Thermal fluctuation energy E^

0,7 X 10"24J

0,7x10-24 J

14x lo*24J

Quantum fluctuation energy Eq


1 x lo"23 J

0,3 x lo"24 J

Josephson coupling energy Ej


32x 10-24 J

4x10-2° J

Charging energy Ec

43 X IQ'24 J

27 x IQ"24 J

2,7 x 10"28 J

Tab. 1: Numerical values for the experimental realization of quantum circuits

Tab. 1: Numerical values for the experimental realization of quantum circuits

Including SET, at least in a phenomenological sense, the whole set of dual quantum circuits can be constructed with the basic circuit elements of Fig. 2:

Flux regime


ideal ring, purely inductive ideally conducting charge undefined flux localized and quantized energy of one flux quantum in the ring: &¡J2L

Transfer element:

Josephson junction; ) < transfer rate: U~<l^f={h!2e)f flux tunnel junction: transfer rate: U-dbf=(h/e)f (requires ideally conducting electrodes for flux concentration

Charge regime


ideal capacitor, purely capacitive ideally insulating flux undefined charge localized and quantized energy of one charge quantum on the capacitor: Qo/2C

Transfer element:

SCT junction (single Cooper pair tunn.): transfer rate: /=Q0/=ef (Bloch oscillations)

SET junction (single electron tunn ): transfer rate: l=Q0f=ef (classical Coulomb blockade)

Fig. 2 Basic circuit elements for the construction of quantum circuits The rf SQUID (a superconducting ring interrupted by a Josephson junction) corresponds to the electron box (an ideal capacitor linked to a SET junction) and the dc SQUID to a SET transistor. In analogy to the single flux quantum (SFQ) logic circuits, a complete logic circuit library can be designed with the basic SET elements. One information bit is carried by the flux quantum ~ \Udt<* 2 mV ps in the first case and by one electron e = \ldt = 0,11 M ps in the second case. The numerical numbers show the extremely low energy consumption of both circuit types. In the case of the SET logic, the extreme smallness of the circuit elements would basically allow ultimate integration. At least the circuits in the flux regime have been realized and have commonly been used in metrology for a long time [Likharev, 1986]. For a few years, many SET devices - for review, see [Devoret and Grabert, 1992] - have been investigated and even Bloch oscillations have been demonstrated [Kuzmin and Haviland, 1991; Kuzmin et al., 1991; Pashkin et al., 1994]. While the Josephson voltage standard, which is based on the counting of the transferred flux quanta with extreme accuracy, has reached the commercial level, its analogue, the current standard on the basis of SET, is still under development and involves extraordinary experimental difficulties.

3 The Josephson voltage standard

At a finite constant dc voltage, an ac supercurrent flows through the barrier of a superconducting tunnel junction the frequency of which is directly related to the dc voltage:

This is caused by flux quanta traversing the junction at a certain rate, e.g. at U = 145 |J.V, 7 x 1010 flux quanta per second pass the junction = 70 GHz). Vice versa, if the junction is rf-biased with a frequency/, a constant voltage appears in the dc characteristic in the form of a step whose width is dependent on the rf power of the bias. Due to the strong non-linearity of the dc characteristic, this happens at higher harmonics, too:

U] = n&0f (n = 0, ± 1, ± 2,...) [Shapiro, 1963]

In other words, the transfer of one single flux quantum is generated by one cycle of the rf bias.

The number of flux quanta transferred per second and, as a result, the dc voltage are determined with high precision by the stability of the frequency of the external microwave supply. In a low-damped tunnel junction with a hysteretic dc characteristic, strongly overlapping constant voltage steps can be generated in the back switching part of the dc characteristic (Fig. 3) which runs along the voltage axis at such a small distance that at low rf power, the steps cross this axis at zero current. In the case of a Nb/Al203/Nb tunnel junction, the zero current steps reach a maximum of about 1 mV. At higher voltages, the steps are shifted off the voltage axis due to additional photon-induced currents across the junction. For stable step operation, the rf and dc currents have to be homogenously distributed over the junction area. This condition restricts the width and the length of the junction [Kautz et al., 1987; Niemeyer, 1989]. As a Josephson junction is a strongly non-linear oscillator, chaotic behaviour is often dominant. To avoid chaos, the Josephson plasma frequency has to be kept smaller than the external microwave frequency [Kautz, 1981a; Kautz, 1981b; Kautz, 1987].

Fig. 3 Dc characteristic of an Nb/A^Oj/Nb tunneljunction a) without and b) with microwave radiation, rf bias frequency: 70 GHz. The constant voltage steps are crossing the current axis.

This condition restricts the junction critical current density and together with the junction size restrictions it leads to a limited critical current and, as a consequence, to a finite current width of the constant voltage step of less than 100 jxA. But with noise-reduced equipment the voltage steps can be directly used as reference voltages in modern quantum standards. The widely overlapping zero current steps have the advantage that to increase the output voltage, series arrays of Josephson tunnel junctions can be used although the junction parameter spread is relatively large [Levinsen et al., 1977], Such arrays with more than 20 000 junctions have been fabricated in the past decade, providing reference voltages up to more than 10 V. Different technologies were tested in the past [Niemeyer et al., 1985a; Hamilton et al., 1985; Niemeyer et al., 1985b; Lloyd et al., 1987], For practical reasons like minor aging, the Nb/Al203/Nb technology has turned out to be optimum. For proper standard operation it was important to integrate the series array into a microwave circuit which provides homogenous microwave coupling to each single junction [Niemeyer et al., 1985; Niemeyer et al, 1984],

To reach this goal, the series array is designed to form a superconducting microstrip line which is terminated by a lossy line to prevent standing wave patterns in the line by a reflected microwave. A fin line antenna matches the microstrip line impedance of about 5 Q to the impedance of a 70 GHz waveguide. To achieve optimum

Fig. 3 Dc characteristic of an Nb/A^Oj/Nb tunneljunction a) without and b) with microwave radiation, rf bias frequency: 70 GHz. The constant voltage steps are crossing the current axis.

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