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arXiv:0806.1643 · Physical Review A

Effect of feedback on the control of a two-level dissipative quantum system

L. C. Wang, X. L. Huang, X. X. Yi

Published Nov 13, 2008• 14 citations
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Abstract

We show that it is possible to modify the stationary state by a feedback control in a two-level dissipative quantum system. Based on the geometric control theory, we also analyze the effect of the feedback on the time-optimal control in the dissipative system governed by the Lindblad master equation. These effects are reflected in the functions ${\ensuremath{\Delta}}_{A}(\stackrel{P\vec}{x})$ and ${\ensuremath{\Delta}}_{B}(\stackrel{P\vec}{x})$, which characterize the optimal trajectories, as well as the switching functions $\ensuremath{\Phi}(t)$ and $\ensuremath{\theta}(t)$, which characterize the switching point in time for the time-optimal trajectory.

DOI
10.1103/physreva.78.052112

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We show that it is possible to modify the stationary state by a feedback control in a two-level dissipative quantum system. Based on the geometric control theory, we also analyze the effect of the feedback on the time-optimal control in the dissipative system governed by the Lindblad master equation. These effects are reflected in the functions {\ensuremath{\Delta}}_{A}(\stackrel{P\vec}{x}) and {\ensuremath{\Delta}}_{B}(\stackrel{P\vec}{x}), which characterize the optimal trajectories, as well as the switching functions \ensuremathΦ(t)\ensuremath{\Phi}(t)\ensuremathΦ(t) and \ensuremathθ(t)\ensuremath{\theta}(t)\ensuremathθ(t), which characterize the switching point in time for the time-optimal trajectory.
We propose an analysis of the time-optimal control of a dissipative two-level quantum system whose dynamics is governed by the Lindblad equation. This simple system allows one to use tools of geometric control theory and to construct its optimal synthesis, i.e., to determine the set of all the optimal trajectories starting from a given initial point. We study different processes such as conversion of a pure state into a mixed state and purification of a mixed state. In particular cases, we show that dissipation is not undesirable and can help accelerating the control.
We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.