Breaking the Exponential: Decoherence-Driven Power-Law Spontaneous Emission in Waveguide Quantum Electrodynamics

Abstract

We investigate the spontaneous emission of a two-level system coupled to a photonic waveguide, showing that dynamical dephasing in the photon modes profoundly alters the decay law. In the absence of dephasing, the emitter displays conventional exponential decay followed by a long-time power-law tail -- observable only at extremely low survival probabilities. Strikingly, when dephasing is introduced, a robust power-law decay emerges already at short times, driven by photon diffusion in the dynamically disordered environment rather than spectral edge effects. These results reveal a novel, decoherence-induced mechanism for non-exponential spontaneous emission in waveguide QED platforms.

Figures (4)

• Figure 1: (a) Schematic of a semi-infinite array of coupled optical cavities with a two level atom placed inside the edge chain resonator (index $n=0$). $J$ is the hopping rate of photons between adjacent resonators of the array, $\omega_0$ is the atomic resonance frequency, and $g_0$ the atom-photon coupling rate. Dephasing of the cavity photon modes with a rate $\gamma$ is introduced by fluctuations of the cavity resonance frequencies $\omega_n(t)=\omega_c+ \delta \omega_n(t)$ around a mean value $\omega_c$. (b) Numerically-computed behavior of the survival probability of the atom, $P_s(t)$, in the spontaneous decay process in the absence of dephasing (solid blue curve), and approximate exponential decay law (dashed red curve) with a decay rate $\Gamma= 2 g_0^2/J$ as given by the Fermi golden rule. Parameter values are $\lambda=g_0/J=0.3$ and $\omega_0=\omega_c$. The inset in (b) depicts the decay curves on a logarithmic vertical scale, clearly showing long-time deviations from an exponential decay.
• Figure 2: (a) Damped vacuum Rabi oscillations induced by dephasing effects for $J=0$, $\omega_0=\omega_c$.and for a few increasing values of the dephasing rate $\gamma /g_0$: Curve 1: $\gamma /g_0=0$; curve 2: $\gamma /g_0=1$; curve 3: $\gamma /g_0=2$; curve 4: $\gamma /g_0=8$; curve 5: $\gamma /g_0=20$. (b,c) Behavior of the eigenvalues $\lambda$ versus the dephasing rate $\gamma / g_0$ (real and imaginary parts) of the relaxation matrix entering in Eqs.(18-21). An exceptional point, corresponding to the coalescence of two of the four eigenvalues and corresponding eigenvectors, is observed at the critical value $(\gamma/g_0)_c=8$.
• Figure 3: (a) Numerically-computed survival probability $P_s(t)$, plotted on a vertical logarithmic scale as a function of normalized time $Jt$ for $g_0/J=0.3$, $\omega_c=\omega_0$ and for several increasing values of the dephasing rate $\gamma /J$. Curve 1: $\gamma /J=0$; curve 2: $\gamma /J=0.1$; curve 3: $\gamma /J=1$; curve 4: $\gamma /J=3$; curve 5: $\gamma /J=10$. (b) Survival probability $P_s(t)$ multiplied by $\sqrt{Jt}$, plotted on a linear scale versus normalized time $Jt$, shown for the strong dephasing regime ($\gamma /J=3$ and $\gamma/J=10$). The curves plateau to a non-zero stationary value, indicating a power-law decay of the form $P_s(t) \sim 1/ \sqrt{Jt}$.
• Figure 4: (a) Schematic of a classical random walk on a semi-infinite line describing the spontaneous emission process of the two-level atom (at the left edge of the line) under strong dephasing of the photon modes. The hopping rates $\mathcal{R}$ and $\mathcal{Q}$ are given by Eq.(28). (b) Survival probability $P_s(t)$ versus normalized time $Jt$ in the strong dephasing regime ($g_0/J=0.3$ and $\gamma /J=10$). The solid grey curve corresponds to the numerical results obtained by solving the quantum master equation [Eqs.(10-12)], the dashed red curve, almost overlapped with the solid one, is the theoretical prediction based on the classical random walk approximation [Eq.(29)], and the purple dotted curve is the asymptotic power-law decay curve given by Eq.(33).