The Mpemba effect is an anomalous phenomenon in which the time to thermalize does not depend monotonically on the distance from equilibrium, allowing systems with an initially larger temperature difference to thermalize before those with a smaller one. The rarity of the effect makes it hard to find parameters that produce it, complicating the design of experiments and the development of applications. Here, we find necessary and sufficient conditions for the Markovian Mpemba effect in three-level systems and explain its underlying physical mechanisms. Based on our understanding of the three-level case, we develop an efficient algorithm to determine the parameters for the effect in $N$-level systems. This protocol could open the door for the realization of Mpemba experiments and related applications in large systems.
Efficient protocol for the Markovian Mpemba effect in N-level systems
The Mpemba effect is an anomalous phenomenon in which the time to thermalize does not depend monotonically on the distance from equilibrium, allowing systems with an initially larger temperature difference to thermalize before those with a smaller one. The rarity of the effect makes it hard to find parameters that produce it, complicating the design of experiments and the development of applications. Here, we find necessary and sufficient conditions for the Markovian Mpemba effect in three-level systems and explain its underlying physical mechanisms. Based on our understanding of the three-level case, we develop an efficient algorithm to determine the parameters for the effect in N-level systems. This protocol could open the door for the realization of Mpemba experiments and related applications in large systems.
The Mpemba effect is a thermodynamic anomaly in which a system farther away in temperature from equilibrium thermalizes before one that is initially closer. The effect has been experimentally observed across a wide range of systems, including water, colloids, and trapped ions. It has recently been the focus of numerous studies aimed at understanding its mechanisms and developing multiple applications. Despite extensive work in the field, clearly determining which types of systems exhibit the Mpemba effect remains an open question. To address this, we derive simple necessary conditions on the transition rates for the Mpemba effect in a Markovian 3-level system and show that they can be applied to study the Mpemba effect in an N-level system. Multiple time scales govern thermalization in these systems. This allows the evolution to occur more quickly across larger temperature differences, explaining the Mpemba effect. We apply our protocol to evaluate which types of systems exhibit the Mpemba effect and, in doing so, explain why the Mpemba effect in Markovian systems remains a thermodynamic anomaly. In particular, due to the maximum entropy principle, our conditions allow us to discard the sub-Ohmic and Ohmic spectra. The latter describes a wide range of physical and chemical phenomena, which will not exhibit the Mpemba effect. Moreover, our results provide a clear path to determine the minimal physical requirements for the Mpemba effect, and we apply them to understand its underlying mechanisms better. Finally, our protocol could help identify relevant parameters for experiments, numerical simulations and diverse applications.
In recent years, the quantum Mpemba effect (QME) which occurs when an out-of-equilibrium system reaches equilibrium faster than another that is closer to equilibrium has attracted significant attention from the scientific community as an intriguing and counterintuitive phenomenon. It generalizes its classical counterpart, extending the concept beyond thermal equilibration. This paper approaches the QME for Markovian open quantum systems from different perspectives. Firstly, we propose a physical mechanism based on decoherence-free subspaces. Secondly, we show that a macroscopic enhancement of the decay rate that scales linearly with system size can be obtained, leading to an extreme version of the phenomenon on Markovian open quantum systems. Thirdly, we study the Mpemba effect through the unravelings of Davies maps, revealing subtleties in the figures of merit which are chosen for obtaining the QME. Lastly, we propose a microscopic model for a better understanding of thermal bath dynamics in this context.
We study the nature of thermalization dynamics and the associated preparation (simulation) time under the repeated interaction protocol uncovering an 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 × 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.
Quantum thermometry provides a key capability for nanoscale devices and quantum technologies, but most existing strategies rely on probes initialized near equilibrium. This equilibrium paradigm imposes intrinsic limitations: sensitivity is tied to long-time thermalization and often cannot be improved in fast, noisy, or nonstationary settings. In contrast, the \textit{Mpemba effect}, the counterintuitive phenomenon where hotter states relax faster than colder ones, has mostly been viewed as a thermodynamic anomaly. Here, we bridge this gap by proving that Mpemba-type inversions generically yield a finite-time enhancement of the quantum Fisher information (QFI) for temperature estimation, thereby converting an anomalous relaxation effect into a concrete metrological resource. Through explicit analyses of two-level and Λ-level probes coupled to bosonic baths, we show that nonequilibrium initializations can transiently outperform both equilibrium strategies and colder states, realizing a \emph{metrological Mpemba effect}. Our results establish anomalous relaxation as a general design principle for nonequilibrium quantum thermometry, enabling ultrafast and nanoscale sensing protocols that exploit, rather than avoid, transient dynamics.
The Mpemba effect is a counterintuitive physical phenomenon where a hot system cools faster than a warm one. In recent years, theoretical analyses of the Mpemba effect have been developed for microscopic systems and experimentally verified. However, the conventional theory relies on a specific choice of distance measure to quantify relaxation speed, leading to several theoretical ambiguities. In this Letter, we derive a rigorous quantification of the Mpemba effect based on thermomajorization theory, referred to as the thermomajorization Mpemba effect. This approach resolves all existing ambiguities and provides a unification of the conventional Mpemba effect across all monotone measures. Furthermore, we demonstrate the generality of the thermomajorization Mpemba effect for Markovian dynamics, rigorously proving that it can occur in any temperature regime with fixed energy levels.
The Mpemba effect (MpE), where a far-from-equilibrium state of a system relaxes faster compared to a state closer to it, is a well-known counterintuitive phenomenon in classical and quantum systems. Various system-specific theories have been proposed to explain this anomalous behavior in driven systems, though the fundamental mechanism of MpE in undriven systems, where MpE was first observed, remains unresolved. This paper provides a generic model of MpE for a quantum spin-1/2 system following Markovian relaxation dynamics, regardless of system structure or environment. The key lies in the overlap of initial states with the fast relaxation mode; here, the constituents create a fast decay mode via interaction through the shared environment to show MpE, indicating that MpE happens due to the collective behavior of the system. I also show that a system with anisotropic relaxation naturally exhibits MpE, even without a shared environment among the particles.