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Journal of Mathematical Physics

Integrability and superintegrability of the generalized n-level many-mode Jaynes–Cummings and Dicke models

T. Skrypnyk

Published Oct 1, 2009• 21 citations
Publisher

Abstract

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.

DOI
10.1063/1.3205453

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We construct integrable n -level generalizations of Jaynes–Cummings and Dicke models corresponding to the simple (reductive) Lie algebras of rank n . We show that for each such Lie algebra there exist many integrable Jaynes–Cummings and Dicke-type models each of which is associated with the reductive subalgebras containing Cartan subalgebra and is obtained via reduction from Jaynes–Cummings or Dicke-type models with a maximal number of bosons. We diagonalize the constructed Jaynes–Cummings and Dicke-type Hamiltonians in the physically most interesting case of using a nested 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.
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.
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.
We consider quantum integrable systems associated with reductive Lie algebra g l ( n ) and Cartan-invariant non-skew-symmetric classical r -matrices. We show that under certain restrictions on the form of classical r -matrices “nested” or “hierarchical” Bethe ansatz usually based on a chain of subalgebras g l ( n ) ⊃ g l ( n − 1 ) ⊃ . . . ⊃ g l ( 1 ) is generalized onto the other chains or “hierarchies” of subalgebras. We show that among the r -matrices satisfying such the restrictions there are “twisted” or Z p -graded non-skew-symmetric classical r -matrices. We consider in detail example of the generalized Gaudin models with and without external magnetic field associated with Z p -graded non-skew-symmetric classical r -matrices and find the spectrum of the corresponding Gaudin-type hamiltonians using nested Bethe ansatz scheme and a chain of subalgebras g l ( n ) ⊃ g l ( n − n 1 ) ⊃ g l ( n − n 1 − n 2 ) ⊃ g l ( n − ( n 1 + . . . + n p − 1 ) ) , where n 1 + n 2 + . . . + n p = n .