Anomalous Doppler effect in two-component Bose-Einstein condensates

TL;DR

The paper addresses the anomalous Doppler effect in a binary Bose-Einstein condensate with interspecies coupling and counterflow, showing a non-kinematic Doppler shift of sound modes. It derives analytic Doppler shifts from collisionless superfluid hydrodynamics, $c_{d/s}^\pm = c_{d/s}^0 \pm (v_T^0 + \delta_{d/s} w^0)$, with $\delta_d = -\delta_s = [g_{ds} + g m_z]/\sqrt{(g + g_{ds} m_z)^2 - \Delta (1 - m_z^2)}$ and $m_z = s_z^0/n^0$. It proposes a dynamic protocol based on coupled Gross-Pitaevskii equations and a selective perturbation scheme to measure the two Doppler-shifted modes via the density-density response, including a precession method to extract $\Omega_i$ from $\delta n(x,t)$. It also analyzes the Andreev-Bashkin drag, showing how drag modifies the rest-frame speeds and the Doppler shifts, and argues that the effects are detectable in currently accessible cold-atom setups and extend to other superfluid mixtures.

Abstract

We show that two-component Bose-Einstein condensed mixtures, in presence of a persistent current, exhibit a non trivial Doppler shift of the sound velocities. The peculiarity is due to the inter-species interaction and the possibility of generating a counter-flow persistent current. Analytic predictions are derived by using superfluid hydrodynamics. While the existence of anomalous Doppler shifts at finite temperature has been discussed a long time ago in the case of superfluid Helium-4, an experimental verification of the effect is still missing. For this reason, we also propose a protocol for the measurement of the Doppler shifts, based on the density-density response function. The dynamical protocol is simulated by means of coupled Gross-Pitaevskii equations.

Figures (2)

• Figure 1: The Doppler shift of the density-like sound relative to the velocity of the first component as a function of interspecies interaction coupling $g_{12}$ for $v_2=0$ and (a)$g_{ds}=0$ and (b)$g_{ds}/g=0.1$ . Each solid line corresponds to a HD prediction for different values of polarization $m_z$. The symbols represent results of the GPE simulation, whose details are contained in the next section. The $g_{12}$ and $m_z$ dependence of the Doppler shift of the spin-like sound mode ($\Delta c_s$) is the same as of the density-like mode, but mirrored with respect to the $y=0.5$ line. The vertical dashed line marks the stability condition (\ref{['eq:def_delta']}).
• Figure 2: Precession of the density profile $n(x)-\langle n\rangle$ with selectively excited density-like mode (upper panel) and spin-like mode (lower panel). In each panel, the solid line indicates the trajectory of the density maximum at stroboscopic times, as given by the hydrodynamic formula (\ref{['eq:gpe_doppler']}). The results correspond to the following gas parameters: $m_z=0.1$, $g_{ds}/g = 0.1$, $g_{12}/g=0.55$ and $v_T^0 = w^0 = 3\times 2\pi \hbar /(mL)$, for which $\delta_d = 0.67$ and $\delta_s = 0.33$. The dashed line presents the Doppler shift for the same gas parameters but with interspecies interaction $g_{12}=0$, where the density-like (spin-like) mode corresponds to the oscillation of the density of the first (second) component.