Three-state Ratchet

The next step up in complexity and functional richness involves the addition of a third state to the ratchet cycle:

$$f(t) = \begin{cases}0, & 0 \le (t \bmod T) < \frac{T}{3} \\ \frac... ...{2T}{3} \\ -\frac{L}{3}, & \frac{2T}{3} \le (t \bmod T) < T \end{cases} \quad .$$ (13)

This three-state protocol (27) consists of cyclic displacements of the landscape by one third of a lattice constant. Unlike the two-state symmetric thermal ratchet, it has a deterministic limit, an explanation of which helps to elucidate its operation in the stochastic limit.

If the width, $\sigma$, of the individual wells is comparable to the separation $L/3$ between manifolds in consecutive states, then a particle localized at the bottom of a well in one state is released near the edge of a well in the next. Provided $V_0$ is large enough, the particle falls to the bottom of the new well during the $T/3$ duration of the new state. This process continues through the sequence of states, and the particle is transferred deterministically forward from manifold to manifold. This deterministic process is known as optical peristalsis (29), and is useful for reorganizing fluid-borne objects over large areas with simple sequences of generic holographic trapping patterns.

Assuming the individual traps are strong enough, optical peristalsis transfers objects forward at speed $v = L/T$. If, on the other hand, $\beta V_0 \lesssim 1$, particles can be thermally excited out of the forward-going wave of traps and so will travel forward more slowly. This is an example of a deterministic machine's efficiency being degraded by thermal fluctuations. It contrasts with the two-state thermal ratchet, which has no effect in the deterministic limit and instead relies on thermal fluctuations to induce motion.

The three-state protocol enters its stochastic regime when the inter-state displacement of manifolds, $L/3$, exceeds the individual traps' width, $\sigma$. Under these conditions, a particle that is trapped in one state is released into the force-free region between traps once the state changes. If the particle diffuses rapidly enough, it might nevertheless fall into the nearest potential well centered a distance $L/3$ away in the forward-going direction within time $T/3$. The fraction of particles achieving this will be transferred forward in each step of the cycle. This stochastic process resembles optical peristalsis, albeit with reduced efficiency. There is a substantial difference, however.

An object that does not diffuse rapidly enough to reach the nearest forward-going trap in time $T/3$ might still reach the trap centered at $-L/3$ in the third state by time $2T/3$. Such a slow-moving object would be transferred backward by the ratchet at velocity $v = - L/(2T)$. Unlike the two-state ratchet, whose directionality is established unambiguously by the sequence of states, the three-state ratchet's direction appears to depend also on the transported objects' mobility.

Figure 4: Flux reversal in a symmetric three-state optical thermal ratchet. (a) As a function of cycle period for fixed inter-manifold separation, $L$. (b) As a function of inter-manifold separation $L$ for fixed cycle period $T$.

This prediction is borne out by the experimental observations (27) in Fig. 4. The discrete points in Fig. 4(a) show the measured flux of 1.53  $\unit{\mu m}$ diameter silica spheres as a function of the cycle period $T$ with the inter-manifold separation fixed at $L = 6.7~\ensuremath{\unit{\mu m}}\xspace$. Flux reversal at $T/\tau \approx 0.1$ does not result from special symmetry considerations because the spatiotemporal evolution described by Eqs. (1) and (13) violates the conditions in Eqs. (4) and (5) for all values of $T$. Rather, this reflects a dynamical transition in which rapidly diffusing particles are driven in the forward while slowly diffusing particles drift backward. The origin of this transition in thermal ratchet behavior is confirmed (1) by the observation of a comparable transition induced by varying the inter-manifold separation $L$ for fixed cycle period $T = 6~\unit{sec}$, as plotted in Fig. 4(b) (27).

Figure 5: Calculated ratchet-induced drift velocity as a function of cycle period $T$ for representative values of the inter-manifold separation $L$ ranging from the deterministic limit, $L = 6.5~\sigma$ to the stochastic limit $L = 13~\sigma$.

The solid curves in Fig. 4 are fits to Eq. (12) using Eq. (13) to calculate the propagator. The fit values, $\beta V_0 = 8.5 \pm 0.08$ and $\sigma = 0.53 \pm 0.01~\ensuremath{\unit{\mu m}}\xspace$ are consistent with values obtained for the two-state ratchet, given a higher laser power of 2.5 mW/trap. The crossover from deterministic optical peristalsis with uniformly forward-moving flux at small $L$ to stochastic operation with flux reversal at larger separations is captured in the calculated drift velocities plotted in Fig. 5.

Whereas flux reversal in the two-state ratchet is mandated by the protocol, flux reversal in the three-state ratchet depends on properties of diffusing objects through the detailed structure of the probability distribution $\rho(x)$ under different operating conditions. The three-state optical thermal ratchet therefore provides the basis for sorting applications in which different fractions of a mixed sample are transported in opposite directions by a single time-evolving optical landscape. This builds upon previously reported ratchet-based fractionation techniques which rely on unidirectional motion (30,3,1).