Systems Further from Balance Can Relax Faster Than Those Near Equilibrium

A counterintuitive phenomenon known as the quantum Mpemba effect occurs in chaotic quantum systems. Thomas Martin Müller and colleagues at the The Abdus Salam International Centre for Theoretical Physics (ICTP) in collaboration with University of Cologne and University of Naples Federico II, reveal how systems evolving towards thermal equilibrium can exhibit differing relaxation speeds, with a system initially closer to equilibrium actually relaxing more slowly than one starting further away. The work, focused on two chaotic spin chains, establishes a strong mechanism for the quantum Mpemba effect arising from the interplay between conservation laws and hydrodynamic relaxation within closed quantum systems. Understanding these dynamics provides key insight into the thermalisation processes of complex quantum systems and challenges conventional expectations of relaxation behaviour.

Modelling quantum relaxation using finite size numerical simulations of spin chains

Finite size numerical simulations revealed subtle relaxation dynamics within closed quantum systems. The technique modelled the behaviour of chaotic spin chains, effectively creating a controlled environment to observe how quantum states evolve after a disturbance, or ‘quench’. These spin chains, composed of interacting quantum spins, serve as a tractable yet representative model for studying many-body quantum dynamics. By discretising time into small steps and utilising a second-order Trotterization method, a mathematical shortcut for approximating the time evolution operator, the system’s progression towards thermal equilibrium could be tracked with reasonable accuracy. Trotterization allows the complex evolution of the system to be broken down into a series of simpler, more manageable steps, albeit at the cost of introducing a small degree of numerical error. Simulations were performed on chains containing up to N sites, allowing scientists to examine finite size effects and confirm that the observed behaviour wasn’t simply an artefact of the model’s limitations. Increasing the number of sites allows for a more accurate representation of the thermodynamic limit, where finite size effects become negligible.

This careful scaling is key for establishing the durability of the quantum Mpemba effect and ensuring the results are not merely a consequence of the chosen system size. Quantifying the trace distance, entanglement entropy and entanglement asymmetry revealed how closely the system approached thermal equilibrium, demonstrating finite size effects and power-law relaxation regimes. The trace distance, a measure of distinguishability between quantum states, provides a quantitative assessment of how quickly the system loses memory of its initial condition. Entanglement entropy, quantifying the degree of quantum correlations, and entanglement asymmetry, describing the distribution of entanglement, further characterise the thermalisation process. These simulations focused on initial states relaxing towards the infinite temperature state, a common benchmark for thermalisation representing complete randomness, and employed periodic boundary conditions to minimise edge effects. The conserved quantity, either magnetization or energy depending on the model, remained consistent throughout the process, validating the approach to a unique stationary state and confirming the integrity of the simulations. With time discretised into steps of 0.05, the simulations balanced accuracy and computational cost on systems of up to six sites, representing a compromise between precision and computational feasibility.

Hydrodynamic Relaxation Speeds Dictate Quantum Mpemba Effect in Chaotic Spin Chains

Entanglement entropy measures now demonstrate power-law decay with exponents differing by up to two orders of magnitude between initial states, a disparity previously unobserved in chaotic spin chains. This substantial variation in decay rates is a direct consequence of the interplay between conservation laws and the system’s initial conditions. The observation of such a large difference in exponents is particularly noteworthy, as it signifies a significant deviation from the expected behaviour of chaotic systems. This allows for the strong realisation of the quantum Mpemba effect. Systems initially closer to equilibrium can relax more slowly than those starting further away, despite converging on the same final state. This counterintuitive result highlights the non-monotonic relationship between initial proximity to equilibrium and relaxation time.

Differing hydrodynamic relaxation speeds, determined by conservation laws and the specific initial conditions, enable a clear observation of the Mpemba effect for the first time in these systems. Hydrodynamic relaxation refers to the process by which local equilibrium is established within the system, driven by the transport of energy and other conserved quantities. Some decays were as quick as t⁻⁵, while others exhibited slower relaxation, indicating a wide range of timescales for thermalisation. This observation challenges established expectations, clarifying that initial conditions profoundly influence relaxation speeds, even when systems ultimately reach the same final state. The power-law exponents governing these decays provide valuable insights into the underlying mechanisms driving the relaxation process. This suggests that the system’s ability to dissipate energy and establish equilibrium is strongly dependent on its initial configuration.

However, the current study confines itself to specific chaotic spin chain models, namely the Floquet and Mixed Field Ising models, leaving open whether this quantum Mpemba effect extends to systems with different symmetries or in higher dimensions. The choice of these models was motivated by their relative simplicity and well-understood properties, but it is crucial to investigate whether the observed effect is robust and generalisable. It is important to acknowledge that these findings stem from simulations of specific spin chain models; extending this effect to more complex systems remains an open challenge. Nevertheless, demonstrating this counterintuitive behaviour within well-defined quantum systems provides a strong benchmark for future investigations and encourages exploration of the Mpemba effect in a wider range of physical systems.

Initial states dictate thermalisation rates in simulated quantum spin chains

Understanding how closed quantum systems settle into thermal equilibrium is important for advancing quantum technologies, particularly in areas such as quantum information processing and quantum materials. This research establishes a clear mechanism for the quantum Mpemba effect, a counterintuitive phenomenon where a system nearer to its final, stable state relaxes to equilibrium more slowly than one starting further away. Within chaotic spin chains, interconnected quantum systems exhibiting unpredictable behaviour, differing hydrodynamic relaxation speeds govern this behaviour. The ability to predict and control thermalisation rates is crucial for maintaining the coherence of quantum states and preventing decoherence, a major obstacle in building practical quantum devices.

Fundamental conservation laws determine these speeds, offering a strong explanation beyond previous numerical studies. The conservation of energy, momentum, or other relevant quantities constrains the possible pathways for relaxation, leading to distinct hydrodynamic behaviours. The current study, however, confines itself to specific chaotic spin chain models, leaving open the question of whether this effect extends to systems with different symmetries or in higher dimensions. While this research provides a strong benchmark for future investigations, extending the quantum Mpemba effect to more complex systems remains an open challenge. Future work could explore the effect in systems with long-range interactions, disorder, or in higher-dimensional lattices, potentially revealing new insights into the nature of thermalisation and the limits of the Mpemba effect in the quantum realm.

The research demonstrated the quantum Mpemba effect in two chaotic spin chains, revealing that a system initially closer to equilibrium can relax slower than one starting further away. This occurs because differing hydrodynamic relaxation speeds, governed by fundamental conservation laws, dictate how these closed quantum systems reach the same final state. This finding provides a strong benchmark for future investigations into thermalisation processes and encourages exploration of the effect in a wider range of physical systems. The authors suggest further work could investigate systems with more complex interactions or in higher dimensions.