Completely positive dynamical semigroups and quantum resonance theory

Abstract

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

Quantum Markovian master equations: Resonance theory shows validity for all time scales

“…In [18] we give a short (two page) outline of a proof of the Results 1 and 2 presented in the current work. The paper [18] focuses on the construction of an asymptotically exact markovian approximation, which is part of Result 3 of the present publication. However, there is a gap in the proof of the main result in [18].…”

“…However, there is a gap in the proof of the main result in [18]. This is explained in an erratum to [18], where it is also announced that we can still show the result in its full strength for the dynamics of the populations of the system (but not the coherences). We give the corresponding precise statement and proof of it here in (1.32).…”

“…and m = 1, 2 and g 1 (Σ) = e iαḡ 1 (Σ) for an arbitrary phase α and angular function g 1 . The precise regularity condition involves analyticity of g and is given in (2.22) below (see also [18]). It is readily seen that analyticity of the form factor (in the sense of (2.22)) leads to exponential time decay of the correlation function C β (t).…”

“…The paper [18] focuses on the construction of an asymptotically exact markovian approximation, which is part of Result 3 of the present publication. However, there is a gap in the proof of the main result in [18]. This is explained in an erratum to [18], where it is also announced that we can still show the result in its full strength for the dynamics of the populations of the system (but not the coherences).…”

“…Given an eigenvalue e of L 0 (the eigenvalues of L 0 and of L S are the same), we denote by P e the associated spectral projection and we define the level shift operators (compare with (2.26), One also shows that (compare with (2.34), and see [18], Proposition 3.2)…”