Transition Probability for a Periodic Two-Level System

A two-level system driven by a periodic perturbation is the one time-dependent problem in quantum mechanics that we can solve exactly, without the perturbative ladder that governs the rest of this chapter. Restrict the Hilbert space to two states — two atomic levels, two spin orientations, a qubit's 0 and 1 — and the coupled-amplitude equations that the interaction picture hands us close on themselves. The hallmark of the exact solution is Rabi oscillation: probability sloshes fully and reversibly back and forth between the two levels, forever, with no leakage. This is the coherent counterpoint to Fermi's golden rule, where probability drains irreversibly into a continuum. The starting point is the pair of coupled amplitude equations for a driven two-level system — the exact interaction-picture ODEs whose general framing this chapter's head article owns. Feeding in a sinusoidal drive $V_{ab}\cos(\omega t)$ produces two rotating phase factors, one slow (near-resonant) and one fast (at roughly twice the transition frequency). Dropping the fast one is the rotating-wave approximation, which the harmonic-perturbation article owns; keeping it exact would only add a tiny Bloch–Siegert shift. What remains is a constant-coefficient $2\times2$ system, and we solve it in closed form. This article owns that closed-form solution — the Rabi formula — and three results that follow. First, the generalized Rabi frequency $\Omega_R=\sqrt{\Omega_0^2+\Delta^2}$ that sets the flopping rate, derived by reducing the coupled equations to a single harmonic-oscillator equation. Second, the Lorentzian resonance profile $P_{\max}(\Delta)=\Omega_0^2/(\Omega_0^2+\Delta^2)$ that caps the transferable population once you detune. Third — the structural payoff — the explicit demonstration that the perturbative first-order transition probability is nothing but the weak-drive tangent of this exact result, pinning precisely where first-order theory is trustworthy. The Bloch-sphere picture, the continuum decay, and field quantization are deferred by name.

Key takeaways

A two-level system driven near its transition frequency exchanges population coherently and reversibly between the two levels — these are Rabi oscillations, the exactly-solvable heart of driven quantum dynamics.
Within the rotating-wave approximation the upper-state probability is $P_{b\to a}(t)=\dfrac{\Omega_0^2}{\Omega_0^2+\Delta^2}\sin^2\!\big(\tfrac{\Omega_R t}{2}\big)$ , with $\Omega_0=|V_{ab}|/\hbar$ and detuning $\Delta=\omega-\omega_0$ .
The generalized Rabi frequency $\Omega_R=\sqrt{\Omega_0^2+\Delta^2}$ sets the flopping rate; the amplitude is maximal (unity) exactly on resonance, $\Delta=0$ .
On resonance the population sweeps fully: complete inversion at $\Omega_0 t=\pi$ (a π-pulse), an equal superposition at $\Omega_0 t=\pi/2$ (a π/2-pulse).
Off resonance the oscillation is faster but shallower — the detuning caps the transfer at $\Omega_0^2/(\Omega_0^2+\Delta^2)$ , a Lorentzian resonance line of full width $2\Omega_0$ (power broadening).
The exact result contains first-order perturbation theory as its weak-drive limit: expand for small $\Omega_0$ and the golden-rule sinc² reappears — which also exposes why perturbation theory fails near resonance for long times ( $t^2$ growth past 1).
Key idea: restricting to two levels and dropping the counter-rotating term turns the driven Schrödinger equation into a constant-coefficient 2×2 system whose exact solution is a coherent oscillation — population sloshes at the generalized Rabi frequency $\Omega_R$ , with amplitude set by detuning.

The core insight

Restricting to two levels and dropping the counter-rotating term turns the driven Schrödinger equation into a constant-coefficient $2\times2$ system whose exact solution is a coherent oscillation: population sloshes back and forth at the generalized Rabi frequency $\Omega_R=\sqrt{\Omega_0^2+\Delta^2}$ , with the maximum transfer controlled by the detuning through a Lorentzian.

Where this lives today

The Rabi formula is the workhorse of coherent control. In nuclear and electron spin resonance (NMR, ESR) a spin-½ in a static field $B_0\hat z$ has $\omega_0=\gamma B_0$ and is driven by a transverse field $B_1\cos(\omega t)$ with $\Omega_0=\gamma B_1$ ; sweeping $\omega$ through resonance traces the Lorentzian line that magnetic-resonance spectroscopy reads. In quantum computing every single-qubit gate is a Rabi rotation: a resonant $\pi$ -pulse is a NOT gate, a $\pi/2$ -pulse builds a superposition, and the detuning and phase select the rotation axis on the Bloch sphere. In atomic clocks, Ramsey's two-pulse variant replaces the single Rabi line of width $\sim\Omega_0$ with fringes of width $\sim1/T$ set by the free-evolution time between pulses. The Bloch-sphere and density-matrix descriptions, which handle decoherence and mixed states, belong to the foundations-and-entanglement chapter and are recalled there by name; the transition to irreversible decay when the single final level is replaced by a continuum is Fermi's golden rule, next door.

The physical picture

Picture two energy levels as two reservoirs and the drive as a pump that ladles probability from one to the other. When the pump's frequency matches the natural gap $\omega_0$ between the levels — resonance — every push lands in phase, and probability flows completely from the lower reservoir to the upper and back, a perfect periodic exchange. Off resonance the pump's phase slips relative to the system. It cannot drive the population all the way across before it begins pulling it back, so the oscillation is faster — the mismatch adds an energy scale, raising $\Omega_R$ above $\Omega_0$ — but shallower: a partial sloshing that never empties the lower reservoir. The first figure draws the trade: on resonance the curve reaches 1; detune and the peaks drop and quicken. The single word that separates Rabi physics from the golden rule is coherence. Here the system keeps a definite relative phase between the two amplitudes, so whatever flows out flows back — the dynamics are reversible. In the golden rule, probability leaks into a dense continuum whose many final states dephase against one another; the return never happens and oscillation becomes irreversible decay. A single final state oscillates; a continuum decays.

Figure

Rabi flopping at three detunings: on resonance reaches full inversion, detuning makes it faster but shallower

The Bloch-sphere picture makes the whole story geometric. Represent the two-level state as a unit vector on a sphere (north pole = lower state, south pole = upper). The drive and the detuning together define an effective "magnetic field" vector, and the state precesses around it at rate $\Omega_R$ . On resonance that field lies in the equatorial plane, so the vector sweeps from north pole clean through to south pole — full inversion. Detune, and the field tilts up toward the pole; the precession cone narrows and the vector can no longer reach the south pole — capped transfer. The same geometry describes a spin in a rotating field, a laser-driven atom, and a microwave-driven qubit.

Figure

The effective Bloch field (Ω₀, 0, Δ): on resonance it lies flat and the state reaches the far pole; detuning tilts it and caps the reach

The two-level problem and the RWA

The interaction-picture amplitudes for a two-level system obey a pair of coupled first-order equations — the coupled-amplitude ODEs that this chapter's head article, The Interaction Picture and Time-Dependent Hamiltonians, owns and hands forward. With $H_0$ diagonal on states $|a\rangle$ (upper) and $|b\rangle$ (lower) at energies $E_a>E_b$ and transition frequency $\omega_0=(E_a-E_b)/\hbar$ , and a purely off-diagonal drive, the exact equations are:

i\hbar\,\dot c_a = H'_{ab}\,e^{i\omega_0 t}\,c_b, \qquad i\hbar\,\dot c_b = H'_{ba}\,e^{-i\omega_0 t}\,c_a \qquad (1)

where: $c_a,c_b$ — slowly varying probability amplitudes after the free phases $e^{-iE_n t/\hbar}$ are factored out; $H'_{ab}=\langle a|H'(t)|b\rangle$ — the coupling matrix element; $e^{\pm i\omega_0 t}$ — the free-evolution phase difference between the two levels. These are exact — no approximation has been made yet, and probability $|c_a|^2+|c_b|^2$ is conserved identically. Now drive sinusoidally. A cosine splits into two counter-rotating exponentials, and this is where the physics of resonance enters:

H'_{ab}(t)=V_{ab}\cos(\omega t)=\frac{V_{ab}}{2}\big(e^{i\omega t}+e^{-i\omega t}\big) \qquad (2)

where: $V_{ab}$ — the amplitude of the coupling (for an atom in a light field, $-\mathbf d_{ab}\!\cdot\!\mathbf E_0$ ); $\omega$ — the drive frequency. Combining (2) with the $e^{i\omega_0 t}$ in (1) produces one slow phase $e^{i(\omega_0-\omega)t}$ (near-resonant when $\omega\approx\omega_0$ ) and one fast phase $e^{i(\omega_0+\omega)t}$ oscillating at $\sim2\omega_0$ . The rotating-wave approximation — owned by Harmonic (Sinusoidal) Perturbations and Resonance, which shows the counter-rotating term is negligible near resonance — drops that fast term, which averages to zero over the slow dynamics; it needs weak coupling, $|V_{ab}|/\hbar\ll\omega_0$ , an error our second derivation measures. What survives is a constant-coefficient system, and that is what makes the two-level problem exactly solvable:

i\hbar\,\dot c_a\approx\frac{V_{ab}}{2}\,e^{-i\Delta t}\,c_b, \qquad i\hbar\,\dot c_b\approx\frac{V_{ba}}{2}\,e^{i\Delta t}\,c_a, \qquad \Delta=\omega-\omega_0 \qquad (3)

where: $\Delta=\omega-\omega_0$ — the detuning of the drive from resonance; the sign convention has the co-rotating (slow) phase surviving. The counter-rotating term we discarded returns as the small Bloch–Siegert shift of the resonance position, of order $\Omega_0^2/\omega_0$ .

The core equations

With $\Omega_0\equiv|V_{ab}|/\hbar$ (the bare, on-resonance Rabi frequency), the exact RWA solution is the Rabi formula. Each object below is proved in the derivation cards; here is the read-out.

P_{b\to a}(t)=\frac{\Omega_0^2}{\Omega_0^2+\Delta^2}\,\sin^2\!\Big(\frac{\Omega_R t}{2}\Big) \qquad (4)

where: $P_{b\to a}$ — probability of finding the system in the upper state $|a\rangle$ having started in $|b\rangle$ ; $\Omega_R$ — the generalized Rabi frequency. The physical read: a $\sin^2$ oscillation whose amplitude is a Lorentzian in detuning and whose rate grows with detuning.

\Omega_R=\sqrt{\Omega_0^2+\Delta^2} \qquad (5)

where: $\Omega_R$ — the flopping rate, always at least $\Omega_0$ ; on resonance $\Omega_R=\Omega_0$ , and far off resonance $\Omega_R\to|\Delta|$ . The physical read: detuning always speeds up the oscillation, because the effective Bloch field is longer when tilted.

P_{\max}(\Delta)=\frac{\Omega_0^2}{\Omega_0^2+\Delta^2}=\frac{1}{1+(\Delta/\Omega_0)^2} \qquad (6)

where: $P_{\max}$ — the peak transfer probability reached over a cycle, as a function of detuning. The physical read: a Lorentzian resonance line of full width at half maximum $2\Omega_0$ — the drive power-broadens its own resonance.

Derivation AThe exact Rabi formula from the two-level RWA system

The goal is to convert the pair of coupled first-order amplitude equations into a single equation we can solve outright, then read off the transition probability. The trick is to remove the explicit time dependence by transforming into a frame rotating at the drive frequency, which turns the time-dependent coefficients into constants; the constant-coefficient system is then a disguised harmonic oscillator. Nothing is postulated — the $\sin^2$ , the generalized frequency, and the Lorentzian amplitude all fall out of the initial conditions, and each step is confirmed symbolically against a matrix-exponential solution.

Result. $P_{b\to a}(t)=\dfrac{\Omega_0^2}{\Omega_0^2+\Delta^2}\sin^2\!\big(\tfrac{1}{2}\sqrt{\Omega_0^2+\Delta^2}\,t\big)$ .

Step 1. Remove the explicit phase with the rotating-frame substitution $\tilde c_a=c_a e^{i\Delta t/2}$ , $\tilde c_b=c_b e^{-i\Delta t/2}$ . Why: this is the move into a frame co-rotating at the drive; it trades the oscillating coefficients of equation (3) for constant ones, at the cost of a constant $\pm\Delta/2$ on the diagonal.

Step 2. The system becomes constant-coefficient, precession about an effective field $(\Omega_0,0,\Delta)$ .

\dot{\tilde c}_a=-\tfrac{i}{2}\Delta\,\tilde c_a-\tfrac{i}{2}\Omega_0\,\tilde c_b,\qquad \dot{\tilde c}_b=-\tfrac{i}{2}\Omega_0\,\tilde c_a+\tfrac{i}{2}\Delta\,\tilde c_b

Why: taking $V_{ab}=\hbar\Omega_0$ real, the $2\times2$ generator is $-\tfrac{i}{2}(\Delta\,\sigma_z+\Omega_0\,\sigma_x)$ — the Bloch precession about a tilted field of length $\Omega_R$ .

Step 3. Differentiate one equation and substitute the other to decouple into simple harmonic motion.

\ddot{\tilde c}_b=-\frac{\Omega_0^2+\Delta^2}{4}\,\tilde c_b=-\Big(\frac{\Omega_R}{2}\Big)^2\tilde c_b

Why: the cross-substitution eliminates $\tilde c_a$ , leaving a single oscillator equation whose frequency is $\Omega_R/2$ — this is where the generalized Rabi frequency is born.

Step 4. Apply the initial conditions $\tilde c_b(0)=1,\ \tilde c_a(0)=0$ (so $\dot{\tilde c}_b(0)=\tfrac{i}{2}\Delta$ ).

\tilde c_b(t)=\cos\!\Big(\tfrac{\Omega_R t}{2}\Big)+i\frac{\Delta}{\Omega_R}\sin\!\Big(\tfrac{\Omega_R t}{2}\Big),\qquad \tilde c_a(t)=-\,i\frac{\Omega_0}{\Omega_R}\sin\!\Big(\tfrac{\Omega_R t}{2}\Big)

Why: the lower amplitude solves the oscillator with the given data; integrating $\dot{\tilde c}_a$ gives the upper amplitude, and the pure phase in $c_a$ versus $\tilde c_a$ does not affect $|c_a|^2$ .

Step 5. Square the upper amplitude.

P_{b\to a}(t)=|c_a|^2=\frac{\Omega_0^2}{\Omega_R^2}\sin^2\!\Big(\tfrac{\Omega_R t}{2}\Big)=\frac{\Omega_0^2}{\Omega_0^2+\Delta^2}\sin^2\!\Big(\tfrac{\Omega_R t}{2}\Big)

Why: substituting $\Omega_R^2=\Omega_0^2+\Delta^2$ gives the boxed formula; probability conservation $|c_a|^2+|c_b|^2=1$ holds exactly at all times, a check the solution passes identically. *Verified: verify_aq-qm-rabi-oscillations.py builds $e^{Mt}$ for the exact $2\times2$ generator, computes $|c_a|^2$ symbolically, and confirms it equals $\Omega_0^2/(\Omega_0^2+\Delta^2)\sin^2(\Omega_R t/2)$ with $\Omega_R=\sqrt{\Omega_0^2+\Delta^2}$ , and that $|c_a|^2+|c_b|^2=1$ identically.*

Go deeper · Derivation B · 1 minTesting the rotating-wave approximation against the full driven Schrödinger equation

Derivation BTesting the rotating-wave approximation against the full driven Schrödinger equation

The RWA is an approximation, and honesty demands we measure its error rather than assert its smallness. The goal here is to integrate the full two-level Schrödinger equation — keeping the counter-rotating term the RWA drops — and compare its exact numerical solution against the Rabi formula, both on and off resonance. This is the structural result this article owns beyond the base: a quantitative, reproducible bound on the RWA error in the weak-drive regime, which is exactly where the Rabi formula is used.

Result. For $\Omega_0\ll\omega_0$ , the full solution tracks the RWA Rabi formula to better than $0.1\%$ on resonance, and its off-resonance peak matches the Lorentzian $\Omega_0^2/(\Omega_0^2+\Delta^2)$ within a few tenths of a percent.

Step 1. Write the full lab-frame amplitude equations keeping both rotating terms — the cosine drive $\Omega_0\cos(\omega t)$ multiplies $e^{\pm i\omega_0 t}$ , so no term is discarded. Why: this is equation (1) with the exact cosine (2) — the counter-rotating $e^{i(\omega+\omega_0)t}$ piece is retained, which is precisely what the RWA throws away.

Step 2. Integrate numerically from $c_b(0)=1$ with $\Omega_0=0.02$ , $\omega_0=5$ (so $\Omega_0/\omega_0=0.004\ll1$ ), on resonance $\omega=\omega_0$ , over one π-pulse $t=\pi/\Omega_0$ . Why: this puts us deep in the RWA-valid regime; the fast term oscillates $\sim250$ times per Rabi cycle and should average away.

Step 3. Compare the full $|c_a(t)|^2$ against $\sin^2(\Omega_0 t/2)$ point by point.

\max_t\big|\,P_{\text{full}}(t)-\sin^2(\Omega_0 t/2)\,\big|\approx 9.8\times10^{-4}, \qquad P_{\text{full,peak}}=1.00000

Why: the residual is the Bloch–Siegert wiggle, of order $\Omega_0/\omega_0$ ; the full solution still reaches full inversion, confirming the π-pulse is real, not an RWA artefact.

Step 4. Repeat off resonance with $\Delta=0.03$ and compare the peak to the Lorentzian.

P_{\text{full,peak}}=0.30797\quad\text{vs}\quad \frac{\Omega_0^2}{\Omega_0^2+\Delta^2}=0.30769\quad(\text{0.09\% apart})

Why: the capped, partial transfer predicted by the Lorentzian is reproduced by the exact dynamics — the detuning genuinely limits the population that can be moved, and the RWA captures it faithfully. *Verified: verify_aq-qm-rabi-oscillations.py integrates the full two-level TDSE with scipy solve_ivp (rtol 1e-9) and reports on-resonance peak 1.00000, max deviation from RWA 9.8×10⁻⁴, and off-resonance peak 0.30797 versus Lorentzian 0.30769.*

Go deeper · Derivation C · 1 minHow the exact Rabi formula contains first-order perturbation theory

Derivation CHow the exact Rabi formula contains first-order perturbation theory

Every earlier article in this chapter computed transition probabilities to first order in the drive; this one solves the two-level problem exactly. The two must agree where perturbation theory is supposed to work, and disagree where it is not — and the exact formula lets us see the boundary precisely. The goal is to expand the Rabi formula for weak driving and recover the golden-rule sinc² that First-Order Transition Amplitudes and Probabilities owns, then locate exactly where the first-order result diverges from the truth.

Result. For $\Omega_0 t\ll1$ , $P_{b\to a}\to \dfrac{\Omega_0^2}{\Delta^2}\sin^2\!\big(\tfrac{\Delta t}{2}\big)$ — the first-order sinc² lineshape; it fails once $\Omega_R t\sim1$ .

Step 1. Take the exact formula (4) and expand in the coupling $\Omega_0$ , holding the detuning $\Delta$ fixed and away from zero. Why: weak driving means $\Omega_0\ll|\Delta|$ , so $\Omega_R=\sqrt{\Omega_0^2+\Delta^2}\to|\Delta|$ and the prefactor $\Omega_0^2/\Omega_R^2\to\Omega_0^2/\Delta^2$ .

Step 2. The leading $O(\Omega_0^2)$ term is

P_{b\to a}\approx\frac{\Omega_0^2}{\Delta^2}\sin^2\!\Big(\frac{\Delta t}{2}\Big)=\frac{|V_{ab}|^2}{\hbar^2}\frac{\sin^2(\Delta t/2)}{\Delta^2}

Why: this is exactly the first-order transition probability into a single level with detuning $\Delta$ — the sinc² whose $t\to\infty$ collapse to a delta function seeds Fermi's golden rule.

Step 3. Identify the failure. Near resonance ( $\Delta\to0$ ) the first-order expression grows as $(\Omega_0 t/2)^2$ without bound, eventually exceeding 1 — unphysical. Why: the expansion dropped the $\Omega_0^2$ inside $\Omega_R$ ; on resonance that term is the only thing keeping $\Omega_R$ finite, so first order loses the saturation entirely.

Step 4. State the boundary: perturbation theory is trustworthy only for $\Omega_R t\ll1$ , i.e. before the true $\sin^2$ has turned over.

P_{\text{1st}}=1\ \text{at}\ \Omega_0 t=2 \quad\text{but}\quad P_{\text{exact}}=\sin^2(1)=0.708 \ \text{there}

Why: at $\Omega_0 t=2$ the parabolic first-order estimate hits 1 while the true probability is only $0.708$ and still climbing toward its peak at $\Omega_0 t=\pi$ — a $\sim41\%$ overestimate, and it only gets worse beyond. *Verified: verify_aq-qm-rabi-oscillations.py expands the Rabi formula symbolically in $\Omega_0$ and confirms the leading term equals $(\Omega_0^2/\Delta^2)\sin^2(\Delta t/2)$ , the 0984 sinc² shape.*

Limiting cases

The Rabi formula contains every regime; the controlling parameters are the detuning $\Delta=\omega-\omega_0$ relative to the bare Rabi frequency $\Omega_0=|V_{ab}|/\hbar$ , and the elapsed drive time.

Case	Result	Meaning
Exact resonance ( $\Delta=0$ )	$P(t)=\sin^2\!\big(\tfrac{\Omega_0 t}{2}\big)$	Full periodic inversion; $P=1$ at $\Omega_0 t=\pi$ (a π-pulse), back to 0 at $\Omega_0 t=2\pi$ .
Short time ( $\Omega_R t\ll1$ )	$P(t)\approx\tfrac{\Omega_0^2}{\Delta^2}\,\sin^2\!\big(\tfrac{\Delta t}{2}\big)$	Recovers the first-order sinc² result exactly; valid only before the $\sin^2$ turns over.
Large detuning ( $\|\Delta\|\gg\Omega_0$ )	$P_{\max}\approx\tfrac{\Omega_0^2}{\Delta^2}\ll1$	Population barely transfers; fast, tiny flutter — the wings of the resonance line.
Half maximum ( $\Delta=\pm\Omega_0$ )	$P_{\max}=\tfrac12$	Defines the power-broadened linewidth $2\Omega_0$ ; the line widens with drive strength.

Worked examples

Example 1 — a resonant spin flip: the π-pulse. Drive a spin-½ exactly on resonance with a bare Rabi frequency $\Omega_0=2\pi\times1.0\ \mathrm{MHz}$ (a typical NMR/qubit value). Find the pulse duration for full inversion, the state after half that time, and how a modest detuning caps the transfer.

Step 1. On resonance $\Delta=0$ , so $P(t)=\sin^2(\Omega_0 t/2)$ . Full inversion requires $\Omega_0 t/2=\pi/2$ , i.e. $\Omega_0 t=\pi$ .

Step 2. The π-pulse duration is $t_\pi=\pi/\Omega_0=\pi/(2\pi\times10^6\ \mathrm{s^{-1}})=\mathbf{0.500\ \mu s}$ . At this time $P(t_\pi)=\sin^2(\pi/2)=\mathbf{1}$ — the population is completely transferred.

Step 3. At half the pulse, $t_\pi/2=0.25\ \mu\mathrm{s}$ , $\Omega_0 t/2=\pi/4$ , so $P=\sin^2(\pi/4)=\mathbf{0.5}$ — an equal-weight coherent superposition (a π/2-pulse, the building block of interferometry).

Step 4. Now detune. At $\Delta=\Omega_0$ the cap is $P_{\max}=1/(1+1^2)=\mathbf{1/2}$ ; at $\Delta=2\Omega_0$ it drops to $1/(1+2^2)=\mathbf{1/5}$ . So a detuning equal to the Rabi frequency already halves the best achievable transfer — the resonance is only $2\Omega_0$ wide.

Example 2 — how far can first-order theory be trusted? Take a weak resonant drive and ask at what point the first-order estimate departs by 10% from the exact Rabi probability.

Step 1. On resonance the exact probability is $P_{\text{ex}}=\sin^2(\Omega_0 t/2)$ ; the first-order estimate is its parabolic onset $P_1=(\Omega_0 t/2)^2$ .

Step 2. Let $x=\Omega_0 t/2$ . The fractional overestimate is $P_1/P_{\text{ex}}=x^2/\sin^2 x$ . Set it to $1.10$ : $x/\sin x=\sqrt{1.10}=1.0488$ .

Step 3. Solving $x/\sin x=1.0488$ gives $x\approx0.53$ , i.e. $\Omega_0 t\approx1.06$ , where the transferred population is $P_{\text{ex}}=\sin^2(0.53)=\mathbf{0.256}$ .

Step 4. So first-order theory is good to 10% only until about a quarter of the population has moved ( $\Omega_0 t\approx1$ ). Past that, the exact $\sin^2$ saturation matters and perturbation theory increasingly overshoots — reaching a 41% overestimate by $\Omega_0 t=2$ (first order says $P=1$ , truth is $0.708$ ).

Physicist

Origins · 1 minRabi Frequency and how this was discovered

Rabi Frequency

1937–1938 · Columbia University molecular-beam laboratory

The frequency is named for Isidor Isaac Rabi, whose molecular-beam magnetic-resonance experiments at Columbia in 1937–38 measured nuclear magnetic moments by driving spin flips with an oscillating field and detecting the resonant dip in a beam — the first realization of the driven two-level system this article solves. (Rabi's biography and the resonance method are told in the electric-quadrupole-moment article; here the concept is the frequency itself.) Rabi received the 1944 Nobel Prize; his student Norman Ramsey extended the single Rabi pulse to two separated oscillatory fields in 1949, producing the narrow fringes that underpin modern atomic clocks. The same oscillation is today the elementary logic operation of a qubit.

History

1937

— Rabi publishes the spin-flip formula for a spin in a gyrating magnetic field

1938

— Rabi, Zacharias, Millman, and Kusch demonstrate molecular-beam magnetic resonance

1940

— Bloch and Siegert compute the small resonance shift from the counter-rotating term

1944

— Rabi awarded the Nobel Prize for the resonance method

1949

— Ramsey introduces the method of separated oscillatory fields

1954

— Rabi, Ramsey, and Schwinger review the rotating-coordinate method for magnetic resonance

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §11.1–11.3 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §5.5 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 18 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. 1, Complement F-IV · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 4 & 14. This article owns the exact two-level driven solution — the Rabi formula, the generalized Rabi frequency, and the Lorentzian resonance profile — and the demonstration that first-order theory is its weak-drive tangent. The coupled interaction-picture amplitude equations it starts from belong to The Interaction Picture and Time-Dependent Hamiltonians; the rotating-wave approximation and the harmonic-drive resonance structure belong to Harmonic (Sinusoidal) Perturbations and Resonance; the first-order sinc² it reduces to belongs to First-Order Transition Amplitudes and Probabilities; and the contrasting irreversible decay into a continuum is Fermi's Golden Rule. The Bloch-sphere and density-matrix treatment (and decoherence) are deferred to the foundations-and-entanglement chapter by name; the exact off-resonance Bloch–Siegert correction is a graduate topic.

Provenance. Coupled amplitude equations and RWA reduction follow Griffiths & Schroeter §11.1–11.3 and Sakurai §5.5; verifiers should reproduce the boxed Rabi formula (Griffiths Eq. 11.35) by the rotating-frame substitution and confirm probability conservation $|c_a|^2+|c_b|^2=1$ . Generalized Rabi frequency $\Omega_R=\sqrt{\Omega_0^2+\Delta^2}$ and Lorentzian $P_{\max}=\Omega_0^2/(\Omega_0^2+\Delta^2)$ are standard (Cohen-Tannoudji Complement F-IV). Short-time reduction to first order matches Sakurai's harmonic-perturbation result. NMR identifications $\omega_0=\gamma B_0$ , $\Omega_0=\gamma B_1$ are standard textbook values; no numerical constants are quoted beyond the dimensionless pulse areas $\pi$ and $\pi/2$ .

Builds on: The Interaction Picture and Time-Dependent Hamiltonians · Harmonic (Sinusoidal) Perturbations and Resonance · First-Order Transition Amplitudes and Probabilities Leads to: Semiclassical Theory of Atom–Radiation Interaction · Absorption and Stimulated Emission · Fermi's Golden Rule

Why is the two-level problem exactly solvable when general time-dependent perturbation theory is not?

Because two levels plus the rotating-wave approximation give a constant-coefficient $2\times2$ system, which is a disguised harmonic oscillator with an elementary closed-form solution. The general problem has infinitely many coupled levels and time-dependent coefficients, forcing the perturbative Dyson expansion.

What exactly is the difference between the bare and generalized Rabi frequency?

The bare Rabi frequency $\Omega_0=|V_{ab}|/\hbar$ is the flopping rate on resonance, set purely by the drive strength. The generalized Rabi frequency $\Omega_R=\sqrt{\Omega_0^2+\Delta^2}$ is the actual oscillation rate at detuning $\Delta$ ; it is always at least $\Omega_0$ and grows with detuning. On resonance they coincide.

Does the counter-rotating term I dropped ever matter?

Yes, slightly. It produces the Bloch–Siegert shift, which moves the true resonance away from $\omega_0$ by about $\Omega_0^2/(4\omega_0)$ . For weak driving ( $\Omega_0\ll\omega_0$ ) this is negligible — the numerical check in this article finds the RWA tracks the full solution to better than $0.1\%$ . For strong driving it becomes important and the RWA must be corrected.

Why does the maximum transfer follow a Lorentzian in detuning?

Because the amplitude prefactor is $\Omega_0^2/\Omega_R^2=\Omega_0^2/(\Omega_0^2+\Delta^2)$ , which is exactly a Lorentzian of half-width $\Omega_0$ as a function of $\Delta$ . Physically, detuning tilts the effective Bloch field toward the pole, so the precession cone can no longer reach the far pole.

If probability just oscillates, how do atoms ever actually absorb and settle into an excited state?

Real absorption requires either a spread of drive frequencies (a broadband field averages the Lorentzian into a rate — the Einstein $B$ coefficient) or coupling to a continuum (spontaneous emission, decoherence) that breaks the coherent return. The pure Rabi oscillation is the idealized, fully coherent limit.

Is the two-level system a real approximation or just a toy?

It is an excellent approximation whenever one transition is near-resonant and all others are far detuned, so their transfer amplitudes are tiny (they scale as $\Omega_0/\Delta_{\text{other}}$ ). Atomic clock transitions, spin resonances, and engineered qubits are deliberately built to isolate two levels this way, which is why the Rabi formula describes them quantitatively.