Quantum Systems Mimic Heat Flow Between Hot and Cold Reservoirs

Purvaash Panduranghan-Udhayashankar of SISSA and INFN, and colleagues, investigated thermalisation in a closed quantum system simulating interactions with thermal reservoirs. The group explored a tripartite geometry involving finite and semi-infinite chains, modelling a temperature quench and offering a quantum analogue of the classical Mpemba effect. By analysing free-fermion models, the XX chain and the transverse-field Ising chain, they derived analytical predictions for the rate of thermalisation using generalised hydrodynamics and a Gaussian representation of the system’s dynamics. The analysis offers a thorough analytical characterisation of the thermalisation process in boundary-driven systems, and importantly demonstrates the absence of the Mpemba effect in this specific setting.

Frobenius distance thresholds define complete thermalisation characterisation in spin chains

The Frobenius distance, a matrix norm quantifying the dissimilarity between two quantum states, now possesses a quantifiable threshold of 0.11. This threshold enables complete analytical characterisation of thermalisation processes previously considered intractable. The Frobenius distance, defined as the square root of the sum of the squares of the differences between the elements of two matrices representing the density matrices of the quantum states, provides a sensitive measure of how much a system has evolved towards equilibrium. Definitive demonstration of the absence of anomalous equilibration acceleration surpasses prior methods, which lacked the necessary precision to conclusively rule out such effects. Researchers at SISSA and INFN utilised a tripartite geometry, a finite chain linked to two semi-infinite ‘thermal baths’, to model a temperature quench and explore the quantum analogue of the classical Mpemba effect. This geometry allows for a controlled study of how energy flows into and out of a central system, mimicking the influence of a large thermal environment.

A ‘temperature quench’ using this geometry allowed scientists to explore the quantum Mpemba effect, where a hotter system can sometimes relax faster than a colder one. This counterintuitive phenomenon, observed in classical systems under specific conditions, has prompted investigations into its potential quantum counterpart. Free-fermion models were utilised, transforming complex dynamics into manageable Gaussian behaviours. This simplification is achieved by mapping the interacting spin system onto a system of non-interacting fermionic particles, significantly reducing the computational complexity. Combining these with generalised hydrodynamics, a framework describing the long-time, large-scale behaviour of out-of-equilibrium systems, predicted the Frobenius distance with unprecedented accuracy. Analytical characterisation is now complete up to a Frobenius distance of 0.14 for the XX spin chain and 0.18 for the transverse-field quantum Ising chain. These values represent the limits within which the analytical predictions accurately reflect the system’s behaviour, providing a robust benchmark for comparison with numerical simulations or experimental data.

Analytical free-fermion models provide a thermalisation benchmark for complex quantum systems

Pinpointing the precise rate of thermalisation remains elusive, despite its fundamental importance to understanding how isolated quantum systems settle into equilibrium. Thermalisation is crucial for understanding the emergence of statistical mechanics from quantum mechanics, and for interpreting experimental observations in condensed matter physics and quantum information theory. This research delivers a complete analytical description for specific conditions, relying on simplified ‘free-fermion’ models that assume particles move without interaction. These models, while mathematically tractable, represent an idealisation of real-world systems. Real materials, however, are far more complex. This limitation raises an important tension, as strongly interacting quantum systems, where particles constantly influence each other, may deviate sharply from these predictions, potentially obscuring or even reversing the observed lack of a Mpemba effect. The interactions introduce correlations between particles, making the system’s dynamics significantly more challenging to analyse.

This work establishes a strong baseline understanding of thermalisation, offering a rigorously solved example against which more complicated scenarios can be tested. The ability to analytically solve a non-trivial quantum system provides valuable insights into the fundamental mechanisms governing thermalisation. Such analytical tools prove valuable for interpreting experiments and developing approximations applicable to strongly interacting systems, even when direct comparisons are not always possible. By understanding the behaviour of the simplified free-fermion model, researchers can gain intuition and develop strategies for tackling more complex systems. A complete analytical characterisation of thermalisation, the process by which a system reaches equilibrium, is provided within a specific quantum model. This characterisation is achieved through a combination of generalised hydrodynamics and a Gaussian representation of the system’s dynamics, allowing for precise predictions of the system’s evolution.

A closed quantum system, comprising a central chain linked to two ‘thermal baths’, was explored to understand energy distribution and the potential for a paradoxical effect. The thermal baths, represented by semi-infinite chains held at a fixed temperature, serve as reservoirs of energy, driving the system towards equilibrium. Despite complex quantum dynamics mimicking heat transfer, this specific setup does not exhibit the Mpemba effect. The absence of this effect suggests that the conditions required for its emergence in quantum systems may be more restrictive than previously thought. The Frobenius distance proved key to tracking this energy distribution and establishing a threshold for accurate analysis, allowing detailed understanding of the system’s evolution towards equilibrium and confirming the absence of accelerated relaxation. The precise quantification of the Frobenius distance enabled researchers to definitively rule out the Mpemba effect within the defined analytical framework, providing a rigorous test of this intriguing phenomenon in a quantum context.

Researchers demonstrated complete analytical characterisation of thermalisation within a specific quantum model, utilising a central chain connected to two thermal baths. This work provides valuable insights into the fundamental mechanisms governing how systems reach equilibrium, offering tools for interpreting experiments on more complex scenarios. By precisely quantifying the system’s evolution using the Frobenius distance, the study confirmed the absence of the Mpemba effect under these conditions. The authors suggest this analytical framework can be used to test more complicated scenarios and improve understanding of strongly interacting systems.