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Journal of Physics A Mathematical and General

Exact solution and Berry phase for the dissipative two-mode optical system

Bang‐Pin Hou, Shun-Jin Wang, Wei-Can Yu, W L Sun

Published Aug 5, 2004• 1 citation
Publisher

Abstract

We have explored the dynamical algebraic structure of the dissipative two-mode system. On the basis of the algebraic structure, we have obtained the exact eigensolutions of the Liouvillian superoperator of its master equation by using an algebraic dynamical method. Then we investigate the Berry phase of the system within the density-matrix formalism.

DOI
10.1088/0305-4470/37/33/013

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We have explored the dynamical algebraic structure of the dissipative two-mode system. On the basis of the algebraic structure, we have obtained the exact eigensolutions of the Liouvillian superoperator of its master equation by using an algebraic dynamical method. Then we investigate the Berry phase of the system within the density-matrix formalism.
A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.
We examine the role of Berry's geometrical phase [Proc. R. Soc. London, Ser. A 392, 45 (1984)] in the coherent excitation of atoms. The dynamics of a two-level atom prepared in a superposition of its eigenstates will be modified in a nontrivial fashion by the Berry phase. We extend Berry's formalism to treat the Liouville equation for the density matrix. This extension allows us to discuss statistical mixtures and to incorporate Berry's phase within the well-known Bloch-vector description of optical resonance. Furthermore, our density-matrix formalism provides the means of investigating Berry's phase in open, dissipative systems.
We introduce algebraic approach for superoperators that might be useful tool for investigation of quantum (bosonic) multi-mode systems and its dynamics. In order to demonstrate potential of proposed method we consider multi-mode Liouvillian superoperator that describes relaxation dynamics of a quantum system (including thermalization and intermode coupling). Considered algebraic structure of superoperators that form Liouvillian and their algebraic properties allows us to diagonilize multi-mode Liouvillian to find its spectrum. Also it allows to derive linear by mean number of thermal (environmental) photons approximation for time-evolution superoperator that keeps amount of considered dimensions in Fock space finite (assuming initial amount of dimensions finite) that might be helpful regarding entanglement dynamics problems. Conjugate Liouvillian is considered as well in order to perform analysis in Heisenberg picture, it can be implemented for multi-time correlation functions derivation.