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September 13, 2017
Software
Written by: Anton Potočnik
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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.

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
ω_g-e		Maximal qubit ground (g) to first excited state (e) transition frequency
ω_e-f		Maximal qubit first (e) to second excited state (f) transition frequency
ω_g-f		Maximal qubit ground (g) to second excited state (f) transition frequency
α		Qubit anharmonicity == E_c

Formulas

$E_c = \frac{e_0^2}{2C_\Sigma}$	$\Phi_0 = 2.067834\cdot10^{-15}\,\mathrm{Wb}$
$E_J = \left(\frac{\Phi_0}{2\pi}\right)^2\frac{1}{L_J}$	$\Delta = 176\cdot10^{-6}\,\mathrm{V}$
$E_J = \frac{\Phi_0}{2\pi}I_c$	$e_0 = 1.60218\cdot10^{-19}\,\mathrm{As}$
$I_c = \frac{\pi\Delta}{2 R}$	$h = 2\pi\hbar = 6.62607 \cdot10^{-34}\,\mathrm{Js}$
$L_J = \frac{\Phi_0}{2\pi}\frac{1}{I_c}$
$\hbar\omega_{ge} = \sqrt{8 E_c E_J}-E_c$
$\hbar\omega_{ef} = \sqrt{8 E_c E_J}-2E_c$
$\hbar\omega_{gf}/2 = \sqrt{8 E_c E_J}-1.5E_c$

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