Semiclassical approach to long time propagation in quantum chaos: predicting scars

Genuine Many-Body Quantum Scars along Unstable Modes in Bose-Hubbard Systems

The notion of many-body quantum scars is associated with special eigenstates, usually concentrated in certain parts of Hilbert space, that give rise to robust persistent oscillations in a regime that globally exhibits thermalization. Here we extend these studies to many-body systems possessing a true classical limit characterized by a high-dimensional chaotic phase space, which are not subject to any particular dynamical constraint. We demonstrate genuine quantum scarring of wave functions concentrated in the vicinity of unstable classical periodic mean-field modes in the paradigmatic Bose-Hubbard model. These peculiar quantum many-body states exhibit distinct phase-space localization about those classical modes. Their existence is consistent with Heller's scar criterion and appears to persist in the thermodynamic long-lattice limit. Launching quantum wave packets along such scars leads to observable long-lasting oscillations, featuring periods that scale asymptotically with classical Lyapunov exponents, and displaying intrinsic irregularities that reflect the underlying chaotic dynamics, as opposed to regular tunnel oscillations.

Impact of chaos on precursors of quantum criticality

Excited-state quantum phase transitions (ESQPTs) are critical phenomena that generate singularities in the spectrum of quantum systems. For systems with a classical counterpart, these phenomena have their origin in the classical limit when the separatrix of an unstable periodic orbit divides phase space into different regions. Using a semiclassical theory of wave propagation based on the manifolds of unstable periodic orbits, we describe the quantum states associated with an ESQPT for the quantum standard map: a paradigmatic example of a kicked quantum system. Moreover, we show that finite-size precursors of ESQPTs shrink as chaos increases due to the disturbance of the system. This phenomenon is explained through destructive interference between principal homoclinic orbits.

Short-periodic-orbit method for excited chaotic eigenfunctions

An alternative method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wave functions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wave functions is only of the order of the ratio t_{H}/t_{E}, with t_{H} the Heisenberg time and t_{E} the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynamics.

Quantization and interference of a quantum billiard with fourfold rotational symmetry

Systems with discrete symmetries are highly important in quantum mechanics. We consider a two-dimensional quantum billiard with fourfold rotational symmetry, where the eigenenergies and eigenstates can be grouped into four symmetry subspaces. Unlike the threefold rotational symmetry case, here the interference of the scarring states on the fundamental domain orbits (FDO) is clean, that they either interfere constructively or annihilate completely. We shall show the complex behavior of the interference revealed in the length spectra for eigenenergies that belong to a particular symmetry subspace and combinations of different symmetry subspaces. We then provide detailed analysis of phase accumulation along the FDOs, which are the keys to determine the interference and could explain the enhancement or annihilation of the peaks well. The quantization condition for the scarring states belonging to different symmetry subspaces is discussed and used to reveal the time-reversal symmetry broken for a particular subspace. An experimental scheme to observe such complex behaviors is also proposed.

Strong quantum scarring by local impurities

We discover and characterise strong quantum scars, or quantum eigenstates resembling classical periodic orbits, in two-dimensional quantum wells perturbed by local impurities. These scars are not explained by ordinary scar theory, which would require the existence of short, moderately unstable periodic orbits in the perturbed system. Instead, they are supported by classical resonances in the unperturbed system and the resulting quantum near-degeneracy. Even in the case of a large number of randomly scattered impurities, the scars prefer distinct orientations that extremise the overlap with the impurities. We demonstrate that these preferred orientations can be used for highly efficient transport of quantum wave packets across the perturbed potential landscape. Assisted by the scars, wave-packet recurrences are significantly stronger than in the unperturbed system. Together with the controllability of the preferred orientations, this property may be very useful for quantum transport applications.

Theory of short periodic orbits for partially open quantum maps

We extend the semiclassical theory of short periodic orbits [M. Novaes et al., Phys. Rev. E 80, 035202(R) (2009)PLEEE81539-375510.1103/PhysRevE.80.035202] to partially open quantum maps, which correspond to classical maps where the trajectories are partially bounced back due to a finite reflectivity R. These maps are representative of a class that has many experimental applications. The open scar functions are conveniently redefined, providing a suitable tool for the investigation of this kind of system. Our theory is applied to the paradigmatic partially open tribaker map. We find that the set of periodic orbits that belongs to the classical repeller of the open map (R=0) is able to support the set of long-lived resonances of the partially open quantum map in a perturbative regime. By including the most relevant trajectories outside of this set, the validity of the approximation is extended to a broad range of R values. Finally, we identify the details of the transition from qualitatively open to qualitatively closed behavior, providing an explanation in terms of short periodic orbits.

Semiclassical propagator to evaluate off-diagonal matrix elements of the evolution operator between quantum states

We present a powerful semiclassical expression to evaluate off-diagonal matrix elements of the evolution operator between quantum states constructed in the neighborhood of unstable short periodic orbits, which is valid up to the Heisenberg time. The expression is much easier to evaluate than the Van Vleck propagator and consists of a sum over the set of heteroclinic orbits, where each term of the series is computed by canonical invariants. Here we introduce relevant canonical invariants of heteroclinic orbits and with them at hand, the semiclassical expression is derived. Finally, our formula is successfully verified in the hyperbola billiard.

Ergodicity and quantum correlations in irrational triangular billiards

Pseudochaotic properties are systematically investigated in a one-parameter family of irrational triangular billiards (all angles irrational with π). The absolute value of the position correlation function C(x)(t) decays like ~t(-α). Fast (α≈1) and slow (0<α<1) decays are observed, thus indicating that the irrational triangles do not share a unique ergodic dynamics, which, instead, may vary smoothly between the opposite limits of strong mixing (α=1) and regular behaviors (α=0). Upgrading previous data, spectral statistical properties of the quantized counterparts are computed from 150000 energy eigenvalues numerically calculated for each billiard. Gaussian orthogonal ensemble spectral fluctuations are observed when α≈1 and intermediate statistics are found otherwise. Our irrational billiards have zero Kolmogorov-Sinai entropy and essentially infinity genus. Thus, differently from previous works on rational (pseudointegrable) enclosures, our results provide a missing classical-quantum correspondence regarding the ergodic hierarchy for a set of nonchaotic systems that might enjoy the strong mixing property.