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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Polynomial su pd (2) Lie algebra techniques in solving the second-harmonic generation model: comparison of exact and approximate calculations

V. P. Karassiov, Alexander A. Gusev, Sergue Il'ich Vinitsky

Published Jun 4, 2002• 0 citations
Publisher

Abstract

We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of the polynomial Lie algebra supd(2). These schemes, based on diagonal representations of the model evolution operator (due to diagonalizing the model Hamiltonian) and realized in the algorithmic forms, enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.

DOI
10.1117/12.469594

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We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of the polynomial Lie algebra supd(2). These schemes, based on diagonal representations of the model evolution operator (due to diagonalizing the model Hamiltonian) and realized in the algorithmic forms, enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.
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.
Polynomial Lie Algebras E^R1P(u(n);(m))\hat E_{R_1 }^\mathcal{P} (u(n);(m))E^R1​P​(u(n);(m)) in Nonlinear Models of Quantum Optics: Basic Ideas and Cluster Dynamics in the Heisenberg Picture
We consider here the exact solution of a nonrelativistic quantum system composed of a two-level atom interacting with a laser with arbitrary large frequency and intensity. We use the analogy of this system and a ℏ/2-spin particle interacting with a time-dependent magnetic field. A systematic use of the dynamical symmetry underlying the physical system is made. Actually the Hamiltonian is a Hermitian element of the SU(2) Lie Algebra. The exact Temporal Evolution Operator in terms of a Generalized Displacement Operator of the group is constructed. It is possible to develop a nonperturbative method that allows us to solve exactly the model for any value of the relevant frequencies (Rabi, Laser, and Atom Frequencies) and in so doing the usual Rotating Wave and Small Interaction approximations are unnecessary. The properties of periodicity of this model and the phenomenon of harmonic generation are considered by using Floquet Analysis. We find that in addition to the so far well-known spectrum composed by odd harmonics, this model generates another type of noninteger harmonics whose frequencies and amplitudes are determined for any value of the relevant parameters of the system.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.