Exact solution of non-adiabatic model Hamiltonians in solid state physics and optics

Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamiltonians

“…Probably the simplest model for which these solutions have been found is the Rabi Hamiltonian (RH), which describes a two-level atom interacting with a single-mode bosonic field via a dipole interaction [2]. The Juddian solutions of the RH were first discovered by Reik and co-workers [3], where they were seen to occur at the level crossings in the energy schema of the system. This turns out to be a general and important feature of these solutions.…”

The quantum Rabi model: solution and dynamics

“…In the Bargmann-Fock space, the bosonic creation and annihilation operators have the form [48,49,50,51,52,53] …”

“…These points were first noticed by Judd [44] and are known as Judd or Juddian solutions, sometimes also called isolated exact solutions. They have been discussed by a number of authors and can be obtained via different approaches (see, e.g., [44,51,53,83,84,85,86,87]. For example, more recently it was shown that the exceptional points and related constraint polynomials can be obtained from a system of coupled Bethe Ansatz type equations [64].…”

Time evolution of the Rabi Hamiltonian from the unexcited vacuum

“…Lo et al [10] have given an analysis of the validity of the CI method. Reik and others [12] have adapted Judd's method [11] for the Jahn-Teller system for use with the Rabi Hamiltonian. Here, the Hamiltonian is translated into the Bargmann representation [13] Variational results have also been provided by Bishop et al [14] and by Benivegna and Messina [15].…”