Rabi cycle ($P_{g\to e}(t)={\sin }^2(\Omega t/2)$ on resonance)

A Rabi oscillation is a coherent oscillation of populations in a driven two-level quantum system.

A resonant drive can transfer a quantum state from one energy eigenstate to another and then bring its population back. This is unitary time evolution: the drive exchanges energy with the selected two-level system, while total probability stays one. One complete Rabi cycle has population period $2\pi /\Omega$; halfway through, a Rabi pi pulse transfers the whole population in the ideal model. This makes Rabi oscillations useful for calibrating resonant quantum control, preparing quantum superpositions and supplying the pulses of Ramsey interferometry.

A circularly driven spin becomes time-independent after a rotation $R=e^{-i\omega t{S}_z/\hslash }$. The transformed Hamiltonian operator is $R^{†}HR-i\hslash R^{†}\dot{R}$, so the longitudinal frequency is shifted by $-\omega$. This is an exact change of frame for a circular drive. The rotating-wave approximation is a separate approximation needed when a drive also contains a counterrotating component.

For positive gyromagnetic ratio in static field $B\hat{z}$, transverse components $B_1\cos (\omega t+\phi )$ and $-B_1\sin (\omega t+\phi )$ rotate with the spin precession. Their rotating-frame Hamiltonian operator is time independent, with transverse Rabi angular frequency $\Omega =\gamma B_1$. A linearly polarized field of peak amplitude $B_{\mathrm{peak}}$ contains two oppositely rotating components and gives resonant $\Omega =|\gamma |B_{\mathrm{peak}}/2$ in the rotating-wave approximation. Negative gyromagnetic ratio requires opposite rotation sense.

The coherent coupling frequency defined by the chosen driven two-level system Hamiltonian. With $H=\hslash (-\Delta {\sigma }_z+\Omega {\sigma }_x)/2$, resonance gives population oscillations proportional to ${\sin }^2(\Omega t/2)$. Detuning replaces the oscillation frequency by $\sqrt{{\Omega }^2+{\Delta }^2}$ and reduces its contrast. Stating the Hamiltonian avoids factor-of-two ambiguities in the definition of $\Omega$.

In the Driven two-level Rabi model, the transition probability is $P(t)={\Omega }^2{\sin }^2(\sqrt{{\Omega }^2+{\delta }^2}t/2)/({\Omega }^2+{\delta }^2)$. Maximizing over time at each detuning gives a Lorentzian line shape, with first maximizing time $\pi /\sqrt{{\Omega }^2+{\delta }^2}$ and full width at half maximum $2|\Omega |$ in angular frequency. The amplitude parameter $\Omega$ must use the specified Rabi angular frequency convention.

This envelope differs from a spectrum obtained with a fixed pulse duration: optimizing the time separately removes the sine-squared factor, while fixing time retains it. A finite linewidth here describes detuning-limited transfer under coherent driving, without requiring dissipative decay.

Choose an isolated pair of energy levels, with ground state $|g⟩=(0,1{)}^T$ and excited state $|e⟩=(1,0{)}^T$. Use the Pauli matrices in the ordered basis $(|e⟩,|g⟩)$, so ${\sigma }_z=\mathrm{diag}(1,-1)$. After subtracting the mean energy, prescribe the laboratory Hamiltonian operatorThe unperturbed energy gap is $\hslash {\omega }_0$, the drive has angular frequency $\omega$, and $\varphi$ fixes its oscillation phase at the chosen time origin. All of ${\omega }_0$, $\omega$, the Rabi angular frequency $\Omega$ and the Rabi detuning $\delta$ are in radians per second; ordinary frequency is angular frequency divided by $2\pi$. In particular, $\Omega$ here is the coefficient multiplying the cosine in $H_{\mathrm{lab}}/\hslash$, not twice that coefficient.

Write $|{\psi }_{\mathrm{lab}}⟩=R(t)|{\psi }_{\mathrm{rot}}⟩$ with $R(t)=\exp (-i\omega t{\sigma }_z/2)$. Transforming the Schrödinger equation gives $H_{\mathrm{rot}}=R^{†}{H}_{\mathrm{lab}}R-i\hslash R^{†}\dot{R}$. The rotating-wave approximation drops terms at angular frequency $2\omega$, leavingThis approximation requires $\Omega ,|\delta |\ll {\omega }_0,\omega$, and an interaction duration over which omitted corrections remain small. It also assumes that other energy levels can be neglected. Resonance alone does not justify either assumption. An exactly co-rotating drive has no discarded counter-rotating component; the linearly oscillating laboratory drive above does.

The Pauli matrix multiplication law gives $K^2={\Omega }_{\mathrm{eff}}^{2}I$. Separating even and odd terms of the matrix exponential therefore proveswith the continuous limit $U=I$ when $\Omega =\delta =0$. Starting from $|g⟩$, the upper probability amplitude is $-i{e}^{-i\varphi }(\Omega /{\Omega }_{\mathrm{eff}})\sin ({\Omega }_{\mathrm{eff}}t/2)$. The Born rule givesThe rotating transformation only changes the quantum phases of these basis states, so their laboratory populations are the same. Rabi detuning tilts the rotation axis of the Bloch vector: it increases the oscillation rate to ${\Omega }_{\mathrm{eff}}$ but caps the transferred population at ${\Omega }^2/({\Omega }^2+{\delta }^2)$. A fixed drive oscillation phase cancels from this single-pulse population when the initial state is $|g⟩$. It still fixes the relative quantum phase of a resulting quantum superposition and matters when another pulse supplies a phase reference.

For a rotating-frame two-state Hamiltonian operator $H=\hslash \mathbf{W}\cdot \mathbf{\sigma }/2$, the Bloch vector obeys $\dot{\mathbf{R}}=\mathbf{W}\times \mathbf{R}$. Thus $\mathbf{W}$ specifies a rotation axis and angular speed. With detuning defined as drive frequency minus transition frequency and the lower state at positive $R_z$, a phase-zero transverse drive can give $\mathbf{W}=(\Omega ,0,\delta )$. Changing basis or detuning convention requires changing the corresponding signs consistently.

For an initial pole of the Bloch sphere, one fixed-amplitude harmonic pulse detuned by $\Delta$ transfers at most ${\Omega }^2/({\Omega }^2+{\Delta }^2)$ of the population. A pure equatorial state requires half transfer, so some pulse duration reaches the equator exactly when $|\Delta |\le \Omega$. Adjusting drive phase then selects its transverse azimuth. Exact inversion to the opposite pole is unavailable for nonzero detuning with that single fixed pulse. This is a restriction on the specified controls, rather than on composite-pulse or frequency-swept state preparation.

With $H_0=-\hslash {\omega }_0{\sigma }_z/2$ and a co-rotating transverse magnetic drive of phase $\phi$, an initial spin-up state becomes, up to common phase, $\cos (\Omega \tau /2)|\uparrow ⟩+i{e}^{-i(\phi +{\omega }_0\tau )}\sin (\Omega \tau /2)|\downarrow ⟩$. A quarter-turn pulse therefore prepares laboratory $+x$ when ${\phi }_{\mathrm{end}}=\pi /2$ and $+y$ when ${\phi }_{\mathrm{end}}=0$. Dropping the free-precession phase instead specifies targets in rotating axes. Transverse laboratory targets subsequently precess if the static field is retained.

The Rabi angular frequency is the on-resonance population oscillation rate in the specified Driven two-level Rabi model. The rotating-frame off-diagonal Hamiltonian operator entry has magnitude $\hslash \Omega /2$. Stating this convention avoids a factor-of-two ambiguity between drive coefficients and oscillation rates. The generalized rate off resonance is ${\Omega }_{\mathrm{eff}}=\sqrt{{\Omega }^2+{\delta }^2}$.

The Rabi detuning is the unperturbed transition angular frequency minus the drive angular frequency in this convention. Its sign controls the longitudinal rotation of the Bloch vector; the single-pulse population formula is even in $\delta$. Reversing the definition of Rabi detuning requires reversing its sign in the rotating-frame Hamiltonian operator as well.

For a constant-amplitude square pulse on resonance, a Rabi pi pulse lasts ${\tau }_{\pi }=\pi /\Omega$ and gives $|g⟩\mapsto -i{e}^{-i\varphi }|e⟩$ in the rotating frame. A half-duration pulse prepares $(|g⟩-i{e}^{-i\varphi }|e⟩)/\sqrt{2}$. The quantum pulse area $\Omega t$ determines these rotations only under the stated resonant, constant-axis assumptions; a detuned drive rotates about a different axis.

As an original numerical illustration, choose $\Omega =2\pi \times 1000\mathrm{rad}{\mathrm{s}}^{-1}$ and hence ${\tau }_{\pi }=0.500\mathrm{ms}$. These are chosen model parameters, not historical apparatus settings. Holding that duration fixed givesThus $P_{\pi }(0)=1$, $P_{\pi }(1)=\frac{1}{2}{\sin }^2(\pi /\sqrt{2})\simeq 0.3166$, and $P_{\pi }(\sqrt{3})=0$. The last zero comes from completing a full rotation about the tilted axis, rather than switching the drive off. At zero Rabi detuning, doubling the pulse duration restores the ground-state population; continuous driving does not keep transferring population irreversibly.

For the Driven two-level Rabi model at fixed resonant pulse duration $T_{\pi }=\pi /\Omega$, put $x=\delta /\Omega$. Its transfer probability isThe sine-squared numerator differs from one only at order $x^4$, while the denominator gives $1-x^2+O(x^4)$. Thus the leading population-transfer error is $x^2$. For a target transfer error $ϵ$, the frequency mismatch in cycles per second should satisfy $|\delta \nu |\lesssim \Omega \sqrt{ϵ}/(2\pi )$ in this small-detuning regime. This is not a complete characterization of every coherent gate error.

In 1938, Isidor Isaac Rabi, Jerrold Zacharias, Sidney Millman and Polykarp Kusch introduced molecular-beam magnetic resonance. Their 1939 account describes a heated oven, vacuum slits, inhomogeneous A/B magnetic fields with opposite gradients, and a uniform static C magnetic field. Conductors supplied transverse oscillatory radiofrequency (RF) fields. Transitions spoiled refocusing, reducing the tungsten-filament detector signal. Alignment compared A-only and B-only attenuation; C-field sweeps at fixed RF frequency limited oscillator pickup. Matching frequency/field ratios across compounds checked nuclear assignments. Magnetic field calibration limited absolute magnetic moments; weak intervening fields caused unwanted transitions.^[ref]

The readout has a useful logical structure: A and the apertures admit trajectories correlated with a magnetic moment projection; B tests whether that projection has survived C. The two deflectors compensate for unchanged states, whereas a changed projection experiences a different force in B. A dip followed by recovery as the static magnetic field passes through resonance is therefore a population-sensitive signal. A comparison with RF absent or far from resonance is the natural baseline in this functional model; RF pickup, drift or background loss must not be mistaken for a state transition. A dip alone does not establish a pure initial quantum state or resolve an entire time-domain Rabi cycle.

This beam method measures transmitted particles after state-dependent deflection. The later bulk nuclear magnetic resonance methods of Felix Bloch and Edward Mills Purcell instead use a macroscopic sample and detect nuclear induction or RF absorption; NMR spectroscopy builds on that bulk setting. The common resonant coupling does not make their preparations or readouts interchangeable.^[ref][ref]

For an ideal single transit, let the oscillatory magnetic field occupy a region of length $L$, take the longitudinal speed $v$ constant, and neglect edge fields. Each molecule then experiences one square pulse of duration $\tau =L/v$. This is one contiguous interaction region, distinct from the two separated pulses of Ramsey interferometry. The static magnetic field sets the selected transition gap; the transverse drive sets Rabi angular frequency. A change of RF frequency or static magnetic field changes Rabi detuning.

For an original ideal detector model, suppose every successful transition removes an otherwise accepted molecule and the incident beam occupies the lower state of the selected pair. With accepted no-transition signal $S_0$, constant background $S_{\mathrm{bg}}$, and effective loss contrast $0\le \eta \le 1$, writeFor identical trajectories $\overline{P}=P_{g\to e}(L/v)$, giving an intensity dip where transfer is largest. For real trajectory variation, a normalized joint probability density $w(v,ϵ,{\Omega }^{\prime })$ gives the static-offset modelHere $ϵ$ is the trajectory's transition-frequency offset and ${\Omega }^{\prime }$ its drive strength, each in angular-frequency units. The weighting is for molecules accepted by the apparatus, rather than an assumed unfiltered oven distribution. In particular, designing $\Omega L/v_0=\pi$ does not produce a Rabi pi pulse for every speed: at $0.8v_0$ and $1.2v_0$ the resonant transfer probabilities are respectively ${\sin }^2(5\pi /8)\simeq 0.8536$ and ${\sin }^2(5\pi /12)\simeq 0.9330$.

Spread in transit times, static magnetic fields and drive amplitudes can reduce or smear oscillations without quantum decoherence on any individual trajectory. Even the single-transit square pulse has finite spectral width of order $1/\tau$ in angular frequency. If the magnetic field varies substantially along one path, its Hamiltonian operator changes during the passage and the constant-offset average above must be replaced by a time-ordered exponential. Actual molecules may also have several coupled energy levels; the isolated pair and perfect loss discriminator are explicit idealizations of this model.