Oscillatory and dissipative dynamics of complex probability in non-equilibrium stochastic processes

TL;DR

This work extends the classical master equation by introducing complex transition rates, yielding complex conditional probabilities and pre-thermal oscillations in two-state and multi-state systems. For purely imaginary rates, the dynamics do not equilibrate and exhibit persistent oscillations, linking to concepts reminiscent of quantum scars; in higher dimensions, the behavior is governed by the real and imaginary parts of the rate matrix $\mathcal{W}$ and can be tuned via driving. The authors analyze various driving scenarios, including external pumping of probability and rotations in probability space, showing regimes of phase-conserved dynamics versus dissipation or disintegration when driven oppositely. They connect the classical treatment to quantum contexts through the Fermi-Golden rule and discuss experimental realizations in ultracold atomic systems, as well as future directions involving noise-assisted stability and non-Hermitian driven-dissipative dynamics.

Abstract

For a Markov and stationary stochastic process described by the well-known classical master equation, we introduce complex transition rates instead of real transition rates to study the pre-thermal oscillatory behaviour in complex probabilities. Further, for purely imaginary transition rates we obtain persistent infinitely long lived oscillations in complex probability whose nature depends on the dimensionality of the state space. We also take a peek into cases where we perturb the relaxation matrix for a dichotomous process with an oscillatory drive where the relative sign of the angular frequency of the drive decides whether there will be dissipation in the complex probability or not.

Figures (4)

• Figure 1: Panel (a) shows that for $\lambda_{1}=2.5, \lambda=1.5$ and mean transition rate $\lambda=2$ in units of inverse time, the conditional probabilities forget about the initial state memory, $P(1,0)=0.8, P(2,0)=0.2$ to equilibriate to their thermal values. Panel (b) shows that for such a choice of $\lambda_{1}$ and $\lambda_{2}$, th Gershgorin discs lie in the second and third quadrant of the complex plane. The eigenvalues of the $\mathcal{W}$ matrix lie in the union of these two circles signifying that the real part of the eigenvalues are always negative. In this case the eigenvalues are 0 and $-2\lambda$.
• Figure 2: Panel (a) shows oscillations in the conditional probabilities in the pre-thermal regime for complex transition rates with $\omega=100$ and $\lambda_{1}=2.5, \lambda_{2}=1.5$ for a dichotomous process. Inset shows oscillations on a smaller time scale. Panel (b) shows persistent long time oscillations for imaginary transition rates with $\omega=5$ for a dichotomous process. Panel (c) shows oscillations for two specific states having initial probabilities $0.2$ (red) and $0.09$ (blue) for a N=10 state system with imaginary transition rates with $\omega=10$. Real part of the complex probabilities have been plotted. Panel (d) shows the Gershgorin discs in case of a dichotomous process with transition rates $\lambda_{1}+i\omega=1.5+i5$ and $\lambda_{2}+i\omega=2.5+i5$. The eigenvalues are $0,-2(\lambda+i\omega)$, where $\lambda$ is the mean transition rate.
• Figure 3: Panel (a) shows rate of probability exchange between two states in a dichotomous process when the transition rates are complex. Panels (b) and (c) show that probability has to pumped "in" to both the arms to maintain the given transition rates via an external drive. Panel (d) shows that unlike the two previous cases, probability needs to be pumped "in" from one arm and pumped "out" from the other via an external drive to maintain the given transition rates.
• Figure 4: Panel (a) shows that both $Re[P(1,t)]$ and $Re[P(2,t)]$ decay to zero which means the system eventually becomes unstable to the drive and disintegrates/melts. Here, $\omega=0.9$ and $\lambda=1$ with $\lambda_{1}=\lambda_{2}$. Panel (b) shows the Gershgorin disc pairs for three cases: $\omega > \lambda$ when the circles are disjoint and the corresponding eigenvalues are complex (green), $\omega=\lambda$ when the circles just touch at value $-\lambda$ on the real axis (blue) and when $\omega<\lambda$ where there are two distinct real eigenvalues (red).