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Spatial state Stokes-operator squeezing and entanglement for optical beams M. T. L. Hsu,1,2 W. P. Bowen,3 and P. K. Lam1 1ARC COE for Quantum-Atom Optics, Australian National University, Canberra, ACT 0200, Australia. 2Present address: Department of Chemistry, Stanford University, Palo Alto, CA 94305, U.S.A. 3School of Physical Sciences, University of Queensland, Brisbane, QLD 4072, Australia. (Dated: January 30, 2009) The transverse spatial attributes of an optical beam can be decomposed into the position, momentum and orbital angular momentum observables. The position and momentum of a beam is directly related to the quadrature amplitudes, whilst the orbital angular momentum is related to the polarization and spin variables. In this paper, we study the quantum properties of these spatial variables, using a representation in the Stokes-operator basis. We propose a spatial detection 9 schemetomeasureallthreespatialvariablesandconsequently,proposeaschemeforthegeneration 0 of spatial Stokesoperator squeezing and entanglement. 0 2 PACSnumbers: 42.50,42.30 n a J I. INTRODUCTION rangingfromthegenerationofcounter-rotatingsuperpo- 0 sitionsinBose-Einsteincondensates[22,23],andtransfer 3 Squeezed and entangled bright optical fields are es- of orbital angular momentum from an atomic ensemble sential resourcesin the continuous variable quantum op- to a light field [24], to optical angular momentum trans- h] tics and quantum information communities [1]. To date, fer to trapped particles [25, 26, 27]. It should be noted p the vast majority of research has been focused on fields that light with orbital angular momentum has also been - which exhibit non-classical features on their amplitude applied to achieve super-resolution imaging of molecules nt and phase quadratures. Recently, however, a number of andproteinsinbiologicalsystems,viatechniquessuchas a papers have been published on squeezing and entangle- stimulatedemissiondepletion(STED)[28]. Non-classical u mentofotherfieldvariables,andinparticularthe polar- orbital angular momentum states offer the prospect to q ization [2] and spatial structure [3, 4, 5, 6]. These vari- improvetheseprocesses,andtogeneratenon-classicalor- [ ablescanbedirectlyrelatedtomomentumcomponentsof bital angular momentum states in macroscopic physical 1 thefield. Thequadratureamplitudesareassociatedwith systems. v the transverse linear momentum of the field, whilst the Ref. [5] proposed EPR entanglement of the position- 3 polarization states and spatial states considered to date momentum observables of optical fields, and this has re- 1 are respectively associated to the spin angular momen- cently been experimentally observed [6]. In these ref- 8 4 tumandthetransverseangular momentum. Weimmedi- erences, entanglement was present only for one trans- . ately see why interaction of a polarizationsqueezed field verse axis of the beam, and analysis only required con- 1 withanatomicensemblecanyieldatomicspinsqueezing sideration of the TEM00 and TEM01 modes. One of the 0 9 [7],andwhy spatiallyentangledlightcanbe usedto test strengthsofentanglementinthe spatialdomainisaccess 0 theEPRparadoxwithpositionandmomentumvariables to higher dimensional spaces. Here we extend the work : [5] as was originallyproposed by Einstein, Podolsky and of Refs. [5, 6] to include both transversebeam axes, in a v Rosen [8]. formalism easily extended to higher order TEM modes. i X The correspondence of non-classical polarization and In the process, orbital angular momentum is introduced asanentanglementvariableinadditiontobeamposition r spatialstateswithnon-classicalmomentumstatesimme- a diatelyraisesthequestionofwhethernon-classicalorbital and momentum. These quantum variables can be repre- angular momentum states can also be generated. Such sented on a spatial version of the Poincaré sphere com- stateshavebeenunderinvestigationforsometimeinthe monly used for analysis of polarization quantum states. We propose a scheme to measure their signal and noise discrete variable regime. Techniques have been estab- lished to detect the orbital angular momentum proper- properties, via a spatial Stokes detection scheme; and finally introduce a scheme to generate spatial Stokes- ties of single photons [9, 10, 11]; and discrete-variable multi-dimensional entanglementbetween orbitalangular operator squeezing and entanglement. momentumstateshasbeenproposed[12,13]anddemon- strated[14]. However,todate,non-classicalorbitalangu- II. SPATIAL STOKES OPERATORS lar momentum states have not been investigated for the continuous variable regime, that is relevant in many applications[15,16,17,18,19,20,21]. Weinvestigatethese Spatialquantumstatesexistinaninfinite dimensional continuousvariable orbitalangularmomentumquantum Hilbert space, which may be conveniently expanded in a states, proposing techniques for both generationand de- basisofTEMpq modes. Aftersuchanexpansionthepos- tection. Many applications have been proposed for op- itive frequency part of the electric field operator, ˆ+(r), E tical orbital angular momentum in the classical domain, canbewrittenexplicitlyintermsofthedouble-subscript 2 sum where θ is the phase difference between the TEM and pq TEM modes. Asis the casewithpolarization,the spa- ∞ qp ˆ+(r)=i ¯hω aˆ u (r) (1) tial Stokes operators obey commutation relations. By E r2ǫ V pq pq using the commutation relations of the photon annihi- 0 pX,q=0 lation and creation operators, [aˆ ,aˆ ] = δ δ , the mn jk mj nk where aˆ and u (r) are the photon annihilation opera- Stokes-operator commutation relations can be found to pq pq be tor and the normalizedtransversebeam amplitude function,respectively,associatedwiththeTEM mode. The pq Sˆ,Sˆ =2iǫ Sˆ (4) photonannihilationoperatoraˆ canbeformallydefined i j ijk k pq h i with the projection of the transverse beam amplitude function with the positive frequency part of the electric where ǫijk is the Levi-Civita tensor. field operator, given by Each Stokes operator characterizes a different spatial property of the field. Sˆ represents the beam inten- 0 −∞ sity,whilsttheStokesvector(Sˆ ,Sˆ ,Sˆ )characterizesits aˆpq =ZZ∞ dxdyEˆ+(x,y)upq(x,y). (2) transverse momentum. Sˆ1 and1Sˆ22res3pectively quantify the relative proportions of horizontal to vertical trans- Withinthisbasis,itisnaturaltoconsiderpairsofmodes verse momentum, and 45◦ to 45◦ diagonal transverse with amplitude function that correspond to a physical momentum; whilst Sˆ weights−left to right orbital angu- 3 π/2 rotation in the transverse plane. That is, modes of lar momentum. the formTEM and TEM , with p and q interchanged The spatial Poincaré sphere can be fully spanned by pq qp (p = q). Pairs of this form are naturally described by a spatiallyoverlappingtwoorthogonalTEM andTEM pq qp 6 setofspatialStokesoperatorsSˆ(p,q)wherei 0,1,2,3 , modes. The diagonal modes, for example, are given by thatcanberepresentedbyaPoiincarésphere∈[2{9]indirec}t u4p5q◦(r)=uqp(r)−upq(r)andu4qp5◦(r)=uqp(r)+upq(r), analogy to polarization Stokes operators [30], as shown and respectively yield Stokes vectors oriented along the in Fig. 1. negativeandpositiveSˆ2axisofthePoincarésphere. The Laguerre-Gauss modes LG with orbital angular mo- 0q mentumqand q,aregivenbyu+l(r)=u (r)+iu (r) (a) S3 (b) S3 andu−l(r)=u−(r) iu (r),yie0lqdingStok0eqsvectorqs0ori- 0q 0q − q0 ented along the Sˆ axis. For example, the intensity dis- 3 tributions for the diagonal and Laguerre-Gauss modes, generated from the combinations of TEM and TEM 10 01 modes, are shown in Fig. 2. S1 S2 S1 S2 (A) = + FIG.1: (a) Poincaré sphererepresentation based on thespa- tial modes TEM10 and TEM01. S1 is the Stokes variable for (B) = - the u01(r) and u10(r) modes, S2 is the Stokes variable for the u4051◦(r) and u4150◦(r) modes, and S3 is the Stokes variable for the u+1(r) and u−1(r) modes. (b) Poincaré sphere 01 01 representation for the spatial Stokes operators. The shaded area indicates the quantum noise associated with the mean (C) +1 = + i amplitude of theStokes operator. Using the definition of the classical Stokes parameters (D) -1 = - i [29], corresponding quantum mechanical spatial Stokes operators are defined as Sˆ = aˆ† aˆ +aˆ† aˆ FIG. 2: Intensity distribution for the modes given by (A) 0 pq pq qp qp u4051◦(r)=u01(r)+u10(r),(B)u4150◦(r)=u01(r) u10(r),(C) Sˆ1 = aˆ†pqaˆpq−aˆ†qpaˆqp u+011(r)=u01(r)+iu10(r),and(D)u−011(r)=u01−(r)−iu10(r). Sˆ = aˆ† aˆ eiθ +aˆ† aˆ e−iθ 2 pq qp qp pq Sˆ = iaˆ† aˆ e−iθ iaˆ† aˆ eiθ 3 qp pq − pq qp Sincethediagonalandorbitalmodescanbegenerated (3) purely from two orthogonal TEM and TEM modes, pq qp 3 onlybychangingthephasebetweenthemodes,thesesets (a) S0 (b) S1 ofmodesobeytheSU(2)groupproperties. Atthispoint, + - werestrictouranalysistotheTEM andTEM modes, 10 01 Input beam Input beam forillustrativepurposes. Inprinciple,ouranalysisisvalid for all orthogonal TEM and TEM modes. pq qp MS MS (c) S2 (d) S3 - - Input beam Input beam III. SPATIAL STOKES DETECTION MS MS Now that the spatial Stokes operators have been de- (e) (f) Output Output fined, we consider a system for their detection. Analo- beam beams gous to polarization Stokes operator detection, the spa- MS M CL tial Stokes detection requires two separate photodiodes, CL a modal phase shifter, and a mode separator. Input M 50:50 BS 2f beam Input f, 2 A modal phase shifter is a device that introduces a 50:50 BS M beam relative phase shift between the two orthogonal TEM 10 andTEM modes. Themodalphaseshifter canbe con- 01 structed using a pair of cylindrical lenses with variable FIG. 3: Measurements of (a) Sˆ0, (b) Sˆ1, (c) Sˆ2 and (d) Sˆ3. (e) An example of a mode separator (MS) is an asymmetric lens separation, as shown in Fig. 3 (f). The pairing of Mach-Zehnderinterferometer. (f)Theπandπ/2modalphase cylindrical lenses introduces an astigmatic Gouy phase shifterscouldbeconstructedusingapairofcylindricallenses shiftbetweentheTEM andTEM modes. Inorderto 10 01 with lens separation given by 2f and √2f, respectively. f is introduce π and π/2 modal phase shift, the lens separa- thefocal lengthofthecylindricallenses, Misamirror, 50:50 tion is givenby 2f and√2f, respectively,where f is the BS is a symmetric non-polarizing beam-spliter. focal length of the cylindrical lens. Ref. [21] contains a detailed analysis of the π and π/2 modal phase shifters. A mode separator (MS) is a device that separates the two orthogonal TEM and TEM modes and can be 10 01 constructed using an asymmetric Mach-Zehnder interferometer [31], as shown in Fig. 3 (e). Due to the odd modes, are therefore given by and even numbers of reflections in each interferometer arm,differentinterferenceconditionsfor the TEM and 01 TEM modes are present. The resulting outputs from 10 the interferometer is a separation of even and odd (defined along one spatial axis) spatial modes [31]. Sˆ = α2 +α2 =N h 0i 10 01 Measurementsofthe totalsignalandnoise,whichcor- Sˆ = α2 α2 respondtoameasurementofSˆ ,aregivenbythe sumof h 1i 10− 01 0 Sˆ = 2α α cosθ thephotocurrentsasshowninFig.3(a). Measurementof h 2i 10 01 Sˆ1 involvestakingthesubtractionbetweenthephotocur- hSˆ3i = 2α10α01sinθ (5) rentoutputs correspondingto mode componentsTEM 10 and TEM , as shown in Fig. 3 (b). Measurement of Sˆ 01 2 involves subtraction of the diagonal mode components and is obtained by phase shifting one mode by π with where α and α are the mean amplitude terms cor- respect to the other and then taking the subtraction of 01 10 the photocurrent signals, given in Fig. 3 (c). Sˆ is mea- responding to modes u01(r) and u10(r), respectively. 3 Sˆ =N isthetotalmeannumberofphotons, Sˆ isthe sured by decomposing the Laguerre-Gaussmode into its 0 1 hdiffeirence in the mean number of photons in thheiu (r) TEM andTEM modesusingπ andπ/2modalphase 10 shifte1r0s, as shown0i1n Fig. 3 (d). andu01(r)modes, Sˆ2 isthedifferenceinthemeannum- berofphotonsinthheui45◦(r)andu45◦(r)modesand Sˆ The photon annihilation operators in Eq. (3) can be 01 10 h 3i is the difference in the mean number of photons in the writtenintheformaˆ =α +δaˆ ,whereα describes pq pq pq pq u+1(r) and u−1(r) modes. the mean amplitude part and δaˆ is the quantum noise 01 01 pq operator. Using the linearized formalism, where second ordertermsinthefluctuationoperatorareneglected(i.e. The Stokes operators of Eq. (3) can be expanded in α 2 δaˆ2 ), the mean amplitudes of the Stokes terms of quadrature operators with the general form |oppeqr|ato≫rs i|nh Epqq.i(|3), in terms of the TEM and TEM Xˆφ = e−iφδaˆ +eiφδaˆ† , and their variances are then 10 01 apq pq pq 4 given in general by phasedifferenceθ =π/2. Thecombinationofthisspatial squeezed beam with a bright TEM beam would simul- 00 h(δSˆ0)2i = α210h(δXâ+10)2i+α201h(δXâ+01)2i taneouslyenablesub-shotnoiserelativemeasurementsof +2α α δXˆ+ δXˆ+ transverse momentum between the horizontal and verti- 10 01h a10 a01i calaxes,aswellasleft- andright-handedorbitalangular h(δSˆ1)2i = α210h(δXâ+10)2i+α201h(δXâ+01)2i momentum of a TEM00 beam [4]. 2α α δXˆ+ δXˆ+ To achievequantum enhancedmeasurements oftrans- − 10 01h a10 a01i versemomentumbetweenthediagonalandanti-diagonal h(δSˆ2)2i = α210h(δXâ−0θ1)2i+α201h(δXâθ10)2i axes, as well as between left- and right-handed orbital +2α10α01hδXâ−0θ1δXâθ10i asqnugeuelzairngmoofmSêntaunmd Soˆf.aTThEisMc0a0nbbeeamac,hiwevoeudldbyreqovueirre- h(δSˆ3)2i = α210h(δXâ−0θ1+π2)2i+α201h(δXâθ1−0π2)2i lappingquadra2turesqu3eezedTEM10 andTEM01 modes, +2α10α01hδXâ−0θ1+π2δXâθ1−0π2i (6) aantdsqhu(eδeXˆzeaπ40d1)q2uia<dr1a)tuarnedaθng=leπo/f4π./4 (i.e. h(δXâπ410)2i<1 Phase squeezed TEM and TEM modes (i.e. 10 01 where Xˆ+ = Xˆφ=0 and Xˆ− = Xˆφ=π/2 are respec- (δXˆ− )2 < 1 and (δXˆ− )2 < 1) will yield squeez- apq apq apq apq h a10 i h a01 i tively the amplitude and phase quadrature operators of ing of (δSˆ )2 only. 2 h i the TEM mode. pq Non-classicalopticalorbitalangularmomentumstates can be constructed by spatially overlapping a set of or- V. SPATIAL STOKES ENTANGLEMENT thogonal non-classical TEM fields, in analogy to the pq work of Refs. [5, 6] for transverse spatial entanglement. Spatial Stokes entanglement can be generated as is In such a scenario, no correlations should exist between shown in Fig. 4. Two squeezed TEM beams, labeled 10 the quadratures of the different input fields, such that respectivelybythesubscriptsxandy,arecombinedwith hδXâφ110δXâφ021i = 0, ∀{φ1,φ2}. Making this assumption, a π/2 relative phase shift on a 50:50 beam-splitter. The which we will adopt henceforth, the variances of the beamsafterthebeam-splitter,alsoinTEM modes,ex- 10 Stokes operators are simplified to hibit the usual quadrature entanglement. Two spatial mode combiners, each consisting of a spatial mode sep- (δSˆ )2 = α2 (δXˆ+ )2 +α2 (δXˆ+ )2 h 0 i 10h a10 i 01h a01 i arator as shown in Fig. 3 (e) in reverse, are then used (δSˆ )2 = α2 (δXˆ+ )2 +α2 (δXˆ+ )2 to combine the quadrature entangled beams with bright h 1 i 10h a10 i 01h a01 i coherentTEM beams,labelledhereaˆ andaˆ ,re- (δSˆ )2 = α2 (δXˆ−θ)2 +α2 (δXˆθ )2 01 01,x 01,y h 2 i 10h a01 i 01h a10 i spectively. As we will show here, the resulting output (δSˆ )2 = α2 (δXˆ−θ+π2)2 +α2 (δXˆθ−π2)2 . beams are entangled in the spatial Stokes operator ba- h 3 i 10h a01 i 01h a10 i sis. Experimental verification of the spatial Stokes en- (7) tanglementcanbeperformedusingthedetectionscheme described in the preceding sections. IV. SPATIAL STOKES SQUEEZING cboehaemrent a01,x mdeevaicseurement - Si,x An examination of Eq. (7) shows that simultaneous MS a) b) squeezed squeezing of at least two spatial Stokes operators is pos- OPA beam a10,x sible,usingtwoquadraturesqueezedinputfields. Hence, 50:50 BS enstpaantgialeld p p/2 MMSS +/-D 2xySi a spatial Stokes squeezed state can be used to enhance OPA squeezed a10,y beam relativemeasurementsofmomentumvariablesalongmul- MS teixpalme paxleess.. This is best illustrated by considering a few cboehaemrent a01,y - Si,y Simultaneous squeezing of Sˆ , Sˆ and Sˆ can be 0 1 2 achieved through the in-phase (θ = 0) spatial over- FIG. 4: Scheme to generate and characterize spatial Stokes entanglement. 50:50BSisasymmetricnon-polarizingbeam- lap of amplitude squeezed TEM and TEM modes (i.e. (δXˆ+ )2 < 1 and (δXˆ+ 1)20 < 1). Th0i1s spatial splitter, π and π/2 are modal phaseshifters. h a10 i h a01 i squeezed state could be combined with a bright TEM 00 beam to allow enhanced measurements of the transverse To verify the existence of spatial Stokes entanglement momentum of the TEM beam along any axis on the 00 we will use the generalized version of the Duan insepa- Poincaré sphere within the plane formed by Sˆ and Sˆ . 1 2 rability criterion [32] given in Ref. [33]. This criterion Simultaneous squeezing is exhibited along the horizon- provides a sufficient condition for entanglement and is tal/vertical axis and the diagonal/anti-diagonalaxis. given by SqueezingofSˆ ,Sˆ andSˆ canbeachievedbyoverlap- 0 1 3 pingamplitudesqueezedTEM andTEM modeswith ∆2 Aˆ+∆2 Bˆ <2[δAˆ,δBˆ] (8) 10 01 x±y x±y | | 5 where Aˆ and Bˆ are two general observables, and An inspection of Eqs. (12) to (14) show that in the ∆2 ˆ = min (δ ˆ δ ˆ )2 . The degree of insepa- limit considered here only spatial Stokes entanglement raxb±ilyitOy (Aˆ,Bˆ)hisOthxe±n giOveyn biy [33] between Sˆ2 and Sˆ3 can be realistically attained. This is I because the inseparability criteria of Eqs. (12) and (13) ∆2 Aˆ+∆2 Bˆ scale with respect to α01/α10. Since α01/α10 1, the (Aˆ,Bˆ)= x±y x±y , (9) correlation in Xˆ+ and Xˆθ has to be signific≫antly re- I 2[δAˆ,δBˆ] a01 a10 | | ducedbelowone,inordertomaintaininseparability. On the other hand, we see that after setting θ =0, Eq. (14) where (Aˆ,Bˆ)<1 indicates an inseparable state. I reduces to Assuming for simplicity and symmetry (i.e. both beams x and y are identically interchangeable) that α = α = α and α = α = α , and us- 01,x 01,y 01 10,x 10,y 10 ing Eqs. (4) and (5), we find the inseparability criteria (Sˆ ,Sˆ )= 1 ∆2 Xˆ+ +∆2 Xˆ− . (15) between spatial Stokes operators given by I 2 3 4(cid:16) x±y a10 x±y a10(cid:17) So that quadrature entanglement between the TEM 10 ∆2 Sˆ +∆2 Sˆ modes is transformed directly into spatial Stokes entan- (Sˆ ,Sˆ ) = x±y 1 x±y 2 glementbetween the Sˆ andSˆ spatialStokes operators. I 1 2 8α α sinθ 2 3 10 01 | | ∆2 Sˆ +∆2 Sˆ (Sˆ ,Sˆ ) = x±y 1 x±y 3 3 1 I 8α α cosθ 10 01 | | ∆2 Sˆ +∆2 Sˆ (Sˆ ,Sˆ ) = x±y 2 x±y 3. (10) I 2 3 4α2 α2 | 10− 01| The correspondence between spatial Stokes and VI. CONCLUSION quadrature entanglement becomes obvious when we ex- press the spatial Stokes operator conditional variance ∆2x±ySî in terms of quadrature operators. Making the We have identified the relevant spatial modes for op- assumption that α10 α01 and using Eq. (6), gives tical beam position, momentum and orbitalangular mo- ≪ mentum. Weformalizedthequantumpropertiesofthese ∆2x±ySˆ1 = α201∆2x±yXâ+01 observablesusingtheStokes-operatorformalismandpre- ∆2 Sˆ = α2 ∆2 Xˆθ sentedagraphicalrepresentationofthesevariablesusing x±y 2 01 x±y a10 the Poincaré sphere. A spatial Stokes detection scheme ∆2 Sˆ = α2 ∆2 Xˆθ−π2. wasdescribedandschemestogeneratespatialStokesop- x±y 3 01 x±y a10 (11) erator squeezing and entanglement were proposed. Substituting these expressions into Eq. (10), gives α (Sˆ ,Sˆ ) = 01 ∆2 Xˆ+ +∆2 Xˆθ I 1 2 8α10 sinθ (cid:16) x±y a01 x±y a10(cid:17) | | (12) (Sˆ ,Sˆ ) = α01 ∆2 Xˆ+ +∆2 Xˆθ−π2 Acknowledgments I 3 1 8α10 cosθ (cid:16) x±y a01 x±y a10 (cid:17) | | (13) (Sˆ ,Sˆ ) = 1 ∆2 Xˆθ +∆2 Xˆθ−π2 . We would like to thank Hans-A. Bachor for fruitful I 2 3 4(cid:16) x±y a10 x±y a10 (cid:17) discussions. This work was supported by the Australian (14) Research Council Centre of Excellence Programme. [1] N. J. Cerf, and G. Leuchs, Quantum information with S. M. Arakelian, and A. S. Chirkin, Appl. Phys. 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