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The unloaded quality factor of NbTiN and Ta quarterwave resonators vs applied microwave photon number in the resonator. Bath temperatures are 60 mK (closed symbols) and 310 mK (open symbols). Central line width is and gap width is . Frequencies of the resonators used are 3.7 (▼), 4.2 (◆), and 6.2 GHz for the NbTiN, and 3.2 (◼), 4.5 (●), and 5.0 (▲) GHz for Ta. The solid lines are fits using Eq. (1). The quality factors of the metal surfaces (dashed), exposed substrate surface (dotted) and a fixed loss term (dashed-dotted) are shown for the 3.7 GHz NbTiN data. The right inset shows the microwave loss in the single photon regime vs . The left inset shows the resonator geometry.
(a) The charge distribution (red denotes a positive charge, blue a negative, and green a neutral one), electric fields (red arrows), and magnetic fields (blue arrows) in the coplanar waveguide geometry. (b) The power dependence using Eq. (1) for a TLS distribution placed on the exposed substrate surface, top metal surface, substrate–metal (sub-met) interface and etched metal edges. , and at low intensity each surface is assumed to limit the to . (c) The normalized contribution to loss of the dielectric layers vs central line width , for (solid) and (dashed).
The unloaded quality factor vs applied photon number of NbTiN quarterwave resonators with the standard geometry and and (6.2 GHz), and with grooves etched in the exposed Si substrate with and (4.7 GHz), and and at frequencies of 4.2 (◼), 4.4 (●), and 5.2 (▲) GHz. Bath temperature is 60 mK. The left inset is a scanning electron microscope image from the standard coplanar waveguide design, the right inset shows the etched grooves near the open end of the resonator ( in both images). The cross section of the etched resonators is outlined in Fig. 2(a).
The quality factor of the dielectric layer containing TLS, its saturation field, and the additional loss factor for the superconducting metals and for their Si substrates, used for fitting the data in Fig. 1, using Eq. (1) and . Calculations have been done for and .
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