Geometry and restoration of the quantum Mpemba effect beyond weak-coupling regime in the spin-boson model

Abstract

Understanding relaxation dynamics in open quantum systems is a central problem in nonequilibrium quantum physics. Here we investigate the quantum Mpemba effect in the spin-boson model. In the weak-coupling Markovian regime we show that the occurrence of the effect strongly depends on the choice of distance measure at low temperature: while it appears in the trace distance, it can disappear in the quantum relative entropy. Going beyond the weak-coupling approximation, numerically exact simulations of the full system-bath dynamics reveal that increasing coupling enhances the effect in the trace distance and restores it in the quantum relative entropy. For all spin-bath couplings prior to delocalized-localized quantum phase transition, we uncover a simple geometric structure of the effect on the Bloch sphere: within the excited-state hemisphere, pairs of states related by rotations generically exhibit relaxation-order inversion. These results highlight the role of geometry and system-environment correlations in anomalous quantum relaxation.

Figures (5)

• Figure 1: Geometric structure of the quantum Mpemba effect on the Bloch sphere. The sphere is partitioned into ground-state (light) and excited-state (dark) hemispheres. Colored curves represent dynamical trajectories for $\alpha=0.6$ obtained from MPS simulations for different initial states with fixed Bloch-vector modulus. The red and green dots are the ground states for $\alpha=0.6$ and $\alpha=0$ respectively. These last two dots roughly show that as $\alpha \to\alpha_c^-$ the ground state of the system tends to maximally mixed state.
• Figure 2: Quantum relative entropy with respect to the stationary state as a function of time for three initial states at high temperature ($T=10$ in units of $\Delta$), labeled by their Bloch vectors $\mathbf{r}_1=(-1,0,0)$, $\mathbf{r}_2=(-\frac{\sqrt{3}}{2},0,\frac{1}{2})$, and $\mathbf{r}_3=(0,0,1)$. The occurrence of a quantum Mpemba effect is signaled by the finite time crossing of the curves, calculated solving Lindblad equation.
• Figure 3: Quantum relative entropy with respect to the stationary state as a function of time for three initial states of the spin--boson model at zero temperature, labeled by the qubit Bloch vectors $\mathbf{r_i} = (-\mathrm{sin}(\phi_i),0,-\mathrm{cos}(\phi_i))$, with $\phi_i\in \{0,0.1\pi,0.2\pi, 0.3 \pi, 0.4\pi\}$. (a) The graph of the quantum relative entropies do not intersect, signaling the absence of the effect at zero temperature (in this panel the Lindblad solution was used). (b), (c), (d) show the relative entropy computed with numerical simulations.
• Figure 4: Trace distance with respect to the stationary state as a function of time for different initial states at zero temperature, labeled by their Bloch vectors $\mathbf{r_i} = (-\mathrm{sin}(\phi_i),0,-\mathrm{cos}(\phi_i))$, with $\phi_i\in \{0,0.1\pi,0.2\pi, 0.3 \pi, 0.4\pi\}$. The effect occurs for every value of the coupling with the environment. Panel (a) was derived using the Lindblad solution, while the others using the numerical solutions.
• Figure 5: Time evolution of the Bloch components $r_x$, $r_y$, and $r_z$ obtained from numerical simulations at coupling strength $\alpha = 0.6$. Different curves correspond to initial states parameterized by $\phi$, with Bloch vectors $\mathbf{r}_i = (-\sin\phi_i, 0, -\cos\phi_i)$. The dynamics highlights a two-stage relaxation process: a rapid initial decay dominated by population dynamics ($r_x$), followed by a slower evolution governed by coherences ($r_y$, $r_z$).