Algorithm for finding clusters with a known distribution and its application to photon-number resolution using a superconducting transition-edge sensor
Zachary H. Levine, Thomas Gerrits, Alan L. Migdall, Daniel V. Samarov, Brice Calkins, Adriana E. Lita, Sae Woo Nam
Improving photon-number resolution of single-photon sensitive detectors is important for many applications, as is extending the range of such detectors. Here we seek improved resolution for a particular superconducting transition-edge sensor (TES) through better processing of the TES output waveforms. With that aim, two algorithms to extract number resolution from TES output waveforms are compared. The comparison is done by processing waveform data sets from a TES illuminated at nine illumination levels by a pulsed laser at 1550 nm. The algorithms are used to sort the individual output waveforms and then create clusters associated with individual photon numbers. The first uses a dot product with the waveform mean (for each illumination level), while the second uses K-means clustering modified to include knowledge of the Poisson distribution. The first algorithm is shown to distinguish adjacent peaks associated with photon numbers up to 19, whereas the second algorithm distinguishes photon numbers up to 23, using the same data.
Improving photon-number resolution of single-photon sensitive detectors is important for many applications, as is extending the range of such detectors. Here we seek improved resolution for a particular superconducting transition-edge sensor (TES) through better processing of the TES output waveforms. With that aim, two algorithms to extract number resolution from TES output waveforms are compared. The comparison is done by processing waveform data sets from a TES illuminated at nine illumination levels by a pulsed laser at 1550 nm. The algorithms are used to sort the individual output waveforms and then create clusters associated with individual photon numbers. The first uses a dot product with the waveform mean (for each illumination level), while the second uses K-means clustering modified to include knowledge of the Poisson distribution. The first algorithm is shown to distinguish adjacent peaks associated with photon numbers up to 19, whereas the second algorithm distinguishes photon numbers up to 23, using the same data.
As part of an effort to extend fundamental single-photon measurements into the macroscopic regime, we explore how best to assign photon-number uncertainties to output waveforms of a superconducting transition edge sensor and how those assignments change over that extended dynamic range. Three methods are used. At the lowest photon numbers (up to 20 photons), the widths of histogram peaks of individual waveforms are used to determine the uncertainty. From 100 to 1000 photons, mean waveforms are used to create a photon-number scale. The photon-number uncertainty of the detector in this range is given by the excess of the total variance of the photon number obtained from individual waveforms on this scale beyond the shot noise due to the source. In the midrange (from 10 to 100 photons), including a range where the two other methods do not produce definitive results, we fit waveforms to several adjacent mean waveforms to estimate the photon-number uncertainty. A one-standard-deviation uncertainty in photon number of no more than ±1 is found for pulses of up to 100 photons.
Superconducting transition-edge sensors (TES) are extremely sensitive microcalorimeters used as photon detectors with unparalleled energy resolution. They have found application from measuring astronomical spectra through to determining the quantum property of photon-number, n ^ = a ^ † a ^ , for energies from 0.6-2.33eV. However, achieving optimal energy resolution requires considerable data acquisition – on the order of 1GB/min – followed by post-processing, which does not allow access to energy information in real time. Here we use a custom hardware processor to process TES pulses while new detections are still being registered, allowing photon-number to be measured in real time as well as reducing data requirements by orders-of-magnitude. We resolve photon number up to n = 16 – achieving up to parts-per-billion discrimination for low photon numbers on the fly – providing transformational capacity for applications of TES detectors from astronomy through to quantum technology.
Transition-edge sensors (TES’s) are are extremely sensitive calorimeters able to measure energies of the order of a few electronvolts. They are now well-known in the quantum technology community because of their superb combination of high-efficiency— higher than 95% for photons in the near infrared range [1]—essentially zero dark counts, and photon-number resolution. One of the main challenges when working with TES’s is to extract the photon-number information from the continuous output signal of the detectors. The usual procedure is to accumulate signals over some time—typically ~2 GB per minuite—and then use post-processing techniques to obtain photon-number resolution from the recorded signals [2] Here we introduce an FPGA circuit that analyses TES signals in real-time—recording only specific characteristics such as signal area, height, and length—allowing near realtime photon-number resolution and reducing the memory requirements by orders-of-magnitude. Using this new capability, we are able to optimise the number-resolution of the detector to the range we are interested in for each new experiment. In a preliminary study, we calibrated the TES with a weakly-pulsed 820 nm diode laser, and using just the area data from the FPGA were able to discriminate up to 15 photons. We are able to accurately discriminate an n=2 photon Fock state with parts-per-billion precision, dropping to parts-per-hundred precision at n=15 [3]. It is well known that there are physical limits to the precision with which an image can be formed. There are ways in which this limit can be circumvented, for example using super-resolution techniques that exploit the physical structure of the object, or object illumination with entangled states of light. However, in many applications—for example when the object is very far away—we cannot directly interact with the object, or illuminate it with entangled light: the quantum state of the light field is all that is accessible to the observer. Given a finite size imaging system in the far field—i.e., systems with a finite effective numerical aperture—we show the best way to extract the spatial characteristics of the light source. We implement a general imaging method by measuring the complex degree of coherence using linear optics and our photon-number-resolving detectors. In the absence of collective or entanglement-assisted measurements, our method is optimal over a large range of practically relevant values of the complex degree of coherence [4]. We measure the size and position of a small distant source of pseudo-thermal light, and show that our method outperforms the traditional imaging method by an order of magnitude in precision. Additionally, we show that a lack of photon-number resolution in the detectors has only a modest detrimental effect on measurement precision, further highlighting the practicality of this method as a way to gain significant imaging improvements in a wide range of imaging applications. References [1] A. E. Lita, A. J. Miller, and S. W. Nam, Optics Express 16, 3032 (2008). [2] G. Brida, et al., New Journal of Physics 14, 085001, (2012). [3] L. Assis, et al., preprint (2020) [4] L. A. Howard, et al., Physical Review Letters 123, 143604 (2019).
We compare methods for signal classification applied to voltage traces from transition edge sensors (TES) which are photon-number resolving detectors fundamental for accessing quantum advantages in information processing, communication, and metrology. We quantify the effect of numerical analysis on the distinction of such signals. Furthermore, we explore dimensionality reduction techniques to create interpretable and precise photon-number embeddings. We demonstrate that the preservation of local data structures of some nonlinear methods is an accurate way to achieve unsupervised classification of TES traces. We do so by considering a confidence metric that quantifies the overlap of the photon-number clusters inside a latent space. Furthermore, we demonstrate that for our dataset previous methods such as the signal's area and principal component analysis can resolve up to 16 photons with confidence above 90% whereas nonlinear techniques can resolve up to 21 photons with the same confidence threshold. In addition, we showcase implementations of neural networks to leverage information within local structures, aiming to increase confidence in assigning photon numbers. Finally, we demonstrate the advantage of some nonlinear methods to detect and remove outlier signals.
Typically, transition edge sensors resolve photon number of up to 10 or 20 photons, depending on the wavelength and TES design. We extend that dynamic range up to 1000 photons, while maintaining sub-shot noise detection process uncertainty of the number of detected photons and beyond that show a monotonic response up to ≈ 6 · 10(6) photons in a single light pulse. This mode of operation, which heats the sensor far beyond its transition edge into the normal conductive regime, offers a technique for connecting single-photon-counting measurements to radiant-power measurements at picowatt levels. Connecting these two usually incompatible operating regimes in a single detector offers significant potential for directly tying photon counting measurements to conventional cryogenic radiometric standards. In addition, our measurements highlight the advantages of a photon-number state source over a coherent pulse source as a tool for characterizing such a detector.
Transition-Edge Sensors (TESs) are microcalorimeters that measure the energy of incident single photons by the resistance increase of a superconducting film biased within the superconducting-to-normal transition. TES are able to detect single photons from IR to X-ray with an intrinsic energy resolution and photon-number discrimination capability. Metrology, astronomy and quantum \ncommunication are the fields where these properties can be particularly useful. In this work, we report about characterization of different TESs based on Ti films. Single photons have been detected from 200nm to 800 nm working at transition temperature Tc ∼ 100 mK. Using a pulsed laser at 690nm we have demonstrated the capability to resolve up to five photons.