Step 9. Lever B (quantum): visibility protocol → bounds on \(S_\Phi\)

What you do: run a two-path interferometer (or any phase-coherent sequence) with a known time pattern \(f(t)\) (Ramsey, Mach–Zehnder, echo). Measure the fringe visibility \(V(T)\) and turn it into a bound on the lapse-fluctuation spectrum \(S_\Phi(\Omega)\).

Core laws (from Steps 4–5): with Gaussian lapse noise and path indicator \(f(t)\),

\[ V=\exp\!\Big[-\tfrac12\,\omega^2\!\iint f(t)\,C_\Phi(t,t')\,f(t')\,dt\,dt'\Big] \quad\Rightarrow\quad -\ln V=\frac{\omega^2}{2}\!\int_{0}^{\infty}\frac{d\Omega}{2\pi}\,S_\Phi(\Omega)\,|F(\Omega)|^2, \]

where \(F(\Omega)=\int f(t)e^{i\Omega t}dt\). For a rectangular interrogation of duration \(T\): \(|F(\Omega)|^2=\big(2\sin(\Omega T/2)/\Omega\big)^2\).

Simple (Markovian) bound: if low frequencies dominate, \(\Gamma_\phi\simeq \tfrac{\omega^2}{2}S_\Phi(0)\) and over time \(T\),

\[ S_\Phi(0)\;\lesssim\;\frac{2\,|\ln V|}{\omega^2\,T}. \]

At \(T=1\) s and \(\omega/2\pi=10^{15}\) Hz, 1% visibility loss gives \(S_\Phi(0)\lesssim5\times10^{-34}\) s.

Platforms & variants:

Atom interferometer (MZ/Ramsey): choose \(T\), compute \(|F|^2\), constrain \(S_\Phi\) via the integral above. A driven variant places a controlled, shot-noise photon flux at distance \(r\) to imprint a calculable \(S_\Phi\) via stress-tensor fluctuations.
Co-located optical clocks: cross-correlate two clocks’ fractional-frequency noise; in quiet operation you bound \(S_\Phi(0)\) via \(\Gamma_\phi\). With a modulated flux you test the classical template (Lever A).
Optomechanical cat states: off-diagonals decay as \(\exp[-\Gamma_\phi t]\) with \(\Gamma_\phi\) from the same \(\omega^2 S_\Phi(0)/2\) law—purely temporal (not spatial) decoherence.

Reporting + filters (what to publish): quote \(V(T)\) with errors, and provide (i) the filter-weighted bound from the full integral and (ii) the Markovian \(S_\Phi(0)\) bound. Use the measured \(f_{\text{meas}}(t)\) to form \(F(\Omega)\); finite pulse rise/fall simply reshapes \(|F|^2\) and the matched-filter formulas remain valid.

Noise budget (why ground labs are hard): three contributors sum into the practical \(S_\Phi(\Omega)\): (1) Newtonian gravity-gradient from mass density noise (dominant at sub-Hz), (2) photon shot/thermal stress-tensor, (3) TT vacuum modes (tiny in lab band). Typical hierarchy: at \(\sim1\) Hz, \(S_\Phi\sim10^{-32}\) s from Newtonian noise; at \(\sim1\) kHz, \(S_\Phi^{\text{vac}}\sim10^{-45}\) s. Always provide a “source-off” baseline.

Glossary (for this step)

Symbol	Name	Meaning (units)	Typical value/example	Metaphor
\(V\)	Visibility	Fringe contrast \([0,1]\) (–)	Measured vs. \(T\)	“Stripe sharpness”
\(f(t)\)	Path indicator	+1/−1 sequence defining arms/pulses	Ramsey/MZ/echo patterns	“Which road when”
\(F(\Omega)\)	Filter	FT of \(f\); weights \(S_\Phi\) (s)	Rectangular ⇒ sinc²	“Sieve for noise tones”
\(S_\Phi(\Omega)\)	Lapse PSD	One-sided spectrum of \(\delta\Phi\) (s)	Bound via \(-\ln V\)	“Noise color of time”
\(\omega\)	Clock frequency	Internal line \(E/\hbar\) (rad·s\(^{-1}\))	Optical \(\sim 2\pi\times10^{15}\) Hz	“Metronome speed”
\(T\)	Interrogation time	Pulse separation / dwell (s)	Set by sequence	“How long you listen”

Next: Step 10 — Case study & triggers: applying the template to a Galactic supernova and why Sgr A* flares are sub-threshold.

← Step 8 Step 10 →