Josephson Junctions as Single Microwave Photon Counters: Simulation and Characterization

Abstract

Detection of light dark matter, such as axion-like particles, puts stringent requirements on the efficiency and dark-count rates of microwave-photon detectors. The possibility of operating a current-biased Josephson junction as a single-microwave photon-detector was investigated through numerical simulations, and through an initial characterization of two Al junctions fabricated by shadow mask evaporation, done in a dilution refrigerator by measuring escape currents at different temperatures, from 40 mK up to the Al transition temperature. The escape dynamics of the junctions were reproduced in the simulation, including the dissipative effects. Inhibition of thermal activation was observed, leaving the macroscopic quantum tunneling as the dominant effect well beyond the crossover temperature.

1. Introduction

1.1. Detection of Axions

In the 70s, an extension of the standard model of particle physics was advanced to explain the absence of CP violation in the strong interaction (strong CP problem) [1,2]. The theory predicts the existence of additional particles called axions [3,4]—sub-eV particles with feeble interactions with ordinary matter, which could be abundantly produced non-thermally in the early universe [5]. These characteristics make them good candidates for explaining the composition of cold dark matter (CDM) in the universe [6,7,8], an issue that modern cosmology and particle physics are still tackling.

Most of the experiments are sensitive to the axion–photon coupling constant $g_{a\gamma \gamma }$, which is proportional to the axion mass $m_a$. The region of interest for QCD axions is the yellow band in the $m_a$-$g_{a\gamma \gamma }$ plane shown in Figure 1.

Axions are an example of weakly interacting slim particles (WISPs), a category of particles emerging from many extensions of the standard model. WISPs as axion-like particles (ALPs) have a more general relationship between $m_a$ and $g_{a\gamma \gamma }$.

The most exploited detection principle of axions and ALPs is the haloscope proposed by Sikivie [14,15], whose description is addressed in a review: [16]. Examples of running experiments, also shown by the vertical lines in Figure 1, are ADMX [17,18], HAYSTAC [19], CAPP-8T [20], CAPP-9T [21], ORGAN [22], and QUAX [23,24,25,26]; other proposed experiments include RADES [27,28], MADMAX [29], BRASS [30], and KLASH [31,32,33].

Since very tiny signals are involved in the game, searches of axions and ALPs will benefit from the latest developments in microwave quantum technologies, such as single microwave-photon detectors based on superconducting elements. In fact, as discussed in the next section, photon detectors may have better signal-to-noise ratios than quantum amplifiers, especially if they have low dark-count rates.

1.2. Single Photon Counters as Detectors

In the last two decades, attention to single photon detection has grown due to the demanding applications in quantum computing. Several techniques have been developed to detect single photons, such as quantum non-demolition measurements [34,35,36,37,38,39], switching detectors [40,41,42], hot electron detectors [43,44], and quantum dot detectors [45]. Belonging to switching detectors in the microwave range of radiation are Josephson junction (JJ)-based photon counters [41] and transmon qubits [35,46].

The best performances of currently running axion experiments were achieved with quantum-limited devices, such as superconducting quantum interference devices (SQUIDs) and Josephson parametric amplifiers (JPAs) [17,18,19,26,47], but efforts have been put toward counting single photons from axion interactions [42,46,48,49]. The reason is clear from [50]: working at higher frequencies while having a good scan rate at the same time requires single photon counters, which have better noise performances with respect to linear amplifiers above about 10 GHz. As discussed in [42], dark counts of single photon counters may be a limiting factor: in dark matter axion searches, a single photon counter will be competitive with a quantum-limited amplifier if the dark count ${\nu }_{dc}$ is below the intrinsic axion width $\Delta {\nu }_a\simeq {\nu }_a/{10}^6$ corresponding to 10 kHz at 10 GHz. Devices based on qubits have dark-counts rates ${\nu }_{dc}\sim p_{err}/t_{qubit}$ where $p_{err}\sim 10$% is a typical qubit readout error and $t_{qubit}\sim 10\mathsf{\mu }$s is a typical qubit decoherence time, corresponding to ${\nu }_{dc}\sim 10$ kHz. On the contrary, switching devices, such as those based on current-biased Josephson junctions (CBJJ) [41,42,48], have the potential to reach dark counts at mHz rates. However, previous realization of a photon counter based on CBJJ [41] was operated in a regime with far higher dark counts, and no clear evidence exists that dark counts can be lowered without affecting the device efficiency. Within the SIMP project [51], we are developing a microwave single-photon detector based on CBJJ to couple to a haloscope, with the aim of probing the axion existence. In this paper we describe the device and its response to photon excitations obtained from simulations of different experimental conditions, showing that single photons can induce the switch of a JJ. We describe the first step in the characterization of a JJ with parameters optimized to detect photons originating from an axion conversion and derive the dark counts expected for a photon detector. We finally discuss the improvement needed to keep both high efficiency and a low dark-count rate.

2. A CBJJ as a Photon Detector

The electrodynamics of a CBJJ is accurately described in terms of the resistively shunted junction (RSJ) model [52], an equivalent electrical model whose circuit components are related to specific junction physical characteristics. With reference to Figure 2, the capacitor $C_J$ represents the capacitance between the junction electrodes, the resistor $R_J$ represents the tunneling of normal electrons (quasiparticles), and the component $J_J$ represents the tunneling of superconductive electrons (Cooper pairs). The components $I_s$, $I_b$, and $I_n$ represent current sources that will be detailed below. The current $I_J$ through the component $J_J$ and the corresponding voltage drop $V_J$ are related to $\phi$, the phase difference between the macroscopic wavefunctions of the two superconductors, by the Josephson equations:

$$\begin{array}{ccc}\hfill I_J& =& I_{0}sin\phi ,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill V_J& =& \frac{\hslash }{2e}\frac{d\phi }{dt}.\hfill \end{array}$$

(2)

$I_0$ is the maximum Cooper pair current that can flow through the $J_J$ component, and ℏ and e are the reduced Planck constant and electron charge, respectively. A stationary phase, and correspondingly, a zero average voltage is possible, according to the Josephson equations, Equations (1) and (2), when a dc current below $I_0$ flows in the $J_J$ element. Depending on the values of the junction parameters ($I_0,R_J,C_J$), a finite-voltage state is also possible. If the dc bias current overcomes $I_0$, the phase cannot be stationary and a finite-voltage state is the only possibility. From the experimental point of view, this behavior is reflected in the occurrence of hysteresis in the current–voltage characteristic of the JJ, for certain dc bias current intervals, where two voltage states (zero and finite-valued) can be observed. Such hysteresis is at the basis of the use of a JJ as a detector. It is worth noting that in a typical tunnel-type JJ, the resistance of $R_J$ strongly depends both on voltage and temperature. However, as the overall effect of the resistor is to introduce dissipation in the system, its nonlinearity is often not considered in the presence of moderate or weak damping. Another important effect of the quasiparticle current, modeled by $R_J$, is the presence of random charge fluctuations, which can be represented, using the fluctuation dissipation theorem, by a noise current source, indicated with $I_n$ in Figure 2, whose spectral power density is assumed to be frequency independent (Johnson noise) [53].

In order to use a JJ as a detector, it is biased with a dc current just below $I_0$ through a suitable source, indicated with $I_b$ in Figure 2. The occurrence of an external additional current (i.e., the signal to be detected) can induce the switching of the junction from the zero to the finite-voltage state. Of course, current noise can induce a switching, contributing to false alarms or dark counts in detector language. In the case of a simple CBJJ, the phase difference $\phi$ can be considered as a particle moving in a one-dimensional tilted cosine potential [54]; see Figure 3. The tilt of the potential corresponds to the normalized bias current flowing through the junction. The zero-voltage state corresponds to the confinement of the particle in a potential well of barrier height $\Delta U$ [54], where it can oscillate at the characteristic plasma frequency, ${\omega }_J$. Given enough energy, the particle can escape from this metastable state and roll down the potential slope, giving rise to a finite-voltage state [54]. At temperatures such that $k_{B}T\gg \hslash {\omega }_J$, the escape rate is dominated by the thermal activation (TA) process [55]:

$${\mathsf{\Gamma }}_T\sim \frac{{\omega }_J}{2\pi }exp\left(-\frac{\Delta U}{k_{B}T}\right),$$

(3)

whereas at lower temperatures, it is dominated by macroscopic quantum tunneling (MQT) [55]:

$${\mathsf{\Gamma }}_q\sim \frac{{\omega }_J}{2\pi }exp\left(-\frac{7.2\Delta U}{\hslash {\omega }_J}\right)$$

(4)

which provides an irreducible contribution to the dark-count rate.

3. Device Simulation

We investigated the dynamics of a CBJJ in the presence of a microwave pulse, representing the process of single-photon absorption, by means of numerical simulations of the model equations [56]. We estimated in this way the parameters needed to make the junction switch in the presence of a signal.

3.1. Isolated CBJJ

To model a weak microwave field (weak because carrying few photons) coupled to the junction, we consider a deterministic current source, $I_s$ in Figure 2, generating a properly shaped current pulse, representing the absorption of a single microwave photon. The following equations refer to the circuit of Figure 2a and describe the dynamics of an isolated junction where we imagine the current source directly coupled to the junction.

By denoting $I_J$ as the current through the $J_J$ element, $I_{R_J}$ as the current through the resistor, and $I_{C_J}$ as the current through the capacitor, we write the following current balance equation:

$$I_{C_J}+I_{R_J}+I_J=I_b+I_s\left(t\right)+I_n\left(t\right).$$

(5)

By using the constitutive relations of the resistor and capacitor, and the Josephson relations, we obtain the following second-order differential equation:

$$C_J\frac{\hslash }{2e}\frac{d^2\phi }{d{t}^2}+\frac{1}{R_J}\frac{\hslash }{2e}\frac{d\phi }{dt}+I_{0}sin\phi =I_b+I_s\left(t\right)+I_n\left(t\right),$$

(6)

which, due to the presence of the stochastic noise term, is a Langevin equation [57]. By defining a normalized time $\tau ={\omega }_{J}t$, where

$${\omega }_J=\sqrt{2e{I}_0/C_J\hslash }$$

(7)

is the Josephson plasma frequency, Equation (6) is re-written as:

$$\frac{d^2\phi }{d{\tau }^2}+\alpha \frac{d\phi }{d\tau }+sin\phi ={\gamma }_b+{\gamma }_s\left(\tau \right)+{\gamma }_n\left(\tau \right),$$

(8)

where the parameters are defined as

$$\alpha =\frac{1}{R_J{C}_J{\omega }_J},{\gamma }_b=\frac{I_b}{I_0},{\gamma }_s=\frac{I_s}{I_0}.$$

(9)

The statistical properties of the noise term ${\gamma }_n$ are:

$$\begin{array}{ccc}\hfill \langle {\gamma }_n\left(\tau \right)\rangle & =& 0,\hfill \\ \hfill \langle {\gamma }_n\left(\tau \right),{\gamma }_n\left({\tau }^{\prime }\right)\rangle & =& 4D\delta (\tau -{\tau }^{\prime }),\hfill \end{array}$$

(10)

where $D=k_B\mathcal{T}{\omega }_J/\left(R_J{I}_0^2\right)$ is the normalized noise intensity ($k_B$ is the Boltzmann constant and $\mathcal{T}$ the physical temperature), $\delta \left(\right)$ is the Dirac delta function, and the parentheses $\langle \rangle$ represent ensemble averages.

3.2. CBJJ Coupled to an RC Circuit

If the JJ is small in size, its capacitance and conductance can be very small too, thereby making it possible for external parasitic effects to modify its dynamics. To investigate these effects, we considered a model of a JJ loaded with a series parasitic RC circuit; see Figure 2c. In this case, Equation (5) becomes:

$$I_{C_J}+I_{R_J}+I_J+I_{R_p}=I_b+I_s\left(t\right)+I_n\left(t\right),$$

(11)

where $I_{R_p}$ is the current flowing in the parasitic resistor $R_p$. The JJ voltage $V_J$ is related to $I_{R_p}$ by:

$$V_J=R_p{I}_{R_p}+\frac{1}{C_p}\int I_{R_p}dt.$$

(12)

By repeating the same procedure as before, we obtain the following two normalized differential equations:

$$\begin{array}{c}\hfill \frac{d^2\phi }{d{\tau }^2}+\alpha \frac{d\phi }{d\tau }+sin\phi ={\gamma }_b+{\gamma }_s\left(\tau \right)+{\gamma }_n\left(\tau \right)+{\gamma }_R,\end{array}$$

(13)

$$\begin{array}{c}\hfill \frac{d{\gamma }_R}{d\tau }+{\alpha }_{RC}{\gamma }_R+{\alpha }_{int}\frac{d^2\phi }{d{\tau }^2}=0,\end{array}$$

(14)

where

$${\alpha }_{RC}=\frac{1}{R_p{C}_p{\omega }_J},{\alpha }_{int}=\frac{1}{R_p{C}_J{\omega }_J},{\gamma }_R=\frac{I_{R_p}}{I_0}.$$

(15)

3.3. CBJJ Coupled to a Transmission Line

The simplest design for a photon detector operating at microwave frequency based on a JJ is a transmission line (TL), such as a coplanar waveguide, terminated with a JJ. When the JJ is coupled to a TL with a characteristic impedance $Z_{TL}$, as shown in Figure 2b, signal reflection back to the TL has to be considered. The overall effect is modeled by a modified effective junction resistance, given by the values of the parallel resistors $R_J$ and $Z_{TL}$. In Figure 2b, the current source $I_s$ is given by an ideal generator $I_g$ with an impedance $R_g$ in parallel. The latter has to be equal to $Z_{TL}$ to ensure impedance matching. To relate the simulation results to this particular case, we compare Equation (6) with the equation for the flux variable $\varphi =\phi ({\varphi }_0/2\pi )$ in a TL terminated by a parallel LC (a linearized JJ) [58]:

$$C_J\ddot{\varphi }+\frac{\varphi }{L_J}+\frac{1}{Z_{TL}}\dot{\varphi }=2\frac{1}{Z_{TL}}{\dot{\varphi }}^{in}=2I^{in},$$

(16)

where $L_J$ is the Josephson inductance and the term $1/Z_{TL}$ models the aforementioned signal reflection on the TL. Moreover, the current $I_s$ has to be interpreted as twice the input current $I^{in}$. The input peak-current due to a single photon on the waveguide with a Gaussian wavepacket of time duration ${\sigma }_t$ is

$$I_{\mathrm{peak}}^2=\frac{\hslash {\omega }_J}{Z_{TL}}\frac{2}{\sqrt{2\pi }{\sigma }_t}.$$

(17)

Then, the amplitude of the signal current $I_s$ in Equation (6) corresponding to a single photon is:

$$I_s^{\mathrm{photon}}=2\sqrt{\frac{\hslash {\omega }_J}{Z_{TL}}\frac{2}{\sqrt{2\pi }{\sigma }_t}}.$$

(18)

For a 10 GHz photon with ${\sigma }_t=600$ ps on a 50 $\mathsf{\Omega }$ TL, this corresponds to about 26 nA.

We ran several simulations with the CBJJ either isolated or coupled to a TL excited by Gaussian current pulses of about 600 ps in length. We considered values of the critical current ranging from few hundred nanoamperes to a few microamperes, and junction capacitance ranging from a fraction of a picoFarad to a few picoFarads. The ratio $I_b/I_0$, about 0.8, was set in such a way as to keep the estimated rate of the MQT from the ground level by a few hertz at most. With the isolated junction we observed switching currents $I_{switch}$ between 20 and 50 nA corresponding to about 1–4 photons, estimated as $N_{\gamma }={(I_{switch}/I_s^{\mathrm{photon}})}^2$. On the contrary, with the junction coupled to a 50 $\mathsf{\Omega }$ TL, these values increased to a range between 200 to 300 nA corresponding to about 100 photons. The increase of number of photons is directly proportional to the relaxation rate of the junction ${\gamma }_{TL}={\omega }_J/Q={\omega }_J{Z}_J/Z_{TL}=2\pi /Z_{TL}{C}_J$, where $Z_J=\sqrt{L_J/C_J}$, highlighting the need for a proper circuit for matching the CBJJ to the TL. Matching of a device to a TL is discussed in [39,59] where the detection efficiency is expressed as

$$P_{\mathrm{R}}=\frac{4{\gamma }_{TL}{\gamma }_{\mathrm{sw}}}{{({\gamma }_{TL}+{\gamma }_{\mathrm{sw}})}^2}.$$

(19)

${\gamma }_{\mathrm{sw}}$ is the switching rate of the excited JJ to the resistive state. Perfect matching condition ${\gamma }_{TL}={\gamma }_{\mathrm{sw}}$ implies that the escape to the resistive state must be as fast as the relaxation of the junction to the ground level. This prevents us from using MQT transitions from the first excited level as a detection mechanism: an escape caused by MQT from the first excited level, $E_1$ in Figure 3, equal to ${\gamma }_{TL}\sim$ 1–10 GHz, would induce a dark-count rate due to escape from the ground level of about ${\gamma }_0\sim {10}^{-3}{\gamma }_{\mathrm{sw}}\sim$ 1–10 MHz. On the contrary, while keeping low dark-count rates, two or more photons may induce the escape if absorbed by the junction within the relaxation time $1/{\gamma }_{TL}$.

4. Measurements

The study of the escape mechanisms of a CBJJ allow understanding and characterizing the escape rates in the absence of excitation signals, and therefore, of the dark counts. Much literature on measurements and interpretations of the escape processes has been published and underlines different dynamics, such as the TA regime, MQT, and phase diffusion (PD) with multiple retrapping processes [54,60,61,62,63,64,65,66,67,68]. The experimental conditions to highlight the diverse regimes and transitions between them have been defined. The regimes will depend on the comparison between the Josephson energies ($E_J$), related to the Josephson critical current $I_0$; the Coulomb energy $E_C$, depending on the capacity $C_J$; the plasma frequency ${\omega }_J$ proportional to $\sqrt{I_0/C_J}$ ratio; the quality factor $Q={\omega }_J{R}_J{C}_J$, proportional to the losses; the potential tilt $E_J\left(I/I_0\right)$, controlled by the bias current I; and temperature [54,60,61,62,63,64,65,66,67,68]. This rich phenomenology of the escape process makes it ideal for a DC characterization of a JJ. In general, the typical escape dynamic’s experimental setup consists of a slow ramping of the bias current across the junction up to the value of the critical current [61]. This ensures both that the JJ stays at the temperature of the cryostat thermal bath at milli-Kelvins, and that the superconducting state does not dissipate during the escape measurements before the final switching. Then, the data analyzed are the switching current distributions produced after countless repetitions of the process. Finally, measurements as a function of temperature can be a diagnostic tool for determining the processes of TA, tunneling, and phase diffusion regimes that regulate the escape process.

4.1. Fabrication Parameters from Simulations

The JJs we tested were fabricated at IFN-CNR by shadow mask evaporation (Figure 4), with electron beam lithography on a copolymer/PMMA bilayer (thickness of about $1\mathsf{\mu }$m for each layer) over a Si substrate, with two evaporation angles of Al at 25${}^{\circ }$ and 90${}^{\circ }$ (thickness of about 30 nm), and with 5 min of oxidation at 5 mbar (expected thickness in the order of 1 nm). Junctions with different areas were fabricated: $2\mathsf{\mu }\mathrm{m}\times 2\mathsf{\mu }\mathrm{m}$ ($I_0\simeq 300$ nA and $C\simeq 200$ fF, expected from design parameters) and $2\mathsf{\mu }\mathrm{m}\times 4\mathsf{\mu }\mathrm{m}$ ($I_0\simeq 600$ nA and $C\simeq 400$ fF).

4.2. Setup

We tested the devices in a range of temperatures between 50 mK and about 1 K in a Leiden Cryogenics MCK 50–100 dilution refrigerator. We used a 4-terminal measurement scheme (Figure 5), with a bias resistance $R=14.63\mathrm{k}\mathsf{\Omega }$ at room temperature, twisted/shielded phosphor bronze cryogenic lines (about $25\mathsf{\Omega }$ per line), and EMI low-pass filters at room temperature (bandpass 600 kHz). An Agilent 33220A waveform generator was used, together with two low noise preamplifiers EGG 5113 and a NI USB 6366 acquisition board.

This experimental setup allowed us to acquire current-voltage characteristics. We measured a gap $\Delta \simeq 200\mathsf{\mu }$V and critical currents within a factor 2 with respect to the estimated ones (Figure 6).

In order to measure the escape probabilities, we applied a sawtooth waveform with frequency $f=314$ Hz, amplitude $I_{pp}=670$ nA and offset $I_{off}=250$ nA. In each period, during the slow growth (with slope $dI/dt=210\mathsf{\mu }\mathrm{A}/\mathrm{s}$), sudden transition to the voltage state occurred at random values of current $I_c$. In the fast decrease region, the junction resets to the superconducting state. We acquired a series of $N=5000$ transition values $I_c$ by using the voltage jumps as trigger signals. These values were arranged in a histogram to evaluate the escape probability $P\left(I\right)$. In Figure 7 we report the experimental escape probabilities at different temperatures ranging from about 40 to 850 mK. Data analysis is discussed in the next section.

5. Results and Discussion

5.1. Results

From the 5000 transition values $I_c$ we constructed escape-probability densities. Before doing this, we applied a fast Fourier transform to the measured $I_c$ series and identified a narrow peak at 100 Hz and minor peaks at lower frequencies, attributed to electronic noise in the current-measurement setup. We calculated the peaks’ frequencies, phases, and amplitudes from the Fourier analysis, and filtered out the noise from the original $I_c$ series. We thereby obtained a 20% improvement in resolution. We show the experimental switching distributions $P\left(I\right)$ obtained after filtering the data for each temperature in Figure 7a,b—the smallest and the largest junctions, respectively.

We extracted the mean $\langle I_c\rangle$ and width $\sigma$ from each distribution as an arithmetic mean and a standard deviation. These are plotted as a function of temperature in Figure 8a,b, for the two junctions.

The mean critical current values (empty blue squares) follow qualitatively the typical BCS behavior [52] as a function of temperature. The widths (full red squares) present a peculiar feature. The constant behavior of $\sigma$ at low temperatures was expected from quantum tunneling processes (Equation (4)), whereas its decrease at higher temperatures, above about 400 mK, suggests that TA (Equation (3)) is suppressed. We attribute the decreasing regime partially to the decrease of the critical current with increasing temperature, although a further contribution due to retrapping is not excluded.

5.2. Interpretations of Results

To investigate the effect of a dissipative environment on the escape by TA, we ran simulations of the JJ in parallel to the dissipative RC circuit shown in Figure 2c and described in Section 3.2. In our experimental setup, the parasitic circuit is represented by the dc lines carrying the bias current to the junction. The driving equations of the simulations were obtained from Equations (11)–(15) by setting the current source $I_s\left(t\right)$ to zero. Qualitatively, in the RCSJ model, energy dissipation induced a relaxation rate

$${\mathsf{\Gamma }}_{\mathrm{rel}}=\frac{1}{\tau }=\frac{1}{R{C}_J}=\frac{{\omega }_J}{Q},$$

(20)

where R is the parallel between $R_J$ and $R_p$. If relaxation is faster than the TA, the latter is suppressed:

$$\frac{{\omega }_J}{2\pi }exp\left(-\frac{\Delta U}{k_{B}T}\right) < \frac{{\omega }_J}{Q},$$

(21)

corresponding to the condition

$$R<2\pi Z_{J}exp\left(\frac{\Delta U}{k_{B}T}\right).$$

(22)

Thus, even for temperatures such that $k_{B}T\sim \Delta U$, TA is suppressed if $R<2\pi e{Z}_J\sim 20Z_J$. For the described junction, $Z_J$ is $\sim 50 \mathsf{\Omega }$; therefore, an external resistance equal to about 1 k$\mathsf{\Omega }$ would dominate in parallel to $R_J=100$ k$\mathsf{\Omega }$, causing dissipation. On the contrary, the MQT process, which is not affected by dissipation, would become the dominant switching effect even at temperatures higher than the expected crossing temperature. This was confirmed by our simulations, whose results for different simulation parameters are shown in Figure 9. The inclusion of the RC circuit strongly modifies the qualitative behavior of $I_c$ and $\sigma$ as a function of the temperature. In Figure 10 we show the comparison of the experimental results obtained for the $2\mathsf{\mu }\mathrm{m}\times 2\mathsf{\mu }\mathrm{m}$ junction and results of a simulation run with parameters $I_0=143$ nA, $C_J=0.2$ pF, $R_J=100$ k$\mathsf{\Omega }$, $C=2$ pF, and $R_p=1$ k$\mathsf{\Omega }$, where only MQT and dissipation were included. Both the simulated distributions of $I_c$ and $\sigma$ are in good agreement with the measured ones. As anticipated, a further contribution from a retrapping process at a higher temperature could take into account the observed residual difference between the experimental and simulated values of $\sigma$.

5.3. Dark-Count Rate

Finally, we investigate the dark-count rate performances of the $2\mathsf{\mu }\mathrm{m}\times 2\mathsf{\mu }\mathrm{m}$ junction to predict how it would behave if used as a single photon counter. Following [69], we calculated the escape rate from the current distribution of the JJ taken at the lowest temperature (50 mK). The result as a function of the bias current is shown in Figure 11, where the blue dots represent the escape rate values extracted from data. The dashed red line is the escape rate expected from MQT only. It reproduced the data for currents up to 120 nA, where the current distribution at 50 mK has its maximum (Figure 7a). The deviation at higher currents is attributable to a Gaussian resolution effect with standard deviation $\sigma =1.1$ nA, as confirmed by the data’s agreement with the green solid line obtained by applying such a smearing to the theoretical current distribution of the MQT before deriving the escape rate.

To determine the working point for a photon detector based on a similar JJ ($I_0=140$ nA, $C_J=0.2$ pF), we relied on the simulation of an “isolated” junction with $R_J=5$ k$\mathsf{\Omega }$ driven by a resonant rf Gaussian pulse of frequency 6.5 GHz, duration ${\sigma }_t=370$ ps, and peak current $I_{max}=21$ nA corresponding to one photon. The simulated junction switched for a bias current $I_b\simeq 91$ nA (${\gamma }_b\simeq 0.65$) when the junction plasma frequency was about 6.5 GHz. By extrapolating the escape rate measured in our data (Figure 11) to the value ${\gamma }_b\simeq 0.65$, we estimated for the photon detector a dark-count rate of about 1 mHz.

It should be noted that in simulations with $R_J<1.5$ k$\mathsf{\Omega }$, dissipation inhibits junction switches, as observed in the TA process in our data and as expected from the discussion in Section 5.2. As already noted in Section 3.3, when a junction is coupled to a TL, the 50 $\mathsf{\Omega }$ line impedance acts as a dissipative medium, preventing the junction from switching. Finally, when ${\gamma }_b\simeq 0.65$, the height of the potential well is estimated to be 28 GHz, corresponding to about four energy levels, showing that the current pulse must induce a strong modification of the potential to cause the junction switch.

6. Conclusions

We have investigated the escape mechanism of a CBJJ both with simulations and through measurements of the escape currents in two Al JJs. We highlighted the effect of the dissipative mechanism introduced by the current leads—in particular, the suppression of escape induced by TA. While this suppression leads to a reduction in the escape rates and to dark counts above the crossover temperature, it also implies a low quality factor with consecutive reductions in signal efficiency, and this must be taken into account in the design of a microwave photon counter. We in fact showed with simulations that a current pulse with peak current $I_{max}=21$ nA, corresponding to a single photon, induces the switch of an isolated junction of critical current $I_0=140$ nA when biased with a current $I_b\simeq 91$ nA, and that the expected dark-count rate would be at the mHz-level. This value was confirmed by the escape rates extracted from our data. However, when the junction was not isolated but coupled to a TL, the simulation showed that switches were inhibited, and larger current pulses were required. The JJ must then be resonantly matched with the TL with a bandwith estimated to be in the order of $1/R_j{C}_j\sim 1$ GHz.