Thermal state preparation by repeated interactions at and beyond the Lindblad limit

TL;DR

This work analyzes thermal state preparation via repeated system–ancilla collisions beyond the standard stroboscopic-Lindblad limit, using a three-level system as a tractable testbed. It derives exact and approximate motion equations in two regimes—the SL limit and the long-collision $J\tau1$ regime—and provides closed-form expressions for the minimal simulation time $T_{\text{sim}}$ (or $n^*$) to reach a thermal state, including a Mpemba-like effect where starting from a high-temperature (maximally mixed) state can, counterintuitively, thermalize faster to a colder bath. The Mpemba phenomenon is explained through the Liouvillian spectrum and extends to higher dimensions and randomized interactions, with the effect becoming more pronounced as the system size grows. These results offer practical guidance for resource estimation in quantum algorithms that prepare thermal states and motivate exploring RI protocols beyond the SL limit for scalable open-system control. The analysis reveals robust Mpemba-like behavior across regimes, dimensions, and interaction types, underscoring the need for new strategies in high-dimensional thermal-state preparation.

Abstract

We study the nature of thermalization dynamics and the associated preparation (simulation) time under the repeated interaction protocol uncovering a generic anomalous, Mpemba-like trend. As a case study, we focus on a three-level system and analyze its dynamics in two complementary regimes, where the system-ancilla interaction strength is either large or small. Focusing on the estimation of the simulation time, we derive closed-form expressions for the minimum number of collisions, or minimal simulation time, required to achieve a thermal state, which is within $ε$ distance to the target thermal state. At zero temperature, we analytically identify a set of points (interaction strength $\times$ their duration) that minimize the simulation time. At nonzero temperature, we observe a Mpemba-like effect: Starting from a maximally mixed state, thermalization to an intermediate-temperature state takes longer than to a lower-temperature one. We provide an accurate analytical approximation for this phenomenon and demonstrate its occurrence in larger systems and under randomized interaction strengths. The prevalence of the Mpemba effect in thermal state preparation presents a significant challenge for preparing states in large systems, an open problem calling for new strategies.

Figures (13)

• Figure 1: An illustration of thermal state preparation via the repeated interaction protocol, representing the cooling of a hot (red) system towards the state of the cold (blue) bath of ancilla qubits. (a) The RI process starts by preparing a product state of the system and an ancilla. (b) After one collision, the system evolves into $\rho_S^{(1)}$, while the first ancilla is discarded. (c) After $n$ collisions, the system is cooler. Discarded ancilla heats up to some extent due to heat exchange with the system.
• Figure 2: Dynamics of the ground-state population and selected coherences for spin-$s$ systems interacting with thermal qubit ancillas at two different temperatures, $\beta=1$ (top panels) and $\beta=10$ (bottom panels). We compare two system sizes, $d=3$ (spin-$1$, solid lines) and $d=10$ (spin-$9/2$, dashed lines), and two interaction regimes: the $J\tau1$ limit (red for the top panel and light blue for the bottom panel) and stroboscopic-Lindblad limit (brown for the top panel and blue for the bottom panel). Panels (a) and (b) show the evolution of the ground-state population of the system, $p_1$, with dotted lines indicating the corresponding thermal state of the system. Panels (c)-(h) show the decay of coherences $|c_{12}|$, $|c_{13}|$ and $|c_{23}|$. All simulations were performed under the following conditions: $\omega=1$, $J\tau=1$ with $J = 10^{-3}$ and $\tau = 10^3$ for the $J\tau1$ limit and $J = 10$ with $\tau = 10^{-2}$ for the stroboscopic-Lindblad limit. The initial state for both systems is given by a random initial state $\rho_S^{(0)}$.
• Figure 3: Minimum number of collisions required for the thermalization of a system starting from a completely mixed state and targeting a low temperature state ($\beta=10$) within trace distance $\epsilon$, using (a) $J=10$ and (b) $J=10^{-3}$. We present results for system with $d=3$ and $d=10$ states; the minimum number of collisions required to thermalize either systems follows the same pattern. In panel (a), the shaded region corresponds to the SL limit where $J\tau \xrightarrow[]{}0$ but $J^2\tau = \Gamma$ with $\Gamma \sim 1$. In panel (b), the zone around the first minimum corresponds to the $J\tau1$ limit. (c) A zoom-in on the region around $J\tau \sim1$, showing that the minimum occurs precisely at $J\tau = \pi/2$. We present results at low (light blue) and intermediate (red) ancilla (target thermalization) temperatures; in both cases, the minimum takes place at $J\tau = \pi/2$. We use the following settings: The initial state of the system is maximally mixed, $\omega=1$, and $\epsilon=10^{-4}$.
• Figure 4: $J\tau1$ limit. Minimum number of collisions, $n^*$, required for a three-level system to thermalize within $\epsilon$ to temperature $\beta^{-1}$ from a completely mixed state. Different curves correspond to $J\tau = \pi/2$ (dark green), $J\tau = \pi/4$ (medium green) $J\tau = \pi/8$ (light green). The solid lines correspond to numerical simulations; dotted lines ($n^*_{\text{th}}$) depict the analytical result at zero temperature predicted by Eq. (\ref{['eq:nstar-value']}). Energy splitting, identical for the qutrit system and the ancilla is given by $\omega = 1$; the precision is set to $\epsilon = 10^{-4}$.
• Figure 5: (a) Total simulation time $T_{\text{sim}}$ in the stroboscopic-Lindblad limit as a function of the inverse temperature $\beta$. The solid line corresponds to the numerical results, the dotted line represents the asymptotic expression Eq. (\ref{['eq:nstar-value']}), and the dashed brown curve corresponds to the analytic result in Eq. (\ref{['eq:Mpemba_Tsim_lambda2']}), correctly describing the behavior for intermediate temperatures. We fixed the energy splitting to $\omega=1$, the precision to $\epsilon=10^{-4}$, and $J= 10$ with $\tau=10^{-2}$ to efficiently capture the SL limit for $\Gamma=1$. (b) Blue diamonds correspond to the numerical results for $\beta = 10$ with varying $\epsilon$. As the required precision increases (i.e., smaller $\epsilon$), the simulation time $T_{\text{sim}}$ grows logarithmically. The dotted lines represent the analytical result given by Eq. (\ref{['eq:Mpemba_Tsim_lambda2']}). (c) Eigenvalues of the Liouvillian at finite temperature, with $\lambda_1 = 0$, and $\lambda_2 \geq \lambda_3$.