Rabi Oscillation

Definition

A two-level quantum system driven by a field tuned to its transition frequency does not simply jump to the upper state — it cycles smoothly back and forth between the two states. That coherent cycling is a Rabi oscillation, and its rate is the Rabi frequency.

Quantitatively, if the system starts in the ground state and the drive is exactly on resonance, the probability of finding it excited at time t is:

P_excited(t) = sin²(Ω·t / 2)

where Ω is the Rabi frequency. On resonance it is set by the strength of the coupling:

Ω = d·E / ℏ

Here d is the transition dipole matrix element (how strongly the two states couple), E is the amplitude of the driving field, and ℏ is the reduced Planck constant. The key insight: the flopping rate Ω is set by the field strength, not by the photon energy. A brighter laser or stronger RF field flops the atom faster.

How it works

Start with an atom (or spin, or qubit) that has just two relevant energy levels: a ground state |g⟩ and an excited state |e⟩ separated by energy ΔE, so the natural transition frequency is ω = ΔE/ℏ.

Now apply an oscillating field — light, microwaves, or an RF magnetic field — whose frequency matches ω. In a frame rotating at the drive frequency (the "rotating frame," where the fast carrier oscillation is subtracted out), the math collapses to something beautifully simple: the state vector just rotates steadily about an axis at angular rate Ω.

The most natural way to picture this is the Bloch sphere. Any pure state of a two-level system is a point on the surface of a unit sphere: the north pole is |g⟩, the south pole is |e⟩, and the equator holds equal superpositions with different phases. A resonant drive tips the Bloch vector away from the north pole and spins it around a horizontal axis. As it sweeps from pole to pole, the projection onto the excited state rises and falls — and that projection squared is exactly P_excited = sin²(Ωt/2).

Three moments along the rotation deserve names:

• Ωt = π/2 — the pi/2-pulse. The Bloch vector reaches the equator. P_excited = 1/2: an equal superposition of ground and excited. This is the operation that seeds quantum interference.
• Ωt = π — the pi-pulse. The vector reaches the south pole. P_excited = 1: a complete population inversion. The atom is now certainly in the excited state.
• Ωt = 2π — the 2pi-pulse. The vector returns to the north pole. P_excited = 0 again, and the cycle repeats forever (in the ideal, lossless case).

Worked example — calibrating a superconducting qubit

Suppose you run a transmon qubit with a transition frequency of 5 GHz and you drive it with a resonant microwave tone. You sweep the pulse duration and record the excited-state probability. You find the first full inversion (P_excited = 1) occurs at a pulse length of 20 ns. What is the Rabi frequency, and how long is a pi/2-pulse?

Full inversion means Ωt = π at t = 20 ns:

Ω = π / t_pi = π / (20 × 10⁻⁹ s) ≈ 1.57 × 10⁸ rad/s
f_Rabi = Ω / 2π ≈ 25 MHz

So the qubit flops 25 million times per second under this drive — a glacial pace compared to the 5 GHz carrier, which is the whole point: Rabi flopping is the slow envelope riding on a fast oscillation.

The pi/2-pulse needs Ωt = π/2, so it is exactly half the pi-time:

t_(π/2) = t_pi / 2 = 10 ns

If you now want a faster gate, double the microwave amplitude. Because Ω = dE/ℏ is linear in field amplitude, doubling E doubles Ω and halves the pi-time to 10 ns. (In practice you eventually hit limits — leakage to a third level, and the rotating-wave approximation breaking down — but over a wide range, the rule "more power → faster gate" holds.)

Variants and regimes

The single formula P_excited = sin²(Ωt/2) is the on-resonance, lossless ideal. Real and richer cases branch off from it.

Regime	Condition	Behaviour	Max P_excited
On resonance, no loss	Δ = 0, Γ = 0	Clean undamped flopping at Ω	1 (full inversion)
Detuned (off resonance)	Δ ≠ 0	Faster flopping at Ω′ = √(Ω²+Δ²)	Ω²/(Ω²+Δ²) < 1
Far off resonance	Δ ≫ Ω	Tiny fast wobble, no real transfer	≈ 0
Strong driving with loss	Ω ≫ Γ	Many oscillations, slowly decaying	≈ 1 early, → 1/2
Weak driving with loss	Ω ≲ Γ	Overdamped: single rise, no flopping	→ Ω²/(Ω²+Γ²)
Saturation	Ω ≫ Γ, long time	Steady state, populations equalize	1/2

The detuned case is worth dwelling on. When the drive misses the transition by Δ, the generalized Rabi frequency is:

Ω′ = √(Ω² + Δ²)
P_excited(t) = (Ω² / Ω′²)·sin²(Ω′·t / 2)

Two things change at once: the oscillation gets faster (Ω′ > Ω) but shallower — it can no longer reach full inversion. Sweeping Δ and plotting the peak transfer traces a Lorentzian centered on resonance. This is the operating principle of magnetic resonance spectroscopy: the location of the peak is the transition frequency.

Common pitfalls and misconceptions

• Confusing the Rabi frequency with the optical/transition frequency. Ω (the flopping rate, MHz–GHz) is many orders of magnitude smaller than ω (the carrier, often hundreds of THz for optical transitions). Ω is the envelope; ω is the carrier inside it.
• Thinking the atom "absorbs and then emits." Rabi flopping is fully coherent and reversible — the atom is in a definite superposition the whole time, not randomly absorbing photons. Spontaneous emission is a separate, incoherent process that damps the oscillation.
• Forgetting the factor of 2. P_excited = sin²(Ωt/2), not sin²(Ωt). The Bloch vector rotates at Ω, but probability is the square of an amplitude that effectively rotates at Ω/2. A pi-pulse is Ωt = π, giving sin²(π/2) = 1.
• Assuming a stronger drive is always better. Past a point, strong driving excites unwanted transitions (leakage in transmons) and the rotating-wave approximation fails, distorting the clean sin² curve.
• Expecting full inversion off resonance. Any detuning caps the maximum transfer below 1. To reliably invert a population whose frequency you don't know precisely, use an adiabatic chirp instead of a fixed pi-pulse.
• Ignoring decoherence. In the lab the oscillation amplitude decays. Reading the pi-time off a decaying Rabi curve still works, but you must fit the decaying envelope, not just find the first peak.

Applications

• Quantum computing — single-qubit gates. Every X (NOT) gate is a pi-pulse; every Hadamard-like operation builds on a pi/2-pulse. Gate calibration starts by measuring a Rabi curve and extracting the pi-time. Superconducting, trapped-ion, photonic, and spin qubits all share this recipe.
• Nuclear magnetic resonance (NMR) and MRI. Rabi's original 1937 discovery. RF pulses flip nuclear spins; the 90° (pi/2) and 180° (pi) pulses of every MRI sequence are Rabi rotations of proton spins in the body's magnetic field.
• Atomic clocks. Ramsey interferometry — two pi/2-pulses separated by a free-evolution gap — is a refined Rabi technique that defines the SI second via the cesium transition.
• Laser cooling and state preparation. Pi-pulses deterministically pump atoms into chosen states; the same coherent control prepares qubits before computation.
• Quantum sensing. NV centers in diamond use Rabi oscillations of electron spins to measure magnetic fields with nanoscale resolution.
• Spectroscopy. The detuning dependence of the Rabi amplitude maps out transition frequencies and lifetimes.

Derivation and performance analysis

Start from the two-level Hamiltonian in the rotating frame, after applying the rotating-wave approximation. With detuning Δ between drive and transition, it reads (in units where the basis is {|g⟩, |e⟩}):

H = (ℏ/2) · [  -Δ    Ω  ]
              [   Ω    Δ  ]

This is a constant 2×2 Hermitian matrix, so the state simply precesses about the axis defined by (Ω, 0, Δ) at angular rate equal to the generalized Rabi frequency:

Ω′ = √(Ω² + Δ²)

Solving the time-dependent Schrödinger equation for an initial ground state gives the excited-state probability directly:

P_excited(t) = (Ω² / (Ω² + Δ²)) · sin²(Ω′·t / 2)

Set Δ = 0 and this reduces to the headline formula P_excited = sin²(Ωt/2). The first inversion happens at t_pi = π/Ω — the pi-time that sets the speed of a single-qubit gate.

Practical takeaways on speed and fidelity:

• Gate speed scales with field amplitude. Since Ω = dE/ℏ, the pi-time t_pi = π/Ω = πℏ/(dE) shrinks linearly as you crank up the drive. A 25 MHz Rabi frequency gives a 20 ns inversion; a 50 MHz drive gives 10 ns.
• Coherence sets the fidelity ceiling. You can only fit a useful number of oscillations into the coherence time T₂. The ratio Ω·T₂ — roughly the number of clean flops you can do — is the figure of merit. Good superconducting qubits today reach hundreds to thousands.
• Detuning is a precision knob. Even a small Δ caps inversion at Ω²/(Ω²+Δ²); to keep gate error below 1% you typically need |Δ| ≲ Ω/10.
• Saturation in noisy systems. When loss dominates (Ω ≲ Γ), the population just climbs to a steady 1/2 — useful for spectroscopy, useless for gates. The transition between flopping and saturation happens right around Ω ≈ Γ.