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DOE OSTI2021

The Role of Quantum Science Concepts in Enhancing Sensing and Imaging Technologies: Applications for Biology: Proceedings of a Workshop

Quantum concepts hold the potential to enable significant advances in sensing and imaging technologies that could be vital to the study of biological systems. The workshop Quantum Science Concepts in Enhancing Sensing and Imaging Technologies: Applications for Biology, held online March 8–10, 2021, was organized to examine the research and development needs to advance biological applications of quantum technology. Hosted by the National Academies of Sciences, Engineering, and Medicine, the event brought together experts working on state-of-the-art, quantum-enabled technologies and scientists who are interested in applying these technologies to biological systems. Through talks, panels, and discussions, the workshop facilitated a better understanding of the current and future biological applications of quantum-enabled technologies in fields such as microbiology, molecular biology, cell biology, plant science, mycology, and many others. The workshop was organized around three main themes. The first, quantum in biology, examined quantum concepts that are hypothesized to be important for life processes and that researchers are working to observe through biological imaging and sensing. The second, quantum for biology, addressed ways to use quantum concepts to enhance technologies for biological imaging and sensing. The third, biology for quantum, offered a wider discussion of how the frontiers of biological imaging and sensing could enable future study using quantum concepts, tools, or technologies. Throughout the workshop, participants identified a wide range of emerging approaches and opportunities at the intersection of quantum physics and biological sensing and imaging. During the workshop, there were some differences in how each speaker defined the term quantum. During one of the panels, Prem Kumar offered thoughts on what phenomena are classical versus quantum, explaining that techniques get progressively more quantum as you move from just having superposition to having superposition with measurement and entanglement. Another explanation from Clarice Aiello delineates the definition into several levels. This includes a base level of “quantum-ness,” which reflects that all matter is made of atoms, and when these particles are isolated they behave based on quantum mechanical principles. A second level is related to quantum coherence, where a single quantum object might be found in a coherent superposition state. A final level, which she described as the quantum-entangled level, involves multiple quantum systems which are entangled among themselves. Overall, the workshop touched on concepts such as superposition, entanglement, and squeezing and their potential implications for communication, computing, and simulation, in addition to the workshop’s main focal area, biological sensing and imaging. At the opening of the workshop, Thorsten Ritz of the University of California, Irvine, identified two questions at the heart of quantum biology: Is the machinery of life quantum mechanical, and can quantum mechanics be used to study the machinery of life in new ways? Participants highlighted systems in which researchers have explored these questions, from the vast array of molecular interactions involved in biological processes such as photosynthesis, to the mechanics involved in cellular functions such as differentiation and aggregation, to the role of oscillating magnetic fields in flight orientation among birds. Sensing and imaging technologies are crucial to biological research; these technologies could both enhance the study of quantum effects and be enhanced by quantum concepts. A critical challenge in biological research is to develop imaging and sensing tools that do not damage or interfere with the often fragile and fleeting systems being studied. Attendees discussed a variety of established and emerging technologies that could enhance noninvasive biological imaging, including single- and two-photon spectroscopy, single-molecule spectroscopy, quantum illumination, ghost imaging, and cryo-electron microscopy. One example came from Marlan Scully who gave a keynote address on the first day of the workshop. Scully emphasized the use of different laser technologies, which exhibit coherence and other quantum properties, in moving toward real-world biological applications, such as the detection of SARS-CoV-2. Both tools and theory will play an important role in advancing quantum biology research and applications. Several participants suggested theorists and experimentalists should work in tandem to understand and model biological processes. While physics often reduces systems to their simplest forms for fundamental insights, participants also noted the value of observing and understanding biological systems in all their “messiness,” capturing both the inner workings of biological systems and the complex interactions that occur within and between organisms. In discussions among participants, several attendees stressed the need to match emerging tools with the right scientific questions. Rather than developing quantum technologies as “a hammer looking for a nail,” participants emphasized a focus on exploring the problems these technologies are best suited to address. For example, it is important to consider the size of the phenomenon being studied, the timescales that are important in answering the scientific question, and other relevant considerations. Every tool along the spectrum from classical to quantum involves its own set of trade-offs. For example, Ted Laurence of the Lawrence Livermore National Laboratory said that quantum measurements, such as single photon counting, fluorescence transitions, and lasers, can take longer to produce the same results as classical measurements. These quantum measurements, however, do not require calibration and can enable new research questions to be answered. Prem Kumar, Northwestern University, noted that, despite their promise, quantum approaches should not be used simply for the sake of using quantum, especially in situations where classical approaches better meet the needs of the researcher. Understanding and applying quantum concepts could enable advances in a wide range of application areas including energy, synthetic biology, medicine, and sustainability. For example, Michelle O’Malley, University of California, Santa Barbara, described how improved noninvasive imaging approaches could help capture the complex interactions and functions involved in the breakdown of organic matter by microbial communities and lead to new technologies for capturing valuable products from plant waste. Several other participants discussed needs in tracking the movement of metabolites and molecules in microbial communities for insights into nutrient cycling in environments such as soil. Margaret Ahmad, Sorbonne University, discussed potential opportunities to leverage the magnetic properties of cryptochromes to advance new treatment approaches for diseases such as COVID-19 and cancer. Looking toward the future development of the field, participants discussed challenges to advancing quantum biology that arise from disciplinary disconnects between physicists and biologists. The siloing of academic research disciplines represents a significant barrier to progress. Disconnects in terminology, motivations and priorities, and structural barriers to collaborative work underscore the need for concerted efforts to bridge these divides. Attendees and speakers offered suggestions for resolving these divergences, establishing a shared language, moving the field forward, and fostering meaningful feedback between disciplines. Overall, participants stressed a need for balance, open communication, collaboration, unity, and clear dialogue on trade-offs between quantum and classical approaches. Keiko Torii, The University of Texas at Austin, said that the best collaborations happen when the project provides mutual advantages that can show off everyone’s talents, each team finds the work interesting, and partners develop a camaraderie to pursue new knowledge. To enable near- and long-term opportunities in this space, participants suggested that exploratory, high-risk funding could improve existing instrumentation to explore quantum enhancement collaboratively. They also emphasized the need for collaboration between quantum physicists and sensing/imaging scientists, which could be advanced through a dedicated quantum biology investigator program. People, even more than technology, will be crucial to the future of quantum biology. Participants explored training, education, and workforce needs to further develop this burgeoning field and cultivate the next generation of scientists. While many programs are still in their nascent stages, participants highlighted examples of approaches and programs being developed at various types of institutions to engage students and professional scientists in quantum biology research. Several participants stressed the need for an inclusive approach, spanning disciplines as well as communities to foster a diverse field fueled by the intellectual contributions of a wide range of people, including historically under-resourced schools and students. To increase awareness and excitement about quantum physics and related areas of biology, attendees suggested capitalizing on the “buzz” around quantum. Several participants emphasized the need to start early, introducing students to quantum concepts and their appealing “weirdness” in K–12 education. Engaging students early—before they become entrenched in traditional disciplinary siloes as typically happens in graduate school—could help to galvanize interest in the area and foster a generation of scientists with the interdisciplinary mindset and skills needed to advance this interdisciplinary field. While these efforts could be advanced at many levels and across multiple sectors, several participants suggested a national quantum biology center could be a valuable hub to coordinate and support quantum biology education and workforce development across academia, industry, and government.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2024

Dynamical edge modes and entanglement in Maxwell theory

Abstract Previous work on black hole partition functions and entanglement entropy suggests the existence of “edge” degrees of freedom living on the (stretched) horizon. We identify a local and “shrinkable” boundary condition on the stretched horizon that gives rise to such degrees of freedom. They can be interpreted as the Goldstone bosons of gauge transformations supported on the boundary, with the electric field component normal to the boundary as their symplectic conjugate. Applying the covariant phase space formalism for manifolds with boundary, we show that both the symplectic form and Hamiltonian exhibit a bulk-edge split. We then show that the thermal edge partition function is that of a codimension-two ghost compact scalar living on the horizon. In the context of a de Sitter static patch, this agrees with the edge partition functions found by Anninos et al. in arbitrary dimensions. It also yields a 4D entanglement entropy consistent with the conformal anomaly. Generalizing to Proca theory, we find that the prescription of Donnelly and Wall reproduces existing results for its edge partition function, while its classical phase space does not exhibit a bulk-edge split.

Physics↗
DOE OSTI2022

Partial densities of states from x-ray absorption and Auger electron spectroscopy: Use of core-hole memory

Ag L 3 edge x-ray absorption and L 3 −M 4,5 M 4,5 Auger electron spectra have been measured and simulated for a variety of x-ray electric-field polarization and Auger electron emission directions. The theory relies on density-functional theory, use of the Bethe–Salpeter equation, atomic multiplet theory, and a simplified model for the Auger line shape and its dependence on photon energy. We also demonstrate that partial densities of states for dT 2g , dE g , and sA g symmetry partial-wave channels at the Ag site in the solid can be deduced from the experimental measurements with only atomic theoretical input, i.e., with no solid-state theory other than assuming a Fermi edge.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Dilute Fermi gas at fourth order in effective field theory

Using effective field theory methods, we calculate for the first time the complete fourth-order term in the Fermi-momentum or k F a s expansion for the ground-state energy of a dilute Fermi gas. The convergence behavior of the expansion is examined for the case of spin one-half fermions and compared against quantum Monte-Carlo results, showing that the Fermi-momentum expansion is well-converged at this order for |k F a s | ≲ 0.5.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Sprain energy consequences for damage localization and fracture mechanics

The 2023 smooth Lagrangian Crack-Band Model (slCBM), inspired by the 2020 invention of the gap test, prevented spurious damage localization during fracture growth by introducing the second gradient of the displacement field vector, named the “sprain,” as the localization limiter. The key idea was that, in the finite element implementation, the displacement vector and its gradient should be treated as independent fields with the lowest ( C 0 ) continuity, constrained by a second-order Lagrange multiplier tensor. Coupled with a realistic constitutive law for triaxial softening damage, such as microplane model M7, the known limitations of the classical Crack Band Model were eliminated. Here, we show that the slCBM closely reproduces the size effect revealed by the gap test at various crack-parallel stresses. To describe it, we present an approximate corrective formula, although a strong loading-path dependence limits its applicability. Except for the rare case of zero crack-parallel stresses, the fracture predictions of the line crack models (linear elastic fracture mechanics, phase-field, extended finite element method (XFEM), cohesive crack models) can be as much as 100% in error. We argue that the localization limiter concept must be extended by including the resistance to material rotation gradients. We also show that, without this resistance, the existing strain-gradient damage theories may predict a wrong fracture pattern and have, for Mode II and III fractures, a load capacity error as much as 55%. Finally, we argue that the crack-parallel stress effect must occur in all materials, ranging from concrete to atomistically sharp cracks in crystals.

Science & Technology - Other Topics↗
DOE OSTI2019

Transistor-Based Work-Function Measurement of Metal–Organic Frameworks for Ultra-Low-Power, Rationally Designed Chemical Sensors

A classic challenge in chemical sensing is selectivity. Metal–organic frameworks (MOFs) are an exciting class of materials because they can be tuned for selective chemical adsorption. Adsorption events trigger work-function shifts, which can be detected with a chemical-sensitive field-effect transistor (power ≈microwatts). Here, several case studies were used towards generalizing the sensing mechanism, ultimately towards our metal-centric hypothesis. HKUST-1 was used as a proof-of-principle humidity sensor. The response is thickness independent, meaning the response is surface localized. ZIF-8 is demonstrated to be an NO 2 -sensing material, and the response is dominated by adsorption at metal sites. Finally, MFM-300(In) shows how standard hard–soft acid–base theory can be used to qualitatively predict sensor responses. This paper sets the groundwork for using the tunability of metal–organic frameworks for chemical sensing with distributed, scalable devices.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Turbulence theories and statistical closure approaches

When discussing research in physics and in science more generally, it is common to ascribe equal importance to the three components of the scientific trinity: theoretical, experimental, and computational studies. This review will explore the future of modern turbulence theory by tracing its history, which began in earnest with Kolmogorov’s 1941 analysis of turbulence cascade and inertial range [A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 30, 299, (1941); 32, 19, (1941)]. The 80th Anniversary of Kolmogorov’s landmark study is a welcome opportunity to survey the achievements and evaluate the future of the theoretical approach of turbulence research. Over the years, turbulence theories have been critically important in laying the foundation of our understanding of the nature of turbulent flows. In particular, the Direct Interaction Approximation (DIA) [R.H. Kraichnan, J. Fluid Mech., 5, 497 (1959)] and its subsequent development, known as the statistical closure approach, can be identified as perhaps the most profound single advancement. The remarkable success of the statistical closure has furnished a platform to study such essential concepts as the energy transfer process and interacting scales, and the roles of the straining and sweeping motions. More recently, the quasi-Lagrangian formulation of V. L’vov & I. Procaccia and Kraichnan’s solvable passive scalar model provided powerful ways to explore another fundamental aspect of turbulent flows, the phenomena of intermittency, and the associated anomalous scaling exponents. In the meantime, the theory of fluid equilibria has been developed to describe the large-scale structures that can emerge from turbulent cascades of two-dimensional and geophysical flows at a later time. And yet, despite all these successes, analytical treatments suffer from mathematical complexities. As a result, the utility of theoretical approaches has been limited to relatively idealized flows. On the other hand, in recent decades, computational abilities and experimental facilities have reached an unprecedented scale. Looking beyond the horizon, the imminent deployment of exascale supercomputers will generate complete datasets of the entire flow field of key benchmark flows, allowing researchers to extract additional measurements concerning fully developed, complex turbulent flow fields far beyond those available from the statistical closure theories. Some other developments that could potentially influence the future course of turbulence theories include the advancement of machine learning, artificial intelligence, and data science; likely disruptions arising from the advent of quantum computation; and the increasingly prominent role of turbulence research in providing more accurate climate scientific data. Finally, turbulence theorists can leverage these developments by asking the right questions and developing advanced, sophisticated frameworks that will be able to predict and correlate vast amounts of data from the other two components of the trinity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Comments on foliated gauge theories and dualities in 3+1d

We investigate the properties of foliated gauge fields and construct several foliated field theories in 3+1d that describe foliated fracton orders both with and without matter, including the recent hybrid fracton models. These field theories describe Abelian or non-Abelian gauge theories coupled to foliated gauge fields, and they fall into two classes of models that we call the electric models and the magnetic models. We show that these two classes of foliated field theories enjoy a duality. We also construct a model (using foliated gauge fields and an exactly solvable lattice Hamiltonian model) for a subsystem-symmetry protected topological (SSPT) phase, which is analogous to a one-form symmetry protected topological phase, with the subsystem symmetry acting on codimension-two subregions. We construct the corresponding gauged SSPT phase as a foliated two-form gauge theory. Some instances of the gauged SSPT phase are a variant of the X-cube model with the same ground state degeneracy and the same fusion, but different particle statistics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

A classical, non-singular, bouncing universe

We present a model for a classical, non-singular bouncing cosmology without violation of the null energy condition (NEC). The field content is General Relativity plus a real scalar field with a canonical kinetic term and only renormalizable, polynomial-type self-interactions for the scalar field in the Jordan frame. The universe begins vacuum-energy dominated and is contracting at t=-∞. We consider a closed universe with a positive spatial curvature, which is responsible for the universe bouncing without any NEC violation. An Rφ 2 coupling between the Ricci scalar and the scalar field drives the scalar field from the initial false vacuum to the true vacuum during the bounce. The model is sub-Planckian throughout its evolution and every dimensionful parameter is below the effective-field-theory scale M P , so we expect no ghost-type or tachyonic instabilities. This model solves the horizon problem and extends co-moving particle geodesics to past infinity, resulting in a geodesically complete universe without singularities. We solve the Friedman equations and the scalar-field equation of motion numerically, and analytically under certain approximations.

Astronomy & Astrophysics↗
DOE OSTI2020

Comments on D3-brane holography

We revisit the idea that the quantum dynamics of open strings ending on N D3-branes in the large N limit can be described at large ‘t Hooft coupling by classical closed string theory in the background created by the D3-branes in asymptotically flat spacetime. We study the resulting thermodynamics and compute the Hagedorn temperature and other properties of the D3-brane worldvolume theory in this regime. We also consider the theory in which the D3-branes are replaced by negative branes and show that its thermodynamics is well behaved. We comment on the idea that this theory can be thought of as an irrelevant deformation of $ \mathcal{N} $ = 4 SYM, and on its relation to $ T\overline{T} $ deformed CFT 2 .

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Solving the homogeneous Bethe-Salpeter equation with a quantum annealer

The homogeneous Bethe-Salpeter equation (hBSE), describing a bound system in a genuinely relativistic quantum-field theory framework, was solved for the first time by using a D-Wave quantum annealer. After applying standard techniques of discretization, the hBSE, in ladder approximation, can be formally transformed in a generalized eigenvalue problem (GEVP), with two square matrices: one symmetric and the other nonsymmetric. The latter matrix poses the challenge of obtaining a suitable formal approach for investigating the GEVP by means of a quantum annealer, i.e., to recast it as a quadratic unconstrained binary optimization problem. A broad numerical analysis of the proposed algorithms, applied to matrices of dimension up to 64, was carried out by using both the simulated-annealing package and the D-Wave . The numerical results very nicely compare with those obtained with standard classical algorithms, and also show interesting scalability features. Published by the American Physical Society 2024

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Nonlinear Spectroscopy of Collective Modes in an Excitonic Insulator

The nonlinear optical response of an excitonic insulator coupled to lattice degrees of freedom is shown to depend in strong and characteristic ways on whether the insulating behavior originates primarily from electron-electron or electron-lattice interactions. Linear response optical signatures of the massive phase mode and the amplitude (Higgs) mode are identified. Upon nonlinear excitation resonant to the phase mode, a new in-gap mode at twice the phase mode frequency is induced, leading to a huge second harmonic response. Excitation of in-gap phonon modes leads to different and much smaller effects. A Landau-Ginzburg theory analysis explains these different behaviors and reveals that a parametric resonance of the strongly excited phase mode is the origin of the photoinduced mode in the electron-dominant case. Here, the difference in the nonlinear optical response serves as a measure of the dominant mechanism of the ordered phase.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

On the implications of positive W mass shift

We investigate the phenomenological implications of the recent W mass measurement by the CDF collaboration, which exhibits tension with the standard model (SM) electroweak fit. Performing the fit to the electroweak observables within the SM effective field theory, we find that the new physics that contributes either to the determination of the electroweak vacuum expectation value, or to the oblique parameters, can improve the agreement with data. The best description is obtained from a fit where flavor universality is not required in the new physics operators, with 2 to 3 σ indications for several nonzero Wilson coefficients. We point out that top partners with order TeV masses could lead to the observed shift in the W mass.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Effective Field Theory for jet substructure in heavy ion collisions

I develop an Effective Field Theory (EFT) framework to compute jet substructure observables for heavy ion collision experiments. As an example, I consider dijet events that accompany the formation of a weakly coupled long lived Quark Gluon Plasma (QGP) medium in a heavy ion collision and look at an observable insensitive to jet selection bias: the simultaneous measurement of jet mass along with the transverse momentum imbalance between the jets that are groomed to remove soft radiation. Treating the jet as an open quantum system, I write down a factorization formula within the SCET (Soft Collinear Effective Theory) framework in the forward scattering regime. The physics of the medium is encoded in a universal soft field correlator while the jet-medium interaction is captured by a medium induced jet function. The factorization formula leads to a Lindblad type equation for the evolution of the reduced density matrix of the jet in the Markovian approximation. The solution for this equation allows a resummation of large logarithms that arise due to the final state measurements imposed while simultaneously summing over multiple incoherent interactions of the jet with the medium.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Modeling iterative reconstruction and displacement field in the large scale structure

The next generation of galaxy surveys like the Dark Energy Spectroscopic Instrument and Euclid will provide datasets orders of magnitude larger than anything available to date. Our ability to model nonlinear effects in late time matter perturbations will be a key to unlock the full potential of these datasets, and the area of initial condition reconstruction is attracting growing attention. Iterative reconstruction developed by Marcel Schmittfull et al. is a technique designed to reconstruct the displacement field from the observed galaxy distribution. The nonlinear displacement field and initial linear density field are highly correlated. Therefore, reconstructing the nonlinear displacement field enables us to extract the primordial cosmological information better than from the late time density field at the level of the two-point statistics. This paper will test to what extent the iterative reconstruction can recover the true displacement field and construct a perturbation theory model for the postreconstructed field. We model the iterative reconstruction process with Lagrangian perturbation theory (LPT) up to third order for dark matter in real space and compare it with N-body simulations. Here we find that the simulated iterative reconstruction does not converge to the nonlinear displacement field, and the discrepancy mainly appears in the shift term, i.e., the term correlated directly with the linear density field. On the contrary, our 3LPT model predicts that the iterative reconstruction should converge to the nonlinear displacement field. We discuss the sources of discrepancy, including numerical noise/artifacts on small scales, and present an ad hoc phenomenological model that improves the agreement.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Toward Addressing the Challenge to Predict the Heat Capacities of RDX and HMX Energetic Materials

Availability of heat capacity as function of pressure and temperature is an essential prerequisite for development of a computational multiscale strategy capable to address the evolution of microstructure and energy release in advanced high energy density materials. In the case of 1,3,5-trinitro-1,3,5-triazinane (RDX) and 1,3,5,7-tetranitro-1,3,5,7-tetrazocane (HMX) systems as two of the most studied energetic materials, there are substantial gaps in experimental data, with available heat capacities values distributed only in a region close to standard ambient conditions. In this study we demonstrate how these major experimental limitations can be addressed in the case of the RDX and HMX systems based on the combined use of classical and quantum mechanical calculations. We show that by considering ideal gas properties evaluated using quantum mechanical methods, and residual properties obtained from molecular simulations using fully flexible atomistic force field models, excellent agreement can be obtained for the predicted heat capacities to the most recent experimental values. An important advantage of the current computational methodology is that it allows evaluation of both constant-volume and constant-pressure heat capacities for a broad interval of temperatures and pressures, which encompasses solid and liquid phases conditions. In the case of the solid α and γ phases of RDX and the β phase of HMX, the predicted results follow closely both the available experimental data at standard ambient conditions and the results obtained using density functional theory calculations at high pressures, a regime where experimental data are not available. A perspective to expand the current methodology is also discussed.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Interfacial spin–orbit torques

Spin–orbit torques offer a promising mechanism for electrically controlling magnetization dynamics in nanoscale heterostructures. While spin–orbit torques occur predominately at interfaces, the physical mechanisms underlying these torques can originate in both the bulk layers and at interfaces. Classifying spin–orbit torques based on the region that they originate in provides clues as to how to optimize the effect. While most bulk spin–orbit torque contributions are well studied, many of the interfacial contributions allowed by symmetry have yet to be fully explored theoretically and experimentally. To facilitate progress, we review interfacial spin–orbit torques from a semiclassical viewpoint and relate these contributions to recent experimental results. In this study, we show the relationship between different interface transport parameters within the same model. For charges and spins flowing perpendicular to the interface, interfacial spin–orbit coupling both modifies the mixing conductance of the magnetoelectronic circuit theory and gives rise to spin memory loss. For in-plane electric fields, interfacial spin–orbit coupling gives rise to torques described by spin–orbit filtering, spin swapping, and precession. In addition, these same interfacial processes generate spin currents that flow into the non-magnetic layer. For in-plane electric fields in trilayer structures, the spin currents generated at the interface between one ferromagnetic layer and the non-magnetic spacer layer can propagate through the non-magnetic layer to produce novel torques on the other ferromagnetic layer.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗