🔬 condensed matter

Extensive entanglement between coupled Tomonaga-Luttinger liquids in and out of equilibrium

This paper theoretically demonstrates that extensive quantum entanglement between coupled Tomonaga-Luttinger liquids persists at experimentally accessible finite temperatures and in non-equilibrium regimes, providing a viable strategy for its detection in 1D Bose gas experiments.

Original authors: Taufiq Murtadho, Marek Gluza, Nelly H. Y. Ng

Published 2026-08-07

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CC BY 4.0

Original authors: Taufiq Murtadho, Marek Gluza, Nelly H. Y. Ng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible dance floor. In the world of classical physics—the world of baseballs, cars, and coffee cups—dancers move independently. If you watch one dancer, you know nothing about what their neighbor is doing unless they talk or bump into each other. But in the quantum world, the rules change completely. Here, dancers can become "entangled," a spooky connection where two particles share a single existence, no matter how far apart they are. If you check one, you instantly know the state of the other. This isn't just a party trick; it's the fundamental difference between the quantum and classical worlds and the secret sauce for future super-computers.

The big challenge for scientists is that while we can see this dance in small groups of particles, it gets incredibly hard to spot when you have a massive crowd. Usually, heat acts like a chaotic mosh pit, shaking the dancers apart and destroying the delicate entanglement. Most experiments have focused on cutting a system in half like slicing a loaf of bread to see if the two halves are connected. But what if the connection isn't about the slices, but about the two entire loaves dancing side-by-side? This is the question tackled in a new study by researchers at Nanyang Technological University in Singapore. They wanted to know if two parallel streams of ultra-cold atoms, known as Tomonaga-Luttinger liquids, could stay entangled even when things get a bit warm, and if we could actually catch them in the act.

The paper focuses on two specific ways to set up this atomic dance: first, by letting two streams of atoms tunnel (or leak) into each other, and second, by taking a single stream and coherently splitting it into two, like a beam splitter for atoms. The researchers used advanced math to calculate a measure called "logarithmic negativity," which acts like a detector for entanglement. They found that entanglement does survive, but only if the temperature is kept below a very specific threshold. For the tunneling scenario, this "entanglement zone" exists at temperatures between 1 and 4 nanokelvin (nK). While this is incredibly cold, it is within reach of current experiments. Even more exciting, the "splitting" scenario allows for entanglement at slightly warmer temperatures, between 30 and 60 nK, which is much easier to achieve in a lab.

The study reveals that this entanglement isn't just a tiny, fleeting spark; it is "extensive," meaning the amount of entanglement grows linearly with the length of the gas. If you double the length of your atomic stream, you double the entanglement. This is a big deal because it suggests that large-scale quantum connections are possible, not just in the frozen ground state of a system, but in the messy, warm, real-world conditions we can actually create. The researchers also looked at "mutual information," which measures all connections, both quantum and classical. They found that while the quantum entanglement fades away as the temperature rises, the total connection (mutual information) stays surprisingly steady. This suggests that the initial "memory" of how the atoms were split is preserved in the system's dynamics, preventing it from settling into a generic, uninteresting state too quickly.

In short, the paper suggests that by looking at parallel atomic gases rather than sliced segments, and by using clever splitting techniques, we can detect extensive quantum entanglement at temperatures that are achievable today. It doesn't claim to have built a quantum computer yet, but it provides a clear roadmap and a "yes, it's possible" signal for experimentalists. The authors argue that the threshold temperatures they calculated are accessible in current and near-term experiments, offering a concrete strategy to witness this quantum magic in a many-body system that is warm and out of equilibrium.

Technical Summary: Extensive Entanglement between Coupled Tomonaga-Luttinger Liquids in and out of Equilibrium

Problem Statement
While quantum entanglement is a fundamental signature distinguishing quantum from classical physics, experimentally witnessing and characterizing it in quantum many-body systems remains a significant challenge. This difficulty is exacerbated in systems described by effective quantum field theories (QFTs), where the vast Hilbert space contrasts with the limited number of accessible experimental observables. Existing literature on entanglement in one-dimensional (1D) systems predominantly focuses on "transversal cuts," partitioning the system across the 1D axis. For such partitions, ground-state entanglement typically follows an area law (or logarithmic growth in gapless systems), and at finite temperatures, entanglement is often diluted by thermal noise, making it difficult to detect.

This work addresses the gap in characterizing entanglement between two parallel, spatially separated Tomonaga-Luttinger liquids (TLLs) using a "longitudinal cut." This partition divides the system along the 1D axis, separating the two parallel gases (A and B) rather than segments of a single gas. The authors investigate whether this configuration allows for the observation of extensive entanglement (scaling linearly with subsystem length) at finite temperatures and in non-equilibrium regimes, specifically in 1D Bose gases.

Methodology
The study utilizes a theoretical framework based on coupled 1D Bose gases, modeled as TLLs with quantum pressure corrections. The system is described by a Hamiltonian that includes a tunnel-coupling term, which, in the strong coupling limit, maps to a sine-Gordon model that can be approximated as a Gaussian model (Klein-Gordon field) in the antisymmetric sector.

Key methodological steps include:

Modeling: The system is analyzed in two distinct scenarios:
- Thermal Equilibrium: Two gases coupled via tunneling at a finite temperature.
- Non-Equilibrium: Two gases generated by the coherent splitting of a single-well trap into a double-well.
Gaussian State Formalism: The state of the system is characterized by a covariance matrix ( $\Gamma$ ) in momentum space. The authors decompose the fields into symmetric (+) and antisymmetric (−) modes, which decouple the Hamiltonian.
Entanglement Measures:
- Logarithmic Negativity ( $E_N$ ): Used as the primary measure for entanglement in mixed Gaussian states. It is calculated via the symplectic eigenvalues of the partially transposed covariance matrix. A positive $E_N$ indicates entanglement.
- Mutual Information ( $I(A:B)$ ): Used to quantify total correlations (quantum + classical). It is calculated from the von Neumann entropies of the reduced and joint covariance matrices.
Analytical and Numerical Calculation: The authors derive analytical expressions for $E_N$ and $I(A:B)$ by solving for the symplectic eigenvalues of the covariance matrix. These results are validated against numerical diagonalization.

Key Contributions and Results

Threshold Temperature for Entanglement:
- Tunnel-Coupled Equilibrium: The authors analytically derive a threshold temperature ( $T^*$ ) below which the system remains entangled. For realistic experimental parameters (e.g., $^{87}$ Rb atoms), $T^*$ is found to be in the range of 1–4 nK. While this is lower than temperatures typically achieved in current experiments, it is within the realm of near-term capabilities.
- Coherent Splitting: In the non-equilibrium scenario following coherent splitting, the threshold temperature is significantly higher, ranging from 30 to 60 nK. This range is explicitly identified as accessible in current experimental setups.
Extensive Scaling:
- The study confirms that both logarithmic negativity and mutual information scale extensively (linearly) with the subsystem length ( $L$ ) in the longitudinal partition. This contrasts with the transversal cut, where entanglement is typically bounded or logarithmic.
- The linear scaling is driven by the number of accessible momentum modes increasing with system length, rather than a simple scaling with atom number.
Quantum-to-Classical Crossover:
- The authors investigate the behavior of correlations near the threshold temperature. They find that while logarithmic negativity ( $E_N$ ) decays to zero as temperature increases, the mutual information ( $I(A:B)$ ) remains approximately constant.
- This suggests that as the system approaches $T^*$ , quantum entanglement is transformed into classical correlations, preserving the total shared information.
Conservation of Mutual Information in Prethermalization:
- In the coherent splitting scenario, the initial mutual information established by the splitting process is shown to be conserved during the subsequent TLL dynamics.
- The authors argue that this conservation prevents the system from relaxing into certain generalized Gibbs ensembles (GGE) that would otherwise be reached during prethermalization, as the initial correlations constrain the relaxation dynamics.
Parameter Dependence:
- Tunneling Strength ( $J$ ): Stronger tunneling increases the threshold temperature and the magnitude of entanglement in the equilibrium case.
- Squeezing ( $r$ ): In the coherent splitting case, entanglement exhibits non-monotonic behavior with respect to the squeezing parameter $r$ . Entanglement is robust for both large ( $r > 1$ ) and small ( $r < 0.4$ ) squeezing but is suppressed in the intermediate regime ( $0.4 < r < 1$ ). This is attributed to the competition between phase-field correlations (dominant at $r>1$ ) and density-field correlations (dominant at $r<1$ ).

Significance and Claims
The paper claims to extend the study of entanglement between coupled TLLs beyond the ground state to finite temperatures and out-of-equilibrium regimes. Its primary significance lies in:

Experimental Strategy: Proposing the longitudinal cut as a more natural and viable strategy for experimentally detecting extensive entanglement in 1D Bose gases, leveraging well-understood mechanisms like tunneling and coherent splitting.
Feasibility: Demonstrating that the conditions for observing extensive entanglement (specifically in the coherent splitting scenario) are achievable with current and near-term experimental parameters.
Theoretical Insight: Providing a framework to distinguish between quantum and classical correlations in many-body systems by comparing logarithmic negativity and mutual information, and highlighting the role of initial correlations in constraining non-equilibrium dynamics.

The work does not propose new experimental hardware but rather provides a theoretical roadmap and analytical tools to interpret existing and future experiments in parallel 1D quasicondensates.

Technical Summary: Extensive Entanglement between Coupled Tomonaga-Luttinger Liquids in and out of Equilibrium

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