Master Equations For Open Systems Research Articles

Gauge-independent emission spectra and quantum correlations in the ultrastrong coupling regime of open system cavity-QED

Abstract A quantum dipole interacting with an optical cavity is one of the key models in cavity quantum electrodynamics (cavity-QED). To treat this system theoretically, the typical approach is to truncate the dipole to two levels. However, it has been shown that in the ultrastrong-coupling regime, this truncation naively destroys gauge invariance. By truncating in a manner consistent with the gauge principle, we introduce master equations for open systems to compute gauge-invariant emission spectra, photon flux rates, and quantum correlation functions which show significant disagreement with previous results obtained using the standard quantum Rabi model. Explicit examples are shown using both the dipole gauge and the Coulomb gauge.

Equivalence of stochastic formulations and master equations for open systems

Several stochastic formulations have been developed to investigate the quantum dynamics of open systems. We show the equivalence between the Liouville equation based on the stochastic decoupling of the system-bath interaction and the non-Markovian quantum state diffusion approach. The procedure we use provides not only a means of calculating the unknown functional derivative in the latter, but also a possibility of developing more efficient theoretical schemes. Further, we demonstrate that the stochastic decoupling approach can feasibly be used to derive master equations for the spontaneously decaying multistate systems, which are difficult to obtain by other available methods.

Memory Effects and Nonequilibrium Correlations in the Dynamics of Open Quantum Systems

We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.

Decoherence-induced surface hopping

A simple surface hopping method for nonadiabatic molecular dynamics is developed. The method derives from a stochastic modeling of the time-dependent Schrödinger and master equations for open systems and accounts simultaneously for quantum mechanical branching in the otherwise classical (nuclear) degrees of freedom and loss of coherence within the quantum (electronic) subsystem due to coupling to nuclei. Electronic dynamics in the Hilbert space takes the form of a unitary evolution, intermittent with stochastic decoherence events that are manifested as a localization toward (adiabatic) basis states. Classical particles evolve along a single potential energy surface and can switch surfaces only at the decoherence events. Thus, decoherence provides physical justification of surface hopping, obviating the need for ad hoc surface hopping rules. The method is tested with model problems, showing good agreement with the exact quantum mechanical results and providing an improvement over the most popular surface hopping technique. The method is implemented within real-time time-dependent density functional theory formulated in the Kohn-Sham representation and is applied to carbon nanotubes and graphene nanoribbons. The calculated time scales of non-radiative quenching of luminescence in these systems agree with the experimental data and earlier calculations.

Master-equation approach to muonium depolarization in solids

The theory of master equations for open systems is applied to describe the spin relaxation of muonium in solids. An equation of motion for the density matrix of muonium is derived which takes into account the interaction of the bound electron with the host. The calculated muon relaxation rates depend strongly on the applied magnetic field. The range of validity of the Markov approximation is investigated and the results are compared with those of previous theories.

On the structure of the generalized master equation for open systems

A quantum open system S, interacting with a reservoir R is considered. The kernel of the generalized master equations (G.M.E.'s), usually deduced to describe the dynamics of S, is expressed in terms of time correlation functions of the bath operators involved in the interaction between S and R, calculated on a suitable equilibrium state of R. Such equilibrium state “enters” into the kernel in a rather artificial way, e.g. via the projection operator used to deduce the G.M.E. In this paper the structure of the kernel of the G.M.E. for open systems is justified from first principles, showing that the presence of the equilibrium state of R in the kernel arises in quite a natural way. The thermodynamic limit of R plays an essential role in connection with this result.

Interference of coherent and incoherent interactions in the Master Equation for open systems

The limits of validity of a well known type of Master Equation for open systems are discussed. Using Zwanzig's projector formalism we derive a “Generalized Master Equation” which describes interference processes between coherent and incoherent interactions. The conditions under which this equation reduces to the known ones is given and discussed for laser systems.