Unpolarized states and hidden polarization

We capitalize on a multipolar expansion of the polarization density matrix, in which multipoles appear as successive moments of the Stokes variables. When all the multipoles up to a given order $K$ vanish, we can properly say that the state is $K\mathrm{th}$-order unpolarized, as it lacks of polarization information to that order. First-order unpolarized states coincide with the corresponding classical ones, whereas unpolarized to any order tally with the quantum notion of fully invariant states. In between these two extreme cases, there is a rich variety of situations that are explored here. The existence of hidden polarization emerges in a natural way in this context.

Figure 1

Second-order degree of polarization as a function of the purity, for the first-order unpolarized states (4.5).

Figure 2

$Q$ function for three-photon first-order pure unpolarized states (4.13).

Figure 3

$Q$ function for axially symmetric three-photon second-order unpolarized states with maximal purity. In the left, we represent the state $3/4|3/2,1/2\rangle \langle 3/2,1/2|+1/4|3/2,-3/2\rangle \langle 3/2,-3/2|$, while in the right the state $1/3|3/2,3/2\rangle \langle 3/2,3/2|+1/2|3/2,-1/2\rangle \langle 3/2,-1/2|+1/6|3/2,-3/2\rangle \langle 3/2,-3/2|$ is plotted.