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Physical Review A

Gross-Pitaevskii Equation for a Strongly Interacting Superfluid Boson System

Donald H. Kobe

Published Feb 1, 1972• 5 citations
Publisher

Abstract

The Gross-Pitaevskii equation for the order parameter of a superfluid boson system is derived in a form which is applicable to a system in which the two-body potential has a strong repulsive core, as is the case for liquid helium. The equation of motion for the order parameter, defined as the matrix element of the field annihilation operator between the states of an ($N\ensuremath{-}1$)-particle and an $N$-particle system, is obtained. The term describing the average potential due to the other particles is expanded in a perturbation series, in which the coherentstate representation is used. The resulting expansion is the same as using the Bogoliubov approximation and Wick's theorem, and shows that the Bogoliubov expansion is exact in the limit of an infinite number of particles. The perturbation expansion is partially resummed to give a Hartree field in which the potential has been replaced by the $T$ matrix, which gives a finite average effective field.

DOI
10.1103/physreva.5.854

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•Feb 26, 2008•2 citations
The Gross-Pitaevskii equation for the order parameter of a superfluid boson system is derived in a form which is applicable to a system in which the two-body potential has a strong repulsive core, as is the case for liquid helium. The equation of motion for the order parameter, defined as the matrix element of the field annihilation operator between the states of an (N\ensuremath−1N\ensuremath{-}1N\ensuremath−1)-particle and an NNN-particle system, is obtained. The term describing the average potential due to the other particles is expanded in a perturbation series, in which the coherentstate representation is used. The resulting expansion is the same as using the Bogoliubov approximation and Wick's theorem, and shows that the Bogoliubov expansion is exact in the limit of an infinite number of particles. The perturbation expansion is partially resummed to give a Hartree field in which the potential has been replaced by the TTT matrix, which gives a finite average effective field.
We derive the time-independent Gross-Pitaevskii equation at zero temperature for condensed bosons, which form as bound-fermion pairs when the mutual fermionic attractive interaction is sufficiently strong, from the strong-coupling limit of the Bogoliubov-de Gennes equations that describe superfluid fermions in the presence of an external potential. Three-body corrections to the Gross-Pitaevskii equation are also obtained by our approach. Our results are relevant to the recent advances with ultracold fermionic atoms in a trap.
The relative-density amplitude profile of a quantized vortex line in He II is calculated from a generalized Gross-Pitaevskii (GP) equation. The ordinary GP equation is a Hartree nonlinear integrodifferential equation for the order parameter of an interacting system of bosons. The generalization is made by replacing the two-body potential in the average field by a local TTT matrix, which is calculated from a realistic interatomic potential for helium. In configuration space the TTT matrix is similar to the potential, except in the repulsive region where it is much softer. The only adjustable parameter in the generalized GP equation is the condensate fraction for which a theoretical value of 0.10 and an experimental value of 0.024 are used. In all cases the density profile is a monotonically increasing function with an effective core radius of from 3.7 to 4.7 \AA{}.
We investigate the properties of the one-dimensional Bose gas at zero temperature, for which exact results exist for some model systems. We treat the interactions between particles in the gas with an approximate form of the many-body T-matrix, and find a form of Gross-Pitaevskii equation valid in 1D. The results presented agree with the exact models in both the weakly and strongly interacting limits, and interpolate smoothly between them. We also investigate the use of mean-field treatments of trapped BECs to describe the 1D system in the strongly interacting limit. We find that the use of the many-body T-matrix to describe interactions leads to qualitative agreement with the exact models for some physical quantities. We indicate how the standard mean-field treatments need to be modified to extend and improve the agreement.
The issue of adequate describing a gas of complex particles, composed of an even number of fermions, as point bosons is studied in alkali metals. In the low-density approximation we obtain the equation for the complex order parameter Φ(r1, r2) representing the wave function of atoms with taking into account the internal motion of the valence electrons. For superfluid systems formed by particles with internal degrees of freedom, this equation replaces the Gross-Pitaevskii equation. It is shown that, in general, exchange effects should be considered in the same approximation as effects of the direct interaction of atoms with each other. In particular, in the case of only the Coulomb interaction the neglect of exchange effects leads to qualitatively incorrect results. The problem of spontaneous polarization in superfluid systems is studied. The expression for the electric polarization of the inhomogeneous superfluid system is obtained.
Consider a system of NNN bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj))N^2V(N(x_i-x_j))N2V(N(xi​−xj​)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HNH_NHN​ denote the Hamiltonian of the system and let ψN,t\psi_{N,t}ψN,t​ be the solution to the Schrodinger equation. Suppose that the initial data ψN,0\psi_{N,0}ψN,0​ satisfies the energy condition ≤CN>. \leq C N >. ≤CN>. and that the one-particle density matrix converges to a projection as N→∞N \to \inftyN→∞. Then, we prove that the kkk-particle density matrices of ψN,t\psi_{N,t}ψN,t​ factorize in the limit N→∞N \to \inftyN→∞. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential VVV. In \cite{ESY}, we proved the same statement under the condition that the interaction potential VVV is sufficiently small; in the present work we extend this result to large potentials.