Coplanar Waveguide Resonators for Circuit Quantum Electrodynamics

M. Göppl,A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff Department of Physics, ETH Zürich,CH-8093, Zürich,Switzerland. (Dated: July 25, 2008) We have designed and fabricated superconducting coplanar waveguide resonators with fundamental frequencies from 2 to 9 GHz and loaded quality factors ranging from a few hundreds to a several hundred thousands reached at temperatures of 20 mK. The loaded quality factors are controlled by appropriately designed input and output coupling capacitors. The measured transmission spectra are analyzed using both a lumped element model and a distributed element transmission matrix method. The experimentally determined resonance frequencies, quality factors and insertion losses are fully and consistently characterized by the two models for all measured devices. Such resonators find prominent applications in quantum optics and quantum information processing with superconducting electronic circuits and in single photon detectors and parametric amplifiers.

I. INTRODUCTION sign, fabricate and characterize CPW resonators with well defined resonance frequency and coupled quality fac- Superconducting coplanar waveguide (CPW) res- tors. The resonance frequency is controlled by the resonators find a wide range of applications as radi- onator length and its loaded quality factor is controlled ation detectors in the optical, UV and X-ray fre- by its capacitive coupling to input and output trans- quency range1,2,3,4,5, in parametric amplifiers6,7,8, mission lines. Strongly coupled (overcoupled) resonators for magnetic field tunable resonators7,9,10 and with accordingly low quality factors are ideal for perform- in quantum information and quantum optics ing fast measurements of the state of a qubit integrated 12,29 experiments11,12,13,14,15,16,17,18,19,20,21,22. into the resonator . On the other hand, undercoupled In this paper we discuss the use of CPWs in the con- resonators with large quality factors can be used to store text of quantum optics and quantum information pro- photons in the cavity on a long time scale, with potential 30 cessing. In the recent past it has been experimentally use as a quantum memory . demonstrated that a single microwave photon stored in The paper is structured as follows. In Sec. II we dis- a high quality CPW resonator can be coherently coupled cuss the chosen CPW device geometry, its fabrication to a superconducting quantum two-level system11. This and the measurement techniques used for characteriza- possibility has lead to a wide range of novel quantum op- tion at microwave frequencies. The dependence of the tics experiments realized in an architecture now known as CPW resonator frequency on the device geometry and its circuit quantum electrodynamics (QED)11. The circuit electrical parameters is analyzed in Sec. III. In Sec. IV QED architecture is also successfully employed in quan- the effect of the resonator coupling to an input/output tum information processing23 for coherent single qubit line on its quality factor, insertion loss and resonance fre- control11, for dispersive qubit read-out12 and for cou- quency is analyzed using a parallel LCR circuit model. pling individual qubits to each other using the resonator This lumped element model provides simple approxima- as a quantum bus17,20. tions of the resonator properties around resonance and Coplanar waveguide resonators have a number of ad- allows to develop an intuitive understanding of the de- vantageous properties with respect to applications in cir- vice. We also make use of the transmission (or ABCD) cuit QED. CPWs can easily be designed to operate at matrix method to describe the full transmission spec- frequencies up to 10 GHz or higher. Their distributed el- trum of the resonators and compare its predictions to ement construction avoids uncontrolled stray inductances our experimental data. The characteristic properties of and capacitances allowing for better microwave prop- the higher harmonic modes of the CPW resonators are erties than lumped element resonators. In comparison discussed in Sec. V. to other distributed element resonators, such as those based on microstrip lines, the impedance of CPWs can be controlled at different lateral size scales from millime- II. DEVICE GEOMETRY, FABRICATION AND ters down to micrometers not significantly constrained MEASUREMENT TECHNIQUE by substrate properties. Their potentially small lateral dimensions allow to realize resonators with extremely The planar geometry of a capacitively coupled CPW large vacuum fields due to electromagnetic zero-point resonator is sketched in Figure 1a. The resonator is 24 arXiv:0807.4094v1 [cond-mat.supr-con] 25 Jul 2008 fluctuations , a key ingredient for realizing strong cou- formed of a center conductor of width w = 10 µm sepa- pling between photons and qubits in the circuit QED ar- rated from the lateral ground planes by a gap of width chitecture. Moreover, CPW resonators with large inter- s = 6.6 µm. Resonators with various center conduc- nal quality factors of typically several hundred thousands tor lengths l between 8 and 29 mm aiming at funda- 25,26,27,28 can now be routinely realized . mental frequencies f0 between 2 and 9 GHz were de- In this paper we demonstrate that we are able to designed. These structures are easily fabricated in opti- 2

FIG. 1: (Color online) (a) Top view of a CPW resonator with finger capacitors (l.h.s.) and gap capacitors (r.h.s.). (b) Cross section of a CPW resonator design. Center conductor and lateral ground metallization (blue) on top of a double layer substrate (grey/yellow). Parameters are discussed in the main text. cal lithography while providing sufficiently large vacuum field strengths24. The center conductor is coupled via FIG. 2: (Color online) Optical microscope images of an Al gap- or finger capacitors to the input and output trans- coplanar waveguide resonator (white is metallization, grey is mission lines. For small coupling capacitances gap capac- substrate). The red squares in the upper image indicate the positions of the input/output capacitors. Also shown are mi- itors of widths wg = 10 to 50 µm have been realized. To achieve larger coupling, finger capacitors formed by from croscope images of finger- and gap capacitor structures. The labels D, E, H, I, K refer to the device ID listed in Tab. II. one up to eight pairs of fingers of length lf = 100 µm, width wf = 3.3 µm and separation sf = 3.3 µm have been designed and fabricated, see Fig. 1. The resonators are fabricated on high resistivity, un- the refrigerator as well as one or two room temperature doped, (100)-oriented, thermally oxidized two inch sili- amplifiers with 35 dB gain each. Low Q resonators were characterized without additional amplifiers. con wafers. The oxide thickness is h2 = 550 nm ±50 nm determined by SEM inspection. The bulk resistivity of The measured Q of undercoupled devices can vary the Si wafer is ρ > 3000 Ω cm determined at room tem- strongly with the power applied to the resonator. In our perature in a van-der-Pauw measurement. The total measurements of high Q devices the resonator transmis- thickness of the substrate is h1 = 500 µm ±25 µm. A sion spectrum looses its Lorentzian shape at drive powers cross-sectional sketch of the CPW resonator is shown in above approximately −70 dBm at the input port of the Fig. 1b. resonator due to non-linear effects32. At low drive pow- The resonators were patterned in optical lithography ers, when dielectric resonator losses significantly depend using a one micron thick layer of the negative tone re- on the photon number inside the cavity28,33, measured sist ma-N 1410. The substrate was subsequently met- quality factors may be substantially reduced. We ac- allized with a t = 200 nm ±5 nm thick layer of Al, elec- quired S21 transmission spectra at power levels chosen tron beam evaporated at a rate of 5 A/sec˚ and lifted-off to result in the highest measurable quality factors, i.e. at in 50◦ C hot acetone. Finally, all structures were diced high enough powers to minimize dielectric loss but low into 2 mm × 7 mm chips, each containing an individual enough to avoid non-linearities. This approach has been resonator. The feature sizes of the fabricated devices de- chosen to be able to focus on geometric properties of the viate less than 100 nm from the designed dimensions as resonators. determined by SEM inspection indicating a good control over the fabrication process. Altogether, more than 80 Al CPW resonators covering a wide range of different coupling strengths were designed III. BASIC RESONATOR PROPERTIES and fabricated. More than 30 of these devices were care- fully characterized at microwave frequencies. Figure 2 A typical transmission spectrum of a weakly gap ca- shows optical microscope images of the final Al resonators pacitor coupled (wg = 10 µm) CPW resonator of length with different finger and gap capacitors. l = 14.22 mm is shown in Figure 3a. The spectrum Using a 40 GHz vector network analyzer, S21 transmis- clearly displays a Lorentzian lineshape of width δf cen- sion measurements of all resonators were performed in a tered at the resonance frequency f0. Figure 3b shows 31 pulse-tube based dilution refrigerator system at tem- measured resonance frequencies f0 for resonators of dif- peratures of 20 mK. The measured transmission spectra ferent length l, all coupled via gap capacitors of widths are plotted in logarithmic units (dB) as 20 log10 |S21|. wg = 10 µm. Table I lists the respective values for l and High Q resonators were measured using a 32 dB gain f0. For these small capacitors the frequency shift induced high electron mobility transistor (HEMT) amplifier with by coupling can be neglected, as discussed in a later sec- noise temperature of ∼ 5 K installed at the 4 K stage of tion. In this case the resonator‘s fundamental frequency 3

ID Coupling Cκ (fF) f0 (GHz) QL A 8 + 8 finger 56.4 2.2678 3.7 · 102 B 7 + 7 finger 48.6 2.2763 4.9 · 102 C 6 + 6 finger 42.9 2.2848 7.5 · 102 D 5 + 5 finger 35.4 2.2943 1.1 · 103 E 4 + 4 finger 26.4 2.3086 1.7 · 103 F 3 + 3 finger 18.0 2.3164 3.9 · 103 G 2 + 2 finger 11.3 2.3259 9.8 · 103 H 1 + 1 finger 3.98 2.3343 7.5 · 104 I 10 µm gap 0.44 2.3430 2.0 · 105 J 20 µm gap 0.38 2.3448 2.0 · 105 K 30 µm gap 0.32 2.3459 2.3 · 105 L 50 µm gap 0.24 2.3464 2.3 · 105

FIG. 3: (Color online) (a) Transmission spectrum of a 4.7 TABLE II: Properties of the different CPW resonators whose GHz resonator. Data points (blue) were fitted (black) with a transmission spectra are shown in Fig. 4. Cκ denotes the Lorentzian line. (b) Measured f0 (red points) of several res- simulated coupling capacitances, f0 is the measured resonance onators coupled via wg = 10 µm gap capacitors with different frequency and QL is the measured quality factor. l together with a fit (blue line) to the data using Eq. (1) as fit function and eff as fit parameter. to be34,35

0 f0 (GHz) l (mm) µ0 K(k0) L` = , (2) 2.3430 28.449 4 K(k0) 3.5199 18.970 K(k ) C = 4 0 . (3) 4.6846 14.220 ` 0 eﬀ 0 K(k0) 5.8491 11.380 7.0162 9.4800 Here, K denotes the complete elliptic integral of the ﬁrst 8.1778 8.1300 kind with the arguments w TABLE I: Designed values for resonator lengths l and mea- k0 = , (4) sured resonance frequencies f , corresponding to the data w + 2s 0 q shown in Fig. 3. 0 2 k0 = 1 − k0. (5)

For non magnetic substrates (µeff = 1) and neglecting kinetic inductance for the moment L is determined by the f0 is given by ` CPW geometry only. C` depends on the geometry and eff . Although analytical expressions for eff exist for dou- c 1 ble layer substrates deduced from conformal mapping34, f0 = √ . (1) eff 2l the accuracy of these calculations depends sensitively on the ratio between substrate layer thicknesses and the di- √ mensions of the CPW cross-section36 and does not lead Here, c/ eff = vph is the phase velocity depending on to accurate predictions for our parameters. Therefore, the velocity of light in vacuum c and the effective permit- −10 −1 we have calculated C` ≈ 1.27 · 10 Fm using a finite tivity eff of the CPW line. eff is a function of the waveg- element electromagnetic simulation and values 1 = 11.6 uide geometry and the relative permittivities 1 and 2 of (see Ref. 37) for silicon and 2 = 3.78 (see Ref. 37) for substrate and the oxide layer, see Fig. 1b. Furthermore, silicon oxide for our CPW geometry and substrate. From 2l = λ0 is the wavelength of the fundamental resonator this calculation we find eff ≈ 5.22 which deviates only mode. The length dependence of the measured reso- by about 3% from the value extracted from our measure- nance frequencies f0 of our samples is well described by ments. The characteristic impedance of a CPW is then Eq. (1) with the effective dielectric constant eff = 5.05, given by Z = pL /C which results in a value of 59.7 Ω see Fig. 3b. 0 ` ` √ for our geometry. This value deviates from the usually The phase velocity vph = 1/ L`C` of electromagnetic chosen value of 50 Ω as the original design was optimized waves propagating along a transmission line depends on for a different substrate material. the capacitance C` and inductance L` per unit length of In general, for superconductors the inductance L` is the line. Using conformal mapping techniques the geo- the sum of the temperature independent geometric (mag- m metric contribution to L` and C` of a CPW line is found netic) inductance L` and the temperature dependent 4

FIG. 4: (Color online) S21 transmission spectra of 2.3 GHz resonators, symmetrically coupled to input/output lines. The left part of the split plot shows spectra of ﬁnger capacitor coupled resonators whereas on the right hand side one can see spectra of gap capacitor coupled resonators. The data points (blue) were ﬁtted (black) with the transmission matrix method, see text.

k kinetic inductance L` (see Ref. 38). For superconduc- sured transmission spectra are shown in Fig. 4. The left k tors, L` refers to the inertia of moving Cooper pairs hand part of Fig. 4 depicts spectra of resonators cou- and can contribute significantly to L` since resistivity pled via finger capacitors having 8 down to one pairs of is suppressed and thus charge carrier relaxation times fingers (devices A to H). The right hand part of Fig. 4 k 2 are large. According to Ref. 35, L` scales with λ (T ), shows those resonators coupled via gap capacitors with where λ(T ) is the temperature dependent London pen- gap widths of wg = 10, 20, 30 and 50 µm (devices I to etration depth which can be approximated as35 λ(0) = L) respectively. The coupling capacitance continuously −3p p 1.05·10 ρ(Tc)/Tc K m/Ω at zero temperature in the decreases from device A to device L. The nominal val- local and dirty limits. In the dirty (local) limit the mean ues for the coupling capacitance Cκ obtained from EM- free path of electrons lmf is much less than the coherence simulations for the investigated substrate properties and length ξ0 =hv ¯ f /π∆(0), where vf is the Fermi velocity geometry are listed in table II. The resonance frequency of the electrons and ∆(0) is the superconducting gap en- f0 and the measured quality factor QL = f0/δf of the ergy at zero temperature39. The clean (nonlocal) limit respective device is obtained by fitting a Lorentzian line occurs when lmf is much larger than ξ0 (see Ref. 39). shape Tc = 1.23 K is the critical temperature of our thin film −9 aluminum and ρ(Tc) = 2.06·10 Ω m is the normal state δf FLor(f) = A0 2 2 , (6) resistivity at T = Tc. Tc and ρ(T ) were determined in a (f − f0) + δf /4 four-point measurement of the resistance of a lithograph- ically patterned Al thin film meander structure from the to the data, see Fig. 3a, where δf is the full width half same substrate in dependence on temperature. The re- maximum of the resonance. With increasing coupling ca- sulting residual resistance ratio (RRR300 K/1.3 K) is 8.6. pacitance Cκ, Fig. 4 shows a decrease in the measured Since our measurements were performed at temperatures (loaded) quality factor QL and an increase in the peak well below Tc, λ = λ(0) approximately holds and we find transmission, as well as a shift of f0 to lower frequencies. λ(0) ≈ 43 nm for our Al thin films (compared to a value of In the following, we demonstrate how these character- 40 nm, given in Ref. 40). Using the above approximation istic resonator properties can be fully understood and k shows that L` is about two orders of magnitude smaller modeled consistently for the full set of data. m −7 −1 than L` = 4.53·10 Hm legitimating the assumption A transmission line resonator is a distributed device m L` ≈ L` made in Eq. (2). Kinetic inductance effects in with voltages and currents varying in magnitude and Niobium resonators are also analyzed in Ref. 25. phase over its length. The distributed element representation of a symmetrically coupled resonator is shown in Fig. 5a. R`, L` and C` denote the resistance, inductance IV. INPUT/OUTPUT COUPLING and capacitance per unit length, respectively. According to Ref. 41 the impedance of a TL resonator is given by To study the effect of the capacitive coupling strength 1 + i tan βl tanh αl on the microwave properties of CPW resonators, twelve ZTL = Z0 (7) 2.3 GHz devices, symmetrically coupled to input/output tanh αl + i tan βl lines with different gap and finger capacitors have been Z0 ≈ π . (8) characterized, see Table II for a list of devices. The mea- αl + i (ω − ωn) ω0 5

FIG. 5: (Color online) (a) Distributed element representation of symmetrically coupled TL resonator. (b) Parallel LCR oscillator representation of TL resonator. (c) Norton equivalent of symmetrically coupled, parallel LCR oscillator. Symbols are explained in text.

α is the attenuation constant and β = ωn/vph is the phase The small capacitor Cκ transforms the RL = 50 Ω ∗ 2 constant of the TL. The approximation in Eq. (8) holds load into the large impedance R = RL/k with k = when assuming small losses√ (αl 1) and for ω close to ωnCκRL 1. For symmetric input/output coupling the ωn. Here, ωn = nω0 = 1/ LnC is the angular frequency loaded quality factor for the parallel combination of R of the n-th mode, where n denotes the resonance mode and R∗/2 is number (n = 1 for the fundamental mode). C + 2C∗ Around resonance, the properties of a TL resonator can Q = ω∗ (16) be approximated by those of a lumped element, parallel L n 1/R + 2/R∗ LCR oscillator, as shown in Fig. 5b, with impedance C ≈ ωn (17) −1 1/R + 2/R∗ 1 1 ZLCR = + iωC + (9) iωLn R with the n-th resonance frequency shifted by the capac- R itive loading due to the parallel combination of C and ≈ , (10) 2C∗ 1 + 2iRC(ω − ωn) 1 and characteristic parameters ω∗ = . (18) n pL (C + 2C∗) 2L l n L = ` , (11) n 2 2 ∗ ∗ n π For ωn ≈ ωn with C + 2C ≈ C, the Norton equivalent C`l expression for the loaded quality factor Q is a parallel C = , (12) L 2 combination of the internal and external quality factors Z0 R = . (13) 1 1 1 αl = + , (19) QL Qint Qext The approximation Eq. (10) is valid for ω ≈ ωn. The LCR model is useful to get an intuitive understanding with of the resonator properties. It simplifies analyzing the nπ effect of coupling the resonator to an input/output line Qint = ωnRC = , (20) 2αl on the quality factor and on the resonance frequency as ∗ ωnR C discussed in the following. Qext = . (21) The (internal) quality factor of the parallel LCR os- 2 p cillator is defined as Qint = R C/Ln = ωnRC. The The measured loaded quality factor QL for devices A quality factor QL of the resonator coupled with capaci- to L is plotted vs. the coupling capacitance in Fig. 6a. tance Cκ to the input and output lines with impedance QL is observed to be constant for small coupling capac- Z0 is reduced due to the resistive loading. Additionally, itances and decreases for large ones. In the overcoupled the frequency is shifted because of the capacitive loading regime (Qext Qint), QL is governed by Qext which 2 of the resonator due to the input/output lines. To under- is well approximated by C/2ωnRLCκ, see dashed line in stand this effect the series connection of Cκ and RL can Fig. 6. Thus, in the overcoupled regime the loaded qual- −2 be transformed into a Norton equivalent parallel connec- ity factor QL ∝ Cκ can be controlled by the choice tion of a resistor R∗ and a capacitor C∗, see Figs. 5b, c, of the coupling capacitance. In the undercoupled limit with (Qext Qint) however, QL saturates at the internal 5 2 2 2 quality factor Qint ≈ 2.3 · 10 determined by the intrin- ∗ 1 + ωnCκRL R = 2 2 , (14) sic losses of the resonator, see horizontal dashed line in ωnCκRL Fig. 6a. Radiation losses are expected to be small in CPW 42 ∗ Cκ resonators , resistive losses are negligible well below the C = 2 2 2 . (15) 25 1 + ωnCκRL critical temperature Tc of the superconductor and at 6

FIG. 7: (Color online) (a) Comparison of Cκ values extracted from the measured quality factors using Eqs. (14, 19, 21) to the EM-simulated values for Cκ (red points) for devices A to H. The blue curve is a line through origin with slope one. (b) Dependence of f0 on Cκ. The mapped LCR model prediction given by Eqs. (15, 18) is shown (blue line) for resonators coupled via finger capacitors together with the measured values FIG. 6: (Color online) (a) Dependence of QL on Cκ. Data for f0 (red points). points (red) are measured quality factors. These values are compared to QL predictions by the mapped LCR model (solid blue line) given by Eqs. (14, 19, 21). (b) Dependence of L0 on Cκ. Data points (red) show measured L0 values. The using Eqs. (19, 22), thus allowing to roughly estimate values are compared to the mapped LCR model (solid blue internal losses even of an overcoupled cavity. line) given by Eqs. (14, 21, 22). Dashed lines indicate the limiting cases for small and large coupling capacitances (see For the overcoupled devices A to H the coupling induced text). resonator frequency shift as extracted from Fig. 4 is in good agreement with calculations based on Eqs. (15, 18), ∗ see Fig. 7b. For C ≈ Cκ and C Cκ one can Taylor- ∗ frequencies well below the superconducting gap. We be- approximate ωn as ωn(1−Cκ/C). As a result the relative ∗ lieve that dielectric losses limit the internal quality factor resonator frequency shift is (ωn − ωn)/ωn = −Cκ/C of our devices, as discussed in References 33 and 28. for symmetric coupling. Figure 7b shows the expected Using Eqs. (14, 19, 21), Cκ has been extracted from linear dependence with a maximum frequency shift of 5 the measured value of Qint ∼ 2.3 · 10 and the measured about 3% over a range of 60 fF in Cκ. loaded quality factors QL of the overcoupled devices A to H, see Fig. 7. The experimental values of Cκ are in As an alternative method to the LCR model which is good agreement with the ones found from finite element only an accurate description near resonance we have an- calculations, listed in table II, with a standard deviation alyzed our data using the transmission matrix method41. of about 4%. Using this method the full transmission spectrum of the The insertion loss CPW resonator can be calculated. However, because of g the mathematical structure of the model it is more in- L = −20 log dB (22) 0 g + 1 volved to gain intuitive understanding of the CPW devices. of a resonator, i.e. the deviation of peak transmission All measured S21 transmission spectra are consistently from unity, is dependent on the ratio of the internal fit with a single set of parameters, see Fig. 4. The trans- to the external quality factor which is also called the mission or ABCD matrix of a symmetrically coupled TL coupling coefficient g = Qint/Qext (see Ref. 41). The is defined by the product of an input-, a transmission-, measured values of L0 as extracted from Fig. 4 are and an output matrix as shown in Fig. 6b. For g > 1 (large Cκ) the resonator is overcoupled and shows near unit transmission (L0 = 0). The resonator is said to be critically coupled for g = 1. ! ! ! ! AB 1 Zin t11 t12 1 Zout For g < 1 (small Cκ) the resonator is undercoupled and = , (23) the transmission is significantly reduced. In this case L0 CD 0 1 t21 t22 0 1 2 is well approximated by −20 log(2ωnQintRLCκ/C), see dashed line in Fig. 6b, as calculated from Eqs. (14, 21, 22). Qext and Qint can be determined from QL and L0 with input/output impedances Zin/out = 1/iωCκ and the 7 transmission matrix parameters

t11 = cosh (γl), (24)

t12 = Z0 sinh (γl), (25)

t21 = 1/Z0 sinh (γl), (26)

t22 = cosh (γl). (27)

Here, γ = α + iβ is the TL wave propagation coefficient. The resonator transmission spectrum is then defined by the ABCD matrix components as 2 S21 = . (28) A + B/RL + CRL + D Here, R is the real part of the load impedance, account- L FIG. 8: Measured quality factors for the overcoupled res- ing for outer circuit components. α is determined by Qint onator D vs. mode number n (red points) together with the and l and β depends on eff as discussed before. Accord- prediction of the mapped LCR model given by Eqs. (18, 21) ing to Eqs. (2, 3) Z0 is determined by eff , w and s. The (solid blue line). The inset shows the S21 transmission spec- attenuation constant is α ∼ 2.4 · 10−4 m−1 as determined trum of resonator D with fundamental mode and harmonics. 5 from Qint ∼ 2.3 · 10 . The measured data (blue) is compared to the S21 spectrum For gap capacitor coupled devices, the measured data (black) obtained by the ABCD matrix method. fits very well, see Fig. 4, to the transmission spectrum calculated using the ABCD matrix method with eff = 5.05, already obtained from the measured dependence of f0 on measured data for fundamental and harmonic modes. In the resonator length, see Fig. 3. For finger capacitor cou- the case of resonators coupled via finger capacitors simu- pled structures however, see Fig. 1a, approximately 40% lated values for Cκ deviate by only about 4%. About 40% of the length of each 100 µm finger has to be added to the of the capacitor finger length has to be added to the to- length l of the bare resonators in order to obtain good tal resonator length to obtain a good fit to the resonance fits to the resonance frequency f0. This result is inde- frequency. pendent of the number of fingers. The ABCD matrix The resonator properties discussed above are consis- model describes the full transmission spectra of all mea- tent with those obtained from measurements of addi- sured devices very well with a single set of parameters, tional devices with fundamental frequencies of 3.5, 4.7, see Fig. 4. 5.8, 7.0 and 8.2 GHz. The experimental results presented in this paper were obtained for Al based resonators on an oxidized silicon substrate. The methods of analysis V. HARMONIC MODES should also be applicable to CPW devices fabricated on different substrates and with different superconducting So far we have only discussed the properties of the materials. The good understanding of geometric and fundamental resonance frequency of any of the measured electrical properties of CPW resonators will certainly fos- resonators. A full transmission spectrum of the overcou- ter further research on their use as radiation detectors, pled resonator D, including 5 harmonic modes, is shown in quantum electrodynamics and quantum information in Fig. 8. The measured spectrum fits well to the ABCD processing applications. matrix model for the fundamental frequency and also for higher cavity modes, displaying a decrease of the loaded quality factor with harmonic number. The dependence Acknowledgments of the measured quality factor QL on the mode number n is in good agreement with Eqs. (19, 21) and scales ap- We thank P. Fallahi for designing the optical lithogra- 2 proximately as C/2nω0RLCκ. phy mask used for fabricating the devices and L. Frunzio and R. Schoelkopf at Yale University for their continued collaboration on resonator fabrication and characteriza- VI. CONCLUSIONS tion. We also thank the Yale group and the group of M. Siegel at the Institute for Micro- and Nanoelectronic In summary, we have designed and fabricated symmet- Systems at University of Karlsruhe for exchange of ma- rically coupled coplanar waveguide resonators over a wide terials and preparation of thin films. Furthermore we range of resonance frequencies and coupling strengths. acknowledge the ETH Zürich FIRST Center for Micro- We demonstrate that loaded quality factors and reso- and Nanoscience for providing and supporting the device nance frequencies can be controlled and that the LCR- fabrication infrastructure essential for this project. We and ABCD matrix models are in good agreement with acknowledge discussions with J. Martinis and K. Lehn- 8 ert and thank D. Schuster for valuable comments on the Fund (SNF) and ETH Zürich. P. J. L. was supported by manuscript. This work was supported by Swiss National the EC with a MC-EIF.

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