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arXiv:2403.18081

Antiscarring in Chaotic Quantum Wells

Joonas Keski-Rahkonen, Anton M. Graf, Eric J. Heller

Published Mar 26, 2024• 0 citations
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Abstract

Chaos plays a crucial role in numerous natural phenomena, but its quantum nature has remained large elusive. One intriguing quantum-chaotic phenomenon is the scarring of a single-particle wavefunction, where the quantum probability density is enhanced in the vicinity of a classical periodic orbit. These quantum scars illustrate the quantum suppression of classical chaos, offering a unique way to explore the classical-quantum relationship beyond conventional limits. In this study, we establish an ergodicity theorem for slacking a group of adjacent eigenstates, revealing the aspect of antiscarring -- the reduction of probability density along a periodic orbit generating the corresponding scars. We thereafter apply these two concepts to variational scars in a disordered quantum well, and finally discuss their broader implications, suggesting potential experimental approaches to observe this phenomenon.

DOI
10.48550/arxiv.2403.18081

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Chaos plays a crucial role in numerous natural phenomena, but its quantum nature has remained large elusive. One intriguing quantum-chaotic phenomenon is the scarring of a single-particle wavefunction, where the quantum probability density is enhanced in the vicinity of a classical periodic orbit. These quantum scars illustrate the quantum suppression of classical chaos, offering a unique way to explore the classical-quantum relationship beyond conventional limits. In this study, we establish an ergodicity theorem for slacking a group of adjacent eigenstates, revealing the aspect of antiscarring -- the reduction of probability density along a periodic orbit generating the corresponding scars. We thereafter apply these two concepts to variational scars in a disordered quantum well, and finally discuss their broader implications, suggesting potential experimental approaches to observe this phenomenon.
We reveal a feature of quantum scarring in systems with many particles: quantum scars, living densely near an unstable periodic orbit, must be compensated by corresponding antiscarred states suppressed there to establish the uniformity of the whole. The uniformity of the underlying phase space is linked to early-time dynamics---a regime beyond the predictions of random matrix theory and encapsulated in the eigenstate stacking theorem. By extending the domain of the stacking theorem, we apply our theory to a chaotic spinor Bose-Einstein condensate, whose quantum scar dynamics have recently been observed in the laboratory. Our work uncovers how scarring of some eigenstates affects the rest of the chaotic and thermal spectrum in quantum systems with many particles.
We predict and numerically observe strong periodic orbit effects in the properties of weakly open quantum systems with a chaotic classical limit. Antiscars lead to a large number of exponentially narrow isolated resonances when the single-channel (or tunneling) opening is located on a short unstable orbit of the closed system; the probability to remain in the system at long times is thus exponentially enhanced over the random matrix theory prediction. The distribution of resonance widths and the probability to remain are quantitatively given in terms of only the stability matrix of the orbit on which the opening is placed. The long-time remaining probability density is nontrivially distributed over the available phase space; it can be enhanced or suppressed near orbits other than the one on which the lead is located, depending on the periods and classical actions of these other orbits. These effects of the short periodic orbits on quantum decay rates have no classical counterpart, and first appear on times scales much larger than the Heisenberg time of the system. All the predictions are quantitatively compared with numerical data.
We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic systems, have recently regained interest for their role in non-ergodic dynamics, as they retain memory of their initial states. We elucidate these features of quantum scars within the same framework of this toy model. The integrable part of the model consists of two large spins, with a classical counterpart, which we combine with a random matrix to induce ergodic behavior. Scarred states can be selectively generated from the integrable spin Hamiltonian by protecting them from the ergodic states using a projector method. Deformed projectors mimic the ‘quantum leakage’ of scarred states, enabling tunable mixing with ergodic states and thereby controlling the degree of scarring. In this simple model, we investigate various properties of quantum scarring and shed light on different aspects of many-body quantum scars observed in more complex quantum systems. Notably, the underlying classicality can be revealed through the entanglement spectrum and the dynamics of ‘out-of-time-ordered correlators’.
Quantum scars are tracks of enhanced probability densities along unstable classical periodic orbits in quantum chaotic systems. They are a visual representation of classical-quantum correspondence as well as an example of quantum suppression of chaos. A new type of quantum scarring was recently discovered in radially symmetric two-dimensional potential wells perturbed by localized impurities. These perturbation-induced scars show potential to be utilized in controlled applications in classically chaotic quantum systems due to the ability to efficiently propagate wave packets along the scars. In this thesis, the focus is on special linearly shaped perturbation-induced quantum scars which correspond to the classical motion of a ball bouncing between two walls. These bouncing ball quantum scars have potential to be used as efficient channels in quantum transport, but their behavior is still mostly unexplored. This thesis studies how bouncing ball scars could be controlled by changing the perturbation parameters and how stable these scars are against external noise. Both of these features are relevant when utilization in experimental applications is considered. Bouncing ball scars are studied in a radially symmetric two-dimensional (2D) potential well perturbed by a singular Gaussian-like potential bump or a dip, which are placed into the same location. For the stability calculations, randomly placed smaller Gaussian bumps are also placed into the potential with a constant uniform mean density in addition to the main perturbation. Imaginary time propagation algorithm is then applied to solve the 2D Schrödinger equation in our model systems. Bouncing ball scars are detected from the solved eigenstates by integrating the probability densities of the states inside narrow detection boxes. The results indicate that the singular potential bump and the dip affect the same states of the clean system in different ways. The bump seems to pin the bouncing ball scars onto itself, when the dip causes the scars to avoid it. This means that the different perturbation types have the opposite scar orientation preferences. The abundance of the bouncing ball scars seems to also be connected to the perturbation parameters, and the results indicate that the parameters could be optimized in order to maximize the abundance. The stability results show impressive robustness against external noise. The number of bouncing ball scars remain nearly constant with weaker noise and even when the noise bumps were 10% from the main perturbation size, a notable amount of bouncing ball scars persist. It also seems that the scars retain their orientations even in the noisy systems.
Quantum many-body scars (QMBSs) represent a weak ergodicity-breaking phenomenon that defies the common scenario of thermalization in closed quantum systems. They are often regarded as a many-body analog of quantum scars (QSs)—a single-particle phenomenon in quantum chaos—due to their superficial similarities. However, unlike QSs, a clear connection between QMBSs and classical chaos has remained elusive. It has nevertheless been speculated that in an appropriate semiclassical limit, QMBSs should have a correspondence to weakly unstable periodic orbits. In this paper, I present a counterexample to this conjecture by studying a bosonic model with a large number of flavors. The dynamics of out-of-time-ordered correlators (OTOCs) suggest that QMBS do not display chaotic behavior in the semiclassical limit. In contrast, chaotic dynamics are expected for initial states not associated with QMBS. Interestingly, the anomalous OTOC dynamics persist even under weak perturbations that eliminate the scarred eigenstates, suggesting a certain robustness in the phenomenon.
A quantum scar-an enhancement of a quantum probability density in the vicinity of a classical periodic orbit-is a fundamental phenomenon connecting quantum and classical mechanics. Here we demonstrate that some of the eigenstates of the perturbed two-dimensional anisotropic (elliptic) harmonic oscillator are strongly scarred by the Lissajous orbits of the unperturbed classical counterpart. In particular, we show that the occurrence and geometry of these quantum Lissajous scars are connected to the anisotropy of the harmonic confinement, but unlike the classical Lissajous orbits the scars survive under a small perturbation of the potential. This Lissajous scarring is caused by the combined effect of the quantum (near) degeneracies in the unperturbed system and the localized character of the perturbation. Furthermore, we discuss experimental schemes to observe this perturbation-induced scarring.