# Confused about fluorescence at resonance

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Malamala
Hello! I am a bit confused about a resonance signal that is obtained by measuring the fluorescence signal from overlapping a laser beam with some atoms. Based on the signal shape, the maximum number of counts corresponds to the resonant frequency of the transition (ignoring for this questions effects that might skew the shape of the signal). But I am not totally sure I understand why. If we use the laser at exactly the resonant frequency (we can ignore the linewidth of the laser), the electrons will have the highest probability of doing a transition to the upper level. However, a laser is basically an oscillating electric field (consider only electric dipole transitions). So the electron that we want to move to the upper state will have a high probability of transitioning to the upper state, but it will also have an equally big probability of going from the upper to the lower state (by stimulated emission) so the electron will oscillate between the 2 states. At resonance this oscillation will be higher (as the probability is higher), but why would one expect that when the laser is off and we measure the fluorescence signal, the electrons will end up in the upper level? Isn't it equally likely to be in the lower level too?

Mentor
I'm not sure I understand what kind of experiment your are considering, so my answer may be off.

Yes, a laser resonant with a transition will lead to Rabi cycling between the ground and the excited state, but you have to compare the Rabi frequency with the duration of the experiment. If the laser is not intense, you will probably never reach a the point where the population of the excited state is being lowered by the laser.

Gold Member
I'm also not sure I understand the the questions.
However, if you saturate an ensemble of two-level systems (which is the simplest case) you will indeed end up in a situation where on average 50% of the population is in the ground state. However, this still leaves the other 50% so you will still see a signal.

If you are working with a single two-level system (say a qubit or a single ion) you will indeed have Rabi oscillations and "saturation" is not really a thing, the state of the system when you turn off the laser will depend on total areas of the pulse (amplitude*time) you've applied, regardless of how long the pulse is (*) . Hence, you could indeed be unlucky and not see anything.

(*) Assuming the lifetime of the excited state is still much longer than the pulse.

Malamala
I'm also not sure I understand the the questions.
However, if you saturate an ensemble of two-level systems (which is the simplest case) you will indeed end up in a situation where on average 50% of the population is in the ground state. However, this still leaves the other 50% so you will still see a signal.

If you are working with a single two-level system (say a qubit or a single ion) you will indeed have Rabi oscillations and "saturation" is not really a thing, the state of the system when you turn off the laser will depend on total areas of the pulse (amplitude*time) you've applied, regardless of how long the pulse is (*) . Hence, you could indeed be unlucky and not see anything.

(*) Assuming the lifetime of the excited state is still much longer than the pulse.
@f95toli @DrClaude thank you for your replies! I am actually new to this, so I am not totally sure I know what I mean either. Sorry for that! So one thing I think is getting me confused is that, from the derivations that I have seem for Rabi oscillations, it looks like the state being populated (assuming a 2 level system) is oscillating with time, so the final position depends on the time you stop the pulse. However from time dependent perturbation theory (from my QM class, Griffiths book) I remember that if you send a laser towards a 2 level system, you reach a steady state, where the rate of going to the upper state is equal to the rate of going to the lower state, so the number of electrons in the upper state will become (on average) constant. But as far as i understand, in the case of Rabi oscillations, you move all the electrons at once i.e. they are all in the upper state then lower state and so on, so you never reach a steady state. I am not totally sure what am I missing, or how to combine the 2 pictures (which seem to be contradicting). Thank you!

Gold Member
You need to be very careful about which model you are using and which physical situation you are trying to understand.

One thing to keep in mind is that QM is all about probabilities. If you measure a single system only once you will always find it in either the excited or ground state. The equation for Rabi oscillation tells you the probability to detect the state in either of these two states. This means that an experimental curve is always an average created by either measuring a single TLS many, many times or (less often) by measuring lots of TLS (an ensemble) at once.

The "basic" Rabi model is for a single two-level system (TLS) and does not take decoherence into account (T1 and T2 are both infinite). In this context the latter means that if you pump your two-level to the excited state it will stay there forever.
A real TLS will spontaneously relax (transition back to the ground state) with some characteristic time T1 and de-phase (time T2). This also means that the amplitude of real Rabi oscillations will decay with a time constant T1 (which btw how T1 is measured).

Now, if you apply lots of power you can make sure that the system is immediately excited again once it has decayed. If you plot the power dependence you will find that you then reach a point where the dependence becomes nearly flat and this is what is known as saturation. If you work through the math you will find that the amount of power that is needed for saturation depends on T1*T2.
But again, this makes no sense in the basic Rabi model since there it is assumed that T1 and T2 are infinite.

This is probably the source of your confusion.

In lasers and similar systems you are typically dealing with ensembles of TLS but math is similar. However, instead of measuring the same system many times, you are now usually measuring lots of identical systems.

Now, you can observe regular Rabi oscillations in ensembles of TLS as well but the reasons for why this works are often complicated (you need to take interactions between different TLS into account) and goes well beyond the basic model.

• DrClaude