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arXiv:physics/0111170 · Physical Review A

Theory of a slow-light catastrophe

Ulf Leonhardt

Published Apr 5, 2002• 19 citations
arXivHTMLPDF

Abstract

In diffraction catastrophes such as the rainbow, the wave nature of light resolves ray singularities and draws delicate interference patterns. In quantum catastrophes such as the black hole, the quantum nature of light resolves wave singularities and creates characteristic quantum effects related to Hawking radiation. This paper describes the theory behind a recent proposal [U. Leonhardt, Nature (London) 415, 406 (2002)] to generate a quantum catastrophe of slow light.

DOI
10.1103/physreva.65.043818

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In diffraction catastrophes such as the rainbow, the wave nature of light resolves ray singularities and draws delicate interference patterns. In quantum catastrophes such as the black hole, the quantum nature of light resolves wave singularities and creates characteristic quantum effects related to Hawking radiation. This paper describes the theory behind a recent proposal [U. Leonhardt, Nature (London) 415, 406 (2002)] to generate a quantum catastrophe of slow light.
Although slow light (electromagnetically induced transparency) would seem an ideal medium in which to institute a ``dumb hole'' (black hole analogue), it suffers from a number of problems. We show that the high phase velocity in the slow light regime ensures that the system cannot be used as an analogue displaying Hawking radiation. Even though an appropriately designed slow-light setup may simulate classical features of black holes---such as horizon, mode mixing, ``Bogoliubov'' coefficients, etc.---it does not reproduce the related quantum effects.
The island proposal guarantees a unitary Page curve for evaporating black holes, implying that operators deep behind the horizon must be fully reconstructable from asymptotic Hawking radiation. However, because the interior is causally disconnected and highly dynamical, standard entanglement wedge reconstruction methods, relying on global isometries or continuous modular flow, fail to provide this mapping. We resolve this by constructing a protocol for island reconstruction using the algebraic Petz map. By using modular theory, we reflect island operators into the accessible complement, transport them to the boundary, and map them into the radiation algebra. Furthermore, we demonstrate that this purely algebraic derivation naturally yields the structure of Papadodimas-Raju mirror operators. This framework provides a rigorous operational dictionary for the island formula, proving that information trapped inside a black hole can be decoded.
Light occupies a special position in our attempts to understand nature both classically and quantum mechanically. We recall that Newton, who made so many fundamental contributions to optics, championed a particle description of light and was not favorably disposed to the wave picture of light. However, the beautiful unification of electricity and magnetism achieved by Maxwell clearly showed that light was properly understood as the wave-like undulations of electric and magnetic fields propagating through space. The central role of light in marking the frontiers of physics continues on into the twentieth century with the ultraviolet catastrophe associated with black-body radiation on the one hand and the photoelectric effect on the other. Indeed, it was here that the era of quantum mechanics was initiated with Planck's introduction of the quantum of action that was necessary to explain the black-body radiation spectrum. The extension of these ideas led Einstein to explain the photoelectric effect, and to introduce the photon concept. It was, however, left to Dirac to combine the wave-and particlelike aspects of light so that the radiation field is capable of explaining all interference phenomena and yet shows the excitation of a specific atom located along a wave front absorbing one photon of energy. In this chapter, following Dirac, we associate each mode of the radiation field with a quantized simple harmonic oscillator, this is the essence of the quantum theory of radiation.
The slow-light effect, which implies reduction of the group velocity of light by many orders of magnitude, is considered to be one of the most exciting discoveries in quantum optics of the last decade. The physics underlying the effect is related to an extremely steep dispersion of the medium in a very narrow region of its transparency, while all the manifestations of the effect are usually reduced just to a small pulse delay. Perhaps this is the reason why, after the discovery of the slow-light effect, some of the simplest nonlinear phenomena associated with a retarded optical response have been implicitly revised and assigned to reduction of the light group velocity. In this lecture, a few examples of common inconsistencies in the area of the slow-light physics will be presented.
What light is and how to describe it has always been a central subject in physics. As our understanding has increased, so have our theories changed: geometrical optics, wave optics and quantum optics are increasingly sophisticated descriptions, each referring to a larger class of phenomena than its predecessor. But how exactly are these theories related? How and when wave optics reduces to geometric optics is a rather simple problem. Similarly, how quantum optics reduces to wave optics has also been considered to be a very simple business. It is not so. As we show here the classical limit of quantum optics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.
Black holes offer an exciting area to explore the nature of quantum gravity. The classic work on Hawking radiation indicates that black holes should decay via quantum effects, but our ideas about how this might work at a technical level are incomplete. Recently Almheiri-Marolf-Polchinski-Sully (AMPS) have noted an apparent paradox in reconciling fundamental properties of quantum mechanics with standard beliefs about black holes. One way to resolve the paradox is to postulate the existence of a ``firewall'' inside the black hole horizon which prevents objects from falling smoothly toward the singularity. A fundamental limitation on the behavior of quantum entanglement known as ``monogamy'' plays a key role in the AMPS argument. Our goal is to study and apply many-body entanglement theory to consider the entanglement among different parts of Hawking radiation and black holes. Using the multipartite entanglement measure called negativity, we identify an example which could change the AMPS accounting of quantum entanglement and perhaps eliminate the need for a firewall. Specifically, we constructed a toy model for black hole decay which has different entanglement behavior than that assumed by AMPS. We discuss the additional steps that would be needed to bring lessons from our toy model to our understanding of realistic black holes.