Quantum chaotic scattering

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Pierre Gaspard (2014), Scholarpedia, 9(6):9806. doi:10.4249/scholarpedia.9806 revision #143296 [link to/cite this article]
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Curator: Pierre Gaspard

Quantum chaotic scattering refers to the scattering of quantum-mechanical or other waves in deterministic systems without quenched disorder, yet manifesting chaotic classical motion in the zero-wavelength limit. Chaotic scattering typically generates irregular energy spectra of scattering resonances, hindrance to wavepacket decay with respect to the classically expected behavior, as well as speckle diffraction patterns. Quantum chaotic scattering has been studied in atomic, molecular, and nuclear systems, in mesoscopic electronic and optical devices, in microwave and acoustic cavities, among other systems (see Figure 1).

Figure 1: Examples of potentials manifesting quantum chaotic scattering: (a) Potential for the scattering of an electron on the screened Coulomb potentials of three atoms in a molecule. (b) Potential of an electron in ballistic transport in a four-lead mesoscopic device. (c) Potential of three atoms in the collinear chemical reaction A+BC\(\,\rightleftharpoons\,\)AB+C. \(r_{\rm A-BC}\) is the distance between the atom A and the diatomic molecule BC. \(r_{\rm AB-C}\) is the distance between the diatomic molecule AB and the atom C.


Historical developments

The interest for chaotic scattering rose in the late eighties with the discovery of chaotic dynamics in open classical systems modelling satellite encounters (Petit and Hénon 1986, Hénon 1988), unimolecular chemical reactions (Noid et al. 1986), potential scattering (Eckhardt and Jung 1986, Jung 1986, Jung and Scholz 1987), or bouncing balls (Eckhardt 1987). These studies revealed the genericity of chaotic behavior among scattering systems where integrability turns out to be exceptional. A review of classical chaotic scattering is given in Ref. (Seoane and Sanjuán 2013).

In the eighties also, the field of quantum chaology had undergone fundamental advances in our knowledge of bounded quantum systems and their irregular discrete energy spectra (Berry 1987, Haake 2001). The Bohr-Sommerfeld quantization rule had been extended to classically chaotic systems with semiclassical periodic-orbit theory (Gutzwiller 1990) and the quantum spectral statistics of classically chaotic bounded systems was conjectured by Bohigas, Giannoni, and Schmit (1984) to have the universality predicted by random matrix theory (Mehta 1967, Porter 1965).

At the crossroad between these lines of research, the question arose whether there might exist signatures of classical chaos in wave scattering. A natural approach to address this issue with some generality is to use the semiclassical method, which extends results already obtained for an abstract model that had been exactly solved in terms of the Riemann zeta function (Gutzwiller 1983, Series 1987, Wardlaw and Jaworski 1989). For scattering systems, the observable quantities are the cross sections, the transmission probabilities, the quantum time delay, or the scattering resonances and their lifetime. Between the late eighties and the beginning of the nineties, signatures of chaos were indeed discovered in the fluctuations of these quantities as the energy or an external magnetic field is varied. An early result (Blümel and Smilansky 1988) showed that the characteristic energy scale of these fluctuations is related to the rate of escape from the chaotic scatterer in a way that is reminiscent of the phenomenon of Ericson fluctuations first observed in nuclear physics during the sixties (Ericson 1960). These considerations appeared to be of great importance for conductance fluctuations in ballistic electron microconductors (Jalabert et al. 1990, Baranger et al. 1993a, Baranger et al. 1993b, Lin et al. 1993, Nakamura and Harayama 2004). Moreover, it was proved with the semiclassical approach that the quantum lifetimes of a classically chaotic scatterer can be longer than classically expected (Gaspard and Rice 1989a, Gaspard and Rice 1989b, Gaspard and Rice 1989c). This effect is a manifestation of the quantum suppression of chaos. Since dynamical chaos provides multiple paths to wave propagation, the interference between these paths is causing the hindrance of the classically expected escape. These fluctuation effects have been experimentally demonstrated in ballistic electron transport through mesoscopic semiconductor devices (Marcus et al. 1992, Marcus et al. 1993, Chang et al. 1994, Chan et al. 1995), as well as in microwaves cavities (Stöckmann 1999, Kuhl et al. 2005, Lu et al. 1999, Lu et al. 2000, Pance et al. 2000, Barkhofen et al. 2013). Similar effects have also been investigated in atomic and molecular systems (Main and Wunner 1992, Stania and Walther 2005, Xu et al. 2008, Reid and Reisler 1994).

A complementary approach is based on random matrix theory (Lewenkopf and Weidenmüller 1991, Fyodorov et al. 1997). If the semiclassical approach is suitable for systems with a few degrees of freedom, the random matrix approach can also be extended to many-body systems. The random matrix approach has been developed since the pioneering work by Wigner in the fifties to understand irregular features in the energy spectra of nuclear scattering processes, especially in the regime of isolated or weakly overlapping resonances (Porter 1965, Alt et al. 1995, Mitchell et al. 2010). Since the early nineties, random matrix theory has also been used to understand the quantum fluctuations of electronic conductance in mesoscopic devices (Beenakker 1997). Recently, the Bohigas-Giannoni-Schmit conjecture has been explained on the basis of periodic-orbit semiclassical theory (Sieber et al. 2001, Sieber 2002, Müller et al. 2004, Müller et al. 2005, Heusler et al. 2007, Müller et al. 2009). This fundamental result justifies the extension of random matrix theory to weakly open chaotic scattering systems. The random matrix approach has been systematically developed in particular for nuclear systems (Mitchell et al. 2010), microwave resonators (Stöckmann 1999, Kuhl et al. 2005, Dietz et al. 2010), and ultracold atomic or molecular collisions (Mayle et al. 2012, Mayle et al. 2013, Frisch et al. 2014).

One of the most recent developments is the evidence for a fractal Weyl law in the distribution of the scattering resonances. In analogy with Weyl's law for the distribution of energy eigenvalues in bounded quantum systems, the number of scattering resonances in a given energy range has been shown to scale with energy in a way that reveals the fractal dimension of the set of classical orbits composing the chaotic scatterer (Lin and Zworski 2002, Lu et al. 2003, Nonnenmacher and Zworski 2005, Shepelyansky 2008, Eberspächer et al. 2010, Potzuweit et al. 2012). This scaling law is fundamental in order to establish the universality of the scattering fluctuation properties on small energy scales.

Theoretical results obtained on quantum chaotic scattering are presented here below, together with selected examples and applications. Since this topic is deeply rooted in quantum scattering, some elements of this theory are first summarized in the following section, before coming to chaotic scattering.

Quantum scattering

Wave equation and propagator

In quantum mechanics, the scattering of a nonrelativistic particle of mass \(m\) on obstacles is described by Schrödinger's wave equation \[ i\, \hbar \, \partial_t \, \psi = \hat H \, \psi \, , \] where \(i=\sqrt{-1}\), \(\hbar\) is Planck's constant, \(t\) the time, \(\psi=\psi({\mathbf r},t)\) the wavefunction, \({\mathbf r}\in{\mathbb R}^d\) the position of the particle in a space of dimension \(d\), and \(\hat H\) the Hamiltonian operator. This latter is given by \[ \hat H = - \frac{\hbar^2}{2m}\, \nabla^2 + V({\mathbf r}) \] with a potential \(V({\mathbf r})\) featuring the energy landscape caused by the obstacles and the Laplacian operator \(\nabla^2=\partial_{\mathbf r}^2\). If the obstacles are located at finite distance from each other, the potential may be assumed to vanish at large distances from the obstacles, \(\lim_{{\mathbf r}\to\infty} V({\mathbf r})=0\). Accordingly, the Hamiltonian operator naturally decomposes as \(\hat H=\hat H_0+\hat V\) into the Hamiltonian \(\hat H_0\) of the free particle moving at large distance from the obstacles and the potential \(\hat V\). Examples of such potentials are depicted in Figure 1.

We notice that there exist other scattering problems such as the ballistic transport of electrons in semiconducting circuits composed of several connected leads in the form of rectilinear waveguides. In this case, the potential has asymptotically certain constant shapes in the direction of each lead. This is also the case for triatomic chemical reactions, A+BC\(\,\rightleftharpoons\,\)AB+C, in one dimension. Here, the potential is asymptotically equal to the potential of the diatomic molecule AB (resp. BC) as the distance between AB and C (resp. A and BC) goes to infinity, which is shown at right hand in Figure 1. In such multichannel systems, the Hamiltonian decomposes as \(\hat H=\hat H_{0c}+\hat V_{c}\) into the asymptotically free Hamiltonian \(\hat H_{0c}\) and the interaction potential \(\hat V_{c}\) for every possible channel \(c\) (Goldberger and Watson 2004, Joachain 1975, Newton 2002, Taylor 2000).

In billiards, the potential is infinitely steep at the boundaries so that the Hamiltonian is the one of a free particle outside the obstacles and suitable boundary conditions should be specified at the borders of the obstacles (Stöckmann 1999).

These considerations also apply mutatis mutandis to other wave phenomena: electronic waves in crystals or on surfaces, electromagnetic or acoustic waves in various media such as photonic crystals, microlasers, elastic media, etc... (Rex et al. 2002, Søndergaard et al. 2002, Wirzba et al. 2005, Lebental et al. 2007, Wang et al. 2013).

Because it is Hermitian, the Hamiltonian operator has a spectrum composed of discrete eigenvalues at negative energies \(\{E_b\}\) corresponding to the bound states of the particle in the potential and a continuous spectrum at the positive energies \(0<E<\infty\) associated with the unbound states asymptotically described by \[ \psi_t({\mathbf r}) \simeq_{r\to\infty} {\rm e}^{-\frac{i}{\hbar} E t} \left[ {\rm e}^{\frac{i}{\hbar} {\mathbf p}\cdot{\mathbf r}} + \frac{f({\mathbf n};E)}{r^{d/2}} \, {\rm e}^{\frac{i}{\hbar} p\, r}\right] , \] where \(\mathbf p\) is the momentum vector of the incident particle, \(p=\Vert{\mathbf p}\Vert=\sqrt{2mE}\) its magnitude, \(r=\Vert{\mathbf r}\Vert\) the radial distance from the center of the scatterer, \({\mathbf n}={\mathbf r}/r\) a unit vector pointing from this center to the point \(\mathbf r\) of observation, and \(d\) the dimension of space. The scattering amplitude \(f({\mathbf n};E)\) gives the differential cross section in the direction of the unit vector \(\mathbf n\) as \[ \frac{d\sigma}{d\Omega} = \vert f({\mathbf n};E)\vert^2 \, , \tag{1} \] where \(d\Omega\) is the \((d-1)\)-dimensional angular element.

The time evolution of a wavepacket from some initial condition \(\psi_0\) is described by the unitary operator \[ \psi_t = {\rm e}^{-\frac{i}{\hbar} \hat H t} \psi_0 \equiv \hat K(t) \, \psi_0 \, , \tag{2} \] which defines the propagator. By Fourier analysis and using the Heaviside function \(\theta(t)\) selecting the positive times, the retarded propagator can be expressed as \[ \hat K_{\rm ret}(t) = \theta(t)\, \hat K(t) = \frac{1}{2\pi i} \int_{C_+} dz \, {\rm e}^{-\frac{i}{\hbar} z t} \, \frac{1}{z-\hat H} \] in terms of the resolvent of the Hamiltonian, \((z-\hat H)^{-1}\), integrated along the contour \(C_+\) at \(z=E+i0\) extending from \(E=+\infty\) to \(E=-\infty\) in the space of complex energy \(z\), as depicted in Figure 2. This space is a Riemann surface with a structure determined by the dispersion relation \(E=p^2/(2m)\) of the quantum waves at large distance where the Hamiltonian reduces to the one of a free particle, \(\hat H_0=-\hbar^2\nabla^2/(2m)\). Since the square root giving the momentum \(p=\sqrt{2mE}\) has two determinations for complex energies, the Riemann surface is composed of two sheets attached to each other along a branch cut taken as the positive real energy axis and ending at the branch point \(E=0\), as shown in Figure 2. The contour \(C_+\) is taken slightly above the real energy axis in order to obtain the retarded propagator at positive times. The advanced propagator at negative times is defined with another contour \(C_-\) (not shown) slightly below the real energy axis.

Scattering resonances and lifetimes

Besides the poles of the bound states at the negative real energies \(\{E_b\}\) on the first Riemann sheet, the resolvent \((z-\hat H)^{-1}\) of the Hamiltonian operator is known to have further poles at complex energies on the second Riemann sheet defining the scattering resonances \(\{E_r={\cal E}_r-i\Gamma_r/2\}\) and the anti-resonances \(\{E_r^*={\cal E}_r+i\Gamma_r/2\}\) associated by time reversal (Goldberger and Watson 2004, Joachain 1975, Newton 2002, Taylor 2000). The energy of each resonance is given by the real part \({\cal E}_r>0\) and its width \(\Gamma_r>0\) by the imaginary part. These poles control the dependence of the scattering amplitude on the energy \[ f({\mathbf n};E) \simeq f_0({\mathbf n};E) + \sum_r \frac{c_r({\mathbf n})}{E-{\cal E}_r+i\Gamma_r/2} \, , \tag{3} \] where \(f_0({\mathbf n};E)\) is a possible smooth background and \(c_r({\mathbf n})\) are coefficients given by the residues of the poles. Near an isolated resonance, the cross section would present a Lorentzian profile \[ \frac{d\sigma}{d\Omega} =\vert f({\mathbf n};E)\vert^2 \sim \left\vert \frac{1}{E-{\cal E}_r+i\Gamma_r/2}\right\vert^2 = \frac{1}{(E-{\cal E}_r)^2+\Gamma_r^2/4} \] with a maximum located at the resonance energy \({\cal E}_r\) and a width at mid-height equal to \(\Gamma_r\). However, the resonances are not isolated in general and they may interfere with the background function \(f_0\) giving Fano profiles (Bohm 1986) or with each other if they overlap (Mitchell et al. 2010).

Figure 2: Riemann surface of complex energy \(E={\rm Re}\, E+i\, {\rm Im}\, E\) with the real energies \(\{E_b\}\) of the bound states and the complex energies of the scattering resonances \(\{E_r\}\) ruling the forward semigroup at positive time \(t>0\) and the scattering anti-resonances \(\{E_r^*\}\) ruling the backward semigroup at negative time \(t<0\). See the text for explanations about the integration contours \(C_+\) and \(C'_+\).

In order to decompose the time evolution in terms of the resonances, the contour \(C_+\) is deformed towards negative values of \({\rm Im}\, z={\rm Im}\, E\) into the new contour \(C'_+\) shown in Figure 2. This deformation has the virtue of encircling every pole to pick up the contributions of the bound states and the resonances. In the deformation, the contour should go around the branch point at \(z=E=0\), coming from \(z=-i\infty\) to \(z=0\) on the second Riemann sheet and going back from \(z=0\) to \(z=-i\infty\) but on the first Riemann sheet. The contribution of the branch point gives a power law decay as \(t^{-d/2}\), which is expected because the wavepacket may have an arbitrarily slow dynamics due to its components at arbitrarily small energy near \(E=0\) (Rosenfeld 1965). Accordingly, the survival amplitude of the wavepacket can be decomposed as \[ \begin{align*} &\langle\psi_0\vert\psi_t\rangle = \langle\psi_0\vert {\rm e}^{-\frac{i}{\hbar} \hat H t}\vert\psi_0\rangle \\ &= \sum_b {\rm e}^{-\frac{i}{\hbar} E_b t} \vert\langle \varphi_b\vert\psi_0\rangle\vert^2 + \sum_r {\rm e}^{-\frac{i}{\hbar} E_r t} \vert\langle \varphi_r\vert\psi_0\rangle\vert^2 + O(t^{-d/2}) \, . \end{align*} \] Here, the residue of every pole defines the corresponding eigenfunction: \[ \hat H \, \varphi_{j} = E_{j}\, \varphi_{j} \] for a bound state \(j=b\), or the generalized eigenfunction for a scattering resonance \(j=r\).

Supposing that the initial wavepacket has its energy components concentrated around the energy \({\cal E}_r\) of the isolated resonance \(E_r={\cal E}_r-i\Gamma_r/2\) and negligible components away from this energy, the survival probability of the wavepacket would present an exponential decay \[ {\cal P}(t) = \vert\langle\psi_0\vert\psi_t\rangle\vert^2 \sim \left\vert {\rm e}^{-\frac{i}{\hbar} E_r t} \right\vert^2 = \left\vert {\rm e}^{-\frac{i}{\hbar} {\cal E}_r t} {\rm e}^{-\frac{\Gamma_r}{2\hbar}\, t} \right\vert^2 = {\rm e}^{-\frac{\Gamma_r}{\hbar}\, t} ={\rm e}^{-\frac{t}{\tau_r}} \, , \] showing that the lifetime \(\tau_r\) is related to the width of the resonance according to \[ \tau_r = \frac{\hbar}{\Gamma_r} \, , \] where \(\hbar\) is Planck's constant. The larger the width, the shorter the lifetime.

The scattering process is thus described in detail thanks to the spectrum of resonances and the associated generalized eigenfunctions. The spectrum of resonances is to an open scattering system what the spectrum of energy eigenvalues is to a bounded system, from which particles neither enter nor exit.

In general, the resonances interfere with the background scattering amplitude \(f_0({\mathbf n};E)\) or among themselves in the regime of strongly overlapping resonances, which is of special interest for quantum chaotic scattering.

Scattering matrix and cross sections

A characterization of the scattering process can be established in terms of the unitary scattering operator \[ \hat S = \lim_{t\to\infty} {\rm e}^{\frac{i}{\hbar} \hat H_0 t/2}{\rm e}^{-\frac{i}{\hbar} \hat H t}{\rm e}^{\frac{i}{\hbar} \hat H_0 t/2} \, , \tag{4} \] where \(\hat H_0\) is the asymptotic part of the Hamiltonian operator \(\hat H=\hat H_0+\hat V\) (Goldberger and Watson 2004, Joachain 1975, Newton 2002, Taylor 2000, Thirring 1979). The scattering operator maps incoming onto outgoing states according to \[ \psi_{\rm out} = \hat S \, \psi_{\rm in} \, . \] This operator can be decomposed onto the eigenstates of the asymptotic Hamiltonian as \[ \hat S = \int_0^{\infty} dE \; \delta(E-\hat H_0) \; \hat S(E) \, , \] which defines the \(S\)-matrix \[ \hat S(E) = 1 - 2\pi i \; \delta(E-\hat H_0) \, \hat T(E+i0) \] in terms of the transition operator \[ \hat T(z) = \hat V + \hat V \, \frac{1}{z-\hat H}\, \hat V \, . \] Therefore, the \(S\)-matrix has poles at the same complex energies as the scattering resonances. The scattering amplitude (3) for a transition \(a\to b\) from the initial state \(a\) toward the final state \(b\) is given in terms of the corresponding matrix elements of the transition operator giving the cross-sections as \[ \vert f_{ba}\vert^2 = \frac{(2\pi)^4}{\hbar v_a} \, \rho_b(E) \, \vert T_{ba}\vert^2 \, , \tag{5} \] where \(v_a\) is the incident particle velocity and \(\rho_b(E)\) the density of final states at the energy \(E\) (Goldberger and Watson 2004, Joachain 1975, Newton 2002, Taylor 2000).

The cross sections (1) or (5) fluctuate as the outgoing direction \(\mathbf n\) or the energy \(E\) is varied. An important question is to understand the statistics of these fluctuations and to determine whether they are in correspondence with the underlying spectrum of the scattering resonances.

Quantum time delay

Signatures of chaotic scattering may also manifest themselves in the time delay undergone by the scattered wave due to its interaction with the obstacle at a given energy \(E\). This concept was first published by Wigner (Wigner 1955) and is related to the \(S\)-matrix by (Smith 1960, Thirring 1979) \[ {\cal T}(E)= \frac{\hbar}{i} \, {\rm tr}\, \frac{d}{dE} \, \ln \hat S(E) \, . \] The time delay can equivalently be expressed in terms of the resolvents of the full and asymptotic Hamiltonian operators as (Thirring 1979) \[ {\cal T}(E)= -2\,\hbar\, {\rm Im} \, {\rm tr} \left(\frac{1}{E-\hat H + i 0}-\frac{1}{E-\hat H_0 + i 0}\right) =2\pi\hbar \,D(E) \, , \tag{6} \] which shows that it is proportional to the difference \(D(E)\) between the level densities of the full and asymptotic Hamiltonian operators bounded by a large enough domain to enclose the scatterer (Balian and Bloch 1974, Gaspard and Rice 1989b). The time delay is thus determined by the resonance spectrum and fluctuates accordingly. For this reason, the time delay is often used in theoretical studies of quantum chaotic scattering (Wardlaw and Jaworski 1989, Lewenkopf and Weidenmüller 1991, Gaspard 1993, Brouwer et al. 1997, de Carvalho and Nussenzveig 2002).

Classical chaotic scattering

General presentation

The classical scattering of a particle is said to be chaotic if the particle may be trapped forever in a set of orbits where the periodic orbits are unstable of saddle type and proliferate exponentially at arbitrarily large periods. This set of trapped orbits is invariant with respect to the classical time evolution.

In such a chaotic invariant set, there exist countably many periodic orbits, which are dense among uncountably many nonperiodic orbits. Sensitivity to initial conditions manifests itself in the chaotic invariant set, which forms a fractal. Orbits may escape from its vicinity up to infinity. In this regard, the invariant set acts as a repeller in phase space. By time-reversal symmetry, orbits may also converge from infinity towards the fractal set, which is thus a chaotic saddle (Tél and Gruiz 2006, Seoane and Sanjuán 2013).

If such a chaotic saddle exists on a given energy shell, it will continue to exist on neighboring energy shells because of its topological robustness. Nevertheless, there may also exist Kolmogorov-Arnold-Moser (KAM) elliptic islands containing stable periodic orbits of center type and which undergo bifurcations causing ruptures in the topological continuity of the invariant set (Arnold 1963, Moser 1968). The elliptic islands are typically bordered by zones of chaotic motion.

Figure 3: Elastic scattering of a particle on one or several disks of radius \(a\) fixed in the plane. The scatterer is composed of (a) one disk; (b) two disks; (c) three disks centered at the vertices of an equilateral triangle; (d) four disks centered at the vertices of a square. In the scatterers with several disks, the distance between next-neighboring disks is equal to \(R\).

As examples, let us consider the classical scattering of a particle moving in the plane and bouncing in elastic collisions on one or several immobile disks, as illustrated in Figure 3 (Eckhardt 1987, Gaspard and Rice 1989a, Gaspard 1998, Tél and Gruiz 2006). The disks have the same radius \(a\) and are fixed in the plane in a configuration that defines the scatterer. The kinetic energy of the particle is conserved during the collision process.

The scattering on a single disk is well known and characterized by the differential cross section \[ \frac{d\sigma}{d\theta} = \frac{a}{2}\left\vert\sin\frac{\theta}{2}\right\vert \] of incident orbits which are deflected by an angle in the interval \((\theta,\theta+d\theta)\) with \(-\pi\leq\theta\leq+\pi\). However, this scatterer has no trapped orbit.

The situation already changes with the two-disk scatterer in Figure 3b, which contains a unique trapped orbit. This unstable periodic orbit is bouncing forever along the line joining the centers of both disks. The motion on this unstable periodic orbit manifests sensitivity to initial conditions. Nevertheless, this motion is not chaotic because the invariant set contains a single periodic orbit.

The invariant set becomes chaotic for the scatterer of Figure 3c formed by three disks centered at the vertices of an equilateral triangle. Here, the unstable periodic orbits proliferate exponentially because, at every collision on a disk, the particle may go to either of the two other disks, or escape to infinity. If the disks are not too close to each other, i.e., if \(R> 2.0482142 \, a\) (Hansen 1993, Sano 1994), the exponential proliferation goes as \(2^n\) with the number \(n\) of bounces in the period. This open billiard provides a nice example of chaotic scattering with a so-called hyperbolic invariant set, meaning that all the trapped orbits are unstable of hyperbolic (saddle) type (Eckhardt 1987, Gaspard and Rice 1989a, Gaspard 1998, Tél and Gruiz 2006).

The proliferation goes as \(3^n\) in the scatterer with four disks at the vertices of a large enough square (see Figure 3d).

Remarkably, hyperbolic invariant sets also exist for Hamiltonian motion in smooth potentials over significant ranges of energy (Klein and Knauf 1992, Burghardt and Gaspard 1994, Burghardt and Gaspard 1995, Gaspard et al. 1995, Gaspard and Burghardt 1997, Burghardt and Gaspard 1997). However, in smooth potentials, bifurcations may happen at critical energies where KAM elliptic islands could appear, greatly complexifying the phenomenon of chaotic scattering (Bleher et al. 1989, Bleher et al. 1990, Ding et al. 1990, Burghardt and Gaspard 1994, Burghardt and Gaspard 1995, Gaspard et al. 1995, Gaspard and Burghardt 1997, Burghardt and Gaspard 1997).

The quantitative formulation of classical chaotic scattering is presented in the following subsections.

Classical scattering theory

In a system with \(f\) degrees of freedom, classical motion is ruled by Hamilton's equations \[ \left\{ \begin{matrix} \frac{d{\mathbf r}}{dt} = +\frac{\partial H}{\partial{\mathbf p}} \, ,\\ \\ \frac{d{\mathbf p}}{dt} = -\frac{\partial H}{\partial{\mathbf r}} \, , \end{matrix} \right. \tag{7} \] where \(t\) denotes the time, \({\mathbf r}\in{\mathbb R}^f\) the positions, \({\mathbf p}\in{\mathbb R}^f\) the canonically conjugated momenta, and \(H({\mathbf r},{\mathbf p})\) the Hamiltonian function. The phase space of all the variables is of dimension \(2f\), \({\mathbf X}=({\mathbf r},{\mathbf p})\in{\mathbb R}^{2f}\). By Cauchy's theorem, the solutions of Hamilton's equations are uniquely determined by their initial conditions, which defines a flow in phase space, \({\mathbf X}_t=\pmb{\Phi}^t({\mathbf X}_0)\). Since the Hamiltonian function does not depend on time, it is a constant of motion, which represents the total energy. Every orbit is thus contained in an energy shell \(H({\mathbf r},{\mathbf p})=E\) of dimension \(2f-1\).

For a particle undergoing elastic collisions on immobile disks, the Hamiltonian dynamics is composed of free flights \({\mathbf r}_j=({\mathbf p}_{j}/m)(t_j-t_{j-1})+{\mathbf r}_{j-1}\) between the collisions and specular reflections \({\mathbf p}_{j+1}={\mathbf p}_j-2({\mathbf n}_j\cdot{\mathbf p}_j){\mathbf n}_j\) at elastic collisions where \({\mathbf n}_j\) is a unit vector normal to the wall at the collision point \({\mathbf r}_j\) with \(j\in{\mathbb Z}\).

For a nonrelativistic particle of mass \(m\) moving in the potential \(V({\mathbf r})\), the Hamiltonian function is given by \[ H({\mathbf r},{\mathbf p}) = \frac{{\mathbf p}^2}{2m} + V({\mathbf r}) \] and Hamilton's equations (7) reduce to Newton's equations \[ m \, \frac{d^2{\mathbf r}}{dt^2} = - \frac{\partial V}{\partial{\mathbf r}} \, . \] Here, we consider the general case of an anisotropic potential \(V({\mathbf r})\) so that angular momentum is not necessarily conserved. Since the potential vanishes at large distances, the orbits follow asymptotically straight lines as the particle is incoming or outgoing a collision. These straight orbits are the solutions \({\mathbf X}_t=\pmb{\Phi}_0^t({\mathbf X}_0)\) of Hamilton's equations for the free-particle Hamiltonian function \(H_0={\mathbf p}^2/(2m)\). The scattering process is characterized by the relation between the incoming and outgoing straight orbits: \[ {\mathbf X}_{\rm out} = \pmb{\Sigma}({\mathbf X}_{\rm in})=\lim_{t\to\infty} \pmb{\Phi}_0^{-t/2}\circ\pmb{\Phi}^t\circ \pmb{\Phi}_0^{-t/2}({\mathbf X}_{\rm in}) \, , \] which is the classical analogue of the quantum scattering operator (4) (Narnhofer 1980, Narnhofer and Thirring 1981).

On a given energy shell, the incoming straight orbits are specified by \(f-1\) impact parameters \({\mathbf s}\in{\mathbb R}^{f-1}\), and the outgoing straight orbits by \(f-1\) angles \(\pmb{\Omega}\) or the corresponding unit vector \(\mathbf n\). The relation \({\mathbf n}={\mathbf n}({\mathbf s})\) between the impact parameters \({\mathbf s}\) of the incoming orbit and the direction \(\mathbf n\) of the outgoing orbit gives in particular the scattering cross section \(d\sigma/d\Omega\) as the ratio between \(d\sigma=d^{f-1}s\) and \(d\Omega=d^{f-1}\Omega\).

This relation is obtained from the generating function defined by the reduced action \[ \tilde S(E) = - \int {\mathbf r}\cdot d{\mathbf p} \, , \tag{8} \] where the integral is carried out over a scattering orbit on some energy shell \(H=E\). The time delay taken by the particle due to its interaction with the scatterer is given by \[ {\cal T} = \frac{\partial \tilde S}{\partial E} \, . \] This time is negative, as it should to satisfy the causality principle.

For the scattering of a particle of mass \(m\) on a single disk of radius \(a\), the impact parameter \(s\) (which is the distance between the incoming orbit and the parallel straight line going through the center of the disk) is related to the deflection angle \(\theta\) by the relation \(s=a\cos(\theta/2)\). The reduced action has the expression \(\tilde S=-2mav\vert\sin(\theta/2)\vert\) where \(v=\sqrt{2E/m}\) is the velocity at the kinetic energy \(E\). The time delay is thus given by \({\cal T}=-(2a/v)\vert\sin(\theta/2)\vert\). If the particle is not deflected, \(\theta=0\) and the time delay vanishes. If the particle velocity is completely reversed, \(\theta=\pm\pi\) and \(v{\cal T}=-2a\), which is consistent with the shortening of the path length with respect to a straight orbit without collision.

In the case of the two-disk scatterer of Figure 3b, there exist two values of the impact parameter \(s\), for which the incoming orbit converges asymptotically to the unstable periodic orbit. The first one is incident on the upper disk and the second on the lower disk. Accordingly, the time-delay function \({\cal T}(s;E)\) of the impact parameter \(s\) has two asymptotes where the time delay becomes infinite. For initial conditions coinciding with these asymptotes, the particle is finally trapped in the invariant set.

Figure 4: Time-delay function \(T(y_0)=\vert{\cal T}(y_0)\vert\) for the chaotic scattering on the three-disk scatterer for \(R=3.5\) and \(a=1\) versus the impact parameter \(s=y_0\) of the incident orbits and a zoom on a piece of it. The impact parameter \(s=y_0\) is defined with respect to the center of the three-disk system. [Adapted from Ref. (Gaspard and Rice 1989a).]

For the three-disk scatterers of Figure 3c, the time-delay function has infinitely many such asymptotes, as shown in Figure 4. These asymptotes reveal the fractal character of the invariant set, which is therefore chaotic.

The chaotic and fractal character of the invariant set can also be revealed by taking a Poincaré surface of section in the Hamiltonian flow on an energy shell. In a system with \(f\) degrees of freedom, this surface of section is of dimension \(2f-2\). For every initial condition in the surface of section, the time taken by an orbit to escape at a large enough distance from the scatterer is computed forward and backward in time giving the escape functions \({\cal T}_{\pm}({\mathbf s};E)\). The escape function \({\cal T}_{+}({\mathbf s};E)\) is infinite on the stable manifolds of the orbits that are trapped forever, while the escape function \({\cal T}_{-}({\mathbf s};E)\) is infinite on their unstable manifolds (Eckmann and Ruelle 1985, Burghardt and Gaspard 1994, Burghardt and Gaspard 1995, Gaspard et al. 1995, Gaspard and Burghardt 1997, Burghardt and Gaspard 1997). Therefore, the plot of the function \(\vert{\cal T}_{+}({\mathbf s};E)\vert+\vert{\cal T}_{-}({\mathbf s};E)\vert\) allows us to localize the fractal set of trapped orbits in the surface of section.

Figure 5: (a) A few shortest periodic orbits of the collinear dynamics of IHgI in the position space \((r_1,r_2)\) of the interatomic distances IHg and HgI. The periodic orbits are in correspondence with the symbolic dynamics based on the alphabet \(\{0,1,2\}\). (b) Density plot of the sum of the absolute values of the escape times for the forward and backward dynamics in a Poincaré section transverse to the fractal set of the trapped orbits. \(q\) is defined transverse to the diagonal line \(r_1=r_2\) and \(p\) is the canonically conjugated momentum. (c) Schematic representation of the corresponding ternary fractal set. [Adapted from Ref. (Burghardt and Gaspard 1995).]

This procedure is used in Figure 5 for the fractal set of the Hamiltonian flow of the collinear dissociation of the triatomic molecule ABA \(\to\) AB \(+\) A. In a frame moving with the center of mass, this system has \(f=2\) degrees of freedom and its Hamiltonian function is given by \[ H=\frac{p_1^2+p_2^2}{2\mu_{\rm AB}}-\frac{p_1p_2}{m_{\rm B}} + V(r_1,r_2) \] with a potential as the one depicted at right hand in Figure 1 and the reduced mass \(\mu_{\rm AB}\) of the diatomic molecule AB (Burghardt and Gaspard 1994, Burghardt and Gaspard 1995). The analysis shows that the invariant set is a ternary fractal where the motion is fully hyperbolic. The unstable periodic orbits proliferate exponentially as \(3^n\) with the number \(n\) of passages in the surface of section. The shortest periodic orbits at the beginning of this exponential proliferation are depicted in Figure 5a.

Escape from the scatterer

If the impact parameters \(\mathbf s\) of the incoming orbits are statistically distributed according to the probability density \(\rho_0({\mathbf s})\), the corresponding time delays \(\{t=-{\cal T}({\mathbf s};E)\}\) at the energy \(E\) have the probability density: \[ p(t;E) = \int d{\mathbf s} \, \rho_0({\mathbf s}) \, \delta\left[ t+{\cal T}({\mathbf s};E)\right] \, . \tag{9} \] Since the time delay is an estimation of the time taken by the particle to escape from the vicinity of the scatterer, this probability density characterizes the process of escape from the invariant set of trapped orbits. An alternative procedure is to take a distribution \(\rho_0({\mathbf X})\) of initial conditions inside the scatterer and to get the probability density \(p(t;E)\) to escape at time \(t\) from a large enough domain enclosing the scatterer.

If the invariant set is hyperbolic as for scatterers with two or more disks, this distribution decays exponentially, which defines the escape rate \[ \gamma_{\rm cl}(E)= \lim_{t\to\infty} -\frac{1}{t} \, \ln \, p(t;E) \, . \]

However, if the invariant set is not hyperbolic (for instance, if the disks are replaced by squares with parallel sides or if the invariant set contains KAM elliptic islands), power-law decays \(p(t;E)\sim 1/t^{\alpha}\) are observed with some exponent \(\alpha\), because of stickiness in the neighborhood of nonhyperbolic orbits (Gaspard and Rice 1989a, Bauer and Bertsch 1990). Power-law decays are also observed in the hyperbolic case if the initial conditions are distributed in energy with a non-vanishing probability density at zero energy (Gaspard and Rice 1989a). We notice that if the initial conditions are taken inside the scatterer for the statistics of escape times, rather than at large distances as for the statistics of time delays, there is no escape from initial conditions inside the KAM elliptic islands in systems with \(f=2\) degrees of freedom.

Sensitivity to initial conditions

The sensitivity to initial conditions is characterized by the so-called Lyapunov exponents defined as \[ \lambda({\mathbf X}_0;\delta{\mathbf X}_0) = \lim_{t\to\infty} \frac{1}{t} \, \ln \frac{\Vert\delta{\mathbf X}_t\Vert}{\Vert\delta{\mathbf X}_0\Vert} \] for an infinitesimal perturbation \(\delta{\mathbf X}_t\) of an orbit \({\mathbf X}_t=\pmb{\Phi}^t({\mathbf X}_0)\) of initial conditions \({\mathbf X}_0\) in the phase space \({\mathbf X}\in{\mathbb R}^{2f}\). Therefore, the Lyapunov exponent depends not only on the orbit, but also on the specific direction of the initial perturbation \(\delta{\mathbf X}_0\). The dependence on the orbit disappears in ergodic invariant sets (Eckmann and Ruelle 1985). Accordingly, an ergodic invariant set is characterized by as many Lyapunov exponents as the phase-space dimension: \[ \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_f=0=\lambda_{f+1} \geq \cdots \geq \lambda_{2f} \, . \] If the invariant set does not reduce to a stationary point, the Lyapunov exponent corresponding to the direction of the flow is vanishing. In Hamiltonian systems, another Lyapunov exponent is equal to zero because of energy conservation. Moreover, the symplectic character of Hamiltonian flows implies the pairing rule, \(\lambda_{2f-i+1}=-\lambda_i\) for \(i=1,2,...,2f\), which guarantees that the sum of all the Lyapunov exponents is vanishing in agreement with Liouville's theorem, \(\sum_{i=1}^{2f}\lambda_i=0\).

Since the disk scatterers are systems with two degrees of freedom, they are characterized by a single positive Lyapunov exponent. Going back to the two-disk scatterer, we mentioned that the time-delay function \({\cal T}(s;E)\) has two vertical asymptotes at \(s=s_{\pm}\) where the particle converges towards the unstable periodic orbit characterized by a Lyapunov exponent \(\lambda\). Therefore, the time delay behaves as \({\cal T}\simeq(1/\lambda)\ln\vert s-s_{\pm}\vert\) in the vicinity of each asymptote. Consequently, the probability distribution (9) is exponential \(p(t;E)\sim \exp(-\lambda t)\) for \(t\to\infty\), in which case the escape rate is equal to the Lyapunov exponent of the unstable periodic orbit, \(\gamma_{\rm cl}(E)=\lambda(E)\). However, this is no longer the case if the invariant set is chaotic because of the proliferation of the asymptotes.

Characterization of the chaotic invariant set

At a given energy \(E\), the exponential proliferation of the unstable periodic orbits contained in the invariant set is characterized by the so-called topological entropy \[ h_{\rm top} = \lim_{t\to\infty} \frac{1}{t}\, \ln\left(\mbox{Number}\{ \mbox{periodic orbits of period} \leq t\}\right) \, , \tag{10} \] which is positive in the presence of chaos and zero otherwise (Eckmann and Ruelle 1985, Walters 1981).

The orbits composing a chaotic invariant set are often in correspondence with a symbolic dynamics based on an alphabet of symbols \(\omega_i\). For instance, a periodic orbit of period \(n\) corresponds to the sequence \(\pmb{\omega}=\omega_1\omega_2\cdots\omega_n\), which is repeated infinitely many times as time runs over \(-\infty < t < +\infty\). More generally, the sequence \(\pmb{\omega}=\omega_1\omega_2\cdots\omega_n\) corresponds to all the periodic and nonperiodic orbits that remain close to each other during a time interval equal to the period \(t_{\pmb{\omega}}\).

If the invariant set is hyperbolic, its orbits are unstable so that the neighboring phase-space volumes are exponentially stretched by the factor \[ \vert\Lambda_{\pmb{\omega}}\vert \sim \exp\left( \sum_{\lambda_i>0} \lambda_i \, t_{\pmb{\omega}}\right) >1 \tag{11} \] during the time interval \(t_{\pmb{\omega}}\), where \(\lambda_i\) are their local Lyapunov exponents. These stretching factors can be used in order to give a probability weight to every trapped orbit. Since the probability weight should decrease if the orbits are more unstable, the probability weight of the orbits corresponding to the symbolic sequence \(\pmb{\omega}\) is supposed to be inversely proportional to the stretching factors to some power \(\beta\): \[ \mu_\beta(\pmb{\omega}) = \frac{\vert\Lambda_{\pmb{\omega}}\vert^{-\beta}}{\sum_{\pmb{\omega}}\vert\Lambda_{\pmb{\omega}}\vert^{-\beta}} \, . \tag{12} \] This defines a normalized probability measure \(\sum_{\pmb{\omega}}\mu_\beta({\pmb{\omega}})=1\). We notice that the exponent should take the unit value \(\beta=1\) in order to get the probability weight attributed to the orbits by time averaging in an ergodic Hamiltonian system (Eckmann and Ruelle 1985, Gaspard 1998).

The family of probability measures (12) is characterized by the so-called Ruelle topological pressure (Eckmann and Ruelle 1985): \[ P(\beta) = \lim_{t\to\infty} \frac{1}{t} \, \ln \sum_{\pmb{\omega}:\, t<t_{\pmb{\omega}}<t+\Delta t}\vert\Lambda_{\pmb{\omega}}\vert^{-\beta} \, , \tag{13} \] where the limit \(t\to\infty\) is taken for a large enough value of \(\Delta t\) for \(P(\beta)\) to be independent of both \(t\) and \(\Delta t\). The topological pressure is known to satisfy the following inequality (Walters 1981) \[ P(\beta_1+\beta_2) \leq P(\beta_1)+P(\beta_2) \, . \tag{14} \]

If \(\beta=0\), every orbit has the same weight independently of their instability and the sum gives the number of orbits corresponding to a period \(t\), whereupon the topological pressure is equal to the topological entropy: \[ P(0)= h_{\rm top} \, . \] If \(\beta=1\), the sum gives an estimation of the phase-space volume of initial conditions which have not yet escaped by the time \(t\). Since this phase-space volume is proportional to the probability (9), the topological pressure is related to the escape rate by \[ P(1) = - \gamma_{\rm cl} \, . \tag{15} \] Now, the derivative of the topological pressure with respect to the exponent \(\beta\) at \(\beta=1\) gives the average of the sum of positive Lyapunov exponents by Eq. (11): \[ \frac{dP}{d\beta}(1) = - \lim_{t\to\infty} \frac{1}{t} \sum_{\pmb{\omega}:\, t<t_{\pmb{\omega}}<t+\Delta t} \mu_1(\pmb{\omega}) \, \ln\vert\Lambda_{\pmb{\omega}}\vert = - \sum_{\lambda_i>0} \lambda_i \, . \tag{16} \] The Kolmogorov-Sinai entropy per unit time characterizing the dynamical randomness of chaotic motion in the invariant set is defined as (Eckmann and Ruelle 1985, Walters 1981) \[ h_{\rm KS} = - \lim_{t\to\infty} \frac{1}{t} \sum_{\pmb{\omega}:\, t<t_{\pmb{\omega}}<t+\Delta t} \mu_1(\pmb{\omega}) \, \ln\mu_1(\pmb{\omega}) \, , \tag{17} \] which is always lower than or equal to the topological entropy (10). These characteristic quantities were introduced in analogy with equilibrium statistical mechanics, which explains the terminology. Combining Eqs. (15)-(17), we obtain the formula \[ \gamma_{\rm cl} = \sum_{\lambda_i>0} \lambda_i - h_{\rm KS} \, , \] giving the escape rate as the difference between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy (Eckmann and Ruelle 1985, Kantz and Grassberger 1985). If the invariant set is not chaotic because it contains a single unstable periodic orbit, its Kolmogorov-Sinai entropy is vanishing and the escape rate is given by the sum of positive Lyapunov exponents \(\gamma_{\rm cl} = \sum_{\lambda_i>0} \lambda_i\). The Pesin entropy formula is recovered if the escape rate vanishes.

Furthermore, the fractal character of the invariant set can be quantitatively measured thanks to the concept of Hausdorff dimension (Falconer 1990). For a chaotic system with \(f=2\) degrees of freedom, the invariant set has one unstable direction transverse to the flow in an energy shell, and one corresponding stable direction. The fractal character develops symmetrically in the unstable and stable directions. The Hausdorff dimension of the fractal set in the unstable direction is given by a number contained in the unit interval, \(0\leq d_{\rm H}\leq 1\). This dimension is given by the value of the exponent \(\beta\) such that \[ \sum_{\pmb{\omega}:\, t<t_{\pmb{\omega}}<t+\Delta t}\vert\Lambda_{\pmb{\omega}}\vert^{-\beta} \sim 1 \qquad\mbox{for} \qquad t\to\infty \, . \] Therefore, this Hausdorff dimension is obtained as the root of the topological pressure: \[ P(d_{\rm H})=0 \, . \] The dimension of the fractal set is thus equal to \(2d_{\rm H}+1\) in the tridimensional energy shell and \(2d_{\rm H}+2\) in the four-dimensional phase space. If the invariant set is composed of a single periodic orbit, the Hausdorff dimension \(d_{\rm H}\) is equal to zero in consistency with the vanishing of the topological entropy because the dynamics is not chaotic in this case.

In systems with \(f\) degrees of freedom, the fractal set should have a Hausdorff dimension \(0\leq d_{\rm H}\leq f-1\) in the subspace formed by all the unstable directions and the same dimension in the subspace of the stable directions. Accordingly, its Hausdorff dimension should be equal to \(0\leq 2d_{\rm H}+1\leq 2f-1\) in an energy shell and \(0\leq 2d_{\rm H}+2\leq 2f\) in the full phase space.

The semiclassical approach

Path integrals and the semiclassical approximation

In the position representation, the unitary evolution operator (2) reads \[ \psi_t({\mathbf r}) = \int d{\mathbf r}_0 \, K({\mathbf r},{\mathbf r}_0;t) \, \psi_0({\mathbf r}_0) \, . \] The propagator \(K({\mathbf r},{\mathbf r}_0;t)\) can be written in the form of the Feynman path integral \[ K({\mathbf r},{\mathbf r}_0;t) = \int {\cal D}{\mathbf r}(\tau) \, \exp\left[\frac{i}{\hbar} \int_0^t L({\mathbf r},{\mathbf{\dot r}}) \, d\tau\right] \, , \] where \(L({\mathbf r},{\mathbf{\dot r}})\) is the Lagrangian function of the corresponding classical system with the velocity \({\mathbf {\dot r}}=d{\mathbf r}/dt\).

In the limit where the action of the path \(\{{\mathbf r}(\tau)\}_{\tau=0}^{\tau=t}\) followed by the particle is much larger than Planck's constant \(\hbar\), \[ W=\int_0^t L({\mathbf r},{\mathbf{\dot r}}) \, d\tau =\int_0^t \left[{\mathbf p}\cdot d{\mathbf r} - H({\mathbf r},{\mathbf p}) \, d\tau\right] \; \gg \; \hbar \, , \] the dominant contributions to the path integral are given by the classical orbits, which are known to be the extremal paths of the action satisfying the variational principle \(\delta W=0\). In this semiclassical approximation, the propagator is given by \[ K({\mathbf r},{\mathbf r}_0;t) \simeq \sum_{l} A_{l}({\mathbf r},{\mathbf r}_0;t) \, \exp\left[\frac{i}{\hbar} W_{l}({\mathbf r},{\mathbf r}_0;t)\right] \, , \] where the sum extends over all the classical orbits going from \({\mathbf r}_0\) to \(\mathbf r\) during the time interval \(t\). \(W_{l}\) is their action and \(A_{l}\) their semiclassical probability amplitude \[ A_{l}({\mathbf r},{\mathbf r}_0;t) = \frac{1}{(2\pi i \hbar)^{f/2}} \left\vert\det\left[-\frac{\partial^2W_{l}({\mathbf r},{\mathbf r}_0;t)}{\partial{\mathbf r}\;\partial{\mathbf r}_0}\right]\right\vert^{1/2} \, {\rm e}^{-i\pi\nu_l/2} \, , \] where \(\nu_l\) denotes the number of conjugate points along the classical orbit. At every conjugate point, the rank of the \(f\times f\) matrix \(\frac{\partial{\mathbf r}}{\partial{\mathbf p}_0}=\left[-\frac{\partial^2W_{l}({\mathbf r},{\mathbf r}_0;t)}{\partial{\mathbf r}\;\partial{\mathbf r}_0}\right]^{-1}\) is reduced by one so that neighboring orbits cross each other in position space at conjugate points (Berry and Mount 1972, Gutzwiller 1990).

The probability amplitude behaves as \[ \vert A_{l}({\mathbf r},{\mathbf r}_0;t)\vert\sim \exp\left( -\frac{1}{2}\sum_{\lambda_i>0}\lambda_i \, t\right) \] along an unstable orbit of Lyapunov exponents \(\{\lambda_i\}_{i=1}^{2f}\), but as \(\vert A_{l}({\mathbf r},{\mathbf r}_0;t)\vert\sim \vert t\vert^{-f/2}\) along a stable orbit.

Trace formula for the quantum time delay

The semiclassical approximation has been developed in particular to obtain the level density of bounded quantum systems (Berry and Mount 1972, Gutzwiller 1990). The level density is given in terms of the trace of the Green function, which is the Fourier transform of the propagator. In the semiclassical approximation, the trace selects classical orbits that are closed onto themselves, i.e., periodic orbits and stationary points. This idea is at the origin of periodic-orbit theory for the semiclassical quantization of chaotic systems where all the periodic orbits are unstable of saddle type, isolated from each other, and dense in the invariant set.

For scattering systems, the level density should be replaced by the difference between the level densities of the full and asymptotic Hamiltonian operators bounded by a large enough domain to enclose the scatterer (Balian and Bloch 1974, Gaspard and Rice 1989b). This difference is proportional to the quantum time delay according to Eq. (6). Using periodic-orbit theory, the semiclassical approximation of the quantum time delay is thus given by \[ {\cal T}(E) = \int\frac{d{\mathbf r}\, d{\mathbf p}}{(2\pi\hbar)^{f-1}}\left[\delta(E-H_{\rm cl})-\delta(E-H_{0,{\rm cl}})\right] + O(\hbar^{-f+2}) + 2 \sum_p\sum_{r=1}^{\infty} T_p\, \frac{\cos\left(r\frac{S_p}{\hbar}-r\frac{\pi}{2}\mu_p\right)}{\vert\det({\boldsymbol{\mathsf M}}_p^r-{\boldsymbol{\mathsf 1}})\vert^{1/2}}+O(\hbar) \, , \tag{18} \] where the sum extends over all the prime periodic orbits \(p\) and their repetitions \(r\). Each of them is characterized by its reduced action \(S_p(E)=\oint_p{\mathbf p}\cdot d{\mathbf r}\), its period \(T_p=\partial_E S_p\), its Maslov index \(\mu_p\), and the \((2f-2)\times(2f-2)\) matrix \({\boldsymbol{\mathsf M}}_p\) of the linearized Poincaré map in the vicinity of the periodic orbit \(p\) (Gaspard 1993, Gaspard and Burghardt 1997).

The corrections of order \(\hbar^{-f+2}\) are obtained with the method of Wigner transform (Wigner 1932, Grammaticos and Voros 1979). Methods to obtain corrections in powers of Planck's constant to the periodic-orbit contributions have also been developed (Gaspard 1993, Alonso and Gaspard 1993, Gaspard et al. 1995, Vattay and Rosenqvist 1996, Grémaud 2002).

Formulas similar to Eq. (18) are also known for quantum magnetic and transport properties (Richter 1995, Hackenbroich et al. 1995, Richter et al. 1996).

Zeta function for quantum scattering resonances

For systems with \(f=2\) degrees of freedom, the matrix of the linearized Poincaré map has the two eigenvalues \(\{\Lambda_p,\Lambda_p^{-1}\}\) where the largest one in absolute value gives the stretching factor \(\vert\Lambda_p\vert=\exp(\lambda_pT_p)>1\). The time delay (18) can be written as \[ {\cal T}(E) = {\cal T}_{\rm av}(E)-2\hbar \, {\rm Im}\frac{d}{dE} \ln Z(E) +O(\hbar) \, , \] where \({\cal T}_{\rm av}(E)\) is the smooth part of the time delay given by the first terms of Eq. (18), while the oscillating part due to the periodic orbits can be expressed in terms of the Zeta function (Voros 1988) \[ Z(E) = \prod_p \prod_{m=0}^{\infty} \left( 1 - \frac{{\rm e}^{\frac{i}{\hbar}S_p-i\frac{\pi}{2}\mu_p}}{\vert\Lambda_p\vert^{1/2}\Lambda_p^m}\right) \, . \tag{19} \]

Now, the scattering resonances given by the poles of the resolvent operator correspond to zeroes of the Zeta function. Extending to complex energies \(z\), we have that the contributions of the periodic orbits to the trace of the resolvent are given by \[ {\rm tr}\, \frac{1}{z-\hat H}\biggr\vert_{\rm p.o.} \simeq \frac{d}{dz} \ln Z(z) \simeq \frac{1}{i\hbar} \sum_p\sum_{r=1}^{\infty} T_p \, \frac{{\rm e}^{\frac{i}{\hbar}rS_p-i\frac{\pi}{2}r\mu_p}}{\vert\Lambda_p\vert^{r/2}} \tag{20} \] up to terms of higher inverse powers of the stretching factors. If the Zeta function has a zero, \(Z(z)\propto(z-z_r)\), its logarithmic derivative has a pole at the complex energy \(z_r\), which can be identified as a resonance within the semiclassical approximation. Accordingly, periodic-orbit theory provides a method to obtain the scattering resonances as \[ Z(E_r)=0 \qquad\mbox{for}\qquad E_r={\cal E}_r - i\,\Gamma_r/2 \, . \]

In chaotic systems, the infinite product defining the Zeta function (19) should be expanded and truncated to obtain its true zeroes, which is called the method of cycle expansion (Cvitanovic 1988, Cvitanovic and Eckhardt 1989, Cvitanovic et al. 2013). For instance, in a chaotic system described by a binary symbolic dynamics, the prime periodic orbits correspond to the sequences \(p\in\{0,1,01,001,011,...\}\) and the product over them should be expanded, ordered by cycle length and instability, and truncated order after order to improve the approximation according to \[ \prod_p (1-t_p) = \underbrace{1-t_0-t_1}_{\rm 1st \; order}\underbrace{-(t_{01}-t_0t_1)}_{\rm 2nd \; order}\underbrace{-(t_{001}-t_0t_{01})-(t_{011}-t_{01}t_1)}_{\rm 3rd \; order}+\cdots \tag{21} \] with \(t_p=\exp(\frac{i}{\hbar}S_p-i\frac{\pi}{2}\mu_p)/\vert\Lambda_p\vert^{1/2}\) (Cvitanovic et al. 2013). This expansion and improvements have been successfully applied to the disk scatterers, as well as to atomic and molecular systems (Cvitanovic and Eckhardt 1989, Tanner et al. 1991, Ezra et al. 1991, Gaspard et al. 1993, Tanner et al. 1995, Tanner et al. 2000, Gaspard et al. 1995).

If the invariant set contains a single periodic orbit, the scattering resonances are directly obtained from the factors of the Zeta function. The leading resonances \(E_r={\cal E}_r-i\Gamma_r/2\) have their real energies given by the Bohr-Sommerfeld quantization condition \[ S_p({\cal E}_r) = 2\pi\hbar\left( r + \frac{\mu_p}{4}\right) + O(\hbar^2) \qquad\mbox{with} \qquad r=0,1,2,3,... \] and their width by \[ \Gamma_r = \frac{\hbar}{T_p({\cal E}_r)} \ln \vert\Lambda_p({\cal E}_r)\vert + O(\hbar) \, . \tag{22} \] The larger the instability of the periodic orbit, the broader the resonances and the shorter the lifetimes \(\tau_r=\hbar/\Gamma_r\).

Condition for the formation of a gap in the resonance spectrum

If the sum (20) over the prime periodic orbits and their repetitions converges absolutely, the resolvent should have no pole. Therefore, the domain of complex energies \(z\) where the convergence of the sum is absolute should be empty of scattering resonances in the semiclassical approximation. At complex energies \(z={\cal E}-i\Gamma/2={\cal E}-i\hbar/(2\tau)\), the reduced action can be expanded as \[ S_p(z) = S_p({\cal E}) - \frac{i\hbar}{2\tau} \, T_p({\cal E}) + O(\hbar^2) \tag{23} \] for \(\hbar\to 0\), where \({\cal E}\) is the real part of the complex energy and \(\tau\) the lifetime corresponding to its imaginary part, because \(T_p=\partial_ES_p\). Now, the convergence is absolute if the sum of the absolute values of all the terms corresponding to a period \(T=rT_p\in[t,t+\Delta t]\) is vanishing faster than \(1/t^{1+\epsilon}\) with \(\epsilon>0\) for \(t\to\infty\). With the notation \(\Lambda_T=\Lambda_p^r\) and Eq. (23), the absolute value of all these terms is given by \[ \begin{align*} \sum_{(p,r):\, t<rT_p<t+\Delta t} \left\vert \, T_p \, \frac{{\rm e}^{\frac{i}{\hbar}rS_p-i\frac{\pi}{2}r\mu_p}}{\vert\Lambda_p\vert^{r/2}}\right\vert &\simeq \sum_{(p,r):\, t<T<t+\Delta t} T \, \frac{{\rm e}^{\frac{T}{2\tau}}}{\vert\Lambda_T\vert^{1/2}} \\ &\simeq t \; \exp\left(\frac{t}{2\tau}\right) \sum_{(p,r):\, t<T<t+\Delta t} \frac{1}{\vert\Lambda_T\vert^{1/2}} \\ &\simeq t \; \exp\left(\frac{t}{2\tau}\right) \; \exp\left[t \, P\left(\frac{1}{2};{\cal E}\right)\right] \end{align*} \tag{24} \] in terms of Ruelle's topological pressure (13) at \(\beta=1/2\) for the invariant set on the energy shell \(H({\mathbf r},{\mathbf p})={\cal E}\). The value \(\beta=1/2\) appears because quantum dynamics deals with probability amplitudes roughly going as the square root of the probabilities. Since the convergence is absolute if Eq. (24) is vanishing fast enough, there is no resonance in the domain of complex energies such that \[ \hbar \, P\left(\frac{1}{2};{\cal E}\right)< -\frac{\hbar}{2\tau} = {\rm Im}\, E \, . \] Since the forward time evolution for \(t\to +\infty\) is controlled by the resonances with \({\rm Im}\, E<0\), there is a gap below the real energy axis only if the topological pressure at \(\beta=1/2\) is negative (Ikawa 1988, Gaspard and Rice 1989b, Naud 2005, Nonnenmacher and Zworski 2009, Nonnenmacher 2011). This condition means that the Hausdorff dimension \(d_{\rm H}\) of the invariant set should be lower than \(\beta=1/2\), \(d_{\rm H}<1/2\). In this case, the set of trapped orbits is filamentary. In contrast, no gap is predicted by this argument in the case where the set of trapped orbits is bulky when \(d_{\rm H}>1/2\).

In summary, the resonances \(\{E_r\}\) should satisfy the following conditions: \[ \begin{align} \mbox{filamentary invariant set:} \quad d_{\rm H} < \frac{1}{2} &\qquad {\rm Im}\, E_r < \hbar\, P\left(\frac{1}{2};\,{\rm Re}\, E_r\right) <0 \; , \tag{25} \\ \mbox{bulky invariant set:} \quad d_{\rm H} \geq \frac{1}{2} &\qquad {\rm Im}\, E_r < 0 \leq \hbar\, P\left(\frac{1}{2};\,{\rm Re}\, E_r\right) \, , \end{align} \] as depicted in Figure 6. We notice that the gap (25) is obtained in the semiclassical limit \(\hbar\to 0\) so that some resonances at low energy or low quantum numbers may not satisfy these conditions, but there are too few to modify the statistics over many resonances extending to large quantum numbers.

Figure 6: Schematic representation of Ruelle's topological pressure \(P(\beta)\) versus the exponent \(\beta\), together with its consequence on the spectrum of scattering resonances for (a) filamentary and (b) bulky invariant sets.

No gap exists in weakly open systems such as the three-disk scatterer with \(R<2.83 a\) (Gaspard and Rice 1989b, Barkhofen et al. 2013) or the helium atom and the hydrogen negative ion where the invariant set is unbounded in configuration space (Ezra et al. 1991, Gaspard et al. 1993, Tanner et al. 1995, Tanner et al. 2000).

However, a gap already appears if the invariant set contains a single periodic orbit, in which case the gap is given by Eq. (22). This gap may also exist if the invariant set is chaotic, as it has been shown for the three-disk scatterer if \(R>2.83 a\) (Gaspard and Rice 1989b, Barkhofen et al. 2013), as well as for the collinear dissociation of triatomic molecules (Gaspard and Burghardt 1997) and for quantum graphs (Barra and Gaspard 2001b). Such results have been anticipated for dynamics on surfaces with negative curvature (Patterson 1976, Sullivan 1979, Patterson and Perry 2001).

Chaotic lengthening of quantum lifetimes

In the presence of the gap (25) in the resonance distribution, we may wonder how the quantum lifetimes compare with the classical lifetime given by the inverse of the escape rate and, thus, the topological pressure at \(\beta=1\): \[ \tau_{\rm cl}({\cal E}) = \frac{1}{\gamma_{\rm cl}({\cal E})} = - \frac{1}{P(1;{\cal E})} \, . \] As shown in the following subsection, the classical lifetime characterizes the decay of wavepackets extending in energy over many resonances.

If the invariant set contains a single unstable periodic orbit, the Ruelle topological pressure depends linearly on the exponent \(\beta\) because: \[ \mbox{periodic invariant set:} \qquad P(\beta;{\cal E}) = -\beta\, \sum_{\lambda_i>0} \lambda_i({\cal E}) \, , \] in which case, we find that \[ \mbox{periodic invariant set:} \qquad P\left(\frac{1}{2};{\cal E}\right) = \frac{1}{2}\, P(1;{\cal E}) <0 \, . \] If there exist resonances at the border of the gap, as we have seen with Eq. (22), their lifetime would be given by \[ \mbox{periodic invariant set:} \qquad \tau_r = - \frac{1}{2P\left(\frac{1}{2};{\cal E}_r\right)}= - \frac{1}{P(1;{\cal E}_r)} \, . \] In this case, the quantum lifetimes are equal to the classical lifetime within the semiclassical approximation.

Most remarkably, the situation changes if the invariant set is classically chaotic. Indeed, the topological pressure satisfies the inequality (14) so that \[ P\left(\frac{1}{2};{\cal E}\right)\geq \frac{1}{2}\, P(1;{\cal E}) \, . \] We have just seen that the equality is satisfied if the invariant set contains a single periodic orbit. However, in chaotic scatterers, there is an exponential proliferation of orbits with different stretching factors and the inequality prevails. Typically, the gap in the resonance spectrum is thus smaller than predicted by the classical escape rate, so that there may exist resonances with a lifetime that is longer than the classical lifetime: \[ \mbox{chaotic filamentary invariant set:} \qquad {\rm max}\{\tau_r\}_{{\cal E}_r\simeq{\cal E}} = - \frac{1}{2P\left(\frac{1}{2};{\cal E}\right)} > - \frac{1}{P(1;{\cal E})} = \tau_{\rm cl}({\cal E}) \] for \(P\left(\frac{1}{2};{\cal E}\right)<0\).

Therefore, the quantum lifetimes can be longer than the classical lifetime if the invariant set is chaotic, in which case there is hindrance to wavepacket decay with respect to the classically expected behavior. This effect is analogous to the phenomena of dynamical localization occurring in chaotic diffusion (Casati et al. 1979, Grempel et al. 1984, Shepelyansky 1987) and of Anderson localization in disordered media (Anderson 1958, Anderson 1978). In such phenomena, the system offers different possible paths to the wave propagation and interferences occur between the paths, which is slowing down the wave propagation, hence the lengthening of the lifetimes in quantum chaotic scattering.

Emergence of classical Pollicott-Ruelle resonances

The classical lifetime plays an important role in the time evolution of wavepackets. Even if there exist resonances with a lifetime that is longer than the classical one, these resonances do not necessarily dominate the time evolution of the wavepacket. Indeed, if the wavepacket is broadly distributed over an interval of energy covering many resonances, the early decay of this wavepacket would proceed at the rate that is an average property over these many resonances. A systematic method to describe quantitatively this intuitive idea is provided by extending the calculation that has led to Eq. (18), to the quantum survival probability \[ {\cal P}(t) =\int_D \vert\psi_t({\mathbf r})\vert^2 \, d{\mathbf r} \tag{26} \] of a wavepacket \(\psi_t({\mathbf r})\) covering many resonances from the domain \(D\in{\mathbb R}^f\) of position space. This survival probability can be expressed as \[ {\cal P}(t) = {\rm tr} \, \chi_D(\hat{\mathbf r}) \, {\rm e}^{-\frac{i}{\hbar} \hat H t} \, \hat\rho_0 \, {\rm e}^{+\frac{i}{\hbar} \hat H t} \tag{27} \] in terms of the density operator \(\hat\rho_0=\vert\psi_0\rangle\langle\psi_0\vert\) associated with the initial wavepacket and the indicator function such that \(\chi_D({\mathbf r})=1\) if \({\mathbf r}\) belongs to the domain \(D\) and zero otherwise. In fact, this survival probability is more general than Eq. (26) because the density operator may also describe a statistical mixture.

In order to obtain the semiclassical approximation, the method of Wigner transform should first be used to obtain the smooth quasiclassical part and, subsequently, periodic-orbit theory to obtain the oscillating part, which is the same structure as in Eq. (18). We thus get \[ {\cal P}(t) \simeq \int\frac{d{\mathbf r}\, d{\mathbf p}}{(2\pi\hbar)^{f}} \, \chi_D\, {\rm e}^{\hat L_{\rm cl}t} \rho_{0{\rm W}} + O(\hbar^{-f+1}) + \frac{1}{\pi\hbar} \int dE \sum_p\sum_{r=1}^{\infty} \frac{\cos\left(r\frac{S_p}{\hbar}-r\frac{\pi}{2}\mu_p\right)}{\vert\det({\boldsymbol{\mathsf M}}_p^r-{\boldsymbol{\mathsf 1}})\vert^{1/2}} \oint_p \chi_D\, {\rm e}^{\hat L_{\rm cl}t} \rho_{0{\rm W}}\, dt +O(\hbar^0) \, , \tag{28} \] where \(\hat L_{\rm cl}=\{ H_{\rm cl}, \cdot \}_{\rm Poisson}\) denotes the classical Liouvillian operator and \(\rho_{0{\rm W}}\) the Wigner transform of the initial density operator \(\hat\rho_0\) (Wilkinson 1987). The first terms describe the smooth part of the time evolution which is quasiclassical with corrections of order \(\hbar^{-f+1}\) that can be obtained with the method of Wigner transform (Wigner 1932, Grammaticos and Voros 1979). The last oscillating part contains the contributions of the periodic orbits which are superposed to the smooth quasiclassical part.

The Pollicott-Ruelle resonances (Pollicott 1985, Ruelle 1986a, Ruelle 1986b) can be defined as generalized eigenvalues \[ \hat L_{\rm cl} \, \phi_{n} = s_{n} \, \phi_{n} \, , \] of the classical Liouvillian operator \(\hat L_{\rm cl} \, \phi\equiv\{H,\phi\}\) defined by taking the Poisson bracket with the Hamiltonian function (Balescu 1975). The corresponding eigenstates \(\phi_{n}\) are Gelfand-Schwartz distributions, which have the unstable manifolds of the invariant set for support (Gaspard 1998). The eigenstates \(\tilde\phi_{n}\) of the adjoint operator \(\hat L_{\rm cl}^{\dagger}\) have the stable manifolds of the invariant set as support. The generalized eigenvalues are in general complex \(\{s_{n}={\rm Re}\, s_{n} + i \, {\rm Im}\, s_{n}\}\) and they control the decay of statistical ensembles of classical orbits so that their real part is not positive, \({\rm Re}\, s_{n}\leq 0\). The Pollicott-Ruelle resonances are defined for the classical dynamics on a given energy shell. The quasiclassical part of the survival probability (18) can thus be expanded on the spectrum of Pollicott-Ruelle resonances as (Gaspard and Alonso Ramirez 1992) \[ {\cal P}(t) \simeq\int dE \sum_{n} (\chi_D,\phi_{n,E}) \; {\rm e}^{s_{n,E}\, t} \; (\tilde\phi_{n,E}, \rho_{0{\rm W}}) \sim {\rm e}^{-\gamma_{\rm cl}(E)\, t} \, . \tag{29} \] Since the long-time decay of such classical ensembles is controlled in open systems by the classical escape rate, we may conclude that the leading Pollicott-Ruelle resonances is nothing else than minus the classical escape rate: \[ s_{0,E} = - \gamma_{{\rm cl},E} \, . \tag{30} \] The further Pollicott-Ruelle resonances typically control shorter transients in the decay of the survival probability. The quasiclassical decay of the quantum survival probability has been shown in detail to be indeed controlled by the classical escape rate in hard-disk and hard-sphere billiards (Goussev and Dorfman 2006).

Figure 7: Microwave cavity with \(1/8\) of the four-disk geometry: (a) Experimental autocorrelation functions \(C(\kappa)= \left\langle \vert S_{21}(k-\kappa/2)\vert^2 \vert S_{21}(k+\kappa/2)\vert^2\right\rangle\) of the transmission function \(\vert S_{21}\vert^2\) versus the wavenumber \(\kappa\) in the four-disk system with \(a=5\) cm and \(R=20\) cm. The autocorrelation functions are compared with the Lorentzian \(\left[ 1+(\kappa/\gamma)^2\right]^{-1}\) for a fitted value of the parameter \(\gamma\). The different autocorrelations correspond to averaging over intervals centered on different values \(k_0\). (b) The experimental values of the parameter \(\gamma\) versus the ratio \(R/a\) and compared with the classical escape rate \(\gamma_{\rm cl}\) in cm\(^{-1}\) (solid line) and a lower bound on the quantum escape rate (dot-dashed line). [Reprinted with permission from Ref. (Lu et al. 1999). Copyright 1999 by the American Physical Society.]
Figure 8: Above: the same experimental autocorrelation function \(C(\kappa)\) as in Figure 7 versus the wavenumber \(\kappa\) (cm\(^{-1}\)) for the \(n\)-disk systems in their fundamental domain with \(a=5\) cm: (left) Two-disk system with \(R/a=8\); (center) Three-disk system with \(R/a=4\sqrt{3}\); (right) Four-disk system with \(R/a=4\sqrt{2}\). Below: The decompositions of the autocorrelation functions into Lorentzian peaks. The filled squares give the positions of the so-obtained experimental peaks and the open circles the Pollicott-Ruelle resonances predicted by the classical Liouvillian dynamics. [Reprinted with permission from Ref. (Pance et al. 2000). Copyright 2000 by the American Physical Society.]

It turns out that the Pollicott-Ruelle resonances control any observable quantity that depends on many quantum levels and, in particular, on many quantum scattering resonances. This is the case for the observable quantities used in the statistics of transmission probabilities or cross sections defined as \(\sigma_{ba}(E)=\vert S_{ba}(E)\vert^2\) in terms of the elements \(S_{ba}(E)\) of the \(S\)-matrix, such as their spectral autocorrelation functions: \[ \tilde C_E(\varepsilon) = \left\langle \sigma_{ba}\left(E-\frac{\varepsilon}{2}\right)\, \sigma_{ba}\left(E+\frac{\varepsilon}{2}\right)\right\rangle - \langle\sigma_{ba}(E)\rangle^2 \, . \] The Fourier transform of such quantities are time-dependent observables \[ C_E(t) =\int_{-\infty}^{+\infty} \tilde C_E(\varepsilon) \, {\rm e}^{-i\varepsilon \, t/\hbar} \, d\varepsilon \] defined on the energy shell \(E\) and evolving in time in a similar way as the survival probability (26). Accordingly, they also admit the expansion (29) in terms of the Pollicott-Ruelle resonances \(\{s_{n,E}=-\gamma_{n,E}\pm i\omega_{n,E}\}\) (Agam 2000) \[ C_E(t) \simeq \sum_{n} c_{n} \, \exp(-\gamma_{n,E}\, \vert t\vert) \, \cos\omega_{n,E} t \] with some coefficients \(c_{n}\). The leading term corresponding to Eq. (30) with \(\omega_{0,E}=0\) is controlled by the classical escape rate (Blümel and Smilansky 1988): \[ C_E(t) \simeq \exp(-\gamma_{{\rm cl},E}\, \vert t\vert) \, . \]

By inverse Fourier transform, the spectral autocorrelation function becomes \[ \tilde C_E(\varepsilon) \simeq \sum_{n} \frac{\tilde c_{n}}{(\varepsilon-\hbar\omega_{n,E})^2+(\hbar\gamma_{n,E})^2} + (\varepsilon\to -\varepsilon) \, , \] which shows that the Pollicott-Ruelle resonances can be identified by decomposing the spectral autocorrelation function into Lorentzian peaks. The main Lorentzian peak is given by (Blümel and Smilansky 1988) \[ \tilde C_E(\varepsilon) \sim \frac{1}{\varepsilon^2+(\hbar\gamma_{{\rm cl},E})^2} \, , \] which is reminiscent of the phenomenon of Ericson fluctuations (Ericson 1960).

Experiments on microwave cavities with the shape of \(n\)-disk scatterers have shown in detail that the classical Pollicott-Ruelle resonances can indeed be determined from the statistics of the transmission coefficients of wave scattering (Lu et al. 1999, Lu et al. 2000, Pance et al. 2000). Figure 7 depicts the autocorrelation function of transmission coefficients in these experiments, as well as the width of the Lorentzian peak at the origin as a function of the distance between the disks in the four-disk scatterer, showing the agreement of this width with the classical escape rate (Lu et al. 1999). The Pollicott-Ruelle resonances emerge as further peaks in the autocorrelation functions obtained for the two-, three-, and four-disk scatterers, as demonstrated in Figure 8.

Spectra of quantum scattering resonances and their features

Figure 9: Spectrum of scattering resonances for the three-disk scatterer with \(R/a=6\) with \(a=1\) in the plane of complex wavenumber \(k={\rm Re}\, k+i\, {\rm Im}\, k\). The crosses give the exact resonances obtained by the Korringa-Kohn-Rostoker method (Gaspard and Rice 1989c). The dots show the resonances obtained by the semiclassical periodic-orbit theory up to 6th order in the cycle expansion (21). Because of the C\(_{3v}\) symmetry for the three disks forming an equilateral triangle, the spectrum separates according to the irreducible representations of this group: the filled circles are the A\(_1\) resonances; the open circles the A\(_2\) resonances; and the crossed squares the E resonances. The spectrum has a gap given by the topological pressure per unit length at \(\beta=1/2\). The resonances accumulate around the value \(-\gamma_{\rm cl}/2\) given by the classical escape rate per unit length. [Adapted from Refs. (Gaspard 1993, Gaspard et al. 1995).]
Figure 10: Spectrum of A\(_1\) scattering resonances for the three-disk scatterer with \(R/a=6\) in the plane of complex wavenumber \(k={\rm Re}\, k+i\, {\rm Im}\, k\). The pluses denote the exact resonances obtained with the Korringa-Kohn-Rostoker method and the crosses the semiclassical resonances obtained as the zeroes of the Zeta function (19) up to 12th order in the curvature expansion (Wirzba 1999). This expansion converges only above the boundary of convergence around \({\rm Im}\, k\simeq -0.9/a\). (Courtesy of Andreas Wirzba.)

Going back to the correspondence with the spectrum of the quantum scattering resonances, the emergence of classical behavior is due to the fact that, although the quantum lifetimes may be longer than the classical one, the quantum scattering resonances are denser around the imaginary value of energy that corresponds to the classical escape rate \[ {\rm Im}\, E_r\biggl\vert_{\rm cl} \simeq \frac{\hbar}{2}\, P(1;{\rm Re}\, E_r) = -\frac{\hbar}{2}\, \gamma_{\rm cl}({\rm Re}\, E_r) \, , \tag{31} \] than close to the gap \[ {\rm Im}\, E_r\biggl\vert_{\rm gap} \simeq \hbar\, P\left(\frac{1}{2};{\rm Re}\, E_r\right) \, . \tag{32} \]

This is observed in the spectrum of quantum scattering resonances obtained for the three-disk scatterer (Gaspard and Rice 1989a, Gaspard and Rice 1989b, Gaspard and Rice 1989c, Gaspard et al. 1995, Wirzba 1999) (see Figs. Figure 9 and Figure 10). For such a billiard, the scattering resonances can be computed with the Korringa-Kohn-Rostoker method (Gaspard and Rice 1989c, Decanini et al. 1998a, Decanini et al. 1998b, Wirzba 1999), allowing the comparison with the approximation provided by the semiclassical method based on the Zeta function (19). The accumulation of quantum scattering resonances around the classically expected value (31) is nicely confirmed experimentally in Ref. (Barkhofen et al. 2013). In Figure 10, we notice that the very first resonances near \(k=0\) are found above their semiclassical approximation. This deviation is explained in terms of corrections in powers of \(\hbar\sim k^{-1}\) (Alonso and Gaspard 1993). Besides, the chain structures observed in the resonance spectrum can be explained in relation with the behavior of the resonant states by using the semiclassical cycle expansion (21) (Weich et al. 2014).

Figure 11: Spectrum of scattering resonances for the collinear dynamics of the triatomic molecule IHgI. The resonances obtained by numerical wavepacket propagation are depicted as filled circles. The corresponding classical dynamics undergoes a transition from a single unstable periodic orbit at low energy \(({\rm Re}\, E< 523\) cm\(^{-1})\) to a purely chaotic set of trapped orbits at high energy \(({\rm Re}\, E > 575\) cm\(^{-1})\). A nonhyperbolic regime with KAM elliptic islands exists in between. The dotted open circles give the resonances obtained by semiclassical periodic-orbit theory. The crosses are the resonances obtained by extrapolation from the saddle equilibrium point at zero energy. In the chaotic regime, the dotted line gives the gap in the resonance spectrum at \(\hbar P(1/2;{\rm Re}\, E)\). The dot-dashed line shows the value expected from the classical escape rate, \(\hbar P(1;{\rm Re}\, E)/2=-\hbar\gamma_{\rm cl}({\rm Re}\, E)/2\). The solid lines show the values due to the Lyapunov exponents of the individual periodic orbits \(\{0,1,2\}\). In the periodic regime, all these curves coincide, as explained in the text. [Adapted from Ref. (Gaspard and Burghardt 1997).]

Figure 11 shows the resonance spectrum for collinear dissociation of triatomic molecules in comparison with the values (31) and (32) (Gaspard and Burghardt 1997). The invariant set contains the single periodic orbit \(0\) at low energy where the gap coincides with the value of the classical escape rate. However, beyond a certain energy, the invariant set is chaotic with a purely hyperbolic dynamics. The corresponding symbolic dynamics is based on three symbols \(\{0,1,2\}\) associated with the three periodic orbits of shortest periods. Their periods differ, which induces interferences between the contribution of these three periodic orbits so that the spectrum of resonances shows irregularities and the gap is closer to the real energy axis than predicted by the classical escape rate, as already seen in Figure 9. This phenomenon of quantum lengthening of lifetimes is the quantum signature of the classically chaotic dynamics.

The fractal Weyl law

If the previous considerations are focused on the imaginary part of the resonances, the fractal Weyl law concerns the statistical distribution of their real part.

For billiards such as the disk scatterers, the wavenumber \(k=\sqrt{2mE}/\hbar\) is conveniently used instead of the energy \(E=\hbar^2k^2/(2m)\) because Schrödinger's equation reduces in billiards to Helmholtz' equation, \((\nabla^2+k^2)\psi=0\). In bounded billiards, Weyl's law states that the number of levels \(\{k_n\}\) up to a given wavenumber should scale as \[ \mbox{Number}\{ k_n < k\} \sim k^f \qquad\mbox{for}\quad k\to\infty \, , \] where \(f\) is the number of degrees of freedom, i.e., the dimension of position space, which is half the phase-space dimension \(2f\). The heuristic argument is that an eigenfunction of wavenumber \(k\) has the wavelength \(\lambda=2\pi/k\) and the cavity may accommodate down to the scale fixed by the wavelength as many eigenfunctions as small cells of size equal to this wavelength, \(N(\lambda)\sim\lambda^{-f}\). This argument can be extended to open billiards such as the disk scatterers if the generalized eigenfunctions associated with the resonances are supposed to be supported by the projection onto position space of the fractal invariant set. This projection is notably characterized by the Hausdorff dimension \(0\leq D_{\rm H}/2\leq f\). Accordingly, this heuristic argument suggests that \[ \mbox{Number}\{ k_r={\rm Re}\, k_r + i \, {\rm Im}\, k_r: \; {\rm Re}\, k_r<{\rm Re}\, k,\; {\rm Im}\, k_r>-C\} \sim {\rm Re}\, k^{D_{\rm H}/2} \] for \({\rm Re}\, k\to\infty\) and \(C>0\), where the phase-space Hausdorff dimension \(0\leq D_{\rm H}=2 d_{\rm H}+2 \leq 2f\) is given in terms of the Hausdorff dimension \(0\leq d_{\rm H}\leq f-1\) in the stable or unstable directions transverse to the flow in the energy shell. An important remark is that the Hausdorff dimension refers to the geometrical fractal set of trapped orbits in deterministic dynamical systems.

What has been mathematically proved until now is an upper bound on the number of resonances given in terms of the (upper) Minkowski dimension, a concept that is closely related to the box-counting dimension (Falconer 1990). In a given energy shell \({\cal S}_{\cal E}\) of dimension \(f-1\), the upper Minkowski dimension of the invariant set \({\cal I}_{\cal E}\) of trapped orbits is defined as \[ \overline{\rm dim}_{\rm M}\, {\cal I}_{\cal E} = 2f-1 -{\rm sup}\left\{c: \; \varlimsup_{\epsilon\to 0} \, \epsilon^{-c} \, {\rm vol}\{{\mathbf X}\in{\cal S}_{\cal E}:\; {\rm dist}({\mathbf X},{\cal I}_{\cal E})<\epsilon\}<\infty\right\} \, . \] This dimension is contained in the interval, \(0\leq \overline{\rm dim}_{\rm M}\, {\cal I}_{\cal E}\leq 2f-1\). For standard fractal sets of deterministic systems, the upper Minkowski dimension would coincide with the Hausdorff dimension, \(\overline{\rm dim}_{\rm M}\, {\cal I}_{\cal E}=D_{\rm H}=2d_{\rm H}+1\).

The known result states that \[ \mbox{Number}\{ E_r={\rm Re}\, E_r + i \, {\rm Im}\, E_r: \; {\cal E}-a\hbar \leq {\rm Re}\, E_r\leq {\cal E}+a\hbar,\; -b\hbar\leq {\rm Im}\, E_r\leq 0\}\leq C(a,b;{\cal E}) \; \frac{1}{\hbar^{\mu}} \] for \(\hbar\to 0\) with \[ \mu > \frac{1}{2}\left(\overline{\rm dim}_{\rm M}\, {\cal I}_{\cal E}-1\right) \] and the constants \(a,b,C>0\). This theorem has been proved for potential scattering, for scattering on surfaces of constant negative curvature, as well as for open quantum maps (Sjöstrand 1990, Sjöstrand and Zworski 2007, Nonnenmacher and Zworski 2007, Nonnenmacher 2011, Novaes 2013).

We notice that this result is consistent with Weyl's law for bounded systems where \(\overline{\rm dim}_{\rm M}\, {\cal I}_{\cal E}=2f-1\). Indeed, their average level density is given by \[ n_{\rm av}({\cal E}) \simeq \int \frac{d{\mathbf r}\, d{\mathbf p}}{(2\pi\hbar)^f} \, \delta({\cal E}-H_{\rm cl}) \qquad\mbox{for}\quad \hbar\to 0 \tag{33} \] so that the number of levels in the interval \(\vert E_n-{\cal E}\vert < a\hbar\) is estimated by \(2a\hbar \, n_{\rm av}({\cal E})\sim 1/\hbar^{f-1}\), as it should for \(\hbar\to 0\).

The fractal Weyl law has been observed numerically for potential scattering (Lin and Zworski 2002), for open quantum maps (Nonnenmacher and Zworski 2005, Shepelyansky 2008), and experimentally in microwave cavities (Lu et al. 2003, Potzuweit et al. 2012). Moreover, the generic case of open mixed systems with KAM elliptic islands in chaotic zones has also been studied (Körber et al. 2013).

The theoretical challenge remains to determine with precision the scaling exponent that is appropriate for this statistics, in particular, in high-dimensional billiards (Eberspächer et al. 2010).

The case of open quantum graphs

Quantum graphs are networks of \(B\) bonds connected by vertices (Kottos and Smilansky 1997, Kottos and Smilansky 2000, Kottos and Smilansky 2003). Waves propagate in the bonds according to the one-dimensional Schrödinger equation \[ \left(\frac{d^2}{dx^2}+k^2\right)\psi_b(x)=0 \] and are scatterered at every vertex \(j\) by some local \(S\)-matrix \(\varsigma_{ab}^{j}\) such that \[ \psi_a^{\rm out}(j) = \sum_b \varsigma_{ab}^{j} \, \psi_b^{\rm in}(j) \, . \] The quantization condition for the wavenumber \(k\) is given by the Zeta function: \[ Z(k) =\det\left[ {\boldsymbol{\mathsf 1}}-{\boldsymbol{\mathsf R}}(k)\right]=0 \] in terms of the \(2B\times 2B\) matrix \[ {\boldsymbol{\mathsf R}}(k) = {\boldsymbol{\mathsf T}} \cdot {\boldsymbol{\mathsf D}}(k) \qquad\mbox{with}\qquad D_{ab}(k) = \delta_{ab} \, {\rm e}^{ikl_a} \qquad\mbox{and}\qquad T_{ab}=\varsigma_{ab}^{\overleftarrow{a}} \, \delta_{\overleftarrow{a}\overrightarrow{b}} \, , \] where \(\overleftarrow{a}\) or \(\overrightarrow{b}\) denote the vertex at left or right hand of the bond. Quantum graphs can be bounded with an associated discrete eigenvalue spectrum, or open with semi-infinite bonds extending to infinity. In this latter case, the quantum graph is of scattering type and has an associated spectrum of scattering resonances at complex values of the wavenumber \(\{k_r\}\).

Figure 12: (Above) Open quantum graph with periodic dynamics: (a) the survival probability versus time for a wavepacket with the superposition of wavenumbers shown as the solid line in (b). (b) The corresponding spectrum of scattering resonances in the plane of complex wavenumbers. All the resonances have the same imaginary part at the value predicted by the classical escape rate per unit length. (Below) Open quantum graph with chaotic dynamics: (c) the survival probability versus time for a wavepacket with the superposition of wavenumbers shown as the solid line in (d). The early decay is exponential at the classical escape rate (dashed line). The long-time decay is controlled by the resonance which is the closest to the real wavenumber axis in (d). (d) The corresponding spectrum of scattering resonances in the plane of complex wavenumbers. The spectrum is now irregular. There is no gap for this open graph. [Adapted from Ref. (Barra and Gaspard 2001b).]
Figure 13: Spectrum of scattering resonances in the plane of complex wavenumbers for the linear open quantum graphs described in the text: (a) for the graph with \(n=1\) cell; (b) for the graph with \(n=2\) cells; (c) for the graph with \(n=3\) cells; (d) for the graph with \(n=4\) cells. The topological pressure per unit length \(P(1/2)\) is negative for \(n=1,2\) so that the spectra (a) and (b) have a gap, but it is positive for \(n=3,4\) so that the spectra (c) and (d) have no gap predicted by the semiclassical theory. The value corresponding to the classical escape rate per unit length is also shown. [Adapted from Ref. (Barra and Gaspard 2001b).]

The classical limit of quantum graphs leads to a Markovian process (Barra and Gaspard 2001a). The reason is that the probability amplitude is split when the wave is scattered on a vertex connecting two or more bonds. This behavior is reminiscent of systems with ray splitting (Blümel et al. 1996a, Blümel et al. 1996b). Nevertheless, the classical Markovian process on the graph can be characterized by the Ruelle topological pressure because this function is known to apply to Markovian processes as well as to chaotic dynamics (Gaspard 1998).

For quantum graphs, the topological pressure at \(\beta=1\) gives the classical escape rate and the pressure at \(\beta=1/2\) the gap in the spectrum of quantum scattering resonances.

The role of the classical escape rate is illustrated in Figure 12 depicting the time dependence of the survival probability and the corresponding resonance spectrum in an open quantum graph with a single periodic orbit and another one that is classically chaotic (Barra and Gaspard 2001b). For the classically periodic quantum graph, all the scattering resonances have the same imaginary part at a value given in terms of the classical escape rate \({\rm Im}\, k_r=-\tilde\gamma_{\rm cl}/2\), which coincides with the gap given by the topological pressure at \(\beta=1/2\). In this case, the survival probability is decaying exponentially at the corresponding rate with damped oscillations due to periodicity. In contrast, the classically chaotic quantum graph has an irregular spectrum of scattering resonances distributed around the value corresponding to the classical escape rate. Here, the survival probability has an early exponential decay at the escape rate, followed by the slower exponential decay controlled by the quantum scattering resonance that is the closest to the real wavenumber axis in the range of the wavepacket. However, there is no gap in this case.

In order to see the formation of the gap, further examples of open quantum graphs are considered in Figure 13 (Barra and Gaspard 2001b). They are linear and composed of \(n\) unit cell containing two bonds of lengths \(l_a=0.5\) and \(l_b=\sqrt{2}\). The \(S\)-matrix at a vertex is given by \[ {\boldsymbol{\varsigma}}^{j}= \left( \begin{matrix} i\, \sin\eta_j & \cos\eta_j \\ \cos\eta_j & i\, \sin\eta_j \end{matrix} \right) \] with \(\eta_1=0.1\) and \(\eta_2=(\sqrt{5}-1)/2\). At \(\beta=1/2\), the topological pressure is negative for \(n=1,2\), but positive for \(n=3,4\). Consequently, a gap should form only for \(n=1,2\). This is confirmed by the resonance spectra shown in Figure 13. The early decay of the survival probability is always controlled by the classical escape rate (Barra and Gaspard 2001b).

Concerning the fractal Weyl law, the geometry of quantum graphs remains one-dimensional because the wave cannot extend outside the unidimensional bonds. Therefore, the Minkowski dimension of quantum graph is always equal to \(\overline{\rm dim}_{\rm M}\, {\cal I}_{\cal E}=1\), which is the dimension of the energy shell in a system with a single degree of freedom \(f=1\). Accordingly, the fractal Weyl law is trivially satisfied: \[ \mbox{Number}\{ k_r= {\rm Re}\, k_r +i\, {\rm Im}\, k_r \, : \; {\rm Re}\, k_r < {\rm Re}\, k,\; {\rm Im}\, k_r >-C\} \sim {\rm Re}\, k \] for \({\rm Re}\, k\to\infty\). All the resonances are found in a strip with a large enough constant \(C>0\) (Barra and Gaspard 2001b).

We notice that, in quantum graphs, the Minkowski dimension differs from the value that would be given by the root of the topological pressure \(P(\beta)=0\), because the topological pressure characterizes the Markovian process due to ray splitting at the vertices and is no longer purely geometrical as in deterministic dynamical systems. A similar difference could be expected for leaking cavities with ray splitting.

Magnetoconductance fluctuations in quantum electron transport

Nowadays, quantum electron transport can be studied in semiconducting microstructures of artificial shapes. At low temperature, the conductance of these devices manifests quantum fluctuations as a function of the external magnetic field (as well as the gate voltage that controls the Fermi energy of the conducting electrons). These fluctuations are the feature of interferences between quantum-mechanical waves and are thus reproducible. In the early nineties, different behaviors have been predicted with semiclassical theory for classically chaotic and nonchaotic billiards (Jalabert et al. 1990, Baranger et al. 1993a, Baranger et al. 1993b, Lin et al. 1993, Nakamura and Harayama 2004) and experiments have been carried out to compare the conductance fluctuations of the chaotic stadium and the nonchaotic circle billiards (Marcus et al. 1992, Marcus et al. 1993, Chang et al. 1994, Chan et al. 1995).

According to Landauer scattering approach (Landauer 1957, Büttiker et al. 1985, Nazarov and Blanter 2009), the conductance at low temperature is given by \[ G(E,B) = \frac{e^2}{\pi\hbar} \sum_{ab} \vert T_{ba}(E,B)\vert^2 \] in terms of the transmission amplitudes \(T_{ba}(E,B)\) between the incoming and outgoing modes in the leads. In semiclassical theory, the transmission amplitudes are obtained by summing over the corresponding classical orbits \(\{l\}\) as \[ T_{ba}(E,B) \simeq \sum_{l} \xi_{bal} \, {\rm e}^{i \tilde S_l(E)/\hbar} \, {\rm e}^{-i \frac{\pi}{2}\mu_l} \, {\rm e}^{i eA_lB/\hbar} \, . \] In every term of this sum, \(\xi_{bal}\) is a coefficient associated with the orbit \(l\), \(\tilde S_l(E)\) its action, \(\mu_l\) the corresponding Maslov index, and the last factor represents the Aharonov-Bohm phase due to the external magnetic field \(B\) (Jalabert et al. 1990). This phase depends on the effective area \(A_l\) covered by the classical orbit in the plane perpendicular to the magnetic field and \(e\) is the electric charge.

Figure 14: (Left) Magnetoresistance \(R=1/G\) as a function of the external magnetic field \(B\) for (a) the stadium-shaped and (b) circle-shaped microstructures. (Right) Averaged power spectra (35) for the corresponding stadium-shaped (solid diamonds) and circle-shaped (open circles) microstructures with the autocorrelation function (34) as inset. [Reprinted with permission from Ref. (Marcus et al. 1992). Copyright 1992 by the American Physical Society.]

In chaotic systems, the areas \(\{A_l\}\) are exponentially distributed at large values as \[ P(A) \sim \exp(-\alpha_{\rm cl} \vert A \vert) \] with some constant \(\alpha_{\rm cl}\), although this behavior is not observed for nonchaotic systems such as the open circle billiard. Accordingly, the autocorrelation function of the conductance fluctuations should vary as (Jalabert et al. 1990) \[ \begin{align*} &C_G(\Delta B) = \langle \delta G(E,B+\Delta B) \, \delta G(E,B)\rangle \\ &\sim \left\vert \int dA \, P(A) \; {\rm e}^{i e A \Delta B/\hbar}\right\vert^2 \sim \frac{1}{ \left[1 + \left( \frac{e\Delta B}{\hbar\alpha_{\rm cl}}\right)^2\right]^2} \, . \tag{34} \end{align*} \] Therefore, the Fourier transform of this autocorrelation function is given by \[ \begin{align*} &S_G(f) = \int d\Delta B\; C_G(\Delta B) \; {\rm e}^{i2\pi \Delta B \, f} \\ &\sim \left(1+2\pi\hbar\alpha_{\rm cl} f/e\right) \, \exp\left(-2\pi\hbar\alpha_{\rm cl} f/e\right) \, . \tag{35} \end{align*} \] As seen in Figure 14, this behavior is indeed observed experimentally for the stadium-shaped microstructure although deviations are observed for the circle-shaped one.

Figure 15: Magnetoresistance for (a) 48 stadium- and (b) 48 circle-shaped cavities at the temperature \(T=50\) mK. The cavities are fabricated on a GaAs/Al\(_x\)Ga\(_{1-x}\)As heterostructure crystal. The peak is Lorentzian for the chaotic cavity, but different for the nonchaotic one. [Reprinted with permission from Ref. (Chang et al. 1994). Copyright 1994 by the American Physical Society.]

Another prediction concerns the peak of magnetoresistance \(R(E,B)=1/G(E,B)\) around \(B=0\), which is directly given by (Baranger et al. 1993a, Baranger et al. 1993b) \[ \langle\delta R(E,B)\rangle \sim \int dA \, P(A) \; {\rm e}^{i e A B/\hbar} \sim \frac{1}{ 1 + \left( \frac{eB}{\hbar\alpha_{\rm cl}}\right)^2} \, . \tag{36} \] Figure 15 depicts this quantity after averaging over 48 stadium- and circle-shaped cavities in experiments with semiconducting devices (Chang et al. 1994). The peak is Lorentzian as predicted by Eq. (36) for the chaotic cavity, although it clearly deviates from this prediction for the nonchaotic cavity.

These results show the influence of the classically chaotic dynamics on the interferences of the electronic wave scattering in the semiconducting microstructures.

The random matrix approach


If the scatterer is slightly open, the resonances are expected to have long lifetimes and, thus, to lie close to the real energy axis. This is the case for disk scatterers if the disks are nearly touching each other so that they delimit an internal cavity (Gaspard and Rice 1989c) or for the stadium-shaped billiard with openings coupled to leads (Marcus et al. 1992). These leads may be antenna plugged into a microwave cavity (Stöckmann 1999). Such scatterers also exist in leaking chaotic systems (Altmann et al. 2013).

In these circumstances, the scattering problem can be solved by first considering a fictitious bounded system with closed leads. This fictitious bounded system has a purely discrete eigenvalue spectrum. In a second step, the coupling to the leads is treated as a perturbation, giving an imaginary part to the eigenvalues. This idea has been developed in unimolecular reaction-rate theory (Holbrook et al. 1996), as well as in nuclear reaction theory (Mitchell et al. 2010).

For the dissociation of a system at a given energy \(E\), an essential parameter is given by the number \(\nu(E)\) of open channels in the leads coupled to the quasi bounded region. If \(n_{\rm av}(E)\) denotes the level density (33) of the quasi-bound states, an estimation of the reaction rate is given by \[ \overline{\kappa}(E) \simeq \frac{\nu(E)}{2\pi\hbar\, n_{\rm av}(E)} \simeq \gamma_{\rm cl}(E) \, , \] which is also an estimation of the classical escape rate (Holbrook et al. 1996, Gaspard and Burghardt 1997). The scattering resonances are thus expected to be distributed at complex energies with their imaginary parts around the value (31). Both the average level density \(n_{\rm av}(E)\) and the number \(\nu(E)\) of open channels are specific to the system of interest.

If the bounded reference system is classically chaotic, which is the case for the stadium billiard (Bunimovich 1979), the Bohigas-Giannoni-Schmit conjecture suggests that its discrete energy spectrum, which has the level density \(n_{\rm av}(E)\) over large energy scales, would manifest the universality of random matrix theory over the small energy scale of the average level spacing \(\langle\Delta E\rangle=\langle E_{j+1}-E_j\rangle\simeq n_{\rm av}(E)^{-1}\). As aforementioned, this conjecture is nowadays supported by periodic-orbit semiclassical theory (Sieber et al. 2001, Sieber 2002, Müller et al. 2004, Müller et al. 2005, Heusler et al. 2007, Müller et al. 2009). As long as the energy levels of the bounded reference system are slightly perturbed by the coupling to the leads for both their real and imaginary parts, the statistics of scattering resonances can thus be inferred from random matrix theories (Lewenkopf and Weidenmüller 1991, Stöckmann 1999, Mitchell et al. 2010).

The scattering matrix for a slightly open scattering system

Methods have been systematically developed in order to deduce the \(S\)-matrix for scattering systems, in which a quasi bounded subsystem is coupled to leads (Lewenkopf and Weidenmüller 1991, Stöckmann 1999, Mitchell et al. 2010). The quasi bounded subsystem can be described by a \(N\times N\) Hamiltonian Hermitian matrix \(\boldsymbol{\mathsf H}\) and its coupling to \(\nu\) open channels by the \(N\times\nu\) matrix \(\boldsymbol{\mathsf W}\). The coupling may also induce a shift \(\boldsymbol{\Delta}\) of the energy levels of \(\boldsymbol{\mathsf H}\). The resonant part of the \(S\)-matrix can be shown to be given by the expression: \[ \boldsymbol{\mathsf S}_{\rm res} = \boldsymbol{\mathsf 1}- 2i\pi \boldsymbol{\mathsf W}^{\dagger}\frac{1}{E-\boldsymbol{\mathsf H}-\boldsymbol{\Delta}+i\pi \boldsymbol{\mathsf W}\boldsymbol{\mathsf W}^{\dagger}} \,\boldsymbol{\mathsf W} \, . \] The scattering resonances are thus given by the complex eigenvalues of the effective non-Hermitian Hamiltonian operator: \[ \boldsymbol{\mathsf H}_{\rm eff}=\boldsymbol{\mathsf H}+\boldsymbol{\Delta}-i\pi \boldsymbol{\mathsf W}\boldsymbol{\mathsf W}^{\dagger} \, . \] In general, the propagation at the exterior of the quasi bounded subsystem, i.e., in the leads, may also contribute to the \(S\)-matrix (which is often called the background contribution). The complete \(S\)-matrix can be written as \[ \boldsymbol{\mathsf S}= \boldsymbol{\mathsf O}\, {\rm e}^{i\boldsymbol{\delta}}\, \boldsymbol{\mathsf S}_{\rm res} \, {\rm e}^{i\boldsymbol{\delta}}\, \boldsymbol{\mathsf O}^{\rm T} \, , \] where \(\boldsymbol{\mathsf O}\) is an orthogonal matrix and \(\boldsymbol{\delta}\) the diagonal matrix of phase shifts (Mitchell et al. 2010).

For an isolated energy level, this description reproduces the corresponding Breit-Wigner resonance (Stöckmann 1999). This framework can be extended to systems with \(N\) levels coupled to \(\nu\) open channels.

If the Hamiltonian matrix \(\boldsymbol{\mathsf H}\) has the level density \(n_{\rm av}(E)\), the density of the real parts of the resonances has the same density as long as the wavelength \(\lambda=2\pi\hbar/\sqrt{2mE}\) remains larger than the diameter of the leads. However, the fractal Weyl law suggests that ultimately the density of the real parts of the resonances should scale as the fractal dimension of the trapped set, which should happen if the wavelength becomes significantly shorter than the diameter of the leads.

Regime of non-overlapping resonances

Earlier studies have concerned the regime of weak coupling between the cavity and its leads (Porter 1965). If the cavity is a chaotic billiard such as the stadium-shaped cavity, the spectrum of resonances reflect the irregular eigenvalue spectrum of the closed cavity. Although its level density depends on the geometry of the cavity, its statistical properties on small energy scales follow the predictions of random matrix theory. If the Hamiltonian \(\boldsymbol{\mathsf H}\) is Hermitian, the spacing distribution and the spectral correlation functions are those of the Gaussian orthogonal ensemble for the desymmetrized cavity. Typically, the eigenfunctions are real and their values are distributed according to Gaussian random variables (Stöckmann 1999).

Besides, the levels get a width due to the weak coupling to the leads. Since the coupling to the leads is determined by the eigenfunctions at the position of each lead, the coupling matrix \(\boldsymbol{\mathsf W}=(W_{jc})\) with \(j=1,2,...,N\) and \(c=1,2,...,\nu\) has elements \(W_{jc}\) that may be assumed to be also ruled by Gaussian distributions with the same variance for every channel \(c=1,2,...,\nu\). An isolated energy level \(E_j\) would thus has a width given by \[ \Gamma_j = 2\pi \sum_{c=1}^{\nu} \vert W_{jc}\vert^2 \, , \] obeying the chi-square probability distribution: \[ p_\nu(x)=\frac{d}{dx} \,{\rm Prob}\left\{ \Gamma_j<\alpha x\right\} = \frac{x^{\nu/2-1}}{2^{\nu/2}\, \Gamma(\nu/2)} \, {\rm e}^{-x/2} \] with \(\alpha=2\pi\langle\vert W_{jc}\vert^2\rangle\) and the Gamma function \(\Gamma(\nu/2)\) for normalization to unity (Porter 1965, Stöckmann 1999). The random variable \(x\) has the mean value \(\langle x\rangle=\nu\) equal to the number of open channels and its variance is \({\rm Var}(x)=2\nu\).

In the case where there is only one open channel, we find the Porter-Thomas distribution \[ p_1(x)= \frac{1}{\sqrt{2\pi x}} \, {\rm e}^{-x/2} \, , \] which describes an accumulation of resonances close to the real energy axis (Stöckmann 1999, Kuhl et al. 2005). This distribution has indeed been observed in the radiative linewidths of NO\(_2\) (Georges et al. 1995, Delon and Jost 1999), as well as in the microwave stadium billiard (Alt et al. 1995).

If the number of open channels increases, the chi-square distribution tends towards a Gaussian distribution for the random variable \(z=(x-\nu)/\sqrt{2\nu}\). In this case, the scattering resonances accumulate around their average width \(\langle\Gamma_j\rangle = \alpha\nu\). This behavior mimics the formation of a gap. However, the chi-square distributions still extends from \(x=0\) to \(\infty\), so that there is no true gap in the present approximation.

Since the coupling to the leads is weak, the widths of the resonances are smaller than their spacings and the resonances are said to be non-overlapping. In this regime, further statistical properties such as the distributions of \(S\)-matrix elements have also been investigated for microwave cavities (Dietz et al. 2010), showing good agreement with the predictions of random matrix theories (Fyodorov et al. 2005).

Regime of overlapping resonances

The situation changes as the coupling of the cavity to the leads increases, in which case the widths of the resonances may become larger than their spacings. The system enters into the regime of overlapping resonances, which has been studied by systematic analysis of random matrix models (Lewenkopf and Weidenmüller 1991, Fyodorov et al. 2005, Mitchell et al. 2010). Finally, the phenomenon of Ericson fluctuations manifests itself in the regime of strongly overlapping resonances.

A model has been analyzed in detail by supersymmetric methods (Haake et al. 1992, Lehmann et al. 1995), in which the Hamiltonian matrix is taken in the Gaussian orthogonal ensemble with \[ \langle H_{ij}\rangle = 0 \qquad\mbox{and}\qquad \langle H_{ij}H_{i'j'}\rangle = \frac{\lambda^2}{N} \left(\delta_{ii'}\delta_{jj'} + \delta_{ij'}\delta_{i'j}\right) , \] while the elements of the coupling matrix are also Gaussian \[ \langle W_{jc}\rangle = 0 \qquad\mbox{and}\qquad \langle W_{jc}W_{j'c'}\rangle = \frac{\gamma\lambda}{N} \, \delta_{jj'} \delta_{cc'} \, . \] The analysis of this model shows that the resonances are distributed as a cloud at complex energies with a sharp boundary in the limit \(N,\nu\to\infty\) for \(\nu/N\ll 1\). This boundary forms a gap with respect to the real energy axis and, thus, a gap in the imaginary parts of the resonance energies (Lehmann et al. 1995, Mitchell et al. 2010). This gap is reminiscent of the one obtained with the semiclassical approach, for instance, in the three-disk scatterers if the distance between the disks is large enough. If so the number of open channels is indeed large, which could explain the analogy between the gaps obtained in both approaches.

Perspectives and open issues

Quantum chaotic scattering is a fascinating research topic, knowing important advances at the crossroad of several fields.

If the dynamics of the scatterer is classically chaotic, several results have been obtained for its wave-mechanical properties. The underlying scattering resonances, which determine wavepacket decays as well as the fluctuations of cross sections and transmission probabilities, are statistically distributed in a way characterizing the scattering process. On the one hand, the random matrix approach provides a systematic description of the statistical properties of quantum scattering for slightly open systems from the regime of isolated and non-overlapping resonances up to the regime of strongly overlapping resonances where the phenomenon of Ericson fluctuations manifests itself. In this latter regime, a gap may appear in the distribution of the resonance widths. On the other hand, the semiclassical approach allows us to relate the quantum properties to the classical orbits and, in particular, the periodic orbits that are trapped forever in the scatterer. The distribution of the resonance widths turns out to be determined by the Lyapunov exponents of the periodic orbits, providing a close connection to the chaotic properties. In classically chaotic scatterers, the periodic orbits proliferate exponentially with their period and interferences between the contributions of these periodic orbits induce a lengthening of the quantum lifetimes with respect to the classically expected value. If the system is open enough, a gap appears in the distribution of the resonance widths. This gap is determined in terms of Ruelle's topological pressure, which characterizes the classical chaotic dynamics. In this way, a quantum signature of chaos is established for this gap in the resonance spectrum. Remarkably, such quantum signatures of chaos are observed experimentally in open microwave cavities, in mesoscopic electronic circuits, as well as in atomic, molecular, and nuclear systems.

There are many open questions that are under current investigation. An important issue is to understand the behavior of quantum scattering systems with a classically integrable dynamics (Pichaureau et al. 1999) or with mixed phase-space structures composed of KAM elliptic islands amidst chaotic seas (Körber et al. 2013). These islands may accommodate long-lived resonant states if their area is larger than Planck's constant (Löck et al. 2010). Another fundamental issue is to establish the fractal Weyl law with precision for the exponent controlling the statistics of the real parts of the resonances. An upper bound is known for this exponent in terms of the Minkowski dimension of the fractal set of trapped orbits. However, the precise value of this exponent remains difficult to determine, especially in high-dimensional systems. Besides, the connection between the semiclassical and random matrix approaches could be strengthened to understand better the formation of the gap in the distribution of imaginary parts.

An important trend is the study of high-dimensional and many-particle systems, which have more complex properties of scattering and reaction than in low-dimensional systems. Another direction concerns the transition from finite scatterers towards crystalline or disordered spatially extended systems, such as Lorentz gases composed of hard-disk or Yukawa-potential lattices.

If the fluctuations of the scattering properties have been much studied with respect to energy, wavenumber, or external magnetic field, the properties of fluctuations in the scattering angles have not been much explored in relation to the well-known speckle phenomenon. Although this phenomenon does not require the existence of a chaotic saddle, the inside-outside duality may be addressed in order to see how an internal chaotic dynamics may influence the external speckle pattern (Dietz et al. 1995, Doya et al. 2002).

Among the applications, the semiclassical method could be considered for the theory of extended X-ray absorption fine structure (EXAFS), which is caused by the multiple scattering of a photoemitted electron on neighboring atoms (Gutzwiller 1990, Alonso 1995).

Further interesting properties can be studied for quantum chaotic scattering systems, such as the Casimir forces between scatterers (Wirzba 2008). These properties exist at zero or positive temperatures. In this latter case, the Bose-Einstein or Fermi-Dirac statistics should be taken into account, which is also essential for transport properties between reservoirs. Such considerations are thus of importance for quantum electron transport in mesoscopic semiconducting devices and, in particular, for the full counting statistics of electron transfers across these devices. These properties are also determined by the scattering matrix and we may wonder how they behave in the transition from quantum to classical regimes (Levitov and Lesovik 1993, Gaspard 2013).

In mathematics, quantum chaotic scattering is also considered in relation with the Riemann zeta function and Riemann's hypothesis (Gutzwiller 1983, Series 1987, Wardlaw and Jaworski 1989, Schumayer and Hutchinson 2011).

Acknowledgments. This research is financially supported by the Université Libre de Bruxelles and the Belgian Federal Government under the Interuniversity Attraction Pole project P7/18 "DYGEST".


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See Also

Quantum chaos, Microwave billiards and quantum chaos, Microwave ionization of hydrogen atoms, Semiclassical theory of helium atom, Anderson localization and quantum chaos maps, Cold atom experiments in quantum chaos, Random matrix theory, Loschmidt echo, Chirikov standard map, Kicked Harper model.

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