Coherent beam-beam effects X. Buffat Content Coherent vs. - PowerPoint PPT Presentation

Coherent modes of oscillation Vlasov perturbation theory (5) Rigid bunch model : Each beam centroid position and momentum x 1 ,x' 1 and x 2 ,x' 2 Equation of motion :  ( x i ' ) t + 1 = M lattice ⋅ M BB ( x i ' ) t x i x i Non Linear beam-beam map :  2 −Δ x Δ x' coh =− 2 r 0 N 2 ) 1 4 σ Δ x ( 1 − e γ r Linearized kick :  Δ x ' coh = 4 πξ Δ x 2 Write one turn matrix and find  eigenvalues / eigenvectors

Coherent modes of oscillation Vlasov perturbation theory (5) Rigid bunch model : Vlasov perturbation theory : Each beam phase space distribution Each beam centroid position and momentum x 1 ,x' 1 and x 2 ,x' 2 ( 1 ) , F ( 2 ) F Liouville's thorem : Equation of motion :   { ( 1 ) ∂ F ( 1 ) , H ( F ( 2 ) )]= 0 ( x i ' ) t + 1 = M lattice ⋅ M BB ( x i ' ) t ∂ t +[ F x i x i ( 2 ) ∂ F ( 2 ) , H ( F ( 1 ) )]= 0 ∂ t +[ F Non Linear beam-beam map : Hamiltonian   (lattice + beam-beam) 2 −Δ x Δ x' coh =− 2 r 0 N 2 ) 1 4 σ Δ x ( 1 − e ( 1 ) , Ψ ( 1 ) , F ( 2 ) ,t ) H ( J γ r First order perturbation Linearized kick :   ( i ) = F 0 + F 1 ( i ) Δ x ' coh = 4 πξ F Δ x 2 Formulate the linearized system  Write one turn matrix and find  as a linear operator → find eigenvalues / eigenvectors eigenvalues / eigenfunctions

Coherent mode spectrum Q σ = Q Q π = Q - ξ Rigid bunch :

Coherent mode spectrum Self-consistent Q σ = Q Q π = Q - Y ξ Model :

Coherent mode spectrum Self-consistent Q σ = Q Q π = Q - Y ξ Model :  The Yokoya factor Y is usually between 1.0 and 1.3 depending on the type of interaction (Flat, round, asymmetric, long-range, …) (5)

Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum

Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum

Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum  Both the σ and π mode are outside the incoherent spectrum

Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum  Both the σ and π mode are outside the incoherent spectrum → Absence of Landau damping !

Observations (6) LEP TRISTAN PETRA

Observations (6) LEP TRISTAN PETRA LHC RHIC

Observations (6) LEP TRISTAN PETRA LHC Perfect agreement RHIC  with fully self- consistent models

Observations (6) LEP TRISTAN PETRA LHC Perfect agreement RHIC  with fully self- consistent models  SPPS ?  Tevatron ?

Multiparticle tracking (see K. Li's lectures)  Model the beam distribution with a discrete set of macro- particles  Track the particles, solving for each beam's fields at each interaction  Non-linear beam-beam map  Gaussian fit : soft-Gaussian approximation 2 − x i ( x i ' ) t + 1 = M lattice ⋅ M BB ( x i ' ) t Δ x' i =− 2 r 0 N 1 x i x i 2 ) 2 σ ( 1 − e γ r x i Numerical Poisson solver 

Beam-beam coherent mode spectrum Soft-Gaussian approximation  The soft-Gaussian approximation underestimate the Yokoya factor

Beam-beam coherent mode spectrum Soft-Gaussian approximation  The soft-Gaussian approximation underestimate the Yokoya factor → Need to fully resolve the particles distribution

Beam-beam coherent mode spectrum Self-consistent field solver Soft-Gaussian approximation  The soft-Gaussian approximation underestimate the Yokoya factor → Need to fully resolve the particles distribution

Decoherence : weak-strong  Multiparticle tracking simulation, with a single beam and a fixed beam-beam interaction → weak-strong regime :  Start the simulation with a beam offset with respect to the closed orbit and let it decohere

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong

Decoherence : weak-strong The amplitude detuning due to beam-beam interaction leads to  decoherence identically to other lattice non-linearities d ϵ 1 2 dt = Δ → ϵ 0 2 Decoherence time ~1/ξ 

Decoherence : strong-strong  Similar setup but :  Two independent beams  Non-linear beam-beam map based on the charge distribution  Start the simulation with both beams offset in the same direction with respect to the closed orbit  Let the mode decohere ?

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence : σ-mode

Decoherence of the σ mode

Decoherence of the σ mode

Decoherence of the σ mode

Decoherence of the σ mode

Decoherence of the σ mode  The single particle motion is the linear composition of the centroid position and the position with respect to the centroid position → The single particle motion does not change the coherent force

Decoherence of the σ mode  The single particle motion is the linear composition of the centroid position and the position with respect to the centroid position → The single particle motion does not change the coherent force  The incoherent and coherent motion are decoupled → Absence of decoherence

Decoherence : strong-strong  Identical setup :  Two independent beams  Non-linear beam-beam map based on the charge distribution  Start the simulation with both beams offset in opposite directions with respect to the closed orbit  Let the mode decohere ?

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode

Decoherence of the π mode  Due to the particles frequency spread, the beam distribution is distorted (i.e. non-Gaussian)  The bunch centroids remain out of phase → The coherent force is (almost) unaffected  This is a consequence of the decoupling of the incoherent and coherent motion, as they have different frequencies

Decoherence of the π mode