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Lecture notes series, Institute For Mathematical Sciences

Mathematical Horizons for Quantum Physics

Huzihiro Araki, Berthold‐Georg Englert, L. C. Kwek, Jun Suzuki

Published Jun 1, 2010• 7 citations
Publisher

Abstract

Control of the Molecular Alignment or Orientation by Laser Pulses (A Keller) Quantum Computing and Devices: A Short Introduction (Z-G Zhang et al.) Dynamics of Mixed Classical-Quantum Systems, Geometric Quantization and Coherent States (H R Jauslin & D Sugny) Quantum Memories as Open Systems (R Alicki) Two Mathematical Problems in Quantum Information Theory (A S Holevo) Dissipatively Induced Bipartite Entanglement (F Benatti) Scattering in Nonrelativistic Quantum Field Theory (J Derezinski) Mathematical Theory of Atoms and Molecules (V Bach).

DOI
10.1142/7797

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The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies.
We consider deflection of polarizable molecules by inhomogeneous optical fields and analyze the role of molecular orientation and rotation in the scattering process. It is shown that molecular rotation induces spectacular rainbowlike features in the distribution of the scattering angle. Moreover, by preshaping molecular angular distribution with the help of short and strong femtosecond laser pulses, one may efficiently control the scattering process, manipulate the average deflection angle and its distribution, and reduce substantially the angular dispersion of the deflected molecules. We study the problem both classically and quantum mechanically and arrive at the same conclusions in both treatments. The effects of strong deflecting field on the scattering of rotating molecules are considered by the means of the adiabatic invariants formalism. The suggested control scheme opens new ways for many applications involving molecular focusing, guiding, and trapping by optical and static fields.
Ultracold molecules are expected to find applications in cold chemistry, quantum phases, precision measurements and quantum information. In this thesis three novel applications of cold molecules are studied. First the thesis presents a general method for coherent control of collisions between non-identical particles. It is shown that by preparing two alkali-metal atoms in a superposition of hyperfine states, the elastic-to-inelastic cross section ratio can be manipulated at ultracold temperatures by tuning laser parameters in the presence of a magnetic field. The static field is needed to induce quantum interference between scattering states. Extensions of this scheme for ultracold molecular reactive scattering are discussed. Second, the thesis describes rotational excitons and polarons in molecular ensembles trapped in optical lattices. Rotational excitons can be manipulated using static electric and magnetic fields. For a one-dimensional molecular array with substitutional impurities any localized exciton state can be delocalized by applying a suitable electric field. The electric field induces correlations between diagonal and off-diagonal disorder. It is also shown that the translational motion of polar molecules in an optical lattice can lead to phonons. The lattice dynamics and the phonon spectrum depend on the strength and orientation of a static electric field. An array of polar molecules in an optical lattice can be described by generalized polaron model with tunable parameters including diagonal and off-diagonal exciton-phonon interactions. It is shown that in a strong electric field the system is described by a generalized Holstein model, and at weak electric fields by the Su-Schrieffer-Heeger (SSH) model. The possibility of observing a sharp polaron transition in the SSH model using polar alkali-metal dimers is discussed. Finally, the thesis presents a method to generate entanglement of polar molecules using strong off-resonant laser pulses. Bipartite entanglement between alkali-metal dimers separated by hundreds of nanometers can be generated. Maximally entangled states can be prepared by tuning the pulse intensity and duration. A scheme is proposed to observe the violation of Bell’s inequality based on molecular orientation correlation measurements. It is shown that using a combination of microwave and off-resonant optical pulses, arbitrary tripartite and many-particle states can be prepared.