0198506961.pdf

220 Magnetic trapping, evaporative cooling and Bose–Einstein condensation

10.2 Magnetic trapping

10.2.1 Confinement in the radial direction

The estimate in the introduction shows that magnetic forces have a

significant effect on cold atoms and in this section we examine the specific

configuration used to trap atoms shown in Fig. 10.1. The four parallel

wires arranged at the corners of a square produce a quadrupole magnetic

field when the currents in adjacent wires flow in opposite directions.

Clearly this configuration does not produce a field gradient along the

axis (z-direction); therefore from Maxwell’s relation divB= 0 we deduce

that

dBx

dx

=−

dBy

dy

=b′.

These gradients have the same magnitudeb′, but opposite sign. There-

fore the magnetic field has the form

B=b′(x̂ex−ŷey)+B 0. (10.7)

Here we simply assume thatb′=3Tm−^1 ; that this is a realistic field

gradient can be shown by using the Biot–Savart law to calculate the field

produced by coils carrying reasonable currents (see caption of Fig. 10.1).

In the special case ofB 0 = 0, the field has a magnitude

|B|=b′(x^2 +y^2 )^1 /^2 =b′r. (10.8)

Thus the magnetic energy (eqn 10.2) has a linear dependence on the

radial coordinater=

√

x^2 +y^2. This conical potential has the V-shaped

cross-section shown in Fig. 10.2(a), with a force in the radial direction

of

F=−∇V=−gFμBMFb′̂er. (10.9)

(a) (b)

− 4 − 2024

− 4

− 2

0

2

4

Fig. 10.1(a) A cross-section through four parallel straight wires, with currents into and out of the page as indicated. These
give a magnetic quadrupole field. In a real magnetic trap, each ‘wire’ is generally made up of more than ten strands, each of
which may conduct over 100 amps, so that the total current along each of the four wires exceeds 1000 amps. (b) The direction
of the magnetic field around the wires—this configuration is a magnetic quadrupole.