Optimal control of quantum dissipative dynamics: Analytic solution for cooling the three-levelΛsystem

Abstract

We study the problem of optimal control of dissipative quantum dynamics. Although under most circumstances dissipation leads to an increase in entropy (or a decrease in purity) of the system, there is an important class of problems for which dissipation with external control can decrease the entropy (or increase the purity) of the system. An important example is laser cooling. In such systems, there is an interplay of the Hamiltonian part of the dynamics, which is controllable and the dissipative part of the dynamics, which is uncontrollable. The strategy is to control the Hamiltonian portion of the evolution in such a way that the dissipation causes the purity of the system to increase rather than decrease. The goal of this paper is to find the strategy that leads to maximal purity at the final time. Under the assumption that Hamiltonian control is complete and arbitrarily fast, we provide a general framework by which to calculate optimal cooling strategies. These assumptions lead to a great simplification, in which the control problem can be reformulated in terms of the spectrum of eigenvalues of ρ, rather than ρ itself. By combining this formulation with the Hamilton-Jacobi-Bellman theorem we are able to obtain an equation for the globaly optimal cooling strategy in terms of the spectrum of the density matrix. For the three-level Λ system, we provide a complete analytic solution for the optimal cooling strategy. For this system it is found that the optimal strategy does not exploit system coherences and is a 'greedy' strategy, in which the purity is increased maximally at each instant.

Resonant purification of mixed states for closed and open quantum systems

“…Several strategies have already been proposed, especially in the context of entanglement purification [2,3]. They rely on two mechanisms: measurement or cooling procedures (see [4,5,6,7,8] and references therein). From a general viewpoint, it is necessary to consider how it is possible to affect the dynamics of the system, and what states can be attained during its time evolution.…”

Flag-based control of quantum purity forn=2systems

“…In this example, the bias is saturated in some sense because the relaxation point is on the Bloch sphere; the dissipation cannot be any more biased in that direction. This type of dissipation is also seen in the cooling problem [17], where spontaneous emission of photons causes the system to relax to the ground state. One can extend this to model both absorption and emission of photons in a reservoir at non-zero temperature by having two amplitude-damping channels: one to the ground state |g , and one to the excited state |e .…”

Control of mixed-state quantum systems by a train of short pulses

“…a physical environment or a dissipative system [29,30]. The answer is not obvious because the strategy would involve non-unitary evolution and special attention has to be paid to decoherence [31,32].…”

“…The answer is not obvious because the strategy would involve non-unitary evolution and special attention has to be paid to decoherence [31,32]. Moreover, for an important class of dissipative systems such as laser cooling, the dissipation can increase the purity of the state [29,30,33]. In this case, the optimal target state would be the one defined in Ref.…”