recent advances in the control of spin...
TRANSCRIPT
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: