Mesoscopic fluctuations of the Loschmidt echo

Area of Science:

• Quantum mechanics
• Chaos theory
• Wave packet dynamics

Background:

• Investigating quantum system dynamics and fidelity is crucial for understanding quantum information processing.
• Time reversal of quantum dynamics with perturbed Hamiltonians presents challenges in fidelity reconstruction.
• Semiclassical analysis offers insights into the interplay between classical and quantum behaviors.

Purpose of the Study:

• To analyze the time-dependent variance of wave packet reconstruction fidelity under a perturbed Hamiltonian.
• To explore the semiclassical regime of perturbation and its effect on fidelity.
• To develop a method for extracting the classical Lyapunov exponent (lambda) from fidelity decay.

Main Methods:

• Theoretical investigation of time-dependent variance in quantum fidelity.
• Application of semiclassical approximation to a perturbed Hamiltonian.
• Derivation of fidelity decay terms involving classical (lambda) and quantum (Gamma) exponents.
• Numerical simulations to validate theoretical predictions.

Main Results:

• Wave packet fidelity variance initially increases algebraically up to a critical time t(c).
• Beyond t(c), the variance exhibits exponential decay.
• The decay is characterized by classical (exp[-2lambdat]), quantum (2Planck's(eff) exp[-Gamma t]), and mixed (2 exp[-(Gamma+lambda)t]) terms.
• The method allows extraction of the classical Lyapunov exponent (lambda) over a broader parameter range than average fidelity analysis.

Conclusions:

• The study provides a theoretical framework for understanding fidelity decay in time-reversed quantum dynamics.
• The derived decay formula facilitates more robust extraction of the classical Lyapunov exponent (lambda).
• Numerical simulations confirm the accuracy of the theoretical predictions, validating the approach.