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$\mathcal{PT-}$Symmetric Open Quantum Systems: Information Theoretic Facets

Baibhab Bose, Devvrat Tiwari, Subhashish Banerjee

Abstract

The theory of an $η$-pseudo Hermitian Hamiltonian with $\mathcal{PT}$ symmetry is reviewed and extended to include open system dynamics. A first-principles derivation of the generalized Gorini-Kossakowski-Sudarshan-Lindblad master equation appropriate for a $\mathcal{PT}-$symmetric Hamiltonian is presented. Inspired by a simple light matter interaction open system model, information theoretic quantities like a non-Markovian witness and fidelity are calculated for the $\mathcal{PT-}$symmetric Hamiltonian, and the results are compared with their corresponding Hermitian counterparts. The nature of entanglement between two $\mathcal{PT-}$symmetric and Hermitian open quantum systems is calculated, and the contrast observed.

$\mathcal{PT-}$Symmetric Open Quantum Systems: Information Theoretic Facets

Abstract

The theory of an -pseudo Hermitian Hamiltonian with symmetry is reviewed and extended to include open system dynamics. A first-principles derivation of the generalized Gorini-Kossakowski-Sudarshan-Lindblad master equation appropriate for a symmetric Hamiltonian is presented. Inspired by a simple light matter interaction open system model, information theoretic quantities like a non-Markovian witness and fidelity are calculated for the symmetric Hamiltonian, and the results are compared with their corresponding Hermitian counterparts. The nature of entanglement between two symmetric and Hermitian open quantum systems is calculated, and the contrast observed.

Paper Structure

This paper contains 13 sections, 96 equations, 6 figures.

Table of Contents

  1. Introduction
  2. $\eta$-pseudo Hermiticity of a $\mathcal{PT-}$symmetric Hamiltonian
  3. Dynamics of an open system for a $\mathcal{PT-}$symmetric Hamiltonian
  4. Information-theoretic study of a non-Hermitian open system: comparison with its Hermitian counterpart
  5. The BLP measure
  6. Fidelity
  7. Entanglement between two $\mathcal{PT-}$symmetric open systems
  8. Conclusions
  9. The left eigenvectors
  10. The Hermitian Hamiltonian
  11. The $\eta$-adjoint of the unitary operator $U^{\ddagger}$
  12. Derivation of generalized GKSL master equation for a $\mathcal{PT-}$symmetric system
  13. Redefining Pauli operators in the bi-orthonormal basis

Figures (6)

  • Figure 1: A diagrammatic representation of the open system scheme we apply for a $\mathcal{PT-}$symmetric Hamiltonian.
  • Figure 2: The trace distance is plotted in (a) for the dynamics of $\mathcal{PT-}$symmetric system Hamiltonian $H$ and (b) for the dynamics of Hermitian $H_0$. The two initial states in (a) are the ground state $\rho_{\mathcal{G}}^-$ and the excited state $\rho_{\mathcal{G}}^+$, whereas in (b) the initial states are the ground and the excited states of a two-level Hermitian spin system. The other parameters are $T=10$, $\omega_c=2$, dimension of bath = 10.
  • Figure 3: Variation of the fidelity $F(t)$ with different values of coupling constants $g$ is shown for the dynamics of $\mathcal{PT-}$symmetric $H$ in (a) and for the dynamics of Hermitian $H_0$ in (b). Other parameters are $T=10$, $\omega_c=\omega_0 = 2$.
  • Figure 4: Variation of the fidelity $F(t)$ with different values of the bath frequency $\omega_c$ is shown for the dynamics of $\mathcal{PT-}$symmetric $H$ in (a) and for the dynamics of Hermitian $H_0$ in (b). Other parameters are $T=10$, $g=0.5$, $\omega_0 = 2$.
  • Figure 5: Variation of the fidelity $F(t)$ with different values of temperature of the bath $T$ is shown for the dynamics of $\mathcal{PT-}$symmetric $H$ in (a) and for the dynamics of Hermitian $H_0$ in (b). Other parameters are $g=0.5$, $\omega_c=\omega_0 = 2$.
  • ...and 1 more figures