Quantum Mpemba Effect in a Four-Site Bose-Hubbard Model

Abstract

We investigate relaxation-order inversion, known as the quantum Mpemba effect (QME), in a minimal open many-body system called a one-dimensional four-site Bose--Hubbard chain governed by Lindblad dynamics with local number dephasing. Families of thermal initial states are prepared at a fixed temperature and evolved under a common reference Liouvillian toward the same stationary state. Relaxation is characterized using four complementary diagnostics: trace distance, quantum relative entropy, symmetry-projected entropy imbalance (entanglement asymmetry), and the $\ell_{1}$-norm of coherence in the Fock basis. We find that QME emerges robustly in -the clean interacting regime, where on-site interactions redistribute the overlaps of initial states with slow Liouvillian decay modes, enabling states initially farther from equilibrium to converge faster at late times. In contrast, the noninteracting limit exhibits a monotonic relaxation hierarchy across all metrics. Introducing a linear Stark potential or random on-site disorder suppresses relaxation and eliminates QME signatures by inhibiting transport-assisted mixing and enhancing the dominance of slow modes. Within the explored parameter regime, the Stark field induces significantly stronger retardation than disorder. We further show that symmetry-projected entropy imbalance is particularly sensitive to charge-sector decoherence in reduced subsystems and provides a stringent probe of QME in bosonic platforms. Our results elucidate the essential role of interactions in enabling anomalous relaxation in open lattice systems and connect the suppression of QME under spatial inhomogeneity to localization phenomena in tilted and disordered Bose--Hubbard chains.

Figures (4)

• Figure 1: Absence of the QME in the non-interacting regime ($U=0$). Relaxation dynamics of a dissipative bosonic system under varying hopping strengths $\tau$. A four-site Bose--Hubbard model evolves from thermal initial states under Lindblad dynamics with local dephasing noise (rate $\gamma$). The reference steady state $\hat{\rho}_{ss}$ corresponds to a fixed hopping strength $\tau_\text{ref}$. All four metrics show consistent monotonic relaxation without crossovers: (a) trace distance $D[\hat{\rho}(t), \hat{\rho}_{ss}]$ measuring distinguishability, (b) relative entropy $S[\hat{\rho}(t) || \hat{\rho}_{ss}]$ quantifying information-theoretic divergence, (c) entanglement asymmetry $\Delta S_A$ for bipartition $A$ characterizing particle number symmetry breaking, and (d)$\ell_1$-norm of quantum coherence $\mathcal{C}[\hat{\rho}(t)]$ in the Fock basis. States initially closer to the steady state (lower $\tau$) maintain their proximity advantage throughout evolution, demonstrating conventional thermalization and highlighting the necessity of interactions for QME emergence.
• Figure 2: Emergence of the QME in the interacting regime. Relaxation dynamics of a dissipative bosonic chain with finite on-site interactions under varying hopping strengths $\tau$. A four-site system evolves from thermal initial states under Lindblad dynamics with local dephasing noise at rate $\gamma$. The reference steady state $\hat{\rho}_{ss}$ corresponds to a fixed hopping strength $\tau_{\rm ref}$. All four diagnostics exhibit characteristic crossover behavior indicative of the QME: (a) trace distance $D[\hat{\rho}(t), \hat{\rho}_{ss}]$, showing inversion of the relaxation hierarchy; (b) relative entropy $S[\hat{\rho}(t)|\hat{\rho}_{ss}]$, demonstrating faster convergence for initially more distant states; (c) entanglement asymmetry $\Delta S_A$ for bipartition $A$, revealing enhanced symmetry-restoration dynamics; and (d) $\ell_1$-norm of quantum coherence $\mathcal{C}[\hat{\rho}(t)]$ in the Fock basis, highlighting interaction-modified decoherence pathways. States initially farther from equilibrium (larger $\tau$) overtake closer ones, demonstrating anomalous thermalization enabled by many-body interactions.
• Figure 3: Suppression of relaxation dynamics under Stark potential. Time evolution of quantum metrics for a Bose--Hubbard system with varying Stark field strengths $g$. A four-site system with fixed on-site interaction evolves from thermal initial states under Lindblad dynamics with local dephasing noise (rate $\gamma/U$). The reference steady state $\hat{\rho}_{ss}$ corresponds to a fixed Stark strength $g_\text{ref}$. All four metrics show systematic retardation of thermalization: (a) trace distance $D[\hat{\rho}(t), \hat{\rho}_{ss}]$ exhibits maintained hierarchy with increased separation, (b) relative entropy $S[\hat{\rho}(t) || \hat{\rho}_{ss}]$ shows delayed convergence for stronger potentials, (c) entanglement asymmetry $\Delta S_A$ for bipartition $A$ demonstrates persistent symmetry breaking, and (d)$\ell_1$-norm of quantum coherence $\mathcal{C}[\hat{\rho}(t)]$ reveals coherence suppression under spatial inhomogeneity. Increasing Stark potential strength creates localization barriers that impede thermalization and prevent QME emergence.
• Figure 4: Moderate thermalization suppression under random disorder. Relaxation dynamics of a Bose--Hubbard system with varying disorder amplitudes $\delta$. A four-site system with fixed on-site interaction evolves from thermal initial states under Lindblad dynamics with local dephasing noise (rate $\gamma$). The reference steady state $\hat{\rho}_{ss}$ corresponds to a fixed disorder amplitude $\delta_\text{ref}$. All metrics indicate mild localization effects: (a) trace distance $D[\hat{\rho}(t), \hat{\rho}_{ss}]$ shows modest convergence delays, (b) relative entropy $S[\hat{\rho}(t) || \hat{\rho}_{ss}]$ exhibits maintained relaxation hierarchy, (c) entanglement asymmetry $\Delta S_A$ for bipartition $A$ reflects disorder-induced symmetry breaking, and (d)$\ell_1$-norm of quantum coherence $\mathcal{C}[\hat{\rho}(t)]$ demonstrates slower coherence decay. While random disorder impedes thermalization compared to the clean interacting case, its effects are less severe than Stark potential-induced localization, and no QME signatures are observed.