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Modeling The Gross-Pitaevskii Equation Using The Quantum Lattice Gas Method

Armen Oganesov

Published Jan 1, 2017• 1 citation
Publisher

Abstract

We present an improved Quantum Lattice Gas (QLG) algorithm as a mesoscopic unitary perturbative representation of the mean field Gross Pitaevskii (GP) equation for Bose–Einstein Condensates (BECs). The method employs an interleaved sequence of unitary collide and stream operators. QLG is applicable to many different scalar potentials in the weak interaction regime and has been used to model the Korteweg–de Vries (KdV), Burgers and GP equations. It can be implemented on both quantum and classical computers and is extremely scalable. We present results for 1D soliton solutions with positive and negative internal interactions, as well as vector solitons with inelastic scattering. In higher dimensions we look at the behavior of vortex ring reconnection. A further improvement is considered with a proper operator splitting technique via a Fourier transformation. This is great for quantum computers since the quantum FFT is exponentially faster than its classical counterpart which involves non-local data on the entire lattice (Quantum FFT is the backbone of the Shor algorithm for quantum factorization). We also present an imaginary time method in which we transform the Schrödinger equation into a diffusion equation for recovering ground state initial conditions of a quantum system suitable for the QLG algorithm.

DOI
10.21220/s2-2dcm-7v23

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We present an improved Quantum Lattice Gas (QLG) algorithm as a mesoscopic unitary perturbative representation of the mean field Gross Pitaevskii (GP) equation for Bose–Einstein Condensates (BECs). The method employs an interleaved sequence of unitary collide and stream operators. QLG is applicable to many different scalar potentials in the weak interaction regime and has been used to model the Korteweg–de Vries (KdV), Burgers and GP equations. It can be implemented on both quantum and classical computers and is extremely scalable. We present results for 1D soliton solutions with positive and negative internal interactions, as well as vector solitons with inelastic scattering. In higher dimensions we look at the behavior of vortex ring reconnection. A further improvement is considered with a proper operator splitting technique via a Fourier transformation. This is great for quantum computers since the quantum FFT is exponentially faster than its classical counterpart which involves non-local data on the entire lattice (Quantum FFT is the backbone of the Shor algorithm for quantum factorization). We also present an imaginary time method in which we transform the Schrödinger equation into a diffusion equation for recovering ground state initial conditions of a quantum system suitable for the QLG algorithm.
A new unitary quantum lattice gas (QLG) algorithm is proposed as a mesoscopic unitary perturbative representation of the mean field Gross Pitaevskii equation for Bose–Einstein Condensates (BECs). This consists of an interleaved sequence of unitary collide-stream operators and is tested on the 1D nonlinear Schrödinger (NLS) equation since exact soliton solutions are well known. An earlier QLG algorithm, based on the collision operator has been found to have limited application to spinor-BECs. Here, a new unitary collision operator, based on the recent QLG of Yepez for the Dirac particle, is used to model the 1D NLS soliton–soliton problem. It is found that this new unitary algorithm can handle parameters (soliton amplitudes and speeds) a factor of over 20 greater than those under the previous algorithm.[Re: (v.07) To be submitted to special issue ‘Plasma and Fluid Dynamics: Computational, Experimental and Theoretical Advancement’]
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 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).
By modifying the collision operator in the quantum lattice gas (QLG) algorithm one can develop an imaginary time (IT) integration to determine the ground state solutions of the Schrödinger equation and its variants. These solutions are compared to those found by other methods (in particular the backward-Euler finite-difference scheme and the quantum lattice Boltzmann). In particular, the ground state of the quantum harmonic oscillator is considered as well as bright solitons in the one-dimensional (1D) non-linear Schrödinger equation. The dark solitons in an external potential are then determined. An advantage of the QLG IT algorithm is the avoidance of any real/complex matrix inversion and that its extension to arbitrary dimensions is straightforward.
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
Quantum vortex structures and energy cascades are examined for two dimensional quantum turbulence (2D QT) using a special unitary evolution algorithm. The qubit lattice gas (QLG) algorithm, is employed to simulate the weakly-coupled Bose-Einstein condensate (BEC) governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a very short Poincare recurrence time. This short recurrence time is destroyed as the nonlinear interaction energy is increased. Energy cascades for 2D QT are considered to examine whether 2D QT exhibits the inverse cascades of 2D classical turbulence. In the parameter regime considered, the spectra analysis reveals no such dual cascades---dual cascades being a hallmark of 2D classical turbulence.