A Quantum/classical Entropy Concept for Measuring Phase-space Localization

We present an entropy concept measuring phase-space localization in dynamical systems based on time-averaged phase-space densities. This entropy has a direct classical counterpart; its local scaling with ln h ?1=2 is the information dimension of the underlying invariant sets of the classical dynamics. The proposed entropy concept allows to visualize the global phase-space structure of the quantum dynamics and by comparison with the corresponding classical entropy to detect and quantify quantum localization. 1 The entropy concept Basis quantity is the time-averaged probability density obtained from the time-evolution of an initial coherent state, i.e. j(t=0)i=jp; qi , %(p; q; p 0 ; q 0) = lim T !1 1 T Z T 0 dt jhp 0 ; q 0 j(t)ij 2 : (1) The entropy S(p; q) = ? Z dp dq 2 h %(p; q; p 0 ; q 0) ln %(p; q; p 0 ; q 0) (2) measures then the spreading of this initial Gaussian in phase space. This entropy has two important properties which quantify it as a suitable measure for quantum localization. One advantage is that due to the time-averaging process, the innuence of inherent quantum uctuations on the entropy (2) is reduced to order h. A comprehensive discussions of the uctuation properties and the relation to other measures of localization like Wehrl's entropy can be found in 2. Most important is that the quantum entropy (2) has a direct classical counterpart which can be deened in a entirely analogous manner on basis of time-averaged coarse-grained phase-space distributions. This classical density and thus the classical entropy depends on h as a coarse graining parameter; for details see 3;2. The purely classical entropy had already been introduced by N u~ nez et al. 4 as an indicator of chaos in models of celestial dynamics.