Quantum simulators provide a powerful tool for investigating the complex physics of quantum many-body systems, both statically and dynamically. However, current read-out techniques often limit the potential of these simulators. This work focuses on addressing this limitation, particularly in one-dimensional (1D) superfluids where the dynamics of equilibration and the formation of generalized Gibbs ensembles have been observed, but the lack of direct density fluctuation measurements has hindered further analysis. Previous experiments with 1D superfluids have demonstrated the observation of coherent recurrences in the dynamics of thousands of atoms. However, the inability to measure density fluctuations, in contrast to readily measured phase quadratures, restricts deeper insights into the system's behavior. The ability to measure both density and phase fluctuations would unlock a far more comprehensive understanding of the role of interactions and entanglement dynamics after a quantum quench. This limitation is not unique to 1D superfluids; read-out restrictions represent a significant bottleneck across various quantum simulation platforms. This paper proposes a novel method to overcome this challenge by exploiting coherent non-interacting dynamics induced by a global parameter quench within the system. This approach enables a genuine quantum read-out, allowing for a form of state reconstruction and opens up a new avenue for probing systems previously accessible only via incoherent, 'classical' read-out methods. The approach contrasts with typical many-body tomography, which is often restricted by the large number of required observables and the complexity of control. This work leverages the fact that the dynamics of many large systems can be well-approximated by an effective free field theory, where long-lived modes dominate the dynamics. The method then reconstructs relevant correlation functions by analyzing these long-lived modes' dynamics. This is achieved by demonstrating how non-equilibrium evolution mixes non-commuting operators (quadratures) in such a way that observed correlations provide information about unobserved ones, allowing for their quantitative reconstruction. Inspired by tomographic techniques in quantum optics, the method presented here extends the basic concept to a genuine multi-mode setting and demonstrates its applicability to 1D superfluids. It involves acquiring data at multiple times for numerous modes simultaneously and using semi-definite programming for noise-resilient reconstruction.

The authors reference a number of prior works exploring quantum simulations with ultra-cold atoms and ions (Bloch et al., 2012; Cirac & Zoller, 2012; Blatt & Roos, 2012). Studies on non-equilibrium coherence dynamics in 1D Bose gases (Hofferberth et al., 2007), generalized Gibbs ensembles (Langen et al., 2015), and higher-order correlations (Schweigler et al., 2017) are cited to establish the context for their work. The quantum gas microscope's impact on studying physical phenomena (Endres et al., 2011; Trotzky et al., 2012; Ronzheimer et al., 2013; Schreiber et al., 2015; Cheneau et al., 2012; Kaufman et al., 2016) and related theoretical work on quantum state reconstruction using random or deterministic unitary dynamics (Merkel et al., 2010; Ohliger et al., 2013; Elben et al., 2018; Barthel & Lu, 2018; Hauke et al., 2014; Ardila et al., 2018) are also discussed. The theoretical basis of 1D superfluids, including the Luttinger model and elementary excitations (Cazalilla, 2004; Mora & Castin, 2003), is referenced to ground the experimental setup and analysis. Finally, the authors note similar developments in the context of optical lattices and topological band insulators (Hauke et al., 2014; Ardila et al., 2018; Tarnowski et al., 2019), highlighting the broader relevance of their method.

The experimental setup consists of two adjacent 1D Bose gases created using ultra-cold atoms. The low-energy relative fluctuations in phase and density are described by an effective Hamiltonian, derived from a 1D Gross-Pitaevskii (GP) equation. The Hamiltonian describes phonons as elementary density-phase excitations, satisfying bosonic commutation relations. The density profile is engineered experimentally through the trapping potential, influencing the density-density interaction strength. A numerical scheme is described for obtaining approximate eigenfunctions of the Hamiltonian for various density profiles. The excitations are confined within the finite atomic cloud, resulting in a discrete spectrum. The method then employs a tomographic approach. The experiment measures referenced correlation functions of the relative phase via matter-wave interferometry. The goal is to reconstruct the second moments of the initial state of the quadratures (phase and density fluctuations) of the lowest-lying eigenmodes. The covariance matrix of the initial state, encompassing second moments, is defined and its positive-semidefinite nature ensures the Heisenberg uncertainty principle is satisfied. The measured phase correlations are expressed in terms of the eigenmodes, utilizing a cut-off to account for higher-energy modes' negligible contributions. The time evolution of the Hamiltonian is non-mixing for different eigenmodes, which leads to a relationship between the covariance matrix at different times. The authors then frame the reconstruction as a least-squares recovery problem. Measured phase correlations at various spatial points and times are assembled into a vector. A linear map is defined which takes a trial covariance matrix and outputs predicted phase correlations, leveraging the time evolution relationship. An optimization problem, minimizing the weighted least-squares residues and maintaining positive-semidefiniteness of the covariance matrix, is solved using semi-definite programming. The solution to this convex quadratic problem provides the optimal covariance matrix. The software package CVX is used for the implementation of this optimization. Weighting is incorporated to prioritize more precise measurements. The whole framework extends beyond 1D and continuum systems, making it broadly applicable. The experimental data is from a recent experiment with recurrent dynamics. The average density profile is obtained numerically, along with the relevant eigenmode wave functions. The number of relevant wave functions is determined by experimental resolution limitations. Initially, the two gases are strongly coupled, with the initial state preparation governed by a Hamiltonian incorporating a tunnel coupling term. A quench is performed by suddenly switching off the tunnel coupling. Measurements of phase fluctuations at various times are made to reconstruct the initial state. A Gaussian convolution is used to relate measured values to theoretical continuum predictions, and a cut-off is implemented to account for experimental resolution. The initial state reconstruction is performed and a comparison of the reconstructed and measured phase correlations is done. The reconstructed covariance matrix shows diagonal blocks for phase-phase and density-density correlations, indicating good eigenmode representation and supporting the assumption of a thermal initial state. The analysis compares the diagonal elements of the covariance matrix with predictions from a thermal model, estimating the temperature and tunnel coupling in the state preparation. The reconstruction procedure is then extended to predict the system's dynamics beyond the initial observation times.

The paper's key findings revolve around the successful implementation and application of the novel quantum read-out method. The method successfully reconstructs the full initial state of the system, including both phase and density fluctuations, from measurements of phase fluctuations alone. This is a significant advancement because direct measurements of density fluctuations were previously inaccessible. The reconstructed initial state reveals that phase fluctuations are significantly smaller than density fluctuations, directly reflecting the energetic penalty on phase fluctuations during the state preparation. The near-diagonality of the reconstructed covariance matrix suggests that the system's eigenmodes before and after the quench are closely related and that the effective theory employed accurately captures the system's relevant degrees of freedom. The fitting of the reconstructed state to a thermal model provided estimates of the temperature and tunnel coupling during state preparation. The ability to predict the system's recurrent dynamics using input data from seemingly dephased times, far from the recurrence points, highlights the method's predictive power. The analysis of phonon occupation numbers shows that these numbers remain relatively constant over time, which is consistent with the non-interacting effective model. The observed damping of the oscillations in individual modes, as seen in the time-resolved central moments of phase and density fluctuations, suggests that recurrence damping results from a loss of initial quadrature squeezing within each mode rather than a change in their occupation numbers. The reconstruction also revealed subtle deviations from the assumed thermal model, possibly due to limitations in the experimental resolution or the quench process itself. The study opens doors to analyzing the influence of interactions by examining deviations from the idealized non-interacting model. The successful reproduction of the observed recurrences demonstrates the accuracy and power of the reconstruction method in capturing the underlying dynamics of the system. This capability represents a major step forward in the understanding and analysis of quantum many-body systems.

The results of this study address the long-standing challenge of limited read-out capabilities in quantum simulators. The successful reconstruction of both phase and density fluctuations from phase measurements alone validates the proposed method's effectiveness. The ability to characterize initial states, predict dynamics, and analyze phonon behavior opens new avenues for investigating fundamental quantum phenomena. The insights gained into the system's thermalization and the origin of recurrence damping contribute to a deeper understanding of equilibration processes in 1D superfluids. The generalizability of the method suggests its potential applicability to a wide range of quantum simulation platforms, boosting the capabilities of existing experimental setups. The observed deviations from the idealized model motivate further investigation into the influence of interactions and experimental imperfections. The quantitative analysis of phonon damping rates holds potential for testing and refining theoretical models of interacting bosonic systems. The work provides a powerful tool for studying quantum correlations and dynamics, facilitating deeper exploration of entanglement and other quantum information concepts within the context of quantum simulators.

This paper presents a groundbreaking quantum read-out method enabling the reconstruction of both phase and density fluctuations in 1D superfluids using only phase measurements. This achievement overcomes a major limitation in quantum simulation, providing new insights into the system's initial state, dynamics, and thermalization. The method's predictive power, demonstrated by its ability to predict recurrences from seemingly dephased data, is remarkable. The analysis of phonon occupation numbers and the observation of damped oscillations offer valuable information about interaction effects. The method's generalizability extends its potential impact beyond 1D superfluids, paving the way for enhanced understanding and control in diverse quantum simulation platforms. Future work could focus on further improving the accuracy and extending the applicability of the method to more complex systems and exploring its use in quantum information processing.

While the method demonstrated significant success, certain limitations warrant acknowledgement. The accuracy of the reconstruction is dependent on the experimental resolution and the validity of the effective Hamiltonian used. Higher-order terms beyond the effective model could lead to discrepancies between the reconstructed dynamics and the actual experimental observations, particularly at longer timescales. The influence of the finite decoupling ramp during the quench process can affect the initial state, requiring consideration in the interpretation of the results. Furthermore, the method's reliance on an approximate thermal description of the initial state might introduce uncertainties, especially when deviations from thermal equilibrium are present. The computational demands of semi-definite programming could become limiting for significantly larger systems, requiring algorithmic refinements or more efficient optimization techniques. The current implementation assumes the initial states are approximately Gaussian, which may not always be the case in real-world scenarios.