Reformulating third quantization: identifying dissipative symmetries, connections to phase-space, and links to Keldysh field theory

Evolving an arbitrary initial state of a system described by a Markovian Lindblad master equation requires finding the full Liouvillian eigenvalues and eigenvectors. Recently, Prosen [1] and Prosen and Seligman [CITE] developed third quantization, a technique which allows one to diagonalize the Lindbladian of quadratic fermionic or bosons systems linearly-coupled to a set of baths. However, it is not immediately clear how this approach is connected to other more standard methods.In this work we reformulate third quantization by demonstrating that it is naturally related to other well-known open quantum system formalisms and tools. We first show that all such models exhibit a dissipative symmetry that, once used, allows for a simple diagonalization procedure. We then demonstrate how the Wigner quasi-probability function and Keldysh field theory emerge in our framework. Our reformulation renders third quantization an overall more powerful tool in the study of open quantum systems.

[1] T. Prosen, Third quantization: a general method to solve master equations for quadratic open Fermi systems, New J. Phys. 10, 043026 (2008)

[2] T. Prosen and T. H. Seligman, Quantization over boson operator spaces, J. Phys. A: Math. Theor. 43, 392004 (2010)