Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates

We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multisoliton states emerge as a nonlinear continuation of the appropriate excited eigenstates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states are dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally, we present experimental realizations of multisoliton states including a three-soliton state consisting of two solitons oscillating around a stationary one and compare the relevant results to the predictions of the theoretical mean-field model.

Figure 1

The first and second panels show the soliton trajectories, depicted in the $z,t$ plane (here, $z$ and $t$ are respectively measured in units of the healing length $\xi =\sqrt{\hslash /(m\mu )}$ and $\hslash /\mu$) for symmetric two-soliton collisions for different initial velocities: $\nu =0.2$ (first panel) and $\nu =0.8$ (second panel). In the former (latter) case the solitons are reflected by (transmitted through) each other. The third and fourth panels show, respectively, the soliton trajectories for an asymmetric two-soliton collision for initial velocities, ${\nu }_1=0.5$ and ${\nu }_2=0$, and for a three-soliton collision for initial velocities ${\nu }_1=-{\nu }_3=0.4$ and ${\nu }_2=0$. In all panels, density plots of the wavefunction carrying the solitons as obtained by direct numerical integration of the homogeneous NLS equation are shown. The solid lines correspond to the solution of Eq. (14) (i.e., employing the effective interaction potential). In all cases the normalized density is $n_0=1$.

Figure 2

Collision-induced phase shift of the solitons, in units of $\xi$, as a function of the velocity $\nu$, in units of the speed of sound, $c_s$. Solid lines depict the analytical result of Eqs. (15) and (16) and the dashed lines show the results obtained from Eq. (14), which employs the effective repulsive potential. The agreement between the two approaches is very good for well-separated solitons, i.e., for $\nu ⩽1/2$. The left panel shows the case of a symmetric collision, and the right one the case of an asymmetric collision of a moving soliton with $\nu ={\nu }_1$ and a stationary soliton (i.e., a black soliton with ${\nu }_2=0$; see the bottom left panel of Fig. 1). The dashed vertical line depicts the critical value $\nu =1/2$ and defines the range of applicability of the approach based on the effective repulsive potential.

Figure 3

(Left) The soliton oscillation frequency, in units of ${\nu }_z$, as a function of the oscillation amplitude, in units of $\xi$, for a configuration of one (light gray curve), two (gray curves), and three dark solitons (black curves) (see right panel). Shown are the results obtained by the ordinary differential equations (ODEs), i.e., the equations of motion (14) with the potential in Eq. (17) for the two- and three-soliton states (solid lines), as well as results obtained by direct numerical integration of the partial differential equation (PDE) Eq. (2) (dashed lines). The normalized trap strength is $\Omega \approx 0.06$. (Right) Contour plot depicting the evolution of the density, according to Eq. (2), for the three-dark-soliton configuration, consisting of one stationary central soliton and two oscillating ones with a frequency ${\omega }_{\mathrm{osc}}\approx 0.85\Omega$. The thin solid (green) lines show the path of the density minima of all three solitons calculated by the equation of motion employing the effective potential of Eq. (17) (solid green line). The parameter values are $N\approx 1700$, ${\omega }_z=2\pi \times 53$ Hz, ${\omega }_{\perp }=2\pi \times =890$ Hz (i.e., $\Omega \approx 0.06$) and initial displacement from the trap center $\delta z_0=10\xi$.

Figure 4

(Top) The real part ${\omega }_r$ of the lowest eigenfrequencies of the second nonlinear mode as a function of the number of atoms $N$ (for a harmonic trap with $\Omega =0.1$). Eigenfrequencies possessing negative Krein signature, namely eigenfrequencies of the two anomalous modes, are depicted by solid lines. Eigenfrequencies possessing positive Krein signature are depicted by dashed lines, with the lowest ones starting from ${\omega }_r=0.1$ and ${\omega }_r=0.2$ at $N=0$ corresponding, respectively, to the dipole and quadrupole modes. (Bottom) The imaginary part ${\omega }_i$ of the eigenfrequency pair starting from ${\omega }_r=0.2$ at $N=0$ as a function of the number of atoms $N$. The vertical dotted line at the critical value $N=438$ indicates the splitting of the complex pair into real eigenfrequencies.

Figure 5

Spatiotemporal evolution of the condensate density, in dimensionless units, after the excitation of the two anomalous modes. [(a) and (b)] The in-phase and out-of-phase oscillatory motion of the two dark solitons (for $\Omega =0.1$ and $N=1000$); the oscillation frequencies are identical to the eigenfrequencies of the first and second anomalous modes, respectively. (c) The manifestation of the dynamical instability (for $\Omega =0.1$ and $N\approx 400$) due to the collision of the second anomalous mode with the quadrupole mode: the motion of the solitons excites the quadrupole mode of the system, resulting in a breathing behavior of the condensate.

Figure 6

(Top) The real part of the lowest eigenfrequencies of the third excited mode (i.e., three-dark-soliton state) as a function of the number of atoms $N$ (for a harmonic trap with $\Omega =0.1$). The notation follows the one used in Fig. 4. The bold circle in the top panel indicates the collision of eigenfrequencies of opposite Krein signatures which does not lead to instability (see details in the text). (Bottom) The imaginary parts of the the eigenfrequency pairs starting from ${\omega }_r=0.2$ (blue) and ${\omega }_r=0.3$ (green) at $N=0$ as a function of the number of atoms $N$. The vertical dotted lines indicate the splitting of the respective complex pairs into real ones. After the second splitting, all the eigenfrequencies become real and the three-soliton state is stabilized.

Figure 7

The maximum of the imaginary part of the eigenfrequencies of the third excited state for a fixed number of atoms ($N=1000$) as a function of the trap strength $\Omega$. The insets show the spectral plane for a stable ($\Omega =0.05$) and an unstable ($\Omega =0.18$) case. In the latter case, the complex eigenfrequencies originate from the collision of the second anomalous mode with the quadrupole mode.

Figure 8

Spatiotemporal evolution of the condensate density, in dimensionless units, after the excitation of the three anomalous modes. [(a)–(c)] Stable configurations with $\Omega =0.05$, $N=1000$ are shown: in (a) excitation of the first anomalous mode results in a configuration where the three dark solitons move together in-phase; in (b) excitation of the second anomalous mode results in a configuration where the outer solitons move in opposite directions (each being the mirror image of the other around $z=0$) while the middle one is at rest; in (c) excitation of the third anomalous mode results in a configuration where the center soliton is moving in opposite direction to the outer dark solitons. (d) The excitation of the second anomalous mode associated with a complex eigenfrequency for the unstable case with $\Omega =0.18$ and $N=1000$. Here, a dynamical instability manifests itself due to the collision of the second anomalous mode with the quadrupole mode: the motion of the dark solitons excites the quadrupole mode of the system, resulting in a breathing behavior of the entire condensate.

Figure 9

Increasing the distance between the created solitons can be realized by increasing the ramping-down time of the optical lattice ${\tau }_{\mathit{OL}}$, as shown by numerical simulations of Eq. (2): (a) ${\tau }_{\mathit{OL}}=0$ ms, (b) ${\tau }_{\mathit{OL}}=2$ ms, (c) ${\tau }_{\mathit{OL}}=4$ ms.

Figure 10

(a) Observation of the oscillation of four dark solitons including the creation process. The evolution is averaged over 10 experimental runs. The point in time where the creation process of the solitons is finished is marked by the dashed line. (b) Experimental observation of three dark solitons in a harmonic trap averaged over 16 runs. The creation process of the solitons is not shown in this case. The soliton in the center of the trap is at rest (black soliton) whereas the two outer ones oscillate. The time evolution plots were obtained by integrating the images over their transverse axis, meaning that each vertical line shows the longitudinal density of the BEC at a certain point in time.

Figure 11

Single shots of the longitudinal density of the BEC in arb. units. Increasing the ramping-down time of the intensity of the optical lattice from ${\tau }_{\mathit{OL}}=2$ ms to ${\tau }_{\mathit{OL}}=5$ ms offers the possibility of creating a single soliton. (Left) Single shot of the three soliton state shown in Fig. 10. Note that the stationary soliton in the middle appears with finite density on its minimum in the experimental images. This is due to the limited optical resolution. (Right) Single shot of a single-dark-soliton state.

Figure 12

The dependence of the soliton oscillation frequency, in units of ${\nu }_z$, on the oscillation amplitude, in units of $\xi$, for the two- and three-dark-soliton states. The gray shaded area depicts the range expected by numerical simulations for the two-soliton experiments conducted for $\Omega \approx 0.06$ (circles), $\Omega \approx 0.09$ (diamonds), and $\Omega \approx 0.14$ (squares). There exists a single measurement for the three-soliton state (corresponding to $\Omega \approx 0.09$), which is compared to the theoretical prediction (dashed line).