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Tóm tắt nội dung (trích từ tài liệu gốc): Understanding the quantum world with a tennis racket: How classical mechanics helps control qubits QuSCo Seminar Wednesday, 24 march 2021 Dominique SUGNY Laboratoire Interdisciplinaire Carnot de Bourgogne, Dijon, France. Universit� Bourgogne Franche Comt� Collaboration and Fundings A joint work between mathematicians, physicists and chemists Group of S. J. Glaser (Munich, Germany) Group of P. Mardesic (Dijon, France) Introduction to Quantum control Quantum effects: Atomic orbitals (Probability of 95% to find the electron) Quantum theory: Theoretical basis of modern physics that explains the na
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Understanding the quantum world
with a tennis racket: How classical
mechanics helps control qubits
QuSCo Seminar
Wednesday, 24 march 2021
Dominique SUGNY
Laboratoire Interdisciplinaire Carnot de Bourgogne, Dijon, France.
Universit� Bourgogne Franche Comt�
Collaboration and Fundings
A joint work between mathematicians, physicists and chemists
Group of S. J. Glaser (Munich, Germany)
Group of P. Mardesic (Dijon, France)
Introduction to Quantum control
Quantum effects:
Atomic orbitals (Probability of
95% to find the electron)
Quantum theory:
Theoretical basis of modern physics that explains the nature and behavior of matter
and energy at the atomic level.
Fundamental quantum effects Control theory Quantum technologies
Control theory: Realization of basic operations
Quantum control
Manipulating the quantum dynamics of atoms, molecules and spins with external
electromagnetic fields.
Design of specific electric or magnetic fields
Application of tools of control theory (Optimal control theory) to
quantum physics
A famous example in classical physics:
Apollo and Smart I How do we build up physical
intuition in quantum control ?
Analogy with classical physics
The Tennis Racket Effect
The tennis racket effect
R. H. Cushman and L. M. Bates
Geometric effect that can be observed in any three-dimensional asymmetric
rigid body.
The Tennis Racket Effect
How to control a skate board with the tennis racket effect
According to the tennis racket effect, the Monster Flip is impossible.
It can be shown that it is possible, but with a very low probability....
References about the tennis racket effect
Scientific papers:
M. S. Ashbaugh, C. C. Chiconc and R. H. Cushman, The Twisting Tennis
Racket, J. Dyn. Diff. Eq. 3, 67 (1991).
R. H. Cushman and L. Bates, Global Aspects of Classical Integrable Systems
(Birkhauser, Basel, 1997).
L. Van Damme, P. Mardesic and D. Sugny, The tennis racket effect in a three
dimensional rigid body, Physica D 338, 17 (2017)
P. Mardesic, G. J. Gutierrez Guillen, L. Van Damme and D. Sugny, Phys. Rev.
Lett. 125, 064301 (2020)
Popular studies:
Images des Math�matiques (Mardesic and Sugny, 2019)
Le monde (D. Larousserie, Mardesic and Sugny, 2020)
Movies on Youtube: Physics girls (2019), the Monster Flip....
The Dzhanibekov effect, Wikipedia page...
Classical dynamics of a three-dimensional rigid body
The rotational dynamics of a three-dimensional rigid body is described by an
integrable Hamiltonian system.
The position of the rigid body is given by an element of SO(3).
The three Euler angles are used as coordinates.
Two frames:
A space-fixed frame (X,Y,Z)
A body-fixed frame (x,y,z)
Ref.: V. I. Arnold, Mathematical methods of Classical Mechanics
Axes and moments of inertia
Mass repartition: Inertia matrix
Eigenvectors: Inertia axes
Eigenvalues: Inertia moments
Convention: I z I y I x
Classical dynamics of a three-dimensional rigid body
The dimension of the phase space is 6.
In the absence of outside forces, there are four first integrals (the angular
momentum M and the energy): The Euler top.
For a regular point, the dynamics are restricted to a two-dimensional torus.
In the reduced phase space (Mx,My,Mz), the trajectory is the intersection of
two surfaces:
Rem.: Extension to a n-dimensional rigid body with the Lax pair approach
Classical dynamics of a three-dimensional rigid body
Intersection of a sphere and an ellipsoid.
Classical dynamics of a three-dimensional rigid body
Reduced phase space:
Rotating and oscillating trajectories, separatrix
Four stable and two unstable equilibrium points.
Mathematical description of the tennis racket effect
Definition of a particular set of Euler angles:
2
Mathematical description of the tennis racket effect
Angular momentum: Rotational equivalent of the momentum
Mi Iii
Angular Angular velocity
momentum
Euler's equations: Dynamics of the angular momentum in the frame attached
to the racket
M x ( 1 1 Constants of the motion:
Iy Iz
)M yM z M 2 M 2 M 2
E x y z
M y (1 1 2Ix 2Iy 2Iz
Ix Iz
)M xM z M 2 M x2 M 2 M 2
y z
M z ( 1 1 )M xM y Integrable system (Euler top)
Ix Iy
Mathematical description of the tennis racket effect
Euler's equations: Dynamics of the Euler angles
M x M sin cos the two angles described the dynamics in
M y M sin sin the reduced phase space.
M z M cos
The dynamics of the third angle is given by the angular velocity.
M ( 1 1 ) sin sin cos We introduce the following coefficients
Iy Ix Iy
Iz
M (sin 2 cos2 ) 1
a
Iy Ix
M( 1 sin 2 cos2 ) cos b 1 Iy
Ix
Iz Iy Ix
2I y E
c M2 1
Perfect asymmetric rigid body: ab
Mathematical description of the tennis racket effect
The tennis racket effect is a geometric effect which does not depend
directly on the duration of the process.
We can reduce the dynamics to consider only two angles: d
d
d (a b cos2 )(c b cos2 )
d 1 b cos2
Phase space
Mathematical description of the tennis racket effect
Analogy with a standard planar pendulum:
Phase space
A variation of Phase space
Tennis racket effect A variation of 2
Mathematical description of the tennis racket effect
Robustness of the tennis racket effect against initial conditions:
What is the geometric origin of the tennis racket effect ?
Is it possible to estimate the robustness of the effect ?
3 parameters (a,b,c)
Mathematical description of the tennis racket effect
We consider a symmetric configuration: 0 f
2 2
The new parameter is the defect to a perfect tennis racket effect.
Using a change of variable and the parity of the integral:
1 1 (1 bx)dx
( ) b sin2 x(x )(1 x)(x )
x cos2 ; a ; c
b b
Incomplete elliptic integral depending on the different parameters of the problem.
We study the solution of the following equation:
a,b,c ( ) 2
Mathematical description of the tennis racket effect
1 1 (1 bx)dx
( ) b sin2 x(x )(1 x)(x )
We complexify the x- coordinate and we introduce a Riemann surface:
y2 x(x )(1 x)(x )
The integral is interpreted as an Abelian integral over this surface.
Its behavior is given by the geometry and the singularity of the surface.
Mathematical description of the tennis racket effect
Two different configurations: A pole appears when c goes to 0
sin 2 sin 2
; 0 ~
; 0
~
In the first case, by the Picard-Lefschetz formula, the integration contour is
deformed to itself plus a loop around the singularity.
This property reveals the multi-valued character of the function: a logarithmic
function. No logarithmic divergence in the second case !
Mathematical description of the tennis racket effect
We deduce:
( ) 1 ha,b,c (sin 2 ) 1 ln(sin 2 )
ab
ab
Bounded and analytic function (m is given by the bound of h).
Theorem of the Tennis Racket Effect:
For all c such that: c b exp(2 ab m)
For ab large enough, the equation a,b,c ( ) 2
has a unique solution which verifies:
arcsin( c ) S arcsin(exp( ab m))
b 2
This leads to:
lim S (a, b, c) 0
ab
Mathematical description of the tennis racket effect
Estimation of the robustness of the tennis racket effect:
2
exp( ab )
2
Robustness with respect to the shape of the body
a I y 1;b 1 I y
Iz Ix
Refs.: L. Van Damme et al, Physica D (2017)
The Monster Flip effect
The same analysis can be conducted for the Monster Flip effect.
/ 2 1 b cos2 d
2
i (a b cos2 )(c b cos2 )
We arrive at:
( ) 1 ha,b,c (sin 2 ) 1 ln(sin 2 )
ab
ab
exp( ab)
2
This parameter has to be very small
How to use this effect in the quantum world ?
Formal equivalence between the Euler equations and the Bloch equations:
M 0 Mz / Iz M y / I y M 0 z y
Mz / Iz 0 M x / Ix M z 0 x M
M y / I y Mx / Ix 0 y x 0
Euler equations Bloch equations (spin �,
magnetic resonance)
Quantum state
i External control fields
Identification: i Mi The moments of inertia are free parameters
Ii
How to translate this property into the quantum world ?
Formal equivalence between the Euler equations and the Bloch equations:
M 0 M3 / I3 M 2 / I2 M 0 3 2
M3 / I3 0 M1 / I1 M 3 0 1 M
M 2 / I2 M1 / I1 0 2 1 0
Euler equations Bloch equations
Identification: Specific choice of the control fields (only two fields are available)
Case (a):I21M1 / I1 Case (b): 1 M1 / I1
3 M 3 / I3 2 M 2 / I2
I3
Geometric control of population transfer
We consider the case (a) to illustrate the properties of the control fields.
Without loss of generality, we can set: I1 1
1
I3 k 2 ,k [0,1]
Some standard solutions of the Bloch equation can be recovered from limiting
cases of the tennis racket effect:
k 0 : Pi-pulse 1 sec h( t )
k 1: Adiabatic pulse 1 k2
Separatrix : Allen-Eberly solutions
k tanh( t )
1 k2
How to use this effect in the quantum world ?
A tennis racket effect for a spin � particle:
A trajectory close to the separatrix Trajectories of
the angular
A robust transfer of state for the momentum
qubit
Refs.: L. Van Damme et al, Sci. Rep. (2017)
Robustness of the control process
Evaluation of the robustness in the spin case: Ix 1; I y 1 ;Iz
k2
1(,2) (1 )1,2
3 3 k 0.2; 0.6; 0.9; 0.99
The robustness of the process can be adjusted by choosing appropriate moments
of inertia (parameter k)