Indirect control of the asymptotic states of a dynamical semigroup (original) (raw)

Indirect control of the asymptotic states of a quantum dynamical semigroup

Eprint Arxiv 0709 1675, 2007

In the dynamics of open quantum systems, the interaction with the external environment usually leads to a contraction of the set of reachable states for the system as time increases, eventually shrinking to a single stationary point. In this contribution we describe to what extent it is possible to modify this asymptotic state by means of indirect control, that is by using an auxiliary system coupled to the target system in order to affect its dynamics, when there is a purely dissipative coupling between the two systems. We prove that, also in this restrictive case, it is possible to modify the asymptotic state of the relevant system, give necessary conditions for that and provide physical examples. Therefore, in indirect control schemes, the environmental action has not only a negative impact on the dynamics of a system, it is rather possible to make use of it for control purposes.

On the existence of stationary states for quantum dynamical semigroups

Journal of Mathematical Physics, 2001

We provide two criteria on the existence of stationary states for quantum dynamical semigroups. The first one is based on the semigroup itself, while the second criterion is based on the generator which is in general unbounded and interpreted as a sesquilinear form. These results are illustrated by physical examples drawn from quantum optics.

Invitation to Quantum Dynamical Semigroups

Lecture Notes in Physics, 2002

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.

A comparison of quantum dynamical semigroups obtainable by mixing or partial tracing

Some simple examples of quantum systems are collected to illustrate requirements suffi-cient for the evolution of a subsystem according to a quantum dynamical semigroup. For this, a class of quantum dynamics of a system S coupled to a reservoir R is analyzed in the Hilbert space H SR = H S ⊗ H R , where H R = L 2 (R) and H S = l 2 I , with I standing for a complete at most countable set of pure orthogonal states of S. The Hamiltonian of SR is built of tensor products of multipliers acting on H S and H R . The chosen linear coupling implies the exponential decoherence of the reduced evolution of S if and only if the occupation density in R is of the Cauchy type. Then the system indicates the expo-nential decoherence. On the other hand, since the occupation density in S is discrete, the reduced evolution of R is never governed by a semigroup (unless there is no coupling). In the considered case, the reduced evolution of the subsystem S as well as of the reservoir R can be equivalently...

Dynamics of Open Quantum Systems-Markovian Semigroups and Beyond

Symmetry, 2022

The idea of an open quantum system was introduced in the 1950s as a response to the problems encountered in areas such as nuclear magnetic resonance and the decay of unstable atoms. Nowadays, dynamical models of open quantum systems have become essential components in many applications of quantum mechanics. This paper provides an overview of the fundamental concepts of open quantum systems. All underlying definitions, algebraic methods and crucial theorems are presented. In particular, dynamical semigroups with corresponding time-independent generators are characterized. Furthermore, evolution models that induce memory effects are discussed. Finally, measures of non-Markovianity are recapped and interpreted from a perspective of physical relevance.

Stabilizing generic quantum states with Markovian dynamical semigroups

2009

Based on recent work on the asymptotic behavior of controlled quantum Markovian dynamics, we show that any generic quantum state can be stabilized by devising constructively a simple Lindblad-GKS generator that can achieve global asymptotic stability at the desired state. The applicability of such result is demonstrated by designing a direct feedback strategy that achieves global stabilization of a qubit state encoded in a noise-protected subspace.

Hamiltonian control of quantum dynamical semigroups: stabilization and convergence speed

We consider finite-dimensional Markovian open quantum systems, and characterize the extent to which timeindependent Hamiltonian control may allow to stabilize a target quantum state or subspace and optimize the resulting convergence speed. For a generic Lindblad master equation, we introduce a dissipation-induced decomposition of the associated Hilbert space, and show how it serves both as a tool to analyze global stability properties for given control resources and as the starting point to synthesize controls that ensure rapid convergence. The resulting design principles are illustrated in realistic Markovian control settings motivated by quantum information processing, including quantum-optical systems and nitrogen-vacancy centers in diamond.

A resolution of quantum dynamical semigroups

Eprint Arxiv Math 0505384, 2005

We consider a class of quantum dissipative systems governed by a one parameter completely positive maps on a von-Neumann algebra. We introduce a notion of recurrent and metastable projections for the dynamics and prove that the unit operator can be decomposed into orthogonal projections where each projections are recurrent or metastable for the dynamics.

On the recurrence of Quantum Dynamical Semigroups

The mathematical description of the evolution of Quantum Open Systems seems to reach a suitable formalism within the Theory of Quantum Dynamical Semigroups extensively developed during the last two decades. Moreover, from a probabilistic point of view, this theory provides a natural non commutative extension of Markov Processes. Following that line, we discuss the notion of recurrence and summarize a number of results on the large time behavior of Quantum Dynamical Semigroups.