In analyzing a system of two coupled oscillators, I noticed a rather interesting correspondence between the so-called "rotating wave approximation" (RWA) for solving differential equations and the structure of the algebraic representation of the matrix form of those equations.

## Background

The system of equations under consideration is
$$
\frac{d}{dt}
\left( \begin{array}{c} a \\ a^* \\ b \\ b^* \end{array} \right)
=
i \left(
  \begin{array}{cccc}
    - \omega_a & 0 & -g & g \\
    0 & \omega_a & -g & g \\
    -g & g & - \omega_b & 0 \\
    -g & g & 0 & \omega_b
  \end{array}
\right)
\left( \begin{array}{c} a \\ a^* \\ b \\ b^* \end{array} \right)
\tag{$\star$} \, .
$$
This system is the Hamiltonian version of the system described in [a previous question](https://physics.stackexchange.com/questions/466732/), and can be derived from the Hamiltonian
$$
H = \omega_1 a^* a + \omega_2 b^* b
- g \left( a b - a^* b - a b^* + a^* b^* \right)
$$
using the appropriate Hamiltonian equations of motion
$$
\dot{a} = -i \frac{\partial H}{\partial a^*}
\quad \text{and} \quad
\dot{b} = -i \frac{\partial H}{\partial b^*}
$$
and similarly for $a^*$ and $b^*$.

The parameter $g$ characterizes the strength of coupling between the two oscillators.
Note that if $g=0$, then we have two uncoupled equations
$$
\dot a = -i \omega_a a \quad \text{and} \quad \dot b = -i \omega_b b
$$
with solutions
$$
a(t) = a(0) e^{-i \omega_a t} \quad \text{and} \quad b(t) = b(0) e^{-i \omega_b t}
$$
representing two uncoupled oscillators.

Now note that the matrix in Equation $(\star)$ can be written algebraically as
$$
  -i \sigma_z \otimes
  \left(
    g \sigma_x + \frac{d}{2} \sigma_z + \frac{s}{2} \mathbb{I}
  \right)
  - g (\sigma_y \otimes \sigma_x)
$$
where $s \equiv \omega_1 + \omega_2$ and $d \equiv \omega_1 - \omega_2$.

## Rotating wave approximation

The rotating wave approximation (RWA) comes essentially from noting that while the time dependences of e.g. $a$ and $b^*$ have opposite sign and therefore lead to slow oscillation, the time dependences of e.g. $a$ and $b$ have the same sign and lead to fast oscillation.
Therefore, the terms $ab$ and $a^* b^*$ are dropped from the Hamiltonian.
Doing this leads to an equation like $(\star)$, but with a matrix given by
$$
  -i \sigma_z \otimes
  \left(
    g \sigma_x + \frac{d}{2} \sigma_z + \frac{s}{2} \mathbb{I}
  \right) \, .
$$

## Question

Why is the rotating wave approximation equivalent to dropping the $(\sigma_y \otimes \sigma_x)$ term in the dynamical equation matrix?
What intuitive link is there between the argument we gave for the RWA (or any other argument) and the dropping of the algebraic term $(\sigma_y \otimes \sigma_x)$?