Quantum tunneling Mpemba effect

Abstract

We investigate a quantum tunneling Mpemba effect for a particle in a continuous one-dimensional symmetric double-well potential subject to irreversible boundary loss. The dissipation is described by a complex absorbing potential (CAP) and the corresponding conditional no-jump evolution. Using a biorthogonal spectral representation, we derive the exact non-Hermitian decomposition of the subnormalized density matrix and analyze the corresponding relaxation dynamics. We show that the off-diagonal coefficients exhibit a parity selection rule, leaving same-parity interference terms in general. To isolate the interplay between thermal spectral preparation and mode-dependent decay rates, we introduce a diagonal spectral relaxation measure $S(t,T_i)$. For a quartic double well, numerical calculations reveal finite-time crossings of $S(t,T_i)$ prepared at different initial temperatures, as well as a non-monotonic temperature dependence of selected diagonal spectral weights. These features are explained by the enhanced thermal population of spatially extended, rapidly leaking excited states. In contrast, the trace distance of the normalized conditional state to the asymptotic quasi-stationary state exhibits no crossing, demonstrating that the anomalous ordering is specifically a spectral Mpemba effect rather than a universal acceleration of the complete conditional state. Our continuous-space formulation provides a real-space perspective on anomalous quantum relaxation without invoking universal nodal or topological theorems.