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Collective modes

Another possible experimental signature of the super-Tonks regime can be provided by the study of collective modes. To this aim, we calculate the frequency of the lowest compressional mode of a system of $N$ particles in a harmonic potential $V_{ext}=\sum_{i=1}^N m\omega_z^2z_i^2/2$. We make use of LDA (Sec. 1.6) which allows us to calculate the chemical potential of the inhomogeneous system $\tilde{\mu}$ and the density profile $n(z)$ from the local equilibrium equation $\tilde{\mu}=\mu[n(z)]+m\omega_z^2z^2/2$, and the normalization condition $N=\int_{-R}^R n(z) dz$, where $R=\sqrt{2\tilde{\mu}/(m\omega_z^2)}$ is the size of cloud. For densities $n$ smaller than the critical density for cluster formation, $\mu[n]$ is the equation of state of the homogeneous system derived from the fit to the VMC energies (Fig. 6.1). From the knowledge of the density profile $n(z)$ one can obtain the mean square radius of the cloud $\langle
z^2\rangle=\int_{R}^R n(z)z^2 dz /N$ and thus, making use of the result [MS02]

\begin{displaymath}
\omega^2=-2\frac{\langle z^2\rangle}{d\langle z^2\rangle/d\omega_z^2} \;,
\end{displaymath} (6.3)

one can calculate the frequency $\omega$ of the lowest breathing mode. Within LDA, the result will depend only on the dimensionless parameter $Na_{1D}^2/a_z^2$, where $a_z=\sqrt{\hbar/m\omega_z}$ is the harmonic oscillator length. For $g_{1D}>0$, i.e. in the case of the LL Hamiltonian, the frequency of the lowest compressional mode increases from $\omega=\sqrt{3}\omega_z$ in the weak-coupling mean-field regime ( $Na_{1D}^2/a_z^2\gg 1$) to $\omega=2\omega_z$ in the strong-coupling TG regime ( $Na_{1D}^2/a_z^2\ll 1$). The results for $\omega$ in the super-Tonks regime are shown in Fig. 6.4 as a function of the coupling strength. In the regime $Na_{1D}^2/a_z^2\ll 1$, where the HR model is appropriate, we can calculate analytically the first correction to the frequency of a TG gas (refer to Table 1.1). One finds the result $\omega=2\omega_z [1+(16\sqrt{2}/15\pi^2)(N
a_{1D}^2/a_z^2)^{1/2}+...]$. Fig. 6.4 shows that this expansion accurately describes the frequency of the breathing mode when $Na_{1D}^2/a_z^2\ll 1$, for larger values of the coupling strength the frequency reaches a maximum and drops to zero at $Na_{1D}^2/a_z^2\simeq 0.6$. The observation of a breathing mode with a frequency larger than $2\omega_z$ would be a clear signature of the super-Tonks regime.

Figure 6.4: Square of the lowest breathing mode frequency, $\omega ^2$, as a function of the coupling strength $Na_{1D}^2/a_z^2$ for the LL Hamiltonian ($g_{1D}>0$) and in the super-Tonks regime ($g_{1D}<0$). The dashed line is obtained from the HR expansion (see Table 1.1).
\includegraphics[width=0.6\textwidth]{STfreq.eps}


next up previous contents
Next: Conclusions Up: Beyond Tonks-Girardeau: super-Tonks gas Previous: One-body density matrix and   Contents
G.E. Astrakharchik 15-th of December 2004