Analogue gravity from Bose-Einstein condensates
Carlos Barceló · S Liberati · Matt Visser
Abstract and summary · read the original at the source
In one page
Carlos Barceló, Stefano Liberati and Matt Visser ask how far a cloud of ultracold atoms can be pushed as a stand-in for curved spacetime, and the answer is: remarkably far. Starting from a very general nonlinear Schrödinger equation — one that allows a position-dependent, direction-dependent effective mass and any nonlinearity you like — they show that small ripples in the phase of a Bose-Einstein condensate’s wavefunction obey precisely the wave equation a field obeys in a curved four-dimensional spacetime. The effective metric is generic, it reads off algebraically from the background density, sound speed and flow, and none of it depends on the particular form of the atom-atom interaction. Sound in the condensate plays the part light plays in relativity, and a region where the flow outruns the sound speed is a horizon. The analogy also has an honest floor: below what they call an acoustic Compton wavelength, the ripples stop behaving relativistically and revert to ordinary Schrödinger physics.
Why it matters hereChapter 5’s premise is that the vacuum behaves like a quantum fluid, and this is the paper showing how completely a real laboratory fluid can carry a spacetime metric on its back — groundwork for the same authors’ Analogue Gravity reviews on this site at /library/stm-9d9474c4b5 and /library/stm-a50c456d62, and a working bench for the horizon physics chapter 4 needs.
What it claims
01Linearized excitations of the phase of the condensate wavefunction obey a (3+1)-dimensional d’Alembertian equation coupled to a Lorentzian-signature effective metric, and this is a generic feature, independent of the explicit form of the nonlinear terms in the Schrödinger equation.Abstract; Section 9, summary and discussion, first result
Published and peer-reviewed02The effective acoustic interval is fixed algebraically by the background density, the speed of sound and the background flow velocity, so the metric can be read straight off the condensate rather than solved for.Section 6, Equation 47
Published and peer-reviewed03A negative scattering length — attractive forces between the atoms — is formally equivalent to an imaginary speed of sound and turns the effective metric Euclidean, so manipulating the sign of the scattering length amounts to building an analogue of a signature-changing spacetime.Section 6, first bullet following Equation 47; Section 9, summary
Published and peer-reviewed04At high momenta the eikonal approximation recovers exactly the dispersion relation Bogoliubov derived in 1947, which interpolates between ordinary phonons at wavelengths long compared with the acoustic Compton wavelength and free-particle Schrödinger behaviour at wavelengths short compared with it — so observing a Lorentz-invariant spectrum does not guarantee that the underlying physics is Lorentz invariant.Section 8, Equation 77 and observations 1, 2 and 4, Equations 79 to 81
Published and peer-reviewed05Because the group velocity grows without bound at large momentum, the acoustic horizon in these models is frequency-dependent rather than absolute, and high-momentum modes escape from behind it — the same escape Corley and Jacobson linked to self-amplified instabilities in configurations carrying both an inner and an outer horizon.Section 8, observation 3, Equation 78
What to watch06Mimicking a fully generic gravitational field is not implausible a priori, but it would be technically very challenging, since it requires anisotropy introduced directly into the generalized nonlinear Schrödinger equation through a position-dependent 3-tensor effective mass; the crude Hawking temperature for a ten-micron acoustic hole is about a billionth of a kelvin, only three orders of magnitude below the temperature of the condensate itself.Section 9, summary and discussion, second result and Equation 82
What to watch
Read it · abstract
Abstract
We analyze prospects for the use of Bose–Einstein condensates as condensed-matter systems suitable for generating a generic “effective metric”, and for mimicking kinematic aspects of general relativity. We extend the analysis due to Garay et al, [gr-qc/0002015, gr-qc/0005131]. Taking a long term view, we ask what the ultimate limits of such a system might be. To this end, we consider a very general version of the nonlinear Schrödinger equation (with a 3-tensor position-dependent mass and arbitrary nonlinearity). Such equations can be used for example in discussing Bose–Einstein condensates in heterogeneous and highly nonlinear systems. We demonstrate that at low momenta linearized excitations of the phase of the condensate wavefunction obey a (3+1)-dimensional d’Alembertian equation coupling to a (3+1)-dimensional Lorentzian-signature “effective metric” that is generic, and depends algebraically on the background field. Thus at low momenta this system serves as an analog for the curved spacetime of general relativity. In contrast at high momenta we demonstrate how to use the eikonal approximation to extract a well-controlled Bogoliubov-like dispersion relation, and (perhaps unexpectedly) recover non-relativistic Newtonian physics at high momenta. Bose–Einstein condensates appear to be an extremely promising analog system for probing kinematic aspects of general relativity.
The way in
https://doi.org/10.1088/0264-9381/18/6/312Licence checked directly. The published article is free to read at the publisher but carries no Creative Commons statement, and the preprint on arXiv as gr-qc/0011026, filed 7 November 2000, sits under arXiv’s assumed licence for submissions of 1991 to 2003, which does not grant redistribution. So this page carries the summary, the claims and the authors’ own abstract, and sends the reader to the source. Published as Classical and Quantum Gravity 18, 1137–1156 (2001); the preprint’s own first page carries the title ‘Analog gravity from Bose–Einstein condensates’, written at the Physics Department, Washington University in Saint Louis, SISSA Trieste and the University of Maryland. The summary, the claims and the locators below were written from the full author text; the abstract below is the authors’ own, as filed with the preprint.
How to cite it
Carlos Barceló, S Liberati, Matt Visser (2001) Analogue gravity from Bose-Einstein condensates. doi:10.1088/0264-9381/18/6/312
Where it sits in the curriculum
The vacuum as a quantum fluidThe metric, warp drives and wormholes