Polarization statistics
In classical optics the polarization state of light is characterized by means of the Stokes
parameters (proportional to second order field correlations), that become the mean value of
the Stokes operators in the quantum domain. While this characterization may be enough in
classical optics, it becomes incomplete in the quantum domain, where higher order
correlations are usually crucial. In such a case, the Stokes parameters are not enough or even
may become an unsuitable parametrization. This is the case of polarization squeezed states,
which are actually defined by the fluctuations of the Stokes operators, rather than by their
mean values. On the other hand, the Stokes parameters can vanish for states that are far from
being unpolarized and that are actually extremely useful for their good polarization properties.
In the left-hand side figure we have represented the probability distribution for the electric
field and on the right-hand side the corresponding probability distribution on the Poincarè
sphere (to be defined below) for the state 1 x 1 y (the product of photons in linearly polarized

modes along axes X and Y). For this state we have S  0 while it is clear that this is not
unpolarized light (moreover note that the electric field does not follow an ellipse, in spite of
the fact that this is monochromatic light). The right-hand side figure shows that this is an state
with polarization squeezing with lesser fluctuations along X and larger along Y, Z).
A full characterization of polarization should be provided by a polarization distribution
function defined on the Poincaré sphere. Such a definition is not unique in quantum optics
because the Stokes operators do not commute. In other words, no quantum state can have a
definite state of polarization, and the points on the Poincaré sphere have no definite quantum
correspondence. The definition of polarization distribution we have adopted is the SU(2) Q
function. This is essentially a radial integration of the projection of the state on the coherent
states of the electromagnetic field

1
 n 1
Q( ,  ) 
d drr 3  ,    ,   n 0
n, ,   n,  , 
2 2 0
4
4


where  ,  are two-mode coherent states with   r sin e i e  i ,   r cos e i
and
2
2
n, ,  are the associated SU(2) coherent states. The main advantage of this approach is that
Q ( ,  ) is a legitimate probability distribution arising by projection of the studied state on the
states with minimum joint fluctuations of the Stokes operators (the SU(2) coherent states).
Moreover, it can be easily computed and simplifies calculations and analyses.
In these conditions we have defined a new degree of visibility as the distance D between the
corresponding probability distribution Q ( ,  ) on the Poincaré sphere and the uniform
distribution Qunpol ( , )  1/( 4 ) corresponding fully unpolarized light
1 
4

2
D  4  d Q( , ) 
 4  dQ( , )  1 
1

4 


2
where

1
 dQ( , )
2
is a measure of the area on the Poincaré occupied by Q ( ,  ) . We have applied this definition
to diverse light states such as: SU(2) coherent and squeezed states as well as to number states,
obtaining meaningful results. For example for coherent states we have found that D scales
with the number of photons while for number status it scales as the square root of the number
of photons.
Degree of polarization in quantum optics
A. Luis, Phys. Rev. A 66, 013806 (2002)
In order to better understand quantum and classical polarization we have applied this
formalism to the most common quantum and classical polarization states with practical
relevance. In general these states have Gaussian distributions for the complex amplitudes.
Moreover, we have extended the analysis to include other approaches to polarization
statistics. We have focus in that results must be compatible with classical optics in the proper
limit of large photon numbers. We have shown that not all polarization formalisms satisfy this
natural requirement.
Polarization distributions and degree of polarization for quantum Gaussian light fields
A. Luis, Opt. Commun. 273, 173-181 (2007)
As an interesting case of application of polarization statistics we have considered in the so
called type II unpolarized states. This kind of states satisfy some (not all) of the properties
that characterize fully unpolarized light.
More specifically they satisfy the following properties: i) Invariance under rotations around
the propagation direction, ii) Symmetry under the exchange of dextro and levo circular light
produced by a half wave plate. These properties imply that the Stokes parameters and the
standard second-order degree of polarization vanish, so that this is unpolarized light under
standard second-order criteria.
When computing the polarization distribution on the Poincaré sphere for relevant examples of
type-II unpolarized light we have found that the distribution is far from being uniform so that
these states should not be referred to as unpolarized. As a matter of fact, their distance to the
uniform distribution can be arbitrarily large. This is to say that type-II unpolarized light can
actually have a polarization state as well defined as desired provide that we focus on
polarization properties beyond the second-order Stokes parameters.
Degree of polarization of type-II unpolarized light
A. Luis, Phys. Rev. A 75, 053806 (2007)
To complete the analysis of polarization statistics beyond the quantum Q function we have
considered other possibilities. More specifically we have studied the so called s-ordered
distributions for the complex amplitudes that depend on a real parameter s including the Q
function as the case s  1, the Wigner function as the case s  0 and the P function as the
case s  1. From them we have derived the corresponding marginal distributions removing by
integvration the other variables not describing polarization properties.
We hace compared the result with other distribution analogous to the s-ordered but directly
introduced on the Poincaré sphere, referred to as SU(2) distributions, instead from being
derived after distributions for the complex amplitudes.
In particular, we have focus on the negative of these distributions, spetially the Wigner
function, as a manifestation of the nonclassical character of the corresponding light state. This
is done by computing the area of the Poincaré sphere where the Wigner function takes
negative values, something forbidden by any classical theory.
In the following figures we show the polarization distribution for different SU(2) coherent
states n,0 in the left, a for SU(2) squeezed states n, n in the right. The solid line is the
marginal distribution derived after the Wigner for the complex amplitudes while the dashed
line is the SU(2) Wigner function. Both are plotted in terms of the polar angle  on the
Poincaré sphere since they are invariant under rotations around the north-south pole. It can be
appreciated that all them take negative values. For the SU(2) coherent states the negativity
decreases when the number of photons n increases, while for the SU(2) squeezed status the
negativity increases when the number of photons n increases.
SU(2) coherent status n,0
SU(2) squeezed states n, n
Maybe the most striking result is that the SU(2) Wigner function for quadrature coherent
states take negative values, despite these states being paradigmatic classical states. The
negativity is shown in the next figure where the SU(2) distribution is plotted in dashed line
takes negative values around    / 2 (south pole) as it can be clearly appreciated in the
inset box. On the other hand, the marginal distribution derived from the Wigner for the
complex amplitudes (solid line) is always nonnegative. This, along other similar results also
reported in this web page, confirm that there is still much to be known about the shaky
boundary between the classical and quantum worlds.
Quantum polarization distributions via marginals of quadrature distributions
A. Luis, Phys. Rev. A 71, 053801 (2005)
Nonclassical polarization states
A. Luis, Phys. Rev. A 73, 063806 (2006)
Recently we have extended this formalism to embrace polarization correlations between two
waves. The degree of correlation can be measured as the distance between the joint
distribution Q1,2 (1 ,  2 ) associated to the two waves, and the product of the individual
distribution for each wave Q1 (1) Q2 (2 ) as


D  4 d1d 2 Q1,2 (1 ,  2 ) Q1 (1) Q2 (2 )

2
where Q1 (1)  d 2 Q1,2 (1 ,  2 ) and similarly for Q2 (2 ) . It is worth noting that this
distance assess correlations irrespective of whether their origin is either classical or quantum
(entanglement). We have demonstrated that this formalism properly accounts for the
polarization correlations of different field states.
This approach also reflects naturally the known complementarity between two-photon
polarization correlations and the degree of polarization of the individual photons: maximum
correlation implies minimum degree of one-photon polarization and vice versa, maximum
degree of polarization for the individual photons implies total lack of polarization
correlations, as illustrated in the figure for the state sin  1,0 1,0  cos  e i 0,1 0,1
1
2
1
2
We have also defined a degree of polarization for two waves by generalizing the above
definition for individual waves as
1
P1,2  1 
2
(4 )2 d1d 2 Q1,2 (1, 2 )



After this definition we have demonstrated that the degree of polarization P1,2 and the degree
of correlation D are complementary variables.
Polarization correlations in quantum optics
A. Luis, Opt. Commun. 216, 165 (2003)
With the help of these tools we have compared the maximum polarization correlations that
can be reached with entangled or separable states. We have found that entangled and
separable can reach very similar values of polarization correlations for high photon number.
More specifically, the entangled and separable states with maximum polarization correlations
that we have examined are
  n,0 1 n,0 2  0, n 1 0, n 2 ,
  n,0 1 n,0  n,0 2 n,0  0, n 1 0, n  0, n 2 0, n .
The first one is an entangled pure state while the second is a mixed separable state. In both
cases the polarization state is always the same for the two waves. We have the 50%
superposition of two orthogonal possibilities for such polarization state, for example n
photons vibrating in axis X and 0 photons vibrating in axis Y.
This similarity of polarization correlations is striking since it seems that quantum entangled
status might reach larger correlations than classical separable status. According to the above
result we may say that the correlations are of different nature, but not larger for entangled
states.
Classical and quantum polarization correlations
A. Luis, Phys. Rev. A 69, 023803 (2004)
We have analyzed the polarization of three-dimensional waves, this is to say of waves with
three nontrivial electric-field components. This can be of interest in particular in quantum
optics where no electric field can vanish exactly so that the electric field of any wave is
always a three-dimensional phenomenon.
Firstly we have carried out an analysis in terms of the Stokes parameters. For threedimensional waves there are nine Stokes parameters that can be obtained by developing the
field correlation matrix in the matrix basis given by the Gell-Mann matrices, in the same way
that the Stokes parameters for two-dimensional waves are given by developing the correlation
matrix in the Pauli matrices basis. We have considered two definitions of degree of
polarization in terms of Stokes parameters
8
 Sj
P2 
j 1
S0
2
8
2
 Sj
,
P2 
j 1
8
2
.
 S 2j
j 1
Although the first one is closer to the two-dimensional classical definition it arises that it
gives worst results in the quantum case. For example, quadrature coherent states arbitrarily
close to the vacuum have always maximum degree of polarization. Moreover, maximum
polarization does not coincides with minimum fluctuations of the Stokes parameters
 S j 2 .
8
j 1
All these difficulties are avoided by the second definition so that the states with maximum
degree of polarization and minimum polarization fluctuations are the SU(3) coherent states,
which are the result of applying a SU(3) transformation to states of the form 0,0, n where
there are a definite number of photons in one component and vacuum in the other
components.
The quadrature coherent states, and the SU(2) and SU(3) coherent states satisfy an interesting
relationship. The quadrature coherent states for three field components can be expressed as
sing.-component quadrature coherent states in the number basis replacing number states by
SU(3) coherent states, that in turn can be expressed in the number basis as SU(2) coherent
states replacing number states by SU(2). Coherent states.
Finally we have shown that an arbitrary wave can be decomposed as incoherent superposition
of two fully polarized waves and a fully unpolarized wave.
Quantum polarization for three-dimensional fields via Stokes operators
A. Luis, Phys. Rev. A 71, 023810 (2005)
Also for three-dimensional waves we have addressed the statistical description of polarization
beyond second-order field quantities such as the Stokes parameters. To this end we have
followed the same strategy as for two-dimensional waves. We begin with the Q function
defined by projection on three-component quadrature coherent states and then we remove the
variables that do not enter on the idea of polarization. The result is a marginal Q function
defined by projection on SU(3) coherent states. We can assess the degree of polarization as
the distance between the polarization distribution and the uniform distribution associated to
unpolarized light.
Polarization distribution and degree of polarization for three-dimensional quantum light
fields
A. Luis, Phys. Rev. A 71, 063815 (2005)