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Third-quantized master equations as a classical Ornstein-Uhlenbeck process

Léonce Dupays

TL;DR

This paper builds a bridge between third quantization and semiclassical phase-space descriptions by introducing a new superoperator coherent-state basis that projects the Lindblad master equation onto the $Q$ representation. The resulting dynamics form a multidimensional complex Ornstein-Uhlenbeck process, allowing the drift and diffusion matrices to encode thermodynamic properties and to determine the nonequilibrium steady state via a Lyapunov equation. The authors establish a full quantum-classical correspondence with the Prosen-Seligman basis for the $P$ representation and illustrate the formalism with a squeezed-thermal bath, showing consistent OU structure across representations. The framework opens avenues for classical stochastic simulations of open quantum systems and for extending entropy-based analyses (e.g., Wehrl entropy) within a quantum thermodynamics context.

Abstract

Third quantization is used in open quantum systems to construct a superoperator basis in which quadratic Lindbladians can be turned into a normal form. From it follows the spectral properties of the Lindbladian, including eigenvalues and eigenvectors. However, the connection between third quantization and the semiclassical representations usually employed to obtain the dynamics of open quantum systems remains opaque. We introduce an alternative basis for third quantization that bridges this gap between third quantization and the $Q$ representation by projecting the master equation onto a superoperator coherent-state basis. The equation of motion reduces to a multidimensional complex Ornstein-Uhlenbeck process.

Third-quantized master equations as a classical Ornstein-Uhlenbeck process

TL;DR

This paper builds a bridge between third quantization and semiclassical phase-space descriptions by introducing a new superoperator coherent-state basis that projects the Lindblad master equation onto the representation. The resulting dynamics form a multidimensional complex Ornstein-Uhlenbeck process, allowing the drift and diffusion matrices to encode thermodynamic properties and to determine the nonequilibrium steady state via a Lyapunov equation. The authors establish a full quantum-classical correspondence with the Prosen-Seligman basis for the representation and illustrate the formalism with a squeezed-thermal bath, showing consistent OU structure across representations. The framework opens avenues for classical stochastic simulations of open quantum systems and for extending entropy-based analyses (e.g., Wehrl entropy) within a quantum thermodynamics context.

Abstract

Third quantization is used in open quantum systems to construct a superoperator basis in which quadratic Lindbladians can be turned into a normal form. From it follows the spectral properties of the Lindbladian, including eigenvalues and eigenvectors. However, the connection between third quantization and the semiclassical representations usually employed to obtain the dynamics of open quantum systems remains opaque. We introduce an alternative basis for third quantization that bridges this gap between third quantization and the representation by projecting the master equation onto a superoperator coherent-state basis. The equation of motion reduces to a multidimensional complex Ornstein-Uhlenbeck process.
Paper Structure (21 sections, 163 equations, 1 figure, 3 tables)

This paper contains 21 sections, 163 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Normal form of the Ornstein-Uhlenbeck process
  3. Third quantized basis
  4. Coherent states of the third quantized basis
  5. The $Q$ representation and its characteristic function in the superoperator coherent state basis
  6. Q representation of the eigenvectors
  7. Equation of motion of the covariance matrix
  8. Evolution of the $Q$ representation
  9. Classical-Quantum correspondence in the Prosen-Seligman basis
  10. Squeezed master equation in the three third-quantized bases and their equivalent quasi-probability distributions
  11. Conclusion
  12. Normal form solution
  13. Normal form solution for the adjoint operator
  14. Details on the derivation of the eigenstates of the Ornstein-Uhlenbeck process
  15. Derivation of the Q representation eigenstates
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Different choices of third quantization basis are possible. Introducing a superoperator coherent-state basis allows us to project the third-quantized master equation into its quasiprobability distribution. We introduce a third-quantization basis to obtain the equation of motion for the $Q$ representation. Alternatively, using the basis developed by Prosen and Seligman Prosen_2010 allows us to determine the $P$ representation equation of motion, while the basis by McDonald and Clerk allows us to determine the Wigner function representation McDonald2023. Projecting the dynamical equation onto the quasiprobability distribution provides a clear physical interpretation in terms of an Ornstein-Uhlenbeck process.