Finite temperature excitations of BECs in trapsFinite temperature quantum field theory is a notoriously hard problem. One of the most intriguing consequences of the experimental realization of Bose-Einstein condensation (BEC) was the prospect of quantitative tests of finite temperature quantum field theory. The first measurements were made at JILA, Colorado in 1997 [1], and there have been many theoretical attempts to describe the data, not all of them corresponding to reality!One common approach to finite temperature field theory is to diagonalise the quadratic part of the Hamiltonian using the familiar Bogoliubov transformation to a quasi-particle basis. This has been proven to work well near T=0, however, the cubic and quartic parts of the Hamiltonian quickly become important at finite temperature. There have been many attempts at consistently including these higher order terms, many of which have problems of one sort or another. One particularly successful approach has been to do perturbation theory on these terms in the quasiparticle basis [2], and this approach has apparently resolved all ourstanding issues with the JILA data [3,4]. It remains to apply this most general calculation method to many other experimental systems. The JILA experiment was performed in a harmonic TOP trap, which is cylindrically symmetric with a z-axis frequency a factor of sqrt(8) larger than the radial frequency. Thus the calculation was performed in 2D with the number of points in each dimension comparable. However, many BEC experiments are performed in cigar shapped trapping geometries with aspect ratio ranging from 10-1000! In this situation a similar calculation for such a system is computationly unfeasible. A potential project would be to attempt another method for these systems. We envisage treating the z-axis in the local density approximation and attempting to integrate out this dimension, leaving a 1D problem. This idea is still under development. Evans-Rashid transition in a trapped Bose gas with attractive interactionsAnother interest related to this is studying the possibility of a pairing phase transition in the attractive Bose gas - in some sense 'BCS with bosons'. The possibility of a pairing transition in a Bose gas with a partly attractive model interatomic potential was first considered by Evans and Imry in the context of superfluid helium [5]. Neutron scattering experiments indicated that the fraction of Bose condensed particles was small, and it was suggested that the superfluid properties of the system could be due to a different phase transition somewhat analogous to the BCS transition in superconductors, with an order parameter m = <psi psi>. Dorre et al [6] later developed a model that allowed for both condensate and pairing theories and concluded that for helium the condensate theory had the lower free energy and the higher transition temperature.Later Stoof [7] considered the equation of state for homogeneous alkali gases with an attractive interatomic potential. He found the interesting result that the critical temperature for the pairing transition was higher than that for Bose condensation; however, both these possibilities were preceded by the mechanical collapse of the gas. Jeon et al. [8] recently reached the conclusion that such a system is generally unstable against fluctuations in pairing, and that collapse can occur due to this instability. Mueller and Baym [9] have considered pairing in the trapped gas within the local density approximation, and concluded that the instability towards collapse occurs at lower densities than the pairing instability. We have been applying a sophisticated numerical treatment based on Hartree-Fock Bogoliubov theory to study the possibility of observing the pairing transition in a trapped Bose gas with an attractive delta function interaction. These calculations have suggested that significant pairing can be expected for certain ranges of interaction strength and densities. A potential project would be to perform similar calculations for 2D systems, where it is suggested that the pairing state is more favourable. [1] D. S. Jin et al., Phys. Rev. Lett. 78 , 764 (1997). [2] S. A. Morgan, J. Phys. B 33 , 3847 (2000). [3] S. A. Morgan, M. Rusch, D. A. W. Hutchinson, and K. Burnett, cond-mat/0305535 (2003). [4] S. A. Morgan, cond-mat/0307246 (2003). [5] W. A. B. Evans and Y. Imry, Il Nuovo Cimento 63B, 155 (1969). [6] P. Dorre, H. Haug, and D. B. Tran Thoai, J. Low Temp. Phys. 35, 465 (1979). [7] H. T. C. Stoof, Phys. Rev. A, 49, 3824 (1994). [8] G. S. Jeon, L. Yin, S. W. Rhee, and D. J. Thouless, Phys. Rev. A 66, 011603 (2002). [9] E. J. Mueller and G. Baym, Phys. Rev. A, 62, 053605 (2000). |
||||||||