
QUANTUM CHAOS
Klaus Richter, Peter Schlagheck, Marta Gutiérrez, Arseni Goussev, Cyril Petitjean,
Juan Diego Urbina, and Daniel Waltner
The field of "quantum chaos" comprises the universal properties of quantum systems whose classical counterparts exhibit chaotic dynamics. Generally, classical chaos may arise in nonintegrable systems, where the number of degrees of freedom exceeds the number of constants of motion, and is characterized by a very sensitive dependence of the trajectories on the initial condition. This leads, in the corresponding quantum system, to a strong repulsion between adjacent energy levels in the spectrum, which can be described by the eigenvalues of ensembles of random matrices (RMT) preserving the general symmetries [1]. Classical integrability, on the other hand, is characterized by a Poissonian distribution of the levels [2] (see Fig. 1). Both situations have been quantitatively understood by means of the semiclassical ansatz, where quantities based on the Green function of the system are expressed in terms of trajectories or periodic orbits in the underlying classical system [3,4].


Fig. 1: Examples for chaotic and integrable systems (stadium billiard and circular billiard, respectively). The panels show (from left to right) representative trajectories, a Poincare section of the classical phase space, as well as the distribution of spacings between adjacent quantum eigenenergies. 
In the context of quantum chaos, we are presently focusing on three different topics: 
Spectral statistics of mixed regularchaotic systems
Integrability and full chaoticity are rather exceptional situations. In general, realistic physical systems exhibit both regular and chaotic dynamics, which means that their phase space contains a mixture of "islands" with nearly integrable motion and regions with chaotic motion. One of the main characteristics of the dynamics in such mixed regularchaotic systems are bifurcations, which arise at the coalescence of periodic orbits under variations of the energy or some parameters of the system. Bifurcations are important in the semiclassical theory because they typically contribute with a larger weight than isolated orbits. They may therefore lead to noticeable effects in the spectral statistics [5], which can be observed in longrange correlation functions such as the spectral form factor (the Fourier transform of the autocorrelation function of the energy levels) or the spectral rigidity (the least square deviation of the integrated level density from the best linear fit).



Fig 2: Bifurcation of a straightline orbit in the quartic oscillator (upper panel) at variation of a system parameter α. The lower panel shows the saturation of the spectral rigidity in the corresponding quantum system (dots) together with its semiclassical prediction (solid lines). 
So far, we have studied the role of pitchfork bifurcations on the spectral statistics. We focused here on an autonomous system with two degrees of freedom, namely the twodimensional coupled quartic oscillator, which is fairly close to realistic physical situations. This system exhibits a relevant bifurcation of a short periodic orbit (left panel of Figure 2), which leaves a significant trace in the spectral rigidity (right panel of Figure 2). Indeed, the semiclassical reproduction of the latter requires to include nondiagonal contributions from the two orbits that arise just above the bifurcation, which are important as long as the action difference between those orbits is smaller than Planck's constant hbar. 
Tunneling in presence of classical chaos
As soon as classically forbidden tunneling becomes relevant, the semiclassical description of the quantum system can become rather complicated. This is specifically the case for the dynamical tunneling process between quantum states that are localized on two symmetryrelated regular islands in a mixed regularchaotic phase space. The level splittings between the symmetric and antisymmetric combination of those local eigenmodes exhibit strong fluctuations at variations of system parameters if the islands are separated by a large chaotic layer [6]. At present, no complete and feasible semiclassical theory exists for the quantitative reproduction of such chaosassisted tunneling rates.



Fig 3: Resonanceassisted tunneling in the kicked rotor model. The lower panel shows the level splitting between the states that are localized on the two regular islands displayed in the upper panel. The quantum splittings (fluctuating black line) are indeed well reproduced by the semiclassical prediction (red line) based on the mechanism of resonanceassisted tunneling, using a 10:3 resonance that is prominantly manifested in the islands (red island chains in the left panel). 
We could show, however, that this tunneling process is dominantly governed by nonlinear resonances of the classical dynamics, which induce a coupling of the locally quantized eigenmode to the surrounding chaotic sea [7,8]. A semiclassical description of this resonanceassisted coupling process leads to a steplike decay of the tunneling rate with 1/hbar, which is on average indeed encountered in the true quantum splittings. We recently managed to generalize the theory of resonanceassisted tunneling to open systems, where we could reproduce the lifetimes of nondispersive wave packets in microwavedriven hydrogen [9]. 
Loschmidt echo in chaotic billiards
There are still open issues also in the context of fully chaotic systems. This is, for instance, the case for the decay of the quantum Loschmidt echo [10] which characterizes the sensitivity of the quantum dynamics to perturbations of the Hamiltonian. Formally, this quantity is obtained as follows: one starts with a given initial wavefunction, propagates it for a given time t, then slightly modifies the underlying Hamiltonian, reverses the arrow of time, and propagates the wavefunction backwards till the starting time (see Figure 4). The Loschmidt echo, also known as fidelity, is then defined by the overlap between the initial and the final state obtained in this way. Jalabert and Pastawski have analytically shown [11] that this quantum fidelity decays exponentially with the propagation time for a system with a chaotic classical counterpart, where the decay rate is given by the average Lyapunov exponent of the classical dynamics.



Fig 4: Loschmidt echo in the chaotic Sinai billiard with a local boundary perturbation. In contrast to the generic case for chaotic systems [11], the decay rate of the quantum fidelity is given by the classical escape rate of the corresponding open billiard (without the boundary segment) and not by the Lyapunov exponent of the (closed) Sinai billiard. 
We studied the decay of the Loschmidt echo in chaotic billiards, where the variation of the Hamiltonian is induced by a local deformation of the billiard boundary. For this case, we could show that the Loschmidt echo first follows an exponential decay and then fluctuates around a saturated value [12]. The decay rate is here different from the Lyapunov exponent and equals twice the rate at which classical particles would escape from an open billiard obtained from the original, closed one by removing the perturbationaffected boundary segment. As a matter of fact, the Loschmidt echo decay is independent of the shape of a particular boundary perturbation, and only depends on the length of the perturbed region. Furthermore, our numerical analysis shows that for certain choices of system parameters the exponential decay persists for times even longer than the Heisenberg time. 
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