Transmon Qubit Calculator

With this calculator you can convert between charging energy (E_c) and total capacitance (C_Σ), between Josephson energy (E_J), Josephson inductance (L_J), junction critical current (I_c) 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.

Parameter Description

E_c		Charging energy
C_Σ		Total capacitance
E_J		Junction/SQUID Josephson energy
L_J		Junction/SQUID Josephson inductance
I_c		Junction/SQUID critical current
R		Junction/SQUID room temperature resistance for 30/40 nm thin Al films
f_ge		Maximal qubit ground (g) to first excited state (e) transition frequency
f_ef		Maximal qubit first (e) to second excited state (f) transition frequency
f_gf		Maximal qubit ground (g) to second excited state (f) transition frequency
α		Qubit anharmonicity ≈ E_c
f_res		Readout resonator frequency
Q_res		Readout resonator Q-factor
κ_res		Readout resonator: spectral linewidth
g		Resonator-qubit coupling constant
Δ		Qubit-resonator transition frequency difference
T_Purcell		Qubit’s Purcell decay time
κ_Purcell		Qubit’s Purcell spectral linewidth
χ-shift		Half of the f_res difference between qubit being in the g or e state.
T₁		Energy relaxation time
T₂		Decoherence time
T_φ		Dephasing time
f		Corresponding frequency
df_fwhm		Full width at half maximum – spectral linewidth
n_crit		Critical photon number

Formulas

\Phi_0 = 2.067834\cdot10^{-15}\,\mathrm{Wb}

Tagged under: tools

2 Comments

Vladimir Milchakov November 5, 2020 at 8:24 am Reply

Thank you for providing this page!
I just understood, that I use it quite frequently, because it is convinient for simple transmons calculations.
Tanmoy Bera December 16, 2021 at 1:15 pm Reply

I started using it when I joined my PhD. I still do.

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

Anton Potočnik - research website

Anton Potočnik

Transmon Qubit Calculator

Capacitance

Al Josephson Junction/SQUID

Qubit frequencies

Purcell decay and χ-shift

Coherence times / spectral linewidth

Parameter Description

Formulas

2 Comments

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