Engineering the speedup of quantum tunneling in Josephson systems via dissipation

We theoretically investigate the escape rate occurring via quantum tunneling in a system affected by tailored dissipation. Specifically, we study the environmental assisted quantum tunneling of the superconducting phase in a current-biased Josephson junction. We consider Ohmic resistors inducing dissipation both in the phase and in the charge of the quantum circuit. We find that the charge dissipation leads to an enhancement of the quantum escape rate. This effect appears already in the low Ohmic regime and also occurs in the presence of phase dissipation that favors localization. Inserting realistic circuit parameters, we address the question of its experimental observability and discuss suitable parameter spaces for the observation of the enhanced rate.

Figure 1

(a) Scheme of the current-biased Josephson junction. The capacitance $C_J$ is the intrinsic capacitance of the Josephson junction which is connected in parallel to a shunt resistor $R_S$ and and to a branch containing an external capacitance $C$ and a resistance $R_g$. (b) Sketch of the tilted cosine potential (solid line) $V\left[\phi \right]$ for the superconducting phase difference across the junction and its cubic approximation (dashed line).

Figure 2

Bounce path in the presence of pure charge dissipation $(R_S=\infty )$ in (a) and pure phase dissipation $(R_g=0)$ in (b). ${\phi }_V\left(\tau \right)$ shows the variational treatment (solid line), ${\phi }_B\left(\tau \right)$ corresponds to the numerical values (dots), and ${\phi }_S\left(\tau \right)$ denotes the nondissipative bounce (dashed line). Parameters: $I_b/I_C=0.985, I_C=21\mu \text{A}, C_{\mathrm{tot}}=6\text{pF}$, and $C_J/C\to 0$.

Figure 3

(a) Cubic potential for a fixed ratio $V_0/\left(\hslash {\omega }_I\right)=4$ and different values of $I_b/I_C$. For larger ratios $I_b/I_C$ the barrier becomes narrower and hence steeper. (b) Values for $I_b$ and $I_C$ for a fixed barrier height. Each color combination of dot and line in (b) corresponds to the respective barrier in (a). (c) and (d) Results for $\mathcal{E}$ as a function of dissipation, for fixed $V_0/\left(\hslash {\omega }_I\right)=4$ and different ratios $I_b/I_C$. The respective values for $I_C$ can be read in (b). The symbols display the results using the unperturbed path ${\phi }_B^{\left(S\right)}$ in the dissipative action (undamped-bounce approximation; see text). Panel (c) is for pure phase dissipation: suppression of the tunneling as a function of $R_S^{-1}$, and (d) for pure charge dissipation: enhancement of the tunneling as function of $R_g$. Capacitance $C_{\mathrm{tot}}=6\text{pF}$ and $C_J/C\to 0$.

Figure 4

Landscape plot of $\mathcal{E}$ as a function of both resistances $R_S$ and $R_g$ for $V_0/\left(\hslash {\omega }_I\right)=4$. The red area corresponds to the region where the exponential factor is enhanced due to dissipation; the blue is instead where the tunneling rate is suppressed. The other parameters are $I_b/I_C=0.99, I_C=32\mu \mathrm{A}$, and $C_{\mathrm{tot}}=6$ pF.

Figure 5

(a) Change of the barrier for a fixed $I_C$ and variable $I_b$. (b) Barrier height for different values of $I_b/I_C$ for fixed $I_C=21\mu \text{A}$. The colors correspond to the barriers displayed in (a). (c) and (d) Results for $\mathcal{E}$ for a fixed $I_C=21\mu \text{A}$ and different ratios $I_b/I_C$. The respective values for the ratio $V_0/\left(\hslash {\omega }_I\right)$ can be found in (b). The symbols display the results using the unperturbed path ${\phi }_B^{\left(S\right)}$ in the dissipative action (the undamped-bounce approximation; see text). Panel (c) is for pure phase dissipation: suppression of the tunneling as a function of $R_S^{-1}$, and (d) for pure charge dissipation: enhancement of the tunneling as function of $R_g$. Capacitance $C_{\mathrm{tot}}=6\text{pF}$ and $C_J/C\to 0$.

Figure 6

Enhancement of $\mathcal{E}$ in the presence of both resistors, for ${\tau }_p{\omega }_I^2/\gamma =\text{const}$. The two panels display two different regimes of enhancement (or suppression) of the tunneling. From top to bottom: ${\tau }_p{\omega }_I^2/\gamma =3.43,2.98,2.43$ in (a) and ${\tau }_p{\omega }_I^2/\gamma =6.47,5.61,4.59$ in (b), for $C_J=0$ (solid lines) and $C_J/C=0.02$ (dashed lines). The other parameters are $I_C=21\mu \mathrm{A}$ and $C_{\mathrm{tot}}=6$ pF. In (b) for every $1/R_S$, the value $R_g$ is larger than in (a). Therefore, the enhancement is larger in (b) than in (a). This is also reflected by larger values of ${\tau }_p{\omega }_I^2/\gamma$ for the same $I_b/I_C$.

Figure 7

Enhancement of $\mathcal{E}$ as a function of $R_g$, for different ratios of the two capacitances $C_J/C=0,0.01,0.2,0.5,1$ from top to bottom. The other parameters are $I_b/I_C=0.99,R_S=100\mathrm{\Omega }, I_C=21\mu \mathrm{A}$, and $C_{\mathrm{tot}}=6$ pF.

Figure 8

$\mathrm{\Gamma }$ as obtained by using $K=K_0$ for the prefactor, for $C_J/C=0.02, R_S=100\mathrm{\Omega }$, and different values of $R_g$. The other parameters are $I_C=21\mu \mathrm{A}$ and $C_{\mathrm{tot}}=6$ pF. For $R_g=0$, we also show the uncertainty induced by a $10\%$ experimental uncertainty of the capacitance $C_{\mathrm{tot}}$ (gray area). Increasing $R_g$ leads to a significant enhancement of the escape rate. For larger $I_b/I_C$ the increase due to charge dissipation becomes smaller, as can be seen from the three lines getting closer. For the observation of the effect, the ratio $I_b/I_C$ must be determined with an accuracy of the order of $\delta (I_b/I_C)<0.002$ for $R_g=4\mathrm{\Omega }$.

Figure 9

Transmission line circuit used to calculate the dissipative kernels for the Ohmic resistors in Fig. 1.

Figure 10

Comparison of the three different techniques to calculate the action minimized by the bounce path. We show the results for the exact numerical (dots) and the variational (solid line) results, together with the solution using the nondissipative bounce approximation (dashed line). (a) Pure charge dissipation. (b) Both dissipative couplings and the variational treatment.