Cesium Fountain Clock — Ramsey Phase Portrait (2D)

This is the 2D-native counterpart to the 3D cesium fountain clock simulator. Rather than orbiting a rendered atom cloud tossed through a cavity, it draws the physics that actually sets the clock's precision: a genuine Bloch-vector phase portrait showing the atomic qubit tip onto the equator at the first microwave pulse, precess at a rate set by the detuning during free flight, and tip toward the ground or excited pole at the second pulse — the textbook Ramsey-interferometer derivation, animated directly rather than viewed from a camera. Alongside it, the detected fraction that thermal cloud expansion costs the clock is computed in closed form from a Rayleigh-distributed radial miss-distance instead of scattering discrete particles, and the live fringe curve P(Δf) is recomputed independently from the same fountain timing (launch velocity sets the free-evolution time between pulses; detuning sets the phase accumulated during that time; temperature sets the thermal spread that erodes contrast) used by the 3D version.