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Journal of Modern Optics

The gouy phase shift as a geometrical quantum effect

P. Hariharan, Peter A. Robinson

Published Feb 1, 1996• 69 citations
Publisher

Abstract

The phase shift experienced by a light wave when it passes through a focus (the Gouy phase shift) can be regarded as a geometrical effect arising, via the uncertainty principle, from a modification of the volume of space available for the propagation of the photon. This effect is closely related to the shift in the phase of the wave function of a particle travelling along a tube; however, it does not require the presence of physical boundaries.

DOI
10.1080/09500349608232735

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The phase shift experienced by a light wave when it passes through a focus (the Gouy phase shift) can be regarded as a geometrical effect arising, via the uncertainty principle, from a modification of the volume of space available for the propagation of the photon. This effect is closely related to the shift in the phase of the wave function of a particle travelling along a tube; however, it does not require the presence of physical boundaries.
The transverse spatial structure of a paraxial beam of light is fully characterized by a set of parameters that vary only slowly under free propagation. They specify bosonic ladder operators that connect modes of different orders, in analogy to the ladder operators connecting harmonic-oscillator wave functions. The parameter spaces underlying sets of higher-order modes are isomorphic to the parameter space of the ladder operators. We study the geometry of this space and the geometric phase that arises from it. This phase constitutes the ultimate generalization of the Gouy phase in paraxial wave optics. It reduces to the ordinary Gouy phase and the geometric phase of nonastigmatic optical modes with orbital angular momentum in limiting cases. We briefly discuss the well-known analogy between geometric phases and the Aharonov–Bohm effect, which provides some complementary insights into the geometric nature and origin of the generalized Gouy phase shift. Our method also applies to the quantum-mechanical description of wave packets. It allows for obtaining complete sets of normalized solutions of the Schrödinger equation. Cyclic transformations of such wave packets give rise to a phase shift, which has a geometric interpretation in terms of the other degrees of freedom involved.
We show that the well known geometric phase, the Gouy phase in optics can be defined for in vacuum as well. In particular we show that the underlying physics for the matter waves Gouy phase is the generalized Schroedinger-Robertson uncertainty principle, more specifically, the off diagonal elements of the covariance matrix. Recent experiments involving the diffraction of fullerene molecules and the uncertainty principle are shown to be quantitatively consistent with the existence of a Gouy phase for waves.
Abstract Controlling the evolution of photonic quantum states is crucial for most quantum information processing and metrology tasks. Due to its importance, many mechanisms of quantum state evolution have been tested in detail and are well understood; however, the fundamental phase anomaly of evolving waves, called the Gouy phase, has had a limited number of studies in the context of elementary quantum states of light, especially in the case of photon number states. Here we outline a simple method for calculating the quantum state evolution upon propagation and demonstrate experimentally how this quantum Gouy phase affects two-photon quantum states. Our results show that the increased phase sensitivity of multi-photon states also extends to this fundamental phase anomaly and has to be taken into account to fully understand the state evolution. We further demonstrate how the Gouy phase can be used as a tool for manipulating quantum states of any bosonic system in future quantum technologies, outline a possible application in quantum-enhanced sensing, and dispel a common misconception attributing the increased phase sensitivity of multi-photon quantum states solely to an effective de Broglie wavelength.