Multi-emitter oscillating bound states in Waveguide QED

Abstract

Waveguide quantum electrodynamics platforms have emerged as promising candidates for exploring and implementing non-Markovian quantum phenomena. In this work, we investigate the formation and dynamics of superpositions of bound states in a cavity array waveguide coupled to two spatially separated quantum emitters. By tuning the system parameters, we show that spontaneous emission can drive the system into non-local equilibrium states in which both photonic and emitter populations exhibit persistent oscillations. These states arise from the coexistence of bound states embedded in the energy continuum and bound states outside it, leading to hybrid oscillatory modes. We analytically derive the conditions required for the emergence of these states, numerically simulate their formation through spontaneous emission, and predict their long-time behaviour. Our results demonstrate that such bound-state superpositions enable the generation of emitter-emitter interaction through free evolution, while supporting oscillatory breathing modes of the photon density between the emitters.

Figures (4)

• Figure 1: (a) Schematic representation of a waveguide composed of an infinite cavity array coupled to two quantum emitters. The intercavity hopping amplitude is denoted by $\xi$, and the emitter-cavity coupling strength by $g$. Each cavity has a resonant frequency $\omega_{0}$, while the excitation energy of each emitter is $\Delta$. The two emitters are coupled to cavities separated by $x$ intermediate cavities. (b) Dispersion relation of the cavity-array waveguide (solid blue line). The dashed orange line shows the dispersion relation of a continuous waveguide with similar low-energy behaviour. The cavity-array band is bounded by dotted black lines and highlighted by the green shaded region. Below the lower band edge, we indicate the energies of two hypothetical bound states outside the continuum arising from the coupling of the two emitters shown in (a).
• Figure 2: Regions of parameter space in which the system of two identical emitters coupled to a cavity-array waveguide (see Fig. \ref{['fig:Scheme']}) supports zero, one, or two bound states outside the continuum. Each panel corresponds to a different inter-emitter separation: $x = 5$, $11$, $21$, and $31$ in panels (a)-(d), respectively. The plots show the $\Delta$-$g$ parameter space, indicating the number of BOCs supported by the system. Below the solid curves no BOCs are present, while the regions between the curves, highlighted by the coloured background, correspond to the existence of a single BOC. Above both curves, the system supports two BOCs.
• Figure 3: Illustration of three spontaneous emission processes in which an initially excited emitter relaxes into different superpositions of bound states involving both the emitters and the waveguide field. In all cases, the parameters are chosen such that the lowest-energy BIC participates in the dynamics: $\xi = 1$ and $\Delta = \omega_0 - 2\xi \cos[(x - 1)\pi/x]$, where $x = 31$ is the number of cavities separating the two emitters. Each column corresponds to a different emitter-waveguide coupling strength: $g = 0.02$, $0.05$, and $0.1$ for the left, central, and right columns, respectively. The left column [panels (a)-(c)] shows decay into a superposition of one BIC and one BOC, while the remaining columns correspond to decay into a superposition of one BIC and two BOCs. Panels (a), (d), and (g) display the functions $F_{\pm}(E)$ for the symmetric (green curves) and antisymmetric (red curves) matter sectors. The real and imaginary parts of $F_{\pm}(E)$ are shown as solid and dashed lines, respectively. The solutions of Eq. \ref{['eq:PoleEq']} are marked by circles, colour-coded according to the corresponding $\pm$ sector, and the shaded green region denotes the waveguide energy band. Panels (b), (e), and (h) show the time evolution of the excited-state populations of the emitters. Populations in the ${\vert L(R) \rangle}$ basis are shown with dashed lines, while those in the ${\vert \pm \rangle}$ basis are shown with solid lines. The long-time asymptotic predictions, obtained from the bound-state contributions, are indicated by dotted black lines. Finally, panels (c), (f), and (i) show the spatial distribution of the photon occupation number generated during the spontaneous emission process. The positions of the two emitters within the cavity array are marked by vertical red dashed lines.
• Figure 4: Spontaneous emission leading to oscillatory dynamics between two states involving three emitters and the waveguide field. The emitters are located at positions $n = -16$, $0$, and $16$, corresponding to the left ($L$), central ($C$), and right ($R$) emitters, respectively, and are initially prepared in the fully symmetric state ${\vert \psi \rangle}_{\mathrm{in}} = ({\vert L \rangle} + {\vert C \rangle} + {\vert R \rangle}) / \sqrt{3}$. (a) Time evolution of the excited-state populations of the emitters. The populations of the collective emitter states ${\vert \psi_{+,1} \rangle}$ (green), ${\vert \psi_{+,2} \rangle}$ (lime), and ${\vert \psi_{-} \rangle}$ (red) are shown as solid lines. The populations of the individual emitter states ${\vert L \rangle}$, ${\vert C \rangle}$, and ${\vert R \rangle}$ are shown as a blue dash-dotted line, an orange dashed line, and a brown dotted line, respectively. (b) Time evolution of the emitted waveguide field. The positions of the emitters are indicated by vertical red dashed lines. The emitter-waveguide coupling strength is $g = 0.02$, while the remaining parameters are the same as in Fig. \ref{['fig:LargeFigure']}.