Charge Energy Source Three-Row Scan - Superconducting Qubits: LC Energy, Toy Josephson Coupling, and Readout Step by Step

The charge-energy scan keeps capacitance at 4 and changes the charge source, so each row squares its own charge before dividing by the same denominator. Charges 2, 4, and 6 produce squared charges 4, 16, and 36, giving capacitor-energy rows 1/2, 2, and 9/2. Real charge-energy calibration depends on capacitance extraction, offset charge, readout gain, noise, rounding, and uncertainty.

Three checked capacitor-energy rows make the charge-square relation scanable. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Charge row 1 gives capacitor energy

Charge 2 with capacitance 4 produces a checked capacitor-energy row.

E_C=\frac{Q^{2}}{2C}=\frac{4}{8}=\frac{1}{2}

Charge row 2 gives capacitor energy

Charge 4 with capacitance 4 produces a checked capacitor-energy row.

E_C=\frac{Q^{2}}{2C}=\frac{16}{8}=2

Charge row 3 gives capacitor energy

Charge 6 with capacitance 4 produces a checked capacitor-energy row.

E_C=\frac{Q^{2}}{2C}=\frac{36}{8}=\frac{9}{2}

Charge changes the energy quadratically

The capacitance stays fixed. Each row squares its own charge source before dividing by the same denominator.

\begin{array}{c|c|c}Q&Q Q&E_C\\2&4&\frac{1}{2}\\4&16&2\\6&36&\frac{9}{2}\\\end{array}