Researchers at the Jožef Stefan Institute and University of Ljubljana have discovered a key condition for the quantum Mpemba effect, the counter-intuitive phenomenon where a hotter system cools faster than a colder one. The work demonstrates the effect is only possible when a system’s Hamiltonian commutes with other operators, essentially meaning it is integrable.

This finding links the effect to the mathematical properties of the system itself, not just initial temperature, and challenges conventional thermodynamic intuition. The team used tensor networks and free-fermion techniques to model large system sizes, supporting their claim that multiple conserved quantities allow dynamics to evolve in a multi-dimensional space, enabling faster relaxation.

Integrable Hamiltonians Enable Faster Relaxation

The possibility of a hotter quantum system relaxing to a steady state faster than a colder one depends on the presence of multiple conserved quantities within the system, according to new simulations. Unlike systems where energy is the sole conserved property, those with additional conserved quantities evolve within a more complex, multi-dimensional space of generalized Gibbs ensembles. This expanded dynamic space appears to shorten the relaxation time, enabling the quantum Mpemba effect, a counter-intuitive phenomenon observed in both classical and quantum realms.

The research establishes a clear link between the mathematical properties of a quantum system and its ability to exhibit this unusual cooling behavior. The simulations reveal that chaotic many-body models, limited to energy as their primary conserved quantity, cannot demonstrate the Mpemba effect. However, systems possessing additional approximate conserved quantities, with integrable Hamiltonians representing the extreme case, do allow for this accelerated relaxation.

Researchers modeled systems using tensor networks and free-fermion techniques to validate this claim. When energy is the only conserved quantity, the dynamics are confined to a single-parameter manifold of thermal states. This restriction limits the pathways available for relaxation. In contrast, the presence of additional conserved quantities expands the dynamic possibilities, allowing the system to “drift” within a multi-dimensional space. This drift, described as evolution within a space of generalized Gibbs ensembles, results in a shorter “distance” to the steady state, and thus faster relaxation.

“When projecting this dynamics on one-dimensional distances between the time-dependent state and the steady state, crossings can appear if the state evolves in the multidimensional GGE manifold and cannot appear for dynamics within the one-dimensional thermal manifold,” the study notes. To illustrate this effect, the researchers examined the transverse field Ising Hamiltonian under both chaotic and integrable conditions.

Chaotic parameters, specifically, J = 0.75, $h_{z}$ = 1.0 and $h_{x}$ = 0.3, resulted in a system with only the Hamiltonian as an approximate conserved quantity. This difference in conserved quantities directly correlated with the potential for the Mpemba effect.

Conserved Quantities Define Generalized Gibbs Ensembles

The number of conserved quantities within a quantum system dictates the dimensionality of its dynamic evolution, influencing how quickly it reaches a stable, non-equilibrium state. This expanded dynamic space alters the relaxation process, potentially explaining observations of faster cooling in seemingly paradoxical scenarios. Researchers found that projecting the system’s evolution onto a single measure of distance to its steady state reveals crossings in the multi-dimensional case, impossible within the one-dimensional thermal manifold.

A system’s Hamiltonian must commute with other “extensive operators” to exhibit this behavior, effectively making it integrable. This means the system possesses a greater number of predictable, unchanging properties beyond just energy. The mathematical description of these systems uses Lagrange parameters, one for each conserved quantity, to approximate the time-dependent state as a generalized Gibbs ensemble.

These parameters evolve according to a set of equations that govern the rate of change in each conserved quantity, influenced by the system’s coupling to external baths. The equations reveal how dissipation steers the evolution from thermal states within the multi-dimensional manifold of generalized Gibbs ensembles. The researchers demonstrated this effect through numerical simulations, modeling large system sizes using established computational techniques.

They examined the distance between the time-evolving state and the non-equilibrium steady state, finding that crossings, indicating a reversal in the expected relaxation order, occur when the dynamics unfold within the higher-dimensional space of generalized Gibbs ensembles. The team’s analysis extends to integrable Hamiltonians, where the equations of motion can be interpreted as describing movement within this multi-dimensional space of Lagrange parameters. Even if the system begins in a simple, one-dimensional thermal state, weak dissipation will generally drive it into this higher-dimensional manifold.

This suggests a fundamental link between the number of conserved quantities and the potential for counter-intuitive thermal behavior, offering a pathway to understand the conditions under which a hotter system can indeed cool faster than a colder one. The study’s findings provide a framework for interpreting the dynamics of weakly open quantum systems and offer insights into the role of conserved quantities in shaping their evolution toward a steady state.

Quantum Mpemba Effect: Initial Thermal States

Weakly open quantum systems, those interacting with an environment, can exhibit the Mpemba effect under specific conditions, according to recent analysis of their initial thermal states. Specifically, the Mpemba effect is not generically observed for chaotic weakly open dynamics with a single approximately conserved quantity. The dynamics are constrained to a one-dimensional manifold of thermal states, preventing the necessary conditions for faster relaxation. The team’s work demonstrates that the distance between a time-evolving quantum state and its eventual, non-equilibrium steady state, denoted as $d_{}$ (ℓ)(ρ(t), $ρ_{}$ ∞), provides a critical metric for identifying the Mpemba effect.

Analysis of this distance, calculated for different initial temperatures, shows that for chaotic Hamiltonians, the hotter initial state remains farther from the steady state than its colder counterpart. This is a key distinction, as a crossing of these distance curves, indicating faster relaxation of the hotter system, is absent in these scenarios.

The researchers emphasize the importance of choosing a distance measure that is a monotonic function of the absolute temperature distance to the steady-state temperature |β-β_∞| to avoid false detection of the Mpemba effect. However, the picture changes dramatically when considering systems with multiple conserved quantities.

When projecting the dynamics onto a scalar, the distance between the time-evolved and the non-equilibrium steady state (NESS), crossings can happen in the presence of multiple approximately conserved quantities. The study’s findings suggest that the number of conserved quantities within the system’s Hamiltonian plays an important role in enabling this counter-intuitive phenomenon, and that the dynamics unfold differently depending on whether energy is the sole conserved quantity or one of several.

Lindblad Operators Model Weak System Coupling

When dissipation is weak, the density matrix describing the system splits into a dominant component and a small correction, allowing researchers to parameterize the dynamics using time-dependent chemical potentials. These potentials function as Lagrange parameters governing the nearly conserved quasiparticle occupations, their evolution determined by a scattering equation that captures the influence of Lindblad perturbations. The team employed periodic boundary conditions in their fermionic formulation, noting that the absence of current-inducing Lindblad operators minimized the importance of boundary condition choices for sufficiently large system sizes.

This approach allows for the modeling of complex interactions within the quantum system, providing a framework for understanding the Mpemba effect’s underlying mechanisms. A Hamiltonian with an additional conserved quantity, total magnetization, was examined alongside a weakly dissipative perturbation encoded in bulk Lindblad operators.

In this scenario, the steady state and the dynamics are approximately described by a time-dependent temperature and chemical potential linked to the approximately conserved magnetization. The presence of these approximate conserved quantities appears necessary, though not sufficient, for observing the Mpemba effect and a general criterion for selecting appropriate Lindblad operators remains an open question. The researchers provide numerical results for large system sizes using tensor networks and free-fermion techniques, thereby supporting their claim.

Number of Conserved Quantities Impacts Dynamics (N_C)

Simulations reveal that systems with multiple conserved quantities navigate a more complex dynamic space than those governed solely by energy conservation. The study modeled these dynamics using systems up to L = 80, employing parameters such as J = 1 and $Δ_{j}$ = 1.2 + 0.4·(-1)^j. For Hamiltonians without additional conserved quantities, each dynamics can be approximated by a thermal ensemble with a time-dependent temperature. While the existence of multiple conserved quantities is a necessary condition for the Mpemba effect, it is not, on its own, sufficient.

Symmetry Restoration and Quantum Mpemba Observations

The team’s analysis differentiates between systems based on the number of conserved quantities present. However, when multiple conserved quantities are present, the system’s evolution occurs within a multi-dimensional space of generalized Gibbs ensembles. This expanded space offers more routes for the system to reach equilibrium, potentially accelerating the relaxation process. This dimensionality shift is not merely theoretical; it is demonstrably linked to observable changes in the system’s evolution.

Analysis focused on distances between time-evolved states and the steady state, using the normalized Frobenius distance as a metric. The paper states, outlining the scope of the simulations. The researchers also investigated the impact of different numbers of conserved quantities, specifically, $N_{C}$ = 1, $N_{C}$ approximately equal to L, and $N_{C}$ = 2, on the time dependence of energy density.

These simulations revealed that the presence of additional conserved quantities can significantly alter the system’s dynamic behavior. The team’s work builds on previous observations of the quantum Mpemba effect in isolated systems, which were initially observed in symmetry restoration processes within integrable Hamiltonian models. This new research extends those findings to weakly open systems, providing a more general framework for understanding the phenomenon.

👉 More information
🗞 Conserved quantities enable the quantum Mpemba effect in weakly open systems
✍️ Iris Ulčakar, Rustem Sharipov, Gianluca Lagnese and Zala Lenarčič
🧠 DOI: http://link.aps.org/doi/10.1103/xrfl-drcm

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