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International Journal of Modern Physics B

A COMPARATIVE STUDY OF THE EXPERIMENTAL FEATURES OF THE BOSE–EINSTEIN CONDENSATES OF 7 Li , 23 Na , 41 K , 85 Rb , 87 Rb AND 133 Cs VIA A LINEARLY PERTURBED HARMONIC OSCILLATOR POTENTIAL

G. P. Malik, Vijaya S. Varma

Published Apr 24, 2013• 5 citations

Abstract

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.

DOI: 10.1142/s0217979213500422

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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.

Abstract This chapter describes the main properties of the ideal Bose gas confined by a harmonic potential. It calculates the temperature dependence of the condensate density and identify the value of the critical temperature in terms of the oscillator frequency and the number of atoms. This identification provides an important temperature scale for experiments with ultracold atomic gases. The chapter also discusses the behaviour of the expanding cloud after the release of the trap and how the anisotropy of the gas evolves in time, reflecting the initial anisotropy of the momentum distribution. It finally discusses the possibility of getting Bose–Einstein condensation through adiabatic transformations, working with non-harmonic trapping potentials or with mixtures of two atomic species trapped by different harmonic potentials.

Although atomic Bose gases are experimentally investigated in the dilute regime, interparticle interactions play an important role on the transition temperatures of Bose–Einstein condensation. In this study, Bose–Einstein condensation is handled using fractional calculus for a Bose gas consisting of interacting bosons which are trapped in a three-dimensional harmonic oscillator. In this frame, in order to introduce the nonextensive effect, fractionally generalized Bose–Einstein distribution function which features Mittag–Leffler function is adopted. The dependence of the transition temperature of Bose–Einstein condensation on α (a measure of fractality of space) has been established. The transition temperatures for the dilute 87 Rb , 23 Na and 7 Li atomic gases have been obtained in consistent with experimental data and the nature of the interactions in the Bose–Einstein condensate has been enlightened. In the course of our investigations, we have arrived to the conclusion that for α < 1 attractive interactions and for α > 1 repulsive interactions are predominant.

The experimental realization in 1995 of Bose–Einstein condensation in dilute atomic gases marked the beginning of a very rapid development in the study of quantum gases. The initial experiments were performed on vapours of rubidium, sodium, and lithium. So far, the atoms 1 H, 7 Li, 23 Na, 39 K, 41 K, 52 Cr, 85 Rb, 87 Rb, 133 Cs, 170 Yb, 174 Yb and 4 He * (the helium atom in an excited state) have been demonstrated to undergo Bose–Einstein condensation. In related developments, atomic Fermi gases have been cooled to well below the degeneracy temperature, and a superfluid state with correlated pairs of fermions has been observed. Also molecules consisting of pairs of fermionic atoms such as 6 Li or 40 K have been observed to undergo Bose–Einstein condensation. Atoms have been put into optical lattices, thereby allowing the study of many-body systems that are realizations of models used in condensed matter physics. Although the gases are very dilute, the atoms can be made to interact strongly, thus providing new challenges for the description of strongly correlated many-body systems. In a period of less than ten years the study of dilute quantum gases has changed from an esoteric topic to an integral part of contemporary physics, with strong ties to molecular, atomic, subatomic and condensed matter physics. The dilute quantum gases differ from ordinary gases, liquids and solids in a number of ways, as we shall now illustrate by giving values of physical quantities.

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.

The thermodynamical properties of interacting Bose atoms in a harmonic potential are studied within the mean-field approximation. For weak interactions, the quantum statistics is equivalent to an ideal gas in an effective mean-field potential. The eigenvalue of the Gross–Pitaevskii equation is identified as the chemical potential of the ideal gas. The condensation temperature and density profile of atoms are calculated. It is found that the critical temperature Tc decreases as the interactions increase. Below the critical point, the condensation fraction exhibits a universal relation of N0/N = 1-(T/Tc)γ, with the index γ ≈ 2.3 independent of the interaction strength, the chemical potential, as well as the frequency of the confining potential.

In this work we shall present the overview of our work to study trapped quantum gases in terms of global thermodynamic variables, as well as in terms of coherent modes excitation and investigation of critical phenomena within a condensate. Using cold Na atoms we have produced a sample of Bose‐Einstein condensate atoms, with which we have performed the first experiments to establish an equation of state using global thermodynamic variables. In our system using Rb, we are working to continue on this direction as well as to implement and study the excitation of coherent modes on the confining potential.