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172 R. GROSS AND A. MARX Chapter 4Σ / Σ(γβ L=1/80)10Σ (γβ L=1/80)64200 2 4 6β Lβ L0.60.70.81.01.52.03.04.010.01 0.1 1Figure 4.7: Calculated reduced energy resolution Σ(γβ L ) normalized to Σ(γβ L = 1/80). Inset shows Σ(γβ L =1/80) versus β L (data from D. Kölle et al., Rev. Mod. Phys. 71, 631 (1999)).γβ Lenergy of SQUIDs in units of ¯h ≃ 10 −34 Js. Optimized SQUIDs have a noise energy approaching thequantum limit ¯h. However, these SQUIDs have very low inductance and therefore are not useful for mostapplications requiring optimum magnetic field resolution. Best practical dc SQUIDs have reached anenergy resolution of some 10¯h. 24A recent result of a numerical simulation is shown in Fig. 4.7. Here, the reduced noise energyΣ( f ) = ε( f )2Φ 0 k B TI c R N(4.1.50)is plotted versus the dimensionless parameterβ th = γβ L = 2πk BTI c Φ 02I c LΦ 0= L Φ 2 04πk B T≡LL th. (4.1.51)For γβ L ≤ 0.2 corresponding to L ≤ 1 5 L th, the reduced noise energy is almost constant, while for highervalues of γβ L it increases rapidly. The rapid increase in noise energy arises from the rapid degradationof the transfer function with increasing γβ L = L/L th due to thermal noise rounding of the IVCs.4.1.4 Practical dc-SQUIDsPractical dc-SQUIDs do not only consist of the SQUID loop discussed so far, but also of an antenna anda room temperature electronics as schematically shown in Fig. 4.8. The antenna has both to transfer thequantity that has to be measured into magnetic flux and to couple this flux effectively into the SQUIDloop. The SQUID itself acts as a flux-to-voltage transducer. The room temperature electronics has toamplify the voltage signal as well as to provide the current and flux bias.24 A.A. Jin, T.R. Stevenson, F.C. Wellstood, W.W. Johnson, IEEE Trans. Appl. Supercond. AS-7, 2742 (1997).© Walther-Meißner-Institut