We show that quantum effects modify the decay rate of Poincaré recurrences P (t) in classical chaotic systems with hierarchical structure of phase space. The exponent p of the algebraic decay P (t) ∝ 1/t p is shown to have the universal value p = 1 due to tunneling and localization effects.Experimental evidence of such decay should be observable in mesoscopic systems and cold atoms.PACS numbers: 05.45.+b, 03.65.SqThe general structure of classical phase space in chaotic Hamiltonian systems displays a hierarchical mixture of integrable and chaotic components down to smaller and smaller scales [1]. This complicated structure leads, in particular, to an anomalous power law decay of Poincaré recurrences P (t) and correlations C(t) inside the chaotic components [2,3]. Physically, such slow decay appears due to a decrease, down to zero, of the diffusion rate for a trajectory when it approaches the chaos border determined by some critical invariant curve [2][3][4][5][6]. Typically, P (t) ∼ C(t)/t ∼ t −p with p ≈ 1.5 [2]. As a result the integrated correlation function, which determines the diffusion rate (D ∼ Cdt), can diverge thus leading to a super-diffusive propagation [7]. Such effects are important for electron dynamics in super-lattices where usually the phase space has a mixed structure [8].The above anomalous properties had been studied in great detail for classical systems [2][3][4][5][6][7][8]. However, the question how they are affected by quantum dynamics was not addressed up to now. This problem becomes more and more important not only due to its fundamental nature but also in the light of recent experiments with mesoscopic systems. Indeed, different types of ballistic quantum dots can now be studied in laboratory experiments [9] and the phase space in such systems generally has a mixed structure. Since, the probability to stay in a given region is directly related with P (t) and C(t), its slow decay can significantly affect conductance properties. In particular it has been proposed that such decay should lead to fractal conductance fluctuations [10], the experimental observation of which has been reported recently [11]. According to [10,11] the fractal exponent σ for conductance fluctuations is directly related to the exponent p as σ = 2 − p/2.A different type of systems in which such effects should be observable experimentally is given by cold atoms in external laser fields where the Kicked Rotator model of quantum chaos has been built experimentally [12,13]. Possibilities of experimental investigation of slow probability decay in such systems has been discussed recently [14].The experimental studies of slow power correlation decay in the regime of quantum chaos are also important from the fundamental point of view, since here the typical scale of correlation decay is much larger than the Ehrenfest time scale t E ∼ ln 1/h on which the minimal coherent wave packet spreads over the avaible phase space. To the best of our knowledge the comparison of classical and quantum correlations in such regime has not ...

We study quantum chaos in open dynamical systems and show that it is characterized by quantum fractal eigenstates located on the underlying classical strange repeller. The states with longest life times typically reveal a scars structure on the classical fractal set.