Fate of a gray soliton in a quenched Bose-Einstein condensate

We investigate the destiny of a gray soliton in a repulsive one-dimensional Bose-Einstein condensate undergoing a sudden quench of the nonlinearity parameter. The outcome of the quench is found to depend dramatically on the ratio $\eta$ of the final and initial values of the speed of sound. For integer $\eta$ the soliton splits into exactly $2\eta -1$ solitons. For noninteger $\eta$ the soliton decays into multiple solitons and Bogoliubov modes. The case of integer $\eta$ is analyzed in detail. The parameters of solitons in the out state are found explicitly. Our approach exploits the inverse scattering method and can be easily used for similar quenches in any classical integrable system.

Figure 1

One-soliton solution. The upper panel shows the density distribution ${\left|\Psi \left(x\right)\right|}^2$, and the lower panel shows the phase shift $\mathrm{Arg}\Psi \left(x\right)$ across the soliton. The special case of $\theta =\pi$, called the dark soliton, is shown in the inset.

Figure 2

Dynamics of the one-soliton density profile after the quench $c_s\to 2c_s$. The initial distribution is shown by a dashed line. The evolution after the quench is shown by a solid line. Time is measured in units of correlation time $\tilde{\tau }=\tilde{\xi }/{\tilde{c}}_s$ in the quenched quasicondensate. The top panel shows the splitting of the dark soliton. This splitting results in a symmetric density distribution consisting of one dark soliton and two gray solitons moving with equal speeds in opposite directions. The bottom panel shows how a gray soliton splits into three solitons with different nonzero velocities. The evolution is calculated analytically from Eqs. (18, 19, 20) with the phases given by (8).