Expedited thermalization dynamics in incommensurate systems

TL;DR

This paper investigates nonequilibrium thermalization in a quantum lattice with an incommensurate potential coupled to a Markovian bath, using the Lindblad framework to drive the system toward the infinite-temperature state $\rho_{ss}=\frac{1}{L}{\bf I}$. By analyzing the Liouvillian spectrum and the overlap with the slowest decaying mode $\lambda_2$, the authors reveal a quantum Mpemba-like phenomenon: localized and colder initial states can relax faster to $\rho_{ss}$ than extended or hotter states, despite appearing farther from equilibrium in other measures. The key mechanism is the reduced overlap $\text{Tr}(l_2\rho_0)$ for localized states, which accelerates relaxation even as localization would naively slow dynamics in closed systems. The result demonstrates a robust, size- and parameter-insensitive link between localization and anomalous dissipative thermalization, with potential experimental realizations in cold-atom or photonic platforms and implications for controlling quantum thermalization in disordered environments.

Abstract

We study the thermalization dynamics of a quantum system embedded in an incommensurate potential and coupled to a Markovian thermal reservoir. The dephasing induced by the bath drives the system toward an infinite-temperature steady state, erasing all initial information-including signatures of localization. We find that initially localized states can relax to the homogeneous steady state faster than delocalized states. Moreover, low-temperature initial states thermalize to infinite temperature more rapidly than high-temperature states -- a phenomenon reminiscent of the Mpemba effect, in which hotter liquids freeze faster than colder ones. The slowest relaxation mode in the Liouvillian spectrum plays a critical role in the expedited thermalization for localized or cold initial states. Our results reveal that the combination of disordered structure and environmental dissipation may lead to non-trivial thermalization behavior, which advances both the conceptual framework of the Mpemba effect and the theoretical understanding of nonequilibrium processes in dissipative disordered systems.

Figures (4)

• Figure 1: Mobility Edge Display. (a) The inverse participation ratio (IPR) of the eigenstates of the Hamiltonian, where $m$ denotes the eigenstate index ordered by increasing energy. (b) The site occupation density matrix of the localized state (indexed by $m = 3$), where $j$ denotes the site index. (c) The site occupation density matrix of the extended state (indexed by $m = 16$). Here the system parameters are chosen as $V = 1.4$, $J = 1$, quasiperiodic modulation $\beta = 4\pi^2$, $\alpha = 0.7$, and the system size $L = 35$.
• Figure 2: The time evolution of the Frobenius distance. (a) Temporal evolution of the Frobenius distance $D$ between the extended state (indexed by $m = 16$, $T_e = 9.15$, orange curve) and the thermal state (at temperature $T_t = 0.25$, blue curve). The upper-right inset shows the logarithmic Frobenius distance $\ln D$. The absence of curve intersection clearly indicates no inverse QME occurs. (b) Temporal evolution of the Frobenius distance between the localized state (indexed by $m = 3$, $T_l = 0.22$, dark red curve) and the thermal state (at $T_t = 0.25$, blue curve). The intersection of the two curves confirms the occurrence of inverse QME.
• Figure 3: Temporal evolution of $D$ under different potential strengths $V$. The orange, dark red, and blue curves respectively represent extended states (indexed by $m = 16$), localized states (indexed by $m = 3$), and thermal states at temperature $T_t = 0.3$, with potential strengths $V = 1.2$ (panels (a)-(b)), $V = 1.4$ (panels (c)-(d)), and $V =1.6$ (panels (e)-(f)), where in all cases the temperatures satisfy $T_{l} < T_t < T_{e}$, with $T_{l}$, $T_{t}$, $T_{e}$ and denoting the localized, thermal and extended state temperatures respectively.
• Figure 4: The inverse QME for a lattice with $L = 100$ sites. (a) Real-space density matrix of the extended state (energy index $m = 47$); (b) Real-space density matrix of the localized state (energy index $m = 7$); (c) Temporal evolution of the Frobenius distance between the extended state (orange curve) and thermal state (blue curve); (d) Temporal evolution between the localized state (dark red curve) and thermal state. The initial state temperatures are $T_l = 0.23$, $T_t = 0.3$, and $T_e =12.26$ for the localized, thermal, and extended states respectively. The inset in (d) displays the absolute value of the overlap between initial states and the slowest mode.