A Pockels cell is a crystal (e.g. KDP or lithium niobate) whose refractive index changes linearly with an applied electric field — the linear electro-optic effect. Driving it with voltage V across a crystal of length L imprints an optical phase shift on light passing through:
Δn = -½ n³ r V / d (r = electro-optic coefficient, d = electrode gap)
φ = 2π Δn L / λ ≡ π · V / V_π
Vπ is the half-wave voltage: the drive that produces exactly a π phase shift. One arm of a Mach–Zehnder interferometer carries this phase shifter; the other is a plain reference path (plus any fixed bias φ₀ from a path-length mismatch). After the two arms recombine at the second beamsplitter, single-photon interference sends each photon to one of two detectors with probability set entirely by the total phase φ = φ₀ + πV/Vπ:
P(D0) = cos²(φ/2)
P(D1) = sin²(φ/2)
- Drive voltage V — the electric field across the crystal; each volt adds π/Vπ radians of phase.
- Vπ — sets how sensitive the crystal is; a lower Vπ means the same voltage shifts the phase further.
- Bias φ₀ — a fixed offset from arm-length mismatch, added before the voltage-controlled term. Drag the phase-vector dial in the bottom-right panel to set the total phase directly — the bias slider updates to match, holding the current voltage fixed.
- Faint pulses travel both arms at once (the photon is in a coherent superposition of paths); only when they recombine does a single detector actually click, and which one is random on any given photon — but over many photons the counts converge on cos²(φ/2) and sin²(φ/2).
Real-world relevance: this exact device — an electro-optic phase shifter inside an interferometer — is how phase-encoded quantum key distribution, quantum-optical gates, and integrated photonic quantum processors control and read out a photon's phase.