Take a two-path interferometer and mark which arm you’re on with a simple path indicator \(f(t)\): \(+1\) on arm A, \(-1\) on arm B. The lapse fluctuation \(\delta\Phi(t)\) writes differential phase between the two arms, so the total phase difference is just the weighted integral
\[ \Delta\theta \;=\; \omega\!\int f(t)\,\delta\Phi(t)\,dt . \]
If \(\delta\Phi\) is (zero-mean) Gaussian noise with two-time correlator \(C_\Phi(t,t')=\langle\delta\Phi(t)\delta\Phi(t')\rangle\), then the ensemble-averaged fringe visibility obeys the standard phase-diffusion result
\[ V \;=\; \exp\!\left[-\tfrac12\,\omega^2\!\iint f(t)\,C_\Phi(t,t')\,f(t')\,dt\,dt'\right]. \]
This is the visibility kernel: it tells you exactly how a proposed pulse/timing sequence \(f(t)\) turns lapse fluctuations into a measurable loss of interference contrast. It also makes the scaling explicit: \(-\ln V \propto \omega^2\) — higher internal frequency \(\omega\) means stronger coupling to \(\Phi\).
Intuition: \(f(t)\) is a stencil for “when the two arms feel different clock rates.” Only those time slices contribute. If both arms are identical at some instant (\(f=0\) effectively), that instant doesn’t degrade visibility.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(f(t)\) | Path indicator | +1 on arm A, −1 on arm B; encodes timing/pulses | Ramsey or Mach–Zehnder plateaus | “Which road you’re on” |
| \(\Delta\theta\) | Differential phase | Phase written by \(\delta\Phi\) between arms (rad) | \(\Delta\theta=\omega\!\int f\,\delta\Phi\,dt\) | “Needle separation on two dials” |
| \(V\) | Visibility | Fringe contrast \(\in[0,1]\) (dimensionless) | \(V=\exp[-\tfrac12\omega^2\!\iint f C_\Phi f]\) | “Sharpness of stripes” |
| \(C_\Phi(t,t')\) | Lapse correlator | \(\langle\delta\Phi(t)\delta\Phi(t')\rangle\) (dimensionless) | Stationary case used next | “How time-ripples echo across moments” |
| \(\omega\) | Clock frequency | \(E/\hbar\) or line splitting \(\Delta E/\hbar\) (rad·s\(^{-1}\)) | Optical: \(2\pi\times10^{15}\,\text{Hz}\) | “Metronome speed” |
Next up: Step 5 — Spectral form & filters: turn \(C_\Phi\) into a one-sided spectrum \(S_\Phi(\Omega)\), introduce the filter \(|F(\Omega)|^2\) for a given sequence \(f(t)\), and read off practical dephasing rates and bounds.