Multichannel quantum defect theory: a quantum Poincaré map

The multichannel quantum defect theory (MQDT) can be reinterpreted as a quantum Poincaré map in representation of angular momentum. This has two important implications: we have a paradigm of a true quantum Poincaré map without semiclassical input and we get an entirely new insight into the significance of MQDT. PACS: 05.45.Mt; 33.80.Rv; 03.65.Sq In recent years there has been a rapidly growing interest in the quantum Poincaré map (QPM) [1–10], i.e. the quantization of a classical Poincaré map, for a time independent Hamiltonian system. Bogomolny[1] started out with a semi-classical formulation. Among other things he shows that unitarity of the representation is reached in the limit ~ → 0, while this is only approximate for finite ~ [8]. Prosen [4] gives an elegant general solution to the unitarity problem at the expense of obtaining an infinite matrix for the QPM. The semiclassical approach common to most discussions causes a number of problems that make the use of this new and powerful tool a little obscure. With other words, the quantum Poincaré section implicitly defined by Bogomolny, lacks a paradigmatic example where a quantum treatment can be performed properly throughout and leads to a finite unitary matrix. Multichannel quantum defect theory (MQDT) [11–15] and its classical limit [16–18] will be shown to provide the framework for such a paradigm. Indeed 1 Present address: Fachbereich 7, Physik, Universität G. H. Essen, 45117 Essen 2 Corresponding author; Fax: +33 476 514 544; e-mail: [email protected] Preprint submitted to Elsevier Preprint 5 February 2008 we shall see that a simplified model of the Rydberg molecule allows to construct a classical Poincaré map on the unit sphere, whose exact quantization is provided by MQDT. Thus the result is necessarily entirely quantal, exactly unitary and for finite ~ given in terms of a finite matrix. We shall show that the results commonly derived for MQDT are directly properties of the unitary representation of this classical map as obtained by MQDT. After a short description of the model for a Rydberg molecule and the simplification introduced in Ref. [16], we proceed to give the quantum map for this case explicitly. We illustrate the two important aspects of our result by two applications. First the new interpretation allows modifications of the MQDT method, that prove particularly effective in near integrable systems. Second we proceed to show by way of examples that the properties of this map are relevant to the study of chaos and order in this system. Simplifying to the most basic case, these molecules can be viewed as a rotating system with positive charge and cylindrical symmetry that binds one electron in an orbit that is at large distances hydrogenic. The classical limit of the MQDT is the following classical model [16]: The motion is composed of two consecutive steps. (i) when the electron is far from the molecular core (i.e. most of the time for a Rydberg electron) it feels only the Coulomb part (−1/r) of the potential. Its orbit is hydrogenic and its angular momentum L is fixed in the laboratory reference frame. Meanwhile the core rotates freely with an angular momentum N which is also fixed in the laboratory frame. The total angular momentum J = L+N is always conserved. In the molecular reference frame, the OZ axis is the cylindrical symmetry axis of the core. The core angular momentum N points in a perpendicular direction, taken as the OX axis. The angles θe and φe are the polar and azimuthal angles respectively of the electronic angular momentum L in this frame. During this step, L rotates freely around the OX axis. (ii) during the so called ”collision” step, the electron senses also the cylindrically symmetric short range part of the potential of the core. Aside from the energy and J, the projection of L onto the core axis Λ = L cos θe is conserved due to the cylindrical symmetry of the core. We will add an extra, simplifying, hypothesis, namely that the magnitude L of L remains constant[16]. This is justified for Rydberg Molecules at least for small L’s, but the classical and quantum map with this approximation exist for all L. Thus the collision can be described by a θe–dependent rotation of L around the core axis. The simplest form of this rotation compatible with the symmetry is [16]: δφe = K cos θe, where K is a coupling constant. This simplification is not essential. Notice further that the conservation of the total angular momentum J implies that the molecular core feels a simultaneous recoil which changes the direction and magnitude of N. This change of N in turn entails a change of the rotational energy EN of the core and because of conservation of total energy a change of the energy Ee of the electron. This exchange of energy makes this model much richer than the kicked spin model [19] (which