Light in moving media

Light spirals down into an optical black hole


Confinement to the Black Hole ... to be reserved for cases of Drunkeness, Riot, Violence, or Insolence to Superiors.

-- British Army regulation (1844)


Consider light in a moving medium, say in flowing water. Water is, like glass or other transparent materials, a dielectric medium. Such media reduce the speed of light by a factor that is known as the refractive index. Water, for example, has a refractive index of 1.333, i.e. light travels in water at about 75 percent of the speed of light in vacuum. What happens to light in a medium that is moving? Imagine that the medium consists of countless small droplets that altogether form the flow of the medium. In general, the droplets may move in different directions or with different velocities, constituting, for example, a vortex flow. Imagine that we attach to each droplet a tiny invisible observer that is moving with the droplet. All the comoving observers would agree that in their specific drop light travels along a straight line and with a constant velocity (the speed of light in vacuum divided by the refractive index). However, how would we, the observers at rest, perceive the propagation of light in a flowing liquid? In vacuum, the speed of light is a universal constant c, independent of the relative velocity u of the observer. Einstein's addition theorem of velocities guarantees that c ``plus'' u gives c. In a medium, the speed of light is reduced by the refractive index. In this case, light appears to us as moving faster if it propagates in the direction of the flow and as moving slower if the light swims against the current, in accordance with our normal experience of motion in moving frames. As a consequence, the flowing medium drags light to a degree that was theoretically predicted as early as 1818 by Fresnel (without knowing special relativity) and experimentally observed by Fizeau in 1851 (without having a laser). Formulated in more abstract terms, the light propagation we see consists of both the propagation within the medium and the motion of the medium droplets themselves. We may certainly say that the way we perceive the propagation of light in the medium disagrees in general from the way comoving observers see light. This has a radical consequence, because one could, in principle, use light to synchronize clocks. For example, imagine that we wish to synchronize our clock with a distant master clock that emits light pulses at regular intervals. We can employ a semi-transparent mirror to trigger our clock and, simultaneously, to reflect the light pulse back to the sender. Our clock is then synchronized with the master that takes the average between sending and receiving time and communicates the result to us. Imagine, however, what happens when a moving medium drags the light between the two clocks. The clocks will become desynchronized. In principle, the synchronization problem occurs in all moving droplets. So, formulated radically, the way light measures time in the observer's frame differs from the time measure at comoving frames. The measure of time is mathematically expressed in a space-time metric, and so the flowing medium appears to light as an effective metric. According to Einstein, gravity is mediated by changes in the metric of space and time. Here space-time itself appears as a medium. Consequently, a moving dielectric medium acts on light as an effective gravitational field. In a forgotten 1923 paper [1] Gordon discovered this remarkable connection.

There is yet another interesting analogy. At a sufficiently low flow speed the velocity vector of light in the observer's frame is simply the velocity of light in the medium plus the medium velocity times Fresnel's dragging coefficient. An electrically charged particle in a magnetic field can be viewed in a similar way. The velocity of the particle is, up to prefactors, the sum of the momentum and the magnetic vector potential times the electric charge. In electromagnetism the vector potential forms an abstract vector field that is used to calculate the magnetic field. The velocity of light in the moving medium resembles the momentum of an electrically charged particle and the flow plays the role of the vector potential, providing this otherwise rather abstract quantity with a hydrodynamic archetype. Light in a moving medium behaves like an electrically charged wave or particle (an electron wave, for example) that propagates in a magnetic field, an analogy that was noticed by Hannay [2] in 1976. We rediscovered Gordon's insight into the connection between gravity and moving media and developed the theory of light propagation in more detail [3]. We also showed how Hannay's magnetic analogy finds a natural place in the mathematical framework of general relativity.

Consider a specific example of a flowing fluid, a vortex. Such a vortex is frequently spontaneously formed when water flows out of the drain of a bathtub. (The common myth is that the bathtub vortex rotates in one direction on the northern hemisphere and in the opposite direction on the southern part of the globe, which is wrong.) What happens to light that propagates in the vicinity of a vortex? In terms of Hannay's magnetic model of light in moving media, the fluid vortex turns out to correspond to the magnetic vector potential generated by an extremely thin solenoid. A solenoid is a long and thin spool of wire. When conducting an electric current the coils generate a magnetic field. However, a thin solenoid concentrates the magnetic field in the interior of the spool and does not produce a field outside. Hence the solenoid would not inflict any force on charged particles that, consequently, would pass the solenoid unaffected. Coming back to light in a vortex flow, we conclude from the magnetic analogy that, despite the whirling fluid, light rays remain straight. However, the travel times of rays that propagate with or against the flow differ by a characteristic number (that is proportional to the vortex strength). The light rays differ by a certain phase. Consequently, light waves that enclose the vortex will show a distinct interference pattern, in analogy to the Aharonov-Bohm effect of electrically charged matter waves [4].

Vortices are violent phenomena. A tornado, for example, attracts with ease substantial ``test particles'' such as cars and tears them apart. Can a vortex attract light? Hannay's magnetic model of light in moving media does not predict any force of the vortex on light, note however that the model is only valid for slow flows. When the flow reaches the speed of light in the medium the light will begin to loose maneuverbility. Similarly to fish swimming in a rapid stream that flows faster than the top speed of the fish, light is irresistibly carried away and spirals towards the vortex singularity, like in Edgar Allen Poe's Maelstrom. At a certain point a return is not possible anymore. The points of no return constitute the so-called event horizon. But, as has been said, in order to observe such a phenomenon we would need vortex velocities that are in the vicinity of the speed of light in the medium. Long before becoming superluminal, a normal bathtub vortex will form a hollow core, the ``eye of the hurricane''. (The core is formed because the water surface is sucked into the vortex due to the strongly falling pressure of a rapid flow, the kind of pressure that allows birds and airplanes to fly.) For establishing an optical black hole, the core radius must not exceed the optical event horizon, similar to a star that turns into a black hole only if the Schwarzschild radius exceeds the star's size.

Recently, extraordinary dielectric media have been made [5] that reduce the speed of light to just a few meters per second, making light slow enough to be comparable with earthly measures. Note, however, that these media differ from water in many more ways than the value of the speed of light would indicate. For example, only the group velocity is reduced, whereas water or glass slows down the phase velocity as well. The group velocity quantifies how fast a short light pulse moves, in contrast to the phase velocity that refers to the motion of an extended continuous wave. Nevertheless both types of media behave sufficiently alike if velocity-dependent effects are concerned. To understand this, let us consider in a bit more depth the experimental scheme [5] that generates slow light. Here one uses an auxiliary laser beam to produce a phenomenon that is called electromagnetically-induced transparency [6]. The auxiliary beam manipulates an atomic resonance, i.e. the optical transition between two atomic levels. Normally, light that bears the color of the resonance (that oscillates at the atomic transition frequency) is immediately absorbed by the atoms that constitute the medium. (The missing colors of a piece of matter, say a picture, form the colors of the picture that we see.) In electromagnetically-induced transparency, the auxiliary beam drives the atoms of the medium into a state such that the medium is bleached. Resonant light feels the full strength of the atomic resonance and yet is not absorbed. Note that in a moving medium the atomic resonances are slightly out of tune, depending on their velocity, because of the Doppler effect. The enormous coupling strength of the light-matter interaction, made possible by electromagnetically-induced transparency, increases the sensitivity to such small velocity-dependent detunings. The medium turns out [7] to drag resonant light such that even a tiny quantum vortex can generate an observable optical Aharonov-Bohm effect. An optical black hole could be made in the not-too-distant future [7].

Slow light [5] opens the opportunity to build analogs of astronomical objects in an earthly laboratory. Vortices or other rapid flows might be employed in order to make optical black holes. (The concerned reader should note that optical black holes are safe. They would attract only light and just light that happens to have the right color, and they would be confined to the container of the medium.) Apart from having the pleasure of making quasi-astronomical objects with ones own hands, one might be able to demonstrate analogs of controversial phenomena, such as Hawking radiation. Hawking predicted [8] that black holes radiate due to the quantum effects of a curved space-time. However, the radiation is usually obscured by the notorious microwave background of the universe, and therefore Hawking's phenomenon is probably doomed to remain unobservable. An optical analog would strengthen the confidence in Hawking's theory. Note that Unruh [9] proposed also an acoustical analog of a black hole, a dumb hole (sound in a flowing acoustical medium). Analogies often help to understand difficult issues in physics. Light in moving media may shed light on the public understanding of classical general relativity. We have also reasons to hope that the study of light quanta traveling in quantum matter could establish a prototype theory of quantum gravity. As we have seen, light experiences a material medium as an effective metric, but also the reverse is true [10]: Atomic matter waves see light as forming a space-time metric, i.e. as an effective gravitational field. Bose-Einstein condensates form quantum media. Such condensates have been made in alkali vapors since 1995 [11], and the experimental progress achieved is truly spectacular. Bose-Einstein condensates could thus become a testable prototype model of quantum gravity.

References

  1. W. Gordon, Ann. Phys. (Leipzig) 72, 421 (1923).
  2. J. H. Hannay, Cambridge University Hamilton prize essay 1976 (unpublished), see also R. J. Cook, H. Fearn, and P. W. Milonni, Am J. Phys. 63, 705 (1995).
  3. U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301 (1999).
  4. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959), see also M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect, (Springer, Berlin, 1989).
  5. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
  6. P. L. Knight, B. Stoicheff, and D. Walls (eds.), Phil. Trans. R. Soc. Lond. A 355, 2215 (1997); S. E. Harris, Phys. Today 50(7), 36 (1997); M. O. Scully and M. Zubairy, Quantum Optics, (Cambridge University Press, Cambridge, 1997).
  7. U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000).
  8. S. M. Hawking, Nature 248, 30 (1974); Commun. math. Phys. 43, 199 (1975).
  9. W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981), M. Visser, Class. Quantum Grav. 15, 1767 (1998); L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000); Phys. Rev. A 63, 023611 (2001).
  10. U. Leonhardt, Phys. Rev. A 62, 012111 (2000).
  11. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).


Ulf Leonhardt and Paul Piwnicki got interested in the optical Aharonov--Bohm effect when U.L. visited the University of Bristol in fall 1997. We thank Daniel Andre, Sir Michael Berry, Balasz Gyorffy, John Hannay, Jon Keating, Susanne Klein, and Duncan O'Dell for their hospitality and for fruitful and pleasant conversations. We are grateful to Harry Paul for a helpful correspondence and Benita Finck von Finckenstein, Michael Nieto, and Martin Wilkens for conversations on the Aharonov-Bohm effect. We thank Salvatore Antoci, Carsten Henkel, Björn Hessmo, Daniel James, Gerd Leuchs, Rodney Loudon, Peter Milonni, Wolfgang Schleich, and Stig Stenholm for discussions on the optics of moving dielectrics and on related subjects. U.L. gratefully acknowledges the support of the Alexander von Humboldt Foundation and of the Göran Gustafsson Stiftelse. P.P. was partially supported by the research consortium Quantum Gases of the Deutsche Forschungsgemeinschaft.