Confused about Ramsey interferometry

[kelly0303]

Hello! Assume we have a 2 level system with the frequency between the 2 levels $\omega_0$ and ignore the lifetime of the upper state. In order to measure the transition frequency using Ramsey technique, you apply 2 $\pi/2$ pulses separated by a time $T$. We have that the probability of finding an atom in the excited sate under the interaction with light (assuming strong field i.e. not using perturbation theory) is given by: $$\frac{\Omega^2}{W^2}sin^2(Wt/2)$$ where $\Omega$ is the Rabi frequency and $$W^2=\Omega^2+(\omega-\omega_0)^2$$ with $\omega$ being the laser frequency (which we scan in order to find $\omega_0$). If we are on resonance, i.e. $W=\Omega$, a $\pi/2$ pulse is a pulse of duration $t=\frac{\pi}{2\Omega}$ such that, starting in the ground state, the atom becomes an equal superposition of upper and lower states. However in the Ramsey technique, we don't know $\omega_0$ beforehand (this is what we want to measure). So I am not sure I understand how we can still create an equal superposition of upper and lower level using a $\pi/2$ pulse. Assuming we use a pulse of duration $t=\frac{\pi}{2W}$, from the equation above we get that the population of the upper state is $$\frac{\Omega^2}{2W^2}$$ which is not $1/2$ as in the resonance case. How do we still get equal superposition of upper and lower case when we are not on resonance? Thank you

 

[Twigg]

All the current Ramsey spectroscopy experiments I can think of off-hand are looking for interesting miniscule shifts in otherwise boring structures (tiny shifts of $\omega_0$). In other words, you wouldn't use Ramsey spectroscopy to characterize a transition for the first time if you had other options. Even so, an imperfect $\pi/2$ pulse isn't useless. You'll still see a fringe, just a lower quality fringe. Bad state preparation doesn't change the fringe frequency in an ideal 2-level system.

 

[kelly0303]

All the current Ramsey spectroscopy experiments I can think of off-hand are looking for interesting miniscule shifts in otherwise boring structures (tiny shifts of $\omega_0$). In other words, you wouldn't use Ramsey spectroscopy to characterize a transition for the first time if you had other options. Even so, an imperfect $\pi/2$ pulse isn't useless. You'll still see a fringe, just a lower quality fringe. Bad state preparation doesn't change the fringe frequency in an ideal 2-level system.

Thanks a lot for the reply! So, in principle one measures $\omega_0$ for the first time in other ways. Actually, while reading about this I got a bit confused about Rabi's technique, too. Initially I thought that in order to measure a transition (the bulk part of it) one can, for example, scan the laser frequency in a given region (for example predicted by theory) and search for a peak in, say, the fluorescence of the sample. However, based on the Rabi oscillation formula above (I will assume the lifetime is long enough that we can ignore it), even if you are perfectly on resonance, if the time of the laser atom interaction is $\frac{2\pi}{\Omega}$, the probability of the upper state being excited is zero. So even if you are on resonance, you would see no fluorescence. In practice, for a new transition, one doesn't know the transition frequency, or the Rabi frequency, so how do they actually perform the search for the transition frequency? Do they scan both the length of the interaction time and the frequency of the laser? Thank you!

 

[f95toli]

There is no single answer to this since it very much depends on the system and how fast you can adjust and measure different parameters.
That said, for single 2-level systems where you only have very rough idea of the the frequency continuous-wave spectroscopy tends to be the fastest; i.e. you use a constant tone and apply so much power that you can be sure that the system would end up up in a 50:50 distribution when you are on resonance.

"Tuning up" a 2-level system (a qubit) can be very time consuming and there are multiple parameters that needs to be optimised. These days, this is often done using various clever optimisation techniques and when dealing with more complex system machine-learning approaches can also be useful.

For many real-life experiments you are actually dealing with ensembles of 2-level systems meaning there is a distribution of parameters; this means that in e.g. pulsed electron spin resonance you are likely to see some sort of response (echo) of say a Hahn sequence as long as you are reasonably close to resonance and your pulse is "hard" (high amplitude) enough.

 

[kelly0303]

There is no single answer to this since it very much depends on the system and how fast you can adjust and measure different parameters.
That said, for single 2-level systems where you only have very rough idea of the the frequency continuous-wave spectroscopy tends to be the fastest; i.e. you use a constant tone and apply so much power that you can be sure that the system would end up up in a 50:50 distribution when you are on resonance.

"Tuning up" a 2-level system (a qubit) can be very time consuming and there are multiple parameters that needs to be optimised. These days, this is often done using various clever optimisation techniques and when dealing with more complex system machine-learning approaches can also be useful.

For many real-life experiments you are actually dealing with ensembles of 2-level systems meaning there is a distribution of parameters; this means that in e.g. pulsed electron spin resonance you are likely to see some sort of response (echo) of say a Hahn sequence as long as you are reasonably close to resonance and your pulse is "hard" (high amplitude) enough.

Thank you for your reply! I am a bit confused by the first paragraph. Why does having a lot of power means you are in a 50:50 case? If we are in a case where the laser-atom interaction is much bigger than the lifetime of the upper state, the population of the upper state is given by: Ω2/4δ2+Ω2/2+Γ2/4 where δ is the detuning and Γ is the lifetime of the upper state. If we have a huge power, so Ω is huge, we indeed end up with a 50:50 population as you said. However, my question was in the opposite regime, in which the lifetime is a lot longer than the laser-atom interaction, in which the population of the upper state on resonance is given by: sin2(Ωt2) So in this case, having a huge power won't bring the system in a 50:50 state. So my question is about this case, in which, even on resonance and with huge power, you can end up with zero atoms in the upper state. How do you deal with this in practice?

 

[Twigg]

Thank you for your reply! I am a bit confused by the first paragraph. Why does having a lot of power means you are in a 50:50 case?

I think what f95toli is trying to say is that it doesn't pay to start characterizing a qubit with coherent pulses. (If the jargon isn't clear, by coherent pulses I mean laser pulses that are faster than the lifetime.) Think about it from an experimentalist's point of view: you just got a job and a lab, your peers are breathing down your neck, you have grand plans for some qubit, but first things first you need to find a major transition. Are you going to splurge on a high-power, quickly-modulated pulsed laser setup for coherent pulses that takes a month to get right? Nah, you're going to buy a cheap continuous-wave diode off craigslist and tune it around until you see a lineshape. The outside world will limit what power and timescale you work in for initial spectroscopy. That's why you expect 50:50 saturation: the cheapest and fastest way to get it done works on a much larger timescale than the lifetime.

 

[f95toli]

Twigg is correct, when finding the resonance frequency the idea is often NOT to attempt any form of pulsed operation but just a constant tone. If you work through the math you will see that in most situations this will result in 50:50 saturation if you apply enough power.
Note that I say "most" here, in the real world there are of course still issues. One thing that can happen is that you accidentally start pumping your system into higher levels or multi-photon excitation which can obscure the signal, If you the system lifetime is very long you can also end up "bleaching" the signal meaning you won't see anything as you sweep the frequency.

I should insert a caveat here. I have no practical experience of doing Ramsey (or any other form of coherent manipulation) using lasers; all the systems I work with (solid state qubits and various spin ensembles)are manipulated using microwaves. There could be issues that I am not aware of around using lasers for this.

 

[kelly0303]

Twigg is correct, when finding the resonance frequency the idea is often NOT to attempt any form of pulsed operation but just a constant tone. If you work through the math you will see that in most situations this will result in 50:50 saturation if you apply enough power.
Note that I say "most" here, in the real world there are of course still issues. One thing that can happen is that you accidentally start pumping your system into higher levels or multi-photon excitation which can obscure the signal, If you the system lifetime is very long you can also end up "bleaching" the signal meaning you won't see anything as you sweep the frequency.

I should insert a caveat here. I have no practical experience of doing Ramsey (or any other form of coherent manipulation) using lasers; all the systems I work with (solid state qubits and various spin ensembles)are manipulated using microwaves. There could be issues that I am not aware of around using lasers for this.

@f95toli @Twigg thanks a lot for the explanations! Also I just realized my latex didn't work in my previous post, sorry for that! Just to summarize, to make sure I got it right (ignoring some of the issues that might appear in the real world, that you mentioned). In order to do Ramsey spectroscopy, you need to know approximately the resonant frequency $\omega_0$ and the Rabi frequency $\Omega$, in order to be able to produce a $\pi/2$ pulse. For the frequency, one would use a continuous wave laser, adjust the frequency, and keep each frequency on for a time long enough such that the time is bigger than the expected lifetime of the system. In the case when the interaction time is much bigger than the lifetime, the population of the higher level is given by: $$\frac{\Omega^2/4}{\delta^2+\Omega^2/2+\Gamma^2/4}$$, with $\Gamma$ being the lifetime and $\delta$ the detuning. If the power is very high, hence $\Omega$ is very high the ratio converges to 1/2 so we get the 50:50 population you mentioned. Is this correct? Now, once we know the resonant frequency, by fitting that function to our counts vs frequency, we can switch to a pulsed laser, and send pulses of different lengths at the resonant frequency, and from there we can extract the Rabi frequency using: $$\sin^2(\frac{\Omega t}{2})$$Once we know both the Rabi frequency and the resonant frequency we can apply the Ramsey method in order to get a more accurate value of the resonant frequency. Is my understanding correct? I have one more question. I expected that the longer the interaction time, the narrower the peak and hence the better you can measure the resonant frequency. However in both of the above equations (assuming they are right), increasing the measurement time doesn't seem to help. In the first case the tine doesn't even appear in the equation, while in the second case we have an oscillatory behavior so a longer time won't help with narrowing down the line profile. Where does this Fourier limit comes into play? Thank you!

 

[f95toli]

Yes, that is correct. In the systems I work with Ramsey is often the last experiment you try since you should ideally have already calibrated your pulses really well before it makes sense to even start.
When working with qubits Ramsey is mostly used for the last "fine tuning" of the frequency, e.g. for a qubit with a resonance frequency of say 6 GHz (quite a typical value) Ramsey would only be used when you are at most a few MHz from that value.

How accurately you can determine the frequency is typically not limited by the measurement system at all but by broadening of the line itself, so under normal circumstances the interaction time does not really come into play. I am sure it is possible to model it, but I can't think of a practical situation where it would matter.

You will of course have the "usual" influence of averaging of measurement data etc but those times are much, much longer than any timescales of the system itself for a real system (see any Allan deviation plot)

That said, what is known as the "line splitting factor" does depend on how you measure the system so in very, very stable systems (say frequency references) this does become a factor. However, the analysis depends on the details of how you measure. For an old fashioned Pound loop (later developed into Pound-Drever-Hall locking) the line splitting factor can be ~10000 or so

 

[Twigg]

Quick note before I get to your next question. Whether it's microwaves or lasers, power broadening can complicate the initial measurement of ω0. It's easier to pinpoint the resonance in a narrow lineshape than in a broad one. There is such a thing as too much power!

Also, to tune even an approximate Ramsey pulse, you need to know the transition dipole moment, μ=ℏΩ/E. Without knowing the dipole moment, you don't know the Rabi rate so you can't tune the phase to pi/2. I've never done it myself, but I'm guessing the easiest way is to just drive the system with a known field for a set time and see how much Rabi flopping you get.

In a spectroscopic measurement, it's not the interaction time but the sampling time that gives you better frequency measurements. Each point in the initial spectroscopy is one more sample you can use to fit a model function to the lineshape (Lorentzian in the ideal case). Those points take measurement time. Likewise, if you want to measure the frequency of a Ramsey fringe, you want to take measurements at the longest possible time (in this case, sampling time and interaction time are roughly equivalent). The need for long time has less to do with Fourier analysis and more to do with how you measure frequency from sinusoidal data. Frequency is the total phase elapsed divided by the time over which it elapsed: $f = \frac{\Delta \phi}{\Delta t}$. The bigger you make $\Delta \phi$ (and by correlation also $\Delta t$), the less uncertainty there is in your measured frequency.