Quantum field theory of photons with orbital angular momentum - Core

PHYSICAL REVIEW A 73, 013805 共2006兲

Quantum field theory of photons with orbital angular momentum G. F. Calvo, A. Picón, and E. Bagan Grup de Física Teórica & IFAE, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain 共Received 27 July 2005; published 9 January 2006兲 A quantum-field-theory approach is put forward to generalize the concept of classical spatial light beams carrying orbital angular momentum to the single-photon level. This quantization framework is carried out both in the paraxial and nonparaxial regimes. Upon extension to the optical phase space, closed-form expressions are found for a photon Wigner representation describing transformations on the orbital Poincaré sphere of unitarily related families of paraxial spatial modes. DOI: 10.1103/PhysRevA.73.013805

PACS number共s兲: 42.50.Ct, 03.70.⫹k, 03.67.Lx

I. INTRODUCTION

Photons are among the main carriers of information. This information can be encoded in their energy, linear momentum, and polarization state. In recent years, another degree of freedom for photons has been recognized: their orbital angular momentum 共OAM兲. In 1992 Allen and co-workers 关1兴 showed that optical paraxial cylindrical beams having an azimuthal phase dependence of the form exp共il␾兲 carry a discrete OAM of lប units per photon along their propagation direction. This angular momentum produces a mechanical effect 共induces a torque兲 when suitable light patterns interact with matter; it can be transferred from spatial beams containing phase dislocations on their axis 共e.g., optical vortices兲 to suitable trapped microscopic particles in optical tweezers 关2兴. At the quantum level, considerable interest has been brought for quantum information processing exploiting single and entangled photons prepared in a superposition of states bearing a well defined OAM 关3–12兴. Indeed, one of their main distinguishing features is that, at variance with two-level quantum states, or qubits, OAM photon eigenstates involve the more general case of d-level 共d 艌 2兲 quantum states or qudits. Such a generalization to multidimensional states would allow to extend quantum coding alphabets without the need to increase the number of entangled photons, providing also a more secure quantum cryptography. Since fewer photons are needed, the multidimensional approach reduces the decoherence associated to many photon entanglement. Moreover, a striking consequence of such a higher dimensional encoding is that violation of local realism for two maximally entangled qudits is stronger than for two maximally entangled qubits, and increases with d 关6,13兴. For quantum computation applications, OAM photon eigenstates could even enable to optimize certain quantum computing architectures, where a compromise between the number of required qubit and qudit states exists 关14兴. The aim of this paper is to develop a general quantization scheme of field operators in both the nonparaxial and paraxial regimes of light propagation. Within the nonparaxial regime, the obtained operators possess the suitable phase structure that will straightforwardly allow us to proceed, at a later stage, to paraxial field operators. In this regime, the field operators can conveniently be expressed in terms of eigenstate modes of the paraxial orbital and spin angular mo1050-2947/2006/73共1兲/013805共10兲/$23.00

mentum operators. Our approach is further extended to the optical phase space. The paper is organized as follows: Section II reexamines the problem of the separation of the angular momentum of a classical electromagnetic field into orbital and spin angular momentum components and provides a general approach as to whether such a decomposition is physically meaningful. Section III adapts the previous approach into the field quantization formalism. Section IV presents a powerful scheme, based on the Wigner representation, to describe geometric transformations of photons prepared in states bearing OAM. Conclusions of the paper are drawn in Sec. V.

II. CLASSICAL APPROACH REVISITED

Energy, linear momentum, and angular momentum constitute the key physical quantities that characterize the electromagnetic field configurations. First of all, they are constants of motion. Their conservation can be cast as a continuity equation relating a density and a flux 共tensor兲 density, or current, associated to the conserved quantity. The total free electromagnetic angular momentum J共r0兲 in a given volume V with respect to a point r0 is defined by 关15兴

J共r0兲 = ␧0

冕

V

d3r共r − r0兲 ⫻ 共E ⫻ B兲 = J共0兲 − r0 ⫻ P, 共1兲

where r is the position vector, E and B are the electric and magnetic fields, and P denotes the total linear momentum of the free electromagnetic field. We shall focus our study on J共0兲 ⬅ J. Notice that J is defined in an analogous manner as the angular momentum of a system of massive particles. It can be expressed as the integral of an angular momentum density which is equal to the cross product of r with the linear momentum density ␧0共E ⫻ B兲. The conservation of J is guaranteed if the flux of angular momentum through the surface S enclosing V vanishes 共e.g., for fields that decay sufficiently fast when V becomes large兲. This is reflected by the integral form of the continuity equation for the i component of the angular momentum 关16兴

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©2006 The American Physical Society

PHYSICAL REVIEW A 73, 013805 共2006兲

CALVO, PICÓN, AND BAGAN

⳵Ji =− ⳵t

冕

S

M lidSl ,

共2兲

−1 2 where M li = ⑀ijkr j关␦kl共␧0E2 + ␮−1 0 B 兲 / 2 − ␧0EkEl − ␮0 BkBl兴 is the angular momentum flux tensor 关66兴. It is well known that in classical mechanics the total angular momentum of a system of point matter particles can be separated into two contributions. A first one describes the angular momentum associated with the center-of-mass motion of the whole system, while the second one refers to the relative angular momentum of the constituent particles with respect to the total center of mass. The center-of-mass contribution is thus dependent on the choice of the reference frame, whereas the relative part is independent. Both quantities obey independent evolution equations. In quantum mechanics, in addition to these two contributions 共which one could consider as giving rise to a purely orbital angular momentum兲, there is an intrinsic or spin angular momentum 共independent of the choice of a reference frame兲 with no classical analog for point particles. Now, the situation for the free electromagnetic field is more subtle. A similar separation of J into strict orbital and spin angular momentum vectors is known to be impossible because no reference frame exist for the photon in which it is at rest 关15,17,18兴. It is, however, feasible to decompose J into two observables 关19兴, whose physical meaning will be provided below, as

J = L + S,

j

冕

V

⬜ d3rE⬜ j 共r ⫻ ⵜ兲A j ,

A共r,t兲 = 兺 ␴

冕

d 3k ⫻ 关⑀␴共k兲␣␴共k兲ei共k·r−c兩k兩t兲 共16␲3␧0c兩k兩兲1/2 共6兲

+ c.c.兴,

where ␣␴共k兲 are the complex amplitudes corresponding to the two circular polarization unit vectors ⑀␴共k兲 共␴ = + 1 for right-handed and ␴ = −1 for left-handed兲. They satisfy ⑀␴共k兲 · ⑀␴* ⬘共k兲 = ␦␴␴⬘ and k · ⑀␴共k兲 = 0. Expansion 共6兲 is a solution of the d’Alembert wave equation. Inspired by the work of Aiello and Woerdman 关23兴, we rewrite Eq. 共6兲 in a way useful to derive its paraxial limit. The essential feature of any paraxial field is that it can be represented as an envelope field modulating a carrier plane wave with wave vector k0. Without loss of generality we take k0 = k0uz, with k0 ⬎ 0 and unitary vector uz, which implies that the carrier plane wave propagates along the positive z direction. Let us introduce the trivial identity

冕

共3兲

⬁

0

dk0

eik0共z−ct兲 ␦关k0 − f共k兲兴 = 1, eik0共z−ct兲

共4兲 A共r,t兲 =

S = ␧0

冕

V

冕

⬁

0

d3rE⬜ ⫻ A⬜ ,

共7兲

where the function f共k兲 ⬎ 0, which at this stage can be any arbitrary function, will be specified below. Upon multiplication of Eq. 共6兲 by Eq. 共7兲 and rearranging, we have

with L = ␧0 兺

We begin by considering in the Coulomb gauge the wellknown expansion in the continuous plane-wave basis of the vector potential A 共which is henceforth assumed to be transverse兲,

共5兲

where the symbol ⬜ denotes the transverse component of the fields 共recall that transverse components of any field F satisfy ⵜ · F⬜ = 0兲. Since L and S involve the transverse part of the vector potential A, they are gauge invariant. Notice that, at variance with L, S is independent of the definition of the origin of the coordinate system. As it occurs for J, one may show that both L and S satisfy continuity equations similar to Eq. 共2兲. Within a purely classical description of paraxial light propagation, the total optical angular momentum along the propagation direction can be decomposed into spin and OAM contributions, each associated with polarization and phase distribution of the beam, respectively 关1,20兴 共see also the excellent reviews and references therein on this subject 关21,22兴兲. As we will prove below, these two contributions correspond to the paraxial versions of L and S along that same direction. Our framework is completely general, thus allowing us to envisage the proper quantization of the fields at a later stage.

dk0eik0共z−ct兲Ak0共r,t兲.

共8兲

Now, by imposing Ak0共r , t兲 to obey the paraxial wave equation, we obtain the explicit dependence of f共k兲,

f共k兲 =

kz + 冑kz2 + 2q2 , 2

共9兲

where we have denoted k = q + kzuz, q and kz being the transverse 共in the x-y plane兲 and longitudinal 共along the z axis兲 wave vectors, respectively. Equation 共9兲 yields a dispersion relation between k0 and k which gives rise to the constraint imposed by the argument f共k兲 in the delta function of Eq. 共7兲. Namely, ␦关k0 − f共k兲兴 = ␦关kz − 共k0 − 共q2 / 2k0兲兲兴共1 + ␽2兲, with ␽ = q / 冑2k20 a parameter that governs the degree of paraxiality. The above transformations enable us to map the variable kz, defined in the whole real axis, into the more convenient positive variable k0. In this way, one may easily carry the integration with respect to kz and cast Eq. 共6兲 in a form that displays both the paraxial and nonparaxial contributions in a more transparent fashion,

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A共r,t兲 = 兺 ␴

冕 冕 冋 ⬁

d 2q

dk0

0

共1 + ␽2兲2

16␲ ␧0ck0冑1 + ␽ 3

4

⫻兵⑀␴关q,k0共1 − ␽ 兲兴␣␴关q,k0共1 − ␽ 兲兴e 2

2

册

1/2

A P共r,t兲 =

ik0共z−ct兲

⫻e

⫻ exp关iq · r⬜ − ik0␽2z − ick0共冑1 + ␽4 − 1兲t兴 + c.c.其. 共10兲 Isotropy of free space allows us to choose the unit polarization vectors ⑀␴关q , k0共1 − ␽2兲兴 in a variety of ways. In what follows, it will be convenient to write them as

⑀␴关q,k0共1 − ␽2兲兴 =

e

−i␴␸

冑2

冋

共1 − ␽ 兲 2

u␳

冑1 + ␽4 − i␴u␸ − uz

冑

册

共11兲

Sz =

2␽ , 1 + ␽4 2

␴

冕

⬁

0

dk0 共16␲ ␧0ck0兲1/2 3

⫻ eik0共z−ct兲ei共q·r⬜−k0␽

2z兲

冕

d2q关⑀␴␣␴共q,k0兲

+ c.c.兴,

共12兲

where we have only retained the quadratic dependence on ␽ in the second phase factor of Eq. 共10兲 as the relevant paraxial contribution. The polarization vectors ⑀␴ = 共ux − i␴uy兲 / 冑2 are now independent of q and k0 共ux and uy are the unit vectors along x and y directions兲. Notice that the structure of the q integrand on the right-hand side of Eq. 共12兲 resembles the well-known paraxial angular spectrum 关24兴. One may exploit this fact and expand, rather than in transverse plane-wave components, into a different transverse state basis: that of Laguerre-Gaussian 共LG兲 modes, LGl,p共r⬜ , z ; k0兲, and their Fourier-transformed profiles, LGl,p共q兲, at z = 0 共see the Appendix for details兲. Using their closure relation, one has ei关q·r⬜−k0␽

2z兴

* = 兺 LGl,p 共q兲LGl,p共r⬜,z;k0兲.

共13兲

l,p

Since the LG modes constitute a complete, infinitedimensional basis for the solutions of the paraxial wave equation, any spatial beam satisfying the paraxial wave equation can therefore be represented in the LG basis in terms of an infinite expansion with complex amplitudes ␣␴,l,p defined by

␣␴,l,p共k0兲 =

冕

* d2qLGl,p 共q兲␣␴共q,k0兲.

Hence one may cast Eq. 共12兲 as

共14兲

冕

dk0 关⑀␴␣␴,l,p共k0兲 共16␲ ␧0ck0兲1/2 3

0

ik0共z−ct兲

LGl,p共r⬜,z;k0兲 + c.c.兴.

共15兲

The paraxial electric and magnetic fields follow from Eq. 共15兲 via the well-known relations E P = −⳵A P / ⳵t and B P = ⵜ ⫻ A P. It is now possible to show, by employing our convenient representation of these paraxial fields into expressions 共4兲 and 共5兲, that the z components of L and S are given by Lz =

where u␳ = q / q, u␸ = uz ⫻ q / q, and ␸ is the polar angle in cylindrical coordinates. The circular polarization vectors ⑀␴关q , k0共1 − ␽2兲兴 are both orthogonal to q + uzk0共1 − ␽2兲, as they should be. It is important to emphasize that the field 共10兲 is exactly equal to the starting expansion 共6兲. It still obeys the d’Alembert equation for all times t ⬎ 0. However, it now exhibits the suitable structure to perform the paraxial approximation for which ␽ Ⰶ 1. In this limit, Eq. 共10兲 reduces to A P共r,t兲 = 兺

兺 ␴,l,p

⬁

兺l ␴,l,p

冕

兺 ␴,l,p

冕

␴

⬁

dk0兩␣␴,l,p共k0兲兩2 ,

共16兲

dk0兩␣␴,l,p共k0兲兩2 .

共17兲

0 ⬁

0

That is, within the paraxial approximation, the total angular momentum Jz along the beam propagation direction, i.e., along z, can be decomposed into the so-called orbital Lz and spin Sz angular momenta which are related to the azimuthal phase dependence of the LG mode basis and their corresponding circular polarization state, respectively. Equations 共16兲 and 共17兲 generalize the well-known results of Allen and co-workers 关1兴 with the remarkable feature that they now possess the appropriate form to carry out the field quantization. Notice also that, since the total energy of any paraxial spatial beam is H P = ␧0 兰 d3r关E2P + c2B2P兴 / 2 = 兺␴,l,p兰⬁0 dk0ck0兩␣␴,l,p共k0兲兩2, in units of ប, the ratio Lz / H P can be conceived from a semiclassical point of view as the OAM per photon.

III. PARAXIAL QUANTIZATION

In the previous section we have derived general classical paraxial expressions for the orbital and spin angular momenta starting from the continuous plane-wave expansions of the fields in free space. In this section we shall undertake the paraxial quantization of the fields. This problem has been considered by several authors in the past 共see Ref. 关25兴 and references therein兲. There, the approach was approximated, often requiring perturbation expansions which did not provide any clear and manageable expressions for the paraxial quantum modes, and, more important, they resorted to the quasimonochromatic approximation which is unsuitable for describing such nonclassical processes as the propagation of broadband entangled photons generated by spontaneous parametric down conversion. The framework developed in the previous section can easily be adapted for field operators 共see also Ref. 关23兴兲. The essential step is to substitute the set of complex amplitudes ␣ by creation and annihilation operators. At a suitable stage we will find more convenient to introduce a different set of creation and annihilation operators: those associated with the LG modes. ˆ 共r , t兲 in The expansion for the vector potential operator A terms of continuous plane-wave mode operators 关recall Eq. 共6兲兴 is

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冕 冉

ˆ 共r,t兲 = 兺 A

d 3k

␴

ប 3 16␲ ␧0c兩k兩

⫻关⑀␴共k兲aˆ␴共k兲e

冊

i共k·r−c兩k兩t兲

1/2

iប Lˆ = − 兺 2 j,␴,␴

+ H.c.兴.

共18兲

The annihilation and creation operators aˆ␴ and aˆ␴† satisfy the usual canonical commutation rules 关aˆ␴共k兲 , aˆ␴† 共k⬘兲兴 ⬘ = ␦␴␴⬘␦共3兲共k − k⬘兲. Using again the trivial identity 共7兲, we can derive the quantized version of Eq. 共10兲, ˆ 共r,t兲 = 兺 A ␴

冕 冕 冋 ⬁

d 2q

dk0

0

ប共1 + ␽2兲2

16␲3␧0ck0冑1 + ␽4

册

1/2

⫻兵⑀␴关q,k0共1 − ␽2兲兴aˆ␴关q,k0共1 − ␽2兲兴eik0共z−ct兲

⫻exp关iq · r⬜ − ik0␽2z − ick0共冑1 + ␽4 − 1兲t兴 + H.c.其. 共19兲 The unit polarization vectors ⑀␴关q , k0共1 − ␽2兲兴 are given by Eq. 共11兲. Equation 共19兲, together with its classical counterpart 共10兲, constitute the first main results of this work. We stress that the field operator 共19兲 is exactly equal to expansion 共18兲, and thus it obeys the d’Alembert wave equation for any time t ⬎ 0. We may now build the paraxial version of Eq. 共19兲. As mentioned previously, this corresponds to the limit with ␽ Ⰶ 1. Recalling the transformation 共13兲 for the LG modes basis, Eq. 共19兲 reduces to the form ˆ 共r,t兲 = A P

兺

␴,l,p

冕 冉 ⬁

dk0

0

ប 3 16␲ ␧0ck0

冊

1/2

⫻ ⑀ j,␴⬘共k兲aˆ␴⬘共k兲e−ic兩k兩t − H.c.其, and

␴

* d2qLGl,p 共q兲aˆ␴共q,k0兲,

共21兲

which, by employing the commutation relations 关aˆ␴共q , k0兲 , aˆ␴† ⬘共q⬘ , k0⬘兲兴 = ␦␴␴⬘␦共2兲共q − q⬘兲␦共k0 − k0⬘兲, and the orthonormalization conditions for the LG modes, satisfy the commutators † 关aˆ␴,l,p共k0兲,aˆ␴⬘,l⬘,p⬘共k0⬘兲兴 = ␦␴␴⬘␦ll⬘␦ pp⬘␦共k0 − k0⬘兲.

k d3k aˆ␴† 共k兲aˆ␴共k兲. k

共23兲

共24兲

ˆ and Sˆ is a true Notice, however, that neither component of L angular momentum operator since they do not fulfill the usual SU共2兲 commutation relations. Instead, they read as 关Sˆi , Sˆ j兴 = 0, 关Lˆi , Sˆ j兴 = iប␧ijkSˆk, and 关Lˆi , Lˆ j兴 = iប␧ijk共Lˆk − Sˆk兲. Nevertheless, it is still possible to choose any two commuting components Lˆi and Sˆi, and thus construct simultaneous eigenstates of these operators. Within the paraxial approximation discussed above, we may concentrate ourselves on the two commuting operators Lˆz and Sˆz. Upon inverting Eq. 共21兲, yielding aˆ␴共q , k0兲 = 兺l,pLGl,p共q兲aˆ␴,l,p共k0兲, we find from Eqs. 共23兲 and 共24兲 that the paraxial OAM and spin operators are finally given by Lˆz = ប 兺 l ␴,l,p

冕

Sˆz = ប 兺 ␴ ␴,l,p

共20兲

again independent of q and k0. At this point, we have introduced the LG mode annihilation operators

冕

冕

Sˆ = ប 兺 ␴

兩␺典 =

where the circularly polarized vectors ⑀␴ = 共ux − i␴uy兲 / 冑2 are

aˆ␴,l,p共k0兲 =

d3k兵⑀*j,␴共k兲aˆ␴† 共k兲eic兩k兩t关k ⫻ ⵜk兴

⬁

0

冕

dk0aˆ␴† ,l,p共k0兲aˆ␴,l,p共k0兲,

共25兲

dk0aˆ␴† ,l,p共k0兲aˆ␴,l,p共k0兲.

共26兲

⬁

0

This means that the most general paraxial one-photon state can then be described as consisting of arbitrary superpositions of eigenstates of Lˆz and Sˆz,

关⑀␴aˆ␴,l,p共k0兲

⫻ eik0共z−ct兲LGl,p共r⬜,z;k0兲 + H.c.兴,

⬘

冕

冕

⬁

0

dk0C␴,l,p共k0兲aˆ␴† ,l,p共k0兲兩0典,

共27兲

where 兩0典 is the vacuum state. The complex coefficients C␴,l,p共k0兲 satisfy the normalization condition 兺␴,l,p兰⬁0 dk0兩C␴,l,p共k0兲兩2 = 1, and can be interpreted as the probability amplitudes for finding the photon in an eigenstate 兩␴ , l , p , k0典 共Fock state兲 with circular polarization ␴, wave vector k0 along the z axis and corresponding to a LG mode having indices l and p, that is, with a well defined spin and OAM in the direction of the z axis. Interestingly enough, if one uses Eq. 共21兲 and rewrites the one-photon state 共27兲 as 兩␺典 =

共22兲

We have now at our disposal all the necessary ingredients to examine the quantization of angular momentum. For massless particles, such as the photon, the only physically meaningful component of the total angular momentum operator Jˆ is the one along their propagation direction 关17兴. Of course, one may formally quantize the decomposition of Jˆ ˆ + Sˆ where the operators L ˆ and Sˆ are given by Eqs. 共4兲 and =L 共5兲 with the classical fields replaced by field operators. They can be cast in the form 关26兴

兺 ␴,l,p

兺冕 冕 ⬁

␴,l,p

dk0

0

d2qf ␴,l,p共q,k0兲aˆ␴† 共q,k0兲兩0典,

共28兲

where f ␴,l,p共q , k0兲 = C␴,l,p共k0兲LGl,p共q兲, one can conceive f ␴,l,p共q , k0兲 as the components of the paraxial photon wave function in momentum representation. This implies that, within the paraxial approximation, the energy density of a single photon can be localized in the transverse plane orthogonal to their main propagation direction 共z axis兲 with a Gaussian-dependence falloff. Such a spatial localization is not in contradiction with the exponential 共but less than Gaussian兲 falloff localization limit shown by Bialynicki-

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Birula 关27兴 which applies to three 共and thus to nonparaxial photons兲 rather than to two spatial dimensions 共see also Refs. 关28–31兴 for closely related discussion on this兲. In fact, the Paley-Wiener theorem still affects the maximum localization of the energy density along the propagation direction. Moreover, the characteristic length that controls the transverse spatial extension of the LG mode cannot certainly take arbitrarily small values, but those which are compatible with the paraxial approximation, i.e., much larger than the photon wavelength ␭ = 2␲ / k0. It is important to point out that although the described paraxial quantization has been carried out for LG modes, which constitute the natural basis for operator Lˆz, it is straightforward to show that one could have chosen as well other paraxial modes such as the Hermite-Gaussian 共in Cartesian coordinates兲 and the Ince-Gaussian 关32兴 共in elliptical coordinates兲. Moreover, our nonparaxial expression 共19兲 could readily accommodate photon states in Bessel 共diffraction-free兲 modes 关33,34兴 and oblate spheroidal mode solutions of the Helmholtz equation 关35兴, to name just a few examples.

IV. PHASE-SPACE PICTURE OF OAM PHOTON STATES

Phase space, which is a fundamental concept in classical mechanics, remains useful when passing to quantum mechanics. In a similar fashion with probability density distribution functions in classical systems governed by Liouville dynamics, quasiprobability distributions have been introduced in quantum mechanics. They can provide a description of quantum systems at the level of density operators 共although not at the level of state vectors兲. Among them, the Wigner function stands out because it is real, nonsingular, yields correct quantum-mechanical operator averages in terms of phase-space integrals, and possesses positivedefinite marginal distributions 关36–38兴. It is, however, only positive for Gaussian pure states, according to the HudsonPiquet theorem. By exploiting the analogy between classical and quantum mechanics with geometrical and wave optics, Wigner distributions have been developed in the context of classical wave optics of both coherent and partially coherent light fields 关39–41兴, where they are Fourier-related to the cross-spectral densities. Particularly outstanding has been the symplectic invariant approach by Simon and Mukunda 关42兴, which has been applied to anisotropic Gaussian Schell-model beams via the relation between ray-transfer matrices of first-order optical systems and unitary 共metaplectic兲 operators acting on wave amplitudes and cross-spectral densities. It is well known that a convenient way to visualize the transformation of qubits is provided by the Poincaré 共also known as the Bloch兲 sphere representation. For polarization states, the north and south poles correspond to right- 兩␴ = + 1典 and left-handed 兩␴ = −1典 circularly polarized eigenstates, respectively. More generally, any completely polarized state can be described as a linear superposition of left- and righthanded circular polarization in the form 共up to a global phase兲

␪ ␪ 兩␪, ␸典 = cos 兩␴ = + 1典 + ei␸ sin 兩␴ = − 1典, 2 2

共29兲

which, on the Poincaré sphere, corresponds to a point on the surface having angular coordinates ␪ and ␸. In an analogous manner with the above picture for polarization states, a Poincaré sphere was introduced by Padgett and Courtial to represent paraxial first-order-mode spatial beams carrying OAM 关43兴. Its underlying SU共2兲 symmetry was subsequently shown 关44兴. In this picture, the poles of the sphere correspond to LG modes with radial-node number p = 0 and topological charge l = ± 1 共plus and minus standing for the north and south poles, respectively兲. Hence, in complete analogy with Eq. 共29兲, any state on the first-order-mode sphere can be written as

␪ ␪ 兩␪, ␸典l=1,p=0 = cos 兩l = 1,p = 0典 + ei␸ sin 兩l = − 1,p = 0典. 2 2 共30兲 This superposition generates other structurally stable states. For example, in the equatorial plane of the sphere, combinations with ␪ = ␲ / 2 and ␸ = 0 共␪ = ␲ / 2, ␸ = ␲兲 give rise to Hermite-Gaussian modes 兩nx , ny典 having indexes nx = 1, ny = 0 共nx = 0, ny = 1兲. Though the first-order orbital Poincaré sphere constitutes an elegant framework to represent families of states bearing OAM and their transformation, as points and paths connecting these points on the sphere, higher-order modes cannot be described by Eq. 共30兲. That is, states on higher-order Poincaré spheres involve more complex superpositions of LG mode states. A generalization would provide, among the many possibilities, direct means to visualize the development of geometric phases in optical beams 关45,46兴 and photon states. Recently, we were able to carry out the above-mentioned generalization to all higher-order orbital Poincaré spheres, and extend the concept of geometric phases in paraxial optical beams under continuous mode transformations 关47兴. Via an SU共2兲 Lie-group operator algebra, we mapped spin coherent states 关48兴 onto families of spatial modes carrying OAM and belonging to such generalized Poincaré spheres. Remarkably, these families of spatial modes could be represented in a compact form by resorting to the Wigner function formalism, allowing us to reveal their hidden symmetries. Our aim here is to explicitly show this generalization by constructing sets of OAM photon states 兩␪ , ␸典l,p as points 共␪ , ␸兲 on orbital Poincaré spheres Ol,p 共which can be labeled by l and p or equivalently by l and the sphere order N = 2p + 兩l兩 艌 0兲. The continuum of states on Ol,p can be generated † 兩0典 共fixed on the poles of from LG mode states 关67兴 兩l , p典 = aˆl,p the sphere兲 through the following unitary operations: ˆ · u 兲兩l,p典 ⬅ U ˆ 共␪, ␸兲兩l,p典. 兩␪, ␸典l,p = exp共− i␪L ␸

共31兲

ˆ is a true angular momentum operator 共not to be Here, L confused with Lˆ , see below兲 and u␸ = 共−sin ␸ , cos ␸ , 0兲 a unit vector in the equatorial plane of the sphere. The action of the ˆ on states 兩l , p典 can be interunitary 共metaplectic兲 operator U

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FIG. 1. Orbital Poincaré sphere of second-order modes. The poles of the sphere correspond to Laguerre-Gaussian modes with l = ± 2 and p = 0 共positive and negative signs for north and south poles, respectively兲. The states in the equatorial plane 共␪ = ␲ / 2兲 with ␸ = 0, and ␸ = ␲ yield the Hermite-Gaussian modes having indexes nx = 2, ny = 0, and nx = 0, ny = 2 共not shown兲, respectively.

preted as a counterclockwise rotation of ␪ about u␸ that takes the axis containing the poles into the direction with unit vector ur = 共cos ␸ sin ␪ , sin ␸ sin ␪ , cos ␪兲. The important observation is that such a rotation gives rise to multidimensional superpositions of LG mode states that can be cast as l

兩␪, ␸典l,p =

p

兺 兺 Cl⬘,p⬘共␪, ␸ ;l,p兲兩l⬘,p⬘典,

共32兲

l⬘=−l p⬘=0

where 2p⬘ + 兩l⬘兩 = 2p + 兩l兩 ⬅ N. The complex coefficients Cl⬘,p⬘共␪ , ␸ ; l , p兲 depend on the point 共␪ , ␸兲 on the Nth-order Poincaré sphere Ol,p. One may easily verify that Eq. 共32兲 includes the particular case of Eq. 共30兲 with C1,0共␪ , ␸ ; 1 , 0兲 C0,0共␪ , ␸ ; 1 , 0兲 = 0 and C−1,0共␪ , ␸ ; 1 , 0兲 = cos ␪ / 2, = ei␸ sin ␪ / 2. Figure 1 depicts representative modes associated to their angular orientations on the second-order-mode Ol,p sphere 共N = l = 2兲. The 共␪ , ␸兲 distribution of states 共31兲 on Ol,p can be obtained by employing the Wigner function representation. In the optical phase space, let r⬜ = 共x , y兲 and p = 共px , py兲 denote the transverse position and momentum 共normalized wave vector兲 variables, respectively, and rˆ ⬜, pˆ be the associated canonical Hermitian operators. The only nonvanishing commutation relations among these operators are 关xˆ , pˆx兴 = 关yˆ , pˆy兴 = i⑄. The reduced wavelength ⑄ = ␭ / 共2␲兲 = 1 / k0 plays here the optical analog of ប. For convenience, we arrange the phase space variables and the canonical operators in column vectors ␨ = 共x , y , px , py兲, and ␨ˆ = 共xˆ , yˆ , pˆx , pˆy兲. In terms of ␨ˆ , ˆ are the components of the operator L 2 2 2 2 2 ˆ = xˆ − yˆ + 共pˆx − pˆy 兲w0 , L x 8⑄2 2w20

共33a兲

2 ˆ = xˆyˆ + pˆx pˆyw0 , L y 4⑄2 w20

共33b兲

ˆ = xˆ pˆy − yˆ pˆx . L z 2⑄

共33c兲

They satisfy the usual SU共2兲 angular momentum commutaˆ 兴 = i⑀ L ˆ ˆ ,L tors 关L i j ijk k 共at variance with the components of opˆ erator L兲. Of these SU共2兲 generators, only Lz represents real ˆ and spatial rotations on the transverse x-y plane, whereas L x ˆ Ly represent simultaneous rotations in the four-dimensional ˆ produces rotations in the x-p and y-p phase-space: L x x y ˆ gives rise planes by equal and opposite amounts, whereas L y to rotations in the x-py and y-px planes by equal amounts. ˆ does actually correspond 共it is proportional兲 Hence only L z to component Lˆz of Eq. 共25兲. There is, however, no analog ˆ and the x and y compoˆ and L correspondence between L x y ˆ ˆ ˆ and L nents of the operator L, respectively. Operators L x y necessarily involve a change of the spatial modes. The Wigner representation of a photon in a pure state ␺共p兲 = 具p 兩 ␺典 is 1 共2␲⑄兲2

W共␨兲 =

冕

⬁

d2␰ exp共ir⬜ · ␰/⑄兲

−⬁

冉 冊冉 冊

1 1 ⫻ ␺ p + ␰ ␺* p − ␰ . 2 2

共34兲

As shown recently in Ref. 关47兴, it is possible to obtain the Wigner representation of states 兩␪ , ␸典l,p without explicitly calculating the integrals in Eq. 共34兲. To this end, we invoke two remarkable properties: 共i兲 On account of the Stone–von Neumann theorem, unitary operators whose generators are ˆ 共␪ , ␸兲兴 induce linear canonical quadratic in ␨ˆ 关such as U transformations, T : ␨ˆ ⬘ → T␨ˆ , in the optical phase space; 共ii兲 under the action of T the Wigner function experiences a simple point transformation W共␨兲 → W⬘共␨兲 = W共T−1␨兲 关42兴. In our case, the linear canonical transformation generated by the quadratic operators 共33兲 results from the relation T␨ˆ ˆ , and reads as ˆ −1␨ˆ U =U

T=

冢

c␪

0

0

c␪

z 0s ␪c ␸

z 0s ␪s ␸

s ␪c ␸ z0

c␪

0

s ␪c ␸ s ␪s ␸ − z0 z0

0

c␪

s ␪s ␸ z0 −

− z 0s ␪s ␸ z 0s ␪c ␸

−

冣

,

共35兲

where c␪ = cos共␪ / 2兲, s␪ = sin共␪ / 2兲, c␸ = cos ␸, s␸ = sin ␸ 共recall that z0 = w20 / 共2⑄兲 is the Rayleigh range and w0 the mode width at z = 0兲. Notice that T has the form of a symplectic ray-transfer matrix of a generally anisotropic first-order system 关49,50兴; it is an element of the symplectic group Sp共4 , R兲, that is, det T = 1, and, under transposition, T⌳T t

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= T t⌳T = ⌳, where ⌳ is a real antisymmetric nonsingular four-dimensional symplectic metric matrix ⌳=

冉

02⫻2

12⫻2

− 12⫻2 02⫻2

冊

共36兲

.

Also, the action of T is independent of the chosen states at ␪ = ␸ = 0, that is, besides 兩l , p典, one could have chosen other state vectors, for instance, Hermite-Gaussian states 兩nx , ny典 with sphere-order N = nx + ny = 2p + 兩l兩. The key point is thus to observe that owing to the unitary relation 共31兲 between states belonging to the same sphere Ol,p, knowledge of the Wigner function of any given state on Ol,p allows one to determine the Wigner function of all states on that same sphere. LG states constitute the convenient choice here. Using their Wigner representation 关51兴, together with property 共ii兲 and Eq. 共35兲, the found normalized Wigner function is 关47兴 共− 1兲 −Q e 0L共N−l兲/2共Q0 − 4Q · ur兲 ␲ 2⑄ 2 N

Wl,p共␨ ; ␪, ␸兲 =

⫻ L共N+l兲/2共Q0 + 4Q · ur兲, 2

+ 共p2x + p2y 兲z20兴 / w2,

兩 l⬘,p⬘具␪⬘, ␸⬘兩␪, ␸典l,p兩2 = 共2␲⑄兲2

冕

⬁

−⬁

d4␨Wl⬘,p⬘共␨ ; ␪⬘, ␸⬘兲Wl,p共␨ ; ␪, ␸兲

冤 兺 冢 冣冢 冣冉 冊 冥 共N−l兲/2

=

共− 1兲

k

k=0

⫻ ␶ ␦l,l⬘␦ p,p⬘ , N

N+l 2

N−l 2

k

k

1−␶ ␶

ˆ 典 ⬅ 具␪, ␸兩L ˆ 兩␪, ␸典 = 具L l,p l,p

2

k

共38兲

␶ = cos2关共␪ − ␪⬘兲 / 2兴cos2关共␸ − ␸⬘兲 / 2兴 where parameter 2 2 + cos 关共␪ + ␪⬘兲 / 2兴sin 关共␸ − ␸⬘兲 / 2兴 共notice that 0 艋 ␶ 艋 1兲. Equation 共38兲 implies the following: 共i兲 any two states belonging to different spheres are mutually orthogonal; 共ii兲 if l ⬎ 0 and p = 0, only states corresponding to antipodal points are mutually orthogonal. However, if l ⬎ 0 and p ⬎ 0, addi-

冕

⬁

l d4␨L共␨兲Wl,p共␨ ; ␪, ␸兲 = ur . 2 −⬁ 共39兲

Via the Heisenberg-Robertson uncertainty relation and using ˆ of operators 共33兲 Eq. 共39兲 we obtain that the variances ⌬L i satisfy the following inequalities: 1 ˆ 典兩. ⌬Li⌬L j 艌 兩⑀ijk具L k 2

共37兲

Lm共␩兲 are the mth order where Q0 = 2关x + y Laguerre polynomials, and the quadratic polynomials Q共␨兲 ˆ , L ˆ , and L ˆ in Eqs. 共33兲 by ⬅ 共Qx , Qy , Qz兲 follow from L x y z ˆ replacing ␨ → ␨. When ␪ = 0 共␪ = ␲兲 one recovers from Eq. 共37兲 the Wigner function of LG states 兩l , p典 共兩−l , p典兲. If ␪ = ␲ / 2 and ␸ = 0 共␪ = ␲ / 2 and ␸ = ␲兲 one obtains the Wigner function of Hermite-Gaussian states 兩nx , N − nx典 共兩N − ny , ny典兲. Equation 共37兲 is a strictly positive and angle-independent Gaussian distribution only when l = p = 0 关in this case its associated Poincaré sphere becomes degenerated, i.e., all points 共␪ , ␸兲 on the sphere represent the same Gaussian mode state兴. Moreover, though Wl,p共␨ ; ␪ , ␸兲 does not explicitly contain the propagation variable z, its spatial evolution along z can be fully described by applying a Galilean boost r⬜ → r⬜ − zp. The orthogonality relations 共scalar product兲 satisfied by states 兩␪ , ␸典l,p and 兩␪⬘ , ␸⬘典l⬘,p⬘ are given by the overlap integral of their associated Wigner functions 2

tional points exist on the sphere 共apart from the antipodal兲 where their associated states are also orthogonal 关e.g., if p = 1, Eq. 共38兲 vanishes when ␶ = 共l + 1兲 / 共l + 2兲兴; 共iii兲 when l = 0, antipodal points no longer correspond to orthogonal states but to identical states. ˆ 兩␪ , ␸典 may be easily The expectation value l,p具␪ , ␸兩L l,p evaluated with the help of the Wigner function 共37兲. Since the operators 共33兲 do not involve products of noncommuting canonically conjugated operators, their corresponding phasespace representation in the Wigner-Weyl ordering is simply given by replacing ␨ˆ → ␨ in Eqs. 共33兲. Therefore

共40兲

In particular, states on the sphere equator 共with ␪ = ␲ / 2兲 yield ⌬Lx⌬Ly 艌 0. Moreover, the OAM carried by states 兩␪ , ␸典l,p follows immediately from Eq. 共39兲. The key observation is to notice that the operator Lˆz, given by Eq. 共25兲, for a given ˆ polarization ␴ and wave vector k0, is related to operator L z ˆ ˆ ˆ by Lz = 2បLz. The result, l,p具␪ , ␸兩L兩␪ , ␸典l,p = lប cos ␪, has a very simple geometrical interpretation. It is the projection of the unit vector ur corresponding to 兩␪ , ␸典l,p along the vertical axis of Ol,p. Hence arbitrary states on any sphere bear fractional OAM in units of ប. The limiting cases, being represented by LG and Hermite-Gaussian states, have the wellknown lប and zero values, respectively 关1,21兴. V. CONCLUSIONS

We have presented a detailed description of classical optical beams and single-photon states bearing OAM. Our quantum-field-theory formalism generalizes previous studies on this subject and, via phase-space methods, highlights the inherent symmetries of unitarily related families of paraxial spatial modes, of which, those carrying integer OAM 共in units of ប兲 constitute one particular subset. The Nth-order orbital Poincaré sphere representation enables us to visualize mode transformations. These transformations correspond to first-order optical systems with symplectic ray-transfer matrix T 关given by Eq. 共35兲兴 and evidence that it is possible to manipulate single photons prepared in superpositions of OAM states, and thus implement single qubit gates 共which correspond to particular rotations on the Poincaré sphere兲. Combination of these gates with two-photon quantum gates exploiting OAM entangled states generated from parametric down conversion 关3,6–8,10,52–55兴 constitutes a realistic and a fascinating possibility towards quantum computation with linear optical networks 关56–60兴. One could also exploit the multidimensional Hilbert structure of OAM photon states for

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CALVO, PICÓN, AND BAGAN

quantum key distribution 共QKD兲 protocols. At variance with QKD schemes employing polarization states, which only allow transmission of one key bit per photon and require the reference frames of the sender and receiver to be aligned with each other 关61兴, OAM photon states are invariant under rotations along their propagation direction. The continuous reference frame alignment monitorization for polarization states may not seem to strong of a restriction for groundbased stations, but it could be an important limitation on a moving station such as a satellite. The distortions created by atmospheric turbulence on OAM photon states could be corrected using two-dimensional filtering techniques that have been proposed for image recovering 共adaptive optics兲 under various degradation mechanisms 关62兴. To conclude, we also wish to point out that detection of phenomena related to highly energetic photons 关63兴 共x rays and gamma rays兲 carrying OAM could be of interest for astrophysical measurements of distant cosmic entities 共pulsars, quasars, black holes兲 through Compton scattering experiments.

ACKNOWLEDGMENTS

We gratefully acknowledge financial support from the Spanish Ministry of Science and Technology through Project No. BFM2002-02588. G.F.C. gratefully acknowledges the Spanish Ministry of Education and Science for a Juan de la Cierva grant.

2ik0

LGl,p共r, ␾,z;k0兲 =

冉 冊 冉 冊 册

冋

兩l兩

L兩l兩 p

2r2 w2共z兲

k 0r 2 r2 + i⌽G共z兲 , + il␾ + i 2R共z兲 w 共z兲 2

共A3兲

where w共z兲 = w0冑1 + 共z / z0兲2 with w0 being the width of the mode at z = 0, R共z兲 = z关1 + 共z0 / z兲2兴 is the phase-front radius, z0 = k0w20 / 2 is the Rayleigh range, ⌽G共z兲 = −共2p + 兩l兩 + 1兲arctan共z / z0兲 is the Gouy phase 关65兴, and L兩l兩 p 共x兲 are the associated Laguerre polynomials p

L兩l兩 p 共x兲

=

共兩l兩 + p兲!

兺 共− 1兲m 共p − m兲!共兩l兩 + m兲!m! xm . m=0

共A4兲

The indices l = 0 , ± 1 , ± 2 , . . . and p = 0 , 1 , 2 , . . . correspond to the winding 共or topological charge兲 and the number of nonaxial radial nodes of the mode. The wave front 共equal phase surface兲 forms in space part of a helicoidal surface given by l␾ + kz = const. The topological charge attributed to this wave-front manifold is positive 共l ⬎ 0兲 for right-handed helicoids, and vice versa. Of equal importance are the normalized Fouriertransformed LG modes, which, at z = 0, are given by

For completeness, we provide here a resume of the main expressions and properties of Laguerre-Gaussian 共LG兲 modes 共see also Ref. 关64兴兲. Starting from Maxwell’s equation in vacuum, one obtains the scalar wave equation

for any of the field components F共r , t兲 共potential vector, electric and magnetic fields, etc兲. The paraxial approximation assumes that if, under propagation along one given direction 共e.g., the z axis兲, the field components evolve in an essentially plane-wave fashion modulated by some slowly varying amplitude u共r兲, so that F共r , t兲 = u共r兲eik0共z−ct兲 共k0 is the wave vector along the z axis兲, then, u共r兲 is a solution of the 2 u = 0. Depending on paraxial wave equation 2ik0共⳵u / ⳵z兲 + ⵜ⬜ the particular geometry considered, one may distinguish several families of complete, orthogonal set of solutions which include the well-known Hermite-Gaussian beams 共in Cartesian coordinates兲, the LG beams 共in cylindrical coordinates兲 and the more recently discovered Ince-Gaussian beams 关32兴 共in elliptical coordinates兲. LG modes satisfy the paraxial wave equation, which, in cylindrical coordinates, reads as

冑

冑2r 1 2p! ␲共兩l兩 + p兲! w共z兲 w共z兲

⫻exp −

LGl,p共␳, ␸兲 =

共A1兲

共A2兲

Their whole z-propagating normalized profiles are

APPENDIX

1 ⳵ 2F ⵜ 2F = 2 2 , c ⳵t

⳵ u ⳵ 2u 1 ⳵ u 1 ⳵ 2u + + + = 0. ⳵z ⳵r2 r ⳵r r2 ⳵␾2

冑

冉 冊 冉 冊 冉 冊 冋 册

w20 p! w 0␳ 2␲共兩l兩 + p兲! 冑2

⫻ exp −

兩l兩

L兩l兩 p

w20␳2 2

w20␳2 ␲ exp il␸ − i 共2p + 兩l兩兲 , 4 2 共A5兲

where ␳ and ␸ denote the frequency-space cylindrical coordinates. They fulfill the orthogonality conditions * 共q兲LGl⬘,p⬘共q兲 = ␦ll⬘␦ pp⬘. With the help of their clo兰d2qLGl,p * sure relation 兺l,pLGl,p 共q兲LGl,p共q⬘兲 = ␦共2兲共q − q⬘兲 one has the following identity: ei关q·r⬜−k0␽

2z兴

= =

冕 冕

d2q⬘ei关q⬘·r⬜−k0␽ z兴␦共2兲共q − q⬘兲 2

d2q⬘ei关q⬘·r⬜−k0␽

* = 兺 LGl,p 共q兲 l,p

冕

2z兴

冋兺

* LGl,p 共q兲LGl,p共q⬘兲

l,p

册

d2q⬘ei关q⬘·r⬜−k0␽ z兴LGl,p共q⬘兲 2

* 共q兲LGl,p共r⬜,z;k0兲, = 兺 LGl,p l,p

where in the last step we employed the propagation formula of the angular spectrum 关24兴. This proves Eq. 共13兲.

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CALVO, PICÓN, AND BAGAN 关62兴 S. I. Sadhar and A. N. Rajagopalan, J. Opt. Soc. Am. A 22, 604 共2005兲; C. Paterson, Phys. Rev. Lett. 94, 153901 共2005兲. 关63兴 A. G. Peele et al., Opt. Lett. 27, 1752 共2002兲. 关64兴 M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 共1997兲. 关65兴 S. Feng and H. G. Winful, Opt. Lett. 26, 485 共2001兲. 关66兴 Here, Roman letter subindices run through the three components x, y, and z, and repeated indices are summed over all these three components. We also employ the usual Kronecker

delta ␦ij and the Levi-Civita tensor ⑀ijk, which takes the value +1 if ijk = 123, 231, 312, the value −1 if ijk = 321, 213, 132, and is zero otherwise. 关67兴 For simplicity, we assume that all states 具l , p典 have equal polarization ␴ and propagate with wave vector k0 along z. Also, without loss of generality, we consider all Ol,p having l 艌 0. Spheres with l ⬍ 0 exhibit a state configuration identical to those with l ⬎ 0, after inversion with respect to their centers.

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