Master equation

About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.

Preservation of positivity by dynamical coarse graining

Abstract: We compare different quantum master equations for the time evolution of the reduced density matrix. The widely applied secular approximation (rotating wave approximation) applied in combination with the Born-Markov approximation generates a Lindblad-type master equation ensuring for completely positive and stable evolution and is typically well applicable for optical baths. For phonon baths however, the secular approximation is expected to be invalid. The usual Markovian master equation does not generally preserve positivity of the density matrix. As a solution we propose a coarse-graining approach with a dynamically adapted coarse-graining time scale. For some simple examples we demonstrate that this preserves the accuracy of the integro-differential Born equation. For large times we analytically show that the secular approximation master equation is recovered. The method can in principle be extended to systems with a dynamically changing system Hamiltonian, which is of special interest for adiabatic quantum computation. We give some numerical examples for the spin-boson model of cases where a spin system thermalizes rapidly, and other examples where thermalization is not reached.

Determination of the spectral gap for Kac's master equation and related stochastic evolution

Abstract: We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model due to Mark Kac of energy conserving collisions with one dimensional velocities. It is also sufficiently robust to provide qualitatively sharp bounds also in the case of more physically realistic momentum and energy conserving collisions in three dimensions, as well as a range of related models.

Out-of-equilibrium structures in strongly interacting Rydberg gases with dissipation

Abstract: The nonequilibrium dynamics of a gas of cold atoms in which Rydberg states are off-resonantly excited is studied in the presence of noise. The interplay between interaction and off-resonant excitation leads to an initial dynamics where aggregates of excited Rydberg atoms slowly nucleate and grow, eventually reaching long-lived metastable arrangements which then relax further on much longer time scales. This growth dynamics is governed by an effective Master equation which permits a transparent and largely analytical understanding of the underlying physics. By means of extensive numerical simulations we study the many-body dynamics and the correlations of the resulting nonequilibrium states in various dimensions. Our results provide insight into the dynamical richness of strongly interacting Rydberg gases in noisy environments, and highlight the usefulness of these kinds of systems for the exploration of soft-matter-type collective behavior.

Coherent control of open quantum dynamical systems

Abstract: A systematic analysis of the behavior of the quantum Markovian master equation driven by coherent control fields is proposed. Its irreversible character is formalized using control-theoretic notions and the sets of states that can be reached via coherent controls are described. The analysis suggests to what extent (and how) it is possible to counteract the effect of dissipation.

Fourth-order quantum master equation and its Markovian bath limit

Abstract: Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.