Observation of area laws in an interacting quantum field simulator

TL;DR

This work experimentally demonstrates an area-law scaling of information in an interacting quantum field simulator built from two tunnel-coupled one-dimensional Bose-Einstein condensates that realize a sine-Gordon field. Using a data-driven, model-agnostic mutual-information estimator on spatially resolved relative-phase measurements and quantifying non-Gaussianity with relative entropy $S[f||f^G]$, the authors show that $I(A:B)$ scales with boundary area and decays exponentially with separation $d$ across a range of interaction strengths and finite temperature. They map how the information content and non-Gaussianity depend on the underlying length scales $\ell_J$, $\lambda_T$, and their ratio $q=\lambda_T/\ell_J$, extracting a correlation length $\ell_{\text{fit}}$ in the $3$–$8$ $\mu$m range. The results establish a universal, data-driven toolkit for probing information in high-dimensional quantum systems, transferable to other platforms and potentially illuminating information transport, thermalization, and analogue curved-spacetime phenomena in quantum matter.

Abstract

Information shared between parties quantifies their correlation. The encoding of correlations across space and time characterises the structure, history, and interactions of systems. One of the most fundamental properties that emerges from studies of information is the area law, which states that information shared between spatial subregions typically scales with the area of their boundary rather than their volume. In non-interacting, quantum many-body systems, where Gaussian statistics apply, the scaling of information measures is well understood. Within interacting systems, the readout of information measures is impeded by the complexity of state reconstruction. As such, no measurements beyond small quantum systems (e.g., composed of few, localised particles) have been made. Here, we fill this gap by experimentally demonstrating the area law of mutual information in an ultra-cold atom simulator of quantum fields with tuneable interaction strength. Our results detail the scaling of mutual information with subsystem volume, boundary area, and separation between spatial regions at finite temperature. Moreover, we quantify the total effect of non-Gaussian correlations using an information-theoretic measure - relative entropy. Our presented approach is data-driven, model agnostic, and readily applicable to other platforms and observables, thus constituting a universal toolkit for probing information in high-dimensional quantum systems and its role in shaping quantum matter and spacetime.

Figures (10)

• Figure 1: Estimating information and its scalinga. A $1$--dimensional system of length $L$ along $z$ is considered to be formed by spatial subregions $A$ ($z\leq\ell$) and $B$ ($z>\ell$). Spatially resolved measurements $\hat{O}\left(z\right)$ populate a vector space of collected samples $\hat{O}\left(\bm{z_A}\right)$ describing subsystem $A$. b. Schematics showing sample correlations in low-dimensional ($D=2$) vector spaces. Top row: patterns formed by mock samples; bottom row: intuitive understanding of shared information. The diagrams range from uncorrelated (left) to fully correlated (right). Within the top row, vertical (orange) and horizontal (green) stripes illustrate $n_A$ and $n_B$ used for computing mutual information as in equation \ref{['eq:MI_estimator']}. c. Diagrammatic representation of information scaling. Top row: area law scaling exemplified by short-range correlations, insensitive to boundary position, resulting in flat scaling of mutual information with subsystem size (right). Bottom row: volume law scaling, dominated by long-range correlations, sensitive to boundary position, resulting in an extensive scaling of mutual information with subsystem size (right).
• Figure 2: Model-agnostic extraction of information in an interacting quantum field simulator.a. Schematic of the experiment consisting of two, $1$--dimensional superfluid ${}^{87}\text{Rb}$ condensates, trapped to harmonic potentials under the atom chip and coupled by tunable tunnelling at rate $n_{1\text{D}}J$. b. Example phase extraction, showing (left) $2$D projected atomic density, (right) its single $z$ value slice with performed fitting (see methods section), and (bottom) the resulting phase profile over the full $z$ extent. This provides samples of the emergent quantum field being simulated. The illustrated process is repeated $N_s$ times at constant temperature $T$ and coupling strength $J$, to obtain statistics. c. Collection of observed phase profiles among various effective interaction regimes. Interaction strength determines phase locking as quantified by the coherence factor $\langle \cos\left(\varphi\right) \rangle$. At three representative values of $\langle \cos\left(\varphi\right) \rangle$, we plot the experimental phase profiles. The data respects dynamics of: a weakly interacting field (left : light purple), a strongly interacting, massive field (middle : purple), a non-interacting, massive Klein-Gordon field (right : dark purple). Intuition building diagrams of quasi-particle mobility in the potential $U\left(\varphi\right)$ (black) at thermal energy $E$ (grey) are provided above. One phase profile is marked out in black. d. The spatial profiles are coarse-grained to six pixels labelled by $\tilde{z}_i$. A selection of pixels into two spatial subsystems ($\tilde{z}_2$ and $\tilde{z}_3$) is made by grouping the collected data. The black profile displays the effect of this process; additionally, it serves as an example of data preparation. e. Data cloud constructed from two subsystems formed by pixels $\tilde{z}_2$ and $\tilde{z}_3$. The previously outlined black phase profile constructs the data point shown by a black circle, from local values of relative phase marked by a star ($\varphi\left(\tilde{z}_2\right)$) and a diamond ($\varphi\left(\tilde{z}_3\right)$). Neighbour searches are then used to estimate mutual information between chosen subsystems as shown in Fig. \ref{['fig:Figure_1']}.b.
• Figure 3: Area-law behaviour across interaction regimes. Scaling of information within the same three datasets as in Fig. \ref{['fig:Figure_2']} (light purple, purple and dark purple). a-c. Mutual information with varying volume; done by displacing the ($0$--dimensional) boundary, as shown above the panels. d-f. Mutual information with varying boundary area; done by defining subsystems through disjoint intervals of constant volume ($V_A = V_{B} = 3 \mathrm{d} \tilde{z}$). Points at different subsystem realisations that yield the same number of boundaries are offset along the $x$-axis for clarity. Additional shading (pink) highlights regions of area law scaling. g-i. Mutual information with increasing subsystem separation; done by separating subsystems $A$ and $B$ (size of five $z_i$ pixels each, coarse-grained to $\tilde{z}_{A/B}$). Red lines (dashed) are a guide fit of exponential decay $a e^{-d/\ell_{\text{fit}}} + b$, showcasing the associated correlation length-scale $\ell_{\text{fit}}$. In all panels, the errors are estimated using a delete-$d$ Jackknife at $95\%$ confidence intervals. Simulations (shaded regions) are interpolated to match experimental $\langle \cos\left(\varphi\right) \rangle$ at a constant $\lambda_T=15 \, µ m$ and include statistical as well as interpolation error. Additionally, shaded simulations in d-f include the spread over different realisations of the same area. Simulation results are computed using $N_s=2,000$. Both simulations and data analysis use $k=2$ within neighbour search algorithms.
• Figure 4: Interaction strength and temperature dependence of information.a. Decay of mutual information with increasing effective mass parameter $q$. The system is kept at a constant spatial division with $B=\{\tilde{z}_6\}$ and $A$ from the remainder (as indicated on the y-axis label). b. Relative entropy measurements of non-Gaussianity as in equation \ref{['eq:rel_S_of_non_gaussty']}. In both panels (a-b), shading indicates simulations, with colours specifying the experimental variation with temperature Schweigler2017 (low temperature shown in blue, high temperature in red). The points shown in black are experimental results, and empty circles indicate not sufficiently converged experimental results (see extended data section). Axis markers drawn in three shades of purple indicate datasets as in Fig. \ref{['fig:Figure_2']} and Fig. \ref{['fig:Figure_3']}. Both simulations and data analysis use $k=2$ within neighbour search algorithms.
• Figure 5: Extended datasets. Same analysis and presentation as Fig. \ref{['fig:Figure_3']}, extended to all $11$ collected datasets. Note, the $0$th dataset suffers from low data sample size, and poor interpolation of simulations as it lies at the edge of the coherence parameter's range, i.e., $\langle\cos\left(\varphi\right)\rangle\approx0.00$. Compare to Fig. \ref{['fig:Figure_4']}.a.