What is the characteristics time scale of a quantum system, in context of the Markovian approximation?

Question

In the theory of open quantum system, we make the markovian approximation when the timescale of the memory of the reservoir is small. But this timescale is measured with respect to the characteristic time scale of the quantum system.

How to calculate this characteristic timescale for an arbitrary quantum system with given energy eigenvalues?

What I tried: For a two state system, the answer appears to be as following. Assume a two state system with energy difference $\Delta E$. Therefore, the characteristic time scale of the system is given as $T = \frac{2 \pi \hbar}{\Delta E}$. So, if the timescale for the memory of the reservoir is $T'\ll T$ then the markovian approximation is valid.

But how to generalize this method for a several state quantum system? As an example, lets take a harmonic oscillator with frequency $\omega$. We already know the answer- $T=\frac{2 \pi}{\omega}$. But how to derive this answer? We note that after this time $T$, the wave-function of the system repeats itself.

But let us take a quantum system with unevenly placed energy levels. The wave-function may never repeat itself. What is the characteristic time for such a system? Is it calculated using the energy difference between ground and first excited state? Or using the least energy gap in the energy spectrum? Or is it somehow calculated using the temperature T of the reservoir? Like, using the energy gap at energy near $E \approx K_B T$?

Answer 1

What I tried: For a two state system, the answer appears to be as following. Assume a two state system with energy difference ΔE. Therefore, the characteristic time scale of the system is given as T=2πℏΔE. So, if the timescale for the memory of the reservoir is T′≪T then the markovian approximation is valid.

In general, one generalizes this argument and considers a "typical" time scale ($T = 2\pi \hbar/\Delta E$) determined by a "typical" energy difference ($\Delta E)$. In the case of an harmonic oscillator, all possible energy differences $n\hbar \omega$ can be found in the spectrum, so then which one should we consider? In this case we need to think what we expect will happen in the system dynamics. If there is no driving and the temperature is low ($\hbar \omega/K_B T < 1$), do we expect transitions between Fock states, say, $|n=0\rangle$ and $|n=10\rangle$?. We shouldn't, so then we know that $10\hbar \omega$ is not a typical transition energy in our dynamics. Most likely we could see transitions between states with one quantum $\hbar \omega$ of energy difference. Then, as you pointed out, $T=2\pi/\omega$ would be a good guess.

The same line of reasoning should be applied for any quantum system: we need some insight of the relevant energy scale of the transitions occurring in the dynamics.