Recoherence, adiabaticity, and Markovianity in Gaussian maps

TL;DR

This work analyzes how recoherence, adiabaticity, and non-Markovianity interrelate in Gaussian open-system dynamics using a toy model of two linearly coupled harmonic oscillators with a time-dependent coupling. By developing perturbative and adiabatic expansions and introducing the Bures velocity as a non-Markovian measure, the authors identify a critical coupling ξ_c = ω_S ω_E separating oscillatory from decohering behavior and show recoherence arises under slow, adiabatic turn-off, while decoherence is non-perturbative and non-Markovian. They demonstrate that, regardless of parameter choices, the dynamics cannot be captured by a Markovian map, and they propose an optimal Markovian approximation by minimizing the Bures velocity. The results provide a framework to interpret quantum-to-classical transitions in cosmology and offer quantitative tools to approximate non-Markovian Gaussian dynamics in practical settings. Overall, the paper illuminates the nuanced roles of adiabaticity and memory effects in decoherence and recoherence within Gaussian quantum systems with potential cosmological applications.

Abstract

Motivated by the recent discovery of situations where cosmological fluctuations recohere during inflation, we investigate the relationship between quantum recoherence (late-time purification after a transient phase of decoherence), adiabaticity, and Markovianity. To that end, we study a simple setup of two linearly-coupled harmonic oscillators, and compute the purity of one oscillator when the interaction is switched off. We find that there exists a critical value for the coupling strength below which the purity oscillates and above which it decays exponentially. This decay cannot be captured by perturbation theory; hence, decoherence is always a non-perturbative phenomenon. When the interaction is turned off, the purity either freezes to its value prior to the turn-off, or it smoothly goes back to a value very close to one (recoherence). This depends on the rate at which the turn-off occurs. We thus develop a new adiabatic-expansion scheme and find complete recoherence at any finite order in the inverse turn-off time. Therefore, decoherence is always a non-adiabatic effect. The critical value of the turn-off time above which recoherence takes place is then expressed in terms of the other time scales of the problem. Finally, we show that the dynamics of the system is never Markovian, even when decoherence takes place. We introduce a new measure of Markovianity dubbed the Bures velocity and use it to optimise Markovian approximations.

Figures (15)

• Figure 1: Coupling function for a few values of $\tau$ with $\xi_0=1$ and $t_0 =10$ (left panel), and for a few values of $t_0$ with $\xi_0=1$ and $\tau=1$ (right panel).
• Figure 2: Purity as a function of time for a few values of the coupling strength $\xi_0$, with $\omega_{\mathrm{S}} =1$, $\omega_{\mathrm{E}} =2$, and with $\tau=1$ and $t_0 =10$ (this corresponds to the orange line in the right panel of \ref{['fig.xi']}).
• Figure 3: Same as in \ref{['fig.gs.cases']} where the supercritical cases are displayed in the left panel with a logarithmic scale, and the subcritical cases in the right panel. In the supercritical regime, the purity decays as $e^{-\vert\omega_1\vert t}$.
• Figure 4: Purity as a function of time for a few values of $\omega_{\mathrm{E}}/\omega_{\mathrm{S}}$, with $\tau=1$ and $t_0 =10$, for $\xi_0=1.1\xi_{\mathrm{c}}$ (supercritical regime, left panel) and $\xi_0=0.1\xi_{\mathrm{c}}$ (subcritical regime, right panel)
• Figure 5: Criticality ($\psi=1$, blue line) and perturbativity ($g_{\mathrm{p}}=0.1$, orange line) in the parameter space $(w,\psi)$. Several points are labelled, which correspond to the specific cases discussed in \ref{['sec:purity:expansion:instantaneous']} and \ref{['sec:ISOSO:discussion']}.