Universal quantum Hawking evaporation of integrable two-dimensional solitons

Abstract

We show that any soliton solution of an arbitrary two-dimensional integrable equation has the potential to eventually evaporate and emit the exact analogue of Hawking radiation from black holes. From the AKNS matrix formulation of integrability, we show that it is possible to associate a real spacetime metric tensor which defines a curved surface, perceived by the classical and quantum fluctuations propagating on the soliton. By defining proper scalar invariants of the associated Riemannian geometry, and introducing the conformal anomaly, we are able to determine the Hawking temperatures and entropies of the fundamental solitons of the nonlinear Schrödinger, KdV and sine-Gordon equations. The mechanism advanced here is simple, completely universal and can be applied to all integrable equations in two dimensions, and is easily applicable to a large class of black holes of any dimensionality, opening up totally new windows on the quantum mechanics of solitons and their deep connections with black hole physics. their energy and mass is radiated away. This outward flux of radiation can be measured at infinity with a characteristic temperature known as the Hawking temperature [4]. Since Hawking's seminal work other thermodynamical properties of black holes have been described in detail, perhaps most significantly the entropy, which in four dimensions turns out to be proportional to the black hole surface area, an observation due to Bekenstein which predates Hawking's work [5]. This result has had major implications for modern physics, and as its study requires a combination of general relativity, quantum field theory, thermodynamics and information theory, it is considered a promising route towards a deeper understanding of quantum gravity [6].Much literature has been devoted, experimentally and theoretically [7][8][9][10][11], to the possible detection of Hawking radiation in classical or semiclassical analogue systems (typically optical, hydrodynamical or based on quantum condensates), due to the considerable (and probably insurmountable) difficulty of detecting it in a real astrophysical setting. In light of Salam and Strathdee's conjecture about the equivalence between solitons and black holes, it is suggestive to imagine that the exact analogue of Hawking radiation can be directly seen in the physics of solitons. It is however important to establish a precise, unambiguous mathematical and physical connection between the two objects, something that we shall do in the present paper for the first time.In this paper, following Salam's steps, we push the analogy between black holes and solitons to a new level. We show that any soliton solution of a two-dimensional integrable nonlinear evolution equation potentially possesses a Hawking temperature which is determined solely by the geometrical properties of an internal surface connected to the specific soliton, called the integrable surface. The curvature of this surface is turned into Hawking radiation due to the existence of a quantum anomaly, well-known in two-dimens...

Quantum Decay of an Optical Soliton

“…This is not an inconsistency -as was pointed out in the beginning of this section, we have neglected the ∆v sol terms in Eq. (15). These terms provide the necessary counter terms to account for the total change in photon number, as they must by conservation of Eq.…”

“…Optical quantum soliton evaporation has previously been studied by way of computing an approximated power spectrum for a soliton which is initially in a fundamental soliton state [14]. Similarly, geometric approaches have been utilized in calculating the temperature of an optical soliton [15,16]. We would also like to highlight the extensive work of Malomed and collaborators regarding the analysis of decaying optical and non-optical solitons subject to perturbations in classical and quasiclassical quantized frameworks [17][18][19][20][21][22][23][24].…”