Polarization versus photon spin Alfredo Luis∗ and Alfonso Rodil Departamento déOptica, Facultad de Ciencias Fı́sicas, Universidad Complutense 28040 Madrid, Spain ∗ alluis@fis.ucm.es http://www.ucm.es/info/gioq/alfredo.html Abstract: We examine whether the Stokes parameters of a two-mode electromagnetic field results from the superposition of the spins of the photons it contains. To this end we express anyn-photon state as the result of the action on the vacuum ofn creation operators generating photons which can have may different polarization states in general. We find that the macroscopic polarization holds as sum of the single-photon Stokes pa- rameters only for the SU(2) orbits of photon-number states. The states that lack this property are entangled in every basis of independent field modes, so this is a class of entanglement beyond the reach of SU(2) transformations. © 2014 Optical Society of America OCIS codes:(270.0270) Quantum optics; (260.5430) Polarization. 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Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys.12, 073025 (2010). 17. P. Bruno, “Quantum geometric phase in Majorana’s stellar representation: mapping onto a many-body Aharonov- Bohm phase,” Phys. Rev. Lett.108, 240402 (2012). 18. O. Giraud, P. Braun, and D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys.12, 063005 (2010). #200245 - $15.00 USD Received 28 Oct 2013; revised 15 Dec 2013; accepted 15 Dec 2013; published 15 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001569 | OPTICS EXPRESS 1569 1. Introduction From a quantum perspective there is the widespread idea that polarization is the spin of the photon. Accordingly, the polarization of a light beam should result from the combination of the spins of the photons it contains [1, 2]. This polarization-spin connection is reinforced when expressing polarization by the Stokes operators, since they are formally equivalent to an angular momentum [3]. In this work we examine in more detail the representation of two-mode polarization as the superposition of the spin of individual photons. To this end in Sec. 2 we express anyn-photon state as the result of the action on the vacuum ofn creation operators, generating photons with many different polarization states in general. Then, in Sec. 3 we investigate whether the Stokes parameters of anyn-photon state is the sum of the Stokes parameters of then individual photons that appear in the expression derived in Sec. 2. We find that this is true only for the SU(2) orbits of photon-number states. We provide also a simple criterion to determine whether a given state is in the SU(2) orbit of a photon-number state via the Stokes-operators covariance matrix. We illustrate these results with some relevant examples. 2. Quantum polarization of a two-mode field In a typical mode decomposition of the transverse electromagnetic field in terms of plane waves of wave-vectork we have E(r , t) ∝ ∑ k,± ak,±εk,±ei(kr−ωkt), (1) where the complex two-dimensional vectorsεk,± with ε∗k,+ ·εk,− = 0 express the vibration state of the field mode with complex amplitude operatorak,±, the subscript± representing circular polarization for example. Throughout we will consider a two-mode approach with a singlek, so no subscriptk will be necessary from now on. In the most general terms quantum polarization is addressed in terms of the Stokes operators [3] S0 = a† +a++a† −a−, Sz = a† +a+−a† −a−, Sx = a† −a++a† −a† +, Sy = i ( a† −a+−a−a† + ) , (2) so that the Stokes parameters are their mean valuessj = 〈Sj〉. The Stokes operators satisfy the commutation relations of an angular momentum [Sx,Sy] = 2iSz, (3) and its cyclic permutations, with S2 = S0(S0+2), [S,S0] = 0, (4) whereS= (Sx,Sy,Sz). Thus there is a complete formal equivalence between the subspaceHn of fixed total photon numbern with an spin j = n/2. In particular, a single photonn = 1 is equivalent to an spin 1/2. The Stokes operators are also the infinitesimal generators of SU(2) transformations [4, 5] U = exp(iθu ·S/2) , (5) with θ a real parameter, andu a unit three-dimensional real vector. These are all linear, energy preserving transformations of the field amplitudes, embracing very fundamental optical opera- tions such as lossless beam splitters, phase plates, and all linear interferometers [6]. It can be seen that the action ofU onS is a rotationR of angleθ and axisu [5] U†SU = RS, (6) #200245 - $15.00 USD Received 28 Oct 2013; revised 15 Dec 2013; accepted 15 Dec 2013; published 15 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001569 | OPTICS EXPRESS 1570 whereRtR= RRt = I , the superscriptt denotes matrix transposition, andI is the 3×3 identity matrix. Throughout, by SU(2) invariance we mean that two field states connected by a SU(2) transformation are fully equivalent concerning polarization statistics, leaving aside their mean polarization state. Equivalently, SU(2) invariance means that the conclusions which one could draw are independent of which polarization basis one chooses. In order to link quantum field states with individual-photon properties we demonstrate in the Appendix A that any state withn photons|ψn〉 ∈Hn can be expressed as the result of the action on the vacuum ofn creation operators generating photons that in general will have many with different polarization states. This is |ψn〉= N Πn m=1a† m|0,0〉, (7) whereN is a normalization constant,|0,0〉 is the two-mode vacuum state, and the complex amplitude operatorsam are am = cosθma++e−iφm sinθma−, (8) whereθm andφm are independent parameters. In appendix A we show the close relation of expressions (7) and (8) with the Majorana representation of spins [7]. The action of each creation operatora† m on the vacuum generates the single-photon pure state |εm〉= a† m|0,0〉= cosθm|1,0〉+eiφm sinθm|0,1〉, (9) where|n+,n−〉 are the photon-number states of modesa±. These states|εm〉 are actually SU(2) coherent states [5, 8]. Their Stokes parameters are sz = cos(2θm), sx = sin(2θm)cosφm, sy = sin(2θm)sinφm, s0 = 1, (10) so they reach the maximum of the standard definition of degree of polarization,P = |s|/s0 = 1. Thus, we may say thata† m creates photons with the polarization state expressed by the two- dimensional complex vectorεm εm = cosθmε++eiφm sinθmε−. (11) Note that in general the polarization states of the photons appearing in Eqs. (7) and (9) are not orthogonal,ε∗ℓ · εm 6= 0 for ℓ 6= m. 3. Polarization versus single-photon spins The question to be addressed is whether the Stokes parameters of an-photon state are the sum of the Stokes parameters of the individual photons: 〈S〉n = n ∑ m=1 〈S〉1,m, (12) or, equivalently, 〈Sz〉n = n ∑ m=1 〈Sz〉1,m, 〈 a† +a− 〉 n = n ∑ m=1 〈 a† +a− 〉 1,m , (13) where〈S〉n are the Stokes parameters in then−photon state|ψn〉 ∈ Hn in Eq. (7), while〈S〉1,m are the Stokes parameters (10) of the corresponding one-photon states|εm〉 in Eq. (9). In the transit from Eqs. (12) to (13) we have replaced a pair of real equations forSx andSy by a single complex equation forSx+ iSy = 2a† +a−. We are going to demonstrate the following proposition: The property (12) holds exclusively for the SU(2) orbits of the photon-number states|n+,n−〉, i. e., for every state of the form U |n+,n−〉, whereU is any SU(2) unitary transformation (5). #200245 - $15.00 USD Received 28 Oct 2013; revised 15 Dec 2013; accepted 15 Dec 2013; published 15 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001569 | OPTICS EXPRESS 1571 3.1. Proof of the proposition The proposition can be demonstrated via induction, starting with the simplest nontrivial case with two photonsn= 2. 3.1.1. Two photons Let us take full advantage of the SU(2) invariance stated above considering without loss of generality the properly normalized two-photon state |ψ2〉= N a† α ( cosθa† α +eiφ sinθa† β ) |0,0〉, (14) this is |ψ2〉= 1√ 1+ cos2 θ (√ 2cosθ |2,0〉+eiφ sinθ |1,1〉 ) , (15) whereaα ,β are two arbitrary field modes with orthogonal polarizationsε∗α · εβ = 0, and|n,m〉 the associated photon-number basis. On the other hand, the corresponding single-photon states are, in the same photon-number basis, |ε1〉= |1,0〉, |ε2〉= cosθ |1,0〉+eiφ sinθ |0,1〉. (16) After replacinga+,− by aα ,β the equalities (13) read, 4cos2 θ 1+ cos2 θ = 2cos2 θ , 2eiφ cosθ sinθ 1+ cos2 θ = eiφ cosθ sinθ . (17) It can be easily seen that these equalities are satisfied only whenθ = 0,π/2 modulusπ , for any φ . This means that the photons must have either the same polarization state,ε1 = ε2 = εα for θ = 0, or orthogonal polarization states forθ = π/2,ε1 = εα , ε2 = εβ . In other words, Eq. (12) holds just for the SU(2) orbits of the number states|2,0〉 and|1,1〉. 3.1.2. n+1 photons Now we assume that Eqs.(13) hold for an state|ψn〉 with n photons, this is|ψn〉 = |n+,n−〉, modulus SU(2) transformations. Then we add another photon in an arbitrary polarization state |ψn+1〉= N ( cosθa† ++eiφ sinθa† − ) |n+,n−〉, (18) this is |ψn+1〉= N ( √ n++1cosθ |n++1,n−〉+ √ n−+1eiφ sinθ |n+,n−+1〉 ) , (19) whereN = 1/ √ 1+n+cos2 θ +n−sin2 θ . Next we examine whether Eqs. (13) hold as 〈Sz〉n+1 = 〈Sz〉n+ 〈Sz〉ε , 〈 a† +a− 〉 n+1 = 〈 a† +a− 〉 n + 〈 a† +a− 〉 ε , (20) where the subscriptsn+ 1, n, andε refer to the states|ψn+1〉, |ψn〉, and |ε〉 = cosθ |1,0〉+ eiφ sinθ |0,1〉, respectively. An straightforward calculation implies that Eqs. (20) are equivalent to (n++1)(n+−n−+1)cos2 θ+(n−+1)(n+−n−−1)sin2 θ 1+n+ cos2 θ+n− sin2 θ = n+−n−+ cos2 θ − sin2 θ , (n++1)(n−+1)eiφ sinθ cosθ 1+n+ cos2 θ+n− sin2 θ = eiφ sinθ cosθ . (21) #200245 - $15.00 USD Received 28 Oct 2013; revised 15 Dec 2013; accepted 15 Dec 2013; published 15 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001569 | OPTICS EXPRESS 1572 It can be easily seen that these equalities are satisfied only whenθ = 0,π/2 modulusπ , for any φ , so that either|ψn+1〉= |n++1,n−〉, or |ψn+1〉= |n+,n−+1〉. This completes the proof of the proposition. This is that the polarization-sum property (12) holds only for the SU(2) orbits of all number states. These are the states that result from the addition to the vacuum of photons either with the same or orthogonal polarization states. 3.2. Sum property and covariance matrix Let us provide a simple criterion to determine whether a given state satisfies the sum property (12) or not. We demonstrate that property (12) holds if and only if detM = 0, whereM is the covariance matrix of Stokes-operators [4] Mℓ,m = 1 2 (〈SℓSm〉+ 〈SmSℓ〉)−〈Sℓ〉〈Sm〉 . (22) The diagonal elements ofM are the variances of the Stokes operatorsS, while the variance of any other Stokes componentSu = u ·S is computed as(∆Su) 2 = utMu, whereu is any unit three-dimensional real vector. The states satisfying Eq. (12) are SU(2) transforms of the eigenstates ofSz, which are |n+,n−〉. The number states|n+,n−〉 have the covariance matrix M = (n++n−+2n+n−)   1 0 0 0 1 0 0 0 0   , (23) with detM = 0. Under SU(2) transformations (5) we haveM → RtMR, so that the determinant is preserved det(RtMR) = 0. Thus, if the state satisfies Eq. (12) then detM = 0. The reverse is also true. If the state has detM = 0 thenM has a vanishing eigenvalue, say Mu = 0, and the variance of the corresponding Stokes componentSu = u ·S vanish∆Su = 0. SinceSu andSz can be always related by an SU(2) transformation we get that the states with detM = 0 are SU(2) transforms of the eigenstates ofSz, so that the sum property (12) is fulfilled. 3.3. Sum property and entanglement Let us note that for two-mode field states with exactlyn photons the only states that factorize as product of single-mode states are the number states|n,m〉 for any polarization-orthogonal mode basis. This is to say that all the states that satisfy the sum property (12) can be rendered factorized by an SU(2) transformation. The other way round, the states that lack property (12) are entangled states ofn photons that cannot be rendered factorized by any choice of polarization-orthogonal mode basis. This means that condition (12) reveals a definite class of entanglement beyond the reach of devices performing SU(2) transformations. 3.4. Examples Let us consider three relevant examples. 3.4.1. SU(2) coherent states All the SU(2) coherent states satisfy property (12) since they can be actually defined as the SU(2) orbit of the number states|n,0〉 [5]. This is to say that all the photons are in the same polarization state. They can be regarded as the output of an ideal polarizer since we can always find a mode which is in the vacuum state. Moreover, the SU(2) coherent states are considered as the most classical states regarding spin properties [8]. #200245 - $15.00 USD Received 28 Oct 2013; revised 15 Dec 2013; accepted 15 Dec 2013; published 15 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001569 | OPTICS EXPRESS 1573 3.4.2. Twin-number states On the other hand, we can consider a typical example of nonclassical states satisfying property (12) as the orbits of the twin-photon number statesU |n,n〉 [9]. In this case half of the photons are in one polarization state while the other half are in the orthogonal polarization state. These states have found a lot of attention by their good properties in quantum metrology and they can be regarded as the limiting case of large SU(2) squeezing [10]. 3.4.3. N00N states Finally let us consider a relevant family of nonclassical states that do not satisfy property (12). These are the SU(2) orbits of the so-called N00N states, which are proper examples of Schrödinger-cat states [11, 12, 13] |ψn〉= 1√ 2 (|n,0〉− |0,n〉), n> 2, (24) with covariance matrix [4] M =   n 0 0 0 n 0 0 0 n2   , (25) so that detM 6= 0 as expected. Following the program outlined in the Appendix the factorized form (7) of these states is |ψn〉 ∝ Πn m=1 ( a† +−ei2πm/na† − ) |0,0〉. (26) This is to say that there are no two photons in the same polarization state. All them have the sameθm = π/4 but differentφm = 2πm/n. 4. Conclusions We have shown that every state can be regarded as the result of the action on the vacuum of creation operators generating photons with different spin states. Then we have found the states whose polarization Stokes vector results from the sum of the spins of the individual photons it contains. These are the SU(2) orbits of number states and correspond to the addition of photons either in the same or in orthogonal polarization modes. This is that the Stokes vectors of all the photons are either parallel or antiparallel. Moreover, we have shown that the states that lack such sum property have a distinguished entanglement behavior since they are entangled for every choice of field modes. This is entanglement that cannot be reached from factorized states via SU(2) transformations. A. Multi-photon states as photon-added states Let us demonstrate that everyn-photon state|ψn〉 ∈ Hn can be expressed in the form (up to a normalization constant) |ψn〉 ∝ Πn m=1a† m|0,0〉, (27) wheream are in Eq. (8). This is equivalent to say that there arek complex numberξm such that |ψn〉 ∝ a†n−k − Πk m=1 ( a† +− ξma† − ) |0,0〉, (28) whereξm = −eiφm tanθm, and we have singled out the potentialn− k photons with cosθm = 0, so that all thek parametersξm are finite. #200245 - $15.00 USD Received 28 Oct 2013; revised 15 Dec 2013; accepted 15 Dec 2013; published 15 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001569 | OPTICS EXPRESS 1574 The existence and uniqueness of factorization (28) can be demonstrated by projecting|ψn〉 on the two-mode Glauber coherent states|α+,α−〉, with a±|α+,α−〉= α±|α+,α−〉, 〈α+,α−|ψn〉 ∝ α∗n − Πk m=1 (x− ξ j)e− |α+ |2+|α−|2 2 , x= α∗ + α∗ − . (29) On the other hand, every|ψn〉 can be expressed in the photon number basis as |ψn〉= k ∑ m=0 cm|n+ = m,n− = n−m〉, (30) for suitablecm andk. Using that each photon-number state|n〉 can be expressed as then-times action on the vacuum state|0〉 of the corresponding creation operator |n〉= 1√ n! a†n|0〉, (31) we get |ψn〉= k ∑ m=0 cm √ m!(n−m)! ( a† + )m( a† − )n−m |0,0〉. (32) Projecting from the left on Glauber coherent states|α+,α−〉 we have 〈α+,α−|ψn〉= k ∑ m=0 cm √ m!(n−m)! α∗m + α∗n−m − e− |α+|2+|α−|2 2 , (33) so that after extracting a common factorα∗n − we get 〈α+,α−|ψn〉= α∗n − P(x)e− |α+ |2+|α−|2 2 , (34) where P(x) = k ∑ m=0 cm √ m!(n−m)! xm, x= α∗ + α∗ − . (35) The key point is that for anyn-photon state〈α+,α−|ψn〉 is a complex polynomial of the com- plex variableα∗ +/α∗ − of degreek ≤ n. Thus, comparing Eqs. (29) and (34), the equality in Eq. (27) is the standard factorization of a complex polynomialP(x) in terms of itsk rootsξm, maybe degenerate. Thus, the factorization in Eq. (27) always exists and is unique. It is worth noting that this way of expressing quantum states in Eqs. (27) and (28) is actually equivalent to the Majorana representation of angular momentum states [7], where we can take advantage of the formal similarity between the subspaceHn of n photons and an angular mo- mentum j = n/2, and also to the fully symmetric states ofn qubits. In the Majorana approach angular-momentum states are represented by the zeros of the wave-function in the coherent- state basis. These are the zerosξm of 〈α+,α−|ψn〉 in Eqs. (29) or (34), sometimes referred to as vortices, or constellation of Majorana stars. This representation is currently being used in quantum information science [14, 15, 16], and other areas [17, 18]. Acknowledgments A. L. acknowledges support from projects FIS2012-35583 of the Spanish Ministerio de Economı́a y Competitividad and QUITEMAD S2009-ESP-1594 of the Consejerı́a de Edu- cación de la Comunidad de Madrid. #200245 - $15.00 USD Received 28 Oct 2013; revised 15 Dec 2013; accepted 15 Dec 2013; published 15 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001569 | OPTICS EXPRESS 1575