Rotating-Wave Approximation

The rotating-wave approximation keeps near-resonant couplings and neglects rapidly oscillating terms after an exact frame change. The discarded terms produce small rapid motion and cumulative shifts; accuracy therefore depends on the drive strength, bandwidth, observation time, and measured quantity.

Required background. Rotating frames and the interaction picture supply the exact transformations; harmonic perturbations identifies the resonant phases. The two-level model supplies the Pauli operators. The driven two-level notebook compares the approximation with laboratory-frame evolution.

Driven Two-Level System

Take $\omega_0,\omega>0$ and real $\Omega$, all angular frequencies. Consider a two-level system with

$$H_0 = \frac{\hbar\omega_0}{2}\sigma_z$$

and a linearly oscillating drive

$$V(t) = \hbar\Omega\cos(\omega t)\sigma_x.$$

Using

$$\sigma_x=\sigma_++\sigma_-, \qquad \cos(\omega t) = \frac12 \left(e^{i\omega t}+e^{-i\omega t}\right),$$

the interaction-picture perturbation is

$$V_I(t) = \frac{\hbar\Omega}{2} \big[ \sigma_+e^{i(\omega_0+\omega)t} +\sigma_+e^{i(\omega_0-\omega)t} +\sigma_-e^{-i(\omega_0-\omega)t} +\sigma_-e^{-i(\omega_0+\omega)t} \big].$$

Near resonance,

$$\Delta=\omega_0-\omega$$

is small compared with $\omega_0+\omega$. The slowly rotating terms are

$$\sigma_+e^{i\Delta t}, \qquad \sigma_-e^{-i\Delta t}.$$

The counter-rotating terms oscillate with angular frequency approximately $2\omega_0$ and average to small corrections when the drive is weak.

RWA Hamiltonian

Dropping the fast terms gives

$$V_I^{\mathrm{RWA}}(t) = \frac{\hbar\Omega}{2} \left( \sigma_+e^{i\Delta t} +\sigma_-e^{-i\Delta t} \right).$$

With $|\psi_{\rm lab}\rangle=R(t)|\psi_{\rm rot}\rangle$ and $R(t)=e^{-i\omega t\sigma_z/2}$, the exact generator is $R^\dagger HR-i\hbar R^\dagger\dot R$. After dropping its $2\omega$ terms,

$$H_{\mathrm{RWA}} = \frac{\hbar\Delta}{2}\sigma_z +\frac{\hbar\Omega}{2}\sigma_x,$$

This convention fixes the sign of the detuning. The Hamiltonian generates Rabi oscillations with generalized frequency

$$\Omega_R = \sqrt{\Omega^2+\Delta^2}.$$

On exact resonance, $\Delta=0$, the population oscillates at the drive amplitude scale $\Omega$ in this convention.

Why Fast Terms Average Out

A rapidly oscillating contribution has integrals of the form

$$\int_0^T dt\,B(t)e^{i\Lambda t}, \qquad \lvert\Lambda\rvert\gg \frac1T,$$

where $B(t)$ varies slowly compared with $e^{i\Lambda t}$. Integration by parts gives the explicit bound

$$\left\|\int_0^T B(t)e^{i\Lambda t}\,dt\right\| \le\frac{\|B(T)\|+\|B(0)\|+\int_0^T\|\dot B(t)\|\,dt}{|\Lambda|}.$$

The endpoint terms are the rapid motion left by switching the field on and off. A small oscillation amplitude does not by itself bound the accumulated slow phase at long times.

For the driven two-level system, the fast frequency is roughly

$$\Lambda\approx\omega_0+\omega.$$

Thus the counter-rotating contribution is small when

$$\frac{\Omega}{\omega_0+\omega}\ll1.$$

This is an averaging argument, not a symmetry argument. The dropped terms are not forbidden; they are perturbatively small under specified conditions.

Validity Conditions

The rotating-wave approximation is typically controlled when the following conditions hold:

• The drive amplitude is weak compared with the fast frequency scale.
• The detuning is small compared with the fast frequency; a required spectroscopic tolerance must also exceed the omitted frequency shift, or that shift must be included.
• The drive envelope changes slowly over an optical, microwave, or Larmor cycle.
• The observation time is not so long that small counter-rotating effects accumulate beyond the desired accuracy.
• Other transitions are sufficiently far away or forbidden by selection rules.

A compact diagnostic is

$$\lvert\Omega\rvert, \lvert\Delta\rvert \ll \omega_0+\omega.$$

The relevant dimensionless error estimate depends on the observable. For spectroscopy, even a small frequency shift can matter.

Bloch–Siegert Shift

The counter-rotating coupling virtually mixes states separated by $\hbar(\omega_0+\omega)$. With the drive convention above, its leading contribution is

$$\delta H=\frac{\hbar\Omega^2}{4(\omega_0+\omega)}\sigma_z, \qquad \delta\omega_{\rm BS}=\frac{\Omega^2}{2(\omega_0+\omega)}.$$

Thus the effective detuning becomes $\Delta+\delta\omega_{\rm BS}$ and the resonance moves upward to $\omega\simeq\omega_0+\Omega^2/(4\omega_0)$ at weak near-resonant drive. Each level shifts oppositely; the transition shift is twice the positive level shift divided by $\hbar$. Stating the Hamiltonian avoids the common factor-of-four ambiguity between drive-amplitude conventions (Bloch and Siegert, 1940).

The omitted rapid motion is of order $\Omega/(\omega_0+\omega)$; the omitted slow phase can be of order $T\delta\omega_{\rm BS}$. These control different errors. The Magnus expansion and high-frequency methods organize corrections, provided their own scale assumptions hold.

Relation to Harmonic Perturbations

In first-order transition theory, harmonic perturbations produce terms with phase factors

$$e^{i(\omega_{fi}-\omega)t} \quad\text{and}\quad e^{i(\omega_{fi}+\omega)t}.$$

The rotating-wave approximation keeps the phase-matched term and discards the term whose phase rotates rapidly. Thus the RWA can be viewed as a controlled refinement of the resonance logic used in transition-rate calculations.

The same idea extends beyond two levels. In a multilevel system, each term must be judged in the rotating frame appropriate to its transition frequency. A term that is fast for one transition may be resonant for another.

Relation to Effective Hamiltonians

The RWA is an effective-Hamiltonian construction in time rather than in energy-subspace projection. A fast scale is removed, and its leading effect is encoded in a simpler generator of slow dynamics.

Compared with the Schrieffer–Wolff transformation, the RWA is less about eliminating a high-energy subspace and more about eliminating rapidly oscillating Fourier components. Both require a small ratio and both can generate higher-order shifts when pushed beyond leading order.

Adiabatic Elimination removes a fast amplitude after the rotating frame has been chosen. The RWA may first make the couplings slowly varying, but it does not by itself justify removing an excited state or cavity mode.

The Magnus expansion gives a systematic language for this averaging viewpoint: a periodically driven Hamiltonian can be replaced, over one period, by an effective exponential generator plus corrections involving commutators of the time-dependent Hamiltonian.

Exercises

1. Circular versus linear driving

Replace the linear drive by $V_c=(\hbar\Omega/2)(\sigma_x\cos\omega t+\sigma_y\sin\omega t)$. Evaluate the exact rotating-frame Hamiltonian. Is an RWA needed in this ideal two-state model?

Solution

Since $R\sigma_xR^\dagger=\sigma_x\cos\omega t+\sigma_y\sin\omega t$, the rotating drive is exactly $\hbar\Omega\sigma_x/2$. Together with the frame term, the generator is exactly $\hbar(\Delta\sigma_z+\Omega\sigma_x)/2$. There is no counter-rotating component and no Bloch–Siegert shift from it. In a real atom the two-level projection remains an additional approximation.

2. A small amplitude error and a resolved frequency shift

Let $\Omega/(\omega_0+\omega)=10^{-3}$. Estimate the counter-rotating motion and the leading frequency shift as fractions of the fast frequency. What further parameter is needed to control long-time phase error?

Solution

The rapid amplitude correction is of order $10^{-3}$, while $\delta\omega_{\rm BS}/(\omega_0+\omega)=5\times10^{-7}$. A spectrometer can resolve the latter even if the former seems small. One also needs $T\delta\omega_{\rm BS}$ relative to the allowed phase error; the small amplitude ratio alone does not control arbitrarily long evolution.

References

• F. Bloch and A. Siegert, “Magnetic Resonance for Nonrotating Fields”, Physical Review 57, 522–527 (1940).
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley, 1992.
• I. I. Rabi, “Space quantization in a gyrating magnetic field,” Physical Review 51, 652-654, 1937.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, 1990.