## Polarization versus photon spin

Optics Express, Vol. 22, Issue 2, pp. 1569-1575 (2014)

http://dx.doi.org/10.1364/OE.22.001569

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### 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 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

## 1. Introduction

1. M. Born and E. Wolf, *Principles of Optics*, 7 (Cambridge University, 1999). [CrossRef]

*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

**k**we have where the complex two-dimensional vectors

*ε*

_{k}_{,±}with

*a*

_{k}_{,±}, the subscript ± representing circular polarization for example. Throughout we will consider a two-mode approach with a single

**k**, so no subscript

**k**will be necessary from now on.

*s*= 〈

_{j}*S*〉. The Stokes operators satisfy the commutation relations of an angular momentum and its cyclic permutations, with where

_{j}**S**= (

*S*,

_{x}*S*,

_{y}*S*). Thus there is a complete formal equivalence between the subspace

_{z}*ℋ*of fixed total photon number

_{n}*n*with an spin

*j*=

*n*/2. In particular, a single photon

*n*= 1 is equivalent to an spin 1/2. The Stokes operators are also the infinitesimal generators of SU(2) transformations [4

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]

*θ*a real parameter, and

**u**a unit three-dimensional real vector. These are all linear, energy preserving transformations of the field amplitudes, embracing very fundamental optical operations such as lossless beam splitters, phase plates, and all linear interferometers [6

6. A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. **7**, 153–160 (1995). [CrossRef]

*U*on

**S**is a rotation

*R*of angle

*θ*and axis

**u**[5

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]

*R*=

^{t}R*RR*=

^{t}*I*, the superscript

*t*denotes matrix transposition, and

*I*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.

*n*photons |

*ψ*〉 ∈

_{n}*ℋ*can be expressed as the result of the action on the vacuum of

_{n}*n*creation operators generating photons that in general will have many with different polarization states. This is where

*𝒩*is a normalization constant, |0, 0〉 is the two-mode vacuum state, and the complex amplitude operators

*a*are where

_{m}*θ*and

_{m}*ϕ*are independent parameters. In appendix A we show the close relation of expressions (7) and (8) with the Majorana representation of spins [7

_{m}7. E. Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento **9**, 43–50 (1932). [CrossRef]

*n*

_{+},

*n*

_{−}〉 are the photon-number states of modes

*a*

_{±}. These states |

*ε*〉 are actually SU(2) coherent states [5

_{m}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]

8. O. Giraud, P. Braun, and D. Braun, “Classicality of spin states,” Phys. Rev. A **78**, 042112 (2008). [CrossRef]

*𝒫*= |

**s**|/

*s*

_{0}= 1. Thus, we may say that

*ε*Note that in general the polarization states of the photons appearing in Eqs. (7) and (9) are not orthogonal,

_{m}*ℓ*≠

*m*.

## 3. Polarization versus single-photon spins

*n*-photon state are the sum of the Stokes parameters of the individual photons: or, equivalently, where 〈

**S**〉

*are the Stokes parameters in the*

_{n}*n*–photon state |

*ψ*〉 ∈

_{n}*ℋ*in Eq. (7), while 〈

_{n}**S**〉

_{1,}

*are the Stokes parameters (10) of the corresponding one-photon states |*

_{m}*ε*〉 in Eq. (9). In the transit from Eqs. (12) to (13) we have replaced a pair of real equations for

_{m}*S*and

_{x}*S*by a single complex equation for

_{y}*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

*n*= 2.

#### 3.1.1. Two photons

*a*are two arbitrary field modes with orthogonal polarizations

_{α}_{,}_{β}*n*,

*m*〉 the associated photon-number basis. On the other hand, the corresponding single-photon states are, in the same photon-number basis, After replacing

*a*

_{+,−}by

*a*the equalities (13) read, It can be easily seen that these equalities are satisfied only when

_{α}_{,}_{β}*θ*= 0,

*π*/2 modulus

*π*, for any

*ϕ*. This means that the photons must have either the same polarization state,

*ε*

_{1}=

*ε*

_{2}=

*ε*for

_{α}*θ*= 0, or orthogonal polarization states for

*θ*=

*π*/2,

*ε*

_{1}=

*ε*,

_{α}*ε*

_{2}=

*ε*. In other words, Eq. (12) holds just for the SU(2) orbits of the number states |2, 0〉 and |1, 1〉.

_{β}#### 3.1.2. *n* + 1 photons

*ψ*〉 with

_{n}*n*photons, this is |

*ψ*〉 = |

_{n}*n*

_{+},

*n*

_{−}〉, modulus SU(2) transformations. Then we add another photon in an arbitrary polarization state this is where

*n*+ 1,

*n*, and

*ε*refer to the states |

*ψ*

_{n}_{+1}〉, |

*ψ*〉, and |

_{n}*ε*〉 = cos

*θ*|1, 0〉 +

*e*sin

^{iϕ}*θ*|0, 1〉, respectively. An straightforward calculation implies that Eqs. (20) are equivalent to

*θ*= 0,

*π*/2 modulus

*π*, for any

*ϕ*, so that either |

*ψ*

_{n}_{+1}〉 = |

*n*

_{+}+ 1,

*n*

_{−}〉, or |

*ψ*

_{n}_{+1}〉 = |

*n*

_{+},

*n*

_{−}+ 1〉.

### 3.2. Sum property and covariance matrix

*M*= 0, where

*M*is the covariance matrix of Stokes-operators [4

4. A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A **77**, 022105 (2008). [CrossRef]

*M*are the variances of the Stokes operators

**S**, while the variance of any other Stokes component

*S*=

_{u}**u**·

**S**is computed as (Δ

*S*)

_{u}^{2}=

**u**

^{t}M**u**, where

**u**is any unit three-dimensional real vector.

*S*, which are |

_{z}*n*

_{+},

*n*

_{−}〉. The number states |

*n*

_{+},

*n*

_{−}〉 have the covariance matrix with det

*M*= 0. Under SU(2) transformations (5) we have

*M*→

*R*, so that the determinant is preserved det(

^{t}MR*R*) = 0. Thus, if the state satisfies Eq. (12) then det

^{t}MR*M*= 0.

### 3.3. Sum property and entanglement

*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.

*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.

### 3.4. Examples

#### 3.4.1. SU(2) coherent states

*n*, 0〉 [5

**6**, 2211–2237 (1972). [CrossRef]

8. O. Giraud, P. Braun, and D. Braun, “Classicality of spin states,” Phys. Rev. A **78**, 042112 (2008). [CrossRef]

#### 3.4.2. Twin-number states

*U*|

*n*,

*n*〉 [9

9. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. **71**, 1355–1358 (1993). [CrossRef] [PubMed]

10. C. Brif and A. Mann, “Nonclassical interferometry with intelligent light,” Phys. Rev. A **54**, 4505–4518 (1996). [CrossRef] [PubMed]

#### 3.4.3. N00N states

11. N. D. Mermin, “Extreme quantum entanglement in a superposition of macroscopically distinct states,” Phys. Rev. Lett. **65**, 1838–1840 (1990). [CrossRef] [PubMed]

12. Ph. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature **429**, 158–161 (2004). [CrossRef] [PubMed]

13. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature **429**, 161–164 (2004). [CrossRef] [PubMed]

4. A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A **77**, 022105 (2008). [CrossRef]

*M*≠ 0 as expected. Following the program outlined in the Appendix the factorized form (7) of these states is This is to say that there are no two photons in the same polarization state. All them have the same

*θ*=

_{m}*π*/4 but different

*ϕ*= 2

_{m}*πm/n*.

## 4. Conclusions

## A. Multi-photon states as photon-added states

*n*-photon state |

*ψ*〉 ∈

_{n}*ℋ*can be expressed in the form (up to a normalization constant) where

_{n}*a*are in Eq. (8). This is equivalent to say that there are

_{m}*k*complex number

*ξ*such that where

_{m}*ξ*= −

_{m}*e*

^{iϕm}tan

*θ*, and we have singled out the potential

_{m}*n*−

*k*photons with cos

*θ*= 0, so that all the

_{m}*k*parameters

*ξ*are finite.

_{m}*ψ*〉 on the two-mode Glauber coherent states |

_{n}*α*

_{+},

*α*

_{−}〉, with

*a*

_{±}|

*α*

_{+},

*α*

_{−}〉 =

*α*

_{±}|

*α*

_{+},

*α*

_{−}〉, On the other hand, every |

*ψ*〉 can be expressed in the photon number basis as for suitable

_{n}*c*and

_{m}*k*. Using that each photon-number state |

*n*〉 can be expressed as the

*n*-times action on the vacuum state |0〉 of the corresponding creation operator we get Projecting from the left on Glauber coherent states |

*α*

_{+},

*α*

_{−}〉 we have so that after extracting a common factor

*n*-photon state 〈

*α*

_{+},

*α*

_{−}|

*ψ*〉 is a complex polynomial of the complex variable

_{n}*k*≤

*n*. Thus, comparing 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

*ξ*, maybe degenerate. Thus, the factorization in Eq. (27) always exists and is unique.

_{m}7. E. Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento **9**, 43–50 (1932). [CrossRef]

*ℋ*of

_{n}*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

*ξ*of 〈

_{m}*α*

_{+},

*α*

_{−}|

*ψ*〉 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

_{n}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]

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | Ch. Brosseau, |

3. | J. Schwinger, |

4. | A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A |

5. | F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A |

6. | A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. |

7. | E. Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento |

8. | O. Giraud, P. Braun, and D. Braun, “Classicality of spin states,” Phys. Rev. A |

9. | M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. |

10. | C. Brif and A. Mann, “Nonclassical interferometry with intelligent light,” Phys. Rev. A |

11. | N. D. Mermin, “Extreme quantum entanglement in a superposition of macroscopically distinct states,” Phys. Rev. Lett. |

12. | Ph. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature |

13. | M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature |

14. | A. R. Usha Devi, Sudha, and A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process |

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. |

16. | M. Aulbach, D. Markham, and M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. |

17. | P. Bruno, “Quantum geometric phase in Majorana’s stellar representation: mapping onto a many-body Aharonov-Bohm phase,” Phys. Rev. Lett. |

18. | O. Giraud, P. Braun, and D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. |

**OCIS Codes**

(260.5430) Physical optics : Polarization

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: October 28, 2013

Revised Manuscript: December 15, 2013

Manuscript Accepted: December 15, 2013

Published: January 15, 2014

**Citation**

Alfredo Luis and Alfonso Rodil, "Polarization versus photon spin," Opt. Express **22**, 1569-1575 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1569

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### References

- M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University, 1999). [CrossRef]
- Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
- J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).
- A. Rivas, A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008). [CrossRef]
- F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972). [CrossRef]
- A. Luis, L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995). [CrossRef]
- E. Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento 9, 43–50 (1932). [CrossRef]
- O. Giraud, P. Braun, D. Braun, “Classicality of spin states,” Phys. Rev. A 78, 042112 (2008). [CrossRef]
- M. J. Holland, K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993). [CrossRef] [PubMed]
- C. Brif, A. Mann, “Nonclassical interferometry with intelligent light,” Phys. Rev. A 54, 4505–4518 (1996). [CrossRef] [PubMed]
- N. D. Mermin, “Extreme quantum entanglement in a superposition of macroscopically distinct states,” Phys. Rev. Lett. 65, 1838–1840 (1990). [CrossRef] [PubMed]
- Ph. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004). [CrossRef] [PubMed]
- M. W. Mitchell, J. S. Lundeen, A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004). [CrossRef] [PubMed]
- A. R. Usha Devi, Sudha, A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process 11, 685–710 (2012). [CrossRef]
- T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, E. Solano, “Operational families of entanglement classes for symmetric N-qubit states,” Phys. Rev. Lett. 103, 070503 (2009). [CrossRef] [PubMed]
- M. Aulbach, D. Markham, M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010). [CrossRef]
- 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]
- O. Giraud, P. Braun, D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010). [CrossRef]

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