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

Rotating wave approximation

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

\[\begin{aligned} \hat{H}_1(t) = \hat{V} \cos\!(\omega t) = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) \end{aligned}\]

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

As an example, consider a two-level system consisting of states \(\ket{g}\) and \(\ket{e}\), with a resonance frequency \(\omega_0 = (E_e \!-\! E_g) / \hbar\). From the derivation of time-dependent perturbation theory, we know that the state \(\ket{\Psi} = c_g \ket{g} + c_e \ket{e}\) evolves as:

\[\begin{aligned} i \hbar \dv{c_g}{t} &= \matrixel{g}{\hat{H}_1(t)}{g} \: c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} \: c_e(t) \: e^{- i \omega_0 t} \\ i \hbar \dv{c_e}{t} &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t) \end{aligned}\]

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

\[\begin{aligned} \dv{c_g}{t} &= \frac{1}{i \hbar} \matrixel{g}{\hat{H}_1}{e} \: c_e \: e^{- i \omega_0 t} \\ \dv{c_e}{t} &= \frac{1}{i \hbar} \matrixel{e}{\hat{H}_1}{g} \: c_g \: e^{i \omega_0 t} \end{aligned}\]

We now insert the general \(\hat{H}_1\) defined above, and define \(V_{eg} \equiv \matrixel{e}{\hat{V}}{g}\) to get:

\[\begin{aligned} \dv{c_g}{t} &= \frac{V_{eg}^*}{i 2 \hbar} \Big( e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t} \Big) \: c_e \\ \dv{c_e}{t} &= \frac{V_{eg}}{i 2 \hbar} \Big( e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t} \Big) \: c_g \end{aligned}\]

At last, here we make the rotating wave approximation: since \(\omega\) is assumed to be close to \(\omega_0\), we argue that \(\omega \!+\! \omega_0\) is so much larger than \(\omega \!-\! \omega_0\) 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 (\(\omega \!+\! \omega_0\)) oscillations have a tiny amplitude compared to the slow (\(\omega \!-\! \omega_0\)) ones. Furthermore, since they average out to zero over most realistic time intervals, the fast terms can be dropped, leaving:

\[\begin{aligned} \boxed{ \begin{aligned} e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t} &\approx e^{i (\omega - \omega_0) t} \\ e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t} &\approx e^{- i (\omega - \omega_0) t} \end{aligned} } \end{aligned}\]

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

\[\begin{aligned} \dv{c_g}{t} &= \frac{V_{eg}^*}{i 2 \hbar} c_e \: e^{i (\omega - \omega_0) t} \\ \dv{c_e}{t} &= \frac{V_{eg}}{i 2 \hbar} c_g \: e^{- i (\omega - \omega_0) t} \end{aligned}\]

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 \(\omega_0\) from the above equations, i.e. multiplying them by \(e^{i \omega_0 t}\) and \(e^{- i \omega_0 t}\) respectively, which can be regarded as a rotation.

Relative to this rotation, when we split the wave \(\cos\!(\omega t)\) 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.


© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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