Polarized Atoms Visualized by Multipole Moments
An ensemble of spin-polarized atoms is characterized by its density matrix 
, which can be decomposed into atomic multipole moments 
that transform under rotations as irreducible tensors of rank 
. This Demonstration gives a graphical visualization of the longitudinal multipole moments 
(left) and of the corresponding populations of the magnetic sublevels (right).
Contributed by: Gianni Di Domenico (Université de Neuchâtel) and Antoine Weis (Université de Fribourg) (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The polarization of an ensemble of atoms with angular momentum 
is described by its density matrix that has 
elements. The state of polarization can be represented by a probability surface
that represents the probability of finding the ensemble in the stretched state 
when the quantization axis is oriented along (
, 
).
The density matrix can be expanded into multipole moments
,
where the 
are irreducible tensor operators with rank 
and 
. The , called multipole moments or state multipoles, transform under rotations as the spherical harmonics 
The moments are called longitudinal moments, while the moments 
are called transverse moments or coherences. An ensemble of particles with spin can have multipole moments of rank 
.
The longitudinal moments can be expressed in terms of the populations 
of particles in the magnetic sublevel 
and conversely the sublevel populations depend on the state multipole moments according to
.
A given single multipole moment is represented by the probability surface
.
This Demonstration visualizes the probability surfaces of ensembles with spin 
that have a pure longitudinal multipole moment in addition to the (trivial) monopole moment of value 
. You can vary the value of between its extreme values, and the corresponding populations of the sublevel populations are shown as vertical bars.
An ensemble with nonvanishing multipole moments 
is said to be oriented, while an ensemble with non-vanishing multipole moments 
is said to be aligned.
Permanent Citation
https://demonstrations.wolfram.com/PolarizedAtomsVisualizedByMultipoleMoments/
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