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1 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY1 Loss Dependence on Geometry and APPLIED Power inSuperconducting Coplanar ResonatorsMoe S. Khalil, F. C. Wellstood, and Kevin D. OsbornAbstract The loss in superconducting microwave resonators atlow-photon numbers and low temperatures is not well understoodbut has implications for achievable coherence times in supercon-ducting qubits. We have fabricated single-layer resonators witha high quality factor by patterning a superconducting aluminumfilm on a sapphire substrate. Four resonator geometries werestudied with resonant frequencies ranging from 5 to 7 GHz: aquasi-lumped element resonator, a coplanar strip waveguide res-onator, and two hybrid designs that contain both a coplanar stripand a quasi-lumped element.

2 Transmitted power measurementswere taken at 30 mK as a function of frequency and probe find that the resonator loss, expressed as the inverse of theinternal quality factor, decreases slowly over four decades ofphoton number in a manner not merely explained by loss from aconventional uniform spatial distribution of two-level systems inan oxide layer on the superconducting surfaces of the Terms Dielectric loss, superconducting microwave res-onators, superconducting quantum computing, two-level INTRODUCTIONTHE growing interest in superconducting quantum com-puting and microwave kinetic inductance detectors(MKIDS) has motivated the extensive study of superconductingresonators at milli-Kelvin temperatures in the limit of lowphoton numbers [1] [9].

3 Thin-film resonators operating in thislimit are potentially useful for photon detection [10] and ascomponents for storing and transferring information betweenqubits [11] [13]. It has been found that in the single-photonregime, some resonators are limited by amorphous dielectricloss due to two-level system (TLS) defects and that this typeof loss can be an important source of decoherence in super-conducting phase qubits [14]. However, the loss in coplanarresonators on crystalline dielectrics such as sapphire and crys-talline silicon is less clear and is more difficult to locate dueto non uniform field distributions. Many such resonators havebeen studied and the phase noise [1], [2], and loss [7], [9], areoften modeled as a surface distribution of previous research in this area has focused onquasi-one-dimensional cavity resonators, such as coplanarManuscript received August 03, 2010; accepted October 20, S.

4 Khalil iswith the Laboratory for Physical Sciences, College Park, MD20740 USA and with the Center for Nanophysics and Advanced Materials, De-partment of Physics , University of Maryland, College Park, MD 20740 USA(e-mail: C. Wellstood is with the Joint Quantum Institute and also with the Centerfor Nanophysics and Advanced Materials, Department of Physics , University ofMaryland, College Park, MD 20740 USA (e-mail: D. Osborn is with the Laboratory for Physical Sciences, College Park, MD20740 USA (e-mail: Object Identifier transmission line resonators [11] [13]. Lumpedelement devices are less popular but are sometimes used inquantum information, like in qubits [15], [16], and in Josephsonjunction resonators [17].)))

5 There is also growing interest in cou-pling quasi-lumped element resonators to qubits [18] becausethe lack of harmonic modes reduces loss from the Purcelleffect [19]. When coupling to a qubit, it s been found that thesymmetry of the qubit must be considered, since the type ofcoupling to it may affect coherence [20].Here we present measurements on the internal quality factor, of four distinct coplanar superconducting resonators be-tween 5 and 7 GHz, which include both quasi-lumped andquasi-one-dimensional cavity transmission-line symmetric shape induces inductive, rather than capaci-tive, coupling to RESONATORDESIGN ANDFABRICATIONAll resonators were fabricated with 100 nm thick sputteredaluminum films on c-plane sapphire wafers.

6 The aluminum waspatterned with positive photoresist and wet etched in a bath com-posed mainly of phosphoric and nitric of the resonators is a quasi-lumped element resonator,composed of a meandering quasi-lumped inductor (QLL) and aninterdigital quasi-lumped capacitor (QLC), shown in Fig. 1(a),with an inductance and a capacitance of approximately 2 nHand pF respectively. Another is a mm long shortedlength coplanar strip (CPS) resonator, shown in Fig. 1(b). Unlikein the quasi-lumped (QL) resonator the electric field in the CPSis distributed across the length of the resonator rather than beingconfined to a single element. The two others have both a CPSelement and a QL element, Fig. 1(c) and (d).

7 The four resonators were embedded in the ground plane ofthe same 50coplanar waveguide (CPW) and were coupledinductively (rather than capacitively) to that waveguide (seeFig. 2). Since these resonators primarily modify the trans-mission near resonance, they form a four-notch band-blocktransmission point connecting the two nominally symmetric halvesof the resonator (circle 1 in Fig. 1) is at the current anti-nodeand voltage node, and the sides of the resonators far from thecoplanar waveguide (circle 2 in Fig. 1) are at the voltage anti-nodes. The fundamental resonance frequency is antisymmetricin all four resonators. The effective capacitive coupling at thismode is weak, due to the design of the can be understood from the nearly symmetric shape ofthese structures, which implies the capacitances to either side ofthe resonators are approximately equal.

8 As a result, the capac-itance cannot couple to antisymmetric modes. Since the lowest1051-8223/$ 2010 IEEEThis article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of TRANSACTIONS ON APPLIED SUPERCONDUCTIVITYFig. 1. Optical images of four resonators measured (gray/white is aluminummetal and black is sapphire substrate). Circle 1 labels the current anti-node andvoltage node of resonators and Circle 2 labels one of the two current nodes andvoltage anti-nodes. (a) Quasi-lumped (QL) resonator at GHz. (b) Coplanarstrip (CPS) resonator at GHz. (c) Quasi-lumped inductor with a CPS (QLL-CPS) resonator at GHz. (d) Quasi-lumped capacitor with a coplanar strip(QLC-CPS) resonator at mode is antisymmetric we expect only inductive cou-pling between the resonators and the CPW.

9 An example of thisis shown in Fig. 2(f), where a capacitive network couples voltageV to a resonator. With a symmetric design,,,,, the coupling cannot excite structure of the resonance was confirmed with an EMsimulator. It should be pointed out that while the symmetry ofthe resonances is the same for all four resonators, the dimensionof resonances is not. The CPS resonator is a quasi-one-dimen-sional cavity, while the QL resonator acts like a quasi-zero-di-Fig. 2. (a) Schematic drawing of an arbitrary resonator inductively coupled toa coplanar waveguide. (b) (e) Schematic drawing of each of the four types ofresonators. (b) QL. (c) CPS. (d) QLL-CPS. (e) QLC-CPS. (f) Schematic of thecapacitive coupling to these resonators.

10 Due to the symmetry of these resonators , , , and , capacitive coupling cannotexcite an antisymmetric resonance. As a result, only inductive coupling remains.(g) Norton equivalent of a resonator coupled to the cavity. This is clear when simulating the full spec-trum of these structures, because for the CPS resonator higher-order harmonics occur every half wavelength but for the QL res-onator no higher order modes occur until frequencies above the resonance frequency each of the four resonators canbe represented by an equivalent lumped LC circuit. The Nortonequivalent circuit is shown in Fig. 2(g). The internal loss compo-nents such as TLSs discussed earlier, radiation, or metal loss canbe represented (for the purposes of this paper) by the resistor Rand act as a loss component lowering internal quality coupling of the resonator to the CPW can be represented bya resistorand acts as a loss component with coupling qualityfactor.