Orthogonal sequencing multiplexer for superconducting nanowire single-photon detectors with RSFQ electronics readout circuit

Matthias Hofherr; Olaf Wetzstein; Sonja Engert; Thomas Ortlepp; Benjamin Berg; Konstantin Ilin; Dagmar Henrich; Ronny Stolz; Hannes Toepfer; Hans-Georg Meyer; Michael Siegel

doi:10.1364/OE.20.028683

1. Introduction

During the last years, it was demonstrated that superconducting nanowire single-photon detectors (SNSPD) have unique properties, which make them interesting for applications in the field of quantum communication, spectroscopy, astronomy, etc. The main effort in SNSPD research has been to understand the detection mechanisms to find methods to optimize quality and reproducibility of the detectors [1–4]. Some investigations have already been done on multi-detector element systems, but in these cases, all detector readouts were realized by independent microwave cabling, coupled to roomtemperature, which leads to a limitation of the number of detector elements [5, 6]. Large multi-detector element systems could increase the field of applications by realizing fast and highly resolved imaging. Therefore, seminal concepts are necessary, which focus on the scalability to high numbers of detector elements, keeping fast response times and high count rates and which are compatible with the restrictions of the cryogenic environment. A possible method is the use of SQUID amplifiers for the detector readout and digital multiplexing techniques [7]. However, the response time of SQUID signals is much longer than the original response time of the SNSPD pulses [8].

It has already been demonstrated that rapid single flux quantum (RSFQ) electronics is a promising candidate to meet the requirements needed for single-photon multi-detector element systems [9–12]. RSFQ electronics can be operated at the same cryogenic conditions as the detector chip. Moreover, an integration of the detector array on the RSFQ chip is possible, because of comparable technological conditions, which leads to short wiring and less attenuation in the signal interconnects. Another advantage is that RSFQ electronics provides an even higher time resolution than required by the detector signals [13]. Logic operations can be realized in RSFQ electronics with high speed and very low power dissipation, thus various functions for data pre-processing and data rate reduction could be implemented at cryogenic temperatures. As a result, the RSFQ circuit offers several possibilities to reduce the number of wires leading into the cryostat, in contrast to a multi-channel setup with separate readout microwave cable for each detector element.

Terai et al demonstrated for a 4-pixel detector that RSFQ electronics allows to readout detector elements and merge the signals to one readout line [12]. They showed that in this case it is possible to read out the detectors with the same detection efficiency (DE) and comparable timing jitter than a classical single pixel setup. Yamashita et al. presented that a spatial reconstruction can be reached by using the time-correlated single-photon counting measurement method [13]. However, this technique is limited on the one side to pulsed laser applications and to smaller arrays, because the maximum number of pixels is given by the relation of the laser pulse repetion rate and the minimum detector jitter.

However, a study of multi-pixel SNSPDs for imaging in continuous wave (cw) light is not given so far. The idea of imaging, which can be used for highly sensitive camera system applications, is to get information about the momentary mean photon rate per pixel. The information is determined from periodically repeated time gated measurements. An important task in developing applicable readout concepts for spatial resolution of detector pixels is therefore a method to readout detectors arrays while cw light is applied. We suggest a method for multi-pixel SNSPD readout by using only one digital readout line in combination with an orthogonal sequencing multiplexer (OSM) to extract the average count rate of each pixel. This method is based on a detector bias switching scheme, applied with longer “on” times for all detectors compared to a time multiplex approach. We explain the used measurement method and propose some ideas for further scaling of the pixel number. We experimentally rate the multiplexing method and the frame conditions in terms of resolvability and the accuracy using a four pixel SNSPD array with a directly connected four channel pulse merger RSFQ readout.

2. SNSPD fabrication

A multi-pixel SNSPD can be designed as a long meander cut into separate sections with individual connection lines that act as individual detectors [6]. Fabricating several arrays would directly lead to a detector element matrix scheme. The main effort for scaling the pixel number in future is to handle the number of signal and ground lines on the SNSPD chip connecting the SNSPD with the RSFQ input, because they interrupt the active area of the detector chip. For our first experiment, we fabricated a thin NbN film four-detector device with a distance of 500 µm between the detectors (see Fig. 1 ) to allow different illumination of the detector elements.

Fig. 1 (a) Photo image of a four-detector element array of SNSPDs, the position of SNSPDs is marked by circles, (b) SEM image of a single detector element structure.

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The NbN film was deposited by DC reactive magnetron sputtering of a pure niobium target in argon/nitrogen atmosphere onto R-plane sapphire substrate at 750°C. Details can be found in [14]. The current RSFQ chip is desinged for higher input currents than standard 4 to 5 nm thick SNSPDs can provide based on their critical current I_C. For that reason, we decided for this experiment to use a thicker film of 10 nm thickness, because optimization of the detection efficiency was not in the focus of this experiment. The detectors were patterned as a meander line with a nominal width of about 90 nm and a filling factor of about 50%. The meander covers an area of 3.5 x 4 µm² per detector element. The patterning was made by electron-beam lithography and Ar⁺ ion-milling technique.

Each of the four detectors is connected to a separate coplanar readout line. The typical superconducting parameters were measured: the critical currents I_C of all 4 detectors were about 140 µA at 4.2 K, the critical temperature T_C was 14 K.

In future, also 4 nm thick films could be used in connection with this RSFQ chip, by using parallel stripe meanders, which offer higher critical currents keeping the typical high detection efficiencies of 4 nm films [15].

3. RSFQ electronics

RSFQ electronics uses a single flux quantum (SFQ) Φ₀ = 2.067 10⁻¹⁵ Vs, given by natural constants, as a data carrier [16]. Thus, RSFQ electronics is intrinsically digital and because of the low operation temperature within the range of T = 4 K, it is a low noise circuit technique. The Josephson junction is the active component in RSFQ electronics. Depending on the circuit complexity, the timing jitter of this electronics family is in the order of pico-seconds [13]. Furthermore, RSFQ circuits provide a high-speed operation in the upper GHz range of complex circuits at low temperature and keep nevertheless a low power consumption.

The SNSPD is an asynchronous digital detector. This feature spares an extensive analog to digital conversion but requires a conversion of the data carrier. Recent experiments demonstrated the triggering of a SFQ by the voltage pulse of a SNSPD [9, 10]. A typical input interface for RSFQ circuits is the DC/SFQ converter [17]. We used a modified version of this cell as interface between SNSPD and RSFQ circuit [9].

Once the SNSPD voltage pulse is converted into a SFQ signal, any logic operation as well as data processing can be realized. The state-of-the-art fabrication process at AIST Japan allows a circuit complexity of about 20,000 Josephson junctions per chip [18]. Consequently, the number of junctions, required to readout one detector, should be as low as possible to enable large detector arrays. The main goal of utilizing RSFQ electronics is to decrease the number of rf cables into the cryogenic environment and keeping a minimum of the system complexity. Therefore, we implemented and fabricated for the following experiments a RSFQ based SFQ pulse merger as shown in Fig. 2(a) .

Fig. 2 (a) Block diagram of a binary-tree pulse merger utilized in the experiments (DC/SFQ: input interface, JTL: Josephson transmission line, CB: Confluence buffer, SFQ/DC: output interface). This design is used in the later experiment (b) Proposal of a serial pulse merger, which allows better routing for larger arrays with passive transmission lines (MSL: micro stripe line, PTL: superconducting passive transmission lines).

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This circuit consists of several DC/SFQ converters which generate single-flux quanta, Josephson transmission lines (JTL) which transfer the data, and confluence buffers (CB) which combine two input lines to one output line. This asynchronous merger combines four input channels to one single output channel (see Fig. 2(a)).

The data output stream is transferred to a SFQ/DC converter, which generates a voltage output signal. The output signal toggles its state depending on the bias operation point between 0 µV and up to about 350 µV for each incoming detector pulse. Figure 3 shows an oscilloscope screen shot, wherein the upper blue curve is the reference signal of the laser diode illuminating the SNSPD, which shows if the laser diode is on or not, and the green curve is the output of the RSFQ circuit amplified by 80 dB. A room temperature amplifier with a bandwidth between DC and 50 kHz was used. In this example, three photons were detected per period.

Fig. 3 Oscilloscope screen shot of the RSFQ output signal: Proof of principle of one detector element readout. The upper trace demonstrates the switching of the photon source (blue). Only during the laser light “on” phase level transitions of the RSFQ output voltage indicate photon count events (green). The RSFQ output was amplified by 80 dB.

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The important advantage of the RSFQ merger is the small circuit complexity. Indeed, if we would implement more pixels, the binary-tree architecture (Fig. 2(a)) of the pulse merger can be restricted due to layout constrains, because this structure only allows to put the inputs close to each other. A better distribution of the location of the inputs on the chip with this structure can only be reached by inserting several JTLs, which produce further power dissipation. Especially, if we think of an integration or hybrid solution of the SNSPD array on top of the RSFQ chip, an alternative solution is the serial pulse merger illustrated in Fig. 2(b), which allows a homogenous distribution of the inputs on the RSFQ chip using passive transmission line connections (PTL) [19]. The PTLs enable a fast and passive data transmission over wide distances on the chip. The circuit block in Fig. 2(b) is easily scalable to higher pixel numbers.

In the proposal of Fig. 2(b) the number of components for an i-pixel array is (i-1) confluence buffers, i DC/SFQ converters and i of each PTL interface cells. This results in 11 Josephson junctions per detector. Assuming that each detector counts 10⁶ photons per second the dynamic power consumption of the RSFQ circuit in both circuits in Fig. 2 is only 5 pW per detector. However, the static power consumption for the conventional RSFQ electronics designed for a 2.5 mV supply voltage is 2.5 µW per channel. This number could be reduced to about 250 nW by using low-power RSFQ circuits [20]. An even further reduction to only the dynamic power is possible by applying recently developed new circuit techniques [21].

This RSFQ readout is not able to resolve multiple photons at exactly the same time. Our simulation of the RSFQ circuit yields that the shortest resolvable time between two count events of different detectors amounts to 15 ps, which is more than two orders of magnitude faster than a normal single-pixel detector could provide [8]. The limiting factor in our system is the bandwidth of the room temperature amplifier. A more advanced RSFQ output interface was already tested up to 3 GHz bandwidth [22]. To reach real simultaneous multiple-photon counting the second proposal from Terai et al. has to be realized [23], which is quite more complex in design, has a higher heat dissipation scaling with the number of channels and requires more cable connections to room temperature.

4. Multiplexing technique

If a spatial resolution of the pixel is not necessary, all detectors can be biased continuously and the whole system operates as a single pixel with enlarged detection area. The maximum possible count rate in this mode exceeds the values that a single detector could provide.

In cases, where spatial resolution of the counts is required, the readout concept has to be enhanced. To address this spatial issue, we propose the use of an orthogonal sequencing multiplexer for the detector channels to enable a post-processing estimation of the spatial origin of the counts. This is an adaption of a communication code-division multiple access approach (CDMA [24], ). Our multiplexer is not based on digital data modulation, but it extracts the analog information of counts per predefined time and not just a photon flux “available/non-available” state. The extraction of count events is also the difference to the CDMA SQUID-multiplexer of Irvin et al. [7].

Orthogonal sequencing means, that each detector is switched “on” or “off” with a certain code sequence, which also encloses the “on” state of more than one detector at the same time. Under the assumption that the code sequences are linear independent, this is a scheme to estimate the original counts of each detector element integrated over a certain time. The code sequence is assigned to each detector as an individual signature. Convenient orthogonal sequence functions are the Walsh functions. These functions can be iteratively generated by using the Hadarmard matrix H_2n (see [24] for details):

\begin{array}{l} H_{2 n} = [\begin{matrix} H_{n} & H_{n} \\ H_{n} & - H_{n} \end{matrix}] \in {1, - 1}^{^{2 n \times 2 n}} \\ with the starting matrix: H_{2} = (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}) . \end{array}

We use the rows H_2n(i), i>1, to define the code sequences. The parameter 2n is the range of the Hadamard-Matrix and defines the number of clock cycles t_clk of one frame of the Walsh function. A H_2n matrix allows to readout m = 2n-1 detectors. For our experiments, we used the eighth-order Hadamard matrix (2n = 8), therefore maximum m = 7 detectors can be readout. Table 1 gives four Walsh functions used for the experiments described in section 6.

Table 1. , One frame of switching code sequences for four detectors, defined by the H₈ Walsh functions. The H₈ matrix can principally readout up to seven detectors.

View Table

The states in the sequences are defined as “1” and “-1”. State “1” is considered as “detector is on” and “-1” as “detector is off”. The response of a single detector delivers certain counts during state “on” and no counts during the state “off”. All states are synchronously clocked.

The multiplexing/demultiplexing is mathematically described by:

c_{e x t}^{T} = c^{T} \cdot M \cdot H_{2 n},

where the vector c_ext contains in each element c_ext,_X the extracted counts of the detector element X

\in

{1,2,3,4,…,2n-1} during one recording frame T_frame = 2n∙t_clk, which is proportional to the photon flux. The matrix M describes the coding of the channels which is applied by the switching of the bias currents. M can be formed from a H_2n matrix by replacing −1 to 0. The H_2n matrix in (2) extracts the estimated counts during T_frame. Equation (2) can be written as

\begin{array}{l} c_{e x t}^{T} = (0, c_{1}, c_{2}, c_{3}, ... c_{n - 1}) \cdot H_{2 n} {- 1 \to 0} \cdot H_{2 n} = \\ (0, c_{1}, c_{2}, c_{3}, ... c_{n - 1}) \cdot (\begin{matrix} 2 n & 0 & \dots & \begin{matrix} \dots & 0 \end{matrix} \\ n & n & 0 & \begin{matrix} 0 & ⋮ \end{matrix} \\ \begin{matrix} ⋮ \\ ⋮ \end{matrix} & \begin{matrix} 0 \\ ⋮ \end{matrix} & \begin{matrix} ⋱ \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} 0 & ⋮ \end{matrix} \\ \begin{matrix} ⋱ & 0 \end{matrix} \end{matrix} \\ n & 0 & \dots & \begin{matrix} 0 & n \end{matrix} \end{matrix}) = {(\begin{matrix} \begin{matrix} n \cdot (c_{1} + c_{2} + \dots + c_{n - 1}) \\ n \cdot c_{1} \end{matrix} \\ n \cdot c_{2} \\ n \cdot c_{3} \\ \begin{matrix} ⋮ \\ n \cdot c_{n - 1} \end{matrix} \end{matrix})}^{T} . \end{array}

Each element c_ext, _X

\in

{1,2,3,4,…,2n-1} is only dependent on the counts c_X with identical index and therefore from only one detector element. The original counts are multiplied by the constant prefactor n. However, the RSFQ readout system delivers a toggling output Out(t) (see Fig. 3), so the matrix scheme has to be modified:

c_{t_{c l k}, X \in {1, 2, 3, 4}} = \frac{1}{n} \int_{0}^{T} H_{2 n}_{, X} (t) \cdot O u t (t) d t = \frac{1}{n} \sum_{i = 1}^{2 n}_{C o u n t} H_{2 n}_{, X} (i \cdot t_{c l k}) \cdot \sum_{τ = (i - 1) t_{c l k}}^{τ = i t_{c l k}} O u t (τ)

The vector

c_{t_{c l k}}

contains the counts during the time t_clk, which is the n-th part of c_ext. The

\sum_{C o u n t}

function describes the counting of the asynchronous digital toggling edges of the RSFQ output module during one clock time t_clk. H_2n,X is the row of detector X of Table 1.

The mean counts per second for one imaging frame is than defined:

C R_{X} = \frac{2 n \cdot c_{t_{c l k}, X \in {1, 2, ..., 2 n - 1}}}{t_{c l k}}

5. Measurement setup

Cryogenic electronics

The cryogenic part is a hybrid two-chip system consisting of the SNSPD multi-pixel chip and the RSFQ chip. Both chips are mounted side by side on one common brass plate (see Fig. 4 ).

Fig. 4 Mounting of the SNSPD array next to the RSFQ electronics. The SNSPD chip (left) has a size of 3 mm x 3 mm. Each of the 4 detectors is connected to a separate coplanar readout line and bonded to the RSFQ chip. The RSFQ chip (right) has a size of 5 mm x 5 mm. The complete readout circuit is symmetrically designed twice on the chip.

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The system operates at 4.2 K. A light beam is transmitted from room temperature to the detector chip at the cold stage by means of a multimode fiber. To apply the bias current I_B to each detector element, several cryogenic inductance coils are used as part of a filter stage.

The RSFQ chip is biased by dc current sources. The number of bias lines scales with the functionality of the RSFQ but not with the number of channels, because the operation point of all RSFQ input channels is controlled by one common bias current. The difference in current level between SNSPD bias and RSFQ bias is in the range of two orders of magnitude. Therefore, to minimize the influences of potential shifts of the dc cables, both chips need separate ground connections to the bias sources. A Cryoperm cylinder magnetically shields the earth magnetic field.

The cryogenic electronics is extended by a room temperature part. A scheme of the complete readout setup is given in Fig. 5 . The room temperature components, which are required for the multiplexer are the programmable switch controller and the FPGA (Field Programmable Gate Array) based data acquisition system.

Fig. 5 Scheme of the whole measurement setup. The SNSPD array and the RSFQ readout are operating at 4.2 K. The bias is provided by low noise current sources. The current switching and the data aquisition are made by a combined programmable switch controller and FPGA system.

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Bias current switching

The presented multi-pixel approach requires a bias current switching. The bias current I_B is produced by four low noise battery-powered current sources in this first approach. I_B can be set individually for each detector. For activating or deactivating the SNSPDs, an analog CMOS switch is used. In operation, the bias current source of a detector is set to a fixed current value and the detector is deactivated by shorting the current source. Since the bias current is switched “off” in “off” state, there are even no dark counts possible. That means dark counts are handled identical to a photon event from the multiplexing view and increase therefore the calculated count rates.

The current switching frequency is f_switch = 1/ t_clk = f_frame∙2n. The frequency f_frame is the frame rate of the imaging system and we remember 2n is the number of clock cycles per frame and in principal the number of detector, as well. Even if one assumes a future kilo-pixel array, the bias currents have to be switched with a frequency of only 25 kHz to obtain a video frame rate of 25 frames per second. Twisted pair wires can easily handle such low frequencies. Moreover, the measurement of low photon fluxes requires even lower imaging frame rates then 25 s⁻¹(see next section). Of course, thousand twisted pair lines connected to room temperature increase the thermal coupling and will probably heat the detector to an inapplicable operation point. Brandel et al. [25] recently suggested polar bias switches by RSFQ/SQUID technology. These cryogenic switches allow an operation of the switching electronics in combination with a set of RSFQ based pattern generators in the cryogenic environment. It is not necessary to set the bias current I_B of a detector to zero to switch it “off”. It would be enough to reduce I_B to a level of less than 0.5I_C to set the count rate to a negligible value. This fact reduces the requirements for the switches and all detectors can be supplied by one single voltage biasing line.

In the current setup, the switchings are triggered by a clock signal from the FPGA. The swiching pattern can be programmed individually for each detector channel in the switch controller. The user can freely define the number of repetitions of the switching pattern. The used analog CMOS switches have a very low charge injection to prevent the detectors from unwanted latching in the moment of switching, which was veryfied in the experiment.

Data acquisition

For real time data readout, a FPGA of the type Cyclone 3 from Altera was used. The room temperature electronics was battery powered. The FPGA operates with an internal clock frequency of 50 MHz. Its data input is realized by a 18-bit analog to digital converter (ADC) with a practical maximum sampling rate of 500 kS/s. In the recent configuration, the input noise of the ADC allows a reliable detection of a voltage difference of 150 µV.

In order to cover the voltage range for the detection of toggling events of the RSFQ output, a differential amplifier is used at the input of the ADC. Its bandwidth reaches from DC to 250 kHz with a gain of 40 dB. With this input configurations the system can record up to 200,000 counts per second. After AD-conversion the FPGA provides an edge detection to identify the toggle events of the RSFQ output signal. An internal 32 bit counter was used to determine the number of toggle events representing the number of detected photons. The results will be transmitted from the FPGA to a computer via Ethernet protocol and a glass-fiber connection in real-time. Several parameters such as the discriminator level for the edge detection, the sampling rate of the ADC and the gate time of the measurement can be defined. The readout electronics is able to do long time measurements with direct graphical control under real-time conditions.

6. Experiments

Photon statistics with classical readout and RSFQ readout

The multiplexing method will be demonstrated using the four pixel SNSPD chip and the four channel RSFQ electronics.

In a first experiment, the correct operation of each channel is tested separately. An oscilloscope screenshot of such a measurement for one detector is shown in Fig. 3. The method is described in detail elsewhere [9]. All detectors operate during the complete experiment in a bias current regime, where dark counts can be neglected.

For later analysis it is important to know, if the measured and extracted probability distribution of counts is dominated by the photon statistic or by the system inaccuracy. The photon statistic should follow a Poisson distribution [26]. That means for the single photon regime, that for a mean number µ of count events, which can be interpreted as expectancy value, the standard deviation σ has to be $\sqrt{µ}$ . In the environment of a single-photon system, the mean value µ can be described as:

µ = D E_{X} \cdot n_{phX} \cdot Δ t

DE_X describes the detection efficiency of the detector X

\in

{1,2,3,4,…,2n-1} and n_phX is the current photon flux on the detector X, Δt is the gate time of the measurement. We neglect to differ between the types of detection efficiencies as system detection efficiency, detector detection efficiency, et cetera, because the absolute value is not necessary as we will see in this section.

To estimate the capability of the RSFQ readout, a comparison with a classical pulse counter based SNSPD readout is helpful. Both, the classical readout system and the RSFQ based readout allow gated time measurements of the input events. We used as benchmark sample a 4 nm thick single pixel detector for comparison. The single detector was irradiated with 400 nm light at 0.9 I_C and the detection efficiency was 10%. The optical power on the active area of the detector was ≈34 aW. We first measured the counts for different gate times Δt with the classical readout based on a pulse counter electronics SR620. For each Δt the measurement was repeated 30 times to get statistically significant data.

The results are presented in Fig. 6 (black squares). The mean value of the counts increases linearly with the measurement time, as expected. The standard deviation (open black squares) is identical to the square root (half open black squares) of the mean value of counts as expected from Poission distribution. The values of the standard readout fit to the theoretical predictions. We compare this to the RSFQ readout. The illumination of the multi pixel chip is done by a 650nm cw laser source for all measurements. Since the 10 nm thick film has a lower detection efficiency DE, we increased the photon flux n_ph on the system to get a similar mean value µ (see Eq. (6)). In Fig. 6, the results of detector channel 3 is demonstrated. The mean value (red cycles) varies linearly with the gating time as in the classical setup. Moreover the standard deviation (open red cycles) follows the Poisson distribution (the square root of the mean value is plotted by the half open cycles). So the RSFQ system is consistent with the classical readout and the system works in the single photon regime as well. The lower detection efficiency of the 10 nm thick multi-pixel chip can be compensated by a higher photon flux on the chip without changing the photon statistic. That means that the multiplexing scheme, which is analyzed in the following with lower detection efficiency is also valid for detectors with higher detection efficiency.

Fig. 6 Comparison of the photon statistic between classical pulse counter based readout and the RSFQ readout. Presented is the mean value µ and the standard deviation σ for different gate times Δt. For the standard readout a 4 nm thick single pixel detector is used as benchmark. In the RSFQ case, one channel of the 10 nm thick multi-pixel chip is used. Both readouts show the characteristics as expected from the Poission distribution.

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For all detector elements the probability distribution of the counts is shown in Fig. 7 .

Fig. 7 Measured switching probability distribution of the count numbers for all four detector elements readout by the RSFQ electronics. Each detector was analyzed separately and a fit with a Gaussian distribution is shown.

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Due to the distance between the detector elements on the chip and a non-homogeneous spatial radiation spot, each detector element was illuminated with a different photon flux. In this measurement the gate time is fixed to Δt = 100 ms. The count events are Gaussian distributed. This agrees with theoretical predictions, because the Poisson distribution converge to a Gaussian distribution for larger mean values. We can conclude that the RSFQ readout delivers for all channels the dynamics based on the Poisson photon statistic.The photon statistics that produce a deviation from the mean value of the counts per time interval is one of the reasons, why all types of time gated multiplexer always measure inaccurate numbers. Since the relation between mean value and standard deviation $µ / \sqrt{µ} = \sqrt{µ}$ increases with gating time Δt (see Eq. (6)), the count rates become more accurate for longer measurement times Δt, because the count rate calculation normalizes µ and σ of $c_{t_{c l k}}$ by t_clk (see Eq. (5)), which leads to a constant count rate for all measurement times and a decreasing standard deviation.

Dependence on the number of pixels

We check the influence on the counts depending on the number of active detectors.

The complete chip is again illuminated by a random but fixed photon flux on each detector. The number of overall count events is plotted in Fig. 8 for Δt = 50 ms. The mean value of counts increases with the number of active detectors as well as the standard deviation. Therefore, the accuracy of the overall count rate increases for the same reason as described in the last paragraph, but for the single pixel the higher standard deviation of the whole system influences the minimum resolvable count rate as we will see in the next paragraphs.

Fig. 8 Mean value of overall counts and standard deviation for different numbers of active detectors. The standard deviation increases with increasing number of active detectors.

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OSM: Dependence on measurement time

We want to continue this analysis with the orthogonal sequencing multiplexer. The orthogonal sequence of Table 1 was programmed in the switch controller and triggered by a user defined switching time t_clk. All detectors were biased by a current of 0.9I_C.

Figure 9 shows the typical count pattern if an orthogonal bias switching is applied to the system. The switching time was t_clk = 50 ms. The eight states of the Walsh code and the repetion of the pattern can be clearly recognized.

Fig. 9 Five measured periods of the orthogonal sequencing multiplexer. The detectors are activated according to Table 1. Each biasing sequence is applied for a time t_clk = 50 ms. Each point describes the counts during the time t_clk.

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We expand this measurement by applying different switching frequencies to the setup. Using Eq. (4) the detector count vector $c_{t_{c l k}}$ can be extracted.

As we can see in Fig. 10 , detector 3 has the highest count rate and detector 4 the lowest. The linear dependency on the measurement time for all channels, which we already recognized at the single channel analysis, can also be seen in the orthogonal sequencing case. The deviations from the linear slope come from slight variations in the optical flux, because the used multimode fiber has a certain optical mode profile in the beam spot, which slightly changed during the measurement.

Fig. 10 Orthogonal sequencing multiplexer: Linear dependency of the extracted counts on the switching interval t_clk. The absolute value is proportional to the momentary photon flux on each detector element

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OSM: Quantitative comparison to single pixel readout

To verify the accordance of the counts of the orthogonal sequencing multiplexer with a single pixel measurement, we exemplary analyze three measurement situations more in detail. Each measurement was done with a different illumination on the chip which therefore leads to different counts and causes variation in the overall number of counts. In Fig. 11 , each colour represents one of the four detectors. The solid columns represent the mean value µ of the extracted counts during t_clk based on several tens of frame cycles. The striped columns represent the standard deviation σ of the extracted values. The used measurement setup cannot be calibrated for the real photon number on each detector element to calculate the detection efficiency. Therefore, for each orthogonal sequence measurement we also measured the single detector sequence for all four detectors as a reference like in a time domain multiplexing approach. The deviation from this reference count number is marked as error bar on each mean value column. The size of the error bars lets conclude that the deviation is marginal and not systematically shifted. It is mainly produced by the already mentioned slight variation in the photon flux during the measurement.

Fig. 11 Mean value µ and standard deviation σ of orthogonal sequencing measurements for different photon flux conditions. The error bars mark the deviation from a single pixel reference measurement. The standard deviation in each of the three measurements is almost identical for all detector elements. The common standard deviation varies with the overall count number.

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The measured mean value of the orthogonal sequencing multiplexer is therefore, as expected from the theory, very close to a time gated single channel measurement. The standard deviation will be analysed in the next paragraph.

OSM: Accuracy, limitations and count rate analysis

We concentrate now on the accuracy of the demultiplexed counts. For optimal extraction the following conditions must be fulfilled:

1. The detector bias switching sequences follow the Walsh functions exactly;
2. The values of the count rates of all detectors are constant during T_frame, because the vector c in Eq. (3) is assumed to be time invariant during one Walsh cycle;
3. The standard deviation of $c_{t_{c l k}}$ is zero;

Condition 1 is met by the switching of the bias currents with the correct coding sequence. Condition 2 cannot be fulfilled completely, because in an imaging system we normally have a dynamic photon flux. Each imaging frame is a progress in time and delivers a change in the count numbers. That means, the OSM can average fluctuations in the photon flux that appear during the frame time T_frame, but is not able to compensate a monotone drift of photon flux during T_frame. Even if the variation of the photon flux is small and negligible during T_frame, each count measurement has a certain count statistics as we already saw. This deviation is in conflict to condition 3 and will influence the orthogonality.

We have to balance between to influences: The measurement time for each pixel is in the orthogonal sequencing case T_frame/2 and is independent of the number of pixels. Compared to a time multiplexing measurement, where the measurement time is inversely related to the number of pixels, this is a time enlargement for the activation of all detector elements and would increase the accuracy._. However, merging several detector signals to one output leads to a correlated cross talk, which degrades the orthogonality.

Figure 10 demonstrates how this influences the accuracy. For all three measurements the standard deviation of the mean counts is calculated. For each illumination condition, the standard deviations for all detector elements are almost identical. An orthogonal sequencing multiplexer produces therefore a standard deviation level, which is similar for all detector elements. This is a logical conclusion, because the counts of all detector elements are inserted with positive or negative sign for exactly half the measurement time in Eq. (4), which leads to a summation of the standard deviation of all detector elements. The deviation level is always between the standart deviations that a time domain multiplexer would deliver for the highest and the lowest count rate. The exact mathematical context can be found in [27].

Since the orthogonal sequencing multiplexer averages the measurement over the time n·t_clk, a detector element with high count rate will profit from this multiplexing method, if detector elements with lower count rate exist at the same time. A detector element with low count rate will be disadvantaged. Furthermore, the common standard deviation level is a limitation for the absolute resolvability of low count rates. It restricts the dynamic range of possible count numbers, because detector elements with count numbers lower than the common standard deviation level, cannot be extracted correctly any more for a fixed time t_clk, as described in the following condition:

µ_{X} > \sqrt{σ_{l e v e l}^{2}} \to D E_{X} \cdot n_{p h X} \cdot n t_{clk} > \sqrt{σ_{D 1}^{2} + σ_{D 2}^{2} + ...}

Typical CDMA multiplexers use the time enlargement to increase the signal-to-noise-ratio (SNR). However, the innaccuracy in our case is not a type of uncorrelated noise. The standard deviations

σ_{X}

in Eq. (7), which are the standard deviations of each detector element based on the active time nt_clk, are a part of the signal and therefore correlated to the counts. Moreover the summation of

σ_{X}^{2}

scales with the number of detectors (see also Fig. 8 and Eq. (7)). Concerning further scaling, the possible number of resolvable detector elements is therefore a trade-off, between the measurement time t_clk and the requested resolvabale minimum and maximum photon flux

n_{p h}

.

Finally, the count rates are evaluated. We selected the counts based on the measurement times t_clk in Fig. 10 and calculated the mean count rate and the standard deviation of the mean count rate (see Fig. 12 ).

Fig. 12 Calculated count rates for a frame T based on different measurement times t_clk. The count rates are almost independent of the switching time t_clk, but the standard deviation decreases with increasing measurement time t_clk.

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We expect that the count rates should be identical for all measurement times t_clk. A slight variation can be seen, which comes from the already mentioned flux change during measurement (especially detector 1 and 2 at t_clk = 0.5s), but a consistency is nevertheless obvious. The standard deviation decreases for longer measurement times t_clk, which fits to the prediction of the Poisson distribution and the the analysis of the single pixel measurements of Fig. 6. We can conclude, that longer measurement times or alternatively higher detection efficiencies improve the accuracy level, which is predicted by Eq. (6).

The standard deviation could be even further decreased. In communication theory, methodes like successive interference cancelation [28] or whitening concepts [29] are provided. Especially whitening procedures, which try to reorthogonalize the measured data are interesting to improve the accuracy of the measurements in a postprocess although keeping the measurement time constant.

7. Conclusion

We suggest an imaging concept for a count rate multiplexer system by using a multi-pixel superconducting single-photon detector and a RSFQ pulse merger and discuss several approaches for scaling of the pixel number. This concept is based on an orthogonal Walsh sequenced switching of the detector bias currents and enables local count rate extraction of a multi-pixel detector by one readout line. The decoding algorithm which allows resolving the original local count rates of a multi-pixel array is explained. We fabricated a four-detector element SNSPD chip and a four-channel RSFQ readout electronics and demonstrated the operation of the multiplexing method with this hybrid SNSPD/RSFQ system. The RSFQ readout increases the maximum count rate of two orders of magnitude compared to a classical SNSPD readout and delivers the same photon statistic. We reveal the conditions that limit the resolvability of the count rates depending on the photon distribution and variation of photon flux on different detector elements. The orthogonal sequencing multiplexer approach delivers the expected counts compared to a single pixel measurement and the standard deviation of the counts is homogenized to a common value for all pixels, which means an improvement of the accuracy for high count rates at the expense of the accuracy of low count rates compared to a single pixel measurement. We showed that longer measurement times reduce the standard deviation of the count rates.

In future, the integration of the switching electronics in the cryogenic area comes into the focus. Moreover, designs for enlarged number of pixel on both the SNSPD chip and the RSFQ chip has to be fabricated and tested. In this context several approaches could be tried to decrease the common standard deviation level by several signal post-processing approaches

Acknowledgments

This work is partly supported by the Deutsche Forschungsgemeinschaft (DFG) Centre of Functional Nanostructures (Project No. A4.3). We thank Marco Schulz, Andre Krueger, Torsten Krause, Alexander Stassen and Karlheinz Gutbrod for technical assistance and we acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Karlsruhe Institute of Technology.

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Orthogonal sequencing multiplexer for superconducting nanowire single-photon detectors with RSFQ electronics readout circuit

Abstract

1. Introduction

2. SNSPD fabrication

3. RSFQ electronics

4. Multiplexing technique

5. Measurement setup

Cryogenic electronics

Bias current switching

Data acquisition

6. Experiments

Photon statistics with classical readout and RSFQ readout

Dependence on the number of pixels

OSM: Dependence on measurement time

OSM: Quantitative comparison to single pixel readout

OSM: Accuracy, limitations and count rate analysis

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (12)

Tables (1)

Equations (7)

Optics Express

Clock cycle index	1	2	3	4	5	6	7	8
H_{8,Detector 1}	1	1	1	1	−1	−1	−1	−1
H_{8,Detector 2}	1	1	−1	−1	−1	−1	1	1
H_{8,Detector 3}	1	1	−1	−1	1	1	−1	−1
H_{8,Detector 4}	1	−1	−1	1	1	−1	−1	1
Detector 5-7	…	…	…	…	…	…	…	…