Variational Approaches to the Evolution and Control of Strongly Driven Quantum Systems.

Ongoing experimental advances, especially the advent of high intensity pulsed lasers, have produced the need for theoretical methods for modelling the behavior of quantum systems under the influence of strong driving fields. Despite decades of effort developing approaches adapted to strong driving, no single method enjoys the position of preeminence held by lowest order perturbation theory in the weak field regime. Three methods well suited to the time evolution of strongly driven systems are discussed and illustrated through numerical calculation. The impulse representation in which the time dependent Schrodinger equation is solved by switching to a shifted momentum space representation is described and applied to the problem of a hydrogen atom driven by a Stark kick for which the approach yields an exact solution. Using this result convenient exact closed form expressions for exctitation probabilities from the ground state are derived. Secondly, the time dependent variational method (TDVM) for obtaining approximate solutions to the Schrodinger equation by stationarizing an appropriately defined variational functional is discussed. A simple form of the equations of motion for time dependent variational parameters is derived and applied to the calculation of the radial wavefunction following the beta decay of tritium. Accurate values for time dependent observables are obtained by approximating the wavefunction as a superposition of the ground states of the Z = 1 and Z = 2 hydrogenic atoms. Finally, the control of quantum dynamics is considered and two methods are presented for deriving external fields for the purpose of driving a quantum system into a desired behavior. With the inverse control method a procedure is described for producing exact tracking of a time dependent observable. The method is applied to a two-level atom and is shown to lead to non-unique values of the driving field which exhibit undesirable numerical anomalies. The more robust method of optimal control is presented for maximizing a chosen observable at a single point in time. The method is applied to the hydrogen atom and an external field is obtained for the production of a bound state gaussian wave packet.