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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

Soliton dynamics and nonlinear quantum mechanics

Tullio A. Minelli, A. Pascolini

Published Mar 1, 1980• 5 citations
Publisher
DOI
10.1007/bf02759268

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•Apr 27, 2012•69 citations
The nonlinear Schrödinger equation (NLSE) model with an external harmonic potential is one of the most important in modern science. This model makes it possible to analyze a variety of nonlinear phenomena, in nonlinear optics and laser physics, biophysics and in the theory of Bose-Einstein condensation of atoms. It is shown that the main specific feature of the dynamics of dark GP matter wave solitons in a parabolic trap is the formation of solitons with dynamically changing form-factors producing the periodic variation in the modulation depth (the degree of “blackness”) of dark solitons. In general, the period of dark soliton oscillations in trapping potential depends on the specific conditions of the experiment and does not coincide with the oscillation period of a linear quantum-mechanical oscillator. In the case of an immobile pedestal in the trap, the oscillation period of the black soliton considerably increases because of the periodic transformation of the black soliton to the gray one and vice versa. Surprisingly, that if the dark soliton is superimposed on the base pedestal oscillating in the trap and displaced from the trap center, the oscillation period of the dark soliton coincides with the period of oscillations of the linear harmonic oscillator, while the dynamics of the dark soliton is similar to that of a classical particle obeying the Newton mechanics laws.
We numerically study the classical and quantum dynamics of an atomic bright soliton in a highly elongated one-dimensional harmonic trap with a Gaussian barrier. In the regime of the recent experiment by Dyke et al the system realizes a coherent nonlinear soliton beam-splitter and interferometer whose accuracy we analyze. In the case of tighter radial trap confinement and enhanced quantum fluctuations, a split soliton can represent a spin-squeezed, or alternatively, a fragmented condensate with reduced phase coherence that can be measured by interfering the split soliton by the barrier. We also find large quantum mechanical uncertainties in the soliton's position and momentum due to nonlinear interaction with the barrier.
We numerically study the classical and quantum dynamics of an atomic bright soliton in a highly elongated one dimensional harmonic trap with a Gaussian barrier. In the regime of the recent experiment by Dyke et al the system realizes a coherent nonlinear soliton beam-splitter and interferometer whose accuracy we analyze. In the case of tighter radial trap confinement and enhanced quantum fluctuations, a split soliton can represent a spin-squeezed, or alternatively, a fragmented condensate with reduced phase coherence that can be measured by interfering the split soliton by the barrier. We also find large quantum mechanical uncertainties in the soliton’s position and momentum due to nonlinear interaction with the barrier.