🔥 Transmon relaxation and cavity dephasing due to nonequilibrium quasiparticles
The transmon is constructed from a superconductor and uses the josephson effect to add non-linearities to the system. One limiting feature of this design is quasiparticle induced noise.
Relaxetion
Starting with the easy-to-calculate part of the problem is the relaxation of the qubit due to the quasiparticles. We want to calculate $\tau_{e \rightarrow g} = \frac{1}{\Gamma_{e \rightarrow g}}$ (relaxation time from the excited to ground state). From Catelani’s paper, the relaxation time between an initial state $\lvert i \rangle$ and final state $ \lvert f \rangle$ due to the quasiparticles is given by:
\[\Gamma_{i \rightarrow f} \vert \langle f \lvert \sin{\frac{\hat{\phi}}{2}} \lvert i \rangle \lvert\ S_{qp} (\omega_{if})\]With
⏰ Coherence - \(S_{qp} (\omega) = \frac{16 E_j}{\pi} \int_0^\infty \frac{1}{\sqrt{x}\sqrt{x+\omega/\Delta}} \lbrack f_E((1+x)\Delta) (1-f_E((1+x)\Delta+\omega)) \rbrack dx\)
Here $f_E(\epsilon)$ is the energy distribution function. At low temperatures $T \ll \Delta$ and thermal equilibrium we can approximate it as $f_E(\epsilon) = e^{-\epsilon/T}$, which would reduce the expression of $S_{qp}$ to
\[S_{qp} = \frac{16 E_j}{\pi} e^{\Delta/T}e^{\omega/2T}K_0 \left( \frac{|\omega|}{2T} \right)\]…
Weakly anharmonic qubit
\[\Gamma_{n \rightarrow n-1} = n\frac{\omega_p^2}{\omega_{10}} \frac{x_{qp}}{2\pi} \sqrt{\frac{2\Delta}{\omega_{10}}}(1+\cos{\phi_0})\] \[\omega_p = \sqrt{8 E_C E_J}\] \[\omega_{10} = \sqrt{8E_C (E_L + E_J\cos{\phi_0})}\]From nature paper:
\[\Gamma = x_{qp} \frac{\omega}{\pi}\sqrt{\frac{2\Delta}{\hbar\omega}}\]import numpy as np
import matplotlib.pyplot as plt
import qutip as qt
from scipy.special import kn
plt.style.use('ggplot')
k = 8.617e-5 # eV/K
hbar = 6.58e-10 # eV us
# k = 1
# hbar = 1
Tc = 5
delta = 1.76*k*Tc
T = 20e-3
Ej = 14.79e3 # MHz
El = 0
Ec = 341.0 # MHz
phi0 = 0
n = 1
x_qp = 4e-8 # x_qp ~ 10^-6, worst case scenerio x_qp < 10^-4
w10 = np.sqrt(8*Ec*(El+Ej*np.cos(phi0)))
wp = np.sqrt(8*Ec*Ej)
# w10 = np.linspace(10,20,1000)
gamma = n*((wp**2)/w10)*(x_qp/(2*np.pi))*np.sqrt(2*delta/(w10*hbar))*(1+np.cos(phi0))
gamma = (w10*x_qp/np.pi)*np.sqrt(2*delta/(hbar*w10))
# S = (16*Ej/np.pi)*np.exp(delta/T)*np.exp(omega/(2*T))*kn(0, np.abs(omega)/(2*T))
# plt.figure(figsize=(10,7))
# plt.plot(omega, gamma);
# gamma
print(f"""\
\33[1m\33[34mCalculated:\33[0m
->\33[1m Decay-rate\33[0m = {gamma:.2e} 1/us
->\33[1m Lifetime\33[0m = {1e-3/gamma:.2f} ms
->\33[1m Qubit-freq\33[0m = {w10/1e3:.2f} GHz
\33[1m\33[34mGiven:\33[0m
->\33[1m delta\33[0m = {delta}
->\33[1m Ec\33[0m = {Ec} MHz
->\33[1m Ej\33[0m = {Ej/1e3} Ghz
->\33[1m El\33[0m = {El} MHz
->\33[1m x_qp\33[0m = {x_qp}
->\33[1m phi0\33[0m = {phi0}
->\33[1m w_p\33[0m = {wp/1e3:.2f} GHz
""")
[1m[34mCalculated:[0m
->[1m Decay-rate[0m = 1.54e-03 1/us
->[1m Lifetime[0m = 0.65 ms
->[1m Qubit-freq[0m = 6.35 GHz
[1m[34mGiven:[0m
->[1m delta[0m = 0.000758296
->[1m Ec[0m = 341.0 MHz
->[1m Ej[0m = 14.79 Ghz
->[1m El[0m = 0 MHz
->[1m x_qp[0m = 4e-08
->[1m phi0[0m = 0
->[1m w_p[0m = 6.35 GHz