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 Fri Mar 5 14:36:00 2021, aaron, DailyProgress, Lab Work, learned something about measuring clocks Mon Mar 8 12:11:52 2021, aaron, DailyProgress, Lab Work, allan covariance Thu Apr 15 16:28:20 2021, shruti, Update, PSOMA, setup to do both locking and incoherent 3CH Thu Apr 22 20:25:49 2021, shruti, Update, PSOMA, setup to do both locking and incoherent 3CH
Message ID: 2674     Entry time: Fri Mar 5 14:36:00 2021     Reply to this: 2677
 Author: aaron Type: DailyProgress Category: Lab Work Subject: learned something about measuring clocks

enter about 2pm

### daq

• chatted with Radhika, who is going to get our particle counter logging again.
• We found the services script that Duo and others had worked on in controls@cominaux:~/services, but many of the corresponding files in /etc/systemd/system were null linked.
• In the process, I remembered that I'd tried to initialize the x1oma model again, but was unsuccessful. Consequently, cymac crashes during starting 'all' frontends.
• Regardless, one could always caget the slow channels to see what's up (eg caget C5:PEM-COUNT_05UM returns the 5 um particle count). The data may not be saved to frames by cymac1, but at least the particle counter seems to be talking with cominaux.
• untangling cymac1
• Undid the steps I had taken to try to install x1oma:
• Removed x1oma entry from /etc/advligorts/systemd_env
• removed x1oma entry from /etc/advligorts/master
• Manually restart rts-local_dc and rts-daqd services using systemctl
• Rebooting cymac after these changes seems to have resolved the issues with frontends, but I still have white boxes for the GPS signal on X0DAQ (attachment 1).
• meanwhile spirou's cds-workstation package is broken (no medm, for example).

### Three Corner Hat to Allan Covariance [aka Cross-variance, aka Groslambert Variance]

A conversation with Rana had me thinking again about the 2-channel, 'coherent hat' measurement, where one compares the beat note between a laser of interest (X) and two other lasers (N, S). It felt like there should be enough information there. If I know $(\phi_{X}-\phi_{S})^2\equiv \Delta_{SX}$ and likewise for $\Delta_{NX}$, I ought to know the difference between I and Q.

Indeed I found this paper comparing the two methods... looks promising! The authors have further work quantifying the confidence intervals for the so-called Allan covariance.

The method is a natural extension of the Allan variance, where the phase noise on an oscillator as a function of frequency (1/t) is estimated by the autocorrelation of the oscillator's frequency at varying time separations (t). "Allan covariance" then is the same method applied to a set of different oscillators, where the oscillator frequencies are cross-correlated. I'll adapt the equations below from the paper to use more familiar notation.

Let $\phi_i$ denote the phase error (additonal phase relative to the carrier frequency) on the i'th laser (N, S, and X, for our cryo lab measurement). We want to estimate the phase noise of the TeraXion laser (\phi_X) by measuring the phase errors of the beat note between X and the N and S lasers (\phi_{NX}, \phi_{SX}). To see why the covariance of $\phi_{NX}$ and $\phi_{SX}$ gives a good estimate of the variance of $\phi_X$, consider

$\phi_{NX} = \phi_N -\phi_X+\theta_{NX}$

$\phi_{SX} = \phi_S -\phi_X+\theta_{SX}$

$\implies \langle \phi_{NX}\phi_{SX}\rangle = 2\langle\phi_N\phi_S\rangle + 2\langle\phi_X^2\rangle -\langle\phi_N\phi_X\rangle - \langle\phi_S\phi_X\rangle + \langle\phi_S\theta_{NX}\rangle + \langle\phi_N\theta_{NX}\rangle - \langle\phi_X\theta_{SX}\rangle - \langle\phi_X\theta_{NX}\rangle$

Since the expectations of all the uncorrelated terms are 0, we can conclude that as long as the measurements were taken simultaneously,

$\langle\phi_X^2\rangle_\tau\to \langle \phi_{NX}\phi_{SX}\rangle$

We should (and the paper does) use the phase errors $\phi$ to define a corresponding frequency error at each timestep $k$ set by our sampling period $\tau$, $f_i(k) \equiv \frac{\phi_i(k)-\phi_i(k-1)}{\tau}$. The frequency error is then $\Delta_i(k) = f_i(k)-f_i(k-1)$. The Allan variance is the variance of that frequency error, while the Allan covariance is the covariance of $\Delta_i$ with $\Delta_k$. Averaging the residuals makes explicit the dependence on $\tau$ and according to David Allan "effectively modulates the bandwidth in the software allowing one to distinguish between white-noise phase modulations (PM) and flicker-noise PM." I'm a bit unclear why it is necessary to take the second difference (that is, correlating frequencies separated by \tau rather than directly phases separated by \tau), but I think this paper from Allan goes into a bit more depth. My best guess is, it's because flicker phase noise (which asymptotes with infinite slope at low frequencies) will show up as a DC offset that is consequently not averaged away in terms like $\langle\phi_N\phi_S\rangle$. Taking a second difference means rather than negating this DC offset once (as I did above by defining $\phi_i$ as a phase error), phase noise below the sampling rate is discarded with each measurement time step.

With the frequency noise estimate from Allan (co)variance and some knowledge of the power law dependence of the noise, one can derive a corresponding spectral density. The Allan (co)variance with a particular sampling rate \tau corresponds to sampling the frequency-domain noise distribution with a function related to the Fourier transform of the chosen time-domain estimator. See the paper on converting between time and frequency domain noise estimates from Allan (linked above and here).

### measurement

Very exciting! Sketch of the measurement with equipment on our tables in attachment 2. Whoops, we only have a free space 1811, not a fiber coupled one. Between that and the floppy path to PSOMA table, I'm going to think further on this measurement and try it next week. Maybe I can pick up some spare fiber parts from the 40m when I stop by to search for a transformer. In the meantime, yesterday's log has been updated with the current noise spectra.

exit Fri Mar 5 20:47:39 2021

 Attachment 1: Screenshot_from_2021-03-05_15-14-24.png  37 kB
 Attachment 2: 8B3029C0-A1D3-4044-B238-68D545187BE5.jpeg  394 kB  Uploaded Fri Mar 5 19:45:48 2021
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