Propagation of dark soliton from superfluid core to Mott-insulator shell and superfluid shell

We study dynamics of phase imprinted solitons of ultracold bosons in an optical lattice with a harmonic trap, which shows the superfluid (SF) and Mott-insulator (MI) shell structures. In the uniform Bose-Hubbard system, he soliton generated by the phase imprinting method is known to propagate in three different ways in the uniform system [Konstantin V. Krutitsky, Phys. Rev. A 82, 033618 (2010).], such as the in-phase soliton, out-of-phase soliton, and wavelet. In this presentation, we discuss the propagation of phase imprinting soliton from the inner superfluid core to Mott-insulator shell and outer superfluid shell. All the three types of solitonic excitations are deformed into an in-phase soliton before the excitation irrupts into the Mott shell, because of the spatial dependence of the effective chemical potential. Interestingly, in the in-phase soliton and out-of-phase soliton regimes, the excitation does not penetrate into the outer SF shell, and is reflected by its surface. By inducing a repulsive potential at the center of the trap, however, we can inject the wavelet to the MI shell, and permeate the excitation into the outer SF shell.