Two-state Ratchet

The simplest optical ratchet protocol involves a two-state cycle (24),

$$f(t) = \begin{cases}0 & 0 \le (t \bmod T) < T_1 \\ \frac{L}{3} & T_1 \le (t \bmod T) < T. \end{cases}$$ (6)

In this case, $T_2 = T - T_1$. This is implemented by alternately shifting the optical trap array by one-third of a lattice constant to the right and then returning it to its initial position. Spacing the manifolds so that $L/3 \gg \sigma$ ensures that their potential wells do not overlap. Consequently, any particles localized in tweezers in one state are released into a force-free region when the landscape abruptly shifts. They subsequently diffuse freely unless they find and fall into another manifold of traps, or perhaps are recaptured when the initial state is projected again.

This protocol explicitly satisfies the symmetry condition in Eq. (4) when the two states are of equal duration, $T_1 = T_2 = T/2$. This particular operating point therefore should create a flux-free nonequilibrium steady-state, with particles being juggled back and forth between neighboring manifolds of traps. Breaking spatiotemporal symmetry by setting $T_1 \ne T_2$ does not guarantee a flux, but at least creates the possibility.

Figure 2: Flux induced by a two-state holographic optical ratchet. Discrete points show measured mean drift speed as a function of $T_2$ for $T_1 = 3~\unit{sec}$. The solid curve is a fit to the data for $\beta V_0 = 2.75$ and $\sigma = 0.65~\ensuremath{\unit{\mu m}}\xspace$. Other curves show how the induced drift depends on $T/\tau$, with optimal flux obtained for $T/\tau = 0.193$.

The data in Fig. 2 demonstrate that this possibility is borne out in practice. The discrete points in Fig. 2 show the measured average drift velocity, $\ensuremath{\mathcal{V}}\xspace$ $v$, for an ensemble of colloidal silica spheres 1.53  $\unit{\mu m}$ in diameter dispersed in a 40  $\unit{\mu m}$ thick layer of water between a coverslip and a microscope slide (24). The spheres are roughly twice as dense as water and rapidly sediment into a free-floating layer above the coverslip (25). The holographic optical tweezer array was projected into the layer's midplane to minimize out-of-plane fluctuations, with an estimated power of $1~\unit{mW/trap}$. Roughly 30 spheres were in the trapping domain at any time, so that reasonable statistics could be amassed in 10 minutes despite the very large fluctuations inherent in thermal ratchet operation. This number is small enough, moreover, to minimize the rate of collisions among the particles.

Given the spheres' measured diffusion coefficient of $D = 0.33~\unit{\ensuremath{\unit{\mu m}}\xspace ^2/sec}$, the time required to diffuse the inter-manifold separation of $L = 5.2~\ensuremath{\unit{\mu m}}\xspace$ is $\tau = L^2/(2D) = 39~\unit{sec}$. This establishes a natural velocity scale, $L/\tau$, in which $v$ is presented. These data were acquired with $T_1 = 3~\unit{sec}$ and $T_2$ varying from 0.8 sec to 14.7 sec.

As anticipated, the ratchet-induced flux vanishes at the point of spatiotemporal symmetry, $T_2 = T_1$, and is non-zero otherwise. The vanishing point signals a reversal in the direction of the drift velocity, with particles being more likely to advance from the wells in the longer-lived state toward the nearest manifold in the shorter-lived state. This trend can be understood as resulting from the short-duration state's biasing the diffusion of particles away from their localized distribution in the long-lived state.

To make this qualitative argument more concrete, we calculate the steady-state velocity for particles in this system by considering the evolution of the probability density $\rho(x,t)$ for finding a particle within $dx$ of position $x$ at time $t$. The Fokker-Planck equation associated with Eq. (2) is (26,27):

$$\frac{\partial \rho(x,t)}{\partial t} = D\, \left[\frac{\partial^... ...c{\partial}{\partial x} \left\{ \rho(x,t) \, V^\prime(x-f(t)) \right\} \right],$$ (7)

where the prime denotes a derivative with respect to the argument. Equation (7) is formally solved by the master equation

$$\rho(x,t + T) = \int P(x, T \vert x_0, 0) \, \rho(x_0, t) \, dx_0$$ (8)

for the evolution of the probability density, with the propagator

$$P(x, t \vert x_0, 0) = \exp\left(\int^{t} L(x,t^\prime) \, dt^\prime\right) \, \delta(x-x_0),$$ (9)

describing the transfer of particles from $x_0$ to $x$ under the Liouville operator

$$L(x,t) = D \, \left(\frac{\partial^2}{\partial x^2} + \beta \frac{\partial}{\partial x} V^\prime(x - f(t)) \right).$$ (10)

From Eq. (8), it follows that the steady-state particle distribution $\rho(x)$ is an eigenstate of the propagator,

$$\rho(x) = \int P(x, T \vert x_0,0) \, \rho(x_0) \, dx_0,$$ (11)

associated with one complete cycle. The associated steady-state flux is (27)

$$v = \int \frac{x - x_0}{T} \, \rho(x_0) \, P(x,T \vert x_0,0) \, dx \, dx_0.$$ (12)

Figure 3: Stochastic resonance in the two-state optical thermal ratchet for $\sigma /L = 0.125$. (a) Dependence on cycle period $T$ in units of the diffusive time scale $\tau$ for $\beta V_0 = 2.5$ at the optimal duty cycle $T_2/T_1 = 0.3$. (b) Dependence on well depth for the optimal cycle rate $T/\tau = 0.193$ and duty cycle.

The solid curve in Fig. 2 is a fit of Eq. (12) to the measured particle fluxes for $\beta V_0 = 2.5$ and $\sigma = 0.65~\ensuremath{\unit{\mu m}}\xspace$. The additional curves in Fig. 2 show how $v$ varies with $T_1/T_2$ for various values of $T/\tau$ for these control parameters. The induced flux, $v$, plotted in Figure 3(a), falls off as $1/T$ in the limit of large $T$ because the particles spend increasingly much of their time localized in traps. It also vanishes in the opposite limit because the diffusing particles cannot keep up with the landscape's evolution. The optimal cycle period at $T/\tau \approx 0.2$ constitutes an example of stochastic resonance (11,12). Although a particle's diffusivity controls the speed with which it traverses the ratchet, its direction is uniquely determined by $T_2/T_1$.

No flux results if the traps are too weak. Increasing the potential wells' depths increases the maximum attainable flux, but only up to a point. If the traps are too strong, particles also become localized in the short-lived state, and the ratchet approaches a deterministic flux-free limit in which particles simply hop back and forth between neighboring manifolds. This behavior is shown in Fig. 3(b).

Different objects exposed to the same time-evolving optical intensity pattern experience different values of $V_0$ and $\sigma$ (28,22), and also can have differing diffusive time scales, $\tau$. Such differences establish a dispersion of mean velocities for mixtures of particles moving through the landscape that can be used to sorting the particles. Despite this method's symmetry and technical simplicity, however, the two-state protocol is not the most effective platform for such practical applications. A slightly more elaborate protocol yields a thermal ratchet whose deterministic limit transports material rapidly and whose stochastic limit yields flux reversal at a point not predicted by the symmetry selection rules in Eqs. (4) and (5).