STATISTICAL PROPERTIES OF UNDULATOR RADIATION

Two experiments were carried out to study the statistical properties of undulator radiation in the Integrable Optics Test Accelerator (IOTA) storage ring at Fermilab. The first experiment studied the turn-to-turn fluctuations in the power of the radiation generated by an electron bunch. The magnitude of these fluctuations depends on the 6D phase-space distribution of the electron bunch. In IOTA, we demonstrated that this effect can be used to measure some electron bunch parameters, small transverse emittances in particular. In the second experiment, a single electron was stored in the ring, emitting a photon only once per several hundred turns. In this regime, any classical interference-related collective effects were eliminated, and the quantum fluctuations could be studied in detail to search for possible deviations from the expected Poissonian photon statistics. In addition, the photocount arrival times were used to track the longitudinal motion of a single electron and to compare it with simulations. This allowed us to determine several dynamical parameters of the storage ring such as the rf cavity phase jitter and the dependence of the synchrotron motion period on amplitude.


INTRODUCTION
In our previous experiments [1][2][3][4] with an electron bunch we showed that turn-to-turn fluctuations var(N ) of the number of detected undulator radiation photons per turn N have two contributions: (1) a Poissonian contribution equal to ⟨N ⟩, due to the discrete quantum nature of light, and (2) a collective contribution ∝ ⟨N ⟩ 2 , related to the interference between the fields generated by the electrons in the bunch. In this paper, we get rid of the collective contribution by considering a single electron circulating in the Integrable Optics Test Accelerator (IOTA) storage ring at Fermilab in order to thoroughly study the quantum fluctuations and verify that they follow the Poissonian photostatistics var(N ) = ⟨N ⟩, predicted by [5][6][7][8]. This research is motivated by the surprising observation of sub-Poissonian photostatistics (var(N ) < ⟨N ⟩) in synchrotron radiation reported in Ref. [9] in a similar experiment setting. In addition, we will use the recorded detection times to study the synchrotron motion of a single electron in IOTA [10], similar to previous experiments in Novosibirsk [11,12]

APPARATUS
In our experiment, a single electron circulated in IOTA with a revolution period of 133 ns and an energy of 96.4 MeV. The undulator parameter is u = 1.0 with the number of periods u = 10.5 and the period length u = 5.5 cm. The wavelength of the fundamental was 1 = u (1 + 2 u /2)/(2 2 ) = 1.16 µm, where = 188.6 is the Lorentz factor. The second harmonic was in the visible wavelength range. We used a Single Photon Avalanche Diode (SPAD) [13] as a detector, which was mostly sensitive to the visible light with detection efficiency of up to 65%. Two edge-pass filters were used to only collect the radiation between 550 nm and 800 nm. The radiation was focused on the sensitive area of the detector (180 µm) by a single focusing lens with a focal distance of 180 mm, see Figs. 1(a),(b). The radiation was collected in a large angle > 1/ . The SPAD detector produced a 10-ns-long TTL pulse at each detection event. Its dead time (20 ns) was shorter than the IOTA revolution period (133 ns). Our data acquisition system [ Fig. 1(c)] allowed us to record the revolution number and the arrival time relative to the IOTA revolution marker for each detection event for as long as 1 minute at a time.

PHOTOSTATISTICS MEASUREMENTS
In the optimal focusing, the measured photocount rate was 24.7 kHz, or one photocount per 304 revolutions in IOTA (on average). The dark count rate of the SPAD detector was 108 Hz. In addition, we used a 5-ns-long gate around the expected detection arrival time, which allowed us to reduce the effective dark count rate to 4.0 Hz.
Before any analysis of the photostatistics, it was important to realize that the SPAD detector is binary. It produces the same type of pulses (TTL, 10-ns-long) no matter how many photons are detected per one pass. The collected turnby-turn data can be represented as a sequence of zeros and ones only. Therefore, we had to alter our original expectation of Poissonian photostatistics to a sequence of Bernoulli trials, i.e., there is a probability of a detection at every revolution, and a probability (1 − ) of no detection. In our case, = (3.29 ± 0.02) × 10 −3 . Figure 2 illustrates the comparison between the expectation (for a sequence of Bernoulli trials) and the measurement for (a) the distribution of interarrival times and for (b) the distribution of the number of photocounts in a certain time window. In both cases, the 2 goodness-of-fit test [14, p. 637] results in a P-value [14, p. 140] above the conventional 0.05 threshold. This means that the null hypothesis (exponential or binomial distribution, respectively) cannot be rejected.  Some measurements were carried out with an upgraded setup consisting of two SPAD detectors separated by a beam splitter [15]. In this case, the photon number resolution was improved, since there were three possible outcomes for each pass: 0, 1, or 2 detection events. Still, so far we have not observed anything unusual. There was no statistically significant correlation or anticorrelation in the two detectors.  We decided to compare the measured arrival times with a simulation of the synchrotron motion. In our simulation [16] we use the following transformation of the relative energy deviation and the rf phase of a single electron from turn to turn + 1,

SYNCHROTRON MOTION STUDIES
where is the electron charge, 0 = 2 = 96.4 MeV, is the electron mass, is the speed of light, = √︁ 1 − 1/ 2 is the relativistic velocity parameter, 0 = 380 V is the rf voltage amplitude, = 4 is the rf harmonic number, = − 1/ 2 = 0.070 83 is the phase slip factor (variation of due to variation of is negligible), = 0.070 86 is the momentum compaction factor, J = 2.64 is the longitudinal damping partition number [17, p. 445], is the radiation energy loss at th turn, is the rf cavity phase jitter at the th turn. We model as a random variable following a normal distribution with a standard deviation . We refer the reader to [17,Eq. (3.28)] for the symplectic part of the transformation. The derivation of the synchrotron damping term, − ⟨ ⟩ J / 0 , is described in [17, pp. 438-445]. The quantum excitation term, −( − ⟨ ⟩)/( 2 0 ), is considered in [18]. The energy kick at the synchronous phase compensates for the average energy loss due to the synchrotron radiation, i.e., 0 sin = ⟨ ⟩. The average emitted energy per turn in an isomagnetic ring is [17, pp. 434-435] ⟨ ⟩ = 8 /9 = 10.9 eV, where is the fine-structure constant, = 3ℏ 3 /(2 ) = 2.8 eV is the critical energy [18,Eq. (11)], = 70 cm is the electron trajectory radius in the dipole magnets, ℏ is the reduced Plank constant. The isomagnetic ring approximation works well in IOTA, the radiation in the undulator is negligible  compared to the bending magnets. The average number of photons emitted per turn in an isomagnetic ring is [18] ⟨N ⟩ = 5 / √ 3 = 12.5. To simulate the number of emitted photons at th revolution, we use a Poisson random number generator with the expectation value ⟨N ⟩ = 12.5 . To simulate the energies of these photons, we use the Monte Carlo generator described in Ref. [18]. The sum of these energies gives .
Using our computer code we can generate the data points as in Fig. 3 for a long interval of time, e.g., 1 minute. By fitting the data with short pieces of sinusoidal curves (as in Fig. 3) one can plot the synchrotron motion amplitude as a function of time, see Fig. 4. We cannot compare the measurement and the simulation in this way directly, because it is a stochastic process. However, we can compare the distributions of the synchrotron motion amplitudes. Figure 5(a) illustrates such a comparison, where the simulation results are presented at three different values of the rms rf phase jitter . We can conclude that in IOTA ≈ 6.0×10 −5 rad. We considered more values of than illustrated in Fig. 5(a). Further, using the same piecewise sinusoidal fit we can plot the synchrotron motion period as a function of the syn-chrotron motion amplitude, see Fig. 5(b). Every point in Fig. 5(b) is calculated from a 25-ms-long interval of time. The measured and the simulated synchrotron motion periods agree rather well, which shows that we understand the parameters of the IOTA ring well.

CONCLUSION
To conclude, in our experiment with a single electron and a binary photon detector, we have not yet observed any deviations of the undulator radiation photostatistics from a memoryless Bernoulli process. Our measurements with two SPAD detectors did not reveal any correlation or anticorrelation in the detected photon pairs. Currently, the next phase of experiments is under preparation, namely, the Mach-Zehnder interferometry of undulator radiation [15]. The detection arrival times can be used to study the synchrotron motion of a single electron and to infer some dynamical parameters of the storage ring, such as the rms rf cavity phase jitter and the synchrotron motion period as a function of amplitude. This diagnostics of the longitudinal motion can complement the diagnostics of the transverse motion [19,20] and facilitate a complete 3D tracking of a single electron in a ring.