Quantum Landau damping in dipolar Bose-Einstein condensates

We consider Landau damping of elementary excitations in Bose-Einstein condensates (BECs) with dipolar interactions. We discuss quantum and quasiclassical regimes of Landau damping. We use a generalized wave-kinetic description of BECs which, apart from the long-range dipolar interactions, also takes into account the quantum fluctuations and the finite-energy corrections to short-range interactions. Such a description is therefore more general than the usual mean-field approximation. The present wave-kinetic approach is well suited for the study of kinetic effects in BECs, such as those associated with Landau damping, atom trapping, and turbulent diffusion. The inclusion of quantum fluctuations and energy corrections changes the dispersion relation and the damping rates, leading to possible experimental signatures of these effects. Quantum Landau damping is described with generality, and particular examples of dipolar condensates in two and three dimensions are studied. The occurrence of roton-maxon excitations, and their relevance to Landau damping, are also considered in detail. The present approach is mainly based on a linear perturbative procedure, but the nonlinear regime of Landau damping, which includes atom trapping and atom diffusion, is also briefly discussed.

Figure 1

Dispersion relation of elementary phonon modes in a quasi-2D dipolar condensate, displaying a roton-maxon configuration. The phase velocity ${\displaystyle v_{\varphi }=\omega /k}$ decreases in the roton region and increases in the maxon region, thus changing the value of the corresponding Landau damping rates. The dotted line depicts the mean-field situation (${\displaystyle Q=0}$ and ${\displaystyle \chi =0}$). The stabilization of the roton instability due to the LHY quantum correction is represented for ${\displaystyle Q=0.2g/\sqrt{n_0}}$ (dot-dashed line). The inclusion of the finite range of the atom collisions enhances the roton instability, here depicted for ${\displaystyle \chi =0.08g/l_z^2}$ (solid line). The dashed line represents the imaginary part of ${\displaystyle \omega }$ in Eq. (28). In all situations, we have set ${\displaystyle {\epsilon }_{dd}=8.2}$.

Figure 2

Unstable reduced distribution, ${\displaystyle G_0\left(v\right)}$, of a thermal dipolar condensate in the presence of a stream of atoms, obtained from Eq. (40) for ${\displaystyle n_b=0.2n_0}$ and ${\displaystyle v_b=1.7c_{2D}}$ (solid line). For comparison, a stable thermal distribution is depicted (dotted line). In both cases, we have set ${\displaystyle T=0.4\mu /k_B}$.

Figure 3

Estimation of the quantum Landau damping due to the Heisenberg uncertainty. Real (top) and imaginary (bottom) parts of the dispersion relation in Eq. (28). Depicted are the mean-field case ${\displaystyle Q=\chi =0}$ (dotted line); the LHY correction, ${\displaystyle Q=0.3g/\sqrt{n_0}}$ and ${\displaystyle \chi =0}$ (dot-dashed line); and its combination with the finite range of the atomic collisions, ${\displaystyle Q=0.3g/\sqrt{n_0}}$ and ${\displaystyle \chi =0.08g/l_z^2}$ (solid line). The circles mark the position of the roton minimum in the different situations. In all cases, we have set ${\displaystyle {\epsilon }_{dd}=8.2}$ and a small effective temperature of ${\displaystyle T_{\mathrm{eff}}=0.01m{c}_{2D}^2/k_B}$, which is much less than the critical temperature for condensation ${\displaystyle T_c}$.