An Iterative Method for Extreme Optics of Two-Level Systems

Abstract

We formulate the problem of a two-level system in a linearly polarized laser field in terms of a nonlinear Riccati-type differential equation and solve the equation analytically in time intervals much shorter than half the optical period. The analytical solutions for subsequent intervals are then stuck together in an iterative procedure to cover the whole scale time of the laser pulse. Very good quality of the iterative method is shown by recovering with it a number of subtle effects met in earlier numerically calculated photon--emission spectra from model molecular ions, double quantum wells, atoms, and semiconductors. The method is used to describe novel, carrier-envelope offset phase effects in the region of extreme nonlinear optics, i.e., when two--level systems are exposed to pulses of only a few cycles in duration and strength ensuring the Rabi frequency to approach the laser light frequence.

Extreme Nonlinear Optics in Terms of Riccati-Type Equation

“…(4) we have given an iterative algorithm for solving the Riccati-type equation for the ratio of population amplitudes of a two-level system in a pulsed oscillating field. The approach based on the Riccati-type equation [12][13][14][15]17] is an alternative to the commonly used one exploiting the optical Bloch equations [1-7, 10, 11, 19-21]. Our algorithm suffers no approximations, like the rotating-wave [5] and adiabatic [18] approximations.…”

“…Due to the above approximation of piecewise constant field, the analytical solution of Eq. (2) at the end of the j-th interval, r(t = j∆) = r j , turns out to be interrelated with that at the beginning of the j-th interval, r(t = (j − 1)∆) = r j−1 , in the following way [17]:…”