Determination of fundamental constants in quantum electrical metrology. François Piquemal

Transcription

1 Determination of fundamental constants in quantum electrical metrology François Piquemal

2 I) Introduction OUTLINE 1) Towards a quantum SI 2) Quantum electrical metrology II) Determination of R K α and K J2 R K h 1) Thompson Lampard calculable capacitor 2) Watt balance III) Quantum Metrological Triangle - determination of Q X e 1) Uncertainty thresholds 2) Determination of the charge quantum IV) QMT experimental set-up 1) Electron counting capacitance standard: Q = CV 2) Cryogenic current comparator: U = RI 3) Overall CCC based set-up at LNE and first results V) Conclusion

3 The present SI the website: bipm.org «The seven base units provide the reference used to define all the measurement units of the SI. As science advances, and methods of measurement are refine, their definitions have to be revised.» 1960: Definition of the meter based on the radiation wavelength between 2p 10 and 5d 5 of krypton /68: Definition of the second «Mise en pratique»: Cesium clock 1983: New definition of the meter by fixing the speed of light in vacuum «Mise en pratique»: Laser 1889: PtIr International Prototype of kilogram The only remaining artefact used to define a base unit in the SI! The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. It follows that the mass of the international prototype of the kilogram, m(k) is always 1 kg exactly Stability of the unit of mass?

4 Evolution of national prototypes over 120 years M (µg) / /92 Possible drift of IPK: per century!

5 The present SI 1948: Definition of the ampere by fixing the magnetic constant µ 0 The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to newton per meter of length. It follows that the magnetic constant µ 0, also known as the permeability of free space is 4π H/m exactly Biot-Savart Law : v B = µ 0 4π v I r ˆ dl B = µ I 0 r 2 2πd v F = I ( ) Lorentz Force Law : µ 0 = 4π dl v B v F dl = IB = µ 0 I2 2πd π 10-7 N/A (N/m) µ 0 /2π (A 2 /m) This definition of the ampere is difficult to be realised with enough accuracy. Today, its realization derives from the volt and the ohm ampere balance: the best accuracy was parts in 10 6

6 Towards a Quantum SI: a new SI well matched with the present knowledge of physics Fixing h and e and keep floating M(IPK) and µ 0 But, considering that the uncertainties from measurements were not at the required level, the Comité International des Poids et Mesures (CIPM ) called for Preparative steps towards new definitions of the kilogram, the ampere, the kelvin and the mole in terms of fundamental constants for possible adoption by the 24 th CGPM in 2011 and recommended that NMIs Constant Present SI CODATA 2006 ppb uncertainty 2011 quantum SI CODATA 2006 ppb uncertainty M (IPK) exact 50 h 50 exact N A 50 exact e 25 exact K J (2e/h) 25 exact R K (h/e 2 ) 0.68 exact F 25 exact m e m p u, m u 50 (amu) 1.4 (amu)... should pursue vigorously their work presently underway aimed at providing the best possible values of the fundamental constants needed for the redefinitions now being considered

7 Chain of SI realisations of electrical units and materialization meter, kilogram, second µ 0 c 2 watt farad ohm henry volt siemens coulomb ampere

8 Chain of SI realisations of electrical units and materialization Equivalence between electrical and mechanical powers Watt balance watt meter, kilogram, second µ 0 c 2 farad Thompson-Lampard theorem Calculable capacitor 10 pf ohm henry volt siemens coulomb ampere

9 Cross calculable capacitance standard Theorem of A. Thompson and D. Lampard (1956): For a cylindrical system of 4 isolated electrodes of infinite length and placed in vacuum, exp(-π γ 13 /ε 0 ) + exp(-π γ 24 /ε 0 ) = 1 In the case of a perfect symmetry with identical γ ij γ 13 = γ 24 = γ = (ε 0 ln2)/π = pf/m C = γ L 1 4 γ 13 γ LNE, five-electrodes capacitor 2000 From the farad to the ohm METRE Calculable capacitor QHE QHE R K 2012 a) d) f) e) b) c) 1 pf 10 pf Capacitance bridges 100 pf 1000 pf 10 nf SECOND R C ω = 1 ω (rd/s) R 100 Ω Calculable CCC resistor DC D C Quadrature bridge AC 10 kω

10 Chain of SI realisations of electrical units and materialization Equivalence between electrical and mechanical powers Watt balance watt meter, kilogram, second µ 0 c 2 farad Thompson-Lampard theorem Calculable capacitor 10 pf ohm henry volt siemens coulomb ampere

11 Watt balance: monitoring the kilogram electrically B F z I Static P m v B Dynamic ε speed v LNE balance F z = mg = -I Φ/ z ε = - Φ/ t = - Φ/ z v Equivalence between mechanical and electrical power F z v = εi mgv = εv/r

12 Chain of SI realisations of electrical units and materialization Equivalence between electrical and mechanical powers Watt balance watt meter, kilogram, second µ 0 c 2 farad Thompson-Lampard theorem Calculable capacitor 10 pf ohm henry volt siemens coulomb ampere

13 Chain of SI realisations of electrical units and materialization Equivalence between electrical and mechanical powers Watt balance watt meter, kilogram, second µ 0 c 2 farad Thompson-Lampard theorem Calculable capacitor 10 pf ± 10 V Josephson effect volt ohm siemens Quantum Hall effect henry 100 Ω 1 ΜΩ coulomb ampere

14 Quantum Hall resistance standards Klaus von Klitzing, 1980 At low temperature and under high magnetic field, the Hall resistance of twodimensional electron gas (2DEG) exhibits plateaux centred on quantized values: R H (i) = h/ie 2 = R K /i, where i is an integer, R K the von Klitzing constant. LEP 514 Hall bar sample GaAs/AlGaAs heterostructure 2DEG E 0 Undoped GaAs Ionized donors n doped AlGaAs E 1 E 2 + E Potential well 2DEG thickness nm n-type GaAs, N D # cm n-type Al x Ga 1-x As, N D # cm -3, x = 28% 40 Al x Ga 1-x As, x = 28% non dopée 14.5 undoped GaAs 600 Substrat GaAs

15 Josephson voltage standards Brian Josephson 1962 Quantum effects occurring between two superconducting electrodes separated by a small region where the superconductivity is weakened: thin insulating film By applying microwave radiation, voltage steps occurred at values V = n f h/2e = n f / K J where n is an integer, f a frequency and K J the Josephson constant Microwave radiation I n = 1 Oxyde layer 2 nm w L V = (h/2e) f mv I (ma) V = (h/2e) f 145 µv at f = 70 GHz Tunnel type Josephson junction 1 V or 10 V Josephson standards 2000 or > junctions in series array Array of 2000 Josephson junctions in series

16 Quantum standards: universality and high reproducibility (1/2) Unique representation of the volt and the ohm: The international comparisons of complete JE and QHE systems show a high level of consistency: from a few to a few Comparison of 10 V JAVS (r Lab-rBIPM)/r BIPM (10-9 ) BIPM.EM-K12 Bilateral comparison of QHRS with BIPM BNM-LCIE METAS PTB NPL RH R H (2) / 100 Ω 10k-100 kω / 100 Ω Ω / 1 Ω NIST 99-04

17 Quantum standards: universality and high reproducibility (2/2) Universality tests U J / f and i R H (i) independent of materials at a level of In microbridge planar Nb/Cu/Nb junction J.S. Tsai et al Si-MOSFET GaAs/AlGaAs A. Hartland et al 1991, B. Jeckelmann et al 1995 few 10-9 Graphene GaAs/AlGaAs A. Tzalenchuk et al I s L s V 1 V 2 I 1 I 2 L 1 L hf 2 These remarkable results from comparison and universality test do not prove that the phenomenological constants are exactly 2e/h and h/e 2 but they strengthen our confidence in the equalities K J = 2e/h and R K = h/e 2 in addition to strong theoretical arguments. If corrections exist, they will be probably of fundamental nature.

18 Chain of SI realisations of electrical units and materialization Equivalence between electrical and mechanical powers Watt balance watt meter, kilogram, second µ 0 c 2 farad Thompson-Lampard theorem Calculable capacitor 10 pf ± 10 V Josephson effect volt ohm siemens Quantum Hall effect henry 100 Ω 1 ΜΩ coulomb ampere

19 Watt balance: monitoring the kilogram electrically B F z I Static P m v B Dynamic ε speed v LNE balance F z = mg = -I Φ/ z ε = - Φ/ t = - Φ/ z v Equivalence between mechanical and electrical power F z v = εi mgv = εv/r

20 Watt balance: monitoring the kilogram electrically B F z I Static P m v B Dynamic ε speed v LNE balance F z = mg = -I Φ/ z ε = - Φ/ t = - Φ/ z v Equivalence between mechanical and electrical power F z v = εi mgv = εv/r - ε and V in terms of Josephson effect: ε = n 1 f 1 /K J, V = n 2 f 2 /K J - R in terms of quantum Hall effect: R = R K /i 4mgv h =, assuming K J = 2e/h and R K = h/e A 2 m(ipk) = h A 4gv A mgv = K 2 J R K where A = n 1 f 1 n 2 f 2 i Monitoring the kilogram in term of h

21 I) Introduction OUTLINE 1) Towards a quantum SI 2) Quantum electrical metrology II) Determination of R K α and K J2 R K h 1) Thompson Lampard calculable capacitor 2) Watt balance III) Quantum Metrological Triangle - determination of Q X e 1) Uncertainty thresholds 2) Determination of the charge quantum IV) QMT experimental set-up 1) Electron counting capacitance standard: Q = CV 2) Cryogenic current comparator: U = RI 3) Overall CCC based set-up at LNE and first results V) Conclusion

22 R K = h/e 2 and K J = 2e/h Strong theoretical arguments supporting von Klitzing and Josephson relations: R K = h/e 2 and K J = 2e/h Today, from validation tests: R K = h/e 2 (1 + [24 ± 18] 10-9 ) K J = 2e/h (1 + [238 ± 720] 10 9 ) The large uncertainty on K J comes from the present discrepant values of Γ p h-90 (lo) and V m (Si) Calc. capacitor Watt balance P. Mohr, B.N. Taylor, D.B. Newell, Rev of Mod Phys., vol.80, 2008

23 Thompson Lampard calculable capacitor Douglas Geoffrey Lampard von Klitzing constant R K & fine structure constant α Klaus von Klitzing

24 Calculable capacitor & determination of R K α (1/8) Theorem of A. Thompson and D. Lampard (1956): For a cylindrical system of 4 isolated electrodes (by means of infinitesimally small gaps) of infinite length and placed in vacuum, 1 4 γ 13 γ exp(-π γ 13 /ε 0 ) + exp(-π γ 24 /ε 0 ) = 1 and γ 13 = γ 24 = γ = (ε 0 ln2)/π if perfect symmetry The T.L. theorem has been extended to a system of five electrodes by N. Elnekavé (BNM-1973) In such a configuration if one connect successively two adjacent electrodes, the five electrodes system is equivalent to five different four electrodes T.L. capacitors by circular permutation. - Redundant values - Information on the degree of perfection Electrode axis are on the vertices of a regular pentagon if 5 π γ = ln ε γ 25 γ 14 γ γ γ = γ = γ 13 = pf/m 5 1

25 Calculable capacitor & determination of R K α (2/8) The challenges to be tackled for the «mise en pratique» of the theorem: 1) Infinite length of the electrodes The theorem is valid only for a uniform field repartition (out of extremity effect) 1 3 and 4 Thompson finds out a solution consisting in introducing a movable guard electrode in the inter-electrode cavity C 1 = L 1 γ + δ extr(1) 1 3 and 4 C 2 = L 2 γ + δ extr(2) 1 L 1 3 and 4 L 2 C = C 1 C 2 = (L 1 L 2 ) γ if δ extr(1) = δ extr(2)

26 α=0 0, 1 0, = = = n z z C C n n k n δ α π α = = = z z C C k δ π α α z z Calculable capacitor & determination of R K α (3/8) 2) Null thickness inter-electrodes insulator It can be demonstrated that the error δ due to the non null thickness of the insulator, varies as a function of angle, α, between the electrodes tangents, at their contact to the insulator (of thickness z) z α

27 Calculable capacitor & determination of R K α (4/8) 3) Cylindrical system A set of electrodes with straightness < 100 nm over the useful length (σ < ) dedicated apparatus for the measurement of the cylindricality defect Electrodes in vertical position, with a positionning better than 100 nm Lateral shifting of the movable guard < 50 nm (σ < ) Coaxiality between cross section & movable axis < 0.1 µrad Main frame Movable guard assembly Capacitor structure Calculable capacitor in vacuum chamber

28 Present situation (5/8) The overall mechanical elements are realised and assembled Setting elements, measurement ring, Lampard structure, vacuum chamber in-situ control system of the position of the electrodes Measurement ring Remains to do: The moving guard system Lampard Structure assembling of the of the whole mechanical parts with dummy electrodes

29 Calculable capacitor & determination of R K α (6/8) 4) Electrical measurement chain ω wagner tare: 1,25 pf (for LVb) quad. AT2 1 pf 10 phase 7 or 0 D Mobile guard a b a : in b : out 1pF inj. -1 LVb or 10 pf Capacitance bridges Calculable capacitor 1 pf 10 pf 100 pf 1000 pf 10 nf METRE Calculable resistor R C ω = 1 ω (rd/s) R D C QHE QHE R K 100 Ω CCC DC AC 10 kω SECOND Quadrature bridge - New transfer resistors 10 kω, 20 kω and 40 kω with very low parameters frequency coeff. < per khz; temp coeff = ± C -1, drift /day. - New calculable Haddad type resistors 1 kω - New two stage transformers have been built, compared to the previous ones: increased maximum operating voltage and a lower ratio corrections. Calibration of transformers in progress

30 Fabrication of a new set of electrodes with straightness < 100 nm (7/8) Fabrication of a new set of electrodes The new set of electrodes is made of amagnetic stainless steel Dimensions : diameter 75.5 mm, length : 450 mm After grinding the cylindricality defect is about 1.5 µm To reduce this defect to a value close to 100 nm, the cylinders are manually lapped and polished Fabrication of specific lapping-tool The best results were obtained with grinding-tool made of corundum (abrasive) with shellac and rosin for bonding. The tool is shaped to fit the cylinders.

31 µ m Measurements of the straightness and the parallelism of the generating lines of electrodes with σ < 25 nm (8/8) - The measurement of the straightness and parallelism of the generating lines of the electrodes with a measuring machine specifically built - Currently improved for a target uncertainty < 25 nm Portal structure After the grinding stage ±1 µm Mobile frame Sensors support ring Electrode under test Reference cylinders After 30 hours of lapping µ m ±100 nm The five electrodes are within ± 100 nm Further polishing to reach < ± 50 nm

32 Present status on calculable capacitor in the world R K at one part in NMIA: - 2 sets of 4 electrodes have been fabricated for BIPM and NRC - 2 other sets of electrodes are being made for NMIA and NIM - Fabrication techniques suitable for meeting the requirements of cylindricity deviation < 100 nm over the useful length (50 mm) - BIPM: - The assembling of the calculable capacitor is in progress - LNE: - Mechanical structure has been designed - New set of 5 electrodes is being finished - Assembling of the calculable capacitor in progress - NIST: - On-going development of short calculable capacitor - Manufacturing cylinders (40 mm diam. and 160 mm in length) with straightness deviation less than 100 nm - Design of Fabry Perot Interferometry to measure short displacement of the moving guard - NRC: - 4 electrodes from NMIA but the assembling has not started (WB!)

33 Watt balance based on moving coil Brian Kibble Planck constant h Max Planck

34 Watt balance & determination of K J2 R K h (1/8) NPL NIST METAS LNE BIPM

35 Watt balance & determination of K J2 R K h (2/8) A watt balance is composed of: Requirements - A magnetic source to create B Permanent or superconducting coil Flux magnitude to be maximised - A force sensor & suspension (incl. the moving coil) Coil geometry & structure optimized in terms of A.turns, orientation of wires, size, rigidity & weight - A device to displace the moving coil during the dynamic phase - A device to measure or control its position and velocity - Electrical standards to measure U and I To move at a known velocity along a defined trajectory Carefully designed interferometer Programmable Josephson junction array & resistance calibrated against QHE - A test mass Weak magnetic suscept., a very good surf. stability, a high density, a very good homogeneity, no porosity, cavities - A gravimeter Absolute gravimeter, g at a ref. point near the WB - Alignment devices for the magnetic field and the moving coil (magnet-coil assembly) Laser interferometry

36 Watt balance & determination of K J2 R K h (3/8) Linear Motor Guiding stage Flexible strips Moving part Position detector Beam Interferometer and position sensors Suspending Moving coil Magnetic circuit LNE watt balance Static Phase: M = 500g (1 kg, 250 g) I = 5mA R = 200 Ω Dynamic phase: Total range: 80 mm Useful range: 40 mm v = 2 mm/s, V = 1 V Coil: 640 turns, Φ = 266 mm Magnet: B = 0.94 T gap e = 9 mm, h = 90 mm working in vacuum

37 Watt balance & determination of K J2 R K h (4/8) Linear Motor Guiding stage Flexible strips Moving part Position detector Beam Interferometer and position sensors Suspending Moving coil Magnetic circuit

38 Watt balance & determination of K J2 R K h (5/8) Linear Motor Guiding stage Flexible strips Moving part Position detector Beam Interferometer and position sensors Suspending Moving coil Magnetic circuit

39 Watt balance & determination of K J2 R K h (6/8) Linear Motor Guiding stage Flexible strips Moving part Position detector Beam Interferometer and position sensors Suspending Moving coil Magnetic circuit

40 Watt balance & determination of K J2 R K h (7/8) Linear Motor Guiding stage Flexible strips Moving part Position detector Beam Interferometer and position sensors Suspending Moving coil Magnetic circuit Fe Magnets (Sm 2 Co 17 ) Fe + core FeCo

41 Watt balance & determination of K J2 R K h (8/8) Linear Motor Guiding stage Flexible strips Moving part Position detector Beam Interferometer and position sensors Suspending Absolute gravimeter based on cold atoms Moving coil Magnetic circuit δg(µgal) D-MOT - Raman 1 Raman 2 3 D-MOT 10 7 atoms in 50 ms T atoms ~ 2µK Temps (heures) Fluctuations of g recorded by the gravimeter between 5 and 10 October Détection a et b π/2 π π/2 Interféromètre σ g (µgal) 10 1 Miroir Φ = k eff.g.t 2 S. Merlet; PhD thesis; Observatoire de Paris; 5 July Temps (s)

42 Present status on watt balances in the world value of h at one part in 10 8 NPL: -WB transferred to NRC - Problem found on the structure to suspend the mass standard - Proposition to increase the uncertainty NIST: - Some improvements - Pb with 1 kg CuAu mass of NPL (due to its behaviour) - Pb in 2008 on 500 g mass measurements, OK on 500 g Si mass of NPL (ongoing studies) - Uncertainty unchanged - Start of a new project METAS:- Results in Start a new WB BIPM: - Work in progress, first results in 2010 LNE: NIM: MSL: - On-going assembly - Joule balance under development - Shaping a project on a WB based on a pressure balance INRIM:- Investigation on an electromagnetic pendulum

43 Determination of N A based on silicon sphere Vm(Si) and comparison with present WB values of h No N A value proposed at CPEM 2010 Reduced uncertainty for natural Si Deviation between IRMM and PTB on enriched Si (probably due to the determination of the isotopic ratio) The available values are not official! u = 10-5 [(h/10-34 J.s) ]x10 5

44 I) Introduction OUTLINE 1) Towards a quantum SI 2) Quantum electrical metrology II) Determination of R K α and K J2 R K h 1) Thompson Lampard calculable capacitor 2) Watt balance III) Quantum Metrological Triangle - determination of Q X e 1) Uncertainty thresholds 2) Determination of the charge quantum IV) QMT experimental set-up 1) Electron counting capacitance standard: Q = CV 2) Cryogenic current comparator: U = RI 3) Overall CCC based set-up at LNE and first results V) Conclusion

45 Chain of SI realisations of electrical units and materialization meter, kilogram, second µ 0 c 2 Single electron tunneling Watt balance watt farad Calculable capacitor 10 pf ± 10 V Josephson effect volt ohm siemens Quantum Hall effect henry 100 Ω 1 ΜΩ coulomb < 1 na Single electron tunneling ampere

46 Single electron tunneling: towards quantum current standard Electron box U C J + - Tunnel junction n Metallic island q = Q G - Q = n e Coulomb Blockade of the tunneling events when n - 1/2 < C G U/e < n + 1/2 Average box charge [e] <n> 4.0 Experiment Theory Coulomb staircase C g V g [e] C G U/e C G Gate capacitor - Thermal fluctuations of n are negligible if k B T << e 2 /C i - Quantum fluctuations of n negligible when R j >> R K The wave function of electron in excess on the island is well localised 3-junctions electron pump Two modulation signals transistor at frequency f phase-shifted by Φ =π/2 I +V/2 n 1 U 1 C 1 n 2 U 2 C 2 -V/2 I = e f For f = 100 MHz I = 16 pa AnimPompe.exe e (0,1) Minimum energy states of of the the island pump As as a function of of U 1 and 1 and U 2 2 Unwanted effects and error corrections -e -e C 2 U 2 P N e (0,0) (1,0) C 1 U 1 (n 1,n 2 )

47 The quantum metrological triangle (QMT) experiment By means of SET devices such as electron pumps, hybrid turnstile, a current standard with quantized amplitude is available : I = e f The experiment originally proposed by K. Likharev and A. Zorin in 1985 consists of applying Ohm s law, U = R I directly to the quantities issued from ac JE, QHE and SET. f U =n (h/2e) f I = e f SET pump I = e f R H V J J f J U Josephson effect Quantum Hall effect SET effect I I = (e 2 /h)u

48 Closing the triangle: first way U = R I As for JE, K J = (2e/h) JE and QHE, R K = (h/e 2 ) QHE, one can define a phenomenological constant: Q X = e SET Dimensionless product to be measured JE U =n K J -1 f J R K K J Q X = 2 if : Exactness?! R K = h/e 2 K J = 2e/h Q X = e U R K K J Q X = (n i/g) ( f J / f SET ) DC R I G = N P /N S gain of the CCC QHE R = R K /i SET I = Q X f SET

49 The QMT experiment Another approach to close the triangle is to apply Q = CV Charging a capacitor electron per electron with a pump measuring the voltage drop with Josephson voltage standard, calibrating the capacitance by means of QHR standard Q = Ne SET pump controlled by an electrometer C V J J f J Electron counting capacitance standard (ECCS)

50 Closing the triangle: second way Q = C U ECCS: Charging a capacitor electron per electron by a SET pump and measuring the voltage drop with Josephson standard C ECCS = K J Q X /[(n/n) f J ] C ECCS compared to C X calibrated by means of QHR standard via quadrature bridge (RCω = 1) JE U =n K J -1 f J U R K K J Q X = (n/n)(c ECCS /C X ) ( f J / f q ) i Quadrature bridge AC C Q QHE C = i/(2πf q R K ) SET N Q X

51 Aims of the QMT (1/3) To confirm with a very high accuracy that these three different effects of condensed matter physics give the free space values of constants 2e/h, h/e 2 and e. The ultimate target uncertainty is one part in 10 8 If there is no deviation, our confidence on the three phenomena to provide us with 2e/h, h/e 2 and e will be strengthened. If deviation occurs, some other works both experimental and theoretical will have to be done.

52 Different uncertainty thresholds for closing the QMT First critical test of validity for SET: Uncertainty of 1 ppm Neither the JE nor the QHE is questionable at that uncertainty level recently completed by NIST with σ = 9.2 parts in 10 7 Second uncertainty level lies between a few parts in 10 7 and 2 parts in 10 8 Resulting information will be mainly relevant for the JE and the SET

53 Aims of the QMT (2/3) To confirm with a very high accuracy that these three effects of condensed matter physics give the free space values of constants 2e/h, h/e 2 and e. The ultimate target uncertainty is one part in 10 8 If there is no deviation, our confidence on the three phenomena to provide us with 2e/h, h/e 2 and e will be strengthened. If deviation occurs, some other works both experimental and theoretical will have to be done. To determine the elementary charge e or, in other words, the charge quantum brought by the SET devices Indeed, one can show that by combining the three experiments QMT, calculable capacitor and watt balance, a determination of e involved in SET devices without assuming that R K = h/e 2 and K J = 2e/h is possible.

54 Determination of the charge quantum (1/3) 1897: J.J. Thomson Discovery of the electron : R.A. Millikan Charged oil droplets method e/e ref direct values ajdusted values u r F = N A e 1928 : E. Bäcklin The value e is derived from known value of Faraday constant and the determination of N A by x-ray method > 1940, e is derived from a complex calculation and is no more related to an experiment. Least-Squares Adjustments of the Constants (R.T. Birge, 1929) A precise knowledge of only a limited number of constants and relations was needed to evaluate the values of a large group with sufficient accuracy 1952: Dumond and Cohen Increasing number of data 1973: 1 st LSA from CODATA e ref = C

55 Determination of the charge quantum (2/3) In the framework of the LSA by the CODATA (>1973), e is no more an adjustable constant and its value is obtained from the relation giving α: µ 0 c α = e = 2αh 2h/e 2 µ 0 c CODATA 2006: e = C and σ = α from a e, h/m and R K (Calc. capacitor + QHE) h via K J2 R K (WB + QHE + JE) In fact, values of e can be derived directly from experiment, but with a larger uncertainty

56 Determination of the charge quantum (3/3) Two direct values independent of the QHE and the JE See N. Feltin F. Piquemal, EPJST 172, (e - e CODATA 2006 ) x 10 6 Direct value LSA&CODATA e direct e = [α 3 A r (e)m u /(µ 0 R N A )]1/2 - A r (e): 2006 CODATA - M u : = 10 3 kg.mol 1 exactly. - R : 2006 CODATA - α : a e and h/m Cs, Rb - N A N A = V m (Si)/(8 1/2 d ) e CODATA 2006 = (40) C Comparisons: Q X e, R K h/e 2, K J 2e/h Q X Q X value from QMT at NIST with ECCS σ =

57 Aims of the QMT (3/3) To confirm with a very high accuracy that these three effects of condensed matter physics give the free space values of constants 2e/h, h/e 2 and e. The ultimate target uncertainty is one part in 10 8 If there is no deviation, our confidence on the three phenomena to provide us with 2e/h, h/e 2 and e will be strengthened. If deviation occurs, some other works both experimental and theoretical will have to be done. To determine the elementary charge e or, in other words, the charge quantum brought by the SET devices Last but not least, to establish whether the SET can achieve a high level quantization and lead to a natural mise en pratique of the future definition of the ampere New quantum electrical standards

58 SET standards New definition of the ampere based on fixed value of e: The ampere is the electrical current scaled such that the elementary charge is exactly C An alternative (but explicit unit definition instead of explicit constant definition): The ampere is the electrical current equivalent to the flow of exactly 1/( ) elementary charges per second. It follows that this definition fixes the elementary charge as exactly A s New quantum electrical standards: - Quantum current standard for the calibration domain < 1 na Direct traceability for electrical current instead of the present route from JAVS&QHRS Q = Ne - Natural capacitance standard Quantum standard like JAVS & QHR C = Q / V (2e 2 /h) f electron pump C V

59 Electron counting capacitance standard E. Williams et al, J. of NIST, ) Pumping electrons on and off cryogenic capacitor Cryogenic Capacitor C cryo 1.8 pf N2 N1 V FB island 1 ff V V FB (V) V e Time (s) M. Keller et al, Science 1999 C J 0.2 ff Each V gives a value of C cryo = Ne/ V 2) Comparing capacitances V G1 V G2 electrometer 1.6 khz C cryo = 1.8 pf V G5 7-J pump 10 pf stray capacitance V 1 ND N2 N1 island T < 100 mk V G6 V 2 C ref = 10 pf

60 1) NIST Present status of Q = CV triangle experiments First comparison of the C cryo values by counting electrons and by a capacitance bridge M. Keller et al, Science, vol 285, 1999 Q X /e 1 = (-0.09 ± 0.92) 10-6 M. Keller et al, Metrologia ppm 2) PTB Progress on electron counting capacitance standard

61 U = RI triangle experiments based on CCC Principle of the CCC Harvey 1972 δφ few µφ 0 /Hz 1/2 (typ.) I 1 I 2 I B=0 (a) Application of the Ampère s law : (a) B.dl = 0 = µ 0 (I 1 +I 2 -I) I = I 1 +I 2 SQUID X X Superconducting shield Pick-up coil Φ = f(i) I I ampere - turn balance : I = 0 N 2 I 2 = N 1 I 1 G = I 2 /I 1 = N 1 /N 2 I 2 I 1 Β = f(i) Supercurrent Secondary I = N 1 I 1 -N 2 I 2 Primary winding winding N N 1 I 2 I 2 1

62 Quantum metrological triangle set-up at LNE 10 MHz rubidium reference Waveform generator DC SQUID 70 GHz source π/2 δφ Flux Transformer INT FB R FB V out EXT FB 30 mk SET pump N 1 turns CCC N 2 turns Calibrated 10 kω V R V d 4.2 K 4.2 K INT FB: non accurate mode EXT FB: accurate mode V J DAC based Bias current source Program. JAVS

63 Results on 3-junctions R pump (SQUID in int FB) 3 junctions R pump fabricated by the PTB A. Zorin et al., JAP2000 S. Lotkhov et al, APL, vol.78, 2001 source R Cr (25-50 kω) drain V g1 V g2 gates Measurement of current steps up to 100 MHz! B. Steck et al, Metrologia 2008 I (pa) 40 fa

64 Measurements, Fall 2009 Spring 2010 (SQUID in ext FB) Q X = (N 2 /N 1 )(n J f J /K J V d )/(f SET R) MHz e/e = (Q X - e CODATA )/e CODATA S. Sassine et al. CPEM 10 (Qx-e codata )/e codata 3.0E E E E MHz MHz Systematic deviation reproducible at a level of part in 10 6 Failures of the electron pump and the experimental set-up 1.0E bias voltage, V b (µv) N Q x (C) E E E E E E E E E frequency (MHz) Frequency dependence! Qx (C) PN P frequency (MHz)

65 V- Conclusion Advances in quantum electrical metrology: strongly contribute in the approaching reform of the SI - by enhancing our confidence on QHE, JE and SET to provide free space values of h/e 2, 2e/h and e - by improving knowledge on fundamental constants, in particular α, h and e towards a fully quantum SI intrinsically in correlation with advances in nanotechnology At fundamental level, the metrology changes: from the usual macroscopic measurand (emf, resistance ) to the detection, the control and the counting of quanta (Φ 0, h/e 2, e ) A growing link/interaction between atomic and condensed matter physics (e.g. graphene)

66 Acknowledgment The Quantum Electrical Metrology group at the LNE Single electron tunneling: Quantum Hall effect: Josephson effect: Calculable capacitor: L. Devoille, S. Djordjevic, N. Feltin, O. Séron, S. Sassine T. Charron W. Poirier, F. Schopfer, D. Leprat, J. Guignard S. Djordjevic, O. Séron L. Dupont, P. Gournay, O. Thévenot The French watt balance project team: LNE, INM, SYRTE G. Genevès, F. Bielsa, P. Espel, T. Madec, G. Mann, P-A. Meury, P. Juncar, P. Pinot, S. Macé, Z. Silvestri, F. Pereira Dos Santos, D. Holleville, A. Landragin, A. Clairon, S. Merlet Collaboration CEA-Saclay & Grenoble, LPN/CNRS Marcoussis, PTB Braunschweig&Berlin + On-going European projects JOSY, Nanospin, REUNIAM, ULQHE, emass

67 Thanks a lot for your attention! Спасибо за ваше внимание! (Spacibo za vaché vnimanié)