Figure 1

Shell circuit models: (a) general case; (b) dissipation “on the junction,” and (c) dissipation “on the shell.”Reuse & Permissions

Figure 2

(Color online) (a) A three-dimensional plot of the 2D potential for ${\gamma }_b=0.95$ and $\beta =7$. Shown is also a possible escape path from the minima; (b) the evolution of the phase under thermal noise at $T=75\mathrm{mK}$ (thin curve) and $T=150\mathrm{mK}$ (thick curve) on the 2D phase plane for the circuit of Fig. 1b. The (red) thick curve is ${\phi }_s=F_{\beta }\left({\phi }_j\right)$, the stars mark the positions of the minima (on the right-hand side) and the saddle point (on the left-hand side); in the inset (at right) are shown the jumps after the switching event over the next determinations of $F_{\beta }$; (c) the evolution of the phase under rf-drive in the “on the shell” circuit case on the 2D phase plane at $T=0\mathrm{mK}$ [the other curves and symbols are similar to the ones in plot (b)]. An extremal path between minimum and saddle point is also shown [thick curve (violet)].Reuse & Permissions

Figure 3

(Color online) The normalized exact low resonance frequency from Eq. (A7) (full curve) and approximate Eq. (8) (dotted curve) for $\beta =7$ and three different values of $\chi$, respectively, from the top 1.0, 0.25, and 0.05.Reuse & Permissions

Figure 4

(Color online) Analytical and numerically calculated scaled quality factors $q$ for a Josephson junction with shell circuit ($\beta =7$, $\chi =0.05$, $I_0=1.3\mu \mathrm{A}$): (a) model with dissipation “on the junction” ${\alpha }_j=0.025$; the full thick (red) curve is the analytical expression $q_{1,j}$, i.e., Eq. (11), the dashed thick (red) curve corresponds to $q_{1,j}∕3$; (b) model with dissipation “on the shell” ${\alpha }_s=0.025$; the full thick (red) curve is the analytical expression $q_{1,s}$, i.e., Eq. (12), the dashed thick (red) curve corresponds to $q_{1,s}∕3$. In both plots the dots represent numerically calculated $q$’s using the interval $\Delta {\gamma }_2∕2$ in Eq. (14) and the error bars are calculated using the minimal interval between two ${\gamma }_b$ values, i.e., the dimension of the bins in the histogram. The squares are relative to $\Omega =0.061$, 0.063, 0.0203, and 0.021 at $T=15\mathrm{mK}$, circles to $\Omega =0.065$, 0.067, 0.022, and 0.0223 at $T=75\mathrm{mK}$ and triangles to $\Omega =0.069$, 0.071, 0.023, and 0.0236 at $T=150\mathrm{mK}$. The thin curve (green) is the scaled $q$ without the shell circuit $(\beta =0)$. The insets show the frequency vs bias current plots from Eq. (8) for both the $n=1$ and $n=3$ subharmonic. The dots represent the position of the observed resonance current (bold arrows on the left marking the beginning of plateau of constant enhancement in Figs. 7, 9).Reuse & Permissions

Figure 5

(Color online) Deterministic critical current in the “on the junction” model at $T=0$ at different rf-amplitudes vs frequency of rf-drive $\Omega$. From top to below: (a) ${\gamma }_{ac}=0.005$; (b) 0.01; (c) 0.05; (d) 0.1; (e) 0.2; (f)0.3. Thin dotted lines present some frequency dependence on the bias current at ${\omega }_{pr\pm }$ and some subharmonics. In the inset is reported the observed current depression $\delta \gamma ({\gamma }_{ac},0)$ in both off-resonant (엯), with a detuning of $\delta \Omega =+0.0012$ above the $n=3$ resonance frequency, and at resonance frequency (◻). The crosses (×) are the fitted value of off-resonant depression $\delta \gamma ({\gamma }_{ac},T)$ evaluated by fitting of escape rate at finite temperature $T=15\mathrm{mK}$ (see the insets of Fig. 9). Thin grey line represents the straight line $\delta \gamma ={\gamma }_{ac}$.Reuse & Permissions

Figure 6

(Color online) Numerical simulated histograms of switching events in presence of rf-drive for a Josephson junction with a shell circuit ($\beta =7$, $\chi =0.05$, $I_0=1.3\mu \mathrm{A}$). The histograms show the resonant activation at the fundamental frequency $(n=1)$: (a) Dissipation on the junction ${\alpha }_j=0.025$ at $T=15\mathrm{mK}$ from right to left the rf-amplitudes are (a1) ${\gamma }_{ac}=0$, (a2) $1.5\times {10}^{-4}$, (a3) $2.5\times {10}^{-4}$, (a4) $3.5\times {10}^{-4}$, (a5) $5\times {10}^{-4}$, (a6) $7\times {10}^{-4}$; (b) dissipation on the shell ${\alpha }_s=0.025$ at $T=15\mathrm{mK}$ from right to left the rf-amplitudes are (b1) ${\gamma }_{ac}=0$, (b2) $2.5\times {10}^{-4}$, (b3) $3.5\times {10}^{-4}$, (b4) $6.5\times {10}^{-4}$, (b5) $8.5\times {10}^{-4}$, (b6) $10.5\times {10}^{-4}$; (c) dissipation on the junction ${\alpha }_j=0.025$ at $T=150\mathrm{mK}$ from right to left the rf-amplitudes are (c1) ${\gamma }_{ac}=0$, (c2) $1.0\times {10}^{-4}$, (c3) $3.5\times {10}^{-4}$, (c4) $4.5\times {10}^{-4}$, (c5) $5.5\times {10}^{-4}$, (c6) $6.5\times {10}^{-4}$; (d) dissipation on the shell ${\alpha }_s=0.025$ at $T=150\mathrm{mK}$ from right to left the rf-amplitudes are, respectively, (d1) ${\gamma }_{ac}=0$, (d2) $4\times {10}^{-4}$, (d3) $6\times {10}^{-4}$, (d4) $8\times {10}^{-4}$, (d5) $9\times {10}^{-4}$, (d6) $1\times {10}^{-3}$.Reuse & Permissions

Figure 7

(Color online) Enhancement of the escape rate, calculated from numerical data, due to resonant activation at the fundamental frequency $(n=1)$ for a Josephson junction with a shell circuit ($\beta =7$, $\chi =0.05$, $I_0=1.3\mu \mathrm{A}$): (a) Dissipation on the junction ${\alpha }_j=0.025$ at $T=15\mathrm{mK}$ for two values of rf-amplitude ${\gamma }_{ac}$; (b) dissipation on the shell ${\alpha }_s=0.025$ at $T=15\mathrm{mK}$ for two values of rf-amplitude ${\gamma }_{ac}$; in the main plots the bold arrow marks the bias position of resonance current as calculated from Eq. (8), the other thin arrows indicate the intervals $\Delta \gamma$ and $\Delta {\gamma }_1$ used for calculation of quality factor. In the insets some numerically calculated escape rates $\Gamma$ for both “on the junction” and “on the shell” models are shown. In the insets the escape rate reported with top curve (with triangles) is given by a (relatively) high value of rf-amplitude $({\gamma }_{ac}=7.5\times {10}^{-4})$. The slope $d\ln \Gamma ∕d{\gamma }_b$ is very similar to thick curves (with squares) which shows the zero-rf escape rate and an escape rate for small rf-amplitude.Reuse & Permissions

Figure 8

(Color online) Numerical simulated histograms of switching events in the presence of a rf-drive for a Josephson junction with shell circuit ($\beta =7$, $\chi =0.05$, $I_0=1.3\mu \mathrm{A}$). The histograms show the resonant activation at the subharmonic frequency $(n=3)$: (a) dissipation on the junction ${\alpha }_j=0.025$ at $T=15\mathrm{mK}$ from right to left the rf-amplitudes are, respectively, (a1) ${\gamma }_{ac}=0$, (a2) 0.0075, (a3) 0.01, (a4) 0.012, (a5) 0.0135, (a6) 0.015, and (a7) 0.0175; (b) dissipation on the shell ${\alpha }_s=0.025$ at $T=15\mathrm{mK}$ from right to left the rf-amplitudes are, respectively, (b1) ${\gamma }_{ac}=0$, (b2) 0.01, (b3) 0.01125, (b4) 0.012, (b5) 0.015, (b6) 0.0175, and (b7) 0.02; (c) dissipation on junction ${\alpha }_j=0.025$ at $T=150\mathrm{mK}$ from right to left the rf-amplitudes are (c1) ${\gamma }_{ac}=0$, (c2) 0.01, (c3) 0.015, (c4) 0.02, (c5) 0.025, (c6) 0.03, and (c7) 0.035; (d) dissipation on the shell ${\alpha }_s=0.025$ at $T=150\mathrm{mK}$ from right to left the rf-amplitudes are (d1) ${\gamma }_{ac}=0$, (d2) 0.015, (d3) 0.02, (d4) 0.025, (d5) 0.028, (d6) 0.032, and (d7) 0.036.Reuse & Permissions

Figure 9

(Color online) Enhancement of the escape rate, calculated from numerical data, due to resonant activation at the subharmonic frequency $(n=3)$ for a Josephson junction with shell circuit ($\beta =7$, $\chi =0.05$, $I_0=1.3\mu \mathrm{A}$): (a) dissipation on the junction ${\alpha }_j=0.025$ at $T=15\mathrm{mK}$; (b) dissipation on the shell ${\alpha }_j=0.025$ at $T=15\mathrm{mK}$. In the main plots the bold arrows (on the right-handside) mark the bias position of the resonance current as calculated from Eq. (8), the bold arrow (on the left-handside) indicate the position of resonance current defined as the current value marking the beginning of plateau of constant enhancement; the thin arrows marks the intervals $\Delta \gamma$ and $\Delta {\gamma }_1$ which are used for calculating the quality factor (see Fig. 7). In the insets some numerically calculated escape rate $\Gamma$ for both “on the junction” and “on the shell” models are also shown. In the insets the escape rate reported with top curve at left (with pluses) is given by a (relatively) high value of rf-amplitude $({\gamma }_{ac}=0.03)$. Also in this case the slope $d\ln \Gamma ∕d{\gamma }_b$ is very similar to the thick curve showing the zero-rf escape rate and the thick curve with squares that show the escape rate for small rf-amplitude. The thin lines represent a fit of the escape rate in off-resonance conditions. The depression of critical current $\delta ({\gamma }_{ac},T)$ is, respectively, equal to 0.009 and 0.014 for the “on the junction” dissipation model and 0.0085 and 0.016 for the “on the shell” dissipation model. The diamonds (◆) in the inset of (a) are the averaged values of escape rate over 5000 switching events calculated using the expression of barrier given by $\Delta U\left({\phi }_j^{\max }\right)$.Reuse & Permissions