Alice’s disturbing adventures in Wonderland tumbled upon her like a string of accidents as she wandered a world of chaos. Rules were never what they seemed and shifted whenever they wanted. She even met a cat who grinned ear-to-ear and could disappear entirely, or almost entirely, leaving only its grin.
The vanishing Cheshire Cat reminds us of another famous cat—Arnold’s Cat—that introduced the ideas of stretching and folding of phase-space volumes in non-integrable Hamiltonian systems. But when Arnold’s Cat becomes a Quantum Cat, a central question remains: What happens to the chaotic behavior of the classical system … does it survive the transition to quantum mechanics? The answer is surprisingly like the grin of the Cheshire Cat—the cat vanishes, but the grin remains. In the quantum world of the Cheshire Cat, the grin of the classical cat remains even after the rest of the cat vanished.
The Cheshire Cat fades away, leaving only its grin, like a fine filament, as classical chaos fades into quantum, leaving behind a quantum scar.
The Quantum Mechanics of Classically Chaotic Systems
The simplest Hamiltonian systems are integrable—they have as many constants of the motion as degrees of freedom. This holds for quantum systems as well as for classical. There is also a strong correspondence between classical and quantum systems for the integrable cases—literally the Correspondence Principle—that states that quantum systems at high quantum number approach classical behavior. Even at low quantum numbers, classical resonances are mirrored by quantum eigenfrequencies that can show highly regular spectra.
But integrable systems are rare—surprisingly rare. Almost no real-world Hamiltonian system is integrable, because the real world warps the ideal. No spring can displace indefinitely, and no potential is perfectly quadratic. There are always real-world non-idealities that destroy one constant of the motion or another, opening the door to chaos.
When classical Hamiltonian systems become chaotic, they don’t do it suddenly. Almost all transitions to chaos in Hamiltonian systems are gradual. One of the best examples of this is the KAM theory that starts with invariant action integrals that generate invariant tori in phase space. As nonintegrable perturbations increase, the tori break up slowly into island chains of stability as chaos infiltrates the separatrixes—first as thin filaments of chaos surrounding the islands—then growing in width to take up more and more of phase space. Even when chaos is fully developed, small islands of stability can remain—the remnants of stable orbits of the unperturbed system.
When the classical becomes quantum, chaos softens. Quantum wave functions don’t like to be confined—they spread and they tunnel. The separatrix of classical chaos—that barrier between regions of phase space—cannot constrain the exponential tails of wave functions. And the origin of chaos itself—the homoclinic point of the separatrix—gets washed out. Then the regular orbits of the classical system reassert themselves, and they appear, like the vestige of the Cheshire Cat, as a grin.
The Quantum Circus
The empty stadium is a surprisingly rich dynamical system that has unexpected structure in both the classical and the quantum domain. Its importance in classical dynamics comes from the fact that its periodic orbits are unstable and its non-periodic orbits are ergodic (filling all available space if given long enough). The stadium itself is empty so that particles (classical or quantum) are free to propagate between reflections from the perfectly-reflecting walls of the stadium. The ergodicity comes from the fact that the stadium—like a classic Roman chariot-race stadium, also known as a circus—is not a circle, but has a straight stretch between two half circles. This simple modification takes the stable orbits of the circle into the unstable orbits of the stadium.
A single classical orbit in a stadium is shown in Fig 1. This is an ergodic orbit that is non-periodic and eventually would fill the entire stadium space. There are other orbits that are nearly periodic, such as one that bounces back and forth vertically between the linear portions, but even this orbit will eventually wander into the circular part of the stadium and then become ergodic. The big quantum-classical question is what happens to these classical orbits when the stadium is shrunk to the nanoscale?
Simulating an evolving quantum wavefunction in free space is surprisingly simple. Given a beginning quantum wavefunction A(x,y,t0), the discrete update equation is
Perfect reflection from the boundaries of the stadium are incorporated through imposing a boundary condition that sends the wavefunction to zero. Simple!
A snap-shot of a wavefunction evolving in the stadium is shown in Fig. 2. To see a movie of the time evolution, see my YouTube episode.
The time average of the wavefunction after a long time has passed is shown in Fig. 3. Other than the horizontal nodal line down the center of the stadium, there is little discernible structure or symmetry. This is also true for the mean squared wavefunction shown in Fig. 4, although there is some structure that may be emerging in the semi-circular regions.
On the other hand, for special initial conditions that have a lot of symmetry, something remarkable happens. Fig. 5 shows several mean-squared results for special initial conditions. There is definite structure in these cases that were given the somewhat ugly name “quantum scars” in the 1980’s by Eric Heller who was one of the first to study this phenomenon [1].
One can superpose highly-symmetric classical trajectories onto the figures, as shown in the bottom row. All of these classical orbits go through a high-symmetry point, such as the center of the stadium (on the left image) and through the focal point of the circular mirrors (in the other two images). The astonishing conclusion of this exercise is that the highly-symmetric periodic classical orbits remain behind as quantum scars—like the Cheshire Cat’s grin—when the system is in the quantum realm. The classical orbits that produce quantum scars have the important property of being periodic but unstable. A slight perturbation from the symmetric trajectory causes it to eventually become ergodic (chaotic). These scars are regions with enhanced probability density, what might be termed “quantum trajectories”, but do not show strong interference patterns.
It is important to make the distinction that it is also possible to construct special wavefunctions that are strictly periodic, such as a wave bouncing perfectly vertically between the straight portions. This leads to large-scale interference patterns that are not the same as the quantum scars.
Quantum Chaos versus Laser Speckle
In addition to the bouncing-wave cases that do not strictly produce quantum scars, there is another “neutral” phenomenon that produces interference patterns that look a lot like scars, but are simply the random addition of lots of plane waves with the same wavelength [2]. A snapshot in time of one of these superpositions is shown in Fig. 6. To see how the waves add together, see the YouTube channel episode.
By David D. Nolte, Aug. 14, 2022
YouTube Video
Read more about the history of chaos theory in Galileo Unbound from Oxford Press:
References
[1] Heller E J, Bound-state eigenfunctions of classically chaotic hamiltonian-systems – scars of periodic-orbits, Physical Review Letters 53 ,1515 (1984)
[2] Gutzwiller M C, Chaos in classical and quantum mechanics (New York: New York : Springer-Verlag, 1990)
Galileo Unbound 
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