Aharonov-Bohm Effect

The two-slit setup, with a solenoid in the middle

Start with an electron biprism interferometer: a coherent electron beam splits in two, the partial beams travel symmetrically around an obstacle, and recombine on a detector screen to make interference fringes. Now place a thin, infinitely long solenoid between the two paths. Inside the solenoid is a magnetic field B parallel to the axis carrying total flux Φ; outside, B = 0 exactly. The electrons travel only outside.

Classically, the electrons feel no force — the Lorentz force F = −ev×B is zero where B = 0. So a classical physicist predicts: solenoid current makes no difference, and the interference pattern is unchanged.

Quantum mechanics disagrees. The wavefunction along each path picks up a phase given by the line integral of the vector potential A:

ψ_path(x) = ψ_free(x) · exp[i(e/ℏ) ∫_path A · dl]

When the two paths recombine, the relative phase between them is the line integral around the closed loop they form. By Stokes' theorem:

Δφ = (e/ℏ) ∮ A · dl = (e/ℏ) ∫∫ B · dA = e Φ / ℏ

The enclosed flux Φ is the flux inside the solenoid, which is non-zero even though the electrons never visited that region. So changing the solenoid current — without changing anything along the electron's path — shifts the interference pattern.

Worked example — one full fringe shift

What current through a 100-turn-per-cm solenoid of radius 1 µm makes the interference pattern shift by exactly one fringe?

• One fringe corresponds to Δφ = 2π, so Φ = 2πℏ/e = h/e = Φ₀ = 4.136 × 10⁻¹⁵ Wb.
• Solenoid flux: Φ = μ₀ n I A, with n = 10⁴ turns/m, A = π(10⁻⁶)² = 3.14 × 10⁻¹² m².
• Solve for I: I = Φ / (μ₀ n A) = (4.14 × 10⁻¹⁵) / (4π × 10⁻⁷ × 10⁴ × 3.14 × 10⁻¹²) ≈ 105 mA.

So a sub-amp current through a micron-scale solenoid is enough to shift the pattern by a full fringe. Tonomura's experiment used a sub-micron ring magnet with permanent magnetisation; the trapped flux was a few flux quanta and the fringes shifted by detectable fractions.

Why the phase is gauge-invariant

The vector potential is not unique. The gauge transformation A → A + ∇χ leaves B = ∇×A unchanged, so changes A without changing any classical observable. Does this kill the Aharonov-Bohm phase?

No, because the line integral ∮A·dl around a closed loop is gauge-invariant: ∮∇χ·dl = χ(end) − χ(start) = 0 for a closed loop. Equivalently, the enclosed flux Φ = ∫∫(∇×A)·dA = ∫∫B·dA is gauge-invariant. So although the phase along each individual open path depends on the gauge, the relative phase between two paths forming a closed loop does not. This is the precise sense in which the Aharonov-Bohm phase is observable: it shows up only in interference, where relative phases matter, never in single-path expectation values.

Classical vs quantum and topological vs dynamical

Aspect	Classical EM	Quantum mechanics (AB)
What feels the field	Force F = qE + qv×B	Phase φ from A and scalar potential
Field-free region	No effect on charge	Still observable if A loops enclose flux
Role of A	Convenience for computing B	Physical degree of freedom
Phase formula	Not applicable	Δφ = (e/ℏ)∮A·dl = eΦ/ℏ
Path-dependence	Force is local	Phase is topological — depends only on enclosed flux
Gauge transformations	Leave force invariant	Leave loop integral invariant; shift open-path phase
Modern interpretation	B is field strength	A is U(1) connection; phase is holonomy

JavaScript — Aharonov-Bohm phase shift

// Constants
const e = 1.602176634e-19;   // electron charge (C)
const hbar = 1.054571817e-34; // reduced Planck (J·s)
const h = 2 * Math.PI * hbar; // Planck (J·s)
const PHI0 = h / e;           // magnetic flux quantum (Wb)
console.log(`Φ₀ = h/e = ${PHI0.toExponential(3)} Wb`);

// Aharonov-Bohm phase for enclosed flux Φ
function abPhase(flux_Wb) {
  return (e / hbar) * flux_Wb;  // radians
}

// One fringe shift = 2π
console.log(`Phase for Φ = Φ₀:`, abPhase(PHI0).toFixed(4), '(should be 2π ≈ 6.283)');
console.log(`Phase for Φ = Φ₀/2:`, abPhase(PHI0 / 2).toFixed(4), '(should be π ≈ 3.142)');

// Solenoid flux given geometry
function solenoidFlux(current_A, turns_per_m, radius_m) {
  const mu0 = 4 * Math.PI * 1e-7;  // T·m/A
  const area = Math.PI * radius_m * radius_m;
  return mu0 * turns_per_m * current_A * area;  // Wb
}

// Worked example: 1 µm radius solenoid, 10⁴ turns/m
const I_for_one_fringe = PHI0 / (4*Math.PI*1e-7 * 1e4 * Math.PI * (1e-6)**2);
console.log(`Current for one fringe: ${(I_for_one_fringe * 1000).toFixed(1)} mA`);

// Fringe pattern with AB shift
function fringeIntensity(x_m, wavelength_m, slit_sep_m, screen_dist_m, phase_shift_rad) {
  // Two-slit interference with extra AB phase
  const pathDiff = (slit_sep_m * x_m) / screen_dist_m;
  const kineticPhase = 2 * Math.PI * pathDiff / wavelength_m;
  return Math.cos((kineticPhase + phase_shift_rad) / 2) ** 2;
}

// Without flux vs with one full flux quantum
for (let x of [-2e-6, -1e-6, 0, 1e-6, 2e-6]) {
  const I_0 = fringeIntensity(x, 50e-12, 1e-6, 0.5, 0);
  const I_Phi = fringeIntensity(x, 50e-12, 1e-6, 0.5, 2*Math.PI);
  console.log(`x = ${(x*1e6).toFixed(1)} µm: I(Φ=0) = ${I_0.toFixed(3)}, I(Φ=Φ₀) = ${I_Phi.toFixed(3)}`);
}
// Note: a 2π shift moves each maximum onto the position of the previous maximum -- visually identical
// to no shift, but the pattern envelope tells the difference. For half-fringe Δφ = π, maxima ↔ minima.

Where the Aharonov-Bohm effect matters

• Foundations of gauge theory. The textbook example that gauge connections — not just curvatures — are physical. Sets the language used by every modern gauge theory, including the Standard Model.
• Topological phases of matter. Integer and fractional quantum Hall states are protected by Aharonov-Bohm-like flux insertions through holes in the sample geometry (Laughlin's argument).
• Mesoscopic conductors. Persistent currents in metal rings, conductance oscillations with period Φ₀ in single-loop interferometers ("AB oscillations"), and Altshuler-Aronov-Spivak Φ₀/2 oscillations in disordered rings.
• Superconductors and SQUIDs. Flux quantisation in superconducting loops is the bosonic analog (with paired electrons, Φ₀ replaced by h/2e). SQUIDs use Aharonov-Bohm physics to make femtotesla magnetometers.
• Quantum computing. Topological qubits proposed in non-Abelian anyon systems use generalised Aharonov-Bohm braiding phases to encode information immune to local noise.
• Cosmic strings and monopoles. Hypothetical cosmic strings carry magnetic flux that would produce Aharonov-Bohm scattering of charged particles; one of the few observational handles on string-like topological defects.

Common mistakes

• "It's just fringe fields." Tonomura's superconducting torus pinned B to zero outside to parts in 10⁻⁸ and still saw the shift. Real, not residual.
• "A is not gauge-invariant, so its line integral can't be physical." The closed-loop integral ∮A·dl is gauge-invariant — that is what's observed.
• Confusing it with Faraday induction. Faraday requires dΦ/dt ≠ 0. The Aharonov-Bohm effect works with static Φ.
• Treating the phase as dynamical. It depends only on which homotopy class of paths you took (how many times around the flux), not on speed or shape — topological, not dynamical.
• "It violates locality." It is local in A but non-local in B. Quantum field theory accepts A as a local field; no FTL signalling.
• Thinking the effect requires solenoids. Any topologically non-trivial flux configuration works: rings, tori, cosmic strings, monopole strings.