A qubit prepared on the Bloch sphere is driven by a continuous microwave field. Three "modulators" shape that field before it reaches the qubit: the phase modulator sets which equatorial axis the drive rotates the state around, the amplitude modulator sets how fast that rotation happens (the Rabi rate), and the frequency modulator sets the detuning between the drive and the qubit's own transition frequency, which tilts the rotation axis toward the poles. The Bloch vector precesses around the resulting effective axis exactly like a spinning top around a tilted magnetic field.
H = (Δ/2)σz + (Ω/2)(cosφ·σx + sinφ·σy)
dr/dt = Ω_eff × r − decoherence·(rx,ry,0)
Ω_eff = (Ω cosφ, Ω sinφ, Δ), |Ω_eff| = √(Ω²+Δ²)
- Phase φ — rotates the drive axis in the equatorial plane; at φ=0 the drive is a pure X-rotation, at φ=90° a pure Y-rotation.
- Amplitude Ω — the Rabi rate; higher amplitude spins the Bloch vector faster around the drive axis, so a fixed pulse length flips more of the population.
- Frequency detuning Δ — how far the drive frequency sits from resonance; nonzero detuning tilts the rotation axis toward the pole, so the vector can no longer swing all the way to the opposite pole (incomplete population transfer).
- Decoherence 1/T2 — dephasing shrinks the equatorial (x,y) components of the Bloch vector over time, pulling the trajectory toward the polar axis and washing out interference.