Grow a quantum dot into a ring — a nanoscale toroid — and thread a magnetic field through the hole. The electron orbits in a region with zero field, yet its energy still oscillates with the flux through the centre. That is the Aharonov–Bohm effect: in quantum mechanics the potential is real even where the field is not. Sweep the field below and watch the spectrum breathe.
This instrument computes the real one-dimensional ring spectrum E(l,Φ) live, with energy periodic in the flux quantum Φ₀ = h/e — the measured signature. Physics as represented in my training data; a teaching model, with its simplifying assumptions stated at the end.
Lower trace: ground-state energy as flux sweeps 0→3 Φ₀. The cusps where it touches zero are where the electron's angular momentum l jumps to the next integer — one jump per flux quantum. That period is the Aharonov–Bohm oscillation.
Confine an electron to a thin ring of radius R and only its angular motion survives. Its allowed states carry integer angular momentum l = …,−1,0,+1,…. Thread magnetic flux Φ through the hole and the electron — even in a field-free region — accumulates a quantum phase set by the enclosed flux. The energies become:
This is the textbook 1-D Aharonov–Bohm ring. The instrument evaluates it exactly: at each flux it picks the integer l that minimises the energy (the ground state), and the trace is that minimum vs. flux — reproducing the measured Φ₀-periodic oscillation and the persistent current that changes sign at each cusp.
Each angular-momentum state l traces a parabola in flux, centred at Φ = l·Φ₀. The visible spectrum on the ring view is that whole family at once; the bright lower envelope — the lowest parabola at every flux — is the ground state, and its scalloped shape is the Aharonov–Bohm oscillation. Shrink the radius and every parabola steepens (energy scales as 1/R², the same confinement law as the dot), but the Φ₀ period never changes — it is set by fundamental constants alone, not by the ring.
Independent of radius, material, or temperature. h/e is a constant of nature, which is why AB oscillations are such a clean experimental fingerprint.
At each half-integer flux two angular-momentum states are degenerate; the ground state switches l, and the persistent current reverses direction.
Smaller ring → larger energy scale, exactly as quantum confinement dictates. The dot's size-tuning law lives here too.
−∂E/∂Φ of the ground state; absolute magnitudes in real rings depend on the full level occupation.Background: Y. Aharonov & D. Bohm, "Significance of Electromagnetic Potentials in the Quantum Theory" (Phys. Rev., 1959); M. Büttiker, Y. Imry & R. Landauer on persistent currents (1983); A. Lorke et al. on self-assembled InAs quantum rings (Phys. Rev. Lett., 2000).