Synchronization of Bloch Oscillations in a Strongly Coupled Pair of Small Josephson Junctions: Evidence for a Shapiro-like Current Step

Bloch oscillations are a fundamental phenomenon linking the adiabatic transport of Cooper pairs to time. Here, we investigate synchronization of the Bloch oscillations in a strongly coupled system of sub-100 nm $\mathrm{Al}/{\mathrm{AlO}}_{\mathrm{x}}/\mathrm{Al}$ Josephson junctions in a high-Ohmic environment composed of highly inductive meanders of granulated aluminum and high-Ohmic titanium microstrips. We observe a pronounced current mirror effect in the coupled junctions and demonstrate current plateaus, akin to the first dual Shapiro step in microwave experiments. These findings suggest that our circuit design holds promise for realizing protected Bloch oscillations and precise Shapiro steps of interest for current metrology.

Figure 1

A symmetric circuit with two coupled Josephson junctions allowing for the synchronization of Bloch oscillations. The signature of this is the current mirror effect with $I_1=-I_2$, appearing due to transport via Cooper pair dipoles indicated in red, see main text for details. Note that, in order to achieve coherent Bloch oscillations, the bias resistances must exceed the resistance quantum with $R_{\mathrm{e}}\gg R_{\mathrm{Q}}=h/4e^2\approx 6.45\mathrm{k}\mathrm{\Omega }$.

Figure 2

(a) Optical picture of high-impedance circuitry. Both current arms include titanium microstrips, $20\times 0.16\times 0.15\mathrm{\mu }{\mathrm{m}}^3$ long/wide/thick each, as well as $0.14\mathrm{\mu }\mathrm{m}$ wide meandered lines of grAl. (b) False-colored SEM image of the SQUID and single junction structure. Comparing to Fig. 1, the junction (SQUID) plays the role of the first (second) current arm. (c) Close-up view of a single junction marked by the dashed circle in (b).

Figure 3

(a) dc junction voltage $U_{\mathrm{JJ}}=\overline{U_1(t)}$ measured as a function of current $I_{\mathrm{JJ}}$ and magnetic flux ${\mathrm{\Phi }}_{\mathrm{ext}}$ at $I_{\mathrm{SQ}}=0$. The green (orange) vertical line marks weak (strong) magnetic frustration of the SQUID, for which the cross-sectional $I\text{−}V$ curves are plotted in (b). The dashed lines in (b) depict the dc voltage $U_{\mathrm{SQ}}=\overline{U_2(t)}$ induced across the SQUID. (c) Equivalent circuit as a single junction shunted via the Bloch capacitance realized by the SQUID. (d) Simulation of the $I\text{−}V$ curves. The values of the parameters are those of Figs. 4 and 4.

Figure 4

(a),(b) Voltage $U_{\mathrm{JJ}}$ as a function of the two bias currents for the two frustration cases. (c),(d) The simulated data with Johnson-Nyquist noise at a constant temperature of $k_{\mathrm{B}}T=0.25e{V}_{\mathrm{c}1}$ in the regime of (c) symmetric Josephson energies and Bloch critical voltages $V_{\mathrm{c}}\equiv V_{\mathrm{c}1}=V_{\mathrm{c}2}=V_{+}$ and (d) the single junction regime $V_{\mathrm{c}2}=V_{+}=0$.

Figure 5

(a) Current in the junction plotted vs junction voltage at $f\approx 0.5$ and $I_{\mathrm{SQ}}$ varying from 0 to 0.5 nA (bottom to top). (b) Simulated data for the synchronized Bloch oscillations in the regime of symmetric Josephson energies for an equidistant series of SQUID current values $I_{\mathrm{SQ}}$.

Figure 6

Simulation of the time-resolved synchronization of Cooper pair transfers in the regime of symmetric Josephson energies with asymmetric bias voltages $V_{\mathrm{b}1}=5V_{\mathrm{c}1}$, $V_{\mathrm{b}2}=-4V_{\mathrm{c}1}$. (a) Voltage drop $U_1$ ($U_2$) across the junction (SQUID). (b) A small common node current $I_{\mathrm{out}}$ through the lower biasing resistor indicates a current mirror effect.