2-Level Rabi Story

The simplest nontrivial quantum system has two states, \(|g\rangle\) and \(|e\rangle\). When a near-resonant field couples them, probability flows coherently between the states — a phenomenon called Rabi oscillation. This story powers NMR, atomic clocks, trapped-ion and superconducting qubits, and even simple spin control in solids.

We will build intuition step by step: start from the driven Hamiltonian, move to a rotating frame, see the motion on the Bloch sphere, then add detuning, damping, and pulse design (\(\pi\) and \(\pi/2\) pulses, Ramsey).

Let the bare splitting be \(\hbar\omega_0\) and drive with a classical field at frequency \(\omega_d\). In the \(\{|e\rangle,|g\rangle\}\) basis and under dipole coupling the Hamiltonian can be written \[ H(t)=\frac{\hbar\omega_0}{2}\sigma_z + \hbar\Omega\cos(\omega_d t+\phi)\,\sigma_x , \] where \(\Omega\) is the on-resonance coupling strength (Rabi rate) and \(\sigma_i\) are Pauli matrices.

Transform to the frame rotating at \(\omega_d\) and apply the rotating-wave approximation (RWA) to drop fast terms. The effective Hamiltonian becomes \[ H_{\rm RWA}=\frac{\hbar}{2}\big(\Delta\,\sigma_z + \Omega\,\sigma_x\big),\qquad \Delta=\omega_0-\omega_d . \] Now the problem is time-independent and transparent.

With \(H_{\rm RWA}\), the state precesses around a static “effective field” \(\boldsymbol{\Omega}_{\rm eff}=(\Omega,0,\Delta)\) with magnitude \[ \Omega_R=\sqrt{\Omega^2+\Delta^2}\,, \] the generalized Rabi frequency. Starting in \(|g\rangle\), the excited-state probability is \[ P_e(t)=\frac{\Omega^2}{\Omega_R^2}\sin^2\!\Big(\frac{\Omega_R t}{2}\Big). \] On resonance (\(\Delta=0\)), this reduces to \(P_e(t)=\sin^2(\Omega t/2)\).

The Bloch vector \(\mathbf{r}\) rotates by angle \(\theta=\Omega t\) around the drive axis (on resonance). A pulse of duration \(t_\pi=\pi/\Omega\) flips \(|g\rangle\to|e\rangle\) (a \(\pi\)-pulse); \(t_{\pi/2}=\pi/(2\Omega)\) makes an equal superposition (a \(\pi/2\)-pulse). Two \(\pi/2\) pulses separated by free evolution produce Ramsey fringes that measure detuning and dephasing precisely.

Off resonance, oscillations slow to \(\Omega_R\) and their amplitude shrinks by \(\Omega^2/\Omega_R^2\). In steady-state drive with relaxation, the excited-state probability vs. detuning is a Lorentzian whose width grows with drive strength (power broadening) — a key trade-off in spectroscopy and qubit control.

Real systems are not perfectly isolated. Longitudinal relaxation (energy loss) is set by \(T_1\); transverse dephasing by \(T_2\le 2T_1\). The Bloch equations add damping terms: \[ \dot{\mathbf{r}}=\mathbf{r}\times\boldsymbol{\Omega}_{\rm eff} -\bigg(\frac{r_x}{T_2},\frac{r_y}{T_2},\frac{r_z-r_z^{\rm eq}}{T_1}\bigg). \] Damping reduces visibility and sets how long coherent rotations remain accurate.

NMR/ESR spins: RF or microwave fields drive Zeeman-split spins; readout via induction or fluorescence. Trapped ions & neutral atoms: Lasers drive internal transitions; state readout uses electron-shelving photons. Superconducting qubits: Microwave pulses shape fast \(\pi\) and \(\pi/2\) rotations; dispersive readout measures state.

Calibrating a \(\pi\)-pulse is as simple as sweeping pulse length and picking the first maximum of \(P_e\). Ramsey sequences reveal detuning and dephasing; spin-echo cancels slow noise by inserting a \(\pi\)-pulse between two \(\pi/2\) pulses.

Drive amplitude sets \(\Omega\); frequency error sets \(\Delta\). Calibrate both. Gaussian or DRAG-shaped pulses reduce leakage and AC-Stark shifts in fast control. Average many shots to beat quantum projection noise; shield and filter to tame dephasing.