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Through case study analysis of physicists working in quantum systems, we establish that knowledge of systems engineering (SE) will benefit scientists by ensuring that experimental apparatus is robust and fully meets the experiment requirements. We suggest a significant change in the training of early career research scientists to ensure that they have a strong appreciation of the systems approach and experience of applying SE techniques. Two case studies were used: the first was a detailed analysis of increasing levels of SE in the development of a three-dimensional (3-D) printer for fabricating superconducting nanocircuitry. Applying SE techniques improved record keeping, reduced the risk of failure modes, and took better account of future development through which scientific discovery may be exploited in practical devices. The second case study focused on a group of 30 quantum physics Ph.D. students undertaking a short-course in basic SE, and used their opinions to check and add confidence to the findings of the first case study. A variety of tools were used in both studies and, in both the cases, the Functional Failure Means Effects Analysis technique was considered by participants to be most useful. Recommendations are made for the future training of early career scientists to include some light-weight SE.

Three new graph invariants are introduced which may be measured from a quantum graph state and form examples of a framework under which other graph invariants can be constructed. Each invariant is based on distinguishing a different number of qubits. This is done by applying alternate measurements to the qubits to be distinguished. The performance of these invariants is evaluated and compared to classical invariants. We verify that the invariants can distinguish all non-isomorphic graphs with 9 or fewer nodes. The invariants have also been applied to `classically hard' strongly regular graphs, successfully distinguishing all strongly regular graphs of up to 29 nodes, and preliminarily to weighted graphs. We have found that although it is possible to prepare states with a polynomial number of operations, the average number of preparations required to distinguish non-isomorphic graph states scales exponentially with the number of nodes. We have so far been unable to find operators which reliably compare graphs and reduce the required number of preparations to feasible levels., Comprehensive update including a study of scalability of algorithms. 12 pages, 7 figures

In this work we show how constructing Wigner functions of heterogeneous quantum systems leads to new capability in the visualization of quantum states of atoms and molecules. This method allows us to display quantum correlations (entanglement) between spin and spatial degrees of freedom (spin-orbit coupling) and between spin degrees of freedom, as well as more complex combinations of spin and spatial entanglement. This is important as there is growing recognition that such properties affect the physical characteristics, and chemistry, of atoms and molecules. Our visualizations are sufficiently accessible that, with some preparation, those with a nontechnical background can gain an appreciation of subtle quantum properties of atomic and other systems. By providing insights and modeling capability, our phase-space representation will be of great utility in understanding aspects of atomic physics and chemistry not available with current techniques.

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems., Comment: Major update - removal of unnecessary Fourier transform from definition of the Weyl function which allowed generalisation of the work to a complete framework of phase-space quantum mechanics. This manuscript is completely re-written and improved to reflect this. 2 figures, 16 pages

Recent progress in Affective Computing (AC) has enabled integration of physiological cues and spontaneous expressions to reveal a subject’s emotional state. Due to the lack of an effective technique for evaluating multimodal correlations, experience and intuition play a main role in present AC studies when fusing affective cues or modalities, resulting in unexpected outcomes. This study seeks to demonstrate a dynamic correlation between two such affective cues, physiological changes and spontaneous expressions, which were obtained by a combination of stereo vision based tracking and imaging photoplethysmography (iPPG), with a designed protocol involving 20 healthy subjects. The two cues obtained were sampled into a Statistical Association Space (SAS) to evaluate their dynamic correlation. It is found that the probability densities in the SAS increase as the peaks in two cues are approached. Also the complex form of the high probability density region in the SAS suggests a nonlinear correlation between two cues. Finally the cumulative distribution on the zero time-difference surface is found to be small (

High fidelity models, which support accurate device characterization and correctly account for environmental effects, are crucial to the engineering of scalable quantum technologies. As it ensures positivity of the density matrix, one preferred model for open systems describes the dynamics with a master equation in Lindblad form. The Linblad operators are rarely derived from first principles, resulting in dynamical models which miss those additional terms that must generally be added to bring the master equation into Lindblad form, together with concomitant other terms that must be assimilated into an effective Hamiltonian. In first principles derivations such additional terms are often cancelled (countered), frequently in an ad hoc manner. In the case of a Superconducting Quantum Interference Device (SQUID) coupled to an Ohmic bath, the resulting master equation implies the environment has a significant impact on the system's energy. We discuss the prospect of keeping or cancelling this impact; and note that, for the SQUID, measuring the magnetic susceptibility under control of the capacitive coupling strength and the externally applied flux, results in experimentally measurable differences between models. If this is not done correctly, device characterization will be prone to systemic errors., 5 pages, 3 figures

In this work we construct Wigner functions for hybrid continuous and discrete variable quantum systems. We demonstrate new capabilities in the visualization of the interactions and correlations between discrete and continuous variable quantum systems, where visualizing the full phase space has proven difficult in the past due to the high number of degrees of freedom. Specifically, we show how to clearly distinguish signatures that arise due to quantum and classical correlations in an entangled Bell-cat state. We further show how correlations are manifested in different types of interaction, leading to a deeper understanding of how quantum information is shared between two subsystems. Understanding the nature of the correlations between systems is central to harnessing quantum effects for information processing; the methods presented here reveal the nature of these correlations, allowing a clear visualization of the quantum information present in these hybrid discrete-continuous variable quantum systems. The methods presented here could be viewed as a form of quantum state spectroscopy.

A means of unifying some semiclassical models of computational chemistry is presented; these include quantized Hamiltonian dynamics, quantal cumulant dynamics, and semiclassical Moyal dynamics (SMD). A general method for creating the infinite hierarchy of operator dynamics in the Heisenberg picture is derived together with a general method for truncation (or closure) of that series, and in addition, we provide a simple link to the phase space methods of SMD. Operator equations of arbitrary order may be created readily, avoiding the tedious algebra identified previously. Truncation is based on a simple recurrence formula which is related to, but avoids the more complex contractions of, Wick's theorem. This generalized method is validated against a number of trial problems considered using the previous methods. We also touch on some of the limitations involved using such methods, noting, in particular, that any truncation will lead to a state which is in some sense unphysical. Finally, we briefly introduce our quantum algebra package QuantAL which provides an automated method for the generation of the required equation set, the initial conditions for all variables from any start, and all the higher order approximations necessary for truncation of the series, at essentially arbitrary order.

In this paper we consider the modelling and simulation of open quantum systems from a device engineering perspective. We derive master equations at different levels of approximation for a Superconducting Quantum Interference Device (SQUID) ring coupled to an ohmic bath. We demonstrate that the master equations we consider produce decoherences that are qualitatively and quantitativly dependent on both the level of approximation and the ring's external flux bias. We discuss the issues raised when seeking to obtain Lindbladian dissipation and show, in this case, that the external flux (which may be considered to be a control variable in some applications) is not confined to the Hamiltonian, as often assumed in quantum control, but also appears in the Lindblad terms., 11 pages, 4 figures

Non-contact imaging photoplethysmography (iPPG) to detect pulsatile blood microcirculation in tissue has been selected as a successor to low spatial resolution and slow scanning blood perfusion techniques currently employed by clinicians. The proposed iPPG system employs a novel illumination source constructed of multiple high power LEDs with narrow spectral emission, which are temporally modulated and synchronised with a high performance sCMOS sensor. To ensure spectrum stability and prevent thermal wavelength drift due to junction temperature variations, each LED features a custom-designed thermal management system to effectively dissipate generated heat and auto-adjust current flow. The use of a multi-wavelength approach has resulted in simultaneous microvascular perfusion monitoring at various tissue depths, which is an added benefit for specific clinical applications. A synchronous detection algorithm to extract weak photoplethysmographic pulse-waveforms demonstrated robustness and high efficiency when applied to even small regions of 5 mm2. The experimental results showed evidences that the proposed system could achieve noticeable accuracy in blood perfusion monitoring by creating complex amplitude and phase maps for the tissue under examination.

One solution to the global challenge of increasing ocular disease is a cost-effective technique for rapid screening and assessment. Current ophthalmic imaging techniques, e.g. scanning and ocular blood flow systems, are expensive, complex to operate and utilize invasive contrast agents during assessment. The work presented here demonstrates a simple retinal imaging photoplethysmography (iPPG) system with the potential to provide screening, diagnosis, monitoring and assessment that is non-invasive, painless and radiationless. Time series of individual retinal blood vessel images, captured with an eye fundus camera, are processed using standard filtering, amplitude demodulation and principle component analysis (PCA) methods to determine the values of the heart rate (HR) and respiration rate (RR), which are in compliance with simultaneously obtained measurements using commercial pulse oximetry. It also seems possible that some information on the dynamic changes in oxygen saturation levels (SpO2) in a retinal blood vessel may also be obtained. As a consequence, the retinal iPPG modality system demonstrates a potential avenue for rapid ophthalmic screening, and even early diagnosis, against ocular disease without the need for fluorescent or contrast agents.

The demand of non-invasive ocular screening is rapidly growing due to an increase of age related eye diseases worldwide. An indeed in-depth understanding of optical properties is required to elucidate nature of retinal tissue. The research aims to investigate an effective biomedical engineering approach to allow process region of interests (ROIs) in eyes to reveal physiological status. A dynamic opto-physiological model (DOPM) representing retinal microvascular circulation underlying a diffusion approximation to solve radiative transport theorem (RTT) has being developed to interpret patho-physiological phenomena. DOPM is being applied in imaging photoplethysmography (iPPG) to extract PPG signals from a series of 2D matrix images to access blood perfusion and oxygen saturation distributions. A variation of microvascular circulation could be mapped for an effectively diagnostic screening. The work presents mathematical modelling based ten layers of ocular tissue tested with four set of controlled parameters demontrated detection ratio between normal tissue damage or abnormal tissue and significant change of AC signal amplitude in these tissues. The result shows signicant change of AC signal amplitude in abnormal tissue. The preliminary results show extractable PPG signals from eye fundus video; experimented at five ROIs: whole fundus, optical disk, main vein vessel, lesion area and affected area. The outcome shows optical disk region gave a better performance compared to whole fundus region and main vein vessel. The robustness, miniaturization and artefact reduction capability of DOPM to discriminate oxygenation levels in retina could offer a new insight to access retinal patho-physiological status.

For electromigration in short aluminum interconnects terminated by tungsten vias, the well known “short-line” effect applies. In a similar manner, for longer lines, early failure is determined by a critical value Lcrit for the length of polygranular clusters. Any cluster shorter than Lcrit is “immortal” on the time scale of early failure where the figure of merit is not the standard t50 value (the time to 50% failures), but rather the total probability of early failure, Pcf. Pcf is a complex function of current density, linewidth, line length, and material properties (the median grain size d50 and grain size shape factor σd). It is calculated here using a model based around the theory of runs, which has proved itself to be a useful tool for assessing the probability of extreme events. Our analysis shows that Pcf is strongly dependent on σd, and a change in σd from 0.27 to 0.5 can cause an order of magnitude increase in Pcf under typical test conditions. This has implications for the web-based two-dimensio...

The application of the weakest-link or failure-unit model to electromigration failure is discussed in relation to a simple model that should describe the early failures in fine-line, as-patterned aluminium reasonably well. The earliest failures are expected to be due to the presence of at least one polygranular cluster longer than some critical (Blech) length, while the next earliest are assumed to arise from linked shorter clusters. Using a simple model of the microstructure, the probability density function for the number of each of these types of failure-unit is determined as a function of the linewidth. For this simple model the overall failure distribution may be calculated essentially exactly using the Theory of Runs; consequently we can compare these exact results with some common approximations. It is demonstrated through Jensen's inequality that a common approximate method is in fact a rigorous upper bound for the failure distribution. The model exhibits the expected increase in median-time-to-failure and deviation-in-time-to-failure values with decreasing linewidth when forcing a lognormal distribution. Although this would suggest an eventual decrease in reliability with scale reduction, the exact results show that this is not the case.

We aim here to describe electromigration-induced interconnect failure using a one-dimensional microstructure model (already one-dimensional models are used for the stress evolution). One might expect such a model to be reasonably successful because either (i) line widths are well below grain-boundary diameters (e.g. in as-patterned aluminium) or (ii) sidewall or topside surfaces provides the dominant diffusion path (e.g. in post-pattern annealed aluminium or copper). Our simple model is based on the Theory of Runs from probability theory and consequently may be solved analytically. As a test, we demonstrate that the model is able to reproduce the cluster length statistics of the two-dimensional simulator MIT/EmSIM for narrow metal (near-bamboo) stripes to a very good approximation. The importance of developing such a model is that, because of its analytical nature, it may access important areas where statistics are difficult for the simulator, such as rare events?e.g. time-to-first-failure rather than average events such as the mean-time-to-failure?or impossible events such as ?d ? 0.27. In addition, it should allow some light to be thrown on the validity of the failure-unit model used to obtain interconnect lifetimes.

A theoretical study is made of how the transport-time to state-lifetime ratio ( t / s ) varies as a function of sheet carrier density in semiconductor quantum-well alloys at low temperature. Various sources of ionized impurity scattering are considered, in addition to alloy scattering, using a self-consistent multiple-scattering theory proposed originally by Gold and Gotze. To begin, we present the theory of Gold and Gotze in its most transparent form. We then examine electron transport in In1-x Gax As quantum wells and hole transport in Si1-x Gex quantum wells. The interplay between alloy scattering and ionized impurity scattering leads to interesting behaviour, and to a significant reduction in the value of t / s as compared to systems where alloy scattering is not a factor. Multiple-scattering effects become progressively more important as the sheet carrier density decreases, and indications are given as to when lowest-order transport theory ceases to be adequate.

We consider a class of random geometric series with an underlying tree-like structure that has a number of applications in statistical physics. Convergence criteria for these series are discussed and consistency with different criteria known to hold in the one-dimensional limit is established. A multiplicative, two-component model of percolation on a Cayley tree is defined and analysed. The order of the percolation transition and certain critical exponents are altered compared to conventional bond percolation.

We consider a simple model of transport on a regular tree, whereby species evolve according to the drift-diffusion equation, and the drift velocity on each branch of the tree is a quenched random variable. The inverse of the steady-state amplitude at the origin is expressed in terms of a random geometric series whose convergence or otherwise determines whether the system is localized or delocalized. In a recent paper [P. C. Bressloff et al., Phys. Rev. Lett. 77, 5075 (1996)], exact criteria were presented that enable one to determine the critical phase boundary for the transition, valid for any distribution of the drift velocities. In this paper we present a detailed derivation of these criteria, consider a number of examples of interest, and establish a connection with conventional percolation theory. The latter suggests a wider application of the results to other models of statistical processes occurring on treelike structures. Generalizations to the case where the underlying tree is irregular in nature are also considered.

The initial value problem of Burgers' equation on a branching structure consisting of N semi-infinite line segments radiating from a common junction is solved in terms of a set of N linear integral equations of Volterra type involving N + 1 unknowns, coupled by a single nonlinear constraint ensuring current conservation at the junction. We outline an iterative scheme for solving these integral equations, and illustrate their application to physical problems by considering a simple phenomenological model of road traffic.

We investigate the localization-delocalization transition for the drift-diffusion equation on a regular tree with quenched random drift velocities on its branches. The inverse of the steady-state amplitude at the origin is expressed in terms of a random geometric series whose convergence or otherwise determines the critical phase boundary. We establish exact criteria for localization valid for an arbitrary distribution of the drift velocities. The phase transition is shown to be first order except in the percolation limit.

We show that classical localization occurs for the drift - diffusion equation on an ordered Cayley tree when the drift velocity v on each branch of the tree exceeds a critical value , where z is the coordination number, D is the diffusion constant and L is the segment length. For the asymptotic decay of the delocalized state exhibits conventional diffusive behaviour, whereas at the critical point there is anomalous behaviour in the form of a critical slowing-down. A necessary condition for localization in the presence of randomly distributed drift velocities is also derived.

We present an integral equation reformulation of the drift–diffusion equation for modeling void buildup in integrated circuit interconnects due to the electromigration of aluminum atoms. The method can deal with arbitrary time‐dependent current stress and offers a number of advantages over the conventional finite‐difference approach. In particular, the boundary conditions are included in an intuitive and straightforward fashion. By way of illustration we identify the conditions under which the time‐to‐failure scales as r−2 for unidirectional square‐wave stress of duty cycle r.

The problem of current induced electromigration in VLSI interconnects, under an arbitrary time-dependent stress, is considered within the drift/diffusion model. It is shown that, by transforming into the convected frame and solving the resulting moving boundary problem, the vacancy build-up may be followed by solving two coupled integral equations. The important large-time behavior is obtained using standard asymptotic techniques. A series solution and an approximate small-time solution are also derived. It is found that, for a unidirectional periodic stress, the equivalent dc current, appropriate to EM reliability tests is the periodic average value. In addition a design rule for arbitrary time-dependent stress is suggested.

We use a `sum-over-paths' method to derive an explicit expression for the Green's function of the diffusion equation on an arbitrary tree, making connection with the conventional Laplace approach but emphasizing the real-time nature of the technique. This provides a general framework for studying both discrete-space and continuous-space diffusion processes on complex one-dimensional topologies.

The (one‐dimensional) electromigration boundary‐value problem is considered for the case of a single blocking boundary with a constant vacancy supply at the other boundary. Using the drift/diffusion model expressed by the Fokker–Planck equation, we find that the saturation time (tsat) increases exponentially with current density (j) and not as j−2, as has been suggested. However, it is not the saturation time which determines the lifetime (tbd); it is the time to reach some critical vacancy concentration (c*). In agreement with experimental results and numerical calculations, we find that tbd∼j−2. We also find that tbd∼c*.

Neglecting elastic scattering in the inversion process for reconstructing composition–depth profiles from angle-resolved Auger/XPS spectra leads to several effects that are likely to be misinterpreted. For example, there appears to be an erroneous diffusion of the substrate material to the surface in the case of a simple overlayer. In addition, the thickness of the overlayer is likely to be overestimated by typically ∼20% for Auger/XPS energies.

The possible extraction of inelastic mean free path (IMFP or λi) data from elastic backscattering experiments is analyzed within the transport approximation. It is found that the characteristic length obtained by this method is not the IMFP, λi, as previously thought, but a length Λ=[λ−1i(λ−1i+λ−1tr)]−1/2, where λtr, the transport mean free path, is the length associated with large‐angle deflection. It is significant that this length Λ is also the length which characterizes the overlayer experiment for an ultrathin overlayer (thickness≪IMFP) and at an exit angle of ∼45°, and is also the length obtained using two‐stream methods [V. M. Dwyer, Surf. Sci. 291, 261 (1993)].

The relationship between parametric drift and latent damage in ESD gate-stressed MOSFETs is studied. Sub-breakdown damage causes minor characteristic distortion, which may remain undetected until failure. However, such damage is only significant within a narrow stress-voltage window. Oxide breakdown may cause straightforward malfunction (i.e. catastrophic failure ) or degraded transistor action. Degraded devices can degenerate further under working voltages (0–10 V), providing a latent failure mechanism. Degradation phenomena are attributed to the intrusion of polysilicon gate-material into the oxide and channel regions. Catastrophically failed and degraded devices are modelled using the PSpice circuit simulation system. The effects of degradation upon CMOS logic operation are also examined.

This paper presents a study of electrical waveforms which characterise the behaviour of the gate Schottky diode of a GaAs MESFET (MEtal Schottky FET) under electrostatic discharge (ESD) conditions. It is demonstrated that the ESD breakdown threshold is highly dependent on the external circuit elements of the ESD pulsing apparatus and pulse transmission medium and therefore dependent on the circuit model used for ESD threshold assessment. It is also established that a GaAs MES device under forward bias subthreshold/threshold ESD stress will exhibit a region of current controlled negative differential resistance (NDR) which is not observed in the DC characteristics. Beyond this NDR region a region of positive differential resistance (PDR) exists and it is whilst operating within this region that thermal breakdown can occur. In contrast, the reverse bias threshold is highly voltage dependent and the DC avalanche voltage must be attained during an ESD pulse for impact ionisation to occur. Continued operation in this avalanche condition, which is possible with ESD pulses greater than ≈-100 V, allows localised heating sufficient to initialise thermal runaway.

The depth distribution function (DDF) for Auger/XPS spectroscopy obtained in the two-stream approximation contains little useful information. This situation may be improved considerably by invoking the principles of reciprocity and solving instead the reciprocal problem. Doing so produces an approximate DDF of the same two-exponential form as the approximate DDF obtained in the transport approximation lending further credence to this result. There are many forms of the two stream method. However, from the literature it is clear that the delta-Eddington version gives the best combination of accuracy and simplicity for the reciprocal problem. Two forms of the delta-Eddington version are compared to the DDF from the transport approximation, leading to the conclusion that the three DDF's which include elastic scattering are remarkably close to one another and substantially different from the DDF used in standard quantitative procedures. Using these forms of the DDF should allow quantitative Auger/XPS spectroscopy to include elastic scattering in a simple yet accurate way.

The effects of elastic scattering on the information available from angle-resolved electron spectroscopies on non-crystalline materials is studied. These effects force the consideration of elastic scattering cross-sections which are dependent on atomic concentration and therefore on depth. This is done by discretizing the transport equation within a two-stream/transport approximation. The variation of inelastic mean free paths with composition is also included

The experimentally measured angular distribution of electrons elastically backscattered from polycrystalline or amorphous solids exhibts a strong resemblance to the differential cross-section for single-atom elastic scattering. Consequently most analytical attempts to describe the problem, which involve some simplifying approximations to this scattering cross-section, are inadequate in the modelling of backscattered electrons. The present paper represents an attempt to derive a simple analytical approximation that is able to describe such events. This is done by including the first scatteing event exactly and including multiple scattering events using a sort of ‘rectilinear + diffusion’ theory, which has proved to be successful for modelling the depth distribution function used in Auger and x-ray spectroscopies. For the cases considered (Ag, Al and Cu at 1 kV), the results compare very well with both experimental data and recent Monte Carlo calculations.

A recent model of particle transport in plane-parallel media, including both quasielastic and inelastic scattering, has been proposed [Goulet et al., Phys. Rev. A 41, 6006 (1990)] which claims to be fully three dimensional, to include the effects of multiple scattering, and to avoid the normal difficulties associated with such problems. The present paper demonstrates that a transport problem is being solved and that the method used is of comparable accuracy to elementary diffusion theory. It is shown that the model, applied here to the low-energy-electron transmission and reflection spectroscopy problem, is valid only under a fairly stringent assumption on the interface scattering at the boundaries of the layer. The model is rederived for this case in terms of standard transport theory. A comparison of the two models points to an ambiguity in the interface reflectivity in the previous model. In addition the method presented here leads to a more satisfactory application of the Snell-Descartes law and a full solution in terms of known functions.

We consider the interpretation of some theoretical and experimental work regarding electromigration voiding in nanoindented, single crystal aluminum lines. A recently suggested voiding criterion of a critical accumulated flux divergence is found, in fact, to be identical to the widely accepted critical stress criterion. The inclusion of the stress dependence of the atomic diffusion coefficient is shown to be vital when the steady state is characterized by J≠0, such as in the case of a void growing at a constant rate. It is found, for example, that the stress required for steady void growth, within single crystal Al lines, is probably significantly smaller than previously suggested.