Two-Level Rabi Oscillations

A two-level system driven on resonance does not sit still — it cycles coherently between its two levels. These Rabi oscillations are the workhorse of qubit control: every single-qubit gate is a carefully timed Rabi flop.

The driving Hamiltonian

In a frame rotating with the drive (the "rotating frame"), a resonant drive of strength $\Omega$ — the Rabi frequency — reduces to the time-independent Hamiltonian

\hat H = \frac{\hbar\Omega}{2}\,\sigma_x.

The evolution operator is the exponential of a Pauli matrix. Using $e^{-i\theta\sigma_x/2} = \cos\tfrac{\theta}{2}\,I - i\sin\tfrac{\theta}{2}\,\sigma_x$ ,

U(t) = e^{-i\hat H t/\hbar} = e^{-i(\Omega t/2)\sigma_x} = \cos\!\frac{\Omega t}{2}\,I - i\sin\!\frac{\Omega t}{2}\,\sigma_x.

This is exactly the rotation gate $R_X(\theta)$ with angle $\theta = \Omega t$ . Time evolution under an $X$ -type drive is an $R_X$ rotation — the link between physics and the gate set is direct.

Population transfer

Apply $U(t)$ to the ground state $|0\rangle$ :

|\psi(t)\rangle = \cos\!\frac{\Omega t}{2}\,|0\rangle - i\sin\!\frac{\Omega t}{2}\,|1\rangle.

The probability of finding the system in the excited state is the Rabi formula:

P_1(t) = \sin^2\!\frac{\Omega t}{2}.

It oscillates fully between $0$ and $1$ at the Rabi frequency. Some landmark times:

| $\Omega t$ | State | $P_1$ | Name | |------------|-------|-------|------| | $0$ | $|0\rangle$ | $0$ | start | | $\pi/2$ | $\tfrac{1}{\sqrt2}(|0\rangle - i|1\rangle)$ | $1/2$ | $\pi/2$ -pulse | | $\pi$ | $-i|1\rangle$ | $1$ | $\pi$ -pulse (full flip) | | $2\pi$ | $-|0\rangle$ | $0$ | full cycle |

Off resonance (briefly)

If the drive is detuned from the transition by $\delta$ , the oscillation frequency increases to the generalised Rabi frequency $\Omega' = \sqrt{\Omega^2 + \delta^2}$ , and the population no longer reaches $1$ : $P_1(t) = \tfrac{\Omega^2}{\Omega'^2}\sin^2(\Omega' t/2)$ . Perfect flopping requires exact resonance, which is why qubit calibration hunts for the frequency that maximises the transfer amplitude.

Try it

Prepare the qubit state after a quarter Rabi period, $\Omega t = \pi/2$ , by applying $R_X(\pi/2)$ to $|0\rangle$ . The grader checks the full statevector: you should get $\tfrac{1}{\sqrt2}(|0\rangle - i|1\rangle)$ , with each level carrying probability $1/2$ .

Run your code to see the quantum state.