Abstract

We demonstrate the ability to calibrate a variable optical attenuator directly at the few-photon level using a superconducting Transition Edge Sensor (TES). Because of the inherent linearity of photon-number resolving detection, no external calibrations are required, even for the energy of the laser pulses, which ranged from means of 0.15 to 18 photons per pulse at the detector. To verify this method, calibrations were compared to an independent conventional calibration made at much higher photon fluxes using analog detectors. In all cases, the attenuations estimated by the two methods agree within their uncertainties. Our few-photon measurement determined attenuations using the Poisson-Influenced K-Means Algorithm (PIKA) to extract mean numbers of photons per pulse along with the uncertainties of these means. The robustness of the method is highlighted by the agreement of the two calibrations even in the presence of significant drifts in the optical power over the course of the experiment.

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References

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2014 (1)

2012 (1)

2011 (1)

2010 (1)

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[Crossref]

2009 (2)

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Inst. and Meth. A 610, 183–187 (2009).
[Crossref]

A. P. Worsley, H. B. Coldenstrodt-Ronge, J. S. Lundeen, P. J. Mosley, B. J. Smith, G. Puentes, N. Thomas-Peter, and I. A. Walmsley, “Absolute efficiency estimation of photon-number-resolving detectors using twin beams,” Opt. Express 17, 4397–4411 (2009).
[Crossref] [PubMed]

2007 (1)

2005 (1)

H. K. Lo, X. F. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref] [PubMed]

2002 (1)

J. H. Lehman and C. L. Cromer, “Optical trap detector for calibration of optical fiber powermeters: coupling efficiency,” Appl. Opt. 31, 6531–6536 (2002).
[Crossref]

2000 (2)

G. Eppeldauer, “Noise-optimized silicon radiometers,” J. Res. Natl. Inst. Stand. Technol. 105, 209–219 (2000).
[Crossref]

A. Czitrovszky, A. Sergienko, P. Jani, and A. Nagy, “Measurement of quantum efficiency using entangled photons,” Laser Phys. 10, 86–89 (2000).

1999 (1)

A. Migdall, “Correlated-photon metrology without absolute standards,” Phys. Tod. 52, 41–46 (1999).
[Crossref]

1987 (1)

Avella, A.

Beaumont, A. R.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Inst. and Meth. A 610, 183–187 (2009).
[Crossref]

Brida, G.

Calkins, B.

Carlin, J. B.

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, FL, 2014).

Chen, K.

H. K. Lo, X. F. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref] [PubMed]

Cheung, J. Y.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Inst. and Meth. A 610, 183–187 (2009).
[Crossref]

Chunnilall, C. J.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Inst. and Meth. A 610, 183–187 (2009).
[Crossref]

Coldenstrodt-Ronge, H. B.

Cromer, C. L.

J. H. Lehman and C. L. Cromer, “Optical trap detector for calibration of optical fiber powermeters: coupling efficiency,” Appl. Opt. 31, 6531–6536 (2002).
[Crossref]

Czitrovszky, A.

A. Czitrovszky, A. Sergienko, P. Jani, and A. Nagy, “Measurement of quantum efficiency using entangled photons,” Laser Phys. 10, 86–89 (2000).

Datta, A.

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” arXiv:1502.07649 (2015).

Degiovanni, I. P.

Dunson, D. B.

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, FL, 2014).

Eppeldauer, G.

G. Eppeldauer, “Noise-optimized silicon radiometers,” J. Res. Natl. Inst. Stand. Technol. 105, 209–219 (2000).
[Crossref]

Fujiwara, M.

Gelman, A.

M. D. Hoffman and A. Gelman, “The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo,” J. Machine Learning Research (2014).

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, FL, 2014).

Genovese, M.

Gerrits, T.

Glebov, B. L.

Gottlieb, D.

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society of Industrial and Applied Mathematics, Philadelphia, PA, 1977). P. 159.

Gramegna, M.

Haus, H. A.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, Berlin, 2000). Sect. 9.2.
[Crossref]

Hiemstra, T.

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” arXiv:1502.07649 (2015).

Hoffman, M. D.

M. D. Hoffman and A. Gelman, “The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo,” J. Machine Learning Research (2014).

Humphreys, P. C.

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” arXiv:1502.07649 (2015).

Ireland, J.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Inst. and Meth. A 610, 183–187 (2009).
[Crossref]

Jani, P.

A. Czitrovszky, A. Sergienko, P. Jani, and A. Nagy, “Measurement of quantum efficiency using entangled photons,” Laser Phys. 10, 86–89 (2000).

Klein, R.

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[Crossref]

Kolthammer, W. S.

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” arXiv:1502.07649 (2015).

Lehman, J. H.

J. H. Lehman and C. L. Cromer, “Optical trap detector for calibration of optical fiber powermeters: coupling efficiency,” Appl. Opt. 31, 6531–6536 (2002).
[Crossref]

Levine, Z. H.

Lita, A. E.

Lo, H. K.

H. K. Lo, X. F. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref] [PubMed]

Lolli, L.

Lundeen, J. S.

Ma, X. F.

H. K. Lo, X. F. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref] [PubMed]

Metcalf, B. J.

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” arXiv:1502.07649 (2015).

Midgall, A. L.

Migdall, A.

A. Migdall, “Correlated-photon metrology without absolute standards,” Phys. Tod. 52, 41–46 (1999).
[Crossref]

Migdall, A. L.

Monticone, E.

Mosley, P. J.

Nagy, A.

A. Czitrovszky, A. Sergienko, P. Jani, and A. Nagy, “Measurement of quantum efficiency using entangled photons,” Laser Phys. 10, 86–89 (2000).

Nam, S. W.

Orszag, S. A.

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society of Industrial and Applied Mathematics, Philadelphia, PA, 1977). P. 159.

Portesi, C.

Puentes, G.

Rajteri, M.

Rarity, J. G.

Rastello, M. L.

Ridley, K. D.

Rubin, D. B.

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, FL, 2014).

Samarov, D. V.

Sasaki, M.

Sergienko, A.

A. Czitrovszky, A. Sergienko, P. Jani, and A. Nagy, “Measurement of quantum efficiency using entangled photons,” Laser Phys. 10, 86–89 (2000).

Smith, B. J.

Stern, H. S.

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, FL, 2014).

Tapster, P. R.

Taralli, E.

Thomas-Peter, N.

Thornagel, R.

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[Crossref]

Traina, P.

Ulm, G.

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[Crossref]

Vehtari, A.

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, FL, 2014).

Walmsley, I. A.

A. P. Worsley, H. B. Coldenstrodt-Ronge, J. S. Lundeen, P. J. Mosley, B. J. Smith, G. Puentes, N. Thomas-Peter, and I. A. Walmsley, “Absolute efficiency estimation of photon-number-resolving detectors using twin beams,” Opt. Express 17, 4397–4411 (2009).
[Crossref] [PubMed]

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” arXiv:1502.07649 (2015).

White, M.

White, M. G.

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Inst. and Meth. A 610, 183–187 (2009).
[Crossref]

Worsley, A. P.

Appl. Opt. (3)

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

J. Res. Natl. Inst. Stand. Technol. (1)

G. Eppeldauer, “Noise-optimized silicon radiometers,” J. Res. Natl. Inst. Stand. Technol. 105, 209–219 (2000).
[Crossref]

Laser Phys. (1)

A. Czitrovszky, A. Sergienko, P. Jani, and A. Nagy, “Measurement of quantum efficiency using entangled photons,” Laser Phys. 10, 86–89 (2000).

Metrologia (1)

R. Klein, R. Thornagel, and G. Ulm, “From single photons to milliwatt radiant power in electron storage rings as radiation sources with a high dynamic range,” Metrologia 47, R33–R40 (2010).
[Crossref]

Nucl. Inst. and Meth. A (1)

A. R. Beaumont, J. Y. Cheung, C. J. Chunnilall, J. Ireland, and M. G. White, “Providing reference standards and metrology for the few photon counting community,” Nucl. Inst. and Meth. A 610, 183–187 (2009).
[Crossref]

Opt. Express (2)

Phys. Rev. Lett. (1)

H. K. Lo, X. F. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref] [PubMed]

Phys. Tod. (1)

A. Migdall, “Correlated-photon metrology without absolute standards,” Phys. Tod. 52, 41–46 (1999).
[Crossref]

Other (10)

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” arXiv:1502.07649 (2015).

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, FL, 2014).

The mention of commercial products does not imply endorsement by the authors’ institutions nor does it imply that they are the best available for the purpose.

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society of Industrial and Applied Mathematics, Philadelphia, PA, 1977). P. 159.

OZ Optics Limited, “Digital variable attenuator,” http://www.ozoptics.com/ALLNEW_PDF/DTS0007.pdf , 2013, p. 3.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, Berlin, 2000). Sect. 9.2.
[Crossref]

R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria (2014).

Stan Development Team, Stan: A C++ Library for Probability and Sampling, Version 2.5.0 (2014).

Stan Development Team, Stan Modeling Language User’s Guide and Reference Manual, Version 2.5.0 (2014).

M. D. Hoffman and A. Gelman, “The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo,” J. Machine Learning Research (2014).

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Figures (6)

Fig. 1
Fig. 1 Schematic representation of variable attenuator calibration setup. A pulsed laser acted as the light source. It also provided an electronic synchronization signal used to trigger data acquisition. The laser light was coupled into a single-mode optical fiber connected to the computer-controlled variable attenuator under test. The light was sent to the detector — either an analog detector or a TES — using a second optical fiber of the same type. When the TES was used, the laser is attenuated by a nominal 70 dB by a fixed neutral density filter. Analog electronic signals were read from the detector by the data acquisition system.
Fig. 2
Fig. 2 The ideal response for photons with n = 9 to n = 18 as derived from mean photon numbers of 〈n〉 = 12.08 (thick red dashed lines) and 〈n〉 = 15.17 (thin blue solid lines). The responses are almost entirely independent of the mean photon number although those with the higher mean photon number come slightly later. The black line indicates half of the maximum value on the falling edge. The differences in the time Δt between the two cases at the half maximum is given as a function of the exact photon number n are shown in the inset.
Fig. 3
Fig. 3 A histogram of the density in each row as a function of both the effective photon number and the mean photon number. Each row is normalized to its peak value. For each mean photon number the response to a few nearby exact photon numbers may be determined. The visibility of the peaks is seen to become small in the low 20s, similar the our earlier report [8].
Fig. 4
Fig. 4 The mean photon number 〈n〉 which maximizes the likelihood function is given for each data set as a function of the time of acquisition in the experiment. The repeated measurement sequence pattern of the attenuator setting (labeled) increasing from 30 dB to 3 dB and the decreasing back to 30 dB (blue points) is evident. (Only the values between 5 dB and 23 dB are presented here.) The data were acquired over a 168 minute period with each point taken in 9.3 s including dead time.
Fig. 5
Fig. 5 The data from Fig. 4 are modeled (as described in text and in Appendix A) by a constant attenuation for each set of conditions and a time varying optical power, which is plotted here. The systematic variation is attributed to changes in the source optical power over the course of the experiment. The result yields a slow intensity variation which is fit by a quartic function and shown in the figure. The coefficients of the fit are given in Table 1. The red and blue points correspond to those of Fig. 4 corrected for the constant attenuation for each set of conditions.
Fig. 6
Fig. 6 Attenuation values as measured with the TES and those measured with an analog detector for (a) attenuation values stepped in an increasing direction; and (b) attenuation values stepped in a decreasing direction. The solid red and blue circles refer to the values derived from few-photon (TES) data, and the open green and brown circles refer to the analog data. The 5 dB mean value for the analog data is set to 0 and the few-photon data are rigidly shifted to achieve the least squared difference. There are no further free parameters in panels (a) and (b); note that the rigid shifts affect both panels together. The uncertainties shown represent 95% credible intervals. The green and brown circles are the values derived from the analog measurement, with the corresponding uncertainties shown representing 95% prediction intervals. (See Appendix A for definitions.) For comparison, 0.01 dB is equivalent to a difference of 0.23%.

Tables (1)

Tables Icon

Table 1 Coefficients for the fit to the optical power over the time period from tmin = 0 min to tmax = 168 min are given with 95% credible intervals. The expansion is I ( t ) = 1 + k = 1 4 c k T k ( x ) where x = 2(t/tmax) − 1 (hence, −1 ≤ x ≤ 1) and the Tk(x) are Chebyshev polynomials of the first kind [13]. The expansion is truncated after four coefficients because zero lies in the 95% credible interval for both the fifth (0.0007 ± 0.0028) and sixth coefficients (−0.0009 ± 0.0028).

Equations (1)

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z r ( t ) = α r + k = 1 4 c k T k ( u ) + ε r ( t ) ,

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