Abstract

We show that appropriate nonclassical photon states can attain Heisenberg-limited sensitivity in polarimetric measurements, in which the squared amplitudes of the photon states are measured. These photon states, which can be constructed from the two-mode Fock states, are found as the eigenstates of a Hermitian operator, which is closely related to the quantum Stokes operators. The algebraic property of the Hermitian operator governs the interferometric behavior in polarimetric interactions and provides the frequency of the interferometric fringes scaled by the total number of photons of the state. Thus, the polarimetry is able to achieve Heisenberg-limited sensitivity.

© 2007 Optical Society of America

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  1. C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
    [CrossRef]
  2. B. Yurke, S. L. McCall, and J. R. Klauder, "SU(2) and SU(1,1) interferometers," Phys. Rev. A 33, 4033-4054 (1986).
    [CrossRef] [PubMed]
  3. B. Yurke, "Input states for enhancement of Fermion interferometer sensitivity," Phys. Rev. Lett. 56, 1515-1517 (1986).
    [CrossRef] [PubMed]
  4. 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]
  5. C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
    [CrossRef] [PubMed]
  6. A. Kuzmich and L. Mandel, "Sub-shot-noise interferometric measurements with two-photon states," Quantum Semiclassic. Opt. 10, 493-500 (1998).
    [CrossRef]
  7. A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, D. N. Klyshko, and S. P. Kulik, "Polarization state of a biphoton: quantum ternary logic," Phys. Rev. A 60, R4209-R4212 (1999).
    [CrossRef]
  8. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, "Optimal frequency measurements with maximally correlated states," Phys. Rev. A 54, R4649-R4652 (1996).
    [CrossRef] [PubMed]
  9. Z. Y. Ou, J.-K. Rhee, and L. J. Wang, "Observation of four-photon interference with a beam splitter by pulsed parametric down-conversion," Phys. Rev. Lett. 83, 959-962 (1999).
    [CrossRef]
  10. R. A. Campos, C. C. Gerry, and A. Benmoussa, "Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements," Phys. Rev. A 68, 023810 (2003).
    [CrossRef]
  11. T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutuary orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
    [CrossRef] [PubMed]
  12. E. Collett, "Stokes parameters for quantum systems," Am. J. Phys. 38, 563-574 (1970).
    [CrossRef]
  13. A. Luis and L. L. Sánchez-Soto, "Phase-difference operator," Phys. Rev. A 48, 4702-4708 (1993).
    [CrossRef] [PubMed]
  14. N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, "Polarization squeezing and continuous-variable polarization entanglement," Phys. Rev. A 65, 052306 (2002).
    [CrossRef]
  15. In case of birefringence, the operator Â2 can also be expressed by a real matrix. The operator Â=−i(âxdaggerây+âydaggerâx) has pure imaginary elements in the matrix representation. The matrix representation of  is then written by the form, i× (a real matrix). This shows that all the elements of Â2 are real.
  16. In case of birefringence, all elements of the vector Âe are pure imaginary. Hence in the following discussion, the notation Âe should read Âe/i.
  17. P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
    [CrossRef]
  18. H. Mikami, Y. Li, and T. Kobayashi, "Generation of the four-photon W state and other multiphoton entangled states using parametric down-conversion," Phys. Rev. A 70, 052308 (2004).
    [CrossRef]

2004

H. Mikami, Y. Li, and T. Kobayashi, "Generation of the four-photon W state and other multiphoton entangled states using parametric down-conversion," Phys. Rev. A 70, 052308 (2004).
[CrossRef]

2003

R. A. Campos, C. C. Gerry, and A. Benmoussa, "Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements," Phys. Rev. A 68, 023810 (2003).
[CrossRef]

2002

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, "Polarization squeezing and continuous-variable polarization entanglement," Phys. Rev. A 65, 052306 (2002).
[CrossRef]

2001

P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
[CrossRef]

2000

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutuary orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
[CrossRef] [PubMed]

1999

Z. Y. Ou, J.-K. Rhee, and L. J. Wang, "Observation of four-photon interference with a beam splitter by pulsed parametric down-conversion," Phys. Rev. Lett. 83, 959-962 (1999).
[CrossRef]

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, D. N. Klyshko, and S. P. Kulik, "Polarization state of a biphoton: quantum ternary logic," Phys. Rev. A 60, R4209-R4212 (1999).
[CrossRef]

1998

A. Kuzmich and L. Mandel, "Sub-shot-noise interferometric measurements with two-photon states," Quantum Semiclassic. Opt. 10, 493-500 (1998).
[CrossRef]

1996

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, "Optimal frequency measurements with maximally correlated states," Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

1993

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]

A. Luis and L. L. Sánchez-Soto, "Phase-difference operator," Phys. Rev. A 48, 4702-4708 (1993).
[CrossRef] [PubMed]

1987

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

1986

B. Yurke, S. L. McCall, and J. R. Klauder, "SU(2) and SU(1,1) interferometers," Phys. Rev. A 33, 4033-4054 (1986).
[CrossRef] [PubMed]

B. Yurke, "Input states for enhancement of Fermion interferometer sensitivity," Phys. Rev. Lett. 56, 1515-1517 (1986).
[CrossRef] [PubMed]

1981

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

1970

E. Collett, "Stokes parameters for quantum systems," Am. J. Phys. 38, 563-574 (1970).
[CrossRef]

Am. J. Phys.

E. Collett, "Stokes parameters for quantum systems," Am. J. Phys. 38, 563-574 (1970).
[CrossRef]

Opt. Commun.

P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
[CrossRef]

Phys. Rev. A

H. Mikami, Y. Li, and T. Kobayashi, "Generation of the four-photon W state and other multiphoton entangled states using parametric down-conversion," Phys. Rev. A 70, 052308 (2004).
[CrossRef]

A. Luis and L. L. Sánchez-Soto, "Phase-difference operator," Phys. Rev. A 48, 4702-4708 (1993).
[CrossRef] [PubMed]

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, "Polarization squeezing and continuous-variable polarization entanglement," Phys. Rev. A 65, 052306 (2002).
[CrossRef]

B. Yurke, S. L. McCall, and J. R. Klauder, "SU(2) and SU(1,1) interferometers," Phys. Rev. A 33, 4033-4054 (1986).
[CrossRef] [PubMed]

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, D. N. Klyshko, and S. P. Kulik, "Polarization state of a biphoton: quantum ternary logic," Phys. Rev. A 60, R4209-R4212 (1999).
[CrossRef]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, "Optimal frequency measurements with maximally correlated states," Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

R. A. Campos, C. C. Gerry, and A. Benmoussa, "Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements," Phys. Rev. A 68, 023810 (2003).
[CrossRef]

Phys. Rev. D

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

Phys. Rev. Lett.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutuary orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
[CrossRef] [PubMed]

Z. Y. Ou, J.-K. Rhee, and L. J. Wang, "Observation of four-photon interference with a beam splitter by pulsed parametric down-conversion," Phys. Rev. Lett. 83, 959-962 (1999).
[CrossRef]

B. Yurke, "Input states for enhancement of Fermion interferometer sensitivity," Phys. Rev. Lett. 56, 1515-1517 (1986).
[CrossRef] [PubMed]

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]

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt.

A. Kuzmich and L. Mandel, "Sub-shot-noise interferometric measurements with two-photon states," Quantum Semiclassic. Opt. 10, 493-500 (1998).
[CrossRef]

Other

In case of birefringence, the operator Â2 can also be expressed by a real matrix. The operator Â=−i(âxdaggerây+âydaggerâx) has pure imaginary elements in the matrix representation. The matrix representation of  is then written by the form, i× (a real matrix). This shows that all the elements of Â2 are real.

In case of birefringence, all elements of the vector Âe are pure imaginary. Hence in the following discussion, the notation Âe should read Âe/i.

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Figures (1)

Fig. 1
Fig. 1

Scheme for polarimetric measurement using nonclassical photon states.

Tables (1)

Tables Icon

Table 1 Eigenvalues and Eigenstates of A ̂ 2 for the First Several Photon Numbers

Equations (23)

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S ̂ 0 = a ̂ x a ̂ x + a ̂ y a ̂ y = n ̂ x + n ̂ y ,
S ̂ 1 = a ̂ x a ̂ x a ̂ y a ̂ y = n ̂ x n ̂ y ,
S ̂ 2 = a ̂ x a ̂ y + a ̂ y a ̂ x ,
S ̂ 3 = i ( a ̂ x a ̂ y a ̂ y a ̂ x ) ,
[ S ̂ i , S ̂ j ] = 2 i S ̂ k e i j k , i , j , k = 1 , 2 , 3 ,
S ̂ 2 S ̂ 1 2 + S ̂ 2 2 + S ̂ 3 2 = S ̂ 0 2 + 2 S ̂ 0 .
λ = ( e , A ̂ 2 e ) = ( A ̂ e , A ̂ e ) = ( A ̂ e , A ̂ e ) = A ̂ e 2 0 ,
e = A ̂ e ( λ ) 1 2 ,
e = A ̂ e ( λ ) 1 2 .
U ̂ 3 ( θ ) = exp ( θ A ̂ ) = k = 0 1 k ! ( θ ) k A ̂ k .
U ̂ 3 ( θ ) = k = 0 1 ( 2 k ) ! ( θ ) 2 k A ̂ 2 k + k = 0 1 ( 2 k + 1 ) ! ( θ ) 2 k + 1 A ̂ 2 k + 1 .
U ̂ 3 ( θ ) e = k = 0 1 ( 2 k ) ! ( θ ) 2 k A ̂ 2 k e + k = 0 1 ( 2 k + 1 ) ! ( θ ) 2 k + 1 A ̂ 2 k + 1 e = k = 0 1 ( 2 k ) ! ( θ ) 2 k λ k e + k = 0 1 ( 2 k + 1 ) ! ( θ ) 2 k + 1 λ k ( λ ) 1 2 e = cos [ ( λ ) 1 2 θ ] e + sin [ ( λ ) 1 2 θ ] e .
e 1 = ( 1 0 0 ) = 0 , N , e 2 = ( 0 1 0 ) = 1 , N 1 , , e N + 1 = ( 0 0 1 ) = N , 0 .
A i , j = m , N m A ̂ n , N n = [ ( n + 1 ) ( N n ) ] 1 2 δ m , n + 1 [ n ( N n + 1 ) ] 1 2 δ m , n 1 = [ j ( N j + 1 ) ] 1 2 δ i , j + 1 [ i ( N i + 1 ) ] 1 2 δ j , i + 1 ,
A i , j 2 = k = 1 N + 1 A i , k A k , j = [ j ( N j + 1 ) ( j + 1 ) ( N j ) ] 1 2 δ i , j + 2 + [ i ( N i + 1 ) ( i + 1 ) ( N i ) ] 1 2 δ j , i + 2 [ ( i 1 ) ( N i + 2 ) + i ( N i + 1 ) ] δ i , j .
A ̂ = [ 0 1 1 0 ] , A ̂ 2 = [ 1 0 0 1 ] .
A ̂ = [ 0 2 1 2 0 2 1 2 0 2 1 2 0 2 1 2 0 ] , A ̂ 2 = [ 2 0 2 0 4 0 2 0 2 ] .
1 2 1 2 ( 1 0 1 ) and ( 0 1 0 )
1 2 1 2 ( 1 0 1 )
A ̂ = [ 0 3 1 2 0 0 3 1 2 0 2 0 0 2 0 3 1 2 0 0 3 1 2 0 ] ,
A ̂ 2 = [ 3 0 2 × 3 1 2 0 0 7 0 2 × 3 1 2 2 × 3 1 2 0 7 0 0 2 × 3 1 2 0 3 ] .
1 2 ( 0 3 1 2 0 1 ) and 1 2 ( 1 0 3 1 2 0 )
1 2 ( 0 1 0 3 1 2 ) and 1 2 ( 3 1 2 0 1 0 )

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