Nonuniform Bose-Einstein condensate: I. An improvement of the Gross-Pitaevskii method

A nonuniform condensate is usually described by the Gross-Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz (Psi) over cap (r, t) = Psi ( r , t ) . Proceeding from a more accurate operator ansatz (Psi) over cap (r, t) = (a) over cap (0)Psi(r , t)/root N , where N is the number of Bose particles, we find the equation i h partial derivative Psi(r,t)/partial derivative t = -h(2)/2m partial derivative(2)Psi(r , t)/partial derivative r(2) + (1 - 1/N)2c Psi(r , t)|Psi( r , t )|(2) , which we call the GP(N ) equation. It differs from the GP equation by the factor (1 - 1/N). We compare the accuracy of the GP and GP(N) equations by analysing the ground state of a one-dimensional system of point bosons with repulsive interaction (c > 0) and zero boundary conditions. Both equations are solved numerically, and the system energy E and the particle density profile rho(x) are determined for the mean particle density (rho) over bar (x) = 1, different values of N and of the coupling constant gamma = c/(rho) over bar. The solutions are compared with the exact ones obtained by the Bethe ansatz. The results show that the GP and GPN equations equally well describe the many-particle system (N greater than or similar to 100) being in the weak coupling regime (N-2 <<gamma less than or similar to 0.1). But for the few-boson system (N less than or similar to 10) with gamma <= N- 2 the solutions of the GP(N) equation are in much better agreement with the exact ones. Thus, the multiplier ( 1 - 1/N) allows one to describe few-boson systems with high accuracy. This means that it is reasonable to extend the notion of Bose-Einstein condensation to few-particle systems.