Rabi Oscillations

Equivalent forms

\Omega' = \sqrt{\Omega^2 + \Delta^2}

P_e(t) = \sin^2(\Omega t/2) \quad (\Delta = 0)

One sine squared governs atomic clocks, MRI pulses, and quantum computers — the entire control theory of two-level physics.

Symbol	Name	SI unit	Dimension	Range
$P_e$	Excited-state probabilityoutput Probability of finding the system in the upper level at time t	dimensionless	1	0 – 1
$Omega$	Rabi frequency Drive strength: dipole coupling times field amplitude over hbar	rad/s	T^-1	100000 – 100000000
$Delta$	Detuning Drive frequency minus transition frequency	rad/s	T^-1	-100000000 – 100000000
$t$	Pulse duration Time the drive has been applied	s	T	0 – 0.00001

Unit systems

Symbol	SI	CGS	Imperial
$P_e$	dimensionless	dimensionless	dimensionless
$Omega$	rad/s	rad/s	rad/s
$Delta$	rad/s	rad/s	rad/s
$t$	s	s	s

Where it holds

Two-level system in the rotating-wave approximation: drive near resonance and Omega much smaller than the transition frequency. Decoherence (T1, T2) damps the oscillations; strong drive adds Bloch-Siegert shifts.

Dimensional analysis

P_e -> dimensionless probability

(Omega^2/(Omega^2+Delta^2)) * sin^2(...) -> dimensionless * dimensionless; the sine's argument

[T^-1]*[T] = 1

Probability is a pure number; the sine argument is an angle.

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Discovery

Isidor Isaac Rabi · 1937

Rabi worked out the response of a spin to a rotating magnetic field, then used the resonance to measure nuclear magnetic moments in molecular beams — winning the 1944 Nobel Prize. The same flopping now clocks atomic time and executes every gate on every qubit platform.

Try this

How do you flip a qubit with a flash of microwaves — and stop at exactly halfway?

Drive a two-level quantum system with a resonant field and its population doesn't just jump — it cycles sinusoidally between ground and excited state. Time the pulse right and you write any superposition you want. Every quantum gate ever executed is a chapter of this formula.

Research status: stable

Real-world applications

Single-qubit gates (X, Y, Hadamard) on every quantum-computing platform
Atomic clocks: Ramsey interferometry built from pi/2 Rabi pulses
MRI flip-angle control: the RF pulse is literally a Rabi rotation
Coherent control of chemical reactions with shaped laser pulses

Common misconceptions

Rabi frequency is set by the drive amplitude, not the transition frequency — they differ by many orders of magnitude
Off-resonant driving oscillates FASTER (at Omega'), but with smaller amplitude — detuning never slows the flopping
The oscillation is coherent population cycling, not repeated absorption-then-spontaneous-emission

Experimental verification

Observed in every qubit platform: superconducting transmons show Rabi chevrons as drive detuning is scanned; trapped ions reach 99.99% pi-pulse fidelity; cesium fountain clocks use Ramsey's separated pi/2-pulses, a direct descendant. Vacuum Rabi oscillations with single photons confirmed the quantized-field version (cavity QED, Haroche Nobel 2012).

Derivation

Write the state as c_g|g> + c_e|e> and apply a drive coupling the levels.

In the frame rotating at the drive frequency (rotating-wave approximation), the Hamiltonian becomes time-independent:

H =

(hbar/2)*(-Delta*sigma_z + Omega*sigma_x).

Its eigenvalues are split by hbar*Omega' with

\Omega ' = \sqrt{\Omega ^2 + \Delta ^2}

; starting from |g>, the interference of the two dressed states yields

P_e(t) = (\Omega ^2/\Omega '^2)*\sin ^2(\Omega '*t/2)

.

Limiting cases

\Delta = 0

(resonance)⟶

P_e = \sin ^2(\Omega *t/2)

, reaches 1Full population transfer — a pi-pulse

(t = pi/\Omega )

is a perfect bit flip.

|Delta| >> Omega⟶ P_e <

= \Omega ^2/\Delta ^2

<< 1, fast shallow wiggleFar off resonance the system barely responds — frequency selectivity of quantum control.

t = pi/\Omega

at resonance (pi-pulse)⟶

P_e = 1

: NOT gateHalf that duration (pi/2-pulse) lands on an equal superposition — the workhorse of Ramsey interferometry.

What if…

What if you stop the drive at a quarter period (pi/2-pulse)?

The system freezes in an equal superposition (|g> + |e>)/sqrt(2) — the initialization step of Ramsey interferometry and most quantum algorithms.

What if decoherence acts during the pulse?

Oscillations decay toward $P_e = 1/2$ with the qubit's T2 envelope — counting visible Rabi cycles is a standard coherence diagnostic.

What if the field is quantized and contains zero photons?

Vacuum Rabi oscillations: the atom exchanges a single quantum with the empty cavity mode at frequency 2g — the hallmark of cavity QED.

1

Pi-pulse duration for a transmon qubit

Given ·

Omega:: 6283185
Delta:: 0

Find · t_pi

Steps

Resonant: $P_e = \sin ^2(\Omega *t/2) = 1$ when $\Omega *t/2 = pi/2$
$t_pi = pi/\Omega = 3.14159 / 6.283185e_{6}$
$= 5.0e-7\,\mathrm{s} = 0.5$ us

Result ·

t_pi = pi/\Omega = 0.5

microseconds — the duration of an X gate at a 1 MHz Rabi frequency.

2

Maximum transfer when detuned by one Rabi frequency

Given ·

Delta:: Omega

Find · max P_e and oscillation frequency

Steps

$\Omega ' = \sqrt{\Omega ^2 + \Omega ^2} = \sqrt{2}*\Omega$
amplitude $= \Omega ^2/\Omega '^2 = 1/2$
$P_e(t) = 0.5*\sin ^2(\sqrt{2}*\Omega *t/2)$ , never exceeding 0.5

Result ·

Max P_e = \Omega ^2/(\Omega ^2+\Delta ^2) = 1/2

, oscillating at

\Omega ' = \sqrt{2}*\Omega

— faster but only half as deep.

Larmor Precession→Spin-1/2 Operators→Time-Dependent Schrödinger Equation→Photon Energy (Planck Relation)→