Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.
Two types of working memory (WM) have recently been proposed: (i) active WM, relying on sustained neural firing, and (ii) activity-silent WM, for which firing returns to baseline, yet memories may be retained by short-term synaptic changes. Activity-silent WM in particular might also underlie the recently discovered phenomenon of non-conscious WM, which permits even subliminal stimuli to be stored for several seconds. However, whether both states support identical forms of information processing is unknown. Theory predicts that activity-silent states are confined to passive storage and cannot operate on stored information. To determine whether an explicit reactivation is required before the manipulation of information in WM, we evaluated whether participants could mentally rotate brief visual stimuli of variable subjective visibility. Behaviorally, even for unseen targets, subjects reported the rotated location above chance after several seconds. As predicted, however, at the time of mental rotation, such blindsight performance was accompanied by (i) neural signatures of consciousness in the form of a sustained desynchronization in alpha/beta frequency and (ii) a reactivation of the memorized information as indicated by decodable representations of participants’ guess and response. Our findings challenge the concept of genuine non-conscious “working” memory, argue that activity-silent states merely support passive short-term memory, and provide a cautionary note for purely behavioral studies of non-conscious information processing.
Abstract. We prove an analogue of Shnirelman, Zelditch and Colin de Verdiè-re's Quantum Ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the coupling parameter, in particular the equidistribution holds for both the weak and strong coupling quantizations of the point scatterer.
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