Induced dipole moment (adiabatic) following the rotating E-field

[BillKet]

Hello! Assume I have a 2 level system, where the 2 levels have opposite parity. If I apply an electric field, I will get an induced dipole moment. For now I want to keep it general, so the induced dipole moment can be very large, too. Let's say that I start rotating this electric field in the x-y plane, such that the field is given by:

$$E(t) = E_0(\cos{(\omega t)}\hat{x} + \sin{(\omega t)}\hat{y} )$$

I want to describe the behavior of the dipole in the lab frame. I assume that for small rotation frequencies, $\omega$, I am in an adiabatic regime and the dipole would follow the electric field, while at high frequencies, the dipole would not be able to follow. However, I am not sure how to derive these in practice starting from the Hamiltonian. Also, I am not sure what we mean by small/large frequencies in this case. Would I need to compare to the spacing between the 2 levels of opposite parity, or with the Rabi frequency of the field i.e. $dE_0$ where $d$ is the matrix element between the 2 levels of opposite parity. Can someone help me with this, or point me towards some derivations? Thank you!

 

[Twigg]

Would I need to compare to the spacing between the 2 levels of opposite parity

If your rotation frequency is comparable to the level splitting, then you have a resonant process and you can use the rotating wave approximation.

or with the Rabi frequency of the field i.e. dE0 where d is the matrix element between the 2 levels of opposite parity

This isn't really a limit on the adiabatic theorem. If the system has a large transition dipole moment ($d$ in your post), then you can transform into the rotating frame (so that your rotating state is now a stationary state) and then use the adiabatic theorem. For comparison, this same strategy is used to derive the theory of adiabatic rapid passage (ARP) in atoms. In short, the Rabi frequency is irrelevant.

while at high frequencies, the dipole would not be able to follow

You already have the right idea, as shown in the above quote. The limit has to do with the internal structure of the dipole, not the strength of the applied field (i.e., the Rabi rate and level spacing). This limit is specific to the nature of the two-level system and is contained in the polarizability $\alpha = |\vec{d}|/|\vec{E}|$.

In the adiabatic limit, you can describe the induced dipole by means of the static polarizability (aka DC polarizability), which you take as constant in frequency. Then your Hamiltonian works out to $H = -\alpha_{0} |\vec{E}|^2$, which is time-invariant. In reality, your two-level system will have some off-resonant excitation between the two opposite parity states, and this will show up as a pole in the dynamic polarizability $\alpha(\omega)$. (The static polarizability is simply the low-frequency limit of the dynamic polarizability.) Other resonances due to internal structure of the two-level system will also cause poles in the dynamic polarizability. These poles are what limit the applicability of the adiabatic theorem.

 

[BillKet]

If your rotation frequency is comparable to the level splitting, then you have a resonant process and you can use the rotating wave approximation.This isn't really a limit on the adiabatic theorem. If the system has a large transition dipole moment ($d$ in your post), then you can transform into the rotating frame (so that your rotating state is now a stationary state) and then use the adiabatic theorem. For comparison, this same strategy is used to derive the theory of adiabatic rapid passage (ARP) in atoms. In short, the Rabi frequency is irrelevant.You already have the right idea, as shown in the above quote. The limit has to do with the internal structure of the dipole, not the strength of the applied field (i.e., the Rabi rate and level spacing). This limit is specific to the nature of the two-level system and is contained in the polarizability $\alpha = |\vec{d}|/|\vec{E}|$.

In the adiabatic limit, you can describe the induced dipole by means of the static polarizability (aka DC polarizability), which you take as constant in frequency. Then your Hamiltonian works out to $H = -\alpha_{0} |\vec{E}|^2$, which is time-invariant. In reality, your two-level system will have some off-resonant excitation between the two opposite parity states, and this will show up as a pole in the dynamic polarizability $\alpha(\omega)$. (The static polarizability is simply the low-frequency limit of the dynamic polarizability.) Other resonances due to internal structure of the two-level system will also cause poles in the dynamic polarizability. These poles are what limit the applicability of the adiabatic theorem.

Sorry, I am not sure I understood your answer. For concreteness, in my case the Rabi frequency is about $10$kHz, the spacing between the levels is about $1$ kHz, while the external frequency (the freq of the rotating field) is about $100$ kHz. Given that the external frequency is so high, I assume I can't assume the dipole moment of the molecule follows the rotating field (as it was the case in, for example, the HfF$^{+}$ experiment). But I am not totally sure how to proceed on calculating the induced dipole moment as a function of time. If I were in the limit where the external frequency was infinity, then the dipole moment wouldn't follow at all, so I would have no induced dipole moment in the lab frame (right?). But I would like to obtain a general formula, such that the infinity limit and the adiabatic case are extreme cases.

 

[Twigg]

Gotcha. Thanks for the clarification

I have a suggestion, it just might not be accurate. I've never used it personally. Try equation A.16 of this appendix. Again, this ignores any other structure in your system (like the DC tail of a microwave resonance), which would add as a background to the polarizability. If you had a pure two-level system with only the 1kHz resonance, then I think A.16 would be valid. But it sounds like you have a molecule in mind?

 

[BillKet]

Gotcha. Thanks for the clarification

I have a suggestion, it just might not be accurate. I've never used it personally. Try equation A.16 of this appendix. Again, this ignores any other structure in your system (like the DC tail of a microwave resonance), which would add as a background to the polarizability. If you had a pure two-level system with only the 10kHz resonance, then I think A.16 would be valid. But it sounds like you have a molecule in mind?

A two level system is enough for me, but I am not sure that would be enough tho. I had in my mind something along the lines of finding the state of the system as a function of time by fully solving the TDSE for the 2x2 system (starting in the ground state for example), and getting $|\psi{(t)}>$. Then in order to get the dipole moment orientation I would need to compute something of the form

$$\frac{<\psi{(t)}|D|\psi{(t)}>}{<0|D|1>}$$

where $|0>$ and $|1>$ would be my 2 states of opposite parityy. But I wouldn't expect to have a nice closed form, I was thinking of solving the TDSE numerically.