Rabi Oscillation
(Ω · (Δ · (1 − cos(sqrt(Ω² + Δ²) · u)) · cos(12u + t) + sqrt(Ω² + Δ²) · sin(sqrt(Ω² + Δ²) · u) · sin(12u + t))/(Ω² + Δ²), (Δ² + Ω² · cos(sqrt(Ω² + Δ²) · u))/(Ω² + Δ²), Ω · (Δ · (1 − cos(sqrt(Ω² + Δ²) · u)) · sin(12u + t) − sqrt(Ω² + Δ²) · sin(sqrt(Ω² + Δ²) · u) · cos(12u + t))/(Ω² + Δ²))
Open in the app The dials and keys named below are the app's.
What it draws
The curve is the direction of one spin over time, drawn on the Bloch sphere. The top of the sphere is spin up and the bottom is spin down. The height is the chance of finding the spin up minus the chance of finding it down, so the chance of finding it flipped is (1 − y)/2.
Two magnetic fields act on it. A strong static field along the vertical splits the two levels, and the spin precesses about it at 12 + Δ radians per unit of u. A weak field of strength Ω turns in the horizontal plane at 12. The difference Δ is the detuning between the drive and the splitting.
Here u is the time, and one turn of u is 2π. The t in the formula only turns the drive's starting phase, so the whole spiral turns about the vertical.
Resonance
Seen from a frame that turns with the drive, the fast precession disappears. What remains is a slow rotation about a tilted axis, (Ω, 0, Δ), at the rate W = √(Ω² + Δ²). The spiral is that slow rotation with the fast turning put back.
On resonance, Δ = 0, the tilted axis lies flat and the spin swings over the pole. At the opening Ω = 1 it is fully flipped at u = π and back up at u = 2π, wrapping 12 times on the way.
Off resonance the axis tilts and the spin only reaches Ω²/(Ω² + Δ²) flipped: Rabi's formula. Because the drive turns rather than oscillates, the formula is exact here. It matches the spin's equation of motion integrated directly to about 5·10⁻¹¹.
History
Wolfgang Pauli gave the electron its two-component wave equation in 1927. Isidor Rabi worked out a spin in a turning magnetic field in 1937 and used the resonance to measure nuclear moments in molecular beams, which won him the 1944 Nobel Prize in Physics. The same flip is how MRI excites a nucleus and how a qubit gate turns a bit.
Try
- Set Δ to 1. The spiral stops at the equator: at most half flipped, since 1/(1 + 1) = 1/2.
- Set Δ to 2. It stays near the top, never more than a fifth flipped, and it cycles W = √5 ≈ 2.24 times in one turn of u.
- Set Ω to 2 with Δ at 0. The flip runs twice as fast, two full cycles in one turn of u.
- Change every 12 in the x and z expressions to 3. The drive is slower, so the spiral opens up to 3 wraps per cycle, and the precession is now 3 + Δ.