⚛️ quantum physics

Multi-emitter oscillating bound states in Waveguide QED

This paper demonstrates that in a waveguide QED system with two spatially separated emitters, spontaneous emission can drive the formation of non-local equilibrium states characterized by persistent oscillations in both photonic and emitter populations, arising from the hybridization of bound states embedded in and outside the energy continuum.

Original authors: Sergi Terradas-Briansó, Carlos A. González-Gutiérrez, Iván Huarte, David Zueco, Luis Martin-Moreno

Published 2026-02-19

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CC BY 4.0

Original authors: Sergi Terradas-Briansó, Carlos A. González-Gutiérrez, Iván Huarte, David Zueco, Luis Martin-Moreno

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a long, endless hallway made of perfectly connected rooms. This is your waveguide. Inside this hallway, there are two special "actors" (the quantum emitters) standing far apart from each other. These actors are like excited atoms that want to relax, but they can only do so by shouting a message (a photon) into the hallway.

Usually, when an actor shouts, the sound travels down the hallway and disappears forever. This is what we call "decay." But in this paper, the authors discovered a way to trap the sound so it never leaves, creating a strange, magical state where the actors and the sound dance together forever.

Here is the story of how they did it, explained simply:

1. The Setup: A Hallway with Echoes

Think of the hallway not as an empty corridor, but as a series of rooms (cavities) connected by doors. The sound can hop from room to room, but it can only hop at certain speeds (this is the energy band).

The two actors are placed in specific rooms. When one actor gets excited, it tries to send a photon into the hallway. However, because of the specific distance between the actors and the "rules" of the hallway, the sound waves they send out can interfere with each other.

2. The Magic Trick: Trapping the Ghost

Normally, if you shout in a hallway, the sound fades. But here, the authors found a way to create "Bound States."

The "Outside" Ghost (BOC): Imagine the actor is so close to the edge of the hallway's allowed sounds that the sound gets stuck right next to the actor, unable to escape. It's like a ghost haunting a specific room.
The "Inside" Ghost (BIC): This is even weirder. Imagine the actor is shouting a sound that should travel down the hallway, but because of a perfect interference pattern (like two waves canceling each other out), the sound gets trapped inside the flow of traffic. It's a ghost that lives inside a busy crowd but never gets bumped by anyone.

3. The Dance: Oscillating Bound States

The real magic happens when the system has both types of ghosts at the same time.

Think of it like a trampoline.

If you have one trampoline, you just bounce up and down.
But if you have two trampolines connected by a spring, you can create a complex dance. One person bounces, then the spring pulls the other person, then they swap energy.

In this paper, the "spring" is the connection between the two trapped states (one inside the crowd, one outside). When the first actor starts to relax, it doesn't just fade away. Instead, the energy gets caught in a superposition (a mix) of these two trapped states.

The Result:

The "Breathing" Mode: The photon (the sound) gets trapped in the space between the two actors. It doesn't leave; it just expands and contracts, like a lung breathing. It pulses back and forth between the two actors forever.
The "Swap" Mode: If the conditions are just right, the energy doesn't just pulse; it actually travels from the left actor to the right actor, then back again, in a perfect, endless loop. The actors are essentially talking to each other without ever losing a word.

4. Why This Matters

In the real world, quantum systems are fragile. Usually, as soon as you try to use them, they lose their energy to the environment (decoherence). This is like trying to have a conversation in a noisy, windy storm.

This paper shows that by carefully tuning the "hallway" (the waveguide) and the distance between the actors, you can create a quiet room inside the storm.

The energy stays trapped.
The actors can exchange information (entanglement) without it leaking out.
The system settles into a "steady state" where it keeps oscillating forever, rather than dying out.

The Big Picture Analogy

Imagine two people standing on opposite sides of a giant, echoing canyon.

Normal Physics: If one person throws a ball, it flies across and hits the other side, then falls off the cliff. The game is over.
This Paper's Physics: The canyon is shaped so perfectly that when the ball is thrown, it gets caught in a magical loop. It bounces between the two people, getting bigger and smaller, never falling off the cliff. The two people are now locked in a permanent, rhythmic dance, connected by the ball that never leaves their hands.

Why Should We Care?

This isn't just a cool physics trick. It's a blueprint for building quantum computers and quantum networks.

It shows us how to make quantum bits (qubits) talk to each other without losing their information.
It proves that we can create "quantum mirrors" that trap light and matter together, allowing us to build new types of devices that store and process information in ways we couldn't before.

In short, the authors found a way to turn a "leaky" quantum system into a "perfectly sealed" one, where energy dances forever instead of disappearing.

1. Problem Statement

Waveguide Quantum Electrodynamics (WQED) platforms are powerful for exploring non-Markovian quantum phenomena, particularly the formation of bound states. Traditionally, these are categorized into:

Bound States Outside the Continuum (BOCs): States where the emitter energy lies outside the waveguide's photonic band, leading to localized, non-decaying states.
Bound States in the Continuum (BICs): States where the emitter energy lies within the photonic band but remains localized due to destructive interference, preventing decay into propagating modes.

While previous research demonstrated that "giant atoms" (emitters coupled to multiple points in a waveguide) can form oscillating bound states (superpositions of BICs or BIC-BOC pairs), this requires specific, often complex, multi-point coupling geometries. The authors address the question: Can oscillating bound states emerge in a simpler system of two spatially separated standard emitters coupled to a cavity-array waveguide, and can spontaneous emission drive the system into these non-local equilibrium states?

2. Methodology

The authors employ a combination of analytical derivation and numerical simulation to study a system of two identical two-level quantum emitters coupled to a 1D cavity-array waveguide.

System Model:
- The waveguide is modeled as an infinite array of coupled cavities with hopping amplitude $\xi$ and resonant frequency $\omega_0$ .
- Two emitters are coupled to specific cavities separated by $x$ intermediate cavities.
- The Hamiltonian is a spin-boson type model under the Rotating Wave Approximation (RWA).
- The analysis is restricted to the single-excitation manifold.
Analytical Framework:
- The authors use the resolvent operator method ( $G(E) = (E-H)^{-1}$ ) to obtain exact time evolution.
- They project the resolvent onto the basis of maximally entangled symmetric ( $|+\rangle$ ) and antisymmetric ( $|-\rangle$ ) emitter states. This diagonalizes the problem, decoupling the dynamics of the two sectors.
- Bound State Identification: Bound states correspond to the poles of the Green's function, found by solving $F_{\pm}(E) = E - \varepsilon_{\pm} - \Sigma_{\pm}(E) = 0$ , where $\Sigma_{\pm}(E)$ are the self-energy terms derived analytically.
- Time Evolution: The amplitude of the excited state $c_e(t)$ is calculated via the inverse Laplace transform (Fourier transform) of the resolvent, separating contributions from the continuum (decaying) and poles (bound states).
Numerical Simulation:
- The dynamics are simulated for various coupling strengths ( $g$ ) and inter-emitter separations ( $x$ ).
- They track the population of individual emitters ( $|L\rangle, |R\rangle$ ), the collective states ( $|+\rangle, |-\rangle$ ), and the spatial distribution of photon occupation numbers.

3. Key Contributions

Novel Platform for Oscillating Bound States: The paper introduces a configuration distinct from "giant atoms." It demonstrates that two standard, spatially separated emitters coupled to a cavity array can naturally support superpositions of bound states (BICs and BOCs) without requiring multi-point coupling to a single emitter.
Spontaneous Emission as a Driver: The authors show that spontaneous emission from a single initially excited emitter can drive the system into a non-local equilibrium state. Instead of decaying to the ground state, the system relaxes into a steady superposition of bound modes.
Analytical Conditions for Coexistence: They derive the precise parameter regimes (emitter energy $\Delta$ $Δ$ and coupling strength $g$ $g$ ) required for the coexistence of:
- One BIC and one BOC.
- One BIC and two BOCs.
Scaling to Multi-Emitters: The study extends the analysis to three emitters, demonstrating that while the dynamics become more complex, coherent oscillations between parity-defined collective states persist.

4. Key Results

Formation of Oscillatory Steady States:
- When the system parameters are tuned such that bound states exist, the initial excitation does not fully decay. Instead, the system settles into a state where both emitter populations and photon densities exhibit persistent, non-decaying oscillations.
- Weak Coupling ( $g=0.02$ ): The system supports one BIC and one BOC (both in the antisymmetric $|-\rangle$ sector). The steady state exhibits a "breathing mode" where the photon density oscillates between the two emitters, while the total excitation remains trapped.
- Intermediate/Strong Coupling ( $g=0.05, 0.1$ ): The system supports one BIC and two BOCs (one in $|+\rangle$ , two in $|-\rangle$ ). The energy splitting between these states creates complex beat frequencies. The dynamics shift from a simple breathing mode to a coherent exchange of population between the left and right emitters (the excitation swaps back and forth).
Role of Symmetry and Interference:
- The existence of BICs is dictated by the interference condition $k(E)x = m\pi$ .
- The coexistence of BICs and BOCs leads to hybrid oscillatory modes. The oscillation frequency is determined by the energy difference between the bound states ( $\Delta E = E_i - E_j$ ).
- The system effectively creates a decoherence-free subspace where radiative decay is suppressed, but coherent emitter-emitter interaction is preserved.
Spatial Dynamics:
- In the weak coupling regime, a "trapped" photonic mode forms between the emitters, acting as a tunable supercavity.
- In the multi-bound-state regime, the photon density exhibits rich spatiotemporal structures, oscillating in synchrony with the matter degrees of freedom.
Three-Emitter Extension:
- For three emitters initialized in a symmetric state, the system evolves into coherent oscillations between even-parity collective states, while odd-parity modes remain unpopulated. This confirms the robustness of the phenomenon in small arrays.

5. Significance

Experimental Feasibility: The proposed setup (two emitters in a cavity array) is experimentally accessible using current superconducting circuit architectures or photonic crystal waveguides, unlike the more complex giant-atom configurations.
New Route for Interaction: The results provide a mechanism to generate long-range emitter-emitter interactions via free evolution, mediated by the waveguide, without external driving fields.
Quantum Control: The ability to tune system parameters to switch between "breathing modes" and "population swapping" offers a new degree of control for quantum information processing, such as storing and transferring quantum states between distant nodes.
Fundamental Physics: The work deepens the understanding of non-Markovian dynamics, specifically how structured reservoirs can stabilize non-trivial, oscillating quantum states that defy standard exponential decay laws.

In conclusion, the paper establishes that multi-emitter oscillating bound states are a generic feature of WQED systems with structured reservoirs, achievable through simple spatial separation and parameter tuning, opening new avenues for creating stable, entangled quantum states.