Selected for a Viewpoint in Physics PHYSICAL REVIEW X 6, 041051 (2016)

Practical Quantum Realization of the Ampere from the Elementary Charge

J. Brun-Picard, S. Djordjevic, D. Leprat, F. Schopfer, and W. Poirier* LNE-Laboratoire national de métrologie et d’essais, 78197 Trappes, France (Received 8 June 2016; revised manuscript received 30 September 2016; published 12 December 2016) 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 ef f milliampere range are accurately quantized in terms of J ( 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.

DOI: 10.1103/PhysRevX.6.041051 Subject Areas: Condensed Matter Physics, Interdisciplinary Physics, Quantum Physics

I. INTRODUCTION these fundamental constants expressed in the units : 2: −1 : Measurements rely on the International System of Units kg m s and A s, respectively, will set the definitions (SI) [1], which is a consistent system historically con- of the kilogram [3] and of the ampere [4] without structed on seven base units, namely, the meter (m), the specifying the experiment for their realizations. The mass h kilogram (kg), the second (s), the ampere (A), the kelvin unit, bound to , will no longer be realized by the (K), the mole (mol), and the candela (cd), all other units international prototype of the kilogram suspected to be being formed as products of powers of the base units. The drifting with time but rather, for example, using the watt SI has evolved following the scientific knowledge with the balance experiment [7,8]. Similarly, the ampere will be e aim of decreasing the uncertainty in the measurements but realized from the elementary charge and the frequency −1 also with the aim of universality. This is best illustrated by fðs Þ and no longer from Ampère’s force law [9], which the history of the definition of the meter, which was first relates electrical units to mechanical units and thereby 7 based on an artifact, then based on a reference to an atomic limits the relative uncertainty to a few parts in 10 [10]. transition, and more recently related to the second through a A direct way to realize the ampere from the elementary fixed value of the speed of light c expressed in the unit charge and the frequency f, illustrated in Fig. 1(a), is based m:s−1. This success has guided the choice for the future on single-electron tunneling (SET) devices [11] in meso- revision of the SI [2–6], in which the definitions of the scopic systems at very low temperatures, where the charge seven base units will be based on constants ranging from quantization manifests itself because of the Coulomb fundamental constants of nature to technical constants [6]. blockade [12]. Among SET devices, electron pumps [13] Quantum mechanics will be fully exploited by fixing the transfer a precise number nQ of charge Q ≡ e at each cycle values of the Planck constant h and of the elementary of a control parameter, which is synchronized to an external e f charge . From the definitions of the second and the meter, frequency P so that the amplitude of the output current is n Qf n ef ideally equal to Q P, i.e., theoretically equal to Q P. *[email protected] The first electron pumps consisted of small metallic islands in series, isolated by tunnel junctions. In a seven-junction Published by the American Physical Society under the terms of device, an error rate per cycle of 1.5 × 10−8 was measured the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to for frequencies in the MHz range [14]. The accuracy of the author(s) and the published article’s title, journal citation, such devices, obtained by charging a cryogenic capacitor and DOI. with a precise number of electrons [15], was then

2160-3308=16=6(4)=041051(15) 041051-1 Published by the American Physical Society J. BRUN-PICARD et al. PHYS. REV. X 6, 041051 (2016)

(a) (b) 10-3

-4 Best calibration and 10 measurement -5 capabilities (CMC) [21] 10 [17] [19] -6 10 [16] 10-7 [20] Programmable quantum Single electron tunneling current generator (PQCG) 10-8 Relative uncertainty devices 10-9 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Current (A) (c) (d)

5x10-8 PQCG I / 0 PQCG I -8 Δ -5x10 -3 -2 -1 0 1 2 3

IPQCG (mA)

FIG. 1. Practical realization of the ampere from quantum electrical effects. (a) Illustration of the two ways to generate a current from the frequency fðs−1Þ 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 A, by charging a capacitor (Physikalisch-Technische Bundesanstalt), and from 10−10 up to 10−1 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 μA up to 10 mA. (c) Principle of the ΔI =I PQCG. (d) Relative deviation PQCG 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. demonstrated with a relative uncertainty of 9.2 × 10−7 for can be understood as the transfer of an integer number of flux currents below 1 pA [16]. A similar experiment reached quanta ϕ0 ¼ h=2e per microwave cycle due to the circulation a relative uncertainty of 1.66×10−6 with a five-junction of a current of Cooper pairs. The QHE links the current I to the U R −1 R-pump [17] [Fig. 1(b)]. Recently, alternative electron voltage through the constant K .Thisquantumphe- pumps based on tunable barriers in a nonadiabatic regime nomenon manifests itself in a device based on a two- were proposed as a trade-off between accuracy and increased dimensional electron gas under a perpendicular magnetic current, as reviewed in Refs. [11,18]. In most recent devices field, by the quantization of the Hall resistance at values R =i i operating at very low temperatures (T ≤ 0.3 K) and under K K, where K is an integer. This comes from the existence B ≥ 14 i high magnetic fields ( T), the quantization of the at Fermi energy of K one-dimensional ballistic chiral states current was demonstrated with relative measurement uncer- [25] of conductance e2=h each at the device edges, and the 1 2 10−6 f 945 tainties of . × at 150 pA ( P ¼ MHz) [19] and absenceofdelocalizedstatesinthebulkduetotheopeningofa 2 10−7 f 545 × at 90 pA ( P ¼ MHz)[20] [Fig. 1(b)]. gapintheenergyspectrum(thedensityofstatesisquantizedin As illustrated in Fig.1(a),thefuturedefinitionoftheampere Landau levels). From the application of Ohm’s law to these can also be realized by applying Ohm’s law to the quantum two quantum standards, the frequency f can therefore be I K R −1≡ voltage and resistance standards that are based on the converted in a current with the constant ð J × KÞ e=2 K 2e=h R Josephson effect (JE) [22] and the quantum Hall effect . It is planned that the relationships J ¼ and K ¼ (QHE) [23], two gauge-invariant macroscopic quantum h=e2 will be assumed in the new SI. This is a reasonable effects that involve the Josephson and the von Klitzing assumption since no measurable deviation has been predicted K ≡ 2e=h R ≡ h=e2 constants, J and K , respectively. More by quantum mechanics [26–30] and no significant deviation precisely, the ac JE converts the frequency f of an electro- has been demonstrated experimentally, either by independent U K −1 magnetic wave to a voltage with the constant J .This determinations of the constants in the relations [11,31,32] or effect is characterized by the appearance of quantized voltage by universality tests. Several experiments have indeed dem- n K −1f steps (Shapiro steps) [24], at values J J J in the current- onstrated the universality of the JE and the QHE with relative voltage characteristic of a Josephson junction irradiated by a measurement uncertainties below 2 × 10−16 [33–35] and f n 10−10 – microwave field of frequency J, where J is an integer. This [36 38], respectively. In the future SI, the Josephson

041051-2 PRACTICAL QUANTUM REALIZATION OF THE AMPERE … PHYS. REV. X 6, 041051 (2016) voltage standard (JVS) and the quantum Hall resistance (Appendix A2)of10−8 (1 standard deviation or 1 s.d.). In standard (QHRS) would therefore become realizations of principle, this new standard is programmable and versatile; the ohm and the volt with relative uncertainties below 10−9, i.e., it can generate currents over a wide range of values, only limited by their experimental implementation and no extendingfrom10mAdownto1 μA. The PQCG uncertainty K R 4 10−7 longer by the uncertainties on J and K of × [39] and budget result reported in Fig. 1(b) by the red dashed-dotted 1 × 10−7 [40], respectively, in the present SI (Appendix A1). line shows that the accuracy is unchanged in the whole current Applying Ohm’s law to those standards in the new SI would range. The PQCG is a primary quantum current standard, i n ef =2 resultinacurrentstandardthatcan beexpressed as K J J , accurate over a wide current range, able to greatly improve the where e has an exact value. The expectation is to reach an current measurement traceability by reducing uncertainties of accurate quantum current standard realizing the ampere 2 orders of magnitude compared to those declared in the best definition with a high level of reproducibility and universality, CMCs. This is demonstrated by the calibration of a digital which directly benefits from that of the JVS and the QHRS. ammeter (DA) on several current ranges from 1 μAto5mA, This goal was unattainable with the former definition estab- with measurement uncertainties only limited by the device lished in 1948. National Metrology Institutes already apply under test. More fundamentally, the PQCG is able to imple- Ohm’s law to secondary voltage and resistance standards, ment the future ampere definition in terms of the elementary K R e 10−8 traceable to J and K, for the current traceability. However, charge with a target uncertainty of . This will rely on the the uncertainty claimed in their calibration and measurement adoption of the solid-state quantum theory in the planned new capabilities (CMC) for currents, reported in Fig. 1(b),is,in SI. In this context, demonstrating the equivalence of the two practice, not better than 10−6 [21].Above1 μA, this limita- quantum realizations of the ampere described in Fig. 1(a), tion is mainly caused by higher calibration uncertainties of called closing the metrological triangle [43], is a challenging secondary standards and the lack of accurate and stable true experiment of great interest. Improvements in its realization currentsources, while below thiscurrentlimit, the uncertainty, are expected from the PQCG and the quantum circuit methods which increases steadily towards lower current levels, is reported here. instead due to a lack of sensitivity of the measurement techniques. These CMCs emphasize the advantage of an accurate reference current standard able to deliver high II. PROGRAMMABLE QUANTUM currents in order to optimize and shorten the current trace- CURRENT GENERATOR ability in NMIs over a wide range of values above 1 μA. In the range of lower currents, a traceability improvement might be A. Realization expected by exploiting not only SET devices as quantum The experimental scheme of the PQCG, described current standards but also accurate current amplifiers to make in Fig. 2(a), aims at realizing the principle shown in thelinkbetween lowcurrentsandhigher-current references.A Fig. 1(c). The PQCS is built from a programmable recent example of low-noise amplifiers that operates at room Josephson voltage standard (PJVS) [44], which is used to temperature is an ultra-low-noise current amplifier (ULCA) maintain the quantized voltage at the terminals of a quantum [20,41,42], stable within 10−7 over one week and having a Hall resistance standard (QHRS) [45,46] of resistance −6 R R =2 ≡ h=2e2 typical long-term drift of 5 × 10 per year. The downscaling H ¼ K . The PJVS is based on a 1-V series approach also further motivates the development of current array of SINIS Josephson junctions [47], where S, I, and N standards with large values. correspond to superconductor, insulator and normal metal f ∼ 70 Here, we report on a programmable quantum current respectively, operating at frequencies J GHz. The generator (PQCG), linked to the elementary charge e, array is divided into segments that can be individually which is built from an application of Ohm’s law to quantum biased on the n ¼ 0 or n ¼1 Shapiro steps by a standards combined in an original quantum circuit [Fig. 1(c)]. programmable bias source. The quantized voltage steps U n K−1 f ≡ n h=2e f n In brief, it is based on a current source locked, by means of are given by J ¼ Jð J Þ J Jð Þ J, where J a highly accurate cryogenic amplifier of gain G using a is now the number of biased junctions on the first Shapiro magnetic coupling, to a multiple or fraction value of a step and it can be as large as several thousands [Fig. 2(b) and I programmable quantum current standard (PQCS) used as a Appendix A3a]. The current PQCS circulating in the reference. The PQCS is the current circulating in a closed Josephson array, of a few tens of μA, is well below the U n K−1f circuit formed by a Josephson voltage J ¼ J J J applied current amplitude of the Shapiro steps and ensures a perfect R R =2 to a quantum Hall resistance standard H ¼ K using a quantization of the QHRS (Appendix A3b). The PJVS and special connection scheme, which drastically reduces the two- the QHRS are individually checked following the usual wires series resistance (r ≃ 0) thanks to the QHE properties technical guidelines [48,49]. and allows its accurate detection by the amplifier. Figure 1(d) A simple connection of the PJVS to the QHRS would U =R demonstrates that currents generated by the PQCG from not allow realizing J H with the highest accuracy because 0.7 to 2.2 mA are perfectly quantized in terms of of the large value of the two-wires series resistance [sym- K R −1 ≡ e=2 r ð J × KÞ , with a combined standard uncertainty bolized by in Fig. 1(c)] caused by the connecting links. A

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(a) (b) 1.03757 PJVS#A n = 7168 1.03756 J n = 1 1.03755 (V) J

U 0.00001 n = 0 Optical link 0.00000 -0.00001 To PJVS#ref -0.3 0.0 0.3 1.8 2.1 2.4 2.7 IBias (mA) Vout (c) 15 LEP 514, T = 1.3 K 2 f J 10 (k ) SQUID (k ) 1 H electronics xx I = 10.8 T R PQCS 5 R B UJ 0 0 02468101214 I PQCG B (T)

(d) 5 10 PQCS with damping circuit 4 NJK NJK Load ) 10 Bare CCC 1/2 Calculation 103 /Hz 0 102

r (µ 2 i 101 r S 3 N D IPQCG 100 10-1 100 101 102 103 104 105 f (Hz) C RD (e) r3' D QHR 1.11148 r1' r1 1.11146

(mA) 0.00002

r PQCG 0.00000 4 RH I -0.00002 0 100 200 300 400 Time (s)

FIG. 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 (NJK turns) on the low potential side. A third terminal (dotted I I line) connected at the top of the QHE setup further reduces the cable contribution to the current PQCS. The current 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 V I I source using out. Note that PQCG is locked to the current PQCS of the PQCS by means of the CCC, which is used as an accurate adder amplifier of NJK=N gain. A current divider injecting a fraction β of IPQCG in a CCC winding of NDiv (¼ 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 1 1 Ω N 1600 series with a . -k resistor is connected to a CCC winding of D (¼ ) turns. (b) PJVS#A output voltage as a function of the bias I f 70 T 4 2 R current Bias for the total number of Josephson junctions at J ¼ GHz and ¼ . K. (c) Hall resistance H and longitudinal resistance Rxx measured, using a current of 10 μAatT ¼ 1.3 K, as a function of B in the GaAs=AlGaAs-based Hall bar device 1=2 (LEP514). The device is used at B ¼ 10.8 T. (d) The noise amplitude spectral density (expressed in μϕ0=Hz ), 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 IPQCG recorded on a digital ammeter (at 1.1 mA), obtained for nJ ¼ 3073, NJK ¼ 129, and N ¼ 4.

R i r r0 =R multiple series connection of the QHRS [50,51], a technique H in the QHE regime, a small current ¼ð 1 þ 1Þ H × I that exploits fundamental properties of the QHE, is imple- PQCS circulates this time in the connection link probing the mented to reduce their effect. Each superconducting pad of Hall voltage. This results in a small voltage, no more than r r0 r =R I the PJVS is connected to two QHRS terminals located along ð 1 þ 1Þ 2 H × PQCS, which adds to the Hall voltage R I an equipotential edge of the Hall bar [Fig. 2(a)]. Because of H × PQCS. This gives a relative correction to the quantized the chirality of the Hall edge states for the given magnetic- r=R r r0 r =R2 Hall resistance, H,ofð 1 þ 1Þ 2 H (typically field direction, I essentially flows in the link of 9 10−8 r r0 =R PQCS × ), much lower than ð 1 þ 1Þ H (typically r r0 4 Ω −4 resistance ( 1 þ 1) (typically about ). This gives rise 3 × 10 ) for a single connection. For some measurements, r r0 I to a voltage ð 1 þ 1Þ × PQCS. Because of the edge equi- a third terminal was connected at the top of the QHE cryostat potentiality and knowing that the two-terminal resistance is [dotted line in Fig. 2(a)] to further reduce the correction. In

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I U =R 1 − α 2 I general, PQCS can be written as ð J HÞð Þ¼ The current source is therefore locked to PQCS, and it n K R −1f 1 − α α I Jð J KÞ Jð Þ, where is a small relative correction generates a quantized current PQCG theoretically equal to GI G N =N that is calculated by taking into account the resistance of all PQCS, where ¼ð JK Þ spans 2 orders of magnitude connections (Appendix A4). In our experiments, this above or below the unity gain. The CCC relies on a correction α of no more than 3 × 10−7 is determined with magnetic coupling between windings, and it allows a high −9 an uncertainty of uα ¼ 2.5 × 10 . The validity of this electrical insulation between the PQCS and the external I relationship giving PQCS assumes a perfect equipotentiality circuit connected to the devices under test that provides in the superconducting pads where voltage and current protection from an alteration of the PQCS quantization. G is terminals of the QHRS are connected. The quantized current the main control parameter determining the range of the I can be rewritten as I ≡ n ef ð1 − αÞ, a form output current. The fine programmability of the quantized PQCS PQCS J J I n f which is very similar to the expression of the current current PQCG can be achieved either by changing J or J. generated by the electron pumps, nQef . This reflects that Another option consists in tuning the CCC gain using a P β both ways of realizing the ampere, i.e., from SET devices calibrated current divider that derives a fraction of the N 16 and from the application of Ohm’s law to quantum standards current in a fourth CCC winding ( Div ¼ turns). In this case, the resulting CCC gain can be expressed as [Fig. 1(a)], are theoretically equivalent. It also points out that G N = N βN β I n β ¼½ JK ð þ DivÞ. Here, can be varied over a PQCS corresponds to the circulation of J elementary −5 n range 5 × 10 and is determined with a standard uncer- charges per cycle of the external frequency. Because J −9 f n f tainty uβ ¼ 0.5 × 10 . The accuracy in realizing the gain and J are orders of magnitude larger than Q and P, the PQCS, and consequently the PQCG can generate higher depends on the feedback electronics. To avoid significant currents than SET devices. error caused by the finite value of the amplifier gain of The accuracy of the PQCG also relies on the detection the SQUID electronics, the total ampere.turn value in the and the amplification with gain G of the quantized current CCC is nominally strongly reduced by a fine-tuning of V I the external current source using out as indicated in PQCS while keeping the accuracy. This is achieved using a cryogenic current comparator (CCC) [52] (see Appendix A Fig. 2(a). Moreover, the gain of the feedback loop is set at 3c), which is able to accurately compare currents with a the largest value possible to keep the SQUID locked −11 during on-off switchings of the current (controlled by the relative uncertainty of a few 10 . The CCC accuracy on-off switchings of U ). In these conditions, the relative relies on Ampère’s theorem and the perfect diamagnetism J quantization error of the PQCG, related to the finite open of the superconductive toroidal shield (Meissner effect), in loop gain, is lower than 0.5 × 10−9 (see Appendix A5). which several superconducting windings of a variable number of turns are embedded. Its current sensitivity relies on a dc superconducting quantum interference device B. Noise, current uncertainty, and stability S I (SQUID) detecting the flux generated by the screening The noise of the PQCG current, I, originates from PQCS current circulating on the shield [Fig. 2(a)]. More precisely, and the gain Gβ. It manifests itself in the flux detection by the N two windings of an equal number of turns JK (128 or 129) SQUID of the CCC amplifier. Note that SI can be expressed are inserted in the connections to the QHRS on the low- S =I f 1= n ef γ =N in relative values by ð I Þð Þ¼½ ð J JÞð CCC JKÞ× potential side of the circuit, i.e., the grounded PJVS side, in SϕðfÞ, where SϕðfÞ is the flux noise amplitude density order to detect the sum of the currents in the two windings, detected by the SQUID and γ ¼ 8 μA:turn=ϕ0 is I CCC hence PQCS. It is essential to cancel the leakage current that the flux to ampere.turn sensitivity of the CCC. This could alter the accurate equality of the total currents expression shows that the larger the number of Josephson circulating in the QHRS and in these two windings. This n N junctions J and the number of turns JK, the better the is achieved by placing high- and low-potential cables (high- signal-to-noise ratio. Sϕ results from the SQUID noise insulation R > 1 TΩ resistance) connected to the QHRS S f L SQUIDð Þ, the Johnson-Nyquist noise of the QHRS resis- inside two separated shields, which are then twisted S f tance, and some external noise extð Þ captured by the together and connected to ground. In this way, direct measurement circuit. Note that the noise of the external leakage currents short circuiting the QHRS, the most current source servo-controlled by the SQUID is of no troublesome, are canceled. Other leakage currents are concern in the operation frequency bandwidth (< 1 kHz) redirected to ground. They lead to a relative error on the S f ofqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the SQUID feedback. In these conditions, ϕð Þ¼ detected current, which is negligible, of no more than ðr1 þ S f 2 4k T =R N =γ 2 S f 2 r0 =R ∼ 4 10−12 SQUIDð Þ þ½ð B Þ Hð JK CCCÞ þ extð Þ , where 1Þ L × [51]. A third CCC winding, with number of turns N (chosen between 1 and 4130), is T ¼ 1.3 K is the QHRS temperature. The flux noise density 1=2 connected to an external battery-powered and low-noise generated by the QHRS, of ∼1 μϕ0=Hz , is well below the 1=2 current source servo-controlled by the feedback voltage of base noise (∼10 μϕ0=Hz ) measured by the SQUID I the dc SQUID; it delivers the current PQCG so that the total operating in the bare CCC (Appendix A3c), as reported N I − NI 0 ampere.turn in the CCC is zero ( JK PQCS PQCG ¼ ). in Fig. 2(d) (black curve).

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2G n R K −1 Experimentally, the PQCG involves three quantum measurement to its expected expression β Jð K JÞ × f 1 − α I devices placed in independent cryogenic setups. The Jð Þ. Experimentally, the current PQCG generated by quantum devices are connected together using long the PQCG, operated with PJVS#A (Appendix A3a), is shielded cables made of twisted pairs. Given the high determined by measuring the voltage difference ΔV (using sensitivity of the SQUID to electromagnetic noise, achiev- a EM N31 nanovoltmeter) between the voltage drop at the ing a stable and accurate operation of the PQCG was a terminals of a very stable 100-Ω resistance standard R100, N challenge. To ensure the SQUID stability using JK as R 2 5 10−9 calibrated in terms of K with an uncertainty of . × , large as 129, it was necessary to connect a damping circuit and the reference voltage Vref of a second PJVS (PJVS#ref) to a fifth CCC winding (N 1600) in order to avoid the J D ¼ (Appendix A3a), linked to K and operated synchronously amplification of the current noise in the PQCS loop at the J at the same frequency f [Fig. 3(a)]. For experimental resonance frequency of the CCC. Note that the noise J convenience, the frequencies of both PJVS were kept spectrum SϕðfÞ measured at the output of the SQUID constant, while the voltage balance on the null detector presents a damped resonance at 1.6 kHz, much broader was done by selecting an appropriate number of Josephson than the self-resonance of the bare CCC around 13 kHz junctions for both PJVS and by fine-tuning the CCC gain [Fig. 2(d)]. Its amplitude is in good agreement with the G R β. Then, the experimental procedure consists in finding Johnson-Nyquist noise emitted by the resistor D in the β damping circuit, which is shown as the blue dashed line in the fraction 0 corresponding to equilibrium, i.e., for I Vref=R I Fig. 2(d) (Appendix A6). At frequencies between 0.1 Hz PQCG ¼ J 100.Thismeansthat PQCG is measured and 6 Hz, the noise level remains low and flat, with an through an identification with the reference quantized 1=2 Vref=R amplitude of ∼20 μϕ0=Hz higher than the level in the current, J 100, which is itself perfectly known in terms R K −1 bare CCC, indicating, however, that some external extra of ð K JÞ . The accuracy of the PQCG is then expressed S f ΔI =I I − noise [ extð Þ] couples to the measurement circuit. Below by the relative deviation PQCG PQCG ¼ð PQCG 1=f G I =I I 0.1 Hz, excess noise corresponding to a typical β0 PQCSÞ PQCG between the measured current PQCG frequency dependence of the power spectral density Vref =R G I (¼ J 100) and the current β0 PQCS calculated from S2 SQUID is observed. β0. In practice, β0 is determined from the successive The noise SϕðfÞ manifests itself in the current measurements of two small voltages ΔV1 and ΔV2, obtained uA β β β measurement by a relative standard uncertainty PQCG (type for two settings 1 and 2 chosen above and below 0, A evaluation) that can be evaluated by a statistical analysis of respectively, and the linear relationship β0 ¼ β1 þðβ2 −β1Þ× series of observations (Appendix A2). The other significant ΔV1=ðjΔV1jþjΔV2jÞ (Appendix A8). Moreover, series of contributions to the relative uncertainty of the PQCG, which on-off-on cycles illustrated in Fig. 3(b) areusedtosubtract are evaluated by nonstatistical methods (type B evaluation), voltage offsets and truncate the 1=f noise of the CCC at the 1=τ come from the cable correction (uα) and the current divider repetition frequency 0 of the cycles. The noise of the calibration (uβ) since components coming from frequency, measurements is analyzed with the help of the Allan QHRS, CCC, electronic feedback, and current leakage are deviation [53] (Appendix A9), which allows us to distin- negligible. These contributionsqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi result in a relative standard guish between the different types of noise according to the B 2 2 exponent of its power dependence with time. The efficiency uncertainty u ≃ uα uβ ×N =N (Appendix A7). PQCG þð Div Þ of the 1=f noise rejection procedure is demonstrated by −9 In our experiments, it varies from only 2.5 × 10 (when the Fig. 3(c), which reports the typical time dependence of the 8 4 10−9 σ ΔV =Vref current divider is not used) up to . × (for relative Allan deviation of the voltage ð Þ J [or, N =N ¼ 16). The combined standard uncertainty of σ I =I τ−1=2 Div qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi equivalently, of the current ð PQCGÞ PQCG]. The I uA 2 uB 2 PQCG can then be calculated from PQCG þ PQCG . behavior is typical of a white noise regime, and it legitimates Figure 2(e) shows a series of on-off switchings of the the calculation of experimental standard deviation of the I mean for the 11-cycle time series (792 s duration) to evaluate current PQCG at 1.1 mA amplitude, as recorded by a digital the standard uncertainties (type A evaluation) uΔ and ammeter. The current value was obtained for n ¼ 3073, V1 J u ΔV ΔV N 129 N 4 G 129=4 ΔV2 ,of 1 and 2, respectively. These uncertainties, of JK ¼ , and ¼ ( ¼ ). This figure demon- 2 5 10−8 Vref n 1549 strates the capability of the PQCG to generate large no more than about . × of J for J ¼ ,are u u currents, i.e., in the milliampere range. It also reveals then combined with β1 and β2 (evaluated by type B u β the low-noise level and the stability of the PQCG, notably methods) to determine the standard uncertainty β0 of 0 characterized by the absence of SQUID unlocking at on-off (Appendix A8). The latter is the main contribution to the ΔI =I switchings of the current. measurement uncertainty of PQCG PQCG (AppendixA10). A 10−8 uncertainty is typically achievable for n ¼ 3072 and III. ACCURACY MEASUREMENTS OF PQCG J an experiment duration of 1600 s. ΔI =I The accuracy of the PQCG is determined by measuring Figure 3(d) shows the relative deviation PQCG PQCG as I n ∝ I the generated current PQCG and then comparing this a function of the number Jð PQCSÞ of biased Josephson

041051-6 PRACTICAL QUANTUM REALIZATION OF THE AMPERE … PHYS. REV. X 6, 041051 (2016)

(a) (d)

(b)

(c)

(e)

FIG. 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 KJRK, supplies a calibrated 100-Ω resistor R100, and its voltage is compared to the Vref voltage J 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, β1 (black dots) and β2 (red dots), of the current divider. Note that τ0 ¼ 72 s is the duration of one on-off-on cycle, and τw ¼ 12 s is the waiting time before recording after the current switching. (c) Relative Allan −1=2 deviation calculated from a series of 49 on-off-on cycles of IPQCG plotted as a function of time τðsÞ.Aτ fit shows good agreement below τ ¼ 1000 s and corresponds to a relative Allan deviation of 2.5 × 10−8 for the duration of the time series used in this work, τ 792 ΔI =I n I Series ¼ s. (d) PQCG PQCG as a function of J (or 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 ΔIPQCG=IPQCG (over four hours) obtained (with nJ ¼ 3074, N ¼ 2, NJK ¼ 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.).

I 2 10−8 n junctions at four different amplitudes of PQCG in the mA than × in relative value, whatever the value of J, n 4098 I range. Note that maintaining the output current for different except for J ¼ and PQCG at 1.5 mA. n N n > 3074 J requires varying the number of turns in order to keep Indeed, for values of J , i.e., a current n =N I > 35 μ the ratio J constant. All reported data represent the PQCS A, one observes an increased dispersion of I arithmetic mean value of measurements carried out at the experimental data for PQCG with significant deviations different moments. The data show no significant deviation from theoretical values. These deviations are not clearly within combined (including type A and type B uncertainty understood at the present time. The usual individual contributions) relative standard uncertainties (1 s.d.) of less quantization tests of both PJVS#A and PJVS#ref and of

041051-7 J. BRUN-PICARD et al. PHYS. REV. X 6, 041051 (2016) the QHRS have confirmed that these deviations were not IV. USING PQCG FOR CURRENT TRACEABILITY caused by a lack of voltage quantization of the voltage These high-accuracy measurements have validated the steps nor of the quantum Hall resistance plateau. PQCG as a quantum current standard. However, it remains Nevertheless, the discrepancies increased with time and important to demonstrate that the PQCG, once checked were sensitive to room-temperature cycling of PJVS#A. An using quick quantization criteria, can be used to calibrate a alteration of the perfect equipotentiality in the supercon- commercial digital ammeter (DA) over several current ducting pads of PJVS#A caused by the circulation of the I current might be considered but will need further ranges. This has been realized by replacing the load in PQCS Fig. 2(a) by a precision DA (a HP3458A multimeter— investigation. Appendix A11) with the low-potential input connected to For n ≤ 3074, such accuracy deterioration has also been J ground. Connecting the DA directly to the output of the observed occasionally after a long period of operation PQCG is an extra challenge because of the sensitivity of (several hours or a day), the phenomenon being less quantum devices (in particular, the SQUID) to the environ- pronounced and more rare at low n . However, the J mental noise. This has been possible at the expense of quantization of the current I was always fully restored PQCG shunting the differential input by a 100-nF highly insulated by a room-temperature cycling of PJVS#A, the Josephson PTFE capacitance that short circuits some digital noise array through which I circulates. To illustrate the time PQCS generated by the DA. Prior to the calibration, the adjustment n reproducibility of the current quantization at low J after a procedure recommended by the manufacturer was followed. cycling, Fig. 3(e) shows successive measurements of We then performed current measurements in the DA ΔI =I PQCG PQCG, carried out over four hours. All results ranges from 10 mA down to 1 μA by changing the gain G 10−8 are close to zero within a relative uncertainty of for (N ¼ 128 and N spanning from 1 to 4130) while using the I n 3074 N 129 JK PQCG at 2.2 mA and J ¼ ( JK ¼ ). This figure highest number of Josephson junctions possible when n has ΔI =I J also demonstrates the independence of PQCG PQCG as a to be reduced below 3072, in order to optimize the signal-to- I function of Bias, i.e., the current used to bias the junctions noise ratio. Two quantization criteria have been identified. of PJVS#A, over 0.1 mA from the voltage step center at The first one is the independence of the output current as 2.2 mA. This property is a necessary quantization criterion. a function of the current biasing the Josephson array. Following these considerations, Fig. 1(d) was elaborated Figure 4(a) shows three quantized current steps, which 10−6 I ∼ 1 1 by averaging, for each output current value, the data are flat within a few at Meas . mA, 0.37 mA, n ≤ 3074 reported in Fig. 3(d) obtained at different J .For and 0.18 mA. These current steps were obtained by varying each current value from 0.7 to 2.2 mA, no significant the bias current of Josephson array segments of PJVS#B ΔI =I relative deviation PQCG PQCG is observed considering (Appendix A3a) containing 3072, 1024, and 512 Josephson −8 the combined measurement uncertainty of 1 × 10 (1 s.d.). junctions, respectively. Note that the operating current Moreover, the experimental standard deviation of all data margins are the same as those of the corresponding voltage −9 points over the whole range amounts to only 8 × 10 . steps. The second quantization criterion is the independence ΔI =I n Finally, the weighted mean of PQCG PQCG is equal to of the calibration results obtained with different values of J −9 ð6 6Þ × 10 . These results demonstrate the quantization for the same output current. This is illustrated in Fig. 4(b), K R −1 ≡ e=2 accuracy of the PQCG in terms of ð J × KÞ , which shows the relative deviation, obtained from eight on- −8 within a 10 relative uncertainty in the mA range. off-on cycles over about 15 minutes, between the measured I I I I GI The current value PQCG results not only from the PQCS current Meas and the quantized current PQCG ¼ PQCS (see G N =N I value but also from the amplification gain ¼ JK , Appendix A11 for calculation of PQCG values). In the I which is highly accurate and can span 2 orders of 1-mA range, the same currents PQCG have been generated magnitude above or below the unity gain (N is between by biasing both n (N ¼ 4)andn =2 (N ¼ 2) Josephson N I J J 1 and 4130, JK is fixed). Therefore, PQCG is quantized junctions. The relative deviations are in agreement within the with the same accuracy over a wide range of current values measurement uncertainties (see Appendix A11), which G I accessible by changing , while PQCS remains below confirms that the current generated by the PQCG is 35 μ n ≤ 3074 I I A, i.e., J . This upper limit for PQCS is close to independent of the value of PQCS, provided that it is lower = 35 μ n ≤ 3074 the current value used to bias the GaAs AlGaAs-based than A( J ). QHRS in optimized resistance calibration and thus does not The lowest uncertainty measurements of Fig. 4(b) show restrict the PQCG use. Moreover, the relative current that the DA is accurate and linear within a relative uncer- S =I G N −7 density noise I does not depend on ( JK is fixed) tainty of 5 × 10 , which is better than the manufacturer n I n 3074 but only on J (i.e., PQCS). Considering J ¼ , which specifications (see Appendix A11). The same measurements gives the best signal-to-noise ratio, one concludes that the were performed on the 1-μA range. The results show a I 3 10−6 PQCG can accurately generate currents with a combined significant deviation from PQCG of about × and a relative measurement uncertainty of 10−8 in the whole higher dispersion of the data points, due to the accuracy range from 1 μA up to 10 mA, as illustrated in Fig. 1(b). limitation and bigger instability in this range, as can be

041051-8 PRACTICAL QUANTUM REALIZATION OF THE AMPERE … PHYS. REV. X 6, 041051 (2016)

(a) deduced from the manufacturer specifications. However, it is -5 5x10 important to note that a relative uncertainty of ∼2 × 10−7 is nJ = 3072, PJVS#B (A) G = 128/4 achieved for measurements at the top of both ranges nJ = 1024 Mid I presented in Figs. 4(b) and 4(c).Notethatthisisthecase

)/ n = 512 0 J for all ranges studied in this paper, as is demonstrated in Mid I

- Fig. 4(d), which reports the relative uncertainties of the n 3072 N measurements performed with J ¼ andbyvarying Meas I ( -5x10-5 from 1 up to 4130. Figure 4(d) shows that the current noise 3.2 3.4 3.6 3.8 4.0 4.2 of the PQCG is independent of N and that the measurement

IBias (mA) uncertainty is dominated by the noise of the DA (red data (b) points) (Appendix A11). This result can be expected from -6 3x10 n = 3072, n = 1536, n = 768 the DA specifications and the low noise of the PQCG J J J 10−8

PQCG N = 2 demonstrated in Sec. III (about uncertainty for similar /I

) N = 4 n measurement time and J value). More surprisingly, for 0 n ≤ 1536 J , the PQCG noise appears to overcome the one of PQCG I

- n the DA in the 1-mA range since dividing J byafactorof2

Meas for a given current value doubles the measurement uncer- I ( -3x10-6 tainties (a few 10−7) reported in Fig. 4(b).Thisisnotthecase -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 1 μ n I (mA) in the - A range when decreasing J by a factor of 10 (c) PQCG [Fig. 4(c)]. This can be explained by the noise spectrum of 6x10-6 the PQCG [Fig. 2(d)] and the bandwidth of the measure- nJ = 3072, nJ = 1536, nJ = 304 -6 PQCG ments, which depends on the current ranges, due to the 3x10

/I N = 465 ) N = 4130 presence of the 100-nF capacitance forming a low-pass filter 0 with the input resistance of the DA, with cutoff frequency PQCG I - -6 decreasing at lower-current ranges (Appendix A11). This -3x10

Meas filter prevents the medium-frequency noise from the damp- I ( -6x10-6 ing circuit from overcoming the noise of the DA for the low- -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 current ranges. Hence, to fully benefit from the low noise at I (µA) PQCG low frequencies of the PQCG (as demonstrated in the PQCG (d) 10-5 accuracy measurements) when calibrating the DA in all

DA with PQCG (nJ = 3072) current ranges, improvements of the filtering will be carried DA alone -6 out in the future. Simultaneously, cooling down the damping 10 R resistor D responsible for the Johnson-Nyquist noise will

-7 significantly decrease the PQCG noise at medium frequen- 10 cies and will also be implemented in the future.

Relative uncertainty 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 V. DISCUSSION I (A) PQCG Returning to Ohm’s law, which is the basis for the definition of the resistance unit, we developed a quantum FIG. 4. Calibration of a digital ammeter using the PQCG and identification of quantization criteria. (a) Relative deviation current standard from the quantum Josephson voltage and Hall resistance standards that are combined in an original of IMeas [the current generated by the PQCG and measured by the DA (HP3458A)] to IMid (the current measured at the center of the quantum circuit, with the aim of universality, accuracy, and I n 3072 step) as a function of the biasing current Bias for J ¼ , simplicity [54]. The PQCG reported here is able to generate 1024, and 512 junctions of PJVS#B. The independence of the currents from 1 μA up to 5 mA values, which are quantized I −1 −8 output current as a function of Bias is a first quantization criterion. in terms of ðK × R Þ with a 10 relative standard I I I J K (b) Relative deviation of Meas to PQCG as a function of PQCG in uncertainty. This universal and versatile quantum current the mA range and (c) in the μA range. The agreement of n n =2 standard improves the accuracy of the current sources of 2 the measurements performed using J and J is a second orders of magnitude compared to CMCs. It opens the way quantization criterion. (d) Relative uncertainty of the measured current (blue squares) and of the DA (open red dots) as a function to a renewed metrology of the electrical current, which will also rely on the development of more stable current transfer of IPQCG over four decades of current. The DA noise dominates over that of the PQCG. Error bars are standard uncertainties standards. As a first proof of its impact, we showed that the (1 s.d.). PQCG, after identifying quantization criteria, can be used

041051-9 J. BRUN-PICARD et al. PHYS. REV. X 6, 041051 (2016) to efficiently calibrate a digital ammeter with measurement metrological triangle [43,59], which consists in comparing K R −1 Q uncertainties only limited by the device under test. the ampere realizations in terms of both ð J × KÞ and , Many improvements and extensions of the PQCG can be is an important and long-awaited experiment. It indeed leads R K Q further considered. First, one can expect a noise reduction, to the direct measurement of the product K × J × , typically by a factor of 10, by increasing the number of theoretically equal to 2. Validating this equality with a ampere.turns in the CCC, which can be achieved by a larger measurement uncertainty down to 10−8 would strengthen the N number of turns JK of the detection windings (up to 1600) confidence in the description, in terms of h and e only, of the I and also by a higher current PQCS (for example, by constants involved in the three solid-state quantum physics n increasing J values while preserving the accuracy). In phenomena. In this perspective, the PQCG used as a any case, the damping circuit of the CCC resonances reference and the quantum ammeter built on the PQCS should be adapted and refined. In this work, the multiple could be used to accurately measure SET-based current R K −1 connection of the QHRS was successfully implemented sources [19,20,60] in terms of ð K × JÞ ,inamoredirect using different cable configurations (Appendix A4), thanks way than in previous experiments [61]. to a correct evaluation of the cable correction in the PQCG expression. Nevertheless, the implementation of a complete ACKNOWLEDGMENTS triple connection of the QHRS will make the cable We wish to acknowledge F. Lafont for fruitful discussions correction negligible and therefore further simplify the in the early stage of the experiment design, D. Estève and Y. PQCG use. From all of these improvements, the target 10−9 De Wilde for critical reading and comments, and R. Behr for uncertainty of should be reached. In addition, the useful advice in using Josephson arrays. This research was availability of graphene-based quantum resistance stan- partly supported by the project 15SIB08 e-SI-Amp funded dards operating in relaxed experimental conditions [37] by the European Metrology Programme for Innovation and should allow the implementation of the quantum voltage, Research (EMPIR). The EMPIR initiative is co-funded by resistance, and current standards, as well as their combi- the European Union’s Horizon 2020 research and initiative nation, in a unique, compact, cryogen-free setup. This programme and the EMPIR participating states. would constitute a major step towards the realization of a universal and practical quantum generator or multimeter. APPENDIX A: METHODS More generally, the principle of using the PQCS as a reference for building the PQCG is seminal and can be 1. Volt, ohm, and ampere representations exploited for other experiments or instruments [51].For The uncertainties of 4 × 10−7 [39] and 1 × 10−7 [40] on instance, a quantum current generator working in the ac K R the determinations of J and K in SI units, respectively, do regime can be developed using pulse-driven Josephson not allow us to benefit from the high reproducibility of the standards [55,56], ac QHRS [57], and current transformers Josephson and quantum Hall effects for the traceability of [58]. This perspective cannot be considered with single- the volt and the ohm. To overcome this limitation, conven- K R electron sources in the present state-of-the-art case. Very tional values for J and K were recommended in 1990 by accurate and sensitive comparisons of quantum Hall the Comité International des Poids et Mesures (CIPM) [39] resistance can be performed by opposing, by means of for the traceability of the voltage and the resistance in the CCC, the PQCS currents obtained from two different calibration certificates based on the implementation of QHRS polarized by the same Josephson voltage reference. these quantum effects. These constants are exact and given K 483597 9 = R 25812 807 Ω This novel comparison technique could be used to test the by J−90 ¼ . GHz V and K−90 ¼ . . K R K K universality of the QHE from the integer to the fractional They are related to J and K through J ¼ K−90× 1 4 10−7 R R 1 1 10−7 regime. Finally, a quantum ammeter [51] can be realized by ð × Þ and K ¼ K−90ð × Þ. The volt- K R directly comparing the current delivered by an external age and the resistance traceable to J−90 and K−90 give source to the PQCS using the CCC. representations of the volt and the ohm and not realization More fundamentally, the PQCG can implement the of the unit volt and the unit ohm (SI). Note that the current planned new definition of the ampere with a target uncer- realized by application of Ohm’s law from the representa- 10−8 e K R tainty of since it is linked to the elementary charge . tions of the volt and the ohm based on J−90 and K−90 Our work therefore provides an essential piece of the revised gives a representation of the ampere, not spoiled by the K R SI founded on constants of physics. This achievement will uncertainties of J and K. rely on the adoption of the fundamental relationships for the The aim of the new SI, which notably adopts exact quantum Hall and Josephson effects in the future SI, which is values for h and e and a new definition of the ampere from e K h=2e also necessary for the realization of the kilogram from the , is to solve this problem. If the relationships J ¼ h R h=e2 Planck constant using the watt balance experiment [7]. and K ¼ are adopted, the constants involved in the Indeed, it relies on comparing the mechanical power with the Josephson effect and the quantum Hall effect will no longer electrical power, calibrated itself from the quantum voltage have uncertainties. As a consequence, the Josephson and resistance standards. In this context, the closure of the voltage standard and the quantum Hall resistance standard

041051-10 PRACTICAL QUANTUM REALIZATION OF THE AMPERE … PHYS. REV. X 6, 041051 (2016) will become SI realizations of the volt and ohm. The TABLE I. PJVS device characteristics. combination of these two quantum effects, as proposed in this paper, will lead to a SI realization of the ampere. PJVS#A PJVS#B PJVS#ref IC (mA) 1.4 1.6 3.5 ΔI @n 0 2. Uncertainty vocabulary Bias (mA) ¼ 0.6 1.1 3.1 ΔIBias (mA) @n ¼ 1 0.6 0.8 1.0 This section gives the definitions from the Guide to the IMid (mA) 2.4 3.65 4.4 n Expression of Uncertainty in Measurement (GUM) [62] of J 7168 8191 1920 f the metrological terms used in the main text. J (GHz) 70 70.111 70 P Uncertainty (of measurement): parameter, associated RF (mW) 30 10 65 with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed b. QHRS device to the measurand. Standard uncertainty: uncertainty of the result of a The Hall resistance standard, based on an eight-terminal = measurement expressed as a standard deviation. Hall bar made of a GaAs AlGaAs semiconductor hetero- Type A evaluation (of uncertainty): method of evaluation structure (LEP514), was produced at the Laboratoire of uncertainty by the statistical analysis of series of Electronique de Philips [63]. Figure 2(c) shows the Hall R R observations. resistance H and the longitudinal resistance per square xx B Type B evaluation (of uncertainty): method of evaluation measured as a function of . The metrological quality of of uncertainty by means other than the statistical analysis of the sample was checked following the technical guidelines B 10 8 T 1 3 series of observations. [49].At ¼ . T, ¼ . K, and for currents below 60 μ R R =2 Combined standard uncertainty: standard uncertainty of A, H is perfectly quantized at K within a relative 1 10−10 the result of a measurement when that result is obtained uncertainty of × , and the two-dimensional electron R ≤ 10 μΩ from the values of a number of other quantities, equal to the gas is dissipationless ( xx ) [37]. The resistance of 0 1 Ω positive square root of a sum of terms, the terms being the the eight contacts is lower than . . variances or covariances of these other quantities weighted according to how the measurement result varies with c. CCC device changes in these quantities. The cryogenic adder amplifier is based on a CCC that is usually used in a bridge performing accurate resistance 3. Quantum devices comparisons. More precisely, it is made of 15 windings with a. PJVS devices 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 The three PJVS used in this work (PJVS#A, PJVS#B, 3 μϕ = 1=2 = = = = Quantum Design Inc. dc SQUID having a - 0 Hz base and PJVS#ref) are based on 1-V Nb Al AlOx Al white noise [64]. Figure 2(d) shows the noise spectral density AlOx=Al=Nb Josephson junction series arrays fabricated Sϕ measured at the output of the SQUID as a function of the at the Physikalisch-Technische Bundesanstalt (PTB) frequency for the CCC alone (no winding connected). The [35,47]. PJVS#A (data shown in Figs. 2 and 3) and 1=2 bottom white noise level is around 10 μϕ0=Hz , indicating PJVS#B (data shown in Fig. 4) were used in the PQCS. that some external noise is captured. Considering the CCC PJVS#ref was used for the opposition voltage Vref in the J ampere.turn gainofG ¼ 8 μA:turn=ϕ0, this corresponds accuracy measurements of the PQCG in Fig. 3. The CCC to a 80-pA:turn=Hz1=2 current sensitivity. At frequencies Josephson arrays are subdivided into 14 smaller array above 10 kHz, intrinsic electrical resonances of the CCC due segments. PJVS#A follows a sequence 256=512=3072= to its high inductance are observable. From a few kilohertz 2048=1024=128=1=1=2=4=8=16=32=64; PJVS#B and down to 6 Hz, Fig. 2(d) displays peaks that are caused by PJVS#ref follow the sequence 4096=2048=1024=512= mechanical and acoustic resonances. At lower frequencies, 256=128=1=1=2=4=8=16=32=64. For PJVS#ref, only a Sϕ features white noise down to 0.1 Hz, and for even lower few segments were used, corresponding to a maximum 1=f1=2 1=f number of junctions of 1920. Table I sums up the character- frequencies, it rises according to because of the I ΔI SQUID noise. istics of the arrays, where C is the critical current, Bias the current amplitude of the Shapiro steps, and P the RF α microwave power applied at the array. Note that a 4. Cable corrections Josephson junction is missing in PJVS#ref but acts as a The use of multiple series connections to the QHRS perfect short circuit [47]. The I-V characteristics for the drastically reduces the positive correction to the quantized total number of junctions of the arrays have been system- Hall resistance caused by the resistance of these connec- atically checked before and after the current measurements. tions [50,51]. It results in a negative relative correction α n ef The microwave synthesizer is locked to a 10-MHz refer- added to the quantized current J J, leading to I n ef 1 − α ence, delivered by a GPS rubidium frequency standard. PQCS ¼ J Jð Þ. Considering the link resistances

041051-11 J. BRUN-PICARD et al. PHYS. REV. X 6, 041051 (2016) r r0 r r r0 r T 300 1, 1, 2, 3, 3, and 4, as indicated in Fig. 2(a), one D ¼ K leads to the circulationffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of a noise current of δi f jC 2πfp4k T R = 1 R jC 2πf− calculates, using a Ricketts and Kemeny model [65] of the density ð Þ¼f½ D B D D ½ þ D D α r r0 r =R2 r r0 r =R2 L C 2πf 2 C 100 Hall bar, ¼ f½ð 1 þ 1Þ 2 Hg þ f½ð 3 þ 3Þ 4 Hg D Dð Þ g, where D ¼ nF is the capacitance of L 70 for the double connection scheme, and α is reduced to α ¼ the damping circuit and D ¼ mH is the inductance of r r =R2 r r =R2 the winding of N ¼ 1600 turns. This results in a flux noise f½ð 1Þ 2 Hg þ f½ð 3Þ 4 Hg if a third terminal is con- D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SD f N C 2πf 4k T R = nected at the top of the QHE setup. For the double densitypffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of modulus j ϕ ð Þj ¼ f½ D D B D D α 1 94 10−7 2 2 2 connection scheme, we determined ¼ . × for ½γ ð1 − L C ð2πfÞ Þ þðR C 2πfÞ g character- N 128 α 2 99 10−7 N 129 CCC D D D D JK ¼ and ¼ . × for JK ¼ . With a ized by two main frequency ranges only because α 1 16 10−7 N 128 pffiffiffiffiffiffiffiffiffiffiffiffi third terminal connected, ¼ . × for JK ¼ 1=R C is close to 1= L C : (i) for f ≪ 1= α 2 20 10−7 N 129 D D D D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and ¼ . × for JK ¼ . Since the total resis- ð2πR C Þ, jSDðfÞj ¼ ½ðN C 2πf 4k T R Þ=γ ; mΩ D D ϕ pffiffiffiffiffiffiffiffiffiffiffiffi D D B pDffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD CCC tance of each connection is measured with a 50- f ≫ 1= 2π L C SD f N 4k T R = −9 (ii) for ð D DÞ, j ϕ ð Þj ¼ ½ð D B D DÞ uncertainty, α is determined with a uα ¼ 2.5 × 10 relative γ L 2πf SD f standard uncertainty. CCC D . Figure 2(d) shows that the j ϕ ð Þj fitting function (blue dashed line) adjusts the experimental 5. Impact of the feedback settings detected noise (red) very well in the frequency range from on the PQCG accuracy 10 Hz up to 10 kHz. This extra noise manifests itself differently in the measurements according to the frequency To keep the PQCG stable, even during the on-off bandwidth of the detector. V switching of the current, the control voltage out was well adjusted so that the nominal ampere.turn value remained close to zero, and the feedback gain of the SQUID was 7. Type B standard uncertainty budget of the PQCG reduced so that the closed-loop gain (CLG) increased from Table II presents the different contributions to the type B 0.75 V=ϕ0 (the value in internal feedback mode operation) uncertainty of the PQCG current that were evaluated: the to 4.2 V=ϕ0 for N values from 1 up to 465. For N ¼ 4130, cable correction α (Appendix A4), the feedback electronics the CLG was further increased up to 8.4 V=ϕ0. One can (Appendix A5), the CCC accuracy (Appendix A1c), the expect a quantization error of the PQCG resulting from the QHRS accuracy (Appendix A1b), current leakage finite amplification gain (FAG) of the SQUID electronics (Sec. II), the frequency accuracy, and the calibration of (in V=ϕ0) that leads to a nonzero ampere.turn value in the current divider fraction β (Sec. II). The two main the CCC. The relative current error is given by contributions come from the cable correction and the ΔI =I − = ΔadjI =I PQCG PQCG ¼ CLG ðCLG þ FAGÞ PQCG PQCG, current divider calibration. In principle, the PQCG should ΔadjI G I − Iadj be implemented without using the current divider, as for the where PQCG ¼ β PQCS PQCG is the deviation ammeter calibration, adjusting the current with the number between the target quantized current GβI and the PQCS of biased junctions n or the frequency only. In this case, Iadj ΔadjI J adjustment current PQCG. Note that PQCG can be the total type B relative uncertainty, essentially caused by determined from the SQUID output voltage, which is equal the cable correction, is evaluated to about 2.5 × 10−9. NΔadjI =γ to PQCG ×CLG CCC. Since the SQUID electronics Let us note that this contribution should be canceled amplifier is based on an integrator, we get FAG ∝ 1=f, by the implementation of a triple connection of the where f is the measurement frequency. The error caused by QHRS, as planned in the future. In the accuracy test, the FAG is therefore nulled in the dc limit. To estimate the the current divider is used, and it gives a contribution to error on the quantized current generated by the PQCG in 2 10−9 N =N 1 the type B uncertainty from × ( Div ¼ )to normal operation, we performed accuracy measurements 8 × 10−9 (N =N ¼ 4). using the usual on-off switching frequency while inten- Div Iadj tionally shifting PQCG from adjustment. It turns out that ΔadjI =I 10−3 TABLE II. Type B standard uncertainty budget of PQCG. increasing PQCG PQCG up to leads to relative −8 errors ΔI =I amounting to 2.3 1.3 × 10 . u 10−9 uB 10−9 PQCG PQCG ð Þ Contribution ð Þ Sensitivity PQCG ð Þ This corresponds to FAG ∼ 4.3 × 104 ×CLG∼ 1.8 × Cable correction uα ¼ 2.5 1 2.5 105 =ϕ ΔadjI =I −9 −9 V 0. Since PQCG PQCG is maintained below Electronic feedback <0.5 × 10 1 <0.5 × 10 2 × 10−5 in accurate operations of the PQCG, we deduce CCC accuracy <1 × 10−10 1 <1 × 10−10 I <1 10−10 <1 10−10 a relative error on the current generated, PQCG of QHRS × 1 × <1 10−11 <1 10−11 ð4.6 2.6Þ × 10−10, i.e., lower than 10−9. Current leakage × 1 × Frequency <1 × 10−11 1 <1 × 10−11 Current divider (CD) uβ ¼ 0.5 N =N 0.5 × N =N 6. Noise generated by the damping circuit Div Div Total (without CD) 2.5 The Johnson-Nyquist noise of the R ¼ 1.1 kΩ resis- D Total (with CD) 8.4 (N ¼ 1) tance of the damping circuit placed at room temperature

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M−1 8. Determination of β0 realizing equilibrium 1 X 2 2 σqðτ ¼ mτ0Þ¼ ðq¯ i j 1 − q¯ i jÞ ; For the accuracy measurements of PQCG, the current 2ðM − 1Þ þ þ þ I i¼1 divider was used to adjust PQCG so that the voltage balance Vref=R is realized. At equilibrium, J 100 can then be compared where the M samples measured at τ ¼ τ0 are extended by N = N β N I β to its theoretical value ½ JK ð þ 0 DivÞ PQCS, where 0 reflection at both ends to form a virtual array q¯ . The original I N q¯ q¯ i 1 M is the fraction of PQCG injected by the divider in the Div- data are in the center, where i ¼ i for ¼ to ,andthe turn winding. In practice, to simplify the calibration extended data for j ¼ 1 to M − 1 are equal to q¯ 1−j ¼ q¯ j and ΔV1 q¯ q¯ measurement chain, two sets of nonzero voltages Mþj ¼ Mþ1−j. Let us remark that the total Allan variance and ΔV2 obtained for two fractions β1 and β2, respectively, can be calculated for an analysis time τ up to half the total below and above β0 are measured [see Fig. 3(b)]. Here, β0 measurement time. is given by β0 ¼β1 þðβ2 −β1Þ×ΔV1=ðjΔV1jþjΔV2jÞ. Figure 3(c) shows the relative total Allan deviation −6 Depending of the measurement, β0 is between a few 10 σ I =I or σ ΔV =Vref, which was calculated from −4 2 ð PQCGÞ PQCG ð Þ J and a few 10 . Its uncertainty uβ is given by uβ ¼ I 0 0 a series of 49 on-off-on cycles of PQCG using the software u2 ΔV 2 u2 ΔV 2 = ΔV ΔV 2 β −β 2 −1=2 ½ð β1 j 2j þ β2 j 1j Þ ðj 1jþj 2jÞ þ½ð 2 1Þ × STABLE 32. The τ dependence confirms the white noise ΔV 2u2 ΔV 2u2 = ΔV ΔV 4 u ðj 2j ΔV1 þj 1j ΔV2 Þ ðj 1jþj 2jÞ , where ΔV1 of data and legitimates the calculation of the experimental and uΔV are the experimental standard deviations of the deviation of the mean for the 11-cycle time series to 2 u mean (type A evaluation) calculated from the measurements evaluate the standard uncertainties ΔV (type A evaluation). ΔV ΔV u u of 1 and 2, and β1 and β2 are the calibration standard uncertainties of β1 and β2. Taking into account that jΔV1j ≃ 10. Measurement standard uncertainty of jΔV2j and that β1 and β2 are strongly correlated quantities, ΔIPQCG=IPQCG in accuracy quantization u 0 5 10−9 the first term contributes to β0 by . × . tests of the PQCG For the experiments consisting in testing the accuracy 9. Allan deviation of the PQCG, the relative combined standard uncer- u ΔI =I The Allan variance [53] is the two-sample variance that tainty ð PQCGÞ PQCG of the relative deviation of relies on three hypotheses: The distribution law of data is the generated current to its theoretical expectation, ΔI =I normal, the power spectral density can be decomposed into PQCG PQCG, is calculated from the combination, powers of the frequency, and the time between data is using the propagation law of uncertainties [62],from u constant, equal to τ0, without dead time. The advantage of the cable correction α (Appendix A4), the current u this variance over the classical variance is that it converges divider fraction β0 (Appendix A8) realizing equilib- for most of the commonly encountered kinds of noise, rium (main contribution that combined type A and whereas the classical variance does not always converge to type B components), and the 100-Ω resistor uR100= R 2 5 10−9 u ΔI =I ≃ a finite value. q100ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ . × .Theresultis ð PQCGÞ PQCG Considering a measurement performed during a time u2 u N =N 2 uR =R 2 α þð β0 × Div Þ þð 100 100Þ . All uncertain- T ¼ Mτ0, where M is the total number of samples, and q¯ i the ith average of the samples calculated over an analysis ties reported in the figures of Secs. I and III are time τ ¼ mτ0, where m can be varied up to M=2, the Allan combined standard uncertainties (1 s.d.). variance is defined as 11. Calibration of the DA: Accuracy and 1 MX−1 σ2 τ mτ q¯ − q¯ 2 measurement uncertainties qð ¼ 0Þ¼2 M − 1 ð iþ1 iÞ . ð Þ i¼1 The DA is a HP3458A multimeter. Prior to calibrations, the DA has been adjusted by using a 10-kΩ resistor The Allan variance allows us to differentiate noise types standard and a 10-V Zener voltage standard calibrated in according to the exponent of its power dependence with terms of R and K , respectively. The manufacturer −1 K J time. As an example, white noise manifests itself by a τ specifications of the apparatus concerning the accuracy dependence. This corresponds to an Allan deviation, of current measurements are the following: (10-ppm read- −1=2 σqðτ ¼ mτ0Þ, characterized by a τ dependence. In this ing + 4-ppm range) in the 1-mA range and (10-ppm reading case, the Allan deviation is an unbiased estimator of the true + 40-ppm range) in the 1 μA range. The 100-nF capaci- deviation. tance forms a low-pass filter, with the input resistance of the f In this paper, we have used the total Allan variance DA resulting in cutoff frequencies C depending on the f 16 (TOTAVAR) calculated with the software STABLE 32 Version current range of the DA: C ¼ kHz for the 1-mA range 100 Ω f 35 1 μ 1.5. The total Allan variance is similar to the Allan variance ( - input resistance) and C ¼ Hz for A and has the same expected value, but it offers improved (45.2-kΩ input resistance). The current values measured I I confidence at long averaging times. It is defined as Meas by the DA are compared to PQCG values that are

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K R calculated using J−90 and K−90 (Appendix A1). In [13] H. Pothier, P. Lafarge, C. Urbina, D. Estève, and M. H. Figs. 4(b)–4(d), the error bars correspond to relative type Devoret, Single-Electron Pump Based on Charging Effects, A uncertainties evaluated by experimental standard devia- Eur. Phys. Lett. 17, 249 (1992). tions of the mean calculated from eight on-off-on cycles [14] M. W. Keller, J. M. Martinis, N. M. Zimmerman, and (about 15 minutes). In these experiments, the relative type A. H. Steinbach, Accuracy of Electron Counting Using a B uncertainty contributions of the PQCG, reduced to 7-Junction Electron Pump, Appl. Phys. Lett. 69, 1804 (1996). uB u 2 5 10−9 PQCG ¼ α ¼ . × , are not included because they [15] M. W. Keller, A. L. Eichenberger, J. M. Martinis, and N. M. are negligible compared to the type A uncertainty con- Zimmermann, A Capacitance Standard Based on Counting tribution. In Fig. 4(d), the relative uncertainties (red bars) of Electrons, Science 285, 1706 (1999). the DA are calculated from the dispersion of the uncer- [16] M. W. Keller, N. M. Zimmermann, and A. L. Eichenberger, tainties of several measurements performed with the DA Uncertainty Budget for the NIST Electron Counting entries connected to the 100-nF capacitance only and using Capacitance Standard ECCS-1, Metrologia 44, 505 the same protocol based on eight on-off-on cycles (about (2007). 15 minutes). [17] B. Camarota, H. Scherer, M. W. Keller, S. V. Lotkov, G. Willenberg, and J. Ahlers, Electron Counting Capacitance Standard with an Improved Five-Junction R-Pump, Metrologia 49, 8 (2012). [18] B. Kaestner and V. Kashcheyevs, Non-adiabatic Quantized [1] BIPM, The International System of Units, 8th ed. (2006), Charge Pumping with Tunable-Barrier Quantum Dots: A www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf. Review of Current Progress, Rep. Prog. Phys. 78, 103901 [2] CIPM, Resolution 1 of the 25th CGPM (2014), http://www (2015). .bipm.org/en/CGPM/db/25/1/. [19] S. P. Giblin, M. Kataoka, J. D. Fletcher, P. See, T. J. B. M. [3] I. M. Mills, P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. Janssen, J. P. Griffiths, G. A. C. Jones, I. Farrer, and D. A. Williams, Redefinition of the Kilogram: A Decision Whose Ritchie, Towards a Quantum Representation of the Ampere Time Has Come, Metrologia 42, 71 (2005). Using Single Electron Pumps, Nat. Commun. 3, 930 [4] I. M. Mills, P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. Williams, Redefinition of the Kilogram, Ampere, Kelvin and (2012). Mole: A Proposed Approach to Implementing CIPM [20] F. Stein et al., Validation of a Quantized-Current Source Recommendation 1, Metrologia 43, 227 (2006). with 0.2 ppm Uncertainty, Appl. Phys. Lett. 107, 103501 [5] M. J. T. Milton, J. M. Wiliams, and S. J. Bennett, (2015). Modernizing the SI: Towards an Improved, Accessible [21] BIPM, Calibration and Measurement Capabilities Electric- and Enduring System, Metrologia 44, 356 (2007). ity and Magnetism: DC Current (Key Comparison Data- [6] BIPM, Draft of the Ninth SI Brochure (2015), p. 4, http:// base, 2016), http://kcdb.bipm.org/AppendiXC/country_list www.bipm.org/utils/common/pdf/si‑brochure‑draft‑2016 .asp?Sservice=EM/DC.3.2. .pdf. [22] B. D. Josephson, Possible New Effects in Superconductive [7] B. P. Kibble, A Measurement of the Gyromagnetic Ratio of Tunnelling, Phys. Lett. 1, 251 (1962). the Proton by the Strong Field Method, Atomic Masses and [23] K. V. Klitzing, G. Dorda, and M. Pepper, New Method Fundamental Constants (Plenum Press, New York, 1976), for High-Accuracy Determination of the Fine-Structure Vol. 5, pp. 545–551. Constant Based on Quantized Hall Resistance, Phys. [8] M. Stock, Watt Balance Experiments for the Determination Rev. Lett. 45, 494 (1980). of the Planck Constant and the Redefinition of the Kilo- [24] S. Shapiro, Josephson Currents in Superconducting Tun- gram, Metrologia 50, R1 (2013). neling: The Effect of Microwaves and Other Observations, [9] The ampere is the constant current which, if maintained in Phys. Rev. Lett. 11, 80 (1963). two straight parallel conductors of infinite length, of [25] M. Buttiker, Absence of Backscattering in the Quantum Hall negligible circular cross section, and placed 1 meter apart Effect in Multiprobe Conductors, Phys. Rev. B 38, 9375 in vacuum, would produce a force equal to 2.10−7 Newton (1988). per meter of length between these conductors. See BIPM, [26] F. Bloch, Josephson Effect in a Superconducting Ring, The International System of Units (SI), 8th ed. (2006), Phys. Rev. B 2, 109 (1970). p. 113, www.bipm.org/utils/common/pdf/si_brochure_8_en [27] R. B. Laughlin, Quantized Hall Conductivity in Two Di- .pdf. mensions, Phys. Rev. B 23, 5632 (1981). [10] BIPM, The International System of Units (SI), Appendix 2, [28] D. J. Thouless, Topological Interpretations of Quantum (2006), http://www.bipm.org/en/publications/mises‑en‑ Hall Conductance, J. Math. Phys. 35, 5362 (1994). pratique/electrical‑units.html. [29] A. A. Penin, Quantum Hall Effect in Quantum Electrody- [11] J. P. Pekola, O. P. Saira, V. Maisi, A. Kemppinen, M. namics, Phys. Rev. B 79, 113303 (2009); 81, 089902(E) Möttönen, Y. A. Pashkin, and D. Averin, Single-Electron (2010). Current Sources: Toward a Refined Definition of the [30] A. A. Penin, Measuring Vacuum Polarization with Ampere, Rev. Mod. Phys. 85, 1421 (2013). Josephson Junctions, Phys. Rev. Lett. 104, 097003 (2010). [12] H. Grabert and M. H. Devoret, Single Charge Tunneling [31] B. N. Taylor, P. J. Mohr, and D. B. Newell, CODATA Coulomb Blockade Phenomena in Nanostructures (Plenum Recommended Values of the Fundamental Physical Con- Press, New York, 1991). stants: 2010, Rev. Mod. Phys. 84, 1527 (2012).

041051-14 PRACTICAL QUANTUM REALIZATION OF THE AMPERE … PHYS. REV. X 6, 041051 (2016)

[32] P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATA [49] F. Delahaye and B. Jeckelmann, Revised Technical Guide- Recommended Values of the Fundamental Physical lines for Reliable DC Measurements of the Quantized Hall Constants: 2014, Rev. Mod. Phys. 88 (2016). Resistance, Metrologia 40, 217 (2003). [33] J. S. Tsai, A. K. Jain, and J. E. Lukens, High-Precision Test [50] F. Delahaye, Series and Parallel Connection of Multi- of the Universality of the Josephson Voltage-Frequency terminal Quantum Hall-Effect Devices, J. Appl. Phys. 73, Relation, Phys. Rev. Lett. 51, 316 (1983). 7914 (1993). [34] A. K. Jain, J. E. Lukens, and J. S Tsai, Test for Relativistic [51] W. Poirier, F. Lafont, S. Djordjevic, F. Schopfer, and L. Gravitational Effects on Charged Particles, Phys. Rev. Lett. Devoille, A Programmable Quantum Current Standard 58, 1165 (1987). from the Josephson and the Quantum Hall Effects, J. Appl. [35] I. Y. Krasnopolin, R. Behr, and J. Niemeyer, Highly Precise Phys. 115, 044509 (2014). Comparison of Nb=Al=AlOx=Al=AlOx=Al=Nb Josephson [52] I. K. Harvey, A Precise Low Temperature DC Ratio Trans- Junction Arrays Using a SQUID as a Null Detector, former, Rev. Sci. Instrum. 43, 1626 (1972). Supercond. Sci. Technol. 15, 1034 (2002). [53] T. Witt, Allan Variances and Spectral Densities for DC [36] F. Schopfer and W. Poirier, Quantum Resistance Standard Voltage Measurements with Polarity Reversals, IEEE Accuracy Close to the Zero-Dissipation State, J. Appl. Phys. Trans. Instrum. Meas. 54, 550 (2005). 114, 064508 (2013). [54] B. P. Kibble, In Metrology Simpler Is Better, IEEE Instrum. [37] R. Ribeiro-Palau, F. Lafont, J. Brun-Picard, D. Kazazis, A. Meas. Mag. 13, 43 (2010). Michon, F. Cheynis, O. Couturaud, C. Consejo, B. Jouault, [55] S. P. Benz, S. B. Waltman, A. E. Fox, P. D. Dresselhaus, A. W. Poirier, and F. Schopfer, Quantum Hall Resistance Rfenacht, J. M. Underwood, L. A. Howe, R. E. Schwall, and Standard in Graphene Devices Under Relaxed Experimen- C. J. Burroughs, One-Volt Josephson Arbitrary Waveform tal Conditions, Nat. Nanotechnol. 10, 965 (2015). Synthesizer, IEEE Trans. Appl. Supercond. 25, 1300108 [38] T. J. B. M. Janssen, N. E. Fletcher, R. Goebel, J. M. (2015). Williams, A. Tzalenchuk, R. Yakimova, S. Kubatkin, S. [56] O. F. Kieler, R. Behr, R. Wendisch, S. Bauer, L. Palafox, and Lara-Avila, and V. I. Falko, Graphene, Universality of the J. Kohlmann, Towards a 1 V Josephson Arbitrary Wave- Quantum Hall Effect and Redefinition of the SI System, form Synthesizer, IEEE Trans. Appl. Supercond. 25, New J. Phys. 13, 093026 (2011). 1400305 (2015). [39] T. Quinn, News from the BIPM, Metrologia 26, 69 (1989). [57] F. J. Ahlers, B. Jeannneret, F. Overney, J. Schurr, and B. M. [40] CIPM, Procès-Verbaux des Séances du Comité International Wood, Compendium for Precise AC Measurements of the des Poids et Mesures, 89th meeting, 101 (2000), http://www Quantized Hall Resistance, Metrologia 46, R1 (2009). .bipm.org/en/committees/cipm/cipm‑2000.html. [58] B. P. Kibble and G. H. Rayner, Coaxial AC Bridges (Adam [41] D. Drung, C. Krause, U. Becker, H. Scherer, and F. J. Hilger, Bristol, 1984). Ahlers, Ultrastable Low-Noise Current Amplifier: A Novel [59] H. Scherer and B. Camarota, Quantum Metrology Triangle Device for Measuring Small Electric Currents with High Experiments: A Status Review, Meas. Sci. Technol. 23, Accuracy, Rev. Sci. Instrum. 86, 024703 (2015). 124010 (2012). [42] D. Drung, C. Krause, S. P. Giblin, S. Djordjevic, F. [60] X. Jehl, B. Voisin, T. Charron, P. Clapera, S. Ray, B. Roche, Piquemal, O. Séron, F. Rengnez, M. Götz, E. Pesel, and M. Sanquer, S. Djordjevic, L. Devoille, R. Wacquez, and M. H. Scherer, Validation of the Ultrastable Low-Noise Vinet, Hybrid Metal-Semiconductor Electron Pump for Current Amplifier as Travelling Standard for Small Direct Quantum Metrology, Phys. Rev. X 3, 021012 (2013). Currents, Metrologia 52, 756 (2015). [61] L. Devoille, N. Feltin, B. Steck, B. Chenaud, S. Sassine, S. [43] K. K. Likharev and A. B. Zorin, Theory of the Bloch-Wave Djordevic, O. Séron, and F. Piquemal, Quantum Metrologi- Oscillations in Small Josephson Junctions, J. Low Temp. cal Triangle Experiment at LNE: Measurements on a Phys. 59, 347 (1985). Three-Junction R-Pump Using a 20000:1 Winding Ratio [44] R. Behr, O. Kieler, J. Kohlmann, F. Müller, and L. Palafox, Cryogenic Current Comparator, Meas. Sci. Technol. 23, Development and Metrological Applications of Josephson 124011 (2012). Arrays at PTB, Meas. Sci. Technol. 23, 124002 (2012). [62] BIPM, Evaluation of Measurement Data—Guide to the [45] B. Jeckelmann and B. Jeanneret, The Quantum Hall Effect Expression of Uncertainty in Measurement (JCGM, 2008), as an Electrical Resistance Standard, Rep. Prog. Phys. 64, http://www.bipm.org/utils/common/documents/jcgm/JCGM_ 1603 (2001). 100_2008_E.pdf. [46] W. Poirier and F. Schopfer, Resistance Metrology Based on [63] F. Piquemal, G. Genevès, F. Delahaye, J. P. André, J. N. the Quantum Hall Effect, Eur. Phys. J. Spec. Top. 172, 207 Patillon, and P. Frijlink, Report on a Joint BIPM-Euromet (2009). Project for the Fabrication of QHE Samples by the LEP, [47] F. Mueller, R. Behr, L. Palafox, J. Kohlmann, R. Wendisch, IEEE Trans. Instrum. Meas. 42, 264 (1993). and I. Krasnopolin, Improved 10 V SINIS Series Arrays for [64] F. Lafont et al., Quantum Hall Resistance Standards from Applications in AC Voltage Metrology, IEEE Trans. Appl. Graphene Grown by Chemical Vapour Deposition on Supercond. 17, 649 (2007). Silicon Carbide, Nat. Commun. 6, 6806 (2015). [48] R. Behr et al., Analysis of Different Measurement Setups for [65] B. W. Ricketts and P. C. Kemeny, Quantum Hall a Programmable Josephson Voltage Standard, IEEE Trans. Effect Devices as Circuit Elements, J. Phys. D 21, 483 Instrum. Meas. 52, 524 (2003). (1988).

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