Estimation of equilibration time scales from nested fraction approximations.

Area of Science:

• Quantum mechanics
• Statistical mechanics

Background:

• Estimating equilibration time (τ) for quantum systems is crucial for understanding their dynamics.
• Traditional methods can be computationally intensive and challenging to implement.

Purpose of the Study:

• To develop a practical and accurate method for estimating the equilibration time of quantum observables.
• To demonstrate the mathematical equivalence of the recursive method and Mori formalism for autocorrelation functions.

Main Methods:

• Utilizing the recursive method to iteratively evaluate Lanczos coefficients.
• Employing the Mori formalism by solving coupled differential equations.
• Proposing an approximation scheme based on a few Lanczos coefficients to estimate τ.
• Developing a numerical scheme to assess the precision of the estimation.

Main Results:

• The recursive method and Mori formalism are mathematically equivalent for autocorrelation functions.
• The proposed approximation scheme accurately estimates τ with minimal input (a few Lanczos coefficients).
• The method's precision can be reliably estimated.
• The approach is validated across various relaxation dynamics.

Conclusions:

• The developed method provides a practical and accurate way to quantify the equilibration time of isolated quantum systems.
• This offers a significant advancement in addressing a notoriously difficult problem in quantum dynamics.