Calculator

Quantum LC resonator

Select two independent rows to solve the oscillator. Values mirror the uploaded notebook formulas and run fully in the browser.

Circuit

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Result

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Parameter	Value	Unit

Hamiltonian

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\hat{H}_{\rm LC} = \frac{\hat{q}^{2}}{2C} + \frac{\hat{\phi}^{2}}{2L}

\hat{q} = i q_{\rm zpf}\left(\hat{a}^{\dagger} - \hat{a}\right),\quad \hat{\phi} = \phi_{\rm zpf}\left(\hat{a}^{\dagger} + \hat{a}\right)

Formulas

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\phi_0 = \frac{\hbar}{2e}

E_L = \frac{\phi_0^2}{L}

E_C = \frac{e^2}{2C}

Z = \sqrt{\frac{L}{C}}

\frac{E_L}{E_C} = \frac{\hbar^2}{2e^4Z^2}

\omega = \frac{1}{\sqrt{LC}} = \frac{1}{\hbar}\sqrt{8E_LE_C}

n_{\rm zpf} = \frac{q_{\rm zpf}}{2e} = \sqrt{\frac{\hbar}{8e^2Z}}

\varphi_{\rm zpf} = \frac{\Phi_{\rm zpf}}{\phi_0} = \sqrt{\frac{2e^2Z}{\hbar}}

Adapted from log/widget_demo/LC_resonator.ipynb.