Markovian master equations

Davies, E. B.

doi:10.1007/bf01608389

Cited by 812 publications

(867 citation statements)

References 14 publications

(16 reference statements)

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“…Concretely, we shall assume that L is of the Davies type, obtained under the assumption of weak system-bath coupling. 21 We are interested in searching for sufficient conditions that the Liouvillian dynamics (1) must satisfy in order to preserve the TI phase. In the absence of dissipation, we know that a key ingredient is that the TI Hamiltonian H s is a band Hamiltonian that satisfies the Bloch theorem and can be decomposed as H s = k∈B.Z.…”

Section: Band Liouvillian Dynamicsmentioning

confidence: 99%

Density-matrix Chern insulators: Finite-temperature generalization of topological insulators

2013

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Thermal noise can destroy topological insulators (TI). However, we demonstrate how TIs can be made stable in dissipative systems. To that aim, we introduce the notion of band Liouvillian as the dissipative counterpart of band Hamiltonian, and show a method to evaluate the topological order of its steady state. This is based on a generalization of the Chern number valid for general mixed states (referred to as density-matrix Chern value), which witnesses topological order in a system coupled to external noise. Additionally, we study its relation with the electrical conductivity at finite temperature, which is not a topological property. Nonetheless, the density-matrix Chern value represents the part of the conductivity which is topological due to the presence of quantum mixed edge states at finite temperature. To make our formalism concrete, we apply these concepts to the two-dimensional Haldane model in the presence of thermal dissipation, but our results hold for arbitrary dimensions and density matrices.

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Section: Band Liouvillian Dynamicsmentioning

confidence: 99%

Density-matrix Chern insulators: Finite-temperature generalization of topological insulators

2013

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“…In some situations where the system-environment interaction is weak, measured by a small coupling constant λ, one can implement a (time-dependent) perturbation theory, λ = 0 giving the unperturbed (uncoupled) case. For certain systems it has been shown [7] that for all a > 0, lim λ→0 sup 0 λ 2 t<a V (t) − e t(L 0 +λ 2 K) = 0,…”

Section: The Issuementioning

confidence: 99%

“…While it is known that the 'Davies generator' L 0 + λ 2 K, describing the weak coupling limit, generates a dynamical semigroup [7,9], this is not known for the generator M(λ) emerging from the dynamical resonance theory. In the present paper, we construct a dynamical semigroup T t satisfying…”

Section: The Issuementioning

confidence: 99%

Completely positive dynamical semigroups and quantum resonance theory

Könenberg

Merkli

2017

Lett Math Phys

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Abstract. Starting form a microscopic system-environment model, we construct a quantum dynamical semigroup for the reduced evolution of the open system. The difference between the true system dynamics and its approximation by the semigroup has the following two properties: It is (linearly) small in the system-environment coupling constant for all times, and it vanishes exponentially quickly in the large time limit. Our approach is based on the quantum dynamical resonance theory. The issueDue to the entanglement of an open system with its surroundings, its dynamics V (t) : ρ 0 → ρ t , mapping an initial system density matrix ρ 0 to its value at time t, is not a semigroup in time. For each fixed t, the mapping V (t), called a dynamical map, is a linear, completely positive, trace preserving transformation. 1 Under certain assumptions, one can approximate the dynamics of an open system by a continuous one-parameter semigroup of dynamical maps, called a quantum dynamical semigroup [5,11]. The dynamics given by such a semigroup has two important features: (i) is it markovian due to the semigroup property and (ii) it maps density matrices into density matrices due to its trace and positivity preserving quality. Complete positivity of the dynamical semigroup implies its positivity preservation, but not vice-versa. It is a crucial physical property which ensures that the dynamics of initially entangled systems interacting with an environment is well defined [1,3]. The semigroup property is particularly convenient since the spectral analysis of the generator L of the semigroup yields dynamical properties of the system, such as the final state(s) and convergence speeds. Controlling the remainder in the approximation V (t)ρ 0 ≈ e tL ρ 0 rigorously is difficult. Microscopic derivations, passing from a full (hamiltonian) model of system plus environment and tracing out the environment degrees of freedom, involve approximations (Born, Markov, rotating wave) that are hard to deal with mathematically. In some situations where the system-environment interaction is weak, measured by a small coupling constant λ, one can implement a (time-dependent) perturbation theory, λ = 0 giving the unperturbed (uncoupled) case. For certain systems it has been shown [7] that for all a > 0, lim

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“…In the present review we are not going to reproduce involved rigorous derivations which can be found in the literature [34][35][36]. Our aim is to discuss physical assumptions behind different Markovian regimes and proper approximation schemes based on the relevant order of perturbation which yield mathematically consistent results.…”

Section: Markovian Limitsmentioning

confidence: 99%

“…The details can be found in [34] see also the related idea of stochastic limit in [33]. MME of the type (65) possess several interesting properties.…”

Section: Weak Coupling Limitmentioning

confidence: 99%

Invitation to Quantum Dynamical Semigroups

Alicki

2002

Dynamics of Dissipation

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The theory of quantum dynamical semigroups within the mathematically rigorous framework of completely positive dynamical maps is reviewed. First, the axiomatic approach which deals with phenomenological constructions and general mathematical structures is discussed. Then basic derivation schemes of the constructive approach including singular coupling, weak coupling and low density limits are presented in their higly simplified versions. Two-level system coupled to a heat bath, damped harmonic oscillator, models of decoherence, quantum Brownian particle and Bloch-Boltzmann equations are used as illustrations of the general theory. Physical and mathematical limitations of the quantum open system theory, the validity of Markovian approximation and alternative approaches are discussed also.

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Markovian master equations

Cited by 812 publications

References 14 publications

Density-matrix Chern insulators: Finite-temperature generalization of topological insulators

Density-matrix Chern insulators: Finite-temperature generalization of topological insulators

Completely positive dynamical semigroups and quantum resonance theory

Invitation to Quantum Dynamical Semigroups

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