Exponentially Accelerated Approach to Stationarity in Markovian Open Quantum Systems through the Mpemba Effect

Ergodicity breaking and slow relaxation are intriguing aspects of nonequilibrium dynamics both in classical and quantum settings. These phenomena are typically associated with phase transitions, e.g., the emergence of metastable regimes near a first-order transition or scaling dynamics in the vicinity of critical points. Despite being of fundamental interest the associated divergent timescales are a hindrance when trying to explore steady-state properties. Here we show that the relaxation dynamics of Markovian open quantum systems can be accelerated exponentially by devising an optimal unitary transformation that is applied to the quantum system immediately before the actual dynamics. This initial “rotation” is engineered in such a way that the state of the quantum system no longer excites the slowest decaying dynamical mode. We illustrate our idea—which is inspired by the so-called Mpemba effect, i.e., water freezing faster when initially heated up—by showing how to achieve an exponential speeding-up in the convergence to stationarity in Dicke models, and how to avoid metastable regimes in an all-to-all interacting spin system.

Figure 1

Mpemba effect in a Markovian open quantum system. (a) We consider a quantum system, initially prepared in some pure state $|\psi ⟩$ and subject to Markovian open quantum dynamics. Generically, the timescale for the approach to stationarity is contained in the dynamical generator $\mathcal{L}$ and is related to the slowest decaying excitation mode. Before the time evolution starts, we apply a unitary operation $U$ to the quantum state $|\psi ⟩$, which deexcites such a slow mode. (b) After applying the unitary operation the system dynamics is not affected by long-lived excitation and approaches the stationary state in a “more direct” way. (c) Sketch of the slow relaxation (blue line), contrasted with the accelerated one emerging after the applying the unitary (red line). The $y$ axis is in logarithmic scale.

Figure 2

Dissipative Dicke model. (a) Overlap $c(s)$ of a rotated initial random state as a function of $s$. According to Eq. (9), $c(s)$ can interpolate between the eigenvalue ${\alpha }_1$ of ${\ell }_2$—which we take here to be the largest one—and the eigenvalue ${\alpha }_n$—which we take to be the smallest. For the choice of parameters in this figure, $\omega =1$, $g=1$, $\kappa =1$ (all in units of $\mathrm{\Omega }$), and $N=40$ spins, ${\alpha }_1\approx 1.00$, and ${\alpha }_n\approx -0.70$. The optimal value for which the overlap $c(s)$ is zero is given by $\overline{s}=\arctan (\sqrt{|{\alpha }_1/{\alpha }_n|})\approx 0.87$. The dashed line shows the overlap $⟨\psi |{\ell }_2|\psi ⟩$ of the initial random state with the decaying mode $r_2$. (b) Distance between the time-evolved state and the stationary state ${\rho }_{\text{SS}}$. We compare the case of an initial random state (black line) with the time evolution ensuing after the application of the rotation $U$ (see main text for discussion). While in the original case, the approach to stationarity is governed by the eigenvalue ${\lambda }_2$, the application of $U$ leads to an exponentially faster convergence to the steady state with the rate given by $|\mathrm{Re}({\lambda }_3)|$.

Figure 3

All-to-all interacting spin model. (a) Overlap $c(s)$ of a rotated initial random state as a function of $s$. We take ${\alpha }_1$ to be the largest eigenvalue of ${\ell }_2$ and ${\alpha }_n$ the smallest. For the choice of parameters in this figure, $\mathrm{\Delta }=-1$, $V=3$, $\kappa =1$ (all in units of $\mathrm{\Omega }$) and $N=40$ spins, we have ${\alpha }_1\approx 2.71$ and ${\alpha }_n\approx -0.02$. The optimal value $\overline{s}$ is thus $\overline{s}=\arctan (\sqrt{|{\alpha }_1/{\alpha }_n|})\approx 1.49$, which makes the overlap $c(s)$ zero. The dashed line shows the initial overlap $⟨\psi |{\ell }_2|\psi ⟩$. (b) Distance between the time-evolved state and the stationary state ${\rho }_{\mathrm{SS}}$ of the interacting spin system. We compare the case of the initial random state $\rho ={\rho }_0$ with the one obtained after the rotation $\rho =U{\rho }_0{U}^{†}$. In this case, the transformation $U$ prevents the system from entering a metastable regime which would slow down the approach to stationarity.