Interaction-induced crossover versus finite-size condensation in a weakly interacting trapped one-dimensional Bose gas

We discuss the transition from a fully decoherent to a (quasi)condensate regime in a harmonically trapped weakly interacting one-dimensional (1D) Bose gas. By using analytic approaches and verifying them against exact numerical solutions, we find a characteristic crossover temperature and crossover atom number that depend on the interaction strength and the trap frequency. We then identify the conditions for observing either an interaction-induced crossover scenario or else a finite-size Bose-Einstein condensation phenomenon characteristic of an ideal trapped 1D gas.

Figure 1

Equation of state of the uniform weakly interacting 1D Bose gas for three different values of the temperature parameter $t=2T{\hslash }^2∕m{g}^2$. The exact numerical result (solid line) is compared with the behavior in the quasicondensate regime (dash-dotted lines) and with the ideal Bose gas result of Eq. (1) (dashed lines). The straight dotted lines correspond to the classical (Boltzmann) ideal gas.

Figure 2

Density profiles of a 1D Bose gas in a harmonic trapping potential for five different values of the ratio ${\mu }_0∕T$ and a fixed value of the temperature parameter $t=2{\hslash }^{2}T∕m{g}^2={10}^5$. The exact numerical solution (solid line) is compared with the ideal Bose gas distribution (dashed line), classical Boltzmann distribution (dotted line), and Thomas-Fermi distribution in the Gross-Pitaevskii regime (dash-dotted line). The resulting values of the dimensionless ratio $N\hslash \omega ∕T$, following the exact solutions, are also shown. The distance from the trap center $z$ is in units of $R_T={(2T∕m{\omega }^2)}^{1∕2}$. All calculations are done within the LDA using the equation of state for the homogeneous gas shown in Fig. 1, with ${\mu }_0$ and $n\left(0\right)$ in (b)–(e) being the same as $\mu$ and $n$ indicated by the points (b)–(e) in Fig. 1.

Figure 3

Peak density $n_0$ (in units of $mg∕{\hslash }^2$) of a trapped gas versus $N\hslash \omega ∕T$ for three values of $t=2{\hslash }^{2}T∕m{g}^2$. The three black dots show the respective crossover values of $N_{\mathrm{co}}\hslash \omega ∕T$ from Eq. (6). The different lines are as in Fig. 1.