Quantum isoperiodic stable structures and directed transport

Quantum Lyapunov exponent in dissipative systems

The out-of-time order correlator (OTOC) has been widely studied in closed quantum systems. However, there are very few studies for open systems and they are mainly focused on isolating the effects of scrambling from those of decoherence. Adopting a different point of view, we study the interplay between these two processes. This proves crucial in order to explain the OTOC behavior when a phase space contracting dissipation is present, ubiquitous not only in real life quantum devices but in the dynamical systems area. The OTOC decay rate is closely related to the classical Lyapunov exponent-with some differences-and more sensitive in order to distinguish the chaotic from the regular behavior than other measures. On the other hand, it is revealed as a generally simple function of the longest lived eigenvalues of the quantum evolution operator. We find no simple connection with the Ruelle-Pollicott resonances, but by adding Gaussian noise of ℏ_{eff} size to the classical system we recover the OTOC decay rate, which is a consequence of the correspondence principle put forward in Phys. Rev. Lett. 108, 210605 (2012)10.1103/PhysRevLett.108.210605 and Phys. Rev. E 99, 042214 (2019)10.1103/PhysRevE.99.042214.

Effects of chaotic dynamics on quantum friction

By means of studying the evolution equation for the Wigner distributions of quantum dissipative systems we derive the quantum corrections to the classical Liouville dynamics, taking into account the standard quantum friction model. The resulting evolution turns out to be the classical one plus fluctuations that depend not only on the ℏ size but also on the momentum and the dissipation parameter (i.e., the coupling with the environment). On the other hand, we extend our studies of a paradigmatic system based on the kicked rotator, and we confirm that by adding fluctuations only depending on the size of the Planck constant we essentially recover the quantum behavior. This is systematically measured in the parameter space with the overlaps and differences in the dispersion of the marginal distributions corresponding to the Wigner functions. Taking into account these results and analyzing the Wigner evolution equation we conjecture that the chaotic nature of our system is responsible for the independence on the momentum, while the dependence on the dissipation is provided implicitly by the dynamics.

Three-dimensional classical and quantum stable structures of dissipative systems

We study the properties of classical and quantum stable structures in a three-dimensional (3D) parameter space corresponding to the dissipative kicked top. This is a model system in quantum and classical chaos that gives a starting point for many body examples. We are able to identify the influence of these structures in the spectra and eigenstates of the corresponding (super)operators. This provides a complementary view with respect to the typical two-dimensional parameter space systems found in the literature. Many properties of the eigenstates, like its localization behavior, can be generalized to this higher-dimensional parameter space and spherical phase space topology. Moreover, we find a 3D phenomenon-generalizable to more dimensions-that we call the coalescence-separation of (q)ISSs, whose main consequence is a marked enhancement of quantum localization. This could be of relevance for systems that have attracted a lot of attention very recently.

Signatures of classical structures in the leading eigenstates of quantum dissipative systems

By analyzing a paradigmatic example of the theory of dissipative systems-the classical and quantum dissipative standard map-we are able to explain the main features of the decay to the quantum equilibrium state. The classical isoperiodic stable structures typically present in the parameter space of these kinds of systems play a fundamental role. In fact, we have found that the period of stable structures that are near in this space determines the phase of the leading eigenstates of the corresponding quantum superoperator. Moreover, the eigenvectors show a strong localization on the corresponding periodic orbits (limit cycles). We show that this sort of scarring phenomenon (an established property of Hamiltonian and projectively open systems) is present in the dissipative case and it is of extreme simplicity.

Proliferation of stability in phase and parameter spaces of nonlinear systems

In this work, we show how the composition of maps allows us to multiply, enlarge, and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using Hénon maps with distinct parameters, we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis, and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system.

Classical counterparts of quantum attractors in generic dissipative systems

In the context of dissipative systems, we show that for any quantum chaotic attractor a corresponding classical chaotic attractor can always be found. We provide a general way to locate them, rooted in the structure of the parameter space (which is typically bidimensional, accounting for the forcing strength and dissipation parameters). In cases where an approximate pointlike quantum distribution is found, it can be associated with exceptionally large regular structures. Moreover, supposedly anomalous quantum chaotic behavior can be very well reproduced by the classical dynamics plus Gaussian noise of the size of an effective Planck constant ℏ_{eff}. We give support to our conjectures by means of two paradigmatic examples of quantum chaos and transport theory. In particular, a dissipative driven system becomes fundamental in order to extend their validity to generic cases.

Transient chaos - a resolution of breakdown of quantum-classical correspondence in optomechanics

Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient chaos associated with quantum trajectories. The transient behavior is consistent with chaos in the classical limit, while the long term evolution of the quantum system is regular. Transient chaos thus serves as a bridge for the quantum-classical transition (QCT). Strikingly, as the system transitions from the quantum to the classical regime, the average chaotic transient lifetime increases dramatically (faster than the Ehrenfest time characterizing the QCT for isolated quantum systems). We develop a physical theory to explain the scaling law.

Correspondence behavior of classical and quantum dissipative directed transport via thermal noise

We systematically study several classical-quantum correspondence properties of the dissipative modified kicked rotator, a paradigmatic ratchet model. We explore the behavior of the asymptotic currents for finite ℏ_{eff} values in a wide range of the parameter space. We find that the correspondence between the classical currents with thermal noise providing fluctuations of size ℏ_{eff} and the quantum ones without it is very good in general with the exception of specific regions. We systematically consider the spectra of the corresponding classical Perron-Frobenius operators and quantum superoperators. By means of an average distance between the classical and quantum sets of eigenvalues we find that the correspondence is unexpectedly quite uniform. This apparent contradiction is solved with the help of the Weyl-Wigner distributions of the equilibrium eigenvectors, which reveal the key role of quantum effects by showing surviving coherences in the asymptotic states.

Classical to quantum correspondence in dissipative directed transport

We compare the quantum and classical properties of the (quantum) isoperiodic stable structures [(Q)ISSs], which organize the parameter space of a paradigmatic dissipative ratchet model, i.e., the dissipative modified kicked rotator. We study the spectral behavior of the corresponding classical Perron-Frobenius operators with thermal noise and the quantum superoperators without it for small ℏ(eff) values. We find a remarkable similarity between the classical and quantum spectra. This finding significantly extends previous results-obtained for the mean currents and asymptotic distributions only-and, on the other hand, unveils a classical to quantum correspondence mechanism where the classical noise is qualitatively different from the quantum one. This is crucial not only for simple attractors but also for chaotic ones, where just analyzing the asymptotic distribution is revealed as insufficient. Moreover, we provide with a detailed characterization of relevant eigenvectors by means of the corresponding Weyl-Wigner distributions, in order to better identify similarities and differences. Finally, this model being generic, it allows us to conjecture that this classical to quantum correspondence mechanism is a universal feature of dissipative systems.

Quantum-classical transition and quantum activation of ratchet currents in the parameter space

The quantum ratchet current is studied in the parameter space of the dissipative kicked rotor model coupled to a zero-temperature quantum environment. We show that vacuum fluctuations blur the generic isoperiodic stable structures found in the classical case. Such structures tend to survive when a measure of statistical dependence between the quantum and classical currents are displayed in the parameter space. In addition, we show that quantum fluctuations can be used to overcome transport barriers in the phase space. Related quantum ratchet current activation regions are spotted in the parameter space. Results are discussed based on quantum, semiclassical, and classical calculations. While the semiclassical dynamics involves vacuum fluctuations, the classical map is driven by thermal noise.

Chaos in kicked ratchets

We present a minimal one-dimensional deterministic continuous dynamical system that exhibits chaotic behavior and complex transport properties. Our model is an overdamped rocking ratchet with finite dissipation, that is periodically kicked with a δ function driving force, without finite inertia terms or temporal or spatial stochastic forces. To our knowledge this is the simplest model reported in the literature for a ratchet, with this complex behavior. We develop an analytical approach that predicts many key features of the system, such as current reversals, as well as the presence of chaotic behavior and bifurcation. Our analytical approach allows us to study the transition from regular to chaotic motion as well as a tangent bifurcation associated with this transition. We show that our approach can be easily extended to other types of periodic driving forces. The square wave is shown as an example.

Quantum parameter space of dissipative directed transport

Quantum manifestations of isoperiodic stable structures (QISSs) have a crucial role in the current behavior of quantum dissipative ratchets. In this context, the simple shape of the ISSs has been conjectured to be an almost exclusive feature of the classical system. This has drastic consequences for many properties of the directed currents, the most important one being that it imposes a significant reduction in their maximum values, thus affecting the attainable efficiency at the quantum level. In this work we prove this conjecture by means of comprehensive numerical explorations and statistical analysis of the quantum states. We are able to describe the quantum parameter space of a paradigmatic system for different values of ℏ(eff) in great detail. Moreover, thanks to this we provide evidence on a mechanism that we call parametric tunneling by which the sharp classical borders of the regions in parameter space become blurred in the quantum counterpart. We expect this to be a common property of generic dissipative quantum systems.

Temperature resistant optimal ratchet transport

Stable periodic structures containing optimal ratchet transport, recently found in the parameter space dissipation versus ratchet parameter by [A. Celestino et al. Phys. Rev. Lett. 106, 234101 (2011)], are shown to be resistant to reasonable temperatures, reinforcing the expectation that they are essential to explain the optimal ratchet transport in nature. Critical temperatures for their destruction, valid from the overdamping to close to the conservative limits, are obtained numerically and shown to be connected to the current efficiency, given here analytically. A region where thermal activation of the rachet current takes place is also found, and its underlying mechanism is unveiled. Results are demonstrated for a discrete ratchet model and generalized to the Langevin equation with an additional external oscillating force.

Circular ratchets as transducers of vertical vibrations into rotations

Granular ratchets are well-known devices that when driven vertically produce a counterintuitive horizontal transport of particles. Here we report the experimental observation of a complementary effect: the striking ability of circular ratchets to convert their vertical vibration into their own rotation. The average revolution speed shows a maximum value for an optimal tooth height. With no special effort the rotation speed could be maintained steady during several hours. Unexpected random arrests and reversals of the velocity were also observed abundantly.