Rotating Wave Approximation

A two-level atom driven by a classical light field lives, at first sight, in a hopelessly time-dependent world: its interaction energy $-\mathbf{d}\cdot\mathbf{E}(t)$ swings up and down at the optical frequency $\omega$ , hundreds of trillions of times a second, and a time-dependent Hamiltonian has no stationary states to hang an analysis on. The rotating wave approximation is the single move that tames this. It takes the two-level structure from aq-optics-stimulated-emission and the coupling strength $\mathbf{d}\cdot\mathbf{E}_0$ from aq-optics-transition-dipole-moment, expands the driving cosine into two counter-rotating phasors, and throws away the one that never comes close to conserving energy. The reward is out of all proportion to the effort. Expanding $\cos\omega t$ into $e^{+i\omega t}+e^{-i\omega t}$ and the dipole into its raising and lowering parts $\sigma_\pm$ gives four coupling terms. Two, viewed in a frame rotating with the drive, evolve slowly at the detuning $\Delta=\omega-\omega_0$ ; the other two whirl at $\omega+\omega_0\approx2\omega_0$ and average to almost nothing. Discard the fast pair and the Hamiltonian becomes time-independent — a $2\times2$ matrix you diagonalize by hand. Out drop coherent Rabi oscillations, the optical Bloch equations, dressed states, and — once the field is quantized — the Jaynes-Cummings model at the heart of cavity QED. The approximation is not free: the discarded terms push the resonance up by the tiny Bloch-Siegert shift $\delta\omega\approx\Omega^2/(4\omega_0)$ and, in ultrastrong-coupling circuit QED, remake the ground state itself. Knowing the validity window — $\Omega,\,|\Delta|\ll\omega,\,\omega_0$ — is what separates using the RWA from misusing it.

Key takeaways

The rotating wave approximation (RWA) drops the rapidly oscillating counter-rotating terms in the atom-field coupling, keeping only the near-resonant co-rotating terms.
It is justified when the Rabi frequency $\Omega$ and the detuning $\Delta=\omega-\omega_0$ are both small next to the optical frequencies, so the discarded terms oscillate at $\omega+\omega_0\approx2\omega_0$ and average away.
In a rotating frame the two-level Hamiltonian becomes time-independent, giving the exact Rabi solution: the excited population oscillates at $\Omega_R=\sqrt{\Omega^2+\Delta^2}$ .
On resonance a pulse of area $\Omega t=\pi$ fully inverts the atom; off resonance the flopping is faster but never reaches the top.
The neglected terms leave the small Bloch-Siegert shift $\sim\Omega^2/\omega_0$ and, in ultrastrong cavity coupling, virtual-photon dressing of the ground state.
Key idea: in a frame rotating with the drive, the atom-field coupling splits into a slow co-rotating part (kept) and a fast counter-rotating part oscillating at $\omega+\omega_0$ (discarded); dropping the latter makes the Hamiltonian time-independent and exactly solvable.

The core insight

In a frame rotating with the drive, the atom-field coupling splits into a slowly varying co-rotating part (kept) and a fast counter-rotating part oscillating at $\omega+\omega_0$ (discarded); dropping the latter makes the Hamiltonian time-independent and exactly solvable.

Where this lives today

The rotating wave approximation is the silent partner in nearly every experiment that manipulates a quantum two-level system coherently. In a superconducting-qubit processor each single-qubit gate is a shaped microwave pulse whose length and phase are set in the rotating frame — an $X$ gate is a $\pi$ -pulse, a Hadamard a $\pi/2$ rotation about a tilted axis — with the control electronics synthesizing the drive at the qubit frequency so the hardware itself lives in the RWA's rotating frame. The same mathematics runs trapped-ion and neutral-atom computers, atomic clocks (Ramsey's separated-oscillatory-fields method is the rotating frame applied twice), and every MRI scanner, whose spin-echo sequences are Rabi $\pi/2$ and $\pi$ pulses on nuclear spins. Where the approximation is instead studied for its failure, the frontier is ultrastrong and deep-strong coupling in circuit QED, with $g/\omega_0$ reaching 0.1 to above 1. There the discarded counter-rotating terms become the dominant new physics: the ground state fills with virtual photons and the spectrum leaves the neat Jaynes-Cummings ladder for the full quantum Rabi model. Measuring the Bloch-Siegert shift and those virtual photons is now a live program — a direct look at exactly the terms a century of near-resonant optics was right to throw away.

The physical picture

Split the linearly polarized drive $E_0\cos\omega t$ , as any linear oscillation can be split, into two circular fields rotating in opposite senses, and climb into a frame that spins with the atom at $\omega_0$ . One circular component almost stops — its phase drifts only at the detuning $\Delta=\omega-\omega_0$ — while the other whirls past at $\omega+\omega_0$ , nearly twice the optical frequency. The atom feels a strong sustained push from the slow one and only rapid buffeting from the fast one, and over any interval long compared with an optical cycle that buffeting integrates to essentially zero. The RWA is the licence to say: ignore the buffeting, keep the push.

Figure

Linear drive resolved into two counter-rotating phasors. In the frame rotating at the drive frequency one phasor is nearly stationary, drifting only at the detuning Δ = ω − ω₀ (kept), while the other spins at ω + ω₀ ≈ 2ω₀ (dropped) — for an optical transition ≈ 4.8×10¹⁵ rad/s, one turn every ≈ 1.3 fs

Energetically the kept terms describe the only near-resonant processes: absorb a photon while the atom climbs to $|e\rangle$ , or emit one while it falls to $|g\rangle$ , both conserving energy to within $\hbar\Delta$ . The discarded terms — climb while emitting, fall while absorbing — are off the energy shell by about $2\hbar\omega_0$ , possible only as fleeting virtual excursions whose net effect near resonance averages to a negligible second-order correction. The mathematical payoff is the whole point: a time-dependent problem becomes time-independent after the rotating-frame transformation, a $2\times2$ matrix you can diagonalize by hand. Out drops the Rabi formula — clean, coherent population flopping between the levels. The second figure makes it concrete for a real optical drive.

Figure

Rabi flopping predicted by the RWA for a drive of Ω = 2π × 10 MHz. On resonance (Δ = 0, blue) the excited-state population reaches unity — full inversion — at the π-pulse time t = π/Ω = 50 ns. Detuned by Δ = Ω (amber) the oscillation is faster, at Ω_R = √2 Ω = 2π × 14.1 MHz, but tops out at Ω²/(Ω²+Δ²) = 1/2

The cost is a small phase. The counter-rotating terms shift the resonance (the Bloch-Siegert shift) and, for a quantized field, dress even the ground state with virtual photons — both second order in $\Omega/\omega_0$ and negligible for an ordinary optical drive (for the 10 MHz example the shift is a fraction of a hertz on a $3.8\times10^{14}$ Hz transition). They matter only when the coupling approaches the transition frequency, exactly where the RWA is not allowed.

The core equations

Everything begins with the semiclassical Hamiltonian of a two-level atom in a classical field: a bare atomic energy splitting the levels by $\hbar\omega_0$ , plus the electric-dipole coupling $-\mathbf{d}\cdot\mathbf{E}(t)$ , which flips the atom between levels and carries the awkward $\cos\omega t$ time dependence the whole approximation exists to remove.

H = \tfrac{1}{2}\hbar\omega_0\,\sigma_z - \mathbf{d}\cdot\mathbf{E}_0\,\cos\omega t\;(\sigma_+ + \sigma_-) \qquad (1)

where: $\omega_0$ — atomic transition (angular) frequency; $\omega$ — drive frequency; $\sigma_z=|e\rangle\langle e|-|g\rangle\langle g|$ — Pauli inversion operator; $\sigma_+=|e\rangle\langle g|$ and $\sigma_-=|g\rangle\langle e|$ — raising and lowering operators; $\mathbf{d}$ — transition dipole moment; $\mathbf{E}_0$ — real field amplitude. The diagonal piece stores energy, the off-diagonal piece exchanges it with the field, modulated at the optical frequency. The strength of that exchange is the Rabi frequency, set by how strongly the dipole couples to the field — the single number governing how fast population sloshes between the levels, against which everything downstream is measured.

\Omega = -\frac{\mathbf{d}\cdot\mathbf{E}_0}{\hbar} \qquad (2)

where: $\Omega$ — on-resonance Rabi frequency, the rate of coherent population exchange. A larger dipole or brighter field means a faster flop; doubling the amplitude doubles $\Omega$ and halves the $\pi$ -pulse time. Now expand $\cos\omega t=\tfrac12(e^{i\omega t}+e^{-i\omega t})$ and pass to the interaction picture, where the atomic operators pick up free phases $\sigma_\pm\to\sigma_\pm e^{\pm i\omega_0 t}$ . The coupling separates into two families by whether the field and atomic phases add or cancel.

H_{\text{int}} \propto \underbrace{\sigma_+ e^{-i\Delta t}+\sigma_- e^{+i\Delta t}}_{\text{co-rotating (slow)}} \;+\; \underbrace{\sigma_+ e^{+i(\omega+\omega_0)t}+\sigma_- e^{-i(\omega+\omega_0)t}}_{\text{counter-rotating (fast)}} \qquad (3)

where: $\Delta=\omega-\omega_0$ — detuning. The co-rotating terms crawl at $\Delta$ ; the counter-rotating terms race at $\omega+\omega_0\approx2\omega_0$ . The RWA keeps the first pair and discards the second, since a term oscillating at $2\omega_0$ contributes a vanishing time-average over any timescale longer than an optical cycle. Making the approximation and rotating into the drive frame collapses the whole time-dependent problem to a constant $2\times2$ matrix — the heart of the method: a diagonal detuning term and a constant off-diagonal drive.

H_{\text{RWA}} = -\hbar\Delta\,|e\rangle\langle e| + \tfrac{1}{2}\hbar\Omega\big(\sigma_+ + \sigma_-\big) \qquad (4)

where: $\Delta=\omega-\omega_0$ — detuning, sitting on the diagonal; $\tfrac12\hbar\Omega$ — the constant coupling driving $|g\rangle\leftrightarrow|e\rangle$ . No time is left in this operator, which is precisely why it can be diagonalized once and for all. Diagonalizing $H_{\text{RWA}}$ gives a splitting set by $\Omega$ and $\Delta$ in quadrature — the generalized Rabi frequency, the actual rate at which population oscillates when the drive is detuned.

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

where: $\Omega_R$ — generalized Rabi frequency, the observed flopping rate off resonance. Detuning always speeds up the oscillation ( $\Omega_R\ge\Omega$ ) while shrinking its amplitude — inseparable effects, both in the Rabi formula below. Solving the Schrödinger equation for the constant $H_{\text{RWA}}$ from the ground state gives the Rabi formula — the exact closed-form population dynamics that is the RWA's signature result.

P_e(t) = \frac{\Omega^2}{\Omega^2+\Delta^2}\,\sin^2\!\left(\frac{\Omega_R t}{2}\right) \qquad (6)

where: $P_e$ — probability of finding the atom excited, given it started in $|g\rangle$ . The prefactor $\Omega^2/(\Omega^2+\Delta^2)$ caps the oscillation: 1 on resonance (full inversion reachable), $\tfrac12$ at $|\Delta|=\Omega$ , tending to zero far off. The atom flops fastest exactly when it flops least completely. The price of the approximation is a small, computable shift of the resonance left by the discarded terms — the Bloch-Siegert shift. It scales as drive strength squared over the optical frequency, invisible for weak drives and unavoidable for very strong ones.

\delta\omega_{\text{BS}} \approx \frac{\Omega^2}{4\,\omega_0} \qquad (7)

where: $\delta\omega_{\text{BS}}$ — upward shift of the apparent resonance from the counter-rotating terms. Second order in $\Omega/\omega_0$ : negligible for optical Rabi frequencies, but a famous, measurable correction in strongly driven NMR and superconducting circuits. Finally, quantizing the field and applying the same RWA produces the Jaynes-Cummings model — the workhorse of cavity and circuit QED, the quantum sibling of $H_{\text{RWA}}$ with photon operators for the classical amplitude.

H_{JC} = \tfrac12\hbar\omega_0\,\sigma_z + \hbar\omega\, a^\dagger a + \hbar g\big(\sigma_+ a + \sigma_- a^\dagger\big) \qquad (8)

where: $a,a^\dagger$ — photon annihilation and creation operators; $g$ — single-photon (vacuum) coupling. The kept term $\sigma_+ a+\sigma_- a^\dagger$ conserves excitation number, splitting the Hilbert space into decoupled two-state ladders with dressed states at vacuum Rabi splitting $2g\sqrt{n+1}$ .

Derivation AFrom the full coupling to the time-independent RWA Hamiltonian

The goal is to turn the time-dependent driven two-level Hamiltonian into a constant matrix, tracking exactly which terms are dropped and why. We move to the interaction picture so the atomic operators carry their own phases, expand the cosine, sort the four terms by how fast their net phase turns, discard the two that turn at $\sim2\omega_0$ , and rotate into the drive frame to absorb the residual slow phase. The approximation enters at exactly one labelled step; everything else is an exact change of basis.

Step 1. Factor the Rabi frequency out of the coupling, $H_{\text{int}}=\tfrac12\hbar\Omega(\sigma_++\sigma_-)(e^{i\omega t}+e^{-i\omega t})$ , and transform to the interaction picture generated by $H_0=\tfrac12\hbar\omega_0\sigma_z$ , under which $\sigma_\pm\to\sigma_\pm e^{\pm i\omega_0 t}$ . Why: stripping off the trivial free evolution leaves only the atom-field exchange, and it is here that "fast" and "slow" become well-defined labels.

Step 2. Multiply out the four products. Two combine matching phases into slow factors $\sigma_+e^{-i\Delta t}$ and its conjugate; two combine into fast factors $\sigma_+e^{+i(\omega+\omega_0)t}$ and its conjugate, turning at $\omega+\omega_0\approx2\omega_0$ . Why: the messy four-term coupling has sorted itself into a near-resonant pair and an off-resonant pair, and only their timescales now matter.

Step 3. Apply the approximation: drop the fast pair. Their contribution $\int e^{\pm i(\omega+\omega_0)t}\,dt$ oscillates around zero with amplitude $\sim\Omega/(\omega+\omega_0)$ over any interval long compared with $1/\omega_0$ — negligible when $\Omega\ll\omega_0$ . Why: this is the one and only approximation; the RWA's entire validity rests on the smallness of $\Omega/\omega_0$ and $|\Delta|/\omega_0$ .

Step 4. Undo the residual slow phase with $U=\exp(-\tfrac{i}{2}\omega t\,\sigma_z)$ , cancelling the $e^{\pm i\Delta t}$ factors and adding $-\tfrac12\hbar\omega\sigma_z$ ; with the bare term this yields the diagonal $-\hbar\Delta|e\rangle\langle e|$ . Why: rotating with the drive finally removes all time dependence, giving the constant $H_{\text{RWA}}=-\hbar\Delta|e\rangle\langle e|+\tfrac12\hbar\Omega(\sigma_++\sigma_-)$ with eigenvalues $\tfrac{\hbar}{2}(-\Delta\pm\Omega_R)$ .

Go deeper · Derivation B · 1 minRabi flopping from the RWA Hamiltonian

Derivation BRabi flopping from the RWA Hamiltonian

The goal is to extract the closed-form population dynamics — the Rabi formula — from the constant matrix built above, so that "exactly solvable" becomes an explicit $P_e(t)$ . We expand the state in the bare basis, insert $H_{\text{RWA}}$ into the Schrödinger equation to get two coupled amplitude equations, decouple them into one oscillator equation whose frequency is the generalized Rabi frequency, apply the ground-state initial condition, and take the on-resonance limit to recover the $\pi$ -pulse picture that governs coherent control.

Step 1. Write $|\psi\rangle=c_g|g\rangle+c_e|e\rangle$ and substitute into $i\hbar\,\partial_t|\psi\rangle=H_{\text{RWA}}|\psi\rangle$ , giving the coupled pair $i\dot c_g=\tfrac12\Omega\,c_e$ and $i\dot c_e=-\Delta\,c_e+\tfrac12\Omega\,c_g$ . Why: projecting onto the two basis states turns operator dynamics into ordinary differential equations for numbers — possible only because $H_{\text{RWA}}$ carries no time dependence.

Step 2. Differentiate the $c_e$ equation and eliminate $c_g$ , yielding $\ddot c_e+i\Delta\dot c_e+\tfrac14\Omega^2 c_e=0$ , with characteristic roots $\tfrac{i}{2}(-\Delta\pm\Omega_R)$ and $\Omega_R=\sqrt{\Omega^2+\Delta^2}$ . Why: decoupling exposes the natural oscillation frequency; the generalized Rabi frequency emerges as an eigenfrequency, not an assumption, already carrying the quadrature of drive and detuning.

Step 3. Impose $c_g(0)=1$ , $c_e(0)=0$ , solve for $c_e(t)$ , and form $P_e=|c_e|^2=\dfrac{\Omega^2}{\Omega^2+\Delta^2}\sin^2\!\big(\tfrac12\Omega_R t\big)$ . Why: squaring the amplitude converts the solution into the measurable population, delivering the Rabi formula with its detuning-dependent ceiling built in.

Step 4. Set $\Delta=0$ : the ceiling becomes 1 and $P_e=\sin^2(\Omega t/2)$ , reaching unity at $\Omega t=\pi$ (a $\pi$ -pulse, full inversion) and one-half at $\Omega t=\pi/2$ (a $\pi/2$ -pulse). Why: these pulse areas are the elementary gates of magnetic resonance and qubit control — the whole logic of coherent manipulation sits in this one line.

Limiting cases

Case	Result	Meaning
Near resonance, weak drive	$\lvert\Delta\rvert,\ \Omega \ll \omega,\ \omega_0$	Counter-rotating terms average to zero over a Rabi period; the RWA is excellent and the Rabi formula essentially exact.
Exact resonance	$\Delta=0,\ P_e=\sin^2(\Omega t/2)$	Full inversion reachable; $\pi$ -pulse at $t=\pi/\Omega$ , $\pi/2$ -pulse at $t=\pi/(2\Omega)$ — the basis of coherent control.
Strong drive	$\Omega \sim \omega_0$	RWA breaks down; the Bloch-Siegert shift $\sim\Omega^2/(4\omega_0)$ grows large and multiphoton resonances appear.
Far off resonance	$\lvert\Delta\rvert \gtrsim \omega_0$	Both term families matter; the full time dependence must be kept to get the AC-Stark shift right.
Ultrastrong cavity coupling	$g/\omega_0 \gtrsim 0.1$	Jaynes-Cummings fails; the quantum Rabi model is required and the ground state acquires virtual photons.

Common trapThe RWA is always a safe approximation because the terms it drops are tiny

The dropped terms are tiny only in a specific regime; the RWA is a resonance-and-weak-coupling approximation, not a universal one. Its validity rests on two small ratios at once, $\Omega/\omega_0$ and $|\Delta|/\omega_0$ . Push either toward unity and the counter-rotating terms stop averaging away: at strong drive ( $\Omega\sim\omega_0$ ) they give a large Bloch-Siegert shift and multiphoton resonances; far off resonance ( $|\Delta|\gtrsim\omega_0$ ) they contribute comparably to the kept terms, so level shifts come out wrong; and in ultrastrong cavity QED ( $g/\omega_0\gtrsim0.1$ ) they reshape the ground state, replacing Jaynes-Cummings with the full quantum Rabi model. The terms the RWA throws away are precisely the physics of these regimes.

Worked examples

1 · The π-pulse time for an optical qubit (confidence builder)

A laser drives a two-level atom on exact resonance ( $\Delta=0$ ) with a Rabi frequency $\Omega=2\pi\times10\ \mathrm{MHz}$ . How long must the pulse last to fully invert the atom, and how good is the RWA if the transition is the rubidium D2 line at 780 nm?

Step 1. On resonance the Rabi formula reduces to $P_e=\sin^2(\Omega t/2)$ , which first reaches 1 when $\Omega t/2=\pi/2$ , i.e. $\Omega t=\pi$ — the $\pi$ -pulse condition $t_\pi=\pi/\Omega$ .

Step 2. With $\Omega=2\pi\times10^{7}\ \mathrm{s^{-1}}$ , $t_\pi=\pi/(2\pi\times10^{7})=5.0\times10^{-8}\ \mathrm{s}=\mathbf{50\ ns}$ . A pulse of exactly this length inverts $|g\rangle\to|e\rangle$ with unit probability; a 25 ns pulse ( $\pi/2$ ) leaves an equal superposition.

Step 3. Check the RWA. The transition frequency is $\omega_0=2\pi c/\lambda=2\pi(2.998\times10^{8})/(780\times10^{-9})=2.42\times10^{15}\ \mathrm{rad/s}$ , so $\Omega/\omega_0=(6.28\times10^{7})/(2.42\times10^{15})=2.6\times10^{-8}$ . The counter-rotating terms turn at $\omega+\omega_0\approx4.8\times10^{15}\ \mathrm{rad/s}$ , one cycle every 1.3 fs, while the atom takes 50 ns to flip. The π-pulse lasts 50 ns and the RWA is superb, in error only at the $10^{-8}$ level.

2 · Detuning kills the contrast

The same drive, $\Omega=2\pi\times10\ \mathrm{MHz}$ , is now detuned by $\Delta=\Omega$ . Find the generalized Rabi frequency, the largest excited-state population reached, and when it first reaches it.

Step 1. The oscillation frequency is $\Omega_R=\sqrt{\Omega^2+\Delta^2}=\sqrt{2}\,\Omega=2\pi\times14.1\ \mathrm{MHz}$ — detuning has sped the flopping up by $\sqrt2$ .

Step 2. The amplitude is the prefactor $\Omega^2/(\Omega^2+\Delta^2)=\Omega^2/(2\Omega^2)=1/2$ , so $P_e(t)=\tfrac12\sin^2(\Omega_R t/2)$ climbs no higher than one-half — the atom never fully inverts once detuned by a full Rabi frequency.

Step 3. The first maximum is at $\Omega_R t/2=\pi/2$ , i.e. $t=\pi/\Omega_R=t_\pi/\sqrt2=50/\sqrt2\ \mathrm{ns}=35.4\ \mathrm{ns}$ . The atom flops faster ( $\Omega_R=2\pi\times14.1$ MHz), reaches at most $P_e=0.5$ , and first hits it at 35.4 ns — detuning buys speed at the cost of contrast.

Physics of the portrait

The rotating-frame trick and the flopping it predicts are so bound up with one experimenter that his name is attached to both the frequency and the oscillation — a story from 1930s molecular beams to the qubits of the 2020s.

Origins · 1 minIsidor Isaac Rabi and how this was discovered

Isidor Isaac Rabi

1898–1988 · molecular-beam magnetic resonance and the two-level atom

Isidor Isaac Rabi, raised on New York's Lower East Side, built the molecular-beam magnetic-resonance method that first made a two-level quantum system dance on command. In his 1937 paper on "space quantization in a gyrating magnetic field" he solved a spin-½ driven by a rotating magnetic field exactly — no approximation needed, since a purely rotating field has only the co-rotating component — and out came the flopping formula and resonance condition that carry his name. Real coils, though, produce a linearly oscillating field, a sum of two counter-rotating components; what the neglected one does was answered in 1940 by Felix Bloch and Arnold Siegert, whose small shift is exactly the correction the RWA leaves behind. Rabi, with Norman Ramsey and Julian Schwinger, formalized the "use of rotating coordinates" in 1954. He won the 1944 Nobel Prize for the resonance method, which seeded nuclear magnetic resonance, the atomic clock, and MRI. The through-line to today's driven qubits is unbroken: every $\pi$ -pulse is a Rabi flop, and every rotating-frame analysis his coordinate change in modern dress.

History

1937

— Rabi solves a spin-½ in a rotating magnetic field, obtaining the flopping formula and resonance condition

1940

— Bloch and Siegert compute the resonance shift left by the counter-rotating component of a linear field

1954

— Rabi, Ramsey, and Schwinger formalize the rotating-coordinate (rotating-frame) method as a general tool

1963

— Jaynes and Cummings quantize the field within the RWA, creating the model that bears their names

2010

— circuit-QED experiments reach ultrastrong coupling, where the counter-rotating terms visibly break the RWA

Going further

Texts. Foot, Atomic Physics, ch. 7 (RWA and Rabi flopping) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 2 · Allen & Eberly, Optical Resonance and Two-Level Atoms · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 14 · Cohen-Tannoudji, Dupont-Roc, and Grynberg, Atom-Photon Interactions (dressed states, quantized RWA).

Provenance. The RWA Hamiltonian, the Rabi formula, and the interaction-picture bookkeeping are standard from Foot (ch. 7), Loudon (ch. 2), and Allen & Eberly; the Jaynes-Cummings model is from Jaynes and Cummings (1963); the Bloch-Siegert shift from Bloch and Siegert (1940). A verifier should recompute: (i) the interaction-picture phases $e^{-i\Delta t}$ versus $e^{i(\omega+\omega_0)t}$ ; (ii) the eigenvalues $\tfrac{\hbar}{2}(-\Delta\pm\Omega_R)$ of $H_{\text{RWA}}$ ; (iii) the Rabi formula from the amplitude equations; (iv) the $\pi$ -pulse time $t_\pi=\pi/\Omega=50$ ns and $\Omega/\omega_0\approx2.6\times10^{-8}$ for the 780 nm example; (v) the Bloch-Siegert scaling $\Omega^2/(4\omega_0)$ . No non-standard numbers are asserted.

Prerequisites. Electric Dipole Approximation · Transition Dipole Moment · Stimulated Emission. Leads to. Radiative Lifetime and Natural Linewidth · Lorentzian Line Shape · Resonant Absorption

Ask the tutor: Ask me why a detuned drive can never fully invert the atom, or to estimate the Bloch-Siegert shift for a given transition and drive.

Why is it called the "rotating wave" approximation?

Because it is cleanest in a frame that rotates with the drive. A linearly oscillating field is the sum of two circular fields turning in opposite senses. Step into a frame spinning at the drive frequency and the co-rotating component becomes nearly stationary — a slowly turning "rotating wave" — while the other spins at roughly twice the optical frequency. The approximation keeps the slow rotating wave and discards the fast one, hence the name. The phrase is inherited from magnetic resonance, where Rabi's original problem used a genuinely rotating radiofrequency field, so the kept term literally was a rotating wave.

Is the rotating wave approximation the same as the dipole approximation?

No — they are independent simplifications at different stages. The electric-dipole approximation concerns space: it assumes the field is uniform across the atom, giving the coupling $-\mathbf{d}\cdot\mathbf{E}(t)$ that lets you write equation (1). The RWA concerns time: given that coupling with its $\cos\omega t$ factor, it drops the fast-oscillating half. You can make the dipole approximation without the RWA (keeping all four terms and solving numerically), and the RWA logic applies to non-dipole couplings too. They are often used together, but neither implies the other.

Do the discarded terms violate energy conservation?

Not really — they describe virtual, not real, processes. The kept terms pair excitation with absorption and de-excitation with emission, conserving energy to within $\hbar\Delta$ . The discarded terms pair excitation with emission and de-excitation with absorption, each off the energy shell by about $2\hbar\omega_0$ . Energy-time uncertainty allows such excursions for a time $\sim1/\omega_0$ , after which they must reverse, so they cannot build a real transition. Dropping them asserts not that they never happen, but that these fleeting virtual events average to a negligible correction near resonance — their residue being exactly the Bloch-Siegert shift and virtual-photon dressing.

When exactly can I trust the RWA?

Two conditions, both small ratios to the optical frequencies. First, weak coupling: $\Omega\ll\omega_0$ , so the counter-rotating terms (amplitude $\Omega/2$ , detuning $\sim2\omega_0$ ) contribute only at order $\Omega/\omega_0$ . Second, near resonance: $|\Delta|\ll\omega_0$ , so the kept terms really are the slow ones. For optical transitions both ratios are typically $10^{-7}$ or smaller and the RWA is essentially exact. It fails when $\Omega$ or $g$ climbs toward $\omega_0$ (strong and ultrastrong driving), or when you work far off resonance to compute AC-Stark shifts, where the dropped terms contribute comparably.

What is the difference between the Jaynes-Cummings and the quantum Rabi model?

Both describe a two-level atom coupled to one quantized field mode; the difference is the RWA. The full quantum Rabi model keeps all four coupling terms, including the counter-rotating $\sigma_+a^\dagger+\sigma_-a$ , which change excitation and photon number together and so do not conserve total excitation. Jaynes-Cummings applies the RWA and keeps only $\sigma_+a+\sigma_-a^\dagger$ ; that conservation law block-diagonalizes the Hamiltonian into decoupled two-state ladders, making it exactly solvable. For weak coupling the two agree. In ultrastrong coupling ( $g/\omega_0\gtrsim0.1$ ) they diverge: the quantum Rabi ground state contains virtual photons that Jaynes-Cummings misses entirely.

Is moving to the rotating frame itself an approximation?

No — the rotating-frame transformation is exact. It is just a time-dependent change of basis (a unitary $U=e^{-i\omega t\sigma_z/2}$ ); making it shifts the diagonal energies and rotates the coupling without discarding anything. The approximation is a separate step that merely looks natural in that frame: once you rotate with the drive, the co-rotating term becomes stationary and the counter-rotating term visibly fast, and only then do you drop the fast one. Keep every term in the rotating frame and you have made no error at all — just a harder, time-dependent problem.