Multislip Friction with a Single Ion

A trapped ion transported along a periodic potential is studied as a paradigmatic nanocontact frictional interface. The combination of the periodic corrugation potential and a harmonic trapping potential creates a one-dimensional energy landscape with multiple local minima, corresponding to multistable stick-slip friction. We measure the probabilities of slipping to the various minima for various corrugations and transport velocities. The observed probabilities show that the multislip regime can be reached dynamically at smaller corrugations than would be possible statically, and can be described by an equilibrium Boltzmann model. While a clear microscopic signature of multislip behavior is observed for the ion motion, the frictional force and dissipation are only weakly affected by the transition to multistable potentials.

Figure 1

(a) Schematic of the experimental setup. A ${}^{174}{\mathrm{Yb}}^{+}$ ion is trapped by a linear Paul trap, located $135\mu \mathrm{m}$ above the surface of a lithographic microchip. The chip generates radial trapping with a rf field and axial trapping with a dc harmonic potential. An axial optical-lattice standing wave is produced by 370 nm light, blue detuned from the ${{}^{2}S}_{1/2}\to {{}^{2}P}_{1/2}$ atomic transition by 12.6 GHz. Ion fluorescence is collected during laser cooling. (b) Stick-slip friction in a periodic optical potential. As the harmonic trap is translated at speed $v$, it drags the ion along the sinusoidal optical-lattice potential, causing the ion to slip. The excess energy acquired during a slip is dissipated by continuous laser cooling. (c) Combined harmonic-sinusoidal potential for different values of the corrugation parameter $\eta$. The potential energy landscape is drawn just before the slipping point, where the leftmost minimum vanishes and a potential with initially $n$ local minima has $n-1$ available minima to which the ion can slip. Note that for $n=1$, there is no energy barrier and, consequently, no stick-slip friction.

Figure 2

Measured ion fluorescence as a function of trap translation, indicating single-slip vs multislip behavior. After the ion is prepared in the global energy minimum, the harmonic trap is translated at constant velocity across the optical standing wave. This forces the ion to slip over local potential maxima, resulting in fluorescence peaks. The fluorescence traces are averaged over an exposure time of 300 seconds, corresponding to roughly $1.5\times 10^5$ realizations of the experiment. Error bars are statistical and indicate one standard deviation. (a) Fluorescence trace indicating single-slip behavior for a potential with $n=2$ minima ($\eta =3.2$). Fluorescence peaks are of equal height, indicating that the ion always slips to the adjacent minimum (lattice period $a=185\mathrm{nm}$). There is a small variation of peak heights due to the finite recooling time (see the text). [(b) and (c)] Fluorescence traces indicating multislip behavior in a potential with $n=3$ and $n=4$ minima, respectively ($\eta =6.4$ and $\eta =9.6$). Fluorescence peaks vary in height, indicating that an ion can slip into one of multiple local minima, and therefore may not always slip when a given minimum disappears. Note the hysteretic delay of the peaks in these time traces compared to Fig. 2. This is due to the greater static friction force $F_{\text{static}}$ exerted on the ion by the optical lattice at higher $\eta$. [See Ref. [31] and Fig. 4].

Figure 3

Multislip probabilities vs corrugation parameter $\eta$ for different transport velocities $v$. Data points are the extracted slipping probabilities $p_A$ (slip to next minimum, red squares), $p_B$ (slip to second-next minimum, blue circles), and $p_C$ (slip to third minimum, green diamonds). Velocities are reported as functions of the system’s recooling rate ${\gamma }_c$ and thermal hopping rate ${\gamma }_{\mathrm{th}}$, the rate at which the ion hops over a barrier due to its finite temperature (see the main text). Error bars are statistical and indicate one standard deviation. The multislip regime is distinguished by non-negligible values of $p_B$. Curves are calculated from a theoretical model. Vertical dash-dotted lines separate regimes with different numbers of minima [as in Fig. 1].

Figure 4

(a) Measured static friction force vs $\eta$ for three different drive velocities ($v/a=0.17$ ${\gamma }_c$ for green circles, $v/a=0.36$ ${\gamma }_c$ for blue squares, and $v/a=0.78$ ${\gamma }_c$ for black diamonds). The velocity dependence of the slope is due to thermal hopping, which reduces friction for slow transport [30]. The red theory curve is a realization of the Prandtl-Tomlinson model with no free parameters, without thermal hopping. (b) Energy dissipated per slip vs $\eta$. Data points are calculated from experimentally determined values for $p_i$, while the curves use the model values for $p_i$ from Fig. 3. Both use the calculated energy landscape. Vertical dash-dotted lines separate regimes with different numbers of minima.