224 Magnetic trapping, evaporative cooling and Bose–Einstein condensation
field has the formBpinch(z)=Bpinch(0) +d^2 Bz
dz^2z^2
2. (10.13)
This gives a corresponding minimum in the magnetic energy and hence
a harmonic potential along thez-axis. Typically, the Ioffe trap has an
axial oscillation frequencyωzan order of magnitude lower thanωr(=
ωx=ωy), e.g.ωz/ 2 π=15Hzandωr/ 2 π= 250 Hz (see Exercise 10.1).
Thus the atoms congregate in a cigar-shaped cloud along thez-axis. The
curvature of the magnetic field alongzdepends only on the dimensions
of the pinch coils and their current. Therefore a uniform field alongz
doesnotaffectωz, but it does changeωrthrough the dependence onB 0
in eqn 10.12. The pairs of compensation coils shown in Fig. 10.3 create
a uniform field along thez-axis that opposes the field from the pinch
coils. This allows experimenters to reduceB 0 and make the trap stiff in
the radial direction.^5(^5) In practice, the compensation coils
need not have exactly the Helmholtz
spacing; the current in these coils to-
gether with pinch coil current gives
two experimental parameters that al-
low the adjustment of the magnitude
and field curvature alongzto any de-
sired value (limited by the maximum
current through the pairs of coils). Nei-
ther pair of these coils gives a field gra-
dient, by symmetry.
To load the approximately spherical cloud of atoms produced by op-
tical molasses, the Ioffe trap is adjusted so thatωrωz. After loading,
an increase in the radial trapping frequency, by reducing the bias field
B 0 (see eqn 10.12), squeezes the cloud into a long, thin cigar shape. This
adiabatic compression gives a higher density and hence a faster collision
rate for evaporative cooling.
10.3 Evaporative cooling
Laser cooling by the optical molasses technique produces atoms with a
temperature below the Doppler limit, but considerably above the recoil
limit. These atoms can easily be confined in magnetic traps (as shown
in Section 10.1) and evaporative cooling gives a very effective way of
reducing the temperature further. In the same way that a cup of tea
loses heat as the steam carries energy away, so the cloud of atoms in a
magnetic trap cools when the hottest atoms are allowed to escape. Each
atom that leaves the trap carries away more than the average amount of
energy and so the remaining gas gets colder, as illustrated in Fig. 10.6.
A simple model that is useful for understanding this process (and for
quantitative calculations in Exercise 10.4) considers evaporation as a
sequence of steps. At the start of a step the atoms have a Boltzmann
distribution of energiesN(E)=N 0 exp(−E/kBT 1 ) characteristic of a
temperatureT 1. All atoms with energies greater than a certain value
E>Ecutare allowed to escape, whereEcut=ηkBT 1 and typically the
parameterηlies in the rangeη= 3–6; this truncated distribution has
less energy per atom than before the cut so that, after collisions between
the atoms have re-established thermal equilibrium, the new exponential
distribution has a lower temperatureT 2 <T 1.^6 The next step removes(^6) Temperature is only defined at ther-
mal equilibrium and in other situations
the mean energy per atom should be
used.
atoms with energies aboveηkBT 2 (a lower energy cut-off than in the first
step) to give further cooling, and so on.^7 Formanysmallstepsthismodel
(^7) An exponential distribution extends
to infinity, and so for any value ofEcut
(orη) there is always some probability
of atoms having a higher energy; how-
ever, the removal of a very small frac-
tion of atoms has a little effect when av-
eraged over the remaining atoms. Ex-
ercise 10.4 compares different depths of
cut.
gives a reasonable approximation to real experiments where evaporation
proceeds by a continuous ramping down ofEcutthat cuts away atoms at