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arXiv:physics/0406149

Complex States of Simple Molecular Systems

R. Englman, Asher Yahalom A

Published Jun 29, 2004• 0 citations
arXivHTMLPDF

Abstract

A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.

DOI
10.48550/arxiv.physics/0406149

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A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.
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Theories with indefinite metric have a long and very rich history. Such theories are being used time to time in searching the solutions for many outstanding problems in physics over the several decades. However much attentions were not paid for these as they were not mathematically consistent with usual quantum theories. Towards the end of last century this field of research received huge boost, when attempts had been made to accommodate certain class of non-Hermitian theories in rigorous mathematical formulation. Subsequently it has been shown that certain specific types of non-Hermitian theories can lead to fully consistent quantum theories with complete real spectrum, unitary time evolution and probabilistic interpretation in a modified Hilbert space. Due to this important realisation, theories with complex Hamiltonian have become the topic of frontier research over last two decades and enormous applicationswere found in various branches of physics. In this brief note we try to review certain aspects of such theories where we have contributed. In particular, we discuss PT phase transition in non-relativistic, relativistic and quantum field theoretic systems and some new features of scattering in complex systems.
Abstract The general representation theory of quantum mechanics requires one to represent the states of a dynamical system by functions of a set of real variables q'r, each of which has a domain consisting of either discrete points or a continuous range of points, or possibly both together. A dynamical variable is represented by a function of two such sets of real variables q'r and q"r forming a generalized “matrix". In this paper we shall show that in certain cases it is advantageous to consider some of our variables q'r as complex variables and to suppose the representatives of states and dynamical variables to depend on them in accordance with the theory of functions of a complex variable. In the usual theory the domains of the s are the eigenvalues of certain observables q"r. This significance of the q"s of course gets lost when we consider them as complex variables, but we have, however, some beautiful mathematical features appearing instead, and we gain a considerable amount of mathematical power for the working out of particular examples.
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Coherent control is a technique to manipulate the amplitudes and phases of wave functions by exploiting the coherent property of light. It is of great importance for the development of quantum technologies including quantum information processing and chemical reaction control as well as for addressing fundamental problems in quantum mechanics such as the quantum–classical boundary. In this chapter, we describe the development and applications of coherent control based on wave-packet interferometry with attosecond precision, which has been made possible by tuning the delay between two laser pulses on the attosecond time scale. This technique has been applied to several different hierarchies from simple to complex quantum systems: (i) isolated iodine molecules in the gas phase, (ii) molecular crystals of para-hydrogen, and (iii) single crystals of bismuth. In system (i), we have controlled and visualized the interference of vibrational wave packets in the isolated iodine molecules, and that wave-packet interference has been utilized to develop a molecular computer, which executes ultrafast Fourier transform. In system (ii), we have manipulated vibrational wave-functions delocalized in a para-hydrogen crystal. The long coherence lifetime of para-hydrogen crystals is a promising resource to preserve quantum information in bulk solids. In system (iii), we have controlled the excitation ratio of two orthogonal coherent phonon modes, whose coherent superposition has allowed for active control of ultrafast two-dimensional atomic motions in a single crystal of bismuth. Those atomic motions have been visualized by density functional theory calculations. Finally, the coherent control of an ensemble of ultracold Rydberg atoms is briefly discussed as future perspectives for exploring the quantum–classical boundary.