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Journal of Physics A Mathematical and Theoretical

Integrable modifications of Dicke and Jaynes–Cummings models, Bose–Hubbard dimers and classicalr-matrices

T. Skrypnyk

Published Apr 28, 2010• 18 citations
Publisher

Abstract

We consider quantum integrable systems associated with non-skew-symmetric sl (2)-valued classical r -matrices. For a special class of such r -matrices we construct a one-parametric family of integrable modifications of the ‘two-level one-mode’ Jaynes–Cummings–Dicke Hamiltonians, with non-uniform coupling constants, containing additional Kerr- and Stark-type nonlinearities. We also construct a family of integrable Bose–Hubbard-type dimers and a family of integrable models that unifies Bose–Hubbard- and Jaynes–Cummings–Dicke-type models and may be called a ‘two-level, two-mode’ Jaynes–Cummings–Dicke model or a ‘spin generalization’ of a Bose–Hubbard dimer. We diagonalize the constructed models with the help of the algebraic Bethe ansatz technique in any irreducible representation of the sl (2) ⊕ N spin algebra.

DOI
10.1088/1751-8113/43/20/205205

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We consider quantum integrable systems associated with non-skew-symmetric sl (2)-valued classical r -matrices. For a special class of such r -matrices we construct a one-parametric family of integrable modifications of the ‘two-level one-mode’ Jaynes–Cummings–Dicke Hamiltonians, with non-uniform coupling constants, containing additional Kerr- and Stark-type nonlinearities. We also construct a family of integrable Bose–Hubbard-type dimers and a family of integrable models that unifies Bose–Hubbard- and Jaynes–Cummings–Dicke-type models and may be called a ‘two-level, two-mode’ Jaynes–Cummings–Dicke model or a ‘spin generalization’ of a Bose–Hubbard dimer. We diagonalize the constructed models with the help of the algebraic Bethe ansatz technique in any irreducible representation of the sl (2) ⊕ N spin algebra.
Using the technique of classical r -matrices and quantum Lax operators we construct the most general form of the quantum integrable ‘two-level, one-mode’ spin-boson Jaynes–Cummings–Dicke-type model. We consider in detail two subclasses of this model and the corresponding examples associated with concrete classical r -matrices with spectral parameters. We show that the so-called ‘diagonal’ in the root basis r -matrices produces integrable Jaynes–Cummings–Dicke-type models with the ‘rotating wave approximation’, while all other types of classical r -matrices produce integrable JCD models without the rotating wave approximation. We construct an explicit example of an integrable Hermitian Jaynes–Cummings–Dicke-type Hamiltonian containing both ‘rotating’ and ‘counter-rotating’ terms. In the case of the diagonal r -matrices we find the spectrum of the corresponding quantum Hamiltonians using the algebraic Bethe ansatz.
Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.
Abstract We consider quantum integrable models based on the Lie algebra gl ( n ) and non-skew-symmetric classical r -matrices associated with Z 2 -gradings of gl ( n ) of the following type: gl ( n ) = gl ( n ) 0 ¯ + gl ( n ) 1 ¯ , where gl ( n ) 0 ¯ = gl ( n 1 ) ⊕ gl ( n − n 1 ) . Among the considered models are Gaudin-type models with an external magnetic field, used in nuclear physics to produce proton–neutron Bardeen–Cooper–Schrieer-type models, n -level many-mode Jaynes–Cummings–Dicke-type models of quantum optics, matrix generalization of Bose–Hubbard dimers, etc. We diagonalize the constructed models by means of the ‘generalized’ nested Bethe ansatz.
We analyze symmetries of the integrable generalizations of Jaynes–Cummings and Dicke models associated with simple Lie algebras g and their reductive subalgebras gK [T. Skrypnyk, “Generalized n-level Jaynes-Cummings and Dicke models, classical rational r-matrices and nested Bethe ansatz,” J. Phys. A: Math. Theor. 41, 475202 (2008)]. We show that their symmetry algebras contain commutative subalgebras isomorphic to the Cartan subalgebras of g, which can be added to the commutative algebras of quantum integrals generated with the help of the quantum Lax operators. We diagonalize additional commuting integrals and constructed with their help the most general integrable quantum Hamiltonian of the generalized n-level many-mode Jaynes–Cummings and Dicke-type models using nested algebraic Bethe ansatz.
We construct quantum integrable systems associated with non-skew-symmetric gl(2)-valued classical r-matrices. We find a new explicit multiparametric family of such the non-skew-symmetric classical r-matrices. We consider two classes of examples of the corresponding integrable systems, namely generalized Gaudin systems with and without an external magnetic field. In the case of arbitrary r-matrices diagonal in a standard gl(2)-basis, we calculate the spectrum of the corresponding quantum integrable systems using the algebraic Bethe ansatz. We apply these results to a construction of integrable fermionic models and obtain a wide class of integrable Bardeen–Cooper–Schrieffer (BCS)-type fermionic Hamiltonians containing the pairing and electrostatic interaction terms. We also consider special cases when the corresponding integrable Hamiltonians contain only pairing interaction term and are exact analogs of the “reduced BCS Hamiltonian” of Richardson.