Finite-time Revivals Demonstrate Robust Quantum Dynamics from Equilibrium States

The complex behaviour of quantum systems following a disturbance is a fundamental question in physics, and recent work by Debarghya Chakraborty and Dario Rosa, both at the ICTP South American Institute for Fundamental Research and UNESP, sheds new light on this phenomenon. They demonstrate a novel method for constructing ‘scar states’ within quantum systems, characterised by strong connections between the components governing each side of the system. This approach predicts predictable, repeating patterns, known as finite-time revivals, whenever the system begins in a common, stable state, and importantly, the behaviour remains consistent regardless of the specific details of the system’s components. By applying this framework to a specific model of quantum chaos, the double-scaled SYK model, the researchers analytically and numerically confirm these predictions, revealing a surprisingly robust and predictable form of quantum motion even in complex systems.

To facilitate the construction of finite-time revivals, the research develops an interaction term that supports a tower of equally-spaced energy eigenstates. This approach generates revivals whenever the system begins in a purified equilibrium state, and the resulting dynamics remain largely independent of the specific details of the system’s components. By examining the two-sided chord states of the double-scaled SYK model, the team finds an approximate realization of this framework, allowing for analytical study of the revival dynamics. The results demonstrate rigid motion for wavepackets localized on the spectrum of a single SYK copy, and these findings are confirmed through numerical simulations for systems of finite size.

Hamiltonian Derivation and Quantum Chaos Analysis

This document establishes the mathematical framework and derives recursion relations for wavefunctions, defining key parameters and calculating the partition function and correlation functions to describe the system’s statistical properties. It establishes completeness and orthogonality relations for the basis states used in the calculations and derives a crucial quantity, the return amplitude, to measure the probability of returning to the initial state after a certain time. The analysis simplifies calculations by setting a parameter to zero, allowing for explicit results involving Bessel functions and Taylor expansions. These calculations provide explicit formulas for the return amplitude and other quantities in the random matrix theory limit, utilizing q-analogues of mathematical functions, Hermite polynomials, Bessel functions, Jacobi theta functions, and concepts like the partition function and correlation functions to rigorously analyze the system. This document provides a rigorous mathematical foundation for the research, demonstrating the authors’ ability to perform complex calculations and connect their results to established theories in quantum chaos and random matrix theory. The use of q-deformation and the connection to the SYK model suggest potential implications for understanding quantum gravity and the nature of spacetime.

Bipartite System Hosts Zero-Energy Scarred Eigenstates

Scientists have developed a novel framework for constructing quantum many-body scars, creating a tower of equally-spaced energy levels within a bipartite system. This work begins with a system divided into two parts, where the uncoupled dynamics of each side are perfectly anticorrelated. Initial analysis reveals that diagonal eigenstates of this system, specifically those with correlations between the left and right sides, are zero-energy states, termed “scarred eigenstates”. The team focused on the infinite temperature thermofield double state, known as the “rainbow state”, as a crucial component of this construction.

To introduce dynamics, researchers employed a Krylov-like approach, defining a subspace generated by applying integer powers of the left Hamiltonian to the rainbow state. This subspace forms a zero-energy component of the total Hamiltonian, with states exhibiting equilibrium behavior when traced over either the left or right side. The key achievement is the construction of an interaction term that lifts these zero-energy states into a tower of equally-spaced energy levels, thereby supporting non-trivial dynamics. Experiments demonstrate that this framework gives rise to finite-time revivals whenever the system is initialized in a purified equilibrium state.

Further investigation of the two-sided chord states of the double-scaled Sachdev-Ye-Kitaev (DSSYK) model reveals an approximate realization of this framework. Analytical studies of the revival dynamics show rigid motion for wavepackets localized on the spectrum of a single DSSYK copy. These predictions were rigorously tested numerically for systems of finite size, demonstrating excellent agreement with the analytical results. The framework is largely independent of the specific details of the Hamiltonians defining each partition, highlighting its generality and robustness. Measurements confirm a clear relationship between the energy gap among the equally spaced eigenstates and the period of the observed revivals.

Predictable Revivals in Correlated Quantum Systems

This research establishes a novel framework for constructing so-called “scar states” in complex quantum systems, specifically bipartite systems where two parts exhibit strong correlations. By carefully designing interactions between these parts, the team demonstrated the creation of a series of equally-spaced energy levels, leading to predictable, repeating dynamics known as finite-time revivals. These revivals occur even when the system begins in a mixed state, suggesting a robustness to initial conditions and a departure from typical thermal behavior. The researchers validated this framework by applying it to a specific model, the two-sided chord states of the double-scaled SYK model, and analytically studying the resulting revival dynamics.

They found that wavepackets localized within this model exhibit rigid motion, and numerical simulations confirmed these analytical predictions for systems of finite size. Furthermore, the team investigated the behavior of the system as the strength of interactions approaches zero, revealing that revivals can be preserved as long as the system’s initial state remains confined to a limited range of configurations. This led to the definition of a “length operator” which connects the system to concepts in JT gravity, offering potential links between quantum mechanics and gravity. The authors acknowledge that the developed code, while demonstrating interesting properties, may not be immediately practical due to its nonlocal nature. They suggest that future research could focus on assessing the inherent stability of this code and exploring its potential relevance to holographic principles.