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Nonlinear dynamics

Chaotic billiard lasers

The chaotic motion of light rays gives microlasers surprising emission properties, enhancing quantum tunnelling by many orders of magnitude and producing highly directional output beams.

Physicists have worried for some time about what happens when chaos and quantum mechanics meet. Chaos refers to a generic effect of nonlinear forces in classical physics: the final state of a system depends with exponential sensitivity on the initial conditions. In a practical sense, therefore, it is unpredictable — the 'butterfly effect' made famous in fiction and film. Conversely, quantum mechanics is governed by linear equations, but to reproduce classical behaviour on macroscopic scales it must somehow contain the seeds of chaos.

Albert Einstein pointed out in 1917 that the rules then established by Niels Bohr and Arnold Sommerfeld to connect quantum and classical dynamics would be inapplicable if the classical motion had properties that we now term chaotic1. For the past few decades, physicists across many sub-fields (nuclear, atomic, condensed-matter and optical physics) have been trying to tease out the signatures of classical chaos in the study of microscopic systems2. Writing in Physical Review Letters, Shinohara et al.3 demonstrate clearly one such effect — known as chaos-assisted tunnelling — by studying the emission properties of special micrometre-sized lasers.

Shinohara and colleagues' study was motivated by extensive theoretical work in chaos theory relating to 'dynamical billiards'. These are among the simplest and most easily visualized systems for studying chaos, consisting of a two-dimensional enclosure with reflecting walls and negligible friction, in which a point mass is constrained to move in a straight line until it hits the boundary — at which point it obeys the familiar rule that the angle of incidence is equal to the angle of reflection. Of course, real billiard tables have straight sides, so the motion of the ball is rather simple as it bounces off multiple walls. However, unless they are highly symmetrical (circular, for example), other billiard shapes give rise to a mixture of unpredictable chaotic motion, or regular predictable motion, depending on the starting position and momentum of the mass. Physicists and mathematicians going back to Lord Kelvin4 have learned a great deal about chaotic motion and its onset by studying dynamical billiards because of their varied shape.

It was thus natural for physicists to ask, and to model, what would happen if the billiard was very small and the mass was a quantum particle5. How would its quantum wavefunction and quantum dynamics reflect the possibility of chaotic motion in the billiard? A number of clear differences were found compared with highly symmetrical ('non-chaotic') quantum systems, for example pseudo-random behaviour of energy levels and wavefunctions, as well as a new avenue for quantum tunnelling — a process in which a quantum particle mysteriously transits a region that cannot be traversed according to the laws of classical mechanics. The 'quantum billiard' was a wonderful theoretical model, but experimentally it was difficult to find real physical systems to which the model applied. Artificial atoms known as quantum dots have been analysed with some success as chaotic quantum systems, but these systems typically contain many interacting electrons, and the electrons are not confined by 'hard walls' of a known shape so as to make a simple connection to billiards. This situation changed in 1997 when Nöckel and Stone6 pointed out that certain types of laser cavity are realizations of quantum billiards.

Typical laser cavities are formed by arrangements of mirrors, which trap the light that is generated by pumping an atomic gain medium with some external energy source, often an electrical current. The pumped gain medium then amplifies the light as it bounces back and forth in the cavity before eventually escaping. However, the push for small on-chip lasers to enable integrated optical circuits has given rise to new, cylindrically shaped laser cavities, in which the light is almost completely trapped by total internal reflection. If such cylindrical cavities are smoothly deformed so that their cross-section is roughly oval-shaped (Fig. 1), the result is an 'asymmetric resonant cavity' (ARC), in which a thin cross-sectional slice contains the gain medium, and photons bounce around within this slice, behaving like quantum particles in a two-dimensional billiard of the corresponding shape. However, there is one crucial and interesting difference: the photons can sometimes refract out at the boundary instead of reflecting back in, so these systems are leaky quantum billiards. How would their behaviour differ from the corresponding 'classical behaviour', which in the case of light means the behaviour expected from ray optics?

Figure 1: Cross-sectional shape of the 'asymmetric resonant cavity' microlaser of Shinohara and colleagues3.
figure1

The laser is 'pumped' by an electrical current so that most of the light rays generated follow a rectangular periodic orbit (blue), which is totally internally reflected according to the usual law of specular reflection in ray optics. Conventional 'quantum tunnelling' would lead to light emission at the corners of the rectangle (dashed blue arrow; similar processes would occur at all four corners). But the authors3 find that light is actually emitted by 'chaos-assisted tunnelling', whereby the light rays violate the specular reflection law of ray optics by a small amount (dashed green arrow; again a similar process would occur at each corner) and subsequently obey ray optics along a chaotic trajectory (green; simplified for clarity) until they escape by refraction at points far from the corners of the rectangle and near to the horizontal axis of the asymmetric resonant cavity. All of the emission occurs in four highly directional beams approximately parallel to the vertical axis of the cavity, in contrast to the output directions at approximately 45° to that axis expected from conventional tunnelling.

Surprisingly, in many cases the chaotic, classical motion of light rays can be used to predict how these laser cavities will emit light, with little need to take into account quantum effects6,7. For these cavities, the periodic ray orbits that are intended to trap the light are unstable, and chaos carries rays away, causing them to fall below the angle of total internal reflection and refract out in a surprisingly regular manner7. The work of Shinohara et al.3 provides a dramatic exception to this classical ray mechanism for emission, and shows how chaos and quantum effects can work cooperatively.

In their experiment, a specific shape of ARC microlaser was designed that had a periodic ray orbit in the shape of a rectangle (Fig. 1); only light in the vicinity of this orbit would be strongly amplified owing to the authors' pumping scheme. Unlike previous ARC cavities, this rectangular orbit was stable, meaning that rays close to it would not behave chaotically, but would remain in the orbit's vicinity indefinitely, undergoing an oscillatory motion. Moreover, the angles of incidence at the bounce points of the ray orbit were all larger than the critical angle for total internal reflection, so that no light rays would escape the laser at all if ray optics were valid for this system. Not to worry, quantum mechanics steps in here and tells us that there is some small probability that a photon can 'tunnel' out of the cavity, despite the law of total internal reflection for rays. So an experienced quantum physicist would not be surprised to find some weak emission of light near the bounce points of the orbit, in the tangential direction (Fig. 1).

But that is not at all what is seen in Shinohara and colleagues' set-up3. Instead, the light escapes far from the bounce points and ends up travelling in several highly directional beams perpendicular to the major axis of the ARC. This is because chaos alters familiar quantum behaviour. A light ray travelling on or near to the rectangular orbit is unaffected by the chaotic motion that it would undergo if it could get just a little further away. But the photon 'wavefunction' has some small chance of sneaking further away, into the 'chaotic sea'. In fact, this small chance is still much larger than the probability that the photon will tunnel directly out of the cavity, because it involves a smaller violation of classical mechanics. However, once the photon makes this small quantum detour, it is quickly carried out of the cavity by chaotic motion, causing it to fall rapidly below the critical angle for total internal reflection. This process, called 'chaos-assisted tunnelling', was predicted many years ago6,8,9, but has been observed in only a few experiments10,11, and in none as dramatically as in the experiment of Shinohara and colleagues3.

There is one last puzzle with the authors' observations. Why doesn't the chaotic motion of the photon lead to essentially random transmission in all directions? The reason is that the full pseudo-random behaviour of chaotic billiards develops only after many bounces. As noted above, it has previously been shown6,7 that highly directional emission is typical from these leaky chaotic cavities, and that the favoured emission directions can be predicted from the study of few short, unstable periodic orbits in the chaotic sea. This 'unstable manifold' theory7 was used by Shinohara et al.3 to explain the origin of the brightest emission points near the major axis of the ARC (Fig. 1) and the highly directional beams perpendicular to this axis seen in the experiment. It is this directional emission property that has motivated the study and design of ARC microlasers as potentially useful on-chip light sources for integrated optical circuits12,13. Studies such as that of Shinohara and colleagues exemplify the gratifying confluence of fundamental and technological interest in these systems.

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Stone, A. Chaotic billiard lasers. Nature 465, 696–697 (2010). https://doi.org/10.1038/465696a

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