Author(s):

Lieb, Élliott H; Seiringer, Robert; Yngvason, Jakob

Article Title: 
Onedimensional behavior of dilute, trapped Bose gases

Affiliation 

Abstract: 
Recent experimental and theoretical work has shown that there are conditions in which a trapped, lowdensity Bose gas behaves like the onedimensional deltafunction Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantummechanical phenomenon because it is not necessary to have a trap width that is the size of an atom  as might have been supposed  but it suffices merely to have a trap width such that the energy gap for motion in the transverse direction is large compared to the energy associated with the motion along the trap. Up to now the theoretical arguments have been based on variational  perturbative ideas or numerical investigations. In contrast, this paper gives a rigorous proof of the onedimensional behavior as far as the ground state energy and particle density are concerned. There are four parameters involved: the particle number, N, transverse and longitudinal dimensions of the trap, r and L, and the scattering length a of the interaction potential. Our main result is that if r/L → 0 and N → ∞ the ground state energy and density can be obtained by minimizing a onedimensional density functional involving the LiebLiniger energy density with coupling constant ∼ a/r 2. This density functional simplifies in various limiting cases and we identify five asymptotic parameter regions altogether. Three of these, corresponding to the weak coupling regime, can also be obtained as limits of a threedimensional GrossPitaevskii theory. We also show that BoseEinstein condensation in the ground state persists in a part of this regime. In the strong coupling regime the longitudinal motion of the particles is strongly correlated. The GrossPitaevskii description is not valid in this regime and new mathematical methods come into play.

Journal Title:

Communications in Mathematical Physics

Volume: 
244

Issue 
2

ISSN:

14320916

Publisher:

Springer

Date Published:

20040101

Start Page: 
347

End Page:

393

URL: 

DOI: 
10.1007/s0022000309933

Open access: 
yes (repository) 