Two level system under electric field without RWA

A
Thread starter kelly0303
Start date Mar 23, 2023
Tags

Electric Electric field Field System

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.

Mar 23, 2023

kelly0303

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!

1. How does the electric field affect the energy levels of a two level system without the Rotating Wave Approximation (RWA)?

The electric field causes a splitting of the energy levels in a two level system without the RWA. This is because the electric field introduces a time-dependent term in the Hamiltonian, leading to non-constant energy levels.

2. What is the difference between a two level system under electric field with and without RWA?

In a two level system under electric field without RWA, the energy levels are not equally spaced and there is a non-zero probability of transitions between the levels. However, with the RWA, the energy levels are equally spaced and transitions between them are not possible.

3. How does the electric field strength affect the energy splitting in a two level system without RWA?

The energy splitting in a two level system without RWA is directly proportional to the strength of the electric field. As the electric field strength increases, the energy levels become further apart.

4. Is it possible to observe the effects of the electric field on a two level system without RWA experimentally?

Yes, it is possible to observe the effects of the electric field on a two level system without RWA experimentally. This can be done through techniques such as spectroscopy, where the energy levels of the system can be measured and compared to theoretical predictions.

5. How does the Rabi frequency change in a two level system under electric field without RWA?

The Rabi frequency, which is a measure of the strength of the coupling between the energy levels, is affected by the electric field in a two level system without RWA. As the electric field strength increases, the Rabi frequency also increases, leading to a faster oscillation between the energy levels.