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

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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
2.3 Microwave ResonatorsFigure 2.11: Ilustration of CPW resonators (a) λ/2 resonator (b) λ/4 resonator.a short-circuited λ/2- resonator would behave as a serial circuit. The resonance frequencydescribed in (2.27) can be derived by using the phase velocity v ph (2.12) divided by twotimes the resonator length Df = v ph2D . (2.31)A so called antiresonance can be achieved using a transmission line which is short circuitedat one end with a resonance at λ/4, as shown in 2.11(b). In contrast to a short circuitedλ/2 resonator this resonator behaves as a parallel circuit. This is due to the fact that atransmission line of length λ/4 behaves as an impedance converter and therefore transformsa serial circuit into a parallel circuit.When capacitively coupling an λ/2 open circuited transmission line resonator that normallylooks like a parallel RLC circuit near resonance at one end, it will look like a seriesRLC circuit near resonance. This is because the series coupling capacitor has the effect ofinverting the driving point impedance of the resonator (like a quarter wavelength line does).Coupling the λ/2 open circuited resonator on either side, it behaves as a parallel circuit.The quality factor of a microwave resonator can be calculated byQ = β2α , (2.32)where β = 2π λ= 2π f rv phis the propagation constant and α is the attenuation constant discussedin subsection 2.1.6 for normal conducting media and in subsection 2.2.2 for the superconductingcase.21
Page 4 and 5: Contents5.2.3 R27 on Silicon . . .
Page 6 and 7: 1 Motivation and Outlinethe antinod
Page 8 and 9: 2 Theoretical Description of a CPW
Page 14: 2 Theoretical Description of a CPW
Page 19 and 20: 2.2 Including Superconductivity in
Page 21 and 22: 2.2 Including Superconductivity in
Page 23: 2.3 Microwave Resonators2.3.2 Quali
Page 27 and 28: 2.3 Microwave ResonatorsFigure 2.13
Page 29 and 30: 2.4 The Scattering MatrixThe first
Page 31 and 32: 3 Designs Procedure for the Resonat
Page 33 and 34: 3.3 ConnectorsFigure 3.1: Coupling
Page 35 and 36: 4 Fabrication and Measurement Setup
Page 37 and 38: 4.2 Measurement SetupDevelop Resist
Page 39 and 40: 4.2 Measurement SetupFigure 4.3: Me
Page 41 and 42: 4.3 Calibration4.3 CalibrationIn or
Page 43 and 44: 5 MeasurementsThe first part of thi
Page 45 and 46: 5.1 First Experimentsname design W
Page 47 and 48: 5.2 Measurements - Design 1Figure 5
Page 49 and 50: 5.2 Measurements - Design 1The adva
Page 51 and 52: 5.2 Measurements - Design 15.2.2 R2
Page 53 and 54: 5.2 Measurements - Design 1Figure 5
Page 55 and 56: 5.3 Measurements - Yale DesignFigur
Page 57 and 58: 5.3 Measurements - Yale DesignFigur
Page 59 and 60: 5.4 Measurement - Design 2: R36 on
Page 61 and 62: 5.5 Discussion of the Experimental
Page 63 and 64: 5.5 Discussion of the Experimental
Page 65 and 66: 6 SimulationsThis chapter gives a b
Page 67 and 68: 6.1 Simulations in SonnetFigure 6.4
Page 69 and 70: 6.1 Simulations in SonnetFigure 6.5
Page 71 and 72: 6.2 Simulations Done at the Institu
Page 73 and 74: 7 Conclusion and OutlookThree centr
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A Specimen OverviewDesigns for Rose
Page 77 and 78:
B Masks73
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B.2 Mask for Probing StationB.2 Mas
Page 81 and 82:
Bibliography[1] INFORMATIK: Quanten
Page 83 and 84:
List of Figures2.1 Different realiz
Page 85 and 86:
List of Figures6.6 Forward and back
Page 87 and 88:
List of Tables2.1 Values where (2.1
Page 89:
DanksagungVor allem möchte ich mic
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Design, Fabrication and Characterization of a Microwave Resonator ...

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