## Modified detector tomography technique applied to a superconducting multiphoton nanodetector |

Optics Express, Vol. 20, Issue 3, pp. 2806-2813 (2012)

http://dx.doi.org/10.1364/OE.20.002806

Acrobat PDF (1858 KB)

### Abstract

We present an experimental method to characterize multi-photon detectors with a small overall detection efficiency. We do this by separating the nonlinear action of the multiphoton detection event from linear losses in the detector. Such a characterization is a necessary step for quantum information protocols with single and multiphoton detectors and can provide quantitative information to understand the underlying physics of a given detector. This characterization is applied to a superconducting multiphoton nanodetector, consisting of an NbN nanowire with a bowtie-shaped subwavelength constriction. Depending on the bias current, this detector has regimes with single and multiphoton sensitivity. We present the first full experimental characterization of such a detector.

© 2012 OSA

## 1. Introduction

1.. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature **409**, 46–52 (2001). [PubMed]

*photon-number resolved detection,*where the detector is able to distinguish precisely the number of photons, and

*threshold detection*, where the detector is merely able to distinguish between the cases ‘N photons or more’ and ‘fewer than N photons’ [2].

4.. M. K. Akhlaghi, A. H. Majedi, and J. S. Lundeen, “Nonlinearity in Single Photon Detection : Modeling and Quantum Tomography,” Opt. Express **19**, 21305 (2011). [PubMed]

*η*is the system detection efficiency) which would result in an overwhelming number of free parameters leading to a strongly overdetermined system.

_{sde}6.. D. Bitauld, F. Marsili, A. Gaggero, F. Mattioli, R. Leoni, S. J. Nejad, F. Lévy, and A. Fiore, “Nanoscale optical detector with single-photon and multiphoton sensitivity,” Nano Lett. **10**, 2977–81 (2010). [PubMed]

## 2. NbN Nanodetectors

6.. D. Bitauld, F. Marsili, A. Gaggero, F. Mattioli, R. Leoni, S. J. Nejad, F. Lévy, and A. Fiore, “Nanoscale optical detector with single-photon and multiphoton sensitivity,” Nano Lett. **10**, 2977–81 (2010). [PubMed]

6.. D. Bitauld, F. Marsili, A. Gaggero, F. Mattioli, R. Leoni, S. J. Nejad, F. Lévy, and A. Fiore, “Nanoscale optical detector with single-photon and multiphoton sensitivity,” Nano Lett. **10**, 2977–81 (2010). [PubMed]

**10**, 2977–81 (2010). [PubMed]

8.. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science **328**, 879–81 (2010). [PubMed]

## 3. Experimental setup

*μ*A.

*μ*m at 1500 nm. The readout electronics consist of a bias-tee (Minicircuits ZNBT-60-1W+), an amplifier chain and a pulse counter.

*μ*A, up to the critical current. Power stability during each run was monitored by a power meter which receives a pick-off beam from a beam splitter in the fiber leading to the experiment. Finally, the 2-dimensional set of count rates

*C*(

*I*,

_{b}*N*) is rearranged and normalized by the repetition rate of the laser to yield the detection probability per pulse

*R*(

*N*) at fixed bias current

*I*.

_{b}*R*= 10

^{−6}to

*R*= 1. This required varying the input power over 5 orders of magnitude, typically from 20 pW to 5

*μ*W input power into the cryostat. At a repetition rate of 20 MHz the largest input power corresponds to

*N*= 2 * 10

^{6}incident photons per pulse. Since the detection efficiency of our detector is low (order 10

^{−4}), it was not necessary to introduce further attenuation, as is usually done in detector tomography experiments [2].

## 4. Effective Photon Detector Characterization (EPDC)

*ρ*, the probability

*R*to observe a click is: where Π

*is the POVM operator of having a click, and*

_{click}*p*is the probability of a click occuring given a Fock state with i photons as input.

_{i}*c*(

_{i}*N*) to reconstruct the set of probabilities

*p*, either by a maximum likelyhood algorithm [2] or a simple curve fit [12]. Since we are dealing with a detector that saturates, i.e. that always clicks at high input power, the problem is simplified by reasoning from the case that the detector doesn’t click [13]. Since there are only two possible outcomes, this gives: where N is the mean photon number. The case

_{i}*p*

_{0}= 0,

*p*

_{i}_{>0}= 1 is applicable to any one-photon threshold detector, such as an APD with unity detection efficiency [2].

*linear loss parameter η*that describes the probability of for each photon to participate in the nonlinear process. Since coherent states remain coherent under attenuation, the EPDC function then becomes: where {

*p*} and

_{i}*η*are the free parameters. Since the POVM description is complete [3, 13] and we have added a parameter, we have now created a function that is overdetermined by one parameter. However, we can choose a solution based on physical grounds. Since we know our detector has threshold-like behaviour, it is reasonable to assume that for some large number of photons

*i*the probability

_{max}*p*with which the detector will click is arbitrarily close to 1. Furthermore, once we have found such an

_{imax}*i*, we can assume that

_{max}*p*

_{j>imax}= 1 for all

*j*>

*i*, since otherwise we would have the unphysical case that adding photons makes it less likely that the detector clicks. We can then create a series of candidate solutions by fitting Eq. 6 to our measured count rates as a function of input photon number, truncating the sum at various values

_{max}*i*. This gives a series of candidate solutions parameterized by {

_{max}*η*,

*p*

_{0}...

*p*

_{imax}}. The solution we pick is the one that fits our data and has the minimum

*i*, since this is the one that requires the fewest parameters to explain our data.

_{max}*p*is the quantity of interest for multiphoton detection. This approach is particularly relevant for detectors with a large linear loss component, since if this loss is not taken into account separately it would dominate the characterization of the detector.

_{i}## 5. Result

*i*(see legend). For each fit the reduced (i.e. normalized to the number of data points minus the number of fit parameters)

_{max}*χ*

^{2}[14] are shown in the bar diagrams in the insets of the figure. We take the fit that explains the data with the smallest number of parameters as the most physically realistic solution. This choice is indicated by the arrows in the bar diagrams. By repeating this algorithm over a range of bias currents, we can completely characterize how the response of the detector to a given number of photons varies with the bias current.

*p*and

_{i}*η*describe the operation of the detector, independent of power. We therefore conclude that we have obtained a complete description of the detector behaviour.

*p*

_{0}= 4 * 10

^{−4},

*p*= 1 for

_{i}*i*> 0, reproducing the expected result.

## 6. Discussion

*p*obtained from the fit represent the nonlinear action of our detection system, which is the physical property of interest. Since there are no other nonlinear elements in the detection system, we can unambiguously attribute the behaviour of the

_{i}*p*to the NbN nanodetector. It should be noted that the result presented here is consistent with earlier results on these detectors [6

_{i}**10**, 2977–81 (2010). [PubMed]

*p*(

_{i}c_{i}*N*), where

*c*(

_{i}*N*) is the probability of having N photons. From this we can see that each

*p*will be most dominant in the range of powers where the probability of having the corresponding number of photons is highest. For example, at 17

_{i}*μ*A the detector has

*p*

_{1}= 0.06 and

*p*

_{2}= 0.37, meaning that at low powers (

*η*

*N*< 0.16), where the one-photon contribution from the state is dominant, the detector will respond mostly to single photons, but at higher powers (

*ηN*> 0.16) the response will be dominated by the two-photon events. This quantifies the change of detection regimes reported in measurements of count rate as a function of power [6

**10**, 2977–81 (2010). [PubMed]

*R*< 10

_{dark}^{−4}) in our measurement. However, we note that for dark counts the assumption

*p*

_{i}_{+1}>

*p*, holds [13], since otherwise it would be the case that illuminating the detector makes it less likely to click. Therefore, our model is compatible with the presence of dark counts.

_{i}*η*fluctuates between 1 × 10

^{−4}and 1.5 × 10

^{−4}. Normalizing to the estimated effective area of the detector of 100 nm by 150 nm and the beam size, we obtain an intrinsic detection efficiency of 5–7%. While it should be noted that this is only a rough estimate, it is higher than the value of 1% reported in [6

**10**, 2977–81 (2010). [PubMed]

*p*completely describes the behaviour of the detector.

_{i}16.. D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, Loss-Tolerant Characterization of Nonclassical Photon Statistics,” Phys. Rev. Lett. **97**, 043602 (2006) [PubMed]

## 7. Conclusion

20.. A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Phot. **2**, 302–306 (2008).

4.. M. K. Akhlaghi, A. H. Majedi, and J. S. Lundeen, “Nonlinearity in Single Photon Detection : Modeling and Quantum Tomography,” Opt. Express **19**, 21305 (2011). [PubMed]

## 8. Note

## References and links

1.. | E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature |

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

3.. | 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. |

4.. | M. K. Akhlaghi, A. H. Majedi, and J. S. Lundeen, “Nonlinearity in Single Photon Detection : Modeling and Quantum Tomography,” Opt. Express |

5.. | M. Hofherr, D. Rall, K. Ilin, M. Siegel, A. Semenov, H.-W. Hübers, and N. A. Gippius, “Intrinsic detection efficiency of superconducting nanowire single-photon detectors with different thicknesses,” J. Appl. Phys. |

6.. | D. Bitauld, F. Marsili, A. Gaggero, F. Mattioli, R. Leoni, S. J. Nejad, F. Lévy, and A. Fiore, “Nanoscale optical detector with single-photon and multiphoton sensitivity,” Nano Lett. |

7.. | G. N. Goltsman, 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. |

8.. | I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science |

9.. | A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, V. Anant, K. K. Berggren, G. N. Goltsman, and B. M. Voronov, “Constriction-limited detection efficiency of superconducting nanowire single-photon detectors,” Appl. Phys. Lett. |

10.. | A. Gaggero, S. J. Nejad, F. Marsili, F. Mattioli, R. Leoni, D. Bitauld, D. Sahin, G. J. Hamhuis, R. Noetzel, R. Sanjines, and A. Fiore, “Nanowire superconducting single-photon detectors on GaAs for integrated quantum photonic applications,” Appl. Phys. Lett. |

11.. | 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. Optic. |

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

13.. | T. Amri, “Quantum Behavior of Measurement Apparatus,” arXiv:1001.3032v2 (2011). |

14.. | R. J. Barlow, |

15.. | Y. Yamamoto and A. Imamoglu, |

16.. | D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, Loss-Tolerant Characterization of Nonclassical Photon Statistics,” Phys. Rev. Lett. |

17.. | G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. |

18.. | A. Semenov, A. Engel, H.-W. Hübers, K. Il’in, and M. Siegel, “Spectral cut-off in the efficiency of the resistive state formation caused by absorption of a single-photon in current-carrying superconducting nano-strips,” Eur. Phys. J. B |

19.. | M. K. Akhlaghi and A. H. Majedi, “Semiempirical Modeling of Dark Count Rate and Quantum Efficiency of Superconducting Nanowire Single-Photon Detectors,” IEEE T. Appl. Supercon. |

20.. | A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Phot. |

21.. | O. Haderka, M. Hamar, and J. Perina, “Experimental multi-photon-resolving detector using a single avalanche photodiode,” Eur. Phys. J. D |

22.. | P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-number-resolving detectors,” New J. Phys. |

23.. | E. A. Dauler, A. J. Kerman, B. S. Robinson, J. K. W. Yang, B. Voronov, G. Gol’tsman, S. A. Hamilton, and K. K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Optic. |

**OCIS Codes**

(040.0040) Detectors : Detectors

(270.4180) Quantum optics : Multiphoton processes

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Detectors

**History**

Original Manuscript: October 24, 2011

Revised Manuscript: December 21, 2011

Manuscript Accepted: January 6, 2012

Published: January 23, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

J. J. Renema, G. Frucci, Z. Zhou, F. Mattioli, A. Gaggero, R. Leoni, M. J. A. de Dood, A. Fiore, and M. P. van Exter, "Modified detector tomography technique applied to a superconducting multiphoton nanodetector," Opt. Express **20**, 2806-2813 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2806

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### References

- E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature409, 46–52 (2001). [PubMed]
- A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: theory and practice,” New J. Phys.11, 093038 (2009).
- 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 (2008).
- M. K. Akhlaghi, A. H. Majedi, and J. S. Lundeen, “Nonlinearity in Single Photon Detection : Modeling and Quantum Tomography,” Opt. Express19, 21305 (2011). [PubMed]
- M. Hofherr, D. Rall, K. Ilin, M. Siegel, A. Semenov, H.-W. Hübers, and N. A. Gippius, “Intrinsic detection efficiency of superconducting nanowire single-photon detectors with different thicknesses,” J. Appl. Phys.108, 014507 (2010).
- D. Bitauld, F. Marsili, A. Gaggero, F. Mattioli, R. Leoni, S. J. Nejad, F. Lévy, and A. Fiore, “Nanoscale optical detector with single-photon and multiphoton sensitivity,” Nano Lett.10, 2977–81 (2010). [PubMed]
- G. N. Goltsman, 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 (2001).
- I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science328, 879–81 (2010). [PubMed]
- A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, V. Anant, K. K. Berggren, G. N. Goltsman, and B. M. Voronov, “Constriction-limited detection efficiency of superconducting nanowire single-photon detectors,” Appl. Phys. Lett.90, 101110 (2007).
- A. Gaggero, S. J. Nejad, F. Marsili, F. Mattioli, R. Leoni, D. Bitauld, D. Sahin, G. J. Hamhuis, R. Noetzel, R. Sanjines, and A. Fiore, “Nanowire superconducting single-photon detectors on GaAs for integrated quantum photonic applications,” Appl. Phys. Lett.97, 151108 (2010).
- 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. Optic.56, 432 (2009).
- G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Full quantum characterization of superconducting photon counters” arXiv: 1103.2991 (2011).
- T. Amri, “Quantum Behavior of Measurement Apparatus,” arXiv:1001.3032v2 (2011).
- R. J. Barlow, Statistics (Wiley, 1989).
- Y. Yamamoto and A. Imamoglu, Mesoscopic quantum optics (Wiley, 1999).
- D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, Loss-Tolerant Characterization of Nonclassical Photon Statistics,” Phys. Rev. Lett.97, 043602 (2006) [PubMed]
- G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett.95, 6 (2005).
- A. Semenov, A. Engel, H.-W. Hübers, K. Il’in, and M. Siegel, “Spectral cut-off in the efficiency of the resistive state formation caused by absorption of a single-photon in current-carrying superconducting nano-strips,” Eur. Phys. J. B47, 495–501 (2005).
- M. K. Akhlaghi and A. H. Majedi, “Semiempirical Modeling of Dark Count Rate and Quantum Efficiency of Superconducting Nanowire Single-Photon Detectors,” IEEE T. Appl. Supercon.19, 361–366 (2009).
- A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Phot.2, 302–306 (2008).
- O. Haderka, M. Hamar, and J. Perina, “Experimental multi-photon-resolving detector using a single avalanche photodiode,” Eur. Phys. J. D28, 11 (2003).
- P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-number-resolving detectors,” New J. Phys.9, 233–233 (2007).
- E. A. Dauler, A. J. Kerman, B. S. Robinson, J. K. W. Yang, B. Voronov, G. Gol’tsman, S. A. Hamilton, and K. K. Berggren, “Photon-number-resolution with sub-30-ps timing using multi-element superconducting nanowire single photon detectors,” J. Mod. Optic.56, 13 (2008).

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