Thermodynamics of the quantum Mpemba effect (2403.16959v3)

Abstract: We investigate the quantum Mpemba effect from the perspective of non-equilibrium quantum thermodynamics by studying relaxation dynamics of quantum systems coupled to a Markovian heat bath, which are described by Davies maps. Starting from a state with coherences in the energy eigenbasis, we demonstrate that an exponential speedup to equilibrium will always occur if the state is transformed to a diagonal state in the energy eigenbasis, provided that the spectral gap of the generator is defined by a complex eigenvalue. When the transformed state has a higher nonequilibrium free energy, we argue using thermodynamic reasoning that this is a \textit{genuine} quantum Mpemba effect. Furthermore, we show how a unitary transformation on an initial state can always be constructed to yield the effect and demonstrate our findings by studying the dynamics of both the non-equilibrium free energy and the irreversible entropy production in single and multi-qubit examples.

• The paper demonstrates that eliminating quantum coherence via unitary transformation accelerates thermalization, marking a genuine quantum Mpemba effect when non-equilibrium free energy is higher.
• The study employs Davies maps to model weak system-bath interactions and uses numerical examples in single and many-body systems, including the transverse-field Ising model.
• The results offer insights for quantum computing and dissipative state engineering by revealing how controlled relaxation dynamics can optimize state preparation.

Thermodynamics of the Quantum Mpemba Effect

The paper examines the quantum Mpemba effect through the lens of non-equilibrium quantum thermodynamics. Utilizing Davies maps, which describe a system's evolution when weakly coupled to a Markovian heat bath, the authors investigate relaxation dynamics when a quantum state is driven to equilibrium. While the classical Mpemba effect—a counterintuitive observation where a hotter system cools faster than a cooler one—has long been studied with classical systems, this work extends the framework to quantum scenarios. More specifically, the research focuses on quantum systems that exhibit a unique relaxation pattern due to quantum coherence effects.

Characterization of the Quantum Mpemba Effect

In the quantum context, the Mpemba effect is examined by considering the role of quantum coherences. The paper begins with states possessing coherence in the energy eigenbasis—i.e., off-diagonal terms that represent elements of superposition. The core finding is that when transformed into a diagonal state (removing coherence), these states achieve an exponential speed-up towards equilibrium, contingent upon the spectral gap of the generator being defined by a complex eigenvalue.

To identify circumstances under which this speed-up represents a genuine Mpemba effect, the paper introduces a thermodynamic perspective based on the non-equilibrium free energy. The quantum Mpemba effect is recognized as genuine when, after a unitary transformation, the state exhibits a higher non-equilibrium free energy relative to the original state. This transient increase signifies that an exponential speed-up has occurred from the modified state to thermal equilibrium faster than from the original state.

Implications and Numerical Illustration

The theoretical framework is corroborated through numerical illustrations in the paper, highlighting single and multiple qubit systems. Applying a unitary transformation can nullify overlaps with the slowest decaying eigenmodes of the Davies generator, directly linking population inversion with enhanced thermalization rates.

For single qubit systems, removing overlap with the lowest eigenmodes directly transforms the system into a state that harmonizes the Mpemba effect by exponentially accelerating relaxation without coherence. Extending this to many-body systems, such as the transverse-field Ising model with open boundary conditions, demonstrates that similar dynamics occur, albeit on longer timescales. The work identifies nuances in how the overlap with eigenmodes evolves, impacting the timing and degree of observable Mpemba effects across different quantum systems.

Theoretical and Practical Implications

The paper chiefly contributes to understanding anomalous thermal relaxation in quantum systems governed by Davies maps. By exploring non-equilibrium free energy as a metric for defining and studying the quantum Mpemba effect, the research strengthens the theoretical connection between classical and quantum thermodynamics. This offers insight relevant for thermodynamics-driven state preparation processes like quantum computing and dissipative quantum state engineering where a precise control over relaxation dynamics is essential.

Speculation on Further Developments

This paper invites future exploration into potential explanations of ultra-fast thermalization in open quantum systems. Future research might probe deeper into efficient state preparation for quantum information processing applications. Furthermore, exploring how coupling strength and environmental interactions influence the Mpemba effect both theoretically and experimentally could yield richer insights into quantum thermodynamics.

In conclusion, the integration of coherent and incoherent quantum states in paper enriches the understanding of quantum thermal dynamics, positioning the quantum Mpemba effect as a fundamental phenomenon with implications that extend beyond classical intuition.