Tensor network methods for the Gross-Pitaevskii equation on fine grids

TL;DR

This paper demonstrates how tensor network methods can efficiently simulate the Gross-Pitaevskii equation on large grids, enabling detailed analysis of complex cold atomic gas dynamics that are difficult for traditional numerical methods.

Contribution

It introduces tensor network techniques, including matrix product operators, for simulating the Gross-Pitaevskii equation on fine grids, improving computational feasibility for turbulent and short-range interaction phenomena.

Findings

Tensor networks enable large-scale simulations of the Gross-Pitaevskii equation.

Spectral methods with tensor networks improve equilibrium and dynamic state calculations.

Efficient simulation of vortex formation and non-equilibrium dynamics in cold gases.

Abstract

The Gross-Pitaevskii equation and its generalisations to dissipative and dipolar gases have been very useful in describing dynamics of cold atomic gases, as well as polaritons and other nonlinear systems. For some of these applications the numerically accessible grid spacing can become a limiting factor, especially in describing turbulent dynamics and short-range effects of dipole-dipole interactions. We explore the application of tensor networks to these systems, where (in analogy to related work in fluid and plasma dynamics), they allow for physically motivated data compression that makes simulations possible on large spatial grids which would be unfeasible with direct numerical simulations. Analysing different non-equilibrium cases involving vortex formation, we find that these methods are particularly efficient, especially in combination with a matrix product operator representation of the quantum Fourier transform, which enables a spectral approach to calculation of both equilibrium states and time-dependent dynamics. The efficiency of these methods has interesting physical implications for the structure in the states that are generated by these dynamics, and provides a path to describe cold gas experiments that are challenging for existing methods.