Cesium Fountain Clock — Ramsey Phase Portrait (2D)

Fountain Setup

Launch velocity v₀ 4.0 m/s

Microwave detuning Δf 0.00 Hz

Cloud temperature 5 µK

Readouts

Free evolution T

0.00 s

Fringe width

0 mHz

Detected fraction

Line Q (f₀·T)

Excited-state probability P(Δf)

How it works (2D method)

This is the 2D-native counterpart to the 3D cesium-fountain simulator. Instead of orbiting a rendered atom cloud flying through a cavity, it draws the two things that actually determine a fountain clock's precision as genuine 2D diagrams:

Bloch phase portrait — the atomic hyperfine state is a real quantum two-level (qubit) system. The first microwave π/2 pulse tips the Bloch vector from the south pole (ground state) onto the equator; during free flight it precesses in the equatorial plane at angular rate 2π·Δf; the second π/2 pulse converts that accumulated phase into a population tilt toward the north (excited) pole. This is the textbook Ramsey Bloch-vector derivation, animated directly — not a camera view of anything.
Analytic detection fraction — rather than scattering discrete instanced atoms and counting how many drift outside a 3D aperture, the transverse thermal spread is modelled in closed form: initial cloud radius and thermal velocity both contribute independent Gaussian components to the transverse position at the second pass, so the radial miss-distance is Rayleigh-distributed and the detected fraction is its exact CDF at the aperture radius.

Free evolution time:  T = t_down − t_up
Bloch vector (x,y,z): ground=(0,0,-1) --π/2--> (0,1,0)
  --free flight, phase φ=2πΔf·T--> (-sinφ,cosφ,0)
  --π/2--> (-sinφ, 0, cosφ);  P_excited = (1-cosφ)/2
Radial spread:  σ_R² = σ0² + σv²·t_down²   (σ0 = cloud r0 / 2, σv = thermal scale)
Detected fraction:  1 − exp(−r_aperture² / (2σ_R²))   (Rayleigh CDF)

Both panels are driven by the identical fountain timing used in the 3D version (same launch parabola, same 9,192,631,770 Hz hyperfine splitting), so every slider still traces out the same physical clock — only the representation is genuinely 2D.

Ground-state pole Excited-state pole Equatorial precession (free flight) π/2 pulse event