Rotating Wave Approximation

The rotating wave approximation (RWA) simply states that any terms in the Hamiltonian oscillating faster than a user-specified cutoff, \(\omega_{\text{cutoff}}\), are neglected.

Looking at the general Hamiltonian, we see that these oscillating terms will arise from the product of \(e^{-i\Delta_{nmjk}t}\) and the matrix elements of \(D(\alpha)\). Because we are computing these via Laguerre polynomials, it is straightforward to derive a condition on what terms must be dropped.

Let's first consider one of these oscillating terms for when \(m\geq n\). Plugging in \(\alpha = i\eta e^{i\nu t}\) into first order Lamb-Dicke approximation, we get

\[ \begin{align} e^{-i\Delta_{nmjk}t} D_{mn} &= e^{-i\Delta_{nmjk}t} \sqrt{\frac{n!}{m!}}(i\eta e^{i\nu t})^{m-n}e^{-\eta^2 /2}L_n^{(m-n)}(\eta^2)\\ &= \sqrt{\frac{n!}{m!}}(i\eta)^{m-n}e^{i[(m-n)\nu - \Delta_{nmjk}]t- \eta^2/2}L_n^{(m-n)}(\eta^2) \end{align} \]

So we find that these Hamiltonian matrix elements rotate at frequency \((m-n)\nu-\Delta_{nmjk}\). We find the same condition on this 'combined' frequency for the \(m>n\) case as well

\[ \begin{align} e^{-i\Delta_{nmjk}t} D_{mn} &= e^{-i\Delta_{nmjk}t} \sqrt{\frac{m!}{n!}}(-i\eta e^{-i\nu t})^{n-m}e^{-\eta^2 /2}L_m^{(n-m)}(\eta^2)\\ &= \sqrt{\frac{m!}{n!}}(-i\eta)^{m-n}e^{-i[(n-m)\nu + \Delta_{nmjk}]t- \eta^2/2}L_m^{(n-m)}(\eta^2) \end{align} \]

Thus, the condition on the RWA can be written as

\[ \text{If } |(m-n)\nu - \Delta_{nmkj}| < \omega_{\text{cutoff}} \text{ , } D_{mn} \rightarrow 0 \]