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Bucket Shaking Stops Bunch Oscillations In The Tevatron

Bunches in the Tevatron are known to exhibit longitudinal oscillations which persist indefinitely. These oscillations are colloquially called"dancing bunches". Although the dancing bunches do not cause single bunch emittance growth or beam loss at injection, it leads to bunch lengthening at collisions. In operations, a longitudinal damper has been built which stops this dance and damps out coupled bunch modes. Recent theoretical work predicts that the dance can also be stopped by an appropriate change in the bunch distribution. This paper shows the Tevatron experiments which support this theory.

For the inductive impedance, the incoherent tune shift decreases with the bunch length as ∆Ω ∝ ℓ −3 . The combined action of this decrease with increasing nonlinear tune spread δΩ ∝ ℓ 2 results in ℓ 5 in Eq. 1. This high sensitivity on the bunch length indicates that approximations of the bunch profile or arbitrary assumptions about the eigenfunctions can lead to significant errors in the calculated LLD threshold because they can change the effective bunch length. For example, for a full bucket of a single-harmonic RF system with an inductive impedance above transition, the threshold relative tune shift ∆Ω/Ω was found to be as low as 10% for the Hofmann-Pedersen distribution, and just ∼ 1% for a model of the Tevatron coalesced [15] bunch [3]. In terms of bunch population, the two thresholds differ by almost two orders of magnitude. It turns out that the onset of LLD is highly sensitive to the steepness of the distribution function at low amplitudes: the flatter the distribution, the more stable it is. This prediction appears to be generally correct when the bare RF synchrotron frequency monotonically decreases with amplitude and the wake field is repulsive, i.e. the wake lowers the incoherent synchrotron frequencies. This conclusion agrees with Ref. [13], where the LLD threshold was calculated for several distributions with the inductive impedance above and below transition. It was shown there that below transition LLD is sensitive to the edges of the distribution, while above transition, it is sensitive to the flatness of the bunch core.
As was discussed in Ref. [3], in the case of a sinusoidal RF system, any combination of inductance, wall resistivity, or high order cavity modes above transition, or space charge below transition will shift the incoherent spectrum down to lower frequency and the coherent mode emerges above it. Since the incoherent frequencies of low amplitude particles are close to the mode frequency, their wieght in the mode dominates. Hence, for a single-harmonic RF and a repulsive wake function, the discrete mode causes dipole motion of the bunch center while its tails remain still. This is the behavior of the bunches in the Tevatron [1].

A. Flattening out the distribution for particles with small amplitudes
To flatten out the bunch distribution at small amplitudes, resonant shaking of the RF phase was suggested for the Tevatron [5], with an idea to use anomalous diffusion within a controlled phase space area; see Ref. [16][17][18] and references therein.
Let it be assumed that the RF phase is modulated at a frequency Ω m , which is close to the synchrotron frequency Ω s . Let the amplitude of the modulation φ m (t) adiabatically grow from zero, then stay a while at some value φ 0 and then adiabatically decrease to zero.
To prevent excitation of the tail particles and the coherent modes, the process has to be adiabatic. However, even when the process is generally adiabatic, i.e. when |dφ m /dt| ≪ Ω s φ 0 , the adiabaticity for some particles is going to be broken. Indeed, resonant RF shaking results in either one or two stable fixed points (SFPs) inside the bucket. In the last case, there is an inner separatrix between the two SFPs and when the modulation amplitude changes, the separatrix moves and some particles cross it. Separatrix crossing is a non-adiabatic process resulting in classical chaos and anomalous diffusion.
Thus, the phase space density can be changed only in the case of two SFPs which occur when the modulation frequency is lower than the synchrotron frequency, Ω m < Ω s , and the modulation amplitude is lower than its bifurcation value, φ m < φ b = 3.08ǫ 3/2 with ǫ = 1 − Ω m /Ω s . When the modulation amplitude grows from zero to its bifurcation value, and when it comes back to zero later, the irreversible change of the phase space density occurs for the phase space area with action J ≤ J lim , where and J bucket is the bucket acceptance. For dimensionless variables associated with the unperturbed Hamiltonian H(z, p) = p 2 /2 + 1 − cos z, the acceptance J bucket = 8/π. The numerical factor "6" in Eq. (2) was approximated from a numerical solution discussed below and it is about two times larger than the separatrix border at zero amplitude. After this adiabatic cycle, the phase space density becomes nearly constant for the entire area J < J lim , provided that the shaking amplitude crosses its bifurcation value, i.e.
It is worth mentioning that the adiabatically ramped modulation does not excite any coherent motion when the modulation is turned off. Thus, to make a flat phase space density within a certain action J lim , the adiabatic RF phase modulation has to be applied slightly below the synchrotron frequency, ǫ = 0.16J lim /J bucket , and its amplitude must cross the bifurcation, Eq. (3).
A simulation of how the bunch distribution is modified with RF phase shaking has been done using the following map where z n and p n are the coordinate and momenta respectively in dimensionless units, t n is the time variable in radians of the synchrotron oscillation, ∆t is its numerical step. The amplitude of the RF phase modulation φ m (t) was taken as a trapezoid similar to Fig. 4.
• ∆t = 0.01 radians which is small enough for the results to be independent of its specific value.
• Initial phase space density is assumed to be F (J) ∝ (J max − J) 2 with the emittance J max set close to the bucket acceptance.
The simulation results before and after shaking are shown in Fig This simulation shows that the adiabaticity of shaking is really important: after every cycle, the Hamiltonian goes to zero. The irregular features of this plot probably reflect the chaotic nature of the anomalous diffusion responsible for the flattening of the distribution.

III. EXPERIMENT
The block diagram of the phase modulation hardware used for shaking the beam is shown in Fig. 3. A signal generator generates a sine wave where its amplitude and frequency can be programmed and its output is fed into a phase shifter module. The phase shifter modulates the Tevatron low level RF (LLRF) and the result is fed into the Tevatron high level RF (HLRF). Essentially, the block diagram produces the following where f HLRF is the phase modulated signal sent to the HLRF, A is the amplitude of the signal sent to the HLRF, f LLRF is the frequency from the LLRF, θ is an arbitrary phase, and the signal generator produces the amplitude φ m and frequency f m for the phase modulation.
The time evolution of the bunch during the experiment are measured using the Sampled Bunch Display (SBD) [19]. Its block diagram is shown in Fig. 3. The SBD measures the bunch profile from a resistive wall current monitor with an oscilloscope that has a 2 GHz bandwidth and the collected data is processed with a LabView program which calculates the following parameters: • bunch centroid, • bunch current, • rms bunch length.
These parameters are then returned to the control system and can be plotted to give Figures 5   and 9. Furthermore, the snapshots of the bunch from the resistive wall signal can also be downloaded. The SBD trigger has been set up to take five consecutive snapshots of the bunch that are 1 s apart. These snapshots are presented in the figures below.  Fig. 4. The maximum amplitude of the shake has been tested for φ 0 = 1 • , 2 • and 3 • respectively and experimentally, φ 0 = 3 • has been found to produce the best effect for the duration of the shake. Figure 5 shows the shake duration and the behavior of the bunch current, centroid, and rms bunch length before and after shaking. The beam current drops by about 2.3% and the rms bunch length grows by about 1.8% after being shaken. The beam current drop is not surprising because of the filled bucket. And the change in rms bunch length comes from the shape change which is clearly seen in Fig. 6. After the shake is turned off, a nice divot structure forms which confirms the prediction previously discussed in section II A.

Contrast to Dampers
The bunch distribution after it has been shaken can be contrasted to the distribution when dampers are used instead to stop the dancing. The before and after distributions are shown in Fig. 7. The effect of dampers on the bunch distribution is to make it more triangular. This can be contrasted to the shaking technique shown in Fig. 6 where the distribution becomes more rotund. Also, after the dance stops and the dampers are turned off, the bunches do not start dancing again even after the dampers have been off for 5 min.
At first glance, the stability of the bunch after the dampers are off contradicts the described theory of LLD. Indeed, according to this theory, the LLD threshold is lowered when   the distribution function becomes more steep and thus the beam distribution shown in Fig. 7 has to be less stable after the dampers are turned off than it was before they were on. A resolution for this seemingly contradictory observation relies on the necessity to distinguish between LLD and instability. Indeed, if Landau damping is lost, the growth rate is determined by the coupled bunch wake forces. If these forces are weak enough, the instability takes too long to grow and so it cannot be observed. There are two types of long range wake fields that can be considered as possible candidates for driving the longitudinal coupled bunch instability (LCBI): cavity modes and the resistive wall wake. Direct calculations show that the resistive wall wake is extremely weak and has to be removed from this list. Even for 36 Tevatron bunches, the calculated resistive wall LCBI growth time is ∼ 10 days and so the only remaining candidate is the RF cavity modes. According to Ref. [20], the LCBI observations at the top energy for 36 proton bunches can be explained by higher order modes at 311 MHz with caveats: the calculated growth time using the rigid bunch approximation is an order of magnitude faster than the measured one. There can be two reasons for this discrepancy: the first is a decreased Q-value compared to its measured value done in 2000 [20,21] and the second is that the rigid bunch approximation overestimates both the threshold and the growth rate of the instability by several times.

Initial Bunch Shape Effects
The number of shaking cycles required to stop the bunch varied from case to case. Most likely, this is due to the non-optimized detuning of the shake frequency and some variations  In this experiment five bunch coalescing is used rather than the usual seven. The initial bunch distribution between bunch 1 and 2 are quite different because the Main Injector has not been tuned up for five bunch coalescing. Therefore, the random effects of untuned coalescing has made bunch 1 dance a lot more than bunch 2 before shaking is applied. The bucket is about a factor of two larger than the beam size at 980 GeV, and thus allows the beam to freely change shape without being constrained by the bucket edges. A φ m ramp has been created so that there are no abrupt changes in the RF as shown in Fig. 4. Previous experiments have shown that sudden turn-ons can cause some beam loss even though the bucket is large compared to the beam size.
In this experiment, the total time the phase is ramped is 3 s. The rise and fall time of the ramp has been chosen to be 1 s because it is slow compared to the synchrotron period of 29 ms. The flattop period can be varied, but for this experiment it has been set to 1 s.
The modulation frequency f m has been set to the measured synchtron tune 34.75 Hz and the bunch is shaken seven times with the φ m ramp. Figure 9 shows the seven φ m ramps and the behavior of the bunch current, centroid, and rms bunch length for the duration of the experiment. The beam current is constant throughout the experiment but the rms bunch length grows by about 18% (from 1.67 ns to 1.97 ns) at the end of the experiment. It is interesting to notice that the rms bunch length grows after each shake because of the shape change. A comparison of the bunch shapes before and after the shakes shows that the rms bunch length growth comes from the flatter core of the bunch while its tails remain unchanged.   Figure 11 shows both the the bunch shape evolution and the spectrum from the phase detector after the first, third, fifth and seventh shakes. It is clear from these plots that after the first shake the amplitude of the dance has increased by about 14 dB relative to the amplitude before shaking. And after each subsequent shake, the amplitude becomes smaller, which after the seventh shake, the dance amplitude has decreased by 14 dB relative to the amplitude before shake. The shape of the bunch after the seventh shake has clearly changed. Figure 12 superimposes the before and after the seventh shake snapshots where the shape change is clearly evident.

IV. CONCLUSION
Similar ideas for bunch distribution flattening have been suggested and implemented in the KEK-PS [22,23] and the KEK Photon Factory [24]. This technique is also routinely applied in the CERN SPS to blow up the longitudinal emittance for stabilizing the beam [25].
However, in all these cases, narrow band RF noise around the synchrotron sidebands are excited. In the KEK-PS and SPS, the RF perturbation is applied to the voltage amplitude while at the KEK Photon Factory, noise is applied to the RF phase. The experiments described in this paper take a different approach: instead of noise, the RF phase is excited at the synchrotron frequency, and its amplitude is ramped adiabatically. This technique works because anomalous diffusion flattens the bunch distribution. It is also possible that this technique is able to finely regulate the width where the distribution is flattened while keeping the remaining distribution untouched. Unfortunately, due to the lack of machine studies time and the shutdown of the Tevatron [26], there was no opportunity to pursue these ideas further.
All the Tevatron experiments discussed here show that an RF phase modulation that is ramped to an amplitude of a few degrees for a duration of a few seconds can flatten the low amplitude distribution of the beam. In some cases, a divot formsà la computer simulations.
These beam studies show that stabilization really does happen and confirms the proposal that resonant RF shaking can stop the beam from dancing.