Exactly solvable models and investigation of level crossing

T. A. Shahverdyan, Tigran A. Ishkhanyan, Artur Mikhailovich Ishkhanyan

Abstract We present a new exactly integrable time-dependent constant-amplitude level-crossing two-state quantum model for which the detuning of the excitation laser field varies with time over a restricted interval. This field configuration is a member of one of the eleven independent classes of general Heun two-state models. We prove that this is the only non-classical unconditionally solvable field configuration among the general Heun classes, solvable in terms of finite sums of Gauss hypergeometric functions. Each of the two fundamental solutions that compose the general solution of the problem is written as an irreducible linear combination of two ordinary hypergeometric functions.

D. Ya. Petrina

Tigran A. Ishkhanyan

We introduce a new exactly integrable level-crossing model of quantum semiclassical two-state problem for which the analytic solution is written in terms of the Kummer confluent hypergeometric functions. This is a constant-amplitude field-configuration describing an asymmetricin- time level-crossing process for which the laser field frequency detuning is given in terms of the Lambert-W function. The general solution of the problem for this model is written as a linear combination, with arbitrary constant coefficients, of two fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. We present the fundamental solutions and analyze the behavior of the system in the external field defined by the specific field configuration.

Tigran A. Ishkhanyan, A. Papoyan, Artur Mikhailovich Ishkhanyan +1 more

Abstract We introduce a new unconditionally solvable level-crossing two-state model given by a constant-amplitude optical field configuration for which the detuning is an inverse-square-root function of time. This is a member of one of the five families of bi-confluent Heun models. We prove that this is the only non-classical exactly solvable field configuration among the bi-confluent Heun classes, solvable in terms of finite sums of the Hermite functions. The general solution of the two-state problem for this model is written in terms of four Hermite functions of a shifted and scaled argument (each of the two fundamental solutions presents an irreducible combination of two Hermite functions). We present the general solution, rewrite it in terms of more familiar physical quantities and analyze the time dynamics of a quantum system subject to excitation by a laser field of this configuration.