Fermi-Bose Transformation for the Time-Dependent Lieb-Liniger Gas

Exact solutions of the Schrödinger equation describing a freely expanding Lieb-Liniger gas of delta-interacting bosons in one spatial dimension are constructed. We demonstrate that for any interaction strength the system enters a strongly correlated regime during such expansion. The asymptotic form of the wave function is shown to have the form characteristic for “impenetrable-core” bosons. Exact solutions are obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function that obeys the Schrödinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator that depends on the strength of the interaction and the number of particles.

Figure 1

The correlation properties of the initial state and the evolution of the SP density for various interaction strengths, for $N=3$ particles. (Left column) The probability density $|{\psi }_{B,c}(0,x_2,x_3,0){|}^2$ for $c=1$, 3, 10. (Right column) The SP density at $t=0$ (black solid line), $t=0.5$ (red dotted line), and $t=1$ (blue dot-dashed line), for $c=5$, 10, 50.

Figure 2

The evolution of correlation properties during free expansion. (a)–(c) Contours of the probability density $|{\psi }_{B,c}(0,x_2,x_3,t){|}^2$ for $c=0.1$, at times $t=0,0.8$, and 2; the labels are ${\eta }_i=x_i/b(t)$. (d) The asymptotic form of the SP density ${\rho }_{\infty ,c}(\eta )$ for $c=0.1$, 3, and 10.