Graduate School in Nonlinear Science

 
MIDIT                              OFD                          CATS
Modelling, Nonlinear Dynamics      Optics and Fluid Dynamics    Chaos and Turbulence Studies
and Irreversible Thermodynamics    Risø National Laboratory     Niels Bohr Institute and
Technical University of Denmark    Building 128                 Department of Chemistry
Building 321                       P.O. Box 49                  University of Copenhagen
DK-2800 Lyngby                     DK-4000 Roskilde             DK-2100 Copenhagen Ø
Denmark                            Denmark                      Denmark


FINITE TEMPERATURE CORRELATIONS IN THE TRAPPED AND
UNTRAPPED BOSE-EINSTEIN GASES
IN EACH OF D=1, D=2 AND D=3 SPACE DIMENSIONS

by R.K. Bullough
UMIST, Manchester
UK


AUTHORS: R K Bullough, UMIST, Manchester with N M
Bogoliubov, C Malyshev, and V S Kapitonov, St Petersburg,and J
Timenon, U of Jyvaeskylae, Finland.


MIDIT-seminar 518



Tuesday May 6, 2003, 14.00 h
at IMM, Bldg. 305, Room 053, DTU

Abstract:
I develop an internally consistent and comprehensive formulation of the theory of finite temperature thermal Green's functions for the 2-point correlations (and indeed for the arbitrary n- point correlations) describing a weakly coupled Bose gas held in thermal equilibrium in a magnetic trap at temperatures T in the range 0 less than about T less than about T sub zero where T sub zero is greater than about T sub c and T sub c is the transition temperature arising from a realisable change in the asymptotics of the 2-point correlation functions. For d = 3 space dimensions T sub c is a Bose-Einstein Condensation (BEC) temperature (and is about 60 nano-Kelvin for super 87 Rb vapour) and the asymptotics behaves as that of a condensate held in the trap with features of long range order but confined in actuality by the trap potential. The analytical methods used are methods of functional integration in each of d = 1, d = 2 and d= 3 space dimensions and the general approach is to express the thermal Green's functions at finite temperatures as solutions of appropriate nonlinear partial differential equations whose solutions approximate the thermal Green's functions in mean field approximations of temperature dependent generalised Gross-Pitaevsky types. The thermal Green's functions are defined as equilibrium Green's functions by working in the so-called Matsubara representation in which ordinary time t is at formal level replaced by minus i tau (i = root minus 1), and tau is then a "thermal time". In this Lecture I give most attention to the case d=1, and this ONE space dimensional case is exceptional in that the the thermal Green's functions can be evaluated EXACTLY when but only when the trap potential V(x) is identically zero. For these exact results we can (and I do) use the Quantum Inverse Method (method of algebraic Bethe Ansatz. We find (for d = 1) rather miraculously that results for the asymptotics of the 2-point functions obtained via the Inverse Method when V(x) is identically zero are IDENTICAL with those found via the approximated (at the mean field level of approximation mentioned) thermal Green's functions when V(x) is also identically zero. This exact equivalence apparently totally JUSTIFIES the approximated methods of functional integration for d = 1 and V(x) identically zero and the latter methods extend directly to d =1 and V(x) not identically zero and to d = 2, V(r) zero or not zero, and to d =3 V(r) zero or not zero (r = vectors in R to power d). I regard the general theory as relatively tough Mathematical Physics (primarily due to my colleagues in St Petersburg) Even so for d = 3 we find a good measure of agreement with with the experimental results of Immanuel Bloch, Ted Haensch and Tilman Esslinger published in Nature 404, 161 (2000). However we also predict very particular results from the presence of the magnetic trap potential specifically arising from the breakdown of translational invariance by this potential which were neither noted or explored in thse experiments. The model of the weakly coupled Bose gas model we use is the REPULSIVE QUANTUM Nonlinear Schroedinger model in each of d = 1, d =2 and d = 3 space dimensions. For and only for d = 1 and translational invariance the quantum NLS models repulsive AND attractive are both QUANTUM completely integrable and eg the quantum inverse method can be used to solve each of them at finite temperatures in the fashion I shall SKETCH early in my Lecture: the quantum single solitons predicted in and only in the attractive case with d = 1 have been observed but not yet the multi quantum soliton solutions.