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arXiv:2410.17223

Semiclassical origin of suppressed quantum chaos in Rydberg chains

Markus Müller, Ruslan Mushkaev

Published Oct 22, 2024• 0 citations
arXivHTMLPDF

Abstract

The surprisingly long-lasting oscillations observed in the dynamics of highly excited states of chains of Rydberg atoms defy the expectation that interacting systems should thermalize fast. The phenomenon is reminiscent of wavepackets in quantum billiards that trace classical periodic orbits. While analogs of the associated scarred eigenfunctions have been found for Rydberg chains, an underlying classical limit hosting periodic orbits has remained elusive. Here we generalize the Rydberg (pseudospin $S=1/2$) system to a chain of arbitrary spin $S$. Its classical limit features unexpectedly stable periodic orbits that are essential to understand the emergence of robust, parametrically suppressed quantum chaoticity, with semi-classical coherence times diverging as $\sqrt{S}$. The classical limit successfully explains several empirical features of the quantum limit.

DOI
10.48550/arxiv.2410.17223

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The surprisingly long-lasting oscillations observed in the dynamics of highly excited states of chains of Rydberg atoms defy the expectation that interacting systems should thermalize fast. The phenomenon is reminiscent of wavepackets in quantum billiards that trace classical periodic orbits. While analogs of the associated scarred eigenfunctions have been found for Rydberg chains, an underlying classical limit hosting periodic orbits has remained elusive. Here we generalize the Rydberg (pseudospin S=1/2S=1/2S=1/2) system to a chain of arbitrary spin SSS. Its classical limit features unexpectedly stable periodic orbits that are essential to understand the emergence of robust, parametrically suppressed quantum chaoticity, with semi-classical coherence times diverging as S\sqrt{S}S​. The classical limit successfully explains several empirical features of the quantum limit.
The relaxation of few-body quantum systems can strongly depend on the initial state when the system’s semiclassical phase space is mixed; i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit. The time-dependent variational principle (TDVP) allows one to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly entangled variational states. We demonstrate that such dynamical systems generally possess mixed phase space. When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model. For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals. As a proof of principle, we identify new types of “quantum many-body scars,” i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions. Intriguingly, the initial states that give rise to most robust revivals are typically entangled states. On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to a surprising slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions. Moreover, the mixed phase space classical variational equations allow one to find slowly thermalizing initial conditions in interacting models. Our results shed light on a link between classical and quantum chaos, pointing toward possible extensions of the classical Kolmogorov-Arnold-Moser theorem to quantum systems.
The relaxation of few-body quantum systems can strongly depend on the initial state when the system's semiclassical phase space is mixed; i.e., regions of chaotic motion coexist with regular islands.In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit.The time-dependent variational principle (TDVP) allows one to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly entangled variational states.We demonstrate that such dynamical systems generally possess mixed phase space.When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model.For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals.As a proof of principle, we identify new types of "quantum many-body scars," i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions.Intriguingly, the initial states that give rise to most robust revivals are typically entangled states.On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to a surprising slowdown of thermalization.Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions.Moreover, the mixed phase space classical variational equations allow one to find slowly thermalizing initial conditions in interacting models.Our results shed light on a link between classical and quantum chaos, pointing toward possible extensions of the classical Kolmogorov-Arnold-Moser theorem to quantum systems.
Quantum chaos and the eigenstate thermalization hypothesis are based on the assumption of the validity of random matrix theory description on the spectrum and eigenstates. They provide the foundation and descriptions for the typical dynamics and thermalization in generic closed quantum systems. In this thesis, we investigate situations where the systems show atypical dynamics or anomalous thermalization, conflicting with the usual expectations from quantum chaos and eigenstate thermalization hypothesis. We first examine weak thermalization in a nonintegrable spin chain. The system shows long-lived strong oscillations and relaxes to the thermal equilibrium weakly. We identify the dynamics describable by quasiparticles and recognize the oscillation frequency to be the quasiparticle mass gap. We also estimate the damping time for the oscillations. Next, we study prethermalization, a phenomenon where a system relaxes to an intermediate almost-equilibrium stage before reaching the true thermal equilibrium. We study a nonintegrable spin chain in the strong coupling limit, where an almost-conserved quantity emerges and gives rise to the prethermalization. We also study a newly proposed diagnostic for quantum chaos: out-of-time-ordered correlators. Contrasting to the chaotic systems, we inspect their behaviors in various noninteracting integrable models. Finally, we dig into the quantum many-body scar states in the PXP model which describes a Rydberg atom chain. These special states do not satisfy the random matrix theory description nor the eigenstate thermalization hypothesis, therefore defying quantum chaos.
Recent experiments on Rydberg-atom quantum simulators have observed surprising signatures of nonergodic quantum dynamics, which has been attributed to the emergence of ``many-body quantum scars.'' Here, the authors present a detailed study of the eigenstate properties of strongly interacting Rydberg-atom chains, identifying distinct families of quantum scarred states that are responsible for the revivals in the quench dynamics. The results show that quantum scarred states have distinct features, such as subthermal entanglement and anomalous expectation values of local observables. The presence of scarred eigenstates leaves an imprint on the rest of the many-body spectrum, leading to a weak breakdown of ergodicity.
A recent proposal by Hallam et al. suggested using the chaotic properties of the semiclassical equations of motion, obtained by the time dependent variational principle (TDVP), as a characterization of quantum chaos. In this paper, we calculate the Lyapunov spectrum of the semiclassical theory approximating the quantum dynamics of a strongly interacting Rydberg atom array, which lead to periodic motion. In addition, we calculate the effect of quantum fluctuations around this approximation, and obtain the escape rate from the periodic orbit. We compare this rate to the rate extracted from the exact solution of the quantum theory, and find an order of magnitude discrepancy. We conclude that in this case, chaos in the TDVP equations does not correpond to phsyical properties of the system. Our result complement those of Ho et al. regarding the escape rate from the semiclassical periodic orbit.
We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.