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*.