Anton Potočnik

                            at ETH Zürich

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.

Capacitance

Al Josephson Junction/SQUID

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

Qubit

ωg-e/2π = GHz
ωe-f/2π = GHz
ωg-f/2/2π = GHz
α/2π = GHz
Schematic of a transmon qubit.

 

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
ωg-e   Maximal qubit ground (g) to first excited state (e) transition frequency
ωe-f   Maximal qubit first (e) to second excited state (f) transition frequency
ωg-f   Maximal qubit ground (g) to second excited state (f) transition frequency
α   Qubit anharmonicity == Ec

 

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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