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Physical Review A

Interference in hyperbolic space

S. Chaturvedi, Gerard J Milburn, Zhongxi Zhang

Published Mar 1, 1998• 4 citations
Publisher

Abstract

The interference in a phase space algorithm of Schleich and Wheeler [Nature 326, 574 (1987)] is extended to the hyperbolic space underlying the group SU(1,1). The extension involves introducing the notion of weighted areas. Analytic expressions for the asymptotic forms for overlaps between the eigenstates of the generators of su(1,1) thus obtained are found to be in excellent agreement with the numerical results.

DOI
10.1103/physreva.57.1529

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The interference in a phase space algorithm of Schleich and Wheeler [Nature 326, 574 (1987)] is extended to the hyperbolic space underlying the group SU(1,1). The extension involves introducing the notion of weighted areas. Analytic expressions for the asymptotic forms for overlaps between the eigenstates of the generators of su(1,1) thus obtained are found to be in excellent agreement with the numerical results.
We extend the interference in the phase-space algorithm of Wheeler and Schleich [W. P. Schleich and J. A. Wheeler, Nature 326, 574 (1987)] to the case of a compact, spherical topology in order to discuss the large j limits of the angular momentum marginal probability distributions. These distributions are given in terms of the standard rotation matrices. It is shown that the asymptotic distributions are given very simply by areas of overlap in the classical spherical phase-space parametrized by the components of angular momentum. The results indicate the very general validity of the interference in phase-space concept for computing semiclassical limits in quantum mechanics.
We apply geometric phase ideas to coherent states to shed light on interference phenomenon in the phase space description of continuous variable Cartesian quantum systems. In contrast to Young's interference characterized by path lengths, phase space interference turns out to be determined by areas. The motivating idea is Pancharatnam's concept of "being in phase" for Hilbert space vectors. Applied to the overcomplete family of coherent states, we are led to preferred one-dimensional integral representations for various states of physical significance, such as the position, momentum, Fock states and the squeezed vacuum. These are special in the sense of being "in-phase superpositions". Area considerations emerge naturally within a fully quantum mechanical context. Interestingly, the Q-function is maximized along the line of such superpositions. We also get a fresh perspective on the Bohr-Sommerfeld quantization condition. Finally, we use our exact integral representations to obtain asymptotic expansions for state overlaps and matrix elements, leading to phase space area considerations similar to the ones noted earlier in the seminal works of Schleich, Wheeler and collaborators, but now from the perspective of geometric phase.
We apply geometric phase ideas to coherent states to shed light on interference phenomenon in the phase space description of continuous variable Cartesian quantum systems. The motivating idea is Pancharatnam's concept of two Hilbert space vectors being ’’in phase”. This leads to special sums and integrals of vectors which are locally in-phase superpositions. Geometric phase considerations naturally lead to the emergence of phase space areas. Applied to the overcomplete family of coherent states, we are led to preferred in-phase integral representations for various states of physical significance, such as the position, momentum and Fock states as well as the squeezed vacuum state. Interestingly, the Husimi-Kano Q function is maximized along the line of such superpositions. We also get a fresh perspective on the Bohr-Sommerfeld quantization condition. Finally, we use our exact integral representations to obtain asymptotic expansions for state overlaps and matrix elements, leading to phase space area considerations similar to the ones noted earlier in the seminal works of Schleich, Wheeler and collaborators, but now from a geometric phase perspective.