1. Introduction

Using the heterodyne technique, one could achieve a fine frequency resolution limited by the stability and linewidth of the Local Oscillator (LO). Such a technique was originally developed for radio-waves and later on spread into high frequency radio-astronomy. Simultaneously, its application has been promoted for optics [1], with the first experimental proof demonstrated in the before-laser age [2]. With invention of the laser, heterodyne analysis of the signals has been widely exploited in optics, including visible and IR waves, in numerous experiments [3–5], and applications in spectroscopy [6,7], space research [8–10], LIDAR [11,12], optics communication systems [13,14], optical tomography [15–17]. With aim to develop a sensitive tool for the astrophysical observations, heterodyne detection was demonstrated with the use of a p-i-n diode at the telecommunication wavelength [18]. The best sensitivity was achieved in expense of enlarged integration time and quite narrow intermediate frequency (IF) bandwidth which was compensated by the LO sweep. However, a diode–based heterodyne receivers require a significant amount of the LO power [19,20]. Another approach was presented in [21], where a superconducting Hot-Electron-Bolometer mixer made from a thin superconducting film of NbN was used for coherent detection of near-IR radiation. Integration of NbN detector with a plasmonic nano-antennae [22,23] allowed to decrease significantly the LO power preserving a relatively large IF bandwidth which is determined by the hot-electrons relaxation time in the film [24]. However, the LO power required was still quite noticeable (a few tens of microwatts) and sensitivity was strongly dependent on the radiation absorption by the superconducting film. This approach required also special care to be taken of the IF signal processing, including the need for a Low Noise Cryogenic Amplifier followed by a room temperature amplification chain.

An alternative detection approach which combines precise timing resolution of heterodyne technique with the sensitivity of the single quanta detectors has been introduced in [21] and reintroduced in [25]. To achieve such a combination, a photon counter is simultaneously irradiated by the LO and a weak signal beam, which are differ in frequency slightly. Due to the interference between the LO and signal waves the power absorbed by the counter oscillates with the intermediate frequency (which is the difference between the LO and signal frequencies). This variation of the power in time produces the variation of photon flux that can be recorded by the photon counter. Afterwards one is able to restore the incoming signal frequency knowing reference laser frequency and retrieving the intermediate frequency from the recorded power variation, quite analogous to a classical mixer operation. Experimental study of the concept was done with use of a Geiger-mode operated avalanche photodiode (APD) array in [25–27]. Usage of the APD array allowed shortening the system dead-time significantly compared to the operation of a single pixel APD. The laser power was split into two beams, one serving as the signal and the other being the LO shifted in frequency by an acousto-optical modulator. This technique was shown to suit LADAR requirements. However, use of an APD for detection yields some disadvantages, namely, high dark count rate, long dead time and afterpulsing, leading to a limited counting rate [28,29].

In contrast, state-of-the-art superconducting nanowire single-photon detectors (SSPDs or SNSPDs) which respond to each incident photon by a short voltage pulse due to destroying of the superconducting state [30–32] outperform their counterparts in terms of sensitivity and counting rate, demonstrate very low dark count rates (about 0.1 s⁻¹), and quite short dead time, typically ~5 ns [31,32]. Several SNSPD design modifications were demonstrated to be capable for the photon-number resolving [33,34]. When integrated with resonating structures enhancing photon absorption [35], SNSPDs demonstrate above 90% detection efficiency [36] making them useful in a number of applications, such as free-space optical communication [37,38], quantum information processing [39–41], quantum key distribution [42], ranging [43], and life sciences [44,45]. Moreover, arranging SNSPDs into an array may lead [46,47] to even larger system count rates achieved and facilitate spatial resolution which would be beneficial for many practical applications. Recently SNSPDs integrated with optical waveguides [48,49] on a single chip were demonstrated providing a possible solution for on-chip quantum information processing and the quantum computer implementation [50,51].

In this paper we present implementation of the photon-counting heterodyne detection with use of a single SNSPD coupled to a single-mode optical fiber.

2. Experiment

For our work, we have exploited a typical meander-shaped SNSPD patterned by e-beam lithography on a 4 nm thick NbN film, with the latter being deposited by reactive DC magnetron sputtering on top of a high resistivity silicon substrate [52]. Physics of the SNSPD response to the incident radiation is thoroughly addressed in [31,32], including introduction to recently developed models. The nanowire width was 80 nm and spacing between the nanowires was 120 nm. Figure 1 depicts our experimental setup. The device was installed on top of a single mode optical fiber and a double-wall insert with vacuum insulation was used to bring the bath temperature down to 1.7 K, facilitating the filtering effect of the cooled single mode optical fibers [53]. One laser serves as the LO irradiating at 1550 nm wavelength, and the other one plays a role of the signal. Each laser is a fiber-coupled single mode distributed feedback laser packaged in a standard butterfly case with a thermo-electric cooler allowing for precise laser chip temperature control, allowing for smooth tuning of the LO wavelength. The width of the laser emission line was estimated with the homodyne technique using a delay fiber-optics line of a few kilometers, and is ~1 MHz, defined as FWHM.

 

Fig. 1 A schematic view of the experimental setup (see the details in the text). Insets: (a) interference of the LO and signal electric fields incident on the detector which represent the electric field beating, (b) schematic representation of distribution of pulses with time, (c) IF power spectrum in 1 MHz to 3 GHz window due to direct response of the SNSPD to incident radiation of a light source, (d) a train of pulses registered by the oscilloscope, (e) SEM image of the SNSPD chip and its central part – meandered NbN film.

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Mechanically controlled attenuators and fiber-optics polarization controllers are used to adjust laser power and polarization, respectively. With the use of a fiber-based beam splitter the wave fronts are combined and directed to the SNSPD. The SNSPD is biased by an under-critical current through a bias-tee. In our experiment we used bias current of 24 µA providing system detection efficiency (which includes detector-to-fiber coupling efficiency, probabilities of photon absorption and generation of pulse by the detector) of ~(8 ± 2) %, measured independently. An absorbed photon destroys superconductivity for a short time giving the rise to resistance and leading to a voltage pulse which can be further amplified and recorded by an oscilloscope or any other electronics. When the SNSPD is illuminated by both LO and signal simultaneously (which are operated with a slight difference in their wavelength) the waves interfere and the electric field oscillates (as schematically shown in the inset (a) in Fig. 1) with the frequency which is the difference between the LO and signal frequencies. This frequency is traditionally called the “intermediate frequency”. Photon count rate is proportional to the incident power and thus also oscillates with the intermediate frequency resulting in a train of pulses with varied density of the pulses, as depicted schematically in the inset (b) in Fig. 1. The voltage pulses from the SNSPD produced in response to the photon absorption are amplified by the room temperature amplification chain of two amplifiers with total gain of 48 dB and bandwidth of 2 GHz, and the IF signal is further fed in to either an RF signal analyzer or digital oscilloscope. Optical power of the lasers was measured independently for both lasers by optical spectrum analyzer with the instrument accuracy of ~10% and uncertainty of a single measurement of about 5%. Attenuation down to 1 pW was done with help of a manually controlled fiber-optics attenuator. Even though our system was not free from “true” power fluctuations caused by the laser output power fluctuations and uncertainty in the measuring power, we assume that these do not significantly affect the data analysis as for the calculation we used mean values of the power, obtained with relative standard error of about 1%. An average of the measured data was also used when determining the IF power for graphical representation of the data in Fig. 3. Relative standard error was about 5% caused by combination of the optical power measurement uncertainty addressed above, fluctuation of the laser frequency, the SNSPD intrinsic detection efficiency dependence on the bias current. Fluctuation of the IF chain gain was less than +/−0.05 dB, which did not affect the IF power measurement with the SA and, actually, did not play a role in the statistical analysis of the photo-counts.

3. Analysis

In principle, one can analyze time-resolved photon statistics to obtain the intermediate frequency f_IF and the power of the down-converted signal P_S. Theoretical basis of the concept with a rigorous statistical analysis was disclosed in details in [54]. As beatings emerge due to interference between the two light beams of the two sources, and if the field can be treated classically, the power irradiating the detector at a specific moment of time t is:

Here, the coefficient _{$\alpha \equiv \frac{2\sqrt{P_S{P}_{LO}}}{P_S+P_{LO}}$} determines the relative depth of the modulation. Finite spectral width of the beatings can be accounted for by stochastic slow time variations of the phase φ.

In case when the detector resolves single photons, the field should be considered as a quantized one. In this case the power given by Eq. (1) has the meaning of the average over ensemble of measurements and determines probability to detect a photon per unit time:

with η being the detection efficiency. To extract power of the signal and its spectrum from the sequence of the pulses, one can use Fourier-analysis: the power spectrum density (PSD) of the Fourier-transformed voltage trace from the detector containing many photon counts is

where s_IF(f) is the power spectrum of the beatings (having maximum at f_IF and normalized such as _{${\displaystyle \int s_{IF}\left(f\right)}df=1$}, ε(f) is the energy spectrum density of a single voltage pulse from the detector, r_S, r_LO are the count rates which were caused by the signal and local oscillator lasers independently, i.e. without interference, r_D denotes the rate of dark counts of the detector, and the term S_el(f) is introduced to account for spectral density of additional electrical noise (added by amplifiers and the IF chain), which plays a role if the IF signal is processed by the spectrum analyzer (but would be absent if only time moments of the photon counts are recorded and the Fourier analysis is applied). Energy spectrum density of the single voltage pulse from the detector can be expressed through the Fourier transform of the single voltage pulse (v(f)) and the input impedance of the spectrum analyzer R as _{$\epsilon \left(f\right)={\left|v\left(f\right)\right|}^2/R$}. The signal contribution is presented by the first term in Eq. (3), and the other terms proportional to r_S,LO,D should be considered as a statistical noise, related to the fluctuation of photon and dark counts. According to Eq. (3), gain bandwidth is limited by the energy spectrum of the pulse ε(f) and we shall proof this experimentally below. Term “gain bandwidth” originates from the traditional characterization of heterodyne receivers at THz frequencies. It is used as the response or conversion bandwidth. Use of “gain bandwidth” here helps to underline that SNSPD exhibits intrinsic gain which drops down with increase of the IF, as discussed below.

Unlike that for APDs, gain and bandwidth of SNSPD are independent, and term Gain-Bandwidth-Product is not applicable to this type of detectors.

One can use the RF spectrum analyzer to perform Fourier transformation in hardware. Alternatively, only the moments of time when the counts are registered can be recorded and then processed in line with the schematic sketched above. In this case a proper post-processing is required but obtained IF signal is altered neither by the pulse spectrum, which is eliminated from the analysis, nor by the electrical noise, which affects averaged power (processed by the SA) rather than the pulses itself (processed by the oscilloscope/fast digital counter) as long as the pulse amplitude is sufficiently higher than the continuum output from the detector. Both approaches should produce same result. We tested both methods in our experiments and the results are compared in Fig. 2.

 

Fig. 2 (a) SNSPD output signal processed with the RF spectrum analyzer with resolution bandwidth (RBW) set to 300 kHz. Red curve is obtained when SNSPD is illuminated by both LO and signal lasers. The signal at the IF is ~2 MHz wide peak (marked P_peak(f)). The noise floor (green curve marked as P_noise(f)) is essentially the spectrum of the single-photon response pulse of the SNSPD. (b) The same result is obtained by mathematical Fourier-analysis of the 1-ms-long trace of the SNSPD pulses recorded with the digital oscilloscope. Solid (green) curve is the fit by the Cauchy-Lorentzian distribution formula.

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First we used the power spectrum analyzer to obtain the spectrum shown in Fig. 2(a). When SNSPD is irradiated by either the LO or signal laser, output of the spectrum analyzer is the spectrum of a single SNSPD pulse only with its width defined by the duration of the pulse and the electronics bandwidth. When both the LO and signal lasers are in operation, one can observe a peak (red curve marked “LO + signal” in Fig. 2(a)) in the spectrum which is located exactly at the frequency which is the difference between the LO and signal laser frequencies. In our case the LO and signal are shifted in frequency by 10 MHz. The spectrum of a single source represented by a green curve and marked “1 source” in Fig. 2(a) acts as a noise floor for the signal.

Then, we used a 1 Gs/sec digital oscilloscope to acquire an oscillogram of 1 ms long with time resolution of 1 ns. With mathematical post-processing, we determined moments of time t_k at which the photo-counts occurred, calculated Fourier transform as the sum of exponents with arguments proportional to those times ∑_kexp(2πift_k), and then obtained power spectrum shown by red line in Fig. 2(b) which reproduces reasonably well the analog spectrum obtained with the RF spectrum analyzer. The green solid line is the fit by the Cauchy-Lorentzian distribution formula: f(x,x₀,γ) = [1/(π∙γ)]∙[γ²/[(x-x₀)² + γ²]] with x₀ the peak frequency, 2γ ≈2 MHz the peak width, which equals to doubled width of the laser emission line if identical lasers with lorentzian profile of the line are used for the LO and signal, as it was in our case. One can see that both methods produce similar results, meanwhile mathematical processing of the digital data from the oscilloscope gives the result with already eliminated spectrum of the voltage pulse. Below we present the results obtained by analog processing with use of the RF spectrum analyzer.

To characterize signal and noise properties of our receiver, we measured dependence of output PSD vs signal power at f = f_IF = 15 MHz and different LO powers. Figure 3 presents an example of such dependence measured at 15.6 pW LO power. Red points correspond to the PSD at f = f_IF, whereas blue points are PSD at the same frequency but with changed f_IF, such as the peak is shifted away, i.e., blue points correspond to PSD_N and red ones correspond to PSD_peak = PSD_S(f_IF) + PSD_N(f_IF). Hence, to calculate PSD_S, one has to subtract PSD_N from PSD_peak.

 

Fig. 3 IF power vs input signal power for 15.6 pW LO power measured at RBW 300 kHz. Lines are guides for an eye. Inset shows the ratio of PSD of output signal to PSD of noise (left vertical axis) and corresponding SNR (right axis) for minimal levels of signal power. Line is the quantum limit of noise for η = 0.08.

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To estimate absolute gain of our detector, one needs to know signal power and gain of the IF chain. As one can extract from data shown in Fig. 3, at signal laser power of 1 pW the output power in the band RBW = 300 kHz is 2 × 10⁻⁸ W, and the total power, in the band Δf = 2 MHz, is P_S = 1.3 × 10⁻⁸ W. Accounting for gain of the IF chain (48 dB) and detection efficiency η = 0.08, one finds that our SNSPD exhibits intrinsic gain of ~10 dB, which we relate to the operation principles of the SNSPD: as a photon is absorbed, SNSPD generates a voltage pulse whose amplitude is determined by the Joule power released while the SNSPD is in the resistive state. This is a kind of internal amplification mechanism allowing one to increase significantly the gain which essentially results in the system sensitivity, eliminating the need for a low-noise amplification stage in the IF chain.

As a measure of noise, one can use ratio of PSD of the output signal (PSD_S) at the frequency f_IF (corresponding to its maximum) to PSD of the noise (PSD_N). According to Eq. (3), this ratio is

Substituting here Eq. (4) one comes to

The inset in Fig. 3 demonstrates near quantum limited level of noise achieved at the minimum power of signal on the input P_Sin (below 1 pW). The quantum limit of noise given by inequalities in the RHS of Eq. (4) or Eq. (5) is represented by straight line. Rate of counts, figuring there, was expressed through P_Sin as r_S = η × P_Sin/hν, where hν is the photon energy.

Another important characteristic of a practical heterodyne receiver is the intermediate frequency bandwidth, which determines maximum separation between the LO and incident signal in frequency allowing yet for an efficient signal acquisition. We have measured both gain and SNR intermediate frequency bandwidths of the SNSPD based heterodyne receiver. By tuning the LO wavelength (while the amplitude and frequency of the signal source remained unchanged) we measured the PSD_S(f_IF). By definition, this quantity is proportional to frequency-dependent gain. We defined Gain Bandwidth (GBW) as the frequency f_3dB where the Gain is reduced by a factor of 2. The measurement results are summarized in Fig. 4 for SNSPDs of two sizes: 7 × 7 µm² and 3 × 3 µm². Smaller detector has shorter pulse duration (and, as a consequence, shorter dead time) due to a lower kinetic inductance [55], and, hence, a wider GBW, as shown in the Fig. 4(a). For 7 × 7 µm² detector GBW is measured to be 65 MHz (blue triangles in Fig. 4(a)). To proof that the gain bandwidth is limited by the spectrum width of the single-photon pulse of the SNSPD we performed Fourier transform of 7 × 7 µm² SNSPD single-photon pulse (shown in the inset in Fig. 4(a)). The calculated spectrum is plotted in Fig. 4(a) as open magenta squares. The 3 × 3 µm² detector has the GBW of 140 MHz due to its shorter photo-response pulse duration.

 

Fig. 4 (a) Gain bandwidth (GBW) measured for SNSPDs of two sizes: 7×7 µm² (blue triangles) and #2: 3×3 µm² (red crosses), which differ in photo-response pulse duration. The GBW is limited by the duration of the single-photon response pulse: magenta squares is the Fourier transform of the single-photon response of 7×7 µm² SNSPD, shown in the inset. (b) Signal-to-noise ratio (SNR) bandwidth for 3×3 µm² SNSPD. (c) Signal-to-noise ratio (SNR) bandwidth for 7×7 µm² SNSPD. Experimental points are fitted by Eq. (6). Actual level of the SNR plateau before its decay is determined by power of the LO and signal lasers, which were different for the two detectors. In order to emphasize comparison on the SNR-bandwidth we use relative-SNR rather than actual SNR values.

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Similarly, the Signal-to-Noise Bandwidth (SNR-BW) is defined as the intermediate frequency at which the SNR is reduced by a factor of 2, with the SNR(f) given by Eq. (5). Figure 4(b,c) shows the measured SNR-bandwidth of ~1400 MHz and ~1200 MHz for the same pair of detectors. We directly compare measured SNR to Eq. (6). To perform the comparison, one has to know r_S, r_LO, S_el(f) and ε(f). S_el(f) was obtained from distinct set of measurements of PSD on the output of spectral analyzer at different input power of single LO laser on input, including zero power. It was found to be nearly independent on frequency in the range of interest, from 0 to 1500 MHz, and equal to 4.5 × 10⁻¹⁶ W/Hz. ε(f) was obtained from Fourier-transform of digitized voltage pulse of the SNSPD, its dependence on frequency is presented in Fig. 3(a) and its absolute value at f = 50 MHz was 2 × 10⁻²² J/Hz.

Then we calculated r_S + r_LO, dividing measured PSD_N(f) at 50 MHz, where it was several order of magnitude greater than S_el(f) (thus the latter can be neglected) by ε(f). Analogously, we calculated the product r_Sr_LO by dividing measured PSD_S(f) by (ε(f)/Δf). We repeated this calculation for several greater values of frequency, taken from the range where frequency dependence of PSD_N,S(f) coincided with ε(f) (i.e. no nonlinear distortions of the pulse occurred) and at the same time S_el(f)<< PSD_N(f), and proved independence of the extracted rates on the frequency. The resulted plots are presented in Fig. 4(b,c) as solid curves.

Hence, we demonstrated that measured values are dominated by our measurement setup system noise. An ultimate limit on SNR BW of any photon-counter based heterodyne receiver is set by the photon-counter jitter. With increase of the IF, the time uncertainty dτ of the pulse appearance in regard to the photon absorption, i.e. jitter, becomes comparable with the periodicity of the beatings (2π/f_IF) leading to their averaged amplitude decrease at IF approaching 1/2πdτ and its complete vanishing at IF >> 1/2πdτ. For the SNSPD, probability distribution of pulses was found to have Gaussian shape and corresponding jitter, defined as FWHM, was shown to be less than 20 ps [56], which would correspond to a cut-off frequency f_cut-off = 2∙ln2/πdτ ≈22 GHz, which opens room for improvement of the heterodyne system presented in the paper.

4. Conclusion

To conclude, we have demonstrated that NbN thin film based superconducting nanowire single photon detector can be operated in the heterodyne regime. In this regime it requires local oscillator power of the order of a few picowatts to resolve subpicowatt optical signals. We demonstrated 65 MHz gain bandwidth for 7 × 7 µm² device limited by single-photon pulse duration, and 140 MHz for 3 × 3 µm² device with a shorter response pulse. The SNR bandwidth was ~1400 and ~1200 MHz for 7 × 7 µm² and 3 × 3 µm² device respectively. Although our system had somewhat poor detection efficiency, it still allowed us to demonstrate the coherent mode sensitivity limited by the quantum noise only. This opens a wide possibility for development of coherent detector arrays. Combining the advantages of heterodyne method and single-photon detection, it is possible to produce power spectral analysis of weak signals with high spectral resolution and sensitivity at the quantum noise limit.