- #1
Malamala
- 311
- 27
Hello! I have a 3 level system with energies ##E_0##, ##E_1## and ##E_2## and 2 time varying electric fields, the first one connects ##E_0## and ##E_1## while the second one connects ##E_0## and ##E_2## with Rabi frequencies ##\Omega_1## and ##\Omega_2## and the frequencies of the electric fields are ##\omega_1## and ##\omega_2## such that ##|E_0-E_1-\omega_1|<<|E_0-E_1|## and ##|E_0-E_2-\omega_2|<<|E_0-E_2|##. So my Hamiltonian looks like:
$$
\begin{pmatrix}
E_0 & \Omega_1\cos\omega_1 t & \Omega_2\cos\omega_2 t\\
\Omega_1\cos\omega_1 t & E_1 & 0\\
\Omega_2\cos\omega_2 t t & 0 & E_2
\end{pmatrix}
$$
If I had only ##E_0##, ##E_1##, I would use this matrix:
$$
U = \begin{pmatrix}
1 & 0\\
0 & e^{i\omega_1 t}
\end{pmatrix}
$$
to go in a frame rotating with the electric field and remove all the explicit time dependence from the problem. But I am not sure what to do for a 3 level system (i.e. can I remove all the time dependence?). In principle I can use this for a 3 level system:
$$
U = \begin{pmatrix}
1 & 0 & 0\\
0 & e^{i\omega_1 t} & 0\\
0 & 0 & e^{i\omega_2 t}
\end{pmatrix}
$$
which would give a time-independent Hamitlonian (and that is a valid unitary transformation), but I am not sure this is ok physically. Basically, if I use this, the 2 different excited states seem to be in different rotating frames, which is confusing to me. What is the right way to proceed? Can I actually go to a time independent frame that makes physical sense?
$$
\begin{pmatrix}
E_0 & \Omega_1\cos\omega_1 t & \Omega_2\cos\omega_2 t\\
\Omega_1\cos\omega_1 t & E_1 & 0\\
\Omega_2\cos\omega_2 t t & 0 & E_2
\end{pmatrix}
$$
If I had only ##E_0##, ##E_1##, I would use this matrix:
$$
U = \begin{pmatrix}
1 & 0\\
0 & e^{i\omega_1 t}
\end{pmatrix}
$$
to go in a frame rotating with the electric field and remove all the explicit time dependence from the problem. But I am not sure what to do for a 3 level system (i.e. can I remove all the time dependence?). In principle I can use this for a 3 level system:
$$
U = \begin{pmatrix}
1 & 0 & 0\\
0 & e^{i\omega_1 t} & 0\\
0 & 0 & e^{i\omega_2 t}
\end{pmatrix}
$$
which would give a time-independent Hamitlonian (and that is a valid unitary transformation), but I am not sure this is ok physically. Basically, if I use this, the 2 different excited states seem to be in different rotating frames, which is confusing to me. What is the right way to proceed? Can I actually go to a time independent frame that makes physical sense?