Analog Black Holes and Energy Extraction by Super-Radiance from Bose Einstein Condensates (BEC) with Constant Density
aa r X i v : . [ c ond - m a t . o t h e r] D ec Analog Black Holes and Energy Extraction bySuper-Radiance from Bose Einstein Condensates(BEC) with Constant Density
Betül Demirkaya ∗ , Tekin Dereli † , Kaan Güven ‡ , Department of Physics, Koç University, 34450 Sarıyer, İstanbul, Turkey
December 6, 2018
Abstract
This paper presents a numerical study of the acoustic superradiancefrom the single vortex state of a Bose-Einstein condensate (BEC).The draining bathtub model of an incompressible barotropic fluid isadopted to describe the vortex. The propagation of the velocity poten-tial fluctuations are governed by the massless scalar Klein-Gordon waveequation, which establishes the rotating black-hole analogy. Hence, theamplified scattering of these fluctuations from the vortex comprise thesuperradiance effect. Particular to this study, a coordinate transforma-tion is applied which enables the identification of the event horizon andthe ergosphere termwise in the metric. Thus, the respective spectralsolutions can be obtained asymptotically at either boundary. Fur-ther, the time-domain calculations of the energy of the propagatingperturbations and the independently performed reflection coefficientcalculations from the asymptotic solutions of the propagating pertur-bations are shown to be in very good agreement. While the formersolution provides the full dynamical behavior of the superradiance, thelatter method gives the frequency spectrum of the superradiance as afunction of the rotational frequency of the vortex. Hence, a compre-hensive analysis of the superradiance effect can be conducted withinthis workframe. ∗ [email protected] † [email protected] ‡ [email protected] Introduction
Analogies in physics enable us to observe a particular phenomenon withthe same characteristic features in different systems pertaining to disparatemechanisms and space-time-energy scales. The present study takes on suchan analogy between a black hole and a vortex state of a Bose-Einstein con-densate and focuses on the Hawking radiation, the superradiance of lightfrom black holes, in the form of an acoustic superradiance of sound from avortex. The analogy initiated by Unruh’s calculations showed the equiva-lence between the background solution of velocity perturbations on a perfectbarotropic, irrotational Newtonian fluid and the Klein-Gordon field propa-gating in a 4-dimensional pseudo-Riemannian manifold, in which the speedof sound plays the role of speed of light [1],[2].The superradiance phenomenon is the amplification of the waves scat-tered from a black-hole in the presence of ergoregion and it is character-ized by a reflection coefficient larger than unity [3], [4]. Because this phe-nomenon occurs in the space-time background of rotating black holes,theanalogy could be set for a rotating acoustic black-hole in a liquid [5], [6].The theoretical and computational investigation of the superradiance in var-ious analogous systems have been reported by a number of studies. Basakand Majumdar introduced a DBT model of a water vortex and studied theconditions under which the density fluctuations of the fluid exhibit amplifiedscattering from the water vortex [7],[8]. The phenomena is also investigatedwidely for optical systems [9],[10],[11],[12], relativistic fluids [13], shallowwater systems [14] .On the other hand, the experimental studies emerged only within thelast few years. Experimental realizations of horizons were reported in wa-ter channels [15], atomic Bose-Einstein condensates (BECs) [16]. Recently,rotational superradiant scattering in a water vortex flow is reported [17].Acoustic black hole in in a needle-shaped BEC of 87Rb is achieved and re-cently spontaneous Hawking radiation, stimulated by quantum vacuum fluc-tuations, emanating from an analogue black hole in an atomic Bose-Einsteincondensate is reported [18],[19].Motivated by the recent experimental progress in BEC systems, wepresent here a consolidating study of the temporal and spectral featuresof the scattering process from a BEC vortex with constant background den-sity. We primarily adopt the draining bathtub model(DBT) introduced byVisser [20], to describe the acoustic black-hole. The time domain solutionsare obtained by solving the Klein-Gordon equation for the propagation ofacoustic waves, whereas the spectral analysis of the superradiance is con-1ucted by asymptotic solutions of the waves at the event horizon and theergosphere. The time-domain solutions are obtained by implementing thenumerical techniques described mainly in Ref.s [21], [22]. In particular, ourstudy demonstrates a very good agreement between the full time-domainand asymptotic frequency domain solutions and reveals spectral features tounderstand the dependence of superradiance to the rotational speed and tothe frequency of the incident waves. The paper is organized as follows: Sec-tion 2 describes briefly the BEC system and gives a theoretical formulationleading to the main (Klein-Gordon) equation. Section 3 and 4 are devotedto the implementation and computation of the time-domain solutions. Sec-tion 5 presents the asymptotic solutions in the frequency domain. The lastsection discusses the main results and concludes the paper.
We begin by a brief description of the Bose-Einstein condensate as the phys-ical system of interest. A quantum system of N interacting bosons in whichmost of the bosons occupy the same single particle quantum state, the sys-tem can be described by a Hamiltonian of the form; H = Z dx ˆΨ † ( t, x ) " − ~ m ∇ + V ext ( x ) ˆΨ( t, x )+ 12 Z dxdx ′ ˆΨ † ( t, x ) ˆΨ † ( t, x ′ ) V ( x − x ′ ) ˆΨ( t, x ′ ) ˆΨ( t, x ) . (2.1)Here V ext is an external potential and V ( x − x ′ ) is the interatomic two-bodypotential, m is the mass of the bosons and ˆΨ † ( t, x ) is the boson field operatorwhich includes the classical contribution ψ ( t, x ) plus excitations ˆ ϕ .In the non relativistic limit most of the atoms lie on the ground stateand the interatomic interaction is taken as V ( x − x ′ ) = U δ ( x − x ), U =4 aπ ~ /m , where the constant a is called the scattering length. Closed-formequation for weakly interacting bosons, with the potential defined aboveleads to the time dependent Gross-Pitaevskii(GP) equation: i ~ ∂ψ∂t = − ~ m ∇ + V ext + U | ψ | ! ψ ( r, t ) . (2.2)Here in hydrodynamic form the wave function can be written in terms of itsmagnitude and phase: ψ ( r, t ) = √ ρe iS . (2.3)2hen the density of particles is given by ρ ( t, r ) = | ψ ( t, r ) | and the back-ground fluid velocity is defined as ~υ = ( ~ /m ) ∇ S . A general review on BECanalogy can be found in [23], [24].Fluid velocity for the DBT model is defined to have a tangential andradial components, ~υ = υ ˆ φ + υ ˆ r = − Ar ˆ r + Br ˆ φ (2.4)where A and B are constants to be determined.By linearizing the GP equation around some background ρ = ρ + ρ and S = S + S , we reach two equations defining the density fluctuationsand the phase fluctuations: ∂ρ ∂t + ~ m ∇ · ( ρ ∇ S ) + ∇ · ( ρ υ ) = 0 , (2.5) ∂ t S = − υ · ∇ S − U ~ ρ + ~ m D ρ , (2.6)where D is given by D = 12 √ ρ ∇ ρ √ ρ − ρ ρ / ∇ √ ρ . (2.7)Hydrodynamic approximation (quasiclassical approximation) where D = 0is justified by pointing out that D is relatively small compared to otherterms. The pressure term is of the order U ρ/R while the quantum pressureterm is of the order ~ /mR , where R is the spatial scale [25]. This impliesthat for R >> ~ √ mU ρ (2.8)hydrodynamic approximation hold, that gives the healing length parameter.Therefore, the approximation leading to the KG equation is given in its finalform as ∂∂t (cid:20) ρ c (cid:18) ∂S ∂t + ~υ · ∇ S (cid:21)(cid:19) − ∇ · ( ρ ∇ S ) + ∇ · (cid:20) ρ c (cid:18) ∂S ∂t + ~υ · ∇ S (cid:19)(cid:21) = 0(2.9)where the speed of sound is defined by c = p ρU /m . The stationary, axiallysymmetric metric associated with this configuration will be; ds = ρ c " − ( c − A + B r ) dt + 2 Ar dtdr − Bdtdφ + dr + r dφ + dz . (2.10)3or a constant density profile, for which the speed of sound is constant,the resulting equation is the massless Klein-Gordon wave equation for linearperturbations of the velocity potential, or phase of the wave function. The coordinate transformation indicated below is particularly useful to min-imize the number of off-diagonal components of the metric, leaving only one,which helps in analyzing the asymptotic behavior, and to reveal the eventhorizon and the ergosphere. dt = dt ∗ − g ∗ dr dφ = dφ ∗ − h ∗ dr r = r ∗ z = z ∗ , (2.11)where h = − ( AB ) / ( r ( A − c r )) and g = − ( Ar ) / ( A − c r ). We dropthe *-superscript in the following part of the formulation. The line equationtakes the form ds = ρ c − − A + B c r ! dt + − A c r ! − dr − Bdφdtc + r dφ + dz . (2.12)Now it is easy to see the distinction between the event horizon and the ergo-sphere. From the definitions, as for the Kerr black hole in general relativity,the radius of the ergosphere is given by the vanishing of g and the coor-dinate singularity of the metric signifies the event horizon. For the DBTmodel, they read as r event = A/c, r ergo = ( A + B ) / /c > r event . (2.13) In order to solve the Eq.2.9, first we write the line element in the form; ds = − α dt + γ ij ( dx i + β i dt )( dx j + β j dt ) , (3.1)where α = c , γ ij = diag (1 , r ,
1) and β i = ( A/r, − B/r , S ( r, t ), we introduce twoconjugate fields;Φ = ∂ Ψ ∂x i Π = − α (cid:18) ∂ Ψ ∂t − β i Φ i (cid:19) , (3.2)4here Ψ = ψ ( t, r ) e imφ e ikz , Π = π ( t, r ) e imφ e ikz and Φ = φ ( t, r ) e imφ e ikz and ( k, m ) are the axial and azimuthal wave numbers [22], [26]. In this work,in accordance with the BEC vortex stability conditions, we consider theazimuthal wave numbers of m = 0 and 1 only. For m = 2, the vortex woulddecay into two separate vortices of m =1 [27], [28]. Then our hyperbolicsystem reads, ∂ t π + c∂ r φ − Ar ∂ r π = − imBπ /r + c ( k + m /r ) ψ − cφ /r∂ t ψ − Ar ∂ r ψ = − imBψ /r − cπ ∂ t φ + c∂ r π − Ar ∂ r φ = 2 imBψ /r − ( A + imB ) φ /r . (3.3)The remaining first order set of coupled PDEs are much easier to handlethan the hyperbolic PDE above that we start with. The numerical challenges of using a constrained evolution scheme is mostlyabout avoiding constraint violations and other possible numerical issueswhich may be associated to solver type and settings, element type and size,meshing, tolerances, etc. All of these ingredients must be fine tuned in thecomputation to get proper results. However, we still have the freedom totry different interior boundary conditions because excision, i.e by placingthe boundary inside the horizon and excises its interior from the computa-tional domain. In theory at least, nothing physical inside the black hole caninfluence any of the physics outside the horizon [29].This main section is organized as follows: We first calculate the time evo-lution of the perturbations of the velocity potential by solving the equations3.3 and the energy of the perturbations given further in Eq.4.4. Based onthe ranges of the model parameters Ω and ω , superradiant and superradiantcases are demonstrated for comparison.The initial value is chosen as a Gaussian pulse centered at r , modulatedby a monochromatic wave [26]: ψ (0 , r ) = Aexp h − ( r − r + ct ) /b − iω ( r − r + ct ) /c i . (4.1)5
20 40 60 80 100 120 r/a -0.6-0.4-0.200.20.40.6 R e t=30t=50 t=90 t=0 t=130 Figure 1: Snapshots of the time evolution of the perturbation, for the casem=0 as a function of distance r from the vortex, for r =50a and b =10a with ω =0.7c/a, Ω=1.4c/a 6
20 40 60 80 100 120 r/a -0.500.511.5 R e t=50 t=30 t=90 t=0 t=130 Figure 2: Snapshots of the time evolution of the perturbation, for the casem=1 as a function of distance r from the vortex. The parameters used arein Fig. 1The equation system 3.3 is integrated numerically using Matlab PDEsolver by modifying equation format and boundary conditions accordingly[21]. The set of equations 3.3 allows us to decompose Π , Φ and ψ intocharacteristic fields that propagate along null ray u + ∝ Π + Φ u − ∝ Π − Φ (4.2)At large distances purely outgoing wave is implemented such that u − = 0, π = φ . While outer boundary condition has to be well-behaved for thecalculation, simulation ends before the wave reaches the outer boundary,therefore nature of the reflections produced when a wave passes through theboundary is largely irrelevant. And for the inner boundary, because of theexcision, no boundary condition is set. The computational spatial (radial)and time domain are set as 0 . < r < , < t < r = 0 . , ∆ t = 0 .
05, respectively. Inner boundary is set accordingto the constraint values at the event horizon. While the waves are free topropagate inside the horizon, reflected waves from the r = 0 should noteffect the expected result, such that inner boundary should not be too close7o the singularity r = 0 or to the event horizon r = 1 a . The domain issufficiently large to achieve steady state solutions, whereas the discretizationsteps provide good accuracy for the solution and for the constraint equations.The incident wave is a cylindrically imploding Gaussian wave, centeredat r = 50 a with a width of b = 10 a and azimuthal wavenumber k = 0 . /a .Here, c is the propagation speed of sound in the condensate and a is locationof the event horizon. Both parameters are scaled to unity. We note that thelocation of the incident wave should be chosen numerically far enough sothat the scattering outcome is independent from the location of the incidentwave. The angular speed of the vortex is Ω. In the present calculations, weconsider values of Ω up to Ω = 4 c/a . The frequency of the incident wave is ω = Ω / m = 0) case goes to zero while in the superradiant case(m=1) it gets amplified through backscattering. t c/a C ( - ) t c/a C ( - ) Figure 3: Constraint violations at event horizon(a) and outer boundary( r =80 a )(b) for superradiance (m=1). The parameters used are in Fig. 1. Dottedlines signify the time frames(t=30,50,90,130) shown in Fig1 and Fig.28
30 60 90 120 150 t c/a C ( - ) t c/a C ( - ) Figure 4: Constraint violations at event horizon(a) and outer boundary( r =80 a )(b) for non-superradiance (m=0). The parameters used are in Fig. 1In order to check the quality of the numerical analysis, we monitor theconstraint value C, from the definition of Φ, Eq.3.2 C = | ∂ r ψ − φ | . (4.3)The constraint values should be closer to zero and not increase in time suchthat any unphysical waves, backscattered radiation would not overpowerthe actual results. Since the reflected wave can only reach approximately to r = 80 /a at the time t = 130 c/a , as seen in the Fig1 and Fig.2, r = 80 /a point is chosen to calculate the constraint violations for the outer boundary.Even though the outer boundary for the simulation is at r = 150 /a , wherethe wave can not reach during the computation time.It shows from the Fig.4 and Fig.3 that the constrained value, C, doesnot grow indefinitely in time and remains under a certain value. At theinner horizon, scaled to 1 a , constraint values shown to be larger than outerboundary,reaches the maximum value around t = 50 c/a , when the wave ispartially absorbed by the event horizon. But still remains small enoughthat the violations are negligible. In addition, we observe that for largerfrequencies, we have to keep an eye on the inner constraint violations moreclosely to check that results are meaningful, since the simulations becomeunstable much faster.The time variation of the energy of wave packet is given by E ( t ) = ( ρ ~ / M ) Z π dφ Z H dz Z r max ( ∇ ψ ) rdr. (4.4)9
30 60 90 120 t c/a -0.200.20.40.60.811.21.41.6 E ( t ) / E ( ) m=0m=1 Figure 5: Time evolution of the energy gain of the wave packet, superradiantm=1 case and non-superradiant m=0 case. The parameters used are in Fig.1 Figure 5 shows the time evolution of the energy of the wave, normalizedby the energy of the incident wave, for the non-radiant (dashed blue curve)and for the superradiant (solid red curve) cases respectively. Note that thewave arrives to the event horizon near t = 35 c/a. In the non-radiant case,all the impinging energy is lost to the vortex sink. In the superradiant caseconditions the scattering process extracts energy from the ergosphere andthe energy of the backscattered wave exceeds its incident value. Also, wedid not calculate the total energy densities but the energy densities per unitlength in z-direction [25]. 10igure 6: Density fluctuations, ρ in r-t plane for superradiance case, m=1(left) and m=0 (right).Figure 7: Closeups of the data in Fig.6.The time evolution of the density fluctuations associated with the acous-tic wave propagation are plotted Fig.6 for superradiant and non-radiantcases. Figure 7 give the detailed view of the propagation of fluctuationsnear the event-horizon. Sudden increase in the density fluctuations for thesuperradiance case, shown in Fig. 7, stays inside the event horizon, r=1.Evidently, the numerical treatment of the region beyond the event hori-zon is inherently prone to numerical instabilities. We found that at largetime scales after the scattering event, noise fluctuations emerge within theevent horizon, which can propagate into the real space and render the sim-ulation results unacceptable. Furthermore, this time scale decreases withincreasing omega. Thus, simulation times are adjusted to avoid this prob-lem. Fortunately, the existence of the event horizon allows non-strict bound-ary conditions so that the numerical instability contained withing the event11 / =4=1.4 Figure 8: The Energy at t final = 150 c/a normalized to its initial value E ( t = 0) as a function of ω/ Ω where 0 < ω < Ω i .The parameters used inthe calculations are Ω i = 1 .
4, 4, r = 50 a , b = 10 a .horizon (i.e. r < r event ) does not affect the results in the real space domain.The amplification factor of the reflected wave as a function of the ratio ω/ Ω is plotted in Fig.8, for Ω = 1 .
4, and Ω = 4, respectively. The am-plification increases rather monotonically up to a certain ω/ Ω ratio, whichdepends on the particular value of Ω. After that, the amplification decreasesrapidly to unity as ω/ Ω approaches unity. Thus, the maximum superradi-ance occurs at a particular ω of the incident wave, in relation to the Ω of thevortex. In the next section, we will analyze this behavior in the frequencydomain. In this section we analyze the Klein Gordon equation (Eq.2.9) in the fre-quency domain. Using separation of variables, the formal solution of theKG is expressed as ψ = e − iωt e imφ e ikz P ( r ) , (5.1)12here k and m are the axial and azimuthal wave numbers, respectively. Toavoid polydromy problems [21], that is to make ψ single valued, m should betaken as an integer and k a real number defined by the boundary conditionsalong the z axis.By inserting (5.1) into (2.9), we obtain a second order ODE for the radialpart: d Pdr + A + r c + 2 iA ( Bm − r ω ) r ( r c − A ) ! dPdr + iABm − B m + c m r + 2 Bmωr − r ω + c k r r ( r c − A ) ! P = 0 . (5.2)We substitute P = R ( r ) H ( r ∗ ) with a Regge-Wheeler tortoise coordinate, r ∗ , which will map r ∈ [ r H , ∞ ] to r ∗ ∈ [ −∞ , + ∞ ]: r ∗ = Z r r − A /c dr. (5.3)Then after a few careful calculations [30], the radial equation takes the finalform d H ( r ∗ ) dr ∗ + ω c − V ( r ) ! H ( r ∗ ) = 0 , (5.4)where V = k (1 − A r c ) − A c r − A (cid:0) m − / (cid:1) + B m c r − r c (cid:16) c − m c − Bω (cid:17) . (5.5)Near the event horizon and at the far field ( r → + ∞ ), the asymptoticsolutions are given by the harmonic functions, H ( r ∗ ) = e iω + r ∗ c + Re − iω + r ∗ c , r ∗ → + ∞ (5.6) H ( r ∗ ) = T e − i ( ω − Ω m ) r ∗ c , r ∗ → −∞ (5.7)where ω = ω − k c and B = Ω A /c . The equality of the Wronskian ofthese solution at asymptotics gives1 − | R | = (cid:18) ω − m Ω ω + (cid:19) (cid:12)(cid:12)(cid:12) T (cid:12)(cid:12)(cid:12) (5.8)13here R and T are the amplitudes of the reflection and transmission co-efficients of the scattered wave. It shows that when the superresonancecondition, ω < m Ω, is satisfied, reflection coefficient has a magnitude largerthan unity [31],[32]. / | R | = 0.5= 0.7= 0.9= 1 Figure 9: Reflection coefficients as a function of ω , calculated in the range0 < ω < m Ω i . Parameters are m = 1, and Ω i =0.5, 0.7, 0.9, 1.14 | R | = 2= 3= 4= 5= 6 Figure 10: Reflection coefficients as a function of ω , calculated in the range0 < ω < m Ω i . Parameters are m = 1, and Ω i =2, 3, 4, 5, 6.Eq.5.8 reveals the superradiance condition clearly (i.e ω < m Ω) and givesthe full spectral behavior of the reflection coefficient. Thus, we can obtainthe reflection coefficient through the Fourier components of the asymptoticfar field solution, which is obtained through Eq.5.3 and Eq.5.4. Figure 9and Fig.10 show the reflection coefficient as a function of incident wavefrequency for different values of the angular speed of the vortex (Ω) (thatis the horizontal axis represents multiple ranges 0 < ω < Ω i ). Fig.9 is forΩ <
1, Fig.10 shows the range 2 < Ω <
6. Here, we used the same modelparameters as in the time-domain solution presented in the previous section.15 / | R | -domaint-domain Figure 11: Reflection coefficients of the scattered wave with parametersgiven under Fig.1 calculated in the time domain and the frequency domainIn the frequency domain, coordinate transformation allowed us to carrythe calculations outside the event horizon with the asymptotic solutions andreflection coefficient defined in Eq.5.6 is calculated. But, in the time domain,no coordinate transformation is applied, and inner boundary for radius iskept inside the event horizon, r = 1 a , thus allowing the wave propagatefreely inside the horizon. Reflection coefficient is calculated at sufficientlyfar away from the horizon, and inside the horizon is dismissed from thecalculation vie excision technique. If we compare the reflection calculatedin the time domain for a given initial wave with the one in the frequencydomain with given asymptotic solutions, results do not differ. Figure 11shows the comparison between two solution methods. Superradiance phenomena is the analog of the Penrose process for rotatingblack holes. Energy extraction from the black hole analogy, i.e. the vortexdefined in BEC is shown to be possible by examining the scattering process.Acoustic superradiance defined as amplification of the reflection coefficientto values greater then one.Although the superradiant experiments focuses on the shallow water16ave, water vortex or optical systems rather than 2D-BEC vortex, theoreti-cally shown that it is possible and due to quantum nature it may be a greatcandidate for other phenomenas like Hawking Radiation [14],[19],[15].In this work, we investigated the amplified scattering of acoustic wavespropagating in a BEC, from a vortex state with a constant background den-sity, by obtaining both time-domain and asymptotic frequency domain solu-tions numerically. Time-domain study amounts for solving the Klein-Gordonequation which governs the propagation of sound waves in the presence ofvortex in an analogy to scalar field propagation in curved space-time of ablack-hole. It is worth to note that the classical (macroscopic) wave func-tion of the BEC represents the classical space-time of General Relativity onlywhen probed at long-enough wavelengths such that it behaves purely hydro-dynamically. The major outcome of the study is to demonstrate a good spec-tral agreement of the superradiance (reflection coefficient) as obtained fromfull time-domain calculations and from the asymptotic frequency domaincalculations. This strengthens the validity of the spectral analysis basedonly to the asymptotic solutions, which can be calculated with significantlyless computational resource compared to that required by the time-domaincalculations. The frequency spectrum analysis gives further insight to thesuperradiance condition that is given in terms of the modulation frequencyof the incident wave and the angular speed of the vortex as ω < m
Ω. Themaximum superradiance shows a gradual increase with increasing Ω. For agiven Ω, superradiance is maximized for ω/ Ω ≈ . − .
69 when Ω < ω/ Ω ≈ . − .
85 when Ω >
1. Typically, a strongly modulated Gaussianpulse is able to acquire more energy through scattering. As a final note, thetheoretical and computational formulation presented in this work is suitablefor the implementation of different background density profiles, which canpotentially extend the exploration beyond the constant background-densityapproximation. This will be pursued in subsequent studies.
Acknowledgement
Betül Demirkaya is supported by TUBİTAK-BİDEB 2211 National Schol-arship Program for PhD Students.
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