Rabi oscillations (or Rabi cycles) are periodic oscillations in the state of a driven two-level quantum system. When a qubit is driven by a resonant electromagnetic field, its population continuously oscillates between $\lvert 0\rangle$ and $\lvert 1\rangle$ at the Rabi frequency $\Omega$. The phenomenon is named after Isidor Isaac Rabi, who first observed it in 1937.

When driven exactly on resonance (drive frequency $\omega = \omega_0$ where $\omega_0$ is the qubit transition frequency) with coupling strength $\Omega$, the probability of finding the qubit in $\lvert 1\rangle$ starting from $\lvert 0\rangle$ is:

$$P_1(t) = \sin^2\!\left(\frac{\Omega t}{2}\right), \qquad P_0(t) = \cos^2\!\left(\frac{\Omega t}{2}\right)$$

On the Bloch sphere, Rabi oscillations correspond to the Bloch vector rotating continuously about an axis in the equatorial plane. Starting from the north pole $\lvert 0\rangle$, the state sweeps along a great circle passing through $\lvert 1\rangle$ at the south pole and back. The axis of rotation and the speed are determined by the drive amplitude and frequency.

Pulses of specific duration implement important quantum gates. A $\pi$ pulse (duration $t = \pi/\Omega$) completely inverts the qubit from $\lvert 0\rangle$ to $\lvert 1\rangle$, implementing the X gate. A $\pi/2$ pulse (duration $t = \pi/(2\Omega)$) creates an equal superposition, implementing a rotation equivalent to the Hadamard gate. These pulses are the fundamental building blocks of qubit control in superconducting qubits, trapped ions, and NMR systems.

When driven off resonance with detuning $\delta = \omega - \omega_0 \neq 0$, the oscillation frequency increases to the generalized Rabi frequency $\tilde\Omega = \sqrt{\Omega^2 + \delta^2}$, and the maximum probability of reaching $\lvert 1\rangle$ drops below 1 to $\Omega^2/(\Omega^2 + \delta^2)$.