Figure 1

(a) Energy-momentum diagram for the linear matter waves in optical lattice: $n=0$ Brillouin zone is shown. (b) Dependence of the radiation quasimomentum, $k_r$, from the soliton quasimomentum, $k_s$. The full diagonal line in (b) corresponds to $k_r=k_s$. Dots in (b) mark $(k_s,k_r)$ pairs measured from the modeling of Eq. (1). Dashed vertical lines in (a) and (b) mark the $\pm k_0$ points with ${ϵ}^{\prime \prime }=0$. $\mathrm{<ruby><rb>\beta </rb><rt>べーた</rt></ruby>}=0.33$.Reuse & Permissions

Figure 2

Results of the direct numerical modeling of Eq. (1). (a),(c) Square roots of the atomic density as functions of $x$. The inset in (a) shows fine details of the spatial profile of the radiation. (b),(d) Corresponding Fourier spectra. (a) and (b) are obtained for the initial solitonic wave packet with $k_s=0.87$ and $\mathrm{<ruby><rb>\kappa </rb><rt>かっぱ</rt></ruby>}=0.01$. (c) and (d) are obtained for the initial Gaussian wave packets with the zero central momentum and lattice moving with velocity $2k_s=1.74$. Integration time $t=700$ corresponds to 0.035 s. The physical number of particles in the soliton shown in (a) is $\simeq {10}^3$, its velocity is $3\times {10}^{-3}\mathrm{m}/\mathrm{s}$, and width at the half height is $\simeq 4\mathrm{<ruby><rb>\mu </rb><rt>みゅー</rt></ruby>}\mathrm{m}$.Reuse & Permissions

Figure 3

(a) Logarithm of the particle transfer rate from the solitonic to the radiation part of the wave function as a function of the soliton quasimomentum for $\mathrm{<ruby><rb>\kappa </rb><rt>かっぱ</rt></ruby>}=0.01$ and 0.006. The dashed vertical line marks the ${ϵ}^{\prime \prime }=0$ point. (b) Temporal evolution of the normalized number of particles $N_{\mathrm{r}\mathrm{a}\mathrm{d}}$ in the radiation component of the field (left axis) and corresponding dynamics of the soliton quasimomentum (right axis). $N_{\mathrm{r}\mathrm{a}\mathrm{d}}$ is calculated as the integral of $|\mathrm{<ruby><rb>\psi </rb><rt>ぷさい</rt></ruby>}{|}^2$ over the tail behind the soliton. The estimate for the physical number of particles in the radiation tail is given by $3\times {10}^3{N}_{\mathrm{r}\mathrm{a}\mathrm{d}}$. The inset shows the soliton decay rate $\mathrm{<ruby><rb>\gamma </rb><rt>がんま</rt></ruby>}$ as a function of $t$: solid line, numerical modeling; dots, theoretical results.Reuse & Permissions