Mitglied der HelmholtzGemeinschaft Yu Senichev Spin Decoherence in

Mitglied der Helmholtz-Gemeinschaft Yu. Senichev Spin Decoherence in Multipole Fields 01 November 2020

“Tomas-Bargmann, Michel, Telegdi” equation with EDM term Source of decoherence 01 November 2020 EDM signal Folie 2

The EDM search methods in Storage Ring: 1. “Magic” method with initial spin orientation in ring S║p; S┴E; S={0, 0, Sz} and E={Ex, 0, 0} 2. Resonant method with initial spin orientation in ring S║B; S={0, Sy , 0} and B={0, By, 0} 01 November 2020 Folie 3

“Magic” method in purely electrostatic ring In purely electrostatic ring the spin of particle with “magic energy” rotates with the same angular frequency as the momentum and it tilts up in the YZ plane due to the EDM with angular rate 01 November 2020 Folie 4

In resonant method* the spin frequency is parameterized : using RF flipper. In case of parametric resonance when we shall observe the resonant build up: 01 November 2020 *A. Lehrach, B. Lorentz, W. Morse, N. Nikolaev and F. Rathmann Folie 5

Spin decoherence In purely magnetostatic ring w/o RF the spin decoherence is In magnetostatic ring with RF electric or magnetic field In purely electrostatic ring In all options the energy spread is main source of spin decoherence: 01 November 2020 Folie 6

Spin decoherence in electrostatic ring RF cavity was first obvious step to increase SCT: , where is longitudinal tune. In case of the pure electrostatic ring the spin component leading to the decoherence is: Since spin tune the spin vibrates within a very narrow angle Φmax with frequency. The value Φmax depends on the frequency ratio and Φmax ~10 -6 01 November 2020 Folie 7

RF on: Second order approach of spin tune versus Δp/p In the second approach versus momentum the average tune spin is not zero At (Δp/p)max=10 -4 and an axial particle the number of turns for SCT is ~6 107 turns, that is ~180 sec. The code COSY infinity simulation 01 November 2020 Folie 8

RF on/off at Δp/p=10 -4 RF off: spin coherence time ~1 msec RF on: spin coherence time ~100 -200 sec To achieve a longer spin coherence time we have to use either the sextuple field of deflector or special independent sextupoles. 01 November 2020 Folie 9

Spin decoherence in magnetostatic ring w/o RF In purely magnetostatic ring w/o RF the spin decoherence is In this ring we have no nonlinear term and we should expect fully compensated decoherence of spin tune. However, we observe in COSY ring the strong decoherence with RF cavity as well. It turned out the decoherence can be due to the displacement of the average energy level 01 November 2020 Folie 10

Orbit lengthening effects Momentum deviation is described by eq: Thus due to the betatron oscillation, the square term of momentum compaction factor and the slip factor by the value : 01 November 2020 dependent on the equilibrium level energy is shifted Folie 11

Orbit lengthening due to MCF second order and spin decoherence If the equilibrium energy for depends on the particle parameters the spin tune spread turns has incoherent spread It reduces the spin coherence time SCT. For example, let us consider the case with the spin coherence time (SCT) limited by 1000 seconds (~ turns) and : For the momentum deviation we have: At exceed we get , 01 November 2020 , and the rms momentum spread should not the value , and reducing the second order of MCF up to. Folie 12

Orbit lengthening due to betatron motion and spin decoherence Now let us estimate the restriction for the emittance value: At should be , , mm mrad , and both emittances From these estimations we can conclude that the contribution to the spin tune decoherence is the same for the rms values of emittance and momentum spread. 01 November 2020 Folie 13

Orbit lengthening in multipole field Let us suppose that we have multipoles in the ring: Following Courant-Snyder formalism with the new variables we can reduce the equation to the form (with y=0 just for simpler explanation): 01 November 2020 Folie 14

Tune shift and orbit lengthening in multipole field Using the Landau method of successive approximations: quadrupole 01 November 2020 sextupole octupole decapole Folie 15

Tune shift and orbit lengthening in sextupole field Orbit lengthening due to sextupoles placed in non-zero dispersion: and zero-dispersion Now: 01 November 2020 Folie 16

Sextupole Families Required number of sextupole families to compensate x, y, ∆p/p deviation and both chromaticities is 5! In the electrostatic ring we need additionally to compensate the nonlinear dependence of the spin tune versus energy. 01 November 2020 Folie 17

Spin decoherence in magnetostatic ring with RFB solenoid field In magnetostatic ring with RF solenoid - RF frequency in cavity - phase deviation from synchronous particle in cavity - new t, z-coordinate 01 November 2020 Folie 18

Resonance with RFB solenoid field For easier understanding let us consider the equation for horizontal component by reducing the equation to the canonical form: At resonance 01 November 2020 Folie 19

Spin decoherence in magnetostatic ring with RFB field Spin decoherence is defined by 1. 2. 01 November 2020 Folie 20

Conclusion: -in both magnetostatic and electrostatic rings we have similar dependence spin tune spread versus momentum and axis deviation; -in both rings the spin decoherence can be compensated by sextupoles; -in both ring we need five families of sextupoles; -in both rings we have residual uncompensated decoherence of spin tune; 01 November 2020 Folie 21