Fluxonium LCAO

Fluxonium wave function approximation by linear combination of atomic orbitals (LCAO), i.e., wave function basis in the fluxonium potential wells.

import numpy as np
import scipy.constants as pyc

from bfqcircuits.core import fluxonium as flx
from bfqcircuits.core import fluxonium_LCAO as flc

import matplotlib.pyplot as plt

fluxonium = flx.Fluxonium()
fluxonium_LCAO = flc.FluxoniumLCAO()

L = 2e-6
C = 2e-15
Ej = 20.0

fluxonium.set_parameters(L=L, C=C, Ej=Ej)
fluxonium.calc_hamiltonian_parameters()

fluxonium_LCAO.set_parameters(L=L, C=C, Ej=Ej)
fluxonium_LCAO.calc_hamiltonian_parameters()
print(fluxonium_LCAO.__repr__())

L = 2.0000e-06
C = 2.0000e-15
Ec = 7.7481e+01
El = 8.1731e-02
Ej = 2.0000e+01
Ejs = 0.0000e+00
Ejd = 0.0000e+00
ℏω = 2.5165e+00
ℏωⱼ = 3.9365e+01
u = 8.8023e+00
Z = 4.9003e+00
flux_zpf = 6.2446e-01
charge_zpf = 1.2743e-01

fig, ax = plt.subplots(2, 2, figsize=(8, 6), constrained_layout=True)
ax[0, 0].axis("off")

fluxonium.set_parameters(p_ext=1.0 * np.pi, N=60)
fluxonium.diagonalize_hamiltonian()
fluxonium.plot_fluxonium(ax[1, 0], 7, x_range=3.0, fill_between=False)

fluxonium_LCAO.set_parameters(p_ext=1.0 * np.pi)
fluxonium_LCAO.create_states([(-1, 2), (-1, 2), (-1, 2), (0, 1), (0, 1), (0, 1)])
fluxonium_LCAO.plot_states(ax[0, 1])

fluxonium_LCAO.diagonalize_hamiltonian()
fluxonium_LCAO.plot_fluxonium(ax[1, 1], 7, x_range=3.0, fill_between=False)

print(fluxonium.E[1] - fluxonium.E[0])
print(fluxonium_LCAO.E[1] - fluxonium_LCAO.E[0])

plt.show()

1.4916318279086056
1.5140829477194617

Convergence

fig, ax = plt.subplots(1, 2, figsize=(12, 5))

fluxonium.convergence_sweep(70)
fluxonium.plot_convergence_sweep(ax[0], 4)

ax[0].set_xlim(0, 60)
ax[0].set_yscale("log")
ax[0].set_ylim(1e-4, 10)

###
states = [(-1, 2)] * 5
states.extend([(0, 1)] * 5)   # with more states it becomes numerically unstable
fluxonium_LCAO.create_states(states)
print(fluxonium_LCAO.N)

fluxonium_LCAO.convergence_sweep(fluxonium_LCAO.N)
fluxonium_LCAO.plot_convergence_sweep(ax[1], 2)

ax[1].set_xlim(0, 60)
ax[1].set_yscale("log")
ax[1].set_ylim(1e-4, 10)

plt.show()

30

Flux sweep

fig = plt.figure()
ax = fig.add_subplot(111)

fluxonium_LCAO.create_states([(-2, 3)] * 5)
fluxonium_LCAO.sweep_external_flux(np.linspace(0.0, np.pi, 101))
fluxonium_LCAO.substract_groundstate_energy_sweep()
fluxonium_LCAO.plot_energy_sweep(ax, np.arange(4))

fluxonium.sweep_external_flux(np.linspace(0.0, np.pi, 101))
fluxonium.substract_groundstate_energy_sweep()

ax.set_xlabel(r"$\varphi_\text{ext}$ (rad)")

plt.show()

Matrix elements and losses

fig = plt.figure(figsize=(10, 4))
ax = fig.add_subplot(111)

flux_dm, charge_dm = fluxonium_LCAO.calc_dipole_moments_sweep(0, 1)

ax.plot(fluxonium_LCAO.par_sweep, flux_dm, color="C0")
ax.set_xlabel(r"$\varphi_\text{ext}$ (rad)")
ax.set_ylabel(r"$|\langle \psi_m|\phi| \psi_n\rangle |$ ($\Phi_0$)")

axt = ax.twinx()
axt.plot(fluxonium_LCAO.par_sweep, charge_dm, color="C3")
axt.set_ylabel(r"$|\langle \psi_m|q| \psi_n\rangle |$ ($2e$)")

#flux_dm, charge_dm = fluxonium.calc_dipole_moments_sweep(0, 1)
#ax.plot(fluxonium_LCAO.par_sweep, flux_dm, color="C0")
#axt.plot(fluxonium_LCAO.par_sweep, charge_dm, color="C3")

plt.show()

fig = plt.figure(figsize=(10, 4))
ax = fig.add_subplot(111)

sin_mel = fluxonium_LCAO.calc_sin_phi_over_two_sweep(0, 1)
ax.plot(fluxonium.par_sweep, sin_mel)

#sin_mel = fluxonium.calc_sin_phi_over_two_sweep(0, 1)
#ax.plot(fluxonium.par_sweep, sin_mel

ax.set_xlabel(r"$\varphi_\text{ext}$ (rad)")
ax.set_ylabel(r"$|\langle \psi_m|\sin(\varphi /2 + \varphi_\text{ext} / 2)| \psi_n\rangle |$")

plt.show()

Inductive loss

Q_ind = 1e6 # inductive loss quality factor
T = 25.0 # in [mK]

fig = plt.figure(figsize=(10, 4))
ax = fig.add_subplot(111)

G1_TLSs = fluxonium_LCAO.calc_inductive_loss_sweep(1, 0, Q_ind, environment="TLSs")
G1_bosonic = fluxonium_LCAO.calc_inductive_loss_sweep(1, 0, Q_ind, environment="Bosonic", T=T)

ax.plot(fluxonium_LCAO.par_sweep / (2 * np.pi), 1 / G1_TLSs)
ax.plot(fluxonium_LCAO.par_sweep / (2 * np.pi), 1 / G1_bosonic)
ax.axhline(1e6 * Q_ind / (2e9 * np.pi * fluxonium_LCAO.w), color="k")  # harmonic oscillator

ax.set_xlabel(r"$\Phi_\mathrm{ext}$ ($\Phi_0$)")
ax.set_ylabel("$T_1$ (us)")

plt.show()

Capacitive loss

Q_cap = 1e6 # inductive loss quality factor
T = 25.0 # in [mK]

fig = plt.figure(figsize=(10, 4))
ax = fig.add_subplot(111)

G1_TLSs = fluxonium_LCAO.calc_capacitive_loss_sweep(1, 0, Q_cap, environment="TLSs")
G1_bosonic = fluxonium_LCAO.calc_capacitive_loss_sweep(1, 0, Q_cap, environment="Bosonic", T=T)

ax.plot(fluxonium_LCAO.par_sweep / (2 * np.pi), 1 / G1_TLSs)
ax.plot(fluxonium_LCAO.par_sweep / (2 * np.pi), 1 / G1_bosonic)
ax.axhline(1e6 * Q_ind / (2e9 * np.pi * fluxonium_LCAO.w), color="k")  # harmonic oscillator

ax.set_xlabel(r"$\Phi_\mathrm{ext}$ ($\Phi_0$)")
ax.set_ylabel("$T_1$ (us)")

plt.show()

More

• for more examples see Fluxonium notebook.