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arXiv:cond-mat/0202344 · Physical Review A

Topology of the ground state of two interacting Bose-Einstein condensates

Francesco Riboli, Michele Modugno

Published Jun 13, 2002• 113 citations
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Abstract

We investigate the spatial patterns of the ground state of two interacting Bose-Einstein condensates. We consider the general case of two different atomic species (with different mass and in different hyperfine states) trapped in a magnetic potential whose eigenaxes can be tilted with respect to the vertical direction, giving rise to a nontrivial gravitational sag. Despite the complicated geometry, we show that within the Thomas-Fermi approximations and upon appropriate coordinate transformations, the equations for the density distributions can be put into a very simple form. Starting from these expressions we give explicit rules to classify the different spatial topologies that can be produced, and we discuss how the behavior of the system is influenced by the interatomic scattering length. We also compare explicit examples with the full numeric Gross-Pitaevskii calculation.

DOI
10.1103/physreva.65.063614

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We investigate the spatial patterns of the ground state of two interacting Bose-Einstein condensates. We consider the general case of two different atomic species (with different mass and in different hyperfine states) trapped in a magnetic potential whose eigenaxes can be tilted with respect to the vertical direction, giving rise to a nontrivial gravitational sag. Despite the complicated geometry, we show that within the Thomas-Fermi approximations and upon appropriate coordinate transformations, the equations for the density distributions can be put into a very simple form. Starting from these expressions we give explicit rules to classify the different spatial topologies that can be produced, and we discuss how the behavior of the system is influenced by the interatomic scattering length. We also compare explicit examples with the full numeric Gross-Pitaevskii calculation.
We identify all possible classes of solutions for two-component Bose-Einstein condensates (BECs) within the Thomas-Fermi (TF) approximation and check these results against numerical simulations of the coupled Gross-Pitaevskii equations (GPEs). We find that they can be divided into two general categories. The first class contains solutions with a region of overlap between the components. The other class consists of non-overlapping wavefunctions and also contains solutions that do not possess the symmetry of the trap. The chemical potential and average energy can be found for both classes within the TF approximation by solving a set of coupled algebraic equations representing the normalization conditions for each component. A ground state minimizing the energy (within both classes of states) is found for a given set of parameters characterizing the scattering length and confining potential. In the TF approximation, the ground state always shares the symmetry of the trap. However, a full numerical solution of the coupled GPEs, incorporating the kinetic energy of the BEC atoms, can sometimes select a broken-symmetry state as the ground state of the system. We also investigate effects of finite-range interactions on the structure of the ground state.
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 .
We study the dynamics of two interacting Bose-Einstein condensates, by numerically solving two coupled Gross-Pitaevskii equations at zero temperature. We consider the case of a sudden transfer of atoms between two trapped states with different magnetic moments: the two condensates are initially created with the same density profile, but are trapped into different magnetic potentials, whose minima are vertically displaced by a distance much larger than the initial size of both condensates. Then the two condensates begin to perform collective oscillations, undergoing a complex evolution, characterized by collisions between the two condensates. We investigate the effects of their mutual interaction on the center-of-mass oscillations and on the time evolution of the aspect ratios. Our theoretical analysis provides a useful insight into the recent experimental observations by Maddaloni et al. [Phys. Rev. Lett. 85, 2413 (2000)].
Abstract We study harmonically trapped two-species Bose–Einstein condensates within the Gross–Pitaevskii formalism. By invoking the Thomas–Fermi approximation, we derive an analytical solution for the miscible ground state in a particular region of the system’s parameter space. This solution furnishes a simple formula that relates the interspecies interaction strength to the second spatial moments of the density distribution of the minority condensate species. Accompanying numerical simulations confirm the accuracy of the solution and the interaction strength formula for sufficiently large numbers of condensed particles. The introduced formula may provide an additional scheme for determining interspecies scattering lengths that complements existing methods, such as those based on collective-excitation spectroscopy of the two-species condensates or on collisional measurements on thermal samples.