Quantum Pontus-Mpemba Effect Enabled by the Liouvillian Skin Effect

TL;DR

The paper addresses quantum Mpemba-type relaxation in open quantum systems by exploiting the Liouvillian skin effect produced by non-reciprocal dissipation. It demonstrates that a two-step Pontus protocol—coherent preparation of an auxiliary state followed by relaxation under the target Liouvillian—can accelerate approach to a common stationary state without changing the asymptotic decay rate, provided the incoherent hopping is non-reciprocal ($J_{\rm R} \neq J_{\rm L}$). The mechanism hinges on reducing the initial overlap with the slow Liouvillian mode $R_2$ (lowering $|c_2|$ relative to $|c_2'|$), a geometric effect of boundary localization that vanishes at symmetry $J_{\rm R}=J_{\rm L}$. The work connects non-Hermitian spectral geometry to protocol-dependent transient dynamics and outlines experimental paths in ultracold-atom and photonic platforms, with potential extensions to many-body and disordered settings.

Abstract

We unveil a quantum Pontus-Mpemba effect enabled by the Liouvillian skin effect in a dissipative tight-binding chain with asymmetric incoherent hopping and coherent boundary coupling. The skin effect, induced by non-reciprocal dissipation, localizes relaxation modes near the system boundaries and gives rise to non-orthogonal spectral geometry. While such non-normality is often linked to slow relaxation, we show that it can instead accelerate relaxation through a two-step protocol - realizing a quantum Pontus-Mpemba effect. Specifically, we consider a one-dimensional open chain with coherent hopping $J$, asymmetric incoherent hoppings $J_{\rm R} \neq J_{\rm L}$, and a controllable end-to-end coupling $ε$. For $ε=0$, the system exhibits the Liouvillian skin effect, with left and right eigenmodes localized at opposite edges. We compare two relaxation protocols toward the same stationary state: (i) a direct relaxation with $ε=0$, and (ii) a two-step (Pontus) protocol where a brief coherent evolution transfers the excitation across the lattice before relaxation. Although both share the same asymptotic decay rate, the two-step protocol relaxes significantly faster due to its reduced overlap with the slow boundary-localized Liouvillian mode. The effect disappears when $J_{\rm R}=J_{\rm L}$, i.e., when the skin effect vanishes. Our results reveal a clear connection between boundary-induced non-normality and protocol-dependent relaxation acceleration, suggesting new routes for controlling dissipation and transient dynamics in open quantum systems.

Figures (6)

• Figure 1: (a) Schematic of the tight-binding chain made of $L$ sites with coherent ($J$) and asymmetric incoherent($J_R, J_L$) hoppings between adjacent sites. The end sites $n=1$ and $n=L$ of the chain are connected by a coherent hopping at tunable rate $\epsilon$. (b) Illustration of the quantum Pontus-Mpemba effect in phase space. The initial state $\rho_i$ can relax toward the same stationary state $\rho_E$ via two protocols. In the single-step (direct) protocol the system evolves under the Liouvillian $\mathcal{L}$. In the two-step (Pontus) protocol, for $0\le t\le \tau$ the system is evolved under a preparatory Liouvillian $\mathcal{L}_1$ that drives the system toward the auxiliary state $\rho_i^{\prime}$. For $t>\tau$ the system then evolves under the same Liouvillian $\mathcal{L}$ as in the single-step protocol. Both protocols approach the same stationary state $\rho_E$. The Pontus-Mpemba effect arises whenever the time to reach the stationary state is shorter in the two-step protocol.
• Figure 2: (a) Spectrum of the Liouvillian $\mathcal{L}$ in the single-excitation subspace for parameter values $L=11$, $\epsilon=0$, $J_R/J=1$ and $J_L/J_R=1$ (symmetric hopping). The eigenvalues $\lambda_1=0$ and $\lambda_2$, corresponding to the stationary (equilibrium) state $\rho_E$ and to the slowest-decaying state, are highlighted with blue and red circles, respectively. (b) Behavior of the stationary state [modulus of $(\rho_E)_{n,m}$]. (c,d) Behavior of the right eigenvector $R_2$ [modulus of $(R_2)_{n,m}$, panel (c)] and left eigenvector $L_2$ [modulus of $(L_2)_{n,m}$, panel (d)], corresponding to the eigenvalue $\lambda_2$. (e-h) Same as (a-d), but for $J_L/J_R=0.5$ (asymmetric hopping).
• Figure 3: Relaxation dynamics (temporal evolution of the trace distance $D_{tr}$, upper plots, and Hilbert-Schmidt distance $D_{HS}$, lowers plots) under the Liouvillian $\mathcal{L}$ for parameter values $\epsilon=0$, $J_R/J=1$, $L=11$ and for (a) $J_L/J_R=1$ (symmetric incoherent hopping) and (b) $J_L/J_R=0.5$ (asymmetric incoherent hopping). Curves 1 and 2 refer to different initial states (curve 1: $\rho_i=|1 \rangle \langle 1|$, curve 2: $\rho_i^{\prime}=|L \rangle \langle L|$). In the symmetric hopping case owing to mirror symmetry curves 1 and 2 are overlapped yielding the same relaxation dynamics. The insets in panels (b) show the relaxation dynamics on a vertical log scale.
• Figure 4: Numerically-computed relaxation time $t_{rel}$ versus preparation time $\tau$ in the two-step (Pontus) protocol (red circles). The horizontal dashed blue curve is the relaxation time, independent of $\tau$, in the one-step (direct) protocol. Parameter values are as in Fig.3(b). The relaxation time $t_{rel}$ is defined as the time instant such that $D(\rho(t_{tr}), \rho_E)=0.01$. In panel (a), the distance $D$ is the trace distance, whereas in panel (b) the distance $D$ is the Hilbert-Schmidt distance.
• Figure 5: Illustrative relaxation dynamics showing the emergence of the Pontus-Mpemba effect. Panels (a) and (b) display the numerically-computed temporal evolution of the trace distance $D_{tr}(\rho(t), \rho_E)$ [panel (a)] and the Hilbert-Schmidt distance $D_{HS}(\rho(t), \rho_E)$ [panel (b)] for the one-step (direct) protocol (curve 1) and the two-step (Pontus) protocol (curve 2). Parameter values of the Liouvillian $\mathcal{L}$ are $\epsilon=0$, $J_R=J$, $J_L/J_R=0.5$, $L=11$. In the first-step of the Pontus protocol, the Liouvillian is $\mathcal{L}_1$ with $J=J_L=J_R=0$ and $\epsilon=\epsilon_1= \pi/(2 \tau)$ with $\tau=3$ (in units of $1/J$ of first-step protocol). The insets show the relaxation dynamics on a vertical log scale.