Practical Quantum Realization of the Ampere from the Elementary Charge

One major change of the future revision of the International System of Units is a new definition of the ampere based on the elementary charge $e$. Replacing the former definition based on Ampère’s force law will allow one to fully benefit from quantum physics to realize the ampere. However, a quantum realization of the ampere from $e$, accurate to within ${10}^{-8}$ in relative value and fulfilling traceability needs, is still missing despite the many efforts made for the development of single-electron tunneling devices. Starting again with Ohm’s law, applied here in a quantum circuit combining the quantum Hall resistance and Josephson voltage standards with a superconducting cryogenic amplifier, we report on a practical and universal programmable quantum current generator. We demonstrate that currents generated in the milliampere range are accurately quantized in terms of $e{f}_{\mathrm{J}}$ ($f_{\mathrm{J}}$ is the Josephson frequency) with measurement uncertainty of ${10}^{-8}$. This new quantum current source, which is able to deliver such accurate currents down to the microampere range, can greatly improve the current measurement traceability, as demonstrated with the calibrations of digital ammeters. In addition, it opens the way to further developments in metrology and in fundamental physics, such as a quantum multimeter or new accurate comparisons to single-electron pumps.

Popular Summary

Although electrical current can be described by the flow of elementary charges per second, the ampere is still defined in the International System of Units (SI) from an electromechanical force occurring between current-carrying wires, explained by Ampère’s force law. This definition, set in 1948, limits the accuracy of electrical measurements in SI units. A major overhaul is planned in 2018 to put SI units in line with modern physics by fixing the values of some fundamental constants, among them the elementary charge $e$, which will define the ampere. However, a highly accurate quantum realization of the ampere is still missing despite many efforts to develop quantum devices handling electrons one by one. Here, we report on a programmable quantum current generator delivering quantized currents that range from $1\mu \mathrm{A}$ to 5 mA, directly linked to $e$.

This breakthrough relies on the programmable amplification of the quantized current obtained from an accurate application of Ohm’s law to the programmable quantum Josephson voltage standard and the quantum Hall resistance standard combined in an original quantum circuit. Our novel quantum current generator accordingly benefits from the high level of universality and reproducibility of these two quantum standards, which are based on macroscopic quantum phenomena only linked to the Planck constant and the elementary charge $e$. We show that the generated currents are quantized in terms of $e{f}_{\mathrm{J}}$—the Josephson frequency—with uncertainties of only 10 parts in one billion.

Our work improves the accuracy of current standards by 2 orders of magnitude and paves the way for fully quantum-based electrical measurements that will benefit the new SI system.

Figure 1

Practical realization of the ampere from quantum electrical effects. (a) Illustration of the two ways to generate a current from the frequency $f({\mathrm{s}}^{-1})$ and the elementary charge $e$, expressed in amperes in the future SI. The current can be generated either using single-electron tunneling devices or the combination of the Josephson effect and the quantum Hall effect. Quantum effects involve fundamental constants whose uncertainties in the present SI will be reduced to zero in the future SI. (b) Comparison of the relative uncertainty of the current generated by the PQCG in the milliampere range (red diamonds) to state-of-the-art current measurements or generation. These high-accuracy results include the best calibration and measurement capabilities (CMCs) [green squares: from ${10}^{-13}$ up to ${10}^{-11}\mathrm{A}$, by charging a capacitor (Physikalisch-Technische Bundesanstalt), and from ${10}^{-10}$ up to ${10}^{-1}\mathrm{A}$, by applying Ohm’s law (Laboratoire national de métrologie et d’essais)] [21] and the main SET-device results mentioned in the text (blue dots). The red dashed-dotted line shows the estimated relative uncertainties of the current generated by the PQCG from $1\mu \mathrm{A}$ up to 10 mA. (c) Principle of the PQCG. (d) Relative deviation $\mathrm{\Delta }{I}_{\mathrm{PQCG}}/I_{\mathrm{PQCG}}$ of the measured current from the quantized value in the milliampere range. Error bars are combined standard uncertainties (1 s.d.). No significant relative discrepancies can be observed within an uncertainty of ${10}^{-8}$.

Figure 2

Experimental realization of the PQCG. (a) The PQCS is composed of a PJVS biasing a QHRS through double connections, each one incorporating a cryogenic current comparator (CCC) winding ($N_{\mathrm{JK}}$ turns) on the low potential side. A third terminal (dotted line) connected at the top of the QHE setup further reduces the cable contribution to the current $I_{\mathrm{PQCS}}$. The current $I_{\mathrm{PQCG}}$ of the PQCG, generated by an external current source into a winding of $N$ turns, is synchronized and coarsely adjusted by the PJVS programmable bias source using $V_{\mathrm{out}}$. Note that $I_{\mathrm{PQCG}}$ is locked to the current $I_{\mathrm{PQCS}}$ of the PQCS by means of the CCC, which is used as an accurate adder amplifier of $N_{\mathrm{JK}}/N$ gain. A current divider injecting a fraction $\beta$ of $I_{\mathrm{PQCG}}$ in a CCC winding of $N_{\mathrm{Div}}$ ($=16$ turns) allows a fine-tuning of the CCC amplification gain. The damping circuit formed by a 100-nF highly insulated Polytetrafluoroethylene (PTFE) capacitance in series with a $1.1\text{−}\mathrm{k}\mathrm{\Omega }$ resistor is connected to a CCC winding of $N_{\text{D}}$ ($=1600$) turns. (b) PJVS#A output voltage as a function of the bias current $I_{\text{Bias}}$ for the total number of Josephson junctions at $f_{\mathrm{J}}=70\mathrm{GHz}$ and $T=4.2\mathrm{K}$. (c) Hall resistance $R_{\mathrm{H}}$ and longitudinal resistance $R_{xx}$ measured, using a current of $10\mu \mathrm{A}$ at $T=1.3\mathrm{K}$, as a function of $B$ in the $\mathrm{GaAs}/\mathrm{AlGaAs}$-based Hall bar device (LEP514). The device is used at $B=10.8\mathrm{T}$. (d) The noise amplitude spectral density (expressed in $\mu {\varphi }_0/{\mathrm{Hz}}^{1/2}$), measured at the output of the SQUID in internal feedback mode, for the CCC alone (black) and the CCC connected to the PQCS and the damping circuit (magenta) without the external current source. (e) Series of on-off switchings of the current $I_{\mathrm{PQCG}}$ recorded on a digital ammeter (at 1.1 mA), obtained for $n_{\mathrm{J}}=3073$, $N_{\mathrm{JK}}=129$, and $N=4$.

Figure 3

Quantization tests of the PQCG. (a) Scheme of the setup for the accuracy measurements. The PQCG, symbolized by the current source linked to the product of constants $K_{\mathrm{J}}{R}_{\mathrm{K}}$, supplies a calibrated $100\text{−}\mathrm{\Omega }$ resistor $R_{100}$, and its voltage is compared to the voltage $V_{\mathrm{J}}^{\mathrm{ref}}$ of PJVS#ref using a battery-powered nanovoltmeter EM N31 (response time constant about 2 s). (b) Raw data of the voltage null detector for two on-off-on cycles for two settings, ${\beta }_1$ (black dots) and ${\beta }_2$ (red dots), of the current divider. Note that ${\tau }_0=72\mathrm{s}$ is the duration of one on-off-on cycle, and ${\tau }_w=12\mathrm{s}$ is the waiting time before recording after the current switching. (c) Relative Allan deviation calculated from a series of 49 on-off-on cycles of $I_{\mathrm{PQCG}}$ plotted as a function of time $\tau (s)$. A ${\tau }^{-1/2}$ fit shows good agreement below $\tau =1000\mathrm{s}$ and corresponds to a relative Allan deviation of $2.5\times {10}^{-8}$ for the duration of the time series used in this work, ${\tau }_{\text{Series}}=792\mathrm{s}$. (d) $\mathrm{\Delta }{I}_{\mathrm{PQCG}}/I_{\mathrm{PQCG}}$ as a function of $n_{\mathrm{J}}$ (or $I_{\mathrm{PQCS}}$) for several currents (both positive and negative) generated by the PQCG in the mA range, using three different values for $N$. Error bars are combined standard uncertainties (1 s.d.). (e) Successive measurements of $\mathrm{\Delta }{I}_{\mathrm{PQCG}}/I_{\mathrm{PQCG}}$ (over four hours) obtained (with $n_{\mathrm{J}}=3074$, $N=2$, $N_{\mathrm{JK}}=129$) for different bias currents, demonstrating the reproducibility of the current generated at 2.2 mA. Error bars are combined standard uncertainties (1 s.d.).

Figure 4

Calibration of a digital ammeter using the PQCG and identification of quantization criteria. (a) Relative deviation of $I_{\text{Meas}}$ [the current generated by the PQCG and measured by the DA (HP3458A)] to $I_{\mathrm{Mid}}$ (the current measured at the center of the step) as a function of the biasing current $I_{\mathrm{Bias}}$ for $n_{\mathrm{J}}=3072$, 1024, and 512 junctions of PJVS#B. The independence of the output current as a function of $I_{\mathrm{Bias}}$ is a first quantization criterion. (b) Relative deviation of $I_{\text{Meas}}$ to $I_{\mathrm{PQCG}}$ as a function of $I_{\mathrm{PQCG}}$ in the mA range and (c) in the $\mu \mathrm{A}$ range. The agreement of the measurements performed using $n_{\mathrm{J}}$ and $n_{\mathrm{J}}/2$ is a second quantization criterion. (d) Relative uncertainty of the measured current (blue squares) and of the DA (open red dots) as a function of $I_{\mathrm{PQCG}}$ over four decades of current. The DA noise dominates over that of the PQCG. Error bars are standard uncertainties (1 s.d.).