Atom interferometer - 2D space-time diagram

Light-pulse interferometer

A cloud of 87Rb atoms is split by a pi/2 laser pulse, redirected by a pi pulse after T and recombined by a second pi/2 pulse after 2T. The two-photon kick has wave number k_eff = 4π/λ (λ = 780.24 nm), giving a recoil velocity of about 1.2 cm/s. For uniform gravity the phase is Δφ = k_eff g T², and the fraction of atoms in port 1 is P = ½(1 + C sin Δφ), with fringe contrast C.

Shot noise

With N atoms the measured fraction has variance P(1 - P)/N. Near mid-fringe the phase uncertainty is δφ = 1/(C √N), so the acceleration uncertainty per shot is δg = δφ / (k_eff T²): doubling T gains a factor 4, and 100 times more atoms gains 10.

The diagram is drawn in the freely falling frame, where gravity bends both arms equally and only the recoil motion is visible. The page recovers the same phase from a numerically integrated free-fall trajectory, using φ = k_eff (z₀ - 2 z₁ + z₂).

Honest simplifications: uniform gravity (no gradient or Coriolis phase), ideal pulses, a Gaussian approximation to the binomial detection noise, only shot noise (no vibration or laser phase noise), and the anomaly range is limited to small phases so one quadrature is enough.

• Double T and compare the sensitivity per shot: it improves by 4.
• Use Many atoms: the scatter of the estimates shrinks by a factor of 10 for 100 times the atoms.
• Use Poor contrast and watch the second graph curve shift upward.
• Move the gravity anomaly slider and see the yellow marker slide along the fringe.