We report numerical evidence of Hawking emission of Bogoliubov phonons from a sonic horizon in a flowing one-dimensional atomic Bose-Einstein condensate. The presence of Hawking radiation is revealed from peculiar long-range patterns in the density-density correlation function of the gas. Quantitative agreement between our fully microscopic calculations and the prediction of analog models is obtained in the hydrodynamic limit. New features are predicted and the robustness of the Hawking signal against a finite temperature discussed.

We have used the analogy between gravitational systems and non-homogeneous fluid flows to calculate the density-density correlation function of an atomic Bose-Einstein condensate in the presence of an acoustic black hole. The emission of correlated pairs of phonons by Hawking-like process results into a peculiar long-range density correlation. Quantitative estimations of the effect are provided for realistic experimental configurations.PACS numbers: 03.75. Gg, 04.62.+v, 04.70.Dy Hawking's prediction of black holes evaporation is generally regarded as a milestone of modern theoretical physics. Combining Einstein's General Relativity and Quantum Mechanics, Hawking was able to show that black holes are not "black", but emit thermal radiation at a temperature inversely proportional to their mass [1]. This quantum mechanical process is triggered by the formation of a horizon and proceeds via the conversion of vacuum fluctuations into on-shell particles. Unfortunately so far there is no experimental support for this amazing theoretical prediction. The emission temperature (Hawking temperature) for a solar mass black hole is expected to be of the order of 10 −8 K, far below the 3 K cosmic microwave background. Nor evidence has been found so far of a X-ray background from a hypothetical primordial population of low mass black holes (∼ 10 10 Kg) in the final stages of their evaporation [2]. Expectations to directly observe Hawking radiation from mini-black holes formed in colliders like LHC or next generation ones, are based on models where the quantum gravity scale (Planck scale: 10 19 GeV) is lowered down to the TeV scale by the presence of extra-dimensions [3]. It is perhaps fair to say that the prospects to have a direct experimental detection of Hawking radiation from black holes in the near future are not very optimistic.In a remarkable work Unruh [4] showed that Hawking radiation is not peculiar to gravity, but is rather a purely kinematic effect of quantum field theory which only depends on field propagation on a black hole-type curved space-time background. This opens the concrete possibility to study the Hawking radiation process in completely different physical systems. As an example, the propagation of sound waves in Eulerian fluids can be described in terms of the same equation describing a massless scalar field on a curved spacetime characterized by an acoustic metric G µν which is function of the background flow: the curvature of the acoustic geometry is induced by the inhomogeneity of the fluid flow, while flat minkowskian spacetime is recovered in the case of a homogeneous system. In particular, an acoustic black hole (or dumb hole) configuration is obtained whenever a subsonic flow turns supersonic: sound waves in the supersonic region are in fact dragged away by the flow and can not propagate back towards the acoustic horizon separating the supersonic and subsonic regions. Upon quantization, Hawking radiation is expected to appear as a flux of thermal phonons emitted from the horizon at a temperature...

We present a theory of the density correlations that appear in an atomic Bose-Einstein condensate as a consequence of the dynamical Casimir emission of pairs of Bogoliubov phonons when the atomatom scattering length is modulated in time. Different regimes as a function of the temporal shape of the modulation are identified and a simple physical picture of the phenomenon is discussed. Analytical expressions for the density correlation function are provided for the most significant limiting cases. This theory is able to explain some unexpected features recently observed in numerical calculations of Hawking radiation from analog black holes.The dynamical Casimir effect [1] is a very general prediction of quantum field theory: whenever the boundary conditions and/or the dispersion law and/or the background of a quantum field are quickly varied in time, pairs of quanta are generated off the vacuum state by parametric amplification of zero-point noise. The simplest and most celebrated example of dynamical Casimir effect was predicted for an optical cavity whose planeparallel mirrors are made to rapidly oscillate in time along the cavity axis [2,3,4]. Despite the significant effort devoted to these fascinating effects, no experimental observation of the dynamical Casimir effect has yet appeared, the main reason being the difficulty in moving the mirror at a fast enough speed [3]. Alternative schemes to modulate the effective optical length of a cavity on a very fast time-scale by acting on the refractive index of the cavity material have been proposed and shown to give a sizeable intensity of dynamical Casimir emission [5,6,7,8,9,10,11]. Experiments in this direction are in progress in several groups [12,13].Since the original proposal by Unruh [14], the advances in the field of the so-called analog models [15] have pointed out the possibility of simulating the physics of a quantum field on a generic curved space-time in tabletop condensed-matter experiments: the propagation of elementary excitations in spatially and temporally inhomogeneous systems can in fact be recast in terms of a relativistic wave equation on an effective curved spacetime. In the simplest case of acoustic waves in a fluid, the space-time metric is fixed by the spatial and temporal profiles of the sound speed and of the flow velocity. Upon quantization of the resulting field theory, an analog dynamical Casimir effect is then expected to appear whenever the sound speed in a spatially homogeneous system is made to quickly vary in time, which in the language of the analogy corresponds to the expansion or contraction of the underlying universe [16,17]. On the other hand, in the presence of an acoustic horizon separating an upstream region of sub-sonic flow from a downstream one of super-sonic flow, the emission of analog Hawking radiation has been predicted [14,18,19].As proposed in [20,21], a most promising way of experimentally investigating this physics involves the measurement of the correlation function of density fluctuations. A recent numerical e...

The backreaction equations for the linearized quantum fluctuations in an acoustic black hole are given. The solution near the horizon, obtained within a dimensional reduction, indicates that acoustic black holes, unlike Schwarzschild ones, get cooler as they radiate phonons. They show remarkable analogies with near-extremal Reissner-Nordström black holes.PACS numbers: 04.62.+v, 04.70.Dy, 47.40.Ki One of the most surprising and far reaching result for its implications in modern theoretical physics is the prediction made by Hawking [1] that black holes emit thermal radiation at a temperature T H proportional to the surface gravity k of the horizon. For a Schwarzschild black hole of mass M , k = (4M ) −1 and T H = (8πM ) −1 (we have set the velocity of light and Boltzman constant equal to one). Hawking obtained this result using quantum field theory in curved space, a scheme for dealing with the matter-gravity system where matter is quantized according to quantum field theory whereas gravity is treated classically according to Einstein General Relativity. The scale at which this framework becomes unreliable is the Planck length where the description of spacetime as a continuous differentiable manifold probably breaks down. Coming back to black holes, because of the quantum emission, they are unstable. Extrapolating Hawking's result (which is strictly valid only for stationary or static black holes) one can conjecture that as the mass decreases, the hole gets hotter and hotter (being the temperature inversely proportional to the mass) and eventually disappears in a time scale of the order of the initial mass to the third power. A more quantitative analysis can be performed by looking at the first order (in ) corrections g (1) αβ to a classical black hole metric g (0) αβ induced by the quantum emission. These can be calculated using the semiclassical Einstein equations [2]Here G µν is the Einstein tensor evaluated for the quan- * Email addresses: balbinot@bo.infn.it, fagnocchi@bo.infn.it † Email address: fabbria@bo.infn.it ‡ Email address: gpp27@cam.ac.uk tum corrected metric g αβ = g (0) αβ + g (1) αβ and linearized in the perturbation g (1) αβ (of order ). The r.h.s. represents the expectation value of the stress tensor for the quantum matter field evaluated in the classical background g (0) αβ . In a very interesting paper, appeared in 1981, Unruh [3] showed that a thermal radiation similar to the one predicted by Hawking for black holes is expected in a completely (at first sight) different physical scenario, namely a fluid undergoing hypersonic motion. This opened the way for the study of condensed matter analogues of Hawking radiation [4], a rather promising field of research where the connections to the experimental side do not seem so remote, compared to gravity.The Eulerian equations of motion for an irrotational and homentropic fluid flow can be derived from the actionwhere ρ is the mass density, ψ the velocity potential, i.e. − → v = − → ∇ψ, u the internal energy density and a dot means time derivative....

We study acoustic white holes in a steadily flowing atomic Bose-Einstein condensate. A white hole configuration is obtained when the flow velocity goes from a super-sonic value in the upstream region to a sub-sonic one in the downstream region. The scattering of phonon wavepackets on a white hole horizon is numerically studied in terms of the Gross-Pitaevskii equation of mean-field theory: dynamical stability of the acoustic white hole is found, as well as a signature of a nonlinear back-action of the incident phonon wavepacket onto the horizon. The correlation pattern of density fluctuations is numerically studied by means of the truncated-Wigner method which includes quantum fluctuations. Signatures of the white hole radiation of correlated phonon pairs by the horizon are characterized; analogies and differences with Hawking radiation from acoustic black holes are discussed. In particular, a short wavelength feature is identified in the density correlation function, whose amplitude steadily grows in time since the formation of the horizon. The numerical observations are quantitatively interpreted by means of an analytical Bogoliubov theory of quantum fluctuations for a white hole configuration within the step-like horizon approximation.

The quantum stress tensor in the Unruh state for a conformal scalar propagating in a 4D Schwarzschild black hole spacetime is reconstructed in its leading behavior at infinity and near the horizon by means of an effective action derived by functionally integrating the trace anomaly. PACS numbers: 04.70.Dy, 04.62. + v In the mid-1970s Hawking [1] showed that black holes are quantum mechanically unstable: they decay by the emission of thermal radiation at a temperature inversely proportional to their mass; i.e., T H ͑8pM͒ 21 in units whereh c G k B 1. This is one of the most astonishing discoveries of theoretical physics in the second half of the century. Nowadays black hole radiation and its thermodynamical implications, most notably the Bekenstein-Hawking area-entropy formula [2], are among the consistency tests any candidate of quantum gravity theory has to successfully pass in order to be seriously considered. Notwithstanding decades of intensive studies, the evolution and fate of an evaporating black hole (EBH) are still unknown. In the opinion of many people the final answer to this issue has to wait until a complete and self-consistent quantum gravity theory has been found. String theory appears to be the most promising candidate to achieve this goal and many efforts have been devoted to showing its compatibility with black hole radiation. However, one is still far away from understanding, within string theory, how black holes evaporate.A more traditional field theoretical approach to the evolution of black holes driven by the quantum fluctuations of the matter fields relies on the effective action S eff ͑g ab ͒. This is the so-called backreaction, which in mathematical terms is governed by the semiclassical Einstein equationwhere G mn is the Einstein tensor for the metric g ab andis the renormalized expectation value of the stress energy tensor operator for the matter fields propagating on g ab . A quantum state and boundary conditions appropriate to black hole evaporation have to be supplied to Eq. (1). The framework is quantum field theory in curved spacetime [3], a semiclassical approach in which only the mat-ter fields are quantized, whereas gravity is still described classically according to Einstein's general relativity. One expects this approximation to be consistent until the size of the EBH becomes comparable to the Planck length (10 233 cm). At this point one has to move to a genuine quantum gravity theory which unfortunately is still lacking. Even within semiclassical gravity, however, the evolution of an EBH is hard to follow simply because the relevant effective action S eff ͑g ab ͒ is not explicitly known. The only information available on black hole evaporation comes from analytical estimates of ͗T mn ͘ for matter fields propagating in a fixed static Schwarzschild black hole geometry of a given mass M. Selecting a mode basis suitable for black hole evaporation (Unruh modes [4]), the matter fields are expanded in that basis, canonically quantized, and then ͗T mn ͘ is directly calculated by ...

Recently proposed 2D anomaly induced effective actions for the matter-gravity system are critically reviewed. Their failure to correctly reproduce Hawking's black hole radiation or the stability of Minkowski space-time led us to a modification of the relevant "quantum" matter stress energy tensor that allows physically meaningful results to be extracted.

Following a minisuperspace approach to the dynamics of a spherically symmetric shell, a reduced Lagrangian for the radial degree of freedom is derived directly from the Einstein-Hilbert action. The key feature of this new Lagrangian is its invariance under time reparametrization. Indeed, all classical and quantum dynamics is encoded in the Hamiltonian constraint that follows from that invariance. Thus, at the classical level, we show that the Hamiltonian constraint reproduces, in a simple gauge, Israel's matching condition *