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Bogoliubov theory of the Hawking effect in BoseEinstein condensates

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS

J. Opt. B: Quantum Semiclass. Opt. 5 (2003) S42S49 PII: S1464-4266(03)55826-8

Bogoliubov theory of the Hawking effectin BoseEinstein condensatesU Leonhardt 1, T Kiss 1,2,3 and P Ohberg 1,4

1 School of Physics and Astronomy, University of St Andrews, North Haugh,St Andrews KY16 9SS, UK2 ResearchInstitutefor Solid State Physics andOptics, H-1525Budapest, PO Box49, Hungary3 Institute of Physics, University of P ecs, Ifj usag u. 6 H-7624 P ecs, Hungary4 Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK

Received 4 November 2002Published 2 April 2003Online at stacks.iop.org/JOptB/5/S42

AbstractArticial black holes may demonstrate some of the elusive quantumproperties of the event horizon, in particular Hawking radiation. Onepromising candidate is a sonic hole in a BoseEinstein condensate. Weclarify why Hawking radiation emerges from the condensate and how thiscondensed-matter analogue reects some of the intriguing aspects of quantum black holes.

Keywords: Articial black holes, BoseEinstein condensates

1. Introduction

Picture a BoseEinstein condensate owing through a nozzlewhere the condensate exceeds the speed of sound. Supposethat the nozzle is designed such that the trans-sonic ow doesnot become turbulent. One could build such a nozzle, theequivalent of the Laval nozzle [1], out of light, using the dipoleforce between light and atoms to conne the condensate inan appropriate potential. Consider the fate of sound wavespropagating against the current of the trans-sonic condensate.In the subsonic region sound waves may advance against theow, whereas in the supersonic zone they are simply sweptaway. No sound can escape the point where the ow turnssupersonicthe sonic horizon (gure 1). The trans-sonic uidacts as the acoustic equivalent of the black hole [2, 3].

An articial black hole [4] of this kind could beemployed to demonstrate some elusive quantum propertiesof the event horizon in the laboratory, in particular Hawkingradiation[5, 6]. Hawking [7,8] predictedthat theevent horizonemits quanta as if the horizon had a temperature given by thegradient of the gravitational potential. To be more precise, thehorizon should spontaneously emit quantum pairs where oneparticle of each pair falls into the hole and the other escapesinto space, constituting the radiation of the horizon. Both thespectral distribution and the quantum state of the emergingradiation are thermal. For solar-mass or larger black holes theHawking temperature is in the order of 10 7 K or below, whichmakes the effect next to impossible to observe in astronomy.In the case of sonic holes the Hawking temperature is given by

u

Figure 1. Schematic diagram of a sonic horizon. A uid is forced tomove through a constriction where the ow speed u becomessupersonic (dashed line). The constriction may be formed by thewalls of a tube or, if the uid is an alkali BoseEinstein condensate,by a suitable trapping potential. The picture shows a Lavalnozzle [1] where a supersonic uid is hydrodynamically stable.

the velocity gradient at the sonic horizon [2, 3],

k B T = h 2

, (1)

and the emitted quanta are phonons. A velocity gradientof 10 3 Hz would correspond to about 1.2 nK temperature.In order to observe such subtle quantum effects one shouldemploy the best and coldest superuids availableBoseEinstein condensates of dilute gases [9, 10].

Some detailed schemes for sonic black holes in BoseEinstein condensates have been investigated theoretically [1115]. The ultimate design depends on experimental details andon the state of the art in manipulating condensates, a rapidlyevolving eld. In this paper we analyse the general aspectsof the Hawking effect in BoseEinstein condensates. In therst part of the paper we collect and combine the ingredientsof the effect, results that are scattered in the literature. In thesecond part we show how the Hawking effect arises naturally

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within the Bogoliubov theory of the elementary excitationsin BoseEinstein condensates [16, 17]. For the rst time, toour knowledge, we connect the quantum physics of the eventhorizon to the behaviour of a realistic quantum uid.

2. Sound in uids

Consider the propagation of sound in uids moving with owspeed u . Suppose that the ow varies little over the scaleof an acoustic wavelength. In this regime we can describesound propagation in geometrical acoustics (the equivalent of geometrical optics [18] or of the semiclassical approximationin quantum mechanics [19]). Sound rays follow Hamiltonsequations,

dr

dt = k

,dk

dt = r

, (2)

where the dispersion relation between the frequency andthe wavevector k denes the effective Hamiltonian ( r , k ) .Assume that in each uid cell

2 = c2k 2 , (3)where c denotes the speed of sound and refers to thefrequencyin locally comovingframes. In the laboratoryframe, is Doppler shifted,

= u k . (4)In order to see why waves in uids are related to waves ingeneral relativity, we write the dispersion relation (3) in arelativistic form. We introduce the spacetime wavevector

k = (, k ) (5)and the matrix

g = 21 uu c21I + u u

. (6)

The prefactor is an arbitrary non-vanishing function of thecoordinates called the conformal factor. In this notation thedispersion relation appears in the relativistic form

g k k = 0, (7)adopting Einsteins summation convention. Therefore, soundwaves experience the moving uid as an effective spacetimegeometry with the metric g , the inverse matrix of g , given

byg = 2

c2 u 2 uu 1I. (8)

The analogy between sound waves in uids and waves ingeneral relativity [2, 3, 20] turns out to be exact for anirrotational uid [21] with arbitrary density prole 0 , ow uand speed of sound c, where 0, u and c may vary in space andtime. The velocity potential and the density perturbations s of sound obey the linearized equation of continuity and thelinearized Bernoulli equation [22]

t s + (u s + 0 ) = 0, (9)

( t + u

) + c2 s

0 = 0. (10)

As a consequence, the velocity potential of sound obeys theequation [3]

t 0c2

( t + u ) + 0c2

[u t (c2 u 2) ] = 0, (11)which can be written as the relativistic wave equation [2, 3]

D D = 1 g

gg = 0 (12)

with the conformal factor , in d spatial dimensions, chosenas [3]

= 0c3

1/( d 1). (13)

Theassumptionsmadein orderto derive thewave equation(11)are that the uid [22] is irrotational (1) and isentropic (2).BoseEinstein condensates naturally satisfy condition (1).Condition (2) characterizes the hydrodynamic regime of condensates [9, 10]. Here the local pressure depends onlyon the density and on the temperature of the uid and thequantum pressure is negligible [9, 10]. Condition (2) turns outto be violated close to the sonic horizon.

3. Sonic horizon

Consider the propagation of sound waves in the vicinity of the sonic horizon. Focus on the physics in the direction zof the ow at the horizon in a quasi-one-dimensional model.Assume that the speed of sound in the uid is constant. Thewave equation (11) reads explicitly

t ( t + u z) + z[u t (c2 u 2) z] = 0. (14)We obtain the general solution

= 0( t ), = d zc u . (15)The t refer to null coordinates in the frame comovingwith the uid 5 . In these coordinates sound waves propagateexactly as in homogeneous space. In the laboratory framesound waves are accelerated or slowed down by their carrier,the moving uid. An interesting behaviour occurs near thehorizon, say at z = 0, where for small z

u = c + z. (16)The constant describes the velocity gradient of thecondensate at the sonic horizon. We see that

+ = ln( z/ z)

. (17)

Wavepackets localized just before the horizon at z 0 takean exponentially large time to advance against the current. Onthe other side of the horizon, z 0, such waves drift equallyslowly in the direction of the ow. The horizon at z = 0 marksa clear watershed, cutting space into two disconnected regions.In terms of the coordinate (17) these regions are characterized5 In a uid extending in more than one spatial dimension the acoustic spacetime geometry is curved, in general. There is no longer a universal comovingframe, because in this frame the metric is at.

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by the sign of z. For stationary sound waves with frequency we get from the general solution (15)

= Re{ A zi/ eit }. (18)The phase of the wave, (/) ln( z/ z), diverges logarithmi-cally at the horizon where, in turn, the wavenumber k develops

a pole,k =

z

ln z z =

z

, (19)

and the wavelength of sound shrinks beyond all scales,

= 2

k = 0 z. (20)However, when reaches the scale of the healing length of the condensate [9, 10] (also called the correlation length)the hydrodynamic description of sound in BoseEinsteincondensates is no longer valid [9, 10]. The acoustic theoryat the horizon predicts its own demise. Similarly, waves nearthe event horizon of a gravitational black hole are compressedbeyond all scales. New physics beyond the Planck scale mayaffect the event horizon [2326].

4. Bogoliubov dispersion

For BoseEinstein condensates the equivalent of anyhypothetical trans-Planckian physics is well knownBogoliubovs theory of elementary excitations. In thedispersion relation (3) we replace the right-hand side byBogoliubovs famous result [9, 10, 27, 28] 6

2 = c2k 2 1 + k 2

k 2c

. (21)

The parameter k c is the acoustic Compton wavenumber

k c = mc

h = 1

2 , (22)

with m being the atomic mass, also expressed in terms of thehealing length [9, 10] (the correlation length). Typically, is in the order of 10 6 m and c reaches a few 10 3 m s1in BoseEinstein condensates (without exploiting Feshbachresonances). We calculate the group velocity

v = k = u + v , (23)

v =

k = c2 k

1 +

2k 2

k 2c. (24)

Equation (23) shows that the group velocity obeys the Galileanaddition theorem of velocities. Equation (24) expresses thegroup velocity in the uid frame, v , in terms of the frequencyand the wavenumber. The acoustic Compton wavenumber,k c , sets the scale beyond which v deviates signicantlyfrom c. For large wavelengths, sound is communicated byatomic collision, and the product of the condensates densityand the atomic collision strength gives mc 2 [9, 10]. Forwavelengths comparable with or shorter than the healing6 Corley and Jacobson studied the effect of a Bogoliubov-typedispersion onscalar waves in [29, 30].

length, the interaction-free Schr odinger dynamics of theatoms dominates the transport of excitations. Perturbationsof the free wavefunction travel with innite velocity. Sothe acoustic Compton wavenumber, k c , characterizes thecrossover between the speed of sound and the innite speed of perturbations of free matter waves.

Close to the sonic horizon, the wavenumber (19) increasesdramatically, and, in turn, theeffective speedof sound v grows.The horizon, dened as the place where the uid exceeds thespeedof sound,seemsto dissolvelikea mirage. Natureappearsto prevent the existence of an event horizon. However, weshow in section 6 that the horizon still exists, but at a lesswell dened location and for a particular class of elementaryexcitations only. As long as k 2 is much smaller than k 2c we getthe acoustic relation

k u c

. (25)

For the other extreme, where k 2 is much larger than k 2c , onends [29, 30]

k 2k c u 2/ c2 1 + uc2 u 2

. (26)

Consider the turning points z0, the points where the groupvelocity of sound (23) vanishes. If the acoustic dispersionrelation (3) were universally valid the horizon would be theturning point. Therefore | z| | z0| does indicate the spatialscale of the trans-acoustic range around the horizon, whichdenes the spatial delocalization of the horizon. To proceedwe recall that elementary excitations are smallperturbations of the condensate. Their energies h ought to be much smallerthan the mean-eld energy of the condensate, which is in theorder of mc2 (with c being the speed of sound). Therefore,

= h mc 2

, || 1. (27)We expand the solution z0 of v = 0 as a power series in 1/ 3and nd, to leading order, three turning points in the complexplane given by [31]

z0 = c

32

3 12

2/ 3

. (28)

Far away from the horizon we may characterize the fourfundamental solutionsof the dispersionrelation (21) combinedwith the Doppler shift (4) by their asymptotics (25) or (26).However, close to a turning point geometrical acoustics alonedoes not provide a good description of wave propagationany longer. The turning points may cause scattering. Theconnections between elementaryexcitations across the horizonmust be examined with care. Extending z to the complex planerepresentsan elegantwayof analysingthis connection. We ndin the appendix that the acoustic relation (19) remains valid oneither the upper or the lower half of the complex plane.

5. Bogoliubov modes

Elementary excitations are perturbations of the condensate,ripples on the macroscopic wavefunction 0 of the condensedatoms. Theexcitations constitutethe non-condensed partof the

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atomic gas. Todescribeelementaryexcitations, the totalmany-bodyeld operator ofthe atoms is split into twocomponents,the condensate with the mean-eld wavefunction 0 and thenon-condensed part. The mean-eld wavefunction comprisesthe density prole 0 and the ow u , as

0

= 0eiS 0 , u

= h

m

S 0. (29)

The atomic eld operator is split into the condensate and thenon-condensed part according to the relation

= 0 + e iS 0 . (30)The non-condensed part consists of Bogoliubov modes u andv [9, 10, 16, 17],

=

(u a + v a ). (31)The u and v are subject to the BogoliubovdeGennesequations [9, 10, 16, 17]. If one requires that the Bogoliubovmodes satisfy the orthonormality relations

(u u v v ) d z = , (32) (u v v u ) d z = 0, (33)

then the a and a obey the commutation relations of Boseannihilation and creation operators, as a consequence of thefundamental commutator of atoms with Bose statistics

[ ( z), ( z )] = ( z z ). (34)Each pair of u and v characterizes the spatial shape and theevolution of an excitation wave, and the Fock space of the a and a

spans the state space of the excitation quasiparticles.

To see how the Bogoliubov modes are related to soundwaves, we write down the macroscopic wavefunction of thecondensate combined with one of the excitations,

= 0 + e iS 0 (u + v ). (35)We represent as

= eiS , = 0 + s , S = S 0 + s, (36)where s denotes the local density of the sound wave and s isproportional to the velocity potential

=

h

ms. (37)

Assuming that s and s are small perturbations we get

u + v = 0 s2 0

+ is . (38)

Assuming further that u and v are stationary waves withfrequency , we obtain from our solution (18) of thehydrodynamic sound-wave equation and from the linearizedBernoulli equation (10) the Bogoliubov modes

u A 2 z

+ mc

h zi/ eit ,

v A 2 z

mch z

i/

eit

.

(39)

These asymptotic expressions are valid as long as the elemen-tary excitations are sound waves with wavenumber (19). Weshow in the appendix that this is the case sufciently far awayfrom the turning points and on either the upper or the lowerhalf plane. Here the term /( 2 z) in the expressions (39) isalways small compared with mc / h . For real and positive z wehave

( z) i/ = e(2/) zi/ . (40)The sign refers to the two ways in which we may circumventthe trans-acoustic region, on the upper (+) or on the lower() half plane. Modes with the acoustic asymptotics (39)throughout the upper half plane are suppressed on the left-hand side of the horizon and for positive frequencies andenhanced for negative . Modes with the asymptotics (39) onthe lower half plane show the opposite behaviour.

6. Negative energy

Bogoliubov modes are normalized according to the scalarproducts (32) and (33). Let us calculate the norm of themodes (39) of the sonic hole. The energy parameter (27)is small and so is the extension of the trans-acoustic regionaround the horizon, measured roughly by the location of the turning points (28). Consequently, we can neglect thetrans-acoustic contribution to the normalization integral (32).We approximate the Bogoliubov modes by their asymptoticexpressions (39), utilize the relation (40) and get

(u u v v ) d z | A |2

0

+ +0

2k c z

zi( )/ d z

= | A |2(1 e(2/) )4 k c ( ). (41)If we choose modes with the asymptotics (39) on the upperhalf plane the norm is positive and the u and v may serve asproper Bogoliubov modes. We nd the normalized amplitude

A = [(1 e2/ )4 k c ]1/ 2 . (42)Remarkably, the Bogoliubov norm is also positive formodes with negative frequencies. The BogoliubovdeGennesequations have a well known symmetry [16, 17]: if the (u , v )are solutions then the complex-conjugated and interchangedmodes, (v , u ) , are solutions as well. Yet the norm of theconjugate modes is the negative norm of the original (u

, v

) .

In contrast, sonic black holes generate negative-frequencymodes with positive norm, which is the unusual feature thatgives rise to the acoustic analogue of Hawking radiation [6].We also see how the mentioned symmetry of the BogoliubovdeGennes equations [16, 17] appears in our case. If we chosethe u and v with the asymptotics (39) on the lower half planewe would get negative normalization integrals (41) for bothpositive and negative frequencies.

The negative frequencies of the positive-norm Bogoliubovmodes give rise to negative energies in the Hamiltonian of theelementary excitations,

H = h( a+ a + a

a) N d. (43)S45

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Here and later the subscripts refer to positive and negativefrequencies, respectively, and N denotes the density of modes.We see that there is no natural ground state of the elementaryexcitations. In practice, of course, the spectrum is limited bythe requirement (27) that the energies of the excitations oughtto be much smaller in magnitude than the condensates mean-eld energy. We note that the Hamiltonian (43) is invariantunder the Bogoliubov transformations

a = a cosh a

sinh ,

u = u cosh v sinh ,v = v cosh u sinh

(44)

with an arbitraryreal parameter . Inthe case where we choose

tanh = e/ (45)we get a new set of modes, equation (39), with

A

(

z)(

4 k

c)

1/ 2 . (46)

The step function indicates that the primed modes appear oneither the left- or on the right-hand side of the trans-acousticregion. Therefore, despite the trans-Planckian problem, asonic horizon exists, but at a less well dened location, within

| z| | z0|, and the horizon applies to a particular set of modesonly.

7. Hawking effect

The Bogoliubov transformations (44) relate one set of quasiparticles to another one, both representing perfectlyvalid energy eigenvalues, yet their quasiparticle vacua differ(the states |0 or |0 that are annihilated by a anda, respectively). This ambiguity has direct physicalconsequences, because the cloud of non-condensed atoms [28]depends on the vacuum state of the elementary excitations,

0| |0 = (|v+|2 + |v|2) N d, (47)0 | |0 = (|v+|2 + |v|2) N d = 0| |0 . (48)

In general relativity the notion of the vacuum is observerdependent. For example, the vacuum of empty spacein Minkowski coordinates appears as a thermal eld toaccelerated observers [5, 6, 32]. In the case of the black hole, the gravitational collapse has created a state of quantumelds that an inward-falling observer perceives as vacuum,yet an external observer sees as thermal radiation, Hawkingradiation [58]. In our case, the equivalent of the gravitationalcollapse, the formation of the sonic horizon, chooses thequasiparticle vacuum, if the trans-sonic velocity prole hasinitially been created from a condensate without a horizon.Such a process must be sufciently smooth to keep thecondensate intact.

To analyse the quasiparticle vacuum, we use theHeisenberg picture of quantum mechanics where observablesevolve while the quantum state is invariant. We describethe initial (and nal) vacuum state with respect to one set of

continuous modes given before the formation of the horizon.In the Heisenberg picture these modes evolve. We sort theinitial modes into left- and right-moving modes that, closeto complex , are analytic on the upper or on the lowerhalf plane, respectively, because here exp ( ikz ) converges forpositive k on the upper and for negative k on the lower half plane. The upstream modes we are interested in stem fromright-moving modes. The formation of the sonic horizon, asmooth process, cannot fundamentally alter the analyticity of the vacuum modes. In particular, the process can never createnon-analytic modes of the type expressed in equation (46).Consequently, the initial quasiparticle vacuum assumes theanalytic modes of equations (39) and (42).

Given this vacuum state, we determine the quantumdepletion of the condensate. We write the density of the non-condensed atoms in terms of the primed Bogoliubov modes.As we have seen, these modes describe the set of elementaryexcitations that exhibit the sonic horizon. We nd that

0

|

|0

= (

| z

|),

( z) = [(|u +|2 + |v+|2) n () + |v+|2] N d. (49)Here n() denotes the average number of non-condensedatoms per excitation mode,

n() = 1

e2/ 1 = 1

eh/ k B T 1. (50)

The non-condensed atoms are Planck distributed with theHawkingtemperature (1). Therefore, as soonas thecondensateows through the nozzle, breaking the speed of sound,a thermal cloud of atoms is formed. This effect is thesignature of Hawking radiation for sonic holes in BoseEinstein condensates. The thermal cloud due to the Hawkingeffect should be observable when the initial temperature of theatoms is below the Hawking temperature. On the other hand,one could also regard the Hawking effect in the condensate asthe quantum depletion (47) of atoms at zero temperature withrespect to the analytic modes (42) that transcend the horizon.This feature reects the ambiguity of the vacuum in generalrelativity. Using techniques for measuring the population of Bogoliubov modes [33, 34], one could perhaps demonstratethe ambiguity of the vacuum in the laboratory.

Finally we note that within our model sonic black holesare stable, provided of course that the trans-sonic ow is notplagued by hydrodynamic instabilities. In reality, elementaryexcitations interact with each other, giving rise to what isknown as LandauBeliaev damping [3537]. Since a sonichorizon does not have a ground state, this damping mechanismwill lead to the gradual evaporation of the condensate.Therefore, LandauBeliaev damping [3537] plays the role of black-hole evaporation. It is tempting to turn matters aroundand to approach cosmological problems from the perspectiveof condensed-matter physics [4, 3841].

Acknowledgments

We thank M V Berry, I A Brown, L J Garay, T A Jacobson,R Parentani and G E Volovik for discussions. Our work was supported by the ESF programme Cosmology in the

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laboratory, the Leverhulme Trust, the National ScienceFoundation of Hungary (contractno F032346), theMarieCurieProgramme of the European Commission, the Royal Societyof Edinburgh and the Engineering and Physical SciencesResearch Council.

Appendix

In this appendixwe examine theasymptoticsof theBogoliubovmodes on the complex plane. We use the semiclassicalapproximation [31, 42]

u = U exp i k d z it ,v = V exp i k d z it .

(51)

The wavenumber k should obey Bogoliubovs dispersionrelation (21) including the Doppler shift (4). We obtain

Figure A.1. Wavenumbers k of elementary excitations around a sonic black-hole horizon, analytically continued onto the complex plane.The gure shows three roots of the dispersion relation [ h 2k 2/( 2m) + mc 2]2 h2[ (c + z)k ]2 = m2c4 for = 0.1 (mc 2 / h ) and = 0.5 (mc 2 / h ) , with the branch cuts of k chosen according to gure A.2. The top row displays the wavenumber of a sound wave thatpropagates against the current. The picture indicates the characteristic /( z) asymptotics away from the branch points. The two lowerrows display two trans-acoustic branches of k . The fourth root of the dispersion relation is not shown, because it corresponds to the trivialcase of sound waves that propagate with the ow.

four fundamental solutions of this fourth-order equation.FigureA.1 shows thethreebranches of k that arerelevant in ouranalysis. The amplitudes U and V obey the relation [31, 42]

z(U 2 V 2 )v = 0, (52)where v denotes the group velocity (23) of the elementary

excitation. Equation (52) formulates the conservation law of thequasiparticle uxfor stationary states [42] if z is real, whereU 2 V 2 gives |U |2 |V |2 up to a constant phase factor. Therelation (52) can be extended to the complex z plane and tocomplex frequencies [31]. Equation (52) implies that theamplitudes U and V diverge close to a turning point z0 wherev vanishes. Consequently, the semiclassical approximationbreaks down at the turning point [19]. The turning pointcauses signicant scattering, i.e. the conversion of one modewith a given k into two modes, one with wavenumber k andthe other one with a different wavenumber that satises thedispersion relation as well. At the turning point the twobranches coincide. To prove this, we regard for a moment

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A,C B,C B,A

C,A A,B C,B

C,B

C,A

A,B

Figure A.2. Stokes lines of elementary excitations at a sonic black

hole (dotted lines), given the choice of branch cuts made ingure A.1. The pairs of letters indicate which branches of thesuperposition (52) are connected by the lines. The rst letter of eachpair identies the exponentially dominant branch. We construct aBogoliubov mode that is acoustic (component A) on the upper half plane by putting the coefcient cC to zero on the C, A Stokes lineoriginating from the left turning point.

z as a function of k at constant frequencies = 0, denedimplicitly by equations (4), (16) and (21). We getv =

k = u +

k

( 0 uk ) = k zk

. (53)

We see that the function z(k ) reaches extrema at the point z0where v vanishes, i.e. at the turning point. Close to the z0 wendafter some algebra [31], by expanding z into a powerseriesin 1/ 3,

z z0 3(k k 0)2

8, k 0 =

3 4, (54)to leading order. Consequently, each turning point connectstwo branches of the wavenumber k . In general the modeconversion occurs near specic lines in the complex plane,called Stokes lines in the mathematical literature [43]. Stokeslines, originating from the turning point z0 , are dened asthe lines where the differences between the phases k d z of the two connected k branches are purely imaginary. Hereone of the waves is exponentially small compared with theother. We obtain from equation (54) that the differencebetween the two branches is proportional to the square rootof z z0. Consequently, the phase difference is proportionalto ( z z0)3/ 2, giving rise to three Stokeslines from each turningpoint z0, as in the traditional case of Schr odinger waves in onedimension [19, 44]. Figure A.2 shows the Stokes lines for thethree turning points close to the horizon and for the branch cutsof k chosen in gure A.1.

At a Stokes line the phase difference between the twoconnected branches is purely imaginary. One of the wavesexponentially exceeds the other and, within the semiclassicalapproximation, the smaller wave is totally overshadowed bythe larger one, if the larger wave is present. In general,

the Bogoliubov modes consist of a superposition of the fourfundamental solutions that correspond to the four branches of the dispersion relation

u = c A u A + c B u B + cC u C + c D u D ,v = c A v A + c B v B + cC vC + c D v D .

(55)

The u A and v A refer to the k branch where k obeys theasymptotics (19), i.e. where the wavenumber satises thedispersion relation (3) of sounds in moving uids, taking intoaccount the Doppler detuning (4), and where k corresponds toan upstream wave. We call such Bogoliubov modes acousticmodes. When crossing a Stokes line, the exponentiallysuppressed solution may gain an additional component that isproportional to the coefcient of the exponentially enhancedsolution [44]. If we wish to construct Bogoliubov modeswhere only the exponentially smaller component exists inthe vicinity of a Stokes line we must put the coefcient of the larger one to zero. In gure A.2 the pairs of letters

indicate which branches are connected by the lines, and therst letter identies the exponentially dominant branch. Thepicture shows that with the choice of branch cuts made wecan construct a Bogoliubov mode that is acoustic on the upperhalf plane. Trans-acoustic physics is conned to the lowerhalf plane. On the other hand, if we chose other branchcuts of k we may get Bogoliubov modes that are acousticon the lower half plane and trans-acoustic on the upper one.Therefore, according to equation (40), the choice of the k branch determines whether a Bogoliubov mode is larger orsmaller beyond the sonic horizon at the real axis, for z < 0.Branch cuts of k are fairly arbitrary. Given the Bogoliubovmode on the right-hand side of the horizon, we cannot predict

within the semiclassical approximation the amplitude of themode on the left side. Therefore, the two sides are causallydisconnected. Within the semiclassical approximation, thehorizon is a genuine horizon, despite the acoustic analogueof the notorious trans-Planckian problem [23, 24].

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