We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the characteristics of both of these approaches. This allows us to assess the differences between them as well as to clarify which observed features come from the Lindblad or the non-Hermitian part, when it comes to experiment. Using a generic two-mode single-atom laser system as a practical example, we analytically solve the dynamics of the normalized density matrix operator. We study the two-level model in a number of cases (depending on parameters and types of dynamics), compute different observables and study their physical properties. It turns out that one is able not only to describe the different types of damping in dissipative quantum optical systems but also to mimic the undamped anharmonic oscillatory phenomena which happen in quantum systems with more than two levels (while staying within the framework of the analytically simple two-mode approximation).Comment: 13 pages, 9 figures. The article is based on the talks given at the conferences "12th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics" (02-06 July, 2013, Koc University, Istanbul, Turkey) and "Quantum Information Processing, Communication and Control 2" (25-29 November, 2013, KwaZulu-Natal, South Africa

In 1980 Andersen introduced the use of “extended system” as a means of exploring by molecular dynamics simulation the phase space of a physical model according to a desired ensemble distribution different from the standard microcanonical function. Following his original work on constant pressure-constant enthalpy a large number of different equations of motion, not directly derivable from a Hamiltonian, have been proposed in recent years, the most notable of which is the so-called Nosé-Hoover formulation for “canonical” molecular dynamics simulation. Using a generalization of the symplectic form of the Hamilton equations of motion we show here that there is a unique general structure that underlies most, if not all the equations of motion for “extended systems.” We establish a unifying formalism that allows one to identify and separately control the conserved quantity, usually known as the “total energy” of the system, and the phase-space compressibility. Moreover, we define a standard procedure to construct conservative non-Hamiltonian flows that sample the phase space according to a chosen distribution function [Tuckerman et al., Europhys. Lett. 45, 149 (1999)]. To illustrate the formalism we derive new equations of motion for two example cases. First we modify the equations of motion of the Nosé-Hoover thermostat applied to a one-dimensional harmonic oscillator, and we show how to overcome the ergodicity problem and obtain a canonical sampling of phase space without making recourse to additional degrees of freedom. Finally we recast an idea recently put forward by Marchi and Ballone [J. Chem. Phys. 110, 3697 (1999)] and derive a dynamical scheme for sampling phase space with arbitrary statistical biases, showing as an explicit application a demixing transition in a simple Lennard-Jones binary mixture.

We introduce a formalism for time-dependent correlation functions for systems whose evolutions are governed by non-Hermitian Hamiltonians of general type. It turns out that one can define two different types of time correlation functions. Both these definitions seem to be physically consistent while becoming equivalent only in certain cases. Moreover, when autocorrelation functions are considered, one can introduce another function defined as the relative difference between the two definitions. We conjecture that such a function can be used to assess the positive semidefiniteness of the density operator without computing its eigenvalues. We illustrate these points by studying analytically a number of models with two energy levels.