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

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

URL: http://arxiv.org/abs/2304.02367v2
Date: Thu, 3 Aug 2023 12:07:14 GMT
Title: Third quantization for bosons: symplectic diagonalization, non-Hermitian Hamiltonian, and symmetries
Authors: Steven Kim and Fabian Hassler
Abstract summary: 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 $mathcalPT$-symmetry' of an open system.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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 $\mathcal{L}$ 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 $\mathcal{PT}$-`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.

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