# Dyson Series Second Order Term Importance vs. Time

Given a time ordered Dyson series expansion of

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

$${\cal T}\exp\left[-\frac{i}{\hbar}\int_0^tH_I(t')dt'\right]=I-\frac{i}{\hbar}\int_0^t dt' V_0\sin(\omega t')\sigma_1e^{\frac{2i}{\hbar}\Delta\sigma_3t'}+\ldots.$$

where $\sigma_i$ are the Pauli matrices, what is the condition for the second or higher order terms to become non-negligible? I know it scales like $V_0>V_0^2$ for the first to second order, but what is the role of the time? at very long times, the second term must become more and more important, what is that time scale?

(based on this question: Rigorous justification for rotating wave approximation )

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