Interference Visibility

Interference fringes turn complex amplitudes into bright and dark bands. Their contrast is summarized by the visibility \[ V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}} \quad (0\le V\le 1). \] Perfect coherence gives \(V=1\); any leakage of which-path information or phase noise lowers \(V\). Visibility is a laboratory dial that reads “how quantum” a superposition behaves under real conditions.

With two paths \(\psi_1\) and \(\psi_2 e^{i\phi}\), the intensity is \[ I(\phi)=|\psi_1+\psi_2 e^{i\phi}|^2=|\psi_1|^2+|\psi_2|^2+2|\psi_1||\psi_2|\cos\phi. \] Scanning \(\phi\) yields a sinusoid with \[ I_{\max/\min}=(|\psi_1|\pm|\psi_2|)^2,\qquad V=\frac{2|\psi_1||\psi_2|}{|\psi_1|^2+|\psi_2|^2}. \] Balance the paths to maximize \(V\); amplitude mismatch alone reduces contrast even without decoherence.

For path basis \(\{|1\rangle,|2\rangle\}\) with state \(\rho\), the interference term is proportional to the off-diagonal element \(\rho_{12}\). If \(\rho\) is pure, \(|\rho_{12}|=\sqrt{\rho_{11}\rho_{22}}\) and visibility reaches the amplitude-limited bound. Random phases, entanglement with an environment, or explicit which-path tags reduce \(|\rho_{12}|\rightarrow 0\), killing fringes.

In many setups one finds \(V\simeq 2|\rho_{12}|\) when paths are balanced. Thus measuring visibility is a practical proxy for coherence.

Introduce path distinguishability \(D\) (how well an optimal detector could guess the path). For a broad class of interferometers, Englert’s duality relation bounds them: \[ V^2 + D^2 \le 1. \] Tagging the paths increases \(D\) and necessarily lowers \(V\). Eraser experiments recover high \(V\) only inside post-selected subsets that erase the tags.

If the relative phase fluctuates with distribution \(P(\phi)\), the observed signal averages to \[ \langle I\rangle = I_0\big[1+V_0\,\Re\langle e^{i\phi}\rangle\big]. \] For Gaussian phase noise of variance \(\sigma_\phi^2\), \(\langle e^{i\phi}\rangle=e^{-\sigma_\phi^2/2}\), so visibility decays exponentially. Slow drifts can be refocused (spin echo, active stabilization); fast noise sets a fundamental \(T_2^\*\) limit.

Balance amplitudes (neutral density, beam splitter ratio), stabilize phase (common-path optics, feedback), and avoid unintentional which-path tags (polarization, spatial walk-off). In photon experiments, spectral filtering and single-mode fibers enforce indistinguishability; in matter-wave setups, velocity selection and vibration isolation are key.