Improvements of the programmable quantum current generator for better traceability of electrical current measurements

The principle of the programmable quantum current generator is presented in figure 1. It is based on a primary circuit, the programmable quantum current standard (PQCS), highlighted by the light yellow box, where the quantum reference current I_PQCS is circulating. This circuit is magnetically linked by the cryogenic current amplifier to a second circuit that drives the output current of the PQCG, I_PQCG. In the reference loop, at the superconducting pads of the PJVS, quantized voltage steps, ${U}_{\text{J}}={n}_{\text{J}}{f}_{\text{J}}{K}_{\text{J}}^{-1}$ with K_J = (2e/h), are generated when n_J Josephson junctions are dc biased and driven at a microwave frequency f_J. This voltage is applied to the QHRS through a double connection scheme [13, 14]. Doing so, and thanks to the properties of the quantum Hall effect [13, 14], the equivalent resistance R connected to the superconducting pads is equal to R = R_H(1 + α), where R_H is the quantum Hall resistance, which is equal to R_K/2 at Landau level filling factor ν = 2 with R_K = h/e² the von Klitzing constant, and where $\alpha =\mathcal{O}({(r/{R}_{\text{H}})}^{2})$ is a second order correction due to the cable resistances r. This correction, which is significatively reduced compared to the usual first order correction of a simple connection corresponding to r/R_H ∼ 3 × 10⁻⁴, can be determined with a relative uncertainty of a few parts in 10⁹.

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The reference current ${I}_{\text{PQCS}}=\frac{{U}_{\text{J}}}{R}=\frac{{U}_{\text{J}}}{{R}_{\text{H}}}(1-\alpha )$ is circulating into the PJVS. It can also be written as I_PQCS = n_J ef_J(1 − α), which emphasizes the relation to the elementary charge. This current is divided into two separate branches in the double connection to the QHRS. In order to detect and amplify the sum of these currents, we use a CCC, which insures the ampere-turns balance of several superconducting windings by means of the SQUID-based feedback loop. Thanks to the introduction of two CCC windings of identical number of turns, N_JK, into the double connection between the PJVS and the QHRS, the balance between the primary and secondary circuit, N_JK I_PQCS − NI_PQCG = 0, leads to ${I}_{\text{PQCG}}=\frac{{N}_{\text{JK}}}{N}{I}_{\text{PQCS}}$. The gain $G=\frac{{N}_{\text{JK}}}{N}$ is exact within one part in 10⁹ for all windings and within few parts in 10¹⁰ for the windings used in the present study [15]. The current I_PQCG is generated by an external battery-powered low-noise current source which is servo-controlled by the feedback voltage of the SQUID that monitors the flux imbalance. An adjusted pre-balance current (or compensation current), I_comp, driven by the voltage V_out supplied by a digital-to-analog converter of the programmable current bias source of the PJVS, is added to the feedback current of the SQUID to reduce the set-point error due to the finite open loop gain of the SQUID electronics [1]. The current I_PQCG circulates in the load which can be a resistance R_Load, an ammeter or the ULCA.

The PJVS used in this work is the PJVS♯A from reference [1]. It is based on 1 V Nb/Al/AlO_x /Al/AlOx/Al/Nb Josephson junction series arrays fabricated at the PTB [16, 17]. The QHRS is described in reference [18] and the CCC details can be found in [15]. More details are given in appendix A.1.

In our previous work [1], the correct operation of the PQCG was achieved by solving several technical issues. As the PJVS, the QHRS and the CCC are implemented in three different cryostats and connected by long shielded cables, it makes the PQCG very sensitive to external electromagnetic noise. The stability issues are more severe for high ampere-turns values. For instance, the SQUID became unstable for N_JK > 2 when the PQCS circuit was connected. However, increasing N_JK is relevant to improve the signal-to-noise ratio of the PQCG, which is given by the ratio of the flux generated into the SQUID by the primary circuit, N_JK I_PQCS/γ_CCC (γ_CCC being the flux to ampere-turn sensitivity of the CCC), to the total flux noise $\sqrt{{S}_{\phi }^{\text{tot}}(f)}$. To solve this problem, a damping circuit was implemented that stabilizes the CCC operation. Since it cannot be added directly on the reference circuit without introducing systematic errors, a series arrangement of a resistance R_D and a capacitance C_D has been connected to a spare winding of N_D turns [19] as shown in figure 1. However, the damping circuit introduced increased noise in the 100 Hz–10 kHz bandwidth, the consequence of which was not analysed in detailed at that time. Besides, it remained impossible to increase N_JK above 128. Moreover, the low-frequency noise was significantly higher than the SQUID specifications. Finally, the repeatability of the PQCG turned out to be restricted due to flux frequently trapped in the Josephson array during the measurement campaigns. In the following, we report on investigations about noise, stability issues and resonance damping, and we will present the improvements implemented to tackle the previous limitations of the PQCG.

2.2.1. Analysis of the noise of the system

The SQUID is an ultra sensitive magnetic flux-to-voltage converter, and hence interfering electromagnetic signals are critical: both, low-level high-frequency noise and sharp transients, can be detrimental to the operation of the system. In the PQCG, the noise to which the SQUID is exposed can have different origins: (i) direct flux noise picked up by the SQUID, (ii) flux noise coming from the currents that circulate in the CCC windings due to Johnson–Nyquist noise emitted by the resistive elements connected to them, and (iii) external electromagnetic noise picked-up in the circuit through capacitive or inductive couplings. Moreover, the CCC is a complex system of 15 magnetically coupled superconducting windings with distributed capacitances and with inductances L_i ∼ i² L_CCC, where L_CCC is the inductance of the toroidal shield and i the number of turns. For example, for the windings of 2065 turns, L₂₀₆₅ = 85 mH and the equivalent parallel capacitance is 1 nF. The total flux noise $\sqrt{{S}_{\phi }^{\text{tot}}(f)}=\sqrt{{S}_{\text{SQUID}}(f)+{S}_{\text{CCC}}(f)}$ can be deduced from the power spectral density ${S}_{\phi }^{\text{tot}}(f)$, where S_SQUID(f) and S_CCC(f) are the respective contributions from the noise of the SQUID (intrinsic or captured), and from the noise of the CCC and the circuit connected to it. The latter is the most difficult to analyse. Thus, to have a better understanding, a model of the circuit is useful. Here we have used a simplified model of two magnetically coupled series-RLC circuits as a generic model to account for the main features of the spectra (see appendix A.2) in different situations. Because it includes the magnetic coupling between the windings, this model goes beyond our previous simulations [20].

Figure 2(a) compares the noise spectra of the CCC not connected (bare CCC) and connected to the QHRS, without the damping circuit. For the present study, we have performed noise measurements on the output voltage of the dc-SQUID when operating it in internal feedback mode and with the highest bandwidth (0.75 V/ϕ₀ sensitivity and 50 kHz bandwidth). We present the results in terms of the flux noise, $\sqrt{{S}_{\phi }^{\text{tot}}(f)}$, expressed in ${\phi }_{0}/\sqrt{\mathrm{H}\mathrm{z}}$.

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For the bare CCC, two resonances appear at 12.6 kHz and 36.4 kHz. They can be identified as the LC-resonances resulting from the magnetic coupling of the two identical windings of 2065 turns short-circuited by their parasitic capacitances (see figure 2(b) and appendix A.2). Considering that the mutual inductance is given by M₂₀₆₅ = κL₂₀₆₅, where κ < 1 is the coupling factor, the assignment of the LC-resonances to the calculated frequencies ${f}_{0}/\sqrt{1-\kappa }$ and ${f}_{0}/\sqrt{1+\kappa }$, where f₀ is the LC-resonance without coupling, leads to f₀ = 17.2 kHz and κ ∼ 0.8, which are reasonable values for the CCC [21]. The calculated noise spectrum of figure 2(a) is in good agreement with the experimental noise spectrum of the bare CCC down to 10 Hz. At lower frequencies, the noise level can be explained by the dominant contribution of the SQUID noise with a slightly increased white noise level of 5 $\mu {\phi }_{0}/\sqrt{\mathrm{H}\mathrm{z}}$ compared to the SQUID's intrinsic noise.

The noise spectra when the QHRS is doubly connected to the windings of 160 and 1600 (red and green lines of figure 2(a)) have been measured to clarify the problem of the reduced stability when increasing N_JK. In this case, at low frequency, the bottom white noise levels are well explained by considering both contributions of the SQUID noise and the Johnson–Nyquist noise of the QHRS $\frac{{N}_{\text{JK}}}{{\gamma }_{\text{CCC}}}\sqrt{\frac{4kT}{{R}_{\text{H}}}}$, where k is the Boltzmann constant and T the temperature. The latter becomes dominant for 1600 turns and amounts to 15 μϕ₀/$\sqrt{\mathrm{H}\mathrm{z}}$. To avoid this increased noise, an optimum value of N_JK would be about 500 turns. However, the most striking feature is the almost linear increase from 100 Hz up to 10 kHz, on which 50 Hz harmonics are superimposed. This extra noise, the magnitude of which scales with N_JK, is explained by a coupling of the circuit with external electrical noise sources. It can be modelled by a voltage noise source coupled to the circuit through the capacitance to ground of the 15 m long shielded twisted pairs connecting the CCC and the QHRS [22] as shown schematically in figure 2(c). This capacitance has been measured and amounts to 4 nF. To account for the increased noise in the frequency range 100 Hz to 10 kHz, we have simulated the quantum circuit as a simple RL-circuit considering a single winding connected to the QHRS. A 4 nF capacitance in series with a noise source is connected in parallel to the inductance of the winding as shown in figure 2(d). The red and green hatched areas in figure 2(a) correspond to adjustments of the effective white noise voltage source to (30 ± 10) nV ${\mathrm{H}\mathrm{z}}^{-1/2}$, for both N_JK = 160 and 1600. The good agreement confirms that the SQUID instability observed for large values of N_JK comes from the existence of external noise sources that couple to the quantum circuit and excite the high-frequency resonances especially as large N_JK values are used. Moreover, when the battery powered programmable Josephson bias source is connected, its additional digital noise is sufficient to cause the system to become unstable even in internal feedback mode. In future we will improve this situation by reducing the length of the cable between the CCC and the QHRS. For the moment this length is fixed by the current location of the cryostats in the laboratory.

2.2.2. Damping

Given the coupling of external noise to the PQCS, it was therefore crucial to damp the high frequency resonances [1] to stabilize the SQUID operation. The main question is how to choose the parameters of the damping. Experimentally, we observed that high values of N_D enhanced the effect of the damping. The results of a simulation shown in figure 3(a) illustrates the increasing efficiency of the damping on the high frequency peak for three different values of N_D from 16 to 2065. The simulation is based on the same generic model for the CCC, i.e. the series coupled RLC-circuit of in figure 2(b), in which the values of the resistance and the capacitance of one of the circuits have been replaced by the values used in reference [1], R_D = 1 kΩ and C_D = 100 nF (see figure 3(b)).

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Since the first implementation of the damping circuit in the PQCG with N_D = 1600 [1], the winding of 2065 turns has then been favoured [20] not only to achieve a better efficiency but also to allow the use of the 1600-turns winding for N_JK. At intermediate frequency (100 Hz < f < 10 kHz), the damping circuit is responsible for a damped resonance much broader than the self-resonance of the bare CCC [1, 19, 23]. As a first approximation, the center of the broad peak is at ${f}_{\text{D}}\sim 1/(2\pi \sqrt{{L}_{\text{D}}{C}_{\text{D}}})$ and its amplitude, due to the Johnson–Nyquist noise of R_D, is given by $\sim \frac{{N}_{\text{D}}}{{\gamma }_{\text{CCC}}}\sqrt{\frac{4k{T}_{\text{D}}}{{R}_{\text{D}}}}$ [1]. Once the winding has been chosen, N_D is fixed and hence L_D is fixed. The choice of the capacitance C_D is made such that the broad peak is well below 10 kHz so that the damping is efficient at 12 kHz. The last parameter, R_D, fixes the amplitude of the broad resonance and the quality factor $Q\sim \frac{1}{2\pi {f}_{\text{D}}{R}_{\text{D}}{C}_{\text{D}}}$. Noise measurements have been performed for the bare CCC for three different sets of damping parameters [R_D, C_D] (R_D C_D = 10⁻⁴ s), with N_D = 2065 (see figure 4(a)). Three situations are represented corresponding to Q ∼ 2.7, 0.9 and 0.3 respectively. The flux noise has been calculated, for the corresponding parameters, in the case of the damping circuit magnetically coupled to a winding of 2065 turns figure 4(b), and they account very well for the complex shapes of the spectra in the three cases.

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Finally, we have chosen R_D = 1 kΩ and Q ∼ 1 as a good compromise because it suppresses the high quality factor resonance at 12.6 kHz, such that the main goal has been reached, and it limits the amplitude of noise towards the low frequency part. However, the amplitude of the broad peak is about 1 m${\phi }_{0}/\sqrt{\mathrm{H}\mathrm{z}}$. Two possible improvements can be considered, increasing of the resistance by a factor of 2 to reach the critical value Q = 1/2 and/or placing the resistance at 4.2 K. The latter point will be detailed in the present paper. Note that the resonance at 35 kHz did not disappear, it has been broadened and slightly shifted to 36 kHz; for the moment, this is not clearly understood, and the model cannot account for it because only two windings are simulated. It is possible detrimental effect on the stability may be limited anyhow by the damping, which attenuates part of the high-frequency components of the signal during the fast reversals that are likely to excite this resonance.

More unexpectedly, figure 4(a) shows that the low frequency part of the spectrum is enhanced from about 7 $\mu {\phi }_{0}/\sqrt{\mathrm{H}\mathrm{z}}$ to 20 $\mu {\phi }_{0}/\sqrt{\mathrm{H}\mathrm{z}}$ at 1 Hz compared to the bare CCC. This may be a consequence of mixing-down effects due to the high level of broadband noise in the kHz range. The SQUID is used in flux-locked loop (FLL) in order to amplify and linearize the periodic V − ϕ transfer function of the SQUID. The SQUID is then locked at a given working point of the V − ϕ characteristic by the feedback loop. At low frequencies, within the FLL bandwidth, the FLL suppresses the effect of the noise on the SQUID, in other words, the flux 'seen by the SQUID' is constantly compensated by FLL. However, if the noise above the FFL bandwidth is sufficiently strong, then the linearized flux range can be exceeded, leading to non-linear distortions, and consequently to rectified noise which will be superimposed with the low frequency feedback signal. Hence these mixing down effects deteriorate the signal-to-noise ratio [24] but, unless the amplitude of the noise is much higher, the SQUID is still locked at its working point.

This hypothesis has been confirmed recently, as the noise level at low frequency retrieves the expected one, as soon as the peak in the intermediate bandwidth is reduced by a factor of 8.5 by placing the resistance R_D = 1 kΩ at 4.2 K, as shown in figure 4(b). The integrated noise between 100 Hz and 10 kHz can be used to quantify the detrimental noise level. For instance, it is equal to 60 mϕ₀ for R_D at ambient temperature (black line), whereas it falls at 7 mϕ₀ for R_D at 4.2 K (light blue line).

2.2.3. External feedback

In external feedback mode, a CCC winding is used as the feedback path instead of the dedicated modulation coil of the SQUID (internal feedback mode). It has a big impact on the dynamic of the system and the problems of stability are even more severe.

Compared to the internal feedback, the stability of the FLL is ensured by choosing an operation mode (5S) with a reduced bandwidth of 500 Hz as suggested in the documentation of the SQUID [25], i.e. characterized by a cutoff frequency much lower than the high frequency resonances. However, despite both this change and the fine adjustment within 10⁻⁶ of the current ratio I_PQCG/I_PQCS, the system proved to be unstable (i.e. frequent unlocking of the SQUID) for high ampere-turns reversals, which are needed during the PQCG operation to eliminate voltage offsets. This instability is related to the periodic nature of the V − ϕ characteristic of the SQUID, with several equivalent working points, each separated by a flux quantum. Fast reversals of the signal can force the SQUID to jump between adjacent working points if large signals are applied and if they change faster than the FLL can track. These problems are very similar to those encountered in different CCC resistance bridges [15, 19, 26], where two current sources feed the resistors to be compared, each being connected in series with a ratio winding of the CCC. A special care is paid to keep the bridge balanced during current reversals, by matching the filters on both sides of the bridge [26, 27] and by using offset, in-phase and in-quadrature correction circuits [15]. Other authors have implemented wideband feedback [19]. However, in the case considered here, because the two circuits connected to the CCC, the PQCS circuit and the output circuit, are biased by a voltage source and a current source, respectively, these adjustments are more difficult.

This instability has been overcome by modifying the closed-loop feedback gain. Let us detail the original external feedback circuit as illustrated in figure 5(a). The SQUID output voltage (mode 5S, bandwidth 500 Hz) is connected to the input of a high impedance differential amplifier to avoid loading the 500 kΩ resistance at the output of the SQUID electronics with the capacitance of the long cable plugged into the external current source. However a parasitic capacitance C_SQ of 1.6 nF loading the resistance has been measured at the output of the circuit connecting the SQUID to the isolation amplifier. The same closed-loop feedback gain of ${G}_{\text{CLG}}^{0}$ = ${R}_{\text{FB}}\frac{{\gamma }_{\text{CCC}}}{N}$ = 0.75 V/ϕ₀ as in the internal feedback mode of the SQUID has been targeted for a winding of N = 16 turns; therefore, a feedback resistance R_FB = 1.5 MΩ has been chosen. A small capacitance of 200 pF in series with a 20 kΩ resistor is connected in parallel to R_FB to partially compensate the phase shift caused by the 1 kHz low-pass filter (R_F and C_F) present at the output of the current source [15]. Nevertheless, a residual phase shift exists in the feedback circuit [28] that translates into a peak in the frequency response of the feedback loop (dark green line of figure 5(b)), according to the calculated response based on the circuit presented in figure 5(a). From the simulation, it can be deduced that the phase shift comes mainly from the parasitic capacitance, C_SQ. This certainly explains the instability against current reversals in the PQCG. By increasing the closed-loop gain to 4.5 V/Φ₀ for the PQCG, the peak in the frequency response has almost disappeared, as shown by the green line in figure 5(b). This smooth response of the feedback with a slightly reduced bandwidth has solved the problem of stability against high ampere-turns reversals. In the future, a possible improvement would be to reduce the parasitic capacitance. However, the impact of the higher closed-loop gain on the accuracy of the generated current has been tested in [1] and it revealed negligible.

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To reduce the common mode noise on the PQCS circuit, a common mode torus has been placed on the output of the PJVS, on the current leads that supply the QHRS, as depicted in figure 1. A similar solution had been implemented on the output of the voltage-controlled current source [15]. Guided by the reduction of the noise at frequencies below 10 Hz while ensuring the stability even for the highest ampere-turns, i.e. for N_JK = 160 and 1600 when the CCC is connected to the PQCS, we found that the noise and therefore the stability of the PQCG is very sensitive to the position of the ground connection. In [1], in order to reduce the effect of leakage currents redirected to ground, it was connected at 4.2 K at the low voltage pad of the Josephson array and on the current contact of the QHRS. However, the relative error due to the leakage currents has been estimated to be less than 10⁻¹¹, and moving the position to room temperature does not change much from this point of view. On the contrary, the ground connection at room temperature allows testing different positions and their effect on the noise of the circuit. We found that the best position, reducing the low frequency noise from 25 μV/$\sqrt{\mathrm{H}\mathrm{z}}$ to 7 μV/$\sqrt{\mathrm{H}\mathrm{z}}$, is at the head of the CCC as shown in figure 1, on the voltage contact of the QHRS. This certainly short-circuits a noise source $(\tilde {e})$ that couples directly into the windings. The presence of the high inductance of the common mode torus on the current lead might explain that the noise couples less on this lead.

However, in this configuration, the amplitude of the voltage steps of the PJVS were reduced due to an increased noise that perturbed the array. This has been resolved by adding a 1 μF highly isolated capacitance placed between the 0 V of the battery powered Josephson programmable bias source and the ground (see figure 1). This new configuration is much more reliable, the array is less subject to flux trapping than it was previously with the ground connection at low temperature. Moreover, a heating system has been implemented at PTB on the Josephson array chip (see figure 1), and allows to get rid of trapped flux within few minutes.

Finally, after all these modifications, the system has gained stability in internal feedback mode. As shown in figure 6, the PQCS (including the PJVS bias source) can be connected notably to N_JK = 160 (purple line) and 1600 (grey line) and the low frequency noise in both configurations is close to the expected levels. The flux noise of the broad resonance integrated between 100 Hz and 10 kHz is below 10 mϕ₀ in both cases which is close to the 7 mϕ₀ for the bare CCC (see figure 4(b)). However, in external feedback mode, the stability issues still limit the operation to N_JK = 160.

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Compared to reference [1], not only the noise in the intermediate frequency range has been reduced but also the low-frequency noise below 10 Hz. The reduction of the latter by a factor of 2 and the increase of N_JK by a factor of 1.25, may lead to a maximum increase of the signal-to-noise ratio by a factor of 2.5 for the most accurate measurements. However, this potential gain may be reduced if the excess noise (1/f noise) becomes the dominant noise at very low frequencies [29]. Note that the PQCG experiments aiming for highest accuracy are performed at current-reversal frequencies below 100 mHz. Hence, new measurements are needed to confirm the improvement in Type-A uncertainty of the PQCG [1]. Nonetheless, the reduction of the noise in the intermediate frequency range has already improved the Type-A uncertainty of digital ammeter calibrations as will be seen in the next section.

In this last section, we present the results of the direct comparison of the current measurements performed using the ultrastable low-noise current amplifier (ULCA) and the PQCG. The ULCA can be used for two purposes: current measurement or current generation, i.e., for current-to-voltage or voltage-to-current conversion. It is based on a two-stage design: a first stage provides a current gain G_I of 1000 and a second stage provides a current-to-voltage conversion with a transresistance R_IV (typically, 1 MΩ for the 'standard' ULCA). Thanks to this design, the device can be operated either in normal or extended mode [12] which differ by the factor of 1000 in current range; for example, in the case of the 'standard' ULCA, the 5 nA range (normal mode) and the 5 μA range (extended mode) are accessible. Several versions of the ULCA have been developed [12]. Among the second generation of ULCAs, the prototype of a high-accuracy variant (R_{I V} -nominal value 100 kΩ) offers the highest current range compatible with the PQCG (±50 μA in the extended mode).

Thanks to its small size and compact design, the ULCA is perfectly suited as a travelling standard [32, 33]. The use of an ULCA as a travelling standard has been exercised first in an interlaboratory comparison between three national metrology institutes including LNE and PTB in 2015 with a two-channel 'standard' ULCA [32] operated in the 5 nA range. The calibration history of the travelling standard including the effects of lab-to-lab transports are crucial information; for instance, the history of the device which has been used in [32] has been recorded over more than five years [34]. However, comparably reliable information on the stability of the chosen high-accuracy ULCA, especially when travelling, was not available at the time of the first experiments reported below. The shipment from PTB to LNE was done by a commercial carrier and for the return transport, a PTB car has been used.

The comparison has been done in extended mode in the 50 μA range at two current levels 43.75 μA and 21.875 μA. Similarly to the calibration of the DAs, I_PQCG is directly fed into the ULCA (i.e. into the current-to-voltage conversion stage). Different from the work with the DAs described in the previous section, no filter was necessary on the input of the ULCA. Introducing much less noise, the latter did not affect the SQUID's stability. The calibration of the total transimpedance A_TR = G_I R_IV , which is performed at PTB in two steps, has been done before and after travelling to LNE. The gain G_I is directly calibrated by means of a CCC and the transresistance R_IV is traced back to the quantum Hall resistance by means of a CCC-based resistance bridge [32]. Note that for the present comparison with the ULCA operated in extended mode throughout, only the value of R_IV is relevant.

The multiple connection of the PQCG has been implemented. However, to even further reduce the absolute value of the correction, we have 'short circuited' part of the cable resistances by using the 'local triple connection' on one side of the QHRS (dashed connection on figure 1). That is, an additional connection is made between the voltage pad of the QHR and the top of the cryostat rather than directly on the superconducting pad of the PJVS. From the measurements of the resistances of the cables, we have estimated the relative correction α = 1.35 × 10⁻⁷ ± 2 × 10⁻⁹.

The current measured by the ULCA is given by ${I}_{\text{ULCA}}={U}_{{R}_{IV}}/{R}_{IV}$. It is determined from the value of the transresistance R_IV estimated at the time of the comparison from the calibration history determined at PTB (see appendix A.4 in the appendix to this paper) and from the voltage drop ${U}_{{R}_{IV}}$ across R_IV measured on the 10 V range of a digitizing voltmeter (DVM). The latter is routinely calibrated against a 10 V conventional Josephson voltage standard at LNE.

In that way, we compare on one side the current measured by the combination of the ULCA and the DVM–both traceable to PTB's quantum Hall resistor and LNE's Josephson voltage standard, respectively—and the current generated on the other side by the PQCG—involving LNE's PJVS and QHRS—and estimated from the expression ${I}_{\text{PQCG}}=\frac{{N}_{\text{JK}}}{N}\frac{{U}_{\text{J}}}{{R}_{\text{H}}}(1-\alpha )$. Considering the degrees of equivalence of PTB and LNE in the international comparison BIPM.EM-K10.b [35] of Josephson array voltage standards at 10 V, this experiment can be interpreted as a comparison of the realizations of the ampere in the two laboratories. However, without the knowledge of the transresistance, this experiment can be used to determine the value R_IV from I_PQCG and ${U}_{{R}_{IV}}$ (see appendix A.4).

We have performed eighteen measurements within two days. We have applied exactly the same measurement sequence as for DA♯1. Figure 11 shows the relative deviation ΔI/I = (I_ULCA − I_PQCG)/I_PQCG over the two days. These measurements include the tests of the quantization criteria of the PQCG and the consistency check in terms of linearity of the ULCA. A summary of the parameters of the measurements is presented in table 1.

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The Type-A relative uncertainty of each measurement point is <2 × 10⁻⁸. This low Type-A uncertainty allowed us to resolve a drift of 2.5 × 10⁻⁷ over the first 24 h followed by a period of stabilization to within 2.5 × 10⁻⁸, corresponding to the last 9 points in figure 11. The quantization criteria have been checked within the standard deviation of ⩽2.5 × 10⁻⁸. After stabilization, the linearity of the ULCA has been tested at ±21.875 μA and ±43.750 μA. The results are shown in figure 12. The error bars represent only the Type-A standard uncertainty. A slight deviation of the order of 10⁻⁸ can be observed but it is within the stability of the measurements.

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The DVM gain error has been estimated from a two-weeks calibration campaign before the comparison. Unfortunately, we observed a deviation from the known linear drift when recalibrating the instrument one month after the comparison, which might have been due to a temperature regulation problem in the laboratory during this period. Therefore the relative Type-B uncertainty contribution due to this correction has been increased to cover this problem and amounts to 2.3 × 10⁻⁷.

The typical uncertainty of the calibrations of R_IV at PTB is 6 × 10⁻⁹. During the comparison the internal temperature of the ULCA was recorded and was about 21 °C at LNE, whereas the transresistance calibration is given at 23 °C. This temperature difference was accounted for by a correction calculated from the −1.4 × 10⁻⁷ °C⁻¹ temperature coefficient determined at PTB. The corresponding relative uncertainty contribution is 0.2 × 10⁻⁷. An additional uncertainty component has been added in the uncertainty budget, it accounts for the long term stability and the stability against transportation of R_IV . It has been estimated to 2 × 10⁻⁷ (see appendix A.4). Table 2 summarizes the Type-B contributions to the relative standard uncertainty. The DVM calibration and the stability of the output stage of the ULCA with transportation appear as the dominant contributions.

For the final result of the comparison, we have used 9 data points obtained for ±21.875 μA and ±43.750 μA. The experimental standard deviation of the mean of the 9 points is 7 × 10⁻⁹. The final result of the comparison is expressed as the mean value of the 9 points within the total relative combined uncertainty (represented by the red dashed dotted lines in figure 11) (k_c = 1):

$$\begin{equation}[{I}_{\text{ULCA}}-{I}_{\text{PQCG}}]/{I}_{\text{PQCG}}=(-3.7{\pm}3.1){\times}1{0}^{-7}.\end{equation} \tag{ 1 }$$

In this comparison, the realizations of the ampere from both laboratories agree to −3.7 parts in 10⁷ with a combined standard uncertainty of 3.1 parts in 10⁷ (k_c = 1). This result is reported in figure 11. Clearly there is no significant discrepancy between the two current values since the difference is lower than the expanded uncertainty of 6.2 parts in 10⁷ (k_c = 2).

The Josephson array is subdivided into 14 smaller array segments, and follows the sequence 256/512/3072/2048/1024/128/1/1/2/4/8/16/32/64. The quantum Hall resistance standard, based on an eight-terminal Hall bar made of a GaAs/AlGaAs semiconductor heterostructure (LEP514), was produced at the Laboratoire Electronique de Philips [18]. The design of the CCC used here is detailed in [15]. It is composed of 15 windings with the following numbers of turns: 1, 1, 2, 2, 16, 16, 32, 64, 128, 160, 160, 1600, 1600, 2065, and 2065. It is equipped with a Quantum Design Inc. dc SQUID, operating with flux modulation at 500 kHz. The SQUID has a white noise level of 3 $\mu {\phi }_{0}/\sqrt{\mathrm{H}\mathrm{z}}$ and a 1/f corner frequency f_c = 0.3 Hz, where ${\phi }_{0}={K}_{\text{J}}^{-1}$ is the superconducting flux quantum. The flux to ampere-turn sensitivity of the CCC is γ_CCC = 8 μA turn/ϕ₀. Note that in the present study no additional current divider [1] is used to finely adjust the gain between I_PQCG and I_PQCS.

For the model of the noise coming from the CCC at high frequency, we consider two series-RLC circuits (R₁, C₁, L₁, R₂, C₂, L₂) coupled by a mutual inductance $M=\kappa \sqrt{{L}_{1}{L}_{2}}$, where κ is the coupling factor. The parameters are given in the figure 13. The currents circulating in the two circuits i₁ and i₂ are written as a current vector:

$$\begin{equation*}I=\left(\genfrac{}{}{0.0pt}{}{{i}_{1}}{{i}_{2}}\right).\end{equation*}$$

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The noise voltage density ${\tilde {e}}_{1}$ and ${\tilde {e}}_{2}$ of the voltage noise sources, e₁ and e₂, due to the Johnson–Nyquist noise of the resistances R₁ and R₂ are expressed in V ${\mathrm{H}\mathrm{z}}^{-1/2}$ and are given by: ${\tilde {e}}_{1}=\sqrt{4kT{R}_{1}}$ and ${\tilde {e}}_{2}=\sqrt{4kT{R}_{2}}$ respectively and are written as a vector:

$$\begin{equation*}\tilde {E}=\left(\genfrac{}{}{0.0pt}{}{\tilde {{e}_{1}}}{\tilde {{e}_{2}}}\right).\end{equation*}$$

The currents are deduced from the following system of linear equations:

$$\begin{equation*}I={[K]}^{-1}\tilde {E},\end{equation*}$$

where

$$\begin{equation*}K=\left(\begin{matrix}\hfill j\frac{{L}_{1}}{\omega }({\omega }^{2}-j{\gamma }_{1}\omega -{\omega }_{01}^{2})\hfill & \hfill jM\omega \hfill \\ \hfill jM\omega \hfill & \hfill j\frac{{L}_{2}}{\omega }({\omega }^{2}-j{\gamma }_{2}\omega -{\omega }_{02}^{2})\hfill \end{matrix}\right)\end{equation*}$$

and ${\gamma }_{1}={R}_{1}/{L}_{1},{\gamma }_{2}={R}_{2}/{L}_{2},{\omega }_{01}=\sqrt{\frac{1}{{L}_{1}{C}_{1}}}$ and ${\omega }_{02}=\sqrt{\frac{1}{{L}_{2}{C}_{2}}}$.

The screening current δI on the toroidal superconducting shield due to the noise currents is deduced from the relation: $\delta I={N}_{1}{i}_{1}+{N}_{2}{i}_{2}=\eta {\tilde {e}}_{1}+\beta {\tilde {e}}_{2}$, where $\eta ={N}_{1}{K}_{11}^{-1}+{N}_{2}{K}_{12}^{-1}$ and $\beta ={N}_{1}{K}_{12}^{-1}+{N}_{2}{K}_{22}^{-1}$. The noise sources e₁ and e₂ being uncorrelated, the current noise amplitude is given by: $\sqrt{\vert \delta I{\vert }^{2}}=\sqrt{\vert \eta {\vert }^{2}{\tilde {e}}_{1}^{2}+\vert \beta {\vert }^{2}{\tilde {e}}_{2}^{2}}$. Finally, the flux noise contribution is $\sqrt{{S}_{\text{CCC}}(f)}=\sqrt{\vert \delta I{\vert }^{2}}/{\gamma }_{\text{CCC}}$.

The Allan deviation allows the analysis of the noise by differentiating noise types according to the exponent of its power dependence with time. White noise manifests itself by a τ^−1/2 dependence. In the white noise regime, the Allan deviation σ_w is related to the flux power spectral density S_ϕ by:

$$\begin{equation}{\sigma }_{\mathrm{w}}=\sqrt{{S}_{\phi }/2\tau }\end{equation} \tag{ 2 }$$

where τ is the sampling time.

We want to relate the flux power spectral density measured for the PQCG to the relative Allan deviation σ_I /I (=σ_ϕ /ϕ) expected for a series of current measurements obtained from on-off-on cycles that allows us to subtract the offsets. For that, we calculate the standard deviation of an on-off-on cycle by adapting a calculation done for polarity reversals by Drung et al [11].

The value of the current calibration point, corresponding to one on-off-on cycle, is calculated from the average of the 'on' values for which we subtract the average of the 'off' values. We suppose that the standard deviations, σ_on and σ_off, for the 'on' and 'off' polarities, can be calculated from the white noise standard deviations (see equation (2)) corresponding to the duration, τ_on and τ_off = 2τ_on, of the 'on' and 'off' polarities, respectively.

The white noise uncertainty for the on-off-on cycle, σ_cycle, is then given by:

$$\begin{equation}{\sigma }_{\text{cycle}}=\sqrt{{\sigma }_{\mathrm{o}\mathrm{n}}^{2}/2+{\sigma }_{\text{off}}^{2}}={\sigma }_{\mathrm{o}\mathrm{n}}=\sqrt{{S}_{\phi }/2{\tau }_{\mathrm{o}\mathrm{n}}}\end{equation} \tag{ 3 }$$

with ${\sigma }_{\text{off}}^{2}={\sigma }_{\mathrm{o}\mathrm{n}}^{2}/2$.

In figure 7, we plot the relative Allan deviation as a function of the sampling time τ = n_s τ₀, so we want to express σ_cycle as a function of τ₀. Following [11], we define the effective measuring time τ_e, i.e. the sampling time without the waiting time after polarity reversals, here: τ_e = τ₀ − 2τ_w = τ₀(1 − ρ), with ρ = 2τ_w /τ₀ the rejection rate. For the on-off-on cycle, τ_on = τ_e/4. We rewrite σ_cycle as a function of τ₀,

$$\begin{equation}{\sigma }_{\text{cycle}}=\sqrt{2{S}_{\phi }/{\tau }_{\mathrm{e}}}=\sqrt{2{S}_{\phi }/{\tau }_{0}(1-\rho )}.\end{equation} \tag{ 4 }$$

We introduce the factor A' defined as ${A}^{\prime }=\sqrt{2{S}_{\phi }/(1-\rho )}$, such that:

$$\begin{equation}{\sigma }_{\text{cycle}}={A}^{\prime }/\sqrt{{\tau }_{0}}.\end{equation} \tag{ 5 }$$

Finally, the Allan deviation for a sequence of n_s cycles is:

$$\begin{equation}{\sigma }_{\text{sequence}}={A}^{\prime }/\sqrt{{n}_{\mathrm{s}}{\tau }_{0}}.\end{equation} \tag{ 6 }$$

All the parameters used for the calibration measurements and for the Allan deviation are given in table 3. We tried to realize as much as possible similar acquisition times for the two DAs despite the different parameter settings and the different dead times. As can be seen from table 3, the DA♯1 has been used in the mode AUTOZERO ON (internal offset errors due to bias currents of the amplifiers and thermal emfs are subtracted), which is the most accurate mode. However compared to the other DA, which does not have this function, the measuring time has been doubled for DA♯1, in order to have the same effective measuring time in a given polarity. This explains the difference of the timing parameters for the two DAs.

To calculate the relative Allan deviation expected for the PQCG at I_PQCG = 0.918 mA, we have to divide A' by the flux seen by the SQUID due to I_PQCS, i.e. ${A}_{\text{rel}}^{\prime }$. The flux is 460 ϕ₀ (calculated with the parameters N_JK = 160, N = 4, I_PQCS = 23 μA and the flux sensitivity 8 μA turns/ϕ₀). We calculate ${A}_{\text{rel}}^{\prime }$ based on a large range of estimates for $\sqrt{{S}_{\phi }}$ at 20 mHz, which spans from 8 μϕ₀/$\sqrt{\mathrm{H}\mathrm{z}}$ to 50 μϕ₀/$\sqrt{\mathrm{H}\mathrm{z}}$ taking into account the all cases from white noise to 1/f noise.

Based on the previous calculation, it is possible to use the fitting parameters ${A}_{\text{rel}}^{\prime }$ of the relative Allan deviation σ_I /I of the current measurements of the 2 DAs presented in figure 7, to estimate the equivalent current noise $\sqrt{{S}_{I}}={A}_{\text{rel}}^{\prime }\sqrt{(1-\rho )/2}\;{\ast}\;{I}_{\text{PQCG}}$ and eventually to compare with the constructor specifications (table 4).

Table 4. Results of the calculation of ${A}_{\text{rel}}^{\prime }$ for the PQCG when $\sqrt{{S}_{\phi }}$ at 20 mHz is equal to 8 μϕ₀/$\sqrt{\mathrm{H}\mathrm{z}}$ or 50 μϕ₀/$\sqrt{\mathrm{H}\mathrm{z}}$ and results of the calculation of $\sqrt{{S}_{I}}$ for DA♯1 and DA♯2 from the values of ${A}_{\text{rel}}^{\prime }$ obtained from the fit of figure 7.

Instrument	${A}_{\text{rel}}^{\prime }$	$\sqrt{{S}_{\phi }}$	$\sqrt{{S}_{I}}$
PQCG	$2.6{\times}1{0}^{-8}/\sqrt{\mathrm{H}\mathrm{z}}$ a	8 μϕ0/$\sqrt{\mathrm{H}\mathrm{z}}$	—
PQCG	$1.6{\times}1{0}^{-7}/\sqrt{\mathrm{H}\mathrm{z}}$ a	50 μϕ0/$\sqrt{\mathrm{H}\mathrm{z}}$	—
DA♯1	$1.35{\times}1{0}^{-6}/\sqrt{\mathrm{H}\mathrm{z}}$ b	—	0.8 nA ${\mathrm{H}\mathrm{z}}^{-1/2}$ a
DA♯2	$6{\times}1{0}^{-7}/\sqrt{\mathrm{H}\mathrm{z}}$ b	—	0.3 nA ${\mathrm{H}\mathrm{z}}^{-1/2}$ a

^aDenotes a result from a calculation. ^bDenotes a result from a fit.

Figure 14 shows the relative deviation of R_IV from the 100 kΩ nominal value measured at PTB (black dots) as a function of time over a period of more than two years including the period of the comparison. Before transportation, the transresistance followed an exponential decay at PTB (red dashed line in figure 14). However, the first calibration after returning to PTB showed a deviation of 4.8 × 10⁻⁷ from the predicted value, which might be due to mechanical and thermal shocks during transportation. Unfortunately, we cannot clearly assign the discontinuity to the transport to or from LNE. This impacts the estimation of the transresistance, and hence the value of I_ULCA, at the time of the comparison. Here, to make the less assumptions possible, we have considered a linear interpolation at the date of the comparison, between the value of the last calibration before transportation and the first after return to PTB. The estimated uncertainty is calculated from a rectangular probability distribution of total width equal to the difference between these two values (7 × 10⁻⁷). The relative uncertainty on the estimation of R_IV is therefore 2 × 10⁻⁷. From the measurements done at LNE, the transresistance R_IV can be estimated from ΔI/I = (R_IV − 100 kΩ)/100 kΩ. This value can be compared to the value obtained at PTB. It has been added in figure 14. The error bar reflects the total combined uncertainty without the contribution of the stability of the R_IV and is therefore dominated by the contribution due to the calibration of the DVM. On a general point of view, in principle, the transresistance could also be directly calibrated by means of the CCC-based resistance bridge presented in [15] and traced back to the quantum Hall resistance at LNE.

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