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

Quantum Lattice-Gas Algorithm for Quantum Turbulence - CAP Simulations on 12,288 Cores of Cray XT-5 Einstein at NAVO

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

Published Jun 1, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1109/hpcmp-ugc.2009.20

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•Jan 1, 2011•1 citation
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.
A unitary qubit lattice algorithm, which scales almost perfectly to the full number of cores available (e.g., 216000 cores on a CRAY XT5), is used to examine quantum turbulence and its interrelationship to classical turbulence with production runs on grids up to 57603. The maximal grids achievable by conventional CFD for quantum turbulence is just 20483, and artificial dissipation had to be introduced. Our unitary algorithms preserve the Hamiltonian structure of the Gross-Pitaevskii equation which describes quantum turbulence in a zero-temperature Bose-Einstein condensate (BEC). As a result, parameter regimes have been uncovered which exhibit very short Poincare recurrence time, as well as a strong triple cascade structure in the kinetic energy spectrum. Moreover, a detailed examination of the incompressible kinetic energy spectrum has revealed for the first time within a turbulence simulation the κ−17/5 quantum Kelvin wave cascade. By generalizing the unitary entanglement operators on the 2 qubits, a finite temperature BEC system is examined. These unitary qubit lattice algorithms are directly applicable to quantum computers as they become available.
The main topological feature of a superfluid is a quantum vortex with an identifiable inner and outer radius. A novel unitary quantum lattice gas algorithm is used to simulate quantum turbulence of a Bose-Einstein condensate superfluid described by the Gross-Pitaevskii equation on grids up to 5760(3). For the first time, an accurate power-law scaling for the quantum Kelvin wave cascade is determined: k(-3). The incompressible kinetic energy spectrum exhibits very distinct power-law spectra in 3 ranges of k space: a classical Kolmogorov k(-(5/3)) spectrum at scales greater than the outer radius of individual quantum vortex cores and a quantum Kelvin wave cascade spectrum k(-3) on scales smaller than the inner radius of the quantum vortex core. The k(-3) quantum Kelvin wave spectrum due to phonon radiation is robust, while the k(-(5/3)) classical Kolmogorov spectrum becomes robust on large grids.
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.
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.
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.
We examine the energy cascades and quantum vortex structures in two-dimensional quantum turbulence through a special unitary time evolution algorithm. An early attempt at using the Lattice Boltzmann Method proved successful in correctly representing some features of the Nonlinear Schrodinger System (NLS), such as the phase shift following the one-dimensional soliton-soliton collision, as well as the two-dimentional modulation instability. However, to accurately evaluate NLS, the implicit Euler method is required to resolve the time evolution, which is computationally expensive. A more accurate and efficient method, the Quantum Lattice Gas model is employed to simulate the quantum turbulence governed by the Gross-Pitaevskii equation, an equaiton that describes the evolution of the ground state wave function for a Bose-Einstein condensate (BEC). It is discovered that when the ratio of the internal energy to the kinetic energy is below 0.05, an unexpected short Poincare recurrence occurs independent of the initial profile of the wave function. It is demonstrated that this short recurrence is destroyed as the internal energy is strengthened. to compare the two-dimensional quantum turbulence with its classical counterpart, the incompressible energy spectra of quantum turbulence is analyzed. However, the result reveals no sign of dual cascades which is a hallmark of the classical incompressible two-dimensional fluid (inverse energy cascade to large scales with a direct cascade of enstrophy to small scales). It is the spectra of the compressible energy that can exhibits multiple cascades, but this is strongly dependent on the initial condition.