Acoustic superradiance from optical vortices in self-defocusing cavities
Francesco Marino,
1
Marzena Ciszak,
2
and Antonello Ortolan
3
1
European Laboratory for Nonlinear Spectroscopy, Via N. Carrara 1, Sesto Fiorentino, I-50019 Firenze, Italy
2
CNR–Istituto Nazionale di Ottica Applicata, L.go E. Fermi 6, I-50125 Firenze, Italy
3
Laboratori Nazionali di Legnaro, INFN, Viale dell’Università 2, Legnaro, I-35020 Padova, Italy
Received 27 August 2009; published 10 December 2009
In a self-defocusing optical cavity, optical-field perturbations on a vortex background behave as sound waves
in a 2+1 rotating acoustic black-hole spacetime. Numerical integration of the associated Klein-Gordon
equation using typical experimental parameters shows that optical perturbations in the appropriate frequency
range are amplified through the mechanism of superradiance. These results suggest the possibility to observe
this phenomenon in a common nonlinear optical system.
DOI: 10.1103/PhysRevA.80.065802 PACS numbers: 42.65.Jx, 04.70.-s, 04.80.-y
I. INTRODUCTION
Black holes play a central role in general relativity and
quantum field theory in curved spacetime. Unfortunately, the
immense progress in their theoretical understanding cannot
be easily supported by direct experimental investigations.
For this reason, Unruh’s observation of sound waves propa-
gation in inhomogeneous fluids being equivalent to light
propagation in curved spacetime 1 has intrigued both the
astrophysical and condensed matter communities 2. A sur-
face in the flow, where the normal component of the fluid
velocity is inward pointing and equal to the local sound
speed behaves as an event horizon, i.e., sound waves cannot
propagate through this surface in the outward direction. This
kinematical analogy enables laboratory tests of those prob-
lems in black-hole physics which are insensitive to whether
or not the metric satisfies the Einstein equation. For instance,
the emission of Hawking radiation 3 in the form of a ther-
mal bath of phonons is expected to occur 1,4.
Although most of the attention has been focused on the
Hawking process, also other phenomena could shed light on
important aspects of gravitational black holes. This is the
case of superradiant scattering by rotating Kerr black holes,
a wave equivalent of the Penrose process 5. Kerr black
holes are surrounded by the ergosphere, a region where no
physical objects can remain at rest relative to an inertial ob-
server at infinity. A scalar wave, i.e., a solution of the mass-
less Klein-Gordon equation, entering the ergosphere, is re-
flected back and may be amplified leading to a “spin down”
of the black hole 6–8. As Hawking radiation, this phenom-
enon is not expected to be directly observable for actual as-
trophysical situations but it should occur in systems simulat-
ing the appropriate spacetime geometry, as 2+1 vortex
flows with a central sink 9–11. Around the event horizon,
these vortices possess an ergosphere, where the rotating fluid
velocity exceeds the speed of sound. Therefore, immediate
prospects of observing superradiance are currently related to
the experimental implementation of such vortex structures.
So far, experimental proposals have been based on superfluid
3
He 12 and Bose-Einstein condensates BECs13–15
where the spacetime analogy holds for sound wavelengths
larger than the healing length hydrodynamic limit. Suitable
“hydrodynamic” BEC vortices 13 could be realized for in-
stance by outcoupling the condensate to form an atom laser
beam 16.
Recently, it has been shown that rotating acoustic black
holes can be created also in a self-defocusing optical cavity
17. An optical field in self-defocusing media can be de-
scribed in terms of a two-dimensional “photon-fluid” 18,19
on which linear perturbations, i.e., sound waves, experience
an effective curved spacetime depending on the background
flow. Since in an optical cavity the background flow is
“pinned” by the driving field, the injection of a suitable op-
tical vortex beam allows the generation of acoustic ergore-
gions and event horizons.
On the basis of these results, optical-field perturbations on
a vortex background have been studied as an acoustic analog
for field propagation in a 2+1 rotating black-hole space-
time. Numerical integration of the associated Klein-Gordon
equation using typical experimental parameters shows that
optical perturbations in the appropriate frequency range are
amplified through the mechanism of superradiance. These
results suggest that this phenomenon could be observable in
current optical experiments.
II. ACOUSTIC SPACETIME ANALOGY
In a self-defocusing optical cavity, the slowly varying en-
velope of the intracavity field, E, evolves according to 20
t
E =
ic
2kn
0
2
E - i
n
2
n
0
E|E|
2
+ iE - E - E
d
, 1
where k =2n
0
/ is the wave number, c is the speed of light,
E
d
is a coherent driving field of frequency , is the detun-
ing, and = cT / 2n
0
L is the cavity decay rate, where T is the
mirrors transmissivity and L the cavity length.
2
E, defined
with respect to the transverse coordinates x , y, accounts for
diffraction and n
2
|E|
2
is the refractive index change self-
defocusing effect.
The physical conditions in which the spacetime analogy
applies to this system, become evident linearizing Eq. 1
around a background solution, E = E
0
+ e
i
0
, where E
0
=
0
1/2
e
i
0
and E
0
is a complex variable. By locking the
laser frequency at the cavity resonance, = n
2
/ n
0
0
, and in
the eikonal approximation 2, we obtain the dispersion law
PHYSICAL REVIEW A 80, 065802 2009
1050-2947/2009/806/0658024 ©2009 The American Physical Society 065802-1