A AC Stark shift in diatomic molecules

[BillKet]

Hello! I am analyzing data from a diatomic molecules with a rotational constant around $0.2$ cm$^{-1}$, for transitions between rotational levels between the ground $^2\Sigma_{1/2}$ and excited $^2{\Pi}_{1/2}$ electronic state. I was wondering if there is a way to approximate the shift of the measured transitions due to AC Stark shift, as a function of the power of the main (spectroscopy) laser. I imagine that due to the interaction with the other close by (but off resonant) levels, the measured transition might be slighted shifted, but I am not sure how to estimate the magnitude (I don't need a precise value, just an order of magnitude for now). Unfortunately I have almost all my scans at a given power and I can't repeat the experiment so I was wondering of there is a way to estimate it. Thank you!

 

[Zarqon]

For atomic systems the AC Stark shift is usually approximated as Δω_AC = Ω²/4δ, where Ω is the Rabi frequency of the transition (depends on electric field of the laser and the dipole moment of the transition), and δ is the detuning. I don't see any reason why it would be different for molecules, as the difference should be incorporated into the different transition dipole moments.

I got my information about this from "Atomic physics" by Foot (p. 144) in case you want to read more, but I'm sure you can find other sources as well.

 

[BillKet]

For atomic systems the AC Stark shift is usually approximated as Δω_AC = Ω²/4δ, where Ω is the Rabi frequency of the transition (depends on electric field of the laser and the dipole moment of the transition), and δ is the detuning. I don't see any reason why it would be different for molecules, as the difference should be incorporated into the different transition dipole moments.

I got my information about this from "Atomic physics" by Foot (p. 144) in case you want to read more, but I'm sure you can find other sources as well.

Thanks for your reply! But that formula is for a 2 levels system, which in atoms can be a good approximation (as the other levels are very far away). But in a molecule we have lots of closely spaced levels which perturb each other and I am not sure if the same formula apply, or by how much it is off. It might still apply, I just don't have a clear understanding of this effect to know if I can just ignore the other close-by levels.

 

[Twigg]

Sorry for such a late reply.

My understanding is that the Rabi flopping between the molecule will Rabi flop to some extent into each of the near-resonant excited states, and those Rabi floppings will interfere with each other.

Based on that, I would make a super crude guess that if you have N near-resonant excited states, the light shift can be anywhere between 0 in the case of perfect destructive interference (like a magic wavelength in an atomic clock) and N times the average two-level light shift ($N \times \Omega^2 / 4\delta$) in the case of perfect constructive interference. Again, this is an educated guess on my part.

 

[BillKet]

Sorry for such a late reply.

My understanding is that the Rabi flopping between the molecule will Rabi flop to some extent into each of the near-resonant excited states, and those Rabi floppings will interfere with each other.

Based on that, I would make a super crude guess that if you have N near-resonant excited states, the light shift can be anywhere between 0 in the case of perfect destructive interference (like a magic wavelength in an atomic clock) and N times the average two-level light shift ($N \times \Omega^2 / 4\delta$) in the case of perfect constructive interference. Again, this is an educated guess on my part.

Thank you for this! I assume that the levels that can interfere with my transition are levels that can be connected, through an electric dipole interaction, to the 2 levels I am interested in. So even in a molecule, this is quite a small number due to selection rules based on parity and spin. So N would be 3 or 6, or something of that order, if including the spin-rotational coupling. Is that right?

 

[Twigg]

I assume that the levels that can interfere with my transition are levels that can be connected, through an electric dipole interaction, to the 2 levels I am interested in.

I was only thinking of near-resonant states in the $^2\Pi_{1/2}$ electronic manifold. You're right to include near-resonant states in the $^2\Sigma_{1/2}$ manifold as well. However, what matters is the number of transitions, not the number of states. So I think it should be $N_\Sigma \times N_\Pi$ where $N_\Sigma$ is the number of near-resonant ground states and $N_\Pi$ is the number of near-resonant excited states. For example, if you have 2 ground states and 3 excited states, there are 6 possible transitions. I couldn't tell you what $N_\Sigma$ and $N_\Pi$ are, as your knowledge of the molecular structure is probably much better than mine.

Again, this is a huge guess on my part, and of course reality will probably be much, much less than what you'd get in perfect constructive interference.

 

[BillKet]

I was only thinking of near-resonant states in the $^2\Pi_{1/2}$ electronic manifold. You're right to include near-resonant states in the $^2\Sigma_{1/2}$ manifold as well. However, what matters is the number of transitions, not the number of states. So I think it should be $N_\Sigma \times N_\Pi$ where $N_\Sigma$ is the number of near-resonant ground states and $N_\Pi$ is the number of near-resonant excited states. For example, if you have 2 ground states and 3 excited states, there are 6 possible transitions. I couldn't tell you what $N_\Sigma$ and $N_\Pi$ are, as your knowledge of the molecular structure is probably much better than mine.

Again, this is a huge guess on my part, and of course reality will probably be much, much less than what you'd get in perfect constructive interference.

Right, it will be the number of transitions! However for $^2\Sigma_{1/2}$ state, let's say in a $N=2$ state (in Hund case b), won't the only nearby states that can mix with it, in the electric dipole picture be $N=3$ and $N=1$? The other will differ by 2 units of angular momentum? And I guess there is a fixed number for the $^2\Pi_{1/2}$ state, too in general. Of course the splitting will be different in different molecules, but I assume that the number of states should be the same, no?