Weakly-coupled bilinear quantum systems Thomas Chambrion Nabile - - PowerPoint PPT Presentation

Bilinear quantum systems Averaging techniques Examples Conclusion

Weakly-coupled bilinear quantum systems

Thomas Chambrion Nabile Boussaïd (Besançon) and Marco Caponigro (Rutgers) PICOF 2012. April 2–4, 2012

Bilinear quantum systems Averaging techniques Examples Conclusion

Objective of the talk

Most of the quantum systems encountered in practice are governed by PDEs i∂ψ ∂t (x, t) = (−∆ + V (x))ψ(x, t) + u(t)W (x)ψ(x, t) We will try to understand how the properties (controllability) of these infinite dimensional systems can be deduced from the properties of their finite dimensional approximations. In what follows, we neglect decoherence.

Bilinear quantum systems Averaging techniques Examples Conclusion

Outline of the talk

1

Bilinear quantum systems Bilinear Schrödinger equation Good Galerkin approximation

2

Averaging techniques Rotating wave approximation

3

Examples Quantum harmonic oscillator Rotation of a planar molecule

4

Conclusion Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion

Outline of the talk

1

Bilinear quantum systems Bilinear Schrödinger equation Good Galerkin approximation

2

Averaging techniques Rotating wave approximation

3

Examples Quantum harmonic oscillator Rotation of a planar molecule

4

Conclusion Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion

Some examples

A quantum system evolving in Ω, a finite dimensional Riemannian manifold, is described by its wave function ψ in the unit sphere of L2(Ω, C). The system is in the subset ω with probability

• ω |ψ|2dµ. The

time evolution is given by the Schrödinger equation i∂ψ ∂t (x, t) = (−∆ + V (x))ψ(x, t) When submitted to an external field (e.g., a laser) with time variable intensity, ψ satisfies i∂ψ ∂t = (−∆ + V (x))ψ(x, t) + u(t)W (x)ψ(x, t)

Bilinear quantum systems Averaging techniques Examples Conclusion

Some examples

Rotation of a planar molecule Ω = SO(2) ≃ R/2πZ i∂ψ ∂t ψ(θ, t) = −∂θθψ(θ, t) + u(t) cos θψ(θ, t) (Dion, Salomon, Turinici, Sugny,...) Rotation of a molecule in space Ω = S2 i∂ψ ∂t ψ(θ, ν, t) = −∆ψ(θ, ν, t) + u(t) cos θψ(θ, ν, t)

Bilinear quantum systems Averaging techniques Examples Conclusion

Some examples

Harmonic oscillator Ω = R i∂ψ ∂t ψ(x, t) = (−∂xx + x2)ψ(x, t) + u(t)xψ(x, t) (Mirrahimi, Rouchon, Illner,...) Infinite square potential well Ω = (0, π) i∂ψ ∂t ψ(x, t) = ∂xxψ(x, t) + u(t)xψ(x, t) (Beauchard, Coron, Laurent, Nersessyan,...)

Bilinear quantum systems Averaging techniques Examples Conclusion

Abstract form

In the Hilbert space H(= L2(Ω, C)), we consider an unbounded skew-adjoint linear operator A(= −i(∆ + V )), a skew symmetric operator B(= −iW (x)) and the evolution equation dψ dt = (A + u(t)B)ψ For more complicated models, see Morancey (2011).

Bilinear quantum systems Averaging techniques Examples Conclusion

Well-posedness

• dψ

dt = (A + u(t)B)ψ

ψ(0) = ψ0 Well-posedness is very far from obvious when A or B is unbounded (i.e., not continuous), what is the case here. In the presented examples, for every locally integrable u : R → R, we can define the solution t → Υu

t (ψ0). If ψ0 belongs to D(A), then Υu(ψ0) is

absolutely continuous and d dt Υu

t (ψ0) = (A + u(t)B)Υu t (ψ0)

for a.e.t

Bilinear quantum systems Averaging techniques Examples Conclusion

Discrete spectrum

In the presented examples, A has discrete spectrum. There exists a non-decreasing sequence (λn)n∈N in [0, +∞) and an Hilbert basis (ψn)n∈N of H such that Aψn = −iλnφn for every n. Infinite dimensional matrices representation A =        −iλ1 · · · · · · −iλ2 ... . . . ... −iλ3 ... . . . ... ...        bj,k = φj, Bφk, bjk = −bkj

Bilinear quantum systems Averaging techniques Examples Conclusion

Weakly coupled quantum systems

Definition (Weakly coupled system) (A, B) is k-weakly-coupled if A is skew-adjoint with discrete spectrum (−iλn)n∈N, (λn)n∈N is positive, non decreasing and tends to infinity; B is skew-adjoint; for every u in R, D(|A + uB|k/2) = D(|A|k/2); there exists CA,B such that, for every ψ in D(|A|k), |ℜ|A|kψ, Bψ| ≤ CA,B||A|kψ, ψ|

Bilinear quantum systems Averaging techniques Examples Conclusion

Growth of |A|k/2 norms

ψ2

k/2 = |A|k/2ψ2 = ||A|kψ, ψ| =

• n∈N

λk

n|φn, ψ|

The 1/2 norm is the expected value of the energy at ψ. | d dt |A|kψ, ψ| = |2u(t)ℜ|A|kψ, Bψ| ≤ 2|u(t)|CA,B||A|kψ, ψ| By Gronwall’s lemma: ||A|kψ(t), ψ(t)| ≤ e2CA,B

t

0 |u(τ)|dτ||A|kψ(0), ψ(0)|

Bilinear quantum systems Averaging techniques Examples Conclusion

Size of velocity tail

Define πN : H → H, the orthogonal projection on the first N eigenstates

• f A.

(Id − πN)ψ(t)2

r

≤

• n≥N

λr

N|φn, (Id − πN)ψ(t), |2

≤ 1 λk−r

N

• n≥N

λk

n|φn, (Id − πN)ψ(t), |2

≤ 1 λk−r

N

||A|k(Id − πN)ψ(t), (Id − πN)ψ(t)| ≤ e2CA,BuL1 ||A|kψ(0), ψ(0)| λk−r

N

→ 0

Bilinear quantum systems Averaging techniques Examples Conclusion

Good Galerkin approximation

πNψ′(t) = A(N)πNψ(t) + u(t)πNBπNψ(t) + u(t)πNB(1 − πN)ψ(t) Denoting with X (N)

u

(t) the propagator of the N-dimensional system x′ = (A(N) + u(t)B(N))x, πNψ(t) = X (N)

u

(t)πNψ(0) + t X (N)

u

(t, s)u(τ)πNB(1 − πN)ψ(τ)dτ Proposition (Boussaid-Caponigro-Chambrion, 2011) Let (A, B) be k- weakly-coupled and B be bounded relatively to|A|r for r < k. For every ǫ > 0, for every K > 0, for every ψ0, for every s < k, there exists N = N(ǫ, K, ψ0) such that uL1 ≤ K = ⇒ πNΥu

t (ψ0) − X (N) u

(t)πNψ0s < ǫ.

Bilinear quantum systems Averaging techniques Examples Conclusion

Outline of the talk

1

Bilinear quantum systems Bilinear Schrödinger equation Good Galerkin approximation

2

Averaging techniques Rotating wave approximation

3

Examples Quantum harmonic oscillator Rotation of a planar molecule

4

Conclusion Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion

Non degenerate transitions

Definition A transition (j, k), j = k, is non degenerate if bj,k = 0 and, for every l1, l2, |λl1 − λl2| = |λj − λk| = ⇒ {l1, l2} = {j, k} or bl1,l2 = 0.

Bilinear quantum systems Averaging techniques Examples Conclusion

Periodic control laws

Proposition Let (j, k) be a non-degenerate transition of (A, B) and u∗ be a

2π |λj −λk|-periodic function. If

• 2π

|λj −λk |

u∗(τ)ei|λj −λk|τdτ = 0 and

• 2π

|λj −λk |

u∗(τ)ei|λl −λm|τdτ = 0 for every l, m such that |λl − λm| ∈ (N \ {1})|λj − λk|, then there exists T ∗ such that

• φk, X u∗/n

nT ∗ φj

• n→∞

− → 1. T ∗ = πT 2|bj,k|

• T

0 u∗(τ)ei(λj −λk)τdτ

• .

Bilinear quantum systems Averaging techniques Examples Conclusion

Some estimates

L1 norm needed to achieve the transition from level j to k π 2|bjk| 1 Effjk(u∗) with 0 ≤ Effjk(u∗) =

• 2π

|λj −λk |

u∗(τ)ei(λj −λk)τdτ

• 2π

|λj −λk |

|u∗(τ)|dτ ≤ 1. Error estimates (for bounded B) 1 − |φk, X u∗/n

nT ∗ φj| ≤

C(u∗, B) n infl1,l2

• |λl1−λl2|

|λj −λk| − 1

• Error × Time ≤ Const

Bilinear quantum systems Averaging techniques Examples Conclusion

Outline of the talk

1

Bilinear quantum systems Bilinear Schrödinger equation Good Galerkin approximation

2

Averaging techniques Rotating wave approximation

3

Examples Quantum harmonic oscillator Rotation of a planar molecule

4

Conclusion Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion

Quantum harmonic oscillator: Good Galerkin approximation

i∂ψ ∂t ψ(x, t) = (−∂xx + x2)ψ(x, t) + u(t)xψ(x, t) x ∈ R A = −i diag(1/2, 3/2, 5/2, . . .) B = −i       √ 1 . . . √ 1 √ 2 ... √ 2 √ 3 . . . ... √ 3 ...       B is not bounded. However, B is bounded relatively to A. The system is 2-weakly coupled and admits a sequence of Good Galerkin

• approximations. For ψ0 = φ1, ε = 10−4 and K = 3, one finds N = 420.

Bilinear quantum systems Averaging techniques Examples Conclusion

Controllability of the infinite dimensional system?

The quantum harmonic oscillator is not controllable, in any sense (Mirrahimi, Rouchon, Illner,..). But all of its Galerkin approximations are! “Proof” of the approximate controllability of the infinite dimensional system:

1 Find a sequence of Galerkin approximations that are controllable. 2 Prove that these Galerkin approximations are controllable with a

uniformly bounded L1-norm.

3 Use the Good Galerkin Approximation property.

The second step is impossible for the harmonic oscillator.

Bilinear quantum systems Averaging techniques Examples Conclusion

The planar molecule

Let us consider a 2D-planar molecule submitted to a laser i ∂ψ ∂t (θ, t) = −1 2∂2

θψ(θ, t) + u(t) cos(θ)ψ(θ, t)

θ ∈ Ω = R/2π The parity of ψ cannot change (global controllability is hopeless). We restrict ourselves to the odd subspace. An Hilbert basis of L2

• dd(Ω, C) made of eigenvectors of ∆ is

(sin(kθ))k∈N. We try to steer the system from the first odd eigenstate to the second odd eigenstate

Bilinear quantum systems Averaging techniques Examples Conclusion

Good Galerkin approximation

A = −i diag(12, 22, 32, . . . , N2) B = −i          1/2 · · · 1/2 1/2 ... . . . ... ... . . . . . . ... ... . . . . . . 1/2          For ψ0 = φ1, s = 0, ǫ = 10−4 and K = 3, one finds N = 14. For ψ0 = φ1, s = 0, ǫ = 10−6 and K = 3, one finds N = 22.

Bilinear quantum systems Averaging techniques Examples Conclusion

Numerical simulations

u∗(t) = cos3(t), Eff1,2(u∗) = 9π/32 ≈ 0.88, n = 30, N = 22.

50 100 150 200 250 300 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bilinear quantum systems Averaging techniques Examples Conclusion

Numerical simulations: zero efficiency case

u∗(t) = cos2(t), Eff1,2(u∗) = 0, n = 30, N = 22.

50 100 150 200 250 300 0.0e+00 2.0e−06 4.0e−06 6.0e−06 8.0e−06 1.0e−05 1.2e−05 1.4e−05 1.6e−05 1.8e−05

Bilinear quantum systems Averaging techniques Examples Conclusion

Outline of the talk

1

Bilinear quantum systems Bilinear Schrödinger equation Good Galerkin approximation

2

Averaging techniques Rotating wave approximation

3

Examples Quantum harmonic oscillator Rotation of a planar molecule

4

Conclusion Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion

Concluding remarks

Very few results about the structure of the attainable set. Some sufficient criterion for approximate controllability. Some reasonable estimates (L1 norm, time, precision). Constructive methods (control and simulations are possible).

Bilinear quantum systems Averaging techniques Examples Conclusion

Future works

Continuous spectrum. Time minimization: does there exist a minimal transfert time? Taking decoherence into account.