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

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Abstract

We solve the non-stationary Schrödinger 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 Schrödinger equation for the above models has a similar structure and is integrable with a similar technique as Knizhnik–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.

Original languageEnglish (US)
Pages (from-to)323-339
Number of pages17
JournalAnnals of Physics
Volume392
DOIs
StatePublished - May 2018

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

Keywords

  • Gaudin magnets
  • Integrable time-dependent Hamiltonians
  • Knizhnik–Zamolodchikov equations
  • Landau–Zener models

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