Conserved Quantities Enable the Quantum Mpemba Effect in Weakly Open Systems, Accelerating Relaxation to Steady States

The seemingly counterintuitive Mpemba effect, where a hotter system can sometimes freeze faster than a colder one, continues to fascinate scientists and challenge conventional understanding of thermal dynamics. Iris Ulčakar, Rustem Sharipov, and Gianluca Lagnese, along with colleagues at the Jožef Stefan Institute and the University of Ljubljana, now investigate this phenomenon in quantum systems that weakly interact with their environment. Their work reveals that the existence of conserved quantities, properties that remain constant during the system’s evolution, fundamentally determines whether the quantum Mpemba effect can occur. The team demonstrates that when a system possesses multiple conserved quantities, its journey towards a stable state becomes more efficient, potentially explaining how a hotter starting point can lead to faster relaxation, a finding that expands our understanding of non-equilibrium dynamics and the role of conservation laws in quantum systems.

Thermal states and examine when an initially hotter state relaxes to the steady state faster. The research reveals that the number of conserved quantities within the system’s Hamiltonian is critical; the Mpemba effect is possible only when the Hamiltonian possesses multiple conserved quantities or is integrable. This difference arises because the dynamical evolution occurs in spaces of different dimensions. When energy is the only approximately conserved quantity, dissipation confines the dynamics to a single-parameter manifold of thermal states. However, for Hamiltonians with several conserved quantities, the dynamics drift within a multi-dimensional space of generalized Gibbs ensembles.

Fermionic Transformation of the Ising Hamiltonian

This supplemental material provides detailed calculations and explanations supporting the main text’s findings on the quantum Mpemba effect in weakly open systems. It outlines the steps to transform the transverse field Ising Hamiltonian into a form suitable for analysis using free fermionic operators, involving the Jordan-Wigner transformation, Fourier transform, and Bogoliubov transformation. The transformed Hamiltonian is expressed in terms of quasiparticle occupation operators and dispersion, simplifying calculations. The material uses the generalized Gibbs ensemble to approximate the dynamics, parameterizing the reduced density matrix with chemical potentials.

It introduces the concept of the correlation matrix to characterize the reduced density matrix of a Gaussian state and expresses the reduced density matrix in Gaussian form, allowing for efficient distance calculations. It provides a formula for calculating the trace of the product of reduced density matrices and uses these formulas to calculate the Frobenius distance between two reduced density matrices. Applying these formulas to the generalized Gibbs ensemble approximation of the transverse field Ising model simplifies the calculation of distances between states. In essence, the supplemental material provides the mathematical framework and detailed calculations underpinning the main text’s conclusions about the quantum Mpemba effect, demonstrating how to efficiently calculate the dynamics of open quantum systems using free fermionic operators and quantify the distance between different quantum states.

Conserved Quantities Dictate Quantum Cooling Rates

Scientists have achieved a breakthrough in understanding the quantum Mpemba effect, revealing a fundamental principle governing how quickly systems cool under specific conditions. The research focuses on weakly open quantum many-body systems, where dissipation is minimal and a dominant unitary evolution prevails, and demonstrates that the number of conserved quantities within the system’s Hamiltonian is crucial to determining cooling rates. Experiments reveal that systems with only energy as an approximate conserved quantity cannot exhibit the Mpemba effect, while those possessing additional conserved quantities do. The team modeled the dynamics of these systems using both tensor network techniques and free-fermion approaches, allowing for analysis of large system sizes ranging from 80 to 400.

Results demonstrate that the system’s initial state, when prepared in mixed thermal states, evolves differently depending on the Hamiltonian’s symmetries. Specifically, the density matrix of the system decomposes into a dominant component independent of dissipation and a small correction dependent on weak coupling to thermal baths. The researchers parameterized this evolution using Lagrange multipliers, representing each approximately conserved quantity. When the Hamiltonian possesses only one approximate conserved quantity, the system’s evolution is described by a time-dependent Gibbs ensemble. However, with more than one conserved quantity, the evolution shifts to a time-dependent generalized Gibbs ensemble, indicating a more complex dynamic. Measurements confirm that this multi-dimensional evolution within the generalized Gibbs ensemble allows for the possibility of the Mpemba effect, where a hotter initial state relaxes to the steady state faster than a colder one. This work establishes a unifying principle for the quantum Mpemba effect, linking the system’s symmetries to its cooling behavior and providing a foundation for future investigations into non-equilibrium dynamics.

Conserved Quantities Drive Faster Cooling

Researchers have demonstrated a link between the number of conserved quantities within a physical system and the potential for the Mpemba effect, the counterintuitive phenomenon where a hotter system can sometimes cool faster than a colder one. Through numerical simulations of weakly open systems, the team investigated how initial thermal states evolve towards a steady state, focusing on systems with either few or many conserved quantities. Their results indicate that the Mpemba effect is observed in integrable systems, those possessing a large number of conserved quantities, where dynamics occur in a multi-dimensional space of possible states. Conversely, chaotic systems, characterized by only approximate energy conservation, do not exhibit this effect.

The study establishes that the dimensionality of the dynamical evolution plays a crucial role, with systems possessing numerous conserved quantities exhibiting more complex behavior conducive to the Mpemba effect. Specifically, the researchers found that in integrable systems, the distance between the evolving state and the steady state decreases more rapidly for the initially hotter state, leading to faster cooling. They confirmed that the Gibbs and generalized Gibbs ensembles accurately describe the initial stages of this weakly dissipative dynamics. While the crossing times observed remain dependent on system size and the chosen distance metric, a clear Mpemba effect emerges for experimentally relevant, small system sizes.