Roton excitations in a trapped dipolar Bose-Einstein condensate

We consider the quasiparticle excitations of a trapped dipolar Bose-Einstein condensate. By mapping these excitations onto linear and angular momentum we show that the roton modes are clearly revealed as discrete fingers in parameter space, whereas the other modes form a smooth surface. We examine the properties of the roton modes and characterize how they change with the dipole interaction strength. We demonstrate how the application of a perturbing potential can be used to engineer angular rotons, i.e., allowing us to controllably select modes of nonzero projection of angular momentum to become the lowest energy rotons.

Figure 1

Roton fingers in the spectrum of a trapped dipolar BEC. (a), (b) Two views of the quasiparticle excitations of a trapped dipolar condensate mapped against their angular momentum projection $m$ and effective linear momentum ${\langle k_{\rho }\rangle }_j$ [see Eq. (5)]. The individual mapped excitations are represented by dots and separated into two categories: (i) a smooth nonrotonic background part indicated by (blue) dots joined by lines to form a surface; (ii) the roton fingers indicated by (red) dots that extend as linear chains below the nonrotonic background. (c), (d) Two views of the roton fingers for the same case shown in (a), (b). Parameters: $\lambda =20$ and $D=220$.

Figure 2

Roton mode functions in position and momentum space. The $u$ quasiparticle amplitudes for the roton modes are shown for the [(a), (b)] $n=0$ and [(c), (d)] $n=1$ fingers [as identified in Fig. 1d] as a function of angular momentum projection. The rotons are shown in (a), (c) position space along the $\rho$ axis, and in (b), (d) momentum space along the $k_{\rho }$ axis. The scaled condensate amplitude in position space is shown for reference in (a), (c). Note that the quasiparticle amplitudes are of the form $u\left(\mathbf{x}\right)=u(\rho ,z)e^{im\varphi }$, etc., and we plot these along the $x$ axis, i.e., taking $\varphi =0$. Parameters: $\lambda =20$ and $D=220$.

Figure 3

Comparison of quasiparticle modes. (a) The spectrum of excitations mapped to a dispersion for the case in Fig. 1, only showing modes with angular momentum projection $\left|m\right|\le 2$. The $m=1$ phonon, $m=0$ roton, and a nonrotonic background $m=0$ mode (of similar ${\langle k_{\rho }\rangle }_j$ to the roton mode) are identified with arrows. These modes are analyzed below in subplots (b1) and (b2), (c1) and (c2), and (d1) and (d2), respectively. (b1), (c1), (d1) Quasiparticle $u$ (solid blue line) and $v$ (dashed red line) amplitudes, and the scaled condensate amplitudes (light gray line). (b2), (c2), (d2) Corresponding density fluctuations. Parameters: $\lambda =20$ and $D=220$.

Figure 4

Emergence and properties of roton fingers. The energies of the lowest ($m=0$) roton in each finger for (a) $\lambda =20$, (b) $\lambda =40$. Insets: the respective effective linear momenta of the $m=0$ rotons in each finger. (c) The dispersion-relation-like characterization of a $\lambda =40$ dipolar BEC at $D=700$ [this case is indicated by a gray vertical line in (b)]. The $m=0$ rotons at the start of each finger are marked with the symbols used to represent them in (b).

Figure 5

Avoided crossings of phonon (typically low ${\langle k_{\rho }\rangle }_j$) and roton (typically high ${\langle k_{\rho }\rangle }_j$) modes. (a) The spectrum of excitations mapped to a dispersion for $D=197.382$ only showing modes with angular momentum projection $\left|m\right|\le 2$. The $m=0$ phonon and roton modes, which exhibit the avoided crossing, are identified with arrows, as is the location of the finger that the dislocated mode originated from. The values of the (b) quasiparticle energies and (c) effective linear momentum ${\langle k_{\rho }\rangle }_j$ of the two modes identified in (a) as the DDI strength is varied. Insets to (b) and (c) show an enlarged view of the data near the avoided crossing. Gray vertical lines mark the case considered in (a). Note, trap aspect ratio is $\lambda =20$.

Figure 6

Condensate density profile and roton finger structure under the influence of a perturbing potential. (a), (c), (e) The condensate density profile (solid black line) and the total external potential (dashed blue line) for the various cases considered. (b), (d), (f) Lowest two roton fingers obtained from the corresponding quasiparticles. (g) The radial location of the condensate maximum density ${\rho }_{\max }$ and the $m$ value of the lowest energy roton $m_{\mathrm{soft}}$ as $V_0$ increases. Inset: $m_{\mathrm{soft}}/{\rho }_{\max }$ ratio for the data in the main plot. Shaded thick line shows that as $V_0$ increases the ratio remains approximately constant. Arrows indicate the relevant vertical axis for data in subplots. Results for ${\sigma }_0=2a_{\rho }$ and [(a), (b)] $D=251.26$, $V_0=1\hslash {\omega }_{\rho }$; [(c), (d)] $D=274.47$, $V_0=3\hslash {\omega }_{\rho }$; [(e), (f)] $D=285.53$, $V_0=9\hslash {\omega }_{\rho }$. Values of $D$ for all results chosen so that $D\max \left\{\right|{\psi }_0{|}^2\}{a}_{\rho }^3=4.740$, to ensure the system is approximately the same distance from the stability boundary.