The slow-light quantum catastrophe: viability, observation, and the classification of quantum catastrophes

In 2002 Leonhardt proposed a laboratory analogue of a black-hole event horizon in which a control beam shapes the group velocity of probe light in an electromagnetically-induced-transparency medium into a parabolic profile with an interface of zero control intensity; waves freeze at that interface with a logarithmic phase singularity, and the quantum vacuum was predicted to respond by continuously creating photon pairs with a narrow-band, non-Planckian spectrum $\bar n=1/(e^{\pi\mu}+e^{-\pi\mu})^2$, one of exactly three then-known 'quantum catastrophes' (with Hawking radiation and Schwinger pair production). The detailed calculations were published within months; a fundamental critique by Unruh and Schützhold argued that slow-light systems reproduce the classical horizon features but not the quantum ones; and the experimental program migrated to optical fibres and Bose-Einstein condensates, where analogue Hawking radiation and its cross-horizon correlations have since been observed — but never in the slow-light medium itself. The surviving open problem is threefold: a complete quantum theory of the slow-light horizon in a realistic, absorptive, nonlinear EIT medium that settles whether genuine pair creation survives; an experimental route to observing the emitted flux and non-local pair correlations in such a medium; and a systematic classification determining whether more than the three known quantum catastrophes exist.

Background

Classical catastrophe theory describes the singularities that smooth maps can develop — the fold, the cusp, and Thom's other elementary catastrophes. In wave optics these appear as caustics: ray optics predicts infinitely intense light at the rainbow angle, while the wave nature of light resolves the divergence into interference fringes. A black hole event horizon is a wave singularity of a related but stronger kind: for a stationary outside observer, waves freeze at the horizon and oscillate with a logarithmically diverging phase, and the horizon cuts space into disconnected regions. Because the quantum vacuum shares the perspective of an inward-falling observer, it does not occupy the modes seen from outside, and the outside observer therefore detects Hawking radiation — thermal emission with temperature $\hbar c/(4\pi r_s)$ for a Schwarzschild radius $r_s$. Electromagnetically induced transparency (EIT) is a quantum-optical effect in which a strong control beam opens a narrow transparency window in an otherwise absorbing atomic medium and produces very steep dispersion, so that a weak probe pulse propagates with a group velocity $v_g$ essentially proportional to the control intensity; the propagating excitation is a dark-state polariton, a quantum superposition of light and atomic coherence. Since group velocities of metres per second were demonstrated in ultracold gases, room-temperature vapours, and light-storage experiments (Hau et al. 1999; Budker et al. 1999; Liu et al. 2001), EIT became a candidate platform for analogue gravity. Leonhardt's 2002 Nature Letter proposed tuning the control beam from uniform intensity to a profile with a dark stripe: near the interface $Z$ of zero control intensity the control intensity is quadratic in the distance $z$, giving a parabolic group-velocity profile $v_g\simeq c\,z^2/a^2$ with characteristic length $a$, and a group index $\alpha=c/v_g-1=a^2/z^2$ that diverges quadratically. The slow-light wave equation then has solutions $\varphi\propto z\,J_{\pm\nu}(kz)$ (Bessel functions of imaginary order $\nu=i\mu$ when $4a^2(\omega^2-\omega_0^2)>c^2$), which behave as $\zeta^{\,i\mu+1/2}$ with $\zeta=kz$: waves freeze at $Z$ with a logarithmic phase singularity, exactly the structure of waves at a black-hole horizon. Because the vacuum must occupy modes that are analytic in the complex time and space variables while the emitted detector modes are not, the Letter predicted stationary pair creation of slow-light quanta on opposite sides of $Z$, with a non-Planckian spectrum $\bar n=1/(e^{\pi\mu}+e^{-\pi\mu})^2$ at most $1/4$ photons per mode, confined to a narrow frequency window above a critical frequency near the EIT resonance $\omega_0$; for then-realistic parameters this was estimated as a few million photons per second. The Letter closed with a taxonomy: Hawking radiation and the Unruh effect (the thermal radiation seen by a uniformly accelerated detector) correspond to wave-singularity exponent $i\mu$ with a Planck spectrum $1/(e^{2\pi\mu}-1)$, Schwinger pair production of charged particles in an electric field to exponent $i\mu-1/2$ with a Boltzmann spectrum $e^{-2\pi\mu}$, and slow light to exponent $i\mu+1/2$ with the spectrum above — three 'quantum catastrophes' in which quantum physics resolves a wave singularity by particle creation — and asked whether Nature offers more than these three. The Letter also deferred its detailed calculations, gave only an estimate that optical nonlinearity sets in before absorption for sufficiently steep control gradients (the control Rabi frequency, i.e. the frequency scale of the control beam's coupling to the atomic transition, growing at least ~10 MHz per optical wavelength, i.e. $a\approx5\times10^3$ wavelengths), and noted that, unlike gravitational black holes, both sides of this horizon are accessible so the non-local correlations of the photon pairs could be measured.

Problem Statement

Determine whether the slow-light quantum catastrophe proposed by Leonhardt (Nature 415, 406 (2002)) is a real, observable phenomenon, and settle the general taxonomy it opened. Concretely, for an electromagnetically-induced-transparency medium in which a control beam with a parabolic intensity profile creates an interface $Z$ of zero control intensity and hence a group-velocity horizon $v_g\simeq c\,z^2/a^2$ with group index $\alpha=a^2/z^2$: (1) construct the complete quantum theory of the slow-light field at this horizon including absorption, the finite EIT transparency bandwidth, and the nonlinear back-reaction and depletion of the control beam that the original Letter set aside (all three effects must be retained), and settle the dispute raised by Unruh and Schützhold — whether the quantum vacuum of the EIT field is actually forced out of the catastrophic modes so that stationary photon (dark-state polariton) pair creation with the predicted non-Planckian spectrum $\bar n=1/(e^{\pi\mu}+e^{-\pi\mu})^2$ in the narrow band above the critical frequency occurs in a realistic medium, or whether only classical mode mixing survives; (2) given that theory, specify how the experiment could be realized (shaping and stabilizing the control profile with the required gradient of order 10 MHz of control Rabi frequency per wavelength, managing decoherence and loss) and how the emitted narrow-band, non-Planckian photon flux and the non-local correlations between the photon pairs emitted on opposite sides of $Z$ could be detected with a defensible signal-to-noise estimate — or, if pair creation is found not to survive in the realistic medium, the null-result bound and the sensitivity required to exclude the predicted flux; and (3) answer the question the Letter posed explicitly: does Nature offer more than the three known quantum catastrophes (Hawking/Unruh with exponent $i\mu$ and Planck spectrum, Schwinger pair production with $i\mu-1/2$ and Boltzmann spectrum, slow light with $i\mu+1/2$)? That is, establish a systematic classification of wave-singularity exponents and the quantum particle-production spectra they generate, identifying any further examples or proving the list exhaustive. These three parts share one objective — completing and settling the theory of quantum catastrophes that the slow-light proposal initiated.$

References

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