arXiv:1708.02623v2 [gr-qc] 6 Jun 2018 eakbefidn pndtewyt ieinvestiga- wide phenomena. a gravitational analogue to This of way tion the system. opened non-gravitational find finding to a remarkable possible in is radiation it sound that Hawking study showing to fluids, moving model pro- in a Unruh waves as when [14] evaporation 1981 black-hole to posed back dates analog first The physics. ex- quan- nonlinear novel and tum classical furnish for may regimes physics unexplored and analog planations hand, while, other gravity, the quantum suggest on for may interpretations laboratory these emulating the unexpected hand, line in one research phenomena on This Planck-scale advantage: two-fold 25]. a 24, has 18, superradiance[8–15, and radiation of Hawking as confirmations phenomena, for experimental quantum-gravitational looking provide is to scientists systems of community analog broad evidences. a most experimental result, pos- of a the As absence not is the still proposal to is due theoretical it promising, which however, establish efforts, to these sible all non- Despite gravity[4], 2], relativity[6, 7]. the special quantum doubly theories[1, to loop and geometry[5] gravity theory[3], commutative leading string quantum out, as many such carried of been formulation have Nume- attempts mechanics. quantum rous and relativity general fying quantum nonlinear in problems open physics. for novel provide tools can and theories these theories that gravity ex- demonstrate quantum fascinating of numerous tests to perimental way Our the superradiance. open and exis- results horizon event the an inhibits of tence energy-dependence that A showing metric reported, the is metric. analysis energy-dependent quantized em- in fully rainbow optics, those the recent nonlinear as ulate a and media, gases in nonlocal Bose-condensed as in gravity. space- waves hole rainbow of Nonlinear black called analog rotating theory, first quantum-gravity a the near on superra- report time and We horizons event diance. in- holes, The relativity, black general cluding to developments. limited and further were simulations for models inspiration theoretical Laboratory give validate gravity. analogue may of emulation challenge the frameworks is experimental different in phenomena n ftecretmjrcalne npyisi uni- is physics in challenges major current the of One gravitational quantum unobserved the Testing 2 nttt o ope ytm,Ntoa eerhCucl( Council Research National Systems, Complex for Institute unu iuaino ano rvt ynnoa nonlinea nonlocal by gravity rainbow of simulation Quantum 3 eateto hsc,Uiest aina izaeAldo Piazzale Sapienza, University Physics, of Department nvriyo ’qia i eoo1,I600LAul,It L’Aquila, I-67010 10, Vetoio Via L’Aquila, of University ∗ orsodn uhr [email protected] author: Corresponding 1 eateto hscladCeia Sciences, Chemical and Physical of Department .C Braidotti C. M. 1 , 2 ⋆ with omaet h pc-iepoete,a nuiga inducing their as and properties, speed energy-dependence space-time light non-constant the metric affects the form all enclose lc oe hr h ereo nonlocality of rotating degree a allows the nonlinear of nonlocality where space-time defocusing that gravity hole, excitations rainbow a show black the of We in mimic behavior to background the medium. vortex study nonlocal we a paper, on developments. further this for inspiration In an of unexpected be many and known implications furnish results not may are the emulations thermodynamics, inventing of hole and black Some in as Eq.(1), constant. gravitational neeg eedn ercocr ntehydrodynami- the in occurs metric dependent energy an a furnishes metric. This dependent a energy with in ergoregion. space-time fields curved second-quantized the of of quantum-simulation true vanishing enhancement the an shows of and algorithm[28] Runge-Kutta when horizon superradiance event the of hole black weakening increasing the consequent give Numerical of the excitations fading and quantum the scale. and of Planck evidence classical the of to simulations proximity the mines nryo h refligpril.Ti orsod to element: corresponds line invariant This with the metric on particle. energy-dependent free-falling depends an which the 27], geometry, of 2004[26, space-time energy in a Smolin on and relies Magueijo gra- spe- by Rainbow doubly proposed curvature. of vity, “rainbow incorporate generalization to called a relativity[6] gravity is cial quantum theory of This theory gravity”. analogs recent quantum and a classical on of report been we study, never this In has scenario gravity an quantum proposed. numerous knowledge, a our the of To test analogue gravity. as- to quantum phenomenological of frameworks formulations providing to all not However, address pects, mainly [12]. in studies reviewed these as liquids[22], Fermi and 5 6,Bs-isencnests[7 9 20], 19, optics[10, [17, condensates acoustics[14], Bose-Einstein as 16], fields black 15, research of many emulations in reported holes Unruh after authors Many n .Conti C. and ds esatcnieigtecasclrgm n h way the and regime classical the considering start We S-N) i e arn 9 08 oe Italy Rome, 00185 19, Taurini dei Via ISC-CNR), 2 P = E rpeetto yaped-pcrlstochastic pseudo-spectral a by -representation P g µν h lnkeeg.Tefunctions The energy. Planck the ( σ E ulqatmdnmc saaye in analyzed is dynamics quantum Full . ) 2 oo5 08 oe Italy Rome, 00185 5, Moro dx , 3 µ dx ν = t aly − c f n/ra energy-dependent an and/or 2 ( ( dx E/E 0 ) 2 P rity ) + g 2 ( ( dx E/E i ) f σ 3 2 P He ) and deter- , [21] (1) g 2 cal approximation of the normalized nonlocal nonlinear by the background ρ0. Schr¨odinger equation [r = (x, y)] [29–31] The geometrical description of the optical system fails when the background density varies on a scale smaller 1 ı∂ ψ + 2 ψ P R(r) ψ 2ψ =0. (2) ˜ t 2∇xy − ∗ | | than the healing length ξ =1/(2 P Rρ0), i.e. in the high energy limit where E K2. Inq terms of the quantum- Equation (2) describes the field evolution in many phy- gravity analog, the scale∝ of background density variation sical systems as nonlinear optics with thermal[29, 32] or exceeds the analog Planck length ξ, violating the Lorentz re-orientational nonlinearity, Bose-Einstein condensates invariance. Hence, the analogue is self-consistent only (BEC)[33, 34] and plasma-physics[35, 36]. In (2), de- ∗ in the low energy and low-momentum regime for which notes a convolution integral. The form of the kernel K < 1/ξ. R(x, y) depends on the specific physical system and its Following general relativity, a metric determines the in- ˜ Fourier transform is R(Kx,Ky). The field ψ is norma- variant square of a line element, ds2, given by lized such that ψ 2dr = 1, and P measures the | | 2 µ ν strength of the nonlinearity. ds =g ˜µν dx dx = Thanks to the hydrodynamicalR approach, commonly used R˜ for dispersive shock waves [32] and also analog gravity 2 ˜ 2 2 = 2 csR v0 dt + (6) cs !(− − [37], we study the behavior of small excitations on top   of the metric induced by the fluid of light. By writing ıφ 2 2 the field ψ as ψ = ρe with v = φ, Eq. (2) re- + dr + (rdθ) 2vrdrdt 2vθrdθdt . √ − − duces to the continuity equation and∇ the Euler equa- " # ) tion, with bulk pressure [ρ] = ρP R(r) ρ (see Sup- P ∗ Equation (6) is written in polar coordinates (r, θ), with plementary Information). Small excitations in the pho- 2 2 2 1 v0 = vr + v , vr = ∂rφ0 and vθ = ∂θφ0. The local case ton fluid are described by letting ρ = ρ + ǫρ + O(ǫ2) θ r 0 1 [8] is found letting R˜ = 1 in Eq. (5). and φ = φ + ǫφ + O(ǫ2). In a slowly varying back- 0 1 Within the geometrical description validity limit region ground ρ and in the eikonal approximation, we have 0 (K < 1/ξ), we can mimic a black hole through vortex ρ =ρ ¯ eı(Kxx+Kyy Et) and φ = φ¯ eı(Kxx+Ky y Et), with 1 1 − 1 1 − solutions of Eq. (2). This, in addition with the presence E the angular frequency for an inertial observer at infi- of nonlocality, enables to simulate rainbow gravity. nity. The generalized dispersion relation is (see Supple- Vortexes are formed by a dark hole with circular sym- mentary Information) [38–41]: metry and a helical wave front. Their wavefunctions, iφ0(θ) 4 in polar coordinates, is ψ0 = ρ0(r)e , with phase K v 2 ˜ 2 K (E 0) = ρ0P R(K)K + . (3) φ = mθ. The integer m is the winding number, or vor- − · 4 0 p tex charge. The region in proximity of r = 0, where In Eq. (3) v0 = φ0 is the background velocity field. ρ(0) = 0, is called the vortex core. v ∇ When 0 = 0, the high energy limit, i.e., short wave- The form of the metricg ˜µ,ν determines the properties of 2 2 1/2 lengths and large momentum K = Kx + Ky , corre- the analog space-time, as the presence of an event hori- sponds to E K2 as for free particles. Whereas, in the zon and ergoregion. ∝
long wavelength limit (i.e. for small K) E csK, where The analog system presents an event horizon surface if 2 ∂≃ [ρ] the velocity of sound equals the radial velocity of the cs = Pρ0 is the local speed of sound cs P∂ρ . ≡ ρ=ρ0 fluid, i.e. cs = vr. Low energy modes cannot escape In the hydrodynamical regime, we obtain the following from this region. Furthermore, by geometrical consider- equation for the Fourier transformed massless scalar field ations, one can find thatg ˜ 0, and hence c2 < v2, ˜ tt ≥ s 0 φ1 in a 2 + 1 dimensional curved space (see Supplemen- determines the ergoregion, which is the region of space- tary Information): time where low energy modes are dragged in the moving light flow. 1 µν ] ∆φ1 = ∂µ( g˜g˜ ∂ν φ1), (4) In the case of local response function R˜ = 1, this geo- √ g˜ − metry has an ergosurface placed at c = v . In order to − p s θ µν g µν 1 mimic a black hole, it is necessary to introduce a radial whereg ˜ is the covariant metric (˜g )− =g ˜µν , inward velocity vr, by letting φ0(r, θ)=2π r/r0 + mθ, ˜ 2  2 T  which induces an event horizon [8, 42, 43]. In this frame, R (c R˜ v ) v0 g˜ = s 0 (5) 2 p 2 µν 2 − v− − the position of the event horizon rH is π /r0cs and the cs 0 I !  −  1 2 2 2 2 ergoregion rE = 2 rH + rH +4m r0rH . Unfortu- andg ˜ = det(˜gµν ). In (5), I is the 2 2 identity matrix. nately, such vortex solution p with a central sink in a non- Equation (4) gives the analogy between× the light wave linear medium with local response function is not stable propagation and the gravitational field: the light fluc- [44, 45] and hence it does not allow experimental tests of tuations behavior is affected by the metric gµν induced the analogy. Whereas, in the nonlocal case, the stability 3 of the vortex solitons has been reported [46]. Nonloca- with σ the degree of nonlocality. In (11), B = 0 corre- lity opens the way to many theoretical and experimental sponds to a Lorentzian spectral response characteristic developments. of optical media with thermal or re-orientational nonli- In order to provide an analog of a rotating black hole nearities; while, in the case B = 1, the nonlocal response and analyze the effects of a rainbow gravity space-time, is composed by a local and a nonlocal contribution, as we study the propagation of a pulse with group velocity occurs in photonic BEC. [47, 48] Figure 1b shows the v = ∂ ω in a nonlocal medium. We remark that nonlo- horizon area trend for Lorentzian response as function g K A cality provides the stability of the vortex background, al- of the transverse wavevector Kx for different values of lowing mimicking the Kerr-type black hole metric, while, σ. We observe that nonlocality affects the value of the the different frequency components of the pulse spectrum horizon area and hence, wavepackets with different mo- will help in testing the energy space-time dependence of mentum see different areas. In the following, we will see rainbow gravity. that this kind of behavior resembles what happens in a For a narrow-band wavepacket with mean momentum K rainbow gravity scenario. In fig. 1b the horizon area A in a nonlocal medium and in the small K limit, the en- decreases with Kx in the nonlocal case. In Fig. 1c, when ergy E is proportional to K and the metric in Eq. (6) σ = 0 the horizon area saturates at a lower value when 6 A can be written in terms of the energy as in the origi- Kx growths, i.e. the horizon area fades into a volume. nal rainbow gravity theory. (In our units EP = 1.) We In the following, we show that these findings affect the show in the following that our system emulates a rain- black hole thermodynamics. bow gravity theory and the metric can be written as the In this frame, a notable effect that may arise is super- corresponding Kerr black hole metric[26, 27, 49], i.e., radiance, which takes place in proximity of the event horizon of a rotating black hole. Superradiance is the T (r) R(r) r2dθ2 rdθdt 2 2 amplification of radiation excitations due to the angular dsr = 2 dt + 2 dr + 2 +Vθ(r) −f (E) g (E) g(E) f(E)g(E) velocity of the vortex. [11] Being rainbow gravity a re- (7) cent theory, there is not a comprehensive vision about its where the functions T (r), R(r) and V (r) are the el- θ phenomenology, hence the occurrence of superradiance ements of the energy-independent metric matrix. By has still not been fully addressed. Despite this, in the rescaling time and azimuthal coordinates far from the rainbow gravity scenario, we expect that superradiance vortex core, we find that, in our case, the functions f(E) is reduced because of the energy dependent coupling and and g(E) are the fading of the event horizon area. We are not aware c2 c2 v2 c2 c2R˜ v2 of previously reported analysis of analog rainbow super- f(E)= s s − 0 ; g(E)= s s − 0 . (8) 2 2 2 2 radiance. R˜ c R˜ v R˜ cs v0 s − 0 − In order to analyze superradiance, we consider a pertur- 1/2 i(Ωt nθ) Details on the calculus are reported in the Supplementary bation φ1 of the form φ1(z,r,θ) = r− G(r∗)e − Information. Equation (8) shows that, in case of nonlocal such that it is solution to the Klein-Gordon equation (4) response, the position of the event horizon and ergoregion with metric (5), where n is the winding number and Ω the are energy independent, while the horizon area is energy- wave frequency. It is worth to change the coordinate sys- dependent, i.e., in the 2D case we have tem, adopting the ”tortoise coordinate” r∗, which maps 2π 2π the region r [rH , [ to the entire axis. Tortoise co- r ∈ ∞ 1 = dθ√g = dθ = (9) ordinates are defined as dr∗ = (1 rH /r)− dr. Note A 0 r=r 0 g(E) r=r − Z H Z H that r∗ is defined only for r > rH . As r approaches the r 2πr v R˜ event horizon rH , r∗ , while far from the vortex = H θ (10) → −∞ 2 ˜ 2 core (r ) we have r∗ r. After some algebra (see cs scsR v0 → ∞ → − Supplementary Information) we find the Schr¨odinger-like since at r = rH we have that cs = vr . The dependence equation for the radial component G(r∗) of on the energy E is due to nonlocality.| | This is in 2 A ∗ agreement with a key-prediction of rainbow gravity[27]. ∂r G + Veff G =0, (12) Indeed, in the nonlocal case, particles with different e- where the effective potential V is given by nergy see different horizon areas: Fig. 1a shows a sketch eff of the fading of the area of a rotating black hole near the 2 Ω v n Planck energy scale in rainbow gravity. In the standard V = θ + eff − local case (R˜ = 1), there is no energy dependence. cs R˜ rcs R˜ ! 2 In order to describe different physical systems as opti- dr 1 rp n2 p dr 1 H + , (13) cal nonlinear waves and Bose-Einstein condensates, we 3 2 2 − dr∗ 2 Rr˜ − r dr∗ 4r consider a nonlocal response function of the form     ˜ 1 where we assumed ∂rR 0. R˜(K)= B + , (11) ≃ 1+ σ2K2 In order to compute superradiance, we analyze Eq. (12) 4

(b) (a) 20 =0

A 10 =5 =10 0 Event Horizon (c)

rH 20 =0 =5 15 Fading A =10 10 0 0.5 1 k x

FIG. 1. (Color online) (a) Sketch of the black hole in rainbow gravity. Red lines correspond to event horizons in the local (σ = 0-inner horizon) and nonlocal cases (σ =6 0-outer horizon) for a specific K; (b) horizon area A as function of Kx for lorentzian nonlocal response function (B = 0) varying σ; (c) as in (b) for BEC-like response function B = 1.

in two limits r∗ and find that the effective poten- fully developed. Because of this, we resort to numerical tial is → ±∞ simulations in order to suggest specific experimental 2 directions. To our knowledge, simulations of analog Ω superradiance have not been addressed yet. for r Veff = , (14) → ∞ In the following, we first analyze classical superradiance cs R˜ ! 2 in the local and nonlocal case, then we will consider the p Ω vθn fully-quantum counterpart. for r rH Veff = , (15) → − We simulate the propagation of a vortex beam cs R˜ rH cs R˜ ! in a defocusing nonlinear nonlocal medium by and hence we have p p the classical nonlinear Schr¨odinger equation, ∗ ∗ i Ω r i Ω r Eq. (2) (see Methods). The initial condition is √R˜∞c √R˜∞c for r G(r∗)= e s + e− s , 16 r iφ0 → ∞ R ψ0 = N exp[ (r/w0) ] tanh e with wv is the (16) − wv (Ω−nΩ ) ∗ vortex waist and N is a normalization constant. ψ0 i H r   √R˜ c for r r G(r∗)= e H s , (17) includes a finite supergaussian background with waist → H T w0 wv. The vortex velocity v = (vr, vθ) is composed vθ where ΩH = is the angular velocity at the horizon ≫ rH by the radial component vr = π/√rr0, which is linked and and are the reflection and transmission coeffi- to the event horizon, and the azimuthal component cients.R Eq. (17)T accounts only for the ingoing wave at the vθ = m/r. horizon since the outgoing mode can not be considered The considered initial condition ψ0 is not an exact as a physical solution. By the Abel’s theorem, the Wron- solution of Eq. (2). For this reason, it can be written skian of Eqs. (16) and (17) is constant. Hence, equating as the exact solution ψ¯ of Eq. (2) plus an additional the Wronskian computed at the two limits, we get the term ψ′: ψ0 = ψ¯ + ψ′. This fact, in the presence of the relation between and : R T radial velocity field, makes our configuration an optimal ˜ analog for superradiance. Figure 2 shows the intensity 2 Ω nΩH R 2 2 1 = − ∞ . (18) spot profile, its phase and the x section of ψ at t = 0 ˜ | | − |R| Ω s RH |T | (up line panels) and at t = 30 (down line panels). Inset Equation (18) shows that, if the frequency Ω of the of panel 2c shows the positions of the event horizon (x 28) and ergoregion (x 46) of our black hole incident perturbation is in the range 0 < Ω < nΩH and ≃ ≃ R˜∞ analog. Comparing panels 2c and 2f, we see that during if ˜ > 0, the amplitude of the scattered wave is larger RH the beam evolution the intensity near the vortex core than that of the incident one, i.e., > 1. Hence the |R| increases and the emitted radiation profile exhibits field perturbation is superradiantly amplified in analogy with oscillations. Furthermore, as expected for superradiance, superradiant scattering from a rotating black hole. the oscillations’ amplitude decreases moving away from analog black hole. Classical simulations. As previously said, rainbow We then consider the effect of the energy dependent gravity is a recent theory of quantum gravity not yet 5

×10-5 1 (a) (b) (c) Event Horizon 200 200 Ergoregion 0.8 ×10-6 100 100 3.5 2

0.6 |ψ| 3 2 y 0 0 2.5

|ψ| -50 0 50 0.4 x -100 -100 0.2 -200 -200 0 -200 0 200 -200 0 200 -200 0 200

-5 1 ×10 (d) (e) (f) 200 200 0.8 ×10-6 10 100 100 2

0.6 |ψ| 5 2

y 0 0 -50 0 50 |ψ| 0.4 x -100 -100 0.2 -200 -200 0 -200 0 200 -200 0 200 -200 0 200 x x x

FIG. 2. (Color online) (a) intensity field profile for t = 0; (b) phase profile for t = 0; (c) field section along x for y = 0 for t = 0; inset: corresponding enlarged central region of the vortex profile at t = 0, dots mark the location of the event horizon (x ≃ 28 - red) and ergoregion (x ≃ 46 - green); (d) as in (a) for t = 30; (e) as in (b) for t = 30; (f) as in (c) at t = 30; inset: corresponding enlarged central region of the vortex profile at t = 30. (The beam parameters are w0 = 160, wv = 5, m = 3, σ = 0 and P = 105) metric on the Bogoliubov dispersion relation for local equation and nonlocal nonlinearities. We add to the initial ψ0 a classical noise, which mimics fluid excitations. Figure 3 ˆ 1 2 ˆ ˆ ˆ ˆ ı∂tψ + xyψ P R1 ⋆ ψ†ψ ψ =0. (19) shows the excitations spectrum and the event horizon at 2∇ − h i t = 30 in the local and nonlocal cases. Different nonlocal The quantum field ψˆ obeys the equal time commuta- response functions are taken into account. For local ˜ tion operation ψˆ(r,t), ψˆ (r′,t) = δ(r r′). Quan- nonlinearity (R = 1), the dispersion exhibit a linear † − trend as expected for particles trapped in the event tum fields are mathematicallyh describedi by operator di- horizon (Fig. 3a). This corresponds to a preferred space stributions, and hence the equations which govern their location in the (x, y) space (circle in Fig. 3d). In the evolutions are operator equations that can not be nu- nonlocal case (A = 1), Figs. 3b and 3e show the fading merically solved. However, operators distributions can of the event horizon and the destruction of the linear be expressed by phase-space representations that map Bogoliubov spectrum. Figures 3c and 3f show the case operator equations to equivalent stochastic differential A = 0 with the disappearance of the event horizon: the equations. We adopt the positive -representation that P perturbation behaves as free particles homogeneously transforms the Heisenberg (operator) equations of mo- distributed in space (Fig. 3f). tion in a Fokker-Planck equation (FPE).[28] The posi- tivity of the representation allows to map the FPE to Itˆo stochastic differential equations (details are reported in Quantum simulations. So far, we have considered only Methods): classical analogs of the black hole. It is extremely rele- vant to show the possibility of a true quantum simulation ∂ 1 u =+i 2 u iP [R(r) uv] u + √iΓ(1)u (20) of the analog black hole in order to test the rainbow grav- ∂z 2∇xy − ∗ ity in a true quantum scenario. To this aim, we resort ∂ 1 v = i 2 v + iP [R(r) uv] u + √ iΓ(2)v (21) to the second quantized nonlocal nonlinear Schr¨odinger ∂z − 2∇xy ∗ − 6

FIG. 3. (Color online) (a) excitations spectrum in the local case; (b) as in (a) in the nonlocal case (A = 1 and σ = 5); (c) as in (b) for A = 0 and σ = 5; (d) excitations field in the configuration space in the local case; (e) as in (d) for A = 1 and σ = 5; (f) as in (e) for A = 0 and σ =5. (P = 105) where Γ(i)(r, z) is a real Gaussian white noise. We research in quantum gravity by novel experimental and solve (20) and (21) by a second-order pseudo-spectral theoretical emulations. Since several open questions are stochastic Runge-Kutta algorithm.[28] still present about rainbow gravity and competing the- Figure 4 shows the excitation field spectra and intensi- ories, we think that quantum simulations may provide ties of local and nonlocal case for t = 30. Comparing the new surprising insights. panels 4a, 4b and 4c, we see that Figs. 4a and 4b exhibit linear sidebands which are not present in the nonlocal Methods Classical simulations: we simulate the NLS equa- lorentzian spectrum (Fig. 4c). These sidebands can be tion through the split-step Fourier method with noisy initial attributed to the quantum radiation in proximity to the condition. In order to calculate the Bogoliubov dispersion re- event horizon. Superradiance is also present. Panel 4d lation we subtract the evolved noisy field to the unperturbed shows the presence of an ergoregion in the local frame. solution at the same instant of propagation t, obtaining the This region fades progressively with the increasing of evolved noise. We mediate over several (n=20) noise configu- the nonlinear effect (see panels 4e and 4f). Furthermore rations. the central region of the spectrum in Fig. 4a exhibits a Quntum simulations: The system has been simulated with the stochastic Runge-Kutta algorithm [28]: second order in linear dispersion that can be attributed to the radiation the deterministic part and order 1.5 in the stochastic part in proximity of the event horizon. A key difference with 20 disorder averages. The derivatives in the determini- between the classical and the quantum analysis, is that, stic part are computed using the fast Fourier transform (FFT) in the latter case, the spectral content of the noise is algorithm; the second derivatives of the field with respect to much wider, because the quantum noise is continuously the variable x is computed by inverse Fourier transform. The generated upon evolution. This is evident in Fig. 4a, noise realizations have the form dWk ≈ N(0, Nx)p∆z/∆x , with respect to the classical case in Fig. 3a, but the where Nx is the number of points in the discretization of the variable x and N is a random number normally distributed key point is that the spectrum of the trapped quantum between 0 and Nx. excitation fades and then completely disappears in the rainbow gravity case. In addition, in the quantum case we see the vanishing of the ergoregion not evident in the classical case (see Figs. 3e and 4e). In conclusion, we propose the first analog of rainbow [1] C. Kiefer, Quantum Gravity (Oxford University Press, quantum gravity by nonlocal nonlinear waves. We 2012). theoretically describe a black hole as a stable nonlocal [2] V. Mukhanov and S. Winitzki, Introduction to quantum vortex and show the fading of the event horizon and effects in gravity (Cambgridge University Press, 2007). [3] A. M. Polyakov, Gauge Fields and Strings (CRC Press, the inhibition of superradiance in classical and second 1987). quantized frameworks. Our findings can trigger further [4] C. Rovelli and F. Vidotto, Covariant Loop Quantum 7

10 (a) (b) (c)

E 0

−10 −5 0 5 −5 0 5 −5 0 5 kx kx kx 50 (d) (e) (f)

y 0 −50 −50 0 50 −50 0 50 −50 0 50 x x x

FIG. 4. (Color online) (a) quantum spectrum in the local case; (b) as in (a) in the nonlocal case (A = 1 and σ = 5); (c) as in (b) for A = 0 and σ = 5; (d) quantum field in the configuration space in the local case; (e) as in (d) for A = 1 and σ = 5; (f) as in (e) for A = 0 and σ = 5.

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