# Towards Measuring Terahertz Photon Statistics by a Superconducting Bolometer

^{*}

## Abstract

**:**

_{3}crystal placed to a cooled cryostat together with the bolometer NbN film. Results of theoretical approximation of experimental histograms reveal the discrete nature of THz detection by superconducting bolometers and open a way for studying their quantum characteristics. It is shown that bolometer readings per pulse consist of discrete counts (“single charges”), with the mean number linearly dependent on the number of input photons. Contributions of single counts to a total analog reading are statistically distributed according to the normal law, with average values slightly depending on the number of counts in each reading. A general formula is proposed to describe the relationship between continuous statistical distribution of the bolometer readings and discrete quantum statistics of the incident photons.

## 1. Introduction

## 2. Materials and Methods

^{3+}:YLF solid-state laser, generated in KTP crystal (2ω), was used to pump Mg:LiNbO

_{3}crystal with 7.1 mol.% Mg content (LN). The pump wavelength was 523.35 nm, the pump pulse duration and repetition frequency were 10 ns and 7 kHz, correspondingly, and the pump beam waist at the crystal was ~100 μm. The crystal was mounted in He cryostat together with superconducting HEB manufactured by company SCONTEL, Russia [32]. Temperatures of the crystal and the bolometer sensitive NbN element were kept at the same value: 4.8 K. According to data provided by manufacturers, the characteristic response time and noise equivalent power of the bolometer were 50 ps and $2.5\text{}\times {\text{}10}^{-13}\mathrm{W}\cdot {\mathrm{Hz}}^{-1/2}$, respectively. A special planar logarithmic spiral antenna [33] was optimized for the spectral range 0.1–6 THz. The cryostat windows of 20 mm diameter, designed to enter the pump radiation into the cooled chamber and to extract it out after passing the crystal, were equipped with ITO filters [34]. The bolometer was inside three shielding shells of the cryostat. Glass windows were installed on an external aluminum shell that was kept at room temperature. The next shell from a stainless steel was provided by open windows. Finally, a third brass shell was in contact with the cooled board. The ITO filters were installed into the windows in the third shell, and the bolometer sensitive element was attached to the board. All cryostat compartments were pumped out to a pressure of 10

^{−6}mbar. By this arrangement, the temperature of both ITO filters practically coincided with the temperature of the bolometer, and these cooled filters prevented outside thermal radiation at THz frequencies from entering the cryostat [5]. At the same time, not only direct pumping beam, but also optical radiation of Stokes and anti-Stokes frequencies, generated in LN at small angles, could leave the cryostat without any significant absorption [11]. In addition, IR filter (ZitexRG-106) protected the bolometer from pump radiation elastically scattered within the crystal.

_{3}crystal was excited in type-0 geometry. According to phase-matching conditions, the photons of ~1 THz frequency were most effectively generated in this sample at polar angles ${\theta}_{i0}\approx {60}^{\circ}$ with respect to pump direction [4,10]. THz part of PDC radiation was extracted from the crystal through its side face in immediate proximity to the pump beam [5]. A high-resistivity silicon 45° prism was used in order to avoid total internal reflection of THz waves at the output crystal surface. After passing the prism, a part of PDC radiation was selected by a band pass filter (Tydex BPF1.0-24/31) and focused by 5 mm Si hemispherical lens to the bolometer input window. The total spectrum of terahertz photons which can be generated under PDC in the sideways geometry with Mg:LiNbO

_{3}crystal occupies the range from 0.1 THz up to 7.2 THz [35]. A narrow band pass filter with a central frequency of 1 THz and a half-width of 0.15 THz transmitted a part of this wide spectrum. The energy bandgap in NbN is 4.5 meV [36], which corresponds to a frequency of 1.15 THz. However, as it was shown experimentally in [33], the spectral responsivity of our bolometer is the highest in the range 1–2.5 THz. So, the spectral range of THz photons incident on the bolometer in our set-up hits the range of its maximal responsivity. Electric signal from HEB detector was amplified by a broadband cryogenic amplifier associated with the bolometer and then directed to time-gated Boxcar integrator SRS 250. The integrator was gated by pulses from a pin-diode (PD) placed in the lateral pump beam outlet, the integration strobe window was 6 ns.

## 3. Results and Discussion

_{33}is the crystal nonlinear coefficient, $L$ is its length, $\Delta {\omega}_{i}$, $\Delta {\Omega}_{i}$, and $S$ are the spectral band, a solid angle, and an input area of the detection system, correspondingly. Planck’s factor $\langle {N}_{T}\rangle ={\left[\mathrm{exp}(\hslash {\omega}_{i}/{k}_{B}T)-1\right]}^{-1}$ accounts thermal field fluctuations at the crystal temperature $T$. Frequencies of quantum-correlated THz (${\omega}_{i}$) and optical (${\omega}_{s}$) photons are related with the pump frequency ${\omega}_{p}$ via the energy conversation law, $\hslash {\omega}_{s}+\hslash {\omega}_{i}=\hslash {\omega}_{p}$; ${n}_{p},{n}_{s},{n}_{i}$ are the crystal refractive indexes at corresponding frequencies, ${\mathsf{\theta}}_{s0},{\mathsf{\theta}}_{i0}$ are the propagation angles of optical and THz photons generated under exact phase-matching condition. Evidently, the mean THz wave power and, correspondingly, the averaged HEB readings should demonstrate the same dependence on pump power as is predicted by Equation (1). It was shown in our previous study of PDC process for the same LN crystal sample [38] that the numerical relation

## 4. Conclusions

_{3}crystal pumped by 10 ns 523 nm laser pulses. Together with the input elements and NbN film of each bolometer, the crystal was placed in a cooled He cryostat closed from thermal radiation from other external sources. It was shown that statistical distributions of bolometer readings per laser pulse can be approximated with high accuracy under the following assumptions:

- (1)
- any reading consists of a number of discrete single counts with Poisson statistical distribution,
- (2)
- contributions of individual counts (“single charges”) to each total reading are statistically distributed according to the normal law,
- (3)
- the average values of contributions (“mean single charges”) depend on the number of counts in each reading.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematics of the experimental set-up. The pumping radiation for PDC is generated using a series of the pulsed laser (Nd

^{3+}:YLF), the frequency doubling crystal (KTP), and the 1st harmonic cut-off filter (HPF). A small part of the pump beam is diverted by a mirror to a PIN diode (PD) which triggers integrator and boxcar averager modules SR250 (Boxcar integrator) at the beginning of each laser pulse. PDC originates in Mg-doped lithium niobate crystal (Mg:LiNbO

_{3}) equipped by a prism from high-resistivity silicon (Si prism). The crystal and a system for detecting idler THz radiation are placed inside a He cryostat. The detection system contains a pump cut-off filter film (Zitex G-106), a narrow THz band pass filter Tydex BPF1.0-24/31 (TF), a focusing Si hemispherical lens (Si lens), and the NbN superconducting bolometer SCONTEL (HEB). The optical photons (signal radiation) are generated at small angles to the pump beam and leave the cryostat via the same windows as the pump radiation passes (marked by the blue color). In each window there is an external glass filter and an internal ITO filter. The latter is kept at the same temperature as the crystal and HEB.

**Figure 2.**Histograms of the HEB detector readings, recorded experimentally under different laser pump powers (circles), and results of their numerical modeling (curves). The time-averaged pump power was 29.6 mW (red circles), 53.4 mW (magenta circles), 76.6 mW (blue circles), and 92.8 mW (green circles). The transport current of HEB was 64 µA (a) and 69 μA (

**b**). Solid curves: numerical approximation by Equation (7). A dashed curve (

**a**): approximation by Equation (6); the inset shows a plot section in an enlarged scale.

**Figure 3.**The mean number of the bolometer photo counts per pulse (in absolute unites, dots), and the theoretical prediction for the mean number of incident photons (in arbitrary units, a solid curve), both versus the time-averaged pump power. Open dots: results of approximation of experimental histograms by Equation (6), filled dots: by Equation (7). Black circular dots correspond to experimental data obtained under the HEB transport current of 64 µA, red triangular dots correspond to the data under the HEB transport current of 69 μA.

**Table 1.**Results of approximation by two probability distributions, ${P}_{c}(z)$ from Equation (6) and $P(z)$ from Equation (7). The uncertainties are the root-mean-square errors obtained under approximating the experimental dependences by the least squares method.

Bolometer Current | Pump Power | Number of Counts <m> | Mean Single Current <e_{1}>/τ, arb.un. | Single-Current Deviation σ, arb.un. | |||
---|---|---|---|---|---|---|---|

Equation (6) | Equation (7) | Equation (6) | Equation (7) | Equation (6) | Equation (7) | ||

64 μA | 29.6 mW | 7.04 ± 0.14 | 4.58 ± 0.12 | 22.9 ± 0.5 | 46.4 ± 1.9 | 44.9 ± 0.2 | 56.6 ± 1.0 |

53.4 mW | 8.95 ± 0.7 | 9.22 ± 0.06 | 43.8 ± 3.6 | 35.1 ± 0.4 | 37.7 ± 2.6 | 16.4 ± 0.5 | |

76.6 mW | 11.9 ± 5.7 | 13.82 ± 0.03 | 45 ± 23 | 29.7 ± 0.15 | 35 ± 20 | 11.3 ± 0.2 | |

69 μA | 53.4 mW | 8.9 ± 1.2 | 9.07 ± 0.06 | 80 ± 12 | 66.5 ± 0.5 | 66.8 ± 9.2 | 30.3 ± 1.2 |

76.6 mW | 12.4 ± 8.8 | 14.3 ± 0.2 | 71 ± 53 | 48.3 ± 1.0 | 54.7 ± 48.3 | 19.7 ± 1.6 | |

92.8 mW | 17.91 ± 0.03 | 16.1 ± 0.8 | 87.9 ± 0.2 | 156 ± 4 | 19.7 ± 0.6 | 115 ± 4 |

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**MDPI and ACS Style**

Prudkovskii, P.; Leontyev, A.; Kuznetsov, K.; Kitaeva, G.
Towards Measuring Terahertz Photon Statistics by a Superconducting Bolometer. *Sensors* **2021**, *21*, 4964.
https://doi.org/10.3390/s21154964

**AMA Style**

Prudkovskii P, Leontyev A, Kuznetsov K, Kitaeva G.
Towards Measuring Terahertz Photon Statistics by a Superconducting Bolometer. *Sensors*. 2021; 21(15):4964.
https://doi.org/10.3390/s21154964

**Chicago/Turabian Style**

Prudkovskii, Pavel, Andrey Leontyev, Kirill Kuznetsov, and Galiya Kitaeva.
2021. "Towards Measuring Terahertz Photon Statistics by a Superconducting Bolometer" *Sensors* 21, no. 15: 4964.
https://doi.org/10.3390/s21154964