A two-level atom (a qubit) is driven by a near-resonant oscillating field. Watch the population coherently slosh between the ground and excited states while the Bloch vector rotates on its sphere. This is the fundamental mechanism behind every single-qubit gate.
Ω = d·E/ℏ Ω_R = √(Ω² + Δ²) P_e(t) = (Ω²/Ω_R²)·sin²(Ω_R·t/2)
A perfectly timed π pulse is a quantum NOT gate. Add detuning and the atom can never be fully inverted — the price of going off resonance, and the reason atomic clocks are stabilised exactly on resonance.
When a two-level quantum system is driven by an oscillating field tuned near its transition frequency, the population does not simply jump — it flows back and forth coherently between the ground and excited states. This periodic exchange is the Rabi oscillation, and its rate, the Rabi frequency, is set by how hard you drive the atom.
The simulation above solves the two-level Schrödinger equation in the rotating frame, displays the excited-state population over time, and renders the Bloch vector so you can see the rotation directly. Adjust the drive strength and detuning to watch the oscillation speed up and its amplitude shrink.
What is a Rabi oscillation?
A Rabi oscillation is the coherent, periodic transfer of population between the two levels of a quantum system when it is driven by a near-resonant oscillating field. The probability of finding the atom in the excited state swings smoothly between 0 and a maximum and back again.
What is the Rabi frequency?
The Rabi frequency Ω is the rate at which population cycles between the two levels on resonance. It equals the dipole matrix element times the field amplitude divided by ℏ: Ω = d·E/ℏ. Stronger driving gives faster oscillations.
What is detuning?
Detuning Δ is the difference between the drive frequency and the atomic transition frequency. When Δ is zero the drive is on resonance; non-zero detuning speeds up the oscillation but reduces the maximum excited-state population.
The generalised (effective) Rabi frequency is Ω_R = √(Ω² + Δ²). It sets the actual oscillation rate when there is detuning, and the oscillation amplitude is reduced by the factor Ω²/Ω_R².
A π pulse is a drive applied for exactly the time needed to complete half a Rabi cycle, Ω_R·t = π. On resonance it fully inverts the atom, moving all population from the ground state to the excited state — equivalent to a qubit NOT gate.
A π/2 pulse drives the system for a quarter Rabi cycle, Ω_R·t = π/2. On resonance it creates an equal superposition of ground and excited states — the Bloch vector ends up lying in the equatorial plane. It is the building block of Ramsey interferometry.
The Bloch sphere is a geometric picture of a two-level quantum state. The north pole is the ground state, the south pole is the excited state, and every pure state is a point on the surface. Driving the atom rotates the Bloch vector around an axis set by the drive and detuning.
A qubit is a controllable two-level system, so Rabi oscillations are exactly how single-qubit gates are performed. By choosing the pulse length, phase and detuning, an experimenter rotates the qubit to any desired state on the Bloch sphere.
In real systems spontaneous emission and dephasing gradually destroy coherence, so the oscillations damp out and the population settles toward a steady value. This idealised simulation shows the coherent, undamped limit to make the physics clear.
They are named after Isidor Isaac Rabi, who developed the theory of magnetic resonance in molecular beams in the late 1930s. His work won the 1944 Nobel Prize in Physics and underpins NMR, atomic clocks and quantum computing.