Loschmidt Echo for Deformed Wigner Matrices

TL;DR

This paper investigates how the Loschmidt echo decays over time when using deformed Wigner matrices as Hamiltonians that are close but not identical, revealing effects of imperfect time reversal in quantum systems.

Contribution

It introduces a novel analysis of Loschmidt echo decay for deformed Wigner matrices, using two-resolvent laws to handle non-commuting Hamiltonians.

Findings

Derived two-resolvent laws for deformed Wigner matrices.

Quantified the decay rate of Loschmidt echo in this setting.

Provided insights into quantum irreversibility with complex Hamiltonians.

Abstract

We consider two Hamiltonians that are close to each other, $H_1 \approx H_2 $, and analyze the time-decay of the corresponding Loschmidt echo $\mathfrak{M}(t) := |\langle \psi_0, \mathrm{e}^{\mathrm{i} t H_2} \mathrm{e}^{-\mathrm{i} t H_1} \psi_0 \rangle|^2$ that expresses the effect of an imperfect time reversal on the initial state $\psi_0$. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools for our results are two-resolvent laws for such $H_1$ and $H_2$.