0198506961.pdf

236 Magnetic trapping, evaporative cooling and Bose–Einstein condensation

Hence, the number density of atomsn(r)=N|ψ(r)|^2 in the harmonic

potential has the form of an inverted parabola:

n(r)=n 0

(

1 −

x^2

R^2 x

−

y^2

R^2 y

−

z^2

R^2 z

)

, (10.38)

wheren 0 , the number density at the centre of the condensate, is

n 0 =

Nμ

g

. (10.39)

The condensate has an ellipsoidal shape and the density goes to zero at

points on the axes given byx=±Rx,y=±Ryandz=±Rz, defined

by

1

2

Mω^2 xRx^2 =μ, (10.40)

and similarly forRyandRz. In the Thomas–Fermi regime, the atoms

fill up the trap to the level of the chemical potential as illustrated in

Fig. 10.11, just like water in a trough. The chemical potentialμis

determined by the normalisation condition

1=

∫∫∫

|ψ|^2 dxdydz=

μ

g

8 π

15

RxRyRz. (10.41)

A useful form forμis

μ=
ω×

1

2

(

15 Na

aho

) 2 / 5

. (10.42)

The mean oscillation frequency is defined asω=(ωxωyωz)^1 /^3 andaho

is calculated using this frequency in eqn 10.34. Typical values for an

Ioffe trap are given in the following table together with the important

properties of a BEC of sodium, calculated from the key formulae in

eqns 10.42, 10.40 and 10.39 (in that order).

Scattering length (for Na) a 2.9 nm

Radial oscillation frequency (ωx=ωy) ωx/ 2 π 250 Hz

Axial oscillation frequency ωz/ 2 π 16 Hz

Average oscillation frequency ω/ 2 π 100 Hz

Zero-point energy (in temperature units)^12 
ω/kB 2.4 nK

Harmonic oscillator length (forω) aho 2. 1 μm

Number of atoms in condensate N 0 106

Chemical potential μ 130 nK

Radial size of condensate Rx=Ry 15 μm

Axial size of condensate Rz 95 μm

Peak density of condensate n 0 2 × 1014 cm−^3

Critical temperature (for 4× 106 atoms) TC 760 nK

Critical density (for 4× 106 atoms) nC 4 × 1013 cm−^3