Amplitude-Multiplexed readout of single photon detectors based on superconducting nanowires

The realization of large-scale photonic circuit for quantum optics experiments at telecom wavelengths requires an increasing number of integrated detectors. Superconductive nanowire single photon detectors (SNSPDs) can be easily integrated on chip and they can efficiently detect the light propagating inside waveguides. The thermal budget of cryostats poses a limit on the maximum number of elements that can be integrated on the same chip due to the thermal impact of the readout electronics. In this paper, we propose and implement a novel scheme able for an efficient reading of several SNSPDs with only one readout port, enabling the realization of photonic circuits with a large number of modes.

Single photons are promising candidates as quantum bits (qubits) for quantum information applications due to their low decoherence and ease of transmission, both in free space and by means of optical fibers. Small-and medium-scale quantum computing 1 and simulation 2 can be realized combining single-photon sources, single-photon detectors and linear optics. In the last ten years many efforts have been devoted to develop an integrated platform containing all functionalities needed to achieve the tantalizing "quantum supremacy" Several experiments have been performed or proposed exploiting photonic integrated circuits (PICs), including boson sampling, quantum walk and quantum simulation [3][4][5][6][7] . The increasing complexity of the experiments reflects directly in the PIC architectures, requiring an always-increasing number of integrated components. All of the aforementioned experiments have been performed with hybrid setups, i.e. external sources and detectors, and a clear path to achieve a complete integrated platform is still lacking. SNSPDs are the only detectors 8 that showed an on-chip integration feasibility with outstanding performances in terms of detection efficiency, dark count rate and timing resolution in the infrared wavelength range [9][10][11][12][13][14][15][16] . The increasing PIC complexity requires the integration of several tens of SNSPDs, posing new challenges related to the simultaneous readout of different channels. The use of dedicated readout electronics for each detector channel is not a practical solution and has a tremendous impact on the thermal budget allowed by cryostats. To overcome these limitations, several multiplexing schemes have been proposed based on different approaches. A row-column multiplexing scheme enables the readout of planar 2D array N 2 detectors with only 2N channels 17 . Another proposal is based on frequency multiplexing and uses different RF resonators to read out an SNSPD array, where the switching of a portion of the nanowire causes a shift in the resonator operating frequency. Although with this technique it is possible to read out a huge numbers of detectors, each RF tone needs a 3 demultiplexing circuit thus limiting the maximum filling factor achievable by the array 18 . Time domain approaches have been also exploited, where, using a time-tagged multiplexing scheme, the signals coming from two SNSPDs were separated in time using a delay line 19 . This approach requires only a single readout line, but has no photon number resolution and the overall dimensions of the array are dictated by the design of the delay line. Another time tagging scheme was employed to demonstrate a single photon imager using a continuous nanowire delay line. In this work, we implemented a new and simple scheme that is able to read out multiple SNSPDs with only one coaxial cable without the requirement of complex post processing. Our approach consists of a novel version of a spatially multiplexed PNRD (photon number resolving detector) [22][23][24] . The readout scheme (see Figure 1a) is based on the spatial multiplexing of N active elements consisting of a SNSPD in parallel with an on-chip AuPd resistor, of resistance Ri.
When a photon is absorbed in an active element a normal resistance Rn appears in the superconductive nanowire and, being Rn >> Ri by design, all the bias current is diverted to the parallel resistance 25,26 . The position of the photon-absorption event is then encoded in the voltage amplitude of the pulse. Due to the compact readout scheme, we were able to show a 4 proof of concept based on two-element array integrated in a silicon nitride (Si3N4) PIC. In addition to retrieve the specific properties of each SNSPD, the circuit is intrinsically able to implement g (2) (τ) measurements 27,28 . Combining our approach with the use of a cryogenic amplifier, tens of detectors can be read with a single coaxial cable with a minimum thermal impact on the operating temperature.
In order to show the resolution capability of our approach, we design the parallel resistances  To avoid the use of expensive and bulky cryogenic positioners, we align and glue a Pyrex fiber array (FA), composed of 6 single mode fibers to the ports of the PIC, obtaining in this way a fast and reliable light coupling. The alignment is obtained at room temperature by maximizing the light at the control port of the photonic circuit (see Figure 1a) by using a 6-axis manipulator for the FA. The coupling efficiency of the grating coupler is about 15% at room temperature.
Successively, the chip is mounted on a GM refrigerator operating at a temperature T=2.9 K.
During the sample cooling, the transmitted optical power decreases due to the different thermal contractions between the Pyrex (FA) and the silicon substrate, causing a reduction of the coupling efficiency down to ~2%. The readout electronics implements a chain of two bias Tee and two RF amplifiers with a gain of 49 dB in the bandwidth 0.1-500 MHz. The NbN film has a 9 K critical temperature and a critical current density of 3.9 MA/cm 2 at 2.9 K. The travelling 6 photons at 1550 nm wavelength, generated by a 10 ps pulsed laser, are injected through the input port of the PIC.   The scalability of our approach depends on three main parameters: the input resistance of the readout electronics Rout (generally coincides with the RF amplifier impedance), the maximum resistance value RM allowed by the system and the signal to noise ratio (SNR). In our setup RM is limited by the 50 Ω input resistance of the RF amplifier being Rout in parallel with the SNSPD array (see Figure 1 a). This poses a limit to the maximum pulse amplitude Vmax=IB•RM. In the single photon regime, only one detector is firing at once, implying that the only selection rules that are required for the choice of the set of resistances are Ri≠ Ri+1 and ΔR= Ri+1 -Ri large 9 enough to produce voltage signals greater than the amplifier noise. Therefore, the maximum number of channels readable with a single coax cable is Ch= RM/ΔR. Being RM= 50 Ω, the scalability of our scheme can be increased by reducing ΔR as low as possible. Table 1 summarizes three cases, the first row takes into account all the parameters we measured in our experiment, the second row describes how we can improve our scalability using a commercially available cryogenic RF amplifier 31 , while the third one refers to an amplifier with an input impedance of 1 kΩ.
The fit of the histogram (Figure 2 c) provides a variance 2σ= 7.4 mV for all three Gaussian peaks, giving a noise resistance ΔR2σ≈ 1.6 Ω (amplifier gain G= 49 dB), in close agreement with ΔRth= 1.2 Ω calculated from the Johnson-Nyquist voltage noise that is expected for an RF amplifier working at room temperature T= 300 K 32 . As shown in the second row of Tab. 1, a reduction of a factor 15 of the amplifier working temperature corresponds to an increase of SNR of a factor ~ 4. The SNR can be improved also increasing the bias current IB by improving the film quality, by tuning the geometric parameters of the nanowires or by connecting several nanowires in parallel 33,34 . In the latter case, we expect an IB increase of at least a factor two. ΔR2σ and ΔR6σ are the resistance step values considering an experimental noise of 2σ and 6σ given by the width of the Gaussian fits. The RM is the maximum resistance allowed by the system and Ch is the number of channels that can be implemented for each electronic configuration in single photon regime in the cases ΔR2σ and ΔR6σ.
Selecting an appropriate resolution determined by the step resistances, ΔR2σ and ΔR6σ, we can identify the firing element of the array with 68.3% probability and 99.7%, respectively, where the choice strongly depends on the application. For example, if detectors would be used as pixels for spectroscopy or imaging, an uncertainty in the precise position of the firing detector might be tolerated. When error rates have to be minimized, like in quantum computation, it is possible to assign the values of Ri in a strategic way and making use of additional delay lines to separate in time pulses with close voltage amplitudes and thus reject erroneous events.
In the single photon regime and considering that the maximum pulse amplitude is VM= IB•RM, where RM= 50 Ω, we can calculate the maximum number of elements that can be read, given by Ch6σ=50/ΔR6σ= 10 channels at 300 K and Ch6σ= 41 at 20 K. If we consider a less stringent step resolution of R2σ, the maximum number of channels rises to Ch2σ= 30 at 300 K and Ch2σ= 121 at 20 K.
In the multi photon regime, the firing of different channels occurs simultaneously increasing the number N of Gaussian peaks to be discriminated 36 . For example, the number N of levels required to resolve the position of two elements firing simultaneously is N= Ch(Ch-1)/2 36 . In the case of a 50 Ω RF amplifier at T=300 K (first row of Table 1) the maximum number of channel that can be read in a two-photons regime is Ch2σ≈ 8 that becomes ~16 by cooling the amplifier at 20 K. However, the lack of linearity due to the parallel with 50 Ω input impedance of amplifier complicates the selection rules for the resistance steps. The linearity requirements in the multi photons regime can be met by implementing a high-impedance cryogenic readout (row 3 in Tab. 1). In this case, the conditions required for RM are RM << Rout, to preserve the linearity, and RM<Rlat, to ensure the recovery of the nanowire superconductivity, being Rlat the resistance value that causes the latching of the nanowires in a stable normal state [37]. The high input resistance 11 of the amplifier increases ΔRth, resulting in a total number of levels of 105 and 35 that can be equally spaced by the values ΔR2σ and ΔR6σ, respectively.
In conclusion, we demonstrated an easy-to-use and scalable architecture for the readout of