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arXiv:0707.3383 · Physical Review A

Relaxation to equilibrium driven via indirect control in Markovian dynamics

Raffaele Romano

Published Nov 27, 2007• 4 citations
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Abstract

We prove that it is possible to modify the stationary states of a quantum dynamical semigroup, describing the irreversible evolution of a two-level system, by means of an auxiliary two-level system, a quantum probe that can be suitably prepared. The target system and the probe can be initially entangled or uncorrelated. We find that this indirect control of the stationary states is possible, even if there are no initial correlations, under suitable conditions on the dynamical parameters characterizing the evolution of the joint system.

DOI
10.1103/physreva.76.052115

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We prove that it is possible to modify the stationary states of a quantum dynamical semigroup, describing the irreversible evolution of a two-level system, by means of an auxiliary two-level system, a quantum probe that can be suitably prepared. The target system and the probe can be initially entangled or uncorrelated. We find that this indirect control of the stationary states is possible, even if there are no initial correlations, under suitable conditions on the dynamical parameters characterizing the evolution of the joint system.
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Recently, a new approach for the controllability of a two-dimensional quantum system S has been proposed, based on its interaction with an initially uncorrelated two-dimensional probe P whose initial state can be arbitrarily modified. Following this scheme and considering a particular model for the environment, we show that, in some specific cases, the environment-induced entanglement is rich enough to completely control the dynamics of S. Under suitable conditions on the interaction of S, P, and the environment, we prove that the state of S can be driven to an arbitrary target state by varying the initial state of P.
In many experiments, a target quantum mechanical system is controlled and-or measured indirectly using an auxiliary quantum system. In the case of control, this means that the control action only affects the auxiliary system while the evolution of the target system is affected indirectly through the interaction with the auxiliary system. In the case of measurement, only the state of the auxiliary system can be measured, providing information on the initial state of the target system. Indirect controllability is the property of the target system of being driven between two arbitrary states while indirect observability refers to the possibility of extracting all the information on the state of the target system by measuring the auxiliary system. This paper has three main goals. First, we summarize recent notions and results introduced by us on the study of indirect controllability. Then, we present some new technical results on indirect controllability. In particular, we present a counterexample to the converse of a criterion for indirect controllability, which shows that the Lie algebraic condition introduced in this criterion is necessary but not sufficient for indirect controllability. This gives an open problem which is how to strengthen this condition to obtain a necessary and sufficient condition for indirect controllability. Lastly, we present a parallel treatment of indirect observability, give the relevant notions and definitions and relate the properties of indirect controllability and observability. Our final result says that strong notions of indirect controllability and observability are equivalent properties to complete controllability of the system. We discuss examples of physical systems which are controlled and-or measured indirectly. The research is motivated by experimental schemes of current interest
We propose a new technique to control quantum states by exploiting the decoherence due to the coupling with environment.With this technique we can get any target state, in a stable way and in an exponentially small time, as a stationary state of a semigroup (master equation) canonically derived from a microscopic Hamiltonian model.The stationary state of the system depends (i) on the interaction with the environment; (ii) on the initial state of the environment and it "inherits" some properties of it in a sense that will be explained in sec.5 of the present paper.Moreover this is true not only for by thermal or vacuum environments but also for more general non-equilibrium environments.We prove that, by appropriately choosing this control parametes (interaction system-environment, initial environment state) one can drive the system to an arbitrary preassigned quantum state.
A quantum mechanical system S is indirectly controlled when the control affects an ancillary system A and the evolution of S is modified through the interaction with A only. A study of indirect controllability gives a description of the set of states that can be obtained for S with this scheme. In this paper, we study the indirect controllability of quantum systems in the finite dimensional case. After discussing the relevant definitions, we give a general necessary condition for controllability in Lie algebraic terms. We present a detailed treatment of the case where both systems, S and A, are two-dimensional (qubits). In particular, we characterize the dynamical Lie algebra associated with S+A, extending previous results, and prove that complete controllability of S+A and an appropriate notion of indirect controllability are equivalent properties for this system. We also prove several further indirect controllability properties for the system of two qubits, and illustrate the role of the Lie algebraic analysis in the study of reachable states.