Phase Dependent Vectorial Current Control in Symmetric Noisy Optical Ratchets

In this work, we demonstrate single microparticle transport in a symmetric noisy optical ratchet made with a linear array of 20 optical potentials, where each potential is a spatially symmetric low power ($<2.5\mathrm{mW}$) three-dimensional trap. Both the external force $F(t)$ and the depth $V_{0i}(t)$ of the optical potentials are dynamic and change at the same frequency $\nu =2\mathrm{Hz}$. The depths of the individual optical potentials are random (uncorrelated noise) distributed around a mean value $V_0$, $⟨V_{0i}(t)⟩=V_0$, while the external force is periodic and unbiased $⟨F(t)⟩=0$. The system is completely symmetric for times $t\gg 1/\nu$. Directed transport is possible as a result of the symmetry being broken at times on the order of $1/\nu$. We find that the direction and speed of motion (current) are coupled to the phase difference between the noise in the optical potentials and the external periodic force.

Figure 1

(a) Sketch of the system showing the position of the optical traps (separated by $L=2.3\mu \mathrm{m}$, $\sigma =519\mathrm{nm}$) and the position of a particle (blue circle) trapped by the $i$th trap. Once the external rocking force $F(t)$ is introduced, the trapped particle will oscillate around the $i$th trap (dotted circles). (b) Linear optical potential array with random heights $V_{0i}(t_0)$. (c) Dynamics of the experiment: The external force is represented by the blue line (constant during half-period), and the red line represents the change into one of the optical potentials; the first plot is the case for zero phase difference and the second plot is for a phase difference of $\pi /2$.

Figure 2

(a) Measured trajectory of a reference microparticle stuck at the bottom of the microscope coverslip. The red line is part of a speckle pattern that is used to track the update in the SLM (dips). The phase difference is 1.1 rad. (b) Measured trajectories of the trapped microparticle for each cycle. The red line represents the average trajectory over 37 different realizations for a phase difference of 1.1 rad.

Figure 3

(a) Average displacement (per cycle) of the particle as a function of the periodic-force phase $\varphi$. The red round markers correspond to 37 different realizations of the experiment. The blue shaded region is composed of 1000 independent numerical realizations of the experiment, showing that the measured signals are expected within our simplified model. (b) Composite images for phase differences of 1.8, 3.2, and 4.1 rad for a time of 19.5 s. Each video frame is cut to a width of three pixels and each composite image contains 780 frames. The video in the Supplemental Material shows the case of negative displacement (to the left).