Demonstration of dual Shapiro steps in small Josephson junctions

Nat. Commun. 15, 8726 (2024) · DOI: 10.1038/s41467-024-53011-z

License: CC BY 4.0.

Abstract

Bloch oscillations in small Josephson junctions were predicted theoretically as the quantum dual to Josephson oscillations. A significant consequence of this prediction is the emergence of quantized current steps, so-called dual Shapiro steps, when synchronizing Bloch oscillations to an external microwave signal. These steps potentially enable a fundamental standard of current I, defined via the frequency f of the external signal and the elementary charge e, I = ± n × 2ef, where n is a natural number. Here, we realize this fundamental relation by synchronizing the Bloch oscillations in small Al/AlOx/Al Josephson junctions to sinusoidal drives with frequencies from 1 to 6 GHz and observe dual Shapiro steps up to I ≈3 nA. Inspired by today’s voltage standards and to further confirm the duality relation, we investigate a pulsed drive regime and observe an asymmetric pattern of dual Shapiro steps. This work confirms quantum duality effects in Josephson junctions and paves the way towards a range of applications in quantum metrology based on well-established fabrication techniques and straightforward circuit design.

Figures

14 panels with data across 4 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

• panel (a): Energy $E_s(q)$ (arb. units) of the two lowest Bloch bands, $s = 0$ (red) and $s = 1$ (blue), against quasicharge $q/(2e)$ for $E_C/E_J \approx 2.3$ (solid), 1.3 (dashed) and 0.5 (dash-dotted). The print has no energy scale; the vertical axis is the height in the printed frame. The $E_J$ marker is not drawn. data.csv
• panel (b): Ground-state Bloch oscillations of voltage, $\partial E_0/\partial q$, against $q/(2e)$ for $E_C/E_J \approx 2.3$ (solid), 1.3 (dashed) and 0.5 (dash-dotted); vertical axis labelled $E_s$ (arb. units) as printed, with no scale, so it is the height in the printed frame. Curves are clipped at the frame as printed. data.csv

Fig. 2

• panel (a): IV curves without rf drive: voltage $U$ ($\mu$V) against bias current $I$ (nA) for external flux $\Phi/\Phi_0 = 0$, 0.3, 0.41 and 0.45. data.csv
• panel (b): Zoom on the IV curves of (a) for $\Phi/\Phi_0 = 0$ and 0.30: voltage $U$ ($\mu$V) against bias current $I$ (nA). The print draws the voltage axis on the right; it is on the left here. data.csv

Fig. 3

• panel (a): IV curves under sinusoidal rf drive at $p_{drive} = -49$ dBm: voltage $U$ ($\mu$V) against current $I$ (nA) for $f_{drive} = 2.8$ GHz (brown), 4.6 GHz (violet) and 7.8 GHz (blue). Dotted lines mark the first dual Shapiro steps $\pm 2 e f_{drive}$ for 2.8 and 4.6 GHz. data.csv
• panel (b): Differential resistance $R_{diff}$ (k$\Omega$) against current $I$ (nA) and drive frequency $f_{drive}$ (GHz) at $p_{drive} = -49$ dBm. White dashed lines mark the expected dual Shapiro steps $I = \pm n\, 2 e f_{drive}$ for $n = 1$, 2. data.csv
• panel (c): Differential resistance $R_{diff}$ (k$\Omega$) against current $I$ (nA) and drive power $P_{drive}$ (dBm) at $f_{drive} = 3$ GHz. White dashed lines mark the dual Shapiro steps at $\pm n \cdot 0.96$ nA, $n = 1$ to 3. data.csv

Fig. 4

• panel (a): Drive waveforms $U_d(t)$ (V) against time $t$ (ns): symmetric triangular (brown, rect) and the two sawtooth waveforms, + pulse (beige, dashed) and - pulse (black, dash-dotted), named as in (c). data.csv
• panel (b): rf current $I_{rf} \propto dU_d/dt$ (arb. units) against time $t$ (ns) for the three waveforms of (a): rectangular (brown, rect), short positive pulses (beige, dashed, + pulse) and short negative pulses (black, dash-dotted, - pulse). data.csv
• panel (c): Differential resistance $R_{diff}$ (k$\Omega$) against current $I$ (nA) at $f_{drive} = 2$ GHz for the rect, + pulse and - pulse current profiles of (b) and a sinusoidal drive (red) at $p_{drive} = -48$ dBm. data.csv
• panel (d): $R_{diff}$ (k$\Omega$) against current $I$ (nA) while the drive is swept back and forth between positive and negative sawtooth. Vertical axis: waveform sweep step (no ticks in print). Lines mark the curves of (c): rect (brown), + pulse (beige, dashed), - pulse (black, dash-dotted). The polarity colour bar is not drawn. data.csv
• panel (e): $R_{diff}$ (k$\Omega$) against current $I$ (nA) and drive power $P_{drive}$ (dBm) for a pulsed drive at 2 GHz. data.csv
• panel (f): $R_{diff}$ (k$\Omega$) against current $I$ (nA) and drive frequency $f_{drive}$ (GHz) for the pulsed drive. Black dashed line: $I_{dSs} = 2 e f_{drive}$. The inset is hosted as f-inset. data.csv
• panel (f-inset): Inset of (f): current of the first dual Shapiro step $I_{1.dSs}$ (nA) against $f_{drive}$ (GHz) for the pulsed (beige) and sinusoidal (red, offset by 0.1 nA) drives, with error bars and linear fits $I = 2 e f_{drive}$. data.csv

Cite

Fabian Kaap, Christoph Kissling, Victor Gaydamachenko, Lukas Grünhaupt, Sergey Lotkhov. Demonstration of dual Shapiro steps in small Josephson junctions. Nat. Commun. 15, 8726 (2024). https://doi.org/10.1038/s41467-024-53011-z

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