Time-independent approximations for time-dependent optical potentials

Abstract

We explore the possibility of modifying the Lewis-Riesenfeld method of invariants developed originally to find exact solutions for time-dependent quantum mechanical systems for the situation in which an exact invariant can be constructed, but the subsequently resulting time-independent eigenvalue system is not solvable exactly. We propose to carry out this step in an approximate fashion, such as employing standard time-independent perturbation theory or the WKB approximation, and feeding the resulting approximated expressions back into the time-dependent scheme. We illustrate the quality of this approach by contrasting an exactly solvable solution to one obtained with a perturbatively carried out second step for two types of explicitly time-dependent optical potentials.

Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators

“…A detailed analysis of the spectra and eigenfunctions using approximation methods for time-dependent potential [31] is left for future investigations. Moreover, it is well known that Dyson maps are not unique, in the time-dependent as well as time-independent case, and it might therefore be interesting to explore whether additional spectrally equivalent Hamiltonians to H(z, t) can be found in the same fashion for new type of maps.…”

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

“…In future works it is worthwhile to see how one can actually solve the new relativistic Ermakov systems with an explicit time-dependence. For this purpose, the search for quasi-exact solutions [62] can be a fruitful approach.…”

Ermakov-Pinney equation for time-varying mass systems

“…Interest in exploring the dynamics of time-dependent quantum systems started with the method of seeking Hermitian invariants [8][9][10]. Recently, Fring and Tenney [11,12] took up up a systematic study of time-independent approximations for a class of timedependent optical potentials. Their general strategy was to write down an approximate Lewis-Riesenfeld scheme [9] and explore, among other issues, time-dependent potentials with a Stark term.…”

“…The purpose of the this article is to pursue an analog procedure of [11] but consider a time-dependent mass instead of a constant one 1 . The EP equation thus gets modified but its highly nonlinear character precludes us from finding any exact analytical solution.…”