Experimental test of uncertainty relations for quantum mechanics on a circle
J. Rehacek, Z. Bouchal, R. Celechovsky, Z. Hradil, L. L. Sanchez-Soto
aa r X i v : . [ qu a n t - ph ] D ec Experimental test of uncertainty relations for quantum mechanics on a circle
J. ˇReh´aˇcek, Z. Bouchal, R. ˇCelechovsk´y, Z. Hradil, and L. L. S´anchez-Soto Department of Optics, Palacky University, 17. listopadu 50, 772 00 Olomouc, Czech Republic Departamento de ´Optica, Facultad de F´ısica, Universidad Complutense, 28040 Madrid, Spain (Dated: October 31, 2018)We rederive uncertainty relations for the angular position and momentum of a particle on a circle by employ-ing the exponential of the angle instead of the angle itself, which leads to circular variance as a natural measureof resolution. Intelligent states minimizing the uncertainty product under the constraint of a given uncertaintyin angle or in angular momentum turn out to be given by Mathieu wave functions. We also discuss a num-ber of physically feasible approximations to these optimal states. The theory is applied to the orbital angularmomentum of a beam of photons and verified in an experiment that employs computer-controlled spatial lightmodulators both at the state preparation and analyzing stages.
PACS numbers: 03.65.-w, 42.50.Dv, 42.50.Vk, 42.60.Jk
I. INTRODUCTION
Despite the outstanding role that angular variables play inphysics, their proper definition in quantum mechanics is besetby difficulties and requires more care than perhaps might beexpected [1, 2, 3]. Consider, for instance, the simple exampleof a particle moving on a circle of unit radius: the problemsessentially arise then from the periodicity, which prevents theexistence of a well-behaved angle operator, but not of its com-plex exponential, we shall denote by ˆ E .In quantum optics, this topic is by no means purely aca-demic: it turns out to be crucial for a proper understandingof, e.g., the orbital angular momentum (OAM) of light [4].Indeed, as put forward by Allen and coworkers [5], theLaguerre-Gauss modes, typical of cylindrical symmetry, carrya well-defined OAM per photon. Since it is surprisingly sim-ple to generate, control, filter, and detect OAM states of lightexperimentally, researchers have begun to appreciate theirpractical potential for classical [6, 7, 8] and quantum infor-mation applications [9, 10, 11, 12, 13, 14, 15].Intimately linked to the issue of a proper angle descrip-tion it is the question of the associated uncertainty relations.Surprisingly enough, some subtle aspects of these relationsstill remains under discussion. From previous work in thistopic [16, 17, 18, 19, 20, 21, 22, 23, 24] it seems clear that, ifone insists in holding to an angle operator, special care mustbe taken when using the standard variance, since this is a non-periodic measure of spread that makes the angular uncertaintydepend on the π window chosen. Moreover, the associatedcommutation relation depends on the value of the angle dis-tribution at a point, which turns it of somewhat cumbersomehandling.By precise measurements on a light beam, a detailed testof the uncertainty principle for angle and angular momentumhas been recently demonstrated [25, 26]. The idea is to passthe beam through an angular aperture and measure the result-ing angular-momentum distribution [27]. In the same vein,we have presented experimental results [28] that strengthenthe evidence that ˆ E furnishes a correct description of angu-lar phenomena. When a sensible periodic resolution measure(namely, the circular variance) is employed, the associated in- telligent states should minimize two inequalities (one for thecosine and other for the sine), and both cannot be saturated si-multaneously. To bypass this drawback, we have looked at themore physically meaningful notion of constrained intelligentstates; that is, states that minimize the uncertainty product fora given spread either in angle or in angular momentum. Infact, they prove to be Mathieu wave functions, which havebeen attracting great interest in relation with nondiffractingoptical fields [29, 30, 31].In this paper we go one step beyond and present an im-proved experimental setup (that uses computer-controlled spa-tial light modulators both at the state preparation and ana-lyzing stages) to verify in great detail the properties of theseconstrained intelligent states. As a byproduct, we also bringout that ˆ E can be associated with a feasible transformation(a fork-like hologram) that shifts the values of the angularmomentum. Our formulation paves thus the way for a fullquantum processing of vortex beams and provides a bridgebetween the classical theory of singular optics and the realmof quantum optics.The plan of this paper is as follows. In Sec. II we providea comprehensive quantum treatment of angular variables, in-cluding a discussion about the associated coherent and intel-ligent states, as well as various suboptimal states. A feasibleoptical realization of the system under study is provided inSec. III, with special emphasis on the detection of the angular-momentum spectrum. In Sec. IV our setup is shown and ex-perimental results are presented and discussed at length. Fi-nally, the summary of our achievements and suggestions forpossible future upgrades are given in Sec. V. II. THEORYA. Quantum description of rotation angles
We consider rotations of angle φ generated by the angu-lar momentum along the z axis, which for the simplicity weshall denote henceforth as L . Classically, a point particle isnecessarily located at a single value of the periodic coordi-nate φ , defined within a chosen window, e.g., [0 , π ) . Thecorresponding quantum wave function, however, is an objectextended around the unit circle S and so can be directly af-fected by the nontrivial topology.If we treat φ as a continuous variable, the Poisson bracketfor the angle and the angular momentum is { φ, L } = 1 . (2.1)Direct application of the correspondence between Poissonbrackets and commutators, suggests the commutation relation(in units ~ = 1 ) [ ˆ φ, ˆ L ] = i . (2.2)One is tempted to interpret ˆ φ as multiplication by φ , and thento represent ˆ L by the differential operator ˆ L = − i ddφ (2.3)that verifies the fundamental relation (2.2). However, theuse of this operator may entail many pitfalls for the unwary:single-valuedness restricts the Hilbert space to the subspace of π -periodic functions, which, among other things, rules outthe angle coordinate as a bona fide observable [32, 33, 34].A possible solution, proposed by Judge and Lewis [35], is tomodify the angle operator so that it corresponds to multipli-cation by φ plus a series of step functions that sharply changethe angle by π at appropriate points, which coincides withthe classical Poisson bracket of L and a periodic variable.Many of these difficulties can be avoided by simply se-lecting instead angular coordinates that are both periodicand continuous. However, a single such quantity cannotuniquely specify a point on the circle because periodicity im-plies extrema, which excludes a one-to-one correspondenceand hence is incompatible with uniqueness. Perhaps the sim-plest choice [36, 37] is to adopt two angular coordinates, suchas, e. g., cosine and sine. In classical mechanics this is indeedof a good definition, while in quantum mechanics one wouldhave to show that these variables, we shall denote by ˆ C and ˆ S to make no further assumptions about the angle itself, forma complete set of commuting operators. One can conciselycondense all this information using the complex exponentialof the angle ˆ E = ˆ C + i ˆ S , which satisfies the commutationrelation [ ˆ E, ˆ L ] = ˆ E . (2.4)In mathematical terms, this defines the Lie algebra of the two-dimensional Euclidean group E(2) that is precisely the canon-ical symmetry group for the cylinder S × R (i.e., the classicalphase space of the system under study).The action of ˆ E on the angular momentum basis is ˆ E | m i = | m − i , (2.5)and, since the integer m runs from −∞ to + ∞ , ˆ E is a unitaryoperator whose normalized eigenvectors are | φ i = 1 √ π ∞ X m = −∞ e imφ | m i . (2.6) The intuitively expected relationship of a discrete Fourierpair between angle and angular momentum is an immedi-ate consequence of Eqs. (2.3) and (2.6). Indeed, denoting Ψ m = h m | Ψ i and Ψ( φ ) = h φ | Ψ i , it holds Ψ( φ ) = 1 √ π ∞ X m = −∞ e − imφ Ψ m , (2.7) Ψ m = Z π dφ ψ ( φ ) e imφ . There is an appealing physical interpretation beyond thedefinition of ˆ E . Whereas in the case of (2.2) one thinks interms of complementarity between two measurable quantities, ˆ E primarily represents a transformation and (2.4) may be in-terpreted as the complementarity between measurement andtransformation. On the other hand, the action of ˆ E can be castalso in terms of measurement, since any unitary operator maybe generated by an appropriate Hermitian generator. Thereis a twofold goal in the theory of angular momentum and itsconjugate variable: to characterize them either as a transfor-mation or as a measurement. Notice that the pioneering workin Refs. [25] and [26] anticipated the former interpretation.The role of ˆ E as a transformation is determined by the ac-tion on the basis states (2.5). What deserves to be explainedis the possible measurement associated with ˆ E . Although thevectors | φ i provide an adequate description of angle, one musttake into account that realistic measurements are always im-precise. In particular, the measurement of the spectrum wouldrequire infinite energy. In other words, the mathematical con-tinuum of angles will be observed always with a finite reso-lution. In consequence, it could be interesting to extend theprevious formalism by including fuzzy, unsharp or noisy gen-eralizations of the ideal description provided by ˆ E . To this endwe shall use positive operator-valued measures (POVMs), thatare a set of linear operators ˆΛ( φ ) furnishing the correct prob-abilities in any measurement process through the fundamentalpostulate that [38] p ( φ ) = Tr[ˆ ̺ ˆΛ( φ )] , (2.8)for any state described by the density operator ˆ ̺ . Compati-bility with the properties of ordinary probability imposes therequirements ˆΛ( φ ) ≥ , ˆΛ( φ ) = ˆΛ † ( φ ) , Z π dφ ˆΛ( φ ) = ˆ . (2.9)In addition to these basic statistical conditions, some otherrequisites must be imposed to ensure a meaningful descriptionof angle as a canonically conjugate variable with respect ˆ L .We adopt the same axiomatic approach developed previouslyby Leonhardt et al [39] for optical phase. First, we require theshifting property e iφ ′ ˆ L ˆΛ( φ ) e − iφ ′ ˆ L = ˆΛ( φ + φ ′ ) , (2.10)which reflects nothing but the basic feature that an angleshifter is a angle-distribution shifter and imposes the follow-ing form for the POVM [40] ˆΛ( φ ) = 12 π ∞ X m,m ′ = −∞ λ m,m ′ e i ( m − m ′ ) φ | m ih m ′ | . (2.11)We must also take into account that a shift in ˆ L should notchange the phase distribution. But a shift in ˆ L is generated by ˆ E since, according to (2.5), it shifts the angular momentumdistribution by one step. Therefore, we require as well ˆ E ˆΛ( φ ) ˆ E † = ˆΛ( φ ) , (2.12)which, loosely speaking, is the physical translation of the factthat angle is complementary to angular momentum. This im-poses the additional constraint λ m +1 ,m ′ +1 = λ m,m ′ , and thismeans that λ m,m ′ = λ m − m ′ . In consequence, Eq. (2.11) canbe recast as ˆΛ( φ ) = 12 π ∞ X l = −∞ λ ∗ l e − ilφ ˆ E l , (2.13)and the conditions (2.9) are now | λ l | ≤ , λ ∗ l = λ − l . (2.14)Expressing ˆ E in terms of its eigenvectors, we finally arrive atthe more general form of the POVM describing the angle vari-able and fulfilling the natural requirements (2.10) and (2.12): ˆΛ( φ ) = Z π dφ ′ K ( φ ′ ) | φ + φ ′ ih φ + φ ′ | , (2.15)where K ( φ ) = 12 π ∞ X l = −∞ λ l e ilφ . (2.16)The convolution (2.15) shows that this POVM effectively rep-resents a noisy version of the usual projection | φ ih φ | , and thekernel K ( φ ) gives the resolution provided by this POVM. B. Gaussian distributions on a circle
Experience with quantum mechanics of simple systems,such as the free particle and harmonic oscillator, suggests thatGaussian states can be an important tool for a better under-standing of the periodic motion on a circle. Even so, it is re-markable that there is no clear concise definition of the Gaus-sian distribution on a circle and one can only find vague state-ments scattered through the literature.We do not want to enter here into a mathematical treatment,but rather we just try to grasp the properties that make theGaussian distribution on the line to play such a key role inphysics and that we are particularly keen on retaining whenconstructing its circular counterpart. We itemize the most rel-evant ones in our view: 1. The sum of many independent random variables tend tobe distributed following a Gaussian distribution.2. All marginal and conditional densities of a Gaussian areagain Gaussians.3. The Fourier transform of a Gaussian is also a Gaussian.4. The Gaussian distribution maximizes the Shannon en-tropy for a fixed value of the variance.The first property (subject to a few general conditions) isthe central-limit theorem and explains the ubiquity of Gaus-sians in physics: the distribution of the phenomenon understudy does not have to be Gaussian because its average willbe. The second and third ones are responsible for the goodproperties than one assigns to Gaussian states in quantum op-tics. Finally, the last condition bears on the information-basedapproach to quantum theory, but strongly depends on the def-inition of entropy we adopt.In statistics there are two distributions that have been some-how suggested for having good properties on a circle, namely p κ ( φ ) = 12 πI (2 κ ) exp[2 κ cos( φ − µ )] , (2.17) p σ ( φ ) = 1 √ πσ ∞ X k = −∞ exp (cid:20) −
12 ( φ − µ + 2 πk ) σ (cid:21) , (2.18)where I n denotes the modified Bessel function of first kind.The first one is known as the von Mises distribution, while thesecond is the wrapped Gaussian. By a trivial application ofthe Poisson summation formula we can express the latter as p σ ( φ ) = 12 π ϑ (cid:18) φ − µ (cid:12)(cid:12)(cid:12)(cid:12) e σ (cid:19) , (2.19)where ϑ ( ζ | q ) = ∞ X k = −∞ q k e ikζ (2.20)is the third Jacobi theta function [41, 42]. For both distri-butions µ represents the main direction, while σ and κ areparameters related to the concentration [43].From the previous checklist, the wrapped Gaussian satis-fies properties 1 to 3, while the von Mises satisfies property 4when the variance is replaced by its circular version (so it rep-resents the minimally prejudiced angle distribution, given theinformation constraints [44]). Therefore, it is tempting to sidewith the former. Additionally, the Jacobi ϑ function is thesolution of the diffusion equation on a circle with the initialstate being a delta function, which is another way of defininga Gaussian wave function [45]. However, note that if we takesuch a route, Gaussian wave functions do not lead to Gaussianprobability distributions anymore (because the square of a ϑ is not a ϑ function), a limitation that does not apply to thevon Mises.In this respect, it is convenient to make a small detour intothe question of coherent states [46] (recall that for the har-monic oscillator they are precisely Gaussian wave packets).Possible definitions of coherent states for a particle on a cir-cle have been outlined in the literature [47, 48], but they areof very mathematical nature. We prefer to adopt the ideasof Rembieli´nski and coworkers [49] and construct coherentstates | w i as eigenstates of the operator ˆ W = e i ( ˆ φ + i ˆ L ) = e − ˆ L +1 / ˆ E , (2.21)so that ˆ W | w i = w | w i , (2.22)where the complex number w = e iθ − ℓ parametrizes the unitcylinder. Note in passing that ˆ W | m i = e m − / | m − i , (2.23) ˆ W † | m i = e m +1 / | m + 1 i , with [ ˆ W , ˆ W † ] = sinh(1) e L . The projection of the vector | w i onto the basis | m i gives then w m = h m | w i = w − m e − m / , (2.24)while in the angular basis the corresponding expression is w ( φ ) = 1 √ π ϑ (cid:18)
12 ( φ − µ ) (cid:12)(cid:12)(cid:12)(cid:12) e (cid:19) , (2.25)where µ = θ + iℓ .In consequence, we have found three families of states withinteresting properties: (i) states with von Mises probabilitydensity, Eq. (2.17); (ii) states with wrapped Gaussian prob-ability distribution, Eq. (2.18); and (iii) coherent states withwrapped Gaussian amplitude density, Eq. (2.25). Neverthe-less, leaving aside fundamental reasons, for computationalpurposes these three families have very similar angular shapesand give almost indistinguishable numerical results. There-fore, sometimes we will use von Mises states because of theirsimplicity and the possibility of obtaining analytical results. C. Constrained intelligent states
Coherent states for the harmonic oscillator are also min-imum uncertainty wave packets. Given the analogy of ˆ W in Eq. (2.23) with the standard annihilation operator, one istempted to introduce quadrature-like combinations ˆ Q = 1 √ W + ˆ W † ) , ˆ P = 1 √ i ( ˆ W − ˆ W † ) , (2.26)that satisfy the uncertainty principle (∆ ˆ Q ) (∆ ˆ P ) ≥ |h [ ˆ Q, ˆ P ] i| , (2.27)where (∆ ˆ A ) = h ˆ A i − h ˆ A i is the standard variance. Alengthy calculation [3] shows that the coherent states (2.25)obey (2.27) as an equality and so they are indeed minimum packets for the variables (2.26). In fact, they are also mini-mum for more intricate uncertainties [23]. However, the prob-lem is that, at difference of the harmonic oscillator, we do nothave any clear operational prescription of how to measure thequadratures (2.26), so they give no real physical insight intothe statistical description of angle.Let us then turn back to the general commutation relation(2.4). First, we observe that dealing with angle mean and vari-ance in the ordinary way has drawbacks. Consider, for exam-ple, a sharp angle distribution localized at the origin and thesame one shifted by π . Despite the fact that the physical infor-mation they convey is the same, in the later case the variance ismuch bigger. Since angle is periodic but variance is not, it haslittle meaning to consider the angle measurement itself [50].In circular statistics one usually calculates the moments ofthe exponential of the angle [51, 52, 53, 54], that are referredto as circular moments and give rise, e. g., to a circular vari-ance σ φ = 1 − |h e iφ i| , (2.28)where h e iφ i = Z π dφ p ( φ ) e iφ , (2.29)and p ( φ ) is the probability density. It possesses all the goodproperties expected: it is periodic, the shifted distributions P ( φ + φ ′ ) are characterized by the same resolution, and forsharp angle distributions it coincides with the standard vari-ance since |h e iφ i| ≃ h φ i . Moreover, this circular vari-ance coincides with (∆ ˆ E ) = h ˆ E † ˆ E i − h ˆ E † ih ˆ E i , (2.30)which is the natural extension of variance for unitary opera-tors [34].If we use (2.30), the uncertainty relation associated with(2.4) reads (∆ ˆ E ) (∆ ˆ L ) ≥
14 [1 − (∆ ˆ E ) ] . (2.31)Sometimes it probes convenient to express this in terms of thecorresponding Hermitian components ˆ C and ˆ S . We have [ ˆ C, ˆ L ] = i ˆ S, [ ˆ S, ˆ L ] = − i ˆ C , (2.32)while [ ˆ C, ˆ S ] = 0 , so that (∆ ˆ C ) (∆ ˆ L ) ≥ |h ˆ C i| , (2.33) (∆ ˆ S ) (∆ ˆ L ) ≥ |h ˆ S i| . Both inequalities depend on the choice of state used to evalu-ate h ˆ C i and h ˆ S i . So intelligent states need to be distinguishedfrom minimum uncertainty states: there are intelligent statesfor which the right-hand side of Eq. (2.33) is not the obviousminimum value of 0. The condition of intelligence for, say,the first of (2.33), reads as ( ˆ L + iκ ˆ C ) | Ψ i = µ | Ψ i , (2.34) Η η ce ( η , q ) q = 0q = 10q = 100 Η -1-0.500.51 0 2 4 6 8 10q η -1012 ce ( η , q ) q = 0q = 10q = 100 FIG. 1: Plot of the functions ce ( η, q ) (top) and ce ( η, q ) (bottom).On the right, we show two-dimensional sections of these functionsfor the values q = 0 , , , , , and 100. that once expressed in the angle representation can be imme-diately solved to give Ψ( φ ) = 1 p πI (2 κ ) exp( iµφ + κ sin φ ) , (2.35)so that the associated probability is the von Mises distribution.The intelligent states for the the second equation in (2.33) canbe worked out in the same way. However, it is not difficultto prove that both inequalities cannot be saturated simultane-ously [55]. In other words, the fundamental relation (2.31)can never hold as an equality: it is exact, but is too weak.To get an attainable bound we look instead at normalizedstates that minimize the uncertainty product (∆ ˆ E ) (∆ ˆ L ) ei-ther for a given (∆ ˆ E ) or for a given (∆ ˆ L ) , which we callconstrained intelligent states. We use the method of unde-termined multipliers, so the linear combination of variationsleads to [ ˆ L + p ˆ L + ( q ∗ ˆ E + qE † ) / | Ψ i = a | Ψ i , (2.36)where p , q , and a are Lagrange multipliers. Working in theangle representation, the change of variables exp( ipφ )Ψ( φ ) eliminates the linear term from (2.36). In addition, we cantake q to be a real number, since this merely introduces aglobal phase shift. We finally get d Ψ( η ) dη + [ a − q cos(2 η )] Ψ( η ) = 0 , (2.37)where we have introduced the rescaled variable η = φ/ ,which has a domain ≤ η < π and plays the role of po-lar angle in elliptic coordinates. Equation (2.37) is preciselythe standard form of the Mathieu equation, which has manyapplications not only in optics, but also in other branches ofmodern physics [56]. An uncertainty relation of this type hasbeen already investigated by Opatrn´y [57]. In our case, the only acceptable Mathieu functions are those periodic with pe-riod of π or π . The values of a in Eq. (2.37) that satisfy thiscondition are the eigenvalues. We have then two families ofindependent solutions, namely the angular Mathieu functions ce n ( η, q ) and se n ( η, q ) with n = 0 , , , . . . , which are usu-ally known as the elliptic cosine and sine, respectively. Theparity of these functions is exactly the same as their trigono-metric counterparts; that is, ce n ( η, q ) is even and se n ( η, q ) isodd in η , while they have period π when n is even or period π when n is odd. To illustrate these behaviors, in Fig. 1 wehave plotted wave functions ce n ( η, q ) of orders n = 0 and n = 2 .Since the π periodicity in φ requires π periodicity in η ,the acceptable solutions for our eigenvalue problem are the in-dependent Mathieu functions ce n ( η, q ) and se n ( η, q ) , with n = 0 , , . . . . In what follows, we consider only even so-lutions ce n ( η, q ) , although the treatment can be obviouslyextended to the odd ones with analogous results. We take then Ψ n ( η, q ) = r π ce n ( η, q ) , (2.38)where we have made use of the property Z π ce m ( η, q ) ce n ( η, q ) dη = πδ mn , (2.39)to normalize the wave function. Using (2.38) we have (∆ ˆ L ) n = 12 π Z π dη ce ′ n ( η, q )= 14 [ A (2 n )2 n ( q ) − q Θ n ( q )] , (2.40) (∆ ˆ E ) n = 1 − (cid:12)(cid:12)(cid:12)(cid:12) π Z π dη ce n ( η, q ) cos(2 η ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 − | Θ n ( q ) | . To obtain these analytical expressions we have expanded ce n ( η, q ) in Fourier series ce n ( η, q ) = ∞ X k =0 A (2 n )2 k ( q ) cos(2 kη ) , (2.41)and integrate term by term, in such a way that Θ n ( q ) = A (2 n )0 ( q ) A (2 n )2 ( q ) + ∞ X k =0 A (2 n )2 k ( q ) A (2 n )2 k +2 ( q ) . (2.42)The coefficients A (2 n )2 k determine the Fourier spectrum andsatisfy recurrence relations that can be efficiently computed bya variety of methods [58]. In Fig. 2 we have plotted (∆ ˆ L ) n and (∆ ˆ E ) n as functions of the Lagrange multiplier q . D. Intelligent states: asymptotic limits
To better understand Fig. 2 we first concentrate on the limitof small q [i. e., large (∆ ˆ E ) n ]. We expand ce n ( η, q ) in q ( ∆ L ) ce ( η ,q)ce ( η ,q)ce ( η ,q) q ( ∆ E ) ce ( η ,q)ce ( η ,q)ce ( η ,q) FIG. 2: Plot of (∆ ˆ L ) n and (∆ ˆ E ) n for the three lowest order wave-functions Ψ n ( η, q ) in terms of the parameter q . powers of q and retain only linear terms [56] ce ( η, q ) = 1 √ h − q η ) i , ce ( η, q ) = cos(2 η ) − q (cid:20) cos(4 η )12 − (cid:21) , ce n ( η, q ) = cos(2 η ) − q (cid:26) cos[(2 n + 2) η ]2 n + 1 − cos[(2 n − η ]2 n − (cid:27) , n ≥ . (2.43)This leads to (∆ ˆ L ) = 1 − q
48 + O ( q ) , (∆ ˆ E ) = 1 − q
144 + O ( q ) , (2.44) (∆ ˆ L ) n = n − q n −
1) + O ( q ) , (∆ ˆ E ) n = 1 − q n − + O ( q ) , n = 1 showing a quadratic behavior which can be appreciated inFig. 2.In the opposite limit of large q [small (∆ ˆ E ) n ] we take theapproximation in terms of Hermite polynomials [58] ce n ( η, q ) ∝ e − u / H n (cid:18) u √ (cid:19) + O ( q − / ) , (2.45) where u = 2 q / cos η . Apart from constant factors, the states(2.45) look like the harmonic oscillator wave functions and wecan use them to evaluate analytically the variances. The finalexpressions involve the modified Bessel functions I k ( √ q ) , butthe crucial fact is that the following simple asymptotic expres-sions hold: (∆ ˆ L ) n = (4 n + 1)4 √ q + O ( q ) , (2.46) (∆ ˆ E ) n = 4 n + 1 √ q + O ( q − ) , showing a square-root behavior that is also apparent fromFig. 2. The range of moderate values of q , where the tran-sition between the quadratic (small q ) and the square-root(large q ) regions happens, is magnified in the inset. Accord-ing to Eq. (2.46), with increasing q the uncertainty product (∆ ˆ L ) n (∆ ˆ E ) n approaches a constant value depending ex-clusively on the mode index n ; lim q →∞ (∆ ˆ L ) n (∆ ˆ E ) n =(4 n + 1) / . These asymptotic limits, confirmed in Fig. 2,identify the fundamental mode n = 0 as the minimum uncer-tainty state for all the values of the parameter q . Henceforth,we always refer to the fundamental Mathieu mode, unless themode index is explicitly given.Finally, let us note that, from Eq. (2.43), it follows immedi-ately that when q → the probability distribution for this fun-damental mode is p ( φ ) ∝ [1 − q cos( φ ) / ≃ exp( − q cos φ ) ;while when q → ∞ , according to Eq. (2.45), we have H ( u/ √
2) = 1 and p ( φ ) ∝ exp[ − √ q cos ( φ/ ∝ exp( −√ q cos φ ) . We therefore get the interesting result that p ( φ ) ∝ | ce ( η, q ) | ≃ e − q cos φ , q → ,e −√ q cos φ , q → ∞ , (2.47)and hence the optimal states with very sharp and nearly flatangular profiles attain the von Mises shape. E. Suboptimal states
Up to now we have investigated extremal states that will beused in the experiments as an ultimate calibration to asses theperformance of our setup. Here, we compare these extremalstates with suboptimal ones. There are a plenty of possiblecandidates for that: we will select a few examples that canbe easily prepared and intuitively grasp various features of “awell localized angle”.The wedge structure is our first representative. The aper-ture function possesses sharp edges and may be defined in theangle representation as Ψ( φ ) = ( / √ α , | φ | ≤ α/ , , | φ | > α/ , (2.48) α being the opening angle of the wedge. The probability dis-tribution of angular momentum p m = | Ψ m | can be calcu-lated using Eq. (2.7) and one finds (∆ ˆ L ) → ∞ ,p m = α π sin ( mα/ mα/ , [wedge] (∆ ˆ E ) = 1 − α sin ( α/ . (2.49)Similarly to the Fraunhofer diffraction pattern observed be-hind a rectangular slit, the variance (∆ ˆ L ) is infinite due to theheavy tails of the sinc distribution and thus the angle-angularmomentum uncertainty relation is trivially satisfied for (2.48).In spite of this divergence, the experimenter cannot establishthis simple fact from a finite data set: in general, the sampledangular-momentum uncertainty will grow with the size of dataacquired. Most of the detected m fall within the central peak | m | ≤ π/α of the distribution, which tends to regularize theunbounded uncertainty product.As another candidate for a simple single-peaked angulardistribution we choose the state Ψ( φ ) = r πα cos( φ/α ) , | φ | ≤ πα/ , , | φ | > πα/ , (2.50)that is, the positive cosine half-wave stretched to the inter-val − πα/ ≤ φ ≤ πα/ , so that α ≤ . The followingdiscussion is valid even for α > provided the cosine half-wave, which now spans an interval of width larger than π ,is wrapped onto the unit circle. Since the delimiting aperturehas no sharp edges, one can expect more regular results. Astraightforward calculation yields (∆ ˆ L ) = 1 /α ,p m = 4 α cos ( πmα/ π ( m α − , [cosine] (∆ ˆ E ) = 1 −
64 sin ( πα/ π α ( α − . (2.51)At first sight, the angular-momentum distribution in Eq. (2.51)has a sinc-like shape with infinitely many side lobes, whichstrongly resembles the one in Eq. (2.49). However, herethe higher-order contributions to the angular-momentum vari-ance are negligible and both uncertainties appear to be fi-nite. Notice that the parameter α has now a very simplephysical meaning: it is inversely proportional to the angular-momentum uncertainty.Next, we consider the von Mises wave function Ψ( φ ) = 1 p πI (1 /α ) e cos φ/ (2 α ) , (2.52) α being a monotonic function of the angular width. According ∆ E ∆ L ∆ E ∆ E ∆ L ∆ E (a) (b)(c) (d) FIG. 3: Theoretical uncertainty products for angle and angular-momentum variables calculated for (a) wedge angle distribution, (b)cosine distribution; (c) von Mises distribution, and (d) truncated nor-mal distribution. In all panels the solid line denotes the optimal un-certainty product generated by the intelligent Mathieu wave function.In panel (a) the broken and dotted lines correspond to p m truncatedat the first and second minimum, respectively. to Eq. (2.47), the relevant uncertainties, (∆ ˆ L ) = I (1 /α )4 αI (1 /α ) ,p m = I m (1 / α ) I (1 /α ) , [von Mises] (∆ ˆ E ) = 1 − I (1 /α ) I (1 /α ) , (2.53)can be also taken (with an appropriate fitting of the parame-ter α ) as excellent approximations to the uncertainties of theoptimal Mathieu states, especially in the regions of small andlarge variances.Finally, we evaluate the uncertainties of the truncated Gaus-sian Ψ( φ ) = [ π erf ( πα ) /α ] − / e − α φ / , (2.54)which minimizes the uncertainty product for angular mo-mentum and angle, the latter in the sense of ordinary vari-ance [25, 26]. Such states become suboptimal when the vari-ance is replaced by a periodic measure, such as the circularvariance advocated in this paper: (∆ ˆ L ) = α " − √ παe − π α erf( πα ) ,p m = e − m /α n Re h erf (cid:16) πα + im √ α (cid:17)io √ πα erf ( πα ) , [truncated] (∆ ˆ E ) = 1 − e − / (2 α ) n Re h erf (cid:16) πα + i α (cid:17)io erf ( πα ) , (2.55)where erf( z ) is the error function.Figure 3 shows the comparison of these four states withthe optimal Mathieu one. As it has been already mentioned,the measured uncertainty product for the wedge distributiongrows with the size of the data acquired, since more and moreside maxima are sampled [see panel (a)]. On the other hand,the uncertainty product of the cosine distribution in panel (b)is well defined and lies well above the quantum limit givenby the Mathieu profile. In agreement with the asymptoticanalysis of the previous section, the uncertainty product forthe von Mises angular distribution falls very close to the op-timal curve: only at intermediate angular spreads ∆ ˆ E we seea significant deviation from the standard quantum limit, whilefor the truncated Gaussian states the deviation is larger andshifts toward higher values of ∆ ˆ E . The point is whether thecurrently available measurements have sufficient resolution todiscriminate between the optimal and suboptimal states men-tioned above. This question is addressed in the next two sec-tions. III. MEASUREMENT OF THE ORBITAL ANGULARMOMENTUMA. Single vortex beam
The theory presented thus far can be applied to a variety ofphysical systems. Here, we consider a particularly appealingrealization of a planar rotator in terms of optical beams.Light beams can carry angular momentum, which com-prises spin and orbital components that are associated withpolarization and helical-phase fronts, respectively. In gen-eral, the spin and orbital contributions cannot be consideredseparately, but in the paraxial approximation both contribu-tions can be measured and manipulated independently. Weemphasize that this OAM manifests at the macroscopic andsingle-photon levels and therefore paraxial quantum optics isthe most convenient context in which to treat the OAM of lightas a quantum resource.In consequence, we can leave aside the spin part and con-sider the simplest scalar monochromatic beam carrying OAM:this is precisely a vortex beam; i.e., a beam whose phase variesin a corkscrew-like manner along the direction of propagation.The corresponding spatial amplitude can be written as U ( r ) = u ( r ) exp( imφ ) , (3.1)where we have assumed that the beam propagates dominantlyalong the z axis, so we have cylindrical symmetry. Accordingto the representation in Eq. (2.3), (3.1) is an eigenstate of ˆ L with eigenvalue m , which is also known as the topologicalcharge (or helicity) of the vortex. To check this interpretation,note that, the OAM density is also dominantly along the z axis and is given by l = rS φ /c , where S φ is the azimuthalcomponent of the Poynting vector. In a scalar theory, the time-averaged Poynting vector can be calculated computed as S = is ω ( U ∗ ∇ U − U ∇ U ∗ ) , where s is a constant (with units ofm s). The density of the OAM of the vortex beam (3.1) thendepends on its intensity I = | U | and wavefront helicity and can be expressed in terms of its power P as L = 2 ωs mPc . (3.2)If we divide now by the total energy density of the field wefinally get that the OAM per photon can be interpreted pre-cisely as the topological charge m . In this way, light beamsprepared in OAM eigenstates can be used in quantum opticsexperiments in the same way as qudits. B. Principle of the measuring method
A general scheme of our experimental method is sketchedin Fig. 4. A collimated Gaussian beam with complex am-plitude U G illuminates an amplitude mask (with transmis-sion coefficient t A ) performing an angular limitation of thebeam. Immediately behind the mask, the beam transverseprofile has a cake-slice shape given by U A = t A U G . Ac-cording to Eq. (2.7), the field azimuthal amplitude distributionresults in a spread of the spectrum composed of vortex compo-nents with different topological charges and amplitudes. Thebeam propagates toward a spiral phase mask [with transmis-sion t P = exp( − iN φ ) ] introducing a helicity − N . Behindthe phase mask, the transmitted field is Fourier analyzed, soits spatial spectrum U F can be obtained as U F = F [( t A U G ∗ h ) t P ] , (3.3)where F denotes the Fourier transform, h is the impulse re-sponse function of free-space propagation between the ampli-tude and phase masks and ∗ is the convolution product. Thedetected power can be thus interpreted as the spectral intensitycollected at the power meter placed at the back focal plane ofthe lens used for the optical implementation of the Fouriertransform.As we show below, provided the aperture radius of thepower meter is suitably chosen, the measured power can beused for estimating the OAM of the vortex mode with topo-logical charge N in the field behind the amplitude mask.This procedure can be repeated with spiral phase masks ofdifferent topological charges yielding the vortex distribution(i.e., angular-momentum spectrum) of the prepared angular-restricted beam. Several experimental realizations of this ideahave been proposed and realized [26, 27, 28], differing intechnical details and data analysis [59, 60].To put this in quantitative terms, let us first introduce amode decomposition of the amplitude mask t A = ∞ X m = −∞ a m exp( imφ ) , (3.4)where a m are Fourier coefficients. We assume that the waistof the Gaussian beam (of width w ) is placed exactly at themask plane. The transmitted field propagates through freespace, so the complex amplitude U I of the field impingingon the spiral phase mask can be written in the form U I ( r ) = ∞ X m = −∞ a m u m ( r, z ) exp( imφ ) , (3.5) FT -5-4-3-2-1012N ...... -3-2-1012 ...... m FREE SPACEPROPAGATION
Spiral phasemask t p =e -iN f t A AmplitudemaskGaussianbeam Powermeter m m -1012-N-4-N-3-N-2-N-1-N-N+1-N+2
FIG. 4: Principle of the measurement of the angular momentum spectrum of the angular restricted field. where we have used cylindrical coordinates ( r, φ, z ) and u m ( r, z ) = 2 πu i m h exp (cid:16) − ikr z (cid:17) A m ( r, z ) ,A m ( r, z ) = Z ∞ exp( − αr ′ ) J m ( βrr ′ ) r ′ dr ′ , (3.6) h = iλz , α = 1 w + ik z , β = kz . Here J m and u denote the m -th order Bessel function of thefirst kind and a constant amplitude of the Gaussian beam, re-spectively, and z is the distance between amplitude and phasemasks. The integration in (3.6) can be carried out and resultsin A m ( r, z ) = r Qβ r πα exp( − Qr ) × [ I ( m − ( Qr ) − I ( m +1) ( Qr )] , (3.7)where Q = β / (8 α ) .After transmission through the spiral phase mask, theFourier transform of the field is performed optically and thespatial distribution at the back focal plane of the Fourier lenscan be represented by the complex amplitude U F given by U F ( ν, ψ ) = a N [ u N ( ν ) − v N ( ν )]+ ∞ X m = −∞ a m v m ( ν ) exp[ i ( m − N ) ψ ] , (3.8)where u m ( ν ) = 2 π Z ∞ u m ( r, z ) J (2 πνr ) rdr , (3.9) v m ( ν ) = 2 πi ( m − N ) Z ∞ u m ( r, z ) J m − N (2 πνr ) rdr . Here ( ν, ψ ) are polar coordinates in the transverse Fourierplane ( x F , y F ) , defined as ν = p x F + y F / ( λf ′ ) and ψ =arctan( y F /x F ) , f ′ being the lens focal length. The power captured by the circular aperture of the meter placed at thefocal plane of the lens is given by P = Z ν Z π | U F ( ν, ψ ) | νdνdψ , (3.10)where ν = R/ ( λf ′ ) and R denotes the aperture radius. Sub-stituting (3.8) into (3.10), the detected power can be expressedas P = P N + P C , (3.11)where P N = 2 π | a N | Z ν | u N | νdν,P C = 2 π ∞ X m = −∞ m = N | a m | Z ν | v m | νdν . (3.12)This power appears as composed of two terms: the first term, P N , represents the power carried by the vortex mode of topo-logical charge N . The second term, P C , is a crosstalk that rep-resents disturbing contributions of the remaining vortex com-ponents and it can be reduced by a convenient choice of theaperture radius R of the power meter. This possibility fol-lows directly from Eq. (3.8), since | u N (0) | = 0 and hasits maximum in the middle of the receiving aperture, while | v m (0) | = 0 for m = N . The spectral intensity of the vortexwhose power is measured creates a sharply peaked bright spotat the Fourier plane, meanwhile, the spectral intensities con-tributing to the crosstalk power have an annular form. Thesedifferent spatial shapes enable an optimal choice of the radiusof the receiving aperture. In this case, both the power lostof the measured vortex and the influence of the crosstalk areminimized. IV. EXPERIMENTAL RESULTS
To verify our theory, the angle-angular momentum uncer-tainty products were experimentally measured on various light0
PHASESLM HALF-WAVEPLATEFL2 PINHOLEPOWERMETERLASER SPATIALFILTER AMP.SLM 4-F SYSTEMCOLLIMATINGLENS FL1 BS MIRRORMO
Preparation of the state Measurement of the state (a) (b) (a) (b)
HOLOGRAM ON ASLM PHASE PROFILES ON PSLM DETECTED INTENSITY PATTERNSFL3
FIG. 5: Experimental setup for the generation of beams with an ar-bitrary transverse profile and subsequent detection of their angularmomentum spectrum. beams. Given the small difference between the optimal Math-ieu beams and other suboptimal single-peaked angular distri-butions, such a measurement is also an indicator of the resolu-tion attainable with the present commercially available tech-nology.Figure 5 shows our setup. The beam generated by the laser(Verdi V2: 532 nm, 20 mW) is spatially filtered, expandedand collimated by the lens and impinges on the hologram gen-erated by the amplitude spatial light modulator (SLM) (CRLOpto, × pixels). The bitmap of the hologram is com-puted as an interference pattern of the tested state (with the de-sired angular amplitude distribution) and an inclined referenceplane wave. After illuminating the hologram with the colli-mated beam, the Fourier spectrum of the transmitted beam islocalized at the back focal plane of the first Fourier lens FL .It consists of three diffraction orders ( − , , +1) . The unde-sired 0 and − orders are removed by a spatial filter. After in-verse Fourier transformation, performed by the second Fourierlens FL , a collimated beam with the required complex ampli-tude profile U A = t A U G is obtained. This completes the statepreparation.The analysis begins by reflecting the prepared field U A at aphase SLM (Boulder, × pixels), whose reflectivity isproportional to t P ∝ e − iNφ . As it has been discussed in theprevious section, after the Fourier transformation of the re-flected field, the spectral component whose helicity was elim-inated by the phase SLM gives rise to a bright spot (Fig. 5a),while the other components have an annular intensity distri-bution (Fig. 5b). The vortex components of the spiral spec-trum can be subsequently selected by the phase SLM andtheir OAM determined by a power measurement performedwith an optimal aperture size of the power meter. To suppresscrosstalks, the calibrating response functions were acquiredfor each phase mask.After the setup was carefully aligned using Laguerre-Gaussbeams, transverse amplitude distributions of different shapesand angular variances were generated. Each beam was thenscanned for values of helicities in the range of m ∈ [ − , .A typical transverse intensity profile and the correspondingmeasured raw data are shown in Fig. 6.In addition, the response functions were measured for pure m FIG. 6: Preparation and measurement of Mathieu states having cir-cular variances (from top to bottom) ∆ ˆ E = 0 . , . , . , and0.91. Left: computed intensity distribution; right: measured angular-momentum spectrum (in arbitrary units). -4 -2 0 2 400.51 -4 -2 0 2 4 600.51-2 0 2 4 6 m m FIG. 7: Measured response functions of our detection scheme forpure vortex modes of helicities m = 0 , , , and 5. vortex modes (see Fig. 7). For an ideal detection, the purevortex mode with the topological charge N should have a δ -like angular momentum spectrum with a sharp peak at m = N . Any real detection scheme suffers from crosstalks be-tween modes, which tends to broaden the measured spectrain Fig. 6; this effect becomes more pronounced for larger he-licities ( m > ). It can be seen, from comparing Figs. 6 and7, that the reliability of the measured spectrum decreases fromthe center ( m = 0 ) to the edges and that beams of smaller vari-ances have broader angular spectra and vice versa . Hence thereliability of experimentally determined uncertainty productsis expected to increase with variance.1 -4 -2 0 2 4 m FIG. 8: Analysis of measured angular momentum spectrum of aMathieu beam of ∆ ˆ E = 0 . . Raw data (small black circles), de-convoluted data (large gray circles) and best fit with a theoreticaldistribution (+ symbols) are shown. In the next step, the acquired response functions were usedto increase the resolution of our detection scheme. Sincethe measured spectra could be considered to be convolutionsof true angular spectra and known response functions, wecould apply an inverse transformation to minimize the effectof crosstalks. Then, the angular-momentum variances wereestimated by fitting the deconvoluted data to the theoretically-calculated distributions. The family of distributions used forfitting experimental data was parametrized by an overall nor-malization factor and a parameter characterizing the angularwidth of the corresponding state. For example, the fitting pro-cedure applied to data measured on a Mathieu beam yieldedthe value of parameter q , which was then used to determinethe variance of angular momentum via Eq. (2.41). This stageof analysis is illustrated in Fig. 8. Besides getting (∆ ˆ L ) thequality of the best fit was quantified enabling to place error-bars on the resulting uncertainty products.Experimental results are summarized in Fig. 9. Given theresolution of the setup (indicated by error bars), the obtaineduncertainty products fit quite well the theoretical predictions.As anticipated, the resolution is not uniform and gets better inthe region of large variances.The inspection of the upper panels of Fig. 9 shows that themeasurements of the optimal Mathieu and von Mises beamsyield very similar results. This could be expected, sincethe difference between the uncertainty products of these twobeams (see Fig. 3c) is below the resolution of the presentsetup. The cosine and wedge angle distribution in the bottompanels of Fig. 9 can be discriminated from the optimum moreeasily. While the suboptimality of the cosine distribution isconfirmed only for moderate to large variances ∆ ˆ E > . ,the wedge angular shape shows entirely different behavior:the uncertainty product increases with the variance. This ten-dency of the wedge distribution can be readily explained: asthe variance gets larger, more and more side maxima of thesinc-like angular momentum spectrum fall into the detectedwindow m ∈ [ − , , yielding a corresponding grow of theuncertainty product. ∆ E ∆ L ∆ E ∆ E ∆ L ∆ E (a) (b)(c) (d) FIG. 9: Experimentally determined uncertainty products for angleand angular momentum. The following angular distributions of beamamplitude in the transverse plane were measured: (a) Mathieu dis-tribution of Eq. (2.38), (b) von Mises distribution of Eq. (2.52),(c) cosine distribution of Eq. (2.50), and (d) wedge distribution ofEq. (2.48). Experimentally obtained uncertainty products are de-noted by circles. For comparison, theoretical uncertainty productsof the optimal Mathieu angular distribution (solid line), cosine angu-lar distribution (broken line), and wedge angular distributions whoseangular momentum spectra have been truncated at the first, second,etc. minimum (dotted lines) are also shown.
V. CONCLUSIONS
In conclusion, we have formulated rigorous uncertainty re-lations for angle and angular momentum based on circularvariance as a proper statistical measure of angular error. Fun-damental Mathieu states were identified as intelligent statesunder the constraint of given uncertainty either in angle orin angular momentum. In this sense, the Mathieu states pro-vide the optimal distribution of information between the twoobservables with possible applications in information process-ing. An optical test of the uncertainty relations was performedby using spatial light modulators both for the beam prepara-tion and analysis.Although the present experiment nicely confirmed our the-ory, the resolution of the present setup was not sufficient forobserving finer details in the angular-momentum representa-tion of light beams. Further improvements both on the de-tection scheme and on hardware are highly desirable. Ourscheme is conceptually simple but suffers from crosstalks andartifacts, especially at large helicities. New detection schemesbased on direct sensing of beam wavefront could perhapssolve this problem. Concerning beam manipulation, spatiallight modulators used in our experiment, though very flexibleand easy-to-use devices, have also their drawbacks, namelysmall light efficiencies and pixellated structures. A possiblefuture upgrade of the experimental setup lies in employing theoptically-addressed SLM.2
Acknowledgments
We acknowledge discussions with Andrei Klimov, IoannesRigas, and Hubert de Guise. This work was supported by the Czech Ministry of Education, Projects MSM6198959213 andLC06007, the Czech Grant Agency, Grant 202/06/307, andthe Spanish Research Directorate, Grant FIS2005-06714. [1] V. Peˇrinova, A. Lukˇs, and J. Peˇrina,
Phase in Optics (WorldScientific, Singapore, 1998).[2] A. Luis and L. L. S´anchez-Soto, Prog. Opt. , 421 (2000).[3] H. A. Kastrup, Phys. Rev. A , 052104 (2006).[4] L. Allen, S. M. Barnett, and M. J. Padgett, Optical AngularMomentum (Institute of Physics Publishing, Bristol, 2003).[5] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Wo-erdman, Phys. Rev. A , 8185 (1992).[6] Z. Bouchal and R. ˇCelechovsk´y, New J. Phys. , 131 (2004).[7] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko,S. M. Barnett, and S. Franke-Arnold, Opt. Express , 5448(2004).[8] R. ˇCelechovsk´y and Z. Bouchal, New J. Phys. , 328 (2007).[9] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature (London) , 313 (2001).[10] A. Vaziri, G. Weihs, and A. Zeilinger, J. Opt. B , S47 (2002).[11] G. Molina-Terriza, A. Vaziri, J. ˇReh´aˇcek, Z. Hradil, andA. Zeilinger, Phys. Rev. Lett. , 167903 (2004).[12] N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J.Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev.Lett. , 053601 (2004).[13] S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel,G. W. ’t Hooft, and J. P. Woerdman, Phys. Rev. Lett. , 240501(2005).[14] L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. ,163905 (2006).[15] G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. , 305(2007).[16] M. M. Nieto, Phys. Rev. Lett. , 182 (1967).[17] J. Zak, Phys. Rev. (1969).[18] C. T. Whelan, J. Phys. A , L181 (1980).[19] D. Loss and K. Mullen, J. Phys. A , L235 (1992).[20] Y. Ohnuki and S. Kitakado, J. Math. Phys. , 2827 (1993).[21] S. Szab´o, Z. Kis, P. Adam, and J. Janszky, Quantum Opt. , 527(1994).[22] V. A. Kosteleck´y and B. Tudose, Phys. Rev. A , 1978 (1996).[23] K. Kowalski and J. Rembieli´nski, J. Phys. A , 1405 (2002).[24] J. Y. Bang and M. S. Berger, Phys. Rev. D , 125012 (2006).[25] S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial,and M. Padgett, New J. Phys. , 103 (2004).[26] D. T. Pegg, S. M. Barnett, R. Zambrini, S. Franke-Arnold, andM. Padgett, New J. Phys. , 62 (2005).[27] J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, andJ. Courtial, Phys. Rev. Lett. , 257901 (2002).[28] Z. Hradil, J. Rehacek, Z. Bouchal, R. ˇCelechovsk´y, and L. L.S´anchez-Soto, Phys. Rev. Lett. (2006).[29] J. C. Guti´errez-Vega, M. D. Iturbe-Castillo, and S. Ch´avez-Cerda, Opt. Lett. , 1493 (2000).[30] J. C. Guti´errez-Vega, R. M. Rodr´ıguez-Dagnino, M. A. Meneses-Nava, and S. Ch´avez-Cerda, Am. J. Phys. , 233(2003).[31] M. A. Bandr´es, J. C. Guti´errez-Vega, and S. Ch´avez-Cerda,Opt. Lett. , 44 (2004).[32] P. Carruthers and M. M. Nieto, Rev. Mod. Phys , 411 (1968).[33] G. G. Emch, Algebraic Methods in Statistical Mechanics andQuantum Field Theory (Wiley, New York, 1972).[34] J. M. L´evy-Leblond, Ann. Phys. (N.Y.) , 319 (1976).[35] D. Judge and J. Lewis, Phys. Lett. , 190 (1963).[36] W. H. Louisell, Phys. Lett. , 60 (1963).[37] G. W. Mackey, Mathematical Foundations of Quantum Me-chanics (Benjamin, New York, 1963).[38] C. Helstrom,
Quantum Detection and Estimation Theory (Aca-demic Press, New York, 1976).[39] U. Leonhardt, J. A. Vaccaro, B. B¨ohmer, and H. Paul, Phys.Rev. A , 84 (1995).[40] A. Luis and L. L. S´anchez-Soto, Eur. Phys. J. D , 195 (1998).[41] D. Mumford, Tata Lectures on Theta I (Birkhauser, Boston,1983).[42] M. Abramowitz and I. A. Stegun, eds.,
Handbook of Mathemat-ical Functions (Dover, New York, 1984).[43] K. V. Mardia and P. E. Jupp,
Directional Statistics (Wiley,Chichester, 2000).[44] R. Barakat, J. Opt. Soc. Am. A , 1213 (1987).[45] A. Klimov and S. Chumakov, Phys. Lett. A , 7 (1997).[46] A. Perelomov, Generalized Coherent States and their Applica-tions (Springer, Berlin, 1986).[47] J. A. Gonz´alez and M. A. del Olmo, J. Phys. A , 8841 (1998).[48] B. C. Hall and J. J. Mitchell, J. Math. Phys. , 1211 (2002).[49] K. Kowalski, J. Rembieli´nski, and L. C. Papaloucas, J. Phys. A , 4149 (1996).[50] C. R. Rao, Linear Statistical Inference and its Applications (Wi-ley, New York, 1965).[51] E. Breitenberger, Found. Phys. , 353 (1985).[52] J. B. M. Uffink, Ph.D. thesis, University of Utrecht (1990).[53] I. Bialynicki-Birula, M. Freyberger, and W. Schleich, Phys. Scr. T48 , 113 (1993).[54] G. W. Forbes and M. A. Alonso, Am. J. Phys. , 340 (2001).[55] R. Bluhm, V. A. Kosteleck´y, and B. Tudose, Phys. Rev. A ,2234 (1995).[56] N. W. McLachlan, Theory and Application of Mathieu Func-tions (Oxford University Press, New York, 1947).[57] T. Opatrn´y, J. Phys. A , 6961 (1995).[58] D. Frenkel and R. Portugal, J. Phys. A , 3541 (2001).[59] M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, Opt.Lett. , 2285 (2003).[60] G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, andS. Carrasco, J. Eur. Opt. Soc.2