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Journal of Physics A Mathematical and General

A quasiclassical analysis of second-harmonic generation

R. F. Álvarez-Estrada, Á. Gómez Nicola, Luis Lorenzo Sánchez-Soto, Alfredo Luis

Published Jun 21, 1995• 8 citations
Publisher

Abstract

We investigate a quantum two-oscillator model for second-harmonic generation. The total Hamiltonian is the sum of two commuting Hamiltonians with eigenvalues E 0 and E int . The exact determination of these eigenvalues is studied using tridiagonal matrices. We present two general equations in the quasiclassical regime yielding the largest E int for a given E 0 and a representation for both eigenvalues in terms of an additional quantum number. Some numerical analysis shows that both equations are fairly consistent for suitably large quantum numbers with the exact quantum results. An approximate analytical expression for the wavefunction is also given.

DOI
10.1088/0305-4470/28/12/017

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We investigate a quantum two-oscillator model for second-harmonic generation. The total Hamiltonian is the sum of two commuting Hamiltonians with eigenvalues E 0 and E int . The exact determination of these eigenvalues is studied using tridiagonal matrices. We present two general equations in the quasiclassical regime yielding the largest E int for a given E 0 and a representation for both eigenvalues in terms of an additional quantum number. Some numerical analysis shows that both equations are fairly consistent for suitably large quantum numbers with the exact quantum results. An approximate analytical expression for the wavefunction is also given.
We study the energy spectrum of the degenerate χ(2)χ^{(2)}χ(2) Hamiltonian describing second-harmonic generation and parametric down-conversion at large total excitation numbers. Owing to the conservation of the total excitation number, the spectral problem reduces to the diagonalization of finite-dimensional tridiagonal matrices. Following the approach of Alvarez and Alvarez-Estrada, we map this problem onto an effective one-dimensional Schrödinger equation with a double-well potential. We then derive an implicit quantization condition for the energy levels near the top of the potential barrier and obtain two explicit asymptotic formulas applicable in complementary spectral regions. The ranges of applicability of the resulting formulas are established, and their accuracy is compared with exact matrix diagonalization and with the conventional JWKB approximation. The proposed approach provides an accurate analytical description of the energy levels near the center of the spectrum, where the conventional approximation loses accuracy.
Quantum harmonic oscillators coupled through coordinates and momenta, represented by the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}={\ensuremath{\sum}}_{i=1}^{2}(\frac{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{i}^{2}}{2{m}_{i}}+\frac{{m}_{i}{\ensuremath{\omega}}_{i}^{2}}{2}{x}_{i}^{2})+{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\text{int}}$, where the interaction of two oscillators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\text{int}}=i{k}_{1}{x}_{1}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}+i{k}_{2}{x}_{2}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}+{k}_{3}{x}_{1}{x}_{2}\ensuremath{-}{k}_{4}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}$, are found in many applications of quantum optics, nonlinear physics, molecular chemistry, and biophysics. Despite this, there is currently no general solution to the Schr\"odinger equation for such a system. This is especially relevant for quantum entanglement of such a system in quantum optics applications. Here this problem is solved and it is shown that quantum entanglement depends on only one coefficient, $R\ensuremath{\in}(0,1)$, which includes all the parameters of the system under consideration. It has been shown that quantum entanglement can be very large at certain values of this coefficient. The results obtained have a fairly simple analytical form, which facilitates analysis.
Without using the assumption of the Bethe ansatz, we have obtained exact explicit analytical expressions for the energy spectrum and eigenstates in terms of a parameter λ for a model of optical second harmonic generation. We have shown that the parameter λ is determined by the roots of a polynomial of order [( N +1)/4]. Here [ y ] denotes y ' s integer part and N = n a +2 n b with n a and n b being photon numbers of the fundamental and the second harmonic modes respectively. We also obtain explicit analytical expressions for λ for , and explicit analytical expressions for all the eigenstates without showing any interactions between the two modes.
Abstract A formalism is presented to obtain approximate analytic expressions for the eigenstates and eigenvalues of a quantum double oscillator (QDO). The matrix elements of a large class of operators with respect to states of different double oscillators result as finite sums of explicit functions of the respective parameters. Matrix elements between states of a harmonic oscillator and a double oscillator are also determined. The analytic expressions were used to calculate Franck-Condon factors for electronic transitions including double oscillator anharmonicities.
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This chapter begins discussing the classical harmonic oscillator and its solution as a second-order differential equation of the general type. Next, the one-particle quantum harmonic oscillator is presented, its corresponding Hamiltonian functionand wave equation followed by the undimensionalization of this wave equation yielding a second-order differential equation of the general type which leads to the general Hermite equation and Hermite polinomials as partial solution of the wave function. The energy for each wave function is obtained from the recursion relation of the Hermite equation. Afterwards, the numerical analysis of the one-particle harmonic oscillator is given by using the Numerov numerical method. Finally, a corresponding Fortran code for this problem is presented in order to plot the wave function according to a pre-determined quantum number. Plots of the wave function obtained from the Fortran code are also given. In the end, some exercises are given to include other information in the Fortran code.(153words).