Fractional Schrödinger Dynamics and Decoherence
We study the dynamics of the Schrödinger equation with a fractional Laplacian (−Δ)α and the decoherence of the solution is observed. Analytically, we obtain equations of motion for the expected position and momentum in the fractional Schödinger equation, equations that are the fractional counterpart of the well-known Newtonian equations of motion for the standard (α=1α=1) Schrödinger equation. Numerically, we propose an explicit, effective numerical method for solving the time-dependent fractional nonlinear Schrödinger equation—a method that has high order spatial accuracy, requires little memory, and has low computational cost. We apply our method to study the dynamics of fractional Schrödinger equation and find that the nonlocal interactions from the fractional Laplacian introduce decoherence into the solution. The local nonlinear interactions can however reduce or delay the emergence of decoherence. Moreover, we find that the solution of the standard NLS behaves more like a particle, but the solution of the fractional NLS behaves more like a wave with interference effects.
K. Kirkpatrick and Y. Zhang, "Fractional Schrödinger Dynamics and Decoherence," Physica D: Nonlinear Phenomena, vol. 332, pp. 41-54, Elsevier, Oct 2016.
The definitive version is available at https://doi.org/10.1016/j.physd.2016.05.015
Mathematics and Statistics
Center for High Performance Computing Research
Keywords and Phrases
Fractional Schrödinger equation; Fourier pseudo-spectral method; Center of mass; Fractional momentum; Decoherence
International Standard Serial Number (ISSN)
Article - Journal
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