Sonic black holes in dilute Bose-Einstein condensates

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by  L. J. Garay

Sonic black holes in dilute Bose-Einstein condensates
L.J. Garay1,2, J.R. Anglin1,3, J.I. Cirac1, and P. Zoller1
1 Institut fu¨r Theoretische Physik, Universit¨at Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria
2 Instituto de Matem´aticas y F´ısica Fundamental, CSIC, C/ Serrano 121, E-28006 Madrid, Spain
3 Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge Massachussets 02135
(30 May 2000)
Thesonicanalogofagravitationalblackholeindilute-gasBose-Einsteincondensatesisinvestigated.
It is shown that there exist both dynamically stable and unstable configurations which, in the
hydrodynamic limit, exhibit a behavior completely analogous to that of gravitational black holes.
The dynamical instabilities involve creation of quasiparticle pairs in positive and negative energy
states. We illustrate these features in two qualitatively different one-dimensional models, namely,
a long, thin condensate with an outcoupler laser beam providing an “atom sink,” and a tight ring-
1 shaped condensate. We have also simulated the creation of a stable sonic black hole by solving
0 the Gross-Pitaevskii equation numerically for a condensate subject to a trapping potential which
0 is adiabatically deformed. A sonic black hole could in this way be created experimentally with
2 state-of-the-art or planned technology.

n
a PACS number(s): 03.75.Fi, 04.70.Dy, 04.80.-y gr-qc/0005131; Phys. Rev. A 63, 023611 (2001)
J

8
1 I. INTRODUCTION those we use in condensed matter is that in the latter
case the next microscopic level of description is actually

2 Many investigations of dilute gas Bose-Einstein con- well understood.
v densates are directed towards experimentally creating With this fundamental background in mind, it is not
1
nontrivial configurations of the semiclassical mean field, so surprising that condensed matter analogs of nontriv-
3
1 or to predicting the properties of such configurations in ial configurations appearing in relativistic quantum field
5 thepresenceofquantumfluctuations. Suchproblemsare theories and gravitation can be constructed. For ex-
0 hardly peculiar to condensates: the quantum neighbor- ample, 3He has been proposed as a laboratory counter-
0 hoods of interesting classicalbackgrounds are important part of high-energy particle physics. It has been argued
0
areas of researchin most fields of physics. But ultracold that, under appropriate conditions, excitations around
/
c dilute gases are so easy to manipulate and control, both thegroundstateofthesystemmayresembletheparticle-
q
experimentally [1] and theoretically [2], that they may spectrum of gauge theories of high energy physics [3].
-
r allow us to decipher less amenable systems by analogy. These condensed matter systems have also been used to
g
Asanessayinsuchanapplicationofcondensates,inthis simulate topologicaldefects characteristicof gauge theo-
:
v paper we discuss the theoretical framework and propose riesandwhichareconsideredtohaveplayedacosmologi-
Xi anexperimentto createthe analogof a blackhole in the calroleintheearlystagesoftheevolutionoftheuniverse
laboratory and simulate its radiative instabilities. such as monopoles and cosmic strings [3].
r
a It is now commonly believed that, even in the con- The past decade has witnessed an increasing interest
text of elementary particle physics, quantum field the- in simulating gravitational configurations and processes
ory arises from a still unknown underlying structure: it in condensed matter systems in the laboratory. The key
is an effective dynamical theory, describing the low en- observation was originally made by Unruh [4,5] and fur-
ergy limit of collective phenomena of the underlying mi- ther analyzed by Visser [6,7]: phononic propagationin a
croscopic theory. From this viewpoint, our description fluid is described by a wave equation which, under ap-
of (more) fundamental phenomena, such as gravity or propriate conditions, can be interpreted as propagation
electromagnetism, is actually similar to the theoretical in an effective relativistic curved spacetime background,
descriptions of many phenomena of condensed matter. the spacetime metric being entirely determined by the
To understand superfluidity, superconductivity or dilute physical properties of the fluid under study, namely, its
Bose-Einstein condensation, we describe the dynamics density and flow velocity. Unruh urged a specific moti-
of the system in terms of collective modes (quasipar- vation [4] for examining the hydrodynamic analogue of
ticles) whose typical size is much larger than the dis- an event horizon [8], namely that as an experimentally
tances between the particles that constitute the under- and theoretically accessible phenomenon it might shed
lying medium; but even electrons and photons must be some light on the Hawking effect [9] (thermal radiation
considered as the quasiparticles of the deeper theory we from black holes, stationary insofar as the back reaction
do not yet know. In this sense we may say that the is negligible). In particular, one would like to gain in-
major difference between our fundamental theories and sight into the role in the Hawking process of ultrahigh
1
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 2
frequencies [5,10,11]. physics [14], and optics [15]; but the outstanding recent
An event horizon for sound waves appears in princi- experimental progress in cooling, manipulating and con-
ple wherever there is a surface through which a fluid trolling atoms [16] make Bose-Einstein condensates an
flows at the speed of sound, the flow being subsonic especially powerful tool for this kind of investigation.
on one side of the surface and supersonic on the other. The basic challenge of our proposal is to keep the
There is a close analogy between sound propagation on trapped Bose-Einstein gas sufficiently cold and well iso-
a background hydrodynamic flow, and field propagation lated to maintain a locally supersonic flow long enough
in a curved spacetime; and although hydrodynamics is to observe its intrinsic dynamics. Detecting thermal
only a long-wavelength effective theory for physical (su- phonons radiating from the horizons would obviously be
per)fluids, so also field theory in curved spacetime is to adifficultadditionalproblem,sincesuchradiationwould
be considereda long-wavelengthapproximationto quan- be indistinguishable from many other possible heating
tum gravity [5,7]. Determining whether and how sonic effects. This further difficulty does not arise in our pro-
blackholesradiatesound,inafullcalculationbeyondthe posal, however,because the black-hole radiation we pre-
hydrodynamicapproximationorinanactualexperiment, dictisnotquasistationary,butgrowsexponentiallyunder
canthusoffersomesuggestionsaboutblackholeradiance appropriateconditions. Itshouldthereforebeobservable
and its sensitivity to high frequency physics (beyond the in the next generationof atom traps, and may also raise
Planck scale). The possibility that such high frequencies new issues in the theory of gravitationalblack holes.
might have consequences for observably low frequency In this paper, we extend and generalize the results of
phenomena is one of the main reasons that black holes Ref.[17],including amoredetailedanalysisofthemodel
have deserved much attention: there is reason to expect considered there as well as a new qualitatively different
thataneventhorizoncanactasamicroscope,givingusa case. The paper is organized as follows. In Sec. II, we
viewintophysicsonscalesbelowthePlancklength. This show how sonic horizons in dilute Bose-Einsteinconden-
is because modes coming from an event horizon are red- sates may appear in the hydrodynamic approximation,
shifted into the low-energyregimeas they propagateout discuss the regime of validity of such approximation, as
to be observedfarawayfromthe blackhole. Conversely, wellthevalidityofone-dimensionalmodels. SectionIIIis
if we imagine tracking the observed signal back towards devotedtothestudyofsonichorizonsincondensatessub-
its source, the closer we get to the horizon, the shorter ject to tight ring-shapedexternalpotentials. We present
the wave length of the signal must be, until at the very numerical results showing that both stable and unstable
horizonwemusteitherreachinfiniteenergyscales,oren- black holes may be created under realistically attainable
counter a breakdown in general relativity and quantum conditionsincurrentornear-futurelaboratories. Wealso
field theory in curved spacetime [12]. study the nature of the dynamical instabilities that ap-
In understanding this problem, hydrodynamic and pear for certain configurations. In Sec. IV, we discuss a
condensed matter analogs of black holes may offer some different configuration, namely that of a sink-generated
oftheexperimentalguidanceotherwisedifficulttoobtain blackholeinaninfiniteone-dimensionalcondensate,and
in the case of gravity [12]. Under appropriate conditions show that there also exist black-hole configurations, al-
andapproximations(which canbe basicallysummarized thoughthey arenotstable. We summarize andconclude
inthe requirementthe wavelengths ofthe perturbations in Sec. V. The Appendix is devoted to the issues of
be sufficiently large), the propagation of collective fluc- redundancyandnormalizationofthedynamicallyunsta-
tuations (phonons) admits an effective general relativis- bleBogoliubovmodes(associatedwithcomplexeigenfre-
tic description, in terms of a spacetime metric. This quencies).
longwavelengthregimewouldcorrespondanalogicallyto
quantum field theory in curved spacetime. The effective
phononmetricmaydescribeblackholes,asingeneralrel- II. SONIC BLACK HOLES IN CONDENSATES
ativity,andsoaphononHawkingeffectmaybe possible;
and certainly the problem of arbitrarily high frequencies A Bose-Einstein condensate is the ground state of a
at the horizon is also present. But in this case, when at second quantized many body Hamiltonian for N inter-
shortwavelengthsthemetricapproximationisnolonger acting bosons trapped by an external potential V (x)
ext
valid and a more microscopic theory must be used in- [2]. At zero temperature, when the number of atoms is
stead,theaccuratemicroscopictheoryisactuallyknown. large and the atomic interactions are sufficiently small,
AndifthehydrodynamicsystemisadiluteBose-Einstein almostalltheatomsareinthesamesingle-particlequan-
condensate, the microscopic theory is actually tractable tumstateΨ(x,t),evenifthesystemisslightlyperturbed.
enough that we can make reliable calculations from first TheevolutionofΨisthengivenbythewell-knownGross-
principles. As we will argue, trapped bosons at ultralow Pitaevskii equation, which in appropriate units can be
temperature can indeed provide an analogue to a black-
holespacetime. Similaranalogueshavebeenproposedin
othercontexts,suchassuperfluidhelium[13],solidstate
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 3
written as This class of metrics can possess event horizons. For
instance,ifaneffectivesinkforatomsisgeneratedatthe
¯h2 4πa¯h2
i¯h∂ Ψ= 2+V + Ψ2 Ψ, center of a spherical trap (such as by an atom laser out-
t ext
(cid:18)−2m∇ m | | (cid:19) couplingtechnique[18]),andiftheradialpotentialprofile
is suitably arranged, we can produce densities ρ(r) and
where m is the mass of the individual atoms and a is
flow velocities v(x) = v(r)r/r such that the quantity
the scattering length. The wave function of the con- c2 v2 vanishesatarad−iusr =r ,being negativeinside
densate is normalized to the total number of atoms − h
and positive outside. The sphere at radius r is a sonic
d3xΨ(x,t)2 =N. h
| | event horizon completely analogous to those appearing
R Our purposes do not require solving the Gross-
in generalrelativistic black holes, in the sense that sonic
Pitaevskii equation with some given external potential
perturbations cannot propagate through this surface in
V (x);ourconcernisthepropagationofsmallcollective
ext the outwarddirection[4,5,7]. This canbe seenexplicitly
perturbations of the condensate, around a background
by writing the equation for the radial null geodesics of
stationary state
the metric g :
µν
Ψ (x,t)= ρ(x)eiϑ(x)e−iµt/h¯,
s r˙ = v c,
p ± − ±
where µ is the chemical potential. Thus it is only nec-
which can be obtained from setting the proper interval
essary that it be possible, in any external potential that
ds2 = g dxµdxν equal to zero and restricting the al-
can be generated, to create a condensate in this state. µν
lowed motion to the radial direction, so that
Indeed,manyrealistictechniquesfor“quantumstateen-
gineering,” to create designer potentials and bring con- (c2 v2)+2vr˙+r˙2 =0.
densates into specific states, have been proposed, and − −
even implemented successfully [16]; and our simulations The ingoing null geodesic r (t) is not affected by the
−
indicate that currently known techniques should suffice presence of the horizon and can cross it in a finite coor-
to generate the condensate states that we propose. dinate time t. The outgoing null geodesic r (t) on the
+
PerturbationsaboutthestationarystateΨs(x,t)obey other hand needs an infinite amount of time to leave the
the Bogoliubov system of two coupled second order dif- horizon since r˙ =0 at the horizon. The physical mech-
+
ferential equations. Within the regime of validity of the anism of the sonic black hole is quite simple: inside the
hydrodynamic (Thomas-Fermi) approximation [2], these horizon, the background flow speed v is larger than the
two equations for the density perturbation ̺ and the localspeedofsoundc,andsosoundwavesareinexorably
phaseperturbationφintermsofthelocalspeedofsound dragged inwards.
In fact there are two conditions which must hold for
¯h
c(x) 4πaρ(x), this dragged sound picture to be accurate. Wavelengths
≡ m
p larger than the black hole itself will of course not be
and the background stationary velocity draggedin,but merelydiffractedaroundit. And pertur-
bations must have wavelengths
¯h
v ϑ
π¯h π¯h
≡ m∇
λ , .
≫ mc mc 1 v/c
read | − |
p
Otherwise they do not behaveas sound wavessince they
m 4πa¯h
̺˙ = c2 φ+v̺ φ˙ = v φ ̺. lie outside the regime of validity of the hydrodynamic
−∇·(cid:16)4πa¯h ∇ (cid:17) − ·∇ − m
approximation. These short-wavelength modes must be
Furthermore,lowfrequencyperturbationsareessentially described by the full Bogoliubov equations, which allow
justwavesof(zero)sound. Indeed,theBogoliubovequa- signals to propagate faster than the local sound speed,
tions may be reduced to a single second order equation and thus permit escape from sonic black holes. So, to
for the condensate phase perturbation φ. This differen- identifyacondensatestateΨ asasonicblackhole,there
s
tial equationhas the form ofa relativistic waveequation must exist modes with wavelengths larger than these
∂µ(√ ggµν∂νφ) = 0, with g = detgµν, in an effective lower limits (which in terms of the local healing length
−
curved spacetime with the metric gµν being entirely de- ξ(x) ¯h/[mc(x)] read λ 2πξ, 2πξ/ 1 v/c), but
≡ ≫ | − |
termined by the local speed of sound c and the back- also smaller than the black hole size. pEven if such an
ground stationary velocity v. Up to a conformal factor, intermediate range does exist, the modes outside it may
this effective metric has the form still affect the stability of the black hole as discussed be-
low.
(c2 v2) vT
(gµν)= − −v −1 . As it stands, this description is incomplete. The con-
(cid:18) (cid:19)
− densate flows continually inwards and therefore at r =0
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 4
there must be a sink that takes atoms out of the con- havelongbeenclear,inasmuchasthecurrentdilute con-
densate. Otherwise, the continuity equation (ρv) = 0, densatesreallyaretheweaklyinteractingBosegasesthat
∇
which must hold for stationary configurations will be vi- havebeenusedastoymodelsforsuperfluidityforseveral
olated. From the physical point of view, such a sink can decades. The fact that actual critical velocities in liquid
beaccomplishedbymeansofanoutcouplerlaserbeamat helium are generallyfar below the Landaucriticalveloc-
the origin. (Such outcouplers are the basic mechanisms ity isunderstoodtobe due partlyto the rotonfeatureof
for making trapped condensates into “atom lasers,” and the helium dispersion relation, but this is not present in
they have already been demonstrated experimentally by the dilute condensates. Viscosity also arises due to sur-
severalgroups. A tightly focusedlaser pulse changesthe face effects, however, and these may indeed afflict dilute
internal state of the atoms at a particular point in the condensatesaswell. Thepointhereisthatinadditionto
trap,and canalsobe made to give them a largemomen- thebulkphononmodesconsideredbyLandau,andquite
tum impulse. This ejects them so rapidly through the adequately represented in our one-dimensional analysis,
always dilute condensate cloud that they do not signifi- there may inprinciple be surfacemodes, with a different
cantly disturb it; effectively, they simply disappear.) (and generally lower) dispersion curve. If such modes
We have analyzed several specific systems which may exist and are unstable, it is very often the case that, as
be suitable theoretical models for future experiments, theygrowbeyondtheperturbativeregime,theyturninto
and have found that the qualitative behavior is analo- quantized vortices, which can cut through the supercur-
gousinallofthem. Blackholeswhichrequireatomsinks rent and so lower it.
areboththeoreticallyandexperimentallymoreinvolved,
however; moreover, maintaining a steady transonic flow
into a sink may require either a very large condensate
or some means of replenishment. We will therefore first
discussanalternativeconfigurationwhichmaybeexper-
imentally more accessible and whose description is par-
ticularly simple: a condensate in a very thin ring that
effectively behaves as a periodic one-dimensional system
(Fig.1). Underconditionsthatwewilldiscuss,thesuper-
sonic region in a ring may be bounded by two horizons:
ablackholehorizonthroughwhichphononscannotexit,
anda‘whitehole’horizonthroughwhichtheycannoten-
ter. Thenwewillanalyzeanothersimpleone-dimensional FIG. 2. The tight cigar-shaped configuration, with two
model, of a long, straight condensate with an atom sink black hole horizons and a ‘singularity’ where condensate is
at the center (Fig. 2). outcoupled. Arrows indicate condensate flow velocity, with
longer arrows for faster flow.
Whether or not such unstable surface modes actually
existintheBogoliubovspectrumofadilutecondensateis
anissuethathasrecentlybeenanalyzedbothnumerically
and analytically, and it is quite clear that such surface
modes exist only if the confining potentialis quite rough
(whichis notonlyeasyto avoidwith amagneticoropti-
caltrappingfield,butveryhardtoachieve)[19],orifthe
condensate dynamics in the directions perpendicular to
the flow is hydrodynamic. That is, the condensate must
be at least a few healing lengths thick, so that surface
modes decaying on the healing length scale can satisfy
alltherequiredboundaryconditions[20]. Bysayingthat
weareconsideringaneffectivelyone-dimensionalconden-
FIG. 1. The tight ring-shaped configuration, with both
sate, we mean precisely that this is not the case. For
black and white horizons, and no singularity. Arrows indi-
instance, for the tight-ring model, in this regime, the ra-
cate condensate flow velocity, with longer arrows for faster
dialtrapscaleistheshortestlengthscaleintheproblem,
flow.
and the radial trap frequency is the highest frequency;
thiseffectivelymeansthatexcitationsofnontrivialradial
Theexistenceofinstabilitiesthatdonotshowupinthe
one-dimensionalapproximationis animportantquestion modes, including surface modes, are energetically frozen
in condensate physics, which is under active theoretical out. (In the limit of radial confinement within the scat-
and experimental investigation. The essential principles tering length, our model breaks down for other reasons
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 5
— but the scattering length can easily be two orders of on wavelengths shorter than the inner and the outer re-
magnitudesmallerthanthehealinglength). Theissueof gionsareindeedphononicimplies 2π,whichinturn
U ≫
the supercurrentstabilityintightlyconfinedringshaped requires w 1 and 1 b 1/w2. In fact, detailed
traps has been addressed in Ref. [21] where the authors analysis sho≫ws that w >≫5 is s≫ufficient.
arriveatapositive conclusionandalsoclarifythe roleof ∼
finite temperature and possible trap anisotropy.
A. Stability
III. SONIC BLACK/WHITE HOLES IN A RING The mere existence of a black hole solution does not
necessarilymeanthatitisphysicallyrealizable: itshould
In a sufficiently tight ring-shaped external potential also be stable over sufficiently long time scales. Since
of radius R, motion in radial (r) and axial (z) cylin- stability must be checked for perturbations on all wave-
drical coordinates is effectively frozen. We can then lengths, the full Bogoliubov [2] spectrum must be deter-
write the wave function as Ψ(z,r,θ,τ) = f(z,r)Φ(θ,τ) mined. For largeblack holeswithin large,slowlyvarying
and normalize Φ to the number of atoms in the con- condensates, this Bogoliubov problem may be solved us-
densate 2πdθ Φ(θ)2 = N, where with the azimuthal ing WKB methods that closely resemble those used for
0 | |
coordinaRte θ we have introduced the dimensionless time solvingrelativisticfieldtheoriesintrue blackholespace-
τ = (h¯/mR2)t. The Gross-Pitaevskii equation thus be- times [11]. A detailed adaptation of these methods to
comes effectively one-dimensional: the Bogoliubovproblemwillbe presentedelsewhere[22].
The results are qualitatively similar to those we have
i∂ Φ= 1∂2+ + U Φ2 Φ, (1) found for black holes in finite traps with low winding
τ (cid:18)−2 θ Vext N| | (cid:19) number, where we have resorted to numerical methods
because, in these cases, WKB techniques may fail for
where 4πaNR2 dzdrrf(z,r)4 and (θ) is
ext just those modes which threaten to be unstable.
U ≡ | | V
the dimensionless effectRive potential (in which we have Our numerical approach for our three-parameter fam-
already included the chemical potential) that results
ily of black/white holes in the ring-shaped condensate
from the dimensional reduction. The stationary solu-
has been to write the Bogoliubov equations in discrete
tion can then be written as Φ (θ,τ) = ρ(θ)ei dθv(θ) Fourier space, and then truncate the resulting infinite-
s
R
and the local dimensionless angular sppeed of sound dimensional eigenvalue problem. Indeed, writing the
as c(θ) = ρ(θ)/N. Periodic boundary conditions wave funtion as Φ = Φ +ϕei dθv(θ), decomposing the
U s
around thepring require the “winding number” w perturbation ϕ in discrete modRes
2π ≡
(1/2π) dθv(θ) to be an integer.
The Rq0ualitative behavior of horizons in this system is ϕ(θ,τ)= e−iωτeinθAω,nuω,n(θ)+eiω∗τe−inθA∗ω,nvω∗,n(θ),
well representedby the two-parameterfamily of conden- Xω,n
sate densities
and substituting into the Gross-Pitaevskii equation we
N obtain the following equation for the modes u and
ρ(θ)= (1+bcosθ), ω,n
2π vω,n:
where b∈[0,1]. Continuity,∂θ(ρv)=0, then determines ω uω,n = h+np fnp uω,p .
the dimensionless flow-velocity field (cid:18) vω,n (cid:19) Xp (cid:18)−fnp h−np (cid:19)(cid:18) vω,p (cid:19)
w√1 b2
In this equation,
v(θ)= U − ,
2πc(θ)2
1 2π
f = dθe−i(n−p)θc(θ)2,
which depends on w as a third discrete independent pa- np
2π Z
0
rameter. Requiring that Φ (θ,τ) be a stationary solu-
s n2 1 2π
tion to Gross-Pitaevskii equation then determines how h± = δ + dθe−i(n−p)θ
np ± 2 np 2π Z
the trapping potential must be modulated as a function 0
of θ. All the properties of the condensate, including pv(θ) 1v′(θ) c(θ)2+ 1c′′(θ) ,
whether and where it has sonic horizons, and whether ×(cid:20) − 2 ±(cid:18) 2 c(θ) (cid:19)(cid:21)
or not they are stable, are thus functions of , b and w.
U which,aftersomelengthycalculations,canbe writtenas
Forinstance,ifwerequirethatthehorizonsbelocatedat
θh = π/2,whichimposestherelation =2πw2(1 b2), b b
then w±e must have c2 v2 positive forUθ ( π/2,−π/2), fnp = 2Uπ (cid:18)δn,p+ 2δn,p+1+ 2δn,p−1(cid:19),
− ∈ −
zeroatθ = π/2,andnegativeotherwise,providedthat
U < 2πwh2. T±he further requirement that perturbations h±np = 21(n+p)wp1−b2αn−p
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 6
4n2 1 1 b2 More serious are dynamical instabilities, which occur
f + − δ + − β ,
np n,p n−p
±(cid:18) 8 8 (cid:19) for modes with complex eigenfrequencies. Since the Bo-
goliubovtheoryisbasedonaquantizedHamiltonianthat
where
isHermitian,therearecertainlynocomplexenergyeigen-
∞ j values; but the natural frequencies of normal modes can
b j
α = − , indeedbe complex[inwhichcasethe usualrule,thaten-
i (cid:18) 2 (cid:19) (cid:18)(i+j)/2(cid:19)
j≥|i|,Xi+j even ergy eigenvalues are ¯h(n+1/2)times the mode frequen-
∞ b j j cies, simply breaks down]. A detailed discussion of the
β = − (j+1). quantummechanicsofdynamicalinstability is presented
i (cid:18) 2 (cid:19) (cid:18)(i+j)/2(cid:19)
j≥|i|,Xi+j even intheAppendix; forthe purposesofourmaindiscussion
it suffices to note that complex (mode) eigenfrequencies
EliminatingFouriercomponentsaboveasufficientlyhigh areindeedgenuinephysicalphenomena,andbynomeans
cutoff Q has negligible effect on possible instabilities, a numerical artifact. For sufficiently high values of the
which can be shown to occur at relatively long wave- cutoff (e.g., Q 25 in our calculations), the complex
lengths. We face then an eigenvalue problem for the ≥
eigenfrequencies obtained from the truncated eigenvalue
2(Q +1) 2(Q + 1) matrix built out of blocks of the
problembecomeindependentofthecutoffwithinthenu-
×
form
merical error. The existence and rapidity of dynamical
h+np fnp . sinesetFabigil.it3iefsordaepceonndtosuernspiltoivteolyftohne(mUa,xb,imwu).mFoofrtinhsetaabnscoe-,
(cid:18)−fnp h−np (cid:19) lute values of the imaginary part of all eigenfrequencies
The numerical solution to this eigenvalue equation, to- for w = 7, showing that the regions of instability are
gether with the normalization condition long, thin fingers in the ( ,b) plane. Figure 4 shows the
U
aRlldoθw(ued∗ω∗f,rneuqωu′e,nn′c−iesv.ω∗∗R,nevaωl′,nne′)ga=tivδenne′iδgωeωn′f,repqruoevnicdieesstfhoer seiazcehopfotihnet olanrgtehsetdaabsshoeludtecuvravleueofofFitgh.e3in.sIttabililluitsitersatfoers
modes of positive normare alwayspresent, which means the important fact that the size of the imaginary parts,
that blackhole configurationsare energeticallyunstable, which gives the rate of the instabilities, increases start-
as expected. This feature is inherent in supersonic flow, ingfromzero,quiterapidlywithb,althoughtheyremain
sincethespeedofsoundisalsotheLandaucriticalveloc- small as compared with the real parts.
ity. Inasufficientlycoldanddilutecondensate,however,
the time scale for dissipation may in principle be made
verylong,andsotheseenergeticinstabilitiesneednotbe
problematic [23].
FIG. 4. Stability diagram for black/white holes of maxi-
mum size, i.e., along thedashed line of Fig. 3.
FIG.3. Stabilitydiagramforwindingnumberw=7. Solid
dark-grey areas represent the regions of stability. Smaller B. Creation of a black/white hole
plots at higher resolution confirm that the unstable ‘fingers’
are actually smooth and unbroken. Points on the dashed
The stability diagramof Fig. 3 suggests a strategy for
curve are states with horizons at θh = ±π/2, so that the
creating a sonic black hole from an initial stable state.
black/white hole fills half the ring.
Within the upper subsonicregion,the verticalaxisb=0
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 7
corresponds to a homogeneous persistent current in a
ring, which can in principle be created using different
techniques [24]. Gradually changing and b, it is possi-
U
ble to move from such an initial state to a black/white
hole state, along a path lying almost entirely within the
stable region, and only passing briefly through instabili-
tieswheretheyaresufficientlysmalltocausenodifficulty.
Indeed,wehavesimulatedthisprocessofadiabaticcre-
ation of a sonic black/white hole by solving numerically
(using the split operator method) the time-dependent
Gross-Pitaevskiiequation(1)thatprovidestheevolution
of the condensate when the parameters of the trapping
potentialchangesoastomovethecondensatestatealong
various paths in parameter space. One of these paths is
shown in Fig. 3 (light-grey solid line): we start with a
current at w = 7, b = 0, and sufficiently high [Fig.
U
5(a)]; we then increase b adiabatically keeping fixed
U
until an appropriate value is reached [Fig. 5(b)]; finally,
keeping b constant, we decrease adiabatically (which
U
can be physically implemented by decreasing the radius
of the ring trap), until we meet the dashed contour for
black holes of comfortable size [Fig. 5(c)]. Our simu-
lations confirm that the small instabilities which briefly
appear in the process of creation do not disrupt the adi-
abatic evolution. The final quantum state of the con-
densate, obtained by this procedure, indeed represents
a stable black/white hole. We have further checked the
stability of this final configuration by numerically solv-
ing the Gross-Pitaevskii equation (1) for very long pe-
riods of time (as compared with any characteristic time
scale of the condensate) and for fixed values of the trap
parameters. Thisevolutionreflectsthefactthatnoimag-
inaryfrequenciesarepresent,aspredictedfromthemode
analysis, and that the final state is indeed stationary
[Fig. 5(d)]. Once the black/white hole has been cre-
ated, one could further change the parameters ( ,b) so
U
as to move between the unstable ‘fingers’ into a stable
region of higher b (a deeper hole).
C. Quasiparticle pair creation
Instead of navigating the stable region of parameter
space, one could deliberately enter an unstable region
[Fig. 5(e)–5(i)]. In the this case, the black hole should
disappearinanexplosionofphonons,whichmaybeeasy
todetectexperimentally. Suchaneventmightberelated
FIG.5. Simulationofcreation ofastableblack/whitehole
and subsequent evolution into an unstable region. Figures totheevaporationprocesssuggestedforrealblackholes,
a–d are snapshots taken at the initial time (a), at an inter- in the sense that pairs of quasiparticles are created near
mediate time, still within the subsonic region (b), when the the horizon in both positive and negative energy modes.
black/whiteholesofmaximumsizeisapproached(c),andaf- We will explain this point briefly; a more detailed expo-
teralongtimeinthatconfiguration(d). Thentheparameters sition is included in the Appendix.
are changed along thedashed curveof Fig. 3to enteran un- In the language of second quantization, the perturba-
stable region (e) and keptthere(f–i). Itcan beobserved that
a perturbation grows at the black hole horizon and travels
rightwards until it enters thewhite hole horizon.
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 8
tion field operator ϕ satisfies the linear equation present regions of stability and instability in parameter
space and, in this sense, their behavior is analogous to
1 1c′′ i
iϕ˙ = ϕ′′ ivϕ′+ v′+c2 ϕ+c2ϕ†, thatinaring-shapedtrappingpotential. Herewepresent
−2 − (cid:18)2 c − 2 (cid:19) asimplemodelthatexhibitsthemainqualitativefeatures
of more general situations and that can be studied ana-
which,takingintoaccountthat[ϕ(θ),ϕ†(θ′)]=δ(θ θ′),
− lytically. Althoughin this model, we study a condensate
can be written as
of infinite size, in more realistic models or experiments,
it will suffice to take condensates which are sufficiently
iϕ˙ =[ϕ,H],
large, since the stability pattern is not significantly af-
where the Bogoliubov Hamiltonian is fected by the (large but finite) size of the condensate.
1 1c′′ i
H = dθ ϕ†ϕ′′ ivϕ†ϕ′+ v′+c2 ϕ†ϕ
Z (cid:20)−2 − (cid:18)2 c − 2 (cid:19) A. The model
1
+ (ϕ†ϕ†+ϕϕ) . (2) Letusconsideratightcigar-shapedcondensateofinfi-
2 (cid:21)
nite size suchthat the motionin the (y,z)plane is effec-
The Hermiticity of the Bogoliubov linearized Hamil- tively frozen. In appropriate dimensionless units, the ef-
tonian implies that eigenmodes with complex frequen- fectively one-dimensionalGross-Pitaevskiiequation thus
cies appear always in dual pairs, whose frequencies are becomes:
complex conjugate. In the language of second quantiza-
1
tion, the linearized Hamiltonian for each such pair has i∂ Φ= ∂2+ + Φ2 Φ,
τ (cid:18)−2 x Vext U| | (cid:19)
the form
H = (ωA†ω∗,nAω,n+ω∗A†ω,nAω∗,n), with the normalization condition
Xn 1 D
lim dxΦ(x)2 =n.
andtheonlynonvanishingcommutatorsamongtheseop- D→∞2D Z−D | |
erators are [Aω,n,A†ω∗,n′] = δnn′. The asterisk on the
subscript is important: the mode with frequency ω∗ is a Inthisequation, extisthedimensionlesseffectivepoten-
V
differentmodefromtheonewithfrequencyω,andA†ω∗,n tial that results form the dimensional reduction, which
already includes the chemical potential.
is not the Hermitian conjugate of A† . It is therefore
ω,n In order to obtain a black hole configuration, let us
clear that none of these operatorsis actually a harmonic
choosethe potential so thatit producesa profile for
oscillator creation or annihilation operator in the usual Vext
the speed of sound c(x)= ρ(x) of the form
sense. However, the linear combinations U
p
an = √12(Aω,n+Aω∗,n) , bn = √i2(A†ω,n+A†ω∗,n) c(x)= cc00[,1+(σ 1)x/ǫ], |Lx|<<xL<L+ǫ ,
 σc , − L+ǫ|<| x
0
| |
and their Hermitian conjugates are true annhilation and

creationoperators,with the standardcommutationrela- with σ > 1, and a flow velocity in the inward direction.
tions, and in terms of these the Bogoliubov Hamiltonian The continuity equation then provides the flow velocity
becomes profile
H = Re(ω)(a†nan−b†nbn)−Im(ω)(a†nb†n+anbn) . v(x)= v0c20 x ,
Xn (cid:2) (cid:3) −c(x)2 x
| |
This interaction obviously leads to self-amplifying cre- where v is the absolute value of the flow velocity in the
0
ation of positive and negative frequency pairs. Evapora- inner region.
tion through an exponentially self-amplifying instability As it stand this model fails to fulfill the continuity
is not equivalent, however, to the usual kind of Hawk- equationat x=0. In order to take this into account, we
ing radiation [11]; this issue will be discussed in detail will also introduce a sink of atoms at x = 0 that takes
elsewhere. atoms out of the condensate (this can be physically im-
plemented by means of a laser). From the mathematical
point of view, it can be modeled by an additional term
IV. SINK-GENERATED BLACK HOLES
in the equation of the form iEδ(x) which indeed in-
−
duces loss of atoms at x = 0. Equivalently, it can be
Condensates which develop black hole behaviors by represented by boundary conditions of the form
means of flows generated by laser-driven sinks also
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 9
Φ(0+,τ) Φ(0−,τ)=0, IntheintermediateregionsL< x <L+ǫ,thefactors
− x | |
Φ′(0+,τ) Φ′(0−,τ)= 2iEΦ(0,τ), (3) e±2i v in the last terms of Eqs. (4) become 1+ (ǫ).
− − R O
Then the solution of these equations is
which determine the flow velocity inside in terms of the
characteristics of the outcoupler laser, namely, v =E. u =α +β x/ǫ+ (ǫ2),
0 ω,k ω,k ω,k
O
Perturbations φ around this stationary state Φs = v =γ +κ x/ǫ+ (ǫ2) (5)
ω,k ω,k ω,k
√ρei v(x)dx, suchthat Φ=Φs+φ(note thatfor conve- O
niencRewehavechosenadifferentconventionascompared as can be easily seen by defining the variable q =x/ǫ so
withthe ringinwhichΦ=Φ +ϕei v)mustsatisfythe that the equations become
s
R
boundary conditions (3) and the equation
∂2u =∂2v = (ǫ2).
q ω,k q ω,k O
1 x
iφ˙ = φ′′+(c2 v2/2+c′′/2c)φ+c2e2i vφ∗ The singular character of c′′/c at x = L, L+ǫ can
−2 − R | |
be substituted by matching conditions at x =L, L+ǫ
| |
where which will in turn provide the connection formulas be-
tween the modes outside (x >L) and the modes inside
c′′ σ 1 1 | |
= − [δ(x L) δ(x L ǫ)]. (x < L). Furthermore, the symmetry of the problem
c ǫ | |− − σ | |− − | |
allows us to study the region x>0.
Asafurthersimplifyingassumption,wewillassumethat These matching conditions are
v ǫ 1 so that
0 ≪ φ(L+) φ(L−)=0,
−
x L+ǫ v σ 1
dx′v(x′) dx 0 v ǫ 1. φ′(L+) φ′(L−)= − φ(L),
(cid:12)Z (cid:12)≤Z [1+(σ 1)x/ǫ]2 ≤ 0 ≪ − ǫ
(cid:12)(cid:12) L (cid:12)(cid:12) L − φ(L+ǫ+) φ(L+ǫ−)=0,
(cid:12)Let us now(cid:12)expand the perturbation φ in modes −
σ 1
φ′(L+ǫ+) φ′(L+ǫ−)= − φ(L+ǫ).
φ= A u (x)e−iωτ +A∗ v (x)∗eiω∗τ . − − σǫ
ω,k ω,k ω,k ω,k
Xω,kh i These equations together with the form of the modes in
theregionL<x<L+ǫ,providetheconnectionformulas
Then, the modes uω,k(x) and vω,k(x) satisfy, in each re- between the inside and outside modes (from now on we
gion, the Bogoliubov equations will drop the subindex ω):
ωuω,k =−21u′ω′,k+(c2−v2/2)uω,k+c2e2iRxvvω,k, uin,k(L)=−ǫu′out,k(L)+ σ1uout,k(L),
ωvω,k = 12vω′′,k−(c2−v2/2)vω,k−c2e−2iRxvuω,k. (4) u′in,k(L)=σu′out,k(L),
and likewise for the modes v .
in,out
B. Matching conditions
C. Dispersion relation and boundary conditions for
large x
The intermediate regions L< x <L+ǫ, provide the
| |
connection between the perturbation modes in the inner
Ineachoftheregions(insideandoutside),wecanwrite
and outer regions. Once these connection formulas have
beenestablished,inthelimitofsmallǫ,wewillonlyneed u (x)=u ei(k−|v|)(x−L), v (x)=v ei(k+|v|)(x−L).
k k k k
to study the inside and outside modes and their relation
through such formulas. The case of an abrupt horizon, Upon substitution of this expansion into the Bogoliubov
in which the background condensate velocity is steeply equations (4), we obtain, for each region, the following
andlinearlyrampedwithinaveryshortinterval,isobvi- set of algebraic equations:
ously quite special; andit does notparticularlyresemble
the horizon of a large black hole in Einsteinian gravity. h−u +c2v =0, c2u +h+v =0,
k k k k k k
But the connection formula that we derive for this case
will qualitatively resemble those that are obtained, with whereh±k =k2/2+c2±(k|v|+ω). Fortheseequationsto
considerablymoretechnicaleffort,forsmootherhorizons have a solution, the determinant must vanish thus pro-
[22]. AndtheresultswewillobtainfortheglobalBogoli- viding the dispersion relation
ubov spectrum of the condensate black hole will indeed
k4/4+(c2 v2)k2 2ω v k ω2 =0 (6)
be representative of more generic cases. − − | | −
Garay, Anglin, Cirac, and Zoller. Sonic black holes in dilute Bose-Einstein condensates 10
which, for fixed ω, is a fourth-order equation for k. For D. Connection formulas for complex frequencies
each of the four solutions u and v must be related by
k k
Since we are interested in the existence of dynamical
1
vk =hkuk with hk =−c2(k2/2+c2−k|v|−ω). instabilities, we will concentrate in the case in which ω
is complex. Then, as we have seen, the dispersion equa-
The constant coefficients uk can be regarded as normal- tion (6) has four complex solutions for k in each region.
ization constants and will be set to unity. Let us study Inside, all four solutions k , i = 1 4 are in prin-
in,i
the possible solutions to the dispersion relation depend- ciple possible but outside those with I·m··(k ) < 0 will
out
ing on whether ω is real or complex. increaseexponentially. Therefore,uptocorrectionscom-
Complex frequencies.— Inthis caseallfoursolutions ing from the finite size of the condensate, which we ig-
are pure complex, two of them with positive imaginary nore here, only modes associated with k , α = 1,2
out,α
part and two of them with negative one. such that Im(k ) > 0 are allowed. Each mode
out,α
In order to prove this statement, let us first assume uout,α(x)=ei(kout,α−v0/σ2)(x−L) will match a linear com-
that there exists a real solution k for a complex ω =
bination u (x) = F u (x) of modes u (x) in-
̟+iγ. Then the imaginary part of Eq. (6) implies that in,α i αi in,i in,i
side, i.e., P
̟ = k v . Introducing this result into its real part
− | |
leadstoγ2 =−(k4/4+c2k2)whichisimpossibletofulfill uin,α(x)= Fαiei(kin,i−v0)(x−L)
because k is real. So the four solutions are complex.
Xi
Because of continuity, all ω in the upper half complex
ω plane have the same number ofsolutions with positive and similarly for vout,α and vin,α:
imaginary part. Otherwise, for some ω there should ex-
ist a realsolutionthat interpolates between positive and vin,α(x)= Fαihin,iei(kin,i+v0)(x−L).
negative imaginary part solutions, but this is not possi- Xi
ble, as we have seen.
Aftersomestraightforwardcalculations,itcanbeseen
Nowletusconcentrateonsmallfrequencies,i.e.,onfre-
thatthese connecting coefficients F aregivenby F =
quenciesaroundω =0. Forω =0,wehaveadoubleroot αi αi
(M−1) C , where
at k = 0. The other two solutions are k = 2√v2 c2, j ij αj
± − P
which are real for c2 < v2 (i.e., inside) and pure imagi-
1 1 1 1
nary for c2 > v2 (i.e., outside). Let us follow these four
 k− k− k− k− 
solutions when ω = iε. The solutions coming from the M = in,1 in,2 in,3 in,4 ,
h h h h
double root k =0 will now be of the form k =kr+iεki. h ink,1+ h ink,2+ h ink,3+ h ink,4+ 
It is easy to see that kr = 0 and ki = 1/( v c). If  in,1 in,1 in,2 in,2 in,3 in,3 in,4 in,4 
−| |±
c > v , one is positive and one is negative. If c < v ,
| | | |
both of them are negative. On the other hand, the solu- 1/σ iǫk−
ctioonnjsugka=te.±F2o√rvc22−<c2v,2,fowrec2w>ritve2k, a=rekarlr+eaidεykicaonmdplienx- Cα = (1/σ σi−ǫkko−+ut,αou)th,α .
troduce it into Eq. (6). We then see that at first order  − out,α out,α 
 σk+ h 
in ε, k = v /(v2 c2)>0. Thus, we have seen that for  out,α out,α 
i
| | −
ω = iε, we have two solutions with positive imaginary
In these equations,
part and two with negative one in any case (inside and
outside). But if this is so for ω = iε it must be true in k± =k v and k± =k v /σ2.
in,i in,i± 0 out,α out,α± 0
the whole upper ω plane and consequently in the whole
2
complex ω plane.
In the inside region all possible solutions are in prin- E. Boundary conditions at the sink. Complex
ciple allowed but outside we are only left with the two eigenvalues
that have Im(k) > 0, because the other two grow expo-
nentially. We have already found the modes in the inner and
Real frequencies.— Outside (c2 > v2), there are two outerregionaswellastheirrelation. Todeterminewhich
real and two complex conjugate solutions. Of these two (complex-frequency) modes will be present, there only
complex solutions, only one is allowed [the one with remains to impose the boundary conditions dictated by
Im(k) > 0] because the other grows exponentially. In- the presence of the sink at x=0.
side (c2 < v2), for ω > ω there are two real and two As we have already mentioned, the symmetry of the
max
complexconjugatesolutions;forω <ω therearefour system under reflection (x x) allows us to study
max
→ −
real solutions; the value ω =ω is a bifurcating point. only the region x > 0, provided that we study the even
max
and odd perturbations separately. For odd fluctuations