Extended Data Fig. 6: Pulse length dependence of \({{{\boldsymbol{x}}}}_0,{{{\boldsymbol{y}}}}_0\) with \({{{\mathbf{M}}}}_{{{{\mathbf{YIG}}}}}\) along \(- {{{\mathbf{y}}}}\).

a, b, Mean displacement values \(x_0,y_0\) extracted from the trajectory of several skyrmion bubbles with \({{{\mathbf{M}}}}_{{{{\mathrm{YIG}}}}}\) along \(- {{{\mathbf{y}}}}\) (see Fig. 4a,b for details regarding the analysis). Data taken for \(H_z = - 20\) Oe (\(Q = + 1\)) and for both polarities of \({{{\mathbf{J}}}}_x\) (indicated by an arrow). The sign of \(x_0,y_0\) corresponds to the sign of \({{{\mathbf{J}}}}_x.\) Different colours indicate the current density. The error bars are the standard errors of \(x_0\), \(y_0\) calculated from the variance of these magnitudes to the double Gaussian distribution of \(\delta x,\delta y\). As for the case of YIG demagnetized (Fig. 4c,d), \(x_0\) and \(y_0\) tend to finite values when \(t_{{{\mathrm{p}}}} \to 0\). Remarkably, the \(x_0,y_0(t_{{{\mathrm{p}}}} \to 0)\) values are similar for both directions of \({{{\mathbf{J}}}}_{{{\mathrm{x}}}}\) and similar to the ones measured for YIG demagnetized. As \(t_{{{\mathrm{p}}}}\) increases, \(x_0\) and \(y_0\) start to increase from a pulse length threshold value that depends on the amplitude of the current. Larger (smaller) current densities are required for driving skyrmion bubbles with \({{{\mathbf{M}}}}_{{{{\mathrm{YIG}}}}}\) pointing to \(- {{{\boldsymbol{y}}}}\) and \(J_{{{\mathrm{x}}}} < 0\) (\(J_{{{\mathrm{x}}}} > 0\)), which is in agreement with the ratchet effect reported in Figs. 5 and 6 and Supplementary Note 10. c, Velocity of the skyrmion bubbles computed as \(\left| {v_0} \right| = \sqrt {x_0^2 + y_0^2} /t_{{{\mathrm{p}}}}\) from the data shown in a and b. The sign of the velocity is defined by the sign of \(x_0\); the error bars are computed by error propagation. As observed for the mean bubble velocity \(\bar v_{{{{\mathrm{sk}}}}}\) for YIG demagnetized (Fig. 4e), \(v_0\) increase as \(t_{{{\mathrm{p}}}}\) reduces (note that \(v_0(t_{{{\mathrm{p}}}})\) exhibits a stepper increase than \(\bar v_{{{{\mathrm{sk}}}}}(t_{{{\mathrm{p}}}})\) when reducing t_p).