Cesium Fountain Clock (2D)

What the 2D view shows

A side cross-section of the fountain: laser-cooled caesium-133 atoms are launched upward, rise through the microwave cavity, turn at the apex and fall back through the same cavity. Two short microwave pulses, separated by the free-flight time T, make a Ramsey interferometer; the fraction of atoms that end in the upper hyperfine state oscillates with the detuning.

Equations

• Apex height h = v² / (2g), flight time 2v/g (g = 9.80665 m/s²).
• Cavity at height hc: the two crossings are separated by T = (2/g) sqrt(v² - 2 g hc). With the cavity at launch height this is the 3D page's T = 2v/g.
• Ramsey fringes P = ½ (1 + cos(2π Δ T)), full width at half maximum 1/(2T), line quality factor Q = f0/FWHM with f₀ = 9 192 631 770 Hz.
• A cloud at temperature Θ has velocity spread σv = sqrt(kBΘ/m); averaging the fringes over it multiplies the oscillation by exp(-½(2π Δ T′ σv)²), so fringes wash out away from the line centre.
• Short-term stability (white frequency noise only) σy(τ) = FWHM / (π f0 SNR) sqrt(Tc/τ), T_c = flight time + dead time.

Honest simplifications

Two ideal pi/2 pulses, no Rabi-pulse imperfections, no gravity gradient, no collisional, cavity-phase, Zeeman or relativistic shifts and no Dick effect; the stability curve is the textbook white-noise limit, not a measured clock. Atoms that never reach the cavity are counted as not excited. Sideways spread is drawn 25 times too large to be visible. This is an educational, illustrative model and is not affiliated with any laboratory.

What to try

• Raise the launch speed: T grows and the fringe narrows (higher Q).
• Move the cavity up (drag it): T shrinks and the flight time between pulses falls.
• Set detuning to 1/(4T) and read P = 0.5.
• Heat the cloud to 40 uK and detune: the fringe washes out.