Undergrad How do two photons affect Rabi oscillations in a 2-level atomic system?

[kelly0303]

Hello! Assuming we have a 2 level system (e.g. an atom with 2 energy levels) and the lifetime of the upper level can be neglected, if we make the atom interact with a laser at a fixed frequency, we would get Rabi oscillations (assume the laser is on resonance). Would we still get Rabi oscillations if the transition is driven by 2 photons, instead of one? To give a concrete example, assume we have an atom passing through an optical cavity (formed of 2 mirrors in which we have power build-up), and the frequency of the light is half the transition frequency, so the atom would need to interact with 2 photons for the excitation to take place (this is used for example in Doppler free spectroscopy). If we can adjust the time that the atom spends inside the cavity, would we still get Rabi oscillations in this case? And if so, would the treatment of the problem be the same as in the 1 photon case? Thank you!

 

[anuttarasammyak]

Two or multi photon microscopy https://en.wikipedia.org/wiki/Two-photon_excitation_microscopy tells us multi photon process is rare but we can make use of it by advanced technology like lasers. Though I am layman for quantum computing I observe no reason to restrict transition or entanglement by 2 photons instead of higher energy 1 photon.

 

[f95toli]

Yes, you can use 2-photon excitation to get Rabi oscillations (via a "virtual" intermediate level).
The problem is that you need a lot power which means that you can easily get a number of other effects (e.g. accidentally excite the system into higher levels) so the oscillations are rarely "ideal".
This is typically described using Floquet theory.

 

[kelly0303]

Yes, you can use 2-photon excitation to get Rabi oscillations (via a "virtual" intermediate level).
The problem is that you need a lot power which means that you can easily get a number of other effects (e.g. accidentally excite the system into higher levels) so the oscillations are rarely "ideal".
This is typically described using Floquet theory.

Thank you so much for this! I will look into Floquet theory. Probably a detailed answer to this will be found there, but I am still a bit confused conceptually. In the case of one photon excitation (or at least normal Rabi formalism found in classic textbooks), on resonance the transition probability is given by $sin^2(\frac{\Omega t}{2})$, so at certain times, we can have the system fully in the upper state (for a $\pi$ pulse). Ignoring other effects that might appear in practice (i.e. assume we have only a 2 level system) if the Rabi oscillations would be there for 2 photon oscillations, would we still get full population inversion for a $\pi$ pulse too? That seems a bit counterintuitive as, if the formalism would be the same, we would get probability of 1 after the right amount of time even for n-photon oscillations, without any power dependence in the formula describing the oscillations.

 

[f95toli]

I am by no means an expert when it comes to the theory here (I do know that it works in practice since it is something we sometimes do in our experiments). My understanding is that how much "extra" power you need for a two-photon transition depends on the details of the system you are working with so I am not sure there is a simple answer. If I remember correctly the power dependence takes the form of a Bessel function in the simplest case.
I do know that in strongly non-linear systems you can sometimes see transitions corresponding to not just two but several photons (this is something I did as part of my PhD many years ago) although I am not sure if if also in -in practice- possible to get oscillations in this case.

Edit: some quick Googling results in lots of hits see e.g

Gatzke, M., et al. "Microwave multiphoton Rabi oscillations." Physical Review A 48.6 (1993): 4742.

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.48.4742