Device and measurement setup

Figure 1a, b shows the detector and its measurement scheme. We couple the detector to an 8.4-GHz microwave source through a 50-Ω transmission line, which allows us to calibrate the heater power P h incident at the detector input with a decibel level of uncertainty. Essentially all incident heater power is absorbed by the long SNS junction between the leads H and G since the junction is long enough for its impedance to be almost entirely real, 36 Ω, and well matched to the transmission line impedance of Z 0 = 50 Ω. The imaginary part of the impedance, arising from the capacitor C 1 , is roughly −i0.2 Ω and an order of an ohm from parasitic series inductance at 8.4 GHz, and can be therefore neglected. Thus, an increase in P h leads to an increase in the electron temperature T e in the Au x Pd 1−x (x ≈ 0.6) nanowire used as the normal-metal part in the SNS junctions. This in turn results in an increased inductance of the short SNS junctions40,41 between leads P and G, which implies a lower resonance frequency of the effective LC oscillator formed by the short SNS junctions, the on-chip meander inductor L s , and the on-chip parallel plate capacitors C 1 and C 2 . We detect this change by measuring the reflection coefficient at the detector gate capacitor Γ(T e , ω p ) at a fixed probe frequency f p = ω p /2π. See Methods for the extraction of Γ from the detector output voltage at the digitizer V. Furthermore, we have the option to amplify the readout signal with a JPA42 (Fig. 1c, see Methods).

Fig. 1 Measurement setup and device characterization. a–c Simplified detector measurement setup (a) together with micrographs of the superconductor–normal-metal–superconductor (SNS) junctions between leads H, G, and P taken from a similar device (b) and of a similar Josephson parametric amplifier (JPA) (c). The impedance Z(T e ) of the series of short SNS junctions forms a temperature-sensitive resonant circuit together with a meander inductor L s ≈ 1.2 nH and the capacitors C 1 ≈ 87 pF and C 2 ≈ 33 pF. The gate capacitance is C g ≈ 0.87 pF. There is an 8.4-GHz bandpass filter connected to lead H. The scale bar in b indicates 1 μm and in c 15 μm (top) and 400 μm (bottom). d Phase of the reflection coefficient at the gate capacitor Γ as a function of probe frequency f p and power P p , without heating and with the JPA off. e Example of the ensemble-averaged detector output voltage at the digitizer V (top curve) and voltage in the other quadrature (bottom curve), with the JPA on. The blue curves show exponential fits to the rising and falling edges of the signal. f, g Change in the detector output voltage at the digitizer after the heater is turned on (markers, see e) as a function of the finite heater power P h with the JPA off (f) and on (g). The typical error of voltage measurement is of the order of size of the symbols. The (f p , P p ) operation points are indicated in d. The JPA shifts the resonance frequency slightly, thus the highest output is found at slightly different operating point. The bath temperature is T b = 25 mK for all data in this paper Full size image

Characterization experiments

Figure 1d shows the phase of the reflection coefficient at the gate capacitor (see Methods for details) as a function of probe frequency and probe power, at zero heater power. The most striking feature in Fig. 1d is the decreasing resonance frequency as the probe tone begins to significantly heat the electrons in the SNS junctions above P p = −135 dBm. This electrothermal feedback43 can be used to optimize the sensitivity and the time constant of the detector or even induce temperature bistability28 (not visible in Fig. 1d).

The NEP is determined by how noisy the readout signal is relative to the responsivity of the signal to changes in P h (see Methods). Thus in Fig. 1e, we show an example of the detector output voltage V, which is defined as the voltage in the quadrature providing the greatest response to the heater power after amplification (≈103 dB), demodulation, and an optimally chosen phase rotation. In Fig. 1e, we first set P h to a small value (3 aW) for a period of several tens of milliseconds, then turn P h off for a similar period, and finally average over repetitions of this modulation pattern. From such data we extract the quasistatic voltage response at the digitizer and the time constant using exponential fitting functions. Figure 1f, g shows the quasistatic response of the detector output voltage to the heater power up to 3 aW. We define the detector responsivity as the ratio of the voltage response and the corresponding heater power. We also employ this information to choose an appropriate power level for the heater in our experiments discussed below.

Dimensionless susceptibility

To understand the detector response at high probe power, we develop a model for the electrothermal feedback28,44 (see Methods for details). We define a dimensionless susceptibility as

$$\chi = \left. {\frac{{\partial {\mathrm{\Delta }}P}}{{\partial P_{\mathrm{h}}}}} \right|_{\partial _t{\mathrm{\Delta }}P = 0},$$ (1)

where ΔP equals the amount of additional heat flowing from the nanowire electrons to their thermal bath. Therefore, χ is the factor by which the probe-induced electrothermal feedback enhances the heating of the bolometer relative to the externally applied power P h .

Detector responsivity and noise

In Fig. 2a, b, we show the responsivity of the detector output voltage. Note that the NEP is unaffected by the calibration of the gain of the readout circuitry since the measured responsivity and noise are both amplified equally. The responsivity is maximized for probe frequencies close to the resonance. As the probe power is increased, the width of the resonance decreases, leading to a sharp increase in the responsivity. Note that the color scales are different in Fig. 2a, b, since the JPA adds gain in excess of 20 dB.

Fig. 2 Noise equivalent power (NEP) and thermal time constant. a–d Quasistatic responsivity of the probe voltage to the heater power (a, b) and probe voltage spectral density (c, d) as functions of probe frequency and power with the Josephson parametric amplifier (JPA) off (a, c) and on (b, d) averaged over noise frequencies between 20 and 100 Hz. e, f Noise equivalent power as a function of probe frequency and power with the JPA off (e) and on (f) averaged over noise frequencies between 20 and 100 Hz. g, h NEP (black) and thermal time constant (red) with the JPA off at fixed P p = −126 dBm (g) and with the JPA on at P p = −126.5 dBm (h). The error bars indicate the standard error of the mean for the time constant and error arising from heater power calibration for NEP Full size image

Interleaved with the measurements of the responsivity, we also record separate noise spectra for the detector output voltage and for the out-of-phase quadrature at each probe power and frequency. In Fig. 2c, d, we show the voltage noise spectral density across the same range of f p and P p as for the responsivity. Let us first discuss the low-probe-power limit (P p ≲ −132 dBm) with the JPA off in Fig. 2c. In this case, the electrothermal feedback is negligible (χ ≈ 1), ΔP vanishes, and the spectrum is dominated by noise added by the amplifiers in the readout circuitry. Furthermore, the noise power assumes a similar value on and off resonance. However, with the JPA on in Fig. 2d, we consistently observe a peak in the noise near the resonant probe frequency, indicating that amplifier noise is not dominating the signal even at the lowest probe powers shown (−132.5 dB). With the JPA off, the thermal fluctuations of the detector surpass the amplifier noise only at high P p .

NEP and time constant

Figure 2e, f presents the main results of this paper, that is, the measured NEP for an 8.4-GHz input over a range of probe powers and frequencies. We compute the NEP by dividing the voltage spectral density by the quasistatic responsivity and multiplying the result by a factor \(\sqrt {1 + (2\pi \tau f_{\mathrm{n}})^2}\), where f n is the noise frequency. This factor takes into account the fact that the thermal time constant τ decreases the responsivity of the detector with respect to the quasistatic case (see Methods). Figure 2e, f shows the NEP with the JPA on and off, respectively, averaged over noise frequencies from 20 to 100 Hz.

In Fig. 2g we show the NEP and the time constant as functions of the probe frequency at fixed P p = −126 dBm with the JPA off. Figure 2h is measured in identical conditions except that the JPA is on and the probe power is set to −126.5 dBm. The electrothermal feedback is strong and positive (χ ≫ 1) at probe frequencies just below the resonance frequency. In contrast, the electrothermal feedback is strongly negative (χ ≪ 1) at probe frequencies just above the resonance. This is clearly visible in the time constant τ = χτ b , which increases by nearly an order of magnitude as the probe approaches the resonance despite the fact that the bare thermal time constant τ b simultaneously decreases owing to increased electron temperature. Here, τ b denotes the time constant in the absence of electrothermal feedback (see Methods). The lowest NEP of \(20\,{\mathrm{zW}}/\sqrt {{\mathrm{Hz}}}\) in Fig. 2h coincides with the peak of the time constant (1 ms), suggesting that at P p = −126.5 dBm the NEP is optimized at the frequency that maximizes χ.

As the probe frequency exceeds the resonance, the time constant quickly decreases by more than an order of magnitude below 100 μs. In this regime, the positive effect of the JPA is particularly clear: the NEP degrades quickly with increasing probe frequency if the JPA is disabled, but stays roughly constant when it is enabled. The fact that the NEP remains relatively flat at \(60\,{\mathrm{zW}}/\sqrt {{\mathrm{Hz}}}\) with the JPA on (Fig. 2h) is an indication that the internal fluctuations of the detector are limiting the performance instead of amplifier noise. This is an example of the convenient in situ tunability of the SNS detector, that is, we can choose a different trade off between the NEP and the time constant by a small change of the probe frequency or power. We can also tune the time constant and the dynamic range by changing the bath temperature or by applying an additional constant heating power through the heater port. However, such an optimization is left for future work.

Noise analysis

In Fig. 3a, b, we present the full noise spectrum of the output signal at P p = −126 dBm and P p = −126.5 dBm with the JPA off and on, respectively. Above 1 Hz and below 1 kHz, the noise increases up to 14.5 dB above the broadband background set by the amplifier noise for probe frequencies near resonance. Far off-resonance we find only the broadband amplifier noise floor in addition to 1/f n noise. We also observe in Fig. 3b noise peaks at multiples of 1.4 Hz, matching the frequency of vibrations caused by the pulse tube cryocooler. Note that these peaks are clearly visible only when the JPA is on and the probe is far from the resonance, suggesting that the pulse tube noise does not couple directly to the detector, but rather to the amplifiers. At operation points with low NEP, the pulse tube noise is masked by the noise generated by the detector itself.

Fig. 3 Frequency spectra of voltage noise and noise equivalent power (NEP). a, b Spectral density of the noise in the signal quadrature of the down-converted probe tone with the Josephson parametric amplifier (JPA) off (a) and on (b) as functions of the noise frequency f n . The bottommost curve shows the spectral density far off resonance at f p = 539.275 MHz and P p = −132.5 dBm, whereas the green, blue, orange, and yellow curves are measured at P p = −126 dBm (JPA off) and P p = −126.5 dBm (JPA on), and span a narrow frequency range near the resonance. Specifically, the probe frequencies are f p = 540.6125 MHz − δ, where the values of δ are indicated in a. For clarity, the curves have been offset vertically in increments of 10 dBm/Hz. The two peaks above 1 kHz in b are due to the aliased JPA idler. The excess noise at multiples of 1.4 Hz is attributed to the pulse tube (PT) cryocooler. The dashed line indicates a first-order RC filter response with a time constant identical to that in the δ = 0 trace in b. c NEP with the JPA on as a function of the noise frequency at P p = −126.5 dBm and f p = 541.9625 MHz. These data yield 0.3 zJ for the energy resolution estimate of the detector (see text). Discontinuity in the data on all panels near 2 kHz is caused by the fact that we measure the high- and low-frequency noise separately with different time steps Full size image

Predicted energy resolution

In Fig. 3c, we present the NEP measured with the JPA on as a function of the noise frequency at a (f p , P p ) point selected for short time constant and low NEP. From the NEP, we can obtain an estimate for an upper bound on the energy resolution45

$$\varepsilon \approx \left( {\mathop {\smallint }\limits_0^\infty \frac{{4{\mathrm{d}}f}}{{{\mathrm{NEP}}(f)^2}}} \right)^{ - 1/2}.$$

By restricting the above frequency integration below the thermal cutoff frequency 1/(2πτ) = 5.8 kHz, the data in Fig. 3c yields ε = 0.32 zJ = h × 480 GHz, surpassing, for example, the anticipated resolution of the TES-based Fourier transform spectrometer19 specified to have about an octave of resolution in the band of 1.4–9 THz. Increasing the cutoff frequency to 10 kHz yields ε = 0.26 zJ = h × 390 GHz. Here, h denotes the Planck’s constant.

Feasibility for terahertz detection

Inspired by the above-suggested energy resolution, we theoretically study the future feasibility of the SNS bolometer as a THz detector. We simulate a complete experiment, including a possible THz antenna design, aimed at detecting individual photons from a thermal source. The THz coupling scheme is based on a substrate-lens-coupled planar antenna. An extended hemispherical silicon lens (diameter 1 mm) integrated with a double-slot antenna as a feed46 was designed for the center frequency of 1.3 THz (see Fig. 4a). We employ electromagnetic simulations to study the performance in THz detection. Note that at the considered signal frequency range, well exceeding the Bardeen–Cooper–Schrieffer gap frequency, even a fully superconducting bolometer acts as a resistive load, and therefore no separate load resistor is needed. Thus, for simplicity, the bolometer is modeled as a 50-Ω port in the simulations. We have designed band-stop filters at 1.3 THz to prevent the bolometer readout circuitry from interfering with the antenna47. The radiation patterns at 1.3 THz (Fig. 4b) show −3-dB beam widths of about 8°. The detector efficiency is quantified by the effective area A e , peaking to 0.35 mm2 at 1.3 THz (Fig. 4c).

Fig. 4 Terahertz antenna and detection simulations. a Electromagnetic simulation model of an extended hemispherical silicon lens with a double-slot antenna as a feed. b Antenna gain at 1.3 THz as a function of the incident angle in the two principal planes, H-plane (dashed line) and E-plane (solid line). The H-plane (E-plane) is the plane containing the magnetic (electric)-field vector and the direction of maximum radiation49. c Effective detection area as a function of frequency. d Planck spectral irradiance at T = 3.0 K (solid line), and projected frequency-selective surface (FSS) filter transmission (dashed line) for blackbody measurements. e Computed photon rate power spectral density (PSD) corresponding to d. f Computed total detected photon rate (dashed line) and the corresponding detected power (solid line) as a function of the temperature of the blackbody emitter Full size image

To analyze the detected power and photon count rates, we assume a blackbody thermal source represented by the Planck spectral irradiance for a single polarity expressed as B r (f) = (hf3/c2)/{exp[hf/(k B T)] − 1}, where k B is the Boltzmann constant, T is the temperature of the source, and c is the speed of the light. We aim at a detection band above the peak frequency at hf ~ k B T. Thus, we must filter the low-frequency tail very efficiently, as it represents orders of magnitude higher power density in comparison to the band of interest.

Such filtering between the blackbody source and the detector can be achieved, for example, by metal mesh structures acting as frequency-selective surfaces (FSSs)48. Filter systems based on FSS structures have been demonstrated previously in the context of low-power detector experiments: de Visser et al.33 employ a bandpass-filter system with a low-frequency roll-off above 60 dB per octave, and a stop-band rejection of more than 60 dB. For our purposes, we aim to capture the essential features of such filters by defining a high-pass filter with a comparable roll-off, stop-band transmission of −60 dB, and a 3-dB cutoff at 1.3 THz.