Categories: Optics, Physics, Quantum mechanics, Two-level system.

Rotating wave approximation

Consider the following periodic perturbation H^1\hat{H}_1 to a quantum system, which represents e.g. an electromagnetic wave in the electric dipole approximation:

H^1(t)=V^cos⁡(ωt)=V^2(eiωt+e−iωt)\begin{aligned} \hat{H}_1(t) = \hat{V} \cos(\omega t) = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) \end{aligned}

Where V^\hat{V} is some operator, and we assume that ω\omega is fairly close to a resonance frequency ω0\omega_0 of the system that is getting perturbed by H^1\hat{H}_1.

As an example, consider a two-level system consisting of states ∣g⟩\ket{g} and ∣e⟩\ket{e}, with a resonance frequency ω0=(Ee ⁣− ⁣Eg)/ℏ\omega_0 = (E_e \!-\! E_g) / \hbar. From the amplitude rate equations, we know that the general superposition state ∣Ψ⟩=cg∣g⟩+ce∣e⟩\ket{\Psi} = c_g \ket{g} + c_e \ket{e} evolves as:

iℏdcgdt=⟨g∣H^1(t)∣g⟩ cg(t)+⟨g∣H^1(t)∣e⟩ ce(t) e−iω0tiℏdcedt=⟨e∣H^1(t)∣g⟩ cg(t) eiω0t+⟨e∣H^1(t)∣e⟩ ce(t)\begin{aligned} i \hbar \dv{c_g}{t} &= \matrixel{g}{\hat{H}_1(t)}{g} \: c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} \: c_e(t) \: e^{- i \omega_0 t} \\ i \hbar \dv{c_e}{t} &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t) \end{aligned}

Typically, V^\hat{V} has odd spatial parity, in which case Laporte’s selection rule reduces this to:

dcgdt=1iℏ⟨g∣H^1∣e⟩ ce e−iω0tdcedt=1iℏ⟨e∣H^1∣g⟩ cg eiω0t\begin{aligned} \dv{c_g}{t} &= \frac{1}{i \hbar} \matrixel{g}{\hat{H}_1}{e} \: c_e \: e^{- i \omega_0 t} \\ \dv{c_e}{t} &= \frac{1}{i \hbar} \matrixel{e}{\hat{H}_1}{g} \: c_g \: e^{i \omega_0 t} \end{aligned}

We now insert the general H^1\hat{H}_1 defined above, and define Veg≡⟨e∣V^∣g⟩V_{eg} \equiv \matrixel{e}{\hat{V}}{g} to get:

dcgdt=Veg∗i2ℏ(ei(ω−ω0)t+e−i(ω+ω0)t) cedcedt=Vegi2ℏ(ei(ω+ω0)t+e−i(ω−ω0)t) cg\begin{aligned} \dv{c_g}{t} &= \frac{V_{eg}^*}{i 2 \hbar} \Big( e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t} \Big) \: c_e \\ \dv{c_e}{t} &= \frac{V_{eg}}{i 2 \hbar} \Big( e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t} \Big) \: c_g \end{aligned}

At last, here we make the rotating wave approximation: since ω\omega is assumed to be close to ω0\omega_0, we argue that ω ⁣+ ⁣ω0\omega \!+\! \omega_0 is so much larger than ω ⁣− ⁣ω0\omega \!-\! \omega_0 that those oscillations turn out negligible if the system is observed over a reasonable time interval.

Specifically, since both exponentials have the same weight, the fast (ω ⁣+ ⁣ω0\omega \!+\! \omega_0) oscillations have a tiny amplitude compared to the slow (ω ⁣− ⁣ω0\omega \!-\! \omega_0) ones. Furthermore, since they average out to zero over most realistic time intervals, the fast terms can be dropped, leaving:

ei(ω−ω0)t+e−i(ω+ω0)t≈ei(ω−ω0)tei(ω+ω0)t+e−i(ω−ω0)t≈e−i(ω−ω0)t\begin{aligned} \boxed{ \begin{aligned} e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t} &\approx e^{i (\omega - \omega_0) t} \\ e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t} &\approx e^{- i (\omega - \omega_0) t} \end{aligned} } \end{aligned}

Such that our example set of equations can be approximated as shown below, and its analysis can continue (see Rabi oscillation for more):

dcgdt=Veg∗i2ℏce ei(ω−ω0)tdcedt=Vegi2ℏcg e−i(ω−ω0)t\begin{aligned} \dv{c_g}{t} &= \frac{V_{eg}^*}{i 2 \hbar} c_e \: e^{i (\omega - \omega_0) t} \\ \dv{c_e}{t} &= \frac{V_{eg}}{i 2 \hbar} c_g \: e^{- i (\omega - \omega_0) t} \end{aligned}

This approximation’s name is a bit confusing: the idea is that going from the Schrödinger to the interaction picture has the effect of removing the exponentials of ω0\omega_0 from the above equations, i.e. multiplying them by eiω0te^{i \omega_0 t} and e−iω0te^{- i \omega_0 t} respectively, which can be regarded as a rotation.

Relative to this rotation, when we split the wave cos⁡(ωt)\cos(\omega t) into two exponentials, one co-rotates, and the other counter-rotates. We keep only the co-rotating waves, hence the name.

The rotating wave approximation is usually used in the context of the two-level quantum system for light-matter interactions, as in the above example. However, it is not specific to that case, and it more generally refers to any approximation where fast-oscillating terms are neglected.