596,526 research outputs found

### Measurement master equation

We derive a master equation describing the evolution of a quantum system
subjected to a sequence of observations. These measurements occur randomly at a
given rate and can be of a very general form. As an example, we analyse the
effects of these measurements on the evolution of a two-level atom driven by an
electromagnetic field. For the associated quantum trajectories we find Rabi
oscillations, Zeno-effect type behaviour and random telegraph evolution spawned
by mini quantum jumps as we change the rates and strengths of measurement.Comment: 14 pages and 8 figures, Optics Communications in pres

### Secular non-secular master equation

Redfield non-secular master equation governing relaxation of a spin in weak
interaction with a thermal bath is studied. Using the fact that the relaxation
follows the exponential law, we prove that in most cases the semi-secular
approximation is sufficient to find the system relaxation rate. Based on this,
a "secular" form of the non-secular master equation is for the first time
developed which correctly set up one of most fundamental equations in
relaxation investigation. This key secular form allows us to derive a general
formula of the phonon-induced quantum tunneling rate which is valid for the
entire range of temperature regardless of the basis. In incoherent tunneling
regime and localized basis, this formula reduces to the ubiquitous incoherent
tunneling rate. Meanwhile, in eigenstates basis, this tunneling rate is
demonstrated to be equal to zero. From this secular form, we end the
controversy surrounding the selection of basis for the secular approximation by
figuring out the conditions for using this approximation in localized and
eigenstates basis. Particularly, secular approximation in localized basis is
justified in the regime of high temperature and small tunnel splittings. In
contrast, a large ground doublet's tunnel splitting is required for the secular
approximation in eigenstates basis. With these findings, this research lays a
sound foundation for any treatments of the spin-phonon relaxation under any
conditions provided that the non-secular master equation is relevant.Comment: 9 pages, 0 figure

### Derivation of exact master equation with stochastic description: Dissipative harmonic oscillator

A systematic procedure for deriving the master equation of a dissipative
system is reported in the framework of stochastic description. For the
Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and
elementary derivation of the bath-induced stochastic field is presented. The
dynamics of the system is thereby fully described by a stochastic differential
equation and the desired master equation would be acquired with statistical
averaging. It is shown that the existence of a closed-form master equation
depends on the specificity of the system as well as the feature of the
dissipation characterized by the spectral density function. For a dissipative
harmonic oscillator it is observed that the correlation between the stochastic
field due to the bath and the system can be decoupled and the master equation
naturally comes out. Such an equation possesses the Lindblad form in which time
dependent coefficients are determined by a set of integral equations. It is
proved that the obtained master equation is equivalent to the well-known
Hu-Paz-Zhang equation based on the path integral technique. The procedure is
also used to obtain the master equation of a dissipative harmonic oscillator in
time-dependent fields.Comment: 24page

### Stochastic simulation of catalytic surface reactions in the fast diffusion limit

The master equation of a lattice gas reaction tracks the probability of visiting all spatial configurations. The large number of unique spatial configurations on a lattice renders master equation simulations infeasible for even small lattices. In this work, a reduced master equation is derived for the probability distribution of the coverages in the infinite diffusion limit. This derivation justifies the widely used assumption that the adlayer is in equilibrium for the current coverages and temperature when all reactants are highly mobile. Given the reduced master equation, two novel and efficient simulation methods of lattice gas reactions in the infinite diffusion limit are derived. The first method involves solving the reduced master equation directly for small lattices, which is intractable in configuration space. The second method involves reducing the master equation further in the large lattice limit to a set of differential equations that tracks only the species coverages. Solution of the reduced master equation and differential equations requires information that can be obtained through short, diffusion-only kinetic Monte Carlo simulation runs at each coverage. These simulations need to be run only once because the data can be stored and used for simulations with any set of kinetic parameters, gas-phase concentrations, and initial conditions. An idealized CO oxidation reaction mechanism with strong lateral interactions is used as an example system for demonstrating the reduced master equation and deterministic simulation techniques

### Time-convolutionless master equation for quantum dots: Perturbative expansion to arbitrary order

The master equation describing the non-equilibrium dynamics of a quantum dot
coupled to metallic leads is considered. Employing a superoperator approach, we
derive an exact time-convolutionless master equation for the probabilities of
dot states, i.e., a time-convolutionless Pauli master equation. The generator
of this master equation is derived order by order in the hybridization between
dot and leads. Although the generator turns out to be closely related to the
T-matrix expressions for the transition rates, which are plagued by
divergences, in the time-convolutionless generator all divergences cancel order
by order. The time-convolutionless and T-matrix master equations are contrasted
to the Nakajima-Zwanzig version. The absence of divergences in the
Nakajima-Zwanzig master equation due to the nonexistence of secular reducible
contributions becomes rather transparent in our approach, which explicitly
projects out these contributions. We also show that the time-convolutionless
generator contains the generator of the Nakajima-Zwanzig master equation in the
Markov approximation plus corrections, which we make explicit. Furthermore, it
is shown that the stationary solutions of the time-convolutionless and the
Nakajima-Zwanzig master equations are identical. However, this identity neither
extends to perturbative expansions truncated at finite order nor to dynamical
solutions. We discuss the conditions under which the Nakajima-Zwanzig-Markov
master equation nevertheless yields good results.Comment: 13 pages + appendice

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