◬ MIMZY · instrument № 11 · the analog–quantum suite · HARDWARE

The Toroid Tank

Wind a real toroid, pick a capacitor, and watch energy slosh between the magnetic field and the electric field — the LC resonator. Then the bridge: a superconducting qubit is this exact circuit with a nonlinear inductor, run cold enough that the sloshing comes in quanta.

⊙ EDUCATIONAL & SIMULATION — real formulas: L = μ₀μᵣN²A / 2πR · f₀ = 1/2π√(LC) · E = ℏω

Wind the hardware

Turns N 30

Core μᵣ 1 (air)

Major radius R 2.0 cm

Minor radius r 0.50 cm

Capacitor C 100 pF

Loss Rₛ 2.0 Ω

The tank, computed

Inductance L

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Resonance f₀

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Quality factor Q

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Ring-down τ = 2L/Rₛ

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Impedance √(L/C)

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⚛ the quantum bridge

Energy quantum ℏω

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…in micro-eV

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To see quanta, cool below T = ℏω/k_B

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Quanta in a 1 nW ring at f₀ (per s)

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The tank — energy sloshing B ⇄ E

Ring-down scope — strike the tank

The quantum ladder — Eₙ = (n + ½)ℏω at your f₀

How real is this? The inductance formula is the standard toroid result (μ₀μᵣN²A/2πR, A = πr²); resonance, Q, and ring-down are textbook. The quantum panel is also real: an LC resonator is a harmonic oscillator with level spacing ℏω — and a transmon qubit is precisely an LC circuit whose inductor is replaced by a Josephson junction (a nonlinear inductor), cooled to ~10–20 mK so that k_BT ≪ ℏω and single quanta matter. The "cool below" temperature shown is the genuine criterion. What this page does not claim: that a room-temperature toroid behaves quantum-mechanically — at 300 K your tank holds billions of thermal quanta, which is exactly why it behaves classically.