Spontaneous Soliton Formation and Modulational Instability in Bose-Einstein Condensates

The dynamics of an elongated attractive Bose-Einstein condensate in an axisymmetric harmonic trap is studied. It is shown that density fringes caused by self-interference of the condensate order parameter seed modulational instability. The latter has novel features in contradistinction to the usual homogeneous case known from nonlinear fiber optics. Several open questions in the interpretation of the recent creation of the first matter-wave bright soliton train [K. E. Strecker et al., Nature (London) 417, 150 (2002).] are addressed. It is shown that primary transverse collapse, followed by secondary collapse induced by soliton-soliton interactions, produces bursts of hot atoms at different time scales.

Figure 1

Shown is the right half of the axial density of a BEC in a harmonic potential, as determined by (a) numerical solution of the nonlinear Schrödinger equation, and (b) analytical solution of the linear Schrödinger equation (dashed line: exact; solid line: approximate), at $t=0.016\times 2\pi /{\omega }_{\rho }$, after 1D evolution from an initial box state of length $L/l_z=10$ centered at the origin. The formation of solitons first at the edge of the condensate in (a) results from the longer wavelength of the linear fringes, as seen in (b), since MI takes place at a wavelength $\sim 2\pi \xi =0.43l_z$ in this simulation, using the parameters of Strecker et al. [4, 15].

Figure 2

Shown is the phase of the order parameter corresponding to Figs. 1a, 1b. The phase difference $\Delta \varphi$ between solitons may be seen by comparing to the large density peaks of Fig. 1a: $\Delta \varphi$ is highly sensitive to the nonlinearity, and may take on arbitrary values between $0$ and $2\pi$, in contrast to the usual case of MI in fiber optics, where it is restricted to $0$. At later stages in the simulation, the relative phases drift as the solitons decouple.

Figure 3

Shown is the 1D evolution of the axial density of an elongated Bose-Einstein condensate when the interactions are suddenly tuned negative, starting from a boxlike initial state. Here the parameters of the Strecker experiment [4] are used. At early times, small fringes develop as the time slice of Fig. 1 clearly shows; once the MI is seeded, the nonlinearity focuses the fringes to produce solitons, the leading edge of which is seen at the bottom of the figure as a wide triangle shape. Lines visible above this edge are solitons. Their complicated interaction pattern clearly shows many examples of both attractive and repulsive soliton interactions.