Collective excitation of Bose-Einstein condensate of Bose atoms with Pöschl-Teller interaction

Abstract

We investigate the collective excitations of Bose atoms in the condensate phase with finite-range interactions, modeled using the Pöschl-Teller (PT) potential to explicitly account for the finite interaction range. Utilizing Bogoliubov theory, we derive the excitation spectrum and examine its dependence on interaction parameters. Our analysis reveals that the emergence and disappearance of the roton minima are directly influenced by the range of the PT interaction. Furthermore, we calculate the static structure factor and find enhanced correlations in the low-momentum regime, with their nature controlled by the characteristics of the PT potential. We have examined the interaction dependence of sound velocity and compressibility in detail.

Figures (6)

• Figure 1: Excitation spectra for various values of the parameter $\alpha$ of PT interaction potential. Interaction is a long-range type for smaller values of $\alpha$, so a roton minima appears. Whereas for larger values of $\alpha$, the interaction becomes a delta function type; that's why the roton minima disappear. When $U$ is smaller, the sharpness of minima is less as contact interaction dominates. Upper panel: $U=50$, [a] and [b] for different values of $\alpha$, and Lower panel: $U=5$, [c] and [d] for different values of $\alpha$. $\alpha$ is expressed in units of $\xi^{-1}$ and $U$ is expressed in units of $\rho_c g \, \xi$.
• Figure 2: Excitation spectra for various values of the interaction strength $U$ of the PT interaction potential. For $U=0$, there is only contact interaction between atoms, so there is only a phononic mode of spectra. For larger values of $U$, the strength and range of the PT interaction potential increase. As a result, roton minima start to appear in the spectra. The minima become sharper for larger values of $U$. For lower $U$, the sharpness of the roton minima is less when other parameters are the same. $\alpha$ is expressed in units of $\xi^{-1}$ and $U$ is expressed in units of $\rho_c g \, \xi$.
• Figure 3: [a] $\frac{1}{\rho_c g}\frac{dE_k}{d( k\xi)}$ (slope of the excitation spectrum) for different values of $U$ and $\alpha=1$. It is studied that the gap at $k\to0$ increases with increasing $U$, which means the sound velocity increases with $U$. [b] $\frac{1}{\rho_c g}\frac{dE_k}{d( k\xi)}$ for different values of $\alpha$ and $U=50$. Here, the gap remains unchanged for different values of $\alpha$, which means the sound velocity $(v_s)$ is independent of $\alpha$. A negative slope suggests a roton mode in the excitation spectrum.
• Figure 4: variation of roton minima energy ($\Delta$ denotes the roton minima energy, defined as the lowest value of the excitation energy at the roton momentum region) for different values of $U$ and $\alpha$ is shown in figures (a) and (b), respectively. The roton mode appears for $U > 10$ when $\alpha= 1$, and the roton minima energy increases with $U$. The roton minima energy also increases with $\alpha$, and the roton mode disappears for $\alpha> 2.6$ when $U = 50$. Figure (c) shows that sound velocity $(v_s)$ increases with $U$. The sound velocity does not depend upon $\alpha$. $\alpha$ is expressed in units of $\xi^{-1}$ and $U$ is expressed in units of $\rho_c g \, \xi$.
• Figure 5: Phase diagram for the appearance of roton excitations in the $(U, \alpha)$ parameter space. Here $\alpha$ is expressed in units of $\xi^{-1}$ and $U$ is expressed in units of $\rho_c g \, \xi$. The solid curve marks the boundary of roton onset: above the curve the excitation spectrum exhibits a roton minimum (Roton present), while below the curve the spectrum remains phonon-like without rotons (Roton absent). The boundary was obtained from the condition that the slope (figure \ref{['du']}) of the excitation spectrum at finite momentum touches zero with no negative slope present, which signals the onset of roton formation.