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Contractivity of time-dependent driven-dissipative systems

Lasse H. Wolff, Daniel Malz, Rahul Trivedi

Abstract

In a number of physically relevant contexts, a quantum system interacting with a decohering environment is simultaneously subjected to time-dependent controls and its dynamics is thus described by a time-dependent Lindblad master equation. Of particular interest in such systems is to understand the circumstances in which, despite the ability to apply time-dependent controls, they lose information about their initial state exponentially with time i.e., their dynamics are exponentially contractive. While there exists an extensive framework to study contractivity for time-independent Lindbladians, their time-dependent counterparts are far less well understood. In this paper, we study the contractivity of Lindbladians, which have a fixed dissipator (describing the interaction with an environment), but with a time-dependent driving Hamiltonian. We establish exponential contractivity in the limit of sufficiently small or sufficiently slow drives together with explicit examples showing that, even when the fixed dissipator is exponentially contractive by itself, a sufficiently large or a sufficiently fast Hamiltonian can result in non-contractive dynamics. Furthermore, we provide a number of sufficient conditions on the fixed dissipator that imply exponential contractivity independently of the Hamiltonian. These sufficient conditions allow us to completely characterize Hamiltonian-independent contractivity for unital dissipators and for two-level systems.

Contractivity of time-dependent driven-dissipative systems

Abstract

In a number of physically relevant contexts, a quantum system interacting with a decohering environment is simultaneously subjected to time-dependent controls and its dynamics is thus described by a time-dependent Lindblad master equation. Of particular interest in such systems is to understand the circumstances in which, despite the ability to apply time-dependent controls, they lose information about their initial state exponentially with time i.e., their dynamics are exponentially contractive. While there exists an extensive framework to study contractivity for time-independent Lindbladians, their time-dependent counterparts are far less well understood. In this paper, we study the contractivity of Lindbladians, which have a fixed dissipator (describing the interaction with an environment), but with a time-dependent driving Hamiltonian. We establish exponential contractivity in the limit of sufficiently small or sufficiently slow drives together with explicit examples showing that, even when the fixed dissipator is exponentially contractive by itself, a sufficiently large or a sufficiently fast Hamiltonian can result in non-contractive dynamics. Furthermore, we provide a number of sufficient conditions on the fixed dissipator that imply exponential contractivity independently of the Hamiltonian. These sufficient conditions allow us to completely characterize Hamiltonian-independent contractivity for unital dissipators and for two-level systems.
Paper Structure (35 sections, 25 theorems, 127 equations, 4 figures, 1 table)

This paper contains 35 sections, 25 theorems, 127 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Preliminaries and setup
  4. Some surprises
  5. Contractivity of driven Lindbladians with small or slow Hamiltonians
  6. Sufficient conditions for Hamiltonian-independent contractivity
  7. Applying thm orthog and thm eigen to specific dissipators
  8. Unital dissipators
  9. Low-dimensional Lindbladians
  10. Ladder dissipators
  11. Proofs of the Main results
  12. Notation
  13. Proof of thm small H and thm slow H
  14. Proof of thm orthog vects and related results
  15. Proof of cor off diagonal
  16. ...and 20 more sections

Key Result

Theorem 7

Consider the driven Lindbladian $\mathcal{L}_t = - i[H(t), \ \cdot \ ] + \mathcal{D}$, where the Hamiltonian $H(t)$ has a time-dependent part $V(t)$ and constant part $H_0$, i.e. $H(t) = H_0 + V(t)$. Suppose that the constant Lindbladian $- i[H_0, \ \cdot \ ] + \mathcal{D}$ is exponentially contract

Figures (4)

  • Figure 1: Schematic depiction of the setting considered in this paper: a quantum system with dissipation described by jump operators $L_1, L_2 \dots L_M$ is driven with a time-dependent Hamiltonian $H(t)$.
  • Figure 2: (a) Schematic representation of the dissipator from \ref{['counterex:const_H']}. (b) Time-evolution of the expectation value of $I \otimes \sigma^z$ for different initial pure states $\ket{+,1}$ and $\ket{+,0}$, with and without the Hamiltonian added.
  • Figure 3: (a) Dependence of $-\text{Re}(\lambda_2(\mathcal{L}_t))$ on $\phi(t)$ --- it can be seen that $\forall t, -\text{Re}(\lambda_2(\mathcal{L}_t)) > 0.05$. (b) Time-evolution of expectation value of $I \otimes \sigma^z$ for different initial pure states $\ket{+,1}$ and $\ket{+,0}$ when $\phi(t)=2 \pi (1+2t)^3$.
  • Figure 4: $\mu_2$ as a function of the Hilbert space dimension $d$ for the harmonic oscillator, angular momentum and uniform ladder dissipator. When $\mu_2 < 0$, then the dissipator is Hamiltonian-independently contractive by \ref{['thm:thm eigen condition']}.

Theorems & Definitions (51)

  • Definition 1: Exponentially contractive Lindbladian
  • Definition 2: Irreducible Lindbladians
  • Theorem 7
  • Theorem 8
  • Definition 9
  • Theorem 10
  • Corollary 11
  • Theorem 12
  • Proposition 13
  • Proposition 14
  • ...and 41 more