Single-particle density matrix for a time-dependent strongly interacting one-dimensional Bose gas

We derive a $1/c$ expansion for the single-particle density matrix of a strongly interacting time-dependent one-dimensional Bose gas, described by the Lieb-Liniger model ($c$ denotes the strength of the interaction). The formalism is derived by expanding Gaudin’s Fermi-Bose mapping operator up to $1/c$ terms. We derive an efficient numerical algorithm for calculating the density matrix for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential. We have applied the formalism to study contraction dynamics of a localized wave packet upon which a parabolic phase is imprinted initially.

Figure 1

(Color online) The $x$-space density (left, red solid line), and momentum distribution (right, red solid line) of the LL gas with $N=12$ bosons and $c=50$ at different times of the propagation. At the time of 1.3 ms the cloud attains the maximal compression. The corresponding shapes for the TG gas are also shown as black dot-dashed lines. The difference between red solid and black dot-dashed lines corresponds to the $1/c$ correction. The curves are shifted on the ordinate axis for better visibility. See text for details.

Figure 2

(Color online) Time evolution of the two leading occupation numbers (first occupation numbers are shown by squares and the second by circles, lines are present as a guide to the eyes). See text for details.