Energy Transfer Keeps the Total - Superconducting Qubits: LC Energy, Toy Josephson Coupling, and Readout Step by Step

An ideal LC mode can trade energy between capacitor and inductor stores while keeping the total fixed. The equality is exact only for the declared snapshots, lossless lumped elements, and no drive or dissipation; real circuits have damping, parasitic modes, dielectric and conductor loss, drift, and finite measurement precision.

The idealized mode swaps stores while preserving total energy. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One snapshot stores more energy in flux

Snapshot one has capacitor energy 1/2 joule and inductor energy 2 joules.

E_C=\tfrac{1}{2}\ \text{J},\quad E_L=2\ \text{J}

A later snapshot swaps the storage emphasis

Snapshot two has capacitor energy 2 joules and inductor energy 1/2 joule, but the total is still 5/2 joules.

2+\tfrac{1}{2}=\tfrac{5}{2}\ \text{J}