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Physics of Atomic Nuclei

The su q(2) algebra in the off-diagonal basis and applications to quantum optics

Ángel Ballesteros, S. M. Chumakov

Published Dec 1, 2001• 0 citations
Publisher

Abstract

We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su q (2) algebra. The corresponding matrix elements of finite rotations (the q -deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su q (2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.

DOI
10.1134/1.1432904

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We discuss a general and systematic method for obtaining effective Hamiltonians that describe different nonlinear optical processes. The method exploits the existence of a nonlinear deformation of the usual su(2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter, dictated by the process under consideration, becomes small, we immediately get a diagonal effective Hamiltonian that correctly represents the dynamics for arbitrary states and long times. We extend the technique to su(3) and su(N), finding the corresponding effective Hamiltonians when some resonance conditions are fulfilled.
We reelaborate on a general method for diagonalizing a wide class of nonlinear Hamiltonians describing different quantum optical models. This method makes use of a nonlinear deformation of the usual su(2) algebra and when some physical parameter, dictated by the particular model under consideration, becomes small, it gives a diagonal effective Hamiltonian that describes correctly the dynamics for arbitrary states and long times. We extend the technique to NNN-level atomic systems interacting with quantum fields, finding the corresponding effective Hamiltonians when the condition of kkk-photon resonance is fulfilled.
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We re-elaborate on a general method for obtaining effective Hamiltonians that describe different nonlinear optical processes. The method exploits the existence of a nonlinear deformation of the su (2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter (usually related to the dispersive limit) becomes small, we immediately get a diagonal effective Hamiltonian that represents correctly the dynamics for arbitrary states and long times. We apply the technique to obtain how the noise terms in the original model transform under this scheme, providing a systematic way of including damping effects in processes described in terms of effective Hamiltonians.