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arXiv:cond-mat/0406190 · Physical Review Letters

Exactly Solvable Models for Atom-Molecule Hamiltonians

J. Dukelsky, G.G. Dussel, Carlos Esebbag, Stuart Pittel

Published Jul 30, 2004• 67 citations
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Abstract

We present a family of exactly solvable generalizations of the Jaynes-Cummings model involving the interaction of an ensemble of SU(2) or SU(1,1) quasispins with a single boson field. They are obtained from the trigonometric Richardson-Gaudin models by replacing one of the SU(2) or SU(1,1) degrees of freedom by an ideal boson. The application to a system of bosonic atoms and molecules is reported.

DOI
10.1103/physrevlett.93.050403

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We present a family of exactly solvable generalizations of the Jaynes-Cummings model involving the interaction of an ensemble of SU(2) or SU(1,1) quasispins with a single boson field. They are obtained from the trigonometric Richardson-Gaudin models by replacing one of the SU(2) or SU(1,1) degrees of freedom by an ideal boson. The application to a system of bosonic atoms and molecules is reported.
We present the exact solution of the Richardson–Gaudin models associated with the SU (3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU (3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU (3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis–Cummings model for three level atoms interacting with two modes of the quantized electric field.
The interaction between an atom and a one mode external driving field is an ubiquitous problem in many branches of physics and is often modeled using the Rabi Hamiltonian. In this paper we present a series of analytically solvable Hamiltonians that approximate the Rabi Hamiltonian and compare our results to the Jaynes-Cummings model which neglects the so-called counter-rotating term in the Rabi Hamiltonian. Through a unitary transformation that diagonalizes the Jaynes-Cummings model, we transform the counter-rotating term into separate terms representing several different physical processes. By keeping only certain terms, we can achieve an excellent approximation to the exact dynamics within specified parameter ranges.