Resonator geometry and design. (a) A single resonator device is shown that has W = 10 μm and S = 5 μm. The resonator is meandered to keep the chip size small enough to avoid package resonances near the first few harmonics of the resonator. There is a gap of size G = 4 μm in the straight section of this design, not visible at this resolution, which couples the feedline to the resonator. (b) Zoomed-in views of the resonator-to-feedline coupling are shown. The left and center pictures show an L-coupler of length l C of the type used for the multiple resonator devices and single resonator devices, respectively. The right picture shows a gap coupler of size G that has a width equal to that of the resonator and feedline center trace, W. The spacing S between ground planes and center conductor is also shown. (c) A multiple resonator device is shown with four resonators having different lengths for frequency multiplexing. The length difference is chosen so that adjacent resonators have frequencies that differ by ∼100 MHz. The particular device shown has resonators of different widths with W = 3, 10, 25, and 75 μm. The full chip [(a) and (c)] is ∼5 × 15 mm.
(Color online) Internal Q vs circulating power and equivalent average photon number for resonators of different materials for (a) W = 3 μm and (b) W = 10 μm. The symbols indicate the different materials: poly-Nb/wet SiO2/Si; poly-Nb/Si; poly-Nb/sapphire; epi-Nb/sapphire; poly-Al/dry SiO2/Si; poly-Al/sapphire; poly-Al/Si; epi-Al/sapphire; epi-Re/sapphire; TiN/Si. The dashed lines are fits to Eq. (6).
(Color online) Comparison of resonators with varying width. The data are taken using a Nb/SiO2/Si multiplexed resonator device that has four resonators: W = 3, S = 2 μm; W = 10, S = 5 μm; W = 25, S = 15 μm; W = 75, S = 50 μm. (The W = 75 μm resonator data are not shown due to its very low Q at all temperature and power levels). The resonance frequencies are between 2 and 3 GHz. (a) The resonance center frequency vs temperature is shown for W = 3 μm , W = 10 μm , and W = 25 μm . The resonance frequency is expressed as a fractional change from the value at 300 mK. The dotted lines are fits to the data using Eq. (7). It can be seen that the frequency dependence is stronger for narrower resonators. (b) The internal Q vs circulating power is shown for W = 3 μm , W = 10 μm ), and W = 25 μm . The dotted lines are fits to Eq. (6). It can be seen that the loss at low power is larger for narrower resonators. Solid data points appear at the location corresponding to an average single-photon circulating power level; their values correspond to what we obtain by adding in parallel the values extracted from the fits in (a) to the Q 0 values extracted from the fits in (b). Doing this allows us to compare the low-power internal Q value predicted from the frequency vs temperature data with the direct measurement of the low-power internal Q.
(Color online) Internal Q vs the transverse resonator dimension, S+W. The internal Q measured at high excitation power for eight resonators is plotted vs the value of S + W for each resonator. The data are taken using two Re/sapphire multiplexed resonator devices having four resonators each. Each device has resonators with S values of 5, 15, 30, and 50 μm and fixed width: W = 3 μm () and W = 10 μm . It can be seen that Q I decreases as S + W increases and that all the points appear to lie on a common curve. Because the trend is observed for both devices separately, where W is fixed, the most likely explanation is a radiation loss that increases with S + W. The dotted line is a fit to Eqs. (8) and (9) from which we extract a dependence of Q I arising from radiation loss Q rad.
(Color online) Pump/probe measurements. The first four harmonics of a W = 10, S = 5, G = 4 μm Nb/SiO2/Si single resonator device are investigated. The internal Q is plotted vs incident power of a probe microwave tone for the (a) fundamental at 2.1381 GHz, (b) second harmonic at 4.2751 GHz, (c) third harmonic at 6.4116 GHz, and (d) fourth harmonic at 8.5485 GHz. Each plot has five traces corresponding to the cases where either a second high-power pump tone is sent to the resonator with a frequency near the fundamental , second harmonic , third harmonic , fourth harmonic , or where there is no pump tone present . The detuning of the pump tone from its nearby resonance is chosen to be 20, 30, 40, and 50 kHz for the fundamental, second, third, and fourth harmonic, respectively. The pump power value is chosen to be such that the high-power quality factor saturates to its maximum value when the pump is near the same harmonic as the probe; here, the incident pump power is approximately −20 dBm. The data show that in all cases, Q I is maximally enhanced by the presence of the pump whenever the pump frequency is near the same harmonic as the probe; this indicates that the TLS interact resonantly with the pump and probe. We also observe enhancement of Q I when the pump is near a harmonic different from that of the probe. We claim that this enhancement is in general due to a heating effect except in the cases where the probe frequency is near a multiple of the pump frequency. We argue that the cause for the enhancement in these special cases is multiphoton driving of TLS. The instances of n-photon driving are indicated by the black arrows in (b) and (d) for n = 2 and (c) for n = 3.
Fit values for tan δ eff.
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