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

TL;DR

This paper introduces a new basis for third quantization in open quantum systems, transforming the master equation into a multidimensional complex Ornstein-Uhlenbeck process, thus linking third quantization with semiclassical $Q$ representations.

Contribution

It proposes an alternative basis that connects third quantization with semiclassical representations, simplifying the master equation to a well-understood stochastic process.

Findings

The master equation reduces to a multidimensional complex Ornstein-Uhlenbeck process.

The new basis provides a clearer connection between third quantization and semiclassical methods.

Spectral properties of Lindbladians can be analyzed using this approach.

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.