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CPW Kappa Calculation

Let’s start by importing Qiskit Metal:

import qiskit_metal as metal
from qiskit_metal import designs, draw
from qiskit_metal import MetalGUI, Dict, open_docs

Next, let’s import the function “kappa_in” located in the file kappa_calculation.py. This function calculates the photon loss of a CPW resonator which is capacitively coupled to an input transmission line.

# Import the function "kappa_in" from the file kappa_calculation.py
from qiskit_metal.analyses.em.kappa_calculation import kappa_in

The function “kappa_in” takes either three or six arguments, depending on how the lowest resonant frequency of the resonator is handled. In the first case, the resonant frequency of the CPW resonator is calculated numerically (using HFSS, for example) and passed as a floating-point input along with the frequency of interest and the capacitance between the resonator and the transmission line. In the second case, the lowest resonant frequency of the CPW resonator can be estimated by assuming an ideal resonator, in which case some additional inputs are required (1/2 or 1/4 depending on the type of resonator, the resonator length, width of resonator trace, width of resonator gap.)

Here’s a quick sanity check to verify that we only get numerical output from this function in the cases of N=3 or N=6 arguments:

# SANITY CHECK #1
# Let's check that output is only given for three and six arguments
print("Output for N=1 Args:", kappa_in(1.0))
print("Output for N=2 Args:", kappa_in(1.0, 1.0))
print("Output for N=3 Args:", kappa_in(1.0, 1.0, 1.0))
print("Output for N=4 Args:", kappa_in(1.0, 1.0, 1.0, 1.0))
print("Output for N=5 Args:", kappa_in(1.0, 1.0, 1.0, 1.0, 1.0))
print("Output for N=6 Args:", kappa_in(1.0, 1.0, 1.0, 1.0, 1.0, 1.0))
print("Output for N=7 Args:", kappa_in(1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0))

Output for N=1 Args: None
Output for N=2 Args: None
Output for N=3 Args: 1591.5494309189535
Output for N=4 Args: None
Output for N=5 Args: None
Output for N=6 Args: 3234.721973158391
Output for N=7 Args: None

Now, let’s actually calculate the photon loss for a representative CPW resonator with realistic values of input parameters. Here we’ll assume a resonant frequency of 5 GHz, capacitive coupling of 30fF and a CPW resonant frequency of 4 GHz. The calculated value of kappa is the range of 0-1 MHz, as expected.

# SANITY CHECK #2
# Let's check that the magnitude of the output is what we would expect for 3 arguments:
# Input #1: omega = 5GHz = 5E9 Hertz
# Input #2: C_in = 30fF = 30E-15 Farads
# Input #3: omega_n = 4GHz = 4.5E9 Hertz
print("Calculated kappa (in Hz):", kappa_in(5.0E9, 30.0E-15, 4.5E9), "Hz")
print("Calculated kappa (in MHz):", kappa_in(5.0E9, 30.0E-15, 4.5E9)/1.0E6, "MHz")

Calculated kappa (in Hz): 161144.37988054403 Hz
Calculated kappa (in MHz): 0.16114437988054403 MHz

For more information, review the Introduction to Quantum Computing and Quantum Hardware lectures below

Superconducting Qubits I: Quantizing a Harmonic Oscillator, Josephson Junctions Part 1	Lecture Video	Lecture Notes	Lab
Superconducting Qubits I: Quantizing a Harmonic Oscillator, Josephson Junctions Part 2	Lecture Video	Lecture Notes	Lab
Superconducting Qubits I: Quantizing a Harmonic Oscillator, Josephson Junctions Part 3	Lecture Video	Lecture Notes	Lab
Superconducting Qubits II: Circuit Quantum Electrodynamics, Readout and Calibration Methods Part 1	Lecture Video	Lecture Notes	Lab
Superconducting Qubits II: Circuit Quantum Electrodynamics, Readout and Calibration Methods Part 2	Lecture Video	Lecture Notes	Lab
Superconducting Qubits II: Circuit Quantum Electrodynamics, Readout and Calibration Methods Part 3	Lecture Video	Lecture Notes	Lab