Coordinate Analysis for Interpreting the Decoherence in the Coherent NO with Ar Collision: A Physico-mathematical Picture Using the Stereographic Projection and the Cusp Catastrophe

Abstract

We present here a physico-mathematical picture for explaining the unexpectedly large decoherence cross-section (almost 10 times larger than its gas-kinematic cross-section) recently observed by Urena and coworkers in their scattering experiment involving a coherent NO beam with Ar gas. The present topological picture consists of a stereographic projection and the cusp catastrophe theory of Thom, and we find that this model enables us to clarify the origin of the collisional decoherence. From the view of the stereographic projection, we can naturally introduce the wave property originating from the singular point at the “North pole” on the circumference S1 coordinate corresponding to a critical point for the collisional decoherence (condition 1). This picture also predicts the sudden changes of wave-phase collapse due to network interaction in the many-body system (condition 2). Thus it is hoped that the model proposed by Urena et al. based on the dipole-induced dipole interaction in the NO + Ar system could be modified through this picture by including interactions with many Ar atoms in the environment. One way to fill the gap between the single-pair interaction picture and the multiple interaction one would be to employ theoretical calculations by use of the density matrix theory with and without adding the second Ar atom to the NO–Ar system. The cusp catastrophe theory reinforces the necessity of some cooperative network interaction between the coherent NO molecule and many neighboring Ar atoms and provides a qualitative scenario in which the whole system leads to a sudden change of the collisional decoherence of NO as a function of the control parameters (a, b). At this stage, the present physico-mathematical picture cannot give any specific values of the decoherence distance by the theory itself, but it clearly provides us a new topological concept for clarifying the origin of collisional decoherence which is strongly connected with the complexity of the system. Thus it gives us a global guide map toward further clarification of the collisional decoherence phenomenon with the aid of more sophisticated quantum mechanical calculations in the future.