Numerical Implementation of Quantum Fluid Dynamics: A Working Example

Xu-Guang Hu, Tak‐San Ho, HERSCHEL A. RABITZ +1 more

The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation psi = exp(R + iS)/[symbol: see text]. The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density rho = magnitude of psi 2 and on the structure of R = [symbol: see text]/2 ln rho generally being simpler and smoother than rho. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool.

Masashi Himi, Kenji Fukushima

Jeffrey Yepez

Jonathan E. Chen

Several theoretical methods for the computation of quantum dynamical quantities are formulated, implemented, and applied, in the overall context of overcoming the exponential scaling property of quantum mechanics. The convergence of the approximate forward-backward semiclassical dynamics (FBSD) method has been accelerated by working with only the imaginary part of the real-time position and momentum time correlation functions. Simple manipulations allow the computationally favorable limit of a single factorization (“bead”) of the Boltzmann operator to yield greatly improved results. Numerical examples for a one-dimensional anharmonic oscillator and Lennard-Jones fluids are presented. Bohm's trajectory formulation of quantum mechanics is an exact mapping of the time-dependent Schrödinger equation onto classical fluid dynamics, plus one extra term responsible for all non-classical effects, the dreaded quantum potential. A computational procedure has been introduced to stably solve Bohm's hydrodynamic equations of motion in the forward-backward quantum dynamics (FBQD) formulation of time correlation functions, and applied to one-dimensional bound systems at zero and low temperature. A key step in the formulation of this procedure was the use of Hamilton's law of varying action, a generalization of Hamilton's principle of stationary action applicable to initial-value problems. This has resulted in the first reported synthetic computation of non-stationary quantum trajectories for a fully anharmonic oscillator. Success in one dimension has laid the groundwork for future extension to a rigorous quantum-classical scheme, which is briefly discussed. Other unpublished methods for obtaining time correlation functions and propagators in quantum dynamics are also discussed.

Andrew A. Voitiv, Mark E. Siemens, Mark T. Lusk

We propose a method to controllably create free-space, “hard-trapped” two-dimensional fluids with Bessel-Gaussian beam superpositions. We use these systems to experimentally demonstrate several vortex dynamics that resemble dynamics numerically predicted in weakly-interacting quantum fluids. Theoretically, we perform a Bessel-mode decomposition on an initial state containing an arrangement of optical vortices. The experiment is performed using a laser beam that is structured holographically with a spatial light modulator that contains the initial condition of the reconstructed field from the theoretical decomposition. This works allows for the experimental verification of many numerically predicted dynamics or the discovery of new dynamics from experiment.

Chia‐Liang Lin, Jenn‐Kun Kuo, Tsu-Hsu Yen