Formation and Dynamics of Many-Boson Fragmented States in One-Dimensional Attractive Ultracold Gases

The dynamics of attractive ultracold bosonic clouds in one dimension is studied by solving the many-particle time-dependent Schrödinger equation. The initially coherent wave packet can dynamically dissociate into two parts when its energy exceeds a threshold value. Noticeably, the time-dependent Gross-Pitaevskii theory does not show up the splitting. We call the split object fragmenton. It possesses remarkable properties; in particular, it is macroscopically fragmented. A simple static model predicts the existence of fragmented states responsible for the formation and dynamics of fragmentons.

Figure 1

Many-body (left) vs GP (right) dynamics for $N=1000$ attractive (${\lambda }_0=-0.008$) bosons in 1D. The densities are plotted as a function of time for initially-coherent wave packets $\mathrm{sech}[\gamma x]$ of different widths. While the GP dynamics shows breathing oscillations for all initial widths, the many-body dynamics changes dramatically from breathing oscillations (${\gamma }_{\mathrm{I}}=3.0$), to attempts for dynamical splitting (${\gamma }_{\mathrm{II}}=1.2$), and to dynamical dissociation into two parts (${\gamma }_{\mathrm{III}}=1.0$). All quantities are dimensionless.

Figure 2

Evolutions of natural occupation numbers $n_1/N$ and $n_2/N$ of the reduced one-body densities plotted in the left panels of Fig. 1 on a log scale in %. At $t=0$ all initial states are condensed: $n_1/N=100\%$. In the first study (${\gamma }_{\mathrm{I}}=3.0$) the dynamics describe breathing oscillations only. In the second (${\gamma }_{\mathrm{II}}=1.2$) and third (${\gamma }_{\mathrm{III}}=1.0$) studies the systems evolve to be twofold fragmented after a few breathing oscillations. Notice that in GP theory $n_1/N$ is always 100%. All quantities are dimensionless.

Figure 3

Energy diagram $E(n_1)$ of the model system introduced in Eqs. (1, 2) for twofold fragmented $|n_1,n_2⟩$ states $n_1+n_2=N$. There are two branches of solutions bifurcating at $n_1=N/2$ (bifurcation energy $E_{\mathrm{BF}}$). Shown are also the orbitals $\sqrt{n_i/N}{\varphi }_i$ corresponding to $n_1=3N/4$ at both branches and to the GP solutions of $n_1=N$ and $n_2=N$ at energies $E_{\mathrm{GP}}^{\mathrm{GS}}$ and $E_{\mathrm{GP}}^{\mathrm{ES}}$. $E_{\mathrm{I}}$, $E_{\mathrm{II}}$, $E_{\mathrm{III}}$ are the energies of the three cases studied in Fig. 1. To activate a fragmenton the energy of the initial cloud must exceed $E_{\mathrm{BF}}$. All quantities are dimensionless.