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

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  1. M. Born, 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, 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, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
    [CrossRef]
  6. A. Luis, 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, D. Braun, “Classicality of spin states,” Phys. Rev. A 78, 042112 (2008).
    [CrossRef]
  9. M. J. Holland, K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
    [CrossRef] [PubMed]
  10. C. Brif, A. Mann, “Nonclassical interferometry with intelligent light,” Phys. Rev. A 54, 4505–4518 (1996).
    [CrossRef] [PubMed]
  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, 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, A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
    [CrossRef] [PubMed]
  14. A. R. Usha Devi, Sudha, 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, 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, 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, D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010).
    [CrossRef]

2012

A. R. Usha Devi, Sudha, A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process 11, 685–710 (2012).
[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]

2010

O. Giraud, P. Braun, D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010).
[CrossRef]

M. Aulbach, D. Markham, M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010).
[CrossRef]

2009

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]

2008

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

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

2004

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]

1996

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

1995

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

1993

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

1990

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

1972

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

1932

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

Arecchi, F. T.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Aspelmeyer, M.

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]

Aulbach, M.

M. Aulbach, D. Markham, M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010).
[CrossRef]

Bastin, T.

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]

Born, M.

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

Braun, D.

O. Giraud, P. Braun, D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010).
[CrossRef]

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

Braun, P.

O. Giraud, P. Braun, D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010).
[CrossRef]

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

Brif, C.

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

Brosseau, Ch.

Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Bruno, P.

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]

Burnett, K.

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

Courtens, E.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Gasparoni, S.

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]

Gilmore, R.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Giraud, O.

O. Giraud, P. Braun, D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010).
[CrossRef]

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

Godefroid, M.

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]

Holland, M. J.

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

Krins, S.

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]

Lamata, L.

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]

Luis, A.

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

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

Lundeen, J. S.

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]

Majorana, E.

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

Mann, A.

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

Markham, D.

M. Aulbach, D. Markham, M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010).
[CrossRef]

Mathonet, P.

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]

Mermin, N. D.

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

Mitchell, M. W.

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]

Murao, M.

M. Aulbach, D. Markham, M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010).
[CrossRef]

Pan, J.-W.

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]

Rajagopal, A. K.

A. R. Usha Devi, Sudha, A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process 11, 685–710 (2012).
[CrossRef]

Rivas, A.

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

Sánchez-Soto, L. L.

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

Schwinger, J.

J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).

Solano, E.

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]

Steinberg, A. M.

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]

Sudha,

A. R. Usha Devi, Sudha, A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process 11, 685–710 (2012).
[CrossRef]

Thomas, H.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Ursin, R.

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]

Usha Devi, A. R.

A. R. Usha Devi, Sudha, A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process 11, 685–710 (2012).
[CrossRef]

Walther, Ph.

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]

Wolf, E.

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

Zeilinger, A.

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]

Nature

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]

New J. Phys.

M. Aulbach, D. Markham, M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010).
[CrossRef]

O. Giraud, P. Braun, D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys. 12, 063005 (2010).
[CrossRef]

Nuovo Cimento

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

Phys. Rev. A

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

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

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]

Phys. Rev. Lett.

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

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

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]

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]

Quantum Inf. Process

A. R. Usha Devi, Sudha, A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process 11, 685–710 (2012).
[CrossRef]

Quantum Semiclass. Opt.

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

Other

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).

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Equations (35)

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E ( r , t ) k , ± a k , ± ε k , ± e i ( kr ω k t ) ,
S 0 = a + a + + a a , S z = a + a + a a , S x = a a + + a a + , S y = i ( a a + a a + ) ,
[ S x , S y ] = 2 i S z ,
S 2 = S 0 ( S 0 + 2 ) , [ S , S 0 ] = 0 ,
U = exp ( i θ u S / 2 ) ,
U S U = R S ,
| ψ n = 𝒩 m = 1 n a m | 0 , 0 ,
a m = cos θ m a + + e i ϕ m sin θ m a ,
| ε m = a m | 0 , 0 = cos θ m | 1 , 0 + e i ϕ m sin θ m | 0 , 1 ,
s z = cos ( 2 θ m ) , s x = sin ( 2 θ m ) cos ϕ m , s y = sin ( 2 θ m ) sin ϕ m , s 0 = 1 ,
ε m = cos θ m ε + + e i ϕ m sin θ m ε .
S n = m = 1 n S 1 , m ,
S z n = m = 1 n S z 1 , m , a + a n = m = 1 n a + a 1 , m ,
| ψ 2 = 𝒩 a α ( cos θ a α + e i ϕ sin θ a β ) | 0 , 0 ,
| ψ 2 = 1 1 + cos 2 θ ( 2 cos θ | 2 , 0 + e i ϕ sin θ | 1 , 1 ) ,
| ε 1 = | 1 , 0 , | ε 2 = cos θ | 1 , 0 + e i ϕ sin θ | 0 , 1 .
4 cos 2 θ 1 + cos 2 θ = 2 cos 2 θ , 2 e i ϕ cos θ sin θ 1 + cos 2 θ = e i ϕ cos θ sin θ .
| ψ n + 1 = 𝒩 ( cos θ a + + e i ϕ sin θ a ) | n + , n ,
| ψ n + 1 = 𝒩 ( n + + 1 cos θ | n + + 1 , n + n + 1 e i ϕ sin θ | n + , n + 1 ) ,
S z n + 1 = S z n + S z ε , a + a n + 1 = a + a n + a + a ε ,
( n + + 1 ) ( n + n + 1 ) cos 2 θ + ( n + 1 ) ( n + n 1 ) sin 2 θ 1 + n + cos 2 θ + n sin 2 θ = n + n + cos 2 θ sin 2 θ , ( n + + 1 ) ( n + 1 ) e i ϕ sin θ cos θ 1 + n + cos 2 θ + n sin 2 θ = e i ϕ sin θ cos θ .
M , m = 1 2 ( S S m + S m S ) S S m .
M = ( n + + n + 2 n + n ) ( 1 0 0 0 1 0 0 0 0 ) ,
| ψ n = 1 2 ( | n , 0 | 0 , n ) , n > 2 ,
M = ( n 0 0 0 n 0 0 0 n 2 ) ,
| ψ n m = 1 n ( a + e i 2 π m / n a ) | 0 , 0 .
| ψ n m = 1 n a m | 0 , 0 ,
| ψ n a n k m = 1 k ( a + ξ m a ) | 0 , 0 ,
α + , α | ψ n α * n m = 1 k ( x ξ j ) e | α + | 2 + | α | 2 2 , x = α + * α * .
| ψ n = m = 0 k c m | n + = m , n = n m ,
| n = 1 n ! a n | 0 ,
| ψ n = m = 0 k c m m ! ( n m ) ! ( a + ) m ( a ) n m | 0 , 0 .
α + , α | ψ n = m = 0 k c m m ! ( n m ) ! α + * m α * n m e | α + | 2 + | α | 2 2 ,
α + , α | ψ n = α * n P ( x ) e | α + | 2 + | α | 2 2 ,
P ( x ) = m = 0 k c m m ! ( n m ) ! x m , x = α + * α * .

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