quantum metrology triangle · the present si 1948: definition of the ampere by fixing the magnetic...

Quantum Metrology TriangleFrançois Piquemal

The Quantum Electrical Metrology group at the LNE

SET&CCC: Laurent Devoille, Nicolas Feltin, Sami Sassine

QHE&CCC: Wilfrid Poirier, Félicien Schopfer, Laetitia SoukkassianDominique Leprat

JE: Sophie Djordjevic, Olivier Séron, Hervé Ndilimabaka

CalculableCapacitor: Pierre Gournay, Olivier Thévenot, Ludovic Lahousse

For the quantum metrological triangle:

- CollaborationCEA-Saclay & Grenoble, LPN/CNRS Marcoussis, PTB Braunschweig&Berlin

REUNIAM project: METAS, MIKES, VSL, NPL, PTB

Bonn, 4 November 2009

OUTLINE

I) Introduction1) Towards a quantum SI2) Quantum electrical metrology

II) Aims of the Quantum Metrological Triangle1) Uncertainty thresholds2) Determination of the charge quantum3) SET standards

III) QMT via Electron Counting Capacitance Standard: Q = CV

IV) QMT Experimental set-up based on the CCC: U = RI1) CCC: Cryogenic Current Comparator2) Overall set-up at LNE and first results

V) Conclusion

⇒ the website: bipm.org

The present SI

« 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. »

1967/68: Definition of the second« Mise en pratique »: Cesium clock

1983: Definition of the meter by fixingthe speed of light in vacuum« Mise en pratique »: Laser

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 ?

International Prototype of kilogram

1889:PtIr

The only remaining artefact usedto define a base unit in the SI !

The present SI

1948: Definition of the ampereby fixing the magnetic constant µ0

“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 2 × 10-7 newton per meter of length.”It follows that the magnetic constant µ0, also known as the permeability of free space is 4π × H/m exactly

v B = μ0

4π

v I × ˆ r r2

⌠ ⌡ ⎮ dl → B =

μ0I2πd

Biot-Savart Law :

2×10-7(N/m)

v F = I d

v l ×

v B ( )∫ →

Fdl

= IB =μ0I2

2πdLorentz Force Law : → µ0 = 4π×10-7 N/A2

µ0/2π (A2/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 106

Towards a Quantum SI: a new SI well matched with the present knowledge of physics

But, considering that the uncertainties from measurements were not at the required level,the Comité International des Poids et Mesures (CIPM - 2005) 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 24th CGPM in 2011”

and recommended that NMIs

“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”

Constant Present SI CODATA 2006ppb uncertainty

2011 quantum SI CODATA 2006 ppb uncertainty

M (IPK) exact 50 h 50 exact NA 50 exact e 25 exact KJ (2e/h) 25 exact RK (h/e2) 0.68 exact F 25 exact me 50 1.4 mp 50 1.4 u, mu 50 (amu) 1.4 (amu) ...

Fixing h and eand keep floating M(IPK) and µ0

Chain of SI realisations of

electrical units and materialization

farad

µ0c2

siemens

coulomb

ampere

volt

ohm henry

watt

meter, kilogram, second

Watt balance

Equivalence between electrical and mechanical powers

10 pF

Thompson-Lampard theorem

Calculablecapacitor

± 10 V Josephsoneffect

Quantum Hall effect

100 Ω – 1 ΜΩ

Quantum standards: universality and high reproducibility (1/3)

The international comparisons of complete JE and QHE systems show a high level of consistency: from a few 10-11 to a few 10-9.

Comparison of 10 V JAVS

-10

-8

-6

-4

-2

0

2

4

6

8

10

( rLa

b-r B

IPM)/ r

BIP

M (1

0-9)

100-RH

10k-100

100-1

BNM-LCIE 93-12

METAS 94-11

PTB 95-10

NPL 97-12

NIST 99-04

RH(2) / 100 Ω

10 kΩ / 100 Ω

100 Ω / 1 Ω

BIPM.EM-K12

Bilateral comparison of QHRS with BIPM

Unique representation of the volt and the ohm:


Universality tests

These remarkable results from comparison and universality test do not prove that the phenomenological constants are exactly 2e/hand h/e2 but they strengthen our confidence in the equalities KJ = 2e/h and RK = h/e2 in addition to strong theoretical arguments.

If corrections exist, they will be probably of fundamental nature.

UJ / f and i× RH (i) independent of materials

at a level of 2.10-16

In microbridge ⇔ planar Nb/Cu/Nb junctionJ.S. Tsai et al – 1983

1.10-10 Si-MOSFET ⇔ GaAs/AlGaAs A. Hartland et al – 1991,B. Jeckelmann et al – 1995

few 10-9 Graphene ⇔ GaAs/AlGaAs

A. Tzalenchuk et al - 2009

I1 I2

IsV1 V2Ls

L1 L2hf


RK = h/e2 (1 + [24 ± 18] 10-9 ) KJ = 2e/h (1 + [238 ± 720] 10−9)

Today, from validation tests:

Strong theoretical arguments supporting von Klitzing and Josephson relations:

RK = h/e2 and KJ = 2e/h

The large uncertainty on KJ comes from the present discrepant values of Γ’p h-90 (lo) and Vm(Si)

P. Mohr, B.N. Taylor, D.B. Newell, Rev of Mod Phys., vol.80, 2008

Calc. capacitor

Chain of SI realisations of

electrical units and materialization

farad

µ0c2

siemens

coulomb

ampere

volt

ohm henry

watt

meter, kilogram, second

Watt balance

± 10 V Josephsoneffect

100 Ω − 1 ΜΩ

10 pF

Quantum Hall effect

Calculablecapacitor

< 1 nA Single electrontunneling

Single electrontunneling

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 :

The experiment originally proposed by K. Likharev and A. Zorin in 1985consists of applying Ohm’s law, U = R I directly to the quantities issuedfrom ac JE, QHE and SET.

I = e f

U =n (h/2e) f

I

f

U

Josephsoneffect

SETeffect

IQuantum Halleffect

I = e f

I = (e2/h)U

SETpump VJ

I = e f

RH JfJ

Closing the triangle: first way U = R × I

As for JE, KJ = (2e/h)|JE and QHE, RK = (h/e2)|QHE, one can define a phenomenological constant: QX = e|SET

Dimensionless product to be measured

U

R I

JE

QHE SET

DC

U =n KJ-1fJ

I = QX fSETR = RK /i

Exactness ?!

RK KJ QX = 2 if :RK = h/e2

KJ = 2e/hQX = e

RK KJ QX = (n i/G) ( fJ / fSET)

G = NP/NSgain of the CCC

The QMT experiment

Another approach to close the triangle is to apply Q = CV

Charging a capacitor electron per electron with a pumpmeasuring the voltage drop with Josephson voltage standard,calibrating the capacitance by means of QHR standard

VJ

Q = Ne

CSET pump

controlled by an

electrometer

JfJ

⇒ Electron counting capacitance standard (ECCS)

Closing the triangle: second way Q = C × U

U

C Q

JE

QHE SET

AC

U =n KJ-1f’J

C = i/(2πfqRK) N QX

RK KJ QX = (n/N)(CECCS/CX) ( f’J / fq) i

Quadrature bridge

ECCS: Charging a capacitor electron per electron by a SET pump andmeasuring the voltage drop with Josephson standard

CECCS = KJ QX /[(n/N) f’J]

CECCS compared to CX calibrated by meansof QHR standard via quadrature bridge (RCω = 1)

OUTLINE





V) Conclusion

Aims of the QMT (1/3)

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/e2 and e.

The ultimate target uncertainty is one part in 108

If there is no deviation, our confidence on the three phenomena to provide us with 2e/h, h/e2 and e will be strengthened.

If deviation occurs, some other works both experimental and theoretical will have to be done.

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 107

Second uncertainty level lies between7 parts in 107 and 2 parts in 108

Resulting information will be mainly relevant for the JE and the SET






To determine the elementary charge e or, in other words, the charge quantum brought by the SET devicesIndeed, 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 RK = h/e2 and KJ = 2e/h is possible.

Determination of the charge quantum (1/3)

1890 1910 1930 1950 1970 1990 2010

⏐e/eref⏐-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

direct valuesajdusted values

e ref = 1.602 176 5 10-19 C

1910 1930 1950 1970 1990 2010

10-2

10-3

10-4

10-5

10-6

10-7

10-8

u r

> 1940, e is derived from a complex calculation and is no more relatedto an experiment.

1929: Raymond T. Birge⇒ Least-Squares Adjustments of the Constants- Concept of most probable value;- Interconnection of constants by well established relation;- 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- The willing of labs to share their works

1952: Dumond and CohenIncreasing number of data and observational equation

1969: CODATATask group on fund. constants was formed

1973: 1st LSA from CODATA

1897: J.J. ThomsonDiscovery of the electron

1906 - 1913: R.A. MillikanCharged oil droplets method

1928 : E. Bäcklin The value e is derived from known value of Faraday constant and the determination of NA by x-ray method

F = NA e


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:

µ0c2h/e2

In fact, values of e can be derived directly from experiment, but with a larger uncertainty

CODATA 2006: e = 1.602 176 487 C and σ = 2.5 10-8

α from ae, h/m and RK (Calc. capacitor + QHE)

h via KJ2RK (WB + QHE + JE)

α = e = 2αhµ0c√


Two direct values independent of the QHE and the JE

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1980 1985 1990 1995 2000 2005 2010

Direct value LSA&CODATA

e CODATA 2006 = 1.602 176 487(40) 10-19 C

(e - e CODATA 2006 ) x 106

e direct

Q X

e = [α3Ar(e)Mu/(µ0R∝NA)]1/2

- Ar(e): 2006 CODATA- Mu : = 10−3 kg.mol−1 exactly.- R∝ : 2006 CODATA- α : ⇐ ae and h/mCs, Rb- NA ⇐ NA = Vm(Si)/(81/2d220

3)

QX value from QMT

at NIST with ECCS

σ = 0.92 10-6Comparisons: QX ⇔ e,

RK ⇔ h/e2, KJ ⇔ 2e/h






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 levelquantization and lead to a natural “mise en pratique” of the future definitionof the ampere

⇒ New quantum electrical 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 1.602 176 487 × 10-19 C”

An alternative (but explicit unit definition instead of explicit constant definition):“ The ampere is the electrical current equivalent to the flow of exactly1/(1.602 176 487 × 10-19) elementary charges per second”.It follows that this definition fixes the elementary charge as exactly 1.602 176 487 × 10-19 A⋅s

SET standards

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

- Natural capacitance standard

Quantum standard like JAVS & QHR V

Q = Ne

Celectron pumpC = Q / V ≡ (2e2/h) f

OUTLINE





V) Conclusion

electrometerVG1

VG2

7-J pump

VG5

VG6

VFB

Ve

CryogenicCapacitor

Ccryo≈ 1.8 pF

island1 fF

10 pF stray capacitance

N1N2

VFB (V)

Time (s)

600

20+5

-5

0ΔV

VFB (V)

Time (s)

600

20+5

-5

0ΔV

1) Pumping electrons on and offcryogenic capacitor

Ccryo = 1.8 pF

island

T < 100 mK

Cref = 10 pF

ND

V1

V2

N1N2

1.6 kHz Ccryo = 1.8 pF

island

T < 100 mK

Cref = 10 pF

NDND

V1

V2

N1N2

1.6 kHz

2) Comparing capacitances

Principle of ECCS E. Williams et al, Journal of NIST, 1992, M. Keller et al, Science 1999

CJ ≈ 0.2 fF

⇒ Each ΔV gives a value of Ccryo = Ne/ΔV

1 ppm

1) NISTFirst comparison of the Ccryo values by counting electrons and by a capacitance bridge

M. Keller et al, Science, vol 285, 1999

QX/e – 1 = (-0.09 ± 0.92)×10-6

M. Keller et al, Metrologia 2008

Present status of Q = CV triangle experiments

2) PTB ⇒ talk of M. Wulf poster of B. Camarota

Progress on electron counting capacitance standard

OUTLINE





V) Conclusion

Principle of the CCC Harvey 1972

Application of the Ampère’s law :

∫(a) B.dl = 0 = µ0 (I1+I2-I)

I = I1+I2

I1

I2

B=0

I

(a)I1

I2

B=0

I

(a)

ampere - turn balance : I = 0

N2I2 = N1I1

G = I2/I1 = N1/N2

δΦ ≈ few µΦ0/Hz1/2 (typ.)

SQUIDX

X I1

I2Pick-up coil

Φ = f(I)

II

Β = f(I)

Superconductingshield

SupercurrentI = N1I1-N2I2 Primary

windingN1I1

Secondarywinding

N2I2

SQUIDX

X

SQUIDX

X

X

X I1

I2

I1

I2Pick-up coil

Φ = f(I)

II

Β = f(I)Β = f(I)

Superconductingshield

SupercurrentI = N1I1-N2I2 Primary

windingN1I1

Secondarywinding

N2I2

Hergé, “Tintin au Congo”Les éditions du petit 20ème 1931Castermann 1971-1991

« The snake swallowing its tail ! »

Experimental set-ups based on CCC

CCC as current detector

Single detection level

Rcryo = 100 MΩ at 4.2 K: σ<I>/I < 10-7 ⇒ I > 80 pA

in both cases: SQUID operates at δΦ ≈ 0 (Flux Locked Loop)

δI/I = [4kT/Rcryo)]1/2/ISET

SETpump

Rcryo

J

δΙ

OutputISET IR δΦ

SETpump

Rcryo

JJ

δΙ

OutputISET IR δΦN1 N2

CCC as current amplifier

Two detection levels: magnetic flux and voltage

Highly accurate gain: σG < 10-9

CCC (G = 30 000): δI < 1 fA/Hz1/2, tmeas = 10 hours, σ<I>/I < 10-7 ⇒ I > 80 pA

δV

RH JδΦ

GISET

ISET

VSETpump

δV

RH JJδΦ

GISET

ISET

VSETpump

N1 N2

δV/V = ([δI2CCC+ (4kT/G2RH)]/I2

SET+ δV2ND/V2)1/2 ≈ δICCC/ISET

Present CCCs with DC SQUID

CCCs in use

- SCCCth = 4.5 µA⋅t/Φ0, δIth = 0.7 fA/Hz1/2

- SCCCmeas = 5 µA⋅t/Φ0, δImeas = 4 fA/Hz1/2

δImeas = 5 fA/Hz1/2 at 1 Hz connected to R pump

- Winding ratio error : εdc = 3 10-8, εac = 8 10-9

- Self resonances

Φext = 49 mm, Φint = 22 mm, h = 13 mmGmax = 20 000

Spectral density of noise, config. 3

7.4 kHz

3.5 kHz

Spectral density of noise, config. 3Spectral density of noise, config. 3

7.4 kHz

3.5 kHzfres = 3.5 kHz(10 000 turns)

Principle of the QMT set-up at LNE

INT FB: non accurate mode

EXT FB: accurate mode

FluxTransformer

CCCSET pump Program.

JAVS

DC SQUID

N1 turns N2 turns

RFB

70 GHzsource

10 MHz rubidiumreference

Calibrated10 kΩ

Vout

INT FBEXT FB

DAC basedBias current

source

Waveformgenerator

30 mK4.2 K 4.2 K

VR VJ

Vd

≈ π/2δφ

Results on 3-junctions R pump (SQUID in int FB)

40 fA

3 junctions R pump fabricated by the PTB

A. Zorin et al. , JAP2000S. Lotkhov et al, APL, vol.78, 2001

I (pA)B. Steck et al, Metrologia 2008

⇒ Measurement of current steps up to 100 MHz !

RCr(25-50 kΩ)Vg1 Vg2Vg1 Vg2 gates

source drain

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

-125 -75 -25 25 75 125 175

V p (µV)

ΔI /I

Spring 2009: precise measurement of flatness (int FB mode)

⇒ The plateaux are flat over 60 µV (σmean = 2 parts in 105)

⇒ Better and reproducible white noise level 5 fA/Hz1/2

Step +, 10 MHz

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

-200 -150 -100 -50 0 50 100

V p (µV)

ΔI /I Step -

-0.0002

-0.0001

0

0.0001

0.0002

0 20 40 60 80

V p (µV)

ΔI /I

V P+

0.3 fA

-0.0001

0

0.0001

0.0002

0.0003

-80 -60 -40 -20 0V p (µV)

ΔI /I

0.3 fA

⇒ See poster Sami Sassine et al.

-1.E-04

0.E+00

1.E-04

2.E-04

3.E-04

4.E-04

5.E-04

0 10 20 30 40 50 60Number of measurement

(Qx-e

)/e

fSET=23.55 MHznJ=5

fSET=22.65 MHznJ=5

ur=7x10-6

Weighted average

ur=1.3x10-5

-1.E-04

0.E+00

1.E-04

2.E-04

3.E-04

4.E-04

5.E-04

0 10 20 30 40 50 60Number of measurement

(Qx-e

)/e

fSET=23.55 MHznJ=5

fSET=22.65 MHznJ=5

ur=7x10-6

Weighted average

ur=1.3x10-5

Measurements with SQUID in ext FB

B. Steck et al.CPEM’08

0 1 2 3 4 5 6 7 81.1

1.2

1.3

1.4

1.5

1.6

1.7

QX = (N2/N1)(nJfJ/KJ – Vd)/(fSETR)

⇒ Δe/e = (QX - eCODATA)/eCODATA

Number of measurement

Δe/e

(*10

-4)

ur = 4⋅10-6

fSET = 37.69 MHz nJ = 8

Weighted mean

Discrepancies between 2 runs of measurements

-1.2E-04

-8.0E-05

-4.0E-05

0.0E+00

4.0E-05

8.0E-05

1.2E-04

1.6E-04

2.0E-04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

#

Δe

/e 1.5x10-4fSET=37.69 MHznJ=8

V- Conclusion: contribution of the QMT

to test and hopefully to enhance our confidenceon QHE, JE and SET to provide h/e2, 2e/h and e

to improve knowledge on fundamental constants, in particular the elementary charge

⇒ The direct determination of e via QMT, calculable capacitor and watt balance links up with the historical experiment of Millikan, early last century

to give some elements of thought in the frame of the discussion around the reform of the SI

to advocate the “mise en pratique” of the definition of some electrical units with SET standards

Outlook of the QMT

To complete the development of new CCCs with the aim at reaching : δI < 1 fA/Hz1/2

To investigate single charge transport deviceswith I >> 1 pA, for example:

- Hybrid SINIS SET turnstile (MIKES/TKK)- Pump based on Si or GaAs/AlGaAs nanowire (CEA, PTB)

⇒ Current plateaux observed at a level 100 pA

- Parallelization of devices to increase much more the current

To close the triangle- in short term, at a level of 1 part in 106,- in mid term, at a level of parts in 107,- in long term down to part in 108 and less …

…depending on the SCT current sources !

REUNIAM !

Thank you very much !

http://www.cmongout.com/images/images_produit/pains_et_sandwitch/triangles_d-enchiladas.jpg

1889 1946/53 1989/92

+80

+60

+40

+20

0

-20

-40

-60

ΔM (µg)Possible drift of IPK:

3 × 10-8 per century !

Evolution of national prototypes over 120 years

Cross calculable capacitance standardTheorem of A. Thompson and D. Lampard (1956):For a cylindrical system of 4 isolated electrodesof infinite length and placed in vacuum,

exp(-π γ13/ε0) + exp(-π γ24/ε0) = 1

In the case of a perfect symmetry with identical γij

γ13 = γ24 = γ = (ε0ln2)/π = 1.953 549 043 ... pF/m ⇒ ΔC = γΔL

4

12

3

γ13 γ24

4

12

3

γ13 γ24

LNE, five-electrodes capacitor

Quadrature bridge

AC

DC

SECOND

METRE

R C ω = 1 ω (rd/s)R

CD 10 000

Calculablecapacitor

1 pF

Calculableresistor

100 Ω

QHE

10 kΩ

Capacitancebridges

CCC10 pF

100 pF

1000 pF

10 nF

Quadrature bridgeQuadrature bridge

AC

DC

SECOND

METRE

R C ω = 1 ω (rd/s)R

CD 10 000

Calculablecapacitor

1 pF

Calculableresistor

100 Ω

QHE

10 kΩ

Capacitancebridges

Capacitancebridges

CCC10 pF

100 pF

1000 pF

10 nF

QHERK

a) d)

f)

b)e)

c)

2012

2000

From the farad to the ohm

Josephson voltage standards

Quantum effects occurring between two superconducting electrodes separated by a small region where the superconductivity is weakened: thin insulating film

Brian Josephson 1962

⇒ Programmable arrays for DC and AC applicationsSee Review: B. Jeanneret, S. Benz, EPJST 172, 2009

10 V Josephson junction arrays

Steps around 10 V

V = (h/2e) f≈ 145 µV at f = 70 GHz

Oxide layer≈ 2 nm

Microwaveradiation f

I

VOxide layer

≈ 2 nm


I

V

n = 1

1 2 3 4

I (mA)

V = (h/2e) f≈ 145 µV at f = 70 GHz

Oxide layer≈ 2 nm


I

VOxide layer

≈ 2 nm


I

V

n = 1

1 2 3 41 2 3 4

I (mA)

V = f/KJ ≈ 145 µV at f= 70 GHz

KJ = 2e/h, the Josephson constant

V = (h/2e) f≈ 145 µV at f = 70 GHz

Oxide layer≈ 2 nm


I

VOxide layer

≈ 2 nm


I

V

n = 1

1 2 3 4

I (mA)

V = (h/2e) f≈ 145 µV at f = 70 GHz

Oxide layer≈ 2 nm


I

VOxide layer

≈ 2 nm


I

V

n = 1

1 2 3 41 2 3 4

I (mA)

V = f/KJ ≈ 145 µV at f= 70 GHz

KJ = 2e/h, the Josephson constant

Quantum Hall resistance standards Klaus von Klitzing, 1980

LEP 514 Hall bar sampleGaAs/AlGaAs heterostructure

At low temperature and under high magnetic field, the Hall resistance of the 2DEG exhibits plateaux centred on quantized values:

RH (i) = h/ie2 = RK/i, where i is an integer, RK the von Klitzing constant.

⇒ Arrays of Hall bars for scaling up to 1.29 MΩand down to 100 Ω Review: W. Poirier, F. Schopfer EPJST 172, 2009

⇒ Quantum Wheatstone bridge for ultra high precise comparison (10-11)

⇒ Quantum impedance standard: 6 × 10-9 on 10 pFJ. Schurr et al. Metrologia 2009

0

25

50

75

100

125

150

0 1 2 3 4 5 6 7 8 9 10 11B (T)

RH

C3 (B

+) ( Ω

)

05101520253035404550

Rxx

a ( Ω)

T=1,3KI=100 µA

QHARS 129

QHARS 100

RH3RH4

RH1 RH2

V

Iub.

RH3RH4

RH1 RH2

V

Iub.

Watt balance: monitoring the kilogram electrically

LNE balanceStatic

B I

FzP

m

Fz = mg = -I ∂Φ/∂z

speed v

ε

Bv

Dynamic

ε = - ∂Φ/∂t = -∂Φ/∂z v

Equivalence between mechanical and electrical power ⇒ Fzv = εI ⇒ mgv = εV/R

Monitoring the kilogram in term of h

- ε and V in terms of Josephson effect: ε = n1f1/KJ, V = n2f2/KJ

- R in terms of quantum Hall effect: R = RK/i

⇒ h = , assuming KJ = 2e/h and RK = h/e2

⇒ m(IPK) = h

K2JRK

A

4gvA

⇒ mgv =

where A = n1f1 n2f2i

A4mgv

Single electron tunneling: towards quantum current standard

Electron box Coulomb Blockade of the tunneling events when n - 1/2 < CGU/e < n + 1/2

The wave function of electron in excess on the island is well localised

- Thermal fluctuations of n are negligible if kBT << e2/Ci

- Quantum fluctuations of n negligible when Rj >> RK

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

ExperimentTheory

Ave

rage

box

cha

rge

[e]

CgVg [e]

Coulomb staircase<n>

CGU/e

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

ExperimentTheory

Ave

rage

box

cha

rge

[e]

CgVg [e]

Coulomb staircase<n>

CGU/e

CG

Metallic islandq = QG - Q = n e

U

Tunnel junction

-

+

Gatecapacitor

CJ

n

CG

Metallic islandq = QG - Q = n e

U

Tunnel junction

-

+

Gatecapacitor

CJ

n

Unwanted effects and error corrections ⇒ talk of M. Wulf

3-junctions electron pump Two modulation signalsat frequency f phase-shifted by Φ =π/2

For f = 100 MHz ⇒ I = 16 pA

⇒ I = e × f+V/2 -V/2U1

I

n1

C1

U2

n2

C2

+V/2 -V/2U1

I

n1

C1

U2

n2

C2

transistor

Minimum energy states of the islandas a function of U1 and U2

e

e-e

-e

C2U2

PN

C1U1(0,0) (1,0)

(0,1)

(n1,n2)


e

e-e

-e

C2U2

PN

C1U1(0,0) (1,0)

(0,1)

(n1,n2)

Minimum energy states of the pumpAs a function of U1 and U2


e

e-e

-e

C2U2

PN

C1U1(0,0) (1,0)

(0,1)

(n1,n2)


e

e-e

-e

C2U2

PN

C1U1(0,0) (1,0)

(0,1)

(n1,n2)

Minimum energy states of the pumpAs a function of U1 and U2

AnimPompe.exe

Observational equations taken into account in the adjustment of values of fundamental constants by CODATA

A = f(X,Y,..)•

measured quantity

adjustedconstants

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1E-18 1E-15 1E-12 1E-09 1E-06 0.001 1

relative standard uncertainty (1σ)

I (A)

integration bridge

potential difference

RT comparator

SET currentstandard

QHE & JVstandards

Sub-nano ammeters(integration bridge, electronic device)

SET current standard

Toward an improved calibration of small currents

Metrology (elect., ionising radiat.):Calibration of

- low-current standard source- high-value resistance standard

Electronic industry:- Characterization of components

Physical size ⇒ electrical signals such as current

Existing methods

quantum metrology triangle · the present si 1948: definition of the ampere by fixing the magnetic...

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