kappa_calculation

This code calculates the photon loss (kappa) due to the capacitive coupling between CPWs and input/output transmission lines in a quantum circuit.

Two cases are treated: In the first case, three arguments are passed to the function kappa_in and the resonant frequency of the CPW is input as a float. In the second case, six arguments are passed to kappa_in and the frequency of the CPW is calculated assuming an ideal CPW.

Key References:

D. Schuster, Ph.D. Thesis, Yale University (2007) https://rsl.yale.edu/sites/default/files/files/RSL_Theses/SchusterThesis.pdf

T. McConkey, Ph.D. Thesis, University of Waterloo (2018) https://uwspace.uwaterloo.ca/bitstream/handle/10012/13464/McConkey_Thomas.pdf?sequence=3&isAllowed=y

Mohebbi and Majedi, Superconducting Science and Technology 22, 125028 (2009) https://iopscience.iop.org/article/10.1088/0953-2048/22/12/125028/meta

P. Krantz, et al. Physical Review Applied 6, 021318 (2019) https://aip.scitation.org/doi/10.1063/1.5089550

M. Goppl, et al. Journal of Applied Physics 104, 113904 (2008) https://doi.org/10.1063/1.3010859

Functions

kappa_in(*argv)

Linewidth of a CPW resonator capacitively coupled to a matched line.
kappa_in(*argv)[source]

Linewidth of a CPW resonator capacitively coupled to a matched line.

Two call forms:

kappa_in(freq, C_in, freq_res)
kappa_in(freq, C_in, length, res_width, res_gap, eta)

Uses the weak-coupling result (e.g. Goppl et al. 2008) for a resonator coupled at one end through \(C_\mathrm{in}\) to a line of impedance \(Z_0 = 50\,\Omega\):

\[\kappa = \frac{\eta}{\pi}\,\omega^2 C_\mathrm{in}^2 Z_0^2\,\omega_r, \qquad \omega = 2\pi f,\ \omega_r = 2\pi f_r,\]

with \(\eta = 2\) for a half-wave and \(\eta = 4\) for a quarter-wave resonator (whose equivalent capacitance is half that of a half-wave line). The three-argument form assumes a half-wave resonator.

Units. The return value is \(\kappa/2\pi\) in Hz (the full width at half maximum of the resonance); multiply by \(2\pi\) for the energy decay rate in 1/s. Before #1204 the formula was evaluated with \(f\) in place of \(\omega\), which returned a value \((2\pi)^2\) smaller than \(\kappa/2\pi\).

In the six-argument form the resonance frequency is that of an ideal (conformal-mapping) CPW whose effective permittivity makes its impedance 50 Ohm for the given width and gap: \(f_r = c / (\eta\,\ell\sqrt{\epsilon_\mathrm{eff}})\) with \(\sqrt{\epsilon_\mathrm{eff}} = 30\pi K(k_0')/(Z_r K(k_0))\).

Parameters:

  • freq (float) – The frequency at which kappa is evaluated, in Hz (normally the resonance frequency).

  • C_in (float) – Coupling capacitance between the CPW and the feed line (from Q3D), in Farads.

  • freq_res (float) – Resonant frequency of the CPW (from HFSS), in Hz. Three-argument form only.

  • length (float) – Length of the CPW resonator, in meters.

  • res_width (float) – Width of the resonator center trace, in meters.

  • res_gap (float) – Width of the resonator gap, in meters.

  • eta (float) – 2.0 for a half-wavelength resonator; 4.0 for a quarter-wavelength resonator.

Returns:

\(\kappa/2\pi\) in Hz, or None (with a warning) if the number of arguments is neither 3 nor 6.

Return type:

float