Hamiltonian Ratchets Research Articles

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.

Dissipation engineered directional filter for quantum ratchets

We demonstrate transport rectification in a hermitian Hamiltonian quantum ratchet by a dissipative, dynamic impurity. While the bulk of the ratchet supports transport in both directions, the properly designed loss function of the local impurity acts as a direction-dependent filter for the moving states. We analyse this scheme theoretically by making use of Floquet-S-Matrix theory. In addition, we provide the direct experimental observation of one-way transmittance in periodically modulated plasmonic waveguide arrays containing a local impurity with engineered losses.

Hamiltonian Ratchets with Ultra‐Cold Atoms

Quantum‐resonance ratchets have been realized over the last ten years for the production of directed currents of atoms. These non‐dissipative systems are based on the interaction of a Bose‐Einstein condensate with an optical standing wave potential to produce a current of atoms in momentum space. In this paper we provide a review of the important features of these ratchets with a particular emphasis on their optimization using more complex initial states. We also examine their stability close to resonance conditions of the kicking. Finally we discuss the way in which these ratchets may pave the way for applications in quantum (random) walks and matter‐wave interferometry.

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.

Quantum ratchet with photons

Quantum simulation is one of the most important applications of quantum computers. Unlike full-scale quantum computing where hundreds of qubits are required, quantum simulation can be performed at small-scale quantum information processing system. In a recent work [1], Zhang et al. from the University of Science and Technology of China (USTC) have experimentally demonstrated a quantum simulation of an interesting phenomenon with photons, the quantum ratchet. A classical ratchet is a spatially periodic rotor in which the directed motion of particles can be realized without a bias force. Similarly, in a quantum ratchet, a quantum particle is directed evolved under apparent no bias force. Quantum ratchet effect has attracted much attention recently. The effect may exist in many quantum systems [2], and it was demonstrated with cold atoms in optical lattice [3]. This quantum effect plays a crucial role in many related studies, such as dynamical localization, quantum resonance, and quantum chaos, and has potential applications in solving the evolution of wave functions. Researchers in USTC elaborately created a delta-kicked Hamiltonian quantum ratchet [4] in an all-optical system, and this quantum rotor is driven by Hamiltonian chaos, in the absence of noise. Directed motion and quantum resonance are observed. They constructed successfully a delta-kicked potential for light by a phase mirror which is the key of the periodically flashing potential in the experiment and the potential is determined by the etched depth of the phase mirror. The quantum particle is the photons which are reflected between a plane mirror and the phase mirror, and these photons encounter the phase mirror again and again with a certain temporal interval. In order to obtain the momentum information, they use skillfully a cylindrical lens to convert the pattern of the position distribution into that of the momentum distribution on the focal plane of the lens. They eventually successfully observed the quantum ratchet and quantum resonance in the experiment in the momentum and kinetic energy space with photons. Since the experiment is performed using classical light, the effects can be even observed with a naked eye for up to 22 steps. Compared to the massive quantum ratchet, photons have longer wavelength which provides longer spatial period and makes the quantum ratchet easier to control and measure. Light is easy to be coherently controlled and detected. The approach and the result provided here are significant to simulate other quantum phenomena.

Interaction-induced current-reversals in driven lattices

Long-range interactions are shown to cause, as time evolves, consecutive reversals of directed currents for dilute ensembles of particles in driven lattices. These current reversals are based on a general mechanism that leads to an interaction-induced accumulation of particles in the regular regions of the underlying single-particle phase space and to a synchronized single-particle motion as well as enhanced efficiency of Hamiltonian ratchets. Suggestions for experimental implementations using ionized mesoscopic clusters in micromechanical lattices or dipolarly interacting colloidal particles in ac-driven optical lattices are provided.

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 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.

Directed transport in classical and quantum chaotic billiards

We construct an autonomous chaotic Hamiltonian ratchet as a channel billiard subdivided by equidistant walls attached perpendicularly to one side of the channel, leaving an opening on the opposite side. A static homogeneous magnetic field penetrating the billiard breaks time-reversal invariance and renders the classical motion partially chaotic. We show that the classical dynamics exhibits directed transport, owing to the asymmetric distribution of regular regions in phase space. The billiard is quantized by a numerical method based on a finite-element algorithm combined with the Landau gauge and the Bloch formalism for periodic potentials. We discuss features of the billiard eigenstates such as node lines and vortices in the probability flow. Evidence for directed quantum transport, inherited from the corresponding features of the classical dynamics, is presented in terms of level-velocity statistics.

Periodically driven quantum ratchets: Symmetries and resonances

We study the quantum version of a tilting and flashing Hamiltonian ratchet, consisting of a periodic potential and a time-periodic driving field. The system dynamics is governed by a Floquet evolution matrix bearing the symmetry of the corresponding Hamiltonian. The dc-current appears due to the desymmetrization of Floquet eigenstates, which become transporting when all the relevant symmetries are violated. Those eigenstates that mostly contribute to a directed transport reside in phase space regions corresponding to classical resonances. Quantum dynamics leads to the dependence of the average velocity on the initial phase of the ac-field. A resonant enhancement (or suppression) of the dc-current, due to avoided crossings between different Floquet states, takes place upon tuning some control parameters. Our studies are predominantly aimed at experimental realizations of ac-driven quantum ratchets with cold atoms.

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.

Chaotic quantum ratchets and filters with cold atoms in optical lattices: Analysis using Floquet states

Recently, cesium atoms in optical lattices subjected to cycles of unequally spaced pulses have been found to show interesting behavior: they represent an experimental demonstration of a Hamiltonian ratchet mechanism, and they show strong variability of the dynamical localization lengths as a function of initial momentum. The behavior differs qualitatively from corresponding atomic systems pulsed with equal periods, which are a textbook implementation of a well-studied quantum chaos paradigm, the quantum $\ensuremath{\delta}$-kicked rotor ($\ensuremath{\delta}$-QKR). We investigate here the properties of the corresponding eigenstates (Floquet states) in the parameter regime of the recent experiments and compare them with those of the eigenstates of the $\ensuremath{\delta}$-QKR at similar kicking strengths. We show that by studying the properties of the Floquet states we can shed light on the form of the observed ratchet current, as well as variations in the dynamical localization length.

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.

Chaotic Hamiltonian ratchets for pulsed periodic double-well potentials: Classical correlations and the ratchet current

We present analytical derivations of the diffusion rates, ratchet currents, and time scales of a new ratchet in a fully chaotic Hamiltonian system, introduced in Phys. Rev. Lett. 89, 194102 (2002)] with a proposed implementation using atoms in pulsed standing waves of light. The origin of this type of ratchet current is in asymmetric momentum diffusion rates which result when a "double-well" lattice is pulsed with unequal "kick" periods. The form of the new short-time correlations which modify the diffusion rates are derived. The resulting formulas for the classical energy diffusion rates are shown to give good agreement with numerical simulations. A closed analytical formula for the ratchet current is also obtained, which predicts correctly the current magnitudes and current reversals. The characteristic "ratchet time," a classical time scale associated with the momentum-diffusion ratchet is derived analytically. The competition between the ratchet time t(r) , and the quantum break time t(*) is investigated further.

Directed Transport in a Spatially Periodic Magnetic Billiard

As an example of a hamiltonian ratchet in two-dimensional space but without driving, we study directed transport in a spatially periodic billiard immersed in a uniform stationary magnetic field. The magnetic field together with the geometry of the billiard are sufficient to break all symmetries relevant for transport, making an external driving unnecessary. In the high- and low-field limits, the motion becomes respectively pseudo-integrable and integrable; in both cases transport is absent. In the intermediate regime, surfaces of section reveal a mixed phase space, a necessary condition for directed currents. We find directed transport for individual invariant sets, while on average over the entire energy shell, a sum rule excludes finite transport. The dynamical mechanisms underlying directed transport are analyzed in terms of stable and unstable periodic orbits.

Classical and quantum Hamiltonian ratchets.

We explain the mechanism leading to directed chaotic transport in Hamiltonian systems with spatial and temporal periodicity. We show that a mixed phase space comprising both regular and chaotic motion is required and we derive a classical sum rule which allows one to predict the chaotic transport velocity from properties of regular phase-space components. Transport in quantum Hamiltonian ratchets arises by the same mechanism as long as uncertainty allows one to resolve the classical phase-space structure. We derive a quantum sum rule analogous to the classical one, based on the relation between quantum transport and band structure.

Classical and quantum transport in deterministic Hamiltonian ratchets

We study directed transport in classical and quantum area-preserving maps, periodic in space and momentum. On the classical level, we show that a sum rule excludes directed transport of the entire phase space, leaving only the possibility of transport in (dynamically defined) subsets, such as regular islands or chaotic areas. As a working example, we construct a mapping with a mixed phase space where both the regular and the chaotic components support directed currents, but with opposite sign. The corresponding quantum system shows transport of similar strength, associated to the same subsets of phase space as in the classical map.

Classical and quantum transport in deterministic Hamiltonian ratchets

AbstractWe study directed transport in classical and quantum area‐preserving maps, periodic in space and momentum. On the classical level, we show that a sum rule excludes directed transport of the entire phase space, leaving only the possibility of transport in (dynamically defined) subsets, such as regular islands or chaotic areas. As a working example, we construct a mapping with a mixed phase space where both the regular and the chaotic components support directed currents, but with opposite sign. The corresponding quantum system shows transport of similar strength, associated to the same subsets of phase space as in the classical map.