Stationary Hamiltonian transport with dc bias

– The dynamics of a particle in a symmetric periodic potential under the influence of a time-periodic field is characterized by a mixed phase space with regular and chaotic components. An additional external dc bias transforms the chaotic manifold into a domain with unbounded acceleration. We study the stationary transport which originates from the persisting invariant manifolds (regular islands, periodic orbits, and cantori) that are initially embedded in the chaotic manifold. We prove persistence and emergence of transporting islands. The transient dynamics of the accelerated domain separates fast chaotic motion from ballistic type trajectories which stick to the vicinity of the invariant submanifold. Experimental studies with cold atoms in laser-induced optical lattices are ideally suited for testing and observing our findings. The main mechanisms governing Hamiltonian transport are strongly related to a mixed phase space with coexisting regular and chaotic regions [1, 2]. Modern experimental research using manipulations with cold atoms ensembles in optical potentials provides a testing ground to explore this issue [3]. The driven pendulum, a paradigmatic system for the study of dynamical chaos [2], is realized using a periodically modulated optical standing wave, with the possibility to control the strength of dissipation down to arbitrarily small values [3]. Mixed phase space structures have been resolved already with cold atom experiments [4]. Mixed phase space structures are also at the heart of the recently discussed directed chaotic transport in driven Hamiltonian systems, so-called Hamiltonian ratchets [5–7]. Recent experiments [8] study the crossover from dissipative to Hamiltonian ratchets, confirming these theoretical predictions. What will happen to a driven Hamiltonian system when exposed to an additional dc bias? Contrary to the common expectation that the trajectories acquire unbounded acceleration, it has very recently been found in biased Hamiltonian ratchet systems that this does not hold true for all trajectories [9]. Again, the main reason for that finding is a mixed phase space structure of the unbiased system [9]. In this letter we systematically explore the route of obtaining a stationary Hamiltonian transport in the presence of a dc bias. While the chaotic phase space regions transform into accelerated evolution for any arbitrary small value of the dc bias, regular transporting c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10565-4 S. Denisov et al.: Stationary Hamiltonian transport with dc bias 589 submanifolds (unstable periodic orbits, regular islands and cantori with nonzero average velocities), initially embedded in the chaotic region, persist under finite dc bias. Even more, some transporting unstable periodic orbits at zero dc bias lead to the appearance of stable transporting islands for some finite nonzero values of the dc bias. The presence of cantori [10] and the corresponding sticking of chaotic trajectories in the vicinity of regular manifolds has a strong impact on the transient dynamics of chaotic phase space parts which are close to the above-mentioned regular manifolds. We consider the canonical Hamiltonian model of a particle moving in a symmetric (i.e. non-ratchet-type) one-dimensional space-periodic potential U(X) = 1 2π cos(2πx) under the influence of a time-periodic space-homogenous external field E(t) = Eac sin(ωt) [2], to which we add an external dc bias Eb: ẋ = p, ṗ = sin(2πx) + Eac sin(ωt)− Eb. (1) Due to time and space periodicity of the system we can map the original three-dimensional phase space (x, p, t) onto a two-dimensional cylinder, T = (x mod1, p), by using the stroboscopic Poincaré section after each period T = 2π/ω, cf. fig. 1(a). Let us start out with the case of zero dc bias Eb = 0. Then, the phase space of the system is characterized by the presence of a stochastic layer which originates from the destroyed separatrix of the undriven case Eac = 0 [2]. The chaotic layer is confined by transporting KAM-tori, which originate from perturbed trajectories of particles with large kinetic energies. These tori are noncontractible (since they cannot be continuously contracted to a point on T), and separate the cylinder. The stochastic layer is not uniform and contains different regular invariant manifolds. There is an infinite number of periodic orbits (POs) embedded in the layer. Each PO, X̂Tp(t) = {xTp(t), pTp(t)}, is characterized by the period Tp = kT, k = 1, 2, . . ., an integer shifting distance L, and a winding number υ = L/Tp: xTp(t+ Tp) = xTp(t) + L, pTp(t+ Tp) = pTp(t). (2) If υ = 0 then that PO is transporting. POs can be linearly stable or unstable [11]. Towards this goal we linearize the phase space flow around a PO X̂Tp(t), and map it onto itself by integrating over one period Tp. The resulting 2×2 symplectic Floquet matrix M has eigenvalues (Floquet multipliers) λ1 and λ2 with λ1λ2 = 1 [11]. For a stable PO, both multipliers are located on the unit circle, while for an unstable PO, both multipliers are located either both on the negative or positive real axis. Stable POs are always enclosed by regular islands, which are filled by contractible tori, cf. fig. 1(a). If a stable PO is transporting then the corresponding island is also transporting with the same winding number υ. In addition, within the chaotic sea there is an infinite number of unstable POs with different periods and winding numbers. They are unstable with respect to small perturbations and are not isolated from the chaotic layer. Finally, cantori [10] exist which generate semipenetrable barriers in phase space. Equation (1) is invariant under time reversal S : t → −t+ T/2, x → x, p → −p, (3) which changes the sign of the current J = v. Thus, transporting invariant POs and islands appear as symmetry-related pairs in phase space with opposite winding numbers. Consequently, no overall net transport occurs in this driven but unbiased situation [5–7,9]. Let us focus now on the case Eb = 0. In fig. 1(b) we plot the escape time Tesc as a function of initial conditions {x0, p0} for nonzero Eb. The window {x0, p0} coincides with fig. 1(a). The 590 EUROPHYSICS LETTERS Fig. 1 – (a) Poincaré section for the system equation (1), ω = 2π, T = 1, Eac = 5.8, for (a) Eb = 0 and (b) Eb = 0.183. Several POs are shown: one stable (k = 1, L = 1) (squares), and several unstable POs (k = 2, L = 1) (circles), (k = 3, L = 2) (upward triangles), and (k = 3, L = 1) (downward triangles). The island group near p = 0 has zero winding number υ = 0. (b) Time Tesc to be accelerated until p(Tesc) = −10 as a function of the initial conditions in phase space. Trajectories from the islands are not accelerated at all. Inset in (b): Poincaré section zoom of phase space structure showing the transformation of an unstable PO into a stable PO with a surrounding invariant regular island. escape time is reached by a trajectory if p(Tesc) = −10, and we integrate up to t = 200. While most of the chaotic layer in fig. 1(a) is quickly accelerated, regular islands persist and their trajectories are in fact not accelerated at all. Moreover we observe phase space regions with delayed acceleration. These trajectories stick for a large time to the persisting regular islands. Noncontractible KAM-tori do not survive in the biased Hamiltonian system. If there exists at least one accelerating trajectory, then KAM-tori do not persist [9, 12]. Such accelerating trajectories always exist for Eb = 0, both with Eac = 0 and Eac = 0. Let us give analytical proof that POs at Eb = 0 persist for nonzero Eb. Denote a solution of (1) with initial conditions X0 = {x0, p0} at t = 0 by X̂(t,X0, Eb) = S. Denisov et al.: Stationary Hamiltonian transport with dc bias 591 0 0.1 0.2 0.3 0.4 0.5 E b -2 -1 0 1 2 3 4 θ 1e-01 1e+00 1e+01 1e+02 1e+03