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Physical Review A

Method of small rotations and effective Hamiltonians in nonlinear quantum optics

A. B. Klimov, Luis Lorenzo Sánchez-Soto

Published May 5, 2000• 111 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.61.063802

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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.
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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.
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 propose a general perturbative approach to quantum-optical models without the rotating-wave approximation. We show that a generic Hamiltonian describing interaction between two subsystems can be represented as a series of operators corresponding to different transitions between bare energy levels of the whole system. Under certain relations between frequencies of interacting subsystems one of these transitions becomes resonant. The rotating-wave approximation leads to separation of the resonant transition and to appearance of the integral of motion, which makes the problem exactly solvable in this approximation. (Different resonance conditions lead to different integrals of motion.) All of the other terms in these expansion can be considered as a perturbation. They result in dynamic Stark shifts and small corrections to the integrals of motion. All possible resonances are classified, and approximate integrals of motion are found for each resonance. Examples of field-field, field-atom, and atom-atom interactions are considered.