超導量子位元運作的過程中，我們會需要一個共振器來執行色散讀取。利用兩者耦合後，共振器共振頻率的平移來得知量子位元的狀態。然而，當共振器的線寬過寬時，將不利色散讀取，又共振器的線寬與其能量耗散的程度相關，因此製造出一個低損耗的共振器是至關重要的。

即使超導體的電阻為零，在超導共振器中仍存在著諸多如介電耗損、輻射耗損、準粒子耗損等能量耗損機制。在超導共振器運作的10mK環境中，元件內的非晶結構與材料間的介面提供的二能階系統導致介電耗損是主要的耗散機制。當共振器的固有品質因子主要受二能階系統給限制時，我們便能透過分析其固有品質因子而獲得其介電損耗的耗損正切。本論文中，我們使用電子束蒸鍍的鋁薄膜，搭配一道黃光微影製程與一道濕式鋁蝕刻製程來製造U型設計的超導平面共振器。接著置入三維鋁製波導中，放進稀釋致冷機降至10mK進行品質因子量測。

我們從量測結果中觀察到二能階系統的飽和效應，以及共振頻率與固有品質因子隨溫度下降的趨勢。此外，以矽為基板的共振器的固有品質因子在低光子區間可達到10^4；維持相同平均光子數、相同製程、相同鋁薄膜厚度，將矽基板換成藍寶石基板後，固有品質因子可以達到10^6，與矽基板相比，相差了兩個數量級。但我們也同時發現到以目前製程製造的平面超導共振器的固有品質因子會隨著時間下降，過了兩個月就掉了三個數量級。推測有可能是製程不均勻、就地氧化層不夠緻密、以及側壁氧化所導致。

In operating quantum computer based on superconducting qubits, we need a resonator to perform dispersive readout. By coupling the two, we can infer the state of the qubit from the shift of the resonant frequency of the resonator. However, if the linewidth of the resonator is too wide, it will limit the sensitivity of the dispersive readout. The linewidth of the resonator is related to its energy dissipation. Therefore, it is essential to fabricate a low-loss resonator.

Even though the resistance of the superconductor is zero, there are still various energy dissipation mechanisms in superconducting resonators, such as dielectric loss, radiative loss, quasiparticle loss, etc. In 10mK environment where the superconducting resonator operates, two-level systems provided by the amorphous structure and the interfaces in the device cause dielectric loss to be the main dissipation mechanism. If the intrinsic quality factor of the resonator is mainly limited by two-level systems, we can obtain its dielectric loss tangent by analyzing its intrinsic quality factor. In this thesis, we use electron beam evaporation to deposit aluminum thin films, and use photolithography process and wet aluminum etching process to fabricate U-shaped two-dimensional superconducting resonators. Then it was coupled with a three-dimensional aluminum waveguide and cooled down to 10mK for quality factor measurement.

We observe the saturation effect of two-level systems and the trend of resonant frequency and intrinsic quality factor decreasing with temperature from the measurement results. In addition, the intrinsic quality factor of the resonator with silicon substrate can reach 10^4 at low photon regime. Keeping the same average photon number, same process, and same aluminum film thickness, the intrinsic quality factor can reach 10^6 with sapphire substrate, which is two orders of magnitude higher than that of silicon substrate. We also found that the intrinsic quality factor of the two-dimensional superconducting resonator fabricated by the current process will decrease over time. It dropped by three orders of magnitude after two months. It may be caused by uniformity of the fabrication process, insufficient density of in-situ oxidation layer, and sidewall oxidation.

[1] D. M. Pozar, Microwave engineering. John wiley & sons, 2011.
[2] C. R. H. McRae, H. Wang, J. Gao, M. R. Vissers, T. Brecht, A. Dunsworth, D. P. Pap-
pas, and J. Mutus, “Materials loss measurements using superconducting microwave res-
onators,” Review of Scientific Instruments, vol. 91, no. 9, 2020.
[3] V. V. Schmidt and P. Müller, The physics of superconductors: Introduction to fundamentals
and applications. Springer Science & Business Media, 1997.
[4] R. Feynman, “Simulating physics with computers, 1982, reprinted in: Feynman and com-
putation,” 1999.
[5] M. A. Nielsen and I. Chuang, “Quantum computation and quantum information,” 2002.
[6] J. M. Martinis, M. H. Devoret, and J. Clarke, “Energy-level quantization in the zero-
voltage state of a current-biased josephson junction,” Physical review letters, vol. 55,
no. 15, p. 1543, 1985.
[7] Y. Nakamura, Y. A. Pashkin, and J. Tsai, “Coherent control of macroscopic quantum states
in a single-cooper-pair box,” nature, vol. 398, no. 6730, pp. 786–788, 1999.
[8] E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation
theories with application to the beam maser,” Proceedings of the IEEE, vol. 51, no. 1,
pp. 89–109, 1963.
[9] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quan-
tum electrodynamics for superconducting electrical circuits: An architecture for quantum
computation,” Physical Review A, vol. 69, no. 6, p. 062320, 2004.
[10] M. Boissonneault, J. M. Gambetta, and A. Blais, “Dispersive regime of circuit qed:
Photon-dependent qubit dephasing and relaxation rates,” Physical Review A, vol. 79,
no. 1, p. 013819, 2009.
[11] C. Müller, J. H. Cole, and J. Lisenfeld, “Towards understanding two-level-systems in
amorphous solids: insights from quantum circuits,” Reports on Progress in Physics,
vol. 82, no. 12, p. 124501, 2019.
[12] J. Gao, The physics of superconducting microwave resonators. California Institute of
Technology, 2008.
[13] C. E. Murray, “Material matters in superconducting qubits,” Materials Science and Engi-
neering: R: Reports, vol. 146, p. 100646, 2021.
[14] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Microscopic theory of superconductivity,”
Physical Review, vol. 106, no. 1, p. 162, 1957.
[15] J. Zmuidzinas, “Superconducting microresonators: Physics and applications,” Annu. Rev.
Condens. Matter Phys., vol. 3, no. 1, pp. 169–214, 2012.
[16] R. Barends, J. Wenner, M. Lenander, Y. Chen, R. C. Bialczak, J. Kelly, E. Lucero, P. O＇
Malley, M. Mariantoni, D. Sank, et al., “Minimizing quasiparticle generation from stray
infrared light in superconducting quantum circuits,” Applied Physics Letters, vol. 99,
no. 11, 2011.
[17] J. Gao, M. Daal, A. Vayonakis, S. Kumar, J. Zmuidzinas, B. Sadoulet, B. A. Mazin, P. K.
Day, and H. G. Leduc, “Experimental evidence for a surface distribution of two-level
systems in superconducting lithographed microwave resonators,” Applied Physics Letters,
vol. 92, no. 15, 2008.
[18] P. J. Petersan and S. M. Anlage, “Measurement of resonant frequency and quality factor
of microwave resonators: Comparison of methods,” Journal of applied physics, vol. 84,
no. 6, pp. 3392–3402, 1998.
[19] M. Castellanos-Beltran and K. Lehnert, “Widely tunable parametric amplifier based on a
superconducting quantum interference device array resonator,” Applied Physics Letters,
vol. 91, no. 8, 2007.
[20] M. Žemlička, P. Neilinger, M. Trgala, M. Rehák, D. Manca, M. Grajcar, P. Szabó,
P. Samuely, Š. Gaži, U. Hübner, et al., “Finite quasiparticle lifetime in disordered su-
perconductors,” Physical Review B, vol. 92, no. 22, p. 224506, 2015.
[21] D. Zoepfl, P. R. Muppalla, C. Schneider, S. Kasemann, S. Partel, and G. Kirchmair, “Char-
acterization of low loss microstrip resonators as a building block for circuit qed in a 3d
waveguide,” AIP Advances, vol. 7, no. 8, 2017.
[22] G. Calusine, A. Melville, W. Woods, R. Das, C. Stull, V. Bolkhovsky, D. Braje, D. Hover,
D. K. Kim, X. Miloshi, et al., “Analysis and mitigation of interface losses in trenched
superconducting coplanar waveguide resonators,” Applied Physics Letters, vol. 112, no. 6,
2018.
[23] P. J. Petersan and S. M. Anlage, “Measurement of resonant frequency and quality factor
of microwave resonators: Comparison of methods,” Journal of applied physics, vol. 84,
no. 6, pp. 3392–3402, 1998.
[24] S. Probst, F. Song, P. A. Bushev, A. V. Ustinov, and M. Weides, “Efficient and robust
analysis of complex scattering data under noise in microwave resonators,” Review of Sci-
entific Instruments, vol. 86, no. 2, p. 024706, 2015.
[25] N. Chernov and C. Lesort, “Least squares fitting of circles,” Journal of Mathematical
Imaging and Vision, vol. 23, pp. 239–252, 2005.
[26] M. S. Khalil, M. Stoutimore, F. Wellstood, and K. Osborn, “An analysis method for asym-
metric resonator transmission applied to superconducting devices,” Journal of Applied
Physics, vol. 111, no. 5, p. 054510, 2012.