Quantum dark solitons in the one-dimensional Bose gas

Dark and gray soliton-like states are shown to emerge from numerically constructed superpositions of translationally invariant eigenstates of the interacting Bose gas in a toroidal trap. The exact quantum many-body dynamics reveals a density depression with ballistic spreading that is absent in classical solitons. A simple theory based on finite-size bound states of holes with quantum-mechanical center-of-mass motion quantitatively explains the time-evolution and predicts quantum effects that could be observed in ultracold gas experiments. The soliton phase step is found relevant for explaining finite-size effects in numerical simulations. An invariant fundamental soliton width is shown to deviate from the Gross-Pitaevskii predictions in the interacting regime and vanishes in the Tonks-Girardeau limit.

Figure 1

Time evolution of the quantum dark soliton (3) constructed as a superposition of yrast eigenstates of the Lieb-Liniger model with $N=100$ particles at the intermediate interaction strength $\gamma =1$, where $\gamma =c/n_0$ and $n_0=N/L$. The superposition is prepared with $\mathrm{\Delta }P=0.11\pi \hslash n_0$ and $P_0=0.64\pi \hslash n_0$. The solid line tracks the minimum of the dip and the dashed lines on either side of it are displaced by half of the soliton's width, i.e., by $\pm \mathrm{\Delta }X/2$ [see Eq. (6)].

Figure 2

Width $\mathrm{\Delta }X$ of the density depression from simulations (symbols) compared with fits of Eq. (7) (lines). (a) Ballistic growth of $\mathrm{\Delta }X$ in time for $\gamma =1$ with ${\sigma }_{\mathrm{fs}}$ and $M$ fitted. (b) Initial variance $\mathrm{\Delta }{X}^2$ at $t=0$ vs $\mathrm{\Delta }{P}^{-2}$ used for extracting ${{\sigma }_{\mathrm{fs}}}^2$ as the intercept. Interaction strengths $\gamma =0.1,1,10$ are shown by red dashed, blue continuous, and black dotted lines, respectively. Panels (c), (d), and (e) show linear fits of $\mathrm{\Delta }{X}^2$ vs $t^2$ used to extract $M$ for $\gamma =0.1,1,10$, respectively, with $\mathrm{\Delta }P=0.045\pi \hslash n_0$. All panels used $P_0=\pi \hslash n_0$ and $N=100$. The expected quadratic dependence of the variance on $1/\mathrm{\Delta }P$ and $t$ is evident in all parameter regimes.

Figure 3

(a) Dispersion relation of yrast states of the Lieb-Liniger model. Symbols show the excitation energies $E_P^N-E_0^N$ for $N=100$ vs momentum. Thick lines show the approximate formulas for the finite system (10) and thin lines show the dispersion relations $E_{\mathrm{s}}^{\infty }\left(P\right)$ in the thermodynamic limit [34] for comparison. The interaction strengths are $\gamma =0.1$ (dashed red line and red circles), $\gamma =1$ (full blue line & blue circles), and $\gamma =10$ (dash-dotted black line and black circles); the same color code is used in the bottom panels. Bottom panels: particle number depletion (c) and phase step (d) with finite-size corrections for $N=100$.

Figure 4

Length scales of the quantum dark soliton. (a) Fundamental soliton width ${\sigma }_{\mathrm{fs}}$ and minimum center-of-mass wave-packet width ${\sigma }_{0,\min }\approx 0.8n_0^{-1}$ vs coupling strength $\gamma =c/n_0$ for $P_0=\pi \hslash n_0$. Limiting analytical approximations: ${\sigma }_{\mathrm{fs}}/\xi \to \pi /\sqrt{6}$ from GP theory for $\gamma \ll 1$ and Eq. (12) for $\gamma \gg 1$ (black dashed lines). Magenta dashed lines: ${\sigma }_{0,\min }/\xi$ with $\xi \sim 1/\left(n_0\sqrt{2\gamma }\right)$ for $\gamma \ll 1$ and $\xi \sim n_0^{-1}{\pi }^{-1}(1+8/3\gamma )$ for $\gamma \gg 1$. (b) Numerical data for ${{\sigma }_{\mathrm{fs}}}^2$ (multiplied by $c^2{N}_{\mathrm{d}}^2$) vs the particle number depletion $N_{\mathrm{d}}$ from Eq. (11) (with finite-size corrections). Data from different momenta and coupling strengths collapse onto the same curve and deviate from the result ${\sigma }_{\mathrm{GP}}^2$ (dashed line) for the classical dark soliton only near the Tonks-Girardeau limit where $N_{\mathrm{d}}\to -1$. The width ${\sigma }_{\mathrm{fs}}$ was extracted from simulation data with $N=100$.