Third quantization for bosons: symplectic diagonalization, non-Hermitian Hamiltonian, and symmetries
arXiv:2304.02367 · doi:10.1088/1751-8121/acf177
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.
arXiv comment18 pages, 1 figure
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