Abstract

Assuming two quantum states of spontaneous parametric downconversion carrying orbital angular momentum, one may ask the question what is the minimum probability of error in identifying between two of these biphoton states by an arbitrary physical measurement over the biphoton state generated. While correctly chosen geometries may lead to perfect distinguishability of modes, it is worth noticing that experimental subtleties may lead to a poor mode distinguishability. We discuss the case where a restricted range instead of the needed range of wave vectors is collected by the experimental setup. These considerations may be useful for some applications, e.g., cryptography.

© 2008 Optical Society of America

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References

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  1. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  4. G. A. Barbosa, Phys. Rev. A 76, 033821 (2007).
    [CrossRef]
  5. H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett. 85, 286 (2000).
    [CrossRef] [PubMed]
  6. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  7. C. W. Helstrom, Quantum Detection and Estimation Theory, R.Bellman, ed. (Academic, 1976), p. 113, Eq. (2.34).

2007 (1)

G. A. Barbosa, Phys. Rev. A 76, 033821 (2007).
[CrossRef]

2004 (1)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

2000 (1)

H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett. 85, 286 (2000).
[CrossRef] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Arnaut, H. H.

H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett. 85, 286 (2000).
[CrossRef] [PubMed]

Barbosa, G. A.

G. A. Barbosa, Phys. Rev. A 76, 033821 (2007).
[CrossRef]

H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett. 85, 286 (2000).
[CrossRef] [PubMed]

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Courtial, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory, R.Bellman, ed. (Academic, 1976), p. 113, Eq. (2.34).

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Padgett, M. J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

G. A. Barbosa, Phys. Rev. A 76, 033821 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett. 85, 286 (2000).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

C. W. Helstrom, Quantum Detection and Estimation Theory, R.Bellman, ed. (Academic, 1976), p. 113, Eq. (2.34).

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Figures (2)

Fig. 1
Fig. 1

Cryptographic setup for quantum key distribution with OAM. One of the conjugate photons from SPDC generated with a pump mode with OAM l is sent from A to the user B. User A randomly detects a photon with OAM l A ; the conjugate photon is sent to B. A Dove sorter diverts the incoming photon to the detector port l B . Over a public channel, A informs B the obtained value of l A . B obtains l = l A + l B that, in principle, is also known by A; l becomes a shared key in the OAM alphabet with M elements.

Fig. 2
Fig. 2

P e in binary identification of Laguerre wave states with total biphoton angular momenta l 1 and l 2 under poor collecting conditions.

Equations (11)

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E ̂ ( + ) ( r , t ) = k l E ( k ) e k a ̂ k e i ( k r ω k t ) ,
E ̂ ( + ) ( r , t ) = l [ k U l , k * l E ( k ) e k a ̂ k ] × [ k U k , l e i ( k r ω k t ) ] l c ̂ l U l ( r , t ) .
ψ ( t ) = 0 + σ , σ d 3 k d 3 k A k , σ ; k , σ l E ( ) ( ω k ) l E ( ) ( ω k ) × T ( Δ ω ) ψ ̃ l p ( Δ k ) a ̂ ( k , σ ) a ̂ ( k , σ ) 0 ,
ψ l p ( ρ , ϕ , z ) = A l p 1 + ( z z R ) 2 [ ρ 2 w ( z ) ] l L l p [ 2 ρ 2 w ( z ) 2 ] exp [ ρ 2 w ( z ) 2 ] exp [ i ( k ρ 2 z 2 ( z 2 + z R 2 ) + l arctan ( y x ) ) ] × exp [ i ( 2 p + l + 1 ) arctan ( z z R ) ] .
ψ ̃ l p ( Δ k ) = π A l p ( i 2 ) l ( z R k P ) 1 + ( l 2 ) e i l ( π 2 ) × ρ k l L l p [ ( z R k P ) ρ k 2 ] e ( z R 2 k P ) ρ k 2 e i l arctan ( Δ k y Δ k x ) × z 0 l c 2 z 0 + l c 2 e i Δ k z z e ( 1 + l + 2 p ) arctan ( z R z ) × e i ( 1 + l + 2 p ) arctan ( z ( 2 z R ) ) d z .
ψ ̃ l p ( Δ k ) = π A l p ( 1 ) i ( i 2 ) l ( l c z R k P ) e i ( p π z 0 Δ k z ) × e i l arctan ( Δ k y Δ k x ) e ξ 2 ξ l 2 L l p ( ξ ) sin ( l c Δ k z 2 ) ( l c Δ k z 2 ) .
ψ ( t ) = sin θ p m k p m 2 sin θ p m k p m 2 A k p m ; k p m l E ( ) ( ω k p m ) l E ( ) ( ω k p m ) ψ ̃ l 0 ( Δ k p m ) a ̂ l l ( k p m ) a ̂ l ( k p m ) 0 .
ψ ̃ l p ( Δ k ) = [ π 2 l i l + 1 l c A l p z R k P e i l arctan ( Δ k y Δ k x ) ] × e ξ 2 ξ l 2 L l p ( ξ ) 1 p m 1 p m g l p ( Δ k x , Δ k y , 0 ) e ξ 2 ξ l 2 L l p ( ξ ) 1 p m 1 p m .
ψ ̃ l p ( Δ k ) n = ψ ̃ l p ( ξ ) n = i ( 1 ) l e ξ 2 ξ l 2 L l p ( ξ ) 0 e ξ ξ l L l p ( ξ ) 2 d ξ .
ψ l j p j ψ l k p k 0 ψ ̃ l j p j ( ξ ) n ψ ̃ l k p k ( ξ ) n d ξ .
P e = 1 2 ( 1 1 4 p 0 p 1 ψ 0 ψ 1 2 ) .

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