polarization distributions and degree of polarization for quantum gaussian light fields

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Optics Communications 273 (2007) 173–181

Polarization distributions and degree of polarizationfor quantum Gaussian light fields

Alfredo Luis *

Departamento de Optica, Facultad de Ciencias Fısicas, Universidad Complutense, 28040 Madrid, Spain

Received 30 July 2006; received in revised form 8 December 2006; accepted 11 January 2007

Abstract

We examine polarization properties of electromagnetic field states with Gaussian complex-amplitude distributions, such as quadra-ture coherent and squeezed states, thermal chaotic states, and two-mode squeezed vacuum states. We compute their polarization distri-bution and we apply to them diverse measures of degree of polarization and polarization fluctuations. This allows us to investigate themain properties of the degrees of polarization introduced so far.� 2007 Elsevier B.V. All rights reserved.

PACS: 42.50.Dv; 03.65.Ca; 42.25.Ja

Keywords: Polarization; Quantum fluctuations

1. Introduction

Polarization is a key ingredient of light with a largenumber of applications, both in the quantum and classicalrealms. The amount of polarization is usually measured bythe degree of polarization, which can be regarded also as ameasure of polarization fluctuations.

The classic definition of degree of polarization is theintensity-normalized length of the Stokes-parameters vec-tor (the first moment of the Stokes variables). This providesa convenient measure in the classical domain, where mostsources produce thermal-chaotic light. In such a case, theStokes parameters provide complete information aboutthe polarization statistics. For more complex light states,specially in quantum optics, the proper assessment ofpolarization properties requires statistical evaluationsbeyond first moments, as it is the case of polarizationsqueezing for example [1–7]. This has prompted some noveldefinitions and generalizations of the degree of polariza-tion, both in the quantum and classical domains [8–26].

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.01.016

* Tel.: +34 913945011; fax: +34 913944683.E-mail address: [email protected].

The purpose of this work is to develop a comparativestudy of these polarization measures, recalled in Sections 2and 3. In Section 4 we apply all them to quantum states withGaussian complex-amplitude distributions. More specifi-cally, we study the polarization distribution, the fluctuationsof the Stokes parameters, and the degree of polarization ofquadrature coherent states, thermal-chaotic states, squeezedcoherent states, and the two-mode squeezed vacuum [27].These examples are interesting since they are the field statesproduced by current light sources used in technological andscientific applications, and they cover most of the relevantquantum and classical optical properties.

2. Polarization distribution

Polarization properties are conveniently addressed bymeans of the Stokes operators S0, S

S0 ¼ ay1a1 þ ay2a2; Sy ¼ iðay2a1 � ay1a2Þ;Sx ¼ ay2a1 þ ay1a2; Sz ¼ ay1a1 � ay2a2;

ð1Þ

where a1, a2 are the complex amplitude operators for twofield modes [1]. Their mean values are the classic Stokesparameters hSi. These operators are formally equivalent

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174 A. Luis / Optics Communications 273 (2007) 173–181

to an angular momentum satisfying the commutations rela-tions ½S0;S� ¼ 0, ½Sx; Sy � ¼ 2iSz, and its cyclic permutations.

The lack of commutation between the Stokes operatorsmeans that they cannot take definite values simultaneously.The electric vector cannot describe a definite ellipse in thesame sense that quantum particles cannot follow definitetrajectories [28].

Since no quantum field state can have a definite value ofS, the statistical description of polarization is convenient inthe quantum domain from the very beginning. In recentworks we have addressed the derivation of polarization dis-tributions for quantum light fields following different strat-egies [9,12,29–32]. The use and relative merits of differentdistributions depend on the particular problem to be con-sidered. Concerning the degree of polarization and theassessment of polarization fluctuations we will use thepolarization Q function for the reasons explained below,which is defined as [9,29]

QðXÞ ¼X1n¼0

nþ 1

4phn;Xjqjn;Xi; ð2Þ

where q is the density matrix for the two-mode field understudy, jn;Xi are the SU(2) coherent states [33]

jn;Xi ¼Xn

m¼0

n

m

� �1=2

cosh2

� �m

sinh2

� �n�m

e�im/jmi1jn� mi2;

ð3Þ

jmi1jn� mi2 are the product of photon number states in thecorresponding mode, and X ¼ ðh;/Þ, with h and / the po-lar and azimuthal angles, respectively, on the Poincaresphere. The SU(2) coherent states are eigenstates of the to-tal number operator S0jn;Xi ¼ njn;Xi.

The polarization distribution QðXÞ can be obtained alsoby removing from the quadrature Q function Qða1; a2Þ thevariables irrelevant for the specification of polarization,[9,12,29] where

Qða1; a2Þ ¼1

p2ha1jha2jqja1ija2i; ð4Þ

and ja1ija2i are the quadrature coherent states

jai ¼ e�jaj2=2X1n¼0

anffiffiffiffin!p jni: ð5Þ

One of the main reasons for using this approach to polar-ization is that the QðXÞ function is always nonnegative forall states, being a truly probability distribution obtained byprojection on the SU(2) coherent states, which can be re-garded as the states with minimum polarization fluctua-tions [9].

It is worth noticing in (2) that the matrix elements of qconnecting subspaces of different total photon number n donot contribute to QðXÞ. This is consistent with the fact thatin classical optics polarization and intensity are in principleindependent concepts: the form of the ellipse described bythe electric vector (polarization) versus the size of theellipse (intensity). Nevertheless, for particular relevant

cases, such as the Gaussian field states, some definite rela-tions between degree of polarization and intensity fluctua-tions arises [34,35]. This is because of the small number ofparameters characterizing these states. In quantum optics,a similar potential independence of polarization and inten-sity variables would be supported by the commutation ofany function of the Stokes operators f ðSÞ with the totalnumber of photons, ½f ðSÞ; S0� ¼ 0, so that the matrix ele-ments of q connecting subspaces of different total photonnumber n do not contribute to hf ðSÞi. This fact is repro-duced by all polarization approaches considered so far inquantum and classical regimes, with the only exception ofthe degrees of polarization introduced in Ref. [13].

Finally we derive a suitable one-mode approximationvalid for the case in which the polarization distributionQðXÞ is concentrated around a single point of the Poincaresphere. Without loss of generality we can consider thathSxi ¼ hSyi ¼ 0 so that the Stokes vector hSi points tothe north pole. The approximation to be developed holdsif the coherent amplitude of one of the modes, say a1,ha1i ¼ a ¼

ffiffiffi�np� 1 (assumed to be real for simplicity) is

large enough so that hay1a1i ’ �n� hay2a2i and we canreplace in (1) the operator a1 by the variable a so that

Sx ’ffiffiffiffiffi2�np

X ; Sy ’ffiffiffiffiffi2�np

Y ; Sz ’ �n; ð6Þwhere X, Y are the quadratures of the mode a2

a2 ¼1ffiffiffi2p ðX þ iY Þ; ð7Þ

which can be regarded as Cartesian coordinates of the tan-gent plane to the sphere at the point hSi=hS0i. This linear-ization of the Stokes operators is equivalent to the oneperformed in Ref. [36].

We complete the approximation by relating the spheri-cal coordinates ðh;/Þ on the sphere, with the Cartesianðx; yÞ and polar ðr;/Þ coordinates in the tangent plane,where r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pand we have used the same azimuthal

angle / for the sphere and the tangent plane. Since S2 ’ �n2

and using (6) we get

tan h ’ sin h ’ h ’

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2

x þ S2y

q�n

’ffiffiffi2

�n

rr; ð8Þ

and

tan / ¼ Sy

Sx¼ y

x; ð9Þ

with

dX ¼ sin hdhd/ ’ 2

�nr dr d/ ¼ 2

�ndxdy: ð10Þ

This allows us to obtain the proper relation between thedistributions in the two sets of variables in the form

QðXÞdX ’ ~Qðx; yÞdxdy; ð11Þleading to

QðXÞ ’ �n2

~Qðx; yÞ: ð12Þ

A. Luis / Optics Communications 273 (2007) 173–181 175

3. Polarization fluctuations and degree of polarization

In this section we recall the definitions available in theliterature concerning the degree of polarization of quantumand classical two-dimensional transverse light fields. Allformalisms to be considered are SU(2) invariant. Thismeans that the degree of polarization of the states q andU ycqU c are the same, where Uc is any SU(2) transformation(i.e., a rotation on the Poincare sphere generated by theStokes operators [37]). This will allow us to simplifyexpressions.

3.1. Stokes parameters

As a first measure of polarization fluctuations we canconsider [38]

ðDSÞ2 ¼ ðDSxÞ2 þ ðDSyÞ2 þ ðDSzÞ2

¼ hS0ðS0 þ 2Þi � hSi2 P 2hS0i; ð13Þ

where we have used the identity S2 ¼ S0ðS0 þ 2Þ.The equal-ity is reached exclusively by the SU(2) coherent states forwhich jhSij ¼ hS0i and DS0 ¼ 0.

The classic definition of degree of polarization

P S ¼jhSijhS0i

; ð14Þ

is closely related to DS,

ðDSÞ2 ¼ hS0i2ð1� P 2SÞ þ 2hS0i þ ðDS0Þ2; ð15Þ

which shows explicitly that DS contains polarization andintensity fluctuations. Equivalently

P S ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihS0ðS0 þ 2Þi � ðDSÞ2

qhS0i

: ð16Þ

We can note that PS depends exclusively on the first mo-ment of the Stokes operators. This is enough for thermalchaotic fields, whose polarization distributions are fullydetermined by hSi, so that hSi ¼ 0 is equivalent to uniformdistribution QðXÞ ¼ 1=ð4pÞ, as illustrated by the examplein Section 4.2 below, while there are classical and quantumlight fields with hSi ¼ 0 and QðXÞ 6¼ 1=ð4pÞ as revealed byhigher-order moments of the Stokes operators and illus-trated by the example in Section 4.4 below [39,40]. In quan-tum optics moments of the Stokes operators beyond thefirst one are crucial, as illustrated by the concept of polar-ization squeezing for example [1–7].

As a particular example beyond Gaussian fields we notethat for the factorized state jwi1j0i2, where jwi1 is arbitraryand j0i2 is the vacuum, we get P S ¼ 1 for all jwi1 6¼ j0i1 sothat when jwi1 ! j0i1 we have P S ¼ 1 for field states asclose as desired to the quantum two-mode vacuumj0i1j0i2 with uniform polarization distributionQðXÞ ¼ 1=ð4pÞ [41–43].

A slight modification of definition (14) is obtained by

replacing hS0i byffiffiffiffiffiffiffiffiffihS2i

qin the denominator [8]

P 0S ¼jhSijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

hS0ðS0 þ 2Þip ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðDSÞ2

hS0ðS0 þ 2Þi

s; ð17Þ

so that for every state jwi1j0i2 we get P 0S < 1 and P 0S ! 0when jwi1 ! j0i1.

3.2. Distance to the unpolarized distribution

The degree of polarization can be measured in terms ofthe distance between the polarization distribution QðXÞand the uniform polarization distribution QuðXÞ ¼ 1=ð4pÞassociated to unpolarized light [9]

DQ ¼ 4pZ

dX QðXÞ � 1

4p

� �2

¼ 4pZ

dXQ2ðXÞ � 1: ð18Þ

Since DQ ranges from 0 to 1 we normalize it defining thedegree of polarization as

P Q ¼D

1þ D¼ 1� RQ

4p; ð19Þ

so that 1 P P Q P 0, being

RQ ¼1R

dXQ2ðh;/Þ; ð20Þ

a measure of the effective area of the Poincare sphere occu-pied by QðXÞ.

This and similar definitions have been already used asmeasures of localization, uncertainty and information indifferent contexts being an example of Renyi entropy[44–55], with properties such as the ones listed in Ref.[44].

The states with maximum degree of polarization whenthe intensity is kept fixed are the SU(2) coherent states[9]. By construction, this degree of polarization is indepen-dent of intensity fluctuations since the intensity variable isremoved when defining QðXÞ. Moreover, PQ depends on allthe moments of the Stokes operators.

3.3. Gibbs entropy

A degree of polarization PG in terms of the entropy ofpolarization (Shannon, Wehrl, or Gibbs–Boltzmannentropy) has been recently considered in the frameworkof propagation of classical optical waves in nonlinearmedia defined as [10]

P G ¼ 1� eW G

4p; ð21Þ

where

W G ¼ �Z

dXQðXÞ log QðXÞ: ð22Þ

Being introduced originally in the classical domain, wehave translated it to the quantum domain in terms of QðXÞ.

This approach is qualitatively and quantitatively verysimilar to the preceding one, as it will be confirmed whenapplying them to the Gaussian states in Section 4. Closely


related measures such as the Kullback relative entropyhave been proposed and analyzed within the classicaldomain in Refs. [14–17].

3.4. Distinguishability

Another definition of the degree of polarization recentlyintroduced is (for pure states jwi for simplicity)

P D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�mincjhwjU cjwij2

q; ð23Þ

where Uc are arbitrary SU(2) transformations [11]. This is ameasure of the minimum overlap between jwi and its ro-tated counterparts so this is a kind of distinguishabilityor sensitivity to SU(2) transformations.

As a particular example different from Gaussian fields wemay consider a polarization distribution different from zeroexclusively in an hemisphere of the Poincare sphere, say thenorthern hemisphere. From a classical perspective, the scalarproduct in (23) tends to be the overlap between the rotatedQcðXÞ and the original QðXÞ polarization distributions. Inour hemispheric example the minimum overlap vanishesfor a p rotation around any axis on the equatorial plane

jhwjU cjwij2 /Z

dXQcðXÞQðXÞ ¼ 0; ð24Þ

so that relation (23) predicts P D ¼ 1. Thus, we get that apolarization distribution spanning uniformly on a wholehemisphere would be maximally polarized.

As a further example, in Ref. [11] it is stated that P D ¼ 1for any pure state jwi with definite total photon numberS0 ¼ 2 or S0 ¼ 2k þ 1 for any integer k. For example,because of this we have P D ¼ 1 for arbitrary superpositionof SU(2) coherent states located at arbitrary, randompoints of the Poincare sphere.

3.5. Distance to unpolarized density matrices

A recently introduced degree of polarization is a Hilbertspace counterpart of PQ in terms of the distance between qand the set qu of unpolarized density matrices

P B ¼ infutr½ðq� quÞ2�; ð25Þ

where qu are all field states with uniform polarization dis-tribution QuðXÞ ¼ 1=ð4pÞ [13]. After some elaboration thisapproach leads to

P B ¼ trq2 �X1n¼0

p2n

nþ 1; ð26Þ

where pn is the probability of total photon number S0 ¼ n.It can be appreciated in (26) that PB depends exclusively

on the purity of the state and on the statistics of the inten-sity pn.

Moreover, this approach depends on matrix elements ofq connecting subspaces with different total photon number,which contradicts the commutation of polarization proper-ties with the total photon number ½f ðSÞ; S0� ¼ 0.

More specifically, the above definitions PS, P 0S, PQ, PG,and PD, assign the same degree of polarization to the fol-lowing pure and mixed states, for example,

jwi ¼X1n¼0

ffiffiffiffiffipnp jni1j0i2; q ¼

X1n¼0

pnjni1hnj � j0i2h0j; ð27Þ

where jni are number states. These two states have exactlythe same mean values of any function of the Stokes opera-tors f ðSÞ

hwjf ðSÞjwi ¼ tr½qf ðSÞ�; ð28Þ

while definition (25) leads to very different degrees of polar-ization for jwi and q,

P BðwÞ ¼ 1�X1n¼0

p2n

nþ 1; P BðqÞ ¼

X1n¼0

nnþ 1

p2n: ð29Þ

This is illustrated in Sections 4.1 and 4.2, showing thatP B ! 0 for intense beams after a linear polarizer.

3.6. Fidelity

Another degree of polarization introduced in [13]

P F ¼ 1� suputrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qu

pqffiffiffiffiffiqu

pq� �; ð30Þ

is defined in terms of fidelity between the studied field stateq and the field states qu with uniform polarization distribu-tion QuðXÞ ¼ 1=ð4pÞ.

This definition also assigns different degrees of polariza-tion to the states (27), namely

P FðwÞ ¼ 1� supu

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX1n¼0

Pnpn

nþ 1

s; ð31Þ

P FðqÞ ¼ 1� supu

X1n¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

pn

nþ 1

r; ð32Þ

where we have expressed the unpolarized states in the form

qu ¼X1n¼0

Pn

nþ 1I n; ð33Þ

where In is the identity matrix in the subspace of fixed totalphoton number S0 ¼ n.

The maximum fidelity between jwi and qu occurs forPN ¼ 1, Pn6¼N ¼ 0, where N is the integer such thatpn=ðnþ 1Þ takes its maximum value,

pn

nþ 16

pN

Nþ 1; ð34Þ

for all n, leading to

P FðwÞ ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pN

Nþ 1

r: ð35Þ

On the other hand, the maximum fidelity between q and qu

occurs for


Pn ¼1P1

n0¼0pn0ðn0þ1Þ

pn

nþ 1; ð36Þ

leading to

P FðqÞ ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX1n¼0

pn

nþ 1

s: ð37Þ

We can appreciate that for every q2 6¼ q we have alwaysP FðwÞ > P FðqÞ.

4. Degree of polarization for Gaussian fields

In this section we compute the polarization distributionfor the most relevant states with Gaussian complex-ampli-tude distributions. We apply to them all the measures ofpolarization fluctuations and degree of polarizationrecalled in the preceding section.

4.1. Quadrature coherent state

By using the SU(2) invariance we can consider withoutloss of generality the state jai1j0i2 where j0i2 is the vacuumand jai1 is a quadrature coherent state a1jai1 ¼ ajai1 withreal a ¼

ffiffiffi�np

. The quadrature Q function is

Qða1; a2Þ ¼1

p2e�ja1�aj2�ja2j2 : ð38Þ

The product of quadrature coherent states is a Poissoniansuperposition of SU(2) coherent states [56]

jai1j0i2 ¼ e��n=2X1n¼0

�nn=2ffiffiffiffin!p jni1j0i2; ð39Þ

so that

QðXÞ ¼ e��nX1n¼0

�nn

n!Qn;0ðXÞ; ð40Þ

where Qn;0ðXÞ is the polarization distribution of a SU(2)coherent state jni1j0i2 centered at the north pole

Qn;0ðXÞ ¼nþ 1

4pcos

h2

� �2n

: ð41Þ

The above summation can be performed leading to [57]

QðXÞ ¼ 1

4p�n cos2 h

2þ 1

� �e��n sin2 h

2: ð42Þ

For large enough �n� 1 this distribution is concentratedaround the north pole so that sin h ’ h, cos h ’ 1 andQðXÞ can be approximated in the form

QðXÞ ’ �n4p

e��nh2=4: ð43Þ

The same polarization distribution (42) corresponds to thephase-averaged coherent state

q ¼ 1

2p

Z p

�pdujaeiui1haeiuj � j0i2h0j

¼ e��nX1n¼0

�nn

n!jni1hnj � j0i2h0j: ð44Þ

This is a particular example of the states (27) with pn thePoissonian distribution

pn ¼�nn

n!e��n: ð45Þ

For the states (39) and (44) we have

�n ¼ jhSij ¼ hS0i; ðDSÞ2 ¼ 3hS0i; ð46Þ

and

P S ¼ 1; P 0S ¼ffiffiffiffiffiffiffiffiffiffiffi

�n�nþ 3

r: ð47Þ

According to DS the fluctuations of the Stokes operatorsare above the minimum, but when hS0i ! 1 we have

DShS0i! 0; P 0S ’ 1� 3

2�n; ð48Þ

while P S ¼ 1, irrespectively, of �n.On the other hand PQ, PG, and PD are well behaved, as

illustrated in Fig. 1a.For PQ we have the exact result

P Q ¼ 1� 4�n1þ 2�nþ 2�n2 � e�2�n

’ 1� 2

�n; ð49Þ

where the last equality holds for �n� 1.For PG an approximate expression valid for �n� 1 can

be derived from (43) or also from the single mode approx-imation in Section 4.3, leading to

P G ’ 1� e

�n: ð50Þ

A numerical evaluation of the exact expression has beenrepresented in Fig. 1a.

Concerning PD, we have that for every SU(2) coherentstate, say jni1j0i2, there is another SU(2) coherent stateorthogonal to it, j0i1jni2, located at the antipodal pointon the Poincare sphere. Thus the minimum scalar productin (23) occurs for U cjai1j0i2 ¼ j0i1jai2, and the only non-vanishing contribution to the scalar product comes fromthe projection of jai1j0i2 on the two-mode vacuumj0i1j0i2 (which is invariant under any SU(2) transforma-tion), leading to

P D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e�2�np

: ð51Þ

It can be appreciated in Fig. 1a that PD tends to one muchfaster than P 0S, PQ, and PG.

On the other hand, we have that PB is very different forthe pure state (39) and the phase-averaged state (44), asdiscussed in the preceding section. The differences betweenthese two cases are very clear in Fig. 1b, where for the

Fig. 1. (a) PQ (solid line), PG (dashed line), and PD (dotted line), asfunctions of �n for quadrature coherent states. (b) The upper solid anddashed lines are P BðwÞ and P FðwÞ, respectively, for the pure state (39),while the lower solid and dashed lines refer to P BðqÞ and P FðqÞ,respectively, for the mixed state (44).


pure state (39) we have P BðwÞ ! 1 when �n!1, while forthe phase averaged state (44) we have the completelyopposite result P BðqÞ ! 0 when �n!1, which corre-sponds, for example, to the steady state regime of a sin-gle-mode laser well above threshold after a perfect linearpolarizer [58].

Finally, Fig. 1b also illustrates the difference betweenP FðwÞ and P FðqÞ for the polarization-equivalent states(39) and (44), respectively.

4.2. Chaotic light

Let us consider the case of thermal-chaotic light repre-sented by the product of density matrices [27,58]

q ¼ q1 � q2; ð52Þbeing

qj ¼1

1þ �nj

X1n¼0

�nj

1þ �nj

� �n

jnijhnj; ð53Þ

for j ¼ 1; 2. The corresponding quadrature Q function is

Qða1; a2Þ ¼1

p2ð1þ �n1Þð1þ �n2Þexp � ja1j2

1þ �n1

� ja2j2

1þ �n2

!:

ð54ÞThe polarization distribution can be obtained as

QðXÞ ¼ 1

1þ �n1

1

1þ �n2

X1n¼0

�Xn

m¼0

�n1

1þ �n1

� �m�n2

1þ �n2

� �n�m

Qm;n�mðXÞ; ð55Þ

where Qm;n�mðXÞ is the polarization distribution of theproduct of number states jmi1jn� mi2,

Qm;n�mðXÞ ¼nþ 1

4p

n

m

� �sin

h2

� �2ðn�mÞ

cosh2

� �2m

; ð56Þ

leading to

QðXÞ ¼ 1

4p1� N 2

ð1� N cos hÞ2; ð57Þ

with

N ¼ �n1 � �n2

2þ �n1 þ �n2

¼ �n2þ �n

P S; ð58Þ

being

P S ¼jhSijhS0i

¼ �n1 � �n2

�n1 þ �n2

; �n ¼ hS0i ¼ �n1 þ �n2; ð59Þ

and we have assumed without loss of generality that�n1 P �n2.

It is worth noting that the polarization distribution (57)is formally identical to the polarization distribution for aclassical thermal-chaotic field [59]. The only difference isthat in the classical case the parameter N is given simplyby N class ¼ P S. This means that QðXÞ becomes identical tothe classical distribution when �n� 1.

The fluctuations of the Stokes operators are

ðDSÞ2 ¼ �n21 þ �n2

2 þ 4�n1�n2 þ 3�n1 þ 3�n2: ð60Þ

In (59) we have already computed PS. Concerning P 0S weget

P 0S ¼�n1 � �n2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2�n21 þ 2�n2

2 þ 2�n1�n2 þ 3�n1 þ 3�n2

p : ð61Þ

For the simplest case �n2 ¼ 0, �n ¼ �n1 we get

ðDSÞ2 ¼ �n2 þ 3�n; P S ¼ 1; P 0S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�n2�nþ 3

r: ð62Þ

More specifically, for the case �n2 ¼ 0, �n ¼ �n1 !1, thatcorresponds to a classical intense light beam after a perfectlinear polarizer, we get

DShS0i! 1; P 0S !

1ffiffiffi2p : ð63Þ


On the other hand, PS assigns maximum polarizationP S ¼ 1 for states that when �n1; �n2 ! 0 are as close as de-sired to the two-mode vacuum state j0i1j0i2, for whichQðXÞ ¼ 1=ð4pÞ.

It is possible to derive exact expressions for PQ and PG

P Q ¼4N 2

N 2 þ 3; ð64Þ

P G ¼ 1� e2 1� N1þ N

� �1=N

; ð65Þ

where N is defined in (58).In Fig. 2a we can appreciate that P Q; P G ! 1 for �n2 ¼ 0

and �n ¼ �n1 !1 and P Q; P G ! 0 when �n! 0.Concerning PD for the case �n2 ¼ 0 and �n ¼ �n1 we have

that q is a superposition of SU(2) coherent states jni1j0i2so that the only contribution to PD comes from the projec-tion of q on the two-mode vacuum j0i1j0i2

P D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

ð1þ �nÞ2

s: ð66Þ

Fig. 2. (a) PQ (solid line), PG (dashed line), and PD (dotted line), asfunctions of �n for chaotic fields with �n2 ¼ 0. (b) PB (solid line) and PF

(dashed line) as functions of �n for chaotic fields with �n2 ¼ 0.

Concerning PB and PF, for �n2 ¼ 0 and �n ¼ �n1 we have that

P B ¼X1n¼0

nnþ 1

p2n; P F ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX1n¼0

pn

nþ 1

s; ð67Þ

with

pn ¼1

1þ �n�n

1þ �n

� �n

: ð68Þ

It can be seen in Fig. 2b that P B ! 0 for �n� 1 which is thecase of a classical intense thermal-chaotic light after a per-fect linear polarizer.

4.3. Bright coherent squeezed state

For quadrature coherent squeezed states we can obtainsome meaningful results if we consider the one-modeapproximation developed in Section 2 valid for the casein which the field state is distributed around a single pointof the Poincare sphere

QðXÞ ’ �n=2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2r2

x þ 1Þð2r2y þ 1Þ

q exp � x2

2r2x þ 1

� y2

2r2y þ 1

!;

ð69Þwhere DX ¼ rx, DY ¼ ry , are the standard deviations of thequadrature operators X, Y in (7) satisfying the uncertaintyrelation

DXDY ¼ rxry P1

2; ð70Þ

the equality being reached exclusively by unsqueezed distri-butions rx ¼ ry ¼ 1=

ffiffiffi2p

. Within this approximation wehave

hSzi ’ hS0i ’ �n; ð71Þ

and for definiteness we will assume a Poissonian distribu-tion for Sz so that DSz ’

ffiffiffi�np

.After (6) the polarization fluctuations may be expressed

in terms of DX , DY in the form

ðDSÞ2 ’ 2�n½ðDX Þ2 þ ðDY Þ2� þ �n P 3�n; ð72Þwhere the lower bound is derived from the fact that theuncertainty relation (70) implies that

ðDX Þ2 þ ðDY Þ2 P 1: ð73Þ

The minimum DS is obtained again for unsqueezed distri-butions rx ¼ ry ¼ 1=

ffiffiffi2p

. In other words, quadraturesqueezing implies in the bright limit the increase of polari-zation fluctuations above the minimum.

The increase of polarization fluctuations caused bysqueezing is not reflected neither by PS nor P 0S, since theleading terms of the quantitiesffiffiffiffiffiffiffiffiffihS2i

q’ jhSij ’ hS0i ’ �n; ð74Þ

are independent of rx, ry .


On the other hand, for PQ and PG we get

P Q ’ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2r2


q�n

; ð75Þ

P G ’ 1�effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2r2


q2�n

: ð76Þ

It can be appreciated in both expressions that largersqueezing implies lesser degree of polarization.

It can be further noticed that for the unsqueezed caserx ¼ ry ¼ 1=

ffiffiffi2p

the expressions (75), (76) are consistentwith (49), (50) obtained for bright coherent states via a dif-ferent procedure.

On the other hand it seems that PD displays a differentbehavior. To see this let us consider that mode a1 is in acoherent state jai1 with real a ¼

ffiffiffi�np

, and mode a2 is in asqueezed vacuum state jni2. If there were no squeezingjni2 ¼ j0i2 the minimum overlap between the original androtated state occurs when U cjai1j0i2 ¼ j0i1jai2 leading to

jhwjU cjwijmin ¼ jh0jaij2 ¼ e��n: ð77Þ

If now we squeeze the mode a2 we have that for the sameUc of (77) we get U cjai1jni2 ¼ jni1jai2 and

jhwjU cjwij ¼ jhnjaij2

¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2r2


q exp � 2�n2r2

x þ 1

� �: ð78Þ

For rx < 1=ffiffiffi2p

we get that the overlap is lesser than theunsqueezed minimum (77). Therefore, in this case thesqueezing increases the degree of polarization PD.

4.4. Two-mode squeezed vacuum

An example of state with Gaussian field distribution andstrong nonclassical properties is the two-mode squeezedvacuum [27]

jni ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�n=2þ 1

p X1n¼0

�n=2

�n=2þ 1

� �n=2

jni1jni2; ð79Þ

where jni are number states, and �n ¼ hS0i, leading to aquadrature Q function

Qða1; a2Þ ¼1

p2ð�n=2þ 1Þ

� exp �ja1j2 � ja2j2 þffiffiffiffiffiffiffiffiffiffiffi

�n�nþ 2

rða�1a�2 þ a1a2Þ

� �:

ð80Þ

The polarization distribution can be computed as

QðXÞ ¼X1n¼0

1

�n=2þ 1

�n=2

�n=2þ 1

� �n

Qn;nðXÞ; ð81Þ

where Qn;nðXÞ is the polarization distribution for the prod-uct of number states jni1jni2

Qn;nðXÞ ¼2nþ 1

4p22n

2n

n

� �sin2n h: ð82Þ

The polarization distribution admits a suitable Gaussianapproximation valid for large �n. To this end we firstapproximate Qn;nðXÞ in the form

Qn;nðXÞ ’1

2p

ffiffiffinp

re�nðh�p=2Þ2 : ð83Þ

Using this approximation in (81) and replacing the n-sumby an integral we get

QðXÞ ’ffiffiffiffiffiffiffiffi�n=2

p4p

1

½1þ ð�n=2Þðh� p=2Þ2�3=2: ð84Þ

Let us note that in this case the three Stokes parametersvanish simultaneously hSi ¼ 0 while QðXÞ 6¼ 1=ð4pÞ when�n 6¼ 0.

The fluctuations of the Stokes operators are

ðDSÞ2 ¼ 2�n2 þ 4�n; ð85Þand in this case, when hS0i ! 1,

DShS0i!

ffiffiffi2p

; ð86Þ

On the other hand P S ¼ P 0S ¼ 0 for all �n.The approximation (84) allows us to derive the follow-

ing expressions for PQ and PG

P Q ’ 1� 16ffiffiffi2p

3pffiffiffi�np ; P G ’ 1� e3

ffiffiffi2p

8ffiffiffi�np ; ð87Þ

that both tend to maximum degree of polarization when�n!1.

Concerning the evaluation of PD we note that QðXÞ islocated around the equator being independent of the azi-muthal angle /. Thus it is advantageous to factorize Uc

in terms of the Euler angles in the form

U c ¼ eic0Sz eicSx eic00Sz : ð88ÞThe two-mode squeezed vacuum is eigenstate of Sz, withSzjni ¼ 0, so that for every c

eicSz jni ¼ jni; ð89Þand

hnjU cjni ¼ hnjeicSx jni: ð90ÞThis is further simplified if we use a different set of fieldmodes

b� ¼1ffiffiffi2p ða1 � a2Þ; ð91Þ

so that the two-mode squeezed vacuum factorizes into theproduct of two single-mode squeezed vacuum statesjni ¼ jfiþj � fi�, where the squeezing for modes bþ andb� takes place along orthogonal directions. Since Sx ¼byþbþ � by�b� we get

hnjU cjni ¼ hfjeicbyþbþ jfih�fje�icby�b� j � fi; ð92Þ


where j � fi is a state in mode b�, respectively. It can beseen that these overlaps are minimum when c ¼ p=2 so thateicSx jfiþj � fi� ¼ j � fiþjfi� and

P D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jhfj � fij4

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

ð1þ �nÞ2

s: ð93Þ

We can conclude that PQ, PG, and PD predict that the de-gree of polarization tends to the maximum as the meannumber of photons increases.

5. Conclusions

We have examined the polarization properties of Gauss-ian states focusing on the degree of polarization. We hopethat the results obtained might be of interest concerningthe concepts of quantum polarization and its relation withthe more familiar polarization in classical optics. In princi-ple, it seems reasonable that the approaches to quantumpolarization should be compatible with classical polariza-tion in the spirit of the correspondence principle. Neverthe-less, it must be also taken into account that classicalquantum correspondences involve particular families ofstates, while the potential applications of quantum polariza-tion might involve states without classical analog. Let usalso point out that inconsistencies and difficulties can beexpected when characterizing something as complicated aspolarization (specially in the quantum domain) by a singlenumber.

Acknowledgement

I thank the anonymous Referees for their constructivecomments that have helped me to improve the manuscript.

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