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## Quantum Electrical Standards and the Metrological Triangle

All electrical measurements below 10 MHz at NPL are traceable to two quantum standards: the quantum Hall effect (QHE) resistance standard and the Josephson voltage standard (JVS). For details of the traceability above 10 MHz, see RF & Microwave Primary Metrology.

The quantum Hall and Josephson effects are believed to be fundamental physical effects, which are the same everywhere, and at all times. This allows measurement laboratories worldwide to maintain the same standard ohm and volt with an agreement of 10 parts per billion (ppb), without the need to transport artefact standards.

The quantum Hall resistance R and Josephson voltage V are given by

R = RK / i, with RK = h/e2 (1 + ΔR)                    (1)

V = n1f1 / KJ, with KJ = 2e/h (1 + ΔV)                    (2)

Where

f1 is a frequency (See Time and Frequency for information on how this is measured).
n1 and i are integers.
h and e are fundamental constants: Planck’s constant, and the charge on the electron.

Adoption of the quantum standards proceeded in four stages[1]:

1. Theoretical arguments were developed, indicating that ΔR = ΔV = 0.
2. Experimental studies showed that each equation (1) and (2) on its own describes a universal effect. For example, RK in two QHE devices made from different semiconductor materials is the same to within 1 ppb. This is strong, but not conclusive evidence, for deduction 1.
3. Measurements of RK and KJ in SI units were performed at different laboratories, for example using the NPL Watt Balance. These measurements had uncertainties of roughly 100 ppb, much higher than in step 2.
4. Based on an analysis of the measurements in step 3, the International Committee of Weights and Measures (CIPM) published standard values of RK and KJ, for routine use. These numbers were published in 1990 and are known as RK-90 and KJ-90.

In the SI system, the base electrical unit is the ampere. The QHE ohm and Josephson volt are linked to the ampere via the difficult experiments in step 3, with a relatively high uncertainty. In consequence, the QHE and JVS are referred to as ‘representations’ of the SI ohm and volt. To address this inconsistency and other issues, the International Committee of Weights and Measures (CIPM) has recommended the study of proposals to re-define some of the SI units in 2011.

A third quantum electrical standard, based on single electron transport, or Quantum Phase Slip is under development. This will yield a current given by

I = n2 f2 KI, with KI = e (1 + ΔR)                    (3)

Joining the three quantum standards together in one experiment will measure the product RKKJKI. This is expected to be exactly 2 if the Δ terms in equations (1) to (3) are zero. Any deviation from the expected value will indicate a flaw in our understanding of one or more of these quantum effects. This experiment is known as the 'metrological triangle', and its outcome will be an important input into the CIPM deliberations on the future of the SI. It is one of the high priorities in fundamental metrology today.

### Reference

1. ‘The route to Atomic and Quantum Standards’, J. Flowers, Science 306, 1324 (2004).
Last Updated: 19 Aug 2010
Created: 19 Aug 2010

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