Interplay of coupling, residual, and quasiparticle losses for the frequency- and temperature-dependent quality factor of superconducting resonators

Abstract The overall, loaded quality factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> quantifies the loss of energy stored in a resonator. Here we discuss on general grounds how <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> of a planar microwave resonator made of a conventional superconductor should depend on temperature and frequency. We consider contributions to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> due to dissipation by thermal quasiparticles ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi>QP</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ), due to residual dissipation ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi>Res</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ), and due to coupling ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ). We present experimental data obtained with superconducting stripline resonators fabricated from lead (Pb), with different center conductor widths and different coupling gaps. We probe the resonators at various harmonics between 0.7 GHz and 6 GHz and at temperatures between 1.5 K and 7 K. We find a strongly frequency- and temperature-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , which we can describe by a lumped-element model. For certain resonators at lowest temperatures we observe a maximum in the frequency-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi>Res</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> match, and here the measured <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> can exceed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> .