Dielectric Black Holes Induced by a Refractive Index Perturbation and the Hawking Effect
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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Heriot Watt Pure PHYSICAL REVIEW D 83, 024015 (2011) Dielectric black holes induced by a refractive index perturbation and the Hawking effect F. Belgiorno,1 S. L. Cacciatori,2,3 G. Ortenzi,4 L. Rizzi,2 V. Gorini,2,3 and D. Faccio5 1Dipartimento di Fisica, Universita` di Milano, Via Celoria 16, IT-20133 Milano, Italy 2Department of Physics and Mathematics, Universita` dell’Insubria, Via Valleggio 11, IT-22100 Como, Italy 3INFN sezione di Milano, via Celoria 16, IT-20133 Milano, Italy 4Dipartimento di Matematica e Applicazioni, Universita` di Milano-Bicocca, Via Cozzi 53, IT-20125 Milano, Italy 5School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh, Scotland EH14 4AS, UK (Received 22 March 2010; revised manuscript received 18 November 2010; published 13 January 2011) We consider a 4D model for photon production induced by a refractive index perturbation in a dielectric medium. We show that, in this model, we can infer the presence of a Hawking type effect. This prediction shows up both in the analogue Hawking framework, which is implemented in the pulse frame and relies on the peculiar properties of the effective geometry in which quantum fields propagate, as well as in the laboratory frame, through standard quantum field theory calculations. Effects of optical dispersion are also taken into account, and are shown to provide a limited energy bandwidth for the emission of Hawking radiation. DOI: 10.1103/PhysRevD.83.024015 PACS numbers: 04.62.+v, 04.70.Dy I. INTRODUCTION experiments involving Hawking analogue radiation has been explored [11–23]. Recently Philbin et al. proposed One of the most intriguing predictions of quantum fields an optical analogue in which a soliton with intensity I, in curved spacetime geometries is the production of propagating in an optical fibre, generates through the non- Hawking radiation. In 1974 S. Hawking predicted that linear Kerr effect a refractive index perturbation (RIP), black holes emit particles with a thermal spectrum. ¼ Therefore a black hole will evaporate, shedding energy n n2I, where n2 is the Kerr index [24]. The same under the form of a blackbody emission [1–3]. However, mechanism has also been generalized to a full 4D geometry it turns out that for a typical stellar mass black hole the by Faccio et al. [25] and has led to the first experimental temperature of this radiation is so low ( 10 nK) that it has observation of quanta emitted from an analogue horizon no hope of being directly detected. Notwithstanding, [26]. The RIP modifies the spacetime geometry as seen by Hawking radiation does not actually require a true gravi- copropagating light rays and, similarly to the acoustic tational event horizon but, rather, it is essentially a kine- analogy, if the RIP is locally superluminal, i.e. if it locally matical effect, i.e. it requires some basic ‘‘kinematical’’ travels faster than the phase velocity of light in the ingredients but no specific underlying dynamics (see e.g. medium, an horizon is formed and Hawking radiation is [4–6]). What suffices is the formation of a trapping horizon to be expected. in a curved spacetime metric of any kind, and the analysis Here we take into consideration the Hawking effect in of the behavior of any quantum field therein (see also the dielectric black holes induced by a RIP which propagates discussion in [6]). The quanta of this field will then be with constant velocity v. We first show that, under suitable excited according to the prediction of Hawking. On this conditions, we can remap the original Maxwell equations basis, a number of analogue systems have been proposed, for nonlinear optics into a geometrical description, in anal- for the first time by W. Unruh [7] and later by other ogy to what is done in the case of acoustic perturbations in researchers (see e.g. [8] and references therein), that aim condensed matter. Then we provide a model in which the at reproducing the kinematics of gravitational systems. presence of the Hawking radiation can be deduced even Most of these analogies rely on acoustic perturbations without recourse to the analogous model characterized by a and on the realization of so-called dumb holes: a liquid curved spacetime geometry. Hawking radiation is a new or gas medium is made to flow faster than the velocity of phenomenon for nonlinear optics, never foreseen before, acoustic waves in the same medium so that at the transition which could legitimately meet a sceptical attitude by the point between sub and supersonic flow, a trapping horizon nonlinear optics community, because of a missing descrip- is formed that may be traversed by the acoustic quanta, viz. tion of the phenomenon by means of the standard tools of phonons, only in one direction [8]. Unfortunately the pho- quantum electrodynamics. As a consequence, it is impor- non blackbody spectrum is expected to still have very low tant to provide also a corroboration of the analogue picture temperatures, thus eluding direct detection (see e.g. [9]). A simply by using ‘‘standard’’ tools of quantum field theory parallel line of investigation involves effective geometry (without any reference to the geometrical picture). for light, which has been introduced by Gordon [10] and We then take into consideration the effects of optical extended also to nonlinear electrodynamics; black hole dispersion, which give rise to relevant physical consequen- metrics have been introduced and the possibility to perform ces on the quantum phenomenon of particle creation. 1550-7998=2011=83(2)=024015(17) 024015-1 Ó 2011 American Physical Society F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) We first introduce a 2D reduction of the model in the group velocity horizons, which are shown to be involved presence of dispersion, and see how the dispersion relation with different spectral bandwidths, and with very different is affected by the frequency-dependence of the refractive qualitative behavior regarding their action on photons. index. Then we discuss both phase velocity horizons and Section V is devoted to the conclusions. group velocity horizons, which can both play a relevant Finally, we have added three Appendixes. The first es- role in the physical situation at hand; their qualitative tablishes a correspondence between our black hole metric difference and the possibility to discriminate between and the acoustic one. The second one relates the tempera- them in experiments (numerical and/or laboratory-based) ture to the standard conical singularity of the Euclidean is considered. version of the metric. The third establishes the relevant The structure of the paper is as follows. In Sec. II we analytic properties of the Bogoliubov coefficients. introduce our idealized model of a RIP propagating, in a nonlinear Kerr medium, with a constant velocity v in the x II. STATIC DIELECTRIC BLACK HOLE direction and infinitely extended in the transverse y and z directions. In the eikonal approximation, the model is We consider a model-equation which is derived from Þ embodied in a suitable wave equation for the generic nonlinear electrodynamics (with 2 ¼ 0 and 3 0)in component of the electric field propagating in the nonlinear the eikonal approximation for a perturbation of a full- medium affected by the RIP. We describe this propagation nonlinear electric field propagating in a nonlinear Kerr as taking place in an analogue spacetime metric, written in medium. In order to make the forthcoming analysis as the pulse frame, where the metric is static and displays two simple as possible we replace the electric field with a scalar field È (cf. e.g. Schwinger’s analysis of sonoluminescence horizons xþ and xÀ for propagating photons, analogues to a black hole and a white hole horizons, respectively. By [27]), for which we obtain the wave equation [28] 2 assigning to the black hole horizon a surface gravity in a n ðxl À vtlÞ 2 2 2 2 @t È À @x È À @yÈ À @z È ¼ 0: (1) standard way, we calculate the evaporation temperature Tþ c2 l l of the RIP in the latter frame, which turns out to be proportional to the absolute value of the derivative of the Coordinates in the lab frame are denoted by tl, xl, y, z (the RIP’s profile, evaluated at x . We also show that T is labels of y, z are omitted because they will not be involved þ þ in the boost relating the lab frame with the pulse frame). conformally invariant, as expected for the consistency of Here, nðx À vt Þ is the refractive index, which accounts the model. l l for the propagating RIP in the dielectric. The latter can be In Sec. III, we tackle the model from a different per- obtained by means of an intense laser pulse in a nonlinear spective, namely, within the framework of quantum field dielectric medium (Kerr effect). In our model the RIP does theory. We construct a complete set of positive frequency not depend on the transverse coordinates, namely, it is solutions Èk of the wave equation in the lab frame, in l infinitely extended in the transverse dimensions. This ap- terms of which we quantize the field in the standard way. proximation is necessary in order to carry out the calcu- Then, we compare the Èk with the corresponding plane l lations below, which allow us to draw a tight analogy wave solutions of the wave equation in the absence of the between the Hawking geometrical description and the RIP. This enables us to calculate the Bogoliubov coeffi- quantum field theoretical treatment. Such an infinitely cients, in terms of which we express the expectation value extended RIP is clearly an idealization, which cannot be of the number operator of the outgoing quanta in the in obtained in an actual experiment.