Phase Transitions and Virtual Exceptional Points in Quantum Emitters Coupled to Dissipative Baths

TL;DR

We address how a single two-level emitter coupled to a uniformly lossy semi-infinite lattice relaxes under non-Hermitian bath dynamics, revealing dynamical phase transitions in spontaneous emission. By exact resolvent (Laplace) analysis, the dynamics are governed by poles of the (analytic) self-energy on the first and second Riemann sheets, with bound-state poles on the first sheet and resonant poles on the second sheet; coalescence of resonance poles on the second sheet defines virtual exceptional points that induce dynamical transitions. Across three coupling regimes, the pole structure restructures with dissipation, yielding an optimal loss rate (e.g., $\gamma=\gamma_{c1}$ in weak/moderate and $\gamma=\gamma_{c2}$ in strong) where relaxation is fastest. The results show that the bath's spectral features, particularly resonance coalescence, govern emitter relaxation and highlight dissipation engineering as a practical tool for quantum technologies.

Abstract

Controlling atom-photon interactions in engineered environments is central to quantum optics and emerging quantum technologies. Non-Hermitian (NH) photonic baths, where dissipation fundamentally reshapes spectral and dynamical properties, provide versatile platforms for such control. Here we investigate the relaxation dynamics of a single two-level quantum emitter coupled to the edge of a semi-infinite dissipative bosonic lattice with uniform loss. Despite the simplicity of this bath, we uncover rich dynamical phase transitions, i.e. qualitative changes in spontaneous emission decay as system parameters are varied. In particular, we establish the existence of an optimal dissipative environment for accelerated spontaneous emission. The phase transitions are traced to spectral restructuring of the resolvent, in some cases governed by the coalescence of resonance states on the second Riemann sheet. We identify these coalescences as virtual exceptional points (EPs) of resonance origin, providing a conceptual bridge with EP physics while highlighting distinctive features of infinite-dimensional NH systems. More broadly, our results illustrate how the specific nature of dissipation -- whether uniform losses, staggered losses, or dephasing -- can profoundly impact emitter relaxation, pointing to dissipation engineering as a versatile tool for quantum technologies.

Figures (13)

• Figure 1: Schematic of a two-level quantum emitter coupled to a dissipative bosonic bath consisting of a semi-infinite array of coupled optical cavities. The emitter is placed inside the edge resonator ($n=1$). Here, $J$ denotes the photon hopping rate between adjacent resonators, $g_0$ the atom-photon coupling strength, and $\gamma_n$ the photon loss rates in each resonator.
• Figure 2: Contour paths in the complex $s$-plane for Eq. \ref{['eq:Laplace']}. The solid segment $\mathbf{I}$ along ${\rm Re}(s)=-\gamma$ is the branch cut of the self-energy $\Sigma(s)$. The Bromwich path $\mathbf{B}$ can be deformed into the Hankel paths $h_1$ and $h_2$, and the pole contributions $s_k$ (first Riemann sheet, bound states) and $S_k$ (second Riemann sheet, resonant states). The shaded region indicates the domain of analytic continuation for $\hat{c}_a(s)$ on the second Riemann sheet.
• Figure 3: (a) Behavior of the real and imaginary parts of the two poles $s_{p_1}$ and $s_{p_2}$ versus loss rate $\gamma$ in the weak coupling regime ($g_0/J=0.8$). The dashed curve, corresponding to ${\rm Re}(s)=-\gamma$, separates the two Riemann sheets. Pole $s_{p_2}$, with positive real part, is located on the second Riemann sheet and does not impact on the relaxation dynamics. Pole $s_{p_1}$ is located on the second Riemann sheet for $\gamma < \gamma_{c_1} \equiv g_0^2/J=0.64 J$, corresponding to a resonant state, whereas it is located on the first Riemann sheet for $\gamma> \gamma_{c_1}$, corresponding to an atom-photon bound state. (b) Loci of the two poles $s_{p_{1,2}}$ in complex $s$ plane as the loss rate $\gamma$ is varied from $\gamma=0$ to $\gamma=3J$.
• Figure 4: (a) Integration contour in complex $s$ plane in the weak coupling regime and for $\gamma< \gamma_{c_1}$. The contribution to the decay law $c_a(t)$ comes from the two Hankel paths $h_{1,2}$ and from the pole $S_1=s_{p_1}$ on the second Riemann sheet (resonant state). (b) Numerically computed behavior of the survival probability $P_s(t)=|c_a(t)|^2$ versus normalized time $Jt$ for parameter values $g_0/J=0.6$ and $\gamma/J=0.05$.
• Figure 5: (a) Integration contour in complex $s$ plane in the weak coupling regime and for $\gamma> \gamma_{c_1}$. (b) Numerically computed behavior of the survival probability $P_s(t)=|c_a(t)|^2$ versus normalized time $Jt$ for parameter values $g_0/J=0.6$ and $\gamma/J=0.5$. Note that the pole $s_{p_1}$ has now crossed the branch cut and is located on the first Riemann sheet. The contribution to the decay law $c_a(t)$ comes from the two Hankel paths $h_{1,2}$ and from the pole $s_1=s_{p_1}$ (bound state). Note the different long-time relaxation behavior of the survival probability in (b) as compared to Fig.4(b).