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ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2007, Vol. 104, No. 1, pp. 30­46. © Pleiades Publishing, Inc., 2007. Original Russian Text © V.P. Karassiov, S.P. Kulik, 2007, published in Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, 2007, Vol. 131, No. 1, pp. 37­53.

ATOMS, MOLECULES, OPTICS

Polarization Transformations of Multimode Light Fields
V. P. Karassiova,* and S. P. Kulikb,**
a

Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia b Moscow State University, Moscow, 119992 Russia * e-mail: vpk43@mail.ru ** e-mail: skulik@qopt.phys.msu.su
Received July 19, 2006

Abstract--The concept of P-quasispin is used as a framework for developing a formalism for polarization transformation by means of lossless polarization-changing optical elements for both classical and quantum multimode light fields. As an example, a classification of transformations of this kind is presented for two-mode biphoton fields that implement the information-theoretic concepts of qutrit and ququart (quantum systems in three- and four-dimensional Hilbert states, respectively). PACS numbers: 42.25.Ja, 42.50.-p DOI: 10.1134/S1063776107010049

1. INTRODUCTION Polarization is a fundamental property of electromagnetic waves. Polarization of light is remarkable in that its formal description is similar to that developed later for two-level quantum systems. Anyone who has an idea of the characteristics of a two-mode classical field,1 such as Jones vector, Stokes parameters, degree of polarization, or PoincarÈ sphere, can easily see their relationship to spinors, Pauli matrices, purity criterion for (two-level) quantum systems, Bloch sphere, and so on. Many quantum experiments on photon polarization states are directly analogous to classical experiments based on birefringence or polarizer action. However, intuitive (and frequently naÎve) analogies between classical and quantum polarization states lead to incorrect interpretation of experimental results. In particular, the concepts of degree of polarization and purity of state, which are equivalent for classical monochromatic plane waves, as well as concept of state of a two-level system, must be refined as applied to multiphoton and multimode fields [1­ 4]. Adequate description of light polarization is required largely because multiphoton and/or multimode polarization entanglement is used in quantum cryptography [5]. To date, two-photon polarization-transforming mechanisms have been investigated in a number of experiments by using two spatial or frequency modes (e.g., see [6­8] or [9­12], respectively). In those studies, polarization transformations were performed either on the total state (simultaneously on both spatiotemporal modes) or independently on each mode. In another class of experiments (so-called conditional transformations), the result of a measurement per1

formed on one spatiotemporal mode determined the polarization transformation of other modes [13]. This made it possible to achieve a desired change in the polarization state [14]. These and other experiments stimulated a detailed analysis of general polarization transformations of various quantum states belonging to several spatiotemporal modes. The analysis was motivated by the fact that knowledge of the polarization state of a light wave provided only an incomplete characterization of the field. Therefore, a closed-form description of light polarization should be based on (a) a complete characterization of light fields including definitions of fundamental dynamical variables and (b) a procedure of reduction to a characterization in terms of polarization parameters only. Note that, even though studies of polarization had a long history, no consistent conceptual framework had been developed, and semiphenomenological (operational) approaches were used in both classical [15­19] and quantum [20­23] optics. This shortcoming was eliminated in [24­30] by introducing a concept of P-quasispin based on symmetry considerations, which elucidated the nature of polarization and provided a consistent quantum framework for its description, as well as for solving a number of problems in quantum polarization optics. In this paper, we use this framework to develop a formalism for polarization transformation of multimode light fields by means of lossless polarization-changing optical elements. To ensure a better understanding, we premise the main presentation with a brief restatement of some fundamental elements of classical and quantum polarization optics. On the one hand, we can thus highlight the existing analogies between operational characterizations of polarization
30

The modes correspond to the two polarization degrees of freedom.


POLARIZATION TRANSFORMATIONS OF MULTIMODE LIGHT FIELDS

31

in classical and quantum optics. On the other hand, this obviates the need to formally translate the concepts and definitions that have clear physical meanings in the classical framework into the quantum theory, where they are of limited scope. The main conclusions of this study are illustrated by a discussion of single- and twomode biphoton fields, which provides a basis for determining the quantum polarization states in Hilbert spaces of dimensions D = 3 and 4. 2. POLARIZATION OF CLASSICAL LIGHT In classical optics, an adequate description of light polarization was developed only for monochromatic and quasi-monochromatic plane waves [15, 17, 18, 30]. 2.1. Consider the standard stochastic model in which the electric field E(t) (treated as a fundamental dynamical variable) is a random analytic signal: E(t ) = ex Ex(t ) + ey Ey(t ), E
= x, y

Among these, of primary importance for classical polarization optics is the coherency matrix [17, 18] ^ C = E ( t ) E * ( t ) ^ 0 =
,

, = x, y

1 = -2


i=0

3

^ i ( t ) i ,
i ,

(4)

^, I

^

i = 1, 2, 3

=

,

where the Pauli matrices ^1 1 0 , 0 ­1 ^2 0 ­i , i 0

^3 0 1 10 act on the space C p of polarization indices and the ^ coefficients i of the decomposition of C in these 3 matrices are called Stokes parameters [17]. 2.3. Using the properties of the Pauli matrices (see [17] and Appendix A.1), the Stokes parameters can be expressed as the expected values [30]
2

( t ) = A ( t ) e

i ( t ) i t

(1)

e.

Here, ex and ey are the (complex) polarization basis vectors, E = x, y(t) denotes the Jones vector components,2 eit is a fast-oscillating factor containing the carrier frequency , Ax and Ay are slowly varying amplitudes, and x and y are the corresponding phases. 2.2. When both amplitudes and phases are statistically independent, a statistical description of field states can be formulated in terms of the probability distribution ( A x, A y ; x, y ) =
f

^^ i Tr [ C i ] =

, = x , y





i ,

E , E * = i ( { E } )
f

(5)

( i ( { E } ) ) ( { A ; } )



= x, y



d A d



of the classical Stokes vector components
= x, y



( A ) ( ) .
a ph

(2)

i


,



i ,

E E* ,

All observables that can be measured experimentally are defined in terms of Glauber's correlation functions * E 1 ( t 1 ) ... E n ( t n ) E n + 1 ( t E 1 ( t 1 ) ... E n ( t n ) E *n + 1 ( t
f n+1

which are bilinear functions of the Jones vector components
1, 0 2 2 ( { E } ) = E x E* - E y E* = A x - A y , + x+ y

* ) ... E n + m ( t ) ... E *n + m ( t

n+m

) ) (3)

(6)



2 ( { E } ) = E x E * + E y E * = 2 A x A y cos ( x ­ y ) , (7) y x 3 ( { E } ) = i [ E x E* ­ E y E* ] y x = 2 A x A y sin ( x ­ y ) and satisfy the relation 0 = 1 + 2 + 3 .
2 2 2 2

n+1

n+m

â ( A x, A y ; x, y ) dA x dA y d x d y .
2

(8)

Whereas it may seem that the Jones vector is defined in terms of the field vector E in the real three-dimensional space, it actually spans the space C p of polarization indices , which lies at the heart of the concept of P-quasispin [4]. Unfortunately, an almost complete disregard of this distinction has vitiated the understanding and modeling of polarization in classical optics.
2

(9)

3

Note that attempts to introduce generalized Stokes parameters for arbitrary wavefronts (e.g., see [19, 25]) have failed. Vol. 104 No. 1 2007

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Relation (9) defines the PoincarÈ sphere, which provides a convenient graphic representation of polarization states and transformations performed on them: SP = { {
2 i = 1, 2, 3

ples, we write out the matrices describing phase shift ^ between the Jones vector components ( u ps ) and rota^ tion of a polarization plane ( u pr ): i =e 0 ­i 0e = cos 0 + i sin 1 , ^ ^

} = 0 n ,
2 2

n ( n 1 = sin cos , n
tr

(10)

^ u

ps



= sin sin , n 3 = cos ) : n = 1 } , where the column vector (a, b, c)tr is the transpose of the row vector (a, b, c). 2.4. The Stokes parameters are used to define the degree of polarization 1 P cl = ---^ 0 ^ u
pr

(14)

^2 ^0 = cos ­ sin = cos + i sin . sin cos

i = 1, 2, 3



1/2

Furthermore, the common phase shift of the Jones vector components and the field E(t) is described by the matrix i i ^ 0 =e 0 =e , i 0e



2 i

,

(11) ^ u
phs

(15)

which plays a key role in classical polarization optics [17, ^ 18]. It is invariant under the group U(2) = { u (a0, a) : ^^ ^^ u u = u u = ^ } of the polarization transformations I defined in terms of Pauli matrices as ^ u ( a 0, a ) = a
2 0

which is an element of the subgroup ^ U ( 1 ) = { u ( a0 = e , 0 ) } U ( 2 ) ^ = { u ( a 0, a ) } = U ( 1 ) â SU ( 2 ) , ^ SU ( 2 ) = { u ( a 0 = cos ', i sin ' n ) } . A polarization transformation is defined by the angular coordinates (, ) of the rotation axis on the PoincarÈ sphere and the angle ' of rotation about it. However, the polarization transformers used in experiments are generally characterized by transmittance t = cos + i sin cos2 and reflectance r = i sin sin2, where = (/)(no ­ ne)d, the polarizer rotation angle is measured from the vertical direction, d is the geometric thickness of the polarizer plate, and no and ne denote the ordinary and extraordinary refractive indices of the plate material. The change from polarization transformer parameters (t, r) or (, ) to parameters (', n) in the PoincarÈ space is described by the following relations [32]: cos ' = Re t ­ r ,
2 2 i


i=0

3

^i ^0 ^ ai = a0 + a ,

(12)

+ a a * = 1,

* a 0 a j + a 0 a * + i klj a k a l* = 0 , j

where the coefficients ai quantify the polarizationtransforming properties of particular devices [16, 17].4 In physical terms, the invariance of P cl under SU(2) transformations means that the degree of polarization of a quasi-monochromatic plane wave cannot be changed by means of lossless polarization-changing devices (such as retarders, rotators, or beamsplitters). However, the invariance breaks down when the field has a multimode structure. The polarization transformation by a series of optical elements represented by matrices having the form of (12) is quantified by the product of these matrices (group elements) according to multiplication rules (A.2): ^ ^ ^ u ( a 01, a 1 ) u ( a 02, a 2 ) = u ( a 01, a ) , a 0 = a 01 a 02 + a 1 a 2 , a = { a j = a 01 a j 2 + a 02 a j 1 + i klj a k 1 a l 2 } . ^ The explicit forms of the matrices u (a0, a) representing optical elements can be found in [16, 17, 31]. As exam4

(13)

n 1 sin cos = 2 sin 'Re t Im r , n 2 sin sin = ­ 2 sin 'Re t Re r , n 3 cos = sin 'Im t .
2

(16)

However, the group properties of polarization transformations ^ { u (a0, a)} were not emphasized in standard texts on polarization optics [16, 17].

In particular, a retarder plate made from a birefringent ­1 ^^^ material performs the transformation u pr u ps u pr repreVol. 104 No. 1 2007

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sented by the matrix t r . ­r * t * (17)

In the most widespread case of plane-wave radiation (which may not be quasi-monochromatic), the har^ monic-oscillator expansion of E has the form [30, 34] ^ ^ ^ E(z; t ) = ex Ex(z; t ) + ey Ey(z; t ) ^ ^ = ex[ Ex1( z; t ) + Ex2( z; t ) + ... ] ^ ^ + ey[ Ey1( z; t ) + Ey2( z; t ) + ... ], (22)

2.5. Polarization transformations of the elements of the coherency matrix and the Stokes parameters are defined in terms of matrices (12) and Jones vectors, E
u({ai})



^ u E = =

= x, y


3

u ( { i } ) E ai ,
i



^ (­) ^ (+) ^ El ( z ; t ) = El ( z ; t ) + El ( z ; t ) , (18) 2 z ^ (+) E l ( z ; t ) = i ---------------l a l exp i l -- ­ t , -^ c V
^ (+) ^ (­) El ( z ; t ) = ( El ( z ; t ) ) ,

E
i=0

by using relations (4)­(8), (12), (13), and (14). According to these rules, 1, 2, 3({E}) and 0({E}) are transformed as components of three-dimensional SU(2) vectors and SU(2) scalars under transformations (12), respectively:
i = 1, 2, 3

( { E } )

u({ai})

^ U

i = 1, 2, 3

=

j = 1, 2, 3



U ij ( { a l } ) j ,
1

(19)

where the creation and annihilation operators for pho ^ ^ tons of frequency l and polarization , a l and a l , play a key role (analogous to that of the amplitudes Ax, y , the phases x, y , and the Jones vector components) in describing quantized radiation. Both quantum optical observables and quantum states are determined in terms of these operators [4, 34]. 3.1.2. In particular, the Hamiltonian Hf and momentum Pf of a 2m-mode electromagnetic field are expressed as ^ Hf =

1 1 U ij ( { a l } ) = -2

k, l = 1, 2, 3 2 0



^^^^ a k a * Tr ( i k j l ) l
ij


l l=1

m

^ nl ,


^ Pf =

= (a +i

­ aa * )

(20)

= x, y

k
l l=1

m

^ nl ,

= x, y

(23)

^ ^^ nl al al . The corresponding Hilbert space of 2m-mode field states is the tensor product

l = 1, 2, 3 0



( a 0 a * ­ a l a * ) lij + a i a * + a j a * , l 0 j i



u({ai}) 0

^ U 0 = 0 .

(21)

LF ( 2 m ) =


l m

m

LF ( 2 )
l

3. DESCRIPTION OF PHOTON POLARIZATION IN QUANTUM OPTICS 3.1. General Remarks and the Concept of P-Quasispin 3.1.1. In quantum optics (viewed as a subfield of quantum electrodynamics), the standard harmonic^ oscillator expansions of the vector potential A and the ^ electric field strength E operators are used instead of (1) as a fundamental model for describing the quantized (photon) structure of electromagnetic radiation in terms of plane-wave modes [4, 20, 30, 33, 34].

= Span |{ n x, l, n y, l }



l = 1 = x, y

l ^ ( a l ) |0
n l

(24)

of two-mode Fock spaces of the form L F (2) = Span{|{nx, l, ny, l}} [4, 34], where the basis states |{nx, l, ny, l} are the eigenstates of the Hamiltonian oper^ ator H f (and the momentum operator Pf): ^ H f |{ n x, l, n y, l } = H f |{ n x, l, n y, l } , Hf =
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l l=1

m

nl .

(25)

= x, y

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KARASSIOV, KULIK

^ Here, the subscript l in photon number operators n l and the corresponding eigenvalues nl are associated with "spatiotemporal," rather than "polarization," degrees of freedom. 3.1.3. To characterize polarization states, the "classical" plane-wave Stokes operators are defined by analogy with (6)­(8): ^
cl 1, 0

Here, the plus and minus signs are the polarization indices in the helicity basis [4, 33]. This basis is best suited for representation of polarization observables and lossless transformations in polarization optics in terms of ^ ^ the total photon number operator n = , l n l and the



(t)

^ P-quasispin component operators P i = 1, 2, 3 , which satisfy the commutation relations of the Lie algebra u(2) = su(2) + u(1): ^^ ^ ^^ ^ [ P 1, P 2 ] = iP 3 , [ P 2, P 3 ] = iP 1 , ^^ ^ ^ ^ [ P 3, P 1 ] = iP 2 , [ P = 1, 2, 3, n ] = 0. (32)

(­) ^ (­) ^ (+) ^ (+) = [ E x ( z, t ) E x ( z, t ) - E y ( z, t ) E y ( z, t ) ] , +^

(26)

^ (t)
cl 2

^ (­) ^ (+) ^ (­) ^ (+) = [ E x ( z, t ) E y ( z, t ) + E y ( z, t ) E x ( z, t ) ] , ^ cl 3 ( t ) ^ (­) ^ (+) ^ (­) ^ (+) = i [ E x ( z, t ) E y ( z, t ) ­ E y ( z, t ) E x ( z, t ) ] . Substituting (22), we obtain the expression ^ =
cl i

(27)

(28)

3.1.5. For plane-wave modes (with well-defined wavevectors k and frequencies ), which play the role of basis elements in the standard harmonic-oscillator ^ expansions of light fields [33], the operators P i = 1, 2, 3 are proportional to the "classical" Stokes operators: ^ ^ cl P i = i ­2/2. We have the expressions 1 1 ^ -^^ -^ ^ ^^ P 1 = -- ( n x ­ n y ) = -- ( a + a ­ + a ­ a + ) , 2 2 (33)

j, k = 1



m

j k e

i ( k ­ j )


,



i ^^ , a j a k

, (29)

z = -- ­ t , c

j =

2 j --------------- . V

When m 2, time-dependent operators (29) cannot be used to characterize polarization states of photons. Indeed, because they contain fast-oscillating factors of the form e k j , the field will exhibit beats unrelated to polarization behavior, which are determined by the spatiotemporal structure of the field. 3.1.4. These shortcomings in the characterization of photon polarization states are eliminated by invoking a concept of P-quasispin based on the (gauge) invariance of light fields under U(2)P polarization transformations [4, 35], which are defined by complex rotations of polarization basis states e(i) in the spaces L Pspin (i) of "polarization spinors" (e+(i), e­(i))tr, e
=± i( ­ )

1 1 ^ -^^ -^ ^ ^^ P 2 = -- ( n x ' ­ n y ' ) = -- ( a x a y + a y a x ) 2 2 1 -^^ ^^ = i -- ( a ­ a + ­ a + a ­ ) , 2 1 1 ^ -^^ ^^ -^ ^ P 3 = -- ( n + ­ n ­ ) = i -- ( a y a x ­ a x a y ) 2 2

(34)

(35)

in terms of the differences of photon number operators ^ ^^ n = a a in the standard basis pairs of orthogonal polarization modes ( = x, y correspond to the rectilinear basis; = x', y' correspond to the diagonal basis; and = +, ­ correspond to the helicity basis). The corresponding pairs of ladder operators are related by unitary transformations: ^ ^ a x ­ ia y ^ a + = ----------------- , 2 ^ ^ ^ ^ ­ ax + ay ax + iy ^ ^ a y ' = -------------------- , a ­ = ------------------ . 2 2 ^ ^ ax + ay ^ a x ' = --------------- , 2

(i)

~ e ( i ) =

= +, ­



u e ( i ) , (30)

= +, ­



u* ' u



=

(36)

'

^ and polarization-spinor operators a ^ a
5
=±

=±

,5 (31)

(i)

~ a ( i ) =

= +, ­



^ u a ( i ) .

Expressions for the matrices ||u|| in the two representations discussed above are given by (A.9) and (A.10) in Appendix 2.

^ Note that the operator P 3 scales with the helicity oper^ ^ ator ( P 3 = ( S · k)c/2), which makes the helicity basis particularly suitable for use in general theoretical anal^ cl yses [33]. Furthermore, the Stokes operator 0 is pro^ portional to the total photon number operator n =
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^ n



and is related to the P-quasispin component

^ operators P as follows: ^ cl 0 -----2 ^ cl 0 ^^ nn ^2 ^ ------- + 1 = -- -- + 1 = P + P 22 2
2 3

The quantum counterpart (40) of (10) redefines the classical PoincarÈ sphere in terms of quantum expecta^ tion values P i : ^ 2 S P P ( u ) :

(37)

^2 ^2 ^2 ^ P P1 + P2 + P

^ n ^ P = -- , 2

i = 1, 2, 3



^2 P i ( u ) =

i = 1, 2, 3



^2 P i . (41)

^2 where P is the Casimir of the group SU(2)P of polarization transformations [4, 30]6 ^ ^ SU ( 2 ) P = U ( u ) = exp ( ­ i u P ) = exp ­ i


(38)


j=1

3

^ u j P j

^ ^ Note that T (n) U (u = (­sin , cos , 0)tr) is the operator of rotation on both spheres. 3.1.7. Many elements of the P-quasispin formalism ^ (including the formulas for U (u) can be extended to an arbitrary m-mode beam with a spectrum of wavevectors ^ kj by using both components P i k j of "partial" Pj-qua^ ^ P i k j of the total sispins and components P i = P-quasispin characterizing cooperative behavior of polarization states of a multiphoton field [30]. Note that ^ cl whereas the Stokes operators i are defined in terms ^ (±) ^ of the field operators E , the operators P i are defined in terms of their "filtered" Fourier transforms: ^ a
V
1/3



j

^ ^ = exp ( ­ i ' n P ) = U ( ', n ) , ^ ^ ^ ^ n P ( P 1 sin cos + P 2 sin sin + P 3 cos ) ^ ^^ = T ( n ) P3 T ( n ) , where ^ T ( n ) exp ­ -- e 2
­i

(39) â

l

V = ­ i --------------2 l (42)
­i l z / c

1/3


0

^ (+) ( e* E ( z ; t = 0 ) ) e

dz ,

i ^ ^ P + + -- e P ­ 2

^ ^ = exp { ­ i ( ­ P 1 sin + P 2 cos ) } , ^ ^ ^ P ± P 1 ± iP 2 . 3.1.6. The group SU(2) is a subgroup of the group ^ ^ U ( 2 ) P = { exp ( iu o n ) exp ( ­ i u P ) } , which is the quantum counterpart of the unitary group ^ of polarization transformations U(2) = { u (a0 , a)} in classical optics. The operator analog (37) of relation (9) ^2 defines a corresponding PoincarÈ sphere S P [29]: ^ ^ ^2 ^ ^2 ^ ^ ^ SP P( u ) U ( u )PU ( u ) : U ( u )P U ( u ) =
6

(+) *^ where the dot product e · E (z; t = 0) represents the effect of a polarization filter. Furthermore, relation (37) ^ does not hold any longer, and the operator P of total Pquasispin is an independent dynamical variable.

3.2. Polarization Basis in the Hilbert Space LF(2m) of Multiphoton Field States 3.2.1. The P-quasispin formalism is applied to describe the diversity of multiphoton polarization states as follows. In the 2m-mode Hilbert space LF ( 2 m ) = Span |{ n i +, n i ­ }



m

j = 1 = ±

n j ^ ( a j ) |0 ,

(43)

(40)

i = 1, 2, 3



^ P (u) =

2 i

i = 1, 2, 3



^ ^ P = P .
2 i 2

a polarization basis {|P; µ; } is defined as the set of ^2 ^ eigenstates of the commuting operators P and P 3 : ^2 P | P ; µ ; = P ( P + 1 ) | P ; µ ; , ^ P 3 | P ; µ ; = µ | P ; µ ; , 2 P = 0, 1, ..., ,
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Note that this group was implicitly used in [21­23] to determine unpolarized quantum states of light without invoking the concept of P-quasispin. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

(44)

µ P,


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KARASSIOV, KULIK

where the SU(2)P-invariant composite index is introduced to represent the respective degenerate eigenvalues P and µ of the total P-quasispin and its projection. The introduction of basis (44) is equivalent to the decomposition LF ( 2 m ) = L( P) =

Arbitrary (pure) polarization states of such a field are represented by superpositions of the form
P=1

r1 r3 = ------ |2 +, 0 ­ + r 2 |1 +, 1 ­ + ------ |0 +, 2 ­ , 2 2

(48)


2P = 0



L( P), (45)

which are called qutrits by analogy with qubits. The quasispin representation of a qubit (with P = 1/2 and µ = ±1/2) has the form
P = 1/2


µ,

c

P µ,

| P ; µ ;

= r 1 |1 +, 0 ­ + r 2 |0 +, 1 ­ .

(0)

(0)

(49)

Span { |P ; µ ; : P = const } of the space LF(2m) into the infinite direct sum of SU(2)-invariant subspaces L(P) corresponding to particular P-quasispin values. By analogy with decomposition of the state space in the well-known "point" Dicke model, decomposition (45) can be used to study correlated polarization in multimode light [28, 30]. 3.2.2. In the simplest case of a single spatiotemporal mode (m = 1), basis states |P; µ; can be obtained by renumbering the two-mode Fock states |n+, n­ in the helicity basis: |P ; µ = |n + = P + µ, n ­ = P ­ µ 1 n+ n­ ^ ^ = ----------------------- ( a + ) ( a ­ ) |0 . [ n +! n ­! ] (46)

In the case of an arbitrary number m of spatiotemporal modes, the polarization basis may be either of the following: (i) the tensor product


j=1

m

| P j ; µ j
j

of basis sets (46) in the spaces L F (2) associated with individual modes is well suited for analyzing their respective polarization states independently; in particular, the basis set of two-mode (m = 2) two-photon (n = 2) states is 1 = |1 +1, 0 ­1 |1 +2, 0 ­2 , 2 = |0 +1, 1 ­1 |0 +2, 1 ­2 , 3 = |0 +1, 1 ­1 |1 +2, 0 ­2 , 4 = |1 +1, 0 ­1 |0 +2, 1 ­2 . (50)

(51)

For example, the basis set of two-photon states of a single spatiotemporal mode is 1 2 -^ |2 +, 0 ­ = ------ ( a + ) |vac = |P = 1, µ = 1 , 2 1 2 -^ |0 +, 2 ­ = ------ ( a ­ ) |vac = |P = 1, µ = ­ 1 , 2 ^^ |1 +, 1 ­ = a + a ­ |0 = |P = 1, µ = 0 .


(ii) the basis set of states |P; µ; obtained as a combination of "partial" states |Pj ; µj by using SU(2) Clebsch­Gordan coefficients C P1, µ1, P2, µ2 is well suited for analyzing polarization-entangled states; in this case, the composite indices are expressed in terms of "partial" (Pj = nj /2) and "intermediate" (Pij , Pijk, ...) P-quasispins [4, 26], which are determined by using the Clebsch­Gordan coefficients from the standard quantum mechanical addition rules for angular momentum [36]:
P, µ

(47)

nm n1 n2 P ; µ ; P 1 = ---- , P 2 = ---- , ... , P m = ----- ; P ij, P ijk, ... 2 2 2 =

(52)

µ1 + µ2 = µ



C

P ij, µ ij P i, µ i, P j, µ

j

...C

P ijk, µ ijk P ij, µ ij, P k, µ

k

C

P, µ P ijk ..., µ ijk ..., P m, µ

m

|P 1 ; µ 1 |P 2 ; µ 2 ... |P m ; µ m .

3.2.3. In the case of m = 2, we have

=

µ1 + µ2 = µ



C

P, µ P 1, µ 1, P 2, µ

2

| P 1 ; µ 1 | P 2 ; µ 2 (53)

n1 n2 P ; µ ; P 1 = ---- , P 2 = ---- 2 2

1 = P ; µ ; ' n = n 1 + n 2, = -- ( n 1 ­ n 2 ) 2
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^ ( P­ )

P­µ

^ ( E 21 )

P­

2P ^ ( a 11 ) ( X 12 )

n /2 ­ P

|0 ,

and are similar to (50): |P = 1; µ = 1 ( µ 1 = 1/2, µ 2 = 1/2 ) ; { 1, 1 } =C
1, 1 ^^ 1/2, 1/2 ; 1/2, 1/2 a +1 a +2

where the composite index ' is associated with the quantum numbers of the group SU(2) generated by ^ E
12

^^ |0 = a +1 a +2 |0 ,


=

=±



^^ a1 a2 ,



^ E


21

=

=±



^^ a2 a1 , (54)



|P = 1; µ = ­1 ( µ 1 = ­1/2, µ 2 = ­1/2 ) ; { 1, 1 } =C
1, ­ 1 ^^ 1/2, ­ 1/2; 1/2, ­ 1/2 a ­ 1 a ­ 2

(57)

1 ^ E 0 = -2

=±



^^ |0 = a ­1 a ­2 |0 .


^^ [ a1 a

1

^^ ­ a2 a2 ] ,


and with the quantum numbers of the group SU(1,1) generated by ^ X
12

Note that entangled states are naturally described in the framework of the present formalism by using P-quasispin symmetries. In particular, the entangled states of a system with m = 2 and n = 2 correspond to zero value of the projection P3. 3.3. Operational Characterization of Quantum Polarization States of Light in the P-Quasispin Formalism 3.3.1. A conventional operational characterization of quantum polarization states is obtained by using the quantum averages of operators (26)­(28) in formulas analogous to expression (11) for degree of polarization. 3.3.2. An additional characterization of quantum polarization states can be given in terms of the ^i ( P i ) of the P-quasispin component moments ^ operators P i or the characteristic function P({ui}) ({u }). By analogy with the definitions of polarizaU i tion basis sets in LF(2m), operational characteristics of polarization can in principle be defined both for individual and for collective spatiotemporal modes. The most important ones are the following [30]: (1) the collective P-quasispin degree of polarization

^^ ^^ = a +1 a ­2 ­ a ­1 a +2 ,





^ X

12

^ = ( X 12 ) , (55)

1 n -- + 1 = -2 2

=±



^^ [ a1 a


1

^^ + a 2 a 2 ] + 1.

These groups commute with each other and with the group SU(2)P [25, 30]. It follows from (53) that certain collective states in the basis set of two-mode two-photon states are maximally entangled; i.e., they cannot be represented as the direct products of separate mode polarization states. In particular, if each mode is in the single-photon state, then we have the basis states |P = 0 ; µ = µ 1 + µ 2 = 0; = 1/2, 1/2, ( ' = 2, 0 ) =C ­C
0, 0 1/2, ­ 1/2; 1/2, 1/2



i

|1/2, ­ 1/2 (56a)

0, 0 1/2, 1/2 ; 1/2, ­ 1/2

|1/2, 1/2

1 = ------ [ a +1 a ­2 ­ a +2 a ­1 ] |0 2 1 = ------ [ 1 +1 1 ­2 ­ 1 ­1 1 +2 ] , 2 |P = 1 ; µ = µ 1 + µ 2 = 0; = 1/2, 1/2, ( ' = 2, 0 ) =C +C
1, 0 1/2, ­ 1/2; 1/2, 1/2

i = 1, 2, 3 P = 2 ------------------------------- , ^ n



^ P i

2

^ n=



j=1

m

^ n j, (58a)

^ nj =

=±



^ n j,

^ n

j

^^ = a ja j;

|1/2, ­ 1/2 (56b)

(2) the P-quasispin component "noise"
2 ^2 ^2 P i = P i ­ P i ;

1, 0 1/2, 1/2 ; 1/2, ­ 1/2

|1/2, 1/2

(59)

1 = ------ [ a +1 a ­2 + a +2 a ­1 ] |0 2 1 = ------ [ 1 +1 1 ­2 + 1 ­1 1 +2 ] . 2 The remaining basis states (with P = 1 and µ = ±1) are separable into polarization states of individual modes

(3) the total P-quasispin "noise" P =
2


i=1

3

^ 2 2 ^ 2 n ^ P i = P ­ --------- ( P n ) , 4
2

(60)

which gives an operational meaning to the squared ^2 P-quasispin operator P . Note that P-quasispin degree
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of polarization (58a) in the case of m = 1 equals a "classical" and/or a partial one (the latter corresponds to an individual spatiotemporal mode). In the general case of m 2, the P-quasispin degree of polarization is an independent quantity defined for fields with arbitrary spectra in the spatial and frequency domains. However, the "classical" degree of polarization 1 P cl = ----------^ cl 0

i = 1, 2, 3



^ cl i ,

2

ization for the four-mode coherent states |{±i = 1, 2} in the case of two spatiotemporal modes (m = 2). 3.3.3. Since |P; µ; are the eigenstates of the ^2 squared P-quasispin operator P and the projection P3 on the z axis, and the remaining two projections (P1, 2) ^ of the operator P have no nonzero diagonal matrix elements, we have µ P = ------------------------------------------ , P1 + P2 + ... + Pm and hence 11 -P P = 1; µ = ± 1 ; = --, -- = 1, 2 2 11 -P P = 0; µ = 0 ; = --, -- = 0. 2 2 (62) (63) P = P ( P + 1 ) ­ µ , (61)
2 2

(58b)

calculated by substituting the quantum expectations ^ cl (averages) i of the operators given by (29), is obviously a function of time and is different from (58a). Operationally, if T (k ­ j )­1 (T is the measurement time), then a steady degree of polarization can be defined as 1 P st = ----------cl ^ 0 â

i = 1, 2, 3



2 ^ cl i = 2 m


j=1 2 j

m

^ j n j

­1

(58c)

i = 1, 2, 3 j = 1



P i k j

,

It can readily be shown that these values of degree of polarization are equal to those calculated by using (58c). Note that both characteristics given by (61) are independent of the (quantum) numbers . 3.3.4. The coherent states of a two-mode field can be parameterized as follows:
+i

where the overbar denotes the time-averaged value of operator (29) in the Heisenberg representation: ^ =
cl i

i = r i e i cos ---i , 2



­i

= ri e

i ( i + i )

sin ---i . 2

(64)


,



i ,


l=1

m

(

2 ^^ l ) al al

.

^ Then, we can use the results of the calculations of P i presented in [27] to obtain r 1 + r 2 + 2 r 1 r 2 cos 1 cos 2 cos ( 1 ­ 2 ) P = -------------------------------------------------------------------------------------------------- , 2 2 r1 + r2 (65) 32 2 2 P = -- ( r 1 + r 2 ) . 4
4 4 22

Note that the steady degree of polarization also differs from the quasispin one in the general case, because it is calculated as a weighted average of partial contributions. As examples, we calculate the degree of polarization and the total polarization noise for the basis states |P; µ; given by (50) and (56) and the degree of polar24 24

The expression for the steady degree of polarization found by using (58b),

1 r 1 + 2 r 2 + 2 1 2 r 1 r 2 cos 1 cos 2 cos ( 1 ­ 2 ) P st = ---------------------------------------------------------------------------------------------------------------------------- , 2 2 1 r1 + 2 r2
22

(66)

disagrees with (65) and can easily be verified experimentally. 3.4. Polarization Transformations of Important Quantities and Photon Characteristics Polarization transformations can be divided into two classes, which are defined in terms of transformations of creation and annihilation operators covariant under the "partial" and complete groups, respectively.

The transformations covariant under "partial" groups are performed on each jth spatiotemporal mode independently: SU ( 2 )
j P

^ j j = U ( u = { u i } ) = exp ­ i


i

j ^ u i P i k j .
No. 1

(67)

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The collective transformations covariant under the complete group can be reduced to phase-uniform transformations performed on individual spatiotemporal modes: ^ ^ SU ( 2 ) P = U ( u = { u i } ) = exp ( ­ iu P ) = exp ­ i

3.4.3. Transformations of the polarization basis {|P; µ; } are defined as | P ; µ ; = ^ |P ; µ ; ( ', n ) U ( ', n ) |P ; µ ;


i

^ u i P i .

(68)

µ ' = ­P P µ, µ '



P

U

P µ, µ '

( ', n ) |P ; µ '; ,

(72)

where the matrix elements U (A.7)­(A.13).

(', n) are given by

3.4.1. The transformations of polarization-spinor operators are defined as ^ a
±j ^ ^ ^^ a ± j ( ', n ) U ( ', n ) a ± j U ( ', n )

4. POLARIZATION TRANSFORMATIONS IN BIPHOTON OPTICS 4.1. Biphoton optics deals with parametric downconversion models described by interaction Hamiltonians of the form7 ^ H

=

µ=±



^ U

1/2 ±, µ

^ ( ', n ) a µ j ,
1/2 ±, µ

(69)

^ where the matrix elements U expression (A.9).

(', n) are given by

bf

=

i, j = 1 , = ±



m

^^ ^^ [ g ij a i a i + ( g ij ) a i a i ] . (73)




*

3.4.2. The transformations of polarization-vector operators are defined as ^ V
+ = +, 0, ­

^ The quadratic operator H
bf F

bf

is defined on the Fock

^ =V

+ = +, 0, ­

( ', n ) (70)

^ ^ U ( ', n ) V =

+ = +, 0, ­

^+ U ( ', n )

µ = 0, ±



^ U

1 , µ

^+ ( ', n ) V µ ( ', n ) , n) are given the transformausing the operof rotations on

^1 where the matrix elements U , µ (', by (A.11). As an example, we write out tions of the P-quasispin components by ^ ^ ator T (n) U (u = (­sin , cos , 0)tr) the PoincarÈ sphere [30], P
= 1, 2, 3

spaces L (2m) LF(2m) obtained by applying Hamiltonian operator (73) to the vacuum vector |0 (as in standard models of parametric down-conversion [37­39]) or bf to another particular state vector | in LF(2m) [30, 39]. In the simplest model, a pair of photons is generated in a nonlinear crystal (with (2) 0) pumped by a laser beam. The highest biphoton intensity is achieved when the phase matching conditions kP = ki + ks and P = i + s are satisfied. By convention, a biphoton consists of idler and signal photons, which corresponds to m = 2 in (73): LF ( 2 ) =
bf

^ ^ P ( n ) = T ( n ) P T ( n ) :

P = 0, 1



L ( P ) LF ( 2 ) , c |P ; µ .
P µ

2 2 P 1 ( n ) = P 1 cos -- ­ cos 2 sin -2 2 2 ­ P 3 cos sin ­ P 2 sin 2 sin -- , 2 2 -^ P 2 ( n ) = ­ P 1 sin 2 sin -- ­ P 3 sin sin 2 2 2 ^ + P 2 cos -- + cos 2 sin -- , 2 2

L( P) = (71)

(74)


µ

4.2. For two-mode biphoton fields, we can classify the elements of Hamiltonian (73) with respect to the group ^ SU ( 2 ) P = U ( u = { u i } ) = exp ­ i
7


j

^ u i P j

,

^ ^ ^ ^ P 3 ( n ) = P 1 sin cos + P 2 sin sin + P 3 cos .

This simple discrete formulation can readily be generalized to describe continuous models [37, 40]. Vol. 104 No. 1 2007

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^ where P j = P j kl are the components of the total l P-quasispin operator. As a result, we have two classes of elements: the class of P vectors
2 ^+ ^+ ^ V [ ii ] = ( V + [ ii ] 2 ( a +i ) , + 2 ^+ ^^ ^ ^ V 0 [ ii ] a +i a ­i, V ­ [ ii ] 2 ( a ­i ) ) , 2 ^+ ^+ ^ V [ ss ] = ( V + [ ss ] 2 ( a +s ) , + 2 ^+ ^^ ^ ^ V 0 [ ss ] a +s a ­s, V ­ [ ss ] 2 ( a ­s ) ) , ^+ ^+ ^^ V [ is ] = ( V + [ is ] a +i a +s ,



^+ V 0 [is] state (+); ^+ X [is] Bell state (­).

1 ------ {|1+i , 1­s + |1­i , 1+s}, the triplet Bell 2 1 ------ {|1+i , 1­s ­ |1­i , 1+s}, the singlet 2

(75)

(76)

Since the spatiotemporal modes are degenerate, the scalar component does not belong to the qutrit space; hence, neither does the antisymmetric (singlet) Bell state. It is obvious that inverse transformations of the ^+ ^+ V [is] components and X is [is] into quadratic forms of ^ the field operators a example, ^^ a +i a
­s i, s

1 ^+ -^ ^ ^^ V 0 [ is ] ------ ( a +i a ­s + a +s a ­i ) , 2
^+ ^^ V ­ [ is ] a ­i a ­s )

(77)

can also be performed; for

and the P-scalars 1 ^+ -^ ^ ^^ X [ is ] ------ ( a +i a ­s ­ a ­i a +s ) . 2 (78)

^^ a ­i a



+s

1 ^+ ^+ = ------ [ V 0 [ is ] + X is ] , 2 1 ^+ ^+ = ------ [ V 0 [ is ] ­ X is ] . 2

(79)

The following sets of three elements in (75) and (76) correspond to polarization transformations for the indi^+ vidual modes and define the qutrit space: V + [ii ], ^+ ^+ ^+ ^+ V 0 [ii ], V ­ [ii ] for the idler mode and V + [ss], V 0 [ss], ^+ V ­ [ss] for the signal mode. In the notation used in [10, 11, 41, 42], the vectors defined by (75) and (76) correspond to the following basis states:8 ^+ |2+, 0­, two photons in the right circular V + [ii ] polarization mode; ^+ |1+, 1­, single photons in the right and V 0 [ii ] left circular polarization modes; ^+ V ­ [ii ] |0+, 2­, two photons in the left circular polarization mode. The following set of four elements in (77) and (78) corresponds to polarization transformations for both ^+ ^+ modes and defines the ququart space: V + [is], V 0 [is], ^+ ^+ V ­ [is], X [is]. These operators are associated with the following basis states: ^+ V + [is] |1+i , 1+s, single right circularly polarized photons in each idler mode; ^+ V ­ [is] |1­i , 1­s, single left circularly polarized photons in each signal mode;
8

In the experiments on polarization ququarts recently reported in [10, 11], a different (rectilinear) basis was ^^ ^^ employed: a x i a x s |0 , a x i a y s |0 , The change from operators (36) to ^+ ^+ ^+ from V + [is], V ­ [is], V 0 [is], and by means of the matrices


^^ ^^ a y i a x s |0 , a y i a y s |0 . the basis constructed ^+ X [is] is performed


1/2 ­ i /2 1/2 i /2 GK = 1/ 2 0 0 i/ 2 ­ GK = = 1 x i, 1 1 x i, 1 1 y i, 1 1 y i, 1

­ 1/2 ­ 1/2 , 0 1/ 2 i/ 2 0 i/2 i/2

xs ys xs ys

(80)

The change from the rectilinear basis used in [10, 11] to the circular basis can easily be performed; the corresponding transformations can be found in [36].

|1 ­i, 1 ­s 1 ------ ( |1 +i, 1 ­s + |1 ­i, 1 +s ) . 2 1 ------ ( |1 +i, 1 ­s ­ |1 ­i, 1 +s ) 2 |1 +i, 1 +s
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The inverse transformations are GK = = ­ 1/2 i 1/2 i 0 1 / 2 i , ­ 1/2 i 1/2 i 0 ­ 1 / 2 i ­ 1/2 ­ 1/2 1 / 2 0 1/2 1/2 i 1/ 2 0 |1 ­i, 1 ­s 1 ------ ( |1 +i, 1 ­s + |1 ­i, 1 +s ) 2 1 ------ ( |1 +i, 1 ­s ­ |1 ­i, 1 +s ) 2 |1 +i, 1 +s = 1 x i, 1 1 x i, 1 1 y i, 1 1 y i, 1 .

and any number of modes. The discussion in the next subsection is restricted to polarization transformations of two-mode biphoton fields. 4.3. In accordance with Section 3.4, polarization transformations on a frequency-nondegenerate (twomode) biphoton field are of two types. 4.3.1. Collective polarization transformations of the ^+ ^+ components V [is]( = ­, 0, +) of the P-vectors V [is] covariant under the complete group SU(2)P of the total P-quasispin are defined by (70) and are performed by means of zeroth-order retarders. In the case of two (or more) spatially separated modes, a common transformation is performed by passing both beams through similarly oriented identical retarder plates [6, 8]. 4.3.2. Independent action of the "partial" groups ^ j j j SU ( 2 ) P = U ( u = { u i } ) = exp ­ i j ^ u i P i k j

G

KK

(81)

xs ys xs ys


i

The classification introduced here, relying directly on the concept of P-quasispin, leads to the correspond^ ing classification of the vectors V 0, +, ­ [is]|0 L(P = 1) ^+ and X |0 L(P = 0) (since |0 is a P-scalar) [25, 26]: ^ U ( u = { u i } ) |0 = |0 . (82)

It should be noted that the present classification of polarization states of the two-mode biphoton field based on the concept of P-quasispin is not unique. As alternative examples, one may consider the basis sets constructed from the maximally entangled Bell states
(±)

^ also transforms the operators V [ii ] for qutrits as ^+ P-vectors according to (70), while the operators V [is] ^+ (with s i ) and X is for ququarts are transformed as bispinors. In this case, the creation and annihilation oper ^ ^ ators a ± j and a ± j for spatiotemporal individual modes are transformed according to (A.9). In experiments on frequency-nondegenerate biphoton fields, these transformations are performed by using dispersive birefringent materials in polarization transformers. When two modes are spatially separate, each beam is passed through retarder plates with different optical thicknesses and orientations . Examples of such transformations are given in Appendix 3. It is important that when the "partial" groups j SU ( 2 ) P act independently, transformations (A.14)­



(±)

1 -^ ^ ^^ = ------ [ a x 1 a x 2 ± a y 1 a y 2 ] |0 , 2 1 -^ ^ ^^ = ------ [ a x 1 a y 2 ± a y 1 a x 2 ] |0 2

(83)

and the separable states of a photon pair ^^ ^^ ^^ ^^ a x 1 a x 2 |0, a x 1 a y 2 |0, a y 1 a x 2 |0, a y 1 a y 2 |0 ,


^+ (A.16) mix P-vectors V [is] which agrees with the notion of entity. As a result of collective j of the "partial" groups SU ( 2 ) P ,

^ and P-scalars X [is], a ququart as an integral (phase-uniform) action the noninvariant three-



(84)

which may be suitable for specific applications. However, the P-quasispin-based classification appears to be the most natural one from a group-theoretic perspective and can be extended to fields with arbitrary statistics

^+ ^+ ^+ ^+ component vectors V [is] = ( V + [is], V ­ [is], V 0 [is]) ^+ and invariant singlets X [is] are extracted from the two-mode polarization state. Since polarization transformers produce similar effects in all modes, the vec^+ ^+ tors V [is] reduce to qutrits; i.e., the operators V [is], ^+ ^+ V [ii ], and V [ss] are transformed a similarly.
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Note that classification (75)­(78) applies to the most general case of a two-mode two-photon Fock state of light. However, when the parametrically conjugated (spatial or frequency) modes of a biphoton field created by spontaneous parametric down-conversion are detected, each mode can contain only one photon, as dictated by the energy and momentum conservation laws. Therefore, the actual vectors in (75) and (76) correspond to frequency-degenerate biphoton generation in collinear geometry, in which case m = 1. Regimes in which the number of photons generated in an individual mode is two, four, six, and so on can definitely be described in the framework of the P-quasispin formalism.9 Detailed analysis of these regimes requires special treatment and goes beyond the scope of the present study. Four-photon polarization states generated in two spatial modes by parametric down-conversion were experimentally investigated in [43]. 5. CONCLUSIONS The (partial and collective) transformations of biphoton light analyzed here demonstrate the suitability of the classification of states based on the P-quasispin formalism: as a biphoton beam in an arbitrary initial polarization state passes through a series of zerothorder retarders, its singlet component remains invariant while the other three are transformed according to (A.14) and (A.15). An excellent example illustrating the invariance of singlets is the class of states belonging to the so-called decoherence-free subspace [44]. Robustness of these states under collective and partial transformations performed on two spatial modes of biphoton light was demonstrated in [6, 7]. Note that a breakdown of invariance of the scalar (singlet) component of a frequency-nondegenerate ququart can be interpreted as an indicator of the dispersive effect of the polarization-transformer material. Indeed, it follows from (81) and (82) that when a phase-preserving transformation is performed on both idler and signal modes, ^+ the invariance of X [is] entails the invariance of the quantity c2 ­ c3, where c2 and c3 are the respective ^ amplitudes of the ququart basis states a
^ yiaxs ^ xiays

In summary, the present analysis shows that the concept of P-quasispin can be used to overcome the restrictions of classical optics in understanding and characterization of polarization of light. In particular, correct generalization of the classical degree of polarization is possible only in this conceptual framework. Moreover, the group-theoretic meaning is elucidated of both Stokes parameters and polarization transformations, which are formally defined in terms of Pauli matrices in classical optics. Further development of the concept of P-quasispin will certainly facilitate analysis and classification of entangled states of multimode n-photon fields in the context of implementation of partial, cluster, and collective transformations. ACKNOWLEDGMENTS This work was supported by the Russian Federal Program "Research and Development in the Priority Fields of Science and Technology" (State Contract no. 2006-RI-19.0/001/593), the Russian Foundation for Basic Research (project nos. 06-02-16769 and 04-02-17165), and by the State Program for Support of Leading Science Schools (grant NSh-4586.2006.2). APPENDIX 1 Pauli Matrices and "Classical" Polarization Transformations 1. The Pauli matrices acting on the space C P of polarization indices are defined as follows (see [17]): ^0 = 1 0 , 01 3 = 0 1 , 10 ^1 = 1 0 , 0 ­1 ^2 = 0 ­i , i 0 (A.1)
2

|0 and

^j Tr = 2 j 0 . 2. The Pauli matrices obey the multiplication rules ^a^0 ^a ^0^a = =, ^a^b = i where
abc

^ |0 in the rectilinear basis.10 This effect has not a yet been observed. It can be revealed experimentally by statistical reconstruction of quantum states [10, 11, 41, 42] and quantum tomography [30, 47].
9

This can be achieved by appropriate choice of the transformation parameters 'j and nj in (A.14)­(A.16). occurs no matter whether the initial two-mode biphoton field is in an entangled or a separable state.

^c ^0 + ab ,

a, b, c = 1, 2, 3 ,

(A.2)

10This

abc

is the permutation symbol.
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3. The "classical" polarization transformations ^ u (a0 = cos ', a = i sin 'n) SU(2)P are represented in terms of the Pauli matrices as follows [45]: ^0 ^ ^ u ( a 0, a ) = a 0 + a ^ ^ ^0 = cos + i sin ' n = exp [ i ' n ] .

where the parameters ' and n of the resultant transformation are determined by the relations '1 '1 '2 '2 ' cos --- = cos ---- cos ---- ­ ( n 1 n 2 ) sin ---- sin ---- , 2 2 2 2 2 '2 '1 ' n sin --- = n 1 cos ---- sin ---2 2 2 '1 '2 '2 '1 + n 2 cos ---- sin ---- ­ [ n 1 â n 2 ] sin ---- sin ---- . 2 2 2 2 2. The matrix elements U , (', n) for arbitrary P are expressed as follows [45, 46]:
P 2P l

(A.3)

(A.6)

This relation establishes correspondence between the parameters of the classical and quantum SU(2) polarization transformations (cf. (A.2)), and multiplication rules (13) are rewritten as follows [46]: ^ ^ u ( a 01, a 1 ) u ( a 02, a 2 ) ^ '^ '^ = exp [ i 1 n 1 ] exp [ i 2 n 2 ] = exp [ i ' n ] , where the parameters ' and n of the resultant transformation are determined by the relations. cos ' = cos '1 cos '2 ­ ( n 1 n 2 ) sin '1 sin '2 , ' ' ' ' n sin ' = n 1 cos 2 sin 1 + n 2 cos 1 sin 2 ­ [ n 1 â n 2 ] sin '1 sin '2 . (A.4)

U

P ,

( ', n ) = âC


l = 0 m = ­l

l 2l + 1 P ( ­ i ) --------------- l ( ' ) 2P + 1

(A.7)

P, P, ; l, m

4 ------------- Y lm ( , ) , 2l + 1

where C P, ; l, m are the Clebsch­Godran coefficients for the group SU(2), Ylm(, ) are spherical harmonics, APPENDIX 2 and l are the generalized characters of the group SU(2) defined as
P P

P,

Properties of Polarization Transformations in Quantum Optics [46] 1. The multiplication rules for elements are formulated as ^ U ( u = { u i } ) = exp ( ­ i u P ) ^ = exp ( ­ i ' n P ) = U ( ', n ) SU ( 2 ) P : ^ ^ ^ U ( '1, n 1 ) U ( '2, n 2 ) = U ( ', n ) , (A.5)

l i
P

l


m = ­P

e

­ im

C

P, m P, m ; l, 0

= ( ­1 ) l ( ­ ' )
l P

(A.8)

= l ( ' ) ,
P*

l 2P.

For P = 1/2 and 1, matrices (A.7) are specified as follows. 2.1. For polarization spinors,

^ U ( ', n ) = [ U

1/2

1/2 = ±, = ±

( ', n ) ] =

' ' ' cos --- ­ i sin --- cos ­ i exp ( ­ i ) sin --- sin 2 2 2 . ' ' ' ­ i exp ( i ) sin --- sin cos --- + i sin --- cos 2 2 2

(A.9)

In terms of the retarder parameters t = cos + i sin cos 2 and r = i sin sin 2, the spinor transformation is represented by the matrix
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^ 1/2 U ( , ) =
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t r. ­r * t *

(A.10)

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2.2. For polarization vectors, ^ U
P=1

( ', n ) = [ U

1 = +, 0, ­ ; µ = +, 0, ­

( ', n ) ] ' --- sin exp ( ­ i ) 2

cos -' ­ i sin -' cos ---- 2 2

2

' ­ 2 i sin --- sin exp ( ­ i ) 2

­ sin cos -' ­ i sin -' cos --â -- 2 2 1 ­ 2 sin2 -' sin2 -- 2

2

=

' ­ i sin --- sin exp ( ­ i ) 2 ' ' â cos --- ­ i sin --- cos 2 2 ' ­ sin --- sin exp ( i ) 2
2

' + i sin --- sin exp ( ­ i ) 2 ' ' â cos --- + i sin --- cos 2 2 cos -' + i sin -' cos ---- 2 2
2

(A.11) ,

' ­ 2 i sin --- sin exp ( i ) 2 ' ' â cos --- + i sin --- cos 2 2

in terms of t and r, (t) r*
2

U 2 rt
2

0 ,

( ', n ) =

, 0,

.

(A.13) APPENDIX 3

^ U

P=1

( , ) = ­ tr * t 2 ­ r
2

(r)

2

t*r .
2

(A.12) Partial Transformations

­ 2t*r* t*

2.3. For polarization scalars,

1. Transformations of P-vectors are expressed as

^+ V [ is ] ^ =U
1/2 ±, +

SU(2)i â SU(2)s

^+ ^ 1/2 ^ 1/2 ^+ U ( 'i ; n i ) U ( 's ; n s ) V [ is ] : V ± [ is ]
1/2 ±, ­

SU(2)i â SU(2)s


, 1/2 ±, ­

^ U

1/2 ±,

^ (i)U

1/2 ±,

^^ ( s ) ai a
1/2 ±, +

s

^ (i)U

1/2 ±, +

^+ ^ ( s ) V + [ is ] + U
1/2 ±, +

^ (i)U
1/2 ±, ­

1/2 ±, ­

1 ^+ ^+ ^ ( s ) V ­ [ is ] + ------ V 0 [ is ] ( U 2
1/2 ±, ­

1/2 ±, +

^ (i)U

^ (s) + U

1/2 ±, ­

^ (i)U

(s))

(A.14)

1 ^+ ^ + ------ X is ( U 2 where j = i, s; ^+ V 0 [ is ] 1^ = ------ { U 2
1/2 +, +

^ (i)U

^ (s) + U

^ (i)U

1/2 ±, +

(s)),

^ U

1/2 ±,

^ ( j) U

1/2 ±,

( 'j n j ) ,

SU(2)i â SU(2)s

1 -----2


,

^ {U

1/2 +,

^ (i)U

1/2 ­,

^ (s) + U
1/2 +, ­

1/2 ­,

^ (i)U
1/2 ­, ­

1/2 +,

^^ ( s ) } ai a
1/2 ­, ­

s

^ (i)U

1/2 ­, +

^ (s) + U

1/2 ­, +

^ (i)U

1/2 +, +

1^ ^+ ( s ) } V + [ is ] + ------ { U 2

^ (i)U

^ (s) + U

^ (i)U

1/2 +, ­

^+ ( s ) } V ­ [ is ] (A.15)

1^+ ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 + -- V 0 [ is ] { U +, + ( i ) U ­, ­ ( s ) + U ­, + ( i ) U +, ­ ( s ) + U +, ­ ( i ) U ­, + ( s ) + U ­, ­ ( i ) U +, + ( s ) } 2 1^+ ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 ^ 1/2 + -- X [ is ] { U +, + ( i ) U ­, ­ ( s ) + U ­, + ( i ) U +, ­ ( s ) + U +, ­ ( i ) U ­, + ( s ) + U ­, ­ ( i ) U +, + ( s ) } . 2
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 104 No. 1 2007


POLARIZATION TRANSFORMATIONS OF MULTIMODE LIGHT FIELDS

45

2. Transformations of P-scalars are expressed as ^+ X [ is ] 1^ = ------ { U 2
1/2 +, + SU(2)i â SU(2)s

1 -----2


,

^ {U

1/2 +,

^ (i)U

1/2 ­,

^ (s) ­ U
1/2 +, ­

1/2 ­,

^ (i)U
1/2 ­, ­

1/2 +,

^^ ( s ) } ai a
1/2 ­, ­

s

^ (i)U

1/2 ­, +

^ (s) ­ U
1/2 +, + 1/2 +, +

1/2 ­, +

^ (i)U

1/2 +, +

1^ ^+ ( s ) } V + [ is ] + ------ { U 2
1/2 ­, +

^ (i)U

^ (s) ­ U

^ (i)U

1/2 +, ­

^+ ( s ) } V ­ [ is ] (A.16)

1^+ ^ + -- V 0 [ is ] { U 2 1^+ ^ + -- X [ is ] { U 2

^ (i)U ^ (i)U

1/2 ­, ­ 1/2 ­, ­

^ (s) + U ^ (s) ­ U

^ (i)U

1/2 +, ­

^ (s) + U

1/2 +, ­

^ (i)U

1/2 ­, +

^ (s) + U

1/2 ­, ­

^ (i)U

1/2 +, +

(s)}

1/2 ­, +

^ (i)U

1/2 +, ­

^ (s) ­ U

1/2 +, ­

^ (i)U

1/2 ­, +

^ (s) + U

1/2 ­, ­

^ (i)U

1/2 +, +

(s)}.

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Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 104

No. 1

2007