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arXiv:1008.3738 · Nonlinearity

Exact solutions for a family of spin-boson systems

Yuan-Harng Lee, Jon Links, Yao-Zhong Zhang

Published May 24, 2011• 24 citations
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Abstract

We obtain the exact solutions for a family of spin-boson systems. This is achieved through application of the representation theory for polynomial deformations of the su (2) Lie algebra. We demonstrate that the family of Hamiltonians includes, as special cases, known physical models which are the two-site Bose–Hubbard model, the Lipkin–Meshkov–Glick model, the molecular asymmetric rigid rotor, the Tavis–Cummings model and a two-mode generalization of the Tavis–Cummings model.

DOI
10.1088/0951-7715/24/7/004

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We obtain the exact solutions for a family of spin-boson systems. This is achieved through application of the representation theory for polynomial deformations of the su (2) Lie algebra. We demonstrate that the family of Hamiltonians includes, as special cases, known physical models which are the two-site Bose–Hubbard model, the Lipkin–Meshkov–Glick model, the molecular asymmetric rigid rotor, the Tavis–Cummings model and a two-mode generalization of the Tavis–Cummings model.
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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.
We present a family of many-body models which are exactly solvable analytically. The models are an extended n-body interaction Lipkin-Meshkov-Glick model which considers spin-flip terms which are associated with the interaction of an external classical field which coherently manipulates the state of the system in order to, for example, process quantum information. The models also describe a two-mode Bose-Einstein condensate with a Josephson-type interaction which includes n-particle elastic and inelastic collisions. One of the models corresponds to the canonical two-mode Bose-Einstein Hamitonian plus a term which we argue must be considered in the description of the two-mode condensate. Intriguingly, this extra term allows for an exact and analytical solution of the two-particle collision two-mode BEC problem. Our results open up an arena to study many-body system properties analytically.
We have constructed the quasi-exactly-solvable two-mode bosonic realizations of su(2) and su(1, 1) algebra. We derive the relations leading to the conditions for quasi-exact solvability of two-boson Hamiltonians by determining a general procedure which maps the Schwinger representations of the su(2) and su(1, 1) algebras to the Gelfand-Dyson representations, respectively. This mapping allows us to study nonlinear quantum-optical systems in the framework of quasi-exact solvability. Our approach also leads to a simple construction of special functions of two variables which are the most appropriate functions to study quasi-probabilities in quantum optics.
Spin-Boson Mappings in the Formalism of fff-Deformations
We develop a unified algebraic approach to spin-boson transformations based on the formalism of fff-deformed oscillators. In the single-mode case, we show that the Holstein-Primakoff and Dyson-Maleev transformations, together with the interpolating ααα-family, arise as different factorizations of the same algebraically determined object. The standard spin-boson mappings can thus be interpreted as realizations of a common structure, which makes it possible to clearly separate the exact algebraic content on the physical subspace from effects associated with non-Hermiticity, the choice of metric, and extensions beyond the physical subspace. In the two-mode case, the same approach yields both deformed versions of the Jordan-Schwinger transformation and new exact two-mode realizations. Our results provide a unified description of known spin-boson transformations and naturally lead to new bosonic representations.