🔬 mesoscale physics

Initial-Slip Dynamics Enables the Quantum Mpemba Effect

This paper demonstrates that the quantum Mpemba effect in a linearly coupled quantum harmonic oscillator arises from initial-slip dynamics caused by the sudden switch-on of system-bath interactions, which generate transient correlations essential for enabling a system initially farther from equilibrium to relax faster than one closer to it, even beyond weak-coupling and Markovian approximations.

Original authors: M. Crotti, F. Cavaliere, D. Ferraro, G. Benenti, M. Sassetti

Published 2026-09-16

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CC BY 4.0

Original authors: M. Crotti, F. Cavaliere, D. Ferraro, G. Benenti, M. Sassetti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of quantum physics, where particles behave more like waves than solid objects, scientists study how systems settle down after being disturbed. Imagine a cup of hot coffee left on a table; it cools down until it matches the room's temperature. This process, called relaxation, is usually predictable: the hotter the coffee starts, the longer it takes to cool. However, nature sometimes surprises us with exceptions. In a phenomenon known as the Mpemba effect, a system that starts farther from equilibrium can actually reach a steady state faster than one that starts closer to it. While this counterintuitive behavior has been debated in everyday contexts like freezing water, its existence in the quantum realm has remained a mystery. Researchers have long wondered if this effect could happen in the simplest possible quantum systems, starting from ordinary thermal states without any special preparation, and if so, what hidden mechanism drives it.

A team of physicists has now answered these questions by looking at a quantum harmonic oscillator, a fundamental model of a vibrating particle, connected to a surrounding environment of bosonic particles. By using exact mathematical methods that do not rely on common simplifying assumptions, they discovered that the quantum Mpemba effect does indeed occur, but only under a specific condition. The effect is not caused by the initial temperature difference alone, nor by the way the system interacts with its environment over long periods. Instead, the key lies in the very first instant when the connection between the system and its environment is turned on. This sudden switch creates a brief, intense burst of correlation between the two, a transient state that the researchers call "initial-slip dynamics."

The team found that this initial-slip dynamics acts as a hidden engine for the effect. When the connection is established suddenly, the system does not immediately settle into a smooth path toward equilibrium. Instead, it experiences a sharp, short-lived departure from balance, generating a specific type of energy fluctuation. This fluctuation leaves a lasting imprint on how the system relaxes. If the system starts at a higher temperature, this initial burst can actually help it shed energy more efficiently than a cooler system, allowing it to catch up and overtake the slower-relaxing, cooler system. The researchers proved that if this initial-slip contribution is removed from the equations, the effect vanishes completely. The hotter system then behaves exactly as expected, taking longer to cool down than the colder one. This finding rules out the idea that the effect is a result of long-term memory or complex initial preparations, pinpointing the sudden onset of interaction as the sole cause.

The study also mapped out exactly when this effect can be observed. It appears in two distinct regimes depending on how strongly the system is coupled to its environment. In a weak-coupling scenario, the effect is fragile and oscillatory, appearing and disappearing in rhythmic waves as the system evolves. In stronger coupling regimes, the effect becomes robust and stable, governed by a clear condition related to the amplitude of the slowest relaxation mode. The researchers showed that for the effect to happen, the environment must be warm enough and the coupling strong enough to satisfy a precise mathematical threshold. They demonstrated that by tuning these parameters, one can create a situation where a "hot" quantum system consistently reaches equilibrium faster than a "cold" one.

Crucially, the work clarifies why previous attempts to find this effect in simpler models failed. Many standard theories used to describe quantum systems ignore the initial-slip term, assuming the system and environment adjust to each other gradually. The paper shows that this assumption is fatal to the Mpemba effect; without the sudden, non-adiabatic switch-on that generates the initial correlations, the anomaly simply does not exist. This insight suggests that the fleeting, transient moments right after a system is disturbed are far more important for its long-term behavior than previously thought. The findings provide a definitive explanation for the quantum Mpemba effect, showing it is a real, calculable phenomenon driven by the immediate aftermath of turning on a connection, rather than a trick of initial conditions or long-term dynamics.

Technical Summary: Initial-Slip Dynamics Enables the Quantum Mpemba Effect

Problem Statement
The relaxation of open quantum systems toward thermal equilibrium is a central problem in nonequilibrium statistical mechanics. While intuition suggests that a system initially farther from equilibrium should take longer to relax, the Mpemba effect describes the counterintuitive phenomenon where a "hotter" system thermalizes faster than a "colder" one. Previous investigations into the quantum Mpemba effect have largely relied on Markovian approximations, non-Markovian dynamics with engineered initial coherences, or specific nonequilibrium steady states. A fundamental open question remained: Can the quantum Mpemba effect emerge within exact open-system dynamics, starting from simple, uncorrelated thermal states, without ad hoc initial conditions? Furthermore, if it does, what is the underlying physical mechanism?

Methodology
The authors investigate this problem using a quantum harmonic oscillator linearly coupled to a bosonic environment, modeled via the Caldeira–Leggett Hamiltonian. This paradigmatic model of quantum dissipation allows for an exact treatment via the quantum Langevin equation, avoiding weak-coupling and Markovian approximations.

Key methodological steps include:

Exact Dynamics: The system is initialized in a factorized thermal state where the oscillator and bath are uncorrelated, with the oscillator at temperature $T_\alpha$ and the bath at temperature $T$ . The dynamics are derived from the quantum Langevin equation, which includes a memory kernel and a fluctuating noise operator.
Covariance Matrix Evolution: Due to the linearity of the dynamics, the reduced state of the oscillator remains Gaussian at all times, fully characterized by its covariance matrix (CM). The authors derive exact analytical expressions for the time evolution of the CM, decomposing it into a homogeneous part (dependent on initial conditions) and a thermal fluctuation part (dependent on bath correlations).
Distance Metric: The relaxation toward the stationary state is quantified using the Frobenius distance between the time-dependent reduced density matrix and the asymptotic state.
Regime Analysis: The study analyzes three distinct dynamical regimes based on the system-bath coupling strength ( $\gamma_0$ $γ_{0}$ ) and damping:
- Underdamped (UD) weak-coupling.
- Overdamped (OD) moderate-coupling.
- UD strong-coupling.
Controlled Comparison: To isolate the mechanism, the authors explicitly compare the exact dynamics (including the "initial-slip" term arising from the sudden switch-on of the interaction) against a modified dynamics where this term is neglected.

Key Contributions and Results
The paper derives exact analytical criteria for the occurrence of the direct quantum Mpemba effect (where the initially hotter state relaxes faster than the colder one) and identifies the specific dynamical features enabling it:

Exact Criteria Beyond Approximations: The authors establish that the effect can occur in exact open-system dynamics starting from simple thermal states. They identify two distinct behaviors:
- In the UD weak-coupling regime, the effect is "reentrant" and oscillatory. The distance ordering inverts periodically due to the interplay of complex-conjugate relaxation modes, leading to alternating time windows where the Mpemba effect is observed.
- In the OD moderate-coupling and UD strong-coupling regimes, the effect becomes robust. The authors derive a sharp, necessary, and sufficient condition for asymptotic inversion: $B_{th} < -\frac{1}{2}(B_c + B_h)$ , where $B_{th}$ represents the thermal contribution and $B_{c,h}$ represent the homogeneous contributions for the cold and hot initial states. This condition is satisfied when the bath temperature $T$ exceeds a critical value $T^*$ , which decreases as the system-bath coupling $\gamma_0$ increases.
The Role of Initial-Slip Dynamics: The central finding is that the quantum Mpemba effect is enabled by transient system-bath correlations generated immediately after the sudden switch-on of the interaction. These correlations are encoded in the initial-slip term ( $\gamma(t)Q(0)$ $γ (t) Q (0)$ ) of the quantum Langevin equation.
- When the initial-slip term is included, the oscillator energy exhibits pronounced transient oscillations and reaches values significantly higher than in its absence, creating a strong departure from equilibrium.
- Crucially, the authors demonstrate that neglecting the initial-slip contribution completely suppresses the Mpemba effect. In the absence of this term, the distance ordering $D_h(t) > D_c(t)$ is preserved at all times, and no inversion occurs.
Connection to Markovian Limits: The study shows that standard Born-Markov approximations, which typically neglect the initial-slip contribution, fail to capture the effect. The effect persists even in high-temperature regimes provided the initial-slip term is retained in the exact dynamics.

Significance and Claims
The paper claims to resolve the open question of whether the quantum Mpemba effect can emerge from simple thermal initial conditions in exact dynamics. The authors assert that the effect is not an artifact of engineered coherences or specific initial states but is a generic consequence of the non-adiabatic establishment of system-bath interactions.

The significance of the work lies in identifying initial-slip dynamics as the key physical mechanism underlying the anomalous relaxation. By proving that the effect vanishes when this contribution is removed, the authors highlight the long-time persistence of transient correlations generated during the quench. This challenges the conventional reliance on factorized initial states in relaxation studies and suggests that transient system-bath correlations play a fundamental role in determining relaxation pathways in open quantum systems. The results motivate further investigation into initially correlated system-bath states and offer a new perspective on controlling relaxation in quantum thermodynamics and dissipative systems.

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