Interference-Induced Bound States in the Continuum in Optical Giant Atoms

Abstract

The giant atom paradigm, where a single quantum emitter couples to a continuum at multiple discrete points, has enabled unprecedented control over light-matter interactions, including decoherence-free subspaces and chiral emission. However, realizing these non-local effects beyond the microwave regime remains a significant challenge due to the diffraction limit. Here, we theoretically propose a photonic analog of giant atoms operating at optical frequencies, utilizing a quantum emitter resonantly coupled to a pair of spatially separated single-mode cavities interacting with a common 1D photonic continuum. By rigorously deriving the effective non-Hermitian Hamiltonian and integrating out the bath degrees of freedom, we demonstrate that the interference between cavity-mediated emission pathways leads to the formation of robust Bound States in the Continuum (BICs). These interference-induced dark states allow for the infinite trapping of excitation within the emitter-cavity subsystem, effectively shielding it from radiative decay. Our results extend the giant atom toolbox to the optical domain, offering a scalable architecture for integrated quantum photonics and quantum interconnects.

1. Introduction

The field of waveguide quantum electrodynamics (wQED), where quantum emitters interact with a one-dimensional continuum of propagating photons, has become a cornerstone of modern quantum optics [1,2,3]. This platform enables the exploration of fundamental light-matter interactions, ranging from collective radiance [4] and matter-wave dynamics [5,6] to topological and chiral quantum effects [7,8,9,10,11]. Conventionally, these interactions are described within the dipole approximation, treating the emitter as a point-like object that couples to the field at a single spatial location.

However, recent years have witnessed a paradigm shift with the emergence of giant atoms—emitters that couple to the waveguide continuum at multiple discrete points separated by distances comparable to or larger than the transition wavelength [12,13]. This non-local coupling regime was first realized in superconducting circuit architectures, specifically using surface acoustic waves (SAWs) or meandering transmission lines [14,15,16,17,18,19,20,21,22,23]. The resulting phase interference between coupling points leads to exotic phenomena forbidden in small atoms, such as non-exponential decay dynamics [24,25,26] and the formation of decoherence-free subspaces (DFS) where the emitter is protected from radiative loss [27,28,29,30].

The giant-atom configuration has introduced a versatile platform for engineering chirality and non-reciprocity in quantum networks without the need for external magnetic fields [31,32,33,34,35,36,37,38,39,40]. By leveraging the phase interference inherent to non-local coupling, these systems can support unidirectional light-matter interfaces and interference-induced Bound States in the Continuum (BICs) or oscillating bound states, both of which are critical for long-lived quantum memory applications [41,42,43,44,45,46,47,48,49].

While many of these advances were initially pioneered in the microwave domain, there is a burgeoning effort to extend giant-atom physics to the optical regime using structured photonic environments and topological lattices [50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79]. In such platforms, the synthetic engineering of the photonic-bath dispersion allows for the replication of non-local interference effects at optical frequencies. In particular, topological models such as Creutz ladders have been investigated for their ability to support robust edge states and flat bands, providing fertile ground for exploring protected quantum states and chiral transport within integrated photonic architectures [80,81,82,83,84,85,86,87,88,89,90,91,92,93].

Despite these theoretical strides, realizing giant atoms in the optical regime remains a significant challenge due to the diffraction limit; stretching a natural atom to span optical wavelengths is unphysical. Recent works have proposed synthetic frequency dimensions or structured baths to circumvent this [94,95,96]. Building on recent investigations into giant atoms in photonic lattices [97,98], we here propose a scalable Hamiltonian engineering approach to realize optical giant atoms. This architecture circumvents the geometric constraints of direct coupling by employing cavity modes as spatially separated interaction ports, thereby synthesizing the effective non-local Hamiltonian of a giant atom using purely local optical elements. By utilizing a pair of waveguide-coupled cavities as mediators, we demonstrate that non-local interference can create robust BICs, effectively trapping excitation in the emitter-cavity subsystem and extending the giant-atom toolkit to integrated photonics. We rigorously derive the effective non-Hermitian Hamiltonian for this hybrid system, solving the time-dependent Schrödinger equation to map the transition from super-radiant decay to infinite trapping. Furthermore, we extend this formalism to a system of two coupled giant atoms, revealing the existence of collective dark dimer states formed by inter-atom interference. Our results provide a blueprint for implementing non-local quantum optics in integrated photonic platforms, with applications ranging from long-lived quantum memories to protected quantum interconnects.

2. Dynamics of a Single Giant Atom

We consider a hybrid quantum photonic system composed of a two-level quantum emitter (QE) resonantly coupled to two identical single-mode optical cavities. These cavities are coupled to a common one-dimensional waveguide continuum (see Figure 1). The total Hamiltonian of the system, $H_{\mathrm{tot}}$, can be decomposed into the free energy of the constituents ($H_0$), the coherent QE-cavity interaction ($H_{\mathrm{JC}}$), and the dissipative cavity-waveguide coupling ($H_{\kappa }$)

$$H_{\mathrm{tot}}=H_0+H_{\mathrm{JC}}+H_{\kappa }.$$

(1)

The free energy Hamiltonian $H_0$ describes the uncoupled energies of the emitter, the localized cavity modes, and the propagating waveguide modes

$$H_0=\hslash {\omega }_a{\sigma }^{†}\sigma +\sum _{j=1,2}\hslash {\omega }_c{a}_j^{†}{a}_j+{\int }_{-\infty }^{\infty }dk\hslash {\omega }_k{b}_k^{†}{b}_k,$$

(2)

where ${\sigma }^{†}$ ($\sigma$) are the raising (lowering) operators for the QE with transition frequency ${\omega }_a$; $a_j^{†}$ ($a_j$) creates (annihilates) a photon in the j-th cavity with frequency ${\omega }_c$; and $b_k^{†}$ ($b_k$) acts on the waveguide continuum modes with dispersion relation ${\omega }_k$. Around the resonance frequency ${\omega }_c$, we linearize the dispersion as ${\omega }_k\approx {\omega }_c+v_g(k-k_0)$, where $v_g$ is the group velocity.

The coherent interaction between the QE and the cavities is governed by the Jaynes-Cummings model

$$H_{\mathrm{JC}}=\hslash g\sum _{j=1,2}\left(a_j^{†}\sigma +{\sigma }^{†}{a}_j\right),$$

(3)

where g denotes the vacuum Rabi coupling strength, assumed here to be identical for both cavities due to symmetry.

The coupling between the localized cavity modes and the continuum is described by

$$H_{\kappa }=\hslash \sum _{j=1,2}{\int }_{-\infty }^{\infty }dk\sqrt{{\displaystyle \frac{\gamma v_g}{2\pi }}}\left(b_k^{†}{a}_j{e}^{-ik{x}_j}+a_j^{†}{b}_k{e}^{ik{x}_j}\right).$$

(4)

Here, $\gamma$ is the intrinsic decay rate of the cavities into the waveguide. Crucially, the phase factor $e^{-ik{x}_j}$ encodes the spatial position $x_j$ of the j-th cavity, which gives rise to non-local interference effects.

It should be noted that in our architecture, the coupling is not distributed over a continuous spatial profile (e.g., a Gaussian distribution). Instead, the giant atom is synthesized via discrete interaction ports provided by localized single-mode cavities. Consequently, the coupling is point-like at the specific positions $x_j$, and the phase interference is determined solely by the discrete phase factors $e^{-ik{x}_j}$. This discrete coupling scheme bypasses the need for a filling factor or spatial averaging, as the emitter-waveguide interaction is mediated entirely through these localized cavity modes.

To study the system dynamics, we solve the time-dependent Schrödinger equation, $i\hslash {\displaystyle \frac{d}{dt}}|\Psi \left(t\right)\rangle =H_{\mathrm{tot}}|\Psi \left(t\right)\rangle$. We restrict our analysis to the single-excitation subspace, valid for weak excitation regimes. The general ansatz for the state vector is

$$\begin{array}{cc}\hfill |\Psi \left(t\right)\rangle & =c_e\left(t\right)|e,0,0,{\left\{0\right\}}_k\rangle \hfill \\ \hfill & +\sum _{j=1,2}{c}_j\left(t\right)|g,1_j,0_{\overline{j}},{\left\{0\right\}}_k\rangle \hfill \\ \hfill & +{\int }_{-\infty }^{\infty }dk{c}_k\left(t\right)|g,0,0,1_k\rangle ,\hfill \end{array}$$

(5)

where $c_e\left(t\right)$, $c_j\left(t\right)$, and $c_k\left(t\right)$ are the probability amplitudes for the emitter, the j-th cavity, and the waveguide mode k, respectively.

2.1. Coupled Equations of Motion

Projecting the Schrödinger equation onto the basis states yields a system of coupled first-order differential equations (setting $\hslash =1$)

$$\begin{array}{cc}\hfill {\dot{c}}_e\left(t\right)& =-i{\omega }_a{c}_e\left(t\right)-ig\sum _{j=1,2}{c}_j\left(t\right),\hfill \\ \hfill {\dot{c}}_j\left(t\right)& =-i{\omega }_c{c}_j\left(t\right)-ig{c}_e\left(t\right)\hfill \end{array}$$

(6a)

$$\begin{array}{cc}\hfill & -i\int dk\sqrt{{\displaystyle \frac{\gamma v_g}{2\pi }}}{c}_k\left(t\right)e^{ik{x}_j},\hfill \end{array}$$

(6b)

$$\begin{array}{cc}\hfill {\dot{c}}_k\left(t\right)& =-i{\omega }_k{c}_k\left(t\right)-i\sqrt{{\displaystyle \frac{\gamma v_g}{2\pi }}}\sum _{j=1,2}{c}_j\left(t\right)e^{-ik{x}_j}.\hfill \end{array}$$

(6c)

2.2. Elimination of the Reservoir

We formally integrate Equation (6c) to eliminate the waveguide-bath variables. Assuming the waveguide is initially in the vacuum state ($c_k\left(0\right)=0$), we obtain (see Appendix A)

$$c_k\left(t\right)=-i\sqrt{{\displaystyle \frac{\gamma v_g}{2\pi }}}\sum _{l=1,2}{e}^{-ik{x}_l}{\int }_0^{t}d{t}^{\prime }{e}^{-i{\omega }_k(t-t^{\prime })}{c}_l\left(t^{\prime }\right).$$

(7)

Substituting this solution back into Equation (6b) yields

$${\dot{c}}_j\left(t\right)=-i{\omega }_c{c}_j\left(t\right)-ig{c}_e\left(t\right)-{\displaystyle \frac{\gamma v_g}{2\pi }}\sum _{l=1,2}{\int }_0^{t}d{t}^{\prime }{c}_l\left(t^{\prime }\right){\int }_{-\infty }^{\infty }dk{e}^{-i{\omega }_k(t-t^{\prime })}{e}^{ik(x_j-x_l)},$$

(8)

where the inner integral over the waveguide modes k represents the interaction kernel that accounts for the retarded feedback mediated by the 1D continuum. Based on the Weisskopf-Wigner approximation, we have extended the integration limits to $\pm \infty$. Furthermore, we assume a linear dispersion ${\omega }_k\approx {\omega }_c+v_g(k-k_0)$. The resulting integral tunrs out to be (see Appendix B)

$$\begin{array}{cc}\hfill \mathcal{I}& ={\int }_{-\infty }^{\infty }dk{e}^{-i{v}_{g}k(t-t^{\prime })}{e}^{ik(x_j-x_l)}\hfill \\ \hfill & ={\displaystyle \frac{2\pi }{v_g}}\delta \left(t-t^{\prime }-{\displaystyle \frac{|x_j-x_l|}{v_g}}\right)e^{i{k}_0|x_j-x_l|}.\hfill \end{array}$$

(9)

The above Dirac delta function enforces causality, introducing a time delay $\tau =|x_j-x_l|/v_g$ corresponding to the propagation time between cavities.

2.3. Effective Non-Hermitian Hamiltonian

Eliminating the waveguide continuum under the Markov approximation yields an effective non-Hermitian Hamiltonian $H_{\mathrm{eff}}$ governing the evolution of the discrete amplitudes $\mathbf{c}={(c_1,c_2,c_e)}^T$ (see Appendix C). In the rotating frame resonant with the cavity frequency, the dynamics are given by

$${\displaystyle \frac{d}{dt}}\mathbf{c}=-i{\mathbf{H}}_{\mathrm{eff}}\mathbf{c},$$

(10)

where the effective Hamiltonian matrix is

$${\mathbf{H}}_{\mathrm{eff}}=\left(\begin{array}{ccc}-i{\displaystyle \frac{\gamma }{2}}& -i{\displaystyle \frac{\gamma }{2}}{e}^{i\varphi }& g\\ -i{\displaystyle \frac{\gamma }{2}}{e}^{i\varphi }& -i{\displaystyle \frac{\gamma }{2}}& g\\ g& g& 0\end{array}\right),$$

(11)

with $\varphi =k_0(x_1-x_2)=k_{0}L$. Here, the diagonal terms $-i\gamma /2$ represent the individual radiative decay of the cavities. The off-diagonal term $-i(\gamma /2)e^{i\varphi }$ arises from the waveguide-mediated cross-coupling, which is responsible for the formation of the sub-radiant dark state when $\varphi =\pi$ (see below).

While the diagonal terms in Equation (11) specifically account for the radiative decay into the waveguide continuum, realistic optical cavities may also exhibit intrinsic losses, such as material absorption or out-of-plane scattering. These additional dissipative effects can be incorporated by adding an intrinsic decay rate to the diagonal elements of ${\mathbf{H}}_{\mathrm{eff}}$, a formal treatment of which is provided in Appendix E.

While our current analysis is restricted to the single-excitation subspace, the inclusion of optical nonlinearities—such as those arising from the Kerr effect or strong photon-photon interactions—could significantly enrich the system dynamics. In the presence of nonlinearities, a BIC might become power-dependent, potentially enabling the all-optical switching of the giant atom’s radiative decay. Furthermore, such interactions could lead to the formation of multi-photon bound states or non-classical light generation, where the interference-induced shielding provides a protected environment for quantum nonlinear optics.

2.4. Diagonalization and BIC Formation

To elucidate the physical nature of the eigenstates, we transform the cavity basis into symmetric and antisymmetric superpositions, $c_{S,A}=(c_1\pm c_2)/\sqrt{2}$. In this basis, the effective Hamiltonian block-diagonalizes into two distinct sectors due to the parity symmetry of the coupling.

The antisymmetric mode $c_A$ decouples from the emitter [since $g(c_1+c_2)\to 0$] and evolves independently with an effective decay rate ${\Gamma }_A\left(\varphi \right)=\gamma (1-cos\varphi )$. While this mode becomes dark at $\varphi =2\pi m$, it is trivial for quantum information processing as it does not involve the emitter.

The symmetric sector involves the hybridization of the cavity mode $c_S$ and the emitter $c_e$. The reduced $2\times 2$ Hamiltonian governing this subspace is

$${\mathbf{H}}_{\mathrm{Sym}}=\left(\begin{array}{cc}-i\tilde{\gamma }\left(\varphi \right)& \sqrt{2}g\\ \sqrt{2}g& 0\end{array}\right),$$

(12)

where $\tilde{\gamma }\left(\varphi \right)={\displaystyle \frac{\gamma }{2}}(1+e^{i\varphi })$ represents the phase-dependent coupling to the continuum.

2.5. Complex Eigenvalue Analysis

To rigorously demonstrate the formation of the BIC, we solve the characteristic equation for the symmetric sector, ${\lambda }^2+i\tilde{\gamma }\left(\varphi \right)\lambda -2g^2=0$. The complex eigenvalues ${\lambda }_{\pm }$ describing the hybrid polariton modes are given by

$${\lambda }_{\pm }\left(\varphi \right)=-i{\displaystyle \frac{\tilde{\gamma }\left(\varphi \right)}{2}}\pm \sqrt{2g^2-{\left({\displaystyle \frac{\tilde{\gamma }\left(\varphi \right)}{2}}\right)}^2}.$$

(13)

The imaginary part of ${\lambda }_{\pm }$ corresponds to the decay rate of the eigenmodes, while the real part corresponds to the oscillation frequency.

At the critical phase condition $\varphi =(2m+1)\pi$ (corresponding to $L=\lambda /2$), the waveguide coupling term vanishes: $\tilde{\gamma }\left(\pi \right)={\displaystyle \frac{\gamma }{2}}(1-1)=0$. Substituting this into the eigenvalue equation yields purely real solutions

$${\lambda }_{\pm }\left(\pi \right)=\pm \sqrt{2}g.$$

(14)

Since $\mathrm{Im}\left({\lambda }_{\pm }\right)=0$, the lifetime of these states diverges to infinity. This confirms that the system supports a hybrid BIC, where the emitter undergoes vacuum Rabi oscillations with the symmetric cavity mode without any radiative loss to the waveguide (see below).

3. Numerical Results for a Single Giant Atom

We now examine the temporal dynamics of the system by solving the effective equations of motion, Equation (10), for different waveguide-mediated phase delays $\varphi$ (details on the numerical solution of Equation (10) can be found in Appendix D). To fully capture the storage fidelity and dissipation mechanisms, we analyze two primary observables: the population of the quantum emitter, $P_e\left(t\right)={\left|c_e\left(t\right)\right|}^2$, and the total survival probability of the excitation within the discrete emitter-cavity system, $P_{\mathrm{tot}}\left(t\right)=|c_e{\left(t\right)|}^2+{\sum }_j{\left|c_j\left(t\right)\right|}^2$.

3.1. Regime I: The BIC ($\varphi =\pi$)

Setting the inter-cavity distance to $L=\lambda /2$, the dissipative terms in the symmetric sector vanish. The system effectively decouples from the waveguide bath, reducing to a closed Jaynes-Cummings system. For an initially excited emitter [$c_e\left(0\right)=1$], the time evolution of the emitter population is given analytically by

$$P_e\left(t\right)={cos}^2\left(\sqrt{2}gt\right).$$

(15)

This represents vacuum Rabi oscillations with an enhanced frequency $\Omega =\sqrt{2}g$ relative to a single-cavity system, as shown in Figure 2a (orange curve). Crucially, while the emitter population oscillates, the total system population shown in Figure 2b remains clamped at unity [$P_{\mathrm{tot}}\left(t\right)=1$]. This conservation of probability confirms that the energy is not leaking into the waveguide but is merely exchanging coherently between the emitter and the localized cavity modes. This proves the formation of a high-Q dark state (BIC) where the hybrid polariton is physically shielded from the radiative bath by destructive interference at the coupling points.

3.2. Regime II: Super-Radiant Decay ($\varphi =0$)

Conversely, when $L=\lambda$, the interference becomes constructive, maximizing the coupling to the continuum. The effective decay rate of the symmetric mode becomes ${\Gamma }_S=2\gamma$. In this regime, the system behaves as a damped harmonic oscillator. The emitter population approximates as

$$P_e\left(t\right)\approx e^{-\gamma t}{cos}^2\left({\Omega }^{\prime }t\right),$$

(16)

where ${\Omega }^{\prime }=\sqrt{2g^2-{(\gamma /2)}^2}$ is the modified Rabi frequency resulting from the hybridization of the emitter and the symmetric cavity mode. As seen in Figure 2a (blue curve), the excitation is rapidly lost. This is corroborated by the total population dynamics in Figure 2b, where $P_{\mathrm{tot}}\left(t\right)$ drops monotonically to zero. This signifies that the energy is irreversibly radiated into the propagating waveguide modes, demonstrating the switchable nature of the light-matter interface via geometric tuning.

3.3. Regime III: Intermediate Coupling and Partial Decay ($\varphi =\pi /2$)

When the inter-cavity phase delay is set to $\varphi =\pi /2$, the waveguide-mediated cross-coupling becomes purely imaginary and equal to $\gamma /2$. In this regime, the interference between coupling points is only partially destructive, preventing the formation of a perfect BIC while simultaneously avoiding the maximal decay associated with the super-radiant regime.

The effective Hamiltonian governing the symmetric subspace is given by

$${\mathbf{H}}_{\mathrm{Sym}}=\left(\begin{array}{cc}-i{\displaystyle \frac{\gamma }{2}}(1+i)& \sqrt{2}g\\ \sqrt{2}g& 0\end{array}\right).$$

(17)

The complex eigenvalues lead to a decay rate for the emitter population that is reduced compared to the $\varphi =0$ case, as can be observed from Figure 2a. Specifically, the emitter population $P_e\left(t\right)$ exhibits damped oscillations that reach a vanishing steady state, but at a significantly slower rate than the super-radiant decay observed in Regime II. The same effect is corroborated by the total population dynamics in Figure 2b, where $P_{\mathrm{tot}}\left(t\right)$ drops monotonically to zero, but with a much slower rate relative to the case of $\varphi =0$.

4. Dynamics of Two Coupled Giant Atoms

We generalize the formalism to a system of two giant atoms, labeled $\mu \in \{A,B\}$, interacting via the same waveguide continuum (see Figure 3). Each giant atom $\mu$ consists of a quantum emitter (QE_μ) resonantly coupled to a distinct pair of cavities. Specifically, giant atom A comprises QE_A coupled to cavities 1 and 2 (located at $x_1,x_2$), while giant atom B comprises QE_B coupled to cavities 3 and 4 (located at $x_3,x_4$). The separation between the centers of the two giant atoms is denoted by D.

The total Hamiltonian for the two-atom system, $H_{\mathrm{tot}}^{\left(2\right)}$, is an extension of the single-atom case, accounting for the additional degrees of freedom

$$H_{\mathrm{tot}}^{\left(2\right)}=H_0+H_{\mathrm{JC}}+H_{\kappa }.$$

(18)

The free energy Hamiltonian includes both emitters and all four cavity modes:

$$H_0=\sum _{\mu =A,B}\hslash {\omega }_q{\sigma }_{\mu }^{†}{\sigma }_{\mu }+\sum _{j=1}^4\hslash {\omega }_c{a}_j^{†}{a}_j+{\int }_{-\infty }^{\infty }dk\hslash {\omega }_k{b}_k^{†}{b}_k.$$

(19)

The coherent interaction Hamiltonian describes the local coupling of each emitter to its respective cavity pair. Assuming identical coupling strengths g

$$H_{\mathrm{JC}}=\hslash g\left[\sum _{j=1,2}(a_j^{†}{\sigma }_A+\mathrm{h}.\mathrm{c}.)+\sum _{j=3,4}(a_j^{†}{\sigma }_B+\mathrm{h}.\mathrm{c}.)\right].$$

(20)

Note that we assume local addressing, meaning there is no direct coupling between QE_A and cavities 3,4, nor between QE_B and cavities 1,2.

The dissipative coupling to the common waveguide introduces the crucial long-range interactions

$$H_{\kappa }=\hslash \sum _{j=1}^4{\int }_{-\infty }^{\infty }dk\sqrt{{\displaystyle \frac{\gamma v_g}{2\pi }}}\left(b_k^{†}{a}_j{e}^{-ik{x}_j}+\mathrm{h}.\mathrm{c}.\right).$$

(21)

This term mediates interactions between all cavities, regardless of which giant atom they belong to, creating a fully connected network via the continuum.

4.1. Time-Dependent Dynamics

We solve the Schrödinger equation in the single-excitation subspace using the ansatz

$$\begin{array}{cc}\hfill |\Psi \left(t\right)\rangle & =c_A\left(t\right)|e_A,g_B,\mathbf{0},{\left\{0\right\}}_k\rangle +c_B\left(t\right)|g_A,e_B,\mathbf{0},{\left\{0\right\}}_k\rangle \hfill \\ \hfill & +\sum _{j=1}^4{c}_j\left(t\right)|g_A,g_B,1_j,{\left\{0\right\}}_k\rangle \hfill \\ \hfill & +{\int }_{-\infty }^{\infty }dk{c}_k\left(t\right)|g_A,g_B,\mathbf{0},1_k\rangle .\hfill \end{array}$$

(22)

Here, $c_A$ and $c_B$ are the amplitudes for the emitters, and $c_j$ ($j=1,\dots 4$) are the amplitudes for the cavities. Also, $|\mathbf{0}\rangle \equiv |0_1,0_2,0_3,0_4\rangle$ denoting the joint vacuum state of the four localized cavity modes. Following the same Wigner-Weisskopf procedure as in the single-atom case, we integrate out the waveguide modes $c_k\left(t\right)$. This results in a set of coupled differential equations where the waveguide mediates an effective retarded interaction between any two cavities m and n with a delay ${\tau }_{mn}=|x_m-x_n|/v_g$.

In the Markovian limit ($\tau \to 0$), the feedback becomes instantaneous. The dynamics of the discrete system amplitudes $\mathbf{C}={(c_1,c_2,c_3,c_4,c_A,c_B)}^T$ are governed by the effective non-Hermitian Hamiltonian ${\mathbf{H}}_{\mathrm{eff}}^{\left(2\right)}$

$${\displaystyle \frac{d}{dt}}\mathbf{C}=-i{\mathbf{H}}_{\mathrm{eff}}^{\left(2\right)}\mathbf{C}.$$

(23)

The validity of this Markovian approximation holds when the propagation delay $\tau$ is much smaller than the characteristic emitter-cavity coupling time $1/g$ and the decay time $1/\gamma$. If the separation between the cavities is increased such that the delay becomes significant, the system enters the non-Markovian regime, where the retarded feedback can lead to more complex dynamics, such as non-exponential decay and the emergence of multiple bound states.

4.2. Effective Non-Hermitian Hamiltonian

The $6\times 6$ effective Hamiltonian matrix takes the block form

$${\mathbf{H}}_{\mathrm{eff}}^{\left(2\right)}=\left(\begin{array}{cc}{\mathbf{H}}_{\mathrm{cavity}}& {\mathbf{H}}_{\mathrm{int}}\\ {\mathbf{H}}_{\mathrm{int}}^T& {\mathbf{0}}_{2\times 2}\end{array}\right).$$

(24)

The interaction block ${\mathbf{H}}_{\mathrm{int}}$ couples emitters to their cavities

$${\mathbf{H}}_{\mathrm{int}}=\left(\begin{array}{cc}g& 0\\ g& 0\\ 0& g\\ 0& g\end{array}\right).$$

(25)

The cavity-cavity block ${\mathbf{H}}_{\mathrm{cavity}}$ contains the dissipative couplings. Defining $J_{mn}=-i{\displaystyle \frac{\gamma }{2}}{e}^{i{k}_0|x_m-x_n|}$ we obtain

$${\mathbf{H}}_{\mathrm{cavity}}=\left(\begin{array}{cccc}-i{\displaystyle \frac{\gamma }{2}}& J_{12}& J_{13}& J_{14}\\ J_{21}& -i{\displaystyle \frac{\gamma }{2}}& J_{23}& J_{24}\\ J_{31}& J_{32}& -i{\displaystyle \frac{\gamma }{2}}& J_{34}\\ J_{41}& J_{42}& J_{43}& -i{\displaystyle \frac{\gamma }{2}}\end{array}\right).$$

(26)

This matrix reveals that while the emitters are physically separated, the cavities form a fully connected graph via the waveguide. The term $J_{mn}$ represents the phase-dependent interference between any two emission points.

4.3. Diagonalization and BIC Formation

The rich physics of this system arises from the interplay between intra-atom interference (within giant atom A or giant atom B) and inter-atom interference (between giant atom A and giant atom B). We assume identical internal structures for both giant atoms, with intra-atom phase delay $\varphi =k_0|x_2-x_1|=k_0|x_4-x_3|$. The separation between the two atoms is defined by the phase delay $\Phi =k_{0}D$.

We transform the basis into the symmetric (S) and antisymmetric (A) modes for each giant atom

$$c_{S_{\mu }}={\displaystyle \frac{c_{2\mu -1}+c_{2\mu }}{\sqrt{2}}},c_{A_{\mu }}={\displaystyle \frac{c_{2\mu -1}-c_{2\mu }}{\sqrt{2}}},\mu \in \{A,B\}.$$

(27)

This transformation yields two primary regimes of BICs.

4.3.1. Regime A: Local Trivial BICs ($\varphi =\pi$)

If the internal phase of each giant atom is set to $\varphi =\pi$ (i.e., $L=\lambda /2$), then each giant atom individually decouples from the waveguide. The intra-atom terms $J_{12}$ and $J_{34}$ become maximally destructive. In this regime, the system behaves as two independent, non-interacting quantum memories. The excitation stays localized within giant atom A or giant atom B, and the waveguide connection is effectively severed at the source.

4.3.2. Regime B: Collective Dimer BICs ($\varphi =0,\Phi =\pi$)

A non-trivial collective state emerges when the individual atoms are configured to be bright. For example, we set the intra-atom phase delay to $\varphi =2\pi$ (corresponding to an inter-cavity spacing of $L=\lambda$), such that the symmetric mode of each atom strongly couples to the waveguide. Simultaneously, we set the inter-atom phase delay—defined as the distance D between the centers of the two giant atoms—to $\Phi =\pi$ (corresponding to $D=\lambda /2$).

It is important to note that these conditions are geometrically consistent: while the two cavities within a single giant atom are separated by a full wavelength ($L=\lambda$), the centers of the two distinct giant atom units are separated by only half a wavelength ($D=\lambda /2$). This configuration allows the two bright giant atoms to interfere destructively with each other as a single unit, leading to the collective dimer BIC.

In this limit, the effective Hamiltonian for the symmetric modes $S_A$ and $S_B$ simplifies to a coupled $2\times 2$ block:

$${\mathbf{H}}_{\mathrm{dimer}}\approx \left(\begin{array}{cc}-i\gamma & -i\gamma e^{i\Phi }\\ -i\gamma e^{i\Phi }& -i\gamma \end{array}\right).$$

(28)

For $\Phi =\pi$, the off-diagonal coupling becomes $+i\gamma$ and the eigenstates are the superpositions $|{\Psi }_{\pm }\rangle =(|S_A\rangle \pm |S_B\rangle )/\sqrt{2}$. Diagonalizing this block reveals that one superposition is super-radiant (decay rate $2\gamma$), while the other has a decay rate of zero.

This corresponds to a collective BIC or a dark molecular state. The wave emitted by giant atom A interferes destructively with the wave emitted by giant atom B. Consequently, a photon can be delocalized across the entire four-cavity array, hopping between the two distant emitters via virtual waveguide photons, without ever leaking into the continuum. This demonstrates a decoherence-free subspace (DFS) generated purely by geometry.

5. Numerical Results for Two Coupled Giant Atoms

To confirm the existence of BICs and explore the rich dynamics governed by the interplay between intra-atom (local) and inter-atom (global) interference, we numerically solve Equation (23) according to the numerical scheme described in Appendix D. The system is initialized with the first emitter excited and the second in the ground state [$c_A\left(0\right)=1,c_B\left(0\right)=0$], with all cavity modes initially empty. We characterize the evolution of the system by tracking three complementary observables: the total emitter population $P_{\mathrm{emit}}\left(t\right)=|c_A{\left(t\right)|}^2+{\left|c_B\left(t\right)\right|}^2$, the total system survival probability $P_{\mathrm{sys}}\left(t\right)=\sum {\left|c_i\left(t\right)\right|}^2$, and the individual emitter populations $|c_A{\left(t\right)|}^2$ and $|c_B{\left(t\right)|}^2$. The global population dynamics [$P_{\mathrm{emit}}\left(t\right)$ and $P_{\mathrm{sys}}\left(t\right)$] are summarized in Figure 4, while the microscopic distribution of excitation (individual emitter populations $|c_A{\left(t\right)|}^2$ and $|c_B{\left(t\right)|}^2$) is detailed in Figure 5.

The first regime of interest is local independent trapping. When the internal phase delay of each giant atom is tuned to $\varphi =\pi$ (orange curves), each emitter acts as an individual BIC, decoupled from the waveguide. As shown in Figure 4a, the total emitter population oscillates with high amplitude, mimicking single-atom BIC behavior. Crucially, Figure 4b confirms that the total system energy is conserved at unity ($P_{\mathrm{sys}}\approx 1$), proving that no leakage into the waveguide occurs. The microscopic origin of this trapping is revealed in Figure 5b: the population stays entirely localized on the initially excited giant atom A ($|c_A{|}^2\approx 1$), while atom B remains in the vacuum state ($|c_B{|}^2\approx 0$). This confirms that the waveguide-mediated interaction is suppressed at the source, effectively creating two isolated quantum memories.

A fundamentally different mechanism emerges in the collective dark dimer regime. Here, the individual atoms are configured to be radiative ($\varphi =0$), but they are separated by a destructive interference distance $\Phi =\pi$ (green curves). We observe that the total population does not decay to zero; instead, it stabilizes at exactly $P_{\mathrm{emit}}\approx 0.5$ (Figure 4a). This initial 50% loss corresponds to the super-radiant symmetric component decaying into the waveguide. However, the remaining 50% is trapped in the antisymmetric dark state. This result confirms the formation of a dark molecule mediated by the waveguide. The physics of this state are illustrated in Figure 5a, where we see an initial coherent transfer of energy from the source (giant atom A, green solid line) to the target (giant atom B, orange dashed line) until both populations eventually equilibrate to 0.25, signifying the formation of a maximally entangled Bell-like state $|{\Psi }_{-}\rangle =\left(\right|e_A{g}_B\rangle -|g_A{e}_B\rangle )/\sqrt{2}$ which is delocalized across the two distant nodes.

6. Conclusions

In summary, we have theoretically proposed and analyzed a scalable architecture for realizing giant-atom physics in the optical domain. By utilizing a hybrid cavity-QED approach, where a quantum emitter is coherently coupled to a pair of spatially separated cavities, we effectively synthesized the non-local interaction Hamiltonian characteristic of giant atoms. This scheme circumvents the physical size constraints imposed by the diffraction limit, offering a practical route to implement giant-atom protocols using standard integrated photonic components.

Through a rigorous derivation of the effective non-Hermitian Hamiltonian, we mapped the dynamics of the system across different interference regimes. For a single giant atom, we demonstrated that the phase delay between the cavity-mediated emission pathways can be tuned to extinguish radiative losses completely. This results in the formation of a local BIC, where the excitation oscillates indefinitely within the emitter-cavity subsystem, shielded from the waveguide bath.

Furthermore, extending our formalism to a two-atom array revealed the emergence of collective phenomena distinct from single-atom physics. We identified a dark dimer regime where two individually radiative (bright) giant atoms interfere destructively via the waveguide to form a robust, delocalized dark state. This collective BIC traps half of the initial population in an entangled superposition, highlighting the potential of this platform for generating long-range entanglement and decoherence-free subspaces in open quantum systems.

Regarding the experimental feasibility of our proposal, while giant-atom physics was historically pioneered in the microwave regime, the rapid maturation of integrated nanophotonics now offers a viable path for optical implementations. State-of-the-art platforms such as silicon nitride ($S{i}_3{N}_4$) and silicon-on-insulator provide the high-precision fabrication and low-loss environments required to realize the phase-dependent interference effects and BICs described here. Furthermore, the use of photonic crystal waveguides or micro-ring resonators can provide the necessary field enhancement and slow-light effects to facilitate strong coupling. The localized cavities can be implemented as micro-disk or photonic crystal defect resonators, which are routinely fabricated with high quality factors using standard electron-beam lithography and reactive-ion etching. Moreover, suitable quantum emitters for this optical architecture include solid-state systems such as semiconductor quantum dots or color centers in diamond (e.g., nitrogen-vacancy centers), which can be resonantly coupled to the cavity modes with high cooperativity.

Our findings establish a bridge between microwave giant-atom concepts and optical waveguide QED. Future work may explore the integration of this architecture with topological photonic lattices to engineer chiral bound states. Also, a particularly interesting avenue for future research involves exploring the non-Markovian regime, where the time delay between spatially separated interaction ports is no longer negligible. In such cases, the memory effects of the waveguide continuum could be leveraged to engineer non-local correlations or to study the transition between interference-induced trapping and delay-induced oscillating bound states. The proposed setup holds promise for applications in quantum interconnects, distributed quantum computing, and on-chip quantum memories.