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arXiv:1807.04567 · Communications Physics

Quantum read-out for cold atomic quantum simulators

Marek Gluza, Thomas Schweigler, Bernhard Rauer, Christian Krumnow, Jörg Schmiedmayer, Jens Eisert

Published Jan 17, 2020• 27 citations
arXivHTMLPDF

Abstract

Abstract Quantum simulators allow to explore static and dynamical properties of otherwise intractable quantum many-body systems. In many instances, however, the read-out limits such quantum simulations. In this work, we introduce an innovative experimental read-out exploiting coherent non-interacting dynamics. Specifically, we present a tomographic recovery method allowing to indirectly measure the second moments of the relative density fluctuations between two one-dimensional superfluids, which until now eluded direct measurements. Applying methods from signal processing, we show that we can reconstruct the relative density fluctuations from non-equilibrium data of the relative phase fluctuations. We employ the method to investigate equilibrium states, the dynamics of phonon occupation numbers and even to predict recurrences. The method opens a new window for quantum simulations with one-dimensional superfluids, enabling a deeper analysis of their equilibration and thermalization dynamics.

DOI
10.1038/s42005-019-0273-y

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Abstract Quantum simulators allow to explore static and dynamical properties of otherwise intractable quantum many-body systems. In many instances, however, the read-out limits such quantum simulations. In this work, we introduce an innovative experimental read-out exploiting coherent non-interacting dynamics. Specifically, we present a tomographic recovery method allowing to indirectly measure the second moments of the relative density fluctuations between two one-dimensional superfluids, which until now eluded direct measurements. Applying methods from signal processing, we show that we can reconstruct the relative density fluctuations from non-equilibrium data of the relative phase fluctuations. We employ the method to investigate equilibrium states, the dynamics of phonon occupation numbers and even to predict recurrences. The method opens a new window for quantum simulations with one-dimensional superfluids, enabling a deeper analysis of their equilibration and thermalization dynamics.
This thesis summarizes a course of investigation of various aspects of non-equilibrium dynamics in isolated quantum systems which can be controlled to the extent that one can speak of not just realizing but rather simulating a desired physical effect. The first subject considered concerns a general question of relaxation in a large class of physical models. It is rigorously proven that equilibration can occur for arbitrary local observables despite the entire system being perfectly isolated. Various mechanisms responsible for convergence to local equilibrium are highlighted. These involve in particular the memory loss of non-Gaussian correlations following an interaction quench, a notion of Gaussian ergodicity and a proof of the emergence of translation invariance of correlations due to the presence of this symmetry in the Hamiltonian governing the evolution. These results provide the long time and large system size asymptotics facilitating a thermodynamic limit, but at the same time are relevant for state-of-the-art quantum simulation experiments with large numbers of ultra-cold atoms: A related effect has been observed in a one-dimensional phononic quantum field simulator and additionally a method is provided to study relaxation dynamics of this type in optical lattice quantum simulators. Within the second theme explored in this thesis a novel quantum read-out method is proposed and applied in continuous field quantum simulators which allowed for the first time to measure experimentally various thermodynamical properties of one-dimensional quasi-condensates. In particular, tomographic results concerning thermal properties, non-commuting observables, momentum and time-resolved occupation numbers of phonons are presented. Finally, ideas for practical benchmarking of the dynamics of certain closed quantum systems are put forward, based on the concept of a fidelity witness. It is demonstrated that fidelity, despite being a sensitive measure for large systems, can be efficiently estimated for non-equilibrium dynamics in coherent quantum simulators implementing paradigmatic models of condensed matter physics. The method developed has already found an independent application in studies of variational quantum circuits aiming at achieving so-called quantum chemistry accuracy using the Sycamore quantum processor. The fact that all three themes of research laid out in this thesis have found an experimental realization hints at a prognosis for future developments in physics that it will become standard that quantum simulators will realize experimentally novel theoretical ideas on demand and the time between theoretical insights and experimental observations will be dramatically shortened.
Quantum simulations with ultracold atoms in optical lattices open up an exciting path toward understanding strongly interacting quantum systems. Atom gas microscopes are crucial for this as they offer single-site density resolution, unparalleled in other quantum many-body systems. However, currently a direct measurement of local coherent currents is out of reach. In this Letter, we show how to achieve that by measuring densities that are altered in response to quenches to noninteracting dynamics, e.g., after tilting the optical lattice. For this, we establish a data analysis method solving the closed set of equations relating tunneling currents and atom number dynamics, allowing us to reliably recover the full covariance matrix, including off-diagonal terms representing coherent currents. The signal processing builds upon semidefinite optimization, providing bona fide covariance matrices optimally matching the observed data. We demonstrate how the obtained information about noncommuting observables allows one to quantify entanglement at finite temperature, which opens up the possibility to study quantum correlations in quantum simulations going beyond classical capabilities.
Interference upon free expansion gives access to the relative phase between two interfering matter waves. Such measurements can be used to reconstruct the spatially-resolved relative phase, which is a key observable in many quantum simulations of quantum field theory and non-equilibrium experiments. However, in 1D systems, longitudinal dynamics is typically ignored in the analysis of experimental data. In our work, we give a detailed account of various effects and corrections that occur in finite temperatures due to longitudinal expansion. We provide an analytical formula showing a correction to the readout of the relative phase due to longitudinal expansion and mixing with the common phase. Furthermore, we numerically assess the error propagation to the estimation of the gases’ physical quantities such as correlation functions and temperature. We also incorporate systematic errors arising from experimental imaging devices. Our work characterizes the reliability and robustness of interferometric measurements, directing us to the improvement of existing phase extraction methods necessary to observe new physical phenomena in cold-atomic quantum simulators.
Quantum state tomography provides a powerful and versatile framework for characterizing quantum states across diverse quantum systems. However, tomography is still largely restricted to systems with a few degrees of freedom, while for continuous systems, it is typically tailored to quantum-optical platforms. For other many-body systems described by continuous quantum fields such as ultracold atomic gases, tomographic technique is not as well-established. The present thesis addresses this gap by providing new approaches to measurement and tomography of ultracold Bose gases trapped in one (1D) and two dimensions (2D). These systems serve as analog quantum field theory (QFT) simulators and exhibit novel physics absent in higher dimensions. We begin by systematically assessing state-of-the-art method for extracting the relative phase field in parallel 1D Bose gases with matter-wave interference. In particular, we derive a formula for the systematic phase shift due to longitudinal dynamics, a factor that is often ignored in previous modeling of the measurement. We further examine how the single-shot readout error propagates to correlation and distribution functions of the relative phase. Then, we broaden the scope of experimentally accessible quantities by developing a new methodology to extract the common phase field---the sum of the phase fields of the two parallel gases---from the already available density ripple data. We rigorously benchmark this new readout with numerical simulations and experimental data in 1D systems and further generalize it to 2D systems. By combining the relative and common phases information, we improve Gaussian state tomography in parallel 1D Bose gases and apply it to probe new physical phenomena in quantum field simulators. Specifically, we focus on two applications: (i) field-space entanglement detection at finite temperatures and out of equilibrium, and (ii) probing non-linear relaxation in near-integrable quantum systems. This thesis bridges theory and experiment in the simulation of QFTs and reveals new opportunities for exploring many-body quantum effects in ultracold bosonic gases.
Studying the dynamics of quantum many-body systems is often constrained by the limitations in probing relevant observables, especially in continuous systems. A powerful method to gain information about such systems is the reconstruction of local currents from the continuity equation. We show that this approach can be used to extract the total phase fluctuation of adjacent Bose gases. We validate our technique numerically and demonstrate its effectiveness by analyzing data from selected experiments simulating 1D quantum field theories through the phase difference of two parallel 1D Bose gases. This analysis reveals the previously hidden sector of the sum mode of the phase, which is important for studying long-time thermalization and out-of-equilibrium dynamics of the system. Our method is general and can be applied to other cold atom systems with spatial phase gradients, thereby expanding the scope and capabilities of cold-atomic quantum simulators.
Measuring fluctuations in matter's low-energy excitations is the key to unveiling the nature of the non-equilibrium response of materials. A promising outlook in this respect is offered by spectroscopic methods that address matter fluctuations by exploiting the statistical nature of light-matter interactions with weak few-photon probes. Here we report the first implementation of ultrafast phase randomized tomography, combining pump-probe experiments with quantum optical state tomography, to measure the ultrafast non-equilibrium dynamics in complex materials. Our approach utilizes a time-resolved multimode heterodyne detection scheme with phase-randomized coherent ultrashort laser pulses, overcoming the limitations of phase-stable configurations and enabling a robust reconstruction of the statistical distribution of phase-averaged optical observables. This methodology is validated by measuring the coherent phonon response in α-quartz. By tracking the dynamics of the shot-noise limited photon number distribution of few-photon probes with ultrafast resolution, our results set an upper limit to the non-classical features of phononic state in α-quartz and provide a pathway to access non-equilibrium quantum fluctuations in more complex quantum materials.