⚛️ quantum physics

Quantum Mpemba effect of Z2 symmetry restoration in the Dicke model

This paper demonstrates that a quantum Mpemba effect for $\mathbb{Z}_2$ symmetry restoration can occur in the dissipative Dicke model using cat-product initial states, where initial-state mode filtering suppresses slow decay channels to allow more asymmetric states to relax faster than less asymmetric ones, even in the presence of weak Liouvillian symmetry.

Original author: Dongni Chen, Wei Xiong, Peng-Cheng Li, Yee Sin Ang, Ching Hua Lee, Yi Qin

Published 2026-11-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Dongni Chen, Wei Xiong, Peng-Cheng Li, Yee Sin Ang, Ching Hua Lee, Yi Qin

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of physics where light and matter meet, there exists a strange rule about how systems settle down. Usually, if you push a system far away from its resting state, it takes longer to return than if you only nudged it slightly. This is the intuitive order of things: a cup of boiling water takes longer to cool to room temperature than a cup of lukewarm water. However, nature occasionally breaks this rule. There are rare cases where a system starting further from equilibrium actually arrives at its final state faster than one starting closer. This counterintuitive phenomenon is known as the Mpemba effect, named after an observation that hot water can sometimes freeze faster than cold water. While this has been studied in classical fluids, scientists are now asking if it can happen in the quantum world, where particles behave like waves and systems are constantly interacting with their environment. The key to understanding this in quantum systems lies in symmetry, a fundamental property where a system looks the same after a specific change, like flipping a switch. When a system loses this symmetry, it must eventually restore it as it settles down. The question researchers are now tackling is whether a system that starts with a huge break in symmetry can somehow restore that order faster than a system with a smaller break.

A team of researchers has now demonstrated that this quantum version of the Mpemba effect is real, but only under very specific conditions involving how the system is prepared. They studied a theoretical setup called the Dicke model, which describes a collection of atoms interacting with light inside a cavity. In this environment, the system is not isolated; it loses energy to the outside world, a process known as dissipation. The researchers focused on a specific type of symmetry called parity, which relates to whether the total number of excitations in the system is even or odd. In their setup, the rules of the game allow this symmetry to be broken and then restored as the system relaxes. To track this restoration, they used a measure called entanglement asymmetry, which essentially quantifies how much the system's quantum state is out of balance with respect to this even-or-odd rule. By watching how this imbalance changed over time, they could see which systems were restoring their symmetry faster.

The team tested two different ways of preparing the initial state of the system. The first method involved creating a state that is somewhat like a simple, classical wave of light, where the atoms and light are aligned in a straightforward way. When they started with these states, the results were exactly what one would expect. The systems that started with a larger imbalance took longer to fix it, and the systems with a smaller imbalance fixed it sooner. There was no surprise here; the order of arrival was preserved, and no Mpemba effect occurred. This ruled out the idea that any initial state would show this anomaly; the specific structure of the starting point mattered immensely.

The story changed completely when the researchers used a second type of preparation, known as a cat-product state. In this scenario, the system is prepared in a delicate superposition, meaning it exists in two different configurations at once, much like a coin spinning in the air that is both heads and tails until it lands. These two configurations are mirror images of each other regarding the symmetry. When the researchers started with these superposition states, they observed a dramatic reversal. They found that some states which began with a much larger imbalance actually restored their symmetry faster than states that started with a smaller imbalance. The curves representing the two systems crossed over, with the "further away" system overtaking the "closer" one to reach the steady state first. This was a clear signature of the quantum Mpemba effect.

The reason for this surprising behavior lies in how the system's internal machinery filters out the slow parts of the relaxation process. Every quantum system trying to settle down does so by fading away through various channels, some of which are very fast and others that are painfully slow. The researchers discovered that the cat-product states were special because their unique structure acted like a filter. By combining the two mirror-image configurations, the system naturally canceled out its connection to the slowest, most sluggish channels of decay. This forced the system to rely on the faster channels instead. Even though the system started with a bigger mess, it had a clear, fast highway to the finish line. In contrast, the simpler states were stuck in the slow lanes, taking the long way around regardless of how small their initial mess was.

This effect was not a fluke limited to a single, perfectly tuned setup. The researchers mapped out a wide range of starting conditions and found that this reversal of order happened across a broad region of possibilities, not just at one isolated point. They also checked how this played out in different phases of the system, including a state where the atoms and light are strongly coupled and a state where they are weakly coupled. In both cases, the mechanism held true, though the detailed shape of the relaxation curve changed. The study confirms that the quantum Mpemba effect is a robust phenomenon that can arise even when the symmetry is not perfectly preserved by every single interaction, but only by the overall rules of the system. It shows that by carefully engineering the initial quantum state, one can bypass the usual rules of relaxation, allowing a system to heal its own disorder with surprising speed.

Technical Summary: Quantum Mpemba Effect of Z2 Symmetry Restoration in the Dicke Model

Problem Statement
The paper investigates the Quantum Mpemba Effect (QME)—a phenomenon where a system initially farther from equilibrium relaxes to a steady state faster than one initially closer to it—within the context of symmetry restoration in open quantum systems. While QME has been observed in systems with strong conservation laws (e.g., U(1) charge) or non-Hermitian dynamics, its emergence in systems with only a weak discrete $Z_2$ symmetry remains unexplored. In such "weakly symmetric" systems, the full Liouvillian superoperator is covariant under parity conjugation, but individual quantum jumps (dissipation channels) do not conserve Hilbert-space parity sectors. The central question is whether anomalous relaxation ordering (QME) can arise when parity sectors are mixed by quantum jumps, and if so, how the initial state structure influences this process.

Methodology
The authors utilize the dissipative Dicke model, a paradigmatic light-matter system consisting of a single bosonic cavity mode collectively coupled to an ensemble of $N$ two-level atoms.

Model Dynamics: The system is governed by a Lindblad master equation including cavity photon loss ( $L_\kappa = \sqrt{\kappa}a$ ). While the Hamiltonian possesses exact parity symmetry, the photon loss operator connects even and odd parity sectors, rendering the symmetry "weak" (covariant but not conserved by jumps).
Diagnostic Tool: To quantify parity restoration, the authors employ Entanglement Asymmetry (EA), specifically the second-Rényi parity asymmetry defined as $\Delta S^{(2)}(t) = S^{(2)}[\rho_P(t)] - S^{(2)}[\rho(t)]$ , where $\rho_P$ is the parity-projected density matrix. A non-zero EA indicates residual coherence between parity sectors; its decay tracks symmetry restoration.
Initial States: Two classes of initial states are compared:
1. Coherent-product states: $|\psi_{coh}\rangle = |\alpha\rangle \otimes |{\downarrow}\rangle^{\otimes N}$ , where parity breaking is encoded in a classical-like cavity displacement.
2. Cat-product states: $|\psi_{cat}\rangle = \mathcal{N}(|\alpha\rangle + e^{i\theta}|-\alpha\rangle) \otimes |{\downarrow}\rangle^{\otimes N}$ , a coherent superposition of two parity-related coherent states.
Analysis Framework: The relaxation dynamics are analyzed via the spectral decomposition of the Liouvillian superoperator. The time evolution of the density matrix is expanded in terms of Liouvillian eigenmodes, allowing the authors to correlate initial-state overlaps with specific decay rates.

Key Results

Absence of QME in Coherent-Product States: For coherent-product states, the EA decays monotonically. States with larger initial asymmetry (larger $|\alpha|$ ) relax more slowly, and no crossing of relaxation trajectories is observed. This is attributed to the linear scaling of the overlap between the initial state and the leading slow Liouvillian modes with the amplitude $\alpha$ .
Emergence of QME in Cat-Product States: Cat-product states exhibit a robust QME. Over a finite region of the parameter space (specifically for certain ranges of amplitude $\alpha$ and phase $\theta$ ), states with larger initial parity asymmetry (higher initial EA) overtake and restore symmetry faster than states with smaller initial asymmetry.
Mechanism: Initial-State Mode Filtering: The paper identifies the microscopic origin of the QME as initial-state mode filtering.
- In cat-product states, the coherent superposition of $|\alpha\rangle$ and $|-\alpha\rangle$ leads to destructive interference in the projection onto the leading slow Liouvillian modes (which are parity-odd).
- Mathematically, the overlap coefficient $c_k$ for these slow modes scales with a factor of $e^{-2\alpha^2}$ , which suppresses the contribution of the slowest decay channels as $\alpha$ increases.
- Consequently, the relaxation dynamics of highly asymmetric cat states are dominated by faster-decaying subleading Liouvillian modes, accelerating the restoration of parity symmetry compared to less asymmetric states that retain significant overlap with the slow modes.
Regime Dependence: The QME is observed in both the normal ( $\lambda < \lambda_c$ ) and superradiant ( $\lambda > \lambda_c$ ) phases. However, the relaxation profiles differ: the normal phase exhibits nearly single-stage decay, while the superradiant phase shows multi-timescale relaxation due to a more complex hierarchy of low-lying Liouvillian modes. Despite these structural differences, the mode-filtering mechanism remains operative in both regimes.

Significance and Claims
The paper claims to establish that symmetry-restoration QME can arise even in the absence of strong symmetry conservation laws, provided the system possesses a weak Liouvillian symmetry where parity conjugation is a symmetry of the full evolution but not of individual jumps.

The primary contribution is the identification of initial-state mode filtering as a general mechanism for QME in weakly symmetric open systems. By demonstrating that the coherent superposition of parity-related states can selectively suppress the projection onto dominant slow decay channels, the authors provide a spectral explanation for how a state "farther" from equilibrium can relax faster. This work extends the understanding of QME beyond systems with exact conservation laws or extensive symmetry sectors, highlighting the critical role of the interplay between initial-state coherence and the Liouvillian spectral structure.

The authors note that while the detailed relaxation profile (single-stage vs. multi-timescale) depends on the specific Liouvillian structure and dissipation channels, the fundamental crossing mechanism (QME) is robust across parameter regimes and persists even when additional collective atomic decay is included.

Technical Summary: Quantum Mpemba Effect of Z2 Symmetry Restoration in the Dicke Model

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