Simulation of Spin Interference and Echo Simulation of Spin Interference and Echo EffectEffect

Abstract

Successively jumping across a depolarization resonance twice produces interesting spin dynamics effects. The dynamics involves an interplay among four parameters:

depolarization resonance strength epsilon the jump in spin tune ∆ν the time duration between the two jumps the spin tune spread of the beam.

By adjusting these parameters, the beam polarization after the second jump exhibits a wealth of effects:

constructive interference destructive interference spin echo.A simulation code is written to study these effects. Subtleties such as control of numerical noise and how to prepare the beam polarization condition at the launch are resolved. The simulation results are compared with analysis [Chao/Courant, PRST-AB, 2007]. Experiments are proposed for AGS and RHIC.

M. Bai1, A. Chao2, T. Roser1

1 Brookhaven National Laboratory, 2 SLAC, Stanford University

IntroductionIntroduction

Spin motion at the vicinity of a depolarizing resonance ε

While the spin component in the horizontal plane precesses around the vertical direction, the vertical component is given by

€

Py θ( ) =α 0

Ω

€

Ω= α02 + ε0

2

€

dψ

dθ= −

i

2

−Gγ εe ikθ

ε*e−ikθ Gγ

⎡

⎣ ⎢

⎤

⎦ ⎥ψ

Interference-overlap effect of multi resonance Interference-overlap effect of multi resonance crossingcrossing

Polarization after crossing a resonance twice depends on the spin phase advance Ω(θ-θ1) between the two crossings.

€

Py θ( ) ≈A

Ω5[ A2 − ε0

2

( )2

+ 4A2 ε0

2cosΩ(θ1 −θ2) +

2A2 ε0

2[cosΩ(θ −θ2) − cosΩ(θ +θ1 − 2θ2)]

For single pt, the vertical component of final polarization is

Interference-overlap effect of multi resonance Interference-overlap effect of multi resonance crossingcrossing

Destructive case

€

Ω θ2 −θ1( ) = 2kπ

€

Ω θ2 −θ1( ) = kπ +π

2

Constructive case

θ1 θ2

• multiple particles with Gaussian distributed momentum spread• σE/E=0.0002 • ε0=0.005, A=0.01, jump size:0.02

Spin Echo EffectSpin Echo Effect

Shock response from the 1st resonance jump. It precess with frequency Ω

A sudden spin signal occurs at θ2 −θ1 after the second resonance jump

For a beam of polarized particles with a momentum spread of σδ in Gaussian distribution

Spin Echo EffectSpin Echo Effect

Condition:

interference terms are insigficant:

Jump length:

The magnitude of echo signal

€

Py,echo =2Aε0

4

Ω5

A sudden spin signal occurs at θ2 −θ1 after the second resonance jump

€

1

κ≤ N jump ≤

2Ω

2π Aκσ δ€

θ2 −θ1 ≥2Ω

Aκσ δ

€

when A =ε0

2, Py,echo,max = 57%

Spin jump is within resonance width

Numerical Simulation of Spin Echo Numerical Simulation of Spin Echo EffectEffect

• beam energy Gγ=4.6 •Single resonance is induced by an rotating field with a constant spin kick around an axis which rotates at a fixed frequency• a total of 1201 particles with an rms energy spread

•resonance is adiabatically turned on to a strength 0.002 at a fixed tune of 0.599. The jump size is 0.002

€

σE

E= 0.000022

Simulation resultSimulation result

40,000 turns between two jumps

Smearing of Spin Echo EffectSmearing of Spin Echo Effect

The echo signal can be smeared out due to high order dependence of omega on the momentum spread, i.e.

€

Ω= A + κδ( )2

+ ε0

2≈

A2 + ε0

2+

Aκδ

A2 + ε0

2+

ε0

2κδ( )

2

2(A2 + ε0

2)

3

2

€

1

θ2 −θ1

<ε0

2κδ( )

2

2 A2 + ε0

2

( )

3

2

The echo signal is significant damped if

Simulation of Spin Echo SmearingSimulation of Spin Echo Smearing

300,000 turns

Observe echo at AGSObserve echo at AGS

Beam energy: Gγ=4.6

AGS An ac dipole at frequency ~100kHz and 20 Gauss-m ε0 = 0.0001

This requires: A=0.00005 Momentum spread: σδ~ 4x10-6

This is very difficult to obtain In addition, the oscillating field of single ac dipole

drives two resonances instead of one. This introduces additional motions

Oscillating field vs. Rotating fieldOscillating field vs. Rotating field

A single ac dipole was adiabatically ramped to 333Gauss-m with a fixed tune at 0.01 away from spin tune at Ggamma=4.6

Two ac dipoles with a 90 degrees spin rotator in the middle to provide a rotating field. Each ac dipole was adiabatically ramped to 333 Gauss-m

Possible experiments at RHICPossible experiments at RHIC

RHIC spin flipper provides a rotating field with a maximum strength of ε0 = 0.00092

Small spin tune spread because the two full snakes which removes the spin tune dependence on the beam energy

Source of Spin tune spread

difference of horizontal divergence of the beam at the two snakes. This causes a spread of horizontal angle between the two snakes. The spin tune spread is given by

€

Gγ

Δ1+α 2

βε

π

Beam emittance

Twiss parameter at snakes

Simulation of Spin Echo in the presence of full Simulation of Spin Echo in the presence of full snakessnakes

• two snakes apart by 180 degrees

• spin tune spread is achieved through the spread of the angle between the two snake axis •1201 particles. ε0= 0.00092, A=0.00046

• spin tune spread 4d-5

•Tune jump from 0.49956 to 0.50048 in 200

Simulation of Spin Echo in the presence of full Simulation of Spin Echo in the presence of full snakessnakes

70,000 turns

ConclusionConclusion

The interference-overlap effect when a single resonance is jump crossed twice was studied through numerical simulation. The single resonance excited by a rotating field which consists of two ac dipoles and a 90 degree spin rotator in between.

Simulation of using the same rotating field in the presence of two full Siberian snakes also shows an spin echo effect can be generate if the condition is met. This suggests that it may be possible to observe the spin echo effect at RHIC injection

Simulation of using single ac dipole was also done and shows that the extra term excited by the oscillating field introduces extra noises to the spin signal

Work to be done in the future:

Detailed simulation using RHIC lattice

Design and propose experiment at RHIC