利用者:Glayhours/sandbox/傷跡状態

傷跡状態（きずあとじょうたい、英: scar state）は、不安定な古典的な周期軌道の近傍で確率密度が増加するような、古典的にカオスを示す量子系の固有状態である。 周期軌道の不安定性は傷跡状態を特徴付ける重要な点であり、安定な周期軌道の近傍で確率密度が増加する自明な例と決定的に異なる点である。 後者はボーアの対応原理に則した純粋に古典的な現象として理解できる。しかし前者は量子系における干渉が本質的な役割を果たしていると理解できる。As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.

A classically chaotic system is also ergodic, and therefore (almost) all of its trajectories eventually explore evenly the entire accessible phase space. Thus, it would be natural to expect that the eigenstates of the quantum counterpart would fill the quantum phase space in the uniform manner up to random fluctuations in the semiclassical limit. However, scars are a significant correction to this assumption. 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 proving that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. Nonetheless, the quantum ergodicity theorems do not exclude scarring the quantum phase space volume of the scars gradually vanishes in the semiclassical limit.

On the classical side, there is no direct analogue of scars. 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. Scars correspond to nonergodic states which are permitted by the quantum ergodicity theorems. In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system would be without structure. In addition to conventional quantum scars, the field of quantum scarring has undergone its renaissance period, sparked by the discoveries of perturbation-induced scars and many-body scars.

Scar theory

The existence of scarred states is rather unexpected based on the Gutzwiller trace formula which connects the quantum mechanical density of states to the periodic orbits in the corresponding classical system. According to the trace formula, a quantum spectrum is not a result of a trace over all the positions but it is determined by a trace over all the periodic orbits only. Furthermore, every periodic orbit contributes to an eigenvalue, although not exactly equally. It is even more unlikely that a particular periodic orbit would stand out in contributing to a particular eigenstate in a fully chaotic system since altogether periodic orbits occupy a zero-volume portion of the total phase space volume. Hence, nothing seems to imply that any particular periodic orbit for a given eigenvalue could have an significant role compared to other periodic orbits. Nonetheless, quantum scarring proves this assumption to be wrong.

The scarring was first described in 1983 by S. W. McDonald in his thesis on the stadium billiard as an interesting numerical observation. This finding was not thoroughly reported in the article discussion about the wave functions and nearest-neighbor level spacing spectra for the stadium billiard. A year later, E. J. Heller published the first examples of scarred eigenfunctions together with a theoretical explanation for their existence. The results revealed large footprints of individual periodic orbits influencing some eigenstates of the classically chaotic Bunimovich stadium, named as scars by E. J. Heller.

A wave packet analysis was a key in proving the existence of the scars, and it is still a valuable tool to understand them. In the original work of E. J. Heller, the quantum spectrum is extracted by propagating a Gaussian wave packet along a periodic orbit. Nowadays, this seminal idea is known as the linear theory of scarring. Scars stand out to the eye in some eigenstates of classically chaotic systems, but are quantified by projection of the eigenstates onto certain test states, often Gaussians, having both average position and average momentum along the periodic orbit. These test states give a provably structured spectrum that reveals the necessity of scars. However, there is no universal measure on scarring; the exact relationship of the stability exponent $\chi$ to the scarring strength is matter of definition. Nonetheless, there is a rule of thumb: quantum scarring is significant when $\chi <2\pi$, and the strength scales as $\chi ^{-1}$. Thus, strong quantum scars are, in general, associated with periodic orbits that are moderately unstable and relative short. The theory predicts the scar enhancement along a classical periodic orbit, but it cannot precisely pinpoint which particular states are scarred and how much. Rather, it can be only stated that some states are scarred within certain energy zones, and by at least by a certain degree.

The linear scarring theory outlined above has been later extended to include nonlinear effects taking place after the wave packet departs the linear dynamics domain around the periodic orbit. At long times, the nonlinear effect can assist the scarring. This stems from nonlinear recurrences associated with homoclinic orbits. A further insight on scarring was acquired with a real-space approach by E. B. Bogomolny and a phase-space alternative by M. V. Berry complementing the wave-packet and Hussimi space methods utilized by E. J. Heller and L. Kaplan.

The first experimental confirmations of scars were obtained in microwave billiards in the early 1990s. Further experimental evidence for scarring has later been delivered by observations in, e.g., quantum wells, optical cavities and the hydrogen atom. In the early 2000s, the first observations were achieved in an elliptical billiard. In this system, many classical trajectories converge and lead to pronounced scarring at the foci, commonly called as quantum mirages. In addition, recent numerical results indicated the existence of quantum scars in ultracold atomic gases.

Perturbation-induced quantum scars

In addition to the conventional scarring above, a new class of quantum scars was discovered in disordered two-dimensional nanostructures.

Many-body quantum scarring

The area of quantum many-body scars is a subject of active research.

Scars have occurred in investigations for potential applications of Rydberg states to quantum computing, specifically acting as qubits for quantum simulation. The particles of the system in an alternating ground state-Rydberg state configuration continually entangled and disentangled rather than remaining entangled and undergoing thermalization. Systems of the same atoms prepared with other initial states did thermalize as expected. The researchers dubbed the phenomenon "quantum many-body scarring".

The causes of quantum scarring are not well understood. One possible proposed explanation is that quantum scars represent integrable systems, or nearly do so, and this could prevent thermalization from ever occurring. This has drawn criticisms arguing that a non-integrable Hamiltonian underlies the theory. Recently, a series of works has related the existence of quantum scarring to an algebraic structure known as dynamical symmetries.

Fault-tolerant quantum computers are desired, as any perturbations to qubit states can cause the states to thermalize, leading to loss of quantum information. Scarring of qubit states is seen as a potential way to protect qubit states from outside disturbances leading to decoherence and information loss.

See also

• Classical limit
• Time crystal
• Trojan wave packet
• Many body localization (MBL)