Rabi Oscillations

In a quantum two-level system without decay, applying an external oscillating drive can cause population to oscillate continuously between the ground and excited states. Under the rotating-wave approximation, this dynamic is captured by Rabi oscillations.

Worked Example

Consider a system starting in the ground state. Let the drive strength be Omega = 1 and the detuning be Delta = 0.

1. Generalized Rabi frequency: Omega_R = sqrt(1^2 + 0^2) = 1 2. Maximum excitation probability: P_max = 1^2 / 1^2 = 1 3. At time t = pi: P_e = 1 * sin^2(1 * pi / 2) = 1 4. Ground state probability: P_g = 1 - 1 = 0

Thus, the system has completely inverted its population to the excited state, which yields an expected excited probability result of 1.0.

Experiment

You can observe this behavior in the Rabi Oscillations module.

Set the Drive Omega to 1 and Detuning to 1.
Observe that the maximum excitation probability drops to 0.5 (since 1^2 / (1^2 + 1^2) = 0.5), meaning the population can never be fully transferred to the excited state. The oscillation also becomes faster, with a generalized frequency of sqrt(2) ≈ 1.414.

Limitations

This model assumes the rotating-wave approximation, which is valid when the drive frequency is close to the transition frequency and the drive strength is comparatively weak. It also omits spontaneous emission and other decoherence mechanisms (decay). In real open quantum systems, these oscillations damp out over time, leading to a steady-state mixed population rather than continuous undamped oscillation.

References

https://doi.org/10.1103/PhysRev.51.652

This foundational paper by I. I. Rabi introduces the magnetic resonance method for molecular beams, deriving the probability of transition between states for a spin in an oscillating magnetic field. This forms the basis for the exact probability formulas used in this module.

Rabi Oscillations