Spin Evolution and Decoherence Richard Talman Laboratory for - PowerPoint PPT Presentation

1 Spin Evolution and Decoherence Richard Talman Laboratory for Elementary-Particle Physics Cornell University Juelich, October 5, 2017

2 Outline Spin evolution, approximately horizontal orbits Spin evolution through bends Spin decoherence Spin evolution in idealized lattice Storage ring as “Penning-Like Trap” Pseudo-frozen spin magnetic ring parameters for e, p, d and He 3 Magnetic ring spin tune alteration using Siberian snakes Can pseudo-frozen spin satisfy Q s = 0 in magnetic ring? Yes. Can snakes enable EDM measurement in magnetic ring? No.

3 Spin evolution, approximately horizontal orbits (a) 3D spatial orbit (b) 2D planar (bend plane) bend plane projection e 3 ^ β ~ || ∼ actual orbit s sin α − design orbit θ ~ || ~ || ∼ α s s β 0 ^ z e 1 ∼ ϑ α ~ ^ x ^ s y y ~ ^ s y e 2 y x r E 0 r θ 0 ◮ (a) Spatial, 3D orbit and spin evolution. The spin vector s has precessed through angle ˜ α away from its nominal direction along the proton’s velocity. ◮ (b) Projection of figure (a) onto the 2D laboratory horizontal plane. x is the deviation of the (bold face) particle orbit from the (pale face) design orbit. ◮ If the bend plane coincides with the design bend plane (as is always approximately the case) ˆ β β β 0 and ˆ z are identical. ◮ θ is the reference particle global horizontal angle and ϑ is the tracked particle global horizontal angle . ◮ Betatron oscillations cause θ and ϑ to differ (slightly) on a turn by turn basis but, on the average, they are the same

4 ◮ Though the spin vector has three components, only two are independent; s 2 s 2 ˜ � + ˜ ⊥ = 1 . ◮ The angle ˜ α fixes the direction of ˜ s � in the bend plane. Since we are assuming the bend plane and design plane are almost parallel, the tildes can pretty much be ignored. ◮ The bend frame spin coordinates are     ˜ s x − ˜ s � sin ˜ α   =   . s y ˜ ˜ s ⊥ ˜ s z s � cos ˜ ˜ α ◮ The spin precesses about the ˜ y -axis in the bend plane. ◮ These coordinates are ideal for evolving the spins through the (dominant) ring horizontal bending elements.

◮ From particle tracking one knows the laboratory frame vectors r , p , 5 and angular momentum L = r × p , and one also has the spin vector s ; r = r x ˆ x + r y ˆ y + r z ˆ z , p = p x ˆ x + p y ˆ y + p z ˆ z , L = L x ˆ x + L y ˆ y + L z ˆ z , s = s x ˆ x + s y ˆ y + s z ˆ z . Any spin precession in the interior of a bend element occurs in the bend plane. To exploit this reduction from 3D to 2D it is necessary to obtain the spin components in an orthonormal frame having its “y” axis perpendicular to the plane and its “z” coordinate tangential to the orbit. In the (always excellent) paraxial approximation, this transformation is always close to identity. ◮ A right-handed basis triad has axis 3 parallel to p and axis 2 parallel to − L , where L is angular momentum and the negative sign is appropriate for clockwise orbits; e 3 = p x x + p y y + p z p ˆ p ˆ p ˆ z , e 2 = r × p − L , e 1 = e 2 × e 3 .

6 These equations can be re-expressed in terms of known coefficients e 1 = a 11 ˆ x + a 12 ˆ y + a 13 ˆ z e 2 = a 21 ˆ x + a 22 ˆ y + a 23 ˆ z e 3 = a 31 ˆ x + a 32 ˆ y + a 33 ˆ z . The vector s can be expanded as s = ˜ s 1 e 1 + ˜ s 2 e 2 + ˜ s 3 e 3 = ˜ s 1 ( a 11 ˆ x + a 12 ˆ y + a 13 ˆ z ) + . . . = ( a 11 ˜ s 1 + a 21 ˜ s 2 + a 31 ˜ s 3 ) ˆ x + . . . .

7 ◮ The final relation can be expressed in matrix form as     s x ˜ s 1  = R  ,   s y ˜ s 2 s z s 3 ˜ where R is an orthogonal matrix,   a 11 a 21 a 31  .  R = a 12 a 22 a 32 a 13 a 23 a 33 (Aside: the magnitude | det R | of the determinant of R is necessarily 1, but the actual value is ± 1. This sign correlates with the clockwise/counterclockwise orbit ambiguity.) Because R is orthogonal, R − 1 = R T and Eq. (7) can be inverted to give       ˜ s 1 a 11 a 12 a 13 s x   =     . ˜ s 2 a 21 a 22 a 23 s y ˜ s 3 a 31 a 32 a 33 s z This yields the spin components in the bend frame. Their propagation through the bend is described next.

8 Spin evolution through bends ◮ For motion restricted to a single plane (which is implicit in the present discussion) the BMT equation can be solved exactly in closed form. ◮ In this frame any precession of the spin is purely around an axis normal to the plane. Because of ultraweak vertical focusing, (e.g. in the WW-AG-CF lattice) vertical betatron oscillations are negligible for the 2D evolution through electric bend elements. Any betatron oscillations actually present are treated as exactly horizontal. ◮ The initial spin vector is s = − s � sin α ˆ x + s y ˆ y + s � cos α ˆ z . Here s y ˆ y is the out-of-plane component of s , s � is the magnitude of the in-plane projection of s , and α is the angle between the projection of s onto the plane and the tangent vector to the orbit.

9 Jackson gives the rate of change in an electric field E , of the longitudinal spin component as � g β � d β · s ) = − e 2 − 1 dt ( ˆ m p c ( s ⊥ , J · E ) . β Substituting for the spin vector produces � g β � dt ( s � cos α ) = − e d 2 − 1 m p c ( s � sin α E ) . β With the orbit confined to a plane, any precession occurs about the normal to the plane, conserving s y . Since the magnitude of s is conserved it follows that the magnitude s � is also conserved. This allows s � to be treated as constant this equation, which reduces to � g β � d α dt = eE 2 − 1 . m p c β

10 Meanwhile the velocity vector itself has precessed by angle ϑ relative to a direction fixed in the laboratory. The precession rate of ϑ is governed by the equation d ϑ dt = v r = eE p , The independent variable can be switched from t to ϑ by dividing the previous two equations � g � d α γ − g / 2 d θ = 2 − 1 γ . In this step I have also surrepticiously made the replacement ϑ → θ . Though these angles are not instantaneously the same, over long times they advance at the same average rate.

11 Integrating over θ , the bend frame precession advance is the sum of two definite integrals � g � I γ − g ∆ α = 2 − 1 2 I γ i , where � θ � θ d θ ′ γ ( θ ′ ) d θ ′ , I γ = and I γ i = γ ( θ ′ ) . 0 0 To account for fringe fields two more terms, ∆ α FF , in and ∆ α FF , out , need to be included eventually. A quicker yet equivalent formula for the precession occuring in a bend element is to use the so-called “spin tune” in an electric field (derived in an appendix), � g � � Q s ≡ d α E = G β 2 γ − 1 γ − g / 2 � also = 2 − 1 γ . � d θ γ The precession advance in a bend element is then given by ∆ α = Q s ∆ θ, where ∆ θ is the total magnet bend angle. In our EDM ring, with all orbits having very nearly the magic velocity, the incremental precession in any single bend is very small.

12 Re-derivation to include horizontal betatron displacement In estimating spin decoherence we need to account for the transverse position oscillations accompanying potential energy variation. For simplicity we assume the lattice is uniform, with no drift regions. The spin precession angle α , relative to the proton direction, evolves as � g β ( x ) � d α dt = eE ( x ) 1 − . m p c 2 β ( x ) The variables β , γ , and E in this equation are now being allowed to depend on x . Definition of angular momentum and its conservation yield d θ L dt = γ m p r 2 ; (which is valid in bend regions, but would not be in drift regions, where r becomes ambiguous). In this equation the angular momentum L is a constant of the motion (because the force is radial) but γ and r = r 0 + x depend on x . Combining the two previous equations, �� g � � d θ = eE ( x )( r 0 + x ) 2 d α γ ( x ) − g / 2 2 − 1 , Lc β ( x ) γ ( x )

◮ To find the evolution of α over long times, for an individual particle, 13 we need to average this equation. ◮ What makes this averaging difficult is the fact that the final factor has, intentionally, been arranged to cancel for the central, design particle. ◮ The initial factor, though not constant, varies over a quite small range. A promising approximation scheme for this factor is to neglect the (small) rapidly oscillating betatron contibution to x coming from the betatron oscillation, and retain only the off-momentum part x = D x ∆ γ O associated with the slowly varying synchrotron oscillation; �� g � � d α � � ( γ O ) = eE ( D x ∆ γ O )( r 0 + D x ∆ γ O ) 2 γ ( x ) − g / 2 2 − 1 . d θ Lc β ( D x ∆ γ O ) γ ( x ) ◮ No superscript “I” is needed on γ ( x ) in the final factor since only “inside” motion is under discussion. ◮ If the average of � γ � were the inverse of � 1 /γ � the averaging over horizontal betatron oscillation would be easy. But this is not true. ◮ However the above factorization has allowed the averaging over γ O to be deferred.