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Chinese Journal of Chemical Physics

Hamiltonian Reduction of Quantum Systems Controlled by Pulses

Qiang Xu, Yao-xiong Wang, Feng Shuang, HERSCHEL A. RABITZ

Published Aug 1, 2011• 0 citations
Publisher

Abstract

We explores Hamiltonian reduction in pulse-controlled finite-dimensional quantum systems with near-degenerate eigenstates. A quantum system with a non-degenerate ground state and several near-degenerate excited states is controlled by a short pulse, and the objective is to maximize the collective population on all excited states when we treat all of them as one level. Two cases of the systems are shown to be equivalent to effective two-level systems. When the pulse is weak, simple relations between the original systems and the reduced systems are obtained. When the pulse is strong, these relations are still available for pulses with only one frequency under the first-order approximation.

DOI
10.1088/1674-0068/24/04/378-382

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Examining the dynamics of a degenerate two-level system excited by coherent resonant light, we exhibit a transformation of basis states which reduces the problem to a set of independent nondegenerate one- and two-state systems. More generally, reduction of multilevel Hamiltonians requires only that the states can be collected into two sets; within each set the states are degenerate, unlinked by Hamiltonian matrix elements. The remaining Hamiltonian linkages must share the same time dependence, but are otherwise arbitrary. We apply this method to reduction of the angular momentum degeneracy of a two-level atom excited by elliptically polarized light and to multiple-laser resonant excitation of multiple levels into a common level.
Two simple schemes are presented for the selective population transfer in systems where the target state is degenerate with a second level. Both states are coupled with the ground state via a one photon resonance. The schemes use a combination of two laser pulses. The first pulse breaks the degeneracy of the levels by inducing a Stark shift. The second pulse is chirped across the new levels. The idea is illustrated by numerical simulations of the time-dependent Schr\"odinger equation for a four-level system. The efficient population transfer is explained using a dressed state representation. In conclusion, applications for molecules are suggested.
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In closed quantum systems, if controlled systems are not strongly regular or there exists at least one eigenstate that is directly uncoupled to the target state, then such systems control will become degenerate cases. This paper proposes a quantum control method designed by an implicit Lyapunov function to transfer states of the quantum systems in these two degenerate cases, especially the method proposed is suitable for multi-control Hamiltonian systems. The state distance is selected as the Lyapunov function in this paper. Numerical simulation experiments on a 4-level system are done, and the experiment results indicate the effectiveness of the implicit Lyapunov control method proposed for degenerate cases and multi-control Hamiltonians.