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arXiv:0805.0609

Gouy Phase for Fullerene Molecules

I. G. da Paz, Maria Carolina Nemes, Carlos Henrique Monken, Sebastião Pádua, J. G. Peixoto de Faria

Published May 5, 2008• 0 citations
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Abstract

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.

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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.
In this thesis, we study theoretical and experimental aspects of the Gouy phase in matter waves. We show that this phase is related to the covariance between position and momentum and can be indirectly extracted from interferometric experiments. Next, we analyze a diffraction experiment with macro-molecules realized by Zeilingers group [1] and we show that the existence of a Gouy phase is compatible with those experimental data. Similar to the Gouy phase for classical electromagnetic waves, it is shown that a beam of matter waves suffers a phase anomaly around the focus of a quantum lens. An experimental proposal to measure this phase is presented with experimental parameters viable to current technology. We hope this work motivates studies of the theoretical problem, open a few years ago, and forexperiments that lead us to an understanding deeper and more realistic of the Gouy phase for mixed states of matter waves and light. At the end, we present our conclusions.
The Heisenberg uncertainty principle for material objects is an essential corner stone of quantum mechanics and clearly visualizes the wave nature of matter. Here, we report a demonstration of the Heisenberg uncertainty principle for the fullerene molecule C70{\mathrm{C}}_{70}C70​ at a temperature of 900 K. We do this by showing the increase in molecular momentum spread after passage through a narrow slit with a variable width down to 70 nm. We find good quantitative agreement with the theoretical expectation.
Gravitational acceleration has been measured with high precision using macroscopic probes and for a few quantum probes. A recent diffraction experiment (Arndt M et al 1999 Nature 401 680) has demonstrated the wave-particle duality for the C 60 fullerene molecules. In this paper, a new method to measure gravitational acceleration with fullerene molecules is proposed. The g value is deduced from the measurement of the phase shift of a thermal beam in a three-grating Mach–Zehnder interferometer. The experimental setup and the uncertainty sources are discussed. A measurement with a precision of 1 × 10 −4 is feasible with 10 8 thermal fullerene molecules and gratings with 500 nm period. The precision achievable by the proposed method is limited by the velocity spread of the fullerene beam.
Recently there have been experimental results on Poisson spot matter-wave interferometry followed by theoretical models describing the relative importance of the wave and particle behaviors for the phenomenon. We propose an analytical theoretical model for Poisson's spot with matter waves based on the Babinet principle, in which we use the results for free propagation and single-slit diffraction. We take into account effects of loss of coherence and finite detection area using the propagator for a quantum particle interacting with an environment. We observe that the matter-wave Gouy phase plays a role in the existence of the central peak and thus corroborates the predominantly wavelike character of the Poisson's spot. Our model shows remarkable agreement with the experimental data for deuterium (D2{\text{D}}_{2}D2​) molecules.