2,052 research outputs found
Periodic control laws for bilinear quantum systems with discrete spectrum
We provide bounds on the error between dynamics of an infinite dimensional
bilinear Schr\"odinger equation and of its finite dimensional Galerkin
approximations. Standard averaging methods are used on the finite dimensional
approximations to obtain constructive controllability results. As an
illustration, the methods are applied on a model of a 2D rotating molecule.Comment: 6 pages, submitted to ACC 201
Periodic excitations of bilinear quantum systems
A well-known method of transferring the population of a quantum system from
an eigenspace of the free Hamiltonian to another is to use a periodic control
law with an angular frequency equal to the difference of the eigenvalues. For
finite dimensional quantum systems, the classical theory of averaging provides
a rigorous explanation of this experimentally validated result. This paper
extends this finite dimensional result, known as the Rotating Wave
Approximation, to infinite dimensional systems and provides explicit
convergence estimates.Comment: Available online
http://www.sciencedirect.com/science/article/pii/S000510981200286
Which notion of energy for bilinear quantum systems?
In this note we investigate what is the best L^p-norm in order to describe
the relation between the evolution of the state of a bilinear quantum system
with the L^p-norm of the external field. Although L^2 has a structure more easy
to handle, the L^1 norm is more suitable for this purpose. Indeed for every
p>1, it is possible to steer, with arbitrary precision, a generic bilinear
quantum system from any eigenstate of the free Hamiltonian to any other with a
control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm
are computed on an example
Explicit approximate controllability of the Schr\"odinger equation with a polarizability term
We consider a controlled Schr\"odinger equation with a dipolar and a
polarizability term, used when the dipolar approximation is not valid. The
control is the amplitude of the external electric field, it acts non linearly
on the state. We extend in this infinite dimensional framework previous
techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in
finite dimension. We consider a highly oscillating control and prove the
semi-global weak stabilization of the averaged system using a Lyapunov
function introduced by Nersesyan. Then it is proved that the solutions of the
Schr\"odinger equation and of the averaged equation stay close on every finite
time horizon provided that the control is oscillating enough. Combining these
two results, we get approximate controllability to the ground state for the
polarizability system
Approximate Controllability, Exact Controllability, and Conical Eigenvalue Intersections for Quantum Mechanical Systems
International audienceWe study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties. We conclude the article by proving that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems
Approximate controllability of the Schr\"{o}dinger Equation with a polarizability term in higher Sobolev norms
This analysis is concerned with the controllability of quantum systems in the
case where the standard dipolar approximation, involving the permanent dipole
moment of the system, is corrected with a polarizability term, involving the
field induced dipole moment. Sufficient conditions for approximate
controllability are given. For transfers between eigenstates of the free
Hamiltonian, the control laws are explicitly given. The results apply also for
unbounded or non-regular potentials
Integrable Floquet dynamics
We discuss several classes of integrable Floquet systems, i.e. systems which
do not exhibit chaotic behavior even under a time dependent perturbation. The
first class is associated with finite-dimensional Lie groups and
infinite-dimensional generalization thereof. The second class is related to the
row transfer matrices of the 2D statistical mechanics models. The third class
of models, called here "boost models", is constructed as a periodic interchange
of two Hamiltonians - one is the integrable lattice model Hamiltonian, while
the second is the boost operator. The latter for known cases coincides with the
entanglement Hamiltonian and is closely related to the corner transfer matrix
of the corresponding 2D statistical models. We present several explicit
examples. As an interesting application of the boost models we discuss a
possibility of generating periodically oscillating states with the period
different from that of the driving field. In particular, one can realize an
oscillating state by performing a static quench to a boost operator. We term
this state a "Quantum Boost Clock". All analyzed setups can be readily realized
experimentally, for example in cod atoms.Comment: 18 pages, 2 figures; revised version. Submission to SciPos
The transfer matrix: a geometrical perspective
We present a comprehensive and self-contained discussion of the use of the
transfer matrix to study propagation in one-dimensional lossless systems,
including a variety of examples, such as superlattices, photonic crystals, and
optical resonators. In all these cases, the transfer matrix has the same
algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as
well as the group of unimodular real matrices underlying the structure of the
abcd law, which explains many subtle details. We elaborate on the geometrical
interpretation of the transfer-matrix action as a mapping on the unit disk and
apply a simple trace criterion to classify the systems into three types with
very different geometrical and physical properties. This approach is applied to
some practical examples and, in particular, an alternative framework to deal
with periodic (and quasiperiodic) systems is proposed.Comment: 50 pages, 24 figure
- …