Saturation of black hole lasers in Bose-Einstein condensates

To obtain the end-point evolution of the so-called black hole laser instability, we study the set of stationary solutions of the Gross-Pitaevskii equation for piecewise constant potentials which admit a homogeneous solution with a supersonic flow in the central region between two discontinuities. When the distance between them is larger than a critical value, we recover that the homogeneous solution is unstable, and we identify the lowest energy state. We show that it can be viewed as determining the saturated value of the first (node-less) complex frequency mode which drives the instability. We also classify the set of stationary solutions and establish their relation both with the set of complex frequency modes and with known soliton solutions. Finally, we adopt a procedure \`a la Pitaevski-Baym-Pethick to construct the effective functional which governs the transition from the homogeneous to non-homogeneous solutions.Comment: Final version 22 pages, 12 figures v2: added a note on unitarit

Mode mixing in sub- and trans-critical flows over an obstacle: When should Hawking's predictions be recovered?

We reexamine the scattering coefficients of shallow water waves blocked by a stationary counter current over an obstacle. By considering series of background flows, we show that the most relevant parameter is $F_{\rm max}$, the maximal value of the ratio of the flow velocity over the speed of low frequency waves. For subcritical flows, i.e., $F_{\rm max} < 1$, there is no analogue Killing horizon and the mode amplification is strongly suppressed. Instead, when $F_{\rm max} \gtrsim 1.1$, the amplification is enhanced at low frequency and the spectrum closely follows Hawking's prediction. We further study subcritical flows close to that used in the Vancouver experiment. Our numerical analysis suggests that their observation of the "thermal nature of the mode conversion" is due to the relatively steep slope on the upstream side and the narrowness of the obstacle.Comment: 11 pages, 6 figure

Dynamical instabilities and quasi-normal modes, a spectral analysis with applications to black-hole physics

Black hole dynamical instabilities have been mostly studied in specific models. We here study the general properties of the complex-frequency modes responsible for such instabilities, guided by the example of a charged scalar field in an electrostatic potential. We show that these modes are square integrable, have a vanishing conserved norm, and appear in mode doublets or quartets. We also study how they appear in the spectrum and how their complex frequencies subsequently evolve when varying some external parameter. When working on an infinite domain, they appear from the reservoir of quasi-normal modes obeying outgoing boundary conditions. This is illustrated by generalizing, in a non-positive definite Krein space, a solvable model (Friedrichs model) which originally describes the appearance of a resonance when coupling an isolated system to a mode continuum. In a finite spatial domain instead, they arise from the fusion of two real frequency modes with opposite norms, through a process that closely resembles avoided crossing.Comment: 31 pages, 13 figures. Small clarifications, title changed, matches published versio

Controlling and observing nonseparability of phonons created in time-dependent 1D atomic Bose condensates

We study the spectrum and entanglement of phonons produced by temporal changes in homogeneous one-dimensional atomic condensates. To characterize the experimentally accessible changes, we first consider the dynamics of the condensate when varying the radial trapping frequency, separately studying two regimes: an adiabatic one and an oscillatory one. Working in momentum space, we then show that in situ measurements of the density-density correlation function can be used to assess the nonseparability of the phonon state after such changes. We also study time-of-flight (TOF) measurements, paying particular attention to the role played by the adiabaticity of opening the trap on the nonseparability of the final state of atoms. In both cases, we emphasize that commuting measurements can suffice to assess nonseparability. Some recent observations are analyzed, and we make proposals for future experiments.Comment: 26 pages, 17 figure