Third quantization for bosons: symplectic diagonalization, non-Hermitian Hamiltonian, and symmetries

Journal of Physics A Mathematical and Theoretical, vol. 56, pp. 385303

Abstract

Abstract Open quantum systems that interact with a Markovian environment can be described by a Lindblad master equation. The generator of time-translation is given by a Liouvillian superoperator acting on the density matrix of the system. As the Fock space for a single bosonic mode is already infinite-dimensional, the diagonalization of the Liouvillian has to be done on the creation- and annihilation-superoperators, a process called ‘third quantization’. We propose a method to solve the Liouvillian for quadratic systems using a single symplectic transformation. We show that the non-Hermitian effective Hamiltonian of the system, next to incorporating the dynamics of the system, is a tool to analyze its symmetries. As an example, we use the effective Hamiltonian to formulate a -‘symmetry’ of an open system. We describe how the inclusion of source terms allows us to obtain the cumulant generating function for observables such as the photon current.

Authors 2

1. Steven Kim corresponding Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information, RWTH Aachen University, Sommerfeldstrasse 26, Aachen, Nordrhein-Westfalen, 52056, GERMANY

2. Fabian Hassler Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information, RWTH Aachen University, Sommerfeldstrasse 26, Aachen, Nordrhein-Westfalen, 52056, GERMANY