Quantum Rotation Matrices

Brian L. Silver

P. R. Berman

T. Skrypnyk

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.

Otto C. W. Kong

Nikolai L. Manakov, Alexei V. Meremianin, Anthony F. Starace

Our recent results [N. L. Manakov et al., Phys. Rev. A 57, 3233 (1998); 61, 022103 (2000)] on the invariant representations of finite rotation matrices (FRM's) of integer rank j (in terms of tensor products of vectors connected with a space-fixed reference frame) are generalized here for the general case of arbitrary (i.e., integer or half-integer) rank j. This extension is carried out by using new spinor representations of FRM's in terms of specially introduced spinor-annihilation operators. We demonstrate that all widely used, standard representations of FRM's follow as special cases of our invariant representation for particular parametrizations of the rotation parameters. As the simplest application of invariant spinor representations of FRM's, the factorized form of Wigner ${\mathbf{d}}^{j}(\ensuremath{\beta})$ matrices with an arbitrary rank j is obtained as a product of two triangular matrices composed of various powers of $\mathrm{cos}(\ensuremath{\beta}/2).$

James Mc Connell