Rabi oscillations
/ RAH-bee /
Take a single atom with just two relevant levels and drive it with light tuned close to the gap between them. Naively you might expect it to absorb a photon, jump up, and stay there. Instead, if the drive is coherent and the atom does not decay, the probability of being excited swings smoothly up to one and back down to zero, over and over. This coherent cycling is the Rabi oscillation, and it is the fundamental heartbeat of every qubit and atomic clock.
For an ideal two-level system driven near resonance, and within the rotating-wave approximation (which drops the fast counter-rotating term of the drive), the probability of finding the atom excited is P_e(t) = (Omega^2/Omega_R^2) sin^2(Omega_R t/2). Here Omega, the Rabi frequency, is proportional to the product of the atomic dipole moment and the field amplitude and measures how hard the atom is driven; delta = omega - omega_0 is the detuning of the drive from resonance; and Omega_R = sqrt(Omega^2 + delta^2) is the generalized Rabi frequency. On exact resonance (delta = 0) the oscillation is complete, swinging all the way from ground to excited; off resonance the swing is faster but never reaches the top. A pulse lasting long enough to complete half a cycle (a 'pi-pulse') fully inverts the atom; a quarter-cycle 'pi/2-pulse' leaves it in an equal superposition.
Rabi oscillations are the operational basis of quantum control: they define how you rotate a qubit, calibrate the pulses in NMR and atomic clocks, and read the coupling strength of an atom to a field. The instructive contrast is with Fermi's golden rule: that rule gives irreversible, exponential decay into a continuum, whereas Rabi flopping is coherent and reversible because there is only one final state. The honest caveats: the clean sinusoid requires the rotating-wave approximation (valid when Omega is much smaller than omega_0) and, above all, coherence — any spontaneous emission or dephasing damps the oscillation, so real Rabi flopping is a decaying oscillation that eventually settles.
On resonance, a pi-pulse of duration t = pi/Omega drives the atom from ground to fully excited: P_e = sin^2(pi/2) = 1. Double the time to a 2-pi-pulse and it returns all the way to the ground state. This is exactly the operation used to flip a qubit or invert a spin in NMR.
Coherent driving cycles the atom up and down; the pulse area sets where you stop.
Rabi flopping is coherent and reversible, the opposite of golden-rule decay; the difference is a single final state versus a continuum, and any decoherence turns the perfect oscillation into a damped one.