Spin Echo in Synchrotrons

1. Spin Echo in Synchrotrons Alex Chao, SLAC Spin dynamics in crossing a single resonance is well-studied. Notable well-known example is Froissart-Stora 1960. More is needed to complete the understanding, but question arises what happens if we cross a resonance twice. In two resonance crossings, two phenomena occur • The two crossings will interfere with each other • Spin echo

2. Spin motion of a single particle Consider a depolarization resonance spin tune G integer, or integer ± y, etc) Let resonance strength =  Let the spin tune of the particle be G  is deviation from resonance, and is function of time  = (number of turns ) x 

3. Spinor equation of motion Define Then, the particle’s polarization along the vertical y-direction is

4. Initially, we assume the particle is 100% polarized, and adiabatically brought to a launching position  from the resonance. The launching condition at time  is eigen-state, Launching y-polarization

5. The simplest crossing of a resonance is a sudden spin tune jump. Two crossings = double jumps.  is piecewise constant. Launching position jump1 Jump 2

6. Matrix formalism When  = constant , resonance strength , spinor state from  to  (arbitrary , ) transforms by  oscillation frequency of Py near the resonance. Not to be confused with the fast frequency which describes precession of Px, Pz. We have 

7. For the case of double resonance jumps. Apply matrix formalism. (1) Before the first jump: Just a phase rotation from launching condition => Launching Spin state Polarization = constant before the jump.

8. (2) After the first jump: => oscillatory with 1 dc term polarization reduction factor = This is the factor used in conventional calculation of polarization reduction crossing a resonance. Oscillating term is ignored!

9. (3) After the second jump: => Omitted. Contains a dc term and an oscillatory term with frequency 

10. Interference Consider the case when the jumps are from -A to A and back to -A. Polarization depends sensitively on the phase advance between 1 and 

11. A complete destructive interference occurs when

12. A complete constructive interference occurs when

13. One amusing case of constructive interference occurs when final polarization does not oscillate!

14. Note: For a single particle, interference occurs between any two resonance crossings even if they are far apart. Considering resonance crossings as independent individual events, as is done conventionally, apparently gives very wrong answer as the result should depend sensitively on the interference. The resolution lies in the fact that a beam of particles does not behave as a single particle. The beam has an energy spread . Averaged over the energy spread, polarization of the beam is:

15. Launching dc terms The launching & “dc terms” => Polarization reduced by (A2 -|0|2)/ (A2 +|0|2) each time crossing the resonance. Conventional calculation gives correct answer, i.e. it is justified after all ! --- But only if all “the other” oscillatory terms are ignored !

16. shock response to first jump echo terms shock response to second jump interference term • Examine “the other” terms: • They are all oscillatory, and all Gaussian damped by the energy spread. • Which terms play important roles depend on the relative sizes of  and the time between two jumps .

17. When =0, all terms mix together. Result is same as single particle.  Interference between two jumps (constructive example shown)  When ≠0, different terms separate. Each term acquires its own physical meaning. shock response to first jump shock response to second jump echo 

18. Shock responses to first and second jumps are interesting, yielding results beyond Froissart-Stora...  (Krisch et al, SPIN collaboration, 2006) ….. But here we emphasize effects involving interplay of the two jumps…. Interference occurs when Time it takes to smear out due to energy spread Echo occurs when Time spacing between the two jumps Njump = number of turns for the resonance jump to occur

19. Echo • Let  = time between the two jumps. Echo occurs at a time  after the second jump. • There is only one echo. Waiting longer does not give another echo. • Echo signal maximizes when  The maximum value is Py(echo, max) = (4/5)5/2 = 57%. • Echo signal oscillates with frequency .

20. Two proposed experiments COSY 2.1 GeV/c proton = 10-4 (electron cooling) = 4.4 crossing speed fc = 1.5 MHz Njump < 100 Beam initially 100% polarized, brought adiabatically to launching position  below the resonance G, jump across to , wait for time , jump back to , then measure final polarization. Adjustable parameters:  With RF dipole, the jumps are done by switching RF frequency,  controlled by RF dipole strength. The RF dipole is turned on throughout the process.

21. Echo experiment Shock to first jump Shock to second jump Echo • To dramatize the echo, may try to increase  by X1000. Echo should still appear! • Echo signal maximized by  => 57%. • Echo oscillates with . • Polarimeter needs to be gated with 0.5 ms window. Statistics is problem. With 120 up- and 120 down-cycles, may be able to obtain Py = (57 ± 10)%. V. Morozov Private comm.

22. Interference experiment • Final polarization depends sensitively on , indicating strong interference. • Don’t forget the echo signals.

23. Finally, by choosing x, the constructive interference case should yield a final polarization = constant in time (plus an echo):

24. Summary • Matrix formalism applied to the case of double resonance jumps. • Interference effects between the two jumps analyzed. • Echo effect analyzed. • Two experiments proposed to study interference and echo effects.