Numerical method for evolving the Projected Gross-Pitaevskii equation

In this paper we describe a method for evolving the projected Gross-Pitaevskii equation (PGPE) for a Bose gas in a harmonic oscillator potential. The central difficulty in solving this equation is the requirement that the classical field is restricted to a small set of prescribed modes that constitute the low energy classical region of the system. We present a scheme, using a Hermite-polynomial based spectral representation, that precisely implements this mode restriction and allows an efficient and accurate solution of the PGPE. We show equilibrium and non-equilibrium results from the application of the PGPE to an anisotropic trapped three-dimensional Bose gas.

💡 Research Summary

The paper presents a comprehensive numerical framework for solving the Projected Gross‑Pitaevskii Equation (PGPE), which describes the finite‑temperature dynamics of a trapped Bose gas while explicitly restricting the field to a low‑energy “classical” subspace. The authors first motivate the need for a projector: only modes whose single‑particle energies lie below a cutoff ε_cut are treated as highly occupied classical fields; the remaining high‑energy modes form an incoherent reservoir that is not evolved directly. Implementing this projector efficiently is the central challenge because the evolution must preserve all retained modes with high fidelity—standard split‑step or Crank‑Nicolson schemes that work for zero‑temperature GPE (where only the condensate mode is occupied) are insufficient.

The authors adopt a spectral Galerkin approach. They expand the field ψ(x,t) in the eigenbasis {φ_n(x)} of the single‑particle Hamiltonian H₀ (harmonic oscillator for the trapped case, plane waves for the uniform case). The expansion coefficients c_n(t) become the dynamical variables. By projecting the PGPE onto this basis, the equation of motion for each coefficient reads
∂c_n/∂t = –i