Directed anomalous diffusion without a biased field: a ratchet accelerator

Interaction-induced directed transport in quantum chaotic subsystems

Quantum directed transport can be realized in noninteracting, deterministic, chaotic systems by appropriately breaking the spatiotemporal symmetries in the potential. In this work, the focus is on the class of interacting two-body quantum systems whose classical limit is chaotic. It is shown that one subsystem effectively acts as a source of "noise" to the other leading to intrinsic temporal symmetry breaking. Then, the quantum directed currents, even if prohibited by symmetries in the composite system, can be realized in the subsystems. This current is of quantum origin and does not arise from semiclassical effects. This protocol provides a minimal framework-broken spatial symmetry in the potential and presence of interactions-for realizing directed transport in interacting chaotic systems. It is also shown that the magnitude of directed current undergoes multiple current reversals upon varying the interaction strength and this allows for controlling the currents. It is explicitly demonstrated in the two-body interacting kicked rotor model. The interaction-induced mechanism for subsystem directed currents would be applicable to other interacting quantum systems as well.

Hamiltonian Ratchet for Matter-Wave Transport

We report on the design of a Hamiltonian ratchet exploiting periodically at rest integrable trajectories in the phase space of a modulated periodic potential, leading to the linear nondiffusive transport of particles. Using Bose-Einstein condensates in a modulated one-dimensional optical lattice, we make the first observations of this spatial ratchet, which provides way to coherently transport matter waves with possible applications in quantum technologies. In the semiclassical regime, the quantum transport strongly depends on the effective Planck constant due to Floquet state mixing. We also demonstrate the interest of quantum optimal control for efficient initial state preparation into the transporting Floquet states to enhance the transport periodicity.

Ratchet current in nontwist Hamiltonian systems

Non-monotonic area-preserving maps violate the twist condition locally in phase space, giving rise to shearless invariant barriers surrounded by twin island chains in these regions of phase space. For the extended standard nontwist map, with two resonant perturbations with distinct wave numbers, we investigate the presence of such barriers and their associated island chains and compare our results with those that have been reported for the standard nontwist map with only one perturbation. Furthermore, we determine in the control parameter space the existence of the shearless barrier and the influence of the additional wave number on this condition. We show that only for odd second wave numbers are the twin island chains symmetrical. Moreover, for even wave numbers, the lack of symmetry between the chains of twin islands generates a ratchet effect that implies a directed transport in the phase space.

Directed momentum current induced by the PT-symmetric driving

We investigate the directed momentum current in the quantum kicked rotor model with PT-symmetric deriving potential. For the quantum nonresonance case, the values of quasienergy become complex when the strength of the imaginary part of the kicking potential exceeds a threshold value, which demonstrates the appearance of the spontaneous PT symmetry breaking. In the vicinity of the transition point, the momentum current exhibits a staircase growth with time. Each platform of the momentum current corresponds to the mean momentum of some eigenstates of the Floquet operator whose imaginary parts of the quasienergy are significantly large. Above the transition point, the momentum current increases linearly with time. Interestingly, its acceleration rate exhibits a kind of "quantized" increment with the kicking strength. We propose a modified classical acceleration mode of the kicked rotor model to explain such an intriguing phenomenon. Our theoretical prediction is in good agreement with numerical results.

Controlling chaos-assisted directed transport via quantum resonance

We report on the first demonstration of chaos-assisted directed transport of a quantum particle held in an amplitude-modulated and tilted optical lattice, through a resonance-induced double-mean displacement relating to the true classically chaotic orbits. The transport velocity is controlled by the driving amplitude and the sign of tilt, and also depends on the phase of the initial state. The chaos-assisted transport feature can be verified experimentally by using a source of single atoms to detect the double-mean displacement one by one, and can be extended to different scientific fields.

One Versus Two Photon Control of Dynamical Tunneling: Influence of the Irregular Floquet States

A useful approach to control quantum processes involves driving systems with two colored laser fields and varying the relative phase between the fields to control the quantum interferences. A particularly interesting class of bichromatic control schemes involves the so-called M versus N-photon control that results in laser-induced symmetry breaking and leads to directed transport; however, recent studies have shown that the mechanism of laser-induced symmetry breaking has a common classical and quantum origin. In this context, a relevant question is the extent to which such a detailed classical-quantum correspondence holds if the process to be controlled involves quantum tunneling. In this work, we address this issue in terms of controlling dynamical tunneling between field-induced islands of stability in the classical phase space of a model system, a periodically driven pendulum. This is also a paradigmatic model for Hamiltonian ratchets wherein the islands of stability, that is, nonlinear resonances, play a crucial role in the observed directed transport. We compute an appropriate control landscape for the process and show that despite breaking the relevant symmetries, there exist regions in the control landscape where the control fails. The lack of control can be understood in terms of the phase-space nature of the quantum Floquet states that participate in the dynamics of the initial wavepacket. We argue that robust regions of no control arise due to the phenomenon of chaos-assisted tunneling and comment on the possible influence of such regions on the directed transport in the model system.

Nonlinearity effects on the directed momentum current

We investigate the quantum transport dynamics governed by the nonlinear Schrödinger equation with a periodically-δ-kicking potential and discover the emergence of a directed current in momentum space. With the increase of nonlinearity, we find strikingly that the momentum current decreases, reverses, and finally vanishes, indicating that the quantum transport can be effectively manipulated through adjusting the nonlinearity. The underlying dynamic mechanism is uncovered and some important implications are addressed.

Classical and quantum transport in one-dimensional periodically kicked systems

This paper is a brief review of classical and quantum transport phenomena, as well as related spectral properties, exhibited by one-dimensional periodically kicked systems. Two representative and fundamentally different classes of systems will be considered: those satisfying the classical Kolmogorov−Arnol’d−Moser scenario and those that do not. The experimental realization of some of these systems using atom-optics methods will be mentioned.

Quantized adiabatic transport in momentum space

Though topological aspects of energy bands are known to play a key role in quantum transport in solid-state systems, the implications of Floquet band topology for transport in momentum space (i.e., acceleration) have not been explored so far. Using a ratchet accelerator model inspired by existing cold-atom experiments, here we characterize a class of extended Floquet bands of one-dimensional driven quantum systems by Chern numbers, reveal topological phase transitions therein, and theoretically predict the quantization of adiabatic transport in momentum space. Numerical results confirm our theory and indicate the feasibility of experimental studies.

Weak-chaos ratchet accelerator

Classical Hamiltonian systems with a mixed phase space and some asymmetry may exhibit chaotic ratchet effects. The most significant such effect is a directed momentum current or acceleration. In known model systems, this effect may arise only for sufficiently strong chaos. In this paper, a Hamiltonian ratchet accelerator is introduced, featuring a momentum current for arbitrarily weak chaos. The system is a realistic, generalized kicked rotor and is exactly solvable to some extent, leading to analytical expressions for the momentum current. While this current arises also for relatively strong chaos, the maximal current is shown to occur, at least in one case, precisely in a limit of arbitrarily weak chaos.

Directed transport and localization in phase-modulated driven lattices

We explore the dynamics of noninteracting particles loaded into a phase-modulated one-dimensional lattice formed by laterally oscillating square barriers. Tuning the parameters of the driven unit cell of the lattice selected parts of the classical phase space can be manipulated in a controllable manner. We find superdiffusion in position space for all parameters regimes. A directed current of an ensemble of particles can be created through locally breaking the spatiotemporal symmetries of the time-driven potential. Magnitude and direction of the current are tunable. Several mechanisms for transient localization and trapping of particles in different wells of the driven unit cell are presented and analyzed.

Directed chaotic transport in the tokamap with mixed phase space

On the basis of the tokamap, characteristic features of magnetic field lines and zeroth-order guiding-center particle motion in the whole body of a magnetically confined plasma, e.g., a tokamak plasma, are investigated. It is shown that the tokamap exhibits a poloidal transport that can be considered as a Hamiltonian ratchet. In a situation with partially chaotic magnetic field lines the locking of the averaged poloidal velocity occurs to a value that does not depend precisely on the initial conditions. The so-called sum rule predicts the mean velocity in agreement with the observed magnitude. Possible consequences for the onset of poloidal plasma rotation in ergodized plasmas are elucidated.

Chaos and correspondence in classical and quantum Hamiltonian ratchets: a Heisenberg approach

Previous work [Gong and Brumer, Phys. Rev. Lett. 97, 240602 (2006)] motivates this study as to how asymmetry-driven quantum ratchet effects can persist despite a corresponding fully chaotic classical phase space. A simple perspective of ratchet dynamics, based on the Heisenberg picture, is introduced. We show that ratchet effects are in principle of common origin in classical and quantum mechanics, although full chaos suppresses these effects in the former but not necessarily the latter. The relationship between ratchet effects and coherent dynamical control is noted.

Quantum ratchet accelerator without a bichromatic lattice potential

In a quantum ratchet accelerator system, a linearly increasing directed current can be dynamically generated without using a biased field. Generic quantum ratchet acceleration with full classical chaos [J. B. Gong and P. Brumer, Phys. Rev. Lett. 97, 240602 (2006)] constitutes a new element of quantum chaos and an interesting violation of a sum rule of classical ratchet transport. Here we propose a simple quantum ratchet accelerator model that can also generate linearly increasing quantum current with full classical chaos. This model does not require a bichromatic lattice potential. It is based on a variant of an on-resonance kicked-rotor system, periodically kicked by two optical lattice potentials of the same lattice constant, but with unequal amplitudes and a fixed phase shift between them. The dependence of the ratchet current acceleration rate on the system parameters is studied in detail. The cold-atom version of our quantum ratchet accelerator model should be realizable by introducing slight modifications to current cold-atom experiments.

Controlling the ratchet effect for cold atoms

Low-order quantum resonances manifested by directed currents have been realized with cold atoms. Here we show that by increasing the strength of an experimentally achievable delta-kicking ratchet potential, quantum resonances of a very high order may naturally emerge and can induce larger ratchet currents than low-order resonances, with the underlying classical limit being fully chaotic. The results offer a means of controlling quantum transport of cold atoms.

Ratchet effect and the transporting islands in the chaotic sea

We study directed transport in a classical deterministic dissipative system. We consider the generic case of mixed phase space and show that large ratchet currents can be generated thanks to the presence, in the Hamiltonian limit, of transporting stability islands embedded in the chaotic sea. Because of the simultaneous presence of chaos and dissipation the stationary value of the current is independent of initial conditions, except for initial states with very small measure.

Quantum diffusion dynamics in nonlinear systems: a modified kicked-rotor model

Using a simple method analogous to a quantum rephasing technique, a simple modification to a paradigm of classical and quantum chaos is proposed. The interesting quantum maps thus obtained display remarkably rich quantum dynamics. Emphasis is placed on the destruction of dynamical localization without breaking periodicity, unbounded quantum anomalous diffusion in integrable systems, and transient dynamical localization. Experimental realizations of this work are also discussed.

Directed deterministic classical transport: symmetry breaking and beyond

We consider transport properties of a double delta -kicked system, in a regime where all the symmetries (spatial and temporal) that could prevent directed transport are removed. We analytically investigate the (nontrivial) behavior of the classical current and diffusion properties and show that the results are in good agreement with numerical computations. The role of dissipation for a meaningful classical ratchet behavior is also discussed.

Observation of high-order quantum resonances in the kicked rotor

Quantum resonances in the kicked rotor are characterized by a dramatically increased energy absorption rate, in stark contrast to the momentum localization generally observed. These resonances occur when the scaled Planck's constant Planck's [over ]=r/s 4pi, for any integers r and s. However, only the variant Planck's [over ]=r2pi resonances are easily observable. We have observed high-order quantum resonances (s>2) utilizing a sample of low energy, noncondensed atoms and a pulsed optical standing wave. Resonances are observed for variant Planck's [over ]=r/16 4pi for integers r=2-6. Quantum numerical simulations suggest that our observation of high-order resonances indicate a larger coherence length (i.e., coherence between different wells) than expected from an initially thermal atomic sample.

Directed motion for delta-kicked atoms with broken symmetries: comparison between theory and experiment

We report an experimental investigation of momentum diffusion in the delta-function kicked rotor where time symmetry is broken by a two-period kicking cycle and spatial symmetry by an alternating linear potential. We exploit this, and a technique involving a moving optical potential, to create an asymmetry in the momentum diffusion that is due to the classical chaotic diffusion. This represents a realization of a type of Hamiltonian quantum ratchet.

Current behavior of a quantum Hamiltonian ratchet in resonance

We investigate the ratchet current that appears in a kicked Hamiltonian system when the period of the kicks corresponds to the regime of quantum resonance. In the classical analog, a spatial-temporal symmetry should be broken to obtain a net directed current. It was recently discovered that in quantum resonance the temporal symmetry can be kept, and we prove that breaking the spatial symmetry is a necessary condition to find this effect. Moreover, we show numerically and analytically how the direction of the motion is dramatically influenced by the strength of the kicking potential and the value of the period. By increasing the strength of the interaction this direction changes periodically, providing us with un-expected source of current reversals in this quantum model. These reversals depend on the kicking period also, though this behavior is theoretically more difficult to analyze. Finally, we generalize the discussion to the case of a nonuniform initial condition.

Generic quantum ratchet accelerator with full classical chaos

A simple model of quantum ratchet transport that can generate unbounded linear acceleration of the quantum ratchet current is proposed, with the underlying classical dynamics fully chaotic. The results demonstrate that generic quantum ratchet transport can occur with any type of classical phase space structure. The quantum ratchet transport with full classical chaos is also shown to be very robust to noise due to the large linear acceleration afforded by the quantum dynamics. One possible experiment allowing observation of these predictions is suggested.

Laser-induced spatial symmetry breaking in quantum and classical mechanics

Phase-controllable transport in laser-irradiated spatially symmetric systems is shown to arise both quantum mechanically and classically from a common field-driven interference mechanism. Specifically, the quantum-to-classical transition for symmetry breaking in a quartic oscillator driven by an omega+2omega field is studied. For this, a double perturbation theory in the oscillator anharmonicity and external field strength, that admits an analytic classical limit, is carried out in the Heisenberg picture. The interferences responsible for the symmetry breaking are shown to survive in the classical limit and are the origins of classical control. Differences between reflection symmetry that plays a key role in the analysis, and parity that does not, are discussed.

QUANTUM CHAOS MEETS COHERENT CONTROL

▪ Abstract  Coherent control of atomic and molecular processes has been a rapidly developing field. Applications of coherent control to large and complex molecular systems are expected to encounter the effects of chaos in the underlying classical dynamics, i.e., quantum chaos. Hence, recent work has focused on examining control in model chaotic systems. This work is reviewed, with an emphasis on a variety of new quantum phenomena that are of interest to both areas of quantum chaos and coherent control.

Quantum ratchets in dissipative chaotic systems

Using the method of quantum trajectories, we study a quantum chaotic dissipative ratchet appearing for particles in a pulsed asymmetric potential in the presence of a dissipative environment. The system is characterized by directed transport emerging from a quantum strange attractor. This model exhibits, in the limit of small effective Planck constant, a transition from quantum to classical behavior, in agreement with the correspondence principle. We also discuss parameter values suitable for the implementation of the quantum ratchet effect with cold atoms in optical lattices.

Directed chaotic transport in Hamiltonian ratchets

We present a comprehensive account of directed transport in one-dimensional Hamiltonian systems with spatial and temporal periodicity. They can be considered as Hamiltonian ratchets in the sense that ensembles of particles can show directed ballistic transport in the absence of an average force. We discuss general conditions for such directed transport like a mixed classical phase space. A sum rule is derived which connects the contributions of different phase-space components to transport. We show that regular ratchet transport can be directed against an external potential gradient while chaotic ballistic transport is restricted to unbiased systems. For quantized Hamiltonian ratchets we study transport in terms of the evolution of wave packets and derive a semiclassical expression for the distribution of level velocities which encode the quantum transport in the Floquet band spectra. We discuss the role of dynamical tunneling between transporting islands and the chaotic sea and the breakdown of transport in quantum ratchets with broken spatial periodicity.