## Nonlinearity in single photon detection: modeling and quantum tomography |

Optics Express, Vol. 19, Issue 22, pp. 21305-21312 (2011)

http://dx.doi.org/10.1364/OE.19.021305

Acrobat PDF (931 KB)

### Abstract

Single Photon Detectors are integral to quantum optics and quantum information. Superconducting Nanowire based detectors exhibit new levels of performance, but have no accepted quantum optical model that is valid for multiple input photons. By performing Detector Tomography, we improve the recently proposed model [M.K. Akhlaghi and A.H. Majedi, IEEE Trans. Appl. Supercond. **19**, 361 (2009)] and also investigate the manner in which these detectors respond nonlinearly to light, a valuable feature for some applications. We develop a device independent model for Single Photon Detectors that incorporates this nonlinearity.

© 2011 OSA

## 1. Introduction

1. L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. **4**, S176–S183 (2002). [CrossRef]

2. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature **429**, 161–164 (2004). [CrossRef] [PubMed]

3. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. **4**, 282–286 (2008). [CrossRef]

4. O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. **121**, 866–884 (2009). [CrossRef]

5. R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for raman lidar,” Appl. Opt. **48**, 3903–3914 (2009). [CrossRef] [PubMed]

6. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” *Science*313, 1642–1645 (2006). [CrossRef] [PubMed]

7. D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. **28**, 2387–2389 (2003). [CrossRef] [PubMed]

8. G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. **102** (2009). [CrossRef] [PubMed]

9. M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A **53**, R1209–R1212 (1996). [CrossRef] [PubMed]

10. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental observation of nonclassical effects on single-photon detection rates,” Phys. Rev. A **63**, 020102 (2001). [CrossRef]

*P*

_{1}is the quantum efficiency, |

*n*〉 is an n-photon state, and the Click operator is

11. A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature **448**, 784–786 (2007). [CrossRef] [PubMed]

12. T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A **82**, 031802 (2010). [CrossRef]

7. D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. **28**, 2387–2389 (2003). [CrossRef] [PubMed]

13. C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett. **38**, 1578–1579 (2002). [CrossRef]

14. R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics **3**, 696–705 (2009). [CrossRef]

17. M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. **19**, 361–366 (2009). [CrossRef]

18. G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. **79**, 705–707 (2001). [CrossRef]

19. J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. **5**, 27–30 (2009). [CrossRef]

20. H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. **56**, 432–441 (2009). [CrossRef]

## 2. Detector tomography

*α*. We sequentially send

*R*= 10

_{T}^{5}copies of this state into the SNSPD system and record the number of detector Clicks,

*R*

_{1}. This is repeated for a set of D states {|

*α*〉}, increasing

_{i}*α*from

*α*

_{0}= 0 until the detector response is unchanging at

*α*. i.e.

_{D}*∂R*

_{1}/

*∂α*= 0. The estimated Click probability,

*R*

_{1}/

*R*, is equal to the Q-function of the Click POVM operator,

_{T}*Q*(

*α*). This, in itself, completely characterizes the detector response.

21. J. L. F. X. Orgiazzi and A. H. Majedi, “Robust packaging technique and characterization of fiber-pigtailed superconducting nbn nano wire single photon detectors,” IEEE Trans. Appl. Supercond. **19**, 341–345 (2009). [CrossRef]

22. Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt. **56**, 380–384 (2009). [CrossRef]

*μ*A. In Fig. 1(b), we plot the measured response for each of these (blue circles). We expect a standard linear SPD response at 25

*μ*A since this is the normal operation mode. Using Eq. (1), one expects a Q-function of the form

*Q*(

*α*) = 1 – exp (−

*P*

_{1}|

*α*|

^{2}). We estimate

*P*

_{1}using a single data point at

*Q*(

*α*) =

*R*

_{1}/

*R*= 0.1. Indeed, using this

_{T}*P*

_{1}, the resulting predicted response (red line) agrees well with the measured response. Repeating this analysis for 20 and 16

*μ*A, we find that the estimated

*P*

_{1}s, and thus, quantum efficiencies decrease as the bias current decreases. More significantly, the disagreement between the shape of the predicted and measured Click Probability distributions is substantial. Evidently, a SNSPD quickly becomes nonlinear as the bias current is lowered.

*R*

_{1}/

*R*according to the methods given in [24

_{T}24. A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. **11** (2009). [CrossRef]

*N*× 1). We truncate Π at a number state

*N*– 1 that is sufficiently high that Π (

*N*) ≈ 1.

24. A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. **11** (2009). [CrossRef]

*N*– 1, required to span their response was large. Instead of using large matrices in the fitting, we scaled the inputs {|

*α*〉} by a factor

_{i}*k*≪ 1. For each bias current,

*k*is chosen so that the Click Probability is 95% at an average photon number 〈

*n*〉 = 30. This scaled data is shown in Fig. 2(a) (black circles). We plot Π determined from it in Fig. 2(b) (blue circles). Using this Π, in Fig. 2(a) (blue line) we plot the predicted detector response to coherent input states. This fits the scaled data well, confirming the fitting procedure.

*k*loss back into the POVM (i.e. Π

*=*

_{unscaled}*L*

^{−1}Π

*, where*

_{scaled}*L*is the binomial distribution matrix described in [24

24. A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. **11** (2009). [CrossRef]

*α*〉}. For all the bias currents, the difference between the predicted click probability and the raw data (i.e. |

_{i}*R*

_{1}/

*R*– Tr[Π

_{T}*|*

_{unscaled}*α*〉〈

_{i}*α*|]|) is less than 0.15% on average and has a maximum of 1.4%. This indicates that we have accurately estimated the SNSPD POVM using the scaling technique.

_{i}## 3. Generalized model

*O*(

*N*) parameters) that can be difficult to interpret. Tomography can hardly replace the natural ease and intuition that is associated with a model. A good model candidate is the AM model. Indeed, plotting the AM model in Fig. 2(a) (red dashed line) shows that it can be fit to the scaled data.

17. M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. **19**, 361–366 (2009). [CrossRef]

17. M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. **19**, 361–366 (2009). [CrossRef]

**19**, 361–366 (2009). [CrossRef]

*n*number of photons; any less and the detector responds with No Click, any more and it still only outputs one Click. This is the n-photon generalization of the SPD and, hence, we call it an n-photon detector (NPD). If

*m*>

*n*photons impinge on the detector, there are m choose

*n*ways for those

*m*photons to trigger the NPD. Consequently the generalization of Eq. (1) is: where

*P*is the n-photon detection efficiency and

_{n}*n*>

*m*, = 1 for

*n*= 0). This generalization works even for a zero photon detector. We can identify

*P*

_{0}as what is commonly called the ‘dark count probability’ per gate window.

*α*〉 is: This can be rewritten in matrix form as: where

*C*(with dimensions

*D*× 1), includes the D measured statistics,

*R*

_{1}/

*R*; Π (

_{T}*N*× 1) includes the diagonal elements of

*F*contains the D coherent state probes,

*F*= |

_{i,j}*α*|

_{i}^{2}

*exp (−|*

^{j}*α*|

_{i}^{2})/

*j*!;

*E*is a matrix of ones;

*G*is a matrix of binomial coefficients such that

*H*(

*M*× 1) is an unknown matrix which includes the unknown set {

*P*},

_{n}*H*

_{i,1}= ln(1 −

*P*

_{n=i−1}).

*H*by solving the following constrained nonlinear multivariable optimization problem: subject to

*H*≤ 0. The second norm of a matrix is defined as ||

*A*||

_{2}= (Σ

*|*

_{i,j}*A*|

_{i,j}^{2})

^{1/2}. Each element of the expression is normalized to

*C*to give equal weighting to all the points. The constraint of the problem ensures the optimization leads to a physical result for the set {

*P*}. We note the function exp(

_{n}*αx*) is convex on ℝ for any

*α*∈ ℝ [25], which also makes Eq. (6) convex.

*P*} we only keep those elements that change the minimum of Eq. (6) by more than 1% (25

_{n}*μ*A: {

*P*

_{0},

*P*

_{1}}; 20

*μ*A: {

*P*

_{0}, ... ,

*P*

_{4}}; 16

*μ*A: {

*P*

_{1}, ... ,

*P*

_{4}}). These parameters classify the operation of the SNSPD, from a standard SPD at 25

*μ*A to a composite of one, two, three, and four photon detectors at 16

*μ*A.

*P*} we calculate the nonlinear SPD Click POVM operator for the three bias currents in Fig. 2(b) (black dotted line). They agree with the tomography POVMs to within 1% for most of the elements of Π. For 25

_{n}*μ*A and 20

*μ*A the maximum difference is 3% at Π (

*n*= 1) and 6% at Π (

*n*= 3) for 16

*μ*A. This excludes the large error at Π (

*n*= 0), which we attribute to insufficient measured statistics at extremely small mean photon numbers. The Quantum Fidelity (see [24

**11** (2009). [CrossRef]

18. G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. **79**, 705–707 (2001). [CrossRef]

*P*

_{2}at 20

*μ*A and

*P*

_{2},

*P*

_{3}at 16

*μ*A) reconfirms the detector includes some NPDs as its basic detection elements. This rules out the other possible justifications of the observed nonlinearities including heating effects at higher input photon flux.

*P*

_{1}→

*ηP*

_{1}under a preceding optical loss of

*η*, there is no analytic formulae for how {

*P*} transform under loss. By inspection of the scaled and non-scaled model fits (see Fig. 3(b)), however, each element of {

_{n}*P*} that is significant approximately satisfies

_{n}*P*} scale as

_{n}*η*. Consequently, removing any linear optical inefficiency from a nonlinear photon counter makes it more nonlinear. We also note that, as expected,

^{n}*P*

_{0}is not dependent on the optical input as expected.

## 4. Conclusion

## Acknowledgments

## References and links

1. | L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. |

2. | M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature |

3. | J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. |

4. | O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac. |

5. | R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for raman lidar,” Appl. Opt. |

6. | E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” |

7. | D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. |

8. | G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett. |

9. | M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A |

10. | K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental observation of nonclassical effects on single-photon detection rates,” Phys. Rev. A |

11. | A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature |

12. | T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A |

13. | C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett. |

14. | R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics |

15. | G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005). |

16. | A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Physica C |

17. | M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. |

18. | G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. |

19. | J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. |

20. | H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt. |

21. | J. L. F. X. Orgiazzi and A. H. Majedi, “Robust packaging technique and characterization of fiber-pigtailed superconducting nbn nano wire single photon detectors,” IEEE Trans. Appl. Supercond. |

22. | Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt. |

23. | U. Leonhardt, |

24. | A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys. |

25. | S. Boyd and L. Vandenberghe, |

26. | G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011). |

**OCIS Codes**

(040.5570) Detectors : Quantum detectors

(270.5570) Quantum optics : Quantum detectors

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 22, 2011

Revised Manuscript: September 23, 2011

Manuscript Accepted: September 28, 2011

Published: October 12, 2011

**Citation**

Mohsen K. Akhlaghi, A. Hamed Majedi, and Jeff S. Lundeen, "Nonlinearity in single photon detection: modeling and quantum tomography," Opt. Express **19**, 21305-21312 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21305

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### References

- L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt.4, S176–S183 (2002). [CrossRef]
- M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429, 161–164 (2004). [CrossRef] [PubMed]
- J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys.4, 282–286 (2008). [CrossRef]
- O. Daigle, C. Carignan, J. L. Gach, C. Guillaume, S. Lessard, C. A. Fortin, and S. Blais-Ouellette, “Extreme faint flux imaging with an emccd,” Publ. Astron. Soc. Pac.121, 866–884 (2009). [CrossRef]
- R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for raman lidar,” Appl. Opt.48, 3903–3914 (2009). [CrossRef] [PubMed]
- E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313, 1642–1645 (2006). [CrossRef] [PubMed]
- D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett.28, 2387–2389 (2003). [CrossRef] [PubMed]
- G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging particle and wave sensitivity in a configurable detector of positive operator-valued measures,” Phys. Rev. Lett.102 (2009). [CrossRef] [PubMed]
- M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric bell-state analysis,” Phys. Rev. A53, R1209–R1212 (1996). [CrossRef] [PubMed]
- K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental observation of nonclassical effects on single-photon detection rates,” Phys. Rev. A63, 020102 (2001). [CrossRef]
- A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘schrdinger cats’ from photon number states,” Nature448, 784–786 (2007). [CrossRef] [PubMed]
- T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A82, 031802 (2010). [CrossRef]
- C. Xu, L. Mollenauer, and X. Liu, “Compensation of nonlinear self-phase modulation with phase modulators,” Electron. Lett.38, 1578–1579 (2002). [CrossRef]
- R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics3, 696–705 (2009). [CrossRef]
- G. Goltsman, A. Korneev, O. Minaeva, I. Rubtsova, G. Chulkova, I. Milostnaya, K. Smirnov, B. Voronov, A. Lipatov, A. Pearlman, A. Cross, W. Slysz, A. Verevkin, and R. Sobolewski, “Advanced nanostructured optical nbn single-photon detector operated at 2.0 k,” Prog. Biomed. Opt. Imag., Proc. SPIE5732, 520–529 (2005).
- A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Physica C351, 349–356 (2001). [CrossRef]
- M. K. Akhlaghi and A. H. Majedi, “Semiempirical modeling of dark count rate and quantum efficiency of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond.19, 361–366 (2009). [CrossRef]
- G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett.79, 705–707 (2001). [CrossRef]
- J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys.5, 27–30 (2009). [CrossRef]
- H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith, W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, and I. A. Walmsley, “A proposed testbed for detector tomography,” J. Mod. Opt.56, 432–441 (2009). [CrossRef]
- J. L. F. X. Orgiazzi and A. H. Majedi, “Robust packaging technique and characterization of fiber-pigtailed superconducting nbn nano wire single photon detectors,” IEEE Trans. Appl. Supercond.19, 341–345 (2009). [CrossRef]
- Z. Yan, M. K. Akhlaghi, J. L. Orgiazzi, and A. H. Majedi, “Optoelectronic characterization of a fiber-coupled nbn superconducting nanowire single photon detector,” J. Mod. Opt.56, 380–384 (2009). [CrossRef]
- U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, 1997).
- A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: Theory and practice,” N. J. Phys.11 (2009). [CrossRef]
- S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, 2004).
- G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters,” arXiv:quant-ph:1103.2991 (2011).

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