Many-body processes in black and gray matter-wave solitons

We perform a comparative beyond-mean-field study of black and gray solitonic excitations in a finite ensemble of ultracold bosons confined to a one-dimensional box. An optimized density-engineering potential is developed and employed together with phase imprinting to cleanly initialize gray solitons. By means of ab initio simulations with the multiconfiguration time-dependent Hartree method for bosons, we demonstrate that quantum fluctuations limit the lifetime of the soliton contrast, which increases with increasing soliton velocity. A natural orbital analysis reveals a two-stage process underlying the decay of the soliton contrast. The broken parity symmetry of gray solitons results in a local asymmetry of the orbital mainly responsible for the decay, which leads to a characteristic asymmetry of remarkably localized two-body correlations. The emergence and decay of these correlations as well as their displacement from the instantaneous soliton position are analyzed in detail. Finally, the role of phase imprinting for the many-body dynamics is illuminated and additional nonlocal correlations in pairs of counterpropagating gray solitons are observed.

Figure 1

Time evolution of the reduced one-body density ${\rho }_1(x;t)$ for $N=100$ bosons in a box of length $L=20\xi$ and a Lieb-Liniger parameter $\gamma =0.04$. The initial state resembles the properties of (a) a black $\left(\beta =0.0\right)$ and (b) a gray $\left(\beta =0.5\right)$ soliton. The calculations are performed with $M=4$ optimized SPFs. Insets: $M=1$ mean-field simulations.

Figure 2

Time evolution of the relative contrast $c\left(t\right)/c\left(0\right)$ for dark solitons of various soliton velocity to speed-of-sound ratios $\beta =u/s$. Inset: Contrast lifetime ${\tau }_c$ vs $\beta$. All other parameters as in Fig. 1.

Figure 3

(a) Time evolution of the quantum depletion $d\left(t\right)$ for various $\beta$. (b) Evolution of the natural populations ${\lambda }_i$ for $\beta =0.0$ (solid black lines) and $\beta =0.5$ (dash-dotted red lines). The blue dashed lines refer to the natural populations for a $\beta =0.0$ density- but not phase-engineered initial state. Inset: Close-up of the two least dominant natural populations. All other parameters as in Fig. 1.

Figure 4

Comparison of the density evolution of the most and second dominant natural orbital for (a),(b) a black soliton and (c),(d) a gray soliton with $\beta =0.5$. The instantaneous soliton positions $x_t^s$ defined as the minimum of ${\rho }_1(x;t)$ are depicted as white dashed lines. Insets: Phase profile of respective natural orbital. All other parameters as in Fig. 1.

Figure 5

Two-body correlation function $g_2(x_1,x_2;t)$ for a black soliton (first row) and a gray soliton with $\beta =0.5$ (second row) at times $t=0.0$ (first column), $t=2.5\tau$ (second column), and $t=5\tau$ (third column). Dashed lines refer to the instantaneous soliton position $x_t^s$ and the arrows indicate the direction of the soliton movement. All other parameters as in Fig. 1.

Figure 6

Analysis of the bunching center ${\overline{x}}_t$ for various $\beta$: (a) Displacement of ${\overline{x}}_t$ with respect to instantaneous soliton position $x_t^s$. The dashed lines indicate the trajectories of a fictitious sonic excitation emitted at $t=0$ from the soliton position in the negative $x$ direction for the various $\beta$, i.e., $x\left(t\right)-x_t^s=-\sqrt{g{\rho }_{\mathrm{bulk}}}t-x_t^s$, where the bulk density ${\rho }_{\mathrm{bulk}}$ is evaluated directly after the density engineering at $t=0$. (b) The value of the two-body correlation function $g_2({\overline{x}}_t,{\overline{x}}_t;t)$ at the bunching center. The bunching distribution (7) is calculated for $R=4\xi$. All other parameters as in Fig. 1.

Figure 7

Evolution of a density-engineered initial state (no phase imprinting is applied): (a) Reduced one-body density ${\rho }_1(x;t)$, (b) two-body correlation function $g_2(x_1,x_2)$ at time $t=5\tau$, as well as density of the (c) dominant and (d) second dominant natural orbital, with corresponding phase profiles as insets. The corresponding natural populations are depicted in Fig. 3. Dashed lines refer to the instantaneous soliton position $x_t^s$ obtained by linear regression of the position of the local ${\rho }_1(x;t)$ minima and the arrows indicate the direction of the soliton movement. All other parameters as in Fig. 1.