Rigorous justification for rotating wave approximation

Question

Whenever I have encountered the rotating wave approximation, I have seen "the terms that we are neglecting correspond to rapid oscillations in the interaction Hamiltonian, so they will average to 0 in a reasonable time scale" as the justification for its use. However, it is not completely clear to me why does this justify that the Hamiltonian that we obtain is a good approximation of the original one, and I was wondering if there is a more rigorous version of the justification, whether it is for a particular system, or in a more general case.

As an example, something that would be a satisfying answer would be a result of the form "If you consider an arbitrary state of the system and any time t large enough, and evolve the system according to the RWA Hamiltonian, we obtain with high probability a state close to the one we would obtain under evolution of the original Hamiltonian". "t large enough", "close" and "high probability" would preferably have some good quantitative description.

Answer 1

The rotating wave approximation (RWA) is well justified in a regime of a small perturbation. In this limit you can neglect the so-called Bloch-Siegert and Stark shifts. You can find an explanation in this paper. But, in order to make this explanation self-contained, I will give an idea with the following model

$$H=\Delta\sigma_3+V_0\sin(\omega t)\sigma_1$$

being, as usual $\sigma_i$ the Pauli matrices. You can easily work out a small perturbation series for this Hamiltonian working in the interaction picture with

$$H_I=e^{-\frac{i}{\hbar}\sigma_3t}V_0\sin(\omega t)\sigma_1e^{\frac{i}{\hbar}\sigma_3t}$$

producing, with a Dyson series, the following next-to-leading order correction

$${\cal T}\exp\left[-\frac{i}{\hbar}\int_0^tH_I(t')dt'\right]=I-\frac{i}{\hbar}\int_0^t dt' V_0\sin(\omega t')e^{-\frac{i}{\hbar}\Delta\sigma_3t'}\sigma_1e^{\frac{i}{\hbar}\Delta\sigma_3t'}+\ldots.$$

Now, let us suppose that your system is in the eignstate $|0\rangle$ of the unperturbed Hamiltonian. You will get

$$|\psi(t)\rangle=|0\rangle-\frac{i}{\hbar}\int_0^t dt' V_0\sin(\omega t')e^{-\frac{2i}{\hbar}\Delta t'}\sigma_+|0\rangle+\ldots$$ $$=|0\rangle-\frac{1}{2\hbar}\int_0^t dt' V_0\left(e^{i\omega t'-\frac{2i}{\hbar}\Delta t'}-e^{-i\omega t'-\frac{2i}{\hbar}\Delta t'}\right)\sigma_+|0\rangle$$

Now, very near the resonance $\omega\approx2\Delta$, one term is overwhelming large with respect to the other and one can write down

$$|\psi\rangle\approx|0\rangle-\frac{V_0}{2\hbar}t\sigma_+|0\rangle+\ldots.$$

but in the original Hamiltonian this boils down to

$$H_I=V_0\sigma_1\sin(\omega t)\left(\cos(2\Delta t)+i\sigma_3\sin(2\Delta t)\right)$$ $$=\frac{V_0}{2}\sigma_1\left(\sin((\omega-2\Delta)t)+\sin((\omega+2\Delta)t)\right)$$ $$+\frac{V_0}{2}\sigma_2\left(\cos((\omega-2\Delta)t)-\cos((\omega+2\Delta)t)\right)$$ $$\approx \frac{V_0}{2}\sigma_2$$

with all the counter-rotating terms properly neglected with the condition $\omega\approx 2\Delta$ applied. It is essential to emphasize that, as the applied field increases, this approximation becomes even less reliable and it is just the leading order of a perturbation series in a near-resonance regime.

Answer 2

The Magnus expansion is a simple, quantitative way to see why the rotating wave approximation (RWA) works, and what the leading order correction are. Given a time-dependent Hamiltonian $H(t)$, the dynamics over a time interval $T$ can be approximated by the series expansion

$$\bar{H} = \frac{1}{T}\int_0^T H(t) dt - \frac{i}{2T}\int_0^T dt\int_0^t dt' [H(t),H(t')]+...$$

valid if $\int_0^T \text{Tr}[H^2] \ll \pi$. If the Hamiltonian is periodic, as is typically the case in RWA problems, we can set $T$ to be the period. The first term is simply the RWA Hamiltonian, because oscillating terms average to zero over a single period. The second term scales proportional to $T$, so this leading order-correction goes to zero in the high-frequency limit $T\rightarrow 0$.

An advantage of the Magnus expansion over a Dyson series is that you can approximate the dynamics of periodic Hamiltonians with the time evolution operator $\bar{U} = e^{-iHt}$. Because $\bar{H}$ is Hermitian, $\bar{U}$ is exactly unitary, and thus you can use $\bar{U}$ to compute the corrected dynamics out to long times.

Answer 3

I'm completely agree with you both, to invoke the RWA it is mandatory a driving close to resonance, and consider a weak driving strength (weak in comparison with the proper frequency of the system). However, how close? it's a good question. What we need is just consider a detuning which leads to a quasi time-independent behaviour for the propagating terms (for the counter-propagating ones... you already know), it means the detuning should be bounded as: $\delta < 2\Delta$.

Answer 4

This has bugged me for some years, but finally I know a rigorous bound for the finite dimensional case. See Eq. (3.12) in https://arxiv.org/abs/2111.08961

The bound is on the difference between the full unitary evolution and the approximated one, in spectral norm. It scales inverse proportionally to the frequency, so that in the limit of infinite frequency the approximation becomes perfect.

You will see that the bound actually becomes worse roughly proportionally over the total evolution time, rather than better.