I know nothing about physics but I find this fascinating.
However, there're workarounds to beat this SQL. One approach people are working on is called the "spin squeezing", which, one can think of the uncertainties of a measurement mapped onto a 2D plane, for simplicity, a circle. Spin squeezing is used to improve the uncertainty in one axis while sacrificing the other, like squeezing the circle into an eclipse. There's a group in MIT which attempted to show metrological gain from spin squeezing in 2020, which they failed.
Norman Ramsey's biography of Zacharias at http://www.nasonline.org/publications/biographical-memoirs/m... (.pdf link) talks a little about it:
Zacharias at this time became interested in developing atomic clocks and pursued two versions concurrently. One was a cesium atomic beam clock using my (Ramsey's) separated oscillatory field method, well engineered for reliability and commercial applications, including a source and vacuum system that could be operated for years rather than hours. He cooperated with the National Company in developing a commercial clock known as the Atomichron. The availability of this highly successful cesium atomic beam clock contributed greatly to the adoption of atomic time and to the international definition of the second as 9,192,631,770 oscillations of the cesium atom.
His other version had the potential for much greater accuracy but the risk of total failure. A very slow beam of atomic cesium was directed upward and allowed to fall as a fountain, with separated oscillatory field excitation on the way up and down. The half second required for the roundtrip in the fountain was approximately fifty times greater than that for an atom to traverse the oscillatory field region of a conventional atomic beam apparatus, so the resonance width, by the Heisenberg uncertainty principle, would be fifty times narrower with correspondingly increased clock accuracy. Despite valiant efforts by Zacharias and his associates, the fountain experiment failed because the numbers of ultraslow atoms in the beam were far below theoretical predictions, probably due to scattering in the nonequilibrium region between the slits. It is of interest to note that thirty years later, in 1989, Steven Chu succeeded in making an atomic fountain by using the new laser cooling techniques to produce ultra-slow atoms. The atomic fountain with laser cooling is now one of the most promising prospects for increasing the accuracy of clocks and frequency standards.
They attributed their major obstacle as their laser phase noise, and they claimed if they subtract the estimated laser noise from the result they would get sub-SQL. But hey, that's not how things work. If you think you can make it, you just make it.
[1] https://www.optica-opn.org/home/newsroom/2019/december/squee...
Usually other systematic errors are the problem though.
This should be, in some sense, natural to people who understand how a Fourier transform works… basically, when you apply a “window” function to measure a signal over a limited period of time, you end up with the signal smeared in the frequency domain.
So can it be precise in frequency?
Am I misunderstanding?
In order to measure the bandwidth of the signal (and how accurate the clock is), you have to measure the clock over a period of time. The shorter that period of time, the worse your measurement will be. The smaller the bandwidth, the more time you need to measure it.
Note that a signal with 5 Hz bandwidth and 5 Hz center frequency would make a very terrible clock, while a signal with 5 Hz bandwidth and 5 GHz center frequency would make a very nice clock. The bandwidth is the same in both cases, and the amount of time required to measure bandwidth is the same, but they make very different clocks.
My understanding of this from the domain of GPS time, has always been; you can have a good clock (GPS is slow, accurate over a long time) or a good ocillator (fast, inaccurate over along time). But together you can have both.
A signal that goes from 9 Hz to 11 Hz has a center frequency of 10 Hz and a bandwidth of 2 Hz.
But during that time there will be several copies of it built, and they'll run in several locations around the world, and the outputs of those copies will deviate less than some tinysecond per longtime. Attoseconds per month or whatever.
And the useful timescale is more like, how much do they deviate over five seconds?
Because they'll be used for something like astronomical ranging. Say you have two radio telescopes at distant points on the earth, both observing a distant stellar source simultaneously. The long baseline forms one side of a triangle, and the distances from the two observatories to the star form the other two sides. The more precisely you can determine those distances, the better you can resolve the position of the source. So you've got coherent receivers that can measure individual wavefronts in the signal, but how coherent are they, anyway? That depends on the clock they share. But they're on opposite sides of the planet, they don't actually share a clock...
And the measurement period over which you can compensate for other confounding factors (probe and measure the atmospheric distortions in the way, for instance) might be very brief, so the clocks' stability over brief periods is salient.
This is my lay understanding, so if someone more versed in the art wants to chime in, I'm all ears!
But Fourier theory!!!! That's FINE!!! I went carefully through the very careful treatment of Fourier series in the third edition of the W. Rudin Principles of Mathematical Analysis and also his very careful treatment of the Fourier transform in his Real and Complex Analysis, also distributions in his Functional Analysis. Also read and used the Blackman and Tukey The Measurement of Power Spectra, wrote some corresponding software that pleased some people in the US Navy, etc. Then I took a course at the level of the second Rudin book, and in the class when the prof got to the Fourier transform one of the students, with a physics background blurted out "That's the Heisenberg uncertainty principle!"
So, now as I look at the MIT lectures, etc., I expect to see a good, solid, clear connection with the Fourier transform -- but so far I've found NOTHING.
Yup, so, thanks for confirming that, yup, the Heisenberg uncertainty principle is not some weird and obscure feature of nature but, really, just an elementary result of any elementary but precise math derivation of the Fourier transform!!!!
But, looks like I will have to make the connections with the basics of quantum mechanics myself.
Yup, multiply in the time domain is the same as doing a convolution in the frequency domain -- that is, if take the Fourier transform of a function zero outside of a finite interval, then in effect have multiplied the function by a box, 1 in the interval and 0 otherwise, and then for the transform are doing a convolution, that is, a weighted sum, of the transform of the box which is just a version of sin(x)/x, that is, with a peak at the origin and falling off rapidly away from the origin. That is, the weighted sum of the convolution is a "smearing out". So, on to the connection with the physics ..., hopefully.
Yes, it has begun to dawn on me that since
d/dt e^(iwt) = iwe^(iwt)
(frequency w, time t) that
e^(iwt)
is both an eigenvector of simple versions of Schrödinger's equation and also, essentially via Fourier theory, one axis of an orthogonal basis of a linear algebra (or Hilbert space if prefer that since it permits infinitely many orthogonal basis vectors) representation of quite general wave function solutions to the equation. And just by including more basis vectors, can get whatever level of accuracy we might want.
That is, the terns in the Fourier transform act like the coefficients in the linear combination of basis vectors, the linear combination that approximates the wave function solution of Schrödinger's equation.
So, essentially we are into harmonic motion, e.g., swinging pendulums, whether we wanted to be or not.
Then we are ready to guess that for treatment of the electron in a hydrogen atom we will be into something like Sturm-Liouville theory of two point boundary value problems and standing waves, vibrating violin strings, etc. That is, an electron in what it does with a hydrogen nucleus will be like a vibrating violin string.
Thanks.
The weird thing is not that there is a relationship between a function and its Fourier transform. That's pretty "elementary" math, as you observe. The weird thing is that physically meaningful quantities such as position and momentum should be related via Fourier transform in the first place. No amount of math can prove this fact---you need experiments.
By the way, in Heisenberg-type lower bounds, you allow both the function and its Fourier transform to be nonzero everywhere, yielding the result that the Gaussian distribution minimizes the product of the variances. But you can also ask a different question: assuming that I want the function to be band-limited, i.e., I want the Fourier transform to be strictly 0 outside an interval [-B, B], which function minimizes the dispersion in the time domain? This question yields the beautiful theory of the "prolate spheroidal functions", which are far from elementary. This kind of question is useful e.g. in signal processing: if you are allowed to look at N samples of an audio signal, what is the best low-pass filter that you can design?
And thanks for the URL to Feynman's lectures -- I lost my paper copy in a move.
Yup, I understand that early on in quantum mechanics, we will get to both energy and momentum of these particles that have wave functions that evolve as in Schrõdinger's equation. Then I was wondering what math was going to take a wave function and pop out energy and momentum. I began to suspect something that smelled like it popped out of someone's back side. Good to hear that the theory was guessing and experiment confirmed, likely not the first time.
You are farther into digital filtering than I ever got: For a while I was doing such things for the US Navy with data they collected in sea trials. At the time the fast Fourier transform (FFT) was a hot topic. Actually it was later that I studied Fourier theory with some care.
I'm going after quantum mechanics just out of curiosity and with the basic assumption that it's not quite right and I want to check carefully. Maybe it's not really the final theory.
Occasionally I get torqued: E.g., okay, sure, there is a Hilbert space (as I recall, Rudin shows that, really, there is only one) and each wave function is a point in that space, but no way will I easily accept that all the wave function FORM a Hilbert space: That is, via Rudin and more, a Hilbert space is a complete inner product space where here complete means that every Cauchy convergent sequence converges. Then I recall the common, old examples that nice, smooth, likely even infinitely differentiable, functions can converge to a square wave with its jump discontinuities. But the physics people assure me that each wave function is differentiable and also continuous. And that's a point of small irritation -- of course they are continuous; it's an elementary exercise to show that every differentiable function is continuous.
And I got torqued at Feynman where in his Lectures he has that a particle of unknown position has position probability density uniform everywhere -- no it doesn't; it can't; there can be no such density since its integral would not be 1, actually either 0 or infinity.
I don't even like the common integration from minus infinity to infinity: Rudin develops the Riemann integral very carefully but only on closed intervals of finite length, that is, on compact sets. Sure the integral from minus infinity to infinity can be defined as a limit, an improper integral, but then we have a problem: Start with some standard Rudin material that there can be an infinite series that does converge but is conditionally convergent and then with rearrangements can have the series converge to anything might want. Well, the same could hold for integrating from minus infinity to infinity -- the result get can depend on just how the limiting operation is done. E.g., integrate on Monday, Tuesday, and Wednesday, then on Saturday, Friday, Thursday, then Sunday, and continue this pattern for each week. So, without more assumptions, that improper integral is not so good. So, sure, measure theory and the Lebesgue integral clean up this mess, have some assumptions and derivations that do permit integrating from minus infinity to infinity. Right, I can be picky. Uh, who's to say that God is not?
The way this would probably he mediated physically is that sampling the clock changes its energy level, which changes its oscillation frequency.