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Transmon Analytics: Plotting Eigenvalues as a Function of Offset Charge

This demo notebook uses the Hamiltonian Cooper Pair Box (Hcpb) class to recreate the plots of energy as a function of offset charge, originally found in Koch et al. Phys. Rev. A 76, 042319 (2007).

We’ll start by importing the Hcpb class as well as numpy and matplotlib:

from qiskit_metal.analyses.hamiltonian.transmon_charge_basis import Hcpb
import numpy as np
import matplotlib.pyplot as plt

We’ll use the variable x to represent the offset charge (ng, measured in units of Cooper pair charge: 2e) which we’ll have vary between -2 and 2:

# We'll use the variable x to represent the offset charge (ng), which will vary from -2.0 to 2.0
x = np.linspace(-2.0,2.0,101)

Next, we’ll define a value for the Josephson Energy (E_J) as well as the ratio between the Josephson Energy and the Charging Energy. We will see that the ratio of E_J/E_C controls the anharmonicity of the qubit as well as the dispersion.

# Define a value of the Josephson Energy (E_J) as well as the ratio between the E_J and the Charging Energy (E_C):
E_J = 1000.0
ratio = 1.0
E_C = E_J/ratio

Now we will actually use the Hcpb class to calculate the transmon eigenvalues for a given value of offset charge. Note that in the plots found in the original paper, the eigenvalues are normalized by 0->1 transition energy evaluated at an offset charge of ng=0.5, so we first calculate that value and use it for normalization later. We create three empty lists, one for each energy level, then populate the lists with the corresponding eigenvalues for a given offset charge. Lastly, we calculate the minimum value of the lowest energy eigenvalue (called “floor”) and set this to E=0 in our final plots.

# we'll normalize the calculated energies by the 0->1 transition state energy evaluated at the degenercy point (ng=0.5)
H_norm = Hcpb(nlevels=2, Ej=E_J, Ec=E_C, ng=0.5)
norm = H_norm.fij(0,1)

# Next we'll empty lists to the first three eigenvalues (m=0, m=1, m=2):
E0 = []
E1 = []
E2 = []

# For a given value of offset charge (ng, represented by x) we will calculate the CPB Hamiltonian using the previously assigned values of E_J and E_C. Then we calculate the eigenvalue for a given value of m.
for i in x:
    H = Hcpb(nlevels=3, Ej=E_J, Ec=E_C, ng=i)
    E0.append(H.evalue_k(0)/norm)
    E1.append(H.evalue_k(1)/norm)
    E2.append(H.evalue_k(2)/norm)

# define the minimum of E0 and set this to E=0
floor = min(E0)

plt.plot(x, E0 - floor, 'k')
plt.plot(x, E1 - floor, 'r')
plt.plot(x, E2 - floor, 'b')
plt.xlabel("ng")
plt.ylabel("Em/E01")

Text(0, 0.5, 'Em/E01')

You can vary the value of ratio to 5.0, 10.0 and 50.0 to verify that the generated plots match those found in the original paper.

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