Integrable time-dependent Hamiltonians, solvable Landau-Zener models and Gaudin magnets

TL;DR

This paper provides exact solutions for various time-dependent quantum Hamiltonians, revealing their integrability and mapping to Gaudin magnets, with applications in atomic physics and transition probability calculations.

Contribution

It demonstrates the integrability of several time-dependent Hamiltonians and maps Landau-Zener models to Gaudin magnets, extending the understanding of solvable many-body quantum systems.

Findings

Exact solutions for multiple time-dependent Hamiltonians.

Mapping of Landau-Zener models to Gaudin magnets.

Application to molecular production and transition probabilities.

Abstract

We solve the non-stationary Schrodinger equation for several time-dependent Hamiltonians, such as the BCS Hamiltonian with an interaction strength inversely proportional to time, periodically driven BCS and linearly driven inhomogeneous Dicke models as well as various multi-level Landau-Zener tunneling models. The latter are Demkov-Osherov, bow-tie, and generalized bow-tie models. We show that these Landau-Zener problems and their certain interacting many-body generalizations map to Gaudin magnets in a magnetic field. Moreover, we demonstrate that the time-dependent Schrodinger equation for the above models has a similar structure and is integrable with a similar technique as Knizhnikov-Zamolodchikov equations. We also discuss applications of our results to the problem of molecular production in an atomic Fermi gas swept through a Feshbach resonance and to the evaluation of the Landau-Zener transition probabilities.