Vibration Studies for the CLARA Interferometer
Giulio Stancari, Alexander Shemyakin and Aleksandr Romanov
Fermi National Accelerator Laboratory, Batavia, Illinois, USAMay 07, 2024 15:32
FERMILAB-FN-1246-AD
This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.
1 Introduction
The purpose of the CLARA experiment is to study the nature of undulator radiation emitted by single electrons circulating in the IOTA storage ring [1]. The classical and quantum properties of the radiation are investigated by measuring its coherence length, intensity fluctuations and time correlations. The experiment took data in IOTA Run 4 (2022-2023).
The key component of the apparatus is a Mach-Zehnder interferometer (MZI), in which the optical length of one of the arms can be precisely controlled. For some measurements, the fine regulation of the arm length must be smaller than the radiation wavelength. For this reason, the apparatus is particularly sensitive to mechanical vibrations. In this note, we model and measure the effect of vibrations on the performance of the MZI under various conditions. Several improvements of the setup were implemented to minimize systematic distortions of the observed interference patterns.
In this report, we present a mathematical model of the effect of mechanical vibrations on observed detector and coincidence rates. We also describe the measurements that were made to estimate the magnitude and spectra of rate fluctuations and their sources in the CLARA MZI under various conditions. Finally, we estimate the magnitude of arm length fluctuations and we deduce the sensitivity to coincidence-rate variations in our apparatus.
2 The CLARA Experiment and the Mach-Zehnder Interferometer
The CLARA experiment studies the radiation emitted by single or multiple 150-MeV electrons in the SLAC undulator located in the D-Right section of the IOTA ring. For this undulator, the fundamental wavelength is at 630 nm.
Figure 2.1: Layout of the experiment in IOTA.
The key components of the apparatus are a Mach-Zehnder interferometer placed on top of the M4R magnet, 2 detector modules based on single-photon avalanche photodiodes (SPADs) at the interferometer outputs, a digital camera, and electronics to record photocount rates and arrival times.
Figure 2.2: Schematic diagram of the Mach-Zehnder interferometer.
For alignment and commissioning of the interferometer, a 637-nm diode laser (Thorlabs LPS-635-FC, LD in the figure) was used. The current threshold for lasing was 47 mA.
2.1 Fine Scans of MZI Arm Length
One of the main measurement techniques of the CLARA experiment is the scan of the optical length of the bottom interferometer arm by displacing the hollow-roof mirror IM4 in small steps compared to the radiation wavelength. The photocounts in the two detectors SPAD1 and SPAD2 and their coincidences are recorded as a function of the interferometer configuration. The interference patterns reveal the coherence length of the radiation. The value of coincidence rate near the interference condition with respect to its value away from interference gives information about the nature of the radiation (chaotic/thermal, coherent, quantum Fock state, etc.).
A list of available stage scans with diode laser can be found in Table 4.1.
Figure 2.3 shows an example of a stage scan, where the photocount rates are measured as a function of the position of the hollow-roof mirror. More examples can be found in Section 7.
Figure 2.3: Example of measured detection rates during a stage delay scan.
The individual rates show the typical oscillatory behavior with an envelope determined by the coherence time.
3 Model of the Effect of Vibrations on Delay Scans
One of the signatures of quantum radiation states is the Hong-Ou-Mandel dip in the coincidence rates of the output detectors in a Mach-Zehnder interferometer. Mechanical vibrations can generate fluctuations in the lengths of the interferometer arms, which also results in a reduction of coincidence rates near the center of the interference pattern. Quantitatively, how do vibrations affect delay scans? What do we expect?
3.1 Individual Detector Rates
During a delay scan, the photocount event rates \(R_1\) and \(R_2\) in the output detectors D1 and D2 are
\[ \begin{split} R_1 & = C_1 \left[ 1 + V_1 \cdot \exp\left(-\frac{d^2}{2d_c^2}\right) \cdot \cos\left(\frac{2\pi}{\lambda} d\right) \right] \\ & = C_1 \left[ 1 + V_1 \cdot \exp\left(-\frac{x^2}{2x_c^2}\right) \cdot \cos\left(2\pi x\right) \right] \\ R_2 & = C_2 \left[ 1 - V_2 \cdot \exp\left(-\frac{d^2}{2d_c^2}\right) \cdot \cos\left(\frac{2\pi}{\lambda} d\right) \right] \\ & = C_2 \left[ 1 - V_2 \cdot \exp\left(-\frac{x^2}{2x_c^2}\right) \cdot \cos\left(2\pi x\right) \right] \\ \end{split} \]
where \(\lambda\) is the wavelength of the radiation, \(x = d/\lambda\) is the normalized difference in optical path \(d\) and \(x_c = d_c/\lambda\) is the normalized coherence length \(d_c\). \(C_1\) and \(C_2\) are the constant rates away from interference (\(|x| \gg x_c\)). \(V_1\) and \(V_2\) are the visibilities of the interference patterns. For each detector, the maximum and minimum rates near \(x = 0\) are \(R_\mathrm{max} = C (1+V)\) and \(R_\mathrm{min} = C (1-V)\), so that \((R_\mathrm{max}-R_\mathrm{min}) / (R_\mathrm{max}+R_\mathrm{min}) = V\), in accordance with the definition of visibility. The visibility envelope is not necessarily Gaussian and may depend on the nature of the light source.
The two detector rates are equal when \[ \cos\left(2\pi x\right) \cdot \exp\left(-\frac{x^2}{2x_c^2}\right) = \frac{C_2 - C_1}{C_1 V_1 + C_2 V_2}. \] In the case of equal rates away from interference (\(C_1 = C_2\)), this condition reduces to \(x\) being equal to an odd multiple of 1/4.
3.2 Coincidence Rates
For randomly distributed events, the coincidence rate in a time window \(t_w\) is
\[ R_c = R_1 \cdot R_2 \cdot t_w \]
From the measured detector and coincidence rates, using a known coincidence window, one can estimate the degree of second-order coherence \(g^{(2)}\) of the radiation.
For comparisons with experiments, we define a normalized coincidence window \[ W \equiv \frac{R_c}{R_1 \cdot R_2} \] and normalized coincidence rates \[ \begin{split} r & \equiv \frac{R_1 \cdot R_2}{C_1 \cdot C_2} \\ r_c & \equiv \frac{R_c}{R_1 \cdot R_2 \cdot t_w} \simeq g^{(2)}, \end{split} \] For classical radiation (coherent Glauber states), \(r_c = 1\). For quantum light, one can have \(r_c < 1\).
As an example, let us consider a numerical model with the following parameters:
| \(C_1\) [MHz] | \(V_1\) | \(C_2\) [MHz] | \(V_2\) | \(x_c\) | \(t_w\) [ns] |
|---|---|---|---|---|---|
| 0.5 | 0.8 | 0.4 | 0.6 | 3 | 20 |
The corresponding individual photocount rates and coincidence rates as a function of displacement are shown in Figure 3.1.
Figure 3.1: Numerical example of expected detector and coincidence rates vs. optical delay in the interferometer.
One should note that, with different visibilities, the maximum normalized coincidence rate \(\hat{r}\) is larger than 1, as shown in Figure 3.1. The maximum value is \[ \hat{r} = \frac{(V_1+V_2)^2}{4 V_1 V_2}. \] In our numerical example, \(\hat{r} = 1.021\).
3.3 Effect of Delay Fluctuations on Event Rates
Fluctuations in the optical delay \(x\) affect the detector and coincidence rates. This effect has an influence on the interpretation of the data, such as the magnitude of a possible Hong-Ou-Mandel dip in the coincidence rate. Fluctuations in \(x\) smear out the dependence of the detector rates on MZI delay within the coherence region \(x \lesssim x_c\). In addition, the maximum observed coincidence rate is suppressed by fluctuations in \(x\).
To derive quantitative estimates of these effects, we assume that optical delays fluctuate randomly around a fixed value \(x_0\) with standard deviation \(\sigma\). We consider fast oscillations compared to the observation time. We then calculate the corresponding rates and their variances.
In order to evaluate the systematic uncertainty on these estimates, we use two random distributions, both with standard deviation \(\sigma\): a uniform distribution \(\rho_U\) \[ \rho_U(x) = \begin{cases} \frac{1}{\sqrt{12} \cdot \sigma} & \text{for } | x - x_0 | < \sqrt{3} \cdot \sigma \\ 0 & \text{otherwise} \end{cases} \]
and a Gaussian distribution \(\rho_G\) \[ \rho_G(x) = \frac{1}{\sqrt{2\pi} \sigma} \exp{\left[ -\frac{(x-x_0)^2}{2\sigma^2} \right]} \] Averages over these distributions are denoted as \(\left< \right>_{x,U}\) and \(\left< \right>_{x,G}\), respectively.
3.3.1 Average Rates
The average detector and coincidence rates are calculated from the following integrals, for both the uniform and Gaussian distributions:
\[ \begin{split} \left< R_1 \right>_{x, U} & \equiv \int_{x_0 - \sqrt{3} \sigma}^{x_0 + \sqrt{3} \sigma} R_1 \cdot \rho_U \, dx \\ \left< R_1 \right>_{x, G} & \equiv \int_{-\infty}^{\infty} R_1 \cdot \rho_G \, dx \\ \left< r \right>_{x, U} & \equiv \int_{x_0 - \sqrt{3} \sigma}^{x_0 + \sqrt{3} \sigma} r \cdot \rho_U \, dx \\ \left< r \right>_{x, G} & \equiv \int_{-\infty}^{\infty} r \cdot \rho_G \, dx \end{split} \] Although cumbersome, these integrals have general analytical forms. They can also be evaluated numerically.
For example, let us consider the following cases with different visibility, finite coherence length, and variable noise amplitude:
| \(V_1\) | \(V_2\) | \(x_c\) | \(\sigma\) |
|---|---|---|---|
| 0.8 | 0.6 | 3 | 0.00 |
| 0.8 | 0.6 | 3 | 0.03 |
| 0.8 | 0.6 | 3 | 0.06 |
| 0.8 | 0.6 | 3 | 0.09 |
| 0.8 | 0.6 | 3 | 0.12 |
The corresponding distortion of the coincidence curve vs. path difference is shown in Figure 3.2 for the case of Gaussian fluctuations.
Figure 3.2: Distortion of the observed rates vs. optical delay due to random noise in the optical delay (Gaussian model of fluctuations).
In the simple case of maximum visibilities (\(V_1 = V_2 = 1\)) and large coherence lengths (\(x_c \gg 1\)), the rates are
\[ \begin{split} R_1 & = C_1 \left[1 + \cos{(2\pi x)} \right] \\ r & = \left[ 1 + \cos{(2\pi x)} \right] \cdot \left[ 1 - \cos{(2\pi x)} \right] = 1 - \cos^2{(2\pi x)} \end{split} \] The first maximum of the coincidence rate occurs at \(x = 1/4\). Under these simplified conditions, the rates averaged over \(x\) are
\[ \begin{split} \left< R_1 \right>_{x, U} & = C_1 \left[ 1 + \frac{\sqrt{3} \cos{(2\pi x_0)} \sin{(2\sqrt{3}\pi\sigma)}}{6\pi\sigma} \right] \\ \left< R_1 \right>_{x, G} & = C_1 \left\{ 1 - \left[ 1 - 2 \cos{(\pi x_0)}^2 \right] \cdot \exp{(-2\pi^2\sigma^2)} \right\} \\ \left< r \right>_{x, U} & = \frac{1}{2} -\frac{\sqrt{3} \cos{(4\pi {x_0})} \sin{(4 \sqrt{3} \pi \sigma)}}{24\pi\sigma} \\ \left< r \right>_{x, G} & = \frac{1}{2} - \exp{\left(-8\pi^2\sigma^2\right)} \left[ \frac{1}{2} -4\sin{(\pi x_0)^2} + 4\sin{\pi x_0}^4 \right] \end{split} \] In particular, near the first maximum of the coincidence rate at \(x_0 = 1/4\), fluctuations suppress the average coincidence rate by the following amounts:
\[ \begin{split} \left< r \right>_{x, U} & = \frac{\sqrt{3} \cos\left(\sqrt{3} \pi \sigma\right)^{3} \sin\left(\sqrt{3} \pi \sigma\right)}{3 \, \pi \sigma} - \frac{\sqrt{3} \cos\left(\sqrt{3} \pi \sigma\right) \sin\left(\sqrt{3} \pi \sigma\right)}{6 \, \pi \sigma} + \frac{1}{2} \\ \left< r \right>_{x, G} & = \frac{1}{2} \left[ 1 + \exp{\left(-8 \, \pi^{2} \sigma^{2}\right)} \right] \end{split} \]
In Figure 3.3, one can see the effect of uniform or Gaussian \(x\) distributions on the coincidence rate suppression near the maximum at \(x_0 = 1/4\) for this simplified case.
Figure 3.3: Calculated dependence of the average normalized coincidence rate as a function of the amplitude of optical path fluctuations, for maximum visibility and large coherence length.
For instance, if the optical path difference fluctuates by 5% of the radiation wavelength, then the maximum coincidence rate is suppressed by a factor 0.907 (uniform model) or 0.91 (Gaussian model).
3.3.2 Rate Variances
To infer the magnitude of the position fluctuations from the observed event rates, we also calculate the variances and standard deviations of the detector and coincidence rates as a function of \(\sigma\): \[ R_1^\mathrm{rms} \equiv \sqrt{ \left< R_1^2 \right>_x - \left< R_1 \right>_x^2 } \] One can obtain analytical expressions for these variances and standard deviations.
As a numerical example, let’s consider how the standard deviation of the detector rate \(R_1\) varies with the optical delay \(x_0\) and fluctuation amplitude \(\sigma\) for a given visibility \(V_1\), shown in Figure 3.4.
Figure 3.4: Calculated standard deviation of the detector 1 rate as a function of optical delay and fluctuation rms.
As expected, fluctuations are small near the rate extrema (\(x_0 = 0, 1/2, 1, 3/2, \ldots\)) and large where the slope of the detector rate is maximum (\(x_0 = 1/4, 3/4, 5/4, \ldots\)). In this example, small fluctuations are amplified by a factor 4.9: a \(\sigma\) of 5% corresponds to a maximum \(R_1\) rms of 25%. Away from the interference region (\(x_0 \gg x_c\)), the rate is constant and fluctuations in \(x\) have no effect on \(R_1\).
4 Experimental Methods
A few different types of experiments were done to assess the effects of
vibrations on the performance of the interferometer. Some measurements
were done during the testing and commissioning phase
at ESB; others were done in the final location,
after the installation of the MZI in the IOTA tunnel
on top of the M4R dipole magnet.
Data is backed up on beamssrv1 in the iota-fast.bd network drive.
4.1 Event Rates vs. MZI Arm Delay
These are the typical optical delay scans. Event rates in both detectors were recorded at 15 Hz as a function of optical delay. From these measurements, the fringe visibilities and coherence length of the radiation can be estimated. Several scans were done with the laser diode during the commissioning phase. They are listed in Tables 4.1 and 4.2.
| ID | Date | Location | Diode current [uA] | Step [nm] | Steps | Interval [s] | Coinc. window [ns] |
|---|---|---|---|---|---|---|---|
| Scan A | 2023-01-18 13:48:11 | ESB | 25 | 25 | 1000 | 2 | 19.2 |
| Scan B | 2023-01-18 14:36:31 | ESB | 10 | 25 | 1000 | 2 | 19.2 |
| Scan C | 2023-01-19 15:29:18 | ESB | 4000 | 25 | 1000 | 2 | 19.2 |
| Scan D | 2023-01-19 16:35:53 | ESB | 22000 | 25 | 1000 | 2 | 19.2 |
| Scan E | 2023-02-01 16:18:20 | M4R | 20 | 25 | 200 | 1 | 20.7 |
| Scan F | 2023-02-09 11:16:56 | M4R | 18 | 25 | 400 | 1 | 20.7 |
| ID | Date | Location | \(R_1^\mathrm{max}\) [kHz] | \(R_1^\mathrm{min}\) [kHz] | \(R_2^\mathrm{max}\) [kHz] | \(R_2^\mathrm{min}\) [kHz] | \(V_1\) | \(V_2\) |
|---|---|---|---|---|---|---|---|---|
| Scan A | 2023-01-18 13:48:11 | ESB | 1552.87(97) | 63.50(20) | 1437.13(95) | 149.46(30) | 0.92143(24) | 0.81159(36) |
| Scan B | 2023-01-18 14:36:31 | ESB | 224.13(37) | 9.224(74) | 205.24(35) | 20.69(11) | 0.92094(62) | 0.81683(94) |
| Scan C | 2023-01-19 15:29:18 | ESB | 1015.61(78) | 82.95(22) | 946.36(75) | 132.41(28) | 0.84899(39) | 0.75451(49) |
| Scan D | 2023-01-19 16:35:53 | ESB | 955.40(76) | 55.18(18) | 909.27(74) | 131.14(28) | 0.89080(35) | 0.74791(50) |
| Scan E | 2023-02-01 16:18:20 | M4R | 1218.5(14) | 286.15(65) | 1099.7(13) | 293.84(66) | 0.61965(78) | 0.57830(85) |
| Scan F | 2023-02-09 11:16:56 | M4R | 948.8(12) | 14.14(15) | 849.6(11) | 56.68(29) | 0.97064(30) | 0.87491(62) |
Scans were done for different values of the diode laser current.
The relevant e-log entries with further details are listed below:
- 2023-01-18: https://www-bd.fnal.gov/Elog/?orEntryId=231843
- 2023-01-19: https://www-bd.fnal.gov/Elog/?orEntryId=231950
- 2023-02-01: https://www-bd.fnal.gov/Elog/?orEntryId=232749
- 2023-02-09: https://www-bd.fnal.gov/Elog/?orEntryId=233230
4.2 Fast Measurements of Event Rates at Fixed MZI Position
These were the main measurements on which the vibration study is based. Photocount events in the two SPADs were recorded for a few seconds using the picosecond event timer (Picoquant HydraHarp 400), which has a resolution of 1 ps and an accuracy of about 15 ps. With this technique, we were able to detect fluctuations of the detector rates at the millisecond time scale.
The optical delay between the two arms of the interferometer was set at one of two positions:
near the center of interference curve, where detector rates are approximately equal and coincidence rates are maximum. Here detector rates are most sensitive to variations in optical delay.
away from the interference condition, where detector rates should not depend on the difference between arm lengths, to minimize the effects of vibrations and to isolate fluctuations in light intensity or other factors.
For reference, the event timer configuration was as follows:
- Removed IOTA channels (revolution marker, M2L PMT, M3L PMT, injection trigger)
- Ch2 = SPAD1
- Ch3 = SPAD2
- Trigger = -400 mV, delays = 0 ps
- T2 mode (all events recorded with absolute time stamp)
Time-tagged photocount events from SPAD1 and SPAD2 were typically collected in data sets of 10 s each.
Here is the list of the available data sets:
- 2023-01-20, at ESB: https://www-bd.fnal.gov/Elog/?orEntryId=232032.
- 2023-02-02, in IOTA with interferometer on M4R dipole magnet: https://www-bd.fnal.gov/Elog/?orEntryId=232829.
- 2023-02-09, in IOTA after mitigation of vibrations: https://www-bd.fnal.gov/Elog/?orEntryId=233230.
The data sets for vibration analysis are summarized in Table 4.3.
| ID | Time | Location | Diode current [uA] | Interference? | Acquisition time [s] |
|---|---|---|---|---|---|
| 1 | 2023-01-20 14:19:39 | ESB | 22000 | TRUE | 10 |
| 2 | 2023-01-20 14:49:03 | ESB | 41100 | FALSE | 100 |
| 3 | 2023-02-02 10:33:59 | M4R | 18 | TRUE | 10 |
| 4 | 2023-02-02 13:02:18 | M4R | 14 | FALSE | 10 |
| 5 | 2023-02-09 15:53:04 | M4R | 18 | TRUE | 10 |
| 6 | 2023-02-09 15:54:20 | M4R | 18 | FALSE | 2 |
4.3 Measurements of Environmental Acoustic Noise
For a qualitative comparison with the observed rate fluctuations, we recorded environmental acoustic noise for about 10 s both at ESB and in the IOTA enclosure at M4R, using a cell-phone microphone. Acoustic noise may excite vibrations of the MZI components.
2023-02-03: measurement of acoustic noise at ESB. At 15:36, placed cell-phone on the table near Rack 903 were the interferometer was installed before it was moved to IOTA. Recorded about 10 s of environmental noise. (No e-log entry.)
2023-02-10: measurement of acoustic noise at M4R near the location of the interferometer. E-log entry: https://www-bd.fnal.gov/Elog/?orEntryId=233291.
5 Data Analysis and Results
5.1 Event Rates vs. Time at Fixed Delay
The time series of detector counts from the picosecond event timer was subdivided into time bins. For each time bin the event rates were calculated. Coincidence rates within a given time window were calculated offline with a software algorithm. For each data set, the analysis parameters are shown in Table 5.1.
| ID | Time | Location | Diode current [uA] | Interference? | Acquisition time [s] | Time bin [ms] | Coinc. window [ns] |
|---|---|---|---|---|---|---|---|
| 1 | 2023-01-20 14:19:39 | ESB | 22000 | TRUE | 10 | 1 | 20 |
| 2 | 2023-01-20 14:49:03 | ESB | 41100 | FALSE | 100 | 100 | 2000 |
| 3 | 2023-02-02 10:33:59 | M4R | 18 | TRUE | 10 | 1 | 20 |
| 4 | 2023-02-02 13:02:18 | M4R | 14 | FALSE | 10 | 1 | 20 |
| 5 | 2023-02-09 15:53:04 | M4R | 18 | TRUE | 10 | 1 | 20 |
| 6 | 2023-02-09 15:54:20 | M4R | 18 | FALSE | 2 | 1 | 20 |
Because of the strong attenuation filters used at high laser diode intensity, that data set had much lower event rates. Therefore, different time bins and coincidence windows were used in that case. This anomaly turned out to be useful for checking systematics, as described below.
Figure 5.1 shows the time evolution of event rates in the two detectors near the interference condition and away from it, together with the correlation between the two rates.
Figure 5.1: Example of detector rate evolution and correlation
Tables 5.2, 5.3 and 5.2 show a summary of the data sets and of the statistical properties of the measured event rates.
| ID | Time | Location | Interference? | SPAD1 events | SPAD2 events | \(\left< R_1 \right>\) [kHz] | \(\left< R_2 \right>\) [kHz] | Rate correlation |
|---|---|---|---|---|---|---|---|---|
| 1 | 2023-01-20 14:19:39 | ESB | TRUE | 5051811 | 5209153 | 505.2(11) | 520.92(97) | -0.955 |
| 2 | 2023-01-20 14:49:03 | ESB | FALSE | 245276 | 307427 | 2.4528(49) | 3.0743(56) | -0.017 |
| 3 | 2023-02-02 10:33:59 | M4R | TRUE | 4858424 | 4078147 | 485.8(25) | 407.8(22) | -0.971 |
| 4 | 2023-02-02 13:02:18 | M4R | FALSE | 1498719 | 1438736 | 149.87(16) | 143.87(16) | 0.446 |
| 5 | 2023-02-09 15:53:04 | M4R | TRUE | 4230092 | 4832078 | 423.01(57) | 483.21(49) | -0.412 |
| 6 | 2023-02-09 15:54:20 | M4R | FALSE | 938496 | 889920 | 469.25(74) | 444.96(71) | 0.606 |
| ID | \(R_1^\mathrm{rms}\) [kHz] | \(R_2^\mathrm{rms}\) [kHz] | \(R_1^\mathrm{rms} / \left< R_1 \right>\) | \(R_2^\mathrm{rms} / \left< R_2 \right>\) | \(1/\sqrt{\left<N_1\right>}\) | \(1/\sqrt{\left<N_2\right>}\) |
|---|---|---|---|---|---|---|
| 1 | 110.63(78) | 96.82(68) | 0.2190(16) | 0.1859(14) | 0.044491(49) | 0.043814(41) |
| 2 | 0.1538(34) | 0.1762(39) | 0.0627(14) | 0.0573(13) | 0.063852(63) | 0.057033(52) |
| 3 | 249.8(18) | 216.2(15) | 0.5141(45) | 0.5301(47) | 0.04537(12) | 0.04952(13) |
| 4 | 16.43(12) | 15.78(11) | 0.10966(78) | 0.10967(78) | 0.081685(45) | 0.083370(46) |
| 5 | 56.67(40) | 49.16(35) | 0.13396(96) | 0.10173(73) | 0.048621(33) | 0.045492(23) |
| 6 | 33.03(52) | 31.84(50) | 0.0704(11) | 0.0715(11) | 0.046164(36) | 0.047407(38) |
| ID | \(t_w\) [ns] | Coinc. events | \(\left< R_c \right>\) [Hz] | \(W \equiv \frac{\left< R_c \right>}{\left< R_1 \right> \cdot \left< R_2 \right>}\) [ns] | \(r_c = W / t_w\) | \(R_c^\mathrm{rms}\) [Hz] | \(R_c^\mathrm{rms} / \left< R_c \right>\) | \(1/\sqrt{\left<N_c\right>}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 20 | 50274 | 5027(23) | 19.10(10) | 0.9552(51) | 2289(16) | 0.4553(38) | 0.4460(10) |
| 2 | 2000 | 1414 | 14.14(39) | 1875(53) | 0.938(26) | 12.47(28) | 0.882(32) | 0.841(12) |
| 3 | 20 | 31232 | 3123(20) | 15.76(16) | 0.7882(78) | 2033(14) | 0.6508(63) | 0.5658(18) |
| 4 | 20 | 4822 | 482.2(69) | 22.36(32) | 1.118(16) | 692.2(49) | 1.435(23) | 1.440(10) |
| 5 | 20 | 44747 | 4475(22) | 21.89(11) | 1.0946(57) | 2198(16) | 0.4912(42) | 0.4727(12) |
| 6 | 20 | 9162 | 4581(50) | 21.94(24) | 1.097(12) | 2232(35) | 0.4872(94) | 0.4672(25) |
Near the interference region, event rates showed large oscillations (Figure 5.1 and Table 5.2). SPAD1 and SPAD2 rates were anti-correlated, confirming the hypothesis of changes in relative arm length. These effects were significantly mitigated after installation in IOTA. Away from the interference condition, detector rates were positively correlated. This suggests fluctuations in the laser intensity.
Fluctuations were quantified by the standard deviation and range of rates (Table 5.3). For comparison, the expected fluctuations arising from the number of observed events per time bin (\(N_1\) and \(N_2\)) is also shown. From these measurements, one can estimate the rate variations that are not explained by statistical fluctuations alone.
The data sets with larger time bins shows no rate correlations or fluctuations beyond the expected statistical ones, indicating that vibrations were faster than the chosen averaging period.
The statistical properties of coincidence rates are shown in Table 5.4. Fluctuations are dominated by the number of counts per bin, \(N_c\). Still, one can evaluate the magnitude of the normalized coincidence rate suppression in the interference region.
For each data-set pair (interference / no interference), Table 5.5 shows the excess detector rate fluctuations, defined as the relative standard deviation not explained by statistical fluctuations or by changes in laser diode intensity. The excess fluctuations are calculated from the observed detector rate variances near the interference condition, after subtracting the statistical variance and the variance observed away from the interference condition. The \(r_c\) ratio is the normalized coincidence rate in the interference region divided by the one away from interference. It represents the estimated coincidence rate suppression due to vibrations.
| Date | Location | Comment | Data-set pair | \(R_1^\mathrm{rms} / \left< R_1 \right>\) excess | \(R_2^\mathrm{rms} / \left< R_2 \right>\) excess | \(r_c\) ratio |
|---|---|---|---|---|---|---|
| 2023-01-20 | ESB | commissioning | 1, 2 | 0.2141(17) | 0.1805(15) | 1.019(29) |
| 2023-02-02 | IOTA/M4R | after installation | 3, 4 | 0.5068(46) | 0.5230(47) | 0.705(12) |
| 2023-02-09 | IOTA/M4R | after mitigation | 5, 6 | 0.1130(13) | 0.0735(15) | 0.998(12) |
The observed detector rate fluctuations and coincidence rate suppression can be compared with the vibration model described in Section 3. For each data set, we calculate the predicted rates for a given set of optical delay fluctuations \(\sigma\), using the parameters listed in Table 5.6.
| Data set | \(x_0\) | \(x_c\) | \(V_1\) | \(V_2\) | \(\sigma_\mathrm{min}\) | \(\sigma_\mathrm{max}\) | \(\sigma\) points | \(x\) distrib. | Samples |
|---|---|---|---|---|---|---|---|---|---|
| 2023-01-19 | 0.25 | 10 | 0.87 | 0.75 | 0 | 0.25 | 100 | uniform | 10000 |
| 2023-01-19 | 0.25 | 10 | 0.87 | 0.75 | 0 | 0.25 | 100 | Gaussian | 10000 |
| 2023-02-01 | 0.25 | 10 | 0.62 | 0.58 | 0 | 0.25 | 100 | uniform | 10000 |
| 2023-02-01 | 0.25 | 10 | 0.62 | 0.58 | 0 | 0.25 | 100 | Gaussian | 10000 |
| 2023-02-09 | 0.25 | 10 | 0.97 | 0.87 | 0 | 0.25 | 100 | uniform | 10000 |
| 2023-02-09 | 0.25 | 10 | 0.97 | 0.87 | 0 | 0.25 | 100 | Gaussian | 10000 |
Visibilities \(V_1\) and \(V_2\) are taken from delay scans acquired before the event-timer acquisitions at fixed MZI arm positions. However, as visibility may drift with time due to changing interferometer configuration, a systematic uncertainty is introduced in the model predictions.
The left plot in Figure 5.2 shows the predicted suppression in the normalized coincidence rate as a function of the excess in detector-1 rate fluctuations (lines). The points represent the measured values. The plot on the right in Figure 5.2 shows the expected and observed correlation between excess fluctuations in the detector rates.
Figure 5.2: Comparison of observed rate fluctuations with vibration model.
The model can explain the observed behavior within a few percent. A significant contribution to the discrepancy is the uncertainty (10–20%) on the visibility.
From the observed detector rate fluctuations and the model, one can estimate the magnitude \(\sigma\) of the delay fluctuations. Figure 5.3 shows the value of \(\sigma\) that gives rise to a given level of excess fluctuations in the detector rates. The vertical lines correspond to the measured values.
Figure 5.3: Estimate of delay fluctuations from observed detector rates
From this comparison, one estimates \(\sigma = 4\%\) (or about 25 nm) at ESB and \(\sigma = 2\%\) (12 nm) in IOTA/M4R after mitigation. The large fluctuations in IOTA/M4R after installation do not allow to extract a value of \(\sigma\); for this data set, the comparison also suggests that the visibilities were larger than the ones used in the model — an increase of 0.1 in \(V_1\) and \(V_2\) gives a good fit.
5.2 Spectral Analysis of Event Rates
To investigate the nature of rate oscillations, spectral analysis is done on the time series of event rates. Not much activity was observed above 1 kHz (verified up to 50 kHz), hence the choice of 1-ms time bins.
Figure 5.4 shows the Fourier spectra for each data set. The vertical axis corresponds to the relative amplitude of the oscillation.
Figure 5.4: Fourier spectra of detector rates.
Most of the oscillations are concentrated between 100 and 200 Hz, which could correspond to mechanical vibrations of the supports of the optical elements. These region of the spectrum was significantly suppressed thanks to the mitigation intervention.
Away from the interference condition, only the 60-Hz harmonics are significant. This suggests electric noise or fluctuations in the intensity of the light source.
The 60-Hz line was not present at ESB. In the tunnel, at IOTA/M4R, several harmonics of the line frequency are present. The 120-Hz line is less dominant at M4R than it was at ESB.
The following plots show interactive version of the spectra, which can be zoomed in and visualized in more detail.
5.3 Measurements of Environmental Acoustic Noise
As a qualitative investigation on the possible causes of mechanical vibrations, we measured the acoustic noise level with a cell-phone microphone, once at ESB on 2023-02-03 and once at M4R in the IOTA tunnel on 2023-02-10.
The following interactive plot allows one to explore the spectra in more detail.
There were many different sources of acoustic noise, both at ESB and in IOTA at M4R: power supplies, lights, fans, etc. It is plausible that acoustic noise sources could excite vibrations in the optical components. We did not investigate this further.
6 Conclusions
A vibration model for the CLARA Mach-Zehnder interferometer was developed. For a given magnitude of the fluctuations, it predicts the expected averages and standard deviations of the detector and coincidence rates. Measurements were made under different experimental conditions to estimate the magnitude of detector-rate fluctuations, their frequency spectra and their possible sources. Rate fluctuations originated from mechanical vibrations of the optical components in the interferometer with a standard deviation of 10–20 nm. After optimization, the apparatus was sensitive to variations of the coincidence rate at the level of 1%.
7 Appendix
7.1 Diagnostic Plots for the Stage Scans
Details of the detector rates vs. time are shown in Figures 7.1 and 7.2, to test whether the steps in delay stage position were identified correctly.
Figure 7.1: Detector rate vs. time, initial steps in each delay scan.
Figure 7.2: Detector rate vs. time, final steps in each delay scan.
The stage scans with laser diode are plotted in Figure 7.3.
Figure 7.3: Stage scans with laser diode.
7.2 Full Table of Fluctuation Results
| start_time | location | interference | time_bin_s | n_records | rate1_Hz | rate2_Hz | coinc_rate_Hz | coinc_window_calc_ns | rate_correlation | rate1_rms_Hz | rate2_rms_Hz | coinc_rms_Hz | rate1_range_Hz | rate2_range_Hz | coinc_range_Hz | rms_over_mean_1 | rms_over_mean_2 | rms_over_mean_coinc | rel_rms_expected_1 | rel_rms_expected_2 | rel_rms_expected_coinc | range_over_mean_1 | range_over_mean_2 | range_over_mean_coinc |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2023-01-20 14:19:39 | ESB | TRUE | 0.001 | 10558987 | 5.052(11)e5 | 5.2092(97)e5 | 5027(23) | 19.10(10) | -0.9549389 | 1.1063(78)e5 | 96820(680) | 2289(16) | 486010.0 | 425010.0 | 10000 | 0.2190(16) | 0.1859(14) | 0.4553(38) | 0.044491(49) | 0.043814(41) | 0.4460(10) | 0.9620510 | 0.8158908 | 1.989100 |
| 2023-01-20 14:49:03 | ESB | FALSE | 0.100 | 1057010 | 2452.8(49) | 3074.3(56) | 14.14(39) | 1875(53) | -0.0169742 | 153.8(34) | 176.2(39) | 12.47(28) | 690.1 | 850.1 | 50 | 0.0627(14) | 0.0573(13) | 0.882(32) | 0.063852(63) | 0.057033(52) | 0.841(12) | 0.2813565 | 0.2765209 | 3.536068 |
| 2023-02-02 10:33:59 | M4R | TRUE | 0.001 | 9234594 | 4.858(25)e5 | 4.078(22)e5 | 3123(20) | 15.76(16) | -0.9713658 | 2.498(18)e5 | 2.162(15)e5 | 2033(14) | 833010.0 | 718010.0 | 9000 | 0.5141(45) | 0.5301(47) | 0.6508(63) | 0.04537(12) | 0.04952(13) | 0.5658(18) | 1.7145683 | 1.7606280 | 2.881660 |
| 2023-02-02 13:02:18 | M4R | FALSE | 0.001 | 3235462 | 1.4987(16)e5 | 1.4387(16)e5 | 482.2(69) | 22.36(32) | 0.4455555 | 16430(120) | 15780(110) | 692.2(49) | 75000.0 | 73000.0 | 3000 | 0.10966(78) | 0.10967(78) | 1.435(23) | 0.081685(45) | 0.083370(46) | 1.440(10) | 0.5004274 | 0.5073898 | 6.221485 |
| 2023-02-09 15:53:04 | M4R | TRUE | 0.001 | 9360193 | 4.2301(57)e5 | 4.8321(49)e5 | 4475(22) | 21.89(11) | -0.4118499 | 56670(400) | 49160(350) | 2198(16) | 267000.0 | 224010.0 | 10000 | 0.13396(96) | 0.10173(73) | 0.4912(42) | 0.048621(33) | 0.045492(23) | 0.4727(12) | 0.6311919 | 0.4635894 | 2.234787 |
| 2023-02-09 15:54:20 | M4R | FALSE | 0.001 | 1888020 | 4.6925(74)e5 | 4.4496(71)e5 | 4581(50) | 21.94(24) | 0.6062085 | 33030(520) | 31840(500) | 2232(35) | 148010.0 | 141010.0 | 11000 | 0.0704(11) | 0.0715(11) | 0.4872(94) | 0.046164(36) | 0.047407(38) | 0.4672(25) | 0.3154196 | 0.3169049 | 2.401222 |
7.3 Code Timing
| Chunk label | Time [ms] |
|---|---|
| save-results | 706.2 |
| plot-fourier-spectra | 403.6 |
| plot-stage-scans | 360.8 |
| plot-fourier-spectra-interactive | 281 |
| plot-rates-vs-time-detail | 249.5 |
| show-results-table | 130.1 |
| plot-sound-spectra-interactive | 96.65 |
| view-scan-steps-initial | 75.02 |
| view-scan-steps-final | 73.79 |
| plot-stage-scan-example | 70.85 |
| IOTA-layout | 60.12 |
| find-coincidences | 37.81 |
| plot-comparison-with-vibration-model | 37.31 |
| estimate-delay-fluctuations | 28.6 |
| compute-fluctuation-results | 28.14 |
| show-data-sets | 13.47 |
| stage-scan-list | 12.97 |
| stage-scan-data-sets | 11 |
| MZI-schematic | 8.822 |
| show-results-summary-a | 8.051 |
| show-results-summary-b | 7.254 |
| show-results-summary-c | 6.688 |
| model-parameters | 5.653 |
| show-data-set-parameters | 5.397 |
| stage-scan-list-b | 5.378 |
| graphical-parameters | 5.096 |
| show-vibration-results | 4.661 |
| plot-stage-scan-function | 1.36 |
| define-data-sets | 1.324 |
| constants | 1.152 |
| results-summary | 1.037 |
| pre-process-coincidences | 0.108 |
| find-scan-start | 0.09203 |
Total execution time: 2.74 s.
7.4 R Session Information
R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Ventura 13.6.7
Locale: en_US.UTF-8 / en_US.UTF-8 / en_US.UTF-8 / C / en_US.UTF-8 / en_US.UTF-8
Package version:
askpass_1.2.0 backports_1.4.1
base64enc_0.1.3 bigD_0.2.0
bit_4.0.5 bit64_4.0.5
bitops_1.0.7 blob_1.2.4
bookdown_0.37 broom_1.0.5
bslib_0.6.1 cachem_1.0.8
callr_3.7.3 cellranger_1.1.0
cli_3.6.2 clipr_0.8.0
colorspace_2.1-0 commonmark_1.9.0
compiler_4.3.1 conflicted_1.2.0
cpp11_0.4.7 crayon_1.5.2
curl_5.2.0 data.table_1.14.10
DBI_1.2.0 dbplyr_2.4.0
digest_0.6.33 dplyr_1.1.4
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evaluate_0.23 fansi_1.0.6
farver_2.1.1 fastmap_1.1.1
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fs_1.6.3 gargle_1.5.2
generics_0.1.3 ggplot2_3.4.4
glue_1.6.2 googledrive_2.1.1
googlesheets4_1.1.1 graphics_4.3.1
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gtable_0.3.4 haven_2.5.4
highr_0.10 hms_1.1.3
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httr_1.4.7 ids_1.0.1
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jsonlite_1.8.8 juicyjuice_0.1.0
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labeling_0.4.3 lattice_0.21-8
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magrittr_2.0.3 markdown_1.12
MASS_7.3-60 Matrix_1.5.4.1
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Rcpp_1.0.11 reactable_0.4.4
reactR_0.5.0 readr_2.1.4
readxl_1.4.3 rematch_2.0.0
rematch2_2.1.2 reprex_2.0.2
rlang_1.1.2 rmarkdown_2.25
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Rwave_2.6-5 sass_0.4.8
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stringr_1.5.1 svglite_2.1.3
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