We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the non-unitary quantum propagator, and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as → 0 on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates {ψ( )} →0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker map, for which the probability density in position space is observed to have self-similarity properties. In open systems the lack of unitarity of the quantum propagator gives rise to non-orthogonal decaying eigenstates with complex energies (resonances), the imaginary parts of which are interpreted as decay rates. In the case of open chaotic systems the classical mechanics is structured in phase space around fractal sets associated with trajectories that remain trapped for infinite times, either in the future (forward-trapped set K + ) or in the past (backward-trapped set K − ). The mean density of resonances is believed (but not in general proved) to be determined by the fractal dimension of the invariant set K 0 = K + ∩ K − , the classical repeller. This is the fractal Weyl law [4,5,6]. (Note that this is different to the resonance statistics in weakly open systems, for which the size of the opening vanishes in the semiclassical limit [7]).Much less is known about the resonance (or Gamow) eigenstates. These are important in many areas of physics [8] and chemistry [9], because they have marked influence on observable quantities such as scattering cross sections and reaction rates (they are a component of the Siegert pseudostates basis in terms of which the scattering wavefunctions and S matrix, for example, can conveniently be expanded [8]). Following the well established idea that in the semiclassical limit timeindependent quantum properties should be related to time-independent classical sets, it is natural to expect that long-lived eigenstates of open systems should be determined by the structure of K + and K − . This was tested numerically for some right eigenstates of the open kicked rotator in [10], where the term 'quantum fractal eigenstates' was coined.We here significantly extend the notion of quantum fractal eigenstates in several new directions. First, we draw the important distinctions between left and right eigenstates of the non-unitary propagator, and between states that are 'short-lived' and 'long-lived' with respect to the Ehrenfest time. Second, we show that in the semiclassical limit the long-lived left eigenstates concentrate on K + , while the long-lived right eigenstates concentrate on K − . In chaotic systems the eigenstates thus inherit the intricate fractal structure of the underlying classical trapped sets (this property has also been observed by Nonnenmacher and Rubin [11]). Third, we find that in the s...

Abstract.We present a trajectory-based semiclassical calculation of the full counting statistics of quantum transport through chaotic cavities, in the regime of many open channels. Our method to obtain the mth moment of the density of transmission eigenvalues requires two correlated sets of m classical trajectories, therefore generalizing previous works on conductance and shot noise. The semiclassical results agree, for all values of m, with the corresponding predictions from random matrix theory.

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The statistical properties of quantum transport through a chaotic cavity are encoded in the traces T n =Tr͓͑tt † ͒ n ͔, where t is the transmission matrix. Within the random matrix theory approach, these traces are random variables whose probability distribution depends on the symmetries of the system. For the case of broken time-reversal symmetry, we use generalizations of Selberg's integral and the theory of symmetric polynomials to present explicit closed expressions for the average value, and for the variance of T n for all n. In particular, this provides the charge cumulants ͗͗Q n ͘͘ of all orders. We also compute the moments ͗g n ͘ of the conductance g = T 1 . All the results obtained are exact, i.e., they are valid for arbitrary numbers of open channels.

We propose a matrix model which embodies the semiclassical approach to the problem of quantum transport in chaotic systems. Specifically, a matrix integral is presented whose perturbative expansion satisfies precisely the semiclassical diagrammatic rules for the calculation of general counting statistics. Evaluating it exactly, we show that it agrees with corresponding predictions from random matrix theory. This uncovers the algebraic structure behind the equivalence between these two approaches, and opens the way for further semiclassical calculations.

Explicit formulas are obtained for all moments and for all cumulants of the electric current through a quantum chaotic cavity attached to two ideal leads, thus providing the full counting statistics for this type of system. The approach is based on random matrix theory, and is valid in the limit when both leads have many open channels. For an arbitrary number of open channels we present the third cumulant and an example of non-linear statistics.Comment: 4 pages, no figures; v2-added references; typos correcte

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We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest-living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long-lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.