Unpredictable Tunneling of a Classical Wave-Particle Association

A droplet bouncing on a vibrated bath becomes a “walker” moving at constant velocity on the interface when it couples to the surface wave it generates. Here the motion of a walker is investigated when it collides with barriers of various thicknesses. Surprisingly, it undergoes a form of tunneling: the reflection or transmission of a given incident walker is unpredictable. However, the crossing probability decreases exponentially with increasing barrier width. This shows that this wave-particle association has a nonlocality sufficient to generate a quantumlike tunneling at a macroscopic scale.

Figure 1

(a) The measured thresholds for the Faraday instability ${\gamma }_m^F$ (triangles) and for the walking instability of ${\gamma }_m^W$ of a drop of diameter $D=0.78\mathrm{mm}$ (filled circles) as a function of the depth $h$ of the liquid bath. (b) In cells formed of two regions of depths $h_0=4.1\mathrm{mm}$ and $h_1=1.1\mathrm{mm}$, when the system is tuned to a value ${\gamma }_m^W(h_0)>{\gamma }_m>{\gamma }_m^W(h_1)$ (e.g., ${\gamma }_m/g=3.75$), a drop forms a walker in the regions of large depth but remains motionless in the shallow zone. The continuous lines are simple interpolations.

Figure 2

(a) The wave field surrounding a walker of velocity $V=13.7\pm 0.1\mathrm{mm}/\mathrm{s}$ as it approaches a barrier. A cross section of this barrier, of finite thickness $e=2\mathrm{mm}$ and over which the fluid depth is $h_1=1.1\mathrm{mm}$, is shown in the insert (b). In (a) the arrow on the right shows the direction of motion of the walker, and the barrier is located between the light grey arrows. The topography of the liquid interface is measured using the free-surface synthetic schlieren technique [5], which provides a vertical accuracy $5\times {10}^{-4}\mathrm{mm}$. A transmitted wave of weak amplitude can clearly be observed. (c) Measured ratio $A_t/A_i$ of the amplitude of the transmitted wave over that of the incident wave as a function of the barrier thickness. The dashed line is the result of the exponential fit with a characteristic length ${\lambda }_W=2.9\pm 0.2\mathrm{mm}$.

Figure 3

(a) Sketch of the square frame of width $e$ seen from above. The fluid depth above the barrier (in grey) is reduced to $h_1=1.30\pm 0.02\mathrm{mm}$. (b) Probability of escape of walkers out of cavities with walls of thickness $e=2.5$, 3, 4 and 5 mm, respectively, as a function of their velocity. The continuous curves are simple interpolations. (c) Semilogarithmic plot showing, for three different velocities ($V=10.5\mathrm{mm}/\mathrm{s}$, $V=12\mathrm{mm}/\mathrm{s}$, $V=14\mathrm{mm}/\mathrm{s}$), the probability of escape of the walkers as a function of barrier thickness $e$. The lines are exponential fits with characteristic lengths ${\lambda }_{\tau }=0.65$, 0.85, and 0.8 mm, respectively.

Figure 4

The recorded trajectories of the walker inside the square trap of side $L=55\mathrm{mm}$. In (a) $e=4.5\mathrm{mm}$ and $V=9.95\mathrm{mm}/\mathrm{s}$. In (b) $e=2.5\mathrm{mm}$ and $V=9\mathrm{mm}/\mathrm{s}$. The probability of escape $P$ is of the order of 1%. In (c) $e=2.5\mathrm{mm}$ and $V=11.8\mathrm{mm}/\mathrm{s}$. $P\simeq 10\%$. In (d) $e=2.5\mathrm{mm}$ and $V=13.2\mathrm{mm}/\mathrm{s}$. $P\simeq 30\%$.

Figure 5

(a) Superposition of the trajectories of a walker of velocity $V=12\mathrm{mm}/\mathrm{s}$ producing 110 collisions with a barrier with $e=3\mathrm{mm}$ leading to $n=14$ crossings. (b) Detail of 10 consecutive collisions of the previous recording. $A$ and $A^{\prime }$ are the points where we measure the walker’s velocity $V$ before each collision and where all trajectories coincide. (c) Probability of crossing for walkers impinging with normal incidence on barriers of thicknesses 1.6, 2, 3, 4, 5 and 6 mm, respectively, as a function of their velocity. The continuous curves are simple interpolations.