Change of beam distribution due to decoherence in the presence of transverse feedback

The effect of Landau damping is often calculated based on a Gaussian beam distribution in all degrees of freedom. The stability of the beam is however strongly dependent on the details of the distribution. The present study focuses on the change of bunch distributions caused by the decoherence of the excitation driven by an external source of noise, in the presence of both amplitude detuning and a transverse feedback. Both multiparticle tracking simulations and theoretical models show a similar change of the distribution. The possible loss of Landau damping driven by this change is discussed.


Introduction
In synchrotrons, the beam is kept stable partially by Landau damping due to the tune spread within each bunch. The stability diagram in plane j ∈ {x, y} is calculated from [1] 1 where ∆Q coh,j , J j and Q j are the coherent tune shift, action and tune, respectively, in plane j, Q ∈ (−∞, ∞), and Ψ is the distribution. The stability can be changed significantly by a small change of the distribution [2,3,4,5]. In a recent experiment in the LHC, Landau damping was lost due to a noise driven diffusion [6]. Here we will introduce an analytical theory that explains how the distribution changes after an initial offset, due to the combined effects of a tune spread and a transverse feedback. The goal is to find how the distribution changes, and how the stability diagram evolves as a result.

Theory
The calculation consists of 4 steps: (i) Derive an expression for the change of the action for each particle after a kick, taking into account the balance between the tune spread and the transverse feedback; (ii) Consider the change of action as a Wiener process with a drift, and derive the Fokker-Planck equation for the particle density distribution of the bunch [7]; (iii) Solve the Fokker-Planck equation to get the time evolution of the distribution; (iv) Calculate numerically the stability diagram with PySSD [8], as the distribution evolves. This approach has the advantage that it is modular, each step can be modified if necessary. Furthermore, the

Transverse Feedback and Decoherence
We apply normalized, canonical coordinates [9] x = 1 where X is the offset from the design orbit, s is the position in the beamline, α and β are Twiss parameters, ε 0 is the initial beam emittance and φ is the canonical conjugate of J.
If a bunch is kicked by ∆p = k, the action changes to where J 0 and φ 0 are the action and phase of the particle prior to the kick. There exists an expression for the subsequent emittance growth, when there is both a transverse tune spread and a transverse feedback, and we will take a similar approach [10]. We refer to the centroid of the bunch as z = x + i p , where the angle brackets signify the average over the distribution. The tune of the centroid is Q c , and its transverse offset will each turn be reduced by a factor g, called the gain. Assuming a perfect, immediate feedback, the initial centroid offset z 0 = ik will after n turns be with a damping time of τ = 2/g turns. It is assumed that the reduction of the centroid amplitude due to the tune spread is negligible compared to that of the transverse feedback. The position of an individual particle, with a constant tune of Q c + ∆Q, is referred to as y = x + ip. After many turns, when the centroid tends to the origin in the limit ng 1, the position will become where r 0 = x 0 + ip 0 is the position prior to the kick, and y 0 = r 0 + z 0 is the position just after the kick z 0 . The change of the action in the limit ∆Q 1, ng 1 is thus where L, M and N are factors that depend on ∆Q and g. The first term of Eq. (6) is an average growth equal to the result in [10], while the second term is a spread based on the phase of the particle. Equation (6) simplifies to Eq. (3) minus J 0 in the limit g ∆Q, and to 0 in the limit g ∆Q.

Fokker-Planck Equation in Action
When one kick k becomes a coherent white noise source, the change of action in Eq. (6) can be considered a stochastic process, described by the Fokker-Planck equation [7] with drift and diffusion coefficients where τ is the time interval between each kick, and ∆ is the change of action. The normalization of D by J is convenient in the following. This has been derived by Taylor expanding the Master equation [11,12]. The details could not fit here. The probability distribution for the change of action after a kick, derived from Eq. (6), can be written as where F (k) is the probability distribution of the kicks, with a standard deviation, σ k , and an assumed mean of zero. The coefficients U and D are thus In the limit ∆Q 1 one finds that M 2 + N 2 = L 2 . The Fokker-Planck Equation takes the form In the derivation of Eq. (11), the tune offset ∆Q was assumed constant for each particle individually. In general ∆Q depends on J, which is not constant. From considering the actual process, we postulate a time reversal symmetry at the microscopic level, that the probability of going from J a to J b is equal to the probability of going back, or ϕ( By doing a Taylor expansion of ϕ as in [12], assuming small kicks k, the drift coefficient changes to The second term cancels a term in Eq. (11), which becomes the standard diffusion equation For later reference and discussion, we combine Eq. (11) and Eq. (13), by use of a parameter α ∈ {0, 1}, as 10th

Solving the Fokker-Planck Equation
The next step is to solve Eq. (14). If g = 0, this is the diffusion equation with a constant diffusivity D 0 = σ 2 k /2τ . In another extreme limit, g ∆Q and ∆Q → 0, ∂ t Ψ = 0, and the distribution will not change.
In the interesting regime, when there is a balance between the feedback and the detuning, we require a numerical solver. An original code has been written, implemented with the finitevolume-method to ensure mass conservation, and using scipy.integrate.solve ivp to achieve the time integration [13]. The boundary at J = 0 has to be reflective. In the results that will be presented, the boundary at J Max = 24.5 is absorbing, representing an aperture. The centroid tune, Q c , is kept constant.

Decoherence from Landau Octupoles
The Landau octupoles in the LHC cause a tune spread in both transverse planes, relative to the average, given by ∆Q x,y = a x,y ·(J x,y − J x,y ) + b y,x ·(J y,x − J y,x ), a x,y = 520 · I oct · ε x,y,0 , where a j and b j are detuning coefficients dependent on the octupole current, I oct , and geometrical emittance [14]. In a simplified model when b = 0, L 2 takes the shape in Fig. 1.

Results
We will repeatedly study a toy configuration with a = 5 × 10 −3 , b = 0 and g = 0.2. All values are in the horizontal plane. The subscript x has been omitted, since there is no dependence on vertical phase space. A macroparticle simulation has been run. Simulations of this process require > 10 6 macroparticles to reduce the numerical stochastic cooling, a small ratio σ 2 k /g to keep the centroid amplitude low, and > 10 6 turns for the distribution to change. The distribution is plotted as a function of r = √ 2J in Fig. 2a after T turns such that σ 2 k ·T = [0, 1, ..., 8]·25/6 turns. The time is scaled to hours of operation of the LHC, with a noise of σ k = 5.77 × 10 −4 , comparable to the noise in a recent experiment in the LHC [6].
A stochastic process with kick strength solely dependent on the parameters before the kick, is modelled by Eq. (14) with α = 0. A simulation was run with a centered incoherent noise of   14) is not representing the beam dynamics well.
The distribution evolution calculated with α = 1 is presented in Fig. 2c. This evolution is in comparison quite close to the macroparticle simulation. An edge develops at r = √ 2 ≈ 1.4, where ∆Q = 0. The evolution of multiple edges in the simulation can be a numerical artefact. There is also a small nonzero diffusion across the edge due to the centroid oscillation, which is not included in the new theory.
The evolution of the stability diagram, corresponding to the distribution evolution in Fig. 2c, has been calculated with PySSD, and is presented in Fig. 3. The tune spread is calculated with Eq. (15), using I oct = 400 A and a normalized emittance of 2 µm at 6.5 TeV. Both diagrams show an increased stability at large real coherent tune shifts, due to the population of the tails at J x > 5. The horizontal stability diagram eventually cuts into the stability diagram calculated with half the octupole current for the initial distribution, due to the increased gradient and curvature of Ψ at J x ∼ 1.

CONCLUSION
We have shown that a coherent white noise, combined with a transverse feedback and an amplitude dependent detuning, causes an amplitude dependent diffusion, which changes the distribution. The Fokker-Planck equation has here been used to model this process on long time scales. Macroparticle simulations can also be used, and have been run to compare to the new theory, but require high numbers of macroparticles and turns to study the relevant cases.
With detuning due to Landau octupoles, a Gaussian distribution evolves towards a rectangular distribution. Simultaneously, the stability diagram changes. The example configuration studied in this paper show that this can allow instabilities to evolve at more than twice the predicted required octupole current, depending on the mode, over time scales of hours. The stability also increased for large real coherent tune shifts. To study the impact in the LHC, the model will be extended to include wakefields and tune dependence on the vertical action.