Feedback and Time Optimal Control for Quantum Spin Systems. Kazufumi Ito

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1 Feedback and Time Optimal Control for Quantum Spin Systems Kazufumi Ito Center for Research in Scientific Computation North Carolina State University Raleigh, North Carolina IMA, March, NCSU

2 CONTENTS: 1. Model Problems 2. Feedback Solution 3. Quantum Stochastic Control 4. Time Optimal Control 5. Semismooth Newton method 6. Numerical Tests Joint Work with Karl Kunisch, K.F. University of Garz and Qin Zhang. IMA, March, NCSU

3 1. MODEL PROBLEMS Abstract Schrödinger Control System i t Ψ(x, t) = (H + ɛ(t)µ)ψ(x, t) + γ Ψ 2 Ψ, Ψ(x, ) = Ψ (x). Let the internal Hamiltonian H is a positive closed self-adjoint operator on a Hilbert space X. In the presence of an external interaction taken as an electric field modelled by a coupling operator with amplitude ɛ(t) R and a time independent dipole moment operator µ, the new Hamiltonian H = H + m j=1 ɛ j(t)µ j gives rise to the dynamical equations to be controlled, (e.g, µ j Ψ = v j (x) Ψ) Control Problem We consider the control problem of driving the state Ψ(t) to an energy equilibrium state of Ψ of H, i.e. Ψ = e i φ ψ where H ψ = λ ψ. IMA, March, NCSU

4 Linear Schrödinger Equations: γ = t Ψ 1(x, t) = (H + ɛ(t)µ)ψ 2 (x, t) + γ Ψ 2 Ψ 2 t Ψ 2(x, t) = (H + ɛ(t)µ)ψ 1 (x, t) γ Ψ 2 Ψ 1, i.e., for Ψ(x, t) = Ψ 1 (x, t) + i Ψ 2 (x, t) and Ψ = (Ψ 1, Ψ 2 ), d Ψ = AΨ + ɛ(t)bψ dt with ( ) H A =, B = H m Multiple controls: µ(t) = ɛ j (t) µ j. j=1 ( µ µ ) MIld Solution: Ψ(t) = S(t)Ψ + t S(t s)ɛ(s)bψ(s) ds. IMA, March, NCSU

5 Orbit Tracking: V = V (Ψ(t), O(t)) = 1 2 Ψ(t) O(t) 2 X = 1 Re (O, Ψ) H i t O(t) = H O(t), e.g. O(t) = e i(λt θ) ψ, H = λψ d V (Ψ(t), O(t)) = ɛ(t) Im (O(t), µψ(t)) dt ɛ(t) = 1 α (u(t) + β sign(u(t))v γ ), u(t) = Im (O(t), µψ(t)) d dt V (Ψ(t), O(t)) = 1 α ( u(t) 2 + β u(t) V γ ). [K. Beauchard, J.M. Coron, M. Mirrahimi, and P. Rouchon] Implicit Lyapunov control of finite dimensional Schrödinger equations, preprint. [M.Mirrahimi, P. Rouchon, and G. Turinici] Lyapunov control of bilinear Schrödinger equations, Automatica, 41(25), IMA, March, NCSU

6 Optimality: V = V (Ψ, O(t)) satisfies HJB equation V (Ψ, O(t)) + min [(A Ψ + ɛbψ, V Ψ ) t ɛ + 1 α ( u(t) 2 + β u(t) V γ ) + α ɛ + β α sign(u(t))v γ 2 )] =, u(t) = Im (O(t), µψ) Asymptotic Tracking V (Ψ, O) as t Since V (Ψ(t), O(t))+ t (β u(s) 2 + u(s) V (Ψ(s), O(s))) ds = V (Ψ(), O()). V (Ψ(t), O(t)) is monotonically decreasing in t. Using the compactness of the orbit we prove the following invariance principle. IMA, March, NCSU

7 Invariance Principle Suppose T α Im (O (t), µψ (t)) 2 dt =. i d dt Ψ = H Ψ, Ψ (t) = ω Limit = A k e i( λ kt+θ k ) ψ k. k=1 where for µ k k = (ψ k, µψ k ) H and O (τ) = e i(λ k τ θ k ) ψ k, Im (O(t), µψ(t)) = Im ( A k e i((λ k λ k )τ θ k + θ k )) µ k k ). k=1 Ingham s Lemma Assume µ m µ l δ, m l for If T > 2π δ µ k = λ k λ k, k 1, µ k = (λ k λ k ) k k., there exits a constant c, depending on T and δ > such that c m I a m 2 T f(τ) 2 dτ, f(τ) = m I a m e iµ mτ. Thus, A k =, k k and θ k = θ k if µ k k. IMA, March, NCSU

8 Performance V Implicit schme with different β β= β=.3 β=1 Performance V Implicit schme with different power of V β=.1 with V β=.1 with V 1/2.1.1 Performance V Time T explicit feedback solution with different β β=.3 β=.5 β= Time T Desired and Tracked orbits Time T space β = 1/5, Ψ = sin(2π x), Ψ = e 1(x.8)2, µ i Ψ = v i (x)ψ(x) with v 1 (x) = x (x.5) 2, and v 2 (x) = x (x.5) (x.5) 3 2.5(x.5) 4. IMA, March, NCSU

9 Feedback Control of Quantum Spin Systems Observable process: X t = U t XU t, where the unitary operator satisfies the Schrodinger type quantum stochastic differential equation (QSDE) du t = ( ihdt 1 2 L Ldt + LdA t L da t )U t, U = I (1) where A t is the annihilation and A t is the creation operator and L is an atomic operator. The homodyne detection process: dy t = U t (L + L )U + t da t + da t By using Quantum Ito s rule, we get the QSDE for X(t) dx t = j t (L L,H (X))dt + j t ([L, X])dA t + j t ([X, L])dA t, where j t (X) = U t XU t and the (Lindblad) generator given by L L,H (X) = i[h, X] + L XL 1 2 (XL L + L LX). IMA, March, NCSU

10 K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhauser, Basel, 1992 Luc. Bouten, Ramon van Handel, Matthew R James: An Introduction to Quantum Filtering, arxiv:math. OC/-61741v1, 26. M. Mirrahimi, R.A.Handel, Stabilizing Feedback Controls for Quantum Systems, arxiv:math.ph/5166v2, 25. K. Tsumura, Global Stabilization of N-dimensional Quantum Spin Systems via Continuous Feedback. IMA, March, NCSU

11 Follow the quantum filtering theory, the conditional expectation of the observable X, P(X t Y t ) = π t (X) = T r(ρ t X) is given by dρ t =( i[h, ρ t ] + Lρ t L 1 2 ρ tl L 1 2 L Lρ t )dt + (Lρ t + ρ t L T r[ρ t (L + L )]ρ t )dw t where The Brownian process dw t innovation process. = dy t T r[(l + L )ρ t )]dt is the Consider a quantum spin system with fixed angular momentum J. Let H t = u t F y be the control Hamiltonian and L = F z be the coupling operator. Then the corresponding quantum filtering equation is dρ t = iu t [F y, ρ t ]dt 1 2 [F z, [F z, ρ t ]]dt + η(f z ρ t + ρ t F z 2T r[f z ρ t ]ρ t )dw t IMA, March, NCSU

12 where F y and F z are self adjoint angular moment operators along axis y and z. c 1 F y = 1 c 1 c , c 2i m = (2J + 1 m)m c 2J 1 c 2J c 2J F z = J J 1... J + 1 J IMA, March, NCSU

13 For any ɛ >, let ρ f = (1 ɛ)ρ f + ɛ 2J m f ρ Ψm where ρ f, f [ J J] is an equilibrium solution ( dρ t dt ρ t =ρ f α, β > consider the control law: = ). For u t = (α ũ t + βsign(ũ t ) V I ρ f (ρ t )), ũ t = T r(i[f y, ρ t ]ρ f ). Claim: ρ t ρ f a.s. as t. Let V I ρ f = 1 T r(ρρ f ), V II ρ f = 1 (T r(ρρ f )) 2 AV I = u t T r(i[f y, ρ t ]ρ f ) = (α ũ t 2 + β ũ t V I (ρ t )) AV II = 2T r(ρ t ρ f )(α ũ t 2 + β ũ t V I (ρ t )). + 4η(f tr(f z ρ t )) 2 tr(ρ t ρ f ) 2. IMA, March, NCSU

14 1 J=2, η=.1, ε=1 4 1 J=5, η=.1, ε= performance V.6.4 Performace V Time T Time T Time Integration: Splitting of Deterministic and Stochastic terms. Implicit Euler scheme is S-invariant. M. Mirrahimi, R.A.Handel, Stabilizing Feedback Controls for Quantum Systems, arxiv:math.ph/5166v2, 25. IMA, March, NCSU

15 Quantum Control System Spin- 1 2 system ( ) d Ψ1 = dt Ψ 2 d Ψ(t) = ih(t)ψ(t) dt with controlled Hamiltonian: H(t) = H + ( Iz I z ) ( Ψ1 Ψ 2 ) + ɛ(t) where the Pauli spin matrices are given by I x = 1 ( ) 1, I 2 1 z = 1 2 m ɛ i (t) H i. i=1 ( Ix I x ( 1 1 ), ) ( Ψ1 Ψ 2 ) Ψ() = 1 2 (1 1), 1 2 Ψ(τ) O 2 δ, O = (1 ). IMA, March, NCSU

16 Time Optimal control x = (Ψ 1, Ψ 2 ). u(t) = ɛ(t) min τ (1 + ɛ u(s) 2 ) ds subject to d dt x(t) = (A+u(t)B)x(t), u(t) γ x(τ) x 1 2 δ, x() = x X. Necessary optimality condition d dt x(t) = f(x, u), u(t) = sign ɛ(λ(t) Bx(t)). x() = x, x(τ) = x 1 d dt λ(t) = (A + ub)t λ(t) 1 + λ(τ) (Ax(1) + u(1)bx(1)) = sign ɛ (s) = 1 s ɛ s ɛ s ɛ 1 s ɛ IMA, March, NCSU

17 With coordinate change t = τ s, F (z) = for z = (x, λ, u, τ) H 1 (, 1; R n ) H 1 (, 1, R n ) L 2 (, 1, R) R: d x(t) = τ (Ax + ubx) dt u(t) = sign ɛ (λ(t) Bx) x() = x, x(1) = x 1, d dt λ(t) = τ (A + ub)t λ, 1 + λ(1) (Ax(1) + u(1)bx(1)) =. IMA, March, NCSU

18 Semismmoth Newton algorithm z k+1 = z k J 1 F (z k ) where J z + F = is written as d dt x = τ(a x + B u) + τ(ax + Bu) + f = in L2 (, 1, R n ) d dt λ = τa λ + τa λ + g = in L 2 (, 1, R n ) u + G ɛ (B λ)b λ + e = in L 2 (, 1, R n ) x() =, x(1) + 1 = in R n λ(1) (Ax(1) + Bu(1)) + λ(1) (A x + B u) + 2 = in R. with G ɛ (s) = { s ɛ s < ɛ 1 ɛ IMA, March, NCSU

19 Semismooth Function: F : D X Z is called N-differentiable, if there exists a family of mappings G : U L(X, Z) such that lim h F (x + h) F (x) G(x + h)h Z h X = for all x U. Moreover, F is semismooth at x if lim G(x + t h)h exists unifomly in h = 1. t + Theorem (Local) Suppose F is semismooth at x and F (x, h) β h for β > and all h X. and assume for a N-derivative G in a neighborhood of x G(y) 1 2β for all y N(x ). Then the Newton iterates: x k+1 = x k G(x k ) 1 F (x k ) are well-defined and converges to x N(x ). superlineraly in a neighborhood x k+1 x G(x k ) 1 (F (x k ) F (x ) G(x k )(x k x ) o( x k x ). IMA, March, NCSU

20 Solvability (Linear Case d dtx = Ax + Bu): Let ( λ(1), τ) be unknown. λ(t) = e τa (1 t) λ(1) + 1 t e τa (s t) (A λ(s) τ + g(s)) ds x(t) = t e τa(t s) (BG ɛ (b λ(s))b λ(s) + (Ax + Bu) τ + be(s) + g(s)) ds Thus, we obtain the system of linear equations for ( λ(1), τ): Φ 11 Φ 12 λ(1) + q = (2) Φ 21 Φ 22 τ where q R n+1,1 depends on F = (f, g, e, 1, 2 ) linearly. That is, if the matrix Φ R (n+1) (n+1) is invertible, then J is bounded invertible. IMA, March, NCSU

21 Let {t : B λ(t) < ɛ} = i (t i, t i+1 ) then Φ 11 = 1 e τa(1 ti) G( t i )e τa (1 t i ) ɛ i G( ) = e τas BB e τa s ds Reduced Equations and Backward Shooting method Linear Case: λ(t) is linear equation (independent of x) and u can eliminated by u = sign ɛ (B λ). Thus, x( ) is a function of (λ(1), τ) by solving the state equation with x() = x. One can define adjoint-forward shooting method: Find (λ(1), τ) that satisfies Φ(λ(1), τ) = (x(1) x 1, 1 + λ(1) (Ax(1) + Bu(1)) = IMA, March, NCSU

22 Backward Shooting Method We use the scaled transversality condition 1 + τλ(1) (Ax(1) + u(τ)bx(1)) = In this way one can eliminate τ > by this. So, we may employ the backward shooting method: find λ(1) such that Φ(p(1)) = x() = x = in which we solve the coupled system with the terminal condition (x(1), λ(1)) is given. Theorem u ɛ converges to u with minimum norm. For ɛ ˆɛ τ ɛ τˆɛ τ ɛ 1 u ɛ 2 dt τˆɛ 1 uˆɛ 2 dt τ τ ɛ τ (1 + ɛ 2 ) IMA, March, NCSU

23 Necessary Optimality With coordinate change t = τs, min J ɛ (u, τ) = τ 1 (1 + ɛ 2 u(t) 2 ) dt subject to d dt x = τ (Ax + ubx), x() = x and u U ad = {u L 2 (, 1; R m ) : u(t) γ, a.e. }. Let g 1 (u, τ) = Cx(1) y = (Finite Rank), g 2 (u, τ) = x(1) x 2 δ min J ɛ (u, τ) subject to g 1 (u, τ) =, g 2 (u, τ) and u U ad. 1 Regular Point Condition: int{g u (U ad u ɛ ) + g τ (R + τ ɛ )} (3) (ɛ u ɛ, u u ɛ ) dt + (g u (u u ɛ ), µ ɛ ), (g τ, µ ɛ ) = for all u U ad. IMA, March, NCSU

24 AdjointEquation : d dt λ ɛ(t) = (A + ub)λ ɛ (g τ, µ ɛ ) = 1 ((A + u ɛ B)x ɛ, λ ɛ ) dt. (g u (v), µ ɛ ) = τ 1 Thus, the necessary optimality is written as for all u U ad and 1 1 (λ ɛ Bx ɛ, v) ds (ɛu ɛ + λ ɛ Bx ɛ, u u ɛ ) dt. (1 + ɛ 2 u ɛ 2 + (A + u ɛ B)x ɛ, p ɛ )) dt =. IMA, March, NCSU

25 Standard Minimum Norm Problems (unconstrained): subject to x(τ) = x 1. min τ u 2 dt u (t) = B e A (τ t) µ, µ = G(τ) 1 (x 1 e Aτ x ). L Minimum Norm Problem minimize m i=1 γ2 i subject to Necessary Optimality: u i (t) γ i on [, τ], and x(τ) = x 1. u = sign ɛ (λ(t) Bx(t)), and γ i Mixed Problem Min τ γ 2 τ (λ(t) Bx(t)) i dt = IMA, March, NCSU

26 γ δ guess:τ optimal τ # of switch control form / / / / / / ,1,-1 6.1/ ,1,-1 5.1/ ,1,-1 4.1/ ,1,-1 3.1/ ,1,-1 IMA, March, NCSU

27 control time Figure 1: the Trajectory and Control with δ =.1/2.