cQED Calculators

Quick numerical tools for common circuit QED quantities. All formulas are derived from first principles on this site.

Dispersive Shift χ

In the dispersive regime, the qubit–resonator detuning Δ ≫ g gives a frequency shift χ = g²/Δ. Enter g and Δ in MHz.

Coupling g (MHz)

Detuning Δ (MHz)

χ = 10.000 MHz

g/Δ = 0.1000 ⚠ approaching strong coupling — dispersive approx. may break down

Transmon Frequency ω_q

The transmon qubit frequency from circuit quantisation: ω_q ≈ √(8E_J E_C) − E_C. Anharmonicity α ≈ −E_C. Enter energies in GHz.

E_J (GHz)

E_C (GHz)

ω_q = 5.4569 GHz

α ≈ -0.200 GHzE_J/E_C = 100.0 ✓ transmon regime

Purcell Decay Rate γ_P

The Purcell effect gives the qubit an additional decay channel through the resonator: γ_P = g²κ / (Δ² + (κ/2)²). Enter all values in MHz.

Coupling g (MHz)

Detuning Δ (MHz)

Decay κ (MHz)

γ_P = 0.1000 MHz

T_1 (Purcell) ≈ 10.00 μs

Critical Photon Number n_crit

Above n_crit = Δ²/4g² photons, the system leaves the dispersive regime and the qubit becomes strongly dressed again. Enter g and Δ in MHz.

Coupling g (MHz)

Detuning Δ (MHz)

n_crit = 25.00