Impact of two-dimensional decoherence on the measurement of resonance driving terms

In the presence of a tune spread induced by chromaticity or amplitude detuning, decoherence will lead to the damping of the beam centroid motion after a single transverse excitation. This in turn has implications for the analysis of turn-by-turn based optics measurements, as it affects the precision of the spectral analysis. In the past, it has been shown how the effect of decoherence on spectral lines in a single plane can be accounted for. In this paper, this work will be extended to include the effect from both transverse planes.


Introduction
The analysis of the turn-by-turn (TbT) motion of a particle bunch after excitation captured by the beam position monitors around the ring has proven as a reliable method in many accelerators to determine the linear and nonlinear optics (see [1] and references therein).While this method is preferred in large scale machines because, amongst other reasons, it requires less measurement time compared to closed orbit response, the resolution of the harmonic analysis and thereby the accuracy of inferred optics scales with the number of exploitable turns, that is, turns with a sufficiently large oscillation amplitude.
Processes that decrease the oscillation amplitude of the centroid after a single excitation are synchrotron radiation damping and decoherence.In the latter case, the tune spread induced by detuning with amplitude, chromaticity, or space charge leads to the motion of different particles in a bunch to decohere, leading to the damping of the centroid oscillation amplitude [2,3,4].A larger tune spread results in a faster damping time, leading to a lower number of exploitable turns for the analysis and worsening its precision.To limit this effect, a special machine configuration can be used for conducting TbT measurements, as for example was done in ESRF [5,6].Decoherence will also lead to emittance growth [7,8,9] and, in the presence of white noise and a transverse feedback system, it can affect the beam distribution [10].
Decoherence can be overcome by using an ac-dipole to coherently excite the beam with a frequency close to the betatron tune.In this case, the number of exploitable turns is determined by hardware capabilities.Of particular importance for hadron accelerators, the emittance of the beam after the ramp-down is minimally affected under the right settings [11], allowing to reuse the same bunch for multiple measurements.In the past, a formalism has been found to infer the linear optics from the forced oscillations [12], however, deducing the nonlinear optics from analysis of forced motion has proven difficult so far [13].
For inferring the nonlinear optics from TbT measurements after a single excitation, in [14] it was found that depending on the amplitude detuning coefficients, higher order spectral lines show a smaller line amplitude and larger spectral width compared to the single particle case.
There, only the case considering one transverse plane has been studied.In this paper, the studies will be extended to include the effect of cross plane amplitude detuning on decoherence.

Spectral line distribution in the presence of amplitude detuning
In the following, the centroid motion of a Gaussian beam and its spectral decomposition will be described, summarising the derivation from [14].For simplicity, it is assumed that the linear chromaticity is 0, which allows to neglect the longitudinal plane, where non-zero linear chromaticity leads synchrontron side-bands around the principal lines.
To describe the motion and its spectral decomposition, it is convenient to use the resonance basis.It is given by where û and pu are the normalized Courant-Snyder variables, J u the action, ϕ u the angle, and ϕ u,0 the initial angle.Following [15], the evolution of the resonance basis with the number of turns N is then given by in the horizontal plane.Here, Q and ψ represent the tune and initial phase, respectively, with the subscript x denoting the horizontal plane and y the vertical plane.The first term in Eq. ( 2) is due the linear motion, giving the tune line with an amplitude √ 2I u , I u the invariant of motion in the normal form coordinates.The second term is the sum over contributions from perturbations such as coupling, sextupoles and other higher order multipoles of order n.The spectral line amplitude of these perturbing terms is proportional to the Resonance Driving Terms (RDT) f jklm of order n = j + k + l + m and the product of the invariants to some power of the indices j, k, l, and m.As described in [15], an RDT f jklm will excite a resonance (j − k, l − m) and lead to a spectral line defined by (1 − j + k)Q x + (m − l)Q y in the horizontal spectrum, which for simplicity is expressed as H(1 − j + k, m − l).Analogously, in the vertical plane the spectral line is given by V(k − j, 1 − l + m).
In the following, the centroid motion of a beam with a Gaussian distribution is studied, offset by the kick-amplitude A u .The density in the transverse planes is given by where both the action term √ 2I u and kick-amplitude A u are given in units of σ u of the distribution.
The motion of the centroid as a function of the number of turns N is given by the integral of h − x (N ) over the phase space weighted by the particle distribution from Eq. ( 3), as shown below Performing a Fourier transform using the expression of the weighted centroid motion in the with the spectrum of the tune line given by and for the higher order lines.Here, I n refers to the modified Bessel function of the first kind and of order n.To note, the tunes Q x and Q y in the above integrals implicitly depend on the invariants via where Q ′ uv = ∂Qu ∂2Iv gives the amplitude detuning terms.Solving the argument of the Dirac delta function δ in Eqs. ( 6) and ( 7) using the expression of the tunes from Eq. ( 8), the horizontal invariant can be expressed as noting that, as I x is the amplitude, the additional constraint I x ≥ 0 applies.In [14], these integrals are then solved assuming that I y = 0 to then arrive at an analytical expression describing the horizontal spectrum.In the following, the integrals ( 6) and ( 7) will be solved numerically to demonstrate the impact of the different detuning coefficients and nonzero invariant of motion in the other transverse plane on the spectrum.As a reference, the black line uses the same parameters in all three plots.

Decoherence of tune and sextupole line
Solving Eq. ( 6) numerically assuming a horizontally excited beam (A x = 1, A y = 0) and tunes of Q x = 0.28 and Q y = 0.31, in Fig. 1 the spectral distribution is presented for the case where only direct detuning is present (Fig. 1a) and for different levels of cross-detuning (Figs.1b &  1c).First, we note that increasing direct detuning Q ′ xx or cross detuning Q ′ xy will result in the peak spectral amplitude decreasing, while the spectral width increases.For the case shown in Fig. 1a with no cross-detuning, in [14] it was found that the line width increases linearly with the direct detuning Here, σ denotes the width of the line, with the subscripts indicating j = 1, k = 1, l = 0, and m = 0.This distinction is introduced to separate from the line width of a different RDT later on.Observing Eq. ( 8), it is also apparent that, being I x ≥ 0, a larger positive direct detuning will shift the distribution towards higher tunes.Similarly so, a non-zero vertical invariant I y ≥ 0 together with a positive cross-detuning will have the same effect, as is shown in Fig. 1b.For the case of opposite sign between direct and cross detuning and of similar magnitude, the spectral peak is shifted close to the zero-amplitude tune Q x,0 , illustrated in Fig. 1c.
To illustrate the effect of the detuning on decoherence and, in turn, on measurements of RDTs, the specific example of the f 3000 Resonance Driving Term, excited in first order by sextupoles and driving the H(-2,0) line, is used.This specific term has been selected as it allows comparison to analytical estimates as shown below.The scripts with which the analysis has been performed can be found in [16], permitting the reader to study different RDTs, which due to the space constraints could not be discussed here.The case of no cross detuning Q ′ xy = 0 has been selected for benchmarking against the analytical solution for the line width of the sextupole line as found in [14].In solving Eq. ( 7), the factor −2if jklm is omitted.In Fig. 2, a comparison between the spectral line width of the tune line and of the H(−2, 0) line is presented.First, we note that for the same detuning, the sextupole line has a larger line width and thus decoheres faster than the tune line.In the regime where Q ′ xx ≫ Q ′ xy , the line width approaches the analytical solution from Eqs. (10) and (11).As expected, for the case without cross detuning, the ratio of the line widths does not depend on the direct detuning, but only on the kick amplitude A x , set here to A x = 1.When including a cross detuning and Q ′ xx ≪ Q ′ xy , the ratio of the width of the two lines is roughly 15% smaller, however the decoherence is still faster than the case without cross detuning.These findings suggest that for the measurement of RDTs at a given detuning, the number of turns it takes for the sextupole line to decohere can be found based on the decoherence of the tune line.As the data taken after the sextupole line has decohered only adds noise to measurements, the smaller number of turns used in the spectral analysis may improve the measurement quality.
The maximum spectral amplitude of the two lines and the ratio between those is presented in Fig. 3.As demonstrated in Fig. 1 for the tune line, with increasing direct and cross detuning, the maximum line amplitude of the lines decreases.Similar to the case of the line width, two different regimes are found, depending on the ratio between direct and cross detuning.For Q ′ xx ≫ Q ′ xy , the ratio between line amplitudes is found to be 20% below the case where Q ′ xx ≪ Q ′ xy .Again, the final ratio of the maximum amplitudes depends both on the amplitude of the RDT at the given location as well as the kick amplitude A x .Compared to the single particle case where H(−2, 0)/H(1, 0) = 6f 3000 √ I x , an additional correction factor to account for the change in amplitude ratio based on the detuning can be applied to improve RDT measurements.

Conclusion
In this paper, the impact of amplitude detuning on the spectral distribution of the horizontal tune line and a sextupole line has been shown.Two different regimes depending on the level of the direct versus cross detuning are found both when examining the spectral width and maximum spectral amplitude.Based on these results, a correction factor to the measured RDT amplitude can be determined as well as optimizing the number of turns used in the spectral analysis.Multiparticle tracking studies will be required to benchmark these formulas and study the potential reduction of noise in measurements.

Figure 1 :
Figure 1: Spectral distribution of the tune line for different levels of detuning with amplitude.As a reference, the black line uses the same parameters in all three plots.

Figure 2 :
Figure 2: Comparison between the spectral line width of the H(−2, 0) line and the H(1, 0) line for different levels of detuning.

Figure 3 :
Figure 3: Comparison between the peak spectral amplitude of the H(-2,0) line and the H(1,0) line for different levels of amplitude detuning.