Abstract

Spontaneous parametric down conversion (SPDC) has been largely exploited as a tool for absolute calibration of photon-counting detectors, i.e detectors registering very small photon fluxes. In [J. Opt. Soc. Am. B 23, 2185 (2006)] we derived a method for absolute calibration of analog detectors using SPDC emission at higher photon fluxes, where the beam is seen as a continuum by the detector. Nevertheless intrinsic limitations appear when high-gain regime of SPDC is required to reach even larger photon fluxes. Here we show that stimulated parametric down conversion allow one to avoid this limitation, since stimulated photon fluxes are increased by the presence of the seed beam.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Bachor, A guide to experimental quantum optics (Wiley-VCH, New York, 1998).
  2. M. Genovese, "Research on hidden variable theories: A review on recent progresses," Phys. Rep.  413, 319-398 (2005) and references therein.
    [CrossRef]
  3. See, for a general review, N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2000), and references therein.
    [CrossRef]
  4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, "The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study," J. Opt. Soc. Am. B 23, 2185-2193 (2006).
    [CrossRef]
  5. B. Y. Zel�??dovich and D. N. Klyshko, "Statistics of field in parametric luminescence," Sov. Phys. JETP Lett. 9, 40-44 (1969).
  6. D. C. Burnham and D. L. Weinberg, "Observation of Simultaneity in Parametric Production of Optical Photon Pairs," Phys. Rev. Lett. 25, 84-87 (1970).
    [CrossRef]
  7. D. N. Klyshko, "Use of two-photon light for absolute calibration of photoelectric detectors," Sov. J. Quant. Elect. 10, 1112-1116 (1980).
    [CrossRef]
  8. A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).
  9. D. N. Klyshko and A. N. Penin, "The prospects of quantum photometry," Phys. Uspekhi 30, 716-723 (1987).
    [CrossRef]
  10. A. Migdall, "Correlated-photon metrology without absolute standards," Phys. Today 52, 41-46 (1999).
    [CrossRef]
  11. G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000).
    [CrossRef]
  12. V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
    [CrossRef]
  13. G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
    [CrossRef]
  14. G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005).
    [CrossRef]
  15. G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
    [CrossRef]
  16. G. Brida et al., J. Mod. Opt. to appear.
  17. K. Soda, I. Nishio, and A. Wada, "Analysis of noises in photocurrent time-correlation spectroscopy of scattered light," J. Appl. Phys. 47, 729-735 (1976).
    [CrossRef]
  18. R. G. McIntyre, "Multiplication Noise in Uniform Avalanche Diodes," IEEE Trans. Electron. Devices,  ED-13, 164-168 (1966).
    [CrossRef]
  19. The condition is fulfilled if the electronics is the same for the two detector and has a pass band much smaller than the band of the detectors.
  20. D. N. Klyshko and A. V. Masalov, "Photon noise: observation, squeezing, interpretation," Phys. Uspekhi 38, 1203-1230 (1995).
    [CrossRef]

2006 (1)

2005 (2)

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
[CrossRef]

2004 (1)

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

2000 (2)

G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000).
[CrossRef]

See, for a general review, N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2000), and references therein.
[CrossRef]

1999 (1)

A. Migdall, "Correlated-photon metrology without absolute standards," Phys. Today 52, 41-46 (1999).
[CrossRef]

1995 (1)

D. N. Klyshko and A. V. Masalov, "Photon noise: observation, squeezing, interpretation," Phys. Uspekhi 38, 1203-1230 (1995).
[CrossRef]

1993 (1)

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

1987 (1)

D. N. Klyshko and A. N. Penin, "The prospects of quantum photometry," Phys. Uspekhi 30, 716-723 (1987).
[CrossRef]

1981 (1)

A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).

1980 (1)

D. N. Klyshko, "Use of two-photon light for absolute calibration of photoelectric detectors," Sov. J. Quant. Elect. 10, 1112-1116 (1980).
[CrossRef]

1976 (1)

K. Soda, I. Nishio, and A. Wada, "Analysis of noises in photocurrent time-correlation spectroscopy of scattered light," J. Appl. Phys. 47, 729-735 (1976).
[CrossRef]

1970 (1)

D. C. Burnham and D. L. Weinberg, "Observation of Simultaneity in Parametric Production of Optical Photon Pairs," Phys. Rev. Lett. 25, 84-87 (1970).
[CrossRef]

1969 (1)

B. Y. Zel�??dovich and D. N. Klyshko, "Statistics of field in parametric luminescence," Sov. Phys. JETP Lett. 9, 40-44 (1969).

1966 (1)

R. G. McIntyre, "Multiplication Noise in Uniform Avalanche Diodes," IEEE Trans. Electron. Devices,  ED-13, 164-168 (1966).
[CrossRef]

Brida, G.

G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, "The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study," J. Opt. Soc. Am. B 23, 2185-2193 (2006).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
[CrossRef]

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000).
[CrossRef]

G. Brida et al., J. Mod. Opt. to appear.

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, "Observation of Simultaneity in Parametric Production of Optical Photon Pairs," Phys. Rev. Lett. 25, 84-87 (1970).
[CrossRef]

Chekhova, M.

Chekhova, M. V.

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

Genovese, M.

G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, "The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study," J. Opt. Soc. Am. B 23, 2185-2193 (2006).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
[CrossRef]

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000).
[CrossRef]

Ginzburg, V. M.

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

Gramegna, M.

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005).
[CrossRef]

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

Keratishvili, N. G.

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

Klyshko, D. N.

D. N. Klyshko and A. V. Masalov, "Photon noise: observation, squeezing, interpretation," Phys. Uspekhi 38, 1203-1230 (1995).
[CrossRef]

D. N. Klyshko and A. N. Penin, "The prospects of quantum photometry," Phys. Uspekhi 30, 716-723 (1987).
[CrossRef]

D. N. Klyshko, "Use of two-photon light for absolute calibration of photoelectric detectors," Sov. J. Quant. Elect. 10, 1112-1116 (1980).
[CrossRef]

B. Y. Zel�??dovich and D. N. Klyshko, "Statistics of field in parametric luminescence," Sov. Phys. JETP Lett. 9, 40-44 (1969).

Korzhenevich, Y. L.

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

Krivitsky, L.

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
[CrossRef]

Krivitsky, L. A.

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

Kulik, S. P.

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

Lunev, G. V.

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

Malygin, A. A.

A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).

Masalov, A. V.

D. N. Klyshko and A. V. Masalov, "Photon noise: observation, squeezing, interpretation," Phys. Uspekhi 38, 1203-1230 (1995).
[CrossRef]

McIntyre, R. G.

R. G. McIntyre, "Multiplication Noise in Uniform Avalanche Diodes," IEEE Trans. Electron. Devices,  ED-13, 164-168 (1966).
[CrossRef]

Migdall, A.

A. Migdall, "Correlated-photon metrology without absolute standards," Phys. Today 52, 41-46 (1999).
[CrossRef]

Nishio, I.

K. Soda, I. Nishio, and A. Wada, "Analysis of noises in photocurrent time-correlation spectroscopy of scattered light," J. Appl. Phys. 47, 729-735 (1976).
[CrossRef]

Novero, C.

G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000).
[CrossRef]

Penin, A.

Penin, A. N.

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

D. N. Klyshko and A. N. Penin, "The prospects of quantum photometry," Phys. Uspekhi 30, 716-723 (1987).
[CrossRef]

A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).

Rastello, M. L.

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
[CrossRef]

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005).
[CrossRef]

Ruo-Berchera, I.

Sapritsky, V. I.

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

Sergienko, A. V.

A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).

Soda, K.

K. Soda, I. Nishio, and A. Wada, "Analysis of noises in photocurrent time-correlation spectroscopy of scattered light," J. Appl. Phys. 47, 729-735 (1976).
[CrossRef]

Wada, A.

K. Soda, I. Nishio, and A. Wada, "Analysis of noises in photocurrent time-correlation spectroscopy of scattered light," J. Appl. Phys. 47, 729-735 (1976).
[CrossRef]

Weinberg, D. L.

D. C. Burnham and D. L. Weinberg, "Observation of Simultaneity in Parametric Production of Optical Photon Pairs," Phys. Rev. Lett. 25, 84-87 (1970).
[CrossRef]

Zel???dovich, B. Y.

B. Y. Zel�??dovich and D. N. Klyshko, "Statistics of field in parametric luminescence," Sov. Phys. JETP Lett. 9, 40-44 (1969).

IEEE Trans Instrum. Meas. (1)

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005).
[CrossRef]

IEEE Trans. Electron. Devices (1)

R. G. McIntyre, "Multiplication Noise in Uniform Avalanche Diodes," IEEE Trans. Electron. Devices,  ED-13, 164-168 (1966).
[CrossRef]

J. Appl. Phys. (1)

K. Soda, I. Nishio, and A. Wada, "Analysis of noises in photocurrent time-correlation spectroscopy of scattered light," J. Appl. Phys. 47, 729-735 (1976).
[CrossRef]

J. Mod. Opt. (2)

G. Brida et al., J. Mod. Opt. to appear.

G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Eng. (1)

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993).
[CrossRef]

Phys. Rev. A (1)

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

D. C. Burnham and D. L. Weinberg, "Observation of Simultaneity in Parametric Production of Optical Photon Pairs," Phys. Rev. Lett. 25, 84-87 (1970).
[CrossRef]

Phys. Today (1)

A. Migdall, "Correlated-photon metrology without absolute standards," Phys. Today 52, 41-46 (1999).
[CrossRef]

Phys. Uspekhi (2)

D. N. Klyshko and A. N. Penin, "The prospects of quantum photometry," Phys. Uspekhi 30, 716-723 (1987).
[CrossRef]

D. N. Klyshko and A. V. Masalov, "Photon noise: observation, squeezing, interpretation," Phys. Uspekhi 38, 1203-1230 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

See, for a general review, N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2000), and references therein.
[CrossRef]

Sov. J. Quant. Elect. (1)

D. N. Klyshko, "Use of two-photon light for absolute calibration of photoelectric detectors," Sov. J. Quant. Elect. 10, 1112-1116 (1980).
[CrossRef]

Sov. Phys. JETP Lett. (2)

A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).

B. Y. Zel�??dovich and D. N. Klyshko, "Statistics of field in parametric luminescence," Sov. Phys. JETP Lett. 9, 40-44 (1969).

Other (3)

H. A. Bachor, A guide to experimental quantum optics (Wiley-VCH, New York, 1998).

M. Genovese, "Research on hidden variable theories: A review on recent progresses," Phys. Rep.  413, 319-398 (2005) and references therein.
[CrossRef]

The condition is fulfilled if the electronics is the same for the two detector and has a pass band much smaller than the band of the detectors.

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 (1)

Fig. 1.
Fig. 1.

Scheme for absolute calibration of analog detectors by using stimulated PDC. The quantum efficiency of detector 2, collecting the seed beam and the stimulated emission, is estimated by the ratio between the cross-correlation function of the photocurrents and the auto-correlation of the photocurrent i 1.

Equations (47)

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

i j = n q jn f ( t t n ) = d t n q j f ( t t n ) F ̂ j ( t n )
i j ( t ) i k ( t + τ ) = n , m q jn q km f ( t t n ) f ( t t m + τ )
= d t n d t m q j q k f ( t t n ) f ( t t m + τ ) F ̂ j ( t n ) F ̂ k ( t m ) ,
F ̂ j ( t n ) F ̂ k ( t m ) = F ̂ j F ̂ k + F ̂ j δ ( t n t m ) δ jk
+ : δ F ̂ j ( t n ) δ F ̂ k ( t m ) : .
F ̂ j ( t ) η j F ̂ j ( t )
: F ̂ j ( t ) F ̂ k ( t ) : η j η k : F ̂ j ( t ) F ̂ k ( t ) : .
i j = η j q j F j .
i j ( t ) i k ( t + τ ) = i j i k + η j q j 2 𝓕 ( τ ) F j δ j k
+ η j η k q j q k d t n d t m
f ( t t n ) f ( t t m + τ ) : δ F ̂ j ( t n ) δ F ̂ k ( t m ) : ,
a ̂ 1 out ( q , Ω ) = U 1 ( q , Ω ) a ̂ 1 in ( q , Ω ) + V 1 ( q , Ω ) a ̂ 2 in ( q , Ω )
a ̂ 2 out ( q , Ω ) = U 2 ( q , Ω ) a ̂ 2 in ( q , Ω ) + V 2 ( q , Ω ) a ̂ 1 in ( q , Ω )
U k ( q , Ω ) 2 V k ( q , Ω ) 2 = 1 , ( k = 1 , 2 )
U 1 ( q , Ω ) V 2 ( q , Ω ) = U 2 ( q , Ω ) V 1 ( q , Ω ) ,
[ a ^ j o u t ( q , Ω ) , a ^ k o u t ( q , Ω ) ] = δ j , k δ q , q δ Ω , Ω [ a ^ j o u t ( q , Ω ) , a ^ k o u t ( q , Ω ) ] = 0
B ̂ k ( x , t ) = Ω e i Ω t a ̂ k out ( 2 π λ f x , Ω ) .
I ̂ k ( x , t ) = ΩΩ e i ( Ω Ω ) t
a ̂ k out ( 2 π λ f x , Ω ) a ̂ k out ( 2 π λ f x , Ω ) ,
ψ in = 0 1 q , Ω α ( q , Ω ) 2 ,
I ̂ 1 ( x , t ) V ~ 1 ( x , Ω 0 ) 2 α ~ ( x ) 2
I ̂ 2 ( x , t ) U ~ 2 ( x , Ω 0 ) 2 α ~ ( x ) 2
= ( V ¯ 2 ( x , Ω 0 ) 2 + 1 ) α ~ ( x ) 2
U ~ k ( x , Ω ) = U k ( 2 π λ f x , Ω ) ,
V ~ k ( x , Ω ) = V k ( 2 π λ f x , Ω ) ,
α ~ ( x ) = α ( 2 π λ f x ) .
: δ I ̂ k ( x , t ) δ I ̂ j ( x , t ) : = : I ̂ k ( x , t ) I ̂ j ( x , t ) :
I ̂ k ( x , t ) I ̂ j ( x , t )
: δ I ̂ 1 ( x , t ) δ I ̂ 1 ( x , t ) : = 0 ,
: δ I ̂ 2 ( x , t ) δ I ̂ 2 ( x , t ) : Ω e i ( Ω 0 Ω ) ( t t ) α ~ ( x ) 2
V ~ 2 ( x , Ω ) 2 U ~ 2 ( x , Ω 0 ) 2 δ xx
+ c . c .
: δ I ̂ 1 ( x , t ) δ I ̂ 2 ( x , t ) : Ω e i ( Ω 0 + Ω ) ( t t ) α ~ ( x ) 2
U ~ 2 ( x , Ω 0 ) U ~ 2 ( x , Ω )
V ~ 1 ( x , Ω 0 ) V ~ 1 ( x , Ω ) δ xx
+ c . c .
F 1 R 1 d x V ~ 1 ( x , Ω 0 ) 2 α ~ ( x ) 2 ,
F 2 R 2 d x ( V ~ 2 ( x , Ω 0 ) 2 + 1 ) α ~ ( x ) 2 .
: δ F 1 ( t ) δ F 1 ( t ) : 0
: δ F 2 ( t ) δ F 2 ( t ) : Ω e i ( Ω 0 Ω ) ( t t )
R 2 d x V ~ 2 ( x , Ω ) 2 α ~ ( x ) 2 + c . c . ,
: δ F 1 ( t ) δ F 2 ( t ) : Ω e i ( Ω 0 + Ω ) ( t t ) R 1 d x
α ~ ( x ) 2 V ~ 1 ( x , Ω 0 ) V ~ 1 ( x , Ω ) + c . c . .
δ i 1 ( t ) δ i 1 ( t + τ ) = η 1 q 1 2 𝓕 ( τ ) F 1 .
δ i 1 ( t ) δ i 2 ( t + τ ) = 2 η 1 η 2 q 1 q 2 𝓕 ( τ ) F 1
η 2 = 1 2 δ i 1 ( t ) δ i 2 ( t + τ ) δ i 1 ( t ) δ i 1 ( t + τ ) .
η 2 q 2 = 1 2 d τ δ i 1 ( t ) δ i 2 ( t + τ ) i 1 ,

Metrics