## Abstract

Single Photon Detectors are fundamental to quantum optics and quantum information. Superconducting nanowire detectors exhibit high performance in free-running mode, but have a limited maximum count rate. By exploiting a bistable superconducting nanowire system, we demonstrate the first gated-mode operation of these detectors for a large active area single element device at 625MHz, one order of magnitude faster than its free-running counterpart. We show the maximum count rate in gated-mode operation can be pushed to GHz range without a compromise on the active area or quantum efficiency, while reducing the dark count rate.

© 2012 OSA

## 1. Introduction

When a superconducting nanowire is biased close to its critical current (*I _{C}*), a photon induced resistive hotspot, followed by current assisted formation of a resistive bridge, results in a macroscopic voltage pulse [1]. Supreconducting nanowire single photon detectors (SNSPD) provide outstanding performance [2] in a small cryocooler [3], and thus are the most practical cryogenic single photon detector (SPD). Quantum efficiency (QE) equal to 57% at 1550nm [4], dark count rate (DCR) below 1Hz [5] and timing jitter less that 30ps [6] have been reported. Furthermore, it is possible to resolve the number of photons by spatially multiplexing many nanowires in different configurations [6, 7]. Quantum key distribution [8], lunar laser communication link [9], diagnosis of integrated circuits [10] and characterization of single photon sources [11] are among the demonstrated applications.

To date, SNSPDs have been operated in Free-running Mode (FM), i.e. the nanowires are biased by a constant current as shown by the circuit model in Fig. 1(a) [12], where *R _{L}* represents the load impedance and

*L*represents the kinetic inductance associated with the nanowires in the superconducting phase. In the absence of photons, the superconducting nanowires shunt current away from

_{K}*R*, leaving zero voltage across it. The absorption of a single photon leads to formation of a hot resistive bridge across the width of a nanowire which forces the current through the load, hence creating a rising voltage that signals a photon detection.

_{L}For proper FM operation, the system should be mono-stable in the superconducting phase to ensure the resistive bridge will annihilate after formation, and the bias current will return to its initial value. Therefore, the SNSPD will self-reset to its initial state ready to detect the next photon. The smaller the time constant *τ _{e}* =

*L*, the faster the return to the initial state and the higher the maximum count rate (MCR) [12]. However, current biased superconducting thin-film wires are bistable systems in general [13, 14], i.e. in addition to the completely superconducting state, the state with a resistive bridge (hotspot) might also be stable. Bistability should be carefully avoided in FM-SNSPDs, otherwise the detector will latch to the resistive state following a photon detection or dark count and thus cease detecting further photons.

_{K}/R_{L}Keeping the QE at maximum (equivalent to setting the bias current close to *I _{C}*), the monostability condition puts a lower limit on

*τ*,

_{e}*τ*, beyond which the FM-SNSPD will turn into a bistable system and stop working [15,16]. Furthermore,

_{e–min}*τ*scales up with

_{e–min}*L*; for a single element FM-SNSPDs

_{K}*L*is proportional to the active area of the detector; and the larger the active area the easier the coupling of photons to it and thus the higher system QE. All of these put together impose a tradeoff between the system QE and the MCR that severely limits the MCR of typical high system QE single element SNSPDs. [15].

_{K}There have been different approaches to reduce *τ _{e–min}*, while keeping the active area unchanged. It can be reduced by exploiting different materials [17–19], or by changing the geometry of the device from a single long nanowire to some shorter wires that are either connected in parallel [20–22], operating as independent smaller pixels [6] or placed under a nano-antennae [23]. However, keeping the structure and materials unchanged, the tradeoff between the active area and the MCR remains.

In the following, we report the first operation of SNSPDs in Gated-Mode (GM) in which the condition *τ _{e}* >

*τ*can be relaxed and thus the MCR will be enhanced. High count rates are needed for many applications like quantum computing [24] and communication [8], and laser ranging [25].

_{e–min}## 2. Concept and implementation

In the GM, we increase *R _{L}* such that

*τ*<

_{e}*τ*to make an electrically fast but bistable system. However, instead of a constant bias current, we apply an alternating current with a DC offset to the nanowires. At the peak of the current, the superconducting nanowire latches to the resistive state upon a photon detection or a dark count. The latched nanowire will then be reset to superconducting state by the next minima of the current. Therefore, latching to the resistive state which has been a detrimental effect for FM-SNSPDs, would be part of the detection process in the GM. Also, because

_{e–min}*τ*<

_{e}*τ*, a GM-SNSPD will have a higher MCR compared to the same SNSPD operated in FM.

_{e–min}We use the circuit shown in Fig. 1(b) to implement a GM-SNSPD. The voltage from a bias source is split into two paths. One path creates the alternating bias current by using a termination resistor, *R*_{1} = 50Ω and a biasing resistor, *R _{B}* ≫

*R*

_{1}. This current is sensed by

*R*

_{2}= 50Ω together with two amplifiers. The other path undergoes an adjustable delay and attenuation. The voltage difference,

*V*=

_{d}*V*

_{2}–

*V*

_{1}would be small in absence of incoming photons for an appropriately adjusted circuit. However, as illustrated in Fig. 1(c),

*V*peaks whenever the detector latches. We use discriminated

_{d}*V*to count photons. We also make an FM-SNSPD to compare it with the GM operation: a DC bias source is used in the same circuit of Fig. 1(b) but

_{d}*R*

_{1}replaced with a large 100nF capacitor. Such circuit is electrically equivalent to the one in Fig. 1(a) with

*R*=

_{L}*R*+

_{B}*R*

_{2}and thus provides FM operation with minimal changes. The 100nF capacitor and the high electron mobility transistor (HEMT) isolate the sensitive nanowire from noise and small microwave reflections in both coax cables connecting the device, and thus further ensures appropriate operation of the system.

The devices used in this work were fabricated by Scontel, Moscow, Russia. They consist of a single 500*μ*m long, 4nm thick, 120nm wide Niobium Nitride on Sapphire. Active area is 10 × 10*μ*m^{2} (see [26] for a picture). At 4.2K, we determined the kinetic inductance *L _{K}* = 490nH (see methods) and

*τ*= 3.3ns (equivalent to

_{e–min}*R*= 150Ω) (see methods).

_{L}## 3. Experimental characterization

We excite the detectors using an attenuated 1310nm CW laser at a level that makes the measured count rates linearly proportional to the laser intensity, thus ensuring single photon sensitivity [27]. Both the discriminated time binned response of the FM-SNSPD, and the discriminated response of the GM-SNSPD make binary random process *x*, taking 0 for no-click and 1 for click events. The autocorrelation function, Γ(*τ* ≠ 0) of *x* gives the joint probability of two events separated in time by *τ*. We use normalized Γ(*τ* ≠ 0) to check the independency of successive events and therefore to study features like dead time and after pulses.

Figure 2(a) shows the measured Γ(*τ*). For GM: *R _{B}* = 650Ω, the bias is 625MHz sinusoidal with minima and maxima equal to −2

*μ*A and 0.9

*I*respectively (see methods). For FM: allowing a margin to further avoid latching, we set

_{C}*R*= 100Ω, and the DC bias to 0.9

_{L}*I*. The results show for having Γ(

_{C}*τ*) changed by less than 10%,

*τ*should be greater than 22ns in FM, while within the same limits, adjacent gates of the 625MHz GM-SNSPD keep their statistical independency. This shows about one order of magnitude speed increased in GM. Also, Fig. 2(b) is a time histogram of the detection events within a gate period. It shows the QE changes less than 5% for a time window equal to 57ps (about 1/30th of the gating period).

To study the after pulses in GM, we keep the biasing unchanged but excite the detector with a 625MHz/20, 1310nm pulsed laser. The resulting normalized Γ(*τ*) is shown in Fig. 2(c). It shows clear jumps each 20 gating periods, and in between remains flat at a level determined by the dark count probability per gate. An exception occurs at the first gate where it is enhanced by an after pulsing probability of about 0.03%. We attribute this to either an unwanted oscillatory behavior in the biasing current following a photon detection or a temperature rise in the corresponding gate. The possibility of both options will be seen later in the paper.

The QE and DCR of both of the modes are compared in Fig. 3. For the GM we apply 100MHz bias and lock a 200ps, 1310nm pulsed laser to the bias maxima. The QE for GM and FM shows a good agreement. However, as the GM-SNSPD is not always on, its DCR is smaller than that of the FM-SNSPD by more than one order of magnitude. As we could not synchronize our laser to the bias current oscillation at higher frequencies, we excite the detector with a CW laser. In this way, we confirmed both the average QE (i.e. detection probability per gate divided by the average number of photons per gate period) and the DCR stayed unchanged when the bias frequency was shifted to 625MHz.

One important aspect of the FM and the GM operation is their quantitative comparison. From the FM-SNSPD measurements of Fig. 3, we determined the QE and DCR as a function of the bias current (*I _{B}*):

*QE*(

_{FM}*I*) and

_{B}*DCR*(

_{FM}*I*). For

_{B}*DCR*(

_{FM}*I*), we fitted an exponential function to the data points and assumed the function is valid outside the measured range in later calculations. In the GM, the bias current takes the form:

_{B}*I*(

_{B}*t*) =

*I*

_{0}+

*I*

_{1}

*cos*(2

*πt/T*), where

*I*

_{0}and

*I*

_{1}are the DC and AC components of the bias,

*t*labels the time and

*T*is the bias period. The solid line in Fig. 2(b) is a plot of

*QE*(

_{FM}*I*(

_{B}*t*))

*/QE*(

_{FM}*I*

_{0}+

*I*

_{1}), with

*I*

_{0},

*I*

_{1}and

*T*set for the same experimental conditions. The agreement between the curve and the points shows the GM gate shape is determined by the variation of

*I*during a bias period.

_{B}We also quantitatively compare the DCR of the two operation modes. For a GM gate between −*T*/2 < *t* < *T*/2, the probability of registering a dark count between *t* and *t* + *dt* is equal to *DCR _{FM}*(

*I*(

_{B}*t*))

*dt*, provided the detector has not already been latched in the same gate. That is to write:

*I*

_{0},

*I*

_{1}and

*T*set for the same experimental conditions. The agreement between the points and the curve shows for GM-SNSPD the DCR is smaller because the bias current does not always stay at a high level. In fact our GM-DCR is about 20 times smaller than the corresponding DCR in the FM at the cost of having the detector on for about 1/30th of the time. We attribute the small difference between the calculated curve and the measured points to experimental error in adjusting the high frequency bias current of the nanowires.

## 4. Simulations

To explore the limitations on the gating frequency, we developed an electrical model of our SNSPD as shown in Fig. 4(a) (see methods). An approximate version (see Fig. 4(b), plus the thermal model in [28] (see methods) is used to numerically simulate our GM-SNSPD. In the approximate electrical model, *C _{P}* = 0.14pf+0.38pf+44ff+6.3ff= 0.57pf and

*R*=

_{P}*R*+

_{B}*R*

_{2}+ 25, where 25Ω is the resistance seen from the left of

*R*looking to

_{B}*R*

_{1}and the left coax (see Fig. 1(b)). We implemented our time domain solver in Matlab. The initial temperature of the superconducting wire was set to the substrate temperature (4.2K). The initial currents and voltages were set from the steady state response of the circuit of Fig. 4(b) with no resistive hotspot, to a sinusoidal bias current with a negative peak equal to −2

*μ*A and a positive peak equal to 95% of

*I*= 16.9

_{C}*μ*A at 4.2K. To simulate what happens after a photon detection, we add a hotspot with a maximum temperature of 11.0K (slightly higher than the superconducting critical temperature equal to 10.5K), and a length equal to 16nm when the nanowire current is at a peak. We confirmed that even for the smallest

*L*= 6

_{K}*nH*used throughout our simulations, the hotspot grows to sizes considerably larger than its initial 16nm width. Therefore, a rather arbitrary choice of our hotspot’s initial shape does not have a major impact on the simulation results.

We simulate the peaks of current in the gates following photon detection in our GM-SNSPD. The result is shown in Fig. 4(c) for *R _{P}* = 725Ω (equivalent to

*R*= 650Ω) and

_{B}*L*= 490nH. At less than about 300MHz or at about 600MHz the peaks do not change significantly. Indeed, this is how the gating frequencies in the experiments are selected. Therefore, the oscillatory response of an under-damped RLC circuit puts a purely electrical limitation for the gating frequency of our GM-SNSPD.

_{K}To study the ultimate MCR of GM-SNSPDs, assuming *C _{P}* can be reduced to 0.01pf if

*R*is integrated to the SNSPD chip, we repeat the simulation for values of

_{B}*L*ranging from 6nH to 6

_{K}*μ*H. For each

*L*, we choose

_{K}*R*such that it makes a critically-damped RLC circuit. We simulate for the maximum temperature of the nanowire at the center of the gate following a detection gate (when the nanowire current reaches its maximum again). The results are shown in Fig. 4(c). The curves are up to the frequency at which the detector re-latches in the next maxima of the current due to an elevated nanowire temperature. We also confirm that the currents of Fig. 4(c) are horizontally flat for this case. Therefore, for such an integrated GM-SNSPD the MCR would be purely limited by the thermal response of the SNSPD. Notice, increasing

_{B}*L*over three orders of magnitude decreases the MCR by about 33%. Therefore, in contrast with FM-SNSPDs, little compromise on active area is required for achieving high MCR in GM. Also, because FM-SNSPDs with smaller

_{K}*L*offer higher MCR, while the MCR in GM is not a very sensitive function of

_{K}*L*, we expect less MCR improvement for smaller SNSPDs.

_{K}## 5. Methods

#### 5.1. SNSPD electrical model

We used a high frequency electromagnetic simulation software, SONNET, to derive the SNSPD circuit model as shown in Fig. 4(a). Perfect conductor on top of a loss-less substrate with relative permittivity equal to 11.35 [29,30] was used to simulate the gold pads. Surface inductance equal to 90pH (equivalent to London penetration depth equal to 532nm [12] at a thickness of 4nm) was used to simulate the meandering nanowires. In each case we simulated for the phase and amplitude of the S-parameters up to 5GHz. These results are then converted to the circuit model by using the Agilent Advanced Design System (ADS). The 0.14pf capacitor is obtained with the same methods for a small pad on a printed circuit board where the SNSPD was connected.

To test the validity of the resulting model, we put *R _{B}* = 50Ω, disconnect the connection to the HEMT amplifier and connect RF

*to a coaxial cable. We measured the input reflection coefficient using a vector network analyzer (VNA) calibrated at the cryogenic end of the coax while the device was cooled to 4.2K. The agreement between the measurement and what we simulate in ADS using the model in Fig. 4(a), confirmed the model is fairly accurate at least up to 2GHz.*

_{in}#### 5.2. Minimum τ_{e} in free running mode

We adapted the same method reported in [15] to measure the maximum available count rate of the FM-SNSPD while maintaining the QE at its maximum. For different *R _{L}* values, starting from a high bias current that results in a stable latched state, we measured the current at which the nanowire returns to the superconducting state while sweeping the bias current downwards. We observed the return current starts to decrease for

*R*greater than about 150Ω. This value and its associated

_{L}*τ*are what we report as

_{e}*τ*for our FM-SNSPD. We used our FM-SNSPD setup described earlier in section 2, and operated it at 4.2K to do this experiment.

_{e–min}*R*was changed by changing

_{L}*R*at discrete steps.

_{B}#### 5.3. Adjusting the high frequency current

One of the difficulties of operation in GM is accurately applying a high frequency biasing current with a DC offset to a device operating at cryogenic temperature. Using a VNA, we characterized the components of our electrical setup including biasing coaxial cable and all individual electronic components used. The resulting information was all put together in Agilent ADS where we simulated for frequency dependence of the transconductance between the voltage generated by the bias source and the current that flows in the nanowires. This was used to adjust the minima and maxima of the biasing current throughout the experiments. It is better to set the minima of current to zero to ensure the nanowire always returns to the superconducting state following by a latching. We chose a slightly negative value (−2*μ*A) for the minima to allow more room for the effect of noise and also errors in determining the transconductance.

#### 5.4. Thermal Model

The thermal model for our simulations is the same as the one reported by Yang et.al. [28]. It is basically a 1D heat transfer equation for describing the dynamics of a hotspot on the superconducting nanowire. The equation has the form

*x*,

*T*(

*x,t*),

*J*(

*t*),

*ρ*(

*x,t*),

*k*(

*x,t*),

*α*(

*x,t*),

*d*,

*T*and

_{sub}*c*(

*x,t*) are coordinate, temperature, current density, electrical resistivity, thermal conductivity, thermal boundary conductivity, nanowire thickness, substrate temperature and specific heat, respectively. We use the same nonlinear temperature and phase dependencies of the parameters as in [28] with exactly the same numerical values for the parameters. The only difference is that our nanowire width is 120nm which also scales our zero temperature critical current to 24

*μ*A. The thermal model is coupled to the electrical model through the voltage and current across the hotspot shown in Fig. 4(b). We assume the nanowire is long enough that the hotspot never reaches the two ends, and thus the two ends can always be assumed to be at the substrate temperature. We did all the simulations at

*T*equal to 4.2K.

_{sub}## 6. Conclusion

To conclude, SNSPDs can be operated in GM at the same QE that they have in FM, but with an enhanced MCR and reduced DCR. Using a differencing read out technique, we implemented a gated setup and characterized different features. We have shown how irrespective of the value of *L _{K}*, the MCR can be pushed to the GHz range where a purely thermal limitation does not allow faster operation. The work will add a degree of freedom for designing ultra-high speed SPDs for applications like quantum key distribution and laser ranging.

## Acknowledgments

We acknowledge the financial support of OCE, NSERC and IQC. The authors would like to acknowledge Jeff. S. Lundeen and Thomas Jennewein for helpful comments. We also acknowledge Haig Atikian for his help proofreading the manuscript.

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