(t\right)\right]+\gamma \left[B_{n+1}\left(t\right)-B_{n-1}\left(t\right)\right].\hfill \end{array}$$

(12)

2.3. Self-Energy and Bound States for a Giant Atom Coupled to a Creutz Ladder

To assess long-time dynamics, we compute the Laplace transform ${\tilde{C}}_e\left(z\right)$ and introduce the self-energy ${\Sigma }_e\left(z\right)$

$${\tilde{C}}_e\left(z\right)={\displaystyle \frac{1}{z-\Delta -{\Sigma }_e\left(z\right)}},$$

(13)

where ${\Sigma }_e\left(z\right)$ encodes the effect of photon emission and reabsorption via the structured photonic reservoir. For a GA coupled to the upper band of the Creutz ladder, the self-energy takes the form [84]

$${\Sigma }_e\left(z\right)={\displaystyle \frac{g^2}{2\pi }}{\int }_{-\pi }^{\pi }dk{\displaystyle \frac{1+{\sum }_{p\ne p^{\prime }}{e}^{ik(n_p-n_{p^{\prime }})}}{z-{\omega }_{+}\left(k\right)}},$$

(14)

where ${\omega }_{+}\left(k\right)$ is given by Equation (4).

2.4. Flat-Band Limit

When $\gamma =t$, the upper band [Equation (4)] becomes completely flat:

$${\omega }_{+}\left(k\right)=2t,\forall k.$$

(15)

So the denominator in Equation (14) becomes independent of k, and the self-energy simplifies

$$\begin{array}{cc}\hfill {\Sigma }_e\left(z\right)& ={\displaystyle \frac{g^2}{z-2t}}·{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }dk{\left|\sum _{p=1}^P{e}^{ik{n}_p}\right|}^2\hfill \\ \hfill & ={\displaystyle \frac{g^2}{z-2t}}·{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }dk\sum _{p,p^{\prime }}{e}^{ik(n_p-n_{p^{\prime }})}\hfill \\ \hfill & ={\displaystyle \frac{g^2}{z-2t}}\sum _{p,p^{\prime }}\left[{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{e}^{ik(n_p-n_{p^{\prime }})}dk\right]\hfill \\ \hfill & ={\displaystyle \frac{g^2}{z-2t}}\sum _{p,p^{\prime }}{\delta }_{n_p,n_{p^{\prime }}}={\displaystyle \frac{g^{2}P}{z-2t}},\hfill \end{array}$$

(16)

where we used the identity

$${\left|\sum _{p=1}^P{e}^{ik{n}_p}\right|}^2=\sum _{p,p^{\prime }}{e}^{ik\left(n_p-n_{p^{\prime }}\right)}.$$

(17)

and the orthogonality of Fourier modes over the Brillouin zone

$${\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{e}^{ik\left(n_p-n_{p^{\prime }}\right)}dk={\delta }_{n_p,n_{p^{\prime }}}.$$

(18)

Hence, only the diagonal terms survive the integration, and for nondegenerate positions $n_p$, the result reduces to

$${\Sigma }_e\left(z\right)={\displaystyle \frac{g^{2}P}{z-2t}}.$$

(19)

2.5. Bound States

Bound states correspond to the poles of ${\tilde{C}}_e\left(z\right)$, i.e., the solutions to the equation [84,85]

$$z-\Delta -{\Sigma }_e\left(z\right)=0.$$

(20)

Substituting the flat-band self-energy of Equation (19), we obtain

$$z-\Delta -{\displaystyle \frac{g^{2}P}{z-2t}}=0.$$

(21)

Multiplying both sides by $(z-2t)$ yields a quadratic equation

$$z^2-z(\Delta +2t)+2t\Delta -g^{2}P=0.$$

(22)

Its solutions give the bound-state energies

$$z_{\pm }={\displaystyle \frac{1}{2}}\left(\Delta +2t\pm \sqrt{{(\Delta -2t)}^2+4g^{2}P}\right).$$

(23)

3. Results and Discussion

To analyze the energy structure of the giant atom coupled to a flat photonic band, we evaluate the spectral response of the atomic excited state. In particular, we compute the spectral function associated with the retarded Green’s function of the excited state, which provides direct insight into the presence of bound states and their hybridization with the photonic bath.

The spectral function is defined as

$$A\left(\omega \right)=-2\mathrm{Im}\left[{\displaystyle \frac{1}{\omega -\Delta -{\Sigma }_e(\omega +iϵ)}}\right],$$

(24)

where $ϵ>0$ is a small positive broadening parameter and ${\Sigma }_e\left(z\right)$ denotes the self-energy due to coupling to the environment.

Figure 3 shows the analytic spectral function evaluated for different detuning values $\Delta =0,1,2$ (Table 1), using the flat-band self-energy of Equation (19). The two poles of the dressed Green function correspond to bound-state energies $z_{\pm }$ of Equation (23) lying around the flat photonic band ${\omega }_k=2t$. For representative parameters $t=\gamma =1$, $g=0.2$, and $P=2$, the pole positions are

Table 1. Spectral positions of the poles for t = ɤ = 1, g = 0.2, P = 2 and different values of the detuning Δ.

$\mathbf{\Delta }$	${\mathit{z}}_{\mathbf{-}}$	${\mathit{z}}_{\mathbf{+}}$
0	$-0.039$	$2.039$
1	$0.925$	$2.074$
2	$1.915$	$2.085$

Table 1. Spectral positions of the poles for t = ɤ = 1, g = 0.2, P = 2 and different values of the detuning Δ.

$\mathbf{\Delta }$	${\mathit{z}}_{\mathbf{-}}$	${\mathit{z}}_{\mathbf{+}}$
0	$-0.039$	$2.039$
1	$0.925$	$2.074$
2	$1.915$	$2.085$

The energy splitting $|z_{+}-z_{-}|$ increases with $\Delta$, reflecting the enhanced hybridization between the atom and the flat photonic band. The lower pole remains sharp and well-defined, whereas the upper pole lies close to the diverging photonic density of states at $\omega =2t$, and is therefore strongly broadened and weakly visible in the time-domain response.

3.1. Population Dynamics

The excited-state population dynamics shown in Figure 4 reveal distinct signatures of non-Markovian behavior and flat-band interference, governed by the underlying topology of the photonic environment. In the flat-band regime of the Creutz ladder ($\gamma =t$), destructive interference between vertical and diagonal hopping pathways gives rise to a perfectly flat photonic band and compact localized modes (CLM) [91,92]. This geometrically induced frustration enables Aharonov–Bohm (AB) caging, a phenomenon in which photonic excitations remain trapped in local plaquettes due to phase interference across hopping loops [87,88,91,92].

At $\Delta =0.0$, the atomic transition frequency lies at the center of the photonic bandgap, equidistant from the two flat bands located at $\omega =\pm 2t$. In this regime, the excited-state population exhibits minimal decay and persistent oscillations, indicating the formation of a conventional bound state outside the photonic continuum. The absence of propagating decay channels in the gap, combined with the emitter’s nonlocal coupling to multiple lattice sites, results in partial population trapping and coherent revivals. These features reflect the emergence of a nonradiative dressed state supported by interference across the emitter’s connection points, consistent with the analytic bound-state energies predicted in the flat-band limit [cf. Equation (23)] and with vacancy-like confinement phenomena in topological photonic lattices [96].

As the detuning increases to $\Delta =1.0$, the atomic frequency approaches the upper flat photonic band at $\omega =2t$, where the density of states is high but the group velocity vanishes. In this regime, the population undergoes a more pronounced decay, yet the amplitude does not fully vanish and instead saturates around a nonzero value. This behavior signifies partial hybridization between the atom and flat-band photonic modes. Although these modes do not propagate, their spatial structure enables localized reabsorption and delayed feedback—hallmarks of non-Markovianity in frustrated photonic environments [91,92].

At exact resonance, $\Delta =2.0$, the atomic transition lies precisely within the flat photonic band. Here, the population exhibits a fast initial decay followed by irregular revival oscillations. Unlike the near-resonant case, the dynamics do not show clear saturation over the observed time window. This indicates strong hybridization with the degenerate flat-band modes, which leads to substantial reemission and reabsorption cycles. The analytic spectral function [Figure 3] confirms the presence of a pole near $\omega =2t$, signaling strong atom–field dressing and enhanced memory effects. However, full bound-state formation may require slight detuning from exact resonance to mitigate the critical broadening effects induced by the diverging density of states [96].

Overall, these results demonstrate that flat-band photonic reservoirs—such as the Creutz ladder with $\gamma =t$—enable a continuum of dynamical regimes: from long-lived coherent population trapping in a photonic bandgap, to suppressed decay at band edges, to nonradiative resonances within the continuum. These phenomena arise from the unique interplay between synthetic gauge fields, emitter geometry, and flat-band interference, and are in excellent agreement with previous theoretical and experimental studies on Creutz ladders in circuit QED [89,94], cold atom platforms [90], and photonic waveguide arrays [87].

We note that the non-smooth features in the $\Delta =0$ dynamics—most visibly the sharp turnover points and cusp-like extrema—originate from the phase-coherent exchange between the emitter and a discrete set of localized flat-band modes. In the Creutz ladder, the intra-leg terms $\pm i\gamma$ act as Peierls phases, generating a synthetic magnetic flux per plaquette that, at the flat-band condition $\gamma =t$, supports compact localized modes and Aharonov–Bohm caging [87,91,92]. Coupling to such modes confines the emission–reabsorption process to a finite set of phase-locked pathways, so that revival maxima occur abruptly, producing non-analytic structures in time. These turnover points persist under refinement of numerical parameters (see Supplementary Materials) and are therefore a genuine physical signature of interference in a flat-band environment. Departing from the flat-band condition ($\gamma \ne t$) weakens phase locking, allowing dispersive channels to contribute and smoothing out the revival pattern. Evidently, the corresponding population dynamics (see Supplementary Materials, Figure S3) display smooth, exponentially damped behavior without sharp revivals, confirming that the non-analytic features of Figure 4 originate from flat-band interference.

3.2. Entanglement Dynamics and Von Neumann Entropy

To further characterize the degree of atom–field entanglement and the non-Markovian nature of the dynamics, we compute the von Neumann entropy of the atomic reduced state as a function of time:

$$S\left(t\right)=-{\rho }_e\left(t\right)log{\rho }_e\left(t\right)-\left[1-{\rho }_e\left(t\right)\right]log\left[1-{\rho }_e\left(t\right)\right],$$

(25)

where ${\rho }_e\left(t\right)={\left|C_e\left(t\right)\right|}^2$ is the excited-state population. This entropy quantifies the degree of entanglement between the emitter and the photonic reservoir, with $S=0$ corresponding to a pure (factorized) state and $S=log2$ indicating maximal mixedness in the single-excitation subspace.

Figure 5 shows the entropy dynamics for $\Delta =0.0,1.0,$ and $2.0$. These curves reveal distinct entanglement behaviors shaped by the interplay between the emitter frequency, the structured photonic bath, and the synthetic gauge field geometry of the Creutz ladder [88,89,90].

At $\Delta =0.0$ the entropy quickly rises from zero and settles into bounded oscillations. This behavior is consistent with the formation of a bound state and reduced entanglement generation, reflecting effective confinement in a decoherence-free subspace [96]. The oscillatory pattern indicates coherent atom–photon exchange within localized compact states enabled by AB caging [87,95].

For $\Delta =1.0,2.0$, near the upper flat-band edge, the entropy increases more rapidly and stabilizes at a higher plateau. This signifies partial hybridization between the atomic excitation and localized flat-band photonic modes, whose vanishing group velocity and high density of states enhance non-Markovian memory effects [92]. The entropy saturation at intermediate values reflects a quasi-steady entangled state characterized by persistent, but bounded, field correlations.

3.3. Atomic Mixedness and Purity Evolution

Complementary to the von Neumann entropy, the purity of the atomic reduced state provides a direct measure of mixedness and coherence loss. In the single-excitation subspace, the atomic purity is given by

$$P\left(t\right)={\rho }_e^2\left(t\right)+{\left[1-{\rho }_e\left(t\right)\right]}^2=1-2{\rho }_e\left(t\right)\left[1-{\rho }_e\left(t\right)\right],$$

(26)

where ${\rho }_e\left(t\right)={\left|C_e\left(t\right)\right|}^2$ denotes the excited-state population. Purity takes the value $P=1$ for a pure state and reaches a minimum of $P=0.5$ for a maximally mixed state. It serves as a practical indicator of atom–field entanglement and provides complementary insight into the loss and recovery of coherence during the evolution.

Figure 6 presents the time evolution of atomic purity for three representative detunings: $\Delta =0.0$ (blue), $1.0$ (orange), and $2.0$ (green). The resulting behavior reflects distinct regimes of coherence degradation and atom–field dressing, shaped by the spectral properties of the flat-band environment.

At $\Delta =0.0$, where the atomic frequency lies at the center of the bandgap, the purity exhibits persistent and large-amplitude oscillations. This behavior suggests that the atom remains strongly correlated with a localized photonic mode and undergoes repeated cycles of recoherence. The oscillations arise from interference between the two coupling sites in the presence of flat-band frustration, which stabilizes a nonradiative bound state. As a result, the atom periodically reenters nearly pure states, indicating evolution within a coherence-protected subspace.

For $\Delta =1.0,2.0$, the purity decays more significantly and reaches a steady-state value below unity, indicating a stable but entangled atom–photon dressed state. The absence of strong oscillations points to a reduction in recoherence effects and more effective dressing by the flat-band reservoir. This regime corresponds to moderate non-Markovianity, where memory effects persist but do not lead to full revival of coherence.

The purity dynamics reinforce and refine the insights obtained from the entropy and population analyses. Atomic coherence is preserved and periodically restored when the emitter lies in the bandgap. Near and at the flat-band edge, coherence is lost due to strong and irreversible dressing by nonpropagating photonic modes. These observations highlight the capacity of flat-band photonic lattices to engineer coherence protection and dynamical control of entanglement.

3.4. Effect of a Finite Temperature

To assess the robustness of these features against finite-temperature effects, we note that the present calculations assume a photonic lattice initially in its vacuum state ($T=0$), so that only spontaneous emission occurs. At finite temperature, the lattice modes would be thermally populated according to the Bose–Einstein distribution $n_{\mathrm{th}}\left(\omega \right)={[exp(\hslash \omega /k_{\mathrm{B}}T)-1]}^{-1}$, leading to additional stimulated emission and absorption processes. In our tight-binding framework [Equations (10)–(12)], this amounts to extra terms in the equations of motion proportional to $n_{\mathrm{th}}\left[{\omega }_{\pm }\left(k\right)\right]$ for each lattice mode, which incoherently exchange excitations between the atom and the bath. For optical-frequency emitters, $n_{\mathrm{th}}$ is negligible even at room temperature, so the dynamics shown in Figure 4 and Figure 6 remain essentially unchanged. In circuit QED realizations operating at millikelvin temperatures, $n_{\mathrm{th}}\ll 1$, so the interference-induced revivals and cusps persist but with slightly reduced contrast, while the purity saturates to a value set by the trapped bound-state component and the small thermal mixture of the continuum fraction.

3.5. Photon Emission Profiles and Field Localization

The spatial profiles of the emitted photonic wavefunction, shown in Figure 7, provide a direct snapshot of how the giant atom redistributes its excitation across the lattice modes for different detuning values. These profiles are extracted at a fixed long evolution time $t=t_{\max }$, corresponding to the steady-state or late-time behavior observed in the population dynamics.

At $\Delta =0.0$, the field exhibits a strongly localized emission pattern. The total field profile displays a split central peak—an interference-induced two-peak structure. This splitting stems from a lateral shift between the upper- and lower-leg field distributions: $|A_n{|}^2$ is skewed leftward, while $|B_n{|}^2$ is skewed to the right. The resulting asymmetry highlights a form of synthetic chirality, with the photonic field preferentially propagating in opposite directions on the two legs. This effect originates from the complex hopping topology of the Creutz ladder, where a synthetic gauge field induces leg-dependent interference. Despite the flat-band suppression of ballistic propagation, this internal chiral structure gives rise to a staggered and localized emission envelope.

For $\Delta =1.0$ the profiles remain qualitatively similar, though the asymmetry between legs is reduced. The shift between $|A_n{|}^2$ and $|B_n{|}^2$ persists but is less pronounced, resulting in a broad central feature in the total profile with a residual two-peak structure. This suggests partial hybridization with flat-band modes, where the emitter couples more evenly to the symmetric and antisymmetric leg combinations. The persistence of bounded spatial confinement and partial asymmetry underscores the role of flat-band dispersion in modulating both localization and chirality.

At $\Delta =2.0$, where the emitter is exactly resonant with the upper flat band, the upper- and lower-leg profiles are nearly identical and perfectly overlapping. The total field profile consists of a single, sharply peaked central maximum with no sign of substructure or directional asymmetry. This behavior reflects uniform hybridization with all flat-band modes and the restoration of symmetry in the leg occupations. The collapse of the shift and chiral imbalance at exact resonance indicates a transition to a collectively dressed photonic state, where interference effects become homogenized due to full degeneracy.

3.6. Spacetime Evolution of the Photonic Field

To assess the real-time dynamics of photon emission, we show in Figure 8 the spacetime heatmaps of the total photonic intensity $|A_n{\left(t\right)|}^2+{\left|B_n\left(t\right)\right|}^2$ for three detunings: $\Delta =0.0$, $1.0$, and $2.0$. The plots depict the evolution of the photonic wavepacket emitted by a giant atom coupled at two sites of a Creutz ladder under flat-band conditions ($t=\gamma =1$).

All three heatmaps reveal similar emission patterns: the photonic field spreads along both directions of the ladder in a structured and asymmetric manner, with a clear bias toward one propagation direction. This directional asymmetry reflects the chirality imposed by the Creutz ladder’s synthetic gauge field and the interference between vertical and diagonal hopping paths [88,89]. The absence of a sharp light-cone confirms the inhibition of ballistic transport due to flat-band frustration [91,93].

Although the wavefront structure is nearly identical across detunings, the intensity for $\Delta =2.0$ appears slightly weaker. This may be attributed to destructive interference at exact resonance with the flat band, reducing the effective emission strength despite strong hybridization [94,96]. Still, the dominant behavior remains intact: robust, slow, and directionally biased emission.

A prominent feature across all panels is the emergence of a zig-zag interference pattern, characterized by diagonally alternating intensity maxima. This arises from the sublattice structure of the ladder and the presence of synthetic magnetic flux per plaquette [87,95]. The combination of vertical and diagonal hopping paths leads to interference-enhanced emission in chiral directions, producing the observed sawtooth-like spacetime evolution [90,94].

Overall, the persistence of zig-zag propagation and asymmetric spreading across all detunings underscores the dominance of lattice topology and flat-band interference over the detailed spectral position of the emitter. These results highlight the robustness of synthetic gauge-induced photon dynamics and confirm that the spatial transport characteristics of the photonic field are governed primarily by geometry and interference, not by simple dispersive behavior.

We note here that in all the above simulations, we have made use of open boundary conditions, which correspond to the finite extent of actual experimental platforms and avoid the artificial wrap-around inherent to periodic boundary conditions. Under open boundary conditions, the lattice ends act as hard walls, so late-time snapshots (Figure 8) display a sharp reflected front once the emitted wavepacket reaches the edges. All figures in this work were obtained with the emitter placed at the center of a sufficiently long lattice such that the first edge return lies beyond the plotted interval, ensuring that the reported populations, entropies, and purities are unaffected by boundaries.

A final note concerns the advantages and limitations of the present model. Our model combines the nonlocal coupling of a giant atom with the flat-band and synthetic gauge field physics of the Creutz ladder, enabling an exact treatment of non-Markovian dynamics, bound-state formation, and photon localization within a minimal tight-binding framework. An advantage of this approach is its analytical transparency: the lattice spectrum and emitter–field coupling can be computed exactly, allowing us to directly connect spectral features to the observed dynamical behavior. The Creutz ladder geometry further provides a versatile platform to explore frustration-induced localization and topologically tunable bound states, both of which are accessible in state-of-the-art photonic and circuit QED implementations. However, the model also has limitations. The tight-binding description neglects losses, disorder, and multimode effects that may be present in realistic devices, and the current treatment focuses on zero-temperature dynamics. While the formalism can be extended to include these effects, their inclusion may smooth out some of the sharp features (e.g., cusps and revivals) reported here. Nevertheless, the essential nonlocal interference phenomena identified in this work are expected to remain robust under experimentally relevant conditions, highlighting the relevance of our findings for future quantum photonic platforms.

4. Conclusions

We have studied the spontaneous emission dynamics of a giant atom (GA) embedded in a photonic Creutz ladder, focusing on the impact of flat-band physics, synthetic gauge fields, and nonlocal emitter–lattice coupling. By exploiting the Creutz ladder’s ability to host compact localized modes and Aharonov–Bohm caging in the flat-band regime, we demonstrated that atom–photon interactions acquire highly non-Markovian features and exhibit strong signatures of interference-based localization.

Our spectral analysis revealed the emergence of two dressed bound states as poles in Green’s function, whose energies depend sensitively on the atomic detuning and coupling configuration. When the atomic frequency lies in the photonic bandgap, the emitter retains a long-lived excitation with persistent oscillations and suppressed decay. As the detuning increases toward the flat photonic band, the atom–field hybridization leads to enhanced decay and saturation into entangled steady states, reflecting the vanishing group velocity and diverging density of states near the band edge.

The time evolution of the excited-state population, von Neumann entropy, and atomic purity revealed a crossover from coherence-preserving bound-state behavior to strong atom–field dressing. In the flat-band regime, the presence of multiple coupling points enables the formation of interference-protected states that periodically recohere, even in the absence of a conventional bandgap. This behavior is a hallmark of AB caging and compact photonic localization induced by synthetic gauge fields.

We further analyzed the spatial and spacetime structure of the emitted photonic wavefunction. The field profiles display chirality-induced asymmetry between the upper and lower legs of the ladder, even though the bands are completely flat. Emission remains strongly confined near the coupling sites, with persistent zig-zag interference patterns that are robust against changes in atomic detuning. These effects are controlled entirely by the geometry and topology of the lattice, rather than its dispersion properties.

Overall, our results highlight how flat-band frustration, nonlocal coupling, and synthetic magnetic flux collaborate to shape the quantum dynamics of light–matter interaction. The interplay of interference, topology, and non-Markovian feedback enables the generation of long-lived, localized atom–photon states that are highly sensitive to lattice structure but insensitive to small parameter variations. These findings provide new design principles for structured quantum photonic platforms, with potential applications in coherence-preserving quantum memory, chiral photon routing, and topologically protected quantum interfaces.

Similar to recent studies exploring magnetic field control of structured light generation in complex photonic media [98], our approach could be also adapted to engineer photonic states with tunable spatial profiles via synthetic gauge fields in the Creutz ladder. Moreover, by analogy with recent results in cross-stitch lattices—where the relative phase between coupling points enables a giant atom to selectively couple to either flat or dispersive bands with high fidelity [99]—our Creutz ladder setup also offers the potential for phase-tuned selectivity, whereby emission can be directed into distinct flat-band modes or gap-localized bound states on demand.