# 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. | |

T_{1} | Energy relaxation time | |

T_{2} | Decoherence time | |

T_{φ} | Dephasing time | |

f | Corresponding frequency | |

df_{fwhm} | Full width at half maximum – spectral linewidth | |

n_{crit} | Critical photon number |

## 2 Comments

Thank you for providing this page!

I just understood, that I use it quite frequently, because it is convinient for simple transmons calculations.

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