Acoustic analogue of Hawking radiation in quantized circular superflows of Bose-Einstein condensates
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Acoustic analogue of Hawking radiation in quantized circular superflows ofBose-Einstein condensates
Igor Yatsuta , Boris Malomed , , Alexander Yakimenko Department of Physics, Taras Shevchenko National University of Kyiv,64/13, Volodymyrska Street, Kyiv 01601, Ukraine Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,and Center for Light-Matter Interaction, Tel Aviv University, 69978 Tel Aviv, Israel Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica, Chile
We propose emulation of Hawking radiation by means of acoustic excitations propagating ontop of a persistent current in an atomic Bose-Einstein condensate loaded in an annular confiningpotential. The setting admits realization of sonic black and white event horizons. It is foundthat density-density correlations, representing the acoustic analogue of the Hawking radiation, arestrongly affected by the perimeter of the ring-shaped configuration and number of discrete acousticmodes admitted by it. Remarkably, there is a minimum radius of the ring which admits the emulationof the Hawking radiation. We also discuss a possible similarity of properties of the matter-wave sonicblack holes to the known puzzle of the stability of Planck-scale primordial black holes in quantumgravity.
I. INTRODUCTION
Black holes (BHs) are among the most fascinating ob-jects in the Universe, the study of which may help toilluminate an intimate relation between gravity and thequantum theory. Pioneering works [1–4] of Bekensteinand Hawking had allowed to elucidate remarkable quan-tum properties of BHs. It was discovered that BH is nota stationary everlasting object, as it emits black-body ra-diation. The effective temperature of the Hawking radia-tion is extremely low for known astrophysical BHs, whichmakes it practically impossible to observe the emissioneffect, therefore attention has turned to settings whichadmit emulation of this phenomenology in other physicalsettings.Acoustic BHs were first introduced by Unruh [5, 6],who had demonstrated precise formal equivalence be-tween the behavior of sound waves in a fluid flow andthat of a scalar field in curved space-time. Thus, it ispossible to create analogue BH in superfluids where thetransition from subsonic to supersonic flow plays the roleof the event horizon. Following that work, many ex-perimental realizations were proposed for demonstratingacoustic horizons. Behaviors similar to the analog Hawk-ing radiation have been experimentally and theoreticallyexplored in trapped ions [7], optical fibers [8–11], elec-tromagnetic waveguides [12], water tanks [13, 14], ultra-cold fermions [15], exciton-polariton condensates [16] andsuperfluid He [17]. Tremendous progress in physics ofultracold atomic gases [18] makes atomic Bose-Einsteincondensates (BECs) very suitable testbeds for sonic BHs[19, 20]. In this connections, recent experimental real-ization [21–24] of analogs of the Hawking radiation inatomic BEC is an important milestones in simulations ofquantum properties of BHs.Most previous works devoted to sonic BHs inBECs were focused on either cigar-shaped quasi-one-dimensional condensates [22] with a single BH horizon, or using specific absorbing boundary conditions [19, 20].On the other hand, toroidal BECs were used to emulatequantum features of the Hawking radiation, as angularmomentum is quantized in the ring-shaped BEC due tothe periodic boundary conditions. In this work we ad-dress quasi-one-dimensional atomic BEC confined in atoroidal trap. This setting, which is available in exper-iments [25–28], was used for studies of diverse matter-wave patterns maintained by circular flows, i.e., effectsinduced by the angular momentum [29–42].Previous investigations [43, 44] of the event horizon inBEC with toroidal geometry relied upon the use of thesupersonic flow driven by a variable condensate densitydistribution. On the contrary, we here use the uniformcondensate density , while the event horizon is createdby spatiotemporal modulation of the coupling constantwhich is responsible for interatomic interactions in themean-field approximation [18, 19]. The coexistence ofBH and white-hole event horizons in the toroidal geom-etry, gives rise to self-amplifying Hawking radiation andformation of background ripples [45–49]. Extensive in-vestigation of related instabilities has been carried out inthe 1D system with absorbing boundary conditions [50].Here we focus on the Hawking radiation per se, ratherthan the self-amplification effect. As is known [6, 23, 51],the analog Hawking temperature for the effectively one-dimensional flow is T H = ¯ h πk B (cid:18) ∂v∂x − ∂c∂x (cid:19) (cid:12)(cid:12)(cid:12) x = x h , (1)where v ( x ) is the local velocity of the condensate, c ( x ) isthe local speed of sound, and the derivatives are evalu-ated at the position of the horizon, x = x h . To minimizethe impact of the white hole horizon, we introduce a steepgradient of c ( x ) near the BH horizon, and a smoothedgradient near the white hole event horizon, as shown inFig. 1.The main objective of the present work is impact of thequantization of the superflow, imposed by the periodicboundary conditions, on the analog Hawking radiationin the toroidal BEC. The analysis reveals the followingnoteworthy effects: (i) Density-density correlations, rep-resenting the acoustic analog of the Hawking radiation,are strongly affected by the perimeter of the ring and thenumber of discrete acoustic modes available in the ring-shaped geometry. (ii) Hawking radiation vanishes if theradius of the ring falls below some critical value. FIG. 1. A schematic of the ring-shaped Bose-Einstein conden-sate with asymmetric acoustic event horizons. Shown are theblue isosurface of the condensate density n , and the superflow-velocity profile v (the dark blue line). Note that both n and v are spatially uniform. Azimuthally-modulated interactionstrength g ( x, t ) is responsible for the variation of the localsound speed, c ( x ) (the solid black curve) along the ring. Thecondensate density remains constant both in the subsonic (1: v < c ) and supersonic (2: v > c ) regions. The arrow showsthe direction of the persistent current. Acoustic black holehorizon (BHH) and white-hole event horizon (WHH) appearwhen the local sound speed is equal to the superflow velocity. The rest of the paper is organized as follows. Themodel is introduced in Sec. II. In Sec. III we discussinitial fluctuations and the corresponding dispersion re-lation. In Sec. IV we present results for density-densitycorrelations and the acoustic analog of the Hawking ra-diation near the BH horizon. The paper is concludedby Sec. V, where we also discuss possible similarities ofproperties of the analog BHs to some fundamental prob-lems, such as stability of primordial quantum BHs on thePlanck’s scale.
II. THE MODEL
We numerically analyze the acoustic analog of theHawking radiation in the framework of the one-dimensional Gross-Pitaevskii equation (GPE) i ∂ψ∂t = (cid:20) − ∂ ∂x + V ( x, t ) + g ( x, t ) | ψ | (cid:21) ψ, (2)where ψ ( x + L, t ) = ψ ( x, t ) is the wave function of thering-shaped condensate of length L and radius R , while V ( x, t ) and g ( x, t ) are the effective potential and 1D cou-pling constant, respectively. One of efficient methods fordetecting the analog Hawking radiation is the analysisof density-density correlations [52–54]. We use here themean-field dynamical simulations together with samplingof the quantum noise similar to the well-known truncatedWigner approximation (TWA) [55–59]. In our analysisof the quantum fluid dynamics, we consider the evolu-tion of stochastic trajectories satisfying the 1D GPE (2).Similar to the standard TWA, we add extra noise to theinitial wave function, thus taking it as follows: ψ ( x,
0) = e ik x h √ n + 1 √ L X k =0 (cid:0) β k u k e ikx ++ β ∗ k w ∗ k e − ikx (cid:1)i , (3)where k = M v/ ¯ h is determined by the flow velocity v , β k , β ∗ k are coefficients with random values, the summa-tion is performed over wavenumbers k , n is the spatiallyuniform density, and u k , w k are defined as u k , w k = (cid:2) ( E k /ǫ k ) / ± (( E k /ǫ k ) − / (cid:3) / , (4)where E k ≡ k / ǫ k ≡ p E k ( E k + 2 gn ).It is relevant to briefly discuss the derivation of GPE(2) and the meaning of its parameters, as well as themeaning of the noise in Eq. (3), in more detail. Assaid above, we consider the system schematically dis-played in Fig. 1. The azimuthally-modulated interactionstrength, g ( x, t ), where x = Rϕ and ϕ is the azimuthalangle around the symmetry axis of the ring, is responsi-ble for the variation of the local sound speed, c ( x ), alongthe ring. The starting point of the derivation is the 3Dmean-field GPE: i ¯ h ∂ Ψ ∂t = (cid:20) − ¯ h M ∇ + V e ( r , t ) + γ ( r , t ) | Ψ | (cid:21) Ψ , (5)where M is the atomic mass, V e = V trap + V h is theexternal potential that consists of two terms, viz ., thetransverse confinement and the potential that creates thehorizon, γ = 4 π ¯ h a s /M is the coupling constant, a s be-ing the s -wave scattering length, which is supposed to bea function of coordinate and time.The 3D wave functionΨ( r , t ) is related to the number of atoms, N , by the nor-malization condition, N = R | Ψ | d r . We use N = 10 of Rb atoms in our calculations.We consider a thin ring of radius R filled by diluteBEC, with the transverse degrees of freedom frozen bythe tight confinement. Assuming the usual factorizedapproximation [29, 60–62], we setΨ( r , t ) = ψ ( x, t ) φ ( r ⊥ , t ) , (6)where φ ( r ⊥ ) is the function of transverse coordinates(see, e.g. [63]), and the 1D density is normalized to thetotal number of atoms: Z d r ⊥ | φ | = 1 , n ≡ | ψ | = Z d r ⊥ | Ψ | . (7)Then averaging Eq. (5) in the transverse plane leads tothe 1D GPE (2) with the 1D coupling constant definedas g = 4 π ¯ h a s M Z d r ⊥ | φ | . (8)The state of the condensate in the ring with the uni-form density is provided by the plane-wave solution [19]of Eq. (2): ψ = √ n e i ( k x − ω t ) , ¯ hω = (¯ hk ) M + V ( x, t )+ g ( x, t ) n . (9)Potential V ( x, t ) and nonlinearity coefficient g ( x, t ) mustbe mutually matched so as to admit the existence of thestate with uniform density n .Since the wave function must be periodic, ψ ( x ) = ψ ( x + L ), possible values of k , k in Eq. (3) are restrictedby the length of the ring, k , k = 2 πm/L, m ∈ Z . Thisis related to the obvious quantization of the velocity cir-culation: I v d l = 2 πm ¯ h/M , (10)where integer m is the winding number or topologicalcharge. If we solve Eq. (2) on a grid of N p points withspacing h , then k = 2 πm/ ( N p h ), where integer m coin-cides with the winding number.A desirable spatial distribution of the local soundspeeds in regions 1 and 2 (Fig. 1) can be produced bymaking the coupling constant, g in Eq. (5), spatially andtemporally inhomogeneous. This can be implementedby either tuning the scattering length a s by means ofthe Feshbach resonance [18, 19, 64–66], or longitudinallyvarying the transverse confinement [67]. The sound speedin regions 1 and 2 is c , = p g , n/M , where µ , = g , n are the effective local chemical potentials. For the planewave (9) with uniform density n to be a solution at allvalues of time, we need to adjust the axial potential andcoupling constant in the two regions as follows: V ( x, t ) + g ( x, t ) n = V ( x, t ) + g ( x, t ) n , (11)This condition is satisfied all over the ring. Finally, tocast the 1D GPE in the normalized form of Eq. (2), weapply rescaling, t → t/τ , x → x/ξ , ψ → √ ξ ψ, τ =¯ h/µ , where ξ ≡ q ¯ h M/µ is the healing length in re-gion 1.Boundary conditions for Eq. (2) being periodic, weused the split-step fast Fourier transform method [68]for numerical simulations. The transition from g , V to g , V in region σ BHH is smooth, provided by the potentialtaken, locally, as V ( x, t ) = V + ∆ V ( t ) f (cid:18) t − t σ t (cid:19) f (cid:16) x − x BHH σ x (cid:17) , (12)where x <
0, ∆ V ( t ) = ( V − V ) θ ( t − t ), and f ( x ) = 12 [tanh ( x/
2) + 1] . (13) In our simulation we used x BHH = − t = 5, andchange of V ( x, t ) and g ( x, t ) occurs in the course of time σ t . We used Heaviside function θ ( t − t ) to assure that V and g are uniform at t = 0. However, as we verified,our results do not change qualitatively if potential (12)is used without θ ( t − t ) providing σ t ≪ t . Therefore,initial value of V ( x, t ) is V . The initial condition takenas the uniform condensate with constant V, g makes iteasier to add the long-wavelength noise to the entire sys-tem or some part of it, if necessary. Using this methodwith change of V ( x, t ) and g ( x, t ) in time and runninghundreds of simulations, the initial noise added to theuniform condensate inevitably excites all possible long-wavelength modes admitted by inhomogeneous g and V .We have checked that, for σ t ∈ [0 . , σ t = 0 . x > V ( x, t ) = V + ∆ V ( t ) f (cid:16) t − t σ t (cid:17) f (cid:16) x WHH − xσ W HH (cid:17) , (14)that provides mirror symmetry for the potential as longas σ WHH = σ x , x WHH = − x BHH . One can see that thistype of the transition function cannot allow an arbitraryvalue of the slope at the transition point, as it requiresan extremely large length of the ring for large σ W HH and small | ∂c/∂x | ∼ ( σ WHH ) − . To deal with this case,we used, instead of transition function (14), a double-step one, designed as a set of two similar potentials, seeFigs. 3 and 4(b). These two functions gradually carryover into each other, allowing one to manipulate a suffi-ciently smooth slope near WHH. Parameters of the hori-zon in the dimensionless units are similar to those in themodel used in Ref. [19] and close to the experimentalparameters of Ref. [22] for the speeds: c = 1 (1 mm/s), v = 0 . c = 0 . σ x = 0 . σ t = 0 .
5, with the cor-responding dispersion relations for regions far from thehorizon shown in Fig. 2. Numerical values of g , can beestimated from relation g = M c /n and V = 1 . µ . Theinitial ring has the length of L = 233 µ m, τ = 0 .
73 ms, ξ = 0 . µ m, c = 1 mm/s, which is tantamount to L = 320, τ = 1, ξ = 1, and c = 1 in the dimensionlessunits. III. QUANTUM FLUCTUATIONS AND THEDISPERSION RELATION
In real BEC, the presence of quantum and thermalfluctuations, which are not taken into account by GPE,is inevitable. It is possible to include these fluctuationsby adding Gaussian noise to the initial wave function ac-cording to TWA. The evolution of the initial fluctuationsis governed by the Bogoliubov theory [18], and expecta-tion values of symmetrically ordered observables can beobtained by taking the stochastic average over the en-semble of evolved wave functions.The initial state of the unperturbed system is uniform,and the potential step is switched on at time t > k . In the followingsection, we discuss an appropriate choice of this limitvalue in detail, and consider its impact on the analogueHawking radiation.It is straightforward to derive the respective dispersionrelation for the acoustic waves: ω k = vk ± r k (cid:16) k gn (cid:17) , (15)where wavenumbers k should satisfy the same periodiccondition as wave function (9). To briefly explain thederivation of this relation and move forward, we considerthe wave function in a general form, ψ ( x, t ) = e − iω t h √ n e ik x + 1 √ L X k =0 (cid:0) β k u k ( x ) e − iω k t ++ β ∗ k w ∗ k ( x ) e iω k t (cid:1)i , (16)[cf. the initial condition given by Eq. (3)], where theperturbations are assumed to be small in comparison to √ n , hence this expression can be rewritten as ψ ( x, t ) = e − iω t (cid:2) ψ ( x ) + ψ ′ ( x, t ) (cid:3) , with | ψ ( x ) | ≫ | ψ ′ ( x, t ) | . Afterinserting this ansatz in the GPE and linearizing withrespects to ψ ′ ( x, t ), one obtains the set of equations: ω k u k ( x ) = ( ˆ H − ω + 2 gn ) u k ( x ) + g ( ψ ) w k ( x ) ,ω k w k ( x ) = ( ˆ H − ω + 2 gn ) w k ( x ) + g ( ψ ∗ ) u k ( x ) , (17)where ˆ H = − / ∂ xx + ˆ V ( x ). Since u k ( x ) = u k e i ( kx + k x ) / √ L , w k ( x ) = w k e i ( kx − k x ) / √ L and ω = k / V + gn , we obtain:( ω k − k k − k / − gn ) u k − gn w k = 0 ,gn u k + ( ω k − k k + k / gn ) w k = 0 . (18)It is easy to see that dispersion relation (15) follows fromthe consistency condition of system (18), setting n = n .Coefficients u k , w k ( k = 0) are solutions of Eq. (18)such that the corresponding modes u k ( x ) , w k ( x ) satisfythe normalization condition, R L | u k ( x ) | − | w k ( x ) | dx =1. Therefore, expressions for the Bogoliubov coefficientsbecome as mentioned above in (4).To keep the number of atoms constant, we adjusteddensity n in each simulation. The expression that de-fines the number of excited atoms in the uniform conden-sate was found in ref. [57]. At zero temperature, it canbe written as N ′ s = X k =0 (cid:0) | u k | + | w k | (cid:1)(cid:16) β ∗ k β k − (cid:17) + X k =0 | w k | . Therefore, the number of atoms remaining in the groundstates is N c = N − N ′ s , and n = ( N c + 1 / /L. Suchexpressions follow from the relation between the averageover ensemble and quantum average.According to Refs.[56, 57], amplitudes β k , β ∗ k of the random perturbationare distributed as follows: W ( β k , β ∗ k ) = 12 πσ ǫ exp h − | β k | σ ǫ i , where σ ǫ = (4 tanh ( ǫ k / k B T )) − / , T being the tem-perature of the condensate. If we split the ampli-tudes in real and imaginary parts, β k = β xk + iβ yk ,and, accordingly, replace R dβ k dβ ∗ k → R dβ xk dβ yk , thenthe integration of the probability density immediatelygives 1, and we have two real quantities, β xk and β yk , with the same type of the distribution. To cre-ate such a distribution, we used standard randomizingfunctions: β xk = [4 tanh( ǫ k / (2 k b T ))] − / × randn( x ), β yk = [4 tanh( ǫ k / (2 k b T ))] − / × randn( x ) . Clear evidence of the analogue Hawking radiation wasobtained, both in the zero-temperature limit and for T >
0, in Refs. [19, 20]. Here, we use the distribution for T = 0 and focus on this case. IV. ANALOG HAWKING RADIATION NEARBHH (THE BLACK HOLE HORIZON)
To identify the generation of the Hawking radiationfrom the acoustic BH, we used the normalized density-density correlation function: G ( x , x ) = h n ( x ) n ( x ) ih n ( x ) ih n ( x ) i , (19)where averaging is performed over an ensemble of 100GPE simulations. Increase of the number of simula-tions does not tangibly affect the results. All simulationsstarted with the input in the form of the uniform con-densate with quantum noise added to it, as per Eq. (3).An essential role in our investigation plays the number ofmodes which are used in TWA. This number was chosento satisfy some natural restrictions. TWA was provento be correct for dilute Bose condensates if the numberof modes obeys the constraint N > N modes /
2, as wasshown in Ref. [56]. To provide the presence of P -modesand thermality of the outgoing flux, frequencies shouldobey condition ω < ω max , see Fig. 2, as was found inRef. [69]. For a sufficiently low wavenumber the lattercondition also implies that all modes satisfy regime of thelinear dispersion relation, thus maintaining a clear anal-ogy with the Hawking effect near real BH. It is worthto note that the initial noise in Eq. (3) involves sum-mation over negative and positive wavenumbers k , but,in our investigation, only positive wavevectors are rele-vant, as they correspond to atoms moving towards BHH.Moreover, including modes with the negative directionof motion did not make a visible change in the obtained FIG. 2. The dispersion relation for regions far from the horizon. The dispersion curves (a) for c = c (before BHH) and (b)for c = c (after BHH). Panel (a) is also relevant in the whole region until switching on the horizon at t = 5. Mode HR movesagainst the flow and is expected to be the analogue Hawking radiation. Positive- and negative-norm branches correspond topositive and negative signs of frequencies in the quescent condensate. In the condensate with the flow in the supersonic region,the negative-norm branch acquires positive values for some values of k , which results in the emergence of the P mode. Modesin and P belong to different dispersion branches and move in the direction of the flow. Frequency ω max defines the frequencylimit for the analogue Hawking radiation and corresponds to the maximum value of frequency on the negative-norm branchin the supersonic region. Wavenumber k min represents the restriction on the analogue Hawking emission for modes with thepositive direction of motion. correlations, therefore we include only lower positive- k modes in our initial condition.First, we address the symmetric potential with ini-tial parameters given above ( L = 320, x BHH = − x WHH = 50, σ x = σ WHH = 0 . ω < ω max we have to put ω max ≈ .
1. Thus,we are able to generate three modes with the initial fre-quency below ω max . The obtained results for the corre-lation function are illustrated in Fig. 3. Colored linesin the figure correspond to the expected correlations be-tween different particle pairs from the dispersion relationin Fig. 2. The slope of the colored segment relative to the x -axis is determined by expressions for the sound speedof the Bogoliubov excitations:tan( θ y ) = v − c v + c , tan( θ g ) = v − c v + c , tan( θ r ) = v − c v − c , where θ y , θ g , and θ r refer to the yellow, green and redlines, respectively, in Fig. 3. The length of each segmentreflects the expected length of the correlation tongues atthe corresponding times. By tongues we mean the line-shaped correlation regions that have one end located nearBHH and other end growing in time in some certain di-rection.It is seen that, in spite of the quantization of k ,all the correlation tongues are visible for both potentialsand they agree well with the predictions based on thedispersion relations. In Fig. 3 it is seen that the cor-relation pattern alters at later times [in Fig. 3(a) at t ≃ | ∂c/∂x | ∼ ( σ WHH ) − , which is responsible formixing of modes at the horizon. Applying our double-step potential, we could make | ∂c/∂x | and thus avoidthe destructive impact of such effects, by reducing thecheckerboard correlations to just two crosses created bythe scattering on each step. While we observe the correla-tion in both patterns for the current length, the presenceof such decaying | ∂c/∂x | is inevitable for smaller scalesthat are considered below.The next step is to apply the initial noise, which con-tains modes that are located above ω max in the disper-sion relation. The corresponding correlations, producedby 10 lowest modes, are shown in Fig. 4. As above, wesee the same correlations for steep and smoothed slopesat WHH, with a qualitative difference in the checker-board correlations. Moreover, only HR − in correlationtongues (i.e., ones between excitation modes of the HRand in types) are present, and there is no evidence ofP − in or HR − P tongues. Besides, obtained correlationtongues are expanding a bit faster than expected, whichmay be a direct consequence of being beyond the lineardipersion regime.We have performed similar simulations for the eighttimes enlarged region and distance between the horizons( L = 2560 or ≈ .
87 mm, in physical units), keepingthe same parameters of both horizons as above. Thelarger area makes it possible to admits more initial modesthat lie below ω max . We have thus performed the simu-lations for 23 lowest modes belonging to the linear dis-persion regime. In Fig. 5 one can see the entire rangeof the expected correlations, and the anticipated posi-tion of the tongues very well coinciding with the high-lighted lines in the correlation pattern. These corre-lations feature relatively high intensity, which is about4 × − for the HR − P tongues. This value is still ≃ ξ n = 1 /g = 312 . ≫
1, that defines the intensity ofthe correlation signal. No drastic difference is seen be-tween the correlations near BHH for both potential slopesat WHH. This fact is explained by a finite speed of ex-citations and relatively large distance between the twohorizons.
FIG. 3. Snapshots of the correlation function nξ · ( G ( x , x ) −
1) for t = 90 and t = 150 for two different profiles of the lo-cal speed of sound with three initial modes and L = 320.The bottom row represents distribution of the velocity (blueline) and local speed of sound (black curve) along the ring.(a) Symmetric black and white horizons with sharp gradientsof the local speed of sound near both horizons. (b) Speedof sound gradient is smoothed near the second (white) hori-zon. Note that the checkerboard pattern in the supersonicregion being clearly seen in (a), vanishes in (b), when thelocal speed of sound gradually increases near the white hori-zon. Yellow, green, and red lines show expected positions ofP − in, HR − in, HR − P correlation tongues.
The results obtained for the initial noise containingmodes over ω max in the large region ( L = 2560) sharemain properties with two previous simulations: there isno difference between correlation patterns near BHH for FIG. 4. The same as in Fig. 3 (without the bottom panels)for L = 320 but for the inputs with ten initial modes. Notethat checkerboard pattern vanishes for the smoothed whitehorizon as in Fig. 3. two different slopes at WHH, at least in the course of theobservation times, and the correlation pattern, as onemay expect, features solely HR − in correlation tongues.The same happens with modes taken above ω max .The inference is that the possibility to observe the ana-log Hawking correlations (for fixed parameters of the flowand horizon) depends on the number of modes that havetheir frequencies below and above ω max , as predicted bydispersion relation (15) for the uniform density. Fur-thermore, the effect is strongly affected by the size ofthe ring. The first aspect reflects the fact that, to pro-duce the Hawking effect, one needs to have particles withthe frequency from the negative-norm branch of the dis-persion relation in the laboratory reference frame. Thiscriterion was also discovered analytically for the elon-gated one-dimensional condensate with one horizon [69],and, remarkably, it is true for our system as well. Thesecond aspect implies that, by enlarging the length ofthe ring, one increases the number of allowed modes,which, in the limit of the ring with an infinite radiusleads to the continuous frequency spectrum and infinitenumber of modes below ω max . On the other hand, whenthe length of the ring decreases, the discreteness of thesystem’s spectrum makes a great difference in the pre-dicted observations: while some fixed number of modesstay completely within the linear dispersion region in alarge-radius torus, the same set of modes can partiallyor even completely exceed the frequency limit for a smallring. This change is accompanied by the transition fromthree pairs of the correlation tongues to only HR − insurviving one, or even the absence of any correlations,for a sufficiently large number of modes above ω max ).It is expected that, in an elliptically deformed torus, thedependence of the Hawking effect on L remains the same, FIG. 5. Correlation function nξ · ( G ( x , x ) −
1) for steep(a) and smoothed (b) for 23 initial modes and L = 2560. Thebottom row represents velocity (blue line) and local speed ofsound (black line) profiles near WHH. The spatial scale in( x , x ) plane is the same as in Fig. 3. in the first approximation.Another important issue concerning our system withtwo horizons is its stability, which is sensitive to bound-ary conditions and the presence of different horizons [70].For the initial noise that does not strongly violate the fre-quency condition, we have found the steep horizon to besubject to eventual instability, with lifetime t decay ≃ ≈ .
438 s, in physical units) for L = 320. At largertimes, simulations demonstrate that the development ofthe instability near WHH produces soliton-like struc-tures, and, finally, it leads to decay of both horizons.On the other hand, the system with a smoothed horizonshows no instability (at least, in the course of long-termsimulations for t ∼ − P correlations tongue is determined by thenumber of modes below and above the critical value of ω max , the value of the lowest wavenumber is restricted bythe length of the ring. Therefore, there exists a criticalsize making it impossible to observe the Hawking cor-relations, as no Bogoliubov mode may satisfy the con-straint (very roughly speaking, it resembles the knownproperty of modulational instability, which is suppressedby periodic boundary conditions if the ring’s length falls FIG. 6. Vanishing of HR − P correlations for size below L cr for a ring with v = 0 .
61 ( m = 31) and small | ∂c/∂x | nearWHH compared to BHH. Column (a) pertains the case ofthe ring with 3 lowest modes and L = 320, where even thelowest k -mode exceeds the frequency limit, and (b) providessnapshots of week correlations for the case of a larger region( L = 2560) and 10 lowest initial modes. The scale of thecorrelations is the same in all snapshots. below the respective critical value [71]).This size can beevaluated from the condition that the frequency, whichcorresponds to the lowest wavenumber pursuant to Eq.(15), is equal to ω max , see Fig. 2. For our initial param-eters of BHH the critical length of the ring is L cr ≈ µ m, which corresponds to critical radius R cr ≈ µ m.It remains a challenging objective to produce evidenceof the disappearance of the HR − P correlations, as thenoise with the amplitude used in the above simulationssuppresses all correlations at R ≃ µ m. Therefore, wehave changed parameters of the horizon ( v = 0 .
61) soas to make the size of the ring close to L = 320 (whichis good to observe correlations), and conducted simula-tions for a smoothed slope at WHH. In Fig. 6 we observethe absence of HR − P correlations for the three lowestmodes. On the other hand, a larger eight-fold larger re-gion produced visible P − in correlations (and negligibleHR − P ones) for the ten lowest modes.
V. CONCLUSIONS
We have investigated theoretically the possibility togenerate acoustic Hawking radiation in the superfluidring-shaped BEC. For this purpose, we have introducedthe double-step potential that minimizes the emulatedHawking temperature near the WHH (white hole hori-zon), where instabilities may occur. The desirable re-gion with the supersonic flow and uniform density dis-tribution of the condensate may be designed using thespatiotemporally-modulated interaction constant. Thesefeatures make the system considered here more stableand convenient for the analysis of the acoustic analog ofthe Hawking radiation in the rings.We addressed basic properties of the analogue Hawkingradiation in the ring-shaped matter-wave configurationswith different radii. The location and shape of tongue-shaped correlation patterns are well predicted by the dis-persion relation for the quasi-uniform condensate in thelimit of low wavenumbers. They are stable against vari-ations of the spatiotemporal modulation of the potentialand coupling constant.An important circumstance is that the correlation pat-tern is sensitive to the number of modes admitted bythe ring, depending on its radius, below and above thefrequency limit ω max . Varying these parameters, we ob-served different tongue-shaped correlation patterns: fromthree pairs of the tongues to a single pair for a sufficientlylarge number of modes. The discreteness of the frequencyand momentum spectra in the ring produce a dramaticeffect on properties of the Hawking radiation. Below thecritical radius of the ring, even the lowest mode has itsfrequency above the ω max , so that HR − P correlationsdisappear. Thus, no acoustic Hawking radiation takesplace in the ring-shaped superflows with the radius fallingbelow the critical value.It is relevant to relate the minimum radius of sonicblack hole, which can emit the acoustic analogue of theHawking radiation, to real (astrophysical) BHs. While astrophysical BHs lose their mass through the Hawk-ing radiation extremely slowly, the Hawking temper-ature dramatically increases for small BHs: T H =¯ hc / (8 πGk B M ) = 6 . × − ( M ⊙ /M ) K. In this con-nection, it is relevant to mention that quantum effectsare believed to be crucially important for BHs with thePlanck-scale mass. A well-known puzzle for the quantumtheory of gravity is the final fate of such BHs. There aregood grounds to assume [72] that the Hawking radia-tion is suppressed for sufficiently small BHs when theirsize, r g = 2 GM/c , becomes comparable to the Comp-ton wavelength, λ C = h/ ( M c ), associated with the BHof the Planck’s mass, M ∼ m P = p ¯ hc/G . Accord-ingly, small non-radiating primordial black holes createdin great numbers in the early Universe could survive andbecome a key ingredient of dark matter [73–76], as it isseen today.We hope that the minimum radius suppressing theemission of the acoustic Hawking radiation in the ringcan shed new light on the unsolved puzzle of the stabi-lization of the primordial BHs. VI. ACKNOWLEDGMENT
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