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## Magnetic traps in cold-atom physics

Maple

This presentation is about magnetic traps for neutral particles, first achieved for cold neutrons and nowadays widely used in cold-atom physics. The level is that of undergraduate electrodynamics and tensor calculus courses. Tackling this topic within a computer algebra worksheet as shown below illustrates well the kind of advanced computations that can be done today with the Physics package. A new feature minimizetensorcomponents and related functionality is used along the presentation, that requires the updated Physics library distributed at the Maplesoft R&D Physics webpage.

Magnetic traps in cold-atom physics

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

We consider a device constructed with a set of electrical wires fed with constant electrical currents. Those wires can have an arbitrary complex shape. The device is operated in a regime such that, in some region of interest, the moving particles experience a magnetic field that varies slowly compared to the Larmor spin precession frequency. In this region, the effective potential is proportional to the modulus of the field: , this potential has a minimum and, close to this minimum, the device behaves as a magnetic trap.

Figure 1: Schematic representation of a Ioffe-Pritchard magnetic trap. It is made of four infinite rods and two coils.

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Following [1], we show that:

 a) For a time-independent magnetic field   in vacuum, up to order two in the relative coordinates  around some point of interest, the coefficients of orders 1 and 2 in this expansion,  and  , respectively the gradient and curvature, contain only 5 and 7 independent components.
 b) All stationary points of  (nonzero minima and saddle points) are confined to a curved surface defined by .
 c) The effective potential, proportional to , has no maximum, only a minimum.

Finally, we draw the stationary condition surface for the case of the widely used Ioffe-Pritchard magnetic trap.

 Reference
 [1] R. Gerritsma and R. J. C. Spreeuw, Topological constraints on magnetostatic traps,  Phys. Rev. A 74, 043405 (2006)
 The independent components of  and  entering
 The stationary points are within the surface
 has only minima, no maxima
 Drawing the Ioffe-Pritchard Magnetic Trap

MagneticTraps.mw or in pdf format with the sections open: MagneticTraps.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

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