Flux Reversal in a Two-state Symmetric Optical Thermal Ratchet
Abstract.
A Brownian particle's random motions can be rectified by a periodic potential energy landscape that alternates between two states, even if both states are spatially symmetric. If the two states differ only by a discrete translation, the direction of the ratchet-driven current can be reversed by changing their relative durations. We experimentally demonstrate flux reversal in a symmetric two-state ratchet by tracking the motions of colloidal spheres moving through large arrays of discrete potential energy wells created with dynamic holographic optical tweezers. The model's simplicity and high degree of symmetry suggest possible applications in molecular-scale motors.
Until fairly recently, random thermal fluctuations were considered impediments to inducing motion in systems such as motors. Fluctuations can be harnessed, however, through mechanisms such as stochastic resonance (1) and thermal ratchets (2), as efficient transducers of input energy into mechanical motion. Unlike conventional machines, which battle noise, molecular-scale devices that exploit these processes actually requite thermal fluctuations to operate.
This article focuses on thermal ratchets in which the random motions of Brownian particles are rectified by a time-varying potential energy landscape. Even when the landscape has no overall slope and thus exerts no average force, directed motion still can result from the accumulation of coordinated impulses. Most thermal ratchet models break spatiotemporal symmetry by periodically translating, tilting or otherwise modulating a spatially asymmetric landscape (2). Inducing a flux is almost inevitable in such systems unless they satisfy conditions of spatiotemporal symmetry or supersymmetry (3). Even a spatially symmetric landscape can induce a flux with appropriate driving (4); (5); (6); (7). Unlike deterministic motors, however, the direction of motion in these systems can depend sensitively on implementation details.
We recently demonstrated a spatially symmetric three-state thermal ratchet for micrometer-scale colloidal particles implemented with arrays of holographic optical tweezers, each of which constitutes a discrete potential energy well (7). Repeatedly displacing the array first by one third of a lattice constant and then by two thirds breaks spatiotemporal symmetry in a manner that induces a flux. Somewhat surprisingly, the direction of motion depends sensitively on the duration of the states relative to the time required for a particle to diffuse the inter-trap separation (7). The induced flux therefore can be canceled or even reversed by varying the rate of cycling, rather than the direction. This approach builds upon the pioneering demonstration of unidirectional flux induced by a spatially asymmetric time-averaged optical ratchet (8); (9), and of reversible transitions driven by stochastic resonance in a dual-trap rocking ratchet (10); (11).

Here, we demonstrate flux induction and flux reversal in a symmetric two-state thermal ratchet implemented with dynamic holographic optical trap arrays (12); (13). The transport mechanism for this two-state ratchet is more subtle than our previous three-state model in that the direction of motion is not easily intuited from the protocol. Its capacity for flux reversal in the absence of external loading, by contrast, can be inferred immediately by considerations of spatiotemporal symmetry. This also differs from the three-state ratchet (7) and the rocking double-tweezer (10); (11) in which flux reversal results from a finely tuned balance of parameters.
Figure 1 schematically depicts how the
two-state ratchet operates.
Each state consists of a pattern of discrete optical traps,
modeled here as Gaussian wells of width and depth
,
uniformly separated by a distance
.
The first array of traps is extinguished after time
and replaced
immediately with a
second array, which is displaced from the first by
.
The second pattern is extinguished after time
and replaced
again by the first, thereby completing one cycle.
If the potential wells in the second state overlap those in the first, then trapped particles are handed back and forth between neighboring traps as the states cycle, and no motion results. This also is qualitatively different from the three-state ratchet, which deterministically transfers particles forward under comparable conditions, in a process known as optical peristalsis (14); (7). The only way the symmetric two-state ratchet can induce motion is if trapped particles are released when the states change and then diffuse freely.



The motion of a Brownian particle in this system can be described with the one-dimensional Langevin equation
![]() |
(1) |
where is the fluid's dynamic viscosity,
is the
potential energy landscape,
is its derivative, and
is a delta-correlated stochastic force representing thermal noise.
The potential energy landscape in our system is spatially periodic
with period
,
![]() |
(2) |
The time-varying displacement of the potential
energy in our two-state ratchet is described by a periodic function
with period
, which is plotted
in Fig. 2(a).
The equations describing this
traveling potential ratchet can be recast into the form
of a tilting ratchet, which ordinarily would be implemented
by applying an oscillatory external force to objects on an otherwise
fixed landscape.
The appropriate coordinate transformation,
(2),
yields
![]() |
(3) |
where is the effective driving force.
Because
has a vanishing mean,
the average velocity of the original problem is the same as
that of the transformed tilting ratchet
,
where the angle brackets imply both an ensemble average
and an average over a period
.
Reimann has demonstrated (3); (2)
that a steady-state flux, ,
develops in any tilting ratchet that breaks
both spatiotemporal symmetry,
![]() |
(4) |
and also spatiotemporal supersymmetry,
![]() |
(5) |
for any .
No flux results if either of Eqs. (4)
or (5)
is satisfied.
The optical trapping potential depicted
in Fig. 1 is symmetric
but not supersymmetric.
Provided that violates the symmetry condition in
Eq. (4), the
ratchet must induce directed motion.
Although
is supersymmetric, as can be seen in
Fig. 2(b),
it is symmetric only when
.
Consequently, we expect a particle current
for
.
The zero crossing at
furthermore
portends flux reversal on either side of the equality.







We calculate the steady-state velocity for this system
by solving the master equation associated with Eq. (1)
(15); (7).
The probability for a driven
Brownian particle to drift from position to within
of position
during the interval
,
is given by the propagator
![]() |
(6) |
where the Liouville operator is
![]() |
(7) |
and where is the thermal energy scale (15).
The steady-state particle distribution
is an eigenstate of the
master equation
![]() |
(8) |
and the associated steady-state flux is (7)
![]() |
(9) |
The natural length scale in this problem is , the inter-trap
spacing in either state.
The natural time scale,
, is the time required for particles of
diffusion constant
to diffuse this distance.
Figure 3 shows how varies with
for
various values of
for experimentally accessible values of
,
, and
.
As anticipated, the net drift vanishes for
.
Less obviously, the induced
flux is directed from each well in the longer-duration state
toward the nearest well in the short-lived state.
The flux falls off as
in the limit of large
because the
particles spend increasingly much of their time localized in traps.
It also diminishes
for short
because the particles cannot keep up with the
landscape's evolution.
In between, the range of fluxes
can be tuned with
.











We implemented this model
for a sample of 1.53 diameter colloidal
silica spheres (Bangs Laboratories, lot number 5328)
dispersed in water, using potential energy landscapes
created from arrays of holographic optical traps
(12); (16); (13); (7).
The sample was enclosed in a hermetically sealed glass chamber
roughly 40
thick
created by bonding the edges of a coverslip to a microscope slide,
and was allowed to equilibrate to room temperature (
)
on the stage of a Zeiss S100 2TV Axiovert inverted optical microscope.
A
NA 1.4 oil immersion SPlan Apo objective lens was used to
focus the optical tweezer array into the sample and to image the
spheres,
whose motions were captured with
an NEC TI 324A low noise monochrome CCD camera.
The micrograph
in Fig. 4(a)
shows the focused light from a
array of optical traps
formed by a phase hologram projected with a Hamamatsu X7550
spatial light modulator (17).
The tweezers are arranged in twenty-trap manifolds
long
separated by
.
Each trap is powered by an estimated
of laser light at 532 nm.
The particles, which appear in the bright-field micrograph in
Fig. 4(b),
are twice as dense as water and sediment to the lower glass surface,
where they diffuse freely in the plane with a measured
diffusion coefficient
of
.
This establishes the characteristic time scale for the system of
, which is quite reasonable for digital video
microscopy studies.
Out-of-plane fluctuations were minimized by focusing the traps at the
spheres' equilibrium height above the wall (18).
We projected two-state cycles of optical trapping patterns in which the
manifolds in Fig. 4(a)
were alternately displaced in the spheres' equilibrium plane by ,
with the duration of the first state fixed at
and
ranging from 0.8
to 14.7
.
To measure the flux induced by this cycling potential energy landscape for one value
of
,
we first gathered roughly two dozen particles
in the middle row of traps in state 1, as shown in Fig. 4(b),
and then projected up to one hundred periods of two-state cycles.
The particles' motions
were recorded as uncompressed digital video streams for
analysis (19).
Their time-resolved trajectories then were averaged over the
transverse direction into the probability density,
,
for finding particles within
of position
after time
.
We also tracked particles outside the trapping pattern
to monitor their diffusion coefficients and to ensure the absence of
drifts in the supporting fluid.
Starting from this well-controlled initial condition resolves any uncertainties
arising from the evolution of nominally random initial conditions (7).
Figures 4(c) and (d) show the spatially-resolved
time evolution of for
and
.
In both cases, the particles spend most of their time localized in traps,
visible here as bright stripes,
occasionally using the shorter-lived traps as springboards to neighboring
wells in the longer-lived state.
The mean particle position
advances
as the particles make these jumps, with the associated results plotted in
Fig. 4(e).
The speed with which an initially localized state,
, advances differs
from the steady-state speed plotted, in Fig. 3, but still
can be calculated as
the first moment of the propagator,
![]() |
(10) |
Numerical analysis reveals a nearly constant mean speed that agrees quite closely with the steady-state speed from Eq. (9).
Fitting traces such as those in Fig. 4(e) to linear trends
provides estimates for the ratchet-induced flux, which are plotted in
Fig. 4(f).
The solid curve in Fig. 4(f) shows excellent agreement with
predictions of
Eq. (10) for and
.

Our implementation of the two-state ratchet involves updating the
optical intensity pattern to translate the physical landscape.
However, the same principles can be applied to systems in which
the landscape remains fixed and the object undergoes
cyclic transitions between two states.
Figure 5 depicts a model for an active two-state
walker on a fixed physical landscape that is inspired by
the biologically relevant transport of single myosin head groups
along actin filaments (20).
The walker consists of a
head group that interacts with localized potential
energy wells
periodically distributed on the landscape.
It also is
attached to a lever arm that uses an external energy source to
translate the head group by a distance somewhat smaller than
the inter-well separation.
The other end of the lever arm is connected to the payload,
whose viscous drag would provide the leverage necessary to translate
the head group between the extended and retracted states.
Switching between the walker's two states is
equivalent to the two-state translation of the potential
energy landscape in our experiments, and thus would have the
effect of translating the walker in the direction of the shorter-lived
state.
A similar model in which a two-state walker traverses a
spatially asymmetric potential energy landscape yields deterministic
motion at higher efficiency than the present model (21).
It does not, however, allow for reversibility.
The length of the lever arm and the diffusivity of the motor's body
and payload determine the ratio and thus
the motor's efficiency.
The two-state ratchet's direction does not depend on
,
however, even under heavy loading.
This differs from the three-state ratchet
(7), in which
also controls the
direction of motion.
This protocol could be used in the
design of mesoscopic motors based on synthetic macromolecules or
microelectromechanical systems (MEMS).
We are grateful for Mark Ofitserov's many technical contributions. This work was supported by the National Science Foundation through Grant Number DBI-0233971 and Grant Number DMR-0304906. S.L. acknowledges support from a Kessler Family Foundation Fellowship
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