Observation of Flux Reversal in a Symmetric Optical Thermal Ratchet

Abstract

We demonstrate that a cycle of three holographic optical trapping patterns can implement a thermal ratchet for diffusing colloidal spheres, and that the ratchet-driven transport displays flux reversal as a function of the cycle frequency and the inter-trap separation. Unlike previously described ratchet models, the approach we describe involves three equivalent states, each of which is locally and globally spatially symmetric, with spatiotemporal symmetry being broken by the sequence of states.Brownian motion cannot create a steady flux in a system at equilibrium. Nor can local asymmetries in a static potential energy landscape rectify Brownian motion to induce a drift. A landscape that varies in time, however, can eke a flux out of random fluctuations by breaking spatiotemporal symmetry [1,2,3,4]. Such flux-inducing time-dependent potentials are known as thermal ratchets [5,6], and their ability to bias diffusion by rectifying thermal fluctuations has been proposed as a possible mechanism for transport by molecular motors and is being actively exploited for macromolecular sorting [7].Most thermal ratchet models are based on spatially asymmetric potentials. Their time variation involves displacing or tilting them relative to the laboratory frame, modulating their amplitude, changing their periodicity, or some combination, usually in a two-state cycle. Chen demonstrated that a spatially symmetric potential still can induce drift, provided that it is applied in a threestate cycle, one of which allows for free diffusion [8]. This idea since has been refined [9] and generalized [10].The space-filling potential energy landscapes required for most such models pose technical challenges. Furthermore, their relationship to the operation of natural thermal ratchets has proved difficult to establish.This Letter describes the first experimental demonstration of a spatially symmetric thermal ratchet, which we have implemented with holographic optical traps [11,12,13]. The potential energy landscape in this system consists of a large number of discrete optical tweezers [14], each of which acts as a symmetric potential energy well for nanometer-to micrometer-scale objects such as colloidal spheres. We arrange these wells so that colloidal spheres can diffuse freely in the interstitial spaces but are localized rapidly once they encounter a trap. A threestate thermal ratchet then requires only displaced copies of a single two-dimensional trapping pattern. Despite its simplicity, this ratchet model displays flux reversal [5,15] in which the direction of motion is controlled by a balance between the rate at which particles diffuse across the landscape and the ratchet's cycling rate.Often predicted, and inferred from the behavior of some natural molecular motors and semiconductor devices [5], flux reversal has been directly observed in com- paratively few systems. Previous demonstrations have focused on ratcheting of magnetic flux quanta through type-II superconductors in both the quantum mechanical [16] and classical [17] regimes,...

Two-dimensional optical thermal ratchets based on Fibonacci spirals

“…3(a) for the radial and azimuthal components of the induced drift velocity share much in common with results obtained from one-dimensional thermal ratchet models [18]. Traces in this figure show the drift velocity normalized by the natural velocity scale, D/R 0 , for a few representative values of the inter-trap separation, R 0 .…”

“…The particle can only advance through the pattern by diffusing. In one-dimensional ratchets, such a diffusive contribution to the transport process creates possibilities for temperature-dependent flux reversals [5,18,30]. Observing that particles travel along parastichies in the Fibonacci spiral ratchet's deterministic limit suggests that similar reversals should emerge in its stochastic limit.…”

“…If, on the other hand, the pattern advances too rapidly, particles cannot reach the nearest trap in one time step. Rather, they preferentially find the nearest trap after two time steps, which carries them backward along the same path [18]. The drift velocity vanishes for large T because the pattern itself advances slowly.…”

“…The resulting potential energy landscape is transformed into a ratchet potential through its time evolution. To break spatio-temporal symmetry we adopt the three-state protocol introduced originally for studies of one-dimensional ratchets [18,29]. Rather than translating the pattern, as in earlier studies [18,19,29], we rotate it about its center:…”

“…Introduced with James Clerk Maxwell's thought experiments in the 1860's, thermal ratchets recently have enjoyed a resurgence of interest because of their relevance to biological molecular motors, and have been realized experimentally both for molecular [7,8], micrometer-scaled [9][10][11][12][13][14][15][16][17][18][19][20][21] and for quantum objects [22]. Virtually all of these studies, however, have focused on one-dimensional systems [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], or on systems that can be projected onto one dimension [7,21]. Comparatively little attention has been paid to thermal ratchets in two or higher dimensions.…”