Formation of a sonic horizon in isotropically expanding Bose-Einstein
  condensates
YITP-07-16, OCU-PHYS-264, AP-GR-40
Formation of a sonic horizon in isotropically expanding Bose-Einstein condensates
Yasunari Kurita
Osaka City University Advanced Mathematical Institute, Osaka 558-8585, Japan
Takao Morinari
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We propose a simple experiment to create a sonic horizon in isotropically trapped cold atoms
within currently available experimental techniques. Numerical simulation of the Gross-Pitaevskii
equation shows that the sonic horizon should appear by making the condensate expand. The ex-
pansion is triggered by changing the interaction which can be controlled by the Feshbach resonance
in real experiments. The sonic horizon is shown to be quasi-static for sufficiently strong interaction
or large number of atoms. The characteristic temperature that is associated with particle emission
from the horizon, which corresponds to the Hawking temperature in an ideal situation, is estimated
to be a few nK.
PACS numbers: 03.75.Kk, 03.75.Hh, 04.62.+v, 05.30.Jp
I. INTRODUCTION
For the exploration of cosmology and gravitational
physics, it is necessary to have a deep understanding of
quantum filed theory in curved spacetime: It is widely
believed that everything except for the spacetime it-
self should originate from quantum fluctuations in the
early Universe. Quantum effects on curved spacetime,
such as the Hawking radiation, give us theoretical sup-
port for black hole thermodynamics. However, it is ex-
tremely hard to verify such quantum effects experimen-
tally. For instance, the Hawking radiation is thermal
radiation emitted from a dynamically formed stationary
black hole [1]. However, the characteristic temperature
of the thermal radiation, the Hawking temperature, is on
the order of several tens of nanokelvins at most, which is
much lower than the cosmic microwave background radi-
ation temperature. So detecting thermal radiation from
a real black hole is almost impossible.
One way to circumvent this difficulty is to make use of
artificial black holes [2][3]. Unruh showed in his seminal
paper[4] that excitations in a supersonic flow corresponds
to a scalar field equation on a curved spacetime includ-
ing a horizon. Since the phenomenon of the Hawking
radiation can be separated from gravitational physics, it
is possible to detect the corresponding phenomenon in a
fluid system with sonic horizon [4]. The basic idea is to
identify fluid flow with curved spacetime and excitation
modes with fields on the curved spacetime. A black hole
event horizon corresponds to a sonic horizon in a fluid.
For the purpose of investigating the quantum effects, a
quantum fluid should be considered. As such a quantum
fluid, Bose-Einstein condensates (BEC) in trapped cold
atoms [5, 6] are one of the most suitable systems [7]-[9].
A crucial advantage is that one can control scattering
length between atoms by making use of the Feshbach
resonance[10]. In fact, that experimental technique was
used in observing jets and bursts in a collapsing conden-
sate, which is called “Bose-Novae” [11]. An remarkable
explanation of burst and jet phenomena in Bose-Novae
was proposed in [12][13], based on quantum field theory
of particle creation and structure formation in cosmolog-
ical spacetime.
In order to verify the Hawking effect in fluid analogy, it
is necessary to create a stationary sonic horizon because
it is a phenomenon on a dynamically formed stationary
black hole. Although several possibilities have been dis-
cussed so far [7, 8], it seems difficult to realize exactly sta-
tionary sonic horizon in cold atoms. However, if one can
make a quasi-static horizon for high frequency modes,
particle emission from the horizon is also expected. In
this paper, we numerically demonstrate that a quasi-
static horizon is realized without introducing new exper-
imental techniques beyond currently available ones. We
consider an expanding BEC driven by a sudden change of
the interaction. Numerically solving the Gross-Pitaevskii
(GP) equation, we show that a quasi-static horizon ap-
pear.
We note that there have been made great efforts to cre-
ate cosmological geometry using expanding BEC[14]-[21].
In these papers, the analogue models with specific cos-
mological metrics such as Friedmann-Robertson-Walker
(FRW) metric or de Sitter metric were discussed and the
effects of particle creation in these cosmological space-
times were investigated. However, in this paper, we do
not intend to obtain any cosmological analogue model
with well-known analytic metric. But we try to obtain
dynamically formed quasi-static sonic horizon. Further-
more, the sonic horizon should be formed in hydrody-
namic regime of the condensate because the spacetime
analogy is only valid in such regime. The appearance
of a horizon due to expansion of a condensate was no-
ticed in the previous works, for example in [15], and its
formation itself is not surprising. But it is non-trivial
whether the condensate flow at the horizon is in the hy-
drodynamic regime, or not. In this paper, we show that
the quasi-static sonic horizon will appear in the hydrody-
namic regime of the condensate by changing the atomic
interaction instantaneously.
http://arxiv.org/abs/0704.1096v3
II. ANALOGUE SPACETIME IN BEC
In the coherent state path integral formulation, the
action of bosons is given by
i~φ̄∂tφ−
∇φ̄ · ∇φ − Vextφ̄φ
U0(φ̄φ)
, (1)
where Vext is the confining potential and U0 = 4π~
with a the s-wave scattering length. For φ and Vext,
spatial and time dependences are implicit. The saddle
point equation for this action leads to the GP equation:
i~∂tΨ =
∇2 + Vext + U0|Ψ|2
Ψ. (2)
This GP equation governs the dynamics of the conden-
sate whose order parameter is given by Ψ.
Now we consider hydrodynamical approximation. We
denote the bosonic field φ as φ =
ρ0 + ρe
i(ϕ0+ϕ), where√
ρ0 and ϕ0 are the amplitude and the phase of Ψ, re-
spectively. (Namely, Ψ =
ρ0 exp(iϕ0).) The fields ρ
and ϕ describe the non-condensate part of the bosonic
field. If the density gradient is sufficiently smooth over
the scale determined by the local healing length ξ(r, t) ≡
~/(2mρ0U0)
1/2, or, in other words if the conditions,
|ξ∇ρ0/ρ0|2 ≪ 1 and |ξ∇ρ/ρ|2 ≪ 1, (3)
are satisfied, hydrodynamical approximation is justified.
(The condition (3) shall be examined later.) Under the
above condition, the equation for ρ is
ρ = −~(ϕ̇+ v0 · ∇ϕ)/U0, (4)
where v0 =
∇ϕ0 is the background fluid velocity, and
the effective action for ϕ is
Seff =
(ϕ̇+ v0 · ∇ϕ)2 −
(∇ϕ)2
Taking variation with respect to ϕ, we find that the field
equation for ϕ has the form of a propagating wave equa-
tion. Also, the equation for the field ϕ can be expressed
as ∂µ(
−ggµν∂ν)ϕ = 0, where gµν is the inverse of the
following matrix:
gµν ∝
−(c2s − v02) −v0
−v0 1
, (6)
with cs =
ρ0U0/m and g = detgµν . Thus, the equa-
tion is equivalent to an equation for a massless field on
a curved spacetime determined by the metric (6) with cs
the speed of ”light.” Note that in order to interpret the
quantity cs as a velocity, U0 must be positive because, for
negative U0, cs becomes pure imaginary. Hereafter we
consider positive U0, which leads to an effective space-
time with Lorentzian signature.
For the excitation modes of ϕ whose frequencies, say
ω, are much higher than the frequency ωBEC, which is
associated with the condensate motion, the condensate
will be quasi-static. (For moderate changes of the inter-
action, ωBEC turns out to be the trapping harmonic po-
tential frequency ωho, as shall be discussed below .) The
analogy between fields on the curved spacetime and ex-
citation modes on the fluid flow is meaningful only when
the conditions (3) are satisfied. The latter condition in
Eq.(3) turns out to be ω2 ≪ (cs/ξ)2, by using Eq.(4).
Thus, the frequency ω has an upper limit. The former
condition in Eq.(3) is satisfied in the regions far from
the edge of the condensate. (In contrast, if one is very
close to the edge, zero-point oscillations become domi-
nant, and so the former condition in (3) is not satisfied.)
If there exists intermediate region for ω of
ωBEC ≪ ω ≪ cs/ξ, (7)
then the hydrodynamical approximation is justified and
the condensate is quasi-static for excitation modes. Note
that those modes are associated with particle emission
from the horizon if the hydrodynamical flow has a dy-
namically formed sonic horizon. The necessary condition
for the existence of the intermediate region (7) is
ξωBEC
≫ 1. (8)
In the following, we mainly consider condensate satisfy-
ing the above condition.
III. FORMATION OF SONIC HORIZON
Now we investigate sonic horizon formation in an ex-
panding BEC trapped in isotropic harmonic potential,
Vext = mω
2/2, where r is the radial coordinate. Ini-
tially, we set the condensate in a ground state with an
initial atomic interaction ai. At t = 0, the atomic inter-
action is changed suddenly from ai to af (> ai), which
makes the condensate expand. Then, formation of sonic
horizon can be expected. The reason is as follows: The
sound velocity is proportional to square root of the con-
densate density and a decreasing function of r. In con-
trast, the fluid velocity is an increasing function of r
and the condensate expands fast around its edge whereas
v0(r = 0) = 0 due to the boundary condition. Therefore,
at an intermediate radius, v0 exceeds cs and the fluid flow
is transonic. It has a surface satisfying cs = |v0| which
is called a sonic horizon. We should note that the sonic
horizon corresponds to a horizon in the analogue space-
time defined by the metric (6). We also note that the
subsonic region is around the center of the condensate
and inside of the sonic horizon.
In general, if a fluid has a static sonic horizon and a
proper quantum state for an excitation field is realized,
then it is theoretically predicted that the horizon will
emit thermal radiation of the quantum field. As will
 0  5  10  15  20  25  30
af/ai
ai=50a0
ai=200a0
ai=800a0
FIG. 1: Times tc in each simulation are shown in units of
ω−1ho . The horizontal axis is the ratio of af to ai.
be discussed in Appendix A, if the sonic horizon in the
expanding condensate is quasi-static for the field ϕ, the
horizon will emit thermal radiation into the center of the
condensate. The temperature characterizing the thermal
emission (Hawking temperature) is given by the following
formula:
Tpc =
∂r(v0 − cs)|rH , (9)
where rH is the horizon radius and kB is the Boltzmann’s
constant[4][24]. From the above expression, it is found
that the Hawking temperature is determined by gradi-
ent of fluid and sound velocity at the horizon. Thus,
it is important to investigate the velocity gradients at
the horizon. In the derivation of the formula (9), it is as-
sumed that the dynamically formed horizon is static, but
in actual experiments, this assumption is not satisfied ex-
actly. Therefore, the spectrum of the particle is not fully
given by the single Planck’s distribution function, but
rather given by a superposition of the Planck’s distribu-
tion functions with slightly different temperatures. Even
if this is the case, the energy scale of the particle creation
emitted from the dynamically formed horizon is on the
order of Tpc.
We have simulated the expansion of the condensate by
solving numerically (using the Crank-Nicolson scheme)
the time-dependent GP equation. The initial ground-
state wave function is obtained by solving the GP equa-
tion using the steepest descent method for an initial s-
wave scattering length ai and the number of atoms N .
We have computed cs and the radial velocity of the con-
densate via
(Ψ∗Ψ)U0/m, (10)
v0 = ~ [Ψ
∗∂rΨ− (∂rΨ∗)Ψ] /(2mi|Ψ|2), (11)
and searched for parameter sets leading to |v0| > cs.
In the following, we assume that the condensate con-
sists of N = 105 Rb atoms. (The values of atomic inter-
action given below are those in the case of N = 105. If
 0  2  4  6  8  10  12  14
FIG. 2: Sound velocity cs(solid line) and the fluid velocity
v0(dashed line) versus r at t̃ = 0.4 in the case of ai = 200a0
and af = 5ai are shown in units of (~ωho/m)
1/2. The
healing length ξ(dotted line) is shown as well in units of
aho = (~/mωho)
N = 105/n with an integer n, then ai and af should
be multiplied by n.) The initial atomic interaction
is assumed to be ai = 50a0, 200a0 and 800a0 where
a0 = 0.53 × 10−10m is the Bohr radius. The follow-
ing change of the atomic interaction has been simulated:
af/ai = 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30.
Just after t = 0, the condensate begins to expand in
the trapping potential and the expansion is accelerated
for a while. At some time, say t = tc, the expansion
turns to be decelerated. Figure 1 shows tc as a function
of af/ai. It is seen that tc does not depend on the initial
strength of the interaction. For t̃ := ωhot < π/2, the
condensate continues to expand, and at t̃ ≃ π/2, the
condensate starts to collapse. Therefore, we turn off the
trapping potential at t̃ = π/4 and make the condensate
expand freely in order to keep the horizon for a while.
As far as we have investigated, sonic horizon always
appears in the sence of the surface where v0 exceeds cs.
As an example of a sonic horizon, we show Fig. 2 which
is the snapshot at t̃ = 0.4 in the case of ai = 200a0 and
af = 5ai. We see that, around r = 7aho, the fluid velocity
exceeds the sound velocity, and the sonic horizon exists
there. In this case, we find that, at t̃ ≡ ωhot = 0.11,
the horizon appears. Fig. 3 shows the time dependence
of the radius of the horizon, say rH , and the velocity
gradient at the horizon ∂r(v0 − cs)|rH .
If we keep the trapping potential for a long time, an
oscillating behavior of the condensate is observed. The
period of the oscillation is about π in units of ω−1ho and
we find ωBEC ≃ ωho, within a moderate change of the
interaction. This oscillation is just like an oscillation of a
droplet confined in a harmonic potential. Therefore, the
condition (8) can be rewritten as
≫ 1. (12)
FIG. 3: Time dependence of ∂r(v0−cs)rH (solid line) in units
of ωho and the position of the sonic horizon rH (dashed line)
in units of aho = (~/mωho)
in the case of ai = 200a0 and
af = 5ai.
 0  5  10  15  20  25  30
af/ai
ai=50a0
ai=200a0
ai=800a0
FIG. 4: cs/ξωho as a function of af/ai is shown for each
initial scattering length.
Now, we are interested in sonic horizon where (12) is sat-
isfied. The intermediate region (7) exists if, for example,
the following inequality is satisfied:
≥ 22.5. (13)
For this choice of the lower bound, there exists a re-
gion of ω satisfying both conditions of ω ≥ 10ωho and
(cs/ωξ)
2 ≥ 5. The condition (13) ensures hydrodynamic
flow and quasi-static nature of the condensate. Fig. 4
shows cs/ξωho as a function of af/ai. We define horizon
life time as the time interval during which the condition
(13) continues to be satisfied at the horizon. The horizon
life time is shown in Fig. 5. As far as we have investi-
gated, the condensate flow satisfying (13) at sonic horizon
appears only when af ≥ 5ai for ai = 50a0, af ≥ 4ai for
ai = 200a0 and af ≥ 3ai for ai = 800a0.
The Hawking temperature at t = tc and t̃ = 0.79 (just
 0  5  10  15  20  25  30
af/ai
ai=50a0
ai=200a0
ai=800a0
FIG. 5: Horizon life time as a function of af/ai is shown in
units of ω−1ho .
 0  5  10  15  20  25  30
af/ai
ai=50a0
ai=200a0
ai=800a0
FIG. 6: Hawking temperature at t = tc in units of nK. In
the evaluation, we assume ωho = 1400 Hz.
after turning off the trapping potential) are shown in Fig.
6 and Fig. 7, respectively. In the evaluation, we assume
the frequency ωho = 1400 Hz. The Hawking temperature
at t = tc depends on the ratio af/ai almost linearly. In
contrast, for af/ai ≥ 9, the Hawking temperature at
t̃ = 0.79 does not depend on the ratio so much. From the
simulations, the temperature is expected to be a few nK.
For this spherically symmetric trap, one may concern
the three-body recombination loss of condensed atoms.
Now, we check the effect of three-body losses for the given
peak density. This effects may be taken into account by
incorporating the imaginary term describing the inelastic
process in the GP equation [22]
i~∂tΨ =
∇2 + Vext + U0|Ψ|2
Ψ− i~
K3|Ψ|4Ψ,
where K3 denotes three-body recombination loss-rate co-
efficient. Then, the three-body loss is proportional to the
 0  5  10  15  20  25  30
af/ai
ai=50a0
ai=200a0
ai=800a0
FIG. 7: Hawking temperature at t̃ = 0.79 in units of nK,
when just after the trapping potential is turned off. In the
evaluation, we assume ωho = 1400 Hz.
cube of the atomic density
|Ψ|2d3r = −K3
|Ψ|6d3r,
which implies that the three-body loss rate is given by
R3 ≡ K3
|Ψ|6d3r/
|Ψ|2d3r. For the value of K3, we
assume K3 = 2 × 10−28cm6/s, according to [23]. Of
course, high atomic density causes many inelastic pro-
cesses and gives high atomic loss rate. In our numerical
simulation, the upper limit of the loss rate can be esti-
mated by use of the peak density asR3 ≤ 3×10s−1, where
the total atomic number was set to be N =
|Ψ|2d3r =
105. Then, the three-body loss can be ignored because
we consider the time scale of ≤ 10 ms.
In the above evaluation for Hawking temperature, hori-
zon lifetime and R3, we have assumed that the trapping
frequency is ωho = 1400 Hz. Note that ωho is the en-
ergy scale of the system. Therefore, a large value of
ωho is plausible to increase the characteristic tempera-
ture for the particle emission, though the time evolution
process becomes rapid for large ωho. If lower frequency
is assumed, lower temperature, longer horizon lifetime
and fewer three-body loss rate would be expected. As
an example, Fig. 8 shows ωho-dependence of the Hawk-
ing temperature and the horizon lifetime in the case of
ai = 200a0 and af = 10ai.
IV. BOGOLIUBOV SPECTRUM
In the above numerical simulations, we assume there
is no dynamical instability. Now, we check whether there
is dynamical instability or not, within Gaussian approx-
imation. For that purpose, we study the Bogoliubov-de
Gennes equations: the second quantized field equations
 0  200  400  600  800  1000  1200  1400
horizon life time
FIG. 8: ωho-dependence of TH and horizon lifetime in the
case of ai = 200a0 and af = 10ai. The Hawking temperature
is shown in units of nK and horizon lifetime is in units of ω−1ho .
for the excitation fields δφ and δφ are given by
i~∂tδφ =
∇2 + Vext + 2U0|Ψ|2
δφ+ U0Ψ
−i~∂tδφ =
∇2 + Vext + 2U0|Ψ|2
δφ+ U0
The excitation spectrum is computed by performing the
Bogoliubov transformation:
uα (r) bαe
−iEαt/~ − vα (r) b†αeiEαt/~
,(14)
u∗α (r) b
iEαt/~ − v∗α (r) bαe−iEαt/~
.(15)
The energy spectrum Eα is calculated by diagonalizing
the skew symmetric matrix, which is carried out by using
a routine in LAPACK. For the parameter values taken
above, we find that all eigenvalues do not have the imag-
inary parts within numerical errors. Therefore, within
Gaussian approximation, there is no dynamical instabil-
ity. In addition, we find that there is no level crossing.
V. SUMMARY
To summarize, we have proposed an experiment to
create a quasi-static sonic horizon using an expanding
BEC. It has been shown that the dynamically formed
quasi-static sonic horizon is in hydrodynamic regime as
it should be to discuss analogy with curved spacetime in
BEC. Under suitable choices of the interaction parameter
and the confining potential, the characteristic tempera-
ture of the particle emission is expected to be a few nK
for sufficiently strong confining potential. Large num-
ber of atoms or strong atomic interaction improves the
quasi-static nature of the horizon.
Of course, other effect such as cosmological particle
creation can occur in this expanding BEC setup, as dis-
cussed in [14]-[21]. In this paper, we have focused on
how to make dynamically formed quasi-static sonic hori-
zon in the hydrodynamic regime of the condensate flow.
In order to investigate cosmological particle creation ef-
fect and other excitations arising from depletion, we need
a different numerical simulation scheme. The result will
be reported in a future publication.
Furthermore, it is interesting to investigate numeri-
cally the behavior of negative frequency modes with pos-
itive norm which seem to be related to Hawking effect as
was discussed in [26][27]. This point shall be investigated
in a future publication.
Acknowledgments
Y.K. thanks Hideki Ishihara, Ken-ichi Nakao, and
Makoto Tsubota for useful discussions. The authors
thank Michikazu Kobayashi and Takashi Uneyama for
useful comments on numerical simulations. Y.K. was
partially supported by the Yukawa memorial founda-
tion. This work was also supported by the 21st Century
COE ”Center for Diversity and Universality in Physics”
and ”Constitution of wide-angle mathematical basis fo-
cused on knots” from the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) of Japan. The
numerical calculations were carried out on Altix3700 BX2
at YITP in Kyoto University.
APPENDIX A: PARTICLE CREATION
PHENOMENON
Here we focus on spherically symmetric quantum fluc-
tuations by symmetry. At t ≤ 0, the fluid velocity v0 = 0,
and the metric of the initial static effective spacetime is
ds2 ∝ −c2sdt2 + dr2 + r2dΩ2S2 , (A1)
where dΩ2S2 is the element of solid angle on the unit
sphere S2. After the increase of the interaction, the ef-
fective spacetime evolves dynamically as the BEC starts
to expand. Then, the sonic horizon is formed as was
shown by the above numerical simulation. If the effec-
tive spacetime is static, we can introduce a following time
coordinate: τ = t +
v0dr/(c
s − v20), and the effective
spacetime metric becomes
ds2 ∝ −(c2s − v20)dτ2 +
c2sdr
c2s − v20
+ r2dΩ2S2 . (A2)
From this expression, it is found that the horizon is lo-
cated at the surface where the condition cs = |v0| is sat-
isfied. A new coordinate v is introduced as v ≡ τ + r∗
where r∗ ≡
csdr/(c
s − v20), which is a coordinate char-
acterizing ingoing light-like (null) rays in the effective
spacetime.
We assume here that the initial state of the quantum
field ϕ is the vacuum state for the static observer in the
initial effective spacetime. Under the time evolution of
the effective spacetime caused by the expansion of the
condensate, the creation and annihilation operators for
the field ϕ also evolve, and particle creation occurs.
Now we consider an observer who moves along his
or her outgoing geodesic with proper time λ, crossing
the horizon at λ = 0. Hereafter, we term the observer
geodesic observer. If we assume that the horizon is lo-
cated at r = rH , the proper time λ is related to the coor-
dinate v there via λ ≈ −λ0e−
v, where cH ≡ cs(rH),
α ≡ 2cH∂r(v0 − cs)|r=rH and λ0 is a constant. The ingo-
ing mode functions ϕω = e
−iωv have λ-dependence near
the horizon as
ϕω ≈ exp
ln(−λ)
. (A3)
Initially, the state is the vacuum for the static observer
and therefore the geodesic observer would see no exci-
tation at short distance, because there will be no much
higher positive frequency excitations than those deter-
mined by the time scale of the dynamical expansion of
the BEC. If we ignore the short distance cut-off deter-
mined by the healing length, or equivalently, if the lat-
ter condition in Eq.(3) is ignored, this λ-dependence of
the ingoing mode functions implies that the particle cre-
ation from the horizon into the inside of the condensate
has thermal spectrum with the temperature given by (9).
Furthermore, even if the short distance cut-off is taken
into account, it is known that the result does not change
in principle [25].
Therefore, the particle emission from horizon will oc-
cur in the case of expanding condensate, even where the
subsonic region is inside of the horizon.
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http://arxiv.org/abs/cond-mat/0208569
http://arxiv.org/abs/gr-qc/0404063
http://arxiv.org/abs/gr-qc/0308048
