Collective modes of a harmonically trapped one-dimensional Bose gas: The effects of finite particle number and nonzero temperature

Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.

Figure 1

Density profiles (a) in the mean-field regime and (b) in the Tonks-Girardeau regime, where the effective interaction parameters ${\gamma }_{{}_{\mathrm{eff}}}={10}^{-2}$ and 10 have been used, respectively. The results are calculated with $N=8$ (black solid lines), $N=17$ (red dashed lines), and $N=25$ (blue dotted lines). The density $n\left(x\right)$ and position $x$ are taken in units of harmonic oscillator length $a_{{}_{\mathrm{ho}}}=\sqrt{\hslash /\left(m{\omega }_{\mathrm{ho}}\right)}$ and $a_{\mathrm{ho}}^{-1}$, respectively.

Figure 2

Density profile: at $T=0$ predicted by the generalized Gross-Pitaevskii theory (black circles) and at $T=0$ (red dashed lines) and finite temperatures (other colored lines) calculated by the HFBP theory. The results are shown at the effective interaction parameter ${\gamma }_{{}_{\mathrm{eff}}}={10}^{-2}$ in the mean-field regime. The particle number $N$ is fixed at 25. $T_{\mathrm{c}}^0$ is the critical temperature for a 1D ideal Bose gas, which can be estimated as $k_{\mathrm{B}}{T}_{\mathrm{c}}^0=\hslash {\omega }_{\mathrm{ho}}N/ln\left(2N\right)$ [24, 35].

Figure 3

The ratio of the squared breathing-mode frequency ${\omega }_{\mathrm{m}}^2/{\omega }_{\mathrm{ho}}^2$ as a function of the particle number $N$. Our numerical calculation is shown by the black solid line, and the analytic result [Eq. (42)] is shown by the red dashed line. All results are near the noninteracting limit with an interaction strength $g_{{}_{1\mathrm{D}}}={10}^{-3}$ in the trap units.

Figure 4

The ratio of the squared breathing-mode frequency ${\omega }_{\mathrm{m}}^2/{\omega }_{\mathrm{ho}}^2$ as a function of $C_{{}_{\mathrm{N}}}\sqrt{N}$ (black squares) compared with the analytic prediction [Eq. (43)], which is shown by the red dashed line. The inset shows the same ratio as a function of the particle number $N$. Here, we take an interaction strength $g_{{}_{1\mathrm{D}}}={10}^2$.

Figure 5

The ratio of the squared breathing-mode frequency ${\omega }_{\mathrm{m}}^2/{\omega }_{\mathrm{ho}}^2$ at $N=25$ in the weakly interacting regime. We have shown the results at $T=0$ predicted by the generalized Bogoliubov theory (black squares) and by the HFBP theory (red dashed line) and the results at $T=0.8T_{\mathrm{c}}^0$ given by the HFBP theory (blue dotted line).

Figure 6

The ratio of the squared breathing-mode frequency ${\omega }_{\mathrm{m}}^2/{\omega }_{\mathrm{ho}}^2$ as a function of the effective interaction parameter ${\gamma }_{{}_{\mathrm{eff}}}$. ${\gamma }_{{}_{\mathrm{eff}}}$ covers all interaction regimes and varies from $2.2\times {10}^{-3}$ to $2.9\times {10}^2$. We consider three particle numbers: $N=8$ (black solid line), $N=17$ (red dashed line), and $N=25$ (blue dotted line). We have compared our result with a previous theoretical prediction obtained by using a time-dependent modified nonlinear Schrödinger equation (m-NLSE; yellow dot-dashed line) [11] and the experimental data (green squares with error bars) [3].

Figure 7

The ratio of the squared breathing-mode frequency ${\omega }_{\mathrm{m}}^2/{\omega }_{\mathrm{ho}}^2$ as a function of ${log}_{10}{\gamma }_{{}_{\mathrm{eff}}}$. The plot is the same as Fig. 6 but is shown here as a function of the interaction parameter in a logarithmic scale in order to emphasize the particle number dependence in the noninteracting limit.

Figure 8

The frequency of higher-order compressional modes, (a) ${\omega }_{3\mathrm{rd}}/{\omega }_{\mathrm{ho}}$ and (b) ${\omega }_{4\mathrm{th}}/{\omega }_{\mathrm{ho}}$, as a function of ${log}_{10}{\gamma }_{{}_{\mathrm{eff}}}$ at different particle numbers: $N=8$ (black solid line), $N=17$ (red dashed line), and $N=25$ (blue dotted line).