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arXiv:1307.4031

Solution of the Gross-Pitaevskii equation in terms of the associated non-linear Hartree potential

George H. Rawitscher

Published Jul 15, 2013• 0 citations
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Abstract

The Gross-Pitaevskii equation (GP), 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 GP equation 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 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.

DOI
10.48550/arxiv.1307.4031

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The Gross-Pitaevskii equation (GP), 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 GP equation 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 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.
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.
The mean-field properties of trapped, two-dimensional interacting Bose gases of finite size are surveyed numerically. The coupled nonlinear equations of the Gross-Pitaevskii (GP) equation and the Hartree-Fock equation for the condensate and the thermal component, respectively, are calculated by the hybrid method, in which the GP equation and the low-lying excitations less than cut-off energy are calculated quantum mechanically using the basis representation, while the high-lying states are obtained within the local density approximation. Consistent solutions are obtained for the whole range of temperatures including the critical temperature by determining the thermodynamic chemical potential µ from the condition that the total atomic number must be the desired value, instead of identifying µ with the eigenvalue of the GP equation. Comparisons with the results of the semiclassical two-fluid model are presented.
"The time-independent one-dimensional Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate is solved by using Green functionformalism atT=0 K. From the integral GP equation, the analytical solution for the order parameter,©(x), and the chemical potential,π, has been formally obtained as functions of a single dimensionless parameter. The obtained solutions are valid in the whole range ofnegative and positive values of the non-linear interaction term that is used to characterize the condensates of alkali atoms. A function hasbeen introduced in order to describe the accumulated error that allows a comparative analysis with respect to known methods of solutionsuch as the Thomas-Fermi approximation and the “bright” soliton. This study sets universal ranges of parameters where analytical solutionscan be implemented."
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.
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.
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.