Quantum tunneling Mpemba effect

Abstract: The quantum tunneling Mpemba effect is investigated within a continuous one-dimensional symmetric double-well potential open to external environmental sinks at the boundaries ($x=\pm L$). Using a non-Hermitian spectral decomposition of the effective Hamiltonian, we characterize the open-system relaxation dynamics without relying on abstract state-space quenches. We mathematically prove that the non-monotonic behavior of the first non-trivial even-parity spectral coefficient, $a_{2}(T_{i})$, with respect to the initial preparation temperature $T_{i}$ is a universal topological property born from quantum statistical mechanics. Crucially, we demonstrate that this intermediate thermal peak is governed by the Sturm-Liouville oscillation theorem and remains completely invariant with respect to the global system size $L$, contrasting sharply with the boundary-driven classical Mpemba effect. This universal peak arises from the geometric and nodal alignment between highly localized unperturbed states and extended non-Hermitian decay channels. Furthermore, we clarify that while this mechanism is robust, the actual observation of anomalous crossings in the total survival probability trace $S(t,T_{i})$ and the trace distance $\mathcal{D}(t,T_i)$ demand a strict separation of timescales, requiring the over-barrier escape rate to vastly exceed the decay rate of the deep-well tunneling doublet ($Γ{2}\gg Γ{0}$ and $Γ_2\gg Γ_1$). Our continuous formulation successfully bridges real-space classical boundary-driven dissipation with open quantum dynamics, providing novel insights for engineering non-equilibrium states via tailored boundary loss.

• The paper establishes a non-Hermitian biorthogonal spectral framework that demonstrates faster relaxation in hotter initial quantum states.
• It rigorously proves the non-monotonic temperature dependence of spectral coefficients using Sturm-Liouville theory, validated by numerical simulations in a quartic double-well potential.
• The findings indicate that engineered boundary losses can control quantum state relaxation, offering practical insights for quantum control and open-system simulations.

Quantum Tunneling Mpemba Effect: A Continuous Non-Hermitian Formulation

Overview

The paper "Quantum tunneling Mpemba effect" (2607.03845) introduces and rigorously analyzes the quantum analogue of the Mpemba effect in continuous, real-space open quantum systems. The study focuses on a single particle in a one-dimensional symmetric double-well potential that is coupled to external environmental sinks, modeled via complex absorbing potentials (CAPs) at the boundaries. Unlike prior work, which largely addressed either discrete quantum systems or overlooked explicit spatial considerations, this work establishes a mathematically exact, non-Hermitian spectral framework for open quantum relaxation dynamics with spatial tunneling and environmental loss.

Theoretical Formulation

Non-Hermitian Open-Quantum Dynamics

The system consists of a particle in a symmetric double-well, with absorption implemented by boundary CAPs $W(x)$ localized at $x = \pm L$. The effective non-Hermitian Hamiltonian is

$$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$

Relaxation is tracked via the survival probability, $S(t, T_i) = \text{Tr}[\rho(t)]$, where $\rho(t)$ evolves according to the conditional no-jump GKSL equation.

Spectral Decomposition and Topological Origin

Key to the analysis is a biorthogonal spectral expansion of the non-Hermitian Hamiltonian, with right eigenfunctions $\{\phi_n\}$ and left eigenfunctions $\{\chi_n\}$, $\langle \chi_n | \phi_m \rangle = \delta_{nm}$. The relaxation dynamics of $S(t, T_i)$ are governed by

$S(t, T_i) = \sum_k a_k(T_i) e^{-\Gamma_k t}$

with projection coefficients

$x = \pm L$0

where $x = \pm L$1 are the eigenstates of the closed potential, and $x = \pm L$2 their Boltzmann weights at initial temperature.

A significant theoretical advance is the rigorous proof—using the Sturm-Liouville oscillation theorem—of non-monotonic temperature dependence of the first nontrivial even-parity spectral coefficient $x = \pm L$3, regardless of system size $x = \pm L$4. Specifically, $x = \pm L$5 vanishes at $x = \pm L$6 and $x = \pm L$7 but reaches a universal maximum at an intermediate activation temperature. This distinguishes the quantum effect from the classical Mpemba scenario, which is sensitive to boundary positions.

Robustness and Parameter Regimes

The study establishes that the emergence of the quantum Mpemba effect—the faster relaxation of a hotter initial state—arises generically so long as the following are satisfied:

• Sufficient separation between the over-barrier decay rate $x = \pm L$8 and the tunneling doublet decay ($x = \pm L$9); specifically, $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$0.
• Absorption is neither vanishingly weak nor so strong that quantum reflection dominates (i.e., not in the quantum Zeno regime).
• The system supports well-defined localized and extended eigenstates; the effect is suppressed if the barrier vanishes or if spatial confinement is extreme.

Numerical Validation

The analysis is complemented by numerical studies of the quartic symmetric double-well with CAP boundaries, employing finite-difference discretization and direct diagonalization. The results corroborate all theoretical predictions:

• $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$1 for $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$2 exhibits clear non-monotonic peaks as a function of $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$3, with the peak position for $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$4 being invariant with respect to changes in $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$5.
• The survival probability $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$6 and the trace distance $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$7 both demonstrate anomalous crossings indicative of the Mpemba effect. Specifically, at certain times, a system initialized at higher temperature relaxes below the corresponding measure for a colder initialization, which is nontrivial for monotone measures.
• The crossing times in $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$8 are primarily determined by $$H_\text{eff} = H_0 - i W(x) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) - iW(x)$$9 and are almost independent of initial temperature, while in the trace distance, residual quantum coherences introduce mild $S(t, T_i) = \text{Tr}[\rho(t)]$0-dependence.

Theoretical and Practical Implications

Bridging Quantum and Classical Regimes

The work provides a unifying perspective that connects the spatial intuition of classical boundary-driven relaxation with fundamentally quantum-mechanical processes, specifically tunneling and coherent escape through non-Hermitian decay channels. It proves that in quantum systems, the non-monotonicity responsible for the Mpemba effect is topologically protected and independent of macroscopic boundary positioning—a sharp distinction from classical overdamped dynamics (Liu et al., 2 Apr 2026, Liu et al., 2 Jun 2026).

Generalization and Topological Protection

The formalism generalizes to potentials lacking reflection symmetry, as the node structure of the involved eigenstates—asserted by Sturm-Liouville theory—ensures constructive overlap at intermediate temperatures. Thus, the effect persists irrespective of parity structure, and the activation mechanism is robust to moderate asymmetry.

Engineering and Quantum Control Implications

The analytic and computational machinery elucidated here directly informs practical quantum control:

• Boundary loss engineering can be exploited to accelerate quantum state preparation and relaxation, e.g., for open quantum simulators or coherent quantum devices.
• The demonstrated invariance with respect to boundary location implies stable design flexibility for mesoscopic quantum systems intended to show rapid thermalization-like behavior via tailored bath coupling.

Future Directions

Potential directions for future research include:

• Application to many-body open quantum systems where interactions may modify the spectral structure.
• Exploration of transient non-monotonic relaxation (multiple zero crossings) in trace-based state metrics, connecting to the phenomenon of multiple quantum Mpemba effects [Phys. Rev. A 110, 022213 (2024)].
• Investigation of non-Markovian environmental couplings and their impact on anomalous relaxation and coherence effects.

Conclusion

This paper provides the first complete analytic and numerical treatment of the quantum tunneling Mpemba effect in continuous open quantum systems. By establishing a biorthogonal non-Hermitian spectral framework and connecting the phenomenon to robust topological properties of the quantum spectrum, it clarifies the universality and mechanistic origins of anomalous relaxation in quantum statistical mechanics. The results have significant implications for both fundamental understanding and engineering of open quantum systems, and distinguish quantum relaxation from its classical analogues in a rigorous and instructive manner.