Next we turn the general visibility kernel from Step 4 into a frequency-domain formula that’s easy to use with real data. If the lapse fluctuations are stationary (statistics depend only on time difference), their two-time correlator \(C_\Phi(t-t')\) has a one-sided power spectral density \(S_\Phi(\Omega)\). Then the visibility becomes a weighted integral of that spectrum by a filter \(|F(\Omega)|^2\) determined solely by your pulse/sequence \(f(t)\):
\[ -\ln V \;=\; \frac{\omega^2}{2}\int_{0}^{\infty}\!\frac{d\Omega}{2\pi}\;S_\Phi(\Omega)\,\big|F(\Omega)\big|^2, \qquad F(\Omega)=\int f(t)\,e^{i\Omega t}\,dt. \]
This is Eq. (8) in the paper. You pick \(f(t)\) (Ramsey, Mach–Zehnder, spin-echo…), which fixes \(|F|^2\); the integral then tells you how much dephasing to expect for a given \(S_\Phi\).
Concrete filter example: For a simple rectangular interrogation of duration \(T\) (Ramsey/Mach–Zehnder), \(|F(\Omega)|^2=\!\left(\tfrac{2\sin(\Omega T/2)}{\Omega}\right)^{\!2}\). That sinc² envelope shows which \(\Omega\) contribute to dephasing (peaks near \(\Omega\!\sim\!\pi/T\)).
Markovian limit and a back-of-envelope bound: If low frequencies dominate so that \(S_\Phi(\Omega)\approx S_\Phi(0)\), then the dephasing rate is \(\Gamma_\phi \simeq \tfrac{\omega^2}{2}S_\Phi(0)\) and for an interrogation time \(T\):
\[ -\ln V \;\simeq\; \tfrac{1}{2}\omega^2 S_\Phi(0)\,T \;\Rightarrow\; S_\Phi(0) \;\lesssim\; \frac{2\,|\ln V|}{\omega^2 T}. \]
The paper gives a scale: with \(T=1\,\text{s}\) and an optical line \(\omega/2\pi=10^{15}\,\text{Hz}\), a 1% (0.1%) visibility loss implies \(S_\Phi(0)\lesssim 5\times10^{-34}\,\text{s}\) (\(5\times10^{-35}\,\text{s}\)).
Units & conventions checkpoint: They use one-sided \(S_\Phi(\Omega)\) with units of seconds (s) when integrating with \(d\Omega/2\pi\); equivalently, with ordinary frequency \(f\) (Hz), \(S_\Phi(f)\) carries units Hz\(^{-1}\). \(|F(\Omega)|^2\) has units s², making \(-\ln V\) dimensionless, as it must.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(S_\Phi(\Omega)\) | Lapse PSD | One-sided spectrum of \(\delta\Phi\) (s) | Bound via \(-\ln V\) integral | "Noise color of time" |
| \(F(\Omega)\) | Sequence filter | FT of \(f(t)\); \(|F|^2\) weights \(S_\Phi\) (s²) | Rectangular: \(\left(\frac{2\sin(\Omega T/2)}{\Omega}\right)^2\) | "Sieve selecting frequencies" |
| \(\Gamma_\phi\) | Dephasing rate | Visibility decay rate (s\(^{-1}\)) | \(\Gamma_\phi \!\simeq\! \tfrac{\omega^2}{2}S_\Phi(0)\) | "Blur-speed of fringes" |
| \(S_\Phi(0)\) | Zero-freq level | Low-\(\Omega\) plateau of \(S_\Phi\) (s) | \(\lesssim 5\times10^{-34}\) s (1% loss, 1 s, optical) | "DC fog of time" |
| \(\omega\) | Clock frequency | Internal line frequency (rad·s\(^{-1}\)) | Optical: \(2\pi\times10^{15}\) Hz | "Metronome speed" |
| \(f(t)\) | Path indicator | +1/−1 sequence defining arms/pulses | Ramsey/MZ/echo patterns | "Which road when" |