A Description of the Quantum Mpemba Effect using the Steepest-Entropy-Ascent Quantum Thermodynamics Framework

Abstract

The quantum Mpemba effect describes the unusual relaxation of a quantum system where if starting from a state far from equilibrium reaches equilibrium faster than starting from a closer state. In this work, the steepest-entropy ascent quantum thermodynamics framework is used to model this effect in a three-level ion system coupled to a short-life level which acts as an environment. The four-level Hilbert space is reduced to an effective three-dimensional description via the Feshbach projection, where the resulting model parameters are determined by a differential evolution algorithm. Predictions of both the steepest-entropy-ascent and Lindblad frameworks agree well with experimental data for all three initial conditions considered. In addition it is shown that, independently of the Mpemba condition, the relaxation parameter $τ_D$ is an emergent thermodynamic quantity whose step-function time dependence arises from a near-indetermination at the metastable state, resolved by the two-timescale structure of the dissipative dynamics, with a step height $τ_D^{+}/τ_D^{-} (|λ_f|/|λ_s|)\,\mathfrak{r}$ set by the ratio of the linearised relaxation rates at the metastable state times a geometric factor $\mathfrak{r}$ fixed by the curvature of the entropy and free-energy-variance surfaces there. Furthermore, it is also established the correspondence between the Lindblad and steepest-entropy-ascent order parameters as spectral and geometric suppression of the same slow dissipative mode, and it is shown that for isolated systems with a negative nonequilibrium inverse temperature, as realized in the case studied here, the genuine Mpemba free-energy ordering is equivalent to an entropy ordering.

The paper asks whether steepest-entropy-ascent quantum thermodynamics (SEAQT) can describe the quantum Mpemba effect previously measured in a driven, dissipative three-level ion. It compares a nonlinear, energy-conserving SEAQT model with the standard Lindblad description of that experiment. A Feshbach projection supplies an effective Hermitian three-level Hamiltonian. Two Hamiltonian parameters are fitted jointly across three initial preparations, while four parameters specifying a logistic relaxation time $\tau_D(t)$ are fitted separately for each preparation. The resulting SEAQT reduced chi-squares are $29.20$, $12.89$, and $10.78$; constant-$\tau_D$ fits give $53.13$, $30.91$, and $25.99$. The time-dependent model produces an entropy plateau for the accelerated preparation. The entropy-production identity gives $\tau_D=(\sigma_{SS}-C)/(d\langle S\rangle/dt)$. A local quadratic expansion near a partially canonical metastable state relates its proposed step height to a relaxation-rate ratio times a curvature factor. Only one fitted step is resolved: $\tau_D^+/\tau_D^-=5.033$, compared with a rate ratio of $158.46$. A Hamiltonian-dressed population coordinate approximately reproduces the Lindblad slow-mode cancellation criterion, tested using ensembles of 100 randomly generated states. Stated limitations include unresolved relaxation-time asymptotes for two preparations and an oscillation amplitude roughly three times the measured value for one. SEAQT trajectories conserve different initial energies and reach different equilibria, unlike the common Lindblad steady state; free-energy comparisons therefore require fixed energy. Correspondence between the two Mpemba criteria is established numerically for this model, with its extension beyond the qutrit left open.