We show that rapid decoherence, usually associated with chaotic dynamics, is not necessarily a hallmark of nonintegrability: border effects in integrable systems may produce similarly drastic decoherence rates. These can be found when the subsystem under observation possesses an energy limitation as, e.g., in the N-atom Jaynes-Cummings model. We show for this model that special initial coherent wave packets exhibit entropy production rates strikingly similar to the chaotic case. Also, a ~de!localization phenomenon is found to be a function of the proximity to the phase-space border. @S1063-651X~99!01611-6# PACS number~s!: 05.45.Mt, 32.80.Qk The quantum entanglement process, decoherence, and the quantum $ classical transition @1# have recently attracted much attention from physicists, both theoretical and experimental @2#. Creating entangled quantum states in the laboratory is now possible in ion trap experiments @3# and also with atoms in high-Q cavities @4#. Both of them realize a simple situation in which a two-level atom is coupled to a quantized harmonic oscillator by means of the Jaynes-Cummings model ~JCM !@ 5 # .This simple model has a long and frequent history as a convenient laboratory for testing theoretical predictions @6#, being expected nowadays to serve also in practical implementations. The physics of this kind of system, where two or more atomic levels interact with a single-mode electromagnetic field, is usually explored by means of quantities such as the population inversion and the mean number of photons. These quantities revealed, among other phenomena, the existence of collapse and revival regions in the curves of population inversion @6,7#. One could then infer that the field and atom lose their identity in the collapse region, and most closely return to their initial states during the revival. However, if one is concerned with the entanglement of the atomic and fieldsubsystems, the population inversion can be a misleading quantity, particularly with respect to the purity of the quantum state. In fact, the works of Phoenix and Knight @8# and that of Gea-Banacloche @9# have shown that this system may greatly recover its purity during the very collapse interval, at half the revival time. In these papers, instead of population inversions, reduced density operators were used in calculating either the system’s entropy or idempotency defect ~linear entropy!. Here we are interested in the decoherence process of systems constituted by subsystems with dissimilar Hilbert spaces. To this end, we take the N-atom JCM, where N two-level atoms interact with a single-mode field and, in view of the results just cited, we adopt the idempotency defect as a measure of the entanglement between the atomic subsystem and the field one. We note incidentally that, compared with the entropy, the calculations of the linear entropy are easier and convey essentially the same information. From another point of view, we want to explore the quantum $ classical connection and possible differences in the decoherence process in integrable and nonintegrable situations. For these reasons, we choose conditions which can be mimicked over the classical phase space. At this point a distinction is noteworthy between the here-called ‘‘quasiclassical’’ treatment of the JCM @10#—where one treats the atomic part quantum mechanically and the field one classically— and our ‘‘semiclassical’’ treatment—where the classical limit is taken for both atomic and fieldquantum subsystems. In the latter context, theoretical investigations using several models suggest that systems which are chaotic in the classical limit decohere rapidly @11,12#. This is also true for the N-atom JCM @13# in its nonintegrable version @14#. Moreover, for the specificcase of Ref. @14#, a connection between the entanglement process and the associated classical structures has been investigated. One of the main results of that investigation is the presence of some sensitivity to where in the classical phase space one places the center of the initial quantum coherent wave packets. It is now a well accepted fact that the decoherence rate is larger for chaotic systems than for integrable ones. We argue here that this belief that the fastest decoherence is to be attributed to chaotic regimes can be misleading in some cases. Such a phenomenon is particularly conspicuous when a ~smaller! subsystem under observation has a finite Hilbert space, a ~larger! subsystem coupled to it does not have such a restriction, and the global system is prepared in a state with mean energy larger than the amount allowed for the smaller subsystem. A phase-space description then reveals the presence of a ‘‘border’’ associated with the degrees of freedom of the smaller subsystem. In such a situation, initial conditions that drive the classical motion to the proximities of this border lead to decoherence rates strikingly similar to those of typical chaotic situations, even if the system is completely integrable. Moreover, another interesting phenomenon related to this is shown to occur: the proximity of the border tends to delocalize the wave packet, whereas for times when the dynamical evolution dictates a departure from the border there is a clear tendency to relocalize the wave packet in the sense that it recovers quantum coherence. These results are shown by comparing the classical and quantum description of the N-atom JaynesCummings model, whose experimental realization is feasible in cavity QED setups. We present arguments according to