Spin dynamics in electron storage rings: A stochastic differential - PowerPoint PPT Presentation

Spin dynamics in electron storage rings: A stochastic differential equations approach Oleksii Beznosov, James A. Ellison, Klaus Heinemann, UNM, Albuquerque, New Mexico Desmond P. Barber, DESY, Hamburg and UNM 1 July 1, 2020 1 This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Numbers DE-SC0018008 and DE-SC0018370

Outline and motivation Outline Motivation • Spin-orbit dynamics in Lab frame We use 3 approaches • Spin-orbit dynamics in beam frame • Derbenev-Kondratenko (DK) formula for depolarization time • Reduced linear orbital dynamics and nonlinear spin • Bloch equation for polarization density • Derbenev-Kondratenko formula and Bloch • Monte-Carlo spin tracking equation (BE) We are able to base all 3 approaches on • Effective Bloch equation via averaging stochastic differential equations of Itˆ o type

Spin-orbit dynamics in Lab frame SDEs in Lab frame (Cartesian coordinates) ˙ Y = ˜ ˜ f ( t , ˜ g ( t , ˜ Y ) + ˜ Y ) ξ ( t ) , (1) ˙ S = ˜ ˜ Ω( t , ˜ Y ) ˜ + ˜ M ( t , ˜ Y ) ˜ S + ˜ G ( t , ˜ Y ) + ˜ H ( t , ˜ S Y ) ξ ( t ) (2) � �� � � �� � BMT ST effect, BK correction, kinetic polarization S ∈ R 3 and ξ is scalar white noise where ˜ Y ∈ R n , ˜ QoI: Lab-frame polarization vector �� � P ( t ) = � ˜ ˜ S ( t ) � = s ˜ ˜ p ys ( t , ˜ y , ˜ s ) d ˜ s d ˜ y ≡ η ( t , ˜ ˜ y ) d ˜ y (3) p ys = joint probability density of ˜ Y and ˜ where ˜ S and ˜ η = polarization density • Bloch equation for polarization density ˜ η discovered by Derbenev and Kondratenko (DK) (1975) • The complete form of SDE (2) obtained from DK Bloch equation via reverse engineering (2019)

Spin-orbit dynamics in beam frame SDEs in beam frame Y ′ = f ( θ, Y ) + g ( θ, Y ) ξ ( θ ) , (4) S ′ = Ω( θ, Y ) S + M ( θ, Y ) S + G ( θ, Y ) + H ( θ, Y ) ξ ( θ ) (5) � �� � � �� � BMT ST effect, BK correction, kinetic polarization where Y ∈ R n , S ∈ R 3 , where coefficients are 2 π -periodic in θ and ξ is vector white noise QoI: Beam-frame polarization vector �� � P ( θ ) = � S ( θ ) � = sp ys ( θ, y , s ) ds dy ≡ η ( θ, y ) dy (6) where η is polarization density ∝ spin angular momentum density • P ( θ ) ≈ ˜ P ( t r ( θ )), t r ( θ ) is time of reference particle at azimuth θ

Reduced spin-orbit dynamics in beam frame to study spin diffusion Reduced SDEs in beam frame Y ′ = f ( θ, Y ) + g ( θ, Y ) ξ ( θ ) , (7) S ′ = Ω( θ, Y ) S (8) where Y ∈ R n , S ∈ R 3 , coefficients are 2 π -periodic in θ and ξ is vector white noise • Quantity of interest: Beam-frame polarization vector P ( θ ) • Reduced SDEs ignore self polarization effect • Goal: Quantify decay of P ( θ ), i.e., compute depolarization time • Next: we linearize equation for the orbit (7) and linearize Ω( θ, Y ) in Y in (8).

Linearized model in beam frame Reduced orbit linearized SDEs Y ′ = [ A ( θ ) + ε 1 δ A ( θ )] Y + √ ε 1 B ( θ ) ξ ( θ ) , (9) n � S ′ = [Ω 0 ( θ ) + ε 2 Ω j ( θ ) Y j ] S (10) j =1 where Y ∈ R n , S ∈ R 3 , coefficients are 2 π -periodic in θ , B ( θ ) is diagonal matrix and ξ is vector white noise Reduced Bloch equation (Fokker Planck equation + T–BMT) n n � � + ε 1 � � � � B ( θ ) B T ( θ ) jj ∂ 2 ∂ θ η = − ∂ y j ([ A ( θ ) + ε 1 δ A ( θ )] y ) j η y j η + Ω( θ, y ) η. (11) 2 j =1 j =1 • Linearization in Y is simplest approximation which captures the main spin effects • Unlike SLIM here spin is not linearized (synchrotron sidebands are included) • Reduced linearized SDEs and the Bloch equation ∗ are key for our current research ∗ Bloch equation comes from the condensed matter physics

Gaussian beam density and equilibrium Orbit SDE in beam frame We write (9) more generally as Y ′ = A ( θ ) Y + B ( θ ) ξ ( θ ) , Y (0) = Y 0 , (12) with mean and covariance given by m ′ = A ( θ ) m , m (0) = m 0 (13) K ′ = A ( θ ) K + K A T ( θ ) + B ( θ ) B T ( θ ) , K (0) = K 0 (14) • The PSM for A is defined by Ψ ′ = A ( θ )Ψ , Ψ(0) = I n × n • Radiation damping implies Ψ( θ ) → 0 and thus m ( θ ) → 0 as θ → ∞ and � θ � � Ψ − 1 ( θ ′ ) B ( θ ′ ) B T ( θ ′ )Ψ − T ( θ ′ ) d θ ′ Ψ T ( θ ) K ( θ ) = Ψ( θ ) K 0 + 0 There exist unique K 0 such that K 0 = K (2 π ) and thus K ( θ + 2 π ) = K ( θ ) therefore we get � θ Ψ − 1 ( θ ′ ) B ( θ ′ ) B T ( θ ′ )Ψ − T ( θ ′ ) d θ ′ Ψ T ( θ ) =: K per ( θ ) K ( θ ) = Ψ( θ ) −∞ It can be shown that ( K ( θ ) − K per ( θ )) → 0 as θ → ∞ p Y ( θ, y ) ≈ p eq ( θ, y ) = (2 π ) − n / 2 det( K eq ( θ )) − 1 / 2 exp( − 1 2 y T K − 1 eq ( θ ) y ) , for large θ

Derbenev–Kondratenko formula for depolarization time Invariant spin field (ISF) Let ˆ n ( θ, y ) be the unique periodic solution of (11), with ε 1 = 0 � � n n � � ∂ θ ˆ n = − ∂ y j ([ A ( θ ) y ] j ˆ n ) + Ω 0 ( θ ) + ε 2 Ω j ( θ ) Y j n ˆ (15) j =1 j =1 • Since ε 1 small, in spirit of DK, we look for a solution of BE in the form η ( θ, y ) = c ( θ ) p eq ( θ, y )ˆ n ( θ, y ) + ∆ η ( θ, y ) (16) • Beam frame polarization vector � P ( θ ) ≈ c ( θ ) p eq ( θ, y )ˆ n ( θ, y ) dy (17) • Bloch equation for η gives ODE for c and PDE for ∆ η coupled to c c ′ ( θ ) = − ε 1 q ( θ ) c ( θ ) , (18) � � n � 2 q ( θ ) ≡ 1 ∂ ˆ n � � � B jj ( θ ) p eq ( θ, y ) ∂ y j ( θ, y ) dy (19) � � 2 � � j =1

Unsolved questions 1 How does ∆ η affect the depolarization time? 2 When is ∆ η negligible? We have a simple model where question 1 is easy to answer

Toy model Model SDEs Y ′ = [ A + ε 1 δ A ] Y + √ ε 1 B ( ξ 1 ( θ ) , ξ 2 ( θ )) T , (20) 2 � S ′ = [Ω 0 + ε 2 Ω j Y j ] S (21) j =1 where Y ∈ R 2 , S ∈ R 3 and B is diagonal matrix with ξ 1 ( θ ) , ξ 2 ( θ ) statistically independent white noise processes   � 0 � − a 0 − σ 1 0 � � − b 0 A = , δ A = , Ω 0 = σ 1 0 0   b 0 0 − a 0 0 0     0 0 σ 2 0 0 0  , Ω 1 = 0 0 0 Ω 2 = 0 0 − σ 2    − σ 2 0 0 0 σ 2 0 Goal: Compute depolarization time by integrating the BE and comparing with DK formula from previous slides

Toy model: Bloch equation • Goal: Compute depolarization time • Tool 1: Bloch equation for polarization density ∂ θ η = ε 1 a ( ∂ y 1 ( y 1 η ) + ∂ y 2 ( y 2 η )) + b ( ∂ y 1 ( y 2 η ) − ∂ y 2 ( y 1 η )) 2 � � + ε 1 � B 2 11 ∂ 2 y 1 + B 2 22 ∂ 2 η + [Ω 0 + ε 2 Ω j y j ] η (22) y 2 2 j =1 Numerical method ∗ (for long time simulations) • Spectral Chebyshev-Fourier discretization in phase spase • Embedded high order additive Runge-Kutta time evolution ∗ O. Beznosov, K. Heinemann, J.A. Ellison, D. Appel¨ o, D.P. Barber, Spin Dynamics in Modern Electron Storage Rings: Computational Aspects, Proceedings of ICAP18, Key West, October 2018.

Toy model: Derbenev-Kondratenko formula • Goal: Compute depolarization time • Tool 2: Derbenev-Kondratenko formula for depolarization time • Invariant spin field: � 1 2 )( σ 2 y 1 , σ 2 y 2 , σ 1 − b ) T n ( y ) = ˆ (23) ( σ 1 − b ) 2 + σ 2 2 ( y 2 1 + y 2 • Write polarization density η as η ( θ, y ) = c ( θ ) p eq ( y )ˆ n ( y ) + ∆ η ( θ, y ) (24) 2 � � � 2 c ′ ( θ ) = − ε 1 q c ( θ ) , q = 1 ∂ ˆ � n � � B jj p eq ( y ) ∂ y j ( y ) dy � � 2 � � j =1 � � a − a Γ 2 ( y 2 1 + y 2 p eq ( y ) = π Γ 2 exp 2 ) , B 11 = B 22 = Γ

Numerical results • Via Bloch equation � 0 . 4 P ( θ ) = η ( θ, y ) dy via Bloch equation via DK formula 0 . 35 • Via DK 0 . 3 � 0 . 25 P ( θ ) ≈ c (30) e − ε 1 q ( θ − 30) p eq ( y )ˆ n ( y ) dy Polarization 0 . 2 • Damping time is 1 /ε 1 a = 10 0 . 15 0 . 1 0 . 05 0 0 20 40 60 80 100 120 140 160 180 200 θ

Orbital dynamics: Averaging approximation - 1 Goal: Find effective Bloch equation Reduced linearized orbit & nonlinear spin SDEs Y ′ = [ A ( θ ) + ε 1 δ A ( θ )] Y + √ ε 1 B ( θ ) ξ ( θ ) , (25) n � S ′ = [Ω 0 ( θ ) + ε 2 Ω j ( θ ) Y j ] S (26) j =1 There are two different versions of averaging approximation: • (i) ε 2 = 1 (ii) ε 1 = ε 2 We are here doing (i) • Fundamental solution matrix Φ of Hamiltonian part of SDEs is defined by: Φ ′ ( θ ) = A ( θ )Φ( θ ) (27) where Φ( θ ) is quasiperiodic • Transform Y to U to get standard form for averaging: U ( θ ) = Φ − 1 ( θ ) Y ( θ ) (28)

Orbital dynamics: Averaging approximation - 2 SDEs in slowly varying form U ′ = ε 1 D ( θ ) U + √ ε 1 Φ − 1 ( θ ) B ( θ ) ξ ( θ ) , (29) n � S ′ = [Ω 0 ( θ ) + Ω j ( θ )(Φ( θ ) U ) j ] S (30) j =1 where D ( θ ) is quasiperiodic • ODE for m U and ODE for K U : m ′ U = ε 1 D ( θ ) m U U = ε 1 [ D ( θ ) K U + K U D T ( θ )] + ε 1 Φ − 1 ( θ ) B ( θ ) B T ( θ )Φ − T ( θ ) K ′