Fourier-Coefficient based Solution of a quantum optimal control problem

We consider an optimal control problem for the time-dependent Schrödinger equation modeling molecular dynamics. Given a molecule in its ground state, the interaction with a tuned laser pulse can result in an excitation to a state of interest. By these means, one can optimize the yield of chemical reactions. The problem of designing an optimal laser pulse can be posed as an optimal control problem. Our ansatz for solving this problem is to reformulate the optimization problem by Fourier-transforming the electric field of the laser and narrow the frequency band. In this way, we reduce the dimensionality of the control variable. This allows for storing an approximate Hessian and, thereby, we can solve the optimization problem with a quasi-Newton method. Such an implementation provides superlinear convergence. Moreover, such a reduction of the frequency band can make sure that the laser pulse can be realized in the laboratory.

The standard method for tackling this quantum optimal control problem is to use the Krotov method. We have compared our algorithm to a version of the Krotov method for a configuration of the Rb2 molecule. In this comparison, our algorithm shows considerably faster convergence. The figure shows a pulse that is designed by the Fourier-based algorithm for the Rb2 problem.

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References

  1. A Fourier-coefficient based solution of an optimal control problem in quantum chemistry. Katharina Kormann, Sverker Holmgren, and Hans O. Karlsson. In Journal of Optimization Theory and Applications, volume 147, pp 491-506, 2010. (DOI).

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