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Density Matrix Formalism for PT-Symmetric Non-Hermitian Hamiltonians with the Lindblad Equation

Tommy Ohlsson, Shun Zhou

Abstract

In the presence of Lindblad decoherence, i.e. dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities between the eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for PT-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators $L^{}_j$ are pseudo-Hermitian, namely, $ηL^{}_j η^{-1} = L^\dagger_j$ with $η$ being a linear and positive-definite metric, and respect the PT symmetry as well. We demonstrate that the generalized density matrix $ρ^{}_{\rm G}(t) \equiv ρ(t) η$, instead of the normalized density matrix $ρ^{}_{\rm N}(t) \equiv ρ(t)/{\rm tr}\left[ρ(t)\right]$, should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of PT-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence.

Density Matrix Formalism for PT-Symmetric Non-Hermitian Hamiltonians with the Lindblad Equation

Abstract

In the presence of Lindblad decoherence, i.e. dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities between the eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for PT-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators are pseudo-Hermitian, namely, with being a linear and positive-definite metric, and respect the PT symmetry as well. We demonstrate that the generalized density matrix , instead of the normalized density matrix , should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of PT-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence.

Paper Structure

This paper contains 18 sections, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General Formalism
  3. Two-Level System
  4. Generalized Density Matrix
  5. Normalized Density Matrix
  6. Evolution Equations
  7. Transition Probabilities
  8. Strategy for Calculation of Transition Probabilities
  9. The Lindblad Term with $\hbox{\boldmath$A = 0$}$
  10. General Discussion
  11. Specific Examples with $\hbox{\boldmath$A = 0$}$
  12. The Lindblad Term with $\hbox{\boldmath$B = 0$}$
  13. General Discussion
  14. Specific Examples with $\hbox{\boldmath$B = 0$}$
  15. Summary and Conclusions
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Exact numerical (solid curves) and approximate analytical (dotted curves) transition probabilities as functions of time $t$. Left: Transition probabilities for $+$ and $-$ states. Right: Transition probabilities for $a$ and $b$ states. Parameter values: $r = 0.1$, $s = 0.2$, $\varphi = \pi/3$, $\xi = 0.1$.
  • Figure 7: Exact transition probabilities as functions of time $t$ for the general PT-symmetric non-Hermitian Hamiltonian including the Lindblad term with $B = 0$. Left: Transition probabilities for $+$ and $-$ states. Right: Transition probabilities for $a$ and $b$ states. Parameter values: $r = 0.1$, $s = 0.2$, $\varphi = \pi/3$, $\zeta = 0.1$.