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Annual Meeting Optical Society of America

Lorentz-group Berry phases in squeezed light

R. Y. Chiao, Paul G. Kwiat, William A. Vareka, Thomas F. Jordan

Published Jan 1, 1988• 1 citation
Publisher

Abstract

The Berry phases most often observed in experiments are produced by rotations. Berry phases produced by Lorentz transformations are almost as simple mathematically. They have been discussed at length, but there has been no suggestion of observing them in an experiment. Indeed, it has been shown that in one context they can be removed by a canonical transformation; thus they appear to be a property of the mathematical description that could not be observed physically. Here we show that the essential element of these Berry phases could be observed as a change of phase of the electromagnetic field in squeezed states of light or microwaves, as could be produced by degenerate parametric amplifiers. It could be seen as a fringe shift in interference experiments. In this way, a physical effect of the squeezing could be measured without measuring anything like photon statistics or noise.

DOI
10.1364/oam.1988.mr17

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The Berry phases most often observed in experiments are produced by rotations. Berry phases produced by Lorentz transformations are almost as simple mathematically. They have been discussed at length, but there has been no suggestion of observing them in an experiment. Indeed, it has been shown that in one context they can be removed by a canonical transformation; thus they appear to be a property of the mathematical description that could not be observed physically. Here we show that the essential element of these Berry phases could be observed as a change of phase of the electromagnetic field in squeezed states of light or microwaves, as could be produced by degenerate parametric amplifiers. It could be seen as a fringe shift in interference experiments. In this way, a physical effect of the squeezing could be measured without measuring anything like photon statistics or noise.
Squeezed states of the electromagnetic field are characterized by a reduction of fluctuations for one of two quadrature phase amplitudes below the level of fluctuations for the vacuum state. This reduction is a manifestly quantum or nonclassical feature which is of great interest in its own right but also potentially important for precision measurement with sensitivity beyond the shot-noise or vacuum-state limit. We describe three different experiments in which squeezed light has been generated in our laboratory involving parametric downconversion, intracavity frequency doubling, and optical bistability with two-state atoms. By employing the squeezed light produced by a subthreshold optical parametric oscillator, we have achieved improvements in sensitivity beyond the vacuum-state limit for the detection of both phase and amplitude changes of the electromagnetic field. An alternate route to sensitivity beyond the vacuum-state limit is through quantum nondemolition measurement, where we have suggested and are implementing a scheme involving parametric conversion and polarization mixing.
A squeezed state of the electromagnetic field has less noise in one of its two amplitude components than that of a coherent state. This property makes squeezed states potentially useful in low noise and precision measurement. Squeezed states may be generated by, among other things, degenerate parametric amplifiers and four-wave mixers. The reduced noise in one of the amplitude components is most easily observed via a homodyne detector. A number of experimental efforts to generate and detect squeezed states are under way. Here, in describing experimental techniques, the experiments' carried out by Slusher et al. are emphasized. A number of techniques were employed to successfully generate squeezed light. An atomic beam of sodium provided the optical nonlinearity in a cavity-backward four-wave mixer. By employing pairs of Fabry-Perot modes spaced symmetrically about the pump frequency, squeezed light was generated in nondegenerate conditions where Rayleigh scattering of pump light could be avoided. To observe squeezing, a number of nonidealities such as phase jitter and detector inefficiencies also had to be overcome.
Berry's phase of the Lorentz group has been measured in a system of connected nonlinear electronic oscillators. Each oscillator was driven so that it was in the vicinity of a period-doubling bifurcation where the response to a small perturbation transforms similarly to a Lorentz boost of a spinor. By connecting the output of one oscillator to the input of the next, it was possible to make a closed loop on a hyperboloid (the invariant surface of the Lorentz group), the enclosed area of which is Berry's phase. Good agreement between theory and experiment is found.
It is shown that classical linear canonical transformations are possible in the Wigner phase space. Coherent and squeezed states are shown to be linear canonical transforms of the ground-state harmonic oscillator. It is therefore possible to evaluate the Wigner functions for coherent and squeezed states from that for the harmonic oscillator. Since the group of linear canonical transformations has a subgroup whose algebraic property is the same as that of the (2+1)-dimensional Lorentz group, it may be possible to test certain properties of the Lorentz group using optical devices. A possible experiment to measure the Wigner rotation angle is discussed.
The Berry phase acquired by an electromagnetic field undergoing an adiabatic and cyclic evolution in phase space is a purely quantum-mechanical effect of the field. However, this phase is usually accompanied by a dynamical contribution and cannot be manifested in any light-beam interference experiment because it is independent of the field state. We here show that such a phase can be produced using an atom coupled to a quantized field and driven by a slowly changing classical field, and it is manifested in atomic Ramsey interference oscillations. We also show how this effect may be applied to one-step implementation of multiqubit geometric phase gates, which is impossible by previous geometric methods. The effects of dissipation and fluctuations in the parameters of the pump field on the Berry phase and visibility of the Ramsey interference fringes are analyzed.