State-of-State Transition Probabilities and Control of Laser-Induced Dynamical Processes by The (T, T’) Method

Abstract

The time evolution operator for general time-dependent (not necessarily time-periodic) Hamiltonians is given by the (t, t’) method as, $\hat{u}(x,t\prime ,t) = \exp \{ - \tfrac{i}{\hbar }({{\hat{p}}_{{t\prime }}} + H(x,t\prime )(t - {{t}_{0}}))\}$, where ${{\hat{p}}_{{t\prime }}} \equiv (\hbar /i)\partial /\partial t\prime$ and t’ serves as an additional coordinate. Therefore the inner product is defined in the generalized Hilbert space of x and t’. The physical solution is given by ψ(x,t’, t)’ _t’-t where ψ(x, t’,t) = û(x, t', t)Ψo.

A brief review of the (t,t’) method is given emphasizing two advantages of the method: (I) it enables the extension of time-independent scattering theory to time- dependent Hamiltonian; (II) it enables the derivation of time-independent-like closed- form simple expression of the time evolution operator which can be described by global type very high order polynomial expansions.

It is shown how one can use the (t, t’) method to control the state-to-state transition probabilities in laser-induced processes.