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Annual Meeting Optical Society of America

Two-dimensional oscillator, the two-dimensional hydrogen atom, and lossless optical beam splitter: birds of a feather

Sotirios Danakas, P. K. Aravind

Published Jan 1, 1990• 0 citations
Publisher

Abstract

We point out some striking similarities between three simple, yet fundamental, quantum systems: (1) the two-dimensional isotropic oscillator, (2) the two-dimensional hydrogen atom, and (3) the lossless two-port beam splitter. By using the quantum theory of angular momentum, we obtain the unitary transformation connecting the Cartesian and polar eigenfunctions of the two-dimensional oscillator. By similar methods, we also obtain the unitary transformation connecting the polar and parabolic eigenfunctions of the two-dimensional hydrogen atom. Finally, we show that both the preceding transformations are closely related to the unitary transformation that connects the input and output photon statistics at a lossless 50-50 beamsplitter. Thus, a close connection is established between the quantum beam splitter, recently studied by several workers1,2 and two basic quantum systems.

DOI
10.1364/oam.1990.fff8

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We point out some striking similarities between three simple, yet fundamental, quantum systems: (1) the two-dimensional isotropic oscillator, (2) the two-dimensional hydrogen atom, and (3) the lossless two-port beam splitter. By using the quantum theory of angular momentum, we obtain the unitary transformation connecting the Cartesian and polar eigenfunctions of the two-dimensional oscillator. By similar methods, we also obtain the unitary transformation connecting the polar and parabolic eigenfunctions of the two-dimensional hydrogen atom. Finally, we show that both the preceding transformations are closely related to the unitary transformation that connects the input and output photon statistics at a lossless 50-50 beamsplitter. Thus, a close connection is established between the quantum beam splitter, recently studied by several workers1,2 and two basic quantum systems.
Some striking analogies are exhibited between the lossless quantum-mechanical beam splitter and the following two elementary quantum systems: (1) the two-dimensional isotropic oscillator and (2) the two-dimensional hydrogen atom. For example, we show that the unitary transformation connecting the input and output photon statistics at a lossless 50-50 beam splitter is identical to the transformation connecting the polar and parabolic eigenstates of the two-dimensional (2D) hydrogen atom. A similar connection between (Gaussian) laser beams and the eigenstates of the 2D oscillator is pointed out. These analogies are established by identifying the underlying symmetries of these systems and exploiting them by means of group theory.
For optical homodyning, the matrix representation of a lossless beam splitter belongs to the SU(2) group of unimodular second-order unitary matrices. The connection between this group and the rotation group in three dimensions permits the field density operators at the input and output ports of the beam splitter to be related by means of well-known angular-momentum transformations. This, in turn, provides the joint output photon-number distribution, which may be written as a Fourier series in the relative phase shift imparted by the beam splitter, for a general joint state at its inputs. The series collapses to a single term if one of the input fields is diagonal in the number-state representation. If the inputs to both ports are further restricted to be pure number states, the joint, as well as the marginal photon-number distributions, turn out to be directly proportional to the square of Jacobi polynomials in the beam-splitter transmittance. These photon-number probabilities are invariant to a set of physical and time-reversal symmetries. When one of the input photon-number states is the vacuum, the beam splitter simply deletes photons from the other port in Bernoulli fashion, as if they were classical particles. The output photon number is then described by the binomial distribution. If the inputs at the two ports are different number states, neither of which is the vacuum, the photon-number distribution is expressible in terms of summed and weighted products of the results for photomixing with the vacuum. If the inputs at the two ports are identical number states, and a beam splitter of transmittance \ensuremath{\tau}=(1/2 is used, the photon-number distribution assumes a simple but interesting form. It vanishes for odd photon numbers, indicating that the photons assemble in pairs at each output port. Finally, it is shown that homodyning quantum fluctuations can be reduced by using a balanced photomixer for arbitrary input states.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.
The angular momentum states may be realized by the wave functions of two dimensional harmonic oscillators in polar coordinates. Using the generating function of these wave functions, Schwinger's generating function of Clebsh-Gordan in Fock space and Gauss integral, we determine the generating function of the couplings function of two angular momentums in coordinates representations. We deduce from the last function the connection of the wave function of the hydrogen atom and the four dimensional harmonic oscillator. We also find the expression of the relation between the spherical and parabolic wave functions of hydrogen atom. In addition, we find a new bilinear generating function involving the products of Laguerre polynomials.
The lossless beam splitter is classically well understood. The purpose of this work is to enable one to describe the action of a lossless plane-parallel beam splitter on an incident light beam in purely quantum-mechanical terms. Such a description may be obtained in terms of a unitary transformation that connects the incoming radiation-field operator to the outgoing one. This transformation, which we shall derive and discuss here, provides an elegant means to address issues and concepts that are intrinsically quantum-mechanical.
A lossless beamsplitter has certain (complex-valued) probability amplitudes for sending an incoming photon into one of two possible directions. We use elementary laws of classical and quantum optics to obtain general relations among the magnitudes and phases of these probability amplitudes. Proceeding to examine a pair of (nearly) single-mode wavepackets in the number-states n1 and n2 that simultaneously arrive at the splitter's input ports, we find the distribution of photon-number states at the output ports using an argument inspired by Feynman's scattering analysis of indistinguishable Bose particles. The result thus obtained coincides with that of the standard quantum-optical treatment of beamsplitters via annihilation and creation operators â and â†. A simple application of the Feynman method provides a form of justification for the Bose enhancement implicit in the well-known formulas ân=nn−1 and â†n=n+1n+1.