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 any n-photon state as the result of the action on the vacuum of n 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 parameters 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
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 .
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 any n-photon state as the result of the action on the vacuum of n creation operators, generating photons with many different polarization states in general. Then, in Sec. 3 we investigate whether the Stokes parameters of any n-photon state is the sum of the Stokes parameters of the n 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-vector k we have
In the most general terms quantum polarization is addressed in terms of the Stokes operators 4, 5] 6]. It can be seen that the action of U on S is a rotation R of angle θ and axis u 
In order to link quantum field states with individual-photon properties we demonstrate in the Appendix A that any state with n photons |ψn〉 ∈ ℋn can be expressed as the result of the action on the vacuum of n creation operators generating photons that in general will have many with different polarization states. This isappendix A we show the close relation of expressions (7) and (8) with the Majorana representation of spins .
The action of each creation operator on the vacuum generates the single-photon pure state5, 8]. Their Stokes parameters are Eqs. (7) and (9) are not orthogonal, for ℓ ≠ m.
3. Polarization versus single-photon spins
The question to be addressed is whether the Stokes parameters of a n-photon state are the sum of the Stokes parameters of the individual photons: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 for Sx and Sy by a single complex equation for .
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−〉, where U is any SU(2) unitary transformation (5).
3.1. Proof of the proposition
The proposition can be demonstrated via induction, starting with the simplest nontrivial case with two photons n = 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(13) read, 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 stateEqs. (13) hold as Eqs. (20) are equivalent to
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, where M is the covariance matrix of Stokes-operators 
The states satisfying Eq. (12) are SU(2) transforms of the eigenstates of Sz, which are |n+, n−〉. The number states |n+, n−〉 have the covariance matrix(5) we have M → 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 then M has a vanishing eigenvalue, say Mu = 0, and the variance of the corresponding Stokes component Su = u · S vanish ΔSu = 0. Since Su and Sz can be always related by an SU(2) transformation we get that the states with detM = 0 are SU(2) transforms of the eigenstates of Sz, so that the sum property (12) is fulfilled.
3.3. Sum property and entanglement
Let us note that for two-mode field states with exactly n 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 of n 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.
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〉 . 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 .
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 states U|n, n〉 . 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 .
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]4] Appendix the factorized form (7) of these states is
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 every n-photon state |ψn〉 ∈ ℋn can be expressed in the form (up to a normalization constant)Eq. (8). This is equivalent to say that there are k complex number ξm such that
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−〉,Eqs. (29) and (34), the equality in Eq. (27) is the standard factorization of a complex polynomial P(x) in terms of its k roots ξm, 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 , where we can take advantage of the formal similarity between the subspace ℋn of n photons and an angular momentum j = n/2, and also to the fully symmetric states of n 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 zeros ξm 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].
A. L. acknowledges support from projects FIS2012-35583 of the Spanish Ministerio de Economía y Competitividad and QUITEMAD S2009-ESP-1594 of the Consejería de Educación de la Comunidad de Madrid.
References and links
1. M. Born and E. Wolf, Principles of Optics, 7 (Cambridge University, 1999). [CrossRef]
2. Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
3. J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).
4. A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008). [CrossRef]
5. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972). [CrossRef]
6. A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995). [CrossRef]
7. E. Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento 9, 43–50 (1932). [CrossRef]
8. O. Giraud, P. Braun, and D. Braun, “Classicality of spin states,” Phys. Rev. A 78, 042112 (2008). [CrossRef]
14. A. R. Usha Devi, Sudha, and A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process 11, 685–710 (2012). [CrossRef]
15. T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano, “Operational families of entanglement classes for symmetric N-qubit states,” Phys. Rev. Lett. 103, 070503 (2009). [CrossRef] [PubMed]
16. M. Aulbach, D. Markham, and M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010). [CrossRef]
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). [CrossRef]
18. O. Giraud, P. Braun, and D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010). [CrossRef]