Stability analysis of sonic horizons in Bose-Einstein condensates

Abstract

We examine the linear stability of various configurations in Bose-Einstein condensates with steplike sonic horizons. These configurations are chosen in analogy with gravitational systems with a black hole horizon, a white hole horizon, and a combination of both. We discuss the role of different boundary conditions in this stability analysis, paying special attention to their meaning in gravitational terms. We highlight that the stability of a given configuration, not only depends on its specific geometry, but especially on these boundary conditions. Under boundary conditions directly extrapolated from those in standard general relativity, black hole configurations, white hole configurations, and the combination of both into a black hole-white hole configuration are shown to be stable. However, we show that under other (less stringent) boundary conditions, configurations with a single black hole horizon remain stable, whereas white hole and black hole-white hole configurations develop instabilities associated to the presence of the sonic horizons.

Quasi-normal mode analysis in BEC acoustic black holes

“…Integrating (2.4) in an infinitesimal neighborhood of the discontinuity (say x 0) and simplifying with the use of the continuity equation (vc 2 const) leads, for our choice of a piecewise homogeneous background, to the following set of matching conditions [7]:…”

“…(2.4) is neglected, and so upon integration of the remaining terms in these equations, only two matching conditions remain: The above calculation is strictly valid only for completely subsonic profiles. When the left-hand side is supersonic, the hydrodynamic approximation breaks down in the passage through the sonic horizon [7]. Both modes in the supersonic region become outgoing at x !…”

“…But when looking at the full dispersion law, this relation is no longer valid. Note that for our choice of profiles, we have two asymptotic regions, and moreover no prohibition of crossing the horizon in any direction, since the permeability of the horizon is an essential feature of modified dispersion relations [7]. An outgoing mode is then one that has a group velocity v g < 0 for x < 0 and v g > 0 for x > 0, where v g is obtained from the full dispersion relation:…”

“…In general, these corrections have to be taken into account in any treatment involving high energies (high frequencies). In BECs, in particular, in the presence of horizons, the deviations from the geometric or hydrodynamic regime are important even at low energies [7].…”

“…In a previous article [7], the stability of several onedimensional configurations in BECs was studied. These configurations were chosen in analogy with the geometries associated with gravitational black holes.…”