# Smallest energy we can measure

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Hello! Assuming we can bring 2 energy levels very close to each other (e.g. by applying a magnetic field), what is the practical limit (in terms of lab equipment) on the smallest energy difference that we can measure? And what is the relative error on it, that can be obtained? For example if the 2 levels are 100 Hz apart, can I measure that difference in practice (e.g. by doing some sort of Rabi measurement with an RF generator)?

Baluncore
If you measure the frequency of a photon you can calculate it's energy.
E = h·f; where h is the Planck constant.
The photon will have a non-zero frequency if it exists.
You will need to make your measurement near absolute zero to keep the noise down.

If you measure the frequency of a photon you can calculate it's energy.
E = h·f; where h is the Planck constant.
The photon will have a non-zero frequency if it exists.
You will need to make your measurement near absolute zero to keep the noise down.
I am not sure I understand what you mean. I am asking about what frequency we can reach in lab with an RF generator i.e. a numerical value of frequency that can be produced using nowadays equipment (in order to measure the transition, you need to be able to excite the transition).

Baluncore
Thermal excitation and the emission of a photon can be expected to occur close to absolute zero.

Thermal excitation and the emission of a photon can be expected to occur close to absolute zero.
How can you do Rabi oscillations with thermal excitations? I am talking about a controlled way of exciting the system. And again, I would like a numerical value of the frequency we can produce in a controlled way in lab.

Twigg
Gold Member
@kelly0303 There's a lot to unpack here. First, direct Rabi spectroscopy isn't the only way to get a frequency measurement. There's also Ramsey interferometry and beating techniques (not sure there's a good jargon term to describe these).

Another thing is that we can't quote you a "minimum frequency resolution" for modern equipment because you can always average down on instrument noise. The real limit is how long you can make the grad students run the experiment around the clock (literally 24/7 in some cases, with students working in shifts) before they mutiny and make you walk the plank

To determine the ultimate sensitivity of a frequency measurement, you need ALL of the following information:
• Frequency sensitivity per shot: what is the estimated error on a single run of the experiment?
• experimental repetition rate ("rep rate" for short): how fast can you collect a data point?
• total measurement time: how many hours can you devote to collecting data?
• Systematic uncertainty limits: how large are the leading systematics on your measurement? (these don't average down! once you run into systematics, taking more of the same kind of data doesn't help)
These vary wildly from experiment to experiment. For example, cold atom experiments typically have a much slower rep rate (between 1 and 10s) than ion experiments (<<1s), because of the time it takes to load a MOT and the slowness of evaporative cooling. Systematics are unique to each experiment.

If you want to talk about frequency sensitivity per shot, then there are oodles of experiments out there that achieve the "standard quantum limit" aka the "quantum projection noise (QPN) limit" aka "projection noise limit" aka "shot noise". This is a limit set by quantum physics, not instrumentation.

There are some experiments that go beyond the standard quantum limit by utilizing squeezed states. I don't know anything about them really, just that they're out there. I believe LIGO is one example, but my memory could be bad. I still have this paper on my to-read list.

Twigg
Gold Member
Assuming we can bring 2 energy levels very close to each other (e.g. by applying a magnetic field)

For example if the 2 levels are 100 Hz apart, can I measure that difference in practice (e.g. by doing some sort of Rabi measurement with an RF generator)?
To answer this particular example situation, there's several options.

If this is the ground state, I would use a resonant B-field pulse in the x-direction to make a pi/2 pulse, which is very easy to do even with a shoestring budget. With my pi/2 pulses, I'd so Ramsey measurement, in which case even a janky pi/2 pulse won't limit my ultimate uncertainty. Only the decoherence time will be the problem, and that again has nothing to do with instrumentation and everything to do with the physics of the sample being measured.

To answer this particular example situation, there's several options.

If this is the ground state, I would use a resonant B-field pulse in the x-direction to make a pi/2 pulse, which is very easy to do even with a shoestring budget. With my pi/2 pulses, I'd so Ramsey measurement, in which case even a janky pi/2 pulse won't limit my ultimate uncertainty. Only the decoherence time will be the problem, and that again has nothing to do with instrumentation and everything to do with the physics of the sample being measured.
Thanks a lot for all this info! So in terms of the absolute frequency that I can produce with an actual device, there is no lower limit? If the energy difference between the 2 levels is, say 1mHz (and say that the lifetime of the excited state is negligible), is there a device that can produce an EM field at that frequency? Can I apply a ##\pi/2## pulse at 1 mHz? Or at a frequency as low as I want?

Twigg
Gold Member
Can I apply a π/2 pulse at 1 mHz?
Probably not? I would think pink noise (which is friggin everywhere and an unavoidable fact of life) would become a substantial issue as you drive the system at lower and lower frequencies.

Keep in mind that you can do pi/2 pulses at 100kHz and then measure precession frequencies at 1mHz. You can apply whatever Rabi rate you want to do a pi/2 pulse so long as it gets the job done!

Typically, you want your pi/2 pulses to be much faster than the time scale of the precession you wish to measure. Otherwise, you will see non-negligible precession during your pi/2 pulse and who knows where you end up on the Bloch sphere.

Probably not? I would think pink noise (which is friggin everywhere and an unavoidable fact of life) would become a substantial issue as you drive the system at lower and lower frequencies.

Keep in mind that you can do pi/2 pulses at 100kHz and then measure precession frequencies at 1mHz. You can apply whatever Rabi rate you want to do a pi/2 pulse so long as it gets the job done!

Typically, you want your pi/2 pulses to be much faster than the time scale of the precession you wish to measure. Otherwise, you will see non-negligible precession during your pi/2 pulse and who knows where you end up on the Bloch sphere.
Doesn't the ##\pi/2## pulse need to be at the frequency of the transition (or very close to it)? So if the distance between the 2 levels is 1mHz (or something very small), won't the pulse need to be at 1mHz?

Twigg
Gold Member
So in terms of the absolute frequency that I can produce with an actual device, there is no lower limit?
I guess my point is that you are asking two questions that aren't the same thing.

1) What is the lowest frequency we can produce with current technology? (I have no idea on this one. Again, pink noise makes this a very difficult challenge.)

2) What is the smallest frequency we can measure with current technology? (This has an answer, but it doesn't have anything to do with RF generators. It has to do with the sensitivity limits on our best optical clocks. One limiting factor on this happens to be the stability of ultrastable HeNe lasers and their reference cavities.)

Twigg
Gold Member
Doesn't the π/2 pulse need to be at the frequency of the transition (or very close to it)? So if the distance between the 2 levels is 1mHz (or something very small), won't the pulse need to be at 1mHz?
Hmmmm. I know the answer is "no" from experience, but I also know what you're trying to say and I don't know how to bridge the two off the top of my head.

Here's an example of a technique that doesn't use Ramsey interferometry but does measure small frequencies without a generator at that frequency. The frequency of the beats on these decay curves correspond to Zeeman splittings in the excited state manifold.

I guess my point is that you are asking two questions that aren't the same thing.

1) What is the lowest frequency we can produce with current technology? (I have no idea on this one. Again, pink noise makes this a very difficult challenge.)

2) What is the smallest frequency we can measure with current technology? (This has an answer, but it doesn't have anything to do with RF generators. It has to do with the sensitivity limits on our best optical clocks. One limiting factor on this happens to be the stability of ultrastable HeNe lasers and their reference cavities.)
Sorry! Let me rephrase. Say we have 2 levels of opposite parity separate by an energy E. Now if we apply a constant magnetic field, we are able to bring the 2 levels as close as we want (of course there are some limits in practice, but ignore that for now). What I want to do is to measure the energy difference between the 2 levels as a function of the magnetic field. In practice we can bring them very close together (not zero, as I mentioned, but still very close). However if I bring them 1mHz apart (I don't know if this is possible, but it's just an example) but I have no way to measure that splitting in practice, then there is no point in doing that. So I want to know how close can I bring the 2 levels, such that I can still have a way to measure the splitting between them.

Twigg
Gold Member
Sorry I just remembered a couple of important points!

First, when you do a Ramsey sequence, you temporarily increase the energy difference between states during a pi/2 pulse by applying a stronger field. Then, once you are situated on the equator of the Bloch sphere, then you bring the states close together.

Second, all the precision measurements I can think of measure small changes in a much larger baseline frequency. The reason for this noise performance and feasibility.

To clarify, if you want to measure a small frequency shift X, you have two options:
1) Measure X directly by measuring out to very long precession times
2) Design a system where you can switch between precession frequencies f and f+X where f>>X
Method (2) is much more popular than method (1). And every Ramsey measurement I know follows method (2). This means that the pi/2 pulses only need to be (roughly) resonant with f, not with the tiny frequency X.

Method 1 has a couple of harsh issues. Long precession times means you eat a lot of pink noise (aka 1/f noise). Also, it means that if you want to try something new, it's a very slow process because the rep rate is so low.

kelly0303