Detection of noise-corrupted sinusoidal signals with Josephson junctions

We investigate the possibility of exploiting the speed and low noise features of Josephson junctions for detecting sinusoidal signals masked by Gaussian noise. We show that the escape time from the static locked state of a Josephson junction is very sensitive to a small periodic signal embedded in the noise, and therefore the analysis of the escape times can be employed to reveal the presence of the sinusoidal component. We propose and characterize two detection strategies: in the first, the initial phase is supposedly unknown (incoherent strategy), while in the second, the signal phase remains unknown but is fixed (coherent strategy). Our proposals are both suboptimal, with the linear filter being the optimal detection strategy, but they present some remarkable features, such as resonant activation, that make detection through Josephson junctions appealing in some special cases.

Figure 1

A picture of the potential barrier and of the detection procedure. When the phase (depicted as a dashed circle in figure) reaches the top of the barrier from the left, the system is restarted with a suitable initial phase from the bottom (black disk in figure) of the potential well $U\left(\phi \right)$.

Figure 2

The complementary cumulative distribution function (CCDF) of the escape time with signal $\epsilon =0.2$ in the incoherent (dot-dashed line) or coherent (dashed line) case. We also show the response to pure noise (solid line). The initial phase for the coherent strategy is ${\phi }_0=0$. Parameters are: $\gamma =0.5$, $\alpha =0.2$, $\omega =0.86$, $D=0.05$, ${\omega }_0={\left(1-{\gamma }^2\right)}^{1/4}\simeq 0.93$.

Figure 3

Typical estimated ${⟨\tau ⟩}_S$ as a function of $\omega$ $\left(T=2{10}^{5}2\pi /{\omega }_0\right)$ for the incoherent detection strategy. Parameters are: $\gamma =0.5$, $\alpha =0.2$, $D=0.05$, $\epsilon =0.1$. The average escape time in absence of the signal is ${\tau }_0=134.6$.

Figure 4

The dependence of the K-C index as a function of angular frequency for parameters $\gamma =0.5$, $\alpha =0.2$, $D=0.05$, $\epsilon =0.1$ in the case of the incoherent detection strategy. The curves displayed are for different observation time $T=2\pi N/{\omega }_0$, where $N={10}^p$.

Figure 5

The dependence of K-C index as a function of scaled signal amplitude $\epsilon /\sqrt{D}$ (proportional to the SNR) for the incoherent detection strategy. The parameters used are $\gamma =0.5$, $\alpha =0.2$, $\omega ={\omega }_0$, $T=2{10}^{5}2\pi /{\omega }_0$.

Figure 6

Typical estimated ${⟨\tau ⟩}_S$ as a function of $\omega$ $\left(T=2{10}^{5}2\pi /{\omega }_0\right)$ for the coherent detection strategy. Parameters are $\gamma =0.5$, $\alpha =0.2$, $D=0.05$, $\epsilon =0.1$, ${\phi }_0=0$. The average escape time in absence of the signal is ${\tau }_0=134.6$.

Figure 7

The dependence of K-C index as a function of scaled angular velocity for the coherent detection strategy. Parameters are $\gamma =0.5$, $\alpha =0.2$, $D=0.05$, $\epsilon =0.1$, ${\phi }_0=0$. The curves displayed are for different observation time $T=2\pi N/{\omega }_0$, where $N={10}^p$.

Figure 8

The dependence of K-C index as a function of scaled signal amplitude $\epsilon /\sqrt{D}$ (proportional to the SNR) for the coherent detection strategy. The parameters used are $\gamma =0.5$, $\alpha =0.2$, $\omega ={\omega }_0$, $T=2{10}^{5}2\pi /{\omega }_0$.

Figure 9

Scaled estimated average escape time ${⟨\tau ⟩}_S$ in presence of signal, vs scaled period $2\pi /{\tau }_0\omega$ for different values of $D$ in the case of coherent strategy (initial phase ${\phi }_0=0$). The parameters used are $\gamma =0.5$, $\alpha =0.2$, $\epsilon =0.1$, ${\tau }_0$ is the $\epsilon =0$ escape time. The vertical grid line indicates the resonant condition (7).

Figure 10

Effective waveforms for a system prepared with initial phase ${\phi }_0=-\pi /2$, $-\pi /4$, 0, $\pi /4$. The figures show the sinusoidal drive between the initial phase ${\phi }_0$ and the maximum of the signal, $\phi =\pi /2$.

Figure 11

The figures show the JJ potential $U\left(\phi \right)=-\overline{\gamma }\phi -\text{cos}\left(\phi \right)$ as a function of the phase. We consider $\overline{\gamma }=\gamma +\epsilon \text{cos}\left({\phi }_0\right)$ with initial phase ${\phi }_0=0$ (top) or ${\phi }_0-\pi /2$ (bottom); for illustrative purpose $\gamma =0.5$ and $\epsilon =0.1$. The time $P\left({\phi }_0\right)$ is defined as the time between the application of the signal (with initial phase ${\phi }_0=0,-\pi /2$) and the lowest height of the barrier, $\Delta U_{-}$. The first passage time $\tau$ is defined as the time to overcome the maximum of the barrier (the vertical dashed line). The resonance defined by Eq. (8) states that a minimum of the passage time occurs when are equal the time in which the external signal reduces the barrier to the lowest value and the escape time from such barrier, assumed to be static.

Figure 12

The dependence of the average escape time ${⟨\tau ⟩}_S$ (scaled to ${\tau }_0$) for the coherent detection strategy as a function of signal scaled period $2\pi /\omega$ for different values of the initial phase ${\phi }_0$. In panel (a) we show the escape times when the effective waveform is monotonic, i.e., $-\pi /2\le {\phi }_0\le \pi /2$; in panel (b) we show the four limit cases ${\phi }_0=0,\pi /2,\pi ,-\pi /2$. We underline that for ${\phi }_0=\pi$ the non monotonic behavior of the effective waveform (the maximum is not reach in a single ramp-up) leads to a maximum in the escape time. The parameters used are $\gamma =0.5$, $\alpha =0.2$, $\epsilon =0.1$, $D=0.07$.

Figure 13

The dependence of the resonant frequency as a function of $2\pi /P\left({\phi }_0\right)$, proportional to the inverse time between the initial phase and the maximum of the signal, see Fig. 11. The dashed line corresponds to the prediction of Eq. (8). Parameters are $\gamma =0.5$, $\alpha =0.2$, $\epsilon =0.1$, $D=0.07$.