- #1
kelly0303
- 580
- 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!
$$
\begin{pmatrix}
0 & -\Omega\sin(\omega t) \\
-\Omega\sin(\omega t) & \Delta
\end{pmatrix}
$$
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!