Monopole Excitations of a Harmonically Trapped One-Dimensional Bose Gas from the Ideal Gas to the Tonks-Girardeau Regime

Using a time-dependent modified nonlinear Schrödinger equation (MNLSE)—where the conventional chemical potential proportional to the density is replaced by the one inferred from Lieb-Liniger’s exact solution—we study frequencies of the collective monopole excitations of a one-dimensional Bose gas. We find that our method accurately reproduces the results of a recent experimental study [E. Haller et al., Science 325, 1224 (2009)] in the full spectrum of interaction regimes from the ideal gas, through the mean-field regime, through the mean-field Thomas-Fermi regime, all the way to the Tonks-Giradeau gas. While the former two are accessible by the standard time-dependent NLSE and inaccessible by the time-dependent local density approximation, the situation reverses in the latter case. However, the MNLSE is shown to treat all these regimes within a single numerical method.

Figure 1

Top left panel: plot of the steady state density $n_{ss}(z)=|\psi (z){|}^2$ vs $z$ with atom number $N=25$ for ${\gamma }_{\mathrm{eff}}\approx 0.01$ (dotted line), 1 (dashed line), and ${\gamma }_{\mathrm{eff}}\to \infty$ (solid line). Top right panel: the corresponding chemical potential $\mu (z)$. It is noted that $\mu (z)$ for ${\gamma }_{\mathrm{eff}}\approx 0.01$ (dotted line) is vanishingly small. Bottom left panel: plot of the harmonic oscillator energy $E_{\mathrm{HO}}$ (solid line), interaction energy $E_I$ (dashed line), and the total energy $E_K+E_I$ (dotted line) as defined in the text as a function of ${\log }_{10}{\gamma }_{\mathrm{eff}}$. Bottom right panel: the initial width of the wave function $⟨\mathrm{\Delta }{z}^2⟩$ (dotted line), minimum $⟨\mathrm{\Delta }{z}^2⟩$ (dashed line), and maximum $⟨\mathrm{\Delta }{z}^2⟩$ (solid line) attained during the monopole oscillation as a function of ${\log }_{10}{\gamma }_{\mathrm{eff}}$. The harmonic oscillator units are used throughout.

Figure 2

Plot of the monopole oscillation frequency squared $({\omega }_C/{\omega }_D{)}^2$ as a function of ${\gamma }_{\mathrm{eff}}=2\pi /\sqrt{2Nm{\omega }_z/\hslash }|a_{1\mathrm{D}}|$. Inset: same figure as a function of ${\log }_{10}{\gamma }_{\mathrm{eff}}$ to magnify the region near ${\gamma }_{\mathrm{eff}}\approx 0$—the region that lies outside of the range of validity of hydrodynamic equations. Stars with error bars: the experimental data from the Nägerl group [2]. Green squares: numerical simulation, red crosses: sum rules result [11]. It must be mentioned that in the region of ${\gamma }_{\mathrm{eff}}\ll 1$, the result [11] is not expected to match the experiment since it has been built on the hydrodynamic equations a priori. The difference may be understood from the fact that the effect of the quantum pressure term added in deriving the MNLSE becomes more significant in the regime of weak interactions.