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
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
Thus, the condition on the RWA can be written as