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DoD High Performance Computing Modernization Program Users Group Conference (HPCMP-UGC)

Unitary Quantum Lattice Gas Algorithms for Quantum to Classical Turbulence

George Vahala, Jeffrey Yepez, Min Soe, Linda L. Vahala, Sean B. Ziegeler

Published Jun 1, 2010• 0 citations
Publisher

Abstract

Using a set of interleaved unitary collision-stream operators, a three-dimensional (3D) quantum lattice gas algorithm is devised which, on taking moments, recovers the Gross-Pitaevskii (GP) equation. If a zero-temperature Bose-Einstein condensate (BEC) is trapped in an a magnetic well, the evolution of the ground-state wave function satisfies the scalar GP equation, while if the BEC is trapped in an optical trap the ground-state wave function satisfies spin or GP equations. Quantum turbulence is studied in a scalar GP system on 5,7603 grid yielding not only the classical Kolmogorov k-5/3 cascade but also the quantum vortex k-3 spectrum. For a certain class of initial conditions, one finds an intermittent loss of tangled quantum vortices as the vortex cores attain minimal size, and thus prevent the Kelvin wave cascade (due to helical wave-wave coupling on the vortex). A coupled set of GP equations are solved for spin or BEC. Skrymions, which describe topologically-linked quantum vortices, are examined. One finds, for certain initial conditions that the incompressible kinetic energy spectrum for the condensate component of a vortex ring core rapidly departs from the k-3 linear quantum vortex spectrum.

DOI
10.1109/hpcmp-ugc.2010.15

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Using a set of interleaved unitary collision-stream operators, a three-dimensional (3D) quantum lattice gas algorithm is devised which, on taking moments, recovers the Gross-Pitaevskii (GP) equation. If a zero-temperature Bose-Einstein condensate (BEC) is trapped in an a magnetic well, the evolution of the ground-state wave function satisfies the scalar GP equation, while if the BEC is trapped in an optical trap the ground-state wave function satisfies spin or GP equations. Quantum turbulence is studied in a scalar GP system on 5,7603 grid yielding not only the classical Kolmogorov k-5/3 cascade but also the quantum vortex k-3 spectrum. For a certain class of initial conditions, one finds an intermittent loss of tangled quantum vortices as the vortex cores attain minimal size, and thus prevent the Kelvin wave cascade (due to helical wave-wave coupling on the vortex). A coupled set of GP equations are solved for spin or BEC. Skrymions, which describe topologically-linked quantum vortices, are examined. One finds, for certain initial conditions that the incompressible kinetic energy spectrum for the condensate component of a vortex ring core rapidly departs from the k-3 linear quantum vortex spectrum.
The time evolution of the ground state wave function of a zero-temperature Bose-Einstein condensate (BEC) gas is well described by the Hamiltonian Gross-Pitaevskii (GP) equation. Using a set of appropriately interleaved unitary collision-stream operators, a qubit lattice gas algorithm is devised, which on taking moments, recovers the Gross-Pitaevskii (GP) equation under diffusion ordering (time scales as length(2)). Unexpectedly, there is a class of initial states whose Poincaré recurrence time is extremely short and which, as the grid resolution is increased, scales with diffusion ordering (and not as length(3)). The spectral results of J. Yepez et al. [Phys. Rev. Lett. 103, 084501 (2009).] for quantum turbulence are revised and it is found that it is the compressible kinetic energy spectrum that exhibits three distinct spectral regions: a small-k classical-like Kolmogorov k(-5/3), a steep semiclassical cascade region, and a large-k quantum vortex spectrum k(-3). For most evolution times the incompressible kinetic energy spectrum exhibits a somewhat robust quantum vortex spectrum of k(-3) for an extended range in k with a k(-3.4) spectrum for intermediate k. For linear vortices of winding number 1 there is an intermittent loss of the quantum vortex cascade with its signature seen in the time evolution of the kinetic energy E(kin)(t), the loss of the quantum vortex k(-3) spectrum in the incompressible kinetic energy spectrum as well as the minimalization of the vortex core isosurfaces that would totally inhibit any Kelvin wave vortex cascade. In the time intervals around these intermittencies the incompressible kinetic energy also exhibits a multicascade spectrum.
Abstract : A qubit unitary lattice algorithm is developed for the mean field evolution of the ground state of a Bose-Einstein condensate whether in a magnetic or optical trap. The algorithm consists of interleaved collide-stream operators which in diffusionordering recovers the spinor Gross Pitaevskii equations and is ideally parallelized. The algorithm is benchmarked againstexact one dimensional vector inelastic soliton collision solutions. Three dimensional quantum turbulence is examined forscalar Bose Einstein condensates and the compressible kinetic energy spectrum exhibits three different cascades
A quantum lattice gas algorithm, based on interleaved unitary collide-stream operators, is used to study quantum turbulence of the ground state wave function of a Bose-Einstein condensate (BEC). The Gross-Pitaevskii equation is a Hamiltonian system for a compressible, inviscid quantum fluid. From simulations on a 57603 grid it was observed that a multi-cascade existed for the incompressible kinetic energy spectrum with universal features: the large spatial scales exhibit a classical Kolmogorov k -5/3 spectrum while the very small scales exhibit a quantum Kelvin wave cascade k-3 spectrum. Under certain conditions one can explicitly determine the Poincare recurrence of initial conditions as well as the intermittent destruction of the Kelvin wave cascade.
A novel unitary quantum lattice algorithm is developed to explore quantum turbulence. Because of its low memory requirements and its near perfect parallelization to the full 12,288 cores on the Cray XT5, simulations were run up to spatial grids of 5,7603. The Gross-Pitaevskii equation, which describes the ground state of a Bose Einstein condensate (BEC), is solved and it is found that the incompressible kinetic energy spectrum exhibits 3 distinct power laws: classical Kolmogorov k-5/3spectrum at scales much larger than the individual quantum vortex cores, and a quantum Kelvin wave cascade spectrum of k-3at scales of the order of the quantum cores. In the adjoining semiclassical regime, there is a steeper spectral decay transitioning between the classical and quantum regimes. However, its spectral exponent does not seem to be universal. This is the first, first-principle simulation yielding the universal quantum Kelvin cascade exponent.
Quantum vortex structures and energy cascades are examined for two-dimensional quantum turbulence (2D QT) at zero temperature. A special unitary evolution algorithm, the quantum lattice algorithm, is employed to simulate the Bose-Einstein condensate governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a short Poincaré recurrence time. It is demonstrated that such short recurrence times are destroyed by stronger nonlinear interaction. The similar loss of Poincaré recurrence is also seen in the 3D GP equation. Various initial conditions are considered in an attempt to discern if 2D QT exhibits inverse cascades as is seen in 2D classical turbulence (CT). In our simulation parameter regimes, no dual cascade spectra were observed for 2D QT-unlike that seen in 2D CT.
The ground state wave function for a Bose Einstein condensate is well described by the Gross-Pitaevskii equation. A Type-II quantum algorithm is devised that is ideally parallelized even on a classical computer. Only 2 qubits are required per spatial node. With unitary local collisions, streaming of entangled states and a spatially inhomogeneous unitary gauge rotation one recovers the Gross-Pitaevskii equation. Quantum vortex reconnection is simulated - even without any viscosity or resistivity (which are needed in classical vortex reconnection).