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Decoherence is an echo of Anderson localization in open quantum systems

Frederik Ravn Klausen, Simone Warzel

Abstract

We study the time evolution of single-particle quantum states described by a Lindblad master equation with local terms. By means of a geometric resolvent equation derived for Lindblad generators, we establish a finite-volume-type criterion for the decay of the off-diagonal matrix elements in the position basis of the time-evolved or steady states. This criterion is shown to yield exponential decay for systems where the non-hermitian evolution is either gapped or strongly disordered. The gap exists for example whenever any level of local dephasing is present in the system. The result in the disordered case can be viewed as an extension of Anderson localization to open quantum systems.

Decoherence is an echo of Anderson localization in open quantum systems

Abstract

We study the time evolution of single-particle quantum states described by a Lindblad master equation with local terms. By means of a geometric resolvent equation derived for Lindblad generators, we establish a finite-volume-type criterion for the decay of the off-diagonal matrix elements in the position basis of the time-evolved or steady states. This criterion is shown to yield exponential decay for systems where the non-hermitian evolution is either gapped or strongly disordered. The gap exists for example whenever any level of local dephasing is present in the system. The result in the disordered case can be viewed as an extension of Anderson localization to open quantum systems.
Paper Structure (15 sections, 12 theorems, 79 equations, 2 figures)

This paper contains 15 sections, 12 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Local Lindbladians and their Abel-averaged dynamics
  3. Abel-averages and resolvents
  4. Examples: dephasing, dissipative engineering, and disorder
  5. Geometric resolvent equation and criterion for the decay of coherence
  6. Comments on the finite-volume criterion
  7. Applications
  8. Spectral gap in the non-hermitian Hamiltonian
  9. Modifying boundary conditions: examples
  10. Anderson localization for non-hermitian Hamiltonians
  11. Concluding remarks
  12. Properties of local Lindbladians and related operators
  13. Resolvent estimates for non-normal operators
  14. Psuedospectra of maximally dissipative operators
  15. Combes-Thomas for non-normal operators

Key Result

Proposition 2.2

Under Assumption assumption:locality, $\mathcal{L}$ is the generator of a norm-continuous, positivity and trace preserving contraction semigroup $\exp(t \mathcal{L})$ on the Banach space $\mathcal{S}_1(\ell^2(\Lambda))$ of trace-class operators.

Figures (2)

  • Figure 1: Sketch of the contour $\Gamma_\varepsilon$ (dashed) within $\left\{ z \in \mathbb{C} \mid \Re z < \varepsilon \right\}\backslash\sigma(D_{\Lambda_k} )$ which winds anticlockwise once around $\sigma(D_{\Lambda_k})$. By construction $\Gamma_\varepsilon$ is disjoint from and does not wind around $\sigma(- D_{\Lambda_j}^* + \varepsilon)$.
  • Figure 2: The points $x$ and $y$ and the construction of $\Lambda_x, \Lambda_y$ and $\Lambda_1 =\Lambda_x \uplus \Lambda_y$ and $\Lambda_2 = \Lambda \backslash \Lambda_1$. Any term corresponding to $Z \in \partial$ acts non-trivially within a safety corridor of distance $R$ from the two boundaries of $\Lambda_x, \Lambda_y$. The terms corresponding to $Z \in \partial_{\Lambda_1}$ act nontrivially only in the shaded inner boundary of $\Lambda_1$.

Theorems & Definitions (28)

  • Proposition 2.2: Quantum dynamical semigroup of contractions
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7: Boundary Lindbladians
  • Lemma 2.8: Integral representation
  • proof
  • Definition 2.9
  • Proposition 2.10: Coherence bound in terms of admissible kernel
  • ...and 18 more