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.
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