Quantum scar
The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle, whereas in the former, quantum interference is essential.
A classically chaotic system is also ergodic, and therefore (almost) all of its trajectories eventually explore evenly the entire accessible phase space.
Scars can therefore be considered as an eigenstate counterpart of how short periodic orbits provide corrections to the universal spectral statistics of the random matrix theory.
There are rigorous mathematical theorems on quantum nature of ergodicity,[4][5][6] proving that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average.
On the quantum side, they can be interpreted as an eigenstate analogy to how short periodic orbits correct the universal random matrix theory eigenvalue statistics.
In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system would be without structure.
This finding was not thoroughly reported in the article discussion about the wave functions and nearest-neighbor level spacing spectra for the stadium billiard.
[2] The results revealed large footprints of individual periodic orbits influencing some eigenstates of the classically chaotic Bunimovich stadium, named as scars by Heller.
In the original work of Heller,[2] the quantum spectrum is extracted by propagating a Gaussian wave packet along a periodic orbit.
A further insight on scarring was acquired with a real-space approach by E. B. Bogomolny[14] and a phase-space alternative by Michael V. Berry[15] complementing the wave-packet and Hussimi space methods utilized by Heller and L.
Originally, it was stated[2] that certain unstable periodic orbits are shown to permanently scar some quantum eigenfunctions as
Furthermore, it was shown that the scarred states can lead to strong conductance fluctuations in the corresponding open quantum dots via the mechanism of resonant transmission.
[19][20] Further experimental evidence for scarring has later been delivered by observations in, e.g., quantum wells,[21][22][23] optical cavities[24][25] and the hydrogen atom.
[27] Many classical trajectories converge in this system and lead to pronounced scarring at the foci, commonly called as quantum mirages.
For example, a stadium billiard supports these highly nonergodic eigenstates, which reflect trapped bouncing motion between the straight walls.
, but at the same time this result suggests a diminishing percentage of all the states in the agreement with the quantum ergodicity theorems of Alexander Schnirelman, Yves Colin de Verdière, and Steven Zelditch.
Similar structures of an enhanced probability density occur even as random superpositions of plane waves,[31] in the sense of the Berry conjecture.
In the case, the disorder arising from small perturbations (see red dots in the figure) is sufficient to destroy classical long-time stability.
[51] Recently, a series of works[52][53] has related the existence of quantum scarring to an algebraic structure known as dynamical symmetries.
A hallmark of classical ergodicity is the complete loss of memory of initial conditions, resulting from the eventual uniform exploration of phase space.
The concept of a quantum birthmark[58] bridges the short-term effects, such as due to scarring, and the long-term predictions of random matrix theory.
Figure depicts two birthmarks unveiled within the time-averaged probability density of a wavepacket launched under different initial conditions (indicated by the black arrows) in the stadium.
These two cases clearly demonstrate that a quantum system can violate the classical ergodicity assumption in the sense that the probability density becomes uniform even at infinite time.
