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Hybrid quantum-classical dynamics with stationary thermal states

Adrián A. Budini

Abstract

Quantum and classical systems can consistently be coupled via non-unitary time-irreversible mechanisms. In this paper we characterize which kind of corresponding dynamics converge in the stationary regime to a thermal hybrid state, that is, a density matrix that maximizes the hybrid arrangement entropy under the constraints of a canonical ensemble. Introducing a detailed balance condition, it is found that a specific subclass of hybrid Lindblad equations fulfill the demanded requirement. The main theoretical results are exemplified through a set of specific examples that in addition lighten how the thermal state of each subsystem in isolation is affected by their mutual coupling. In particular, a Gaussian thermal state could become a bimodal distribution when increasing the interaction strength of a classical subsystem with a quantum two-level subsystem.

Hybrid quantum-classical dynamics with stationary thermal states

Abstract

Quantum and classical systems can consistently be coupled via non-unitary time-irreversible mechanisms. In this paper we characterize which kind of corresponding dynamics converge in the stationary regime to a thermal hybrid state, that is, a density matrix that maximizes the hybrid arrangement entropy under the constraints of a canonical ensemble. Introducing a detailed balance condition, it is found that a specific subclass of hybrid Lindblad equations fulfill the demanded requirement. The main theoretical results are exemplified through a set of specific examples that in addition lighten how the thermal state of each subsystem in isolation is affected by their mutual coupling. In particular, a Gaussian thermal state could become a bimodal distribution when increasing the interaction strength of a classical subsystem with a quantum two-level subsystem.

Paper Structure

This paper contains 18 sections, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Entropy of hybrid quantum-classical states
  3. Hybrid thermal states
  4. Dynamics that lead to stationary hybrid thermal states
  5. Density matrix evolution with stationary thermal states
  6. Hybrid thermal evolutions
  7. Examples
  8. Dichotomic classical degrees of freedom
  9. Minimal number of mechanisms to achieve thermality
  10. A particular example of hybrid thermal state
  11. Lattice model
  12. Stationary state in the continuous (smooth) limit
  13. Quantum Fokker-Planck-like equation
  14. Summary and conclusions
  15. Derivation of the thermal hybrid evolution
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Energy levels and coupling mechanisms associated to the evolution (\ref{['GeneralTLS']}). Each letter in the squares corresponds to each term in this equation (see text).
  • Figure 2: Probability density $w(x)$ [Eq. (\ref{['PesoX']})] of the classical subsystem as a function of the coordinate $x.$ The characteristic parameters are $\beta \delta E=0.01$ and $\beta \hbar \omega _{0}=0.$ The value of $\hbar \delta \omega /\delta E$ is indicated in each plot. In all cases, the dotted lines correspond to the uncorrelated case where $w(x)=G_{\mathrm{th}}(x)$ [Eq. (\ref{['Gauss']})].