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At least 1,783 records · Page 99

An updated technique to obtain explosive kinetics data on microsecond timescales

There are few techniques available for chemists to obtain time-to-explosion data with known temperature inputs at the early stages of the design and synthesis of new explosives. In the 1960s, a technique was developed to rapidly heat milligram-quantities of confined explosives to ~1000 K on microsecond timescales. Wenograd loaded explosives inside stainless steel hypodermic needles, connected them to a fireset and rapidly discharged a capacitor through the steel. He obtained the temperature by measuring the needle resistance in a Wheatstone bridge arrangement and the time to explosion from a needle rupture. However, owing to the narrow-gauge needles used in the original research, the experiment was only possible with melt-castable explosives; it was never replicated, and modern diagnostics are now available with advances beyond the 1960s. Here, we report the development of the High Explosives Initiation Time (HEIT) test, which utilizes a 250 J pulsed power system to heat the needles. This work extends the Wenograd approach by using optical diagnostics, computational modeling, and advanced techniques to measure needle resistance and needle rupture. Preliminary rate information for pentaerythritol tetranitrate (PETN) will be presented.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗

Probing electronic and dielectric properties of ultrathin Ga 2 O 3 /Al 2 O 3 atomic layer stacks made with in vacuo atomic layer deposition

Ultrathin (1–4 nm) films of wide-bandgap semiconductors are important to many applications in microelectronics, and the film properties can be sensitively affected by defects especially at the substrate/film interface. Motivated by this, an in vacuo atomic layer deposition (ALD) was developed for the synthesis of ultrathin films of Ga 2 O 3 /Al 2 O 3 atomic layer stacks (ALSs) on Al electrodes. It is found that the Ga 2 O 3 /Al 2 O 3 ALS can form an interface with the Al electrode with negligible interfacial defects under the optimal ALD condition whether the starting atomic layer is Ga 2 O 3 or Al 2 O 3 . Such an interface is the key to achieving an optimal and tunable electronic structure and dielectric properties in Ga 2 O 3 /Al 2 O 3 ALS ultrathin films. In situ scanning tunneling spectroscopy confirms that the electronic structure of Ga 2 O 3 /Al 2 O 3 ALS can have tunable bandgaps (E g ) between ~2.0 eV for 100% Ga 2 O 3 and ~3.4 eV for 100% Al 2 O 3 . With variable ratios of Ga:Al, the measured E g exhibits significant non-linearity, agreeing with the density functional theory simulation, and tunable carrier concentration. Furthermore, the dielectric constant of ultrathin Ga 2 O 3 /Al 2 O 3 ALS capacitors is tunable through the variation in the ratio of the constituent Ga 2 O 3 and Al 2 O 3 atomic layer numbers from 9.83 for 100% Ga 2 O 3 to 8.28 for 100% Al 2 O 3 . The high ε leads to excellent effective oxide thickness ~1.7–2.1 nm for the ultrathin Ga 2 O 3 /Al 2 O 3 ALS, which is comparable to that of high-K dielectric materials.

36 MATERIALS SCIENCE↗

Investigating instabilities in magnetized low-pressure capacitively coupled RF plasma using particle-in-cell (PIC) simulations

The effect of a uniform magnetic field on particle transport in low-pressure radio frequency (RF) capacitively coupled plasma (CCP) has been studied using a particle-in-cell model. Three distinct regimes of plasma behavior can be identified as a function of the magnetic field. In the first regime at low magnetic fields, asymmetric plasma profiles are observed within the CCP chamber due to the effect of $\overrightarrow{E}$ × $\overrightarrow{B}$ drift. As the magnetic field increases, instabilities develop and form self-organized spoke-shaped structures that are distinctly seen within the bulk plasma closer to the sheath. In this second regime, the spoke-shaped coherent structures rotate inside the plasma chamber in the -$\overrightarrow{E}$ × $\overrightarrow{B}$ direction, where $\overrightarrow{E}$ × $\overrightarrow{B}$ are the DC electric and magnetic field vectors, respectively, and the DC electric field exists in the sheath and pre-sheath regions. The spoke rotation frequency is in the megahertz range. As the magnetic field strength increases further, the rotating coherent spokes continue to exist near the sheath. The coherent structures are, however, accompanied by new small-scale incoherent structures originating and moving within the bulk plasma region away from the sheath. This is the third regime of plasma behavior. The threshold values of the magnetic field between these regimes were found not to vary with changing plasma reactor geometry (e.g., area ratio between ground and powered electrodes) or the use of an external capacitor between the RF-powered electrode and the RF source. The threshold values of the magnetic field between these regimes shift toward higher values with increasing gas pressure. Analysis of the results indicates that the rotating structures are due to the lower hybrid instability driven by density gradients and electron-neutral collisions. This paper provides guidance on the upper limit of the magnetic field for instability-free operation in low-pressure CCP-based semiconductor deposition and etch systems that use the external magnetic field for plasma uniformity control.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Optimizing protic ionic liquid electrolyte for pre-intercalated Ti 3 C 2 T x MXene supercapacitor electrodes

Electrical double-layer capacitors (EDLCs) are of increasing importance in energy storage from renewable sources. The properties of the electrode and electrolyte materials influence the energy and power densities of EDLCs. We examined the specific capacitance and ion dynamics of a protic ionic liquid confined in pre-intercalated Ti 3 C 2 T x MXene. Our electrochemical measurements demonstrated that the creation of a protic ionic liquid, 1-butyl-3-H-imidazolium bis(trifluoromethanesulfonyl)imide (BuIMH-NTf2), using a mixture of ionic liquid, 1-butyl imidazole (BuIM), and salt, bis(trifluoromethanesulfonyl)imide (HNTf2), in a ratio of 0.8:0.2 led to the optimal capacitance. Remarkably, quasi-elastic neutron scattering measurements revealed increased particle mobility at this composition, attributed to the more efficient accumulation of BuIMH + on the electrode surface. This deposit of additional ions results in fewer BuIM molecules away from the surface, enhancing their mobility due to reduced crowding. This composition-dependent electrochemical behavior will guide the formulation of more efficient protic ionic liquid systems, enabling faster ion transport in energy storage devices.

36 MATERIALS SCIENCE↗

Fault localization in a microfabricated surface ion trap using diamond nitrogen-vacancy center magnetometry

Here, as quantum computing hardware becomes more complex with ongoing design innovations and growing capabilities, the quantum computing community needs increasingly powerful techniques for fabrication failure root-cause analysis. This is especially true for trapped-ion quantum computing. As trapped-ion quantum computing aims to scale to thousands of ions, the electrode numbers are growing to several hundred, with likely integrated photonic components also adding to the electrical and fabrication complexity, making faults even harder to locate. In this work, we used a high-resolution quantum magnetic imaging technique, based on nitrogen-vacancy centers in diamond, to investigate short-circuit faults in an ion trap chip. We imaged currents from these short-circuit faults to ground and compared them to intentionally created faults, finding that the root cause of the faults was failures in the on-chip trench capacitors. This work, where we exploited the performance advantages of a quantum magnetic sensing technique to troubleshoot a piece of quantum computing hardware, is a unique example of the evolving synergy between emerging quantum technologies to achieve capabilities that were previously inaccessible.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Electrical and structural characterization of in situ MOCVD Al2O3/β-Ga2O3 and Al2O3/β-(AlxGa1−x)2O3 MOSCAPs

This study investigates the electrical and structural properties of metal–oxide–semiconductor capacitors (MOSCAPs) with in situ metal-organic chemical vapor deposition-grown Al2O3 dielectrics deposited at varying temperatures on (010) β-Ga2O3 and β-(AlxGa1−x)2O3 films with different Al compositions. The Al2O3/β-Ga2O3 MOSCAPs exhibited a strong dependence of electrical properties on Al2O3 deposition temperature. At 900 °C, reduced voltage hysteresis (∼0.3 V) with improved reverse breakdown voltage (74.5 V) was observed, corresponding to breakdown fields of 5.01 MV/cm in Al2O3 and 4.11 MV/cm in β-Ga2O3 under reverse bias. In contrast, 650 °C deposition temperature resulted in higher voltage hysteresis (∼3.44 V) and lower reverse breakdown voltage (38.8 V) with breakdown fields of 3.69 and 2.87 MV/cm in Al2O3 and β-Ga2O3, respectively, but exhibited impressive forward breakdown field, increasing from 5.62 MV/cm at 900 °C to 7.25 MV/cm at 650 °C. High-resolution scanning transmission electron microscopy (STEM) revealed improved crystallinity and sharper interfaces at 900 °C, contributing to enhanced reverse breakdown performance. For Al2O3/β-(AlxGa1−x)2O3 MOSCAPs, increasing Al composition (x) from 5.5% to 9.2% reduced net carrier concentration and improved reverse breakdown field contributions from 2.55 to 2.90 MV/cm in β-(AlxGa1−x)2O3 and 2.41 to 3.13 MV/cm in Al2O3. The electric field in Al2O3 dielectric under forward bias breakdown also improved from 5.0 to 5.4 MV/cm as Al composition increased from 5.5% to 9.2%. The STEM imaging confirmed the compositional homogeneity and excellent stoichiometry of both Al2O3 and β-(AlxGa1−x)2O3 layers. These findings demonstrate the robust electrical performance, high breakdown fields, and excellent structural quality of Al2O3/β-Ga2O3 and Al2O3/β-(AlxGa1−x)2O3 MOSCAPs, highlighting their potential for high-power electronic applications.

Bhuiyan, A. F. M. Anhar Uddin (ORCID:0000000334212↗

Quantitative characterization of ZrO2 gate dielectric interface with tellurium

Tellurium (Te) has recently emerged as a promising p-type semiconductor that can be processed at low temperatures, compatible with back end of line CMOS integration. Characterization of tellurium–dielectric interfaces is essential for further device advancements. Here, the interface quality of Te with ZrO2 gate dielectric is studied in a metal-oxide semiconductor capacitor structure. The interface trap density (Dit) is measured as a function of atomic layer deposition (ALD) temperature, without the use of a seed layer. Given the low thermal budget of Te, the ALD temperature is shown to be particularly important. The lowest Dit of 5 × 1012 states/cm2 eV is obtained at a low ALD process temperature of 120 °C. To further assess the impact of Dit on device performance, field-effect transistors (FETs) were fabricated. The subthreshold swing and effective hole mobility of the FETs were analyzed in relation to Dit, emphasizing the importance of a defect-minimized interface for enhancing Te transistor performance.

Byeon, Kyeong-Jae↗

High-frequency electrical behavior in V 3 O 5 thin films

Vanadium oxides are known for their metal–insulator transition (MIT), with V 3 O 5 being notable for its transition temperature exceeding room temperature. At about 430 K, this material shows a change in crystal symmetry accompanied with one order of magnitude increase in its electrical conductivity and alterations in its optical properties. Although the property changes during the MIT in V 3 O 5 are less pronounced than those observed in VO 2 , its transition temperature is 90 K higher, making it appealing for applications requiring elevated temperatures. In this article, the high-frequency characteristics were determined in a V 3 O 5 two-terminal device in the range from 5 to 35 GHz. The S-parameters showed that the return loss at room temperature was close to −1.5 dB, and the isolation between ports was approximately −50 dB. At temperatures above the metal–insulator transition, the isolation decreased to around −40 dB at 35 GHz. For S11 and S22, similar behavior was observed at room temperature, with a notable change in the S-parameter phase of the device. This behavior suggests that V 3 O 5 may function well as a capacitor because the considerable change in phase could control the flow of electrical signals in devices. This property also may be used for matching purposes, especially considering its response to temperature changes. Additionally, conductivity calculation from S-parameters shows a decrease of approximately two orders of magnitude at 500 K and one order of magnitude at 300 K compared to DC values. These findings highlight V 3 O 5 potential for integration into radio frequency devices that demand consistent performance in high-temperature environments.

36 MATERIALS SCIENCE↗

Thickness scaling and ferroelectric switching in wurtzite structure zinc magnesium oxide thin films

This study reports the thickness scaling of sputtered ferroelectric Zn 0.61 Mg 0.39 O (ZMO) thin films down to 43 nm. Encapsulated IrO 2 /ZMO/Ir capacitors exhibited switchable polarizations exceeding 50 μC cm −2 and coercive fields that increased from 3.9 to 4.4 MV cm −1 as the thickness decreased. Switching kinetics are best described by the simultaneous non-linear nucleation and the growth model. Bimodal switching is prevalent at low thicknesses, with the fastest switching times measured to be approximately 400 ns. Device encapsulation made ZMO switching kinetics more abrupt, potentially due to changes in the concentration of atmosphere-induced defects such as hydroxides. These results demonstrate stable ferroelectricity and sub-microsecond switching in sub-50 nm wurtzite ZMO, highlighting its potential as a low-voltage ferroelectric for integrated nonvolatile memory applications.

36 MATERIALS SCIENCE↗

Bragg Coherent Imaging of Fluctuating Domains

This review presents two examples of domain dynamics in crystalline solid state samples which were imaged recently with synchrotron X-rays by Bragg Coherent Diffraction methods. Domains are associated with sub-atomic phase shifts in the structure which are revealed by the strong phase contrast associated with Bragg diffraction. At low temperatures, La2-xSrxNiO4 has an antiferromagnetic spin-ordered phase with correlation lengths of 200 nm; its domains are imaged with magnetic Bragg Ptychography to visualize their slow fluctuations. Hydrothermally synthesized nanocrystalline BaTiO3 is found to be made up of ~50 nm domains using Bragg Coherent Diffraction Imaging; electric field induced movement of its internal domain walls is a likely explanation of the nano-material's anomalously large dielectric constant, which is used in the manufacture of multilayer ceramic capacitors

77 NANOSCIENCE AND NANOTECHNOLOGY↗

Dielectric performance of high permitivity nanocomposites: impact of polystyrene grafting on BaTiO 3 and TiO 2

Polymer nanocomposites are a promising concept to improve energy storage density of capacitors, but realizing their hypothetical gains has proved challenging. The introduction of high permittivity fillers often leads to reduction in breakdown strength due to field exclusion, which intensifies the applied electric field within the polymer matrix near nanoparticle interfaces. This has prompted research in developing new nanoparticle functionalization chemistries and processing concepts to maximize particle separation. Herein, we compare the dielectric performance of blended nanocomposites to matrix free assemblies of hairy (polymer-grafted) nanoparticles (HNPs) that exhibit comparable overall morphology. The dielectric breakdown strength of polystyrene-grafted BaTiO3 (PS@BaTiO3) systems was over 40% greater than a blended nanocomposite with similar loading (~25% v/v BaTiO3). Hairy nanoparticles with TiO2 cores followed similar trends in breakdown strength as a function of inorganic loading up to 40% v/v. Dielectric loss for PS@BaTiO3 HNPs was 2-5 times lower than analogous blended films for a wide frequency spectrum (1 Hz to 100 kHz). For BaTiO3 content above 7% v/v, grafting the polymer chains to the nanoparticle significantly improved energy storage density and efficiency, likely due to the polymer canopy mitigating interfacial transport and restricting particle-particle hot-spots by establishing a finite minimum particle separation.

25 ENERGY STORAGE↗

High-fidelity modeling of breakdown in helium: initiation processes and secondary electron emission

Understanding the role of physical processes contributing to breakdown is critical for many applications in which breakdown is undesirable, such as capacitors, and applications in which controlled breakdown is intended, such as plasma medicine, lightning protection, and materials processing. The electron emission from the cathode is a critical source of electrons which then undergo impact ionization to produce electrical breakdown. In this study, the role of secondary electron yields due to photons (γ ph ) and ions (γ i ) in direct current breakdown is investigated using a particle-in-cell direct simulation Monte Carlo model. The plasma studied is a one-dimensional discharge in 50 Torr of pure helium with a platinum cathode, gap size of 1.15 cm, and voltages of 1.2–1.8 kV. The current traces are compared with experimental measurements. Larger values of γ ph generally result in a faster breakdown, while larger values of γ i result in a larger maximum current. The 58.4 nm photons emitted from He(2 1 P) are the primary source of electrons at the cathode before the cathode fall is developed. Of the values of γ ph and γ i investigated, those which provide the best agreement with the experimental current measurements are γ ph = 0.005 and γ i = 0.01. In this work, these values are significantly lower than those in the literature for pristine platinum or for a graphitic carbon film which we speculate may cover the platinum. This difference is in part due to the limitations of a one-dimensional model but may also indicate surface conditions and exposure to a plasma can have a significant effect on the secondary electron yields. The effects of applied voltage and the current produced by a UV diode which was used to initiate the discharge, are also discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Computational and experimental investigation of plasma deflagration jets and detonation shocks in coaxial plasma accelerators

Here, we present a magnetohydrodynamic (MHD) numerical simulation to study the physical mechanisms underlying plasma acceleration in a coaxial plasma gun. Coaxial plasma accelerators are known to exhibit two distinct modes of operation depending on the delay between gas loading and capacitor discharging. Shorter delays lead to a high velocity plasma deflagration jet and longer delays produce detonation shocks. During a single operational cycle that typically consists of two discharge events, the plasma acceleration exhibits a behavior characterized by a mode transition from deflagration to detonation. The first of the discharge events, a deflagration that occurs when the discharge expands into an initially evacuated domain, requires a modification of the standard MHD algorithm to account for rarefied regions of the simulation domain. The conventional approach of using a low background density gas to mimic the vacuum background results in the formation of an artificial shock, inconsistent with the physics of free expansion. To this end, we present a plasma-vacuum interface tracking framework with the objective of predicting a physically consistent free expansion, devoid of the spurious shock obtained with the low background density approach. The interface tracking formulation is integrated within the MHD framework to simulate the plasma deflagration and the second discharge event, a plasma detonation, formed due to its initiation in a background prefilled with gas remnant from the deflagration. The mode transition behavior obtained in the simulations is qualitatively compared to that observed in the experiments using high framing rate Schlieren videography. The deflagration mode is further investigated to understand the jet formation process and the axial velocities obtained are compared against experimentally obtained deflagration plasma front velocities. The simulations are also used to provide insight into the conditions responsible for the generation and sustenance of the magnetic pinch. The pinch width and number density distribution are compared to experimentally obtained data to calibrate the inlet boundary conditions used to set up the plasma acceleration problem.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Total dose effect of Al 2 O 3 -based metal–oxide–semiconductor structures and its mechanism under gamma-ray irradiation

In our work, insights into the total dose response and native point defect behavior in the Al2O3 gate dielectric during irradiation were gained by gamma-ray irradiation experiments and first-principles calculations. It is found that the O vacancy (VO) can act as a hole trap in the Al2O3 gate dielectric during irradiation, leading to the negative shift of the capacitance–voltage (C–V) curves of the Al2O3-based metal–oxide–semiconductor (MOS) structure. Our calculations show that the neutral defect VO becomes a +2 charged center after irradiation, and the positively charged VO is a kind of conductive path for electrons, which contributes to an increase of the leakage current in the irradiated MOS capacitors. Additionally, the trapped holes are accumulated with irradiation doses, which can lower the barrier height of the Al2O3 gate oxide and further cause the increase of the leakage current. The other native point defects in the Al2O3 layer, such as aluminum vacancy (VAl), aluminum interstitial (Ali) and oxygen interstitial (Oi), only act as fixed charge centers during irradiation. Net negative charges existing in the Al2O3 layers before irradiation are mainly induced by the negatively charged defects of VAl and Oi.

Engineering↗

Improving wafer-scale Josephson junction resistance variation in superconducting quantum coherent circuits

Quantum bits, or qubits, are an example of coherent circuits envisioned for next-generation computers and detectors. A robust superconducting qubit with a coherent lifetime of O (100 µs) is the transmon: a Josephson junction functioning as a non-linear inductor shunted with a capacitor to form an anharmonic oscillator. In a complex device with many such transmons, precise control over each qubit frequency is often required, and thus variations of the junction area and tunnel barrier thickness must be sufficiently minimized to achieve optimal performance while avoiding spectral overlap between neighboring circuits. Simply transplanting our recipe optimized for single, stand-alone devices to wafer-scale (producing 64, 1x1 cm dies from a 150 mm wafer) initially resulted in global drifts in room-temperature tunneling resistance of ± 30%. Inferring a critical current variation from this resistance distribution, we present an optimized process developed from a systematic 38 wafer study that results in < 3.5% relative standard deviation (RSD) in critical current ( ) for 3000 Josephson junctions (both single-junctions and asymmetric SQUIDs) across an area of 49 cm 2 . Looking within a 1x1 cm moving window across the substrate gives an estimate of the variation characteristic of a given qubit chip. Our best process, utilizing ultrasonically assisted development, uniform ashing, and dynamic oxidation has shown = 1.8% within 1x1 cm, on average, with a few 1x1 cm areas having < 1.0% (equivalent to < 0.5%). Such stability would drastically improve the yield of multi-junction chips with strict critical current requirements.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

A pulse-burst laser system for Thomson scattering on NSTX-U

A pulse-burst laser system has been built for Thomson scattering on NSTX-U, and is currently being integrated into the NSTX-U Thomson scattering diagnostic system. The laser will be operated in three distinct modes. The base mode is continuous 30 Hz rep rate, and is the standard operating mode of the laser. The base mode will be interrupted to produce a "slow burst" (specified 1 kHz rep rate for 50 ms) or a "fast burst" (specified 10 kHz rep rate for 5 ms). The combination of base mode→ interruption→ burst mode is new and has not been implemented on any previous pulse-burst laser system. Laser pulsing is halted for a set period (~ 1 minute) following a burst to allow the YAG rods to cool; this type of operation is called a heat-capacity laser. The laser is Nd:YAG operated at 1064 nm, q-switched to produce ≥ 1.5 J pulses with ~ 20 ns FWHM. It is flashlamp pumped, with dual-rod oscillator (9 mm) and dual-rod amplifier (12 mm). Variable pulsewidth drive of the flashlamps is accomplished by IGBT (insulated gate bipolar transistor) switching of electrolytic capacitor banks. Direct control of the laser Pockels cell drive enables optimal pulse energy extraction. The laser system has demonstrated compliance with all specifications, and is capable of exceeding design specifications by significant margins, e.g., higher rep rates for longer burst periods. Burst operation of this laser system will be used to capture fast time evolution of the electron temperature and density profiles during events such as ELMs, the L-H transition, and various MHD modes.

47 OTHER INSTRUMENTATION↗

First operation of a multi-channel Q-Pix prototype: measuring transverse electron diffusion in a gas time projection chamber

We report measurements of the transverse diffusion of electrons in P-10 gas (90% Ar, 10% CH 4 ) in a laboratory-scale time projection chamber (TPC) utilizing a novel pixelated signal capture and digitization technique known as Q-Pix. The Q-Pix method incorporates a precision switched integrating transimpedance amplifier whose output is compared to a threshold voltage. Upon reaching the threshold, a comparator sends a 'reset' signal, initiating a discharge of the integrating capacitor. The time difference between successive resets is inversely proportional to the average current at the pixel in that time interval, and the number of resets is directly proportional to the total collected charge. We developed a 16-channel Q-Pix prototype fabricated from commercial off-the-shelf components and coupled them to 16 concentric annular anode electrodes to measure the spatial extent of the electron swarm that reaches the anode after drifting through the uniform field of the TPC. The swarm is produced at a gold photocathode using pulsed UV light. The measured transverse diffusion agrees with simulations in PyBoltz across a range of operating pressures (200–1500 Torr). These results demonstrate that a Q-Pix readout can successfully reconstruct the ionization topology in a TPC.

47 OTHER INSTRUMENTATION↗

Hybrid MEMS-CMOS ion traps for NISQ computing

Surging interest in engineering quantum computers has stimulated significant and focused research on technologies needed to make them manufacturable and scalable. In the ion trap realm this has led to a transition from bulk three-dimensional macro-scale traps to chip-based ion traps and included important demonstrations of passive and active electronics, waveguides, detectors, and other integrated components. At the same time as these technologies are being developed the system sizes are demanding more ions to run noisy intermediate scale quantum (NISQ) algorithms, growing from around ten ions today to potentially a hundred or more in the near future. To realize the size and features needed for this growth, the geometric and material design space of microfabricated ion traps must expand. In this paper we describe present limitations and the approaches needed to overcome them, including how geometric complexity drives the number of metal levels, why routing congestion affects the size and location of shunting capacitors, and how RF power dissipation can limit the size of the trap array. Finally, we also give recommendations for future research needed to accommodate the demands of NISQ scale ion traps that are integrated with additional technologies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗