Inverse square root level-crossing quantum two-state model

TL;DR

This paper introduces a new exactly solvable two-state quantum model with an inverse-square-root detuning, expanding the class of bi-confluent Heun models and providing explicit solutions for quantum dynamics under this field.

Contribution

It presents the first non-classical, exactly solvable two-state model with inverse-square-root detuning, solved via Hermite functions, and analyzes its quantum dynamics.

Findings

The model is uniquely solvable within bi-confluent Heun classes.

Explicit solutions are expressed in terms of Hermite functions.

The time evolution of the quantum system under this field is characterized.

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.