Researchers Discover Entanglement Between Tomonaga-Luttinger Liquids Below Threshold Temperature

Entanglement, a uniquely quantum phenomenon, fundamentally separates the quantum world from classical physics, yet observing it in complex systems remains a significant challenge. Taufiq Murtadho, Marek Gluza, and Nelly H. Y. Ng, from Nanyang Technological University, investigate entanglement between two interacting, one-dimensional quantum gases known as Tomonaga-Luttinger liquids. Their theoretical work demonstrates extensive entanglement, meaning the amount of entanglement grows proportionally with the size of the system, both when these gases are connected via quantum tunnelling and after being rapidly separated. Importantly, the researchers identify a temperature threshold below which this strong entanglement persists, bringing the experimental detection of extensive entanglement in many-body systems closer to reality, and revealing how initial quantum correlations can be preserved even as the system evolves over time. This research extends our understanding of entanglement beyond simple scenarios and provides a pathway towards observing this crucial quantum property in more complex, real-world systems at practical temperatures.

While entanglement is routinely observed for few-body systems, it presents significantly greater challenges when witnessed in quantum many-body systems. This work theoretically studies entanglement between two parallel and spatially separated Tomonaga-Luttinger liquids, theoretical models describing interacting electrons in one dimension. The research focuses specifically on 1D Bose gases as a realisation of these liquids and investigates two experimentally relevant situations: tunnel-coupled gases at finite temperatures and gases after coherent splitting. In both scenarios, the logarithmic negativity, a measure of entanglement, is analytically calculated, identifying a threshold temperature below which the system exhibits entanglement.

Entanglement Measures via Logarithmic Negativity and Mutual Information

This appendix provides the mathematical derivations for key results presented in the main body of the paper, specifically calculating the logarithmic negativity and mutual information, another measure of correlation between quantum systems. It covers two scenarios: equilibrium tunnel-coupled gases, where two Bose gases are weakly connected and have reached thermal equilibrium, and coherently split gases, created by splitting a single Bose gas into two spatially separated parts. The appendix logically begins by defining the covariance matrix, which describes the quantum correlations between different modes of the Bose gas, then calculates the symplectic eigenvalues of this matrix, crucial for determining the entanglement measures. Using these eigenvalues, the logarithmic negativity and mutual information are calculated, and the threshold temperature below which entanglement occurs is derived.

The covariance matrix captures all second-order correlations between quantum variables, while field operators describe the quantum field of the Bose gas, related to particle creation and annihilation. Particle number operators count particles in each mode, and symplectic eigenvalues are eigenvalues of a matrix related to the covariance matrix, used to determine entanglement properties. Logarithmic negativity is a measure of entanglement particularly useful for mixed states, and mutual information quantifies the total correlation between two systems. The calculations rely on mathematical operations like partial transposition and consider concepts like thermal equilibrium and coherent splitting.

The appendix derives formulas for the logarithmic negativity and mutual information for both equilibrium tunnel-coupled gases and coherently split gases, also finding the threshold temperature below which entanglement occurs. Equations define the elements of the covariance matrix and provide the symplectic eigenvalues, allowing for the calculation of the threshold temperature for entanglement in both scenarios. This appendix provides a rigorous mathematical treatment of entanglement in Bose gases, deriving formulas for the logarithmic negativity and mutual information and finding the threshold temperature below which entanglement occurs. The results are important for understanding the fundamental properties of quantum systems and for developing new quantum technologies.

Logarithmic Negativity in Coupled Bose Gases

Researchers have demonstrated the existence of logarithmic negativity, a unique form of quantum entanglement, between two parallel, one-dimensional Bose gases. This phenomenon is fundamentally distinct from classical physics and challenging to observe in many-body systems. This work theoretically investigates entanglement between these Tomonaga-Luttinger liquids, focusing on scenarios relevant to current experiments: tunnel-coupled gases at finite temperatures and systems created after coherent splitting. The team analytically calculated logarithmic negativity, identifying a critical temperature below which entanglement emerges, a threshold accessible with existing experimental capabilities.

The findings reveal that this entanglement scales extensively with the length of the system, meaning that larger systems exhibit proportionally greater entanglement, a characteristic termed a ‘boundary law’. This contrasts with typical expectations for entanglement in one-dimensional systems, where entanglement growth is often limited by an ‘area law’. Importantly, the study demonstrates that initial correlations established during coherent splitting are preserved as the system evolves, preventing it from reaching certain predicted equilibrium states. This research addresses a significant gap in understanding entanglement in coupled systems, extending previous work beyond ground states to encompass finite temperatures and dynamic, out-of-equilibrium conditions.

By highlighting achievable parameter regimes, the team provides a clear strategy for experimental detection of this extensive entanglement, opening new avenues for quantum simulation and the study of interacting quantum fields. The results are particularly relevant to condensed matter physics, offering insights into systems with ladder geometries and the interplay between inter- and intra-chain coupling. Recent advances in experimental techniques, including outcoupling measurement and common phase interferometry, now provide the necessary tools to probe and verify these theoretical predictions.

Entanglement Survives Finite Temperatures in Luttinger Liquids

This research investigates entanglement between two Tomonaga-Luttinger liquids, theoretical models describing interacting electrons in one dimension. The study demonstrates the existence of entanglement even at finite temperatures, a challenging condition for maintaining these delicate quantum correlations, and explores how this entanglement arises in both systems connected by tunneling and those created by splitting a single system into two. Researchers calculated a measure of entanglement called logarithmic negativity, identifying a specific temperature threshold below which entanglement is present, and this threshold is potentially achievable in current experiments. The findings extend our understanding of entanglement in many-body systems, particularly in scenarios where the subsystems are not identical, and provide a pathway for experimentally detecting this extensive entanglement.

The authors acknowledge that their analysis relies on a harmonic approximation, valid when the tunneling strength dominates, and that further investigation is needed to fully understand the behavior of the system in other regimes. Future work could explore the impact of stronger interactions and different system geometries, potentially revealing new insights into the nature of quantum correlations in condensed matter physics. The team also notes that their approach could be valuable for studying prethermalization, a phenomenon where a system appears to reach equilibrium before truly doing so, by demonstrating how initial correlations are preserved during the system’s evolution.