Quantum Many-Body Mpemba Effect through Resonances

Abstract

Relaxation towards equilibrium is often assumed to be slower when a system starts farther from equilibrium, but this intuition fails in the Mpemba effect. Recent advances in controllable quantum platforms have enabled the exploration of its quantum analogue, the quantum Mpemba effect (QME), yet its microscopic origin remains largely unclear. Here we provide a general framework for understanding the QME in closed quantum many-body chaotic systems by reformulating the equilibration process of local subsystems in terms of Ruelle-Pollicott (RP) resonances. We show that suppressing the initial-state overlap with the dominant RP resonant mode accelerates subsystem equilibration and thereby yields the QME. We further uncover that a novel type of strong QME can occur via complete translation-symmetry breaking of initial states. We substantiate our predictions using the prototypical kicked Ising chain and exotic yet experimentally relevant initial states inspired by number theory. These findings cast the QME in closed many-body systems into a unified framework with open-system analogues and provide experimentally accessible signatures on state-of-the-art quantum platforms.

Figures (5)

• Figure 1: Quantum Mpemba effect through Ruelle-Pollicott resonances. Late-time behaviour of a local subsystem in a closed quantum many-body chaotic system is controlled by the RP resonances, shown schematically as eigenvalues $\lambda$ in the complex plane (insets). The resonant mode closest to the unit circle decays most slowly and sets the bottleneck for subsystem equilibration. Therefore, suppressing its initial-state overlap, encoded in the colour intensity of the spectral points in the insets, accelerates local equilibration and yields the QME. In addition, when the translational symmetry is completely broken by the initial state, equilibration can be further accelerated, giving rise to a strong QME.
• Figure 2: Truncated propagator and RP resonances.a, Under the truncated propagator $\bigoplus_k \mathcal{E}_{k,r}$, any local observable increases its spatial extent strictly within the light cone until its support length reaches the cutoff $r$. b, As $r$ increases, the eigenvalues of $\mathcal{E}_{k,r}$ approach the unit circle because its original counterpart is unitary. The RP resonances are the eigenvalues that remain strictly inside the unit circle even in the limit $r\to\infty$.
• Figure 3: The QME in the dynamics of the kicked Ising chain.a, The Bures distance $D(0)$ for the initial state \ref{['eq:Psi_0']}. It quantifies how far the system is from equilibrium at $t=0$. b, Weight $|c_{1,0}|$ of slowest decay mode (the dominant RP resonance) for the same initial state. As $|c_{1,0}|$ decreases, the subsystem equilibrates faster. c, Time evolution of the Bures distance under the dynamics of the kicked Ising chain. The solid curves correspond to the time evolution from the initial state in equation \ref{['eq:Psi_0']}. Inset shows the results for large times, in which the asymptotic forms of $D(t)$ predicted by equation \ref{['eq:rho_S(t)_asymptotic']} are plotted as the dotted curves. We set $(\tau,h_x,h_z)=(0.65,0.9,0.8)$, subsystem size $\ell=4$, and system size $L=24$ with periodic boundary conditions for all the plots. To obtain the solid curve in b and the dotted curves in the inset of c, we evaluate $\lambda_{1,k}$, $c_{1,k}$, and $\Tr_{}[\rho(0)V_{1,k}^\mathrm{R}]$ by numerically diagonalising $\mathcal{E}_{k,r}$ with $r=12$ (see Methods). The values of $\theta$ used in c are indicated by the filled circles in a and b.
• Figure 4: Strong QME from complete breaking of translational symmetry.a, Schematic contrast in the profile of the overlap function $\mathrm{Tr}[\rho(0)V^{\rm R}_{\alpha,k}]$. For an initial state with translational symmetry, the overlap develops discrete delta-function peaks at selected momenta, resulting in the purely exponential equilibration. By contrast, when translational symmetry is completely broken in the initial state, the overlap is instead spread over a continuum of momenta, which results in the accelerated equilibration with factor $t^{-1/2}$ in the case of the kicked Ising chain. b, Time evolution of $D(t)$ for the kicked Ising model starting from the initial states constructed from Legendre sequences (solid curves, different primes $p$), which decay faster than that for a translationally invariant initial state \ref{['eq:Psi_0']} (dashed curve, $\theta=\pi/2$). Inset represents the results for late times, in which the asymptotic forms of $D(t)$ predicted by equation \ref{['eq:rho_S_NTI_KI']} are plotted as the dotted curves. To obtain the dotted curves in b, we evaluate $\lambda_{1,k}$ and $\Tr_{}[\rho(0)V_{1,k}^\mathrm{R}]$ by numerically diagonalising $\mathcal{E}_{k,r}$ with $r=12$. We set $(\tau,h_x,h_z)=(0.65,0.9,0.8)$, subsystem size $\ell=4$, and system size $L=24$ with periodic boundary conditions for all the plots.
• Figure sm-1: The time evolution of the Bures distance $D(t)$ for the dynamics starting from the initial state \ref{['eq:rho(0) de Bruijn']} generated from the de Bruijn sequence with $r_d=5$, which corresponds to $L=2^{r_d}=32$ qubits. We find that the decay of $D(t)$ is consistent with Eq. \ref{['eq:KI_Laplace']} with $\gamma=1$, which corresponds to the faster decay than that discussed in the main text.