A more accurate analysis of Bose-Einstein condensation in harmonic traps

Abstract

Using the Euler-Maclaurin summation we calculate analytically the internal energy for non-interacting bosons confined within a harmonic oscillator potential. The specific heat shows a sharp λ-like peak indicating a condensation into the ground state at a well-defined transition temperature. Full agreement is obtained with direct numerical calculation of the same quantities. When the number of trapped particles is very large and at temperatures near and above the transition temperature, the results also agree with previous approximate calculations. At extremely low temperatures both the specific heat and the number of particles excited from the condensate are exponentially suppressed.Bose-Einstein condensation has now been experimentally demonstrated in magnetic traps of rubidium[1], lithium [2] and most recently sodium[3] gases. To a good approximation one describes the trapping potentials as 3-dimensional anisotropic oscillators which in the rubidium experiments[1] have frequencies typically around ω = 500 − 1000 Hz. For the sake of simplification we will here ignore the anisotropy and the weak interactions between the alkali gas atoms. The energy levels of one particle are then simply given as ε n =hωn where the quantum number n takes the values n = 0, 1, 2 . . . when we drop the zero-point energy. Since each energy level has a degeneracy g n = (n + 1)(n + 2)/2, the total number N of particles in such a trap at temperature T and chemical potential µ is given by the Bose-Einstein distribution aswhere β = 1/k B T . The ground state has quantum number n = 0 and thus contains N 0 = λ/(1 − λ) particles where λ = exp (βµ) is the fugacity. We then havewhere the number of particles in the higher states isg n λe −bn 1 − λe −bn1

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

“…10(a) demonstrate. The cusp-like structure of the heat capacity that was predicted theoretically for a number of the confining potentials [35,36,[40][41][42][43] and observed experimentally for, e.g., the dilute Bose gas of 87 Rb atoms [44], for the infinite number of particles, N = ∞, turns at T = T cr into the discontinuity that is a manifestation of the phase transition; in our case, it is a transition from the BE condensate to the normal phase of the noninteracting particles in the linear potential. This justifies the definition of the critical temperature from Eq.…”

Optimized perturbation theory for charged scalar fields at finite temperature and in an external magnetic field

“…To our knowledge, one of the few works that we are aware of that have previously made use of the EM formula before were for example Refs. [44][45][46]. In Ref.…”

“…I, this is exactly the region and order in the approximation that lead to the largest errors. In [45] the EM formula was used to obtain an expression for the effective potential for vector bosons and to study pair production in an external magnetic field, while in [46] it was used to obtain analytical expressions for the internal energies for non-interacting bosons confined within a harmonic oscillator potential. None of these previous works have assessed the reliability of the use of the EM formula.…”

Hartree-Fock Approach for Rotating Bose-Einstein Condensation in One-dimensional Deep Optical Lattice

“…The latter is obtained as usual [17,30,31] by setting in Eq. (23) equal to zero, thus (24) This result enabled us to investigate the effects of the and alpha on . Indeed, in Fig.…”

“…The effect of the rotation is to change the shape of the distribution function of the thermal atoms so that, the thermal density out of the condensate takes the form [23,24], (7) After substituting the Hamiltonian and [ doing the p integration by making the change of variables [25], the integral in Eq. (7) takes the same form as in the absence of rotation with an effective frequencies and (8) where is the thermal de-Broglie wavelength and is the effective fugacity.…”