Photon echoes of molecular photoassociation

Revivals of optical coherence of molecular photoassociation driven by two ultrashort laser pulses are addressed in the Condon approach. Based on textbook examples and numerical simulation of KrF excimer molecules, a prediction is made about an existence of photon echo on free-bound transitions. Delayed rise and fall of nonlinear polarization in the half-collisions are to be resulted from the resonant quantum states interference whether it be in gas, liquid or solid phases.Comment: 15 pages and 5 figures presented at ICONO '98'(Moscow, 1998): Fundamental Aspects of Laser-Matter Interaction, New Nonlinear Optical Materials and Physics of Low-Dimensional Structure

On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale

We study the joint laws of a continuous, uniformly integrable martingale, its maximum, and its minimum. In particular, we give explicit martingale inequalities which provide upper and lower bounds on the joint exit probabilities of a martingale, given its terminal law. Moreover, by constructing explicit and novel solutions to the Skorokhod embedding problem, we show that these bounds are tight. Together with previous results of Az\'ema & Yor, Perkins, Jacka and Cox & Ob{\l}\'oj, this allows us to completely characterise the upper and lower bounds on all possible exit/no-exit probabilities, subject to a given terminal law of the martingale. In addition, we determine some further properties of these bounds, considered as functions of the maximum and minimum.Comment: 19 pages, 4 figures. This is the authors' accepted version of the paper which will appear in Stochastic Processes and their Application