USING STANDARD PRA S

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Received 15 July 1999; published 10 February 2000. In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU2 that serves.
PHYSICAL REVIEW A, VOLUME 61, 034101

Connection between two Wigner functions for spin systems Sergey M. Chumakov,1 Andrei B. Klimov,1,2 and Kurt Bernardo Wolf 1,3

1

Centro Internacional de Ciencias, Avenida Universidad 1001, 62210 Cuernavaca, Morelos, Mexico Departamento de Fı´sica, Universidad de Guadalajara, Corregidora 500, 44100 Guadalajara, Jalisco, Mexico 3 Centro de Ciencias Fı´sicas, Universidad Nacional Auto´noma de Me´xico, Apartado Postal 48-3, Cuernavaca, Morelos 62251, Mexico 共Received 15 July 1999; published 10 February 2000兲 2

In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU共2兲 that serves to analyze two-particle spin states on a sphere. Recent work by our group has included the definition of an apparently distinct Wigner function on generic Lie groups whose natural range has the dimension of the group and serves for all square-integrable representations; for the SO共3兲 case this entails a three-dimensional ‘‘metaphase’’ space. Both have the fundamental properties covariance and completeness. Here we show how the former is obtained as a restriction of the latter. PACS number共s兲: 03.65.Bz, 42.50.Fx I. CONSTRUCTION OF AGARWAL

This relation is important because it allows the formulation of a measurement theory.

In Ref. 关1兴, Agarwal proposed the construction of a Wigner quasiprobability distribution function for SO共3兲 irreducible representation 共irrep兲 classified states, which has been much used to describe two interacting systems of fixed spin S 关2兴. First one considers the irreducible tensor 关polarization operator, cf. 关3兴, Eq. 共2.4共6兲兲兴 S兲 ⫽ Tˆ 共L,M



S

2L⫹1 S,m ⬘ 兩 S,m ⬘ 典 C S,m;L,M 具 S,m 兩 , 2S⫹1 m,m ⬘ ⫽⫺S



共1兲

S,m ⬘ where C S,m;L,M is the Clebsch-Gordan coefficient that couples two representations of spin S to a total spin 0⭐L ⭐2S and projection M. 关We note that operators only of integer spin L participate, representing SO共3兲 rather than SU共2兲.兴 Then, for a density 共Hilbert-Schmidt operator兲 ␳, the corresponding Wigner-Agarwal type-⍀ function 关4兴 is defined on the sphere ( ␪ , ␾ )苸S 2 through 2S

W A⍀ 共 ␳ 兩 ␪ , ␾ 兲 ⫽

L

兺 兺

L⫽0 M ⫽⫺L

S兲 ⍀ L,M Y L,M 共 ␪ , ␾ 兲 * Tr共 Tˆ 共L,M ␳ 兲,

共2兲

where Y L,M ( ␪ , ␾ )⫽Y L,⫺M ( ␪ , ␾ ) * is the spherical harmonic, Tr is the operator trace, and ⍀⫽ 兵 ⍀ L,M 其 are complex numbers. The simplest analogue of the Wigner function with desirable properties is obtained for ⍀ L,M ⫽1, while other sets ⍀ yield other quasi- or distribution functions. 关The operator 共2兲 * .兴 In particular, acting on ␳ is self-adjoint for ⍀ L,M ⫽⍀ L,⫺M when ⍀ L,M is independent of M, the Wigner function is manifestly covariant under SO共3兲, because similarity transformations of ␳ by a rotation will devolve a geometric transformation of the sphere coordinates 共␪, ␾兲. Also 共for ⍀ ⫽1), using Clebsch-Gordan identities, one can easily prove that the overlap 共or completeness兲 relation holds:



s2

sin ␪ d ␪ d ␾ W IA 共 ␳ 兩 ␪ , ␾ 兲 W IA 共 ␶ 兩 ␪ , ␾ 兲 ⫽Tr共 ␳ ␶ 兲 .

1050-2947/2000/61共3兲/034101共3兲/$15.00

共3兲

II. WIGNER FUNCTION ON A GENERAL LIE GROUP

A rather general construction of a covariant Wigner function has been made on unimodular, exponential-type Lie groups 关5兴, in particular on SU共2兲 关6兴, and can be generalized to nonunimodular 共such as the affine兲 groups 关7兴. Basically, one builds an operator that is the integral over the group manifold of the operator exp关yជ •(xជ⫺Jជ)兴, where Jជ is the vector of its generators, yជ are its polar coordinates, and xជ is a vector whose components are the meta-phase-space coordinates over a real space with the dimension of the group manifold. Here we write the SO共3兲 Wigner function as W ␻B 共 ␳ 兩 xជ 兲 ⫽

冕␻

共 y 兲 dy



yជ ជ d e iyជ •xជ Tr共 e ⫺iyជ •J ␳ 兲 , s2 y

共4兲

where ␻ (y) is a weight function of the length y⫽ 兩 yជ 兩 , one of the polar coordinates of the group; this ‘‘radius’’ labels conjugation classes. Instead of 共0, 2␲兲, the ‘‘radius’’ y must have the range 共⫺␲, ␲兲 to ensure the hermiticity of the Wigner function, W ␻B ( ␳ † 兩 xជ )⫽W ␻B ( ␳ 兩 xជ ) * . In the SO共3兲 manifold of polar coordinates, we recall that diametrically opposite points are identified; hence y actually ranges over a circle. In polar coordinates, the left- and right-invariant measure over the group is ␻ (y)⫽ 21 sin2 21 y. Note that the Wigner function 共4兲 is a function of xជ 苸R3 , in contradistinction to the Wigner-Agarwal function 共2兲, which is a function of the sphere ( ␪ , ␾ )苸S2 only, and that all spins 共irreducible representations兲 of SO共3兲 are included. It is easy to see also that, for any weight ␻, the general Wigner function 共4兲 is covariant. The purpose of this Brief Report is to show in what sense both Wigner functions are equivalent, and in what do they differ. III. EQUIVALENCE BETWEEN THE TWO WIGNER FUNCTIONS

First we recall two generating functions that are valid for numbers and for SO共3兲 operators in their (2S⫹1) 2 -dimensional reducible representation 关3兴; the latter

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is reduced into the irreducibles L⫽0,1,...,2S. They are e

iyជ •xជ





L

兺 兺 4 ␲ i L j L共 xy 兲 Y L,M L⫽0 M ⫽⫺L 2S



e ⫺iyជ •J ⫽

g LS 共 y 兲 ⫽

L

兺 兺

g LS 共 y 兲 Y L,M

L⫽0 M ⫽⫺L

2 冑␲ 共 ⫺i 兲 L

冉冊 冉冊

yជ xជ * Y L,M , 共5兲 y x

冉冊

where j L (z) is the spherical Bessel function of order L, and g LS (y) is written in terms of the generalized characters of the group, which are 关see Ref. 关3兴, Sec. 2.4, Eqs. „2.4共8兲… and „4.15共38兲…兴

␹ LS 共 y 兲 ⫽







d d cos 21 y



L

1 2

sin共 S⫹ 21 兲 y sin 21 y)

2S

L

兺 兺 h LS共 x 兲 Y L,M L⫽0 M ⫽⫺L A

.

L

h LS 共 x 兲 ⫽4 ␲ i L

冉冊

yជ * S兲 Tr共 Tˆ 共L,M ␳兲 y

冉冏冊

⫽W h S 共 x 兲 ␳





0

x ␯ dxh LS 共 x 兲 ⫽4 ␲ i L

xជ , x

共7兲

共8兲

共9兲

dy ␻ 共 y 兲 j L 共 xy 兲 ␹ LS 共 y 兲 ,





⫺␲

dy ␻ 共 y 兲 ␹ LS 共 y 兲

1 4 ␲ 2 2 ␯ ⌫„ 2 共 L⫹ ␯ ⫹1 兲 …







0

dy ␹ LS 共 y 兲 ␹ L ⬘ 共 y 兲 ⫽2 ␲ ␦ L,L ⬘ S

2S⫹1 . 2L⫹1

共12兲

共10兲

dx x ␯ j L 共 xy 兲

␻共 y 兲 ⫽ dy ␯ ⫹1 ␹ LS 共 y 兲 . 1 冑2S⫹1 ⌫„ 2 共 L⫺ ␯ ⫹2 兲 … ⫺ ␲ y







冉冏冊

x ␯ dx W ␻B 共 ␳ 兩 xជ 兲 ⫽W A1 ␳ S, ␯

xជ , x

0⬍ ␯ ⬍1.

共14兲

The conclusion of this development is that the new, group-theoretically motivated Wigner function 共4兲, which is defined over R3 and is a functional of a density 共HilbertSchmidt兲 operator, when restricted to spin S, yields the covariant Wigner-Agarwal function 共2兲, which is defined over the sphere, endowing it with a radius-dependent type ⍀ L (x). As shown in Eq. 共10兲, it is the spherical Hankel transform of order L of the weight function ␻ (y) times the generalized characters 共7兲 of the group. IV. OVERLAP FORMULA AND THE WEIGHT FUNCTION

The freedom we have assumed for the weight function ␻ (y) of the conjugation classes of the group 共and their harmonic transform function over the irreducible representations兲 is curtailed when we demand that the general Wigner function satisfy the overlap condition 共3兲. Indeed, when we use Eq. 共8兲 to compute this integral over R3 in polar coordinates, we find



R3

dxជ W ␻B 共 ␳ 兩 xជ 兲 W ␻B 共 ␶ 兩 xជ 兲 2S



L

兺 兺

L⫽0 M ⫽⫺L







0

S兲 S兲 ␳ 兲 Tr共 Tˆ 共L,⫺M ␶兲 共 ⫺1 兲 M Tr共 Tˆ 共L,M

共15兲

x 2 dx 关 h LS 共 x 兲兴 2 ,

where we have used the integration over the sphere of the two spherical harmonics to eliminate two of the sums. Now we can compute the remaining integral over x, replacing the square of Eq. 共10兲 and noting the Parseval relation between norms of two functions related by the Hankel transform,







0

共11兲

2S 1 2L ⬘ ⫹1 ⌫„ 2 共 L ⬘ ⫺ ␯ ⫹2 兲 … S y ␯ ⫹1 ␹ 共 y 兲, 2 ␯ 共 2 ␲ 兲 3 L ⬘ ⫽0 冑2S⫹1 ⌫„ 21 共 L ⬘ ⫺ ␯ ⫹1 兲 … L ⬘ 共13兲

which will be independent of L. Then, the relation between the general and the Wigner-Agarwal functions 共4兲 and 共2兲 is

0

where we have used the integral over the sphere yជ /y of two spherical harmonics to eliminate two of the sums. The form of Eq. 共8兲 is that of a Wigner-Agarwal function 共2兲 of type ⍀ L,M ⫽h LS (x). We note that the independence of ⍀ on the projection number M is a necessary and sufficient condition for covariance. We now show that, if we integrate the general Wigner function W ␻B ( ␳ xជ ) over x⭓0 with measure x ␯ dx, 0⬍ ␯ ⬍1, we can find a weight function ␻ (y) such that it yields the Wigner-Agarwal function. In Eq. 共8兲, this integral affects only the factor h LS (x), yielding



␻ S, ␯ 共 y 兲 ⫽

y

The generalized characters are functions that include the group characters in the way that the associated Legendre polynomials include the ordinary Legendre ones. Now, replacement of Eqs. 共5兲 and 共6兲 into Eq. 共4兲 yields the relation between the two Wigner functions in the form W ␻B 共 ␳ 兩 xជ 兲 ⫽



This allows us to extract the function ␻ S (y) such that Eq. 共11兲 is 兰 ⬁0 x ␯ dxh LS (x)⫽1, namely 共6兲

共 2S⫹1 兲共 2S⫺L 兲 ! L sin 共 2S⫹L⫹1 兲 !



⫺␲

yជ * 共 S 兲 Tˆ L,M , y

␹ LS 共 y 兲 ,

冑2S⫹1

共See Ref. 关8兴, Eq. 6.561.14.兲 An orthogonality relation for the generalized characters 关which does not appear in Ref. 关3兴, but can be proven from its Eq. 4.15共2兲兴 is

x 2 dx 兩 h LS 共 x 兲 兩 2 ⫽

32␲ 4 2S⫹1



dy 关 ␻ 共 y 兲 ␹ LS 共 y 兲兴 2 . y2

共16兲

When ␻ (y) is such that this integral is a constant, then the

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right-hand side of Eq. 共15兲 is Tr 共␳␶兲, and the usual overlap formula 共3兲 holds. This occurs when ␻ (y)⫽y sin 21 y, independently of the assumed bound on the spins S. While the Wigner-Agarwal function was introduced prompted by physical considerations in quantum optics, the Wigner function 共4兲 was built in Ref. 关5兴 for polychromatic wave optics, but with the view to extend its definition to

general Lie groups. Several particular cases were studied 关6兴, among them the SU共2兲 case that appears in this report. A more comprehensive mathematical treatment in 关7兴 shows that for the affine group, the proposed Wigner function matches the concept of hyperimage in wide-band radar imaging 关9兴. It is good to see the relation between these apparently disjoint functions.

关1兴 G. S. Agarwal, Phys. Rev. A 24, 2889 共1981兲; see also R. L. Stratonovich, Zh. Eksp. Teor. Fiz. 关 Sov. Phys. JETP 31, 1012 共1956兲兴. 关2兴 J. P. Dowling, G. S. Agarwal, and W. P. Schleich, Phys. Rev. A 49, 4101 共1994兲; G. S. Agarwal, R. R. Puri, and R. P. Singh, ibid. 56, 2249 共1997兲. 关3兴 D. A. Varshalovich, A. N. Moskalev, and V. K. KhersonskiŽ, Quantum Theory of Angular Momentum 共World Scientific, Singapore, 1988兲. 关4兴 R. Gilmore, in The Physics of Phase Space, edited by Y. S. Kim and W. W. Zachary, Lecture Notes in physics Vol. 278 共Springer, Berlin, 1988兲, p. 211; P. Fo¨ldi, M. G. Benedict, and A. Czirja´k, Acta Phys. Slov. 48, 335 共1998兲. 关5兴 K. B. Wolf, Opt. Commun. 132, 343 共1996兲. 关6兴 K. B. Wolf, N. M. Atakishiyev, and S. M. Chumakov, Wigner

distribution function for finite signals 关in Photonic Quantum Computing, Proceedings of SPIE, edited by S. P. Hotaling and A. R. Pirich 共SPIE, Bellingham, WA, 1997兲, Vol. 3076, pp. 196–206兴; N. M. Atakishiyev, S. M. Chumakov and K. B. Wolf, J. Math. Phys. 39, 6247 共1998兲; S. M. Chumakov, A. Frank and K. B. Wolf, Phys. Rev. A 60, 1817 共1999兲. 关7兴 S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf 共unpublished兲. 关8兴 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Sums, Series and Products 共Nauka, Moscow, 1971兲. 关9兴 J. Bertrand and P. Bertrand, Int. J. Imaging Syst. Technol. 5, 39 共1994兲; J. Bertrand and P. Bertrand, IEEE Trans. Geosci. Remote Sens. 34, 1144 共1996兲; J. Bertrand and P. Bertrand, J. Math. Phys. 39, 4071 共1998兲.

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