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Bogoliubov Theory of acoustic Hawking radiation in Bose-Einstein Condensates

A. Recati,1, 2 N. Pavloff,3 and I. Carusotto1

1Dipartimento di Fisica, Universita di Trento and CNR-INFM BEC Center, I-38050 Povo, Trento, Italy2Physik-Department, Technische Universitat Munchen, D-85748 Garching, Germany

3 LPTMS, Universite Paris Sud, UMR8626, F-91405 Orsay, France

We apply the microscopic Bogoliubov theory of dilute Bose-Einstein condensates to analyze quan-tum and thermal fluctuations in a flowing atomic condensate in the presence of a sonic horizon. Forthe simplest case of a step-like horizon, closed-form analytical expressions are found for the spectraldistribution of the analog Hawking radiation and for the density correlation function. The peculiarlong-distance density correlations that appear as a consequence of the Hawking emission featuresturns out to be reinforced by a finite initial temperature of the condensate. The analytical resultsare in good quantitative agreement with first principle numerical calculations.

PACS numbers: 03.75.Kk, 04.62.+v, 04.70.Dy

I. INTRODUCTION

Thanks to the impressive advances in the cooling andmanipulation of ultracold atomic gases, experiments arenow able to address novel regimes where the physics ofcoherent matter wave is strongly affected by zero pointquantum fluctuations [1, 2, 3, 4, 5]. In particular Bose-Einstein condensed (BEC) atomic vapors appear as a ver-satile and efficient tool for observing a very fascinatingmanifestation of quantum fluctuations, namely Hawkingradiation from acoustic black holes, the so-called dumb

holes [6, 7].

Hawking radiation is a most celebrated, yet still un-observed prediction of quantum field theory on curvedspace-times, which consists of the conversion of vacuumfluctuations into observable radiation in the vicinity of ablack-hole horizon [8]. Elaborating on the mathematicalanalogy between the propagation of sound waves in in-homogeneous and moving media and the propagation ofquantum fields on a curved spacetime background, Unruhpredicted in 1981 the occurrence of an analog Hawkingemission of sound in any system showing a sonic horizon,i.e., an interface between a sub-sonic and a super-sonicregion [9].

Very recently, the experimental realization of a dumbhole-like configuration in a flowing atomic Bose-Einsteincondensate was presented in Ref.[14]. From a more gen-eral standpoint, a dumb-hole configuration for surfacewaves on a tank of moving water was realized in [10]:the reported observation of the conversion of positive-frequency waves into negative frequency ones can be con-sidered as a classical analog of the Hawking effect [11].The observation of a horizon in a microstructured opticalfiber was reported in [12]. The possibility of simulatingacoustic black holes using a ring-shaped chain of trappedions was considered in [13].

So far, much of the theoretical work on the analogHawking radiation in atomic Bose-Einstein condensateshas used some gravitational analogy to characterize theproperties of the emission. Our point of view is differ-ent and aims at developing a microscopic understanding

of analog Hawking radiation starting from the generaltheory of quantum fluctuations in condensed matter sys-tems and without referring to any gravitational analogy.A similar approach has been adopted in the few last yearsby other researchers: A first step in this direction can befound in Ref. [7] where a general theory of analog Hawk-ing radiation based on Bogoliubov theory is presentedbut no detailed analysis of the observable consequencesis carried out. This point of view has been pushed fur-ther in the very recent papers [17, 18], where the generalframework is developed in full detail and numerical re-sults for some observable quantities are also discussed forthe case of a smooth sub- to super-sonic flow transition.

In the present paper we report a completely analyticalstudy of Hawking radiation from dumb-holes in atomicBose-Einstein condensates. Our calculations are basedon a direct application of the standard Bogoliubov the-ory of dilute condensates and no explicit reference is evermade to the gravitational analogy. In order to makethe problem analytically tractable, we consider a sim-plest step-like configuration where the transition betweenthe sub-sonic to the super-sonic regions occurs on a veryshort length scale. Even if the surface gravity of this con-figuration is formally infinite, still a thermal-like Hawk-ing emission is found at a temperature fixed by the heal-ing length. Closed analytical formulas for the densitycorrelations are extracted, which are found in excellentagreement with the numerical simulations of [16]. Theseresults extend the analytical understanding of the ana-log Hawking radiation to the sharp interface limit oppo-site to the hydrodynamic regime previously considered inthe literature. Even if the singularity at the step makesthe analogy with gravitational physics strictu sensu nolonger valid, still the remarkable properties of the re-sulting emission support our choice of calling it analog

Hawking radiation.The paper is organized as follows. In Sec. II we present

the physical system under consideration and we reviewthe Bogoliubov description of quantum fluctuations inthe presence of a sonic horizon. In Sec. III we derive ana-lytical formulas for the main observable quantities such asthe emission spectrum and the density correlation func-

2

tion. These formulas are shown to successfully comparewith the numerical results of [16]. The effect of a finiteinitial temperature is discussed. Conclusions are drawnin Sec. IV.

II. THE PHYSICAL SYSTEM AND THE

BOGOLIUBOV DESCRIPTION

The system we consider is sketched in Fig.1. A Bose-Einstein condensate is flowing along a one-dimensionalatomic waveguide. For simplicity, the transverse trap-ping is assumed to be tight enough for a one-dimensionaldescription to be accurate. The flow velocity is directedalong the positive-x direction, v(x) > 0. The density pro-file n(x) and/or velocity field v(x) of the condensate showa significant spatial modulation in the region aroundx = 0. Far from this interface region, both the densityand the flow velocity tend to their up-stream (x→ −∞)and down-stream (x→ +∞) asymptotic values nu,d andvu,d. The flow is kept stationary in time by means of asuitable external potential Vext(x) and/or spatial mod-ulation of the atom-atom interaction constant g(x), sothat the condensate wavefunction ψ0(x) is a solution ofthe stationary Gross-Pitaevskii equation

HGPψ0 =

[

− ~2

2m∂2

x + Vext(x) + g(x) |ψ0|2]

ψ0 = µψ0,

(1)at a chemical potential µ. A dumb-hole configuration isrealized when the up-stream region is subsonic vu < cuand the down-stream region is instead super-sonic vd >cd. Here, cu,d =

√

gu,dnu,d/m is the speed of sound andgu,d = g(x → −∞,+∞) are the asymptotic values ofthe interaction constant in the u, d-regions, respectively.For later purposes, we also introduce the correspondingupstream (downstream) healing length ξu,d = ~/(mcu,d),chemical potential µu,d = mc2u,d, and condensate wave-

vector ku,d = mvu,d/~.

In this dumb-hole configuration a sound wave emittedin the down-stream region is dragged away by the flowwithout being able to reach the up-stream region. Thedown-stream region is thus the sonic analog of the interiorof a gravitational black hole, and the transition pointwhere the velocity of the flow is exactly equal to thespeed of sound is the analog of the event horizon. Weconventionally locate this point at x = 0.

Provided the condensate is everywhere diluten(x) ξ(x) ≫ 1, small fluctuations on top of the conden-sate can be described by the Bogoliubov theory of dilutecondensates [19]. In particular, the elementary excita-tions correspond to the eigenvectors of the Bogoliubovoperator:

L =

(

HGP − µ+ g|ψ0|2 gψ20

−gψ∗20 −HGP + µ− g|ψ0|2

)

. (2)

A. Bogoliubov modes in a homogeneous system

Within each homogeneous u, d region far from the in-terface, the eigenvectors of operator (2) at an eigenenergy~ω are plane waves of the form

(

u(x)w(x)

)

=

(

u(x) eiku,dx

w(x) e−iku,dx

)

= eikx

(

Uk eiku,dx

Wk e−iku,dx

)

.

(3)The wavevector k and frequency ω satisfy the Bogoliubovdispersion relation ω = ω±

B (k) with

ω±B (k) = vu,d k ± cu,d k

√

1 +1

4(k ξu,d)2 . (4)

As usual in Bogoliubov theory [19], the upper (lower) signin Eq. (4) refers to the positive (negative) norm branch,for which |Uk|2 − |Wk|2 = +1 (−1).

Let us restrict our attention to the positive frequencymodes (ω > 0). The negative frequency modes can in factbe obtained from the positive frequency ones by simplyreverting the sign of k and exchanging the values of theBogoliubov coefficients U and W ; corresponding modesthen have opposite norms.

In the up-stream sub-sonic region, for any ω > 0 thereare two positive norm modes satisfying the Bogoliubovequation ω = ω+

B (k) with real wavevectors kinu , kout

u re-spectively. These modes correspond to propagating planewaves. In the Bogoliubov dispersion shown in Fig.2(left panel), they correspond to the branches indicatedas u-in and u-out. Throughout the whole paper, modeswill be labelled as in-going (“in”) if their group veloc-ity vg = dω/dk points toward the horizon and out-going(“out”) in the opposite case. In addition to the positivenorm modes, a second pair of negative norm of modes ex-ist with complex wavevectors satisfying the Bogoliubovequation ω = ω−

B (k): these correspond to exponentiallygrowing or decreasing evanescent waves [20].

In the down-stream super-sonic region, two positive-norm real solutions of the Bogoliubov dispersion at kin

d1and kout

d1 exist for any ω > 0: In the Bogoliubov disper-sion of Fig.2 (right panel), these wavevectors correspondto the d1-in and d1-out branches. On the other hand,two cases have to be distinguished in the negative normsector, depending on whether ω is larger or smaller thanthe critical value Ω = ω−

B (K) with K defined by

(ξd K)2 = −2 +v2

d

2 c2d+

vd

2 cd

√

8 +v2

d

c2d. (5)

A pair of real solutions kind2 and kout

d2 to the negative-normdispersion relation ω = ω−

B (k) exist as long as ω < Ωand correspond to the d2-in and d2-out branches of Fig.2(right panel). The critical frequency Ω is the maximumof the d2-in and d2-out branches. For ω > Ω, thesereal solutions are again replaced by a pair of complexsolutions corresponding to evanescent waves. Of course,the d2 branches do not exist if the condensate flow iseverywhere sub-sonic.

3

upstream v < cu u

downstream

x

v > cd d

u−in

u−out d2−out

d1−out0

FIG. 1: (color online) Sketch of the one dimensional dumb hole configuration considered in this paper. A BEC is flowing inthe positive x direction. The horizon is located at x = 0. The flow is uniform in the asymptotic regions far from the horizon,with a velocity which is sub-sonic in the upstream region and supersonic in the downstream one. The wiggly arrows illustrateone among the different scattering processes described by Eq.(10): an incident u-in wavepacket is partially reflected onto theu-out branch and partially transmitted on the d1-out and d2-out branches. The chosen labelling of the modes is explained inthe text and illustrated in Fig.2).

−2 −1 0 1 2 kξd

d2−ind1−outd1−in

Ω

−Ω−0.5 0 0.5

kξu

−0.5

0

0.5

ω /

µ u

u−out d2−outu−in

FIG. 2: (color online) The Bogoliubov mode dispersion relations (4) in the asymptotic upstream (left panel) and in thedown-stream region (right panel). Solid (dashed) lines correspond to modes with a positive (negative) Bogoliubov norm.

B. The scattering solution

Far from the interface, the generic eigenfunction of theBogoliubov operator (2) at a frequency ω is built by su-perimposing within each u, d region all available propa-gating (that is non-evanescent) plane waves at the givenfrequency ω:

(

u(x)w(x)

)

u,d

=

∑

i∈in

(

Ukini

Wkini

)

eikini x

√

4π|ving,i|

βini +

∑

j∈out

(

Ukoutj

Wkoutj

)

eikoutj x

√

4π|voutg,j |

βoutj

u,d

. (6)

Here, plane waves have been separated into in-going(in) and out-going (out) ones according to the sign oftheir group velocity. Most remarkable among these so-lutions are the so-called scattering solutions, that de-scribe a plane-wave excitation originating from infinity(either upstream or downstream) on a well defined in-going mode, impinging on the horizon, and then leavingagain towards infinity as a superposition of the differentout-going branches: only a single in-going βin

i amplitude

is then non-vanishing, while two (if ω > Ω) or three (ifω < Ω) out-going βout

j components have a finite value,describing reflected and transmitted waves.

As a specific example, in the case where the in-goingi = d1, d2 wave originates from the down-stream region,the scattering solution has the form

(

u(x)w(x)

)

d

=

(

Ukini

Wkini

)

eikini x

√

4π|ving,i|

βini +

∑

j∈out

(

Ukoutj

Wkoutj

)

eikoutj x

√

4π|voutg,j |

βoutj , (7)

in the down-stream region and the form

(

u(x)w(x)

)

u

=

(

Ukoutu

Wkoutu

)

eikoutu x

√

4π|voutg,u |

βoutu . (8)

in the up-stream region. The two out-going componentsin Eq. (7) correspond to the reflected part in respectivelythe d1 and d2 channels; of course, a d2 component canonly be present if ω < Ω. On the other hand, only asingle outgoing transmitted component is present in the

4

upstream region. Similar expressions can be straightfor-wardly written in the case where the in-going u-in waveoriginates from the up-stream region.

The expressions (7) and (8) based on the plane-waveexpansion are only valid in the asymptotic regions farfrom the interface. For any given realization of the dumb-hole configuration the complete scattering solution for allx can be obtained by solving the full Bogoliubov problem(2). A numerical solution for the case of a smooth inter-face was presented in [18]. An analytical solution for asimplest step-like case will be presented in Sec.II C.

In general, the linear relation between the amplitudesβin

i and βoutj of in- and out-going waves can be written

in a compact matricial form [7]. In the case of a genericflow that remains sub-sonic in both asymptotic regions,as well as for ω > Ω in a dumb-hole configuration, thisrelation has the two-by-two form:

(

βoutu (ω)βout

d1 (ω)

)

= S(ω)

(

βinu (ω)βin

d1(ω)

)

(9)

which only involves positive-norm modes. On the otherhand, the negative-norm d2 mode appears as soon as fre-quencies ω < Ω are considered. In this case, the matricialrelation has the three-by-three form:

βoutu (ω)βout

d1 (ω)βout

d2 (ω)

= S(ω)

βinu (ω)βin

d1(ω)βin

d2(ω)

. (10)

With the chosen normalization of the solution (6), theS matrix connecting the β coefficients in the asymptoticregions is unitary in the Bogoliubov metric η inheritedby the norm of the corresponding plane-wave modes

S(ω)†η S(ω) = η . (11)

More specifically, η = diag(1, 1,−1) in the case ω < Ω,whereas η is just the standard 2 × 2 identity matrix η =diag(1, 1) for ω > Ω.

Physically, the square moduli |Sij |2 of the S-matrixelements give the trasmission/reflection coefficients for aj-ingoing mode which scatters into an i-outgoing mode.The property (11) ensures total energy conservation. Itis interesting to note that a closely related approach wasused in the context of quantum evaporation from super-fluid 4He [21].

As an illustrative example, for each quantum that in-cides on the system from the d1 in-going mode withan energy ~ω, one has |Sud1|2 transmitted quanta inthe u-out mode, |Sd1d1|2 reflected quanta in the d1-out mode, and |Sd2d1|2 quanta reflected in the d2-outmode. While the energy of the transmitted u-out and re-flected d1-out quanta is positive and equal to ~ω, theenergy of a quantum of the d2-out Bogoliubov modeis negative and equal to −~ω [19]. As a result, en-ergy conservation then recovers the unitarity condition1 = |Sud1|2 + |Sd1d1|2 − |Sd2d1|2.

C. The step like geometry

All the theory reviewed so far is not limited to a par-ticular realization of dumb-hole and can be applied togeneric configurations as long as the density and flow ve-locity tend to constant values in the asymptotic regionsfar from the interface. A numeric solution for a smoothinterface was indeed reported in [18].

In the present paper, we restrict our attention to amodel step-like structure with Vext(x) = V u

extΘ(−x) +V d

extΘ(x) and g(x) = guΘ(−x) + gdΘ(x) for which ana-lytical formulas for the components of the S-matrix canbe found in closed form. As usual, Θ is here the Heavy-side step-function. Provided one imposes [16]

V uext + µu = V d

ext + µd , (12)

the time-independent Gross-Pitaevskii equation (1) hasa solution in the plane-wave form

ψ0(x) =√n0 exp (ik0x) (13)

that describes a flow with uniform density nu,d = n0

and velocity vu,d = v0 = ~k0/m. If one chooses cd <v0 < cu, a dumb-hole configuration is obtained. Thevalue of the potential jump V u

ext − V dext is then fixed by

the condition (12). This model configuration has theremarkable advantage of allowing for analytical insightwithout having to solve any differential equation. In afuture publication we will present the analysis of otherrealistic dumb-hole configurations [22].

In the step like geometry, the expressions (7) and (8)for the scattering solution can be extended to the wholex > 0 and x < 0 regions simply by including evanescentmodes as well [20]. A complete scattering solution is thenformed by the superposition of five two dimensional col-umn vectors of the form (3): one of these correspondsto the incoming (i = u, d1, d2) mode and the four othersare outgoing waves (two reflected and two transmitted).Depending on the value of ω, some of the outgoing wavesmay be evanescent: in this case, between the two con-jugate complex wavevectors that solve the Bogoliubovdispersion (4), only the wavevector value giving an expo-nentially decaying wave at large distance from the hori-zon has to be considered.

Across the step-structure at x = 0 the column vec-tor eigenfunction of L and its first derivative have to becontinous. This provides the four matching conditionsthat are necessary to determine the amplitudes of thefour outgoing waves as a function of the amplitude βin

i ofthe in-going i one. The coefficients of the non evanescentoutgoing waves correspond to the matrix elements Sji

with j = u, d1, d2. As usual, the d2 element only existsfor ω > Ω. Note that correct inclusion of the evanescentwaves is here crucial to be able to fulfill the four matchingconditions.

5

0.2 0.4 0.6 0.8 1ω/µ

u

0

1

2

3

4

5

6

7|S

ud1|2

|Sd2d1

|2

|Sd1d1

|2

0.970.980.99

1

0 0.1 0.2 0.3 0.40

1

2

3

4

2 4 6 80

0.010.020.030.04

b)

a.1)

a.2)

FIG. 3: Transmission and reflection coefficients for a d1 in-going mode on a step-like dumb hole configuration. |Sud1|

2

(dotted curve) corresponds to the transmission on the u-outmode; |Sd1d1|

2 (solid curve) and |Sd2d1|2 (dashed curve) cor-

respond to the reflection on the d1-out and d2-out modesrespectively. The parameters v0/cu = 0.75, v0/cd = 1.5 usedhere are the same as in Ref. [16]. Above Ω (Ω/µu = 0.316for the chosen set of parameters) one recovers (insets a.1) anda.2) the usual case of a single transmitted (u-out) and a sin-gle reflected (d1-out) waves. respectevely. Inset b) displays acomparison of the exact |Sd1d1|

2 coefficient (solid line) withits low energy approximation (14) (dot-dashed line).

D. Transmission and reflection

The typical behaviour of the scattering coefficients fora d1 ingoing wave is plotted in Fig.3 as a function of thefrequency ω. For ω > Ω, the transmitted u-out wave andthe reflected d1-out wave are only present. As usuallyexpected in wave mechanics, the transmission (reflection)coefficient increases (decreases) as a function of ω, see theinsets (a1-a2). Energy conservation imposes that the sumof these two coefficients is always equal to unity.

For ω < Ω also the (negative-norm) reflected moded2-out is involved in the dynamics. Remarkably, allthree scattering coefficients diverge as 1/ω in the lowω limit. Nonetheless, energy conservation is ensuredby the η-unitarity of the S-matrix, which now imposes|Sud1|2 + |Sd1d1|2 − |Sd2d1|2 = 1.

Straightforward algebraic manipulations lead to a sim-ple analytical formula for the reflection coefficient in thelow-ω limit,

|Sd1d1|2 ≃ cucd

(cu − v0)(cu − cd)

(cu + v0)(cu + cd)

(

v20

c2u− c2dc2u

)3/2µu

2~ω:

(14)the excellent agreement between this low energy approx-imation and the exact Bogoliubov result is apparent inFig. 3(b).

The wave mechanics encoded in the S-matrix is physi-cally illustrated in Fig.4 where we show the result of a nu-

-2×10-4

-1×10-4

0

1×10-4

2×10-4

-2×10-4

-1×10-4

0

1×10-4

2×10-4

δn(x

) / n

0

-1000 0 1000 2000 3000

x / ξu

-2×10-4

-1×10-4

0

1×10-4

2×10-4

u-outd1-out

d1-outd2-outu-out

d1-in ( a )

( b )

( c )

FIG. 4: A d1 ingoing wave packet is incident on a step-likesonic horizon located at x = 0 (upper panel). When thecentral frequency omega of the wave packet is smaller thatOmega, a single transmitted u wavepacket and two d1, d2 re-flected wave packets are visible (middle panel). A single utrasmitted and a single d1 reflected wave packet are insteadvisible for ω > Ω (lower panel). Same system parameters asin Fig.3. Wave-packet carrier wave-vector q ξu = −1.2 (upperand middle panels), q ξu = −1.35 (lower panel).

merical solution of the time-dependent Gross-Pitaevskiiequation that describes the scattering of a incident d1in-going wave packet (upper panel) onto the black holehorizon. For a low initial energy, the in-going wave packetsplits into a u-out transmitted one (central panel), anda pair of d1, d2 reflected ones: Among these, the biggerand slower wavepacket corresponds to the negative-normd2-out mode, while the other one corresponds to the d1-out mode. As expected, for an in-going wavepacket en-ergy above Ω (lower (c) panel), only the u transmittedand the d1 reflected wavepackets are visible. A relatedphenomenology was experimentally observed for surfacewaves in a water tank in [10] and theoretically discussedin [13] for a chain of trapped ions. In [24], this nega-tive norm reflected mode was interpreted in terms of abosonic analog of Andreev reflection.

E. Quantization of the modes

In the presence of a macroscopically occupied conden-sate, the full Bose field operator can be expanded as

ψ(x) = ψ0(x) + δψ(x) . (15)

The coherent condensate is described by a classical fieldψ0 that evolves according to the Gross-Pitaevskii equa-tion. Quantum fluctuations are described by the opera-

tor term δψ whose dynamics is ruled by the Bogoliubovoperator (2).

A most favourable basis to our purposes consists of thescattering solutions at a given frequency ω that we have

6

discussed in Sec.II B. Within each ω sub-space, the threescattering modes corresponding to the I = u, d1, d2 in-going modes form an ortho-normal basis (of course, onehas to restrict to I = u, d1 for ω > Ω) in the Bogoliubovη-metric,

∫

dx (uIω(x)∗uJω′(x) − w∗Iω(x)wJω′ (x))

= sIδIJδ(ω − ω′) (16)∫

dx (uIω(x)vJω′(x) − vIω(x)uJω′(x)) = 0 . (17)

The sI coefficient gives the sign of the Bogoliubov normof the mode: in our case, one has sI = +1 for I = u, d1and sI = −1 for I = d2. Spatial integration is over thewhole space.

The final form of the quantum field is obtained bythe standard Bogoliubov quantization prescription [19]:the amplitudes corresponding to positive-norm modes be-come destruction operators while amplitudes of negative-norm modes become creation operator,

δψ(x) =

∫ ∞

0

dω∑

I=u,d1

[

uIω(x) aI(ω) + w∗Iω(x) aI(ω)†

]

+

∫ ∞

0

dω[

ud2,ω(x) a†d2(ω) + w∗d2,ω(x) ad2(ω)

]

. (18)

Expectation values of physical observables (e.g. the den-sity correlation function) are then straightforwardly eval-uated by imposing the suitable boundary conditions onthe expectation values of products of in-going operators,

aI and a†I .If one is interested in correlation functions involving

the out-going modes only, a reformulation of (18) inthe input-output language [23] can be used, as proposedin [7]. In our case, the input-output relations consist of alinear relation connecting the operators of the out-going

modes bu,d1,d2 to the in-going au,d1,d2 ones via the S ma-trix (10) introduced in the previous subsection:

bu(ω)

bd1(ω)

b†d2(ω)

= S(ω)

au(ω)ad1(ω)

a†d2(ω)

. (19)

As a consequence of their negative Bogoliubov norm, thed2 modes appear in both the left- and the right-hand sideof (19) as creation operators rather than destruction ones.As we shall see in Sec.III A, this simple fact is the mathe-matical origin of the Hawking emission in our formalism.From a quantum optical perspective, Hawking emissioncan then be interpreted as parametric down-conversionof Bogoliubov sound waves by the horizon.

III. OBSERVABLES

The expression (18) for the quantum field in the Bo-goliubov approximation and the input-output relation

(19) are the starting point for the calculation of physicalobservables of the system: in particular, our attentionwill be focussed on two most remarkable ones, namelythe spectral distribution of the Hawking emission andthe long-distance behaviour of the correlation functionof density fluctuations.

A. Emission spectrum

In the gravitational context, the only observable quan-tity is the Hawking radiation outside the black hole. Oneof the most remarkable feature is that this radiation isthermal at a temperature univocally determined by thesurface gravity of the black hole.

In our condensed matter context, the Hawking emis-sion from the dumb hole corresponds to the phonons thatare emitted into the upstream region on the u-out branch.Assuming that no correlation between the different in-going modes exist, the emission spectrum, that is thenumber of phonons emitted per unit time and per unitbandwidth into the u-out branch is straightforwardly cal-culated from the input-output relation (19):

dIoutu

dt dω= 〈b†u(ω) bu(ω)〉 =

= |Suu|2 I inu + |Sud1|2 I in

d1 + |Sud2|2 (I ind2 + 1) . (20)

For a system initially at zero-temperature, all the

I inu,d1,d2 = 〈a†u,d1,d2(ω) au,d1,d2(ω)〉 = 0, yet a finite Hawk-

ing emission exists as a consequence of the +1 term aris-

ing from the 〈ad2(ω) a†d2(ω)〉 expectation value that en-codes quantum fluctuations.

Restricting our attention to the low-frequency part ofthe spectrum, an analytical form can be found for theemission spectrum:

dIoutu

dt dω

∣

∣

∣

∣

T=0

= |Sud2(ω)|2 ≃

(cu − v0)

(cu + v0)

c2u(c2u − c2d)

(

v20

c2u− c2dc2u

)3/22µu

~ω. (21)

It is remarkable to note that this formula still gives thetypical the 1/ω thermal behavior of Hawking radiationeven though one is not allowed to use the gravitationalanalogy: in the present step-like case, the surface gravityis in fact formally infinite. The effective temperature ofthe emission, i.e. the coefficient of the 1/ω term, is heredetermined by the microscopic physics of the condensatethat fixes the cut-off frequency Ω and the chemical po-tential µu,d.

B. Density correlations

As first remarked in Ref.[15], the density correlationfunction appears to be the most promising tool for iden-tifying Hawking radiation: in particular, this quantity

7

was at the heart of the numerical observation of [16]. Anexample of such a calculation is reproduced in Fig.5. Thedark regions correspond to antibunching, and the brightones to bunching. The strongest feature in this figure isthe dark stripe along the x = x′ line: as its origin is wellknown and corresponds to the antibunching due to therepulsive inter-atomic interaction [25], we will not discusit further. In the present paper, we will rather focus ourattention onto the other features of the correlation plot,labeled as u − d2, u − d1 and d1 − d2, that encode theinformation on the Hawking emission.

Within Bogoliubov theory, the density correlationfunction in the stationary state

G(2)(x, x′) =〈ψ†(x)ψ†(x′)ψ(x′)ψ(x)〉〈ψ†(x)ψ(x)〉〈ψ†(x′)ψ(x′)〉

− 1 . (22)

can be expanded in its ω components as:

G(2)(x, x′) =

∫

R+

G(2)(ω, x, x′) dω . (23)

Each ω component has to be evaluated imposing suit-able boundary conditions on the expectation values ofin-going operator averages. In the following of the discus-sion, it will reveal useful to separate the zero temperature

contribution G(2)0 from the thermal one G

(2)th

G(2)(ω, x, x′) = G(2)0 (ω, x, x′) +G

(2)th (ω, x, x′) . (24)

The former term only involves the zero-point fluctuationsof the in-going modes

n0G(2)0 (ω, x, x′) =

1

4π

[

∑

I=u,d1

w∗Iω(x) rIω(x′)

+ u∗d2ω(x) rd2ω(x′) + c.c.]

, (25)

while the latter includes the initial populations

〈a†I(ω) aI(ω)〉:

n0G(2)th (ω, x, x′) =

1

4π

∑

I=u,d1,d2

[r∗Iω(x) rIω(x′) + c.c.]×

× 〈a†I(ω) aI(ω)〉 . (26)

In both these formulas, we have set rI = uI + wI .

C. Zero Temperature density correlations

1. In-Out correlation

Let us first consider the in-out zero-temperature cor-relations between the inside (x′ > 0) and the outside(x < 0) regions. For x and x′ far from the interface,it is legitimate to only keep the terms with a stationaryphase and neglect the fast oscillating ones. Making use

FIG. 5: (color online) Plot of the rescaled density correlation

n0 ξu [G(2)0 (x, x′) − 1] for an initial zero temperature. The

data are the same as the one presented in [16], but the colorscale has been modified in order to make the positive u-d1correlation clearly visible. This signal was overlooked in theoriginal paper given its relative weakness. The dashed linesidentify the different feature of the correlation function asdiscussed in the text. The white dotted line corresponds tothe cut x′ − x = 39.5 ξu used in Figs. 7 and 6.

of the η-unitarity of the scattering matrix and settingRk = Uk +Wk, the simple expression

n0G(2)0 (ω, x, x′) =

1

4 π

∑

l=d1,d2

Rkoutu

(ω)Rkoutl

(ω)√

|voutg,u v

outg,l |

×

ei[koutu (ω)x−kout

l (ω)x′]S∗ud2(ω)Sld2(ω) + c.c.

, (27)

is obtained for the long-distance zero-temperature corre-lation function. As all the S-matrix elements appearingin Eq.(27) involve the d2 mode, it is apparent that anon-zero long-distance correlation is only possible in adumb-hole configuration: in terms of the input-outputpicture of (19), the two terms in (27) originate from thequantum fluctuations of the ingoing d2 mode that arescattered in either the u or the l = d1, d2 modes.

As the integrals in (23) are dominated by the small ωpart of the integral, simple analytical expressions can beworked out by extrapolating the low-ω asymptotic formof the S-matrix elements

S∗ud2(ω)Sld2(ω) ≃ Alµu

~ω(28)

to the whole spectrum and then cutting the integral at Ω.In this hydrodynamic approximation, kout

j ≃ ω/Vj (j =u, d1, d2) and the velocities Vj of the outgoing modes areVu = v0 − cu < 0, Vd1 = v0 + cd > 0, Vd2 = v0 − cd > 0.As typical of Bogoliubov theory [19], the Rk coefficientsgiving the density weight of the Bogoliubov modes scaleas

√ω. The in-out correlation function then takes the

8

simple form

n0G(2)0 (x, x′) ≃

∑

l=d1,d2

Al c2u Ω sinc

(

Ω( xVu

− x′

Vl))

4π|VuVl|√cucd

, (29)

where sinc(x) = sinx/x and the Al coefficient has thefollowing explicit form

Ad1 (d2) =

√

cucd

(cu − v0)

(cu + v0)

(

v20

c2u− c2dc2u

)3/2cu

cd + (−)cu(30)

in terms of the microscopic parameters of the system.From the explicit expression (29), it is immediate to

see that the density correlation is maximum along thepair (l = d1, d2) of straight lines

x

Vu=x′

Vl(31)

that exit in opposite directions x < 0, x′ > 0 from thehorizon position x = x′ = 0. The absence of lateral shiftfrom the horizon position x = x′ = 0 is a consequence ofthe reality of the Al coefficients. As a consequence of theinequality cu > v0 > cd, the sign of the d2 contribution ispositive, while the one of the d1 contribution is negative.The d2 contribution is always stronger than the d1 oneby a significant factor

|Ad2/Vd2||Ad1/Vd1|

=(cu − cd)(v0 − cd)

(cd + cu)(v0 + cd)(32)

The d2 contribution corresponds to the Hawking signa-ture anticipated in [15] and numerically observed in [16].Once the horizon is formed, the quantum fluctuation ofthe incoming negative-norm d2 mode start being con-verted into correlated pairs that emerge from the horizonin the u and d2 modes. The two phonons are simulta-neously created at the horizon and then propagate awayat speeds Vu = v0 − cu < 0 and Vd2 = v0 − cd > 0. Attime t after their emission, they are therefore located atx = Vut < 0 and x′ = Vd2t > 0, which explains the corre-lation between the density fluctuations at points verifyingx/Vu = x′/Vd2.

This simple argument can be straightforwardly ex-tended to explain the geometry of the other d1 feature.This feature went overlooked in [16] because of its weakintensity, but was mentioned in [18]. When the numeri-cal data of [16] are plotted with a suitable color scale asdone in Fig.5, the u-d1 correlation is perfectly visible asa positive correlation tongue.

A quantitative comparison of the prediction (27) withthe numerical results is illustrated in Fig.6 where cutsof the density correlation function along the line x′ −x = 39.5 ξu (marked by a white dashed line in Fig.5) areshown using the same set of parameters as in Ref.[16].In spite of the approximations made, the agreement isremarkable.

The accurateness of hydrodynamical approximation(29) is finally validated in Fig.7. As shown in the main

0 20 40 60

(x+x’)/2ξu

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

n 0ξ u G0(2

) (x,x

+39

.5ξ u)

FIG. 6: Cut of the zero-temperature density correlation func-

tion G(2)0 along the x′−x = 39.5 ξu line. Solid line: numerical

result from [16]. Dashed lines: prediction of Eqs.(27) and (33)for respectively the u− d2 and the d1− d2 contributions. Onthis scale the u − d1 contribution is hardly visible.

-10 0 10x/ξ

u

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

nξu G

0(2) (x

,x+

39.5

ξ u)

-10 0 10 20-0.0002

0

0.0002

0.0004

0.0006

20 30 40 50 60

-0.001

-0.0005

0 b)a)

FIG. 7: (Color online) Cut of the density correlation function

G(2)0 along the x′ − x = 39.5 ξu line. Black solid lines are

the prediction of (27) and (33). Red dashed lines are the hy-drodynamical approximations (29) and (36). The main panelshows the u− d2 feature in the in-out region. Inset (a) showsthe u − d1 feature in the in-out region. Inset (b) shows thed1 − d2 feature in the in-in region.

panel, the agreement for the l = d2 feature is quitegood even though the hydrodynamic approximation isnot fully able to reproduce the asymmetry of the peaknor its slightly back-shifted position with respect to thestraight line (31). Both these facts can be traced back tothe quick deviation from a linear dispersion that is visi-ble in Fig.2 (right panel) for the d2 Bogoliubov branch.This interpretation is confirmed by the excellent agree-ment which is instead found for the l = d1 contribution(inset (a) of Fig.7): the discrepancy of the dispersions ofboth the u and the d1 modes from the hydrodynamic ap-proximation is in fact very small in the whole frequencyrange up to ω = Ω.

9

2. In-In correlation

The in-in correlation function for a pair of points x, x′

both located inside x, x′ > 0 the dumb hole can be calcu-lated along the same lines. In this case, a single contribu-tion appears that originates from the correlation betweenthe d1 and the d2 modes,

n0G(2)0 (ω, x, x′) =

1

4 π

Rkoutd1

(ω)Rkoutd2

(ω)√

|voutg,d1v

outg,d2|

×

× ei(koutd1 (ω)x−kout

d2 (ω)x′)S∗d1d2(ω)Sd2d2(ω) + c.c.

)

+ (x↔ x′). (33)

Even though it did not appear in the analytical calcu-lations of [15] based on the gravitational analogy, thiscontribution was observed and discussed in the numeri-cal paper [16].

The same hydrodynamical approximation that led to(29) can be used to obtain an approximate expression tothis d1 − d2 feature as well. For small ω, the S-matrixproduct can be expanded as

S∗d1d2(ω)Sd2d2(ω) ≃ Bµu

~ω(34)

with

B = − cu2cd

(cu − v0)

(cu + v0)

(

v20

c2u− c2dc2u

)3/2

. (35)

Extrapolating this expression to the whole ω spectrumand cutting the integral at Ω, one immediately gets tothe expression

n0G(2)0 (x, x′) ≃

B Ω c2u sinc(

Ω( xVd1

− x′

Vd2))

4πVd1Vd2 cd+

+ (x↔ x′) . (36)

D. Finite Temperature density correlations

One of the major issues in view of an experimentalobservation of the analog Hawking radiation is the roleof temperature. To this purpose, a naive, but extremelyconstraining requirement is often imposed that the tem-perature of the system should be (much) smaller thanthe Hawking temperature of the emission. The numericalsimulations of [16] have instead shown that correlationsare robust with respect to temperature, which supportsonce more their promise in view of an actual experiment.In the present section we extend the theory of correla-tions developed in the previous section to the case of acondensate at a finite initial temperature.

In particular, we focus our attention on the same dy-

namical situation already considered in [16]: The con-densate initially has a uniform density n0, flow velocity

v0 and sound velocity cu and is at thermal equilibrium inthe moving frame at a temperature T [26]. The horizonis then (adiabatically) switched on by ramping down thescattering length in the downstream region.

Within each semi-infinite uniform region, the popu-lation of each k mode is adiabatically preserved dur-ing the formation of the dumb hole. The initial popu-

lation 〈a†IaI〉 of each I = u, d1, d2 mode at wave vec-tor kI is given by the thermal population n(ΩI) =[exp(~ΩI/kBT ) − 1]−1 corresponding to its frequency

~ ΩI =

√

~2k2I

2m

(

~2k2I

2m+ 2mc2u

)

. (37)

in the comoving frame before the creation of the horizon.For each ω, the wavevector kI of the corresponding in-going mode has to be determined by inverting the relationω = ω±

B (kI).The density correlation function at a finite tempera-

ture can then be written as a sum of different terms,

G(2)(ω, x, x′) = G(2)0 (ω, x, x′) [1 + n(Ωd1) + n(Ωd2)]

+ FSH(ω, x, x′) + FNSH(ω, x, x′). (38)

Here, n(ω) = [exp(~ω/kBT )− 1]−1 is the thermal law ata temperature T .

The first term in the r.h.s. of (38) is straightforwardlyinterpreted as a thermal enhancement of the zero-pointsignal described in the previous subsection, i.e. a stim-

ulated Hawking emission. Other scattering processes pe-culiar to the T 6= 0 case are responsible for the othercontributions, which can be separated in a u − d2 termdue to the presence of a sonic hole,

n0FSH(ω, x, x′) =Rkout

uRkout

d2ei(kout

u x−koutd2 x′)

4π√

|voutg,uv

outg,d2|

×

S∗uu Sd2u(n(Ωu) − n(Ωd1)) + c.c. , (39)

and a u − d1 term that instead persists even in absenceof the sonic hole:

n0FNSH(ω, x, x′) =Rkout

uRkout

d1

4π√

|voutg,uv

outg,d1|

ei(koutu x−kout

d1 x′)×

S∗uu Sd1u(n(Ωu) − n(Ωd1)) + c.c. . (40)

In the low-ω limit the S-matrix element products in-volved in the new terms (39) and (40) tend to finite values

S∗uu(ω)Sd1(d2)u(ω) ≃ −

√

cucd

cu − v0cu + v0

cd + (−)v0cu + v0

, (41)

while the thermal population is proportional to 1/ω,

n(Ωu) − n(Ωd1) ≃v0 + cucu

kBT

~ω. (42)

10

Combining these two facts, it is immediate to see thatin the hydrodynamic approximation the contributions of(39) and (40) to respectively the u−d1 and u−d2 featuresthen the same form as the zero-temperature one (29).

Summing up all the terms, the only effect of a finiteinitial temperature on the d1 − d2 feature is a bosonicstimulation factor 1 + n(Ωd1) + n(Ωd2). In particular,this feature is not affected by the FSH and FNSH terms.

The u − d2 feature is affected by this multiplicativefactor as well as by the additional term FSH : remark-ably, the zero-point (29) and thermal (39) contributionshave opposite signs. Within the hydrodynamic approxi-mation, the ratio of their contributions to the peak cor-relation signal has the simple expression

− cu(v0 − cd)(cu − cd)

(v20 − c2d)

3/2

kBT

µu. (43)

As usual, the zero-point contribution dominates at lowtemperature while it is overwhelmed by the thermal oneat high temperature. For the parameters used for thefinite temperature numerical simulations in Ref. [16], theT = 0 was still the most relevant one.

The effect of the thermal occupation on the u−d1 fea-ture is even more dramatic. Also in this case, the (am-plified) zero-point positive feature due to the l = d1 termin (29) and the thermal contribution (40) have oppositesigns. As the involved scattering process is not limited tothe ω < Ω window, the term (40) gets contributions fromhigh frequencies and is for this reason generally larger inmagnitude than (39). For this reason, the zero-point con-tribution only dominates at very low temperatures andis quickly overwhelmed by the thermal one: this effectis clearly visible when comparing the T = 0 numericalsimulations shown in Fig.6 with the finite temperatureones shown in Fig.6(a) of Ref. [16].

When the flow remains everywhere subsonic, all long-distance features of the density correlation function dis-appear but for the positive u − d1 tongue at finite tem-perature. This fact was apparent in Figs.4(a) and 6(b) ofRef.[16] and corresponds within the present formalism tothe FNSH term. Its physical origin can be traced back tothe finite reflectivity of the interface region even in theabsence of an horizon and the different initial occupation

of the u and d1 incident modes.Before concluding, it is important to stress that a dif-

ferent configuration was considered in [18], where thedumb hole was assumed to be in thermal equilibriumin the comoving frame after completion of the horizonformation process. Of course, this configuration can bestill described by Eq.(38), but the initial populationsn(Ωd1,d2) have to be computed using in (37) the cd in-stead of cu for the downstream region.

IV. CONCLUSION

In this work we have made use of the Bogoliubov the-ory of dilute Bose-Einstein condensates to study the ana-log Hawking emission that is emitted by an acoustic blackholes. Our framework does not rely on any gravitationalanalogy and is able to provide closed-form analytical for-mulas for the emission spectrum and the density corre-lation function of a simplest step-like configuration. Al-though the surface gravity is in this case formally infinite,the low-energy part of the emission spectrum still showsa thermal character, but the temperature is found to de-pend on the microscopic properties of the condensate.The extension of our theory to the case of a non-vanishinginitial temperature is discussed. We have shown that thisgeneral framework is able to quantitatively reproduce theresults of first principle numerical calculations [16], andto provide a physical understanding of the various fea-tures that appear in the density-density correlation func-tion. Generalization of the present theory to more gen-eral dumb-hole configurations is in progress [22].

Acknowledgments

We are grateful to F. Sols, I. Zapata, R. Parentani,and W. Zwerger for fruitful discussions. Continuous ex-changes with R. Balbinot and A Fabbri are warmly ac-knowledged. This work was supported by the IFRAF In-stitute, by Grant ANR-08-BLAN-0165-01, by MinisterodellIstruzione, dellUniversita e della Ricerca (MIUR) andby the EuroQUAM FerMix program.

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