Quantum theory of the lossless beam splitter

Sudhakar Prasad, Marlan O. Scully, W. Martienssen

Sotirios Danakas, P. K. Aravind

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.

Francese Pi, Gaspar T. Orriols

The energy balance in the superposition of light waves with a lossless beam splitter is analyzed along with its connection with some general relations between the parameters describing beam-splitter effects on the two incoming waves. These general relations are derived either from the electromagnetic theory or from energy conservation considerations. The apparent paradox of a symmetric beam splitter producing complementary outputs is explained.

Stephen Mark Barnett, John Jeffers, Jonas Söderholm

Sotirios Danakas, P. K. Aravind

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.

Franck Laloë, William J. Mullin