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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

High-angular-momentum Rydberg states in a room-temperature vapor cell for dc electric-field sensing

We prepare and analyze Rydberg states with orbital quantum numbers ℓ ≤ 6 using three-optical-photon electromagnetically induced transparency (EIT) and radio frequency (rf) dressing, and employ the high- ℓ states in electric-field sensing. Rubidium-85 atoms in a room-temperature vapor cell are first promoted into the 25 F 5 / 2 state via Rydberg-EIT with three infrared laser beams. Two rf dressing fields then (near-)resonantly couple the 25 F , 25 H ( ℓ = 5 ) , and 25 I ( ℓ = 6 ) Rydberg states. The dependence of the rf-dressed Rydberg-state level structure on rf powers, rf and laser frequencies is characterized using EIT. Furthermore, we discuss the principles of dc-electric-field sensing using high- ℓ Rydberg states and experimentally demonstrate the method using test electric fields of ≲ 50 V/m induced via photo-illumination of the vapor-cell wall. We measure the highly nonlinear dependence of the dc-electric-field strength on the power of the photo-illumination laser. Numerical calculations, which reproduce our experimental observations well, elucidate the underlying physics. Our paper is relevant to high-precision spectroscopy of high- ℓ Rydberg states, Rydberg-atom-based electric-field sensing, and plasma electric-field diagnostics. Published by the American Physical Society 2024

Duspayev, Alisher (ORCID:0000000153225762)↗
DOE OSTI2020

Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling

In magic angle twisted bilayer graphene (TBG), electron-electron interactions play a central role, resulting in correlated insulating states at certain integer fillings. Identifying the nature of these insulators is a central question, and it is potentially linked to the relatively high-temperature superconductivity observed in the same devices. Here, we address this question using a combination of analytical strong-coupling arguments and a comprehensive Hartree-Fock numerical calculation, which includes the effect of remote bands. The ground state we obtain at charge neutrality is an unusual ordered state, which we call the Kramers intervalley-coherent (K-IVC) insulator. In its simplest form, the K-IVC order exhibits a pattern of alternating circulating currents that triples the graphene unit cell, leading to an “orbital magnetization density wave.” Although translation and time-reversal symmetry are broken, a combined “Kramers” time-reversal symmetry is preserved. Our analytic arguments are built on first identifying an approximate U⁡(4) × U⁡(4) symmetry, resulting from the remarkable properties of the TBG band structure, which helps select a low-energy manifold of states that are further split to favor the K-IVC state. This low-energy manifold is also found in the Hartree-Fock numerical calculation. We show that symmetry-lowering perturbations can stabilize other insulators and the semimetallic state, and we discuss the ground state at half-filling and give a comparison with experiments.

Bultinck, Nick↗
DOE OSTI2024

Learning Quantum States and Unitaries of Bounded Gate Complexity

While quantum state tomography is notoriously hard, most states hold little interest to practically minded tomographers. Given that states and unitaries appearing in nature are of bounded gate complexity, it is natural to ask if efficient learning becomes possible. In this work, we prove that to learn a state generated by a quantum circuit with G two-qubit gates to a small trace distance, a sample complexity scaling linearly in G is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by G gates to a small average-case error scales linearly in G . While sample-efficient learning can be achieved, we show that under reasonable cryptographic conjectures, the computational complexity for learning states and unitaries of gate complexity G must scale exponentially in G . We illustrate how these results establish fundamental limitations on the expressivity of quantum machine-learning models and provide new perspectives on no-free-lunch theorems in unitary learning. Together, our results answer how the complexity of learning quantum states and unitaries relate to the complexity of creating these states and unitaries. Published by the American Physical Society 2024

Zhao, Haimeng (ORCID:0000000166751489)↗
DOE OSTI2025

Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems

We establish a holographic duality between d -dimensional mixed-state symmetry-protected topological (mSPT) phases and ( d + 1 ) -dimensional subsystem symmetry-protected topological (SSPT) states. Specifically, we show that the reduced density matrix of the boundary layer of a ( d + 1 ) -dimensional SSPT state with subsystem symmetry S and global symmetry G corresponds to a d -dimensional mSPT phase with strong S and weak G symmetries. Conversely, we demonstrate that the wave function of an SSPT state can be constructed by replicating the density matrix of the corresponding lower-dimensional mSPT phase. This mapping links the density matrix in lower dimensions to the entanglement properties of higher-dimensional wave functions, providing an approach for analyzing nonlinear quantities and quantum information metrics in mixed-state systems. Our duality offers a new perspective for studying intrinsic mSPT phases that are unique to open quantum systems, without pure-state analogues. We show that strange correlators and twisted Rényi- N correlators can diagnose these nontrivial phases and we explore their connection to strange correlators in pure-state SSPT phases. Furthermore, we discuss several implications of this holographic duality, including a method for preparing intrinsic mSPT phases through the duality. Published by the American Physical Society 2025

Sun, Shijun (ORCID:0000000168880030)↗
DOE OSTI2020

Demonstration of a third-order hierarchy of topological states in a three-dimensional acoustic metamaterial

Classical wave systems have constituted an excellent platform for emulating complex quantum phenomena, such as demonstrating topological phenomena in photonics and acoustics. Recently, a new class of topological states localized in more than one dimension of a D -dimensional system, referred to as higher-order topological (HOT) states, has been reported, offering an even more versatile platform to confine and control classical radiation and mechanical motion. Here, we design and experimentally study a 3D topological acoustic metamaterial supporting third-order (0D) topological corner states along with second-order (1D) edge states and first-order (2D) surface states within the same topological bandgap, thus establishing a full hierarchy of nontrivial bulk polarization–induced states in three dimensions. The assembled 3D topological metamaterial represents the acoustic analog of a pyrochlore lattice made of interconnected molecules, and is shown to exhibit topological bulk polarization, leading to the emergence of boundary states.

Science & Technology - Other Topics↗
DOE OSTI2024

Arbitrary State Preparation via Quantum Walks Software

SF-24-077 Our software generates a circuit that prepares a specified target quantum state. It generates circuits with CNOT counts of O(mn), where m is the number of nonzero amplitudes of the state and n is the number of qubits. The method does not require ancillas. It is designed for sparse states, but works for any quantum state. Our method is based on the new perspective of dynamic continuous time quantum walks for quantum state preparation. This is an intuitive perspective where a graph is first constructed to traverse the basis states. The method works by 1.) construct a graph of the basis states, 2.) transform the graph into quantum walks, and 3.) construct the final gate based circuit optimized with our control reduction method.

Liu, Ji↗
DOE OSTI2021

A Possible Observation of Sigma-nn Continuum Structures and A Bound Sigma-NN State Using the (e, e 'K+) Reaction

The E12-17-003 (e, e'K+) experiment was carried out in the experimental Hall A facility of Jefferson Lab in November 2018. The experiment aimed at providing the experimental data for the unknown Lambda-n interaction by measuring the bound state or the resonance state as indicated by the HypHI experiment. The so obtained Lambda-n interaction was assumed to solve the existing charge symmetry breaking problem in the case of Lambda-N interaction. The experiment used Hall A high resolution spectrometers (both left and right spectrometers) positioned symmetrically at constant angle 13.2 deg. each. A 25 cm long aluminum cylinder filled with H-3 gas was used as a production target. The major calibration data were taken with the H target contained in an identical aluminum cylinder. The electroproduction of Lambda and Sigma^0 via p(e, e'K+)Lambda, Sigma^0 reactions was used to calibrate the absolute energy scale with the known masses of the Lambda and Sigma^0. Then by using the (e, e'K+) Sigma-reaction, two possible Lambda-nn resonance states and one NN bound state were observed. The observed states have the energy resolution about 1.6 MeV (FWHM), however, great r statistics are required to solidly confirm the observed states. The A = 3 and 4 Sigma bound states have been predicted long ago but only the A = 4 Sigma hypernucleus, that is He-4/Sigma, was found in the (K-, pi-) reaction. A careful Monte Carlo study was conducted to study the A (nuclear mass number) dependence on the missing mass resolution by using the identical experimental conditions. The simulated and experimentally obtained Lambda and Sigma^0 channels agreed within 100 keV in sigma, where sigma is the standard deviation. The intrinsic missing mass resolution of A = 3 (3/Lambda-n) resonance was predicted about sigma = 0.67 MeV with a natural width of about 0.6 MeV. However, due to the low statistics the precision does not permit sufficient constrain on the determination of the Lambda-n interaction. The reason of having such low statistics is due to the use of unoptimized Hall A system which was definitely feasible but not ideal for this experiment. In addition, the cross section was found to be much smaller than expected.

Pandey, Bishnu↗
DOE OSTI2022

Impacts of phase state and water content on secondary organic aerosol formation and partitioning (Final Technical Report)

Secondary organic aerosols (SOA) account for a major fraction of particulate matter in the atmosphere, affecting climate, air quality, and public health. SOA formation and evolution are highly complex processes involving both chemical reactions and transport of molecules in air, at particle surfaces, and within particles. The state of the matter, or “phase state,” comprising SOA can vary from a liquid, over an amorphous semi-solid, to a glassy solid, depending on chemical composition, water content, relative humidity (RH), and temperature. The occurrence of glassy and amorphous semi-solid states can pose limitations on the rate of transport of molecules, affecting gas-particle interactions and challenging the treatment of SOA in atmospheric models. The objective of this project was to improve fundamental understanding of the interplay of the phase state of particles and water content on the evolution of SOA formation. We have developed a method to estimate the glass transition temperature (T g ) of organic compounds to predict the phase state and viscosity of SOA. This method has been applied to high resolution mass spectrometry data of various types of SOA, achieving a good agreement with viscosity measurements. In addition, the viscosity estimation method was implemented into a regional model CMAQ to simulate diel and seasonal variations of SOA phase state over the U.S. Applying kinetic multilayer modeling, we analyzed laboratory experiments and ARM field measurements (e.g., HI-SCALE) on kinetic limitations of amine uptake. We have also conducted numerical simulations to estimate equilibration timescales of SOA partitioning. Overall, we have evaluated the impacts of phase state and water content on SOA lifecycle that should contribute to reducing the uncertainty of SOA representation in regional climate and air quality predictions. The project has yielded 17 publications, in which DE-SC0018349 is explicitly acknowledged. Please see the list of publications as below, followed by summary of research activities.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Small Hydropower Interconnections: Small Hydropower in the United States

Small hydropower projects, which we define as generators below 20 MW in capacity have been the predominant source of hydropower growth over the past decade and create the most cost-effective and environmentally permissible avenues for new hydropower installation in the United States (DOE 2016; Johnson et al. 2018). Small hydropower developers across the United States have found that interconnecting these projects with the grid can be challenging due to unexpected costs and schedule overruns. Understanding the interconnection challenges and improving the process may allow more small hydropower projects to be successful. Noting these challenges, the U.S. Department of Energy Water Power Technologies Office enlisted Pacific Northwest National Laboratory (PNNL) and Oak Ridge National Laboratory (ORNL) to investigate the small hydropower interconnection landscape across the United States. To begin to analyze the existing interconnection processes and challenges facing small hydropower, the state of small hydropower development in the U.S. must first be described to understand the characteristics of the industry. The first in a series, this paper presents the state of small hydropower projects in the U.S. to describe their type, location, and size based on data extracted from the HydroSource database (ORNL 2020). The following papers in the series will detail the variety of state interconnection processes to connect power generators with the grid (“Small Hydropower Interconnections: State Interconnection Processes”), analyze these interconnection processes (“Small Hydropower Interconnections: Analysis of Interconnection Processes”), and present best practices in interconnection processes (“Small Hydropower Interconnections: Best Practices”) that will help overcome barriers to future small hydropower development.

13 HYDRO ENERGY↗
DOE OSTI2023

Experiences of Member States in Building a Nuclear Security Infrastructure for New Nuclear Power Programmes

A nuclear power programme has many benefits, but it is a major undertaking that requires careful planning and preparation as well as a substantial investment in time. It also necessitates the establishment of a sustainable national infrastructure with committed and sustained financial and human resources. While nuclear power is not unique in this respect, it is different from other sources of energy because of the risks associated with the use of nuclear material, as well as the need for compliance with international legal instruments, internationally accepted nuclear safety standards, nuclear security guidance, and safeguards requirements. The IAEA’s Milestones in the Development of a National Infrastructure for Nuclear Power, IAEA Nuclear Energy Series No. NG-G-3.1 (Rev. 1), defines a phased approach that identifies 19 infrastructure issues that should be addressed in each of the three phases of the development of a nuclear infrastructure for a nuclear power programme. The publication Evaluation of the Status of National Nuclear Infrastructure Development IAEA Nuclear Energy Series NG-T-3.2 (Rev. 2) provides a methodology to determine the status of the infrastructure conditions covering all 19 issues identified in the Milestones Approach. A companion IAEA Implementing Guide, Establishing the Nuclear Security Infrastructure for a Nuclear Power Programme, IAEA Nuclear Security Series No. 19, provides guidance on the recommended actions to be taken by a State to establish an effective national nuclear security infrastructure for a nuclear power programme using the Milestones Approach. This publication is intended to provide guidance based on the experiences and good practices of Member States with embarking nuclear power programmes as well as Member States that are expanding their nuclear power programmes. The guidance is in the form of case studies, which highlight challenges, issues, and solutions identified by Member States as lessons learned for new nuclear power programmes. This TECDOC is designed to present the experiences of Member States to assist other Member States in developing a nuclear security programme. It addresses the key actions necessary to establish an effective national nuclear security infrastructure for a nuclear power programme consistent with the three phases of the IAEA Milestones Approach. The Member States that provided case studies in this document are at different stages in the development of a nuclear power programme.

42 ENGINEERING↗
DOE OSTI2023

State Energy Justice Roundtable Series: Energy Justice Metrics

The National Association of Regulatory Utility Commissioners (NARUC), National Association of State Energy Officials (NASEO), and National Governors Association (NGA) hosted a State Energy Justice Roundtable (Roundtable) in April 2022. Participants included federal and state decision-makers, members of community-based organizations, and subject-matter experts. The Roundtable members explored current state efforts to articulate and incorporate energy justice concerns into energy-related decision-making. Participants established connections with one another to better understand the current landscape of existing resources, learn about emerging efforts, and identify ongoing support opportunities for advancing energy justice.This paper is one of five authored by the host organizations on topics that were the focus of the Roundtable. Each paper summarizes key themes, emerging efforts, and group takeaways that were discussed at the Roundtable and should assist state members in developing and meeting their own state goals around energy justice. The papers all include the same discussions of background, introduction, and reading list so they can be read separately. Each paper is written from the perspective of one association and includes options for its members to take actions that could support more equitable state energy policies and programs. The five papers cover: (1) Participation in decision making (NARUC), (2) Customer affordability and arrearages (NARUC), (3) Energy justice metrics (NARUC), (4) Equity in clean energy research and development (NASEO), and (5) Equitable distributed energy resource (DER) access (NGA). The resources and recommendations listed in these papers are not meant to be exhaustive, as this field of study continues to evolve. Although this brief is focused on electricity, energy justice considerations extend to all energy needs and services, including the impact of energy extraction, processing, and distribution functions.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2023

State Energy Justice Roundtable Series: Customer Affordability and Arrearages

The National Association of Regulatory Utility Commissioners (NARUC), National Association of State Energy Officials (NASEO), and National Governors Association (NGA) hosted a State Energy Justice Roundtable (Roundtable) in April 2022. Participants included federal and state decision-makers, members of community-based organizations, and subject-matter experts. The Roundtable members explored current state efforts to articulate and incorporate energy justice concerns into energy-related decision-making. Participants established connections with one another to better understand the current landscape of existing resources, learn about emerging efforts, and identify ongoing support opportunities for advancing energy justice. This paper is one of five authored by the host organizations on topics that were the focus of the Roundtable. Each paper summarizes key themes, emerging efforts, and group takeaways that were discussed at the Roundtable and should assist state members in developing and meeting their own state goals around energy justice. The papers all include the same discussions of background, introduction, and reading list so they can be read separately. Each paper is written from the perspective of one association and includes options for its members to take actions that could support more equitable state energy policies and programs. The five papers cover: (1) Participation in decision making (NARUC) (2) Customer affordability and arrearages (NARUC) (3) Energy justice metrics (NARUC) (4) Equity in clean energy research and development (NASEO) (5) Equitable distributed energy resource (DER) access (NGA). The resources and recommendations listed in these papers are not meant to be exhaustive, as this field of study continues to evolve. Although this brief is focused on electricity, energy justice considerations extend to all energy needs and services, including the impact of energy extraction, processing, and distribution functions.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2023

State Energy Justice Roundtable Series: Participation in Decision Making

The National Association of Regulatory Utility Commissioners (NARUC), National Association of State Energy Officials (NASEO), and National Governors Association (NGA) hosted a State Energy Justice Roundtable (Roundtable) in April 2022. Participants included federal and state decision-makers, members of community-based organizations, and subject-matter experts. The Roundtable members explored current state efforts to articulate and incorporate energy justice concerns into energy-related decision-making. Participants established connections with one another to better understand the current landscape of existing resources, learn about emerging efforts, and identify ongoing support opportunities for advancing energy justice. This paper is one of five authored by the host organizations on topics that were the focus of the Roundtable. Each paper summarizes key themes, emerging efforts, and group takeaways that were discussed at the Roundtable and should assist state members in developing and meeting their own state goals around energy justice. The papers all include the same discussions of background, introduction, and reading list so they can be read separately. Each paper is written from the perspective of one association and includes options for its members to take actions that could support more equitable state energy policies and programs. The five papers cover: • Participation in decision making (NARUC) • Customer affordability and arrearages (NARUC) • Energy justice metrics (NARUC) • Equity in clean energy research and development (NASEO) • Equitable distributed energy resource (DER) access (NGA). The resources and recommendations listed in these papers are not meant to be exhaustive, as this field of study continues to evolve. Although this brief is focused on electricity, energy justice considerations extend to all energy needs and services, including the impact of energy extraction, processing, and distribution functions.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2024

Advanced Algorithms for Scrutiny of Mandatory State Reports Declarations to the IAEA (Final Project Report)

In compliance with their Comprehensive Safeguards Agreements, based on INFCIRC/153 (corrected) (International Atomic Energy Agency, 1972), States Party to the Treaty on the NonProliferation of Nuclear Weapons (NPT) are obligated to declare to the International Atomic Energy Agency (IAEA) all changes in their nuclear material inventory as well as movement of the material across boundaries of IAEA recognized material balance areas (MBA), inventories and nuclear material balances. This project addresses capabilities to detect irregularities in State reports, thus ensuring their accuracy and completeness, and in the broader context, States’ compliance with safeguards obligations of the NPT. A recent study (Henzl et al., 2022) (lead by this project’s PI) demonstrated how analysis of dynamic correlations in nuclear material movement within the entire fuel cycle of a State (viewed as a single system) can reveal variances consistent with and indicative of “irregular” activities. Expanding on this concept, novel ways to analyze State declarations themselves—again, for the State as a whole entity—will help the IAEA draw accurate safeguards conclusions and trust the validity and authenticity of Stategenerated declaration reports. Introducing new declaration analyses capabilities explored in this project will help to provide credible assurance of both the non-diversion of nuclear material from declared activities and of the absence of undeclared nuclear material and activities in the State in general.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗
DOE OSTI2020

Near-optimal ground state preparation

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Nearly-frustration-free ground state preparation

Solving for quantum ground states is important for understanding the properties of quantum many-body systems, and quantum computers are potentially well-suited for solving for quantum ground states. Recent work [1] has presented a nearly optimal scheme that prepares ground states on a quantum computer for completely generic Hamiltonians, whose query complexity scales as δ − 1 , i.e. inversely with their normalized gap. Here we consider instead the ground state preparation problem restricted to a special subset of Hamiltonians, which includes those which we term "nearly-frustration-free": the class of Hamiltonians for which the ground state energy of their block-encoded and hence normalized Hamiltonian α − 1 H is within δ y of -1, where δ is the spectral gap of α − 1 H and 0 ≤ y ≤ 1 . For this subclass, we describe an algorithm whose dependence on the gap is asymptotically better, scaling as δ y / 2 − 1 , and show that this new dependence is optimal up to factors of log ⁡ δ . In addition, we give examples of physically motivated Hamiltonians which live in this subclass. Finally, we describe an extension of this method which allows the preparation of excited states both for generic Hamiltonians as well as, at a similar speedup as the ground state case, for those which are nearly frustration-free.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Soil organic matter diagenetic state informs boreal forest ecosystem feedbacks to climate change

The fate of soil organic carbon (SOC) in boreal forests is dependent on the integrative ecosystem response to climate change. For example, boreal forest productivity is often nitrogen (N) limited, and climate warming can enhance N cycling and primary productivity. However, the net effect of this feedback on the SOC reservoir and its longevity with climate change remain unclear due to difficulty in detecting small differences between large and variable carbon (C) fluxes needed to determine net changes in soil reservoirs. The diagenetic state of SOC – resulting from the physicochemical and biological transformations that alter the original biomolecular composition of detrital inputs to soil over time – is useful for tracing the net response of SOC at the timescales relevant to climate change not usually discernible from fluxes and stocks alone. Here, we test two hypotheses using a mesic boreal forest climate transect: (1) the SOC diagenetic state is maintained across this climosequence, and (2) the maintenance of the SOC diagenetic state is a consequence of coupled soil C and N cycling, signifying the role of enhanced N cycling supporting SOC inputs that maintain SOC stocks within the warmer-climate forests. Shifts in nonvascular to vascular plant inputs with climate observed in these and other boreal forests highlighted the need to carefully separate biogeochemical indicators of SOC source from those signifying diagenetic alteration. We thus evaluated and applied lignin biomarkers to assess the diagenetic alteration of SOC in these boreal forest organic soils and directly compared the lignin diagenetic state with that of soil organic nitrogen (SON) assessed through amino acid composition. The lignin diagenetic state remained constant across the climate transect, indicating a balance between the input and removal of lignin in these mesic boreal forests. When combined with previous knowledge of these forest ecosystems, including the diagenetic state of SON and direct measures of C fluxes and stocks, the results indicate a coupled increase in C and N cycling with climate warming that supports forest productivity and maintains SOC stocks. This balance could markedly shift as other factors begin to limit forest productivity (e.g., trace nutrients, water) with further climate change or affect forest nutrient allocation (e.g., forest age or compositional change). Further application of the approach presented here could be used to detect the limits of this and other ecosystem–climate feedbacks, by providing a tractable and parameterizable index of the lignin state across large spatial scales, necessary for ecosystem-scale parameterizations.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Microscopic Derivation of Transition-state Theory for Complex Quantum Systems

The decay of quantum complex systems through a potential barrier is often described with transition-state theory, also known as RRKM theory in chemistry. Here we derive the basic formula for transition-state theory based on a generic Hamiltonian as might be constructed in a configuration-interaction basis. Two reservoirs of random Hamiltonians from Gaussian orthogonal ensembles are coupled to intermediate states representing the transition states at a barrier. Under the condition that the decay of the reservoirs to open channels is large, an analytic formula for reaction rates is derived. Here, the transition states act as independent Breit–Wigner resonances which contribute additively to the total transition probability, as is well known for electronic conductance through resonant tunneling states. It is also found that the transition probability is independent of the decay properties of the states in the second reservoir over a wide range of decay widths.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗