Design, Fabrication and Characterization of a Microwave Resonator ...

2 Theoretical Description of a CPW ResonatorThe quality factor of external circuitsQ ext = ωCG extcan be obtained for circuits coupled capacitively to the environment usingG ext =2R LCcouple 2 ω21 + R 2 L C2 couple ω2C couple is the coupling capacity, R L , the load resistance is 50 Ω for matched circuits.Three cases of coupling can be distinguished.• Q√Q ext< 1: undercoupled to feedline, C couple < √ π2Qωr⇒ QZ L ≈ Q0• QQ ext= 1 : critically coupled to feedline• Q√Q ext> 1 : overcoupled to feedline, C couple > √ π2Qωr⇒ QZ L ≈ Q ext0Coupling of a feed line to the resonator has the effect of lowering its resonant frequency[19]:ω rω L = √(2.29)1 + 2 C coupleCFrom the above list one can see that Q L can be tuned by varying the coupling capacitor.The rate of measurement is determined by the decay rateκ = ω rQ L= ∆ f (2.30)of a photon in the cavity.This is the predominant source of dissipation. Tnno have a rate of measurement of≈ 1 MHz at ≈ 10 GHz one needs loaded quality factors of Q L ≈ 10 4 . This means, a photonleaks out of the cavity in T r = 1/κ ≈ 100 ns.The rate of measurements has to be faster than qubit decay rates.2.3.4 Distributed ResonatorsFor microwave frequencies, lumped circuit elements are usually unattainable, so distributedelements are used. The reason is that the dimensions of the resonator are comparable to thewavelength. Therefore, voltage and current can not be averaged any more. In this thesis, weuse resonators built of superconducting CPWs.In Figure 2.11 two different resonator types are shown. A practical resonator that is oftenused in circuits (also in this work) is the transmission line, open circuited on either end, asshown in Figure 2.11(a). As indicated in this figure the electric and magnetic fields have tofulfill the boundary conditions. According to reference [18] such a resonator will behave asa parallel resonant circuit if the length amounts to λ/2, or a multiple of λ/2. In contrast,20