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The Journal of Chemical Physics

Optimal pump-dump control: Linearization and symmetry relation

YiJing Yan, Jianshu Cao, Zhenwen Shen

Published Sep 1, 1997• 13 citations
Publisher

Abstract

A generalized eigenequation is derived for the optimal pairs of phase-unlocked pump-dump fields in the control of pure state molecular systems in the weak response regime. The associated eigenvalues are further correlated to the yields of pump-dump control, thus the globally optimal pair of phase-unlocked pump-dump fields is identified unambiguously. Presented is also a hierarchy of reduction from the general Liouville-space density matrix control formulation in the strong response regime to various linearized control equations in the weak response regime. Application to the control of a molecular ro-vibronic level in the ground electronic surface reveals a novel symmetry relation between the optimal pump field and its counterpart of the optimal dump field.

DOI
10.1063/1.474686

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A generalized eigenequation is derived for the optimal pairs of phase-unlocked pump-dump fields in the control of pure state molecular systems in the weak response regime. The associated eigenvalues are further correlated to the yields of pump-dump control, thus the globally optimal pair of phase-unlocked pump-dump fields is identified unambiguously. Presented is also a hierarchy of reduction from the general Liouville-space density matrix control formulation in the strong response regime to various linearized control equations in the weak response regime. Application to the control of a molecular ro-vibronic level in the ground electronic surface reveals a novel symmetry relation between the optimal pump field and its counterpart of the optimal dump field.
We combine theories of optimal pump-dump control and the related transient probe absorption spectroscopy in order to elucidate the relation between these two optical processes and the possibility of experimental realization. In the weak response regime, we identify the globally optimal pair of pump-dump control fields, and further propose a second-order difference detection scheme to monitor the wave packets dynamics that is jointly controlled by both the pump and dump fields. The globally optimal solution serves also as the initial input for the iterative search for the optimal control fields in the strong response regime. We use a model I2 molecule to demonstrate numerically the pump-dump control and the detection of a highly vibrationally excited wave packet focusing dynamics on the ground X surface in both the weak and strong response regimes. The I2B surface serves as the intermediate to assist the pump-dump control and the optical detection processes. Demonstrated in the strong response regime are the optimal pair of pump-dump molecular-π pulses that invert nearly total population onto the predefined target region within a half period of vibration motion.
An alternative perturbative formulation is introduced to reduce the task of designing optimal controls of molecular motion. The method provides multiple optimal control fields as eigenvectors corresponding to the positive eigenvalues of a matrix {Amn{\mathit{A}}_{\mathit{mn}}Amn​}. Analytical equations for the matrix elements Amn{\mathit{A}}_{\mathit{mn}}Amn​ are derived. Numerical examples are given for simple three- and four-state molecular systems. The method is shown to be computationally simple and efficient in providing good quality control results. \textcopyright{} 1996 The American Physical Society.
This paper considers the optimal control of quantum systems. The controlled quantum systems are described by the probability-density-matrix-based Liouville–von Neumann equation. Using projection operators, the states of the quantum system are decomposed into two sub-spaces, namely the 'main state' space and the 'remaining state' space. Since the control energy is limited, a solution for optimizing the external control force is proposed in which the main state is brought to the desired main state at a certain target time, while the population of the remaining state is simultaneously suppressed in order to diminish its effects on the final population of the main state. The optimization problem is formulated by maximizing a general cost functional of states and control force. An efficient algorithm is developed to solve the optimization problem. Finally, using the hydrogen fluoride (HF) molecular population transfer problem as an illustrative example, the effectiveness of the proposed scheme for a quantum system initially in a mixed state or in a pure state is investigated through numerical simulations.
We consider an ensemble of closed finite-level systems described by a density matrix where the goal is to find an optimal control field to maximize the expectation value of an observable. The eigenvalues of the initial density matrix are assumed to depend on an experimental parameter (e.g., the temperature), whereas the eigenvalues of the observable may depend on an additional application-specific experimental parameter. We show that an optimal control will remain optimal for all such experimental parameters, if the relative ordering, by magnitude, of the eigenvalues of the initial density matrix as well as of the observable is unaltered regardless of the parameter values. More generally, we show like invariance of a control associated with any particular critical point on the corresponding control landscape. The invariance of control laser fields with respect to temperature is illustrated for vibrational excitation of diatomic molecules and photoassociation of atoms.
Mechanisms and optimal experimental conditions in coherent control still intensely stimulate debates. In this work, a phase-only control mechanism in an open quantum system is investigated experimentally and numerically. Several parameterizations for femtosecond pulse shaping (combination of chirp and multipulses) are exploited in transient absorption of a prototype organic molecule to control population and vibrational coherence in ground and excited states. Experimental results are further numerically simulated and corroborated with a four-level density-matrix model, which reveals a phase-only control mechanism based on the interaction between the tailored phase of the excitation pulse and the induced transient absorption. In spite of performing experiment and numerical simulations in the linear regime of excitation, the control effect amplitude depends non-linearly on the excitation energy and is explained as a pump-dump control mechanism. No evidence of single-photon control is observed with the model. Moreover, our results also show that the control effect on the population and vibrational coherence is highly dependent on the spectral detuning of the excitation spectrum. Contrary to the popular belief in coherent control experiments, spectrally resonant tailored excitation will lead to the control of the excited state only for very specific conditions.
Optimal quantum control in dissipative environments is formulated combining the optimal control theory from the Wilson group with the projection operator technique. In the weak response limit, a set of formally exact equations defines the optimal weak field for arbitrary target states and arbitrary system-bath interactions. Perturbative approximations are then made to derive general equations for weakly dissipative non-Markovian systems, which are amenable to analytical and/or numerical solutions. Potential applications of these results are discussed.