- #1
Malamala
- 345
- 28
Hello! I have a simple 2 level system with the energy difference between the 2 levels, ##\Delta## and I applying a time varying electric field, transferring population from the ground level (initially populated), to the excited level (initially with zero population). The simple Hamiltonian for this is:
$$
2\pi\times\begin{pmatrix}
0 & -\Omega\cos(2\pi\omega t) \\
-\Omega\cos(2\pi\omega t) & \Delta
\end{pmatrix}
$$
where, in my case, the Rabi frequency of the field ##\Omega## is comparable to the energy difference between the 2 levels, ##\Delta##, but ##\omega \ll \Delta, \Omega##. So I am very off-resonant, but I have a lot of power in my field.
For some specific values relevant to my problem, I will assume: ##\Delta = 15,000## MHz, ##\Omega = 10,000## MHz, ##\omega = 35## MHz. Here is my question. When I solve the time dependent Schrodinger equation (TDSE) with this Hamiltonian, for 1 second (I see this effect in general for other times, too), I get a population in the ground state at the end of: ##0.6924873010111454##. If I do the same, but I change ##\Omega## from ##10,000## to ##10,001## I get instead ##0.4847252419701252##, which is a huge difference for a change in the electric field of only ##1/10,000##.
However, if I do the same thing but I use ##\sin(2\pi\omega t)## instead of ##\cos(2\pi\omega t)## in the Hamiltonian, the population goes from ##0.9994980316477893## to ##0.9994977756145095##. I tried other times, too, to check if this holds, true and it seems to be the case i.e. when using ##cos## the population transfer changes a lot for small changes in the electric field, relative to the ##sin## case. Is this effect real (I can't find any mistake in my code and I don't think I am missing any physics, as it's such a simple 2 level system)? And if so, why do I have this huge difference in behavior?
$$
2\pi\times\begin{pmatrix}
0 & -\Omega\cos(2\pi\omega t) \\
-\Omega\cos(2\pi\omega t) & \Delta
\end{pmatrix}
$$
where, in my case, the Rabi frequency of the field ##\Omega## is comparable to the energy difference between the 2 levels, ##\Delta##, but ##\omega \ll \Delta, \Omega##. So I am very off-resonant, but I have a lot of power in my field.
For some specific values relevant to my problem, I will assume: ##\Delta = 15,000## MHz, ##\Omega = 10,000## MHz, ##\omega = 35## MHz. Here is my question. When I solve the time dependent Schrodinger equation (TDSE) with this Hamiltonian, for 1 second (I see this effect in general for other times, too), I get a population in the ground state at the end of: ##0.6924873010111454##. If I do the same, but I change ##\Omega## from ##10,000## to ##10,001## I get instead ##0.4847252419701252##, which is a huge difference for a change in the electric field of only ##1/10,000##.
However, if I do the same thing but I use ##\sin(2\pi\omega t)## instead of ##\cos(2\pi\omega t)## in the Hamiltonian, the population goes from ##0.9994980316477893## to ##0.9994977756145095##. I tried other times, too, to check if this holds, true and it seems to be the case i.e. when using ##cos## the population transfer changes a lot for small changes in the electric field, relative to the ##sin## case. Is this effect real (I can't find any mistake in my code and I don't think I am missing any physics, as it's such a simple 2 level system)? And if so, why do I have this huge difference in behavior?
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