457 research outputs found
Dynamic Homotopy and Landscape Dynamical Set Topology in Quantum Control
We examine the topology of the subset of controls taking a given initial
state to a given final state in quantum control, where "state" may mean a pure
state |\psi>, an ensemble density matrix \rho, or a unitary propagator U(0,T).
The analysis consists in showing that the endpoint map acting on control space
is a Hurewicz fibration for a large class of affine control systems with vector
controls. Exploiting the resulting fibration sequence and the long exact
sequence of basepoint-preserving homotopy classes of maps, we show that the
indicated subset of controls is homotopy equivalent to the loopspace of the
state manifold. This not only allows us to understand the connectedness of
"dynamical sets" realized as preimages of subsets of the state space through
this endpoint map, but also provides a wealth of additional topological
information about such subsets of control space.Comment: Minor clarifications, and added new appendix addressing scalar
control of 2-level quantum system
Quantum Control Landscapes
Numerous lines of experimental, numerical and analytical evidence indicate
that it is surprisingly easy to locate optimal controls steering quantum
dynamical systems to desired objectives. This has enabled the control of
complex quantum systems despite the expense of solving the Schrodinger equation
in simulations and the complicating effects of environmental decoherence in the
laboratory. Recent work indicates that this simplicity originates in universal
properties of the solution sets to quantum control problems that are
fundamentally different from their classical counterparts. Here, we review
studies that aim to systematically characterize these properties, enabling the
classification of quantum control mechanisms and the design of globally
efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry,
Vol. 26, Iss. 4, pp. 671-735 (2007
Control of quantum phenomena: Past, present, and future
Quantum control is concerned with active manipulation of physical and
chemical processes on the atomic and molecular scale. This work presents a
perspective of progress in the field of control over quantum phenomena, tracing
the evolution of theoretical concepts and experimental methods from early
developments to the most recent advances. The current experimental successes
would be impossible without the development of intense femtosecond laser
sources and pulse shapers. The two most critical theoretical insights were (1)
realizing that ultrafast atomic and molecular dynamics can be controlled via
manipulation of quantum interferences and (2) understanding that optimally
shaped ultrafast laser pulses are the most effective means for producing the
desired quantum interference patterns in the controlled system. Finally, these
theoretical and experimental advances were brought together by the crucial
concept of adaptive feedback control, which is a laboratory procedure employing
measurement-driven, closed-loop optimization to identify the best shapes of
femtosecond laser control pulses for steering quantum dynamics towards the
desired objective. Optimization in adaptive feedback control experiments is
guided by a learning algorithm, with stochastic methods proving to be
especially effective. Adaptive feedback control of quantum phenomena has found
numerous applications in many areas of the physical and chemical sciences, and
this paper reviews the extensive experiments. Other subjects discussed include
quantum optimal control theory, quantum control landscapes, the role of
theoretical control designs in experimental realizations, and real-time quantum
feedback control. The paper concludes with a prospective of open research
directions that are likely to attract significant attention in the future.Comment: Review article, final version (significantly updated), 76 pages,
accepted for publication in New J. Phys. (Focus issue: Quantum control
Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry
We discuss homotopy properties of endpoint maps for affine control systems.
We prove that these maps are Hurewicz fibrations with respect to some
topology on the space of trajectories, for a certain . We study critical
points of geometric costs for these affine control systems, proving that if the
base manifold is compact then the number of their critical points is infinite
(we use Lusternik-Schnirelmann category combined with the Hurewicz property).
In the special case where the control system is subriemannian this result can
be read as the corresponding version of Serre's theorem, on the existence of
infinitely many geodesics between two points on a compact riemannian manifold.
In the subriemannian case we show that the Hurewicz property holds for all
and the horizontal-loop space with the topology has the
homotopy type of a CW-complex (as long as the endpoint map has at least one
regular value); in particular the inclusion of the horizontal-loop space in the
ordinary one is a homotopy equivalence
Winding around non-Hermitian singularities
Non-Hermitian singularities are ubiquitous in non-conservative open systems. Owing to their peculiar topology, they can remotely induce observable effects when encircled by closed trajectories in the parameter space. To date, a general formalism for describing this process beyond simple cases is still lacking. Here we develop a general approach for treating this problem by utilizing the power of permutation operators and representation theory. This in turn allows us to reveal a surprising result that has so far escaped attention: loops that enclose the same singularities in the parameter space starting from the same point and traveling in the same direction, do not necessarily share the same end outcome. Interestingly, we find that this equivalence can be formally established only by invoking the topological notion of homotopy. Our findings are general with far reaching implications in various fields ranging from photonics and atomic physics to microwaves and acoustics
The Topological Field Theory of Data: a program towards a novel strategy for data mining through data language
This paper aims to challenge the current thinking in IT for the 'Big Data' question, proposing - almost verbatim, with no formulas - a program aiming to construct an innovative methodology to perform data analytics in a way that returns an automaton as a recognizer of the data language: a Field Theory of Data. We suggest to build, directly out of probing data space, a theoretical framework enabling us to extract the manifold hidden relations (patterns) that exist among data, as correlations depending on the semantics generated by the mining context. The program, that is grounded in the recent innovative ways of integrating data into a topological setting, proposes the realization of a Topological Field Theory of Data, transferring and generalizing to the space of data notions inspired by physical (topological) field theories and harnesses the theory of formal languages to define the potential semantics necessary to understand the emerging patterns
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