Historical research guide · 0495 · 2010
Black hole lasers in Bose–Einstein condensates
S Finazzi · R Parentani
In one minute
Stefano Finazzi and Renaud Parentani ask what happens inside a Bose–Einstein condensate — an ultracold gas that moves as one quantum fluid — when its flow crosses the speed of sound twice. Each crossing is a sonic horizon: sound cannot swim back upstream, exactly as light cannot climb out of a black hole. Two horizons then face each other, and the supersonic region between them becomes a resonant cavity. Working from the Bogoliubov–de Gennes equation, and leaning on the gravitational analogy for none of their results, the authors show that this cavity holds a discrete, finite set of complex-frequency modes that grow with time. Hawking radiation emitted at one horizon is bounced back and amplified — a black hole laser. They compute the frequencies by semi-classical methods, confirm them numerically, and predict the density-correlation pattern the effect would print into the gas. Applied to the Technion condensate of June 2010, the growth comes out about ten times too slow to see, and they name the factor that would close the gap.
Connection
This is chapter 5 in its purest laboratory form: a fluid whose excitations obey the same wave equation as light in curved spacetime, so a horizon becomes something you build on a bench rather than something you wait to see in the sky. It gives chapter 4 a measurable analogue of the metric itself, with a named condition — a tenfold increase in surface gravity — under which the amplification becomes observable.
From the source
A Bose–Einstein condensate whose stationary flow crosses the speed of sound twice carries a pair of sonic horizons — a white-hole horizon and a black-hole horizon — and the supersonic region trapped between them behaves as a resonant cavity.Section 2.1, Black-hole–white-hole geometries
The whole analysis is built on the Bogoliubov–de Gennes equation, so although the analogy with light propagating between a black and a white horizon is manifest, none of the results depends on that gravitational analogy.Section 5, Conclusions, first paragraph
Without periodic boundary conditions the spectrum of bound modes splits into real-frequency modes that are only elastically scattered plus a discrete and finite set of pairs of complex-frequency modes; those modes are dynamically unstable because scattering at each sonic horizon mixes positive and negative norm, and their growth is self-amplified Hawking radiation — the laser effect.Sections 3.1 and 3.2; Conclusions, second paragraph
The real part of each complex frequency obeys a Bohr–Sommerfeld condition, which is what makes the set discrete, while the imaginary part is fixed by the norm of the scattering coefficients across the horizons — semi-classical predictions the authors then reproduce by numerically solving the same equation.Section 3.3 and Section 4.2; figures 3 to 5
At early times the phonon flux leaving the two-horizon system behaves very much like the flux a lone black-hole horizon would emit — standard Hawking radiation — while at late times a single most unstable mode dominates both the flux and the equal-time density–density correlation pattern, which carries very specific signatures in the supersonic region.Sections 4.3 and 4.4; figure 11
Applied to the black-hole–white-hole flow realised at the Technion in June 2010, the model finds only a few unstable modes and an instability time scale about ten times longer than the lifetime of the condensate; raising the surface gravity by a factor of ten, at fixed dimensionless parameters, would bring the two time scales together and make the laser effect observable.Section 4.5, The Technion experiment; figure 14
The way in
https://doi.org/10.1088/1367-2630/12/9/095015How to cite it
S Finazzi, R Parentani (2010) Black hole lasers in Bose–Einstein condensates doi:10.1088/1367-2630/12/9/095015