Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Loschmidt Echo for Deformed Wigner Matrices

László Erdős, Joscha Henheik, Oleksii Kolupaiev

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 $\mathfrak{M}(t) := |\langle ψ_0, \mathrm{e}^{\mathrm{i} t H_2} \mathrm{e}^{-\mathrm{i} t H_1} ψ_0 \rangle|^2$ that expresses the effect of an imperfect time reversal on the initial state $ψ_0$. 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 $H_2$.

Loschmidt Echo for Deformed Wigner Matrices

Abstract

We consider two Hamiltonians that are close to each other, , and analyze the time-decay of the corresponding Loschmidt echo that expresses the effect of an imperfect time reversal on the initial state . 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 and .

Paper Structure

This paper contains 22 sections, 18 theorems, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Scenario I: Two deformations of a Wigner matrix
  4. Scenario II: Perturbation by a Wigner matrix
  5. Short-time parabolic decay in Scenario I: Proof of Theorem \ref{['thm:echo1']} (i)
  6. Asymptotic decay in Scenario I: Proof of Theorem \ref{['thm:echo1']} (ii)
  7. Step (i): Global law with two deformations
  8. Step (ii): Preliminary bounds on the stability operator and the shift
  9. Step (iii): Contour integration of the deterministic approximation
  10. Stability operator and shift: Proofs for Section \ref{['subsec:stabopshift']}
  11. Bound on the stability operator: Proof of Proposition \ref{['prop:stab']}
  12. Properties of the shift: Proof of Lemmas \ref{['lem:gen_shift']}--\ref{['lem:Im']}
  13. Contour integration: Proof of technical lemmas from Section \ref{['subsec:contint']}
  14. The second line of \ref{['eq:I_final']} is negligible: Proof of Lemma \ref{['lem:2nd_term']}
  15. Cutting tails in the first line of \ref{['eq:I_final']}: Proof of Lemma \ref{['lem:tails']}
  16. ...and 7 more sections

Key Result

Theorem 2.4

Let $W$ be a Wigner matrix satisfying Assumption ass:Wigner, and $D_1, D_2\in \mathbf{C}^{N \times N}$ be bounded, tracelessIf $D_1$ or $D_2$ had a non-zero trace, it could be absorbed by a simple (scalar) energy shift. Hermitian matrices, i.e. $\lVert D_j\rVert\le L$ for some $L > 0$ and $\langle D

Figures (3)

  • Figure 1: Illustrated is the typical behavior of the Loschmidt echo in its three consecutive phases: Short-time parabolic decay, intermediate-time asymptotic decay, and long-time saturation. In both of our main results \ref{['eq:firstthm']}--\ref{['eq:secondthm']}, the decay parameters $\gamma$ and $\Gamma$ generally satisfy $\gamma \sim \Gamma \sim \langle (H_1 - H_2)^2\rangle$; cf. \ref{['eq:gammaGammageneral']}.
  • Figure 2: Schematic behavior of the overlap $\mathfrak{P}_t(s)$ from \ref{['eq:def_P']} for $s\in [0,2t]$. At the midpoint, $s=t$, typically $\mathfrak{P}_t(t) \ll \mathfrak{P}_t(2t)$, which indicates a partial recovery between time $t$ and $2t$ of the original complete overlap at time $s=0$.
  • Figure 3: Sketch of the contours $\gamma_1$ (dashed) and $\gamma_2$ (full) from \ref{['eq:contdec1']}--\ref{['eq:contdec2']}. The union of the spectra of $H_1$ and $H_2$ is indicated in blue.

Theorems & Definitions (30)

  • Remark 2.3: Sufficient condition for Assumption \ref{['ass:M_bound']}
  • Theorem 2.4: Averaged Loschmidt echo with two deformations
  • Corollary 2.5: Averaged Loschmidt echo process
  • proof : Proof of Corollary \ref{['cor:2bumps']}
  • Corollary 2.6: Scrambled averaged Loschmidt echo with two deformations
  • proof : Proof of Corollary \ref{['cor:echo1']}
  • Remark 2.7: Averaging of the Loschmidt echo
  • Theorem 2.10: Loschmidt echo with a single deformation
  • Corollary 2.11: Scrambled Loschmidt echo with a single deformation
  • Proposition 4.1: Average two resolvent global law
  • ...and 20 more