Loschmidt Echo for Deformed Wigner Matrices

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 $\mathfrak{M}(t):=|⟨\ddot{I}{ˆ}_0,{\mathrm{e}}^{\mathrm{i}t{H}_2}{\mathrm{e}}^{-\mathrm{i}t{H}_1}\ddot{I}{ˆ}_0⟩{|}^2$ that expresses the effect of an imperfect time reversal on the initial state $\ddot{I}{ˆ}_0$. 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 $H_2$.

27 pages, 3 figures