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arXiv:1404.2586 · Physical Review A

Simple model for interactions and corrections to the Gross-Pitaevskii equation

Hagar Veksler, Shmuel Fishman, Wolfgang Ketterle

Published Aug 13, 2014• 36 citations
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Abstract

One of the assumptions leading to the Gross-Pitaevskii equation (GPE) is that the interaction between atom pairs can be written effectively as a $\ensuremath{\delta}$ function so that the interaction range of the particles is assumed to vanish. A simple model that takes into account the extension of the interparticle potential is introduced. The correction to the GPE predictions for the energy of a condensate confined by a harmonic trap in the Thomas-Fermi regime is estimated. Although it is found to be small, we believe that in some situations it can be measured using its dependance on the frequency of the confining trap. Due to the simplicity of the model, it may have a wide range of applications.

DOI
10.1103/physreva.90.023620

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cond-mat.quant-gasCross-listedarXiv:2609.13497•Sep 15, 2026•0 citations
One of the assumptions leading to the Gross-Pitaevskii equation (GPE) is that the interaction between atom pairs can be written effectively as a \ensuremathδ\ensuremath{\delta}\ensuremathδ function so that the interaction range of the particles is assumed to vanish. A simple model that takes into account the extension of the interparticle potential is introduced. The correction to the GPE predictions for the energy of a condensate confined by a harmonic trap in the Thomas-Fermi regime is estimated. Although it is found to be small, we believe that in some situations it can be measured using its dependance on the frequency of the confining trap. Due to the simplicity of the model, it may have a wide range of applications.
We present an effective potential and the corresponding modified Gross-Pitaevskii equation, which account for the energy dependence of the two-body scattering amplitude through an effective-range expansion. For the ground-state energy of a trapped condensate, the theory leads to what we call a shape-dependent confinement correction that improves agreements with diffusion Monte Carlo calculations. The theory illustrates, for relatively strong confinement and/or high density, how the shape dependence on atom-atom interaction can come into play in a many-atom quantum system.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas-Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. Finally, in Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4πħ 2 a / m . In coordinate space this corresponds to a contact interaction U 0 δ( r – r ′), where r and r ′ are the positions of the two particles. To investigate the energy of man y-body states we adopt a Hartree or mean-field approach, and assume that the wave function is a symmetrized product of single-particle wave functions.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas–Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. In Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. Finally, in Sec. 6.5 we show how the magnetic dipole–dipole interaction, which is long-ranged and anisotropic, may be included in the Gross–Pitaevskii equation and determine within the Thomas–Fermi approximation its effect on the density distribution. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4π ħ 2 a/m .
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.
The Gross-Pitaevskii equation (GPE), that describes the wave function of a number of coherent Bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GPE consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a spectral method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained. This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.
We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.