A Description of the Quantum Mpemba Effect using the Steepest-Entropy-Ascent Quantum Thermodynamics Framework

1. Problem Statement

The Quantum Mpemba Effect is a counter-intuitive phenomenon where a quantum system initially in a "hotter" (higher energy or further from equilibrium) state relaxes to a steady state faster than a system starting from a "colder" state. While traditionally observed in classical thermodynamics, recent experiments have demonstrated this in open quantum systems.

Existing theoretical explanations primarily rely on the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, analyzing the spectral properties of the Liouvillian superoperator. Specifically, the effect is often attributed to the suppression of the slowest decay mode (eigenmode) of the Liouvillian when the initial state is orthogonal to the corresponding left eigenmatrix.

However, the GKSL framework is based on the Born-Markov approximation and describes open systems interacting with an environment. There is a need for a first-principles thermodynamic framework that can describe irreversible relaxation, entropy production, and the Mpemba effect from a purely thermodynamic perspective, ensuring consistency with the laws of thermodynamics without relying solely on phenomenological master equations.

2. Methodology

The authors employ the Steepest-Entropy-Ascent Quantum Thermodynamics (SEAQT) framework to model the dynamics of a three-level quantum system (a qutrit) previously studied experimentally by Zhang et al. [7].

A. System Modeling and Dimensionality Reduction

• Experimental Setup: The study models an ion with internal states $|0\rangle$, $|1\rangle$, and $|2\rangle$, coupled to a short-lived auxiliary state $|P\rangle$. The decay of $|P\rangle$ induces dissipation in the system.
• Feshbach Projection: To compare with experimental data (which focuses on the three-level subspace), the authors project the full four-dimensional Hilbert space (including $|P\rangle$) onto a three-dimensional effective Hilbert space. This yields an effective Hamiltonian ($\hat{H}_{eff}$) that captures the essential dynamics and coupling strengths without explicitly simulating the fast-decaying $|P\rangle$ state.
• Initial Conditions: Three distinct initial states are considered:
  1. $|0\rangle\langle 0|$ (Ground state).
  2. $|2\rangle\langle 2|$ (Excited state).
  3. $|s_{ME}\rangle\langle s_{ME}|$ (A specific state optimized to minimize overlap with the slowest decay mode, theoretically inducing the Mpemba effect).

B. The SEAQT Framework

The dynamics are governed by the SEAQT equation of motion:
$$\frac{d\hat{\rho}}{dt} = -\frac{i}{\hbar}[\hat{H}, \hat{\rho}] - \frac{1}{\tau_D}\hat{D}$$

• The first term represents unitary (Hamiltonian) evolution.
• The second term represents irreversible dissipation driven by the steepest entropy ascent principle.
• Dissipative Operator ($\hat{D}$): Constructed to maximize entropy production subject to conservation laws (energy and probability). It depends on the non-equilibrium free energy operator $\hat{f} = \hat{H} - \beta^{-1}\hat{S}$.
• Relaxation Time ($\tau_D$): A critical parameter representing the internal relaxation rate. The authors investigate two scenarios:
  1. $\tau_D$ as a constant.
  2. $\tau_D$ as a time-dependent functional ($\tau_D(t)$), modeled using a logistic (sigmoid) function to capture transitions between different dynamical regimes (fast vs. slow decay).

C. Dynamical Order Parameter

The paper defines a SEAQT-specific order parameter, $\Phi_{SEAQT}(\hat{\rho}) = \sigma_{\hat{F}\hat{F}}$ (the variance of the non-equilibrium free energy).

• If $\Phi_{SEAQT} = 0$, entropy production vanishes initially, leading to accelerated relaxation (the Mpemba effect).
• If $\Phi_{SEAQT} \neq 0$, the system undergoes a slower relaxation with metastable plateaus.

D. Parameter Optimization

Since the effective Hamiltonian parameters and the relaxation function parameters are not fully known, the authors use Machine Learning (Differential Evolution algorithm) to optimize these parameters. The objective is to minimize the Mean Squared Error (MSE) between the model's predicted population dynamics and the experimental data from Zhang et al.

3. Key Contributions

1. Thermodynamic Description of Mpemba Effect: The paper successfully describes the quantum Mpemba effect using the SEAQT framework, providing a thermodynamic alternative to the standard Liouvillian spectral analysis.
2. Definition of Non-Equilibrium Thermodynamic Quantities: The authors derive expressions for non-equilibrium heat capacity and non-equilibrium free energy within the SEAQT formalism. They show that the inverse temperature parameter $\beta$ naturally emerges from the covariance of energy and entropy operators.
3. State-Dependent Relaxation Dynamics: The study demonstrates that the relaxation parameter $\tau_D$ can be effectively modeled as a time-dependent function. This captures the physical reality that dissipation rates change as the system moves from a metastable plateau to the global equilibrium.
4. Validation of Feshbach Projection: The work validates the use of Feshbach projection to reduce a 4-level open system to an effective 3-level system while maintaining accuracy in predicting experimental population dynamics.

4. Results

• Agreement with Experiment: Both the SEAQT model (with constant and time-dependent $\tau_D$) and the standard Lindblad model accurately reproduce the experimental population dynamics for all three initial conditions.
• Observation of Mpemba Effect: The SEAQT model confirms that the initial state $|s_{ME}\rangle$ (where the free energy variance is near zero) exhibits accelerated relaxation compared to the other states.
• Thermodynamic Trajectories:
  • Lindblad Model: The system evolves toward a Non-Equilibrium Steady State (NESS) inside the energy-entropy diagram, reflecting open-system dissipation.
  • SEAQT Model: The system evolves toward a stable thermodynamic equilibrium state (Gibbs state) located on the convex curve of the energy-entropy diagram. Crucially, the SEAQT model preserves the mean energy (first law of thermodynamics) throughout the evolution for the isolated system approximation.
• Entropy and Heat Capacity:
  • Entropy increases monotonically in all cases.
  • The non-equilibrium heat capacity starts near zero and stabilizes at a positive constant value as the system reaches equilibrium.
  • The inverse temperature parameter $\beta$ converges to the thermodynamic inverse temperature at equilibrium.
• Dynamics of $\tau_D$: For the time-dependent case, $\tau_D(t)$ shows that dissipation is strongest at early times and weakens as the system approaches equilibrium, consistent with a system transitioning from a fast decay channel to a slower one.

5. Significance

This work is significant because it bridges the gap between quantum information theory (Liouvillian spectra) and non-equilibrium thermodynamics (entropy production).

• It demonstrates that the Mpemba effect is not merely a spectral artifact of the Liouvillian but can be understood as a consequence of thermodynamic fluctuations (specifically, the suppression of free energy variance).
• It establishes the SEAQT framework as a robust tool for modeling quantum relaxation processes, offering a unified description that satisfies the First and Second laws of thermodynamics by construction.
• The introduction of a state-dependent relaxation time via machine learning provides a more physically realistic model for complex quantum systems where dissipation rates are not constant, offering a pathway to better control and predict quantum thermalization.