Evidence of macroscopic quantum tunneling from both wells in a $\phi$ Josephson junction

We study $\mathrm{Nb}-{\mathrm{AlO}}_x$-Nb Josephson junctions (JJs) with a phase-discontinuity $\kappa$ created by a pair of current injectors attached to one of the Nb electrodes. For $\kappa \approx \pi$ the Josephson potential energy $U$ as a function of the average phase $\psi$ across the JJ has the form of a $2\pi$-periodic double-well potential. Thus, the device behaves as a $\phi$ JJ with degenerate ground state phases $\psi =\pm \phi$ (the value of $\phi$ depends on the system parameters). Experimentally, the existence of two wells of the potential is confirmed by the observation of two different critical currents $I_{\mathrm{c}\pm }$, corresponding to the escape from different wells. We investigate the escape of the Josephson phase from both wells by collecting statistics of the switching currents. The histogram of switching current exhibits two peaks corresponding to $I_{\mathrm{c}\pm }$. The dependence of the width ${\sigma }_{+}$ and ${\sigma }_{-}$ of each peak on the bath temperature $T$ indicates the transition from thermal activation to macroscopic quantum tunneling (MQT) at $T^{★}\approx 260\mathrm{mK}$ as $T$ decreases. We argue that the observed saturation value of ${\sigma }_{+}$ and ${\sigma }_{-}$ below $T^{★}$ is indeed related to quantum tunneling rather than to parasitic noise in the system, as the histogram width can be reduced by tuning the value of $\kappa$ away from $\pi$. The comparison of the experimental escape rate $\mathrm{\Gamma }$ with theoretical predictions further confirms MQT of the phase $\psi$ from both wells.

Figure 1

Sketch of the investigated Josephson junction with two current injector leads attached on the top electrode. The blue line represents the spatial variation of the phase along the junction length.

Figure 2

(a) The CPR of the simulated JJ for $\kappa =\pi$ calculated with the dynamical method (blue) and with the static method (red). The dotted line represents the unstable part of the CPR. (b) Josephson potential $U\left(\psi \right)$ (normalized to the Josephson energy $E_{\mathrm{J}}=I_{\mathrm{c}}{\mathrm{\Phi }}_0/2\pi$) for $\kappa =0$ and $\kappa =\pi$. At $\kappa =\pi$ the wells have different depths, but this can be tuned by either $\kappa /2\pi$ (c) or $f$ (d). (e) $U\left(\psi \right)/E_{\mathrm{J}}$ for different injector positions $x_0$ indicated by the numbers. For each curve, the phase discontinuity $\kappa$ is adjusted to ${\kappa }_{\mathrm{sym}}, f=0$.

Figure 3

Solutions $\mu \left(x\right)$ red lines on top of the grayscale plot $u(\mu ,x)$ of the corresponding normalized Josephson energy density profile given by Eq. (5). The grayscale shows the range of $u$. (a) $x_0=0.02$ (vertical dashed line), $\kappa =\pi$; (b) $x_0=0.02, \kappa ={\kappa }_{\mathrm{sym}}=1.028\pi$; (c) $x_0=0.10, \kappa =0.68\pi$ (${\phi }_0\approx 0.3\pi$ state). In all the plots, the normalized applied magnetic flux $f=0$.

Figure 4

Optical image of a JJ, for which the data are reported. It is equipped with three injector pairs, but only one of them, $I_{\mathrm{inj}1}$, was used for the experiment reported here.

Figure 5

(a) Experimental (blue line) and simulated (orange line) calibration of the injectors of a $40\mu \mathrm{m}$ long JJ with $1.5\mu \mathrm{m}$ wide injectors located at $X_0=-0.125\mu \mathrm{m}$ at $T=4.2\mathrm{K}$ and $H=0$. In (b) a zoom around the first minimum is displayed. The dotted line indicates the value of $\kappa$ (or $I_{\mathrm{inj}}$) for which the potential has symmetric $\pm \phi$ wells. (c) The $I-V$ curve at $I_{\mathrm{inj}}^{\mathrm{sym}}=3.67\mathrm{mA}$ showing two distinct critical currents is plotted.

Figure 6

(a) Several switching current histograms taken at $I_{\mathrm{inj}}^{\mathrm{sym}}=3.71\mathrm{mA}$ and at different temperatures. Each histogram has two peaks corresponding to the escape from the $-\phi$ and the $+\phi$ well. Each peak is somewhat below $I_{c0+}$ and $I_{c0-}$, the fluctuation-free critical currents of the $\phi$ JJ. (b) The widths ${\sigma }_{\pm }$ of the $I_{\mathrm{c}-}$ and $I_{\mathrm{c}+}$ peaks in (a) as a function of temperature $T$ is shown.

Figure 7

The dependences ${\sigma }_{\pm }\left(\kappa \right)$ at $T=20\mathrm{mK}$. The dotted line indicates the symmetry point occurring at $I_{\mathrm{inj}}^{\mathrm{sym}}=3.71\mathrm{mA}$.

Figure 8

Double logarithm of the normalized inverse escape rate $\mathrm{lnln}({\omega }_p/2\pi \mathrm{\Gamma })$ for different temperatures in the range $30\mathrm{mK}<T<600\mathrm{mK}$ for the $I_{c+}$ peak of Fig. 6 [panels (a) and (b)] and for the $I_{c-}$ peak of Fig. 6 [panels (c) and (d)]. In the graphs, the points represent the experimental data and the lines the fits obtained by using Eq. (7) (TA-fit) and Eq. (8) (MQT-fit). One can clearly observe that for both peaks (both energy wells $\pm \phi$) the TA formula fits very well the experimental points above $T^{★}$, but it does not below $T^{★}$. The opposite occurs for the MQT formula, proving quantum tunneling below the crossover, where the standard deviations ${\sigma }_{\pm }$ saturate. In panels (b) and (d), the curves were manually shifted by $-0.5$ to display them better. Since there is no dependence on temperature in ${\mathrm{\Gamma }}_q$, the curves sit almost at the same position in the diagram.

Figure 9

(a) Modulation of the critical current $I_{\mathrm{c}}$ with injector current $I_{\mathrm{inj}}$ at different temperatures. The vertical line indicates $I_{\mathrm{inj}}^{\mathrm{sym}}=3.71\mathrm{mA}$. (b) Mean switching currents vs temperature at $I_{\mathrm{inj}}^{\mathrm{sym}}=3.71\mathrm{mA}$.