Anton Potočnik

                            at imec

Transmon Qubit Calculator

With this calculator you can convert between charging energy (Ec) and total capacitance (CΣ), between Josephson energy (EJ), Josephson inductance (LJ), junction critical current (Ic) or room temperature junction resistance (R), and from charging and Josephson energies calculate relevant qubit transition frequencies. Other calculations are added below including Qubit’s Purcell decay, chi-shift, conversion between spectral linewidth and coherence times, etc.

All frequencies are in natural units.


Al Josephson Junction/SQUID

Ec/2π = GHz
CΣ = fF
EJ/2π = GHz
LJ = nH
Ic = nA
R = Ω

Qubit frequencies

fge GHz
fef GHz
fgf/2 ≈ GHz
α GHz
Schematic of a transmon qubit.

Purcell decay and χ-shift

fres = GHz
Qres = k
κres = MHz
g/2π = MHz
Δ = fgefres = GHz
TPurcell μs
κPurcell kHz
χ-shift ≈ MHz

Coherence times / spectral linewidth

T1 = μs
T2 = μs
Tφ = μs
f = GHz
dffwhm = kHz
Q-factor = M

Parameter Description

Ec Charging energy
CΣ Total capacitance
EJ Junction/SQUID Josephson energy
LJ Junction/SQUID Josephson inductance
Ic Junction/SQUID critical current
R Junction/SQUID room temperature resistance for 30/40 nm thin Al films
fge Maximal qubit ground (g) to first excited state (e) transition frequency
fef Maximal qubit first (e) to second excited state (f) transition frequency
fgf Maximal qubit ground (g) to second excited state (f) transition frequency
α Qubit anharmonicity ≈ Ec
fres Readout resonator frequency
Qres Readout resonator Q-factor
κres Readout resonator: spectral linewidth
g Resonator-qubit coupling constant
Δ Qubit-resonator transition frequency difference
TPurcell Qubit’s Purcell decay time
κPurcell Qubit’s Purcell spectral linewidth
χ-shift Half of the fres difference between qubit being in the g or e state.
T1 Energy relaxation time
T2 Decoherence time
Tφ Dephasing time
f Corresponding frequency
dffwhm Full width at half maximum – spectral linewidth
ncrit Critical photon number


E_c = \frac{e_0^2}{2C_\Sigma}\Phi_0 = 2.067834\cdot10^{-15}\,\mathrm{Wb}
E_J = \left(\frac{\Phi_0}{2\pi}\right)^2\frac{1}{L_J}\Delta_0 = 176\cdot10^{-6}\,\mathrm{V}
E_J = \frac{\Phi_0}{2\pi}I_ce_0 = 1.60218\cdot10^{-19}\,\mathrm{As}
I_c = \frac{\pi\Delta}{2 R}h = 2\pi\hbar = 6.62607 \cdot10^{-34}\,\mathrm{Js}
L_J = \frac{\Phi_0}{2\pi}\frac{1}{I_c} df_\mathrm{fwhm} = 1/(\pi T_2)
hf_{ge} \approx \sqrt{8 E_c E_J}-E_cQ = f/df_\mathrm{fwhm} = \pi f T_2 = 2\pi f T_1
hf_{ef} \approx \sqrt{8 E_c E_J}-2E_c1/T_{2} = 1/(2T_1) + 1/T_\varphi
\kappa_\mathrm{Purcell} \approx \kappa(g/\Delta)^2T_\mathrm{Purcell} = 1/(2\pi\kappa_\mathrm{Purcell})
\chi\mathrm{-shift} \approx g^2/\Delta - g^2/(\Delta - E_c)n_\mathrm{crit} \approx \Delta^2/(2g)^2

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  1. Vladimir Milchakov Reply

    Thank you for providing this page!
    I just understood, that I use it quite frequently, because it is convinient for simple transmons calculations.

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