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arXiv:1304.3979 · Journal of Physics A Mathematical and Theoretical

Solving the two-mode squeezed harmonic oscillator and thekth-order harmonic generation in Bargmann–Hilbert spaces

Yao-Zhong Zhang

Published Oct 24, 2013• 17 citations
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Abstract

We analyze the two-mode squeezed harmonic oscillator and the k th-order harmonic generation within the framework of Bargmann–Hilbert spaces of entire functions. For displaced single-mode squeezed and two-mode squeezed harmonic oscillators, we derive exact closed-form expressions for their energies and wave functions. For the k th-order harmonic generation with k ⩾ 3, our result indicates that it does not have eigenfunctions and is thus ill-defined in the Bargmann–Hilbert space.

DOI
10.1088/1751-8113/46/45/455302

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We analyze the two-mode squeezed harmonic oscillator and the k th-order harmonic generation within the framework of Bargmann–Hilbert spaces of entire functions. For displaced single-mode squeezed and two-mode squeezed harmonic oscillators, we derive exact closed-form expressions for their energies and wave functions. For the k th-order harmonic generation with k ⩾ 3, our result indicates that it does not have eigenfunctions and is thus ill-defined in the Bargmann–Hilbert space.
Recently Zhang (2013 J. Phys. A: Math. Theor. 46 455302) proposed an analytical approach to solve the time-independent Schrödinger equation for the single-mode and two-mode squeezed harmonic oscillators in the Bargmann space of entire functions. In this comment we show that the eigenfunctions of these two systems exist in closed form and are expressed in terms of the Hermite polynomials. Moreover, since both oscillators exhibit the SU(1,1) dynamical symmetry, the eigenvalue problem can be tackled in a unified manner. In the Hilbert space of analytic functions of a complex variable in the unit disc, the energy eigenvalue equations involve first-order ordinary differential equations only, so we can easily solve these equations to obtain simple closed-form solutions.
In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.
In this paper we generalize our previous results of the generalized one-mode harmonic oscillator to the generalized two-mode case. Systematic use is made of the SO(3,2) dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes. A complete classification of the exact solutions is made and we derive them explicitly whenever possible. The relevant results on algebraic decomposition, coherent state generators and classification of the solutions are shown in tables. A plethora of soluble Hamiltonians already treated in the literature, which appear to be particular cases of the general formalism presented herein, are analysed as well as new cases, which to the authors' knowledge, have not yet been considered.
The spectra and generalized eigenfunctions of the hyperbolic and parabolic generators of the standard representation of SU(1,1) in the one-mode boson Hilbert space are derived. The eigenfunctions are given in three different forms, corresponding to the coordinate, photon number, and Fock-Bargmann representations of the state vectors. The possible spectra of general second degree Hamiltonians are determined. Some corresponding results in the two-mode case are also given. - In the Appendix we prove completeness and orthonormality relations for the polynomials giving the number representation expansion coefficients of the generalized eigenfunctions of the hyperbolic generator (= squeezing generator). These polynomials are special cases of Pollaczek polynomials.