Two level system under electric field without RWA

In summary, the conversation discusses a Hamiltonian with energy splitting ##\Delta##, Rabi frequency ##\Omega##, and driving field frequency ##\omega## in the regime where ##\delta << \Omega \leq \omega##. The speaker is looking for an analytical formula for population transfer from the ground state to the excited state, without using the RWA or adiabatic approximations. They are seeking alternative methods to numerical integration.
  • #1
kelly0303
563
33
Hello! I have the following Hamiltonian:
$$
\begin{pmatrix}
0 & -\Omega\sin(\omega t) \\
-\Omega\sin(\omega t) & \Delta
\end{pmatrix}
$$
where ##\Delta## is the energy splitting between the 2 levels, ##\Omega## is the Rabi frequency of the driving field and ##\omega## is the frequency of the driving field. I am in the regime where ##\delta << \Omega \leq \omega##. For example, some values I encounter are ##\delta = 2\pi \times 10##, ##\Omega = 2\pi \times 5000##, ##\omega = 2\pi \times 10000##. I obviously can't use the RWA approximation, and definitely I can't use the adiabatic approximation. I was wondering if there is any approximation I can use to get an approximate analytical formula for the population transfer (Assume we start in the ground state and care about the population in the excited state). Is there such a formula? Or is there anything I can do (other than numerical integration)? Thank you!
 

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