Dressed states for a 3 level system

[BillKet]

Hello! If we have a 2 level system (I will call the states g and e for ground and excited), and a laser field (which can have any detuning relative to the spacing between g and e), it can be shown that that the total number of particles is conserved under the laser-atom interaction hamiltonian, hence when solving for the energy levels one need to look only at pairs of the form [(g,n), (e,n-1)], where n is the number of photon quanta, which gives a 2x2 Hamiltonian that can be solved for and the energies of the system in the presence of the field are obtained.

Can someone point me (or help me understand) towards the same derivation for more than 2 levels (say 3 for now)? In the 3 levels case, we have g, e1, and e2, where e1 and e2 are the 2 excited levels (assume e1<e2). However, I am not sure what states form the irreducible subspace (the equivalent of the [(g,n), (e,n-1)] above). I assume that we have (g,n) but what are the others? I would expect (e1,n-1) and (e2, n-1) basically corresponding to the laser connecting the ground state to e1 and e2 directly, respectively. But I would also imagine (e2,n-2), basically as we use a photon to go from g to e1 and a second one to go from e1 to e2, instead of going from g to e2 directly as above. But of course (e2, n-1) and (e2, n-2) don't have the same number of quantas, so I am not sure if this logic holds. Is (e2,n-2) not allowed? What are the right states in this 3 level case? Thank you!

 

[Twigg]

But of course (e2, n-1) and (e2, n-2) don't have the same number of quantas, so I am not sure if this logic holds. Is (e2,n-2) not allowed?

As you point out, there are some two-photon resonant pathways. Usually, some of them will be energetically forbidden. For example, if the energy $E_{e1} - E_g$ is $200 \mathrm{THz}$ but $E_{e2} - E_{e1}$ is 1 GHz, then you won't be driving the $g \rightarrow e_1 \rightarrow e_2$ two-photon transition very fast.

If you happen to have a scenario where $E_{e1} - E_g \approx E_{e2} - E_{e1}$, then it's fair to start thinking about resonant 2-photon (1-color) transitions. But that's a pretty rare coincidence.

 

[BillKet]

As you point out, there are some two-photon resonant pathways. Usually, some of them will be energetically forbidden. For example, if the energy $E_{e1} - E_g$ is $200 \mathrm{THz}$ but $E_{e2} - E_{e1}$ is 1 GHz, then you won't be driving the $g \rightarrow e_1 \rightarrow e_2$ two-photon transition very fast.

If you happen to have a scenario where $E_{e1} - E_g \approx E_{e2} - E_{e1}$, then it's fair to start thinking about resonant 2-photon (1-color) transitions. But that's a pretty rare coincidence.

Actually I realized that my assumption was a bit wrong. Assuming we are working in the dipole approximation, due to parity constraints I can't have all the transitions mentioned above i.e. g-e1, g-e2 and e1-e2. So for now, let's say that only g-e1 and e1-e2 are allowed. In this case should I expect to have (g,n), (e1,n-1) and (e2,n-1)? Would this be all in the single photon, electric dipole picture? I am not sure how to count the excitations, as e2 is one excitation away from e1, but 2 excitations away from g, so I am not 100% sure if it counts as (e2,n-1) or (e2,n-2). But based on what you said, if we are using just 1 photon transitions, it should be (e2,n-1).

For concreteness, in my case g and e1 are quite close (rotational levels in a molecule), while e2 is much higher (electronic transition).

 

[Cthugha]

What you have in mind seems to be a simplified version of electromagnetically induced transparency, where you simply remove one of the two beams and do not really have a dark state. It might help to have a look at an introduction on EIT. If I get your system right, you essentially consider a ladder-type configuration.

The right way to count depends on what transitions are allowed. If the only path to e2 is via the excited state e1 and direct excitation from g towards e2 is impossible, then indeed (e2,n-2), (e1,n-1) and (g,n) would form a manifold. However, as you consider that the level spacings are very different, you would drive these transitions with very different detunings.

If you look at figure 6 of this review paper:
https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.77.633
Is this what you had in mind?

 

[BillKet]

What you have in mind seems to be a simplified version of electromagnetically induced transparency, where you simply remove one of the two beams and do not really have a dark state. It might help to have a look at an introduction on EIT. If I get your system right, you essentially consider a ladder-type configuration.

The right way to count depends on what transitions are allowed. If the only path to e2 is via the excited state e1 and direct excitation from g towards e2 is impossible, then indeed (e2,n-2), (e1,n-1) and (g,n) would form a manifold. However, as you consider that the level spacings are very different, you would drive these transitions with very different detunings.

If you look at figure 6 of this review paper:
https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.77.633
Is this what you had in mind?

Thank you for your reply! I won't get a chance to look over that paper until later, so I just wanted to briefly describe my setup in case it helps for now. I have 3 levels, g, e1 and e2 and the electric dipole allowed transitions are g->e1 and e1->e2. The spacing between g and e1 is on the order of GHz while the spacing between e1 and e2 is on the order of hundreds of THz. I have a high power laser on, that is higher than the e1->e2 transition by about 150 THz. What I do in the experiment is simply to measure, using a second, weak, probe laser, the transition between e1 and e2 (my system is more complicated but I simplified it for now, I can go into more details if needed). What I want to calculate, knowing my spacing, and my laser power (and Rabi frequency) is by how much is the spacing between e1 and e2 changing due to the high power laser (this would be just a simple AC Stark shift calculation), but also to do the presence of other nearby states, in this case g. Thank you!

 

[Twigg]

What I want to calculate, knowing my spacing, and my laser power (and Rabi frequency) is by how much is the spacing between e1 and e2 changing due to the high power laser (this would be just a simple AC Stark shift calculation), but also to do the presence of other nearby states, in this case g.

With a detuning of 150THz, you are outside the rotating wave approximation. As such, I wouldn't call this a "simple" Stark shift calculation.

I would start with the usual dipole approximation Hamiltonian: $$\mathcal{H} = \left( \begin{matrix} 0 & 0 & \Omega_{13} \cos(\omega t) \\ 0 & \Delta_{21} & \Omega_{23} \cos(\omega t) \\ \Omega^{*}_{13} \cos(\omega t) & \Omega^{*}_{23} \cos(\omega t) & \Delta{31} \end{matrix}\right) $$
($\Omega_{nm}$ means Rabi rate (can be complex, depending on polarizations) between $|n\rangle$ and $|m\rangle$, and $\Delta_{nm}$ means the difference in energy between $|n\rangle$ and $|m\rangle$).
One way you could solve for the Stark-shifted energies would be to do numerical simulation where you turn the laser field on adiabatically, and plot the final energy of the system vs applied field. There may be more efficient methods, I'm not very familiar with calculations beyond the rotating wave approximation.

If the rotating wave approximation is good enough for you, then you can transform the above Hamiltonian to a time-independent version in the interaction picture, just as you would a two-level system.

 

[BillKet]

@Twigg @Cthugha thanks for your help! I came across this paper, which is not exactly what I asked for initially, but their setup is similar and I think it will help me understand the AC Stark shift better. The relevant part is in section D. They are doing a 3 step collinear resonant ionization measurement and they are investigating the effects of lasers power and the delay between the 3 laser pulses on the measured transitions. In one setup, they basically overlap the spectroscopy laser (the first step, connecting g->e1) and the second step laser (connecting e1->e2), while the ionization laser doesn't overlap at all. So for the purpose of my question we can ignore the last laser AC Stark shift effect (this is in Fig. 6). I would like to derive analytically the effect of having 2 lasers acting on a 3 level system (we are ignoring the other states for now), but right now I tried to do a simpler approximation. As the second laser is so much stronger and it connects e1 and e2, if we work in a 2 level scheme picture (ignoring the ground state and the first laser), one would expect that the e1 will be split by $\pm \Omega$, relative to its normal location (where $\Omega$ is the Rabi frequency of the second laser). This follows from the AC Stark shift derivation for 2 levels on resonance with the laser. Now if we add the first laser (which has a much lower power), I would imagine that the signal (ignoring the hyperfine splitting) will be 2 lines separated by $2\Omega$. If we account for the hyperfine splitting, we would expect 2 set of hyperfine spectra, separated by $2\Omega$ (this is exactly the same logic that applies to the Mollow triplet). In order to estimate $\Omega$, I need the transition dipole moment between e1 and e2. For now I will assume it is 1 Debye (please let me know if this is very off). For the electric field, we have the formula

$$\frac{P}{A} = \frac{c\epsilon_0}{2}E^2$$

where A is the laser beam area. From the laser pulse power (100 $\mu J$), assuming a Gaussian shape, I got a peak power of $10^4$W. For the area I got $0.00008$ m$^2$ (this is using table IV). Plugging this in the equation above, I obtain $E = 3\times 10^5$ V/m. Then for the Rabi frequency, I get:

$$\Omega = \frac{dE}{\hbar} = 10^{10} Hz$$

which is 10 GHz which is huge! They barely see changes at the level of MHz so something about my understanding of the AC Stark effect is very wrong. I would really appreciate if someone can tell me what I did wrong in the above derivation. Thank you!