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At least 91 records · Page 5

Structure of Charge Density Waves in La1.875Ba0.125CuO4

Although charge-density wave (CDW) correlations exist in several families of cuprate superconductors, they exhibit substantial variation in CDW wavevector and correlation length, indicating a key role for CDW-lattice interactions. We investigated this interaction in La1.875Ba0.125CuO4 using single crystal x-ray diffraction to collect a large number of CDW peak intensities, and determined the Cu and La/Ba atomic distortions induced by the formation of CDW order. Within the CuO2 planes, the distortions involve a periodic modulation of the Cu-Cu spacing along the direction of the ordering wave vector. The charge ordering within the copper-oxygen layer induces an out-of-plane breathing modulation of the surrounding lanthanum layers, which leads to a related distortion on the adjacent copper-oxygen layer. Our result implies that the CDW-related structural distortions do not remain confined to a single layer but rather propagate an appreciable distance through the crystal. This leads to overlapping structural modulations, in which CuO2 planes exhibit distortions arising from the orthogonal CDWs in adjacent layers as well as distortions from the CDW within the layer itself. We attribute this striking effect to the weak c-axis charge screening in cuprates and suggest this effect could help couple the CDW between adjacent planes in the crystal.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Diethylenetriamine-functionalized graphene oxide: Insights into ion adsorption and applications in rare earth element separation

The growing demand for critical minerals and materials requires atom- and energy-efficient, selective separations to overcome the challenges posed by the similar chemical and physical properties of the rare earth elements (REEs) and their low concentrations in unconventional domestic feedstocks. Here, in this study, we developed diethylenetriamine-functionalized graphene oxide (DETA-GO) as a membrane material for REE adsorption and separation. Synthesis conditions were optimized to maximize nitrogen incorporation while also preserving GO dispersibility for facile membrane fabrication. We investigated the mechanism of amine functionalization, the nitrogen-bonding configurations, the organization of the interlayer transport channels, and the resulting effects on ion and water transport for REE separations. Neat-GO and DETA-GO multilayer laminate membranes were fabricated by vacuum filtration onto polymer supports. To investigate the effects of amine functionalization, the membranes were characterized using scanning electron microscopy, Raman, Fourier transform infrared, and X-ray photoelectron spectroscopy, as well as grazing-incidence X-ray diffraction measurements. Ion permeation experiments with representative lanthanum (La 3+ ) and ytterbium (Yb 3+ ) solutions revealed enhanced ion adsorption and water transport through DETA-GO membranes compared to neat-GO. The strong affinity of the membranes for multivalent REEs was also validated with conductivity and inductively coupled plasma mass spectrometry measurements. Atomistic insight into the role of amine functionalization in modulating nanochannel architecture and long-term stability, optimizing adsorption sites, and regulating REE and water transport was obtained using classical molecular dynamics simulations. Collectively, our joint experimental and theoretical study demonstrates the potential of DETA-GO membranes for selective REE separations, offering insights into ion-binding mechanisms, water-transport properties, and nanochannel optimization for the recovery of critical materials from aqueous feedstocks.

Adsorbent↗

Effective field theories for collective excitations of atomic nuclei

Collective modes emerge as the relevant degrees of freedom that govern low-energy excitations of atomic nuclei. These modes—rotations, pairing rotations, and vibrations—are separated in energy from non-collective excitations, making it possible to describe them in the framework of effective field theory. Rotations and pairing rotations are the remnants of Nambu–Goldstone modes from the emergent breaking of rotational symmetry and phase symmetries in finite deformed and finite superfluid nuclei, respectively. The symmetry breaking severely constrains the structure of low-energy Lagrangians and thereby clarifies what is essential and simplifies the description. The approach via effective field theories exposes the essence of nuclear collective excitations and is defined with a breakdown scale in mind. This permits one to make systematic improvements and to estimate and quantify uncertainties. Effective field theories of collective excitations have been used to compute spectra, transition rates, and other matrix elements of interest. In particular, predictions of the nuclear matrix element for neutrinoless double beta decay then come with quantified uncertainties. This review summarizes these results and also compares the approach via effective field theories to well-known models and ab initio computations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

On the nature of two-photon transitions for a collection of molecules in a Fabry–Perot cavity

Here, we investigate the effect of a cavity on nonlinear two-photon transitions of a molecular system and we analyze how such an effect depends on the cavity quality factor, the field enhancement, and the possibility of dephasing. We find that the molecular response to strong light fields in a cavity with a variable quality factor can be understood as arising from a balance between (i) the ability of the cavity to enhance the field of an external probe and promote multiphoton transitions more easily and (ii) the fact that the strict selection rules on multiphoton transitions in a cavity support only one resonant frequency within the excitation range. Although our simulations use a classical level description of the radiation field (i.e., we solve Maxwell–Bloch or Maxwell–Liouville equations within the Ehrenfest approximation for the field–molecule interaction), based on experience with this level of approximation in the past studies of plasmonic and polaritonic systems, we believe that our results are valid over a wide range of external probing.

74 ATOMIC AND MOLECULAR PHYSICS↗

Metal–organic framework (MOF) materials as polymerization catalysts: a review and recent advances

Synthetic polymers are ubiquitous across both the industrial and consumer segments of the world economy. Catalysts enable rapid, efficient, selective, and even stereoselective, formation of desired polymers from any of a host of candidate monomers. While numerous molecular catalysts have been shown to be effective for these reactions, separation of the catalysts from reaction products is typically difficult – a potentially problematic complication that suggests instead the use of heterogeneous catalysts. Many of the most effective heterogeneous catalysts, however, comprise supported collections of reaction centres that are decidedly nonuniform in their composition, siting, and activity. Nonuniformity complicates atomic-scale evaluation of the basis for catalytic activity and thus impedes scientific hypothesis-driven understanding and development of superior catalysts. In view of the fundamental desirability of structural and chemical uniformity at the meso, nano, and even atomic scale, crystallographically well-defined, high-porosity metal–organic frameworks (MOFs) have attracted attention as model catalysts and/or catalyst-supports for a wide variety of chemical transformations. In the realm of synthetic polymers, catalyst-functionalized MOFs have been studied for reactions ranging from coordination-mediated polymerization of ethylene to visible-light initiated radical polymerizations. Nevertheless, many polymerization reactions remain to be explored – and, no doubt, will be explored, given the remarkable structural and compositional diversity of attainable MOFs. Noteworthy emerging studies include work directed toward more sophisticated catalytic schemes such as polymer templating using MOF pore architectures and tandem copolymerizations using MOF-supported reaction centres. Finally, it is appropriate to recognize that MOFs themselves are synthetic polymers – albeit, uncoventional ones.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Adsorption separation of heavier isotope gases in subnanometer carbon pores

Isotopes of heavier gases including carbon ( 13 C/ 14 C), nitrogen ( 13 N), and oxygen ( 18 O) are highly important because they can be substituted for naturally occurring atoms without significantly perturbing the biochemical properties of the radiolabelled parent molecules. These labelled molecules are employed in clinical radiopharmaceuticals, in studies of brain disease and as imaging probes for advanced medical imaging techniques such as positron-emission tomography (PET). Established distillation-based isotope gas separation methods have a separation factor ( S ) below 1.05 and incur very high operating costs due to high energy consumption and long processing times, highlighting the need for new separation technologies. Here, we show a rapid and highly selective adsorption-based separation of 18 O 2 from 16 O 2 with S above 60 using nanoporous adsorbents operating near the boiling point of methane (112 K), which is accessible through cryogenic liquefied-natural-gas technology. A collective-nuclear-quantum effect difference between the ordered 18 O 2 and 16 O 2 molecular assemblies confined in subnanometer pores can explain the observed equilibrium separation and is applicable to other isotopic gases.

36 MATERIALS SCIENCE↗

On-demand indistinguishable single photons from an efficient and pure source based on a Rydberg ensemble

Single photons coupled to atomic systems have shown to be a promising platform for developing quantum technologies. Yet a bright on-demand, highly pure, and highly indistinguishable single-photon source compatible with atomic platforms is lacking. In this work, we demonstrate such a source based on a strongly interacting Rydberg system. The large optical nonlinearities in a blockaded Rydberg ensemble convert coherent light into a single collective excitation that can be coherently retrieved as a quantum field. We simultaneously observe a fully single-mode (spectral, temporal, spatial, and polarization) efficiency up to 0.098(2), a detector-background-subtracted g ( 2 ) = 5.0 ( 1.6 ) × <#comment/> 10 − <#comment/> 4 , and indistinguishability of 0.980(7), at an average photon production rate of 1.18 ( 2 ) × <#comment/> 10 4 s − <#comment/> 1 . All of these make this system promising for scalable quantum information applications. Furthermore, we investigate the effects of contaminant Rydberg excitations on the source efficiency and observed single-mode efficiencies up to 0.18(2) for lower photon rates. Finally, recognizing that many quantum information protocols require a single photon in a fully single mode, we introduce metrics that take into account all degrees of freedom to benchmark the performance of on-demand sources.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

On single-crystal total scattering data reduction and correction protocols for analysis in direct space

Data reduction and correction steps and processed data reproducibility in the emerging single-crystal total-scattering-based technique of three-dimensional differential atomic pair distribution function (3D-ΔPDF) analysis are explored. All steps from sample measurement to data processing are outlined using a crystal of CuIr 2 S 4 as an example, studied in a setup equipped with a high-energy X-ray beam and a flat-panel area detector. Computational overhead as pertains to data sampling and the associated data-processing steps is also discussed. Various aspects of the final 3D-ΔPDF reproducibility are explicitly tested by varying the data-processing order and included steps, and by carrying out a crystal-to-crystal data comparison. Situations in which the 3D-ΔPDF is robust are identified, and caution against a few particular cases which can lead to inconsistent 3D-ΔPDFs is noted. Although not all the approaches applied herein will be valid across all systems, and a more in-depth analysis of some of the effects of the data-processing steps may still needed, the methods collected herein represent the start of a more systematic discussion about data processing and corrections in this field.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Imaging nuclear shape through anisotropic and radial flow in high-energy heavy-ion collisions

Most atomic nuclei exhibit ellipsoidal shapes characterized by quadrupole deformation β 2 and triaxiality γ, and sometimes even a pear-like octupole deformation β 3 . The STAR experiment introduced a new ‘imaging-by-smashing’ technique to image the nuclear global shape by colliding nuclei at ultra-relativistic speeds and analyzing outgoing debris. Features of nuclear shape manifest in collective observables like anisotropic flow vn and radial flow via mean transverse momentum [p T ]. We present new measurements of the variances of v n (n = 2, 3, and 4) and [p T ], and the covariance of $v$$^{2}_{n}$ with [p T ], in collisions of highly deformed 238 U and nearly spherical 197 Au. Ratios of these observables between the two systems effectively suppress common final-state effects, isolating the strong impact of uranium’s deformation. By comparing results with state-of-the-art hydrodynamic model calculations, we extract β 2U and γ U values consistent with those deduced from low-energy nuclear structure measurements. Measurements of v 3 and its correlation with [p T ] also provide the first experimental suggestion of a possible octupole deformation for 238 U. These findings provide significant support for using high-energy collisions to explore nuclear shapes on femtosecond timescales, with implications for both nuclear structure and quark-gluon plasma studies.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Molecular Driving Force for Facet Selectivity of Sequence-Defined Amphiphilic Peptoids at Au–Water Interfaces

Shape-controlled colloidal nanocrystal syntheses often require facet-selective solution-phase chemical additives to regulate surface free energy, atom addition/migration fluxes or particle attachment rates. Because of their highly tunable properties and robustness to a wide range of experimental conditions, peptoids represent a very promising class of next generation functional additives for control over nanocrystal growth. However, understanding the origin of facet selectivity at the molecular level is critical to generalizing their design. In this work, we employ molecular dynamics simulations and biased sampling methods and report stronger selectivity to Au(111) than to Au(100) for Nce3Ncp6, a peptoid that has been shown to assist the formation of five-fold twinned Au nanostars. We find that facet-selectivity is achieved through synergistic effects of both peptoid-surface and solvent-surface interactions. Moreover, the amphiphilic nature of Nce3Ncp6 together with the order of peptoid-peptoid and peptoid-surface binding energies, i.e., peptoid-Au(100) < peptoid-peptoid < peptoid-Au(111), further amplifies its distinct collective behavior on different Au surfaces. Our studies provide a fundamental understanding of the molecular origin of facet-selective adsorption and highlight the possibility of future designs and uses of sequence-defined peptoids for predictive syntheses of nanocrystals with designed shapes and properties.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Mechanisms of dissolution from gibbsite step edges elucidated by ab initio molecular dynamics with enhanced sampling

Surface reactions underpin our collective understanding of mineral dissolution. This impacts a range of predictive geochemical models (for weathering, the fate and transport of metals) as well as industrial utilization of minerals. Yet atomistic details are rarely known due to the complex mineral/fluid interfacial environment. There is significant need to understand the mechanistic details of dissolution reactions and how they depend on surface morphology and solution conditions. This work utilizes surface pit models in conjunction with changes to solution composition that mimic pH to explore surface detachment of aluminate from gibbsite, which is a primary source of Al in soils and within the industrial processing of aluminum. Ab initio molecular dynamics simulations with enhanced sampling has been used to explore the detailed process of the detachment, the results of which indicate two potential pathways that are differentiated based upon the extent of water hydration. Here, the heights of the energy barriers depend upon the local morphology which influence the number of bridges (quasi-)simultaneously broken (1 or 2) or the Al-O coordination of the neighboring aluminum atoms (5 or 6) at the armchair edge. pH effects are significant, with a nearly 50% reduction in barrier height under alkaline conditions that are relevant to geothermal fluids and Al extraction from minerals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Single-photoelectron collection efficiency in 4D ultrafast electron microscopy

In femtosecond (fs) 4D ultrafast electron microscopy (UEM), a tradeoff is made between photoelectrons per packet and time resolution. One consequence of this can be longer-than-desirable acquisition times for low-density packets, and particularly for low repetition rates when complete photothermal dissipation is required. Thus, gaining an understanding of photoelectron trajectories in the gun region is important for identifying factors that limit collection efficiency (CE; fraction of photoelectrons that enter the illumination system). Here, we continue our work on the systematic study of photoelectron trajectories in the gun region of a Thermo Fisher/FEI Tecnai Femto UEM, focusing specifically on CE in the single-electron regime. Using General Particle Tracer, calculated field maps, and the exact architecture of the Tecnai Femto UEM, we simulated the effects of fs laser parameters and key gun elements on CE. The results indicate CE strongly depends upon the laser spot size on the source, the (unbiased) Wehnelt aperture diameter, and the incident photon energy. The CE dispersion with laser spot size is found to be strongly dependent on aperture diameter, being nearly dispersionless for the largest apertures. A gun crossover is also observed, with the beam-waist position being dependent on the aperture diameter, further illustrating that the Wehnelt aperture acts as a simple, fixed electrostatic lens in UEM mode. This work provides further insights into the operational aspects of fs 4D UEM.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

A statistical mechanics framework for polymer chain scission, based on the concepts of distorted bond potential and asymptotic matching

To design increasingly tough, resilient, and fatigue-resistant elastomers and hydrogels, the relationship between controllable network parameters at the molecular level (bond type, non-uniform chain length, entanglement density, etc.) to macroscopic quantities that govern damage and failure must be established. Many of the most successful constitutive models for elastomers have been rooted in statistical mechanical treatments of polymer chains. Typically, such constitutive models have used variants of the freely jointed chain model with rigid links. However, since the free energy state of a polymer chain is dominated by enthalpic bond distortion effects as the chain approaches its rupture point, bond extensibility ought to be accounted for if the model is intended to capture chain rupture. To that end, a new bond potential is supplemented to the freely jointed chain model (as derived in the u FJC framework of Buche and Silberstein (2021) and Buche et al. (2022)), which we have extended to yield a tractable, closed-form model of single chain behavior that should be amenable to continuum-level constitutive model development. Inspired by the asymptotically matched u FJC model response in both the low/intermediate chain force and high chain force regimes, a simple, quasi-polynomial bond potential energy function is derived. This bond potential exhibits harmonic behavior near the equilibrium state and anharmonic behavior for large bond stretches tending to a characteristic energy plateau (akin to the Lennard-Jones and Morse bond potentials). Using this bond potential, approximate yet highly-accurate analytical functions for bond stretch and chain force dependent upon chain stretch are established. Then, using this polymer chain model, a stochastic thermal fluctuation-driven chain rupture framework is developed. This framework is based upon a force-modified tilted bond potential that accounts for distortional bond potential energy, allowing for the derivation and subsequent calculation of the dissipated chain scission energy. Here, the cases of rate-dependent and rate-independent scission are accounted for throughout the rupture framework. The impact of Kuhn segment number on chain rupture behavior is also investigated. The model is fit to single chain mechanical response data collected from atomic force microscopy tensile tests for validation and to glean deeper insight into the molecular physics taking place. Due to their analytical nature, this polymer chain model and the associated rupture framework can in the future be implemented in finite element models accounting for fracture and fatigue in polydisperse elastomer networks.

36 MATERIALS SCIENCE↗

Parallel algorithms for hyperdynamics and local hyperdynamics

Hyperdynamics (HD) is a method for accelerating the timescale of standard molecular dynamics (MD). It can be used for simulations of systems with an energy potential landscape that is a collection of basins, separated by barriers, where transitions between basins are infrequent. HD enables the system to escape from a basin more quickly while enabling a statistically accurate renormalization of the simulation time, thus effectively boosting the timescale of the simulation. In [Kim, Perez, Voter, J Chem Phys, 139:144110, 2013)1, a local version of HD was formulated, which exploits the intrinsic locality characteristic typical of most systems to mitigate the poor scaling properties of standard HD as the system size is increased. In this paper, we discuss how both HD and local HD can be formulated to run efficiently in parallel. We have implemented these ideas in the LAMMPS MD code, which means HD can be used with any interatomic potential LAMMPS supports. Together, these parallel methods allow simulations of any size to achieve the time acceleration offered by HD (which can be orders of magnitude), at a cost 3-5x that of standard MD. As examples, we performed two simulations of a million-atom system to model the diffusion and clustering of Pt adatoms on a large patch of Pt(100) surface for 80 and 160 μs.

74 ATOMIC AND MOLECULAR PHYSICS↗

Automated Energy-Dispersive X-ray Spectroscopy Analysis for Multi-Modal Few-Shot Learning

Scanning transmission electron microscopy (STEM) is a powerful tool that allows for the atomic-scale analysis of a materials’ structure, chemistry, and defect domains (Akers et al. 2021). The current generation of microscopes generate vast amounts of data, surpassing the limits of effective manual analysis traditionally performed by domain experts (Spurgeon et al. 2021). While recent strides in machine learning have significantly enhanced the processing of large and intricate datasets acquired through electron microscopy, the prevalent use of proprietary software packages for initial data collection poses a challenge. In many cases, these software packages act as a ‘black box’, constraining user functionality and hindering the output of data in a format that is conducive to seamless integration into machine learning models. This work addresses these challenges by adapting HyperSpy, an open-source Python library, for the analysis and quantification of raw energy dispersive spectroscopy (EDS) data acquired through STEM. The modified HyperSpy code successfully facilitates user-defined segmentation of the data, enabling the integration of atomic %, weight %, and raw EDS spectra for each segmented region into an existing few-shot machine learning model. While initial results reveal discrepancies in quantified atomic and weight percentages when compared to proprietary software, ongoing efforts aim to rectify this issue by refining the fit of the HyperSpy model to the EDS spectra. Overall, this research underscores the potential of open-source tools like HyperSpy to enhance the accessibility of analytical tools, fostering a transparent and user-friendly environment for seamlessly incorporating electron microscopy data into machine learning models.

36 MATERIALS SCIENCE↗

Search for subsolar-mass black hole binaries in the second part of Advanced LIGO’s and Advanced Virgo’s third observing run

ABSTRACT We describe a search for gravitational waves from compact binaries with at least one component with mass $0.2$–$1.0 \, \mathrm{M}_\odot$ and mass ratio q ≥ 0.1 in Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) and Advanced Virgo data collected between 2019 November 1, 15:00 utc and 2020 March 27, 17:00 utc. No signals were detected. The most significant candidate has a false alarm rate of $0.2 \, \rm {yr}^{-1}$. We estimate the sensitivity of our search over the entirety of Advanced LIGO’s and Advanced Virgo’s third observing run, and present the most stringent limits to date on the merger rate of binary black holes with at least one subsolar-mass component. We use the upper limits to constrain two fiducial scenarios that could produce subsolar-mass black holes: primordial black holes (PBH) and a model of dissipative dark matter. The PBH model uses recent prescriptions for the merger rate of PBH binaries that include a rate suppression factor to effectively account for PBH early binary disruptions. If the PBHs are monochromatically distributed, we can exclude a dark matter fraction in PBHs $f_\mathrm{PBH} \gtrsim \, 0.6$ (at 90 per cent confidence) in the probed subsolar-mass range. However, if we allow for broad PBH mass distributions, we are unable to rule out fPBH = 1. For the dissipative model, where the dark matter has chemistry that allows a small fraction to cool and collapse into black holes, we find an upper bound fDBH < 10−5 on the fraction of atomic dark matter collapsed into black holes.

79 ASTRONOMY AND ASTROPHYSICS↗

Path integral Monte Carlo approach to the structural properties and collective excitations of liquid $$^3{\text {He}}$$ without fixed nodes

Abstract Due to its nature as a strongly correlated quantum liquid, ultracold helium is characterized by the nontrivial interplay of different physical effects. Bosonic $$^4{\text {He}}$$ 4 He exhibits superfluidity and Bose-Einstein condensation. Its physical properties have been accurately determined on the basis of ab initio path integral Monte Carlo (PIMC) simulations. In contrast, the corresponding theoretical description of fermionic $$^3{\text {He}}$$ 3 He is severely hampered by the notorious fermion sign problem, and previous PIMC results have been derived by introducing the uncontrolled fixed-node approximation. In this work, we present extensive new PIMC simulations of normal liquid $$^3{\text {He}}$$ 3 He without any nodal constraints. This allows us to to unambiguously quantify the impact of Fermi statistics and to study the effects of temperature on different physical properties like the static structure factor $$S({\mathbf {q}})$$ S ( q ) , the momentum distribution $$n({\mathbf {q}})$$ n ( q ) , and the static density response function $$\chi ({\mathbf {q}})$$ χ ( q ) . In addition, the dynamic structure factor $$S({\mathbf {q}},\omega )$$ S ( q , ω ) is rigorously reconstructed from imaginary-time PIMC data. From simulations of $$^3{\text {He}}$$ 3 He , we derived the familiar phonon–maxon–roton dispersion function that is well-known for $$^4{\text {He}}$$ 4 He and has been reported previously for two-dimensional $$^3{\text {He}}$$ 3 He films (Nature 483:576–579 (2012)). The comparison of our new results for both $$S({\mathbf {q}})$$ S ( q ) and $$S({\mathbf {q}},\omega )$$ S ( q , ω ) with neutron scattering measurements reveals an excellent agreement between theory and experiment.

97 MATHEMATICS AND COMPUTING↗

S-QGPU: Shared quantum gate processing unit for distributed quantum computing

We propose a distributed quantum computing (DQC) architecture in which individual small-sized quantum computers are connected to a shared quantum gate processing unit (S-QGPU). The S-QGPU comprises a collection of hybrid two-qubit gate modules for remote gate operations. In contrast to conventional DQC systems, where each quantum computer is equipped with dedicated communication qubits, S-QGPU effectively pools the resources (e.g., the communication qubits) together for remote gate operations, and, thus, significantly reduces the cost of not only the local quantum computers but also the overall distributed system. Our preliminary analysis and simulation show that S-QGPU's shared resources for remote gate operations enable efficient resource utilization. When not all computing qubits (also called data qubits) in the system require simultaneous remote gate operations, S-QGPU-based DQC architecture demands fewer communication qubits, further decreasing the overall cost. Alternatively, with the same number of communication qubits, it can support a larger number of simultaneous remote gate operations more efficiently, especially when these operations occur in a burst mode.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗