Bose-Einstein condensates have been, by now, observed in as many as nine different atomic assemblies of bosons. Such a condensate is quantum mechanical interacting system whose ground state properties can be studied theoretically by solving the appropriate non-linear Gross-Pitaevskii-Ginzburg GPG equation. One can now study the change in the behavior of Bose-Einstein condensate by introducing a localized impurity which interacts with the condensate as a function of position of impurity in the condensate. The introduction of such an impurity can be mimicked by simply allowing an intensely focused laser beam to interact with the condensate. This would lead to alteration of ground state properties of the condensate as it would now interact with a potential of type V Sech2(r/w) where, V and w are amplitude and width of the impurity potential, respectively. The modified GPG equation in the presence of localized impurity potential as function of position in the condensate, has been numerically solved to obtain its various ground state properties as function of position, such as total energy per particle, chemical potential, kinetic, harmonic trap potential and two-body interaction energies per particle in addition to energy associated with impurity potential, correlation length, healing length etc. We have studied the behavior of the above-mentioned ground state properties as the position of localized impurity is changed in the condensate from core to peripheral position. While the total, harmonic oscillator potential and impurity energies decrease as the position of localized impurity is displaced from core of the condensate to its periphery, the value of two-body inter-particle interaction energy increases. Further, the values of chemical potential and total energy per particle shows decrease by ~ 9% and ~ 17% respectively, leading to the inference that the stability of condensate increases as the localized impurity is moved away from the core of the condensate.
Bose-Einstein condensates have been, by now, observed in as many as nine different atomic assemblies of bosons. Such a condensate is quantum mechanical interacting system whose ground state properties can be studied theoretically by solving the appropriate non-linear Gross-Pitaevskii-Ginzburg GPG equation. One can now study the change in the behavior of Bose-Einstein condensate by introducing a localized impurity which interacts with the condensate as a function of position of impurity in the condensate. The introduction of such an impurity can be mimicked by simply allowing an intensely focused laser beam to interact with the condensate. This would lead to alteration of ground state properties of the condensate as it would now interact with a potential of type V Sech2(r/w) where, V and w are amplitude and width of the impurity potential, respectively. The modified GPG equation in the presence of localized impurity potential as function of position in the condensate, has been numerically solved to obtain its various ground state properties as function of position, such as total energy per particle, chemical potential, kinetic, harmonic trap potential and two-body interaction energies per particle in addition to energy associated with impurity potential, correlation length, healing length etc. We have studied the behavior of the above-mentioned ground state properties as the position of localized impurity is changed in the condensate from core to peripheral position. While the total, harmonic oscillator potential and impurity energies decrease as the position of localized impurity is displaced from core of the condensate to its periphery, the value of two-body inter-particle interaction energy increases. Further, the values of chemical potential and total energy per particle shows decrease by ~ 9% and ~ 17% respectively, leading to the inference that the stability of condensate increases as the localized impurity is moved away from the core of the condensate.
Quantum dynamics of impurities in a bath of bosons is a long-standing problem in solid-state, plasma, and atomic physics. Recent experimental and theoretical investigations with ultracold atoms have focused on this problem, studying atomic impurities immersed in an atomic Bose–Einstein condensate (BEC) and for various relative coupling strengths tuned by the Fano−Feshbach resonance technique. Here, we report extensive numerical simulations on a closely related problem: the collision between a bosonic impurity consisting of a few 41 K atoms and a BEC of 87 Rb atoms in a quasi one-dimensional configuration and under a weak harmonic axial confinement. For small values of the inter-species interaction strength (regardless of its sign), we find that the impurity, which starts from outside the BEC, simply causes the BEC cloud to oscillate back and forth, but the frequency of oscillation depends on the interaction strength. For intermediate couplings, after a few cycles of oscillation the impurity is captured by the BEC, and strongly changes its amplitude of oscillation. In the strong interaction regime, if the inter-species interaction is attractive, a local maximum (bright soliton) in the BEC density occurs where the impurity is trapped; if, instead, the inter-species interaction is repulsive, the impurity is not able to enter the BEC cloud and the reflection coefficient is close to one. However, if the initial displacement of the impurity is increased, the impurity is able to penetrate the cloud, leading to the appearance of a moving hole (dark soliton) in the BEC.
Ground state properties of Bose-Einstein condensate of 20000 23 Na atoms confined in isotropic and highly anisotropic magnetic traps in the presence of three-body interaction, in addition to the two-body and hard-core interactions, have been theoretically studied by solving modified highly non-linear Gross-Pitaevskii-Ginzburg (GPG) equation. There is increasingly significant change in the chemical potential, total and differential energies per particle as the aspect ratio. λ, is increased from 0.1 to 1.0, with respect to the corresponding values when only two-body and hard-core interactions are considered. The condensate order parameter changes drastically and becomes quasi 1-dimensional, as λ is varied from 1 to 0.1.
Various ground state properties such as chemical potential, differential and total energy per particle etc. of Bose-Einstein condensate of 10000 85 Rb atoms with varying repulsive self-interaction energy have been reported considering not only two-body interaction but also including the higher order terms of the low-density energy expansion of homogeneous Bose gas in the Ginzburg, Pitaevskii and Gross (GPG) equation. These include hard-core approximation of the bosons neglected earlier and the three-body interaction terms. The study is more general as it includes the terms beyond logarithm in energy density expansion. It is also shown that such a consideration does not violate the lower bound predicted earlier in which the 'constant' beyond logarithm term in the three-body interaction was neglected.
Amplifying on a proposal by O'Dell et al. for the realization of Bose-Einstein condensates of neutral atoms with attractive 1∕r interaction, we point out that the instance of self-trapping of the condensate, without an external trap potential, is physically best understood by introducing appropriate ``atomic'' units. This reveals a remarkable scaling property: the physics of the condensate depends only on the two parameters N2a∕au and $\ensuremath{\gamma}∕{N}^{2}$, where N is the particle number, a the scattering length, au the ``Bohr'' radius, and $\ensuremath{\gamma}$ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, and for energies, sizes, and peak densities, and compare with previous variational results. We point out the existence of a second solution of the extended Gross-Pitaevskii equation for negative scattering lengths, with and without trapping potential, which is born together with the ground state in a tangent bifurcation. This indicates the existence of an unstable collectively excited state of the condensate for negative scattering lengths.
Using the density-functional theory, the Ginzburg Pitaevskii Gross (GPG) equation for Bose-Einstein (BE) condensate, confined in a magnetic trap, is modified to include contribution from three-body collisions in the strongly interacting regime a>>l, 'a' is the scattering length and 'l' being the characteristic low energy length scale. This generalized GPG equation has been solved numerically using the analytically derived Thomas-Fermi order parameter, which also includes three-body interactions. The order parameter, chemical potential, extent of correlation and other ground state properties are computed when the aspect ratio, λ, is varied from1.0 to 0.05 (λ represents the anisotropy of the magnetic trap). As λ is varied from 1.0 to 0.05, the condensate shape changes from isotropic three-dimensional (3-D) to highly anisotropic quasi one-dimensional (1-D). The stability of the BE condensate increases with decrease in λ, which is also borne out by the behavior of chemical potential and the total energy per particle, as there is a decrease of about four times for a=5000 a0 as well as for a=7000 a0, 'a0' being the Bohr radius. The extent of correlations, however, increases by more than five folds, showing that quasi 1-D BE condensate is highly correlated. Both two- and three-body interaction energies show a decrease with decrease in λ: three-body interaction energy staying below two-body interaction energy for a=5000 a0 while for a=7000 a0, a cross-over occurs between the two at λ ~ 0.35. As one goes from 3-D to quasi 1-D, the percentage difference for various physical quantities, computed between only two-body interactions and when both two- and three-body interactions are considered, shows a decrease, suggesting that the effect of three-body collisions become increasingly less significant in agreement with the recent study.
In this paper, we present a theoretical study of a Bose-Einstein condensate of interacting bosons in a quartic trap in one, two, and three dimensions. Using Thomas-Fermi approximation, suitably complemented by numerical solutions of the Gross-Pitaevskii equation, we study the ground sate condensate density profiles, the chemical potential, the effects of cross-terms in the quartic potential, temporal evolution of various energy components of the condensate, and width oscillations of the condensate. Results obtained are compared with corresponding results for a bose condensate in a harmonic confinement.