Projected Gross–Pitaevskii Equation for Ring-Shaped Bose–Einstein Condensates

Abstract

We propose an alternative implementation of the projected Gross–Pitaevskki equation adapted for a numerical modeling of the atomic Bose–Einstein condensate trapped in a toroidally shaped potential. We present an accurate efficient scheme to evaluate the required matrix elements and to calculate tthe ime evolution of the matter wave field. We analyze the stability and accuracy of the developed method for equilibrium and nonequilibrium solutions in a ring-shaped trap with an additional barrier potential corresponding to recent experimental realizations.

Superflow decay in a toroidal Bose gas: The effect of quantum and thermal fluctuations

“…( 2) with γ = 0) onto this single-particle basis is detailed explicitly in Ref. [85], so we will not describe it here. Implementing the number-damping reservoir interaction terms in this basis is a straightforward extension of their method.…”

“…( 5) are not analytically known. However, recent work by Prikhodko et al [85] has demonstrated that an approximate single-particle basis may be used:…”

“…As described in Ref. [85], this basis is approximately orthonormal provided it is truncated such that the highest-energy mode vanishes as r → 0. In this limit, this basis diagonalises the single-particle Hamiltonian with the energy spectrum E (r) α = ħ hω r (α +…”

“…( 2) in a basis of approximate single-particle modes of the toroidal trap, extending the approach outlined for the projected GPE in Ref. [85]. A notable benefit of this approach is that the energy cutoff can be consistently defined in this singleparticle basis, which diagonalises the Hamiltonian at high energies.…”

Superflow decay in a toroidal Bose gas: The effect of quantum and thermal fluctuations

“…( 2) in a basis of approximate single-particle modes of the toroidal trap, extending the approach outlined for the projected GPE in Ref. [84]. A notable benefit of this approach is that the energy cutoff can be consistently defined in this single-particle basis, which diagonalises the Hamiltonian at high energies.…”