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Frontiers of Physics

New application of non-Hermitian Hamiltonian operator in solving master equation for laser process

Junhua Chen, Hong-Yi Fan

Published Dec 1, 2012• 1 citation
Publisher

Abstract

We find that laser process can be equivalently described by a symplectic evolution in the context of thermo field dynamics, and the corresponding coherent state evolution for the corresponding master equation is recognized. More interestingly, this embodies a new application of non-Hermitian Hamiltonian operator which can well expose the entanglement between the system and its environment.

DOI
10.1007/s11467-012-0274-8

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We find that laser process can be equivalently described by a symplectic evolution in the context of thermo field dynamics, and the corresponding coherent state evolution for the corresponding master equation is recognized. More interestingly, this embodies a new application of non-Hermitian Hamiltonian operator which can well expose the entanglement between the system and its environment.
We investigate the time evolution of a quantum state in the laser channel with cavity gain coefficient g and loss coefficient κ by virtue of the thermo entangled state representation. This is a new approach for solving master equations. We exactly calculate the evolving result of the photon-added coherent state (PACS) governed by the master equation describing the laser process, and the normal ordered form of the density operator in the laser channel is derived, based on which we calculate the Wigner function analytically and graphically. The corresponding law of the PACS evolution in the laser channel is evaluated.
By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation ?|, which can arrange master equations of density operators ?(t) in quantum statistics as state-vector evolution equations due to the elegant properties of ?|. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of the decay constant ? we find that the matrix element of ?(t) at time t in ?| representation is proportional to that of the initial ?0 in the decayed entangled state ?e??t| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ? = ?(d2?/?)?|?D(?), which is different from all the previous known representations.
We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach.The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems.We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position.Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space.We present analytical solutions in L 2 (R) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to a GKSL master equation for the full density matrix.