Abstract Synchronisation of the Bloch oscillations in a small Josephson junction (JJ) under the microwave radiation leading to the current quantisation at a current equal to Cooper-pair charge × frequency × integer has been proposed as the effect dual to the Shapiro steps. Experimental confirmation of this phenomenon was delayed for a long time until last year's breakthrough when the current quantisation was demonstrated in the superconducting nanowires. The compact high impedance environment of the nanowires played a key role in the experiment. Direct observation of the current quantization in the JJs would answer the fundamental question of the Bloch oscillations and open a more feasible road for the metrological application. Here we place JJs in the high impedance environment and demonstrate dual Shapiro steps for frequencies up to 24 GHz (I=7.7 nA). The current quantisation exists, however, only in a narrow range of JJ parameters, the critical current and capacitance. We carry out a systematic study to explain this by invoking the model of the JJ in presence of the thermal noise. The findings are important for the fundamental physics and application to quantum metrology.
Abstract Synchronisation of the Bloch oscillations in a small Josephson junction (JJ) under the microwave radiation leading to the current quantisation at a current equal to Cooper-pair charge × frequency × integer has been proposed as the effect dual to the Shapiro steps. Experimental confirmation of this phenomenon was delayed for a long time until last year's breakthrough when the current quantisation was demonstrated in the superconducting nanowires. The compact high impedance environment of the nanowires played a key role in the experiment. Direct observation of the current quantization in the JJs would answer the fundamental question of the Bloch oscillations and open a more feasible road for the metrological application. Here we place JJs in the high impedance environment and demonstrate dual Shapiro steps for frequencies up to 24 GHz (I=7.7 nA). The current quantisation exists, however, only in a narrow range of JJ parameters, the critical current and capacitance. We carry out a systematic study to explain this by invoking the model of the JJ in presence of the thermal noise. The findings are important for the fundamental physics and application to quantum metrology.
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.
The quantum Josephson voltage standard is well established across the metrology community for many years. It relies on the synchronization of the flux tunneling in the S/I/S Josephson junctions (JJ) with the microwave radiation (MW) of frequency f such that V=Φ0fm, where m = 0, 1, 2, …. The phenomenon is called the Shapiro steps. Together with the quantum Hall resistance standard, the voltage standard forms the foundation of electrostatic metrology. The current is then defined as the ratio of the voltage and resistance. Realization of the quantum current standard would close the electrostatic metrological triangle of voltage–resistance–current. The current quantization I=2efm, the inverse Shapiro steps, was recently shown using the superconducting nanowires and small JJ. The effect is a synchronization of the MW with the Cooper pair tunneling. This paves the way to combine the JJ voltage and current standards on the same chip and demonstrate feasibility of the multi-standard operation. We show the voltage and current quantization on the same chip up to frequency of 10 GHz, corresponding to the amplitudes ∼ 20.67 μV and ∼ 3.2 nA, respectively. The accuracy of the voltage and current quantization, however, is relatively low, 35 and 100 ppk, respectively. We discuss measures to optimize the JJs, circuit, and environment to boost the amplitude and accuracy of the standards.
Electrical quantum metrology using superconducting circuits relies on phase synchronization effects of Josephson oscillations with ac signals, resulting in equidistant voltage steps and forming the basis of today’s voltage standards. Because charge and phase are conjugate variables, the dual effect of equidistant current steps should also exist and was indeed predicted theoretically. It relies on so-called Bloch oscillations, which are oscillations of voltage occurring when a current-biased Josephson junction is embedded in an environment with characteristic impedance larger than the superconducting resistance quantum h/4e^2. In addition, Josephson junctions of lateral size on the order of tens of nm are necessary, such that their Josephson and charging energy are approximately equal. Synchronizing such Bloch-oscillations with an ac signal completes the duality with Shapiro steps, and results in the occurrence of steps in IV-measurements at I=2ef with e being the elementary charge, and f the frequency of the external ac signal. To emphasize the duality with Shapiro (voltage) steps, such current steps are called dual-Shapiro steps. Together with the established standards for voltage and resistance, a new standard for current based on dual-Shapiro steps would relate the ampere, volt and ohm through fundamental constants and frequency. If successfully realized with metrological precision this would close the quantum metrological triangle and might even enable the combination of all three electric quantum relations in one single device. Here we show our efforts towards the realization of quantized currents with metrological applicability. Based on advances in fabrication technology and measurement capabilities, we demonstrate quantized current steps up to I≈2 nA by synchronizing Bloch oscillations with external ac signals up to 6 GHz [1,2]. In addition, we demonstrate a pulsed driving scheme resulting in an asymmetric pattern of dual-Shapiro steps featuring increased differential resistance [2]. Finally, we discuss the effects limiting the precision of our quantized current steps and present our efforts for tackling them. [1] F. Kaap et al., “Synchronization of Bloch Oscillations in a Strongly Coupled Pair of Small Josephson Junctions: Evidence for a Shapiro-like Current Step,“ Phys. Rev. Lett. 132, 027001, 2024. [2] F. Kaap et al., “Demonstration of dual Shapiro steps in small Josephson junctions,“ Nature Communications 15, 8726, 2024.
A resistively shunted Josephson junction with a DC-bias exhibits Josephson oscillations that link voltage and frequency through fundamental constants. These oscillations can synchronize their frequency to an external microwave signal achieving the metrological precision that underlies the modern Josephson voltage standard. Its electromagnetic dual, the phase-slip junction, exhibits Bloch oscillations that connect current to frequency in an analogous way. A current standard based on synchronized Bloch oscillations could close the quantum metrology triangle, providing a consistent test of the underlying fundamental relations between voltage, current, and resistance. The experimental realization, however, remains challenging, as external microwave lines introduce parasitic capacitances that can destroy the high-impedance environment required for Bloch oscillations. This work explores the use of on-chip synchronization to stabilize both Josephson and Bloch oscillations without the need for an external microwave drive. We review the theoretical framework of the synchronization between two coupled oscillators and the dynamics of superconducting circuits with a focus on quantum metrology. Building on this, we analyze a recent experiment on the mutual synchronization of Bloch oscillations in a pair of ultrasmall Josephson junctions. Our model in terms of correlated phase-slip junctions realized in the ground state of the device shows good agreement with the experimental results, confirming synchronized Bloch oscillations. At larger currents, nonadiabatic transitions wash out the coherence of the Bloch oscillations and therefore the synchronization features. Motivated by these findings, we propose an alternative circuit setup where the Bloch oscillations in a single phase-slip junction are stabilized by synchronization to on-chip Josephson oscillations. We study the back-action of the Bloch oscillations on the Josephson oscillations and identify a parameter regime in which it is minimized, rendering the latter an effectively rigid driving signal. Moreover, we find that the small remaining back-action further enhances the thermal stability of the synchronized Bloch oscillations compared to the externally driven case. Without a back-action mechanism, the coherence of Josephson oscillations is severely limited by thermal fluctuations. We relate this limitation to the thermodynamic uncertainty relation (TUR) in the field of stochastic thermodynamics. Recent findings show, that the TUR can be broken by a classical pendulum clock. We propose a circuit that mimics the dynamics of a pendulum clock as a flexible on-chip microwave source for high impedance environments. Due to an internal synchronization, the clock circuit emits coherent radiation beyond the limits of the TUR. Finally, we explore the use of symmetry breaking drives as a resource for nonreciprocal transport in superconducting circuits. We demonstrate that a biharmonic microwave signal applied to a conventional Josephson junction induces an ideal supercurrent diode effect with a tunable direction.
We propose a superconducting circuit based on the Bloch transistor, a quantum device consisting of two small-capacitance Josephson junctions, connected in series and having a small superconducting island in between. This device is driven by two dc electrical sources controlling Josephson oscillations of frequency fJ=2eVJ/h, related to the average transistor voltage VJ, and Bloch oscillations of frequency fB=IB/2e, related to the average current IB injected into the transistor island. We show that due to the Bloch transistor properties, these two types of oscillations are coupled and can mutually phase-lock, i.e., fJ=fB. This leads to the formation of a current step on the current-voltage curve at IB=2efJ, which is similar to the dual Shapiro step, appearing under microwave irradiation of frequency f on the current-voltage curve of a small Josephson Junction at current I=2ef. Moreover, the Bloch transistor transconductance IB/VJ takes the fundamental value of 1/RQ, where RQ=h/4e2 is the resistance quantum. The obtained results pave the way to an alternative quantum standard of resistance, based on the superconducting circuit and operating without applying a strong magnetic field.
We demonstrate the synchronization of current quantization in a parallel array of weakly coupled Josephson Junctions operating in the regime of the coherent quantum phase slip. The first quantized current step on the voltage-current characteristic of 69 parallel Josephson junctions, under microwave excitation at a frequency of f = 19.325 GHz, is observed at 426 nA. Experiments show that accuracy of quantization does not degrade with increasing number of combined Josephson junctions. This demonstration addresses the issue of quantized current amplitude requirements adopted for a practical quantum standard, while leaving the question of accuracy for further research. At present, the quantum current standard of accepted metrological accuracy in the triangle of electrical units is under development. The other two, the Volt and Ohm standards based on the Josephson and Quantum Hall effects, respectively, are already well established in metrology.