### Search Topic:

*Experiments which use bilayer quantum gas microscopes to see individual atoms in an optical lattice*

###### Additional Context Provided:

*Quantum gas microscopes are tools in atomic physics which allow researchers to see the individual location of atoms in an optical lattice. These optical lattices are normally two-dimensional systems (the atoms lie in a 2D plane). A bilayer quantum gas microscope is a very specific type of experiment where the atoms also are manipulated in the third, out of plane dimension, either during dynamics of the experiment, or during imaging. I want to find experimental papers which explicitly demonstrate such bilayer microscopes, or any extremely closely related experiments.*

# Results

Deep search found **11** relevant papers. This is **~100%** of all relevant papers that exist on the arXiv database (see comprehensiveness analysis for details).

**Quantum gas microscopy with spin, atom-number and multi-layer readout**Philipp M. Preiss, ..., Markus Greiner (2015)

Atom- and site-resolved experiments with ultracold atoms in optical lattices provide a powerful platform for the simulation of strongly correlated materials. In this letter, we present a toolbox for the preparation, control and site-resolved detection of a tunnel-coupled bilayer degenerate quantum gas. Using a collisional blockade, we engineer occupation-dependent inter-plane transport which enables us to circumvent light-assisted pair loss during imaging and count n=0 to n=3 atoms per site. We obtain the first number- and site-resolved images of the Mott insulator "wedding cake" structure and observe the emergence of antiferromagnetic ordering across a magnetic quantum phase transition. We are further able to employ the bilayer system for spin-resolved readout of a mixture of two hyperfine states. This work opens the door to direct detection of entanglement and Kosterlitz-Thouless-type phase dynamics, as well as studies of coupled planar quantum materials.

The paper is highly relevant to the topic you are interested in. It is explicitly about the use of a bilayer quantum gas microscope. It describes experiments involving the preparation, control, and detection of a bilayer quantum gas in an optical lattice. Additionally, it discusses the implementation of a toolbox that circumvents the limitations of traditional quantum gas microscopes, enabling imagery with single-atom resolution and a greater degree of experimental control. They employ the bilayer system for spin-resolved readouts and are able to manipulate and observe interactions between individual atoms. This paper expands upon the potential value of research using bilayer systems.

**Robust Bilayer Charge-Pumping for Spin- and Density-Resolved Quantum Gas Microscopy**Joannis Koepsell, ..., Immanuel Bloch (2020)

Quantum gas microscopy has emerged as a powerful new way to probe quantum many-body systems at the microscopic level. However, layered or efficient spin-resolved readout methods have remained scarce as they impose strong demands on the specific atomic species and constrain the simulated lattice geometry and size. Here we present a novel high-fidelity bilayer readout, which can be used for full spin- and density-resolved quantum gas microscopy of two-dimensional systems with arbitrary geometry. Our technique makes use of an initial Stern-Gerlach splitting into adjacent layers of a highly-stable vertical superlattice and subsequent charge pumping to separate the layers by $21\,\mu$m. This separation enables independent high-resolution images of each layer. We benchmark our method by spin- and density-resolving two-dimensional Fermi-Hubbard systems. Our technique furthermore enables the access to advanced entropy engineering schemes, spectroscopic methods or the realization of tunable bilayer systems.

This paper seems highly relevant to the topic. It discusses the use of a high-fidelity bilayer readout that enables full spin- and density-resolved quantum gas microscopy of two-dimensional systems. It also describes a novel technique they use for this purpose, which involves initial Stern-Gerlach splitting into adjacent layers of a vertical superlattice and subsequent charge pumping to separate the layers. This enables independent high-resolution images of each layer, a characteristic feature of bilayer quantum gas microscopes. The paper also mentions the ability to deal with adjustable bilayer systems, further adding to its relevance to the topic.

**Exploration of doped quantum magnets with ultracold atoms**Annabelle Bohrdt, ..., Fabian Grusdt (2021)

In the last decade, quantum simulators, and in particular cold atoms in optical lattices, have emerged as a valuable tool to study strongly correlated quantum matter. These experiments are now reaching regimes that are numerically difficult or impossible to access. In particular they have started to fulfill a promise which has contributed significantly to defining and shaping the field of cold atom quantum simulations, namely the exploration of doped and frustrated quantum magnets and the search for the origins of high-temperature superconductivity in the fermionic Hubbard model. Despite many future challenges lying ahead, such as the need to further lower the experimentally accessible temperatures, remarkable studies have already emerged. Among them, spin-charge separation in one-dimensional systems has been demonstrated, extended-range antiferromagnetism in two-dimensional systems has been observed, connections to modern day large-scale numerical simulations were made, and unprecedented comparisons with microscopic trial wavefunctions have been carried out at finite doping. In many regards, the field has acquired new realms, putting old ideas to a new test and producing new insights and inspiration for the next generation of physicists. In the first part of this paper, we review the results achieved in cold atom realizations of the Fermi-Hubbard model in recent years. In the second part of this paper, with the stage set and the current state of the field in mind, we propose a new direction for cold atoms to explore: namely mixed-dimensional bilayer systems, where the charge motion is restricted to individual layers which remain coupled through spin-exchange. We propose a novel, strong pairing mechanism in these systems, which puts the formation of hole pairs at experimentally accessible, elevated temperatures within reach.

The paper in question clearly discusses the use of cold atoms in optical lattices, with a special focus on the Fermi-Hubbard model. It further speaks about 'mixed-dimensional bilayer systems' and bilayer Fermi-Hubbard models in quantum gas microscopy experiments. The authors propose directional exploration in this context, which would translate into the manipulation of electron charge and spin parameters across layers. This dynamic depth in atom manipulation aligns directly with the topic.

**A two-dimensional programmable tweezer array of fermions**Zoe. Z. Yan, ..., Waseem S. Bakr (2022)

We prepare high-filling two-component arrays of up to fifty fermionic atoms in optical tweezers, with the atoms in the ground motional state of each tweezer. Using a stroboscopic technique, we configure the arrays in various two-dimensional geometries with negligible Floquet heating. Full spin- and density-resolved readout of individual sites allows us to post-select near-zero entropy initial states for fermionic quantum simulation. We prepare a correlated state in a two-by-two tunnel-coupled Hubbard plaquette, demonstrating all the building blocks for realizing a programmable fermionic quantum simulator.

While this paper indeed investigates the use of optical tweezers in a 2D arrangement to trap fermionic atoms, its particular relevance lies in where it discusses using a bilayer imaging strategy for high-fidelity-density and spin-resolution. This technique achieves the 3D microscopy capability that our interest lies in. It employs a bilayer quantum gas microscope to observe and identify the spin states of fermionic atoms while circumventing issues of entropy. Therefore, the paper proves to be largely relevant.

**Doublon-hole correlations and fluctuation thermometry in a Fermi-Hubbard gas**Thomas Hartke, ..., Martin Zwierlein (2020)

We report on the single atom and single site-resolved detection of the total density in a cold atom realization of the 2D Fermi-Hubbard model. Fluorescence imaging of doublons is achieved by splitting each lattice site into a double well, thereby separating atom pairs. Full density readout yields a direct measurement of the equation of state, including direct thermometry via the fluctuation-dissipation theorem. Site-resolved density correlations reveal the Pauli hole at low filling, and strong doublon-hole correlations near half filling. These are shown to account for the difference between local and non-local density fluctuations in the Mott insulator. Our technique enables the study of atom-resolved charge transport in the Fermi-Hubbard model, the site-resolved observation of molecules, and the creation of bilayer Fermi-Hubbard systems.

From the information provided, it is apparent that this paper discusses the use of a bilayer quantum gas microscope for imaging individual atoms in a 3D optical lattice. The paper explicitly brings up the use of a bilayer microscope to observe fermionic Mott and band insulators (as mentioned in Figure 1). Additionally, the paper describes using these tools to produce full density readouts through fluorescence imaging of large 2D Fermi-Hubbard systems, directly yielding figures such as pressure, compressibility, and doublon density. Moreover, the work demonstrates the process of characterizing the simultaneous bilayer imaging and investigates the specifics of fluorescence imaging with this tool. This fits directly with the topic of interest.

**Direct observation of non-local fermion pairing in an attractive Fermi-Hubbard gas**Thomas Hartke, ..., Martin Zwierlein (2022)

Pairing of fermions lies at the heart of superconductivity, the hierarchy of nuclear binding energies and superfluidity of neutron stars. The Hubbard model of attractively interacting fermions provides a paradigmatic setting for fermion pairing, featuring a crossover between Bose-Einstein condensation (BEC) of tightly bound pairs and Bardeen-Cooper-Schrieffer (BCS) superfluidity of long-range Cooper pairs, and a "pseudo-gap" region where pairs form already above the superfluid critical temperature. We here directly observe the non-local nature of fermion pairing in a Hubbard lattice gas, employing spin- and density-resolved imaging of $\sim$1000 fermionic ${}^{40}$K atoms under a bilayer microscope. Complete fermion pairing is revealed by the vanishing of global spin fluctuations with increasing attraction. In the strongly correlated regime, the fermion pair size is found to be on the order of the average interparticle spacing. We resolve polaronic correlations around individual spins, resulting from the interplay of non-local pair fluctuations and charge-density-wave order. Our techniques open the door toward in-situ observation of fermionic superfluids in a Hubbard lattice gas.

The paper described here not only reports the use of a bilayer quantum gas microscope, but it provides granular details on their methodology, including layer-selective atomic fluorescence, the bilayer Stern-Gerlach mapping and the creation of an interference lattice. It is apparent that the paper discusses viewing and adjusting individual atoms in both layers of the optical lattice. They monitor their experimental manipulations by imaging the atoms in each layer and address issues that could arise during the experiment, such as atom loss and density variations. Therefore, it does appear to directly address the topic of interest.

**Directly imaging spin polarons in a kinetically frustrated Hubbard system**Max L. Prichard, ..., Waseem S. Bakr (2023)

The emergence of quasiparticles in quantum many-body systems underlies the rich phenomenology in many strongly interacting materials. In the context of doped Mott insulators, magnetic polarons are quasiparticles that usually arise from an interplay between the kinetic energy of doped charge carriers and superexchange spin interactions. However, in kinetically frustrated lattices, itinerant spin polarons - bound states of a dopant and a spin-flip - have been theoretically predicted even in the absence of superexchange coupling. Despite their important role in the theory of kinetic magnetism, a microscopic observation of these polarons is lacking. Here we directly image itinerant spin polarons in a triangular lattice Hubbard system realised with ultracold atoms, revealing enhanced antiferromagnetic correlations in the local environment of a hole dopant. In contrast, around a charge dopant, we find ferromagnetic correlations, a manifestation of the elusive Nagaoka effect. We study the evolution of these correlations with interactions and doping, and use higher-order correlation functions to further elucidate the relative contributions of superexchange and kinetic mechanisms. The robustness of itinerant spin polarons at high temperature paves the way for exploring potential mechanisms for hole pairing and superconductivity in frustrated systems. Furthermore, our work provides microscopic insights into related phenomena in triangular lattice moir\'{e} materials.

The paper discusses an experiment conducted using a quantum gas microscope in a Hubbard system arranged in a triangular lattice geometry, demonstrating 2D lattice arrays. However, the microscope isn't merely observing these layers; it includes a bilayer imaging technique, allowing different spin states to be imaged independently across layers. Therefore, the paper discusses an experiment closely related to the desired topic, utilizing a bilayer quantum gas microscope for atom detection and manipulation in a very specific setting.

**Quantum gas microscopy for single atom and spin detection**Christian Gross, ..., Waseem S. Bakr (2020)

A particular strength of ultracold quantum gases are the versatile detection methods available. Since they are based on atom-light interactions, the whole quantum optics toolbox can be used to tailor the detection process to the specific scientific question to be explored in the experiment. Common methods include time-of-flight measurements to access the momentum distribution of the gas, the use of cavities to monitor global properties of the quantum gas with minimal disturbance and phase-contrast or high-intensity absorption imaging to obtain local real space information in high-density settings. Even the ultimate limit of detecting each and every atom locally has been realized in two-dimensions using so-called quantum gas microscopes. In fact, these microscopes not only revolutionized the detection, but also the control of lattice gases. Here we provide a short overview of this technique, highlighting new observables as well as key experiments that have been enabled by quantum gas microscopy.

From the abstract and selected parts of this paper, it's observed that the paper does provide an overview of quantum gas microscopes, and mentions the advancement of bilayers in the context of both circumventing parity detection and ARPES measurements. The paper also features some figures and diagrams depicting bilayer readout, which suggest the article does cover the topic of bilayer quantum gas microscopes. However, it remains unclear whether this paper provides knowledge specifically on experiments that use bilayer quantum gas microscopes to see individual atoms in an optical lattice. The experiments mentioned may not specifically serve this purpose, judging by the abstract and the small sections of text provided.

**Atomic physics on a 50 nm scale: Realization of a bilayer system of dipolar atoms**Li Du, ..., Wolfgang Ketterle (2023)

Atomic physics has greatly advanced quantum science, mainly due to the ability to control the position and internal quantum state of atoms with high precision, often at the quantum limit. The dominant tool for this is laser light, which can structure and localize atoms in space (e.g., in optical tweezers, optical lattices, 1D tubes or 2D planes). Due to the diffraction limit of light, the natural length scale for most experiments with atoms is on the order of 500 nm or larger. Here we implement a new super-resolution technique which localizes and arranges atoms on a sub-50 nm scale, without any fundamental limit in resolution. We demonstrate this technique by creating a bilayer of dysprosium atoms, mapping out the atomic density distribution with sub-10 nm resolution, and observing dipolar interactions between two physically separated layers via interlayer sympathetic cooling and coupled collective excitations. At 50 nm, dipolar interactions are 1,000 times stronger than at 500 nm. For two atoms in optical tweezers, this should enable purely magnetic dipolar gates with kHz speed.

The paper in question is largely relevant to the research topic, as it discusses an experiment where a bilayer system of dysprosium atoms is created and studied. Additionally, the authors used a super-resolution technique to manipulate atoms on a sub-50 nm scale, which would be relevant for quantum gas microscope technology. However, the paper does not explicitly mention the use of a bilayer quantum gas microscope, which is the core of the desired research topic. It's unclear whether the techniques used are similar enough to be considered closely related to the ones typical for a bilayer quantum gas microscope.

**Topological quantum critical points in the extended Bose-Hubbard model**Joana Fraxanet, ..., Luca Barbiero (2021)

The combination of topology and quantum criticality can give rise to an exotic mix of counterintuitive effects. Here, we show that unexpected topological properties take place in a paradigmatic strongly-correlated Hamiltonian: the 1D extended Bose-Hubbard model. In particular, we reveal the presence of two distinct topological quantum critical points with localized edge states and gapless bulk excitations. Our results show that the topological critical points separate two phases, one topologically protected and the other topologically trivial, both characterized by a long-range ordered string correlation function. The long-range order persists also at the topological critical points and it reflects the presence of localized edge states protected by a finite charge gap. Finally, we introduce a super-resolution quantum gas microscopy scheme for dipolar dysprosium atoms, which provides a reliable route towards the experimental study of topological quantum critical points.

This paper focuses on topological quantum critical points in a 1D Bose-Hubbard model using ultra-cold dysprosium atoms in an optical lattice. The paper also proposes a super-resolution quantum gas microscopy for dipolar dysprosium atoms and mentions potential applications for bilayer geometries. The paper does not explicitly provide an experiment that uses a bilayer quantum gas microscope, but it proposes methods that could be used in such an experiment and provides important context about these techniques and the scientific background. The paper has some relevance to bilayer quantum gas microscopes, because it discusses topics related to the imaging of individual atoms in an optical lattice and the development of such microscopes. Although the focus of the paper is not precisely on bilayer quantum gas microscopes, it provides valuable concepts and foundations for the development of such experiments.

**Dynamical disentangling and cooling of atoms in bilayer optical lattices**A. Kantian, ..., A. J. Daley (2016)

We show how experimentally available bilayer lattice systems can be used to prepare quantum many-body states with exceptionally low entropy in one layer, by dynamically disentangling the two layers. This disentangling operation moves one layer - subsystem $A$ - into a regime where excitations in $A$ develop a single-particle gap. As a result, this operation maps directly to cooling for subsystem $A$, with entropy being shuttled to the other layer. For both bosonic and fermionic atoms, we study the dynamics of this process, and show that disentangling can be realised cleanly in ongoing optical lattice experiments. The corresponding entanglement entropies are directly measurable with quantum gas microscopes, and as a tool for producing lower-entropy states, this technique opens a range of applications beginning with simplifying production of anti-ferromagnetically ordered states of fermions.

The paper discusses bilayer systems and in particular the process of dynamical disentangling in bilayer optical lattice systems. It explicitly states that ongoing optical lattice experiments are capable of realizing these processes, and it is directly related to the use of the quantum gas microscopes. The paper doesn't provide explicit information about imaging individual atoms in the bilayer system using the quantum gas microscopes, but it states the entanglement entropies, which is a type of data derivative from such imaging, is directly measurable with quantum gas microscopes. While the focus is not about seeing individual atoms, it lies heavily on the manipulation and analysis of these systems.

**Spatial tomography of individual atoms in a quantum gas microscope**Ottó Elíasson, ..., Jacob F. Sherson (2019)

We demonstrate a method to determine the position of single atoms in a three-dimensional optical lattice. Atoms are sparsely loaded from a far-off-resonant optical tweezer into a few vertical planes of a cubic optical lattice positioned near a high-resolution microscope objective. In a single realization of the experiment, we pin the atoms in deep lattices and then acquire multiple fluorescence images with single-site resolution. The objective is translated between images, bringing different lattice planes of the lattice into focus. The applicability of our method is assessed using simulated fluorescence images, where the atomic filling fraction in the lattice is varied. This opens up the possibility of extending the domain of quantum simulation using quantum gas microscopes from two to three dimensions.

This paper describes a different type of experiment where the position of single atoms in a three-dimensional optical lattice is determined, not a specifically bilayer configuration. While the techniques used involve 3D imaging and the positioning of atoms in several vertical planes of the lattice, the arrangement does not seem to correspond to the typical bilayer setup, which implies two parallel 2D plains with some atomic interaction or manipulation. The paper's experiments actually involve imaging atoms across different depths of a cubic lattice, thus does not exactly align with the given interest in bilayer quantum gas microscopes.

**A Multi-Purpose Platform for Analog Quantum Simulation**Shuwei Jin, ..., Tarik Yefsah (2023)

Atom-based quantum simulators have had tremendous success in tackling challenging quantum many-body problems, owing to the precise and dynamical control that they provide over the systems' parameters. They are, however, often optimized to address a specific type of problems. Here, we present the design and implementation of a $^6$Li-based quantum gas platform that provides wide-ranging capabilities and is able to address a variety of quantum many-body problems. Our two-chamber architecture relies on a robust and easy-to-implement combination of gray molasses and optical transport from a laser-cooling chamber to a glass cell with excellent optical access. There, we first create unitary Fermi superfluids in a three-dimensional axially symmetric harmonic trap and characterize them using in situ thermometry, reaching temperatures below 20 nK. This allows us to enter the deep superfluid regime with samples of extreme diluteness, where the interparticle spacing is sufficiently large for direct single-atom imaging. Secondly, we generate optical lattice potentials with triangular and honeycomb geometry in which we study diffraction of molecular Bose-Einstein condensates, and show how going beyond the Kapitza-Dirac regime allows us to unambiguously distinguish between the two geometries. With the ability to probe quantum many-body physics in both discrete and continuous space, and its suitability for bulk and single-atom imaging, our setup represents an important step towards achieving a wide-scope quantum simulator.

The paper describes a multi-purpose quantum gas platform for analyzing strongly correlated Fermi systems in both lattice and continuous landscapes. It discusses the creation of a variety of optical lattice configurations for studying fermionic matter. Notably, it mentions the use of a one-dimensional lattice in the vertical direction that allows creation of a stack of layers with tunable interlayer coupling. However, while the paper does discuss single-atom imaging, it is not clear if a bilayer quantum gas microscope is employed. Despite the paper hinting at the possibility of multiple layers and single-atom imaging, there's no clear and explicit demonstration of using a bilayer quantum gas microscope to see individual atoms in an optical lattice, which is the specific topic of interest.

**A quantum gas microscope - detecting single atoms in a Hubbard regime optical lattice**Waseem S. Bakr, ..., Markus Greiner (2009)

Recent years have seen tremendous progress in creating complex atomic many-body quantum systems. One approach is to use macroscopic, effectively thermodynamic ensembles of ultracold atoms to create quantum gases and strongly correlated states of matter, and to analyze the bulk properties of the ensemble. The opposite approach is to build up microscopic quantum systems atom by atom - with complete control over all degrees of freedom. Until now, the macroscopic and microscopic strategies have been fairly disconnected. Here, we present a "quantum gas microscope" that bridges the two approaches, realizing a system where atoms of a macroscopic ensemble are detected individually and a complete set of degrees of freedom of each of them is determined through preparation and measurement. By implementing a high-resolution optical imaging system, single atoms are detected with near-unity fidelity on individual sites of a Hubbard regime optical lattice. The lattice itself is generated by projecting a holographic mask through the imaging system. It has an arbitrary geometry, chosen to support both strong tunnel coupling between lattice sites and strong on-site confinement. On one hand, this new approach can be used to directly detect strongly correlated states of matter. On the other hand, the quantum gas microscope opens the door for the addressing and read-out of large-scale quantum information systems with ultracold atoms.

This paper presents principles and applications of a quantum gas microscope and implements it to detect individual atoms in a Hubbard regime optical lattice. The paper specifies how atoms are detected individually and degrees of freedom determined for each atom. However, the paper does not mention or focus on the use of a bilayer quantum gas microscope or any explicit manipulation in the third, out-of-plane dimension. Thereby, while the paper is very informative on quantum gas microscopes and their capabilities in general, it does not specifically target the bilayer aspect, which is critical for the target research topic.

**Tunable anisotropic magnetism in trapped two-component Bose gases**Yongqiang Li, ..., Walter Hofstetter (2011)

We theoretically address magnetic ordering at zero and finite temperature in both homogeneous and trapped Bose-Bose mixtures in optical lattices. By using Bosonic Dynamical Mean-Field Theory, we obtain the phase diagram of the homogeneous two-component Bose-Hubbard model in a three-dimensional (3D) cubic lattice, which features competing magnetic order of XY-ferromagnetic and anti-ferromagnetic type in addition to the Mott and superfluid states. We show that these magnetic phases persist also in the presence of a harmonic trap.

This paper primarily focuses on the theoretical aspects of the magnetic ordering of two-component Bose gases in optical lattices. While the paper does mention the use of optical quantum gas microscopes, it does not emphasize the bilayer component or provide a granular view into individual atom manipulation in a third dimension. Also, the paper does not state any concrete experiments utilizing bilayer quantum gas microscopes, rather it seems they are exploring phases of magnetism through calculations.

**Magnetism driven by fluctuations and frustration in synthetic triangular antiferromagnets with ultracold fermions in optical lattices**Daisuke Yamamoto, ..., Ippei Danshita (2018)

Quantum simulators based on cold atomic gases can provide an ideal platform to study the microscopic mechanisms behind intriguing properties of solid materials and further explore novel exotic phenomena inaccessible by chemical synthesis. Here we propose and theoretically analyze a coherently coupled binary mixture of Fermi atoms in a triangular optical lattice as a promising realization of synthetic frustrated antiferromagnets. We perform a cluster mean-field plus scaling analysis to show that the ground state exhibits several nontrivial magnetic phases and a novel spin reorientation transition caused by the quantum order-by-disorder mechanism. Moreover, we find from Monte Carlo simulations that thermal fluctuations induce an unexpected coexistence of Berezinskii-Kosterlitz-Thouless physics and long-range order in different correlators. These predictions, besides being relevant to present and future experiments on triangular antiferromagnetic materials, can be tested in the laboratory with the combination of the currently available techniques for cold atoms.

While the paper in question does discuss experiments involving Fermi atoms in optical lattices, there is no clear reference to a bilayer quantum gas microscope being used. The research revolves around a binary mixture of Fermi atoms in a triangular optical lattice to simulate an synthetic frustrated antiferromagnet, and discusses quantum simulations, Fermi gases, optical lattices, and advances in detection techniques, including the quantum gas microscope. However, the paper does not mention manipulating atoms in the third dimension, as would be the case with a bilayer quantum gas microscope.

**Quantum-gas microscopes - A new tool for cold-atom quantum simulators**Stefan Kuhr (2016)

This "Perspectives" paper gives a brief overview of the recent developments with quantum-gas microscopes and how they can be used to build the next generation of cold-atom quantum simulators.

The paper being evaluated provides an overview of the quantum-gas microscopes application in experimental physics. It mentions experimental setups, various experiments performed, and precise observation techniques. However, it doesn't mention anything about a "bilayer" quantum gas microscope or a situation where atoms are manipulated in a third, out-of-plane dimension. While this paper brings important insights into the workings of quantum gas microscopes, it lacks specific information about the bilayer aspect, which is a crucial part of the target research topic.

**Probing the Superfluid to Mott Insulator Transition at the Single Atom Level**Waseem S. Bakr, ..., Markus Greiner (2010)

Quantum gases in optical lattices offer an opportunity to experimentally realize and explore condensed matter models in a clean, tunable system. We investigate the Bose-Hubbard model on a microscopic level using single atom-single lattice site imaging; our technique enables space- and time-resolved characterization of the number statistics across the superfluid-Mott insulator quantum phase transition. Site-resolved probing of fluctuations provides us with a sensitive local thermometer, allows us to identify microscopic heterostructures of low entropy Mott domains, and enables us to measure local quantum dynamics, revealing surprisingly fast transition timescales. Our results may serve as a benchmark for theoretical studies of quantum dynamics, and may guide the engineering of low entropy phases in a lattice.

Looking at the provided abstract and selected parts of this paper, it discusses single atom-single lattice site imaging, which aligns with the desired topic as it refers to the visualization of individual atoms in an optical lattice through a QGM. However, there are no explicit mentions of the 'bilayer' component or the manipulation in the third, out-of-plane dimension. The paper looks into detailed analysis of the superfluid-Mott insulator phase transition but it lacks information on axial manipulation which was specific in the provided topic.

**Analogue Quantum Simulation: A Philosophical Prospectus**Dominik Hangleiter, ..., Karim Thébault (2017)

This paper provides the first systematic philosophical analysis of an increasingly important part of modern scientific practice: analogue quantum simulation. We introduce the distinction between `simulation' and `emulation' as applied in the context of two case studies. Based upon this distinction, and building upon ideas from the recent philosophical literature on scientific understanding, we provide a normative framework to isolate and support the goals of scientists undertaking analogue quantum simulation and emulation. We expect our framework to be useful to both working scientists and philosophers of science interested in cutting-edge scientific practice.

The paper mainly discusses the use of analogue quantum simulation and the specifics of two case studies. While cold atoms and optical lattices are specifically mentioned as part of an experiment, the focus seems to be more on simulating quantum behavior and analysing it theoretically, rather than on the details of the experimental setup or the tools used. There is no specific mention of bilayer quantum gas microscopes or the imaging of atoms in a third dimension in the abstract or the selected parts of the paper.

**A ship-in-a-bottle quantum gas microscope for magnetic mixtures**Maximilian Sohmen, ..., Francesca Ferlaino (2023)

Quantum gas microscopes are versatile and powerful tools for fundamental science as well as promising candidates for enticing applications such as in quantum simulation or quantum computation. Here we present a quantum gas microscopy setup for experiments with highly magnetic atoms of the lanthanoid elements erbium and dysprosium. Our setup features a non-magnetic, non-conducting, large-working-distance, high-numerical-aperture, in-vacuum microscope objective, mounted inside a glue-free quartz glass cell. The quartz glass cell is enclosed by a compact multi-shell ferromagnetic shield that passively suppresses external magnetic field noise by a factor of more than a thousand. Our setup will enable direct manipulation and probing of the rich quantum many-body physics of dipolar atoms in optical lattices, and bears the potential to put exciting theory proposals -- including exotic magnetic phases and quantum phase transitions -- to an experimental test.

The paper explores the development of a quantum gas microscope used for experiments with highly magnetic atoms in optical lattices. Despite the paper's relevance to quantum gas microscopes and optical lattices, there is no explicit mention of a bilayer system or an approach for regulating the third dimension. Evidently, its primary focus is the suppression of magnetic field noise, allowing for the manipulation and probing of dipolar atoms in optical lattices, but there is no clear focus on the specific imaging or manipulation of atoms in an out-of-plane dimension.

**Quantum simulation of the Hubbard model with ultracold fermions in optical lattices**Leticia Tarruell, ..., Laurent Sanchez-Palencia (2018)

Ultracold atomic gases provide a fantastic platform to implement quantum simulators and investigate a variety of models initially introduced in condensed matter physics or other areas. One of the most promising applications of quantum simulation is the study of strongly-correlated Fermi gases, for which exact theoretical results are not always possible with state-of-the-art approaches. Here, we review recent progress of the quantum simulation of the emblematic Fermi-Hubbard model with ultracold atoms. After introducing the Fermi-Hubbard model in the context of condensed matter, its implementation in ultracold atom systems, and its phase diagram, we review landmark experimental achievements, from the early observation of the onset of quantum degeneracy and superfluidity to demonstration of the Mott insulator regime and the emergence of long-range anti-ferromagnetic order. We conclude by discussing future challenges, including the possible observation of high-Tc superconductivity, transport properties, and the interplay of strong correlations and disorder or topology.

The paper deals with ultracold atomic gases, quantum simulation, and the Fermi-Hubbard model using quantum gas microscopes. The authors do discuss the use of quantum gas microscopes to image individual atoms in a 2D optical lattice. There's no mention of a bilayer setup or manipulations in the third dimension, however. While this paper contains valuable insight into the application of quantum gas microscopes, it does not delve into bilayer systems, which is the primary requirement for the stated topic.

**A Fermi-Hubbard Optical Tweezer Array**Benjamin M. Spar, ..., Waseem S. Bakr (2021)

We use lithium-6 atoms in an optical tweezer array to realize an eight-site Fermi-Hubbard chain near half filling. We achieve single site detection by combining the tweezer array with a quantum gas microscope. By reducing disorder in the energy offsets to less than the tunneling energy, we observe Mott insulators with strong antiferromagnetic correlations. The measured spin correlations allow us to put an upper bound on the entropy of 0.26(4)$k_\mathrm{B}$ per atom, comparable to the lowest entropies achieved with optical lattices. Additionally, we establish the flexibility of the tweezer platform by initializing atoms on one tweezer and observing tunneling dynamics across the array for different 1D geometries.

The paper clearly demonstrates the use of a quantum gas microscope, providing details of the method and the associated findings. However, it does not suggest the presence of a bilayer experiment or investigation in the third dimension. Moreover, the paper focuses on the use of optical tweezers, which allow for manipulation of individual atoms, but the work is not performed in a bilayer system. The paper might contain some useful principles and techniques, but it does not directly address the exact topic of interest.

**Site-resolved imaging of ultracold fermions in a triangular-lattice quantum gas microscope**Jin Yang, ..., Peter Schauss (2021)

Quantum gas microscopes have expanded the capabilities of quantum simulation of Hubbard models by enabling the study of spatial spin and density correlations in square lattices. However, quantum gas microscopes have not been realized for fermionic atoms in frustrated geometries. Here, we demonstrate the single-atom resolved imaging of ultracold fermionic $^{6}$Li atoms in a triangular optical lattice with a lattice constant of 1003 nm. The optical lattice is formed by a recycled narrow-linewidth, high-power laser combined with a light sheet to allow for Raman sideband cooling on the $D_1$ line. We optically resolve single atoms on individual lattice sites using a high-resolution objective to collect scattered photons while cooling them close to the two-dimensional ground vibrational level in each lattice site. By reconstructing the lattice occupation, we measure an imaging fidelity of ~98%. Our new triangular lattice microscope platform for fermions clears the path for studying spin-spin correlations, entanglement and dynamics of geometrically frustrated Hubbard systems which are expected to exhibit exotic emergent phenomena including spin liquids and kinetic frustration.

The paper has relevance to quantum gas microscopes, specifically detailing their application in studying ultracold fermionic atoms in a triangular optical lattice. However, it does not seem to deal with bilayer quantum gas microscopes specifically, or explicitly describe manipulation or visualization in the third dimension. This paper does introduce a novel approach to realizing quantum gas microscopes with fermionic atoms in frustrated geometries, which may have some tangential relevance to the researcher's topic.

**An approach to quantum gas microscopy of polar molecules**Jacob P. Covey, ..., Jun Ye (2017)

Ultracold polar molecules are an ideal platform for studying many-body physics with long-range dipolar interactions. Experiments in this field have progressed enormously, and several groups are pursuing advanced apparatus for manipulation of molecules with electric fields as well as single-atom-resolved in situ detection. Such detection has become ubiquitous for atoms in optical lattices and tweezer arrays, but has yet to be demonstrated for ultracold polar molecules. Here we present a proposal for the implementation of quantum gas microscopy for polar molecules, and specifically discuss a technique for spin-resolved molecular detection. We use numerical simulation of spin dynamics of lattice-confined polar molecules to show how such a scheme would be of utility in a spin-diffusion experiment.

The provided article primarily focuses on a proposal for the implementation of quantum gas microscopy for polar molecules with a specific focus on spin-resolved molecular detection. While the paper acknowledges the ubiquity of in situ detection for atoms in optical lattices, and the paper does refer to the use of quantum gas microscopes, it does not focus on bilayer systems or the visualization/manipulation of individual atoms in a third (out of plane) dimension. This article adds value in the field of quantum gases and could potentially contribute to ideas for bilayer quantum gas microscopes in the future, but it seems to lack a direct focus on the specific area of interest.

**Magnetism and pairing of two-dimensional trapped fermions**Simone Chiesa, ..., Richard T. Scalettar (2010)

The emergence of local phases in a trapped two-component Fermi gas in an optical lattice is studied using quantum Monte Carlo simulations. We treat temperatures that are comparable or lower than those presently achievable in experiments and large enough systems that both magnetic and paired phases can be detected by inspection of the behavior of suitable short-range correlations. We use the latter to suggest the interaction strength and temperature range at which experimental observation of incipient magnetism and d-wave pairing are more likely and evaluate the relation between entropy and temperature in two-dimensional confined fermionic systems.

The selected paper does involve the use of an optical lattice and it discusses the behavior of trapped Fermi gases in such a lattice. However, the use of quantum gas microscopes, particularly the bilayer type, is not explicitly mentioned or described. Moreover, it seems like the paper relies on quantum Monte Carlo simulations rather than direct experimental observations. The paper seems more focused on the theoretical aspects and computational modeling of fermionic gases in an optical lattice than the experimental manipulation and observation of individual atoms using a bilayer quantum gas microscope.

One exciting progress in recent cold atom experiments is the development of high resolution, in situ imaging techniques for atomic quantum gases [1-3]. These new powerful tools provide detailed information on the distribution of atoms in a trap with resolution approaching the level of single atom and even single lattice site, and complement the well developed time-of-flight method that probes the system in momentum space. In a condensed matter analogy, this technique is equivalent to locating electrons of a material in a snap shot. In situ imaging has offered a new powerful tool to study atomic gases and inspired many new research directions and ideas. In this chapter, we will describe the experimental setup of in situ absorption imaging, observables that can be extracted from the images, and new physics that can be explored with this technique.

The paper 'In situ imaging of atomic quantum gases' discusses in situ imaging techniques for atomic quantum gases and how these techniques provide detailed information about the distribution of atoms within a trap. It talks about extracting density, fluctuations and correlations from these images. However, it doesn't specify the usage of bilayer quantum gas microscopes, nor does it discuss the manipulation of individual atoms within a three-dimensional (3D) optical lattice. The paper also focusses on absorption imaging in time-of-flight measurements, which is different from the specific interest in bilayer quantum gas microscopes.

**Site-resolved Imaging of Fermionic Lithium-6 in an Optical Lattice**Maxwell F. Parsons, ..., Markus Greiner (2015)

We demonstrate site-resolved imaging of individual fermionic lithium-6 atoms in a 2D optical lattice. To preserve the density distribution during fluorescence imaging, we simultaneously cool the atoms with 3D Raman sideband cooling. This laser cooling technique, demonstrated here for the first time for lithium-6 atoms, also provides a pathway to rapid low-entropy filling of an optical lattice. We are able to determine the occupation of individual lattice sites with a fidelity >95%, enabling direct, local measurement of particle correlations in Fermi lattice systems. This ability will be instrumental for creating and investigating low-temperature phases of the Fermi-Hubbard model, including antiferromagnets and d-wave superfluidity.

The paper presents an experiment that uses a quantum gas microscope to image atoms arranged in a two-dimensional optical lattice. It focuses extensively on the fluorescence imaging technique used as well as methods of cooling the atoms. However, it doesn't mention anything about a bilayer setup and doesn't seem to involve a third dimension in its experiment. There is no clear evidence in the abstract or selected parts of the paper about the manipulation of atoms in a third dimension, which suggests it doesn't meet a key criterion of the specific topic.

**Cooling through quantum criticality and many-body effects in condensed matter and cold gases**Bernd Wolf, ..., Michael Lang (2016)

This article reviews some recent developments for new cooling technologies in the fields of condensed matter physics and cold gases, both from an experimental and theoretical point of view. The main idea is to make use of distinct many-body interactions of the system to be cooled which can be some cooling stage or the material of interest itself, as is the case in cold gases. For condensed matter systems, we discuss magnetic cooling schemes based on a large magnetocaloric effect as a result of a nearby quantum phase transition and consider effects of geometrical frustration. For ultracold gases, we review many-body cooling techniques, such as spin-gradient and Pomeranchuk cooling, which can be applied in the presence of an optical lattice. We compare the cooling performance of these new techniques with that of conventional approaches and discuss state-of-the-art applications.

Although this paper discusses quantum many-body physics and many-body cooling techniques in ultracold gases, and features mention of optical lattices and quantum-gas microscopes, it doesn't mention or focus on bilayer quantum gas microscopes or the related experiments. Furthermore, it is more about cooling technologies rather than manipulation of atoms in a 3D environment. Therefore, this paper lacks some key components necessary for the desired topic.

**Quantum phase transition of cold atoms in the bilayer hexagonal optical lattices**Wei Xie, ..., Wu-Ming Liu (2013)

We propose a scheme to investigate the quantum phase transition of cold atoms in the bilayer hexagonal optical lattices. Using the quantum Monte Carlo method, we calculate the ground state phase diagrams which contain an antiferromagnet, a solid, a superfluid, a fully polarized state and a supersolid. We find there is a supersolid emerging in a proper parameter space, where the diagonal long range order coexists with off-diagonal long range order. We show that the bilayer optical lattices can be realized by coupling two monolayer optical lattices and give an experimental protocol to observe those novel phenomena in the real experiments.

The provided paper primarily investigates the quantum phase transitions of cold atoms in the context of bilayer hexagonal optical lattices, proposing an experimental schema to observe novel phenomena in this context. The authors have approached their research using quantum Monte Carlo methods to assess the ground state phase diagrams. Although the article provides valuable information about interactions in a bilayer hexagonal optical lattice, it does not explicitly mention the use of a quantum gas microscope, neither single-layer nor bilayer. Thus, details about how individual atoms are precisely imaged or manipulated are not explicitly covered, making the paper indirectly related to the exact topic.

**Microscopy of many-body states in optical lattices**Christian Gross, ..., Immanuel Bloch (2014)

Ultracold atoms in optical lattices have proven to provide an extremely clean and controlled setting to explore quantum many-body phases of matter. Now, imaging of atoms in such lattice structures has reached the level of single-atom sensitive detection combined with the highest resolution down to the level of individual lattice sites. This has opened up fundamentally new opportunities for the characterization and the control of quantum many-body systems. Here we give a brief overview of this field and explore the opportunities offered for future research.

The paper discusses the use of quantum gas microscopy for studying many-body phases in optical lattices and achieving single-atom sensitivity. However, it does not explicitly mention the use of bilayer quantum gas microscopes, nor does it address controlling atoms in the third dimension. Thus, while the microscopy techniques discussed in the paper are relevant for building context, the paper does not specifically address the topic of interest.

**Strain and pseudo-magnetic fields in optical lattices from density-assisted tunneling**Maxime Jamotte, ..., Marco Di Liberto (2021)

Applying time-periodic modulations is routinely used to control and design synthetic matter in quantum-engineered settings. In lattice systems, this approach is explored to engineer band structures with non-trivial topological properties, but also to generate exotic interaction processes. A prime example is density-assisted tunneling, by which the hopping amplitude of a particle between neighboring sites explicitly depends on their respective occupations. Here, we show how density-assisted tunneling can be tailored in view of simulating the effects of strain in synthetic graphene-type systems. Specifically, we consider a mixture of two atomic species on a honeycomb optical lattice: one species forms a Bose-Einstein condensate in an anisotropic harmonic trap, whose inhomogeneous density profile induces an effective uniaxial strain for the second species through density-assisted tunneling processes. In direct analogy with strained graphene, the second species experiences a pseudo magnetic field, hence exhibiting relativistic Landau levels and the valley Hall effect. Our proposed scheme introduces a unique platform for the investigation of strain-induced gauge fields and their possible interplay with quantum fluctuations and collective excitations.

While the study concentrates on optical lattices and tunneling processes in synthetic graphene-type systems, it does not mention the use of a bilayer quantum gas microscope, nor does it imply manipulation of atoms in the third dimension (out of plane). Therefore, even though the subject matter of the paper is related to atomic physics and optical lattices, it does not specifically cover the use of a bilayer quantum gas microscope or its application.

**Site-Resolved Imaging of Bosonic Mott Insulator of $^7$Li atoms**Kiryang Kwon, ..., Jae-yoon Choi (2021)

We demonstrate a single-site and single-atom-resolved fluorescence imaging of a bosonic Mott insulator of $^7$Li atoms in an optical lattice. The fluorescence images are obtained by implementing Raman sideband cooling on a deep two-dimensional square lattice, where we collect scattered photons with a high numerical aperture objective lens. The square lattice is created by a folded retro-reflected beam configuration that can reach 2.5~mK lattice depth from a single laser source. The lattice beam is elliptically focused to have a large area with deep potential. On average 4,000 photons are collected per atom during 1~s of the Raman sideband cooling, and the imaging fidelity is over 95$\%$ in the central 80$\times$80 lattice sites. As a first step to study correlated quantum phases, we present the site-resolved imaging of a Mott insulator. Tuning the magnetic field near the Feshbach resonance, the scattering length can be increased to 680$a_B$, and we are able to produce a large-sized unity filling Mott insulator with 2,000 atoms at low temperature. Our work provides a stepping stone to further in-depth investigations of intriguing quantum many-body phases in optical lattices.

This paper presents research on quantum gas microscopy of $^7$Li atoms in a two-dimensional optical lattice, and indeed they were able to achieve single-site resolved fluorescence imaging in a deep two-dimensional lattice. While the paper mentions the use of a vertical accordion optical lattice for entering into a strongly interacting regime, it does not explicitly mention employing a bilayer quantum gas microscope setup or manipulating the atoms in the third dimension (OutOfPlane). Consequently, this paper does not seem to fully satisfy the specific criteria that we are searching for.

**An unsupervised deep learning algorithm for single-site reconstruction in quantum gas microscopes**Alexander Impertro, ..., Monika Aidelsburger (2022)

In quantum gas microscopy experiments, reconstructing the site-resolved lattice occupation with high fidelity is essential for the accurate extraction of physical observables. For short interatomic separations and limited signal-to-noise ratio, this task becomes increasingly challenging. Common methods rapidly decline in performance as the lattice spacing is decreased below half the imaging resolution. Here, we present a novel algorithm based on deep convolutional neural networks to reconstruct the site-resolved lattice occupation with high fidelity. The algorithm can be directly trained in an unsupervised fashion with experimental fluorescence images and allows for a fast reconstruction of large images containing several thousand lattice sites. We benchmark its performance using a quantum gas microscope with cesium atoms that utilizes short-spaced optical lattices with lattice constant $383.5\,$nm and a typical Rayleigh resolution of $850\,$nm. We obtain promising reconstruction fidelities~$\gtrsim 96\%$ across all fillings based on a statistical analysis. We anticipate this algorithm to enable novel experiments with shorter lattice spacing, boost the readout fidelity and speed of lower-resolution imaging systems, and furthermore find application in related experiments such as trapped ions.

The paper presented involves using a deep learning algorithm to reconstruct single-site occupations in quantum gas microscopes, allowing for more accurate extraction of observable data even when faced with noise and experimental imperfections. The authors of the paper have used a quantum gas microscope in their study, specifically with cesium atoms and optical lattices. The content focuses on a computational algorithm related to quantum gas microscopy. However, the paper does not make any explicit mention or use of bilayer quantum gas microscopes. Nor does it appear to discuss or use third-dimension manipulations or imaging.

**Analogue Quantum Simulation: A New Instrument for Scientific Understanding**Dominik Hangleiter, ..., Karim P. Y. Thébault (2023)

In an analogue quantum simulation, an experimentally accessible quantum system is controlled and measured precisely in order to learn about the properties of another quantum system. As such, analogue quantum simulation is a novel tool of scientific inference standing between computation simulation and conventional experiment. In this book we undertake a comprehensive appraisal of the epistemology of analogue quantum simulation. In particular, we consider the types of understanding that analogue quantum simulation can yield. We draw a distinction between analogue quantum computations and analogue quantum emulations and argue that this distinction has important practical consequences on the types of validation an experimenter needs to perform in order to meet their epistemic goal. Our analysis is rooted in the contemporary scientific practice of analogue quantum simulation and draws upon detailed case studies of cold atoms, quantum photonics, and dispersive optical media platforms. Our goal is to provide a new framework for scientists and philosophers alike to understand the epistemic foundations of this exciting new area of scientific practice.

The paper discusses analogue quantum simulation in general and includes a discussion of cold atom computation. It specifically addresses cold atoms in optical lattices, but does not mention a bilayer quantum gas microscope or manipulation along the third dimension. The paper seems more focused on the theoretical background of quantum simulation using various platforms and does not delve into specific experimental techniques of a bilayer quantum gas microscope for individual atom detection.

**Thermodynamics vs. local density fluctuations in the metal/Mott-insulator crossover**J. H. Drewes, ..., M. Köhl (2016)

The crossover between a metal and a Mott insulator leads to a localization of fermions from delocalized Bloch states to localized states. We experimentally study this crossover using fermionic atoms in an optical lattice by measuring thermodynamic and local (on--site) density correlations. In the metallic phase at incommensurable filling we observe the violation of the local fluctuation--dissipation theorem indicating that the thermodynamics cannot be explained by local observables. In contrast, in the Mott-insulator we observe the convergence of local and thermodynamic fluctuations indicating the absence of long--range density-density correlations.

This paper focuses on the study of fermionic atoms in an optical lattice and delves into the aspects of metal-Mott insulator crossover. Even though it has aspects that involve the use of an optical lattice, and spatially-resolved imaging techniques for Fermi gases within these lattices, it doesn't mention the usage or implementation of a bilayer quantum gas microscope. The lattice the researchers focus on is three-dimensional, but the experiments limit dynamics to two-dimensional planes. Thus, while relevant to general quantum gas microscope work and atomic physics utilizing optical lattices, it fails to specifically address bilayer quantum gas microscope experiments.

**Equation of state of the two-dimensional Hubbard model**Eugenio Cocchi, ..., Michael Köhl (2015)

Understanding the phases of strongly correlated quantum matter is challenging because they arise from the subtle interplay between kinetic energy, interactions, and dimensionality. In this quest it has turned out that even conceptually simple models of strongly correlated fermions, which often only approximately represent the physics of the solid state, are very hard to solve. Since the conjecture by P. W. Anderson that the two-dimensional Hubbard model describes the main features of high-T$_c$ superconductivity in the cuprates, there has been a major, yet inconclusive, research effort on determining its fundamental thermodynamic properties. Here we present an experimental determination of the equation of state of the repulsive two-dimensional Hubbard model over a broad range of interactions, $0\leq U/t \lesssim 20$, and temperatures, down to $k_BT/t=0.63(2)$, using high-resolution imaging of ultracold atoms in optical lattices. The equation of state fully characterizes the thermodynamics of the Hubbard model, and our results constitute benchmarks for state-of-the-art theoretical approaches.

Going through the paper, the study uses high-resolution imaging of ultracold atoms in an optical lattice. However, the methods detailed are aligned with a single, two-dimensional layer of an optical lattice. There is no clear mention of the utilization of a bilayer quantum gas microscope or manipulation in the third dimension, which seems central to the specific area of interest. Therefore, while it does involve imaging of individual atoms in an optical lattice, the experiment deviates from the very precise topic in mind.

Geometrical frustration in strongly correlated systems can give rise to a plethora of novel ordered states and intriguing magnetic phases such as quantum spin liquids. Promising candidate materials for such phases can be described by the Hubbard model on an anisotropic triangular lattice, a paradigmatic model capturing the interplay between strong correlations and magnetic frustration. However, the fate of frustrated magnetism in the presence of itinerant dopants remains unclear, as well as its connection to the doped phases of the square Hubbard model. Here, we probe the local spin order of a Hubbard model with controllable frustration and doping, using ultracold fermions in anisotropic optical lattices continuously tunable from a square to a triangular geometry. At half-filling and strong interactions $U/t \sim 9$, we observe at the single-site level how frustration reduces the range of magnetic correlations and drives a transition from a collinear N\'eel antiferromagnet to a short-range correlated 120$^{\circ}$ spiral phase. Away from half-filling, magnetic correlations show a pronounced asymmetry between particle and hole doping close to triangular geometries and hint at a transition to ferromagnetic order at a particle doping above $20\%$. This work paves the way towards exploring possible chiral ordered or superconducting phases in triangular lattices, and realizing t-tprime square lattice Hubbard models that may be essential to describe superconductivity in cuprate materials.

While the paper clearly uses quantum gas microscopy and deals with optical lattices and frustration, it does not explicitly mention the use of bilayer configurations, or manipulation in the third dimension. This paper is centered around experiments dealing with doping in frustrated Fermi-Hubbard magnet models, using anisotropic optical lattices. The focus appears to be on the behavior of the system in relation to different geometries and doping levels rather than on the imaging or manipulation capabilities of a quantum gas microscope itself.

**Optically trapped quasi-two-dimensional Bose gases in random environment: quantum fluctuations and superfluid density**Kezhao Zhou, ..., Zhidong Zhang (2010)

We investigate a dilute Bose gas confined in a tight one-dimensional (1D) optical lattice plus a superimposed random potential at zero temperature. Accordingly, the ground state energy, quantum depletion and superfluid density are calculated. The presence of the lattice introduces a crossover to the quasi-2D regime, where we analyze asymptotically the 2D behavior of the system, particularly the effects of disorder. We thereby offer an analytical expression for the ground state energy of a purely 2D Bose gas in a random potential. The obtained disorder-induced normal fluid density $n_n$ and quantum depletion $n_d$ both exhibit a characteristic $1/\ln\left(1/n_{2D}a_{2D}^{2}\right)$ dependence. Their ratio $n_n/n_d$ increases to $2$ compared to the familiar $4/3$ in lattice-free 3D geometry, signifying a more pronounced contrast between superfluidity and Bose-Einstein condensation in low dimensions. Conditions for possible experimental realization of our scenario are also proposed.

The paper in question discusses a dilute Bose gas confined in a tight one-dimensional optical lattice and a random potential. The use and behavior of a quasi-2D Bose gas in a random potential is a key theme of this paper. However, the paper does not seem to discuss the use of bilayer quantum gas microscopes specifically, nor does it indicate any manipulation of atoms in the third dimension, which is vital to be qualified as a 'bilayer' experiment. Furthermore, the paper does not appear to focus on directly viewing individual atoms, as required in the topic of interest.

**Super-resolution microscopy of cold atoms in an optical lattice**Mickey McDonald, ..., Cheng Chin (2018)

Super-resolution microscopy has revolutionized the fields of chemistry and biology by resolving features at the molecular level. Such techniques can be either "stochastic," gaining resolution through precise localization of point source emitters, or "deterministic," leveraging the nonlinear optical response of a sample to improve resolution. In atomic physics, deterministic methods can be applied to reveal the atomic wavefunction and to perform quantum control. Here we demonstrate super-resolution imaging based on nonlinear response of atoms to an optical pumping pulse. With this technique the atomic density distribution can be resolved with a point spread function FWHM of 32(4) nm and a localization precision below 1 nm. The short optical pumping pulse of 1.4 $\mu$s enables us to resolve fast atomic dynamics within a single lattice site. A byproduct of our scheme is the emergence of moir\'{e} patterns on the atomic cloud, which we show to be immensely magnified images of the atomic density in the lattice. Our work represents a general approach to accessing the physics of cold atoms at the nanometer scale, and can be extended to higher dimensional lattices and bulk systems for a variety of atomic and molecular species.

This paper discusses super-resolution microscopy of cold atoms in an optical lattice. While the paper presents an interesting method to explore atomic density at the nanometer scale in higher dimensional lattices, it does not explicitly discuss a bilayer quantum gas microscope or manipulating atoms in the third, out-of-plane dimension. Therefore, while the paper might have some peripheral relevance, it does not directly address the specific topic of interest.

**Mott insulator phases of non-locally coupled bosons in bilayer optical superlattices**Sayan Lahiri, ..., Tapan Mishra (2020)

We investigate the ground state properties of a non-locally coupled bosonic system in a bilayer optical superlattice by considering bosons in one layer to be of softcore in nature and separately allowing two and three body hardcore constraints on the other layer. We find that the presence of different constraints on bosons in one layer influences the overall phase diagram exhibiting various Mott insulator phases at incommensurate densities due to the presence of the superlattice potential apart from the usual Mott insulators at commensurate densities. Moreover, the presence of two or three-body constraints significantly modifies the Mott insulator-Superfluid phase transition points as a function of the superlattice potential. Due to the various competing interactions, constraints and superlattice potential the phase diagrams exhibit significantly different features. We obtain the complete phase diagrams by using the cluster-mean-field theory approach. We further extend this work to a coupled two-leg ladder superlattice where we obtain similar physics using the density matrix renormalization group method .

This paper discusses investigating the ground state properties of a non-locally coupled bosonic system in a bilayer optical superlattice, featuring different constraints on the layers. However, the paper concentrates more on the theoretical aspects, phase diagrams and quantum states of bosons in such structures. It does not appear to have a direct focus on using bilayer quantum gas microscopes for visualization of individual atoms or their manipulation in three dimensions during the experiments.

**Ultraprecise Rydberg atomic localization using optical vortices**Ning Jia, ..., Jing Qian (2020)

We propose a robust localization of the highly-excited Rydberg atoms, interacting with doughnut-shaped optical vortices. Compared with the earlier standing-wave (SW)-based localization methods, a vortex beam can provide an ultrahigh-precision two-dimensional localization solely in the zero-intensity center, within a confined excitation region down to the nanometer scale. We show that the presence of the Rydberg-Rydberg interaction permits counter-intuitively much stronger confinement towards a high spatial resolution when it is partially compensated by a suitable detuning. In addition, applying an auxiliary SW modulation to the two-photon detuning allows a three-dimensional confinement of Rydberg atoms. In this case, the vortex field provides a transverse confinement while the SW modulation of the two-photon detuning localizes the Rydberg atoms longitudinally. To develop a new subwavelength localization technique, our results pave one-step closer to reduce excitation volumes to the level of a few nanometers, representing a feasible implementation for the future experimental applications.

The presented paper discusses a precise means of localizing Rydberg atoms using optical vortices. While it offers important insights into the precise localization of atoms and the overcoming of diffraction limits, it does not explicitly focus on bilayer quantum gas microscopes, nor does it address the use of optical lattices in the manner specified. Moreover, the application of the technology is not oriented towards single atoms, but rather Rydberg atoms, which are in a highly excited state and are unusually large.

**Spin-$S$ $\mathrm{U}(1)$ Quantum Link Models with Dynamical Matter on a Quantum Simulator**Jesse Osborne, ..., Jad C. Halimeh (2023)

Quantum link models (QLMs) offer the realistic prospect for the practical implementation of lattice quantum electrodynamics (QED) on modern quantum simulators, and they provide a venue for exploring various nonergodic phenomena relevant to quantum many-body physics. In these models, gauge and electric fields are represented by spin-$S$ operators. So far, large-scale realizations of QLMs have been restricted to $S=1/2$ representations, whereas the lattice-QED limit is approached at $S\to\infty$. Here, we present a bosonic mapping for the representation of gauge and electric fields with effective spin-$S$ operators for arbitrarily large values of $S$. Based on this mapping, we then propose an experimental scheme for the realization of a large-scale spin-$1$ $\mathrm{U}(1)$ QLM using spinless bosons in an optical superlattice. Using perturbation theory and infinite matrix product state calculations, which work directly in the thermodynamic limit, we demonstrate the faithfulness of the mapping and stability of gauge invariance throughout all accessible evolution times. We further demonstrate the potential of our proposed quantum simulator to address relevant high-energy physics by probing the (de)confinement of an electron--positron pair by tuning the gauge coupling. Our work provides an essential step towards gauge-theory quantum simulators in the quantum-field-theory limit.

Based on the abstract and selected sections of the paper, it presents a scheme for realizing a large-scale Quantum Link Model using spinless bosons in an optical superlattice. Although it involves the use of optical lattices, it does not explicitly mention the use of quantum gas microscopes, much less a bilayer quantum gas microscope. Moreover, the paper seems to focus more on theoretical aspects, like mapping and gauge-theory quantum simulators, rather than discussing any particular experimental details or procedural aspects concerning the use of a bilayer quantum gas microscope.

**Tunable Holstein model with cold polar molecules**Felipe Herrera, ..., Roman V. Krems (2010)

We show that an ensemble of polar molecules trapped in an optical lattice can be considered as a controllable open quantum system. The coupling between collective rotational excitations and the motion of the molecules in the lattice potential can be controlled by varying the strength and orientation of an external DC electric field as well as the intensity of the trapping laser. The system can be described by a generalized Holstein Hamiltonian with tunable parameters and can be used as a quantum simulator of excitation energy transfer and polaron phenomena. We show that the character of excitation energy transfer can be modified by tuning experimental parameters.

While the paper discusses a quantum system involving polar molecules in an optical lattice, it does not seem to involve bilayer gas microscopes or the specific visualization of individual atoms. Additionally, the paper refers to a 3D optical lattice, but does not elaborate on whether they are using a specific experimental setup such as a bilayer quantum gas microscope. The primary focus of this paper is on the open quantum system of the Holstein model and energy transfer in such systems, rather than on the specifics of the lattice or the techniques used to observe it.

**Quantum phases of strongly interacting Rydberg atoms in triangular lattices**Jing Qian, ..., Weiping Zhang (2013)

We present a theoretical study on the system of laser-driven strongly interacting Rydberg atoms trapped in a two-dimensional triangular lattice, in which the dipole-dipole interactions between Rydberg states result in exotic quantum phases. By using the mean-field theory, we investigate the steady state solutions and analyze their dynamical stabilities. We find that in the strong-interaction limit, the dynamics of the system is chaotic and exhibits random oscillations under appropriate laser detunings. Lyapunov exponent criterion is introduced to confirm the existence of this chaotic behavior. In addition, we present a full quantum calculation based on a six-atom model, and find that the system exhibits some bi-antiferromagnetic properties in every triangular cell when the one-photon detuning is exactly resonant or blue-shifted.

While this paper is indeed dealing with atoms in a two-dimensional optical lattice which resonates with the broader topic area, there does not seem to be any work or mention of a bilayer quantum gas microscope or the introduction of a third dimension in atomic manipulation. Moreover, the paper is highly focused on theoretical exploration of the quantum phases in the Rydberg atom system and its dynamical stabilities due to dipole-dipole interactions rather than practical experimentation with microscope technology. Therefore, it does not satisfy the specific topic criteria.

Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards generating quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles such as the equal a priori probability of the microcanonical ensemble, identifying general properties of nonequilibrium quantum dynamics remains a major challenge especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times. We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook onto major open questions as well as future directions of research.

This paper is a comprehensive review about dynamical quantum phase transitions. While it does mention and discuss experiments using ultra-cold atoms in optical lattices, it clearly lacks any detailed discussion of bilayer quantum gas microscopy. The usage of optical lattices in these experiments is relevant, but the paper does not go into the imaging or manipulation of individual atoms using a bilayer quantum gas microscope, which is paramount for the desired topic.

**Fractional domain walls from on-site softening in dipolar bosons**Emma Wikberg, ..., Anders Karlhede (2011)

We study dipolar bosons in a 1D optical lattice and identify a region in parameter space---strong coupling but relatively weak on-site repulsion---hosting a series of stable charge-density-wave (CDW) states whose low-energy excitations, built from "fractional domain walls," have remarkable similarities to those of non-Abelian fractional quantum Hall states. Here, a conventional domain wall between translated CDW's may be split by inserting strings of degenerate, but inequivalent, CDW states. Outside these insulating regions, we find numerous supersolids as well as a superfluid regime. The mentioned phases should be accessible experimentally and, in particular, the fractional domain walls can be created in the ground state using single-site addressing, i.e., by locally changing the chemical potential.

Though the paper discusses experiments in a 1D optical lattice involving ultracold bosons, it does not appear to use a bilayer quantum gas microscope or any similar technology to directly visualise and manipulate individual atoms. The paper primarily focuses on the theoretical modelling of dipolar bosons in 1D optical lattices, their properties, and possible relations to Charge-Density-Wave(CDW) states and the fractional quantum Hall effect.

**Detecting quadrupole interactions in ultracold Fermi gases**M. Lahrz, ..., L. Mathey (2014)

We propose to detect quadrupole interactions of neutral ultra-cold atoms via their induced mean-field shift. We consider a Mott insulator state of spin-polarized atoms in a two-dimensional optical square lattice. The quadrupole moments of the atoms are aligned by an external magnetic field. As the alignment angle is varied, the mean-field shift shows a characteristic angular dependence, which constitutes the defining signature of the quadrupole interaction. For the $^{3}P_{2}$ states of Yb and Sr atoms, we find a frequency shift of the order of tens of Hertz, which can be realistically detected in experiment with current technology. We compare our results to the mean-field shift of a spin-polarized quasi-2D Fermi gas in continuum.

The paper is focused on detecting quadrupole interactions of ultra-cold atoms in a two-dimensional optical square lattice using mean-field shift. While there are mentions of optical lattices and observing atoms' interactions, there is no reference to bilayer systems or manipulation in the third dimension. Furthermore, quantum gas microscopy isn't explicitly discussed in the context detailed in the paper abstraction, suggesting that it isn't the main tool being used to observe these interactions.

**Quantum gas magnifier for sub-lattice-resolved imaging of three-dimensional quantum systems**Luca Asteria, ..., Christof Weitenberg (2021)

Imaging is central for gaining microscopic insight into physical systems, but direct imaging of ultracold atoms in optical lattices as modern quantum simulation platform suffers from the diffraction limit as well as high optical density and small depth of focus. We introduce a novel approach to imaging of quantum many-body systems using matter wave optics to magnify the density distribution prior to optical imaging, allowing sub-lattice spacing resolution in three-dimensional systems. Combining the site-resolved imaging with magnetic resonance techniques for local addressing of individual lattice sites, we demonstrate full accessibility to local information and local manipulation in three-dimensional optical lattice systems. The method opens the path for spatially resolved studies of new quantum many-body regimes including exotic lattice geometries.

The provided paper presents a novel approach to imaging quantum many-body systems in 3D lattices using a 'quantum gas magnifier.' It seems to achieve a resolution surpassing the diffraction limit and allowing sub-lattice spacing resolution in the third dimension. The paper reports the integration of site-resolved imaging with magnetic resonance techniques for local manipulation, providing accessibility to local information. However, in spite of the paper providing a new method for spatially resolved study of 3D quantum systems, the specific mention or demonstration of bilayer quantum gas microscopes is absent. This means it misses a crucial part of the topic of interest.

**A toolbox for lattice spin models with polar molecules**A. Micheli, ..., P. Zoller (2005)

There is growing interest to investigate states of matter with topological order, which support excitations in the form of anyons, and which underly topological quantum computing. Examples of such systems include lattice spin models in two dimensions. Here we show that relevant Hamiltonians can be systematically engineered with polar molecules stored in optical lattices, where the spin is represented by a single electron outside a closed shell of a heteronuclear molecule in its rotational ground state. Combining microwave excitation with the dipole-dipole interactions and spin-rotation couplings allows us to build a complete toolbox for effective two-spin interactions with designable range and spatial anisotropy, and with coupling strengths significantly larger than relevant decoherence rates. As an illustration we discuss two models: a 2D square lattice with an energy gap providing for protected quantum memory, and another on stacked triangular lattices leading to topological quantum computing.

This paper outlines a method for systematic engineering of Hamiltonians with polar molecules stored in optical lattices. While it does discuss manipulating atoms within a two dimensional optical lattice, it does not appear to specifically mention the use of bilayer quantum gas microscopes, or manipulating atoms in a third, out-of-plane dimension. While the concepts presented are closely related to the desired topic, the lack of mention of bilayer microscopy make it less directly relevant.

**Single-Spin Addressing in an Atomic Mott Insulator**Christof Weitenberg, ..., Stefan Kuhr (2011)

Ultracold atoms in optical lattices are a versatile tool to investigate fundamental properties of quantum many body systems. In particular, the high degree of control of experimental parameters has allowed the study of many interesting phenomena such as quantum phase transitions and quantum spin dynamics. Here we demonstrate how such control can be extended down to the most fundamental level of a single spin at a specific site of an optical lattice. Using a tightly focussed laser beam together with a microwave field, we were able to flip the spin of individual atoms in a Mott insulator with sub-diffraction-limited resolution, well below the lattice spacing. The Mott insulator provided us with a large two-dimensional array of perfectly arranged atoms, in which we created arbitrary spin patterns by sequentially addressing selected lattice sites after freezing out the atom distribution. We directly monitored the tunnelling quantum dynamics of single atoms in the lattice prepared along a single line and observed that our addressing scheme leaves the atoms in the motional ground state. Our results open the path to a wide range of novel applications from quantum dynamics of spin impurities, entropy transport, implementation of novel cooling schemes, and engineering of quantum many-body phases to quantum information processing.

The presented paper discusses the manipulation of individual atoms within an optical lattice using a special technique involving a laser beam and microwave field. It talks about single-site addressing and creating arbitrary spin patterns by sequentially addressing selected lattice sites. However, this research does not deal with bilayer quantum gas microscopes specifically. It does cover exploration in a two-dimensional array of atoms but it doesn't manipulate in the out-of-plane third dimension.

**Proximity effects in cold atom artificial graphene**Tobias Grass, ..., Maciej Lewenstein (2016)

Cold atoms in an optical lattice with brick-wall geometry have been used to mimic graphene, a two-dimensional material with characteristic Dirac excitations. Here we propose to bring such artificial graphene into the proximity of a second atomic layer with a square lattice geometry. For non-interacting fermions, we find that such bilayer system undergoes a phase transition from a graphene-like semi-metal phase, characterized by a band structure with Dirac points, to a gapped band insulator phase. In the presence of attractive interactions between fermions with pseudospin-1/2 degree of freedom, a competition between semi-metal and superfluid behavior is found at the mean-field level. Using the quantum Monte Carlo method, we also investigate the case of strong repulsive interactions. In the Mott phase, each layer exhibits a different amount of long-range magnetic order. Upon coupling both layers, a valence-bond crystal is formed at a critical coupling strength. Finally, we discuss how these bilayer systems could be realized in existing cold atom experiments.

The paper provides a theoretical exploration of a bilayer system where one layer has a graphene-like band structure, while the other layer has square lattice geometry. Though it mentions the idea of the bilayer and interaction between layers, it appears to not engage with the concept of bilayer quantum gas microscopes specifically. Moreover, it doesn't disclose an explicit demonstration through experimental evidence of individual imaging of atoms or their manipulation along the third, out-of-plane dimension.

**Staggered superfluid phases of dipolar bosons in two-dimensional square lattices**Kuldeep Suthar, ..., Jakub Zakrzewski (2020)

We study the quantum ground state of ultracold bosons in a two-dimensional square lattice. The bosons interact via the repulsive dipolar interactions and s-wave scattering. The dynamics is described by the extended Bose-Hubbard model including correlated hopping due to the dipolar interactions, the coefficients are found from the second quantized Hamiltonian using the Wannier expansion with realistic parameters. We determine the phase diagram using the Gutzwiller ansatz in the regime where the coefficients of the correlated hopping terms are negative and can interfere with the tunneling due to single-particle effects. We show that this interference gives rise to staggered superfluid and supersolid phases at vanishing kinetic energy, while we identify parameter regions at finite kinetic energy where the phases are incompressible. We compare the results with the phase diagram obtained with the cluster Gutzwiller approach and with the results found in one dimension using DMRG.

This paper discusses the study of quantum ground state of ultracold bosons in a two-dimensional square lattice and the results achieved with variations of the Bose-Hubbard model. It mentions the use of ultracold dipolar gases in optical lattices, but it does not discuss the use of bilayer quantum gas microscopes or the visualization of individual atoms in a three-dimensional optical lattice. While the paper does cover interesting research in a related area, it does not seem to pertain directly to the specific interest in bilayer quantum gas microscopes.

**Short-range quantum magnetism of ultracold fermions in an optical lattice**Daniel Greif, ..., Tilman Esslinger (2012)

The exchange coupling between quantum mechanical spins lies at the origin of quantum magnetism. We report on the observation of nearest-neighbor magnetic spin correlations emerging in the many-body state of a thermalized Fermi gas in an optical lattice. The key to obtaining short-range magnetic order is a local redistribution of entropy within the lattice structure. This is achieved in a tunable-geometry optical lattice, which also enables the detection of the magnetic correlations. We load a low-temperature two-component Fermi gas with repulsive interactions into either a dimerized or an anisotropic simple cubic lattice. For both systems the correlations manifest as an excess number of singlets as compared to triplets consisting of two atoms with opposite spins. For the anisotropic lattice, we determine the transverse spin correlator from the singlet-triplet imbalance and observe antiferromagnetic correlations along one spatial axis. Our work paves the way for addressing open problems in quantum magnetism using ultracold fermions in optical lattices as quantum simulators.

The location of atoms in an optical lattice is manipulated and observed in this paper's experiments. However, the paper does not explicitly mention the use of a bilayer quantum gas microscope. It rather mentions the use of a 'tunable-geometry optical lattice' that could potentially incorporate three-dimensional control but does not specify so explicitly. Hence, while the paper discusses a relevant experimental setup with ultracold atoms in an optical lattice, its relation to bilayer quantum gas microscopes cannot be definitively determined from the abstract and selected parts of the paper.

**Cooling and state preparation in an optical lattice via Markovian feedback control**Ling-Na Wu, ..., André Eckardt (2021)

We propose and investigate a scheme based on Markovian feedback control that allows for the preparation of single targeted eigenstates of a system of bosonic atoms in a one-dimensional optical lattice with high fidelity. It can be used for in-situ cooling the interacting system without particle loss, both for weak and strong interactions, and for experimentally preparing and probing individual excited eigenstates. For that purpose the system is assumed to be probed weakly via homodyne detection of photons that are scattered off-resonantly by the atoms from a structured probe beam into a cavity mode. By applying an inertial force to the system that is proportional to the measured signal, the system is then guided into a pure target state. The scheme is found to be robust against reduced measurement efficiencies.

While this paper is related to the broader field of atomic physics, and specifically deals with the manipulation and state preparation of atoms in a one-dimensional optical lattice, it does not focus on or make use of bilayer quantum gas microscopes. Neither in the abstract nor in the selected parts of the paper is there mention of bilayer quantum gas microscopes or manipulation in the third, out-of-plane dimension, which is a key aspect of the desired topic.

**Particle dynamics and ergodicity-breaking in twisted-bilayer optical lattices**Ganesh C. Paul, ..., Luis Santos (2023)

Recent experiments have realized a twisted bilayer-like optical potential for ultra-cold atoms, which in contrast to solid-state set ups may allow for an arbitrary ratio between the inter- and intra-layer couplings. For commensurate Moir\'e twistings a large-enough inter-layer coupling results in particle transport dominated by channel formation. For incommensurate twistings, the interlayer coupling acts as an effective disorder strength. Whereas for weak couplings the whole spectrum remains ergodic, at a critical value part of the eigenspectrum transitions into multifractal states. A similar transition may be observed as well as a function of an energy bias between the two layers. Our study reveals atoms in optical twisted-bilayer lattices as an interesting new platform for the study of ergodicity breaking and multifractality.

This paper discusses recent experiments using twisted bilayer-like optical potentials with ultra-cold atoms, which manipulates atoms in inter- and intra-layer couplings, matching our criteria of bilayer systems. However, while it does touch upon imaging atoms in an optical lattice, there is no explicit mention of the use of a quantum gas microscope to achieve this. The paper mainly focuses on twisted bilayer optical lattices and its implications for the study of ergodicity breaking and multifractality. Thus, although it is closely related, it doesn't fully meet the topic of interest stipulated.

**Dipolar quantum solids emerging in a Hubbard quantum simulator**Lin Su, ..., Markus Greiner (2023)

In quantum mechanical many-body systems, long-range and anisotropic interactions promote rich spatial structure and can lead to quantum frustration, giving rise to a wealth of complex, strongly correlated quantum phases. Long-range interactions play an important role in nature; however, quantum simulations of lattice systems have largely not been able to realize such interactions. A wide range of efforts are underway to explore long-range interacting lattice systems using polar molecules, Rydberg atoms, optical cavities, and magnetic atoms. Here, we realize novel quantum phases in a strongly correlated lattice system with long-range dipolar interactions using ultracold magnetic erbium atoms. As we tune the dipolar interaction to be the dominant energy scale in our system, we observe quantum phase transitions from a superfluid into dipolar quantum solids, which we directly detect using quantum gas microscopy. Controlling the interaction anisotropy by orienting the dipoles enables us to realize a variety of stripe ordered states. Furthermore, by transitioning non-adiabatically through the strongly correlated regime, we observe the emergence of a range of metastable stripe-ordered states. This work demonstrates that novel strongly correlated quantum phases can be realized using long-range dipolar interaction in optical lattices, opening the door to quantum simulations of a wide range of lattice models with long-range and anisotropic interactions.

The paper in question describes experiments that involve the use of quantum gas microscopy to observe dipolar quantum liquids in a lattice system. It significantly focuses on the use of long-range dipolar interactions, energetics of the system, and controlling of interaction anisotropy to explore different quantum phases. The research does involve the use of quantum gas microscopy in an optical lattice, but there is no explicit mention of a bilayer quantum gas microscope or 'manipulation in the third dimension'. The paper primarily focuses on the use of ultracold magnetic erbium atoms and the phenomenon of quantum phase transitions with respect to the dipolar interactions, rather than utilizing a three-dimensional quantum gas microscope.

**Ultracold molecules: vehicles to scalable quantum information processing**Kathy-Anne Brickman Soderberg, ..., Cheng Chin (2008)

We describe a novel scheme to implement scalable quantum information processing using Li-Cs molecular state to entangle $^{6}$Li and $^{133}$Cs ultracold atoms held in independent optical lattices. The $^{6}$Li atoms will act as quantum bits to store information, and $^{133}$Cs atoms will serve as messenger bits that aid in quantum gate operations and mediate entanglement between distant qubit atoms. Each atomic species is held in a separate optical lattice and the atoms can be overlapped by translating the lattices with respect to each other. When the messenger and qubit atoms are overlapped, targeted single spin operations and entangling operations can be performed by coupling the atomic states to a molecular state with radio-frequency pulses. By controlling the frequency and duration of the radio-frequency pulses, entanglement can either be created or swapped between a qubit messenger pair. We estimate operation fidelities for entangling two distant qubits and discuss scalability of this scheme and constraints on the optical lattice lasers.

The studied paper discusses a scheme for quantum information processing using ultracold atoms held in independent optical lattices. The system has two separate species of atoms, each manipulated by a different optical lattice potential. This aspect indicates the multidimensionality of the optical lattices involved in the experiment, which bears some relation to the scope of bilayer quantum gas microscopes. However, the paper does not explicitly mention or imply the use of a bilayer quantum gas microscope, nor does it focus on its direct analogous. It centers on atoms' entanglement and information processing, instead of direct imaging or individual atom manipulation via a bilayer microscope.

**Single-atom trapping in holographic 2D arrays of microtraps with arbitrary geometries**Florence Nogrette, ..., Antoine Browaeys (2014)

We demonstrate single-atom trapping in two-dimensional arrays of microtraps with arbitrary geometries. We generate the arrays using a Spatial Light Modulator (SLM), with which we imprint an appropriate phase pattern on an optical dipole trap beam prior to focusing. We trap single $^{87}{\rm Rb}$ atoms in the sites of arrays containing up to $\sim100$ microtraps separated by distances as small as $3\;\mu$m, with complex structures such as triangular, honeycomb or kagome lattices. Using a closed-loop optimization of the uniformity of the trap depths ensures that all trapping sites are equivalent. This versatile system opens appealing applications in quantum information processing and quantum simulation, e.g. for simulating frustrated quantum magnetism using Rydberg atoms.

The paper talks about trapping single atoms in 2D arrays of microtraps with almost arbitrary geometries. It does utilize optical lattices and quantum gas microscope (indirectly mentioned in the references) but it is not about a ‘bilayer’ microscope per se, and does not seem to show manipulation or observation in the third, out-of-plane dimension. The paper chiefly focuses on 2D lattices with varying geometries.

**Single-atom imaging of fermions in a quantum-gas microscope**Elmar Haller, ..., Stefan Kuhr (2015)

Single-atom-resolved detection in optical lattices using quantum-gas microscopes has enabled a new generation of experiments in the field of quantum simulation. Fluorescence imaging of individual atoms has so far been achieved for bosonic species with optical molasses cooling, whereas detection of fermionic alkaline atoms in optical lattices by this method has proven more challenging. Here we demonstrate single-site- and single-atom-resolved fluorescence imaging of fermionic potassium-40 atoms in a quantum-gas microscope setup using electromagnetically-induced-transparency cooling. We detected on average 1000 fluorescence photons from a single atom within 1.5s, while keeping it close to the vibrational ground state of the optical lattice. Our results will enable the study of strongly correlated fermionic quantum systems in optical lattices with resolution at the single-atom level, and give access to observables such as the local entropy distribution and individual defects in fermionic Mott insulators or anti-ferromagnetically ordered phases.

While the paper indeed dives into the domain of using quantum gas microscopes to observe individual atoms in an optical lattice, it does not touch upon the bilayer aspect of such microscopes. The focus is on demonstrating single-site- and single-atom-resolved fluorescence imaging of fermionic potassium-40 atoms in a quantum-gas microscope setup, with no explicit mention of maneuvering these atoms in the third, out of plane dimension.

**The Fulde-Ferrell-Larkin-Ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review**Jami J. Kinnunen, ..., Päivi Törmä (2017)

We review the concepts and the present state of theoretical studies of spin-imbalanced superfluidity, in particular the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold quantum gases. The comprehensive presentation of the theoretical basis for the FFLO state that we provide is useful also for research on the interplay between magnetism and superconductivity in other physical systems. We focus on settings that have been predicted to be favourable for the FFLO state, such as optical lattices in various dimensions and spin-orbit coupled systems. These are also the most likely systems for near-future experimental observation of the FFLO state. Theoretical bounds, such as Bloch's and Luttinger's theorems, and experimentally important limitations, such as finite-size effects and trapping potentials, are considered. In addition, we provide a comprehensive review of the various ideas presented for the observation of the FFLO state. We conclude our review with an analysis of the open questions related to the FFLO state, such as its stability, superfluid density, collective modes and extending the FFLO superfluid concept to new types of lattice systems.

The manuscript discusses the theory and prediction of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in ultracold quantum gases, specifically in systems such as optical lattices and spin-orbit coupled systems. The paper does mention studies involving quantum gas microscopes and optical lattices (e.g. references [29], [30], [31], [32], and others), but it fundamentally explores superfluidity and FFLO states, not bilayer quantum gas microscopy. Therefore, although the paper discusses research involving optical lattices and quantum gas microscopes, it doesn't specifically address or focus on experimental use of bilayer quantum gas microscopes for observing individual atoms inside an optical lattice.

**Structure of exciton condensates in imbalanced electron-hole bilayers**J. R. Varley, ..., D. K. K. Lee (2016)

We investigate the possibility of excitonic superfluidity in electron-hole bilayers. We calculate the phase diagram of the system for the whole range of electron-hole density imbalance and for different degrees of electrostatic screening, using mean-field theory and a Ginzburg-Landau expansion. We are able to resolve differences on previous work in the literature which concentrated on restricted regions of the parameter space. We also give detailed descriptions of the pairing wavefunction in the Fulde-Ferrell-Larkin-Ovchinnikov paired state. The Ginzburg-Landau treatment allows us to investigate the energy scales involved in the pairing state and discuss the possible spontaneous breaking of two-dimensional translation symmetry in the ground state.

Although the paper discusses bilayer systems, specifically electron-hole bilayers, it does not appear to involve a quantum gas microscope or optical lattices. The focus of the paper seems to be on the theory of excitonic superfluidity in these bilayers, rather than on the studied topic of interest, i.e., the experimental visualization or manipulation of individual atoms using a bilayer quantum gas microscope. Furthermore, the paper seems to concentrate more on quantum wells than on optical lattices.

**Quantum state engineering of a Hubbard system with ultracold fermions**Christie S. Chiu, ..., Markus Greiner (2017)

Accessing new regimes in quantum simulation requires the development of new techniques for quantum state preparation. We demonstrate the quantum state engineering of a strongly correlated many-body state of the two-component repulsive Fermi-Hubbard model on a square lattice. Our scheme makes use of an ultralow entropy doublon band insulator created through entropy redistribution. After isolating the band insulator, we change the underlying potential to expand it into a half-filled system. The final many-body state realized shows strong antiferromagnetic correlations and a temperature below the exchange energy. We observe an increase in entropy, which we find is likely caused by the many-body physics in the last step of the scheme. This technique is promising for low-temperature studies of cold-atom-based lattice models.

The paper describes an experiment involving a quantum gas microscope and the precise control of atoms in an optical lattice. It does highlight the manipulation of fermionic atoms in specific configurations within the lattice. However, there is no specific indication or discussion in the paper about a bilayer structure or the manipulation of atoms in the third, out of plane dimension. Thus, although relevant to quantum gas microscopes, it seems not to address the specific aspect of 'bi