It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because in the semi-classical limit, → 0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the fourpoint correlator, C(t), for the classical and quantum kicked rotor -a textbook driven chaotic system -and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are in general distinct quantities, corresponding to the logarithm of phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K → 0, while the OTOC's growth rate may decrease much slower showing higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time tE: transitioning from a time-independent value of t −1 ln C(t) at t < tE to its monotonous decrease with time at t > tE. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.Introduction. -One of the central goals in the study of quantum chaos is to establish a correspondence principle between classical and quantum dynamics of classically chaotic systems [1][2][3][4][5][6][7]. Several previous works [7][8][9][10][11] have attempted to recover fingerprints of classical chaos in quantum dynamics. In particular, Aleiner and Larkin [12] showed the existence of a semiclassical "quantum chaotic" regime attributed to the delay in the onset of quantum effects (due to weak localization) revealing the key measure of classical chaos -the Lyapunov exponent (LE). Recently, the subject of quantum chaos has been revived by the discovery of an unexpected conjecture that puts a bound on the growth rate of an outof-time-ordered four-point correlator (OTOC) [13,14]. OTOC was first introduced by Larkin and Ovchinnikov to quantify the regime of validity of quasi-classical methods in the theory of superconductivity [15]. The growth rate of OTOC appears to be closely related to LE. Recent works have proposed experimental protocols to probe OTOC in cold atom and cavity QED setups [16]. Several recent preprints have employed OTOC as a probe to characterize many-body-localized systems [17].

The out-of-time-ordered correlator has been proposed as an indicator of chaos in quantum systems due to its simple interpretation in the semiclassical limit. In particular, its rate of possible exponential growth at → 0 is closely related to the classical Lyapunov exponent. Here we explore how this approach to quantum chaos relates to the random-matrix theoretical description. To do so, we introduce and study the level statistics of the logarithm of the out-of-time-ordered operator,Λ(t) = ln − [x(t),px(0)] 2 /(2t), that we dub the "Lyapunovian" or "Lyapunov operator" for brevity. The Lyapunovian's level statistics is calculated explicitly for the quantum stadium billiard. It is shown that in the bulk of the filtered spectrum, this statistics perfectly aligns with the Wigner-Dyson distribution. One of the advantages of looking at the spectral statistics of this operator is that it has a well-defined semiclassical limit where it reduces to the matrix of uncorrelated classical finite-time Lyapunov exponents in a partitioned phase space. We provide a heuristic picture interpolating these two limits using Moyal quantum mechanics. Our results show that the Lyapunov operator may serve as a useful tool to characterize quantum chaos and in particular quantum-toclassical correspondence in chaotic systems, by connecting the semiclassical Lyapunov growth at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference effects.

Dual kinetic balance (DKB) technique was previously developed to eliminate spurious states in the finite-basis-set-based solution of the Dirac equation in central fields. In the present paper, it is extended to the Dirac equation for systems with axial symmetry. The efficiency of the method is demonstrated by the calculation of the energy spectra of hydrogenlike ions in presence of static uniform electric or magnetic fields. In addition, the DKB basis set is implemented to solve the timedependent Dirac equation making use of the split-operator technique. The excitation and ionization probabilities for the hydrogenlike argon and tin ions exposed to laser pulses are evaluated.