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Physical Review A

Analytic, one-dimensional model of anN-boson condensate exhibiting saturation

A. C. Merchant, M. P. Isidro Filho

Published Feb 1, 1988• 2 citations
Publisher

Abstract

Application of the variational Hartree method to a system of N bosons interacting via Skyrme-type forces allows us to describe the single-boson wave function for the ground-state condensate in terms of the nonlinear time-independent Schr\"odinger equation, ${\mathit{\ensuremath{\nabla}}\phantom{\rule{0ex}{0ex}}}^{2}$\ensuremath{\varphi}=${\mathrm{\ensuremath{\varepsilon}}}_{1}$\ensuremath{\varphi}-A\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{2}$\ensuremath{\varphi}+B\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{4}$\ensuremath{\varphi}. In one dimension, not only is \ensuremath{\varphi} found by analytic integration of this equation, but the single-boson separation energy, the Hartree potential energy, the kinetic energy, and the mean-square radius are also all evaluated in closed form. The resulting expressions show that the model exhibits the property of saturation.

DOI
10.1103/physreva.37.1371

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The variational Hartree method is applied to a system of N bosons interacting via Skyrme-type attractive and repulsive forces. When the system condenses into its ground state, the wave function for a single boson, \ensuremath{\varphi}, may be described by means of the nonlinear, time-independent Schr\"odinger equation, ${\mathrm{\ensuremath{\nabla}}}^{2}$\ensuremath{\varphi}=${\mathrm{\ensuremath{\epsilon}}}_{1}$\ensuremath{\varphi}-A\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{2}$\ensuremath{\varphi}+B\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{4}$\ensuremath{\varphi}. In one dimension, this equation has a single, analytic, bound state solution and exhibits the property of saturation. In the physically more interesting case of three dimensions with spherical symmetry, no analytic solutions are known, so that a numerical solution must be resorted to, and this shows that saturation is again obtained. However, an analytic approximation to the three-dimensional wave function, which is very accurate for values of ${\ensuremath{\epsilon}}_{1}$B/${A}^{2}$>0.1, is deduced and studied. Approximate analytic expressions for the Hartree potential energy, kinetic energy, and mean square radius of the system are thereby derived, and applications to finite nuclei and infinite nuclear matter are considered.

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Application of the variational Hartree method to a system of N bosons interacting via Skyrme-type forces allows us to describe the single-boson wave function for the ground-state condensate in terms of the nonlinear time-independent Schr"odinger equation, \ensuremath∇2{\mathit{\ensuremath{\nabla}}\phantom{\rule{0ex}{0ex}}}^{2}\ensuremath∇2\ensuremath{\varphi}=\ensuremathε1{\mathrm{\ensuremath{\varepsilon}}}_{1}\ensuremathε1​\ensuremath{\varphi}-A\ensuremath{\Vert}\ensuremath{\varphi}\ensuremath∥2{\mathrm{\ensuremath{\Vert}}}^{2}\ensuremath∥2\ensuremath{\varphi}+B\ensuremath{\Vert}\ensuremath{\varphi}\ensuremath∥4{\mathrm{\ensuremath{\Vert}}}^{4}\ensuremath∥4\ensuremath{\varphi}. In one dimension, not only is \ensuremath{\varphi} found by analytic integration of this equation, but the single-boson separation energy, the Hartree potential energy, the kinetic energy, and the mean-square radius are also all evaluated in closed form. The resulting expressions show that the model exhibits the property of saturation.
The variational Hartree method is applied to a system of N bosons interacting via Skyrme-type attractive and repulsive forces. When the system condenses into its ground state, the wave function for a single boson, \ensuremath{\varphi}, may be described by means of the nonlinear, time-independent Schr\"odinger equation, ${\mathrm{\ensuremath{\nabla}}}^{2}$\ensuremath{\varphi}=${\mathrm{\ensuremath{\epsilon}}}_{1}$\ensuremath{\varphi}-A\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{2}$\ensuremath{\varphi}+B\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{4}$\ensuremath{\varphi}. In one dimension, this equation has a single, analytic, bound state solution and exhibits the property of saturation. In the physically more interesting case of three dimensions with spherical symmetry, no analytic solutions are known, so that a numerical solution must be resorted to, and this shows that saturation is again obtained. However, an analytic approximation to the three-dimensional wave function, which is very accurate for values of ${\ensuremath{\epsilon}}_{1}$B/A2{A}^{2}A2>0.1, is deduced and studied. Approximate analytic expressions for the Hartree potential energy, kinetic energy, and mean square radius of the system are thereby derived, and applications to finite nuclei and infinite nuclear matter are considered.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
Several properties of trapped hard-sphere bosons are evaluated using variational Monte Carlo techniques. A trial wave function composed of a renormalized single-particle Gaussian and a hard-sphere Jastrow function for pair correlations is used to study the sensitivity of condensate and noncondensate properties to the hard-sphere radius and the number of particles. Special attention is given to diagonalizing the one-body density matrix and obtaining the corresponding single-particle natural orbitals and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single-particle orbital with the highest occupation. The effect of interaction on other quantities such as the ground-state energy, the mean radial displacement, and the momentum distribution is calculated as well. Results are compared with mean-field theory in the dilute limit.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
We use the assumption that the potential for the AAA-boson system can be written as a sum of pairwise acting forces to decompose the wave function into Faddeev components that fulfill a Faddeev type equation. Expanding these components in terms of potential harmonic (PH) polynomials and projecting on the potential basis for a specific pair of particles results in a two-variable integro-differential equations suitable for AAA-boson bound-state studies. The solution of the equation requires the evaluation of Jacobi polynomials ${P}_{K}^{\ensuremath{\alpha},\ensuremath{\beta}}(x)$ and of the weight function W(z)W(z)W(z) which give severe numerical problems for very large AAA. However, using appropriate limits for $A\ensuremath{\rightarrow}\ensuremath{\infty}$ we obtain a variant equation which depends only on the input two-body interaction, and the kernel in the integral part has a simple analytic form. This equation can be readily applied to a variety of bosonic systems such as microclusters of noble gasses. We employ it to obtain results for $A\ensuremath{\in}(10\ensuremath{-}100)$ 87Rb^{87}\mathrm{Rb}87Rb atoms interacting via interatomic interactions and confined by an externally applied trapping potential Vtrap(r){V}_{\mathrm{trap}}(r)Vtrap​(r). Our results are in excellent agreement with those previously obtained using the potential harmonic expansion method (PHEM) and the diffusion Monte Carlo (DMC) method.