Optimal control of quantum lambda systems with an occupancy cost?

Abstract

We consider the problem of population transfer optimal control for a quantum Lambda system where the control couples pairwise only the lowest two energy levels to the highest level. The cost to be minimized expresses a compromise between minimizing the energy of the control and the average population in the highest level (occupancy), which is the one mostly subject to decay. Such a problem admits a group of symmetries, that is, a Lie group acting on the state space, which leaves dynamics, cost and initial and final conditions unchanged. By identifying a splitting of the tangent bundle into a vertical (tangent to the orbits) and horizontal (complementary) subspace at every point (a connection), we develop a symmetry reduction technique. In this setting, the problem reduces to a real problem on the sphere S2 for which we derive several properties and provide a practical method for the solution. We also describe an explicit numerical example. (c) 2024 Elsevier Ltd. All rights reserved