Temporal focusing and evolution of two-dimensional Gauss–Hermite quantum states

Abstract

Standard quantum wave packets typically undergo immediate and irreversible dispersion in free space. This study investigates the spatiotemporal evolution of two-dimensional correlated Gauss–Hermite states, which exhibit a "contractive phase" through engineered initial position–momentum anti-correlations (C < 0), where C is the parameter that characterizes the initial anti-correlation between the position and momentum observables of the wave packet. Leveraging the mathematical isomorphism between the time-dependent Schr¨odinger equation and the paraxial wave equation, we develop a generalized two-dimensional analytical framework and demonstrate the resulting "temporal lens" effect. Numerical simulations of the n = m = 1 mode reveal that the correlation parameter C acts as a temporal scheduler, shifting the focal point at which the packet reaches its absolute minimum-uncertainty bound while preserving its inherent nodal topology. Specifically, simulations for C = –0.57 and C = –1.15 show radial-uncertainty Δr reductions of 13.5% and 34.6%, respectively, relative to their initial states. These findings confirm that the spatiotemporal evolution of matter waves can be controlled precisely by shaping the initial phase curvature, thereby providing a mechanism for scheduling maximum-localization events in free-space quantum systems.