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arXiv:1209.2603 · Physical Review A

Equilibrium solutions for immiscible two-species Bose-Einstein condensates in perturbed harmonic traps

R. W. Pattinson, T. P. Billam, S. A. Gardiner, Daniel McCarron, H. W. Cho, Simon L. Cornish, Nicholas G. Parker, Nick P. Proukakis

Published Jan 23, 2013• 51 citations
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Abstract

We investigate the mean-field equilibrium solutions for a two-species immiscible Bose-Einstein condensate confined by a harmonic confinement with additional linear perturbations. We observe a range of equilibrium density structures, including ball and shell formations and axially or radially separated states, with a marked sensitivity to the potential perturbations and the relative atom number in each species. Incorporation of linear trap perturbations, albeit weak, are found to be essential to match the range of equilibrium density profiles observed in a recent ${}^{87}$Rb-${}^{133}$Cs Bose-Einstein condensate experiment [McCarron et al., Phys. Rev. A 84, 011603(R) (2011)]. Our analysis of this experiment demonstrates that sensitivity to linear trap perturbations is likely to be an important factor in interpreting the results of similar experiments in the future.

DOI
10.1103/physreva.87.013625

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We investigate the mean-field equilibrium solutions for a two-species immiscible Bose-Einstein condensate confined by a harmonic confinement with additional linear perturbations. We observe a range of equilibrium density structures, including ball and shell formations and axially or radially separated states, with a marked sensitivity to the potential perturbations and the relative atom number in each species. Incorporation of linear trap perturbations, albeit weak, are found to be essential to match the range of equilibrium density profiles observed in a recent 87{}^{87}87Rb-133{}^{133}133Cs Bose-Einstein condensate experiment [McCarron et al., Phys. Rev. A 84, 011603(R) (2011)]. Our analysis of this experiment demonstrates that sensitivity to linear trap perturbations is likely to be an important factor in interpreting the results of similar experiments in the future.
In this Thesis we study steady state solutions and dynamical evolutions of two– component atomic Bose–Einstein Condensates. We initially investigate the equilibrium properties of condensate mixtures in harmonic trapping potentials at zero temperature. Subsequently we simulate the coupled growth of these condensates by inclusion of damping terms. Finally, we investigate the evolution of coupled Bose gases via the so-called classical–field method. A recent experiment [D. J. McCarron et al., Phys. Rev. A, 84, 011603(R) (2011)] achieved Bose–Einstein Condensation of a two–species 87Rb–133Cs phase segregated mixture in harmonic trapping potentials. Depending on relative atom numbers of the two species, three distinct regimes of density distributions were observed. For these experimental parameters, we investigate the corresponding time–independent ground state solutions through numerical simulations of the coupled Gross–Pitaevskii equations. By including experimentally relevant shifts between the traps, we observe a range of structures including ‘ball and shell’ formations and axially/radially separated states. These are found to be very sensitive to the trap shifts. For all three experimental regimes, our numerical simulations reveal good qualitative agreement. The observed experimental profiles cannot be guaranteed to be fully equilibrated. This, coupled with the rapid sympathetic cooling of the experimental system, leads to a situation where growth may play a determining factor in the density structures formed. To investigate this further, we introduce phenomenological damping to describe the associated condensate growth/decay, revealing a range of transient structures. However, such a model always predicts the predominance of one condensate species over longer evolution times. Work undertaken by collaborators with the more elaborate Stochastic Projected Gross–Pitaevskii equations, which can describe condensate formation by coupling to a heat bath, predicts the spontaneous formation of dark–bright solitons. Motivated by this, we show how the presence of solitons can affect the condensate distribution, thus highlighting the overall dynamical role in the emerging patterns. Finally, we use classical field methods to analyse the evolution of non trapped Bose gases from strongly nonequilibrium initial distributions. The contrast between miscible (overlapping) and immiscible (phase segregated) components gives rise to important distinctions for condensate fractions and the formation of domains and vortices. In addition, splitting the particles of a single component thermalised state into two components is investigated. We then study the effects of suddenly quenching the strength of the interspecies interactions. Under suitable conditions, this quench generates isotropic vortex tangles. While this tangle subsequently decays over time, we propose how a repeat sequence of quenches at regular intervals could be employed to drive the tangle, thereby potentially providing a novel route to the generation of quantum turbulence.
Abstract Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.
We introduce a phenomenological mean-field model to describe the growth of immiscible two-species atomic Bose-Einstein condensates towards some equilibrium. Our model is based on the coupled Gross-Pitaevskii equations with the addition of dissipative terms to account for growth. While our model may be applied generally, we take a recent Rb-Cs experiment [McCarron et al., Phys. Rev. A 84 011603(R) (2011)] as a case study. As the condensates grow, they can pass through ranging transient density structures which can be distinct from the equilibrium states, although such a model always predicts the predominance of one condensate species over longer evolution times.
We show that the observed features of the above-named Bose–Einstein condensates can be understood via an effective confining potential of the form of: [Formula: see text] where T denotes the temperature, m the mass of an atom of the trapped gas, ω the geometric mean of the three frequencies used for confinement, k the Boltzmann constant and b a dimensionless perturbation parameter. Such an exercise is undertaken because Tcs calculated via earlier treatments based solely on an r2-potential lead to a mismatch with the experimental values. We fix b by substituting the density of states corresponding to V(r, T) into the equation for the number of excited atoms N } exc(T) and appealing to the experimental data at T = Tc. The values of b thus found are: 1.3426 (7 Li ), 1.8420 (23 Na ), 0.4998 (41 K ), 0.3486 (85 Rb ), 1.5332 (87 Rb ) and 1.2430 (133 Cs ). While these are used to calculate N exc (T) for each of the condensates at T = Tc/2 and Tc/10, we also report on: (a) the variation of b for each condensate for some selected values of the pair (N } exc, Tc) and (b) the possibility of realizing the state (N exc , pTc; p (a number) ≫1) for all of these condensates with a unique value of b, even though the parameter-sets {m, ω, N exc , Tc} characterizing them differ widely. Attention is drawn to diverse fields where T-dependent Hamiltonians have found useful application.
We study the dilute fermion gas with pairing between two species and unequal concentrations in a harmonic trap using the mean-field theory and the local density approximation. We found that the system can exhibit a superfluid shell structure sandwiched by the normal fermions. This superfluid shell structure occurs if the mass ratio is larger than a certain critical value, which increases from the weak-coupling BCS region to the strong-coupling Bose-Einstein condensation (BEC) side. In the strong-coupling BEC regime, the radii of the superfluid phase are less sensitive to the mass ratios and are similar to the case of pairing with equal masses. However, the lighter leftover fermions are easier to mix with the superfluid core than the heavier ones. A partially polarized superfluid can be found if the majority fermions are lighter, whereas phase separation is still found if they are heavier.