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. I suggest that a quantum catastrophe can be made in the laboratory using frozen light with a parabolic profile of the group velocity.

Lecture on quantum catastrophes

Catastrophes [1] are at the heart of many fascinating optical phenomena. The most prominent example of such a catastrophe is the rainbow. Light rays from the Sun enter water droplets floating in the air. After two refractions and one reflection inside each drop the rays reach an observer. Above a critical observation angle no rays arrive, whereas below the angle two rays strike the observer. A bright bow, the rainbow, appears at the critical angle, because here the cross section of light rays diverges [2]. (The critical angle depends on the refractive index that varies with the frequency of light in dispersive media such as water, giving rise to the rainbow colors.) The direction of a light ray is proportional to the gradient of the phase. The rainbow thus represents a singularity of a gradient map, a catastrophe in the sense of Thom [3] and Arnol'd [4]. Structurally stable singularities of gradient maps fall into distinct classes, depending on the number of control parameters involved [3,4]. Structural stability is the key to Nature's way of focusing light [5] in the caustics created by ray catastrophes. Yet the wave nature of light smooths the harsh singularities of rays. Simultaneously, characteristic interference effects appear. For example, the pairs of light rays below the rainbow create a delicate pattern of supernumerary arcs [1] that are visible under favorable weather conditions (when the floating droplets are nearly uniform in size [2]). Every class of diffraction catastrophes generates its distinct interference structure [1].

*Descartes' illustration of the formation of the rainbow: rays
such as ABCDE form the primary bow, and FGHIKE the secondary one.
*

*The bright side of the rainbow (below the primary bow) shows a
delicate interference pattern.
*

Catastrophe optics describes the wave properties of ray singularities. In the hierarchy of physical concepts, wave optics refines and embraces ray optics, and quantum optics rules above wave optics. So, what would be the quantum effects of wave catastrophes [6]? First, what are quantum catastrophes? It might be a good idea to begin with an example, the black hole [7]. When a star collapses to a black hole an event horizon is formed, cutting space into two disconnected regions. Seen from an outside observer, time stands still at the horizon, freezing all motion. A light wave would freeze as well, propagating with ever-shrinking wavelength. [In mathematical terms [8], a monochromatic light wave of frequency $\omega$ oscillates as $\Theta(r-r_s)(r-r_s)^{i\mu}$ when the radius $r$ approaches the horizon $r_s$, with $\mu=2r_s\,\omega/c$]. A logarithmic phase singularity will develop. Potential quantum effects of such a wave singularity are effects of the quantum vacuum. The gravitational collapse [7] of the star into the black hole has swept along the vacuum. The vacuum thus shares the fate of an inward-falling observer. Yet such an observer would not notice anything unusual at the event horizon. In mathematical terms, the vacuum modes are analytic across the horizon [8,9]. On the other hand, the modes perceived by an outside observer are essentially non-analytic, because they vanish beyond the horizon where the observer has no access to. Consequently, the observer does not see the electromagnetic field in the vacuum state. Instead, the observer notices the quanta of Hawking radiation [10] with a Planck spectrum. The quantum vacuum does not assume catastrophic waves, hence resolving so the associated wave singularity and, simultaneously, generating quantum radiation with a characteristic spectrum. At the heart of such a catastrophe lies a time-dependent process, for example the gravitational collapse in the case of the black hole [7]. The process has disconnected the spatial regions where waves can propagate and has created a logarithmic phase singularity at the interface. Any time-dependent phenomenon will generate some radiation, as long as the process lasts. In remarkable contrast, a quantum catastrophe creates quanta continuously.

*Schematic picture of Waves near the event horizon of a black
hole, seen from an outside observer.
The wavelength shrinks proportionally to the radial distance
from the event horizon.
*

I propose an experiment [11] that could demonstrate an optical analog of Hawking's effect in the laboratory. The experiment is based on Electromagnetically-Induced Transparency (EIT) [12]. In EIT a control beam determines the properties of slow-light polaritons in a suitable medium. In particular, the group velocity of the slow light is inversely proportional to the intensity of the control light. Light can be substantially slowed down or frozen completely [13]. I propose the following scenario: first, illuminate uniformly the EIT medium with control light, then change the control intensity to a parabolic profile, creating s slow-light catastrophe. In turn, the polariton field sets out to deplete the control beam, in an attempt to alter the intensity profile that has caused the wave catastrophe in the first place, yet in vain. The control beam continuously replenishes the parabolic intensity profile, driving a stationary production of polariton pairs. The two polaritons of each pair are created on opposite sides near the horizon, they depart at a snail's pace, accelerate gradually and emerge as detectable photons. The Hawking radiation of a black hole [10] follows a similar scenario [8]. Here the gravitational collapse [7] has triggered a quantum catastrophe at the event horizon, causing pair creation lasting as long as the hole possesses gravitational energy [8,10]. One particle of each pair falls into the black hole, whereas the other escapes into space and appears as thermal radiation [10]. In the optical case [11], and in contrast to gravitational holes, one can explore the other side beyond the horizon and, for example, measure the correlations of the generated photon pairs. Both cases are triggered by catastrophic events with lasting consequences.

*Schematic diagram of the proposed
experiment. A beam of control light with intensity I_c generates
Electromagnetically-Induced Transparency in a medium,
strongly modifying its optical properties for a second field of
slow light. When an initially uniform control intensity is turned
into the parabolic profile shown in the figure, the slow-light
field suffers a quantum catastrophe. To slow-light waves, the
interface Z of zero control intensity cuts space into two
disconnected regions and creates a logarithmic phase singularity,
in analogy to the effect of an event horizon. The quantum
vacuum of slow light cannot assume such catastrophic waves. In
turn, pairs of slow-light quanta, propagating in opposite
directions away from Z, are emitted with a
characteristic spectrum. *

The quantum radiation of a slow-light catastrophe resembles
Hawking radiation but also exhibits some interesting differences.
The emitted spectrum is not Planckian, whereas a black hole
appears as a black-body radiator.
The differences between the two spectra can be traced back to two
different classes of wave catastrophes. In both cases, waves
oscillate at an horizon in the form *z^p*, but the exponents
*p* are different in a characteristic way, see the table.
Note that Unruh's effect [9] of radiation seen by an
accelerated observer is of Hawking-class as well [8] and
so are most of the proposed artificial black holes [14]
Remarkably, Schwinger's pair production of charged particles in
electrostatic fields [15] is accompanied by a subtle
wave catastrophe of a different type and
leads to a Boltzmannian spectrum, see the table.
It might be interesting to find out whether
more than three types of quantum catastrophes can occur.

*In each type of quantum catastrophe a wave develops a
singularity with a characteristic exponent. Quantum physics
resolves the singularity, and produces particle pairs with a
characteristic spectrum (average particle number).
*

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I am very grateful to Sir Michael Berry, Lene Vestergaard Hau, Malcolm Dunn, Tamas Kiss, Patrik Ohberg, Renaud Parentani, Paul Piwnicki, and Matt Visser for inspiring conversations. In particular, Sir Michael raised the question whether wave singularities exhibit interesting quantum effects. The Discussion Meeting on Artificial Black Holes at the Royal Institution in London, 19 January 2001, was an important inspiration as well, and I thank the participants and the organizers. My research has been supported by the ESF Programme Cosmology in the Laboratory and by the University of St Andrews.