**Quantum metrology and fundamental constants:** international school at Les Houches Physics Centre, 1st to 12 October 2007.

> Highlights (291 Ko)

The first objective of this study is to ensure the **conservation of the volt**, by developing a quantum voltage standard based on the **Josephson effect** and by developing and improving instruments used in the calibration chain.
The second objective of the study is to develop **research into new applications** of the Josephson effect in metrology of the volt.

Fig. 1 : Josephson junction

The Josephson effect [1] is one of the notable effects of **superconductivity**, a macroscopic quantum phenomenon which appears in certain metals at very low temperatures. In the superconducting state, electrons are attracted by each other and form bound pairs, called Cooper pairs. The Josephson effect occurs when these pairs of electrons tunnel through a thin insulating barrier placed between two superconductors, the whole being called **Josephson junction** (Fig.1).

**DC Josephson effect**: If no voltage is applied to the junction terminals, a direct current - a current of Cooper pairs Ij, - flows through the junction up to a critical value Ic, which depends on the geometry, temperature and magnetic field (Fig.2a). **AC Josephson effect**: If a direct voltage is applied to the junction terminals, the current of the electron pairs crossing the junction oscillates at a frequency which depends solely on the applied voltage V and fundamental constants (the electron charge ee and the Planck constanth) :

Conversely, if an AC voltage of frequency *f _{a}* is applied to the junction terminals by microwave irradiation, the current of Cooper pairs tends to

The exactitude of the voltage-frequency relation ** V = (h/2e) f** and its independence from experimental conditions (temperature, polarization current, junction materials) have been tested on many occasions with an uncertainty level of up to 10

Fig.2a - 2b : Current-voltage characteristics

a) When no voltage is applied to the junction terminals, a direct current flows through the junction up to Ic,.

Above this value, a direct voltage develops at the junction terminals and the electron pair current IJ oscillates at frequency f = 2e V/h ;

b) Current-voltage characteristics of a Josephson junction under microwave irradiation for different microwave power: Shapiro steps

The definition of the volt in the International System of Units (SI) is as follows: "The volt is the electromotive force between two points of a conductor carrying a current of 1 ampere when the power dissipated between the two points is 1 watt" [3]. **Realization of the volt** in the SI system rests on experiments comparing an electrostatic force with a mechanical force, but the uncertainties obtained by this method are much too great to meet the requirements of modern instrumentation. Conversely, the stability of voltage references based on the Josephson effect depends solely on frequency stability, which can easily reach 10^{-12}. For this reason, National Metrology Laboratories started using the AC Josephson effect as a **representation of the volt** and adopted *K _{J}*,

Fig.4 : Josephson measurement bench

It is based on a weighted average of the *K _{J}* values obtained by SI realizations of the volt and other experimental methods before 1990.

At LNE, as in most National Metrology Laboratories, we have developed a measurement system based on the Josephson effect (Fig. 3) in order to calibrate standard cells (or Weston cells) and Zener diode voltage references by comparison with Josephson junction arrays over the -10 V à +10 V range.A direct comparison of two Josephson arrays in 1994 showed that the total measurement uncertainty of the system for a voltage of 1.018V is 1.7.10^{-10} [4].

Fig. 5: Diagram of a microstrip line

integrating several Josephson junctions

Development of voltage standards based on the Josephson effect is closely linked to the advances in junction manufacture and nanotechnologies (nanofabrication techniques such as thin layer deposition and microlithography). These voltage standards, which at present go up to 10 V, take the form of **superconducting integrated circuits** containing thousands of Josephson junctions connected in series and integrated in a microwave transmission line (Fig. 4). Each junction has a surface area of a few µm² and is only a few nanometers thick. The two major difficulties lie in the need to ensure a great homogeneity of the critical currents Ic across the entire junction array, with no more than a few % dispersion, and a uniform power distribution across the microwave circuit (working frequencies may vary between 1 and 80 GHz).

Equivalent diagram of the junction in a circuit

**In the 1980s**, it was not yet possible to manufacture overdamped Josephson junction arrays (Fig. 7) that were sufficiently uniform for easy polarization of the entire array on the same voltage step. For this reason, the first arrays - comprising several thousand junctions connected in series and rising to 1 V and later 10 V - were built with underdamped (**SIS - Superconductor-Insulator-Superconductor**) Josephson junctions whose I-V characteristic is **hysteretic** (Fig.6). The voltage steps crossing the zero current axis made it possible to obtain quantized voltages for a large number of junctions at **zero polarization current** and circumvented the problem of junction uniformity. These arrays present a major disadvantage, however, as the steps are highly unstable and difficult to select.

Advances in nanotechnologies over the last 10 years have finally made it possible to produce arrays of several thousand non-hysteretic junctions of SNS (Superconductor-Normal metal-Superconductor) or SINIS (Superconductor- Insulator-Normal metal-Insulator-Superconductor) type. The advantage of these arrays is that the voltage steps can be selected precisely and very quickly, opening up numerous applications in AC voltage. A new architecture has been developed for these arrays, based on a distribution of Josephson junctions in binary sequences called segments (Fig. 8a). Each of the segments irradiated at frequency f can be polarized individually on steps *n* = 0, ± 1, by applying a polarization current *I* = 0, *± Ip* (Fig.8b). The output voltage of the array is the sum of the voltages developed by each segment. It rises to a maximum of ± *N f / K _{J-90}*, where

Fig. 8 :

a) Architecture of binary arrays and

b) I-V caracteristic of the array on steps n = 0,+/-1. Each segment irradiated
at frequency f can be polarized individually on steps n = 0, ± 1 by applying a polarization current *I*= 0, ± Ip.

c) Sine curve of 1.25 V amplitude and 100 Hz frequency obtained from a 1 V binary array

- [1] B.D. Josephson, Phys. Lett., 1, 251 (1962)
- [2] J-S. Tsai et al., Phys. Rev. Lett.,51, 316 (1983)
- [3] C. A. Hamilton, Rev. Scien. Inst., 71, 3611 (2000)
- [4] D. Reyman et al., Metrologia 31, 35-37 (1994)
- [5] C. A. Hamilton, IEE Trans. Instrum. Meas., 44, 223 (1995)
- [6] O. Monnoye, Congrès de métrologie, Toulon, (2003)

Tel : 01 30 69 21 57

- J.P. Lo-Hive, S. Djordjevic, P. Cancela, F. Piquemal, R. Behr, C. Burroughs and H. Seppä, "Characterisation of binary Josephson series arrays of different types at BNM-LNE and comparisons with conventional SIS arrays", IEEE T.I.M., Special issue CPEM'2002, Vol. 52, pp. 516-520, April 2003
- R. Behr et al, "Analysis of different measurement set-ups for a programmable Josephson voltage standard", IEEE T.I.M., Special issue CPEM'2002, Vol. 52, pp. 524-528, April 2003
- F. Piquemal, "Point sur les étalons électriques fondamentaux dans le système SI", Groupe de travail sur les unités de base, Académie des Sciences, 31 mars 2003
- J.P. Lo-Hive, D. reymann, G. Geneves, "Using 10 V Josephson voltage standards to estimate the uncertainty of Zener voltage references as travelling standards ", IEEE T.I.M., Special issue CPEM'1998, Vol. 48, pp. 253-256, April 1999
- J.P. Lo-Hive, G. Genevès, "Comparaisons d'étalons de tension à effet Josephson", Bulletin du BNM, n° 111, janvier 1998
- D. Reymann, J. P. Lohive and G. Genevès, "A Comparison of One Volt Josephson Junction Array Voltage Standards driven by a common Microwave source", Metrologia, 31, 35-37, 1994

- Generation of AC voltage with programmable arrays
- Participation in the European Euramet BJAPS project (Binary Josephson Array Power Standard)
- Participation in ANR Trimet project (Quantum metrological triangle)