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Journal of Mathematical Physics

The Quantum-Mechanical Scattering Problem. II. Multi-Channel Scattering

Thomas F. Jordan

Published May 1, 1962• 8 citations
Publisher

Abstract

The study reported in an earlier paper of the single channel scattering problem is extended to include the multi-channel case.
DOI
10.1063/1.1724243

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The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.
Abstract Efimov physics in the vicinity of two overlapping narrow Feshbach resonances can be explored within a framework of a three-channel model where a non-interacting open channel is coupled to two closed molecular channels. Here, we determine how it compares to the extended two-channel model, which includes an open channel with finite background scattering and a single molecular channel. We identify the parameter range in which the three-channel model surpasses the extended two-channel model. Furthermore, the three-channel model is extended to include background scattering, and then both models are applied to the experimentally relevant system of bosonic lithium atoms polarized on two different energy levels, with an isolated and two overlapping narrow Feshbach resonances, respectively. We confirm, in agreement with previous studies, that being small, the background scattering length in lithium has a negligible effect on the Efimov features in the case of isolated resonance. However, in the case of overlapping Feshbach resonances, the inclusion of background scattering improves the performance of the theory with respect to the experimentally measured position of the Efimov resonance.
In this paper, two conclusive three-party cyclic assisted cloning protocols in amplitude damping (AD) channel are put forward that, respectively clone three arbitrary unknown single-qubit states and single-qutrit states with the help of a state preparer. Each of our protocols includes three consecutive stages: quantum channel preparation, cyclic quantum teleportation (CQT), and multi-party assisted cloning. The first stage of each protocol proposes the detailed processes of sharing a pure entangled quantum state as a component of a quantum channel in AD channel via entanglement compensation. In second stage, a three-party CQT is presented where three unknown single-qubit states (or single-qutrit states) are reconstructed simultaneously in three different places, respectively, by introducing auxiliary qubits and performing appropriate operations. In the third stage, the state preparer Victor performs one multi-qubit measurement (or one unitary transformation and one multi-qutrit measurement) and informs the three communicators of his outcome, three distinct unknown single-qubit states or their orthogonal complement states (or single-qutrit states) are cloned simultaneously and with probability at three separate locations,respectively. Furthermore, we extend the above protocols from two aspects: (i) the extension to the case of [Formula: see text] participants; (ii) extension to the case of d-dimensional unknown single-qudit state cycle-assisted cloning.