Dominique SUGNY

Laboratoire Interdisciplinaire Carnot de Bourgogne, Dijon

(France).

Nottingham, Thursday, 22th January 2015

Quantum Cybernetics and Control

Recent advances in the control

of spin systems

Collaborations: Gr. B. Bonnard (Mathematics), Gr. S. Glaser (NMR),

M. Lapert, E. Assémat, L. Van Damme.

Optimal control of quantum systems

Quantum Optimal Control

Optimal control theory Quantum Dynamics

- Collaboration between mathematicians, physicists, chemists…

- Open-loop quantum optimal control

Electronics

Economy

Biology

Quantum physics…

Control theory has a large number of

applications:

Lev Pontryagin 1908-1988

The modern history of optimal control

The mathematical framework of optimal control is based on the

Pontryagin Maximum Principle: Generalization of the Euler-

Lagrange principle for controlled systems - 1960

Pontryagin Maximum Principle

The optimal control problem is described through a classical Hamiltonian

dynamics. The PMP only gives a necessary condition of optimality.

ft

dtuxFMin

tuxFx

0

0 ),(

),,(

),(),,( 00 uxFptuxFpH

Adjoint states:

),( 0pp

Hamiltonian equations and maximization condition:

vvtpxHutpxH )],,,,(max[),,,(

Boundary conditions:

0)()( ptporxtx fff

Terminal cost:

))(( ftx

Optimal control of quantum systems

Geometric (analytic) and numerical methods:

Geometric approach if

Numerical approach if

1N

1N

Ex: Pi-pulse

Population transfert in a two-level quantum system

T

dttu0

)(

Geometry: Analytical and geometric methods to solve the

optimal equations

Numerics: Iterative algorithms used to solve the optimal

equations (Krotov, GRAPE)

Optimal control of a two-level dissipative quantum system

Generalization of the standard pi-pulse to the dissipative case

We consider a two-level quantum system whose dynamics is

ruled by the Lindblad equation.

xuyuTzz

zuTyy

zuTxx

yx

x

y

1

2

2

/)1(

/

/

Bloch equation:

Bloch ball: 1222 zyx

Condition on the relaxation parameters: 12 20 TT

A geometric structure: The magic plane

sin

sincos

coscos

rz

ry

rx

)cossin(tan

cossinsincos

)sin1

(cos

sinsincos

21

1

2

12

2

xy

yx

uu

uuTrT

T

r

TT

rr

We introduce the spherical coordinates:

We cannot directly control the r-coordinate

Ref.: D. J. Tannor and A. Bartana, J. Phys. Chem. A 103, 10359 (1999)

A basic geometric structure: The magic plane

If there is no bound on the control fields:

We can move in an arbitrarily short time along the Bloch

sphere.

How to control the dynamics along the radial direction ?

112

sincos2cossincos2

T

r

TT

r

d

rd

What are the characteristic points of the Bloch sphere ?

0r Steady-state ellipsoid

][/ rMinMax Other geometric structure?

Another geometric structure: The magic plane

z-axis:

)212

(112

222

22

T

z

TT

z

zyx

yx

d

rd

Maximum shrinking if:

][/ rMinMax

The magic plane

)(2 12

20

TT

Tzz

00 zz

Maximum shrinking for any

point of the plane

The magic plane and the Steady-state ellipsoid

The magic plane belongs to

the Bloch ball if:

11 0 z

Time-optimal control of the saturation process

The goal is to reach in minimum time the center of the Bloch ball

On the magic plane, the

control fields satisfy:

xuyuT

zz yx

1

)1(0

Optimal synthesis in the unbounded case

This result can be generalized to any target state

Generalization of a Pi-pulse to a two-level dissipative quantum

system

Ref.: PRA 88, 033407 (2013)

Experimental implementation in NMR

Time-optimal solution versus Inversion solution

Gain of 60% in the

control duration

Ref.: PRL 104, 083001 (2010)

Generalization to the control of non-linear dynamics

The radiation damping interaction in NMR:

2

1

2

/)1(

/

kyyuTzz

kyzzuTyy

x

x

Optimal synthesis without the magic plane

Overlap curve in

blue and green

Switching

curve in red

Ref.: PRA 87, 043417 (2013), JCP 134, 054103 (2011)

Generalization to the control of two uncoupled spins

)1()1(

1

)1()1(

)1()1(

2

)1()1(

/)1(

/

yuTzz

zuTyy

x

x

)2()2(

1

)2()2(

)2()2(

2

)2()2(

/)1(

/

yuTzz

zuTyy

x

x

Optimization of the saturation contrast in MRI:

][

0

2)2(2)2(

)1()1(

zyMax

zy

Goal:

Desoxygeneated blood Oxygeneated blood

Ref.: IEEE 57, 1857 (2012)

Experimental implementation in MRI

Geometry of the sample

Application of the optimal sequence

(optimization with GRAPE)

GOCT gives the physical limit of the process

(80% of the limit with GRAPE)

Ref.: Sci. Rep. 2, 589 (2012)

Generalization to the optimization of the signal to noise ratio

Ref.: PRA90, 023411 (2014), JCP (2015) to be published

Cyclic process repeated N times to

improve the SNR in NMR

The initial and the final states (the

same steady state) are not known.

Optimization of the steady state and

of the control field.

p

N

T

My

Q

1

)(Figure of Merit:

)()(

,

N

TM

N

TMQMaxMaxQMax

pp

Two-step optimization process:

We prove the optimality of the

Ernst angle solution in the

unbounded case: Bang pulse ])[arccos(exp

1

)(

T

TdE

Geometric optimal control in quantum computation

Time-optimal control of SU(2) quantum operations

Definition of the optimal control

problem

UtHUi )(yyxxz StStStH )()()(

2

0

22 )()(

)2(1:)(

tt

SUUtU

yx

f

Use of the Euler angles coordinates

Description in terms of the PMP

Examples of projected

optimal trajectories:

zyz iaSibSiaS

f

zf

eeeUb

SiUa

:)(

]exp[:)(

0

Ref.: PRA88, 043422 (2013)

Geometric optimal control in quantum computation

Time-optimal control of a chain of three coupled spins

Definition of the optimal control

problem

UtHUi )(yzzzz StuSSJSSJtH 232232112 )(22)(

ft 0:)(

Only a four-dimensional control space is considered:

CNOT gate in the original space (Khaneja et al, PRA 2002)

The Pontryagin Hamiltonian describes the free rotation of a

three-dimensional rigid body (Euler top):

1,,1

222

3221

3

2

3

2

2

2

1

2

1

IIk

I

I

L

I

L

I

LH

Ref.: PRA90, 013409 (2014), Poster L. Van Damme

Conclusion and outlook

Reference on GOCT:

Geometric OCT:

A powerful tool for simple

quantum systems

We can combine the advantages

of the two approaches

Numerical OCT:

A powerful tool for complex

quantum systems

Open PhD position in Dijon (september 2014)

Optimal control of spin systems with applications in MRI

ANR-DFG Research program: