Abstract

Propagating quadrature squeezed light through a multiple scattering random medium is found to induce pronounced spatial quantum correlations that have no classical analogue. The correlations are revealed in the number of photons transported through the sample that can be measured from the intensity fluctuations of the total transmission or reflection. In contrast, no pronounced spatial quantum correlations appear in the quadrature amplitudes where excess noise above the shot noise level is found.

©2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
    [Crossref] [PubMed]
  2. A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
    [Crossref]
  3. D. Levitz, L. Thrane, M. Frosz, P. Andersen, C. Andersen, S. Andersson-Engels, J. Valanciunaite, J. Swartling, and P. Hansen, “Determination of optical scattering properties of highly-scattering media in optical coherence tomography images,” Opt. Express 12, 249–259 (2004)
    [Crossref] [PubMed]
  4. D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
    [Crossref]
  5. A.A. Chabanov, M. Stoytchev, and A.Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
    [Crossref] [PubMed]
  6. M.C.W. van Rossum and Th.M. Nieuwenhuizen, “Multiple scattering af classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
    [Crossref]
  7. Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
    [Crossref]
  8. M. Patra and C.W.J. Beenakker, “Propagation of squeezed radiation through amplifying or absorbing random media,” Phys. Rev. A 61, 063805 1–8 (2000).
    [Crossref]
  9. P. Lodahl, A.P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901, 1–4 (2005).
    [Crossref]
  10. P. Lodahl, “Quantum noise frequency correlations of multiply scattered light,” Opt. Lett. 31, 110–112 (2006).
    [Crossref] [PubMed]
  11. J. Tworzydlo and C.W.J. Beenakker, “Quantum optical communication rates through an amplifying random medium,” Phys. Rev. Lett. 89, 043902, 1–4 (2002).
    [Crossref]
  12. P. Lodahl and A. Lagendijk, “Transport of quantum noise through random media,” Phys. Rev. Lett. 94, 153905, 1–4 (2005).
    [Crossref]
  13. D.F. Walls and G.J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
  14. F.J.P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous galium phosphide networks,” Science 284, 141–143 (1999).
    [Crossref] [PubMed]
  15. A.A. Chabanov and A.Z. Genack, “Field distribution in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
    [Crossref]
  16. J.W. Goodman, Statistical Optics (John Wiley & Sons, New York, 1985).
  17. R. Loudon, The Quantum Theory of Light (Oxford University Press, New York, 2000)
  18. J.F. de Boer, Optical Fluctuations on the Transmission and Reflection of Mesoscopic Systems, (Ph.D. Thesis, University of Amsterdam, 1995), available on: http://cops.tnw.utwente.nl/pdf/theses/deboer.pdf.
  19. R. Berkovits and S. Feng, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
    [Crossref]

2006 (1)

2005 (2)

P. Lodahl, A.P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901, 1–4 (2005).
[Crossref]

P. Lodahl and A. Lagendijk, “Transport of quantum noise through random media,” Phys. Rev. Lett. 94, 153905, 1–4 (2005).
[Crossref]

2004 (1)

2003 (1)

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

2002 (2)

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

J. Tworzydlo and C.W.J. Beenakker, “Quantum optical communication rates through an amplifying random medium,” Phys. Rev. Lett. 89, 043902, 1–4 (2002).
[Crossref]

2000 (3)

M. Patra and C.W.J. Beenakker, “Propagation of squeezed radiation through amplifying or absorbing random media,” Phys. Rev. A 61, 063805 1–8 (2000).
[Crossref]

A.A. Chabanov, M. Stoytchev, and A.Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref] [PubMed]

A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
[Crossref] [PubMed]

1999 (2)

M.C.W. van Rossum and Th.M. Nieuwenhuizen, “Multiple scattering af classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

F.J.P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous galium phosphide networks,” Science 284, 141–143 (1999).
[Crossref] [PubMed]

1997 (2)

A.A. Chabanov and A.Z. Genack, “Field distribution in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[Crossref]

D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

1994 (1)

R. Berkovits and S. Feng, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
[Crossref]

Andersen, C.

Andersen, P.

Andersson-Engels, S.

Balents, L.

A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
[Crossref] [PubMed]

Baranger, H.U.

A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
[Crossref] [PubMed]

Bartolini, P.

D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

Beenakker, C.W.J.

J. Tworzydlo and C.W.J. Beenakker, “Quantum optical communication rates through an amplifying random medium,” Phys. Rev. Lett. 89, 043902, 1–4 (2002).
[Crossref]

M. Patra and C.W.J. Beenakker, “Propagation of squeezed radiation through amplifying or absorbing random media,” Phys. Rev. A 61, 063805 1–8 (2000).
[Crossref]

Berkovits, R.

R. Berkovits and S. Feng, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
[Crossref]

Bernard, J.C.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Bidel, Y.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Boer, J.F. de

J.F. de Boer, Optical Fluctuations on the Transmission and Reflection of Mesoscopic Systems, (Ph.D. Thesis, University of Amsterdam, 1995), available on: http://cops.tnw.utwente.nl/pdf/theses/deboer.pdf.

Chabanov, A.A.

A.A. Chabanov, M. Stoytchev, and A.Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref] [PubMed]

A.A. Chabanov and A.Z. Genack, “Field distribution in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[Crossref]

Delande, D.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Derode, A.

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

Feng, S.

R. Berkovits and S. Feng, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
[Crossref]

Fink, M.

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

Frosz, M.

Genack, A.Z.

A.A. Chabanov, M. Stoytchev, and A.Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref] [PubMed]

A.A. Chabanov and A.Z. Genack, “Field distribution in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[Crossref]

Goodman, J.W.

J.W. Goodman, Statistical Optics (John Wiley & Sons, New York, 1985).

Hansen, P.

Kaiser, R.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Klappauf, B.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Labeyrie, G.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Lagemaat, J. van de

F.J.P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous galium phosphide networks,” Science 284, 141–143 (1999).
[Crossref] [PubMed]

Lagendijk, A.

P. Lodahl, A.P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901, 1–4 (2005).
[Crossref]

P. Lodahl and A. Lagendijk, “Transport of quantum noise through random media,” Phys. Rev. Lett. 94, 153905, 1–4 (2005).
[Crossref]

F.J.P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous galium phosphide networks,” Science 284, 141–143 (1999).
[Crossref] [PubMed]

D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

Levitz, D.

Lodahl, P.

P. Lodahl, “Quantum noise frequency correlations of multiply scattered light,” Opt. Lett. 31, 110–112 (2006).
[Crossref] [PubMed]

P. Lodahl, A.P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901, 1–4 (2005).
[Crossref]

P. Lodahl and A. Lagendijk, “Transport of quantum noise through random media,” Phys. Rev. Lett. 94, 153905, 1–4 (2005).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University Press, New York, 2000)

Milburn, G.J.

D.F. Walls and G.J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

Miniatura, C.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Mosk, A.P.

P. Lodahl, A.P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901, 1–4 (2005).
[Crossref]

Moustakas, A.L.

A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
[Crossref] [PubMed]

Nieuwenhuizen, Th.M.

M.C.W. van Rossum and Th.M. Nieuwenhuizen, “Multiple scattering af classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

Patra, M.

M. Patra and C.W.J. Beenakker, “Propagation of squeezed radiation through amplifying or absorbing random media,” Phys. Rev. A 61, 063805 1–8 (2000).
[Crossref]

Righini, R.

D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

Rosny, J. de

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

Rossum, M.C.W. van

M.C.W. van Rossum and Th.M. Nieuwenhuizen, “Multiple scattering af classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

Schuurmans, F.J.P.

F.J.P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous galium phosphide networks,” Science 284, 141–143 (1999).
[Crossref] [PubMed]

Sengupta, A.M.

A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
[Crossref] [PubMed]

Simon, S.H.

A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
[Crossref] [PubMed]

Stoytchev, M.

A.A. Chabanov, M. Stoytchev, and A.Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref] [PubMed]

Swartling, J.

Tanter, M.

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

Thrane, L.

Tourin, A.

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

Tworzydlo, J.

J. Tworzydlo and C.W.J. Beenakker, “Quantum optical communication rates through an amplifying random medium,” Phys. Rev. Lett. 89, 043902, 1–4 (2002).
[Crossref]

Valanciunaite, J.

Vanmaekelbergh, D.

F.J.P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous galium phosphide networks,” Science 284, 141–143 (1999).
[Crossref] [PubMed]

Walls, D.F.

D.F. Walls and G.J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

Wiersma, D.S.

D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

Wilkowski1, D.

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

Yon, S.

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

Nature (2)

D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

A.A. Chabanov, M. Stoytchev, and A.Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rep. (1)

R. Berkovits and S. Feng, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
[Crossref]

Phys. Rev. A (1)

M. Patra and C.W.J. Beenakker, “Propagation of squeezed radiation through amplifying or absorbing random media,” Phys. Rev. A 61, 063805 1–8 (2000).
[Crossref]

Phys. Rev. E (1)

A.A. Chabanov and A.Z. Genack, “Field distribution in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[Crossref]

Phys. Rev. Lett. (5)

J. Tworzydlo and C.W.J. Beenakker, “Quantum optical communication rates through an amplifying random medium,” Phys. Rev. Lett. 89, 043902, 1–4 (2002).
[Crossref]

P. Lodahl and A. Lagendijk, “Transport of quantum noise through random media,” Phys. Rev. Lett. 94, 153905, 1–4 (2005).
[Crossref]

P. Lodahl, A.P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901, 1–4 (2005).
[Crossref]

Y. Bidel, B. Klappauf, J.C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski1, and R. Kaiser, “Coherent light transport in a cold strontium cloud”, Phys. Rev. Lett. 88, 203902, 1–4 (2002).
[Crossref]

A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. 90, 014301, 1–4 (2003).
[Crossref]

Rev. Mod. Phys. (1)

M.C.W. van Rossum and Th.M. Nieuwenhuizen, “Multiple scattering af classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

Science (2)

A.L. Moustakas, H.U. Baranger, L. Balents, A.M. Sengupta, and S.H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science 287, 287–290 (2000).
[Crossref] [PubMed]

F.J.P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous galium phosphide networks,” Science 284, 141–143 (1999).
[Crossref] [PubMed]

Other (4)

D.F. Walls and G.J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

J.W. Goodman, Statistical Optics (John Wiley & Sons, New York, 1985).

R. Loudon, The Quantum Theory of Light (Oxford University Press, New York, 2000)

J.F. de Boer, Optical Fluctuations on the Transmission and Reflection of Mesoscopic Systems, (Ph.D. Thesis, University of Amsterdam, 1995), available on: http://cops.tnw.utwente.nl/pdf/theses/deboer.pdf.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Illustration of quadrature noise measurements of light transmitted through a multiple scattering medium. A squeezed state of light is injected through input channel a 0 while vacuum fluctuations contribute to all other open channels (indicated by dotted arrows). The quadratures amplitudes of the light transmitted to channel b can be measured by phase sensitive homodyning with a strong classical local oscillator (LO). By scanning the LO phase either the x or p quadrature can be obtained by subtracting the two photo currents, and their fluctuations can be recorded with a spectrum analyzer. The fluctuations in the photon number is obtained by blocking the LO and adding the two photocurrents.

Fig. 2.
Fig. 2.

Excess fluctuations above the shot noise level defined as Δ x b 2 ¯ ( Δ x b 2 ¯ ) SN 1 as a function of the squeezing parameter s. The excess fluctuations increase when Cl/NL and the squeezing parameter s are increased.

Fig. 3.
Fig. 3.

Total transmission of noise as a function of the squeezing parameter s for quadrature amplitude-squeezed light and ℓ/L = 0.2.

Fig. 4.
Fig. 4.

Total transmission of noise as a function of the squeezing parameter s for quadrature phase-squeezed light and ℓ/L = 0.2.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

a ̂ b = a t a b a ̂ a in + b′ t b′ b a ̂ b′ in .
x ̂ s = x + e s x ̂ ν ,
p ̂ s = p + e s p ̂ ν ,
x ̂ b = a T a b [ cos ϕ a b x ̂ a in sin ϕ a b p ̂ a in ] + b R b′ b [ cos ϕ b b x ̂ b in sin ϕ b b p ̂ b in ] ,
p ̂ b = a T a b [ cos ϕ a b p ̂ a in sin ϕ a b x ̂ a in ] + b R b b [ cos ϕ b b p ̂ b in sin ϕ b b x ̂ b in ] .
a T a b i T a b j [ cos ϕ a b i cos ϕ a b j + sin ϕ a b i sin ϕ a b j ] + b R b b i R b b j [ cos ϕ b b i cos ϕ b b j + sin ϕ b b i sin ϕ b b j ] = δ b i , b j
a T a b 0 T a b 1 [ sin ϕ a b 0 cos ϕ a b 1 cos ϕ a b 0 sin ϕ a b 1 ] + b R b b 0 R b b 1 [ sin ϕ b b 0 cos ϕ b b 1 + cos ϕ b b 0 sin ϕ b b 1 ] = 0
n ̂ b = a ̂ b a ̂ b = a , a t a b * t a b a ̂ a a ̂ a + b , b r b b * r b b a ̂ b a ̂ b + a , b [ t a b * r b b a ̂ a a ̂ b + h . c . ] ,
x ̂ b = T a 0 b [ cos ϕ a 0 b x ̂ a 0 in sin ϕ a 0 b p ̂ a 0 in ] ,
p ̂ b = T a 0 b [ cos ϕ a 0 b p ̂ a 0 in sin ϕ a 0 b x ̂ a 0 in ] .
x ̂ b i a ̂ b j = a T a b i T a b j [ cos ϕ a b i cos ϕ a b j ( x ̂ a in ) 2 cos ϕ a b i sin ϕ a b j x ̂ a in p ̂ a in sin ϕ a b i cos ϕ a b j p ̂ a in x ̂ a in + sin ϕ a b i sin ϕ a b j ( p ̂ a in ) 2 ]
+ b R b b i R b b j [ cos ϕ b b i cos ϕ b b j ( x ̂ b in ) 2 cos ϕ b b i sin ϕ b b j x ̂ b in p ̂ b in sin ϕ b b i cos ϕ b b j p ̂ b in x ̂ b in + sin ϕ b b i sin ϕ b b j ( p ̂ b in ) 2 ]
p ̂ b i p ̂ b j = a T a b i T a b j [ cos ϕ a b i cos ϕ a b j ( p ̂ a in ) 2 + cos ϕ a b i sin ϕ a b j p ̂ a in x ̂ a in + sin ϕ a b i cos ϕ a b j x ̂ a in p ̂ a in + sin ϕ a b i sin ϕ a b j ( x ̂ a in ) 2 ]
+ b R b b i R b b j [ cos ϕ b b i cos ϕ b b j ( p ̂ b in ) 2 + cos ϕ b b i sin ϕ b b j p ̂ b in x ̂ b in + sin ϕ b b i cos ϕ b b j x ̂ b in p ̂ b in + sin ϕ b b i sin ϕ b b j ( x ̂ b in ) 2 ] .
x ̂ b in p ̂ b in = x ̂ a in p ̂ a in = ( i ) ( a ̂ a in + ( a ̂ a in ) ) ( a ̂ a in ( a ̂ a in ) ) = i [ a ̂ a in , ( a ̂ a in ) ] = i ,
x b i , b j = δ b i , b j + T a 0 b i T a 0 b j [ cos ϕ a 0 b i cos ϕ a 0 b j ( e 2 s 1 ) + sin ϕ a 0 b i sin ϕ a 0 b j ( e 2 s 1 ) ] ,
p b i , b j = δ b i , b j + T a 0 b i T a 0 b j [ sin ϕ a 0 b j sin ϕ a 0 b j ( e 2 s 1 ) + cos ϕ a 0 b j cos ϕ a 0 b j ( e 2 s 1 ) ] ,
x b , b Δ x b 2 = 1 + T a 0 b ( cos 2 ϕ a 0 b e 2 s + sin 2 ϕ a 0 b e 2 s 1 ) ,
p b , b Δ p b 2 = 1 + T a 0 b ( sin 2 ϕ a 0 b e 2 s + cos 2 ϕ a 0 b e 2 s 1 ) .
Δ x b 2 ¯ = Δ p b 2 ¯ = 1 + T a 0 b ¯ ( cosh 2 s 1 ) ,
x b i , b j ¯ = p b i , b j ¯ = 0 ,
Δ x b 2 ¯ ( Δ x b 2 ¯ ) SN = 1 + N × L ( cosh 2 s 1 ) .
n ̂ b = a ̂ b a ̂ b = a , a t a b * t a b a ̂ a a ̂ a + b , b r b b * r b b a ̂ b a ̂ b + a , b [ t a b * r b b a ̂ a a ̂ b + h . c . ] ,
n ̂ b = T a 0 b n ̂ a 0 in ,
n b i , b j = [ T a 0 b i δ b i , b j + T a 0 b i δ b i , b j ( F a 0 in 1 ) n ̂ n 0 in , ]
Δ n b 2 = [ T a 0 b + T a 0 b 2 ( F a 0 in 1 ) ] n ̂ n 0 in ,
n b i , b j = T a 0 b i T b 0 b j ( F a 0 in 1 ) n ̂ n 0 in ,
Δ n T 2 = b Δ n b 2 + bi b j b i n b i , b j .
S T = ( Δ n T 2 ¯ ) F a 0 ( Δ n T 2 ¯ ) F a 0 = 1 = b i [ T a 0 b i ¯ + T a 0 b i 2 ¯ ( F a 0 in 1 ) ] + b i b j b i T a 0 b i T a 0 b j ¯ ( F a 0 in 1 ) b i T a 0 b i ¯ ,
S T = 1 + L ( F a 0 in 1 ) .
S R = 1 + ( 1 L ) ( F a 0 in 1 ) .
F a 0 in = p 2 e 2 s + x 2 e 2 s + cosh 4 s 1 x 2 + p 2 + 2 ( cosh 2 s 1 ) .

Metrics