Coarse-grained Picture for Controlling Complex Quantum Systems

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Controlling atomic and molecular processes by laser fields is one of current topics in physics and chemistry, and there are various control schemes applied to such processes. These strategies are known to work when the system to be controlled is rather simple or small. However, in reality, the system can become ‘‘complex,’’ where the dynamics will be described by multi-level–multi-level transitions with a random interaction. Although the laser field can be obtained by optimal control theory (OCT) even for such complex systems, it is difficult to analyze the controlled dynamics because such a field obtained numerically is often too complicated to interpret. Hence it is desirable to have a more analytical point of view. It is well known that the pulse or its generalizations can be employed to control few-level problems. Recently, an analytic result for multi-level control problems between general quantum states has been reported. The scheme is based on STIRAP, and assumes an intermediate state coupled to the initial and target states. Though the scheme can accomplish perfect control, it relies on the energy level picture of a quantum system, so it is difficult to apply the scheme to large systems. In this short note, we propose a new approach to obtain an analytic optimal field for complex quantum systems. Under a ‘‘coarse-grained picture’’ with OCT, which is valid for such a complex system, we derive an analytic expression for the optimal field which steers initial states to target states in a certain limit. By numerically solving Schrodinger equations, we confirm that perfect control is actually achieved. This point is important because the zeroth-order solutions of OCT, which look similar to our result, are not guaranteed to achieve perfect control. Another point is that our final expression does not require detailed information from the energy level picture, so that it is easy to apply to large quantum systems, in principle. We use OCT as a theoretical vehicle. The aim of OCT is to obtain an optimal field ðtÞ which guides the system from an initial state j ii at t 1⁄4 0 to a target state j f i at some specific time t 1⁄4 T . According to the OCT scheme by Zhu et al., the optimal field for the Hamiltonian H1⁄2 ðtÞ 1⁄4 H0 þ ðtÞV is given by