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arXiv:quant-ph/0102050

Lie-type transformations and effective Hamiltonians in nonlinear quantum optics: applications to multilevel systems

A. B. Klimov, Andrés A. Navarro, Luis Lorenzo Sánchez-Soto

Published Feb 9, 2001• 0 citations
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Abstract

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 $N$-level atomic systems interacting with quantum fields, finding the corresponding effective Hamiltonians when the condition of $k$-photon resonance is fulfilled.

DOI
10.48550/arxiv.quant-ph/0102050

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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.
We propose 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 apply as well the technique to three-level systems interacting with quantum fields, showing that it is possible to engineer resonant interactions through nonresonant processes.
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.
A general and systematic method for obtaining eiective Hamiltonians that describe diierent nonlinear optical processes is discussed. 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, a diagonal eiective Hamiltonian is obtained immediately, that correctly represents the dynamics for arbitrary states and long times. The technique is extended to su(3) and su(N), finding the corresponding eiective Hamiltonians when some resonance conditions are fulfilled.
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.
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.
We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras supd(2)su_{pd}(2)supd​(2). These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the supd(2)su_{pd}(2)supd​(2) defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.