Projector Space Optimization in Quantum Control

C. Besse, O. Goubet, T. Goudon & S. Nicaise, Editors

PROJECTOR SPACE OPTIMIZATION IN QUANTUM CONTROL

A

G

G

T

Abstract. We investigate in this work the numerical resolution of a quantum control problem; the specificity of the approach is that, instead of searching directly for the optimal laser intensity that drives the system toward its target, we consider here as main variable the evolution semigroup i.e. the set of propagators indexed with time. The precise form of the generator of the semigroup (e.g. dipolar) is then enforced as a constraint. We present both an algorithm and associated numerical results.

I

Following successful quantum control experiments in the late 90s [1–4, 6, 7, 20] that build on the introduction of the evolutionary paradigm [5] a large number of works [8, 11–13, 17, 18, 21, 23] investigated numerically the control of quantum phenomena with laser fields.

The mean that allows to control the system is thus the laser field, more precisely its intensity. Traditionally, the numerical simulations considered some description of the interaction of the laser and the system, of which the most used is thedipole approximation, and performed optimizations considering the laser intensity as main variable.

We propose in this work a different view of the problem: we place ourselves in the evolution semigroup of unitary propagators and ask that the resulting Hamiltonian be consistent with the chosen approximation type.

The balance of the paper is as follows: we introduce the main notations and the technical choices that determine our problem in Section 1. Our specific optimization algorithm is presented in Sections 2 and 3 followed in Section 4 by numerical results.

1. P

We consider a quantum system evolving under the Schr¨odinger equation (we use atomic units, i.e~= 1):

id

dtψ(x, t) = H(t)ψ(x, t), (1)

ψ(t= 0) = ψinit

whereψ(x, t)is the wavefunction,ψinit is the initial data andH(t)the Hamiltonian of the system. We assume that the system is described as finite dimensional, so that H is aN ×N complex Hermitian matrix with entries in _Cand ψ(x, t)∈CN. The initial dataψinitverifieskψinitk= 1.

∗_{Financial support from INRIA Rocquencourt projet MicMac and OMQP, projet ANR C-Quid and from PICS NSF-CNRS is acknowledged.}

1_{CEREMADE Universit´e Paris Dauphine,Place du Marechal De Lattre De Tassigny, 75016 Paris, France, [email protected]} 2_{CEREMADE Universit´e Paris Dauphine,Place du Marechal De Lattre De Tassigny, 75016 Paris, France, [email protected]}

c

EDP Sciences, SMAI 2009

This finite dimensional configuration represents an approximation of an infinite dimensional system. The same formu-lation of the system given by(1)can be expressed using the time evolution operatorU(t)∈U(N), called the propagator:

id

dtU(t) = H(t)U(t) (2)

U(t= 0) = I

with the propertyψ(x, t) =U(t)ψ(x,0).We recall thatU(N)is the space of unitary matrices

U(N) ={A∈CN×N;AA∗=A∗A=Id}. (3)

In the following we suppose that our system it is submitted to an external interaction taken here as an electric field. In the dipole approximation [9, 16, 19] the HamiltonianH(t)has the following form:

H(t) =H0+ε(t)µ, (4)

whereH0 is the internal Hamiltonian of the system,ε(t) ∈ Ris the intensity of the laser field andµ ∈ RN×N the

(coupling) dipole moment operator.

Both H0 andµ are known, self adjoint operators. The control is hereε(t)which can be chosen to influence the

evolution of the system.

Our goal is to design an algorithm that finds an (optimal) laser fieldε(t)such that, under the controlε(t), the system evolves from the initial stateU(t= 0) =Ito the final stateU(t=T) =UT.

We work in a spaceHN, of the hermitian matrices with the scalar product defined by:

∀A, B∈ HN < A, B >=real(tr(AB∗)), (5)

whereB∗ _{is the transposed-conjugate of the matrix. This scalar product induces a norm on the space of Hermitian} matrices:

∀A∈ HN kAk=

p

< A, A >. (6)

Instead of searching for the intensityε[12,21,22] we take here a different view and search for a pathU(t)in propagator space such that the corresponding Hamiltonian will be of type (4). This leads us to minimize the following functional

Z T 0

kH−ΠH₀+Span(µ)(H)k2dt, (7)

with the constraints:

• H(t) =iU˙(t)U∗(t)for everytin[0, T]

• U∗(t)U(t) =U(t)U∗(t) =Ifor everytin[0, T].

SincekH−ΠH0+Span(µ)(H)k= minξ∈RkH−H0−ξµk=kH−H0−ΠSpan(µ)(H−H0)kour minimization problem

can be written as :



   

   

min U

RT

0 kH(t)−(H0+ ΠSpan(µ)(H(t)−H0))k 2_dt

H(t) =iU˙(t)U∗(t) for every t in [0, T],

U∗_{(t)U(t) =U(t)U}∗_{(t) =Ifor every t in [0, T],} U(t= 0) =I, U(t=T) =UT.

(8)

Note that ifΠSpan(µ)(H(t)−H0)) =ε(t)µthen

(t) = <ΠSpan(µ)(H(t)−H0), µ > < µ, µ > =

Once a path Ue(t) in propagator space has been given, to improve it we will search U(t) of the form U(t) =

e

U(t) exp(iA(t)), withA(t)∈ HN. SinceU(t)U∗(t) =Ue(t)eiA(t)e−iA(t)Ue∗(t) =Ifor everyt∈[0, T], and replacing H(t) =iU˙(t)U∗(t)we obtain a cost functional depending onA(t)with no constraints:

J(U) =Jc e

U(A) =

Z T

0

kiU˙(t)U∗(t)−(H0+ ΠSpan(µ)(iU˙(t)U∗(t)−H0))k2dt (9)

U(t= 0) =I, U(t=T) =UT. (10)

We discretize the interval[0, T]with pointstj=jT /N; this allows to write an approximation forU˙(t)(e.g. by finite differences). For a given integerN, we define the discretization parameter∆T =T /N, andUj,Aj,Hj,εj represent approximations forU(j∆T),A(j∆T),H(j∆T),ε(j∆T). In the following we denoteε= (εj)0≤j≤N,U = (Uj)0≤j≤N, A= (Aj)0≤j≤N,H = (Hj)0≤j≤N. We consider the time discretized version of the cost function defined in(9):

J e

U(A) = ∆T N

X

j=0

ki ˙

\ f

UjeiAjUf\jeiAj ∗

−(H0+ ΠSpan(µ)(i

˙

\ f

UjeiAjUf\jeiAj ∗

−H0))k2 (11)

Aj∈ HN, U0=I, UN =UT.

2. I

In order to minimizeJU(A)with respect toAwe use a gradient type method. At stepk−1, for a givenUk−1 we minimizeJ_Uk−1(A)by takingUk =Uk−1eiA

k

; we can easily verify that the propertyUk_(Uk₎∗ _{=Iholds at every step} k.

To computeU˙k_{we propose a centered finite differences formula:}

H_jk = iU˙_jk(U_jk)∗=iU k

j+1−Uj−k 1

2∆T

(Uk

j+1)∗+ (Uj−k 1)∗

2

= iU k

j+1(Ujk+1)∗+Ujk+1(Uj−k 1)∗−Uj−k 1(Ujk+1)∗−Uj−k 1(Uj−k 1)∗

4∆T

= iU k j+1(U

k

j−1)∗−U

k j−1(U

k j+1)∗

4∆T

= iU k−1

j+1e

iAk_j+1e−iAk

j−1_(Uk−1

j−1)∗−U

k−1

j−1e

iAk_j−1_e−iA k

j+1_(Uk−1

j+1)∗

4∆T (12)

Now we consider an iteration algorithm for solving the minimization problem(8). The iteration procedure is specified as follows:

Stepk≥2

Ak_j = −ρ∂JUk−1(A) ∂Aj

_A=0f or every j= 1, .., N−1 U_jk = U_jk−1eiAkj _{f or every j= 1, .., N}−₁

whereρis a positive constant. Note that the new value of the functionalJ(Uk_)is:

J(Uk) =JUk−1(A) = ∆T

N

X

j=0

kiU˙_jk(U_jk)∗−(H0+ ΠSpan(µ)(iU˙jk(U k j)

∗_−H

The boundary condition in each step are:

U₀k=I, Ak₀= 0, U_Nk =UT,

The corresponding control field at each iteration step can be written as:

k_j = <(H k

j −H0), µ >

< µ, µ > f or every j= 0, .., N.

In the following for simplicity reasons we noteΠSpan(µ)= ΠandP =I−Π(Iis the identity operator).

3. G

We explain in this section how to compute the variation ∂JU k−1(A)

∂Aj _A=0.

Theorem 3.1. The gradient ofJUk−1(A)aroundA= (0,0, ...,0)is given by:

∇J_Uk−1(A)

_A=0 =

−2(U_j−k−₁1)∗(P(H_j−k ₁)(0)−P(H0))Ujk−1−2(U k−1

j ) ∗_(P(Hk

j−1)(0)−P(H0))Uj−k−21+

2(U_jk−1)∗(P(H_jk₊₁)(0)−P(H0))Ujk−+21+ 2(U

k−1

j+2)

∗_(P(Hk

j+1)(0)−P(H0))Ujk−1

j=1,..,N−1(13)

The components forj= 1andj=N−1are:

∇J_Uk−1(A) _A=0

j=1

= −2(U₀k−1)∗(P(H₀k)(0)−P(H0))U1k−1−2(U

k−1 1 )

∗_(P(Hk

0)(0)−P(H0))U0k−1+

2(U₁k−1)∗(P(H2k)(0)−P(H0))U3k−1+ 2(U

k−1 3 )

∗_(P(Hk

2)(0)−P(H0))U1k−1 (14)

∇J_Uk−1(A) _A=0

j=N−1

= −2(U_Nk−₋1₂)∗(P(H_Nk₋₂)(0)−P(H0))UNk−−11−2(U

k−1

N−1)

∗_(P(Hk

N−2)(0)−P(H0))UNk−−13+

2(U_N−k−1₁)∗(P(HNk)(0)−P(H0))UNk−+11 + 2(U

k−1

N+1)

∗_(P(Hk

N)(0)−P(H0))UNk−−11. (15)

Proof. In order to have an explicit formula for the gradient of J_Uk−1(A)we will compute the partial derivatives of

JUk−1(A)with respect toAjfor everyj= 2, ..., NaroundA= (0,0, ...,0).

∂JUk−1(A)

∂Aj

_A=0 = ∆T ∂ ∂Aj

_XN

`=0

kH_{`</sub>k−ΠH<sub>`}k+ ΠH0−H0k2

_A=0

= ∆T ∂ ∂Aj

XN

`=0

< H_{`</sub>k−ΠH<sub>`}k+ ΠH0−H0, H`k−ΠH k

` + ΠH0−H0>

_A=0

= ∆Tn ∂ ∂Aj

< P(H_j−k ₁)(Aj)−P(H0), P(Hj−k 1)(Aj)−P(H0)>+

+ ∂ ∂Aj

< P(H_jk₊₁)(Aj)−P(H0), P(Hjk+1)(Aj)−P(H0)> o

Continuing the computation we obtain

∂ ∂Aj

< P(H_j−k 1)(Aj)−P(H0), P(Hj−k 1)(Aj)−P(H0)>=

∂ ∂Aj

< P(H_j−k 1)(Aj), P(Hj−k 1)(Aj)>+

+2< P(H_j−k 1)(Aj),−P(H0)>+< P(H0), P(H0)>

which at the first order gives:

< P(H_j−k ₁)(Aj+δAj), P(Hj−k 1)(Aj+δAj)>

+2< P(H_j−k ₁)(Aj+δAj),−P(H0)>+< P(H0), P(H0)>=

=< P(H_j−k ₁)(Aj), P(Hj−k 1)(Aj)>+2< P(Hj−k 1)(Aj), P(Hj−k 1) ∂ ∂Aj

H_j−k ₁(δAj)>

+2< P(H_j−k ₁)(Aj),−P(H0)>+2< P(Hj−k 1) ∂ ∂Aj

H_j−k ₁(δAj),−P(H0)>+

< P(H0), P(H0)>+O(δAj)2.

We proceed now by identification to obtain:

∂ ∂Aj

< P(H_j−k ₁)(Aj)−P(H0), P(Hj−k 1)(Aj)−P(H0)>= (16)

= 2< P(H_j−k ₁)(Aj), P(H_j−k ₁)∂H k j−1 ∂Aj

(δAj)>

+2< P(H_j−k ₁)∂H k j−1 ∂Aj

(δAj),−P(H0)>

= 2< P(H_j−k ₁)(Aj)−P(H0), P(Hj−k 1) ∂Hk

j−1 ∂Aj

(δAj)>

= 2< P∗(P(H_j−k ₁)(Aj)−P(H0)), ∂Hk

j−1 ∂Aj

(δAj)>

= 2< P(H_j−k 1)(Aj)−P(H0), ∂Hk

j−1 ∂Aj

(δAj)> .

Using the same method we can also prove:

∂ ∂Aj

< P(H_jk₊₁)(Aj)−P(H0), P(Hjk+1)(Aj)−T(H0)>= (17)

= 2< P(H_jk₊₁)(Aj)−P(H0), ∂Hk

j+1 ∂Aj

(δAj)> .

In order to compute ∂H

k j−1

∂Aj and

∂Hk j+1

H_j−k ₁(Aj+δAj)

_A=0 =

iU k−1

j eiAj+iδAje−iAj−2(U k−1

j−2)∗

4∆T −

−iU k−1

j−2e

iAj−2_e−iAj−iδAj_(Uk−1

j )∗ 4∆T

_A=0

= iU k−1

j (I+iAj+iδAj)(I−iAj−2)(U_j−k−21)∗

4∆T −

iU k−1

j−2(I+iAj−2)(I−iAj−iδAj)(Ujk−1)∗ 4∆T

_A=0

= iU k−1

j (U k−1

j−2)∗−iU

k−1

j−2(U

k−1

j )∗

4∆T +

iU k−1

j iδAj(U k−1

j−2)∗−iU

k−1

j−2(−iδAj)(U

k−1

j )∗

4∆T +O(δAj)

2_.

Again by identification we determine that:

∂Hk j−1(Aj) ∂Aj

(δAj)

_A=0 =i

U_jk−1iδAj(U_j−k−21)∗−iU

k−1

j−2(−iδAj)(Ujk−1)∗

4∆T . (18)

Same arguments can be invoked to prove :

∂Hj+1(Aj) ∂Aj

(δAj)

_A=0=i

U_jk−₊₂1iδAj(Ujk−1)∗−iU k−1

j (−iδAj)(Ujk−+21)∗

4∆T . (19)

Substituting(18),(19)into(16)and respectively(17)we obtain:

∂JUk−1

∂Aj (δAj)

_A=0 = ∆T n

2< P(H_j−k ₁)−P(H0),

−U_jk−1δAj(Uj−k−21)

∗_−Uk−1

j−2(δAj)(Ujk−1) ∗

4∆T >

+2< P(Hjk+1)−P(H0),

+U_jk−₊₂1δAj(U_jk−1)∗+U_jk−1(δAj)(Ujk−+21)∗

4∆T >

o

= −0.5real(tr((P(H_j−k ₁)−P(H0))Ujk−1(δAj)(Uj−k−21)

∗₎

−(P(H_j−k ₁)−P(H0))Uj−k−21(δAj)(Ujk−1) ∗₎₎

+0.5real(tr((P(H_jk₊₁)−P(H0))Ujk−+21(δAj)(U

k−1

j ) ∗₎

+(P(H_jk₊₁)−P(H0))Ujk−1(δAj)(U k−1

j+2)

∗

).

Using the property of the scalar productreal(tr(AB∗)) =real(tr(AB))and the properties of the trace, by identifi-cation we obtain the relation(13)which concludes the proof.

The previous theorem allows to define the setCof the critical points ofJU(A)

C=nA= (A0, A1, ..., AN), Aj∈ HN,∀j= 0, ..N

−2(U

k−1

j−1)

∗_(P(H

j−1)(0)−P(H0)Ujk−1 −2(U_jk−1)∗(P(Hj−1)(0)−P(H0)Uj−k−21+ 2(U

k−1

j )∗(P(Hj+1)(0)−P(H0)Ujk−+21

+2(U_jk−₊₂1)∗(P(Hj+1)(0)−P(H0)Ujk−1= 0, j= 1, ..., N−1

o

FIGURE1. Convergence of the functionalJ towards the minimum value vs the number of iterations.

4. N

In order to demonstrate the efficiency of the algorithm, we choose a five-dimensional test system [14, 15] having internal HamiltonianH0and dipole moment operatorµ:

H0= 

    

1.0 0 0 0 0

0 1.2 0 0 0

0 0 1.3 0 0

0 0 0 2.0 0

0 0 0 0 2.15



    

, µ=



    

0 0 0 1 1

0 0 0 1 1

0 0 0 1 1

1 1 1 0 0

1 1 1 0 0



     .

Numerical simulations have been performed for a final time T = 200, which represents approximately32 times the natural period, at which the system without control oscillates. The system evolves between the initial stateU0 =I5and

the final stateUT =



    

0 0 0 0 1

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

1 0 0 0 0



    

. The functionalJ(U)is positive and as we can see inF ig.1the algorithm

obtains linear convergence towards the minimum point, zero. As soon as the minimum point is reached we also obtain the optimal laser field, which representation is given byF ig.3.

After that we solve the equation(2)[10, 15] withH(t) =H0+εµ, whereεis the laser field given by the optimization

algorithm, starting from the initial pointU0. InF ig.4we represent|U(1,1)|2and|U(5,1)|2as functions of time, and we

observe that the final stateUT is reached.

The algorithm provides satisfying results for a discretization time step∆T <10−1_{. If we choose the value of∆T =}

10−1_{we accelerate the convergence of the functional towards the minimum point, but instead the laser field thus obtained}

doesn’t allow the evolution of our system between the two chosen states.

If we take a small value for∆Tthe error decreases and we can reach the final stateUT , but we need a larger number of iterations. The numerical simulations presented inF ig.1,F ig.2,F ig.3andF ig.4have been performed for∆T = 0.04, andρ= 0.01.

R

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FIGURE2. The value of the gradient normek∇Jkvs the number of iterations.

FIGURE3. Optimized laser field as a function of time.

FIGURE4. The values of|U(1,1)|2_and|U(5,1)|2_{as functions of time.}

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