I. Da Paz

UNIVERSIDADE FEDERAL DO PIAUÍ (BRAZIL), CCN - FÍSICA, Faculty Member

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Gouy phase and quantum interference with cross-Wigner functions for matter-waves

Scientific reports, May 28, 2024

The Gouy phase is essential for accurately describing various wave phenomena, ranging from classi... more The Gouy phase is essential for accurately describing various wave phenomena, ranging from classical electromagnetic waves to matter waves and quantum optics. In this work, we employ phase-space methods based on the cross-Wigner transformation to analyze spatial and temporal interference in the evolution of matter waves characterized initially by a correlated Gaussian wave packet. First, we consider the cross-Wigner of the initial wave function with its free evolution, and second for the evolution through a double-slit arrangement. Different from the wave function which acquires a global Gouy phase, we find that the cross-Wigner acquires a Gouy phase difference due to different evolution times. The results suggest that temporal like-Gouy phase difference is important for an accurate description of temporal interference. Furthermore, we propose a technique based on the Wigner function to reconstruct the cross-Wigner from the spatial intensity interference term in a double-slit experiment with matter waves. Gouy phase is a wave phenomenon that appears when waves are confined transversely to their propagation direction, such as in the case of lens focusing and diffraction through slits. It was first observed for classical light waves 1,2 and it has been recently investigated for the transverse confinement of matter waves 3-6 . The Gouy phase and its properties have been extensively studied 7-10 , and the acquired phase is known to depend on the type of transversal confinement and the geometry of the waves. For instance, line-focusing a cylindrical wave propagating from -∞ to +∞ yields a Gouy phase of π/2 , while point-focusing a spherical wave in the same interval yields a Gouy phase of π 11 . Gaussian matter wave packets diffracting through small apertures generate a Gouy phase of π/4 12 . Experiments were performed with acoustic and water waves 13-15 , surface plasmon-polaritons with non-Gaussian spatial properties 16 , focused cylindrical phonon-polariton wave packets in LiTaO 3 crystals, and for Bose-Einstein condensates and electron waves . Interestingly, the effects of the quantum Gouy phase was also verified with the evolution of two-photon states . Applications, such as in evaluating the resonant frequencies in laser cavities 22 , in phase-matching in strongfield and high-order harmonic generation , and in describing the spatial profile of laser pulses with high repetition rate 26 was shown to be feasible. In addition, an extra Gouy phase appears in optical and matter waves depending on the orbital angular momentum's magnitude . In recent work, it was found that the Gouy phase may cause nonlocal effects that modify the symmetries of self-organization in atomic systems 29 . This phase may also be useful in communication and optical tweezers using structured light 30 . Gouy phases in matter waves also display potential applications and can be used in mode converters in quantum information systems 31 , in the generation of singular electron optics 20 and in the study of non-classical (exotic or looped) paths in interference experiments . Since the Gouy phase of matter waves is directly related to position-momentum correlations 4,34 , we propose in this work to investigate the role of the Gouy phase employing initial position-momentum correlations by using the cross-Wigner transform to analyze the temporal evolution of matter waves. The cross-Wigner transform has been employed in different contexts such as signal processing , and quantum measurement theory . In signal processing, for instance, it is used to analyze the time-frequency content of signals as it allows one to analyze signals in both the time and frequency domains simultaneously . In quantum theory, the cross-Wigner was employed as an analog of the two-state vector formalism 39 for continuous variables 38 and to derive the weak values of an observable from a complex quasi-probability distribution associated with it 37 . Its physical

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