Josephson-Based Scheme for the Detection of Microwave Photons

We propose a scheme for the detection of microwave-induced photons through a current-biased Josephson junction, from the point of view of the statistical decision theory. Our analysis is based on the numerical study of the zero-voltage lifetime distribution in response to a periodic train of pulses, that mimics the absorption of photons. The statistical properties of the detection are retrieved comparing the thermally induced transitions with the distribution of the switchings to the finite voltage state due to the joint action of thermal noise and of the incident pulses. The capability to discriminate the photon arrival can be quantified through the Kumar-Caroll index, which is a good indicator of the signal-to-noise ratio. The index can be exploited to identify the system parameters best suited for the detection of weak microwave photons.

Figure 1

Equivalent electrical model of a current-biased short Josephson junction. The dc bias current, $I_b$, the current source used to mimic a weak microwave field, $I_s$, and the noise current, $I_n(t)$, of the JJ are included in the diagram. $R_J$ represents the quasiparticle tunneling resistance and $C_J$ the junction capacitance.

Figure 2

(a) Train of current pulses used to mimic the effect of photon absorption. The height and the width of each pulse, $A$ and $\delta t$, the distance, $T$, between two consecutive pulses, and the time $T_0$ the first pulse takes to arrive are also indicated. Each experiment consists in $N$ trains of pulses, each sequence spans for a time $P$. (b) Sketch of the tilted washboard potential in the case of three different bias current values.

Figure 3

(a) Application of the detection scheme to the averaged data, assumed to be Gaussian distributed. The threshold $⟨t{⟩}_{\mathrm{th}}$ determines the quantities $\alpha$ and $\beta$, the false detection probability and the probability to miss detection, respectively. (b) Distribution of escape times obtained numerically, with and without the signal and setting the values $D=0.1$, ${\gamma }_b=0.6$, $A=0.23$, $N=2\times {10}^6$, $P={10}^6$, $\alpha =0.025$, $T=1000$, and $\delta t=10$.

Figure 4

Mean switching time, $⟨{\tau }_{\mathrm{sw}}⟩$, as a function of the normalized bias current ${\gamma }_b$ at different values of $A$, in the case of $D=0.01$ (a) and $D=0.1$ (b). The blue dashed curve indicates the average switching time in the case of purely noise-activated escape processes. The values of the other parameters are: $N={10}^3$, $P={10}^6$, $\alpha =0.025$, $T=1000$, and $\delta t=10$. The gray dashed and black dot-dashed lines indicate the inverse Kramers and MQT rates, $r_{\mathrm{TA}}^{-1}(\alpha ,{\gamma }_b,D)$ and $r_{\mathrm{MQT}}^{-1}(\alpha ,{\gamma }_b)$, calculated according to Eqs. (9) and (11), respectively. The vertical cyan dashed line marks the noise amplitude, ${\gamma }_{\mathrm{Kr}}$, at which $r_{\mathrm{TA}}^{-1}(\alpha ,{\gamma }_{\mathrm{Kr}},D)=P$. The legend in (a) refers to both panels.

Figure 5

Conventional and binomial KC indexes, $d_{\mathrm{KC}}$ and ${\tilde{d}}_{\mathrm{KC}}$, as a function of the normalized bias current ${\gamma }_b$ at different values of $A$, in the case of $D=0.01$ [top panels (a) and (c)] and $D=0.1$ [bottom panels (b) and (d)]. The values of the other parameters are: $N={10}^3$, $P={10}^6$, $\alpha =0.025$, $T=1000$, and $\delta t=10$. The legend in (d) refers to all panels.

Figure 6

KC index, $d_{\mathrm{KC}}$, as a function of ${\gamma }_b$, at different values of $N$. The values of the other parameters are $D=0.1$, $A=0.1$, $P={10}^6$, $\alpha =0.025$, $T=1000$, and $\delta t=10$.

Figure 7

KC index, $d_{\mathrm{KC}}$, as a function of $A$, at different values of ${\gamma }_b$ and $D=0.1$, and $D$, at different values of ${\gamma }_b$ and $A$, see (a) and (b), respectively. The vertical dashed lines in (a) indicate the $A$ values chosen to draw the curves in (b). The values of the other parameters are $N={10}^4$, $P={10}^6$, $\alpha =0.025$, $T=1000$, and $\delta t=10$.

Figure 8

KC index, $d_{\mathrm{KC}}$, as a function of ${\gamma }_b$, at (a) different values of $\delta t$ and (b) different values of $ϵ$ and $\delta t=200$ in the case of rounded current pulses. The values of the other parameters are: $D=0.1$, $A=0.1$, $N={10}^3$, $P={10}^6$, $\alpha =0.025$, and $T=1000$.